Cosmology of generalized modified gravity models

We consider general curvature-invariant modifications of the Einstein-Hilbert action that become important only in regions of extremely low space-time curvature. We investigate the far future evolution of the Universe in such models, examining the possibilities for cosmic acceleration and other ultimate destinies. The models generically possess de Sitter space as an unstable solution and exhibit an interesting set of attractor solutions which, in some cases, provide alternatives to dark energy models.

Figure 1

Two phase portraits for the modified gravity model proposed in 7. The left portrait is in the coordinates $(x,v)$, for which an attractor at constant $v=p$ corresponds to a power-law solution with $a(t)\propto t^p$. The right portrait is for the same theory in the $(\dot{H},H)$ plane, with the unstable de Sitter solution at (0, 1).

Figure 2

The values of the various distinguished points as $\alpha$ is varied.

Figure 3

Two phase portraits for the $f(R,P,Q)=-m^6/P$ modification. The left portrait is in the coordinates $(x,v)$, for which an attractor at constant $v=p$ corresponds to a power-law solution with $a(t)\propto t^p$. The right portrait is for the same theory in the $(\dot{H},H)$ plane. There are two late-time power-law attractors corresponding to $p=(4\pm \sqrt{6})/2$. The ($-$) branch, represented by the solid lines, is nonaccelerating, while the ($+$) branch, represented by the dash-dotted line, is accelerating.

Figure 4

Phase portrait for the $f(R,P,Q)=-M^6/Q$ modification in the coordinates $(x,v)$, for which an attractor at constant $v=p$ corresponds to a power-law solution with $a(t)\propto t^p$.

Figure 5

Phase plot for the linearized equations close to the power-law solution in the coordinates $(x,v)$, for which an attractor at constant $v=p$ corresponds to a power-law solution with $a(t)\propto t^p$.