Precise Relic WIMP Abundance and its Impact on Searches for Dark Matter Annihilation

TL;DR

By refining the analytic relic abundance calculation with updated inputs and a careful treatment of g(T), the paper derives a precise relation between ⟨σv⟩ and Ωh^2 for generic WIMPs. It shows strong mass dependence below 10 GeV, increasing ⟨σv⟩ to about 5.2×10^-26 cm^3 s^-1 near 0.3 GeV, while for higher masses ⟨σv⟩ ≈ 2.2×10^-26 cm^3 s^-1, about 40% below the canonical value. This mass-dependent relic cross section alters the interpretation of Fermi-LAT and CMB limits and strengthens certain cosmological constraints. The results provide analytic formulas that agree with full numerical solutions to within a few percent. The work informs current and future indirect-detection analyses and motivates extending gamma-ray searches to lower WIMP masses.

Abstract

If dark matter (DM) is a weakly interacting massive particle (WIMP) that is a thermal relic of the early Universe, then its total self-annihilation cross section is revealed by its present-day mass density. The canonical thermally averaged cross section for a generic WIMP is usually stated as 3*10^-26 cm^3s^-1, with unspecified uncertainty, and taken to be independent of WIMP mass. Recent searches for annihilation products of DM annihilation have just reached the sensitivity to exclude this canonical cross section for 100% branching ratio to certain final states and small WIMP masses. The ultimate goal is to probe all kinematically allowed final states as a function of mass and, if all states are adequately excluded, set a lower limit to the WIMP mass. Probing the low-mass region is further motivated due to recent hints for a light WIMP in direct and indirect searches. We revisit the thermal relic abundance calculation for a generic WIMP and show that the required cross section can be calculated precisely. It varies significantly with mass at masses below 10 GeV, reaching a maximum of 5.2*10^-26 cm^3s^-1 at masses around 0.3 GeV, and is 2.2*10^-26 cm^3s^-1 with feeble mass-dependence for masses above 10 GeV. These results, which differ significantly from the canonical value and have not been taken into account in searches for annihilation products from generic WIMPs, have a noticeable impact on the interpretation of present limits from Fermi-LAT and WMAP+ACT.

Figures (6)

• Figure 1: Evolution of the cosmological WIMP abundance as a function of $x = m/T$. Note that the y-axis spans 25 orders of magnitude. The thick curves show the WIMP mass density, normalized to the initial equilibrium number density, for different choices of annihilation cross section $\langle \sigma v\rangle$ and mass $m$. Results for $m=100\,{\rm GeV}$, are shown for weak interactions, $\langle \sigma v\rangle=2\times10^{-26}\,{\rm cm^3s^{-1}}$, (dashed red), electromagnetic interactions, $\langle \sigma v\rangle=2\times10^{-21}\,{\rm cm^3s^{-1}}$ (dot-dashed green), and strong interactions, $\langle \sigma v\rangle=2\times10^{-15}\,{\rm cm^3s^{-1}}$ (dotted blue). For the weak cross section the thin dashed curves show the WIMP mass dependence for $m=10^3\,$GeV (upper dashed curve) and $m=1\,$GeV (lower dashed curve). The solid black curve shows the evolution of the equilibrium abundance for $m=100\,$GeV. This figure is an updated version of the figure which first appeared in Steigman (1979) Steigman:1979kw.
• Figure 2: The effective number of interacting (thermally coupled), relativistic degrees of freedom, $g$, as a function of the temperature for $1\,{\rm MeV} \leq T \leq 1\,{\rm TeV}$ (adapted from Laine and Schroeder Laine:2006cp).
• Figure 3: Evolution of the departure of the WIMP abundance from the equilibrium abundance, $\Delta$, for $x$ close to $x_*$. The departure from the equilibrium value is shown as a function of $x$, calculated numerically (solid black), and analytically (dashed red) using Eq. (\ref{['eq:delta']}), for an illustrative case with $m=100\,$GeV and $\langle \sigma v\rangle=2.2\times10^{-26}\,{\rm cm^3s^{-1}}$. The analytical approximation ignores $d\Delta/dx$ (see Eq. (\ref{['eq:deltagen']})), leading to an underestimate of $x_*$ by $\sim 2\,\%$. See the text for details.
• Figure 4: The matching point, $x_{*}$ (dashed red), is shown for WIMP masses from 100 MeV to 10 TeV, along with $g_{*}^{1/2}$ (dot-dashed green) and $(\Gamma/H)_{*}$, the ratio of the annihilation rate to the expansion rate evaluated at $T = T_{*}$ without the logarithmic corrections (dotted blue, upper) and with the logarithmic corrections (dotted blue, lower). Also shown (solid black) is $50\,\alpha_{*}$ (see Eq. (\ref{['eq:alphadef']})). See the text for details.
• Figure 5: The thermal annihilation cross section required for $\Omega_{\chi}h^{2} = 0.11$ as a function of the mass for a Majorana WIMP. The solid (black) curve is from numerical integration of the evolution equation and the dashed (red) curve is for the approximate analytic solution in Eq. (\ref{['eq:sigvomegahsq']}). Note that the agreement between analytical and numerical results is better than $\sim3\%$. For comparison, the thin horizontal line shows the canonical value $\langle \sigma v\rangle=3\times10^{-26}\,{\rm cm^3s^{-1}}$.