Conversions in two-component dark sectors: a phase space level analysis

Andrzej Hryczuk; Shiuli Chatterjee

REVIEW 3 major objections 5 minor 1 cited by

Conversions in two-component dark sectors: a phase space level analysis

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper shows that in a two-component dark sector with conversions, the standard kinetic-equilibrium calculation of relic abundances is off by more than 100% in some regions and 20–50% in the GCE-preferred region, with per-constituent…

desk verdict A serious step forward for fBE relic calculations in multi-component dark sectors; the FP-channel disclosure gap is fixable and shouldn't block a solid referee. read the letter →

arxiv 2502.08725 v1 pith:XLPNTCD3 submitted 2025-02-12 hep-ph astro-ph.CO

Shiuli Chatterjee , Andrzej Hryczuk This is my paper · ORCID

classification hep-phastro-ph.CO PACS 95.35.+d

keywords two-componentdarkmatterkineticequilibriumfullBoltzmannequationphase-spacedistributionconversionprocessesrelicabundanceGalacticCentreexcessFokker-Planckapproximation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks whether the standard assumption of kinetic equilibrium distorts relic abundance predictions in a two-component dark sector with conversion processes. It solves the full Boltzmann equation for the phase-space distributions of two pseudoscalar-coupled fermions in a minimal extension of the Coy dark matter model, including annihilations, elastic scatterings, and 2-to-2 conversions. In the parameter regions that reproduce the observed relic abundance and fit the Galactic Centre excess, the standard number-density treatment is off by more than 100% in some places and typically by −20% to +50%; the abundance of each constituent separately can change by up to an order of magnitude. This matters because the predicted gamma-ray flux from the Galactic Centre scales with the square of the per-constituent density, so the preferred fit region shifts when the phase-space treatment is used.

What carries the argument

The load-bearing object is the discretised, coupled full Boltzmann equation for the two phase-space densities $f_{\chi_1}(q,x)$ and $f_{\chi_2}(q,x)$ (the paper's eq. 40). Annihilations enter as a pre-tabulated matrix term $f_{\mathrm{eq}}(A_M f_{\mathrm{eq}}) - f(A_M f)$; elastic scatterings enter either through a Fokker-Planck operator or, where that approximation is unreliable, through the full $t$-dependent collision term written as a matrix $E_M$; conversions enter through a three-index matrix $\hat A$ that couples the two species and is evaluated on the fly because it depends on the unknown distributions. The conversion collision term is the structurally new piece: it is cubic in $f$, involves momenta of both particles, and requires interpolation when a final-state energy falls off the grid.

What would settle it

Recompute the benchmark and GCE-region points using the full, unexpanded elastic-scattering collision term of eq. (20) for $\chi_i + s \to \chi_i + s$ instead of the Fokker-Planck form, and see whether the >100% and 20–50% fBE-versus-nBE deviations persist; a cheaper check would directly compare the Fokker-Planck operator with the full operator at $M_s = 75$–$84$ GeV, $M_\chi = 30$–$44$ GeV and freeze-out temperatures, along the lines of the paper's appendix.

Watch

Extended reading notes

Core claim

The central discovery is that conversions between the two dark matter states do not merely change particle numbers; they imprint a non-thermal shape on the momentum distributions, and that shape feeds back into the rates. The benchmark shows a second bump in the $\chi_2$ distribution generated by $\chi_1\to\chi_2$ conversion, lying at momenta that additionally fuel resonant annihilations, so the final total abundance is lower than the standard treatment predicts (for the benchmark, $\Omega h^2 = 0.126$ with the standard calculation versus $0.097$ at the phase-space level). Across the studied parameter space the fBE-vs-nBE deviation in total abundance exceeds 100% in some regions and is typically −20% to 50% where the model fits the GCE, with per-component deviations reaching an order of magnitude. The preferred GCE-fit region moves accordingly: for the benchmark at conversion strength $c\simeq 1$ the reduced $\chi^2$ degrades from 0.51 (nBE) to 62 (fBE).

Load-bearing premise

The calculation's headline numbers rest on the Fokker-Planck approximation being accurate for the elastic-scattering processes that set when kinetic decoupling happens; the paper's appendix shows this approximation can fail when the scattering partner is as heavy as the dark matter (as the pseudoscalar $s$ is in the GCE-favoured region), so if those rates are misestimated, the size of the reported abundance shifts could change.

Editorial extensions

If this is right

In the studied model, standard number-density relic calculations are not reliable at the percent level: deviations reach more than 100% in some resonant and sub-threshold regions and 20–50% in the GCE-preferred region.
The abundance of the subdominant constituent can change by up to an order of magnitude, so observables proportional to the square of the density, such as gamma-ray fluxes from annihilations, can be affected far more than the total abundance.
The GCE-preferred parameter region shifts when phase-space abundances are used; the benchmark at conversion strength $c\simeq 1$ goes from a good fit (reduced $\chi^2=0.51$) to a bad one (reduced $\chi^2=62$).
Conversion-driven freeze-out can make the final abundance nearly independent of the annihilation parameter $a$, a regime that only the phase-space analysis fully resolves.
The numerical framework developed here can be applied to other two-component models with conversions, removing the kinetic-equilibrium assumption in future relic and indirect-detection predictions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the 20–50% bias is representative, global fits of two-component dark matter that assume kinetic equilibrium may systematically mis-infer couplings by tens of percent; re-running such fits with the phase-space solver could shift best-fit regions beyond current observational precision.
The mechanism identified, conversions feeding a resonant or sub-threshold annihilation channel at specific momenta, should be generic: any model with a conversion amplitude peaking in $s$ and an annihilation resonance at a different momentum is a candidate for similarly large fBE effects, and the same solver could test that directly.
The Fokker-Planck limitation documented for pseudoscalar scattering suggests that the magnitude of the reported deviations may depend on the elastic-scattering treatment; switching to full collision terms for the $s$-mediated scattering in the GCE region would quantify this.
For indirect detection, the factor-change in per-constituent abundances means that gamma-ray constraints from dwarfs or the Galactic Centre should be recomputed with fBE-inferred densities before drawing conclusions about the model's viability.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [colleague],

Quick take: this is a legitimate step forward, not a headline grab. The authors extend DRAKE to solve the full phase-space Boltzmann equation for two dark-sector states including 2↔2 conversions, and apply it to a double-Coy model. That capability is new—earlier fBE two-component studies either omitted conversions or used moment approximations. The derivation of the discretized collision terms is careful and the FP approximation is stress-tested in Appendix A, which is more than most papers do. The concrete results—deviations between fBE and nBE of order tens of percent in most of the GCE-preferred region, with larger per-component effects and a dramatic reversal of the GCE fit at the benchmark point—are plausible and internally consistent.

The soft spots are real but not fatal. First, the paper uses the Fokker-Planck form for 'remaining' elastic scattering channels without naming them or reporting their fractional contribution to the total momentum-transfer rate. Appendix A shows FP can fail badly when the scattering partner is heavier than the DM or the amplitude is t-dependent—conditions that apply to scattering off the pseudoscalar s. In practice those channels likely have small γX and thus contribute little, but we shouldn't have to infer that; the authors should show it. If a full-collision-term rerun for one benchmark shifts the headline numbers, the paper needs a revised figure. Second, the 'more than 100%' total deviation and 'order of magnitude' per-component effects are asserted in the abstract but never pinned to a single displayed point with the parameters and ratio spelled out. Given the few-percent numerical accuracy and ~100-point scans, the headline should be tied to a concrete example. The neglected self-scattering is a known limitation and probably subleading, but again the justification is a statement, not a quantitative check.

None of this undermines the central point—that conversions plus early kinetic decoupling can make nBE quantitatively wrong in this class of models. The GCE-fit reversal may be the most important phenomenological result, and it deserves independent confirmation once the code is public.

Who should read it: anyone computing relic densities in multi-component dark sectors, especially with pseudoscalar mediators. I'd bring it to our reading group and I'd cite it once the code is out.

Recommendation: send it to a serious referee. Condition acceptance on the authors reporting which channels use FP and their γ-fractions, and on a demonstration that the one headline benchmark is robust to using the full elastic collision term.

Referee Report

3 major / 5 minor

Summary. This paper extends the DRAKE framework to solve the full phase-space Boltzmann equations (fBE) for a two-component dark sector consisting of two stable fermions and a pseudoscalar mediator, including annihilation, elastic scattering, and 2-to-2 conversion processes. The model is the "double Coy DM" extension of the Coy DM model, and the authors scan parts of its parameter space that reproduce the observed relic abundance and fit the Galactic Centre excess. They compare fBE results with the standard number-density approach (nBE) and report that departures from kinetic equilibrium change the total relic abundance by roughly -20% to +50% in most of the interesting region, by more than 100% in some regions, and can change individual component abundances by up to an order of magnitude. They further show that these effects can shift the preferred GCE-fit region in the model's parameter space.

Significance. If the quantitative results hold, the paper provides a useful demonstration that the kinetic-equilibrium assumption can fail non-negligibly in multi-component dark sectors with conversion processes, and it offers a numerical framework for future two-component phase-space studies. The main strengths are the detailed derivation of the discretized collision terms for annihilations, elastic scatterings, and conversions; the explicit use of the full t-channel elastic collision term for the channels where the Fokker-Planck approximation is not justified; and the clear identification of conversion-driven freeze-out regimes. The paper is also candid about the limitations of the Fokker-Planck approximation in Appendix A. The principal caveat is that the magnitude of the headline deviations depends on how the subdominant elastic channels are treated and on the neglect of self-scattering, both of which need to be quantified more directly before the central numbers can be taken at face value.

major comments (3)

[§4.2 and Appendix A] The paper does not specify which elastic-scattering partners are treated with the full collision term, eq. (23), and which are treated with the Fokker-Planck approximation, eqs. (17)-(18), nor does it report the fractional contribution of the FP-treated channels to the total momentum-transfer rate. This matters because Appendix A itself states that the FP expansion fails when the scattering-partner mass is comparable to or larger than the DM mass and when the amplitude depends strongly on t; both conditions hold for scattering off the pseudoscalar s in the mass scans (M_s = 75-84 GeV versus M_chi = 30-44 GeV, with amplitudes t^2/(t-M_s^2)^2 in eq. (2)). If s is among the "remaining particles" treated with FP, the kinetic-decoupling epoch that drives the reported fBE-nBE deviations could shift. A channel with small gamma_X can still transfer significant momentum per event, so selection by the largest gamma_X does not by itself guarantee control of the error. Please report the channel decomposition and either evaluate the full collision term for all channels in the benchmark scans or quantify the resulting error on the headline deviation numbers.
[§4.3, last paragraph] Self-scattering of chi1 and chi2 is neglected with only the statement that its rate is smaller and that it may give sub-leading corrections. Since self-scattering directly contributes to kinetic equilibration, and the paper's central claims concern the size of kinetic-equilibrium violations, this neglect is load-bearing. A quantitative estimate of the self-scattering rate relative to the included elastic and conversion rates should be provided at least for the benchmark point and the scan endpoints; otherwise the reported deviations should be regarded as upper bounds whose reduction by self-scattering is unknown.
[§5.1 and §5.4] The GCE-fit regions in figs. 2 and 3 are computed using nBE relic abundances, while the fBE-based GCE fits are shown only in the one-dimensional scans of figs. 4 and 6. Because the central message is that the fBE treatment shifts the GCE-preferred region, the two-dimensional comparison in figs. 2 and 3 does not directly show the fBE-preferred GCE region. The claim of a significantly shifted GCE region would be strengthened by recomputing the GCE fit with fBE abundances over at least one two-dimensional plane, or by explicitly stating that the shift is demonstrated only along the scanned lines.

minor comments (5)

[§4.3, eq. (38)] In the second branch of the definition of Â_i, the block should presumably use (Â_χ2)_{i-N} rather than (Â_χ1)_{i-N}; as written, the χ2 evolution appears to use the χ1 kernel.
[§5.1, fig. 2 caption] The right-panel caption begins "Right panel:." with an extra period and no description; the sentence should be completed.
[§4.2, eq. (26)] The notation "1 (f_eq)^{-1}" and "1 (f_eq)^{-1}" is ambiguous; please define these as diagonal matrices diag(1/f_eq) or use an explicit diagonal-matrix symbol.
[§4.4] The adaptive grid size N is stated to vary between 40 and 100, but no convergence test with respect to N or the q-grid mapping parameters q_A,i and q_B,i is shown. A short convergence study for at least one benchmark would support the claimed few-percent numerical accuracy.
[§4, opening paragraph] The assumption that the pseudoscalar s remains in kinetic equilibrium with the SM bath is stated without a quantitative estimate. Since s is a scattering partner for the DM species, a brief estimate of its relevant scattering or decay rate would make the assumption easier to assess.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: fBE and nBE are computed from the same Lagrangian by independent solution methods; relic-density and GCE constraints are external, not fitted inputs.

full rationale

The paper's central comparison is between the full phase-space Boltzmann solution (fBE, eq. 40) and the standard integrated number-density Boltzmann equation (nBE). The fBE is derived from first principles in Sec. 4: eq. (3) is the Boltzmann equation, eqs. (11)-(14) give the annihilation collision term, eqs. (20)-(28) give the elastic-scattering term (with the full t-channel kernel used for the dominant scattering partners and the Fokker-Planck operator, eq. 17, only for the remaining ones), and eqs. (29)-(39) give the conversion term. The nBE is not used to set any parameter in the fBE; the observed relic abundance is imposed as a contour constraint (black lines) after the fact, and the GCE fit uses the external observed flux from ref. [46]. Thus the quoted deviations (e.g., >100% in total abundance, up to an order of magnitude per component) are genuine outputs of solving the coupled equations, not quantities that reduce to the inputs by construction. The self-citation to DRAKE [7] is methodological: the code is public and its single-component fBE results were independently validated in [7], and the present paper re-derives the two-component collision-term structure in Sec. 4 rather than importing it as an unexamined black box. The FP approximation for subdominant elastic channels is a numerical-accuracy caveat, explicitly flagged in Appendix A, not a circular step: even if the approximation were biased, the bias would be an error in the reported numbers, not a logical equivalence between the reported prediction and an input. The duplicated sentence in the Conclusions noting the difficulty of predicting the error size is an honest limitation and does not indicate circularity. No equation in the paper is defined in terms of the quantity it is used to predict, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 7 free parameters · 7 assumptions · 1 invented entities

The model has six free parameters (two masses, one mediator mass, three couplings) that are scanned and chosen in part to match the observed relic abundance and GCE fit. The numerical method also introduces grid-mapping parameters. Key physical assumptions include Maxwell-Boltzmann statistics, kinetic equilibrium of the mediator, CP invariance, the Fokker-Planck approximation for some elastic scatterings, and neglect of self-scattering. A second dark fermion is introduced without independent evidence.

free parameters (7)

M_chi1 (heavier DM mass) = scanned; examples 31-44 GeV
Dark fermion mass parameter; scanned in Sections 5.1-5.4, with black contours reproducing the observed relic abundance.
M_chi2 (lighter DM mass) = scanned; examples 30.6-38 GeV
Dark fermion mass parameter; scanned together with M_chi1 and M_s to cover kinematic possibilities.
M_s (pseudoscalar mass) = scanned; examples 75-84 GeV
Mediator mass parameter; scanned to produce resonant annihilations and conversions.
lambda_chi1 = scanned; examples 0.0226-1.82
Coupling of chi1 to the pseudoscalar; chosen by hand in scans and constrained by relic density and GCE fits.
lambda_chi2 = scanned; examples 0.0026-0.39
Coupling of chi2 to the pseudoscalar; scanned similarly.
lambda_y = scanned; examples 0.22-3.86
Pseudoscalar coupling to SM fermions (MFV ansatz); scanned as part of the 3-parameter coupling space.
Grid mapping parameters q_A,i and q_B,i = varies per x-interval
Numerical mapping parameters in eq. (4), chosen by hand to cover the relevant momentum ranges for each particle; not physically observable.

assumptions (7)

standard math Maxwell-Boltzmann statistics for DM particles (f_eq = exp(-E/T))
Invoked in Section 4.1; valid in the non-relativistic freeze-out regime.
domain assumption Mediator s remains in kinetic equilibrium with the SM plasma
Assumed at the start of Section 4; justified by the non-suppressed elastic scattering rate of s.
domain assumption CP invariance and neglect of Bose enhancement and Pauli blocking for final states in annihilation collision terms
Used in Section 4.1 following standard treatments; valid for massive final states with f << 1.
domain assumption Fokker-Planck expansion is valid for elastic scattering where used
Sections 4.2 and Appendix A; the expansion is used only for processes not assigned the full elastic collision term, but its validity is checked only indirectly.
domain assumption Self-scattering processes chi_i chi_i -> chi_i chi_i are negligible
Section 4.3 states self-scattering rates have a smaller impact than included processes for most of the parameter space, but this is not demonstrated in detail and could affect kinetic equilibrium.
domain assumption Minimal Flavour Violation ansatz for pseudoscalar couplings to SM fermions
Section 2, model-building choice that fixes the SM coupling structure.
domain assumption Standard astrophysical inputs for the GCE fit: NFW profile, local DM density 0.3 GeV/cm^3, distance 8.33 kpc, 40x40 degree window
Section 5.1; these are standard but not model-independent inputs that affect the GCE fit regions.

invented entities (1)

Second dark fermion chi_2
purpose: Extension of the Coy DM model to a two-component dark sector, enabling conversion processes chi1 chi1 <-> chi2 chi2 that drive the studied phase-space effects.
No independent detection or external evidence; it is a model hypothesis introduced in Section 2. The paper does not provide a falsifiable prediction that would uniquely confirm chi_2 beyond the model's combined fit to relic abundance and GCE.

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Cite this review

Pith. "Pith review of Conversions in two-component dark sectors: a phase space level analysis." pith.science (2026). https://pith.science/paper/XLPNTCD3

@misc{pith2026250208725,
  author       = {Pith},
  title        = {Pith review of: Conversions in two-component dark sectors: a phase space level analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLPNTCD3}},
  note         = {Machine review of arXiv:2502.08725}
}

abstract

Conversions between the states in the dark sector affect not only their number densities but also their momentum distributions. In this work we study a phenomenologically motivated two-component dark matter scenario, based on the Coy Dark Matter model, in order to quantify the effect of conversions on departure from kinetic equilibrium and consequently the relic abundance. We perform a detailed numerical analysis at the level of the phase space distributions of dark sector particles, implementing all the relevant processes, including conversions, elastic scatterings and annihilations. Focusing on the parameter regions that lead to the observed relic abundance and provide a good fit to the Galactic Centre excess, we find that departure from kinetic equilibrium can alter the predictions for the total abundance by more than $100\%$, while in most of the interesting parameter space being in the range from around $-20\%$ to $50\%$. The effect on each dark matter constituent separately can be much larger, even up to an order of magnitude, which can significantly affect the expected present-day gamma ray flux, and consequently phenomenology of the model.

Figures

Figures reproduced from arXiv: 2502.08725 by the authors.

**Figure 2.** Left panel: Results for an illustrative scan over masses Mχ1 and Ms, keeping fixed Mχ2 = 38 GeV. The solid and dashed black lines are contours of constant relic abundance reproducing the observed value, from the standard (nBE) and phase-space level (fBE) calculations, respectively. The teal blue shaded contours indicate the region of parameter space most preferred for a DM explanation to the Galactic Centre excess,… view at source ↗

**Figure 3.** Results for scans over the conversion strength [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Results for a scan over annihilation parameter [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: Evolution of χ1 and χ2 for the benchmark point. Left panel: Yields of DM particles, Yχi ≡ nχi /s where nχi ≡ gχi /(2π) 3 R d 3 ⃗pfχi (p, x) is the number density and s is the entropy density. Right panel: Effective temperature of DM particles Ti ≡ gχi ni(x) R d 3⃗p (2π…

**Figure 6.** Figure 6: Results for a scan over conversion strength [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

**Figure 7.** Figure 7: The differential photon flux for the excess from Galactic Centre, over a 40 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

**Figure 8.** Figure 8: The ratio between the amplitude of the elastic scattering collision term and the one [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Reviving $Z^\prime$ Portal Dark Matter with Conversion Mechanism
hep-ph 2025-12 conditional novelty 4.0 of 10

In a U(1)_{B-L} Z' portal model with two nearly degenerate dark fermions, the conversion mechanism can produce the observed dark matter relic density while evading current collider and direct-detection constraints.

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