Conditions for the cosmological viability of f(R) dark energy models

TL;DR

This paper establishes the conditions under which f(R) gravity models can produce a realistic cosmological evolution, including a matter-dominated era followed by late-time acceleration, by analyzing the geometric properties of the models in the (r, m) plane.

Contribution

It introduces a geometrical classification of f(R) models based on the m(r) curve, identifying criteria for their cosmological viability and ruling out certain popular models.

Findings

Viable f(R) models must have m(r) curves connecting matter and acceleration regions.

Models like f(R)=alpha R^{-n} are not cosmologically acceptable for positive or negative n.

Most f(R) models tend to replace matter era with a t^{1/2} expansion, and phantom attractors are incompatible with matter epochs.

Abstract

We derive the conditions under which dark energy models whose Lagrangian densities f are written in terms of the Ricci scalar R are cosmologically viable. We show that the cosmological behavior of f(R) models can be understood by a geometrical approach consisting in studying the m(r) curve on the (r, m) plane, where m=Rf_{,RR}/f_{,R} and r=-Rf_{,R}/f with f_{,R}=df/dR. This allows us to classify the f(R) models into four general classes, depending on the existence of a standard matter epoch and on the final accelerated stage. The existence of a viable matter dominated epoch prior to a late-time acceleration requires that the variable m satisfies the conditions m(r) approx+0 and dm/dr>-1 at r approx-1. For the existence of a viable late-time acceleration we require instead either (i) m=-r-1, (sqrt{3}-1)/2<m<1 and dm/dr<-1 or (ii) 0<m<1 at r=-2. These conditions identify two regions in the (r, m) space, one for the matter era and the other for the acceleration. Only models with a m(r) curve that connects these regions and satisfy the requirements above lead to an acceptable cosmology. The models of the type f(R)=alpha R^{-n} and f=R+alpha R^{-n} do not satisfy these conditions for any n>0 and n<-1 and are thus cosmologically unacceptable. Similar conclusions can be reached for many other examples discussed in the text. In most cases the standard matter era is replaced by a cosmic expansion with scale factor a propto t^{1/2}. We also find that f(R) models can have a strongly phantom attractor but in this case there is no acceptable matter era.