# Signatures of quark deconfinement through the $r$-modes of twin stars

P. Laskos-Patkos<EMAIL_ADDRESS>Ch.C. Moustakidis<EMAIL_ADDRESS>Department of Theoretical Physics, Aristotle University of Thessaloniki, 54124
Thessaloniki, Greece

###### Abstract

The observation and distinction of two compact stars with an identical mass
but a different radius would be a clear sign of hadron-quark phase transition
in nuclear matter. Motivated by studies searching for significant deviations
in the observables of twin stars, we investigate the differences that manifest
in their $r$-mode instability windows and spin-down evolution. Firstly, we
obtain a set of hybrid equations of state (which predict the existence of a
third stable branch of compact objects) by employing the well-known Maxwell
construction within the phenomenological framework of constant speed of sound
parametrization. Then, we systematically study the influence of certain
parameters, such as the energy density jump (in the resulting hybrid equation
of state) and the crust elasticity, on the deviations appearing in the
$r$-mode instability windows and spin-down evolution of twin stars. We
conclude that two stars with an identical mass and fairly similar spin
frequency and temperature, may behave differently with respect to $r$-modes.
Thus, the future possible detection of gravitational waves (due to unstable
$r$-modes) from a star laying in the stable region of the frequency-
temperature plane would be a strong indication for the existence of twin
stars. Furthermore, we consider current data for the spin frequencies and
temperatures of observed pulsars and compare them to the predictions made from
equations of state employed in this study. We find that, depending on the
transition density and the rigidness of the crust, hybrid equations of state
may be a viable solution for the explanation of existing data.

Neutron stars, Phase transitions, Strange quark matter, $r$-modes

##  I Introduction

Compact stars serve as excellent astrophysical laboratories for the study of
dense nuclear matter [1, 2, 3, 4, 5]. The systematic study of pulsars and the
detection of gravitational waves (GW) have already yielded significant
constraints on the nuclear equation of state (EOS) [6, 7, 8, 9, 10, 11, 12,
13, 14]. A question that still remains unanswered concerns the relevant
degrees of freedom up to densities appearing in neutron star cores [15, 16].
Compact stars could be purely hadronic, but the very dense environment
indicates the possible existence of exotic forms of matter such as deconfined
quarks. The latter opens up new scenarios that predict strange quark stars,
composed purely of strange quark matter, or hybrid stars where a quark core is
surrounded by a mantle of hadronic matter. In practice, the distinction
between neutron, strange and hybrid stars is not an easy task as their radius
around the observed mass region of 1.4 $M_{\odot}$ is rather similar.
Alternative approaches that may assist identifying the phases of nuclear
matter within compact stars include the study of their thermal evolution [17,
18], binary neutron star mergers [19, 20, 21] and phenomena related to
vibration or rotation [22, 23, 24, 25, 26, 27, 28, 29].

The construction of hybrid EOSs often requires us to describe the hadronic and
quark phases separately. Depending on the dynamics of the phase transition,
and mainly on the speed of sound structure in quark matter, a third family of
compact objects may appear in the mass-radius plane [5]. The aforementioned
family of compact stars gives rise to the existence of twin stars, i.e. stars
with an identical mass but a fairly different radius [30, 31, 32, 33, 34].
Recently, the scenario of twin stars drew a lot attention, mainly because of
the discovery of GW and thus the possibility of detecting them [36, 37, 38,
39, 35]. Note that identifying twins would be the smoking gun evidence of
hadron-quark phase transition in compact stars. In a recent study, Lyra et al.
[18] investigated the impact of compactness on the cooling of twin pairs,
finding that only stars with significantly different radius exhibit
considerable deviations in their thermal evolution. In addition, Tan et al.
[40] examined imprints that manifest in binary Love universal relations due to
the existence of a third family. Furthermore, Landry and Chakravarti [41]
argued on the possibility of distinguishing twins, through their tidal
deformabilities, with next-generation GW detectors. In the present work we
study for the first time the deviations in the $r$-mode instability windows of
twin stars and hence the differences that appear in their rotational limits.

It is well-established that relativistic stars may suffer a number of
different instabilities [42, 43]. Among them, the $r$-mode instability
(rotational mode) has been proposed as an explanation for the fact that
neutron stars do not spin up to the theoretically allowed limit known as the
Kepler frequency [44, 45, 46, 47, 43, 48, 49, 50]. The $r$-modes are
oscillations appearing in rotating stars, and their restoring force is the
Coriolis force. In principle, the $r$-mode instability can only take place if
the gravitational-radiation driving timescale is shorter compared to the
timescales of the various dissipation mechanisms that may occur in the neutron
star interior. By equating the driving and damping timescales one obtains the
so-called $r$-mode instability window, which defines a critical frequency
(maximum spin frequency for stable $r$-modes) as a function of temperature
[44].

In the past decades there has been an extensive study of the $r$-modes (and
numerous other types of oscillation) due to the possible detection of their GW
[44, 45, 46, 47, 43, 48, 49, 50, 51, 52, 53, 54, 56, 55, 57]. There are
several studies predicting that accreting stars in low mass x-ray binaries
(LMXBs) may be subject to long-lasting $r$-modes [58, 59]. In particular,
compact stars containing exotic matter, such as deconfined quarks or hyperons,
may be persistent sources of GW emission [60, 61]. In addition, some authors
[62] argue for the existence of a large unobserved population of quiescent
(postaccretion) LMXBs characterized by long-lived ($\sim$109 yr) $r$-mode
emission. Specifically, Chugunov et al. [62] suggested the existence of a new
class of neutron stars, the so-called HOFNARS (hot and fast non accreting
rotators). Such stars retain a high temperature due to heating associated with
unstable $r$-modes. Following the discovery of gravitational radiation from
binary neutron star mergers, the search for GW signals associated with
$r$-modes has started [63, 64, 65]. It is notable that, the absence of a
detection so far has provided the opportunity to set upper limits on the GW
emission and the $r$-mode saturation amplitude [64, 65].

It has been shown that, the $r$-mode instability window of purely neutron
stars is very wide to be compatible with current LMXBs data (assuming that all
observed stars are stable with respect to $r$-modes, e.g. there are no
HOFNARS) [66, 67]. Specifically, a very strong dissipation mechanism, such as
a perfectly rigid crust, is essential for the stabilization of $r$-modes.
Numerous studies have attempted to treat this problem by considering the
presence of exotic degrees of freedom in compact star cores [67, 68, 69, 70,
71, 72]. In particular, it has been shown that the bulk viscosity of hyperon
or deconfined quark matter may be sufficient to stabilize $r$-modes for the
frequencies and temperatures of the observed pulsars [67, 68]. However, it is
important to comment that hyperons are expected to appear in densities of
$2-3n_{0}$ (where $n_{0}=$ 0.16 fm-3, is the nuclear saturation density).
Thus, the fraction of the core where hyperons are present, and hence the
effective damping due to their viscosity, is limited in low mass neutron
stars. Similarly, the width of the $r$-mode instability window of hybrid stars
is determined mainly by the amount of quark matter in their core [26, 73].
Subsequently, the fastest rotating pulsars can only be explained if they are
massive enough [68, 69].

In Ref. [74], the authors employed a set of analytical solutions of the
Tolman-Oppenheimer-Volkov (TOV) equations in order to study the influence of
neutron star bulk properties on the $r$-modes. They found that the instability
window is quite sensitive to the radius of a star [74]. The latter leads to
the conclusion that if twin stars do exist, their instability windows would
deviate due to their radius difference. In addition, taking into account that
the relevant degrees of freedom are different in the center of the two twins,
the damping mechanisms that suppress the growth of the $r$-mode instability
(bulk and shear viscosities) are going to be different as well [22]. This
opens up a new intriguing scenario where two stars with an identical mass, and
similar rotational frequency and temperature profiles, may behave differently
with respect to $r$-modes. In particular, if we assume that a star having
angular velocity $\Omega_{i}$ and temperature $T_{i}$ is stable with respect
to the $r$-modes, then any other (same mass) star with similar temperature and
$\Omega\leq\Omega_{i}$ should be stable as well. However, this is not
necessarily the case if the two stars are twins since their instability
windows are expected to be different. Thus, the future detection of GW due to
unstable $r$-modes, from multiple sources, may allow us to identify a third
family of compact objects.

The motivation of the present study is twofold. Firstly, we wish to
systematically study the parameters (energy density gap, crust elasticity,
transition density) that affect the deviation between the $r$-mode instability
windows of twin stars. In addition, we wish to clarify how these parameters
affect the differences that appear in the spin-down evolution (due to unstable
$r$-modes) of twins. Secondly, we wish to examine if EOSs that predict a third
family of compact objects are a viable solution for the explanation of current
LMXBs data.

This paper is organized as it follows. Section II is devoted to the
presentation of the hadronic models employed in this work and the construction
of hybrid EOSs that predict twin star configurations. In Sec. III we discuss
in detail the $r$-mode instability formalism, while in Sec. IV we present a
simplified model for the spin-down of compact stars (due to unstable
$r$-modes). In Sec. V we present our results and discuss their implications.
Sec. VI contains a summary of our findings.

##  II Hadron-quark phase transition

A hybrid EOS often results from the combination of a low density hadronic
model and a high density quark EOS. The key ingredient for the construction is
the matching process between the two phases. In particular, there are two
widely employed methods in order to obtain hybrid EOSs: a) the Maxwell
construction and b) the Gibbs construction [75]. The main difference of the
aforementioned approaches is the number of charges that are globally conserved
in the system [76]. In the former case, the phase transition is abrupt (i.e.
the two phases are separate), while in the latter scenario a mixed phase is
present.

In the present work, we adopt the Maxwell construction for the description of
the phase transition. According to lattice QCD calculations, this particular
approach is the favored one in the scenario where the surface tension
$\sigma_{s}$ in the hadron-quark crossover is larger than the critical value
of $\mathrm{\sim 40\;MeV\;fm^{-2}}$ and lower than the highest allowed one of
$\mathrm{\sim 100\;MeV\;fm^{-2}}$ [77]. In this case the phase transition is
sharp, resulting in a discontinuity in the energy density. Specifically, the
energy density reads [36, 37, 38, 35]

$\mathcal{E}(P)=\begin{cases}\mathcal{E}_{\rm HADRON}(P),&P\leq P_{\rm tr}\\\
\mathcal{E}(P_{{\rm tr}})+\Delta\mathcal{E}+(c_{s}/c)^{-2}(P-P_{{\rm
tr}}),&P>P_{{\rm tr}}.\end{cases}$ (1)

where $P$ stands for the pressure, $c_{s}$ is the speed of sound and $c$ is
the speed of light. Furthermore, $P_{tr}$ and $\Delta\mathcal{E}$ denote the
transition pressure and the energy density jump, respectively. It is important
to comment that the first line of Eq. (1) refers to the hadronic phase, while
the second one to the quark model. We treat the quark phase using a
phenomenological approach known as the constant speed of sound (CSS)
parametrization [78]. More precisely, the second line of Eq. (1) can be though
as a first order Taylor expansion of the energy density around the transition
pressure. Even though such a treatment lacks a rigorous theoretical basis, it
is widely employed as it is mimics the dynamics of the phase transition and it
also allows an easy construction of EOSs predicting twin star configurations
[78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88].

Following the assumption of recent works [79, 80, 81, 82, 83, 84, 85, 86, 87,
88], the speed of sound is set equal to the speed of light in order to obtain
EOSs consistent with the 2 $M_{\odot}$ constraint. However, it is important to
comment that, according to perturbative QCD (pQCD) calculations, $c_{s}$ in
quark matter tends to $c/\sqrt{3}$ from below at large densities (conformal
limit) [89, 90, 91]. Nevertheless, the applicability of the conformal limit is
reliable for the density range beyond 40$n_{0}$, which is considered to be
well-below the central density of compact stars [92]. Therefore, the speed of
sound in strange quark matter could possibly be larger than $c/\sqrt{3}$ and
then decrease, with increasing baryon density, in order to satisfy the
constraints from pQCD. For a detailed picture concerning the speed of sound
structure of hybrid EOSs the reader is referred to Refs. [92, 93].

A first order phase transition, between hadronic and quark matter, is not
sufficient by itself for the appearance of a third family of compact objects.
In particular, the appearance of twin stars requires the existence of an
unstable region in the $M$-$R$ diagram, where the mass decreases with
increasing central pressure. The condition that needs to be satisfied in order
to obtain a third family was first studied by Seidov [94] and it is formulated
as follows

$3P_{\rm tr}+3\mathcal{E}_{1}-2\mathcal{E}_{2}<0,$ (2)

where $\mathcal{E}_{1}\equiv\mathcal{E}(P_{{\rm tr}})$ and
$\mathcal{E}_{2}\equiv\mathcal{E}(P_{{\rm tr}})+\Delta\mathcal{E}$. Thus, by
reorganizing Eq. (2) one obtains the minimum energy density jump for the
existence of twin star configurations, which is written as

$\Delta\mathcal{E}_{\rm cr}=\frac{1}{2}\mathcal{E}_{\rm tr}+\frac{3}{2}P_{\rm
tr}.$ (3)

For EOSs that predict $\Delta\mathcal{E}\geq\Delta\mathcal{E}_{\rm cr}$ two
distinct stable branches may appear on the $M$-$R$ plane.

The resulting hybrid EOSs ought to be consistent with neutron star
observations. For example, if one assumes that the $\sim$ 1.4 $M_{\odot}$
compact stars involved in GW170817 [12] or in PSR J0030+0451 [95] are purely
hadronic, then the low density sector of the EOS has to satisfy tight
constraints ($\Lambda_{1.4}=190^{+390}_{-120}$ and $R_{1.4}\leq 14$ km, where
$\Lambda$ denotes the dimensionless tidal deformability) [41]. On the other
hand, if these compact objects are hybrid stars, the aforementioned
constraints are lifted from the hadronic part of the EOS. In the present work
we adopt the GRDF-DD2 (simply DD2 from now on for practical purposes) [96] and
the NL3 [97] EOSs for the description of the low density phase. It is worth
commenting that, both of these EOSs have been previously employed in the study
of twin stars [39, 98]. Finally, for the description of the outer crust (in
the case of the NL3 model) the well-known EOS of Baym et al. [99] is employed.

##  III $R$-mode instability formalism

Hydrodynamics and the influence of various dissipative processes define the
time evolution of the r-modes according to the law $e^{i\omega t-t/\tau}$,
where $\omega$ is the real part of the frequency, given by [51]

$\omega=-\frac{(l-1)(l+2)}{l+1}\Omega.$ (4)

In Eq. (4), $\Omega$ is the angular velocity of the unperturbed star and $l$
defines the kind of mode [51]. In the present study, we will consider the case
$l=2$. The imaginary part $1/\tau$ is related to the effects of gravitational
radiation and the various kinds of viscosity (shear, bulk, etc.) [52, 53, 54,
51]. We consider the case of small-amplitude limit where a mode is a driven,
damped harmonic oscillator and the exponential damping timescale is given by

$\displaystyle\frac{1}{\tau(\Omega,T)}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{{}_{GR}}(\Omega)}+\frac{1}{\tau_{{}_{EL}}(\Omega,T)}+\frac{1}{\tau_{{}_{BV}}(\Omega,T)}$
(5) $\displaystyle+$ $\displaystyle\frac{1}{\tau_{{}_{SV}}(\Omega,T)},$

where $\tau_{{}_{GR}}$ is the gravitational radiation timescale,
$\tau_{{}_{EL}}$ is the damping timescale due to viscous dissipation at the
boundary layer of the rigid crust and fluid core and
$\tau_{{}_{BV}}$,$\tau_{{}_{SV}}$ are the bulk and shear viscosity dissipation
timescales respectively [52, 53, 54, 51]. It is notable that there is a battle
between the gravitational radiation, which tends to drive the $r$-mode
unstable, and the various dissipation mechanisms that induce stabilization.
The critical angular velocity $\Omega_{\rm c}$ (or critical spin frequency
$f_{\rm c}=\Omega_{\rm c}/2\pi$), corresponds to the velocity at which the two
mechanisms (amplification and damping) are balanced and it is found through
the equation $1/\tau(\Omega_{c})=0$ [52, 53, 54, 51].

The contribution of gravitational radiation to the imaginary part of the
frequency is given by the following expression [53, 51]

$\displaystyle\frac{1}{\tau_{{}_{GR}}}$ $\displaystyle=$
$\displaystyle-\frac{32\pi
G\Omega^{2l+2}}{c^{2l+3}}\frac{(l-1)^{2l}}{[(2l+1)!!]^{2}}\left(\frac{l+2}{l+1}\right)^{2l+2}$
(6) $\displaystyle\times$
$\displaystyle\int_{0}^{R}\rho(r)r^{2l+2}dr\quad\left({\rm s}^{-1}\right),$

where $\rho(r)$ is the mass density profile of a star.

The bulk viscosity $\xi_{{}_{BV}}$ is the dominant damping mechanism at high
temperatures [51]. It originates from the variations of pressure and density
due to the pulsation modes and in nucleonic matter it is given by the formula
[51]

$\displaystyle\xi_{{}_{BV}}^{H}$ $\displaystyle=$ $\displaystyle 6.0\times
10^{-59}\left(\frac{l+1}{2}\right)^{2}\left(\frac{{\rm
Hz}}{\Omega}\right)^{2}$ (7) $\displaystyle\times$
$\displaystyle\left(\frac{\rho}{{\rm gr\
cm^{-3}}}\right)^{2}\left(\frac{T}{{\rm K}}\right)^{6}\quad({\rm gr\ cm^{-1}\
s^{-1}}).$

For quark matter, the bulk viscosity is mainly determined by the weak process
$d+s\leftrightarrow{}u+s$ [22]. Following the discussion of Refs. [22, 26], we
will use an approximate expression which is appropriate for small oscillations
of the fluid and when $2\pi T\gg\delta\mu=\mu_{s}-\mu_{d}$. Specifically,

$\xi_{{}_{BV}}^{Q}=\frac{\alpha T^{2}}{(\kappa\Omega)^{2}+\beta
T^{4}}\quad{(\rm g\ cm^{-1}\ s^{-1}),}$ (8)

where

$\alpha T^{2}=6.66\times 10^{20}\left(\frac{\mu_{d}}{\rm
MeV}\right)^{3}\left(\frac{m_{s}}{\rm MeV}\right)^{4}T_{9}^{2}\quad{(\rm g\
cm^{-1}\ s^{-3}),}$ $\beta T^{4}=3.57\times 10^{-8}\left(\frac{\mu_{d}}{\rm
MeV}\right)^{6}\left(1+\frac{m_{s}^{2}}{4\mu_{d}^{2}}\right)^{2}T_{9}^{4}\quad{(\rm
s^{-2}),}$

where $T_{9}=T/(10^{9}{\rm K})$, $\mu_{d}$ and $\mu_{s}$ are the chemical
potential of the down and strange quarks respectively, $m_{s}=$ 100 MeV is the
mass of the strange quark and $\kappa=$ 2/3. Since our model for quark matter
does not provide information about the chemical potential profiles we will
rely on the approximate expression $\mu_{d}=235$ MeV $(\rho/\rho_{0})^{1/3}$
[22], which has been employed in numerous $r$-mode studies [22, 24, 60, 71,
52]. In the previous formula $\rho_{0}$ denotes the nuclear density and it is
equal to $2.8\times 10^{14}$ g cm-3. Finally, the bulk viscosity timescale is
given by [51, 56]

$\displaystyle\frac{1}{\tau_{{}_{BV}}}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{690}\left(\frac{\Omega}{\Omega_{0}}\right)^{4}R^{2l-2}\left(\int_{0}^{R}\rho(r)r^{2l+2}dr\right)^{-1}$
(9) $\displaystyle\times$
$\displaystyle\int_{0}^{R}\xi_{{}_{BV}}\left(\frac{r}{R}\right)^{6}\left[1+0.86\left(\frac{r}{R}\right)^{2}\right]r^{2}dr,$

where $\Omega_{0}=\sqrt{\pi G\overline{\rho}}$ and $\overline{\rho}=3M/4\pi
R^{3}$ is the mean density of the star.

The shear viscosity $\eta_{{}_{SV}}$ is the dominant mechanism at low
temperature. This mechanism is due to the momentum transport when particle-
particle scattering processes take place. In particular, the viscosities
associated with the neutron-neutron and electron-electron scattering are given
respectively by [53]

$\eta_{nn}=347\left(\frac{\rho}{{\rm gr\
cm^{-3}}}\right)^{9/4}\left(\frac{T}{{\rm K}}\right)^{-2}\quad({\rm g\
cm^{-1}\ s^{-1}}),$ (10) $\eta_{ee}=6.0\cdot 10^{6}\left(\frac{\rho}{{\rm gr\
cm^{-3}}}\right)^{2}\left(\frac{T}{{\rm K}}\right)^{-2}\quad({\rm g\ cm^{-1}\
s^{-1}}).$ (11)

For quark matter the shear viscosity is dominated by quark-quark scattering in
QCD. Following Ref. [26] we have

$\eta_{q}=5\times
10^{15}\left(\frac{0.1}{\alpha_{s}}\right)^{3/2}\left(\frac{\rho}{\rho_{0}}\right)^{14/9}T_{9}^{-5/3}\quad{(\rm
g\ cm^{-1}\ s^{-1})},$ (12)

where $a_{s}$ is the coupling constant for the strong interaction. In the
present work we will use a typical value of $a_{s}=$ 0.1. The dissipation
timescale due to the shear viscosity is given by [51]

$\displaystyle\frac{1}{\tau_{{}_{SV}}}$ $\displaystyle=$
$\displaystyle(l-1)(2l+1)\left(\int_{0}^{R}\rho(r)r^{2l+2}dr\right)^{-1}$ (13)
$\displaystyle\times$
$\displaystyle\int_{0}^{R}\eta_{{}_{SV}}r^{2l}dr,\quad({\rm s^{-1}}).$

In the special case where the dissipation effect due to the crust has been
included, the corresponding timescale is given by [53]

$\displaystyle\tau_{{}_{EL}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\Omega}\frac{2^{l+3/2}(l+1)!}{l(2l+1)!!{\cal
C}_{l}}\sqrt{\frac{2\Omega R_{c}^{2}\rho_{cr}}{\eta_{cr}}}$ (14)
$\displaystyle\times$
$\displaystyle\int_{0}^{R_{c}}\frac{\rho(r)}{\rho_{cr}}\left(\frac{r}{R_{c}}\right)^{2l+2}\frac{dr}{R_{c}}\quad({\rm
s}).$

In Eq. (14), $R_{c}$ is the core’s radius, while $\eta_{cr}$ and $\rho_{cr}$
are the viscosity and density of the fluid at the crust-core interface
respectively. The factor ${\cal C}_{l}$, for $l=2$, takes the value ${\cal
C}_{2}=0.080411$. The expression of Eq. (14) refers to the case where the
crust is rigid and consequently static in the rotating frame. However, in a
more realistic scenario, the motion of the crust (due to the mechanical
coupling with the core) induces an increase of the timescale $\tau_{{}_{EL}}$
by a factor of $1/{\cal S}^{2}$, where ${\cal S}$ is the slippage factor
defined as ${\cal S}=\Delta v/v$ [100]. In particular, $v$ denotes the
velocity of the core and $\Delta v$ is the difference between the velocities
in the inner edge of the crust and the outer edge of the core [100].

Even though the $r$-mode instability manifests in rotating objects, the
presented formalism treats stars as if they were spherically symmetric. In
principle, the rotational effects on a compact star’s shape are of
$\sim\Omega^{2}$ and they are typically ignored in most calculations of
inertial modes (as in the present study). At the moment, there are not very
detailed calculations, but in the few cases where the $r$-mode spectrum was
studied in fast rotation there were no reports for any significant effect
[101, 102, 103, 104]. More specifically, earlier non-linear [101] and linear
simulations [102] for fast rotating stars did not report considerable
variations. However, it should be pointed out that, these calculations were
performed in the Cowling approximation [105]. More recent simulations, based
on a fully general relativistic code [103, 104], did not demonstrate
significant influence in the spectrum from the presence of higher order
$\Omega$ terms both in the the equilibrium configuration and in the
perturbation equations [106].

It has been shown that the critical frequency $\Omega_{\rm c}$ is quite
sensitive to the radius of a star [74]. More precisely, it has been found that
for relatively low and high values of temperature, $\Omega_{\rm c}$ scales
with the radius as $\Omega_{\rm c}\sim R^{-3/2}$ and $\Omega_{\rm c}\sim
R^{-3/4}$, respectively [74]. The latter leads to the conclusion that, the
$r$-mode instability windows (the areas above $\Omega_{c}(T)$ curves) of twin
stars are going to be different. Furthermore, if one takes into account that
the damping mechanisms in quark matter are, in principal, stronger than those
in hadronic matter (for moderate temperature values [26]), then the
instability window of a hybrid star is expected to be shifted to larger
$\Omega_{\rm c}$ compared to the one of its hadronic twin. Thus, there are two
mechanisms which act additively and may drastically affect the instability
windows of the two different branches. The above findings are essentially a
strong motivation for investigating the possible identification of twin stars
due to the implications of their different instability windows.

Another crucial issue is the limitation of the instability window, at high
frequencies, from the corresponding Kepler angular velocity $\Omega_{\rm K}$
(the maximum rotation frequency of the star). Following Ref. [51], the latter
quantity is approximately given by $\Omega_{\rm K}=\frac{2}{3}\Omega_{0}$. It
is worth mentioning that, several studies, which investigated possible
universal relations for the Kepler frequency, have verified the existence of a
correlation between the angular velocity of a maximally rotating neutron star
and the mean density of the corresponding (equal mass) spherical configuration
$\overline{\rho}$ [107, 108, 109, 110]. It is interesting that in the case of
twin stars the Kepler frequency of the hybrid branch may be even 20 $\%$
higher compared to the hadronic branch, due to the different radius values.
This apparent differentiation at the upper limit of the instability window can
by itself be a criterion for separating the two branches. Connecting the
analysis presented above with the fact that newly born compact stars may
rotate close to their mass shedding limit [111], we conclude that the spin-
down evolution paths of twin stars are going to exhibit distinct deviations.

##  IV Spin-down and cooling

We are now going to present a simplified model to describe the spin-down (due
to unstable $r$-modes) of a hadronic or hybrid star simultaneously with its
cooling. During the phase that the angular momentum is radiated away to
infinity by gravitational waves, the angular velocity of a star evolves as
follows [54]

$\frac{d\Omega}{dt}=\frac{2\Omega}{\tau_{GR}}\frac{\alpha^{2}Q}{1-\alpha^{2}Q},$
(15)

where $\alpha$ is the dimensionless $r$-mode amplitude parameter. This
parameter strongly affects the $r$-mode evolution and usually takes values in
the large interval $\alpha=1-10^{-8}$. Moreover, $\alpha$ in general depends
both on the viscosity (and consequently on the temperature $T$ and cooling
process) and on time. However, following Refs. [54, 112] we consider that
$d\alpha/dt=0$. In addition, the quantity $Q$ is related to the bulk
properties of a star and it is defined as $Q=3\tilde{J}/2\tilde{I}$ where

$\tilde{J}=\frac{1}{MR^{4}}\int_{0}^{R}\rho(r)r^{6}dr,\qquad\tilde{I}=\frac{8\pi}{2MR^{2}}\int_{0}^{R}\rho(r)r^{4}dr.$
(16)

Under the aforementioned assumptions, on can solve Eq. (15) analytically and
obtain [55, 112]

$\Omega(t)=\left(\frac{1}{\Omega^{-6}_{in}-6{\cal D}t}\right)^{1/6},$ (17)

where

${\cal
D}=\frac{2\alpha^{2}Q}{\tilde{\tau}_{GR}(1-\alpha^{2}Q)}\frac{1}{\Omega_{0}^{6}},\quad\tilde{\tau}_{GR}=\left(\frac{\Omega}{\Omega_{0}}\right)^{6}\tau_{GR}.$
(18)

$\Omega_{in}$ is a free parameter that corresponds to the initial angular
velocity and $\tilde{\tau}_{GR}$ is the fiducial gravitational-radiation
timescale.

In order to combine the concurrent processes of spin-down and thermal
evolution of twin stars, we adopt, as a first approximation, the standard
cooling scenario [113]. Standard cooling is the simplest thermal evolution
model, since it only accounts for the energy loss due to neutrino emission via
the modified Urca process [113]. Such an approach has been previously employed
by Owen et al. [54] and the temperature drops according to the law

$T(t)=\left(\frac{t}{t_{c}}+\left(\frac{10^{9}\ {\rm
K}}{T_{i}}\right)^{6}\right)^{-1/6}10^{9}\ {\rm K},$ (19)

Figure 1: Mass-radius diagrams for the DD2 (left panel) and NL3 (right panel)
EOSs. The black solid curves indicate the original EOSs. The solid (dashed)
horizontal line is set to 1.4 (1.2) $M_{\odot}$. The shaded areas correspond
to the constraints from the analysis of the GW170817 event [12, 13]. Each
hybrid EOS is identified from the baryon density $n_{tr}$ where the phase
transition occurs and by the energy density jump. The energy density gap is
given in units of MeV fm-3 which are omitted in the legend for simplicity.

where $T_{i}$ is the initial temperature of the star (a typical value is
$T_{i}\simeq 10^{11}\ {\rm K}$) and $t_{c}$ is the cooling rate parameter
($t_{c}\simeq 1\ {\rm year}$ [54]). We need to highlight that, the present
model is more suited for the case of young isolated neutron stars, as only
neutrino and no photon emission is considered, and it may fail to provide an
accurate description for the cooling of LMXBs. In particular, the thermal
evolution of LMXBs is a far more complicated problem, as there may be
additional mechanisms in action, such as heating due to accretion. For more
details concerning the use of the present model in the $r$-mode evolution, the
reader is referred to the discussion of Ref. [54].

One may argue that the two twins may cool down in a different way, considering
the different cooling mechanisms in hadronic and quark matter. Obviously, a
more elaborate study is necessary if one is interested in an accurate
quantitative description of the cooling process. However, according to the
findings of Lyra et al. [18], the thermal evolution of twin stars is only
distinct when there is a large difference in their compactness. More
precisely, in the case where there is a 10 $\%$ compactness difference (which
is the case for the configurations constructed in the present study), the
thermal evolution of the two twins is nearly identical [18]. From that
perspective, we expect that the selected model will allow a qualitative
comparison for the evolution of twin stars on the $f-T^{\infty}$ plane.

##  V Results and discussion

###  V.1 Mass-radius diagrams

In order to study the differences that manifest in the $r$-mode instability
windows and spin evolution of twin stars we constructed a set of hybrid EOSs,
using the analysis presented in Sec. II. In particular, the low density phase
is described by the DD2 and NL3 EOSs, where for the quark matter a
phenomenological constant speed of sound model is employed. The values of the
energy jump are selected in order to obtain EOSs that are consistent with the
constraints from astrophysical observations. For both hadronic models, the
resulting EOSs predict twin stars with mass of 1.2 or 1.4 $M_{\odot}$.

Figure 1 depicts the mass-radius dependence for the EOSs employed in this
study. In the left panel the hadronic phase is described using the DD2 EOS,
while for the results of the right panel the NL3 model was employed. The solid
black curves stand for the case where no phase transition occurs (i.e. the
$M$-$R$ diagrams for the purely hadronic EOSs). In addition, the shaded areas
correspond to constraints based on the analysis of the GW170817 event [12,
13]. Finally, the horizontal lines are drawn to indicate the twin
configurations with 1.2 and 1.4 $M_{\odot}$.

As it is evident from Fig. 1, increasing the energy density jump results into
a softening of the EOS. Thus, the largest values for $\Delta\mathcal{E}$ are
selected so that EOSs remain consistent with the 2 $M_{\odot}$ constraint.
Furthermore, we need to highlight that as $\Delta\mathcal{E}$ increases the
radius difference of the two twins becomes larger. The latter is expected to
play a critical role concerning the deviation of the $r$-mode instability
windows (see Sec. III) [74]. It is important to note that our analysis does
not include the limiting case where
$\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}$, as in such a scenario the
separation of the two twins is almost negligible. In particular, if the phase
transition occurs in relatively low baryon density a third family may not even
appear [39].

###  V.2 Qualitative analysis

Figure 2 presents the $r$-mode instability windows of 1.4 $M_{\odot}$ twin
stars for the case where $\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$100 MeV
fm-3. The damping mechanism due to the presence of a solid crust is not
included. The results for the hadronic and hybrid stars are indicated using
dashed and solid curves, respectively. Additionally, the horizontal lines
stand for the corresponding Kepler frequencies. Note that in the x axis of the
plot, one would not find the temperature $T$, appearing in the formalism of
Sec. III, but the so-called redshifted temperature which is given by
$T^{\infty}=T\sqrt{1-2C},$ where $C=GM/Rc^{2}$ is the compactness of a star.

Figure 2: Critical spin frequency $f_{c}$ as a function of the redshifted
temperature $T^{\infty}$ ($r$-mode instability windows) for 1.4 $M_{\odot}$
twin stars for the DD2 (blue) and NL3 (red) EOSs. The dashed lines and solid
lines correspond to the hadronic and hybrid twins, respectively. The
horizontal lines denoted the Kepler frequency for its star. The value for the
energy density gap is $\Delta\mathcal{E}_{cr}+$100 MeV fm-3 for both EOSs.

Firstly, we need to underline the sensitivity of the instability window to the
employed EOS. Specifically, by comparing the instability windows of the purely
hadronic configurations one finds that the predicted critical frequency is
lower (in the low temperature region) when the NL3 EOS is employed. This
results from the fact that the radius of a 1.4 $M_{\odot}$ compact star is
smaller when the DD2 model is used [74]. Incidentally, the radius of the
hybrid star constructed using the NL3 EOS coincides with the radius of the
hadronic configuration using the DD2 EOS. The latter results into an overlap
of their instability windows in the low temperature regime. However, the
existence of a quark core, in the hybrid star, leads to significant
differences in the critical frequencies for $T^{\infty}\geq$ 108 K, where the
bulk viscosity plays a crucial role [26].

For a qualitative comparison of the $r$-mode instability windows of twin pairs
one can divide Fig. 2 into three representative regions. In particular, for
$T^{\infty}\leq$ 108 K (where the shear viscosity is the dominant dissipation
mechanism [26]), the radius difference plays a crucial role for the apparent
critical frequency deviations. For 108 K $\leq T^{\infty}\leq$ 1010 K, the
bulk viscosity (of quark matter [26]) is the major damping mechanism and the
trend of the $f_{c}(T^{\infty})$ curve is altered for the hybrid twin. More
precisely, the critical frequency increases and then decreases with
temperature leading to a local maxima. This topological difference derives
from the fact that the bulk viscosity of quark matter is not a monotononic
function of temperature. Finally, for $T^{\infty}\geq$ 1010 K the bulk
viscosity of hadronic matter dominates in both twins and their instability
windows essentially coincide. From an observational perspective, the
differences that appear in the low temperature regime will lead to different
limits on the spin up of accreting pulsars in LMXBs, depending on whether they
are hybrid or purely hadronic. Apart from the deviations appearing in the
critical spin frequencies of twin stars, we need to comment that there is a
$\sim$ 17 $\%$ difference in their Kepler velocities as well. Subsequently, a
young hybrid star can rotate much faster than its hadronic twin.

###  V.3 Energy density jump and crust effects

At this point we wish to systematically study the influence of certain
parameters on the instability window deviations of twin stars. In particular,
we are going to vary the value of the energy density gap and examine its
effects. Furthermore, up to this point, the only dissipative mechanisms
considered in our calculations were the bulk and shear viscosities. Now, we
are also going to include the damping mechanism due to the presence of a
viscous boundary layer. It is interesting that, as the aforementioned
mechanism is strong and common for both twins, the critical frequency
deviations due to different viscosities are expected to be less pronounced.

Firstly, we are going to investigate the importance of the energy density
jump. As we mentioned, $\Delta\mathcal{E}$ is the regulator of the radius
difference between twin configurations. Figure 3(a) depicts the dependence of
$\Delta R$ on $\Delta\mathcal{E}$ for 1.4 $M_{\odot}$ twin stars.
Surprisingly, we find that the aforementioned quantities are connected through
a linear formula. Even though the exact $\Delta R$-$\Delta\mathcal{E}$
relation is sensitive to the low density model, the slopes of the resulting
fitted lines appear to be very similar. It is worth pointing out that, from
the analysis presented in the previous sections, an increment of
$\Delta\mathcal{E}$ will result into larger deviations in the instability
windows due to an increase of $\Delta R$. However, as it is evident from Fig.
3(b), a larger value of $\Delta\mathcal{E}$ also results into a hybrid twin
with a larger quark core fraction $x_{q}=R_{q}/R$ (where $R_{q}$ is the quark
core radius). Hence, the damping due to quark matter’s bulk viscosity is going
to be even more effective. It is noteworthy that $\Delta\mathcal{E}$ and
$x_{q}$ are also linearly dependent and that the slopes of the lines are, once
again, not strongly sensitive to employed hadronic model. The relations
presented in Fig. 3 can be added to the other correlations found in the
detailed analysis of Ref. [98]. Finally, we need to underline that, through
the relations found above, the knowledge of the radius difference of twin
stars may provide important information concerning the phase transition and
the interior of hybrid stars.

Figure 4 depicts the dependence of the critical frequency on temperature, for
1.4 $M_{\odot}$ twin stars constructed using different $\Delta\mathcal{E}$
values. In addition, the crust elasticity $\mathcal{S}$ is varied from 0 to 1
in order to investigate the effects of a viscous boundary layer. Furthermore,
Fig. 4 contains observational data inferred from LMXBs and millisecond
pulsars, where the temperature uncertainties derive from different assumptions
concerning the star’s envelope composition [114]. In particular, the small
filled circles demonstrate the internal temperature when a partially accreted
envelope is considered ($T_{fid}$ in Table 1 of Ref. [114]), while the error
bars stand for the cases of a fully accreted and a pure iron envelope
($T_{acc}$ and $T_{Fe}$ columns in Table 1 of Ref. [114]).

Figure 3: Panel a: Radius difference between 1.4 $M_{\odot}$ twin stars as a function of the energy density gap, Panel b: The quark core fraction of a 1.4 $M_{\odot}$ hybrid star as a function of the energy density jump.  Table 1: The difference in the critical frequencies for 1.4 M⊙ twin stars using the NL3 model for different values of temperature and energy density jump. The damping due to a viscous boundary layer (rigid crust) is not considered for the results presented in this table. $\Delta\mathcal{E}$ (MeV fm-3) |  $\Delta R$ (km) |  $T^{\infty}$ (108 K) |  $\Delta f_{c}$ (Hz)
---|---|---|---
$\Delta\mathcal{E}_{cr}+$100 |  | 1 | 78.98
1.75 | 5 | 179.66
| 10 | 75.14
$\Delta\mathcal{E}_{cr}+$150 |  | 1 | 132.58
2.40 | 5 | 288.95
| 10 | 137.42
$\Delta\mathcal{E}_{cr}+$200 |  | 1 | 181.99
2.99 | 5 | 381.43
| 10 | 192.05
Table 2: The difference in the critical frequencies for 1.4 $M_{\odot}$ twin stars for different values of temperature and crust elasticity. The results were obtained using the NL3 model with $\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$200 MeV fm-3. The results for this EOS, in the case where the crust damping mechanism is not considered, can be found in Table 1 $S$ |  $T^{\infty}$ (108 K) |  $\Delta f_{c}$ (Hz)
---|---|---
0.2 | 1 | 102.279
5 | 199.014
10 | 103.426
1 | 1 | 165.56
5 | 169.475
10 | 129.472
Figure 4: The effect of energy density gap in the deviation of the $r$-mode
instability windows of 1.4 $M_{\odot}$ twin stars for increasing crust
elasticity values. Panel (a) DD2 EOS and
$\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$75 MeV fm-3, (b) DD2 EOS and
$\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$100 MeV fm-3, (c) NL3 EOS and
$\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$100 MeV fm-3, (d) NL3 EOS and
$\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$100 MeV fm-3. Dashed (solid) lines
indicate the hadronic (hybrid) twin. The dotted points correspond to
observational data taken from Ref. [114]. The no crust indication in the
legend corresponds to the case where the damping due to a viscous boundary
layer has not been included.

It is worth pointing out that, the instability window differences are more
pronounced in the case where the NL3 model is employed. This results from the
fact that as the NL3 model is stiffer, it allows the construction of EOSs that
satisfy observational constraints even for large $\Delta\mathcal{E}$ values.
Furthermore, Fig. 4 illustrates the strong impact of $\Delta\mathcal{E}$ on
the resulting $r$-mode instability window of the hybrid twin. In particular,
in the case of $\Delta\mathcal{E}_{cr}$ \+ 200, the spin frequency difference
for the two twins may reach values of $\sim$ 400 Hz (see Fig. 4(d) and Table
1). In accordance to the results present by Lyra et al. [18], we conclude that
the role of the compactness is not only critical concerning the thermal
evolution of twin pairs, but it also significantly affects the $r$-mode
instability window of the hybrid configuration.

The most important effect when the damping due to a solid crust is included,
is that the peak appearing in the instability window of the hybrid star (in a
temperature region around $\sim$ 3$\times$108 K) drops down. However,
depending on the selected $\Delta\mathcal{E}$ value, large $f_{c}$ differences
for the two twins may remain (see Fig. 4 and Table 2).

###  V.4 Comparison with observational data

Figure 5: The $R$-mode instability windows of compact stars in the mass range
1.2$-$1.9 $M_{\odot}$. The EOSs used are: (a) DD2, $n_{tr}=$ 0.32 fm-3,
$\Delta\mathcal{E}_{cr}+$ 150, (b) DD2, $n_{tr}=$ 0.35 fm-3,
$\Delta\mathcal{E}_{cr}+$ 100, (c) NL3, $n_{tr}=$ 0.25 fm-3,
$\Delta\mathcal{E}_{cr}+$ 150, (d) NL3, $n_{tr}=$ 0.27 fm-3,
$\Delta\mathcal{E}_{cr}+$ 150\. The dotted points correspond to observational
data, and they are taken from Ref. [114]. The (h) appearing in the legend
stands for the most massive purely hadronic configuration. The value for the
crust elasticity is considered to be $\mathcal{S}=$ 0.1.

As previously mentioned, it is rather difficult to explain the observational
data in the context of a purely hadronic star. In particular, the not
realistic assumption of a perfectly rigid crust is essential [67]. For that
matter, several studies have investigated the $r$-mode instability window of
compact stars containing exotic forms of matter [67, 68, 69, 70, 71, 72]. In a
recent work, Ofengeim et al. [68, 69] examined if the existence of hyperons in
the core of compact stars can lead to results compatible with current LMXBs
data. What they found is that for neutron stars with $M\leq 1.9$ $M_{\odot}$,
the bulk viscosity of hyperonic matter leads to $r$-mode stabilization in the
$f-T^{\infty}$ regime where the observed neutron stars appear [68, 69].

At this point we wish to examine if the hybrid EOSs constructed in this study
are in accordance to the observed spin frequencies and temperatures in LMXBs.
As it is evident from Fig. 4, the $r$-mode instability window of the hybrid
twin is always narrower. In addition, in a minimal scenario where the effects
of the crust are not included the explanation of the observational data is not
possible. However, depending on the energy density jump, a moderate crust
elasticity value would suffice for the construction of instability windows
compatible with observations. Specifically, for the NL3 model
($\Delta\mathcal{E}_{cr}+$ 200) and a relatively small crust elasticity of
0.2, most of the observed stars lay in the stable region of the $f-T^{\infty}$
plane. Another critical point is that, in all cases there are stars (from the
dataset) that lay in the region between the $f_{c}(T^{\infty})$ curves for the
two twins. Hence, while such stars can be considered stable with respect to
$r$-modes in the framework of the hybrid twin, they would be unstable if they
were purely hadronic. The latter comment is of most importance, as the
detection of GW emission, from stars laying in a $f-T^{\infty}$ region where
$r$-modes are considered to be stable, would be a strong indication of hadron-
quark phase transition.

In Fig. 5 we present the $r$-mode instability windows for compact stars in the
mass range 1.2$-$1.9 $M_{\odot}$ and a relatively low crust elasticity value
$\mathcal{S}=$ 0.1. In panels a and c the twin configurations have a mass of
1.2 $M_{\odot}$, while in panels b and d their mass is 1.4 $M_{\odot}$. In the
first case we find that, the bulk viscosity of quark matter is sufficient to
stabilize $r$-modes for moderately massive compacts stars ($M\leq 1.6$
$M_{\odot}$) in the whole $f-T^{\infty}$ range occupied by the observed stars
in LMXBs. In latter case, where the phase transition occurs at higher baryon
density, more massive compact star configurations (1.8 or 1.9 $M_{\odot}$
depending on the hadronic EOS) are essential for the explanation of current
LMXBs data.

Figure 6: Panel: (a) Spin frequency as a function of time for 1.4 $M_{\odot}$
twin stars constructed with the DD2 EOS, (b) Spin-down rate as a function of
time for 1.4 $M_{\odot}$ twin stars constructed with the DD2 EOS, (c) Spin
frequency as a function of time for 1.4 $M_{\odot}$ twin stars constructed
with the DD2 EOS, (d) Spin-down rate as a function of time for 1.4 $M_{\odot}$
twin stars constructed with the NL3 EOS. In all panels two different values
for $\Delta\mathcal{E}$ were used (see legends).  Figure 7: The spin-down
evolution of 1.4 $M_{\odot}$ twin stars (NL3 EOS, $n_{tr}=$ 0.27 fm-3 and
$\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$ 100 MeV fm-3) in the frequency-
temperature plane for different values of the saturation amplitude. The
initial frequencies for the twins are their corresponding Kepler frequencies.
The blue (red) solid lines indicate the evolution for the hybrid (hadronic)
twin. The blue and red dashed lines denote the $r$-mode instability window of
the hybrid and hadronic star, respectively. The dotted points stand for
observational data taken from Ref. [114].

Another observation that can be made from Fig. 5 is that right after the phase
transition occurs a narrowing of the instability window is evident. Then, as
the mass further increases the instability window becomes wider for low
temperature values ($T^{\infty}\leq$ 108 K). The fact that higher mass
configurations have wider instability windows is a known result from previous
studies [115]. It is interesting that while a higher mass is necessary for the
stabilization of $r$-modes in observed stars with $T^{\infty}\geq$ 108 K, it
fails to provide an explanation for the stars appearing in a lower temperature
regime. However, the low temperature region can be covered by hybrid star
configurations of lower mass. In principle, if a star slightly surpasses a
critical mass, after which a phase transition occurs, then its instability
window will be also slightly different from the one of the most massive purely
hadronic configuration. In contrast, if the structure of the phase transition
predicts the existence of a third family, then stars with mass equal or
slightly larger than the aforementioned critical mass are going to exhibit
considerable deviations in their $r$-mode instability windows. The nontrivial
behavior of stars having narrower instability windows compared to those of
lower mass stars (for low $T^{\infty}$), is characteristic of an EOS
predicting twin configurations.

###  V.5 Spin down and thermal evolution

In Fig. 6 we display the time evolution for the frequency and the
corresponding spin-down rate of 1.4 $M_{\odot}$ twin stars. The upper and
lower panels contain results for the DD2 and NL3 EOSs, respectively. For
comparison reasons, we consider the same initial frequency of 650 Hz for both
twins. Furthermore, in accordance to previous studies [55, 115], the selected
value for the $r$-mode saturation amplitude is $a=$ 2$\times$10-7. As it is
evident from Fig. 6, the spin-down rate is slower in hybrid stars, right after
their birth. Specifically, the higher the energy density gap the lower the
rate. However, after a certain amount of time the spin-down rates of twin
stars converge to the same value. The latter is reflected on the distinct time
evolution of the frequency for the two cases. In particular, a hybrid star
retains its initial rotational frequency for a longer period of time compared
to its hadronic twin.

Usually, it is more convenient to study the spin down evolution of a compact
star on the $f-T^{\infty}$ plane [112]. The latter demands the simultaneous
knowledge of the spin and thermal evolution for a star. By employing the toy
model for the fall of temperature, presented in Sec. IV, we intend to obtain
the different evolution paths of twin stars on the $f-T^{\infty}$ plane. In
addition, instead of considering the same initial frequency for the two twins
we set as initial condition the corresponding Kepler frequencies. The results
presented in Fig. 7 were constructed using the NL3 EOS with $n_{tr}=$ 0.27
fm-3 (hence 1.4 $M_{\odot}$ twins) and
$\Delta\mathcal{E}=\Delta\mathcal{E}_{cr}+$ 100 MeV fm-3. The latter EOS
predicts twins with a $\sim$ 13 $\%$ difference in compactness, and therefore,
a similar thermal evolution is not an unreasonable assumption [18]. For the
crust elasticity a low value of $\mathcal{S}=$ 0.1 was chosen. Moreover, we
consider three different values of the amplitude $\alpha$ since the results
are very sensitive to it. From Fig. 7 it is obvious that there are three main
reasons which differentiate the time evolution of the two branches. The first
one is the different Kepler velocities. The second one is connected to the
spin down rates of the two twins, even though this effect is less pronounced.
The third one is the deviation of the instability windows. In particular, the
unstable region is more extended in the case of the hadronic branch. The
latter is of most importance, as the $r$-mode instability window essentially
sets the resulting frequency of a star as it comes out of the unstable region.
Of course we need to stress out that, the paths presented in Fig. 7 can be
improved if one considers a more realistic cooling process for the two
branches. However, the general picture will not change noticeably and the main
conclusions of the present study are not expected to be significantly altered.

##  VI Conclusion

The present work was dedicated to the study of twin stars and their $r$-mode
instability windows. In particular, we have conducted a detailed investigation
of the parameters that affect the deviation between the instability windows of
twin stars. This is of most importance as two stars with identical mass may
have different rotational limits. More precisely, two stars in the same region
of the frequency-temperature diagram may behave differently with respect to
$r$-modes. Subsequently, the future detection of ($r$-mode) GW emission, from
stars that are considered to be stable with respect to $r$-modes (due to
existing observations), would be a clear sign for the existence of a third
family and hence of hadron-quark phase transition.

Firstly, we studied the influence of the energy density jump
$\Delta\mathcal{E}$ on the deviation between the instability windows of twin
stars. We found that $\Delta\mathcal{E}$ regulates the radius difference
between twin configurations. In addition, hybrid stars predicted from EOSs
with higher $\Delta\mathcal{E}$ exhibit larger quark core fractions. Thus, the
differences in the critical spin frequencies of twins become more pronounced
as the energy density jump increases. Secondly, we took into consideration the
strong and common (for both twins) dissipation mechanism due to the presence
of a viscous boundary layer. What we found is that, the characteristic peak
appearing in the $r$-mode instability windows of hybrid stars (around
$T^{\infty}$ $\sim$ 3 $\times$ 108 K) flattens as the crust elasticity
increases. However, depending on the selected value of $\Delta\mathcal{E}$,
considerable differences in the limiting frequency of the two twins may
remain.

Furthermore, we examined if the EOSs constructed in this study (i.e. EOSs
predicting a third family of compact objects) are a viable option for the
explanation of current LMXBs data. We found that depending on the phase
transition onset (transition density) and also the masses of stars in LMXBs,
our EOSs may be compatible with the existing observational data. In
particular, for EOSs that predict twin stars with 1.2 $M_{\odot}$, the bulk
viscosity of quark matter is adequate to stabilize $r$-modes for moderately
massive stars ($M\leq 1.6~{}M_{\odot}$) in the whole $f-T^{\infty}$ region
occupied by the observed stars in LMXBs. As the critical compact star mass for
the phase transition to occur increases, more massive configurations are
needed for the stabilization of $r$-modes.

Finally, we studied the differences that manifest in the spin-down evolution
of twin pairs. We found that the hybrid star retains its initial spin
frequency for a larger period of time and this is because its spin-down rate
is lower compared to its hadronic twin. Furthermore, we noticed that larger
$\Delta\mathcal{E}$ values result into lower spin-down rates for hybrid stars.
In addition, by employing a simplified cooling model we evaluated the
evolution paths of twins stars on the $f-T^{\infty}$ plane. The resulting path
differences derive from: a) the fact that the Kepler frequencies (initial
conditions) of twin stars are different, b) the different spin evolution,
which is dependent on the bulk properties of a star, c) the different
instability windows of twin stars which essentially control when and with what
frequency a star is going to pass in the $r$-mode stable region.

There are some other issues that a more elaborate study should take into
account such as additional damping mechanisms or a more rigorous treatment of
the thermal evolution. In addition, it would be interesting to explore the
effects of a mixed phase (EOSs constructed with the Gibbs method). Even though
such a study (already in progress) would be more complete from a quantitative
point of view, we do not expect that our main conclusions will be
significantly altered. Finally, we need to highlight that even though there
are a few studies focusing on the $r$-mode instability and hybrid stars [27,
26, 70, 71, 72], this is the first work dealing with the possible existence of
two stars with identical mass and different $r$-mode instability windows. The
future detection of GW associated with unstable $r$-modes may finally allow us
to distinguish twin stars.

## Acknowledgements

The authors would like to thank Professor K. Kokkotas for his useful insight
and comments and mainly for the fruitful discussion about the effect of rapid
rotation on the $r$-mode spectrum.

## References

  * [1] P. Haensel, A.Y. Potekhin, D.G. Yakovlev, Neutron Stars 1: Equation of State and Structure (Springer-Verlag, New York, 2007).
  * [2] Ya. B. Zel’dovich and I.D. Novikov, Stars and Relativity (Dover Publications, INC, Mineapolis New York, 1971).
  * [3] S. Weinberg, Gravitational and Cosmology: Principle and Applications of the General Theory of Relativity (Wiley, New York, 1972).
  * [4] B.F. Schutz. A First Course in General Relativity (Cambridge University Press, Cambridge, 1985).
  * [5] J. Schaffner-Bielich, Compact Star Physics (Cambridge University Press, Cambridge, England, 2020).
  * [6] P. Demorest, T. Pennucci, S. Ransom, M. Roberts, and J. Hessels, Nature (London) 467, 1081 (2010).
  * [7] E. Fonceca, T. T. Pennucci et al., Astrophys. J. 832, 167 (2016).
  * [8] Z. Arzoumanian, Astrophys. J. Suppl. Ser. 235, 37 (2018).
  * [9] J. Antoniadis, P. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch et al., Science 340, 1233232 (2013).
  * [10] H. T. Cromartie, E. Fonseca, S. M. Ransom et al., Nat. Astron. 4, 72 (2020).
  * [11] M. Linares, T. Shahbaz, and J. Casares, Astrophys. J. 859, 54 (2018).
  * [12] B. P. Abbott et al., Phys. Rev. Lett. 119, 161101 (2017).
  * [13] B. P. Abbott et al., Phys. Rev. Lett. 121, 161101 (2018).
  * [14] B. P. Abbott et al., Phys. Rev. X 9, 011001 (2019).
  * [15] H. Heiselberg and M.Hjorth-Jensen, Phys. Rep. 328, 237 (2000).
  * [16] F. Weber, Prog. Part. Nucl. Phys. 54, 1, 193 (2005).
  * [17] S. M. de Carvalho, R. Negreiros, M. Orsaria, G. A. Contrera, F. Weber, and W. Spinella, Phys. Rev. C 92, 035810 (2015).
  * [18] F. Lyra, L. Moreira, R. Negreiros, R. O. Gomes and V. Dexheimer, Phys. Rev. C 107, 025806 (2023).
  * [19] A. Prakash, D. Radice, D. Logoteta, A. Perego, V. Nedora, I. Bombaci, R. Kashyap, S. Bernuzzi, and A. Endrizzi, Phys. Rev. D 104, 083029 (2021).
  * [20] A. Bauswein and S. Blacker, Eur. Phys. J. Special Topics 229, 3595–3604 (2020).
  * [21] E. R. Most, L. J. Papenfort, V. Dexheimer, M. Hanauske, H. Stoecker, L. Rezzolla, Eur. Phys. J. A 56, 59 (2020).
  * [22] J. Madsen, Phys. Rev. D 46, 3290 (1992).
  * [23] J. Madsen, Phys. Rev. Lett. 81, 3311 (1998).
  * [24] J. Madsen, Phys. Rev. Lett. 85, 10 (2000).
  * [25] W. Steiner, Prog. Theor. Phys. Suppl. 174, 48 (2008).
  * [26] P. Jaikumar, G. Rupak and A.W. Steiner, Phys. Rev. D 78 123007 (2008).
  * [27] M. G. Alford, S.Mahmoodifar, and K. Schwenzer Phys. Rev. D 85, 024007 (2012).
  * [28] I. F. Ranea-Sandoval, O. M. Guilera, M. Mariani, M. G. Orsaria, J. Cosmol. Astropart. Phys. 2018, 12, 031 (2018).
  * [29] E.L Bratton, II, Z. Lin, F. Weber, M.G. Orsaria, I.F. Ranea-Sandoval, N. Saavedra, Galaxies 10, 94, (2022).
  * [30] U.H. Gerlach, Phys. Rev. 172, 1325 (1968).
  * [31] J. Kampfer, Phys. A 14, L471 (1981).
  * [32] J. Kampfer, Phys. Lett. 101B, 366 (1981).
  * [33] N.K. Glendenning and C. Kettner, Astron. Astrophys. 353, L9 (2000).
  * [34] K. Schertler, C. Greiner, J. Schaffner-Bielich, and M.H. Thoma, Nucl. Phys. A677, 463 (2000).
  * [35] M. G. Alford, S. Han, and M. Prakash, Phys. Rev. D 88, 083013 (2013).
  * [36] D. Blaschke, D.E. Alvarez-Castillo, and S. Benic, Proc. Sci. CPOD2013 (2013) 063 [arXiv:1310.3803].
  * [37] D.E. Alvarez-Castillo and D. Blaschke, Proceedings of the 17th Conference of Young Scientists and Specialists, Dubna (2013), arXiv:1304.7758.
  * [38] G. Montaña, L. Tolós, M. Hanauske, L. Rezzolla, Phys. Rev. D 99, 103009, (2019).
  * [39] L.Tsaloukidis, P.S. Koliogiannis, A. Kanakis-Pegios and Ch.C. Moustakidis, Phys. Rev. D 107, 023012, (2023).
  * [40] H. Tan, V. Dexheimer, J. Noronha-Hostler, and N. Yunes, Phys. Rev. Lett. 128, 161101 (2022)
  * [41] P. Landry, K. Chakravarti, arXiv:2212.09733 (2022).
  * [42] K.D. Kokkotas and J. Ruoff, Recent Developments in General Relativity, Genoa 2000 (Springer, Milano,2002).
  * [43] N. Andersson, Class. Quantum Grav. 20, R105 (2003)
  * [44] N. Andersson, Astrophys. J. 502, 708 (1998)
  * [45] J. L. Friedman and S. M. Morsink, Astrophy. J. 502, 714 (1998).
  * [46] J. L. Friedman and K. H. Lockitch, Prog. Theor. Phys. Suppl. 136, 121 (1999).
  * [47] N. Andersson and K. D. Kokkotas, Int. J. Mod. Phys. D 10, 381 (2001).
  * [48] K. D. Kokkotas, in Gravity, Astrophysics and Strings ’02, edited by T. Bau, P. P. Fiziev, and M. D. Todorov (St. Kliment Ohridski University Press, Sofia, 2003).
  * [49] N. Andersson, $r$-mode runaway in rapidly rotating neutron stars, Lectures given at the Conference on Gravitational Waves: A Challenge to Theoretical Astrophysics, Trieste, ICTP Lecture Notes Series Vol. III (ICTP, Italy, 2001), p. 297
  * [50] L. Lindblom, Neutron star pulsations and instabilities, lectures given at the Conference on Gravitational Waves: A Challenge to Theoretical Astrophysics, Trieste, ICTP Lecture Notes Series Vol. III (ICTP, Italy, 2001), p. 257
  * [51] L. Lindblom, B.J. Owen, and S.M. Morsink, Phys. Rev. Lett., 80, 4843 (1998).
  * [52] L. Lindblom, G. Mendell, and B. J. Owen. Phys. Rev. D 60, 064006, (1999).
  * [53] L. Lindblom, B.J. Owen, and G. Ushomirsky, Phys. Rev. D 62, 084030 (2000).
  * [54] B.J. Owen, L. Lindblom, C. Cutler, B.F. Schutz, A. Vecchio, and N. Andersson, Phys. Rev. D 58, 084020 (1998).
  * [55] Ch. C. Moustakidis, Phys. Rev. C 91, 035804 (2015).
  * [56] I. Vidana, Phys. Rev. C 85, 045808 (2012).
  * [57] O.P. Jyothilakshmi, P.E.S. Krishnan, P. Thakur, V. Sreekanth, and T. K. Jha, Mon. Not. R. Astron. Soc. 516, 3381 (2022).
  * [58] N. Andersson, K.D. Kokkotas, and N. Stergioulas, Astrophys. J. 516, 307 (1999).
  * [59] A. Reisenegger and A.A. Bonacic, Phys. Rev. Lett. 91, 201103 (2003).
  * [60] N. Andersson, D. I. Jones, and K. D. Kokkotas, Mon. Not. R. Astron. Soc. 337, 4, 1224 (2002)
  * [61] M. Nayyar and B.J. Owen Phys. Rev. D, 73,084001 (2006)
  * [62] A. I. Chugunov, M. E. Gusakov, and E. M. Kantor, Mon. Not. R. Astron. Soc. 445, 385 (2014).
  * [63] S. Caride, R. Inta, B. J. Owen, and B. Rajbhandari, Phys. Rev. D 100, 064013 (2019).
  * [64] B. Rajbhandari, B. J. Owen, S. Caride, and R. Inta Phys. Rev. D 104, 122008 (2021).
  * [65] P. B. Covas, M. A. Papa, R. Prix, and B. J. Owen, Astrophys. J. Lett. 929 L19 (2022).
  * [66] Xia Zhou, Ang Li, and Bao-An Li, Astrophys. J. 910, 62 (2021).
  * [67] B. Haskell, N, Degenaar, and W. C. G. Ho, Mon. Not. R. Astron. Soc 424, 1, 93 (2012).
  * [68] D.D. Ofengeim, M.E. Gusakov, P. Haensel and M. Fortin, Phys. Rev. D 100, 103017 (2019).
  * [69] D.D. Ofengeim, M.E. Gusakov, P. Haensel, and M Fortin, J. Phys. Conf. Ser. 1400, 022029 (2019).
  * [70] M. G. Alford and K. Schwenzer, Phys. Rev. Lett. 113, 251102 (2014).
  * [71] N. Pan, S. Yang, and X. Zheng, Chin. Astron. Astrophys 29 53 (2005).
  * [72] X. Zheng, N. Pan, S. Yang, X. Liu a, M. Kang, and J. Li, New Astron. 12, 165, (2006).
  * [73] M. G. Alford, S. Mahmoodifar, and K. Schwenzer, Phys. Rev. D 85, 024007 (2012).
  * [74] M. C. Papazoglou and C. C. Moustakidis, Astrophys. Space Sci. 361, 98 (2016).
  * [75] G. Baym, T. Hatsuda, T. Kojo6,1, P.D. Powell1, Y. Song and T. Takatsuka, Rep. Prog. Phys. 81 056902 (2018).
  * [76] N. K. Glendenning, Phys. Rev. D 46, 1274 (1992).
  * [77] M. Mariani, M. Orsaria, and H. Vucetich, Astron. Astrophys. 601  A21 (2017).
  * [78] M. G. Alford, G. F. Burgio, S. Han, G. Taranto, and D. Zappalà, Phys. Rev. D 92, 083002 (2015)
  * [79] J.E. Christian, A. Zacchi, and J. Schaffner-Bielich, Phys. Rev. D 99, 023009 (2019).
  * [80] J.E. Christian and J. Schaffner-Bielich, Phys. Rev. D 103, 063042 (2021)
  * [81] J.E. Christian and J. Schaffner-Bielich, Astrophys. J. 935, 122 (2022).
  * [82] S. Han and A.W. Steiner, Phys. Rev. D 99, 083014 (2019).
  * [83] J. J. Li, A. Sedrakian, and M. Alford, Phys. Rev. D 104, L121302 (2021).
  * [84] Z. Sharifi, M. Bigdeli, and D. Alvarez-Castillo, Phys. Rev. D 103, 103011 (2021).
  * [85] V. Paschalidis, K. Yagi, D. Alvarez-Castillo, D.B. Blaschke, and A. Sedrakian, Phys. Rev. D 97, 084038 (2018).
  * [86] M. Alford and A. Sedrakian, Phys. Rev. Lett. 119, 161104 (2017).
  * [87] T. Deloudis, P. Koliogiannis, and Ch. Moustakidis, EPJ Web Conf. 252, 06001 (2021).
  * [88] S. Han and M. Prakash, Astrophys. J. 899, 164 (2020).
  * [89] A. Kurkela, P. Romatschke, and A. Vuorinen, Phys. Rev. D 81, 105021 (2010)
  * [90] K. Hebeler, J.M. Lattimer, C.J. Pethick, and A. Schwenk, Astrophys. J. 773, 11 (2013)
  * [91] A. Kurkela, E.S. Fraga, J. Schaffner-Bielich, and Aleksi Vuorinen, Astrophys. J. 789, 127 (2014)
  * [92] H. Tan, T. Dore, V. Dexheimer, J. Noronha-Hostler, and N. Yunes Phys. Rev. D 105, 023018 (2022)
  * [93] S. Pal, S. Podder, D. Sen, and G. Chaudhuri Phys. Rev. D 107, 063019 (2023)
  * [94] Z.F. Seidov, Sov. Astron., 15, 347 (1971).
  * [95] T. E. Riley et al., Astrophys. J. Lett. 887 L21, (2019).
  * [96] S. Typel, J. Phys. G 45, 114001 (2018).
  * [97] G. A. Lalazissis, J. König, and P. Ring, Phys. Rev. C 55, 540 (1997).
  * [98] D. Sen, N. Alam, and G.Chaudhuri, Phys. Rev. D 106, 083008 (2022).
  * [99] G. Baym; P. Pethick, and P. Sutherland, Astrophys. J, 170, 299 (1971).
  * [100] Y. Levin and G. Ushomirsky, Mon. Not. R. Astron. Soc. 324, 917 (2001).
  * [101] N. Stergioulas and José A. Font, Phys. Rev. Lett. 86, 1148 (2001).
  * [102] E. Gaertig and K. D. Kokkotas, Phys. Rev. D 78, 064063 (2008).
  * [103] C.J. Krüger and K. D. Kokkotas Phys. Rev. Lett. 125, 111106 (2020).
  * [104] C.J. Krüger and Kostas D. Kokkotas Phys. Rev. D 102, 064026 (2020).
  * [105] P. N. McDermott, H. M. Van Horn, and J. F. Scholl, Astrophys. J. 268, 837 (1983)
  * [106] K.D. Kokkotas (private communication).
  * [107] J.L. Friedman, J.R. Ipser, and L. Parker, Phys. Rev. Lett. 62, 3015 (1989)
  * [108] P. Haensel, J.L. Zdunik, M. Bejger, and J. M. Lattimer, Astron. Astroph. 502, 650 (2009)
  * [109] B. Haskell, J.L. Zdunik, M. Fortin, M. Bejger, R. Wijnands and A. Patruno, Astron. Astrophys. 620, A69 (2018)
  * [110] P.S. Koliogiannis and Ch. C. Moustakidis, Phys. Rev. C 101, 015805 (2020)
  * [111] P.S. Koliogiannis and Ch.C. Moustakidis, Astrophys. J. 912, 69 (2021)
  * [112] T.R. Routray, S.P. Patnaik, C. Gonzalez-Boquera, X. Vinas, M. Centelles, and B. Behera, Phys. Scr. 96, 045301 (2021).
  * [113] D Page, J.M. Lattimer, M. Prakash, and A.W. Steiner, Astrophys. J. 155, 623 (2004).
  * [114] M. E. Gusakov, Andrey I. Chugunov, and E. M. Kantor, Phys. Rev. D 90, 063001 (2014).
  * [115] S. Mukhopadhyay, J. Lahiri, D. Atta, K. Imam, and D. N. Basu Phys. Rev. C 97, 065804 (2018).