Effects of cold dark matter decoupling and pair annihilation on cosmological perturbations

TL;DR

The study develops a first-principles framework for the linear evolution of cosmological perturbations in a WIMP dark matter scenario by reducing WIMP–lepton collisions to a Fokker–Planck equation and solving the coupled perturbation equations through kinetic decoupling, electron–positron annihilation, and radiation–matter equality. It provides both detailed numerical results and analytic approximations (free streaming and imperfect-fluid) to quantify damping and the resulting suppression of the transfer function at small scales. A key result is that Silk-like friction during kinetic decoupling, together with subsequent free streaming, sets a minimum halo mass near Earth scales, with a predicted $d n/d\ln M \propto M^{-1/3}$ below $M_d$, for a smooth window; pair annihilation and the radiation–matter transition imprint modest corrections. These findings highlight how sub-solar-scale fluctuations encode WIMP physics and motivate considerations of Earth-mass microhalos for indirect detection, while acknowledging uncertainties from nonlinear evolution and baryonic effects.

Abstract

Weakly interacting massive particles are part of the lepton-photon plasma in the early universe until kinetic decoupling, after which time the particles behave like a collisionless gas with nonzero temperature. The Boltzmann equation for WIMP-lepton collisions is reduced to a Fokker-Planck equation for the evolution of the WIMP distribution including scalar density perturbations. This equation and the Einstein and fluid equations for the plasma are solved numerically including the acoustic oscillations of the plasma before and during kinetic decoupling, the frictional damping occurring during kinetic decoupling, and the free-streaming damping occurring afterwards and throughout the radiation-dominated era. An excellent approximation reduces the solution to quadratures for the cold dark matter density and velocity perturbations. The subsequent evolution is followed through electron pair annihilation and the radiation-matter transition; analytic solutions are provided for both large and small scales. For a 100 GeV WIMP with bino-type interactions, kinetic decoupling occurs at a temperature $T_d=23$ MeV. The transfer function in the matter-dominated era leads to an abundance of small cold dark matter halos; with a smooth window function the Press-Schechter mass distribution is $dn/d\ln M\propto M^{-1/3}$ for $M<10^{-4} (T_d/$10 MeV)$^{-3}$ M$_\odot$.

Figures (12)

• Figure 1: CDM density transfer function versus wavenumber at conformal time $\tau=72\tau_d$. The three oscillating curves assume that kinetic decoupling occurred at $\tau_d$ for three different values of the radiation temperature $T_d$ relative to the CDM particle mass $m_\chi$; the amplitude of the oscillations decreases with increasing $T_d/m_\chi$. The upper, monotonic curve assumes that the CDM is always collisionless and was never coupled to the radiation. The other non-oscillating curve shows a crude model of kinetic decoupling described by a Gaussian cutoff.
• Figure 2: Contour used to evaluate the integral in Eq. (\ref{['f1int']}). The quarter-circle is actually taken to have a much larger radius than shown so that its contributions to the contour integral vanish. The desired path for the integral is along the lower left curve that joins the quarter-circle.
• Figure 3: The auxiliary functions $f_1(x)$ and $f_2(x)$ (the curve with the higher amplitude of oscillation) appearing in Eq. (\ref{['coldlimsol']}), where $\tau\gg\tau_d$ and $x\equiv(k\tau_d/2\sqrt{3})^{4/5}$.
• Figure 4: CDM transfer function and several approximations plotted versus wavenumber at conformal time $\tau=72\tau_d$. In descending amplitude of the second peak, the curves are (1) fluid approximation with $\pi=\sigma=0$, (2) exact solution of the Fokker-Planck equation, (3) imperfect fluid approximation with shear stress but no entropy perturbation; (4) imperfect fluid approximation with both shear stress and entropy perturbations. The plus signs superimposed on the exact solution curve are the solution with $T_d/m_\chi=0$, multiplied by a Gaussian damping factor as described in the text.
• Figure 5: Real space CDM Green's function (Fourier transform of the transfer function) at conformal time $\tau=10^7\tau_d$, approximately at the end of the radiation-dominated era. An initial planar perturbation sends an acoustic wave through the relativistic plasma. This wave travels through the CDM until kinetic decoupling ends; thereafter the wave is frozen in place but grows logarithmically in amplitude. The three curves show the effect of diffusion with increasing CDM temperature.