Thermal decoupling of WIMPs from first principles

TL;DR

The paper provides a complete analytic treatment of WIMP kinetic decoupling from the thermal bath by deriving a master equation for the WIMP temperature $T_\chi(T)$ that applies to arbitrary scattering amplitudes. By matching asymptotic limits, it defines a precise kinetic decoupling temperature $T_{\rm kd}$ and demonstrates how microphysical interactions determine the small-scale cutoff in the linear matter power spectrum via viscous damping and free streaming. The authors illustrate the method with recombination, bino-like neutralino DM, and LKP DM in UED, obtaining explicit $T_{\rm kd}$ expressions and showing typical MeV-scale decoupling, which has direct implications for the formation and survival of microhalos and for indirect detection signals from DM substructure. This framework strengthens the link between particle microphysics and cosmological structure on sub-galactic scales and provides a robust tool for predicting the damping scale across WIMP models, including cases with resonant scattering.

Abstract

Weakly interacting massive particles (WIMPs) are arguably the most natural DM candidates from a particle physics point of view. After their number density has frozen out in the early universe, determining their relic density today, WIMPs are still kept very close to thermal equilibrium by scattering events with standard model particles. The complete decoupling from the thermal bath happens as late as around 1-10 MeV, thereby setting an important cosmological scale that can directly be translated into a small-scale cutoff of the spectrum of matter density fluctuations. We present here a full analytic treatment of the kinetic decoupling process from first principles. This allows an exact determination of the decoupling scale, for an arbitrary WIMP candidate and any scattering amplitude. As an application, we consider the situation of the lightest supersymmetric particle as well as the lightest Kaluza-Klein particle that arises in theories with universal extra dimensions; furthermore, we show that our formalism can also easily be applied to, e.g., the evolution of the non-relativistic electrons into the recombination regime. Finally, we comment on the impacts for the smallest gravitationally bound structures and the prospects for the indirect detection of dark matter.

Figures (5)

• Figure 1: The LSP temperature $T_\chi$ as a function of the background temperature $T$ in the early universe is shown as a solid line, for the case of a Bino with $M_{\tilde{B}}=100~$GeV and $M_{\tilde{L}}=150~$GeV. The rather abrupt transition to a regime with $T_\chi\propto T^2$, where the LSP is completely decoupled from the thermal bath, makes it straightforward to identify the temperature $T_\mathrm{kd}$ of thermal decoupling as given in Eq. (\ref{['Binokd']}); here, it is indicated by a dashed line. For comparison, the (green and blue) dotted lines show the naive estimates for the evolution of $T_\chi$, assuming a sudden decoupling from the thermal bath at the time of last scattering or the relaxation time, respectively (see also Hofmann:2001bi).
• Figure 2: The left part of this figure shows the kinetic decoupling temperature $T_\mathrm{kd}$ for the LKP in the mUED model. From top to bottom, the cutoff scale is set to $\Lambda R=40,\,30,\,20$ and the grey band shows the region consistent with the $2\sigma$ WMAP relic density constraint for a Higgs mass $m_h\lesssim150\,$GeV (for higher Higgs masses, the grey region broadens and shifts to the right, up to $R^{-1}\lesssim1.3\,$TeV Kakizaki:2006dz). In the right panel, we leave the mUED setup and treat the mass splitting between the $B^{(1)}$ and the KK leptons as a single, free parameter $\delta$; from bottom to top, the curves correspond to $\delta=0.01,\,0.05,\,0.2$.
• Figure 3: Phaseplot for the temperature evolution, expressed in the dimensionless quantities $x\equiv\left|a\right|(T/M_\chi)^{n+2}$ and $y\equiv T_\chi/T$. Here, we show the case for $n=0$, but the same type of behaviour is obtained for any other (positive) value of $n$. As can be seen, any departure from thermal equilibrium ($T_\chi=T$) is restored almost immediately for $x\gtrsim10$, forcing $T_\chi$ to follow the solid line which represents our general solution (\ref{['TCDM']}). For smaller temperatures, $\chi$ decouples from the heat bath; in this regime, its temperature is only influenced by the Hubble expansion and thus evolves as $T_\chi\propto T^2$.
• Figure 4: Feynman diagrams for the elastic scattering of the LKP with SM fermions. In principle, there is also a contribution from a $t$-channel Higgs-exchange, but this is heavily suppressed due to the small Yukawa couplings involved.
• Figure 5: The full LKP scattering amplitude (solid lines) vs. the approximative expression that scales with the photon energy as $\omega^2$ (dashed lines). For simplicity, we have shown the case of electron scattering, and only taken into account the contribution from left-handed states. The two situations correspond to a mass splitting $\delta\equiv\left(M_{f_s^{(1)}}-M_{B^{(1)}}\right)/M_{B^{(1)}}$ of 1% (left) and 5% (right), respectively.