# Effect of a High-Precision Semianalytical Mass Function on the Merger Rate
of Primordial Black Holes in Dark Matter Halos

Saeed Fakhry<EMAIL_ADDRESS>Department of Physics, Shahid Beheshti
University, Evin, Tehran 19839, Iran  PDAT Laboratory, Department of Physics,
K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran
Antonino Del Popolo<EMAIL_ADDRESS>Dipartimento di Fisica e
Astronomia, University of Catania , Viale Andrea Doria 6, 95125 Catania, Italy
Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya str. 48,
119017 Moscow, Russia

###### Abstract

In this work, we study the effect of a high-precision semianalytical mass
function on the merger rate of primordial black holes (PBHs) in dark matter
halos. For this purpose, we first explain a theoretical framework for dark
matter halo models and introduce relevant quantities such as halo density
profile, concentration parameter, and a high-precision semianalytical function
namely Del Popolo (DP) mass function. In the following, we calculate the
merger rate of PBHs in the framework of ellipsoidal-collapse dark matter halo
models while considering the DP mass function, and compare it with our
previous study for the Sheth-Tormen (ST) mass function. The results show that
by taking the mass of PBHs as $M_{\rm PBH}=30M_{\odot}$, the DP mass function
can potentially amplify the merger rate of PBHs. Moreover, we calculate the
merger rate of PBHs for the DP mass function as a function of their mass and
fraction and compare it with the black hole mergers recorded by the LIGO-Virgo
detectors during the latest observing run. Our findings show that the merger
rate of PBHs will fall within the LIGO-Virgo band if $f_{\rm
PBH}\gtrsim\mathcal{O}(10^{-1})$. This implies that the DP mass function can
be used to strengthen constraints on the fraction of PBHs.

Primordial Black Hole; Dark Matter; Mass Function; Merger Rate Per Halo

###### pacs:

95.35.+d; 98.80.-k; 04.25.dg; 95.85.Sz

##  I Introduction

The study of gravitational waves (GWs), as cosmological observables, has been
a focus of research for more than a few decades. Hence, in light of GWs and
their detections, a new framework for evaluating astrophysical and
cosmological phenomena has been provided. Binary black hole mergers are
considered as one of the main sources of GWs, whose traces can potentially be
found in the GW detectors Mandel:2021smh ; Olsen:2022pin . There have been
dozens of binary black hole mergers detected by the Laser Interferometer
Gravitational-Wave Observatory (LIGO)-Virgo collaboration in recent years,
which has opened a new door in studying compact objects in the Universe
Abbott:2016blz ; Abbott:2016nmj ; TheLIGOScientific:2016src ; Abbott:2020khf ;
Abbott:2020tfl . Interestingly, binary black hole mergers recorded by the
LIGO-Virgo observatories have been stellar-mass black holes (those with the
mass of $M_{\rm BH}<100M_{\odot}$). This may lead to an informative indication
of the mass distribution of black holes in the Universe as well as their
formation scenario. The formation channel of black holes participating in
merger events recorded by the LIGO-Virgo detectors has not yet been
definitively identified. Their origin may have come through stellar collapse
(possibly via different channels) fishbach2021 ; rodriguez2021 , or they may
have come through the direct collapse of primordial density fluctuations at
the beginning of the Universe bird2016 ; Clesse:2016vqa ; sasaki2016 .

It is fascinating to note that the GW data recorded by the LIGO-Virgo
detectors are well consistent with the merging of primordial black holes
(PBHs). PBHs are one of the potential candidates proposed to describe dark
matter (see e.g., Spergel:1999mh ; MACHO:2000qbb ; Liebling:2012fv ;
Roszkowski:2017nbc ; ADMX:2019uok ; DiGiovanni:2020frc ; DiGiovanni:2021vlu
for other dark matter candidates). The formation of PBHs is predicted to be
due to the nonlinear peaks in primordial density fluctuations. Actually,
primordial density fluctuations may directly collapse into PBHs when they
exceed a threshold value zeldovic1967 ; Hawking:1971ei ; Carr:1974nx . The
evolution of nonlinear density perturbations on superhorizon scales as well as
the threshold amplitude of curvature perturbations for forming black holes
have been considered in several studies niemar1999 ; shibati1999 ;
plonarov2007 ; musco2013 ; young2014 ; bloomfield15 ; allahyari2017 .
Furthermore, PBHs can span a wide range of masses, which makes them distinct
from astrophysical black holes Sasaki:2018dmp .

As a result of their random spatial distribution, PBHs in dark matter halos
might encounter each other and/or other compact objects, form binaries, emit
GWs, and finally merge. In recent years, LIGO-Virgo detectors have detected
GWs triggered by binary black hole mergers LIGOScientific:2018mvr ;
LIGOScientific:2020ibl ; LIGOScientific:2021djp , which has renewed interest
in PBHs. Due to the mass range of black holes associated with binary mergers,
which is often greater than the mass spectrum of astrophysical black holes,
such black holes may have a primordial origin. Since PBHs are possible
candidates for dark matter and can be clustered in dark matter halos, the
structural characteristics of dark matter halos may affect the merger rate of
PBHs. Thus, it has been indicated that some quantities like the halo mass
function, halo concentration parameter, and halo density profile can improve
the accuracy of theoretical models and their prediction of the merger rate of
PBHs Fakhry:2020plg ; Fakhry:2021tzk ; Fakhry:2022uun ; Fakhry:2022zum .

Meanwhile, the halo mass function is one of the most important quantities that
categorizes dark matter halos according to their mass. In other words, the
halo mass function represents the mass distribution of those structures whose
overdensities exceed the threshold value, separate from the expansion of the
Universe, and gravitationally collapse. Therefore, in addition to describing
the nature of dark energy 2011PhRvL.107b1302S ; Yang:2014okp ; Huterer:2017buf
; DiValentino:2019ffd ; Y:2021ybx ; Yengejeh:2022tpa , one of the main
challenges facing any cosmological model is to provide a high-precision mass
function for the dark matter distribution. Using a spherical-collapse model,
Press and Schechter (PS) proposed an analytical model with primordial
fluctuations that evolve from the linear phase to the collapse time ps1974 .
The PS mass function has been proposed as the simplest model, and it is
consistent with observations in many cases. However, at some mass limits, it
quantitatively differs from the numerical results. Consequently, some
improvements have been made to deal with this challenge. Regarding this, one
of the best formalisms was provided by Sheth and Tormen (ST) based on a more
realistic model, which fits the simulation results more accurately
Sheth:1999mn ; Sheth:1999su ; Sheth:2002 . The ST formalism was based on an
ellipsoidal-collapse model with a dynamical threshold value of overdensities.
However, semianalytical approaches indicate that the ST mass function also
overpredicts the number of halos at large masses Reed:2003sq . Moreover, the
study of the evolution of the PS and ST mass functions reveals a worsening of
the situation. It should be noted that the mass functions derived from
simulations lead to convenient approximations in some cases. However, the
black-box nature of simulations, which includes many physical effects, makes
it hard to decipher how these mechanisms affect the shape of the halo mass
function. Therefore, a suitable semianalytical approach is needed to
separately determine the effect of physical processes on the shape of the halo
mass function.

In this work, we investigate the effect of a high-precision semianalytical
mass function on the merger rate of PBHs in dark matter halos. In this
respect, the outline of the work is as follows. In Sec. III, we briefly
discuss an appropriate dark matter halo model and some key quantities such as
the halo density profile, concentration parameter, and a high-precision
semianalytical function known as Del Popolo (DP) mass function. Also, in Sec.
III we calculate the merger rate of PBHs in the ellipsoidal-collapse model
while considering the DP mass function, and compare it with that obtained from
the ST mass function. In addition, we compare the redshift evolution of the
merger rate of PBHs obtained from the DP mass function with the corresponding
results derived from the ST mass function. We also perform our analysis in
terms of various PBH masses and discuss their constraints arising from the
calculations with the DP mass function. Finally, in Sec. IV, we discuss the
results and summarize the findings.

##  II Dark matter halo models

###  II.1 Halo density profile

In the standard model of cosmology, dark matter halos are considered as
nonlinear cosmological structures that have been distributed in the Universe
based on the formation and evolution of hierarchical structures. The
primordial density fluctuations could have exceeded a critical value,
collapsed due to self-gravitational forces, and become qualified to form dark
matter halos. From a physical point of view, such conditions can be described
by a dimensionless quantity known as density contrast, which is derived from
the excursion sets theory. This quantity is defined as
$\delta(r)\equiv(\rho(r)-\bar{\rho})/\bar{\rho}$, where $\rho(r)$ represents
the density of the overdense region at an arbitrary point $r$, and
$\bar{\rho}$ denotes the mean density of the background.

On the other hand, cosmological and structure formation models are constrained
by the inner region of dark matter halos, whose masses can be described by a
radius-dependent function called the density profile. However, as a reliable
criterion, spectroscopic observations of gravitational lensing, x-ray
temperature maps and stellar dynamics in galaxies can predict dark matter
distribution in the central regions of galactic halos Reed:2003hp . In recent
decades, analytical approaches and numerical simulations have been used to
obtain a suitable density profile in such a way that its predictions are
consistent with the relevant observational data ein ; Jaffe:1983iv ;
deZeeuw:1985sk ; Hernquist:1990be ; dehnen ; Navarro:1995iw . Based on
$N$-body simulations in the framework of cold dark matter (CDM) models,
Navarro, Frenk, and White (NFW) presented the following density profile
Navarro:1995iw

$\rho(r)=\dfrac{\rho_{\rm s}}{r/r_{\rm s}(1+r/r_{\rm s})^{2}},$ (1)

where $\rho_{\rm s}=\rho_{\rm crit}\delta_{\rm c}$ is the scaled density of
the halo, $\rho_{\rm crit}$ is the critical density of the background Universe
at a given redshift, and $r_{\rm s}$ is the scale radius of the halo.

On the other hand, through analytical approaches, Einasto found another
suitable definition of the density profile as follows ein

$\rho(r)=\rho_{\rm s}\exp\\{-\dfrac{2}{\alpha}\left[\left(\dfrac{r}{r_{\rm
s}}\right)^{\alpha}-1\right]\\},$ (2)

where $\alpha$ is the shape parameter. The strength of the presented density
profiles is their consistent predictions with the distribution of dark matter
in galactic halos and their rotation curves.

In addition to the density profile, there is another quantity to describe the
central density of galactic halos, called the concentration parameter, which
is defined as follows

$C\equiv\dfrac{r_{\rm vir}}{r_{\rm s}},$ (3)

where $r_{\rm vir}$ is the halo virial radius. The halo virial radius covers a
volume within which the average halo density is $200$ to $500$ times the
critical density of the Universe. Numerical simulations and analytical
approaches imply that to provide a suitable prediction, the concentration
parameter must evolve dynamically with mass and redshift. Many studies have
been conducted to obtain an appropriate concentration parameter prada ; dutton
; okoli ; ludlow . In this regard, one of the successful analytical approaches
to obtain the appropriate mass-concentration-redshift relation is presented in
Ref. okoli , which is structured based on a triaxial collapse.

###  II.2 A high-precision mass function

The standard model of cosmology, also known as the $\Lambda$CDM model, is
successful in fitting a variety of data from intermediate to large scales
(WMAP:2003elm, ; DelPopolo:2007dna, ; 2011ApJS..192…18K, ; delpopolo2013, ;
2013ApJ…779…86S, ; Das:2013zf, ; DelPopolo:2013qba, ; Planck:2015fie, ).

The accurate prediction of the halo mass function111The mass function
represents the dark matter halos mass distribution, or in other terms the
number density of dark matter halos per mass interval, see DelPopolo:2007dna ;
Hiotelis:2005jh ; Hiotelisapp . is another important test of the $\Lambda$CDM
model. The mass function is used to determine the cosmological parameters and
is also of great importance in studying the dark matter distribution and the
formation and evolution of galaxies. Then a simple and accurate high-precision
mass function, that can be applied to different cosmologies and redshifts is a
helpful and valuable asset.

In the simple mass function model proposed by PS, the initial fluctuations are
spherical and have a Gaussian distribution ps1974 . The fluctuation evolution
is followed from the linear phase to the collapse by means of a spherical-
collapse model. At the epoch of virialization, the density contrast,
$\delta(r)$, obtained by means of the linear perturbation theory, has the
value $\delta_{\rm c}\simeq 1.686$ in the case of an Einstein-de Sitter
cosmology. For a density field having a Gaussian probability distribution, the
probability that on a given scale the overdensity exceeds $\delta_{\rm c}$ is
proportional to the number of cosmic structures having a density perturbation
greater than $\delta_{\rm c}$.

Unfortunately, the PS mass function suffers from serious problems. In this
regard, the number of objects in the high mass tail is underpredicted, and the
reverse for the low mass tail of the mass function, see, e.g.,
Efstathiou:1988tk ; Gross:1997wv ; Jenkins:2000bv ; White:2002at . An
improvement in the PS mass function can be obtained by moving from a spherical
to an ellipsoidal collapse Sheth:1999su . While the mass function obtained
with the ellipsoidal collapse is in agreement with ST $N$-body simulations
(Sheth:1999mn, ; Sheth:1999su, ), a deeper analysis of semianalytical models
showed that the ST mass function overpredicts the halo number at large masses
(Lukic:2007fc, ). More in detail the ST mass function gives an overestimation
up to a factor of $\simeq 3$ for the rarest dark matter halos (Reed:2006rw, ;
Lukic:2007fc, ). Comparing with the Bolshoi simulation (2011ApJ…740..102K, ),
at $z=0$ the discrepancy is smaller than $10\%$ in the mass range
$(10^{9}-10^{14})M_{\odot}$. Unfortunately, the discrepancy increases with
redshift and at $z=10$ the ST mass function gives about $10$ times more halos
than simulations.

The extended PS approach, also known as “excursion set formalism”
(1991ApJ…379..440B, ; 1993MNRAS.262..627L, ), often used to model the halo
formation statistics, is based on stochastic processes. In the quoted
formalism the statistics of halos start the statistical properties of the
average overdensity within a window of radius $R_{\rm f}$, namely
$\overline{\delta}(R_{\rm f})$. A Gaussian density field smoothed with a
filter represents the density perturbations. $\overline{\delta}(R_{\rm f})$ vs
$R_{\rm f}$, in a hierarchical Universe is a random walk DelPopolo:2006gn .
When the random walk crosses a threshold value, or barrier, $\delta_{\rm c}$,
a dark matter halo forms. Instead of the filtering radius, sometimes other
quantities are often used, e.g., the mass variance $S$. The first-crossing
distribution is the probability that a random walk first crosses the
threshold, also dubbed “barrier”, between $S$ and $S+dS$. The distribution of
the first crossings defines the “multiplicity function”, which is connected to
the mass function.

The first improvement on the PS threshold was obtained in Ref. Popolo:1998fz ,
finding that the collapse threshold is not constant as in the PS model but
becomes mass dependent. It is given by

$\delta_{\rm cm}=\delta_{\rm c}(z)\left(1+\frac{\beta}{\nu^{\alpha}}\right),$
(4)

where $\alpha=0.585$, and $\beta=0.46$. Also, $\nu=(\delta_{\rm
c}/\sigma)^{2}$, where $\sigma(M,z)$ is the linear root-mean-square
fluctuation of overdensities on a comoving scale including a mass $M$ at
redshift $z$.

In Ref. Sheth:1999su , the threshold is obtained by considering an ellipsoidal
collapse as follows

$\delta_{\rm ec}=\delta_{\rm
c}(z)\left(1+\frac{\beta_{1}}{\nu^{\alpha_{1}}}\right),$ (5)

where $\alpha_{1}=0.615$ and $\beta_{1}=0.485$.

Based on the excursion set formalism, it can be illustrated that Eqs. (4) and
(5) correspond to the following barriers, respectively

$B(M)=\sqrt{a}\delta_{\rm c}(z)\left(1+\frac{\beta}{a\nu^{\alpha}}\right)\;,$
(6)

and

$B(M)_{\rm ST}=\sqrt{a_{1}}\delta_{\rm
c}(z)\left(1+\frac{\beta_{1}}{a_{1}\nu^{\alpha_{1}}}\right)\;.$ (7)

It should be noted that the quoted barriers with an accurate choice of $a$,
and $a_{1}$ give a mass function in agreement with simulations, see
Sheth:1999su ; DelPopolo:2006bk .

To take into account several physical effects, like fragmentation and mergers
(Sheth:1999su, ), tidal torques (Popolo:1998fz, ), dynamical friction, and the
cosmological constant (DelPopolo:2006bk, ), it is necessary to change the
shape of the barrier and add in it the quoted effects. As shown in
DelPopolo:2006shv , an improved threshold, which is directly proportional
through a constant to the barrier is given by

$\displaystyle\delta_{\rm cm2}$ $\displaystyle=\delta_{\rm
co}\left[1+\int_{r_{\rm i}}^{r_{\rm ta}}\frac{r_{\rm ta}L^{2}\cdot{\rm
d}r}{GM^{3}r^{3}}+\frac{\lambda_{\rm o}}{1-\mu(\delta)}+\Lambda\frac{r_{\rm
ta}r^{2}}{6GM}\right]$ $\displaystyle\simeq\delta_{\rm
co}\left[1+\frac{\beta}{\nu^{\alpha}}+\frac{\Omega_{\Lambda}\beta_{2}}{\nu^{\alpha_{2}}}+\frac{\beta_{3}}{\nu^{\alpha_{3}}}\right]\;.$
(8)

In the above equation, $\lambda_{\rm o}=\epsilon_{\rm o}T_{\rm co}$, where
$T_{\rm co}$ is the collapse time of a perturbation without the effect of
dynamical friction (see Appendix A of DelPopolo:2017snx  and Eq. (24) of
1994ApJ…427…72A ). Also, $\epsilon_{o}$ is proportional to the dynamical
friction coefficient $\eta$ (see Appendix A of DelPopolo:2017snx  and Eq. (23)
of 1994ApJ…427…72A ). Moreover, $\mu(\delta)$ is given in Eq. (29) of
Colafrancesco:1994ne  and Appendix A of DelPopolo:2017snx , $\Lambda$ is the
cosmological constant, and $r_{\rm ta}$ is the turn-around radius. The angular
momentum $L$ is calculated as shown in DelPopolo:2006shv ; 2009ApJ…698.2093D ;
2009A&A…502..733D  and in Appendix A of DelPopolo:2017snx . In addition, the
existing constants take values as $\alpha=0.585$, $\beta=0.46$,
$\alpha_{2}=0.4$, $\beta_{2}=0.02$, $\alpha_{3}=0.45$, and $\beta_{3}=0.29$.

The mass function in the excursion set formalism is defined as the comoving
number density of halos in a mass range $(M\mbox{-}M+dM)$

$n(M,z)=\frac{\overline{\rho}}{M^{2}}\left|\frac{d\log{\nu}}{d\log
M}\right|\nu f(\nu)\;,$ (9)

where $\overline{\rho}$ is the background density. The distribution of the
first crossing, namely the “multiplicity function” is indicated with $f(\nu)$.

As shown in Ref. DelPopolo:2017snx , using the barrier given by Eq. (II.2),
taking into account the effects of angular momentum, dynamical friction, and
cosmological constant one can obtain the following relation

$\nu f(\nu)\simeq
A_{2}\sqrt{\frac{a\nu}{2\pi}}l(\nu)\exp{\left\\{-0.4019a\nu^{2.12}m(\nu)^{2}\right\\}},$
(10)

where

$l(\nu)=\left(1+\frac{0.1218}{\left(a\nu\right)^{0.585}}+\frac{0.0079}{\left(a\nu\right)^{0.4}}+\frac{0.1}{\left(a\nu\right)^{0.45}}\right),$

and

$m(\nu)=\left(1+\frac{0.5526}{\left(a\nu\right)^{0.585}}+\frac{0.02}{\left(a\nu\right)^{0.4}}+\frac{0.07}{\left(a\nu\right)^{0.45}}\right).$

Also, relevant constants take values as $A_{2}=0.93702$, and $a=0.707$. By
incorporating Eq. (10) into Eq. (9), one can obtain a high-precision
semianalytical mass function, which we call the Del Popolo (DP) mass function
from now on.

As mentioned earlier, the DP mass function is considered to be a high-
precision function to describe the mass distribution of dark matter halos. In
the following, the relevant arguments for this claim will be discussed. It is
insightful to mention that, a high-precision analytical mass function requires
a precise characterization of the barrier, whose shape depends on the likely
physical effects. Naturally, under such reasoning, one would expect that the
PS mass function with an oversimplified barrier provides a bad fit to the
simulations. In Ref. DelPopolo:2017snx , a relative comparison between
different mass functions Jenkins:2000bv ; Sheth:2002 ; Reed:2003sq ;
2006ApJ…646..881W ; 2007MNRAS.374….2R ; 2010MNRAS.403.1353C ;
2011MNRAS.410.1911C ; 2011ApJ…732..122B ; 2012MNRAS.426.2046A  with the DP
mass function has been performed. This comparison demonstrates that the DP
mass function at $z=0$ is consistent on average to about $3\%$ level with all
other mass functions in the range $-0.55<\log\sigma^{-1}<1.31$, while the
corresponding deviation for other mass functions (e.g., 2011MNRAS.410.1911C ;
2012MNRAS.426.2046A ) is greater than $3\%$. Furthermore, the redshift
evolution of the ratio of the mass function obtained in 2011ApJ…732..122B  to
the DP mass function exhibits a discrepancy of less than $3\%$. On the other
hand, in $z=0$, the ST mass function deviates by about $10\%$ compared to the
simulations, and this deviation changes directly with redshift. In other
words, in $z=6$, the ST mass function predicts the number of halos by $1.5$
times and in $z=10$ by $10$ times more than the simulations DelPopolo:2017snx
. Meanwhile, the DP mass function deviates by less than $3\%$ compared to the
Bolshoi simulations 2011ApJ…740..102K  at different redshifts. In this
comparison, low-mass objects indicate relatively small deviations while high-
mass ends illustrate larger deviations. Accordingly, it can be inferred that
the DP mass function is capable of providing a very good prediction of the
dark matter distribution in galactic halos.

##  III Primordial black hole merger rate

As mentioned earlier, PBHs are attributed to those black holes that could have
formed in the earliest stages of the Universe from the direct collapse of
density fluctuations. Given that PBHs have been randomly distributed in the
Universe, it is expected that they can encounter each other and form binaries.

The first gravitational wave emitted by two merging black holes with similar
masses strengthened the possibility that PBHs could form dark matter.
Accordingly, a Poissonian isocurvature density fluctuation component is
induced by PBHs into the power spectrum of the adiabatic density fluctuations
produced through the inflationary era Afshordi:2003zb . The isocurvature
component arising from PBHs dominates the small scales, leading to significant
modifications in the collapse history, such that it can initially yield much
greater rates of collapse Kashlinsky:2016sdv . Consequently, one can expect
that the initial power spectrum of isocurvature Poissonian fluctuations
induced by PBHs on dark matter is as follows Afshordi:2003zb

$\mathcal{P}_{\rm PBH,i}=n_{\rm PBH}^{-1},$ (11)

where $n_{\rm PBH}$ is the comoving mean density of PBHs. From the formation
time of PBHs to the present day, the isocurvature fluctuations in subhorizon
scales grow by a factor of $3/2(1+z_{\rm eq})$, where $z_{\rm eq}\simeq 4000$
is the redshift corresponding to the matter-radiation equality. Therefore, the
extra power spectrum at redshift $z$ is specified by Afshordi:2003zb

$\mathcal{P}_{\rm PBH}(z)=\dfrac{9}{4}(1+z_{\rm eq})^{2}\,n_{\rm
PBH}^{-1}\,[D(z)]^{-2},$ (12)

where $D(z)$ is the linear growth factor of density fluctuations from redshift
$z$ to the present day. Thus, the total power spectrum on scale $k$ and at
redshift $z$ takes the following form

$\mathcal{P}_{\rm tot}(k,z)=\mathcal{P}_{\rm ad}(k,z)+\mathcal{P}_{\rm
PBH}(z),$ (13)

where $\mathcal{P}_{\rm ad}(k,z)$ is the adiabatic power spectrum of dark
matter. Therefore, the physical effect of the isocurvature power spectrum
caused by PBHs can be evaluated in the linear root-mean-square fluctuation of
overdensities, which has the following form

$\sigma^{2}(M,z)=\dfrac{1}{2\pi^{2}}\int\mathcal{P}_{\rm
tot}(k,z)W(k)k^{2}dk,$ (14)

where $W(k)$ is the top-hat function that depends on the mass $M$ contained in
wavelength $2\pi/k$. Note that in this work, we employ the total power
spectrum $\mathcal{P}_{\rm tot}(k,z)$ in our calculations.

The merger time of PBH binaries that are supposed to form in dark matter halos
relies on the velocity dispersion of the halos (from hours to kiloyears)
OLeary:2008myb . Accordingly, PBH binaries formed via dissipative two-body
encounters have a merger time much shorter than the age of the Universe.
However, nondissipative three-body encounters can also lead to the formation
of PBH binaries in dark matter halos. The binaries formed through such a
channel often do not have sufficient binding energy for the instantaneous
emission of gravitational radiation. Therefore, nondissipative three-body
encounters often yield wide binaries whose merger time is longer than a Hubble
time Quinlan1989 , and naturally, they should not significantly contribute to
the population of BH mergers recorded by the LIGO-Virgo detectors. However,
the Poisson effect implies that PBHs can lead to a possible enhancement of
small-scale structures if they can make a substantial contribution to dark
matter Carr:2018rid . Under these conditions, it is argued that three-body
encounters occurring at high redshifts can make an enormous contribution to
the population of BH mergers associated with the LIGO-Vigo observations, in
such a way that they are comparable to that of two-body dynamical captures
Korol:2019jud ; Trashorras:2020mwn ; Franciolini:2022ewd .

Assume two PBHs with masses $m_{1}$ and $m_{2}$, and relative velocity $v_{\rm
rel}$ at a large distance can encounter each other in dark matter halos.
Consequently, two-body scattering suggests that notable gravitational
radiation propagates at the closest physical separation called the periastron
peters . PBHs will become gravitationally bound and form binary if the emitted
gravitational energy exceeds the kinetic energy of the system. Hence,
periastron has a maximum value as a result of this condition peters

$r_{\rm
mp}=\left[\dfrac{85\pi\sqrt{2}G^{7/2}m_{1}m_{2}(m_{1}+m_{2})^{3/2}}{12c^{5}v_{\rm
rel}^{2}}\right]^{2/7},$ (15)

where $G$ is the gravitational constant and $c$ is the velocity of light. In
addition, the Newtonian limit implies the following relation between the
impact parameter and the periastron Sasaki:2018dmp

$b^{2}(r_{\rm p})=\dfrac{2G(m_{1}+m_{2})r_{\rm p}}{v_{\rm rel}^{2}}+r_{\rm
p}^{2}.$ (16)

Additionally, the tidal forces produced by surrounding compact objects on the
binary can be neglected once $r_{\rm p}\ll b$ is established as strong limits
to gravitational focusing. As a result, the cross-section for the binary
formation can be derived as follows Quinlan1989 ; Mouri2002

$\xi(m_{1},m_{2},v_{\rm rel})=\pi b^{2}(r_{\rm p,max})\simeq\dfrac{2\pi
G(m_{1}+m_{2})r_{\rm p,max}}{v_{\rm rel}^{2}}.$ (17)

Consequently, by inserting Eq. (15) into Eq. (17), one can obtain an explicit
form of the cross-section for the binary formation as

$\xi\simeq
2\pi\left(\dfrac{85\pi}{6\sqrt{2}}\right)^{2/7}\dfrac{G^{2}(m_{1}+m_{2})^{10/7}(m_{1}m_{2})^{2/7}}{c^{10/7}v_{\rm
rel}^{18/7}}.$ (18)

In this work, the events of interest are those consistent with the LIGO-Virgo
sensitivity. Hence, we restrict our analysis to the case where
$m_{1}=m_{2}=M_{\rm PBH}$ is satisfied. In this regard, the binary formation
rate within a galactic halo is given by the following formula bird2016

$\Gamma(M_{\rm vir})=\int_{0}^{r_{\rm vir}}2\pi r^{2}\left(\dfrac{f_{\rm
PBH}\rho_{\rm Halo}}{M_{\rm PBH}}\right)\langle\xi v_{\rm rel}\rangle dr,$
(19)

where $0<f_{\rm PBH}\leq 1$ is the fraction of PBHs that specifies their
contribution to dark matter, $\rho_{\rm halo}$ represents the halo density
profile, and the angle bracket is an average over the PBH relative velocity
distribution in the galactic halo.

Figure 1: Merger rate of PBHs per unit volume and time as a function of halo
mass for Einasto (top) and NFW (bottom) density profiles. Solid (red) lines
show this relation for DP mass function, while dashed (black) lines indicate
it for ST mass function. Figure 2: The ratio of the merger rate of PBHs
derived from the DP mass function to that obtained from the ST mass function
in terms of halo mass.

Based on Ref. prada2012 , one can calculate the halo velocity dispersion by
using its relation to the maximum velocity at $r_{\rm max}$ radius

$v_{\rm disp}=\dfrac{v_{\rm max}}{\sqrt{2}}=\sqrt{\dfrac{GM(r<r_{\rm
max})}{r_{\rm max}}}.$ (20)

Assuming a cutoff at the halo virial velocity and $v_{\rm rel}=v_{\rm PBH}$,
we demand that the relative velocity distribution of PBHs in the galactic halo
corresponds to the Maxwell-Boltzmann statistics:

$P(v_{\rm PBH},v_{\rm disp})=A_{0}\left[\exp\left(-\dfrac{v_{\rm
PBH}^{2}}{v_{\rm disp}^{2}}\right)-\exp\left(-\dfrac{v_{\rm vir}^{2}}{v_{\rm
disp}^{2}}\right)\right],$ (21)

where $A_{0}$ is specified when the condition $4\pi\int_{0}^{v_{\rm
vir}}P(v)v^{2}dv=1$ is satisfied.

There are two distinct mechanisms for the PBH binary formation. We mainly
focus on the PBH binaries formed in dark matter halos in the late-time
Universe. However, the initial clustering may lead to the separation of PBHs
from the Hubble flow and yield the formation of PBH binaries in the early
Universe 2017PhRvD..96l3523A ; 2020PhRvD.101d3015V . As PBH binaries form in
the early Universe, they emit gravitational waves continuously, slowly shrink,
and ultimately merge. Nevertheless, some of them would disrupt, due to tidal
forces of surrounding PBHs, before the completion of the merger phase
2018PhRvD..98b3536K ; 2019JCAP…02..018R . In fact, the orbital parameters of
the formed binaries determine their merger time. Given that PBHs are randomly
distributed in the Universe, the orbital parameters for each formed binary
will be unique. Therefore, some binaries have already merged, some of them
merge in the present-time Universe, and some will merge in the future. Thus,
today’s merger rate will be enormously increased by those PBH binaries
expected to merge in the present-time Universe. Therefore, the LIGO-Virgo
observations can be justified by the fact that PBHs must constitute a very
small fraction of dark matter in the mentioned mechanism of PBH binary
formation 2020PhRvD.102l3524H ; 2021JCAP…03..068H ; 2022PhLB..82937040C ;
2022PhRvD.106l3526F . However, it should be noted that both mechanisms are
still valid and their predictions of the contribution of PBHs to dark matter
are still being validated through gravitational wave detectors.

The total merger event rate per unit time and volume can be specified by
convolving the merger rate of PBHs in every single dark matter halo with the
halo mass function

$\mathcal{R}=\int_{M_{\rm c}}\frac{dn}{dM_{\rm vir}}\Gamma(M_{\rm vir})dM_{\rm
vir},$ (22)

where $M_{\rm vir}$ is the halo virial mass, and $M_{\rm c}\simeq
400M_{\odot}$ is the minimum mass of dark matter halos whose signals can reach
the present-time Universe while containing PBHs with the mass of $M_{\rm
PBH}=30M_{\odot}$ (see Fakhry:2020plg ; Fakhry:2021tzk ; Fakhry:2022uun ;
Fakhry:2022zum  for more details). Due to the exponential decreasing term in
the halo mass function, the upper limit of integration is not significant in
determining the merger rate of PBHs. On the other hand, in low-mass halos,
dark matter density is expected to be higher than that in high-mass halos,
which is consistent with the hierarchical dynamics of halo formation.
Accordingly, the lower limit of integration plays a crucial role in the
current analysis. To calculate the merger rate of PBHs within the context of
the current analysis, we consider the fraction of PBHs as $f_{\rm PBH}=1$.

In Fig. 1, we have indicated the merger rate of PBHs per unit time and volume
for the DP mass function and compared it with that obtained for the ST mass
function Fakhry:2020plg  while taking into account Einasto and NFW density
profiles. To perform these calculations, relevant concentration-mass-redshift
relations obtained in Ref. okoli , hereafter the Okoli-Afshordi concentration-
mass-redshift relations, have been used. According to this figure, the merger
rate of PBHs has an enhancement in smaller halos when considering the DP mass
function as compared to the ST mass function. Meanwhile, in larger halos, the
merger rate of PBHs for the DP mass function is lower than the corresponding
result obtained from the ST mass function. Based on the analysis concerning
the importance of the minimum mass ends in the PBH scenario, it can be
inferred that subhalos can potentially play a decisive role in the merger rate
of PBHs. Therefore, it can be concluded that considering the high-precision DP
mass function may lead to the strengthening of the merger rate of PBHs.

Figure 3: The evolution of the total merger rate of BHs as a function of
redshift for Einasto (top) and NFW (bottom) density profiles. Solid (red)
lines exhibit this relation for the DP mass function, while dashed (black)
lines indicate it for ST mass function. Figure 4: Total merger event rate of
PBHs while considering the DP mass function with respect to the PBH fraction
and mass. The solid (black), dashed (red), and dot-dashed (blue) lines
demonstrate this relation for a PBH mass of $M_{\rm PBH}=10,30$, and
$100M_{\odot}$, respectively. The shaded (green) band represents the total
merger rate of PBHs estimated by the LIGO-Virgo detectors during the latest
observing run, i.e., $(17.9\mbox{-}44)\,{\rm Gpc^{-3}yr^{-1}}$.

In Fig. 2, we have quantified the relative enhancement of the merger rate of
PBHs in terms of halo mass for DP and ST mass functions in the present-time
Universe. As can be seen from this figure, the relative enhancement of the
merger rate of PBHs by considering the DP mass function occurs for halo masses
less than $\sim 10^{12}M_{\odot}$, while a relative reduction of the merger
rate is observed for masses greater than $\sim 10^{12}M_{\odot}$. Even though
in the cumulative analysis, the relative reduction of the merger rate of PBHs
in cluster-sized halos is negligible, this is likely due to the repulsive
effect of the cosmological constant, which opposes accretion and merging
processes at a large scale. Accordingly, as expected, the physical effects
that directly affect the shape of the mass function lead to the strengthening
of the merger rate of PBHs. In addition, assuming the PBH mass to be
$30M_{\odot}$, the amplification of the PBH merger rate when taking into
account the DP mass function is in the range of $(106\pm 13)\%$.

One of the most interesting topics to study has always been the evolution of
the merger rate of BHs. It is also possible to detect GWs at higher redshifts
as the precision of relevant instruments can be improved. Currently, the LIGO-
Virgo detectors are capable of detecting binary mergers up to $z\simeq 0.75$,
which approximately corresponds to a comoving volume of around $50\,{\rm
Gpc^{3}}$. Regarding this, through the halo mass function and the
concentration parameter, Eq. (22) is dependent on redshift. In Fig. 3, we have
depicted the redshift evolution of the merger event rate of PBHs for NFW and
the Einasto density profiles, wherein the relevant results of DP and ST mass
functions have been compared. It is evident that changes in redshift directly
affect the merger rate of PBHs. This is because halo merger trees and
hierarchical dynamics could have led to more subhalos at higher redshifts. As
a result, PBH merger rates in the past were higher than those in the present-
time Universe. Moreover, it can be concluded that the merger rate of PBHs for
the DP mass function is higher than the corresponding results obtained from
the ST mass function in the whole range of the late-time Universe. This is
where the importance of a high-precision mass function comes into play because
it could potentially provide a more accurate prediction of the accumulation of
PBH mergers in the late-time Universe.

Table 1: Total merger rate of PBHs for different PBH masses, i.e., $M_{\rm pbh}=10,20,30,50,$ and $100\,M_{\odot}$, considering the ellipsoidal-collapse halo models in terms of the NFW and the Einasto density profiles, at the present-time Universe $(z=0)$. Amplification percentages and the merger rate of PBHs have been provided while taking into account the DP and ST mass functions. PBH Mass $(M_{\odot})$ | Density Profile | $\mathcal{R}_{\rm DP}(\rm Gpc^{-3}\rm yr^{-1})$ | $\mathcal{R}_{\rm ST}(\rm Gpc^{-3}\rm yr^{-1})$ | Amplification Percentage
---|---|---|---|---
10 | NFW | $79.46$ | $33.41$ | $137\%$
10 | Einasto | $107.5$ | $51.25$ | $109\%$
20 | NFW | $49.60$ | $22.38$ | $121\%$
20 | Einasto | $68.58$ | $35.01$ | $96\%$
30 | NFW | $33.01$ | $15.06$ | $119\%$
30 | Einasto | $46.51$ | $24.03$ | $93\%$
50 | NFW | $25.01$ | $11.49$ | $117\%$
50 | Einasto | $35.70$ | $18.60$ | $91\%$
100 | NFW | $17.64$ | $8.18$ | $115\%$
100 | Einasto | $25.60$ | $13.48$ | $89\%$

So far, our analysis was carried out assuming the mass of the PBHs to be
$30M_{\odot}$, and their contribution to the dark matter to be maximum.
However, it is interesting to calculate the merger rate of PBHs according to
their fractions and masses. In Fig. 4, we have illustrated the merger rate of
PBHs for the DP mass function in terms of the upper bounds on the fraction of
PBHs and their masses. In this calculation, the Einasto density profile is
considered. Moreover, the shaded band indicates the total merger rate of BHs
estimated by the LIGO-Virgo detectors during the latest observing run, i.e.,
$(17.9\mbox{-}44)\,{\rm Gpc^{-3}yr^{-1}}$ LIGOScientific:2021djp . It can be
observed that the merger rate of PBHs is inversely proportional to their
masses but directly proportional to their fractions. This is because the
number density of PBHs changes inversely with their masses. In addition,
despite the theoretical uncertainties, the merger rate of PBHs for the DP mass
function will fall within the LIGO-Virgo sensitivity if $f_{\rm
PBH}\gtrsim\mathcal{O}(10^{-1})$. Meanwhile, compared to the corresponding
results derived from the ST mass function, the present analysis can
potentially improve the constraints on the abundance of PBHs.

In Table 1, we have provided the relevant results of the merger rate of PBHs
in terms of various PBH masses, NFW and Einasto density profiles, and DP and
ST mass functions. Furthermore, we have presented the relative amplification
of the merger rate of PBHs while considering the DP mass function.
Specifically, in comparison with our previous study Fakhry:2020plg , the
results indicate that considering the DP mass function can lead to a relative
strengthening of the merger rate of stellar-mass PBHs. Additionally, the
results exhibit that the relative enhancement of the merger rate of PBHs
changes inversely with their masses. For example, in the mass range of $M_{\rm
PBH}=(10\mbox{-}100)M_{\odot}$, the enhancement of the merger rate of PBH with
the mass of $10M_{\odot}$ is in the range of $(123\pm 14)\%$, while the
corresponding result for PBHs with the mass of $100M_{\odot}$ takes place in
the range of $(102\pm 13)\%$.

##  IV Conclusions

In the standard model of cosmology, also known as the $\Lambda$CDM model, dark
matter halos are fundamental units of cosmological structures that form and
evolve hierarchically based on the cosmological perturbation theory.
Observational data show that dark matter halos have been distributed randomly
throughout the Universe. Therefore, they can be analyzed from a statistical
point of view. In the classification of dark matter halos, special attention
should be paid to their local quantities. To this end, the halo mass function
plays a crucial role. Hence, it is of fundamental importance to determine a
high-precision halo mass function. Using numerical simulations, several
attempts have been carried out to determine the exact shape of the halo mass
function, which have led to appropriate predictions. However, the black-box
nature of numerical simulations makes it difficult to distinguish the role of
physical mechanisms influencing the shaping of the halo mass function. Based
on this, semianalytical approaches can more accurately demarcate physical
effects in explaining the statistical properties of dark matter halos.

In this work, we investigate the effect of a high-precision semianalytical
mass function on the merger rate of PBHs in dark matter halos. For this
purpose, we have initially introduced the relevant theoretical framework for
dark matter halo models. Dark matter distribution in galactic halos can be
well described by the halo density profile. Another crucial factor that
represents the dark matter density in the central region of halos is known as
the concentration-mass-redshift relation. In this work, we have used NFW and
Einasto density profiles as well as Okoli-Afshordi concentration-mass-redshift
relations. Furthermore, we have introduced the semianalytical DP mass
function, which makes use of tidal torque, dynamical friction, and the
cosmological constant to shape the halo mass function and increase its
precision.

Relying on the PBH scenario and in the framework of an ellipsoidal-collapse
dark matter halo model, we have calculated the merger rate of PBHs considering
the DP mass function and compared it qualitatively and quantitatively with the
relevant results of the ST mass function. Our findings demonstrate that,
compared to that obtained for the ST mass function, the merger rate of PBHs
with the DP mass function experiences a relative amplification for halos
smaller than $10^{12}M_{\odot}$, while it has a relative reduction for those
halos grater than $10^{12}M_{\odot}$. Given that smaller halos have a
prominent contribution to the merger rate of PBHs, it can be argued that using
the DP mass function leads to the amplification of the merger rate of PBHs. In
this regard, taking the mass of PBHs as $M_{\rm PBH}=30M_{\odot}$, the present
analysis estimates the amplification of the merger rate to be in the range of
$(106\pm 13)\%$. One has to mention that in these calculations, Poissonian
fluctuations caused by PBHs play a crucial role in strengthening the power
spectrum of dark matter on small scales, and naturally strengthening the
merger rate of PBHs, which is not considered in the previous work associated
with the ST mass function.

We have also calculated the redshift evolution of the merger event rate of
PBHs for the DP mass function and compared it with the corresponding result
obtained from the ST mass functions. The results exhibit that the merger rate
of PBHs for the DP mass function is higher than that derived from the ST mass
function during the late-time Universe. This is due to the importance of a
high-precision mass function that can provide more accurate predictions of PBH
mergers.

Finally, we have calculated the merger rate of PBHs for the DP mass function
as a function of their mass and fraction and compared the results with the BH
mergers estimated by the LIGO-Virgo detectors during the latest observing run,
i.e., $(17.9\mbox{-}44)\,{\rm Gpc^{-3}yr^{-1}}$. Our findings indicate that
the merger rate of PBHs is inversely proportional to their mass but directly
proportional to their fraction. Based on this argument, the results show that
the merger rate of PBHs can be in the range of the LIGO-Virgo sensitivity if
$f_{\rm PBH}\gtrsim\mathcal{O}(10^{-1})$. On the other hand, compared to the
results obtained for the ST mass function, it is deduced that the constraints
on the fraction of PBHs can be strengthened while considering the DP mass
function. Moreover, our results show that the relative amplification of the
merger rate of PBHs changes inversely with their masses.

Although the current analysis improves our previous results of the merger rate
of PBHs in dark matter halos, the conditions considered therein are not
necessarily satisfied. Hence, the analysis presented in this work may include
theoretical uncertainties. For example, possible three-body encounters as more
complicated channels for the binary formation, and binary disruption by tidal
forces of surrounding compact objects. It is hoped that these uncertainties
can be minimized in the near future.

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#  : Automated Simulations of Adversarial Attacks on
Arbitrary Objects in Realistic Scenes

Matthew Hull1, Zijie J. Wang1, Duen Horng (Polo) Chau1

###### Abstract

Deep Learning models, such as those used in an autonomous vehicle are
vulnerable to adversarial attacks where an attacker could place an adversarial
object in the environment, leading to mis-classification. Generating these
adversarial objects in the digital space has been extensively studied, however
successfully transferring these attacks from the digital realm to the physical
realm has proven challenging when controlling for real-world environmental
factors. In response to these limitations, we introduce , an easy-to-use
Python library that is the first-of-its-kind tool for creating attack
scenarios with arbitrary objects and simulating realistic environmental
factors, lighting, reflection, and refraction. enables researchers and
practitioners to swiftly explore various scenarios within the digital realm by
offering a wide range of configurable options for designing experiments and
using differentiable rendering to reproduce physically plausible adversarial
objects. We will demonstrate and invite the audience to try to produce an
adversarial texture on a chosen object while having control over various scene
parameters. The audience will choose a scene, an object to attack, the desired
attack class, and the number of camera positions to use. Then, in real time,
we show how this altered texture causes the chosen object to be mis-
classified, showcasing the potential of in real-world scenarios. is open-
source and available at https://github.com/poloclub/revamp.

## Introduction

Fig. 1:  allows user to easily perturb textures on arbitrary objects with a
variety lighting and environmental settings in realistic scenes. Top Row:
Benign examples for 3 different scenes. Bottom Row: Perturbed texture maps on
an underwater cube with light distortions perturbed as an airplane, a mailbox
on a city street attacked to appear as a stop sign, and a mesa attacked to
appear as a bus. Fig. 2: ’s pipeline can be run with a simple command,
allowing a user to choose a scene, an object to attack, the desired attack
class, and the number of camera positions to use. Differentiable rendering is
used to iteratively render and update the texture of the attacked object.

It is widely recognized that deep neural networks employed in image
recognition and object detection tasks are vulnerable to adversarial attacks
(Goodfellow, Shlens, and Szegedy 2015). While these attacks were initially
produced in the digital space, there is growing interest in exploring the
feasibility of physically realizable adversarial attacks (Chen et al. 2018).
Subsequent work explored various methods to create physical adversarial
objects, including printing adversarial textures onto objects (Chen et al.
2018), projection mapping (Huang and Ling 2022), and marker and sticker
placement (Song et al. 2018). However, transferring successful attacks from
the digital realm to the physical realm remains challenging. Techniques like
Expectation over Transformation (EoT) (Athalye et al. 2018) focus mainly on
limited scaling and rotation transformations, often neglecting real-world
factors such as environmental conditions and lighting.

Experimentation in 3D environments that realistically portray the real world
has strong potential to address the these limitations (Xiao et al. 2019).
However, creating such environments require expertise in 3D modeling and
rendering. Furthermore, traditional 3D rendering lacks support for optimizing
scene parameters and textures due to non-differentiable operations.

To address these challenges, our research make the following major
contributions:

  * •

We introduce , the first-of-its kind open-source Python library for easily
creating attack scenarios with arbitrary objects in realistic scenes. empowers
researchers and practitioners to swiftly explore various attack scenarios,
offering a wide range of configurable options for designing experiments in a
customizable adversarial attack pipeline for creating 3D objects. uses
differentiable rendering, which propagates gradients through the rendering to
optimize the scene parameters to induce successful attacks (Jakob et al.
2022). With , users can select from scenes that replicate real-world
scenarios, or create their own. They can designate textures and parameters to
craft tailored adversarial examples. Our library of scenes support realistic
lighting effects like reflection and refraction, and other realistic
environmental factors (Fig.1). stands for Reshaping Environment by Varying
Adversarial Model’s Parameters.

  * •

Open-Source Implementation: While some prior works (Liu et al. 2019; Xiao et
al. 2019) have explored differentiable rendering for adversarial examples,
they are closed-source. is open-source,111 code:
https://github.com/poloclub/revamp facilitating collaboration and reproducible
research for machine learning (ML) security researchers and practitioners.

## System Design and Implementation

uses the the Hydra configuration management tool (Yadan 2019) to construct and
customize scenarios, providing flexibility by allowing parameter adjustments
through the command line. Each scenario is provided as a YAML configuration
file including a 3D scene, an attackable parameter, render settings, and a
victim model. For example, in a “city” scenario, a camera is focused on a
mailbox, with its texture as the “attackable” parameter (Fig.2). The scenario
configuration specifies the inference model, optimized parameter(s), and
attack type (targeted or untargeted). Camera transformations, such as
orbiting, can also be configured, offering diverse perspectives for exploring
adversarial impacts. Importantly, employs the Mitsuba differentiable renderer
(Jakob et al. 2022) with ray tracing to precisely simulate real-world
phenomena like reflection, shadows, and refraction. This enhances scene
realism and accuracy, enabling more precise evaluation of adversarial attacks
(Fig.2).

## Usage Scenario

Jim is an ML security engineer at a self-driving car company. He wants to
leverage to assess and enhance the object detector’s robustness for potential
deployment in self-driving cars. Interested in investigating how a large
object in a street environment might be misclassified e.g., a stop sign, if
manipulated adversarially, Jim selects a city scene and designates the
mailbox’s texture as the attackable parameter.

To conduct this simulation, Jim executes ’s pipeline with a simple command
(Fig.2). At each step of differentiable rendering, the resulting scene image
is forwarded to the object detector, and the texture is iteratively perturbed
using a PGD $\ell_{2}$ attack (Madry et al. 2018) until a “stop sign”
detection is achieved. outputs rendered images, object detection bounding
boxes, class labels, and perturbed texture maps for each rendering step. Jim
notes substantial perturbation was required before a “stop sign” detection
with low confidence was achieved.

Following the pipeline run, Jim has gained confidence that the object detector
is resistant against a strong perturbation, and proceeds to use the output
texture maps to conduct adversarial training to further strengthen model
robustness. The ease of use of also encourages Jim to quickly expand his
experiment to more realistic scenes such as open highways or unmarked roads in
wooded areas.

## Conclusion

We present , an open-source, easy-to-use Python library that helps ML
researchers create attack scenarios with arbitrary objects while simulating
environmental factors and lighting with differentiable rendering. helps users
explore these scenarios in the digital realm to help close the gap between
digital and physically plausible attacks.

## References

  * Athalye et al. (2018) Athalye, A.; Engstrom, L.; Ilyas, A.; and Kwok, K. 2018.  Synthesizing Robust Adversarial Examples.  In Dy, J.; and Krause, A., eds., _Proceedings of the 35th International Conference on Machine Learning_ , volume 80 of _Proceedings of Machine Learning Research_ , 284–293. PMLR.
  * Chen et al. (2018) Chen, S.-T.; Cornelius, C.; Martin, J.; and Chau, D. H. P. 2018.  ShapeShifter: Robust Physical Adversarial Attack on Faster R-CNN Object Detector.  In _Joint European Conference on Machine Learning and Knowledge Discovery in Databases_ , 52–68. Springer.
  * Goodfellow, Shlens, and Szegedy (2015) Goodfellow, I. J.; Shlens, J.; and Szegedy, C. 2015.  Explaining and Harnessing Adversarial Examples.  In Bengio, Y.; and LeCun, Y., eds., _3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings_.
  * Huang and Ling (2022) Huang, B.; and Ling, H. 2022.  SPAA: Stealthy Projector-based Adversarial Attacks on Deep Image Classifiers.  In _2022 IEEE Conference on Virtual Reality and 3D User Interfaces (VR)_ , 534–542.
  * Jakob et al. (2022) Jakob, W.; Speierer, S.; Roussel, N.; Nimier-David, M.; Vicini, D.; Zeltner, T.; Nicolet, B.; Crespo, M.; Leroy, V.; and Zhang, Z. 2022.  Mitsuba 3 renderer.  Https://mitsuba-renderer.org.
  * Liu et al. (2019) Liu, H.-T. D.; Tao, M.; Li, C.-L.; Nowrouzezahrai, D.; and Jacobson, A. 2019.  Beyond Pixel Norm-Balls: Parametric Adversaries using an Analytically Differentiable Renderer.  In _International Conference on Learning Representations_.
  * Madry et al. (2018) Madry, A.; Makelov, A.; Schmidt, L.; Tsipras, D.; and Vladu, A. 2018.  Towards Deep Learning Models Resistant to Adversarial Attacks.  In _International Conference on Learning Representations_.
  * Song et al. (2018) Song, D.; Eykholt, K.; Evtimov, I.; Fernandes, E.; Li, B.; Rahmati, A.; Tramèr, F.; Prakash, A.; and Kohno, T. 2018.  Physical Adversarial Examples for Object Detectors.  In _12th USENIX Workshop on Offensive Technologies (WOOT 18)_. Baltimore, MD: USENIX Association.
  * Xiao et al. (2019) Xiao, C.; Yang, D.; Li, B.; Deng, J.; and Liu, M. 2019.  MeshAdv: Adversarial Meshes for Visual Recognition.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_.
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