The growth of matter perturbations in f(R) models

We consider the linear growth of matter perturbations on low redshifts in some $f(R)$ dark energy (DE) models. We discuss the definition of dark energy (DE) in these models and show the differences with scalar-tensor DE models. For the $f(R)$ model recently proposed by Starobinsky we show that the growth parameter $\gamma_0\equiv \gamma(z=0)$ takes the value $\gamma_0\simeq 0.4$ for $\Omega_{m,0}=0.32$ and $\gamma_0\simeq 0.43$ for $\Omega_{m,0}=0.23$, allowing for a clear distinction from $\Lambda$CDM. Though a scale-dependence appears in the growth of perturbations on higher redshifts, we find no dispersion for $\gamma(z)$ on low redshifts up to $z\sim 0.3$, $\gamma(z)$ is also quasi-linear in this interval. At redshift $z=0.5$, the dispersion is still small with $\Delta \gamma\simeq 0.01$. As for some scalar-tensor models, we find here too a large value for $\gamma'_0\equiv \frac{d\gamma}{dz}(z=0)$, $\gamma'_0\simeq -0.25$ for $\Omega_{m,0}=0.32$ and $\gamma'_0\simeq -0.18$ for $\Omega_{m,0}=0.23$. These values are largely outside the range found for DE models in General Relativity (GR). This clear signature provides a powerful constraint on these models.Comment: 14 pages, 7 figures, improved presentation, references added, results unchanged, final version to be published in JCA

Constraining f(R)f(R) gravity models with disappearing cosmological constant

The $f(R)$ gravity models proposed by Hu-Sawicki and Starobinsky are generic for local gravity constraints to be evaded. The large deviations from these models either result into violation of local gravity constraints or the modifications are not distinguishable from cosmological constant. The curvature singularity in these models is generic but can be avoided provided that proper fine tuning is imposed on the evolution of scalaron in the high curvature regime. In principle, the problem can be circumvented by incorporating quadratic curvature correction in the Lagrangian though it might be quite challenging to probe the relevant region numerically.Comment: 9 pages and 4 figures, minor clarifications and corrections added, final version to appear in PR

Scalar-Tensor Dark Energy Models

We present here some recent results concerning scalar-tensor Dark Energy models. These models are very interesting in many respects: they allow for a consistent phantom phase, the growth of matter perturbations is modified. Using a systematic expansion of the theory at low redshifts, we relate the possibility to have phantom like DE to solar system constraints.Comment: Submitted to the Proceedings of the Marcel Grossmann Conference MG11, July 2006, Berlin; 3 page

Galileon gravity and its relevance to late time cosmic acceleration

We consider the covariant galileon gravity taking into account the third order and fourth order scalar field Lagrangians L_3(\pi) and L_4(\pi) consisting of three and four $\pi$'s with four and five derivatives acting on them respectively. The background dynamical equations are set up for the system under consideration and the stability of the self accelerating solution is demonstrated in general setting. We extended this study to the general case of the fifth order theory. For spherically symmetric static background, we spell out conditions for suppression of fifth force effects mediated by the galileon field $\pi$. We study the field perturbations in the fixed background and investigate conditions for their causal propagation. We also briefly discuss metric fluctuations and derive evolution equation for matter perturbations in galileon gravity.Comment: 11 pages, no figure, minor clarifications and few refs added, to appear in pr

The Gravitational Horizon for a Universe with Phantom Energy

The Universe has a gravitational horizon, coincident with the Hubble sphere, that plays an important role in how we interpret the cosmological data. Recently, however, its significance as a true horizon has been called into question, even for cosmologies with an equation-of-state w = p/rho > -1, where p and rho are the total pressure and energy density, respectively. The claim behind this argument is that its radius R_h does not constitute a limit to our observability when the Universe contains phantom energy, i.e., when w < -1, as if somehow that mitigates the relevance of R_h to the observations when w > -1. In this paper, we reaffirm the role of R_h as the limit to how far we can see sources in the cosmos, regardless of the Universe's equation of state, and point out that claims to the contrary are simply based on an improper interpretation of the null geodesics.Comment: 9 pages, 1 figure. Slight revisions in refereed version. Accepted for publication in JCAP. arXiv admin note: text overlap with arXiv:1112.477

Constraints on scalar-tensor theories of gravity from observations

In spite of their original discrepancy, both dark energy and modified theory of gravity can be parameterized by the effective equation of state (EOS) $\omega$ for the expansion history of the Universe. A useful model independent approach to the EOS of them can be given by so-called Chevallier-Polarski-Linder (CPL) parametrization where two parameters of it ($\omega_{0}$ and $\omega_{a}$) can be constrained by the geometrical observations which suffer from degeneracies between models. The linear growth of large scale structure is usually used to remove these degeneracies. This growth can be described by the growth index parameter $\gamma$ and it can be parameterized by $\gamma_{0} + \gamma_{a} (1 - a)$ in general. We use the scalar-tensor theories of gravity (STG) and show that the discernment between models is possible only when $\gamma_a$ is not negligible. We show that the linear density perturbation of the matter component as a function of redshift severely constrains the viable subclasses of STG in terms of $\omega$ and $\gamma$. From this method, we can rule out or prove the viable STG in future observations. When we use $Z(\phi) =1$, $F$ shows the convex shape of evolution in a viable STG model. The viable STG models with $Z(\phi) = 1$ are not distinguishable from dark energy models when we strongly limit the solar system constraint.Comment: 19 pages, 20 figures, 2 tables, submitted to JCA

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

We clarify the conditions under which dark energy models whose Lagrangian densities f are written in terms of the Ricci scalar R are cosmologically viable. The existence of a viable matter dominated epoch prior to a late-time acceleration requires that the variable m=Rf_{,RR}/f_{,R} (where f_{,R}=df/dR) satisfies the conditions m(r) approx +0 and dm/dr>-1 at r approx -1 where r=-Rf_{,R}/f. For the existence of a viable late-time acceleration we require instead either (i) m=-r-1, (sqrt{3}-1)/2 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 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. 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. Among several other results, we find that f(R) models can have a strongly phantom attractor but in this case there is no acceptable matter era

Comments on scalar-tensor representation of nonlocally corrected gravity

The scalar-tensor representation of nonlocally corrected gravity is considered. Some special solutions of the vacuum background equations were obtained that indicate to the nonequivalence of the initial theory and its scalar-tensor representation.Comment: 6 pages, refs adde

Global properties of the growth index of matter inhomogeneities in the universe

We perform here a global analysis of the growth index $\gamma$ behaviour from deep in the matter era till the far future. For a given cosmological model in GR or in modified gravity, the value of $\gamma(\Omega_{m})$ is unique when the decaying mode of scalar perturbations is negligible. However, $\gamma_{\infty}$, the value of $\gamma$ in the asymptotic future, is unique even in the presence of a nonnegligible decaying mode today. Moreover $\gamma$ becomes arbitrarily large deep in the matter era. Only in the limit of a vanishing decaying mode do we get a finite $\gamma$, from the past to the future in this case. We find further a condition for $\gamma(\Omega_{m})$ to be monotonically decreasing (or increasing). This condition can be violated inside general relativity (GR) for varying $w_{DE}$ though generically $\gamma(\Omega_{m})$ will be monotonically decreasing (like $\Lambda$CDM), except in the far future and past. A bump or a dip in $G_{\rm eff}$ can also lead to a significant and rapid change in the slope $\frac{d\gamma}{d\Omega_{m}}$. On a $\Lambda$CDM background, a $\gamma$ substantially lower (higher) than $0.55$ with a negative (positive) slope reflects the opposite evolution of $G_{\rm eff}$. In DGP models, $\gamma(\Omega_{m})$ is monotonically increasing except in the far future. While DGP gravity becomes weaker than GR in the future and $w^{DGP}\to -1$, we still get $\gamma_{\infty}^{DGP}= \gamma_{\infty}^{\Lambda CDM}=\frac{2}{3}$. In contrast, despite $G^{DGP}_{\rm eff}\to G$ in the past, $\gamma$ does not tend to its value in GR because $\frac{dG^{DGP}_{\rm eff}}{d\Omega_{m}}\Big|_{-\infty}\ne 0$.Comment: 15 pages, 7 figures; v3: improved presentation, to appear in Phys.Rev.