Cosmological perturbations in generalised dark Lagrangians

We describe a new method to parameterise dark energy theories including massive gravity, elastic dark energy and tensor-metric theories. We first examine a framework to describe any second order Lagrangian which depends on the variation of the metric and find new constraints on the parameters. We extend the method to Lorentz violating theories which depend on the variation of the time and spatial parts of the metric separately. We show how this can describe massive gravity and elastic dark energy, while ruling out the whole class of theories where the Lagrangian depends only on the variation of the time part of the metric. We further generalise our method to tensor-metric theories, both with and without splitting the metric into time and spatial parts. Our method extends existing physics by providing a mechanism to easily evaluate large classes of dark energy theories.


Introduction
The discovery of the accelerated expansion of the universe [1][2][3][4][5][6][7][8] prompted many attempts to explain the phenomenon, caused by an unknown "dark energy", using modifications to Einstein's theory of General Relativity [9].These theories generally started with a specific dark Lagrangian which was then investigated to test its compatibility with both dark energy and other cosmological and solar system observations.Here we follow the method of [9][10][11][12][13][14][15][16][17] and look at a class of dark Lagrangians, so that future experimental results can immediately rule out a whole swathe of models without the need for further calculation.
We use the formalism introduced in [10][11][12][13][14] which draws on the "Post Parameterised Friedmann" (PPF) approach described in [9,[15][16][17].In these papers the Einstein-Hilbert action is modified with a dark Lagrangian such that the modified action is written as and equations of state for dark sector perturbations are found for the entropy perturbation and the anisotropic stress.The recent observation of a neutron star merger by LIGO [18] severely constrained the speed of gravitational waves, and seemingly placed strict restrictions on many modified gravity theories.However, it was suggested in [19] that although these theories predict the speed of gravitational waves to be different from that of light at low energies, this may not necessarily be the case at the high energies seen in neutron star mergers, due to the unknown UV completions of these theories.We find the perturbed fluid variables when L {2} = L {2} (δ L g µν ), in a similar way to [12].We then impose invariance under time reparameterisation and find the equations of motion.We also find evolution equations for the equation of state parameter w and the elastic bulk modulus.We find restrictions on w such that we can have a realistic (positive and subluminal) sound speed.We find the perturbed fluid variables when we have imposed spatial invariance, and we find the conditions under which the entropy is gauge invariant.
In this paper our main focus is on generalising all of these calculations for the case where L {2} is a function of the change in the time-like and spatial parts of the metric separately, not necessarily packaged together as the spacetime metric.
Elastic dark energy (EDE) discussed in [12,20] is a development of Carter-Quintana relativistic elasticity theory [21][22][23][24][25].These models describe the universe as a solid with certain parameters which can be found by observations.By imposing time translational invariance but not spatial invariance, we fulfill the conditions necessary for EDE.
As we are adding a dark energy term to the Einstein-Hilbert action (1.1), we can decompose the stress energy tensor as T µν = T matter µν + T radiation µν + T dark µν , i.e. we have added a source term to the standard model stress energy tensor.

Decomposition of the metric
We impose spatial isotropy by foliating the four-dimensional (4D) space-time, as in [12,26], by threedimensional (3D) sheets with a time-like unit vector, u µ which is everywhere orthogonal to the sheets.The 4D spacetime has metric g µν , and the 3D sheets have spatial metric γ µν = γ (µν) .The (3 + 1) decomposition of the 4D metric is where u µ and γ µν are subject to the orthogonality and normality conditions: Hence we can write the dark term in the stress-energy tensor in an isotropic spacetime as where ρ is the density of the universe due to this dark term and P is the pressure.The pressure and density are related by P = wρ, where w is the equation of state parameter.From now on, when we write T µν , this will refer to the dark stress-energy tensor rather than the overall energy momentum tensor.
2 L {2} (δ L g µν ) We will first look at the case L {2} (δ L g µν ) before generalising to more generic Lagrangians.For any Lagrangian which is a function only of perturbations of the metric and no derivatives thereof, i.e.L {2} = L {2} (δ L g µν ), then from [12], the most general quadratic Lagrangian for dark sector perturbations is where δ L g µν is the metric fluctuation under a perturbation and W µναβ can be thought of as a mass term for the perturbation, not to be confused with the Weyl tensor.From [13], we can use the symmetries of the W tensor: to obtain the most general possible W αβµν :

Contractions of the Eulerian change in the energy-momentum tensor
The deformation vector, which represents possible coordinate changes [14] 1 , has both time-like parts and space-like parts We can find the various contractions of δ E T µ ν by inserting the derivatives of the deformation vector.We use two different variational operators: δ L and δ E , which are linked via the Lie derivative along the diffeomorphism-generating vector ξ µ via where δ L is the Lagrangian perturbation (in co-moving coordinates) and δ E is the Eulerian perturbation (points are fixed in spacetime).L ξ is the Lie derivative in the direction of ξ µ , a vector field representing possible coordinate transformations, so that the Lie derivative acting on a given symmetric tensor field is . We are interested in the perturbed fluid equations, which are derived from the Lagrangian for perturbations, denoted as L (2) .In order to find the perturbed fluid variables, we first need to find the Eulerian change in the stress-energy tensor, δ E T µν .We will work in the synchronous gauge, where perturbations to the metric have spatial components only Working in the synchronous gauge (2.6), we vary (2.1) to obtain the contractions of the stress energy tensor [12,14] ) where we have defined "time" and "space" differentiation as the derivative operator projected along the time and space directions and where K = 3H, (where H is the Hubble parameter) is the trace of the extrinsic curvature tensor K µν ≡ ∇ µ u ν , which satisfies K µν = K (µν) and u µ K µν = 0.

Perturbed fluid variables
The components of the perturbed energy-momentum tensor T µ ν are written as where v µ is the perturbed dark sector velocity and P Π µ ν is the anisotropic part of the stress tensor, which is orthogonal and symmetric.We have dropped the subscript E on the variation of the density and pressure.We can now compare (2.9) and the contractions of the perturbed stress energy tensor (2.7), to obtain the perturbed fluid variables in terms of the deformation vector (2.4), as described in [12,14] where we have defined

Invariance under time reparameterisation
We now want to discover what constraints invariance under changes in time imposes.This is helpful for [12,20] because applying time translational reparameterisation invariance but not spatial reparameterisation invariance leads to elastic dark energy.Hence we set the coefficients of χ, χ and ∇σ χ in (2.10) to zero and therefore obtain ρ + 3H(P + ρ) = 0, Ṗ + (P + 3D W + E W ) H = 0. (2.11) This gives the conservation equation and an equation for the evolution of pressure with time.We can rewrite the perturbed fluid variables (2.10) as ) and the pressure evolution as where we have defined as parameters that can be determined by experiment.β and µ correspond to the elastic bulk modulus and the elastic shear modulus, respectively [12,20].So far, we have summarised previous work.However, as an aside we note that it is possible to place constraints on the parameters using observational data.In [27], it was noted that it might be possible to place constraints on the parameters of the elastic dark energy model, referred to as Time Diffeomorphism Invariant (TDI) models in their paper.The authors used observational data on cosmic shear and CMB lensing which would give constraints for any given value of w, unless w ≈ −1, as a wide .Unfortunately, recent Planck data has shown that w ≈ −1, so these constraint do not apply.
In Appendix A, using 2018 Planck data [28], we find a constraint of −0.004 < β < 0.106 if we assume that the equation of state parameter w of the Lagrangian (2.1), where P = wρ, is constant.If we assume that w = −1 exactly, then we also find constraints of −0.0477 ≤ μ ≤ 0.0599, where we have defined μ = µ/ρ.However,the assumption that we do not have a phantom equation of state, i.e. we require w ≥ −1, gives 0 ≤ μ ≤ 0.0599.
In the previous section, we summarised the derivation of the perturbed fluid variables for theories where the second variation of the Lagrangian depends only on the change in the metric [12] and found new constraints on the values of the elasticity and rigidity parameters.We now move on to extend this work to more general Lagrangians.
If we take the second variation of the Lagrangian as a function only of the change in the time and spatial parts of the metric separately then the most general possible quadratic Lagrangian which is a function of δ L u µ and δ L γ µν is where Using the identities [20] δ it is possible to write (3.2) as where upon comparing (3.2) with (2.1), we can see that the components of where we have defined (3.5) is a very important result -we can rewrite any Lagrangian dependent on the variation of the spatial part of the metric and the time-like part of the metric separately into one dependent on only the variation of the full metric.
In the rest of this section, we will explore how we can use the method of section ?? to find the perturbations of a Lagrangian of the form (2) without any new calculations.

Decomposition of the X,Y and Q tensors
The most general tensors that satisfy the necessary symmetries (3.3) are: Collecting like terms, we obtain Comparing to (2.3), we find (3.10)

Invariance under time reparameterisation
We repeat our calculations from earlier, where we fixed our equations to be the same under a change in time and we obtained (2.13).
Plugging in the values from (3.10), we find an equation for the evolution of the pressure where β g ≡ 1 3 (P + 3D X + E X ).We also find that the coefficients of the W tensor become Next we will examine the specific cases where the Lagrangian depends only on the variation of either the time or the spatial part of the metric.
First, if we make L {2} a function of δ L u µ only, i.e.L {2} is dependent only on the change in the time part of the metric, we obtain W X = W Q = 0, in which case β = P = 0, and using (2.14b) we can then rewrite the perturbed fluid variables from (2.12) δρ = −ρ 1 2 ĥ + ∇α ξ α , (3.13a) We find that the equation of state parameter is Using constraints from Planck [28] which show w ≈ −1, we can therefore rule out this case.While this case might intuitively seem unlikely, 2 we have completely ruled it out without having to examine any specific model.This shows the power of our parameterisation method.
If we make L a function of δ L γ µν only, i.e.L is dependent only on the change in the spatial part of the metric, we get W Y = W Q = 0, which means where we have defined β γ = 1 3 (P + 3D X + E X ).For L {2} (δ L γ µν ) the evolution equation for w (A.2) remains the same but with β now depending on D X and E X rather than D and F , which lead to the same result if we use (3.15).

Summary of L
The analysis of Section 3 shows that the effect of the metric split is simply to change the coefficients as shown in (3.10).
If we have mandated time reparameterisation invariance by decoupling χ, then the only relevant contribution to β and µ come from the spatial part of the metric γ µν , i.e.only the first term in (3.2) has any effect on the system.

Comparison with elastic dark energy theories
We can compare to [20] to find the properties of the dark energy material, as this Lagrangian can be described using elastic dark energy.
We perform a scalar-vector-tensor (SVT) decomposition, where we decompose the perturbation to the metric h ij as [29] where one can decompose a spatial tensor field as scalars can be constructed from vectors and tensors as [30] ∇ and vectors can be constructed from tensors as with the requirement that Q We now find the entropy perturbation, which is defined by The entropy perturbation and scalar anisotropic stress are given by wΓ = 0, where we have defined δ = δρ ρ , and η = − H S L + 1 3 H S T .This is the same result as [20] finds when looking at elastic mediums.

Tensor-metric theories
So far, we have looked at theories where L {2} = L {2} (δg µν , δf µν ).In this section, we now apply our analysis to theories where the dark Lagrangian is a function of both the variation of the metric g µν and of an unspecified non-dynamical symmetric rank-2 tensor f µν [31,32] 3 .
If L {2} = L {2} (δg µν , δf µν ), then the most general quadratic Lagrangian we can have is The tensors obey the following symmetries The decomposition of these tensors, called "coupling tensors" because they prescribe how the fields combine in the Lagrangian, is

Equations of motion
We use (4.3) to find where we have defined k µν ≡ δ E f µν .

Preferred direction
Using u µ and γ µν in the decomposition of the coupling tensors means that we have chosen a preferred direction for the Lagrangian.If we assume there is no preferred direction and only the full metric is seen in the coupling tensors, then (4.3) becomes (5.1c) In order to obtain (5.1a) we must set the coefficients in (4.3) as follows (5.2)

The perturbed fluid variables
When we use the coupling tensors (5.2), then we find that the perturbed fluid variables when χ is decoupled are δρ ) and we can write the evolution equations (4.7) as 6 Choosing a form for f µν

Conformal and disformal choices
An obvious choice for f µν is where A f and B f are constants and φ is a scalar field.However, because this means that the variation of f µν contains the variation of the metric, we must modify (4.1).In fact, this choice means that (4.1) can be rewritten as the second variation of a Lagrangian of the form which was studied in [13].The terms on the second line of (6.2) could be possibly be generated by "Beyond Horndeski" theories [34,35].The Beyond Horndeski theories contain terms in a specific combination to avoid the Ostrogradski instability, which would place constraints on the couplings A, B µ and C µν .

Flat reference metric
Another choice is a flat reference metric, i.e. f µν = η µν , although this choice does not simplify our perturbed fluid variables greatly while we are still using the generalised lagrangian (4.1).

Summary of results
In this paper, we have • summarised the calculations given in [14] for the perturbed fluid variables for a general dark Lagrangian of the form L {2} = L {2} (δ L g µν ) by working in the synchronous gauge and in the perfectly elastic case, both with and without the imposition of time reparameterisation invariance • obtained new constraints on the values of w and µ for a realistic sound speed under various conditions, using data from the Planck satellite • rewritten the perturbed fluid variables for general dark Lagrangians of the form L {2} = L {2} (δ L γ µν , δ L u µ ), L {2} = L {2} (δ L u µ ) and L {2} = L {2} (δ L γ µν ) both in general and when time reparameterisation invariance is imposed • obtained new evolution equations for Ṗ and w for these new Lagrangians and repeated these calculations for tensor-metric theories where L {2} = L {2} (δ L g µν , δ L f µν ), using either a timespatial metric split in the coupling tensors or using only the full metric • found new evolution equations for ρ and P and rewritten the perturbed fluid variables in these theories There are many different modified gravity theories which attempt to explain the accelerated expansion of the universe.By parameterising theories based on the second variation of the Lagrangian, we were able to develop a framework which can very quickly rule out various theories, and indeed we can rule out any theory which is purely a function of the variation of the time part of the metric.Our method could be used both in future models of massive gravity or to rule out large classes of theories when new observational results are found.
We have examined the connections between these theories and elastic dark energy.For our first case, the elastic dark energy framework can be straightforwardly used.We also placed constraints on the parameters of elastic dark energy using Planck results.Future work could examine whether other dark energy models can use this framework.