Cosmological perturbations on an averaged background

TL;DR

The paper investigates how nonlinear inhomogeneities backreact on the average expansion and, importantly, on the linear growth of structure within a relativistic framework. It combines covariant Buchert spatial averaging with covariant gauge-invariant perturbation theory in irrotational dust spacetimes, modeling backreaction as an effective fluid with pressure that yields a two-fluid background, described by ${\mathcal{Q}}_{\mathcal{D}}$ and ${\left\langle \mathcal{R} \right\rangle_{\mathcal{D}}}$ through ${\rho}^{\mathrm{eff}}_{\mathcal{D}}$ and ${p}^{\mathrm{eff}}_{\mathcal{D}}$. The authors derive the CGI perturbation equations, discuss closure conditions for the effective fluid, and analyze four averaged cosmological models (Timescape, GMC, GMP, RZA) to assess the sensitivity of linear growth to backreaction. They find that neglecting backreaction can bias predictions of structure formation, and that the closure choice substantially affects growth, with the comoving effective-fluid scheme generally more stable than barotropic closures, while the M\’eszáros limit is not universally applicable.

Abstract

In relativistic cosmology, the formation of nonlinear inhomogeneities can induce non-negligible backreaction on late-time expansion. Among the important consequences for precision cosmology is the potential impact on the linear growth of large-scale structures. We address this impact by combining covariant spatial averaging with covariant and gauge-invariant perturbation theory. We focus on irrotational dust model spacetimes. The effects of backreaction and nontrivial dynamical curvature on the average cosmological dynamics are formulated as the addition of an effective perfect fluid with pressure. We then introduce an effective background driven by both the averaged dust density and the emergent effective fluid, and derive the general evolution equations for linear perturbations of this system. The residual freedom in this framework amounts to specifying the properties of the effective-fluid perturbations as a closure condition. We analyse two physically motivated choices for this condition. In addition, we clarify the conditions under which the coupling between linear structure growth and perturbations of the effective fluid can be neglected. Finally, we apply this formalism to four examples of averaged cosmological models from the literature, three of which -- intended as effective full descriptions of the largest scales -- have been shown to provide a good fit to observational data. Our results highlight the importance of backreaction effects in shaping linear structure growth in such models. Neglecting these effects may thus lead to biased predictions for the development of large structures, even when the models provide a good description of the general background observables.

Figures (12)

• Figure 1: The $\Omega$ variables (left panel) and the effective fluid EoS parameter and squared sound speed (right panel) in the timescape model, as functions of the rescaled volume-average scale factor $a_{\mathcal{D}}/a_{w,0}$, with $f_{v,0} = 0.737$, and $H_0 = 73.03$ km/(s$\cdot$Mpc). The model has no cosmological constant.
• Figure 2: Dust density perturbation growth within the comoving effective fluid framework, $\Delta^{(d)}_\mathrm{com}$, and the Mészáros approximation, $\Delta^{(d)}_\mathrm{mz}$, for the timescape model, plotted as functions of the rescaled volume-average scale factor $a_{\mathcal{D}}/a_{w,0}$. For the first scenario we also show the effective fluid density perturbations, $\Delta^{(\mathrm{eff})}_\mathrm{com}$. The ICs are set at $z = 20$, seeded using a $\Lambda$CDM-based approximation for $1100 \geq z > 20$ for both the dust density and expansion perturbations. The IC for the effective fluid perturbation at $z=20$ is instead set to zero.
• Figure 3: The growth of dust perturbations, $\Delta^{(d)}_{\mathrm{bar}}$, and effective fluid density perturbations, $\Delta^{(\mathrm{eff})}_{\mathrm{bar}}$, shown in the left and right panels respectively, for the timescape model, plotted as functions of the rescaled volume-average scale factor $a_{\mathcal{D}}/a_{w,0}$, within the barotropic framework. We show the amplitude evolution for three different wavelengths on a spatially flat background ($K_{\mathcal{D}} = 0$), at horizon ($k = 2\pi$) and sub-horizon scales. Here, the wavenumber $k$ is expressed in units of the present-day inverse Hubble length, $H_0/c$, as evaluated in the $\Lambda$CDM model following the best fit of Planck_2018. The ICs (at $z=20$) are seeded via the $\Lambda$CDM approximation at earlier times for $\Delta^{(d)}$ and ${\mathcal{Z}}$, with the same values for all scales for direct comparison, and fixed to zero for the effective fluid perturbations.
• Figure 4: The $\Omega$ variables (left panel) and the effective fluid EoS parameter and squared sound speed (right panel), as functions of the effective scale factor $a_{\mathcal{D}}$, in the GMC model with parameters $\tilde{R}_c =6.76$ Mpc, $\tilde{R}_v =4.57$ Mpc, $H_{{\mathcal{D}},0} = 70.12$ km/(s$\cdot$Mpc), and $\Omega^{{\mathcal{D}},0}_d = 0.316$.
• Figure 5: Dust density perturbation growth within the comoving effective fluid framework, $\Delta^{(d)}_\mathrm{com}$, and the Mészáros approximation, $\Delta^{(d)}_\mathrm{mz}$, for a GMC background model, as functions of the average scale factor $a_{\mathcal{D}}$. For the first scenario we also show the magnitude of the effective fluid density perturbations, $|\Delta^{(\mathrm{eff})}_\mathrm{com}|$, in the inset panel. The ICs for the dust density and expansion perturbations are set at $z = 20$, seeded using the $\Lambda$CDM-based approximation at earlier times as in Sec. \ref{['subsec:ts']} above. The ICs for the effective fluid density perturbations are instead set to zero.