Density perturbations for running vacuum: a successful approach to structure formation and to the $σ_8$-tension

TL;DR

This work analyzes how running vacuum models (RVMs), where the vacuum energy density runs with the expansion via $ρ_Λ(H)=\frac{3}{8πG}\left(c_0+νH^2\right)$, modify linear density perturbations and structure formation. Through a detailed perturbation treatment in both Newtonian and synchronous gauges, they show vacuum fluctuations are negligible on subhorizon scales and derive a modified growth equation that includes a small matter–vacuum coupling ∝ ν. The authors demonstrate, with analytic and numerical methods, that a best-fit ν≈1.6×10^−3 substantially lowers $σ_8$ (by ~8%) and brings $f(z)σ_8(z)$ into better agreement with LSS data, effectively relaxing the $σ_8$ tension relative to ΛCDM, and they compare to XCDM to illustrate the robustness of the result. They also discuss the relative importance of LSS versus weak-lensing data and emphasize that dynamical vacuum dynamics offer a coherent resolution to several cosmological tensions, with the RVM providing a superior fit to the combined data set.

Abstract

Recent studies suggest that dynamical dark energy (DDE) provides a better fit to the rising affluence of modern cosmological observations than the concordance model ($Λ$CDM) with a rigid cosmological constant, $Λ$. Such is the case with the running vacuum models (RVMs) and to some extent also with a simple XCDM parametrization. Apart from the cosmic microwave background (CMB) anisotropies, the most crucial datasets potentially carrying the DDE signature are: i) baryonic acoustic oscillations (BAO), and ii) direct large scale structure (LSS) formation data (i.e. the observations on $f(z)σ_8(z)$ at different redshifts). As it turns out, analyses mainly focusing on CMB and with insufficient BAO+LSS input, or those just making use of gravitational weak-lensing data for the description of structure formation, generally fail to capture the DDE signature, whereas the few existing studies using a rich set of CMB+BAO+LSS data (see in particular Solà, Gómez-Valent & de Cruz Pérez 2015,2017; and Zhao et al. 2017) do converge to the remarkable conclusion that DDE might well be encoded in the current cosmological observations. Being the issue so pressing, here we explain both analytically and numerically the origin of the possible hints of DDE in the context of RVMs, which arise at a significance level of $3-4σ$. By performing a detailed study on the matter and vacuum perturbations within the RVMs, and comparing with the XCDM, we show why the running vacuum fully relaxes the existing $σ_8$-tension and accounts for the LSS formation data much better than the concordance model.

Figures (10)

• Figure 1: Left plot: Squared ratio between the comoving scales $1/k$ and the comoving Hubble horizon, $\mathcal{H}^{-1}=H^{-1}/a$, as a function of the redshift. The used range of comoving wave numbers, $k$, correspond to those that we have observational access to inside the horizon and at which the linear perturbations regime is still valid, namely the modes $0.01\,h{\rm Mpc}^{-1}\leq k\leq 0.2\,h{\rm Mpc}^{-1}$. They are inside the gray band. The lowest modes (corresponding to the largest scales) reentered the horizon far in the past, at $z\simeq 1920$, previously to the decoupling time but already during the MD epoch, whereas the largest modes (smallest scales) reentered the horizon deeply in the radiation-dominated era, at $z\simeq 61.1\times 10^{3}$ (hence out of the plot); Upper-right plot: As in the left plot, but here for a shorter redshift range, up to $z=100$. This is to show that the modes we are focusing our attention on, i.e. those that lie in the gray region, are deeply inside the horizon from $z\sim 100$ up to the present time, i.e. $(\mathcal{H}/k)^2<0.035\ll 1$. This is all the more true at $z\lesssim 10$ since the relevant modes then satisfy $(\mathcal{H}/k)^2<0.004\ll 1$. This legitimates to use the subhorizon approximation at these scales, see the text for further details; Lower-right plot: Similar to the previous cases, now in the narrow range $0\leq z\leq 4$, allowing us to appreciate the existence of a minimum at $z_{{\rm min}}\sim 0.6-0.7$. The latter indicates the transition from a decelerated to an accelerated universe, which causes that the lowest subhorizon modes start exiting the Hubble horizon again. The transition point, defined as the point at which the deceleration parameter vanishes, i.e. $q(a_t)=-1-\dot{H}(a_t)/[a_tH^{2}(a_t)]=0$, can be analytically computed in the RVM: $a_t=\left[\frac{(1-3\nu)\,\Omega_m}{2(\Omega_{\Lambda}-\nu)}\right]^{1/(3(1-\nu))}$. Using the values of the RVM parameters in Table 1 we obtain $z_{t}=a_t^{-1}-1=0.663$.
• Figure 2: Logarithm of the ratio of density perturbations of vacuum and matter as a function of the redshift and for the same comoving wave numbers of Fig. 1. They are again inside the gray band. The effect of vacuum energy perturbations is enhanced at large scales, but even for the largest comoving scales of interest it is negligible in front of the corrections considered in Eq. \ref{['eq:DensityContrastEq']}, see the discussion in Sect. 4.
• Figure 3: Left plot: Evolution of the percentage difference of the matter density contrast in the RVM with respect to the $\Lambda$CDM, $\Delta (z)$, as defined in \ref{['eq:RelDiff']}, in the redshift range $0\leq z\leq 100$; Right plot: The same, but in the smaller redshift range $0\leq z\leq 4$, just to show the region where $\Delta(z)$ becomes negative, near the present time. It is crystal-clear that such differences are always smaller than $2\%$ in absolute value.
• Figure 4: Left plot: The weighted growth rate for the $\Lambda$CDM, the XCDM and the RVM, obtained by using the best-fit values of Table 1. The values of $\sigma_8$ that we obtain for these models are also indicated. We also plot the reconstructed $f(z)\sigma_8(z)$ curve and its $1\sigma$ uncertainty band, both obtained by using the observational data (depicted in green) and the Gaussian processes method (GPM) with Cauchy's kernel, see e.g. (Seikel, Clarkson & Smith 2012) and references therein. Almost identical results are obtained using alternative kernels as the Gaussian or the Matérn ones. This is to show the preference of the data for lower values of this LSS observable; Right plot: The relative (percentage) difference of the weighted growth rate with respect to the concordance model, $\Delta_{f\sigma_8}$, as defined in \ref{['eq:Deltafsigma8']}.
• Figure 5: Left plot: Reconstruction of the $f(z)\sigma_8(z)$ curve of the RVM from the $\Lambda$CDM one. See the text in Sect. 7 for a detailed explanation; Right plot: Curves of $f(z)\sigma_8(z)$ obtained for the $\Lambda$CDM, with different values of $\Omega_m$ and $h$ satisfying the same relation $\omega_m\equiv\Omega_m h^2=0.1412$. This is to show the approximate degeneracy of the LSS results under modifications of $\Omega_m$ and $h$ that fully respect the strong constraint on $\omega_m$ coming from the CMB data, as explained in the text.