# Final parsec evolution in the presence of intermediate mass black holes

Fan Zhang  Gravitational wave and Cosmology Laboratory, Department of
Astronomy, Beijing Normal University, Beijing 100875, China  Advanced
Institute of Natural Sciences, Beijing Normal University at Zhuhai 519087,
China

###### Abstract

In this short note, we draw attention to the possibility that, under favorable
conditions, the final parsec problem could be alleviated by the presence of a
moderate population of intermediate mass black holes in the centers of merged
galaxies.

##  I Introduction

Recent observations have suggested that supermassive black hole (SMBH)
binaries should reach merger well within a Hubble time. This runs counter to
those theoretical computations predicting that after dynamical friction loses
effectiveness, but prior to gravitational wave (GW) emission takes over,
beginning at around one parsec separation, the binary hardening would stall
into a torturously slow pace. Such stalling would prevent merger events to
ever have happened given the age of our Universe, contradicting observational
evidences, thus giving rise to the so-called “final parsec problem”.

Specifically, the GW amplitude inferred by pulsar timing array observations
Agazie et al. (2023); Reardon et al. (2023); EPTA Collaboration et al. (2023);
Xu et al. (2023) is achievable with a short binary lifespan Agazie et al.
(2023). There is, furthermore, the discovery Wang et al. (2023) of a SMBH
binary system just about to enter the GW dominated regime, signifying that the
final parsec does not take longer than a Hubble time. Moreover, the scarcity
of direct observational evidences for SMBH binaries with roughly a parsec
separation, despite extensive surveys looking for them, could be explained if
the binaries do not persist for a long time in a stalled state Burke-Spolaor
(2011). The observational evidences for a diminishment of stalling is
mounting, so we would have to look carefully at theoretical predictions for
omissions.

The most important ingredient when it comes to impacting the evolution of SMBH
binaries is their surrounding gravitational environment. The stellar cluster
enclosing a SMBH in either of the pre-merger galaxies contains diverse
populations, as should the merged core hosting the SMBH binary. Historically,
literature has modeled the steady state number density of main sequence stars,
white dwarfs, neutron stars, and stellar mass black holes (sBHs), seeing them
in rough total population ratios of $1:0.1:0.01:0.001$ Strokov et al. (2023),
and spatially distributed according to inverse power laws with indices
$\alpha\sim 1.4-2.0$, with the heavier population more concentrated towards
the center of gravity due to mass segregation Hopman and Alexander (2006).

However, as will be discussed in Sec. III below, there could well be a non-
negligible population of intermediate mass black holes (IMBHs) lurking in the
galactic centers as well, possibly even incidentally conforming to the
geometric sequence for the abundances of the last paragraph by filling in the
next slot at $0.0001$ times the main sequence stellar population (note, this
optimistic scenario is not far off from the upper limit due to dark matter
considerations discussed in Appendix A). These IMBHs have much greater masses,
and therefore could act quite differently to the already accounted-for species
mentioned above. Subsequently, a proper acknowledgement of their presence may
well yield more substantial alterations to the hardening processes of the SMBH
binaries, possibly to the desirable effect of alleviating the final parsec
problem.

Immediately, we notice that IMBHs would concentrate in an even steeper cuspy
number density profile towards the SMBH binary than even sBHs, due to their
much greater masses. For intuition, imagine the extreme limit where the IMBHs
are just a little lighter than the SMBHs themselves, then they should
essentially be at where the SMBHs are, as they all sink via the same dynamical
friction process. To see the consequence of this crowding, note first that the
radius of influence for each of the SMBH is Strokov et al. (2023)

$\displaystyle r_{\rm inf}\simeq 2.4{\rm pc}\sqrt{\frac{M_{\rm SMBH}}{4\times
10^{6}\text{M}_{\odot}}}\,,$ (1)

which is on the order of tens of parsecs for a say, $10^{8}\text{M}_{\odot}$
SMBH, much larger than the stalling-epoch parsec separation between the SMBHs.
If we regard all IMBHs within $\sim r_{\rm inf}$ to the center of the binary
as being eligible for being scattered by it Sesana et al. (2006), then with
the aforementioned power law cuspy number density profile, it is easy to show
that111After regularizing by cutting off the profile at roughly the binary
separation, which is much larger than the cutoff given in Strokov et al.
(2023) for single SMBH. The numbers are more conservative for larger cutoffs.
$90\%$ of all the IMBHs in the merged core will be available for scattering if
$\alpha=2$ as with sBHs. However, $\alpha$ should be larger for heavier holes,
giving us $99\%$ availability when $\alpha=3$, and another $9$ added at the
tail of the decimals whenever $\alpha$ goes up by one. In other words, even an
overall relative dearth of IMBHs in the core of the merged galaxies would not
preclude them from being relatively more abundant near the SMBHs, where as far
as resolving the final parsec problem is concerned, is the vital playground
where the relevant slingshot mechanism (see Sec. II below) operates.

More importantly, the greater mass means whenever an IMBH gets an energy boost
from the binary via slingshot interactions with the SMBHs, it takes away far
greater energy than a regular stellar mass object would. For intuition,
imagine almost headon slingshot interactions as being analogues to scatterings
between a billiard ball and a marble, the billiard ball would be affected much
more severely if the marble is of a greater mass. Because the binaries harden
via slingshotting (traditionally) stars, it stands to argue that IMBHs would
present interesting alternative scattering agents that may well substantially
hasten the final parsec separation evolution of the SMBH binaries. In Sec. II,
we review in slightly more details the binary hardening process, highlighting
where the presence IMBHs could make a difference. Then in Sec. III, we make
one (but not unique) argument for why there could indeed be IMBHs present
where needed. Subsequently, in Sec. IV, we estimate how much of a difference
these IMBHs could make to the final parsec problem, before finally discussing
how to verify or falsify observationally that such a IMBH population actually
do exist in Sec. V.

##  II Final parsec problem

For a quick overview of the final parsec problem, we begin by recalling the
SMBH inspiral stages outlined in Begelman et al. (1980) (see also Sigurdsson
and Rees (1997); Milosavljević and Merritt (2003); Sigurdsson (2003); Makino
and Funato (2004)). Note we will inherit their terminology of referring to the
scattering agents as stars, although we will switch to IMBHs later on.

  1. 1.

When two galaxies merge, their cores merge by dynamical friction, and then the
SMBHs, as well as the IMBHs, originally contained within the nuclear clusters
of the respective galaxies, sink towards the center also by dynamical
friction, with a timescale (defined as $|r/\dot{r}|$, with $r$ being
separation between the SMBHs; this quantity gives the time it takes til merger
if the current $\dot{r}$ is maintained)

$\displaystyle t_{\bf df}\propto\frac{\sigma_{c}r_{c}^{2}QM_{\rm SMBH}}{\ln
N}\,,$ (2)

where $N$ is the number of stars in the merged core, $\sigma_{c}$ is their
dispersion velocity, $r_{c}$ is the core radius, while $QM_{\rm SMBH}$ is the
mass of the smaller SMBH, with $M_{\rm SMBH}$ being the mass of the heavier
hole and $Q$ being the mass ratio. The timescale is set by the smaller hole,
because it sinks more slowly.

  2. 2.

After the SMBHs get close enough to become gravitational bound to each other
(when mass enclosed by the orbit becomes comparable to that of the holes), the
orbit keeps shrinking efficiently by dynamical friction at the same timescale
$t_{\rm df}\propto r^{0}$. This estimate, based on dynamical friction
experienced by an individual (i.e., stays the same if the other hole is
absent) SMBH sinking towards the center of gravity of the merged galaxy, is
valid when the SMBHs are so far separated that their interactions with the
stars can essentially be seen as the stars being slingshot by individual
solitary passing holes. This picture is valid until such a time that a star’s
encounter with the holes lasts longer than the orbital period of the binary.
After this time, three body interactions between the star and both SMBHs
become important. The energy transfer from the binary onto the stars being
slingshot out becomes less efficient, reflected in a diminished effective
impact parameter (scatterings where the holes contribute to stellar energy
pickup in offsetting ways correspond to the ineffective segments of the impact
parameter, and thus excluded) that picks up an additional dependence on $r$.
However, this only affects the dynamical friction timescale $t_{\rm df}$ via
the logarithm term, so it remains roughly constant.

  3. 3.

Further down the line though, as the binary becomes more compact, the SMBHs
essentially become pinned down at a single spatial location (this is not
entirely accurate though, the binary can wander a bit as a whole Makino and
Funato (2004)), thus can no longer simply randomly run into stars on their
long tours across the cluster. Instead, for slingshot transfers of energy out
of the binary to continue to work, the stars will have to fall towards the
binary in such a way that their closest point of approach is sufficiently
close to the center that they can tell the two members of the binary apart, or
else their transits will merely be like passing close to a single gravitating
mass, and their energies will be conserved (see also discussion in the
previous paragraph). This means only low angular momentum stars on more
elongated, almost radial, orbits, i.e., those in the so-called loss (because
stars in it get flung out of the cluster by the binary) cone, are eligible to
participate.

As stars initially in the loss cones pick up energy during one or several
encounters with the binary and end up getting ejected from the core, the loss
cone gets depleted and needs replenishment through the two-body relaxation
process between the stars. Depending on how efficiently this replenishment
happens, there are two possibilities:

    1. (a)

If the loss cone is replenished by two-body relaxation more quickly than it is
cleared out (the “pinhole” regime), then the timescale of this stage, with a
loss cone kept at full, is given by $t_{\rm sl:rep}\propto r^{-1}$, which
increases as the binary hardens, but not too dramatically. This slowdown of
hardening happens because as the binary continues to harden, the loss cone
“solid angle”222One can rescale the actual specific angular momentum by that
at the same energy if circular orbit Cohn and Kulsrud (1978); Magorrian and
Tremaine (1999), thus yielding an angle-like dimensionless quantity that
offers a measure of how close to radial the stellar orbit is (this rescaled
quantity equals unity minus eccentricity squared). Begelman et al. (1980)
declines as there needs to be a more precise aiming at the center of the
binary (velocity of stars becoming more radial), for the star to be able to
get inside a smaller binary region. As a result, there are simply less stars
in the (kept full) loss cone that come in to drain the binary orbital energy
(note also, the average fractional energy drain during each slingshot
encounter is independent of the binary separation Begelman et al. (1980)).

    2. (b)

Alternatively, if the two-body relaxation time

$\displaystyle t_{\rm rx}\sim\frac{N}{10\ln N}\frac{r_{c}}{\sigma_{c}}$ (3)

of the stellar cluster (more precisely that part of the core stellar
population that is gravitationally dominated by the SMBHs, see discussions in
Cohn and Kulsrud (1978) regarding the zero binding energy boundary) is too
large, which tends to be the case for merged cores containing many stars, the
depletion can be quicker than replenishment (the “diffusion” or thermal
relaxation regime). In this case, the total capacity of how many stars can be
hosted in the loss cone becomes irrelevant. Instead, as soon as a star wanders
into the loss cone, it gets slingshot out, so the timescale $t_{\rm sl:dep}$
is set by the rate of such “accidental” events occurring (and the fractional
energy loss per encounter). This rate, or replenishing flux into the loss
cone, is insensitive to $r$ Yu (2002) (as does the fractional energy loss),
depending on it only via a logarithm (see e.g., discussions in Yu and Tremaine
(2003)), so $t_{\rm sl:dep}$ is approximately $\propto r^{0}$.

Specifically, Begelman et al. (1980) gives the estimate

$\displaystyle t_{\rm sl:dep}\simeq\frac{N}{10\ln N}\left(\frac{M_{\rm
SMBH}}{M_{\rm cl}}\right)^{2}t_{\rm df}\,,$ (4)

where $M_{\rm cl}$ is the total mass of the star cluster. The mass ratio
squared is roughly the aforementioned loss cone solid angle, which when
multiplied with the cluster relaxation time according to Eq. (3), gives the
inverse of the loss cone repopulation rate. Assuming that the relevant
relaxation/recharging timescale associated with $t_{\rm df}$ for when the
binary was wide, is the cluster crossing time $r_{c}/\sigma_{c}$, then one
obtains Eq. (4) as the re-adjusted binary hardening timescale, according to
the changes in the character and relaxation mechanisms of the effective
scattering agents (dynamical friction can also be viewed from a slingshot
perspective).

In more details, when $r$ is large, the SMBHs traverse large distances, which
multiply with the scattering cross section into large volumes that contain
many stars. Thus chance encounters with the SMBHs are the dominant source of
stars to be scattered, and the relaxation for the distribution of these stars
occurs on spatial (rather than angular momentum or energy) redistribution
timescales, which is just the cluster crossing time. This is because sans the
SMBH’s influence, the phase space distribution of stars in the neighborhood of
the would-be SMBH returns to the undisturbed state (e.g., the wake behind the
SMBH dissipates, and stars repossess speeds commensurate to $\sigma_{c}$)
after the stars in the cluster have reshuffled over the crossing timescale, so
it is mostly those stars untainted by past encounters with the SMBH that now
occupy the said neighborhood. In contrast, when $r$ is small, only those stars
more deliberately falling towards the binary on very elliptical orbits will be
able to scatter against it, and changes in the distribution of ellipticity is
accomplished by migration in specific angular momentum, which is conserved if
the stars do not collide with each other, so relaxation can only happen on the
two-body relaxation timescale (more accurately the loss cone replenishment
timescale, as a full loss cone is the “undisturbed” distribution of scattering
stars that we are trying to relax into).

Note though, as compared to the original formula in Begelman et al. (1980), we
have an additional slow varying (against cluster parameters) dimensionless
factor of $10\ln N$ that should have been swept along when substituting in Eq.
(3), but seems to have been suppressed in Eqs. 2 and 3 of Begelman et al.
(1980), perhaps due to it being roughly constant. It is however useful to make
it explicit when doing semi-quantitative estimates though. In particular, the
stalling segment in Fig. 1 of Begelman et al. (1980) only goes up to
$\sim\mathcal{O}(10^{11})$yr, while computation with their Eq. 3, using the
assumed parameters for that figure, should have yielded
$\sim\mathcal{O}(10^{13})$yr. The extra $10\ln N$ factor clarifies this
apparent “discrepancy”.

Furthermore, we would like to draw attention to that fact that despite the
nomenclature (which is borrowed from plasma physics), the “loss cone” is
actually a hyperboloid in the velocity space Cohn and Kulsrud (1978), that
reduces to roughly a cylinder when we are at a part of the orbit far away from
the binary. The radius of this cylinder is the maximum transverse speed (or
specific angular momentum) allowed if the star is to come close enough to the
binary to be flung out. Semi-quantitatively, the radius is $\sim\sqrt{M_{\rm
SMBH}/M_{\rm cl}}\sim\sqrt{r_{h}/r_{c}}$ Begelman et al. (1980) ($r_{h}$ is
the separation when the binary becomes hard) times the maximum allowed speed
for bound orbits. Therefore the relative size of the loss cylinder within the
phase space region for all bound orbits is $\sim M_{\rm SMBH}/M_{\rm cl}$,
linear and not quadratic as in Eq. (4). Alternatively, in the more
“cone”-based nomenclature, the loss cone “opening angle” as defined by Cohn
and Kulsrud (1978), as the ratio between the specific angular momentum at the
edge of the loss cone and that of the circular orbit at the same energy, is
$\sim\sqrt{r_{h}/r_{c}}$, so the solid angle is once again
$\sim(r_{h}/r_{c})$, linear and not quadratic. In short, because the stars do
not travel on straight lines, the opening size for the radial-hugging “solid
angle” isn’t quite the geometric angle one would obtain by pointing the stars
directly at the binary expanse. Instead, it varies with radial speed and
scales with the mass ratio $M_{\rm SMBH}/M_{\rm cl}$ differently. As a result,
the square in Eq. (4) may perhaps be superfluous333Depending on whether we had
guessed correctly the rationales leading to this expression, the original
short letter contains not much details. It is with this apprehension, that we
keep the original power as in Begelman et al. (1980). It is nevertheless
straightforward to evaluate the changes that altering it would bring.. Since
$M_{\rm SMBH}$ isn’t usually much smaller than $M_{\rm cl}$ consisting of
stars, this issue doesn’t matter much for previous studies (adjusting it will
make $t_{\rm sl:dep}$ even longer though), but it could be of some importance
if we transition into other types of scattering agents.

For the sake of prudence then, as a comparison, we also evoke a more recent
estimate for $t_{\rm sl:dep}$, as given by Eq. (38) of Yu and Tremaine (2003)
(note this expression is roughly consistent with Fig. 1 of Begelman et al.
(1980) when $m_{*}\sim\mathcal{O}(1)\text{M}_{\odot}$)

$\displaystyle t_{\rm sl:dep}$ $\displaystyle\approx$ $\displaystyle 6\times
10^{9}{\rm yr}\frac{(1+Q)M_{\rm SMBH}}{3.5\times
10^{6}\text{M}_{\odot}}\times$ (5)
$\displaystyle\times\frac{1\text{M}_{\odot}}{m_{*}}\frac{2\times 10^{-4}{\rm
yr}^{-1}}{n_{\rm diff}}\,,$

which is inversely proportional to the stellar mass $m_{*}$, thus demonstrates
more explicitly one of the motivations for us wanting to make use of IMBHs.
Essentially, because the average ejection speed of stars is dependent only on
the properties of the binary (the gravitational and inertial masses are equal,
thus cancel out in the computation of acceleration for test masses – self-
potential of the stellar population is ignored), thus their kinetic energies
are proportional to the mass of the scattering agents, and IMBHs, being much
more massive than regular stars or sBHs, can potentially drain energy much
more rapidly. Eq. (5) is more explicit in showing that $t_{\rm sl:dep}$ is set
by the fractional energy drain per slingshot, and the rate of slingshot
pallets being fed, the latter is given by $n_{\rm diff}$, which is the
diffusion rate of stars into the loss cone (the numerical values given in the
ratio are typical for stars). It can be computed by e.g., solving a time-
independent Fokker-Planck equation for the phase space stellar distribution in
a spherical galaxy Cohn and Kulsrud (1978); Magorrian and Tremaine (1999),
which ends up being roughly (see Cohn and Kulsrud (1978) for a more precise
numerical scaling law) proportional to $M^{-1}_{\rm cl}$ Milosavljević and
Merritt (2003), so the two terms on the bottom line of Eq. (5) once again
yield a $\mathrel{\vbox{ \offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern
2.0pt\cr\sim\cr\kern-2.0pt\cr}}}N$ dependence (see Makino and Funato (2004)
for confirmation by numerical experimentation).

Because for $\mathcal{O}(1)\text{M}_{\odot}$ stellar masses in the core, $N$
is a very large number, e.g., $\sim\mathcal{O}(10^{9})$ as assumed by Begelman
et al. (1980), we end up being encumbered by a very large $t_{\rm sl:dep}$. In
particular, this stage may take a time longer than the age of the Universe,
giving rise to the final parsec problem. Note however, by tightening some of
the simplifications involved, such as bringing in time dependence due to e.g.,
the initial fast depletion of the loss cone; accounting for the secondary
slingshots (see also Zhang (2020) for some relativistic considerations on
repeated slingshots) of returned stars that haven’t left the galaxy altogether
and not perturbed on their long journeys by complications in the broader
galactic gravitational potential; and allowing Brownian motion of the binary
inside the cluster, it is possible to lessen this problem Milosavljević and
Merritt (2003). These mechanisms (and others, see e.g., Makino and Ebisuzaki
(1994); Valtonen (1996); Armitage and Natarajan (2002); Blaes et al. (2002);
Iwasawa et al. (2006); Berczik et al. (2006)) are likely at work, and the
enlistment of IMBHs may provide the additional enforcement needed to help
close the remaining gap for the more massive SMBH binaries.

  4. 4.

Eventually, GW radiation takes over as the dominant channel of energy loss
that quickly decreases binary separation as Peters (1964)

$\displaystyle t_{\rm gw}\propto Q^{-1}M^{-3}_{\rm
SMBH}r^{4}(1-e^{2})^{7/2}\,,$ (6)

where $e$ is the binary eccentricity (note we have included it only to provide
a convenient reference for later consideration, most discussions in this
section assume circular orbits).

Historically, only populations with stellar masses had been considered as
scattering agents for the slingshots, but IMBHs could be present as well, as
we will argue in the next section. A cluster of IMBHs can have a much smaller
population size, for a total cluster mass that is comparable to stellar
counterparts, thus substantially decrease $t_{\rm sl:dep}$ as per Eqs. (4) and
(5).

##  III IMBHs in galactic centers

IMBHs are expected to form in stellar clusters, including but not confined to
the nuclear clusters Fragione et al. (2022a); Rose et al. (2022); Atallah et
al. (2023); Kritos et al. (2023) at galactic centers. From an abstract point
of view, because the heat capacity of a self-gravitating system is negative
(long range forces like gravity violate the additivity property of
thermodynamics Bouchet and Barré (2006); cf., gravothermal catastrophe Lynden-
Bell and Eggleton (1980)), galaxies or stellar clusters will evolve towards
the most concentrated form of matter distribution, that of black holes, once
their internal energy source in the form of fusion within stars begins to get
exhausted and their energy leakage into the surrounding environment isn’t shut
off (and dark energy doesn’t grow so quickly that it tears everything apart
before collapses had a chance to run their courses). While galaxies form
SMBHs, their miniature cousins, stellar clusters, could form IMBHs on shorter
timescales, and these IMBHs should also participate in the grander scheme of
the collapse of their entire host galaxies, by merging with the central SMBHs,
sooner or eventually. Therefore, while the nuclear clusters are the densest,
thus ideal incubators, those IMBHs born elsewhere could still make their way
to the centers of galaxies Miller (2005); Gnedin et al. (2014); Fragione
(2022). Conveniently, we may well need IMBHs to help explain the presence and
properties of massive stars near the SMBH in our own Galaxy Hansen and
Milosavljević (2003); Gürkan and Rasio (2005); Generozov and Madigan (2020);
Zheng et al. (2020); Naoz et al. (2020) (see also GRAVITY Collaboration et al.
(2020) for observational constraint), or the hypervelocity stars Yu and
Tremaine (2003); Baumgardt et al. (2006); Löckmann and Baumgardt (2008);
Rasskazov et al. (2019). Observationally, there are also hints that our own
Galactic center may indeed host IMBHs Maillard et al. (2004); Oka et al.
(2017).

It is therefore not outrageous to expect there to be a moderate (subdominant
in total mass to stars, see Appendix A) population of IMBHs at the very center
of galaxies, close to where SMBHs reside. In fact, the merger of a cluster of
IMBHs themselves may have built the SMBHs in the first place (see e.g.,
Ebisuzaki et al. (2001); Matsubayashi et al. (2004); Oka et al. (2017); Wirth
and Bekki (2020)). Indeed, creating an additional channel for the growth of
SMBHs apart from galactic mergers (removing the final parsec hurdle also makes
viable the SMBH mergers as a growth path Hopkins et al. (2007)) could allow
SMBHs to grow faster than their host galaxies, explaining the observations by
e.g., Yang et al. (2018). For IMBH accretion to serve as a substantial growth
engine for SMBHs, there needs to be a steady stream of IMBH supplied to the
galactic centers and be captured by the SMBHs (see e.g., Milosavljević and
Merritt (2003); Miller (2005) for discussions on the lack of fatal stalling
for IMBH-SMBH mergers), and possibly more would remain unmerged with the
SMBHs. There would thus be ($M_{\rm S/IMBH}$ are the S/IMBH masses)

$\displaystyle N_{\rm IMBH}=\frac{M_{\rm SMBH}}{M_{\rm
IMBH}}\frac{1-\xi}{\xi}$ (7)

IMBHs close to an SMBH. Note there should also be an overall suppression
factor as SMBH mergers also contribute to the growth of $M_{\rm SMBH}$.
Because we are not sure how galactic mergers have proceeded through cosmic
time, especially in light of recent JWST observations of early massive
galaxies (see e.g., Giménez-Arteaga et al. (2023)), we leave out this factor,
noting only that it could be absorbed by $\xi$.

The parameter $\xi\leq 1$ is the cumulative capture rate obtained by
integrating a likely complicated and evolving survival probability, until a
time that we are interested in, e.g., when two SMBHs are
$\sim\mathcal{O}(1)$pc apart for the discussions in Sec. II. As an example of
why survival probability would change over time, note that when the SMBH has
grown to larger masses, the SMBH-IMBH bound systems would begin to suffer from
the same final parsec issue as per Eq. 5, so survival probability of IMBHs
could increase over time. Counterbalancing this effect though, is the build up
of IMBH population near the SMBH over time, which induces eccentricity for
each other through mutual scattering, that hasten their fate of being
dynamically captured by the SMBH.

The total mass of the cluster of IMBHs is then $\sim M_{\rm SMBH}(1-\xi)/\xi$,
so would be the same order of magnitude as the SMBH mass if SMBH manages to
capture around half of the IMBHs. Fig. 4 of Rose et al. (2022) suggests that
indeed, around two thirds of the IMBHs formed in situ in the nuclear stellar
cluster, from collisions between sBHs and main sequence stars (see Appendix B
for a partial summary of other IMBH formation channels), fall into the SMBH
after evolving for around $10$Gyr, roughly the age of galaxies. Note also that
the sBH seeds and the stars that they consume may be continuously replenished
over time in the real world, so there could also be IMBHs that have not run
for the full $10$Gyr. Because formation happens prior to subsequent merger
with the SMBH, a greater proportion of them would have formed but have not yet
fallen into the SMBH, than those that have been simulated for the full
$10$Gyr, so the actual capture rate would be somewhat lower than $2/3$.

Moreover, of the $500$ seed sBHs (initially situated between $0.001$pc and
$0.1$pc from the SMBH) simulated by Rose et al. (2022), around a third end up
becoming IMBHs after evolving for a full $10$Gyr (inclusive of those that have
merged with the SMBH). The present day ratio of IMBHs and sBHs would be
dependent on the uncertain sBH and stellar mass replenish rates though, as
some of the late-addition sBHs would not have had enough time to grow.
Nevertheless, the middle panel of Fig. 4 in Rose et al. (2022) shows that
IMBHs have already been seen falling into the SMBH as early as $\sim 1$Gyr,
and these merged population grows quickly after the Gyr mark, which is
consistent with the estimate by Arca-Sedda and Gualandris (2018) that it takes
several Gyr for a SMBH to swallow an IMBH. In other words, many of the IMBHs
likely would have formed within a Gyr, and then hang around the SMBH for
several Gyr afterwards, before getting consumed. If this is so, most of the
sBHs, apart from those introduced to the galactic center within the last Gyr,
would have been effective as IMBH growth seeds, so long as they are also
spatially located sufficiently close to the SMBH. Indeed, because the sBHs are
distributed according to a steep cuspy power law with index $\alpha=2$ Hopman
and Alexander (2006), nearly all of the $\sim\mathcal{O}(10^{4})$ sBHs in the
central parsec of e.g., our own Milky Way Morris (1993); Miralda-Escudé and
Gould (2000); Freitag et al. (2006); Hailey et al. (2018) would be eligible
seeds.

More quantitatively, let $t$ be look back time, $\eta(t)$ be the rate at which
sBHs are introduced into the galactic center, $\varphi(t)$ be the percentage
of IMBHs that have grown from seeds introduced at time $t$ that have survived
today, the the ratio $\mathcal{R}$ of IMBHs and sBHs around the SMBH today is
given by (ignoring the depletion of sBHs including those that became IMBHs or
are swallowed by the SMBH, thus this is conservative in the direction that
disfavor IMBHs)

$\displaystyle\mathcal{R}\sim\frac{\int_{0}^{\sim 10-14{\rm
Gyr}}\varphi(t)\eta(t)dt}{\int_{0}^{\sim 10-14{\rm Gyr}}\eta(t)dt}\,.$ (8)

The sBH birth rate should not be heavily back-loaded because the lifespan of
sBH forming massive progenitor stars are measured in mere millions of years.
For simplicity, we can take $\eta$ to be a constant, and then it drops out
from $\mathcal{R}$. The formation and subsequent survival rate $\varphi(t)$ is
zero for $t\lesssim 1$Gyr, and is anchored at around $1/9$ at $\sim 10$Gyr by
Rose et al. (2022). It likely has a peak at sometime before $10$Gyr and after
$1$Gyr when IMBHs have had enough time to grow but not enough time to be
captured by the SMBH yet. Conservatively, if we ignore the peak and then
simply let $\varphi(t)$ be a linearly growing function between $0$ @$1$Gyr to
$1/9$@$10$Gyr, and be $0$ at $<1$Gyr and $>10$Gyr, then $\mathcal{R}\approx
0.036(0.05)$ for integration out to $14(10)$Gyr. If we acknowledge the
potential existence of the peak above $1/9$, but chop it off and use the area
underneath it but above $1/9$ to fill in the regions of $\varphi(t)$ that fall
short of $1/9$ in other parts of the $1$Gyr to $10$Gyr interval, so that
$\varphi(t)$ becomes a constant there, then $\mathcal{R}\approx 0.07(0.1)$. In
short, one has some grounds in arguing for an optimistic ratio of up to
$\mathcal{R}\sim\mathcal{O}(10^{-1})$. So there could be as many as
$\sim\mathcal{O}(10^{3})$ IMBHs present in our Galactic center that are formed
in the nuclear stellar cluster444On the other hand, the IMBH deposition by
globular clusters falling from outside into the Galactic center is very
roughly estimated to occur at a rate of $0.1-1{\rm Gyr}^{-1}$ for the Milky
Way Miller (2005); Arca-Sedda and Gualandris (2018); Strokov et al. (2023), so
this channel is likely unimportant in comparison, unless the IMBHs they bring
in are much larger in mass.. A comparable number of them being absorbed by the
SMBH can indeed suffice for the mass budget of the $M_{\rm SMBH}\sim 4\times
10^{6}\text{M}_{\odot}$ Sgr A*. For other galaxies with larger SMBHs, the
number and/or the individual masses of the IMBHs could be greater.

##  IV Adding IMBHs to the mix

Now consider the scenario where effective scattering agents for the final
parsec stage of the SMBH binary orbital evolution are IMBHs rather than stars.
We now would have $N$ in Eq. (4) being replaced by $N_{\rm IMBH}$ according to
Eq. (7). Assume that $M_{\rm IMBH}\sim\mathcal{O}(10^{5})\text{M}_{\odot}$ as
per the candidate IMBH observed in the Galactic center by Oka et al. (2017)
(heavier IMBHs are also more likely to quickly sink into the vicinities of
SMBHs Wirth and Bekki (2020), but we explore all possibilities in Fig. 1), and
capture efficiency of $\xi\sim 1/2$ as per discussions from Sec. III, then for
the archetypal final parsec problem demonstration, the example shown in Fig. 1
of Begelman et al. (1980) with $M_{\rm
SMBH}\sim\mathcal{O}(10^{8})\text{M}_{\odot}$, this results in $t_{\rm
sl:dep}\sim\mathcal{O}(10)t_{\rm df}$ according to Eq. (4).

Strictly speaking, $t_{\rm df}$ that appears as a formal segment of the
$t_{\rm sl:dep}$ expression in Eq. (4) should now be for the cluster of IMBHs,
rather than stars. It only weakly depends on $N$, via a logarithm (see Eq. 2),
and dispersion velocity should be the same as the stars (both on the order of
the test mass orbital speed as determined by the cluster’s gravitational
potential) if virialization hasn’t happened and smaller if it has progressed
to some extend, while the cluster radius should be smaller due to the greater
concentration of IMBHs towards the center. These changes to the parameters
would further reduce $t_{\rm df}$ and subsequently $t_{\rm sl:dep}$. In fact,
the stalling may even be further reduced in cases where the IMBH masses are
towards the larger end, as the Kozai-Libov effect Kozai (1962); Lidov (1962)
they excite could increase the eccentricity of the SMBH binary (see in
particular Fig. 4 of Naoz (2016) for the effects of a small mass outer
perturber), which in turn causes enhanced bursts of GW during periastron
passages Loutrel (2020) (cf., Eq. 6; note this energy and angular momentum
loss means we won’t see the same indefinite oscillations as with the regular
Kozai-Libov evolution, and the orbit tends to circularize at smaller
separation), in effect expedites the takeover of the $t_{\rm gw}$ stage and
thus beckons a more efficient merger. This is somewhat similar to how sBH-SMBH
mergers happen. The sBHs are too light for dynamical friction to work
effectively, so they get dynamically captured instead, by losing large amounts
of energy when passing close to the SMBH on eccentric orbits, brought into
elongation when they scatter against other objects. Besides possessing more
perturbing power, the more massive IMBHs are fewer in numbers, so there may be
less cancellations among their contributions to this effect, although the
details of this statistics still needs to be worked out, possibly via
numerical simulations.

The impacts of these effects are more difficult to ascertain, so for our
demonstration of principle, we take a conservative route and assume $t_{\rm
df}$ does not change when we swap from stellar to IMBH scattering agents. Then
the final parsec would take only $\sim\mathcal{O}(10^{7})$yr. For
thoroughness, we can also adopt the alternative expression (5) and compare,
where now $M_{\rm SMBH}=10^{8}\text{M}_{\odot}$,
$m_{*}=10^{5}\text{M}_{\odot}$, while $n_{\rm diff}\propto M_{\rm cl}^{-1}$
can be taken to be $10^{-1}$ of the stellar value (ignoring once again the
impact of $\sigma_{c}$ to $n_{\rm diff}$; lower $\sigma_{c}$ due to
virialization gives higher $n_{\rm diff}$ Cohn and Kulsrud (1978) and thus
smaller $t_{\rm sl:dep}$), because $N_{\rm IMBH}\sim\mathcal{O}(10^{3})$
according to Eq. (7) for our $\xi=1/2$ choice, which is roughly $10\%$ that of
sBHs assuming Milky Way is typical in this respect, thus $10^{-4}$ of main
sequence stars (see population ratios in Sec. I). Eq. (5) subsequently yields
$t_{\rm sl:dep}\sim\mathcal{O}(10^{7})$yr, broadly consistent with that
provided by Eq. (4) for this particular choice of $\xi$ and $M_{\rm IMBH}$
(the dependence on $\xi$ differs for Eq. 4 and Eq. 5 though, see Fig. 1).

\begin{overpic}[width=214.64308pt]{deptime.pdf} \end{overpic} Figure 1:  The
dependence of $t_{\rm sl:dep}$ on $\xi$ for a range of $M_{\rm IMBH}$ values.
Solid curves are obtained using Eq. (4), while dashed curves are from Eq. (5).
The grey horizontal dashed line is the age of the Universe.

In comparison, the whole process taking the SMBH binary separation down to the
final parsec would take $\sim\mathcal{O}(10^{6})$yr (note the earlier stages
of dynamical friction are likely still dominated by stars or sBHs, since IMBHs
congregate more towards the center, so $t_{\rm df}$ in this estimate would
still be for that of the stellar population as scattering agents), while the
prior merger and violent relaxation of the cores takes
$\sim\mathcal{O}(10^{8})$yr Begelman et al. (1980). In other words, there is
barely any stalling during the final parsec. We further plot the dependence of
$t_{\rm sl:dep}$ on $\xi$ and $M_{\rm IMBH}$ in Fig. 1, and see that even for
the smallest IMBHs that can easily form via mergers of sBHs, there are $\xi$
ranges that allow SMBH mergers to have occurred.

We must caution though, the statistical nature of much of the underlying
derivations quoted in this note would render the quantitative expressions
suspect for such small IMBH population numbers. In particular, the dependence
of $t_{\rm sl:dep}$ on $N$ could be weaker for smaller values of this
parameter (see Makino and Funato (2004) and references within). The IMBH
population size adopted here could also be overly optimistic (e.g., we have
assumed that IMBH accretion is not subdominant to SMBH mergers, when it comes
to contributing to the growth of SMBH masses). Nevertheless, the qualitative
heuristics should stand, and there is a three orders of magnitude headroom for
further adjustments if merger within a Hubble time is all that is desired.

##  V Discussion

The key uncertainty in the proposed resolution of the final parsec problem is
in the uncertainty of the IMBH population size. Observationally, the IMBHs
settle in a crevice that helps them remain hidden. If they are mostly formed
inside nuclear stellar clusters of massive galaxies, then they would be
heavily obscured from outside view by intervening stars, dust, gas and etc. As
a result, electromagnetic channel observations Greene et al. (2020) would in
general be rather challenging.

As such, GW may offer an enticing addition, as these waves interact extremely
weakly with matter and can thus avoid being blocked. They are also sourced
directly by the holes as opposed to indirectly by environmental elements
influenced by those holes, and thus could offer up more unambiguous evidences.
With the GW approach, there is a tradeoff between signal strength, thus
detection horizon, against population size per galaxy. This is because given a
fixed sensitivity range in GW frequency, a detector will be able to see the
earlier inspiral stages of binaries involving lighter holes, but the plunge
and merger stages of those involving heavier holes (the frequency at merger
scales inverse proportionally with the mass of the heavier hole, see e.g.,
Miller (2005)). Heavier holes are rarer, and merger phases of binaries last
much shorter than the inspiral phases, so per galaxy event rates are lower;
but offsetting this, the mergers of binaries involving heavier holes send out
stronger waves (proportional to the mass of the lighter hole in the binary
Miller (2005)), so we can detect them in further off galaxies. One should
consider both source types.

With the inspiral-focused option, ingenious schemes (e.g., Strokov et al.
(2022, 2023); Fragione et al. (2022b); Fragione and Loeb (2023); Ellis et al.
(2023)) have been devised to utilize the various bound configurations of
SMBHs, IMBHs, and sBHs that fit into the sensitive bands of LISA Amaro-Seoane
et al. (2017) (see also Taiji Luo et al. (2020) that share similar
characteristics). Assume the most optimistic scenario that the IMBH population
is as high as one tenth of that of the sBHs, there should thus be very roughly
one tenth (one hundredth) the number of IMBH-sBH (double IMBH) binaries as
there are double sBH binaries (adopting dynamical capture during chance
encounters as the leading formation mechanism and ignoring cross section and
spatial distribution differences). During the operation time of LISA, there is
expected to be up to $\sim\mathcal{O}(10^{6})$ double sBH systems being
detected in its sensitivity band Sesana (2016), and after adapting for
duration of stay in band according to Eq. (6), also adjusting for detection
horizon, one sees that the probability of LISA detecting binaries involving
IMBHs, in their inspiral stage, should not be discounted.

Alternatively, we can opt for the merger-focused option. The most sensitive
band of LISA lands at around $3\times 10^{-3}$ to $10^{-2}$Hz, so $M_{\rm
SMBH}\sim\mathcal{O}(10^{6})$ and slightly lighter are the most favorable. If
we restrict attention to the merger of IMBHs into SMBHs, whose final stages
also fall within the LISA band, then one notes that around
$\mathcal{O}(10^{3})$ merger events of
$\sim\mathcal{O}(10^{3})\text{M}_{\odot}$ IMBHs would form a
$\sim\mathcal{O}(10^{6})\text{M}_{\odot}$ SMBH, and happening within the $\sim
10$Gyr lifetime of a galaxy would yield, on average, $\mathcal{O}(10^{-7})$
merger events per year per galaxy, or $\mathcal{O}(10^{-6})$ events per galaxy
throughout the longer end estimate of LISA’s operational time of $10$ years.
The estimates by Yu and Lu (2004); Miller (2005) is that there are
$\mathcal{O}(10^{8})$ SMBHs in this lower mass range within the LISA detection
horizon, so the lower limit (mergers involving higher mass SMBH may also be
observable if close enough) to the observable IMBH-SMBH merger events is
$\mathcal{O}(10^{2})$.

Another interesting class of potential merger sources involving IMBHs are the
sBH-IMBH and double IMBH mergers (as opposed to inspirals that we have already
discussed). To see such events though, we would need sensitivities in the
frequency range of $\sim 10^{-2}-10^{1}$Hz. New instruments are required, and
some have been proposed that cover various parts of this range, such as
Tianqin Luo et al. (2016), DECIGO Kawamura et al. (2008), or proposals that
utilize the Moon as a resonant-mass detector Harms et al. (2021); Li et al.
(2023).

All considered, our knowledge of IMBH population size should be revolutionized
in the near future, and if turns out to be too small, the final parsec problem
can not possibly be resolved with the proposal laid out in this note. On the
other hand, small impact parameter slingshots of IMBHs around a SMBH binary
would also emit bursts of GW that are wide band and potentially detectable.
Seeing such events at a sufficient copiousness could well provide direct
support.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
grants 12073005 and 12021003.

##  Appendix A IMBHs as dark matter constituent

Different types of galaxies exhibit markedly different rotation curves (see
e.g., references in Del Popolo and Le Delliou (2022); Freundlich et al.
(2020)), especially on the interior, suggesting that there may be more than
one form of dark matter. Specifically, besides the physically more exotic
(nonbaryonic in the nomenclature of e.g., Carr (1994)) types that must extend
out deep beyond the stellar concentrations, there may also be more mundane
astrophysical contributions (baryonic Carr (1994)) concentrated near the
galactic centers, where baryonic matters tend to congregate, that we simply
haven’t been able to see. In particular, because we are seeing extended cores
(manifesting as a segment of linear growth) in the rotation curves of many
dark matter dominated dwarves, as well as low surface brightness and irregular
galaxies Moore (1994); Flores and Primack (1994); Oh et al. (2011), for whom
the astrophysical sources would more likely to be deficient, it stands to
argue that the more exotic form of dark matter may perhaps assume this
extended core (corresponding to constant exotic dark matter density)
morphology as its universal form, while the higher density dark matter cusps
needed (in fact, as we will discuss below, there isn’t yet a clear necessity
at the current time) to fit the cuspy (more steeply rising near center)
rotation curves of some of the more baryon-abundant luminous galaxies, contain
additional contributions of astrophysical origins, that concentrate at the
center.

This pragmatic _a posteriori_ approach (see Zhang (2019)555In this model, that
half of the spacetime curvature tensor that does not need to be sourced by
matter, and is the carrier of gravitational entropy as per Penrose (1979),
grows from zero at the big bang to larger values later on. Its positive
contribution to the overall entropy budget allows baryonic matter to come out
of thermal equilibrium (which entails a decreasing contribution to entropy
thus needs offsetting). As such, it is the hidden driving force behind
structure formation, while baryons play a more passive role, much like dusts
being swept along into the center of vortices of an unseen underlying ocean.
for an _a priori_ proposal for such an extended core profile; we do not need
any other specific characteristics of that model though) contrasts with the
main thrust of research into the traditional cold dark matter, which has the
opposite problem in the galactic centers of having to wipe out the cusps that
would naturally arise with cold dark matter piling up, but not seen in dwarf
galaxies, by e.g., conjuring up extreme baryonic outflow events that feedback
on the dark matter, towing it away from the center Del Popolo and Le Delliou
(2022). Theirs is in general a much harder problem, as the heat capacity of a
self-gravitating system is negative, so one would need to extract enormous
energy from baryonic interactions and feed it to the supposedly much heavier
(in the cold dark matter scenario) dark matter sector, to prevent its collapse
into steep cusps. And somehow, this energy supply must also persist, to
prevent re-collapse at later times Laporte and Penarrubia (2015). In any case,
looking for even more hidden masses at galactic centers beyond the SMHBs has
not really been a priority.

The astrophysical dark matter species must be difficult to observe, so they
must not emit light. They must also be compact, so they do not block light
from behind. As such, black holes constitute the ideal candidates. The IMBHs
are the least understood, thus least constrained, so is ideal for enlistment
when there is a hidden mass budget needing to be filled. Incidentally, the
IMBH mass range largely overlaps with that allowed for black hole type dark
matter as assessed by Carr (1994), although that review does not concentrate
on galactic centers. In particular, for the aforementioned cusp production, we
do not need IMBHs in the extended dark matter halo, so the inadequacy of
MACHOs (Massive Compact Halo Objects) as dominant dark matter components, as
appraised through, e.g., microlensing techniques Gates et al. (1995), is not
pertinent. Besides evading existing constraints, because we are trying to
create an astrophysical matter cusp, i.e., increasing density towards the
center, it would be natural to look to more massive compact objects that has
this tendency to sink to the center through mass segregation via dynamical
friction. IMBHs are natural candidates in this regard.

It would be desirable then, to be able to estimate the typical IMBH population
size that would be needed to make up for the shortfall when we move to a cored
exotic dark matter profile. However, as it turns out, the detail level and
accuracy of observational data is insufficient, and the only conclusion we can
draw is that the IMBH contribution does not need to dominate the bulge mass.
Specifically, it was noted in Zhang (2019) that a concentrated central
astrophysical mass (monopole like; incl. potential IMBH contributions) at
$\sim 4\times 10^{10}\text{M}_{\odot}$ is sufficient for fitting to the inner
segments of the M31’s (closest spiral galaxy with an unobscured view to the
center) rotation curve, after we migrate to an extended core exotic dark
matter profile. On the other hand, the observational bulge stellar (i.e.,
without IMBH) mass estimates have been reported in the literature Geehan et
al. (2006); Seigar et al. (2008); Chemin et al. (2009); Corbelli et al.
(2010); Tamm et al. (2012) to range from $2.32\times 10^{10}\text{M}_{\odot}$
Chemin et al. (2009) to as high as $6.6\times 10^{10}\text{M}_{\odot}$ Tamm et
al. (2012). As a result, given our current knowledge, even stellar matter
alone could possibly already account for the cusp (see also Bahcall and Wolf
(1976) for a theoretical prediction on their cuspy number density profile),
without needing to invoke the sBH or SMBH black hole mass cusp in the central
parsec, let alone additional contributions from IMBHs. Nevertheless, by
adopting the lower limit of the stellar mass estimates, we can obtain an upper
limit to the total mass of IMBHs in the bulge of M31 at $\lesssim 1.7\times
10^{10}\text{M}_{\odot}$.

##  Appendix B Other IMBH formation channels

There could be many different formation channels (see e.g., Miller and Colbert
(2004); Askar et al. (2023)), all of which should result in a large number of
IMBHs if viable, some are:

  * •

The population III Madau and Rees (2001) and II Spera and Mapelli (2017)
stars, containing little metal, can in theory collapse to form IMBHs at the
lower end of their mass spectrum, at hundreds of solar masses. Because
metallicity of early galaxies is indeed low, as per JWST observations Heintz
et al. (2023), there would have been a large population of the progenitor
stars, giving birth to a large population of small-end IMBHs. They would all
sink to the centers of their host galaxies, where shear crowding enables
mergers, thus the production of heavier IMBHs. Accretion in the gas-rich
environment of the galactic centers could also spur growth of the IMBHs, where
at Eddington limit, it takes only tens of mega years for an e-folding increase
in mass Madau et al. (2014).

In fact, given our current best understanding of stellar physics, there needs
to be an account of where the compact remnants of the vast number of early
massive stars went, and their congregation at galactic centers (they are the
first generation of stars that lived very short lives, so there would be
plenty of time for them to sink into the galactic centers via mass
segregation), partially or wholly merging into, thus seeding the SMBHs, offers
up one possibility.

  * •

The giant molecular clouds that precedes stellar clusters have total masses
matching that of IMBHs McKee and Ostriker (2007), so if there are efficient
cooling processes that act in a relay fashion to collapse a significant
portion of that matter into a single entity, we would end up with IMBHs
reaching even the upper end of $\mathcal{O}(10^{5})\text{M}_{\odot}$.
Typically, as some stars get flung out of the cluster cores into the halos,
taking away energy, the core collapses, until hard stellar binaries form that
can supply additional energy to the cluster by scarifying (hardening)
themselves. The Heggie-Hills law however suggests that generally, hard
binaries tends to merge, while soft binaries gets disrupted, allowing core
collapse to continue, e.g., through runaway collisions between stars, or
repeated mergers of sBHs (a major hurdle to this scenario is if the merger
imparts large kicks to the resulting BH, ejecting it out of the cluster; see
Zhang (2020) however for some complications regarding estimating kick speeds),
or copious transfer of matter into sBHs, all eventually leading to the
plausible formation of IMBHs, as the most compact thus thermodynamically
preferable terminal state.

So far the simulations (see references in Askar et al. (2023)) have largely
been successful in cooking up IMBHs in clusters, but the percentage of success
and the mass ranges are subject to physical conditions in the clusters that
are difficult to ascertain, especially for those clusters residing in the
earlier universe. We do know however, that there are thousands of stellar
clusters in the Milky Way Kharchenko et al. (2013), and still more must have
existed during its long evolution history which would have since dispersed by
intrinsic relaxation or extrinsic galactic tidal field stripping Chernoff and
Weinberg (1990); Gnedin and Ostriker (1997). Although the relevant physics
remain uncertain, it is not impossible that a non-negligible subset of them
can manufacture IMBHs (see e.g., Bahcall and Ostriker (1975) for observational
support of IMBHs inside stellar clusters). The clusters can subsequently sink
into the nucleus of the host galaxies Ebisuzaki et al. (2001); Portegies Zwart
et al. (2006); Arca-Sedda and Capuzzo-Dolcetta (2017) by dynamical friction,
delivering their IMBHs to the galactic core.

  * •

The masses of the massive black holes in a galaxy correlates with the host’s
mass and velocity dispersion, so those massive black holes in dwarf galaxies
may actually be IMBHs when classified according to mass (as opposed to
formation history, e.g., via direct collapse of low metallicity gas clouds
Bromm and Loeb (2003)). As large galaxies have supposedly swallowed many dwarf
galaxies as they grow, these IMBHs should sink to the center of the merged
large galaxy and find their way towards the SMBH there via mass segregation
Bellovary et al. (2019); Volonteri et al. (2020). Because dwarf galaxies are
numerous and close to the larger galaxies, such mergers are frequent, and so
there could be a number of such IMBHs end up surrounding the SMBH.

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