# Oscillation Modes and Gravitational Waves from Strangeon Stars

Hong-Bo Li,1,2 Yong Gao,1,2 Lijing Shao,2,3 Ren-Xin Xu,1,2 and Rui Xu2
1Department of Astronomy, School of Physics, Peking University, Beijing
100871, China
2Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China
3National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100012, China gaoyong.physics@pku.edu.cnlshao<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The strong interaction at low energy scales determines the equation of state
(EOS) of supranuclear matters in neutron stars (NSs). It is conjectured that
the bulk dense matter may be composed of strangeons, which are quark clusters
with nearly equal numbers of $u$, $d$, and $s$ quarks. To characterize the
strong-repulsive interaction at short distance and the nonrelativistic nature
of strangeons, a phenomenological Lennard-Jones model with two parameters is
used to describe the EOS of strangeon stars (SSs). For the first time, we
investigate the oscillation modes of non-rotating SSs and obtain their
frequencies for various parameterizations of the EOS. We find that the
properties of radial oscillations of SSs are different from those of NSs,
especially for stars with relatively low central energy densities. Moreover,
we calculate the $f$-mode frequency of nonradial oscillations of SSs within
the relativistic Cowling approximation. The frequencies of the $f$-mode of SSs
are found to be in the range from $6.7\,$kHz to $8.7\,\rm{kHz}$. Finally, we
study the universal relations between the $f$-mode frequency and global
properties of SSs, such as the compactness and the tidal deformability. The
results we obtained are relevant to pulsar timing and gravitational waves, and
will help to probe NSs’ EOSs and infer nonperturbative behaviours in quantum
chromodynamics.

###### keywords:

stars: oscillations – pulsars: general – gravitational waves –
asteroseismology

††pubyear: 2022††pagerange: Oscillation Modes and Gravitational Waves from
Strangeon Stars–Oscillation Modes and Gravitational Waves from Strangeon Stars

##  1 Introduction

The equation of state (EOS) of nuclear dense matter plays a crucial role in
many astrophysical phenomena associated with neutron stars (NSs; Ozel et al.,
2010; Lattimer & Prakash, 2007; Abbott et al., 2018). Owing to the non-
perturbative properties of the strong interaction at low energy, the EOS of
dense matters at several nuclear densities still remains unknown. Witten
(1984) conjectured that the true ground state of the dense matter is quark
matter composed of almost free $u$, $d$, and $s$ quarks. The pulsar-like
compact objects should be quark stars (QSs) rather than conventional NSs. The
MIT bag model with almost free quarks (Alcock et al., 1986) and the color-
superconductivity state model (Alford et al., 2008) have been used in
literature to study QSs. In 2003, Xu (2003) proposed that the constituting
units of the supranuclear matter could be strange quark clusters, since the
non-perturbative strong interaction may render quarks grouped in clusters.
Each quark cluster is composed of several quarks (including $u$, $d$, and $s$
flavors) condensing in position space rather than in momentum space. A name
“strangeon” is coined to these strange “nucleons” (Xu & Guo, 2017; Lai & Xu,
2017). In this sense, compressed baryonic matter could be in a state of
strangeons, and pulsar-like compact stars could thus be strangeon stars (SSs).

Strangeon matter, similar to strange quark matter, is composed of nearly equal
numbers of $u$, $d$, and $s$ quarks. However, different from strange quark
matter, quarks in strangeon matter are localized inside strangeons due to the
strong coupling between quarks. There are differences and similarities among
NSs, QSs, and SSs. On the one hand, quarks are thought to be localized in
strangeons in SSs, like neutrons in NSs. On the other hand, a strangeon , with
light-flavour symmetry restoration of quark, may contain more than three
valence quarks. In addition, the matter at the surface of SSs is strangeon
matter, i.e., SSs are self-bound by the strong force, like QSs (Xu, 2003).
These properties are fundamental to a few important astrophysical observables.
A sophisticated study on various global parameters of rotating SSs, including
mass, radius, moment of inertia, tidal deformability, quadrupole moments, and
shape parameters, was carried out by Gao et al. (2022).

SSs can account for many current observational facts in astrophysics. The EOS
of SSs could be very stiff to explain the observed massive pulsars (Demorest
et al., 2010; Antoniadis et al., 2013). The magnetospheric activity of SSs was
discussed in Xu et al. (1999). Lu et al. (2019) explained the sub-pulse
drifting of radio pulsars using the properties at the surface of SSs. Also,
pulsar glitches could be the result of star-quakes (Peng & Xu, 2008; Zhou et
al., 2004, 2014), and a detailed modeling of the glitch behaviours confronted
with observations was discussed in Lai et al. (2018b). The model of SSs can be
extended to explain the glitch activity of normal radio pulsars (Wang et al.,
2020). Recent studies (Lai et al., 2019, 2018a, 2021) have investigated the
tidal deformability as well as the ejecta and light curve of merging binary
SSs, showing consistency with the observations of the gravitational wave (GW)
event GW170817 (Abbott et al., 2017) and its multiwavelength electromagnetic
counterparts (Kasliwal et al., 2017; Kasen et al., 2017).

Owing to the difficulties in determining the EOS of pulsar-like compact stars
from first principles, observations from different channels become important
avenues in studying the EOS at high density, which can in turn be used to
constrain microscopic laws (Ozel et al., 2010). In this respect, GW
asteroseismology that deals with oscillation modes offers a promising channel
in the new era of GWs (Andersson & Kokkotas, 1998; Benhar et al., 2004; Doneva
et al., 2013; Andersson, 2019). It is the focus of this study.

Radial oscillations of stellar models were studied in the pioneering works of
Chandrasekhar (1964a, b). Notably, the properties of radial oscillations can
give information about the stability and the EOS of compact stars. The first
exhaustive compilation of radial oscillations for different zero-temperature
EOSs was presented by Glass & Lindblom (1983). In Vaeth & Chanmugam (1992),
the properties of radial modes of quark stars (QSs) were investigated.
Furthermore, the study of radial oscillations of zero-temperature NSs can be
extended to proto-NSs (Gondek et al., 1997). Because the EOS of proto-NSs is
significantly softer than that of zero-temperature NSs, their spectra of the
radial oscillation modes are very different. It is worth noting that Kokkotas
& Ruoff (2001) presented a useful survey on the radial oscillation modes of
NSs for various EOSs. Based on the equations presented by Misner et al.
(1973), they showed that the derivatives in the linear differential equation
can be written in the form of a self-adjoint differential operator. In this
work, we calculate the frequencies of the first three radial modes of SSs
using the method of the self-adjoint differential operator (Kokkotas & Ruoff,
2001). By that we can investigate the properties of radial oscillations of SSs
in detail and study the stability of SSs rigorously.

Nonradial oscillations of relativistic stars were studied in the pioneering
work of Thorne & Campolattaro (1967). The oscillation modes are damped out due
to the emission of GWs, so these oscillation modes are called quasinormal
modes (QNMs). For typical non-rotating relativistic fluid stars, QNMs are
classified in polar and axial categories. The polar modes include the
fundamental ($f$) modes, pressure ($p$) modes, and gravity ($g$) modes. The
axial modes only have the spacetime ($w$) modes, which are directly associated
with the spacetime metric and have no analogy in the Newtonian theory of
stellar pulsations (Kokkotas & Schutz, 1992). A detailed discussion about the
relativistic perturbation equations was given in many works (see e.g.,
Lindblom & Detweiler, 1983; Detweiler & Lindblom, 1985; Chandrasekhar &
Ferrari, 1991; Allen et al., 1998; Kokkotas & Schmidt, 1999). Using the
Cowling approximation (Cowling, 1941), Sotani et al. (2011) calculated
nonradial oscillations of NSs with hadron-quark mixed phase transition.
Besides, Doneva & Yazadjiev (2012) investigated nonradial oscillations of
anisotropic NSs with polytropic EOSs. In Das et al. (2021), the impact of the
dark matter on the f-mode was also studied.

The $f$-mode of NSs, QSs, and SSs is important for several reasons: (i) it
depends on the EOS of compact stars; (ii) it is expected to be excited in many
astrophysical scenarios and leads to efficient GW emission; (iii) its
frequency is lower than other QNMs such as that of the $p$-modes and the
$w$-modes, hence the $f$-mode oscillation is most likely to be detectable with
a third-generation detector like the Einstein Telescope and the Cosmic
Explorer (Punturo et al., 2010; Sathyaprakash et al., 2019; Kalogera et al.,
2021), or even in an optimal case by the current generation LIGO/Virgo/KAGRA
detectors (Abbott et al., 2019b, 2022; Abe et al., 2022). In this work, we
calculate the $f$-mode frequency of SSs in the Cowling approximation and
compare the results with those of NSs and QSs.

GW observation will be a powerful tool to study the EOS of compact stars in
particular in the case that we have good empirical formulas for the QNMs as
functions of stellar parameters. Indeed, universal empirical formulas relating
the dynamical responses of a compact star under external perturbations—such as
the $f$-mode frequency, the tidal and rotational deformations—to its global
physical parameters—such as the mass, the radius, and the moment of
inertia—have been discovered for NSs and QSs. For example, the I-Love-Q
relations, discovered by Yagi & Yunes (2013a, b), relate the moment of inertia
$I$, the tidal deformability $\lambda$, and the spin induced quadrupole moment
$Q$. Another example is the relation for the $f$-mode frequency, the moment of
inertia, and the tidal deformability (see e.g., Chan et al., 2014). Sotani &
Kumar (2021) have investigated various universal relations between several
oscillation modes and the tidal deformability. Inspired by works on the
universal relations for single NSs (Andersson & Kokkotas, 1998; Benhar et al.,
2004; Doneva et al., 2013; Yagi & Yunes, 2013a, b), recent studies have
investigated various universal relations between the binary tidal
deformability and the $f$-mode frequency of the post-merger remnant of a
binary NS system using numerical relativity simulations (Bernuzzi et al.,
2015b; Rezzolla & Takami, 2016; Kiuchi et al., 2020). It is worth noting that,
Krüger & Kokkotas (2020b) and Manoharan et al. (2021) calculated the $f$-mode
frequency for fast rotating NSs without using the Cowling approximation, and
discovered a relation between the pre-merger tidal deformability and the
dominant oscillation frequency (i.e., $f$-mode) of the post-merger remnant of
a binary NS system. Meanwhile, using the universal relation of $f$-mode
(Krüger & Kokkotas, 2020b; Manoharan et al., 2021), Völkel et al. (2021) and
Völkel & Krüger (2022) studied the Bayesian inverse problem of rotating NSs.
In this work, we will study the universal relation between the $f$-mode
frequency and the tidal deformability of SSs, which will be a useful input for
comparisons among NSs, QSs, and SSs.

The paper is organized as follows. In Sec. 2, we introduce the EOS of SSs and
obtain the structure of non-rotating SSs. Based on the background solutions,
in Sec. 3, we integrate the equations of relativistic radial oscillations to
determine the $f$-mode frequency for different EOSs of SSs. In Sec. 4, we
calculate the frequency of nonradial $f$-mode and the tidal deformability of
SSs. New fits of universal relation between them are discussed. Finally, we
summarize our work in Sec. 5.

Throughout the paper, we adopt geometric units with $c=G=1$, where $c$ and $G$
denote the speed of light and the gravitational constant respectively. The
metric signature is $(-\,,+\,,+\,,+)$.

##  2 EQUATION OF STATE and structure of spherical static stars

We assume that the interaction potential between two strangeons is described
by the Lennard-Jones potential (Jones, 1924; Lai & Xu, 2009; Gao et al.,
2022),

$u(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right]\,,$
(1)

where $\epsilon$ is the depth of the potential, $r$ is the distance between
two strangeons, and $\sigma$ is the distance when $u(r)=0$. We note that this
potential has the property of short-distance repulsion and long-distance
attraction.

According to the results of early studies (Xu, 2003; Lai & Xu, 2009; Gao et
al., 2022), the potential energy density is given by

$\rho_{\rm
p}=2\epsilon\left(A_{12}\sigma^{12}n^{5}-A_{6}\sigma^{6}n^{3}\right)\,,$ (2)

where $A_{12}=6.2$, $A_{6}=8.4$, and $n$ is the number density of strangeons.
The total energy density of zero-temperature dense matter composed of
strangeons reads

$\rho=2\epsilon\left(A_{12}\sigma^{12}n^{5}-A_{6}\sigma^{6}n^{3}\right)+nN_{\rm
q}m_{\rm q}\,,$ (3)

where $N_{\rm q}m_{\rm q}$ is the mass of a strangeon with $N_{\rm q}$ being
the number of quarks in a strangeon and $m_{\rm q}$ being the quark mass. In
the above equation the contributions from degenerate electrons and vibrations
of the lattice are neglected. From the first law of thermodynamics, one
derives the pressure

$P=n^{2}\frac{\mathrm{d}(\rho/n)}{\mathrm{d}n}=4\epsilon\left(2A_{12}\sigma^{12}n^{5}-A_{6}\sigma^{6}n^{3}\right)\,.$
(4)

At the surface of SSs, the pressure becomes zero and we obtain the surface
number density of strangeons as
$\big{[}A_{6}/(2A_{12}\sigma^{6})\big{]}^{1/2}$. For convenience, we transform
it to the number density of baryons

$n_{\rm s}=\left(\frac{A_{6}}{2A_{12}}\right)^{1/2}\frac{N_{\rm
q}}{3\sigma^{3}}\,.$ (5)

Figure 1: Relations between mass-energy density $\rho$ and pressure $P$ for
NSs, QSs, and SSs.

For a given number of quarks $N_{\rm q}$ in a strangeon, the EOS of SSs is
completely determined by the depth of the potential $\epsilon$ and the number
density of baryons $n_{\rm s}$ at the surface of the star. An 18-quark
cluster, called quark-alpha (Michel, 1991), can be completely symmetric in
spin, flavor, and color spaces. Therefore, we set $N_{\rm q}=18$ in our
calculation as a reasonable example.

Besides the EOS of SSs, we also consider six EOSs of NSs and QSs for
comparison, including four popular nuclear matter EOSs for NSs, AP4 (Akmal &
Pandharipande, 1997), SLy4 (Douchin & Haensel, 2001), MS0, and MS2 (Mueller &
Serot, 1996), as well as two QS models, the MIT bag model with a bag constant
$B=60\,\rm MeV\,fm^{-3}$ (Alcock et al., 1986) and SQM3 (Lattimer & Prakash,
2001).111Notice that, the nucleonic EOSs, MS0 and MS2, have similar properties
as SSs of higher maximum masses. The corresponding density-pressure relations
for these EOSs are depicted in Fig. 1. We denote the EOSs of SSs using their
values of $n_{\rm s}$ and $\epsilon$. For example, “LX2430” means a surface
baryon number density $n_{\rm s}=0.24\,\rm fm^{-3}$ and a potential depth
$\epsilon=30\,\rm{MeV}$.

We consider the unperturbed relativistic star to be described by a perfect
fluid. The energy-momentum tensor is
$T_{\mu\nu}=(\rho+P)u_{\mu}u_{\nu}+P\,g_{\mu\nu}$. The static and spherically
symmetric metric, which describes an equilibrium relativistic star, is given
by the line element,

${\rm d}s^{2}=-e^{2\Phi}{\rm d}t^{2}+e^{2\Lambda}{\rm d}r^{2}+r^{2}({\rm
d}\theta^{2}+\sin^{2}\theta{\rm d}\phi^{2})\,,$ (6)

where $\Phi$ and $\Lambda$ are metric functions of $r$. A mass function $m(r)$
is defined as $m(r)=r(1-e^{-2\Lambda})/2$, which satisfies

$\frac{{\rm d}m}{{\rm d}r}=4\pi r^{2}\rho\,,$ (7)

where $\rho$ is the energy density. The Tolman-Oppenheimer-Volkoff (TOV)
equations that determine the pressure $P(r)$ and the metric function $\Phi(r)$
are expressed as

$\displaystyle\frac{{\rm d}P}{{\rm d}r}$ $\displaystyle=-(\rho+P)\frac{{\rm
d}\Phi}{{\rm d}r}\,,$ (8) $\displaystyle\frac{{\rm d}\Phi}{{\rm d}r}$
$\displaystyle=\frac{m+4\pi r^{3}P}{r(r-2m)}\,.$ (9)

Integrating Eqs. (7), (8) and (9) combined with the EOS, one obtains the
stellar structure of spherical stars and the spacetime geometry. In Fig. 2, we
show the mass-radius relations for NSs, QSs, and SSs using the aforementioned
EOSs. The EOSs of SSs are very stiff because the strangeons are
nonrelativistic and there is a very strong repulsion at a short inter-cluster
distance (Gao et al., 2022), which leads to the maximal masses over
$3\,M_{\odot}$. In contrast, the quarks are relativistic and nearly free for
QSs, so the EOSs are soft and the maximal masses only reach $2\,M_{\odot}$
marginally. The observations of the massive pulsars, PSRs J0348+0432
(Antoniadis et al., 2013) and J0740$+$6620 (Fonseca et al., 2021), at $\sim
2\,M_{\odot}$ via pulsar timing support the stiff properties of the EOS. More
massive ones (e.g., $\geq 2.5\,M_{\odot}$) are expected in our model for
future discovery. The GWs from the binary NS inspiral, GW170817, gave
constraints on the tidal deformability for the first time (Abbott et al.,
2017, 2018, 2019a), which rules out several stiff EOSs (e.g., EOSs MS0 and
MS2) and models of SSs with very low surface baryonic densities (say, LX2430
and LX2450) at a $90\%$ credible level (see Fig. 18 in Gao et al. (2022)).

Figure 2: Mass-radius relations of SSs with different combinations of the
surface baryonic density $n_{\rm s}$ and the potential depth $\epsilon$. For
comparison, we also show the mass-radius relations for selected NSs and QSs.
The $1$-$\sigma$ regions of the mass measurements in PSRs J0348+0432
(Antoniadis et al., 2013) and J0740+6620 (Fonseca et al., 2021) are
illustrated.

##  3 RADIAL OSCILLATIONS

In this section, we study radial oscillations of SSs. We denote the radial
displacement of a fluid element as $\delta r(r,t)$ and its harmonic
oscillation mode with circular frequency $\omega$ as $\delta
r(r,t)=X(r)e^{i\omega t}$. To obtain the discrete set of oscillation
frequencies of SSs, we adopt the perturbation equations in Kokkotas & Ruoff
(2001). In practice we define a new variable $\zeta=r^{2}e^{-\Phi}X$. The
master equation for radial oscillations is expressed as

$\frac{{\rm d}}{{\rm d}r}\Big{(}\mathcal{P}\frac{{\rm d}\zeta}{{\rm
d}r}\Big{)}+\big{(}\mathcal{Q}+\omega^{2}\mathcal{W}\big{)}\zeta=0\,,$ (10)

where

$\displaystyle r^{2}\mathcal{P}$ $\displaystyle=\Gamma
P\,e^{(\Lambda+3\Phi)}\,,$ $\displaystyle r^{2}\mathcal{Q}$
$\displaystyle=e^{(\Lambda+3\Phi)}(\rho+P)\left[(\Phi^{\prime})^{2}+4\frac{\Phi^{\prime}}{r}-8\pi
e^{2\Lambda}P\right]\,,$ $\displaystyle r^{2}\mathcal{W}$
$\displaystyle=(\rho+P)e^{(3\Lambda+\Phi)}\,.$ (11)

By setting $\eta=\mathcal{P}\zeta^{\prime}$, one obtains the following coupled
differential equations,

$\displaystyle\frac{{\rm d}\zeta}{{\rm d}r}$
$\displaystyle=\frac{\eta}{\mathcal{P}}\,,$ (12) $\displaystyle\frac{{\rm
d}\eta}{{\rm d}r}$
$\displaystyle=-\big{(}\omega^{2}\mathcal{W}+\mathcal{Q}\big{)}\zeta\,.$ (13)

At the center of the star, the boundary condition is
$3\zeta_{0}={\eta_{0}}/{\mathcal{P}_{0}}$, where $\zeta_{0}$ and $\eta_{0}$
are the values of $\zeta$ and $\eta$ at $r=0$ respectively (Kokkotas & Ruoff,
2001). By setting $\eta_{0}=1$, we have $\zeta_{0}={1}/{3\mathcal{P}_{0}}$,
where $\mathcal{P}_{0}=\Gamma P(0)e^{(\Lambda(0)+3\Phi(0))}$. At the star
surface $r=R$, the pressure perturbation must vanish, namely $\Delta P=0$,
which provides another boundary condition, $\Gamma P\zeta^{\prime}=0$.
Equations (12) and (13) with the above two boundary conditions form a two-
point boundary value problem of the Sturm-Liouville type with eigenvalues
$\omega_{0}^{2}<\omega_{1}^{2}<\omega_{2}^{2}<\cdots$ (Shapiro & Teukolsky,
1983), where $\omega_{0}$ is the eigenfrequency of the $f$-mode. If
$\omega_{0}^{2}>0$, all the eigenfrequencies of the oscillation modes are
real, which indicates that the equilibrium stellar model is dynamically stable
(Chandrasekhar, 1964a, b; Misner et al., 1973). The period of the $f$-mode is
given by $\tau_{0}=1/\nu_{0}=2\pi/\omega_{0}$, where $\nu_{0}$ is the ordinary
or temporal frequency. Inversely, $\omega_{0}^{2}<0$ corresponds to an
exponentially growing unstable radial oscillation.

For adiabatic oscillations, the adiabatic index governing the perturbations is
defined by (Kokkotas & Ruoff, 2001)

$\Gamma=\frac{\rho+P}{P}\frac{{\rm d}P}{{\rm d}\rho}\,,$ (14)

which is equal to the adiabatic index governing the equilibrium pressure-
energy density relation. The relation between the adiabatic index $\Gamma$ and
the mass-energy density $\rho$ is shown in Fig. 3. We note that the adiabatic
indices for QSs and SSs are qualitatively different from that of NSs at low
density. Moreover, SSs generally have a larger adiabatic index than NSs and
QSs, indicating that the EOSs of SSs are stiffer (Gao et al., 2022).

Figure 3: Relation between the adiabatic index $\Gamma$ and the mass-energy
density $\rho$ for NSs, QSs and SSs. Figure 4: The frequencies of the
fundamental mode, $\nu_{0}$, and the first two excited modes, $\nu_{1}$ and
$\nu_{2}$, of radial oscillation, as functions of the central density
$\rho_{c}$ for NSs and QSs.

In Fig. 4, we present the $f$-mode frequency $\nu_{0}$ and the frequencies of
the first two excited modes, $\nu_{1}$ and $\nu_{2}$, for SLy4, AP4, and the
MIT bag model. Our results for NSs reproduce the results of Kokkotas & Ruoff
(2001). We observe that $f$-mode becomes unstable (i.e., $\omega^{2}_{0}$
becoming negative) for central densities above $2.83\times
10^{15}\rm{g\,cm^{-3}}$, $2.70\times 10^{15}\rm{g\,cm^{-3}}$, and $2.05\times
10^{15}\rm{g\,cm^{-3}}$ for three EOSs. The instability point corresponds to
maximal masses $2.04\,M_{\odot}$, $2.21\,M_{\odot}$, and $1.96\,M_{\odot}$ for
SLy4, AP4, and the MIT bag model respectively. It is worth noting that the
$f$-mode frequency of the MIT bag model behaves very different from that of
NSs at low central density, rooting in the self-bound and gravity-bound nature
of QSs and NSs respectively.

To further explore the results for QSs, SSs, and NSs, we note that with a low
central density, the star can be approximated as a homogeneous nonrelativistic
star (Shapiro & Teukolsky, 1983), so that the angular frequency $\omega_{0}$
of the $f$-mode reads $\omega_{0}^{2}=4\pi\rho(4\Gamma-3)/3$. Using the
relations between the density and the adiabatic index shown in Fig. 3, we do
expect the frequency $\omega_{0}$ to diverge as the density approaches a
minimal value for QSs and SSs. For NSs, the adiabatic index does not change
significantly as the density decreases. Therefore for NSs, $\omega_{0}$ tends
to zero mildly when the central density of the star is sufficiently low.
Indeed, these points are confirmed in Fig. 4.

In Fig. 5, we show the ordinary frequency $\nu_{0}$ of the $f$-mode versus the
mass of the stars for SSs and one EOS of QSs. The curves of SSs have the same
trend as that of QSs, with $\nu_{0}$ going to zero at their maximal masses.
However, $\nu_{0}$ for SSs is larger than that of QSs for a given mass, which
arises from the fact that SSs’ EOSs are much stiffer than that of QSs.

Figure 5: The frequencies of the fundamental mode $\nu_{0}$ as functions of
the mass $M$ for QSs and SSs.

##  4 nonradial OSCILLATIONS

In this section, we study nonradial oscillations of a non-rotating SS in the
Cowling approximation, in which the spacetime metric is kept to be the static
spherical background solution in the so-called Cowling approximation (Cowling,
1941). The fluid Lagrangian displacement vector is given by

$\xi^{i}=\left(e^{-\Lambda}W,-V\partial_{\theta},-V\sin^{-2}\theta\partial_{\phi}\right)r^{-2}Y_{\ell
m}\,,$ (15)

where $W$ and $V$ are functions of $t$ and $r$, while $Y_{\ell m}$ is the
spherical harmonic function. Then the perturbation of the four-velocity,
$\delta u^{\mu}$, can be written as

$\delta
u^{\mu}=\left(0,e^{-\Lambda}\partial_{t}W,-\partial_{t}V\partial_{\theta},-\partial_{t}V\sin^{-2}\theta\partial_{\phi}\right)r^{-2}e^{-\Phi}Y_{\ell
m}\,.$ (16)

Assuming a harmonic dependence on time, the perturbative variables can be
written as $W(t,r)=W(r)e^{i\omega t}$ and $V(t,r)=V(r)e^{i\omega t}$. We can
obtain the following system of equations for the fluid perturbations (see
Sotani et al., 2011; Doneva & Yazadjiev, 2012; Yazadjiev & Doneva, 2012, for a
detailed variational derivation),

$\displaystyle\frac{{\rm d}W}{{\rm d}r}$ $\displaystyle=\frac{{\rm
d}\rho}{{\rm d}P}\left[\omega^{2}r^{2}e^{\Lambda-2\Phi}V+\frac{{\rm
d}\Phi}{{\rm d}r}W\right]-\ell(\ell+1)e^{\Lambda}V\,,$ (17)
$\displaystyle\frac{{\rm d}V}{{\rm d}r}$ $\displaystyle=2\frac{{\rm
d}\Phi}{{\rm d}r}V-e^{\Lambda}\frac{W}{r^{2}}\,.$ (18)

The boundary condition at the center of the star can be parameterized as,
$W=Ar^{l+1}$ and $V=-{A}r^{l}/{l}$, with $A$ being an arbitrary constant. It
can be obtained by examining the behavior of $W$ and $V$ in the vicinity of
$r=0$. At the surface of the star, the perturbed pressure must vanish, which
provides

$\omega^{2}e^{\Lambda(R)-2\Phi(R)}V(R)+\frac{1}{R^{2}}\frac{{\rm d}\Phi}{{\rm
d}r}\Big{|}_{r=R}W(R)=0\,.$ (19)

In full general relativity, each QNM is characterized by a complex
eigenfrequency $\omega=\omega_{\rm r}+\rm{i}\,\omega_{\rm{i}}$ (Thorne &
Campolattaro, 1967). The real part $\omega_{\rm r}$ corresponds to the mode
frequency, and the imaginary part $\omega_{\rm{i}}$ gives the damping time
$\tau\equiv 1/\omega_{\rm{i}}$ due to GW emission. However, in the Cowling
approximation, we obtain normal modes of oscillation and there is no emission
of GWs. For a non-rotating stellar model, the Cowling approximation leads to a
relative error $\sim 10\%$–$30\%$ for the $f$-mode (Chirenti et al., 2015;
Sotani & Dohi, 2022). For higher modes, the relative error is smaller (Yoshida
& Kojima, 1997).

###  4.1 F-MODE FREQUENCY

Now we calculate the ordinary frequency $\nu_{0}$ of the $f$-mode for the
$l=2$ nonradial oscillation, and study its relation with the mass $M$, the
compactness $C=M/R$, and the dimensionless tidal deformability $\Lambda$ for
NSs, QSs, and SSs.

Figure 6: The frequency $\nu_{0}$ of $f$-mode as a function of mass $M$ for
NSs and QSs (top), and SSs (bottom).

The frequency $\nu_{0}$ versus mass $M$ for NSs and QSs is shown in the top
panel of Fig. 6. By increasing the mass of the star, the frequency $\nu_{0}$
increases significantly for NSs, while it does not change much for QSs. This
can be understood by noticing that QSs are self-bound by strong interaction
and the density in the interior of the star does not change too much as the
mass increases. This is in contrast to NSs which are gravitationally bound.
From the figure, we can see that the values of $\nu_{0}$ at the maximal masses
of NSs and QSs are $2.907$ kHz, $2.823$ kHz, $2.562$ kHz, and
$2.597\,\rm{kHz}$ for SLy4, AP4, SQM3, and the MIT bag model with $\rm
B=60\,MeV\,fm^{-3}$ respectively.

Additionally, the frequency $\nu_{0}$ versus mass $M$ for SSs is shown in the
bottom panel of Fig. 6. We find that the curves of SSs are similar to those of
QSs only that the frequency $\nu_{0}$ for SSs extends a much wider range. The
values of $\nu_{0}$ at the maximal masses of SSs are $6.676$ kHz, $6.832$ kHz,
$7.977$ kHz, and $8.684\,\rm{kHz}$ for the EOSs of SSs with different values
of $n_{\rm s}$ and $\epsilon$ that we use in the figure. Compared with QSs and
NSs, these values are much larger, and it could be an indicator to distinguish
EOSs via GW observations.

Figure 7: The frequency $\nu_{0}$ of the $f$-mode as a function of the
compactness, $C=M/R$, for NSs and QSs (top), and SSs (bottom).

We show in Fig. 7 the relation between the frequency $\nu_{0}$ and the
compactness of the stars. It might be useful to note that the values of the
maximal compactness, $C_{\rm{max}}$, are $0.21$, $0.23$, $0.19$, and $0.19$
for SLy4, AP4, SQM3, and the MIT bag model with $\rm B=60\,MeV\,fm^{-3}$,
respectively. In contrast, the value of $C_{\rm{max}}$ for SSs with different
values of $n_{\rm s}$ and $\epsilon$ is about the same, $C_{\rm{max}}\simeq
0.27$. This maximal value of the compactness represents the limit of how stiff
EOSs of SSs can be due to the repulsive hardcore and the nonrelativistic
nature of strangeons.

###  4.2 UNIVERSAL RELATIONS

To reveal the internal characters of NSs and assist relevant data analysis,
universal relations between the $f$-mode, $p$-mode, and $w$-mode frequencies
and the mass or the radius of NSs have been investigated (Andersson &
Kokkotas, 1996, 1998; Benhar et al., 1999, 2004; Tsui & Leung, 2005).
Motivated by possible observations of the moment of inertia $I$ of NSs, Lau et
al. (2010) used the moment of inertia to replace the compactness and
discovered EOS-independent relations in QNMs of NSs and QSs. Similar results
were shown in Chirenti et al. (2015). These relations can be used to infer the
stellar parameters—mass, radius, and possibly the EOS—from QNM data with
future GW detectors.

Figure 8: Scaled frequency of the $f$-mode as a function of the compactness
$C$. The solid lines represent universal relations in Eqs. (20) and (21).

Using the Cowling approximation, Sotani et al. (2011) calculated nonradial
oscillations of NSs with hadron-quark mixed phase transition, and discovered
an approximate formula. Inspired by the universal relation between the
$f$-mode and the compactness $C$ (Sotani et al., 2011), we show the scaled
frequency of the $f$-mode versus the compactness $C$ for NSs, QSs, and SSs in
Fig. 8. In particular, as shown by the solid lines in the figure, the
universal relation for NSs can be represented by the following empirical
formula,

$M\nu_{0}=a_{\rm I}+b_{\rm I}\Big{(}\frac{M}{R}\Big{)}+c_{\rm
I}\Big{(}\frac{M}{R}\Big{)}^{2}+d_{\rm I}\Big{(}\frac{M}{R}\Big{)}^{3}\,,$
(20)

with $a_{\rm I}=-0.012$, $b_{\rm I}=19.48$, $c_{\rm I}=71.3$, and $d_{\rm
I}=-125$, while for SSs, we found a new universal relation,

$M\nu_{0}=a_{\rm II}+b_{\rm II}\Big{(}\frac{M}{R}\Big{)}+c_{\rm
II}\Big{(}\frac{M}{R}\Big{)}^{2}+d_{\rm
II}\Big{(}\frac{M}{R}\Big{)}^{3}+e_{\rm
II}\Big{(}\frac{M}{R}\Big{)}^{4}+k_{\rm II}\Big{(}\frac{M}{R}\Big{)}^{5}\,,$
(21)

with $a_{\rm II}=-2.795$, $b_{\rm II}=1.941\times 10^{2}$, $c_{\rm
II}=-3.834\times 10^{3}$, $d_{\rm II}=3.789\times 10^{4}$, $e_{\rm
II}=-1.601\times 10^{5}$, and $k_{\rm II}=2.531\times 10^{5}$. We can observe
that the behavior of the $f$-mode frequencies for the SSs is very different
from the NSs and QSs, especially when the compactness is larger than $\sim
0.15$, where the $f$-mode frequency from SSs is much larger than that of QSs
and NSs. It can be an important “smoking gun” signal for SSs.

For tidally deformed relativistic stars, the quadrupole tidal deformability
gives important information about the stellar structure. To characterize the
deformation of the star, one usually defines the tidal deformability via
$Q_{ij}\equiv-\lambda\mathcal{E}_{ij}$, where $\mathcal{E}_{ij}$ is the
external tidal field and $Q_{ij}$ is the induced traceless quadrupole moment
tensor of stars (Hinderer, 2008; Hinderer et al., 2010). The parameter
$\lambda$ is related to the $l=2$ Love number $k_{2}$ via $k_{2}=3\lambda
R^{-5}/2$. Besides, the dimensionless tidal deformability $\Lambda$, defined
as $\Lambda={2}k_{2}C^{-5}/{3}$, is also commonly used. We note that the tidal
deformability is proportional to the fifth power of the radius $R$. Therefore,
constraining or measuring tidal deformability can provide important
information on the EOS (Abbott et al., 2018, 2019a), as well as test gravity
theories (Hu et al., 2021; Xu et al., 2022). The influence of tidal on the
phase of GW in the inspiral stage is predominantly dependent on the Love
number $k_{2}$, and the effect enters at the fifth PN order (Flanagan &
Hinderer, 2008).

Figure 9: The frequency of the $f$-mode as a function of the dimensionless
tidal deformability $\Lambda$ for NSs, QSs and SSs.

In Fig. 9, we display the relation between the frequency of the $f$-mode and
the dimensionless tidal deformability, $\Lambda$, for NSs, QSs, and SSs. It is
seen that the frequency decreases with $\Lambda$. It is understood that the
more compact the star becomes the harder it can be deformed. For SSs, as the
potential depth $\epsilon$ increases and the surface baryonic density $n_{\rm
s}$ decreases, the EOS becomes stiffer, which leads to larger tidal
deformability and $f$-mode frequency.

For binary NSs of masses $M_{a}$ and $M_{b}$, the dimensionless tidal coupling
constant is defined as (Bernuzzi et al., 2014; Bernuzzi et al., 2015a, b),

$\kappa_{2}^{t}=2\left[q\left(\frac{X_{a}}{C_{a}}\right)^{5}k_{2}^{a}+\frac{1}{q}\left(\frac{X_{b}}{C_{b}}\right)^{5}k_{2}^{b}\right]\,,$
(22)

where $q=M_{b}/M_{a}\leq 1$, $X_{a}=M_{a}/(M_{a}+M_{b})$, and $C_{i}$ and
$k^{i}_{2}$ ($i=a,b$) are the compactness and the quadrupole Love number of
each star. If we consider a binary system with non-rotating equal-mass
configuration, the dimensionless tidal coupling constant is given by
$\kappa_{2}^{t}={k_{2}}/{8C^{5}}={3}\Lambda/16$.

Figure 10: Scaled frequency of the $f$-mode $M\nu_{0}$ as a function of the
tidal quadrupolar ($l=2$) coupling constant $\kappa_{2}^{t}$ for NSs, QSs and
SSs. The solid line represents the best power law fit in $\kappa_{2}^{t}$ to
the scaled frequencies of the NSs, QSs and SSs.

Inspired by the universal relation between the dimensionless tidal coupling
constant and the $f$-mode frequency (Chakravarti & Andersson, 2020), the
relation between $M\nu_{0}$ and $\kappa_{2}^{t}$ for NSs, QSs, and SSs are
shown in Fig. 10. For NSs and QSs, we find the scaled frequency of the
$f$-mode approximately satisfies the following relation,

$M\nu_{0}=0.184(\kappa_{2}^{t})^{-0.016}-0.154\,.$ (23)

For SSs, the universal relation is

$M\nu_{0}=0.071(\kappa_{2}^{t})^{-0.266}\,.$ (24)

The universal relations for QSs and SSs will complement that of NSs, and play
a role in GW data analysis (Dietrich et al., 2017).

##  5 Conclusions

In this paper, we use the Lennard-Jones model to describe the EOS of SSs with
two parameters, the number density at the surface of the star $n_{s}$ and the
potential depth $\epsilon$. Compared to the MIT bag model of QSs, the EOS of
SSs is much stiff due to the nonrelativistic nature of the particles and the
compressed repulsive hardcore at a small intercluster distance. Following
earlier work (Lai & Xu, 2009; Gao et al., 2022), we calculate the mass and
radius relation for SSs for different values of $n_{s}$ and $\epsilon$, and
find that the maximal mass of SSs is higher than that of NSs and QSs. This
serves as background solutions for perturbation studies of various oscillation
modes.

To study radial oscillations of SSs, for the first time we calculate the
frequency of the radial modes for SSs with different combinations of $n_{s}$
and $\epsilon$. The results are compared with that of NSs and QSs. We discover
that radial oscillations of SSs are similar to those of QSs but behave very
differently from those of NSs, especially for stars with low central energy
densities or small masses. For QSs and SSs, the frequencies of radial
oscillations tend to infinity when the central energy density approaches the
minimal value $\rho_{\rm min}$, which corresponds to the pressure being zero.
This can be understood by approximating the stars in the nonrelativistic
regime and noticing that the adiabatic index $\Gamma$ for SSs and QSs goes to
infinity as the density decreases to its minimal value.

For nonradial oscillations of SSs, we calculate the frequency of the $f$-mode
for $l=2$ component using the Cowling approximation, and obtain the universal
relations between the $f$-mode frequency and other global parameters of the
spherical SSs. As recently proposed in Gao et al. (2022), where the I-Love-Q
universal relations for SSs were studied, the universal relation of the
$f$-mode frequency for SSs is also ready to be used for various purposes in GW
astrophysics involving compact stars. With application to data in the future,
possible constraints can be set on the parameter space of the Lennard-Jones
model, namely the $n_{s}$-$\epsilon$ plane, using GW observations of the QNMs
from compact stars.

There can be several interesting extensions of our work. First, our study of
nonradial oscillations uses the Cowling approximation (Cowling, 1941), which
considers only the fluid perturbation. In principle, one should also allow the
spacetime metric to be perturbed, and thus one can obtain QNMs instead of
normal modes. Next, we would like to further investigate how dynamical tides
affect the frequency of the $f$-mode in compact binary systems. NSs have
certain spins and the rotation rate may reach extreme values, especially for
nascent or remnant objects following a binary merger. From the perspective of
detecting oscillation modes with GWs, the most relevant scenarios are likely
to involve rapidly rotating NSs. An important step in this direction has been
carried out using perturbation theory in general relativity with the Cowling
approximation (Krüger et al., 2010; Gaertig & Kokkotas, 2011; Doneva et al.,
2013). In the next step, we can study the oscillation modes of rapidly
rotating SSs in the Cowling approximation based on existing work. Recently,
Krüger & Kokkotas (2020b, a) managed to calculate the oscillations and
instabilities of relativistic stars using perturbation theory without the
Cowling approximation. The oscillation spectrum, universal relations involving
$f$-mode, and the critical values for the onset of the secular Chandrasekhar-
Friedman-Schutz instability are studied in great detail. Further, Manoharan et
al. (2021) investigated universal relations for binary NS mergers with long-
lived remnants. By considering the oscillations of the rapidly rotating merger
remnant, they proposed an approach to relate the pre-merger tidal
deformability to the effective compactness of the post-merger remnant. Those
studies are important to probe the EOS of NSs with GW asteroseismology.
Therefore, to study the oscillations of rapidly rotating SSs without the
Cowling approximation is an important goal worth pursuing.

## Acknowledgements

We thank Christian Krüger for useful comments. This work was supported by the
National SKA Program of China (2020SKA0120300, 2020SKA0120100), the National
Natural Science Foundation of China (11975027, 11991053, 11721303), the
National Key R&D Program of China (2017YFA0402602), the Max Planck Partner
Group Program funded by the Max Planck Society, and the High-Performance
Computing Platform of Peking University. Rui Xu is supported by the Boya
Postdoctoral Fellowship at Peking University.

## Data Availability

The data underlying this paper will be shared on reasonable request to the
corresponding authors.

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