# Core Fragmentation in Simplest Superfluid Dark Matter Scenario

Lasha Berezhiani  Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany  Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany  Giordano Cintia  Max-Planck-
Institut für Physik, Föhringer Ring 6, 80805 München, Germany  Max Warkentin
Arnold Sommerfeld Center, Ludwig-Maximilians-Universität, Theresienstraße 37,
80333 München, Germany

###### Abstract

We study the structure of galactic halos within a scalar dark matter model,
endowed with a repulsive quartic self-interaction, capable of undergoing the
superfluid phase transition in high-density regions. We demonstrate that the
thermalized cores are prone to fragmentation into superfluid droplets due to
the Jeans instability. Furthermore, since cores of astrophysical size may be
generated only when most of the particles comprising the halo reside in a
highly degenerate phase-space, the well-known bound on the dark matter self-
interaction cross section inferred from the collision of clusters needs to be
revised, accounting for the enhancement of the interaction rate due to
degeneracy. As a result, generation of kpc-size superfluid solitons, within
the parameter subspace consistent with the Bullet Cluster bound, requires dark
matter particles to be ultra-light.

##  I I. Introduction

The cold dark matter (CDM) paradigm, in which dark matter is incorporated as a
non-relativistic fluid composed of collisionless particles, is in spectacular
agreement with observations at large scales. However, it presents several
challenges when it comes to galactic scale phenomena Bullock:2017xww . Among
those, the mismatch between the Navarro-Frank-White (NFW) density profile in
inner regions of galaxies and clusters obtained from CDM simulations
Navarro:1995iw  and the density profile inferred from observations, known in
the literature as ’the core-cusp problem’, appears to be a central one and its
resolution will most likely affect the status of some of the other puzzles as
well. In particular, there have been claims regarding the excess of the
dynamical friction predicted by CDM Debattista:1997bi ; Sellwood:2016 ;
Fornaxold ; Tremaine , which is a sensitive function of the dark matter
density-profile (a potentially ameliorating discussion within CDM, in some
cases, can be found in Cole ; SanchezSalcedo:2006fa ). Furthermore,
observations seem to indicate a significant correlation between the
gravitational acceleration of baryons and their distribution in disk galaxies
McGaugh:2016leg ; Lelli:2017vgz . This is usually referred to as the mass
discrepancy acceleration relation, which is the generalized version of the
baryonic Tully-Fisher relation McGaugh:2011ac ; LelliBTFR , see also
Famaey:2011kh  and references therein. These are well accounted for by
Milgrom’s empirical law Milgrom:1983ca ; Milgrom:1983pn ; Milgrom:1983zz ,
which was originally suggested as a modification of gravity. The origin of
these observed relations is not yet understood, but seems to be indicative of
some form of interaction between baryons and the dark sector. However, it must
be pointed out that depending on the mass-range for dark matter particles, the
cosmological constraints on such interactions could be significant
Dvorkin:2013cea . Although it is still unclear whether a purely gravitational
interaction present in CDM, without invoking self- or cross-species
interactions, is sufficient for generating such correlations, it seems to be
an unlikely scenario Oman:2015xda  (the possibility of reconciliation has been
argued Read:2016 ).

These and other related observations have motivated extensions of the standard
paradigm, by giving dark matter some additional properties that could be
reflected in a more desirable galactic scale behaviour. One of the directions
that has attracted much attention recently concerns self-interacting dark
matter Spergel:1999mh . If strong enough, interactions could affect the
density distribution of central regions of the halo significantly, due to a
higher number density of dark matter particles. In particular, numerical
simulations have revealed that the density profile starts deviating from the
NFW profile in regions where particles had the chance to interact at least
once throughout the lifetime of the halo Kaplinghat:2015aga . The main effect
of those interactions is the dark matter redistribution, alleviating the
central cusp of the NFW profile. At large distances from the centre of the
halo, where dark matter particles have not had a chance to interact yet due to
low densities, the density profile is similar to the one in CDM.

An interesting version of self-interacting dark matter was considered in
Goodman:2000tg ; Slepian:2011ev 111See also Boehmer:2007um  for the discussion
of the core profile and galactic rotation curves within the Bose-Einstein
condensate dark matter scenario. in the form of a repulsive, sub-eV, scalar
field exhibiting superfluidity in galaxies, motivated by theoretical
simplicity and the capacity for an additional suppression of the dynamical
friction. The considered structure of the dark matter halo had a superfluid
core enclosed by an isothermal envelope. In that scenario, dark matter was
assumed to be in thermal equilibrium which led to the formation of the
superfluid core in the central region, where the temperature of the substance
is subcritical (due to high densities). The presence of repulsive self-
interaction was vital for the stability of the core, as well as for
superfluidity, as we will discuss shortly. The envelope, on the other hand,
was stabilized by the thermal pressure of dark matter particles, with the
junction condition determined in terms of the finite-temperature equation of
state. Imposing the Bullet Cluster constraint and the reproduction of the
observed features of rotational curves, the model was claimed to be ruled out
in Slepian:2011ev . See Sharma:2018ydn  for a more recent discussion of a
spherically symmetric density profile for a thermalized halo, derived from an
improved finite-temperature equation of state. The core profile and galactic
rotation curves within the Bose-Einstein condensate dark matter scenario have
been discussed in Boehmer:2007um  as well. The same scalar dark matter model
has been also considered in the context of its gravitational production at the
end of inflation Peebles:1999fz ; Peebles:1999se .

For attractive bosons, which is the case for axions, one lacks positive
pressure and consequently the condensate formed at subcritical temperatures is
prone to fragmentation Guth:2014hsa . The only possibility for sustaining the
macroscopic homogeneous core is to consider the case of ultra-light particles.
Achieving macroscopic homogeneity in such a way, in the core of the dark
matter halo, was first suggested by Hu:2000ke , and was coined as Fuzzy Dark
Matter. There, the scale of homogeneity is set by the de Broglie wavelength,
and the presence of a kpc-size core requires particles lighter than
$10^{-21}{\rm eV}$. The detailed analysis of phenomenological implications of
such a scenario was performed in Hui:2016ltb  (see May:2021wwp  for recent
numerical simulations of the structure formation). The idea of substructure
formation has been discussed in the context of the superfluid dark matter
scenario as well, see Schiappacasse:2017ham ; Alexander:2019qsh  and
references therein.

Recently, the idea of dark matter superfluidity was revitalized in
Berezhiani:2015pia ; Berezhiani:2015bqa  as a novel mechanism that could be
behind the above-mentioned long-range correlations in galactic dynamics. The
idea is to have a superfluid dark matter core in galaxies and utilize phonons
to mediate an emergent long-range interaction between baryons submerged within
this quantum liquid222The idea of emergent long-range interactions within an
ideal Bose-Einstein condensate and the cosmological implications for the
scalar dark matter scenario was first considered in Ferrer:2000hm ;
Ferrer:2004xj . Recently, the idea was revisited for superfluids
Berezhiani:2018oxf . It was demonstrated that superfluidity shortens the
range, but still keeping it much longer than the Compton wavelength of dark
matter particles. The potential significance of the mechanism for the galactic
dynamics, in the context of the ultra-light dark matter scenario (like Fuzzy
Dark Matter), was studied as well.. It was demonstrated in Berezhiani:2017tth
that in this incarnation of the superfluid dark matter scenario it was more
natural to give up global thermal equilibrium and instead to require
thermalization within the central neighbourhood of a halo. However, there much
freedom was given in specifying the profile by keeping the superfluid pressure
not very closely related to the $2\rightarrow 2$ scattering cross
section.333The reason behind this was the assumption that the thermalization
could have been governed by two-body interactions (i.e. in the disordered
phase the $2\rightarrow 2$ scattering was considered to be dominant), while
the superfluid pressure was considered to be determined by three-body
interactions.

In this work, we study the structure of galactic halos within the simplest
model of dark matter superfluidity combining ideas of Goodman:2000tg ;
Guth:2014hsa ; Schiappacasse:2017ham  and Berezhiani:2017tth  together. The
additional ingredient added to the pre-existing ideas being the proper
application of the Bullet Cluster bound to sub-eV particles, by including the
Bose-enhancement factor in the calculation of scattering rates. The result is
the inevitable fragmentation of the thermalized core into superfluid islands
due to the Jeans instability. It is demonstrated that the extremely light
values for the mass are required to give a kpc-size coherence length.

One of the theoretically simplest interacting dark matter models can be
introduced as a massive scalar field, with quartic self-interaction and
minimally coupled to gravity, with the following action

$\displaystyle S=\int{d^{4}x\sqrt{-g}\left(\frac{1}{16\pi
G}R-|\partial\Phi|^{2}-m^{2}|\Phi|^{2}-\frac{\lambda}{2}|\Phi|^{4}\right)}\,.$
(1)

Here, we have chosen to work with a complex ($U(1)$-invariant) field, due to
manifest particle number conservation. We could have begun with a real scalar,
but considering that we are interested in a non-relativistic substance, the
net result would have been identical; in this limit, both theories flow to the
nonlinear Schrödinger’s action.

Before diving into the description of the superfluid scenario, let us begin by
recapping the Bullet Cluster constraint Markevitch:2003at  for such a theory.
It is usually invoked as the following bound on the scattering cross section
for dark matter particles

$\displaystyle\frac{\sigma}{m}\lesssim 1{\text{ cm}^{2}}/{\text{g}}\,.$ (2)

Strictly speaking, the bound obtained by Markevitch:2003at  is for the
scattering rate, which for the non-degenerate phase-space translates into (2).
Although the latter has been widely applied to various dark matter models,
including the sub-eV mass-range, we will argue that it may not be necessarily
legitimate. Having said this, let us put this caveat aside for a moment by
focussing on $m\gg{\rm eV}$ and translate (2) into a bound on the coupling
constant

$\displaystyle\lambda\lesssim\left(\frac{m}{10~{}{\rm MeV}}\right)^{3/2}\,.$
(3)

In other words, for dark matter heavier than few MeV, the theory has to be
strongly coupled in order to violate the Bullet Cluster bound.

For the superfluid scenario one is mostly interested in sub-eV particles. As
it is well known, such a candidate must be a non-thermal relict and should be
produced via the axion-like vacuum misalignment mechanism. Consequently, it is
expected to be in the form of the condensate on cosmological scales. Requiring
the equation of state due to interaction-pressure to be the one for a non-
relativistic fluid, one arrives at the following bound for the scattering
cross section

$\displaystyle\frac{P}{\rho}\Big{|}_{\text{equality}}=\frac{\lambda\rho\rvert_{\text{equality}}}{8m^{4}}\ll
1\,,\quad\Rightarrow\quad\frac{\sigma}{m}\ll\left(\frac{m}{2\times
10^{-5}\text{ eV}}\right)^{5}\frac{\text{cm}^{2}}{\text{g}}\,.$ (4)

We have used $\rho\rvert_{\text{equality}}\simeq 0.4\text{ eV}^{4}$ as a dark
matter density at matter-radiation equality. Notice that if a fraction of dark
matter is produced by a mechanism other than the vacuum misalignment, an
additional statistical contribution to the pressure would be generated that
could tighten the bound (4). Interestingly, for light-enough particles (4)
seems to compete with the merger constraint. However, the real Bullet Cluster
bound turns out to be even more restrictive than the naively obtained
inequality (2). Due to phase-space degeneracy for sub-eV particles, the
interaction rate is boosted by the bosonic enhancement factor. In particular,
denoting the average degeneracy factor by $\mathcal{N}$, the improved version
of (2) takes the following form for sub-eV dark matter particles of interest

$\displaystyle\frac{\sigma}{m}\mathcal{N}\lesssim 1{\text{
cm}^{2}}/{\text{g}}\,,\quad\Rightarrow\quad\frac{\sigma}{m}\lesssim
10^{-2}\left(\frac{m}{\text{eV}}\right)^{4}\frac{\text{cm}^{2}}{\text{g}}\,.$
(5)

Moreover, this improved bound results in the exclusion of a significant part
of the parameter space, as we are about to show.

The paper is organized as follows. In section II, we overview some of the key
properties of the condensate of interacting particles governed by (1). In
section III, we study the conditions that lead to a superfluid phase
transition in galaxies and clusters. We show that, in order to have kpc-size
superfluid cores, we have to consider a scenario with highly degenerate phase-
space, implying an ultra-light mass range for dark matter particles. In
section IV, we revise the well known bound on dark matter self-interaction
cross section inferred from merging clusters accordingly. Section V is devoted
to the detailed analysis of the full parameter space for a Milky Way-like
halo. We summarize the results in section VI.

##  II II. Superfluid Properties

In a theory of self-interacting bosons, superfluidity may be achieved through
Bose-Einstein condensation. Within the theory given by (1), the condensate of
$\Phi$ particles can be well described by a homogeneous classical field
configuration with finite number density, which spontaneously breaks the U(1)
global symmetry of the Lagrangian. As it is well known, any homogeneous fluid
is susceptible to the gravitational Jeans instability above a certain length-
scale. For the theory at hand, the low-energy spectrum of excitations around
the homogeneous condensate (phonon spectrum) is given by

$\displaystyle\omega_{k}^{2}=-4\pi
G\rho+c_{s}^{2}k^{2}+\frac{k^{4}}{4m^{2}}\,,\qquad
c_{s}^{2}\equiv\frac{\lambda\rho}{4m^{4}}\;,$ (6)

with $G$ standing for the gravitational constant and $\rho$ denoting the
superfluid density444See, e.g., Berezhiani:2019pzd  for the derivation.. The
first term in (6) is a tachyonic contribution responsible for the Jeans
instability, the second term describes the energy cost for exciting the sound
waves and the last one is the kinetic energy of a massive constituent; in
other words, the last term indicates that in order to create an excitation, we
need to make a massive constituent of the superfluid mobile along the way. It
is straightforward to see that, in order to have stable sound waves, $\lambda$
needs to be positive which corresponds to the case of repulsive bosons555In
the opposite case ($\lambda\leq 0$), the only stabilizing contribution to (6)
would have been the last term., which is essential for superfluidity and will
be assumed throughout this work.

As one can easily deduce from (6), for the homogeneous condensate, the modes
softer than the critical momentum-scale $k_{*}$ are unstable; with

$\displaystyle k_{*}^{2}\equiv 2m^{2}c_{s}^{2}\left(-1+\sqrt{1+\frac{4\pi
G\rho}{m^{2}c_{s}^{4}}}\right)\,.$ (7)

The existence of this Jeans scale implies that due to gravity there is an
upper bound on the coherence length for the homogeneous superfluid
configuration, which is the result of the equilibrium between the
gravitational attraction and either the repulsive self-interaction (giving
rise to the sound-speed) or the quantum pressure (à la Fuzzy Dark Matter
scenario).

It is easy to see that, depending on the value of $c_{s}$, which in turn is
determined by the self-interaction strength $\lambda$ (or equivalently by the
scattering cross section $\sigma$), the Jeans momentum has two interesting
limits depending on the magnitude of the dimensionless quantity

$\displaystyle\xi\equiv\frac{m^{2}c_{s}^{4}}{4\pi G\rho}\,.$ (8)

In particular, in the non-interacting limit $\xi\ll 1$ we get

$\displaystyle\lim_{\xi\ll 1}k_{*}^{2}=\sqrt{16\pi G\rho m^{2}}\,,$ (9)

which is the Jeans scale stabilized by the quantum pressure. It corresponds to
the Fuzzy Dark Matter scenario Hu:2000ke ; Hui:2016ltb  and we refer to it as
the degeneracy pressure case. For completeness, let us stress that in this
limit one cannot talk about sound waves anymore, since the dispersion relation
of phonons with wavelength well within the homogeneity domain are highly
dominated by the last term of (6). Although subdominant, the presence of the
sound-term can still give rise to superfluidity by providing an additional
energy cost for excitations.

For the opposite case, with $\xi\gg 1$, we get

$\displaystyle\lim_{\xi\gg 1}k_{*}^{2}=\frac{4\pi G\rho}{c_{s}^{2}}\,,$ (10)

which is the standard result for the Jeans scale for the superfluid. In this
case, the gravitational instability is counteracted and balanced by the
repulsive interactions (regular positive pressure). This is the case we are
interested in, and we refer to it as the interaction pressure case.

Since the magnitude of the parameter $\xi$ defines the nature of the pressure
that sustains the condensate, let us evaluate it at typical galactic
densities,

$\displaystyle\xi=\frac{4\pi M_{\rm pl}^{2}\rho\sigma}{m^{4}}\simeq
10^{27}\left(\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\right)\left(\frac{m}{\rm
eV}\right)^{-3}\,,$ (11)

where we estimated $\rho\simeq 10^{-25}{\rm g/cm}^{3}$, which is the average
dark matter density of inner regions of the Milky Way. This is justified as we
are after the scenario in which the dark matter density profile is altered
significantly only at short scales. In deriving (11), we have used the
relation between the sound speed and the scattering cross section

$\displaystyle c_{s}^{2}=\frac{\rho}{m^{4}}\sqrt{2\pi m^{2}\sigma}\,.$ (12)

As it can be seen from (11), keeping in mind that we do not wish $m$ to be
significantly heavier than eV, the only way one could end up with $\xi\ll 1$
would be to take an extremely small scattering cross section (per mass)
$\sigma/m$.

For $\xi\gg 1$, it is straightforward to derive the self-sustained spherical
density profile of a zero-temperature superfluid soliton by solving the
equation for the hydrostatic equilibrium and Poisson’s equation. For the
quartic interaction at hand, the superfluid equation of state is
$P=\lambda\rho^{2}/8m^{4}$, which leads to the following analytic expression
for the self-sustained density profile chandrabook

$\rho(r)=\rho_{0}\frac{\sin\left({2\pi r/\ell}\right)}{2\pi r/\ell},$ (13)

where $\ell\equiv 2\pi/k_{*}=\sqrt{\frac{\pi\lambda}{4Gm^{4}}}$ is the Jeans
length in the $\xi\gg 1$ limit and $\rho_{0}$ is the central density of the
soliton. Equation (13) shows the equivalence between the Jeans length $\ell$
and the size of the soliton diameter. It must be noted that the density-
independence of $\ell$ is tightly connected with the type of self-interaction
present for the dark matter field; it would not have been the case for any
other form of the potential. It has proven to be convenient to express the
size of the soliton in terms of the cross section Slepian:2011ev

$\displaystyle\ell=2\pi\left(\frac{8\pi M_{\rm
pl}^{4}}{m^{5}}\frac{\sigma}{m}\right)^{1/4}\simeq 2~{}{\rm
kpc}\left(\frac{\sigma/m}{{\rm cm}^{2}/{\rm
g}}\right)^{1/4}\left(\frac{m}{{\rm meV}}\right)^{-5/4}\,.$ (14)

This way, one can get an idea of what it takes to have a macroscopic core.

Therefore, if we take a nearly-zero-temperature homogeneous superfluid of
$\Phi$s (that would have been stable in the absence of gravity), it will break
into superfluid islands of size $\ell$ and the density profile given by (13),
similar to the formation of the cluster of stars from a baryonic cloud.

We would like to finish this section by pointing out that the above discussion
applies to a zero-temperature superfluid. In the dark matter context, the
initial thermalized region has a finite temperature Slepian:2011ev ;
Berezhiani:2015bqa . Therefore, each superfluid soliton will be dressed in an
envelope of normal dark matter particles. The transition area is expected to
be located where the density drops below the critical one; i.e. the density
for which the de Broglie wavelength becomes shorter than the inter-particle
separation. Within the superfluid soliton the densities will be of order of
the galactic values (maybe somewhat larger, as they would be the result of a
collapse), while the transition density can be roughly estimated as
$\rho_{c}\sim m^{4}v^{3}$; with $v$ denoting the characteristic galactic
velocity determining the temperature. Assuming $v\sim 10^{-3}$, one gets
$\rho_{c}\sim 10^{-28}{\rm g/cm}^{3}(m/{\rm eV})^{4}$. In other words, if the
dark matter mass is significantly sub-eV (which will be the range of our
interest, as we will show), then the envelope begins at the distance from the
centre of the soliton where the densities have dropped by more than few orders
of magnitude compared to the core. Taking this into account, it is safe to
assume that the significant fraction of the dark matter within the thermalized
region will be in the form of the superfluid islands. The rest of the matter
will be split between the envelopes (gravitationally bound to solitons) and
the inter-soliton gas.

##  III III. Thermalization and Superfluid Formation

Let us investigate what are the conditions that lead to the formation of a
superfluid in the galactic medium. We would like to begin by mentioning that
it is possible to have an effective condensate without a local thermal
equilibrium, as long as the number of particles within the de Broglie volume
is large. In other words, the high degeneracy enables us to describe the
quantum state by a homogeneous field configuration, the perturbations around
which obey (6). Therefore, the coherence length can be estimated as
Guth:2014hsa

$\displaystyle\ell\simeq{\rm min}\left(2\pi/k_{*},\lambda_{\rm dB}\right)\,,$
(15)

with $\lambda_{\rm dB}$ denoting the de Broglie wavelength. If equilibrium is
not reached, then $\lambda_{\rm dB}$ is determined by characteristic dark
matter velocities obtained from N-body simulations. Using the value of the
virial velocity for a typical galactic halo, it is straightforward to verify
that (15) will always reduce to $\lambda_{\rm dB}$, unless one considers
particle masses even lighter than the one for the Fuzzy Dark Matter scenario.
Therefore, in practice, one needs to invoke thermalization in order to even
hope to get a macroscopic (kpc-size) core for moderately sub-eV particles.

Following what we said, the Bose-Einstein condensation for weakly interacting
bosons sets in if two conditions are satisfied:

  * •

First, the system must reach the equilibrium. This is achieved after particles
had sufficient time to interact and reach the (nearly) maximum entropy state,
since otherwise the applicability of the Bose-Einstein statistics would be
questionable. The time it takes to reach the equilibrium can be estimated as
$t_{\rm eq}>t_{1}$, with $t_{1}$ denoting the time it takes each particle to
scatter at least once. The longer one waits, compared to $t_{1}$, the more
certain one can be for being close to equilibrium. A more precise statement is
beyond the scope of this paper.

  * •

Second, the de Broglie wavelengths of particles must overlap. This corresponds
to the system being colder than the critical temperature $T_{c}\sim
n^{2/3}/m$; with $n$ denoting the particle number density. Physically, what
happens is that at high temperatures, for which the de Broglie wavelength is
shorter than the inter-particle separation, the gas of weekly interacting
particles behaves as a classical system and the Bose-Einstein distribution is
well-approximated by the Boltzmann distribution. Below the critical
temperature, on the other hand, the latter fails to adequately describe the
system, because indistinguishable particles with overlapping wave-packets
start to populate the zero momentum state (in compliance with the Bose-
Einstein statistics). In fact, for $T\ll T_{c}$ almost all particles of the
gas are in the ground state.

Therefore, we expect the dark matter halo to possess few relevant length
scales which are not in a one-to-one correspondence: the first one, the
thermal radius $R_{T}$, identifies the region where interactions are efficient
enough to allow thermal equilibrium. The second one is the degeneracy radius
$R_{\rm deg}$ within which the de Broglie wavelength exceeds the inter-
particle separation. The shortest of these identifies the region where the
phase transition is expected to take place. The last, but not the least, is
$\ell$ connected to the scale below which the condensate is stable.
Understanding the hierarchy between these scales is vital in order to
understand whether the fragmentation takes place or not, namely if galaxies
present a single superfluid core or a collection of superfluid substructures.

The formation of a dark matter halo is a non-linear process and as such it is
challenging (if not impossible) to establish a precise density profile
analytically. For purely gravitationally interacting (standard) dark matter
models, $N$-body simulations reveal the more or less universal density
distribution, known as the NFW profile

$\displaystyle\rho(r)=\frac{\rho_{0}}{\frac{r}{r_{s}}\left(1+\frac{r}{r_{s}}\right)^{2}}\,.$
(16)

The characteristic density $\rho_{0}$ and the scale radius $r_{s}$ vary from
halo to halo. However, according to simulations, there exists a tight relation
between these two parameters, known as the mass-concentration relation
Dutton:2014xda . The size of the halo itself is conventionally defined by the
virial radius $R_{V}$, which represents the radius within which the average
density of the halo (denoted as $\rho_{200}$) is about 200 times the critical
density. For our qualitative analysis we begin with the NFW profile and
examine under what conditions a significant modification of the profile,
followed by a superfluid formation, takes place. As we have already pointed
out, one should expect the aforementioned thermalization and degeneracy
requirements to be more easily satisfied in central (high density) regions.

Depending on the parameters of the model $m$ and $\lambda$, dark matter
particles could become degenerate at densities lower than the ones at which
the equilibrium can be reached. In that case, the interaction rate,
responsible for thermalization, will be assisted by the degeneracy factor that
roughly counts the number of particles in the de Broglie volume. In general,
the relaxation rate for highly degenerate particles can be estimated as
Sikivie:2009qn ; Erken:2011dz

$\displaystyle\Gamma=\frac{\sigma}{m}\rho
v\mathcal{N}\,,\qquad\mathcal{N}={\rm
max}\left\\{1,~{}\frac{\rho}{m}\left(\frac{2\pi}{mv}\right)^{3}\right\\}\,,$
(17)

where $v(r)=\sqrt{\frac{GM(r)}{r}}$ stands for the orbital velocity of dark
matter particles and the velocity dispersion has been assumed to be of order
$v$, while $M(r)$ is the mass of the halo enclosed in an orbit of radius $r$.

Following our earlier discussion, we estimate the thermal radius $R_{T}$ as
the one within which the particles had a chance to scatter at least once
throughout the lifetime of the galaxy; i.e. $R_{T}$ is a radius within which
we have $\Gamma t_{\text{g}}>1$, with $t_{\text{g}}\approx 13~{}{\rm Gyrs}$
being the age of the galaxy.

For completeness, let us note that using $t_{\text{g}}$ as the time-scale for
thermalization implies the assumption that it is possible to ignore the phase-
space reshuffling of $\Phi$ due to dynamical effects within the galaxy. If
this is not justified, the dynamical time $t_{\text{dyn}}=r/v$ is more
appropriate to determine the thermal radius. We will demonstrate in the
appendix that the utilization of $t_{\text{dyn}}$ results in a reduction of
the thermalization radius $R_{T}$ by a factor of few.

Depending on whether the equilibrium is reached while $\mathcal{N}\gg 1$ or
not, there could be two qualitatively distinct cases to consider:

  * (i)

If thermalization is reached at radius $R_{T}$ while $\mathcal{N}\simeq 1$,
then we would have a non-degenerate classical gas of weakly interacting
particles at distances $r>R_{T}$ from the centre of the halo. In other words,
for $r>R_{T}>R_{\rm deg}$ particles are not aware of interactions, for
$R_{T}>r>R_{\rm deg}$ particles had the chance to experience interactions and
as such the distribution will be more fuzzed out compared to the non-
interacting case. Since the particles are non-degenerate in this region, the
profile would resemble the profile one obtains in self-interacting dark matter
models. At $r<R_{\rm deg}<R_{T}$, on the other hand, the Bose-Einstein
condensation would take place and we would expect to see the presence of
superfluid islands of size $\ell$.

  * (ii)

An alternative scenario would be that the high degeneracy is reached at
distances larger than the thermal radius ($R_{\rm deg}>R_{T}$). In this case,
the halo would have a simpler structure. In particular, at $r>R_{T}$ the
density profile would be similar to the one for the non-self-interacting dark
matter, i.e. like NFW, with the possibility of a BEC-like sub-structure at
scales shorter than the de Broglie wavelength. Then we would expect the
superfluid phase transition directly at $r<R_{T}$, populating the
corresponding volume with the aforementioned superfluid islands of size
$\ell$.

Now, we are going to study those two scenarios separately. For definiteness,
the numerical estimates will be performed for a Milky Way-like galaxy with the
total mass $M_{\rm DM}=10^{12}M_{\odot}$ and the concentration parameter
$c=R_{\rm V}/r_{s}=6$.

## Case (i): Non-degenerate Thermalization

Let us focus on the case where the thermalization of $\Phi$s happens in a non-
degenerate setting. The thermal radius $R_{T}$ can be estimated as

$\displaystyle\Gamma
t_{\text{g}}=\frac{\sigma}{m}\rho(R_{T})v(R_{T})t_{\text{g}}=1\,.$ (18)

For a given density profile, this equality gives $R_{T}$ as a function of
$\sigma/m$. Concerning the scaling of (18) with the parameters of the NFW
profile, this is a cumbersome function of $R_{T}$, $\rho_{0}$ and $r_{s}$.
However, the behaviour is simple in limiting cases

$R_{T}\simeq{}r_{s}\Big{(}\rho_{0}r_{s}\sqrt{2\pi
G\rho_{0}}\frac{\sigma}{m}t_{\rm g}\Big{)}^{\gamma}\,,\quad{\rm
with}\quad\gamma=\begin{cases}{{2}},&\mbox{for }R_{T}\ll r_{s}\\\
2/7,&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (19)

Not surprisingly, the overall result for $R_{T}$ is a monotonically increasing
function of the cross section. In other words, for larger $\sigma/m$ the dark
matter particles manage to reach equilibrium at larger radii (i.e. lower
densities). It is easy to find that for the Milky Way-like halo at hand, using
$\rho\simeq 10^{-25}{\rm g/cm}^{3}$ , one can conveniently express the
thermalization radius as:

$\displaystyle R_{T}^{\rm MW}\simeq r_{s}\left(\frac{\sigma/m}{{\rm
cm}^{2}/{\rm g}}\right)^{\gamma}$ (20)

Let us stress that the case $R_{T}\ll r_{s}$ is sensitive to the specific
values of $r_{s}$ and $\rho_{0}$ and could have strongly been affected by
different fits. Moreover, we may see how the strength of the self-interactions
affects more mildly $R_{T}$ in outer regions of the halo: since the density
scales as $r^{-3}$, stronger self-interactions are needed to overcome the
density drop.

So far, we have not said anything about the mass of the dark matter particle.
The equality (18) will successfully provide us with the estimate for $R_{T}$,
as long as the degeneracy factor is small. Examining the expression for this
factor (see (17)) it is easy to see that to have $\mathcal{N}<1$ at the
distance $R_{T}\ll r_{s}$ from the centre,

$\displaystyle m\gtrsim 20~{}{\rm eV}\cdot\left(\frac{\sigma/m}{{\rm
cm}^{2}/{\rm g}}\right)^{-5/4}\,,$ (21)

implying masses significantly greater than eV. One gets a similar constraint
for $R_{T}\gg r_{s}$, albeit with a different power law. The important message
is that this scenario requires masses above eV, if the interaction strength
satisfies the Bullet cluster constraint. This observation together with (14),
and using $\sigma/m\lesssim{\rm cm}^{2}/{\rm g}$, gives us an absolute upper
bound on the size of the superfluid soliton

$\displaystyle\ell\lesssim 5\cdot 10^{-2}~{}{\rm pc}\,.$ (22)

Interestingly enough, the size of the superfluid patches would be an order of
magnitude or so larger than the solar system scale if one were to saturate
this, taking the density of the order of the NFW density at our location.
However, the density could be somewhat larger as these solitons result from a
fragmentation of a locally thermalized dark matter distribution.

## Case (ii): Degenerate Thermalization

It seems that, as long as thermalization happens in a non-degenerate setting,
particles cannot rely on superfluidity to generate kpc-size solitons. This
indicates that we have to explore the possibility of $R_{\rm deg}>R_{T}$, thus
violating (21). In this case we have to replace (18) with its degenerate
counterpart

$\displaystyle\Gamma
t_{\text{g}}=\frac{\sigma}{m}\rho(R_{T})v(R_{T})\mathcal{N}t_{\text{g}}=1\,,\quad\text{with}\quad\mathcal{N}=\frac{\rho}{m}\left(\frac{2\pi}{mv}\right)^{3}\gg
1\,.$ (23)

Here too, the exact expression for $R_{T}$ is cumbersome. However, similar to
the previous case, it can be nicely presented in limiting cases

$R_{T}\simeq
r_{s}\left(\frac{4\pi^{2}\rho_{0}}{Gm^{4}r_{s}^{2}}\frac{\sigma}{m}t_{\rm
g}\right)^{\delta}\,,\quad{\rm with}\quad\delta=\begin{cases}1/3,&\mbox{for
}R_{T}\ll r_{s}\\\ 1/5,&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (24)

For a Milky Way-like halo, the expression reduces to

$\displaystyle R_{T}^{\rm MW}\simeq\begin{cases}60\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{1/3},&\mbox{for }R_{T}\ll r_{s}\\\ 10\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{1/5},&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (25)

Unlike the non-degenerate case, here the size of the thermalized region is
determined by $\left({\sigma}/{m}\right)m^{-4}$, as it was previously
demonstrated in Berezhiani:2017tth . For instance, as one can see from (25),
for the galactic halo in question

$\displaystyle R_{T}\gtrless
r_{s}\,,\quad\Leftrightarrow\quad\frac{\sigma/m}{{\rm cm}^{2}/{\rm
g}}\left(\frac{m}{\rm eV}\right)^{-4}\gtrless 10^{-5}\,.$ (26)

Moreover, since (24) depends only mildly on $r_{s}$ and $\rho_{0}$, this
result is not very sensitive to a fine tuning of the NFW-parameters.

Therefore, it seems there exists a wide range of parameters, for which a
significant fraction of the halo has had enough time to have reached
equilibrium; the parameter space here is even larger than in case (i), since a
specific value of the thermal radius is now generated by different
combinations of $m$ and $\sigma/m$ due to the introduction of $\mathcal{N}$.

##  IV IV. Bullet Cluster Constraint

In this section, we would like to revisit the Bullet Cluster constraint
Markevitch:2003at  for the ultra-light bosonic dark matter candidate. The
system in question is a merger of two clusters, in which the dark matter
component is offset with respect to the gas component. As it is well known,
the comparison of the observed mass distribution with the simulated one does
not seem to indicate the presence of any other dark matter interaction besides
the gravitational one. In other words, the average number of scatterings
experienced by a given dark matter particle from the bullet cluster, while
crossing the target cluster, seems to be less than one

$\langle n_{sc}\rangle<1.$ (27)

The value $\langle n_{sc}\rangle$ is averaged over all the bullet cluster
particles and could be estimated as the product of the interaction rate
$\Gamma$ and the crossing time

$\langle n_{sc}\rangle=\Gamma\frac{2R_{V}}{v_{\text{in-fall}}},$ (28)

where $v_{\text{in-fall}}\simeq 10^{-2}$ is the in-fall velocity and
$R_{\text{V}}$ is the virial radius of the target cluster. Clearly, the bound
on the scattering cross section that can be extracted from (27) and (28)
depends on the nature of dark matter.

For example, assuming non-degenerate dark matter that interacts through 2-body
interactions

$\langle n_{sc}\rangle=2\frac{\sigma}{m}R_{V}\rho.$ (29)

Using $\rho\simeq 10^{-25}{\text{ g}}/{\text{cm}^{3}}$ as an average dark
matter density and $R_{V}\simeq 2$ Mpc, one obtains

$\frac{\sigma}{m}\lesssim 1{\text{ cm}^{2}}/{\text{g}}.$ (30)

The value of the density that we have chosen represents the average density of
the Target Cluster within 500 kpc, obtained fitting the matter distribution
using an NFW profile Clowe:2003tk 666At this distance, the density profile
changes from $1/r^{3}$ to $1/r$: therefore, we are averaging over the region
where the mass scales as $r^{2}$ but not in the part where the mass scales
logarithmically with $r$..

As we have already discussed from the point of view of dark matter
thermalization in galaxies, if particles have a degenerate phase-space, then
the interaction rate is enhanced by $\mathcal{N}$ (17). The same goes for
mergers, if colliding halos are significantly degenerate, then (30) needs to
be reconsidered. It is straightforward to see that for the NFW profile of a
cluster (with $R_{V}\simeq 2~{}{\rm Mpc}$, $R_{V}/r_{s}\simeq 4$ and
$\rho_{0}\simeq 10^{-25}{\text{ g}}/{\text{cm}^{3}}$) the degeneracy factor
exceeds unity everywhere inside the virial radius if $m\ll{\rm eV}$. Notice
that $\mathcal{N}_{\text{cluster}}\simeq 10^{-3}\mathcal{N}_{\text{galaxy}}$,
since the velocities are an order of magnitude or so higher in clusters.
Therefore, if galaxies are strongly degenerate, which is the case for the
parameters of interest, then the scattering rates in clusters are expected to
be enhanced as well.

We can estimate the improved Bullet Cluster bound on the self-interaction
cross section by boosting the interaction rate by a typical (average) value of
the degeneracy factor for a cluster, resulting in

$\frac{\sigma}{m}\lesssim
10^{-2}\left(\frac{m}{\text{eV}}\right)^{4}\frac{\text{cm}^{2}}{\text{g}}\,.$
(31)

We would like to stress that this relation applies if the halos are in the
form of a gaseous medium of particles with $m\ll{\rm eV}$ and the significant
velocity dispersion. If the entire halo of the cluster were to thermalize and
undergo fragmentation into superfluid solitons, without significant leftover
in the form of a dispersed gas, then (31) would need to be ameliorated: in
this case, almost all particles would lie in the ground state and transitions
to excited states would not be enhanced by degeneracy. However, it seems
unlikely not to end up with a significant fraction of particles to remain un-
condensed in the process of thermalization and fragmentation. Here, we take
this qualitative statement for granted and leave more detailed analysis to
future work.

The revised Bullet Cluster constraint (31) leads to the following upper bound
on the thermal radius of the Milky Way

$\displaystyle\left(R_{T}\right)_{\text{Milky Way}}\lesssim 125\text{ kpc}$
(32)

Moreover, it is straightforward to see that the thermal radii of the target
and bullet clusters cannot exceed $\approx 0.5$ Mpc.

Therefore, even if the estimated inequality (31) is saturated, only inner
regions of the dark matter halo are capable of reaching equilibrium; with the
outskirts being unaffected by the presence of self-interactions.

##  V V. Relative Size of thermalized region and coherence length

In this section, we focus on a relation between the Jeans scale $\ell$ and the
thermal radius $R_{T}$. Since we are interested in the superfluid regime of
the theory, let us begin by pointing out that for interaction-pressure
dominance (large $\xi$) $R_{T}$ and $\ell$ (given by (14)) are controlled by
different combinations of $m$ and $\sigma$. Therefore, it may seem possible to
pick the parameter values in such a way as to have kpc-size superfluid islands
while $R_{T}$ could vary from values lower than $\ell$ up to $R_{V}$. We will
demonstrate that the dark matter masses required for $\ell\geq R_{T}$ are so
low that one enters the Fuzzy Dark Matter parameter space. This is equivalent
to a transition from a superfluid whose degrees of freedom are collective
modes (phonons with a linear dispersion relation) to a condensate whose
dynamics is described by almost-free constituents (quadratic dispersion
relation).

It is straightforward to see that $\ell$, given by (14), can be expressed in
terms of $R_{T}$ using (23) as

$\left(\frac{\ell}{2\text{kpc}}\right)\simeq\left[\mathcal{F}\left({R_{T}}/{\text{kpc}}\right)\cdot{\left(\frac{m}{\text{meV}}\right)^{-1}}\right]^{1/4}\
,$ (33)

where $\mathcal{F}$ is determined by the density profile of a halo. The
explicit form of $\mathcal{F}$ may be easily deduced from (23). This equation
shows that for a given $R_{T}$ the Jeans scale depends only mildly on the mass
of the dark matter particle. Because of this, boosting $\ell$ to sufficiently
large values may require lowering $m$ so much that we may end up leaving the
interaction-pressure domination regime and enter the quantum pressure
dominance. In other words, $\ell$, $\xi$ and $R_{T}$ depend on three different
combinations of $m$ and $\sigma/m$. At the same time, the superfluidity
requires $\xi\gg 1$, which is not satisfied by a generic choice of $\ell$ and
$R_{T}$, due to the mild mass-dependence of (33).

Figure 1: Black lines show the slices of the [m,$\frac{\sigma}{m}$]-parameter
space that generate a Jeans scale of $0.1$ kpc/ $2$ kpc/ $6$ kpc in degenerate
regions of the Milky Way dark matter halo, assuming a particle species
governed by the Lagrangian (1). They are obtained evaluating (7) numerically.
The sloped part of the curve represents the superfluid regime of the theory
($\xi\gg 1$) while the flat part ($\xi\ll 1$) represents the region where
particles self-interact so little as to be considered a non-interacting
species. To evaluate $\ell$ we assumed $\rho\simeq 10^{-25}\text{g/cm}^{-3}$,
which is the expected average density of the thermal core before
fragmentation. This value influences only the $\xi\leq 1$ part of the Jeans
scale, which is favoured by small $R_{T}$. Blue/Orange/Red dashed curves
correspond to the parameter space that generates a thermal radius of 0.1 kpc/2
kpc/6 kpc, estimating $R_{T}$ as the radius within which the particles had the
chance to scatter at least once throughout the lifetime of the halo. On the
right of the gray line, the Milky Way dark matter halo is in global thermal
equilibrium. In order to stress that we do not really know what is the
sufficient number of scattering events for reaching equilibrium, coloured
regions show how $R_{T}$ changes varying the sufficient mean number of
scattering events in the interval 1-10. We see that it is impossible to have
an interaction pressure dominated core with $\ell\gtrsim R_{T}$.

Having made those observations, let us focus on narrowing down the parameter
space for which we can get the macroscopic superfluid patches within the
thermalization radius. For this we use the expression for the Jeans scale (7)
that covers both the superfluid and Fuzzy Dark Matter regimes. The result is
given by Fig.1 for a Milky Way-like halo, where different values of the Jeans
scale and the thermal radius are shown. Let us stress that, even if $\ell$ is
density independent in the interaction pressure case (in case of the
$\lambda\Phi^{4}$-potential), this is not true for $\xi\lesssim 1$. Thus,
while the sloped part of solid curves is the same in every astrophysical
structure, this is not true for the turning point and the flat part.

Now, Fig.1 highlights the impossibility of $\ell\gtrsim R_{T}$ for halos
sustained by the interaction pressure: lines corresponding to a given $\ell$
do not intersect lines describing thermal radii with a smaller relative
magnitude in the interaction pressure limit. Therefore, if the dark matter is
a scalar particle with a $\lambda\Phi^{4}$-potential which forms an
interaction-pressure-supported superfluid core, we have

$\ell<R_{T}\ .$ (34)

To show that (34) is not an artefact of our specific definition of the thermal
radius, coloured regions represent the parameter space that generates specific
values of the thermal radius when changing the sufficient number of scattering
events to equilibrate the dark matter halo in the interval 1-10.

##  VI VI. Summary

Let us conclude by reiterating the qualitative tale of dark matter
superfluidity discussed in this work. As dark matter particles begin to clump
to form a halo, they start similarly to CDM. Up until the point when particles
begin to scatter from each other, they are striving towards an NFW density
profile. In regions where densities increase sufficiently for particles to
start experiencing collisions, the evolution of the density profile starts to
depart from its collisionless counterpart. In fact, the regions within which
each dark matter particle has had a chance to scatter few times should be
close to equilibrium. Although efficient interactions tend to homogenize the
density profile, the phase transition and the formation of superfluid droplets
may take place if the de Broglie wavelengths begin to overlap inside
thermalized regions. Both equilibration and the overlapping wave-functions
favour high densities and are hence easier to achieve within central regions.
Denoting the radii of the corresponding boundaries as $R_{T}$ and $R_{\rm
deg}$ respectively, we have demonstrated that a presence of $\sim$kpc-size
superfluid patches enforces the parameter space for which $R_{\rm deg}>R_{T}$.
Within $R_{T}$ the core breaks into superfluid islands of size $\ell$
(determined by the parameters of the model) due to the gravitational Jeans
instability. The dynamics of the droplets plays an important role to shape the
density distribution of the thermal core. As we have shown in the first
section, each island is a superfluid soliton (non-topological) with a
practically homogeneous core.777This is similar to the Bose-Einstein
condensation of bosonic cold-atoms with attractive contact interactions. As a
result, the homogeneous condensate is unstable and fragments into solitons.
For $\ell\ll R_{T}$, the course-grained (over scales larger than $\ell$)
density profile should resemble the one in CDM as superfluid solitons are
expected to behave as weakly interacting effective particles.

In Fig. 2 we combine the limits discussed in this work to understand what is
the parameter space of a dark matter candidate governed by the Lagrangian (1)
that could generate kpc-size solitons in partially thermalized clusters. The
pink region is excluded by the analysis of the collision of degenerate
clusters (leading to an upper limit of the self-interaction strength). For
completeness, let us stress that the results of our qualitative analysis are
sensitive to the pre-thermalization shape of the density profile and in
particular to the values for the parameters of the NFW distribution. Also, the
value we used to estimate the mean density of the thermal core would drop by
one order of magnitude if thermalization is strong enough to affect outer
regions of the galactic halo. However, this would neither change the behaviour
of the theory in the interaction pressure regime, which is density
independent, nor the conclusion on the fragmentation of the halo. In fact, the
transition point between degeneracy and interaction pressure highlighted by
Fig. 1 would move to the right, since stronger self-interactions are necessary
to compensate the smaller mean density of the thermal core.

Moreover, as we pointed out in Section 4, the bound (31) applies only if a
significant fraction of the bullet cluster halos is in the gaseous form. This
constraint would have been downgraded to (30) if both the bullet and the
target clusters were to thermalize completely and undergo the fragmentation
into superfluid solitons, without a significant leftover in the dispersed
phase. On the off chance that this could happen, we have highlighted the
region of the parameter space in Fig. 2 where both the target and bullet
cluster are expected to have thermalized in their entirety. In this case the
density profile of clusters would deviate from the usual NFW profile, and the
analysis of the profile and the substructure of clusters could lead to
constraints on this scenario. The detailed analysis of this scenario being
beyond the scope of this work, we have assumed such a fragmentation to result
in a significant portion of the cluster mass to have remained in the form of a
gaseous medium.

Figure 2: Parameter space for $\Phi$. We focussed on the superfluid regime and
we excluded halos sustained by the quantum pressure (green region). The pink
region is excluded by the bullet cluster constraint while the purple one
represents condensates which were relativistic at matter-radiation equality.
The striped region highlights the parameter space where both the bullet and
target cluster are in global thermal equilibrium. In the left panel, Black
solid lines identify the parameter space that generates a Jeans scale of 0.5
kpc/2 kpc/6 kpc. In the right panel, we plotted the whole parameter space,
highlighting $\ell=10^{-3}$ pc/ 1 pc/6 kpc for reference.

Finally, let us comment on the magnitude of the coupling constant $\lambda$.
Fig. 2 illustrates that the generation of solitons with a diameter
$\ell\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}0.5$ kpc is compatible with the revised bullet cluster bound
only for cross sections $\sigma/m\lesssim 10^{-63}\text{ cm}^{2}/$g and masses
$m\lesssim 10^{-15}$ eV. It is straightforward to verify that this means
considering $\lambda\lesssim 10^{-65}$: the smallness of the coupling constant
involved is a consequence of the extreme enhancement of the interaction rate
by the sub-eV mass of $\Phi$, both through $\mathcal{N}$ and through the mass
dependence of $\sigma/m$.

## Achnowledgements

We would like to thank Justin Khoury for valuable discussions and comments.

## Appendix: Dynamical time as the time-scale for thermalization

Let us show the differences emerging by using dynamical time $t_{\text{dyn}}$
to describe the time-scale in which dark matter thermalizes. Indeed, if
galactic dynamical effects are efficient enough to reshuffle the phase-space
of $\Phi$ significantly, the correct time-scale involved would be
$t_{\text{dyn}}=r/v$. Therefore, focussing on the degenerate case, particles
had the time to interact only if

$\Gamma t_{\text{dyn}}=\mathcal{N}\frac{\sigma}{m}\rho(R_{T})R_{T}=1\ .$ (35)

In this case, $\rho_{0}$, $R_{T}$ and $r_{s}$ enter in equation (35) through
the combination $\rho^{2}R_{T}/v^{3}$. Again, we may extract the following
limits:

$R_{T}\simeq
r_{s}\left(\sqrt{\frac{8\pi^{3}\rho_{0}}{G^{3}}}\frac{\sigma/m}{m^{4}r_{s}^{2}}\right)^{\delta}\,,\quad{\rm
with}\quad\delta=\begin{cases}2/5,&\mbox{for }R_{T}\ll r_{s}\\\ 2/7,&\mbox{for
}R_{T}\gg r_{s}\end{cases}$ (36)

For a Milky Way-like halo, the expression reduces to

$\displaystyle R_{\rm T,dyn}^{\rm MW}\simeq\begin{cases}30\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{2/5},&\mbox{for }R_{T}\ll r_{s}\\\ 10\cdot
r_{s}\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{2/7},&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (37)

We may now compare (25) and (37):

$\displaystyle\frac{R_{\rm T,dyn}^{\rm MW}}{R_{\rm T,g}^{\rm
MW}}\simeq\begin{cases}1/2\left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm
g}}\left(\frac{m}{\rm eV}\right)^{-4}\right]^{1/15},&\mbox{for }R_{T}\ll
r_{s}\\\ \left[\frac{\sigma/m}{{\rm cm}^{2}/{\rm g}}\left(\frac{m}{\rm
eV}\right)^{-4}\right]^{3/35},&\mbox{for }R_{T}\gg r_{s}\end{cases}$ (38)

One can see that $R_{T}$ gets reduced by a factor 2 at most (the $\sigma$ and
$m$ dependence is too mild to affect the ratio by an order one contribution).
In this case, the analogue of Fig. 2 would reveal a similar picture of what we
obtained using $t_{\text{g}}$.

Finally, we report the upper bound from the bullet cluster on the thermal
radius of the Milky Way using $t_{\text{dyn}}$:

$\displaystyle(R^{\text{dyn}}_{T})_{\text{Milky Way}}\lesssim 60\text{ kpc}.$
(39)

As expected, the bound on the thermal radius is tighter than (32) when we use
$t_{\text{dyn}}$ as the time scale for thermalization.

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