# Rescaling Strangeon Stars and its Implications on Gravitational-wave Echoes

Chen Zhang<EMAIL_ADDRESS>The HKUST Jockey Club Institute for Advanced
Study, The Hong Kong University of Science and Technology, Hong Kong, P.R.
China  Yong Gao<EMAIL_ADDRESS>Department of Astronomy, School of
Physics, Peking University, Beijing 100871, China  Kavli Institute for
Astronomy and Astrophysics, Peking University, Beijing 100871, China  Cheng-
Jun Xia<EMAIL_ADDRESS>Center for Gravitation and Cosmology, College of
Physical Science and Technology, Yangzhou University, Yangzhou 225009, China
Renxin Xu<EMAIL_ADDRESS>Department of Astronomy, School of Physics,
Peking University, Beijing 100871, China  Kavli Institute for Astronomy and
Astrophysics, Peking University, Beijing 100871, China

###### Abstract

Solid states of strange-cluster matter called strangeon matter can form
strangeon stars that are highly compact. We show that strangeon matter and
strangeon stars can be recast into dimensionless forms by a simple
reparametrization and rescaling, through which we manage to maximally reduce
the number of degrees of freedom. With this dimensionless scheme, we find that
strangeon stars are generally compact enough to feature a photon sphere that
is essential to foster gravitational-wave (GW) echoes. Rescaling the dimension
back, we illustrate its implications on the expanded dimensional parameter
space, and calculate the GW echo frequencies associated with strangeon stars,
showing that the minimum echo frequency is $\sim 8$ kHz for empirical
parameter space that satisfies the GW170817 constraint, and can reduce to
$\mathcal{O}(100)$ Hertz at the extended limit.

##  I Introduction

Recent gravitational wave (GW) observations from compact binary mergers by
LIGO/Virgo collaborations LIGOScientific:2016aoc ; LIGOScientific:2017bnn ;
LIGOScientific:2018mvr ; TheLIGOScientific:2017qsa ; Abbott:2018wiz ;
Abbott:2020uma ; Abbott:2020khf  have significantly advanced our understanding
of black holes and compact stars. These binary merger events have inspired
many studies on exotic compact objects (ECOs), which are black hole mimickers
that share a similar compactness but lack an event horizon Holdom:2016nek ;
Holdom:2019ouz ; Ren:2019afg ; Holdom:2022fsm ; Mazur:2001fv ; Schunck:2003kk
; Cardoso:2019rvt . Most interests on probes of ECOs are focused on the
distinctive signatures from gravitational wave echoes in the postmerger
signals Ignatev:1978ax ; Abedi:2018npz ; Ferrari:2000sr ; Cardoso:2016rao ;
Cardoso:2016oxy ; Cardoso:2017njb ; Abedi:2016hgu ; Mark:2017dnq ;
Conklin:2017lwb ; Holdom:2020onl ; Conklin:2019fcs ; Conklin:2019smy ;
Holdom:2019bdv ; Abedi:2020sgg ; Dey:2020pth ; Dong:2020odp ; Wang:2018mlp ;
Konoplya:2022zym  111For probes of ECOs using other methods, see Ref.
Holdom:2020uhf ; Asali:2020wup ; Fransen:2020prl ; Mukherjee:2022wws ., in
which a wave that falls inside the gravitational potential barrier (near the
photon sphere) travels to a reflecting boundary of the ECO before returning to
the barrier after some time delay.

We want to explore the possibility of GW echoes in the context of realistic
compact stars, given the detected binary neutron star merger events. To
generate GW echoes, a star object must feature a photon sphere at $R_{P}=3M$,
where $M$ is the object’s mass. The minimum radius for compact stars should be
above the Buchdahl’s limit $R_{B}=9/4M$ Buchdahl:1959zz . Therefore, GW echo
signals are possible if $R_{B}<R<R_{P}$. This compactness criterion excludes
the realistic neutron stars Chandrasekhar ; Pani:2018flj . To achieve ultra-
compact stellar structure, people commonly assumed ad-hoc exotic equations of
state (EOS) Pani:2018flj ; Mannarelli:2018pjb ; Bora:2020cly ; Alho:2021sli ;
Urbano:2018nrs  or modified gravity Volkmer:2021zjx ; Bora:2022qwe ;
Bora:2022dnu .

Quark matter, a state comprised of deconfined free-flowing quarks, can
possibly exist inside the neutron star core (i.e. hybrid stars Alford:2004pf ;
Alford:2013aca ) or the crust (i.e. inverted hybrid stars Ren:2022loa ), or
constitutes entire star called quark star. Strange quark stars Haensel:1986qb
; Alcock:1986hz ; Zhou:2017pha ; Weissenborn:2011qu ; Xu:1999bw ; Yang:2023haz
composed of strange quark matter (SQM) Bodmer:1971we ; Witten ;
Terazawa:1979hq ; Farhi:1984qu  and up-down quark stars Zhang:2019mqb ;
Zhao:2019xqy ; Ren:2020tll ; Cao:2020zxi ; Wang:2021byk ; Yuan:2022dxb ;
Restrepo:2022wqn ; Xia:2020byy ; Xia:2022tvx  composed of up-down quark matter
($ud$QM) Holdom:2017gdc , can be more compact than neutron stars. As Ref.
Zhang:2021fla  has shown, physically-motivated quark stars can feature GW
echoes, but require perturbative QCD corrections Fraga:2001id ; Fraga:2013qra
and color superconductivity Alford:1998mk ; Rajagopal:2000ff ; Lugones:2002va
; Alford:2002kj ; Zhang:2020jmb ; Zhang:2021iah  effects to be compact enough.
It is interesting to explore whether we can have more compact objects from
other physical grounds.

Strangeon matter is similar to strange quark matter where both are composed of
a nearly equal number of $u,d,s$ quarks Xu:2003xe ; Lai:2009cn ; Lai:2017ney ;
Miao:2020cqj . However, strangeon matter has quarks localized as clusters, in
a state more like a solid. Strangeon stars Xu:2003xe ; Zhou:2004ue ;
Zhou:2014tba ; Lai:2009cn ; Lai:2017ney ; Miao:2020cqj ; Lai:2017mjv ;
Lai:2017xys ; Lai:2018ugk ; Lai:2020mlu ; Gao:2021uus  composed of strangeon
matter have intrinsic stiff EOS and large compactness, and they had already
been proposed to support massive pulsars ($>2M_{\odot}$ Lai:2009cn ) before
the announcement of the first massive pulsar PSR J1614-2230 Demorest:2010bx .
It is then natural to explore whether strangeon stars can feature GW echoes.

As for the organization of this paper, we first work out a general
dimensionless rescaling for the Lennar-Jones model of strangeon matter. This
greatly reduced the number of model parameters from three to one, enabling us
to perform a simple but general analysis over the whole parameter space. Then
we apply the rescaling scheme to the studies of strangeon stars and GW echoes.

##  II Dimensionless rescaling of Strangeon

The mass density $\rho$ and pressure density $p$ of zero-temperature dense
matter composed of strangeons derived from Lennard-Jones potential Jones(1924)
; Lai:2009cn  reads

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle
2\epsilon\left(A_{12}\sigma^{12}n^{5}-A_{6}\sigma^{6}n^{3}\right)+nN_{\rm
q}m_{\rm q}\,,$ (1) $\displaystyle p$ $\displaystyle=$ $\displaystyle
n^{2}\frac{\mathrm{d}(\rho/n)}{\mathrm{d}n}=4\epsilon\left(2A_{12}\sigma^{12}n^{5}-A_{6}\sigma^{6}n^{3}\right)\,,$
(2)

where $A_{12}=6.2$, $A_{6}=8.4$, and $n$ is the number density of strangeons.
$N_{\rm q}m_{\rm q}$ is the mass of a strangeon with $N_{\rm q}$ being the
number of quarks in a strangeon and $m_{q}$ being the average constituent
quark mass. The contributions from degenerate electrons and vibrations of the
lattice are neglected due to their expected smallness.

At the surface of strangeon stars, 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, it is
transformed into baryon number density, i.e.,

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

so that the EOS can be rewritten into a form that depends on parameter set
($\epsilon$, $n_{s}$, $N_{q}$):

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{9}\epsilon\frac{A_{6}^{2}}{A_{12}}\left(\frac{{N_{q}}^{4}}{18n_{s}^{4}}n^{5}-\frac{N_{q}^{2}}{n_{s}^{2}}n^{3}\right)+m_{q}N_{q}n,$
(4) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{2}{9}\epsilon\frac{A_{6}^{2}}{A_{12}}\left(\frac{N_{q}^{4}}{9n_{s}^{4}}n^{5}-\frac{N_{q}^{2}}{n_{s}^{2}}n^{3}\right).$
(5)

We find that one can further remove the parameters $n_{s}$ and $N_{q}$ by
doing the following dimensionless rescaling:

$\bar{\rho}=\frac{\rho}{m_{q}\,n_{s}},\,\,\bar{p}=\frac{p}{m_{q}\,n_{s}},\,\bar{n}=\frac{N_{q}\,n}{n_{s}},\,\bar{\epsilon}=\frac{\epsilon}{N_{q}\,m_{q}},\,\,$
(6)

so that

$\displaystyle\bar{\rho}$ $\displaystyle=$
$\displaystyle\frac{a}{9}\bar{\epsilon}\left(\frac{1}{18}\bar{n}^{5}-\bar{n}^{3}\right)+\bar{n},$
(7) $\displaystyle\bar{p}$ $\displaystyle=$
$\displaystyle\frac{2\,a}{9}\bar{\epsilon}\left(\frac{1}{9}\bar{n}^{5}-\bar{n}^{3}\right),$
(8)

where $a=A_{6}^{2}/A_{12}=8.4^{2}/6.2\approx 11.38$. We thus managed to reduce
the parameter degree of freedom from 3 ($n_{s}$, $\epsilon$, $N_{q}$) to
simply 1 ($\bar{\epsilon}$). Besides, we note that the rescaled number density
at zero pressure always remain $\bar{n}=3$. Requiring $\bar{\rho}$ to be
positive at zero pressure set a theoretical upper bound for $\bar{\epsilon}$:
$\bar{\epsilon}_{\rm max}^{\rm theo}=2/a\approx 0.1757$. However, the value of
this upper bound is slightly beyond the empirical expectation
$\bar{\epsilon}_{\rm max}\sim 120\text{MeV}/(3\times 310\rm MeV)\approx 0.13$.
In the following we will adopt the empirical upper bound $\bar{\epsilon}_{\rm
max}^{\rm em}=0.13$ on this physical ground, with additional comments about
results from $\bar{\epsilon}_{\rm max}^{\rm theo}$ at proper places.

##  III ReScaling Strangeon Stars

Inspecting the Tolman-Oppenheimer-Volkoff (TOV) equation Tolman:1939jz ;
Oppenheimer:1939ne ,

$\displaystyle\begin{aligned} \frac{dm}{dr}&=4\pi\rho r^{2}\,,\\\
\frac{dp}{dr}&=(\rho+p)\frac{m+4\pi pr^{3}}{2mr-r^{2}},\,\\\ \end{aligned}$
(9)

we note that the mass and radius can also be rescaled into dimensionless forms
in geometric units ($G=c=1$)222Note that $m_{q}n_{s}$, which is in units $\rm
MeV/fm^{3}$ in natural units, is in the dimension of $[L^{-2}]$ in geometric
units here.:

$\bar{m}=m{\sqrt{m_{q}\,n_{s}}},\quad\bar{r}={r}{\sqrt{m_{q}\,n_{s}}},$ (10)

so that the TOV equation can be converted into the dimensionless form (simply
replace nonbarred symbols with barred ones). Solving the dimensionless TOV
equation, we obtain the results for the rescaled $\bar{M}-\bar{R}$ relation
shown in Fig. 1. One can easily recast it into dimensional form by reversing
the rescaling relation Eq. (10). At $\bar{\epsilon}=\bar{\epsilon}_{\rm
max}^{\rm em}=0.13$, we have $(\bar{M}_{\rm TOV},\bar{R}_{\rm
TOV})\approx(0.149,\,0.348)$. Lifting $\bar{\epsilon}$ to $0.175$ that is
close to $\bar{\epsilon}_{\rm max}^{\rm theo}$, we obtain $(\bar{M}_{\rm
TOV},\bar{R}_{\rm TOV})\approx(1.28,\,2.89)$ correspondingly.

Figure 1: $\bar{M}$-$\bar{R}$ of strangeon stars for various $\bar{\epsilon}$,
sampling $0.0001\sim 0.13$ in equal $\Delta\bar{\epsilon}$ spacing from the
lighter black line to the darker black lines, respectively. Solid dots denote
the maximum mass configurations.

The detailed dependences of maximum compactness $M_{\rm TOV}/R_{\rm TOV}$ and
the maximum (rescaled) mass $\bar{M}_{\rm TOV}$ on the single parameter
$\bar{\epsilon}$ are illustrated more explicitly in Fig 2333Note that here we
have extended $\bar{\epsilon}$ to $\bar{\epsilon}_{\rm max}^{\rm theo}$ to
have a general view.. We find that all $\bar{M}-\bar{R}$ configurations are
compact enough to feature a photon sphere while not exceeding the Buchdahl’s
limit for a large range of $\bar{\epsilon}$ variations.444We examined
$\bar{\epsilon}$ as low as $10^{-8}$ order. Besides, we see clear positive
correlations . As $\bar{\epsilon}$ increases to $\bar{\epsilon}_{\rm max}^{\rm
theo}$, the maximum mass rapidly grows, while the maximum compactness reaches
the Buchdahl’s limit.

Figure 2:  Maximum compactness $C_{\rm max}=M_{\rm TOV}/R_{\rm TOV}$ (left
axis, black) and $\bar{M}_{\rm TOV}$ (right axis, red) of strangeon stars as a
function of $\bar{\epsilon}$.

The rescaled results on tidal deformabilites are shown in Fig 3. We see that
at a given mass, a larger $\bar{\epsilon}$ increases tidal deformability due
to a larger radius, as can be observed from Fig 1. Besides, we also see that a
larger $\bar{\epsilon}$ yields a smaller tidal deformability at the
corresponding maximum mass point due to the associated larger compactness.

Figure 3: $\Lambda$-$\bar{M}$ of strangeon stars for various $\bar{\epsilon}$.
The line-color convention follows that of Fig 1.

##  IV GW Echoes from Strangeon Stars

The effective potential of axial gravitational perturbations $\Psi_{s,l}$ in
curved background has the general form Conklin:2017lwb ; Urbano:2018nrs :

$\displaystyle V(r)$ $\displaystyle=$ $\displaystyle
B(r)\Big{\\{}\frac{l(l+1)}{r^{2}}+\frac{1-s^{2}}{2rA(r)}\left(\frac{B^{\prime}(r)}{B(r)}-\frac{A^{\prime}(r)}{A(r)}\right)$
(11) $\displaystyle+$ $\displaystyle 8\pi(p(r)-\rho(r))\delta_{s,2}\Big{\\}},$

where the azimuthal quantum number $l\geq s$ with $s=0,\pm 1,\pm 2$ for
scalar, vector and tensor modes, respectively. And

$B(r)=e^{2\Phi(r)}\,,\,A(r)=\frac{1}{1-2m(r)/r}$ (12)

as the metric factors of curved line element describing spherical symmetric
spacetime: $ds^{2}=-B(r)dt^{2}+A(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}$. $\Phi(r)$ is solved via

$\frac{d\Phi}{dr}=-\frac{1}{\rho+p}\frac{dp}{dr},$ (13)

together with the TOV equation Eq. (9). Apparently, we can apply rescaling
relation Eq. (10) and Eq. (6), and perform a further rescaling
$\bar{V}=V/(m_{q}\,n_{s})$ to convert the whole program into a dimensionless
form as other barred quantaties. We obtain the effective potential of the
lowest axial gravitational perturbation mode ($l=s=2$) in strangeon star
background shown in Figure 4 555Note that we normalized $\bar{V}(\bar{r})$
respect to $\bar{V}(\bar{r}=3\bar{M})$, and $\bar{r}$ respect to $\bar{M}$,
where the rescaling factors cancel and thus would yield same result for the
dimensional version., which abruptly changes at the star surface, diverges
towards star center with an outside peak around $3\bar{M}$, forming a trapping
cavity for gravitational waves. We see clearly the trend how the trapping
cavity develops and evolves as the parameter $\bar{\epsilon}$ increases.

Figure 4: Radial profiles of effective potentials for axial gravitational
perturbations of the $l=s=2$ mode in strangeon-star background at $M_{TOV}$
points for various $\bar{\epsilon}$. The color convention of black lines
follows that of Fig 1. The red line denotes the
$\bar{\epsilon}=0.175\approx\bar{\epsilon}_{\rm max}^{\rm theo}$ limit.

The characteristic echo time is the light time from the star center to the
photon sphere Cardoso:2017njb ; Cardoso:2016rao ; Cardoso:2016oxy ,

$\tau_{\text{echo}}=\int_{0}^{3M}\frac{dr}{\sqrt{e^{2\Phi(r)}\left(1-\frac{2m(r)}{r}\right)}}\,,$
(14)

We can also do the dimensionless rescaling

$\quad\bar{\tau}_{\text{echo}}={\tau}_{\text{echo}}{\sqrt{m_{q}\,n_{s}}},\,$
(15)

such that Eq. (14) can also be calculated in a dimensionless approach. After
obtaining the echo time, we directly get the GW echo frequency from the
relation Cardoso:2017njb ; Cardoso:2016rao ; Cardoso:2016oxy

$f_{\text{echo}}=\frac{\pi}{\tau_{\text{echo}}},$ (16)

and similarly, we can rescale it into the dimensionless form
$\bar{f}_{\text{echo}}$ via the relation

$\bar{f}_{\text{echo}}=\frac{{f}_{\text{echo}}}{\sqrt{m_{q}\,n_{s}}}.$ (17)

In Fig. 5, we show the results of rescaled GW echo frequencies
$\bar{f}_{\text{echo}}$ versus the rescaled center pressure $\bar{p}_{c}$ for
the stellar configurations of Fig. 1 that can generate echoes. Note that each
curve’s left and right ends are truncated at the point where
$\bar{R}=3\bar{M}$, and at the point of maximum mass, respectively.

As $\bar{\epsilon}$ and the central pressure $\bar{p}_{c}$ increases,
$\bar{f}_{\text{echo}}$ decreases (due to the increasing compactness) with a
lower bound $\bar{f}_{\text{echo}}^{\,\rm min}(\bar{\epsilon})$ set at the
$\bar{p}_{c}$ of the maximum mass point. At
$\bar{\epsilon}=\bar{\epsilon}_{\rm max}^{\rm em}=0.13$, we have the minimum
echo frequency $\bar{f}_{\rm echo}^{\rm min}=0.655$, translating to

$f_{\rm echo}^{\rm min}\approx
2\left(\frac{n_{s}}{0.24\rm\,fm^{-3}}\right)^{1/2}\rm\,kHz\quad\text{for
}\bar{\epsilon}=\bar{\epsilon}_{\rm max}^{\rm em}.$ (18)

Lifting $\bar{\epsilon}$ to $0.175$ that is close to $\bar{\epsilon}_{\rm
max}^{\rm theo}$, we obtain $\bar{f}_{\rm echo}^{\rm min}=0.023$, mapping to

$f_{\rm echo}^{\rm min}\approx
67\left(\frac{n_{s}}{0.24\rm\,fm^{-3}}\right)^{1/2}\rm\,Hz\quad\text{for
}\bar{\epsilon}\approx\bar{\epsilon}_{\rm max}^{\rm theo}.$ (19)

Figure 5: Rescaled echo frequencies $\bar{f}_{\rm echo}$ as functions of
center pressure $\bar{p}_{c}$ of strangeon stars for various $\bar{\epsilon}$.
The color convention of black lines follows that of Fig 1. The red line
denotes the $\bar{\epsilon}=0.175\approx\bar{\epsilon}_{\rm max}^{\rm theo}$
limit.

Interestingly, we see that in this extreme limit, the minimum echo frequencies
lie well within the sensitivity range of LIGO LIGOScientific:2016aoc ;
Urbano:2018nrs .

##  V Dimensional Parameter Space

In Fig 6, we show the derived quantites in dimensional forms ($f_{\rm echo}$,
$M_{\rm TOV}$) and ($\Lambda$, $C$) in dimensional parameter space of
($\epsilon$, $n_{s}$) by rescaling back previous simple dimensionless results
using relations Eq. (6) and Eq. (10).

Figure 6: Physical parameter space for (a) $N_{q}=18$ and (b)$N_{q}=9$. Black
lines denote $f_{\rm echo}/\text{kHz}$, with red lines denoting the maximum
masses $M_{\rm TOV}/M_{\odot}$, green lines for the tidal deformabilities
$\Lambda$ at $1.4\,\rm M_{\odot}$ and blue dotted lines for the maximal
compactness $M_{\rm TOV}/R_{\rm TOV}$.

Figure 6 manifests the apparent scaling behaviour:

  * •

Decreasing $\epsilon$ for given $N_{q}$ or increasing $N_{q}$ for given
$\epsilon$ is equivalent in terms of $\bar{\epsilon}$.

  * •

Dimensionless quantities like the compactness $C=M/R$ should be independent of
$n_{s}$. This explains why the blue dotted lines are flat.

  * •

Dimensional quantities follows the scaling relation dictated by Eq. (6) and
Eq. (10). For example, the maximum mass $M_{\rm TOV}$ (and corresponding
radius $R_{\rm TOV}$) scale as $\sqrt{n_{s}}$.

From Fig 6, we see explicitly that the minimum echo frequency is $\sim 8$ kHz
for parameter space of large $\epsilon$ and small $n_{s}$ while satisfies the
GW170817 tidal deformability constraint $\Lambda({1.4\,M_{\odot}})\lesssim
800$, and can be reduced to $5$ kHz if GW170817 constraint is dropped, i.e.,
if assuming the detected stars in binary merger not strangeon stars.

##  VI Summary

We worked out a first rescaling scheme that enables us to maximally reduce the
number of free parameters into a single parameter $\bar{\epsilon}$ for
strangeon matter. Utilizing this scheme, we demonstrated that strangeon stars
composed of strangeon matter generally have very large compactness with large
$\bar{\epsilon}$ in most of its parameter space. We showed that all strangeon
stars can meet the compactness condition for generating GW echoes, i.e., they
feature a photon sphere within Buchdahl’s limit. The minimum echo frequencies
are of a few kilohertz for the empirical range of $\bar{\epsilon}$, and can
reduce to $O(100)$ Hertz if $\bar{\epsilon}$ is extended to its allowed limit.
We explicitly constructed the corresponding dimensional parameter space of
$\epsilon$ and $n_{s}$ with variations of $N_{q}$ in their empirical range,
and demonstrated that $f_{\text{echo}}^{\rm min}\approx 8\,\rm kHz$ for
realisitic parameter space that satisfy astrophysical constraints like
GW170817, and can reduce to $5$ kHz if the latter constraint is dropped.

It is generally expected that including the star-rotation effect can slightly
reduce the echo frequencies Pani:2018flj ; Urbano:2018nrs . For strangeon
stars, we expect rotation would yield a similar reduction of $f_{\rm echo}$,
potentially reducing frequencies to what LIGO can detect, considering our
obtained $f_{\text{echo}}^{\rm min}\approx 5\sim 8\rm\,kHz$ for the realistic
non-rotating case is not very far from its detection limit. We leave this
interesting possibility for future studies.

###### Acknowledgements.

Acknowledgments.  C. Zhang greatly thank Prof. Renxin Xu for the visit
invitation to Peking University and is very grateful for the hospitality
during the visit. C. Zhang is supported by the Institute for Advanced Study at
The Hong Kong University of Science and Technology. C.J Xia is supported by
National Natural Science Foundation of China (Grant No. 12275234) and National
SKA Program of China (Grant No. 2020SKA0120300). R.X Xu is supported by the
National SKA Program of China (2020SKA0120100).

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