4.3 Equation of state of dark energy

In order to confront viable f (R) models with SN Ia observations, we rewrite Eqs. (4.59View Equation) and (4.60View Equation) as follows:
3AH2 = κ2 (ρ + ρ + ρ ) , (4.92 ) m r DE − 2A H˙ = κ2 [ρm + (4∕3)ρr + ρDE + PDE ], (4.93 )
where A is some constant and
κ2ρDE ≡ (1∕2 )(F R − f) − 3H ˙F + 3H2 (A − F), (4.94 ) 2 2 κ PDE ≡ F¨ + 2H F˙− (1 ∕2)(FR − f) − (3H + 2 ˙H )(A − F ). (4.95 )
Defining ρDE and PDE in the above way, we find that these satisfy the usual continuity equation
˙ρDE + 3H (ρDE + PDE ) = 0. (4.96 )
Note that this holds as a consequence of the Bianchi identities, as we have already mentioned in the discussion from Eq. (2.8View Equation) to Eq. (2.10View Equation).

The dark energy equation of state, wDE ≡ PDE ∕ρDE, is directly related to the one used in SN Ia observations. From Eqs. (4.92View Equation) and (4.93View Equation) it is given by

˙ 2 2 wDE = − 2A-H--+-3AH---+-κ--ρr∕3 ≃ -----we-ff-----, (4.97 ) 3AH2 − κ2(ρm + ρr) 1 − (F ∕A)Ωm
where the last approximate equality is valid in the regime where the radiation density ρ r is negligible relative to the matter density ρm. The viable f (R) models approach the ΛCDM model in the past, i.e., F → 1 as R → ∞. In order to reproduce the standard matter era (3H2 ≃ κ2ρm) for z ≫ 1, we can choose A = 1 in Eqs. (4.92View Equation) and (4.93View Equation). Another possible choice is A = F0, where F0 is the present value of F. This choice may be suitable if the deviation of F 0 from 1 is small (as in scalar-tensor theory with a nearly massless scalar field [58393Jump To The Next Citation Point]). In both cases the equation of state wDE can be smaller than − 1 before reaching the de Sitter attractor [306Jump To The Next Citation Point31Jump To The Next Citation Point587Jump To The Next Citation Point435Jump To The Next Citation Point], while the effective equation of state we ff is larger than − 1. This comes from the fact that the denominator in Eq. (4.97View Equation) becomes smaller than 1 in the presence of the matter fluid. Thus f (R) gravity models give rise to the phantom equation of state of dark energy without violating any stability conditions of the system. See [210Jump To The Next Citation Point41713613] for observational constraints on the models (4.83View Equation) and (4.84View Equation) by using the background expansion history of the universe. Note that as long as the late-time attractor is the de Sitter point the cosmological constant boundary crossing of we ff reported in  [5250Jump To The Next Citation Point] does not typically occur, apart from small oscillations of weff around the de Sitter point.

There are some works that try to reconstruct the forms of f (R) by using some desired form for the evolution of the scale factor a(t) or the observational data of SN Ia [117130442191621252]. We need to caution that the procedure of reconstruction does not in general guarantee the stability of solutions. In scalar-tensor dark energy models, for example, it is known that a singular behavior sometimes arises at low-redshifts in such a procedure [234271Jump To The Next Citation Point]. In addition to the fact that the reconstruction method does not uniquely determine the forms of f (R), the observational data of the background expansion history alone is not yet sufficient to reconstruct f (R) models in high precision.

Finally we mention a number of works [115118Jump To The Next Citation Point119265Jump To The Next Citation Point31951554290] about the use of metric f (R) gravity as dark matter instead of dark energy. In most of past works the power-law f (R) model f = Rn has been used to obtain spherically symmetric solutions for galaxy clustering. In [118] it was shown that the theoretical rotation curves of spiral galaxies show good agreement with observational data for n = 1.7, while for broader samples the best-fit value of the power was found to be n = 2.2 [265]. However, these values are not compatible with the bound |n − 1 | < 7.2 × 10−19 derived in [62Jump To The Next Citation Point160Jump To The Next Citation Point] from a number of other observational constraints. Hence, it is unlikely that f (R) gravity works as the main source for dark matter.


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