A Dark Matter Model with Quadratic Equation of State: Background Evolution and Structure Formation

TL;DR

The paper introduces a Quadratic Density-dependent Dark Matter (QDDM) with $P_{\rm dm} = \alpha \rho_{\rm dm} + \beta \rho_{\rm dm}^2/\rho_0$ to address potential tensions in standard cosmology. It derives the modified background evolution $H(z)$ and $\Omega_{\rm dm}(z)$ through a density-dependent function $F(z)$, and analyzes linear perturbations with a redshift-dependent equation of state and sound speed, computing the observable growth $f\sigma_8(z)$. The results show that while background evolution can remain compatible with $\Lambda$CDM, the growth of structures acquires a distinct imprint—especially for $\beta>0$—due to both the altered expansion history and a scale-dependent DM pressure, offering a potential route to alleviate late-time tensions. The study outlines future work to constrain $\alpha$ and $\beta$ with comprehensive cosmological data, potentially linking the microphysics of dark matter to the observed growth of structure.

Abstract

We propose that dark matter (DM) possesses a quadratic equation of state, which becomes significant at high densities, altering the Universe's evolution during its early stages. We derive the modified background evolution equations for the Hubble parameter $H(z)$ and the DM density parameter $Ω_{\text{dm}}(z)$. We then perturb the governing equations to study the linear growth of matter fluctuations, computing the observable growth factor $fσ_8(z)$. Finally, we compare the model with the latest cosmological data, including Hubble parameter $H(z)$ measurements, and growth factor $fσ_8(z)$ data, up to $z=3$. Our results indicate that the quadratic model, while remaining consistent with background observations, offers a distinct imprint on the growth of structure, providing not only a new phenomenological avenue to address cosmological tensions but also shedding light on the nature of DM.

Figures (3)

• Figure 1: Evolution of scaled Hubble parameter in terms of the cosmological redshift for our model with $\beta > 0$ (red curve) and with $\beta < 0$ (blue curve), compared to the $\Lambda$CDM model (black dashed curve). The left plot is drawn with $\{\alpha,\left|\beta\right|\}=\{10^{-3},10^{-2}\}$, the right one with $\{\alpha,\left|\beta\right|\}=\{10^{-3},10^{-4}\}$. To generate these plots, we have used the cosmological parameters $\{h,\Omega_{b0},\Omega_{\text{dm0}},\Omega_{r0}\}=\{0.6732,0.04939,0.265,9.267\times10^{-5}\}$.
• Figure 2: Evolution of DM density parameter in terms of scale factor for our model with $\beta = 10^{-4}$ (red curve) and with $\beta < 0$ (blue curve), compared to the $\Lambda$CDM model (black dashed curve). To draw this plot, we used $\{\alpha,\left|\beta\right|\}=\{10^{-3},10^{-4}\}$. The values of the other cosmological parameters are the same as those used in Fig. \ref{['fig:H']}.
• Figure 3: Diagram of the growth factor for QDDM with $\{\alpha,\beta\}=\{2\times10^{-4},-10^{-5}\}$ (blue curve) compared to the $\Lambda$CDM result (black curve). To this plot we set $\sigma_{8}(z=0)=0.812$. The values of the other cosmological parameters are the same as those in Fig. \ref{['fig:H']}.