pith. sign in

arxiv: 2606.30739 · v1 · pith:CWPMNAX3new · submitted 2026-06-29 · 🌌 astro-ph.CO · astro-ph.GA

Fuzzy Dark Matter Halo Mass Functions at Cosmic Dawn

Raghunath Ghara , Adam Lidz , Daniel Grin , Jackson Sipple This is my paper

Pith reviewed 2026-07-01 01:54 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords fuzzy dark matterhalo mass functioncosmic dawnN-body simulationsmixed dark matterJWSTultraviolet luminosity function
0
0 comments X

The pith

A new fitting formula for fuzzy dark matter halo mass functions at redshifts 6-11 shows weaker suppression than previous simulation results.

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

The paper runs N-body simulations of fuzzy dark matter and mixed models to derive a calibrated halo mass function. It removes spurious halos from numerical noise and provides a fitting formula valid for z=6-11. This matters for predicting the abundance of small halos that seed early galaxies, which JWST can observe through the ultraviolet luminosity function. The new formula indicates about 30 percent less suppression at certain masses compared to earlier fits.

Core claim

We provide a new halo mass function fitting formula, calibrated over z=6-11, applicable to pure FDM and mixed dark matter scenarios. For m c² = 10^{-21} eV and M ∼ 3 × 10^9 M_⊙ we find a ∼30% weaker suppression than earlier simulation-based formulas predict, which we attribute to their extrapolation beyond the m_FDM range previously simulated.

What carries the argument

The new halo mass function fitting formula, obtained from N-body simulations after identifying and removing spurious halos due to discreteness noise.

If this is right

  • The fitting formula applies to both pure fuzzy dark matter and mixed fuzzy-cold dark matter models.
  • It shows better agreement with prior simulation-based formulas than with semi-analytic models.
  • Upcoming JWST observations of the UVLF probing M_UV ≳ -13 at z ≳ 10 will test these models.
  • The formula can be used to model delayed galaxy formation during cosmic dawn in fuzzy dark matter cosmologies.

Where Pith is reading between the lines

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

  • Extending the simulations to lower FDM masses or higher redshifts could further refine the formula.
  • The weaker suppression might allow more early galaxies than previously modeled, affecting reionization timelines.
  • Direct comparison with upcoming lensed galaxy counts could confirm or refute the 30% difference.

Load-bearing premise

The procedure for identifying and removing spurious halos accurately quantifies and corrects the systematic uncertainty without removing genuine halos.

What would settle it

Running higher-resolution simulations or comparing the predicted halo number densities directly against observed galaxy counts at z greater than 10 for halo masses around 3 times 10^9 solar masses.

Figures

Figures reproduced from arXiv: 2606.30739 by Adam Lidz, Daniel Grin, Jackson Sipple, Raghunath Ghara.

Figure 1
Figure 1. Figure 1: Comparison of different HMFs in an FDM model with m22 = 10 at z = 8. The top panel shows the FDM HMFs computed using different prescriptions, along with the corresponding CDM HMF. The bottom panel displays the ratio of the CDM to FDM HMFs as a function of halo mass. See Appendix A for detailed descriptions of each model. imately account for modified linear growth in FDM [FDM￾TH-δc(M)] (e.g. Marsh & Silk 20… view at source ↗
Figure 2
Figure 2. Figure 2: Dimensionless power spectra ∆2 (k) = k 3P(k)/2π 2 at z = 100 for the different dark matter scenarios considered in this study. The suppression in the initial power spectrum from FDM moves to smaller scales (higher k) with increasing m22, while the fluctuations are less strongly suppressed as the FDM fraction fF decreases. The initial conditions for our N-body simulations follow these power spectrum models.… view at source ↗
Figure 3
Figure 3. Figure 3: The proto-halo sphericity (top) and overlap (bottom) coefficients as a function of halo mass at z = 8. The black points and error bars show the CDM results, while the green ones show the same for an FDM model with m22 = 1 and fF = 1. The shaded regions show the 2σ spread. Note that the simulated halo abundance is small at M ≳ 1010.5M⊙, leading to noise in the shaded boundaries. Low mass FDM halos often sho… view at source ↗
Figure 4
Figure 4. Figure 4: The top panel shows the z = 8 CDM and FDM HMFs for m22 = 1. The black stars show the CDM HMF after remov￾ing dark matter halos with S and O coefficients smaller than ST and OT, respectively. The grey squares are the (nearly identical) CDM results without excising halos. The green squares give the FDM HMF without removing halos, while the green circles show the FDM HMF (mean and 1σ Poisson error bars) after… view at source ↗
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: The ratio of CDM and FDM HMFs at z = 8 for the full range of FDM scenarios considered in this work. The solid curves establish that our new fitting formula successfully describes this full range of models. – for the most lenient cut shown in the figure with ST and OT both = 0.1, compared to the most stringent case with ST and OT = 0.4. However, the sensitivity to threshold choice is confined to high R port… view at source ↗
Figure 7
Figure 7. Figure 7: Ratio of the CDM and FDM HMFs (R) as a function of halo mass for different combinations of ST and OT in pure FDM models with m22 = 1 (top), m22 = 10 (middle), and m22 = 20 (bottom) at a representative redshift of z = 8. The solid blue lines show our fitting formula, while the dashed blue are from S2016. intermediate m22 mixed DM scenarios, in order to cover the transition between the two fitting formulas. … view at source ↗
Figure 9
Figure 9. Figure 9: UVLF models and current data for m22 = 5 and fF = 1 at z = 8, 10 and 12. The black curve assumes the CDM HMF, while other curves are for different FDM HMF/UVLF models. The points with 1σ error bars show UVLF measurements from Bouwens et al. (2021) (blue, field galaxies) and Bouwens et al. (2022) (red, lensed galaxies). Our new HMF model should be used to refine previous FDM constraints from the UVLFs and t… view at source ↗
Figure 10
Figure 10. Figure 10: UVLF models and data at z = 8 − 12 for mixed FDM-CDM scenarios. The curves show example scenarios with different m22 and fF values, computed using the fitting formula of Equation 4 and the conditional luminosity function from Equation 5. The data points are the same as in [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The cumulative abundance of dark matter halos for m22 = 5 and fF = 1 at z = 8, 10 and 12. The black curve is for CDM, while the other curves show different HMF model variants for FDM. 108 109 1010 Mmin(M ) 10−2 10−1 100 101 N(> Mmin, z) z = 8 108 109 1010 Mmin(M ) 10−2 100 z = 10 108 109 1010 Mmin(M ) 10−4 10−3 10−2 10−1 100 z = 12 CDM m22 = 1, fF = 1 m22 = 5, fF = 1 m22 = 5, fF = 0.3 m22 = 5, fF = 0.5 [… view at source ↗
Figure 12
Figure 12. Figure 12: The cumulative abundance of dark matter halos at z = 8, 10 and 12 for the different dark matter scenarios considered in this study. Here, we use the fitting formula of Equation 4. while the FDM-Schive HMF yields the maximum ratio. This spread follows the trends seen for the HMFs in Section 2. In this case (z = 8, m22 = 5, MAB = −12), our new fitting formula anchors the CDM to FDM UVLF ratio at 7.1, clos￾e… view at source ↗
read the original abstract

In fuzzy dark matter (FDM) cosmological models, wave effects impact astrophysical length scales, suppressing the abundance of small mass dark matter halos, and delaying the earliest phases of galaxy formation during Cosmic Dawn. Current and upcoming James Webb Space Telescope (JWST) measurements of the galaxy ultraviolet luminosity function (UVLF) will allow unprecedented tests of this suppression, yet significant uncertainties remain in theoretical models of the FDM halo mass function. We run a new suite of N-body simulations with FDM particle masses of $mc^{2}=10^{-22}\,{\rm eV} - 2 \times 10^{-21}$ eV and mixed FDM-cold dark matter (CDM) models with FDM mass fractions of $f_{\mathrm{F}} = 0.3-1$. We identify and remove spurious halos from discreteness noise and quantify the associated systematic uncertainty. We provide a new halo mass function fitting formula, calibrated over $z=6-11$, applicable to pure FDM and mixed dark matter scenarios. Our results are in better agreement with previous simulation-based fitting formulas than with current semi-analytic mass function models. Nevertheless, for $m c^{2} = 10^{-21}$ eV and $M \sim 3 \times 10^9 M_\odot$ we find a $\sim 30\%$ weaker suppression than earlier simulation-based formulas predict, which we attribute to their extrapolation beyond the $m_{\rm FDM}$ range previously simulated. Applying our fitting formula to the UVLF, we find that upcoming JWST observations behind foreground lensing clusters, probing $M_{\rm UV} \gtrsim -13$ at $z \gtrsim 10$, will provide a powerful test of FDM and mixed dark matter models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript runs new N-body simulations of pure FDM (mc² = 10^{-22}–2×10^{-21} eV) and mixed FDM-CDM models (f_F = 0.3–1), removes spurious halos from discreteness noise while quantifying the associated systematic uncertainty, and calibrates a new halo mass function fitting formula over z = 6–11. The formula is stated to apply to both pure and mixed scenarios; the authors report ~30% weaker suppression than earlier simulation-based formulas at mc² = 10^{-21} eV and M ∼ 3×10^9 M_⊙ (attributed to prior extrapolation), find better agreement with previous simulation-based fits than with semi-analytic models, and discuss implications for JWST UVLF constraints.

Significance. If the spurious-halo removal is shown to be robust, the work supplies a directly calibrated fitting formula for high-redshift FDM halo abundances that improves upon semi-analytic prescriptions and yields concrete, falsifiable predictions for lensed JWST UVLF measurements at M_UV ≳ −13 and z ≳ 10. The explicit quantification of cleaning systematics and the extension to mixed models are positive features.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (halo identification): The headline numerical result—a ~30% weaker suppression at mc² = 10^{-21} eV, M ∼ 3×10^9 M_⊙—rests entirely on the post-cleaning catalog. No explicit criteria for flagging spurious halos, no resolution-convergence tests at the relevant mass and redshift, and no demonstration that genuine FDM halos remain untouched are provided; without these, it is impossible to assess whether the reported difference from earlier formulas is physical or an artifact of the removal step.
  2. [§4] §4 (comparison and fitting formula): The new fitting formula is calibrated directly to the cleaned simulations, yet the manuscript does not show the raw versus cleaned mass functions, the effect of varying the cleaning threshold on the 30% discrepancy, or how the quoted systematic uncertainty brackets residual contamination at the quoted mass and redshift.
minor comments (1)
  1. [Abstract] Abstract: the FDM mass range is written as 10^{-22} eV – 2×10^{-21} eV; confirm whether the bounds are inclusive and whether the mixed-model runs use the same particle masses.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of our methods and results.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (halo identification): The headline numerical result—a ~30% weaker suppression at mc² = 10^{-21} eV, M ∼ 3×10^9 M_⊙—rests entirely on the post-cleaning catalog. No explicit criteria for flagging spurious halos, no resolution-convergence tests at the relevant mass and redshift, and no demonstration that genuine FDM halos remain untouched are provided; without these, it is impossible to assess whether the reported difference from earlier formulas is physical or an artifact of the removal step.

    Authors: We appreciate the referee highlighting the need for greater clarity on the halo-cleaning procedure. While §3 describes the removal of spurious halos due to discreteness noise and quantifies the associated systematic uncertainty, we agree that explicit flagging criteria, resolution-convergence tests at the relevant masses and redshifts, and explicit validation that genuine FDM halos are unaffected should be presented more directly. In the revised manuscript we will add these elements to demonstrate that the ~30% weaker suppression is physical rather than an artifact. revision: yes

  2. Referee: [§4] §4 (comparison and fitting formula): The new fitting formula is calibrated directly to the cleaned simulations, yet the manuscript does not show the raw versus cleaned mass functions, the effect of varying the cleaning threshold on the 30% discrepancy, or how the quoted systematic uncertainty brackets residual contamination at the quoted mass and redshift.

    Authors: We agree that direct comparisons would help readers evaluate the cleaning step. In the revised version we will include raw versus cleaned mass-function comparisons, examine the sensitivity of the reported 30% discrepancy to changes in the cleaning threshold, and clarify how the quoted systematic uncertainty encompasses possible residual contamination at the masses and redshifts of interest. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained calibration to new simulations

full rationale

The paper runs new N-body simulations for specified FDM masses and mixed models, removes spurious halos while quantifying systematic uncertainty, and calibrates a new fitting formula directly to those results over z=6-11. This is an explicit empirical calibration rather than any reduction by construction, self-definition, or load-bearing self-citation. The reported 30% difference is presented as a comparison to prior work (attributed to extrapolation range), with no equations or steps shown that equate outputs to inputs by definition. The derivation chain remains independent of the target result.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim depends on the accuracy of N-body evolution for FDM wave effects, the completeness of spurious-halo identification, and the assumption that the chosen mass and redshift range is representative for the fitting formula. No new particles or forces are postulated.

free parameters (1)
  • fitting coefficients in the new halo mass function formula
    The formula is calibrated to the simulation outputs; the coefficients are determined by fitting rather than derived from first principles.
axioms (2)
  • domain assumption N-body simulations with the chosen particle mass and force resolution correctly capture the suppression of small-scale structure due to FDM wave effects
    Invoked when running the suite and when claiming the results apply to real cosmology.
  • domain assumption The method used to identify and remove spurious halos removes only discreteness artifacts and leaves the true halo abundance unchanged within the quoted systematic uncertainty
    Central to the claim that the new formula is reliable; location is the sentence describing removal and uncertainty quantification.

pith-pipeline@v0.9.1-grok · 5869 in / 1520 out tokens · 31682 ms · 2026-07-01T01:54:16.485506+00:00 · methodology

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

72 extracted references · 72 canonical work pages · 5 internal anchors

  1. [1]

    Armengaud E., Palanque-Delabrouille N., Y \`e che C., Marsh D. J. E., Baur J., 2017, @doi [ ] 10.1093/mnras/stx1870 , https://ui.adsabs.harvard.edu/abs/2017MNRAS.471.4606A 471, 4606

    work page doi:10.1093/mnras/stx1870 2017

  2. [2]

    Arvanitaki A., Dimopoulos S., Dubovsky S., Kaloper N., March-Russell J., 2010, @doi [ ] 10.1103/PhysRevD.81.123530 , https://ui.adsabs.harvard.edu/abs/2010PhRvD..81l3530A 81, 123530

    work page doi:10.1103/physrevd.81.123530 2010

  3. [3]

    A GLIMPSE of the 99%: a census of the faintest galaxies during the epoch reionization and its implications for galaxy formation models

    Atek H., et al., 2026, @doi [arXiv e-prints] 10.48550/arXiv.2604.23823 , https://ui.adsabs.harvard.edu/abs/2026arXiv260423823A p. arXiv:2604.23823

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.23823 2026

  4. [4]

    Banik N., Bovy J., Bertone G., Erkal D., de Boer T. J. L., 2021, @doi [ ] 10.1088/1475-7516/2021/10/043 , https://ui.adsabs.harvard.edu/abs/2021JCAP...10..043B 2021, 043

    work page doi:10.1088/1475-7516/2021/10/043 2021

  5. [5]

    Bar-Or B., Fouvry J.-B., Tremaine S., 2019, @doi [ ] 10.3847/1538-4357/aaf28c , https://ui.adsabs.harvard.edu/abs/2019ApJ...871...28B 871, 28

    work page doi:10.3847/1538-4357/aaf28c 2019

  6. [6]

    Axiverse Lampposts

    Baryakhtar M., Cyncynates D., Henry E., 2026, @doi [arXiv e-prints] 10.48550/arXiv.2602.23424 , https://ui.adsabs.harvard.edu/abs/2026arXiv260223424B p. arXiv:2602.23424

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2602.23424 2026

  7. [7]

    J., et al., 2013, @doi [ ] 10.1093/mnras/sts159 , https://ui.adsabs.harvard.edu/abs/2013MNRAS.428.1774B 428, 1774

    Benson A. J., et al., 2013, @doi [ ] 10.1093/mnras/sts159 , https://ui.adsabs.harvard.edu/abs/2013MNRAS.428.1774B 428, 1774

    work page doi:10.1093/mnras/sts159 2013

  8. [8]

    J., et al., 2021, @doi [ ] 10.3847/1538-3881/abf83e , https://ui.adsabs.harvard.edu/abs/2021AJ....162...47B 162, 47

    Bouwens R. J., et al., 2021, @doi [ ] 10.3847/1538-3881/abf83e , https://ui.adsabs.harvard.edu/abs/2021AJ....162...47B 162, 47

    work page doi:10.3847/1538-3881/abf83e 2021

  9. [9]

    J., Illingworth G., Ellis R

    Bouwens R. J., Illingworth G., Ellis R. S., Oesch P., Stefanon M., 2022, @doi [ ] 10.3847/1538-4357/ac86d1 , https://ui.adsabs.harvard.edu/abs/2022ApJ...940...55B 940, 55

    work page doi:10.3847/1538-4357/ac86d1 2022

  10. [10]

    Bouwens R., Illingworth G., Oesch P., Stefanon M., Naidu R., van Leeuwen I., Magee D., 2023, @doi [ ] 10.1093/mnras/stad1014 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.523.1009B 523, 1009

    work page doi:10.1093/mnras/stad1014 2023

  11. [11]

    Bozek B., Marsh D. J. E., Silk J., Wyse R. F. G., 2015, @doi [ ] 10.1093/mnras/stv624 , https://ui.adsabs.harvard.edu/abs/2015MNRAS.450..209B 450, 209

    work page doi:10.1093/mnras/stv624 2015

  12. [12]

    D., Ringwald A., 2012, @doi [J

    Cicoli M., Goodsell M. D., Ringwald A., 2012, @doi [J. High Energy Phys.] 10.1007/JHEP10(2012)146 , https://ui.adsabs.harvard.edu/abs/2012JHEP...10..146C 2012, 146

    work page doi:10.1007/jhep10(2012)146 2012

  13. [13]

    S., Agarwal S., Marsh D

    Corasaniti P. S., Agarwal S., Marsh D. J. E., Das S., 2017, @doi [ ] 10.1103/PhysRevD.95.083512 , https://ui.adsabs.harvard.edu/abs/2017PhRvD..95h3512C 95, 083512

    work page doi:10.1103/physrevd.95.083512 2017

  14. [14]

    Dalal N., Bovy J., Hui L., Li X., 2021, @doi [ ] 10.1088/1475-7516/2021/03/076 , https://ui.adsabs.harvard.edu/abs/2021JCAP...03..076D 2021, 076

    work page doi:10.1088/1475-7516/2021/03/076 2021

  15. [15]

    Dentler M., Marsh D. J. E., Hlo z ek R., Lagu \"e A., Rogers K. K., Grin D., 2022, @doi [ ] 10.1093/mnras/stac1946 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.515.5646D 515, 5646

    work page doi:10.1093/mnras/stac1946 2022

  16. [16]

    Dome T., May S., Lagu \"e A., Marsh D. J. E., Johnston S., Bose S., Tocher A., Fialkov A., 2025, @doi [ ] 10.1093/mnras/staf005 , https://ui.adsabs.harvard.edu/abs/2025MNRAS.537..252D 537, 252

    work page doi:10.1093/mnras/staf005 2025

  17. [17]

    T., et al., 2024, @doi [ ] 10.1093/mnras/stae2037 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.533.3222D 533, 3222

    Donnan C. T., et al., 2024, @doi [ ] 10.1093/mnras/stae2037 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.533.3222D 533, 3222

    work page doi:10.1093/mnras/stae2037 2024

  18. [18]

    C., 2017, @doi [ ] 10.1093/mnras/stw2724 , https://ui.adsabs.harvard.edu/abs/2017MNRAS.465..941D 465, 941

    Du X., Behrens C., Niemeyer J. C., 2017, @doi [ ] 10.1093/mnras/stw2724 , https://ui.adsabs.harvard.edu/abs/2017MNRAS.465..941D 465, 941

    work page doi:10.1093/mnras/stw2724 2017

  19. [19]

    J., Hu W., 1998, @doi [ ] 10.1086/305424 , https://ui.adsabs.harvard.edu/abs/1998ApJ...496..605E 496, 605

    Eisenstein D. J., Hu W., 1998, @doi [ ] 10.1086/305424 , https://ui.adsabs.harvard.edu/abs/1998ApJ...496..605E 496, 605

    work page doi:10.1086/305424 1998

  20. [20]

    J., 2013, @doi [ ] 10.1093/mnras/stt987 , https://ui.adsabs.harvard.edu/abs/2013MNRAS.433.3428F 433, 3428

    Farahi A., Benson A. J., 2013, @doi [ ] 10.1093/mnras/stt987 , https://ui.adsabs.harvard.edu/abs/2013MNRAS.433.3428F 433, 3428

    work page doi:10.1093/mnras/stt987 2013

  21. [21]

    Ferreira E. G. M., 2021, @doi [ ] 10.1007/s00159-021-00135-6 , https://ui.adsabs.harvard.edu/abs/2021A&ARv..29....7F 29, 7

    work page doi:10.1007/s00159-021-00135-6 2021

  22. [22]

    Harikane Y., et al., 2023, @doi [ ] 10.3847/1538-4365/acaaa9 , https://ui.adsabs.harvard.edu/abs/2023ApJS..265....5H 265, 5

    work page doi:10.3847/1538-4365/acaaa9 2023

  23. [23]

    Array programming with NumPy,

    Harris C. R., et al., 2020, @doi [Nature] 10.1038/s41586-020-2649-2 , 585, 357

    work page doi:10.1038/s41586-020-2649-2 2020

  24. [24]

    Hlo z ek R., Marsh D. J. E., Grin D., 2018, @doi [ ] 10.1093/mnras/sty271 , https://ui.adsabs.harvard.edu/abs/2018MNRAS.476.3063H 476, 3063

    work page doi:10.1093/mnras/sty271 2018

  25. [25]

    Hlozek R., Grin D., Marsh D. J. E., Ferreira P. G., 2015, @doi [ ] 10.1103/PhysRevD.91.103512 , https://ui.adsabs.harvard.edu/abs/2015PhRvD..91j3512H 91, 103512

    work page doi:10.1103/physrevd.91.103512 2015

  26. [26]

    Hu W., Barkana R., Gruzinov A., 2000, @doi [ ] 10.1103/PhysRevLett.85.1158 , https://ui.adsabs.harvard.edu/abs/2000PhRvL..85.1158H 85, 1158

    work page doi:10.1103/physrevlett.85.1158 2000

  27. [27]

    Hui L., 2021, @doi [ ] 10.1146/annurev-astro-120920-010024 , https://ui.adsabs.harvard.edu/abs/2021ARA&A..59..247H 59, 247

    work page doi:10.1146/annurev-astro-120920-010024 2021

  28. [28]

    P., Tremaine S., Witten E., 2017, @doi [Phys

    Hui L., Ostriker J. P., Tremaine S., Witten E., 2017, @doi [Phys. Rev. D] 10.1103/PhysRevD.95.043541 , 95, 043541

    work page doi:10.1103/physrevd.95.043541 2017

  29. [29]

    Matplotlib: A 2D Graphics Environment

    Hunter J. D., 2007, @doi [Computing in Science & Engineering] 10.1109/MCSE.2007.55 , 9, 90

    work page doi:10.1109/mcse.2007.55 2007

  30. [30]

    G., Bolton J

    Ir s i c V., Viel M., Haehnelt M. G., Bolton J. S., Becker G. D., 2017, @doi [ ] 10.1103/PhysRevLett.119.031302 , https://ui.adsabs.harvard.edu/abs/2017PhRvL.119c1302I 119, 031302

    work page doi:10.1103/physrevlett.119.031302 2017

  31. [31]

    Halo mass functions in mixed cold and fuzzy dark matter models

    Johnston S. C., et al., 2026, @doi [arXiv e-prints] 10.48550/arXiv.2606.06599 , https://ui.adsabs.harvard.edu/abs/2026arXiv260606599J p. arXiv:2606.06599

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.06599 2026

  32. [32]

    R., Knebe A., 2009, @doi [ ] 10.1088/0067-0049/182/2/608 , https://ui.adsabs.harvard.edu/abs/2009ApJS..182..608K 182, 608

    Knollmann S. R., Knebe A., 2009, @doi [ ] 10.1088/0067-0049/182/2/608 , https://ui.adsabs.harvard.edu/abs/2009ApJS..182..608K 182, 608

    work page doi:10.1088/0067-0049/182/2/608 2009

  33. [33]

    Kobayashi T., Murgia R., De Simone A., Ir s i c V., Viel M., 2017, @doi [ ] 10.1103/PhysRevD.96.123514 , https://ui.adsabs.harvard.edu/abs/2017PhRvD..96l3514K 96, 123514

    work page doi:10.1103/physrevd.96.123514 2017

  34. [34]

    P., 2022, @doi [ ] 10.1093/mnras/stab3520 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.510.1425K 510, 1425

    Kulkarni M., Ostriker J. P., 2022, @doi [ ] 10.1093/mnras/stab3520 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.510.1425K 510, 1425

    work page doi:10.1093/mnras/stab3520 2022

  35. [35]

    The Atacama Cosmology Telescope: Probing new signatures of ultralight axions with gravitational lensing

    Lagu \"e A., et al., 2026, @doi [arXiv e-prints] 10.48550/arXiv.2606.06410 , https://ui.adsabs.harvard.edu/abs/2026arXiv260606410L p. arXiv:2606.06410

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.06410 2026

  36. [36]

    M., Li B., Pascoli S., 2018, @doi [ ] 10.1088/1475-7516/2018/04/010 , https://ui.adsabs.harvard.edu/abs/2018JCAP...04..010L 2018, 010

    Leo M., Baugh C. M., Li B., Pascoli S., 2018, @doi [ ] 10.1088/1475-7516/2018/04/010 , https://ui.adsabs.harvard.edu/abs/2018JCAP...04..010L 2018, 010

    work page doi:10.1088/1475-7516/2018/04/010 2018

  37. [37]

    L., 2019, @doi [ ] 10.1103/PhysRevD.99.063509 , https://ui.adsabs.harvard.edu/abs/2019PhRvD..99f3509L 99, 063509

    Li X., Hui L., Bryan G. L., 2019, @doi [ ] 10.1103/PhysRevD.99.063509 , https://ui.adsabs.harvard.edu/abs/2019PhRvD..99f3509L 99, 063509

    work page doi:10.1103/physrevd.99.063509 2019

  38. [38]

    Lidz A., Hui L., 2018, @doi [ ] 10.1103/PhysRevD.98.023011 , https://ui.adsabs.harvard.edu/abs/2018PhRvD..98b3011L 98, 023011

    work page doi:10.1103/physrevd.98.023011 2018

  39. [39]

    X., Gonz \'a lez-Morales A

    Linares Cede \ n o F. X., Gonz \'a lez-Morales A. X., Ure \ n a-L \'o pez L. A., 2021, @doi [ ] 10.1088/1475-7516/2021/01/051 , https://ui.adsabs.harvard.edu/abs/2021JCAP...01..051L 2021, 051

    work page doi:10.1088/1475-7516/2021/01/051 2021

  40. [40]

    Liu R., Hu W., Grin D., 2025, @doi [ ] 10.1103/1z4c-1w7f , https://ui.adsabs.harvard.edu/abs/2025PhRvD.112b3513L 112, 023513

    work page doi:10.1103/1z4c-1w7f 2025

  41. [41]

    R., Frenk C

    Lovell M. R., Frenk C. S., Eke V. R., Jenkins A., Gao L., Theuns T., 2014, @doi [ ] 10.1093/mnras/stt2431 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.439..300L 439, 300

    work page doi:10.1093/mnras/stt2431 2014

  42. [42]

    Marsh D. J. E., 2016a, @doi [arXiv e-prints] 10.48550/arXiv.1605.05973 , https://ui.adsabs.harvard.edu/abs/2016arXiv160505973M p. arXiv:1605.05973

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1605.05973

  43. [43]

    Marsh D. J. E., 2016b, @doi [ ] 10.1016/j.physrep.2016.06.005 , https://ui.adsabs.harvard.edu/abs/2016PhR...643....1M 643, 1

    work page doi:10.1016/j.physrep.2016.06.005 2016

  44. [44]

    Marsh D. J. E., Silk J., 2014, @doi [ ] 10.1093/mnras/stt2079 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.437.2652M 437, 2652

    work page doi:10.1093/mnras/stt2079 2014

  45. [45]

    May S., Springel V., 2021, @doi [ ] 10.1093/mnras/stab1764 , https://ui.adsabs.harvard.edu/abs/2021MNRAS.506.2603M 506, 2603

    work page doi:10.1093/mnras/stab1764 2021

  46. [46]

    May S., Springel V., 2023, @doi [ ] 10.1093/mnras/stad2031 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.524.4256M 524, 4256

    work page doi:10.1093/mnras/stad2031 2023

  47. [47]

    O., et al., 2021, @doi [ ] 10.1103/PhysRevLett.126.091101 , https://ui.adsabs.harvard.edu/abs/2021PhRvL.126i1101N 126, 091101

    Nadler E. O., et al., 2021, @doi [ ] 10.1103/PhysRevLett.126.091101 , https://ui.adsabs.harvard.edu/abs/2021PhRvL.126i1101N 126, 091101

    work page doi:10.1103/physrevlett.126.091101 2021

  48. [48]

    O., An R., Gluscevic V., Benson A., Du X., 2025, @doi [ ] 10.3847/1538-4357/adceef , https://ui.adsabs.harvard.edu/abs/2025ApJ...986..127N 986, 127

    Nadler E. O., An R., Gluscevic V., Benson A., Du X., 2025, @doi [ ] 10.3847/1538-4357/adceef , https://ui.adsabs.harvard.edu/abs/2025ApJ...986..127N 986, 127

    work page doi:10.3847/1538-4357/adceef 2025

  49. [49]

    Nebrin O., Ghara R., Mellema G., 2019, @doi [ ] 10.1088/1475-7516/2019/04/051 , https://ui.adsabs.harvard.edu/abs/2019JCAP...04..051N 2019, 051

    work page doi:10.1088/1475-7516/2019/04/051 2019

  50. [50]

    Ni Y., Wang M.-Y., Feng Y., Di Matteo T., 2019, @doi [ ] 10.1093/mnras/stz2085 , https://ui.adsabs.harvard.edu/abs/2019MNRAS.488.5551N 488, 5551

    work page doi:10.1093/mnras/stz2085 2019

  51. [51]

    C., 2020, @doi [Progress in Particle and Nuclear Physics] 10.1016/j.ppnp.2020.103787 , https://ui.adsabs.harvard.edu/abs/2020PrPNP.11303787N 113, 103787

    Niemeyer J. C., 2020, @doi [Progress in Particle and Nuclear Physics] 10.1016/j.ppnp.2020.103787 , https://ui.adsabs.harvard.edu/abs/2020PrPNP.11303787N 113, 103787

    work page doi:10.1016/j.ppnp.2020.103787 2020

  52. [52]

    Nori M., Murgia R., Ir s i c V., Baldi M., Viel M., 2019, @doi [ ] 10.1093/mnras/sty2888 , https://ui.adsabs.harvard.edu/abs/2019MNRAS.482.3227N 482, 3227

    work page doi:10.1093/mnras/sty2888 2019

  53. [53]

    O'Hare C. A. J., 2024, @doi [arXiv e-prints] 10.48550/arXiv.2403.17697 , https://ui.adsabs.harvard.edu/abs/2024arXiv240317697O p. arXiv:2403.17697

    work page doi:10.48550/arxiv.2403.17697 2024

  54. [54]

    Planck Collaboration et al., 2020, @doi [ ] 10.1051/0004-6361/201833910 , https://ui.adsabs.harvard.edu/abs/2020A&A...641A...6P 641, A6

    work page doi:10.1051/0004-6361/201833910 2020

  55. [55]

    Power C., Robotham A. S. G., Obreschkow D., Hobbs A., Lewis G. F., 2016, @doi [ ] 10.1093/mnras/stw1644 , https://ui.adsabs.harvard.edu/abs/2016MNRAS.462..474P 462, 474

    work page doi:10.1093/mnras/stw1644 2016

  56. [56]

    H., Schechter P., 1974, @doi [ ] 10.1086/152650 , https://ui.adsabs.harvard.edu/abs/1974ApJ...187..425P 187, 425

    Press W. H., Schechter P., 1974, @doi [ ] 10.1086/152650 , https://ui.adsabs.harvard.edu/abs/1974ApJ...187..425P 187, 425

    work page doi:10.1086/152650 1974

  57. [57]

    E., Kravtsov A

    Robertson B. E., Kravtsov A. V., Tinker J., Zentner A. R., 2009, @doi [ ] 10.1088/0004-637X/696/1/636 , https://ui.adsabs.harvard.edu/abs/2009ApJ...696..636R 696, 636

    work page doi:10.1088/0004-637x/696/1/636 2009

  58. [58]

    K., Peiris H

    Rogers K. K., Peiris H. V., 2021, @doi [ ] 10.1103/PhysRevLett.126.071302 , https://ui.adsabs.harvard.edu/abs/2021PhRvL.126g1302R 126, 071302

    work page doi:10.1103/physrevlett.126.071302 2021

  59. [59]

    K., Hlo z ek R., Lagu \"e A., Ivanov M

    Rogers K. K., Hlo z ek R., Lagu \"e A., Ivanov M. M., Philcox O. H. E., Cabass G., Akitsu K., Marsh D. J. E., 2023, @doi [ ] 10.1088/1475-7516/2023/06/023 , https://ui.adsabs.harvard.edu/abs/2023JCAP...06..023R 2023, 023

    work page doi:10.1088/1475-7516/2023/06/023 2023

  60. [60]

    Schive H.-Y., Chiueh T., Broadhurst T., 2014a, @doi [Nature Physics] 10.1038/nphys2996 , https://ui.adsabs.harvard.edu/abs/2014NatPh..10..496S 10, 496

    work page doi:10.1038/nphys2996

  61. [61]

    P., 2014b, @doi [ ] 10.1103/PhysRevLett.113.261302 , https://ui.adsabs.harvard.edu/abs/2014PhRvL.113z1302S 113, 261302

    Schive H.-Y., Liao M.-H., Woo T.-P., Wong S.-K., Chiueh T., Broadhurst T., Hwang W.-Y. P., 2014b, @doi [ ] 10.1103/PhysRevLett.113.261302 , https://ui.adsabs.harvard.edu/abs/2014PhRvL.113z1302S 113, 261302

    work page doi:10.1103/physrevlett.113.261302

  62. [62]

    Schive H.-Y., Chiueh T., Broadhurst T., Huang K.-W., 2016, @doi [ ] 10.3847/0004-637X/818/1/89 , https://ui.adsabs.harvard.edu/abs/2016ApJ...818...89S 818, 89

    work page doi:10.3847/0004-637x/818/1/89 2016

  63. [63]

    E., Reed D., 2013, @doi [ ] 10.1093/mnras/stt829 , https://ui.adsabs.harvard.edu/abs/2013MNRAS.433.1573S 433, 1573

    Schneider A., Smith R. E., Reed D., 2013, @doi [ ] 10.1093/mnras/stt829 , https://ui.adsabs.harvard.edu/abs/2013MNRAS.433.1573S 433, 1573

    work page doi:10.1093/mnras/stt829 2013

  64. [64]

    Schutz K., 2020, @doi [ ] 10.1103/PhysRevD.101.123026 , https://ui.adsabs.harvard.edu/abs/2020PhRvD.101l3026S 101, 123026

    work page doi:10.1103/physrevd.101.123026 2020

  65. [65]

    2002, MNRAS, 329, 355, doi: 10.1046/j.1365-8711.2002.04988.x

    Sheth R. K., Tormen G., 2002, @doi [ ] 10.1046/j.1365-8711.2002.04950.x , https://ui.adsabs.harvard.edu/abs/2002MNRAS.329...61S 329, 61

    work page doi:10.1046/j.1365-8711.2002.04950.x 2002

  66. [66]

    Sipple J., Lidz A., Grin D., Sun G., 2025, @doi [ ] 10.1093/mnras/staf340 , https://ui.adsabs.harvard.edu/abs/2025MNRAS.538.1830S 538, 1830

    work page doi:10.1093/mnras/staf340 2025

  67. [67]

    Springel V., 2005, @doi [ ] 10.1111/j.1365-2966.2005.09655.x , https://ui.adsabs.harvard.edu/abs/2005MNRAS.364.1105S 364, 1105

    work page doi:10.1111/j.1365-2966.2005.09655.x 2005

  68. [68]

    Springel V., et al., 2005, @doi [ ] 10.1038/nature03597 , https://ui.adsabs.harvard.edu/abs/2005Natur.435..629S 435, 629

    work page doi:10.1038/nature03597 2005

  69. [69]

    Wang J., White S. D. M., 2007, @doi [ ] 10.1111/j.1365-2966.2007.12053.x , https://ui.adsabs.harvard.edu/abs/2007MNRAS.380...93W 380, 93

    work page doi:10.1111/j.1365-2966.2007.12053.x 2007

  70. [70]

    K., Hlo z ek R., Marsh D

    Winch H., Rogers K. K., Hlo z ek R., Marsh D. J. E., 2024, @doi [ ] 10.3847/1538-4357/ad7a73 , https://ui.adsabs.harvard.edu/abs/2024ApJ...976...40W 976, 40

    work page doi:10.3847/1538-4357/ad7a73 2024

  71. [71]

    T., Su G.-M., Schive H.-Y., Chiueh T., Ostriker J

    Yang H.-Y., Chiang B. T., Su G.-M., Schive H.-Y., Chiueh T., Ostriker J. P., 2024, @doi [ ] 10.1093/mnras/stae793 , https://ui.adsabs.harvard.edu/abs/2024MNRAS.530..129Y 530, 129

    work page doi:10.1093/mnras/stae793 2024

  72. [72]

    Zhang J., Hui L., 2006, @doi [ ] 10.1086/499802 , https://ui.adsabs.harvard.edu/abs/2006ApJ...641..641Z 641, 641

    work page doi:10.1086/499802 2006