Effects of dark energy anisotropic stress on the matter power spectrum

Abstract

We study the effects of dark energy (DE) anisotropic stress on features of the matter power spectrum (PS). We employ the Parametrized Post-Friedmannian (PPF) formalism to emulate an effective DE, and model its anisotropic stress properties through a two-parameter equation that governs its overall amplitude ($g_0$) and transition scale ($c_g$). For the background cosmology, we consider different equations of state to model DE including a constant $w_0$ parameter, and models that provide thawing (CPL) and freezing (nCPL) behaviors. We first constrain these parameters by using the Pantheon, BAO, $H_0$ and CMB Planck data. Then, we analyze the role played by these parameters in the linear PS. In order for the anisotropic stress not to provoke deviations larger than 10% and 5% with respect to the $\Lambda$CDM PS at $k\sim 0.01h∕\text{Mpc}$, the parameters have to be in the range $-0.30<g_0<0.32$, $0\le c_g^2<0.01$ and $-0.15<g_0<0.16$, $0\le c_g^2<0.01$, respectively. Additionally, we compute the leading nonlinear corrections to the PS using standard perturbation theory in real and redshift space, showing that the differences with respect to the $\Lambda$CDM are enhanced, especially for the quadrupole and hexadecapole RSD multipoles.

Introduction

The discovery of the accelerated expansion of the Universe implied the existence of dark energy (DE), that has been extensively confirmed by a two-decade variety of experiments, initially employing Supernovae type Ia [1], [2], then using anisotropies in the CMB from WMAP and Planck data [3], distance measurements of different tracers [4], and clustering of large galaxy surveys, among other probes [5], [6], [7]. However, little is known of the fundamental properties of DE, apart from being a ‘fluid’ that possess negative pressure. In the most successful model a cosmological constant, $\Lambda$, is capable to fit the observations, albeit current tensions exist among a few parameters when measured with different probes [8].

The effects of DE have been widely studied in the context of background cosmological dynamics; most of the work has been devoted to test different equations of state (EoS) for DE to understand the dynamics of the Hubble expansion flow. However, in comparison its perturbative effects are less explored, partially because we expect little deviations at perturbative level, but also because we have no clues on its fundamental origin. One can, for example, treat DE as a barotropic fluid, hence, its sound speed depends only on background quantities, or to consider it as a non-adiabatic fluid to account for its linear effects for which additional hypotheses have to be made about the fluid’s speed of sound [9]. There are many works that study the effect of DE speed of sound in the perturbative dynamics, initially done by [10], [11], [12], [13]. It turns out that the effects of DE clustering result to be small, especially if the DE EoS is close to $-1$, as demanded by observations, and then they are difficult to discern with late-Universe measurements [14], [15]. But, in fact, varying the DE sound speed can induce deviations of up 2% in the matter power spectrum (PS) [16], [17], that should be important in view of the expected constraints from upcoming galaxy surveys, such as DESI [18].

Another possibility is to consider DE anisotropic stress. A homogenous and isotropic symmetric background metric forbids it, but it can be introduced at the perturbed level [19]. Anisotropic stress can also mimic modified gravity (MG) at linear order [20], [21], [22], [23], since it introduces at least a new parameter, and together with DE EoS and sound speed, it yields a modified growth of structures in the Universe. In fact, DE stress generates similar outcomes as those of varying the sound speed of DE, but the detailed behavior depends on the signs of the EoS and stress parameter [24]. From theoretical grounds, one expects DE anisotropic stress to affect the evolution of the metric potentials and this provokes CMB temperature anisotropies at low-multipoles, to be affected through the Integrated Sachs–Wolfe (ISW) effect. In Refs. [24], [25], [26], [27], [28], [29] DE anisotropic stress was analyzed to prove this conclusion using CMB data available at that time, but due to the cosmic variance, CMB constraints are still broad. However, DE stress should affect also matter clustering at large scales. Effects of anisotropic stress on the matter power spectrum (PS) and on the growth function have been studied in several works [24], [25], [30], [31], [32], [33], [34], showing that shear viscosity has an effect on very large scales, but one the other hand it does not change much other cosmological parameter values; for instance, for this latter reason we do not expect that DE shear terms alone can alleviate the current tension in the Hubble constant; see however [35] in which it is proven that adding anisotropic shear to interacting models helps to increase the Hubble constant to release the tension for phantom DE.

In the literature there is a number of works considering different aspects of imperfect fluids, e.g. in connection to second order perturbation in $\Lambda$CDM [36], or related to generalized scalar fields [37], [38]. Also, based on MG, efforts have been put forward to understand how the gravitational effects of the fifth-force (that generates an effective shear term) influence the observables at cosmological scales, changing the clustering properties [39], [40]. Our motivation here, linked to these latter works, is to analyze the anisotropic stress effects on CMB and matter PS since the level of accuracy of future LSS galaxy surveys and probes shall demand detailed understanding of the clustering properties of the matter field. In this way, being able to constrain an hypothetical anisotropic shear, stemming either from DE or MG. Ways to carry out this comparison are discussed e.g. in Refs. [41], [42], [43]. Recently, analysis of recent probes hints for non-zero anisotropic stress [44], that also encourages us to further analyze its clustering properties.

In the present work, we use the Parametrized Post-Friedmannian (PPF) approach [26], [45], [46], though originally motivated to emulate MG models, they naturally introduce an effective DE anisotropic stress term. In this respect, the formalism serves to phenomenologically introduce either DE stress or effective MG stress terms. Viewed as effective DE, we consider specific equations of state and fix the DE speed of sound, to concentrate our analysis on the effects of the anisotropic stress. We analyze the constraints from CMB power spectra and, especially, look for deviations in the PS. Interestingly, we find that DE anisotropic stress is allowed by Planck CMB data, as in Refs. [24], [26], but the linear and nonlinear PS impose tighter constraints to it. We consider different DE EoS, firstly $w=-1$ that emulates $\Lambda$ at background level, then constant $w_0$, and finally, thawing and freezing models, to find out their effects in combination with stress parameters.

The structure of this paper is the following: In Section 2 we motivate DE EoS parametrizations chosen, and in Section 3 we introduce the DE perturbation theory with anisotropic stress, where a specific anisotropic stress phenomenology is adopted. Section 4 shows our results employing different EoS, and Section 5 shows the theory and results for nonlinear perturbation theory to 1-loop. Finally, Section 6 concludes.