Modified gravity with an exponential function of curvature

The role of an exponential function of the scalar curvature in the modified gravity is analyzed. Two models are proposed. A toy model that complies with local and cosmological constraints and gives appropriate qualitative description of the cosmic evolution. The trajectories in the $m-r$ plane, given by $m=-(r+1)(\eta+r)/r$, lead to saddle matter dominant critical point ($r=-1$, $m=0$) that can evolve towards the late time de Sitter attractor at $r=-2$ and $0<m\le 1$. Initial conditions for the case $\eta=0.68$ have proposed, showing that this toy model has an acceptable matter era and gives an approximate qualitative behavior of cosmic evolution. A second viable model, behaves very close to $\Lambda$CDM at early times and can satisfy local and cosmological constraints. It behaves as $R-2\Lambda$ at $R\rightarrow \infty$ and tends to zero at $R\rightarrow 0$, containing flat spacetime solution. The model gives viable trajectories in the $(r,m)$-plane that, as the first model, connect the matter dominated point with a de Sitter attractor at $r=-2$. The cosmic evolution of the main density parameters in this model is consistent with current observations with an equation of state very close to $-1$.


Introduction
So far the most successful dark energy model is the cosmological constant (for review see [1,2,3,4]), despite its main fine-tuning problem, that motivates the seek for alternative models of dynamical nature. Among these models, the modification of gravity that involves a general function f (R), represents an appealing alternative that has been under intense study last years. The function f (R) generalizes the Einstein-Hilbert Lagrangian by adding corrections that are non-linear functions of the curvature, subject to local (solar system) and cosmological (high redshifts) constraints that determine its viability (see [5,6,7,8] for reviews). These corrections may become relevant in a late universe and many types of modifications to the Einstein-Hilbert action have been proposed so far [5,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Among the first and most studied corrections to the Einstein-Hilbert Lagrangian are the corrections of the form R n , but it is well known that corrections with n > 1 that are relevant at early times like in the case of n = 2 leading to de Sitter expansion [28], are negligible small compared to R at the present epoch and not suitable to explain the current accelerated expansion. Models with n ≤ 0 contain instabilities that prevent them from having a matter dominated era [29,30,23] and are also inconsistent with solar system tests. There are also models that attempt to unify early time inflation with late time acceleration [31,32,33,34].
Modified gravity with arbitrary function of the 4-dimensional Gauss-Bonnet invariant has been introduced in [35,36,37]. Any viable model of modified gravity should pass not only the Solar-system tests, that are perhaps the more reliable and challenging, where the average density of matter is high compared with that of the universe, but also should satisfy the cosmological restrictions from high redshift observations. The so called chameleon mechanism is used to pass solar system tests. The purpose of this mechanism is to give a large enough mass to the scalar field (that appears after the conformal transformation in the metric to convert f (R) to the Einstein frame) to avoid measurable corrections to the local gravity phenomena [27,38,39]. A number of works have been devoted to f (R) models that can satisfy both cosmological and local gravity constraints [27,40,41,42,43,44]. Exact cosmological solutions have been studied in [45,46,47,48,49,50,51].
In the present paper we consider an exponential function of the curvature in modified gravity and study its cosmological consequences. It is shown that the two proposed models can satisfy both, local and large scale cosmological constraints. As the criterium to analyze the outcomes of the models we used the (m, r) diagram which shows that the models are viable and contain the matter era followed by a late time solution with accelerated expansion. The first model gives an approximate qualitative description of the cosmic evolution while the second second model is more realistic and behaves very close to the ΛCDM with disappearing cosmological constant at R → 0.
A simple modified gravity model with exponential gravity, that realize early and late time accelerated expansion, was proposed in [52] and observational constraints on this model were studied in [53]. A more general model with exponential and logarithmic corrections was considered in [54] and constant roll inflation with exponential modified gravity was studied in [55]. This paper is organized as follows. In section 2 we present the general features of the f (R) models, including the autonomous system and the relevant critical points for our study in terms of the (r, m) parameters. In section 3 we present the models, showing the conditions for viability and its trajectories in the (r, m)-plane, and some numerical cases of cosmic evolution. In section 4 we present some discussion.

Field equations
Let us start with the following action for modified gravity where κ 2 = 8πG and L m is the Lagrangian density for the matter component which satisfies the usual conservation equation. Variation with respect to the metric gives the equation of motion µν is the matter energy-momentum tensor assumed as T m µν = (ρ + p) u µ u ν + pg µν and f ,R ≡ df dR . The trace of eq. (2.2) gives The time and spatial components of the Eq. (2.2) are given by the following expres- where dot represents derivative with respect to cosmic time. These expressions 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 . In order to be viable, the function f (R) must satisfy the observational evidence both at the local level and at cosmological distances. The first general restrictions can be summarized as follows. Firstly the condition f ,R > 0 is necessary to avoid negative effective Newtonian coupling. On the other hand, the scalar particle associated with f (R), dubbed scalaron with mass (in matter epoch or in the regime requires f ,RR > 0 in order to avoid ghosts and is also a condition of stability under perturbations.
To study the viability of modified gravity as cosmological model it is useful to consider the autonomous system with the following dimensionless variables that can be obtained from Eq. (2.4)) [56,57] (in what follows we will use indistinctly f ,R or which yields the following dynamical system where N = ln a, w = Ω r is the density parameter of the radiation component, and the following quantities help to understand the viability of f (R) models In terms of these variables the effective EoS (2.6) is written as , (2.16) where F 0 is the current value of f ,R .
The critical points of the above autonomous system, in absence of radiation (w = 0), for the model (2.1) can be written in terms of m and there are three important fixed points [56] that we will consider to analyze the viability of our model: the critical point that gives rise to scaling solutions including the matter dominated era given by , (2.17) with the following main parameters and eigenvalues where prime represents derivative with respect to r. And the other two stable fixed points that lead to de Sitter and accelerated solutions with eigenvalues and with the main parameters and the corresponding eigenvalues Form the coordinates y and z for the points P S and P C it can be seen that they are

Model 1.
Firstly we discuss a toy model that satisfies all above discussed requirements, given by the following function where η > 0. First we note that the coefficient of R has the limits The first limit allows the existence of flat spacetime solutions and the second facilitates the consistency with high redshift CMB observations. This function can also be written as where the correctionf (R) and its derivative satisfy the condition (given η > 0) lim R→∞f (R)/R = 0 and lim R→∞f (R) = 0, which are important to recover the General Relativity at early times to satisfy the restrictions from Big Bang nucleosynthesis and CMB, and at at high curvature regime for local system tests. Taking the derivatives of f (R) we find (3.4) indicating that the model satisfies automatically the condition f ,R > 0, necessary to avoid antigravity regime. The second derivative gives which satisfies the condition f ,RR > 0 whenever This last inequality is always satisfied for 0 < η ≤ 1, but given the fact that the last term is positive, it also allows 1 ≤ η < 1 + ηµ 2η /R η . For 0 < η < 1, the correction to the General Relativity from the model (3.1) contains a finite number of positive powers of R, R γ i (0 < γ i < 1) and an infinite number of negative powers of R, while for η > 1 the correction contains only negative powers of R, but in both cases the correction is a regular function at R → 0 and R → ∞.
To continue the analysis we use the parameters m and r for this model, which are given by which gives the following relationship As follows from this expression the model contains the matter dominated point Applying this condition on (3.8) leads to 0 ≤ η < 2 which after the interception with the condition f ,RR > 0 leads to the allowed values for η On the other hand, as follows from the expression (3.7) for r, the physically allowed values of r satisfy the inequality r < −1, which imply that r can approach −1 only form the left, i.e. r → −1 − . This implies at the same time, according to (3.7) and and for m(r = −2) we obtain Then the stability condition of de sitter point at R = R 1 , 0 < m(r = −2) ≤ 1 is satisfied by 0 ≤ η < 2, which is consistent with the restriction discussed above. On he other hand, the restriction on η given by (3.9), imply the restriction which leads to the following inequality between Ricci scalar at de Sitter point and the curvature scale µ 2 In Fig. 2 the cosmic evolution for the model is shown in terms of the e-folding variable N = ln a = − ln(1 + z). The equation of state corresponding to the scenario of Fig.2 is shown in Fig. 3 As can be seen from Figs. 2 and 3 the expansion rate is a bit slower compared to the ΛCDM model.

Local Gravity Constraints.
The effective mass of the modified gravity f (R) model is given by which under the condition m << 1, can be reduced to  where R s is the curvature of the local structure, and we assumed f Rs 1. Making use of the relationship R ∼ H 2 ∼ 8πGρ applied to the current universe (R 0 , ρ 0 ) and to the local structure (R s , ρ s ), we can write R s ∼ H 2 0 ρ s /ρ 0 and the above constraint becomes [57] Applied to the current universe with ∼ H −1 0 , it leads to m(R 0 ) << 1. For a local structure with << H −1 0 one expects that m is even much smaller than the previous case. Thus, for the solar system with ρ s ∼ 10 −23 gr/cm 3 and ∼ 10 13 cm one finds that m << 10 −24 , where we used H −1 0 ∼ 10 28 cm. In order to find which value of µ 2 can satisfy this restriction we use the expression (3.7) for m in terms of the curvature.
In general, for 0 < η < 1 and b << 1, one has for m << b from (3.7) that Taking, for instance η = 0.6, one finds that µ << 10 −17 H 0 , which is much smaller than the Hubble scale today, but in terms of the f (R) mass M gives M >> 10 15 H 0 ∼ 10 −18 ev, which is the expected bound. So, with an adequate choice of the parameter µ the model passes local system tests.
Einstein Frame Potential.
In the Einstein frame, in terms of the equivalent scalar field [58,59] f the potential is given by the following expression From (3.1) and (3.20) one finds the scalar curvature as which gives, from (??), the explicit expression for the scalar field potential Note that the argument of the W -function is well defined for 0 ≤ φ < ∞ (0 < η ≤ 1), Replacing this value in (??) gives the potential at de Sitter point as The shape of the potential in the interval 0 ≤ φ ≤ 2 is shown in Fig. 4.

Model 2.
A second, viable model, has the following form where λ is positive dimensionless and η is real positive. This model satisfies the limits where the first limit leads to consistency with ΛCDM at high redshift, and the limit In order to analyze the stability conditions, f ,R > 0, f ,RR > 0, we first determine the value of λ by fixing the de Sitter point r = −2 at R = R 1 . From (3.28) it is found The restriction λ > 0 can be solved by imposing Using the above expression for λ in (3.29) we find the condition of stability at de Sitter point, 0 < m(r = −2) ≤ 1, as where η > 0.
Analyzing the condition f ,R > 0 for R > R 1 we find, using (3.30) (3.34) which is equivalent to where x = µ 2 /R and x 1 = µ 2 /R 1 . Due to the difficult to solve this inequality with exponentials, we first use the fact that x < x 1 , which allows to change the above inequality by the following Since ηx η 1 < 2 (see (3.31), then we can set and write the inequality as Note that the function e −x η x η is well defined in the real axis and has its maximum value e −1 at x = 1. Therefore, is enough to prove that e −1 < 1 The general stability condition f ,RR > 0, using (3.30) leads to The denominator is positive according to (3.31) and (3.37). Then, in order to satisfy this inequality we need to prove that 1 + η (1 − x η ) > 0, which leads to Using the fact that x < x 1 and assuming the expression (3.37) for x 1 , this inequality can be satisfied if which takes place as follows from the restriction (3.40). Hence the model satisfies the stability conditions f ,R > 0 and f ,RR > 0 for R ≥ R 1 .
Concerning the viability of the model, since m cannot be expressed analytically in terms of r, we resort to the parametric plot of some trajectories in the (r, m)-plane, using (3.28) and (3.29). To this end, we use the variable y = 1/x = R/µ 2 with de Sitter value y 1 = 1/x 1 = R 1 /µ 2 and will consider the representation for x 1 given by (3.37), i.e.
The corresponding expressions for r and m become where we used the expression for λ from (3.30). It can be checked that, at y = (pη/2) 1/η , r takes the value r = −2. In Fig. 5 we present some trajectories in the (r, m)-plane The local gravity constraints can be addressed using the representation for m given by (3.46). Considering, for instance, the solar system one has y s = R s /µ 2 , where R s 10 6 H 2 0 . As discussed before, the solar system constraints demand m << 10 −24 . For the parameters η and p as used in Fig.5, we find that if we set Turning to the Einstein frame we can write the scalar field and the potential, using (3.20) and (3.21), in terms of y = R/µ 2 as where we used (3.30) and (3.44). The behavior of the potential for the trajectories depicted in Fig. 5 is shown in Fig.6. In order to analyze the cosmic evolution of the main density parameters Ω m , Ω DE and Ω r for the model (3.26) one needs to solve the autonomous system (2.9)-(2.13) with appropriate initial conditions. Since there is no explicit expression for m(r), we resort to an approximation, by making a polynomial is replaced by adequate initial conditions. Further detailed analysis of local gravity constrains is needed, and it will be also interesting to study (ongoing work) the constraints on the model coming from the background and matter density perturbations.