Accelerated cosmological models in Ricci squared gravity

Alternative gravitational theories described by Lagrangians depending on general functions of the Ricci scalar have been proven to give coherent theoretical models to describe the experimental evidence of the acceleration of the Universe at present time. In this paper we proceed further in this analysis of cosmological applications of alternative gravitational theories depending on (other) curvature invariants. We introduce Ricci squared Lagrangians in minimal interaction with matter (perfect fluid); we find modified Einstein equations and consequently modified Friedmann equations in the Palatini formalism. It is striking that both Ricci scalar and Ricci squared theories are described in the same mathematical framework and both the generalized Einstein equations and generalized Friedmann equations have the same structure. In the framework of the cosmological principle, without the introduction of exotic forms of dark energy, we thus obtain modified equations providing values of $w_{\mathrm{e}\mathrm{f}\mathrm{f}}<-1$ in accordance with the experimental data. The spacetime bi-metric structure plays a fundamental role in the physical interpretation of results and gives them a clear and very rich geometrical interpretation.

Figure 1

Phase portrait for the plane $(w,n)$, where $w$ is the fluid parameter characterizing matter and $n$ the exponent of $S$ in the Lagrangian. The shaded areas represent physically and mathematically admissible pairs. We notice the following: for $n>1$ we cannot have radiation ($w=\frac{1}{3}$) but dust is allowed; for dust matter ($w=0$) and $n>1$, $w_{\mathrm{e}\mathrm{f}\mathrm{f}}\to -1^{+}$, i.e., $w_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be approached only from above; on the contrary, negative powers ($n<0$) do not allow dust, although any dustlike matter can be allowed for large enough $|n|$. Moreover in this case $w_{\mathrm{e}\mathrm{f}\mathrm{f}}\to -1^{-}$ is possible from below (superacceleration).