Testing the cosmological Euler equation: viscosity, equivalence principle, and gravity beyond general relativity

TL;DR

The paper develops a model-independent framework to test the cosmological Euler equation in the presence of viscous dark matter, EP violations, and gravity beyond GR. It introduces a viscous generalization of the EP estimator, $\tilde{E}_P$, and a model-independent observable, $C_{\rm vis,0}$, measurable from relativistic galaxy clustering with two tracers. The analysis shows viscosity induces scale-dependent growth suppression and can mimic MG effects, but in the small-viscosity limit, EP tests via $\tilde{E}_{P,z}$ remain valid, while $C_{\rm vis,0}$ can be tightly constrained by DESI, Euclid, and especially SKA2 (down to $\sim 10^{-7}$). The work provides a practical observational pathway to constraining DM viscosity with Stage-IV surveys, highlighting degeneracies and the importance of priors on magnification biases for EP-related inferences.

Abstract

We investigate how the cosmological Euler equation can be tested in the presence of viscous dark matter, violations of the equivalence principle (EP), and modifications of gravity, while relying on minimal theoretical assumptions. Extending the previous analysis, we generalize the observable $E_P$, which quantifies EP violation, to $\tilde{E}_P$, discuss the degeneracy between bulk and shear viscosities and EP-violating effects, and explicitly show that the EP can still be tested in the small-viscosity limit. In addition, we identify a model-independent observable, $C_{\rm vis,0}$, which characterizes the present-day dark matter viscosity and can be measured from relativistic galaxy number counts by cross-correlating two galaxy populations. We perform forecasts for three forthcoming Stage-IV surveys: DESI, Euclid, and SKA Phase 2 (SKA2), and find that $C_{\rm vis,0}$ can be tightly constrained, at the level of $\mathcal{O}(10^{-6})$ or better in all cases. Among these surveys, SKA2 provides the tightest constraint, with a $1σ$ uncertainty of $1.08 \times 10^{-7}$ on $C_{\rm vis,0}$.

Figures (2)

• Figure 1: Scale-dependent impact of the viscosity parameter $C_{\rm vis,0}$ on large-scale structure observables. Here we consider only bulk viscosity (with $C_{\rm vis,0} \simeq \bar{\zeta} / 9\Omega_{m,0}$). Results are shown at redshift $z = 0.45$ for a line-of-sight angle $\mu = 1$, with $\Theta = \Gamma = 0, \mu_{\rm G} = \eta = 1$, assuming DESI survey specifications as listed in Table \ref{['tab:DESI_specs']}. Left: scale-dependent growth rate $f(k)$. Right: linear power spectrum $P_{\Delta_B\Delta_B}(k)$, computed using Eq. \ref{['eq:PBB']}.
• Figure 2: Impact of the viscosity parameter $C_{\rm vis,0}$ on the growth of the density contrast of viscous dark matter and baryons, numerically solved from Eqs. \ref{['eq:delta_vis_dm']} and \ref{['eq:delta_baryon']}, evaluated at $k=0.05$$h$/Mpc (left) and $k=0.12$$h$/Mpc (right). We show the ratio of the density contrast $\delta_{\rm b} / \delta_{\rm dm}$. As expected, the growth of viscous dark matter is suppressed relative to that of baryons. For illustrative purpose, we consider only bulk viscosity and set $\Theta=\Gamma=0, \mu_{\rm G}=\eta =1$.