# Information field based global Bayesian inference of the jet transport
coefficient

Man Xie  Key Laboratory of Quark and Lepton Physics (MOE) & Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China  Weiyao
Ke<EMAIL_ADDRESS>Department of Physics, University of California,
Berkeley, California 94720, USA  Nuclear Science Division MS 70R0319, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  Theoretical
Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA  Hanzhong
Zhang<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics
(MOE) & Institute of Particle Physics, Central China Normal University, Wuhan
430079, China  Xin-Nian Wang<EMAIL_ADDRESS>Department of Physics, University
of California, Berkeley, California 94720, USA  Nuclear Science Division MS
70R0319, Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA

###### Abstract

Bayesian statistical inference is a powerful tool for model-data comparisons
and extractions of physical parameters that are often unknown functions of
system variables. Existing Bayesian analyses often rely on explicit
parametrizations of the unknown function. It can introduce long-range
correlations that impose fictitious constraints on physical parameters in
regions of the variable space that are not probed by the experimental data. We
develop an information field (IF) approach to modeling the prior distribution
of the unknown function that is free of long-range correlations. We apply the
IF approach to the first global Bayesian inference of the jet transport
coefficient $\hat{q}$ as a function of temperature ($T$) from all existing
experimental data on single-inclusive hadron, di-hadron and $\gamma$-hadron
spectra in heavy-ion collisions at RHIC and LHC energies. The extracted
$\hat{q}/T^{3}$ exhibits a strong $T$-dependence as a result of the
progressive constraining power when data from more central collisions and at
higher colliding energies are incrementally included. The IF method guarantees
that the extracted $T$-dependence is not biased by a specific functional form.

Introduction– Recent developments in Bayesian statistical inference have
brought fruitful advances in physics research and greatly improved our ability
to constrain complex physical models with ever increasing data. Some prime
examples in high-energy nuclear physics are the nominal study of the equation
of state of the quark-gluon plasma (QGP) [1], simultaneous studies of initial
conditions of nuclear collisions and QGP shear and bulk viscosity [2, 3, 4, 5,
6, 7, 8, 9, 10], heavy quark diffusion parameter [11, 12, 13] and jet
transport coefficient for parton energy loss in the QGP [14, 15]. The pivot of
these analyses is the Bayes’ Theorem: “Posterior $=$ Prior $\times$
Likelihood”. It describes how a prior belief of the distribution of certain
physical parameters will be updated to the posterior distribution after
comparing theory with new experimental data, as encoded in the likelihood
function that quantifies the model’s descriptive power.

Many physical quantities to be inferred from data are unknown functions of
system variables. For example, viscosity depends on temperature and chemical
potential of the QGP. The jet transport coefficient depends on temperature and
parton energy [16, 17, 15]. The nuclear parton distributions can be functions
of the parton momentum fraction, the renormalization scale and the impact-
parameter [18, 19]. In all recent analyses, these unknown functions are
parameterized in intuitive forms either for convenience or motivated by
physical insights. Bayesian inferences are then performed to extract these
parameters from model-data comparisons. A major drawback of explicit
parameterizations is that they often introduce unwarranted long-range
correlations in the prior distributions of physical functions between
different regions of the variable space, where different experimental data
sets are supposed to provide independent constraints. For example, data in
peripheral heavy-ion collisions and at lower colliding energies, which only
probe QGP at low temperatures, can lead to fictitious constraints on physical
quantities at higher temperatures as predetermined by the parameterizations.

In this work, we design a framework for Bayesian inference of unknown
functions based on the idea of the information field (IF) [20, 21, 22]. The
advantage of the IF approach is that it provides a non-parametric
representation of the unknown functional space and eliminates prior bias and
unnecessary long-range correlations in the prior. It can be easily generalized
to higher-dimensional functional inference. Furthermore, it facilitates
sensitivity analyses that map the constraining power of the experimental data
to the variable space of physical functions. As a proof of principle, we apply
this IF approach to the Bayesian inference of the jet transport coefficient
$\hat{q}(T)$ from experimental data on jet quenching, with the temperature $T$
being the system variable.

Jet quenching is the suppression of large transverse momentum ($p_{\rm T}$)
jets and hadrons caused by energy loss of energetic partons (quarks and
gluons) as they propagate through QGP in high-energy heavy-ion (A+A)
collisions [23, 24, 25, 26, 27]. Parton energy loss is controlled by the jet
transport coefficient $\hat{q}$, defined as the transverse momentum broadening
squared per unit length [28, 29, 30, 31, 32, 33, 34]. In QGP, it can depend on
both the local temperature along the jet propagation path and jet energy. Past
efforts in extracting $\hat{q}$ from model comparisons to experimental data
relied on simple fits [35, 36, 37] or Bayesian inferences from single
inclusive hadron or jet spectra [15, 14]. We will carry out the first global
Bayesian inference of $\hat{q}$ with the IF approach from combined data on
suppression of single inclusive hadron [38, 39, 40, 41, 42, 43, 44],
$\gamma$-hadron [45, 46], and di-hadron [45, 46, 47, 48, 49] spectra at both
the Relativistic Heavy-ion Collider (RHIC) and the Large Hadron Collider (LHC)
energies. We will exclude data on reconstructed jets in this study, since they
are also sensitive to the modeling of jet-induced medium response [17, 50, 51,
52, 53, 54].

NLO parton model with energy loss– We use the next-leading-order (NLO) parton
model [55] to calculate single inclusive hadron, di-hadron and $\gamma$-hadron
spectra in proton-proton (p+p) and A+A collisions. Cross sections are
factorized into parton distribution functions (PDF) of the proton or nuclei,
partonic scattering cross sections, and parton fragmentation functions (FF).
The PDF’s of a free nucleon are given by the CT14 parameterization [56]. The
FF’s in vacuum are given by the Kniehl-Kramer-Potter parameterization [57].
The nuclear PDF’s are from the EPPS16 [58] parameterization with a model for
the impact-parameter dependence [59, 60, 61]. In nuclear collisions, medium-
modified FF’s are assumed to be given by the vacuum ones with reduced parton
energy due to energy loss plus fragmentation of radiated gluons [62, 63, 64].

The radiative energy loss of a parton with momentum $p^{\mu}=(E,\vec{p})$
within the higher-twist approach [32, 33] is,

$\displaystyle\frac{\Delta{E}_{a}}{E}$ $\displaystyle=$
$\displaystyle\frac{2C_{A}\alpha_{s}}{\pi}\\!\\!\int_{\tau_{0}}^{\infty}\\!\\!\\!\\!d\tau\\!\\!\int\frac{dl_{\rm
T}^{2}}{l_{\rm T}^{2}(l_{\rm
T}^{2}+\mu_{D}^{2})}\\!\\!\int\\!\\!dz\left[1+(1-z)^{2}\right]$ (1)
$\displaystyle\times\frac{p^{\mu}\cdot
u_{\mu}}{E}\hat{q}_{a}(T(\tau))\sin^{2}\left[\frac{l_{\rm
T}^{2}(\tau-\tau_{0})}{4z(1-z)E}\right],$

where $\mu_{D}$ is the Debye mass, $C_{A}=3$, $\alpha_{s}$ the strong coupling
constant, $l_{\rm T}$ the transverse momentum and $z$ the longitudinal
momentum fraction of the radiated gluon. The time integral is along the jet
path starting at time $\tau_{0}=0.6$ fm/$c$. The jet transport coefficient of
a gluon $\hat{q}_{A}$ is $9/4$ times that of a quark $\hat{q}_{F}$. In this
paper, the jet transport coefficient $\hat{q}(T)$ always refers to that of a
quark. It is assumed to depend on the local temperature $T$ and negligible in
the hadron phase below $T_{c}=165$ MeV. The space-time profiles of $T$ and the
four-velocity of the QGP fluid ($u^{\mu}$) are provided by the CLVisc 3+1 D
hydrodynamic simulations [65, 66, 67] with initial conditions given by the
TRENTo model [68] averaged over 200 events for each centrality bin 111An
overall envelope function in the spatial rapidity is used to generalized the
TRENTo initial condition at middle rapidity to a 3D distribution.. Parameters
in the initial condition and the QGP transport coefficients are fitted to data
on bulk hadron production. Uncertainties on the inferred $\hat{q}$ due to
variations of these medium-related parameters are not the focus of this work
and are not considered.

Bayesian Inference with Information Field – A prior distribution with
sufficient generality is critical for an unbiased extraction of physical
parameters [5]. However, the “generality” is a subtle issue for unknown
functions. Priors according to an explicit parameterization of $\hat{q}(T)$
can introduce unnecessary long-range correlations between different regions of
temperature. While peripheral or low energy collisions can only provide
constraints at low $T$, $\hat{q}(T)$ at high $T$ can only be calibrated by
data from central collisions at high colliding energies. One can invent
parameterizations with higher flexibility [70], but it can become intractable
for the inference of higher-dimensional functions. We will develop an IF
approach in this study to avoid this problem.

The IF approach views the unconditioned function as a random field. The
Gaussian random field ($F$) is widely used, specified by a mean and a
covariance function,

$\displaystyle\langle F(x)\rangle$ $\displaystyle=\mu(x),~{}~{}\langle\delta
F(x)\delta F(x^{\prime})\rangle$ $\displaystyle=C(x,x^{\prime}),$ (2)

where $\delta F(x)=F(x)-\mu(x)$ and higher cumulants vanish. Since
$\hat{q}(T)$ in QGP is proportional to $T^{3}$, we will present the scaled
quantity $F(x)=\ln(\hat{q}/T^{3})$ as a random field in variable $x=\ln(T/{\rm
GeV})$, which guarantees the positivity of $\hat{q}/T^{3}$ 222Additional
constraints can be imposed by other pre-processing procedures.. For the scaled
quantity $\hat{q}/T^{3}$, it is reasonable to assume a prior that has
$\mu=\textrm{const}$. We assume that the correlation function takes the
Gaussian form,

$\displaystyle
C(x,x^{\prime})=\sigma^{2}\exp\left[-(x-x^{\prime})^{2}/2l^{2}\right],$ (3)

where $\sigma$ controls the variation of the random function with respect to
the mean. The correlation length $l$ captures the essence of the IF approach.
It states that the functional values at inputs $x$ and $x^{\prime}$ are
decorrelated from each other for $|x-x^{\prime}|\gg l$. For the analysis of
$\hat{q}(T)$, this translates to a condition that the high-temperature prior
is unaffected by data that are only sensitive to $\hat{q}(T)$ at low
temperature. This is the key to overcoming the problem of unnecessary long-
range correlations that troubles the approach with explicit parameterizations.

Figure 1: (a) 100 prior random functions for $\hat{q}/T^{3}$ from the IF
approach. (b) Prior random functions for $\hat{q}/T^{3}$ at 95% CI that are
restricted to $\hat{q}/T^{3}=4\pm 0.1$ at $0.15<T<0.2$ GeV (High-$T$ var.,
hatched) or $0.3<T<0.4$ GeV (Low-$T$ var., unhatched) as compared to
unconditioned prior (grey).

The IF prior only imposes continuity, differential properties, and local
correlations on realizations of the random function. This is compatible with
most physical quantities unless there is a discontinuity (e.g., at phase
boundaries). The prior distribution in a compact form is

$\displaystyle
P_{0}[F]=\exp\left[\\!-\frac{1}{2}\\!\int\\!\\!dxdx^{\prime}\delta
F(x)C^{-1}(x,x^{\prime})\delta F(x^{\prime})\right].$ (4)

In the field theory language, the prior appears as the partition function of a
free theory with the mean field $\mu(x)$ and the propagator $C(x,x^{\prime})$.
The posterior distribution,

$\displaystyle P_{1}[F]$ $\displaystyle=$ $\displaystyle
P_{0}[F]\exp\left\\{-\ln\mathcal{L}({\rm Model}[F],{\rm Data})\right\\},$ (5)

will be determined by the likelihood function $\mathcal{L}$ that depends on
the pre-processed random function via the model and data comparison. The
marginalized distribution of $y=F(x^{*})$ can be expressed as the path
integral $P(y)=\int[\mathcal{D}F]P_{1}[F]\delta(y-F[x^{*}])$.

In principle, the IF parameters $\mu,\sigma$ and $l$ should be treated as
hyper-parameters and marginalized in the Bayesian analysis. As a first
application of the IF approach, we fix their values to
$\mu=\langle\ln\hat{q}/T^{3}\rangle=1.36$, $\sigma=0.7$ and $l=\ln(2)$. The
values of $\mu$ and $\sigma$ are chosen so that the 95% credible interval (CI)
of the prior covers most of the range $0.8\lesssim\hat{q}/T^{3}\lesssim 15$
from past analyses as shown (gray band) in Fig. 1(b). The choice of $l$ gives
a smooth random function when $T$ varies within a factor of two while
correlations beyond that are suppressed. Smaller values of $l$ can lead to
short range oscillations in the random functions and more samplings are
required to reach the same accuracy in the final results [61].

Figure 2: 95% variation of the ensemble predictions for (a) $I_{\rm
AA}^{\gamma h}(z_{\rm T})$, (b) $I_{\rm AA}^{hh}(z_{\rm T})$ in 0-10% Au+Au
collisions at $\sqrt{s}=0.2$ TeV, (c) $I_{\rm AA}^{hh}(p_{\rm T}^{\rm assoc})$
in 0-10% Pb+Pb collisions at $\sqrt{s}=2.76$ TeV, (d) $R_{\rm AA}^{h}(p_{\rm
T})$ in 0-10% Au+Au at $\sqrt{s}=0.2$ TeV (red), 0-5% and 40-50% Pb+Pb
collisions at $\sqrt{s}=2.76$ TeV, using $\hat{q}/T^{3}$ sampled from two sets
of random functions, High-$T$ var. (hatched) and Low-$T$ var. (unhatched) as
shown in Fig. 1(b).

In our analysis, Gaussian Process (GP) emulators are used to speed up model
predictions as in other studies (e.g., see Refs. [72] and [5] for detailed
descriptions). We select 100 prior random function samples, shown in Fig.
1(a), according to the IF approach. With each $\hat{q}(T)$ sample, we use the
NLO parton model to compute: 1) the nuclear suppression factor $R_{\rm
AA}^{h}(p_{\rm T})$ for single inclusive hadrons defined as the ratio of
normalized (by the number of binary collisions) $p_{\rm T}$ spectra in A+A and
p+p collisions; 2) the nuclear modification factor $I_{\rm AA}^{hh}$ of di-
hadron and 3) $I_{\rm AA}^{\gamma h}$ of $\gamma$-hadron correlation, defined
as the ratio of hadron yield per trigger in A+A and p+p collisions as a
function of the associate hadron $p_{\rm T}^{\rm assoc}$ or fractional
momentum $z_{\rm T}=p_{\rm T}^{\rm assoc}/p_{\rm T}^{\rm trig}$. The 100
samples of $\hat{q}(T)$ and the corresponding model calculations of
$R_{AA}^{h}$, $I_{\rm AA}^{hh}$, and $I_{\rm AA}^{\gamma h}$ for different
centralities at both RHIC and LHC energies are used to train the GP emulators.

Information field assisted sensitivity analysis – To demonstrate the
sensitivity of different observables to the IF priors of $\hat{q}(T)$ in
different temperature ranges, we create two sets of conditional random
functions, as illustrated in Fig. 1(b). In “High-T var.” prior, the random
functions are constrained to $\hat{q}/T^{3}=4\pm 0.1$ at $0.15<T<0.2$ GeV but
are allowed to vary in the high temperature region. The “Low-T var.” prior, on
the other hand, is restricted to $\hat{q}/T^{3}=4\pm 0.1$ at $0.3<T<0.4$ GeV
and can vary at low temperatures almost in the full range of prior values of
$\hat{q}/T^{3}$. It is clear that the prior at high $T$ and low $T$ are indeed
uncorrelated. The distance of decorrelation is controlled by $l$. The
predicted observables with these two sets of priors are shown in Fig. 2. As
expected, the sensitivities (variations) of $R_{\rm AA}^{h}$ and $I_{\rm AA}$
to high-$T$ $\hat{q}$ increase from RHIC to LHC energies, and from peripheral
to central collisions. Furthermore, $I_{\rm AA}^{hh}$ and $I_{\rm AA}^{\gamma
h}$ are slightly more sensitive to high-$T$ $\hat{q}$ than $R_{\rm AA}^{h}$
because of their different geometric bias on the initial jet production
location in the transverse plane [63, 64].

Global Bayesian inference of $\hat{q}(T)$– Using model emulators and all
existing experimental data on $R_{\rm AA}^{h}$, $I^{hh}_{\rm AA}$ and
$I^{\gamma h}_{\rm AA}$ at both RHIC and LHC energies, we have carried out the
Bayesian inference of $\hat{q}(T)$. Shown in Fig. 3 is the final posterior
$\hat{q}/T^{3}$ (red band) in 95% CI as compared to the fit by the JET
Collaboration [35] (symbols) and the JETSCAPE (95% CI, line-hatched) [15] and
LIDO (95% CI, dot-hatched) [14] Bayesian analyses. In addition, we also draw
20 random posterior samples (blue lines) to illustrate the distribution and
correlation of the posterior $\hat{q}/T^{3}$. The IF approach can be extended
in the future to include momentum and virtuality dependence of $\hat{q}$ as in
the JETSCAPE analysis. With separate analyses in different regions of hadron’s
$p_{\rm T}$, we find only mild momentum dependence within $8<p_{\rm T}<100$
GeV/$c$ [61].

Figure 3: $\hat{q}/T^{3}$ from the global IF-based Bayesian analysis at 95% CI
(red) with 20 posterior samples (lines) as compared to results from JET
Collaboration (symbols) [35] and JETSCAPE (line-hatched) [15] and LIDO
analysis (dot-hatched) [14].

The extracted $\hat{q}/T^{3}$ from the IF-based Bayesian inference is
consistent with the JETSCAPE result [15] at 95% CI except at temperatures
close to $T_{c}$ where a stronger temperature dependence arises from the
combined constraint by experimental data in central and peripheral collisions
at different colliding energies. The power of the combined constraint with the
IF-based Bayesian inference is illustrated in Fig. 4 by the posterior
$\hat{q}/T^{3}$ at 95% CI using only $R_{\rm AA}^{h}$ data in (a) Au+Au at
$\sqrt{s}=0.2$ TeV and (b) Pb+Pb collisions at $\sqrt{s}=2.76$ TeV with data
from different range of centralities. Though data from peripheral collisions
only provide meaningful constraints at lower $T$, $\hat{q}$ is progressively
more constrained at higher $T$ as one includes data from more central
collisions which probe higher $T$ regions of QGP. We see the same trend as
data from low to higher colliding energies are included. Because the IF
approach strongly suppresses correlations between high and low-$T$ prior
values of $\hat{q}$, it guarantees that constraints at higher $T$ are imposed
only by combining data from more central collisions and at higher colliding
energies. Including $I_{\rm AA}^{hh}$ and $I_{\rm AA}^{\gamma h}$ in the
analysis does not significantly reduce the uncertainties because of the large
experimental errors [61]. This may be improved with more accurate data in the
future.

Figure 4: $\hat{q}/T^{3}$ as a function of temperature $T$ from IF-based
Bayesian inference at 95% CI using only data on $R_{\rm AA}^{h}(p_{T})$ in (a)
Au+Au at $\sqrt{s}=0.2$ TeV and (b) Pb+Pb collisions at $\sqrt{s}=2.76$ TeV
with different ranges of centralities.

Finally, we compare in Fig. 5 the model emulator predictions with the globally
Bayesian constrained $\hat{q}$ to a representative selection of experimental
data on $R_{\rm AA}^{h}$, $I^{hh}_{\rm AA}$ and $I^{\gamma h}_{\rm AA}$ at
both RHIC and LHC energies. A full comparison can be found in Ref. [61].
Overall, the experimental data over a large range of $p_{\rm T}$, centralities
and colliding energies are well reproduced by the ensemble predictions of the
model emulator at 60% and 95% CI, indicating the global descriptive power of
our analysis.

Figure 5: The same as Fig. 2 except for posterior distributions of observables
at 95% (light blue) and 60% CI (dark blue) from the model emulator with
$\hat{q}/T^{3}$ given in Fig. 3 as compared to a subset of experimental data
[46, 47, 42, 38].

Summary – We developed an Information Field (IF) approach to the Bayesian
inference of unknown physical functions of system variables from experimental
data. It represents a prior functional distribution that is free from long-
range correlations between physical parameters in different regions of the
variable space, which can be independently sensitive to different data sets.
We applied this IF approach to the first global Bayesian inference of the jet
transport coefficient $\hat{q}$ from combined experimental data on $R_{\rm
AA}^{h}$, $I_{\rm AA}^{hh}$ and $I_{\rm AA}^{\gamma h}$ in heavy-ion
collisions at both RHIC and LHC energies. We showed the IF approach can
provide progressive constraints on $\hat{q}$ from low to high $T$ when
experimental data in more central collisions and at higher colliding energies
are included in the analysis. The extracted $\hat{q}/T^{3}$ is consistent with
previous studies but exhibits a stronger temperature dependence. This IF
approach can be extended to also include energy dependence of $\hat{q}$ and
analyses of other properties of QGP such as bulk transport coefficients and
equation of state.

###### Acknowledgements.

We thank Chi Ding for providing the QGP hydro profiles. This work is supported
by NSFC under Grant Nos. 11935007, 11861131009, 11890714 and 12075098, by U.S.
DOE under Contract No. DE-AC02-05CH11231, by US NSF under Grant OAC-2004571
within the X-SCAPE Collaboration, and by the LDRD Program at LANL.
Computations are performed at NSC3 and NERSC.

## References

  * Pratt  _et al._ [2015] S. Pratt, E. Sangaline, P. Sorensen, and H. Wang, Constraining the Eq. of State of Super-Hadronic Matter from Heavy-Ion Collisions, Phys. Rev. Lett. 114, 202301 (2015), arXiv:1501.04042 [nucl-th] .
  * Bernhard  _et al._ [2015] J. E. Bernhard, P. W. Marcy, C. E. Coleman-Smith, S. Huzurbazar, R. L. Wolpert, and S. A. Bass, Quantifying properties of hot and dense QCD matter through systematic model-to-data comparison, Phys. Rev. C 91, 054910 (2015), arXiv:1502.00339 [nucl-th] .
  * Bernhard  _et al._ [2016] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu, and U. Heinz, Applying bayesian parameter estimation to relativistic heavy-ion collisions: Simultaneous characterization of the initial state and quark-gluon plasma medium, Phys. Rev. C 94, 024907 (2016).
  * Bernhard  _et al._ [2019] J. E. Bernhard, J. S. Moreland, and S. A. Bass, Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma, Nature Phys. 15, 1113 (2019).
  * Everett  _et al._ [2021a] D. Everett _et al._ (JETSCAPE), Multisystem Bayesian constraints on the transport coefficients of QCD matter, Phys. Rev. C 103, 054904 (2021a), arXiv:2011.01430 [hep-ph] .
  * Everett  _et al._ [2021b] D. Everett _et al._ (JETSCAPE), Phenomenological constraints on the transport properties of QCD matter with data-driven model averaging, Phys. Rev. Lett. 126, 242301 (2021b), arXiv:2010.03928 [hep-ph] .
  * Nijs  _et al._ [2021a] G. Nijs, W. van der Schee, U. Gürsoy, and R. Snellings, Bayesian analysis of heavy ion collisions with the heavy ion computational framework Trajectum, Phys. Rev. C 103, 054909 (2021a), arXiv:2010.15134 [nucl-th] .
  * Nijs  _et al._ [2021b] G. Nijs, W. van der Schee, U. Gürsoy, and R. Snellings, Transverse Momentum Differential Global Analysis of Heavy-Ion Collisions, Phys. Rev. Lett. 126, 202301 (2021b), arXiv:2010.15130 [nucl-th] .
  * Parkkila  _et al._ [2021a] J. E. Parkkila, A. Onnerstad, and D. J. Kim, Bayesian estimation of the specific shear and bulk viscosity of the quark-gluon plasma with additional flow harmonic observables, Phys. Rev. C 104, 054904 (2021a), arXiv:2106.05019 [hep-ph] .
  * Parkkila  _et al._ [2021b] J. E. Parkkila, A. Onnerstad, F. Taghavi, C. Mordasini, A. Bilandzic, and D. J. Kim, New constraints for QCD matter from improved Bayesian parameter estimation in heavy-ion collisions at LHC, (2021b), arXiv:2111.08145 [hep-ph] .
  * Xu  _et al._ [2018] Y. Xu, J. E. Bernhard, S. A. Bass, M. Nahrgang, and S. Cao, Data-driven analysis for the temperature and momentum dependence of the heavy-quark diffusion coefficient in relativistic heavy-ion collisions, Phys. Rev. C 97, 014907 (2018), arXiv:1710.00807 [nucl-th] .
  * Ke  _et al._ [2018] W. Ke, Y. Xu, and S. A. Bass, Linearized boltzmann-langevin model for heavy quark transport in hot and dense qcd matter, Phys. Rev. C 98, 064901 (2018).
  * Liu  _et al._ [2022] F.-L. Liu, W.-J. Xing, X.-Y. Wu, G.-Y. Qin, S. Cao, and X.-N. Wang, QLBT: a linear Boltzmann transport model for heavy quarks in a quark-gluon plasma of quasi-particles, Eur. Phys. J. C 82, 350 (2022), arXiv:2107.11713 [hep-ph] .
  * Ke and Wang [2021] W. Ke and X.-N. Wang, QGP modification to single inclusive jets in a calibrated transport model, JHEP 05, 041, arXiv:2010.13680 [hep-ph] .
  * Cao  _et al._ [2021] S. Cao _et al._ (JETSCAPE), Determining the jet transport coefficient $\hat{q}$ from inclusive hadron suppression measurements using Bayesian parameter estimation, Phys. Rev. C 104, 024905 (2021), arXiv:2102.11337 [nucl-th] .
  * Casalderrey-Solana and Wang [2008] J. Casalderrey-Solana and X.-N. Wang, Energy dependence of jet transport parameter and parton saturation in quark-gluon plasma, Phys. Rev. C 77, 024902 (2008), arXiv:0705.1352 [hep-ph] .
  * He  _et al._ [2015] Y. He, T. Luo, X.-N. Wang, and Y. Zhu, Linear Boltzmann Transport for Jet Propagation in the Quark-Gluon Plasma: Elastic Processes and Medium Recoil, Phys. Rev. C 91, 054908 (2015), [Erratum: Phys.Rev.C 97, 019902 (2018)], arXiv:1503.03313 [nucl-th] .
  * Lin  _et al._ [2018] H.-W. Lin, W. Melnitchouk, A. Prokudin, N. Sato, and H. Shows (Jefferson Lab Angular Momentum (JAM) Collaboration), First monte carlo global analysis of nucleon transversity with lattice qcd constraints, Phys. Rev. Lett. 120, 152502 (2018).
  * Del Debbio  _et al._ [2022] L. Del Debbio, T. Giani, and M. Wilson, Bayesian approach to inverse problems: an application to NNPDF closure testing, Eur. Phys. J. C 82, 330 (2022), arXiv:2111.05787 [hep-ph] .
  * Bialek  _et al._ [1996] W. Bialek, C. G. Callan, and S. P. Strong, Field theories for learning probability distributions, Phys. Rev. Lett. 77, 4693 (1996).
  * Ensslin [2013] T. Ensslin, Information field theory, AIP Conf. Proc. 1553, 184 (2013), arXiv:1301.2556 [astro-ph.IM] .
  * Lemm [1999] J. C. Lemm, Bayesian field theory: Nonparametric approaches to density estimation, regression, classification, and inverse quantum problems (1999).
  * Gyulassy and Plumer [1990] M. Gyulassy and M. Plumer, Jet Quenching in Dense Matter, Phys. Lett. B 243, 432 (1990).
  * Wang and Gyulassy [1992] X.-N. Wang and M. Gyulassy, Gluon shadowing and jet quenching in A + A collisions at s**(1/2) = 200-GeV, Phys. Rev. Lett. 68, 1480 (1992).
  * Wang [1998] X.-N. Wang, Effect of jet quenching on high $p_{T}$ hadron spectra in high-energy nuclear collisions, Phys. Rev. C 58, 2321 (1998), arXiv:hep-ph/9804357 .
  * Majumder and Van Leeuwen [2011] A. Majumder and M. Van Leeuwen, The Theory and Phenomenology of Perturbative QCD Based Jet Quenching, Prog. Part. Nucl. Phys. 66, 41 (2011), arXiv:1002.2206 [hep-ph] .
  * Qin and Wang [2015] G.-Y. Qin and X.-N. Wang, Jet quenching in high-energy heavy-ion collisions, Int. J. Mod. Phys. E 24, 1530014 (2015), arXiv:1511.00790 [hep-ph] .
  * Baier  _et al._ [1997a] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, Radiative energy loss of high-energy quarks and gluons in a finite volume quark - gluon plasma, Nucl. Phys. B 483, 291 (1997a), arXiv:hep-ph/9607355 .
  * Baier  _et al._ [1997b] R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne, and D. Schiff, Radiative energy loss and p(T) broadening of high-energy partons in nuclei, Nucl. Phys. B 484, 265 (1997b), arXiv:hep-ph/9608322 .
  * Zakharov [1996] B. G. Zakharov, Fully quantum treatment of the Landau-Pomeranchuk-Migdal effect in QED and QCD, JETP Lett. 63, 952 (1996), arXiv:hep-ph/9607440 .
  * Wiedemann [2000] U. A. Wiedemann, Gluon radiation off hard quarks in a nuclear environment: Opacity expansion, Nucl. Phys. B 588, 303 (2000), arXiv:hep-ph/0005129 .
  * Guo and Wang [2000] X.-F. Guo and X.-N. Wang, Multiple scattering, parton energy loss and modified fragmentation functions in deeply inelastic e A scattering, Phys. Rev. Lett. 85, 3591 (2000), arXiv:hep-ph/0005044 .
  * Wang and Guo [2001] X.-N. Wang and X.-F. Guo, Multiple parton scattering in nuclei: Parton energy loss, Nucl. Phys. A 696, 788 (2001), arXiv:hep-ph/0102230 .
  * Majumder [2012] A. Majumder, Hard collinear gluon radiation and multiple scattering in a medium, Phys. Rev. D 85, 014023 (2012), arXiv:0912.2987 [nucl-th] .
  * Burke  _et al._ [2014] K. M. Burke _et al._ (JET), Extracting the jet transport coefficient from jet quenching in high-energy heavy-ion collisions, Phys. Rev. C 90, 014909 (2014), arXiv:1312.5003 [nucl-th] .
  * Liao and Shuryak [2009] J. Liao and E. Shuryak, Angular Dependence of Jet Quenching Indicates Its Strong Enhancement Near the QCD Phase Transition, Phys. Rev. Lett. 102, 202302 (2009), arXiv:0810.4116 [nucl-th] .
  * Xu  _et al._ [2015] J. Xu, J. Liao, and M. Gyulassy, Consistency of Perfect Fluidity and Jet Quenching in semi-Quark-Gluon Monopole Plasmas, Chin. Phys. Lett. 32, 092501 (2015), arXiv:1411.3673 [hep-ph] .
  * Adare  _et al._ [2008] A. Adare _et al._ (PHENIX), Suppression pattern of neutral pions at high transverse momentum in Au$+$Au collisions at $\sqrt{s_{NN}}=$ 200 GeV and constraints on medium transport coefficients, Phys. Rev. Lett. 101, 232301 (2008), arXiv:0801.4020 [nucl-ex] .
  * Adare  _et al._ [2013] A. Adare _et al._ (PHENIX), Neutral pion production with respect to centrality and reaction plane in Au$+$Au collisions at $\sqrt{s_{NN}}$=200 GeV, Phys. Rev. C 87, 034911 (2013), arXiv:1208.2254 [nucl-ex] .
  * Chatrchyan  _et al._ [2012] S. Chatrchyan _et al._ (CMS), Study of high-pT charged particle suppression in PbPb compared to $pp$ collisions at $\sqrt{s_{NN}}=2.76$ TeV, Eur. Phys. J. C 72, 1945 (2012), arXiv:1202.2554 [nucl-ex] .
  * Abelev  _et al._ [2013] B. Abelev _et al._ (ALICE), Centrality Dependence of Charged Particle Production at Large Transverse Momentum in Pb–Pb Collisions at $\sqrt{s_{\rm{NN}}}=2.76$ TeV, Phys. Lett. B 720, 52 (2013), arXiv:1208.2711 [hep-ex] .
  * Aad  _et al._ [2015] G. Aad _et al._ (ATLAS), Measurement of charged-particle spectra in Pb+Pb collisions at $\sqrt{{s}_{\mathsf{{NN}}}}=2.76$ TeV with the ATLAS detector at the LHC, JHEP 09, 050, arXiv:1504.04337 [hep-ex] .
  * Khachatryan  _et al._ [2017] V. Khachatryan _et al._ (CMS), Charged-particle nuclear modification factors in PbPb and pPb collisions at $\sqrt{s_{\mathrm{N}\;\mathrm{N}}}=5.02$ TeV, JHEP 04, 039, arXiv:1611.01664 [nucl-ex] .
  * Acharya  _et al._ [2018] S. Acharya _et al._ (ALICE), Transverse momentum spectra and nuclear modification factors of charged particles in pp, p-Pb and Pb-Pb collisions at the LHC, JHEP 11, 013, arXiv:1802.09145 [nucl-ex] .
  * Abelev  _et al._ [2010] B. I. Abelev _et al._ (STAR), Studying Parton Energy Loss in Heavy-Ion Collisions via Direct-Photon and Charged-Particle Azimuthal Correlations, Phys. Rev. C 82, 034909 (2010), arXiv:0912.1871 [nucl-ex] .
  * Adamczyk  _et al._ [2016] L. Adamczyk _et al._ (STAR), Jet-like Correlations with Direct-Photon and Neutral-Pion Triggers at $\sqrt{s_{{}_{NN}}}=200$ GeV, Phys. Lett. B 760, 689 (2016), arXiv:1604.01117 [nucl-ex] .
  * Aamodt  _et al._ [2012] K. Aamodt _et al._ (ALICE), Particle-yield modification in jet-like azimuthal di-hadron correlations in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV, Phys. Rev. Lett. 108, 092301 (2012), arXiv:1110.0121 [nucl-ex] .
  * Adam  _et al._ [2016] J. Adam _et al._ (ALICE), Jet-like correlations with neutral pion triggers in pp and central Pb–Pb collisions at 2.76 TeV, Phys. Lett. B 763, 238 (2016), arXiv:1608.07201 [nucl-ex] .
  * Conway [2013] R. Conway (CMS), Very High-$p_{T}$ triggered dihadron correlations in PbPb collisions at 2.76 TeV with CMS, Nucl. Phys. A 904-905, 451c (2013).
  * Tachibana  _et al._ [2017] Y. Tachibana, N.-B. Chang, and G.-Y. Qin, Full jet in quark-gluon plasma with hydrodynamic medium response, Phys. Rev. C 95, 044909 (2017).
  * Chen  _et al._ [2018] W. Chen, S. Cao, T. Luo, L.-G. Pang, and X.-N. Wang, Effects of jet-induced medium excitation in $\gamma$-hadron correlation in A+A collisions, Phys. Lett. B 777, 86 (2018), arXiv:1704.03648 [nucl-th] .
  * Kunnawalkam Elayavalli and Zapp [2017] R. Kunnawalkam Elayavalli and K. C. Zapp, Medium response in JEWEL and its impact on jet shape observables in heavy ion collisions, JHEP 07, 141, arXiv:1707.01539 [hep-ph] .
  * Milhano  _et al._ [2018] G. Milhano, U. A. Wiedemann, and K. C. Zapp, Sensitivity of jet substructure to jet-induced medium response, Phys. Lett. B 779, 409 (2018), arXiv:1707.04142 [hep-ph] .
  * He  _et al._ [2019] Y. He, S. Cao, W. Chen, T. Luo, L.-G. Pang, and X.-N. Wang, Interplaying mechanisms behind single inclusive jet suppression in heavy-ion collisions, Phys. Rev. C 99, 054911 (2019), arXiv:1809.02525 [nucl-th] .
  * Owens [1987] J. F. Owens, Large Momentum Transfer Production of Direct Photons, Jets, and Particles, Rev. Mod. Phys. 59, 465 (1987).
  * Hou  _et al._ [2017] T.-J. Hou, S. Dulat, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, CTEQ-TEA parton distribution functions and HERA Run I and II combined data, Phys. Rev. D 95, 034003 (2017), arXiv:1609.07968 [hep-ph] .
  * Kniehl  _et al._ [2000] B. A. Kniehl, G. Kramer, and B. Potter, Fragmentation functions for pions, kaons, and protons at next-to-leading order, Nucl. Phys. B 582, 514 (2000), arXiv:hep-ph/0010289 .
  * Eskola  _et al._ [2017] K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado, EPPS16: Nuclear parton distributions with LHC data, Eur. Phys. J. C 77, 163 (2017), arXiv:1612.05741 [hep-ph] .
  * Wang [2000] X.-N. Wang, Systematic study of high $p_{T}$ hadron spectra in $pp$, $p$ A and A A collisions from SPS to RHIC energies, Phys. Rev. C 61, 064910 (2000), arXiv:nucl-th/9812021 .
  * Hirano and Nara [2004] T. Hirano and Y. Nara, Interplay between soft and hard hadronic components for identified hadrons in relativistic heavy ion collisions at RHIC, Phys. Rev. C 69, 034908 (2004), arXiv:nucl-th/0307015 .
  * [61] M. Xie, W. Ke, Z. Han-Zhong, and X.-N. Wang, In preparation.
  * Wang [2004] X.-N. Wang, Energy dependence of jet quenching and life-time of the dense matter in high-energy heavy-ion collisions, Phys. Rev. C 70, 031901 (2004), arXiv:nucl-th/0405029 .
  * Zhang  _et al._ [2007] H. Zhang, J. F. Owens, E. Wang, and X.-N. Wang, Dihadron tomography of high-energy nuclear collisions in NLO pQCD, Phys. Rev. Lett. 98, 212301 (2007), arXiv:nucl-th/0701045 .
  * Zhang  _et al._ [2009] H. Zhang, J. F. Owens, E. Wang, and X.-N. Wang, Tomography of high-energy nuclear collisions with photon-hadron correlations, Phys. Rev. Lett. 103, 032302 (2009), arXiv:0902.4000 [nucl-th] .
  * Pang  _et al._ [2012] L. Pang, Q. Wang, and X.-N. Wang, Effects of initial flow velocity fluctuation in event-by-event (3+1)D hydrodynamics, Phys. Rev. C 86, 024911 (2012), arXiv:1205.5019 [nucl-th] .
  * Pang  _et al._ [2015] L.-G. Pang, Y. Hatta, X.-N. Wang, and B.-W. Xiao, Analytical and numerical Gubser solutions of the second-order hydrodynamics, Phys. Rev. D 91, 074027 (2015), arXiv:1411.7767 [hep-ph] .
  * Pang  _et al._ [2018] L.-G. Pang, H. Petersen, and X.-N. Wang, Pseudorapidity distribution and decorrelation of anisotropic flow within the open-computing-language implementation CLVisc hydrodynamics, Phys. Rev. C 97, 064918 (2018), arXiv:1802.04449 [nucl-th] .
  * Moreland  _et al._ [2015] J. S. Moreland, J. E. Bernhard, and S. A. Bass, Alternative ansatz to wounded nucleon and binary collision scaling in high-energy nuclear collisions, Phys. Rev. C 92, 011901 (2015), arXiv:1412.4708 [nucl-th] .
  * Note [1] An overall envelope function in the spatial rapidity is used to generalized the TRENTo initial condition at middle rapidity to a 3D distribution.
  * Ke [2019] W. Ke,  _Partonic transport model application to heavy flavor_ , Phd dissertation, Duke University (2019), arXiv:2001.02766 [nucl-th] .
  * Note [2] Additional constraints can be imposed by other pre-processing procedures.
  * Bernhard [2018] J. E. Bernhard,  _Bayesian parameter estimation for relativistic heavy-ion collisions_ , Phd dissertation, Duke University (2018), arXiv:1804.06469 [nucl-th] .


# Comparing measurements and limits on the warm dark matter temperature-to-
mass ratio

Bruce Hoeneisen

( Universidad San Francisco de Quito, Quito, Ecuador
Email<EMAIL_ADDRESS>
20 August 2023 )

###### Abstract

Limits on the “thermal relic” warm dark matter mass are apparently
inconsistent with measurements of the warm dark matter temperature-to-mass
ratio. We try to understand this problem.

##  1 Introduction

Limits on the “thermal relic” warm dark matter mass have been obtained from
the Lyman-$\alpha$ flux power spectrum of quasars and hydro-dynamical
simulations. For example, [1] obtains $m_{x}>550$ eV at $2\sigma$ in 2005, [2]
obtains $m_{x}>3300$ eV at $2\sigma$ in 2013, and [3] obtains $m_{x}>3100$ eV
at 95 % confidence in 2023.

On the other hand, measurements of the non-relativistic warm dark matter
temperature-to-mass ratio have been obtained from the rotation curves of 10
spiral galaxies measured by the THINGS collaboration [4] [5], 46 spiral
galaxies measured by the SPARC collaboration [6] [7], and 11 dwarf spiral
galaxies measured by the LITTLE-THINGS collaboration [8]; and independently,
from galaxy stellar mass distributions [9] [10] [11], galaxy rest frame ultra-
violet luminosity distributions [12] [13], from first galaxies [9] [14], and
from the re-ionization optical depth [12] [14]. These measurements are
consistent (even tho they depend on the different detailed scenarios assumed
for each measurement). A summary of the final results is presented in [15].
Details of the final measurements can be found in [12] and [8].

It turns out that the measured warm dark matter temperature-to-mass ratio
happens to coincide with the “no freeze-in and no freeze-out” scenario of spin
zero warm dark matter particles that decouple early on from the Standard Model
sector. Spin one-half and spin one dark matter are disfavored because they
correspond to a temperature-to-mass ratio larger than measured. As an example,
if spin zero warm dark matter particles couple to the Higgs boson, their mass
is $M_{S}\approx 150\pm 2$ eV. This uncertainty only includes the contribution
from the measured dark matter density. The mass depends mildly ($\approx\pm
50\%$) on the temperature at which the dark matter particles decouple from the
Standard Model sector, the spin of the dark matter particles, and on the
assumed momentum distribution of the non-relativistic dark matter. The range
of measurements and masses for several scenarios is summarized in Table 3 of
[12].

Let me explain. “No freeze-in” means that dark matter reaches thermal and
diffusive, i.e. chemical, equilibrium, in the early universe, with the
particles of the Standard Model of Quarks and Leptons. Dark matter then
decouples from the Standard Model sector while still ultra-relativistic. “No
freeze-out” means that, when dark matter becomes non-relativistic, its self
interactions are so weak that these particles do not annihilate each other. An
example of “no freeze-in and no freeze-out” spin zero dark matter coupled to
the Higgs boson is shown in Figure 1 [15].

Figure 1:  The “no freeze-in and no freeze-out” warm dark matter scenario is
illustrated for spin zero warm dark matter particles coupled to the Higgs
boson. $T$ is the photon temperature, and the $n$’s are particle number
densities. The abbreviations stand for “Electro-Weak Symmetry Breaking”, “Big
Bang Nucleosynthesis”, “EQuivalence” of matter and radiation densities, and
“DECoupling” of photons from Standard Model matter. Dark matter particles
become non-relativistic at $a_{h\textrm{NR}}$. Time advances towards the
right. From [12] [15].

In this note we try to understand the difference between these measurements
and limits. Our conclusion is that the limits on the “thermal relic” warm dark
matter mass do not apply to the colder “no freeze-in and no freeze-out”
scenario of warm dark matter.

##  2 Comparison

The reason for this conclusion is as follows. If $m_{x}\gg 150$ eV, then dark
matter necessarily freezes-in and/or freezes-out, in order to not exceed the
measured dark matter density, see Figure 1. There are several scenarios. In
any case, “thermal relic” dark matter has the photon temperature at the time
of ultra-relativistic or non-relativistic decoupling from the Standard Model
sector. Therefore the “no freeze-in and no freeze-out” warm dark matter is
colder than a “thermal relic”, after $e^{+}e^{-}$ recombination and while
still ultra-relativistic, by a factor less than, or equal to,
$T_{S}/T_{\gamma}=(43/(11g_{\textrm{dec}}))^{1/3}$ [16]. This factor is
$0.345$ for the example of dark matter coupled to the Higgs boson.
$g_{\textrm{dec}}=\sum{N_{b}}+(7/8)\sum{N_{f}}$ at decoupling of ultra-
relativistic “no freeze-in and no freeze-out” dark matter from the Standard
Model sector [16]. $N_{b}$ or $N_{f}$ are the number of boson or fermion spin
polarization’s, respectively. Hence, to compare $M_{S}$ with limits $m_{x}$ it
is necessary to multiply $m_{x}$ by a factor $<0.345^{1.11}=0.31$ (the index
1.11 is taken from equation (2) of [2]).

As an alternative estimate, let us translate the limit $m_{x}>3300$ eV at
$2\sigma$ of [2] into a limit on $M_{S}$. From equation (2) of [2], i.e.

$k_{1/2}\approx
6.5\frac{h}{\textrm{Mpc}}\left(\frac{m_{x}}{1\textrm{keV}}\right)^{1.11}\left(\frac{\Omega_{c}}{0.25}\right)^{-0.11}\left(\frac{h}{0.7}\right)^{1.22},$
(1)

valid for “thermal relic” dark matter (note that $k_{1/2}$ is really a
function of $m_{x}/T_{x}$), we obtain $k_{1/2}>15.64$ Mpc-1 (we take all
cosmological parameters from [16]). This is the comoving wave-vector at which
the warm dark matter free-streaming reduces the cold dark matter power
spectrum by a factor 2, i.e. $\tau^{2}(k_{1/2})=1/2$, with $\tau^{2}(k)\equiv
P_{\textrm{WDM}}(k)/P_{\textrm{CDM}}(k)$. The power spectrum is reduced by a
factor $e$ at $k_{1/e}=k_{1/2}/0.83255>18.79$ Mpc-1. From Figure 4 of [17] we
obtain the comoving power spectrum cut-off at the time of equal matter and
radiation densities (before the non-linear re-generation of small scale
structure):

$k_{\textrm{fs}}(t_{\textrm{eq}})=\frac{1.455}{\sqrt{2}}\sqrt{\frac{4\pi
G\bar{\rho}_{c}(1)a_{\textrm{eq}}}{v_{h\textrm{rms}}(1)^{2}}}$ (2)

(as a cross-check, this expression agrees with a numerical integral).
$v_{h\textit{rms}}(a)=v_{h\textit{rms}}(1)/a$ is the root-mean-square velocity
of non-relativistic dark matter particles at expansion parameter $a$. Equating
$k_{1/e}=k_{\textrm{fs}}(t_{\textrm{eq}})$ we obtain
$v_{h\textit{rms}}(1)<39.89$ m/s. From (28) of [18] we obtain, for the no
freeze-out scenario,

$M_{S}=108\left(\frac{0.76\textrm{
km/s}}{v_{h\textrm{rms}}(1)}\right)^{3/4}\left(\frac{1}{N_{b}}\right)^{1/4}\textrm{
eV}.$ (3)

With $N_{b}=1$, we finally obtain $M_{S}>985$ eV. Note that
$M_{S}/m_{x}=985/3300=0.30$, in good agreement with the estimate of the
previous paragraph.

The free-streaming power spectrum suppression factor relative to the cold dark
matter scenario has the approximate form

$\displaystyle\tau^{2}(k)$ $\displaystyle=$
$\displaystyle\exp{\left(-\frac{k^{2}}{k^{2}_{\textrm{fs}}(t_{\textrm{eq}})}\right)}\qquad\textrm{
if }k<k_{\textrm{fs}}(t_{\textrm{eq}}),$ (4) $\displaystyle=$
$\displaystyle\exp{\left(-\frac{k^{n}}{k^{n}_{\textrm{fs}}(t_{\textrm{eq}})}\right)}\qquad\textrm{
if }k\geq k_{\textrm{fs}}(t_{\textrm{eq}}),$

with $n=2$ at equality of radiation and matter [17]. At lower red-shift,
$\tau^{2}(k)$ develops a non-linear regenerated “tail” with $n$ measured to be
between 0.5 and 0.8 at red-shift $z=8$ [12] [13].

In summary, limits on the “thermal relic” warm dark matter mass are not
directly applicable to the colder “no freeze-in and no freeze-out” warm dark
matter that decouples early on from the Standard Model sector. Three
corrections would need to be applied:

  1. 1.

To correct for the different linear warm dark matter free-streaming cut-offs,
due to their different temperatures, the limit on $m_{x}$ needs to be
multiplied by a factor $\leq 0.30$, depending on the details of the freeze-out
and decoupling.

  2. 2.

An unknown correction needs to be applied due to the “velocity dispersion”
cut-off. The “velocity dispersion” delays and suppresses the collapse of small
scale density fluctuations, and defines the mass of first galaxies, see Figure
6 of [14], and see Table 1 of [12].

  3. 3.

An unknown correction needs to be applied due to the non-linear regeneration
of small scale structure measured in [12] and [13], that has contributions
from baryons that are difficult to include in hydro-dynamical simulations. The
measured $m_{x}$ is very sensitive to $n$, see [19].

Each of these items lowers the limit on $M_{S}$.

##  3 Conclusions

In conclusion, limits on the “thermal relic” warm dark matter mass do not
apply to the colder “no freeze-in and no freeze-out” scenario of warm dark
matter. It is premature to dismiss several independent and consistent
measurements of the warm dark matter temperature-to-mass ratio summarized in
[15].

## References

  * [1] M. Viel, J. Lesgourgues, M. G. Haehnelt, S. Matarrese and A. Riotto, Phys. Rev. D 71, 063534 (2005).
  * [2] Viel, M., Becker, G.D., Bolton, J.S., Haehnelt, M.G., Warm Dark Matter as a solution to the small scale crisis: new constraints from high redshift Lyman-$\alpha$ forest data
  * [3] Villasenor, B., Robertson, B., Madau, P., Schneider, E., New Constraints on Warm Dark Matter from the Lyman-$\alpha$ Forest Power Spectrum
  * [4] Hoeneisen, B. (2019) A Study of Dark Matter with Spiral Galaxy Rotation Curves. International Journal of Astronomy and Astrophysics, 9, 71-96.
  * [5] Hoeneisen, B. (2019) A Study of Dark Matter with Spiral Galaxy Rotation Curves. Part II. International Journal of Astronomy and Astrophysics, 9, 133-141.
  * [6] Hoeneisen, B. (2019) The Adiabatic Invariant of Dark Matter in Spiral Galaxies. International Journal of Astronomy and Astrophysics, 9, 355-367.
  * [7] Hoeneisen, B. (2021) A Study of Three Galaxy Types, Galaxy Formation, and Warm Dark Matter. International Journal of Astronomy and Astrophysics, 11, 489-508.
  * [8] Hoeneisen, B. (2022) Measurement of the Dark Matter Velocity Dispersion with Dwarf Galaxy Rotation Curves. International Journal of Astronomy and Astrophysics, 12, 363-381.
  * [9] Hoeneisen, B. (2019) Simulations and Measurements of Warm Dark Matter Free-Streaming and Mass. International Journal of Astronomy and Astrophysics, 9, 368-392.
  * [10] Hoeneisen, B. (2020) cold Cold or Warm Dark Matter?: A Study of Galaxy Stellar Mass Distributions. International Journal of Astronomy and Astrophysics, 10, 57-70.
  * [11] Hoeneisen, B. (2020) Fermion or Boson Dark Matter? International Journal of Astronomy and Astrophysics, 10, 203-223.
  * [12] Hoeneisen, B. (2022) Measurement of the Dark Matter Velocity Dispersion with Galaxy Stellar Masses, UV Luminosities, and Reionization. International Journal of Astronomy and Astrophysics, 12, 258-272.
  * [13] Hoeneisen, B. (2023), A Study of Warm Dark Matter, the Missing Satellites Problem, and the UV Luminosity Cut-Off. International Journal of Astronomy and Astrophysics, 13, 25-38.
  * [14] Hoeneisen, B. (2022) Warm Dark Matter and the Formation of First Galaxies. Journal of Modern Physics, 13, 932-948.
  * [15] Hoeneisen, B., (2023) A Data Driven Solution to the Dark Matter Problem. European Journal of Applied Sciences. Vol-11(2), 473-481.
  * [16] P.A. Zyla et al. (Particle Data Group), 2020 Review of Particle Physics, Prog. Theor. Exp. Phys. 2020, 083C01 (2020).
  * [17] Boyanovsky, D., de Vega, H.J. and Sanchez, N.G. (2008) The Dark Matter Transfer Function: Free Streaming, Particle Statistics and Memory of Gravitational Clustering. Physical Review D, 78, Article ID: 063546.
  * [18] Paduroiu, S., Revaz, Y., Pfenniger, D. (2015) Structure formation in warm dark matter cosmologies Top-Bottom Upside-Down. https://arxiv.org/pdf/1506.03789.pdf
  * [19] Hoeneisen, B. (2022) Comments on Warm Dark Matter Measurements and Limits. International Journal of Astronomy and Astrophysics, 12, 94-109.