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Cosmology and Fundamental Physics with the Euclid Satellite

A Later Version of this article was published on 12 April 2018

Abstract

Euclid is a European Space Agency medium-class mission selected for launch in 2019 within the Cosmic Vision 2015–2025 program. The main goal of Euclid is to understand the origin of the accelerated expansion of the universe. Euclid will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and red-shifts of galaxies as well as the distribution of clusters of galaxies over a large fraction of the sky.

Although the main driver for Euclid is the nature of dark energy, Euclid science covers a vast range of topics, from cosmology to galaxy evolution to planetary research. In this review we focus on cosmology and fundamental physics, with a strong emphasis on science beyond the current standard models. We discuss five broad topics: dark energy and modified gravity, dark matter, initial conditions, basic assumptions and questions of methodology in the data analysis.

This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the Euclid mission.

Introduction

EuclidFootnote 1 [551, 760, 239] is an ESA medium-class mission selected for the second launch slot (expected for 2019) of the Cosmic Vision 2015–2025 program. The main goal of Euclid is to understand the physical origin of the accelerated expansion of the universe. Euclid is a satellite equipped with a 1.2 m telescope and three imaging and spectroscopic instruments working in the visible and near-infrared wavelength domains. These instruments will explore the expansion history of the universe and the evolution of cosmic structures by measuring shapes and redshifts of galaxies over a large fraction of the sky. The satellite will be launched by a Soyuz ST-2.1B rocket and transferred to the L2 Lagrange point for a six-year mission that will cover at least 15 000 square degrees of sky. Euclid plans to image a billion galaxies and measure nearly 100 million galaxy redshifts.

These impressive numbers will allow Euclid to realize a detailed reconstruction of the clustering of galaxies out to a redshift 2 and the pattern of light distortion from weak lensing to redshift 3. The two main probes, redshift clustering and weak lensing, are complemented by a number of additional cosmological probes: cross correlation between the cosmic microwave background and the large scale structure; luminosity distance through supernovae Ia; abundance and properties of galaxy clusters and strong lensing. To extract the maximum of information also in the nonlinear regime of perturbations, these probes will require accurate high-resolution numerical simulations. Besides cosmology, Euclid will provide an exceptional dataset for galaxy evolution, galaxy structure, and planetary searches. All Euclid data will be publicly released after a relatively short proprietary period and will constitute for many years the ultimate survey database for astrophysics.

A huge enterprise like Euclid requires highly considered planning in terms not only of technology but also for the scientific exploitation of future data. Many ideas and models that today seem to be abstract exercises for theorists will in fact finally become testable with the Euclid surveys. The main science driver of Euclid is clearly the nature of dark energy, the enigmatic substance that is driving the accelerated expansion of the universe. As we discuss in detail in Part 1, under the label “dark energy” we include a wide variety of hypotheses, from extradimensional physics to higher-order gravity, from new fields and new forces to large violations of homogeneity and isotropy. The simplest explanation, Einstein’s famous cosmological constant, is still currently acceptable from the observational point of view, but is not the only one, nor necessarily the most satisfying, as we will argue. Therefore, it is important to identify the main observables that will help distinguish the cosmological constant from the alternatives and to forecast Euclid’s performance in testing the various models.

Since clustering and weak lensing also depend on the properties of dark matter, Euclid is a dark matter probe as well. In Part 2 we focus on the models of dark matter that can be tested with Euclid data, from massive neutrinos to ultra-light scalar fields. We show that Euclid can measure the neutrino mass to a very high precision, making it one of the most sensitive neutrino experiments of its time, and it can help identify new light fields in the cosmic fluid.

The evolution of perturbations depends not only on the fields and forces active during the cosmic eras, but also on the initial conditions. By reconstructing the initial conditions we open a window on the inflationary physics that created the perturbations, and allow ourselves the chance of determining whether a single inflaton drove the expansion or a mixture of fields. In Part 3 we review the choices of initial conditions and their impact on Euclid science. In particular we discuss deviations from simple scale invariance, mixed isocurvature-adiabatic initial conditions, non-Gaussianity, and the combined forecasts of Euclid and CMB experiments.

Practically all of cosmology is built on the Copernican Principle, a very fruitful idea postulating a homogeneous and isotropic background. Although this assumption has been confirmed time and again since the beginning of modern cosmology, Euclid’s capabilities can push the test to new levels. In Part 4 we challenge some of the basic cosmological assumptions and predict how well Euclid can constrain them. We explore the basic relation between luminosity and angular diameter distance that holds in any metric theory of gravity if the universe is transparent to light, and the existence of large violations of homogeneity and isotropy, either due to local voids or to the cumulative stochastic effects of perturbations, or to intrinsically anisotropic vector fields or spacetime geometry.

Finally, in Part 5 we review some of the statistical methods that are used to forecast the performance of probes like Euclid, and we discuss some possible future developments.

This review has been planned and carried out within Euclid’s Theory Working Group and is meant to provide a guide to the scientific themes that will underlie the activity of the group during the preparation of the mission. At the same time, this review will help us and the community at large to identify the areas that deserve closer attention, to improve the development of Euclid science and to offer new scientific challenges and opportunities.

Dark Energy

Introduction

With the discovery of cosmic acceleration at the end of the 1990s, and its possible explanation in terms of a cosmological constant, cosmology has returned to its roots in Einstein’s famous 1917 paper that simultaneously inaugurated modern cosmology and the history of the constant Λ. Perhaps cosmology is approaching a robust and all-encompassing standard model, like its cousin, the very successful standard model of particle physics. In this scenario, the cosmological standard model could essentially close the search for a broad picture of cosmic evolution, leaving to future generations only the task of filling in a number of important, but not crucial, details.

The cosmological constant is still in remarkably good agreement with almost all cosmological data more than ten years after the observational discovery of the accelerated expansion rate of the universe. However, our knowledge of the universe’s evolution is so incomplete that it would be premature to claim that we are close to understanding the ingredients of the cosmological standard model. If we ask ourselves what we know for certain about the expansion rate at redshifts larger than unity, or the growth rate of matter fluctuations, or about the properties of gravity on large scales and at early times, or about the influence of extra dimensions (or their absence) on our four dimensional world, the answer would be surprisingly disappointing.

Our present knowledge can be succinctly summarized as follows: we live in a universe that is consistent with the presence of a cosmological constant in the field equations of general relativity, and as of 2012, the value of this constant corresponds to a fractional energy density today of ΩΛ ≈ 0.73. However, far from being disheartening, this current lack of knowledge points to an exciting future. A decade of research on dark energy has taught many cosmologists that this ignorance can be overcome by the same tools that revealed it, together with many more that have been developed in recent years.

Why then is the cosmological constant not the end of the story as far as cosmic acceleration is concerned? There are at least three reasons. The first is that we have no simple way to explain its small but non-zero value. In fact, its value is unexpectedly small with respect to any physically meaningful scale, except the current horizon scale. The second reason is that this value is not only small, but also surprisingly close to another unrelated quantity, the present matter-energy density. That this happens just by coincidence is hard to accept, as the matter density is diluted rapidly with the expansion of space. Why is it that we happen to live at the precise, fleeting epoch when the energy densities of matter and the cosmological constant are of comparable magnitude? Finally, observations of coherent acoustic oscillations in the cosmic microwave background (CMB) have turned the notion of accelerated expansion in the very early universe (inflation) into an integral part of the cosmological standard model. Yet the simple truth that we exist as observers demonstrates that this early accelerated expansion was of a finite duration, and hence cannot be ascribable to a true, constant Λ; this sheds doubt on the nature of the current accelerated expansion. The very fact that we know so little about the past dynamics of the universe forces us to enlarge the theoretical parameter space and to consider phenomenology that a simple cosmological constant cannot accommodate.

These motivations have led many scientists to challenge one of the most basic tenets of physics: Einstein’s law of gravity. Einstein’s theory of general relativity (GR) is a supremely successful theory on scales ranging from the size of our solar system down to micrometers, the shortest distances at which GR has been probed in the laboratory so far. Although specific predictions about such diverse phenomena as the gravitational redshift of light, energy loss from binary pulsars, the rate of precession of the perihelia of bound orbits, and light deflection by the sun are not unique to GR, it must be regarded as highly significant that GR is consistent with each of these tests and more. We can securely state that GR has been tested to high accuracy at these distance scales.

The success of GR on larger scales is less clear. On astrophysical and cosmological scales, tests of GR are complicated by the existence of invisible components like dark matter and by the effects of spacetime geometry. We do not know whether the physics underlying the apparent cosmological constant originates from modifications to GR (i.e., an extended theory of gravity), or from a new fluid or field in our universe that we have not yet detected directly. The latter phenomena are generally referred to as ‘dark energy’ models.

If we only consider observations of the expansion rate of the universe we cannot discriminate between a theory of modified gravity and a dark-energy model. However, it is likely that these two alternatives will cause perturbations around the ‘background’ universe to behave differently. Only by improving our knowledge of the growth of structure in the universe can we hope to progress towards breaking the degeneracy between dark energy and modified gravity. Part 1 of this review is dedicated to this effort. We begin with a review of the background and linear perturbation equations in a general setting, defining quantities that will be employed throughout. We then explore the nonlinear effects of dark energy, making use of analytical tools such as the spherical collapse model, perturbation theory and numerical N-body simulations. We discuss a number of competing models proposed in literature and demonstrate what the Euclid survey will be able to tell us about them.

Background evolution

Most of the calculations in this review are performed in the Friedmann-Lemaître-Robertson-Walker (FLRW) metric

$${\rm{d}}{s^2} = - {\rm{d}}{t^2} + a{(t)^2}({{{\rm{d}}{r^2}} \over {1 - k{r^2}}} + {r^2}\,{\rm{d}}{\theta ^2} + {r^2}{\sin ^2}\theta \,{\rm{d}}{\phi ^2})\,,$$
(1.2.1)

where a (t) is the scale factor and k the spatial curvature. The usual symbols for the Hubble function H = ȧ/a and the density fractions Ω x , where x stands for the component, are employed. We characterize the components with the subscript M or m for matter, γ or r for radiation, b for baryons, K for curvature and Λ for the cosmological constant. Whenever necessary for clarity, we append a subscript 0 to denote the present epoch, e.g., ΩM,0. Sometimes the conformal time η = ∫ dt/a and the conformal Hubble function \({\mathcal H} = aH = {\rm{d}}a/(a{\rm{d}}\eta)\) are employed. Unless otherwise stated, we denote with a dot derivatives w.r.t. cosmic time t (and sometimes we employ the dot for derivatives w.r.t. conformal time η) while we use a prime for derivatives with respect to ln a.

The energy density due to a cosmological constant with p = − ρ is obviously constant over time. This can easily be seen from the covariant conservation equation \(T_{\mu; v}^v = 0\) for the homogeneous and isotropic FLRW metric,

$$\dot \rho + 3H(\rho + p) = 0\,.$$
(1.2.2)

However, since we also observe radiation with p = ρ/3 and non-relativistic matter for which p ≈ 0, it is natural to assume that the dark energy is not necessarily limited to a constant energy density, but that it could be dynamical instead.

One of the simplest models that explicitly realizes such a dynamical dark energy scenario is described by a minimally-coupled canonical scalar field evolving in a given potential. For this reason, the very concept of dynamical dark energy is often associated with this scenario, and in this context it is called ‘quintessence’ [954, 754]. In the following, the scalar field will be indicated with ϕ. Although in this simplest framework the dark energy does not interact with other species and influences spacetime only through its energy density and pressure, this is not the only possibility and we will encounter more general models later on. The homogeneous energy density and pressure of the scalar field ϕ are defined as

$${\rho _\phi} = {{{{\dot \phi}^2}} \over 2} + V(\phi)\,,\quad {p_\phi} = {{{{\dot \phi}^2}} \over 2} - V(\phi)\,,\quad {w_\phi} = {{{p_\phi}} \over {{\rho _\phi}}}\,,$$
(1.2.3)

and w ϕ is called the equation-of-state parameter. Minimally-coupled dark-energy models can allow for attractor solutions [252, 573, 867]: if an attractor exists, depending on the potential V (ϕ) in which dark energy rolls, the trajectory of the scalar field in the present regime converges to the path given by the attractor, though starting from a wide set of different initial conditions for ϕ and for its first derivative \({\dot \phi}\). Inverse power law and exponential potentials are typical examples of potential that can lead to attractor solutions. As constraints on w ϕ become tighter [e.g., 526], the allowed range of initial conditions to follow into the attractor solution shrinks, so that minimally-coupled quintessence is actually constrained to have very flat potentials. The flatter the potential, the more minimally-coupled quintessence mimics a cosmological constant, the more it suffers from the same fine-tuning and coincidence problems that affect a ΛCDM scenario [646].

However, when GR is modified or when an interaction with other species is active, dark energy may very well have a non-negligible contribution at early times. Therefore, it is important, already at the background level, to understand the best way to characterize the main features of the evolution of quintessence and dark energy in general, pointing out which parameterizations are more suitable and which ranges of parameters are of interest to disentangle quintessence or modified gravity from a cosmological constant scenario.

In the following we briefly discuss how to describe the cosmic expansion rate in terms of a small number of parameters. This will set the stage for the more detailed cases discussed in the subsequent sections. Even within specific physical models it is often convenient to reduce the information to a few phenomenological parameters.

Two important points are left for later: from Eq. (1.2.3) we can easily see that w ϕ ≥ −1 as long as ρ ϕ > 0, i.e., uncoupled canonical scalar field dark energy never crosses w ϕ = −1. However, this is not necessarily the case for non-canonical scalar fields or for cases where GR is modified. We postpone to Section 1.4.5 the discussion of how to parametrize this ‘phantom crossing’ to avoid singularities, as it also requires the study of perturbations.

The second deferred part on the background expansion concerns a basic statistical question: what is a sensible precision target for a measurement of dark energy, e.g., of its equation of state? In other words, how close to w ϕ = −1 should we go before we can be satisfied and declare that dark energy is the cosmological constant? We will address this question in Section 1.5.

Parametrization of the background evolution

If one wants to parametrize the equation of state of dark energy, two general approaches are possible. The first is to start from a set of dark-energy models given by the theory and to find parameters describing their w ϕ as accurately as possible. Only later can one try and include as many theoretical models as possible in a single parametrization. In the context of scalar-field dark-energy models (to be discussed in Section 1.4.1), [266] parametrize the case of slow-rolling fields, [796] study thawing quintessence, [446] and [232] include non-minimally coupled fields, [817] quintom quintessence, [325] parametrize hilltop quintessence, [231] extend the quintessence parametrization to a class of k-essence models, [459] study a common parametrization for quintessence and phantom fields. Another convenient way to parametrize the presence of a non-negligible homogeneous dark energy component at early times (usually labeled as EDE) was presented in [956]. We recall it here because we will refer to this example in Section 1.6.1.1. In this case the equation of state is parametrized as:

$${w_X}(z) = {{{w_0}} \over {1 + b\ln (1 + z)}},$$
(1.2.4)

where b is a constant related to the amount of dark energy at early times, i.e.,

$$b = - {{3{{\bar w}_0}} \over {\ln {{1 - {\Omega _{{\rm{X}},e}}} \over {{\Omega _{{\rm{X}},e}}}} + \ln {{1 - {\Omega _{m,0}}} \over {{\Omega _{m,0}}}}}}\,.$$
(1.2.5)

Here the subscripts ‘0’ and ‘e’ refer to quantities calculated today or early times, respectively. With regard to the latter parametrization, we note that concrete theoretical and realistic models involving a non-negligible energy component at early times are often accompanied by further important modifications (as in the case of interacting dark energy), not always included in a parametrization of the sole equation of state such as (1.2.4) (for further details see Section 1.6 on nonlinear aspects of dark energy and modified gravity).

The second approach is to start from a simple expression of w without assuming any specific dark-energy model (but still checking afterwards whether known theoretical dark-energy models can be represented). This is what has been done by [470, 623, 953] (linear and logarithmic parametrization in z), [229], [584] (linear and power law parametrization in a), [322], [97] (rapidly varying equation of state).

The most common parametrization, widely employed in this review, is the linear equation of state [229, 584]

$${w_X}(a) = {w_0} + {w_a}(1 - a),$$
(1.2.6)

where the subscript X refers to the generic dark-energy constituent. While this parametrization is useful as a toy model in comparing the forecasts for different dark-energy projects, it should not be taken as all-encompassing. In general a dark-energy model can introduce further significant terms in the effective w X (z) that cannot be mapped onto the simple form of Eq. (1.2.6).

An alternative to model-independent constraints is measuring the dark-energy density ρ X (z) (or the expansion history H (z)) as a free function of cosmic time [942, 881, 274]. Measuring ρ X (z) has advantages over measuring the dark-energy equation of state w X (z) as a free function; ρ X (z) is more closely related to observables, hence is more tightly constrained for the same number of redshift bins used [942, 941]. Note that ρ X (z) is related to w X (z) as follows [942]:

$${{{\rho _X}(z)} \over {{\rho _X}(0)}} = \exp \left\{{\int\nolimits_0^z \, {\rm{d}}z\prime \,{{3[1 + {w_X}(z\prime)]} \over {1 + z\prime}}} \right\}\,.$$
(1.2.7)

Hence, parametrizing dark energy with w X (z) implicitly assumes that ρ X (z) does not change sign in cosmic time. This precludes whole classes of dark-energy models in which ρ X (z) becomes negative in the future (“Big Crunch” models, see [943] for an example) [944].

Note that the measurement of ρ X (z) is straightforward once H (z) is measured from baryon acoustic oscillations, and Ω m is constrained tightly by the combined data from galaxy clustering, weak lensing, and cosmic microwave background data — although strictly speaking this requires a choice of perturbation evolution for the dark energy as well, and in addition one that is not degenerate with the evolution of dark matter perturbations; see [534].

Another useful possibility is to adopt the principal component approach [468], which avoids any assumption about the form of w and assumes it to be constant or linear in redshift bins, then derives which combination of parameters is best constrained by each experiment.

For a cross-check of the results using more complicated parameterizations, one can use simple polynomial parameterizations of w and ρDE(z)DE(0) [939].

Perturbations

This section is devoted to a discussion of linear perturbation theory in dark-energy models. Since we will discuss a number of non-standard models in later sections, we present here the main equations in a general form that can be adapted to various contexts. This section will identify which perturbation functions the Euclid survey [551] will try to measure and how they can help us to characterize the nature of dark energy and the properties of gravity.

Cosmological perturbation theory

Here we provide the perturbation equations in a dark-energy dominated universe for a general fluid, focusing on scalar perturbations.

For simplicity, we consider a flat universe containing only (cold dark) matter and dark energy, so that the Hubble parameter is given by

$${H^2} = {\left({{1 \over a}{{{\rm{d}}a} \over {{\rm{d}}t}}} \right)^2} = H_0^2\left[ {{\Omega _{{m_0}}}{a^{- 3}} + (1 - {\Omega _{{m_0}}})\exp \left({- 3\int\nolimits_1^a {{{1 + w(a\prime)} \over {a\prime}}} \,{\rm{d}}a} \right)} \right].$$
(1.3.1)

We will consider linear perturbations on a spatially-flat background model, defined by the line of element

$${\rm{d}}{s^2} = {a^2}[ - (1 + 2A)\,{\rm{d}}{\eta ^2} + 2{B_i}\,{\rm{d}}\eta \,{\rm{d}}{x^i} + ((1 + 2{H_L}){\delta _{ij}} + 2{H_{Tij}})\,{\rm{d}}{x_i}\,{\rm{d}}{x^j}],$$
(1.3.2)

where A is the scalar potential; B i a vector shift; H L is the scalar perturbation to the spatial curvature; \(H_T^{ij}\) is the trace-free distortion to the spatial metric; d η = dt/a is the conformal time.

We will assume that the universe is filled with perfect fluids only, so that the energy momentum tensor takes the simple form

$${T^{\mu \nu}} = (\rho + p){u^\mu}{u^\nu} + p\;{g^{\mu \nu}} + {\Pi ^{\mu \nu}}\,,$$
(1.3.3)

where ρ and p are the density and the pressure of the fluid respectively, uμ is the four-velocity and Πμν is the anisotropic-stress perturbation tensor that represents the traceless component of the \(T_j^i\).

The components of the perturbed energy momentum tensor can be written as:

$$T_0^0 = - (\bar \rho + \delta \rho)$$
(1.3.4)
$$T_j^0 = (\bar \rho + \bar p)({v_j} - {B_j})$$
(1.3.5)
$$T_0^i = (\bar \rho + \bar p){v^i}$$
(1.3.6)
$$T_j^i = (\bar p + \delta p)\delta _j^i + \bar p\;\Pi _j^i.$$
(1.3.7)

Here \(\bar \rho\) and \(\bar p\) are the energy density and pressure of the homogeneous and isotropic background universe, δρ is the density perturbation, δp is the pressure perturbation, υi is the velocity vector. Here we want to investigate only the scalar modes of the perturbation equations. So far the treatment of the matter and metric is fully general and applies to any form of matter and metric. We now choose the Newtonian gauge (also known as the longitudinal gauge), characterized by zero non-diagonal metric terms (the shift vector B i = 0 and \(H_T^{ij} = 0\)) and by two scalar potentials Ψ and Φ; the metric Eq. (1.3.2) then becomes

$${\rm{d}}{s^2} = {a^2}[ - (1 + 2\Psi)\,\;{\rm{d}}{\eta ^2} + (1 - 2\Phi)\,{\rm{d}}{x_i}\,{\rm{d}}{x^i}].$$
(1.3.8)

The advantage of using the Newtonian gauge is that the metric tensor g μν is diagonal and this simplifies the calculations. This choice not only simplifies the calculations but is also the most intuitive one as the observers are attached to the points in the unperturbed frame; as a consequence, they will detect a velocity field of particles falling into the clumps of matter and will measure their gravitational potential, represented directly by Ψ; Φ corresponds to the perturbation to the spatial curvature. Moreover, as we will see later, the Newtonian gauge is the best choice for observational tests (i.e., for perturbations smaller than the horizon).

In the conformal Newtonian gauge, and in Fourier space, the first-order perturbed Einstein equations give [see 599, for more details]:

$${k^2}\Phi + 3{{\dot a} \over a}\left({\dot \Phi + {{\dot a} \over a}\Psi} \right) = - 4\pi G{a^2}\sum\limits_\alpha {{{\bar \rho}_\alpha}} {\delta _\alpha}\,,$$
(1.3.9)
$${k^2}\left({\dot \Phi + {{\dot a} \over a}\Psi} \right) = 4\pi G{a^2}\sum\limits_\alpha {({{\bar \rho}_\alpha} + {{\bar p}_\alpha})} {\theta _\alpha}\,,$$
(1.3.10)
$$\ddot \Phi + {{\dot a} \over a}(\dot \Psi + 2\dot \Phi) + \left({2{{\ddot a} \over a} - {{{{\dot a}^2}} \over {{a^2}}}} \right)\Psi + {{{k^2}} \over 3}(\Phi - \Psi) = 4\pi G{a^2}\sum\limits_\alpha \delta {p_\alpha}\,,$$
(1.3.11)
$${k^2}(\Phi - \Psi) = 12\pi G{a^2}\sum\limits_\alpha {({{\bar \rho}_\alpha} + {{\bar p}_\alpha})} \;{\pi _\alpha}\,,$$
(1.3.12)

where a dot denotes \(d/d\eta, {\delta _\alpha} = \delta {\rho _\alpha}/{\bar \rho _\alpha}\), the index α indicates a sum over all matter components in the universe and π is related to \(\Pi _j^i\) through:

$$(\bar \rho + \bar p)\;\pi = - \left({{{\hat k}_i}{{\hat k}_j} - {1 \over 3}{\delta _{ij}}} \right)\Pi _j^i.$$
(1.3.13)

The energy-momentum tensor components in the Newtonian gauge become:

$$T_0^0 = - (\bar \rho + \delta \rho)$$
(1.3.14)
$$i{k_i}T_0^i = - i{k_i}T_i^0 = (\bar \rho + \bar p)\;\theta$$
(1.3.15)
$$T_j^i = (\bar p + \delta p)\;\delta _j^i + \bar p\Pi _j^i$$
(1.3.16)

where we have defined the variable θ = ik j υj that represents the divergence of the velocity field.

Perturbation equations for a single fluid are obtained taking the covariant derivative of the perturbed energy momentum tensor, i.e., \(T_{v;\mu}^\mu = 0\). We have

$$\dot \delta = - (1 + w)\left({\theta - 3\dot \Phi} \right) - 3{{\dot a} \over a}\left({{{\delta p} \over {\bar \rho}} - w\delta} \right)\quad \quad \quad {\rm{for}}\quad \nu = 0$$
(1.3.17)
$$\dot \theta = - {{\dot a} \over a}\left({1 - 3w} \right)\theta - {{\dot w} \over {1 + w}}\theta + {k^2}{{\delta p/\bar \rho} \over {1 + w}} + {k^2}\Psi - {k^2}\pi \quad {\rm{for}}\quad \nu = i{.}$$
(1.3.18)

The equations above are valid for any fluid. The evolution of the perturbations depends on the characteristics of the fluids considered, i.e., we need to specify the equation of state parameter w, the pressure perturbation δp and the anisotropic stress π. For instance, if we want to study how matter perturbations evolve, we simply substitute w = δp = π = 0 (matter is pressureless) in the above equations. However, Eqs. (1.3.17)(1.3.18) depend on the gravitational potentials Ψ and Φ, which in turn depend on the evolution of the perturbations of the other fluids. For instance, if we assume that the universe is filled by dark matter and dark energy then we need to specify δp and for the dark energy.

The problem here is not only to parameterize the pressure perturbation and the anisotropic stress for the dark energy (there is not a unique way to do it, see below, especially Section 1.4.5 for what to do when w crosses −1) but rather that we need to run the perturbation equations for each model we assume, making predictions and compare the results with observations. Clearly, this approach takes too much time. In the following Section 1.3.2 we show a general approach to understanding the observed late-time accelerated expansion of the universe through the evolution of the matter density contrast.

In the following, whenever there is no risk of confusion, we remove the overbars from the background quantities.

Modified growth parameters

Even if the expansion history, H (z), of the FLRW background has been measured (at least up to redshifts ∼ 1 by supernova data), it is not yet possible yet to identify the physics causing the recent acceleration of the expansion of the universe. Information on the growth of structure at different scales and different redshifts is needed to discriminate between models of dark energy (DE) and modified gravity (MG). A definition of what we mean by DE and MG will be postponed to Section 1.4.

An alternative to testing predictions of specific theories is to parameterize the possible departures from a fiducial model. Two conceptually-different approaches are widely discussed in the literature:

  • Model parameters capture the degrees of freedom of DE/MG and modify the evolution equations of the energy-momentum content of the fiducial model. They can be associated with physical meanings and have uniquely-predicted behavior in specific theories of DE and MG.

  • Trigger relations are derived directly from observations and only hold in the fiducial model. They are constructed to break down if the fiducial model does not describe the growth of structure correctly.

As the current observations favor concordance cosmology, the fiducial model is typically taken to be spatially flat FLRW in GR with cold dark matter and a cosmological constant, hereafter referred to as ΛCDM.

For a large-scale structure and weak lensing survey the crucial quantities are the matter-density contrast and the gravitational potentials and we therefore focus on scalar perturbations in the Newtonian gauge with the metric (1.3.8).

We describe the matter perturbations using the gauge-invariant comoving density contrast Λ M δ M + 3aHθ M /k2 where δ M and θ M are the matter density contrast and the divergence of the fluid velocity for matter, respectively. The discussion can be generalized to include multiple fluids.

In ΛCDM, after radiation-matter equality there is no anisotropic stress present and the Einstein constraint equations at “sub-Hubble scales” kaH become

$$- {k^2}\Phi = 4\pi G{a^2}{\rho _M}{\Delta _M}\,,\quad \quad \Phi = \Psi \,.$$
(1.3.19)

These can be used to reduce the energy-momentum conservation of matter simply to the second-order growth equation

$$\Delta _M^{\prime\prime} + [2 + {(\ln H)\prime}]\Delta _M\prime = {3 \over 2}{\Omega _M}(a){\Delta _M}\,.$$
(1.3.20)

Primes denote derivatives with respect to ln a and we define the time-dependent fractional matter density as Ω M (a) = 8πGρ M (a)/(3H2). Notice that the evolution of Δ M is driven by Ω M (a) and is scale-independent throughout (valid on sub- and super-Hubble scales after radiation-matter equality). We define the growth factor G (a) as Δ = Δ0G (a). This is very well approximated by the expression

$$G(a) \approx \exp \left\{{\int\nolimits_1^a {{{{\rm{d}}a\prime} \over {a\prime}}} [{\Omega _M}{{(a\prime)}^\gamma}]} \right\}$$
(1.3.21)

and

$${f_g} \equiv {{{\rm{d}}\log G} \over {{\rm{d}}\log a}} \approx {\Omega _M}{(a)^\gamma}$$
(1.3.22)

defines the growth rate and the growth index γ that is found to be γΛ ≃ 0.545 for the ΛCDM solution [see 937, 585, 466, 363].

Clearly, if the actual theory of structure growth is not the ΛCDM scenario, the constraints (1.3.19) will be modified, the growth equation (1.3.20) will be different, and finally the growth factor (1.3.21) is changed, i.e., the growth index is different from γΛ and may become time and scale dependent. Therefore, the inconsistency of these three points of view can be used to test the ΛCDM paradigm.

Two new degrees of freedom

Any generic modification of the dynamics of scalar perturbations with respect to the simple scenario of a smooth dark-energy component that only alters the background evolution of ΛCDM can be represented by introducing two new degrees of freedom in the Einstein constraint equations. We do this by replacing (1.3.19) with

$$- {k^2}\Phi = 4\pi GQ(a,k){a^2}{\rho _M}{\Delta _M}\,,\qquad \Phi = \eta (a,k)\Psi \,.$$
(1.3.23)

Non-trivial behavior of the two functions Q and η can be due to a clustering dark-energy component or some modification to GR. In MG models the function Q (a, k) represents a mass screening effect due to local modifications of gravity and effectively modifies Newton’s constant. In dynamical DE models Q represents the additional clustering due to the perturbations in the DE. On the other hand, the function η (α, k) parameterizes the effective anisotropic stress introduced by MG or DE, which is absent in ΛCDM.

Given an MG or DE theory, the scale- and time-dependence of the functions Q and η can be derived and predictions projected into the (Q, η) plane. This is also true for interacting dark sector models, although in this case the identification of the total matter density contrast (DM plus baryonic matter) and the galaxy bias become somewhat contrived [see, e.g., 848, for an overview of predictions for different MG/DE models].

Using the above-defined modified constraint equations (1.3.23), the conservation equations of matter perturbations can be expressed in the following form (see [737])

$$\begin{array}{*{20}c} {\Delta _M\prime= - {{1/\eta - 1 + {{(\ln Q)}\prime}} \over {x_Q^2 + {9 \over 2}{\Omega _M}}}\,{9 \over 2}{\Omega _M}{\Delta _M} - {{x_Q^2 - 3{{(\ln H)}\prime}/Q} \over {x_Q^2 + {9 \over 2}{\Omega _M}}}\,{{{\theta _M}} \over {aH}}}\\ {\theta _M\prime = - {\theta _M} - {3 \over 2}aH{\Omega _M}{Q \over \eta}{\Delta _M},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;}\\ \end{array}$$
(1.3.24)

where we define \({x_Q} \equiv k/{(_a}H\sqrt Q)\). Remember Ω M = Ω M (a) as defined above. Notice that it is Q/η that modifies the source term of the θ M equation and therefore also the growth of Δ M . Together with the modified Einstein constraints (1.3.23) these evolution equations form a closed system for (Δ M , θ M , Φ, Ψ) which can be solved for given (Q, η).

The influence of the Hubble scale is modified by Q, such that now the size of x Q determines the behavior of Δ M ; on “sub-Hubble” scales, x Q ≫ 1, we find

$$\Delta _M^{\prime\prime} + [2 + {(\ln H)\prime}]\Delta _M\prime = {3 \over 2}{\Omega _M}(a){Q \over \eta}{\Delta _M}$$
(1.3.25)

and \({\theta _M} = - aH\Delta _M\prime\). The growth equation is only modified by the factor Q/η on the RHS with respect to ΛCDM (1.3.20). On “super-Hubble” scales, x Q ≪ 1, we have

$$\begin{array}{*{20}c} {\Delta _M\prime = - [1/\eta - 1 + {{(\ln Q)}\prime}]\;{\Delta _M} + {2 \over {3{\Omega _M}}}{{{{(\ln H)}\prime}} \over {aH}}{1 \over Q}{\theta _M},}\\ {\theta _M\prime = - {\theta _M} - {3 \over 2}{\Omega _M}\,aH{Q \over \eta}{\Delta _M}\,.\quad \quad \quad \quad \quad \quad \quad \quad \;}\\ \end{array}$$
(1.3.26)

Q and η now create an additional drag term in the Δ M equation, except if η > 1 when the drag term could flip sign. [737] also showed that the metric potentials evolve independently and scale-invariantly on super-Hubble scales as long as x Q → 0 for k → 0. This is needed for the comoving curvature perturbation, ζ, to be constant on super-Hubble scales.

Many different names and combinations of the above defined functions (Q, η) have been used in the literature, some of which are more closely related to actual observables and are less correlated than others in certain situations [see, e.g., 41, 667, 848, 737, 278, 277, 363].

For instance, as observed above, the combination Q/η modifies the source term in the growth equation. Moreover, peculiar velocities are following gradients of the Newtonian potential, Ψ, and therefore the comparison of peculiar velocities with the density field is also sensitive to Q/η. So we define

$$\mu \equiv Q\,/\,\eta \qquad \Rightarrow \qquad - {k^2}\Psi = 4\pi G{a^2}\mu (a,k){\rho _M}{\Delta _M}\,.$$
(1.3.27)

Weak lensing and the integrated Sachs-Wolfe (ISW) effect, on the other hand, are measuring (Φ + Ψ)/2, which is related to the density field via

$$\Sigma \equiv {1 \over 2}Q(1 + 1/\eta) = {1 \over 2}\mu (\eta + 1)\qquad \Rightarrow \qquad - {k^2}(\Phi + \Psi) = 8\pi G{a^2}\Sigma (a,k){\rho _M}{\Delta _M}\,.$$
(1.3.28)

A summary of different other variables used was given by [278]. For instance, the gravitational slip parameter introduced by [194] and widely used is related through ϖ ≡ 1/η − 1. Recently [277] used \(\left\{{{\mathcal G} \equiv \sum, \,\mu \equiv Q,\,{\mathcal V} \equiv \mu} \right\}\), μQ, \(R = 1/\eta\), while [115] defined R ≡ 1/η. All these variables reflect the same two degrees of freedom additional to the linear growth of structure in ΛCDM.

Any combination of two variables out of {Q, η, μ, Σ, …} is a valid alternative to (Q, η). It turns out that the pair (μ, Σ) is particularly well suited when CMB, WL and LSS data are combined as it is less correlated than others [see 980, 277, 68].

Parameterizations and non-parametric approaches

So far we have defined two free functions that can encode any departure of the growth of linear perturbations from ΛCDM. However, these free functions are not measurable, but have to be inferred via their impact on the observables. Therefore, one needs to specify a parameterization of, e.g., (Q, η) such that departures from ΛCDM can be quantified. Alternatively, one can use non-parametric approaches to infer the time and scale-dependence of the modified growth functions from the observations.

Ideally, such a parameterization should be able to capture all relevant physics with the least number of parameters. Useful parameterizations can be motivated by predictions for specific theories of MG/DE [see 848] and/or by pure simplicity and measurability [see 41]. For instance, [980] and [278] use scale-independent parameterizations that model one or two smooth transitions of the modified growth parameters as a function of redshift. [115] also adds a scale dependence to the parameterization, while keeping the time-dependence a simple power law:

$$\begin{array}{*{20}c} {Q(a,k) \equiv 1 + \left[ {{Q_0}{e^{- k/{k_c}}} + {Q_\infty}(1 - {e^{- k/{k_c}}}) - 1} \right]\,{a^s}\,,} \\ {\eta {{(a,k)}^{- 1}} \equiv 1 + \left[ {{R_0}{e^{- k/{k_c}}} + {R_\infty}(1 - {e^{- k/{k_c}}}) - 1} \right]\,{a^s}\,,\;\;} \\ \end{array}$$
(1.3.29)

with constant Q0, Q, R0, R, s and k c . Generally, the problem with any kind of parameterization is that it is difficult — if not impossible — for it to be flexible enough to describe all possible modifications.

Daniel et al. [278, 277] investigate the modified growth parameters binned in z and k. The functions are taken constant in each bin. This approach is simple and only mildly dependent on the size and number of the bins. However, the bins can be correlated and therefore the data might not be used in the most efficient way with fixed bins. Slightly more sophisticated than simple binning is a principal component analysis (PCA) of the binned (or pixelized) modified growth functions. In PCA uncorrelated linear combinations of the original pixels are constructed. In the limit of a large number of pixels the model dependence disappears. At the moment however, computational cost limits the number of pixels to only a few. Zhao et al. [982, 980] employ a PCA in the (μ, η) plane and find that the observables are more strongly sensitive to the scale-variation of the modified growth parameters rather than the time-dependence and their average values. This suggests that simple, monotonically or mildly-varying parameterizations as well as only time-dependent parameterizations are poorly suited to detect departures from ΛCDM.

Trigger relations

A useful and widely popular trigger relation is the value of the growth index γ in ΛCDM. It turns out that the value of γ can also be fitted also for simple DE models and sub-Hubble evolution in some MG models [see, e.g., 585, 466, 587, 586, 692, 363]. For example, for a non-clustering perfect fluid DE model with equation of state w (z) the growth factor G (a) given in (1.3.21) with the fitting formula

$$\gamma = 0{.}55 + 0{.}05[1 + w(z = 1)]$$
(1.3.30)

is accurate to the 10−3 level compared with the actual solution of the growth equation (1.3.20). Generally, for a given solution of the growth equation the growth index can simply be computed using

$$\gamma (a,k) = {{\ln (\Delta _M\prime) - \ln {\Delta _M}} \over {\ln {\Omega _M}(a)}}\,.$$
(1.3.31)

The other way round, the modified gravity function μ can be computed for a given γ [737]

$$\mu = {2 \over 3}\Omega _M^{\gamma - 1}(a)[\Omega _M^\gamma (a) + 2 + {(\ln H)\prime} - 3\gamma + \gamma \prime \ln \gamma ].$$
(1.3.32)

The fact that the value of γ is quite stable in most DE models but strongly differs in MG scenarios means that a large deviation from γΛ signifies the breakdown of GR, a substantial DE clustering or a breakdown of another fundamental hypothesis like near-homogeneity. Furthermore, using the growth factor to describe the evolution of linear structure is a very simple and computationally cheap way to carry out forecasts and compare theory with data. However, several drawbacks of this approach can be identified:

  • As only one additional parameter is introduced, a second parameter, such as η, is needed to close the system and be general enough to capture all possible modifications.

  • The growth factor is a solution of the growth equation on sub-Hubble scales and, therefore, is not general enough to be consistent on all scales.

  • The framework is designed to describe the evolution of the matter density contrast and is not easily extended to describe all other energy-momentum components and integrated into a CMB-Boltzmann code.

Models of dark energy and modified gravity

In this section we review a number of popular models of dynamical DE and MG. This section is more technical than the rest and it is meant to provide a quick but self-contained review of the current research in the theoretical foundations of DE models. The selection of models is of course somewhat arbitrary but we tried to cover the most well-studied cases and those that introduce new and interesting observable phenomena.

Quintessence

In this review we refer to scalar field models with canonical kinetic energy in Einstein’s gravity as “quintessence models”. Scalar fields are obvious candidates for dark energy, as they are for the inflaton, for many reasons: they are the simplest fields since they lack internal degrees of freedom, do not introduce preferred directions, are typically weakly clustered (as discussed later on), and can easily drive an accelerated expansion. If the kinetic energy has a canonical form, the only degree of freedom is then provided by the field potential (and of course by the initial conditions). The typical requirement is that the potentials are flat enough to lead to the slow-roll inflation today with an energy scale \({\rho _{DE}} \simeq {10^{- 123}}m_{pl}^4\) and a mass scale m ϕ ≲ 10−33 eV.

Quintessence models are the protoypical DE models [195] and as such are the most studied ones. Since they have been explored in many reviews of DE, we limit ourselves here to a few remarks.Footnote 2

The quintessence model is described by the action

$$S = \int {{{\rm{d}}^4}} x\sqrt {- g} \,\left[ {{1 \over {2{\kappa ^2}}}R + {{\mathcal L}_\phi}} \right] + {S_M}\,,\qquad {{\mathcal L}_\phi} = - {1 \over 2}{g^{\mu \nu}}{\partial _\mu}\phi {\partial _\nu}\phi - V(\phi)\,,$$
(1.4.1)

where κ2 = 8πG and R is the Ricci scalar and S M is the matter action. The fluid satisfies the continuity equation

$${\dot \rho _M} + 3H({\rho _M} + {p_M}) = 0\,.$$
(1.4.2)

The energy-momentum tensor of quintessence is

$$T_{\mu \nu}^{(\phi)} = - {2 \over {\sqrt {- g}}}{{\delta (\sqrt {- g} {{\mathcal L}_\phi})} \over {\delta {g^{\mu \nu}}}}$$
(1.4.3)
$$= {\partial _\mu}\phi {\partial _\nu}\phi - {g_{\mu \nu}}\left[ {{1 \over 2}{g^{\alpha \beta}}{\partial _\alpha}\phi {\partial _\beta}\phi + V(\phi)} \right]\,.$$
(1.4.4)

As we have already seen, in a FLRW background, the energy density ρ ϕ and the pressure p ϕ of

$${\rho _\phi} = - T_{0}^{0^{(\phi)}} = {1 \over 2}{\dot \phi ^2} + V(\phi)\,,\quad {p_\phi} = {1 \over 3}T_{i}^{i^{(\phi)}} = {1 \over 2}{\dot \phi ^2} - V(\phi)\,,$$
(1.4.5)

which give the equation of state

$${w_\phi} \equiv {{{p_\phi}} \over {{\rho _\phi}}} = {{{{\dot \phi}^2} - 2V(\phi)} \over {{{\dot \phi}^2} + 2V(\phi)}}\,.$$
(1.4.6)

In the flat universe, Einstein’s equations give the following equations of motion:

$${H^2} = {{{\kappa ^2}} \over 3}\left[ {{1 \over 2}{{\dot \phi}^2} + V(\phi) + {\rho _M}} \right]\,,$$
(1.4.7)
$$\dot H = - {{{\kappa ^2}} \over 2}\left({{{\dot \phi}^2} + {\rho _M} + {p_M}} \right)\,,$$
(1.4.8)

where κ2 = 8πG. The variation of the action (1.4.1) with respect to ϕ gives

$$\ddot \phi + 3H\dot \phi + {V_{,\phi}} = 0\,,$$
(1.4.9)

where V,ϕ = dV/dϕ.

During radiation or matter dominated epochs, the energy density ρ M of the fluid dominates over that of quintessence, i.e., ρ M ρ ϕ . If the potential is steep so that the condition \({\dot \phi ^2}/2 \gg V\left(\phi \right)\) is always satisfied, the field equation of state is given by w ϕ ≃ 1 from Eq. (1.4.6). In this case the energy density of the field evolves as ρ ϕ a−6, which decreases much faster than the background fluid density.

The condition w ϕ < −1/3 is required to realize the late-time cosmic acceleration, which translates into the condition \({\dot \phi ^2} < V\left(\phi \right)\). Hence the scalar potential needs to be shallow enough for the field to evolve slowly along the potential. This situation is similar to that in inflationary cosmology and it is convenient to introduce the following slow-roll parameters [104]

$${\epsilon _s} \equiv {1 \over {2{\kappa ^2}}}{\left({{{{V_{,\phi}}} \over V}} \right)^2}\,,\qquad {\eta _s} \equiv {{{V_{,\phi \phi}}} \over {{\kappa ^2}V}}\,.$$
(1.4.10)

if the conditions ϵ s ≪ 1 and ∣η s ∣ ≪ 1 are satisfied, the evolution of the field is sufficiently slow so that \({\dot \phi ^2} \ll V\left(\phi \right)\) and \(\vert \ddot \phi \vert \ll \vert 3{H\phi}\vert\) in Eqs. (1.4.7) and (1.4.9).

From Eq. (1.4.9) the deviation of w ϕ from −1 is given by

$$1 + {w_\phi} = {{V_{,\phi}^2} \over {9{H^2}{{({\xi _s} + 1)}^2}{\rho _\phi}}}\,,$$
(1.4.11)

where \({\xi _s} \equiv \ddot \phi/\left({3H\dot \phi} \right)\). This shows that w ϕ is always larger than −1 for a positive potential and energy density. In the slow-roll limit, ∣ξ s ∣ ≪ 1 and \({\dot \phi ^2}/2 \ll V\left(\phi \right)\), we obtain 1 + w ϕ ≃ 2ϵ s /3 by neglecting the matter fluid in Eq. (1.4.7), i.e., 3H2κ2V (ϕ). The deviation of w ϕ from −1 is characterized by the slow-roll parameter ϵ s . It is also possible to consider Eq. (1.4.11) as a prescription for the evolution of the potential given w ϕ (z) and to reconstruct a potential that gives a desired evolution of the equation of state (subject to w ∈ [−1, 1]). This was used, for example, in [102].

However, in order to study the evolution of the perturbations of a quintessence field it is not even necessary to compute the field evolution explicitly. Rewriting the perturbation equations of the field in terms of the perturbations of the density contrast δ ϕ and the velocity θ ϕ in the conformal Newtonian gauge, one finds [see, e.g., 536, Appendix A] that they correspond precisely to those of a fluid, (1.3.17) and (1.3.18), with π = 0 and \(\delta p = c_s^2\delta \rho + 3aH\left({c_s^2 - c_a^2} \right)\left({1 + w} \right){\rho ^\theta}/{k^2}\) with \(c_s^2 = 1\). The adiabatic sound speed, c a , is defined in Eq. (1.4.31). The large value of the sound speed \(c_s^2\), equal to the speed of light, means that quintessence models do not cluster significantly inside the horizon [see 785, 786, and Section 1.8.6 for a detailed analytical discussion of quintessence clustering and its detectability with future probes, for arbitrary \(c_s^2\) ].

Many quintessence potentials have been proposed in the literature. A simple crude classification divides them into two classes, (i) “freezing” models and (ii) “thawing” models [196]. In class (i) the field was rolling along the potential in the past, but the movement gradually slows down after the system enters the phase of cosmic acceleration. The representative potentials that belong to this class are

  1. (i)

    Freezing models

    • V (ϕ) = M4+nϕn (n > 0),

    • \(V\left(\phi \right) = {M^{4 + n}}{\phi ^{- n}}{\rm{exp}}\left({\alpha {\phi ^2}/m_{{\rm{pl}}}^2} \right)\).

The former potential does not possess a minimum and hence the field rolls down the potential toward infinity. This appears, for example, in the fermion condensate model as a dynamical supersymmetry breaking [138]. The latter potential has a minimum at which the field is eventually trapped (corresponding to w ϕ = −1). This potential can be constructed in the framework of supergravity [170].

In thawing models (ii) the field (with mass m ϕ ) has been frozen by Hubble friction (i.e., the term \(H\dot \phi\) in Eq. (1.4.9)) until recently and then it begins to evolve once H drops below m ϕ . The equation of state of DE is w ϕ ≃ −1 at early times, which is followed by the growth of w ϕ . The representative potentials that belong to this class are

  1. (ii)

    Thawing models

    • V (ϕ) = V0 + M4−nϕn (n > 0),

    • V (ϕ) = M4 cos2 (ϕ/f).

The former potential is similar to that of chaotic inflation (n = 2, 4) used in the early universe (with V0 = 0) [577], while the mass scale M is very different. The model with n = 1 was proposed by [487] in connection with the possibility to allow for negative values of V (ϕ). The universe will collapse in the future if the system enters the region with V (ϕ) < 0. The latter potential appears as a potential for the Pseudo-Nambu-Goldstone Boson (PNGB). This was introduced by [370] in response to the first tentative suggestions that the universe may be dominated by the cosmological constant. In this model the field is nearly frozen at the potential maximum during the period in which the field mass m ϕ is smaller than H, but it begins to roll down around the present (m ϕ H0).

Potentials can also be classified in several other ways, e.g., on the basis of the existence of special solutions. For instance, tracker solutions have approximately constant w ϕ and Ω ϕ along special attractors. A wide range of initial conditions converge to a common, cosmic evolutionary tracker. Early DE models contain instead solutions in which DE was not negligible even during the last scattering. While in the specific Euclid forecasts section (1.8) we will not explicitly consider these models, it is worthwhile to note that the combination of observations of the CMB and of large scale structure (such as Euclid) can dramatically constrain these models drastically improving the inverse area figure of merit compared to current constraints, as discussed in [467].

K-essence

In a quintessence model it is the potential energy of a scalar field that leads to the late-time acceleration of the expansion of the universe; the alternative, in which the kinetic energy of the scalar field which dominates, is known as k-essence. Models of k-essence are characterized by an action for the scalar field of the following form

$$S = \int {{{\rm{d}}^4}} x\sqrt {- g} p(\phi, X)\,,$$
(1.4.12)

where X = (1/2)gμν μ ϕ∇ ν ϕ. The energy density of the scalar field is given by

$${\rho _\phi} = 2X{{{\rm{d}}p} \over {{\rm{d}}X}} - p\,,$$
(1.4.13)

and the pressure is simply p ϕ = p (ϕ, X). Treating the k-essence scalar as a perfect fluid, this means that k-essence has the equation of state

$${w_\phi} = {{{p_\phi}} \over {{\rho _\phi}}} = - {p \over {p - 2X{p{,_X}}}}\,,$$
(1.4.14)

where the subscript, x indicates a derivative with respect to X. Clearly, with a suitably chosen p the scalar can have an appropriate equation of state to allow it to act as dark energy.

The dynamics of the k-essence field are given by a continuity equation

$${\dot \rho _\phi} = - 3H({\rho _\phi} + {p_\phi})\,,$$
(1.4.15)

or equivalently by the scalar equation of motion

$${G^{\mu \nu}}{\nabla _\mu}{\nabla _\nu}\phi + 2X{{{\partial ^2}p} \over {\partial X\partial \phi}} - {{\partial p} \over {\partial \phi}} = 0\,,$$
(1.4.16)

where

$${G^{\mu \nu}} = {{\partial p} \over {\partial X}}{g^{\mu \nu}} + {{{\partial ^2}p} \over {\partial {X^2}}}{\nabla ^\mu}\phi {\nabla ^\nu}\phi \,.$$
(1.4.17)

For this second order equation of motion to be hyperbolic, and hence physically meaningful, we must impose

$$1 + 2X{{p{,_{XX}}} \over {p{,_X}}} > 0\,.$$
(1.4.18)

K-essence was first proposed by [61, 62], where it was also shown that tracking solutions to this equation of motion, which are attractors in the space of solutions, exist during the radiation and matter-dominated eras for k-essence in a similar manner to quintessence.

The speed of sound for k-essence fluctuation is

$$c_s^2 = {{p{,_X}} \over {p{,_X} + 2Xp{,_{XX}}}}\,.$$
(1.4.19)

So that whenever the kinetic terms for the scalar field are not linear in X, the speed of sound of fluctuations differs from unity. It might appear concerning that superluminal fluctuations are allowed in k-essence models, however it was shown in [71] that this does not lead to any causal paradoxes.

A definition of modified gravity

In this review we often make reference to DE and MG models. Although in an increasing number of publications a similar dichotomy is employed, there is currently no consensus on where to draw the line between the two classes. Here we will introduce an operational definition for the purpose of this document.

Roughly speaking, what most people have in mind when talking about standard dark energy are models of minimally-coupled scalar fields with standard kinetic energy in 4-dimensional Einstein gravity, the only functional degree of freedom being the scalar potential. Often, this class of model is referred to simply as “quintessence”. However, when we depart from this picture a simple classification is not easy to draw. One problem is that, as we have seen in the previous sections, both at background and at the perturbation level, different models can have the same observational signatures [537]. This problem is not due to the use of perturbation theory: any modification to Einstein’s equations can be interpreted as standard Einstein gravity with a modified “matter” source, containing an arbitrary mixture of scalars, vectors and tensors [457, 535].

The simplest example can be discussed by looking at Eqs. (1.3.23). One can modify gravity and obtain a modified Poisson equation, and therefore Q ≠ 1, or one can introduce a clustering dark energy (for example a k-essence model with small sound speed) that also induces the same Q ≠ 1 (see Eq. 1.3.23). This extends to the anisotropic stress η: there is in general a one-to-one relation at first order between a fluid with arbitrary equation of state, sound speed, and anisotropic stress and a modification of the Einstein-Hilbert Lagrangian.

Therefore, we could simply abandon any attempt to distinguish between DE and MG, and just analyse different models, comparing their properties and phenomenology. However, there is a possible classification that helps us set targets for the observations, which is often useful in concisely communicating the results of complex arguments. In this review, we will use the following notation:

  • Standard dark energy: These are models in which dark energy lives in standard Einstein gravity and does not cluster appreciably on sub-horizon scales. As already noted, the prime example of a standard dark-energy model is a minimally-coupled scalar field with standard kinetic energy, for which the sound speed equals the speed of light.

  • Clustering dark energy: In clustering dark-energy models, there is an additional contribution to the Poisson equation due to the dark-energy perturbation, which induces Q ≠ 1. However, in this class we require η = 1, i.e., no extra effective anisotropic stress is induced by the extra dark component. A typical example is a k-essence model with a low sound speed, \(c_s^2 \ll 1\).

  • Explicit modified gravity models: These are models where from the start the Einstein equations are modified, for example scalar-tensor and f (R) type theories, Dvali-Gabadadze-Porrati (DGP) as well as interacting dark energy, in which effectively a fifth force is introduced in addition to gravity. Generically they change the clustering and/or induce a non-zero anisotropic stress. Since our definitions are based on the phenomenological parameters, we also add dark-energy models that live in Einstein’s gravity but that have non-vanishing anisotropic stress into this class since they cannot be distinguished by cosmological observations.

Notice that both clustering dark energy and explicit modified gravity models lead to deviations from what is often called ‘general relativity’ (or, like here, standard dark energy) in the literature when constraining extra perturbation parameters like the growth index γ. For this reason we generically call both of these classes MG models. In other words, in this review we use the simple and by now extremely popular (although admittedly somewhat misleading) expression “modified gravity” to denote models in which gravity is modified and/or dark energy clusters or interacts with other fields. Whenever we feel useful, we will remind the reader of the actual meaning of the expression “modified gravity” in this review.

Therefore, on sub-horizon scales and at first order in perturbation theory our definition of MG is straightforward: models with Q = η = 1 (see Eq. 1.3.23) are standard DE, otherwise they are MG models. In this sense the definition above is rather convenient: we can use it to quantify, for instance, how well Euclid will distinguish between standard dynamical dark energy and modified gravity by forecasting the errors on Q, η or on related quantities like the growth index γ.

On the other hand, it is clear that this definition is only a practical way to group different models and should not be taken as a fundamental one. We do not try to set a precise threshold on, for instance, how much dark energy should cluster before we call it modified gravity: the boundary between the classes is therefore left undetermined but we think this will not harm the understanding of this document.

Coupled dark-energy models

A first class of models in which dark energy shows dynamics, in connection with the presence of a fifth force different from gravity, is the case of ‘interacting dark energy’: we consider the possibility that dark energy, seen as a dynamical scalar field, may interact with other components in the universe. This class of models effectively enters in the “explicit modified gravity models” in the classification above, because the gravitational attraction between dark matter particles is modified by the presence of a fifth force. However, we note that the anisotropic stress for DE is still zero in the Einstein frame, while it is, in general, non-zero in the Jordan frame. In some cases (when a universal coupling is present) such an interaction can be explicitly recast in a non-minimal coupling to gravity, after a redefinition of the metric and matter fields (Weyl scaling). We would like to identify whether interactions (couplings) of dark energy with matter fields, neutrinos or gravity itself can affect the universe in an observable way.

In this subsection we give a general description of the following main interacting scenarios:

  1. 1.

    couplings between dark energy and baryons;

  2. 2.

    couplings between dark energy and dark matter (coupled quintessence);

  3. 3.

    couplings between dark energy and neutrinos (growing neutrinos, MaVaNs);

  4. 4.

    universal couplings with all species (scalar-tensor theories and f (R)).

In all these cosmologies the coupling introduces a fifth force, in addition to standard gravitational attraction. The presence of a new force, mediated by the DE scalar field (sometimes called the ‘cosmon’ [954], seen as the mediator of a cosmological interaction) has several implications and can significantly modify the process of structure formation. We will discuss cases (2) and (3) in Section 2.

In these scenarios the presence of the additional interaction couples the evolution of components that in the standard Λ-FLRW would evolve independently. The stress-energy tensor \({T^\mu}_v\) of each species is, in general, not conserved — only the total stress-energy tensor is. Usually, at the level of the Lagrangian, the coupling is introduced by allowing the mass m of matter fields to depend on a scalar field ϕ via a function m (ϕ) whose choice specifies the interaction. This wide class of cosmological models can be described by the following action:

$${\mathcal S} = \int {{{\rm{d}}^4}x\sqrt {- g} \left[ {- {1 \over 2}{\partial ^\mu}\phi {\partial _\mu}\phi - U(\phi) - m(\phi)\bar \psi \psi + {{\mathcal L}_{{\rm{kin}}}}[\psi ]} \right]},$$
(1.4.20)

where U (ϖ) is the potential in which the scalar field ϕ rolls, Ψ describes matter fields, and g is defined in the usual way as the determinant of the metric tensor, whose background expression is g μν = diag[−a2, a2, a2, a2].

For a general treatment of background and perturbation equations we refer to [514, 33, 35, 724]. Here the coupling of the dark-energy scalar field to a generic matter component (denoted by index α) is treated as an external source Q(α)μ in the Bianchi identities:

$${\nabla _\nu}T_{(\alpha)\mu}^\nu = {Q_{(\alpha)\mu}}\,,$$
(1.4.21)

with the constraint

$$\sum\limits_\alpha {{Q_{(\alpha)\mu}}} = 0.$$
(1.4.22)

The zero component of (1.4.21) gives the background conservation equations:

$${{{\rm{d}}{\rho _\phi}} \over {{\rm{d}}\eta}} = - 3{\mathcal H}(1 + {w_\phi}){\rho _\phi} + \beta (\phi){{{\rm{d}}\phi} \over {{\rm{d}}\eta}}(1 - 3{w_\alpha}){\rho _\alpha}\,,$$
(1.4.23)
$${{{\rm{d}}{\rho _\alpha}} \over {{\rm{d}}\eta}} = - 3{\mathcal H}(1 + {w_\alpha}){\rho _\alpha} - \beta (\phi){{{\rm{d}}\phi} \over {{\rm{d}}\eta}}(1 - 3{w_\alpha}){\rho _\alpha}\,,$$
(1.4.24)

for a scalar field ϕ coupled to one single fluid α with a function β (ϕ), which in general may not be constant. The choice of the mass function m (ϕ) corresponds to a choice of β (ϕ) and equivalently to a choice of the source Q(α)μ and specifies the strength of the coupling according to the following relations:

$${Q_{(\phi)\mu}} = {{\partial \ln m(\phi)} \over {\partial \phi}}{T_\alpha}\,{\partial _\mu}\phi \,,\,{m_\alpha} = {\bar m_\alpha}{e^{- \beta (\phi)\phi}}\,,$$
(1.4.25)

where \({\bar m_\alpha}\) is the constant Jordan-frame bare mass. The evolution of dark energy is related to the trace T α and, as a consequence, to density and pressure of the species α. We note that a description of the coupling via an action such as (1.4.20) is originally motivated by the wish to modify GR with an extension such as scalar-tensor theories. In general, one of more couplings can be active [176].

As for perturbation equations, it is possible to include the coupling in a modified Euler equation:

$${{{\rm{d}}{{\bf{v}}_\alpha}} \over {{\rm{d}}\eta}} + \left({{\mathcal H} - \beta (\phi){{{\rm{d}}\phi} \over {{\rm{d}}\eta}}} \right){{\bf{v}}_\alpha} - \nabla [{\Phi _\alpha} + \beta \phi ] = 0\,.$$
(1.4.26)

The Euler equation in cosmic time (dt = a dτ) can also be rewritten in the form of an acceleration equation for particles at position r:

$${\dot {\bf{v}}_\alpha} = - \tilde H{{\bf{v}}_\alpha} - \nabla {{{{\tilde G}_\alpha}{m_\alpha}} \over r}\,.$$
(1.4.27)

The latter expression explicitly contains all the main ingredients that affect dark-energy interactions:

  1. 1.

    a fifth force ∇[Φ α + βϕ ] with an effective \({\tilde G_a} = {G_N}\left[ {1 + 2{\beta ^2}\left(\phi \right)} \right]\);

  2. 2.

    a velocity dependent term \({\tilde H_{{\rm{V}}\alpha}} \equiv H\left({1 - \beta \left(\phi \right){{\dot \phi} \over H}} \right){{\rm{V}}_\alpha}\)

  3. 3.

    a time-dependent mass for each particle α, evolving according to (1.4.25).

The relative significance of these key ingredients can lead to a variety of potentially observable effects, especially on structure formation. We will recall some of them in the following subsections as well as, in more detail, for two specific couplings in the dark matter section (2.11, 2.9) of this report.

Dark energy and baryons

A coupling between dark energy and baryons is active when the baryon mass is a function of the dark-energy scalar field: m b = m b (ϕ). Such a coupling is constrained to be very small: main bounds come from tests of the equivalence principle and solar system constraints [130]. More in general, depending on the coupling, bounds on the variation of fundamental constants over cosmological time-scales may have to be considered ([631, 303, 304, 639] and references therein). It is presumably very difficult to have significant cosmological effects due to a coupling to baryons only. However, uncoupled baryons can still play a role in the presence of a coupling to dark matter (see Section 1.6 on nonlinear aspects).

Dark energy and dark matter

An interaction between dark energy and dark matter (CDM) is active when CDM mass is a function of the dark-energy scalar field: m c = m c (ϕ). In this case the coupling is not affected by tests on the equivalence principle and solar-system constraints and can therefore be stronger than the one with baryons. One may argue that dark-matter particles are themselves coupled to baryons, which leads, through quantum corrections, to direct coupling between dark energy and baryons. The strength of such couplings can still be small and was discussed in [304] for the case of neutrino-dark-energy couplings. Also, quantum corrections are often recalled to spoil the flatness of a quintessence potential. However, it may be misleading to calculate quantum corrections up to a cutoff scale, as contributions above the cutoff can possibly compensate terms below the cutoff, as discussed in [958].

Typical values of β presently allowed by observations (within current CMB data) are within the range 0 < β < 0.06 (at 95% CL for a constant coupling and an exponential potential) [114, 47, 35, 44], or possibly more [539, 531] if neutrinos are taken into account or for more realistic time-dependent choices of the coupling. This framework is generally referred to as ‘coupled quintessence’ (CQ). Various choices of couplings have been investigated in literature, including constant and varying β (ϕ) [33, 619, 35, 518, 414, 747, 748, 724, 377].

The presence of a coupling (and therefore, of a fifth force acting among dark-matter particles) modifies the background expansion and linear perturbations [34, 33, 35], therefore affecting CMB and cross-correlation of CMB and LSS [44, 35, 47, 45, 114, 539, 531, 970, 612, 42].

Furthermore, structure formation itself is modified [604, 618, 518, 611, 870, 3, 666, 129, 962, 79, 76, 77, 80, 565, 562, 75, 980, 640].

An alternative approach, also investigated in the literature [619, 916, 915, 613, 387, 388, 193, 794, 192], where the authors consider as a starting point Eq. (1.4.21): the coupling is then introduced by choosing directly a covariant stress-energy tensor on the RHS of the equation, treating dark energy as a fluid and in the absence of a starting action. The advantage of this approach is that a good parameterization allows us to investigate several models of dark energy at the same time. Problems connected to instabilities of some parameterizations or to the definition of a physically-motivated speed of sound for the density fluctuations can be found in [916]. It is also possible to both take a covariant form for the coupling and a quintessence dark-energy scalar field, starting again directly from Eq. (1.4.21). This has been done, e.g., in [145], [144]. At the background level only, [235], [237], [302] and [695] have also considered which background constraints can be obtained when starting from a fixed present ratio of dark energy and dark matter. The disadvantage of this approach is that it is not clear how to perturb a coupling that has been defined as a background quantity.

A Yukawa-like interaction was investigated [357, 279], pointing out that coupled dark energy behaves as a fluid with an effective equation of state w ≲ −1, though staying well defined and without the presence of ghosts [279].

For an illustration of observable effects related to dark-energy-dark-matter interaction see also Section (2.11) of this report.

Dark energy and neutrinos

A coupling between dark energy and neutrinos can be even stronger than the one with dark matter and as compared to gravitational strength. Typical values of β are order 50–100 or even more, such that even the small fraction of cosmic energy density in neutrinos can have a substantial influence on the time evolution of the quintessence field. In this scenario neutrino masses change in time, depending on the value of the dark-energy scalar field ϕ. Such a coupling has been investigated within MaVaNs [356, 714, 135, 12, 952, 280, 874, 856, 139, 178, 177] and more recently within growing neutrino cosmologies [36, 957, 668, 963, 962, 727, 179, 78]. In this latter case, DE properties are related to the neutrino mass and to a cosmological event, i.e., neutrinos becoming non-relativistic. This leads to the formation of stable neutrino lumps [668, 963, 78] at very large scales only (∼ 100 Mpc and beyond) as well as to signatures in the CMB spectra [727]. For an illustration of observable effects related to this case see Section (2.9) of this report.

Scalar-tensor theories

Scalar-tensor theories [954, 471, 472, 276, 216, 217, 955, 912, 722, 354, 146, 764, 721, 797, 646, 725, 726, 205, 54] extend GR by introducing a non-minimal coupling between a scalar field (acting also as dark energy) and the metric tensor (gravity); they are also sometimes referred to as “extended quintessence”. We include scalar-tensor theories among “interacting cosmologies” because, via a Weyl transformation, they are equivalent to a GR framework (minimal coupling to gravity) in which the dark-energy scalar field ϕ is coupled (universally) to all species [954, 608, 936, 351, 724, 219]. In other words, these theories correspond to the case where, in action (1.4.20), the mass of all species (baryons, dark matter, …) is a function m = m (ϕ) with the same coupling for every species α. Indeed, a description of the coupling via an action such as (1.4.20) is originally motivated by extensions of GR such as scalar-tensor theories. Typically the strength of the scalar-mediated interaction is required to be orders of magnitude weaker than gravity ([553], [725] and references therein for recent constraints). It is possible to tune this coupling to be as small as is required — for example by choosing a suitably flat potential V (ϕ) for the scalar field. However, this leads back to naturalness and fine-tuning problems.

In Sections 1.4.6 and 1.4.7 we will discuss in more detail a number of ways in which new scalar degrees of freedom can naturally couple to standard model fields, while still being in agreement with observations. We mention here only that the presence of chameleon mechanisms [171, 672, 670, 172, 464, 173, 282] can, for example, modify the coupling depending on the environment. In this way, a small (screened) coupling in high-density regions, in agreement with observations, is still compatible with a bigger coupling (β ∼ 1) active in low density regions. In other words, a dynamical mechanism ensures that the effects of the coupling are screened in laboratory and solar system tests of gravity.

Typical effects of scalar-tensor theories on CMB and structure formation include:

  • enhanced ISW [725, 391, 980];

  • violation of the equivalence principle: extended objects such as galaxies do not all fall at the same rate [45, 464].

However, it is important to remark that screening mechanisms are meant to protect the scalar field in high-density regions (and therefore allow for bigger couplings in low density environments) but they do not address problems related to self-acceleration of the DE scalar field, which still usually require some fine-tuning to match present observations on w. f (R) theories, which can be mapped into a subclass of scalar-tensor theories, will be discussed in more detail in Section 1.4.6.

Phantom crossing

In this section we pay attention to the evolution of the perturbations of a general dark-energy fluid with an evolving equation of state parameter w. Current limits on the equation of state parameter w = p/ρ of the dark energy indicate that p ≈ −ρ, and so do not exclude p < −ρ, a region of parameter space often called phantom energy. Even though the region for which w < −1 may be unphysical at the quantum level, it is still important to probe it, not least to test for coupled dark energy and alternative theories of gravity or higher dimensional models that can give rise to an effective or apparent phantom energy.

Although there is no problem in considering w < −1 for the background evolution, there are apparent divergences appearing in the perturbations when a model tries to cross the limit w = −1. This is a potential headache for experiments like Euclid that directly probe the perturbations through measurements of the galaxy clustering and weak lensing. To analyze the Euclid data, we need to be able to consider models that cross the phantom divide w = −1 at the level of first-order perturbations (since the only dark-energy model that has no perturbations at all is the cosmological constant).

However, at the level of cosmological first-order perturbation theory, there is no fundamental limitation that prevents an effective fluid from crossing the phantom divide.

As w → −1 the terms in Eqs. (1.3.17) and (1.3.18) containing 1/(1 + w) will generally diverge. This can be avoided by replacing θ with a new variable V defined via V = ρ (1 + w) θ. This corresponds to rewriting the 0-i component of the energy momentum tensor as \(i{k_j}T_0^j = V\), which avoids problems if \(T_0^j \ne 0\) when \(\bar p = - \bar \rho\). Replacing the time derivatives by a derivative with respect to the logarithm of the scale factor ln a (denoted by a prime), we obtain [599, 450, 536]:

$$\delta\prime = 3(1 + w)\Phi\prime - {V \over {Ha}} - 3\left({{{\delta p} \over {\bar \rho}} - w\delta} \right)$$
(1.4.28)
$$V\prime = - (1 - 3w)V + {{{k^2}} \over {Ha}}{{\delta p} \over {\bar \rho}} + (1 + w){{{k^2}} \over {Ha}}\left({\Psi - \pi} \right)\,.$$
(1.4.29)

In order to solve Eqs. (1.4.28) and (1.4.29) we still need to specify the expressions for δp and π, quantities that characterize the physical, intrinsic nature of the dark-energy fluid at first order in perturbation theory. While in general the anisotropic stress plays an important role as it gives a measure of how the gravitational potentials Φ and Ψ differ, we will set it in this section to zero, π = 0. Therefore, we will focus on the form of the pressure perturbation. There are two important special cases: barotropic fluids, which have no internal degrees of freedom and for which the pressure perturbation is fixed by the evolution of the average pressure, and non-adiabatic fluids like, e.g., scalar fields for which internal degrees of freedom can change the pressure perturbation.

Parameterizing the pressure perturbation

Barotropic fluids. We define a fluid to be barotropic if the pressure p depends strictly only on the energy density ρ: p = p (ρ). These fluids have only adiabatic perturbations, so that they are often called adiabatic. We can write their pressure as

$$p(\rho) = p(\bar \rho + \delta \rho) = p(\bar \rho) + {\left. {{{{\rm{d}}p} \over {{\rm{d}}\rho}}} \right\vert _{\bar \rho}}\delta \rho + O[{(\delta \rho)^2}].$$
(1.4.30)

Here \(p\left({\bar \rho} \right) = \bar p\) is the pressure of the isotropic and homogeneous part of the fluid. The second term in the expansion (1.4.30) can be re-written as

$${\left. {{{{\rm{d}}p} \over {{\rm{d}}\rho}}} \right\vert _{\bar \rho}} = {{\dot \bar p} \over {\dot \bar \rho}} = w - {{\dot w} \over {3aH(1 + w)}} \equiv c_a^2,$$
(1.4.31)

where we used the equation of state and the conservation equation for the dark-energy density in the background. We notice that the adiabatic sound speed \(c_\alpha ^2\) will necessarily diverge for any fluid where w crosses −1.

However, for a perfect barotropic fluid the adiabatic sound speed \(c_\alpha ^2\) turns out to be the physical propagation speed of perturbations. Therefore, it should never be larger than the speed of light — otherwise our theory becomes acausal — and it should never be negative \(\left({c_\alpha ^2 < 0} \right)\) — otherwise classical, and possible quantum, instabilities appear. Even worse, the pressure perturbation

$$\delta p = c_a^2\delta \rho = \left({w - {{\dot w} \over {3aH(1 + w)}}} \right)\delta \rho$$
(1.4.32)

will necessarily diverge if w crosses −1 and δρ ≠ 0. Even if we find a way to stabilize the pressure perturbation, for instance an equation of state parameter that crosses the −1 limit with zero slope (), there will always be the problem of a negative speed of sound that prevents these models from being viable dark-energy candidates.

Non-adiabatic fluids. To construct a model that can cross the phantom divide, we therefore need to violate the constraint that p is a unique function of ρ. At the level of first-order perturbation theory, this amounts to changing the prescription for δp, which now becomes an arbitrary function of k and t. One way out of this problem is to choose an appropriate gauge where the equations are simple; one choice is, for instance, the rest frame of the fluid where the pressure perturbation reads (in this frame)

$$\hat{\delta p} = \hat{c}_s^2\hat{\delta \rho},$$
(1.4.33)

where now the \(\hat c_s^2\) is the speed with which fluctuations in the fluid propagate, i.e., the sound speed. We can write Eq. (1.4.33), with an appropriate gauge transformation, in a form suitable for the Newtonian frame, i.e., for Eqs. (1.4.28) and (1.4.29). We find that the pressure perturbation is given by [347, 112, 214]

$$\delta p = \hat c_s^2\delta \rho + 3aH(a)\left({\hat c_s^2 - c_a^2} \right)\bar \rho {V \over {{k^2}}}.$$
(1.4.34)

The problem here is the presence of \(c_a^2\), which goes to infinity at the crossing and it is impossible that this term stays finite except if V → 0 fast enough or = 0, but this is not, in general, the case.

This divergence appears because for w = −1 the energy momentum tensor Eq. (1.3.3) reads: Tμν = pgμν. Normally the four-velocity uμ is the time-like eigenvector of the energy-momentum tensor, but now all vectors are eigenvectors. So the problem of fixing a unique rest-frame is no longer well posed. Then, even though the pressure perturbation looks fine for the observer in the rest-frame, because it does not diverge, the badly-defined gauge transformation to the Newtonian frame does, as it also contains \(c_a^2\).

Regularizing the divergences

We have seen that neither barotropic fluids nor canonical scalar fields, for which the pressure perturbation is of the type (1.4.34), can cross the phantom divide. However, there is a simple model [called the quintom model 360, 451] consisting of two fluids of the same type as in the previous Section 1.4.5.1 but with a constant w on either side of w = −1. The combination of the two fluids then effectively crosses the phantom divide if we start with wtot > −1, as the energy density in the fluid with w < −1 will grow faster, so that this fluid will eventually dominate and we will end up with wtot < −1.

The perturbations in this scenario were analyzed in detail in [536], where it was shown that in addition to the rest-frame contribution, one also has relative and non-adiabatic perturbations. All these contributions apparently diverge at the crossing, but their sum stays finite. When parameterizing the perturbations in the Newtonian gauge as

$$\delta p(k,t) = \gamma (k,t)\,\delta \rho (k,t)$$
(1.4.35)

the quantity γ will, in general, have a complicated time and scale dependence. The conclusion of the analysis is that indeed single canonical scalar fields with pressure perturbations of the type (1.4.34) in the Newtonian frame cannot cross w = −1, but that this is not the most general case. More general models have a priori no problem crossing the phantom divide, at least not with the classical stability of the perturbations.

Kunz and Sapone [536] found that a good approximation to the quintom model behavior can be found by regularizing the adiabatic sound speed in the gauge transformation with

$$c_a^2 = w - {{\dot w(1 + w)} \over {3Ha[{{(1 + w)}^2} + \lambda ]}}$$
(1.4.36)

where λ is a tunable parameter which determines how close to w = −1 the regularization kicks in. A value of λ ≈ 1/1000 should work reasonably well. However, the final results are not too sensitive on the detailed regularization prescription.

This result appears also related to the behavior found for coupled dark-energy models (originally introduced to solve the coincidence problem) where dark matter and dark energy interact not only through gravity [33]. The effective dark energy in these models can also cross the phantom divide without divergences [462, 279, 534].

The idea is to insert (by hand) a term in the continuity equations of the two fluids

$${\dot \rho _M} + 3H{\rho _M} = \lambda$$
(1.4.37)
$${\dot \rho _x} + 3H(1 + {w_x}){\rho _x} = - \lambda,$$
(1.4.38)

where the subscripts m, x refer to dark matter and dark energy, respectively. In this approximation, the adiabatic sound speed \(c_a^2\) reads

$$c_{a,x}^2 = {{{{\dot p}_x}} \over {{{\dot \rho}_x}}} = {w_x} - {{\dot{{w_x}}} \over {3aH\left({1 + {w_x}} \right) + \lambda/{\rho _x}}},$$
(1.4.39)

which stays finite at crossing as long as λ ≠ 0.

However in this class of models there are other instabilities arising at the perturbation level regardless of the coupling used, [cf. 916].

A word on perturbations when w = −1

Although a cosmological constant has w = −1 and no perturbations, the converse is not automatically true: w = −1 does not necessarily imply that there are no perturbations. It is only when we set from the beginning (in the calculation):

$$p = - \rho$$
(1.4.40)
$$\delta p = - \delta \rho$$
(1.4.41)
$$\pi = 0\,,$$
(1.4.42)

i.e., Tμνgμν, that we have as a solution δ = V = 0.

For instance, if we set w = −1 and δp = γδρ (where γ can be a generic function) in Eqs. (1.4.28) and (1.4.29) we have δ ≠ 0 and V ≠ 0. However, the solutions are decaying modes due to the \(- {1 \over a}\left({1 - 3w} \right)V\) term so they are not important at late times; but it is interesting to notice that they are in general not zero.

As another example, if we have a non-zero anisotropic stress π then the Eqs. (1.4.28)(1.4.29) will have a source term that will influence the growth of δ and V in the same way as Ψ does (just because they appear in the same way). The (1 + w) term in front of π should not worry us as we can always define the anisotropic stress through

$$\rho (1 + w)\pi = - \left({{{\hat k}_i}{{\hat k}_j} - {1 \over 3}{\delta _{ij}}} \right)\Pi _{\,j}^i\,,$$
(1.4.43)

where \(\Pi _j^i \ne 0\) when ij is the real traceless part of the energy momentum tensor, probably the quantity we need to look at: as in the case of V = (1 + w)θ, there is no need for Π ∝ (1 + w)π to vanish when w = −1.

It is also interesting to notice that when w = −1 the perturbation equations tell us that dark-energy perturbations are not influenced through Ψ and Φ′ (see Eq. (1.4.28) and (1.4.29)). Since Φ and Ψ are the quantities directly entering the metric, they must remain finite, and even much smaller than 1 for perturbation theory to hold. Since, in the absence of direct couplings, the dark energy only feels the other constituents through the terms (1 + w)Ψ and (1 + w)Φ′, it decouples completely in the limit w = −1 and just evolves on its own. But its perturbations still enter the Poisson equation and so the dark matter perturbation will feel the effects of the dark-energy perturbations.

Although this situation may seem contrived, it might be that the acceleration of the universe is just an observed effect as a consequence of a modified theory of gravity. As was shown in [537], any modified gravity theory can be described as an effective fluid both at background and at perturbation level; in such a situation it is imperative to describe its perturbations properly as this effective fluid may manifest unexpected behavior.

f(R) gravity

In parallel to models with extra degrees of freedom in the matter sector, such as interacting quintessence (and k-essence, not treated here), another promising approach to the late-time acceleration enigma is to modify the left-hand side of the Einstein equations and invoke new degrees of freedom, belonging this time to the gravitational sector itself. One of the simplest and most popular extensions of GR and a known example of modified gravity models is the f (R) gravity in which the 4-dimensional action is given by some generic function f (R) of the Ricci scalar R (for an introduction see, e.g., [49]):

$$S = {1 \over {2{\kappa ^2}}}\int {{{\rm{d}}^4}} x\sqrt {- g} f(R) + {S_m}({g_{\mu \nu}},{\Psi _m})\,,$$
(1.4.44)

where as usual Κ2 = 8πG, and S m is a matter action with matter fields Φ m . Here G is a bare gravitational constant: we will see that the observed value will in general be different. As mentioned in the previously, it is possible to show that f (R) theories can be mapped into a subset of scalar-tensor theories and, therefore, to a class of interacting scalar field dark-energy models universally coupled to all species. When seen in the Einstein frame [954, 608, 936, 351, 724, 219], action (1.4.44) can, therefore, be related to the action (1.4.20) shown previously. Here we describe f (R) in the Jordan frame: the matter fields in S m obey standard conservation equations and, therefore, the metric g μν corresponds to the physical frame (which here is the Jordan frame).

There are two approaches to deriving field equations from the action (1.4.44).

  • (I) The metric formalism

    The first approach is the metric formalism in which the connections \(\Gamma _{\beta \gamma}^\alpha\) are the usual connections defined in terms of the metric g μν . The field equations can be obtained by varying the action (1.4.44) with respect to g μν :

    $$F(R){R_{\mu \nu}}(g) - {1 \over 2}f(R){g_{\mu \nu}} - {\nabla _\mu}{\nabla _\nu}F(R) + {g_{\mu \nu}}\square F(R) = {\kappa ^2}{T_{\mu \nu}}\,,$$
    (1.4.45)

    where F (R) ≡ ∂f/∂R (we also use the notation f,R∂f/∂R, f,RR2f/∂R2), and T μν is the matter energy-momentum tensor. The trace of Eq. (1.4.45) is given by

    $$3\,\square F(R) + F(R)R - 2f(R) = {\kappa ^2}T\,,$$
    (1.4.46)

    where T = gμνT μν = − ρ + 3P. Here ρ and P are the energy density and the pressure of the matter, respectively.

  • (II) The Palatini formalism

    The second approach is the Palatini formalism, where \(\Gamma _{\beta \gamma}^\alpha\) and g μν are treated as independent variables. Varying the action (1.4.44) with respect to g μν gives

    $$F(R){R_{\mu \nu}}(\Gamma) - {1 \over 2}f(R){g_{\mu \nu}} = {\kappa ^2}{T_{\mu \nu}}\,,$$
    (1.4.47)

    where R μν (Γ) is the Ricci tensor corresponding to the connections \(\Gamma _{\beta \gamma}^\alpha\). In general this is different from the Ricci tensor R μν (g) corresponding to the metric connections. Taking the trace of Eq. (1.4.47), we obtain

    $$F(R)R - 2f(R) = {\kappa ^2}T\,,$$
    (1.4.48)

    where R (T) = gμνR μν (Γ) is directly related to T. Taking the variation of the action (1.4.44) with respect to the connection, and using Eq. (1.4.47), we find

    $$\begin{array}{*{20}c} {{R_{\mu \nu}}(g) - {1 \over 2}{g_{\mu \nu}}R(g) = {{{\kappa ^2}{T_{\mu \nu}}} \over F} - {{FR(T) - f} \over {2F}}{g_{\mu \nu}} + {1 \over F}({\nabla _\mu}{\nabla _\nu}F - {g_{\mu \nu}}\square F)\quad \quad \quad}\\ {- {3 \over {2{F^2}}}\left[ {{\partial _\mu}F{\partial _\nu}F - {1 \over 2}{g_{\mu \nu}}{{(\nabla F)}^2}} \right]\,.}\\ \end{array}$$
    (1.4.49)

In GR we have f (R) = R − 2Λ and F (R) = 1, so that the term □F (R) in Eq. (1.4.46) vanishes. In this case both the metric and the Palatini formalisms give the relation R = −κ2T = κ2(ρ − 3P), which means that the Ricci scalar R is directly determined by the matter (the trace T).

In modified gravity models where F (R) is a function of R, the term □F (R) does not vanish in Eq. (1.4.46). This means that, in the metric formalism, there is a propagating scalar degree of freedom, ψF (R). The trace equation (1.4.46) governs the dynamics of the scalar field ψ — dubbed “scalaron” [862]. In the Palatini formalism the kinetic term □F (R) is not present in Eq. (1.4.48), which means that the scalar-field degree of freedom does not propagate freely [32, 563, 567, 566].

The de Sitter point corresponds to a vacuum solution at which the Ricci scalar is constant. Since □F (R) = 0 at this point, we get

$$F(R)R - 2f(R) = 0,$$
(1.4.50)

which holds for both the metric and the Palatini formalisms. Since the model f (R) = αR2 satisfies this condition, it possesses an exact de Sitter solution [862].

It is important to realize that the dynamics of f (R) dark-energy models is different depending on the two formalisms. Here we confine ourselves to the metric case only.

Already in the early 1980s it was known that the model f (R) = R + αR2 can be responsible for inflation in the early universe [862]. This comes from the fact that the presence of the quadratic term αR2 gives rise to an asymptotically exact de Sitter solution. Inflation ends when the term αR2 becomes smaller than the linear term R. Since the term αR2 is negligibly small relative to R at the present epoch, this model is not suitable to realizing the present cosmic acceleration.

Since a late-time acceleration requires modification for small R, models of the type f (R) = Rα/Rn (α > 0, n > 0) were proposed as a candidate for dark energy [204, 212, 687]. While the late-time cosmic acceleration is possible in these models, it has become clear that they do not satisfy local gravity constraints because of the instability associated with negative values of f,RR [230, 319, 852, 697, 355]. Moreover a standard matter epoch is not present because of a large coupling between the Ricci scalar and the non-relativistic matter [43].

Then, we can ask what are the conditions for the viability of f (R) dark-energy models in the metric formalism. In the following we first present such conditions and then explain step by step why they are required.

  • (i) f,R > 0 for RR0 (> 0), where R0 is the Ricci scalar at the present epoch. Strictly speaking, if the final attractor is a de Sitter point with the Ricci scalar R1 (> 0), then the condition f,R > 0 needs to hold for RR1.

    This is required to avoid a negative effective gravitational constant.

  • (ii) f,RR > 0 for RR0.

    This is required for consistency with local gravity tests [319, 697, 355, 683], for the presence of the matter-dominated epoch [43, 39], and for the stability of cosmological perturbations [213, 849, 110, 358].

  • (iii) f (R) → R − 2Λ for RR0.

    This is required for consistency with local gravity tests [48, 456, 864, 53, 904] and for the presence of the matter-dominated epoch [39].

  • (iv) \(0 < {{Rf{,_{RR}}} \over {f{,_R}}}\left({r = - 2} \right) < 1\,{\rm{at}}\,r = - {{Rf{,_R}} \over f} = - 2\).

    This is required for the stability of the late-time de Sitter point [678, 39].

For example, the model f (R) = Rα/Rn (α > 0, n > 0) does not satisfy the condition (ii).

Below we list some viable f (R) models that satisfy the above conditions.

$$({\rm{A}})\;f(R) = R - \mu {R_c}{(R/{R_c})^p}\quad \;\;{\rm{with}}\;\;0 < p < 1,\;\;\;\mu, {R_c} > 0\,,$$
(1.4.51)
$$({\rm{B}})\;f(R) = R - \mu {R_c}{{{{(R/{R_c})}^{2n}}} \over {{{(R/{R_c})}^{2n}} + 1}}\quad \quad {\rm{with}}\;\;n,\mu, {R_c} > 0\,,$$
(1.4.52)
$$({\rm{C}})\;f(R) = R - \mu {R_c}\left[ {1 - {{(1 + {R^2}/R_c^2)}^{- n}}} \right]\quad \quad {\rm{with}}\;\;n,\mu, {R_c} > 0\,,$$
(1.4.53)
$$({\rm{D}})\;f(R) = R - \mu {R_c}{\rm{tanh}}\,(R/{R_c})\quad \quad {\rm{with}}\;\;\mu, {R_c} > 0\,.$$
(1.4.54)

The models (A), (B), (C), and (D) have been proposed in [39], [456], [864], and [904], respectively. A model similar to (D) has been also proposed in [53], while a generalized model encompassing (B) and (C) has been studied in [660]. In model (A), the power p needs to be close to 0 to satisfy the condition (iii). In models (B) and (C) the function f (R) asymptotically behaves as \(f\left(R \right) \to R - \mu {R_c}\left[ {1 - {{\left({{R^2}/R_c^2} \right)}^{- n}}} \right]\) for RRc and hence the condition (iii) can be satisfied even for \(n = {\mathcal O}\left(1 \right)\). In model (D) the function f (R) rapidly approaches f (R) → RμR c in the region RR c . These models satisfy f (R = 0) = 0, so the cosmological constant vanishes in the flat spacetime.

Let us consider the cosmological dynamics of f (R) gravity in the metric formalism. It is possible to carry out a general analysis without specifying the form of f (R). In the flat FLRW spacetime the Ricci scalar is given by

$$R = 6(2{H^2} + \dot H)\,,$$
(1.4.55)

where H is the Hubble parameter. As a matter action S m we take into account non-relativistic matter and radiation, which satisfy the usual conservation equations \({\dot \rho _m} + 3{H_{\rho m}} = 0\) and \({\dot \rho _r} + 4{H_{\rho r}} = 0\) respectively. From Eqs. (1.4.45) and (1.4.46) we obtain the following equations

$$3F{H^2} = {\kappa ^2}\,({\rho _m} + {\rho _r}) + (FR - f)/2 - 3H\dot F\,,$$
(1.4.56)
$$- 2F\dot H = {\kappa ^2}[{\rho _m} + (4/3){\rho _r}] + \ddot F - H\dot F\,.$$
(1.4.57)

We introduce the dimensionless variables:

$${x_1} \equiv - {{\dot F} \over {HF}}\,,\quad {x_2} \equiv - {f \over {6F{H^2}}}\,,\quad {x_3} \equiv {R \over {6{H^2}}}\,,\quad {x_4} \equiv {{{\kappa ^2}{\rho _r}} \over {3F{H^2}}}\,,$$
(1.4.58)

together with the following quantities

$${\Omega _m} \equiv {{{\kappa ^2}{\rho _m}} \over {3F{H^2}}} = 1 - {x_1} - {x_2} - {x_3} - {x_4}\,,\qquad {\Omega _r} \equiv {x_4}\,,\qquad {\Omega _{{\rm{DE}}}} \equiv {x_1} + {x_2} + {x_3}\,.$$
(1.4.59)

It is straightforward to derive the following differential equations [39]:

$$x{\prime}_1 = - 1 - {x_3} - 3{x_2} + x_1^2 - {x_1}{x_3} + {x_4}\,,$$
(1.4.60)
$$x\prime_{2} = {{{x_1}{x_3}} \over m} - {x_2}(2{x_3} - 4 - {x_1})\,,$$
(1.4.61)
$$x\prime_{3} = - {{{x_1}{x_3}} \over m} - 2{x_3}({x_3} - 2)\,,$$
(1.4.62)
$$x\prime_{4} = - 2{x_3}{x_4} + {x_1}{x_4},$$
(1.4.63)

where the prime denotes d/d ln a and

$$m \equiv {{{\rm{d}}\ln F} \over {{\rm{d}}\ln R}} = {{R{f_{,RR}}} \over {{f_{,R}}}}\,,$$
(1.4.64)
$$r \equiv - {{{\rm{d}}\ln f} \over {{\rm{d}}\ln R}} = - {{R{f_{,R}}} \over f} = {{{x_3}} \over {{x_2}}}\,.$$
(1.4.65)

From Eq. (1.4.65) one can express R as a function of x3/x2. Since m is a function of R, it follows that m is a function of r, i.e., m = m (r). The ΛCDM model, f (R) = R − 2Λ, corresponds to m = 0. Hence the quantity m characterizes the deviation from the ΛCDM model. Note also that the model, f (R) = αR1+m − 2Λ, gives a constant value of m. The analysis using Eqs. (1.4.60)(1.4.63) is sufficiently general in the sense that the form of f (R) does not need to be specified.

The effective equation of state of the system (i.e., ptot/ρtot) is

$${w_{{\rm{eff}}}} = - {1 \over 3}(2{x_3} - 1)\,.$$
(1.4.66)

The dynamics of the full system can be investigated by analyzing the stability properties of the critical phase-space points as in, e.g., [39]. The general conclusions is that only models with a characteristic function m (r) positive and close to ΛCDM, i.e.,m ≥ 0, are cosmologically viable. That is, only for these models one finds a sequence of a long decelerated matter epoch followed by a stable accelerated attractor.

The perturbation equations have been derived in, e.g., [473, 907]. Neglecting the contribution of radiation one has

$$\begin{array}{*{20}c} {\delta\prime\prime_{m} + \left({{x_3} - {1 \over 2}{x_1}} \right)\delta\prime_{m} - {3 \over 2}(1 - {x_1} - {x_2} - {x_3}){\delta _m}\quad \quad \quad}\\ {= {1 \over 2}\left[ {\left\{{{{{k^2}} \over {x_5^2}} - 6 + 3x_1^2 - 3x\prime_{1} - 3{x_1}({x_3} - 1)} \right\}\delta \tilde F} \right.}\\ {\left. {+ 3(- 2{x_1} + {x_3} - 1)\delta \tilde F\prime + 3\delta \tilde F\prime\prime} \right],\quad \quad \quad}\\ \end{array}$$
(1.4.67)
$$\begin{array}{*{20}c} {\delta \tilde F\prime\prime+ (1 - 2{x_1} + {x_3})\delta \tilde F\prime\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {+ \left[ {{{{k^2}} \over {x_5^2}} - 2{x_3} + {{2{x_3}} \over m} - {x_1}({x_3} + 1) - x\prime_{1} + x_1^2} \right]\delta \tilde F}\\ {= (1 - {x_1} - {x_2} - {x_3}){\delta _m} - {x_1}\delta\prime_{m}\,,\quad \quad \quad \quad}\\ \end{array}$$
(1.4.68)

where \(\delta \tilde F \equiv \delta F/F\), and the new variable x5aH satisfies

$$x\prime_{5} = ({x_3} - 1)\;{x_5}.$$
(1.4.69)

The perturbation δF can be written as δF = f,RRδR and, therefore, \(\delta \tilde F = m\delta R/R\). These equations can be integrated numerically to derive the behavior of δ m at all scales. However, at sub-Hubble scales they can be simplified and the following expression for the two MG functions Q, η of Eq. (1.3.23) can be obtained:

$$\begin{array}{*{20}c} {Q = 1 - {{{k^2}} \over {3({a^2}{M^2} + {k^2})}}}\\ {\eta = 1 - {{2{k^2}} \over {3{a^2}{M^2} + 4{k^2}}}}\\ \end{array}$$
(1.4.70)

where

$${M^2} = {1 \over {3{f_{,RR}}}}.$$
(1.4.71)

Note that in the ΛCDM limit f,RR → 0 and Q,η → 1.

These relations can be straightforwardly generalized. In [287] the perturbation equations for the f (R) Lagrangian have been extended to include coupled scalar fields and their kinetic energy X ≡ −ϕϕμ/2, resulting in a f (R, ϕ, X)-theory. In the slightly simplified case in which f (R, ϕ, X) = f1(R, ϕ) + f2(ϕ, X), with arbitrary functions f1, 2, one obtains

$$\begin{array}{*{20}c} {Q = - {1 \over F}{{(1 + 2{r_1})({f_{,X}} + 2{r_2}) + 2F_{,\phi}^2/F} \over {(1 + 3{r_1})({f_{,X}} + 2{r_2}) + 3F_{,\phi}^2/F}}\,,}\\ {\eta = {{(1 + 2{r_1})({f_{,X}} + 2{r_2}) + 2F_{,\phi}^2/F} \over {(1 + 4{r_1})({f_{,X}} + 2{r_2}) + 4F_{,\phi}^2/F}}\,,\quad \;\;}\\ \end{array}$$
(1.4.72)

where the notation f,X or F,ϕ denote differentiation wrt X or ϕ, respectively, and where \({r_1} \equiv {{{k^2}} \over {{a^2}}}{m \over R}\) and \({r_2} \equiv {{{a^2}} \over {{k^2}}}M_\phi ^2,\,{M_\phi} = - f{,_{\phi \phi}}/2\) being the scalar field effective mass. In the same paper [287] an extra term proportional to Xϕ in the Lagrangian is also taken into account.

Euclid forecasts for the f (R) models will be presented in Section 1.8.7

Massive gravity and higher-dimensional models

Instead of introducing new scalar degrees of freedom such as in f (R) theories, another philosophy in modifying gravity is to modify the graviton itself In this case the new degrees of freedom belong to the gravitational sector itself; examples include massive gravity and higher-dimensional frameworks, such as the Dvali-Gabadadze-Porrati (DGP) model [326] and its extensions. The new degrees of freedom can be responsible for a late-time speed-up of the universe, as is summarized below for a choice of selected models We note here that while such self-accelerating solutions are interesting in their own right, they do not tackle the old cosmological constant problem: why the observed cosmological constant is so much smaller than expected in the first place. Instead of answering this question directly, an alternative approach is the idea of degravitation [see 327, 328, 58, 330], where the cosmological constant could be as large as expected from standard field theory, but would simply gravitate very little (see the paragraph in Section 1.4.7.1 below)

Self-acceleration

DGP. The DGP model is one of the important infrared (IR) modified theories of gravity. From a four-dimensional point of view this corresponds effectively to a theory in which the graviton acquires a soft mass m. In this braneworld model our visible universe is confined to a brane of four dimensions embedded into a five-dimensional bulk. At small distances, the four-dimensional gravity is recovered due to an intrinsic Einstein-Hilbert term sourced by the brane curvature causing a gravitational force law that scales as r−2. At large scales the gravitational force law asymptotes to an r−3 behavior. The cross over scale r c = m−1 is given by the ratio of the Planck masses in four (M4) and five (M5) dimensions. One can study perturbations around flat spacetime and compute the gravitational exchange amplitude between two conserved sources, which does not reduce to the GR result even in the limit m→ 0. However, the successful implementation of the Vainshtein mechanism for decoupling the additional modes from gravitational dynamics at sub-cosmological scales makes these theories still very attractive [913]. Hereby, the Vainshtein effect is realized through the nonlinear interactions of the helicity-0 mode π, as will be explained in further detail below. Thus, this vDVZ discontinuity does not appear close to an astrophysical source where the π field becomes nonlinear and these nonlinear effects of π restore predictions to those of GR. This is most easily understood in the limit where M4, M5 → ∞ and m → 0 while keeping the strong coupling scale Λ = (M4m2)1/3 fixed. This allows us to treat the usual helicity-2 mode of gravity linearly while treating the helicity-0 mode π nonlinearly. The resulting effective action is then

$${{\mathcal L}_\pi} = 3\pi\square\pi - {1 \over {{\Lambda ^3}}}{(\partial \pi)^2}\square\pi \,,$$
(1.4.73)

where interactions already become important at the scale Λ ≪ MPl [593].

Furthermore, in this model, one can recover an interesting range of cosmologies, in particular a modified Friedmann equation with a self-accelerating solution. The Einstein equations thus obtained reduce to the following modified Friedmann equation in a homogeneous and isotropic metric [298]

$${H^2} \pm mH = {{8\pi G} \over 3}\rho \,,$$
(1.4.74)

such that at higher energies one recovers the usual four-dimensional behavior, H2ρ, while at later time corrections from the extra dimensions kick in. As is clear in this Friedmann equation, this braneworld scenario holds two branches of cosmological solutions with distinct properties. The self-accelerating branch (minus sign) allows for a de Sitter behavior H = const = m even in the absence of any cosmological constant ρΛ = 0 and as such it has attracted a lot of attention. Unfortunately, this branch suffers from a ghost-like instability. The normal branch (the plus sign) instead slows the expansion rate but is stable. In this case a cosmological constant is still required for late-time acceleration, but it provides significant intuition for the study of degravitation.

The Galileon. Even though the DGP model is interesting for several reasons like giving the Vainshtein effect a chance to work, the self-acceleration solution unfortunately introduces extra ghost states as outlined above. However, it has been generalized to a “Galileon” model, which can be considered as an effective field theory for the helicity-0 field π. Galileon models are invariant under shifts of the field π and shifts of the gradients of π (known as the Galileon symmetry), meaning that a Galileon model is invariant under the transformation

$$\pi \rightarrow \pi + c + {v_\mu}{x^\mu}\,,$$
(1.4.75)

for arbitrary constant c and υ μ . In induced gravity braneworld models, this symmetry is naturally inherited from the five-dimensional Poincaré invariance [295]. The Galileon theory relies strongly on this symmetry to constrain the possible structure of the effective π Lagrangian, and insisting that the effective field theory for π bears no ghost-like instabilities further restricts the possibilities [686]. It can be shown that there exist only five derivative interactions, which preserve the Galilean symmetry without introducing ghosts. These interactions are symbolically of the form \({\mathcal L}_\pi ^{\left(1 \right)} = \pi\) and \({\mathcal L}_\pi ^{\left(n \right)} = {\left({\partial \pi} \right)^2}{\left({\partial \partial \pi} \right)^{n - 2}}\), for n = 2,… 5. A general Galileon Lagrangian can be constructed as a linear combination of these Lagrangian operators. The effective action for the DGP scalar (1.4.73) can be seen to be a combination of \({\mathcal L}_\pi ^{\left(2 \right)}\) and \({\mathcal L}_\pi ^{\left(3 \right)}\). Such interactions have been shown to naturally arise from Lovelock invariants in the bulk of generalized braneworld models [295]. However, the Galileon does not necessarily require a higher-dimensional origin and can be consistently treated as a four-dimensional effective field theory.

As shown in [686], such theories can allow for self-accelerating de Sitter solutions without any ghosts, unlike in the DGP model. In the presence of compact sources, these solutions can support spherically-symmetric, Vainshtein-like nonlinear perturbations that are also stable against small fluctuations. However, this is constrained to the subset of the third-order Galileon, which contains only \({\mathcal L}_\pi ^{\left(1 \right)}\), \({\mathcal L}_\pi ^{\left(2 \right)}\) and \({\mathcal L}_\pi ^{\left(3 \right)}\) [669].

The Galileon terms described above form a subset of the “generalized Galileons”. A generalized Galileon model allows nonlinear derivative interactions of the scalar field π in the Lagrangian while insisting that the equations of motion remain at most second order in derivatives, thus removing any ghost-like instabilities. However, unlike the pure Galileon models, generalized Galileons do not impose the symmetry of Eq. (1.4.75). These theories were first written down by Horndeski [445] and later rediscoved by Deffayet et al. [300]. They are a linear combination of Lagrangians constructed by multiplying the Galileon Lagrangians \({\mathcal L}_\pi ^{\left(n \right)}\) by an arbitrary scalar function of the scalar π and its first derivatives. Just like the Galileon, generalized Galileons can give rise to cosmological acceleration and to Vainshtein screening. However, as they lack the Galileon symmetry these theories are not protected from quantum corrections. Many other theories can also be found within the spectrum of generalized Galileon models, including k-essence.

Degravitation. The idea behind degravitation is to modify gravity in the IR, such that the vacuum energy could have a weaker effect on the geometry, and therefore reconcile a natural value for the vacuum energy as expected from particle physics with the observed late-time acceleration. Such modifications of gravity typically arise in models of massive gravity [327, 328, 58, 330], i.e., where gravity is mediated by a massive spin-2 field. The extra-dimensional DGP scenario presented previously, represents a specific model of soft mass gravity, where gravity weakens down at large distance, with a force law going as 1/r. Nevertheless, this weakening is too weak to achieve degravitation and tackle the cosmological constant problem. However, an obvious way out is to extend the DGP model to higher dimensions, thereby diluting gravity more efficiently at large distances. This is achieved in models of cascading gravity, as is presented below. An alternative to cascading gravity is to work directly with theories of constant mass gravity (hard mass graviton).

Cascading gravity. Cascading gravity is an explicit realization of the idea of degravitation, where gravity behaves as a high-pass filter, allowing sources with characteristic wavelength (in space and in time) shorter than a characteristic scale r c to behave as expected from GR, but weakening the effect of sources with longer wavelengths. This could explain why a large cosmological constant does not backreact as much as anticipated from standard GR. Since the DGP model does not modify gravity enough in the IR, “cascading gravity” relies on the presence of at least two infinite extra dimensions, while our world is confined on a four-dimensional brane [293]. Similarly as in DGP, four-dimensional gravity is recovered at short distances thanks to an induced Einstein-Hilbert term on the brane with associated Planck scale M4. The brane we live in is then embedded in a five-dimensional brane, which bears a five-dimensional Planck scale M5, itself embedded in six dimensions (with Planck scale M6). From a four-dimensional perspective, the relevant scales are the 5d and 6d masses \({m_4} = M_5^3/M_4^2\) and \({m_5} = M_6^4/M_5^3\), which characterize the transition from the 4d to 5d and 5d to 6d behavior respectively.

Such theories embedded in more-than-one extra dimensions involve at least one additional scalar field that typically enters as a ghost. This ghost is independent of the ghost present in the self-accelerating branch of DGP but is completely generic to any codimension-two and higher framework with brane localized kinetic terms. However, there are two ways to cure the ghost, both of which are natural when considering a realistic higher codimensional scenario, namely smoothing out the brane, or including a brane tension [293, 290, 294].

When properly taking into account the issue associated with the ghost, such models give rise to a theory of massive gravity (soft mass graviton) composed of one helicity-2 mode, helicity-1 modes that decouple and 2 helicity-0 modes. In order for this theory to be consistent with standard GR in four dimensions, both helicity-0 modes should decouple from the theory. As in DGP, this decoupling does not happen in a trivial way, and relies on a phenomenon of strong coupling. Close enough to any source, both scalar modes are strongly coupled and therefore freeze.

The resulting theory appears as a theory of a massless spin-2 field in four-dimensions, in other words as GR. If rm5 and for m6m5, the respective Vainshtein scale or strong coupling scale, i.e., the distance from the source M within which each mode is strongly coupled is \(r_i^3 = M/m_i^2M_4^2\), where i = 5, 6. Around a source M, one recovers four-dimensional gravity for rr5, five-dimensional gravity for r5rr6 and finally six-dimensional gravity at larger distances rr6.

Massive gravity. While laboratory experiments, solar systems tests and cosmological observations have all been in complete agreement with GR for almost a century now, these bounds do not eliminate the possibility for the graviton to bear a small hard mass m ≲ 6.10−32 eV [400]. The question of whether or not gravity could be mediated by a hard-mass graviton is not only a purely fundamental but an abstract one. Since the degravitation mechanism is also expected to be present if the graviton bears a hard mass, such models can play an important role for late-time cosmology, and more precisely when the age of the universe becomes on the order of the graviton Compton wavelength.

Recent progress has shown that theories of hard massive gravity can be free of any ghost-like pathologies in the decoupling limit where MPl → ∞ and m → 0 keeping the scale \(\Lambda _3^3 = {M_{{\rm{Pl}}{m^2}}}\) fixed [291, 292]. The absence of pathologies in the decoupling limit does not guarantee the stability of massive gravity on cosmological backgrounds, but provides at least a good framework to understand the implications of a small graviton mass. Unlike a massless spin-2 field, which only bears two polarizations, a massive one bears five of them, namely two helicity-2 modes, two helicity-1 modes which decouple, and one helicity-0 mode (denoted as π ). As in the braneworld models presented previously, this helicity-0 mode behaves as a scalar field with specific derivative interactions of the form

$${{\mathcal L}_\pi} = {h^{\mu \nu}}\left({X_{\mu \nu}^{(1)} + {1 \over {\Lambda _3^3}}X_{\mu \nu}^{(2)} + {1 \over {\Lambda _3^6}}X_{\mu \nu}^{(3)}} \right)\,.$$
(1.4.76)

Here, h μν denotes the canonically-normalized (rescaled by Mpl) tensor field perturbation (helicity-2 mode), while \(X_{\mu v}^{\left(1 \right)}\), \(X_{\mu v}^{\left(2 \right)}\), and \(X_{\mu v}^{\left(3 \right)}\) are respectively, linear, quadratic and cubic in the helicity-0 mode π. Importantly, they are all transverse (for instance, \(X_{\mu v}^{\left(1 \right)}\, \propto \,{\eta _{\mu v}}\square\pi - {\partial _\mu}{\partial _v}\pi\)). Not only do these interactions automatically satisfy the Bianchi identity, as they should to preserve diffeomorphism invariance, but they are also at most second order in time derivatives. Hence, the interactions (1.4.76) are linear in the helicity-2 mode, and are free of any ghost-like pathologies. Therefore, such interactions are very similar in spirit to the Galileon ones, and bear the same internal symmetry (1.4.75), and present very similar physical properties. When \(X_{\mu v}^{\left(3 \right)}\) is absent, one can indeed recover an Einstein frame picture for which the interactions are of the form

$$\begin{array}{*{20}c} {{\mathcal L} = {{M_{{\rm{Pl}}}^2} \over 2}\sqrt {- g} R + {3 \over 2}\pi \square\pi + {{3\beta} \over {2\Lambda _3^3}}{{(\partial \pi)}^2}\square\pi + {{{\beta ^2}} \over {2\Lambda _3^6}}{{(\partial \pi)}^2}\left({{{({\partial _\alpha}{\partial _\beta}\pi)}^2} - {{(\square\pi)}^2}} \right)}\\ {+ {{\mathcal L}_{{\rm{mat}}}}[\psi, {{\tilde g}_{\mu \nu}}]\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\\end{array}$$
(1.4.77)

where β is an arbitrary constant and matter fields ψ do not couple to the metric g μν but to \({\tilde g_{\mu v}} = {g_{\mu v}} + \pi {\eta _{\mu v}} + {\beta \over {\Lambda _3^3}}{\partial _\mu}\pi {\partial _v}\pi\). Here again, the recovery of GR in the UV is possible via a strong coupling phenomena, where the interactions for π are already important at the scale Λ3MPl, well before the interactions for the usual helicity-2 mode. This strong coupling, as well as the peculiar coupling to matter sources, have distinguishable features in cosmology as is explained below [11, 478].

Observations

All models of modified gravity presented in this section have in common the presence of at least one additional helicity-0 degree of freedom that is not an arbitrary scalar, but descends from a full-fledged spin-two field. As such it has no potential and enters the Lagrangian via very specific derivative terms fixed by symmetries. However, tests of gravity severely constrain the presence of additional scalar degrees of freedom. As is well known, in theories of massive gravity the helicity-0 mode can evade fifth-force constraints in the vicinity of matter if the helicity-0 mode interactions are important enough to freeze out the field fluctuations [913]. This Vainshtein mechanism is similar in spirit but different in practice to the chameleon and symmetron mechanisms presented in detail in the next Sections 1.4.7.3 and 1.4.7.4. One key difference relies on the presence of derivative interactions rather than a specific potential. So, rather than becoming massive in dense regions, in the Vainshtein mechanism the helicity-0 mode becomes weakly coupled to matter (and light, i.e., sources in general) at high energy. This screening of scalar mode can yet have distinct signatures in cosmology and in particular for structure formation.

Screening mechanisms

While quintessence introduces a new degree of freedom to explain the late-time acceleration of the universe, the idea behind modified gravity is instead to tackle the core of the cosmological constant problem and its tuning issues as well as screening any fifth forces that would come from the introduction of extra degrees of freedom. As mentioned in Section 1.4.4.1, the strength with which these new degrees of freedom can couple to the fields of the standard model is very tightly constrained by searches for fifth forces and violations of the weak equivalence principle. Typically the strength of the scalar mediated interaction is required to be orders of magnitude weaker than gravity. It is possible to tune this coupling to be as small as is required, leading however to additional naturalness problems. Here we discuss in more detail a number of ways in which new scalar degrees of freedom can naturally couple to standard model fields, whilst still being in agreement with observations, because a dynamical mechanism ensures that their effects are screened in laboratory and solar system tests of gravity. This is done by making some property of the field dependent on the background environment under consideration. These models typically fall into two classes; either the field becomes massive in a dense environment so that the scalar force is suppressed because the Compton wavelength of the interaction is small, or the coupling to matter becomes weaker in dense environments to ensure that the effects of the scalar are suppressed. Both types of behavior require the presence of nonlinearities.

Density dependent masses: The chameleon. The chameleon [499] is the archetypal model of a scalar field with a mass that depends on its environment, becoming heavy in dense environments and light in diffuse ones. The ingredients for construction of a chameleon model are a conformal coupling between the scalar field and the matter fields of the standard model, and a potential for the scalar field, which includes relevant self-interaction terms.

In the presence of non-relativistic matter these two pieces conspire to give rise to an effective potential for the scalar field

$${V_{{\rm{eff}}}}(\phi) = V(\phi) + \rho A(\phi),$$
(1.4.78)

where V (ϕ) is the bare potential, ρ the local energy density and A (ϕ) the conformal coupling function. For suitable choices of A (ϕ) and V (ϕ) the effective potential has a minimum and the position of the minimum depends on ρ. Self-interaction terms in V (ϕ) ensure that the mass of the field in this minimum also depends on ρ so that the field becomes more massive in denser environments.

The environmental dependence of the mass of the field allows the chameleon to avoid the constraints of fifth-force experiments through what is known as the thin-shell effect. If a dense object is embedded in a diffuse background the chameleon is massive inside the object. There, its Compton wavelength is small. If the Compton wavelength is smaller than the size of the object, then the scalar mediated force felt by an observer at infinity is sourced, not by the entire object, but instead only by a thin shell of matter (of depth the Compton wavelength) at the surface. This leads to a natural suppression of the force without the need to fine tune the coupling constant.

Density dependent couplings

The Vainshtein Mechanism. In models such as DGP and the Galileon, the effects of the scalar field are screened by the Vainshtein mechanism [913, 299]. This occurs when nonlinear, higher-derivative operators are present in the Lagrangian for a scalar field, arranged in such a way that the equations of motion for the field are still second order, such as the interactions presented in Eq. (1.4.73).

In the presence of a massive source the nonlinear terms force the suppression of the scalar force in the vicinity of a massive object. The radius within which the scalar force is suppressed is known as the Vainshtein radius. As an example in the DGP model the Vainshtein radius around a massive object of mass M is

$${r_ \star}\sim{\left({{M \over {4\pi {M_{{\rm{Pl}}}}}}} \right)^{1/3}}{1 \over \Lambda}\,,$$
(1.4.79)

where Λ is the strong coupling scale introduced in section 1.4.7.1. For the Sun, if m ∼ 10−33 eV, or in other words, Λ−1 = 1000 km, then the Vainshtein radius is r* ∼ 102 pc.

Inside the Vainshtein radius, when the nonlinear, higher-derivative terms become important they cause the kinetic terms for scalar fluctuations to become large. This can be interpreted as a relative weakening of the coupling between the scalar field and matter. In this way the strength of the interaction is suppressed in the vicinity of massive objects.

The Symmetron. The symmetron model [436] is in many ways similar to the chameleon model discussed above. It requires a conformal coupling between the scalar field and the standard model and a potential of a certain form. In the presence of non-relativistic matter this leads to an effective potential for the scalar field

$${V_{{\rm{eff}}}}(\phi) = - {1 \over 2}\left({{\rho \over {{M^2}}} - {\mu ^2}} \right){\phi ^2} + {1 \over 4}\lambda {\phi ^4}\,,$$
(1.4.80)

where M, μ and λ are parameters of the model, and ρ is the local energy density.

In sufficiently dense environments, ρ > μ2M2, the field sits in a minimum at the origin. As the local density drops the symmetry of the field is spontaneously broken and the field falls into one of the two new minima with a non-zero vacuum expectation value. In high-density symmetry-restoring environments, the scalar field vacuum expectation value should be near zero and fluctuations of the field should not couple to matter. Thus, the symmetron force in the exterior of a massive object is suppressed because the field does not couple to the core of the object.

The Olive-Pospelov model. The Olive-Pospelov model [696] again uses a scalar conformally coupled to matter. In this construction both the coupling function and the scalar field potential are chosen to have quadratic minima. If the background field takes the value that minimizes the coupling function, then fluctuations of the scalar field decouple from matter. In non-relativistic environments the scalar field feels an effective potential, which is a combinations of these two functions. In high-density environments the field is very close to the value that minimizes the form of the coupling function. In low-density environments the field relaxes to the minimum of the bare potential. Thus, the interactions of the scalar field are suppressed in dense environments.

Einstein Aether and its generalizations

In 1983 it was suggested by Milgrom [659] that the emerging evidence for the presence of dark matter in galaxies could follow from a modification either to how ‘baryonic’ matter responded to the Newtonian gravitational field it created or to how the gravitational field was related to the baryonic matter density. Collectively these ideas are referred to as MOdified Newtonian Dynamics (MOND). By way of illustration, MOND may be considered as a modification to the non-relativistic Poisson equation:

$$\nabla \cdot\left({\mu \left({{{\vert \nabla \Psi \vert} \over {{a_0}}}} \right)\nabla \Psi} \right) = 4\pi G\rho \,,$$
(1.4.81)

where Ψ is the gravitational potential, α0 is a number with dimensions Length−1 and ρ is the baryonic matter density. The number α0 is determined by looking at the dynamics of visible matter in galaxies [783]. The function μ (x) would simply be equal to unity in Newtonian gravity. In MOND, the functional form is only fixed at its limits: μ → 1 as x → ∞ and μx as x → 0.

We are naturally interested in a relativistic version of such a proposal. The building block is the perturbed spacetime metric already introduced in Eq. 1.3.8

$$d{s^2} = - (1 + 2\Psi)\,{\rm{d}}{t^2} + (1 - 2\Phi){a^2}(t)({\rm{d}}{R^2} + {R^2}\,{\rm{d}}{\Omega ^2}){.}$$
(1.4.82)

A simple approach is to introduce a dynamical clock field, which we will call Aμ. If it has solutions aligned with the time-like coordinate tμ then it will be sensitive to Ψ. The dynamical nature of the field implies that it should have an action that will contain gradients of the field and thus potentially scalars formed from gradients of Ψ, as we seek. A family of covariant actions for the clock field is as follows [988]:

$$I[{g^{ab}},{A^a},\lambda ] = {1 \over {16\pi G}}\int {{{\rm{d}}^4}} x\sqrt {- g} \left[ {{1 \over {{\ell ^2}}}F(K) + \lambda ({A^a}{A_a} + 1)} \right],$$

where

$$K = {\ell ^2}{K^{\mu \nu \gamma \delta}}{\nabla _\mu}{A_\nu}{\nabla _\gamma}{A_\delta}$$
(1.4.83)

with

$${K^{\mu \nu \gamma \delta}} = {c_1}{g^{\mu \gamma}}{g^{\nu \delta}} + {c_2}{g^{\mu \nu}}{g^{\gamma \delta}} + {c_3}{g^{\mu \delta}}{g^{\nu \delta}}\,.$$
(1.4.84)

The quantity is a number with dimensions of length, the c A are dimensionless constants, the Lagrange multiplier field λ enforces the unit-timelike constraint on Aa, and F is a function. These models have been termed Generalized Einstein-Aether (GEA) theories, emphasizing the coexistence of general covariance and a ‘preferred’ state of rest in the model, i.e., keeping time with Aμ.

Indeed, when the geometry is of the form (1.4.82), anisotropic stresses are negligible and Aμ is aligned with the flow of time tμ, then one can find appropriate values of the c A and such that K is dominated by a term equal to \(\vert \nabla \Psi {\vert ^2}/a_0^2\). This influence then leads to a modification to the time-time component of Einstein’s equations: instead of reducing to Poisson’s equation, one recovers an equation of the form (1.4.81). Therefore the models are successful covariant realizations of MOND.

Interestingly, in the FLRW limit Φ, Ψ → 0, the time-time component of Einstein’s equations in the GEA model becomes a modified Friedmann equation:

$$\beta \left({{{{H^2}} \over {a_0^2}}} \right){H^2} = {{8\pi G\rho} \over 3}\,,$$
(1.4.85)

where the function is related to and its derivatives with respect to K. The dynamics in galaxies prefer a value α0 on the order the Hubble parameter today H0 [783] and so one typically gets a modification to the background expansion with a characteristic scale H0, i.e., the scale associated with modified gravity models that produce dark-energy effects. Ultimately the GEA model is a phenomenological one and as such there currently lack deeper reasons to favor any particular form of F. However, one may gain insight into the possible solutions of (1.4.85) by looking at simple forms for F. In [991] the monomial case \(F \propto {K^{{n_{ae}}}}\) was considered where the kinetic index n ae was allowed to vary. Solutions with accelerated expansion were found that could mimic dark energy.

Returning to the original motivation behind the theory, the next step is to look at the theory on cosmological scales and see whether the GEA models are realistic alternatives to dark matter. As emphasized, the additional structure in spacetime is dynamical and so possesses independent degrees of freedom. As the model is assumed to be uncoupled to other matter, the gravitational field equations would regard the influence of these degrees of freedom as a type of dark matter (possibly coupled non-minimally to gravity, and not necessarily ‘cold’).

The possibility that the model may then be a viable alternative to the dark sector in background cosmology and linear cosmological perturbations has been explored in depth in [989, 564] and [991]. As an alternative to dark matter, it was found that the GEA models could replicate some but not all of the following features of cold dark matter: influence on background dynamics of the universe; negligible sound speed of perturbations; growth rate of dark matter ‘overdensity’; absence of anisotropic stress and contribution to the cosmological Poisson equation; effective minimal coupling to the gravitational field. When compared to the data from large scale structure and the CMB, the model fared significantly less well than the Concordance Model and so is excluded. If one relaxes the requirement that the vector field be responsible for the effects of cosmological dark matter, one can look at the model as one responsible only for the effects of dark energy. It was found [991] that the current most stringent constraints on the model’s success as dark energy were from constraints on the size of large scale CMB anisotropy. Specifically, possible variation in w (z) of the ‘dark energy’ along with new degrees of freedom sourcing anisotropic stress in the perturbations was found to lead to new, non-standard time variation of the potentials Φ and Ψ. These time variations source large scale anisotropies via the integrated Sachs-Wolfe effect, and the parameter space of the model is constrained in avoiding the effect becoming too pronounced.

In spite of this, given the status of current experimental bounds it is conceivable that a more successful alternative to the dark sector may share some of these points of departure from the Concordance Model and yet fare significantly better at the level of the background and linear perturbations.

The Tensor-Vector-Scalar theory of gravity

Another proposal for a theory of modified gravity arising from Milgrom’s observation is the Tensor-Vector-Scalar theory of gravity, or TeVeS. TeVeS theory is bimetric with two frames: the “geometric frame” for the gravitational fields, and the “physical frame”, for the matter fields. The three gravitational fields are the metric \({\tilde g_{ab}}\) (with connection \({\tilde \nabla _a}\)) that we refer to as the geometric metric, the vector field A a and the scalar field ϕ. The action for all matter fields, uses a single physical metric g ab (with connection ∇ a ). The two metrics are related via an algebraic, disformal relation [116] as

$${g_{ab}} = {e^{- 2\phi}}{\tilde g_{ab}} - 2\sinh (2\phi){A_a}{A_b}\,.$$
(1.4.86)

Just like in the Generalized Einstein-Aether theories, the vector field is further enforced to be unit-timelike with respect to the geometric metric, i.e.,

$${\tilde g^{ab}}{A_a}{A_b} = {A^a}{A_a} = - 1.$$
(1.4.87)

The theory is based on an action S, which is split as \(S = {S_{\tilde g}} + {S_A} + {S_\phi} + {S_m}\) where

$${S_{\tilde g}} = {1 \over {16\pi G}}\int {{{\rm{d}}^4}} x\sqrt {- \tilde g} \,\tilde R,$$
(1.4.88)

where \(\tilde g\) and \(\tilde R\) are the determinant and scalar curvature of \({\tilde g_{\mu v}}\) respectively and G is the bare gravitational constant,

$${S_A} = - {1 \over {32\pi G}}\int {{{\rm{d}}^4}} x\sqrt {- \tilde g} \;[K{F^{ab}}{F_{ab}} - 2\lambda ({A_a}{A^a} + 1)],$$
(1.4.89)

where F ab = ∇ a b − ∇ b a leads to a Maxwellian kinetic term and λ is a Lagrange multiplier ensuring the unit-timelike constraint on A a and K is a dimensionless constant (note that indices on F ab are raised using the geometric metric, i.e., \({F^a}_b = {\tilde g^{ac}}{F_{cb}}\)) and

$${S_\phi} = - {1 \over {16\pi G}}\int {{{\rm{d}}^4}} x\sqrt {- \tilde g} \left[ {\mu \;{{\hat g}^{ab}}{{\tilde \nabla}_a}\phi {{\tilde \nabla}_b}\phi + V(\mu)} \right],$$
(1.4.90)

where μ is a non-dynamical dimensionless scalar field, \({\hat g^{ab}} = {\tilde g^{ab}} - {A^a}{A^b}\) and V (μ) is an arbitrary function that typically depends on a scale B . The matter is coupled only to the physical metric g ab and defines the matter stress-energy tensor T ab through \(\delta {S_m} = - {1 \over 2}\int {{{\rm{d}}^{\rm{4}}}x} \sqrt {- g} {T_{ab}}\,\delta {g^{ab}}\). The TeVeS action can be written entirely in the physical frame [987, 840] or in a diagonal frame [840] where the scalar and vector fields decouple.

In a Friedmann universe, the cosmological evolution is governed by the Friedmann equation

$$3{\tilde H^2} = 8\pi G{e^{- 2\phi}}({\rho _\phi} + \rho),$$
(1.4.91)

where \(\tilde H\) is the Hubble rate in terms of the geometric scale factor, ρ is the physical matter density that obeys the energy conservation equation with respect to the physical metric and where the scalar field energy density is

$${\rho _\phi} = {{{e^{2\phi}}} \over {16\pi G}}\left({\mu {{{\rm{d}}V} \over {{\rm{d}}\mu}} + V} \right)$$
(1.4.92)

Exact analytical and numerical solutions with the Bekenstein free function have been found in [841] and in [318]. It turns out that energy density tracks the matter fluid energy density. The ratio of the energy density of the scalar field to that of ordinary matter is approximately constant, so that the scalar field exactly tracks the matter dynamics. In realistic situations, the radiation era tracker is almost never realized, as has been noted by Dodelson and Liguori, but rather ρ ϕ is subdominant and slowly-rolling and ϕa4/5. [157] studied more general free functions which have the Bekenstein function as a special case and found a whole range of behavior, from tracking and accelerated expansion to finite time singularities. [309] have studied cases where the cosmological TeVeS equations lead to inflationary/accelerated expansion solutions.

Although no further studies of accelerated expansion in TeVeS have been performed, it is very plausible that certain choices of function will inevitably lead to acceleration. It is easy to see that the scalar field action has the same form as a k-essence/k-inflation [61] action which has been considered as a candidate theory for acceleration. It is unknown in general whether this has similar features as the uncoupled k-essence, although Zhao’s study indicates that this a promising research direction [984].

In TeVeS, cold dark matter is absent. Therefore, in order to get acceptable values for the physical Hubble constant today (i.e., around H0 ∼ 70 km/s/Mpc), we have to supplement the absence of CDM with something else. Possibilities include the scalar field itself, massive neutrinos [841, 364] and a cosmological constant. At the same time, one has to get the right angular diameter distance to recombination [364]. These two requirements can place severe constraints on the allowed free functions.

Until TeVeS was proposed and studied in detail, MOND-type theories were assumed to be fatally flawed: their lack of a dark matter component would necessarily prevent the formation of large-scale structure compatible with current observational data. In the case of an Einstein universe, it is well known that, since baryons are coupled to photons before recombination they do not have enough time to grow into structures on their own. In particular, on scales smaller than the diffusion damping scale perturbations in such a universe are exponentially damped due to the Silk-damping effect. CDM solves all of these problems because it does not couple to photons and therefore can start creating potential wells early on, into which the baryons fall.

TeVeS contains two additional fields, which change the structure of the equations significantly. The first study of TeVeS predictions for large-scale structure observations was conducted in [841]. They found that TeVeS can indeed form large-scale structure compatible with observations depending on the choice of TeVeS parameters in the free function. In fact the form of the matter power spectrum P (k) in TeVeS looks quite similar to that in ΛCDM. Thus TeVeS can produce matter power spectra that cannot be distinguished from ΛCDM by current observations. One would have to turn to other observables to distinguish the two models. The power spectra for TeVeS and ΛCDM are plotted on the right panel of Figure 1. [318] provided an analytical explanation of the growth of structure seen numerically by [841] and found that the growth in TeVeS is due to the vector field perturbation.

Figure 1
figure 1

Left: the cosmic microwave background angular power spectrum l (l +1)C l /(2π) for TeVeS (solid) and ΛCDM (dotted) with WMAP 5-year data [689]. Right: the matter power spectrum P (k) for TeVeS (solid) and ΛCDM (dotted) plotted with SDSS data.

It is premature to claim (as in [843, 855]) that only a theory with CDM can fit CMB observations; a prime example to the contrary is the EBI theory [83]. Nevertheless, in the case of TeVeS [841] numerically solved the linear Boltzmann equation in the case of TeVeS and calculated the CMB angular power spectrum for TeVeS. By using initial conditions close to adiabatic the spectrum thus found provides very poor fit as compared to the ΛCDM model (see the left panel of Figure 1). The CMB seems to put TeVeS into trouble, at least for the Bekenstein free function. The result of [318] has a further direct consequence. The difference Φ − Ψ, sometimes named the gravitational slip (see Section 1.3.2), has additional contributions coming from the perturbed vector field α. Since the vector field is required to grow in order to drive structure formation, it will inevitably lead to a growing Φ − Ψ. If the difference Φ − Ψ can be measured observationally, it can provide a substantial test of TeVeS that can distinguish TeVeS from ΛCDM.

Generic properties of dark energy and modified gravity models

This section explores some generic issues that are not connected to particular models (although we use some specific models as examples). First, we ask ourselves to which precision we should measure w in order to make a significant progress in understanding dark energy. Second, we discuss the role of the anisotropic stress in distinguishing between dark energy and modified gravity models. Finally, we present some general consistency relations among the perturbation variables that all models of modified gravity should fulfill.

To which precision should we measure w ?

Two crucial questions that are often asked in the context of dark-energy surveys:

  • Since w is so close to −1, do we not already know that the dark energy is a cosmological constant?

  • To which precision should we measure w ? Or equivalently, why is the Euclid target precision of about 0.01 on w0 and 0.1 on w a interesting?

In this section we will attempt to answer these questions at least partially, in two different ways. We will start by examining whether we can draw useful lessons from inflation, and then we will look at what we can learn from arguments based on Bayesian model comparison.

In the first part we will see that for single field slow-roll inflation models we effectively measure w ∼ −1 with percent-level accuracy (see Figure 2); however, the deviation from a scale invariant spectrum means that we nonetheless observe a dynamical evolution and, thus, a deviation from an exact and constant equation of state of w = −1. Therefore, we know that inflation was not due to a cosmological constant; we also know that we can see no deviation from a de Sitter expansion for a precision smaller than the one Euclid will reach.

Figure 2
figure 2

The evolution of w as a function of the comoving scale k, using only the 5-year WMAP CMB data. Red and yellow are the 95% and 68% confidence regions for the LV formalism. Blue and purple are the same for the flow-equation formalism. From the outside inward, the colored regions are red, yellow, blue, and purple. Image reproduced by permission from [475]; copyright by APS.

In the second part we will consider the Bayesian evidence in favor of a true cosmological constant if we keep finding w = −1; we will see that for priors on w0 and w a of order unity, a precision like the one for Euclid is necessary to favor a true cosmological constant decisively. We will also discuss how this conclusion changes depending on the choice of priors.

Lessons from inflation

In all probability the observed late-time acceleration of the universe is not the first period of accelerated expansion that occurred during its evolution: the current standard model of cosmology incorporates a much earlier phase with ä > 0, called inflation. Such a period provides a natural explanation for several late-time observations:

  • Why is the universe very close to being spatially flat?

  • Why do we observe homogeneity and isotropy on scales that were naively never in causal contact?

  • What created the initial fluctuations?

In addition, inflation provides a mechanism to get rid of unwanted relics from phase transitions in the early universe, like monopoles, that arise in certain scenarios (e.g., grand-unified theories).

While there is no conclusive proof that an inflationary phase took place in the early universe, it is surprisingly difficult to create the observed fluctuation spectrum in alternative scenarios that are strictly causal and only act on sub-horizon scales [854, 803].

If, however, inflation took place, then it seems natural to ask the question whether its observed properties appear similar to the current knowledge about the dark energy, and if yes, whether we can use inflation to learn something about the dark energy. The first lesson to draw from inflation is that it was not due to a pure cosmological constant. This is immediately clear since we exist: inflation ended. We can go even further: if Planck confirms the observations of a deviation from a scale invariant initial spectrum (n s ≠ 1) of WMAP [526] then this excludes an exactly exponential expansion during the observable epoch and, thus, also a temporary, effective cosmological constant.

If there had been any observers during the observationally accessible period of inflation, what would they have been seeing? Following the analysis in [475], we notice that

$$1 + w = - {2 \over 3}{{\dot H} \over {{H^2}}} = {2 \over 3}{\varepsilon _H}\,,$$
(1.5.1)

where \({\epsilon _H} \equiv 2M_{{\rm{Pl}}}^2({{H\prime}/H})^2\) and here the prime denotes a derivative with respect to the inflaton field. Since, therefore, the tensor-to-scalar ratio is linked to the equation of state parameter through r ∼ 24(1 + w) we can immediately conclude that no deviation of from w = −1 during inflation has been observed so far, just as no such deviation has been observed for the contemporary dark energy. At least in this respect inflation and the dark energy look similar. However, we also know that

$${{d\ln (1 + w)} \over {dN}} = 2({\eta _H} - {\varepsilon _H})$$
(1.5.2)

where \({\eta _H} = 2M_{{\rm{Pl}}}^2{H^{\prime\prime}}/H\) is related to the scalar spectral index by 2η H = (n s − 1) + 4ε H . Thus, if n s ≠ 1 we have that either η H ≠ 0 or ε H ≠ 0, and consequently either w ≠ −1 or w is not constant.

As already said earlier, we conclude that inflation is not due to a cosmological constant. However, an observer back then would nonetheless have found w ≈ −1. Thus, observation of w ≈ −1 (at least down to an error of about 0.02, see Figure 2) does not provide a very strong reason to believe that we are dealing with a cosmological constant.

We can rewrite Eq. (1.5.2) as

$$(1 + w) = - {1 \over 6}({n_s} - 1) + {{{\eta _H}} \over 3} \approx 0.007 + {{{\eta _H}} \over 3}\,.$$
(1.5.3)

Naively it would appear rather fine-tuned if η H precisely canceled the observed contribution from n s − 1. Following this line of reasoning, if ε H and η H ; are of about the same size, then we would expect 1 + w to be about 0.005 to 0.015, well within current experimental bounds and roughly at the limit of what Euclid will be able to observe.

However, this last argument is highly speculative, and at least for inflation we know that there are classes of models where the cancellation is indeed natural, which is why one cannot give a lower limit for the amplitude of primordial gravitational waves. On the other hand, the observed period of inflation is probably in the middle of a long slow-roll phase during which w tends to be close to −1 (cf. Figure 3), while near the end of inflation the deviations become large. Additionally, inflation happened at an energy scale somewhere between 1 MeV and the Planck scale, while the energy scale of the late-time accelerated expansion is of the order of 10−3 eV. At least in this respect the two are very different.

Figure 3
figure 3

The complete evolution of w (N), from the flow-equation results accepted by the CMB likelihood. Inflation is made to end at N = 0 where w (N = 0) = −1/3 corresponding to ϵ H (N = 0) = 1. For our choice of priors on the slow-roll parameters at N = 0, we find that w decreases rapidly towards −1 (see inset) and stays close to it during the period when the observable scales leave the horizon (N ≈ 40–60). Image reproduced by permission from [475]; copyright by APS.

Higgs-Dilaton Inflation: a connection between the early and late universe acceleration

Despite previous arguments, it is natural to ask for a connection between the two known acceleration periods. In fact, in the last few years there has been a renewal of model building in inflationary cosmology by considering the fundamental Higgs as the inflaton field [133]. Such an elegant and economical model can give rise to the observed amplitude of CMB anisotropies when we include a large non-minimal coupling of the Higgs to the scalar curvature. In the context of quantum field theory, the running of the Higgs mass from the electroweak scale to the Planck scale is affected by this non-minimal coupling in such a way that the beta function of the Higgs’ self-coupling vanishes at an intermediate scale (μ ∼ 1015 GeV), if the mass of the Higgs is precisely 126 GeV, as measured at the LHC. This partial fixed point (other beta functions do not vanish) suggests an enhancement of symmetry at that scale, and the presence of a Nambu-Goldstone boson (the dilaton field) associated with the breaking of scale invariance [820]. In a subsequent paper [383], the Higgs-Dilaton scenario was explored in full detail. The model predicts a bound on the scalar spectral index, n s < 0.97, with negligible associated running, −0.0006 < d ln n s /d ln k < 0.00015, and a scalar to tensor ratio, 0.0009 < r < 0.0033, which, although out of reach of the Planck satellite mission, is within the capabilities of future CMB satellite projects like PRISM [52]. Moreover, the model predicts that, after inflation, the dilaton plays the role of a thawing quintessence field, whose slow motion determines a concrete relation between the early universe fluctuations and the equation of state of dark energy, 3(1 + w) = 1 − n s > 0.03, which could be within reach of Euclid satellite mission [383]. Furthermore, within the HDI model, there is also a relation between the running of the scalar tilt and the variation of w (a), d ln n s /d ln k = 3w a , a prediction that can easily be ruled out with future surveys.

These relationships between early and late universe acceleration parameters constitute a fundamental physics connection within a very concrete and economical model, where the Higgs plays the role of the inflaton and the dilaton is a thawing quintessence field, whose dynamics has almost no freedom and satisfies all of the present constraints [383].

When should we stop: Bayesian model comparison

In the previous section we saw that inflation provides an argument why an observation of w ≈ −1 need not support a cosmological constant strongly. Let us now investigate this argument more precisely with Bayesian model comparison. One model, M0, posits that the accelerated expansion is due to a cosmological constant. The other models assume that the dark energy is dynamical, in a way that is well parametrized either by an arbitrary constant w (model M1) or by a linear fit w (a) = w0 + (1 − a)w a (model M2). Under the assumption that no deviation from w = −1 will be detected in the future, at which point should we stop trying to measure w ever more accurately? The relevant target here is to quantify at what point we will be able to rule out an entire class of theoretical dark-energy models (when compared to ΛCDM) at a certain threshold for the strength of evidence.

Here we are using the constant and linear parametrization of w because on the one hand we can consider the constant w to be an effective quantity, averaged over redshift with the appropriate weighting factor for the observable, see [838], and on the other hand because the precision targets for observations are conventionally phrased in terms of the figure of merit (FoM) given by \(1/\sqrt {\vert{\rm{Coc}}\left({{w_0},{w_a}} \right)\vert}\) We will, therefore, find a direct link between the model probability and the FoM. It would be an interesting exercise to repeat the calculations with a more general model, using e.g. PCA, although we would expect to reach a similar conclusion.

Bayesian model comparison aims to compute the relative model probability

$${{P({M_0}\vert d)} \over {P({M_1}\vert d)}} = {{P(d\vert {M_0})} \over {P(d\vert {M_1})}}{{P({M_0})} \over {P({M_1})}}$$
(1.5.4)

where we used Bayes formula and where B01P (dM0)/P (dM1) is called the Bayes factor. The Bayes factor is the amount by which our relative belief in the two models is modified by the data, with ln B01 > (< 0) indicating a preference for model 0 (model 1). Since the model M0 is nested in M1 at the point w = −1 and in model M2 at (w0 = −1, w a = 0), we can use the Savage-Dickey (SD) density ratio [e.g. 894]. Based on SD, the Bayes factor between the two models is just the ratio of posterior to prior at w = −1 or at (w0 = −1, w a = 0), marginalized over all other parameters.

Let us start by following [900] and consider the Bayes factor B01 between a cosmological constant model w = −1 and a free but constant effective w. If we assume that the data are compatible with weff = −1 with an uncertainty σ, then the Bayes factor in favor of a cosmological constant is given by

$$B = \sqrt {{2 \over \pi}} {{{\Delta _ +} + {\Delta _ -}} \over \sigma}{\left[ {{\rm{erfc}}\left({- {{{\Delta _ +}} \over {\sqrt 2 \sigma}}} \right) - {\rm{erfc}}\left({{{{\Delta _ -}} \over {\sqrt 2 \sigma}}} \right)} \right]^{- 1}},$$
(1.5.5)

where for the evolving dark-energy model we have adopted a flat prior in the region −1 − Δweff ≤ −1 + Δ+ and we have made use of the Savage-Dickey density ratio formula [see 894]. The prior, of total width Δ = Δ+ + Δ, is best interpreted as a factor describing the predictivity of the dark-energy model under consideration. For instance, in a model where dark energy is a fluid with a negative pressure but satisfying the strong energy condition we have that Δ+ = 2/3, Δ = 0. On the other hand, phantom models will be described by Δ+ = 0, Δ > 0, with the latter being possibly rather large. A model with a large Δ will be more generic and less predictive, and therefore is disfavored by the Occam’s razor of Bayesian model selection, see Eq. (1.5.5). According to the Jeffreys’ scale for the strength of evidence, we have a moderate (strong) preference for the cosmological constant model for 2.5 < ln B01 < 5.0 (ln B01 > 5.0), corresponding to posterior odds of 12:1 to 150:1 (above 150:1).

We plot in Figure 4 contours of constant observational accuracy σ in the model predictivity space (Δ, Δ+) for ln B = 3.0 from Eq. (1.5.5), corresponding to odds of 20 to 1 in favor of a cosmological constant (slightly above the “moderate” threshold. The figure can be interpreted as giving the space of extended models that can be significantly disfavored with respect to w = −1 at a given accuracy. The results for the 3 benchmark models mentioned above (fluid-like, phantom or small departures from w = −1) are summarized in Table 1. Instead, we can ask the question which precision needs to reached to support ΛCDM at a given level. This is shown in Table 2 for odds 20:1 and 150:1. We see that to rule out a fluid-like model, which also covers the parameter space expected for canonical scalar field dark energy, we need to reach a precision comparable to the one that the Euclid satellite is expected to attain.

Figure 4
figure 4

Required accuracy on weff = −1 to obtain strong evidence against a model where −1 − Δweff ≤ −1 + Δ+ as compared to a cosmological constant model, w = −1. For a given σ, models to the right and above the contour are disfavored with odds of more than 20:1.

Table 1 Strength of evidence disfavoring the three benchmark models against a cosmological constant model, using an indicative accuracy on w = −1 from present data, σ ∼ 0.1.
Table 2 Required accuracy for future surveys in order to disfavor the three benchmark models against w = −1 for two different strengths of evidence.

By considering the model M2 we can also provide a direct link with the target DETF FoM: Let us choose (fairly arbitrarily) a flat probability distribution for the prior, of width Δw0 and Δw a in the dark-energy parameters, so that the value of the prior is 1/(Δw0Δw a ) everywhere. Let us assume that the likelihood is Gaussian in w0 and w a and centered on ΛCDM (i.e., the data fully supports Λ as the dark energy).

As above, we need to distinguish different cases depending on the width of the prior. If you accept the argument of the previous section that we expect only a small deviation from w = −1, and set a prior width of order 0.01 on both w0 and w a , then the posterior is dominated by the prior, and the ratio will be of order 1 if the future data is compatible with w = −1. Since the precision of the experiment is comparable to the expected deviation, both ΛCDM and evolving dark energy are equally probable (as argued above and shown for model M1 in Table 1), and we have to wait for a detection of w ≠ −1 or a significant further increase in precision (cf. the last row in Table 2).

However, one often considers a much wider range for w, for example the fluid-like model with w0 ∈ [−1/3, −1] and w a ∈ [−1, 1] with equal probability (and neglecting some subtleties near w = −1). If the likelihood is much narrower than the prior range, then the value of the normalized posterior at w = −1 will be \(2/\left({2\pi \sqrt {\vert{\rm{Cov}}\left({{w_0},{w_a}} \right)\vert} = {\rm{FoM/}}\pi} \right.\) (since we excluded w < −1, else it would half this value). The Bayes factor is then given by

$${B_{01}} = {{\Delta {w_0}\Delta {w_a}{\rm{FoM}}} \over \pi}\,.$$
(1.5.6)

for the prior given above, we end up with B01 ≈ 4FoM/(3π) ≈ 0.4FoM. In order to reach a “decisive” Bayes factor, usually characterized as ln B > 5 or B > 150, we thus need a figure of merit exceeding 375. Demanding that Euclid achieve a FoM > 500 places us, therefore, on the safe side and allows to reach the same conclusions (the ability to favor ΛCDM decisively if the data is in full agreement with w = −1) under small variations of the prior as well.

A similar analysis could be easily carried out to compare the cosmological constant model against departures from Einstein gravity, thus giving some useful insight into the potential of future surveys in terms of Bayesian model selection.

To summarize, we used inflation as a dark-energy prototype to show that the current experimental bounds of w ≈ −1.0±0.1 are not yet sufficient to significantly favor a cosmological constant over other models. In addition, even when expecting a deviation of w = −1 of order unity, our current knowledge of w does not allow us to favor Λ strongly in a Bayesian context. Here we showed that we need to reach a percent level accuracy both to have any chance of observing a deviation of w from −1 if the dark energy is similar to inflation, and because it is at this point that a cosmological constant starts to be favored decisively for prior widths of order 1. In either scenario, we do not expect to be able to improve much our knowledge with a lower precision measurement of w. The dark energy can of course be quite different from the inflaton and may lead to larger deviations from w = −1. This indeed would be the preferred situation for Euclid, as then we will be able to investigate much more easily the physical origin of the accelerate expansion. We can, however, have departures from ΛCDM even if w is very close to −1 today. In fact most present models of modified gravity and dynamical dark energy have a value of w0 which is asymptotically close to −1 (in the sense that large departures from this value is already excluded). In this sense, for example, early dark-energy parameterizations (Ω e ) test the amount of dark energy in the past, which can still be non negligible (ex. [723]). Similarly, a fifth force can lead to a background similar to LCDM but different effects on perturbations and structure formation [79].

The effective anisotropic stress as evidence for modified gravity

As discussed in Section 1.4, all dark energy and modified gravity models can be described with the same effective metric degrees of freedom. This makes it impossible in principle to distinguish clearly between the two possibilities with cosmological observations alone. But while the cleanest tests would come from laboratory experiments, this may well be impossible to achieve. We would expect that model comparison analyses would still favor the correct model as it should provide the most elegant and economical description of the data. However, we may not know the correct model a priori, and it would be more useful if we could identify generic differences between the different classes of explanations, based on the phenomenological description that can be used directly to analyze the data.

Looking at the effective energy momentum tensor of the dark-energy sector, we can either try to find a hint in the form of the pressure perturbation δp or in the effective anisotropic stress π. Whilst all scalar field dark energy affects δp (and for multiple fields with different sound speeds in potentially quite complex ways), they generically have π = 0. The opposite is also true, modified gravity models have generically π ≠ 0 [537]. Radiation and neutrinos will contribute to anisotropic stress on cosmological scales, but their contribution is safely negligible in the late-time universe. In the following sections we will first look at models with single extra degrees of freedom, for which we will find that π ≠ 0 is a firm prediction. We will then consider the f (R, G) case as an example for multiple degrees of freedom [782].

Modified gravity models with a single degree of freedom

In the prototypical scalar-tensor theory, where the scalar φ is coupled to R through F (φ)R, we find that π ∝ (F ′/F)δφ. This is very similar to the f (R) case for which π ∝ (F ′/F)δR (where now F = df/dR). In both cases the generic model with vanishing anisotropic stress is given by F ′ = 0, which corresponds to a constant coupling (for scalar-tensor) or f (R) ∝ R + Λ. In both cases we find the GR limit. The other possibility, δφ = 0 or δR = 0, imposes a very specific evolution on the perturbations that in general does not agree with observations.

Another possible way to build a theory that deviates from GR is to use a function of the second-order Lovelock function, the Gauss-Bonnet term GR2 − 4R μν Rμν + R αβμν Rαβμν. The Gauss-Bonnet term by itself is a topological invariant in 4 spacetime dimensions and does not contribute to the equation of motion. It is useful here since it avoids an Ostrogradski-type instability [967]. In R + f (G) models, the situation is slightly more complicated than for the scalar-tensor case, as

$$\pi \sim\Phi - \Psi = 4H\dot \xi \Psi - 4\ddot \xi \Phi + 4\left({{H^2} + \dot H} \right)\delta \xi$$
(1.5.7)

where the dot denotes derivative with respect to ordinary time and ξ = df/dG (see, e.g., [782]). An obvious choice to force π = 0 is to take ξ constant, which leads to R + G + Λ in the action, and thus again to GR in four spacetime dimensions. There is no obvious way to exploit the extra ξ terms in Eq. (1.5.7), with the exception of curvature dominated evolution and on small scales (which is not very relevant for realistic cosmologies).

Finally, in DGP one has, with the notation of [41],

$$\Phi - \Psi = {{2H{r_c} - 1} \over {1 + H{r_c}(3H{r_c} - 2)}}\Phi \,.$$
(1.5.8)

This expression vanishes for Hr c = 1/2 (which is never reached in the usual scenario in which Hr c → 1 from above) and for Hr c → ∞ (for large Hr c the expression in front of Φ in (1.5.8) vanishes like 1/(Hr c )). In the DGP scenario the absolute value of the anisotropic stress grows over time and approaches the limiting value of Φ − Ψ = Φ/2. The only way to avoid this limit is to set the crossover scale to be unobservably large, \({r_c} \propto M_4^2/M_5^3 \to \infty\). In this situation the five-dimensional part of the action is suppressed and we end up with the usual 4D GR action.

In all of these examples only the GR limit has consistently no effective anisotropic stress in situations compatible with observational results (matter dominated evolution with a transition towards a state with w ≪ −1/3).

Balancing multiple degrees of freedom

In models with multiple degrees of freedom it is at least in principle possible to balance the contributions in order to achieve a net vanishing π. [782] explicitly study the case of f (R, G) gravity (please refer to this paper for details). The general equation,

$$\Phi - \Psi = {1 \over F}\left[ {\delta F + 4H\dot \xi \Psi - 4\ddot \xi \Phi + 4\left({{H^2} + \dot H} \right)\delta \xi} \right],$$
(1.5.9)

is rather complicated, and generically depends, e.g., on scale of the perturbations (except for ξ constant, which in turn requires F constant for π = 0 and corresponds again to the GR limit). Looking only at small scales, kaH, one finds

$${f_{RR}} + 16({H^2} + \dot H)({H^2} + 2\dot H){f_{GG}} + 4(2{H^2} + 3\dot H){f_{RG}} = 0\,.$$
(1.5.10)

It is in principle possible to find simultaneous solutions of this equation and the modified Friedmann (0-0 Einstein) equation, for a given H (t). As an example, the model f (R, G) = R + GnRm with

$$n = {1 \over {90}}\left({11 \pm \sqrt {41}} \right)\;,\quad m = {1 \over {180}}\left({61 \pm 11\sqrt {41}} \right)$$
(1.5.11)

allows for matter dominated evolution, H = 2/(3t), and has no anisotropic stress. It is however not clear at all how to connect this model to different epochs and especially how to move towards a future accelerated epoch with π = 0 as the above exponents are fine-tuned to produce no anisotropic stress specifically only during matter domination. Additionally, during the transition to a de Sitter fixed point one encounters generically severe instabilities.

In summary, none of the standard examples with a single extra degree of freedom discussed above allows for a viable model with π = 0. While finely balanced solutions can be constructed for models with several degrees of freedom, one would need to link the motion in model space to the evolution of the universe, in order to preserve π = 0. This requires even more fine tuning, and in some cases is not possible at all, most notably for evolution to a de Sitter state. The effective anisotropic stress appears therefore to be a very good quantity to look at when searching for generic conclusions on the nature of the accelerated expansion from cosmological observations.

Parameterized frameworks for theories of modified gravity

As explained in earlier sections of this report, modified-gravity models cannot be distinguished from dark-energy models by using solely the FLRW background equations. But by comparing the background expansion rate of the universe with observables that depend on linear perturbations of an FRW spacetime we can hope to distinguish between these two categories of explanations. An efficient way to do this is via a parameterized, model-independent framework that describes cosmological perturbation theory in modified gravity. We present here one such framework, the parameterized post-Friedmann formalism [73]Footnote 3 that implements possible extensions to the linearized gravitational field equations.

The parameterized post-Friedmann approach (PPF) is inspired by the parameterized post-Newtonian (PPN) formalism [961, 960], which uses a set of parameters to summarize leading-order deviations from the metric of GR. PPN was developed in the 1970s for the purpose of testing of alternative gravity theories in the solar system or binary systems, and is valid in weak-field, low-velocity scenarios. PPN itself cannot be applied to cosmology, because we do not know the exact form of the linearized metric for our Hubble volume. Furthermore, PPN can only test for constant deviations from GR, whereas the cosmological data we collect contain inherent redshift dependence.

For these reasons the PPF framework is a parameterization of the gravitational field equations (instead of the metric) in terms of a set of functions of redshift. A theory of modified gravity can be analytically mapped onto these PPF functions, which in turn can be constrained by data.

We begin by writing the perturbed Einstein field equations for spin-0 (scalar) perturbations in the form:

$$\delta {G_{\mu \nu}}=8\pi G\,\delta {T_{\mu \nu}} + \delta U_{\mu \nu}^{{\rm{metric}}} + \delta U_{\mu \nu}^{{\rm{d}}{\rm{.o}}{\rm{.f}}} + \;{\rm{gauge}}\;{\rm{invariance}}\;{\rm{fixing}}\;{\rm{terms}}\,,$$
(1.5.12)

where δT μν is the usual perturbed stress-energy tensor of all cosmologically-relevant fluids. The tensor \(\delta U_{\mu v}^{{\rm{metric}}}\) holds new terms that may appear in a modified theory, containing perturbations of the metric (in GR such perturbations are entirely accounted for by δG μν ). \(\delta U_{\mu v}^{{\rm{d}}{\rm{.o}}{\rm{.f}}{\rm{.}}}\) holds perturbations of any new degrees of freedom that are introduced by modifications to gravity. A simple example of the latter is a new scalar field, such as introduced by scalar-tensor or Galileon theories. However, new degrees of freedom could also come from spin-0 perturbations of new tensor or vector fields, Stückelberg fields, effective fluids and actions based on curvature invariants (such as f (R) gravity).

In principle there could also be new terms containing matter perturbations on the RHS of Eq. (1.5.12). However, for theories that maintain the weak equivalence principle — i.e., those with a Jordan frame where matter is uncoupled to any new fields — these matter terms can be eliminated in favor of additional contributions to \(\delta U_{\mu v}^{{\rm{metric}}}\) and \(\delta U_{\mu v}^{{\rm{d}}{\rm{.o}}{\rm{.f}}{\rm{.}}}\).

The tensor \(\delta U_{\mu v}^{{\rm{metric}}}\) is then expanded in terms of two gauge-invariant perturbation variables \(\hat \Phi\) and \(\hat \Gamma\). \(\hat \Phi\) is one of the standard gauge-invariant Bardeen potentials, while \(\hat \Gamma\) is the following combination of the Bardeen potentials: \(\hat \Gamma = 1/k\left({\dot \hat \Phi + {\mathcal H}\hat \Psi} \right)\). We use \(\hat \Gamma\) instead of the usual Bardeen potential \(\hat \Psi\) because \(\hat \Gamma\) has the same derivative order as \({\hat \Phi}\) (whereas \({\hat \Psi}\) does not). We then deduce that the only possible structure of \(\delta U_{\mu v}^{{\rm{metric}}}\) that maintains the gauge-invariance of the field equations is a linear combination of \(\hat \Phi, \hat \Gamma\) and their derivatives, multiplied by functions of the cosmological background (see Eqs. (1.5.13)(1.5.17) below).

\(\delta U_{\mu v}^{{\rm{d}}{\rm{.o}}{\rm{.f}}{\rm{.}}}\) is similarly expanded in a set of gauge-invariant potentials \(\left\{{{{\hat x}_i}} \right\}\) that contain the new degrees of freedom. [73] presented an algorithm for constructing the relevant gauge-invariant quantities in any theory.

For concreteness we will consider here a theory that contains only one new degree of freedom and is second-order in its equations of motion (a generic but not watertight requirement for stability, see [967]). Then the four components of Eq. (1.5.12) are:

$$- {a^2}\delta G_0^0 = 8\pi {a^2}G\,{\rho _M}{\delta _M} + {A_0}{k^2}\hat \Phi + {F_0}{k^2}\hat \Gamma + {\alpha _0}{k^2}\hat \chi + {\alpha _1}k\dot \hat \chi + {k^3}{M_\Delta}(\dot \nu + 2\epsilon){.}$$
(1.5.13)
$$- {a^2}\delta G_i^0 = {\nabla _i}\left[ {8\pi {a^2}G\,{\rho _M}(1 + {\omega _M}){\theta _M} + {B_0}k\hat \Phi + {I_0}k\hat \Gamma + {\beta _0}k\hat \chi + {\beta _1}\dot \hat \chi + {k^2}{M_\Theta}(\dot \nu + 2\epsilon)} \right]$$
(1.5.14)
$${a^2}\delta G_i^i = 3\,8\pi {a^2}G\,{\rho _M}{\Pi _M} + {C_0}{k^2}\hat \Phi + {C_1}k\dot \hat \Phi + {J_0}{k^2}\hat \Gamma + {J_1}k\dot \hat \Gamma + {\gamma _0}{k^2}\hat \chi + {\gamma _1}k\dot \hat \chi + {\gamma _2}\ddot \hat \chi$$
(1.5.15)
$$+ {k^3}{M_P}(\dot \nu +2 \epsilon)$$
(1.5.16)
$${a^2}\delta \hat G_j^i = 8\pi {a^2}G\,{\rho _M}(1 + {\omega _M}){\Sigma _M} + {D_0}\hat \Phi + {{{D_1}} \over k}\dot \hat \Phi + {K_0}\hat \Gamma + {{{K_1}} \over k}\dot \hat \Gamma + {\epsilon _0}\hat \chi + {{{\epsilon _1}} \over k}\dot \hat \chi + {{{\epsilon _2}} \over {{k^2}}}\ddot \hat \chi$$
(1.5.17)

where \(\delta \hat G_j^i = \delta G_j^i - {{\delta _j^i} \over 3}\delta G_k^k\). Each of the lettered coefficients in Eqs. (1.5.13)(1.5.17) is a function of cosmological background quantities, i.e., functions of time or redshift; this dependence has been suppressed above for clarity. Potentially the coefficients could also depend on scale, but this dependence is not arbitrary [832]). These PPF coefficients are the analogy of the PPN parameters; they are the objects that a particular theory of gravity ‘maps onto’, and the quantities to be constrained by data. Numerous examples of the PPF coefficients corresponding to well-known theories are given in [73].

The final terms in Eqs. (1.5.13)(1.5.16) are present to ensure the gauge invariance of the modified field equations, as is required for any theory governed by a covariant action. The quantities MΔ, M and M P are all pre-determined functions of the background. ϵ and ν are off-diagonal metric perturbations, so these terms vanish in the conformal Newtonian gauge. The gauge-fixing terms should be regarded as a piece of mathematical book-keeping; there is no constrainable freedom associated with them.

One can then calculate observable quantities — such as the weak lensing kernel or the growth rate of structure f (z) — using the parameterized field equations (1.5.13)(1.5.17). Similarly, they can be implemented in an Einstein-Boltzmann solver code such as CAMB [559] to utilize constraints from the CMB. If we take the divergence of the gravitational field equations (i.e., the unperturbed equivalent of Eq. (1.5.12)), the left-hand side vanishes due to the Bianchi identity, while the stress-energy tensor of matter obeys its standard conservation equations (since we are working in the Jordan frame). Hence the U-tensor must be separately conserved, and this provides the necessary evolution equation for the variable \(\hat \chi\) :

$$\delta \left({{\nabla ^\mu}\left[ {U_{\mu \nu}^{{\rm{metric}}} + U_{\mu \nu}^{{\rm{d}}{\rm{.o}}{\rm{.f}}.}} \right]} \right) = 0.$$
(1.5.18)

Eq. (1.5.18) has two components. If one wishes to treat theories with more than two new degrees of freedom, further information is needed to supplement the PPF framework.

The full form of the parameterized equations (1.5.13)(1.5.17) can be simplified in the ‘quasistatic regime’, that is, significantly sub-horizon scales on which the time derivatives of perturbations can be neglected in comparison to their spatial derivatives [457]. Quasistatic lengthscales are the relevant stage for weak lensing surveys and galaxy redshift surveys such as those of Euclid. A common parameterization used on these scales has the form:

$$2{\nabla ^2}\Phi = 8\pi {a^2}G\,\mu (a,k)\,{\bar \rho _M}{\Delta _M},$$
(1.5.19)
$${\Phi \over \Psi} = \gamma (a,k),$$
(1.5.20)

where {μ, γ} are two functions of time and scale to be constrained. This parameterization has been widely employed [131, 277, 587, 115, 737, 980, 320, 441, 442]. It has the advantages of simplicity and somewhat greater physical transparency: μ (a, k) can be regarded as describing evolution of the effective gravitational constant, while γ (a, k) can, to a certain extent, be thought of as acting like a source of anisotropic stress (see Section 1.5.2).

Let us make a comment about the number of coefficient functions employed in the PPF formalism. One may justifiably question whether the number of unknown functions in Eqs. (1.5.13)(1.5.17) could ever be constrained. In reality, the PPF coefficients are not all independent. The form shown above represents a fully agnostic description of the extended field equations. However, as one begins to impose restrictions in theory space (even the simple requirement that the modified field equations must originate from a covariant action), constraint relations between the PPF coefficients begin to emerge. These constraints remove freedom from the parameterization.

Even so, degeneracies will exist between the PPF coefficients. It is likely that a subset of them can be well-constrained, while another subset have relatively little impact on current observables and so cannot be tested. In this case it is justifiable to drop the untestable terms. Note that this realization, in itself, would be an interesting statement — that there are parts of the gravitational field equations that are essentially unknowable.

Finally, we note that there is also a completely different, complementary approach to parameterizing modifications to gravity. Instead of parameterizing the linearized field equations, one could choose to parameterize the perturbed gravitational action. This approach has been used recently to apply the standard techniques of effective field theory to modified gravity; see [107, 142, 411] and references therein.

Nonlinear aspects

In this section we discuss how the nonlinear evolution of cosmic structures in the context of different non-standard cosmological models can be studied by means of numerical simulations based on N-body algorithms and of analytical approaches based on the spherical collapse model.

N-body simulations of dark energy and modified gravity

Here we discuss the numerical methods presently available for this type of analyses, and we review the main results obtained so far for different classes of alternative cosmologies. These can be grouped into models where structure formation is affected only through a modified expansion history (such as quintessence and early dark-energy models, Section 1.4.1) and models where particles experience modified gravitational forces, either for individual particle species (interacting dark-energy models and growing neutrino models, Section 1.4.4.4) or for all types of particles in the universe (modified gravity models).

Quintessence and early dark-energy models

In general, in the context of flat FLRW cosmologies, any dynamical evolution of the dark-energy density (ρDE ≠ const. = ρΛ) determines a modification of the cosmic expansion history with respect to the standard ΛCDM cosmology. In other words, if the dark energy is a dynamical quantity, i.e., if its equation of state parameter w ≠ −1 exactly, for any given set of cosmological parameters (H0, ΩCDM, Ωb, ΩDE, Ωrad), the redshift evolution of the Hubble function H (z) will differ from the standard ΛCDM case HΛ(z).

Quintessence models of dark energy [954, 754] based on a classical scalar field ϕ subject to a self-interaction potential V (ϕ) have an energy density \({\rho _\phi} \equiv {\dot \phi ^2}/2 + V\left(\phi \right)\) that evolves in time according to the dynamical evolution of the scalar field, which is governed by the homogeneous Klein-Gordon equation:

$$\ddot \phi + 3H\dot \phi + {{{\rm{d}}V} \over {{\rm{d}}\phi}} = 0\,.$$
(1.6.1)

Here the dot denotes derivation w.r.t. ordinary time t.

For a canonical scalar field, the equation of state parameter w ϕ ρϕ/p ϕ , where \({p_\phi} \equiv {\dot \phi ^2}/2 - V\left(\phi \right)\), will in general be larger than −1, and the density of dark energy ρ ϕ will consequently be larger than ρΛ at any redshift z > 0. Furthermore, for some simple choices of the potential function such as those discussed in Section 1.4.1 (e.g., an exponential potential V ∝ exp(−αϕ/MPl) or an inverse-power potential V ∝ (ϕ/MPl)α), scaling solutions for the evolution of the system can be analytically derived. In particular, for an exponential potential, a scaling solution exists where the dark energy scales as the dominant cosmic component, with a fractional energy density

$${\Omega _\phi} \equiv {{8\pi G{\rho _\phi}} \over {3{H^2}}} = {n \over {{\alpha ^2}}}\,,$$
(1.6.2)

with n = 3 for matter domination and n = 4 for radiation domination. This corresponds to a relative fraction of dark energy at high redshifts, which is in general not negligible, whereas during matter and radiation domination ΩΛ ∼ 0 and, therefore, represents a phenomenon of an early emergence of dark energy as compared to ΛCDM where dark energy is for all purposes negligible until z ∼ 1.

Early dark energy (EDE) is, therefore, a common prediction of scalar field models of dark energy, and observational constraints put firm bounds on the allowed range of Ω ϕ at early times, and consequently on the potential slope α.

As we have seen in Section 1.2.1, a completely phenomenological parametrization of EDE, independent from any specific model of dynamical dark energy has been proposed by [956] as a function of the present dark-energy density ΩDE, its value at early times ΩEDE, and the present value of the equation of state parameter w0. From Eq. 1.2.4, the full expansion history of the corresponding EDE model can be derived.

A modification of the expansion history indirectly influences also the growth of density perturbations and ultimately the formation of cosmic structures. While this effect can be investigated analytically for the linear regime, N-body simulations are required to extend the analysis to the nonlinear stages of structure formation. For standard Quintessence and EDE models, the only modification that is necessary to implement into standard N-body algorithms is the computation of the correct Hubble function H (z) for the specific model under investigation, since this is the only way in which these non standard cosmological models can alter structure formation processes.

This has been done by the independent studies of [406] and [367], where a modified expansion history consistent with EDE models described by the parametrization of Eq. 1.2.4 has been implemented in the widely used N-body code Gadget-2 [857] and the properties of nonlinear structures forming in these EDE cosmologies have been analyzed. Both studies have shown that the standard formalism for the computation of the halo mass function still holds for EDE models at z = 0. In other words, both the standard fitting formulae for the number density of collapsed objects as a function of mass, and their key parameter δ c = 1.686 representing the linear overdensity at collapse for a spherical density perturbation, remain unchanged also for EDE cosmologies.

The work of [406], however, investigated also the internal properties of collapsed halos in EDE models, finding a slight increase of halo concentrations due to the earlier onset of structure formation and most importantly a significant increment of the line-of-sight velocity dispersion of massive halos. The latter effect could mimic a higher σ8 normalization for cluster mass estimates based on galaxy velocity dispersion measurements and, therefore, represents a potentially detectable signature of EDE models.

Interacting dark-energy models

Another interesting class of non standard dark-energy models, as introduced in Section 1.4.4, is given by coupled dark energy where a direct interaction is present between a Quintessence scalar field ϕ and other cosmic components, in the form of a source term in the background continuity equations:

$${{{\rm{d}}{\rho _\phi}} \over {{\rm{d}}\eta}} = - 3{\mathcal H}(1 + {w_\phi}){\rho _\phi} + \beta (\phi){{{\rm{d}}\phi} \over {{\rm{d}}\eta}}(1 - 3{w_\alpha}){\rho _\alpha}\,,$$
(1.6.3)
$${{{\rm{d}}{\rho _\alpha}} \over {{\rm{d}}\eta}} = - 3{\mathcal H}(1 + {w_\alpha}){\rho _\alpha} - \beta (\phi){{{\rm{d}}\phi} \over {{\rm{d}}\eta}}(1 - 3{w_\alpha}){\rho _\alpha}\,,$$
(1.6.4)

where α represents a single cosmic fluid coupled to ϕ.

While such direct interaction with baryonic particles (α = b) is tightly constrained by observational bounds, and while it is suppressed for relativistic particles (α = r) by symmetry reasons (1 − 3w r = 0), a selective interaction with cold dark matter (CDM hereafter) or with massive neutrinos is still observationally viable (see Section 1.4.4).

Since the details of interacting dark-energy models have been discussed in Section 1.4.4, here we simply recall the main features of these models that have a direct relevance for nonlinear structure formation studies. For the case of interacting dark energy, in fact, the situation is much more complicated than for the simple EDE scenario discussed above. The mass of a coupled particle changes in time due to the energy exchange with the dark-energy scalar field ϕ according to the equation:

$$m(\phi) = {m_0}{e^{- \int \beta (\phi\prime)\,{\rm{d}}\phi\prime}}$$
(1.6.5)

where m0 is the mass at z = 0. Furthermore, the Newtonian acceleration of a coupled particle (subscript c) gets modified as:

$${\dot \vec v_c} = - \tilde H{\vec v_c} - \vec \nabla {\tilde \Phi _c} - \vec \nabla {\Phi _{nc}}\,.$$
(1.6.6)

where \(\tilde H\) contains a new velocity-dependent acceleration:

$$\tilde H{\vec v_c} = H\left({1 - {\beta _\phi}{{\dot \phi} \over H}} \right){\vec v_c}\,,$$
(1.6.7)

and where a fifth-force acts only between coupled particles as

$${\tilde \Phi _c} = (1 + 2{\beta ^2}){\Phi _c}\,,$$
(1.6.8)

while Φ nc represents the gravitational potential due to all massive particles with no coupling to the dark energy that exert a standard gravitational pull.

As a consequence of these new terms in the Newtonian acceleration equation the growth of density perturbations will be affected, in interacting dark-energy models, not only by the different Hubble expansion due to the dynamical nature of dark energy, but also by a direct modification of the effective gravitational interactions at subhorizon scales. Therefore, linear perturbations of coupled species will grow with a higher rate in these cosmologies In particular, for the case of a coupling to CDM, a different amplitude of the matter power spectrum will be reached at z = 0 with respect to ΛCDM if a normalization in accordance with CMB measurements at high redshifts is assumed.

Clearly, the new acceleration equation (1.6.6) will have an influence also on the formation and evolution of nonlinear structures, and a consistent implementation of all the above mentioned effects into an N-body algorithm is required in order to investigate this regime.

For the case of a coupling to CDM (a coupling with neutrinos will be discussed in the next section) this has been done, e.g., by [604, 870] with 1D or 3D grid-based field solvers, and more recently by means of a suitable modification by [79] of the TreePM hydrodynamic N-body code Gadget-2 [857].

Nonlinear evolution within coupled quintessence cosmologies has been addressed using various methods of investigation, such as spherical collapse [611, 962, 618, 518, 870, 3, 129] and alternative semi-analytic methods [787, 45]. N-body and hydro-simulations have also been done [604, 79, 76, 77, 80, 565, 562, 75, 980]. We list here briefly the main observable features typical of this class of models:

  • The suppression of power at small scales in the power spectrum of interacting dark-energy models as compared to ΛCDM;

  • The development of a gravitational bias in the amplitude of density perturbations of uncoupled baryons and coupled CDM particles defined as P b (k)/P c (k) < 1, which determines a significant decrease of the baryonic content of massive halos at low redshifts in accordance with a large number of observations [79, 75];

  • The increase of the number density of high-mass objects at any redshift as compared to ΛCDM [see 77];

  • An enhanced ISW effect [33, 35, 612]; such effects may be partially reduced when taking into account nonlinearities, as described in [727];

  • A less steep inner core halo profiles (depending on the interplay between fifth force and velocity-dependent terms) [79, 76, 565, 562, 75];

  • A lower concentration of the halos [79, 76, 562];

  • Voids are emptier when a coupling is active [80].

Subsequent studies based on Adaptive Mesh Refinement schemes for the solution of the local scalar field equation [561] have broadly confirmed these results.

The analysis has been extended to the case of non-constant coupling functions β (ϕ) by [76], and has shown how in the presence of a time evolution of the coupling some of the above mentioned results no longer hold:

  • Small scale power can be both suppressed and enhanced when a growing coupling function is considered, depending on the magnitude of the coupling time derivative dβ (ϕ)/dϕ

  • The inner overdensity of CDM halos, and consequently the halo concentrations, can both decrease (as always happens for the case of constant couplings) or increase, again depending on the rate of change of the coupling strength;

All these effects represent characteristic features of interacting dark-energy models and could provide a direct way to observationally test these scenarios. Higher resolution studies would be required in order to quantify the impact of a DE-CDM interaction on the statistical properties of halo substructures and on the redshift evolution of the internal properties of CDM halos.

As discussed in Section 1.6.1, when a variable coupling β (ϕ) is active the relative balance of the fifth-force and other dynamical effects depends on the specific time evolution of the coupling strength. Under such conditions, certain cases may also lead to the opposite effect of larger halo inner overdensities and higher concentrations, as in the case of a steeply growing coupling function [see 76]. Alternatively, the coupling can be introduced by choosing directly a covariant stress-energy tensor, treating dark energy as a fluid in the absence of a starting action [619, 916, 193, 794, 915, 613, 387, 192, 388].

Growing neutrinos

In case of a coupling between the dark-energy scalar field ϕ and the relic fraction of massive neutrinos, all the above basic equations (1.6.5)(1.6.8) still hold. However, such models are found to be cosmologically viable only for large negative values of the coupling β [as shown by 36], that according to Eq. 1.6.5 determines a neutrino mass that grows in time (from which these models have been dubbed “growing neutrinos”). An exponential growth of the neutrino mass implies that cosmological bounds on the neutrino mass are no longer applicable and that neutrinos remain relativistic much longer than in the standard scenario, which keeps them effectively uncoupled until recent epochs, according to Eqs. (1.6.3 and 1.6.4). However, as soon as neutrinos become non-relativistic at redshift znr due to the exponential growth of their mass, the pressure terms 1 − 3w ν in Eqs. (1.6.3 and 1.6.4) no longer vanish and the coupling with the DE scalar field ϕ becomes active.

Therefore, while before znr the model behaves as a standard ΛCDM scenario, after znr the non-relativistic massive neutrinos obey the modified Newtonian equation (1.6.6) and a fast growth of neutrino density perturbation takes place due to the strong fifth force described by Eq. (1.6.8).

The growth of neutrino overdensities in the context of growing neutrinos models has been studied in the linear regime by [668], predicting the formation of very large neutrino lumps at the scale of superclusters and above (10–100 Mpc/h) at redshift z ≈ 1.

The analysis has been extended to the nonlinear regime in [963] by following the spherical collapse of a neutrino lump in the context of growing neutrino cosmologies. This study has witnessed the onset of virialization processes in the nonlinear evolution of the neutrino halo at z ≈ 1.3, and provided a first estimate of the associated gravitational potential at virialization being of the order of Φ ν ≈ 10−6 for a neutrino lump with radius R ≈ 15 Mpc.

An estimate of the potential impact of such very large nonlinear structures onto the CMB angular power spectrum through the Integrated Sachs-Wolfe effect has been attempted by [727]. This study has shown that the linear approximation fails in predicting the global impact of the model on CMB anisotropies at low multipoles, and that the effects under consideration are very sensitive to the details of the transition between the linear and nonlinear regimes and of the virialization processes of nonlinear neutrino lumps, and that also significantly depend on possible backreaction effects of the evolved neutrino density field onto the local scalar filed evolution.

A full nonlinear treatment by means of specifically designed N-body simulations is, therefore, required in order to follow in further detail the evolution of a cosmological sample of neutrino lumps beyond virialization, and to assess the impact of growing neutrinos models onto potentially observable quantities as the low-multipoles CMB power spectrum or the statistical properties of CDM large scale structures.

Modified gravity

Modified gravity models, presented in Section 1.4, represent a different perspective to account for the nature of the dark components of the universe. Although most of the viable modifications of GR are constructed in order to provide an identical cosmic expansion history to the standard ΛCDM model, their effects on the growth of density perturbations could lead to observationally testable predictions capable of distinguishing modified gravity models from standard GR plus a cosmological constant.

Since a modification of the theory of gravity would affect all test masses in the universe, i.e., including the standard baryonic matter, an asymptotic recovery of GR for solar system environments, where deviations from GR are tightly constrained, is required for all viable modified gravity models. Such mechanism, often referred to as the “Chameleon effect”, represents the main difference between modified gravity models and the interacting dark-energy scenarios discussed above, by determining a local dependence of the modified gravitational laws in the Newtonian limit.

While the linear growth of density perturbations in the context of modified gravity theories can be studied [see, e.g., 456, 674, 32, 54] by parametrizing the scale dependence of the modified Poisson and Euler equations in Fourier space (see the discussion in Section 1.3), the nonlinear evolution of the “Chameleon effect” makes the implementation of these theories into nonlinear N-body algorithms much more challenging. For this reason, very little work has been done so far in this direction. A few attempts to solve the modified gravity interactions in the nonlinear regime by means of mesh-based iterative relaxation schemes have been carried out by [700, 701, 800, 500, 981, 281, 964] and showed an enhancement of the power spectrum amplitude at intermediate and small scales. These studies also showed that this nonlinear enhancement of small scale power cannot be accurately reproduced by applying the linear perturbed equations of each specific modified gravity theory to the standard nonlinear fitting formulae [as, e.g., 844].

Higher resolution simulations and new numerical approaches will be necessary in order to extend these first results to smaller scales and to accurately evaluate the deviations of specific models of modified gravity from the standard GR predictions to a potentially detectable precision level.

The spherical collapse model

A popular analytical approach to study nonlinear clustering of dark matter without recurring to N-body simulations is the spherical collapse model, first studied by [413]. In this approach, one studies the collapse of a spherical overdensity and determines its critical overdensity for collapse as a function of redshift. Combining this information with the extended Press-Schechter theory ([743, 147]; see [976] for a review) one can provide a statistical model for the formation of structures which allows to predict the abundance of virialized objects as a function of their mass. Although it fails to match the details of N-body simulations, this simple model works surprisingly well and can give useful insigths into the physics of structure formation. Improved models accounting for the complexity of the collapse exist in the literature and offer a better fit to numerical simulations. For instance, [823] showed that a significant improvement can be obtained by considering an ellipsoidal collapse model. Furthermore, recent theoretical developments and new improvements in the excursion set theory have been undertaken by [609] and other authors (see e.g., [821]).

The spherical collapse model has been generalized to include a cosmological constant by [718, 948]. [540] have used it to study the observational consequences of a cosmological constant on the growth of perturbations. The case of standard quintessence, with speed of sound c s = 1, have been studied by [937]. In this case, scalar fluctuations propagate at the speed of light and sound waves maintain quintessence homogeneous on scales smaller than the horizon scale. In the spherical collapse pressure gradients maintain the same energy density of quintessence between the inner and outer part of the spherical overdensity, so that the evolution of the overdensity radius is described by

$${{\ddot R} \over R} = - {{4\pi G} \over 3}({\rho _m} + {\bar \rho _Q} + 3{\bar p_Q})\,,$$
(1.6.9)

where ρ m denotes the energy density of dark matter while \({\bar \rho _Q}\) and \({\bar p_Q}\) denote the homogeneous energy density and pressure of the quintessence field. Note that, although this equation looks like one of the Friedmann equations, the dynamics of R is not the same as for a FLRW universe. Indeed, ρ m evolves following the scale factor R, while the quintessence follows the external scale factor a, according to the continuity equation \({\dot \bar \rho _Q} + 3\left({\dot a/a} \right)\left({{{\bar \rho}_Q} + {{\bar p}_Q}} \right) = 0\).

In the following we will discuss the spherical collapse model in the contest of other dark energy and modified gravity models.

Clustering dark energy

In its standard version, quintessence is described by a minimally-coupled canonical field, with speed of sound cs = 1. As mentioned above, in this case clustering can only take place on scales larger than the horizon, where sound waves have no time to propagate. However, observations on such large scales are strongly limited by cosmic variance and this effect is difficult to observe. A minimally-coupled scalar field with fluctuations characterized by a practically zero speed of sound can cluster on all observable scales. There are several theoretical motivations to consider this case. In the limit of zero sound speed one recovers the Ghost Condensate theory proposed by [56] in the context of modification of gravity, which is invariant under shift symmetry of the field ϕϕ + constant. Thus, there is no fine tuning in assuming that the speed of sound is very small: quintessence models with vanishing speed of sound should be thought of as deformations of this particular limit where shift symmetry is recovered. Moreover, it has been shown that minimally-coupled quintessence with an equation of state w < −1 can be free from ghosts and gradient instabilities only if the speed of sound is very tiny, ∣c s ∣ ≲ 10−15. Stability can be guaranteed by the presence of higher derivative operators, although their effect is absent on cosmologically relevant scales [260, 228, 259].

The fact that the speed of sound of quintessence may vanish opens up new observational consequences. Indeed, the absence of quintessence pressure gradients allows instabilities to develop on all scales, also on scales where dark matter perturbations become nonlinear. Thus, we expect quintessence to modify the growth history of dark matter not only through its different background evolution but also by actively participating to the structure formation mechanism, in the linear and nonlinear regime, and by contributing to the total mass of virialized halos.

Following [258], in the limit of zero sound speed pressure gradients are negligible and, as long as the fluid approximation is valid, quintessence follows geodesics remaining comoving with the dark matter (see also [574] for a more recent model with identical phenomenology). In particular, one can study the effect of quintessence with vanishing sound speed on the structure formation in the nonlinear regime, in the context of the spherical collapse model. The zero speed of sound limit represents the natural counterpart of the opposite case c s = 1. Indeed, in both cases there are no characteristic length scales associated with the quintessence clustering and the spherical collapse remains independent of the size of the object (see [95, 671, 692] for a study of the spherical collapse when c s of quintessence is small but finite).

Due to the absence of pressure gradients quintessence follows dark matter in the collapse and the evolution of the overdensity radius is described by

$${{\ddot R} \over R} = - {{4\pi G} \over 3}({\rho _m} + {\rho _Q} + {\bar p_Q})\,,$$
(1.6.10)

where the energy density of quintessence ρ Q has now a different value inside and outside the overdensity, while the pressure remains unperturbed. In this case the quintessence inside the overdensity evolves following the internal scale factor R, \({\dot \rho _Q} + 3\left({\dot R/R} \right)\left({{\rho _Q} + {{\bar p}_Q}} \right) = 0\) and the comoving regions behave as closed FLRW universes. R satisfies the Friedmann equation and the spherical collapse can be solved exactly [258].

Quintessence with zero speed of sound modifies dark matter clustering with respect to the smooth quintessence case through the linear growth function and the linear threshold for collapse. Indeed, for w > −1 (w < −1), it enhances (diminishes) the clustering of dark matter, the effect being proportional to 1 + w. The modifications to the critical threshold of collapse are small and the effects on the dark matter mass function are dominated by the modification on the linear dark matter growth function. Besides these conventional effects there is a more important and qualitatively new phenomenon: quintessence mass adds to the one of dark matter, contributing to the halo mass by a fraction of order ∼ (1 + w Q m . Importantly, it is possible to show that the mass associated with quintessence stays constant inside the virialized object, independently of the details of virialization. Moreover, the ratio between the virialization and the turn-around radii is approximately the same as the one for ΛCDM computed by [540]. In Figure 5 we plot the ratio of the mass function including the quintessence mass contribution, for the c s = 0 case to the smooth c s = 1 case. The sum of the two effects is rather large: for values of w still compatible with the present data and for large masses the difference between the predictions of the c s = 0 and the c s = 1 cases is of order one.

Figure 5
figure 5

Ratio of the total mass functions, which include the quintessence contribution, for c s = 0 and c s = 1 at z = 0 (above) and z = 1 (below). Image reproduced by permission from [258]; copyright by IOP and SISSA.

Coupled dark energy

We now consider spherical collapse within coupled dark-energy cosmologies. The presence of an interaction that couples the cosmon dynamics to another species introduces a new force acting between particles (CDM or neutrinos in the examples mentioned in Section 1.4.4) and mediated by dark-energy fluctuations. Whenever such a coupling is active, spherical collapse, whose concept is intrinsically based on gravitational attraction via the Friedmann equations, has to be suitably modified in order to account for other external forces. As shown in [962] the inclusion of the fifth force within the spherical collapse picture deserves particular caution. Here we summarize the main results on this topic and we refer to [962] for a detailed illustration of spherical collapse in presence of a fifth force.

If CDM is coupled to a quintessence scalar field as described in Sections 1.4.4 and 2.11 of the present document, the full nonlinear evolution equations within the Newtonian limit read:

$${\dot \delta _m} = - {{\bf{v}}_m}\,\nabla {\delta _m} - (1 + {\delta _m})\,\nabla \cdot{{\bf{v}}_m}$$
(1.6.11)
$$\begin{array}{*{20}c} {{{\dot{\bf{v}}}_m} = - (2\bar H - \beta \,\dot \bar \phi)\,{{\bf{v}}_m} - ({{\bf{v}}_m}\,\nabla){{\bf{v}}_m}}\\ {- {a^{- 2}}\,\nabla (\Phi - \beta \,\delta \phi)\quad \quad}\\ \end{array}$$
(1.6.12)
$$\Delta \delta \phi = - \beta \,{a^2}\,\delta {\rho _m}$$
(1.6.13)
$$\Delta \Phi = - {{{a^2}} \over 2}\,\sum\limits_\alpha \delta {\rho _\alpha}$$
(1.6.14)

These equations can be derived from the non-relativistic Navier-Stokes equations and from the Bianchi identities written in presence of an external source of the type:

$${\nabla _\gamma}T_\mu ^\gamma = {Q_\mu} = - \beta T_\gamma ^\gamma {\partial _\mu}\phi \,,$$
(1.6.15)

where \(T_\mu ^\gamma\) is the stress energy tensor of the dark matter fluid and we are using comoving spatial coordinates x and cosmic time t. Note that v m is the comoving velocity, related to the peculiar velocities by v m = v pec /a. They are valid for arbitrary quintessence potentials as long as the scalar field is sufficiently light, i.e., \(m_\phi ^2\delta \phi = {V^{\prime\prime}}\left(\phi \right)\delta \phi \ll \Delta \delta \phi\) for the scales under consideration. For a more detailed discussion see [962]. Combining the above equations yields to the following expression for the evolution of the matter perturbation δ m :

$${\ddot \delta _m} = - (2\bar H - \beta \,\dot \bar \phi)\,{\dot \delta _m} + {4 \over 3}{{\dot \delta _m^2} \over {1 + {\delta _m}}} + {{1 + {\delta _m}} \over {{a^2}}}\,\Delta {\Phi _{{\rm{eff}}}}\,,$$
(1.6.16)

Linearization leads to:

$${\ddot \delta _{m,L}} = - (2\bar H - \beta \,\dot \bar \phi)\,{\dot \delta _{m,L}} + {a^{- 2}}\,\Delta {\Phi _{{\rm{eff}}}}\,.$$
(1.6.17)

where the effective gravitational potential follows the modified Poisson equation:

$$\Delta {\Phi _{{\rm{eff}}}} = - {{{a^2}} \over 2}{\bar \rho _m}{\delta _m}(1 + 2{\beta ^2})\,.$$
(1.6.18)

Eqs. (1.6.16) and (1.6.17) are the two main equations which correctly describe the nonlinear and linear evolution for a coupled dark-energy model. They can be used, among other things, for estimating the extrapolated linear density contrast at collapse δ c in the presence of a fifth force. It is possible to reformulate Eqs. (1.6.16) and (1.6.17) into an effective spherical collapse:

$${{\ddot R} \over R} = - \beta \,\dot \phi \left({H - {{\dot R} \over R}} \right) - {1 \over 6}\sum\limits_\alpha {[{\rho _\alpha}(1 + 3{w_\alpha})]} - {1 \over 3}\,{\beta ^2}\,\delta {\rho _m}\,.$$
(1.6.19)

Eq. (1.6.19) [611, 962], describes the general evolution of the radius of a spherical overdense region within coupled quintessence. Comparing with the standard case (1.6.9) we notice the presence of two additional terms: a ‘friction’ term and the coupling term β2 δρ m , the latter being responsible for the additional attractive fifth force. Note that the’ friction’ term is actually velocity dependent and its effects on collapse depend, more realistically, on the direction of the velocity, information which is not contained within a spherical collapse picture and can be treated within simulations [77, 565, 76, 562, 75]. We stress that it is crucial to include these additional terms in the equations, as derived from the nonlinear equations, in order to correctly account for the presence of a fifth force. The outlined procedure can easily be generalized to include uncoupled components, for example baryons. In this case, the corresponding evolution equation for δ b , will be fed by Φeff = Φ. This yields an evolution equation for the uncoupled scale factor R uc that is equivalent to the standard Friedmann equation. In Figure 6 we show the linear density contrast at collapse δ c (z c ) for three coupled quintessence models with α = 0.1 and β = 0.05, 0.1, 0.15.

Figure 6
figure 6

Extrapolated linear density contrast at collapse for coupled quintessence models with different coupling strength β. For all plots we use a constant α = 0.1. We also depict δ c for reference ΛCDM (dotted, pink) and EdS (double-dashed, black) models. Image reproduced by permission from [962]; copyright by APS.

An increase of β results in an increase of δ c . As shown in [962], δ c (β) is well described by a simple quadratic fitting formula,

$${\delta _c}(\beta) = 1{.}686(1 + a{\beta ^2})\,,a = 0{.}556\,,$$
(1.6.20)

valid for small β ≲ 0.4 and z c ≥ 5. We recall that a nonlinear analysis beyond the spherical collapse method can be addressed by means of the time-renormalization-group method, extended to the case of couple quintessence in [787].

If a coupling between dark energy and neutrinos is present, as described in Sections 1.4.4 and 2.9, bound neutrino structures may form within these models [180]. It was shown in [668] that their formation will only start after neutrinos become non-relativistic. A nonlinear treatment of the evolution of neutrino densities is thus only required for very late times, and one may safely neglect neutrino pressure as compared to their density. The evolution equations (1.6.16) and (1.6.17) can then also be applied for the nonlinear and linear neutrino density contrast. The extrapolated linear density at collapse δ c for growing neutrino quintessence reflects in all respects the characteristic features of this model and results in a δ c which looks quite different from standard dark-energy cosmologies. We have plotted the dependence of δ c on the collapse redshift z c in Figure 7 for three values of the coupling. The oscillations seen are the result of the oscillations of the neutrino mass caused by the coupling to the scalar field: the latter has characteristic oscillations as it approaches the minimum of the effective potential in which it rolls, given by a combination of the self-interaction potential U (ϕ) and the coupling contribution β (1 − 3w ν )ρ ν . Furthermore, due to the strong coupling β, the average value of δ c is found to be substantially higher than 1.686, corresponding to the Einstein de Sitter value, shown in black (double-dashed) in Figure 7. Such an effect can have a strong impact on structure formation and on CMB [727]. For the strongly coupled models, corresponding to a low present day neutrino mass m ν (t0), the critical density at collapse is only available for z c ≲ 0.2, 1 for β = −560, −112, respectively. This is again a reflection of the late transition to the non-relativistic regime. Nonlinear investigations of single lumps beyond the spherical collapse picture was performed in [963, 179], the latter showing the influence of the gravitational potentials induced by the neutrino inhomogeneities on the acoustic oscillations in the baryonic and dark-matter spectra.

Figure 7
figure 7

Extrapolated linear density contrast at collapse δ c vs. collapse redshift z c for growing neutrinos with β = −52 (solid, red), β = −112 (long-dashed, green) and β = −560 (short-dashed, blue). A reference EdS model (double-dashed. black) is also shown. Image reproduced by permission from [962]; copyright by APS.

Early dark energy

A convenient way to parametrize the presence of a nonnegligible homogeneous dark-energy component at early times was presented in [956] and has been illustrated in Section 1.2.1 of the present review. If we specify the spherical collapse equations for this case, the nonlinear evolution of the density contrast follows the evolution equations (1.6.16) and (1.6.17) without the terms related to the coupling. As before, we assume relativistic components to remain homogeneous. In Figure 8 we show δ c for two models of early dark energy, namely model I and II, corresponding to the choices (Ωm,0 = 0.332, w0 = −0.93, ΩDE,e = 2 · 10−4) and (Ωm,0 = 0.314, w0 = −0.99, ΩDE,e = 8 · 10−4) respectively. Results show δ c (z c = 5) ∼ 1.685 (∼ 5 · 10−2%) [368, 962].

Figure 8
figure 8

Extrapolated linear density contrast at collapse δ c vs. collapse redshift z c for EDE models I (solid, red) and II (long-dashed, green), as well as ΛCDM (double-dashed, black). Image reproduced by permission from [962]; copyright by APS.

Observational properties of dark energy and modified gravity

Both scalar field dark-energy models and modifications of gravity can in principle lead to any desired expansion history H (z), or equivalently any evolution of the effective dark-energy equation of state parameter w (z). For canonical scalar fields, this can be achieved by selecting the appropriate potential V (φ) along the evolution of the scalar field φ (t), as was done, e.g., in [102]. For modified gravity models, the same procedure can be followed for example for f (R) type models [e.g. 736]. The evolution history on its own can thus not tell us very much about the physical nature of the mechanism behind the accelerated expansion (although of course a clear measurement showing that w ≠ −1 would be a sensational discovery). A smoking gun for modifications of gravity can thus only appear at perturbation level.

In the next subsections we explore how dark energy or modified gravity effects can be detected through weak lensing and redshift surveys.

General remarks

Quite generally, cosmological observations fall into two categories: geometrical probes and structure formation probes. While the former provide a measurement of the Hubble function, the latter are a test of the gravitational theory in an almost Newtonian limit on subhorizon scales. Furthermore, possible effects on the geodesics of test particles need to be derived: naturally, photons follow null-geodesics while massive particles, which constitute the cosmic large-scale structure, move along geodesics for non-relativistic particles.

In some special cases, modified gravity models predict a strong deviation from the standard Friedmann equation as in, e.g., DGP, (1.4.74). While the Friedmann equation is not know explicitly in more general models of massive gravity (cascading gravity or hard mass gravity), similar modifications are expected to arise and provide characteristic features, [see, e.g., 11, 478]) that could distinguish these models from other scenarios of modified gravity or with additional dynamical degrees of freedom.

In general however the most interesting signatures of modified gravity models are to be found in the perturbation sector. For instance, in DGP, growth functions differ from those in dark-energy models by a few percent for identical Hubble functions, and for that reason, an observation of both the Hubble and the growth function gives a handle on constraining the gravitational theory, [592]. The growth function can be estimated both through weak lensing and through galaxy clustering and redshift distortions.

Concerning the interactions of light with the cosmic large-scale structure, one sees a modified coupling in general models and a difference between the metric potentials. These effects are present in the anisotropy pattern of the CMB, as shown in [792], where smaller fluctuations were found on large angular scales, which can possibly alleviate the tension between the CMB and the ΛCDM model on small multipoles where the CMB spectrum acquires smaller amplitudes due to the ISW-effect on the last-scattering surface, but provides a worse fit to supernova data. An interesting effect inexplicable in GR is the anticorrelation between the CMB temperature and the density of galaxies at high redshift due to a sign change in the integrated Sachs-Wolfe effect. Interestingly, this behavior is very common in modified gravity theories.

A very powerful probe of structure growth is of course weak lensing, but to evaluate the lensing effect it is important to understand the nonlinear structure formation dynamics as a good part of the total signal is generated by small structures. Only recently has it been possible to perform structure formation simulations in modified gravity models, although still without a mechanism in which GR is recovered on very small scales, necessary to be in accordance with local tests of gravity.

In contrast, the number density of collapsed objects relies only little on nonlinear physics and can be used to investigate modified gravity cosmologies. One needs to solve the dynamical equations for a spherically symmetric matter distribution. Modified gravity theories show the feature of lowering the collapse threshold for density fluctuations in the large-scale structure, leading to a higher comoving number density of galaxies and clusters of galaxies. This probe is degenerate with respect to dark-energy cosmologies, which generically give the same trends.

Observing modified gravity with weak lensing

The magnification matrix is a 2 × 2 matrix that relates the true shape of a galaxy to its image. It contains two distinct parts: the convergence, defined as the trace of the matrix, modifies the size of the image, whereas the shear, defined as the symmetric traceless part, distorts the shape of the image. At small scales the shear and the convergence are not independent. They satisfy a consistency relation, and they contain therefore the same information on matter density perturbations. More precisely, the shear and the convergence are both related to the sum of the two Bardeen potentials, Φ + Ψ, integrated along the photon trajectory. At large scales however, this consistency relation does not hold anymore. Various relativistic effects contribute to the convergence, see [150]. Some of these effects are generated along the photon trajectory, whereas others are due to the perturbations of the galaxies redshift. These relativistic effects provide independent information on the two Bardeen potentials, breaking their degeneracy. The convergence is therefore a useful quantity that can increase the discriminatory power of weak lensing.

The convergence can be measured through its effect on the galaxy number density, see e.g. [175]. The standard method extracts the magnification from correlations of distant quasars with foreground clusters, see [804, 657]. Recently, [977, 978] designed a new method that permits to accurately measure auto-correlations of the magnification, as a function of the galaxies redshift. This method potentially allows measurements of the relativistic effects in the convergence.

Magnification matrix

We are interested in computing the magnification matrix \({{\mathcal D}_{ab}}\) in a perturbed Friedmann universe. The magnification matrix relates the true shape of a galaxy to its image, and describes therefore the deformations encountered by a light bundle along its trajectory. \({{\mathcal D}_{ab}}\) can be computed by solving Sachs equation, see [775], that governs propagation of light in a generic geometry. The convergence κ and the shear γγ1 + 2 are then defined respectively as the trace and the symmetric traceless part of \({{\mathcal D}_{ab}}\)

$${{\mathcal D}_{ab}} = {{{\chi _S}} \over {1 + {z_S}}}\left({\begin{array}{*{20}c} {1 - \kappa - {\gamma _1}} & {- {\gamma _2}}\\ {- {\gamma _2}} & {1 - \kappa + {\gamma _1}}\\ \end{array}} \right).$$
(1.7.1)

Here z s is the redshift of the source and χ s is a time coordinate related to conformal time η S through χ S = η O η S .

We consider a spatially flat (K = 0) Friedmann universe with scalar perturbations. We start from the usual longitudinal (or Newtonian) gauge where the metric is given by

$${g_{\mu \nu}}\,{\rm{d}}{x^\mu}\,{\rm{d}}{x^\nu} = {a^2}[ - (1 + 2\Psi)d{\eta ^2} + (1 - 2\Phi){\delta _{ij}}\,{\rm{d}}{x^i}\,{\rm{d}}{x^j}].$$
(1.7.2)

We compute \({{\mathcal D}_{ab}}\) at linear order in Φ and Ψ and then we extract the shear and the convergence. We find, see [150, 125]

$$\gamma = {1 \over 2}\int\nolimits_0^{{\chi _S}} {\rm{d}} \chi {{{\chi _S} - \chi} \over {\chi {\chi _S}}}\rlap{/} \partial^2(\Phi + \Psi)\,,$$
(1.7.3)
$$\begin{array}{*{20}c} {\kappa = {1 \over 2}\int\nolimits_0^{{\chi _S}} {\rm{d}} \chi {{{\chi _S} - \chi} \over {\chi {\chi _S}}}\bar {\rlap{/} \partial}\rlap{/} \partial (\Phi + \Psi) + {\Phi _S} - \int\nolimits_0^{{\chi _S}} {{{{\rm{d}}\chi} \over {{\chi _S}}}} (\Phi + \Psi)}\\ {+ \left({{1 \over {{{\mathcal H}_S}{\chi _S}}} - 1} \right)\left({{\Psi _S} + {\bf{n}}\cdot{{\bf{v}}_S} - \int\nolimits_0^{{\chi _S}} {\rm{d}} \chi (\dot \Phi + \dot \Psi)} \right),\quad \quad \;}\\ \end{array}$$
(1.7.4)

where n is the direction of observation and v S is the peculiar velocity of the source. Here we are making use of the angular spin raising \(\not \partial\) and lowering \({\bar{\not\partial}}\) operators (see e.g., [560] for a review of the properties of these operators) defined as

$$\rlap{/} \partial_s X \equiv - \sin^s \theta (\partial_\theta + i \csc \theta \partial_\varphi) (\sin^{-s} \theta) \; {}_s X\,, \qquad \;\bar{\rlap{/} \partial} \; {}_s X \equiv - \sin^{-s} \theta (\partial_\theta - i \csc \theta \partial_\varphi) (\sin^{s} \theta) \; {}_s X \,,$$
(1.7.5)

where s X is an arbitrary field of spin s and θ and φ are spherical coordinates.

Eq. (1.7.3) and the first term in Eq. (1.7.4) are the standard contributions of the shear and the convergence, but expressed here with the full-sky transverse operators

$${1 \over {{\chi ^2}}}\;{\rlap{/} \partial^2} = {1 \over {{\chi ^2}}}\left({\partial _\theta ^2 - \cot \theta {\partial _\theta} - {1 \over {{{\sin}^2}\theta}}{\partial _\varphi}} \right) + {{2{\rm{i}}} \over {{\chi ^2}\sin \theta}}({\partial _\theta}{\partial _\varphi} - \cot \theta {\partial _\theta}),$$
(1.7.6)
$${1 \over {{\chi ^2}}}\rlap{/} \partial \bar{\rlap{/} \partial} = {1 \over {{\chi ^2}}}\left({\partial _\theta ^2 + \cot \theta {\partial _\theta} + {1 \over {{{\sin}^2}\theta}}{\partial _\varphi}} \right).$$
(1.7.7)

In the flat-sky approximation, where θ is very small, \({1 \over {{\chi ^2}}}\not \partial {\bar{\not\partial}}\) reduces to the 2D Laplacian \(\partial _x^2 + \partial _y^2\) and one recovers the standard expression for the convergence. Similarly, the real part of \({1 \over {{\chi ^2}}}{\not \partial ^2}\) that corresponds to γ1 reduces to \(\partial _y^2 - \partial _x^2\) and the imaginary part that corresponds to γ2 becomes \({\partial _x}{\partial _y}\).

The other terms in Eq. (1.7.4) are relativistic corrections to the convergence, that are negligible at small scales but may become relevant at large scales. The terms in the first line are intrinsic corrections, generated respectively by the curvature perturbation at the source position and the Shapiro time-delay. The terms in the second line are due to the fact that we measure the convergence at a fixed redshift of the source z S rather that at a fixed conformal time η S . Since in a perturbed universe, the observable redshift is itself a perturbed quantity, this transformation generates additional contribution to the convergence. Those are respectively the Sachs-Wolfe contribution, the Doppler contribution and the integrated Sachs-Wolfe contribution. Note that we have neglected the contributions at the observer position since they only give rise to a monopole or dipole term. The dominant correction to the convergence is due to the Doppler term. Therefore in the following we are interested in comparing its amplitude with the amplitude of the standard contribution. To that end we define κst and κvel as

$${\kappa _{{\rm{st}}}} = \int\nolimits_0^{{\chi _S}} {\rm{d}} \chi {{{\chi _S} - \chi} \over {2\chi {\chi _S}}}\;\rlap{/} \partial\bar{\rlap{/} \partial} (\Phi + \Psi),$$
(1.7.8)
$${\kappa _{{\rm{vel}}}} = \left({{1 \over {{{\mathcal H}_S}{\chi _S}}} - 1} \right){\bf{n}}\cdot{{\bf{v}}_S}\,.$$
(1.7.9)
Observable quantities

The convergence is not directly observable. However it can be measured through the modifications that it induces on the galaxy number density. Let us introduce the magnification

$$\mu = {1 \over {\det {\mathcal D}}} \simeq 1 + 2\kappa \,,\quad {\rm{when}}\quad \vert \kappa \vert, \vert \gamma \vert \ll 1\,.$$
(1.7.10)

The magnification modifies the size of a source: d Ω O = μd Ω S , where d Ω S is the true angular size of the source and d Ω O is the solid angle measured by the observer, i.e. the size of the image. The magnification has therefore an impact on the observed galaxy number density. Let us call \(\bar n\left(f \right)df\) the number of unlensed galaxies per unit solid angle, at a redshift z S , and with a flux in the range [f, f + df ]. The magnification μ modifies the flux measured by the observer, since it modifies the observed galaxy surface. It affects also the solid angle of observation and hence the number of galaxies per unit of solid angle. These two effects combine to give a galaxy number overdensity, see [175, 804]

$$\delta _g^\mu = {{n(f) - \bar n(f)} \over {\bar n(f)}} \simeq 1 + 2(\alpha - 1)({\kappa _{{\rm{st}}}} + {\kappa _{{\rm{vel}}}})\,.$$
(1.7.11)

Here α ≡ − N ′(&gt f c )f c /N (f c ), where N (> f c ) is the number of galaxies brighter than f c and f c is the flux limit adopted. Hence α is an observable quantity, see e.g. [977, 804]. Recent measurements of the galaxy number overdensity \(\delta _g^\mu\) are reported in [804, 657]. The challenge in those measurements is to eliminate intrinsic clustering of galaxies, which induces an overdensity \(\delta _g^\mu\) much larger than \(\delta _g^\mu\). One possibility to separate these two effects is to correlate galaxy number overdensities at widely separated redshifts. One can then measure \(\left\langle {\delta _g^\mu \left({{z_S}} \right)\delta _g^{cl}\left({{z_{{S\prime}}}} \right)} \right\rangle\), where z S is the redshift of the sources and zS < z S is the redshift of the lenses. Another possibility, proposed by [977, 978], is to use the unique dependence of \(\delta _g^\mu\) on galaxy flux (i.e., on α) to disentangle \(\delta _g^\mu\) from \(\delta _g^{cl}\). This method, combined with precise measurements of the galaxies redshift, allows to measure auto-correlations of \(\delta _g^\mu\), i.e., \(\left\langle {\delta _g^\mu \left({{z_S}} \right)\delta _g^\mu \left({{z_{{S\prime}}}} \right)} \right\rangle\), either for z S zS or for z S = zS. The velocity contribution, κvel, has only an effect on \(\left\langle {\delta _g^\mu \left({{z_S}} \right)\delta _g^\mu \left({{z_{{S\prime}}}} \right)} \right\rangle\). The correlations between \(\delta _g^{cl}\left({{z_{{S\prime}}}} \right)\) and v S are indeed completely negligible and hence the source peculiar velocity does not affect \(\left\langle {\delta _g^\mu \left({{z_S}} \right)\delta _g^\mu \left({{z_{{S\prime}}}} \right)} \right\rangle\). In the following we study in detail the contribution of peculiar motion to \(\left\langle {\delta _g^\mu \left({{z_S}} \right)\delta _g^\mu \left({{z_S}} \right)} \right\rangle\).

The two components of the convergence κst and κvel (and consequently the galaxy number overdensity) are functions of redshift z S and direction of observation n. We can therefore determine the angular power spectrum

$$\langle \delta _g^\mu ({z_S},{\bf{n}})\delta _g^\mu ({z_S},{\bf{n\prime}})\rangle = \sum\limits_\ell {{{2\ell + 1} \over {4\pi}}} {C_\ell}({z_S}){P_\ell}({\bf{n}}\cdot{\bf{n\prime}})\,.$$
(1.7.12)

The angular power spectrum C (z S ) contains two contributions, generated respectively by 〈κstκst〉 and 〈κvelκvel〉. The cross-term 〈κvelκst〉 is negligible since κst contains only Fourier modes with a wave vector k perpendicular to the line of sight (see Eq. (1.7.8)), whereas κvel selects modes with wave vector along the line of sight (Eq. (1.7.9)).

So far the derivation has been completely generic. Eqs. (1.7.3) and (1.7.4) are valid in any theory of gravity whose metric can be written as in Eq. (1.7.2). To evaluate the angular power spectrum we now have to be more specific. In the following we assume GR, with no anisotropic stress such that Φ = Ψ. We use the Fourier transform convention

$${\bf{v}}({\bf{x}},\chi) = {1 \over {{{(2\pi)}^3}}}\int {{{\rm{d}}^3}} k\,{\bf{v}}({\bf{k}},\chi){e^{i{\bf{kx}}}}\,.$$
(1.7.13)

The continuity equation, see e.g., [317], allows us to express the peculiar velocity as

$${\bf{v}}({\bf{k}},\chi) = - i{{\dot G(a)} \over {G(a)}}{{\bf{k}} \over {{k^2}}}\delta ({\bf{k}},a)\,,$$
(1.7.14)

where δ (k, a) is the density contrast, G (a) is the growth function, and Ġ (a) its derivative with respect to χ. With this we can express the angular power spectrum as

$$C_\ell ^{{\rm{vel}}}({z_S}) = {{16\pi \delta _H^2{{({\alpha _S} - 1)}^2}\dot G{{({a_S})}^2}} \over {H_0^4{G^2}(a = 1)}}{\left({{1 \over {{{\mathcal H}_S}{\chi _S}}} - 1} \right)^2}\int {\rm{d}} k\,k{T^2}(k)j\prime_{\ell}{(k{\chi _S})^2}\,.$$
(1.7.15)

Here δ H is the density contrast at horizon and T (k) is the transfer function defined through, see e.g., [317]

$$\Psi ({\bf{k}},a) = {9 \over {10}}{\Psi _p}({\bf{k}})T(k){{G(a)} \over a}\,.$$
(1.7.16)

We assume a flat power spectrum, n s = 1, for the primordial potential Ψ p (k). We want to compare this contribution with the standard contribution

$$C_\ell ^{{\rm{st}}}({z_S}) = {{36\pi \delta _H^2{{({\alpha _S} - 1)}^2}\Omega _m^2{\ell ^2}{{(\ell + 1)}^2}} \over {{G^2}(a = 1)}}\int {{{{\rm{d}}k} \over k}} {T^2}(k){\left[ {\int\nolimits_0^{{\chi _S}} {\rm{d}} \chi {{{\chi _S} - \chi} \over {\chi {\chi _S}}}{{G(a)} \over a}{j_\ell}(k\chi)} \right]^2}\,.$$
(1.7.17)

We evaluate \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\) in a ΛCDM universe with Ω m = 0.25, ΩΛ = 0.75 and δ H = 5.7 · 10−5. We approximate the transfer function with the BBKS formula, see [85]. In Figure 9, we plot \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\) for various source redshifts. The amplitude of \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\) depends on (α − 1)2, which varies with the redshift of the source, the flux threshold adopted, and the sky coverage of the experiment. Since (α − 1)2 influences \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\) in the same way we do not include it in our plot. Generally, at small redshifts, (α − 1) is smaller than 1 and consequently the amplitude of both \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\) is slightly reduced, whereas at large redshifts (α − 1) tends to be larger than 1 and to amplify \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\), see e.g., [978]. However, the general features of the curves and more importantly the ratio between \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\) are not affected by (α − 1).

Figure 9
figure 9

Left: The velocity contribution \(C_\ell ^{{\rm{vel}}}\) as a function of for various redshifts. Right: The standard contribution \(C_\ell ^{{\rm{st}}}\) as a function of for various redshifts.

Figure 9 shows that \(C_\ell ^{{\rm{vel}}}\) peaks at rather small , between 30 and 120 depending on the redshift. This corresponds to rather large angle θ ∼ 90–360 arcmin. This behavior differs from the standard term (Figure 9) that peaks at large . Therefore, it is important to have large sky surveys to detect the velocity contribution. The relative importance of \(C_\ell ^{{\rm{vel}}}\) and \(C_\ell ^{{\rm{st}}}\) depends strongly on the redshift of the source. At small redshift, z S = 0.2, the velocity contribution is about 4 · 10−5 and is hence larger than the standard contribution which reaches 10−6. At redshift z S = 0.5, \(C_\ell ^{{\rm{vel}}}\) is about 20% of \(C_\ell ^{{\rm{st}}}\) whereas at redshift z S = 1, it is about 1% of \(C_\ell ^{{\rm{st}}}\). Then at redshift z S = 1.5 and above, \(C_\ell ^{{\rm{vel}}}\) becomes very small with respect to \(C_\ell ^{{\rm{st}}}:C_\ell ^{{\rm{vel}}} \le {10^{- 4}}C_\ell ^{{\rm{st}}}\). The enhancement of \(C_\ell ^{{\rm{vel}}}\) at small redshift together with its fast decrease at large redshift are due to the prefactor \({\left({{1 \over {{{\mathcal H}_{S\chi S}}}} - 1} \right)^2}\) in Eq. (1.7.15). Thanks to this enhancement we see that if the magnification can be measured with an accuracy of 10%, then the velocity contribution is observable up to redshifts z ≤ 0.6. If the accuracy reaches 1% then the velocity contribution becomes interesting up to redshifts of order 1.

The shear and the standard contribution in the convergence are not independent. One can easily show that their angular power spectra satisfy the consistency relation, see [449]

$$C_\ell ^{\kappa \,{\rm{st}}} = {{\ell (\ell + 1)} \over {(\ell + 2)(\ell - 1)}}C_\ell ^\gamma \,.$$
(1.7.18)

This relation is clearly modified by the velocity contribution. Using that the cross-correlation between the standard term and the velocity term is negligible, we can write a new consistency relation that relates the observed convergence \(C_\ell ^{k{\rm{tot}}}\) to the shear

$${{\ell (\ell + 1)} \over {(\ell + 2)(\ell - 1)}}C_\ell ^\gamma = C_\ell ^{\kappa \,{\rm{tot}}} - C_\ell ^{\kappa \,{\rm{vel}}}\,.$$
(1.7.19)

Consequently, if one measures both the shear \(C_\ell ^\gamma\) and the magnification \(C_\ell ^{k\,{\rm{tot}}}\) as functions of the redshift, Eq. (1.7.19) allows to extract the peculiar velocity contribution \(C_\ell ^{k\,{\rm{vel}}}\). This provides a new way to measure peculiar velocities of galaxies.

Note that in practice, in weak lensing tomography, the angular power spectrum is computed in redshift bins and therefore the square bracket in Eq. (1.7.17) has to be integrated over the bin

$$\int\nolimits_0^\infty {\rm{d}} \chi {n_i}(\chi)\int\nolimits_0^\chi {\rm{d}} \chi \prime {{\chi - \chi \prime} \over {\chi \chi \prime}}{{G(\chi \prime)} \over {a(\chi \prime)}}{j_\ell}(k\chi \prime)\,,$$
(1.7.20)

where n i is the galaxy density for the i-th bin, convolved with a Gaussian around the mean redshift of the bin. The integral over χ ′ is then simplified using Limber approximation, i.e.,

$$\int\nolimits_0^\chi {\rm{d}} \chi \prime F(\chi \prime){J_\ell}(k\chi \prime) \simeq {1 \over k}F\left({{\ell \over k}} \right)\;\theta (k\chi - \ell)\,,$$
(1.7.21)

where J is the Bessel function of order . The accuracy of Limber approximation increases with . Performing a change of coordinate such that k = ℓ/ χ , Eq. (1.7.17) can be recast in the usual form used in weak lensing tomography, see e.g., Eq. (1.8.4).

Observing modified gravity with redshift surveys

Wide-deep galaxy redshift surveys have the power to yield information on both H (z) and f g (z) through measurements of Baryon Acoustic Oscillations (BAO) and redshift-space distortions. In particular, if gravity is not modified and matter is not interacting other than gravitationally, then a detection of the expansion rate is directly linked to a unique prediction of the growth rate. Otherwise galaxy redshift surveys provide a unique and crucial way to make a combined analysis of H (z) and f g (z) to test gravity. As a wide-deep survey, Euclid allows us to measure H (z) directly from BAO, but also indirectly through the angular diameter distance D A (z) (and possibly distance ratios from weak lensing). Most importantly, Euclid survey enables us to measure the cosmic growth history using two independent methods: f g (z) from galaxy clustering, and G (z) from weak lensing. In the following we discuss the estimation of [H (z), D A (z) and f g (z)] from galaxy clustering.

From the measure of BAO in the matter power spectrum or in the 2-point correlation function one can infer information on the expansion rate of the universe. In fact, the sound waves imprinted in the CMB can be also detected in the clustering of galaxies, thereby completing an important test of our theory of gravitational structure formation.

Figure 10
figure 10

Matter power spectrum form measured from SDSS [720]

The BAO in the radial and tangential directions offer a way to measure the Hubble parameter and angular diameter distance, respectively. In the simplest FLRW universe the basis to define distances is the dimensionless, radial, comoving distance:

$$\chi (z) \equiv \int\nolimits_0^z {{{{\rm{d}}z\prime} \over {E(z\prime)}}} \,.$$
(1.7.22)

The dimensionless version of the comoving distance (defined in the previous section by the same symbol χ) is:

$${E^2}(z) = \Omega _m^{(0)}{(1 + z)^3} + (1 - \Omega _m^{(0)})\exp \left[ {\int\nolimits_0^z {{{3(1 + w(\tilde z))} \over {1 + \tilde z}}} \,{\rm{d}}\tilde z} \right].$$
(1.7.23)

The standard cosmological distances are related to χ (z) via

$${D_A}(z) = {c \over {{H_0}(1 + z)\sqrt {- {\Omega _k}}}}\sin \left({\sqrt {- {\Omega _k}} \chi (z)} \right)$$
(1.7.24)

where the luminosity distance, D L (z), is given by the distance duality:

$${D_L}(z) = {(1 + z)^2}{D_A}(z).$$
(1.7.25)

The coupling between D A (z) and D L (z) persists in any metric theory of gravity as long as photon number is conserved (see Section 4.2 for cases in which the duality relation is violated). BAO yield both D A (z) and H (z) making use of an almost completely linear physics (unlike for example SN Ia, demanding complex and poorly understood mechanisms of explosions). Furthermore, they provide the chance of constraining the growth rate through the change in the amplitude of the power spectrum.

The characteristic scale of the BAO is set by the sound horizon at decoupling. Consequently, one can attain the angular diameter distance and Hubble parameter separately. This scale along the line of sight (s(z)) measures H (z) through H (z) = c Δz/s(z), while the tangential mode measures the angular diameter distance D A (z) = s/ Δθ (1 + z).

One can then use the power spectrum to derive predictions on the parameter constraining power of the survey (see e.g., [46, 418, 938,