Cosmology of Ricci-tensor-squared gravity in the Palatini variational approach

We consider the cosmology of the Ricci-tensor-squared gravity in the Palatini variational approach. The gravitational action of standard general relativity is modified by adding a function $f(R^{ab}{R}_{ab})$ to the Einstein-Hilbert action, and the Palatini variation is used to derive the field equations. A general method of obtaining the background and first-order covariant and gauge-invariant perturbation equations is outlined. As an example, we consider the cosmological constraints on such theories arising from the supernova type Ia and cosmic microwave background observations. We find that the best fit to the data is a non-null leading-order correction to Einstein gravity, but the current data exhibit no significant preference over the concordance model. The growth of nonrelativistic matter density perturbations at late times is also analyzed, and we find that a scale-dependent (positive or negative) sound-speed-squared term generally appears in the growth equation for small-scale density perturbations. We also estimate the observational bound imposed by the matter power spectrum for the model with $f({\mathbf{R}}^{ab}{\mathbf{R}}_{ab})=\alpha ({\mathbf{R}}^{ab}{\mathbf{R}}_{ab}{)}^{\beta }$ to be roughly $|\beta |\lesssim \mathcal{O}({10}^{-5})$ so long as the dark matter does not possess compensating anisotropic stresses.

Figure 1

The time evolution of the effective equation of state for the $f({\mathbf{R}}^{ab}{\mathbf{R}}_{ab})=\alpha ({\mathbf{R}}^{ab}{\mathbf{R}}_{ab}{)}^{\beta }$ model. For clearness we only show the cases of $\beta =0.2$ (the dashed curve) and 0 ($\Lambda \mathrm{CDM}$, the solid curve). The other parameters defining the cosmology are ${\Omega }_m=0.27$ and ${\Omega }_r=8.5\times {10}^{-5}$. Clearly an accelerating phase following the standard matter-dominated era can be realized within this model. Note that the redshift range $1+z<1$ corresponds to the future of the present epoch, where the Universe evolves into a de Sitter stage.

Figure 2

The constraints on the parameter space of ${\Omega }_m$ and $\beta$ in the present model from joint SNe and CMB shift parameter data sets. The gray and light gray regions represent the 68.3% and 95.4% confidence contours, respectively. The white circle (${\Omega }_m=0.265$, $\beta =0.07$) is the best-fitting parameter of our model, and the star is the concordance $\Lambda \mathrm{CDM}$ model.