Logarithmic corrected $F(R)$ gravity in the light of Planck 2015

In this letter, we consider the theory of $F(R)$ gravity with the lagrangian density $ \pounds = R+\alpha R^2 + \beta R^2 \ln \beta R $. We obtain the constant curvature solutions and find the scalar potential of the gravitational field. We also obtain the mass squared of a scalaron in the Einstein$^,$s frame. We find cosmological parameters corresponding to the recent Plank 2015 results. Finally, we analyze the critical points and stability of the new modified theory of gravity and find that logarithmic correction is necessary to have successful model.


The Model
We begin with the equation (1.1) to modify the Ricci scalar R in the EH action. The function F (R) satisfies the conditions F (0) = 0, corresponding to the flat space-time without cosmological constant. Thus, the action in the Jordan frame becomes, where κ = M −1 pl , and M pl is the reduced Planck mass, and £ m is the matter Lagrangian density. Our main goal is to study the cosmological parameters describing inflation and the evolution of the early universe. However, we can discuss about consequences in the late time. From the equation (1.1) we obtain, F ′ (R) = 1 + γR + 2βR ln βR, F ′′ (R) = λ + 2β ln βR, (2.2) where γ = 2α + β and λ = 2α + 3β. The function F (R) obeys the quantum stability condition F ′′ (R) > 0 for α > 0 and β > 0. This ensures the stability of the solution at high curvature. It follows from the equation (2.2) that the condition of classical stability F ′ (R) > 0 leads to, 1 + (γ + 2β ln βR) R > 0, (2.3)

Constant curvature condition
We consider constant curvature solutions of the equations of motion that follow from the action given by (2.1) without matter. The governing equation is given by [64], and hence, which satisfy 0 < βR < 1. Here, the condition F ′ (R) F ′′ (R) > R, is satisfied, and therefore the model can describe primordial and present dark energy, which are future stable. From the equation (2.2), we obtain, which simplifies to βR < 1 2 . Thus, the solution R 0 = 0 satisfy the equation (1.1) which then imply that the flat space-time is stable. The second constant curvature solution βR 0 ≈ 1 does not satisfy the equation (3.3), and this leads to unstable de Sitter space-time, so describes inflation.

The scalar tensor form
In the Einstein frame corresponding to the scalar tensor theory of gravity, we have the following conformal transformation of the metric [65], In that case the action given by the equation (2.1) with £ m = 0 written as, where ∇ µ is the covariant derivative, andR is determined using the conformal metric in the equation (4.1). The scalar field Φ was found to be, In Fig. 1 we plotted the function κΦ(R) for different values of β and α. From Fig. 1 (b) we can see that increasing α decreases value of the scalar field, while there is no regular behavior with variation of β. However the condition 0 < βR < 1 satisfied in the plots. We can see that the scalar field (multiple by κ) is positive for small R and β > 0.
The potential V was found to be, In Fig. 2 we plotted the function κ 2 V versus R for different values of the parameters. We can see that there is at least an extremum (minimum) obtained via V ′ = 0 which means, R (λ + 2β ln(βR)) κ 2 (1 + γR + 2βR ln(βR)) 2 = 0, (4.5) therefore, 2α + 3β + 2β ln(βR) = 0. (4.6) Thus, using the equation (2.3) and (3.3) in the equation (4.6) with the condition βR < 0.5, we found that the flat space-time is stable with R = 0 and the curvature We also obtain the mass squared of a scalaron, The plot of the function m 2 versus R is given by Fig. 3 which show periodic behavior. One can verify that m 2 < 0 for the constant curvature solution R 0 = 1 β e

Cosmological parameters
We know that the corrections of F (R) gravity model are small as compared with GR for R ≫ R 0 , where R 0 is a curvature at the present time, so we have the following conditions [66], One can investigate that for 0 < α < 1 all inequalities in the equation (5.2) are satisfied. The slow-roll parameters are given by, For the slow-roll approximation we need the conditions ε ≪ 1, and η ≪ 1. One can obtain the slow-roll parameters expressed through the curvature from the equations (4.4)-(4.7) as follows, and η = 2 3 1 − 4βR + β 2 R 2 − 2αβR 2 − 2β 2 R 2 ln(βR) R(λ + 2β ln(βR))(1 + αR + βR ln(βR)) .
The age of the inflation can be obtained by calculating the e-fold number, Here the value R end corresponds to the time of the end of inflation when ε or |η| are close to 1. We find that selected value of βR and β in previous give 50 < N e < 60 to solve the flatness and horizon problems. The index of the scalar spectrum power law due to density perturbations is given by, The tensor-to-scalar ratio is defined by, The Planck experiment results [63] tell that, We can use above data to fix parameters. One can check that our selected values of α and β give good result in agreement with observational data.

Critical points and stability
There are several ways to specify viable conditions of F (R) gravity theories such as positivity of the effective gravitational coupling [67], stability of cosmological perturbations [68], equivalence principle and solar-system constraints [69], and asymptotic behavior of the Λ-CDM in the large curvature regime [53]. Here, we use autonomous equations to investigate critical points and stability of the model. In order to investigate critical points of equations of motion, it is useful to introduce the following dimensionless parameters [43], = − (λ + 2β ln(βR))Ṙ H(1 + γR + 2βR ln(βR)) , (6.1) , (6.2) 3) where H is Hubble parameter, and the dot denote the derivative with respect to the time. The deceleration parameter q is given by q = 1 − x 3 . The critical points for the system of equations can be studied by the investigation of the function m(r). Equations of motion in the absence of the radiation, ρ rad = 0 (x 4 = 0), with the help of above equations can be written in the form of autonomous equations [43]. One can discuss the critical points of the system of equations by the study of the function m(r) which shows the deviation from the ΛCDM model. The plot of the function m(r) is given in by Fig. 6.
and For the other critical point, ), (6.9) we can find x 3 = 1 2 , m ≈ 0, r = −1, and EoS of a matter era is ω ef f = 0 (a = a 0 t

Conclusion
In this letter, we suggested a new model of modified F (R) gravity representing the effective gravity model which can describe the evolution of universe. Usually, the main purpose of F (R) gravity models is to solve the late-time cosmic acceleration. However, it is possible to study inflation. Our main goal is to compute some inflationary parameters and compare with observational data. The constant curvature solutions, βR = 0.420, and βR = 0.919 were obtained corresponding to the flat and the de Sitter space-time, respectively. The de Sitter space-time gives the acceleration of universe and corresponds to inflation. The flat space-time is stable but the de Sitter space-time is unstable in the model and it goes with the maximum of the effective potential. The slow-roll parameters ε, η and the e-fold number of the model were evaluated. The model gives e-fold number 50 < N e < 60 characterizing the age of inflation. Agreement of our results with observational data suggests that the logarithmic corrections are useful and may be necessary to construct successful model. We show by the analysis of critical points of autonomous equations that the standard matter era exists and the standard matter era conditions are satisfied. The model may be alternative to GR, and can describe early-time inflation.
There are also more comprehensive model to study inflation, such as F (R) proportional to polynomial inflation [70] with logarithmic correction. We left this point as future work.