capbtabboxtable[][]

# Comparing Accretion Disks and Dark Matter Spikes in Intermediate Mass Ratio
Inspirals

Niklas Becker<EMAIL_ADDRESS>Institute for Theoretical Physics,
Goethe University, 60438 Frankfurt am Main, Germany  Laura Sagunski
<EMAIL_ADDRESS>Institute for Theoretical Physics, Goethe
University, 60438 Frankfurt am Main, Germany

###### Abstract

Intermediate Mass Ratio Inspirals (IMRIs) will be observable with space-based
gravitational wave detectors such as the Laser Interferometer Space Antenna
(LISA). To this end, the environmental effects in such systems have to be
modeled and understood. These effects can include (baryonic) accretion disks
and dark matter (DM) overdensities, so called spikes. For the first time, we
model an IMRI system with both an accretion disk and a DM spike present and
compare their effects on the inspiral and the emitted gravitational wave
signal. We study the eccentricity evolution, employ the braking index and
derive the dephasing index, which turn out to be complementary observational
signatures. They allow us to disentangle the accretion disk and DM spike
effects in the IMRI system and can be utilized to study environmental effects
in general.

##  I Introduction

The first direct detection of gravitational waves (GWs) has opened a
fundamentally new window into the Universe. The Laser Interferometer
Gravitational-Wave Observatory (LIGO) has seen the first binary black hole
merger, and, together with the LIGO-Virgo-KAGRA (LVK) collaboration, has
already collected a sizable catalogue of compact binary mergers [1, 2]. They
allow new and unprecedented tests of General Relativity and matter at high
densities [3, 4]. On top of that, there are several space-based observatories
planned such as LISA [5], Taiji [6] and TianQuin [7], which will allow for the
detection of GWs at lower frequencies. For these observatories, accurate
waveforms have to be computed to maximize the science yield [8].

While LVK mostly observes solar mass binary mergers, space-based observatories
will be able to detect Extreme/Intermediate mass ratio inspirals (E/IMRIs). In
these systems, a stellar mass object inspirals into a
supermassive/intermediate mass black hole (S/IMBH). Several IMBH candidates
have been detected, but their origin, evolution, and environment is not yet
well understood [9, 10]. To observe these IMRI systems, the environmental
effects have to be understood first, as accurate waveforms are needed for
detection [8].

Meanwhile, dark matter (DM) as predicted by $\Lambda$CDM has continued to
elude detection [11, 12]. While its effects are observed on large scales, such
as structure formation, on small scales, the effects of dark matter are more
uncertain and a plethora of models has been proposed [13]. Around IMBHs, on
small scales, a dark matter halo could grow adiabatically into a dark matter
spike [14, 15]. These spikes have an extremely high local density compared to
the ambient DM density and would gravitationally interact with any object
passing through. During an IMRI, the dark matter spike leaves its imprint by
modifying the orbital evolution. This is one of several possible environmental
effects.

This has first been explored in [16, 17], where the authors predicted a
dephasing of the GW signal due to dynamical friction with the DM spike [18].
If the secondary object is a black hole, it will accrete the DM as it passes
through the spike, which was first explored in [19] and later in [20], where
the accretion effects were found to be subdominant to dynamical friction
effects. Then, [21, 22] looked at eccentric orbits, and found there to be an
eccentrification of the orbits. Afterwards, we have argued in [23] that by
including the phase space distribution of dark matter particles, the system
circularizes. Other spike effects have been studied, such as periastron
precession[24, 25], halo feedback mechanism [26, 27], relativistic corrections
to dynamical friction and spike distribution [28], and spikes around lower
mass primordial black holes [29].

Another source of important environmental effects is the presence of
(baryonic) accretion disks [30, 31]. While the existence of DM spikes around
IMBH is still speculative, the existence of accretion disks around SMBHs is
supported observationally [32, 33], and a strong argument can be made for
their existence around IMBHs as well. The effects of the interaction between
the secondary object and the accretion disk in an IMRI can also affect the
inspiral. While interesting from a physical standpoint, from the perspective
of trying to detect dark matter, these baryonic effects could mimick or
dominate dark matter effects, spoiling its detection. There have been studies
trying to map out disk effects in IMRIs[34, 35, 36] and in this paper, for the
first time, we want to compare the environmental effects of accretion disks
and DM spikes. This allows us to estimate their relative strength and
observational signatures.

The motivation of this paper is to model IMRIs on eccentric Keplerian orbits
with GW emission, dynamical friction with the DM spike, and gas interaction
with the accretion disk. We do not include all relevant effects, such as halo
feedback and relativistic corrections here, but focus on comparing baryonic
and dark matter effects first. More expansive studies are left for future
work.

The structure of the paper is as follows. In section II, we explain the
theoretical framework to model the orbital evolution of the IMRI and its GW
signal. In section III, we present numerical results, and discuss them in
section IV. Finally, conclusions are drawn in section V.

Throughout the paper we adopt geometrized units with $G=c=1$.

##  II IMRI Modeling

The IMRI system consists of a central IMBH $m_{1}$ and a secondary object
$m_{2}$, both of which are assumed to be Schwarzschild black holes. See Fig. 1
for a sketch. The IMBH is surrounded by both an axially symmetric accretion
disk, and a spherically symmetric DM spike. The secondary is assumed to be on
a Keplerian orbit around the central mass. The system emits GWs and is subject
to environmental effects, such as those given by the interactions with the DM
spike and accretion disk. Through these dissipative forces, the secondary
loses orbital energy and angular momentum, leading to an inspiral. In this
section we present the theoretical background and observational signatures.

Figure 1: A sketch of the IMRI system with masses $m_{1}\gg m_{2}$, semimajor
axis $a$, and eccentricity $e$ inside the DM spike $\rho_{\text{dm}}$ and
accretion disk $\rho_{\text{b}}$. Here, the true anomaly $\phi=\pi-\varphi$,
such that $\phi=0$ is the pericenter, and $\phi=\pi$ the apocenter.

###  II.1 Dark Matter Spike

To model the DM spike, we follow our previous publication [23]. The
development and existence of DM spikes has been discussed extensively in the
literature [14, 37, 15, 27].

We consider a system in which the IMBH $m_{1}$ is assumed to be surrounded by
a static, spherically symmetric DM spike. We describe the spike density around
the central mass by a simple power law [27]

$\rho_{\text{dm}}(r)=\rho_{6}\left(\frac{r_{6}}{r}\right)^{\alpha_{\text{spike}}},\quad
r_{\text{in}}<r<r_{\text{spike}}$ (1)

with the radius from the central mass $r$ and the reference radius
$r_{6}=10^{-6}$pc. Following [15], the inner radius is chosen to be
$r_{\text{in}}=4m_{1}$. The spike radius $r_{\text{spike}}$ is the maximal
radius of the spike, which can be obtained by comparing the gravitational
influence of the IMBH to the total spike mass [17]. In this publication, we
always consider $r\ll r_{\text{spike}}$.

The range of the power law index is $1<\alpha_{\text{spike}}<3$, but we focus
on the $\alpha_{\text{spike}}=7/3$ case in this paper, which represents a halo
grown from an Navarro-Frenk-White (NFW) profile [38]. See [17, 23] for an
exploration of different power laws.

The dark matter particles in the halo can be described by an equilibrium phase
space distribution function $f=dN/d^{3}rd^{3}v$, giving the number density per
phase space volume. Since the halo is spherically symmetric,
$f=f(\mathcal{E})$, with $\mathcal{E}$ being the relative energy per unit mass

$\mathcal{E}(r,v)=\Psi(r)-\frac{1}{2}v^{2}.$ (2)

$\Psi(r)$ is the relative Newtonian potential. Close to the IMBH, it is simply
$\Psi(r)=\frac{m_{1}}{r}$. For gravitationally bound particles we have
$\mathcal{E}>0$.

For a spherically symmetric density profile $\rho(r)$, the distribution
function $f(\mathcal{E})$ can be calculated by the Eddington inversion
procedure [39]. For the power law spike, this gives

$\displaystyle f_{\text{spike}}(\mathcal{E})=$
$\displaystyle\frac{\alpha_{\text{spike}}(\alpha_{\text{spike}}-1)}{(2\pi)^{3/2}}\rho_{6}\left(\frac{r_{6}}{m_{1}}\right)^{\alpha_{\text{spike}}}$
$\displaystyle\times\frac{\Gamma(\alpha_{\text{spike}}-1)}{\Gamma(\alpha_{\text{spike}}-\frac{1}{2})}\mathcal{E}^{\alpha_{\text{spike}}-3/2}$
(3)

where $\Gamma(x)$ is the Gamma function.

The density for a given distribution function can be recovered through

$\rho(r)=4\pi\int_{0}^{v_{\text{max}}(r)}v^{2}f\left(\Psi(r)-\frac{1}{2}v^{2}\right)dv,$
(4)

where the escape velocity is given by $v_{\text{max}}(r)=\sqrt{2\Psi(r)}$.

We ignore the relativistic effects on the dark matter spike, derived by [15]
and explored in [28] and leave them for future work.

###  II.2 Accretion Disk

To model the (baryonic) accretion disk, we follow the approach of [35]. We
assume a radiatively efficient, geometrically thin accretion disk model,
employing the so-called $\alpha$ and $\beta$ disk prescriptions derived by
Shakura & Sunyaev [40], see [41, 42] for a review. Here, radiatively efficient
means that we assume the heat generated by viscosity at any given radius is
immediately radiated away.

The disks are parameterized by the viscosity parameter $\alpha$, which is
estimated to be around $\alpha\sim 0.01-0.1$ [43]. The surface density
$\Sigma$ and scale height $H$ of the disk are then given by the
parameterizations [35, 44]

$\displaystyle\Sigma_{\alpha}\left[\frac{\text{kg}}{\text{m}^{2}}\right]=$
$\displaystyle 5.4\cdot
10^{3}\left(\frac{\alpha}{0.1}\right)^{-1}\left(\frac{f_{\text{Edd}}}{0.1}\frac{0.1}{\epsilon}\right)^{-1}\left(\frac{r}{10m_{1}}\right)^{3/2}$
(5) $\displaystyle\Sigma_{\beta}\left[\frac{\text{kg}}{\text{m}^{2}}\right]=$
$\displaystyle 2.1\cdot
10^{7}\left(\frac{\alpha}{0.1}\right)^{-4/5}\left(\frac{f_{\text{Edd}}}{0.1}\frac{0.1}{\epsilon}\right)^{3/5}\left(\frac{m_{1}}{10^{6}M_{\odot}}\right)^{1/5}$
$\displaystyle\crossproduct\left(\frac{r}{10m_{1}}\right)^{-3/5}$ (6)
$\displaystyle H\left[M_{\odot}\right]=$ $\displaystyle
1.5\left(\frac{f_{\text{Edd}}}{0.1}\frac{0.1}{\epsilon}\right)m_{1},$ (7)

where $f_{\text{Edd}}$ is the fraction of the Eddington accretion rate that
the central IMBH is accreting at, and $\epsilon$ describes the efficiency of
mass-energy conversion into luminosity in the disk. We will assume
$f_{\text{Edd}}=\epsilon=0.1$ throughout this paper.

The corresponding disk density is $\rho_{\text{b}}=\Sigma/2H$, and the Mach
number $\mathcal{M}_{a}=r/H$. The simple scalings of the disk models are valid
within approximately $r\leq 10^{3}m_{1}$ [42].

The models originate from the assumption that the central BH accretes at a
steady rate $\dot{M}_{\text{disk}}=3\pi\nu\Sigma$, where $\nu=\alpha
c_{s}^{2}/\Omega$ is the kinematic viscosity, $c_{s}$ the sound speed, and
$\Omega=\sqrt{m_{1}/r^{3}}$ the orbital frequency. The sound speed is in
general given by $c_{s}^{2}=\frac{p_{\text{rad}}+p_{\text{gas}}}{\rho}$ with
the radiation pressure $p_{\text{rad}}$ and the thermal gas pressure
$p_{\text{gas}}$. In $\beta$ disks, the thermal gas pressure is assumed to be
dominant $p_{\text{gas}}\gg p_{\text{rad}}$. While analytic solutions to the
$\alpha$ disk model have some thermal instabilities, it seems to be more
physically realistic and a decent approximation in the radiation dominated
regime [35].

###  II.3 Orbital Evolution

####  II.3.1 Keplerian Orbit

The secondary is assumed to be on a Keplerian orbit around the central IMBH.
We ignore the additional matter contributions by the spike and disk to the
total and reduced mass $\mu$ of the Keplerian system and assume
$m=m_{1}+m_{2}$, $\mu=\frac{m_{1}m_{2}}{m}$. The mass ratio is defined as
$q=\frac{m_{2}}{m_{1}}$. For the separations at play the system is clearly
gravitationally dominated by the IMBH, and the total enclosed mass of the dark
matter and baryon distributions up to the location of the secondary is much
smaller than the mass of the central IMBH,
$m_{\text{dm}}(r=10^{5}r_{\text{isco}}),m_{\text{b}}(r=10^{5}r_{\text{isco}})\ll
m_{1}$. Here, $r_{\text{isco}}$ is the radius of the innermost stable circular
orbit, which is $r_{\text{isco}}=6m_{1}$ for a Schwarzschild black hole.
According to [24], the inclusion of the gravitational influence of the spike
distribution would primarily lead to orbital precession, which we neglect in
this paper.

The Keplerian orbit can be described by two parameters, the semimajor axis $a$
and the eccentricity $e$. For a bound orbit, $0\leq e<1$, where $e=0$
describes a circular orbit.

The orbital energy is given by [45]

$E_{\text{orb}}=-\frac{m\mu}{2a},$ (8)

the angular momentum $L_{\text{orb}}$ by

$e^{2}-1=\frac{2E_{\text{orb}}L_{\text{orb}}^{2}}{m^{2}\mu^{3}},$ (9)

and the mean orbital frequency by

$\mathcal{F}=\frac{1}{2\pi}\sqrt{\frac{m}{a^{3}}}.$ (10)

Throughout one orbit, the radius and the velocity of the secondary at the true
anomaly $\phi$ is given by

$\displaystyle r=$ $\displaystyle\frac{a(1-e^{2})}{1+e\cos{\phi}},$ (11)
$\displaystyle v^{2}=$ $\displaystyle m\left(\frac{2}{r}-\frac{1}{a}\right).$
(12)

####  II.3.2 Dissipative Forces

The secondary is assumed to lose energy on a secular timescale that is much
larger than the orbital timescale. This assumption allows us to use the
Keplerian orbits to calculate the dissipative forces acting on the secondary.
Over many orbits, these lead to a change in the orbital parameters. To model
the dissipative forces, we use the force $F(r,v)$ depending on the separation
$r$ and the velocity $v$ of the secondary.

The energy and angular momentum loss for a given dissipative force can be
obtained by averaging over one orbit with orbital period $T$ [21],

$\displaystyle\left\langle\derivative{E}{t}\right\rangle=$
$\displaystyle\int_{0}^{T}\frac{\mathrm{d}t}{T}\derivative{E}{t}=-\int_{0}^{T}\frac{\mathrm{d}t}{T}F(r,v)v,$
(13) $\displaystyle\left\langle\derivative{L}{t}\right\rangle=$
$\displaystyle\int_{0}^{T}\frac{\mathrm{d}t}{T}\derivative{L}{t}=-\sqrt{ma(1-e^{2})}\int_{0}^{T}\frac{\mathrm{d}t}{T}\frac{F(r,v)}{v}.$
(14)

These integrals can be calculated with the help of Eqs. (11) and (12) and

$\int_{0}^{T}\frac{\mathrm{d}t}{T}G(r(t),v(t))=(1-e^{2})^{\frac{3}{2}}\int_{0}^{2\pi}\frac{\mathrm{d}\phi}{2\pi}\frac{G(r(\phi),v(\phi))}{(1+e\cos{\phi})^{2}},$
(15)

which is valid for an arbitrary function $G(r,v)$ [45].

Therefore, for a given dissipative force $F(r,v)$, we can compute the energy
and angular momentum loss either analytically or numerically using Eqs. (13)
and (14).

The specific dissipative effects considered here are GW emission, dynamical
friction with the DM spike, and gas interaction with the accretion disk. Each
can be modeled as a force which leads to a loss of orbital energy and angular
momentum over secular timescales,

$\displaystyle\derivative{E_{\text{orb}}}{t}=\left\langle\derivative{E_{\text{gw}}}{t}\right\rangle+\left\langle\derivative{E_{\text{dm}}}{t}\right\rangle+\left\langle\derivative{E_{\text{gas}}}{t}\right\rangle,$
(16)
$\displaystyle\derivative{L_{\text{orb}}}{t}=\left\langle\derivative{L_{\text{gw}}}{t}\right\rangle+\left\langle\derivative{L_{\text{dm}}}{t}\right\rangle+\left\langle\derivative{L_{\text{gas}}}{t}\right\rangle.$
(17)

Gravitational Waves The GW emission loss is given by [45]

$\displaystyle\left\langle\derivative{E_{\text{gw}}}{t}\right\rangle=$
$\displaystyle-\frac{32}{5}\frac{\mu^{2}m^{3}}{a^{5}}\frac{1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}}{(1-e^{2})^{7/2}},$
(18) $\displaystyle\left\langle\derivative{L_{\text{gw}}}{t}\right\rangle=$
$\displaystyle-\frac{32}{5}\frac{\mu^{2}m^{5/2}}{a^{7/2}}\frac{1+\frac{7}{8}e^{2}}{(1-e^{2})^{2}}.$
(19)

Dynamical Friction with the DM Spike

The dynamical friction with the DM spike is given by the Chandrasekhar
equation [18, 26]

$F_{\text{dm}}(r,v)=4\pi
m_{2}^{2}\rho_{\text{dm}}(r)\xi(v)\frac{\log\Lambda}{v^{2}}$ (20)

with the Coulomb logarithm $\log\Lambda$. Here, we adopt the value
$\log\Lambda=\sqrt{\frac{m_{1}}{m_{2}}}$ [26]. The factor $\xi(v)$ accounts
for the fact that the particles in the DM spike are moving with different
velocities relative to the secondary[39, 26], because physically, DM particles
only scatter and absorb momentum from the secondary if they are moving with a
velocity that is slower compared to it.

To estimate the density of particles moving slower than the secondary
travelling at $v$, we can use Eq. (4)

$\rho_{\text{dm}}(r)\xi(v)=4\pi\int_{0}^{v}v^{\prime
2}f\left(\Psi(r)-\frac{1}{2}v^{\prime 2}\right)dv^{\prime}.$ (21)

Baryonic Disk Interaction

There is a wide range of models for compact object (CO) – accretion disk
interactions. In this paper, we consider and compare two different models from
different origins, Type–I migration and dynamical friction with the accretion
disk. See section IV.1 for a discussion of applicability. Here, we primarily
want to give the equations governing the two models.

The most commonly employed model originates from planetary formation models
and is called Type–I migration[46]. In planetary migration models, the
important quantity is the torque acting on the secondary body. The equation
for this torque is given by [46]

$\Gamma_{\text{Type--I}}=\Sigma r^{4}\Omega^{2}q^{2}\mathcal{M}_{a}^{2},$ (22)

which can be translated into a force – the language of our model – by

$F_{\text{Type--I}}=\Gamma_{\text{Type--I}}q/r$ (23)

The derivation assumes the creation of density wave resonances in the disk by
the secondary object, which causes a negative torque on the perturber, and
thus an inspiral. Note that this equation is only valid for circular orbits.

The second model we look at is that of dynamical friction with the gas. To
differentiate between the gas and DM dynamical friction, we call this model
Ostriker. The friction force is given by [47, 48]

$F_{\text{Ostriker}}=4\pi m_{2}^{2}\rho_{\text{b}}(r)\frac{I}{v_{rel}^{2}}$
(24)

with

$I=\frac{1}{2}\begin{cases}\log\frac{1-v_{rel}/c_{s}}{1+v/c_{s}}-v_{rel}/c_{s}&\textit{subsonic}\\\
\log(1-(v_{rel}/c_{s})^{-2})+\log\Lambda&\textit{supersonic}\end{cases}$ (25)

Here, $v_{rel}$ refers to the relative velocity between the gas and the
secondary object. For this model, we will assume the secondary object to be
counter-rotating to the disk, to avoid the scenario of $v_{rel}=0$. This model
allows for non-circular orbits.

The quantities $\derivative{E_{\text{gas}}}{t},\derivative{L_{\text{gas}}}{t}$
in Eq. (16),(17) then refer to either Type–I or Ostriker models, which will be
made clear wherever relevant.

####  II.3.3 Orbital Evolution

We want to obtain the secular evolution of the orbital parameters $a(t),e(t)$
under the backreaction of our dissipative forces.

We use Eq. (8) to obtain

$\displaystyle\partialderivative{E_{\text{orb}}}{a}=$
$\displaystyle\frac{m_{2}m_{1}}{2a^{2}}$ (26)
$\displaystyle\derivative{a}{t}=$
$\displaystyle\derivative{E_{\text{orb}}}{t}/\partialderivative{E_{\text{orb}}}{a}.$
(27)

Similarly, the evolution for $e$ is derived from Eq. (9) as

$\derivative{e}{t}=-\frac{1-e^{2}}{2e}\left(\derivative{E_{\text{orb}}}{t}/E_{\text{orb}}+2\derivative{L_{\text{orb}}}{t}/L_{\text{orb}}\right).$
(28)

Combining Eqs. (27) and (28) with Eqs. (16) and (17), we have a system of
differential equations that can be solved numerically.

###  II.4 Gravitational Wave Signal

The IMRI system emits GWs as a result of the change in the quadrupole moment.
The equations governing the gravitational wave signal are given in [23]. Here,
we want to assume that we can measure a single IMRI system and resolve its
frequency evolution $\mathcal{F}(t)$. We want to explore observational
signatures and see if we can distinguish our models.

####  II.4.1 Braking Index

For large masses or large initial semimajor axes, the inspiral might take
$t\gg 10$yrs, making it difficult to observe in its entirety. At the same
time, at larger separations, GW emission might be subdominant to other
dissipative losses, which can dictate the frequency evolution. A possible
signature to consider is the evolution of the orbital frequency $\mathcal{F}$,
which can be measured on shorter timescales [49, 50]. A useful quantity is the
so called braking index, which was first described in the context of neutron
star spin-down [51] but can be applied to any signal with an evolution in
time,

$n_{b}=\frac{\mathcal{F}\ddot{\mathcal{F}}}{\dot{\mathcal{F}}^{2}}.$ (29)

As an example, when circular GW emission losses are dominant,
$\dot{\mathcal{F}}\propto\mathcal{F}^{11/3}$ and therefore $n_{b}=11/3$ [52].

Relating this to the semimajor axis with Eq. (10), we have

$n_{b}=\frac{5}{3}-\frac{2}{3}\frac{a\ddot{a}}{\dot{a}^{2}}.$ (30)

Following the approach of [23], if we model a dissipative force as $F\sim
r^{\gamma}v^{\delta}$, we have, to second order in eccentricity

$\displaystyle\frac{a\ddot{a}}{\dot{a}^{2}}\approx$ $\displaystyle
k_{1}+2ae\derivative{e}{a}\left(\frac{1/2-k_{1}}{1-e^{2}}+\frac{k_{2}}{1+k_{2}e^{2}}\right)$
(31) $\displaystyle\text{with }k_{1}=$ $\displaystyle
2+\gamma-\frac{\delta+1}{2},$ $\displaystyle k_{2}=$
$\displaystyle(3+\gamma^{2}+\gamma(3-2\delta)-2\delta+\delta^{2})/4$

See appendix A for a more detailed computation.

Therefore, in the circular case ($e=0$), if a given dissipative force
dominates the inspiral, the braking index will be this simple algebraic
combination of parameters. For eccentric inspirals, a measurement of the
braking index and the eccentricity evolution $\derivative{e}{a}$ allows to
determine the dissipative force parameters $\gamma$ and $\delta$ that dominate
the inspiral.

####  II.4.2 Dephasing

For smaller orbital separations, the IMRI is dominated by GW emission loss. To
observe the effect that a subdominant dissipative force $F$ has on the
evolution, we can look at the dephasing. To this end, we compare the number of
GW cycles completed in the cases with and without this force present,
following [26]. We can do this for each harmonic individually between some
initial time $t_{\text{i}}$ and final time $t_{\text{f}}$ with

$N(t_{\text{f}},t_{\text{i}})=2\int_{t_{\text{i}}}^{t_{\text{f}}}\mathcal{F}(t)\mathrm{d}t.$
(32)

Setting $t_{\text{f}}=t_{\text{c}}$ as the time of coalescence, we obtain

$\Delta
N(t)=N_{\text{vacuum}}(t_{\text{c}},t)-N_{\text{tot}}(t_{\text{c}},t).$ (33)

where $N_{\text{vacuum}}$ is the phase accumulation where just GW emission is
driving the inspiral, and $N_{\text{tot}}$ is the phase accumulation of a
system where GW emission and another environmental effect – that we will mark
by the letter $F$ – is present. The dissipative forces are additive, such that
the frequency evolution can be written as
$\dot{\mathcal{F}}_{\text{tot}}=\dot{\mathcal{F}}_{\text{vacuum}}+\dot{\mathcal{F}}_{F}$.
When the GW emission is dominant, we can write this such that
$\dot{\mathcal{F}}_{\text{tot}}=\dot{\mathcal{F}}_{\text{vacuum}}(1+\varepsilon)$,
where
$\varepsilon=\frac{\dot{\mathcal{F}}_{F}}{\dot{\mathcal{F}}_{\text{vacuum}}}$.
This gives us the second derivative of the dephasing

$\frac{1}{2}\derivative[2]{\Delta
N}{t}=\dot{\mathcal{F}}_{\text{vacuum}}-\dot{\mathcal{F}}_{\text{tot}}=\varepsilon\dot{\mathcal{F}}_{\text{vacuum}}.$
(34)

In the circular case,
$\dot{\mathcal{F}}_{\text{vacuum}}\sim\mathcal{F}_{\text{vacuum}}^{11/3}$, and
a calculation of $\varepsilon$ gives

$\varepsilon\propto\mathcal{F}_{\text{vacuum}}^{-2-2k_{1}/3}$ (35)

assuming our dissipative force has the form $F\sim r^{\gamma}v^{\delta}$ as in
the previous section. See appendix A as well.

This results in the amount of dephasing being accumulated

$\Delta N\propto\mathcal{F}_{\text{vacuum}}^{(11-2k_{1})/3}.$ (36)

Since $\varepsilon\ll 1$,
$\mathcal{F}_{\text{tot}}(t)\approx\mathcal{F}_{\text{vacuum}}(t)$, a
measurement of

$n_{d}\equiv\derivative{\log\Delta
N}{\log\mathcal{F}_{\text{tot}}}=\mathcal{F}_{\text{tot}}\derivative{\Delta
N}{\mathcal{F}_{\text{tot}}}/\Delta N\approx\frac{11-2k_{1}}{3},$ (37)

what we will call the dephasing index, could reveal the power law behavior of
the dephasing that is accumulated, and therefore the dissipative force at
play.

##  III Results

In this section, we present the results from the numerical integration of the
system of differential equations derived in sec. II.3. The equations have been
implemented in python and numerically solved. The code is publicly available
at: https://github.com/DMGW-Goethe/imripy.

###  III.1 Accretion Disk Only

As a point of comparison, let us first consider an IMRI system with
$\\{m_{1},m_{2}\\}=\\{10^{5}M_{\odot},1M_{\odot}\\}$, in which the central
mass is surrounded by an accretion disk. We model an $\alpha$ and a
$\beta$-disk with the parameters $\alpha=0.1$. In Fig. 2 we plot the density
close to the $r_{\text{isco}}$.

We first want to compare the dissipative force models Type–I and Ostriker. To
get an estimate of the relative impact of the forces, we plot the relative
semimajor axis loss for the forces involved
$\derivative{E_{\text{force}}}{t}/\partialderivative{E_{\text{orb}}}{a}$,
assuming circular orbits at a given radius. Close to $r_{\text{isco}}$, the
dominant force is the GW emission loss. The accretion disk effects become
dominant for $r\gtrsim 10-100r_{\text{isco}}$. The steeper power law in the
density of the $\alpha$ disk compared to the $\beta$ disk is reflected in the
power law behavior of the energy loss curves. It can be seen that the Ostriker
losses are $10^{3}-10^{4}$ orders of magnitude larger that the Type–I losses
around the crossing with the GW losses $r\sim 10-100r_{\text{isco}}$. At
larger orbital separations the differences decrease. Unfortunately, the region
$r\sim 10-100r_{\text{isco}}$ is crucial to understand when modeling an
inspiral, so a difference this large invokes huge uncertainties. See section
IV.1 for further discussion.

Figure 2:  Top: The density $\rho_{\text{b}}$ of the two accretion disk models
$\alpha$ and $\beta$ around an IMBH with $m_{1}=10^{5}M_{\odot}$. Bottom: The
relative impact of the three dissipative forces: GW emission loss, Type–I, and
Ostriker for the two different disk models for a circular orbit at the given
radius.

####  III.1.1 Circular Inspiral

Figure 3: Top: The evolution of the semimajor axis $a$ for 5 different models,
in a vacuum, in $\alpha$ or $\beta$ accretion disks, with Type–I migration or
Ostriker dynamical friction interaction. Bottom: The dephasing, dephasing
index, and braking index as a function of frequency during the inspiral.

To see the impact this has on an inspiral and the frequency evolution, see
Fig. 3. There, we model a circular inspiral with parameters
$\\{m_{1},m_{2},a_{0}\\}=\\{10^{5}M_{\odot},1M_{\odot},500r_{\text{isco}}\\}$.
In the top plots, the semimajor axis is plotted against time. The different
magnitudes of the forces result in different timescales, with Ostriker
inspiraling orders of magnitude faster compared to the Type–I model. Due to
the higher density of the $\alpha$ disk at the most relevant separations, its
inspiral is also faster compared to the one of $\beta$ disk. For the Ostriker
\+ $\alpha$ disk model, the semimajor axis, and with it the frequency
evolution, significantly changes on the order of $\sim 10$ years, which would
probably make the frequency evolution in this regime observable.111For our
values of $a_{0}=500\,r_{\text{isco}}\sim 3\cdot 10^{3}m_{1}$, we are at the
edge of the validity of Eq. (5), so this effect might be exaggerated.
Nevertheless, the trend seems to continue into the range of validity and
warrants further inspection.

In the second plot, which shows the dephasing amount, this trend is actually
reversed. Here, the dephasing effects are stronger for the $\beta$ disk at
late times, due to the crossing in relative impact seen in Fig. 2 at small
separations. For the Type–I interaction, the dephasing is far below one in the
last $5$ years of inspiral, which indicates that these effects will probably
not be observable for these parameters. For the Ostriker model, the dephasing
($\sim 100$) would also hardly be observable over a 5 year period.

Nevertheless, if the effects were to be observable, the dephasing index, shown
in the third plot, would clearly distinguish between the forces. The lines are
shown until $\Delta N<10$ and they clearly converge to the value given by Eq.
(37) for the different models. Complementary, the braking index, shown in the
fourth plot, initially starts out at the value given by Eq. (31) for the
different models and converge to the value given for the GW emission loss.

The two plots of the braking and dephasing index clearly reflect the different
regimes that the inspiral is subject to. Early in the evolution – for low
frequencies – the braking index is constant with the expectation given by Eq.
(31). Then, as the object inspirals and GW losses become important, the
braking index moves in between these values and approaches the $n_{b}=11/3$
value at later times. At the same time, the approximation used to derive Eq.
(37) becomes accurate, and the lines start to converge to the appropriate
values. The two indices are clearly complementary observational probes.

Comparing the two CO accretion disk interaction models, while these results
are certainly not sufficiently realistic, there stark contrast allows to make
some inferences. If the Ostriker description is close to reality, the
inspirals will most likely be much faster and possibly have observationally
relevant effects. On the other hand, for the Type–I model, the inspirals will
be more difficult to observe.

Nevertheless, due to the nature of either forces as seen in Fig. 2, their
effects would probably be more observable at larger separations.

####  III.1.2 Eccentric Inspiral

Figure 4: Top: The evolution of the semimajor axis and eccentricity for 5
different models, in a vacuum, in $\alpha$ or $\beta$ accretion disks, with
Type–I migration or Ostriker dynamical friction interaction. Bottom: The
dephasing, dephasing index, and braking index as a function of frequency
during the inspiral.

If we allow for some small initial eccentricity $e_{0}=0.001$, we see a very
different behavior. The eccentricity evolution is plotted in the second plot
of Fig. 4, and the temporal evolution is from right to left, from large
semimajor axis to $r_{\text{isco}}$.

First, the Type–I migration model is only valid for circular orbits and breaks
down for eccentricities $>0.001$. The small eccentrification seen in the
Type–I model is most likely an artefact of the model being extended too simply
to eccentric orbits and not physical [53]. However, the eccentricity evolution
of the Ostriker model is of interest. In both $\alpha$ and $\beta$ disks, the
eccentricity increases, as dynamical friction tends to do. Of note are the
different scales of the plots, the eccentrification is much stronger for the
$\alpha$ disk. This can be understood with the condition derived in [23]. For
$F\sim r^{\gamma}v^{\delta}$, the eccentrification is proportional to
$\derivative{e}{t}\sim(1-\delta+\gamma)$. For dynamical friction $\delta=-2$,
for the $\beta$ disk $\gamma=-3/5$, while for the $\alpha$ disk $\gamma=3/2$.
So the density distribution acts as an eccentrification moderator in the
$\beta$ disk case, while the $\alpha$ disk density distribution enhances the
eccentrification effects.

The strong eccentrification for the $\alpha$ disk also has an effect on the
dephasing. Higher eccentricity means stronger GW emission loss, so
eccentricity increases the inspiral rate, and therefore the dephasing. The
dephasing at the $5$ year line increases by a factor of $10^{2}-10^{4}$ in the
two models. Also, the dephasing index converges to a different value, as Eq.
(37) was derived for a circular GW loss dominated inspiral. The different
value is now due to the eccentricity increasing the inspiral rate. To tease
out the accretion disk effects, one would need to expand Eq. (37) to eccentric
GW loss inspirals.

Similarly, the breaking index is affected by the eccentricity. This is due to
the $\derivative{e}{a}$ term in Eq. (31), whose behavior causes the small
spike seen the last plot in Fig. 4. Here, the braking index approximation
would allow a measurement of $\gamma,\delta$ and therefore the profile of the
disk ($\alpha$ disk with $\rho_{\text{b}}\sim r^{3/2}$) and nature of the
interaction (dynamical friction with $\delta=-2$).

We can conclude that by modeling the eccentric behavior it allows us to detect
the environmental effect through the larger dephasing at late times, and
additionally extract accurate information about the environmental effect via
the braking index at early times.

###  III.2 Accretion Disk + DM spike

In the following section, we add a DM spike into the picture. For the spike we
take the parameters $\\{\rho_{6},\alpha_{\text{spike}}\\}=\\{1.3\cdot
10^{17}M_{\odot}/\text{pc}^{3},7/3\\}$.

First, we add the DM distribution and relative impact to the previous
comparative plot in Fig. 5. It can be seen that the DM impact is on par with
the Ostriker model in the regime $r\sim 10-10^{2}r_{\text{isco}}$, but does
not rise like the accretion disk effects. At smaller separations, it is
subdominant to the GW loss, but stronger than the accretion disk effects.

To reduce the number of models, we focus on two combinations: The strongest
effects are expected for Ostriker \+ $\alpha$ disk, while the weakest effects
are with Type–I \+ $\beta$ disk. To get an idea of the possible relative
impacts of DM spike vs accretion disks, we want to compare these combinations.

Figure 5: Top: The density $\rho_{\text{b}}$ of the two accretion disk models
$\alpha$ and $\beta$ and the DM spike $\rho_{\text{dm}}$ around an IMBH with
$m_{1}=10^{5}M_{\odot}$. Bottom: The relative impact of the four dissipative
forces: GW emission loss, DM dynamical friction, and Type-I and Ostriker for
the two different disk models for a circular orbit at the given radius.

Figure 6: Top: The evolution of the semimajor axis $a$ for 6 different models,
in a vacuum, with a DM spike, and $\alpha$ disk+Ostriker and $\beta$ disk +
Type–Iinteraction with and without a DM spike. Bottom: The dephasing,
dephasing index, and braking index as a function of frequency during the
inspiral.

####  III.2.1 Circular inspiral

To this end, the evolution for a circular system with
$m_{1},m_{2},a_{0}=\\{10^{5}M_{\odot},1M_{\odot},500r_{\text{isco}}\\}$ is
plotted in Fig. 6. The results seem to be as anticipated. The fastest inspiral
is for Ostriker \+ $\alpha$ disk + DM spike, while the addition of the DM
spike barely matters for the inspiral time, as Ostriker \+ $\alpha$ disk is
the dominant dissipative force initially. This can be seen in the braking
index, where the evolution is dominated by its value. The dephasing on the
other hand is clearly dominated by the DM spike. This can be understood by
looking at Fig. 5. The dephasing is accumulated where GW emission loss
dominates, which is also where the relative impact of the DM spike is much
stronger than the accretion disk effects.

The slowest inspiral (ignoring the vacuum case) is for Type–I \+ $\beta$ disk.
The addition of the DM spike clearly dominates this model. Still, at early
times, the braking index would dominate the DM effects as inferred from Fig.
5, but not for the range of radii seen here.

Overall, even though there are large modeling uncertainties, this implies that
accretion disk and DM spike effects have different regimes of dominance, and
could therefore be distinguished in an actual observation. DM spike effects
are significant at small separations, while at large separations, accretion
disk effects dominate. This is reflected in the braking and dephasing index.

####  III.2.2 Eccentric inspirals

We now want to look at the orbital evolution allowing for eccentric orbits. To
this end, we look at different initial eccentricities with
$e_{0}=\\{10^{-4},10^{-2},10^{-1}\\}$. The same caveats as described
previously apply to the Type–I migration model, so we will not comment on this
model. The results are shown in Fig. 7.

What can be seen again are the strong eccentrification effects of the Ostriker
\+ $\alpha$ disk model. For small initial eccentricity $e_{0}=10^{-4}$ the
moderating effects of the DM spike as explored in [23] can be seen. For higher
initial eccentricity, the influence of DM is too weak initially, so the
eccentricity (almost) saturates $e\to 1$. The higher eccentricity also means
stronger GW emission, which speeds up the inspiral, causing very large
dephasings for the Ostriker \+ $\alpha$ disk model. There is competition
between two effects described previously: DM causes circularization while also
generally causing higher dephasing, whereas the Ostriker \+ $\alpha$ disk
model causes eccentrification and more dephasing through this. The flip can be
seen in the comparison between the dephasing plots of $e_{0}=10^{-4}$ and
$e_{0}=10^{-2}$.

What can be observed is the breakdown of the differentiation power of the
dephasing index. Where eccentricity dominates, the models cannot be
distinguished through the dephasing index. We would need better modeling of
the circularization effects of the GW emission to pick out the forces
involved.

Overall, this reinforces the idea that the inspirals might be very eccentric
if the Ostriker \+ $\alpha$ disk model is accurate. The inspirals would be
much faster and possibly within the lifetime of a spaceborne GW observation
mission.

The initial eccentricity of the secondary most likely depends on the origin,
whether it formed in-situ in the accretion disk, or whether it was captured
[54]. The results here suggest that these different origins could be
distinguished through the strongly different observational signatures.

Figure 7: Top: The evolution of the semimajor axis and eccentricity for 6
different models, in a vacuum, with a DM spike, and $\alpha$ disk+Ostriker and
$\beta$ disk + Type–I interaction with and without a DM spike for different
initial eccentricities $e_{0}=\\{10^{-4},0.1,0.6\\}$ Bottom: The dephasing,
dephasing index, and braking index as a function of frequency during the
inspiral.

####  III.2.3 Varying Central Mass

We now want to compare different central masses of the massive black hole to
compare the impact in different regimes, therefore we vary
$m_{1}=\\{10^{3},10^{4},10^{5}\\}M_{\odot}$. According to Eq. (5) and 6, the
disk profiles change along with $m_{1}$. To allow for a fair comparison we
vary the spike density $\rho_{6}=\\{5\cdot 10^{15},2.5\cdot 10^{16},1.3\cdot
10^{17}\\}M_{\odot}/$pc3 along with $m_{1}$.222Numerically employing the
procedure described in section II.B of [17], we obtain a geometric dependence
of $\rho_{6}$ on the black hole mass $m_{1}$ such that
$\frac{\Delta\log\rho_{6}}{\Delta\log m_{1}}\approx 0.7$. Therefore,
increasing the black hole mass $m_{1}$ by a factor of $\sim 10$ increases
$\rho_{6}$ by a factor of $\sim 5$. This allows us to generalize the
parameters taken from [27].

Unfortunately, the models we employ here begin to break down for larger mass
ratios $q>10^{-5}$. For example, halo feedback processes actually become
important for the dynamical friction with the DM spike[26, 27], which we do
not model here. Also, Type–I migration requires a smaller mass ratio, see sec.
IV.1 for a discussion. Nevertheless, we still believe there is value in these
plots as we will discuss in the following.

Our results are plotted in Fig. 8. It can be seen that the relative strength
between the DM spike and the accretion disk effects does not change for
central mass. The most prominent difference is the time of inspiral and the
frequency range. The effects of the models are qualitatively the same, just
shifted to the new frequency region and on a faster timescale. This implies
that the detectability of these effects increases for smaller $m_{1}$ for the
limited lifetime of a spaceborne GW observatory mission. We can see that the
dephasing effects increase by many orders of magnitude for the last $5$ years
of the inspiral.

So even though these models break down, this is the region where a naive
extension predicts much stronger observable effects. Therefore, we see a
strong motivation to model this region of mass ratios and to better understand
the forces involved. We leave this for future work.

Figure 8: Top: The evolution of the semimajor axis and eccentricity for 6
different models, in a vacuum, with a DM spike, and $\alpha$ disk+Ostriker and
$\beta$ disk + Type–I interaction with and without a DM spike for different
IMBH masses $m_{1}=\\{10^{3},10^{4},10^{5}\\}M_{\odot}$ Bottom: The dephasing,
dephasing index, and braking index as a function of frequency during the
inspiral.

##  IV Discussion

###  IV.1 Type–I migration vs Dynamical Friction in the Accretion Disk

In this subsection we discuss the applicability of the models we employ for
the CO – accretion disk interaction.

The Type–I torque model originates from protoplanetary disk models, where the
mass ratio is sufficiently small, $q<10^{-4}$. In this regime, linear
perturbation theory describes the protoplanetary system very well[55], and is
supported by numerical simulations [46]. For larger mass ratios, or larger
COs, that have a size comparable to the disk, a gap can form in the accretion
disk, reducing the torque experienced by the secondary. This is dubbed Type–II
migration.

Since the physics primarily depends on the mass ratio, an argument for the
applicability of Type–I can be made for IMRIs, where $q\ll 1$. While the
complete picture is more complicated, simulations have shown that this is a
decent approximation to an order of magnitude for IMRIs [56, 34]. This model
has been studied in the context of E/IMRIs in several previous
publications[57, 42, 31, 35, 36]. Additionally, [42] has argued that – in AGN
– both migration types can appear during an inspiral. Further out in the
orbit, the tidal field of the secondary would dominate over the central SMBH,
and clear a gap. At some crossover radius, the tidal influence diminishes so
that the secondary cannot clear a gap anymore, and the inspiral is dominated
by Type–I. See [34] for a discussion of how the torque can change during an
inspiral in IMRIs.

The other model that is commonly applied to these systems, is that of
dynamical friction, which we have previously referred to as Ostriker. Here,
the secondary creates a wake behind itself in the orbit and the resulting
gravitational interaction slows it down [39]. Ref. [58] derives that this
model is relevant for a hypersonic secondary ($v^{2}\ll c_{s}^{2}$) in thick
disk models, in which it is completely embedded inside the disk. The model is
typically used for eccentric orbits [59, 60, 48] but is not strictly necessary
[61, 62]. Simulations show that dynamical friction is an accurate
approximation for a point like secondary, e.g. a black hole, as long as it is
completely embedded in the disk and the mass ratio is small enough $q\lesssim
10^{-5}$ [60]. Even though we have used a thin disk description here, the
secondary might be sufficiently small to be completely embedded, i.e. the
Roche radius is smaller than the disk $r_{\textit{roche}}\ll H$. For circular
prograde orbits (where the secondary would be co-rotating with the disk), this
model breaks down, as the relative velocity with a differentially rotating
disk would be $0$. For this reason, we focused on retrograde orbits relative
to the disk. These models can be relevant in capture scenarios, where the
secondary was not produced inside the disk, but captured from the
intergalactic medium [60].

Comparing the two models, [61] points out that they are closely related. The
forces are proportional by a factor of $F_{\text{Ostriker}}/F_{\text{Type--
I}}\propto(r/H)$, which comes down to the differential torque [63]. This is
the difference between the inner and outer torque on the secondary in orbit.
Type–I migration modeling is sensitive to this torque, while dynamical
friction in the local ballistic approximation is not. This is analogous to
including tidal forces acting on the secondary [61]. This can be seen in Fig.
2, where the difference is a scaling in $r$ (since $H\sim$const here). The
difference becomes more pronounced for small $r$, which is unfortunately where
the effects are most pronounced in an observable inspiral. Ultimately, whether
these differential torques are relevant depends on multiple factors, such as
the density gradient of the disk, the gravitational size (i.e. Roche radius)
of the secondary, its direction and speed relative to the disk. While there is
numerical support by simulations for both models, each simulation has a
limited range of applicability and caveats.

The conclusion in this discussion can only be that more study is needed to
assess which effects are most relevant at these scales.

###  IV.2 Comparison between different environmental effects

Understanding the different environmental effects in E/IMRIs is of great
importance to maximize the science yield of future space based GW detectors
[31, 8].

The two possible environmental effects we have explored and compared are
(baryonic) accretion discs and DM spikes. For simplicity, we also have focused
on a single DM spike model.

The results seem to indicate that DM spikes and accretion disk effects
dominate at different times in the inspiral. At early times and larger
separations, accretion disk effects dominate, while at late times and small
separations, DM spike effects are stronger. These regimes are reflected in the
different braking and dephasing indices. Another factor is the circularization
effect of DM spikes, explored in [23], which competes with the
eccentrification effect of the Ostriker model. This can influence the inspiral
even if DM is subdominant. From these initial considerations we would
carefully conclude that the environmental effects can be distinguished and –
from the perspective of trying to detect DM – the accretion disk effects are
sufficiently different and not superdominant as to allow a detection. If both
accretion disks and DM spikes are common around IMBH, some IMRIs should
reflect the accretion disk effects while they are at large distance, while
others could reflect the DM spike influences when they have smaller
separations.

One aspect left unexplored is that of halo feedback. As the secondary loses
energy and angular momentum to the DM spike, it can actually significantly
deplete the spike locally [26]. This effect is more prominent for larger mass
ratios $q\gtrsim 10^{-4}$. We chose our parameters ($m_{1}\sim
10^{5}M_{\odot}$) such that for most of the systems explored in the previous
section, halo feedback would be negligible. The results where
$m_{1}<10^{5}M_{\odot}$ should be taken with a grain of salt. As explained
previously, the same relative impact between the forces appeared for different
$m_{1}$, which would break down with the inclusion of halo feedback. But we
can make some inferences what including halo feedback processes would mean for
the model. For example, if DM dynamical friction would be the dominant force,
the reduction in the DM density due to the halo feedback might increase the
relevance of the accretion disk effects. On the other hand, if the accretion
disk effects significantly dominate and the halo feedback timescale is larger
than that of the accretion disk effects, halo feedback might be less relevant.
Overall, the DM spike influence would be harder to observe. We leave these
considerations for future studies.

It is important to keep in mind that we only study linear combinations of
environmental effects. In reality, all these environmental effects would be
interacting with each other and possibly deviate from their simplistic
descriptions. Exploring these interactions will have to be done with numerical
simulations.

###  IV.3 Observational Signatures

While there are large modeling uncertainties in these systems, fortunately the
observational signatures differ between these models. For late time inspirals,
where the GW emission dominates, the dephasing is a tool to observe the
environmental effects[31]. The amount of dephasing and the speed at which it
is accumulated depends on the dissipative force. If the derivative of the
dephasing accumulated can be measured, the dissipative force can in principle
be identified by the dephasing index, as given by Eq. (37).

For larger separations of the secondary, an inspiral might still be too far
off to be observable within the mission lifetime, considering the timescales
involved. At the same time, for larger separations environmental effects would
dominate over the GW emission loss, dictating the frequency evolution. If the
evolution of the frequency is observable, i.e. the second derivative
$\ddot{\mathcal{F}}$ is measurable, the braking index can be used to
differentiate between environmental effects as well. According to Eq. (31),
different dissipative forces can result in different frequency evolutions.
Depending on the physical expectations, the forces involved can be inferred.
If at the same time the eccentricity evolution can be measured, the specific
force can be pinned down.

The observability of the second derivative was estimated in [49] and the
braking index for inspiraling binaries in a common envelope in [50]. These
results indicate that it might be observable for some systems, but most of the
observable systems would be stationary. Nevertheless, if the dissipative force
is strong enough, for example as in the Ostriker model in Fig. 3, the
frequency evolution could be sped up for the effect to be observable during
the observational period of LISA. We leave the detectability of the braking
index for different dissipative forces for future study.

Just like in equal mass binary systems, modeling the eccentricity evolution is
important for E/IMRIs [64]. The eccentricity evolution can have large effects
on the frequency evolution and the dephasing. When measured, it can also hint
at the environmental effects at play, as hinted by Eq. (31).

All of these observational tools, the braking index, the amount of dephasing
and dephasing index, and the eccentricity evolution, are complementary. The
braking index is valuable for large separations when the environmental effects
dominate. The dephasing and dephasing index are important late in the
inspiral, when GW emission loss dominates. The eccentricity evolution
complements both of these tools and can help to pin down the environmental
effect(s) involved.

Of course, we employed a simplistic description of the environmental effects
with a force $F\propto r^{\gamma}v^{\delta}$. More general forces could be
dependent on other features, such as spin, tidal deformability, etc. If the
environmental effects are time-dependent, these indices would vary over time,
losing their descriptive value. But even for more complex environmental
effects, such as the halo feedback [26, 27], a new equilibrium can emerge and
the indices become distinct, as can be seen from the phase parameterization
they develop [27]. Eventually, relativistic and post-Newtonian effects will
have to be analyzed in this framework. We will look at generalizations of
these indices in a future study.

During the finalization of this publication, [36] have published their
results. They compared the inspiral waveforms of DM spikes, accretion disks
and scalar clouds and performed a Bayesian analysis to see if these
environments can be distinguished. They find that these environments can in
fact be easily distinguished by their features. Our analytic approximations
might shed a light on these numerical results.

It could be useful to map out different environmental effects, their braking
and dephasing index and eccentrification/circularization effects. These tools
would allow an abstraction of the environmental effects and focus on the
specific impact on the frequency evolution. This could generalize waveform
generation and put a handle on the large parameter spaces that are incurred
when looking at several environmental effects.

As LISA expects to see a plethora of overlapping signals from different source
classes, the collaboration’s strategy is to attempt a global fit with all the
possible source class parameters. Depending on the behavior of these classes
and their predominant physical effects, their braking index might be an
identifier of a class, and could therefore help distinguish between the
different classes.

##  V Conclusions

In this paper, we studied the effect of CO – accretion disk interactions on
IMRIs. We compared two models – Type–I migration and Ostriker dynamical
friction interaction with the accretion disk [47]– and explored the effects
they have on the evolution of the semimajor axis and eccentricity in an
inspiral. Then, we added a dark matter spike and compared the effect with the
two baryonic accretion models.

  * •

The relative impact of Type–I migration and Ostriker dynamical friction
differs by several orders of magnitude. In a Type–I migration scenario, we
expect no eccentricities and negligible dephasing effects. In a retrograde
dynamical friction scenario we expect large eccentricities, large dephasing
and a domination of the frequency evolution for typical IMRI systems.

  * •

In comparison to the DM, the interactions have different regimes of dominance,
DM is more dominant for small separations and accretion disk effects at larger
separations.

  * •

We are able to differentiate between the models individually and in
combination, due to different amount of dephasing, a difference in braking and
dephasing index, and a different impact on the eccentricity evolution.

Which of the two baryonic accretion disk interactions models is more accurate
remains to be studied. We also leave the inclusion of halo feedback effects to
further study.

We would like to emphasize the usefulness of the study of the braking and
dephasing index and eccentrification effects in distinguishing environmental
effects and leave a systematic study of environmental effects and their
properties for future studies.

###### Acknowledgements.

We thank Andrea Derdzinski, Yury Levin and the anonymous referee for
insightful discussions and helpful comments on the manuscript.

##  Appendix A Derivation of braking and dephasing index

Assuming we have a dissipative force of the form
$F(r,v)=F_{0}r^{\gamma}v^{\delta}$, then plugging it into Eq. (13), (27) gives

$\displaystyle\dot{a}=$
$\displaystyle\left\langle\derivative{E_{F}}{t}\right\rangle/\partialderivative{E_{\text{orb}}}{a}$
(38) $\displaystyle=$
$\displaystyle-\frac{2F_{0}}{\mu}a^{k_{1}}(1-e^{2})^{k_{1}+1/2}m^{(\delta-2)/2}$
$\displaystyle\crossproduct\underbrace{\int_{0}^{2\pi}(1+e\cos\phi)^{-(2+\gamma)}(1+2e\cos\phi+e^{2})^{(\delta+1)/2}}_{\approx
1+k_{2}e^{2}}$ (39)

with $k_{1}=2+\gamma-\frac{\delta+1}{2}$ and
$k_{2}=(3+\gamma^{2}+\gamma(3-2\delta)-2\delta+\delta^{2})/4$. The
approximation of the integral is valid to third order in $e$. This equation
here rectifies a wrong positive sign in comparison to [23].

Taking the time derivative gives

$\displaystyle\ddot{a}={}$
$\displaystyle\dot{a}\left(k_{1}\frac{\dot{a}}{a}-2e\dot{e}\frac{k_{1}-1/2}{1-e^{2}}+2e\dot{e}\frac{k_{2}}{1+k_{2}e^{2}}\right)$
$\displaystyle={}$
$\displaystyle\frac{\dot{a}^{2}}{a}\left(k_{1}+2ae\derivative{e}{a}\left(\frac{1/2-k_{1}}{1-e^{2}}+\frac{k_{2}}{1+k_{2}e^{2}}\right)\right)$
(40)

which immediately results in Eq. (31).

If we now assume that (circular) GW emission is the dominant force with
$\dot{\mathcal{F}}_{\text{vacuum}}\propto\mathcal{F}_{\text{vacuum}}^{11/3}$
[52], and our dissipative force is a small addition to that with
$\dot{\mathcal{F}}_{F}$, we can model this as the frequency evolution being
the sum of both contributions
$\dot{\mathcal{F}}_{\text{tot}}=\dot{\mathcal{F}}_{\text{vacuum}}(1+\varepsilon)$.
A calculation of $\varepsilon$ gives

$\displaystyle\varepsilon={}$
$\displaystyle\frac{\dot{\mathcal{F}}_{F}}{\dot{\mathcal{F}}_{\text{vacuum}}}$
$\displaystyle\propto{}$
$\displaystyle\,\frac{1}{\dot{\mathcal{F}}_{\text{vacuum}}}\left(a^{-1/2}\left\langle\derivative{E_{F}}{t}\right\rangle\right)$
$\displaystyle\propto{}$
$\displaystyle\,\frac{1}{\mathcal{F}_{\text{vacuum}}^{11/3}}\left(a^{-1/2+k_{1}-2}\right)$
$\displaystyle\propto{}$
$\displaystyle\,\mathcal{F}_{\text{vacuum}}^{-11/3-2/3(-5/2-2k_{1}/3)}=\mathcal{F}_{\text{vacuum}}^{-2-2k_{1}/3}$
(41)

where we have assumed that $e=0$ and, in the last step, approximate
$\mathcal{F}_{\text{tot}}\approx\mathcal{F}_{\text{vacuum}}$ such that
$\mathcal{F}_{\text{vacuum}}\propto a^{-3/2}$.

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# Computer Vision Aided Blockage Prediction in Real-World Millimeter Wave
Deployments

Gouranga Charan and Ahmed Alkhateeb
School of Electrical, Computer and Energy Engineering - Arizona State
University
Emails: {gcharan<EMAIL_ADDRESS>
###### Abstract

This paper provides the first real-world evaluation of using visual (RGB
camera) data and machine learning for proactively predicting millimeter wave
(mmWave) dynamic link blockages before they happen. Proactively predicting
line-of-sight (LOS) link blockages enables mmWave/sub-THz networks to make
proactive network management decisions, such as proactive beam switching and
hand-off) before a link failure happens. This can significantly enhance the
network reliability and latency while efficiently utilizing the wireless
resources. To evaluate this gain in reality, this paper (i) develops a
computer vision based solution that processes the visual data captured by a
camera installed at the infrastructure node and (ii) studies the feasibility
of the proposed solution based on the large-scale real-world dataset,
DeepSense 6G, that comprises multi-modal sensing and communication data. Based
on the adopted real-world dataset, the developed solution achieves $\approx
90\%$ accuracy in predicting blockages happening within the future $0.1$s and
$\approx 80\%$ for blockages happening within $1$s, which highlights a
promising solution for mmWave/sub-THz communication networks.

###### Index Terms:

computer vision, deep learning, blockage prediction, mmWave, terahertz.

##  I Introduction

Millimeter wave (mmWave) and sub-terahertz communication systems rely on line-
of-sight (LOS) links to achieve sufficient receive signal power. Blocking
these LOS links by the moving objects in the environment may disconnect the
communication session or cause sudden and significant degaradtion in the link
quality. This is due to the high penetration loss of the mmWave/sub-terahertz
signals and the much less receive power of the NLOS links compared to the LOS
ones [1, 2]. All that highly challenges the reliability and latency of the
mmWave/sub-terahertz communication networks. Initial approaches for overcoming
these blockage challenges relied mainly on multi-connectivity [3, 4]. These
solutions, however, generally keep the user connected to multiple
infastructure nodes which underutilizes the wireless network resources. This
motivated the research for more efficient blockage avoidance approaches.

Leveraging machine learning (ML) to address the blockage challenges has gained
increasing interest in the last few years [5, 6, 7]. In [5], the authors
proposed to leverage recurrent neural networks to process the sequence of
beams serving a mobile user and to predict whether or not a future blockage
will happen. Relying only on beam sequences, however, limits the applications
to stationary blockage prediction. Predicting dynamic blockages require more
information about these moving blockages in the environment. In [6, 7], in-
band mmWave and sub-6GHz based wireless scattering signatures were used to
indentify/predict the incoming mmWave link blockges. These solutions, however,
are mainly capable of predicting immediate blockages and are hard to scale to
complex/crowded scenarios. To enable predicting blockages early enough before
they block the links, solutions based on radar and LiDAR sensory data were
proposed for the first time in [8, 9]. Despite their promising results, easy
sensing modality has its advantages and drawbacks. For examlpe, radar data are
mainly suitable for uncrowded scenarios and LiDAR sensors are expensive and
have relatively short range.

Figure 1: This figure illustrates the overall system model where a mmWave/THz
basestation utilizes the captured RGB images to predict the future link
blockage status.

In [10], we proposed to leverage visual data (captured by cameras) to predict
future dynamic blockages. Ths solutions and analysis in [10], though, were
based only on synthetic datasets, and an important question that arises is
whether the promising results in [10] can be achieved in reality? In this
paper, we attempt to answer this question. In particular, the main
contributions of the paper can be summarized as follows:

  * •

Formulating the vision-aided blockage prediction problem in mmWave/THz
wireless networks considering practical visual and communication models.

  * •

Developing a machine learning approach that is capable of (i) pre-processing
the real-world visual data to enhance the blockage prediction performance,
(ii) extracting the relevant features about the scatterers/environment, and
(iii) efficiently predicting future dynamic link blockages.

  * •

Providing the first real-world evaluation of vision-aided blockage prediction
based on our large-scale dataset, DeepSense 6G [11], that consists of co-
existing multi-modal sensing and wireless communication data.

Based on the adopted real-world dataset, the developed solution achieves
$\approx 90\%$ accuracy in predicting blockages happening within a future
prediction interval of $0.1$s and $\approx 80\%$ for a prediction interval of
$1$s. This highlights the potential of leveraging machine learning and visual
data in addressing the critical LOS link blockage challenges. In particular,
the capability to efficiently predict future blockages enable the network to
make proactive beam/basestation switching decisions which enhances the overall
network reliability/latency performance.

##  II System Model

This work considers a communication scenario where a mmWave basestation is
serving a stationary user located in a busy environment with multiple moving
objects, such as vehicles, pedestrians, etc., as shown in Fig. 1. The mmWave
basestation is equipped with an RGB camera to monitor and gather sensing data
about the surrounding environment. This information could potentially be
leveraged to proactively predict future link blockages caused by the moving
objects.

The adopted system model consists of a mmWave basestation equipped with an
$N$-element antenna array and a standard-resolution RGB camera. The
basestation is serving a stationary user that is, for simplicity, considered
to have a single antenna. The basestation uses a pre-defined beam codebook
$\boldsymbol{\mathcal{F}}=\\{\mathbf{f}_{m}\\}_{m=1}^{M}$ to serve the user,
where $\mathbf{f}_{m}\in\mathbb{C}^{N\times 1}$ and $M$ is the total number of
beamforming vectors in the codebook. As will be described in Section II, the
beamforming codebook adopted by the hardware prototype has $64$ beamforming
vectors (i.e., $M=64$) with the azimuth angles uniformly quantized between
$\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$. The communication system further
adopts OFDM transmission with $K$ subcarriers and cyclic prefix of length $D$.
At any time instant $t$, if the basestation uses the beamforming vector
$\mathbf{f}_{m}\in\boldsymbol{\mathcal{F}}$ to serve the user, then the
downlink received signal at the user at the $k$th subcarrier can be expressed
as

$y_{k}[t]=\mathbf{h}_{k}^{T}[t]{\mathbf{f}}_{m}x[t]+n_{k}[t],$ (1)

where $\mathbf{h}_{k}[t]\in\mathbb{C}^{N\times 1}$ is the channel between the
basestation and the user at the $k$th subcarrier, $x[t]$ is a transmitted data
symbol, ${\mathbb{E}}{\left|x[t]\right|^{2}}=P$, with the average transmit
power $P$, and $n_{k}$ is a receive noise sample,
$n_{k}\sim\mathcal{N}_{\mathbb{C}}(0,\sigma_{n}^{2})$.

LOS Blockage: The channel model $\mathbf{h}_{k}$, defined in (1), is generic
and can be expressed as follows at time instant $t$

$\mathbf{h}_{k}[t]=\left(1-b[t]\right)\mathbf{h}_{k}^{\text{LOS}}[t]+\mathbf{h}_{k}^{\text{NLOS}}[t],$
(2)

where $\mathbf{h}_{k}^{\text{LOS}}$ and $\mathbf{h}_{k}^{\text{NLOS}}$ are the
LOS and NLOS channel components. The binary variable $b[t]\in[0,1]$ represents
the link status at time instant $t$, with $b[t]=1$ indicating that the LOS
path is blocked and $b[t]=0$ otherwise.

It is important to note here that for mmWave and sub-THz communication
systems, the LOS channel gain is much greater than the NLOS channel gain [1,
2]. Therefore, LOS link blockages challenge the reliability of these networks.
Next, we provide a formal definition of the proactive vision-aided blockage
prediction problem which is the focus of this work.

##  III Vision-Aided Blockage Prediction:
Key Idea and Problem Formulation

One of the major challenges in the high-frequency wireless communication
networks is the LOS link blockages; the mmWave/THz communication systems
suffer from link disconnection and significant dips in the received SNR when
an object/blockage intersects the LOS path between the basestation and the
user. Re-establishing a LOS connection is usually done in a reactive way,
which incurs critical latency and impacts the reliability of such systems. The
presence of dynamic moving objects in the environment further increases these
reliability/latency challenges. These challenges could potentially be
addressed if these blockages can be proactively predicted [10, 12, 13]. In
order to develop an efficient solution that can proactively predict the
occurrence of such future blockages, it is essential to equip the wireless
network with a sense of its surroundings. This work attempts to do so by
utilizing machine learning and visual data captured by cameras placed at the
basestation to proactively predict future blockages before they happen. In
this section, we will first present the key idea in Section III-A and then
formulate the vision-aided blockage prediction problem in Section III-B.

Figure 2: This figure presents the proposed architecture for vision-aided
future blockage prediction. The raw visual data is first processed to enhance
the images captured during night or low-light scenario. The processed sequence
of images are then passed through the proposed deep neural network
architecture consisting of a object detection model and recurrent neural
network to predict the future blockage.

###  III-A The Key Idea

In a wireless network, the link blockages are often caused by moving objects,
such as cars, trucks, buses, and humans, present in the wireless environment.
Given the dynamic nature of multiple moving objects in a real-wireless
scenario, the task of future blockage prediction becomes extremely
challenging. While the detection of the different objects in the environment,
such as cars and humans, can be well achieved using a single image, the
success of the dynamic blockage prediction task also relies on characterizing
the mobility patterns and geometric features of these objects. For example,
for a vehicle-to-infrastructure use case, the visual data at the
infrastructure needs to characterize the speed/direction of travel and the
size of the various objects in the environment. In order to capture these
additional indicators, which are normally obtained by analyzing a sequence of
images, our proposed solution observes a sequence of $r$ image samples instead
of making the predictions based on just one sample and attempts to predict
future LOS link blockages before they happen. Building upon the key idea
presented here, in the next sub-section, we provide the formal definitions for
the proactive vision-aided blockage prediction problem.

###  III-B Problem Formulation

The main objective of this work is to observe a sequence of camera image
samples captured at the basestation and utilize the sensing data to predict
whether or not the stationary user will be blocked within a window of future
instances. Let ${\mathbf{X}}[t]\in\mathbb{R}^{W\times H\times C}$ denote a
single RGB image of the environment captured at the basestation at time
instant t, where $W$, $H$, and $C$ are the width, height, and the number of
color channels for the image. At any time instant $\tau\in\mathbb{Z}$, the
basestation uses a sequence of RGB images, ${\mathbf{S}}[\tau]$, defined as

${{\mathbf{S}}}[\tau]=\left\\{{\mathbf{X}}[t]\right\\}_{t=\tau-r+1}^{\tau},$
(3)

where $r\in\mathbb{Z}$ is the length of the input sequence or the observation
window to predict future link blockages. In particular, at any given time
instant $\tau$, the goal in this work is for the basestation to observe
${\mathbf{S}}[\tau]$ and predict whether or not the stationary user is going
to be get blocked within a window of $r^{\prime}$ future instances. It is
important to note here that we do not focus on the exact future instance but
consider the entire future window sequence for denoting the future blockage
status. Given ${{\mathbf{S}}}[\tau]$ and the future window $r^{\prime}$, the
future blockage status at time instant $\tau$ can then be expressed as

$s[\tau]=\left\\{\begin{array}[]{ll}1,&b[t]=1,\
t\in\\{\tau+1,\dots,\tau+r^{\prime}\\}\\\ 0,&\text{otherwise}\\\
\end{array}\right.$ (4)

where $0$ indicates that the user remains LOS within the next $r^{\prime}$
future instances and $1$ points towards the occurrence of blockage within the
same window.

In order to predict the future blockage status, we define a function
$f_{\Theta}$ that maps the observed sequence of images, ${\mathbf{S}}[\tau]$
to a prediction (estimate) of the future blockage status, $\hat{s}_{\tau}$.
The function $f_{\Theta}$ can be formally expressed as

$f_{\Theta}:{\mathbf{S}}[\tau]\rightarrow\hat{s}[{\tau}].$ (5)

In this work, we adopt a machine learning model to learn this prediction
function $f_{\Theta}$, that takes in the observed image sequence and predicts
the future blockage status, $\hat{s}[{\tau}]\in\\{0,1\\}$. Here, $\Theta$
represents the parameters of the machine learning model and is learned from a
dataset of labeled sequences. For this, a dataset of independent sample pairs
$\mathcal{D}=\left\\{\left({\mathbf{S}}_{v},s_{v}\right)\right\\}_{v=1}^{V}$
is collected, where $s_{v}$ is the ground-truth future blockage label for the
observed sequence ${\mathbf{S}}_{v}$, and $V$ is the total number of sequence-
label pairs in the dataset. The labeled dataset $\mathcal{D}$ is then used to
optimize the prediction function $f_{\Theta}$ such that it maintains high
fidelity for any samples drawn from this dataset. The optimization problem can
be written as

$f^{\star}_{\Theta^{\star}}=\underset{f_{\Theta}(.)}{\text{argmax}}\quad\prod_{v=1}^{V}\mathbb{P}(\hat{s}_{v}=s_{v}|{\mathbf{S}}_{v}),$
(6)

where the joint probability in (6) is factored out to convey the identical and
independent (i.i.d.) nature of the samples in dataset $\mathcal{D}$. In the
next section, we present the proposed deep learning-based solution for the
vision-aided future blockage prediction task.

##  IV Vision-Aided Blockage Prediction:
A Deep Learning Solution

Guided by the principles mentioned in Section III-A, the blockage prediction
task is divided into two sub-tasks: (i) object detection and (ii) recurrent
prediction. The first sub-task deals with detecting the relevant objects of
interest in the FoV of the basestation. Given the recent advancements in the
field of computer vision and deep learning, this task can be performed by
utilizing convolutional neural network (CNN)-based object detectors such as
the You Only Look Once (YOLO) model [14]. The first stage consisting of the
object detection models extract relevant features from the sequence of images
and provides this as an input to the next stage of the machine learning
pipeline. The objective of the second stage needs is to predict the future
blockages based on these extracted features. Recurrent Neural Networks (RNNs)
are state-of-the-art machine learning models specifically designed to deal
with such sequential learning problem. Therefore, we adopt a recurrent neural
network in the second stage to learn the underlying key indicators from the
extracted features and predict the future link blockages. In Fig. 2, we
illustrate the proposed deep learning-based blockage prediction solution. In
this section, we first present the details of the image enhancement pre-
processing stage adopted to deal with the low-light/dark images. Then, we take
a deeper dive into the developed two-stage vision-aided blockage prediction
solution.

###  IV-A Data Processing (Image Enhancement)

Compared to LiDAR, radars and other sensing modalities, RGB cameras provide a
low-cost, high-resolution, and low-footprint alternative, making it one of the
preferred choices for wireless sensing applications. However, there is major
bottleneck associated with the visual images captured using an RGB camera.
Under low-light conditions, the visual data turns out to be noisy and dark,
making it unsuitable for further computer vision tasks. Fig. 2 shows an image
captured under such low light conditions. The white truck in the first image
and the red sedan in the second are hardly visible highlighting the challenges
associated with such images. In order to develop a robust and reliable
solution that can work in most of the natural lighting conditions, it is
essential to perform some sort of image enhancement to extract the hidden
details and make the low-light images more usable. For the post-processing
stage, we adopt the state-of-the-art MIRNet [15] model developed for low-light
image enhancement. It is a fully-convolutional architecture that learns an
enriched set of features by combining contextual information from multiple
scales, while simultaneously preserving the high-resolution spatial details.
As shown in Fig. 2, that the objects in the low-light images are clearly
visible after the image enhancement post-processing step, which is important
for the performance of the proposed blockage prediction solution.

Figure 3: This figure shows the data collection setup used for DeepSense 6G
Scenarios 17-22. The figure on the left depicts the street view from Unit 1
perspective. It also highlights the location of the transmitter (Unit 2 TX)
and the receiver (Unit 1 RX) during the data collection process. The figure on
the right shows the exact location of Unit 1 and Unit 2 highlighted on the
Google Earth 3D view.

###  IV-B A Two-Stage Deep Learning Model

Here, we present the details of the proposed blockage prediction architecture,
which consists of two key functions, namely object detection and recurrent
prediction.

Object Detection: The first stage of the proposed solution is the object
detection deep learning model. There are two primary goals of this stage: (i)
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the basestation and (ii) extract the coordinates of the bounding boxes placed
around the relevant objects. For this, in our proposed solution, we adopt the
state-of-the-art YOLO object detection model and more specifically the further
improved YOLOv3 architecture [14]. The YOLOv3 detector is a fast and reliable
end-to-end object detection system, designed for real-time processing of
visual data (images and videos). In this work, instead of training the YOLOv3
object detection model from scratch, we utilize the COCO pre-trained model as
it is already capable of detecting most of the relevant objects present in a
wireless environment.

The pre-trained YOLOv3 architecture is particularly selected for the bounding
box detection task in this work. For each image sample, the pre-trained YOLOv3
is used to detect the relevant objects and extract the bounding box
coordinates of the detected objects. In particular, for each detected object
in the image, we extract a $4$-dimensional vector consisting of the bottom-
left coordinates $[x_{1},y_{1}]$ and the top-right coordinates
$[x_{2},y_{2}]$. These coordinates are normalized to be between $[0,1]$. In
order to account for multiple detected objects in the FoV of the basestation,
the extracted bounding boxes are concatenated to form one dimensional vector
${\mathbf{d}}\in\mathbb{R}^{Y\times 1}$, where $Y$ is the number of objects
detected by the YOLOv3 model. It is important to highlight here that the
number of detected objects might not be the same in each data sample, which
results in a variable length vector ${\mathbf{d}}$. This will lead to
inconsistency in the size of the extracted features and create unnecessary
complications for the next stage of the proposed solution pipeline, i.e., the
recurrent predictions. In order to avoid this inconsistency, the extracted
bounding box vector ${\mathbf{d}}$ is further padded with $Z-Y$ zeros to
obtain a fixed size vector $\tilde{\mathbf{d}}\in\mathbb{R}^{Z\times 1}$. The
fixed size bounding box feature vector $\tilde{\mathbf{d}}$ is then provided
as an input to the recurrent network to predict the future link blockage
status.

Recurrent Prediction: The final stage of the proposed solution utilizes
recurrent neural networks to make the final prediction. In this work, we
consider a two-stage Gated Recurrent Unit (GRU), followed by a fully-connected
layer acting as a classifier. More specifically, the model receives a sequence
of $r$ extracted bounding box feature vectors,
$\\{\tilde{\mathbf{d}}[\tau-r+1,\ldots,\tilde{\mathbf{d}}[\tau]]\\}$, as input
and predicts the future link blockage status over a window of $r^{\prime}$
time instance, $\hat{s}[\tau]$.

Figure 4: This figure plots the future-1, future-5, and future-10 future blockage prediction scores (f1-score) for scenarios 17-22 and the combined scenarios. The combined scenario achieves comparable or better prediction accuracy highlighting the gain of having sufficient diversity in the dataset. TABLE I: Number of Data Sequences in the Development Dataset DeepSense 6G Scenarios | Number of Sequences |  Time of the day
---|---|---
Train | Val. | Test
17 | 4176 | 1190 | 638 | Day + Night
18 | 4524 | 1276 | 667 | Day + Night
19 | 11455 | 3277 | 1653 | Day + Night
20 | 5307 | 1508 | 783 | Day
21 | 1131 | 319 | 203 | Day
22 | 377 | 110 | 55 | Day

##  V Testbed Description and Development Dataset

In order to evaluate the performance of the proposed vision-aided blockage
prediction solution, we adopt multiple scenarios from the DeepSense 6G [11]
dataset. DeepSense 6G is a real-world multi-modal dataset enabling sensing-
aided wireless communication applications. It contains co-existing multi-modal
data such as vision, mmWave wireless communication, GPS data, LiDAR, and
Radar, collected in realistic wireless environments. In this section, we
present a brief overview of the scenarios adopted from the DeepSense 6G
dataset followed by the analysis of the final development dataset utilized in
the blockage prediction task.

TABLE II: Design and Training Hyper-parameters Design | Number of GRUs Per Layer ($r$) | $8$
---|---|---
Embedding Dimension ($Z$) | $30$
Hidden State Dimension | $128$
Number of classes | $2$
Training | Optimizer | ADAM
Learning Rate | $1\times 10^{-3}$
Batch Size | $128$
Number of Training Epochs | $100$

DeepSense 6G: [Scenarios 17 - 22] We adopt Scenarios 17-22 of the DeepSense 6G
dataset for evaluating the efficacy of our proposed solution. The hardware
testbed and the exact location used for collecting these data is shown in Fig.
3. The DeepSense testbed $3$ is utilized for this data collection and is
placed on the opposite sides of a $2$-way street with a passing-lane in-
between. The primary components of the adopted testbed are: (i) A stationary
60 GHz omni-directional mmWave transmitter (unit2), (ii) a directional mmWave
receiver (unit1), (iii) an RGB camera. The receiver employs a 16-element
($N=16$) 60 GHz phased array and it receives the transmitted signal using an
over-sampled beam codebook of $64$ pre-defined beams ($M=64$). Unit 1 is also
equipped with a camera and it captures RGB images for the wireless environment
in the FoV of the receiver. The testbed captures data at 12 samples/sec for
scenarios 17-19 and 6.5 samples/sec for scenarios 20-22. Each data sample
consists of an RGB image of the environment and a 64-element mmWave receive
power vector. For more information regarding the data collected testbed and
setup, please refer to [11, 6].

DeepSense 6G: [Development Dataset] The adopted DeepSense scenarios include
diverse data collected during different times of the day (day and night). Each
row in the dataset scenarios consists of a tuple of an RGB image,
${\mathbf{X}}[\tau]$, and the corresponding receive power vector and the
ground-truth link blockage status, $s[{\tau}]$ (manually labelled). To form
the development dataset of the blockage prediction task described in Section
III-B, the offered DeepSense data is further processed using a sliding window
to generate a time-series dataset consisting of $8$ input image samples
($r=8$) and the corresponding future blockage status in a future window of
$r^{\prime}$ samples (we generate 10 such time-series datasets at
$r^{\prime}=1,2,...,10$. In Table I, we present the number of such sequences
in each scenario. The numbers shown in the table corresponds to an input
sequence length of $8$ and the future prediction window length of $10$. In
order to perform an in-depth study, the development dataset per scenario was
further processed to generate datasets for different future prediction window
size. For example, future-1 dataset, consists of data sequences where the
input sequence length is still $8$, but the future prediction window length is
$1$. During these process of generating datasets with different future
prediction window, we ensure that the dataset is balanced {Number of LOS data
sequences $\approx$ Number of NLOS sequences.}. Each of the development
dataset is further divided into training, validation and test sets following a
split of $70-20-10\%$.

Figure 5: Top-1 blockage prediction accuracy and F1-score for different future
prediction intervals observed in the combined case. It is observed that both
the top-1 accuracy and f1-score decreases as we predict further into the
future.

##  VI Performance Evaluation

In this section, we first discuss the neural network training parameters and
the adopted evaluation metrics. Next, we present the numerical evaluation of
the proposed solution.

Experimental Setup: As described in Section IV, this work adopts a pre-trained
object detection model (YOLOv3) to first extract the coordinates of the
bounding boxes from a sequence of image data. The extracted bounding boxes are
then provided as an input to the recurrent neural network proposed earlier in
Section IV. The GRU model is trained using the labeled development dataset
discussed in Section II using a cross-entropy loss function. All the
simulations were performed on a single NVIDIA Quadro 6000 GPU using the
PyTorch deep learning framework. The detailed design and training hyper-
parameters are presented in Table II. We utilize the top-1 accuracy metric as
the primary method of evaluating the proposed solution. The top-1 accuracy is
defined as follows:

$Acc_{top-1}=\frac{1}{U}\sum_{u=1}^{U}\mathbbm{1}\\{\hat{s}_{u}[\tau]=s_{u}[\tau]\\},$
(7)

where $\hat{s}_{u}[\tau]$ and $s_{u}[\tau]$ are the predicted and ground-truth
link blockage status, respectively. $U$ is the total number of samples present
in the validation/test set. $\mathbbm{1}\\{.\\}$ is the indicator function. In
order to study the robustness of the proposed solution, we also utilize the
F1-score metric.

Can visual data predict LOS blockages? For each of the scenarios 17-22, we
evaluate the proposed solution for various future prediction window lengths.
In Fig. 4, we show the F1-score of the future-1, future-5 and future-10
blockage prediction. Note that ‘future-5’ here represents the development
dataset that considers a future blockage prediction window of length $5$ time
instances (i.e., predicting a blockage that will happen in the future 450ms in
scenarios 17-19 and 750ms in scenarios 20-22). In Fig. 4, we observe that for
all the scenarios, the proposed two-staged solution achieves an accuracy of
$\approx 0.88-0.90$ future-1 and future-5 blockage prediction F1-score,
highlighting the high efficiency of the proposed vision-aided blockage
prediction approach. It is observed that there is a slight degradation in the
model’s performance for scenario 22, which could be attributed to the lower
number of samples in the development dataset of this scenario as shown Table
I; the lower number of training samples can often lead to under-fitting and
impede the model’s capability to learn efficiently.

What is the gain of combining the datasets? To evaluate that, we constructed a
combined dataset by combining the training, validation, and test sequences of
the individual scenarios. This increases the size and diversity of the
dataset. As shown in Fig. 4, the model that is trained based on this combined
dataset is generally achiving better than the models trained on the individual
scenario datasets.

How early can a blockage be predicted? To answer this question, we evaluated
the top1-accuracies and F1-scores for different future prediction window
lengths (based on the combined dataset) in Fig. 5. As shown in this figure,
the proposed approach achieves more than $90\%$ prediction accuracy till the
future-5 prediction interval (an average of 600ms before the blockage
happens). Even though the prediction accuracy and F1-score starts degrading
after the future-5 instance, we observe that the model achieves almost $80\%$
accuracy for predicting up to the $10$th future instance. Accuracy being a
holistic metric may not reflect the intricacies of the blockage prediction
task. To develop a deeper insight into the model’s performance, we plot the
confusion matrices in Fig. 6(a) and Fig. 6(b), for future-1 and future-10
combined predictions, respectively. The high precision of $96\%$ and $78\%$
for both cases further highlights the high efficiency of the proposed
architecture in the future blockage prediction task.

(a) Future-1

(b) Future-10

Figure 6: This figure shows the confusion matrices for future-1 and future-10
blockage prediction interval (based on the combined dataset). It is observed
that the model efficiently differentiates between future LOS and NLOS links.

##  VII Conclusion

This paper explores the potential of leveraging visual sensory data for
proactive blockage prediction in a mmWave communication system. We formulate
the vision-aided blockage prediction problem and develop an efficient machine
learning-based solution to predict future blockages. The key takeaways of
evaluating our proposed vision-aided blockage prediction solution based on the
large-scale real-world dataset, DeepSense, can be summarized as follows: (i)
the vision-aided solution achieves high blockage prediction accuracy of more
than $90\%$ for a shorter prediction window, i.e., for predicting future
moving blockages that are within $600$ ms. (ii) For predicting further into
the future (within one second), the proposed solution achieves an average
prediction accuracy of more than $80\%$. These results highlight the potential
gains of leveraging visual data in predicting future link blockages and enable
proactive network management decisions.

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