The Effective Field Theory of Large Scale Structure for Mixed Dark Matter Scenarios

TL;DR

This work extends the EFTofLSS to mixed dark matter by introducing a two-fluid framework in which a non-cold fraction $f_\chi$ is modeled as a fluid with finite sound speed, yielding a characteristic suppression scale $k_s(a)$. It develops linear and nonlinear perturbation theory for the coupled fluids, provides an infrared-safe kernel prescription to compute the 1-loop galaxy power spectrum, and demonstrates its applicability to Planck+BOSS data for ultra-light axions. The analysis finds that including the two-fluid dynamics and associated EFT counterterms can weaken the bounds on the axion energy density relative to single-fluid analyses, highlighting the need for self-consistent beyond-$\Lambda$CDM modeling in LSS inference. The framework sets the stage for robust constraints on mixed DM with upcoming surveys (DESI, Euclid, Rubin) and motivates extensions to other dark-sector scenarios and higher-point statistics.

Abstract

We initiate a systematic study of the perturbative nonlinear dynamics of cosmological fluctuations in dark sectors comprising a fraction of non-cold dark matter, for example ultra-light axions or light thermal relics. These mixed dark matter scenarios exhibit suppressed growth of perturbations below a characteristic, cosmologically relevant, scale associated with the microscopic nature of the non-cold species. As a consequence, the scale-free nonlinear solutions developed for pure cold dark matter and for massive neutrinos do not, in general, apply. We thus extend the Effective Field Theory of Large Scale Structure to model the coupled fluctuations of the cold and non-cold dark matter components, describing the latter as a perfect fluid with finite sound speed at linear level. We provide new analytical solutions wherever possible and devise an accurate and computationally tractable prescription for the evaluation of the one-loop galaxy power spectrum, which can be applied to probe mixed dark matter scenarios with current and upcoming galaxy survey data. As a first application of this framework, we derive updated constraints on the energy density in ultra-light axions using a combination of Planck and BOSS data. Our refined theoretical modeling leads to somewhat weaker bounds compared to previous analyses.

Figures (15)

• Figure 1: Left: The ratio of the linear density perturbations of $\chi$ and $c$, $g = \delta_\chi^{[1]}/\delta_c^{[1]}$, evaluated at zeroth order in the small $f_\chi$ expansion as a function of the $k$-dependent time variable $\tilde{\eta}$, for some illustrative choices of the parameter $\gamma$ that sets the time dependence of the characteristic scale $k_s(a)$. Right: For $\gamma = 2$, the ratios of linear transfer functions $g$ and $h = \Theta_\chi^{[1]}/\delta_c^{[1]}$ at zeroth order in $f_\chi$, while $s = \Theta_c^{[1]}/\delta_c^{[1]}$ is evaluated up to first order with $f_\chi = 0.05$ or $0.15$. One has $s^{(0)} = 1$ identically, whereas $s^{(1)} = -3/5$ for $k \gg k_s(a)$ (namely, $\tilde{\eta}\ll 0$).
• Figure 2: Left: Time evolution of the characteristic wavenumber $k_s(a)$ for two example models described by our framework: an ULA field with $m_a = 10^{-27}\;\mathrm{eV}$ and $f_\chi = 0.1$ (purple line) and a thermal relic massive neutrino with $m_\nu = 1\;\mathrm{eV}$ and $f_\chi \approx 0.08$ (dashed green line). The black curves show the evolution of the Hubble scale $\mathcal{H}$ for the ULA (solid) and massive neutrino (dashed) cosmologies. The dotted horizontal line indicates a representative $k$ mode that evolves differently in the two models. Right: The ratio of linear perturbations $\delta_\chi^{[1]}/\delta_c^{[1]}$ as a function of $k/k_s$, evaluated at $z = 0.5$, for the same cosmologies shown in the left panel. Dashed curves correspond to our universal closed-form approximation $g^{(0)}(\space - \frac{2+p}{\gamma - 1} \log \frac{k}{k_s})$, with $\gamma = 2$ and $p = 2$ ($p = 0$) for ULAs (massive neutrinos). Solid curves show the same ratios as obtained from CLASS.
• Figure 3: Left: Ratio of the linear matter power spectrum to $\Lambda$CDM (solid purple) compared to the ratio of linear perturbations $\delta_\chi^{[1]}/\delta_c^{[1]}$ (solid blue) for the same ULA cosmology as in Fig. \ref{['fig:ks_a_evolution']}: $m_a = 10^{-27}\;\mathrm{eV}$ and $f_\chi = 0.1$. Both ratios are evaluated at $z = 0.5$ and obtained from CLASS. The dot-dashed vertical lines show analytical approximations for the two relevant dynamical scales: $k_{\rm drop} = k_s(z_{\rm eq})$, which determines the beginning of the drop in the linear power spectrum, and $k_s(z)$, which marks the end of the step. Right: Same as in the left panel, but for the massive neutrino cosmology considered in Fig. \ref{['fig:ks_a_evolution']}: $m_\nu = 1\;\mathrm{eV}$ and $f_\chi \approx 0.08$. Here the dot-dashed vertical lines show $k_{\rm drop} \simeq k_s(z_{\rm nr})$ and $k_s(z)$.
• Figure 4: Coefficient of the UV divergence for the $P^{13}_{\delta_c \delta_c}(k)$ loop integral, as a function of $k/k_s(a)$, normalized to the standard $\Lambda$CDM value. The solid blue line shows the numerical result for $p = 0$, obtained by setting $f_\chi=0.1$. The two dotted lines show the analytical limits for $k\ll k_s(a)$ and $k\gg k_s(a)$, calculated using Eqs. \ref{['eq:Fc13_analytical']}. The dashed purple line corresponds to the analytical interpolation given within curly brackets in Eq. \ref{['eq:Pcc-UV']}. The residual discrepancies observed between numerical and analytical results are due to terms of $\mathcal{O}(f_\chi^2) \sim 1\%$, which were neglected in the analytical expansion for small $f_\chi$.
• Figure 5: The ratio $P_{\delta_g \delta_g}^L/P_{\delta_c \delta_c}^L$ calculated at linear order from the bias expansion in Eq. \ref{['eq:delta_g']}, as a function of $\tilde{\eta}$. Solid lines correspond to a set of bias coefficients $(b_c, b_\chi, b_\Theta)$. Dotted lines correspond to a different set, where $b_\Theta = 0$ but $b_c$ and $b_\chi$ have been appropriately shifted, which reproduces well the shape of the galaxy power spectrum. The small residual difference depends on the size of $b_\Theta$.