Pith. sign in

REVIEW 3 major objections 4 minor 62 references

A single filter reproduces halo counts for WDM and dark-acoustic-oscillation models

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 05:42 UTC pith:TFDS4SFQ

load-bearing objection The VSMK filter is a genuine analytic extension, but the central claim of a single parameter set is not demonstrated: the parameters are taken from fits to the same simulations, and the scale-inversion premise is asserted rather than proven. the 3 major comments →

arxiv 2602.01320 v3 pith:TFDS4SFQ submitted 2026-02-01 astro-ph.CO

A Variable-Slope Smooth-k Filter for Modeling Halo Abundances with Damped and Oscillatory Power Spectra

Andreu Rocamora Martorell This is my paper
classification astro-ph.CO
keywords dark matter theoryhalo mass functionPress-Schechter formalismsmooth-k filterwarm dark matterdark acoustic oscillationsETHOSwindow function
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 argues that the standard smooth-k filter cannot simultaneously describe dark matter models whose power spectra combine a small-scale damping cutoff with dark acoustic oscillations, because its single slope couples the two regimes. It introduces the variable-slope smooth-k (VSMK) filter, whose effective logarithmic slope interpolates between a steep slope at low k/kM and a shallower slope at high k/kM. The paper establishes a scale-inversion property: in the Press-Schechter formalism the small-mass halo mass function is fixed by the filter at k/kM << 1, while the intermediate-mass imprint of dark acoustic oscillations is fixed by the filter at k/kM >~ 1. With one parameter set (beta1=4.8, beta2=3.6, mu=2.1, delta=12, c=3.6), the analytic halo mass function matches N-body simulations for both a warm dark matter model and ETHOS-based DAO models at redshifts 5-12. The result matters because it offers a single analytic description of halo abundances across non-cold dark matter scenarios without model-specific recalibration.

Core claim

Within the extended Press-Schechter formalism, the paper establishes that the asymptotic slope of the halo mass function at small masses is controlled by the low-k/kM behavior of the window function, while the smoothing of dark-acoustic-oscillation features at intermediate masses is controlled by the window at k/kM >~ 1. This scale inversion lets a generalized filter of the form W(k,kM)=[1+(k/kM)^{f(k)}]^{-1}, with f(k)=beta2-(beta2-beta1)[1+(mu k/kM)^delta]^{-1}, decouple the two regimes. Comparing analytic predictions with N-body simulations, the paper shows that a single parameter set reproduces both the WDM small-scale suppression and the DAO oscillations in the halo mass function, with

What carries the argument

The central object is the VSMK window function W_VSMK(k,kM)=[1+(k/kM)^{f(k)}]^{-1}, where f(k) interpolates between the asymptotic slopes beta1 for k/kM << 1 and beta2 for k/kM >> 1. The machinery works through the integrand xi(k,kM)=k^2 P(k) partial(W^2)/partial(kM), whose peak location shifts from k/kM ~ 1 for large halo masses to k/kM << 1 for small masses, mapping filter behavior onto distinct HMF regimes: beta1 sets the small-mass power-law slope dn/dlnM ~ M^{(beta1-3)/3}, while beta2 regulates how strongly DAO oscillations survive in the intermediate-mass HMF.

Load-bearing premise

The scale-inversion mapping — that the small-mass HMF is set by filter behavior at k/kM << 1 and the intermediate-mass DAO imprint by behavior at k/kM >~ 1 — must hold for both WDM and DAO spectra at the chosen mu and delta; if the maximum of the integrand xi(k,kM) does not track these regimes, the two slopes are not truly decoupled.

What would settle it

Compute the exact d sigma^2/dkM integral without the peak approximation for a strong DAO model (h_peak=1) and test whether varying beta2 while holding beta1 fixed leaves the small-mass HMF unchanged, and whether varying beta1 leaves the intermediate-mass oscillation amplitude unchanged; any leakage beyond the claimed few-percent level would falsify the decoupling mechanism.

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

If this is right

  • A single VSMK parameter set reproduces halo mass functions for both warm dark matter and DAO-based ETHOS models, removing the need for model-specific SMK recalibration.
  • The small-mass slope of the HMF is controlled exclusively by beta1, and the intermediate-mass DAO imprint by beta2, so the two physical regimes can be fitted independently.
  • The mass-calibration parameter c continues to shift the HMF along the mass axis without altering its shape or the filter's smoothing of oscillations.
  • For ETHOS DAO models at z=5, 8, and 12, the VSMK predictions agree with N-body simulations at the level of Delta <= 0.11 over resolved mass ranges.
  • The VSMK filter reduces to the standard SMK filter when beta1=beta2, making it a minimal extension of the existing framework.

Where Pith is reading between the lines

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

  • If the decoupling holds, future small-scale HMF measurements could separately constrain the free-streaming cutoff (via beta1) and dark-radiation interactions (via beta2), potentially breaking degeneracies that a single-slope filter cannot.
  • The same scale-inversion argument may extend to other Press-Schechter observables, such as halo bias or void abundances, whenever the power spectrum is damped and oscillatory.
  • Because mu and delta are held fixed across models rather than calibrated, a physical mapping from those parameters to the DAO peak scale or half-mode scale could reduce the filter to its two slope parameters alone.
  • The filter is a fitting window rather than a collapse model, so its success suggests that flexible smoothing, not the specific window shape, is what the HMF responds to; simulations reaching lower masses will test whether the beta1 asymptote extrapolates.

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 / 4 minor

Summary. The paper introduces a variable-slope smooth-k (VSMK) filter, Eqs. (3.6)–(3.7), which generalizes the SMK filter by letting the effective logarithmic slope interpolate between β1 at k/kM ≪ 1 and β2 at k/kM ≫ 1. Within the extended Press–Schechter formalism, the paper claims a 'scale-inversion' mechanism: β1 controls the small-mass slope of the halo mass function, dn/dlnM ∝ M^{(β1−3)/3} (Eq. 3.5), while β2 controls the intermediate-mass imprint of dark acoustic oscillations. Using the single parameter set β1 = 4.8, β2 = 3.6, μ = 2.1, δ = 12, c = 3.6, the paper reports agreement with N-body HMFs for a WDM model (Schaeffer & Schneider 2021) and for ETHOS DAO models (Verwohlt et al. 2024; Bohr et al. 2021). The paper concludes that the VSMK filter provides a unified analytic framework for damped and oscillatory power spectra.

Significance. If the claimed decoupling is robust, the VSMK filter is a useful and minimal extension of the SMK filter, with potential applications to non-CDM HMF predictions at small and intermediate masses. The analytic derivation of the small-mass slope in Section 3 and Appendix A is clean and consistent with the SMK limit, and the comparison spans several redshifts and two DAO scenarios. However, the central validation is currently underdetermined: the asymptotic slopes β1 and β2 are taken from the optimal SMK fits to the same simulations used as benchmarks, the transition parameters μ and δ are fixed without a sensitivity analysis, and the scale-inversion premise is only illustrated for two DAO configurations. The manuscript therefore establishes a promising framework, but the advertised universality of the single parameter set is not yet demonstrated.

major comments (3)
  1. [§3 and Appendix A (Eq. 3.2, Figure 5)] The decoupling claim rests on the assumption that, for kM ≫ khm, the dominant maximum of ξ(k,kM) lies at k/kM < 1/μ, and for kM ≲ khm it lies at k/kM > 1/μ. Figure 5 illustrates this only for m_WDM = 1.61 keV and for kpeak = 35 h/Mpc, hpeak = 1; the paper does not map where the maximum crosses k/kM = 1/μ as a function of kM. Because μ and δ set the transition in units of kM while the DAO peak is at a fixed physical kpeak, the separation of regimes is not guaranteed for other ETHOS parameters. A failure occurs if the dominant ξ-peak lies near the transition region for the mass range where DAO features appear, so that both β1 and β2 contribute simultaneously. Please provide a systematic map of the ξ-peak location as a function of kM/khm and a scan over μ and δ to demonstrate that the one-parameter-per-regime mapping holds.
  2. [§4 (parameter choice, Table 2, Figure 4)] The 'single parameter set' claim is weakened by the way the parameters are selected. β1 = 4.8 and β2 = 3.6 are the optimal SMK slopes for the WDM and DAO simulations used as benchmarks, and c = 3.6 is likewise the SMK calibration for the DAO model of [41]. The two asymptotic regimes are therefore reproduced partly by construction; the actual test is whether the transition parameters μ and δ are robust and whether the same set works for models not used for calibration. This is further qualified by the text of Figure 4, which states that the Bohr et al. model requires 'minor variations in c and β2.' Please either (a) present a predictive comparison with an uncalibrated model (e.g., hpeak = 0.4 at multiple redshifts without re-fitting β2 or c) or (b) state explicitly which parameters are re-fit and provide a sensitivity analysis of μ and δ.
  3. [§4 and Data availability statement] The quantitative validation rests on N-body data extracted from published figures, with no published error bars or resolution cuts and no reproducible pipeline. The paper states that 'No new data were generated or analysed' yet presents comparisons to simulation data, and the numerical routines are only 'available from the author upon reasonable request.' This makes it difficult to assess whether the quoted deviations (e.g., ΔVSMK ≤ 0.05 for the DAO case) are robust or sensitive to the extraction procedure. Please make the extracted data and evaluation code available, or at minimum provide a table of the simulation data points and the exact binning/selection criteria used.
minor comments (4)
  1. [§4, Eq. (4.1) and text below Figure 3] The statement 'Δ3.6(n4.8) → M^{-0.4} − 1 as M → 0' appears incorrect: from Eq. (4.1), Δ3.6(n4.8) = n3.6/n4.8 − 1 ∝ M^{-0.4} − 1, which diverges as M → 0, rather than tending to −1. Please correct the limit or the definition of the deviation.
  2. [§2.2, Eq. (2.15)] The ETHOS transfer function is difficult to parse as typeset; there are missing parentheses and the term '√h2/4' is ambiguous. Please rewrite the equation with unambiguous notation (e.g., sqrt(h2)/4 and explicit exponentials).
  3. [Figure 5 caption] The caption labels the two mass scales as 'kM = 30 h Mpc^{-1}' and 'kM = 1000 h Mpc^{-1}' but does not indicate which line style corresponds to which kM; the figure appears to rely on color alone. Please make the legend explicit.
  4. [§4 and Figure 5] The text uses m_WDM = 0.25 keV for the Schaeffer & Schneider comparison (Figure 3) and m_WDM = 1.61 keV in Figure 5; the relation between these models and the ETHOS parameter mapping (Eq. 2.17) should be stated to avoid confusion.

Circularity Check

1 steps flagged

Partial circularity: the two VSMK slope parameters are the previously fitted SMK slopes for the same benchmark simulations, and VSMK reduces to those SMK limits by construction in the regimes claimed to be reproduced.

specific steps
  1. fitted input called prediction [Section 4, Eqs. (3.6)-(3.7), Eq. (3.4), Appendix A]
    "WVSMK = [1 + (k/kM)^{f(k)}]^{-1}, f(k)=β2-(β2-β1)[1+(μ k/kM)^δ]^{-1} ... The VSMK filter approaches a SMK filter with β=β1 for k/kM < 1/µ ∼ O(1), and a SMK filter with β=β2 for k/kM > 1/µ as directly follows from Eqs. (3.6) and (3.7). The values of β1 and β2 are chosen to match the optimal SMK slopes reported in [14, 34] and [41], respectively."

    The claim 'a single parameter set reproduces both regimes simultaneously' is assembled from two previously fitted slopes: β1 is the optimal SMK slope for the WDM simulation of [14] (4.8), and β2 is the optimal SMK slope for the DAO simulations of [41]/[52] (3.6/3.46). By Eqs. (3.6)-(3.7) and the Appendix quote, VSMK equals SMK(β1) for k/kM < 1/µ and SMK(β2) for k/kM > 1/µ. The WDM small-mass regime is controlled by the k/kM≪1 expansion (Eq. 3.4; HMF slope M^(β1-3)/3), so matching WDM is guaranteed once β1 is the fitted WDM slope; the DAO intermediate imprint is assigned to the k/kM≳1 side where f(k)→β2, so matching DAO is likewise inherited. The remaining parameters µ, δ, c are hand-set transition/calibration parameters, not independent predictions. The comparison is therefore a recombinat

full rationale

The VSMK filter is an explicit ansatz, and the derivation of the small-mass asymptotic slope from the k/kM≪1 expansion is internally consistent. There is no self-citation chain, no imported uniqueness theorem, and no machine-checked claim; the main circularity is the parameter-choice step. β1 and β2 are not predicted or independently fitted here; they are lifted from the optimal SMK fits to the very WDM and DAO simulations that serve as benchmarks. Because the filter is constructed to reduce to SMK(β1) and SMK(β2) in the two regimes that are respectively responsible for the WDM small-mass slope and the DAO intermediate-scale imprint, reproducing both regimes with those values is partly guaranteed by construction. The independent content is limited to the transition parameters µ and δ, which are kept fixed, and the visual/sensitivity check that the two limits coexist without spoiling each other. The asserted scale-inversion mechanism in Section 3 and Appendix A is a physical argument rather than a proof, and the limited number of tested kpeak values (two) and the absence of a µ/δ sensitivity scan affect external validity; these are correctness risks, not additional circularity. Overall, the central 'simultaneous reproduction' claim reduces in part to previously fitted inputs, warranting a partial circularity score of 6.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The VSMK model introduces five effective parameters (beta_1, beta_2, mu, delta, c) whose values are set by hand or taken from previous SMK fits to the same simulations. The PS formalism, Sheth-Tormen first-crossing distribution, mass-scale relation, and WDM/ETHOS transfer functions are assumed from prior literature. No new physical entities, particles, or forces are introduced.

free parameters (5)
  • beta_1 = 4.8
    Small-scale logarithmic slope in the VSMK filter; chosen to match the optimal SMK slope for WDM from Schaeffer & Schneider (2021) and Leo et al. (2018), as stated in Section 4.
  • beta_2 = 3.6 (3.46 for the Bohr et al. 2021 case)
    Intermediate-scale slope; chosen to match the optimal SMK slope for DAO from Verwohlt et al. (2024), with 'minor variations' for the Bohr et al. (2021) model.
  • mu = 2.1
    Transition scale in units of kM; kept fixed across all models without sensitivity analysis or a stated first-principles derivation.
  • delta = 12
    Sharpness of the slope transition; kept fixed across all models without sensitivity analysis or a stated first-principles derivation.
  • c = 3.6
    Mass-to-scale calibration parameter in Eq. (2.11); taken from the SMK literature range (3-3.7), shifts the HMF along the mass axis.
axioms (5)
  • domain assumption Press-Schechter/EPS formalism with ellipsoidal collapse and Sheth-Tormen first-crossing distribution (Eqs. 2.2-2.3).
    The entire HMF calculation rests on this framework; the paper does not derive it.
  • domain assumption Mass-scale relation M = (4/3) pi rho_bar (c R)^3 for filters whose real-space counterparts have divergent integrals (Eq. 2.11).
    Needed to convert the filter scale kM to a halo mass; the relation is assumed standard.
  • domain assumption The WDM and ETHOS transfer functions (Eqs. 2.13-2.16) faithfully describe the linear power spectra of the simulated models.
    The comparison to N-body simulations depends on these parametrized transfer functions being accurate; they are adopted from prior literature.
  • domain assumption For kM >> khm, the variance-derivative integrand is dominated by k/kM << 1, yielding dsigma^2/dkM ~ kM^{-beta_1-1}.
    This asymptotic approximation underlies Eq. (3.5); it is argued in Appendix A.1 from the location of the maximum of xi(k,kM), not proven for all beta, mu, delta.
  • domain assumption The scale-inversion mapping: small-mass HMF controlled by filter behavior at k/kM << 1, intermediate-mass DAO features by filter behavior at k/kM >~ 1.
    This is the central physical premise of the VSMK construction, argued heuristically in Section 3 and Appendix A; if it fails, the filter does not decouple the two regimes.

pith-pipeline@v1.3.0-alltime-deepseek · 16059 in / 14193 out tokens · 139791 ms · 2026-08-03T05:42:15.572911+00:00 · methodology

0 comments
read the original abstract

We introduce a variable-slope smooth-$k$ (VSMK) filter within the Press-Schechter formalism to model halo mass functions derived from damped and oscillatory matter power spectra. While the standard smooth-$k$ approach successfully captures small-scale suppression effects, it intrinsically couples these to oscillatory features at intermediate scales. The VSMK filter generalizes this framework by allowing the effective logarithmic slope of the $k$-space window function to vary smoothly between two asymptotic regimes, thereby decoupling the small-scale suppression of halo abundances from the intermediate-scale oscillatory features characteristic of dark acoustic oscillations (DAO). We compare the analytic predictions obtained with the VSMK filter to $N$-body simulations for warm dark matter and ETHOS-based models with DAO, showing that a single parameter set reproduces both regimes simultaneously. The VSMK filter thus provides a unified and flexible analytic framework for modeling halo abundances in non-cold dark matter scenarios with damped and oscillatory power spectra.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 55 linked inside Pith

  1. [1]

    P. J. E. Peebles,Large-scale background temperature and mass fluctuations due to scale-invariant primeval perturbations,Astrophys. J.263(1982) L1

    1982

  2. [2]

    Springel, S

    V. Springel, S. D. M. White, A. Jenkins, C. S. Frenk, N. Yoshida, L. Gao, J. Navarro, R. Thacker, D. Croton, J. Helly, J. A. Peacock, S. Cole, P. Thomas, H. Couchman, A. Evrard, J. Colberg and F. Pearce et al.,Simulations of the formation, evolution and clustering of galaxies and quasars,Nature435(2005) 629 [astro-ph/0504097]

    Pith/arXiv arXiv 2005

  3. [3]

    S. P. Driver, A. S. G. Robotham, D. Obreschkow, J. A. Peacock, I. K. Baldry, S. Bellstedt, J. Bland-Hawthorn, S. Brough, M. Cluver, B. W Holwerda, A. Hopkins, C. Lagos, J. Liske, J. Loveday, S. Phillipps, E. N. Taylor,An empirical measurement of the halo mass function from the combination of GAMA DR4, SDSS DR12, and REFLEX II data,Mon. Not. Roy. Astron. S...

    Pith/arXiv arXiv 2022

  4. [4]

    D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de Naray and A. H. G. Peter,Cold dark matter: controversies on small scales,Proc. Nat. Acad. Sci.112(2015) 12249 [arXiv:1306.0913]

    Pith/arXiv arXiv 2015

  5. [5]

    Newton, M

    O. Newton, M. Cautun, A. Jenkins, C. S. Frenk and J. C. Helly,The total satellite population of the Milky Way,Mon. Not. Roy. Astron. Soc.479(2018) 2853 [arXiv:1809.09625]

    Pith/arXiv arXiv 2018

  6. [6]

    M. G. Walker and J. Peñarrubia,A method for measuring (slopes of) the mass profiles of dwarf spheroidal galaxies,Astrophys. J.742(2011) 20 [arXiv:1108.2408]

    Pith/arXiv arXiv 2011

  7. [7]

    too big to fail

    E. Papastergis, R. Giovanelli, M. P. Haynes and F. Shankar,Is there a “too big to fail” problem in the field?,Astron. Astrophys.574(2015) A113 [arXiv:1412.4909]

    Pith/arXiv arXiv 2015

  8. [8]

    J. S. Bullock and M. Boylan-Kolchin,Small-scale challenges to theΛCDM paradigm,Ann. Rev. Astron. Astrophys.55(2017) 343 [arXiv:1707.04256]

    Pith/arXiv arXiv 2017

  9. [9]

    Shi and G

    X. Shi and G. M. Fuller,New dark matter candidate: nonthermal sterile neutrinos,Phys. Rev. Lett.82(1999) 2832 [hep-ph/9811448]

    Pith/arXiv arXiv 1999

  10. [10]

    Dodelson and L

    S. Dodelson and L. M. Widrow,Sterile neutrinos as dark matter,Phys. Rev. Lett.72(1994) 17 [hep-ph/9303287]

    Pith/arXiv arXiv 1994

  11. [11]

    P. Bode, J. P. Ostriker and N. Turok,Halo formation in warm dark matter models,Astrophys. J.556(2001) 93 [astro-ph/0106001]

    Pith/arXiv arXiv 2001

  12. [12]

    M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese and A. Riotto,Constraining warm dark matter candidates including sterile neutrinos and light gravitinos with WMAP and the Lyman-αforest,Phys. Rev. D71(2005) 063534 [astro-ph/0501562]

    Pith/arXiv arXiv 2005

  13. [13]

    Sameie, A

    O. Sameie, A. J. Benson, L. V. Sales, H. Yu, L. A. Moustakas and P. Creasey,The effect of dark matter–dark radiation interactions on halo abundance: a Press–Schechter approach, Astrophys. J.874(2019) 101 [arXiv:1901.09009]

    Pith/arXiv arXiv 2019

  14. [14]

    Schaeffer and A

    T. Schaeffer and A. Schneider,Dark acoustic oscillations: imprints on the matter power spectrum and the halo mass function,Mon. Not. Roy. Astron. Soc.504(2021) 3773 [arXiv:2101.04191]

    Pith/arXiv arXiv 2021

  15. [15]

    D. J. E. Marsh,Axion cosmology,Phys. Rept.643(2016) 1 [arXiv:1510.07633]

    Pith/arXiv arXiv 2016

  16. [16]

    Schive, T

    H.-Y. Schive, T. Chiueh, T. Broadhurst and K.-W. Huang,Contrasting galaxy formation from quantum wave dark matter withΛCDM,Astrophys. J.818(2016) 89 [arXiv:1508.04621]

    Pith/arXiv arXiv 2016

  17. [17]

    L. Hui, J. P. Ostriker, S. Tremaine and E. Witten,Ultralight scalars as cosmological dark matter,Phys. Rev. D95(2017) 043541 [arXiv:1610.08297]

    Pith/arXiv arXiv 2017

  18. [18]

    Dentler, D

    M. Dentler, D. J. E. Marsh, R. Hlozek, A. Laguë, K. K. Rogers and D. Grin,Fuzzy dark matter and the Dark Energy Survey Year 1 data,Mon. Not. Roy. Astron. Soc.515(2022) 5646 [arXiv:2111.01199]. – 16 –

    Pith/arXiv arXiv 2022

  19. [19]

    Hsueh, W

    J.-W. Hsueh, W. Enzi, S. Vegetti, M. W. Auger, C. D. Fassnacht, G. Despali, L. V. E. Koopmans and J. P. McKean,SHARP-VII. New constraints on the dark matter free-streaming properties and substructure abundance from gravitationally lensed quasars,Mon. Not. Roy. Astron. Soc.492(2019) 3047 [arXiv:1905.04182]

    Pith/arXiv arXiv 2019

  20. [20]

    Shirasaki, T

    M. Shirasaki, T. Ishiyama and S. I. Ando,Virial halo mass function in the Planck cosmology, Astrophys. J.922(2021) 89 [arXiv:2103.12345]

    Pith/arXiv arXiv 2021

  21. [21]

    Sipple, A

    J. Sipple, A. Lidz, D. Grin and G. Sun,Fuzzy dark matter constraints from the Hubble Frontier Fields,Mon. Not. Roy. Astron. Soc.538(2025) 1830 [arXiv:2407.17059]

    Pith/arXiv arXiv 2025

  22. [22]

    Despaliet al.,Detecting low-mass haloes with strong gravitational lensing I: the effect of data quality and lensing configuration,Mon

    G. Despaliet al.,Detecting low-mass haloes with strong gravitational lensing I: the effect of data quality and lensing configuration,Mon. Not. Roy. Astron. Soc.510(2022) 2480 [arXiv:2111.08718]

    Pith/arXiv arXiv 2022

  23. [23]

    Wagner-Carena, J

    S. Wagner-Carena, J. Lee, J. Pennington, S. Aalbers, S. Birrer and R. H. Wechsler,A strong gravitational lens is worth a thousand dark matter halos: inference on small-scale structure using sequential methods,Astrophys. J.975(2024) 297 [arXiv:2404.14487]

    Pith/arXiv arXiv 2024

  24. [24]

    R. E. Keeleyet al.,JWST lensed quasar dark matter survey II. Strongest gravitational lensing limit on the dark matter free-streaming length to date,Mon. Not. Roy. Astron. Soc.535(2024) 1652 [arXiv:2405.01620]

    Pith/arXiv arXiv 2024

  25. [25]

    Vegettiet al.,Strong gravitational lensing as a probe of dark matter,Space Sci

    S. Vegettiet al.,Strong gravitational lensing as a probe of dark matter,Space Sci. Rev.220 (2024) 58 [arXiv:2306.11781]

    Pith/arXiv arXiv 2024

  26. [26]

    Despali, F

    G. Despali, F. M. Heinze, C. D. Fassnacht, S. Vegetti, C. Spingola and R. Klessen et al., Detecting low-mass haloes with strong gravitational lensing – II. Constraints on the density profiles of two detected subhaloes,Astron. Astrophys.699(2025) A222 [arXiv:2407.12910]

    Pith/arXiv arXiv 2025

  27. [27]

    Banik, J

    N. Banik, J. Bovy, G. Bertone, D. Erkal and T. J. L. de Boer,Novel constraints on the particle nature of dark matter from stellar streams,JCAP10(2021) 043 [arXiv:1911.02663]

    Pith/arXiv arXiv 2021

  28. [28]

    Barry, A

    M. Barry, A. Wetzel, S. Chapman, J. Samuel, R. Sanderson and A. Arora,The dark side of FIRE: predicting the population of dark matter subhaloes around Milky Way-mass galaxies, Mon. Not. Roy. Astron. Soc.523(2023) 428 [arXiv:2303.05527]

    Pith/arXiv arXiv 2023

  29. [29]

    Bonaca and A

    A. Bonaca and A. M. Price-Whelan,Stellar streams in the Gaia era,New Astron. Rev.100 (2025) 101713 [arXiv:2405.19410]

    Pith/arXiv arXiv 2025

  30. [30]

    M. G. Jones, M. P. Haynes, R. Giovanelli and C. Moorman,The ALFALFA HI mass function: a dichotomy in the low-mass slope and a locally suppressed knee mass,Mon. Not. Roy. Astron. Soc.477(2018) 2 [arXiv:1802.00053]

    Pith/arXiv arXiv 2018

  31. [31]

    Zhanget al.,The FAST all sky HI survey (FASHI): the first release of catalog,Sci

    C.-P. Zhanget al.,The FAST all sky HI survey (FASHI): the first release of catalog,Sci. China Phys. Mech. Astron.67(2024) [arXiv:2312.06097]

    arXiv 2024

  32. [32]

    J. T. Garland, K. L. Masters and D. Grin,Using HI observations of low-mass galaxies to test ultralight axion dark matter,Mon. Not. Roy. Astron. Soc.535(2024) 1338 [arXiv:2403.04956]

    Pith/arXiv arXiv 2024

  33. [33]

    S. Bose, W. A. Hellwing, C. S. Frenk, A. Jenkins, M. R. Lovell, J. C. Helly and B. Li,The Copernicus Complexio: statistical properties of warm dark matter haloes,Mon. Not. Roy. Astron. Soc.455(2016) 318 [arXiv:1507.01998]

    Pith/arXiv arXiv 2016

  34. [34]

    M. Leo, C. M. Baugh, B. Li and S. Pascoli,A new smooth-kspace filter approach to calculate halo abundances,JCAP08(2018) 001 [arXiv:1801.02547]

    Pith/arXiv arXiv 2018

  35. [35]

    W. H. Press and P. Schechter,Formation of galaxies and clusters of galaxies by self-similar gravitational condensation,Astrophys. J.187(1974) 425

    1974

  36. [36]

    J. R. Bond, S. Cole, G. Efstathiou and N. Kaiser,Excursion set mass functions for hierarchical Gaussian fluctuations,Astrophys. J.379(1991) 440. – 17 –

    1991

  37. [37]

    R. K. Sheth and G. Tormen,An excursion set model of hierarchical clustering: ellipsoidal collapse and the moving barrier,Mon. Not. Roy. Astron. Soc.329(2002) 61 [astro-ph/0105113]

    Pith/arXiv arXiv 2002

  38. [38]

    Lukić, K

    Z. Lukić, K. Heitmann, S. Habib, S. Bashinsky and P. M. Ricker,The halo mass function: high-redshift evolution and universality,Astrophys. J.671(2007) 1160 [astro-ph/0702360]

    Pith/arXiv arXiv 2007

  39. [39]

    Schneider, R

    A. Schneider, R. E. Smith, A. V. Macciò and B. Moore,Non-linear evolution of cosmological structures in warm dark matter models,Mon. Not. Roy. Astron. Soc.424(2012) 684 [arXiv:1112.0330]

    Pith/arXiv arXiv 2012

  40. [40]

    A. J. Benson, A. Farahi, S. Cole, L. A. Moustakas, A. Jenkins, M. Lovell, R. Kennedy, J. Helly and C. Frenk,Dark matter halo merger histories beyond cold dark matter – I. Methods and application to warm dark matter,Mon. Not. Roy. Astron. Soc.428(2013) 1774 [arXiv:1209.3018]

    Pith/arXiv arXiv 2013

  41. [41]

    Verwohlt, C

    J. Verwohlt, C. A. Mason, J. B. Muñoz, F.-Y. Cyr-Racine, M. Vogelsberger and J. Zavala, Separating dark acoustic oscillations from astrophysics at cosmic dawn,Phys. Rev. D110 (2024) 103533 [arXiv:2404.17640]

    Pith/arXiv arXiv 2024

  42. [42]

    Schneider and S

    A. Schneider and S. Trujillo-Gomez,Constraining cosmology with the velocity function of low-mass galaxies,Mon. Not. Roy. Astron. Soc.475(2018) 4809 [arXiv:1706.09461]

    Pith/arXiv arXiv 2018

  43. [43]

    Kulkarni and J

    M. Kulkarni and J. P. Ostriker,What is the halo mass function in a fuzzy dark matter cosmology?,Mon. Not. Roy. Astron. Soc.510(2021) 1425 [arXiv:2011.02116]

    Pith/arXiv arXiv 2021

  44. [44]

    Schneider, R

    A. Schneider, R. E. Smith and D. Reed,Halo mass function and the free-streaming scale,Mon. Not. Roy. Astron. Soc.433(2013) 1573 [arXiv:1303.0839]

    Pith/arXiv arXiv 2013

  45. [45]

    Parimbelli, G

    G. Parimbelli, G. Scelfo, S. K. Giri, A. Schneider, M. Archidiacono, S. Camera and M. Viel, Mixed dark matter: matter power spectrum and halo mass function,JCAP12(2021) 044 [arXiv:2106.04588]

    Pith/arXiv arXiv 2021

  46. [46]

    F. Y. Cyr-Racine, K. Sigurdson, J. Zavala, T. Bringmann, M. Vogelsberger and C. Pfrommer, ETHOS — an effective theory of structure formation: From dark particle physics to the matter distribution of the Universe,Phys. Rev. D93(2016) 123527 [arXiv:1512.05344]

    Pith/arXiv arXiv 2016

  47. [47]

    Coles and F

    P. Coles and F. Lucchin,Cosmology: The Origin and Evolution of Cosmic Structure, John Wiley & Sons, New York (1995)

    1995

  48. [48]

    A. R. Zentner,The excursion set theory of halo mass functions, halo clustering, and halo growth,Int. J. Mod. Phys. D16(2007) 763 [arXiv:astro-ph/0611454]

    Pith/arXiv arXiv 2007

  49. [49]

    H. Mo, F. van den Bosch and S. White,Galaxy Formation and Evolution, Cambridge University Press, Cambridge (2010)

    2010

  50. [50]

    Maggiore and A

    M. Maggiore and A. Riotto,The halo mass function from the excursion set theory. I. Gaussian fluctuations with non-Markovian dependence on the smoothing scale,Astrophys. J.711(2010) 907 [arXiv:0903.1249]

    Pith/arXiv arXiv 2010

  51. [51]

    R. K. Sheth and G. Tormen,Large-scale bias and the peak background split,Mon. Not. Roy. Astron. Soc.308(1999) 119 [astro-ph/9901122]

    Pith/arXiv arXiv 1999

  52. [52]

    S. Bohr, J. Zavala, F.-Y. Cyr-Racine and M. Vogelsberger,The halo mass function and inner structure of ETHOS haloes at high redshift,Mon. Not. Roy. Astron. Soc.506(2021) 128 [arXiv:2101.08790]

    Pith/arXiv arXiv 2021

  53. [53]

    S. T. Brown, I. G. McCarthy, S. G. Stafford and A. S. Font,Towards a universal model for the density profiles of dark matter haloes,Mon. Not. Roy. Astron. Soc.509(2021) 5685 [arXiv:2110.01632]

    Pith/arXiv arXiv 2021

  54. [54]

    Diemer and A

    B. Diemer and A. V. Kravtsov,A universal model for halo concentrations,Astrophys. J.799 (2015) 108 [arXiv:1407.4730]. – 18 –

    Pith/arXiv arXiv 2015

  55. [55]

    A. D. Ludlow, S. Bose, R. E. Angulo, L. Wang, W. A. Hellwing, J. F. Navarro, S. Cole and C. S. Frenk,The mass–concentration–redshift relation of cold and warm dark matter haloes, Mon. Not. Roy. Astron. Soc.460(2016) 1214 [arXiv:1601.02624]

    Pith/arXiv arXiv 2016

  56. [56]

    Lacey and S

    C. Lacey and S. Cole,Merger rates in hierarchical models of galaxy formation,Mon. Not. Roy. Astron. Soc.262(1993) 627

    1993

  57. [57]

    Lewis, A

    A. Lewis, A. Challinor and A. Lasenby,Efficient computation of cosmic microwave background anisotropies in closed Friedmann-Robertson-Walker models,Astrophys. J.538(2000) 473 [astro-ph/9911177] [arXiv:astro-ph/9911177]

    Pith/arXiv arXiv 2000

  58. [58]

    Non-cold

    R. Murgia, A. Merle, M. Viel, M. Totzauer and A. Schneider,“Non-cold” dark matter at small scales: a general approach,JCAP11(2017) 046 [arXiv:1704.07838]

    Pith/arXiv arXiv 2017

  59. [59]

    S. Bohr, J. Zavala, F. Y. Cyr-Racine, M. Vogelsberger, T. Bringmann and C. Pfrommer, ETHOS – an effective parametrization and classification for structure formation: the non-linear regime atz≳5,Mon. Not. Roy. Astron. Soc.498(2020) 3403 [arXiv:2006.01842]

    Pith/arXiv arXiv 2020

  60. [60]

    Archidiacono, S

    M. Archidiacono, S. Bohr, S. Hannestad, J. H. Jørgensen and J. Lesgourgues,Linear scale bounds on dark matter-dark radiation interactions and connection with the small scale crisis of cold dark matter,JCAP11(2017) 010 [arXiv:1706.06870]

    Pith/arXiv arXiv 2017

  61. [61]

    Aghanim, Y

    Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Battye, K. Benabed, J. P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher and C. Burigana et al.,Planck 2018 results – VI. Cosmological pa...

    Pith/arXiv arXiv 2018

  62. [62]

    Z. J. Xu,Dark matter halo mass functions and density profiles from mass and energy cascade, Sci. Rep.13(2023) 16531 [arXiv:2210.01200]. – 19 –

    Pith/arXiv arXiv 2023