Large Scale Structure Tests of Warm Dark Matter

Lawrence M. Widrow; Scott Dodelson; Stephane Colombi

REVIEW

Large Scale Structure Tests of Warm Dark Matter

Stephane Colombi, Scott Dodelson, Lawrence M. Widrow

Astrophysics · Submitted 5 May 1995

Pith's verdict: A careful, honest 1995 test that mostly succeeds at what it claims, but the headline negative result leans on an assumed near-unity bias the authors themselves flag as a plausibility argument.

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arXiv IDastro-ph/9505029 (version 1)

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CitationLawrence M. Widrow, Scott Dodelson, Stephane Colombi. "Large Scale Structure Tests of Warm Dark Matter." arXiv:astro-ph/9505029 (1995). https://pith.science/paper/FKKPDXFN

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Editor's note

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The thing worth knowing: this is the systematic version of the warm dark matter test that the early-80s papers never did. Colombi, Dodelson and Widrow build a one-parameter WDM family, compute the linear transfer functions from the Boltzmann hierarchy, show the transfer function depends only on alpha/m_x (their Eq. 22), and then run matched N-body simulations against CDM and HDM. The scaling result and the Appendix fitting functions are genuinely reusable, and the matched-initial-conditions comparison is a real step beyond the prior literature. The linear analysis is clean and the m = 2 m_0 choice is legitimate model selection against APM excess power, not a fit dressed up as a prediction. No circularity problem here, contrary to what one might worry about from the abstract alone. The reader's report is right on that point and I'd leave it alone. Where the stress-test note lands, mostly: the small-scale verdict rests on comparing simulated mass to galaxy data under b ~ 1 and b_v ~ 1. Section 5.3.4 concedes that saving the model needs antibias b < 1 and b_v < 0.5 and calls that 'rather unlikely' — that is a plausibility argument, not a measurement, and the abstract's 'indicates that WDM... has too much power' is a notch more confident than the section supports. Two things sharpen this. First, the velocity dispersion test, which the abstract singles out as 'particularly evident,' is the most velocity-bias-sensitive of the four and the least statistically robust — five realizations in a 144 Mpc box, dominated by rare clusters, with the authors' own caveat in 5.1 that the PM code may underestimate small-scale velocities by up to 30 percent. Second, the CfA2 measurement of ~1400 km/s, which the authors set aside from the Mo et al. compilation, actually sits close to what the simulations give. That outlier deserves more weight than a parenthetical. The multiplicity function analysis assumes constant M/L = 123 and is called 'rather crude' in 5.4; treat it as suggestive only. Credit where due: the authors are transparent about all of this, and the conclusion section is properly hedged to the Omega = 1, Lambda = 0, Omega_B = 0, h = 0.5, n = 1 context. The classifier's 'limitations: none found' is wrong — abstract:s10 and sections 5.3.4 and 5.4 state them plainly. Who should read it: anyone working on dark matter structure formation or on the history of the WDM/CDM debate. The transfer function fits alone justify a look. This deserves a serious referee, mainly to press on the bias assumptions and the velocity dispersion statistics rather than on the linear work.

Referee report

Major objections (2)

[Abstract (s8-s10); Section 5.3.4] The abstract attaches the caveat 'though this conclusion is based on the simple assumption that light traces mass in groups of galaxies' only to the multiplicity-function sentence (s10), but Section 5.3.4 states that the pairwise velocity dispersion test also requires velocity bias b_v < 0.5 to reconcile WDM with the data. The abstract's sentence 'This is particularly evident in the small-scale pairwise velocity dispersion' (s9) therefore carries no caveat even though the body identifies the same bias dependence. The abstract should either extend the caveat to the velocity dispersion test or soften 'particularly evident.' Relatedly, Section 5.3.4's judgment that antibias b<1 and b_v<0.5 are 'rather unlikely' is a plausibility argument, not a measurement; the paper should quantify how unlikely (e.g., from the cited biasing studies) or soften the language.
[Section 5.3.3, Fig. 9] The velocity dispersion test is presented in the abstract as the most evident failure of WDM, but Section 5.3.3 itself notes that the errorbars 'are quite large especially for r < 10 Mpc where sigma_1 is dominated by rare, large, and hot clusters,' and the measurement rests on five realizations in a 144 Mpc box. The paper also notes that the PM code 'tends to underestimate small scale velocities,' which strengthens the discrepancy but also means the comparison is resolution-limited. Given that this is the least statistically robust of the four small-scale tests, the abstract should not single it out as the most evident; either present a more careful statistical treatment (e.g., cluster-by-cluster comparison or a larger volume) or demote it relative to the power-spectrum and correlation-function tests.

Minor objections (9)

[Section 5.1, Table 1] The PMS particle mass is quoted as M_part = 9.88 x 10^11 M_sun for L_box = 144 Mpc and N_par = 128^3 with Omega_m = 1 and h = 0.5. This is a factor of 10 too large: the same calculation that gives PML's 1.23 x 10^13 M_sun for L_box = 720 Mpc gives approximately 9.9 x 10^10 M_sun for L_box = 144 Mpc (a factor of 125 smaller volume). The quoted value should be corrected.
[Fig. 3] The WDM curve uses Omega_B = 0 while the CDM, MDM, and HDM curves use Omega_B = 0.01. The paper estimates the effect at no more than 10% for EP, but does not show the Omega_B = 0.01 WDM curve or quantify the shift at intermediate k. Since Fig. 3 is the central linear comparison, either show the matched-Omega_B WDM curve or state the expected shift at each k.
[Appendix] The fitting formula is stated valid for k < 0.5 Mpc^-1, but sigma_0.5 (used for the z_gf estimates in Section 4.1) involves k up to approximately 2 Mpc^-1. Either extend the fit or note the extrapolation and its effect on sigma_0.5.
[Section 5.2, Section 6] 'Structures as small as galaxy halos are excluded' is a linear-theory statement about free-streaming suppression, not a simulation result: the PMS resolution (1.125 Mpc, approximately 9.9 x 10^10 M_sun per particle) cannot resolve galaxy halos. Rephrase to distinguish the linear-theory suppression scale from the simulation resolution.
[Section 5.3.1] The claim that the nonlinear power spectrum is 'only weakly dependent on initial conditions' is stronger than Fig. 7 supports: the HDM curve still differs noticeably from CDM and WDM at small scales. Soften to 'much closer than the linear spectra' or quantify the residual differences.
[Section 5.3.4] The statement that 'the power spectrum of this WDM distribution is approximately twice as large as the one directly measured in the full WDM distribution' (for galaxies selected in overdense regions) is a quantitative claim not shown in a figure. Add a figure or a table entry.
[Section 2.4, Eq. (8)] The relation between the sterile-neutrino mass m_2 and the early-decoupled mass m_1 would benefit from a brief derivation or a citation to the source.
[Section 4.2] The paper says the errorbars on the PD data points are 'based on a simple visual estimate of the vertical scatter in their figure 6'; state the estimated scatter (e.g., a factor of 2) so the reader can judge the comparison.
[Abstract] 'pairwize' should be 'pairwise.'

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The paper

The paper claims

Warm dark matter is tested against the observed large-scale structure, and it fits the largest scales while overshooting on small ones.

The reading

The paper asks whether a dark matter particle with a small but non-negligible velocity dispersion — warm dark matter, sitting between the hot and cold extremes — can fit the observed structure of the universe better than either. The authors collapse a whole family of candidate particles into a single parameter, the mass-to-temperature ratio $m_x/T_x$, compute the linear power spectrum for each member of the family, and find that a particle with twice the standard hot-dark-matter neutrino's ratio (about 46 eV) matches the data on scales larger than $8h^{-1}$ Mpc. They then run N-body simulations of that model beside cold and hot dark matter and compare the resulting matter distribution with galaxy catalogues. The result is a split verdict: warm dark matter inherits the pancake-like collapse and smooth final distribution of hot dark matter while also producing virialized rich clusters, but it still carries too much power on small scales, predicts pairwise velocities roughly twice those observed, and makes too many rich clusters. Within the paper's minimal assumptions the model is therefore not a good fit, though the authors note that the small-scale verdict rests on assuming that light traces mass.

What carries the argument

The load-bearing device is the reduction of a three-parameter distribution function, $f_x(v) = [e^{p/ T} + 1]^{-1}$ with parameters $T$, $T$ and $m_x$, to one physically relevant combination. Fixing the relic density ties $T^4$ to $m_x/93$ eV, and a change of integration variable from momentum to $q = p/T_x$ shows that every integral in the Boltzmann and Einstein equations depends on $m_x$ and $T$ only through $m_x/T$, so the transfer function depends only on the ratio of the particle's temperature to its mass. That is what makes a one-parameter family possible at all, and it lets the authors bracket their models between Holtzman's cold and hot dark matter transfer functions and compare directly with mixed dark matter, which interpolates differently. The second piece of machinery is the pair of statistical tests used to pick the best member: the excess-power statistic $EP = 3.4 T_{25}/T_8$, normalized to 1 for standard cold dark matter and requiring about 1.3 from the APM data, and the linear estimate $1+z_{gf} = T_{0.5}$ for the epoch of galaxy-scale collapse. The third is the simulation suite: particle-mesh and tree codes at $128^3$ and $64^3$ resolution in 144 Mpc and 720 Mpc boxes, all sharing the same random phases so that differences between models are not seed noise.

Extended reading notes

Core claim

The central claim is that warm dark matter, defined as any particle whose velocity dispersion matters for structure formation but is smaller than a standard hot-dark-matter neutrino's, occupies a real and previously under-explored middle ground, and that its best-fitting member still fails the small-scale tests. The authors show that the linear power spectrum of their one-parameter family depends only on $m_x/T_x$, so a single number interpolates between hot and cold dark matter; the value $m_x = 2m_0$ (a 46 eV particle in their accounting) gives a reasonable fit to the reconstructed linear power spectrum of Peacock & Dodds and to the excess-power statistic comparing $25h^{-1}$ Mpc with $8h^{-1}$ Mpc. The N-body runs then show that this model collapses first into large pancake-like sheets, leaves a smooth matter distribution with no galaxy-scale halos, and yet forms dense, spherical, virialized clusters that hot dark matter does not produce. Against the data, however, the simulated matter distribution has too much power for $k > 1h^{-1}$ Mpc, the line-of-sight pairwise velocity dispersion comes out more than a factor of two above the observed values, and the group multiplicity function has the wrong shape with too many rich clusters. The authors' own summary is that, within $h=0.5$, $n=1$, $g=1$, $B=0$ and no cosmological constant, warm dark matter does not agree well with the data.

Load-bearing premise

The small-scale failure is judged on the assumption that galaxies trace the dark matter with bias near one and velocity bias near one; the authors themselves note that fitting the data would require antibias with $b < 1$ and $b_v < 0.5$, which they call rather unlikely but do not rule out.

Editorial extensions

If this is right

A warm dark matter particle with $m_x/T_x$ about twice the standard neutrino value is a viable fit to structure on scales above $8h^{-1}$ Mpc, so searches for such a particle are not ruled out by the large-scale power spectrum.
Because the linear transfer function depends only on $m_x/T_x$, any two candidate particles sharing that ratio — an early-decoupled species and an oscillation-produced sterile neutrino, for instance — make identical predictions for large-scale structure and cannot be told apart by these tests.
Warm dark matter produces virialized rich clusters where hot dark matter does not, so cluster abundance is a discriminator between the two scenarios even when their large-scale power spectra are similar.
The model excludes galaxy-scale halos and predicts that the first objects to collapse are pancake-like sheets, which sets a floor on the smallest structures a warm dark matter universe can form.
If the model is right, matching the observed small-scale clustering requires the galaxy distribution to be antibiased relative to the mass, with bias factors below one and velocity bias below about 0.5.

Reading between the lines

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

The paper's negative small-scale verdict and its positive large-scale verdict are not independent: the same 46 eV particle is being judged against two different normalizations of the same data, and the authors show in their Figure 3 that switching between optical and IRAS normalization moves the comparison substantially. A reader should treat the 'fails at small scales' conclusion as conditional o
The nonlinear power spectra of cold and warm dark matter converge as the simulations evolve, which suggests that small-scale clustering statistics are weak discriminators between dark matter candidates in general, and that the discriminating power lives in the velocity field and in cluster abundances rather than in $P(k)$.
The multiplicity-function failure is the most fragile of the three small-scale objections, because it depends on a single assumed mass-to-light ratio of $123\,M_\odot/L_\odot$ applied uniformly across groups of very different richness; a mass-to-light ratio that rises with richness would remove much of the discrepancy without changing the dark matter model.
A clean follow-up would be to measure the velocity bias directly by comparing galaxy and mass velocity fields in a hydrodynamical version of the same pancake simulation, since the whole small-scale objection hinges on a quantity the paper can only argue about by plausibility.

Original abstract

Warm dark matter (WDM) is an intriguing model of structure formation from the point of view of both cosmology and particle physics. We consider a one-parameter family of WDM models. The linear power spectra for these models is calculated and compared with the corresponding spectra for cold dark matter (CDM), hot dark matter (HDM) and mixed dark matter (MDM) as well as the power spectrum derived from observations. Our linear analyses suggest that a model universe dominated by a particle whose mass to temperature ratio $m_x/T_x$ is increased by a factor of two as compared with the standard HDM neutrino gives a reasonable fit to the data on large $(>8h^{-1} {\rm ~Mpc})$ scales. $N$-body simulations for this particular WDM model show features of both HDM and CDM. As in HDM, the first objects to collapse are large pancake-like structures. The final matter distribution is rather smooth and structures as small as galaxy halos are excluded. However, there appear to be virialized rich clusters evident in the CDM but not the HDM simulations. Unfortunately, a simple comparison of the matter distribution and its statistical properties with observations indicates that WDM, like CDM, has too much power at small scales. This is particularly evident in the small-scale pairwize velocity dispersion. The cluster multiplicity function has the wrong shape with too many rich clusters being produced, though this conclusion is based on the simple assumption that light traces mass in groups of galaxies.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,188 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.95 · human participants 0.01 · controlled comparison 0.62 · randomized assignment 0.01 · states quantitative result 0.85 · compares with prior results 0.94 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Warm dark matter (WDM) is an intriguing model of structure formation from the point of view of both cosmology and particle physics. We consider a one-parameter family of WDM models. The linear power spectra for these models is calculated and compared with the corresponding spectra for cold dark matter (CDM), hot dark matter (HDM) and mixed dark matter (MDM) as well as the power spectrum derived from observations. Our linear analyses suggest that a model universe dominated by a particle whose mass to temperature ratio $m_x/T_x$ is increased by a factor of two as compared with the standard HDM neutrino gives a reasonable fit to the data on large $(>8h^{-1} {\rm ~Mpc})$ scales. $N$-body simulations for this particular WDM model show features of both HDM and CDM. As in HDM, the first objects to collapse are large pancake-like structures. The final matter distribution is rather smooth and structures as small as galaxy halos are excluded. However, there appear to be virialized rich clusters evident in the CDM but not the HDM simulations. Unfortunately, a simple comparison of the matter distribution and its statistical properties with observations indicates that WDM, like CDM, has too much power at small scales. This is particularly evident in the small-scale pairwize velocity dispersion. The cluster multiplicity function has the wrong shape with too many rich clusters being produced, though this conclusion is based on the simple assumption that light traces mass in groups of galaxies.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Warm dark matter (WDM) is an intriguing model of structure formation from the point of view of both cosmology and particle physics.	background 1.00	0.05	0.02	asserted 0.98
s1	We consider a one-parameter family of WDM models.	method 0.59	0.86	0.02	asserted 0.88
s2	The linear power spectra for these models is calculated and compared with the corresponding spectra for cold dark matter (CDM), hot dark matter (HDM) and mixed dark matter (MDM) as well as the power spectrum derived from observations.	method 0.86	0.89	0.04	asserted 0.99
s3	Our linear analyses suggest that a model universe dominated by a particle whose mass to temperature ratio $m_x/T_x$ is increased by a factor of two as compared with the standard HDM neutrino gives a reasonable fit to the data on large $(>8h^{-1} {\rm ~Mpc})$ scales.	result 0.97	0.91	0.47	hedged possibility 0.99
s4	$N$-body simulations for this particular WDM model show features of both HDM and CDM.	result 0.91	0.91	0.07	asserted 1.00
s5	As in HDM, the first objects to collapse are large pancake-like structures.	result 0.98	0.80	0.03	asserted 1.00
s6	The final matter distribution is rather smooth and structures as small as galaxy halos are excluded.	result 0.99	0.84	0.08	asserted 0.96
s7	However, there appear to be virialized rich clusters evident in the CDM but not the HDM simulations.	result 0.94	0.53	0.05	hedged possibility 0.78
s8	Unfortunately, a simple comparison of the matter distribution and its statistical properties with observations indicates that WDM, like CDM, has too much power at small scales.	result 0.86	0.86	0.51	asserted 0.55
s9	This is particularly evident in the small-scale pairwize velocity dispersion.	result 0.97	0.79	0.08	asserted 0.93
s10	The cluster multiplicity function has the wrong shape with too many rich clusters being produced, though this conclusion is based on the simple assumption that light traces mass in groups of galaxies.	result 0.67	0.78	0.21	asserted 0.69

What kind of paper the classifier reports this is: simulation study 0.96 · evidence from simulation or computation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Transfer functions for WDM (top) and MDM (bottom) models. k is in units of h Mpc1 . 8

**Figure 2.** Excess power in theories interpolating between HDM and CDM. Solid curve shows how WDM with its free parameter m [lower axis] interpolates. Note how it quickly becomes similar to CDM. Dashed curve shows the interpolation of MDM with its free parameter, the energy density in neutrinos [upper axis]. The observationally prefered valueof EP is 1:3. It is generally accepted that the power in density uctuations on 25 h1 Mpc relative to 8 h1 Mpc is greater in the data than in the standard CDM model. To quantify this, we rst de ne the linear rms density uctuations on a scale R: R h(M=M)…

**Figure 4.** Thin slices Lbox=32 thick extracted from the simulations PML (lefts panels) of physical size Lbox = 720 Mpc and the simulations PMS (right panels) of physical size Lbox = 144 Mpc. The top, middle and bottom panels correspond respectively to the CDM, WDM and HDM model

**Figure 6.** CfA-like redshift slices of the observed galaxy distribution. The observer is located at the bottom of each slice. The slices are 256 Mpc deep (with our choice of the Hubble constant). The top panel represents a slice of the real observed galaxy distribution (courtesy V. de Lapparent). The others panels correspond to arti cial catalogs built from the simulations, taking into account redshift-space distortions and selection e ects (see text). From top to bottom, one passes from CDM to WDM and HDM. The central slices have the same geometry as the CfA slice, i.e. are covering the…

**Figure 8.** Measured two body correlation function in our WDM, CDM and HDM simulations. The dashed line is the power-law t of Davis & Peebles (1983) in the observed galaxy distribution. worsens the situation for WDM. 5.3.2 Correlation function

**Figure 9.** The quantity 1 (de ned in Eq. [27]) as function of separation ` measured in our N-body experiments, compared to measurements in the observed galaxy distribution. The two dashed lines at the bottom left correspond to the measurement of Davis & Peebles (1983) on the CfA1 catalog and the thick vertical segment to a compilation of the more recent measurements of Mo et al. (1993) on various galaxy catalogs (see text). provides another probe of structure on galaxy and cluster scales. Here, v(x) stands for the peculiar velocity of the matter measured in our simulations. 1(r) calculated…

**Figure 10.** The multiplicity function measured in our N-body experiments (curves with errorbars, see text) compared to the measurement in the CfA galaxy catalog by Moore, Frenk & White (1993) (circles). We assume here that groups and clusters have a constant mass to light ratio M=L = 123M =L , where M =L is the mass to light ratio of the sun. The unit of mass choosen to compute the multiplicity is M = 2:88 1011 solar masses. A cluster of mulplicity X has a mass comprised between 2X1M and 2XM. that M=L is the same for all ob jects. We use M=L = 123 M =L as estimated by MFW. For the simulations,…

Showing 7 of the paper's 10 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9505029 by the authors.

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Lyman-alpha Forests cool Warm Dark Matter
astro-ph.CO 2015-12 conditional novelty 4.0 of 10

The Lyman-alpha forest flux power spectrum from 13,821 SDSS-III quasar spectra requires warm dark matter particles heavier than 4.09 keV (thermal relic) or 24.4 keV (Dodelson-Widrow sterile neutrino) at 95% confidence.
Cosmological constraints on a light non-thermal sterile neutrino
astro-ph 2008-12 conditional novelty 4.0 of 10

Cosmological data require a fourth thermalized neutrino to be lighter than about 0.4 eV at 2 sigma, and a 2 eV sterile neutrino survives at 3 sigma only if it is under-abundant or cooler than active neutrinos.
Populating dark sectors with relativistic bubble walls
hep-ph 2024-12 conditional novelty 3.0 of 10

Relativistic bubble walls can pair-produce dark matter much heavier than the phase transition scale, which then free-streams as warm dark matter.
Light sterile neutrinos
hep-ph 2015-07 conditional novelty 3.0 of 10

A review of eV-scale sterile neutrinos argues that reactor, Gallium, and LSND anomalies can be fit by 3+1 short-baseline oscillations, while acknowledging appearance-disappearance and cosmological tensions.
Dark Matter Candidates: A Ten-Point Test
astro-ph 2007-11 conditional novelty 3.0 of 10

The paper defines ten necessary tests for a dark matter candidate and grades sixteen proposed particles against them, concluding that most survive while charged CHAMPs and the first Kaluza-Klein graviton do not.
Dark Matter Self-interactions and Small Scale Structure
hep-ph 2017-05 conditional novelty 2.0 of 10

Self-interacting dark matter with a cross section near 1 cm^2/g that falls mildly with velocity can reconcile galactic cores and the too-big-to-fail problem with cluster-scale constraints.
Sterile neutrinos in cosmology
hep-ph 2017-05 conditional novelty 2.0 of 10

Light sterile neutrinos below about 10 keV can alter early-Universe cosmology, and keV-scale ones are viable dark matter candidates whose decay may explain the unidentified 3.5 keV X-ray line.
Axion Cosmology
astro-ph.CO 2015-10 accept novelty 2.0 of 10

Axions can act as dark matter, dark energy, or dark radiation depending on mass, and existing cosmological data already exclude a single ultralight axion species below about 10^-24 eV from composing all of the dark matter.

How the literature uses it

57 arXiv papers cite this in 79 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States a conflicting result 1 paper

Our results show that the over-clustering problem on sub-galactic scales can be overcome by WDM models, and all the advantages of CDM models are preserved by WDM models.

Warm Dark Matter Model of Galaxy Formation · 0.95

Papers citing it · 55 sentences without a confident label

Uses its method or tool 1 paper

In particular, we use Refs. [ref] to convert from the thermal mass $M$ to the Dodelson-Widrow mass $m_s$ [equation] where $Ω_M$ is the energy density fraction of the thermal WDM component.

One likelihood to bind them all: Lyman-$\alpha$ constraints on non-standard dark matter · Mixed warm and cold dark matter · 0.93

Compares with it 1 paper

A nonzero $Δ$ has a similar (but not the same) role to that of the normalization factor $β$ in the warm dark matter model considered in [ref] , because, in the classical limit, $exp(Δ)$ becomes the normalization factor of the distribution.

Cosmology of Single Species Hidden Dark Matter · Background evolution I - thermal equilibrium case · 0.87

Cites it as support 2 papers

Actually, for these two models, the degeneracy is exact: it can be shown by a change of variable in the background and linear perturbation equations that the two models are strictly equivalent from the point of view of cosmological observables [ref] .

Cosmological constraints on a light non-thermal sterile neutrino · Light sterile neutrino in cosmology: physical effects and parametrization · 0.98
Such estimates are also supported by theoretical and simulation based results [ref] .

Light Dirac neutrino portal dark matter with gauged $U(1)_{B-L}$ symmetry · FIMP type dark matter · 0.99

Cites it as background 18 papers

* Non-Resonant Production (NRP) [ref] , where the sterile neutrino is produced out-of-equilibrium at temperatures $[𝒪(100)]MeV$ from oscillations of active neutrinos with a FD distribution $f(p,(4/11)^1/3 T)$ that is suppressed by the active–sterile mixing angle [ref] .

Cold keV dark matter from decays and scatterings · Small-scale structure constraints · 0.94
The DM particles which have such non-negligible velocities are denoted in the literature as warm (WDM), with typical velocities $V^WDM_eq∼10^-5.$ In the thermal mechanisms [ref] for the production of WDM, the DM candidate mass has to be small, around keV mass scale.

Populating dark sectors with relativistic bubble walls · Production of dark matter via renormalisable operator · 0.86
For theoretical and simulation based studies of dark matter free-streaming properties, one may refer to [ref] .

Freeze-in Dark Matter via Light Dirac Neutrino Portal · Structure formation constraints · 0.97

Discussion

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