Modified Friedmann Equations in R$^{-1}$-Modified Gravity

Abstract

Recently, corrections to Einstein-Hilbert action that become important at small curvature are proposed. We discuss the first order and second order approximations to the field equations derived by the Palatini variational principle. We work out the first and second order Modified Friedmann equations and present the upper redshift bounds when these approximations are valid. We show that the second order effects can be neglected on the cosmological predictions involving only the Hubble parameter itself, e.g. the various cosmological distances, but the second order effects can not be neglected in the predictions involving the derivatives of the Hubble parameter. Furthermore, the Modified Friedmann equations fit the SN Ia data at an acceptable level.

Figures (6)

• Figure 1: The dependence of the critical redshift $z_E$ which characterizes the valid region of perturbation expansion of the field equations on the matter energy density fraction $\Omega_m$
• Figure 2: The dependence of luminosity distance on redshift computed from the first order modified Friedmann equation. The dotted, dashed and solid lines correspond to $\Omega_m=0.1, 0.2, 0.3$, respectively. The little crosses are the observed data
• Figure 3: The difference between the luminosity distances computed from the first order and second order modified Friedmann equations. The dotted, dashed and solid lines correspond to $\Omega_m=0.1, 0.2, 0.3$, respectively. The difference is very small even up to high redshift larger than 1 which is the upper limit for the validity of perturbation expansion of the full field equation. It shows that we can trust the prediction of first order equation in calculating luminosity distance in almost all the redshift region smaller than the critical value $z_E$.
• Figure 4: The dependence of the $\chi^2$ on the parameter $\Omega_m$. It can be seen that $\chi^2$ gets a little smaller for smaller value of $\Omega_m$.
• Figure 5: The dependence of the deceleration parameter $q(z)$ on redshift $z$ for $\Omega_m=0.1,0.2,0.3$ respectively. The second order effects is seen to be large.