Stable, accelerating universes in modified-gravity theories

Modifications to gravity that add additional functions of the Ricci curvature to the Einstein-Hilbert action—collectively known as $f(R)$ theories—have been studied in great detail. When considered as complete theories of gravity they can generate nonperturbative deviations from the general relativistic predictions in the solar system, and the simplest models show instabilities on cosmological scales. Here, we show that it is possible to treat $f(R)=R\pm {\mu }^4/R$ gravity in a perturbative fashion such that it shows no instabilities on cosmological scales and, in the solar system, is consistent with measurements of the parametrized post-Newtonian parameters. We show that such a theory produces a spatially flat, accelerating universe, even in the absence of dark energy and when the matter density is too small to close the Universe in the general relativistic case.

Figure 1

The phase space of perturbative $1/R$ gravity. The value of $\mu$, referenced to the Hubble parameter, is on the horizontal axis and the matter density is on the vertical axis. The unshaded region is where one finds acceleration, and the dashed line is the locus of points for which the spatial curvature is zero. This graph shows that when $(\mu /H{)}^4\simeq -0.2$, cosmic acceleration and a spatially flat universe are consistent with the amount of matter in the Universe inferred by traditional methods.