A frequentist view on the two-body decaying dark matter model

Ad\`ele Poudou; Elsa M. Teixeira; Thomas Montandon; Vivian Poulin

REVIEW 2 major objections 5 minor 3 cited by

A frequentist view on the two-body decaying dark matter model

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper argues that Bayesian constraints on two-body decaying dark matter are distorted by prior volume effects, and that a frequentist analysis of Planck data favours decay parameters that lower $S_8$ to about 0.70.

desk verdict A self-aware frequentist reanalysis of DDM that makes the volume-effect point clearly, but the headline Planck numbers sit on an emulator extrapolation the authors themselves flag. read the letter →

arxiv 2505.20193 v2 pith:IXT7DCDT submitted 2025-05-26 astro-ph.CO hep-phhep-th

Thomas Montandon , Elsa M. Teixeira , Adèle Poudou , Vivian Poulin This is my paper

classification astro-ph.COhep-phhep-th

keywords decayingdarkmatterprofilelikelihoodfrequentistinferenceS8tensionpriorvolumeeffectsKiDS-1000PlanckCMBweaklensing

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 argues that Bayesian constraints on the two-body decaying dark matter (DDM) model are distorted by prior volume effects, because the model's two extra parameters are only weakly pinned down by the data. A frequentist profile-likelihood analysis of Planck CMB, BAO, and supernova data gives a 68% confidence interval centered on a decay half-life of about 7 Gyr and a velocity kick of about 1250 km/s, which differs strongly from the Bayesian intervals. At that best fit the model predicts $S_8 \approx 0.70$, in agreement with the low values measured by KiDS-1000 and DES-Y3 weak-lensing surveys at roughly the 1.5$\sigma$ level. The paper further shows that KiDS-1000 constraints on DDM found in earlier work were driven by priors on the primordial amplitude and spectral index, and that without those priors the data do not constrain the decay parameters. The wider point is that conclusions about whether dark matter is stable depend on the statistical framework, so frequentist intervals provide a necessary prior-independent cross-check.

What carries the argument

The load-bearing tool is the profile likelihood, $L_p(\alpha)=\max_\theta L(\alpha,\theta)$, which fixes the parameters of interest $\alpha$ and maximises over every other parameter, producing prior-independent confidence intervals through $\Delta\chi^2$ thresholds. The analysis applies it to the two-body DDM model, whose nonlinear matter power spectrum is written as $P^{\Lambda\mathrm{DDM}}_{\mathrm{NL}}(k,z) = P^{\Lambda\mathrm{CDM}}_{\mathrm{NL}}(k,z)\times S_{\Gamma,v}(k,z)$, where the suppression factor $S_{\Gamma,v}$ comes from the DMemu emulator trained on N-body simulations. Comparing the resulting 1$\sigma$ frequentist contour with Bayesian posteriors is what exposes the volume effect.

What would settle it

Recompute the frequentist 68% contour using an emulator or N-body calibration that covers $\log_{10}\Gamma\in[-4,-0.5]$ and $\log_{10}v\in[1.5,4]$, and check whether the best fit at $\Gamma=0.10\,\mathrm{Gyr}^{-1}$, $v=1250\,\mathrm{km/s}$ remains a local minimum. If the minimum shifts to the $\Lambda$CDM limit ($\Gamma\to0$ or $v\to0$) or the improvement over $\Lambda$CDM drops below $\Delta\chi^2\approx1$, the claimed $S_8=0.70$ preference would not survive.

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Extended reading notes

Core claim

On its own terms, the central discovery is that the two-body decaying dark matter model is weakly favoured by Planck CMB, BAO, and supernova data once prior volume effects are removed. The profile likelihood yields 68% confidence intervals $t_{1/2} = 6.93^{+7.88}_{-2.85}$ Gyr and $v = 1250^{+1450}_{-1000}$ km/s, with a best fit at $\Gamma = 0.10\,\mathrm{Gyr}^{-1}$ and $v = 1250\,\mathrm{km/s}$ that predicts $S_8 = 0.70$. This value lies below the Planck $\Lambda$CDM value and matches the KiDS-1000/DES-Y3 measurements within about $1.5\sigma$, whereas the Bayesian posterior from the same data keeps $S_8$ near $0.816$ because the prior volume penalises the decay region. The authors conclude that the apparent weakness of DDM in earlier analyses is largely a prior-volume artefact, and that the model genuinely can lower $S_8$ while fitting the CMB.

Load-bearing premise

The result relies on the emulator that predicts how decay changes the clustering of matter being accurate at the Planck best-fit decay rate and velocity kick, even though the paper states that this best-fit and part of the 1$\sigma$ region lie outside the range the emulator was trained on.

Editorial extensions

If this is right

At 68% confidence the frequentist region for $(\Gamma,v)$ is a closed contour around the best fit, while the Bayesian posterior is pulled toward the $\Lambda$CDM limit; the two approaches agree only at 95% confidence, where volume effects become negligible.
If the best fit is correct, roughly three-quarters of the initial DDM particles have decayed by today, and the model predicts $S_8=0.70$, reconciling Planck with KiDS-1000 and DES-Y3 at about $1.5\sigma$ without needing the full-shape galaxy power spectrum.
KiDS-1000 alone, with broad priors on $A_s$ and $n_s$, does not constrain $\Gamma$ and $v$; earlier DDM constraints from that survey were produced by narrow priors on the primordial parameters, so they should not be read as data-driven exclusions.
Combining KiDS-1000 with Planck-informed priors on $A_s$ and $n_s$ excludes large decay rates and velocity kicks but leaves the 68% frequentist region from Planck alive, so multi-scale data are needed to settle the model.
Forecast studies cited in the paper indicate that Euclid will tighten constraints on the DDM parameter space by up to an order of magnitude, providing a definitive test of the scenario.

Reading between the lines

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

Beyond the paper: any weakly constrained new-physics parameter in cosmology is vulnerable to the same prior-volume effect, so Bayesian upper limits from CMB-only analyses should be cross-checked with profile likelihoods before being treated as exclusions.
Beyond the paper: a decisive test is to extend the DMemu emulator or run matched N-body simulations at $(\Gamma,v)=(0.10\,\mathrm{Gyr}^{-1},1250\,\mathrm{km/s})$; if the predicted suppression factor changes materially outside the trained domain, the $S_8=0.70$ prediction would move with it.
Beyond the paper: because KiDS-1000 mostly constrains scales around $k\sim0.3\,h\,\mathrm{Mpc}^{-1}$ rather than the $S_8$ kernel, future model comparisons should use the full lensing likelihood; a compressed $S_8$ statistic alone could misjudge a model that suppresses power at slightly different scales.
Beyond the paper: a discriminating observation would be the scale-dependent matter power spectrum from galaxy clustering or CMB lensing; a step-like suppression at the free-streaming scale would corroborate the frequentist preference, while a smooth CDM-like spectrum would push $\Gamma$ and $v$ back toward the $\Lambda$CDM limit.

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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

The genuinely new thing here is the first frequentist profile-likelihood analysis of the two-body DDM model against Planck+BAO+SN, and it earns its keep. The 2D profile versus Bayesian comparison in Fig. 3 is instructive: at 1σ the frequentist confidence region is a closed contour that prefers decay half-lives around 7 Gyr and velocity kicks around 1250 km/s, while the Bayesian posterior hugs the CDM limit. That contrast is exactly what you need to see to believe there is a prior-volume effect at work. The paper also does a real service by showing that the KiDS-1000 constraints on DDM from Ref. [30] were driven by the priors on As and ns: with broad priors the DDM parameters are unconstrained, and the scale range KiDS actually probes is centered near k~0.3 h/Mpc, not at the S8 kernel. That is a clean, reproducible point that does not depend on any extrapolation. The soft spot is real and load-bearing. The Planck best-fit (log10 Γ ≈ -1, log10 v ≈ 3.1) and part of its 1σ region lie outside the training domain of the DMemu emulator, and the paper says so explicitly in Sec. IV A. The 68% intervals and the S8=0.70 prediction come from that extrapolated region. Given that the improvement over ΛCDM is only Δχ² = -2.66, a modest error in the suppression factor at high Γ could plausibly move the minimum or change the shape of the profile. The authors are honest about this, but they still present Eq. (13) as the main result without a dedicated validation of the emulator in that region. A referee should ask for that. The KiDS profile likelihood is also compromised by an intrinsic-alignment nuisance parameter pinned at its prior boundary, which the appendix discloses; the frequentist KiDS interval is therefore not trustworthy and the paper wisely focuses on Planck for the headline claim. What holds up: the prior-dependence demonstration for KiDS-1000 (broad vs narrow As/ns priors) is robust and does not rely on the extrapolated region. The conclusion that Bayesian and frequentist results agree at 2σ, with volume effects negligible there, is also consistent with the plots. The paper is well-cited and the statistical framing is clear. This is a paper that deserves a serious referee, not a desk reject. The methodological point about prior volume in DDM constraints is important, and the KiDS prior-dependence result alone is worth publishing. The main numerical claim about Planck needs either emulator retraining or a clearly caveated presentation with a validation appendix. I would send it to review and ask for that before acceptance. The right reader is someone working on cosmological constraints on non-standard dark matter or on prior-volume effects in Bayesian analyses; it would also make a good reading-group discussion for its honest treatment of its own limits.

Referee Report

2 major / 5 minor

Summary. The paper presents Bayesian and frequentist profile-likelihood analyses of the two-body decaying dark matter (DDM) model, characterized by decay rate Γ and velocity kick v, using Planck 2018 CMB plus BAO and Pantheon-Plus supernovae ('Planck') and KiDS-1000 cosmic shear. For the Planck data, the authors report a global best fit Γ = 0.10 Gyr^-1, v = 1250 km/s with S8 = 0.70 and Δχ² = -2.66 relative to ΛCDM, and 68% profile-likelihood intervals (Eq. 13) that they argue differ from Bayesian posteriors due to prior volume effects. They also show that KiDS-1000 alone, with broad priors on As and ns, does not constrain the DDM parameters, and that previously reported KiDS-1000 constraints were largely driven by those priors. A central caveat, acknowledged by the authors in Section IV.A, is that the Planck-DDM best fit and part of the corresponding 1σ region lie outside the domain covered by the DMemu nonlinear emulator used for the predictions.

Significance. If valid, the paper would be the first frequentist analysis of the two-body DDM model, providing prior-independent constraints and a concrete illustration that prior volume effects can change the qualitative interpretation of Bayesian DDM constraints. The demonstration of an As-ns degeneracy in KiDS-1000 and the warning against substituting S8 constraints for the full likelihood are useful and well-illustrated contributions. The paper is transparent about its main limitation, and it builds on public codes and published simulations. However, the headline quantitative claims—the 68% intervals in Eq. (13) and the S8 = 0.70 prediction—are obtained in a parameter region where the emulator is explicitly untrained; until that extrapolation is validated or the claims are restricted to the trained domain, the significance remains conditional.

major comments (2)

[Section IV.A and Eqs. (9)–(13)] The central result is not supported within the emulator's trained domain. Eq. (9) gives log10 Γ ∈ [-4, -1.13] for DMemu, while the Planck best fit (Eq. 12) has log10 Γ = -1.0 and the 1σ upper limit in Eq. (13) extends to log10 Γ ≈ -0.57. The paper itself states in Section IV.A that 'the Planck-DDM best-fit and part of the corresponding 1σ frequentist confidence region' lie outside the emulator domain. Since the DDM improvement over ΛCDM is only Δχ² = -2.66 (Eq. 11), a modest error in the extrapolated suppression factor could move the minimum and alter the conclusion that the Bayesian and frequentist intervals 'strongly differ.' Please validate DMemu at the best-fit parameters (e.g., with DDM N-body simulations at Γ = 0.10 Gyr^-1, v = 1250 km/s) or, failing that, restrict the reported profile-likelihood constraints to the trained domain and state that the global best fit cannot be characterized with the current emulator.
[Abstract, Eq. (12), Section III.B.1] The abstract's statement that the frequentist best fit S8 = 0.70 agrees with KiDS-1000 and DES-Y3 at ~1.5σ needs substantiation. A direct comparison of S8 = 0.70 with the KiDS-1000 value S8 = 0.759^{+0.024}_{-0.021} reported in the paper corresponds to a roughly 2.5σ difference. If the intended claim is that the 68% frequentist region in (Γ, v) contains models whose S8 is compatible with the weak-lensing measurements, the paper should state and justify that metric, and report the range of S8 over that region; otherwise the abstract overstates the agreement.

minor comments (5)

[Appendix B, Eq. (B1)] In the sentence following Eq. (B1), 'where Γ0 and v are arbitrary normalization values' should read 'where Γ0 and v0 are arbitrary normalization values', since v0 is the normalization used in the definition.
[Abstract and Section IV.A] The abstract says the scales best measured by KiDS-1000 are 'centred around k ∼ 0.3 h/Mpc', while Section IV.A states 'a kernel centred around k ∼ 0.5 h/Mpc'; these numbers should be reconciled.
[Section IV.A, Fig. 5 reference] The text says 'In Fig. 5, the overlap between the 1σ region and the trained domain of SΓ,v is shown as a purple contour', but the Fig. 5 caption describes KiDS-1000 posterior contours with Planck profile contours overlaid; please specify exactly which panel or overlay is meant.
[Table II and Eq. (B1)] The row for fDDM in Table II ('<2σ −0.16', etc.) is difficult to parse; please state explicitly that these are 95% upper limits on the combination defined in Eq. (B1), and include the normalization values Γ0 and v0 in the table caption.
[Section III.B.1, Eq. (13)] The interval for N0^DDM/Nini is reported without derivation; please state that this ratio is exp(-Γ t0) in the two-body DDM model, or cite the equation where it is defined.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the frequentist analysis is built on external likelihoods and an independently calibrated emulator; the out-of-domain extrapolation is a modeling limitation, not a circular step.

full rationale

The paper's central result, a frequentist profile-likelihood analysis of the two-body DDM model, is derived from external datasets (Planck 2018, BAO, PantheonPlus, KiDS-1000) and from the DMemu nonlinear suppression factor S_Γ,v(k,z), which is calibrated on independent N-body simulations in Ref. [30] and is not fitted to the data analyzed here. The comparison between Bayesian and frequentist intervals is a legitimate statistical exercise applied to the same model and data; it does not define the target conclusion in terms of its own output. The authors' admitted statement that 'the Planck-DDM best-fit and part of the corresponding 1σ frequentist confidence region identified in Section III B lie outside the domain covered by the emulator SΓ,v' (Section IV A) is a substantive modeling limitation that could weaken the robustness of the quoted best-fit and S8 value, but it is not a circularity: extrapolating a calibrated emulator is an assumption, not a reduction of the result to its inputs. Self-citations to the authors' earlier work (e.g., Refs. [25,47,87]) provide the public CLASS implementation and previous constraints, but these are reproducible codes and external comparisons, not uniqueness theorems or fitted parameters renamed as predictions. No step in the derivation chain equates the predicted quantities with the inputs by construction, and no load-bearing argument reduces to an unverified self-citation. Therefore no significant circularity is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The count of free parameters is minimal. The paper itself introduces no new ad hoc numbers, but its central claim depends on the fitted values of Gamma and v and on the accuracy of an externally trained emulator outside its stated range.

free parameters (2)

decay rate Gamma = 0.10 Gyr^-1 (best fit)
One of the two new DDM parameters; constrained by the profile likelihood over Planck+BAO+SN data (Eq. 13).
velocity kick v = 1250 km/s (best fit)
The second new DDM parameter; constrained jointly with Gamma (Eq. 13).

assumptions (4)

domain assumption Flat LCDM with six parameters and Gaussian adiabatic primordial perturbations (power-law spectrum As, ns-1).
Standard cosmological model assumed before adding DDM (Section II).
domain assumption The DMemu emulator S_{Gamma,v} (Ref. [30]) gives accurate nonlinear matter power spectra for the f=1 two-body DDM model over the full parameter range used for Planck.
Used via Eq. (6) for all nonlinear predictions; the authors themselves note the best-fit lies outside the emulator's trained domain.
domain assumption The simulated annealing minimization of Ref. [83] converges to the global maximum of the likelihood.
Profile likelihood in Eq. (10) relies on global maximization over all nuisance parameters; no convergence diagnostics are reported.
domain assumption Kinematic relation epsilon = 1/2(1 - M_WDM^2/M_CDM^2) and v = epsilon c with f=1.
Model definition taken from Refs. [25, 58] (Section II.A).

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Cite this review

Pith. "Pith review of A frequentist view on the two-body decaying dark matter model." pith.science (2026). https://pith.science/paper/IXT7DCDT

@misc{pith2026250520193,
  author       = {Pith},
  title        = {Pith review of: A frequentist view on the two-body decaying dark matter model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXT7DCDT}},
  note         = {Machine review of arXiv:2505.20193}
}

abstract

Decaying dark matter (DDM) has emerged as an interesting framework to extend the $\Lambda$-cold-dark-matter (LCDM) model, as many particle physics models predict that dark matter may not be stable over cosmic time and can impact structure formation. In particular, a model in which DDM decays at a rate $\Gamma$ and imprints a velocity kick $v$ onto its decay products leads to a low amplitude of fluctuations, as quantified by the parameter $S_8$, in better agreement with that measured by some weak lensing surveys. Bayesian analyses have provided mixed conclusions regarding its viability, with a reconstructed clustering amplitude only slightly below the standard LCDM value. In this paper, we contrast previous results with a frequentist analysis of Planck and SDSS BAO data. We find that the $68\%$ confidence level region corresponds to a decay half-life of $6.93^{+7.88}_{-2.85}$Gyr and a velocity kick of $1250^{+1450}_{-1000}$~km/s. These $1\sigma$ constraints strongly differ from their Bayesian counterparts, indicating the presence of volume effect in the Bayesian analysis. Moreover, we find that under the DDM model, the frequentist analysis predicts lower values of $S_8$, in agreement with those found by KiDS-1000 and DES-Y3 at $\sim 1.5\sigma$. We further show that previously derived KiDS-1000 constraints that appeared to exclude the best-fit model from Planck data were driven by priors on the primordial amplitude $A_s$ and spectral index $n_s$. When those are removed from the analysis, KiDS-1000 constraints on the DDM parameters are fully relaxed. It is only when applying Planck-informed priors on $A_s$ and $n_s$ to the KiDS-1000 analysis that one can constrain the model. We note that without such priors, the scales best measured by KiDS-1000 do not exactly match the $S_8$ kernel, so $S_8$ constraints should not be applied directly to a model in place of the full likelihood.

Figures

Figures reproduced from arXiv: 2505.20193 by the authors.

**Figure 2.** FIG. 2. Planck profile likelihood as a function of log [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4. Bayesian posterior density of the power spectrum (in blue) and approximate frequentist 1 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: FIG. 5. Bayesian constraints on the DDM model for [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: FIG. 7. Constraints on the two additional parameters of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. In blue, we show the Bayesian constraints on the mat [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. Isocontours of the best-fit values of [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. 1D marginalised constraints on the combination of [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

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Forward citations

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Testing decaying dark matter models as a solution to the $S_8$ tension with the thermal Sunyaev-Zel'dovich effect

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2, we show the profile likelihood as a function of Γ and v on the first and second columns, respectively, by using Eq

1-dimensional profile In Fig. 2, we show the profile likelihood as a function of Γ and v on the first and second columns, respectively, by using Eq. (10). The top and bottom panels show respectively the ∆χ2 and the derived parameter S8 com- puted at the maximum likelihood θmax. We show the reconstructed value of S8 in the KiDS-1000 analysis as a blue band...

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from KiDS-1000 alone

2-dimensional profile We show in Fig. 3 the comparison between Bayesian and frequentist intervals reconstructed from the analy- sis of Planck data in the 2-dimensional {log10 Γ, log10 v} plane, as well as in the 1-dimensional case for each param- eter. In the frequentist case, shown in purple, we display the 2-dimensional confidence intervals derived with...

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