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This paper argues that the effect of decaying dark matter on halo abundances can be captured by replacing the constant spherical-collapse barrier with a mass-dependent critical overdensity, and provides a closed-form fit whose key mass scal

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2026-08-01 13:00 UTC pith:HTBCVQR5

load-bearing objection A useful, honestly-written semi-analytic DDM halo mass function with a clean two-plateau δ_c(M), but the population-2 gravitating-mass interpolation is the load-bearing spot and validation is weakest exactly where the signal is largest. the 3 major comments →

arxiv 2607.19244 v1 pith:HTBCVQR5 submitted 2026-07-21 astro-ph.CO

Decaying Dark Matter Halo Abundance from a Revised Spherical Collapse Model

Thomas Montandon , Vivian Poulin , Oliver Hahn , Jozef Bucko , Aurel Schneider This is my paper
classification astro-ph.CO
keywords decaying dark matterhalo mass functionspherical collapsecritical densityvelocity kickpress-schechtercluster abundanceN-body validation
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.

The paper tries to establish a fast, physically motivated way to predict how many dark matter halos form when dark matter decays: instead of feeding the suppressed matter power spectrum into the standard halo-abundance machinery, it keeps the standard power spectrum and shifts the collapse threshold itself. The central claim is that this shift has two plateaus — a large-mass limit where daughter particles are retained and the threshold stays near 1.686, and a small-mass limit where all daughters escape and collapse is equivalent to decay into pure radiation — joined by a transition whose characteristic mass is the scale where the kick velocity equals the halo orbital velocity. If correct, this turns a costly per-parameter N-body task into a closed-form calculation, so cluster number counts can constrain the decay lifetime and kick velocity in parameter estimation. The authors validate the prediction against four N-body runs at two redshifts, finding agreement except for the largest kicks at z=0, and trace that residual to the halo-mass definition rather than the collapse dynamics.

Core claim

The discovery is a revised spherical-collapse prescription for decaying dark matter. Decay produces a massive daughter with a velocity kick and a massless radiation component; the collapse shell loses mass, so the linearly extrapolated overdensity needed for collapse at a given redshift rises above the standard universal value. Partitioning daughters into those whose orbits stay inside the halo, those bound but crossing the boundary, and those unbound, the paper derives analytic interior and bound fractions and an effective gravitating mass that interpolates between parent-dominated and daughter-dominated regimes. The resulting mass-dependent critical density shows a small-mass plateau indep

What carries the argument

The mechanism is a decay-modified spherical-collapse equation: a shell evolves under gravity with a gravitating mass that is not the initial mass but the surviving parent mass plus a population-weighted daughter contribution. Every daughter particle is classified at its creation moment under an instantaneous-orbit approximation into one of three populations using the phase-space amplitude and apocenter; the bound fraction and interior fraction become closed-form functions of the kick parameter, halo radius, and bulk flow. The gravitating mass interpolates the boundary-crossing population between parent-dominated and daughter-dominated regimes, producing the gradual 'halo puffing' that soften

Load-bearing premise

Everything rests on the assumption that a daughter particle's fate — fully inside, boundary-crossing, or escaping — can be decided from the potential at the instant it is created, and that boundary-crossing daughters can be represented by a simple interpolated gravitational weight; if that classification or interpolation is wrong, the mass-dependent collapse barrier shifts and the predictions fail in the strongest-signal regime.

What would settle it

For a large-kick model such as a 20 Gyr lifetime with v_k = 2250 km/s at z=0, recompute the halo-by-halo retained-mass ratio in the N-body output using the paper's own matching procedure; if the measured ratio does not fall systematically below the predicted M_coll/M_0 by up to a factor of two at M around 10^14 M_sun/h, then the claimed discrepancy mechanism is not the explanation. Alternatively, implement the paper's proposed orbital-period weighting for boundary-crossing daughters and check whether the z=0 overprediction of the halo mass function disappears; if it does not, the instantaneous

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

If this is right

  • Cluster number counts across mass and redshift can be compared to decaying-dark-matter predictions by evaluating a closed-form critical density, avoiding a new N-body simulation for every parameter point.
  • The pure dark-radiation decay scenario is included automatically: its mass-independent threshold is the small-mass plateau, so constraints on that case separate cleanly from the kick velocity.
  • The transition mass M_1 gives a physical targeting rule: only halos at or below the scale where the kick velocity equals the orbital velocity feel the barrier shift, so surveys need to reach below M_1 to see the effect.
  • Because the suppression grows toward low redshift and high mass, the redshift evolution of the cluster mass function carries independent information beyond the abundance amplitude at a single epoch.
  • The closed-form fitting functions are cheap enough to embed directly in Markov-chain parameter estimation, making joint constraints on the decay rate and kick velocity from halo abundance tractable.

Where Pith is reading between the lines

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

  • Editorial inference: The paper's suggested fix — weighting boundary-crossing daughters by the fraction of their orbit spent inside the halo radius to define a finder-consistent observable mass — is directly testable in the existing N-body outputs; if it removes the z=0 overprediction for the strongest-kick models, the collapse dynamics would be validated even in the extreme regime.
  • Editorial inference: The universal shape parameters of the transition fit were calibrated on a limited grid of lifetimes and kicks; running the collapse differential equations outside that grid, for example lifetimes below 5 Gyr or above 20 Gyr, would test whether universality persists or whether additional physics enters.
  • Editorial inference: Because the small-mass plateau depends only on the decay rate while the transition depends on the kick velocity, combining small-halo probes with cluster abundance could break the lifetime-kick degeneracy that a single mass range leaves unresolved.
  • Editorial inference: The paper's success suggests that treating decay-induced mass loss at the level of collapse dynamics is more robust than trying to build a decaying-dark-matter-specific window function; a direct comparison of the two strategies on a common simulation set would make that point firm.

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

3 major / 5 minor

Summary. The paper presents a semi-analytic Press–Schechter/Sheth–Tormen halo mass function (HMF) for decaying dark matter (DDM) cosmologies. A spherical-collapse model is extended to track the decay of parent particles, the velocity kicks of daughters, and the resulting mass-dependent critical collapse threshold δ_c(M_0) and collapsed-mass mapping M_coll(M_0). The authors derive analytic large- and small-mass limits for δ_c, provide a fitting function for the transition with a characteristic scale M_1, and validate the framework against one ΛCDM and four DDM N-body simulations at z=0 and z≈1.08. Good agreement is found for small-kick models and at z≈1.08; for the two large-kick models at z=0 the predicted HMF overproduces haloes near M~10^14 M_sun/h, which the authors attribute to the difference between their collapsed mass and the halo-finder mass M_200m and defer to future work.

Significance. If correct, the framework would provide a fast, physically motivated route to DDM constraints from cluster counts, avoiding an N-body simulation per parameter point. The analytic large-mass limit (Eq. 40) and the small-mass fit (Eq. 42) are carefully derived/calibrated and checked against the authors' ODE solutions, and the comparison with an independent N-body suite is a real strength. However, the central gravitating-mass prescription for 'population-2' daughters (Eq. 26) is ad hoc and unvalidated, and the validation does not cover the regime where it matters most. The practical claim of an accurate and efficient route to DDM constraints is therefore not yet fully established.

major comments (3)
  1. [Sec. 2.2, Eq. (26); Sec. 5, Fig. 9] The interpolation M_grav = M_p + [f_in/f_bound + (M_d/(M_d+M_p))(1 − f_in/f_bound)] M_d is the only element of the collapse model not derived from the orbital dynamics, and no error estimate is given. The paper itself notes that f_in can formally exceed f_bound in the large-β regime, making Eq. (26) unphysical there. This is not cosmetic: the two largest-kick models (v_k=1250, 2250 km/s) are precisely those where population-2 daughters dominate, and at z=0 the HMF is overpredicted by up to a factor ~2 near M~10^14 M_sun/h. The proposed explanation—the difference between M_coll and M_200m—is plausible but explicitly left for future work ('observable mass'), so the validation does not demonstrate that Eq. (26) is accurate in the regime of largest DDM signal. Please either derive Eq. (26) from a controlled approximation, implement the orbital-period weighting, or quantify the resulting unce
  2. [Secs. 2.1 and 2.4, Eqs. (8), (30), (31)] The framework computes the variance σ(M) from the ΛCDM linear power spectrum and uses the EdS growth factor D∝t^{2/3} for the linear extrapolation to the delayed collapse time. In DDM cosmologies the linear growth factor is itself modified by the decay; the choice to fold all DDM physics into the collapse time is an approximation that is not derived. This could introduce a systematic error in ν_c and may contribute to the residual at large kicks. A concrete test would be to evaluate the HMF with σ computed from the DDM linear power spectrum (and a suitable DDM growth factor) while keeping the modified δ_c, and to compare with the N-body results. As written, the physical interpretation of δ_c as the linearly extrapolated threshold is not unique.
  3. [Sec. 3.3, Eqs. (43), (44), and footnote 3] The transition fit involves six calibrated constants (A, β, γ in Eq. 42; ν, M2/M1, and B in Eqs. 43–44). While the calibration is transparent, the paper does not report the accuracy of Eq. (43) itself (as it does for the plateaus) nor the covariance of the fitted parameters. The claimed 'transparent physical interpretation' of M_1 is weakened by the fact that the Γ̃^{-1/2} dependence is empirical, with a free fit preferring an exponent ≈0.42. Please report fit residuals across the transition mass range and state how the fitted parameters depend on the two-redshift baseline.
minor comments (5)
  1. [Abstract and Sec. 3] The abstract calls M_1 the 'single free parameter' of the transition, but Eqs. (42) and (43) contain additional fitted constants; 'single free parameter per model' would be more precise.
  2. [Sec. 2.1] The statement that using the DDM linear power spectrum would 'double-count' the physics is plausible but not rigorously justified; a sentence explaining why the mass-loss description and the power-spectrum suppression are not independent would help.
  3. [Fig. 9 caption and Sec. 5] The text refers to 'dashed gray lines' for the semi-analytic fit, but the figure caption lists dashed lines; please make the line styles consistent.
  4. [Sec. 3.3, Eq. (45)] The derivation of M_1 uses M_grav ~ M_1 without specifying the prefactor; please clarify what '~' means here and where the factor 2√2/(πG) comes from.
  5. [References] Reference 'Wis locka' should read 'Wisłocka'.

Circularity Check

0 steps flagged

No significant circularity: the DDM collapse inputs are explicit model assumptions, the surrogate fits are calibrated to the paper's own ODE solutions, and the N-body comparison is an independent benchmark not used to set any parameter.

full rationale

The derivation chain is self-contained. The central objects δ_c(M_0) and M_coll(M_0) are obtained by integrating the ODE system (9), (10), (12) with the gravitating-mass prescription (26); Eqs. (40), (42), (43), (44), and (46) are either perturbative solutions or explicitly fitted surrogates calibrated to the paper's own numerical solutions of these ODEs, not to the N-body simulations. The N-body simulations (Sec. 4) are used only as a posterior validation: no constant entering δ_c or M_coll was adjusted to match the simulation HMFs, and the Sheth-Tormen multiplicity parameters are external to this work. The acknowledged limitations—the ad hoc population-2 interpolation in Eq. (26), the instantaneous-orbit approximation, the f_in > f_bound regime, and the z=0 overprediction for the largest kicks (Sec. 5)—are model-accuracy or halo-mass-definition concerns, not circular reductions. No step in the argument defines a prediction in terms of the quantity it is claimed to predict, and the load-bearing citations (Press-Schechter, Sheth-Tormen, Nadler & Benson, Bucko et al. for the N-body implementation) do not smuggle in the target result. Self-citations such as Schneider et al. (2013) for c_R or Bucko et al. (2024)/Montandon et al. (2025) for context are not load-bearing for the central derivation.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The semi-analytical pipeline is a surrogate of the model's own numerics: six calibrated constants (A, β, γ; ν; M_2/M_1; B) plus a chosen Γ̃ exponent. The load-bearing physics rests on two unquantified modeling choices: the instantaneous-orbit classification and the Eq. (26) M_grav interpolation. The N-body comparison is external and was not used to set these constants. No new entities are introduced; the massive daughter and dark radiation are standard DDM ingredients from the cited literature.

free parameters (5)
  • A, β, γ — small-mass plateau fit constants = A=2.3824, β=0.5818, γ=0.5642
    Eq. (42) for δ_c^small(Γ̃), fit to numerical solutions of the small-mass ODE (41); stated 1.5% accuracy over Γ^{-1} ∈ [1,20] Gyr.
  • ν — transition shape exponent = 0.1484
    Eq. (43); calibrated against the full numerical grid and claimed universal across DDM models.
  • M_2/M_1 — second transition mass ratio = 10^1.3795 ≈ 24
    Eq. (43); calibrated against the numerical grid and claimed universal across DDM models.
  • B — M_1 normalization = log₁₀ B = 3.017
    Eq. (44) M_1 = B v_k^3 Γ̃^{-0.5} t_ta; fit across the (Γ, v_k) grid, stated accurate to 10% over the grid.
  • Γ̃ exponent in M_1 scaling = -1/2 (free fit prefers ≈ -0.42)
    Footnote 3: a free fit prefers ≈ −0.42, 'consistent with −1/2 within the scatter given our two-redshift baseline'; the headline −1/2 is a hand choice.
axioms (7)
  • domain assumption Instantaneous-orbit approximation: daughters are classified as interior/bound-crossing/unbound using A² and r²_max evaluated in the potential at the moment of creation, then evolved by class without tracking the time-dependent potential.
    Sec. 2.2: 'we therefore adopt an instantaneous-orbit approximation'. Load-bearing for f_bound, f_in, M_grav, and hence δ_c; error not quantified.
  • ad hoc to paper Gravitating-mass interpolation, Eq. (26): M_grav = M_p + [f_in/f_bound + (M_d/(M_d+M_p))(1 − f_in/f_bound)] M_d.
    Sec. 2.2: 'we adopt the following effective prescription ... Interpolating with the fraction of daughter particles as the natural parameter gives'. Hand-crafted between two limits; no derivation or error estimate.
  • domain assumption ΛCDM linear power spectrum used for σ(M) while all DDM physics is injected through δ_c(M) and M_coll.
    Sec. 2.1: 'we use the ΛCDM linear power spectrum to compute σ(M)'. The paper argues mass loss and power suppression are the same physical effect; the quantitative equivalence of this decomposition is not demonstrated.
  • domain assumption Constant-barrier multiplicity function (PS/ST) used with a mass-dependent ν_c(M) = δ_c(M)/σ(M).
    Sec. 2.1: 'Using the constant-barrier multiplicity function ... with a mass-dependent ν_c(M) is an approximation whose accuracy cannot be guaranteed analytically.' Acknowledged by the authors.
  • domain assumption EdS initial conditions and EdS growth factor for linear extrapolation, rather than a CLASS-based background.
    Secs. 2.3–2.4: 'we adopt EdS initial conditions throughout', and Eq. (31) uses D_EdS ∝ t^{2/3} because 'using the DDM growth factor would double-count it'. Internally justified but a modeling choice.
  • domain assumption Neglect of ρ_Λ in the collapse ODE, Eq. (9).
    Sec. 2.2: ρ_Λ 'remains negligible compared to self-gravity in the collapse dynamics and shifts only the turn-around epoch at the percent level (Percival 2005)'. Cited standard assumption.
  • domain assumption f_DDM = 1 (all CDM decays).
    Sec. 1: 'In this work, we only consider f_DDM = 1.' Explicit scope restriction.

pith-pipeline@v1.3.0-alltime-deepseek · 23754 in / 24855 out tokens · 248040 ms · 2026-08-01T13:00:00.339798+00:00 · methodology

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read the original abstract

We present a semi-analytical framework for the halo mass function (HMF) in decaying dark matter (DDM) cosmologies, in which dark matter decays into a massive daughter particle inheriting a velocity kick $v_k$ and a massless dark radiation component. Building on the Press-Schechter formalism, we encode the DDM physics through a spherical collapse model that explicitly tracks the decay-induced mass loss, yielding a modified, mass-dependent critical collapse threshold $\delta_c(M_0)$ and a mapping $M_{\rm coll}(M_0)$ between the initial Lagrangian mass and the collapsed halo mass. The critical threshold exhibits a characteristic transition between two analytically tractable plateaus: a large-mass limit, where all daughter particles are retained by the halo, and a small-mass limit, where all daughters escape and the collapse is equivalent to that of a dark matter species decaying entirely into dark radiation, making $\delta_c$ independent of $M_0$ and $v_k$. We provide semi-analytical results and fits for both limits and a fitting formula for the transition, whose single free parameter $M_1 \propto v_k^3\,\tilde\Gamma^{-1/2} t_{\rm ta}$ has a transparent physical interpretation: it is the mass scale at which the kick velocity equals the halo orbital velocity. We validate our predictions against a suite of N-body simulations at $z=0$ and $z\approx 1$, finding good agreement across models spanning mild to strong HMF suppression relative to $\Lambda$CDM. Residual deviations for the largest kick velocities at $z=0$ are observed. Via a halo-by-halo comparison between simulations, we trace the discrepancy to the definition of the halo mass when daughter orbits extend beyond the halo boundary. The resulting fitting functions for $\delta_c(M_0,\Gamma,v_k)$ and $M_{\rm coll}(M_0)$ provide an efficient and accurate route to DDM constraints from current and forthcoming probes of the halo mass function.

Figures

Figures reproduced from arXiv: 2607.19244 by Aurel Schneider, Jozef Bucko, Oliver Hahn, Thomas Montandon, Vivian Poulin.

Figure 1
Figure 1. Figure 1: — Matter power spectra for various DDM models. Top panel: dimensionless power spectrum k 3P(k)/2π 2 for ΛCDM (dashed black) and three DDM lifetimes (distinguished by color), with the velocity kick vk varying within each lifetime as indicated by the colorbar shading. Bottom panel: ratio P/PΛCDM, high￾lighting the small-scale power suppression induced by the DDM decay. function. The initial Lagrangian mass o… view at source ↗
Figure 2
Figure 2. Figure 2: — Naive estimation of the halo mass function injecting the DDM linear power spectrum in (8) using a top-hat filter (left panel) and a sharp-k filter (right panel). behaviour is well known in the context of warm dark matter: the top-hat filter in real space has a very broad kernel in k-space and cannot capture the sharp small￾scale suppression in Pm(k), and is typically resolved by adopting a sharp-k filter… view at source ↗
Figure 3
Figure 3. Figure 3: — Ratio of the halo collapse time to the ΛCDM collapse time, tcoll/tΛCDM coll − 1, as a function of the kick velocity vk and the lifetime Γ−1 , for a top-hat overdensity with initial mass M0 = 5 × 1014 M⊙ and δ0 = 0.003. DDM effects are negligible at small kick velocities and long lifetimes (ratio ≃ 1), while short lifetimes combined with large kicks significantly delay. 0 5 10 t[Gyr] 0.0 0.5 1.0 1.5 2.0 2… view at source ↗
Figure 4
Figure 4. Figure 4: — Linear density contrast δlin(t) as a function of cosmic time for ΛCDM and four DDM models, for a top-hat overdensity with initial mass M0 = 5×1014 M⊙ initialised at t0 = 5×10−4 Gyr. The initial overdensity is chosen such that the collapse happens at z = 0 for each model. The horizontal black line marks the standard ΛCDM value δc = 1.686, and the vertical line indicates the desired collapse time z = 0. Th… view at source ↗
Figure 5
Figure 5. Figure 5: — Mass dependence of the collapse threshold δc subtracted by δ EdS c for 5 velocity kicks (colored lines), and 3 lifetime (panels). At small masses, δc plateaus to a constant value that depends primarily on Γ−1 . At large masses, δc converges to a value close to, but slightly larger than δ EdS c for all models. with different vk converge to the same value – and a large-mass plateau close to δ EdS c that de… view at source ↗
Figure 6
Figure 6. Figure 6: — Relative deviation between the large-mass analytical approximation δ analytical c of Eq. (40) and the large-mass-limit nu￾merical result δ numerical c , as a function of the velocity kick vk for five decay lifetimes as indicated in the legend. Note that, for the EdS cycloid, tta = tcoll/2, so the zeroth-order dimensionless collapse time is t˜EdS coll = 2. Consequently, at this order in ϵ, and apart from … view at source ↗
Figure 7
Figure 7. Figure 7: — Small-mass plateau of the collapse threshold as a func￾tion of Γ. ˜ Top panel: DDM excess δc −δ EdS c computed numerically (blue points) and the fitting formula (red dashed, Eq. (42)). Bot￾tom panel: Relative error between the numerical result and the fit. The orange band represents the relevant range of Γ for a collapse ˜ time today. This is precisely the small-mass plateau δ small c (Γ), where ˜ the sa… view at source ↗
Figure 8
Figure 8. Figure 8: — Retained mass fraction Mcoll/M0 at z = 0 as a function of the collapsed mass Mcoll for the four DDM models (one per panel). Solid black lines show the full numerical solution of the collapse system; dashed black lines show the semi-analytical approximation of eqs. (46) and (48) evaluated along the EdS cycloid; the blue horizontal line marks the ΛCDM limit Mcoll = M0. Points show the per-halo mass ratio M… view at source ↗
Figure 9
Figure 9. Figure 9: — Halo mass function for ΛCDM and four DDM models at z = 0 (left) and z = 1.083 (right). Solid lines show the theoretical predictions from the modified Press–Schechter formalism developed in Sec. 2, while dashed lines show the semi-analytic fit obtained in Sec. 3. Points with error bars show the corresponding N-body simulation measurements. Colors distinguish the five models as indicated in the legend. Not… view at source ↗

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