Gravothermal collapse of Self-Interacting Dark Matter halos as the Origin of Intermediate Mass Black Holes in Milky Way satellites Tamar Meshveliani1Jes us Zavala1and Mark R. Lovell1

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Gravothermal collapse of Self-Interacting Dark Matter halos

as the Origin of Intermediate Mass Black Holes in Milky Way satellites

Tamar Meshveliani,1, ∗Jes´us Zavala,1and Mark R. Lovell1

1Centre for Astrophysics and Cosmology, Science Institute,

University of Iceland, Dunhagi 5, 107 Reykjavik, Iceland

(Dated: May 25, 2023)

Milky Way (MW) satellites exhibit a diverse range of internal kinematics, reﬂecting in turn a

diverse set of subhalo density proﬁles. These proﬁles include large cores and dense cusps, which any

successful dark matter model must explain simultaneously. A plausible driver of such diversity is

self-interactions between dark matter particles (SIDM) if the cross section passes the threshold for

the gravothermal collapse phase at the characteristic velocities of the MW satellites. In this case,

some of the satellites are expected to be hosted by subhalos that are still in the classical SIDM

core phase, while those in the collapse phase would have cuspy inner proﬁles, with a SIDM-driven

intermediate mass black hole (IMBH) in the centre as a consequence of the runaway collapse. We

develop an analytical framework that takes into account the cosmological assembly of halos and is

calibrated to previous simulations; we then predict the timescales and mass scales (MBH) for the

formation of IMBHs in velocity-dependent SIDM (vdSIDM) models as a function of the present-day

halo mass, M0. Finally, we estimate the region in the parameter space of the eﬀective cross section

and M0for a subclass of vdSIDM models that result in a diverse MW satellite population, as well

as their corresponding fraction of SIDM-collapsed halos and those halos’ inferred IMBH masses. We

predict the latter to be in the range 0.1−1000 M⊙with a MBH −M0relation that has a similar

slope, but lower normalization, than the extrapolated empirical relation of super-massive black holes

found in massive galaxies.

I. INTRODUCTION

The cold dark matter (CDM) model is highly success-

ful at explaining observations of the large-scale structure

of the Universe (e.g. [1]). However, it has challenges

in matching observations on small scales, such as in the

regime of dwarf galaxies (for a recent review see e.g [2]).

Observationally, these challenges have been established

prominently for dwarf galaxies in the Local Group, and

particularly within the Milky Way (MW) satellites. For

instance, the dynamical mass is dominated by dark mat-

ter (DM) in the inner region of several bright MW satel-

lites, yet is low compared to the inner densities of the

plausible subhalo hosts of MW-analogues found in colli-

sionless CDM simulations; this is the classical too-big-to-

fail (TBTF) problem [3, 4]. Another recurrent challenge

is that several of the MW satellites are best explained

by density proﬁles of constant density, known as “cored”

density proﬁles, rather than the steep inner density slope

found in CDM simulations, referred to as “cuspy” pro-

ﬁles; satellites that are reported to have cored proﬁles

include Fornax, Sculptor, Crater II and Antlia II [5–9].

Overall, it is now well established that the MW satel-

lites have a diverse range of internal kinematics, which

is likely associated with a subhalo population that ex-

hibits a considerable diversity of inner density proﬁles,

from large cores to dense cuspy systems [10–14]. This

diversity is analogous to that of rotation curves observed

in higher mass, gas-rich dwarf galaxies [15, 16].

∗e-mail:tam15@hi.is

It is important to emphasize that such dwarf-scale chal-

lenges are only insurmountable within CDM if no other

physical mechanisms related to gas/stellar (baryonic)

physics are considered. There are in fact several bary-

onic processes that are known to exist that can alleviate

these challenges. For instance, supernova feedback can

inject energy into the inner DM halo, reducing its den-

sity [17]. If impulsive enough, this is an eﬃcient and ir-

reversible cusp-core transformation mechanism in dwarf

galaxies [18–20]. In addition, tidal forces on the satellite

by the MW DM halo and the MW disk can eﬀectively

lower the densities of MW subhalos if their orbits pass

suﬃciently close to the disk [21]. The diverse orbits of

the MW satellites combined with this eﬀect enhance the

diversity of inner DM densities relative to the CDM-only

expectations [22]. However, how eﬃcient these processes

are in creating the observed diversity of the MW satel-

lite population remains uncertain since, for example, the

impact of supernova feedback is expected to be small in

very faint DM-dominated systems with low stellar-mass

ratios [23]. On the other hand, it has been argued that

the tidal ﬁeld of the MW system might not be strong

enough to explain the extremely low densities of bright

satellites such as Crater II and Antlia II [24, 25].

One exciting possibility is that the properties of the

MW satellites provide clues about the DM nature be-

yond CDM. In particular, if DM particles have strong

self-interactions, they can impact the non-linear evolu-

tion of halos, signiﬁcantly reducing their inner densities

[26]. Modiﬁed N-body simulations that incorporate self-

interacting DM (SIDM) have shown that the collisions

experienced by DM particles with each other lead to sig-

arXiv:2210.01817v2 [astro-ph.GA] 24 May 2023

niﬁcant momentum exchange. This process eﬀectively

transfers heat from the dynamically hot outer regions of

the halo to the colder central regions, thus lowering the

central density of halos and creating constant (isother-

mal) density cores [27–34]. SIDM models can create size-

able DM cores and alleviate the classical TBTF problem

if the transfer cross section per unit mass, σT/mχ, is

≳1 cm2g−1at the characteristic scales/velocities of MW

satellites ≲50km/s [35]. It is also possible to allevi-

ate signiﬁcantly the diversity of rotation curves in higher

mass dwarf galaxies (characteristic velocities >50km/s)

if σT/mχ≳2−3 cm2g−1[36, 37].

Constant cross section SIDM models have been con-

strained more strongly at large scales/velocities. Par-

ticularly, σT/mχis required to be ≲0.1−1 cm2g−1at

the scale of clusters based on gravitational lensing, X-

ray morphology, and dynamical analysis in cluster merg-

ers [38–43]. At scales corresponding to massive elliptical

galaxies, previous constraints based on X-ray morphology

have been shown to be weaker than anticipated by DM-

only simulations [44] once baryonic eﬀects are included.

Current simulations including baryons have shown that

σTmχ∼1 cm2g−1is consistent with the morphologies

of elliptical galaxies [45]. With such constraints at larger

scales, a constant cross section SIDM model is already

only narrowly viable as an alternative to CDM to ex-

plain the properties of dwarf galaxies. Recent develop-

ments regarding the diversity in the inner densities of the

MW satellites virtually rule out this possibility since with

such a low cross section, it is not possible to generate very

high density satellites such as Willman I [13, 46–48].

Remarkably, what is needed for SIDM models to remain

an interesting, viable alternative to CDM is to have even

larger cross sections (>10 cm2g−1) at the scale of the

MW satellites in order to trigger the gravothermal col-

lapse phase (see below). Such large cross sections can

be naturally accommodated by particle models with a

velocity-dependent cross section (e.g. through Yukawa-

like interactions, see e.g. [49–51]), where DM behaves as

a collisional ﬂuid on small scales and is essentially colli-

sionless at cluster scales. Long after the core-formation

phase, further DM particle collisions lead to heat outﬂow

from the hotter inner region to the colder outskirts of the

halo. Since gravitaionally bound systems have negative

speciﬁc heat, mass/energy is continuously lost from the

inner region, while the density and temperature continue

to grow in a runaway instability that drives the collapse

of the inner core. This phenomenon is known as the

gravothermal catastrophe [52] and is observed in globu-

lar clusters, where the collapse is mainly halted by the

formation of binary stars, which act as energy sinks [53].

For SIDM halos, the physical mechanism is the same,

but without the formation of bound DM states to act as

energy sinks, the collapse continues, eventually reaching

a relativistic instability that results in the formation of

a black hole [28, 54–57]. If the core-collapse phase has

been reached at the scales of the MW satellites, then the

SIDM predictions become radically diﬀerent with some

of the satellites expected to be hosted by (sub)halos with

SIDM cores, while those in the collapse phase would have

cuspy (collapsed) inner DM regions [13].

Given the problems with constant cross section SIDM

models mentioned above, it has been argued recently that

such models could be reconciled with the MW satellite

population by suggesting that the collapse phase might

be accelerated in the host (sub) halos of MW satellites

by mass-loss via tidal stripping [58], since mass-loss en-

hances the negative temperature gradient in the out-

skirts of the (sub)halo and makes the heat outﬂow more

eﬃcient. Accelerated core-collapse has been invoked

to explain the diversity of the MW’s dwarf spheroidal

galaxies in constant cross section models with σT/mχ≳

2−3 cm2g−1[14, 58–60]. However, Ref. [61] recently

simulated SIDM subhalo satellites as they orbit the MW

system and found that energy gain due to collisions be-

tween particles in the subhalo and the host instead in-

hibits core-collapse in subhalos.

Another study, Ref. [62], showed that subhalos in mod-

els with constant cross sections between 1 and 5 cm2g−1

are not dense enough to match the densest ultra-faint

and classical dwarf spheroidal galaxies in the MW, and

5 cm2g−1is not suﬃcient to enforce collapse even with

the tidal eﬀect of a MW disk and bulge. This seem-

ingly closes the last possibility for velocity-independent

SIDM models (see also discussion in Section II G be-

low). On the other hand, this result motivates the explo-

ration of velocity-dependent SIDM models, where recent

full cosmological simulations with a speciﬁc benchmark

model [13, 63] have shown that cross sections ≳50 cm2/g

at velocities ≲30 km/s naturally result in a diverse

bimodal population of MW satellites, predicting both

cuspy, high velocity dispersion subhalos, consistent with

dense systems (particularly ultra-faint satellites), and

cored, low velocity dispersion subhalos, consistent with

brighter low-density satellites. These results have been

conﬁrmed and expanded to generic velocity-dependent

SIDM models by the recent cosmological simulation suite

TangoSIDM [64].

In this work, we adopt the benchmark SIDM model

presented in [13, 63] to explore the consequences of

gravothermal collapse for the formation of intermediate

mass black holes (IMBHs) in the MW satellite popula-

tion. Our goal is twofold: (i) to compile a simple analyt-

ical framework (calibrated to the simulations in [13, 63])

that provides predictions for the formation timescales

and mass scales of IMBHs in SIDM halos under arbitrary

velocity-dependent cross sections, and (ii) to provide the

range of IMBH masses that is expected given the plau-

sible range of cross sections that produce a diverse MW

satellite population, i.e., a bimodal – core-cusp – satellite

distribution.

This paper is organised as follows. In Section II, we de-

scribe our model for the evolution of SIDM halos. We

start with our adopted primordial halo density proﬁle

and the concentration–mass relation, describe our com-

putation of the threshold time for the cusp-core transfor-

mation, and ﬁnally estimate the timescales and masses of

IMBHs expected in the SIDM model due to gravothermal

collapse. We also include the impact of tidal stripping. In

Section III, we present our results, discuss how they are

impacted by the various properties of the model, and put

our work in the context of other related studies. Finally,

we draw conclusions in Section IV.

II. GRAVOTHERMAL COLLAPSE IN SIDM

HALOS

Our goal in this section is to follow the relevant stages

in the evolution of an SIDM halo: i) formation of the

progenitor cuspy (i.e. CDM-like) halo, ii) development

of the central core and iii) gravothermal collapse of the

core and formation of the black hole. In addition, we

discuss how tidal stripping might aﬀect the gravothermal

collapse timescale.

A. Cosmic evolution of SIDM halos

In an SIDM halo where thermalization occurs due to

close, rare interactions with large momentum transfer,

a relaxation time can be deﬁned due to self-scattering at

the characteristic radius1r−2, which is given by:

tr=λ

aσvel

,(1)

where σvel is the characteristic velocity dispersion, a=

p16/π for hard-sphere scattering of particles with a

Maxwell–Boltzmann velocity distribution [54] and λ−1=

ρ(r−2)σT/mχis the mean free path, which is inversely

proportional to the local density ρ(r−2) and the cross

section per unit mass σT/mχ(evaluated at the character-

istic velocity σvel in the case of velocity-dependent SIDM

models). Therefore, the scattering rate (mean free path)

is higher (shorter) in denser regions. Within the region

where the age of the inner halo is comparable to the re-

laxation time, self-scattering has a signiﬁcant impact on

the inner DM structure turning the cusp into a core.

In CDM, where DM is collisionless, the velocity disper-

sion peaks near the scale radius, r−2. By contrast, in

SIDM elastic scattering leads to momentum exchange be-

tween DM particles, which, given the positive gradient of

the velocity dispersion proﬁle within r−2, eﬀectively re-

sults in heat transfer from the outside-in, up to the radius

1From here on in, we assign the characteristic radius to the scale

radius of the halo, which for the NFW proﬁle is equal to r−2,

the radius at which the logarithm slope of the proﬁle is −2; see

Section II B.

where the velocity dispersion peaks. As a result, a central

isothermal core is formed, which continues to grow until

it is roughly the size of the scale radius and thus reaches

a quasi-equilibrium state. After core formation, subse-

quent collisions lead to momentum/energy ﬂow from the

center to the outskirts of the halo, where the velocity

dispersion proﬁle has a negative slope. Heat loss in the

core results in the infall of DM particles to more tightly

bound orbits, where they experience more interactions

and are heated further due to the negative heat capacity

of the self-gravitating system; a similar phenomenon oc-

curs in globular clusters, [65]. Without energy sinks, the

core suﬀers a runaway instability, transforming the core

into an ever denser cusp, which ultimately results in the

formation of a black hole [54].

An SIDM halo undergoes gravothermal collapse in a

timescale tcoll ≈382tr, as described in Section II F. The

relaxation time depends on the halo mass and time of as-

sembly/formation (described in Sections II B−and II D)

as well as the SIDM cross section at the characteristic

velocity of the halo (Section II E).

B. Primordial density proﬁle

We assume that in the SIDM cosmology DM assembles

into spherical self-gravitating halos in virial equilibrium,

with a primordial structure that is the same as that of

CDM halos. This is a reasonable assumption at suﬃ-

ciently high redshift when the average number of colli-

sions in the center of halos is still well below one per

Hubble time, and thus the structure of the halo has been

aﬀected only minimally. Cosmological simulations have

shown that DM core sizes are only a small fraction of

their value at z= 0 when the Universe is around 1 Gyr

old (z∼5), e.g. [66].

The spherically averaged density proﬁles of equilibrium

collisionless CDM halos are well approximated by a two-

parameter formula known as the Navarro, Frenk & White

(NFW) proﬁle [67, 68]:

ρNFW(r) = ρcrit

δchar

r/r−2(1 + r/r−2)2,(2)

where r−2is the radius at which the logarithmic slope

of the proﬁle is −2, ρcrit is the critical density of the

Universe and the characteristic overdensity δchar is given

by:

δchar =200

k(c),(3)

where k(c) = ln(1 + c)−c/(1 + c) and the concentration

cis deﬁned as c=r200/r−2with r200 being the virial

radius, which is deﬁned in this work as the radius where

the mean density of the halo is 200 times ρcrit.

C. Concentration-Mass relation and formation

redshift

The NFW proﬁle is to ﬁrst order a one free parameter

proﬁle since the virial mass of the halo and its concen-

tration are strongly correlated, with a 1σscatter in log c

of order 0.1 [68, 69]. We use the concentration-mass re-

lation modeled in [70, 71], where the authors link the

enclosed mass proﬁle of a halo at a given time with the

prior mass aggregation history of the halo. In particu-

lar, following [71], we can deﬁne an assembly/formation

redshift of a halo of mass M0at a redshift z0as the red-

shift z−2when the enclosed mass within r−2at z0,M−2,

was ﬁrst assembled into progenitors more massive than

a certain fraction fof M0.M0is deﬁned as the mass

within the virial radius M0= (4π/3)r3

200200ρcrit. The

virial mass of the halo at z−2is equal to M−2and can

be computed from the assembly history:

M−2=M0×erfc δc(z−2)−δc(z0)

p2(σ2(f×M)−σ2(M))!,(4)

The expression in parentheses on the right hand side

corresponds to the collapsed mass fraction in Extended

Press-Schechter theory [72], where δc(z−2) = δc/D(z) is

the redshift dependent critical density for collapse with

the linear growth factor D(z), and σ(M) being the rms

mass variance. For the NFW proﬁle, the mass is con-

nected to the concentration by:

M−2

=k(1)

k(c),(5)

⟨ρ−2⟩

ρcrit(z−2)= 200c3k(1)

k(c).(6)

The key assumption in the model is that the mean density

inside r−2is directly proportional to the critical density

of the Universe at an assembly redshift z−2:

⟨ρ−2⟩

ρcrit(z−2)=CH(z−2)

H(z0)2

,(7)

where Cis a free parameter. Throughout this paper we

use f= 0.02 and C= 575 [73]. Inserting Eqs. 5−6

into Eq. 4, we have a transcendental equation for the

formation redshift zform =z−2as a function of M0, which

can then be used to obtain the concentration c.

D. Threshold time for the cusp-core transformation

As a benchmark case, we set the halo formation time

z−2of an SIDM halo extant at the present day to be the

threshold epoch at which the cusp-core transformation

begins, zcc =zform =z−2. At this epoch, we assume

that the SIDM halo has an NFW proﬁle with a virial

mass equal to the enclosed mass within r−2at z0= 0,

M(z−2) = M−2|z0. The concentration of this primor-

dial SIDM halo is calculated by repeating the method

described in Section II C, but this time setting z0=z−2.

The range of z−2values for the range of present-day halo

masses that we are interested in, 108≤M0≤1012 M⊙,

is given by 6.1≥z−2≥3. As we noticed earlier, given

this relatively high redshift range, our choice of setting

zcc =z−2is reasonable because the eﬀect of collisions in

the inner halo is minimal at early times.

The next step is to develop a method to calculate the rel-

evant timescale for gravothermal collapse (Section II F),

for which we build a simpliﬁed model in which the evolu-

tionary stages of the SIDM halo occur in isolation. This

approach is somewhat diﬀerent to the full cosmological

setting, where halo mergers are an active mechanism of

halo growth with transitory stages that aﬀect the inner

centre of the halo. Although the cuspy NFW proﬁle of

CDM halos is resilient to merger activity [e.g. 74], the sit-

uation might in principle be more complex in an SIDM

scenario with gravothermal collapse for the following rea-

son. In the standard SIDM model without core-collapse,

the merger between a small halo with a larger one is that

of two shallow (core-like) proﬁles with the smaller one

having progressed further in its core development since

it forms earlier; the result of this merger is a DM pro-

ﬁle that is also cored [75]. Thus, we would naively ex-

pect that halo mergers will not delay the cusp-core trans-

formation. However, cosmological mass infall in general

might delay the core-collapse phase by pumping energy

into the central region to stabilize the core [76]. More-

over, in a velocity-dependent SIDM halo with a sharp

diﬀerence between the cross section of low-mass halos to

that of large mass halos, the former are expected to go

through the cusp-core-collapse stages much faster than

the latter, resulting in a scenario in which mergers be-

tween low-mass core-collapsed (cuspy) halos and high-

mass cored halos are possible. This has the potential to

delay the core-collapse phase.

Since our goal is to provide a simple, ﬁrst-order estimate

for the black hole formation time, rather than a com-

prehensive calculation, we assume that in a relatively ex-

treme scenario, a signiﬁcant merger would reset the clock

for the cusp-core-collapse stage. For this event we adopt

the last major merger (LMM), which we deﬁne as a mass

ratio of 10:1 or higher between the two merging halos,

and we label the corresponding redshift as zLMM. In or-

der to calculate zLMM, we use the ﬁtting formula for the

mean merger rate dNm/dξ/dz – in units of mergers per

halo per unit redshift per unit of mass ratio ξ– for a halo

of mass M(z) at redshift zobtained from the combined

Millennium and Millennium II data sets in [77]:

dNm

dξdz (M(z),ξ, z) =

AM(z)

1012M⊙α

ξβexp ξ

ξγ×(1 + z)η,

(8)

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GravothermalcollapseofSelf-InteractingDarkMatterhalosastheOriginofIntermediateMassBlackHolesinMilkyWaysatellitesTamarMeshveliani,1,∗Jes´usZavala,1andMarkR.Lovell11CentreforAstrophysicsandCosmology,ScienceInstitute,UniversityofIceland,Dunhagi5,107Reykjavik,Iceland(Dated:May25,2023)MilkyWay(MW)satelli...

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Gravothermal collapse of Self-Interacting Dark Matter halos as the Origin of Intermediate Mass Black Holes in Milky Way satellites Tamar Meshveliani1Jes us Zavala1and Mark R. Lovell1

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