Probing non-standard gravity with the growth index: a background independent analysis

Measurements of the growth index $\gamma(z)$ provide a clue as to whether Einstein's field equations encompass gravity also on large cosmic scales, those where the expansion of the universe accelerates. We show that the information encoded in this function can be satisfactorily parameterized using a small set of coefficients $\gamma_i$ in such a way that the true scaling of the growth index is recovered to better than $1\%$ in most dark energy and dark gravity models. We find that the likelihood of current data is maximal for $\gamma_0=0.74\pm 0.44$ and $\gamma_1=0.01\pm0.46$, a measurement compatible with the $\Lambda$CDM predictions. Moreover data favor models predicting slightly less growth of structures than the Planck LambdaCDM scenario. The main aim of the paper is to provide a prescription for routinely calculating, in an analytic way, the amplitude of the growth indices $\gamma_i$ in relevant cosmological scenarios, and to show that these parameters naturally define a space where predictions of alternative theories of gravity can be compared against growth data in a manner which is independent from the expansion history of the cosmological background. As the standard $\Omega$-plane provides a tool to identify different expansion histories $H(t)$ and their relation to various cosmological models, the $\gamma$-plane can thus be used to locate different growth rate histories $f(t)$ and their relation to alternatives model of gravity. As a result, we find that the Dvali-Gabadadze-Porrati gravity model is rejected with a $95\%$ confidence level. By simulating future data sets, such as those that a Euclid-like mission will provide, we also show how to tell apart LambdaCDM predictions from those of more extreme possibilities, such as smooth dark energy models, clustering quintessence or parameterized post-Friedmann cosmological models.Comment: 29 pages, 21 figure

Diagnostic of Horndeski Theories

We study the effects of Horndeski models of dark energy on the observables of the large-scale structure in the late time universe. A novel classification into {\it Late dark energy}, {\it Early dark energy} and {\it Early modified gravity} scenarios is proposed, according to whether such models predict deviations from the standard paradigm persistent at early time in the matter domination epoch. We discuss the physical imprints left by each specific class of models on the effective Newton constant $\mu$, the gravitational slip parameter $\eta$, the light deflection parameter $\Sigma$ and the growth function $f\sigma_8$ and demonstrate that a convenient way to dress a complete portrait of the viability of the Horndeski accelerating mechanism is via two, redshift-dependent, diagnostics: the $\mu(z)-\Sigma(z)$ and the $f\sigma_8(z)-\Sigma(z)$ planes. If future, model-independent, measurements point to either $\Sigma-10$ at high redshifts or $\mu-1>0$ with $\Sigma-1<0$ at high redshifts, Horndeski theories are effectively ruled out. If $f\sigma_8$ is measured to be larger than expected in a $\Lambda$CDM model at $z>1.5$ then Early dark energy models are definitely ruled out. On the opposite case, Late dark energy models are rejected by data if $\Sigma1$, only Early modifications of gravity provide a viable framework to interpret data

Phenomenology of dark energy: exploring the space of theories with future redshift surveys

We use the effective field theory of dark energy to explore the space of modified gravity models which are capable of driving the present cosmic acceleration. We identify five universal functions of cosmic time that are enough to describe a wide range of theories containing a single scalar degree of freedom in addition to the metric. The first function (the effective equation of state) uniquely controls the expansion history of the universe. The remaining four functions appear in the linear cosmological perturbation equations, but only three of them regulate the growth history of large scale structures. We propose a specific parameterization of such functions in terms of characteristic coefficients that serve as coordinates in the space of modified gravity theories and can be effectively constrained by the next generation of cosmological experiments. We address in full generality the problem of the soundness of the theory against ghost-like and gradient instabilities and show how the space of non-pathological models shrinks when a more negative equation of state parameter is considered. This analysis allows us to locate a large class of stable theories that violate the null energy condition (i.e. super-acceleration models) and to recover, as particular subsets, various models considered so far. Finally, under the assumption that the true underlying cosmological model is the $\Lambda$ Cold Dark Matter ($\Lambda$CDM) scenario, and relying on the figure of merit of EUCLID-like observations, we demonstrate that the theoretical requirement of stability significantly narrows the empirical likelihood, increasing the discriminatory power of data. We also find that the vast majority of these non-pathological theories generating the same expansion history as the $\Lambda$CDM model predict a different, lower, growth rate of cosmic structures.Comment: v1: 28 pages, 20 pdf figures. v2: 29 pages, minor improvements in the text, figures improve

Constraints on modified gravity from Planck 2015: when the health of your theory makes the difference

We use the effective field theory of dark energy (EFT of DE) formalism to constrain dark energy models belonging to the Horndeski class with the recent Planck 2015 CMB data. The space of theories is spanned by a certain number of parameters determining the linear cosmological perturbations, while the expansion history is set to that of a standard $\Lambda$CDM model. We always demand that the theories be free of fatal instabilities. Additionally, we consider two optional conditions, namely that scalar and tensor perturbations propagate with subliminal speed. Such criteria severely restrict the allowed parameter space and are thus very effective in shaping the posteriors. As a result, we confirm that no theory performs better than $\Lambda$CDM when CMB data alone are analysed. Indeed, the healthy dark energy models considered here are not able to reproduce those phenomenological behaviours of the effective Newton constant and gravitational slip parameters that, according to previous studies, best fit the data.Comment: 21 pages, 8 figures. Added Mu-Sigma plane in Fig.7 plus some changes in the text with respect to the previous version. This is an author-created un-copyedited version of the article published in JCAP. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscrip

Phenomenology of dark energy: general features of large-scale perturbations

We present a systematic exploration of dark energy and modified gravity models containing a single scalar field non-minimally coupled to the metric. Even though the parameter space is large, by exploiting an effective field theory (EFT) formulation and by imposing simple physical constraints such as stability conditions and (sub-)luminal propagation of perturbations, we arrive at a number of generic predictions. (1) The linear growth rate of matter density fluctuations is generally suppressed compared to $\Lambda$CDM at intermediate redshifts ($0.5 \lesssim z \lesssim 1$), despite the introduction of an attractive long-range scalar force. This is due to the fact that, in self-accelerating models, the background gravitational coupling weakens at intermediate redshifts, over-compensating the effect of the attractive scalar force. (2) At higher redshifts, the opposite happens; we identify a period of super-growth when the linear growth rate is larger than that predicted by $\Lambda$CDM. (3) The gravitational slip parameter $\eta$ - the ratio of the space part of the metric perturbation to the time part - is bounded from above. For Brans-Dicke-type theories $\eta$ is at most unity. For more general theories, $\eta$ can exceed unity at intermediate redshifts, but not more than about $1.5$ if, at the same time, the linear growth rate is to be compatible with current observational constraints. We caution against phenomenological parametrization of data that do not correspond to predictions from viable physical theories. We advocate the EFT approach as a way to constrain new physics from future large-scale-structure data.Comment: 24 pages, 7 figure