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REVIEW 2 major objections 8 minor 43 references

An integral-equation Boltzmann solver makes decaying dark matter with massive decay products fast and accurate enough for real parameter fits.

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2026-07-31 06:09 UTC pith:3EMMREXT

load-bearing objection Solid methods paper shipping a usable generic two-body DDM solver; the iteration-seed concern is real but bounded and does not sink the central claim. the 2 major comments →

arxiv 2607.24910 v1 pith:3EMMREXT submitted 2026-07-27 astro-ph.CO

Cosmological evolution with decaying dark matter: an integral-equation approach

Nanoom Lee , Anna Bencke , Marc Kamionkowski , Jos\'e Luis Bernal This is my paper
classification astro-ph.CO
keywords decaying dark mattercosmological perturbationsintegral equationsBoltzmann solvermatter power spectrumCMB lensingnon-cold relicsfree-streaming
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper presents CLASSIER-DDM, a cosmological Boltzmann code that treats dark matter decaying into two lighter particles of any mass by replacing the usual infinite multipole hierarchy with a few integral equations solved iteratively. The method covers both massless dark radiation and warm massive products, needs neither hierarchy truncation nor fluid approximations, and reaches sub-0.1% convergence on the matter and CMB-lensing power spectra with only a modest number of iterations. About a hundred momentum bins already give O(0.1%) accuracy in today's matter power spectrum out to k ~ 3 Mpc^{-1}, and a full evaluation runs in roughly a minute. That speed and accuracy turn previously expensive DDM forecasts into routine likelihood analyses. The same framework also shows how free-streaming decay products suppress structure growth beyond simple mass removal and leave characteristic oscillatory imprints.

Core claim

CLASSIER-DDM evaluates generic two-body decaying-dark-matter perturbations via iteratively solved integral equations, achieving sub-0.1% convergence in Pm(k) and the CMB lensing spectrum across the observationally relevant parameter space, O(0.1%) accuracy up to k ~ 3 Mpc^{-1} with ~100 momentum bins, and O(1 min) runtimes without a Boltzmann hierarchy or fluid approximation, thereby making DDM parameter estimation numerically tractable.

What carries the argument

The integral-equation solution of the collisionless Boltzmann equation for continuously produced decay products: multipole moments are written as time integrals of metric sources convolved with spherical-Bessel kernels (plus an analytic piece sourced at each particle's birth time), evaluated by NUFFT and closed by a few outer iterations on the metric.

Load-bearing premise

The first guess for the total dark-sector density contrast ignores free-streaming of the decay products, so the claimed fast convergence can fail in the joint large-fraction, fast-decay corner unless that seed is improved.

What would settle it

Run the same DDM model (two massive products, f ~ 0.5–1, Gamma ~ 0.01 Mpc^{-1}) with CLASSIER-DDM at N_iter = 2–3 and with a high-ell truncated hierarchy or an independent Boltzmann code; if the matter-power difference exceeds ~0.1% up to k = 3 Mpc^{-1}, the convergence claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • DDM models with two massive decay products can now be included in standard MCMC cosmological analyses at roughly minute-per-evaluation cost.
  • Suppression of Pm(k) and C_L^phi phi is set by free-streaming length (Gamma and kick velocity) and amplitude f, and exceeds the naive 1-f mass-removal factor because free-streaming products weaken gravitational potentials.
  • Oscillatory interference among differently aged momentum bins is a distinctive DDM signature that can be searched for on small scales.
  • The same integral-equation machinery extends immediately to other non-cold relics with non-trivial phase-space distributions.
  • Small-scale probes (Lyman-alpha, weak and strong lensing) become practical for tightening DDM bounds once the linear power spectrum is this accurate and cheap.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the code already recovers the pure dark-radiation limit, existing massless-decay constraints can be re-run with identical systematics control and then smoothly extended into the warm-product plane.
  • The deferred free-streaming seed suggested by the authors would most improve the extreme (f, Gamma) corner that is already disfavored by data, so current observational forecasts are likely robust even without it.
  • Sub-galactic free-streaming DDM models aimed at small-scale structure anomalies can now be evolved linearly to the precision needed before feeding into N-body or semi-analytic halo pipelines.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

Summary. The manuscript presents CLASSIER-DDM, an extension of the integral-equation Boltzmann solver CLASSIER to generic two-body decays of dark matter into two lighter products of arbitrary mass. The authors derive the background phase-space distribution sourced continuously by decays (Sec. II), the synchronous-gauge integral solutions for the ℓ=0,1,2 perturbation multipoles, split into a decay-time-sourced piece and a continuously-sourced convolution (Sec. III, Eqs. 11–14), and describe the numerical implementation: a Picard-like iteration seeded by the rest-mass ansatz of Eq. (17), NUFFT-based convolution with analytic endpoint subtraction, a small-ξ direct convolution (App. B), and an empirical small-scale quasi-static approximation (Sec. IV B 2, Eq. 37). Convergence in the matter and CMB lensing power spectra is mapped over (f, Γ, v_kick) in Fig. 1; runtimes are O(1 min) per evaluation (Table I). Cross-checks include the massless (dark-radiation) limit against CLASS, the v_kick→0 CDM limit, and an external comparison against the code of Ref. [17] without its fluid approximation (App. C).

Significance. If the accuracy and convergence claims hold, this is a useful and timely methods contribution. DDM perturbation evolution has so far required truncated Boltzmann hierarchies, and the only fast public implementation (Ref. [17]) relies on a fluid approximation that App. C shows induces percent-level errors and misses oscillatory features in P_m(k). CLASSIER-DDM removes both limitations, handles the previously untreated case of two massive decay products, is publicly released, and delivers O(1 min) evaluations — the difference between DDM parameter estimation being prohibitive and routine for CMB-S4/DESI/LSST-era analyses; the companion constraints paper [27] already demonstrates this use. The validation strategy is creditable for a methods paper: internal convergence in iterations and momentum bins, two analytic limiting-case checks, and one genuine external-code comparison. The oscillatory interference structure in δ_χ1 (Fig. 2) and its 1/(k v_kick) period scaling is a nice physical byproduct.

major comments (2)
  1. [Sec. V A, Fig. 1] The convergence metric in Fig. 1 is max|ΔX| relative to a reference computed at N_iter=10, i.e., it certifies agreement with the code's own 10th iterate, not with the fixed point of the iteration. This matters precisely where the Eq. (17) seed is worst: at large f and large Γ, where the DDM sector sources a large fraction of the metric perturbations and the seed's omission of free-streaming damping, velocity, and anisotropic-stress contributions can push the contraction factor toward unity. The text acknowledges slow convergence in this corner and defers a damped seed e^{-(kλ_fs)^2} to future work, and argues the corner is observationally disfavored — all fair. But Fig. 1 as constructed cannot distinguish 'converged' from 'converging slowly and still >0.1% off at N_iter=10', and likelihood chains can sample that corner before excluding it. A modest addition would close this: for a few po
  2. [Secs. V, VII, App. C] The accuracy claims for the generic (two massive products) case rest almost entirely on internal consistency tests; the external validation is confined to limiting cases: the massless limit against CLASS dark radiation (Sec. VII), v_kick→0 against ΛCDM (Fig. 3), and a single parameter point (f=1, Γ=H_0, ε_1=0.99, ε_2=0) against Ref. [17]'s hierarchy without the fluid approximation (App. C, Fig. 6). The App. C point is itself near a corner (one product massless, small Γ). Since the paper's central claim is a general-purpose solver, at least one external or independent check at intermediate ε (both products massive, semi-relativistic v_kick ~ 0.01–0.1c) — e.g., against a brute-force high-ℓ_max hierarchy run, even at a single parameter point and over a restricted k range — would substantially strengthen the validation. If no independent code can currently do this comparison, that should be
minor comments (8)
  1. [Sec. III] End of Sec. III: the statement that 'contributions from all higher multipoles are implicitly captured as the full perturbation system reaches self-consistency through the iterative scheme' is correct in spirit but cryptic; one sentence explaining that the ℓ≤2 moments are exact integrals of the full ∆f (which the integral solution determines exactly, mode by mode) would preempt confusion about implicit hierarchy truncation.
  2. [App. C, Fig. 6] Fig. 6 caption and legend say 'Ref.[18]' (twice) but the comparison is with Ref. [17]; please fix the citation labels.
  3. [Secs. I, V B, Table I] Typos: 'maxium multipole' (Sec. I, first paragraph); 'direcy way' (Sec. V B); 'need only to be computed once' (Table I caption) should read 'need only be computed once'.
  4. [Sec. IV B, Eq. (17) footnote] Footnote after Eq. (17) writes ρ_χi−3P_χi = ∫ d³q/(2π)³ f_0 m_i²/ε_i, whereas Eq. (6) defines ρ and P with a 4π∫q²dq T_0 normalization and no (2π)³; please reconcile the normalization conventions so the rest-mass identity is dimensionally and conventionally transparent.
  5. [Sec. V B, Table I] Table I reports runtimes only for N_iter ≤ 3, but Sec. V A says more iterations are needed for f≳0.5 or Γ≳10⁻² Mpc⁻¹; adding the N_iter rows actually used in those regimes (or stating the linear-scaling extrapolation is exact) would make the runtime claim directly comparable to the convergence map.
  6. [Sec. IV B 2, Eq. (37)] The empirical small-scale criterion (Eq. 37, threshold 0.025) is stated to yield ≲0.1% accuracy, but no supporting figure or table is given; a brief quantification (e.g., the error with the approximation on vs. off for one case) would help users calibrate trust in this default.
  7. [Abstract / Sec. IV B] Sec. IV B notes massive neutrinos are evolved with CLASS's fluid approximation while the abstract says the code requires 'no fluid approximation'; the scoping is clear in the text but the abstract could say 'for the DDM decay products' to avoid overclaiming.
  8. [Sec. VI A, Fig. 2] Fig. 2 uses N_q_ddm = 2000 with smoothing for presentation while the code default is 100; stating the default-100 appearance of the same curves (or noting that the oscillations remain resolved) would reassure readers that the physics shown is not an artifact of the presentation settings.

Circularity Check

0 steps flagged

Methods/validation paper: DDM integral equations and accuracy claims are self-contained; no prediction reduces to its inputs by construction.

full rationale

CLASSIER-DDM derives the homogeneous DDM evolution (Sec. II) and the multipole integral solutions (Sec. III, Eqs. 7–15) from the collisionless Boltzmann equation with a continuous production source; those steps are standard and not defined in terms of the claimed Pm(k)/Cφφ outcomes. Numerical claims (sub-0.1% iteration convergence, ~100 q-bins, O(1 min) runtime) are empirical code measurements against higher-Niter references, the massless CLASS limit, the vkick→0 CDM limit, and an external hierarchy comparison without fluid approximation (Appendix C vs Ref. [17]). Self-citations to CLASSIER and the integral-equation series [23–25] supply the prior numerical framework being extended; they do not force the DDM power-suppression results or uniqueness of the model. The 0th-iteration seed (Eq. 17) and Niter=10 reference are validation-design choices that can leave residual absolute error in the large-f/large-Γ corner, but that is a correctness/scope issue, not circularity: nothing is fitted to data and then re-presented as a prediction, and no load-bearing uniqueness theorem is imported from the authors. Companion constraints [27] are separate. No circular step meets the quote-and-reduce standard.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 1 invented entities

Load-bearing content is standard linear cosmological perturbation theory in synchronous gauge plus the authors' prior integral-equation reformulation, applied to a cold parent with two-body decays. Free numerical knobs (bin counts, iteration count, empirical small-scale threshold) control claimed accuracy but are validated rather than fit to cosmological data. No new particle beyond the existing DDM scenario is postulated.

free parameters (5)
  • N_iter (iteration count) = typically 2–3 (reference uses 10)
    Number of self-consistency iterations for decay-product perturbations; chosen to reach sub-0.1% convergence and shown to need only 2–3 in the observationally relevant region (Fig. 1).
  • Nq_ddm (perturbation momentum bins) = 100 (default); 200 used in some convergence tests
    Discretization of decay-product phase space for integral equations; default 100 claimed sufficient for O(0.1%) Pm accuracy to k~3 Mpc^{-1}.
  • Small-scale approximation threshold 0.025 = 0.025
    Empirical cut s(k,q,τ0)(Γ/H0)<0.025 deciding when the quasi-static analytic multipoles replace the full convolution (Eq. 37); set by hand for ≲0.1% accuracy.
  • N(ξ) NUFFT quadrature count = 1000 (k≤0.1 Mpc^{-1}) / 5000 (otherwise)
    k-dependent number of Gauss–Legendre points for ΔG Fourier coefficients (1000 or 5000); tuned for kmax~3 Mpc^{-1}.
  • Background q-grid (Nq_ddm_bg, qmin, qmax) = 1000 bins
    1000 lnq-spaced background bins between 1e-4(p/T0) and min(10 a_decay,1)(p/T0), plus missing-bin correction; numerical resolution choice.
axioms (6)
  • domain assumption Linear cosmological perturbation theory in the synchronous gauge with standard Einstein–Boltzmann coupling for metric sources h, η.
    Used throughout Secs. III–IV; standard CLASS/CLASSIER setting.
  • domain assumption Parent dark matter χ is cold before decay; comoving number decays as exp(−Γt) with energy density ρχ=ρ0_χ a^{-3} e^{-Γt}.
    Sec. II A, Eqs. (1)–(2); defines the DDM background.
  • domain assumption Decays are two-body to fixed masses mχ1, mχ2 with back-to-back momentum from two-body kinematics (Eq. 3); collisionless free-streaming thereafter.
    Sec. II B; multi-body generalization only sketched in Appendix A.
  • standard math Integral-equation solutions along unperturbed geodesics with spherical-Bessel kernels are equivalent to the collisionless Boltzmann equation for the multipoles that enter the stress-energy tensor.
    Secs. III and IV B; inherits justification from Refs. [23–25].
  • ad hoc to paper 0th-iteration DDM density seed uses rest-mass combinations ρχi−3Pχi times δc without free-streaming suppression (Eq. 17).
    Explicit starting approximation in Sec. IV B; authors note a damping factor could improve it but leave that to future work.
  • ad hoc to paper Focus on decays after recombination (zd≲1100) so the convolution needs no super-/near-horizon split used for neutrinos in CLASSIER.
    Stated in Sec. IV B 1; shapes the numerical path though not the formal equations.
invented entities (1)
  • CLASSIER-DDM solver module independent evidence
    purpose: Public numerical implementation that evaluates DDM background and iterative integral-equation perturbations inside CLASSIER/CLASS.
    The scientific object delivered by the paper; not a new physical particle or force, but the artifact on which accuracy and runtime claims rest.

pith-pipeline@v1.2.0-grok45-kimik3 · 24642 in / 3770 out tokens · 71899 ms · 2026-07-31T06:09:40.927410+00:00 · methodology

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read the original abstract

We present CLASSIER-DDM, an extension of the Boltzmann solver CLASSIER that implements the decaying dark matter (DDM) model via its integral-equation approach. The code handles generic two-body decays of a dark matter particle into two lighter decay products with arbitrary masses, naturally encompassing both massless (dark radiation) and massive (warm) decay products, with their perturbations evaluated via integral equations solved iteratively. We describe the numerical implementation in detail, including the background evolution, the iterative perturbation evolution, and a small-scale analytic approximation. A modest number of iterations is sufficient to achieve sub-$0.1\%$ convergence in the matter power spectrum and the CMB lensing power spectrum across the observationally relevant parameter space. We find that a hundred momentum bins for the decay product perturbations are sufficient to achieve $\mathcal{O}(0.1\%)$ accuracy in the matter power spectrum today up to $k \sim 3\,{\rm Mpc}^{-1}$. The code achieves $\mathcal{O}(1\,{\rm min})$ runtimes per evaluation, requiring neither a Boltzmann hierarchy nor any fluid approximation, making parameter estimation with the DDM model numerically tractable. We also discuss the impact of DDM on cosmological observables, focusing on the case of two massive decay products. This work further establishes the integral-equation approach as a versatile and efficient framework for modeling non-cold relics in cosmological perturbation theory.

Figures

Figures reproduced from arXiv: 2607.24910 by Anna Bencke, Jos\'e Luis Bernal, Marc Kamionkowski, Nanoom Lee.

Figure 1
Figure 1. Figure 1: FIG. 1. Convergence of the matter power spectrum today [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Evolution of perturbations in the DDM model for [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Decay product density perturbation [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Fractional deviations from ΛCDM in the matter [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Fractional deviation of the matter power spectrum [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

discussion (0)

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Reference graph

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