Observational bounds on modified gravity models

Modified gravity provides a possible explanation for the currently observed cosmic acceleration. In this paper, we study general classes of modified gravity models. The Einstein-Hilbert action is modified by using general functions of the Ricci and the Gauss-Bonnet scalars, both in the metric and in the Palatini formalisms. We do not use an explicit form for the functions, but a general form with a valid Taylor expansion up to second order about redshift zero in the Riemann-scalars. The coefficients of this expansion are then reconstructed via the cosmic expansion history measured using current cosmological observations. These are the quantities of interest for theoretical considerations relating to ghosts and instabilities. We find that current data provide interesting constraints on the coefficients. The next-generation dark energy surveys should shrink the allowed parameter space for modified gravity models quite dramatically.

Figure 1

Results for general metric-based $f(R)$ gravity: constraints on $\alpha =f_0/H_0^2$, $\beta =f_{R0}$, and $\gamma =f_{\mathrm{RR}0}{H}_0^2$ (bottom panel), obtained from constraints on ${\Omega }_m$ and the derivatives of $\mathcal{H}$ (top panel), using current cosmological data.

Figure 2

$\beta$ as a function of ${\mathcal{H}}^{\prime \prime \prime }$ and ${\mathcal{H}}^{\prime \prime \prime \prime }$ setting ${\mathcal{H}}^{\prime }$ and ${\mathcal{H}}^{\prime \prime }$ to their best values.

Figure 3

Same as Fig. 1 for the $\beta =\frac{1}{3}$ case of the general metric-based $f(R)$ theories.

Figure 4

Same as Fig. 1, for $f(R)$ gravity in the Palatini formalism. The constraints in the bottom panel are about one solution of the three nonlinear equations of motion. There are other solutions, see text for further discussion.

Figure 5

Same as Fig. 1, for $f(R,R_{\mathrm{GB}}^2)$ gravity in the metric formalism.