CMB constraints on the thermal WIMP mass and annihilation cross section

Gary Steigman

Phys. Rev. D 91, 083538 – Published 27 April 2015

Abstract

A thermal relic, often referred to as a weakly interacting massive particle (WIMP), is a particle produced during the early evolution of the Universe whose present (relic) abundance depends only on its mass and its thermally averaged annihilation cross section (annihilation rate factor) $⟨ σ v ⟩_{ann}$ . Late time WIMP annihilation has the potential to affect the cosmic microwave background (CMB) power spectrum. Current observational constraints on the absence of such effects provide bounds on the mass and the annihilation cross section of relic particles that may be, but need not be, dark matter candidates. For a WIMP that is a dark matter candidate, the CMB constraint sets an upper bound to the annihilation cross section, leading to a lower bound to its mass that depends on whether or not the WIMP is its own antiparticle. For a self-conjugate WIMP, $m_{\min} = 50 f GeV$ , where $f \leq 1$ is an electromagnetic energy efficiency factor. For a non–self-conjugate WIMP, the minimum mass is a factor of two larger. For a WIMP that is a subdominant component of the dark matter density there is no bound on its mass and the upper bound to its annihilation cross section imposed by the CMB transforms into a lower bound to its annihilation cross section. These results are outlined and quantified here using the latest CMB constraints for a stable, symmetric (equal number of particles and antiparticles), WIMP whose annihilation is s-wave dominated, and for particles that are, or are not, their own antiparticle.

Received 5 March 2015

DOI:https://doi.org/10.1103/PhysRevD.91.083538

Authors & Affiliations

Gary Steigman^*

Center for Cosmology and AstroParticle Physics, Ohio State University, 191 West Woodruff Avenue, Columbus, Ohio 43210, USA and Department of Physics, Ohio State University, 191 West Woodruff Avenue, Columbus, Ohio 43210, USA

^*steigman.1@osu.edu

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Vol. 91, Iss. 8 — 15 April 2015

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Images

Figure 1
The s-wave, thermal annihilation cross section (rate factor), $⟨ σ v ⟩_{DM} = ⟨ σ v ⟩_{ann} \equiv ⟨ σ v ⟩_{χ}$ , as a function of the relic particle mass, $m_{χ}$ , required for the WIMP to account for the observed dark matter density ( $Ω_{χ} h^{2} = Ω_{DM} h^{2} = 0.12$ ). The top (purple) curve is for a NS particle ( $χ \neq \bar{χ}$ ) and the bottom (blue) curve is for a S particle ( $χ = \bar{χ}$ ). Only along the curves does the WIMP account for the DM ( $Ω_{χ} = Ω_{DM}$ ). Although the regions above the curves are allowed, in these regions the WIMP is a subdominant contributor to the DM ( $Ω_{χ} < Ω_{DM}$ ). The regions below the curves are forbidden since, in these regions, the relic WIMP mass density exceeds the DM density ( $Ω_{χ} > Ω_{DM}$ ).
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Figure 2
The annihilation rate factors $⟨ σ v ⟩_{χ} = ⟨ σ v ⟩_{DM}$ from Fig. 1 for a WIMP that is the DM are shown as functions of the WIMP mass, along with the CMB constraints on late time annihilations, $⟨ σ v ⟩_{CMB}$ (shown in black) for $f = 1$ (solid) and $f = 0.2$ (dashed). The energy injection from late time annihilations is constrained by $⟨ σ v ⟩_{CMB}$ in that for $Ω_{χ} = Ω_{DM}$ the total annihilation cross section satisfies $⟨ σ v ⟩_{χ} = ⟨ σ v ⟩_{DM} \leq ⟨ σ v ⟩_{CMB}$ , leading to a lower bound on the WIMP mass where the curves cross. But, for $Ω_{χ} < Ω_{DM}$ , there is no constraint on the WIMP mass and $⟨ σ v ⟩_{χ} < ⟨ σ v ⟩_{CMB} (Ω_{χ} / Ω_{DM})^{2}$ . See Sec. 3a for details.
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Figure 3
The minimum annihilation rate factor consistent with the CMB constraint for $f = 1$ and the mass density constraint $Ω_{χ} \leq Ω_{DM}$ is shown as a function of the WIMP mass. The regions above the curves, corresponding to $Ω_{χ} \leq Ω_{DM}$ , are allowed, while the regions below the curves, corresponding to $Ω_{χ} > Ω_{DM}$ , are excluded. The upper (purple) curve is for a NS WIMP ( $χ \neq \bar{χ}$ ) and the lower (blue) curve is for a S WIMP ( $χ = \bar{χ}$ ). For $m_{χ} \geq m_{\min}$ , ${⟨ σ v ⟩}_{\min} = ⟨ σ v ⟩_{DM}$ , while for $m_{χ} < m_{\min}$ , ${⟨ σ v ⟩}_{\min} = (⟨ σ v ⟩_{DM})^{2} / ⟨ σ v ⟩_{CMB} > ⟨ σ v ⟩_{DM}$ , corresponding to $Ω_{χ} < Ω_{DM}$ . See the text for details.
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Figure 4
The two panels show the results for the minimum annihilation rate factor for a S thermal relic, ${⟨ σ v ⟩}_{\min}$ , satisfying the CMB constraint and $Ω_{χ} \leq Ω_{DM}$ , as in Fig. 3. In both panels the dotted curves account for the logarithmic corrections to ${⟨ σ v ⟩}_{\min}$ discussed in the text. The regions above the solid (dotted) curves are allowed, while the regions below them are excluded. In the left-hand panel the solid (dotted) curve is for all masses and for $f = 1$ . The dashed curve shows the extension of ${⟨ σ v ⟩}_{DM}$ to lower masses, $m_{χ} < m_{\min}$ , illustrating that ${⟨ σ v ⟩}_{\min} > ⟨ σ v ⟩_{DM}$ in this mass range. The right-hand panel compares ${⟨ σ v ⟩}_{\min}$ for $f = 1$ (solid) and $f = 0.2$ (dashed).
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Figure 5
The left-hand panel shows (for $f = 1$ ) the ratio of the minimum annihilation rate factor to the DM annihilation rate factor, ${⟨ σ v ⟩}_{\min} / ⟨ σ v ⟩_{DM}$ , as a function of the WIMP mass. The regions above the curves, corresponding to $⟨ σ v ⟩_{χ} \geq {⟨ σ v ⟩}_{\min} \geq ⟨ σ v ⟩_{DM}$ , are allowed, while the regions below the curves are excluded. The right-hand panel shows the variation of the ratio of mass densities, $Ω_{\max} / Ω_{DM}$ , with WIMP mass (also for $f = 1$ ). For the ratio of mass densities, the allowed regions are restricted to lie below the curves ( $Ω_{χ} \leq Ω_{\max} \leq Ω_{DM}$ ). In both panels the dotted curves account for the logarithmic corrections to ${⟨ σ v ⟩}_{\min}$ discussed in the text.
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