# Testing black holes in non-linear electrodynamics from the observed quasi-
periodic oscillations

Indrani<EMAIL_ADDRESS>1
1Department of Physics and Astronomy, National Institute of Technology,
Rourkela-769008, India

###### Abstract

Quasi-periodic oscillations (QPOs), in particular, the ones with high
frequencies, often observed in the power spectrum of black holes, are useful
in understanding the nature of strong gravity since they are associated with
the motion of matter in the vicinity of the black hole horizon. Interestingly,
these high frequency QPOs (HFQPOs) are observed in commensurable pairs, the
most common ratio being 3:2. Several theoretical models are proposed in the
literature which explain the HFQPOs in terms of the orbital and epicyclic
frequencies of matter rotating around the central object. Since these
frequencies are sensitive to the background spacetime, the observed HFQPOs can
potentially extract useful information regarding the nature of the same. In
this work, we investigate the role of regular black holes with a Minkowski
core, which arise in gravity coupled to non-linear electrodynamics, in
explaining the HFQPOs. Regular black holes are particularly interesting as
they provide a possible resolution to the singularity problem in general
relativity. We compare the model dependent QPO frequencies with the available
observations of the quasi-periodic oscillations from black hole sources and
perform a $\chi^{2}$ analysis. Our study reveals that most QPO models favor
small but non-trivial values of the non-linear electrodynamics charge
parameter. In particular, black holes with large values of non-linear
electrodynamics charge parameter are generically disfavored by present
observations related to QPOs.

##  1 Introduction

The theorems of Hawking and Penrose [1] state that black hole (henceforth as
BH) singularities are an unavoidable feature of general relativity. However,
it is believed that spacetime singularities should not exist in Nature and a
suitable theory of quantum gravity can address this issue. In this regard
various quantum gravity models have been put forward [2, 3, 4, 5, 6, 7, 8, 9,
10]. In the absence of a well established quantum gravity model one can
address the black hole singularity problem classically by studying black holes
with regular cores. Such black holes have horizons such that their metric and
curvature invariants are non-singular at all points in spacetime. Motivated by
quantum arguments [11, 12, 13] it was proposed that the central region of the
black hole at $r\simeq 0$ should be de-Sitter like which was studied
extensively [14, 15, 16]. These works propound that quantum fluctuations
prevent the unlimited increase of spacetime curvature during the stellar
collapse process giving rise to black holes with regular cores.

The first regular black hole solution with a de-Sitter core was proposed by
Bardeen [17] which turns out to be a solution of Einstein’s equations coupled
to non-linear electrodynamics. Later Ayon-Beato and Garcia [18] explained that
the physical source that gives rise to the Bardeen solution corresponds to the
gravitational field of a nonlinear magnetic monopole of a self-gravitating
magnetic field. Several other regular black hole solutions of gravity coupled
to non-linear electrodynamics have been obtained later [18, 19, 20]. Since
then there has been a lot of research in the direction of investigating
regular black holes [21, 22, 23, 24, 25, 26, 27, 28].

In this work, we study a class of regular black hole in non-linear
electrodynamics with an asymptotically Minkowski core. Such a metric arises as
the solution of Einstein’s equations with an anisotropic fluid resembling the
Maxwell stress tensor far from the source [29]. The electric field from such a
source asymptotically resembles the Coulomb field, is bound at all distances,
and the corresponding geometry is like that of Reissner-Nordstrom (RN) metric
far from the source. The motivation for exploring such black hole solution
stems from the fact that the mass function has an exponential convergence
factor which makes the quantum gravity model finite to all orders upto the
Planck scale [30]. Further, study of a finite quantum gravity is important as
it can address the cosmological constant problem [31] and can eliminate the
divergences arising in flat space quantum field theories.

Since black holes observed in nature are in general rotating the stationary
and axi-symmetric counterpart of the aforesaid regular solution with Minkowski
core have been worked out by applying the Newman-Janis algorithm [32] and
other methods [33, 34, 35]. These algorithms also enable one to derive regular
black hole solutions inspired from non-commutative geometry with a regular de-
Sitter toroidal core [36]. Since we will be interested in exploring the
aforesaid metric in the context of astrophysical observations we work with the
rotating counterpart in this work [37]. As expected, the rotating solution
resembles the Kerr-Newman metric far from the source. When the radiating
counterpart of this spacetime is considered one obtains generalization of
Carmeli’s spacetime as well as Vaidya’s spacetime in suitable limits.

In this work our goal is to extract signatures of the non-linear
electrodynamics charge parameter from the quasi-periodic oscillations (QPOs)
observed in the power density spectrum of black holes. QPOs generally appear
as peaks in the power spectrum of Low-Mass X-ray binaries (LMXRBs) which
includes black hole and neutron star sources. Although rare, certain active
galactic nuclei (AGNs) also exhibit QPOs in the power spectrum. It is believed
that QPOs, in particular, the ones with high frequencies (henceforth referred
as HFQPOs) encapsulate information regarding the nature of strong gravity
associated with these compact objects and hence these can serve as effective
tools to study the alternatives to general relativity[38, 39]. The QPO
frequencies are inversely proportional to the mass of the compact objects,
such that for neutron stars HFQPOs $\sim$ kHz, for stellar-mass black holes
they are $\sim$ hundreds of Hz while for supermassive black holes they are in
the mHz order [40, 41]. The order of magnitude of the HFQPOs can be directly
attributed to the timescales associated with the motion of matter close to the
compact objects. This can be directly seen from the fact that the dynamical
time scale $t_{d}\sim r^{3}/GM\sim$ milli-seconds for neutron stars and
stellar-mass BH sources when $r<10~{}r_{g}$ is considered [40, 41, 42, 43].
This was predicted in the 1970s which later received confirmation with the
launch of NASA’s Rossi X-Ray Timing Explorer satellite. Thus, QPOs provide
unique opportunity to explore the nature of strong gravity and dense matter.
Since we intend to investigate the nature of strong gravity, i.e. to study the
role of the NED charge parameter in explaining the observed QPOs we will
concentrate only on black hole sources in this work.

The paper is organized in the following way: In 2 we discuss the black hole
solution considered in this work which arises when gravity is coupled to non-
linear electrodynamics. The motion of test particles in such a spacetime is
studied in 3 and the dependence of the epicyclic frequencies on the background
spacetime is derived. 4 is dedicated in reviewing the various theoretical
models proposed to explain the observed QPOs. In 5 we compare the various
model dependent QPO frequencies with observations and perform a $\chi^{2}$
analysis to derive the observationally favored NED charge parameter. We
conclude with a summary of our results and discuss some avenues for future
work in 6.

##  2 Regular rotating black holes in non-linear electrodynamics with a
Minkowski core

In this work we consider regular black holes in non-linear electrodynamics
with an asymptotically Minkowski core. The action corresponding to non-linear
electrodynamics coupled to Einstein gravity is given by [44, 18, 45, 13, 46,
47],

$\displaystyle S=\int
d^{4}x\sqrt{-g}\bigg{(}\frac{\mathcal{R}}{16\pi}-\frac{L(F)}{4\pi}\bigg{)}$
(1)

where $\mathcal{R}$ is the Ricci scalar and $L(F)$ is the Lagrangian density
associated with non-linear electrodynamics such that
$F=F^{\mu\nu}F_{\mu\nu}/4$ is the Faraday invariant and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field strength tensor with $A_{\mu}$ the gauge field. The
Maxwell theory is retrieved in the weak field limit when $L(F)=F$. Varying the
action with respect to the metric yields the Einstein’s equations with $L(F)$
as the source,

$\displaystyle
G_{\mu\nu}=2(L_{F}F_{\mu}^{\sigma}F_{\nu\sigma}-g_{\mu\nu}L(F))$ (2)

where $L_{F}=\frac{\partial L}{\partial F}$ and $G_{\mu\nu}$ is the Einstein
tensor. On the other hand, varying the action with respect to $A_{\mu}$ yields
the equation of motion corresponding to non-linear electrodynamics,

$\displaystyle\\{L_{F}F^{\mu\nu}\\}_{;\mu}=0~{}~{}~{}~{}(^{*}F^{\mu\nu})_{;\nu}=0$
(3)

where ${}^{*}F^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ is the
Hodge-dual of $F^{\mu\nu}$. When the Lagrangian density corresponding to non-
linear electrodynamics is considered to be,

$\displaystyle L(F)=Fe^{-\alpha(2q^{2}F)^{1/4}}$ (4)

with $\alpha=qc^{2}/(2G\tilde{\mathcal{M}})$ ($q$ being the charge and
$\tilde{\mathcal{M}}$ the mass of the black hole), the static, spherically
symmetric and asymptotically flat solution of 2 assumes the form,

$\displaystyle
ds^{2}=-\Bigg{(}1-\frac{2\tilde{m}(r)}{r}\Bigg{)}dt^{2}+\frac{dr^{2}}{\Bigg{(}1-\frac{2\tilde{m}(r)}{r}\Bigg{)}}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})$ (5)

with the mass function

$\displaystyle\tilde{m}(r)=\tilde{\mathcal{M}}e^{-\kappa/r}$ (6)

where $\kappa=\frac{q^{2}c^{2}}{2G\tilde{\mathcal{M}}}$ has dimensions of
length. The above spacetime has two horizons [48, 44]. The outer horizon
corresponds to
$r_{+}=2\tilde{\mathcal{M}}e^{W_{0}\big{(}-\frac{\kappa}{\tilde{\mathcal{M}}}\big{)}}$
while the inner horizon corresponds to
$r_{-}=2\tilde{\mathcal{M}}e^{W_{-1}\big{(}-\frac{\kappa}{\tilde{\mathcal{M}}}\big{)}}$,
where $W_{0},W_{-1}$ corresponds to the Lambert W function. When $r=\kappa$
the two horizons merge resulting in the extremal black hole scenario.

The above metric arises from the solution of Einstein’s equations with the
source,

$\displaystyle T^{0}_{0}=-\rho(r)=\frac{-\tilde{M}\kappa}{4\pi
r^{4}}e^{-\kappa/{r}};$ $\displaystyle
T^{1}_{1}=-\rho(r)=\frac{-\tilde{M}\kappa}{4\pi r^{4}}e^{-\kappa/{r}};$
$\displaystyle T^{2}_{2}=T^{3}_{3}=\frac{\tilde{M}\kappa}{4\pi
r^{4}}(1-\frac{\kappa}{2r})e^{-\kappa/r}$ (7)

The energy density and pressure described by 7 tends to zero as $r\to\infty$
and are non-singular at $r=0$. We further note that the above energy momentum
tensor satisfies the weak energy condition and goes over to the Maxwell stress
tensor far from the horizon [49].

Since black holes in astrophysics are in general rotating, studying rotating
black hole solutions of the above Einstein’s equations are more important.
Such a solution is generated by applying the Newman Janis algorithm [32, 33,
34, 35] and turns out to be

$\displaystyle ds^{2}$
$\displaystyle=-\bigg{(}1-\frac{2\tilde{m}(r)r}{\tilde{\Sigma}}\bigg{)}dt^{2}-\frac{4a\tilde{m}(r)r}{\tilde{\Sigma}}\sin^{2}\theta
dtd\phi+\frac{\tilde{\Sigma}}{\Delta}dr^{2}$
$\displaystyle+\tilde{\Sigma}d\theta^{2}+\bigg{(}r^{2}+a^{2}+\frac{2\tilde{m}(r)ra^{2}}{\tilde{\Sigma}}\sin^{2}\theta\bigg{)}\sin^{2}\theta
d\phi^{2}$ (8)

where,

$\displaystyle\tilde{\Sigma}=r^{2}+a^{2}\cos^{2}\theta~{}{,}~{}\Delta=r^{2}+a^{2}-2\tilde{m}(r)r$
(9)

and $\tilde{m}(r)$ is the mass function mentioned in 6 such that
$lim_{r\rightarrow\infty}\tilde{m}(r)=\tilde{\mathcal{M}}$ and $a$ is the spin
parameter of the black hole. Since it is computationally easier to handle
dimensionless quantities we scale $\kappa$ and $r$ in 6 by the gravitational
radius $r_{g}=GM/c^{2}$. Thus the dimensionless metric parameters correspond
to the squared charge to mass ratio
$k=\kappa/r_{g}=\frac{q^{2}c^{4}}{2G^{2}\tilde{\mathcal{M}}^{2}}$ and the spin
parameter $a\equiv a/r_{g}$.

The above metric goes over to the Kerr-Newman spacetime when $r>>\kappa$. It
is important to note that the above black hole solution does not have the
curvature singularity at $r=0$ and has an asymptotically Minkowski core. In
such a situation the energy density $\rho(r)\to 0$ as $r\to 0$ unlike a de-
Sitter core where the energy density becomes constant at the core. It turns
out that the curvature tensor and invariants in the above spacetime are
considerably simpler than the Bardeen metric and has several physically
interesting features defined by the Lambert W function [50, 51, 52, 53, 54,
55, 56, 49].

In the absence of nonlinear electrodynamics when $\kappa=0$ the above black
hole solution reduces to the Kerr metric. The event horizon of the above
metric is obtained by solving for the roots of $g^{rr}=\Delta=0$ which gives,

$\displaystyle r^{2}+a^{2}-2re^{-k/r}=0$ (10)

For the metric in 2 to represent a black hole real, positive horizons must be
present. This sets the physically allowed range of $k$ to $0\lesssim k\lesssim
0.7$ [57].

In the next section we discuss the derivation of the epicyclic frequencies of
the accreting test particles in the above spacetime.

##  3 Epicyclic frequencies of test particles in a stationary axisymmetric
black hole spacetime

This section investigates motion of massive test particles around a rotating
black hole described by a stationary, axisymmetric spacetime whose metric is
given by,

$\displaystyle
ds^{2}=g_{tt}dt^{2}+2g_{t\phi}dtd\phi+g_{\phi\phi}d\phi^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}~{},$
(11)

We assume that the above spacetime has reflection symmetry such that
$g_{\mu\nu}(r,\theta)=g_{\mu\nu}(r,-\theta)$. Stationarity and axisymmetry
implies $\partial_{t}$ and $\partial_{\phi}$ are Killing vectors such that the
specific energy $E$ and the specific angular momentum $L$ of the test
particles are conserved. The test particles moving in such a spacetime move in
an effective potential given by,

$\displaystyle V(r,\theta)=(g^{tt}-2lg^{t\phi}+l^{2}g^{\phi\phi})$ (12)

which can be obtained from the invariance of the rest mass of the test
particles such that,

$\displaystyle\dot{r}^{2}g_{rr}+\dot{\theta}^{2}g_{\theta\theta}+E^{2}V(r,\theta)=-1$
(13)

where $l=L/E$ is the impact parameter in 12. If we consider circular orbits in
the equatorial plane the above equation reduces to

$\displaystyle E^{2}V(r_{0},\pi/2)=-1$ (14)

where the radius of the circular orbit is denoted by $r_{0}$. The radial
Euler-Lagrange equation leads to a quadratic equation for the angular
frequency

$\displaystyle g_{tt,r}+2\Omega g_{t\phi,r}+\Omega^{2}g_{\phi\phi,r}=0~{}.$
(15)

which when solved gives

$\displaystyle\Omega=\frac{-g_{t\phi,r}\pm\sqrt{g_{t\phi,r}^{2}-g_{tt,r}g_{\phi\phi,r}}}{g_{\phi\phi,r}}$
(16)

where $\pm$ sign represents prograde and retrograde orbits respectively. We
next consider slight perturbations in the motion of the test particle from the
circular orbit and the equatorial plane which are denoted by,

$\displaystyle r(t)\simeq r_{\rm 0}+\delta
r_{0}~{}e^{i\omega_{r}t}~{};\qquad\theta(t)\simeq\frac{\pi}{2}+\delta\theta_{0}~{}e^{i\omega_{\theta}t}~{}.$
(17)

where $\omega_{r}$ and $\omega_{\theta}$ correspond to radial and vertical
epicyclic frequencies respectively. Substituting 17 in 13 and Taylor expanding
$U(r,\theta)$ about $r=r_{0}$ and $\theta=\pi/2$ we get,

$\displaystyle-\delta
r_{0}^{2}\omega_{r}^{2}(u^{t})^{2}g_{rr}-\delta\theta^{2}\omega_{\theta}^{2}(u^{t})^{2}g_{\theta\theta}+E^{2}\bigg{[}V(r_{0},\pi/2)+\frac{1}{2}\frac{\partial^{2}V}{\partial
r^{2}}\Bigg{|}_{r=r_{0},\theta=\pi/2}\delta
r^{2}+\frac{1}{2}\frac{\partial^{2}V}{\partial\theta^{2}}\Bigg{|}_{r=r_{0},\theta=\pi/2}\delta\theta^{2}\bigg{]}=-1$
(18)

In 18 we retain terms upto quadratic order such that the motion in the radial
and the vertical direction are uncoupled. Using 14 in 18 and by equating the
coefficients of $\delta r^{2}$ and $\delta\theta^{2}$ on both sides of 18 we
obtain radial and the vertical epicyclic frequencies of the test particles in
the above spacetime which are respectively given by,

$\displaystyle\omega_{r}^{2}=\frac{c^{6}}{G^{2}\tilde{M}^{2}}\frac{(g_{tt}+\Omega
g_{t\phi})^{2}}{2g_{rr}}\frac{\partial^{2}V}{\partial
r^{2}}\bigg{|}_{r=r_{0},\theta=\pi/2}$ (19)
$\displaystyle\omega_{\theta}^{2}=\frac{c^{6}}{G^{2}\tilde{M}^{2}}\frac{(g_{tt}+\Omega
g_{t\phi})^{2}}{2g_{\theta\theta}}\frac{\partial^{2}V}{\partial\theta^{2}}\bigg{|}_{r=r_{0},\theta=\pi/2}$
(20)

In 20 the radial and vertical epicyclic frequencies are multiplied by the
factor $(c^{6}/G^{2}\tilde{M}^{2})$ so that they have dimensions of frequency
squared. We note that these are functions of the black hole mass $\tilde{M}$,
the radius at which these oscillations are generated $r_{em}$ and the metric
parameters $k$ and $a$.

1 below enlists some BH sources where QPOs have been discovered. We are
chiefly interested in high frequency QPOs in this work which for stellar mass
black holes $\sim$ hundreds of Hz while for supermassive black holes the QPO
frequencies $\sim$ mHz[42, 58, 59, 40, 41, 43]. This can be understood in the
following way. The deviation of the $g_{tt}$ component of the metric from the
Minkowski spacetime corresponds to the Newtonian potential. In our case, if we
consider the spherically symmetric metric given by 5 the Newtonian potential
is given by,

$\displaystyle\Phi=\frac{G\tilde{\mathcal{M}}}{r}e^{-k/r}$ (21)

The force per unit mass can thus be evaluated,

$\displaystyle
F=-\frac{d\Phi}{dr}=\frac{G\tilde{\mathcal{M}}}{r^{2}}e^{-k/r}(1-k/r)=\frac{v^{2}}{r}$
(22)

Therefore the velocity is given by:

$\displaystyle
v\simeq\sqrt{\frac{G\tilde{\mathcal{M}}}{r}\bigg{(}1-\frac{k}{r}\bigg{)}}e^{-k/2r}$
(23)

Taking $r\simeq 8r_{g}$ and $k\simeq 0.2$ we note that

$\displaystyle v\simeq 0.988*\sqrt{\frac{G\tilde{\mathcal{M}}}{r}0.975}$ (24)

The dynamical timescale associated to matter rotating around the black hole is
therefore given by,

$\displaystyle t_{d}=\frac{r}{v}\sim
1.025\sqrt{\frac{r^{3}}{G\tilde{\mathcal{M}}}}$ (25)

which for a $10M_{\odot}$ black hole at $r\simeq 8r_{g}$ turns out to be
$t_{d}\sim 1.11ms$. The corresponding frequency is therefore $\sim$ hundreds
of Hertz.

Theoretical models aimed to explain these QPOs are dependent on the epicyclic
frequencies which are inversely proportional to the black hole mass which
explains the reason for lower frequencies in supermassive BHs [42, 58, 59].
Apart from HFQPOs, low frequency QPOs (LFQPOs) are also observed in some
sources. We denote the observed HFQPOs by $\nu_{u1}$ and $\nu_{u2}$ while the
low frequency QPO is denoted by $\nu_{L}$.

We note that RE J1034+396 galaxy exhibits only a single QPO in its power
spectrum [42, 60, 61, 62]. From 1 we note that HFQPOs in the BH sources
generally appear in the ratio of 3:2 and we will consider only those sources
in 1 (namely, the first five sources) which exhibit the 3:2 ratio HFQPOs.
Therefore the data related to RE J1034+396 galaxy will not be used for
subsequent analysis.

$\rm Source$ | $\rm Mass$ | $\nu_{u1}\pm\Delta\nu_{u1}$ | $\nu_{u2}\pm{\Delta}\nu_{u2}$ | $\nu_{L}\pm\Delta\nu_{L}$
---|---|---|---|---
| $\rm(\rm M_{\rm BH~{}}in~{}M_{\odot})$ | $(\rm in~{}Hz)$ | $\rm(\rm in~{}Hz)$ | $\rm(\rm in~{}Hz)$
$\rm GRO~{}J1655-40$ |  $\rm 5.4\pm 0.3$ [63] |  $\rm 441\rm\pm 2$ [64] |  $\rm 298\rm\pm 4$ [64] |  $\rm 17.3\pm\rm 0.1$ [64]
$\rm XTE~{}J1550-564$ |  $\rm 9.1\pm 0.61$ [65] | $\rm 276\rm\pm 3$ | $\rm 184\pm 5$ | $-$
$\rm GRS~{}1915+105$ |  $\rm 12.4^{+2.0}_{-1.8}$ [66] | $\rm 168\pm 3$ | $\rm 113\pm 5$ | $\rm-$
$\rm H~{}1743+322$ |  $\rm 8.0-14.07$ [67, 68, 69] | $\rm 242\pm 3$ | $\rm 166\pm 5$ | $\rm-$
$\rm Sgr~{}A^{*}$ | $\rm(3.5-4.9)$ | $\rm(1.445\pm 0.16)$ | $\rm(0.886\pm 0.04)$ | $-$
|  $\rm~{}\times 10^{6}$ [70, 71] |  $\rm~{}\times 10^{-3}$ [58, 72] |  $\rm~{}\times 10^{-3}$ [58, 72] | $-$
$\rm REJ1034+396$ | $\rm(1-4)~{}\times 10^{6}$ | $\rm(2.5-2.8)\rm~{}\times 10^{-4}$ | $-$ | $-$
| [42, 73, 74, 75] | [42, 60, 61, 62] |  |
Table 1: Black hole sources where high frequency QPOs (HFQPOs) are observed $\rm Model$ | $\nu_{1}$ | $\nu_{2}$ | $\nu_{3}$
---|---|---|---
$\rm Relativistic~{}Precession~{}Model~{}(kinematic)$ [76, 43, 77] | $\rm\nu_{\phi}$ | $\rm\nu_{\phi}-\nu_{r}$ | $\rm\nu_{\phi}-\nu_{\theta}$
$\rm Tidal~{}Disruption~{}Model~{}(kinematic)$ [78, 79, 80] | $\rm\nu_{\phi}+\nu_{r}$ | $\rm\nu_{\theta}$ | $-$
$\rm Parametric~{}Resonance~{}Model~{}(resonance)$ [81, 82, 83] | $\rm\nu_{\theta}$ | $\rm\nu_{r}$ | $-$
$\rm Forced~{}Resonance~{}Model~{}1~{}(resonance)$ [81] | $\rm\nu_{\theta}$ | $\rm\nu_{\theta}-\nu_{r}$ | $-$
$\rm Forced~{}Resonance~{}Model~{}2~{}(resonance)$ [81] | $\rm\nu_{\theta}+\nu_{r}$ | $\rm\nu_{\theta}$ | $-$
$\rm Keplerian~{}Resonance~{}Model~{}1~{}(resonance)$ [84] | $\rm\nu_{\phi}$ | $\rm\nu_{r}$ | $-$
$\rm Keplerian~{}Resonance~{}Model~{}2~{}(resonance)$ [84] | $\rm\nu_{\phi}$ | $\rm 2\nu_{r}$ | $-$
$\rm Keplerian~{}Resonance~{}Model~{}3~{}(resonance)$ [84] | $\rm 3\nu_{r}$ | $\rm\nu_{\phi}$ | $-$
$\rm Warped~{}Disk~{}Oscillation~{}Model~{}~{}(resonance)$ | $\rm 2\nu_{\phi}-\nu_{r}$ | $\rm 2(\nu_{\phi}-\nu_{r})$ | $-$
$\rm Non-axisymmetric~{}Disk~{}Oscillation~{}Model~{}1~{}(resonance)$ | $\rm\nu_{\theta}$ | $\rm\nu_{\phi}-\nu_{r}$ | $-$
$\rm Non-axisymmetric~{}Disk~{}Oscillation~{}Model~{}2~{}(resonance)$ [85, 86, 87] | $\rm 2\nu_{\phi}-\nu_{\theta}$ | $\rm\nu_{\phi}-\nu_{r}$ | $-$
Table 2: Theoretical models explaining the HFQPOs in black holes.

##  4 Theoretical Models explaining quasi-periodic oscillations in the black
hole power spectrum

This section is dedicated in reviewing some of the existing theoretical models
proposed to explain the observed HFQPOs in BHs [76, 43, 77, 78, 79, 80, 81,
82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93]. These models mainly aim to
explain the commensurability of the QPO frequencies. 2 presents the
mathematical expressions for the model dependent QPO frequencies where the
upper and lower HFQPOs are denoted by $\nu_{1}$ and $\nu_{2}$ while the low
frequency QPO is denoted by $\nu_{3}$. From the table it is evident that the
theoretical QPO frequencies are linear combinations of the angular frequency,
the radial and the vertical epicyclic frequencies respectively given by
$\nu_{\phi}=\frac{\omega_{\phi}}{2\pi}$, $\nu_{r}=\frac{\omega_{r}}{2\pi}$ and
$\nu_{\theta}=\frac{\omega_{\theta}}{2\pi}$. Since the theoretical QPO
frequencies depend purely on the background metric and not on the complex
accretion physics, QPOs can potentially extract more accurate information
about the background spacetime compared to other available observations, e.g.,
the iron line or the continuum-fitting methods.

  * •

Parametric resonance model:  The 3:2 ratio of the observed twin-peak HFQPOs
(with frequencies $\rm\nu_{u1}$ and $\rm\nu_{u2}$) in BH and NS sources
indicates that QPOs might be a consequence of resonance between various
oscillation modes in the accretion disk [94, 81, 95]. When we considered
circular, equatorial motion of test particles and slight perturbations $\delta
r$ and $\delta\theta$ in the orbit we tacitly assumed that the perturbations
obey the equations of simple harmonic motion such that

$\displaystyle\delta\ddot{r}+\omega_{r}^{2}\delta
r=0~{};\qquad\delta\ddot{\theta}+\omega_{\theta}^{2}\delta\theta=0~{}.$ (26)

with frequencies $\omega_{r}=2\pi\nu_{r}$ and
$\omega_{\theta}=2\pi\nu_{\theta}$ respectively. This implies that the motion
in the radial and the vertical direction are uncoupled which is applicable to
both thin disks as well as to more general accretion flow models, e.g.
accretion tori [94, 81] The presence of dissipation and pressure effects
necessitate including forcing terms in 26

$\displaystyle\delta\ddot{r}+\omega_{r}^{2}\delta r=\omega_{r}^{2}F_{r}(\delta
r,\delta\theta,\delta\dot{r},\delta\dot{\theta})~{};\qquad\delta\ddot{\theta}+\omega_{\theta}^{2}\delta\theta=\omega_{\theta}^{2}F_{\theta}(\delta
r,\delta\theta,\delta\dot{r},\delta\dot{\theta})~{}.$ (27)

which are in general some non-linear functions of their arguments. The forms
of $F_{r}$ and $F_{\theta}$ are determined by the accretion flow model [82,
96].

In the parametric resonance model it is assumed that the radial epicyclic
motion triggers the vertical epicyclic motion since random fluctuations in
thin disks are expected to have $\delta r\gg\delta\theta$ [81, 94, 82, 97,
83]. In this scenario 27 assume the form,

$\displaystyle\delta\ddot{r}+\omega_{r}^{2}\delta
r=0~{}~{}~{}~{}~{}~{}\delta\ddot{\theta}+\omega_{\theta}^{2}\delta\theta=-\omega_{\theta}^{2}\delta
r\delta\theta$ (28)

such that $\delta r=B\cos(\omega_{r}t)$ (where $B$ is a constant) and the
equation for $\delta\theta$ assumes the form of the Matthieu equation [98]
given by

$\displaystyle\delta\ddot{\theta}+\omega_{\theta}^{2}(1+B\delta
r)\delta\theta=0$ (29)

and is excited when [83, 98, 82]

$\displaystyle\frac{\nu_{r}}{\nu_{\theta}}=\frac{2}{n}~{},\qquad\rm
where~{}n\in~{}positive~{}integers$ (30)

For rotating non-singular BHs $\nu_{\theta}>\nu_{r}$ [72] (where
$\nu_{\theta}=\omega_{\theta}/2\pi$ and $\nu_{r}=\omega_{r}/2\pi$) and
therefore $n=3$ gives rise to the strongest resonance which naturally explains
the observed $3:2$ ratio of the HFQPOs. This result has been confirmed by
analytical calculations and numerical simulations [83, 96, 82].

  * •

Forced Resonance Model: Thin disks or nearly Keplerian disks are more likely
to exhibit parametric resonance [94, 81, 95]. In a more realistic flow
however, couplings between $\delta r$ and $\delta\theta$ are expected in
addition to parametric resonance, arising from pressure, viscous or magnetic
stresses present in the accretion flow leading to non-zero forcing terms [97].
Numerical simulations confirm that pressure couplings often lead to a resonant
forcing of vertical oscillations by radial oscillations [94, 99]. Since the
physics of accretion that leads to such resonant couplings is not very clearly
understood the forcing terms are given by some mathematical ansatz, e.g.,

$\displaystyle\delta\ddot{\theta}+\omega_{\theta}^{2}\delta\theta=-\omega_{\theta}^{2}\delta
r\delta\theta+\mathcal{F_{\theta}}(\delta\theta)$ (31)

such that $\delta r=Bcos(\omega_{r}t)$ while $\mathcal{F_{\theta}}$
corresponds to the non-linear terms in $\delta\theta$. It can be shown that 31
has solutions of the form,

$\displaystyle\frac{\nu_{\theta}}{\nu_{r}}=\frac{m}{n}\rm~{}~{}~{}~{}~{}where~{}m~{}and~{}n~{}are~{}natural~{}numbers$
(32)

The presence of non-linear couplings lead to forced resonances, e.g. $m:n=3:1$
and $m:n=2:1$, apart from the $3:2$ parametric resonance which permits
resonance between combinations of frequencies, e.g. $\nu_{\theta}-\nu_{r}$,
$\nu_{\theta}+\nu_{r}$. The forced resonance model comprises of $3:1$ and
$2:1$ forced resonances. In $3:1$ forced resonance model (denoted by Forced
Resonance Model 1 or FRM1), the lower HFQPO is given by
$\nu_{2}=f_{-}=\nu_{\theta}-\nu_{r}$ while the upper HFQPO is given by
$\nu_{1}=\nu_{\theta}$. For $2:1$ forced resonance model the upper and lower
high frequency QPOs are denoted by $\nu_{1}=f_{+}=\nu_{\theta}+\nu_{r}$ while
$\nu_{2}=\nu_{\theta}$ respectively.

  * •

Keplerian resonance model:  In Keplerian resonance model one considers
resonance between the orbital motion and the radial epicyclic motion [97, 100,
94, 84]. Keplerian resonance might arise under two possible circumstances: (a)
trapping of non-axisymmetric g-mode oscillations induced by a corotation
resonance in the inner region of relativistic thin accretion disks [100] and
(b) when a pair of spatially separated coherent vortices with opposite
vorticities oscillating with radial epicyclic frequencies couple with the
spatially varying orbital angular frequencies [97, 101]. However, it was
realized that g-mode oscillations are dampened by corotation resonance [102,
103] such that models invoking Keplerian resonance may not be very useful in
explaining the HFQPOs in microquasars. Keplerian model consists of resonance
between (i) $\nu_{1}=\nu_{\phi}$ and $\nu_{2}=\nu_{r}$ (which we denote as
Keplerian Resonance Model 1 or KRM1), (ii) $\nu_{1}=\nu_{\phi}$ and
$\nu_{2}=2\nu_{r}$ (which we denote as Keplerian Resonance Model 2 or KRM2)
and (iii) $\nu_{1}=3\nu_{r}$ and $\nu_{2}=\nu_{\phi}$ (which we denote as
Keplerian Resonance Model 3 or KRM3).

  * •

Warped disk oscillation model: This model which assumes a somewhat unusual
disk geometry [86, 104] explains the HFQPOs in terms of non-linear resonances
between the relativistic disk deformed by a warp with various disk oscillation
modes [105, 106, 107, 108, 109]. Such resonances comprise of horizontal
resonances inducing g-mode and p-mode oscillations as well as vertical
resonances which can induce only the g-mode oscillations [106]. The origin of
such resonances can be attributed to the non-monotonic variation of the radial
epicyclic frequency with the radial distance $r$ [106]. According to this
model the upper high frequency QPO is given by $\nu_{1}=2\nu_{\phi}-\nu_{r}$
while the lower HFQPO is given by $\nu_{2}=2(\nu_{\phi}-\nu_{r})$.

  * •

Non-axisymmetric disk oscillation model: Non-axisymmetric Disk-Oscillation
models consider various combinations of non-axisymmetric disc oscillation
modes as the origin of the HFQPOs [110, 111, 112, 86, 87]. These models which
are variants of the Relativistic Precession model [86] include non-geodesic
effects in the accretion flow by modelling the flow in terms of a slightly
non-slender pressure-supported perfect fluid torus [113, 114, 87]. Two
variants of non-axisymmetric disk-oscillation models are proposed in the
literature: (i) the first (denoted by NADO1) [110, 111, 112]) assumes
resonance between the vertical epicyclic frequency ($\nu_{1}=\nu_{\theta}$)
with $m=-1$ non-axisymmetric radial epicyclic frequency
($\nu_{2}=\nu_{\phi}-\nu_{r}$) while the second (denoted by NADO2) considers
resonance between the $m=-1$ non-axisymmetric radial epicyclic frequency
($\nu_{2}=\nu_{\phi}-\nu_{r}$) with the $m=-2$ non-axisymmetric vertical
epicyclic frequency ($\nu_{1}=2\nu_{\phi}-\nu_{\theta}$) [85, 86, 87] where
$m$ refers to the azimuthal wave number of the non-axisymmetric perturbation.
However, the physical mechanisms inducing couplings between an axisymmetric
and a non-axisymmetric mode or between the pairs non-axisymmetric modes are
yet not very well-understood [115, 86].

  * •

Relativistic precession model: Models discussed so far are examples of
resonance models. Relativistic precession model [76, 43] is a kinematic model
which considers local motion of plasma in the accretion disk as the origin of
QPOs. This model was initially proposed to address the HFQPOs in neutron star
sources and then subsequently extended for black holes [77]. In this model the
upper and lower high frequency QPOs are associated with the orbital anugular
frequency $\nu_{\phi}$ and the periastron precession frequency
$\nu_{\phi}-\nu_{r}$. This model also addresses the observed low frequency QPO
in black holes in terms of the nodal precession frequency
$\nu_{\phi}-\nu_{\theta}$.

  * •

Tidal disruption model: Another example of a kinematic model is the Tidal
Disruption Model [78, 79, 80] which aims to explain the observed HFQPOs in
terms of plasma orbiting the central object which may get tidally stretched
forming ring-like features along the orbit giving rise to the modulation in
the observed flux in the black hole power spectrum. According to this model
$\nu_{1}=\nu_{\phi}+\nu_{r}$ and $\nu_{2}=\nu_{\phi}$.

We note from the above discussion that the theoretical QPO frequencies depend
on the metric parameters $a$, $k$, the black hole mass $\tilde{M}$ and the
radius at which the QPO frequencies are generated $r_{em}$. Therefore we can
directly constrain $k$ and not $q$ from the observations. Once we know $k$ we
can calculate $q$ by using the previously determined masses. This is an
artifact of the dependence of the Lagrangian density on mass. In this work $k$
is denoted to be the non-linear electrodynamics (NED) charge parameter.

The QPO models we have considered here assume that the resonances involved in
giving rise to the QPOs in the power spectrum are generated at the same
circular orbit given by $r_{\rm em}$ [116, 104, 87]. This assumption holds
true for the kinematic and the resonant models which we have discussed above.
Apart from these there also exist certain diskoseismic models which are based
on the assumption that the oscillatory modes giving rise to the HFQPOs are
emitted at different radii of the accretion disk [88, 89, 90].
Magnetohydrodynamic simulations [117, 118, 119] reveal that such models cannot
explain the 3:2 HFQPOs adequately and therefore those models are not
considered in this work. Further, we do not aim to extract the mass of the
black hole sources from the observed QPO frequencies but rather use the
previously estimated masses obtained from other independent observations,
e.g., optical/NIR photometry (see 1).

##  5 Estimating the magnetic monopole charge from the observed QPOs

In this section we aim to derive the most favored value of $k$ from
observations related to QPOs. In order to accomplish this we compare the
observed QPO frequencies given in 1 with the model dependent QPO frequencies
in 2 and calculate the joint-$\chi^{2}$ which is given by,

$\displaystyle\chi^{2}(k)=\sum_{j=1}^{5}\frac{\\{\nu_{\textrm{u1}{,j}}-\nu_{1}(k,a_{\rm
min},\tilde{M}_{\rm min},r_{\rm em,min})\\}^{2}}{\sigma_{\nu_{\rm
u1},j}^{2}}+\sum_{j=1}^{5}\frac{\\{\nu_{\textrm{u2}{,j}}-\nu_{2}(k,a_{\rm
min},\tilde{M}_{\rm min},r_{\rm em,min})\\}^{2}}{\sigma_{\nu_{\rm
u2},j}^{2}}~{},$ (33)

where $\nu_{1}$ and $\nu_{2}$ are the model dependent QPO frequencies,
$\nu_{u1}$ and $\nu_{u2}$ are the observed QPO frequencies with errors given
by $\sigma^{2}_{\nu_{\rm u1}}$ and $\sigma^{2}_{\nu_{\rm u2}}$ respectively.
From 2 it is clear that the theoretical QPO frequencies depend on the orbital
frequency and the epicyclic frequencies which are functions of the metric
parameters $k$, $a$, $\tilde{M}$ and the emission radius $r_{\rm em}$. Since
our goal is to determine the most favored value of $k$ we do not minimize the
joint-$\chi^{2}$ with respect to all the four parameters. Instead we divide
them into two categories [120] namely, (a) the “interesting parameters” (which
is $k$) and (b) the “uninteresting parameters” (here $a$, $\tilde{M}$ and
$r_{\rm em}$), derived from $\chi^{2}$ minimization for various choices of the
“interesting parameters”. We note that we do not estimate the black hole mass
from the present analysis but use masses of these sources independently
obtained from optical/NIR photometry (see 1).

$\rm Comparison$ |  $\rm GRO~{}J1655-40$ |  $\rm XTE~{}J1550-564$ |  $\rm GRS~{}1915+105$ |  $\rm H~{}1743+322$ |  $\rm Sgr~{}A^{*}$
---|---|---|---|---|---
$\rm of~{}mass$ |  |  |  |  |
$\rm estimates$ |  |  |  |  |
$\rm(\rm in~{}M_{\odot})$ |  |  |  |  |
|  |  |  |  |
$\rm Previous$ |  $\rm 5.4\pm 0.3$ [63] |  $\rm 9.1\pm 0.61$ [65] |  $\rm 12.4^{+2.0}_{-1.8}$ [66] |  $\rm 8.0-14.07$ |  $\rm(3.5-4.9)\rm\times 10^{-3}$
$\rm constraints$ |  |  |  |  [67, 68, 69] |  [70, 71]
|  |  |  |  |
$\rm RPM$ |  $\rm 5.13~{}(k\sim 0)$ |  $\rm 9.3~{}(k\sim 0)$ |  $\rm 13.99~{}(k\sim 0)$ |  $\rm 12.06~{}(k\sim 0)$ |  $\rm 4.0\times 10^{6}~{}(k\sim 0)$
|  |  |  |  |
$\rm TDM$ |  $\rm 7.0~{}(k\sim 0.1)$ |  $\rm 6.8~{}(k\sim 0.1)$ |  $\rm 8.6~{}(k\sim 0.1)$ |  $\rm 8.3~{}(k\sim 0.1)$ |  $\rm 4.4\times 10^{6}~{}(k\sim 0.1)$
|  |  |  |  |
$\rm PRM$ |  $5.1\rm~{}(k\sim 0.1)$ |  $8.49\rm~{}(k\sim 0.1)$ |  $\rm 13.5~{}(k\sim 0.1)$ |  $\rm 10.98~{}(k\sim 0.1)$ |  $\rm 3.5\times 10^{6}~{}(k\sim 0.1)$
|  |  |  |  |
$\rm FRM1$ |  $5.1\rm~{}(k\sim 0.4)$ |  $\rm 9.61~{}(k\sim 0.4)$ |  $\rm 11.5~{}(k\sim 0.4)$ |  $\rm 9.37~{}(k\sim 0.4)$ |  $\rm 3.7\times 10^{6}~{}(k\sim 0.4)$
|  |  |  |  |
$\rm FRM2$ |  $\rm 7.1~{}(k\sim 0.3)$ |  $\rm 7.29~{}(k\sim 0.3)$ |  $\rm 8.45~{}(k\sim 0.3)$ |  $\rm 7.75~{}(k\sim 0.3)$ |  $\rm 3.66\times 10^{6}~{}(k\sim 0.3)$
|  |  |  |  |
$\rm KRM1$ |  $\rm 5.1(k\sim 0.1)$ |  $\rm 8.49(k\sim 0.1)$ |  $\rm 12.7(k\sim 0.1)$ |  $\rm 8.1(k\sim 0.1)$ |  $\rm 3.5\times 10^{6}(k\sim 0.1)$
|  |  |  |  |
$\rm KRM2$ |  $\rm 5.36~{}(k\sim 0)$ |  $\rm 8.94~{}(k\sim 0)$ |  $\rm 14.1~{}(k\sim 0)$ |  $\rm 13.08~{}(k\sim 0)$ |  $\rm 4.8\times 10^{6}~{}(k\sim 0)$
|  |  |  |  |
$\rm KRM3$ |  $5.67\rm~{}(k\sim 0)$ |  $\rm 9.65~{}(k\sim 0)$ |  $\rm 12.71~{}(k\sim 0)$ |  $\rm 10.33~{}(k\sim 0)$ |  $\rm 4.2\times 10^{6}~{}(k\sim 0)$
|  |  |  |  |
$\rm WDOM$ |  $\rm 5.62~{}(k\sim 0.2)$ |  $\rm 9.45~{}(k\sim 0.2)$ |  $\rm 11.89~{}(k\sim 0.2)$ |  $\rm 11.43~{}(k\sim 0.2)$ |  $\rm 3.5\times 10^{6}~{}(k\sim 0.2)$
|  |  |  |  |
$\rm NADO1$ |  $\rm 5.54~{}(k\sim 0.4)$ |  $\rm 9.42~{}(k\sim 0.4)$ |  $\rm 11.99~{}(k\sim 0.4)$ |  $\rm 9.55~{}(k\sim 0.4)$ |  $\rm 3.5\times 10^{6}~{}(k\sim 0.4)$
|  |  |  |  |
$\rm NADO2$ |  $\rm 5.12~{}(k\sim 0.3)$ |  $\rm 9.14~{}(k\sim 0.3)$ |  $\rm 13.39~{}(k\sim 0.3)$ |  $\rm 12.26~{}(k\sim 0.3)$ |  $\rm 4.9\times 10^{6}~{}(k\sim 0.3)$
Table 3: The above table presents the mass estimates of the BH sources considered in 1 from $\chi^{2}$ minimization. The previously estimated masses are also reported.  $\rm Comparison$ |  $\rm GRO~{}J1655-40$ |  $\rm XTE~{}J1550-564$ |  $\rm GRS~{}1915+105$ |  $\rm H~{}1743+322$ |  $\rm Sgr~{}A^{*}$
---|---|---|---|---|---
$\rm of~{}spin$ |  |  |  |  |
$\rm estimates$ |  |  |  |  |
|  |  |  |  |
$\rm Previous$ |  $\rm a\sim 0.65-0.75$ [121] |  $\rm-0.11<a<0.71$ [122] |  $a\sim\rm 0.98$ [123] |  $a=\rm 0.2\pm 0.3$ [124] |  $\rm a\sim 0.9$ [125]
$\rm constraints$ |  $\rm a\sim 0.94-0.98$ [126] |  |  $a\sim\rm 0.7$ [127] |  |  $\rm a\sim 0.5$ [128]
|  $\rm a=0.29\pm 0.003$ [64] |  $\rm a=0.55^{+0.15}_{-0.22}$ [122] |  $\rm a\sim 0.6-0.98$ [129] |  |  $\rm a=0.9959\pm 0.0005$ [130]
|  |  |  $\rm a\sim 0.4-0.98$ [131] |  |  $\rm a\lesssim 0.1$ [132]
|  |  |  |  |
$\rm RPM$ |  $\rm 0.3~{}(k\sim 0)$ |  $\rm 0.4~{}(k\sim 0)$ |  $\rm 0.3~{}(k\sim 0)$ |  $\rm 0.5~{}(k\sim 0)$ |  $\rm 0.97~{}(k\sim 0)$
|  |  |  |  |
$\rm TDM$ |  $\rm 0.1~{}(k\sim 0.1)$ |  $\rm 0.2~{}(k\sim 0.1)$ |  $\rm-0.3~{}(k\sim 0.1)$ |  $\rm-0.3~{}(k\sim 0.1)$ |  $\rm 0.9~{}(k\sim 0.1)$
|  |  |  |  |
$\rm PRM$ |  $\rm 0.8~{}(k\sim 0.1)$ |  $\rm 0.8~{}(k\sim 0.1)$ |  $\rm 0.8~{}(k\sim 0.1)$ |  $0.89\rm~{}(k\sim 0.1)$ |  $\rm~{}0.9(k\sim 0.1)$
|  |  |  |  |
$\rm FRM1$ |  $\rm-0.1~{}(k\sim 0.4)$ |  $\rm 0.1~{}(k\sim 0.4)$ |  $\rm-0.3~{}(k\sim 0.4)$ |  $-0.1\rm~{}(k\sim 0.4)$ |  $\rm 0.58~{}(k\sim 0.4)$
|  |  |  |  |
$\rm FRM2$ |  $-0.1\rm~{}(k\sim 0.3)$ |  $\rm-0.1~{}(k\sim 0.3)$ |  $\rm~{}-0.4(k\sim 0.3)$ |  $\rm-0.3~{}(k\sim 0.3)$ |  $\rm 0.695~{}(k\sim 0.3)$
|  |  |  |  |
$\rm KRM1$ |  $\rm 0.8997~{}(k\sim 0.1)$ |  $\rm 0.8997~{}(k\sim 0.1)$ |  $\rm 0.899~{}(k\sim 0.1)$ |  $\rm 0.89~{}(k\sim 0.1)$ |  $\rm 0.8997~{}(k\sim 0.1)$
|  |  |  |  |
$\rm KRM2$ |  $\rm 0.32~{}(k\sim 0)$ |  $\rm 0.36~{}(k\sim 0)$ |  $\rm 0.32~{}(k\sim 0)$ |  $\rm 0.6~{}(k\sim 0)$ |  $\rm 0.97~{}(k\sim 0)$
|  |  |  |  |
$\rm KRM3$ |  $\rm 0.22~{}(k\sim 0)$ |  $\rm 0.3~{}(k\sim 0)$ |  $\rm 0~{}(k\sim 0)$ |  $\rm 0.22~{}(k\sim 0)$ |  $\rm 0.99~{}(k\sim 0)$
|  |  |  |  |
$\rm WDOM$ |  $\rm 0.0~{}(k\sim 0.2)$ |  $\rm 0.1~{}(k\sim 0.2)$ |  $\rm-0.3~{}(k\sim 0.2)$ |  $\rm 0.1~{}(k\sim 0.2)$ |  $\rm 0.79~{}(k\sim 0.2)$
|  |  |  |  |
$\rm NADO1$ |  $\rm 0.0~{}(k\sim 0.4)$ |  $\rm 0.1~{}(k\sim 0.4)$ |  $\rm-0.3~{}(k\sim 0.4)$ |  $\rm-0.1~{}(k\sim 0.4)$ |  $\rm 0.585~{}(k\sim 0.4)$
|  |  |  |  |
$\rm NADO2$ |  $\rm 0.0~{}(k\sim 0.3)$ |  $\rm 0.1~{}(k\sim 0.3)$ |  $\rm 0.0~{}(k\sim 0.3)$ |  $\rm 0.2~{}(k\sim 0.3)$ |  $\rm 0.6~{}(k\sim 0.3)$
Table 4:  The above table presents the spin estimates of the BH sources
considered in 1 from $\chi^{2}$ minimization. The spin measurements obtained
from earlier estimates are also reported.

To determine the most favored value of $k$ we adopt the following procedure:

  1. 1.

We choose a given QPO model as mentioned in 2.

  2. 2.

We next choose a source from 1.

  3. 3.

Then we select a value of $k$ which automatically fixes the allowed values of
spin such that the central singularity is covered by a horizon. We vary the
spin in this allowed range.

  4. 4.

For the chosen $k$ and spin we vary the black hole mass between
$(M_{BH}-\Delta M_{BH})\leq M_{BH}\leq(M_{BH}+\Delta M_{BH})$) where $\Delta
M_{BH}$ is the error in the mass measurement (given in 1).

  5. 5.

Further, for a given combination of $k$, $a$ and $M_{BH}$ we vary
$r_{ms}(k,a)\leq r_{\rm em}\leq r_{ms}(k,a)+20r_{g}$.

  6. 6.

The value of $M_{BH}$, $a$ and $r_{\rm em}$ that minimizes the chi-square for
the given $k$ are denoted by $M_{\rm m}$, $a_{\rm m}$ and $r_{\rm m}$ for the
chosen source.

  7. 7.

We repeat steps 3-6 for the other sources keepinf $k$ fixed.

  8. 8.

Then we repeat steps 3-7 for the remaining values of $k$ keeping the QPO model
fixed. This enables us to calculate the magnitude of the joint-$\chi^{2}$ for
the chosen QPO model which we plot as a function of $k$. The magnitude of $k$
where the joint-$\chi^{2}$ minimizes is denoted by $k_{min}$ and the value of
the minimum $\chi^{2}$ is denoted by $\chi^{2}_{min}$. The $M_{m}$ and $a_{m}$
for each source corresponding to $k_{min}$ are denoted by $M_{min}$ and
$a_{min}$ and are reported in 3 and 4 respectively.

  9. 9.

Models like RPM can also explain the low-frequency QPO observed in GRO
J1655-40, when the form of $\chi^{2}$ is given by,

$\displaystyle\chi^{2}(k)$
$\displaystyle=\sum_{j}\frac{\\{\nu_{\textrm{u1}{,j}}-\nu_{1}(k,a_{\rm
min},M_{\rm min},r_{\rm em,min})\\}^{2}}{\sigma_{\nu_{\rm
u1},j}^{2}}+\sum_{j}\frac{\\{\nu_{\textrm{u2}{,j}}-\nu_{2}(k,a_{\rm
min},M_{\rm min},r_{\rm em,min})\\}^{2}}{\sigma_{\nu_{\rm u2},j}^{2}}$
$\displaystyle\hskip 113.81102pt+\frac{\\{\nu_{L,{\rm GRO}}-\nu_{3}(k,a_{\rm
min},M_{\rm min},r_{\rm em,min})\\}^{2}}{\sigma_{\nu_{L},\rm GRO}^{2}}~{}.$
(34)

  10. 10.

We next choose another model from 2 and repeat steps 2-9. This gives us the
variation of $\chi^{2}$ with $k$ for each of the QPO models.

  11. 11.

When the number of interesting parameters is one, the confidence intervals
(i.e., $\Delta\chi^{2}$ from $\chi^{2}_{\rm min}$) associated with 68%, 90%
and 99% confidence levels are equal to 1, 2.71 and 6.63 [120]. The variation
of $\chi^{2}$ with $k$ for each of the QPO models is illustrated in 1 and 2.
It is important to note that in the present analysis we have assumed all the
black holes to possess similar NED charge parameter $k$. Therefore, when we
consider $k$ as the interesting parameter we mean this to be the average
charge associated with the black holes. This assumption is relevant as long as
$k$ varies in a very small range which in the present case is $0\lesssim
k\lesssim 0.7$ such that taking an average is meaningful. Moreover since
$k\geq 0$, if $k=0$ is favored observationally, it implies that the black
holes in the sample possess no NED charge. Since we are observationally
constraining the squared charge to mass ratio $\kappa/r_{g}$ we do not face
difficulties in considering black holes with largely different masses, i.e. if
mass increases the charge increases proportionally such that the magnitude of
$k$ remains similar and in the range in the range $0\lesssim k\lesssim 0.7$
for the black holes in the sample. This allows us to consider a data sample
comprising of four stellar mass and one supermassive black hole in this work.

This method is quite general and is not restricted to only regular black
holes. The analysis works perfectly fine for regular black holes as well as
for black holes with $r=0$ curvature singularity as long as the metric is
characterized by three parameters, namely, mass, spin and some charge. We have
used this method to constrain the tidal charge parameter of braneworld black
holes [133] and the magnetic monopole charge parameter of Bardeen black holes
where Bardeen black holes are regular black holes with a de Sitter core [134].
This analysis can be further extended to more general regular black holes. In
each case we estimate the dimensionless charge parameter (which is the charge
scaled with the black hole mass) from the observations.

Therefore, method of analysis would not differ if we consider regular black
holes with de Sitter cores. The assumption of Minkowski core comes in the
choice of the mass function. For Bardeen black holes or braneworld black holes
the mass functions are different. Each time the mass function is characterized
by some charge parameter which has some physical interpretation. An external
observer perceives the black hole through the spacetime curvature created by
the metric parameters, namely, mass, spin and the charge, on which the
epicyclic frequencies depend. If the metric comprises of more than three
parameters then depending on the situation at hand the $\chi^{2}$ can be
characterized by more than one interesting parameters [120] in which case we
will be required to generate contour plots of $\chi^{2}$ to estimate the
observationally favored magnitude of the charges which quantify the departure
from GR.

Figure 1: The above figure demonstrates the variation of $\chi^{2}$ with the
non-linear electrodynamics charge parameter $k$ assuming the following QPO
models: (a) Relativistic Precession Model (RPM), (b) Tidal Disruption Model
(TDM), (c) Parametric Resonance Model (PRM), (d) 3:1 Forced Resonance Model
(FRM1) and (e) 2:1 Forced Resonance Model (FRM2). The black, blue and magenta
lines denote respectively the 68%, 90% and 99.7% confidence intervals
corresponding to $\Delta\chi^{2}=1$, $\Delta\chi^{2}=2.71$ and
$\Delta\chi^{2}=6.63$ from $\chi^{2}_{min}$. Figure 2: The above figure
demonstrates the variation of $\chi^{2}$ with the non-linear electrodynamics
charge parameter $k$ assuming the following models: (a) Keplerian Resonance
Model 1, (b) Keplerian Resonance Model 2, (c) Keplerian Resonance Model 3, (d)
Warped Disc Oscillation Model, (e) Non-axisymmetric Disc Oscillation Model 1
and (f) Non-axisymmetric Disc Oscillation Model. The black, blue and magenta
lines denote respectively the 68%, 90% and 99.7% confidence intervals
corresponding to $\Delta\chi^{2}=1$, $\Delta\chi^{2}=2.71$ and
$\Delta\chi^{2}=6.63$ from $\chi^{2}_{min}$.

Although we do not extract the black hole mass from the present analysis we
provide independent estimates of spin since previous estimates of spin are
derived using general relativity and different methods of spin measurements
assuming general relativity yield largely disparate results (e.g., GRO
J1655-40 [64]). In this regard we mention that earlier estimates of spin for
GRS 1915+105, GRO J1655-40 and Sgr A* give rise to results with large
discrepancies, e.g for GRS 1915+105, the Fe-line method gives $a\sim 0.6-0.98$
[129] while from Continuum Fitting Method the spin turns out to be
intermediate ($a\sim 0.7$ [127]) as well as maximal ($a\sim 0.98$ [123]). With
revised mass and inclination the spin of GRS 1915+105 turns out to be
$0.4<a<0.98$. From 4 we note that our results based on PRM and KRM1 are
consistent with previous measurements [123, 129, 131]. Spin measurements of
the source XTE J1550-564 gives $-0.11<a<0.71$ from the Continuum Fitting
method while Fe-line method yields a more stringent constraint
$a=0.55^{+0.15}_{-0.22}$ [122]. Our findings reveal that apart from PRM and
KRM1 all the models yield spin in the range predicted by the Continuum Fitting
method. Based on the Fe-line method and the Continuum-Fitting method the spin
of GRO J1655-40 turns out to be $0.94<a<0.98$ [126] and $0.65<a<0.75$ [121]
respectively while from QPO data assuming RPM the spin of the source is
$a=0.290\pm 0.003$ [64]. From our analysis we find that our spin estimates
based on RPM is in agreement with earlier spin measurements based on QPO
related observations. Using Continuum-Fitting method [124] the spin of
H1743-322 turns out to be $a=0.2\pm 0.3$ (with 68% confidence) and $a<0.92$
(with 99.7% confidence). From 4 we note that models like RPM, FRM1, KRM3,
WDOM, NADO1 and NADO2 yield results in agreement with previous findings. For
the source Sgr A* the spin is obtained from its radio spectrum as well as
motion of S2 stars. Since the radio spectrum of Sgr A* is difficult to model
[135, 125, 128] diverse values of spin are obtained, e.g., $a\sim 0.9$ [125],
$a\sim 0.5$ [128]. Investigating the motion of S2 stars near Sgr A* reveal the
source has a spin $a\lesssim 0.1$ [132]. From the study of X-ray light curve
of Sgr A* one finds that the object has a maximal spin with ($\rm a=0.9959\pm
0.0005$) [130]. From our analysis we find that models like RPM, KRM3, WDOM and
NADO1 yield a maximal spin as obtained in [130].

In 1 and 2 we plot $\chi^{2}$ as a function of $k$ for the QPO models
discussed in 4. In particular, 1 depicts the variation of $\chi^{2}$ with $k$
for (a) Relativistic Precession Model (RPM), (b) Tidal Disruption Model (TDM),
(c) Parametric Resonance Model (PRM), (d) 3:1 Forced Resonance Model (FRM1)
and (e) 2:1 Forced Resonance Model (FRM2) while 2 illustrates the variation of
$\chi^{2}$ with $k$ for (a) Keplerian Resonance Model 1 (KRM1) (b) Keplerian
Resonance Model 2 (KRM2) (c) Keplerian Resonance Model 3 (KRM3) (d) Warped
Disk Oscillation Model (WDOM) (e) Non-axisymmetric Disk Oscillation Model 1
(NADO1) and (f) Non-axisymmetric Disk Oscillation Model 2 (NADO2). The value
of $k$ where minimum $\chi^{2}$ is attained represents the one most favored by
observations. We denote this value of $k$ by $k_{min}$. We also plot the
confidence intervals, namely, with one interesting parameter, the $1-\sigma$
confidence interval corresponds to $\chi^{2}=\chi^{2}_{min}+1$ (plotted with
black dashed line), $2-\sigma$ confidence interval corresponds to
$\chi^{2}=\chi^{2}_{min}+2.71$ (plotted with blue dashed line) while the
$3-\sigma$ confidence interval corresponds to $\chi^{2}=\chi^{2}_{min}+6.63$
(plotted with magenta dashed line).

From 1 and 2 we note that models like Relativistic Precession Model (RPM),
Parametric Resonance Model (PRM) and Keplerian Resonance Model 1 (KRM1)
establish stringent constraints on $k$. According to RPM, $\chi^{2}$ attains
its minimum value when $k\simeq 0$ and discards $k\gtrsim 0.03$ outside
3-$\sigma$ confidence interval, for PRM $\chi^{2}_{min}$ is attained at
$k\simeq 0.15$ and eliminates $k=0$ outside 2-$\sigma$ confidence interval but
includes general relativity within 3-$\sigma$. In particular PRM eliminates
$k\gtrsim 0.3$ outside 3-$\sigma$ confidence interval. For KRM1, $k\simeq
0.15$ corresponds to $\chi^{2}_{min}$ while $k\gtrsim 0.35$ is eliminated
outside 3-$\sigma$ confidence interval. This is interesting because PRM and
RPM which discard large values of the charge parameter $k$ are the most widely
used QPO models. In particular, the commensurability of QPO frequencies can be
most naturally explained by resonance models like PRM. While KRM1 also puts
strong bounds on $k$, it is difficult to realize the resonant couplings
assumed in KRM1 in the accretion scenario.

The remaining QPO models do not establish such strong bounds on $k$. The
minimum $\chi^{2}$ for the remaining QPO models correspond to (i) $k\simeq
0.2$ for the Tidal Disruption Model (TDM) (discards $k\gtrsim 0.65$ outside
3-$\sigma$ confidence interval), (ii) $k\simeq 0.35$ for the 3:1 Forced
Resonance Model (discards $k\gtrsim 0.65$ outside 3-$\sigma$ confidence
interval), (iii) $k\simeq 0.25$ for the 2:1 Forced Resonance Model (discards
$k\gtrsim 0.65$ outside 3-$\sigma$ confidence interval), (iv) $k\simeq 0$ for
the Keplerian Resonance Model 2 (discards $k\gtrsim 0.65$ outside 3-$\sigma$
confidence interval), (v) $k\simeq 0$ for the Keplerian Resonance 3 (discards
$k\gtrsim 0.6$ outside 3-$\sigma$ confidence interval), (vi) $k\simeq 0.25$
for the Warped Disk Oscillation Model (discards $k\gtrsim 0.65$ outside
3-$\sigma$ confidence interval), (vii) $k\simeq 0.45$ for the Non-axisymmetric
Disk Oscillation Model 1 (discards $k\gtrsim 0.6$ outside 3-$\sigma$
confidence interval) and (viii) $k\simeq 0.25$ for Non-axisymmetric Disk
Oscillation Model 2. The above discussion reveals that all the QPO related
observations generically discard large values of the charge parameter $k$.

##  6 Concluding Remarks

In this work we study regular black holes with an asymptotically Minkowski
core which arise in gravity coupled to non-linear electrodynamics. Such black
holes are associated with a non-linear electrodynamics (NED) charge parameter.
Study of regular black holes is important as these can evade the curvature
singularity at $r=0$, an otherwise unavoidable feature in general relativity.
Moreover, the exponential convergence factor arising in the mass function
makes the quantum gravity model finite to all orders upto the Planck scale.
Here we aim to decipher the signatures of the NED charge parameter from the
observed quasi-periodic oscillations in black holes. Theoretical models aimed
to address these peaks in the black hole power spectrum explain them in terms
of local or collective motion of plasma near the marginally stable circular
orbit. Thus, QPOs turn out to be a cleaner probe to the background spacetime
compared to the continuum spectrum or the Fe-line which also depend on the
complex physics of the accretion flow.

We review the various kinematic and resonant models of QPO proposed in the
literature and compare the model based QPO frequencies with the available
observations. In particular, we evaluate error estimator like the $\chi^{2}$
which enables us to derive the observationally favored charge parameter and
the spin corresponding to each of the black holes. Our analysis reveal that
most QPO models favor the general relativistic scenario or small values of
$k$. Particularly, models like RPM, PRM and KRM1 establish stringent
constraints on the NED charge parameter, e.g, PRM and KRM1 favor small but
non-trivial values of the charge parameter, viz, $k\simeq 0.15$ but eliminates
large values of $k$ e.g. $k\gtrsim 0.35$ outside 3-$\sigma$ confidence
interval. The Relativistic Precession model on the other hand favor $k=0$ and
discard $k\gtrsim 0.03$ outside 99.7% confidence level. This is interesting
because the most extensively used QPO models like RPM and PRM impose very
strong bounds on the NED charge parameter. We further note that all the QPO
models studied here discard large values of the charge parameter. It is
important to note that the observationally preferred magnitude of $k$
corresponds to the average charge of the black holes and in the present
analysis we assume that the black holes considered here have similar charges.
A separate analysis considering black holes with different charges can be done
in which case the number of interesting parameters will be two i.e., charge
and spin and one needs to consider the sources individually to obtain
constraints on the observationally favored magnitude of these two parameters.
We leave this analysis for a future work.

Although it might seem from the present analysis that large values of the non-
linear electrodynamics charge parameter are disfavored by observations related
to QPOs, these results have certain limitations. First, our findings are
purely model dependent, i.e., despite observing QPOs for decades there is no
common concensus on the correct choice of the QPO model. While some believe
that HFQPOs in black holes should be explained by a single QPO model which is
yet to be determined, others believe that the correct choice of the QPO model
may be source dependent. Second, there are very few black holes which exhibit
HFQPOs in their power spectrum, and hence our results are limited due to poor
statistics. Moreover, the present data has large errors, which further limits
our analysis. This can however be overcome with the launch of the ESA
(European Space Agency) X-ray mission LOFT (Large Observatory for X-ray
Timing) which aims to improve the precision of data by an order of magnitude.
However, with the available data sample, precision and models, the present
analysis provides a possible framework which may be used to constrain several
alternatives to general relativity, with QPO related observations.

## Acknowledgement

Research of I.B. is funded by the Start-Up Research Grant from SERB, DST,
Government of India (Reg. No. SRG/2021/000418).

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