∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: Department of Physics, Indian Institute of Technology,
Hyderabad 502285, India

# BBN Constraints on $f(Q,T)$ Gravity

Snehasish Bhattacharjee  ID e1,addr1

###### Abstract

$f(Q,T)$ gravity is a novel extension of the symmetric teleparallel gravity
where the Lagrangian $L$ is represented through an arbitrary function of the
nonmetricity $Q$ and the trace of the energy-momentum tensor $T$ fqt . In this
work, we have constrained a widely used $f(Q,T)$ gravity model of the form
$f(Q,T)=Q^{n+1}+mT$ from the primordial abundances of the light elements to
understand its viability in Cosmology. We report that the $f(Q,T)$ gravity
model can elegantly explain the observed abundances of Helium and Deuterium
while the Lithium problem persists. From the constraint on the expansion
factor in the range $0.9425\lesssim Z\lesssim 1.1525$, we report strict
constraints on the parameters $m$ and $n$ in the range $-1.13\lesssim
n\lesssim-1.08$ and $-5.86\lesssim m\lesssim 12.52$ respectively.

###### Keywords:

Big-Bang Nucleosynthesis $f(Q,T)$ Gravity

###### pacs:

04.50.kd.

##  1 Introduction

The $\Lambda$CDM cosmological model has been remarkably successful in
expounding the dynamics and evolution of the Universe from seconds after the
big bang to the present accelerated expansion. However, the model presumes the
Universe is statistically homogenous and isotropic at large scales, and that
the two mysterious entities termed the dark matter and the dark energy exists
in substantial proportions, and that the law of gravity is well described by
General Relativity (GR). Nevertheless, the model is incomplete, is evident
from the fact that hitherto, no conclusive evidence has been found to confirm
the existence of dark matter and dark energy. Furthermore, the model cannot
explain the existence of matter-antimatter asymmetry and it is becoming
increasingly difficult to incorporate GR into quantum field theory to get a
complete theory of reality 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 . Moreover,
new models beyond the $\Lambda$CDM model are being continuously proposed to
alleviate numerous tensions between observational data at different scales 11
; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 .
With this in mind, many alternatives have surfaced which aim to explain the
effects of dark matter and dark energy by modifying or extending GR. Extended
theories of gravity are motivated by the fact that Einstein’s GR does not
provide a complete understanding of several gravitational effects at the UV
and IR scales. In addition to introducing theories such as the $f(R)$ gravity,
which are essentially simple geometric extensions of GR, many alternative
theories of gravity identify the gravitational field to be best described by
variable(s) other than the Ricci scalar. One of the cornerstones of GR and
many extended theories of gravity is the assumption of the Equivalence
principle 1  which leads towards coinciding the causal structure and the
geodesic and secures the Levi-Civita connection ift .
An approach to understanding gravity employing variables other than the Ricci
scalar could turn out to be useful. For instance, the Ricci scalar could be
replaced by the torsion scalar if the gravitational interactions are expressed
in terms of tetrads. Such a theory of gravity is commonly termed teleparallel
gravity, in which the affinities and not the Equivalence principle plays the
most important role. The most successful formulation of a teleparallel gravity
is the $f(T)$ gravity which replaces the Ricci scalar $R$ in the Einstein-
Hilbert action with the Torsion scalar $T$.
Recently, fqt  proposed a novel extension of the symmetric teleparallel
gravity called the $f(Q,T)$ gravity for which the Lagrangian $L$ is
represented through an arbitrary function of the non-metricity $Q$ and the
trace of the energy-momentum tensor $T$. The gravitational equations are
obtained through the variation of both the metric and the connection with the
coupling between matter and geometry accompanying non-conservation of the
energy-momentum tensor fqt . $f(Q,T)$ gravity has been employed to
successfully explain the matter-antimatter asymmetry b1  and the late-time
acceleration b2 .
These alternative theories of gravity are formulated with the primary
objective of being self-consistent and furnish a complete picture of the
evolution of the Universe from the early phase of inflation to the period of
structure formation to the late-time acceleration (readers may refer to20 ; 21
; 22  a comprehensive discussion). Cosmography 23 ; 24  and Big-Bang
Nucleosynthesis (BBN) 25  are capable of providing powerful constraints on
several of these alternate cosmological models. In particular, since the
chemical abundances of the primordial light elements such as Deuterium ($D$),
Helium ($He$), and Lithium ($Li$) have been ascertained to high precision, BBN
offers stringent constraints on alternate theories of gravity since modified
theories of gravity must explain the abundances of these metals to validate
their applicability and efficiency.
In this work, we plan to constrain $f(Q,T)$ gravity from the primordial
abundances of the light elements to understand its viability in cosmology. The
abundances of the aforementioned light elements have been estimated through
various observational techniques. The abundance of Deuterium has been
estimated from the absorption lines of gas clouds 34 ; 35 ; 36 ; 37 ; 38 ,
while that for the Helium from the emission lines of the nearby ionized
Hydrogen regions in metal-poor starburst galaxies 30 ; 31  and finally for
Lithium from the atmospheres of very metal-poor stars 32 ; 33 . This technique
has been successfully exercised to constrain scalar-tensor gravity e32 ,
$f(T)$ gravity 25 ; ift , $f(R)$ gravity e28 ; e29 ; e30 , $f(R,T)$ gravity b3
, Brans-Dicke cosmology with varying $\Lambda$ e33 , massive gravity e34  and
higher dimensional dilation gravity e2 .
The manuscript is organized as follows: In Sec 2 we provide an overview of
$f(Q,T)$ gravity. In Sec 3, we delineate the concept of Big Bang
Nucleosynthesis and constrain the model parameters of the $f(Q,T)$ gravity
model, and in Sec 4 we present the conclusions.

##  2 Overview of $f(Q,T)$ Gravity

The action in $f(Q,T)$ gravity reads fqt

$S=\frac{1}{16\pi}\int\sqrt{-g}\left[f(Q,T)+\mathcal{L}_{M}\right]d^{4}x$ (1)

where $g$ denote the metric scalar, and $\mathcal{L}_{M}$ denote the matter
Lagrangian.
Varying (1) with respect to the metric tensor generates the following field
equation fqt

$8\pi
T_{\epsilon\varepsilon}=f_{T}\left(T_{\epsilon\varepsilon}+\Theta_{\epsilon\varepsilon}\right)+f_{Q}\left(2Q^{\alpha\beta}_{\epsilon}P_{\alpha\beta\varepsilon}-P_{\epsilon\alpha\beta}Q_{\varepsilon}^{\alpha\beta}\right)-\frac{2}{\sqrt{-g}}\bigtriangledown_{\alpha}\left(f_{Q}\sqrt{-g}P^{\alpha}_{\epsilon\varepsilon}\right)-\frac{1}{2}fg_{\epsilon\varepsilon}$
(2)

where,

$T_{\epsilon\varepsilon}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta
g^{\epsilon\varepsilon}},\hskip 14.45377ptf_{\epsilon}=\frac{\partial
f}{\partial\epsilon},\hskip
14.45377pt\Theta_{\epsilon\varepsilon}=g^{\epsilon\varepsilon}\frac{\delta
T_{\epsilon\varepsilon}}{\delta g^{\epsilon\varepsilon}}$ (3)

and the superpotential $P^{\alpha}_{\epsilon\varepsilon}$ is given as fqt

$P^{\alpha}_{\epsilon\varepsilon}=\frac{1}{4}\left[Q^{\alpha}g_{\epsilon\varepsilon}+2Q^{\alpha}_{(\epsilon\varepsilon)}-\delta^{\alpha}_{(\epsilon
Q\varepsilon)}-Q^{\alpha}_{\epsilon\varepsilon}-\tilde{Q}^{\alpha}g_{\epsilon\varepsilon}\right]$
(4)

where

$Q_{\alpha}=Q_{\alpha\varepsilon}^{\varepsilon},\hskip
18.06749pt\text{and}\hskip
18.06749pt\tilde{Q}_{\alpha}=Q^{\epsilon}_{\alpha\epsilon}.$ (5)

Let us consider a FLRW geometry of the form

$ds^{2}=-N^{2}(t)dt^{2}+a^{2}(t)\sum_{j=1,2,3}\left(dx^{j}\right)^{2}$ (6)

where $N(t)$ denote the lapse function, and $a(t)$ denote the scale factor. It
may be noted that the lapse function is unity for a FRW background.
Substituting (6) in (2), the Friedman equations in $f(Q,T)$ gravity reads fqt

$8\pi\rho=-\frac{2\tilde{G}}{1+\tilde{G}}\left(\dot{F}H+F\dot{H}\right)+\frac{f}{2}-6FH^{2},$
(7)

and

$8\pi p=2\left(\dot{F}H+F\dot{H}\right)-\frac{f}{2}+6FH^{2},$ (8)

where,

$\tilde{G}=\frac{f_{T}}{8\pi},\hskip 18.06749ptF=f_{Q}.$ (9)

For this work, we shall set the functional form of $f(Q,\mathcal{T})$ to the
following fqt

$f(Q,T)=Q^{n+1}+mT$ (10)

where $n$, and $m$ are free parameters. Using (10), (7), and (8), the
expression of Hubble parameter $H(t)$ is given as fqt

$H(t)=\frac{H_{0}\left(16\pi-m(\gamma-4)\right)(n+1)}{3\gamma
H_{0}(t-t_{0})(m+8\pi)-\left(m\gamma-4(\beta+4\pi)\right)(n+1)},$ (11)

where $H_{0}$, and $t_{0}$ denote respectively the present value of the Hubble
parameter and the current age of the Universe, and $\gamma$ is the barotropic
EoS parameter.

##  3 Big Bang Nucleosynthesis in $f(Q,T)$ Gravity

We shall now attempt to constrain the model parameters $m$ and $n$ of the
$f(Q,T)$ gravity model from the primordial abundances of the light elements
(i.e, $D$, $He$, and $Li$). We are restricting the analysis to a radiation-
dominated Universe ($i.e,\gamma=4/3$) since the phenomena of BBN transpired
when the Universe was dominated by radiation. The main idea behind this
technique is to obtain a suitable parameter range of $m$, and $n$ for which
the theoretical primordial abundances could be consistent with observations.
To be more precise, we are concerned with the ratio of the Hubble parameter 11
obtained for the $f(Q,T)$ gravity model ($H_{f(Q,T)}$) to the Hubble parameter
for the standard cosmological model (i.e, the Hubble parameter of GR
($H_{GR}$)) in the radiation dominated Universe. We shall define the ratio as

$Z=\frac{H_{f(Q,T)}}{H_{GR}}.$ (12)

It may be noted that the primordial abundances of the light elements are
highly sensitive to the baryon density and on the rate of the expansion of the
Universe (i.e, on the Hubble parameter) e37 ; e38 . The baryon density is
expressed as

$\eta_{10}=10^{10}\eta_{B}=10^{10}\frac{\eta_{B}}{\eta_{\gamma}},$ (13)

where, $\frac{\eta_{B}}{\eta_{\gamma}}$ denote the baryon-to-photon ratio and
$\eta_{10}\simeq 6$ e39 .
$Z\neq 1$ symbolizes the expansion of the Universe to be governed by a non-
standard cosmological model and indicates a non-standard expansion factor.
Such cases could transpire if GR turns out to be an incorrect theory of
gravity or if there exists additional species of neutrinos other than the
three standard types confirmed observationally. However, in this work, we are
interested in exploring GR modifications and therefore we shall set the total
neutrino species to be equal to three.

###  3.1 Helium (${}^{4}He$) abundance in $f(Q,T)$ gravity

The production of Helium is a three step process. In the first step, a
Deuterium (${}^{2}De$) atom is produced through a neutron ($n$) and a proton
($p$). This is followed by the production of a lighter Helium isotope
(${}^{3}He$) atom along with a Tritium (${}^{3}T$). the equations can be
illustrated as follows:

$n+p\rightarrow^{2}De+\gamma;\hskip
10.84006pt^{2}De+^{2}De\rightarrow{{}^{3}He}+n;\hskip
10.84006pt^{2}De+^{2}De\rightarrow{{}^{3}T}+p$ (14)

Finally, the the heavier helium atom ${}^{4}He$ is produced through ${}^{3}T$,
${}^{2}De$, and ${}^{3}He$ as follows:

${}^{3}T+^{2}De\rightarrow^{4}He+n;\hskip
10.84006pt^{3}He+^{2}De\rightarrow^{4}He+p.$ (15)

The numerical best fit equation to estimate the primordial abundance of
${}^{4}He$ can be expressed as e40 ; e41

$Y_{p}=0.2485\pm 0.0006+0.00016\left[100(Z-1)+(\eta_{10}-6)\right].$ (16)

For a standard expansion factor ($i.e,Z=1$), the primordial abundance of
${}^{4}He$ reads $Y_{p}=0.2485\pm 0.0006$ while astrophysical observations
ascertained the same to lie in the range $0.2449\pm 0.0040$ e42 . Therefore,
by setting $\eta_{10}=6$ we can write,

$0.2449\pm 0.0040=0.2485\pm 0.0006+0.0016\left[100(Z-1)\right].$ (17)

Upon solving this, the constraint on the expansion factor reads $Z=1.0475\pm
0.105$.

###  3.2 Deuterium (${}^{2}H$) abundance in $f(Q,T)$ gravity

The production of a Deuterium atom takes place via a neutron and a proton as
follows

$n+p\rightarrow^{2}D+\gamma.$ (18)

The primordial abundance of Deuterium is estimated from the following
numerical best fit equation e37

$y_{Dp}=2.6\left(1\pm
0.0600\right)\left[\frac{6}{\eta_{10}-6(Z-1)}\right]^{1.6}.$ (19)

Similar to the previous case, we set $\eta_{10}=6$. For standard expansion
factor $Z=1$, we end up with a theoretical estimate of $y_{Dp}=2.6\pm 0.1600$.
however, observations reveal the constraints on Deuterium to lie in the range
$y_{Dp}=2.550\pm 0.0300$ e42 . Therefore, plugging in the numbers, we end up
with the following equation

$2.550\pm 0.030=2.60(1\pm
0.0600)\left[\frac{6}{\eta_{10}-6(Z-1)}\right]^{1.6}.$ (20)

The solution to this equation imposes a strict constraint on $Z$ as
$Z=1.0620\pm 0.4440$.

###  3.3 Lithium (${}^{7}Li$) abundance in $f(Q,T)$ gravity

The primordial abundance of Lithium is inconsistent with the theoretical
predictions of the $\Lambda$CDM model. The abundance of Lithium is estimated
to lie between 2.4 to 4.3 times the theoretical predictions e2 ; e43 . The
baryon density parameter $\eta_{10}$ which describes elegantly the abundances
of both Helium and Deuterium, fails to predict the same for Lithium. This is
sometimes termed the Lithium problem and hints at the existence of new physics
beyond the standard model e2 .
The numerical best-fit equation describing the Lithium abundance can be
expressed as e37

$y_{Lip}=4.82(1\pm
0.100)\left[\frac{1}{6}\left\\{3(1-Z)+\eta_{10}\right\\}\right]^{2}.$ (21)

The observational abundance of Lithium lie in the range $y_{Lip}=1.600\pm
0.300$ e42 . Plugging this into the above equation constraints $Z$ in the
range $Z=1.960025\pm 0.076675$.

###  3.4 Results

It is evident that the constraints on the expansion factor for the Helium and
Deuterium abundances are similar and would allow the parameters $m$ and $n$ to
be consistent with the observations of these elements. However, the $f(Q,T)$
gravity model cannot explain the Lithium abundance given the large discrepancy
between observations and theoretical predictions. From Table 1, we find that
the theoretical estimate for the abundances of Helium and Deuterium for the
$\Lambda$CDM and the $f(Q,T)$ gravity model are fairly close and falls well
within the current observational constraints and therefore allows the
parameters $m$ and $n$ to be constrained strictly.
We find that for $0.9425\lesssim Z\lesssim 1.1525$, the constraints on the
parameters $m$ and $n$ are $-1.13\lesssim n\lesssim-1.08$ and $-5.86\lesssim
m\lesssim 12.52$ respectively.

Table 1: The theoretical predictions for the abundances of ${}^{2}He$,
${}^{2}H$ and ${}^{7}Li$ in $\Lambda$CDM model and in $f(Q,T)$ gravity model
along with observational constraints.

Models/Observations |  $Y_{p}$ |  $y_{Dp}$ |  $y_{Lip}$
---|---|---|---
Observational data |  $0.2449\pm 0.0040$ e42  |  $2.550\pm 0.030$ e42  |  $1.600\pm 0.300$ e42
$f(Q,T)$ Gravity |  $0.2487\pm 0.0174$ |  $2.8756\pm 0.1715$ |  $5.29230\pm 0.5300$
$\Lambda$CDM |  $0.2485\pm 0.0006$ |  $2.600\pm 0.1600$ |  $4.8200\pm 0.4800$

Figure 1: $m$ as a function of $Z$ for a fixed $n=1.11$. The plot is drawn for
$0.9425\lesssim Z\lesssim 1.1525$. Figure 2: $n$ as a function of $Z$ for a
fixed $m=0.1$. The plot is drawn for $0.9425\lesssim Z\lesssim 1.1525$.

##  4 Conclusions

$f(Q,T)$ gravity is a novel extension of the symmetric teleparallel gravity
where the Lagrangian $L$ is represented through an arbitrary function of the
nonmetricity $Q$ and the trace of the energy-momentum tensor $T$ fqt .
$f(Q,T)$ gravity has been very successful in explaining the matter-antimatter
asymmetry b1  and the late-time acceleration b2 .
In this work, we constrained a widely used $f(Q,T)$ gravity model of the form
$f(Q,T)=Q^{n+1}+mT$ from the primordial abundances of the light elements to
understand its viability in cosmology. We report that the $f(Q,T)$ gravity
model explains gracefully the observed abundances of Helium and Deuterium
while the Lithium problem persists. From the constraint on the expansion
factor in the range $0.9425\lesssim Z\lesssim 1.1525$, we report stringent
constraints on the parameters $m$ and $n$ in the range $-1.13\lesssim
n\lesssim-1.08$ and $-5.86\lesssim m\lesssim 12.52$ respectively. Therefore,
in addition to explaining the matter-antimatter asymmetry and the late time
acceleration, $f(Q,T)$ gravity also explains the abundances of Helium and
Deuterium and therefore is turning out to be a viable alternative to the
$\Lambda$CDM cosmological model.
In future work, we shall try to investigate the growth of density fluctuations
and configurational entropy in $f(Q,T)$ gravity to understand the efficiency
and applicability in explaining the growth of cosmic structures and the
accelerated expansion of the Universe in greater detail.

## Acknowledgments

I thank Shantanu Desai for reading the manuscript and for the fruitful
discussions. I also thank DST, New-Delhi, Government of India for the
provisional INSPIRE fellowship selection [Code: DST/INSPIRE/03/2019/003141].

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Vol.0 (20xx) No.0, 000–000

11institutetext: Physics Department, King Fahd University of Petroleum and
Minerals, Dhahran 31261, Saudi Arabia<EMAIL_ADDRESS>;
<EMAIL_ADDRESS>
Received 20xx month day; accepted 20xx month day

# Charged Particles Capture Cross-Section by a Weakly Charged Schwarzschild
Black Hole

A. M. Al Zahrani  11 A. Al-Jama 221122

###### Abstract

We study the capture cross-section of charged particles by a weakly charged
Schwarzschild black hole. The dependence of the maximum impact parameter for
capture on the particle’s energy is investigated numerically for different
values of the electromagnetic coupling strength between the particle and the
black hole. The capture cross-section is then calculated. We show that the
capture cross-section is independent of the electromagnetic coupling for
ultra-relativistic particles. The astrophysical implications of our results
are discussed.

###### keywords:

stars: black holes — Stars, accretion, accretion disks — Physical Data and
Processes, black hole physics — Physical Data and Processes

##  1 Introduction

Studying the capture cross-section of black holes is central to understand how
matter interacts with them. It helps us understand the process of matter
accretion by a black hole which in turn determines how its mass, angular
momentum and charge evolve. It can also help us understand the environment
near black holes. Moreover, scrutinizing capture cross-section can be used to
test theories of gravity in strong gravitational fields.

Astrophysicists generally assume black holes are electrically neutral. This is
because they would quickly attract oppositely charged matter to balance out
any access charge. However, there are compelling reasons why weakly charged
black holes might exist as discussed in Al Zahrani (2021, 2022); Zajaček &
Tursunov (2019); Zajaček et al. (2019, 2018); Carter (1973) and the references
therein. The differences in how a black hole accretes electrons and protons
within its plasma environment, influenced by radiation, could render it
charged. Also, the spin of a black hole in the presence of a magnetic field
can induce the accretion of charged particles. In fact, using the EHT
observations, it was inferred that Sgr A* and M87 can be charged Ghosh & Afrin
(2023); Kocherlakota et al. (2021). The black hole’s charge is weak in the
sense that it has no tangible effect on spacetime, but its effect on charged
particle dynamics is prominent.

There are numerous astrophysical scenarios wherein charged particles are drawn
into black holes. Stars within the Roche limits near black holes often
contribute matter through tidal interactions. Additionally, stars emit streams
of charged particles as stellar winds. Highly energetic charged particles,
resulting from supernovae, gamma-ray bursts, and bipolar jets from compact
objects, frequently find their way into the vicinity of black holes. These
processes collectively enrich the environment around black holes with a
significant population of charged particles.

The concept of capture cross-sections has been explored extensively for
various black hole types. Foundational treatment which examine photon and
neutral particle capture by Schwarzschild black holes was given in several
monographs, such as Frolov & Zelnikov (2011). Further work addressed capture
cross-sections of charged and neutral particles by Kerr-Newman black holes,
including the implications for black hole spin and charge evolution Young
(1976). Capture by Reissner-Nordström black holes was also investigated
Zakharov (1994). In the context of higher-dimensional black holes, studies
have focused on calculating photon critical impact parameters for
Schwarzschild-Tangherlini black holes Connell & Frolov (2008); Tsukamoto
(2014); Singh & Ghosh (2018); Bugden (2020). The capture cross-section for
massive particles was determined in Ahmedov et al. (2021). Additionally,
research extends to particle capture in Myers-Perry rotating spacetime which
describes rotating black holes in five-dimensions Gooding & Frolov (2008).
Moreover, wave capture cross-sections have been studied for various black hole
configurations (see Anacleto, et al. (2023) and the references within).

In this research, we examine the capture cross-section of charged particles by
a weakly charged Schwarzschild black hole and discuss the astrophysical
consequences of our findings. The paper is organized as follows: In Sec. 2, we
review the dynamics of charged particles in the background of a weakly charged
black hole. We then review the capture cross-section of neutral particles in
Sec. 3. The capture cross-section of charged particles is calculated for
different coupling strengths and particle energies in Secs. 4. Finally, we
summarize our main findings and discuss their astrophysical consequences in
Sec 5. We use the sign conventions adopted in Misner et al. (1973) and
geometrized units where $c=G=k=1$, where $k$ is the electrostatic constant.

##  2 Charged Particles near a Weakly Charged Schwarzschild Black Hole

Here, we review the dynamics of charged particles near a weakly charged black
hole. The spacetime geometry around a black hole of mass $M$ and charge $Q$ is
described by the Schwarzschild Reissner-Nordströn metric which reads Misner et
al. (1973)

$ds^{2}=-hdt^{2}+h^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}{\theta}d\phi^{2},$
(1)

where $h=1-r_{S}/r+Q^{2}/r^{2}$ and $r_{S}=2M$ is the Schwarzschild radius.
The electromagnetic 4-potential is

$A_{\mu}=-\frac{Q}{r}\delta_{\mu}^{0}.$ (2)

However, when the charge is weak we can ignore the curvature due to it and use
the Schwarzschild metric, which reads Misner et al. (1973)

$ds^{2}=-fdt^{2}+f^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}{\theta}d\phi^{2},$
(3)

where $f=1-r_{S}/r$ and $r_{S}=2M$ is the Schwarzschild radius. This weak
charge approximation is valid unless the charge creates curvature comparable
to that due to the black hole’s mass. This happens when

$Q^{2}\sim M^{2}.$ (4)

In conventional units, the weak charge approximation fails when

$Q\sim\frac{G^{1/2}M}{k^{1/2}}\sim
10^{20}\>\frac{M}{\>M_{\odot}}\>\text{coloumbs}.$ (5)

This charge is way greater than the greatest estimated change on any black
hole. Although the black hole charge is tiny, its effect on charged particles
dynamics is profound because it is multiplied by the charge-to-mass ratio of
these particles ($\sim 10^{21}\>{\text{m}}^{-1}$ for electrons and $\sim
10^{18}\>{\text{m}}^{-1}$ for protons).

The Lagrangian describing a charged particle of charge $q$ and mass $m$ in a
spacetime described by a metric $g_{\mu\nu}$ and an electromagnetic field
produced by a 4-potential $A^{\mu}$ reads Chandrasekhar (1983)

$L=\frac{1}{2}mg_{\mu\nu}u^{\mu}u^{\nu}+qu^{\mu}A_{\mu},$ (6)

where $u^{\mu}\equiv dx^{\mu}/d\tau$ is the particle’s 4-velocity and $\tau$
is its proper time. In our case, the Lagrangian becomes

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{1}{2}m\left[-f\left(\frac{dt}{d\tau}\right)^{2}+f^{-1}\left(\frac{dr}{d\tau}\right)^{2}+r^{2}\left(\frac{d\theta}{d\tau}\right)^{2}+r^{2}\sin^{2}{\theta}\left(\frac{d\phi}{d\tau}\right)^{2}\right]$
(7) $\displaystyle-qQ\frac{dt}{d\tau}.$

This Lagrangian is cyclic in $t$ and $\phi$, which means that the particle’s
energy and azimuthal angular momentum are constants of motion. The specific
energy and azimuthal angular momentum are, respectively, given by

$\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle-\frac{1}{m}\frac{\partial
L}{\partial\left(\frac{dt}{d\tau}\right)}=f\frac{dt}{d\tau}+\frac{qQ}{mr},$
(8) $\displaystyle\ell$ $\displaystyle=$
$\displaystyle\frac{1}{m}\frac{\partial
L}{\partial\left(\frac{d\phi}{d\tau}\right)}=r^{2}\sin^{2}{\theta}\frac{d\phi}{d\tau}$
(9)

Combining these equations with the normalization condition
$g_{\mu\nu}u^{\mu}u^{\nu}=-1$ and solving for $dr/d\tau$ give

$\left(\frac{dr}{d\tau}\right)^{2}=\left({\cal
E}-\frac{qQ}{mr}\right)^{2}-f\left[r^{2}\left(\frac{d\theta}{d\tau}\right)^{2}+\frac{\ell^{2}}{r^{2}\sin^{2}{\theta}}+1\right].$
(10)

In the equatorial plane where $\theta=\pi/2$, the equation becomes

$\left(\frac{dr}{d\tau}\right)^{2}=\left({\cal
E}-\frac{qQ}{mr}\right)^{2}-f\left(\frac{\ell^{2}}{r^{2}}+1\right).$ (11)

Let us rewrite the last equation in a dimensionless form. We first introduce
the following dimensionless quantities:

${\cal T}=\frac{\tau}{r_{S}},\hskip 28.45274pt\rho=\frac{r}{r_{S}},\hskip
28.45274pt{\cal L}=\frac{\ell}{r_{S}}.$ (12)

Equation 11 then becomes

$\left(\frac{d\rho}{d{\cal T}}\right)^{2}=\left({\cal
E}-\frac{\alpha}{\rho}\right)^{2}-f\left(\frac{{\cal
L}^{2}}{\rho^{2}}+1\right),$ (13)

where

$\alpha=\frac{qQ}{mr_{S}}.$ (14)

The parameter $\alpha$ represent the relative strength of the electromagnetic
force to the Newtonian gravitational force. We can rewrite Eq. 13 as

$\left(\frac{d\rho}{d{\cal T}}\right)^{2}=({\cal E}-V_{+})({\cal E}-V_{-}),$
(15)

where

$V_{\pm}=\frac{\alpha}{\rho}\pm\sqrt{f\left(\frac{{\cal
L}^{2}}{\rho^{2}}+1\right)},$ (16)

is an effective potential. It is $V_{+}$ that corresponds to physical, future-
directed motion and hence will be used in all of the analyses below. Without
loss of generality, we will consider ${\cal L}>0$ only.

It was estimated in Ref. Zajaček et al. (2019) that the charge of Sgr A* is
$10^{8}-10^{15}$ coulomb. Using the lower limit of charge, the coupling
constant for electrons $\alpha_{e}$ and protons $\alpha_{p}$ near Sgr A*,
which has a mass of $M=4.3\times 10^{6}M_{\odot}$ according to Ref. GRAVITY
Collaboration (2023), are

$\displaystyle\alpha_{e}\sim 10^{9},$ (17) $\displaystyle\alpha_{p}\sim
10^{6}.$ (18)

##  3 Capture Cross-Section of Neutral Particles

Before we tackle the main problem, let us find the capture cross-cross section
for neutral particles first. Setting $\alpha=0$, the effective potential
$V_{+}$ reduces to

$V_{+}=\sqrt{f\left(\frac{{\cal L}^{2}}{\rho^{2}}+1\right)}.$ (19)

Capture occurs whenever the particle’s energy is greater than the maximum of
$V_{+}$. The function $V_{+}$ is at an extremum when $dV_{+}/d\rho=0$ or

$\rho^{2}+(3-2\rho){\cal L}^{2}=0,$ (20)

which gives the position of the extrema in terms of ${\cal L}$ as

$\rho_{\pm}={\cal L}^{2}\pm{\cal L}\sqrt{{\cal L}^{2}-3},$ (21)

where ${\cal L}\in[\sqrt{3},\infty)$. When ${\cal L}=\sqrt{3}$ ($\equiv{\cal
L}_{\text{min}}$), $\rho_{+}$ and $\rho_{-}$ meet at a saddle point.
Inspecting $d^{2}V_{+}/d\rho^{2}$ reveals that $\rho_{-}$ corresponds to the
position of the local maximum of $V_{+}$. In terms of ${\cal L}$, the escape
condition ${\cal E}=V_{+}|_{\rho=\rho_{-}}$ becomes

${\cal E}=\sqrt{\frac{2}{27}}\left[{\cal L}\left(\sqrt{{\cal L}^{2}-3}+{\cal
L}\right)-\frac{3\sqrt{{\cal L}^{2}-3}}{{\cal L}}+9\right]^{1/2},$ (22)

Inverting this equation gives

$\displaystyle{\cal L}=\left[\frac{27{\cal E}^{4}-36{\cal E}^{2}+\left(9{\cal
E}^{2}-8\right)^{3/2}{\cal E}+8}{8\left({\cal E}^{2}-1\right)}\right]^{1/2}$
(23)

The impact parameter $b$ is defined as the perpendicular distance between the
center of force and the incident velocity Goldstein et al. (2001). It can be
written as

$b=\frac{{\cal L}}{\cal P}=\frac{{\cal L}}{\sqrt{{\cal E}^{2}-1}},$ (24)

where $\cal P$ is the specific linear momentum. The maximum impact parameter
for capture $b_{\text{max}}$ is given by

$b_{\text{max}}=\frac{\left[27{\cal E}^{4}-36{\cal E}^{2}+\left(9{\cal
E}^{2}-8\right)^{3/2}{\cal E}+8\right]}{2\sqrt{2}\left({\cal
E}^{2}-1\right)}^{1/2}.$ (25)

The capture cross-section ${\sigma}_{\text{cap}}$ is given by

${\sigma}_{\text{cap}}=\pi b_{\text{max}}^{2}=\frac{\pi}{8}\frac{27{\cal
E}^{4}-36{\cal E}^{2}+\left(9{\cal E}^{2}-8\right)^{3/2}{\cal
E}+8}{\left({\cal E}^{2}-1\right)^{2}}.$ (26)

Figure 1: The maximum impact parameter for capture $b_{\text{max}}$ vs. ${\cal
E}$ for a neutral particle. Figure 2: The capture cross-section
${\sigma}_{\text{cap}}$ vs. ${\cal E}$ for a neutral particle.

Figures 1 and 2 are plots of $b_{\text{max}}$ and the capture cross-section
$\sigma_{\text{cap}}$ vs. ${\cal E}$, respectively. For ultra-relativistic
particles (${\cal E}\gg 1$),

$b_{\text{max}}=\frac{3\sqrt{3}}{2}+\frac{\sqrt{3}}{2{\cal
E}^{2}}+{\mathcal{O}}\left(\frac{1}{{\cal E}^{3}}\right).$ (27)

The corresponding capture cross-section $\sigma_{\text{cap}}$ is therefore

$\sigma_{\text{cap}}=\frac{27\pi}{4}+\frac{9\pi}{2{\cal
E}^{2}}+{\mathcal{O}}\left(\frac{1}{{\cal E}^{3}}\right).$ (28)

For a slowly moving particle with speed $v\ll 1$,

${\cal E}\approx 1+\frac{v^{2}}{2},$ (29)

and thus

$b_{\text{max}}=\frac{\sqrt{2}}{\sqrt{{\cal E}-1}}+{\mathcal{O}}(\sqrt{{\cal
E}-1})=\frac{2}{v}+{\mathcal{O}}(v),$ (30)

and the capture cross-section becomes

$\sigma_{\text{cap}}=\frac{4\pi}{v^{2}}+{\mathcal{O}}(v^{0}).$ (31)

##  4 Capture Cross-Section of Charged Particles

We will now follow the same procedure we used for the neutral particle.
However, analytic expressions are not viable in this case and we will resort
to numerical solutions, except in the ultra-relativistic particle case. The
structure of the effective potential $V_{+}$ is generically similar to the
neutral particle’s. The effect of $\alpha$ is to raise (lower) the peak of
$V_{+}$ for positive (negative) $\alpha$. The effective potential $V_{+}$ is
at an extremum when

$2\alpha\sqrt{(\rho_{\pm}-1)\left({\cal
L}^{2}+\rho_{\pm}^{2}\right)\rho_{\pm}}-{\cal
L}^{2}(3-2\rho_{\pm})-\rho_{\pm}^{2}=0.$ (32)

The extremum is a maximum when

$\frac{\alpha\left[{\cal
L}^{2}(1-2\rho_{\pm})+(3-4\rho_{\pm})\rho_{\pm}^{2}\right]}{\sqrt{(\rho_{\pm}-1)\left({\cal
L}^{2}+\rho_{\pm}^{2}\right)\rho_{\pm}}}+2\rho_{\pm}-2{\cal L}^{2}<0.$ (33)

To be consistent with the notation of the previous section, we let $\rho_{+}$
correspond to the minimum of $V_{+}$ and $\rho_{-}$ correspond to the maximum.
Here, ${\cal L}_{\text{min}}$ (the value at which $\rho_{-}$ and $\rho_{+}$
meet) depends on the value of $\alpha$. The two parameters are related by the
relation

$-\alpha^{8}+6\alpha^{4}{\cal L}_{\text{min}}^{2}\left({\cal
L}_{\text{min}}^{2}-3\right)-8\alpha^{2}{\cal L}_{\text{min}}^{4}\left({\cal
L}_{\text{min}}^{2}+9\right)+3{\cal L}_{\text{min}}^{4}\left({\cal
L}_{\text{min}}^{2}-3\right)^{2}=0.$ (34)

Figure 3 is a plot of ${\cal L}_{\text{min}}$ vs $\alpha$. When $\alpha=1/2$,
${\cal L}_{\text{min}}$ approaches zero. This is because $V_{+}$ ceases to
have a local minimum for $\alpha\geq 1/2$. Physically, this limit corresponds
to the case when the Coulomb repulsion becomes too strong for stable orbits to
exist as discussed in Ref. Al Zahrani (2021).

Figure 3: The value of ${\cal L}$ at which $\rho_{+}$ and $\rho_{-}$ meet
(${\cal L}_{\text{min}}$) vs. the electromagnetic coupling parameter $\alpha$.

Figure 4 shows how $b_{\text{max}}$ depends on ${\cal E}$ for several negative
values of the coupling parameter $\alpha$. The effect of increasing $|\alpha|$
is to increase the values of $b_{\text{max}}$ for all energies. This is
expected because the Coulombs attraction makes it easier for a charges
particle to get captured. In all cases, $b_{\text{max}}$ is a monotonic
function of ${\cal E}$. In the ultra-relativistic limit, $b_{\text{max}}$
approaches $3\sqrt{3}/2$, the limit in the neutral particle case, for any
finite value of $\alpha$, provided that $\alpha$ is not too large compared to
${\cal E}$.

Figure 4: The maximum impact parameter for capture $b_{\text{max}}$ vs. ${\cal
E}$ for a charged particle with $\alpha=0$ (black), $\alpha=-1$ (blue),
$\alpha=-2$ (green), $\alpha=-3$ (red).

Figure 5 shows how $b_{\text{max}}$ depends on ${\cal E}$ for several values
of $\alpha$ between $0$ and $0.5$. In this range, there is competition between
that gravitational ’attraction’ and the Coulomb repulsion. The curves have
richer structure. They falls quickly as ${\cal E}$ goes beyond $1$ and reach a
minimum. After that, the curves rise and reach $3\sqrt{3}/2$ asymptotically.

Figure 5: The maximum impact parameter for capture $b_{\text{max}}$ vs. ${\cal
E}$ for a charged particle with $\alpha=0$ (black), $\alpha=0.1$ (blue),
$\alpha=0.3$ (green), $\alpha=0.5$ (red).

Figure 6 shows how $b_{\text{max}}$ depends on ${\cal E}$ for several positive
values of $\alpha$ greater than $0.5$. Generally, $b_{\text{max}}$ becomes
smaller as $\alpha$ increases. This is expected because the greater the
Coulomb repulsion the more difficult it is for a charged particle to be
captured. In fact, there is a threshold energy ${\cal E}_{\text{thr}}$ below
which capture cannot occur. It is given by

${\cal E}_{\text{thr}}=\alpha+\frac{1}{4\alpha}.$ (35)

This equation is valid for $\alpha\geq 0.5$ only. Fig. 7 shows how ${\cal
E}_{\text{thr}}$ vary with $\alpha$.

The capture cross-section $\sigma_{\text{cap}}$ corresponding to Figs. 4, 5
and 6 is shown in Figs. 8, 10 and 9, respectively. In all cases,
$\sigma_{\text{cap}}$ vs. ${\cal E}$ curves inherent the features of the
$b_{\text{min}}$ vs. ${\cal E}$ curves.

Figure 6: The max impact parameter for capture $b_{\text{max}}$ vs. ${\cal E}$
for a charged particle with with $\alpha=0$ (black), $\alpha=1$ (blue),
$\alpha=2$ (green), $\alpha=3$ (red). Figure 7: The energy threshold for
escape ${\cal E}_{\text{thr}}$ vs. electromagnetic coupling parameter
$\alpha$. Figure 8: The capture cross-section $\sigma_{\text{cap}}$ vs. ${\cal
E}$ for a charged particle with $\alpha=0$ (black), $\alpha=-1$ (blue),
$\alpha=-2$ (green), $\alpha=-3$ (red). Figure 9: The capture cross-section
$\sigma_{\text{cap}}$ vs. ${\cal E}$ for a charged particle with $\alpha=0$
(black), $\alpha=0.1$ (blue), $\alpha=0.3$ (green), $\alpha=0.5$ (red). Figure
10: The capture cross-section $\sigma_{\text{cap}}$ vs. ${\cal E}$ for a
charged particle with $\alpha=0$ (black), $\alpha=1$ (blue), $\alpha=2$
(green), $\alpha=3$ (red).

For ultra-relativistic particles, we can write $b_{\text{max}}$ as

$b_{\text{max}}=\frac{3\sqrt{3}}{2}-\frac{\sqrt{3}\alpha}{{\cal
E}}+\frac{9-2\alpha^{2}}{6\sqrt{3}{\cal
E}^{2}}+{\mathcal{O}}\left(\frac{1}{{\cal E}^{3}}\right).$ (36)

The corresponding capture cross-section is then

$\sigma_{\text{cap}}=\frac{27\pi}{4}-\frac{9\pi\alpha}{{\cal
E}}+\frac{\left(4\alpha^{2}+9\right)\pi}{2{\cal
E}^{2}}+{\mathcal{O}}\left(\frac{1}{{\cal E}^{3}}\right).$ (37)

These limiting results are in agreement with our numerical findings.

##  5 Conclusion

We have studied the capture cross-section of charged particles by a weakly
charged Schwarzschild black hole. We have shown that a trace charge on the
black hole can have prominent effects.

When the Coulomb force between a charged particle and the black hole is
attractive, it enlarges the capture cross-section significantly. This is
expected since the Coulomb attraction enhances the capture of charged
particles. However, when the Coulomb force between a charged particle and the
black hole is repulsive, it shrinks the capture cross-section significantly.
When the electromagnetic coupling strength is below a critical value, capture
is possible for all values of the particle’s energy. When the electromagnetic
coupling strength is above the critical value, there is a minimum value of the
particle’s energy below which capture is impossible. This is because the
Coulomb repulsion surpasses the gravitational attraction unless the particle’s
radial momentum is large enough.

Our results emphasizes the assertion that charged black holes will favorably
accretes charges of the opposite sign. However, it is still possible for the
black hole charge to grow if the plunging charged particles are energetic
enough to the limit that the capture cross-section becomes independent of the
sign of the charges. Moreover, the fact that the electromagnetic coupling
constant is three orders of magnitudes greater for electrons than protons
suggests that it is relatively easier for a black hole to accumulate positive
charge than negative charge.

It will be an astrophysically interesting to study the energies of charged
particles near an astrophysical black hole to understand better how the black
hole’s charge evolves. The problem can be astrophyically more viable when
other astrophysical black holes, such as rotating black holes, are studied (in
progress).

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# Rescaling CNN through Learnable Repetition of Network Parameters

Arnav Chavan∗, Udbhav Bamba∗, Rishabh Tiwari∗ and Deepak K. Gupta
Transmute AI Lab, Texmin Hub, IIT Dhanbad, India
<EMAIL_ADDRESS>All authors contributed equally.

###### Abstract

Deeper and wider CNNs are known to provide improved performance for deep
learning tasks. However, most such networks have poor performance gain per
parameter increase. In this paper, we investigate whether the gain observed in
deeper models is purely due to the addition of more optimization parameters or
whether the physical size of the network as well plays a role. Further, we
present a novel rescaling strategy for CNNs based on learnable repetition of
its parameters. Based on this strategy, we rescale CNNs without changing their
parameter count, and show that learnable sharing of weights itself can provide
significant boost in the performance of any given model without changing its
parameter count. We show that small base networks when rescaled, can provide
performance comparable to deeper networks with significantly reduced model
size.

Our repetition strategy scales well for large datasets (_e.g._ , ImageNet).
Moreover, it tends to learn important features that help the model to
generalize well across tasks. For example, backbone network of a RepeatNet
model scaled on the task of image classification is shown to work well as an
encoder for the task of segmentation. The relevance of learnable repetitions
weight sharing is further highlighted through the example of group-equivariant
CNNs. We show that the significant improvements obtained with group-
equivariant CNNs over the regular CNNs on classification problems could only
partly due to the added equivariance property, and part of it comes from the
learnable repetition of network weights. For Rot-MNIST dataset, we show that
up to 40% of the relative gain reported by state-of-the-art methods for
rotation equivariance could actually be due to just the learnt repetition of
weights.

##  1 Introduction

One of the major reasons for the popularity of deep learning-based models is
their ability to easily adapt to different problems, ranging from simple input
to output mapping to those involving extremely complex underlying
distributions. A recent popular strategy to improve such models is to employ a
base model and scale it in size, thereby improving its discriminative power.
Due to largely increased number of parameters, such approaches have been able
to achieve state-of-the-art (SOTA) performance on popular benchmark datasets
such as CIFAR10/100 [13] and ImageNet [5].

A common way to scale up CNNs is to make its layers wider or the network
itself deeper. For example, VGG [22] increases the model depth through
stacking multiple layers together. However, vanishing gradients [8] harm the
performance of such stacked models due to which going deeper beyond a certain
limit produces adverse affects on performance. Residual Networks (ResNets)
[10] overcome the outlined issue through adding residual connections for
better gradient flow, thus enabling the construction of even deeper models.
Wide ResNets [28] take the orthogonal direction of making the layers wider
through scaling the number of channels in each convolutional layer by a
variable parameter. This approach brings in significant boost in the
performance of the models. Most current SOTA models in the field of computer
vision now use compound scaling which involves uniformly scaling up the width,
depth and resolution of the models [24].

It is evident from the developments outlined above that the deeper and wider
models exhibit stronger discriminative power and can better extract the rich
feature representations contained in images. Thus it is of major interest to
identify whether such gains are actually due to the increased optimization
parameters or whether part of it is due to the physical size of the scaled
model. The answer being latter would imply that if we can design a mechanism
to upscale the model, we might be able to improve its performance to some
extent even without the need of increased optimization parameters. In this
paper, we investigate the importance of network scaling at fixed parameter
count. To this end, we first demonstrate that a well-drafted scaling policy of
an existing architecture can already improve its performance for the same
number of optimization parameters. Referred henceforth as ‘RepeatNet’, our
scaling strategy is based on the principle that transformed version of the
learned weights from one part of the network, when used again in another part,
results in scaled-up architectures that improve the model performance. For
example, a VGG4 model trained for CIFAR-100 dataset, when scaled up using the
simplest linear variant of RepeatNet to match the physical size of VGG11,
shows a massive absolute improvement of approximately 19%.

We explore two ways of scaling the network architecture: _depth scaling_ and
_width scaling_ by repeating different projections of same set of filters.
While the linear projections can be used to effectively scale the depth of an
existing network, it is degenerate for width scaling (later discussed in
Section 3.3). For more efficient and universal scaling of the existing CNN
models, we further propose two nonlinear scaling strategies ($s$-RepeatNet and
$f$-RepeatNet) that overcome the limitations of the linear scaling. Our
$s$-RepeatNet approach draws inspiration from the way nonlinear activation
functions improve the representation capability of the networks. It adapts the
popular swish activation function [20] to create transformed variants of the
learned weights that can be effectively deployed in other parts of the
network. The $f$-RepeatNet variant learns sign flipping of the repeated
weights, with optimization objective similar to BiRealNet [17] to increase the
representation capability of the scaled models.

The efficacy of RepeatNet is demonstrated on several popular CNN architectures
and benchmark datasets. We show that RepeatNet provides significant boost in
performance with negligible increase of network parameters compared to the
respective base models. Another important aspect of deep learning models is
their transferability across different tasks. In this regard, we also train
scaled-up CNN architectures on ImageNet dataset [5] for the task of
classification, and show that the trained backbone performs well for the
downstream tasks. We show that pre-trained scaled up versions of ResNet-18
model when used as an encoder for the segmentation model, achieves an increase
of more than 5% in the mIOU metric on PASCAL-VOC dataset [6].

We further demonstrate using a popular example from the field of deep learning
(rotation-equivariant CNN or RE-CNN [26]) that ignoring the impact of
repetition of weights can lead to erroneous conclusions when comparing
different models. While several variants exists, we focus on the RE-CNN model
presented in [26]. Their strategy uses steerable filters [3] that compute
orientation dependent responses for multiple orientations efficiently through
sharing of weights [27, 26]. It is very common to compare the performance of
RE-CNNs and other similar models with regular CNNs for the same number of
optimization parameters. However, since RE-CNNs employ weight sharing, the
realized parameter set and eventually the FLOPs are higher during inference
time when compared with an equivalent CNN model. This raises the question
whether the gain in performance observed with RE-CNN on datasets such as Rot-
MNIST [14] is purely due to the induced property of rotation equivariance of
the model, or whether part of it is due to repetition of weights. We address
this concern later in the paper.

To summarize, the contributions of this paper can be outlined as follows.

  * •

We demonstrate through numerical experiments that the gain observed in model
performance due to the use of increased number of optimization weights could
partly be accounted to scaling up of the physical size of the network itself.

  * •

For efficient sharing of weights across the network, we present _RepeatNet_ ,
a learnable filter transformation strategy that can be used to effectively
scale CNNs along the width and depth dimensions. For increased flexibility of
representations in weight sharing, we present two nonlinear transformation
strategies ($s$-RepeatNet and $f$-RepeatNet).

  * •

We demonstrate through experiments on popular CNN architectures and datasets
(including ImageNet) that RepeatNet-scaled architectures achieve significant
boost in performance. Moreover, we demonstrate that features obtained from
RepeatNet generalize well across tasks - the backbone of a RepeatNet
architecture designed for classification can also be used for segmentation
tasks.

  * •

Understanding the influence of weight sharing in a network is important, and
we demonstrate through a popular example that ignoring the impact of network
scaling can lead to erroneous conclusions in deep learning studies.

##  2 Related Works

Scaling the network is one among the most popular ways to improve the
performance of CNN models. Several previous works have addressed this aspect
in the past. For example, VGG architectures [22] are scaled by increasing the
depth of the architecture. Another class of architectures is WideResNet [28]
that are scaled by making the network wider. Recent works have shown that a
more efficient strategy to scale up a network is to grow along depth as well
as its width. Referred as EfficientNets [24], such models use a mix of depth,
width and resolution scaling, and are shown to have a much better performance
per parameter ratio in comparison to ResNets.

The interplay of parameter and model physical size in model’s performance is
also visible in the field of pruning. For deep learning models, several works
have shown that most often the networks are overparameterized, and it is
possible to achieve similar performance with reduced number of parameters [16,
7, 25, 15, 4]. A sub-class of pruning, referred as unstructured pruning
reduces the number of parameters required to define an architecture, but it
still preserves the model size [23, 9, 19, 18]. Unstructured pruning
strategies have been shown to work well, which also indicates that beyond the
number of parameters, the physical size of the model as well plays role in
controlling its performance. However, a clear understanding on this aspect is
still missing. Nevertheless, such approaches do not harness the improvements
brought in through repetition of the parameters. These approaches are
orthogonal to our repetition strategy, and we believe that these could be
combined with our repetition approach presented in this paper.

Scaling up a network through weight sharing is another popular approach that
is frequently adopted. ShaResNet shares the weights of convolutional layers
between residual blocks in ResNets operating at the same spatial scale [1].
Similarly, DACNNs share the same kernel weights across layers using additional
$1\times 1$ convolutional layers, thereby reducing the overall model
parameters [11]. While these works present the general concept of weight
sharing, they either don’t consider the correlation between shared parameters,
require additional layers to counter it or are limited to sharing at same
spatial scales. Weight sharing is also observed in several variants of group-
equivariant networks [3, 27]. While these networks focus on making network
architectures robust against certain transformations, an implicit network
scaling occurs due to multiple filters blocks used to account for several
orientations. The methods generally ignore the affect of network scaling, and
we show in this paper that part of their improvement could solely be due to
network scaling.

$n$$m$$k$$k$$n\times\gamma_{2}$$\mathcal{\psi_{\boldsymbol{\beta}}(\cdot)}$$m\times\gamma_{1}$
Figure 1: Schematic representation of our RepeatNet strategy showing the
generation of child filter sets (right) from the parent filter sets (left).

##  3 Methodology

In this section, we first present the core principles of the trainable filter
repetitions. We then explain the procedure to scale any neural network based
on this principle.

###  3.1 Filter Repetition

In simple terms, our filter repetition approach involves taking a _parent_
filter set from a particular convolutional layer and transforming it to
different _child_ sets through multiple repetitions across the input and
output channel dimensions. Same child filter used at different parts of the
network can have adverse effects on the performance, especially because of the
varying receptive field at different depths. To circumvent this issue and
improve the learning capability of the model, each child filter set can be
passed through a learnable transformation function.

Let $\mathcal{F}\in\mathbb{R}^{k\times k\times m\times n}$ denote a filter
block, comprising a set of convolutional filters of size $k\times k$, where
$m$ and $n$ correspond to the number of input and output channels,
respectively. We define filter repetition as the mapping
$\mathcal{F}_{p}\xrightarrow{\mathcal{\psi_{\boldsymbol{\beta}}(\cdot)}}\mathcal{F}_{c}$,
where $\mathcal{F}_{p}$ and $\mathcal{F}_{c}$ denote the parent and child
filter blocks, respectively, and $\psi_{\boldsymbol{\beta}}(\cdot)$ is a
learnable nonlinear transformation function. In this setting, the weights in
the parent block are trainable and learnt during the training process. The
weights of the child block $\mathcal{F}_{c}$ depend on $\mathcal{F}_{p}$ as
well as an additional parameter set $\boldsymbol{\beta}$. Note that
$\boldsymbol{\beta}$ is generally a very small set of parameters or a set of
boolean variables, and the added memory burden is negligible. For example, for
one of the configurations proposed in this paper, length of this parameter set
is 2 for every mapping.

Fig. 1 provides a schematic representation of how these repetitions are
performed. Here, the trainable filter block
$\mathcal{F}_{p}\in\mathbb{R}^{k\times k\times m\times n}$ is mapped to
another larger block of dimensions $k\times k\times\tilde{m}\times\tilde{n}$.
We propose to define network architectures such that $\tilde{m}$ and
$\tilde{n}$ are integer multiples of $m$ and $n$, respectively. We define
these multiplicities as $\gamma_{1}=\frac{\tilde{m}}{m}$ and
$\gamma_{2}=\frac{\tilde{n}}{n}$, thereby implying
$\gamma_{1}\times\gamma_{2}$ repetitions in Fig. 1. Let $\mathcal{F}_{c,i}$
denote the $i^{\text{th}}$ child filter block to be mapped with one of the
repetitions. In simple terms, the goal is then to fill the entire set of
weights in $\mathcal{F}_{c,i}$ using a dedicated parameter set
$\boldsymbol{\beta}_{i}$ and the parent set $\mathcal{F}_{p}$.

Since the diversity across a set of child blocks obtained from a single parent
depends heavily on the respective $\boldsymbol{\beta}$ set, the choice of this
mapping function needs to be made cautiously. We show later in this paper that
with a proper choice of $\psi_{\boldsymbol{\beta}}$, a large network (_e.g._ ,
VGG11) can be expressed using only a very small fraction of its parameters at
the expense of limited drop in performance of the model.

###  3.2 Network Scaling

Repetition of weights can be used to scale the existing CNN networks in two
different ways: _width scaling_ and _depth scaling_. Details related to these
are provided below.

Width scaling. This scaling strategy involves increasing the number of filters
through performing repetitions within the same convolutional layer. This is
equivalent to having a smaller parent filter set, repeating it multiple times
and concatenating all the repetitions to form the filters for a certain
convolutional layer. This effectively allows a smaller convolutional layer to
take in a larger number of input channels and return a larger number of output
channels at the same time. Networks scaled this way are wider than their
original counterparts with almost the same number of parameters.

Depth scaling. In this strategy, repetitions are performed across different
convolutional layers of the network. An entire convolutional layer constitutes
the parent set and it is repeated to generated child sets where each child
filter set itself forms a different convolutional layer of the network. Parent
filter set is itself a convolutional layer which is repeated accordingly to
form multiple child set, each of which is then used as a convolutional layer
of the network. Networks scaled through this strategy are much deeper than
their original counterparts with almost the same number of parameters which
effectively increases the receptive field of the network.

###  3.3 Learnable Filter Transformation

Linear projections. A simple yet effective way of scaling the existing
networks is to directly use the parents filters at multiple child locations,
thereby setting $\mathcal{F}_{c,i}=\mathcal{F}_{p}$. However, the scope of
this strategy is limited to depth scaling, and it cannot be used to make
networks wider.

In general, expressing an entire filter block $\mathcal{F}_{c}$ with another
parent block $\mathcal{F}_{p}$ is expected to limit the representation
capability of the resultant weights. However, due to the nonlinear activations
applied on the feature output at very hidden layer, the linear projects are
expected to help in depth scaling of the network to some extent. We show later
through experiments that for some cases, even the linear scaling boosts model
performance significantly.

However, linear scaling cannot be used to effectively make networks wider.
Repeating the weights within the same filter block is equivalent to
concatenating the output feature map with its exact copies, thereby adding no
useful information.

Nonlinear filter transformation. To overcome the limitation of linear
repetition, we employ learnable transformations functions that remove the
correlation between the filter sets and increase the diversity across the
filter sets obtained from repetition. We propose two different ways to build
learnable nonlinear repetitions:

_$s$ -RepeatNet. _This strategy relies on a nonlinear projection that
differentiates the outputs of two different child filter sets. The
transformation function $\mathcal{\psi_{\boldsymbol{\beta}}(\cdot)}$ is
defined as

$\psi_{\boldsymbol{\beta}}(x)=\frac{\beta_{1}x}{1+e^{\beta_{2}x}},$ (1)

where $\boldsymbol{\beta}=\\{\beta_{1},\beta_{2}\\}$ is the pair of additional
parameters to be learnt for each repetition. Further, the term $x$ is used to
denote every element of the parent filter set $\mathcal{F}_{p}$. The above
transformation function is inspired by the swish activation function [20],
with the addition of the extra parameter $\beta_{1}$. Without this parameter,
only a very limited set of projections can be learnt. Adding $\beta_{1}$
alleviates this restriction and adds extra flexibility for the repeated
kernels to independently scale themselves. The parameter set
$\boldsymbol{\beta}$ is also treated as the normal trainable parameters of the
network similar to the filters, and these are also updated iteratively through
back-propagation.

As outlined earlier, a total of $\gamma_{1}\times\gamma_{2}$ repetitions of a
parent filter set occur. This would lead to the negligible addition of only
$2\times\gamma_{1}\times\gamma_{2}$ extra training parameters.

_$f$ -RepeatNet. _This strategy is similar to the linear projections, but with
the addition of a learnable sign-flipping function. The motivation for using
this flip function comes from the findings presented in [12]. The gist of our
approach is that new child filters can be created from parent weights through
learning to flip signs in different parts of the parent filter block. This
would imply that $\mathcal{F}_{c,i}=\mathcal{B}_{i}\odot\mathcal{F}_{p}$,
where $\mathcal{B}_{i}\in\\{0,1\\}^{k\times k\times m\times n}$ is the boolean
mask matrix used to create the $\mathcal{F}_{c,i}$, and $\odot$ denotes the
elementwise multiplication operation. A value of 1 denotes that the
corresponding weight parameter in $\mathcal{F}_{p}$ needs to be multiplied by
$-1$ when substituting in $\mathcal{F}_{c,i}$; 0 denotes otherwise. Note that
although the size of this matrix is equal to $\mathcal{F}_{c,i}$ in terms of
number of parameters, it stores only single-bit information which leads to
drastic reduction in model size. To jointly learn sign-flippings along with
the model parameters using gradient descent methods, we use the optimization
paradigm as proposed in BiRealNet [17].

##  4 Experiments

|  | CIFAR-10 | CIFAR-100 | Tiny ImageNet
---|---|---|---|---
Method | Scale factor | Acc. (%) | Rel. Cont. (%) | Acc. (%) | Rel. Cont. (%) | Acc. (%) | Rel. Cont. (%)
Dense | $1\times$ | 91.19 | - | 63.35 | - | 47.37 | -
$2\times$ | 93.60 | - | 72.45 | - | 55.72 | -
$4\times$ | 94.91 | - | 77.64 | - | 59.94 | -
$8\times$ | 95.69 | - | 80.19 | - | 62.89 | -
Linear | $2\times$ | 91.13 | -2.49 | 63.12 | -2.53 | 47.03 | -4.07
$4\times$ | 91.50 | 8.33 | 63.33 | -0.14 | 46.35 | -8.11
$8\times$ | 91.46 | 6.00 | 62.65 | -4.16 | 47.03 | -2.19
$s$-RepeatNet | $2\times$ | 91.70 | 21.16 | 63.65 | 3.30 | 48.53 | 13.89
$4\times$ | 91.72 | 14.25 | 65.42 | 14.49 | 49.55 | 17.34
$8\times$ | 92.13 | 20.89 | 66.70 | 19.89 | 50.52 | 20.30
$f$-RepeatNet | $2\times$ | 91.08 | 2.69 | 65.14 | 10.09 | 50.40 | 36.29
$4\times$ | 92.67 | 39.78 | 70.81 | 52.20 | 56.80 | 75.02
$8\times$ | 93.18 | 44.22 | 75.24 | 70.61 | 60.05 | 81.70
Table 1: Performance scores obtained on the classification task of various
datasets using the base ResNet-16 model, the scaled up versions obtained using
RepeatNet as well as the dense counterparts. Dense networks are the full-
precision fully trainable networks with same network architecture as the
corresponding RepeatNet model. Here, ‘Rel. Cont.’ denotes the fraction of
improvement that the RepeatNet model obtains over the base model compared to
the equivalent dense model.

In this section, we conduct various sets of numerical experiments to
demonstrate the contribution of model’s physical size in its performance. The
first set uses repeated width scaling to increase model’s width while
maintaining the parameters. The second set shows a similar setup but to assess
the importance of depth in the model. Next, we also study the transferability
of our scaled model architectures on the downstream task of semantic
segmentation. We also show an example problem from the domain of group
equivariant CNNs [2, 27], where our experiments reveal that the performance
gains claimed to be obtained from inducing group-equivariance could partly be
due to the repetition of weights in leading to its scale-up.

###  4.1 Rescaling Network Architectures

We demonstrate the efficacy of RepeatNet for depth and width scaling on VGG
[22] and ResNet [10] models, respectively on CIFAR-10, CIFAR-100, Tiny-
ImageNet and ImageNet datasets. All the models are trained using stochastic
gradient descent method with an initial learning of 0.1, a momentum value of
0.9 and weight decay of 1e-3. For models trained on CIFAR-10 and CIFAR-100
datasets, we use 160 epochs and the learning rate is decayed by a factor of
0.1 after 80 and 120 epochs. Further, model are trained on Tiny-ImageNet and
ImageNet datasets for 90 epochs and the learning rate is decayed by a factor
of 0.1 after every 30 epochs.

Width Scaling. We study the influence of width scaling on two different ResNet
models. Using RepeatNet, we scale the base ResNet models on CIFAR-10,
CIFAR-100, TinyImageNet and ImageNet datasets. Architectures are made wider
through repetitions while ensuring that the model size is still preserved.
Table 1 reports performance scores for repetitions applied on ResNet-16 model
for CIFAR and TinyImageNet datasets. We report scores for the base model (no
scaling) as well as the various RepeatNet strategies discussed earlier. We
scale the network by factors of $2\times$, $4\times$ and $8\times$. A general
observation is that scaling the networks and making them wider improves its
performance. Interestingly, we have observed on the TinyImageNet dataset that
around 80% of the performance gain can be achieved by $f$-RepeatNet at
8$\times$ scaling than what would have been obtained using the dense
counterpart. For other cases as well, we see significant improvements at
8$\times$ scaling. An exception is the linear scaling approach where for some
instances, it is observed that the performance drops slightly compared to the
corresponding base model.

Method | Scale factor | Acc (%) | Rel. Cont. (%)
---|---|---|---
Dense | 1$\times$ | 66.24 | -
2$\times$ | 72.67 | -
4$\times$ | 76.6 | -
$f$-RepeatNet | 2$\times$ | 69.96 | 57.85
4$\times$ | 72.41 | 59.56

Table 2: Performance scores obtained on the task of classification of Imagenet
samples obtained using ResNet-18, its RepeatNet variants as well as the
corresponding dense scaled-up models. Here, ‘Rel. Cont.’ denotes the relative
contribution of RepeatNet and can be interpreted as the gain obtained by using
RepeatNet normalized by the total gain observed by the corresponding dense
architecture.

The efficacy of our RepeatNet strategy is better demonstrated through
applicability on the large-scale ImageNet-1k dataset. Table 2 shows the
performance values for the ResNet-18 architecture obtained on this dataset.
Similar to the previous experiments, we observe that models made wider using
RepeatNet delivered significantly improved performance. The RepeatNet models
are able to improve the performance of the base model (1$\times$) by
approximately 60% of what could be achieved by scaling as a dense model.

(a) CIFAR-10

(b) CIFAR-100

(c) Tiny-ImageNet

Figure 2: Performance scores for Base VGG (4-8) and RepeatNet on CIFAR-100
(a), CIFAR-100 (b) and Tiny-ImageNet (c) for different choices of number of
layers. Here, all the RepeatNet models are scaled upto VGG11.

Depth Scaling. We analyze here the applicability of RepeatNet for depthwise
scaling of CNN networks. For depth scaling, we take VGG111Unlike the standard
VGG networks, our models have only one fully-connected layer at the end.
models of different sizes spanning from VGG4 to VGG8. Further, depth scaling
is performed using RepeatNet strategy on each of our VGG models such that
their sizes become equivalent to that of VGG11. Performance scores for the VGG
models and the respective RepeatNet versions are reported in Fig. 2. It is
observed that depth scaling with RepeatNet strategy can significantly improve
the model performance on CIFAR10, CIFAR100 and Tiny-ImageNet datasets. Note
that repetitions in smaller models are more than in bigger models, thus
showing higher relative performance gain in the smaller ones.

Figure 3: Example image samples and the the corresponding ground-truth segmentation masks from Pascal-VOC dataset as well as the masks predicted using ResNet-18 and its scaled variants obtained with RepeatNet.  Method | Scale factor | mIOU (%) | Acc. (%)
---|---|---|---
| 1$\times$ | 59.5 | 89.4
Dense | 2$\times$ | 64.7 | 91.0
| 4$\times$ | 68.0 | 91.8
$f$-RepeatNet | 2$\times$ | 62.1 | 90.2
4$\times$ | 64.5 | 90.7
Table 3: Performance scores for the task of segmentation on Pascal-VOC dataset
obtained using encoders from the base ResNet-18 models, its scaled up variants
obtained with RepeatNet as well as the corresponding dense counterparts. Here,
the memory requirements of RepeatNet models are the same as the base model,
while those for the dense models scale by the corresponding ‘Scale factor’
term.

###  4.2 Rescaling Transferability

Due to the lack of enough training samples on a downstream task, and the
limited computational budget available, model weights trained on one task are
often used as initialization for some other task. For example, various
segmentation models use encoders that are extracted from the backbone of
classification models trained on ImageNet or other large-scale datasets. We
study here whether the architectures obtained after scaling with RepeatNet are
also transferable to other downstream tasks. We train ResNet-18 model for the
classification task on ImageNet-1k and use the backbone as an encoder for a
segmentation model. Segmentation performance is evaluated on the Pascal-VOC
dataset. We perform 2$\times$ and 4$\times$ width scaling of the base
architecture and compare with the respective dense counterpart. For all the
cases, a common U-Net [21] architecture with a 4 layer decoder is used. We
apply 1x1 convolutions as skip connections between encoder and decoder to map
wide activations of the encoder to their normal counterparts as input to the
decoder. This allows for a fair comparison as all the models have exactly
similar decoder network.

Table 3 shows the results of segmentation performance obtained using the base
ResNet-18 model as well as the RepeatNet variants. To highlight the
significance of the performance gains obtained with RepeatNet, we also report
scores obtained with the dense architectures for every scaled up architecture
obtained with RepeatNet. We see that Repeatnet architectures obtained with
2$\times$ and 4$\times$ scaling improve the mean IoU scores of the base model
by 2.6% and 5%, respectively. This accounts for 50% and 58.8% of the total
improvements obtained with the corresponding dense architectures. However, it
is important to note that while the model size of the RepeatNet architectures
is approximately the same as the base model, it becomes $\sim$ 4$\times$ and
$\sim$ 16$\times$ for the respective 2$\times$ and 4$\times$ scaled dense
architectures. Clearly, the improvements reported in Table 3 show that the
RepeatNet architectures are transferable across tasks.

For qualitative analysis, we also show in Fig. 3 a few randomly chosen image
samples from Pascal-VOC dataset. For these samples, we show the ground-truth
masks, those predicted using ResNet-18 model, results of RepeatNet-scaled
variants as well as those with the dense counterparts. First observation is
that the segmentation results improve for the scaled up networks, be it the
RepeatNet models or the dense ones. Moverover, the results of RepeatNet seem
visually similar to those obtained with the dense networks at same scale.

Rot-Equivariant CNN |  $s$-RepeatNet |
---|---|---
Group |  Acc. (%) | Used | Acc.(%) | Rel. Cont. (%)
- | - | ✗ | 92.6 | -
C4 | 96.5 | ✓ | 93.2 | 15.4
C8 | 96.8 | ✓ | 94.4 | 42.9
C16 | 96.3 | ✓ | 93.5 | 24.3

Table 4: Quantitative assessment of Relative Contribution (Rel. Cont.) in
accuracy observed purely due to learned repetition of weights presented as a
percentage of overall gain observed when using rotation-equivariant CNNs
instead of vanilla CNNs for the task of classification of the rot-MNIST
digits. Note that the RepeatNet models are chosen such that the inference
FLOPs are approximately similar to the respective RE-CNN model. The model
denoted with ‘✗’ indicates the vanilla CNN model with number of training
parameters equal to the RE-CNN models and is used as a baseline to evaluate
the performance gains for other models.

###  4.3 Role in Group-Equivariant Networks

In the experiments above, we have shown that it is possible to improve the
performance of any existing CNN model through scaling up its architecture
using RepeatNet. This also implies that for deep learning problems where
sharing of weights occurs, it is important that network scaling is also taken
into consideration while deducing conclusions. In this regard, we consider
here an example problem from the domain of group-equivariant CNNs. We
particularly focus on Rotation-Equivariant Steerable CNN, referred further as
RE-CNN. This approach induces rotation-equivariance into the CNN architecture
through restricting the choice of weights of the network to be build from the
combination of a certain class of basis functions, _e.g._ , Gaussian radial
basis.

We conduct here an experiment with RE-CNN to analyze whether the gain in
performance observed on rot-MNIST dataset is purely due to the inbuilt
property of rotation equivariance of the model, or whether part of it is due
to repetition of weights. Note that while we particularly focus on rotation-
equivariant models in this paper, the idea holds for other transformation
groups as well. For RE-CNN, we use steerable filter CNN 222Rotation
equivariant CNN is implemented in this paper using the e2cnn python library
available at https://github.com/QUVA-Lab/e2cnn. implementation as described in
[26] and experiment with rotation groups of 4, 8 and 16. The choice of model
architecture is the same as described in [26], except that the number of
channels in each convolutional layer are reduced by a factor of 2.

Performance scores for these models are reported in Table 4. Further, we use
the baseline vanilla CNN model with same number of optimization parameters and
upscale its size to create three $s$-RepeatNet variants that are equivalent to
the three RE-CNN models in terms of the inference FLOPs. Training as well as
testing are performed on rot-MNIST dataset. For the training of all models, we
follow the training strategy as described in [26].

From Table 4, we see that both, RE-CNN as well as RepeatNet models improve the
classification performance compared to the baseline. Interestingly, with
RepeatNet scaling, up to 42.9% of the gain is already observed than what is
obtained with RE-CNN. Given that weight sharing already improves the
performance, it is implied that part of the performance gain in RE-CNN could
actually be due to the implicit weight sharing and the resultant scaling
involved in its formulation. Clearly, it would be unfair to make a comparison
between RE-CNN and vanilla CNN at same number of optimization parameters. We
argue that any such comparison should only be performed at fixed number of
FLOPs.

##  5 Discussion

In this paper, we presented _RepeatNet_ , a novel strategy to scale CNNs and
make them deeper and wider without adding extra optimization parameters to the
model. We demonstrated through numerical experiments that our strategy of
scaling models using learnable nonlinear transformation functions can
significantly improve the performance of the existing models. The results
presented in the study reveal that performance gained from scaled networks is
only partly due to the addition of extra parameters, and part of it comes from
the increased physical size of the model itself. The relevance of these
insights is further supported from the observations on group-equivariant CNNs,
where we showed that upto 40% of the performance gain credited to the implicit
property of group equivariance in the model, could actually be due to just
weight sharing.

Method | Scale factor | Mean Acc. | Std Dev
---|---|---|---
Dense | 1$\times$ | 46.61 | 0.37
Linear | 4$\times$ | 47.21 | 0.27
8$\times$ | 47.03 | 0.34
$s$-RepeatNet | 4$\times$ | 49.48 | 0.80
8$\times$ | 51.49 | 0.50
$f$-RepeatNet | 4$\times$ | 57.22 | 0.34
8$\times$ | 60.14 | 0.45
Table 5: Mean accuracies (in %) and standard deviation scores computed over 10
runs for the base ResNet-16 model and its RepeatNet variants on TinyImageNet
dataset.

While we have reported massive absolute improvements with network scaling, the
significance of these results can only be validated based on the error
observed in the performance across multiple runs. In this regard, we performed
10 runs for several variants of RepeatNet for ResNet-16 model on the Tiny-
ImageNet dataset. Table 5 reports the details of the different runs including
mean accuracy score and the deviation. We observe that the deviations in mean
accuracy scores are very small for the base model (1$\times$) as well as its
RepeatNet variants. The maximum is reported for $s$-RepeatNet with a standard
deviation score of 0.80. However, compared to the difference in accuracy
scores for the base model and the RepeatNet variants, it is clear that the
deviation is small, especially for our $s$-RepeatNet and $f$-RepeatNet models.
This demonstrates that the improvements gained by RepeatNet are significant.
In future, it would be of interest to conduct a similar study on other
experiments as well.

Limitations. We have shown that scaling networks using RepeatNet leads to
improved model performance. However, there are are several limitations as
well. First, there is still some gap in performance scores between RepeatNet
and the respective dense implementations. We believe that with more improved
nonlinear projection strategies, this gap can be narrowed down further. We
further speculate that a hybrid of depth and width scaling might work better
that both of these individually, however, this has not been explored, and it
could be a future direction to explore further.

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