Cosmological Limits on Strong Dark Forces

TL;DR

This work demonstrates that cosmology can place powerful constraints on long-range dark forces between dark-matter particles by analyzing both background evolution and perturbations in a Yukawa-coupled fermionic DM model. A dynamical mediator field $\phi$ exhibits attractor-driven evolution and multiple regimes, causing the dark sector equation of state to transiently differ from CDM and in some cases mimic dark radiation or dark energy, which leaves observable imprints on expansion history. The authors derive a Meszaros-like perturbation equation with an effective gravity $G_{\rm eff}$ and show scale-dependent enhancements of DM perturbations that compete with data from CMB, Lyman-$\alpha$, and ultrafaint dwarfs, yielding the strongest cosmological limits to date on DM self-interactions at $\lambda_\phi \lesssim 100$ kpc. They also analyze repulsive dark forces, finding even tighter BBN- and CMB-based constraints, illustrating the broad potential of cosmological probes to constrain non-gravitational dark-sector interactions with substantial phenomenological relevance.

Abstract

We showcase cosmology's ability to constrain long-range forces between dark matter particles. Specifically, we consider a fermionic dark matter interacting via a Yukawa-coupled light scalar, focusing on regimes where the dark forces are stronger than gravitational and yet unconstrained. We show that the dark sector dynamics, both at the background and perturbation levels, is far richer than what can be captured with just the static interparticle Yukawa potential. The background dynamics includes an attractor that funnels a wide range of initial conditions onto an evolution unique to each parameter space. In a large swath of parameter space beyond existing limits, the dark sector deviates drastically from cold dark matter in observable epochs. We rule out this parameter space using existing constraints on dark-sector equation of state and small-scale cosmic perturbations, thus setting the strongest constraints yet on dark matter self-interactions at length scales shorter than 100 kpc. In addition, we briefly discuss repulsive dark forces and place cosmological limits that are stricter than in the attractive case.

Figures (15)

• Figure 1: Effective potential of $\phi$, c.f. Eq. \ref{['eq:Veff']}. The labels "$\phi_0$" and "$\phi_*$" mark the locations of the linear finite-density minimum $\phi_0=m_\chi/g$ and the quadratic finite-density minimum $\phi_*=gn_\chi/m_\phi^2$. In the $\phi_*/\phi_0=2$ case, the $\phi_*$ exists mathematically (the dotted line shows what $V_{\rm eff}$ would be if the $|\phi_0-\phi|$ in Eq. \ref{['eq:Veff']} is replaced with $\phi_0-\phi$ without the absolute value) but in this case $\phi_*$ cannot be reached, because before $\phi$ can get there, the slope of the linear potential flips sign at $\phi=\phi_0$.
• Figure 2: Evolution of the background scalar field $\phi$ as a function of the scale factor $a$ for $\lambda_\phi=1\text{ kpc}$ and $\alpha_{\chi\chi}=1$ (solid blue), $\alpha_{\chi\chi}=10^{2.5}$ (solid green), $\alpha_{\chi\chi}=10^6$ (solid red). Here, we set $a_i=10^{-9}$, $\phi(a_i)=0$, and $\dot{\phi}(a_i)=0$, and numerically evolve $\phi$ using the equation of motion Eq. \ref{['eq:EoM']}. The gray dashed line is the linear finite-density minimum $\phi_0\equiv m_\chi/g$ defined in Eq. \ref{['eq:phi0']}. The dotted lines are the quadratic finite-density minima $\phi_*\equiv gn_\chi/m_\phi^2$, defined in Eq. \ref{['eq:phistar']}, corresponding to the values of $\alpha_{\chi\chi}$ matching their colors. The dark green labels "$\phi_{\rm rise}$" and "$\phi_{\rm DM}$" indicate, respectively, where in the evolution of the green line the $\phi$ field tracks the attractor solution $\phi_{\rm rise}\equiv (3\alpha_{\chi\chi}f_\chi/4)\phi_0(a/a_{\rm eq})$, defined in Eq. \ref{['eq:phirise']}, and oscillates as CDM, with an amplitude scaling as $\propto a^{-3/2}$. The colored bands labeled "BBN" and "CMB" indicate the very rough points where BBN and recombination take place.
• Figure 3: Types of solutions of $\phi$ at matter-radiation equality. For example in the red region $\phi$ is at $\phi_0\equiv m_\chi/g$, the linear finite-density minimum. In the green region $\phi$ is at $\phi_*\equiv gn_\chi/m_\phi^2$, the quadratic finite-density minimum. In the purple region it is at $\phi_{\rm rise}\equiv (3\alpha_{\chi\chi}f_\chi/4)\phi_0(a/a_{\rm eq)}$, the attractor solution initially tracked by $\phi$ for a wide range of initial conditions. And in the blue region it is at $\phi_{\rm DM}$ which is a CDM-like solution where $\phi$ oscillates with its bare mass with an amplitude that scales as $a^{-3/2}$. In the red region, the dark sector behaves as dark energy to the left of the dotted line and as dark radiation to the right of the dotted line. Also shown are the regions ruled out by Bullet Cluster observation Graham:2024hah and small-coupling analysis of the CMB data Bottaro:2024pcb. The CMB limit has a ceiling because Ref. Bottaro:2024pcb employs a small-$\alpha_{\chi\chi}$ approximation in their analysis. We show a rough approximation to this ceiling as a dashed-gray upper boundary, corresponding to $\alpha_{\chi\chi}\sim 0.3$
• Figure 4: Evolution of the instantaneous dark sector equation of state $w_{\rm dark}=p_{\rm dark}/\rho_{\rm dark}$ (solid gray) as a function of the scale factor $a$. The dark sector pressure $p_{\rm dark}$ and energy density $\rho_{\rm dark}$ are defined in Eqs. \ref{['eq:rhodark']}&\ref{['eq:pdark']}. The widths and amplitudes of the red bars represent the four scale-factor bins 1, 2, 3, 4 (see Eq. \ref{['eq:abins']}) used in our analysis and the averaged equation of state $\bar{w}_{\rm dark}^{(i)}$ within them. The black lines mark the boundaries of the allowed ranges of $\bar{w}_{\rm dark}^{(i)}$; we consider parameter space with red bars protruding the black lines as ruled out.
• Figure 5: The evolution of the DM density contrast $\delta_\chi=\delta n_\chi/n_\chi$ (solid black line, normalized to its initial, superhorizon value) as a function of the scale factor $a$ (normalized to that at matter-radiation equality, $a_{\rm eq}$). Here, we follow the mode with comoving wavenumber $k=1\text{ Mpc}^{-1}$, and set the dark-force parameters as indicated in the plots. The gray dashed line marks the point where this mode becomes subhorizon. The black solid lines represent the evolution of $\delta_\chi$ as dictated by Eq. \ref{['eq:Meszaroslike']}, to be contrasted with the evolution in the absence of dark forces ($\alpha_{\chi\chi}=0$) as given by Eq. \ref{['eq:deltaLambdaCDM']} and shown in dashed green lines. The light red region shows the region where $\phi$ is expected to closely track $\phi_*$, in which the $\Theta_*$ in Eqs. \ref{['eq:Meszaroslike']} and \ref{['eq:phistarwindow']} is equal to unity. The light blue region indicates where the dark-force range is longer than the physical wavelength $a/k$ of the mode. The top (bottom) plot corresponds to the case where the dark force is long-range $\lambda_\phi\gg a/k$ (short-range $\lambda_\phi\ll a/k$) when $\phi\approx \phi_*$. It can be seen that enhanced growth occurs in the light red region in either case. Furthermore, $\delta_\chi$ appears to continue growing relative to the $\Lambda$CDM one even outside of the pink region. This is due to the speed $\dot{\delta}_\chi$ acquired in the light red region.