Modified gravity with $f(R)=\sqrt{R^2-R_0^2}$

In this work we propose a modified gravity with the action $f(R)=\sqrt{R^2-R_0^2}$ instead of the Einstein-Hilbert action to describe the late time acceleration of the Universe. We obtain the equation of the modified gravity both in the metric and Palatini formalisms. An asymptotic solution of gravity equations corresponding to a constant Ricci scalar causes a late time acceleration of the Universe. We do a conformal transformation in the action of the modified gravity and obtain the equivalent minimally coupled scalar-tensor gravity. The equivalent Brans-Dicke gravity of this model is also studied. To examine this model with observation, the perihelion precession of the Mercury is compared with our prediction and we put an upper constraint of $R_0<H_0^2$. This range for $R_0$ is also in agreement with the cosmological acceleration at the present time. Finally we show that this action has instability for the small perturbations of the metric in vacuum solution in which adding a quadratic term of Ricci scalar can stabilize it.

Figure 1

Dynamics of $y=R/R_0$ in terms of $\tau =t{R}_0^{1/2}$ for the radiation dominate epoch. This dynamic results from the metric formalism for the field equation. The action for the modified gravity is taken as $f(R)=\sqrt{R^2-R_0^2}$.

Figure 2

Numerical solution of Eqs. (25, 26). Here we omit the Ricci scalar between the two equations and obtain a direct relation between the Hubble parameter and scale factor in the logarithmic scale.

Figure 3

Numerical solution of scale factor in terms of cosmic time, normalized to the $R_0^{-1/2}$.

Figure 4

The dependence of the scalar field on the Ricci scalar in Brans-Dicke correspondence to the modified gravity.

Figure 5

Dependence of potential on the scalar field results from the conformal transformation from Jordan to the Einstein frame in $f(R)=\sqrt{R^2-R_0^2}$ gravity.

Figure 6

Numerical solution of Eq. (49) for the perturbation of Ricci scalar around vacuum solution.