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micrOMEGAs 6.0: N-component dark matter

G. Alguero, G. Belanger, F. Boudjema, S. Chakraborti, A. Goudelis, S. Kraml, A. Mjallal, A. Pukhov

arXiv:2312.14894v2hep-ph

TL;DR

Existing micrOMEGAs versions did not fully cover multi-component dark matter evolution and related production effects. This version generalizes the framework to N-component DM, improves freeze-in treatment, and extends photon-spectrum data.

  • Problem

    Previous micrOMEGAs versions required broader treatment of multi-component dark matter evolution and chemical-equilibrium conditions within dark sectors.

  • Method

    The code generalizes Boltzmann equations to multiple DM components, co-scattering, and unstable dark-sector decays, while adding thermal-mass integration cuts and cosmological-abundance routines.

  • Results

    The new version computes abundances for N-component WIMP and FIMP models, including mixed WIMP–FIMP scenarios, and extends photon spectra to DM masses of 110 MeV with light-meson channels.

  • Takeaways & Limitations

    micrOMEGAs 6 supports combined multi-component DM observables and incorporates co-scattering and thermally corrected freeze-in calculations within a unified code framework.

  • Takeaways & Limitations

    Massless gauge-boson-initiated co-scattering processes are excluded because their thermal corrections are treated as small and negative.

Abstract

from arXiv · show

micrOMEGAs is a numerical code to compute dark matter (DM) observables in generic extensions of the Standard Model of particle physics. We present a new version of micrOMEGAs that includes a generalization of the Boltzmann equations governing the DM cosmic abundance evolution which can be solved to compute the relic density of N-component DM. The direct and indirect detection rates in such scenarios take into account the relative contribution of each component such that constraints on the combined signal of all DM components can be imposed. The co-scattering mechanism for DM production is also included, whereas the routines used to compute the relic density of feebly interacting particles have been improved in order to take into account the effect of thermal masses of t-channel particles. Finally, the tables for the DM self-annihilation - induced photon spectra have been extended down to DM masses of 110 MeV, and they now include annihilation channels into light mesons.

PROGRAM SUMMARY

micrOMEGAs 6 expands the code beyond one- or two-component dark matter to handle multi-component, weakly or feebly interacting candidates. It also adds co-scattering, improves freeze-in calculations with thermal masses, and extends photon-spectrum tables to 110 MeV and light-meson channels.

  • Co-scattering is implemented as an additional dark matter production mechanism.
  • It supports multi-component models containing weakly interacting, feebly interacting, or both types of dark matter particles.
  • Freeze-in relic-density routines account for thermal masses of particles exchanged in t- or u-channel diagrams.
  • Photon-spectrum tables extend to dark matter masses of 110 MeV and include annihilation channels into light mesons.
  • The new version computes relic densities for more than two dark matter components in generic Standard Model extensions.

1 Introduction

The introduction motivates extending single-equation relic-density calculations because dark sectors may contain multiple stable particles, chemically disconnected sectors, or feebly interacting states. micrOMEGAs 6 addresses these cases with generalized Boltzmann equations and broader observable constraints.

  • Multi-component dark matter requires an evolution equation for each component when several stable particles contribute to the cosmic abundance.
  • Separate Boltzmann equations are needed when chemical equilibrium fails within a dark-sector set, including when inelastic scattering affects relic-density determination.
  • Feebly interacting particles cannot reach thermal equilibrium and may be produced through freeze-in decays or bath-particle scattering.
  • The generalized framework includes multiple WIMPs, FIMPs, or mixed WIMP–FIMP models, with conversion-driven freeze-out and decay terms.
  • Users can specify thermal-equilibrium particle sets and check whether chemical equilibrium persists during cosmic evolution.
  • Direct and indirect detection rates include all components and can be rescaled by their relative contributions to the total dark matter density.

2 Multi-component DM

micrOMEGAs 6 introduces a framework for computing relic abundances in multi-component dark matter models by solving coupled evolution equations for chemically equilibrated sectors. The implementation includes decays, co-scattering, mixed WIMP/FIMP scenarios, and validation against existing routines.

  • Framework: The generalized equations cover multiple WIMP components, FIMPs, and mixed WIMP/FIMP models, including decay and co-scattering transitions between sectors.Co-scattering includes inelastic scattering and decay processes that transfer particles between sectors.
  • Thermodynamics: The thermodynamic treatment relates entropy and energy densities to the temperature evolution and Hubble expansion, while noting a low-temperature validity limitation for one relation.The entropy–energy relation is not valid for T ≤1 MeV because photons and neutrinos have different temperatures.
  • Framework: The framework divides particles into dark sectors whose members share chemical equilibrium, assigning an abundance evolution equation to each sector.The formulation assumes kinetic equilibrium with the Standard Model bath and uses entropy density to express the evolution equations.
  • Evolution equations: The code extends relic-density calculations to co-scattering and includes statistical, decay, and annihilation terms in the multi-component evolution framework.For multi-component calculations, spin-statistical distribution effects are generally ignored for computational efficiency, although they can matter for FIMPs.
  • Validation: In the Z5M validation, darkOmega2 and darkOmegaN agree for Y1 and Y2 until ϕ2 decays, after which darkOmegaN transfers its abundance to ϕ1.The post-decay relation is Y1(with decay) = Y1(w/o decay) + 3Y2(w/o decay).
  • Validation: For mixed WIMP/FIMP scenarios, darkOmegaNTR agrees with darkOmegaFI for the FIMP relic abundance until the decay of ϕ2 around x = 10^4.The comparison is made while neglecting statistical distribution effects in the validation setup.

3 The role of decay and co-scattering processes in maintaining chemical equilibrium.

The treatment tests whether decays and co-scattering maintain chemical equilibrium within a dark sector. A sector is retained as one unit only when the interaction rate is sufficiently large relative to the Hubble rate and freeze-out scale.

  • Motivation: Chemical equilibrium within a dark sector can fail when couplings are small, especially when co-scattering drives dark matter formation.The relevant equilibration processes are decays and inelastic scattering involving dark particles and Standard Model states.
  • Method: The method partitions a sector into subsets A and B and evaluates decay and co-scattering terms connecting them.For subset A, the abundance equation is constructed using only decay and co-scattering contributions.
  • Equilibrium criterion: Thermal equilibrium corresponds to ∆≪1, where ∆ measures the deviation from equilibrium.
  • Equilibrium criterion: The chemical-equilibrium condition is ΓA→B/H ≫ Xf.8; checkTE evaluates the minimum ratio over possible sector partitions.If the condition fails, the dark sector must be split into subsectors before calculating the relic density.
  • Low-temperature regime: At low temperatures, ΓA→B/H ≫1 is sufficient because co-annihilation becomes irrelevant when (MA −MB)/T ≫1.After freeze-out, YB is approximately constant in the corresponding evolution equation.

4 Some issues with singularities

The section addresses singularities in t-channel calculations, especially those caused by light or stable exchanged particles. It shows how temperature-dependent masses and transverse-momentum cuts can regulate these effects while preserving gauge invariance, and notes exclusions for problematic massless gauge-boson co-scattering.

  • 4.1 t-channel vector exchange and freeze-in: Light t-channel exchange can produce power or logarithmic high-energy singularities, affecting freeze-in abundances and their reheating-temperature dependence.Spin-1 exchange gives a cross section scaling as 1/M^2, while spin-1/2 exchange scales as log(s/M^2)/s.
  • 4.1 t-channel vector exchange and freeze-in: Thermal masses of exchanged bath particles provide a temperature-dependent cutoff that makes the high-energy cross section decrease with temperature.The effective mass is Meff(T)^2 = M^2 + FM(T), with FM model-dependent.
  • 4.1 t-channel vector exchange and freeze-in: Replacing zero-temperature mediator masses directly with thermal masses can violate gauge invariance and unitarity in gauge-boson exchange processes.The paper uses a temperature-dependent pT cut instead of modifying masses inside generated amplitudes.
  • 4.1 t-channel vector exchange and freeze-in: The leading and subleading singularities depend on the regulators only through the combination 2µ^2 + c, motivating an effective temperature-dependent mass or cut.This identification is verified for more involved spin-1 processes and tested for spin-1/2 exchange.
  • 4.2 t-channel poles and thermally-induced widths: Stable t-channel exchange can create physical-region poles and divergent angular integrals, while a plasma thermal width is expected to regularize them.The example involves ˜µχ → µh with neutralino exchange and also exhibits double-counting with a related annihilation process.
  • 4.3 Infrared problem in co-scattering processes: Massless gauge-boson-initiated co-scattering is excluded because its thermal-width corrections are small and negative within the freeze-out expansion used.The calculation expands in T/Mχ2, with T/M ≈ 1/20 during freeze-out.

5 Other improvements

micrOMEGAs 6 broadens annihilation calculations beyond two-body final states and retains Sommerfeld enhancements for selected interactions. These additions improve treatment near thresholds while using approximations to keep computations practical.

  • 5.1 3-body and 4-body final states: Off-shell 2→3 and 2→4 contributions are added for heavy states beyond W and Z when two-body annihilation is kinematically suppressed.The functions vSigmaPlus23 and vSigmaPlus24 calculate user-defined processes for inclusion in the relic-density annihilation cross section.
  • 5.1 3-body and 4-body final states: The multi-body treatment approximates four-body rates using diagrams with one off-shell particle, with a K-factor available to rescale three-body results near threshold.Users can also compute four-body processes involving two off-shell particles, although these are computationally expensive.
  • 5.1 3-body and 4-body final states: Three-body final states reduce the relic density near threshold and reproduce the four-body-inclusive result within a few percent while being at least three times faster.This comparison was performed in the Inert Doublet Model and a U1 leptoquark model.
  • 5.1 3-body and 4-body final states: Including multi-body final states slightly above threshold shifts the relic density by a few percent for Γ/M = 1% and by 15% for Γ/M ≈10%.The shift depends on the resonance width.
  • 5.2 Sommerfeld enhancement: micrOMEGAs follows the standard prescription for Sommerfeld enhancement from light-boson exchange between non-relativistic incoming particles.The enhancement routine uses relative velocity, mediator mass, reduced mass, and coupling parameters.
  • 5.2 Sommerfeld enhancement: Sommerfeld enhancement for pseudo-scalar and axial-vector interactions is treated as negligible and is not included.The implemented treatment covers scalar and vector interactions.

6 Improved limits on DM

The paper improves experimental and cosmological-limit calculations for multi-component and light-mass dark matter. It adds low-mass photon spectra, indirect-detection channels, and tools for assessing long-lived-particle and structure-formation constraints.

  • 6.1 Direct detection: Direct-detection limits in multi-component dark matter require combining each component’s signal according to its relative relic-density contribution.A recasting procedure for XENON-1T, PICO-60, CRESST-III, and DarkSide-50 is described, with one-component limits generally reproduced within 10%.
  • 6.2 Indirect detection: Photon-spectrum tables now cover dark-matter masses from 110 MeV to 2 GeV using analytical leptonic and hadronic spectra.Previous tables covered 2 GeV to 5 TeV using Pythia 6.
  • 6.2 Indirect detection: The updated spectra include e+e− and µ+µ− channels, radiative muon decays, and improved low-energy electron spectra.For sub-GeV dark matter above the muon threshold, radiative muon decay dominates over FSR before FSR becomes more important at higher masses.
  • 6.2 Indirect detection: Meson-annihilation spectra incorporate photon production from pion and kaon FSR, radiative decays, and subsequent decay chains.The resulting tables are used by CalcSpectrum when pion and kaon states are identified by their PDG codes.
  • 6.3 Cosmological constraints: micrOMEGAs 6 does not compute long-lived-particle cosmological constraints but provides quantities useful for estimating their impact.These include LLP abundances, decay modes, visible hadronic energy injection, and free-streaming lengths.
  • 6.3 Cosmological constraints: An indicative Lyman-α bound is λFS < 0.5 Mpc, especially when a large fraction of dark matter arises through the superWIMP mechanism.Accurate structure-formation constraints generally require dedicated N-body simulations.

7 Description of routines

The routines described here implement sector-based evolution for N-component dark matter, support feebly interacting particles and co-scattering, and expose controls for initialization, evolution, cross sections, and diagnostics.

  • Sector definition and setup: N-component relic-density calculations divide particles into sectors within which chemical equilibrium holds, assigning one abundance equation to each sector.A FIMP is placed in a sector of its own.
  • Sector definition and setup: The defThermalSet function lets users override default sector assignments and accommodate models without particle names beginning with '~'.Particles can be moved between sectors explicitly, with particle-antiparticle assignments handled automatically.
  • Thermal-equilibrium checks: The checkTE routine tests whether decay and co-scattering processes maintain chemical equilibrium in a selected sector.It returns the minimum Γ/H(T) found across tested particle subsets, with modes that isolate decay or co-scattering.
  • Feebly interacting particles: The toFeebleList function assigns particles to the feebly interacting sector, which is excluded from freeze-out relic-density routines.Particles not listed are assumed to be in thermal equilibrium with the Standard Model at high temperatures.
  • Abundance evolution: darkOmegaN and darkOmegaNTR solve the thermal abundance evolution equations and return the total relic density together with sector-level abundances or contributions.The routines evolve over a user-defined temperature interval and store relative sector contributions in fracCDM or fracCDMN.
  • Diagnostics and controls: The routines provide error flags for missing thermal-equilibrium starts, stiff differential equations, invalid integrands, recursion depth, and loss of numerical precision.darkOmegaN returns NAN when specified evolution errors occur.
  • Diagnostics and controls: Users can selectively exclude terms from the N-component evolution equation through the ExcludedForNDM keyword string.Examples include excluding DM decays or 1,1↔0,0 processes.

8 Examples

The example demonstrates a two-component relic-density calculation combining a FIMP and a WIMP, while listing the processes contributing to freeze-in.

  • The sample output calculates relic densities for two dark-matter components: ~x1 as a FIMP and ~~x2 as a WIMP.
  • The calculation compares darkOmegaNTR with the single-component routines darkOmega for the WIMP and darkOmegaFI for the FIMP.
  • The example lists the channels contributing to freeze-in.
  • The model setup includes component masses of 100.000 for ~x1 and 350.000 for ~~x2.
  • Decays of odd-sector particles are included in the Boltzmann equations, with the displayed reactions dominated by ~~x2 → ~x1, ~x1 and W/Z-related processes.
  • The FIMP abundance can be recomputed with Maxwell-Boltzmann statistics by uncommenting NOSTATISTICS, recompiling, and rerunning the code.

9 Conclusion

micrOMEGAs 6 extends the code to N-component dark matter and improves freeze-in, detection, and low-mass photon-spectrum calculations. The release is intended for dark-matter phenomenology beyond the standard WIMP paradigm.

  • micrOMEGAs 6 computes abundances for N-component dark matter containing multiple WIMPs, FIMPs, or combinations of WIMPs and FIMPs.
  • The generalized Boltzmann equations include co-scattering and decays of unstable dark-sector particles.
  • The freeze-in routine accounts for thermal masses of t/u-channel particles through an integration cut shown equivalent to a thermal mass in a Standard Model process.
  • Detection-rate routines combine contributions from all dark-matter components, allowing constraints on their combined signal.
  • 110 MeV is the lower mass limit of the extended annihilation-induced photon-spectrum tables, which add pion and kaon final states.
  • micrOMEGAs 6 is available through Zenodo and can be used for dark-matter phenomenology beyond the standard WIMP paradigm.
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