Modified gravity with disappearing cosmological constant

New corrections to General Relativity are considered in the context of modified $f(R)$ gravity, that satisfy cosmological and local gravity constraints. The proposed models behave asymptotically as $R-2\Lambda$ at large curvature and show the vanishing of the cosmological constant at the flat spacetime limit. The chameleon mechanism and thin shell restrictions for local systems were analyzed, and bounds on the models were found. The steepness of the deviation parameter $m$ at late times leads to measurable signal of scalar-tensor regime in matter perturbations, that allows to detect departures form the $\Lambda$CDM model. The theoretical results for the evolution of the weighted growth rate $f\sigma_8(z)$, from the proposed models, were analyzed.


Introduction
Among all models of dark energy, the ΛCDM is the simplest and the most accurate in terms of consistency with observational data (for review see [1,2,3,4]). However its non-dynamic behavior gives rise to a single possible cosmological scenario in which the fine-tuning problem cannot be solved. This motivates the development of different approaches to the dark energy problem with models that have dynamical nature and avoid the introduction of a cosmological constant. Among these models, modified gravity f (R) models stand out, especially after the recent discovery of gravitational waves [5] and the measurement of their speed with great accuracy, which led to the discarding of several scalar-tensor models and models belonging to the class of Horndeski or Galilean theories [6,7,8]. An attractive feature of modified gravity models is that they lead to cosmic accelerated expansion without introducing a dark energy matter component. These f (R) models contain non-linear in R corrections to the General Relativity that must pass various restrictions ranging from cosmological to those imposed by local gravity phenomena (see [9,10,11,12,13,14] for reviews).
Most f (R) models pass cosmological constraints, but the main obstacle for being successful is the fulfillment with the more stringent local gravity constraints. Compliance with these local constraints renders many models indistinguishable from ΛCDM and probably it is not possible to distinguish them from ΛCDM through observations, at least with the current precision available. So it is important to consider models that maintain a balance between behaving like General Relativity (GR) in local phenomena and showing signals of modified gravity at other scales, which can be measured with the next improvement in observational capacity. Thus for instance, there can be differences in the dynamics of perturbations that lead to interesting signatures that can be observed in the near future.
In the present paper we propose f (R) models that satisfy the stability conditions f (R) > 0, f (R) > 0, comply with cosmological and local gravity constraints and can lead to signals of scalar-tensor regime measurable at late times. Corrections to the GR of two types are considered. Corrections of the form e −g 1 (R) where the function g 1 (R) is a positive definite function that satisfies the asymptotic behavior, g 1 (R → ∞) → 0 and g 1 (R → 0) → ∞. The second type of corrections are of the form (1−e −g 2 (R) ), where g 2 (R) is a positive definite function that satisfies g 2 (R → ∞) → ∞ and g 2 (R → 0) → 0. The first limit leads to an effective cosmological constant while the second leads to disappearing cosmological constant in the flat space time limit.
Hence the accelerated expansion is explained as a geometrical effect. This paper is organized as follows. In section 2 we present the general features of the f (R) models. In section 3 we present some models that give viable cosmologies, and analyze their behavior under large curvature regime and at late times. In section 4 we analyze the restrictions from matter density perturbations. Some discussion is given un section 5.

General field equations and constraints
The modified gravity is described by a general action of the form where dot represents derivative with respect to cosmic time, F = f ,R = ∂f /∂R and ρ and p are the energy density and pressure for the matter component represented as a perfect fluid (in what follows we will use indistinctly f ,R or F = f ,R ). The field equation (2.2) can be written in more compact form by defining the effective energy density as follows where The Eqs. (2.2) and (2.3) lead to the following effective equation of state (EoS) where ρ and p include both matter and radiation components, i.e. ρ = ρ m + ρ r and p = p m + p r . Defining the modified density parameters Ω m and Ω r as where F 0 is the current value of F that is used to rewrite Eqs. (2.2) and (2.3) as w DE can also be written in terms of the redshift as whereH = H/H 0 and the subscript "0" stands for present values.
In general, the function f (R) can be written as the linear term that describes the Einstein gravity plus a non-linear function of R that describe the deviations from Einstein gravity that must be negligible (compared to the curvature) in the early universe and become relevant at late times to account for accelerated cosmic expansion.
Any suitable f (R) model must comply with the absence of ghost instabilities and must be stable under matter perturbations at high curvature regime [56,37], that are satisfied if the conditions take place throughout the whole period of evolution of the universe. In modified gravity, due to the non-linear correction to R, there is a propagating scalar degree of freedom f ,R whose dynamics follows from the trace equation given by where the right hand side of this equation is represented as the derivative of an effective potential V ef f with respect to the scalar field f ,R . Then the mass of f ,R can be defined as (2.14) Since viable models satisfy f ,R ≈ 1, then and at high curvature (in matter epoch for instance) Rf ,RR << 1, and this mass can be approximated as This mass allows to define the corresponding Compton wavelength, λ C = 2π/M , that mediates the interaction due to the extra scalar degree of freedom also called scalaron. In regions of high density (compared to background density) where GR is dominant, the scalaron mass acquires large values (compared to the corresponding background value) giving rise to the so called chameleon mechanism [74,75] which will be discussed in the next section.
On the other hand, the cosmological viability of an f (R) model imply the consistency with all observational evidence on late time accelerated expansion and also consistency with the high redshift universe where the GR is valid. For its analysis it is useful to resort to the parameters r and m defined as The most stringent constraints are related to the local gravity systems where the curvature is much larger than that of the background. In local systems, as well as at high curvature, the model must be practically indistinguishable from GR, which implies for an f (R) model that f ,R (R ) 1 (or lim R→∞ f (R)/R = 1) and f ,RR (R ) << R −1 , where R is the typical curvature of the local system which satisfies R >> R b , where R b is the background curvature. This also applies when R >> R 0 (R 0 is the current curvature) and restrictions from Big Bang nucleosynthesis and the Cosmic Microwave Background appear. Note that the cosmological value of the product Rf ,RR at current, low curvature Universe, is not necessarily too close to zero, since the viability of f (R) models allows current values of the deviation parameter m(R 0 ) O(1).

Background evolution
To solve numerically the field equations we use the variables introduced in [45,50] and work with the e-fold variable ln a, where (') indicates d/d ln a. Note that if we assume where and for y R it is found The equations (2.2) and (2.3) can be written in the standard form which allows to write the EoS of dark energy and the effective EoS in terms of y H and y R as The background evolution can be analyzed by solving the Eq. (2.21) numerically, which allows to find y H as function of the redshift.
Here we discuss some f (R) models that meet all required conditions of stability, cosmological viability, satisfy local gravity constraints and leave their trace on the evolution of matter density perturbations. We introduce the following models Corrections of the type e −g 1 (R) We can define a class of modified gravity models that are represented by functions of the form where the function g(R) is positive definite and satisfies the asymptotic behavior These type of models lead to the absence of cosmological constant in the flat spacetime limit. The simplest choice for g(R) that satisfies these conditions is the monomial which corresponds to the model proposed in [72,73]. The next simple case is given by the following function with η > 0 and α > 0, which leads the f (R) model where λ > 0. This model behaves asymptotically as So, the cosmological constant disappears in the flat spacetime limit. In the regime µ 2 << R this model behaves as HS and Starobinsky models and also coincides with the three-parameter HS model (c 2 = 1 in HS [45]) for α = 1.
The model (3.5) can also be written in the form To analyze the stability conditions we write the first and second derivatives of (3.5) (3.10) with η > 0 and α > 0, a sufficient condition for f ,R > 0 is the following and the condition f ,RR > 0 leads to This inequality is satisfied, independently of R, in the cases α = 1/η or αη < 1.
Depending on R, f ,RR > 0 is satisfied if The de Sitter curvature from r(R ds ) = −2 can be found by fixing λ, which gives (3.14) The condition for λ > 0 is accomplished if which is valid for any y ds , or depending on y ds Given η > 1 and assuming that y ds >> 1 (as in fact takes place for the initial conditions we will use), λ can be approximated as Replacing (3.14) in (3.5) and using the Eqs. (2.16) we find (setting R = µ 2 y) To find the stability condition at the de Sitter point, we evaluate m(y ds ) obtaining Then, the condition of stability (0 < m(r = −2) ≤ 1) can be accomplished, consistently with (3.16), if the following inequalities are satisfied Numerical analysis shows that models (3.5) with η < 1 satisfy cosmological and local gravity constraints, but these last constraints imply too small values of m(r) at current or late times (m << 10 −6 ), making it very difficult to detect measurable differences with the ΛCDM model. More attractive are the results obtained in the case η > 1.
From (3.5), (3.7) it follows that λµ 2 should be compared to the observed value of the cosmological constant On the other hand, using the density parameter for the cosmological constant, Ω Λ = Λ/(3H 2 0 ), we arrive at the following relation from (3.17)  During matter dominated epoch or at high-curvature regime, when R >> µ 2 , a good approximation for the deviation parameter m will be given by the expression and from (3.19), r simplifies to which allows to write explicitly m(r) as To have an estimation of the effect α in the behavior m at late times, we consider the current value of the background curvature R 0 which is given by (using (2.20)) Then, from (3.23) or (3.25) it follows that (using (3.22)) . (3.27) So that at late times α may affect the order of magnitude of m impacting in the steepness of m(r). On the other hand, and given that y >> y ds , in a high curvature regime the parameter α is not so relevant in the expression (3.23) (unless it is large enough), so the power η becomes the dominant parameter.
Another case of the model (3.5) is obtained by setting α = 1/η, giving The parameters m and r are given by the simple expressions where λ is fixed by the de Sitter solution Note that λ > 0 without restrictions. From the previous analysis, applied to the case α = 1/η, it can be seen that a sufficient condition for f ,R > 0 is accomplished It is worth noticing that the HS model [45] corresponds to the function

31)
Corrections of the type (1 − e −g 2 (R) ) Another important class of models can be generated by functions f R) of the type where the function g 2 (R) satisfies the asymptotic limits The simplest choice for these models is the function which leads to where λ > 0, 0 < η < 1 and µ 2 < R. The best known example is the exponential model [49,76] that corresponds to η = 1. As in the case of models (3.1), these models lead to the disappearance of cosmological constant in the flat space-time limit. As will be shown below, the rapid zero trend of the exponential model (η = 1) can be substantially attenuated considering models with η < 1, while all local gravity and cosmological restrictions are respected.
For r and m we find, setting R = yµ 2 Note that m > 0 provided that the conditions for f ,R > 0 hold. It also follows for showing that all trajectories contain the matter-dominated point (r = −1, m = 0).
The de Sitter attractor is fixed by solving the equation r(y ds ) = −2 with respect to λ, which gives From this expressions follows that λ > 0 given that 0 < η < 1. Replacing λ in (3.39) and evaluating at y ds gives To check the conditions of stability at de Sitter point, 0 < m(r = −2) ≤ 1, it is useful to expand the exponential, that yields which clearly satisfies 0 < m(y ds ) < 1 provided that 0 < η < 1.
If one assumes that y ds >> 1, then from (3.41) follows that λ ≈ y ds /2. The following approximation is valid for the deviation parameter m in the R >> µ 2 -regime, as seen and for the parameter r from (3.38), the same expression given by the Eq. (3.24) is obtained, which allows to write explicitly m(r) as The initial redshift is z i = 6.39 for al cases. In all cases the evolution of Ω DE is indistinguishable from that of ΛCDM. Comparing with Fig. 1 it can be seen that the phantom behavior is more perceptible than in the model (3.5).
some examples of the evolution of w DE and Ω DE

Starobinsky and Hu-Sawicki models
Other types of viable models are generated by the g 2 (R) function The corresponding deviation parameters are given by Performing the same analysis as with the previous models, we find the following approximate expression for m in the regime R >> µ 2 m ≈ αη (αη + 1) y ds 2y ηα+1 , (3.50) which in fact is also valid at late times whenever y ds >> 1. This expression depends only on the product αη, which leads to degeneracy. Then we can find equivalent models (under the regime µ 2 << R) by setting one of the parameters to 1. Setting η = 1 in (3.47) we find a model that gives the same results as the Starobinsky model [70]. Setting α = 1 in (3.47) gives the HS model (with c 2 = 1 in [45]). Note also that the function g 2 (R) = ln 1 + λ 1 gives the HS model.
To estimate the difference between the model (3.5) and the HS model we can use the almost model-independent initial condition encoded in the amplitude of the cosmological fieldf ,R = f ,R −1. Then, starting from the same initial condition, the behavior of the models can be followed. If we set the initial condition |f ,R (R 0 )| =f R 0 , then we can determine the constants α in (3.5) and c 2 in HS, which is described by which for a given η defines α in terms off R 0 where W is the Lambert function and y 0 = 12 Ω m0 − 9 . Then, replacing α(f R 0 ) in (3.49) we find the current value of the deviation parameter m 0 . Likewise, for the model (3.51) we find (Ω Λ = 1 − Ω m0 )

Chameleon Mechanism
To avoid conflict with local gravity tests, such as the solar system, an important effect named chameleon mechanism [74,75] can be used. This effect is due to the propagation of the scalar field (scalaron) associated to f (R), whose effective potential is described by the trace equation (2.13). From the chameleon mechanism follows that the scalar field mass m φ depends on the local matter density, being large for highdensity environments and reducing to smaller values for low-density environments.
In the cosmological background, for instance, m φ ∼ H 0 . In the Solar system this chameleon field gives important information about the strength of the force it mediates and the post-Newtonian parameter γ, which can be used to test the viability of f (R) gravity models. This scalar field that appears in the Einstein frame after the scale transformation of the metric, with the factor couples to the matter and gives rise to the scalar field potential The coupling to the matter Lagrangian gives rise to the effective potential [74,75,77] where ρ is the matter density in the Einstein frame, β = 1/ √ 6 is a constant universal coupling between matter and the scalaron φ that originates in the conformal transformation. In the Solar system we consider the Sun as a spherically symmetric object of radius r S and mass M S surrounded by background matter at much lower density. We will assume that it has constant density ρ S (for r < r S ) and outside the body (r > r S ) the density ρ B satisfies ρ B << ρ S (ρ B ≈ 10 −24 g/cm 2 is the local homogeneous matter density in our Galaxy). The gravitational potential on the surface of the body is Φ S = GM S /r S , where M S = (4/3)πr 3 S ρ S . Then the effective potential (3.57) for the Solar system evolves in two different density environments, presenting two different minima at the field values denoted as φ S and φ B , i.e.
where the in high-density region (r < r S ) the scalar field acquires the mass m 2 S = V ef f (φ S ), whereas in the background region m 2 B = V ef f (φ B ). In spherically symmetric spacetime the scalar field obeys the equation of motion ((r is the radial distance in spherical coordinates)) Solving this equation with the appropriate boundary conditions, it was found in [74,75] that the exterior solution for large bodies like the Sun or the Earth develops a thin shell, giving rise to the following expression for the scalar field (r > r S ) where the thin-shell parameter (∆r S /r S ), given by must be much smaller than unity in order to suppress the chameleon effect.
Applied to the above considered f (R) models, in order to estimate the thin-shell parameters we need to find the corresponding fields φ S and φ B from the Eqs. (3.58) and (3.59). Taking into account that the condition µ 2 << R follows, the effective potential for the models (3.5) and (3.35) can be written respectively as (we will use V ef f for the model (3.35) and likewise for the corresponding scalar fields) Note that in this last case the curvature R cannot be expressed explicitly in terms of the scalar field, but the scalar field can be expressed in terms of R via the conformal transformation (3.55). Using the approximation φ << M p , we find the following solutions for the minima of the potentials (3.63) and (3.64) Note that in all above expressions we can neglect λ compared to ρ S,B /(µ 2 M 2 p ) (2λ ∼ y ds ∼ 28, while, for instance, ρ B /(M 2 p µ 2 ) ∼ ρ B /ρ 0 ∼ 10 5 ). It can also be observed that the denser the region, the smaller the scalar field. Hence as ρ S >> ρ B then φ , which leads to the following approximation for the thin-shell parameters from The result (3.69) for the model (3.5) is similar to the results obtained for the HS and Starobinsky models [68], which applies to models whose deviation parameter m (Jordan) frame. Then it was found in [77] that under the chameleon mechanism the spherically symmetric solution in the JF can be written as where the condition λ B ∼ m −1 B >> r S was used. Then the post-Newtonian parameter γ can be approximated as From this expression and using the experimental restriction on γ [78] it is found Then using the result (3.69) we find the following restriction for the model (3.5) The chameleon mechanism applied to the Earth leads to important restrictions that also allow to avoid possible violations of the equivalence principle [74,75]. To estimate the thin-shell conditions for the Earth, it is considering the Earth as a solid sphere of radius r E ≈ 6×10 3 km and homogeneous density ρ E ≈ 10 gr/cm 3 . First we note that there are two environments surrounding the earth (ignoring the influence of the Sun, the Moon and the other planets): the atmosphere which is approximated as a 10 kmthick layer with density ρ atm ∼ 10 −3 g/cm 3 and the homogeneously distributed matter in our Galaxy with density ρ B . The gravitational potentials Φ E = ρ E r 2 E /(6M 2 p ) and Φ atm = ρ atm r 2 atm /(6M 2 p ) are related as Φ atm ≈ 10 −4 φ E (r atm ≈ r E ). In order for the atmosphere to have a thin-shell, it thickness should be less than 10 km, i.e. ∆r atm /r atm ≈ ∆r atm /r E < 10 −3 . Following the same guidelines as for the Solar system, and denoting the values that minimize V ef f as φ E , φ atm , φ B for ρ E , ρ atm and ρ B respectively, we find ∆r and φ atm << φ B , which gives the following upper bound imation, deep inside the Hubble radius (k 2 >> a 2 H 2 ) and using the quasi-static approximation, the evolution of mater perturbations during the matter dominance is controlled by the equation [79,80,11] δ m + 2Hδ m − 4πG ef f ρ m δ m 0 (4.1) where δ m δρ m /ρ m and G ef f is the effective gravitational coupling where M in given by (2.18). Note that the variation of the effective Newtonian coupling affects the expansion rate, which depends on G, and is critical for the process of where δ m evolves as δm ∝ t 2/3 during the matter dominance. At latter times the perturbations can enter the scalar-tensor regime that takes place for M 2 << k 2 /a 2 with the effective gravitational coupling G ef f 4G/(3f ,R ) 4G/3, and the evolution of δ m is different, behaving as δ m ∝ t ( √ 33−1)/6 [79,80,11]. From (4.2) we can see that the transition from GR regime to the modified gravity or scalar-tensor regime occurs which allows us to find from (4.3) the transition redshift z k where we used the approximation for m valid during matter dominated epoch (µ 2 << R) µ 2 = Ω m0 H 2 0 (see (2.20)) and the conditions (3.17) and (3.21) valid for both models. Some values of m(z ≈ 0) and z k are listed in tables I and II.  f ,R at current epoch (m(z = 0)), where we have used Ω m0 ≈ 0.3. Note that with the increase of η to the next integer, α must be increased by orders of magnitude, in order to satisfy the bound m(z ≈ 0) > 10 −6 . The scalar-tensor regime starts later for larger η. Note that for z k smaller than the order of unity, the Universe does not enter the scalar-tensor regime during the matter dominated epoch.

The Growth of Matter Perturbations.
The growth of large scale structure in the universe provides an important test which can reveal a deviation from the ΛCDM model especially at late times. The Eq. (4.1) for the fractional matter density perturbation δ m can be written in terms of the e-fold variable N = ln a as follows is the growth rate and G ef f is given in (4.2) which carries a scale-dependence, and Ω m (a) can be read off from Eq. (2.2) by rewriting it in the form Then where we have neglected the radiation. Note also that for viable f (R) models at high is the growth index of matter perturbations [92,93,76]. In order to integrate the eq. (4.11) in the matter dominated epoch we use the fact that in the high redshift region the model (3.5) is close to the ΛCDM model, and therefore we can assume that the background expansion is well approximated by the ΛCDM model. Using (4.7) in G ef f given in (4.2), we show in Fig. 4  The f σ 8 Tension.
An important cosmological test for dark energy and modified gravity models that has been intensively analyzed lately is the weighted growth rate, expressed as f σ 8 (a), in connection with the discordance found between CMB and LSS observations. σ 8 (a) is the matter 'power spectrum normalization on scales of 8h −1 M pc, and the product f σ 8 (a) is independent of the bias factor between the observed galaxy spectrum and the underlying matter power spectrum [89]. The values of σ 8 predicted by the ΛCDM model lead to an exceeding structure formation power compared to LSS observations.
where, according to the results illustrated in Figs  widely known R 2 -term [55] that is compatible with the latest Planck data [108], then this term will dominate at curvatures typical of the inflation regime and with the adequate coupling will be irrelevant at late times. Then adding the term R 2 6M 2 where M ≈ 10 −5 M p ∼ 10 13 Gev, at large curvature one can neglect the dark energy component and the dominant Lagrangian becomes which gives the well known results for the scalar spectral index n s and the tensor-toscalar ratio r:

Discussion
The main challenge of the modified gravity models in the explanation of late time accelerated expansion is the fulfillment of cosmological and the strictest local gravity constraints, while maintaining its own signatures that differentiate them from the standard ΛCDM model. In the present paper we propose models that comply with all these requirements and can lead to measurable signs of scalar-tensor regime from matter density perturbations. Two types of models were considered: models of the type f (R) = R − λµ 2 e −g 1 (R) , where the positive definite function g 1 (R) satisfies the asymptotic behavior g 1 (R → ∞) → 0 and g 1 (R → 0) → ∞, and models of the type f (R) = R − λµ 2 1 − e −g 2 (R) where the positive definite function g 2 (R) behaves as g 2 (R → ∞) → ∞ and g 2 (R → 0) → 0. The first limit leads to an effective cosmological constant while the second leads to disappearing cosmological constant in the flat space time limit.
Particularly we considered the models (3.5) and (3.35). In Figs. 1 and 2 we show the background evolution of the models, where for the assumed parameters, they present a mildly phantom behavior in the past, being more marked for the model