# Gravitational wave radiation from the magnetar-driven supernovae

Lang Xie  National Astronomical Observatories, Chinese Academy of Sciences,
Beĳing 100101, China  School of Astronomy, University of Chinese Academy of
Sciences, Beĳing 100049, China  Hong-Yu Gong  Key Laboratory of Dark Matter
and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences,
Nanjing 210034, China  School of Astronomy and Space Science, University of
Science and Technology of China, Hefei, Anhui 230026, China  Long LI
Department of Astronomy, University of Science and Technology of China, Hefei
230026, China  Da-Ming Wei  Key Laboratory of Dark Matter and Space Astronomy,
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210034,
China  School of Astronomy and Space Science, University of Science and
Technology of China, Hefei, Anhui 230026, China  J. L. Han  National
Astronomical Observatories, Chinese Academy of Sciences, Beĳing 100101, China
School of Astronomy, University of Chinese Academy of Sciences, Beĳing 100049,
China

###### Abstract

Rapidly spinning magnetars are potential candidates for the energy source of
supernovae (SNe) and gamma-ray bursts and the most promising sources for
continuous gravitational waves (GWs) detected by ground-based GW detectors.
Continuous GWs can be radiated from magnetars due to magnetic-induced
deformation or fluid oscillations, compatible with magnetic dipole (MD)
radiation for spin-down energy. In this paper, we investigate the diverse
light curves of magnetar-driven SNe in the scenario that the spin-down is
dominated by GW radiation and/or MD radiation. By simulating the light curves
of SNe and employing the Markov chain Monte Carlo method, we constrain the
parameters of the magnetars and SN explosions and show that the signature of
GW radiation may be indicated by the bolometric luminosity curves of SNe Ic-BL
2007ru and 2009bb. We find that the ellipticity of magnetars in the order of
$10^{-3}$ can be induced by the magnetic field of $\sim 10^{16}\mbox{ G}$. If
such continuous GWs associated with SNe can be detected in the future by the
Advanced LIGO and Virgo detectors, this would be a smoking gun for a magnetar
engine powering SNe.

gravitational waves — supernovae: general — stars: magnetars

##  1  Introduction

A long-lived magnetar can be formed by the core collapse of a massive star,
which can produce a transient event, such as a gamma-ray burst (GRB) and a
supernova (SN). The rotational energy of the magnetar is a reservoir for the
afterglow emission of SN and GRB with diverse evolutionary features, such as
short GRB having extended emission or GRB with X-ray plateau (Usov, 1992; Dai
& Lu, 1998a, b; Zhang & Mészáros, 2001; Metzger et al., 2011; Gompertz et al.,
2013; Rowlinson et al., 2013; Metzger & Piro, 2014; Lü & Zhang, 2014; Sarin et
al., 2020; Li & Dai, 2021; Xie et al., 2022a, b). The high kinetic energy or
high luminosity SNe are generally interpreted as an extra injection of spin-
down energy from the magnetar engine (Kasen & Bildsten, 2010; Woosley, 2010;
Inserra et al., 2013; Metzger et al., 2015; Kashiyama et al., 2016; Kasen et
al., 2016; Wang et al., 2016; Moriya et al., 2017; Nicholl et al., 2017; Omand
& Sarin, 2024).

Many observations show that some broad-lined Type Ic (Ic-BL) SNe and
superluminous SNe (SLSNe) have peak luminosities and radiant energies tens of
times higher than those of typical SNe (Gal-Yam, 2012; Nicholl, 2021). Various
models have been applied in the literature to interpret the striking kinetic
energy and luminosity. One popular model involves the energy injection from a
rapidly spinning magnetar (Woosley, 2010; Kasen & Bildsten, 2010). The basic
picture is that winds from a magnetar driven by magnetic dipole radiation
interact with the SNe ejecta and deposit their spin energy into the SN ejecta.
Such a magnetar-driven model has successfully explained the light curves of
SNe Ic-BL (Greiner et al., 2015; Cano et al., 2016; Wang et al., 2016) and
SLSNe (i.e, (Dessart et al., 2012; Chatzopoulos et al., 2013; Inserra et al.,
2013; Nicholl et al., 2013; Howell et al., 2013; Nicholl et al., 2017; Yu et
al., 2017; Liu et al., 2017; Villar et al., 2018; Wang et al., 2019)), as well
as the nebular spectral observations of hydrogen-poor SLSNe (Jerkstrand et
al., 2017; Quimby et al., 2018; Nicholl et al., 2019). Other models for
powering the energetic SNe involve the radioactive decay of massive amounts of
56Ni (Barkat et al., 1967; Heger & Woosley, 2002; Gal-Yam et al., 2009), the
jets launched by the center engine for SNe Ic-BL (Lazzati et al., 2012; Gilkis
et al., 2016; Barnes et al., 2018; Eisenberg et al., 2022; Corsi et al.,
2023), the interaction of SN ejecta with the circumstellar medium (Blinnikov &
Sorokina, 2010; Chevalier & Irwin, 2011; Ginzburg & Balberg, 2012; Moriya &
Maeda, 2012), and an accreting black hole (i.e., (MacFadyen et al., 2001;
Dexter & Kasen, 2013; Li et al., 2020)).

In the magnetar-driven SNe scenario, a magnetar with a strong magnetic field
of $B\sim 10^{15}\mbox{ G}$, a rapid rotation of initial spin period
$P_{0}\sim 1\mbox{ ms}$, and high temperatures is born after the core collapse
of massive star (Duncan & Thompson, 1992). The rapid rotation of the magnetar
can induce a large deformation due to the magnetic pressure or fluid
oscillations (Owen et al., 1998; Lai & Shapiro, 1995; Bonazzola & Gourgoulhon,
1996; Andersson, 1998; Lindblom et al., 1998; Cutler, 2002; Stella et al.,
2005; Haskell et al., 2008; Alford & Schwenzer, 2015; Lasky et al., 2017),
implying that the magnetar could emit the remarkable GWs. The GW quadrupole
radiation and magnetic dipole (MD) radiation torque could act together for
magnetar spin-down (Shapiro & Teukolsky, 1983; Zhang & Mészáros, 2001). The GW
energy loss would therefore affect the spin evolution and also electromagnetic
outflows of the magnetar. In other words, if a deformed magnetar powers an SN,
the GW-energy loss could result in less rotational energy supplied to the SNe,
and then lead to a change in the shape of the light curve, indicated by a
lower peak luminosity and a slower decay rate. A magnetar-driven system should
produce a unique luminosity evolution, different from only MD radiation
losses.

Non-axisymmetrically deformed magnetars are potential continuous GW sources
for the Advanced Laser Interferometer gravitational wave Observatory (aLIGO)
detector (Aasi et al., 2015a) and the Advanced Virgo detector (Acernese et
al., 2015). The aLIGO has searched for continuous GWs from various types of
isolated neutron stars (NSs), such as known pulsars (Abbott et al., 2019a),
post-burst magnetars (Abbott et al., 2019b, 2022), accreting NSs and young NSs
in SN remnants (Abbott et al., 2019c, d). However, limited by the sensitivity
at the designed frequencies, no GWs have yet been detected from these newly
formed compact objects.

Detecting GWs from magnetar-driven SNe could help us constrain the structure
and composition of NSs and provide theoretical traction for future ground-
based GW telescopes. We explore in this paper possible indications of the GW
radiation in SNe electromagnetic light curves. We develop a model for
magnetar-driven SNe by considering the GW radiation and the effect on the
bolometric luminosity evolution of SNe. From the SN luminosity curves, we can
constrain the characteristics of magnetars, i.e., the initial spin period
$P_{0}$, the dipole magnetic field strength $B$, and the ellipticity of the
magnetar $\epsilon$, as well as the parameters of SN explosions. Through the
best-fitting for parameters of the magnetar obtained by the Markov chain Monte
Carlo (MCMC) method, we derive the GW characteristic amplitude of the magnetar
and then evaluate the GW detectability of future ground-based GW telescopes.
This paper is organized as follows. We investigate the effect of the GW
energy-less on the bolometric luminosity evolution of SNe under the assumption
of the magnetar model in section 2. In section 3, we take some SNs as examples
and fit their SN observation data with the magnetar model. In Section 4, we
analyze the detectability of GWs from magnetars that power SNe. The discussion
and conclusions are given in Section 5.

##  2 A magnetar model with gravitational wave emission

Some authors (Inserra et al., 2013; Nicholl et al., 2017) have used magnetar
models to explain SNe light curves and constrain the parameters of magnetars
and the ejecta mass. However, the effect of GW radiation is usually neglected.

A rapidly spinning magnetar can be produced after the core collapse of a
massive star. The rotational energy of the magnetar provides a huge energy
reservoir for SNe, which greatly enriches the features of the SNe optical
emission. However, not all of the rotational energy is used to heat the SN
ejecta, part of which may be lost via the significant GW radiation because of
non-axisymmetric deformation. Thus, the total rotational energy of the
magnetar, $E_{\rm rot}=I\Omega^{2}/2$, is released through both GW radiation
and MD radiation, and the spin-down evolution can then be written as (Shapiro
& Teukolsky, 1983; Zhang & Mészáros, 2001)

$\dot{E}_{\rm rot}=I\Omega\dot{\Omega}=L_{\rm EM}+L_{\rm GW},$ (1)

where $I$ is the moment of inertia of a neutron star, $\Omega$ and
$\dot{\Omega}$ are the spin velocity and its time derivative. The
electromagnetic luminosity $L_{\rm EM}$ generated by the magnetar spin-down is
given by Spitkovsky (2006))

$L_{\rm EM}=-\frac{B^{2}R^{6}\Omega^{4}\sin^{2}\theta}{6c^{3}},$ (2)

Here $B$ is the magnetic dipole field strength on the neutron star surface,
$\theta$ is the angle between the magnetic axis and the spin axis, and $R$ and
$c$ are the radius of the NS and the speed of light, respectively. In this
paper, we adopt the typical parameters for a neutron star with a mass of
$M=1.4\,M_{\sun}$, a radius of $R=10$ km, and the moment of $I=1\times
10^{45}{\rm gcm^{2}}$, and also assume that the neutron star has evolved to an
orthogonal rotator ($\theta=\pi/2$).

Various causes for the NS deformation have been proposed in the previous
literature. First, super-strong magnetic fields exist inside the NS due to
differential rotation and magnetic dissipation instability. The anisotropic
pressure generated by strong magnetic fields then deforms the NS into an
ellipsoid (Bonazzola & Gourgoulhon, 1996; Cutler, 2002; Stella et al., 2005).
It has been argued that magnetic-induced deformation plays an important role
in the early evolution of magnetars (Corsi & Mészáros, 2009; Dall’Osso et al.,
2015; Lasky et al., 2017; de Araujo et al., 2016). Second, the NS accretion
can also lead to the deformation (Haskell et al., 2015; Zhong et al., 2019;
Sur & Haskell, 2021). The accreted material flows towards the poles of the NS,
compresses and obscures the local magnetic field, and thus forms “mountains”
on the surface of the NSs. Third, the rapid spin of young NSs undergoes
unstable oscillatory modes that couple to the gravitational field (Andersson,
1998; Lindblom et al., 1998; Alford & Schwenzer, 2015). The spinning magnetars
with non-axisymmetric deformations possess quadrupole moments, which would
emit significant GW as being (Usov, 1992; Zhang & Mészáros, 2001))

$L_{\rm GW}=-\frac{32}{5}\frac{GI^{2}}{c^{5}}\epsilon^{2}\Omega^{6}.$ (3)

Here $G$ is the gravitational constant, and $\epsilon$ is the ellipticity of
the star. Considering the Eq.1, one can find the spin velocity $\Omega$
evolving:

$\frac{d\Omega}{dt}=-\alpha\Omega^{3}-\beta\Omega^{5},$ (4)

where $\alpha\equiv B^{2}R^{6}/6c^{3}I$, $\beta\equiv
32GI\varepsilon^{2}/5c^{5}$. The evolution of spin velocity $\Omega$ depends
on the combination of MD radiation and GW radiation. Here are three possible
kinds of spin-down evolution of magnetars: (1) If the magnetar spins down only
due to MD radiation and the contribution of GW radiation is negligible, the
electromagnetic luminosity evolves as $L_{\rm EM}=L_{\rm EM,0}(1+t/\tau_{\rm
em})^{-2}$, where $\tau_{\rm em}=\frac{1}{2\alpha\Omega_{0}^{2}}$ is the
timescale of MD radiation. The electromagnetic luminosity shows a plateau at
$t<\tau_{\rm em}$, then decays as $L_{\rm EM}\propto t^{-2}$ for $t>\tau_{\rm
em}$. (2) If the GW radiation dominates the spin-down, the electromagnetic
luminosity can be expressed as $L_{\rm EM}=L_{\rm EM,0}(1+t/\tau_{\rm
gw})^{-1}$, where $\tau_{\rm gw}=\frac{1}{2\beta\Omega_{0}^{4}}$ is the
timescale of GW radiation. The electromagnetic luminosity also exhibits a
plateau before decaying as $L_{\rm EM}\propto t^{-1}$ for $t>\tau_{\rm gw}$.
(3) If both the GW radiation and MD radiation act together for the magnetar
spin-down, the electromagnetic luminosity would show a change from $L_{\rm
EM}\propto t^{-1}$ to $L_{\rm EM}\propto t^{-2}$ after the plateau phase. The
radiation efficiency of GW and MD depends on the spin frequency (i.e., GW for
$\Omega^{6}$ vs MD for $\Omega^{4}$), so the GW radiation should dominate the
early evolution of the electromagnetic luminosity and the MD dominates the
later evolution, causing the diversity of magnetar-driven SNe.

With the above basic ideas, we simulate how magnetars interact with SNe ejecta
in different spin-down processes and how the different SNe light curves can be
found. Our models are built based on the magnetar-driven SNe theory proposed
by Kasen & Bildsten (2010)(see also Kashiyama et al. (2016); Ho (2016); Omand
& Sarin (2024) and Kasen et al. (2016)).

Figure 1: The evolution of bolometric luminosity for a magnetar-driven
supernovae in different spin-down modes, simulated with parameters of
$B=1\times 10^{14}\mbox{ G}$, $P_{0}=1\mbox{ ms}$, $\epsilon=2\times 10^{-3}$,
$\kappa$ = 0.1 cm2 g-1, ${\kappa_{\gamma}}$ = 0.1 cm2 g-1, $E_{\rm
sn}=10^{51}\mbox{ erg}$, $M_{\rm ej}=1\,M_{\sun}$ (solid lines) and $M_{\rm
ej}=3\,M_{\sun}$ (dashed lines). The blue solid and dashed lines are plotted
for the MD radiation spin-down; the green solid and dashed lines for the GW
radiation spin-down; and the orange solid and dashed lines by both GW and MD
radiations.

A relativistic wind generated by the MD radiation from the magnetar goes
outward and heats the SN ejecta. Approximating the ejecta at time $t$ in the
form of a uniform sphere with a mass of $M_{\rm ej}$ and a radius of $R_{\rm
ej}$, with a velocity of $v_{\rm sc}$ in the volume of $V\approx 4\pi R_{\rm
ej}^{3}/3$ and the density of $\rho=M_{\rm ej}/V$, one can describe the
evolution of the internal energy as being (Kasen et al., 2016)

$\frac{\partial E_{int}}{\partial t}=\xi L_{\rm EM}-L_{\rm
rad}-P\frac{\partial V}{\partial t}.$ (5)

The first term on the right is the electromagnetic luminosity injected by the
magnetar due to spin-down. Here $\xi$ is the efficiency of thermalization of
electromagnetic energy into radiation. The second term is the thermal
radiation luminosity of SNe. The third term is the energy loss due to the
expansion of the ejecta, $P$ is the radiation pressure that is related to the
internal energy by $P=E_{int}/3V$, ${\partial V}/{\partial t}=4\pi R_{\rm
ej}^{2}v_{\rm sc}$, and then $-P({\partial V}/{\partial t})=E_{\rm int}/t$.
The $\xi$ evolves with time and is related to the gamma-ray leakage of the
ejecta. Hence, we denote $\xi$ as

$\xi=1-e^{-At^{-2}},$ (6)

where

$A=\frac{3\kappa_{\gamma}M_{\rm ej}}{4\pi v_{\rm sc}^{2}}$ (7)

is parameter of gamma-ray leakage (Wang et al., 2015), $\kappa_{\gamma}$
represents the opacity of gamma-ray. The radiated luminosity of SN is given by
(Kasen & Bildsten, 2010; Kotera et al., 2013)

$\displaystyle L_{\rm rad}=$ $\displaystyle\frac{E_{int}c}{\tau R_{\rm
ej}}=\frac{E_{int}t}{t_{\rm diff}^{2}},{\rm(\tau<1),}$ (8) $\displaystyle=$
$\displaystyle\frac{E_{int}c}{R_{\rm ej}},{\rm(\tau>1)},$ (9)

where the optical depth of the ejecta is $\tau=3\kappa M_{\rm ej}/{4\pi R_{\rm
ej}^{2}}$, and the radiative diffusion timescale is

$\displaystyle t_{\rm diff}$ $\displaystyle=$
$\displaystyle\left(\frac{3\kappa M_{\rm ej}}{4\pi cv_{\rm sc}}\right)^{1/2},$
(10)

here $\kappa$ is the opacity of ejecta. The expansion velocity of the ejecta
is given by (Kasen & Bildsten, 2010)

$\displaystyle v_{\rm sc}$ $\displaystyle=$ $\displaystyle\left[\frac{2(E_{\rm
EM}(\Omega)+E_{\rm sn})}{M_{\rm ej}}\right]^{1/2},$ (11)

where $E_{sn}$ is the initial kinetic energy of ejecta, $E_{\rm EM}$ is the
energy of MD radiation integrated $L_{\rm EM}$ over time.

Figure 2: The same as Figure 1 but the evolution of supernova bolometric
luminosity caused by different NS ellipticities for the spin-down by both the
GW and MD radiations or by merely the GW radiation.

Kasen & Bildsten (2010) and Woosley (2010) were the first to develop a
simplified model of magnetar-driven SNe and showed that the magnetar models
can reproduce the timescales and brightness of SNe within reasonable
parameters. Modeling the light curves of magnetar-driven SNe has to choose the
magnetar spin evolution channel and get the solution of the internal energy
evolution and the velocity of ejecta. Fig 1 demonstrates the magnetar-driven
SNe light curves for three different spin-down modes, the magnetar MD
radiation spin-down (blue line), the GW radiation spin-down (green line), and
both (orange line), calculated for the initial parameters such as the magnetic
field strength of $B=1\times 10^{14}\mbox{ G}$, the initial spin period of
$P_{0}=1\mbox{ ms}$, the ellipticity of $\epsilon=2\times 10^{-3}$, the
opacity of $\kappa$ = 0.1 cm2 g-1, and the gamma-ray opacity of
${\kappa_{\gamma}}$ = 0.1 cm2 g-1, the initial explosion energy of $E_{\rm
sn}=10^{51}\mbox{ erg}$, and the ejecta mass of $1\,M_{\sun}$ (solid lines)
and $3\,M_{\sun}$ (dashed lines).

In magnetar-driven SNe scenarios, the light curve evolution after its peak
luminosity can be described as $L_{rad}\propto t^{-\alpha}$. The value of
$\alpha$ depends on the declining index of the electromagnetic luminosity
injected by the magnetar. Often $\alpha=2$ is taken for a magnetar to spin
down due to only the MD radiation, and $\alpha=1$ for the GW radiation spin-
down, or some values in between for the combined cases.

The evolution of the bolometric luminosity of the SN differs due to the
ellipticity of a deformed NS, as shown in Fig 2. A larger ellipticity would
cause a lower peak luminosity of the SN. The greater the loss of the GW
energy, the smaller the amount of electromagnetic energy provided to the SN
for a smaller luminosity peak. In other words, the low-luminosity SNe is more
likely to have a previously unknown GW radiation. When the MD radiation is
involved for NSs with strong magnetic fields, only a large ellipticity can
cause a significant decrease of the bolometric luminosity peak. The GW energy
loss causes a dwarf peak of the SNe light curve and a slower decay rate of the
SNe light curve tail. Note, however, that the observed $t^{-2}$ tail in some
SN light curves cannot be directly considered as a signature of the presence
of a magnetar engine, since the 56Co radioactive decay could exhibit a similar
tail (Inserra et al., 2013).

##  3 Fitting with supernova light curves

We construct a magnetar model containing three spin-down modes and obtain the
best-fitting parameters and the posterior parameter distributions by employing
the emcee python package and the MCMC method (Foreman-Mackey et al., 2013).
The model contains the following seven free parameters: dipole magnetic field
strength $B$, initial spin period $P_{0}$, NS ellipticity $\epsilon$, ejecta
mass $M_{\rm ej}$, initial SN explosion energy $E_{\rm sn}$, opacity $\kappa$
and gamma-ray opacity $\kappa_{\gamma}$. The prior distribution of the
parameters is set in the form of log-uniform or uniform distributions to
ensure the parameters are in sufficiently large intervals, as seen in Table 1.
The values of ${\rm\chi^{2}}/{\rm dof}$ are adjusted for the best-fitting
model of the SN light curves.

Three criteria are adopted for the choice of observation data. First, there
must be sufficient observational data during the rise and decay of the light
curve, and the observational data needs to extend beyond $100$ days. Second,
the later phase of the light curve exhibits a slow decay tail with a decay
rate between $-1$ to $-2$. This feature is consistent with what we expected
from magnetar models, where enlarged decay rates for more rotational energy
loss due to the GW radiation. Finally, SNe with the complex evolved behaviors
in the decay phase are excluded from our sample. We here work on two cases, SN
2009bb and SN 2007ru.

Table 1: The free parameters, units, and prior distributions of the magnetar model and the best-fitting results of the SNe 2007ru and 2009bb.  | Definition | Unit | Prior | Values of SN 2007ru | Values of SN 2009bb
---|---|---|---|---|---
Magnetar |  |  |  |  |
$B$ | magnetic field strength |  $10^{14}$ G | $[0.1,50]$ | $12.55^{+0.32}_{-0.30}$ | $13.52^{+0.40}_{-0.37}$
$P_{0}$ | initial spin period | ms | $[1.0,40]$ | $25.19^{+0.50}_{-0.51}$ | $26.86^{+0.45}_{-0.45}$
$\epsilon$ | ellipticity | $10^{-3}$ | $[0.1,20]$ | $4.79^{+3.58}_{-3.16}$ | $5.33^{+3.29}_{-3.58}$
$M_{\rm ej}$ | ejecta mass | M⊙ | $[0.1,30]$ | $1.51^{+0.74}_{-0.48}$ | $1.08^{+0.96}_{-0.53}$
$E_{sn}$ | initial kinetic energy of ejecta |  $10^{50}$ cm s-1 | $[0.1,100.0]$ | $29.08^{+14.21}_{-16.62}$ | $28.79^{+14.92}_{-16.80}$
$\kappa$ | optical opacity | cm2g-1 | $[0.01,0.2]$ | $0.03^{+0.03}_{-0.01}$ | $0.04^{+0.05}_{-0.02}$
$\log\kappa_{\rm\gamma}$ | gamma-ray opacity | cm2g-1 | $[-2,2]$ | $1.13^{+0.27}_{-0.48}$ | $0.72^{+0.53}_{-0.67}$
$\chi^{\rm 2}$/dof |  |  |  | $32/23$ | $101/43$

###  3.1 SN 2009bb

SN 2009bb is a Type Ic-BL SNe at $d_{L}\approx 40\mbox{ Mpc}$, first detected
by the CHilean Automated Supernova sEarch (CHASE) on 2009 March 21. The
optical and spectroscopic observations are carried out by Pignata et al.
(2011). The peak absolute magnitude of SN 2009bb is
$M_{B}=-18.3\pm{0.44}\mbox{ mag}$, placing it in the bright end of the SNe Ic
sequence. The Very Large Array (VLA) and the Chandra telescopes have also
detected luminous radio emission and weak X-ray emission associated with SN
2009bb (Soderberg et al., 2010). Radio observations from SN 2009 suggest that
the explosion required very high energy and a relativistic jet, which all
point to an active central engine. In the magnetar model, the injected energy
from the magnetar spin-down wind is dissipated either by channeling the jet or
thermalizing the ejecta (Wang et al., 2024). We distribute the energy between
the jet and the thermalized ejecta by introducing an efficiency factor $\xi$
in Eq 5. We use a single power-law function to fit the light curve after the
luminosity peak, and obtain the decay slope of $-1.65$, which suggests that
the magnetar spin-down is caused by a combination of GW quadrupole and MD
torque. We used the magnetar models with different spin modes to fit the
luminosity curve of SN 2009bb, and the results are shown in Fig 3. The most
suitable model for SN 2009bb is the hybrid magnetar model with the GW and MD
acting together on the spin-down. The best-fitting results are shown in Table
1. In addition, we match the observed data of SN 2009bb with the 56Ni cascade
decay model and find that the 56Ni model can explain its early light curve,
but it does not agree with the later data, since the decay rate of the light
curve is slower than expected by the 56Ni model.

Modeling the light curve of SN 2009bb, we derive parameters for magnetar and
SN explosions, such as $B\sim 1.35\times 10^{15}\mbox{ G}$, $P_{0}\sim
26.86\mbox{ ms}$, $\epsilon\sim 5.33\times 10^{-3}$, $M_{\rm ej}\sim
1.08M_{\sun}$, $E_{\rm sn}\sim 2.91\times 10^{51}\mbox{ erg}$. The values of
$B$ and $P_{0}$ are reasonable parameters for magnetar-driven SNe Ic-BL
(Kashiyama et al., 2016). The corner plots in Fig 5 show that the ellipticity
cannot be tightly constrained, so we give the upper and lower limits of
$1\sigma$ confidence interval. The main reason for this may be that the energy
lost by GW radiation is not prominent compared to the energy lost by MD
radiation. The value of $E_{\rm sn}$ is comparable to the energy of a
neutrino-driven SN explosion. Having such a good fitting to the light curve,
we argue that SN 2009bb may be powered by a new-born magnetar with
deformation.

Figure 3: Bolometric luminosity curves of SNe 2007ru and 2009bb together with
model fits. Red solid lines represent the fitting results of the magnetar
model with both GW and MD acting together, yellow dashed lines and black
dashed lines indicate the fitting results of the magnetar models for the MD
and GW spin-downs, respectively. The results of the 56Ni model are shown as
orange dashed lines. Data for SN 2007ru are obtained from Sahu et al. (2009)
and Wang et al. (2017), and data for SN 2009bb are taken from Pignata et al.
(2011).

###  3.2 SN 2007ru

SN 2007ru is a nearby (redshift $z=0.01546$) SNe Ic-BL that was detected on
2007 December 2 by the Himalayan Faint Object Spectrograph Camera (HFOSC)
(Sahu et al., 2009). It peaked at $8$ days after its discovery, and the peak
magnitude is $M_{V}\approx-19.09\mbox{ mag}$, making it one of the brightest
SNe Ic. Its light curve shows no obvious sign of undulation, decaying with a
slope of $-1.75$ after the peak luminosity, which is consistent with the
scenario for both GW and MD acting together for the magnetar spin-down.

By assuming that the SN 2007ru was powered by 56Ni cascade decay, Sahu et al.
(2009) has inferred the 56Ni mass and ejecta mass of $M_{\rm Ni}\approx
0.4M_{\sun}$ and $M_{\rm ej}\approx 1.3M_{\sun}$, respectively. The ratio
between $M_{\rm Ni}$ and $M_{\rm ej}$ is larger than the upper limit for
CCSNe, which implies that the energy source is unlikely to be 56Ni decay. Fig
3 shows the fitting results for different spin-down models and 56Ni model. The
best-fitting model to SN 2007ru is the magnetar model with a combination of GW
radiation and MD radiation. The other models cannot reproduce its early or
late data. These results suggest that the bolometric luminosity evolution of
SN 2007ru contains a contribution from GW radiation. Omand & Sarin (2024) used
the optical emission from SN 2007ru to constrain the braking index of the NSs
under the assumption of the magnetar model and achieved results consistent
with ours.

Comparison of SN 2007ru observations with magnetar models allows us to
constrain the parameters of magnetar and SN explosion as follows $B\sim
1.26\times 10^{15}\mbox{ G}$, $P_{0}\sim 25.19\mbox{ ms}$, $\epsilon\sim
4.79\times 10^{-3}$, $M_{\rm ej}\sim 1.05M_{\sun}$, $E_{\rm sn}\sim
27.81\times 10^{50}\mbox{ erg}$. The derived best-fit ellipticity has a large
uncertainty. The values of $B$ and $P_{0}$ are consistent with previous
results for the magnetar models presented by (Wang et al., 2017), but there is
a discrepancy in the mass of the ejecta. This difference is caused by the fact
that they took a fixed value for the optical depth, i.e., $\kappa$=0.1 cm2 g-1
and then got the mass of ejecta of $M_{\rm ej}$, smaller than that of the
SLSNe but consistent with stripped SNe. Our fitting results suggest that SN
2007ru is produced by a stripped SNe Ic-BL and probably powered by a rapidly
spinning, deformed magnetar.

Note that so far no magnetar candidates for a GW-dominated spin-down have ever
been found in SNe observations for two possible reasons. One is that GW-
dominated spin-down may not play a significant role in the evolution of
magnetars; the other is that the number of SNe observed is currently
insufficient to find a better case. We find that the ellipticity of magnetar
should be around $10^{-3}$ for modeling the light curves of SNe 2009bb and
2007ru. This value is consistent with the conclusions of Moriya & Tauris
(2016) and Ho (2016) on the upper limit of the ellipticity obtained by a
simple comparison of the timescales of GW radiation and MD radiation for the
magnetar-driven SNe. Previously magnetar-driven GRB models have also taken an
ellipticity around $10^{-3}$ (Lasky & Glampedakis, 2016; Xie et al., 2022a),
which may suggest that the deformed magnetars are the sources for both GRBs
and SNe Ic-BL.

Figure 4: Left panel: Frequency evolution of the predicted GW for the SNe
2007ru and 2009bb. The bands are the error range of the GW frequency. Right
panel: The GW amplitude evolution for two SNe by assuming that the center
engine is a magnetar. The bands with colors represent the error ranges. The
designed sensitivity of O3, O4, and O5 of aLIGO are shown as light grey,
olive, and blue curves, respectively. The projected sensitivity of the ET is
shown as a black dashed curve.

##  4 Detectability of the GW signals from the deformed magnetars

The current aLIGO and Virgo detectors have discovered more than 90 cases of
compact star mergers with high confidence, which have opened a new era of GW
astronomy. In addition to the merger of NSs and/or black holes, the continuous
GWs from isolated deformed NSs are also important targets for aLIGO and Virgo
detectors. Post-collapse magnetars are considered to be the most promising
class of NSs for detecting continuous GWs due to their extremely strong
magnetic fields and fast spinning. If the magnetars are the central engines of
SNe, they would emit observable GWs due to non-axisymmetric deformation.

The GW strain from spinning magnetar with a mass quadrupole moment is (Corsi &
Mészáros, 2009; Howell et al., 2011; Dall’Osso et al., 2015)

$\displaystyle h(t)=\frac{4G\Omega(t)^{2}}{c^{4}d}I\epsilon,$ (12)

Here $d$ is the distance from the source. For long-lived GW transient sources,
the corresponding optimal matched filter signal-to-noise ratio is (Corsi &
Mészáros, 2009)

$\displaystyle\rho^{2}_{max}=\int^{+\infty}_{0}\left(\frac{h_{c}}{h_{rms}}\right)^{2}d(\ln
f),$

where $h_{c}=fh(t)\sqrt{dt/df}$ is the GW amplitude,
$h_{rms}=\sqrt{fS_{h}(f)}$ is the noise curve of detector, $S_{h}(f)$ is the
power spectral density of the detector noise. Using Eq 12 to solve for
$h_{c}$, one can obtain the GW amplitude of the magnetar as being (Corsi &
Mészáros, 2009)

$h_{c}=\frac{1}{d}\sqrt{\frac{5GIf}{2c^{3}}},$ (13)

where $f=\Omega/\pi$ is the frequency of the GW signal.

According to Eq 13 we can see that the $h_{c}$ is not only related to the
evolution of the stellar spin frequency but also depends on the distance to a
source. By fitting the SNe luminosity curve with the magnetar model, we can
derive the evolution of the spin frequency and hence the $h_{c}$. The left
side of Fig 4 shows the frequency evolution of the predicted GW for the SN
2007ru and SN 2009bb. From Eq 4 and Eq 13 it can be seen that the fractional
uncertainty in the magnetar parameters will be directly converted to the
uncertainty on the GW frequency and $h_{c}$. The corner plots show that the
ellipticity has a large uncertainty, which increases the uncertainties of the
inferred GW frequency and $h_{c}$. We plot the $h_{c}$ curves of two SNe on
the right side of Fig 4, together with the sensitivity curves of aLIGO and
Einstein detector (ET) (Abbott et al., 2018). The uncertainties of the
magnetar parameters have a small effect on the evolution of the $h_{c}$ curve.
One can find that the GWs from SN 2009bb and SN 2007ru can reach the detection
threshold of the aLIGO and ET detectors. On the other hand, for magnetars not
yet detected by ground-based detectors for GWs, we can give an upper limit on
the $\epsilon$ by comparing the sensitivity of GW detectors and $h_{c}$. Note
that our estimate of the detectability of the GW signal is based on the
assumption of an optimally matched filter, which may not be feasible for the
detection of the proposed signals. The $h_{c}$ curves in Fig 4 represent the
most optimistic upper limit for GW detection.

##  5 Conclusion and Discussion

A rapidly spinning magnetar can be formed after the SN explosion, emitting
continuous GWs due to the magnetic-induced deformation or fluid oscillation
excited by fast spinning. We explore the GW radiation in the magnetar-driven
SNe model and try to find evidence of GW radiation in the SN light curves.
Both MD radiation and GW radiation can act for the spin-down of magnetars. The
GW energy loss can lead to a reduced MD radiative energy to heat the SN,
causing a reduced peak luminosity of the SN and a fast slowing down, and also
a low decay rate of light curve tails. We made magnetar models with GW
emission and reproduced the light curves of SN 2009bb and SN 2007ru. We
derived the parameters of deformed magnetars by fitting the observed light
curves of SNe, and also estimated the detectability of the GWs from deformed
magnetars. Our results suggest that the GWs from SN 2009bb and SN 2007ru can
reach the detection threshold of the aLIGO and ET detectors. If GWs associated
with SNe are detected in the future, this would be a smoking gun that
magnetars can power SNe.

The effect of long-lived magnetar remnants on SN has been explored in detail
(e.g. Inserra et al., 2013; Kasen et al., 2016). The rotational energy
reservoir of magnetars has been used to explain many different features of SN,
such as the high peak luminosity, the large ejecta kinetic energy, and the
tail decaying with $t^{-2}$. However, in some cases, SNe with such a tail may
be attributed to the magnetar plus 56Ni model. The magnetar model explains the
early fast rise and peak luminosity, while the 56Ni model explains the late
tails, depending on if sufficiently massive Ni is synthesized. It may be
difficult to synthesize more than $0.2M_{\sun}$ of 56Ni (Nishimura et al.,
2015; Suwa & Tominaga, 2015) in the SN with a magnetar. Our simulations
suggest that a magnetar model plus 56Ni model does have some difficulty in
explaining the overly flat decay of the tail (slope much less than $-2$),
especially for decays longer than 300 days because the decay rate of 56Co
gradually deviates over time.

By modeling the SN bolometric luminosity curve, we derive the ellipticity of
the magnetar to be $\epsilon\sim 10^{-3}$. The current GW detection has set an
upper limit of the ellipticity of $\sim 10^{-4}-10^{-6}$ for pulsar (Abbott et
al., 2019a), and $\sim 10^{-3}$ for young NSs (Aasi et al., 2015b, 2016);
while the electromagnetic observations from the GRB X-ray plateau constrain
the ellipticity of magnetars to $\sim 10^{-3}-10^{-4}$ (Lasky & Glampedakis,
2016; Xie et al., 2022a). Note, however, that new-born magnetars may have
greater non-axisymmetric deformation than old pulsars, and thus may produce
stronger GW emission.

The ellipticity of a magnetar inferred from two SNe are consistent with those
inferred from the GRB observations, implying that both GRB and some SNe Ic-BL
may have similar central engines. The deformation of NSs may be induced by
anisotropic stresses caused by the internal magnetic field (Cutler, 2002;
Mastrano et al., 2011; Lander, 2014). If NS distortion is induced by the pure
toroidal magnetic field ($B_{\rm t}$), the relation between the $\epsilon$ and
$B_{\rm t}$ can be described as $\epsilon\approx 1.6\times 10^{-4}({B_{\rm
t}}/{10^{16}\,{\rm G}})^{2}$ (Cutler, 2002; Stella et al., 2005; Dall’Osso et
al., 2009). It can be seen that the $B_{t}$ ought to reach $\sim 2\times
10^{16}\mbox{ G}$ for the $\epsilon$ of $\sim 10^{-3}$, which implies that a
stronger toroidal field strength is required, at least $1-2$ orders of
magnitude larger than the dipole field ($B\sim 10^{14}-10^{15}\mbox{ G}$).
Theoretically, the toroidal magnetic field of magnetar can be amplified to
$\sim 10^{16}-10^{17}\mbox{ G}$ due to the $\alpha-\Omega$ dynamo mechanism
(Duncan & Thompson, 1992). This conclusion is supported by previous works on
constraining the internal field strength of soft gamma-ray repeating bursts
(see, (Ioka, 2001; Corsi & Owen, 2011)). Other channels than magnetic-induced
NS deformation, such as secular bar-mode instability (also known as unstable
f-mode oscillation), can also play an important role in nascent magnetars.
Lasky & Glampedakis (2016) proposed that the value of the maximum ellipticity
induced by f-mode depends on the saturation energy of the mode. The maximum
energy of the f-mode is $E_{\rm mode}\sim 10^{-6}Mc^{2}$, which corresponds to
the effective ellipticity of $\sim 2\times 10^{-3}$. In addition, NS accretion
can also induce distortion. The material around the accretion disc will flow
into the poles of the NS due to the gravitational effect to form mountains. In
this scenario, the magnitude of the ellipticity value of the NS depends on the
mass of the accreting material (Zhong et al., 2019). Sur & Haskell (2021)
suggested that for accreting NSs located at $1\mbox{ Mpc}$, the GW amplitudes
at kHz frequencies can reach the detection threshold of aLIGO.

Even though aLIGO has not detected continuous GWs from isolated NSs, we are
optimistic about the prospect of detecting GWs from SNe. If continuous GWs
from the NSw can be detected by future GW detectors, then we can constrain the
ellipticity $\epsilon$ of the NS by Eq 12. Furthermore, the duration and
frequency of continuous GWs can be obtained from their waveforms. According to
Eq 2, 3 and 13, the mass-radius relation can be constrained and then combined
with the inferred ellipticity $\epsilon$ to provide valuable information for
constraining the NS deformation after the SN explosion.

Figure 5: The corner plot of the magnetar model for supernova 2007ru. The
$1\sigma$ confidence level of the parameters is shown as the vertical dashed
lines.  Figure 6: The corner plot of the magnetar model for supernova 2009bb.

##  Acknowledgements

We thank the anonymous referee for helpful comments to improve this paper. The
authors are supported by the National Natural Science Foundation of China
(Nos. 11988101, 11833009, 12073080, 12233011, 11933010, 11921003 and 12303050)
and by the Chinese Academy of Sciences via the Key Research Program of
Frontier Sciences (No. QYZDJ-SSW-SYS024), China Postdoctoral Science
Foundation (grant No. 2023M743397), and the Fundamental Research Funds for the
Central Universities.

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