Thermally Averaged Annihilation Cross Section

• Thermally averaged annihilation cross section is defined as the average of the product of the annihilation cross section and relative velocity over a Maxwell–Boltzmann distribution.
• It is computed using methods such as velocity expansion, full relativistic integration, and incorporates effects like Sommerfeld enhancement and resonant structures.
• Its accurate determination is crucial for predicting dark matter relic abundance, guiding indirect detection strategies, and constraining cosmological models.

The thermally averaged annihilation cross section, customarily denoted ⟨σv⟩, is a central object in the paper of thermal relics and the indirect detection of dark matter (DM). It encodes the average product of the annihilation cross section and the relative velocity between annihilating particles, weighted over their velocity distribution in a thermal bath. This quantity determines the DM freeze-out abundance, sets indirect detection rates, and enters cosmological and astrophysical constraints across a broad class of models including Weakly Interacting Massive Particles (WIMPs), Strongly Interacting Massive Particles (SIMPs), asymmetric DM, and more exotic frameworks such as those involving Sommerfeld enhancement or non-standard cosmologies.

1. Formal Definition and General Properties

The thermally averaged annihilation cross section ⟨σv⟩ for a 2→2 reaction between self-conjugate, non-relativistic dark matter particles χ of mass $m$ is given by

$$\langle\sigma v\rangle = \frac{1}{n_{\rm eq}^2} \int \frac{d^3p_1}{(2\pi)^3} \frac{d^3p_2}{(2\pi)^3} f({\bf p}_1) f({\bf p}_2) \, \sigma(v_{\rm rel})\, v_{\rm rel}$$

where $f({\bf p})$ is the Maxwell–Boltzmann distribution, and $v_{\rm rel}$ is the relativistic relative velocity,

$$v_{\rm rel} = \frac{\sqrt{(p_1 \cdot p_2)^2 - m^4}}{p_1 \cdot p_2}$$

The generalization to n→2 processes (e.g., 3→2 for SIMPs) involves integrating over $n$-particle joint distributions under center-of-mass constraints, with the thermal average containing $v^{n-1}$ (Choi et al., 2017, Choi et al., 2017).

For practical purposes, the velocity expansion

$\sigma v = a + b v^2 + \cdots$

is often employed, with $a$ corresponding to s-wave (velocity-independent), and $b$ to p-wave (velocity-squared) annihilation. The Maxwell–Boltzmann thermal average yields

$\langle\sigma v\rangle = a + \frac{6b}{x} + \cdots$

where $x = m/T$.

2. Relativistic and Non-Relativistic Computation

A fully relativistic formulation is essential for accuracy near freeze-out or for light dark matter. The invariant result, as shown by Gondolo & Gelmini and analyzed in (Cannoni, 2015), is

$$\langle\sigma v\rangle = \frac{1}{8 m^4 T K_2^2(m/T)} \int_{4m^2}^\infty ds\, \sigma(s)\, (s-4m^2)\, \sqrt{s}\, K_1(\sqrt{s}/T)$$

$K_{1,2}$ are modified Bessel functions.

In the large-$x$, or non-relativistic, limit, this reproduces the velocity expansion. A key insight is that expansions must be made only after performing the correct relativistic average to avoid errors and unphysical artifacts such as those induced by using the non-invariant Møller velocity (Cannoni, 2015).

For p-wave processes, the cross section takes the schematic form $\sigma v = b v^2$, so

$\langle \sigma v \rangle = \frac{6b}{x}$

In SIMP models involving 3→2 annihilation, the thermal average is

$$\langle\sigma v^2\rangle = \frac{1}{2} x^3 \int_0^\infty d\eta\, \eta^2 e^{-x\eta} (\sigma v^2)(\eta)$$

where $\eta = \frac{1}{2}(v_1^2 + v_2^2 + v_3^2)$, and the integral structure is dictated by SO(9) invariance and phase-space constraints (Choi et al., 2017, Choi et al., 2017).

3. Beyond the Simple Velocity Expansion: Resonances and Velocity Effects

For cross sections with strong velocity dependence or resonant enhancements, as with s-channel poles (Breit–Wigner), standard expansions can be inaccurate. The general approach replaces the velocity expansion of the cross section with a resonance form,

$$\sigma v (\eta) = \sum_{\ell=0}^\infty \frac{b_R^{(\ell)}}{\ell!} \eta^\ell \frac{\gamma_R}{(\epsilon_R - \tfrac{2}{3}\eta)^2 + \gamma_R^2}$$

where $\epsilon_R$ characterizes the detuning of the resonance and $\gamma_R$ its width (for 3→2), and auxiliary functions $G_\ell(z_R;x)$ encode the thermal convolution (Choi et al., 2017). In the narrow-width limit,

$$\langle\sigma v^2\rangle_R \simeq \frac{27}{16}\pi x^3 \epsilon_R^2 e^{-3x\epsilon_R/2} \sum_{\ell=0}^\infty \frac{b_R^{(\ell)}}{\ell!} \left(\frac{3}{2}\epsilon_R \right)^\ell$$

with strict dependence on the Boltzmann suppression $e^{-E_R/T}$. Resonant enhancements near threshold can increase $\langle\sigma v\rangle$ by orders of magnitude even for small changes in mass parameters, with relic abundances altered accordingly.

4. Sommerfeld Enhancement and Non-Perturbative Thermal Effects

The presence of long-range attractive forces (e.g., massless mediators) between DM particles induces Sommerfeld enhancement, leading to a cross section with non-analytic velocity dependence, $S_\ell(v)$: $$S_s(v) = \frac{2\pi\alpha/v}{1-\exp(-2\pi\alpha/v)}$$ The thermally averaged quantity requires integrating this factor against the velocity-weighted cross section,

$$\langle\sigma v\rangle \simeq a\, B_s(\alpha\sqrt{\pi x}), \qquad B_s \textrm{ (analytic fit)}$$

where $B_s$ interpolates between the perturbative and non-perturbative limits, and is accurately given by rational functions for both s- and p-wave annihilation (Iminniyaz et al., 2010). Accurate implementation of this effect modifies the predicted relic abundance even at the percent level and is essential for models with light mediators or strong coupling.

For heavy WIMPs in QCD or coannihilation scenarios, large Sommerfeld enhancements ($\mathcal{O}(10)$–$100$) due to bound state formation and non-perturbative effects have been confirmed both in resummed perturbation theory and on the lattice (Kim et al., 2019).

5. Integration into the Boltzmann Equation and Relic Density Prediction

The thermally averaged annihilation cross section enters the Boltzmann equation: $$\frac{dY}{dx} = -\frac{s}{xH} [1 + \frac{1}{3} \frac{d \ln g_s}{d \ln T}] \langle\sigma v\rangle (Y^2-Y_{\rm eq}^2)$$ $Y = n/s$ is the comoving yield, $s$ the entropy density, $H$ the Hubble rate, and $g_s$ the entropy degrees of freedom. The freeze-out temperature $x_f$ is found by the implicit condition $n_{\rm eq}(x_f) \langle\sigma v\rangle = H(x_f)$, with all coefficients and temperature dependencies set by $g_*$ and the annihilation cross section (Steigman et al., 2012).

The present-day relic density is thus

$$\Omega_\chi h^2 \approx \frac{1.07\times 10^9\,\mathrm{GeV}^{-1}}{g_*^{1/2} M_{\rm Pl}} \frac{x_f}{\langle\sigma v\rangle}$$

For $m \gtrsim 10$ GeV, the required $\langle\sigma v\rangle \simeq 2.2 \times 10^{-26}$ cm$^3$ s$^{-1}$, notably about 40% below older canonical values. For $m \lesssim 10$ GeV, QCD crossover-induced changes in $g_*$ cause $\langle\sigma v\rangle$ to peak near $5.2 \times 10^{-26}$ cm$^3$ s$^{-1}$ at $m \sim 0.3$ GeV (Steigman et al., 2012).

Similar relations hold for 3→2 processes,

$$Y_\infty^2 \simeq \frac{H(x_f)}{s(x_f) \langle\sigma v^2\rangle} \frac{x_f^2}{2}$$

and higher-order scenarios, with solution techniques depending strongly on the velocity and resonance structure.

6. Model Dependence, Cosmological Constraints, and Observational Implications

The velocity dependence of $\langle\sigma v\rangle$ critically shapes indirect detection signals and relic density predictions:

• s-wave (velocity-independent): Dominant in canonical WIMPs; both freeze-out and late-time annihilation rate are set by constant $\langle\sigma v\rangle$.
• p-wave (velocity-suppressed): $\langle\sigma v\rangle \propto 1/x$; suppressed in low-velocity environments such as the present universe, relaxing indirect detection constraints (Lopes et al., 2016).
• Resonant enhancement: Enhances annihilation at kinetic energies matching a mediator mass, possibly leading to observable cosmic-ray or gamma-ray excesses; present for both 2→2 and 3→2 reactions.
• Sommerfeld and bound-state effects: Enhance annihilation at low velocities; must be incorporated for precision relic density or indirect detection calculations, especially for models with light mediators or large gauge groups (Iminniyaz et al., 2010, Kim et al., 2019).

Empirical constraints from indirect detection (e.g., Fermi-LAT, cosmic microwave background (CMB), weak lensing/gamma-ray cross-correlation) are interpreted in terms of $\langle\sigma v\rangle$ integrated over astrophysical velocity distributions (Shirasaki et al., 2014, Steigman, 2015). For symmetric s-wave WIMPs making up all dark matter, CMB power spectra impose $m_\chi \gtrsim 50$ GeV (for $f=1$), and for subdominant fractions invert the constraint to a lower bound on $\langle\sigma v\rangle$ (Steigman, 2015).

7. Advanced Scenarios and Cosmological Extensions

Non-standard cosmologies such as bouncing universes introduce qualitatively new regimes for thermal relics. In such scenarios, a constant (s-wave) $\langle\sigma v\rangle$ can result in either the standard freeze-out $\Omega_\chi \propto 1/\langle\sigma v\rangle$, a non-thermal regime $\Omega_\chi \propto \langle\sigma v\rangle$, or "thermal & weak-freeze-out" scenarios, where the final abundance is independent of $\langle\sigma v\rangle$ and fixed entirely by early-universe dynamics (Li, 2014).

In asymmetric dark matter (ADM), the fate of the relic density and the required $\langle\sigma v\rangle$ strongly depend on the initial particle-antiparticle asymmetry and can greatly exceed the canonical WIMP value in regimes with small surviving anti-DM abundance. The analytic relic abundance formula includes a logarithmic dependence on the asymmetry and annihilation cross section (Bell et al., 2014).

Table: Summary of Key Thermally Averaged Cross Section Formulas

Scenario/Process	Formula for ⟨σv⟩ (Key scaling)	Reference
2→2 s-wave, non-relativistic	$a$	(Steigman et al., 2012)
2→2 p-wave, non-relativistic	$\frac{6b}{x}$	(Steigman et al., 2012)
3→2 (SIMP, non-resonant)	$a_0 + 3a_1 x^{-1} + 6a_2 x^{-2} + \cdots$	(Choi et al., 2017)
Resonant enhancement (2→2, Breit–Wigner, narrow)	$$2\sqrt{\pi} \gamma_R \epsilon_R^{1/2} x^{3/2} e^{-x\epsilon_R}$$	(Choi et al., 2017)
Resonant enhancement (3→2, Breit–Wigner, narrow)	$$(27\pi/16) \epsilon_R^2 x^3 e^{-3x\epsilon_R/2} \cdots$$	(Choi et al., 2017)
Sommerfeld, s-wave	$a \times B_s(\alpha\sqrt{\pi x})$	(Iminniyaz et al., 2010)
Freeze-out relic density (s-wave, WIMP)	$$\langle\sigma v\rangle \simeq 2.2 \times 10^{-26}\,\mathrm{cm}^3/\mathrm{s}$$ (for $m>10$ GeV)	(Steigman et al., 2012)

Conclusion

The thermally averaged annihilation cross section ⟨σv⟩ is an indispensable theoretical object in the calculation of dark matter relic densities, indirect detection rates, and cosmological constraints. Its accurate computation requires careful treatment of velocity distributions, resonance structures, non-perturbative enhancements, and cosmological context. Analytical and semi-analytical methods exist for standard (WIMP) scenarios, while more sophisticated treatments are necessary in the presence of resonances, strong interactions, or non-standard cosmological evolution. Continued improvement in both theoretical modeling and observational sensitivity motivates the ongoing refinement in the computation and interpretation of ⟨σv⟩ across dark matter research.