11institutetext:  CAS Key laboratory for computational Astrophysics, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China
22institutetext: School of Astronomy, Institute for Research in Fundamental
Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran 33institutetext: School of
Astronomy and Space Science, University of Chinese Academy of Sciences,
Beijing 100049, China

# Testing Ghasemi-Nodehi-Bambi metric parameters with quasi-periodic
oscillations

M. Ghasemi-Nodehi _email:_<EMAIL_ADDRESS>Youjun Lu 1133 Ju Chen 1133 &
Chao Yang 1133

(Received: date / Revised version: date)

###### Abstract

QPOs are seen as peak features in the X-ray power spectral density of stellar
mass black holes and neutron stars, of which frequencies can be measured with
high precision. These QPO frequencies are believed to be related to
fundamental frequencies of test particles, which are mainly determined by the
background metric. We consider the metric introduced in [Ghasemi-Nodehi and
Bambi, Eur. Phys. J. C 76, 290 (2016)]. The fundamental frequencies in this
metric are sensitive to some of the metric parameters but insensitive to other
parameters, which means, the differences in fundamental frequencies in this
metric and those in the Kerr ones can be significant for small changes of some
but not all parameters around the Kerr value. By comparing with the QPO
observations of GRO J1655-40, we find that only one parameter of the Ghasemi-
Nodehi-Bambi metric can be strongly constrained, but other parameters cannot.
We also use nested algorithm to investigate whether better constraints on the
Ghasemi-Nodehi- Bambi metric parameters can be obtained from QPO observations
of multiple objects by simulations. We find that four parameters can be
strongly constrained while other parameters cannot . Our results suggest that
QPOs may be important tools for testing the Kerr metric.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Introduction

Einstein theory of General Relativity (GR) was born more than 100 years ago
and successfully passed test in the weak gravity field regime einstein1916 ;
t1 ; t2 ; t3 ; t4 . According to GR, astrophysical black holes (BHs) are
described by the Kerr metric with only two parameters, i.e., mass and spin k1
; k2 . However, the nature of the Kerr BH is still to be verified. Important
test of GR in the strong gravity field regime can be from direct observational
confirmation of the Kerr metric, via either electromagnetic wave e1 ; e2  or
gravitational wave g1 ; g2  observations. The electromagnetic tests in the
literature mainly make use of the iron line method ka  and/or continuum
fitting method cfm  by utilizing X-ray data r1 ; r2 . Black hole shadow also
offers a unique tool to probe the event horizon and nature of BHs, which has
been demonstrated by recent observations of M87 through the Event Horizon
Telescope (EHT) evh ; evh1 .

There have also been a number of attempts to generalize the Kerr solution by
phenomenological parameterization p1 ; p2 ; p3 ; p4 ; p5 ; p6 ; p7 ; p8 ; p9 ;
p10 ; p11 ; p12 ; p13 ; p14 ; p15 ; p16 . The idea is mostly to write a more
general metric that includes the Kerr one as its special case, similar to the
PPN of the Schwarzschild metric t1 . Generally, parameters of the non-Kerr
metric may be degenerated in testing strong gravity fields. Therefore, the
Kerr metric cannot be verified only by the detection of its characteristic
properties, and it would be proofed only when all deformation parameters of
the non-Kerr metric are required to vanish by observations.

Quasi-periodic oscillations (QPOs), peak features in the X-ray power spectral
density of stellar mass BHs and neutron stars, are recently recognized as
promising tools to test GR in the strong gravity field regime q1 ; q2 ; q3 ;
q4 ; q5 ; q6 ; q7 . QPOs have been detected in four systems GRO J1655-40, XTE
J1550-564, GRS 1915+105, and H1743-322, they are not sensitive to the
properties of accretion flows. The exact physical mechanism responsible for
the production of the QPOs is not clear, though a number of different
scenarios have been proposed, including relativistic precession model rpm ,
diskoseismology models dm , resonance models rm , and p-mode oscillations of
an accretion torus rezz . In most scenarios, QPO frequencies are related to
the characteristic orbital frequencies of test particles, which are determined
only by the BH metric. The main advantage of using QPOs with respect to other
methods is that the QPO frequencies can be measured with high accuracy. Thus,
QPOs can provide a unique tool to probe the spacetime and geometry around the
central compact objects.

One of us proposed a new parameterization to Kerr metric in Reference p15
(hereafter the GB metric). The Kerr case would be recovered by setting all the
parameters involved in the GB metric equal to $1$. Any parameter different
from $1$ indicates a deviation from the Kerr case. Assuming the GB metric, we
have studied the BH shadow in the same paper and X-ray reflection spectroscopy
in ghasemi . BH shadow studies show that only $4$ parameters involved in the
GB metric have weak impact on the shadow shape while other parameters have
almost no impact, and thus it will be difficult to constrain the GB metric by
using BH shadow measurements. In the paper ghasemi , one of us studied the
X-ray reflection spectroscopy resulting from accretion systems to constrain
the parameters in the GB metric. Our results show that all parameters except
one can be potentially well constrained with near future X-ray missions. In
the present paper, we investigate the impact of GB parameters of Ref. p15  on
QPOs. Our results show significant parameter degeneracy in constraining the GB
parameters.

The content of this paper is as follows. In Section 2, we provide the
theoretical framework of our calculations. Results and discussions are written
in Section 3. Summary and conclusions are presented in section 5. Throughout
the paper, we use units in which $G_{N}=c=1$.

##  2 Theoretical Framework

Three fundamental frequencies characterize the equatorial orbit of a test
particle rotating around a BH, i.e., the Keplerian frequency $\nu_{\phi}$,
radial epicyclic frequency $\nu_{r}$, and vertical epicyclic frequency
$\nu_{\theta}$. These frequencies only depend on the BH metric and radius of
orbit.

The general case can be studied by the line element

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

that is for a generic stationary and axisymmetric spacetime and the metric
components are independent of $t$ and $\phi$.

There are two conserved quantities associated to $t$ and $\phi$, specific
energy at infinity, $E$, and z-component of specific angular momentum at
infinity, $L_{z}$. The $t$ and $\phi$ components of 4-velocity of a test
particle are bookbambi

$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{Eg_{\phi\phi}+L_{z}g_{t\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}$
(2) $\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-\frac{Eg_{t\phi}+L_{z}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\,.$
(3)

One may introduce $V_{\rm eff}$ from the conservation of the rest mass
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$,

$\displaystyle g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}=V_{\rm
eff}(r,\theta,E,L_{z})\,,$ (4)

where

$\displaystyle V_{\rm
eff}=\frac{E^{2}g_{\phi\phi}+2EL_{z}g_{t\phi}+L^{2}_{z}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}-1\,.$
(5)

Geodesic equation can also be written as

$\displaystyle\frac{d}{d\lambda}\left(g_{\mu\nu}\dot{x}^{\nu}\right)=\frac{1}{2}\left(\partial_{\mu}g_{\nu\rho}\right)\dot{x}^{\nu}\dot{x}^{\rho}\,.$
(6)

Circular orbit requires that $\dot{r}=\dot{\theta}=\ddot{r}=0$, so the radial
component is

$\displaystyle\left(\partial_{r}g_{tt}\right)\dot{t}^{2}+2\left(\partial_{r}g_{t\phi}\right)\dot{t}\dot{\phi}+\left(\partial_{r}g_{\phi\phi}\right)\dot{\phi}^{2}=0\,.$
(7)

This equation gives us the orbital angular velocity

$\displaystyle\Omega_{\phi}=\frac{\dot{\phi}}{\dot{t}}=\frac{-\partial_{r}g_{t\phi}\pm\sqrt{\left(\partial_{r}g_{t\phi}\right)^{2}-\left(\partial_{r}g_{tt}\right)\left(\partial_{r}g_{\phi\phi}\right)}}{\partial_{r}g_{\phi\phi}}\,,$
(8)

where the sign is ($+/-$) for corotating/counterrotating orbits and orbital
frequency can be calculated as $\nu_{\phi}=\Omega_{\phi}/2\pi$.

Considering $\dot{r}=\dot{\theta}=0$ in
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$, we have

$\displaystyle\dot{t}=\frac{1}{\sqrt{-g_{tt}-2g_{t\phi}\Omega_{\phi}-g_{\phi\phi}\Omega^{2}_{\phi}}}\,.$
(9)

We find $E$ and $L_{z}$ since $-E=g_{tt}\dot{t}+g_{t\phi}\dot{\phi}$ and
$L_{z}=g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}$,

$\displaystyle E$ $\displaystyle=$
$\displaystyle-\frac{g_{tt}+g_{t\phi}\Omega_{\phi}}{\sqrt{-g_{tt}-2g_{t\phi}\Omega_{\phi}-g_{\phi\phi}\Omega^{2}_{\phi}}}\,,$
(10) $\displaystyle L_{z}$ $\displaystyle=$
$\displaystyle\frac{g_{t\phi}+g_{\phi\phi}\Omega_{\phi}}{\sqrt{-g_{tt}-2g_{t\phi}\Omega_{\phi}-g_{\phi\phi}\Omega^{2}_{\phi}}}\,.$
(11)

Substituting $E$ and $L_{z}$ in equation (4) together with metric component
gives us the $V_{\rm eff}$.

From equation (4) we consider small perturbation around circular orbit along
radial and vertical directions at linear order. If $\delta_{r}$ and
$\delta_{\theta}$ are the small displacements around the mean orbit,
$r=r_{0}+\delta_{r}$ and $\theta=\pi/2+\delta_{\theta}$, and neglecting terms
$O(\delta_{r}^{2})$ and $O(\delta_{\theta}^{2})$, we have following
differential equations

$\displaystyle\frac{d^{2}\delta_{r}}{dt^{2}}+\Omega_{r}^{2}\delta_{r}=0,$ (12)
$\displaystyle\frac{d^{2}\delta_{\theta}}{dt^{2}}+\Omega_{\theta}^{2}\delta_{\theta}=0\,,$
(13)

where

$\displaystyle\Omega^{2}_{r}=-\frac{1}{2g_{rr}\dot{t}^{2}}\frac{\partial^{2}V_{\rm
eff}}{\partial r^{2}},$ (14)
$\displaystyle\Omega^{2}_{\theta}=-\frac{1}{2g_{\theta\theta}\dot{t}^{2}}\frac{\partial^{2}V_{\rm
eff}}{\partial\theta^{2}}\,.$ (15)

The radial epicyclic frequency and the vertical epicyclic frequency are,
respectively, $\nu_{r}=\Omega_{r}/2\pi$ and
$\nu_{\theta}=\Omega_{\theta}/2\pi$.

In this paper, we consider relativistic precession model that recent studies
seem to support it Motta1 ; Motta2 . This model first proposed to explain the
frequency of the QPOs in power spectral density of neutron star and then is
extended to the QPOs of the stellar mass BH rpm . This model simply relates
the observed frequencies with the orbital motion of a test particle. One can
find the periastron precession frequency, $\nu_{p}$, and nodal precession
frequency from fundamental frequencies,

$\displaystyle\nu_{p}$ $\displaystyle=$ $\displaystyle\nu_{\phi}-\nu_{r}\,,$
(16) $\displaystyle\nu_{n}$ $\displaystyle=$
$\displaystyle\nu_{\phi}-\nu_{\theta}\,.$ (17)

These frequencies can be measured observationally and compared with the
fundamental frequencies computed from the background metric. This provides a
unique opportunity with high accuracy to probe the strong gravity regime and
test possible deviations from the Kerr solution of GR.

In Ref p15  one of us introduced 11 parameters, $b_{i}$ as follows,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2b_{1}Mr}{r^{2}+b_{2}a^{2}\cos^{2}\theta}\right)dt^{2}$
(18)
$\displaystyle-\frac{4b_{3}Mar\sin^{2}\theta}{r^{2}+b_{4}a^{2}\cos^{2}\theta}dtd\phi+\frac{r^{2}+b_{5}a^{2}\cos^{2}\theta}{r^{2}-2b_{6}Mr+b_{7}a^{2}}dr^{2}$
$\displaystyle+\left(r^{2}+b_{8}a^{2}\cos^{2}\theta\right)d\theta^{2}$
$\displaystyle+\left(r^{2}+b_{9}a^{2}+\frac{2b_{10}Ma^{2}r\sin^{2}\theta}{r^{2}+b_{11}a^{2}\cos^{2}\theta}\right)\sin^{2}\theta
d\phi^{2}\,.$

Kerr case recovers for all $b_{i}=1$. Any differences from $b_{i}=1$ show
deviations from general relativity. $b_{1}$ can be set as one because it is
coefficient of mass, $b_{1}M$ and also $b_{3}=1$ in the same way asymptotic
specific angular momentum is $b_{3}a$. We also have $b_{6}$ close to one from
solar system experiment. So we set $b_{1}=b_{3}=b_{6}=1$ and we do not
consider these three parameters in our QPO calculations.

##  3 Results

QPOs showing in the X-ray power spectral density of stellar mass BHs and
neutron stars provide precise information about the background metric. QPOs
are frequencies measured for a BH system are believed to be determined by the
BH metric but not the accretion flow, though it may model dependent. In this
paper, we adopt the relativistic precession model and assume that QPO
frequencies are related to the characteristic orbital frequencies of test
particles. These characteristic frequencies are solely determined by the BH
metric. We also assume that the GB metric describes the BH metric, for which
the Kerr metric is a special case.

We first calculate frequencies $\nu_{\phi}$, $\nu_{r}$, $\nu_{\theta}$,
$\nu_{p}$, and $\nu_{n}$ for the GB metric with different settings of the
metric parameters. Figures 1 and 2 show $\nu_{\phi}$, $\nu_{p}$, and $\nu_{n}$
as a function of radius $r$ for GB BHs with mass $5.4M_{\odot}$, dimensionless
spin parameter $a_{*}=a/M=0.9$, but different $b_{i}$ (not too much different
from the Kerr case $b_{i}=1$). As seen from these two Figures, different
settings of $b_{2}$ or $b_{4}$ (e.g., $b_{2}$ or $b_{4}=1.2$ and $5$) may lead
to substantial difference in the resulting $\nu_{n}$ but negligible
differences in $\nu_{\phi}$ and $\nu_{p}$ (top left and right panels of Fig.
1); different settings of $b_{7}$, $b_{9}$, or $b_{10}$ (e.g., $1.2$ and $5$)
may lead to significant difference in the resulting $\nu_{p}$ but not
$\nu_{\phi}$ and $\nu_{n}$ (bottom-right panel of Fig. 1, top-right or bottom-
left panel of Fig. 2); different settings of $b_{5}$, $b_{8}$, or $b_{11}$ may
only lead to little differences in all resulting frequencies (bottom-left
panel of Fig. 1, top-left or bottom-right panel of Fig. 2). We emphasize here
that different settings of $b_{i}$ may lead to significantly different
innermost stable circular orbits (ISCOs) and thus the frequency curves cut at
different $r/M$, which are most significant in the top-right and bottom-left
panels of Figure 2 for $b_{9}$ and $b_{10}$, respectively. Such cutoffs are
crucial in obtaining constraints on $b_{i}$ if QPOs are originated from the
inner edge of accretion disks.

In the Kerr metric, the three fundamental frequencies can be derived
analytically as,

$\displaystyle\nu_{\phi}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi}\right)\frac{M^{1/2}}{r^{3/2}\pm
aM^{1/2}}\,,$ (19) $\displaystyle\nu_{r}$ $\displaystyle=$
$\displaystyle\nu_{\phi}\left(1-\frac{6M}{r}\pm\frac{8aM^{1/2}}{r^{3}/2}-\frac{3a^{2}}{r^{2}}\right)^{1/2}\,,$
(20) $\displaystyle\nu_{\theta}$ $\displaystyle=$
$\displaystyle\nu_{\phi}\left(1\mp\frac{4aM^{1/2}}{r^{3/2}}+\frac{3a^{2}}{r^{2}}\right)^{1/2}\,.$
(21)

The Schwarzschild case would recover by imposing $a=0$, i.e.,

$\displaystyle\nu_{\phi}$ $\displaystyle=$
$\displaystyle\nu_{\theta}=\frac{1}{2\pi}\frac{M^{1/2}}{r^{3/2}}\,,$ (22)
$\displaystyle\nu_{r}$ $\displaystyle=$
$\displaystyle\nu_{\phi}\left(1-\frac{6M}{r}\right)^{1/2}\,.$ (23)

In the Kerr metric $\nu_{\theta}\geq\nu_{r}$, and in the Schwarzschild metric
$\nu_{\phi}=\nu_{\theta}>\nu_{r}$. In our calculation for $b_{i}=1.2$, similar
to the Kerr case and Schwarzschild case all $\nu_{\theta}$s are greater than
$\nu_{r}$, but for the case $b_{2}=5$, $\nu_{r}\geq\nu_{\theta}$. This is
possible, for example, if the ISCO is marginally vertically stable,
$\nu_{\theta}=0$ at the ISCO. Thus, we have $\nu_{r}>\nu_{\theta}$. In order
to show differences with respect to the Kerr case, we plot Figure 3. This
figure shows the impact of different GB parameters on the shape of
$\nu_{\phi}$ (top left panel), $\nu_{p}$ (top right panel), $\nu_{n}$ (bottom
panel). For all cases, the mass and dimensionless spin parameter are set to
$5.4\,M_{\odot}$ and $0.9$, respectively. It is obvious that $\nu_{\phi}$
curves for GB metric with $b_{i}=5$ ($i=2,4,5,7,8,9,10$, and $11$,
respectively) are almost the same as that for the Kerr case with all $b_{i}=1$
(top-left panel), while $\nu_{p}$ curves for the cases with $b_{7}=5$,
$b_{9}=5$, or $b_{10}=5$ are significantly different from those for other
cases (almost identical; top-right panel), and $\nu_{n}$ curves for the cases
with $b_{2}=5$, $b_{4}=5$, $b_{9}=5$, or $b_{10}=5$ are significantly
different from those for other cases (almost identical; bottom panel).

According to Figures 1, 2 and 3, we conclude that the GB metric can be
distinguished from the Kerr one if any of the GB parameters $b_{2}$, $b_{9}$,
$b_{10}$ is substantially different from $1$. The reason is that some of the
resulting fundamental frequencies from such a GB metric are substantial
different from those of the Kerr one. The GB parameters $b_{4}$ and $b_{7}$
may also be constrained but with less significance since the differences in
the fundamental frequencies are smaller.

We note here that $b_{11}$ may also be constrained since the QPO frequencies
may be only significantly affected by $b_{11}$ when BH spin is extremely high,
which is discussed later in Section 3.2 and can be seen from Figure 5.

Other GB parameters like $b_{5}$ and $b_{8}$ cannot be constrained because of
the negligible differences in fundamental frequencies for different settings
of those $b_{i}$.

Figure 1: Fundamental frequencies of a test particle on a circular orbit at
different radii in the GB metric. Top-left, top-right, bottom-left, and bottom
right panels show those frequencies obtained for a GB BH with
$(a/M,b_{2})=(0.9,1.2)$ and $(0.9,5)$, $(a/M,b_{4})=(0.9,1.2)$ and $(0.9,5)$,
$(a/M,b_{5})=(0.9,1.2)$ and $(0.9,5)$, and $(a/M,b_{7})=(0.9,1.2)$ and
$(0.9,5)$, respectively. All other not mentioned $b_{i}$ in each case equal to
$1$. In each panel, purple long-dashed, green short-dotted, and red dotted
lines show $\nu_{\phi}$, $\nu_{p}$, and $\nu_{n}$ for case $b_{i}=1.2$,
respectively. The purple, green, and red solid lines show $\nu_{\phi}$,
$\nu_{p}$, and $\nu_{n}$ for case $b_{i}=1.2$, respectively. The mass and
dimensionless spin parameter of the black hole are fixed at $5.4M_{\odot}$ and
$0.9$ .

Figure 2: Legend similar to Figure 2, while top-left, top-right, bottom-left,
and bottom right panels show those frequencies obtained for a GB BH with
$(a/M,b_{8})=(0.9,1.2)$ and $(0.9,5)$, $(a/M,b_{9})=(0.9,1.2)$ and $(0.9,5)$,
$(a/M,b_{10})=(0.9,1.2)$ and $(0.9,5)$, and $(a/M,b_{11})=(0.9,1.2)$ and
$(0.9,5)$, respectively.

Figure 3: Impact of all GB parameters on the shape of $\nu_{\phi}$ (top left
panel), $\nu_{p}$ (top right panel), $\nu_{n}$ (bottom panel). The spin
parameter is $0.9$ and mass is considered as $5.4M_{\odot}$. See the text for
more details.

###  3.1 QPO observations for GRO J1655-40

Rossi X-ray Timing Explorer (RXTE) mission observations of GRO J1655-40 have
revealed that it has two high frequency QPOs and one low frequency QPO Motta1
; gro2 . These frequencies are measured with high accuracies, i.e.,
$\nu_{1}=441^{+2}_{-2}$ Hz, $\nu_{2}=298^{+4}_{-4}$ Hz, and
$\nu_{3}=17.3^{+0.1}_{-0.1}$ Hz. If we assume that these frequencies
correspond to $\nu_{\phi}$, $\nu_{p}$, and $\nu_{n}$, respectively, and the
central BH is described by the Kerr metric, these frequencies should be
determined by three unknown parameters $M$, $a$, and $r$ as seen from
Equations (19)-(21). Therefore, strong constraints should be able to obtain on
the mass $M$ and dimensionless spin $a$ of the BH.

If the BH metric in GRO J1655-40 is described by the GB metric rather than the
Kerr metric, the determined QPO frequencies may be used to put constraint on
the GB parameters $b_{i}$s. For simplicity, we assume that only one of the
$b_{i}$s is different from that of the Kerr case, i.e., $b_{i}\neq 1$ and
$b_{j}$=1 with $j\neq i$, in order to get a meaningful constraint. For each
given $i$ ($2$, $4$, $5$, $7$, $8$, $9$, $10$, or $11$), we calculate
$\chi^{2}$ as follows to obtain a constraint on the $b_{i}$

$\displaystyle\chi^{2}(a_{*},b_{i})$ $\displaystyle=$
$\displaystyle\left(\frac{\nu_{\phi}-441}{2}\right)^{2}+\left(\frac{\nu_{p}-298}{4}\right)^{2}$
(24) $\displaystyle+\left(\frac{\nu_{n}-17.3}{0.1}\right)^{2}\,,$

where $\nu_{\phi},\nu_{n}$, and $\nu_{p}$ are, respectively, the orbital,
nodal, and periastron frequency. The numbers $441$, $298$, and $17.3$ are the
QPO frequencies measured from the X-ray observations of GRO J1655-40, and $2$,
$4$, and $0.1$ are the corresponding $1$-$\sigma$ errors $\sigma_{i}$ here. We
set $M=5.4\,M_{\odot}$ here, which is obtained from independent optical
observations opt . Here we adopt the $\chi^{2}$ statistics to obtain such
constraints.

We first minimize $\chi^{2}$ here over parameter $r$ and then fix $r$ to
obtain constraint on $b_{i}$.111Set $r=r_{\rm ISCO}$, we may get a much worse
fitting. Figure 4 shows the contour levels for parameters $b_{2}$ (top-left
panel), $b_{9}$ (top-right panel), and $b_{10}$ (bottom panel), respectively.
As seen from this Figure, $b_{2}$ can be well constrained to a value of
$2.2^{+0.395}_{-0.523}$. The minimum of $\chi^{2}$ is $0.02$ and
$\Delta\chi^{2}$ for $1\sigma,2\sigma$, and $3\sigma$ confidence levels are
$1$, $4$, and $9$, respectively. The fitting results suggest that the BH in
GRO J1655-40 may be better described by the GB metric with $b_{2}\neq 1$, but
the significance is less than $2\sigma$ and a Kerr metric is also compatible
with the data. This figure also shows degeneracies between $b_{9}$ and $a$ and
between $b_{10}$ and $a$.

Figure 4: Contour levels of parameters of GB metric. Contour levels for
parameters $b_{2}$ (top left panel), $b_{4}$ (top right panel), $b_{5}$
(bottom left panel), $b_{7}$ (bottom right panel). The brown, orange and
lighter orange are for $1-\sigma$, $2-\sigma$ and $3-\sigma$ respectively. See
the text for more details.

###  3.2 Mock QPO observations

The QPO observations of a single object have already enabled some constraints
on the GB metric. If there were QPO observations for many more objects, it may
be possible to get better constraints. In this subsection, we investigate
whether better constraints can be obtained from QPO measurements of multiple
objects and whether the GB metric can be distinguished from the Kerr metric.
To do this, we first generate $10$ Kerr BHs, each with a randomly assigned
mass and spin. We also generate another mock GB BH sample, each with a
randomly assigned mass and spin but a fixed $b_{2}=5$. For both mock samples,
we calculate $\nu_{\phi}$, $\nu_{p}$, and $\nu_{n}$ for each sample object. We
assume that these frequencies are correspondingly the QPO frequencies that can
be measured with typical errors of $1\%$, similar to that of GRO J1655-40, and
thus we get mock QPO observations. We adopt the Bayesian method to investigate
whether whether $b_{2}$ can be strongly constrained as follows.

For a given set of $n$ observations $\vec{d}$, from Bayes’ theorem:

$p(\vec{\theta}|\vec{d})\propto p(\vec{d}|\vec{\theta})p(\vec{\theta}),$ (25)

where $p(\vec{\theta}|\vec{d})$ is the posterior probability distributions of
the parameters $\theta$ being considered, $p(\vec{\theta})$ represent the
prior information on the parameters, and $p(\vec{d}|\vec{\theta})$ is the
likelihood of the observations, i.e.,

$\ln p(\vec{d}|\vec{\theta})\propto\chi^{2}=\sum_{j=1}^{n}\chi^{2}_{j}.$ (26)

For QPO observations of each object $j$, $\chi^{2}_{j}$ can be obtained
similarly as before:

$\chi^{2}_{j}=\left(\frac{\nu^{j}_{\phi}-\nu^{j}_{\phi_{o}}}{\sigma^{j}_{\nu_{\phi_{o}}}}\right)^{2}+\left(\frac{\nu^{j}_{p}-\nu^{j}_{p_{o}}}{\sigma^{j}_{\nu_{p_{o}}}}\right)^{2}+\left(\frac{\nu^{j}_{n}-\nu^{j}_{n_{o}}}{\sigma^{j}_{\nu_{n_{o}}}}\right)^{2}\,,$
(27)

where $\nu^{j}_{\phi}$, $\nu^{j}_{p}$, and $\nu^{j}_{n}$ are the frequencies
from the GB metric with given $b_{i}$, ($\nu^{j}_{\phi_{o}}$,
$\nu^{j}_{p_{o}}$, $\nu^{j}_{n_{o}}$) and
($\sigma^{j}_{\nu_{\phi_{o}}},\sigma^{j}_{\nu_{p_{o}}},\sigma^{j}_{\nu_{n_{o}}}$)
are the mock frequencies and its corresponding uncertainties, respectively.

We first mock the observation data by assuming the Kerr metric, and fit the
data with the generalized GB metric. We only vary one of the GB parameters
$b_{i}$ each time. For each of those cases that only one $b_{i}$ is considered
to be free, there are totally 21 parameters
$\vec{\theta}=\\{b_{i},M_{1},...M_{10},a_{1},...,a_{10}\\}$ in the model.

In Mock data production, we also assume $r=r_{\rm ISCO}$, for simplicity. We
sample the posterior with nested sampling algorithm using dynesty Speagle2019
. The prior for all the parameters are set to be flat with
$b_{i}\in[0,7]$222Here we adopt a narrow range for $b_{i}$ as we assume that
the deviation from the Kerr metric is small. In principle, a larger range of
$b_{i}$ can be adopted, which may give a looser constraint on $b_{i}$ compared
to the present one., $M_{j}\in[4,11]M_{\odot}$ and $a_{j}\in[0.01,0.99]$ with
$j=1,...,10$.

The result shows that $b_{2}$, $b_{9}$, and $b_{10}$ can be well constrained
as expected from previous analysis shown by Figure 1, 2, and 3. Note that
$b_{7}$ cannot be well constrained due to little frequency difference at ISCO
induced by the change of $b_{7}$.

However, $b_{4}$ also cannot be well constrained though significant
differences in $\nu$ can be found for cases with different $b_{4}$ (see Fig.
1), which may be due to strong degeneracies with spin $a$ and mass $M$.

The constraint on $b_{11}$ is quite good, despite the small frequencies
difference induced by the change of $b_{11}$ shown in Figure 1. The reason may
be seen from Figure 5, which shows the impact of $b_{10}$ (left panel) and
$b_{11}$ (right panel) on QPO frequencies at ISCO as a function of the BH
spin. Apparently, the impact of $b_{10}$ gradually increases with increasing
spin, while the impact of $b_{11}$ is significant only when the BH spin is
high ($>0.9$). and become comparable with $b_{10}$ at high spin. In the mock
QPO data, there is only one mock BH with $a=0.93$, which is main contributor
to the constraint on $b_{11}$. As we see for other parameters like $b_{5}$ in
Fig. 1 and $b_{8}$ in Fig. 2 are overlap with each other and it is hard to
constrain.

Figure 5: Dependence of the QPO frequencies at ISCO on the BH spin for a GB
metric with given $b_{10}$ (left panels) and $b_{11}$ (right panels),
respectively. Top panels show the resulting QPO frequencies for $b_{10}$ or
$b_{11}=1,1.2$, and $5$, respectively, while bottom panels show QPO frequency
differences between the cases only with $b_{10}\neq 1$ or $b_{11}\neq 1$ and
the Kerr case with all $b_{i}=1$. The BH mass is set to $5.4M_{\odot}$ here,
the same as previous those in previous Figures. See the text for more details.

To summarize, we have shown that the GB parameters $b_{2}$, $b_{9}$, $b_{10}$,
and $b_{11}$ can be well constrained if using QPO frequency measurements with
a precision of $1\%$ for even only ten mock BH systems. For such a case, the
reconstructed values of these $b_{i}$ parameters, and the mass and spin of
each mock BH are listed in Table 1. If the number of measurements is larger,
then better constraints may be obtained. One should note that the QPO
frequencies are assumed to be originated at $r_{\rm ISCO}$ when generating the
mock data and performing the fitting.

In realistic cases, the QPO phenomenon may not occur exactly at $r_{\rm
ISCO}$, and this uncertainty may reduce the constraint power. As a comparison,
in our study of GRO J1655-40 above, we minimized $\chi^{2}$ over $r$ to first
obtain the radius for the QPO frequencies to occur. We defer detailed study on
that the QPO phenomenon occurs at $r$ different from ISCO to a future work.

Table 1: Inferred values of the GB parameter from the mock observations generating by assuming the Kerr metric.  | Input | Best-fit333Only $b_{2}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting. | Best-fit444Only $b_{9}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting. | Best-fit555Only $b_{10}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting. | Best-fit666Only $b_{11}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting.
---|---|---|---|---|---
$b_{2}$777All other $b_{i},i\neq 2$, are also set to 1. | 1 | $1.022^{+0.042}_{-0.060}$ | $\cdots$ | $\cdots$ | $\cdots$
$b_{9}$888All other $b_{i},i\neq 9$, are also set to 1. | 1 | $\cdots$ | $0.995^{+0.010}_{-0.676}$ | $\cdots$ | $\cdots$
$b_{10}$999All other $b_{i},i\neq 10$, are also set to 1. | 1 | $\cdots$ | $\cdots$ | $0.996^{+0.004}_{-0.003}$ | $\cdots$
$b_{11}$101010All other $b_{i},i\neq 11$, are also set to 1. | 1 | $\cdots$ | $\cdots$ | $\cdots$ | $1.260^{+0.340}_{-0.518}$
$a_{*0}$ | $0.75$ | $0.757^{+0.007}_{-0.009}$ | $0.750^{+0.006}_{-0.077}$ | $0.753^{+0.003}_{-0.004}$ | $0.754^{+0.003}_{-0.003}$
$a_{*1}$ | $0.95$ | $0.957^{+0.018}_{-0.017}$ | $0.948^{+0.008}_{-0.152}$ | $0.941^{+0.011}_{-0.009}$ | $0.956^{+0.012}_{-0.010}$
$a_{*2}$ | $0.37$ | $0.374^{+0.003}_{-0.003}$ | $0.371^{+0.004}_{-0.010}$ | $0.376^{+0.002}_{-0.003}$ | $0.374^{+0.002}_{-0.002}$
$a_{*3}$ | $0.65$ | $0.648^{+0.004}_{-0.006}$ | $0.643^{+0.006}_{-0.050}$ | $0.646^{+0.003}_{-0.003}$ | $0.646^{+0.003}_{-0.003}$
$a_{*4}$ | $0.16$ | $0.162^{+0.001}_{-0.002}$ | $0.161^{+0.002}_{-0.002}$ | $0.162^{+0.001}_{-0.002}$ | $0.162^{+0.001}_{-0.001}$
$a_{*5}$ | $0.08$ | $0.080^{+0.001}_{-0.001}$ | $0.080^{+0.001}_{-0.001}$ | $0.080^{+0.001}_{-0.001}$ | $0.080^{+0.001}_{-0.000}$
$a_{*6}$ | $0.50$ | $0.494^{+0.004}_{-0.004}$ | $0.490^{+0.005}_{-0.022}$ | $0.493^{+0.002}_{-0.003}$ | $0.493^{+0.003}_{-0.002}$
$a_{*7}$ | $0.07$ | $0.069^{+0.001}_{-0.001}$ | $0.069^{+0.001}_{-0.001}$ | $0.069^{+0.000}_{-0.001}$ | $0.069^{+0.000}_{-0.001}$
$a_{*8}$ | $0.15$ | $0.148^{+0.052}_{-0.001}$ | $0.148^{+0.001}_{-0.001}$ | $0.148^{+0.002}_{-0.001}$ | $0.148^{+0.002}_{-0.001}$
$a_{*9}$ | $0.44$ | $0.445^{+0.003}_{-0.004}$ | $0.441^{+0.004}_{-0.017}$ | $0.444^{+0.003}_{-0.003}$ | $0.444^{+0.003}_{-0.003}$
$M_{0}/M_{\odot}$ | $5.4$ | $5.486^{+0.093}_{-0.107}$ | $5.445^{+0.051}_{-0.050}$ | $5.449^{+0.053}_{-0.051}$ | $5.452^{+0.054}_{-0.053}$
$M_{1}/M_{\odot}$ | $7.9$ | $8.167^{+0.913}_{-0.626}$ | $7.830^{+0.197}_{-0.437}$ | $7.658^{+0.281}_{-0.198}$ | $8.098^{+0.593}_{-0.368}$
$M_{2}/M_{\odot}$ | $9.4$ | $9.445^{+0.067}_{-0.066}$ | $9.452^{+0.066}_{-0.061}$ | $9.441^{+0.065}_{-0.064}$ | $9.441^{+0.069}_{-0.065}$
$M_{3}/M_{\odot}$ | $7.8$ | $7.787^{+0.070}_{-0.111}$ | $7.758^{+0.068}_{-0.072}$ | $7.768^{+0.069}_{-0.076}$ | $7.771^{+0.068}_{-0.073}$
$M_{4}/M_{\odot}$ | $6.2$ | $6.221^{+0.036}_{-0.034}$ | $6.223^{+0.035}_{-0.035}$ | $6.221^{+0.036}_{-0.035}$ | $6.221^{+0.034}_{-0.046}$
$M_{5}/M_{\odot}$ | $8.4$ | $8.405^{+0.047}_{-0.045}$ | $8.411^{+0.047}_{-0.046}$ | $8.406^{+0.046}_{-0.138}$ | $8.406^{+0.046}_{-0.047}$
$M_{6}/M_{\odot}$ | $8.4$ | $8.275^{+0.069}_{-0.058}$ | $8.285^{+0.070}_{-0.062}$ | $8.268^{+0.060}_{-0.055}$ | $8.268^{+0.059}_{-0.055}$
$M_{7}/M_{\odot}$ | $5.3$ | $5.293^{+0.027}_{-0.027}$ | $5.295^{+0.028}_{-0.027}$ | $5.292^{+0.028}_{-0.028}$ | $5.292^{+0.028}_{-0.027}$
$M_{8}/M_{\odot}$ | $7.5$ | $7.532^{+0.042}_{-0.042}$ | $7.537^{+0.044}_{-0.044}$ | $7.533^{+0.042}_{-0.042}$ | $7.532^{+0.042}_{-0.043}$
$M_{9}/M_{\odot}$ | $7.2$ | $7.238^{+0.055}_{-0.051}$ | $7.248^{+0.055}_{-0.053}$ | $7.283^{+0.052}_{-0.047}$ | $7.186^{+0.051}_{-0.048}$

We also consider mock data by assuming the GB metric but not the Kerr ones.
For each one $b_{i}$, we generate the mock data by setting this $b_{i}$ to
$5$, and fit the data with the GB metric. we only vary one $b_{i}$ parameter
each time. Similar to the Kerr case, only $b_{2}$, $b_{9}$, $b_{10}$, and
$b_{11}$ can be relatively well constrained, though in most cases the
constraints are not as good as those in the Kerr case. The reconstructed
parameters can be found in Table 2.

Table 2: Values of the GB parameter inferred from the mock observations generating by assuming a GB metric with only $b_{2}\neq 1$ but $=5$.  | Input | Best-fit111111Only $b_{2}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting. | Best-fit121212Only $b_{9}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting. | Best-fit131313Only $b_{10}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting. | Best-fit141414Only $b_{11}$ is taken as a free parameter and other $b_{i}$ are fixed at $1$ in the model fitting.
---|---|---|---|---|---
$b_{2}$151515All other $b_{i},i\neq 2$, are set to $1$. | 5 | $4.993^{+0.006}_{-0.010}$ | $\cdots$ | $\cdots$ | $\cdots$
$b_{9}$161616All other $b_{i},i\neq 9$, are set to $1$. | 5 | $\cdots$ | $4.934^{+0.101}_{-0.065}$ | $\cdots$ | $\cdots$
$b_{10}$171717All other $b_{i},i\neq 10$, are set to $1$. | 5 | $\cdots$ | $\cdots$ | $5.091^{+0.063}_{-0.245}$ | $\cdots$
$b_{11}$181818All other $b_{i},i\neq 11$, are set to $1$. | 5 | $\cdots$ | $\cdots$ | $\cdots$ | $5.709^{+0.946}_{-2.371}$
$a_{*0}$ | $0.75$ | $0.751^{+0.005}_{-0.003}$ | $0.550^{+0.250}_{-0.057}$ | $0.592^{+0.162}_{-0.078}$ | $0.749^{+0.007}_{-0.006}$
$a_{*1}$ | $0.95$ | $0.949^{+0.002}_{-0.001}$ | $0.964^{+0.019}_{-0.020}$ | $0.408^{+0.517}_{-0.013}$ | $0.865^{+0.032}_{-0.024}$
$a_{*2}$ | $0.37$ | $0.396^{+0.037}_{-0.032}$ | $0.910^{+0.033}_{-0.544}$ | $0.952^{+0.029}_{-0.578}$ | $0.370^{+0.002}_{-0.002}$
$a_{*3}$ | $0.65$ | $0.650^{+0.007}_{-0.003}$ | $0.635^{+0.066}_{-0.064}$ | $0.630^{+0.085}_{-0.083}$ | $0.650^{+0.003}_{-0.003}$
$a_{*4}$ | $0.16$ | $0.160^{+0.002}_{-0.002}$ | $0.164^{+0.002}_{-0.002}$ | $0.161^{+0.001}_{-0.001}$ | $0.158^{+0.001}_{-0.001}$
$a_{*5}$ | $0.08$ | $0.080^{+0.001}_{-0.001}$ | $0.080^{+0.001}_{-0.001}$ | $0.079^{+0.001}_{-0.001}$ | $0.078^{+0.001}_{-0.001}$
$a_{*6}$ | $0.50$ | $0.248^{+0.213}_{-0.005}$ | $0.520^{+0.273}_{-0.027}$ | $0.515^{+0.314}_{-0.042}$ | $0.513^{+0.003}_{-0.004}$
$a_{*7}$ | $0.07$ | $0.070^{+0.530}_{-0.000}$ | $0.068^{+0.001}_{-0.001}$ | $0.068^{+0.001}_{-0.001}$ | $0.069^{+0.001}_{-0.001}$
$a_{*8}$ | $0.15$ | $0.151^{+0.001}_{-0.002}$ | $0.150^{+0.001}_{-0.001}$ | $0.149^{+0.001}_{-0.001}$ | $0.153^{+0.001}_{-0.001}$
$a_{*9}$ | $0.44$ | $0.400^{+0.044}_{-0.006}$ | $0.439^{+0.403}_{-0.012}$ | $0.859^{+0.054}_{-0.404}$ | $0.445^{+0.003}_{-0.003}$
$M_{0}/M_{\odot}$ | $5.4$ | $5.442^{+0.060}_{-0.043}$ | $5.431^{+0.052}_{-0.048}$ | $5.238^{+0.049}_{-0.047}$ | $5.336^{+0.104}_{-0.070}$
$M_{1}/M_{\odot}$ | $7.9$ | $7.904^{+0.078}_{-0.068}$ | $7.873^{+0.070}_{-0.054}$ | $7.677^{+0.174}_{-0.069}$ | $5.820^{+0.604}_{-0.349}$
$M_{2}/M_{\odot}$ | $9.4$ | $9.670^{+0.412}_{-0.308}$ | $9.289^{+0.165}_{-0.101}$ | $9.797^{+0.091}_{-0.239}$ | $9.441^{+0.057}_{-0.055}$
$M_{3}/M_{\odot}$ | $7.8$ | $7.770^{+0.070}_{-0.062}$ | $7.887^{+0.075}_{-0.068}$ | $7.746^{+0.065}_{-0.056}$ | $7.802^{+0.058}_{-0.065}$
$M_{4}/M_{\odot}$ | $6.2$ | $6.230^{+0.044}_{-0.039}$ | $6.314^{+0.038}_{-0.038}$ | $6.175^{+0.036}_{-0.036}$ | $6.138^{+0.034}_{-0.033}$
$M_{5}/M_{\odot}$ | $8.4$ | $8.296^{+0.045}_{-0.045}$ | $8.340^{+0.048}_{-0.048}$ | $8.256^{+0.045}_{-0.043}$ | $8.357^{+0.045}_{-0.043}$
$M_{6}/M_{\odot}$ | $8.4$ | $6.743^{+1.665}_{-0.066}$ | $8.333^{+0.076}_{-0.079}$ | $8.354^{+0.099}_{-0.097}$ | $8.623^{+0.064}_{-0.074}$
$M_{7}/M_{\odot}$ | $5.3$ | $5.332^{+3.766}_{-0.033}$ | $5.323^{+0.030}_{-0.029}$ | $5.229^{+0.028}_{-0.028}$ | $5.274^{+0.027}_{-0.027}$
$M_{8}/M_{\odot}$ | $7.5$ | $7.546^{+0.044}_{-0.044}$ | $7.468^{+0.044}_{-0.042}$ | $7.381^{+0.042}_{-0.041}$ | $7.620^{+0.043}_{-0.043}$
$M_{9}/M_{\odot}$ | $7.2$ | $6.937^{+0.332}_{-0.306}$ | $7.218^{+0.062}_{-0.119}$ | $7.499^{+0.109}_{-0.122}$ | $7.303^{+0.061}_{-0.054}$

We may compare these results with those using BH shadow and iron line methods
as follows. BH shadow p15  shows very weak dependent on the parameters
$b_{2}$, $b_{8}$, $b_{9}$ and $b_{10}$. Parameters $b_{4}$, $b_{5}$, $b_{7}$
and $b_{11}$ do not produce any impact on shadow boundary and thus cannot be
constrained. According to the iron line studies in ghasemi , all GB parameters
except $b_{11}$ can be constrained with near future X-ray mission, though
$b_{5}$ and $b_{8}$ are relatively more difficult to constrain. Different
methods may be complementary to each other and put constraints on different
deformation GB parameters, therefore, they can be combined together to help
break the degeneracies and finally pin down the metric of black holes.

##  4 Discussions

In this paper, we simply assume that QPOs are related to the fundamental
frequencies at the same radius in the geodesic models for QPOs rpm . However,
we note that the QPOs may not occur at the same radius even they are indeed
determined by those fundamental frequencies as that in the geodesic models
StuchlikApJ . It is also possible that the QPOs may be related to other
frequencies, such as that proposed in the epicycle resonance (ER) model
Torok11 . Therefore, our demonstrations on the constraining power of QPOs on
the metric parameters $b_{i}$ are valid when the geodesic models correctly
explain the QPO phenomena and our results may be model dependent. The study
presented in our paper may be extended to those cases that the QPOs are due to
other frequencies, such as those in the ER model, which is beyond the scope of
this paper as currently it is still clear which model can explain the QPOs the
best. With many QPO measurements in future, one could combine different QPO
models and BH metrics to obtain strong constraints on both the QPO model and
the metric parameters.

We also note that there are various works in literature on constraining
alternative metrics for black holes. Below we give a short summary for them
and compare some of them with ours presented in this paper.

Reference q1  checked braneworld Kerr BH when bulk-space influence is
described by a single, brany tidal charge parameter, $b$. Similar to Kerr-
Newman solution in GR in which the square of the electric charge $Q^{2}$ is
replaced by a tidal charge $b$. The behavior of radial and vertical epicyclic
frequencies is qualitatively similar to Kerr and brany Kerr BHs. But there are
strong differences in the case of naked singularities. The vertical epicyclic
frequency could be even lower than radial one. In the structure of radial
profile the number of local extrema could be higher in comparison with
standard Kerr naked singularity. Also, the radial epicyclic frequency has no
zero point for some special family of brany naked singularity. Reference q2
introduced the Jahannsen and Psaltis (JP) metric and studied their QPOs. In
their quasi Kerr space time, quadrupole moment is a free parameter in addition
to Mass and spin. They show that for moderate spin, the Keplerian frequency is
independent of small deviations of the quadrupole moment from Kerr value. They
also showed that the epicyclic frequencies shows significant variations.
Reference q4  study QPOs in the space time of rotating braneworld gravity.
Such a BH carries a tidal charge as imprint of the extra fifth dimension. The
metric contains $\beta$ parameter as tidal charge. They found that large
enough value of the positive tidal charge is not supported by observations of
high frequency QPOs. But for large enough negative tidal charge, the
braneworld BH are similar to the Kerr case. They also showed that over-
rotating braneworld and extreme Kerr BH can not be distinguished in the high
frequency QPOs observations. Reference q5  considered Bardeen BH metric where
parameter $g$ in the metric can be introduced as the magnetic charge of a non-
linear electromagnetic field. He also considered JP metric where metric has an
infinite number of deformation parameter $\epsilon_{k}$. He fixes the mass and
found the spin parameter $a/M=0.279$ and $g/M<0.56$ at the $68\%$ C.L.. For
second metric he found $a/M=0.27$ and $\epsilon_{3}=0.5$ at the $68\%$ C.L..
In the reference q6 , they studied Einstein-Dilaton-Gauss-Bonnet (EDGB) theory
with QPOs. $\alpha/M^{2}$ characterizes the theory. They considered low
spinning BH in this work and showed the LOFT observation can put constrain on
the parameter $\alpha/M^{2}$ of EDGB. Reference q7  considered Konoplya and
Zhikendao metric. The metric is obtained by deforming the Kerr metric by
adding a static deformation to mass. They found constraint on spin parameter
and deformation parameter $\delta r/r_{Kerr}$.
If we compare our work with these above mentioned work, we measure QPO
frequencies and use MCMC method in addition to directly check data with
observations.

##  5 Conclusions

In this paper, we investigate whether QPOs can be used to constrain the
parameters of the GB metric introduced in Ref. p15  and possible deviations
from the Kerr one.

We adopt the general relativistic precession model to relate the QPO
frequencies to those fundamental ones, including the periastron and nodal
precession frequencies, of test particle at the inner edge of accretion disks
around black holes. We find that at least some of the QPO frequencies
resulting from the GB metric with $b_{2}$, $b_{4}$, $b_{9}$, $b_{10}$, or
$b_{11}$ significantly deviating from $1$ can be substantially different from
those from the Kerr case, though there are no such differences for other
$b_{i}$s.

Considering the case of GRO J1655-40 with three accurately measured QPO
frequencies, we find that the GB parameter $b_{2}$ is required to be
$2.2^{+0.395}_{-0.523}$ but other GB parameters cannot be well constrained.
Although such a constraint on $b_{2}$ seems to indicate that a GB metric fits
to the QPO data better, but the significance is smaller than $2$-$\sigma$. By
generating mock samples of $10$ black hole systems and each black hole with
three QPO frequency measurements, we use the nested algorithm to reconstruct
the input $b$ parameter(s) by using these mock samples. We find that $b_{2}$,
$b_{9}$, $b_{10}$, and $b_{11}$ can be well reconstructed, while $b_{5}$,
$b_{7}$, and $b_{8}$ cannot, as expected. However, $b_{4}$ cannot also be well
constrained due to parameter degeneracies among $b_{4}$, $M$, and $a$. The
results presented in this paper suggest that strong constraints on a few GB
parameters can be obtained by using the QPO frequency measurements of only
about ten or more black hole systems.

Acknowledgments This work is partly supported by the National Key Program for
Science and Technology Research and Development (Grant No. 2016YFA0400704),
the National Natural Science Foundation of China under grant No. 11690024 and
11390372, and the Strategic Priority Program of the Chinese Academy of
Sciences (Grant No. XDB 23040100). M. G.-N. also acknowledge support from the
China Postdoctoral Science Foundation, Grant No. 2017LH021.

## References

  * (1) Einstein, A. 1916, Annalen der Physik, 354, 769
  * (2) C. M. Will, “The Confrontation between General Relativity and Experiment,” Living Rev. Rel. 17 (2014) 4 [arXiv:1403.7377 [gr-qc]].
  * (3) Will C M 1993 Theory and Experiment in Gravitational Physics (Cambridge University Press) ISBN 0521439736
  * (4) Stairs I H 2003, “Testing General Relativity with Pulsar Timing,”  Living Reviews in Relativity 6
  * (5) Wex N 2014 “Testing Relativistic Gravity with Radio Pulsars” Frontiers in Relativistic Celestial Mechanics, vol 1 ed Kopeikin S (De Gruyter) ISBN 9783110345667 [arXiv:1402.5594]
  * (6) R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics,” Phys. Rev. Lett. 11 (1963) 237.
  * (7) R. P. Kerr, “Gravitational collapse and rotation,”
  * (8) J. R. Gair, M. Vallisneri, S. L. Larson and J. G. Baker, “Testing General Relativity with Low-Frequency, Space-Based Gravitational-Wave Detectors,” Living Rev. Rel. 16, 7 (2013) [arXiv:1212.5575 [gr-qc]].
  * (9) N. Yunes and X. Siemens, “Gravitational-Wave Tests of General Relativity with Ground-Based Detectors and Pulsar Timing-Arrays,” Living Rev. Rel. 16, 9 (2013) [arXiv:1304.3473 [gr-qc]].
  * (10) D. Psaltis, “Probes and Tests of Strong-Field Gravity with Observations in the Electromagnetic Spectrum,” Living Rev. Rel. 11, 9 (2008) [arXiv:0806.1531 [astro-ph]].
  * (11) C. Bambi, “Testing black hole candidates with electromagnetic radiation,” Rev. Mod. Phys. 89, no. 2, 025001 (2017) doi:10.1103/RevModPhys.89.025001 [arXiv:1509.03884 [gr-qc]].
  * (12) Manko, V. S., & Novikov, I. D. 1992, Classical and Quantum Gravity, 9, 2477
  * (13) K. Glampedakis and S. Babak, “Mapping spacetimes with LISA: Inspiral of a test-body in a ‘quasi-Kerr’ field,” Class. Quant. Grav. 23, 4167 (2006) [gr-qc/0510057].
  * (14) S. J. Vigeland and S. A. Hughes, “Spacetime and orbits of bumpy black holes,” Phys. Rev. D 81, 024030 (2010) [arXiv:0911.1756 [gr-qc]].
  * (15) S. J. Vigeland, “Multipole moments of bumpy black holes,” Phys. Rev. D 82, 104041 (2010) doi:10.1103/PhysRevD.82.104041 [arXiv:1008.1278 [gr-qc]].
  * (16) S. Vigeland, N. Yunes and L. Stein, “Bumpy Black Holes in Alternate Theories of Gravity,” Phys. Rev. D 83, 104027 (2011)
  * (17) T. Johannsen and D. Psaltis, “A Metric for Rapidly Spinning Black Holes Suitable for Strong-Field Tests of the No-Hair Theorem,” Phys. Rev. D 83, 124015 (2011) [arXiv:1105.3191 [gr-qc]].
  * (18) V. Cardoso, P. Pani and J. Rico, “On generic parametrizations of spinning black-hole geometries,”  Phys. Rev. D 89, 064007 (2014) doi:10.1103/PhysRevD.89.064007 [arXiv:1401.0528 [gr-qc]].
  * (19) L. Rezzolla and A. Zhidenko, “New parametrization for spherically symmetric black holes in metric theories of gravity,” Phys. Rev. D 90, no. 8, 084009 (2014) [arXiv:1407.3086 [gr-qc]].
  * (20) N. Lin, N. Tsukamoto, M. Ghasemi-Nodehi and C. Bambi, “A parametrization to test black hole candidates with the spectrum of thin disks,” Eur. Phys. J. C 75, no. 12, 599 (2015), [arXiv:1512.00724 [gr-qc]].
  * (21) R. Konoplya, L. Rezzolla and A. Zhidenko, “General parametrization of axisymmetric black holes in metric theories of gravity,” Phys. Rev. D 93, no. 6, 064015 (2016) [arXiv:1602.02378 [gr-qc]].
  * (22) C. Bambi, J. Jiang and J. F. Steiner, “Testing the no-hair theorem with the continuum-fitting and the iron line methods: a short review,” Class. Quant. Grav. 33, no. 6, 064001 (2016) [arXiv:1511.07587 [gr-qc]].
  * (23) K. Yagi and L. C. Stein, “Black Hole Based Tests of General Relativity,” Class. Quant. Grav. 33, no. 5, 054001 (2016) [arXiv:1602.02413 [gr-qc]].
  * (24) T. Johannsen, “Testing the No-Hair Theorem with Observations of Black Holes in the Electromagnetic Spectrum,” Class. Quant. Grav. 33, no. 12, 124001 (2016) [arXiv:1602.07694 [astro-ph.HE]].
  * (25) K. Yagi and L. C. Stein, “Black Hole Based Tests of General Relativity,” Class. Quant. Grav. 33, 054001 (2016) [arXiv:1602.02413 [gr-qc]].
  * (26) M. Ghasemi-Nodehi and C. Bambi, “Note on a new parametrization for testing the Kerr metric,” Eur. Phys. J. C 76, no. 5, 290 (2016), [arXiv:1604.07032 [gr-qc]].
  * (27) V. Cardoso and L. Gualtieri, “Testing the black hole ‘no-hair’ hypothesis,” Class. Quant. Grav. 33, no. 17, 174001 (2016) [arXiv:1607.03133 [gr-qc]].
  * (28) C. Bambi, “Testing the Kerr black hole hypothesis,” Mod. Phys. Lett. A 26, 2453 (2011) [arXiv:1109.4256 [gr-qc]];
  * (29) C. Bambi, “Testing the space-time geometry around black hole candidates with the available radio and X-ray data,” Astron. Rev. 8, 4 (2013) [arXiv:1301.0361 [gr-qc]].
  * (30) A. C. Fabian, M. J. Rees, L. Stella and N. E. White, “X-ray fluorescence from the inner disc in Cygnus X-1,” Mon. Not. Roy. Astron. Soc. 238, 729 (1989); A. C. Fabian, K. Iwasawa, C. S. Reynolds and A. J. Young, “Broad iron lines in active galactic nuclei,” Publ. Astron. Soc. Pac. 112, 1145 (2000) [astro-ph/0004366]; C. S. Reynolds and M. A. Nowak, “Fluorescent iron lines as a probe of astrophysical black hole systems,” Phys. Rept. 377, 389 (2003) [astro-ph/0212065].
  * (31) S. N. Zhang, W. Cui and W. Chen, “Black hole spin in X-ray binaries: Observational consequences,” Astrophys. J. 482, L155 (1997) [astro-ph/9704072]; L. -X. Li, E. R. Zimmerman, R. Narayan and J. E. McClintock, “Multi-temperature blackbody spectrum of a thin accretion disk around a Kerr black hole: Model computations and comparison with observations,” Astrophys. J. Suppl. 157, 335 (2005) [astro-ph/0411583]; J. E. McClintock et al., “Measuring the Spins of Accreting Black Holes,” Class. Quant. Grav. 28, 114009 (2011) [arXiv:1101.0811 [astro-ph.HE]].
  * (32) http://www.eventhorizontelescope.org/science/index.html
  * (33) T. Johannsen, C. Wang, A. E. Broderick, S. S. Doeleman, V. L. Fish, A. Loeb and D. Psaltis, “Testing General Relativity with Accretion-Flow Imaging of Sgr A*,” Phys. Rev. Lett. 117, no. 9, 091101 (2016) [arXiv:1608.03593 [astro-ph.HE]].
  * (34) L. Stella and M. Vietri, “Lense-Thirring precession and QPOS in low mass x-ray binaries,” Astrophys. J. 492, L59 (1998) [astro-ph/9709085]; L. Stella and M. Vietri, “Khz quasi periodic oscillations in low mass x-ray binaries as probes of general relativity in the strong field regime,” Phys. Rev. Lett. 82, 17 (1999) [astro-ph/9812124]; L. Stella, M. Vietri and S. Morsink, “Correlations in the qpo frequencies of low mass x-ray binaries and the relativistic precession model,” Astrophys. J. 524, L63 (1999) [astro-ph/9907346].
  * (35) C. A. Perez, A. S. Silbergleit, R. V. Wagoner and D. E. Lehr, “Relativistic diskoseismology. 1. Analytical results for ’gravity modes’,” Astrophys. J. 476, 589 (1997) [astro-ph/9601146]; A. S. Silbergleit, R. V. Wagoner and M. Ortega-Rodriguez, “Relativistic diskoseismology. 2. Analytical results for C modes,” Astrophys. J. 548, 335 (2001) [astro-ph/0004114]; S. Kato, “Basic Properties of Thin-Disk Oscillations,” Publ. Astron. Soc. Jap. 53 1 (2001).
  * (36) M. A. Abramowicz and W. Kluzniak, “A Precise determination of angular momentum in the black hole candidate GRO J1655-40,” Astron. Astrophys. 374, L19 (2001) [astro-ph/0105077]; M. A. Abramowicz, V. Karas, W. Kluzniak, W. H. Lee and P. Rebusco, “Non-Linear Resonance in Nearly Geodesic Motion in Low-Mass X-Ray Binaries,” Publ. Astron. Soc. Jap. 55 467 (2003); G. Torok, M. A. Abramowicz, W. Kluzniak and Z. Stuchlik, “The orbital resonance model for twin peak kHz quasi periodic oscillations in microquasars,” Astron. Astrophys. 436, 1 (2005).
  * (37) L. Rezzolla, S. ’i. Yoshida, T. J. Maccarone and O. Zanotti, “A New simple model for high frequency quasi periodic oscillations in black hole candidates,” Mon. Not. Roy. Astron. Soc. 344, L37 (2003) [astro-ph/0307487]; J. D. Schnittman and L. Rezzolla, “Quasi-periodic oscillations in the x-ray light curves from relativistic tori,” Astrophys. J. 637, L113 (2006) [astro-ph/0506702].
  * (38) Z. Stuchlik and A. Kotrlova, “Orbital resonances in discs around braneworld Kerr black holes,” Gen. Rel. Grav. 41, 1305 (2009) [arXiv:0812.5066 [astro-ph]].
  * (39) T. Johannsen and D. Psaltis, “Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum. III. Quasi-Periodic Variability,” Astrophys. J. 726, 11 (2011) [arXiv:1010.1000 [astro-ph.HE]].
  * (40) C. Bambi, “Probing the space-time geometry around black hole candidates with the resonance models for high-frequency QPOs and comparison with the continuum-fitting method,” JCAP 1209, 014 (2012) [arXiv:1205.6348 [gr-qc]].
  * (41) A. N. Aliev, G. D. Esmer and P. Talazan, “Strong Gravity Effects of Rotating Black Holes: Quasiperiodic Oscillations,” Class. Quant. Grav. 30, 045010 (2013) [arXiv:1205.2838 [gr-qc]].
  * (42) C. Bambi, “Testing the nature of the black hole candidate in GRO J1655-40 with the relativistic precession model,” Eur. Phys. J. C 75, no. 4, 162 (2015) [arXiv:1312.2228 [gr-qc]].
  * (43) A. Maselli, L. Gualtieri, P. Pani, L. Stella and V. Ferrari, “Testing Gravity with Quasi Periodic Oscillations from accreting Black Holes: the Case of the Einstein-Dilaton-Gauss-Bonnet Theory,” Astrophys. J. 801, no. 2, 115 (2015) [arXiv:1412.3473 [astro-ph.HE]].
  * (44) C. Bambi and S. Nampalliwar, “Quasi-periodic oscillations as a tool for testing the Kerr metric: A comparison with gravitational waves and iron line,” EPL 116, no. 3, 30006 (2016) [arXiv:1604.02643 [gr-qc]].
  * (45) M. Ghasemi-Nodehi and C. Bambi, “Constraining the Kerr parameters via X-ray reflection spectroscopy,” Phys. Rev. D 94, no. 10, 104062 (2016) [arXiv:1610.08791 [gr-qc]].
  * (46) C. Bambi, “Black Holes: A Laboratory for Testing Strong Gravity,”
  * (47) S. E. Motta, T. M. Belloni, L. Stella, T. Muñoz-Darias and R. Fender, “Precise mass and spin measurements for a stellar-mass black hole through X-ray timing: the case of GRO J1655-40,” Mon. Not. Roy. Astron. Soc. 437, no. 3, 2554 (2014) [arXiv:1309.3652 [astro-ph.HE]].
  * (48) S. E. Motta, T. Muñoz-Darias, A. Sanna, R. Fender, T. Belloni and L. Stella, “Black hole spin measurements through the relativistic precession model: XTE J1550-564,” Mon. Not. Roy. Astron. Soc. 439, 65 (2014) [arXiv:1312.3114 [astro-ph.HE]].
  * (49) P. Casella, T. Belloni and L. Stella, “The ABC of low-frequency quasi-periodic oscillations in black-hole candidates: Analogies with Z-sources,” Astrophys. J. 629, 403 (2005) doi:10.1086/431174 [astro-ph/0504318].
  * (50) M. E. Beer and P. Podsiadlowski, “The quiescent light curve and evolutionary state of gro j1655-40,” Mon. Not. Roy. Astron. Soc. 331, 351 (2002) [astro-ph/0109136].
  * (51) Speagle, J. S. 2019, arXiv e-prints, arXiv:1904.02180
  * (52) Z. Stuchlík, & M. Kološ, “Controversy of the GRO J1655-40 Black Hole Mass and Spin Estimates and Its Possible Solutions”, Astrophys. J. 825, 13 (2016)
  * (53) G. Török, Kotrlová, A., E. Šrámková, et al. “Confronting the models of 3:2 quasiperiodic oscillations with the rapid spin of the microquasar GRS 1915+105”, Astron. & Astrophy. 531, A59 (2011)