First-Principles Formalism for Simulating Self-Interacting Dark Matter

Abstract

It is plausible that the dark matter particles have non-gravitational interactions among themselves. If such self interactions are large enough, they could leave an imprint on the morphology of galaxies. These effects can be studied with numerical simulations, which serve as the primary tool to predict the non-linear evolution of galactic structure. A standard assumption is that the course-grained phase-space distribution of the macroscopic simulation particles follows the same evolution equation as that of the fundamental dark matter particles. This Letter tests this assumption directly for the case of frequent dark matter scatterings, demonstrating that this is not generically true. Specifically, we develop a first-principles map from a microscopic particle physics description of self-interacting dark matter to a representation of macroscopic simulation particles for theories in the short-mean-free-path regime. Using this procedure, we show the emergence of an effective force between the simulation particles and derive their interaction cross section, which depends on the one from fundamental particle physics. This work provides the first explicit map from particle physics to simulation, which will facilitate exploring the phenomenological implications for galactic dynamics.

Figures (3)

• Figure 1: The shaded regions of parameter space to the right of the colored lines denote the SMFP regime, where DM scatterings can be implemented as a collective long-range force between simulation macro-particles. The dashed colored lines are evaluated at the stage in which the halo reaches the maximal core size, with $(\rho_c,\, v_c) \simeq (2.4\space \rho_s,\,0.64\space v_{\rm max})$, while the solid colored lines are defined at a later stage of core collapse, when $( \rho_c,\, v_c) \simeq (100\space \rho_s,\,0.75\space v_{\rm max})$. The orange, green and blue lines refer to virial halo masses of $10^8$, $10^{10}$, and $10^{12}~{\rm M}_{\odot}$, respectively. In the parameter space above the black dashed line, the Born approximation holds and the Yukawa model can explain the observed DM abundance via the irreducible thermal freeze-out into the mediator. A benchmark point with $m_{\rm dm}=1\,\text{GeV}$ is represented by a star. The gray dash-dotted line shows the constraints on constant cross sections from galaxy groups from Sagunski:2020spe, re-interpreted for the velocity-dependence predicted by \ref{['eq:Bornxs']}.
• Figure 2: This figure provides an illustration of the coarse-graining procedure introduced in this Letter. We start with a micro-particle physics model of SIDM. Assuming we are in the short mean-free-path (SMFP)/frequent collision regime, we can approximate the SIDM as a fluid described by distribution functions (DFs) that satisfy a Boltzmann equation. We then partition the phase space described by the DFs into cells, which we interpret as the mathematical precursors to the simulation macro-particles. The final step is to coarse grain these cells, which yields an effective force that describes the impact of the DM self interactions between the simulation macro-particles. A mathematical summary is given in \ref{['eq:scheme']} in the appendix.
• Figure 3: This shows the dependence of the effective macro-particle [black solid line] and micro-particle [red dashed line] cross sections on $v_{\rm cir} \sim \sqrt{G M(r)/r} \sim \sqrt{2} v_{\rm 1D}$ for an isothermal halo profile. We set $\sigma_{0}/m_{\rm dm} = 1.4\times 10^4\,\text{cm}^2/\text{g}$ and $\omega = 40\,\text{km}/\text{s}$. This benchmark model is represented by a star in \ref{['fig:smfp']}.