Conditions for the cosmological viability off(R)dark energy models

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) ≈ +0 and dm/dr > −1 at r ≈ −1. For the existence of a viable late-time acceleration we require instead either (i) m = −r − 1 , ( √ 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) = αR −n and f = R + αR −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 ∝ 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.

A minimal set of invariants as a systematic approach to higher order gravity models

“…where we can see a logarithmic action of the form from [47] can be shown to have similar behavior to the exponential action on cosmological scales as we add more higher terms to return the series to its pure infinite form also with a pure de Sitter phase.…”

“…As we add more and more terms, we return to the pure exponential model above with an infinite number of terms, meaning n → ∞. Now, looking at the solutions given in (48) and taking their limit when n → ∞, we see that in the last two solutions, the larger solution tends to infinity, or pure de Sitter (as for example noted for f (R) models in [47]) and the smaller solution goes to zero. This only leaves the other pair of solutions there, depending only on β, not n, which due to a 1 < 5/6 give complex solutions just as we saw for the exponential action and the asymptotic equation (57).…”

“…Using some guidance from previous studies of f (R) models [47], we continue the study of inverse functions of the invariant (−R 2 /3 − 8R1) in the basis {R, R1} by considering the addition of an exponential function,…”

Revisiting the screening mechanism inf(R)gravity

“…One of the simplest attempts is the so-called fðRÞ gravity, in which the Ricci curvature R in the Einstein-Hilbert action of GR is replaced by an arbitrary function of R [5][6][7][8][9][10][11][12][13][14][15][16]. fðRÞ gravity introduces a new scalar field degree of freedom that has profound impacts on cosmology.…”

Simulations of Dark Energy Cosmologies

“…These models are severely constrained by Solar System tests of general relativity (Hu & Sawicki, 2007). In addition to these constraints there are several limits on the functional form of f (R), for example, 1 + f (R) > 0 for all R so that the effective gravi- (Amendola et al, 2007). Scalar-tensor theories, first introduced by Brans and Dicke (Brans & Dicke, 1961), feature a scalar field in the Einstein-Hilbert action which is non-minimally coupled to the matter fields in the so called Einstein frame.…”