Precise relic WIMP abundance and its impact on searches for dark matter annihilation

Gary Steigman, Basudeb Dasgupta, and John F. Beacom

Phys. Rev. D 86, 023506 – Published 3 July 2012

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. This result for a generic WIMP is usually stated as $⟨ σ v ⟩ \approx 3 \times 10^{- 26} {cm}^{3} s^{- 1}$ , 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 $5.2 \times 10^{- 26} {cm}^{3} s^{- 1}$ at $m \approx 0.3 GeV$ , and is $2.2 \times 10^{- 26} {cm}^{3} s^{- 1}$ 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 $WMAP + ACT$ .

Received 2 May 2012

DOI:https://doi.org/10.1103/PhysRevD.86.023506

Authors & Affiliations

Gary Steigman^1,2,3,*, Basudeb Dasgupta^1,†, and John F. Beacom^1,2,3,‡

¹Center for Cosmology and AstroParticle Physics, Ohio State University, 191 W. Woodruff Avenue, Columbus, Ohio 43210, USA
²Department of Physics, Ohio State University, 191 W. Woodruff Avenue, Columbus, Ohio 43210, USA
³Department of Astronomy, Ohio State University, 140 W. 18th Avenue, Columbus, Ohio 43210, USA

^*steigman.1@osu.edu
^†dasgupta.10@osu.edu
^‡beacom.7@osu.edu

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Vol. 86, Iss. 2 — 15 July 2012

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Images

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 $⟨ σ v ⟩$ and mass $m$ . Results for $m = 100 GeV$ are shown for weak interactions, $⟨ σ v ⟩ = 2 \times 10^{- 26} {cm}^{3} s^{- 1}$ , (dashed red), electromagnetic interactions, $⟨ σ v ⟩ = 2 \times 10^{- 21} {cm}^{3} s^{- 1}$ (dot-dashed green), and strong interactions, $⟨ σ v ⟩ = 2 \times 10^{- 15} {cm}^{3} s^{- 1}$ (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) [11].Reuse & Permissions
Figure 2
The effective number of interacting (thermally coupled), relativistic degrees of freedom, $g$ , as a function of the temperature for $1 MeV \leq T \leq 1 TeV$ (adapted from Laine and Schroeder [45]).Reuse & Permissions
Figure 3
Evolution of the departure of the WIMP abundance from the equilibrium abundance, $Δ$ , 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. (12), for an illustrative case with $m = 100 GeV$ and $⟨ σ v ⟩ = 2.2 \times 10^{- 26} {cm}^{3} s^{- 1}$ . The analytical approximation ignores $d Δ / d x$ [see Eq. (11)], leading to an underestimate of $x_{*}$ by $\sim 2 %$ . See the text for details.Reuse & Permissions
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 $(Γ / H)_{*}$ , 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 $50 α_{*}$ [see Eq. (18)]. See the text for details.Reuse & Permissions
Figure 5
The thermal annihilation cross section required for $Ω_{χ} h^{2} = 0.11$ 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. (20). Note that the agreement between analytical and numerical results is better than $\sim 3 %$ . For comparison, the thin horizontal line shows the canonical value $⟨ σ v ⟩ = 3 \times 10^{- 26} {cm}^{3} s^{- 1}$ .Reuse & Permissions
Figure 6
Comparison of the Fermi-LAT limits on the thermal WIMP annihilation cross section into particular channels (the regions above the colored curves are ruled out at 95% C.L.) with the relic annihilation cross section calculated for $Ω h^{2} = 0.11$ (solid black). Constraints from the Milky Way for $u \bar{u}$ (dotted blue); constraints from dwarf galaxies for $b \bar{b}$ (dashed red), and $τ^{+} τ^{-}$ (dot-dashed green); constraints from cosmology for $μ^{+} μ^{-}$ (dot-dot-dashed yellow).Reuse & Permissions

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