Sound Mode and Scale-Dependent Growth in Two-Fluid Dynamical Dark Energy

Abstract

We investigate the effects of dynamical dark energy (DDE) on the growth of cosmic structure using a two-fluid model. This framework allows the dark energy equation of state to smoothly cross the phantom divide, in agreement with recent DESI results. In this effective description, DDE supports propagating perturbations that behave like sound waves. These perturbations induce a scale dependence in the growth of matter fluctuations and in halo bias, which can be exploited to test the dynamical nature of dark energy at the level of its fluctuations. For cluster-sized halos, the amplitude of the scale-dependent halo bias is comparable to that produced by massless neutrinos in $Λ$CDM. Using a Fisher forecast for a multi-tracer analysis of the power spectrum (P) and bispectrum (B) of galaxy number counts, we find that bispectrum information is essential to detect the scale dependence induced by the DDE sound mode. For a survey of volume $V\sim 10\, h^{-3}{\rm Gpc}^3$ at redshift $z=0.5 - 1$, a two-tracer P+B analysis could detect this scale dependence if the sound speeds of the dark energy fluids are in the range $c_s^2\sim 10^{-2} - 10^{-4}$. Lower sound speeds cause halos to experience a gravitational drag force through the excitations of sound waves. This effect impacts measurements of the growth rate inferred from cluster-sized halos at the 10\% level if one of the fluids has a very low sound speed $c_s^2\sim 10^{-5}$. Larger sound speeds $c_s^2 > 10^{-2}$ could be probed with optimal weighting schemes that reduce shot noise and increase the effective bias.

Figures (10)

• Figure 1: Redshift evolution of the total EOS parameter $w_t(z)$, following Eq. \ref{['eq:wt']}, with the parameters of Table \ref{['tab:hu_params']}. The dashed line represents the phantom divide at $w=-1$, while the dashed-dotted line represents the normalization epoch $a=a_n$.
• Figure 2: The dependence of the EOS parameters $(w_+,w_-)$ on the CPL parameters $(w_0,w_a)$ using the two-fluid approach discussed in the text. The color scale represents either $w_+$ (top panel) or $w_-$ (bottom panel). In each panel, the two circles indicate the CPL parameter values of the Phantom and Quintom model considered here. For the Phantom model, we have added the uncertainties on $(w_0,w_a)$ reported in DESI:2025zgx since, in this case, our choice of $(w_0,w_a)$ is motivated by the DESI DR2 results.
• Figure 3: Redshift evolution of the dimensionless energy densities of both DE fluids, as well as the sum in both the Phantom and the Quintom model. The black dashed line indicates $\Omega = 0$. Although the total energy density remains positive, the $w_-$ fluid in the Phantom model develops a negative energy density at late times (which is not directly measurable in the effective framework adopted here).
• Figure 4: The scale-dependent, linear growth rate $D(k,z)$ of matter fluctuations, normalized to the $\Lambda$CDM solution, is shown at redshift $z=0.5$ for several choices of $\hat{c}_\pm^2$.
• Figure 5: Scale-dependent bias computed in the separate universe approach (filled circles with a thin interpolating line) and in the local Lagrangian approximation (thick curves). Results are shown at $z=0.5$ for four different choices of sound speeds assuming a DM halo mass $M=10^{14}\ {\rm M_\odot/{\it h}}$.