# Revisiting Flat Rotation Curves in Chern-Simons Modified Gravity

Waleed El Hanafy<EMAIL_ADDRESS>Centre for Theoretical Physics,
The British University in Egypt, P.O. Box 43, El Sherouk City, Cairo 11837,
Egypt  Mahmoud Hashim<EMAIL_ADDRESS>Centre for Theoretical
Physics, The British University in Egypt, P.O. Box 43, El Sherouk City, Cairo
11837, Egypt  G.G.L. Nashed<EMAIL_ADDRESS>Centre for Theoretical Physics,
The British University in Egypt, P.O. Box 43, El Sherouk City, Cairo 11837,
Egypt

###### Abstract

We revisit slow rotating black hole (BH) solutions in Chern-Simons modified
gravity (CSMG) by considering perturbative solution about Schwarzschild BH. In
particular, the case when nondynamical CSMG with noncanonical CS scalar is
considered. We provide a new solution different from the previously obtained
one [1] which we refer to as KMT model. The present solution accounts for
frame dragging effect which includes not only radial dependence as in the KMT.
Nevertheless, it reduces to KMT as a particular case. We show that the tidal
gravitational force (Kretschmann invariant) associated to the present solution
contains a term of order $1/r^{3}$ additional to Schwarzschild but absent from
directional divergence, unlike KMT model which diverges along the axis of
symmetry. We derive the corresponding circular velocity of a massive test
particle in which the KMT velocity is recovered in addition to an extra term
$\propto r$. We investigate possible constraints on KMT and the present
solutions from the observed rotation curve of UGC11455 galaxy as an example.
We show that perturbation solutions cannot physically explain the flattening
of galactic rotation curves.

##  1 Introduction

The general relativity (GR) theory is no doubt setting the solid bases which
formulate the contemporary approaches to understand nature of gravity. The
astonishing idea that gravity is a curvature of pseudo-Riemann spacetime has
received recognition over the years by passing many observational tests like
planetary motion in the solar system [2], bending of light [3], binary pulsars
[4, 5] (see also [6]), gravitational waves detection due to binary compact
objects merging by Laser Interferometer Gravitational-Wave Observatory (LIGO)
and Virgo Collaborations [7, 8, 9] and recently BH shadows by Event Horizon
Telescope [10]. However, it struggles with inconsistent description of gravity
on the scales of galaxies. The observed velocities at large radii from the
galactic centres are inconsistent with the gravitational field from the
visible light from stars only, which are mostly clustered in the middle of
galaxies. Unlike the solar system planetary motion, the velocities of the
stars away from the centre of a galaxy have almost same velocities of the
stars near the core, which results in a flattening rotational velocity curve
at large distances within galaxies [c.f. 11, 12, 13, 14].

One way to explain the stellar motion is to assume that the galaxies are
embedded in large Cold Dark Matter (CDM) haloes as noted early by Zwicky in
1933. The temperature anisotropy of the cosmic microwave background (CMB)
spectrum as detected by Planck provides a strong evidence supporting existence
of CDM [15]. Weakly interacting massive particles (WIMPs) within an energy
scale GeV–TeV have been suggested as candidates to represent CDM from particle
physics perspective. The hypothesised WIMPs perfectly match the CDM abundance
observed today [16]. However, still no experimental evidences support this
hypothesis, in fact, it is mostly excluded by large hadron collider (LHC)
[17], or other detectors [c.f., 18]. In this sense, recent experiments
significantly narrow possibilities to explain CDM as particles.

Another way to explain the flat rotation curve is to investigate possible
modification of gravity at these scales. This could be reasonable; since the
assumed CDM could not mediate electromagnetic force and it can be seen only
through its gravitational effect on stellar motion. It has been suggested that
the Newtonian acceleration law at weak gravity regimes should be modified as
noted by [19, 20] in Modified Newtonian Dynamics (MOND). Within this frame, it
is possible to gain some successes, e.g. the explanation for the flat galactic
rotation curves and the Tully-Fisher relation without the need for CDM
particles. Notably, the longstanding struggle of MOND explaining CMB power
spectra has been reinvestigated for a new relativistic MOND theory, with some
phenomenological requirements, the recent version of the theory agrees with
the observed CMB and matter power spectra on linear cosmological scales [21].
On the other hand, a sample of 152 galaxy rotation curves included by Spitzer
Photometry and Accurate Rotation Curves (SPARC) database, has been used to
distinguish CDM halos, modified gravity and modified inertia [22]. It has been
shown that CDM halos predict a systematically deviating relation from the
observed one. All aspects of rotation curves are most naturally explained by
modified gravity. Several modified gravity theories have been introduced to
explain the flattening of the rotational velocity curves of spiral galaxies
[c.f., 23, 24, 25, 26, 27, 28, 29, 30, 31, 22, 32]. Remarkably, one could
relate BH solutions to rotation curves of galaxies following a simple
approach: Solve the field equation of the suggested gravity theory to obtain
BH vacuum solution. Obtain the first integral of Euler-Lagrange equation and
deduce the circular motion $v_{circ}$ of a test particle of mass $m$. In the
case of galaxies, the space is not exactly vacuum, it rather has visible
baryons with mass distribution $M(r)$, that is similar to stellar models.
However, the visible matter pressure in the galaxy is very small, the
relativistic effects of the pressure of the baryonic matter are negligible.
Therefore, the setup is consistent with Tolman-Oppenheimer-Volkof (TOV)
equation at weak field limit, whereas $M(r)$ for each galaxy is governed by
$M(r)/r=V^{2}_{\text{bar}}=\sum_{i}\Upsilon_{i}V_{i}^{2}$, where $i$ denotes
the bulge, disk and gas components. Thus, one uses observational data of the
velocity contributions $V_{i}$ for each component to fit the parameter space
including the coefficients $\Upsilon_{i}$ and additional parameters due to
modified gravity. It remains unclear how modified gravity can explain Bullet
Cluster with no dark matter [33, 34], see also [35].

CSMG is one of the well inspired modifications of GR, since it is motivated by
anomaly cancellation in particle physics, string theory and loop quantum
gravity. In particle physics, motivated by the CS modification of
electrodynamics [36], gravitational ABJ anomaly can be cancelled by adding CS
term into Einstein-Hilbert action [37]. In string theory, the coupling of a
gauge field to the string results in a gravitational anomaly in the effective
low energy 4-dimensional GR [38], it has been shown that a CS term should
arise via Greens-Schwarz anomaly cancelling mechanism [39]. In the effort to
quantise GR within the loop quantum gravity framework by considering discrete
spacetime, the investigation of parity and charge-parity conservation leads to
CS modification [40]. For more details see the review [41].

In general, there are two distinct settings in which the CSMG has been
formulated, those are non-dynamical and dynamical CSMG. The pseudo CS scalar
field is understood as a wholly external quantity in the former case, while it
has dynamics in the latter fulfilling its own equation of motion. Although
Kerr metric is not a solution in CSMG [c.f., 42, 41], spinning BH solutions
can be found. We may classify the existing solutions as follows: (i) Exact
solutions using a non-dynamical CS scalar, it has been shown that, for the
standard choice of the CS scalar, the field equations eliminate all solutions
except Schwarzschild [42]. On the other hand, for a noncanonical choice of the
CS scalar, mathematical rotating BH solutions can be found. (ii) Exact
solutions using a dynamical CS scalar have not been found yet as far as we
know. (iii) The slow-rotating approximation using a non-dynamical CS scalar
has been studied in [43] for a noncanonical CS scalar, where the circular
velocity reads $v_{circ}\sim\sqrt{M/r}+\text{const}$ far from the galactic
centre. This solution has been proposed to explain the flattening of the
rotation curve due to the existence of the additional constant. In the same
line, a generalised choice of the scalar field has been suggested [44], where
the circular velocity $v_{circ}\sim\sqrt{M/r}-M/r^{2}$, which cannot explain
the galactic rotation curve without dark matter. (iv) The slow-rotating
approximation using a dynamical CS scalar has been studied in [44], which
shows that the full dynamical solution introduces extra terms that are highly
suppressed by large inverse powers of the radius. Consequently, it would not
be able to describe the flattening of the rotational velocity curve. On the
other hand, CSMG modifies early universe leaving testable features on the
observed CMB polarisation on the superhorizon scales at low multipole moments
[c.f., 45, 46]. Notably, these corrections are practically hidden due to large
cosmic variance at these scales.

In the present work, we revisit slow rotating BHs solution in CSMG using the
perturbation expansion around the Schwarzschild. In particular the
nondynamical CSMG with noncanonical CS scalar which has been proposed by [1]
and will be referred to as KMT model in the present study. As just mentioned
above, this slow-rotating BH solution has been advocated to explain the
flatness of rotational velocity curves of spiral galaxies [43]. However, it
never be confronted with observational data. We organise the paper as follows:
In Sec. 2, we briefly review the nondynamical CS modification of GR gravity.
We further apply the field equations to stationary and axially symmetric
spacetime configuration which represents gravitational field of slowly
rotating BH. We obtain a new CS solution which reduces the previously obtained
solution by KMT as a special case. We discuss the physical features of the new
solution in comparison to the known Kerr solution and KMT model by comparing
Kretschmann invariant in each case. In Sec. 3, we determine the geodesic
motion of a massive test particle in the gravitational field of the obtained
CS solution. Then, we obtain the CS rotational velocity of the circular and
equatorial motion which contains an additional term, proportional to the
radial distance, in comparison to KMT model. In Sec. 4, we use the rotation
curve of UGC11455 galaxy to investigate possible constraints on the parameter
space of the KMT and the present model. This shows clearly the failure of KMT
model to fit the curve in principle. Although the present model shows good
results, it cannot fulfil the perturbation approach assumption to interpret
the rotation curve. This conclusion should be valid for any perturbative
solution around Schwarzschild BH. In Sec. 5, we summarise and conclude the
obtained results.

##  2 Slowly Rotating Black Hole in General Relativity and Chern-Simons
Gravity

We give a brief review on Chern-Simons modifications of the GR theory. In
particular, nondynamical CSMG with noncanonical CS scalar as discussed by [1].
For more general framework with dynamical CSMG, see the review [41]. Unless
otherwise noted, we use the geometric units111The geometric units will be used
in Sections 2–3 to write the equations in simplified forms, while the standard
physical units will be restored in Section 4 for practical purposes. in which
the gravitational constant and the speed of light are set to $G=c=1$, whereas
the action can be written as

$I=\frac{1}{16\pi}\int
d^{4}x\left(-\sqrt{-g}R+\frac{1}{4}l\,\vartheta\,\,^{\ast}{R}{{}^{\sigma}}{{}_{\tau}}{{}^{\mu\nu}}R{{}^{\tau}}_{\sigma\mu\nu}\right).$
(1)

The first term gives Einstein–Hilbert action with metric determinant $g$ and
Ricci scalar $R$. The second term gives CS modifications with a coupling
constant $l$, an external quantity $\vartheta$ acts as a coupling field in CS
modified gravity and dual Riemann tensor
${}^{\ast}{R}{{}^{\sigma}}{{}_{\tau}}{{}^{\mu\nu}}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}R^{\tau}{{}_{\sigma\alpha\beta}}$
where $\varepsilon$ denotes Levi-Civita tensor. For constant $\vartheta$, the
GR theory is reproduced as a particular case [41].

The variation with respect to the metric tensor $g_{\mu\nu}$ gives rise to the
following field equations

$\displaystyle G^{\mu\nu}+lC^{\mu\nu}$ $\displaystyle=$ $\displaystyle-8\pi
T^{\mu\nu},$ (2) $\displaystyle 0=\nabla_{\mu}C^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{8\sqrt{-g}}v^{\nu}\,\
{}^{\ast}{R}{{}^{\sigma}}{{}_{\tau}}{{}^{\mu\lambda}}R{{}^{\tau}}_{\sigma\mu\lambda},$
(3)

where $G^{\mu\nu}$ is Einstein tensor, $T^{\mu\nu}$ is the stress-energy
tensor, $v_{\alpha}=\partial_{\alpha}\vartheta$ is called embedding vector and
$C^{\mu\nu}$ is Cotton tensor,

$\displaystyle C^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\sqrt{-g}}\Bigg{[}\Bigg{.}v_{\sigma}\left(\varepsilon^{\sigma\mu\alpha\beta}\nabla_{\alpha}R{{}^{\nu}}{{}_{\beta}}+\varepsilon^{\sigma\nu\alpha\beta}\nabla_{\alpha}R{{}^{\mu}}{{}_{\beta}}\right)+v_{\tau\sigma}\left({}^{\ast}{R}^{\tau\mu\sigma\nu}+^{\ast}{R}^{\tau\nu\sigma\mu}\right)\Bigg{.}\Bigg{]},$
(4)

with $v_{\tau\sigma}=\nabla_{\sigma}v_{\tau}$ and $\nabla$ is the covariant
derivative associated to Levi-Civita linear connection. By Bianchi identity
$\nabla_{\mu}G^{\mu\nu}\equiv 0$ and by requiring the matter conservation
$\nabla_{\mu}T^{\mu\nu}=0$, the free-divergence Cotton tensor constraint
should be imposed, namely Equation (3). For a solution to fulfil CS gravity,
it should satisfy Equations (2) and (3). Therefore, those are the basic
equations which govern CSMG in the present context.

It is worth mentioning that for a spherically symmetric spacetime the CS
solution is nothing but Schwarzschild solution, since that the Cotton tensor
vanishes identically. For a slowly rotating BH, we consider perturbation
around the Schwarzschild solution, where the expansion parameter $\epsilon$ is
related to the BH angular momentum $J$, i.e. $J\sim O(\epsilon)$. Thus, the
spacetime configuration gives rise to a stationary and axisymmetric metric in
a spherical polar coordinates ($t$, $r$, $\theta$, $\phi$) [1, 43]

$ds^{2}=-A(r)dt^{2}+\frac{dr^{2}}{A(r)}+r^{2}(d\theta^{2}+\sin^{2}{\theta}\,d\phi^{2})-2r^{2}\omega(r,\theta)d\phi
dt,$ (5)

where the time coordinates $-\infty<t<\infty$, the radial distance $0\leq
r<\infty$, the zenith (inclination) angle $0\leq\theta\leq\pi$, the azimuthal
angle $-\pi\leq\phi<\pi$, the metric potential $A(r)\equiv 1-\frac{2M}{r}$ and
$\omega(r,\theta)$ represents dragging of local inertial frames, i.e. Lense-
Thirring effect. Independent of the choice embedding coordinate, it has been
shown that no rotating BH solution can be found up to the first order whereas
the embedding coordinate is timelike, i.e. $\vartheta\propto t$, unless the
limit $M/r\to 0$ is fulfilled [1]. On the contrary, for the choice
$\vartheta\propto r\cos\theta$, i.e. the embedding coordinate is spacelike,
rotating BH solution can be found for arbitrary BH mass. In the following
subsections, we review a previously obtained solution limited to a particular
choice for the frame-dragging, then we obtain the generalised solution with a
comparison to the GR solutions in each case.

###  2.1 Case I: KMT model

For a spacelike embedding coordinate $\vartheta=r\cos\theta/\lambda_{0}$, the
embedding vector is written as $v_{\alpha}$ = ($0$, $\cos\theta/\lambda_{0}$,
$-r\sin\theta/\lambda_{0}$, $0$) [1]. Applying the free-divergence Cotton
tensor (3) to the spacetime (5), we obtain the following constraint up to
$O(\epsilon)$

$\omega_{,r\theta}=0,$ (6)

where the sub-indices $,r$ and $,\theta$ denote the derivatives with respect
to those coordinates. It has been shown that for the particular choice
$\omega(r,\theta)\to\omega(r)$ the above constraint is satisfied.
Additionally, it can be shown that the field equations (2) reduce to one
equation

$r(r-2M)\omega_{,rr}+4(r-2M)\omega_{,r}+2\omega=0.$ (7)

Therefore, the solution has been given as [1, 43]

$\omega(r)=\frac{C_{1}}{r^{2}}\left(1-\frac{2M}{r}\right)+\frac{C_{2}}{r^{3}}\left[r^{2}-2Mr-4M^{2}+4M(r-2M)\ln(r-2M)\right],$
(8)

where the constants $C_{1}$ and $C_{2}$ are of order $O(\epsilon)$ whereas
$\epsilon\equiv J/Mr$ with a total angular momentum $J$. In fact, the choice
of the drag effect to be radial $\omega(r,\theta)\to\omega(r)$ satisfies
$\omega_{,r\theta}\equiv 0$ identically. It, additionally, makes the Cotton
tensor components to vanish up to $O(\epsilon)$. Thus, the field equation (7)
is identical to the GR field equation $G^{t\phi}=0$, whereas the other
components vanish identically. In this sense, we conclude that the particular
choice of the frame drag to have a radial dependence only hides the CS gravity
contribution rendering it equivalent to GR gravity up to $O(\epsilon)$. This
has been already noted by [1, 43], we refer to the solution (8) as KMT model.
In the following subsection, we drop this constraint seeking for a new CS
gravity solution distinguished from the GR one.

###  2.2 Case II: CSMG model

For a spacelike embedding coordinate $\vartheta=r\cos\theta/\lambda_{0}$, we
write the embedding vector
$v_{\alpha}=(0,\cos\theta/\lambda_{0},-r\sin\theta/\lambda_{0},0)$. Applying
the field equations (2) to the spacetime (5), up to the order $O(\epsilon)$,
we obtain the non-vanishing components ($t\phi$), ($rr$), ($r\theta$),
($\theta\theta$), ($\phi\phi$) and ($tt$), respectively, as follows

$\displaystyle
r(r-2M)\omega_{,rr}+\omega_{,\theta\theta}+4(r-2M)\omega_{,r}-\cot\theta\,\omega_{,\theta}+2\omega=0,\qquad$
(9) $\displaystyle
r(r^{2}-5rM+6M^{2})\omega_{,rr}+(r-3M)\omega_{,\theta\theta}-3M(r-2M)\cot\theta\omega_{,r\theta}+(r-3M)\left[4(r-2M)\omega_{,r}-\cot\theta\omega_{,\theta}+2\omega\right]=0,\qquad$
(10)
$\displaystyle(r-3M)\cot\theta\omega_{,\theta\theta}+r^{2}(r-2M)\cot\theta\omega_{,rr}+\left[(r-6M)-\sec^{2}\theta(r-3M)\right]\omega_{,\theta}+2r\cot\theta\left[2(r-2M)\omega_{,r}+\omega\right]=0,\qquad$
(11) $\displaystyle
r(r-2M)\omega_{,rr}+\omega_{,\theta\theta}+3M\tan\theta\omega_{,r\theta}+4(r-2M)\omega_{,r}-\cot\theta\omega_{,\theta}+2\omega=0,\qquad$
(12) $\displaystyle
r^{2}\tan\theta(r-2M)\omega_{,rrr}+\omega_{,\theta\theta\theta}+r(r-2M)\omega_{,rr\theta}+r\tan\theta\omega_{,r\theta\theta}-r\tan\theta\left[(r-2M)\sec^{2}\theta-2(3r-5M)\right]\omega_{,rr}$
$\displaystyle-\frac{1+\cos^{2}\theta}{\sin\theta\cos\theta}\omega_{,\theta\theta}+(3r-5M)\omega_{,r\theta}-2\tan\theta\left[2(r-2M)\sec^{2}\theta-3r\right]\omega_{,r}+2\frac{1+\sin^{2}\theta}{\sin^{2}\theta}\omega_{,\theta}-2\sec\theta\csc\theta\omega=0,\qquad$
(13) $\displaystyle
r^{2}\sin\theta\left[4M^{2}+r(r-4M)\right]\omega_{,rrr}+\cos\theta(r-2M)\omega_{,\theta\theta\theta}+r\cos\theta\left[4M^{2}+r(r-4M)\right]\omega_{,rr\theta}+r\sin\theta(r-2M)\omega_{,r\theta\theta}$
$\displaystyle+r\sin\theta(6r^{2}-23rM+22M^{2})\omega_{,rr}+\left[\sin\theta(r-3M)-\sec\theta(r-2M)\right]\omega_{,\theta\theta}+\cos\theta(3r^{2}-11rM+10M^{2})\omega_{,r\theta}$
$\displaystyle+2\sin\theta(3r^{2}-8rM+4M^{2})\omega_{,r}+\cos\theta\left[(2r-3M)+\sec^{2}\theta(r-2M)\right]\omega_{,\theta}+2M\cos\theta\cot\theta\omega=0.\qquad$
(14)

In addition, the free-divergence Cotton tensor, namely equation (3), up to
$O(\epsilon)$, gives rise to

$\omega_{,r\theta}=0.$ (15)

On the contrary to the previous solution (KMT) [1], we take the general
solution of the above differential equation
$\omega(r,\theta)=\omega_{r}(r)+\omega_{\theta}(\theta)$. Then, we find the
solution of the system, namely Equations (9)–(14), as follows

$\displaystyle\omega(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{C_{1}}{r^{2}}\left(1-\frac{2M}{r}\right)+\frac{C_{2}}{r^{3}}\left[r^{2}-2Mr-4M^{2}+4M(r-2M)\ln(r-2M)\right]$
(16) $\displaystyle+$ $\displaystyle\frac{C_{3}}{r^{3}}\Bigg{\\{}r^{3}(1-\cos
2\theta)-6M\left[5M^{2}-(r-3M)^{2}\right]+24M^{2}(r-2M)\ln(r-2M)\Bigg{\\}},$

Clearly, the above solution differs from the one given in Equation (8) by an
additional term which contains the constant $C_{3}$. The obtained solution,
Equation (16), is a new one which generalises the previously obtained solution
[1], i.e. Equation (8). Consequently, we evaluate the cross term of the metric
component

$\displaystyle g_{t\phi}$ $\displaystyle=$ $\displaystyle
C_{1}\left(1-\frac{2M}{r}\right)+\frac{C_{2}}{r}\left[r^{2}-2Mr-4M^{2}+4M(r-2M)\ln(r-2M)\right]$
(17) $\displaystyle+$ $\displaystyle\frac{C_{3}}{r}\Bigg{\\{}r^{3}(1-\cos
2\theta)-6M\left[5M^{2}-(r-3M)^{2}\right]+24M^{2}(r-2M)\ln(r-2M)\Bigg{\\}}.$

At the limit $r\gg M$, we have

$g_{t\phi}=C_{1}+C_{2}r+2C_{3}r^{2}\sin^{2}\theta.$ (18)

The above equation indicates that the frame-dragging effect works in the whole
space as some terms diverge as $r$ increases.

In the previous subsection, we clarified that the CS and GR provide the same
solution once the frame dragging effect is chosen to be radial function
$\omega(r)$. In order to check the novelty of the just obtained solution by
accounting for CS terms in comparison to GR, assuming the general case that
the dragging effect is $\omega(r,\theta)$, we write the GR field equation up
to $O(\epsilon)$ as follows

$(r-2M)r\omega_{,r\theta}+\omega_{,r\theta}+4(r-2M)\omega_{,r}-\cot\theta\omega_{,\theta}+2\omega(r,\theta)=0.$
(19)

The above equation gives the component $G^{t\phi}=0$, whereas other components
vanish identically. This leads to the solution

$\displaystyle\omega(r,\theta)$ $\displaystyle=$
$\displaystyle\sin^{2}\theta\left[\tilde{C}_{1}(r-2M)^{\frac{-3+\alpha}{2}}F_{1}+\tilde{C}_{2}(r-2M)^{\frac{-3-\alpha}{2}}F_{2}\right]\left[\tilde{C}_{3}F_{3}+\tilde{C}_{4}F_{4}\cos\theta\right].$
(20)

where $\tilde{C}_{i}$ are integration constants and

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\text{hypergeom}\left(\left[{\frac{3-\alpha}{2}},{\frac{5-\alpha}{2}}\right],[1-\alpha],-\frac{2M}{r-2M}\right),$
$\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle\text{hypergeom}\left(\left[{\frac{3+\alpha}{2}},{\frac{5+\alpha}{2}}\right],[1+\alpha],-\frac{2M}{r-2M}\right),$
$\displaystyle F_{3}$ $\displaystyle=$
$\displaystyle\text{hypergeom}\left(\left[{\frac{3+\alpha}{4}},{\frac{3-\alpha}{4}}\right],\left[\frac{1}{2}\right],\cos^{2}\theta\right),$
$\displaystyle F_{4}$ $\displaystyle=$
$\displaystyle\text{hypergeom}\left(\left[{\frac{5+\alpha}{4}},{\frac{5-\alpha}{4}}\right],\left[\frac{3}{2}\right],\cos^{2}\theta\right),$
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\sqrt{9-4c_{1}},\,c_{1}\text{is an arbitrary constant}.$

Obviously, the GR field equation (19) coincides with one of the CS field
equations, the ($t\phi$)-component, namely Equation (9). Consequently, the CS
solution (16) is a GR solution as well. On the contrary, the GR solution (20)
does not fulfil CS field equations (10)–(15). Therefore, we ignore the GR
solution of rotating BH since the exact Kerr BH solution is well known.

We conclude that solution (16) provides a new solution which manifests novel
features of CS gravity, while it still satisfies the GR gravity. The
conclusion is not true for the opposite direction. In the following section,
we discuss some physical features of interest for the obtained solution in
comparison to some other rotating BH solutions.

###  2.3 Physical Features of Solution

In practice, it is always useful to quantify the amount of curvature as a
function of position near the BH to examine the genuine tidal force of gravity
for any solution. We use a quadratic scalar invariant, that is Kretschmann
scalar $R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}$ for that
purpose, and also to compare different solutions. We begin with the well known
Kerr metric, with $g^{(K)}_{t\phi}=-2J_{0}\sin^{2}\theta/r$ where $J_{0}\sim
O(\epsilon)$, then

$R_{(K)}^{\alpha\beta\gamma\delta}R_{(K)\alpha\beta\gamma\delta}\simeq\frac{48M^{2}}{r^{6}}-\frac{144J_{0}^{2}}{r^{8}}(2+\cos
2\theta).$ (21)

The first term is identical to Schwarzschild BH while the second term is
originated by the frame-dragging effect. Clearly, the latter is of order
$r^{-8}$, which decays more rapidly than the Schwarzschild part.

For the slowly rotating BH solution (8), assuming that
$\omega(r,\theta)\to\omega(r)$, we calculate Kretschmann invariant up to
$O(\epsilon)$

$\displaystyle R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}$
$\displaystyle\simeq$
$\displaystyle\left(R^{(0)\alpha\beta\gamma\delta}+R^{(1)\alpha\beta\gamma\delta}\right)\left(R^{(0)}{{}_{\alpha\beta\gamma\delta}}+R^{(1)}{{}_{\alpha\beta\gamma\delta}}\right)$
(22) $\displaystyle\simeq$
$\displaystyle\frac{48M^{2}}{r^{6}}-\frac{4C_{2}^{2}}{r^{4}\sin^{4}\theta},$

where $R^{(0)\alpha\beta\gamma\delta}$ refers to the zeroth order
$\epsilon$-quantity constructed from Schwarzschild spacetime and
$R^{(1)\alpha\beta\gamma\delta}$ refers to the first order $\epsilon$-quantity
constructed from the frame-dragging effect. This is in agreement with the
obtained value in [43]. In comparison to Kerr solution the first term is
common and originated by Schwarzschild metric, while the second term,
$O(r^{-4})$, is due to rotation which decays more slowly than the
corresponding term in Kerr and than Schwarzschild term as well. The solution
is singular along the axis of rotation ($\theta=0$, $\pi$), it is
asymptotically flat otherwise.

In the present study, dropping the frame-dragging assumption
$\omega(r,\theta)\to\omega(r)$, we similarly calculate Kretschmann invariant
for the generalised solution (16). Therefore, we obtain

$R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}\simeq\frac{48M^{2}}{r^{6}}+\frac{192MC_{3}^{2}}{r^{3}}\left(\frac{1}{3}+\cos
2\theta\right).$ (23)

Again the first term is associated with Schwarzschild solution, while the
second term, $O(r^{-3})$, is associated to the rotation. In comparison to Kerr
spacetime, Eq. (21), the frame dragging effect decays more slowly than the
corresponding term in Kerr and than Schwarzschild term as well. On the
contrary to the KMT case when $\omega\equiv\omega(r)$, the behaviour of
Kretschmann invariant is asymptotically flat everywhere with no singularity
along the rotational axis as in the KMT case, namely Eq. (22). It is also
expected to produce stronger dragging effect than the corresponding term in
Eq. (22). In the following section, we investigate possible consequences of
the obtained solution on astrophysical scales.

##  3 Rotation curve of Spiral Galaxies

As a first step towards confronting the CS model, which is characterised by
the spacetime metric (5) with the CS solution (16), with rotational curve
observational data, we obtain the trajectories of a massive test particle with
mass $m$ in the present spacetime configuration. These are determined by the
geodesic equations

$\frac{d^{2}x^{\alpha}}{d\tau^{2}}+\Gamma{{}^{\alpha}}{{}_{\mu\nu}}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=0,$
(24)

where $\tau$ is the proper time characterises the above trajectory and
$\Gamma$ is Levi-Civita linear connection. For the $\theta$-component, we set
$x^{\alpha}=\theta$ which yields

$\frac{d^{2}\theta}{d\tau^{2}}+\frac{2}{r}\frac{dr}{d\tau}\frac{d\theta}{d\tau}-\sin\theta\cos\theta\left[\left(\frac{d\phi}{d\tau}\right)^{2}-\omega(r,\theta)\frac{d\phi}{d\tau}\frac{dt}{d\tau}\right]=0.$
(25)

Taking initially the inclination angle $\theta=\theta_{0}=\pi/2$ and
$d\theta_{0}/d\tau=0$, we have $d\theta/d\tau=0$. This shows that the motion
of a test particle is planer motion in a good agreement with motion of
particles on the galaxy disk. Without loss of generality, we restrict
ourselves to circular motion of a test particle with mass $m$ in the edge-on
disk222The stellar motion in the galactic disk can be, in practice, corrected
due to different inclination angles as will be shown in Section 4., i.e.
inclination angle $\theta=\pi/2$, to confront the present model with the
observed rotational velocity data of spiral galaxies. Since the spacetime
metric does not depend on time nor azimuth angle, the components of the four
momenta $p_{t}=-mE$ and $p_{\phi}=mL$ are conserved on a trajectory where $E$
and $L$ are energy and angular momentum per unit mass. Thus, it is easy to
show that the first integral of Euler-Lagrange equation of $t$, $\phi$ and $r$
components can be written as

$\displaystyle\frac{dt}{d\tau}$ $\displaystyle=$ $\displaystyle\frac{E-\omega
L}{A(r)+r^{2}\omega(r)^{2}},$ (26) $\displaystyle\frac{d\phi}{d\tau}$
$\displaystyle=$ $\displaystyle\frac{L}{r^{2}}+\omega(r)\frac{E-\omega
L}{A(r)+r^{2}\omega(r)^{2}},$ (27) $\displaystyle\frac{dr}{d\tau}$
$\displaystyle=$
$\displaystyle\sqrt{\left(1+\frac{L^{2}}{r^{2}}\right)A(r)+\frac{(E-\omega(r)L)^{2}A(r)}{A(r)+r^{2}\omega(r)^{2}}}.$
(28)

For the static limit $\omega\to 0$, the above equations reduce to
Schwarzschild solution. For a circular orbit, using Equations (26) and (27),
we obtain the circular velocity

$v_{circ}=r\frac{d\phi}{dt}=\frac{A(r)L}{r(E-\omega(r)L)}+\frac{r\omega(r)E}{E-\omega(r)L},$
(29)

where $E$ and $L$ are determined using the circular motion constraints, i.e.
$\frac{dr}{d\tau}=0=\frac{d^{2}r}{d\tau^{2}}$. By virtue of Equation (28), we
determine

$\displaystyle E$ $\displaystyle=$
$\displaystyle\pm\frac{r-2M}{\sqrt{r(r-3M)}}+\sqrt{\frac{r^{2}M}{r-3M}}\left[\omega(r)+\frac{r(r-2M)}{2(r-3M)}\omega_{,r}\right]+O(\epsilon^{2}),$
$\displaystyle L$ $\displaystyle=$ $\displaystyle\pm
r\sqrt{\frac{M}{r-3M}}+\frac{r^{4}(r-2M)}{2\sqrt{r(r-3M)^{3}}}\omega_{,r}+O(\epsilon^{2}).$
(30)

Figure 1: A schematic plot of the total rotational velocity curve (31)
associated with individual term contribution where the longdashed curve
represents exactly Schwarzschild contribution ($C_{2}=0$ and $C_{3}=0$), i.e.
$v_{circ}=\sqrt{M/r}$, the dashdotted curve represents KMT model ($C_{2}\neq
0$, $C_{3}=0$), the dotted curve represents the velocity where $C_{2}=0$ and
$C_{3}\neq 0$ and the solid curve represents the total velocity curve which
gives rise to the present CS model.

Substituting into Equation (29), where the frame-dragging effect in CS gravity
is obtained by (16), we give the circular velocity

$\displaystyle v_{circ}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{M}{r}}+\frac{C_{2}}{2r^{2}}\left[\right.r^{2}+4M^{2}+8M^{2}\ln(r-2M)+4rM\left.\right]$
(31) $\displaystyle+$
$\displaystyle\frac{C_{3}}{r^{2}}\Bigg{\\{}12M^{3}-24M^{3}\ln(r-2M)-3r^{2}M-2r^{3}12rM^{2}\Bigg{\\}}.$

For $M/r\ll 1$, at the distant outer regions of the rotational curve,
following same approximations used in KMT model [43], the circular velocity
reduces to

$v_{circ}\approx\sqrt{\frac{M}{r}}+\frac{C_{2}}{2}+C_{3}(2r-7M).$ (32)

The above equation is the basic equation to calculate the rotational velocity
of a massive particle in CSMG. Clearly, it reduces to KMT model where
$C_{3}=0$. Indeed, the constant term associated to $C_{2}$ has been advocated
to explain the flat part of a rotation curve with no need to dark matter [43].
However, this claim has been never tested with observational data [41]. The
additional linear term in $r$ associated to the constant of integration
$C_{3}$ could be potentially useful to describe galaxy rotation curves, while
the first term in Equation (32) represents the Keplerian rotation curve
(Schwarzschild solution). In Fig. 1, we give a schematic plot showing
individual contribution of each term to the total velocity curve. We note that
we exaggerate Schwarzschild radius $r=2M$, which determines the $r$-intercept
of the curve, for better plotting purposes. The constant term $C_{2}$ alone
seems to be not able to describe the outer region flat part of the curve
unless it has a large value, which in turn pushes the curve peak to higher
values not consistent with a reasonable mass choice. On the contrary, a
reasonable choice of “small” value of the constant $C_{2}$ renders the
velocity curve almost compatible with the Keplerian curve at the outer region.
In the following section, we confront the obtained CS solution (32) with
observational data in comparison to KMT model [43].

##  4 Testing CSMG with UGC11455 galaxy rotation curve

In this section, we give an example to constrain the parameter space of the
KMT and the present models by fitting the rotation curve of UGC11455 galaxy as
obtained by SPARC database [29]. The UGC11455 galaxy is characterised by a
non-bulge profile, which simplifies the fitting procedure by reducing the
parameter space (i.e. to include only the stellar disk component). The number
of points are enough to ensure high resolution for the fitting purpose, where
the effective radius $R_{\rm eff}=10.06$ kpc and scale radius $r_{s}=5.93$
kpc. This galaxy is classified as a High Surface Brightness (HSB), with
effective surface brightness $\Sigma_{\rm eff}=571.26L_{\odot}$pc-2; this is
to ensure that the galaxy is dominated by stellar matter rather than dark
matter around the centre. Additionally, its disk inclination $i>30^{\circ}$,
which allows to ignore the bias due to galaxy’s disk inclination. However, we
account for the disk inclination by modifying the observed velocities by a
factor of $1/\sin(i)$.

###  4.1 Methodology

For CSMG model, we mainly use the rotational velocity equation (32) giving up
the geometric units assumption $G=c=1$. It proves convenient to write the free
parameters of a model in dimensionless forms, therefore we use the
transformation $C_{2}\to cC_{2}$ and $C_{3}\to H_{0}C_{3}$ to obtain $C_{2}$
and $C_{3}$ dimensionless, where $H_{0}$ is Hubble constant with a dimension
[T-1]. We note that the CS solution (16) of the metric (5) has been obtained
for vacuum. However, galaxies have visible baryons matter with mass
distributions, $M(r)$, up to large radial distances. Therefore, baryons matter
with pressure $p(r)$ and density $\rho(r)$ profiles, i.e. equation of state
($p\equiv p(\rho)$), need to be imposed into the field equations similar to
interior solution of stellar models where the metric potential, i.e.

$g_{tt}=A(r)=1-\frac{2GM(r)}{c^{2}r},$ (33)

and the mass function

$M(r)=4\pi\int_{\xi=0}^{r}\rho(\xi)\xi^{2}d\xi.$ (34)

However, in the present case, the relativistic effect of visible matter
pressure in the galaxies is almost negligible. In this sense, the above
solution, namely Equation (33), is a good approximation to the full
hydrostatic equilibrium configuration given by the TOV equation at weak field
limit. Thus, we rewrite the rotational velocity (32) up to $O(r^{-1})$ as
follows

$v_{circ}\approx\sqrt{\frac{GM(r)}{r}}+\frac{c}{2}C_{2}+2rH_{0}C_{3}\left(1-\frac{7}{2}\frac{GM(r)}{c^{2}r}\right),$
(35)

where the mass distribution of the baryons visible matter for every galaxy is
the following parameterised form [29]

$\frac{GM(r)}{r}=V_{\text{bar}}^{2}=\Upsilon_{\rm bulge}V_{\rm
bulge}^{2}+\Upsilon_{\rm disk}V_{\rm disk}^{2}+V_{\rm gas}|V_{\rm gas}|,$ (36)

with $\Upsilon_{\rm bulge,disk}$ being two free parameters (mass-to-light
ratios) associated to the contributions of the bulge velocity, $V_{\rm
bulge}$, and the disk velocity, $V_{\rm disk}$, to the baryon velocity. This
is in addition to the gas contribution $V_{\rm gas}$, where absolute value is
used since the gas near the centre may have opposite rotational direction. We
remind that no bulge is assumed in the present study to simplify the fitting
procedure. The above formula is appropriate with static spherically symmetric
solution. However, it is still useful in our procedure, since the Keplerian
term in the circular velocity (35) is dominant. Plugging (36) into (35), we
finally write the CS rotational velocity

$V_{\rm CS}:=v_{circ}=V_{\rm
bar}+\frac{c}{2}C_{2}+2rH_{0}C_{3}\left(1-\frac{7}{2}\frac{V^{2}_{\rm
bar}}{c^{2}}\right),$ (37)

where the frame dragging effect introduces $C_{2}$ and $C_{3}$ as free
parameters of the CS model to be fit with observational data.

We use Markov Chain Monte Carlo (MCMC) method to fit the observed rotation
curves of UGC11455 galaxy. Additionally, we use the emcee python package [47]
to map the posterior distribution of the fitting parameters. The priors are
taken such that lognormal probability distribution function (PDF) is imposed
on $\Upsilon_{\rm disk}$ around fiducial value 0.5 $M_{\odot}/L_{\odot}$ with
standard deviation of 0.1 dex as suggested by stellar population models [48,
49]. The priors of the tested models, $\vec{\alpha}=\\{\alpha_{i}\\}$, are
flat priors, where $C_{2}\in[0,10]$ and $C_{3}\in[-140,420]$. For KMT model,
we set $C_{3}=0$. We use the likelihood function,

${\cal L}(\chi)=\exp(-\frac{1}{2}\chi^{2}),$ (38)

where $\chi^{2}$ is defined as

$\chi^{2}=\sum_{i=1}^{N}\left(\frac{V_{\rm obs}(R_{i})-V_{\rm
CS}(\Upsilon_{\rm disk},\vec{\alpha},R_{i})}{\sigma_{V_{\rm
obs}(R_{i})}}\right)^{2},$ (39)

and $V_{\rm obs}$ is the $N$th observed rotational velocity of the UGC11455
galaxy and $\sigma_{V}$ is the corresponding uncertainty of $V_{\rm obs}$ as
provided by SPARC database [29]. The MCMC chains are initialised with 164
random walkers using the standard emcee affine-invariant sampler. The number
of steps are set to 1000 during the burn-in period for each walker. Then, we
reset the sampler and run longer chains with number of steps equal to 4000 to
ensure chains convergence.

###  4.2 Results

In this subsection, we mainly present the obtained results by applying the
methodology mentioned above to constrain the parameter space of the KMT and
CSMG models. Nevertheless, we also explain the failure of the KMT model to fit
real rotation curves data as previously discussed by the schematic plot in
Fig. 1. We do this by setting $C_{3}=0$ in the circular velocity given by Eq.
(37) to constrain the free parameter $C_{2}$. Then, we apply the above
mentioned procedure to UGC11455 galaxy rotation curve, and consequently we
obtain the velocity curve as shown by Fig. 2. In this case, the best fit of
the parameter $C_{2}=0.06746$ ($\sim 2.02$ km/s), which is too small to
maintain the flat part ($250-300$ km/s) of the rotation curve. On another
word, the KMT model (dotted curve) is practically reproduces the Keplerian
velocity curve. On the other hand, if one requires $C_{2}$ to be larger in
order to fit the flatten region of the curve, it would much shift the curve
upward spoiling up the inner region $R<10$ kpc by requiring large mass-to-
light ratio $\Upsilon_{\rm disk}$, which results in nonphysical masses above
reasonable values.

Figure 2: The rotational velocity curve of the galaxy UGC11455. Obviously the KMT model cannot fit the data at large distances which are supposed to be subjected to the modified gravity. This leads to the conclusion that the KMT model cannot fit galaxy rotation curves in practice. The CSMG fits the observed curve well where the outer region is dominated by the $C_{3}$-term in Equation (37). Table 1: A comparison between KMT solution and the present study. | KMT | present solution
---|---|---
frame-dragging | Equation (8) | Equation (16)
At $r\gg M$ | $g_{t\phi}=C_{1}+C_{2}r$ | $g_{t\phi}=C_{1}+C_{2}r+2C_{3}r^{2}\sin^{2}\theta$
Cotton tensor | vanishes | does not vanish
comment | GR solution | CS contributes non-trivially
Kretschmann scalar | $\sim\frac{48M^{2}}{r^{6}}-\frac{4C_{2}^{2}}{r^{4}\sin^{4}\theta}$ | $\sim\frac{48M^{2}}{r^{6}}+\frac{192MC_{3}^{2}}{r^{3}}(1/3+\cos 2\theta)$
comment | diverges at $\theta=0$ and $\pi$ | asymptotically flat everywhere
circular velocity | $\sim\sqrt{M/r}+C_{2}/2$ | $\sim\sqrt{M/r}+C_{2}/2+C_{3}(2r-7M)$
rotation curve | does not fit | fits well where the perturbation
|  | term dominates, which violates
|  | the model assumption.

Next, we turn our attention to the CSMG solution, namely Equation (37). For
the same rotation curve of UGC11455 galaxy, the best fit values of the free
parameters are $\Upsilon_{\rm disk}=0.543\pm 0.011$ $M_{\odot}/L_{\odot}$ in
agreement with the predicted value as suggested by stellar population models,
while $10^{-4}C_{2}=0.0347^{+0.035}_{-0.0056}$ and $C_{3}=18.30\pm 0.62$ with
$\chi^{2}_{\nu}:=\chi^{2}/{\rm dof}=3.82329$, where dof is the degrees of
freedom of the model. We note that the $C_{3}$-term is associated with the
radial distance. On the contrary to the KMT, we find that the CSMG model
(dashdotted curve) fits the data well as shown by Fig. 2, since the
$C_{3}$-term contribution to the outer region of the velocity curve is
considerable and has the dominant effect. However, this violates the model
assumption, where the perturbation around Schwarzschild BH should be small
with respect to the leading term. We summarise the results of the present
study in Table 1, which traces the main features of KMT solution and the
present one.

##  5 Summary and conclusion

In the present work, we revisited slow rotating BHs solution in CSMG using the
perturbation expansion around the Schwarzschild. In particular, the
nondynamical CSMG with noncanonical CS scalar, which has been proposed by [1]
to explain the flatness of rotational velocity curves of spiral galaxies [43].
The present study showed that the KMT model assumption, which requires the
frame dragging to have a radial dependence only, derives the Cotton tensor to
vanish up to first order. In effect, the CSMG and GR solutions are identical,
and then CSMG does not provide any new insights. However, by dropping the KMT
assumption, we showed that the full frame dragging should have radial as well
as zenith angle dependence.

Unlike KMT model, the present CS solution differs from GR and reduces to KMT
as a particular case. Indeed, the CS solution fulfils GR field equations,
since $G_{\mu\nu}=0$ is satisfied. We note that the opposite conclusion is not
true. On another word, the GR solution does not necessarily satisfy the
divergence-free Cotton tensor in CSMG, which is the case in the present study.
We showed that the tidal gravitational force in terms of Kretschmann scalar,
$K$, of the new solution has an additional term, $\propto 1/r^{3}$, associated
to the leading term of Schwarzschild solution, i.e. $K=48M^{2}/r^{6}$.
Notably, the present solution provides an asymptotically flat Kretschmann
invariant everywhere and does not diverge along any specific direction. This
is unlike KMT, in which Kretschmann invariant diverges along $\theta=0$ and
$\theta=\pi$ due to presence of $1/(r\sin\theta)^{4}$ term. In addition, the
present CS solution predicts frame dragging effect, which decays more slowly
than the corresponding term in Kerr, $\propto 1/r^{8}$, and than Schwarzschild
term, $\propto 1/r^{6}$, as well. In this sense, the new solution should
modify the static spherically symmetric solution at large distances, and then
it worths to be investigated via astrophysical observations of rotational
velocities on the galactic scales.

We confronted the KMT and CSMG models with the rotational velocity curve of
UGC1145 galaxy using MCMC method to fit the parameter space of each model. The
UGC1145 galaxy is characterised by HSB, no bulge, more than seven points and
inclination $i>30^{\circ}$ to ensure that the galaxies are dominated by
stellar matter rather than dark matter around the centre with good statistical
resolution and no bias due to galaxy’s disk inclination. We concluded that the
KMT model reproduces the Keplerian velocity, since $C_{2}$ term contributes to
the velocity by a small fraction $\sim 2$ km/s, and consequently it cannot fit
the rotation curve. On the other hand, the present solution fits the data well
with reasonable $\chi^{2}_{\nu}$, where the contribution of $C_{3}$ becomes
dominant at the outer region of the velocity curve. However, this contribution
violates the model assumption by assuming the correction to Schwarzschild
solution to be small with respect to the leading term, i.e. the Keplerian
velocity. This conclusion should be valid to any perturbative solution around
Schwarzschild BH, which should be small. Therefore, it would fail to explain
the flatten part of the velocity curve at large distances from the galactic
centre.

In conclusion, we may say that CS gravity is running out of options to explain
galactic rotation curves. However, we may suggest two different approaches to
the problem: (i) In presence of dark matter, CS modified gravity may require
different amount of dark matter than the standard $\Lambda$CDM scenario, which
could be tested on cosmological scales with clustering rates of galaxies.
Similar treatments in generic framework of modified gravity can be found in
[50]. On the contrary, if the analysis requires $C_{2}$ and $C_{3}$ to be
null, this may provide a strict constraint on deviations from Schwarzschild BH
solution even with noncanonical choices of the CS scalar within CSMG. This is
in agreement with Ref. [42], where the standard choice of the CS scalar is
applied. (ii) It has been argued that physical spinning BH solution may exist
in CSMG, if energy-momentum conservation is broken [42]. Then, we may suggest
another approach by considering matter-CS nonminimal coupling, which should
induce extra force orthogonal to the four-velocity rendering the motion of
test particles non-geodesic [51]. In general, the Newtonian limit of such
scenario obtains an acceleration law similar to MOND. We leave this task for a
future study.

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# Short-lived repeating fast radio bursts from tidal disruption of white
dwarfs by intermediate-mass black holes

Jing-Tong Xing, Tong Liu
Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China
E-mail<EMAIL_ADDRESS>
###### Abstract

The origin of repeating fast radio bursts (RFRBs) is still a mystery. We
propose that short-lived RFRBs might be triggered from the tidal disruption of
white dwarfs (WDs) by intermediate-mass black holes (BHs). In this model, we
show that the remnant WD clusters after tidal collapse cuts the magnetic lines
on the BH accretion discs, and during each fall of the clump, so that
electrons are torn from the surface of the mass and instantly accelerated to
the relativistic energy. The subsequent movement of these electrons along
magnetic field lines will result in coherent curvature radiation. This short-
lived radio transients might accompany with the accretion process. The
luminosity and the timescale can be estimated to be $L_{\mathrm{tot}}\sim
1.96\times 10^{40}~{}{\rm erg~{}s^{-1}}$ and $\Delta t\sim 1.14~{}{\rm ms}$,
respectively, which are consistent with the typical properties of RFRBs.
Moreover, the total event rate of our model for generating RFRBs might be as
high as $\sim 10~{}\rm{yr^{-1}~{}Gpc^{-3}}$.

###### keywords:

accretion, accretion discs - black hole physics - fast radio bursts -
transients: tidal disruption events - white dwarfs

††pagerange: Short-lived repeating fast radio bursts from tidal disruption of
white dwarfs by intermediate-mass black holes–LABEL:lastpage

##  1 Introduction

Fast radio bursts (FRBs) are millisecond radio transient sources, dating back
as early as 2007 when an unknown bright single pulse was detected in archival
data from the Parkes Telescope (Lorimer et al., 2007), marking the beginning
of research in this area. At present, with the continuous improvement of the
accuracy of observation instruments, the observed data of FRBs are also
increasing (e.g. Lin et al., 2020; CHIME/FRB Collaboration et al., 2021; Li et
al., 2021), but the origin of FRBs has not been solved. According to the
observational properties, FRBs can be divided into two types: repeating FRBs
(RFRBs) and one-off FRBs. The two types of FRBs differ in the interpretation
models of origin. To model RFRBs, the most important thing is to physically
explain the typical duration and the burst luminosity of FRBs. Available
evidences suggest that FRB core engines are compact, and that the high
luminosity of radio signals requires coherent radiation (e.g. Katz, 2014; Luan
& Goldreich, 2014). These requirements have led some authors to associate
compact objects such as white dwarfs (WDs), neutron stars (NSs), and black
holes (BHs). Relevant models include but not limited to the young, rapidly
rotating pulsars (e.g. Lyutikov, Burzawa, & Popov, 2016), the pulsars that
produce huge pulses (e.g. Connor, Sievers, & Pen, 2016; Cordes & Wasserman,
2016), an accreting NS-WD binary with strong dipole magnetic fields (e.g. Gu
et al., 2016; Gu, Yi, & Liu, 2020; Chen et al., 2022), the BH-NS or NS-NS
mergers (e.g. Mingarelli, Levin, & Lazio, 2015; Wang et al., 2016), a pulsar
passing through the asteroid belt (e.g. Dai et al., 2016; Dai & Zhong, 2020),
the collapse of a rotating supermassive NS to a BH (e.g. Falcke & Rezzolla,
2014; Zhang, 2014), and the mergers of charged BHs (e.g. Liu et al., 2016;
Zhang, 2016). The multiband radiations and multimessenger signals are
inevitable in these models.

The concept of tidal disruption events (TDEs) originated in the late 1970s
(e.g. Hills, 1975). In TDEs, the central objects are generally compact
objects. When the tidal star passes the tidal radius of the central object,
the tidal star is either tidally disrupted or swallowed by the central object,
depending on the ratio of the tidal disruption radius to the pericenter
distance (e.g. Gezari, 2021). The BH-TDEs can produce the bright
electromagnetic radiation flares visible from radio to X-ray even $\gamma$-ray
(e.g. Lu, Huang, & Zhang, 2008; Rosswog, Ramirez-Ruiz, & Hix, 2009; MacLeod et
al., 2016; Gezari, 2021). Previous studies have shown that in TDEs, tidal
stars are disintegrated by BH into debris streams, some of which return to the
BH to form accretion streams that provide energy for the luminous mechanism of
tidal events (e.g. Rees, 1988). An accretion disc formed by a BH with a high
spin generates a magnetic field through the Blandford-Znajek (BZ) mechanism
(Blandford & Znajek, 1977), and the BZ jet power and the magnetic field
strength of the disc are related to the accretion rate (e.g. Liu, Gu, & Zhang,
2017; Zhang, 2018). Here we propose that a WD is tidally disintegrated by an
intermediate-mass BH (IMBH). If a mass of WD fragments falling into the IMBH
could cut the magnetic fields of the accretion disc, which might be trigger a
RFRB.

In the following, we present the IMBH-WD TDE model and show the related
properties of RFRBs in Section 2. The conclusions and discussion are made in
Section 3.

Figure 1: Schematic diagram of tidal capture model. (i) A typical WD is
tidally captured and destroyed by an IMBH at the tidal collapse radius
$R_{\mathrm{t}}$, with a portion of the destroyed material escaping the BH at
high speeds, and the rest being accreted by the BH and undergoing orbital
circularization. (ii) The WD fragments retreat to the destruction radius
$R_{\mathrm{b}}$ and are warped by tidal forces and destroyed again. (iii) The
broken fragments cut the field lines of the accretion disc at the interaction
radius $R_{\mathrm{a}}$, inducing RFRBs through coherent curvature radiation.

##  2 Model

We consider that as the WD approaches the tidal radius of an IMBH, it is
captured and destroyed by the BH’s powerful tidal forces into a debris stream,
some of which is accreted to form an accretion disc and eventually falls into
the BH, while other debris flies away from the edge of the BH with speeds
$\geq 10^{4}~{}\rm km~{}s^{-1}$ (e.g. Rees, 1988; Rosswog, Ramirez-Ruiz, &
Hix, 2009). The formation of a transient accretion disc has such a high
accretion rate that the inner region of the disc is dominated by radiation
pressure. The accretion materials should inherit the magnetic fields of WDs,
then a mass of WD fragments cutting the magnetic fields of the disc can
provide energy of RFRBs.

###  2.1 WD-IMBH TDEs

We set a typical WD with a mass of $M_{\mathrm{wd}}=0.5~{}M_{\mathrm{\odot}}$
and a radius of $R_{\mathrm{wd}}=1.1\times 10^{4}~{}{\rm km}$, which has a
tidal radius of (e.g. Rees, 1988)

$R_{\mathrm{t}}\approx
R_{\mathrm{wd}}\left(\frac{M_{\mathrm{BH}}}{M_{\mathrm{wd}}}\right)^{\frac{1}{3}}=1.39\times
10^{10}{M_{\mathrm{3}}}^{\frac{1}{3}}~{}{\rm cm},$ (1)

where $M_{3}=M_{\mathrm{BH}}/(10^{3}~{}M_{\odot})$. Then, a part of the broken
part is tidally captured by the BH, and another part will be ejected at high
speed, as shown in Figure 1. The captured part will form a transient accretion
disc. For an IMBH, the Schwarzschild radius is
$r_{\mathrm{s}}=2GM_{\mathrm{BH}}/c^{2}\sim 2.96\times
10^{8}M_{\mathrm{3}}~{}\rm cm$, the event horizon radius is expressed as
$r_{\mathrm{h}}=r_{\mathrm{s}}[1+(1-a_{\mathrm{*}}^{2})^{1/2}]/4\sim
1.06\times 10^{8}M_{\mathrm{3}}~{}\rm cm$, and the spin dimensionless constant
$a_{\mathrm{*}}=0.9$ is taken in our model.

According to Rosswog, Ramirez-Ruiz, & Hix (2008), for the tidal collapse model
of WD and IMBH, the initial material enters a circular orbit and forms an
accretion surface, and the time for the initial material to return to the BH
is $\leq t_{\mathrm{a}}\sim
180(a/340r_{\mathrm{s}})^{3/2}M_{\mathrm{3}}^{-1/2}~{}\rm s$, where $a\sim
340r_{\mathrm{s}}M_{\mathrm{3}}^{-1/3}r_{\mathrm{wd,9}}M_{\mathrm{wd,0.5}}^{-2/3}$
is the semi-major axis of the orbit, and
$r_{\mathrm{wd,9}}=R_{\mathrm{wd}}/(10^{9}~{}\rm cm)$ and
$M_{\mathrm{wd,0.5}}=M_{\mathrm{wd}}/(0.5~{}M_{\odot})$. The viscous effect
then releases enough binding energy to power the light-emitting mechanism.
Over time, the vicinity of the BH will be replenished only by the injection of
matter, and the accretion rate will gradually evolve, the evolution of the
accretion rate over time is given by (Rees, 1988)

$\dot{M}=7.94\times
10^{-4}\left(\frac{M_{\mathrm{BH}}}{M_{\mathrm{\odot}}}\right)^{-\frac{1}{2}}\left(\frac{t}{t_{\mathrm{D}}}\right)^{-\frac{5}{3}}~{}{M_{\mathrm{\odot}}~{}\rm
s^{-1}},$ (2)

where $t_{\mathrm{D}}$ is the dynamic or orbital time scale given by (Sanders
& van Oosterom, 1984)

$t_{\mathrm{D}}=\left(\frac{GM_{\mathrm{BH}}}{R_{\mathrm{t}}^{3}}\right)^{-\frac{1}{2}}=4.47~{}{\rm
s}.$ (3)

We consider that the magnetic field is frozen in the disc plasma and
transmitted to the BH through the inward motion of the plasma. The dynamics on
the disc affects the value of the magnetic field, and the plasma with low
magnetic pressure will be swallowed by the BH, while the plasma with high
magnetic pressure will be pushed back to the disc and continue to be
modulated. Only when the magnetic pressure of the disc and the pressure of the
inner region of the disc are balanced, the magnetic field can maintain the
strength suitable for the inner region of the disc. At the inner radius, the
velocity of the plasma falling into the BH $v_{r}\sim c$. One can use this
relation to estimate the magnetic strength (e.g. Ghosh & Abramowicz, 1997;
Lee, Wijers, & Brown, 1999; Liu, Gu, & Zhang, 2017), then we obtain

$\frac{B^{2}}{8\pi}=P_{\mathrm{in}}\sim\rho_{\mathrm{in}}c^{2}\sim\dot{M}\frac{c}{4\pi
r_{\mathrm{ms}}^{2}},$ (4)

where $r_{\mathrm{ms}}$ is the innermost stable orbit of the accretion disc,
expressed as
$r_{\mathrm{ms}}=r_{\mathrm{s}}[3+Z_{2}-\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}]/2\sim
3.44\times 10^{8}M_{\mathrm{3}}~{}\rm cm$ with
$Z_{1}=1+(1-a_{*}^{2})^{1/3}[(1+a_{*})^{1/3}+(1-a_{*})^{1/3}]$ and
$Z_{2}=\sqrt{3a_{*}^{2}+Z_{1}^{2}}$ (e.g., Bardeen, Press, & Teukolsky, 1972;
Novikov, 1998).

To simplify the calculation, we assume that the magnetic fields of the disc
change with the radius, then the accretion rate can be substituted into the
above formula to obtain the magnetic field changes with the radius and time,
i.e.,

$B=1.91\times
10^{11}R_{\mathrm{9}}^{-1}M_{\mathrm{3}}^{-\frac{1}{4}}t_{\mathrm{0}}^{-\frac{5}{6}}~{}{\rm
G},$ (5)

where $R_{9}=R/(10^{9}~{}\rm cm)$ is the dimensionless distance of the
fragment to the BH and $t_{\mathrm{0}}=t/(10^{0}~{}\rm s)$. From the above
formula, we can see that the evolution of the magnetic fields follows
$t^{-5/6}$ in the accretion process.

###  2.2 RFRBs

We set a WD fragment with radius $r_{\mathrm{0}}$, mass $m_{\mathrm{0}}$, and
density $\rho_{\mathrm{0}}$. When the fragment moves towards the center of the
BH as shown in Figure 1, it will be tidally distorted by the BH at the
breaking radius $R_{\mathrm{b}}$. The radius of breakage is given by (Colgate
& Petschek, 1981)

$R_{\mathrm{b}}=\left(\frac{\rho_{\mathrm{0}}r_{\mathrm{0}}^{2}GM_{\mathrm{BH}}}{s}\right)^{\frac{1}{3}}=9.45\times
10^{9}m_{\mathrm{27}}^{\frac{2}{9}}\rho_{\mathrm{7}}^{\frac{1}{9}}s_{\mathrm{19}}^{-\frac{1}{3}}M_{\mathrm{3}}^{\frac{1}{3}}~{}{\rm
cm},$ (6)

where $m_{27}=m_{0}/(10^{27}~{}{\rm g})$ and
$\rho_{7}=\rho_{\mathrm{0}}/(10^{7}{\rm g~{}cm^{-3}})$. We set $10^{27}$ g as
the typical WD fragment mass because for a larger one it is already shredded
into smaller clumps before reaching the tidal radius. We take
$s_{\mathrm{19}}=s/(10^{19}~{}\rm dyn~{}cm^{-2})$ in our model, and its
typical value is set between the estimated tensile strength of different types
of WDs (e.g. Swift et al., 2022). The fragment mass, density, and tensile
strength have their own parameter spaces and they are also related to each
other. Here we just set the typical values to estimate the timescale and
luminosity of FRBs. Although the typical values of these parameters are not
completely certain that they will be produced in an IMBH-WD-TDE, the clumps
between $\sim 10^{24}$ and $\sim 10^{28}~{}\rm g$ are likely to produce FRBs
in our model. The broken fragments will be lengthwise stretched into a
cylindrical-like shape with radius $r_{\mathrm{0}}$ and length
$l_{\mathrm{0}}=2r_{\mathrm{0}}$ (e.g. Colgate & Petschek, 1981; Dai et al.,
2016). After that, it continuously broken into a series of small fragments as
they approach the event horizon of the BH, where the time difference between
the front and rear of fragments falling into the BH can be estimated as (e.g.
Dai et al., 2016)

$\Delta
t\approx\frac{12r_{\mathrm{0}}}{5}\left(\frac{2GM_{\mathrm{BH}}}{R_{\mathrm{b}}}\right)^{-\frac{1}{2}}=1.14m_{\mathrm{27}}^{\frac{4}{9}}\rho_{\mathrm{7}}^{-\frac{5}{18}}s_{\mathrm{19}}^{-\frac{1}{6}}M_{\mathrm{3}}^{-\frac{1}{3}}~{}{\rm
ms}.$ (7)

It can be noted that this time difference is not only independent of the
radius of the star, but also has a weak correlation with other parameters, and
is numerically consistent with the typical duration of an FRB. The magnetic
fields of the accretion disc must be affected during the falling of the
debris.

The surface of the debris after tidal distortion will generate a strong
induced electric field
$\textbf{E}=-\textbf{v}_{\mathrm{ff}}\times\textbf{B}/c$, where
$v_{\mathrm{ff}}=\left({2GM_{\mathrm{BH}}}/{R}\right)^{1/2}$ is the velocity
of the debris in free fall under the gravity of the BH. When the magnetic
field of the accretion disc is so strong that the magnetic pressure dominates
the pressure in the inner region, the material at the edge of the BH’s event
horizon will quickly fall into the central BH in free fall (e.g. Lynden-Bell &
Pringle, 1974; Cheng & Lu, 2001). We take the free fall time
$t_{\mathrm{ff}}\sim R_{\mathrm{b}}/v_{\mathrm{bf}}$ describing the
variability of accretion rate, and set
$v_{\mathrm{bf}}=\left({2GM_{\mathrm{BH}}}/{R_{\mathrm{b}}}\right)^{1/2}$,
which is used to calculate the magnetic field of the debris passing through
the inner region of the accretion disc. It is worth mentioning that this
timescale is the largest one for generating FRBs in our model. During the
timescale of $t_{\mathrm{ff}}$, the values of the magnetic and electric fields
in the inner region of the accretion disc can be given due to changes in the
rate of the accretion disc, i.e.,

$B=1.18\times
10^{11}m_{\mathrm{27}}^{-\frac{5}{18}}\rho_{\mathrm{7}}^{-\frac{5}{36}}s_{\mathrm{19}}^{\frac{5}{12}}R_{\mathrm{9}}^{-1}M_{\mathrm{3}}^{-\frac{1}{4}}~{}{\rm
G},$ (8)

and

$E=6.44\times
10^{10}m_{\mathrm{27}}^{-\frac{5}{18}}\rho_{\mathrm{7}}^{-\frac{5}{36}}s_{\mathrm{19}}^{\frac{5}{12}}R_{\mathrm{9}}^{-\frac{3}{2}}{M_{\mathrm{3}}}^{\frac{1}{4}}~{}{\rm
volt~{}cm^{-1}}.$ (9)

Here we adopt the method in Dai et al. (2016), then assume that the magnetic
energy density obtained by the magnetic dipole moment $\mu\sim
B(R_{\mathrm{b}})R_{\mathrm{b}}^{3}$ at $R_{b}$ is equal to the kinetic energy
density $\rho v_{\mathrm{af}}^{2}/2$ at the interaction radius, i.e.,

$R_{\mathrm{a}}=\left(\frac{2\mu^{2}}{8\pi\rho
v_{\mathrm{af}}^{2}}\right)^{\frac{1}{6}}=3.19\times
10^{8}m_{\mathrm{27}}^{\frac{1}{15}}\rho_{\mathrm{7}}^{-\frac{1}{6}}s_{\mathrm{19}}^{-\frac{1}{10}}M_{\mathrm{3}}^{-\frac{1}{30}}~{}{\rm
cm},$ (10)

where $v_{\mathrm{af}}=\sqrt{2GM_{\mathrm{BH}}/R_{\mathrm{a}}}$. It should be
noted that $R_{\mathrm{a}}$ is obtained under the assumption of magnetic
moment $\mu$, and $r_{\mathrm{h}}<R_{\mathrm{a}}<R_{\mathrm{b}}$ is necessary
so that the effect of the magnetic field on the fall of the fragments is
negligible. In addition, $R_{\mathrm{a}}$ is close to the event horizon
radius, so the time it takes for the debris to cross $R_{\mathrm{a}}$ and
release the FRB can be approximated to $\Delta t$. These results make our
model self-consistent.

Because the electric field on the surface of the fragment has a parallel
component to the magnetic field, the electrons are accelerated to the
relativistic energies and torn apart from the fragment’s surface (e.g. Galvez
& Borovsky, 1991; Neubert et al., 1992). Motion along a magnetic inductance
line of curvature $\rho_{\mathrm{c}}$ generates coherent curvature radiation
(e.g. Ruderman & Sutherland, 1975), and the maximum Lorentz factor of the
accelerating electron can be obtained by assuming that the synchrotron
radiation power $\sigma_{\mathrm{T}}c\gamma_{\mathrm{max}}^{2}B^{2}/6\pi$ and
the accelerating power of the electron $eEc$ are equal (e.g. Dai et al., 2016;
Dai, 2020), thus

$\gamma_{\mathrm{max}}=\chi\left(\frac{6\pi
eE}{\sigma_{\mathrm{T}}B^{2}}\right)^{\frac{1}{2}}=250.32\chi
m_{\mathrm{27}}^{\frac{5}{36}}\rho_{\mathrm{7}}^{\frac{5}{72}}s_{\mathrm{19}}^{-\frac{5}{24}}R_{\mathrm{9}}^{\frac{1}{4}}{M_{\mathrm{3}}}^{\frac{3}{8}},$
(11)

where $\sigma_{\mathrm{T}}$ is the Thomson scattering cross section. $\chi$ is
defined as the proportional coefficient between $\gamma$ and the maximum
Lorentz factor (e.g. Dai, 2020), and the conditions need to be met $\chi\leq
1$. We assume that $\chi\sim 0.2$, substitute it into Equation (12) to get
$\gamma\sim 50.06$.

We note that the breaking radius $R_{\mathrm{b}}\sim 9.45\times 10^{9}~{}{\rm
cm}$ is less than the tidal radius $R_{\mathrm{t}}\sim 1.39\times
10^{10}~{}{\rm cm}$ and greater than the interaction radius
$R_{\mathrm{a}}\sim 3.19\times 10^{8}~{}{\rm cm}$, which means that our model
is self-consistent. Next, we discuss the radiative properties of coherent
radiation. For a ultrarelativistic electron moving along a magnetic inductance
line with radius of curvature $\rho_{\mathrm{c}}$ and Lorentz factor $\gamma$,
the characteristic frequency is given by (e.g., Dai et al., 2016)

$\nu_{\mathrm{curv}}=\frac{3c\gamma^{3}}{4\pi\rho_{\mathrm{c}}}=4.43m_{\mathrm{27}}^{\frac{7}{20}}\rho_{\mathrm{7}}^{\frac{3}{8}}s_{\mathrm{19}}^{-\frac{21}{40}}R_{\mathrm{9}}^{\frac{3}{4}}M_{\mathrm{3}}^{\frac{139}{120}}~{}{\rm
GHz},$ (12)

where the radius of curvature is $\rho_{\mathrm{c}}\sim 0.635R_{\mathrm{a}}$
given by Yang & Zhang (2018), which is consistent with the characteristic
frequency of typical FRBs. In theory, the WD fragments would be laterally
compressed into a series of smaller fragments after the breaking radius and
continue to move along the closed magnetic field lines. The ultrarelativistic
electrons carried by the debris produce the coherent curvature radiation,
emitting a bright radio burst, and forming a sector with an angle of
$\theta=r(R_{\mathrm{a}})/R_{\mathrm{a}}$. We consider a region surrounded by
a circuit and divide the region into slices of coherent radiation, the
luminosity of the coherent curvature radiation is given by (e.g., Dai et al.,
2016)

$\displaystyle L_{\mathrm{tot}}$ $\displaystyle\approx
P_{\mathrm{c}}N_{\mathrm{e,tot}}^{2}N_{\mathrm{slice}}^{-1}$ (13)
$\displaystyle\approx 1.96\times
10^{40}m_{\mathrm{27}}^{\frac{17}{20}}\rho_{\mathrm{7}}^{-\frac{11}{8}}s_{\mathrm{19}}^{\frac{29}{40}}R_{\mathrm{9}}^{-\frac{19}{4}}{M_{\mathrm{3}}}^{\frac{229}{120}}~{}{\rm{erg~{}s^{-1}}},$

where $P_{\mathrm{c}}$ is the curvature radiation power of a ultrarelativistic
electron, $N_{\mathrm{e,tot}}$ is the total number of electrons in the
emission region, and $N_{\mathrm{slice}}$ is the number of coherent slices. It
can be seen from the formula that the total luminosity $L_{\mathrm{tot}}$ has
a strong correlation with $R$.

According to the law of conservation of energy, we know that the coherent
curvature radiation discussed above is actually provided by the gravitational
binding energy of the BH. The total released gravitational energy, $E_{G}\sim
10^{47}~{}\rm erg$, is definitely higher than the energy of FRBs. From
Equation (5) we can see that $B\propto t^{-\frac{5}{6}}$, for our selection of
$t_{\mathrm{ff}}$, we get the minimum value of the magnetic field, this means
that Equation (13) is the minimum luminosity limit for our model to produce
FRBs. For the existing observations, this power meets the luminosity range of
the FRB, which also indicates that in addition to the fragment parameters
selected here, with the change of the distance between the fragment and the BH
event horizon, the change of the size of the tidal fragment, and the evolution
and configuration of the magnetic fields also has the possibility of producing
FRB in different directions.

In addition, the RFRBs induced by the IMBH-WD TDE might be last in the whole
accretion process, especially the early stage with high accretion rate and
consequentially strong magnetic fields.

###  2.3 Event rate estimation

In our model, we believe that if FRBs are to be generated in TDEs, then the
mass range of the BHs should be satisfied $\sim
10^{3}-10^{5}M_{\mathrm{\odot}}$. Once the BH is too massive $(\geq
10^{5}M_{\mathrm{\odot}})$, the WD is swallowed whole instead of being tidally
destroyed. Assuming that the total number of IMBHs in the galaxy is $N$, the
proportion of BHs in the mass range is $k$, and the rate of TDEs between IMBHs
and WDs is $\Gamma_{\mathrm{t}}$, then we can figure out the probability of
TDEs in the galaxy (e.g. Lu, Huang, & Zhang, 2008), i.e.,
$\Gamma_{\mathrm{TDE}}=k\Gamma_{\mathrm{t}}N$. We use assumption in MacLeod et
al. (2016) that the IMBH mass function is flat with a lower mass
$(M_{\mathrm{BH}}<10^{6}M_{\mathrm{\odot}})$, the event rate can be estimate
as

$\Gamma_{\mathrm{TDE}}=k\Gamma_{\mathrm{t}}N\sim
10{\dot{N}}_{\mathrm{IMBH,-6}}n_{\mathrm{IMBH,7}}~{}\rm{yr^{-1}~{}Gpc^{-3}},$
(14)

where ${\dot{N}}_{\mathrm{IMBH,-6}}={\dot{N}}_{\mathrm{IMBH}}/(10^{-6}~{}\rm
yr^{-1})$ and $n_{\mathrm{IMBH,7}}=n_{\mathrm{IMBH}}/(10^{7}~{}\rm Gpc^{-3})$.
This rate can be roughly considered as the FRB event rate.

Observational multi-band radiations of TDEs are generally considered to be
emitted from the accretion discs and only a few fraction is driven by the jets
(e.g. Lei & Zhang, 2011; Lei, Zhang, & Gao, 2013; Lei et al., 2016; Alexander
et al., 2020). In our model, FRBs might be radiated along the magnetic field
lines as shown in Figure 1, so one can be cautiously optimistic for the
discovery on IMBH-WD TDEs associated with FRBs.

##  3 Conclusions and discussion

In this _Letter_ , we present a model of the tidal disintegration of WD by
IMBH for a unified explanation of the origin of extragalactic RFRBs. The main
conclusions are as follows:

  1. 1.

A WD ($\sim 0.5~{}M_{\odot}$) is tidally captured by the IMBH ($\sim
10^{3}~{}M_{\odot}$) at the tidal radius $R_{\mathrm{t}}\sim 1.39\times
10^{10}~{}{\rm cm}$, during the capture process, a transient accretion disc is
formed around the BH, and the accretion rate changes with time,
$\dot{M}\propto t^{-5/3}$. The magnetic field strength near the BH event
horizon is large enough to trigger FRBs.

  2. 2.

Due to the tidal shear of the IMBH, the WD fragments after the tidal collapse
in the accretion disc are again distorted at the breaking radius
$R_{\mathrm{b}}\sim 9.45\times 10^{9}~{}{\rm cm}$ into a series of smaller
fragments. The time difference between the leading and trailing fragments of
this series of fragments reaches the edge of the BH’s event horizon is $\Delta
t\sim 1.14~{}{\rm ms}$, which is consistent with the typical pulse duration of
RFRBs.

  3. 3.

Within an emission region surrounded by a circuit, the surface electrons of
the fragments are accelerated to the ultrarelativistic energy by the strong
magnetic field, and are torn from the surface to move along the magnetic
inductance line, resulting in curvature radiation. The electrons carried by a
series of fragments produce coherent curvature radiation with a characteristic
frequency of $\mu_{\mathrm{curv}}\sim 4.43~{}{\rm GHz}$. The luminosity can be
estimated to be $L_{\mathrm{tot}}\sim 1.96\times 10^{40}~{}{\rm
erg~{}s^{-1}}$, which meets the typical luminosity of FRBs.

  4. 4.

RFRBs induced by IMBH-WD TDEs are short-lived, multiple, and irregular
transients throughout the accretion processes. Their event rate might be $\sim
10~{}\rm{yr^{-1}~{}Gpc^{-3}}$.

Actually, besides IMBH-WD TDEs, the trigger sources of fast radio transients,
i.e., high-velocity compact fragments cutting magnetic lines, exist in some
multibody systems, such as the magnetars surrounded by the unstable asteroid
belts (e.g. Dai et al., 2016; Wu, Zhao, & Wang, 2023). The IMBH-WD TDEs are
usually accompanied by the generation of high-energy transients, and are also
the main goals of the space-based gravitational wave projects, such as LISA,
TianQin, and Taiji. The future multiband and multimessenger detections might
discover the RFRBs induced by TDEs.

In TDEs, except the parts accreted by BH, those clumps that managed to escape
or were not swallowed by the BH are also of interest, and there has been some
related work discussed the ultimate fate of these clumps (e.g. Ryu et al.,
2020; Bortolas et al., 2023). Our future work will consider which clumps are
trapped by BH but not falling into BH to form compact planets, as well as the
possibility that ejected clumps to form high-speed wandering planets.

## Acknowledgements

We thank anonymous referee for helpful suggestions that improved the
manuscript, and also thank Wei-Min Gu, De-Fu Bu, Ang Li, Wei-Hua Lei, Xiao-Yan
Li, Bao-Quan Huang, and Ali Shad for helpful discussion. This work was
supported by the National Natural Science Foundation of China under grants
12173031 and 12221003.

## Data availability

The data underlying this article will be shared on reasonable request to the
first author.

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