Exponential gravity

We investigate a $f(R)$ modification of gravity that is exponential in the Ricci scalar $R$ to explain cosmic acceleration. The steepness of this dependence provides extra freedom to satisfy solar system and other curvature regime constraints. With a parameter to alleviate the usual fine-tuning of having the modification strengthen near the present, the total number of parameters is only one more than $\Lambda \mathrm{CDM}$. The resulting class of solutions asymptotes to $w=-1$ but has no cosmological constant. We calculate the dynamics in detail, examine the effect on the matter power spectrum, and provide a simple fitting form relating the two.

Figure 1

The phase space dynamics of the effective dark energy parameters $w^{\prime }\mathrm{\text{−}}w$ is unlike any standard scalar field. It crosses between the phantom and normal parts of the phase space (often a signal of modified gravity), and “sublimates” rather than thaws from the initial frozen state. Successive oscillations about $\Lambda$ are barely visible here, showing rapid damping to the asymptotic cosmological constant. Thick portions of the curves show the evolution for $a\le 1$.

Figure 2

The Ricci scalar curvature, normalized by the present matter density, evolves very rapidly from the high redshift, high curvature state to a constant asymptotic state (for $c>1$). As $c$ gets larger than unity, $R/m^2$ approaches a constant given by $12({\Omega }_m^{-1}-1)$ (i.e. 30.86 for ${\Omega }_m=0.28$ as here) and the modification factor $e^{R/r}$ always stays large. The influence of the modification occurs near the present for any allowed values of $r$, without additional fine-tuning.

Figure 3

The effective dark energy equation of state evolution is shown for various values of $c$. As $c$ gets large, the expansion history becomes indistinguishable from $\Lambda \mathrm{CDM}$.

Figure 4

The long-term history of the effective dark energy equation of state is evolution from a cosmological constant state ($w=-1$), deviation to both the phantom ($w<-1$) and normal ($w>-1$) sides, and rapid damping to a future cosmological constant state.

Figure 5

The derivative $f_R$ (solid black curves) and the structure parameter $B$ (dashed red), related to the Compton wavelength of the scalar part of the gravity modification, are plotted vs $c$. The upper pair is evaluated at the present and the lower pair is at $a=0.5$. Note that $f_R$ (and $B$) has a very steep dependence on scale factor $a$, so structures that might nominally have $f_R\approx 4\times {10}^{-4}$ today may have formed when $f_R<{10}^{-6}$ and so not exhibit observable deviations from general relativity. The maximum deviation of the effective equation of state from a cosmological constant (dotted blue) closely follows $B(a=1)/2$ (dot-dash magenta).

Figure 6

The matter power spectrum in $f(R)$ theory deviates from the $\Lambda \mathrm{CDM}$ result in a scale-dependent manner. In the exponential gravity case, a wide range of values of $c$ have suppressed deviations and can be consistent with observations. The lowest curve has the same value $c=3$ as the top curve, but plots the power spectrum deviation at $a=0.5$, showing the effect of the steepness of the model.