# Gravitational waves from non-radial perturbations in glitching pulsars

Joydev Lahiri1 and D. N. Basu2 Variable Energy Cyclotron Centre, 1/AF Bidhan
Nagar, Kolkata 700064, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Rossby mode (r-mode) perturbations in pulsars as a steady gravitational
wave (GW) sources have been explored. The time evolution and the intensity of
the emitted GWs in terms of the strain tensor amplitude have been estimated
with the approximation of slow rotation adopting the equation of state derived
using the Skyrme effective interaction with NRAPR parameter set. The core of
the neutron star has been considered to be $\beta$-equilibrated nuclear matter
composed of neutrons, protons, electrons and muons, which is surrounded by a
solid crust. Calculations have been made for the critical frequencies, the
evolution of frequencies and frequency change rates with time as well as the
fiducial viscous and gravitational timescales, across a broad spectrum of
pulsar masses. Our findings reveal that the r-mode instability region is
associated with rotating young and hot pulsars. Furthermore, it is noteworthy
that pulsars with low $L$ value emit gravitational radiation and fall within
the r-mode instability region if the primary dissipative mechanism is shear
viscosity along the crust-core interface boundary layer. The r-mode
perturbation amplitude increases because of GW emissions, in contrast to other
non-radial perturbations which transport to infinity the star’s angular
momentum. Thus the presence of these stellar perturbations implies a non-
negative rate of change in transfer of rotational angular momentum. This
observation suggests that for a glitching pulsar, the GW emission intensity
evolves increasingly with time till the angular frequency diminishes to a
value that is below a crucial threshold, after which the compact star ceases
to emit radiation.

Keywords: Nuclear EoS; Pulsars; Core-crust transition; Crustal MoI; r-mode
instability.

###### pacs:

21.65.-f, 26.60.-c, 04.30.-w, 26.60.Dd, 26.60.Gj, 97.60.Jd, 04.40.Dg, 21.30.Fe

##  I Introduction

The rapid rotations of accreting isolated neutron star (NS) which generate
quasi-normal modes act as a sensitive probe for the effects of general
relativity such as gravitational waves (GWs) and the properties of ultradense
matter. The variations in the period of rotation of compact stars with time
can reveal their internal fluctuations such as the core-crust coupling and
decoupling. The GWs from glitching pulsars due to their rotational
instabilities can explore the high density behavior of an Equation of State
(EoS). The experimental detections of these GWs by LIGO and VIRGO, recently,
provide new vistas in asteroseismology.

The instabilities in rotation of compact stars appear in distinct flavors, but
they have one common feature in general. That is, these can directly be
correlated to the unstable modes Andersson1998 ; Andersson2003 ; Freidman1998
; Provost1981 ; Andersson2001 ; Bondarescu2007  of oscillations. Here the
r-mode instability has been studied in the context of NSs using EoS obtained
from the effective nucleon-nucleon (NN) interaction described by Skyrme
(NRAPR) parameter set. Anderson’s Andersson1998  numerical explorations of
r-mode oscillation in NSs, supplemented by Friedman and Morsink’s Freidman1998
theoretical work, have provided novel pathways for investigating GWs released
by compact stars using cutting-edge detection technology. The detection of GWs
may offer potential insights into the spin up of cold accreting NSs as well as
the spin down mechanism observed in hot young compact stars.

The r-mode oscillation is caused by perturbations in the velocity field of a
star due to small disturbances in its density. Whereas the Coriolis effects
provide a weak restoring force in a rotating star, in a non-rotating star the
r-modes are neutral rotational motions. The frequency of r-mode always appears
with reversed signs in the rotating and inertial frames. This implies that
when observed from the inertial frame, an observer will perceive the modes as
prograde while they will appear retrograde in the rotating system. The
frequency of the mode, to the leading order, is given by Andersson1999 ;
Papaloizou1978

$\sigma=\frac{\left(l-1\right)\left(l+2\right)}{l\left(l+1\right)}\Omega$ (1)

As $0<\sigma<\Omega$ $\forall$ $l\geq 2$, with $\Omega$ being the star’s
angular velocity in the inertial frame, by Chandrasekhar-Friedman-Schutz (CFS)
mechanism the r-modes are destabilized and are unstable due to the GW
emissions. The radiation of GWs resulting from r-modes originates from mass
currents that vary over time, making it a gravitational equivalent to the
magnetic monopole radiation. In NSs, compared to any other mode, the
instability due to gravitational radiation for $l=2$ quadrupole r-mode is
stronger. Further, these modes exist with velocity perturbation iff $l=m$
Provost1981 ; Andersson1999 . In spite of a decrease in energy $E_{inl}$ in
the inertial-frame, the GW emissions cause increase in the energy $E_{rot}$ of
the mode in the rotating frame. This baffling effect can be understood from
the following relation,

$E_{rot}=E_{inl}-\Omega J$ (2)

with $J$ being the star’s angular momentum. It is clear from above that if
both $J$ and $E_{inl}$ decrease, it is quite possible that $E_{rot}$ may
increase. Expressing in terms of expansion in angular velocity $\Omega$, these
r-mode frequencies, in leading order is Papaloizou1978 ; Lindblom1998

$\omega=-\frac{\left(l-1\right)\left(l+2\right)}{l+1}\Omega.$ (3)

As the viscosity Lindblom1987  opposes the GW emission, the instability in the
mode grows. To ensure the instability to be pertinent, it should increase
faster than the viscous dampening. Thus the time scale for gravitation driven
instability should be sufficiently short compared to time scale of viscous
damping . The time dependence pf r-mode amplitude evolves as $e^{i\omega
t-t/\tau}$ due to the joint influence ordinary hydrodynamics and various
dissipative processes. The effects of viscosity, gravitational radiation, etc.
Lindblom2000 ; Lindblom1998 ; Owen1998  dictate the imaginary part of the
frequency $1/\tau$. The associated time-scales of different processes involve
the actual physical parameters of NS. While computing these physical
parameters, nuclear physics plays a significant role in limiting the
uncertainties in nuclear EoS. However, our current understanding of nuclear
EoS is highly uncertain when it comes to high isospin asymmetric dense matter.
Hence correlating the r-mode observables to the EoS predictions, which were
obtained by systematically varying the physical properties, helps constraining
the uncertainties associated with EoS.

##  II Non-radial perturbations in rotating neutron stars

The competition between the gravitational radiation and the dissipative
influence of viscosity results in the evolution of the r-modes. Therefore the
present work necessarily concerns the study of the effects of radiation upon
the energy evolution of the modes. Aforementioned phenomenon can be described
through the following integral

$\widetilde{E}=\frac{1}{2}\int{\left[\varrho\delta\vec{v}.\delta\vec{v}^{*}+\left(\frac{\delta
p}{\varrho}-\delta\Phi\right)\delta\varrho^{*}\right]}d^{3}r,$ (4)

for the fluid perturbation Lindblom1998 ; Lindblom1999 , which contains the
mass density profile $\varrho(r)$ of the star. The quantities $\delta\vec{v}$,
$\delta p$, $\delta\Phi$ and $\delta\varrho$ represent the velocity, the
pressure, the gravitational potential and the density perturbations,
respectively, caused by oscillation of the mode. The r-mode dissipative time
scale can be expressed by Lindblom1998 ,

$\frac{1}{\tau_{i}}=-\frac{1}{2\widetilde{E}}\left(\frac{d\widetilde{E}}{dt}\right)_{i},$
(5)

where the subscript ’$i$’ has been used for indexing the various dissipative
mechanisms like viscous drags (shear and bulk) and GW emissions.

The expression of the energy appearing in Eq.(4) for the mode can be
simplified using the lowest order expression of r-mode $\delta\vec{v}$ and
$\delta\rho$ Lindblom1998 ; Vidana2012  to the following one-dimensional
integral

$\widetilde{E}=\frac{1}{2}\alpha_{r}^{2}R^{-2l+2}\Omega^{2}\int^{R}_{0}\varrho(r)r^{2l+2}dr,$
(6)

where R represents the NS radius, the amplitude of the mode is given by the
dimensionless quantity $\alpha_{r}$, $\Omega$ represents angular velocity of
rotation of NS and the radial dependence of the mass density of NS is given by
$\varrho(r)$.

###  II.1 Dissipative time scales in glitching pulsars

As the expression for $(\frac{d\widetilde{E}}{dt})$ arising because of
gravitational radiation Thorne1980 ; Owen1998  and viscous drags Lindblom1991
; Owen1998 ; Lindblom2000  are rather well known, evaluation of the imaginary
part $\frac{1}{\tau}$ can be facilitated using Eq.(5). It is advantageous to
break up $\frac{1}{\tau}$ as

$\frac{1}{\tau(\Omega,T)}=\frac{1}{\tau_{GR}(\Omega,T)}+\frac{1}{\tau_{SV}(\Omega,T)}+\frac{1}{\tau_{BV}(\Omega,T)},$
(7)

where $1/\tau_{GR}$, $1/\tau_{SV}$ and $1/\tau_{BV}$ are, respectively, the
contributions of gravitational radiation, shear viscosity and bulk viscosity
which can be given by Owen1998 ; Lindblom2000

$\displaystyle\frac{1}{\tau_{GR}}=-\frac{32\pi
G\Omega^{2l+2}}{c^{2l+3}}\frac{(l-1)^{2l}}{[(2l+1)!!]^{2}}\left(\frac{l+2}{l+1}\right)^{(2l+2)}$
$\displaystyle\times\int^{R_{c}}_{0}\varrho(r)r^{2l+2}dr,$ (8)
$\displaystyle\frac{1}{\tau_{SV}}=\left[\frac{1}{2\Omega}\frac{2^{l+3/2}(l+1)!}{l(2l+1)!!I_{l}}\sqrt{\frac{2\Omega
R_{c}^{2}\varrho_{c}}{\eta_{c}}}\right]^{-1}$
$\displaystyle\times\left[\int^{R_{c}}_{0}\frac{\varrho(r)}{\varrho_{c}}\left(\frac{r}{R_{c}}\right)^{2l+2}\frac{dr}{R_{c}}\right]^{-1},$
(9)

where $G$ is the gravitational constant, $c$ is the velocity of light;
$R_{c}$, $\varrho_{c}$, $\eta_{c}$ are, respectively, at the outer edge of the
core the radius, the density and the shear viscosity of the fluid.

The time scale of shear viscosity described in Eq.(9) was determined by
examining the dissipation due to shear viscosity within the boundary layer
separating the solid crust and liquid core. This calculation assumes that the
crust is inflexible and hence stationary within a rotating frame Lindblom2000
.

The crustal movement resulting from its mechanical involvement with the core
causes $\tau_{SV}$ to rise by $(\frac{\Delta v}{v})^{-2}$. The factor
$(\frac{\Delta v}{v})^{-2}$ is calculated by dividing the difference between
the velocities of the outer edge of the core and the inner edge of the crust
by the core’s velocity Levin2001 .

The impact of a solid crust on r-mode instability was initially calculated by
Bildsten and Ushomirsky Bildsten2000 . Their findings revealed that the shear
dissipation within the viscous boundary layer resulted in reduction of the
viscous damping time scale exceeding $10^{5}$ in accreting old NSs and
surpassing $10^{7}$ in young and hot rotating NSs. For $l=2$, the $I_{l}$ in
Eq.(9) has a value of $I_{2}=0.80411$ Lindblom2000 .

Moreover, for stellar temperatures below $10^{10}$ K the bulk viscous
dissipation is rather insignificant and shear viscosity in this temperature
range is the dominant dissipative mechanism. Since in this work the
temperature range has been restricted to $T<10^{10}$ K, the shear dissipative
mechanism alone has been considered. Present studies are similar to the one
performed by Moustakidis Moustakidis2015 , where the primary investigation
focused on the instability boundary and related quantities (e.g. critical
angular frequency, temperature etc.) of a NS by using the Skyrme effective
interaction with NRAPR parameter set St05 ; Ak98 , while considering the
influence of both the equation of state (EoS) and gravitational mass.

###  II.2 The Rossby-mode instability in rotating pulsars

The instability in the temperature range $T\leq 10^{10}$ K has been studied,
where as mentioned above, the dominant dissipative mechanism at the boundary
layer is the shear viscous drag and its time scale is given by Eq.(9) with
$\eta_{c}$ being the viscous coefficient of the fluid. The dominant
contribution to the shear, in the range of temperatures $T\geq 10^{9}$ K, is
primarily from the neutron-neutron (nn) scattering and below $T\leq 10^{9}$,
it is mainly the electron-electron (ee) scattering that contributes to shear
Lindblom2000 . Therefore,

$\frac{1}{\tau_{SV}}=\frac{1}{\tau_{ee}}+\frac{1}{\tau_{nn}},$ (10)

where $\tau_{ee}$ and $\tau_{nn}$ can be evaluated from Eq.(9) by using the
corresponding values of $\eta_{SV}^{ee}$ and $\eta_{SV}^{nn}$ given by
Flowers1979 ; Cutler1987

$\eta_{SV}^{ee}=6\times 10^{6}\varrho^{2}T^{-2}~{}~{}~{}~{}~{}({\rm g/cm/s}),$
(11)

$\eta_{SV}^{nn}=347\varrho^{9/4}T^{-2}~{}~{}~{}~{}~{}({\rm g/cm/s}),$ (12)

where CGS system of units has been used for all the above quantities while the
temperature T is in Kelvin. It is often useful to factorize $\tau_{SV}$ and
$\tau_{GR}$ by defining fiducial time scales $\widetilde{\tau}_{SV}$ and
$\widetilde{\tau}_{GR}$ which facilitate transparent visualization of the
roles of temperature and the angular velocity on different time scales. Hence,
the fiducial shear viscous time scale $\widetilde{\tau}_{SV}$ can be defined
as Lindblom2000 ; Lindblom1998 ,

$\tau_{SV}=\widetilde{\tau}_{SV}\left(\frac{\Omega_{0}}{\Omega}\right)^{1/2}\left(\frac{T}{10^{8}K}\right),$
(13)

while the fiducial time scale for gravitational radiation
$\widetilde{\tau}_{GR}$ can be defined as,

$\tau_{GR}=\widetilde{\tau}_{GR}\left(\frac{\Omega_{0}}{\Omega}\right)^{2l+2},$
(14)

where $\Omega_{0}=\sqrt{\pi G\bar{\rho}}$ and $\bar{\rho}=3M/4\pi R^{3}$ is
the mean density of NS of radius $R$ and mass $M$. Therefore, neglecting bulk
viscosity contributions, Eq.(7) reduces to

$\frac{1}{\tau(\Omega,T)}=\frac{1}{\widetilde{\tau}_{GR}}\left(\frac{\Omega}{\Omega_{0}}\right)^{2l+2}+\frac{1}{\widetilde{\tau}_{SV}}\left(\frac{\Omega}{\Omega_{0}}\right)^{1/2}\left(\frac{10^{8}K}{T}\right).$
(15)

Dissipative effects cause the mode to decay exponentially as $e^{-t/\tau}$
i.e. the mode is stable, as long as $\tau>0$. It is evident from Eqns.(8) and
(9) that $\widetilde{\tau}{SV}$ is positive while $\widetilde{\tau}{GR}$ is
negative. As a result, viscosity has a stabilizing effect while gravitational
radiation pushes these modes towards instability. Since the gravitational
radiation is proportional to $\Omega^{2l+2}$, its impact on $1/\tau$ is very
low for small $\Omega$. It implies that viscosity will dominate and the mode
will be stable for sufficiently small angular velocities. On the contrary, for
large values of $\Omega$ the gravitational radiation dominates and drives the
mode towards instability. For a particular mode $l$ and a given temperature
$T$, the condition $\frac{1}{\tau(\Omega_{c},T)}=0$ defines the critical
angular velocity $\Omega_{c}$. As a consequence, the equation of critical
velocity for a particular mode with index $l$ and temperature $T$ can be
expressed as a polynomial of degree $l+1$ in $\Omega_{c}^{2}$, resulting in a
unique value of $\Omega_{c}$ for each mode. The $l=2$ being the smallest of
these contributes the most, and hence studies have been done only for this
mode in the present work. For the $l=2$ mode, the critical angular velocity
$\Omega_{c}$ can be given by

$\left(\frac{\Omega_{c}}{\Omega_{0}}\right)=\left(-\frac{\widetilde{\tau}_{GR}}{\widetilde{\tau}_{SV}}\right)^{2/11}\left(\frac{10^{8}K}{T}\right)^{2/11}.$
(16)

The mass shedding limit restricts the angular velocity that a NS can attain as
maximum. This maximum angular velocity $\Omega_{K}$, called the Kepler
velocity is given approximately by$\Omega_{K}\approx\frac{2}{3}\Omega_{0}$.
Therefore, there exists a critical temperature below which the viscous drag
completely suppresses the gravitational radiation. The above mentioned
critical temperature $T_{c}$ can be given by Lindblom2000

$\frac{T_{c}}{10^{8}K}=\left(\frac{\Omega_{0}}{\Omega_{c}}\right)^{11/2}\left(-\frac{\tilde{\tau}_{GR}}{\widetilde{\tau}_{SV}}\right)\approx(3/2)^{11/2}\left(-\frac{\widetilde{\tau}_{GR}}{\widetilde{\tau}_{SV}}\right).$
(17)

From Eq.(13) and Eq.(14), the critical angular velocity $\Omega_{c}$ can now
be expressed as

$\left(\frac{\Omega_{c}}{\Omega_{0}}\right)=\frac{\Omega_{K}}{\Omega_{0}}\left(\frac{T_{c}}{T}\right)^{2/11}\approx(2/3)\left(\frac{T_{c}}{T}\right)^{2/11},$
(18)

in terms of $T_{c}$, the critical temperature. Hence, once the EoS of NS is
ascertained, the calculations of essential physical quantities to characterize
the r-mode instability can be carried out.

Because of the fact that the angular momentum is radiated to infinity by the
gravitational radiation, the angular velocity evolves with time. Its time
evolution, as per the approach of Owen et al. Owen1998 , can be given by

$\frac{d\Omega}{dt}=\frac{2\Omega}{\tau_{GR}}\frac{\alpha_{r}^{2}Q}{1-\alpha_{r}^{2}Q},$
(19)

where the dimensionless quantities $\alpha_{r}$ is the r-mode amplitude and
$Q=3\widetilde{J}/2\widetilde{I}$ where

$\widetilde{J}=\frac{1}{MR^{4}}\int^{R}_{0}\varrho(r)r^{6}dr$ (20)

and

$\widetilde{I}=\frac{8\pi}{3MR^{2}}\int^{R}_{0}\varrho(r)r^{4}dr.$ (21)

The dimensionless quantity $\alpha_{r}$ can be obtained from ’thermal’
equilibrium or ’spin’ equilibrium or may be treated as free parameter with
values ranging from $1$ to $10^{-8}$. The solution for the angular frequency
$\Omega(t)$ can be obtained by solving the equation represented by Eq.(19) as

$\Omega(t)=\left(\Omega^{-6}_{in}-\textsl{C}t\right)^{-1/6},$ (22)

by considering that the heat extracted by the neutrino emission is same
Bondarescu2009 ; Moustakidis2015  as that generated by the shear viscosity
under ideal conditions, where

$\textsl{C}=\frac{12\alpha_{r}^{2}Q}{\widetilde{\tau}_{GR}\left(1-\alpha_{r}^{2}Q\right)}\frac{1}{\Omega_{0}^{6}},$
(23)

and $\Omega_{in}$, whose value conforms to initial angular velocity, has been
treated as a free parameter. Using Eq.(22) and Eq.(23), the spin down rate
given by Eq.(19) reduces to

$\frac{d\Omega}{dt}=\frac{\textsl{C}}{6}\left(\Omega^{-6}_{in}-\textsl{C}t\right)^{-7/6}.$
(24)

The spin of a NS decreases continuously till it reaches $\Omega_{c}$. The time
$t_{c}$ elapsed since a NS evolves from the initial angular velocity
$\Omega_{in}$ to $\Omega_{c}$ which is its minimum value and can be given as

$t_{c}=\frac{1}{\textsl{C}}\left(\Omega_{in}^{-6}-\Omega_{c}^{-6}\right).$
(25)

##  III Equation of State for dense nuclear matter

T. H . R. Skyrme developed the Skyrme interaction in 1959 Sk56 . In the year
1972, Vautherin and Brink did Hartree-Fock (HF) calculations for spherical
nuclei using Skyrme’s density-dependent effective NN interaction. They found a
remarkable description of doubly-closed shell nuclei ground-state properties
Va72 . Since Skyrme’s pioneering work, and the Brink and Vautherin
parametrization of the original interaction, constant attempts have been made
by various groups to modify the Skyrme-type effective NN interaction
parameters and confine Skyrme interaction parameter sets in order to better
recreate experimental results. The analytical simplicity of the interactions
is a main advantage allows one to determine the parameters that include
fundamental properties.

###  III.1 Skyrme interaction and equation of state

The standard form of the Skyrme interaction Sk56 ; Ch97  is given by

$\displaystyle V({\bf r_{1}},{\bf r_{2}})$ $\displaystyle=$ $\displaystyle
t_{0}(1+x_{0}P_{\sigma})\delta({\bf r})$ $\displaystyle+$
$\displaystyle\frac{1}{2}t_{1}(1+x_{1}P_{\sigma})[{\bf
P}^{{}^{\prime}2}\delta({\bf r})+\delta({\bf r}){\bf P}^{2}]$ $\displaystyle+$
$\displaystyle t_{2}(1+x_{2}P_{\sigma})[{\bf
P}^{{}^{\prime}}\bm{\cdot}\delta({\bf r}){\bf P}]$ $\displaystyle+$
$\displaystyle\frac{1}{6}t_{3}(1+x_{3}P_{\sigma})[\rho({\bf
R})]^{\gamma}\delta({\bf r})$ $\displaystyle+$ $\displaystyle
iW_{0}\bm{\sigma}\bm{\cdot}[{\bf P}^{{}^{\prime}}\times\delta({\bf r}){\bf
P}]$

where $\rho$ is the number density, ${\bf r}={\bf r_{1}}-{\bf r_{2}}$, ${\bf
R}=\frac{1}{2}[{\bf r_{1}}+{\bf r_{2}}]$, ${\bf
P}=\frac{1}{2i}[\bm{\nabla}_{1}-\bm{\nabla}_{2}]$, ${\bf P}^{{}^{\prime}}$ is
the complex conjugate of ${\bf P}$ acting on the left,
$\bm{\sigma}=\bm{\sigma}_{1}+\bm{\sigma}_{2}$ and
$P_{\sigma}=\frac{1+\bm{\sigma}_{1}\bm{\cdot}\bm{\sigma}_{2}}{2}$. In the
above equation the first term represents central term, second and third terms
represent non-local term, fourth term represents density dependent term and
last term represents spin-orbit term. The Skyrme interaction thus collectively
contains 9 parameters. These are $t_{0}$, $t_{1}$, $t_{2}$, $t_{3}$, $x_{0}$,
$x_{1}$, $x_{2}$, $x_{3}$ and $\gamma$. Using density functional the energy
per particle $\epsilon_{b}$ of an asymmetric infinite nuclear matter (ANM) can
be represented as

Figure 1: Plots of isobaric incompressibility Kτ versus SNM incompressibility
K∞ (Kinf). Present results for NRAPR interaction is compared with other
interactions Pi09 ; Sa07 . The rectangular area encloses K∞ values range
225-270 MeV Sh09  together with Kτ values equal to -370$\pm$120 MeV Chen09 .

$\displaystyle\epsilon_{b}(\rho,\alpha)$ $\displaystyle=$
$\displaystyle\frac{3\hbar^{2}}{10m}\Big{(}\frac{3\pi^{2}}{2}\Big{)}^{2/3}\rho^{2/3}F_{5/3}$
$\displaystyle+$
$\displaystyle\frac{1}{8}t_{0}\rho[2(x_{0}+2)-(2x_{0}+1)F_{2}]$
$\displaystyle+$
$\displaystyle\frac{1}{48}t_{3}\rho^{\gamma+1}[2(x_{3}+2)-(2x_{3}+1)F_{2}]$
$\displaystyle+$
$\displaystyle\frac{3}{40}\Big{(}\frac{3\pi^{2}}{2}\Big{)}^{2/3}\rho^{5/3}[t_{1}(x_{1}+2)+t_{2}(x_{2}+2)]F_{5/3}$
$\displaystyle+$
$\displaystyle\frac{3}{40}\Big{(}\frac{3\pi^{2}}{2}\Big{)}^{2/3}\rho^{5/3}\frac{1}{2}[t_{2}(2x_{2}+1)-t_{1}(2x_{1}+1)]F_{8/3}$

where $\alpha=(N-Z)/A=1-2x_{p}$, $x_{p}=Z/A=\rho_{p}/\rho$ is the proton
fraction, $F_{m}(x_{p})=2^{m-1}[x_{p}^{m}+(1-x_{p})^{m}]$ or
$F_{m}(\alpha)=\frac{1}{2}[(1+\alpha)^{m}+(1-\alpha)^{m}]$.

The pressure $P(\rho,\alpha)$ in ANM can be defined in terms of energy per
particle or chemical potential as

$\displaystyle P(\rho,\alpha)$ $\displaystyle=$
$\displaystyle\rho^{2}\frac{\partial\epsilon_{b}(\rho,x_{p})}{\partial\rho}$
$\displaystyle=$
$\displaystyle\mu_{n}\rho_{n}+\mu_{p}\rho_{p}-H(\rho,\alpha),$

where $H(\rho,\alpha)$ represents energy density of ANM and is related to
energy per particle $\epsilon_{b}(\rho,\alpha)$ of ANM as
$H(\rho,\alpha)=\rho\epsilon_{b}(\rho,\alpha)$ and
$\mu_{n(p)}~{}\Big{(}=\frac{\partial
H(\rho,\alpha)}{\partial\rho_{n(p)}}\Big{)}$ is the neutron (proton) chemical
potential. The pressure $P(\rho,\alpha)$ in ANM may then be expressed as

$\displaystyle P(\rho,\alpha)$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{5m}\Big{(}\frac{3\pi^{2}}{2}\Big{)}^{2/3}\rho^{5/3}F_{5/3}$
$\displaystyle+$
$\displaystyle\frac{1}{8}t_{0}\rho^{2}[2(x_{0}+2)-(2x_{0}+1)F_{2}]$
$\displaystyle+$
$\displaystyle\frac{1}{48}t_{3}(\gamma+1)\rho^{\gamma+2}[2(x_{3}+2)-(2x_{3}+1)F_{2}]$
$\displaystyle+$
$\displaystyle\frac{1}{8}\Big{(}\frac{3\pi^{2}}{2}\Big{)}^{2/3}\rho^{8/3}[t_{1}(x_{1}+2)+t_{2}(x_{2}+2)]F_{5/3}$
$\displaystyle+$
$\displaystyle\frac{1}{8}\Big{(}\frac{3\pi^{2}}{2}\Big{)}^{2/3}\rho^{8/3}\frac{1}{2}[t_{2}(2x_{2}+1)-t_{1}(2x_{1}+1)]F_{8/3}$

Figure 2: Plots of pressure $P$ versus $\rho/\rho_{0}$ for dense nuclear
matter.

The outputs of 240 Skyrme interaction parameter sets, in 4 domains where
experimentally or empirically derived 11 constraints exist, have been examined
by Dutra et al. Du12 . These domains consist of a detailed systematic analyses
of the symmetric nuclear matter (SNM) (4 sets named SM1, SM2, SM3, SM4 in
Dutra et al.) properties, pure neutron matter (PNM) (2 sets named PNM1, PNM2
in Dutra et al.) properties and both the SNM and the PNM (5 sets named MIX1,
MIX2, MIX3, MIX3 and MIX5 in Dutra et al.) properties. These domains have been
covered by quite a few selected macroscopic constraints. It was observed that
only six satisfy all the constraints out of the 240 Skyrme models whereas 66
satisfy all the properties except one. In 10 out of the 66, even those having
only one failure have their magnitudes off by $\leq$ 5$\%$. The final list
includes 16 consistent models (the CSkP set) consisting of GSkI, GSkII,
KDE0v1, LNS, MSL0, NRAPR, Ska25s20, Ska35s20, SKRA, SkT1, SkT2, SkT3, Skxs20,
SQMC650, SQMC700 and SV-sym32. These models satisfy a host of criteria
extracted from the macroscopic properties of nuclear matter in the
neighborhood of the nuclear saturation density at zero temperature and their
density dependence obtained from the liquid drop model, experiments with giant
resonances and heavy-ion collisions. Further curtailment in this number to 5
ensues with the application of the constraints like density dependence of the
proton and the neutron effective masses, Landau parameters of SNM and PNM,
$\beta$-equilibrated matter and observational data on low- and high-mass cold
NSs. These five types of Skyrme models have been termed collectively as the
CSkP∗ set consisting of KDE0v1, LNS, NRAPR, SKRA and SQMC700.

Figure 3: The $\beta$-equilibrium electron ($e^{-}$), muon ($\mu^{-}$) and
proton ($p^{+}$) fractions plotted against the baryon density.
The Direct-URCA proton fraction threshold ${\rm x}_{\rm DU}$ is also
shown.

As extrapolation to densities above the regions of pressures, which show
consistencies with the experimental flow data, is required for describing the
NS structure, constraints like maximum mass and the corresponding central
density of high-mass NSs put further restrictions on the Skyrme models. The
radio pulsars, which are NSs with masses $>$ 1.8 $M_{\odot}$, are critical
probes of nuclear astrophysics in extreme conditions. These massive NSs have
extremely high gravitational fields inside, leading to substantially higher
gravitational binding energies, than inside commonly found 1.4 $M_{\odot}$
NSs. The mass measurement sets an upper limit on this maximum density of 10
times the saturation density De10 . The maximum mass NS with central density
in line with observation is not simulated by any of the CSkP∗ models except
NRAPR and KDE0v1 parameter sets. For the present work, the EoS for
$\beta$-equilibrated NS matter has been derived by using the Skyrme
interaction with NRAPR parameter set St05  provided in Table-I.

Figure 4: Plot of slowly rotating neutron star mass versus central density obtained using the present nuclear Equation of State. The unstable branch of TOV solution is shown in part by the dotted line. Table 1: Values of the nine parameters of nuclear matter for the Skyrme interaction corresponding to NRAPR St05  $\gamma$ | $t_{0}$ | $x_{0}$ | $t_{1}$ | $x_{1}$ | $t_{2}$ | $x_{2}$ | $t_{3}$ | $x_{3}$
---|---|---|---|---|---|---|---|---
0.14416 | -2719.7 | 0.16154 | 417.64 | -0.047986 | -66.687 | 0.027170 | 15042 | 0.13611
Nuclear matter properties at saturation density
$\rho_{0}$ ($\mathrm{fm}^{-3}$) |  $\epsilon_{b}(\rho_{0})$ (MeV) |  $K_{\infty}(\rho_{0})$ (MeV) |  $K_{\tau}$ (MeV) | $\frac{m^{*}}{m}(\rho_{0},k_{f_{0}})$ |  $E_{sym}(\rho_{0})$ (MeV) |  $L(\rho_{0})$ (MeV)
0.1606 | -15.86 | 225.65 | -385.32 | 0.69 | 32.79 | 59.63

It is possible to experimentally ascertain the incompressibility of SNM by
analyzing the compressional modes of nuclei, specifically through the
isoscalar giant monopole resonance (ISGMR) and isoscalar giant dipole
resonance (ISGDR). The inconsistencies found in the self-consistency of HF-RPA
calculations Sh06  for the strength functions of ISGMR and ISGDR lead to
shifts in the calculated centroid energies. These shifts may be larger than
the experimental uncertainties in magnitudes. In fact, the low values of
$K_{\infty}$ in the range of 210-220 MeV were predicted because of not using a
completely self-consistent calculations with the Skyrme effective NN
interactions Sh06 . When this drawback is corrected, the SLy4 type Skyrme
parmetrizations predict $K_{\infty}\sim 230-240$ MeV Sh06 . The relativistic
value of $K_{\infty}\sim 250-270$ MeV $\sim$ MeV is somewhat on the higher
side. The experimental data of ISGMR, however, suggests $K_{\infty}$ to be
$\approx$ 240 $\pm$ 20 MeV.

The ISGDR data usually predict smaller values Lu04 ; Yo04  of $K_{\infty}$.
The extraction of the value of $K_{\infty}$ is more problematic from ISGDR due
to various reasons. In particular, at excitation energies Sh06  exceeding 30
MeV for 116Sn and 26 MeV for 208Pb, the maximum cross-section for ISGDR
reduces sharply with high excitation energies. This decline is particularly
steep and may bring the cross-section below the detection limits of current
experimental techniques. The latest experimental values Yo05  for the
incompressibility $K_{\infty}$ of SNM indicate that it is quite similar to the
estimates of the relativistic mean field (RMF) model Pi09 . The results of
microscopic calculations using Gogny effective interactions Bl80 , which
include nuclei where pairing correlations are important, reproduce
experimental data on heavy nuclei for which SNM incompressibility $K_{\infty}$
is approximately 220 MeV Bl95 . Hence, one can infer that the NRAPR value for
$K_{\infty}=$ 225.7 MeV is a satisfactory theoretical outcome and falls within
the suitable range of 225-270 MeV Chen09  for the incompressibility of SNM,
which is also supported by previous research Vr03 ; Sh09 .

The advanced quantum molecular dynamics transport model can be employed to
simulate the collisions of 112Sn and 124Sn nuclei. These simulations reproduce
isospin diffusion data from two different observables and the ratios of proton
and neutron spectra. By comparing these data to calculations performed over a
range of NSEs at $\rho_{0}$ and different representations of the density
dependence of the NSE, the constraints on the density dependence of the
nuclear symmetry energy (NSE) Kl06  at subnormal density can be determined
Ts09 . With these constraints Ts09 , the results for $K_{\infty}$, $L$,
$E_{sym}(\rho_{0})$ and $K_{\tau}$ CBS09  for NRAPR are consistent. In
Table-I, the values of SNM incompressibility $K_{\infty}$, symmetry energy
$E_{sym}(\rho_{0})$, slope of nuclear symmetry energy $L$ and isobaric
incompressibility $K_{\tau}$ MeV are tabulated.

The reasonable value of incompressibility still remains controversial Sh06 .
Fig.-1 compares the present calculation (NRAPR) of $K_{\tau}$ versus
$K_{\infty}$ with predictions from several other models such as SkI3, SkI4,
SLy4, SkM, SkM*, DDM3Y, FSUGold, NL3, Hybrid Pi09 , NLSH, TM1, TM2, DDME1, and
DDME2, as reported in Table I of Ref. Sa07 . The range of magnitude of current
$K_{\infty}$ values, which fall between 225-270 MeV according to Sh09 , and
$K_{\tau}$ values of $-370\pm 120$ MeV from CBS09  are enclosed by a dotted
rectangle. Although NRAPR, SkI3, SkI4, SLy4, DDM3Y BCS08  and FSUGold are
inside the rectangular region, unlike NRAPR the magnitude of $L$ value
obtained using SkI3 is 100.49 MeV that is substantially larger than the
acceptable limit of $58.9\pm 16$ MeV Wa09 ; Ag13 ; Li13 ; Ag17  while except
for NRAPR and DDM3Y none can reach $\sim$ 2M⊙. A different review Kl17
discovered that $E_{sym}(\rho_{0})$ equals 31.7 $\pm$ 3.2 MeV and $L$ equals
58.7 $\pm$ 28.1 MeV, with $L$ having a notably greater error than
$E_{sym}(\rho_{0})$. Nonetheless, DDME2 is relatively near to the rectangular
area for which $L$ is 51 MeV.

It should be noted that the incompressibility value of 271.76 MeV for SNM
calculated by RMF-NL3 La97 ; La99  is significantly greater when compared to
the value of 225.7 MeV obtained from the NRAPR model. In Fig.-2, the pressure
$P$ of dense nuclear matter as functions of $\rho/\rho_{0}$ is shown. The PNM
is depicted by the unbroken line, while the broken line represents
$\beta$-equilibrated NS matter, and the dotted line represents the A18 model
using the Akmal et al. Ak98  variational chain summation. The areas enclosed
by the continuous and broken lines correspond to the pressure regions for
neutron matter (NM), with soft NM and stiff NM corresponding to the pressures
from asymmetry terms with weak and strong density dependence, respectively,
consistent with experimental flow data Da02 . The present EoS obtained for
NRAPR interaction parameters Du11  agrees well with the experimental flow data
Da02  despite that these parameters have been tuned to reproduce the
properties of finite nuclei. This justifies its extrapolation to high density
since its high density behavior looks phenomenologically confirmed from
Fig.-2.

###  III.2 The $\beta$-equilibrated $npe\mu$ neutron star matter

The direct URCA process which has a significant influence on the nuclear EoS
causes rapid cooling of NSs La91 ; Pe92 . In recent years, attention has been
drawn to the direct URCA process in NSs which may be the primary mechanism for
its rapid cooling. This can, however, occur only when the $\beta$-equilibrium
proton fraction $x_{p}$ in the star is $\geq{\rm x}_{\rm DU}=1/9$, when only
electrons are considered, and $\geq{\rm x}_{\rm
DU}=1/[1+\\{1+\frac{x_{e}^{1/3}+(1-x_{e})^{1/3}}{2}\\}^{3}]$, when both
electrons and muons are considered where ${\rm x}_{\rm DU}$ is the direct URCA
proton fraction threshold. At very high densities or relativistic energies the
leptonic masses can be neglected compared to their kinetic energies and then
electronic number density $\rho_{e}$ $\approx$ muonic number density
$\rho_{\mu}$ implying $x_{e}=\rho_{e}/(\rho_{e}+\rho_{\mu})\approx\frac{1}{2}$
and then $x_{p}\geq{\rm x}_{\rm DU}=1/[1+(1+2^{-1/3})^{3}]\approx~{}0.148$. In
case of the direct URCA process, the reactions which occur are the neutron
decay $n\rightarrow p+e^{-}+\bar{\nu_{e}}$, electron capture
$p+e^{-}\rightarrow n+\nu_{e}$, neutron decay $n\rightarrow
p+\mu^{-}+\bar{\nu_{\mu}}$ and muon capture $p+\mu^{-}\rightarrow
n+\nu_{\mu}$. It would be interesting to know whether present EoS favors or
disfavors direct URCA process. In our study, the lepton energy per particle
$\epsilon_{L}(\rho,x_{p})$ is given by the relativistic, ideal Fermi-gas
expression; in addition to $e^{-}$, $\mu^{-}$ is also considered as and when
they are energetically favored. At $\beta$-equilibrium, one has
${\partial\over\partial
x_{p}}\left(\epsilon_{b}(\rho,x_{p})+\epsilon_{L}(x_{p})\right)=0$, where
$\epsilon_{b}(\rho,x_{p})$ is the baryonic energy per particle including the
rest masses. In Fig.-3, the $\beta$-equilibrium proton fraction thus obtained
in the NS matter is displayed as a function of the baryon density invoking the
condition of charge neutrality.

Figure 5: The mass versus radius plot for the slowly rotating
neutron stars. In the case of the Vela pulsar, if $\frac{\Delta I}{I}>1.4\%$,
then the permissible values of the mass and the radius
are located on the right side of the curve which can be
described by the condition $\frac{\Delta I}{I}=0.014$ (considering $\rho_{t}=$
0.083 fm-3 and the corresponding P${}_{t}=$ 0.545 MeV fm-3).
The unstable branch of TOV solution is shown in part
by the dotted line.

Through the application of the Skyrme NRAPR effective NN interaction, the
investigation of $\beta$-equilibrated dense matter stability in NSs has
yielded the determination the inner edge location of their crusts and the
density and pressure of their core-crust transition. The intrinsic stability,
which pertains to the stability of any single phase, is maintained by the
convexity of $\epsilon(\rho,x)$, as established by At14 ; La07 .

This requirement can be expressed thermodynamically as
$V_{thermal}=\rho^{2}\Big{[}2\rho\frac{\partial\epsilon_{b}}{\partial\rho}+\rho^{2}\frac{\partial^{2}\epsilon_{b}}{\partial\rho^{2}}-\rho^{2}\frac{(\frac{\partial^{2}\epsilon_{b}}{{\partial\rho\partial
x}})^{2}}{\frac{\partial^{2}\epsilon_{b}}{\partial x^{2}}}\Big{]}$. By
imposing $V_{thermal}=0$, the condition for core-crust transition is obtained.
The incompressibility $K_{\infty}$ of SNM, the nuclear symmetry energy
$E_{sym}(\rho_{0})$ at saturation density $\rho_{0}$, the values for both the
slope $L$ and the isospin-dependent component $K_{\tau}$ of the isobaric
incompressibility have been presented in Table-I. These results align well
with the constraints obtained from various measurements, such as the isotopic
dependence of the giant monopole resonance in even-A Sn isotopes, the
thickness of the neutron skin of nuclei, and the analysis of experimental data
on isospin diffusion and isotopic scaling in heavy-ion collisions at
intermediate energies.

Figure 6: The fiducial timescales plotted as a function of neutron star
gravitational masses obtained using NRAPR EoS.

The instability known as the r-mode can arise when the gravitational-radiation
driving time scale of the mode is shorter than the dissipation time scales of
various mechanisms present within a NS’s interior. The nuclear EoS has two
distinct impacts on the r-mode time scales. Firstly, it defines the radial
distribution of mass density, which is the fundamental ingredient in the
integrals that are relevant to the r-mode. Secondly, the EoS specifies the
transition density from the neutron star core to its crust and the radius of
the core, which act as the upper limits of these integrals. Using the NRAPR
EoS, the Tolman-Oppenheimer-Volkoff equation has been solved to produce two
plots, namely Fig.-4, which shows the relationship between mass and central
density, and Fig.-5, which illustrates the mass-radius relationship of slowly
rotating neutron stars. For a particular value of $\frac{\Delta I}{I}$, the
mass-radius relation can be obtained using the density $\rho_{t}$ and pressure
Pt at the core-crust transition. Such mass-radius relations are then included
in Fig.-5 for $\frac{\Delta I}{I}=0.014,0.016~{}{\rm and}~{}0.070$. The
constraint $\frac{\Delta I}{I}>1.4\%$ for Vela pulsar, allows masses and
corresponding radii to be to the right of the curve defined by $\frac{\Delta
I}{I}=0.014$ (for $\rho_{t}$ = 0.083 fm-3 and Pt = 0.545 MeV fm-3) Du11 . On
the basis of glitch activity of Vela pulsar, a recent observational data Ho12
suggests somewhat higher estimate for $\frac{\Delta I}{I}>1.6\%$. However,
this small difference neither affects inferences drawn nor requires any new
concept to extend partially the neutron superfluidity into the core.
Nonetheless, when considering the occurrence of crustal entrainment resulting
from the Bragg reflection of unbound neutrons by the lattice ions, the
fraction of the moment of inertia attributed to the crust increases
significantly (from 1.4-1.6$\%$ to 7$\%$) An12 ; Ch13 . The only reasonable
limitation that can be put on Vela pulsar’s mass is that it should exceed
$\sim 1$ M⊙ in conformity with the simulations of core-collapse supernova.
This study, therefore, implies that the crust is not enough if entrainment is
considered since the mass of Vela pulsar would be below 1 M⊙ (Fig.-5), in
accordance with other studies An12 ; Ch13 ; Ho15 ; De16 ; Do16 . However, the
crust is enough to explain the data of Vela glitches in absence of
entrainment.

A recent conjecture suggests a strong correlation between the slope of the
symmetry energy $L$ and the density at which the core transits to the crust.
This correlation is predicted to be independent of the relationship between
$K_{\tau}$ and $L$ Du10 . However, there was no observed correlation between
the transition pressure and $L$ Du10 . The maximum NS mass for the EoS
obtained using the NRAPR Skyrme set is $\sim$ 1.9445 $M_{\odot}$ with a
corresponding radius of $\sim$ 9.9267 kms where $M_{\odot}$ is the solar mass.
It reconciles with the current observations on the massive compact stars
$\sim$2 M⊙ De10 ; An13 .

Figure 7: The reduced critical angular frequency plotted as a function of
temperature for neutron stars of different gravitational masses.

##  IV Gravitational waves from rotating compact stars

###  IV.1 Constraints from Thermal Equilibrium

The gravitational radiation, in a steady-state, pumps energy into the r-mode
at a rate given by Mahmoodifar2013

$W_{d}=(1/3)\Omega\dot{J}_{c}=-2\widetilde{E}/\tau_{GR}.$ (30)

This amount of energy, under thermal steady-state dissipates totally within
the star. The amount of energy $L_{\nu}$ will be lost due to the neutrino
emission and the rest due to emission of photons $L_{\gamma}$ Moustakidis2015
; Bondarescu2009  both of which will get radiated from the star’s surface. It
is important to mention that when the mode becomes saturated Alford2014 , the
thermal steady-state is independent of the cooling mechanism. Thus the thermal
steady-state condition is $W_{d}=L_{\nu}+L_{\gamma}$ assuming that during
quiescence all the emitted energy is entirely due to dissipation of r-mode
within the star. The $J_{c}$ appearing in the Eq.(30) is the canonical angular
momentum of the mode which is expressed as
$J_{c}=-\frac{3}{2}\widetilde{J}MR^{2}\Omega\alpha_{r}^{2}$, where
$\widetilde{J}$ and $\widetilde{I}$ are the dimensionless quantities defined
by Eqs.(20,21), respectively Mu18 . Using Eq.(5) and the explicit expression
for $J_{c}$, the amplitude at saturation $\alpha_{th}$ in thermal equilibrium
is then given by

$\alpha_{th}=\Big{[}\frac{-\tau_{GR}4\pi R^{2}\sigma
T_{eff}^{4}}{\widetilde{J}M}\Big{]}^{1/2}\frac{1}{\Omega R}$ (31)

where $\sigma$ is the Stefan’s constant.

Figure 8: The critical angular frequency plotted as a function of temperature
for neutron stars of different gravitational masses. The observational data
Ha12  included in Table-IV are represented by square dots.

###  IV.2 Constraints from Spin Equilibrium

The NSs are sources of extremely high gravitational pull towards its center
due to very high compactness. As a consequence, these stars can acquire mass
from the companion if they belong to a binary system. Large angular momentum
finally gets absorbed since any particle around the star can be free only at
angular velocities reaching the Keplerian limit. While conserving the total
angular momentum, the NS gains angular momentum by absorbing such particles of
the companion star which is rotating with speed much lower than the Keplerian
angular velocity. Therefore, a NS gradually spins up as it accumulates mass
from its binary companion at a rate of $\dot{M}$. Hence the outburst
properties and the observed spin-up rates may be used directly to constrain
the amplitude of r-mode Brown2000 ; Ho2011 . Hence one has

$2\pi I\dot{\nu}\Delta=\frac{2J_{c}}{\tau_{GR}}$ (32)

where $I=MR^{2}\widetilde{I}$ is the moment of inertia. The aforementioned
equation displays the effect of r-mode perturbation on the right side,
specifically the torque resulting from gravitational emission causing spin-
down. The quantity $\dot{\nu}$ is the spin-up rate during outburst and
$\Delta=(t_{o}/t_{r})$ is the ratio of the outburst duration $t_{o}$ to the
recurrence time $t_{r}$. Using Eq.(32) and the explicit expression for
$J_{c}$, the amplitude at saturation $\alpha_{sp}$ in spin equilibrium
follows:

Figure 9: The critical temperature plotted as a function
of gravitational masses of neutron stars.

$\alpha_{sp}=\Big{[}\frac{-\tau_{GR}2\pi\widetilde{I}\dot{\nu}\Delta}{3\widetilde{J}\Omega}\Big{]}$
(33)

###  IV.3 Gravitational Wave Amplitudes

The angular momentum gets transferred as a NS accretes mass from its companion
causing increase in its rate of rotation. Ultimately it surpasses
$\Omega_{c}$, its critical value. The NS begins to emit GW at this epoch due
to the perturbation of r-mode. The GW emission fuels this perturbation due to
r-mode because of CFS mechanism, resulting in an increase in the amplitude
$\alpha_{r}$ until it saturates. As described earlier, its saturation value
may be estimated either from the ’thermal equilibrium’ or from the ’spin
equilibrium’. The NS spins down to the region of stability by emitting GWs
which take away the energy and the angular momentum with it. In the present
calculations the emitted GW intensity radiated by NSs has been estimated and
compared with the observational data of LIGO-Virgo collaboration (as Virgo was
down during O1, only LIGO data was used) Abbott2019 . This has been described
in terms of the strain tensor amplitude $h_{0}$ that is related to the
amplitude $\alpha_{r}$ of the $r$-mode by Owen2009 ; Owen2010  as

Figure 10: Plots of the evolving frequencies as a function of time.

$\displaystyle
h_{0}=\sqrt{\frac{8\pi}{5}}\frac{G}{c^{5}}\frac{1}{r}\alpha_{r}\omega^{3}MR^{3}\widetilde{J},$
(34)

where the angular velocity of the star $\Omega$ is related to the r-mode
angular frequency $\omega$ by the relation
$\omega=-\frac{\left(l-1\right)\left(l+2\right)}{l+1}\Omega$. It is important
to mention here that the signature of continuous emission of GW because of the
perturbation due to r-mode is contradistinct from GW emission because of the
ellipticity of a star as described in Ref.Owen2010 . In case of r-mode
perturbation, the GW emission is commanded by the mass quadrupole moment while
for ellipticity, it is dominated by the mass current quadrupolar moment. In
the present work, using data collected from the advanced LIGO interferometers,
upper limits of the amplitude of the intrinsic strain tensor and other
relevant parameters Abbott2019  have been predicted.

##  V Theoretical Calculations

It may be observed from Eq.(8) and Eq.(9) that the integral
$\int_{0}^{R_{c}}\varrho(r)r^{6}dr$ is of prime importance for estimating
various times scales. This integral can be expressed in a dimensionless form
by rewriting it in terms of energy density $\varepsilon(r)=\varrho(r)c^{2}$ as

$I({R_{c}})=\int_{0}^{R_{c}}\left[\frac{\varepsilon(r)}{\rm
MeVfm^{-3}}\right]\left(\frac{r}{\rm km}\right)^{6}d\left(\frac{r}{\rm
km}\right)$ (35)

From Eq.(8) and Eq.(14), the fiducial gravitational radiation timescale
$\widetilde{\tau}_{GR}$ can be given by

$\widetilde{\tau}_{GR}=-0.7429\left[\frac{R}{\rm
km}\right]^{9}\left[\frac{1M_{\odot}}{M}\right]^{3}\left[I(R_{c})\right]^{-1}({\rm
s})$ (36)

where $M$ in $M_{\odot}$ and $R$, $r$ are in km.

Figure 11: Plots of the evolving spin-down rates as a function of time.

The electron-electron and neutron-neutron scattering fiducial shear viscous
timescales $\widetilde{\tau}_{SV}$ may be obtained from Eq.(9), Eqs.(11-13) as

$\displaystyle\widetilde{\tau}_{ee}=0.1446\times 10^{8}\left[\frac{R}{\rm
km}\right]^{3/4}\left[\frac{1M_{\odot}}{M}\right]^{1/4}\left[\frac{\rm
km}{R_{c}}\right]^{6}$ $\displaystyle\times\left[\frac{\rm
g~{}cm^{-3}}{\varrho_{t}}\right]^{1/2}\left[\frac{\rm
MeVfm^{-3}}{\varepsilon_{t}}\right]\left[I(R_{c})\right]({\rm s})$ (37)

$\displaystyle\widetilde{\tau}_{nn}=19\times 10^{8}\left[\frac{R}{\rm
km}\right]^{3/4}\left[\frac{1M_{\odot}}{M}\right]^{1/4}\left[\frac{\rm
km}{R_{c}}\right]^{6}$ $\displaystyle\times\left[\frac{\rm
g~{}cm^{-3}}{\varrho_{t}}\right]^{5/8}\left[\frac{\rm
MeVfm^{-3}}{\varepsilon_{t}}\right]\left[I(R_{c})\right]({\rm s})$ (38)

where core-crust transition density $\varrho_{t}$ has been expressed in units
of g cm-3 while the energy density $\varepsilon_{t}$ at the transition has
been expressed in units of MeV fm-3.

##  VI Results and discussion

The fiducial timescales versus the gravitational masses of NSs have been
plotted in Fig.-6 for the NRAPR EoS. While the timescale for gravitational
radiation falls off rapidly, it is observed that the timescales for viscous
damping increase almost linearly with increasing mass. From Eq.(16), the
dependence of the critical angular velocity $\Omega_{c}$ on temperature for
$(l=2)$ r-mode may be explored by using the fiducial timescales of
gravitational radiation and shear viscous drag. The dimensionless ratio
$\frac{\Omega_{c}}{\Omega_{0}}$ has been shown in Fig.-7 as a function of
temperature for distinct NS masses obtained using the NRAPR EoS. These plots
define the boundaries of the window of the r-mode instability. The NSs whose
angular frequency $>\Omega_{c}$ lie above the plots and possess unstable
r-modes. Hence these NSs reduce their angular frequencies by emitting GWs.
Once their angular frequencies reach $\Omega_{c}$ they reach the region
underneath the plots when the resulting r-mode stability causes cessation of
GW emissions. In Fig.-7, which depicts the instability windows, the fiducial
shear viscous timescale $\widetilde{\tau}_{ee}$ of Eq.(44) is put in Eq.(16)
in place of $\widetilde{\tau}_{SV}$ in the temperature range T $\leq$ $10^{9}$
K and $\tau_{nn}$ of Eq.(45) is substituted for $\widetilde{\tau}_{SV}$ in the
temperature range T $>$ $10^{9}$ K.

The frequencies of spinning and the temperatures of the core (either measured
or the estimated upper limits) of the observed Low Mass X-ray Binaries (LMXBs)
and Millisecond Radio Pulsars (MSRPs) Ha12  have been collected in Table-II
and their positions have been shown in Fig.-8 along with critical frequency
versus temperature plots. The Fig.-8 implies that all of the observed NSs lie
in the stable region of the r-mode oscillations defined using NRAPR EoS with a
rigid crust and have comparatively small amplitudes of r-mode oscillations,
which is consistent with the fact that GW emission from these NSs due to
r-mode instability have not been observed.

Table 2: The frequencies of spinning and the temperatures of the cores (both the measured values and the estimated upper limits) of the observed Ha12  Millisecond Radio Pulsars (MSRPs) and Low Mass X-ray Binaries (LMXBs). Stellar origin |  $\nu$ (s-1) | $T_{core}(10^{8}K)$
---|---|---
Aql X-1 | $550$ | $1.08$
4U 1608-52 | $620$ | $4.55$
KS 1731-260 | $526$ | $0.42$
MXB 1659-298 | $556$ | $0.31$
SAX J1748.9-2021 | $442$ | $0.89$
IGR 00291+5934 | $599$ | $0.54$
SAX J1808.4-3658 | $401$ | $<0.11$
XTE J1751-305 | $435$ | $<0.54$
XTE J0929-314 | $185$ | $<0.26$
XTE J1807-294 | $190$ | $<0.27$
XTE J1814-338 | $314$ | $<0.51$
EXO 0748-676 | $552$ | $1.58$
HETE J1900.1-2455 | $377$ | $<0.33$
IGR J17191-2821 | $294$ | $<0.60$
IGR J17511-3057 | $245$ | $<1.10$
SAX J1750.8-2900 | $601$ | $3.38$
NGC 6440 X-2 | $205$ | $<0.12$
Swift J1756-2508 | $182$ | $<0.78$
Swift J1749.4-2807 | $518$ | $<1.61$
J2124-3358 | $203$ | $<0.17$
J0030+0451 | $205$ | $<0.70$

Figure 12: Plots of the evolving spin-down rates as a function of frequencies.

For a fixed temperature $T$, $\frac{\Omega_{c}}{\Omega_{0}}$ rapidly decreases
with increasing mass as seen in Fig.-7. Eq.(18) implies that when
$\Omega_{c}=\Omega_{K}$, $T=T_{c}$ and $\Omega_{K}$ grows with mass leading to
a fall in $T_{c}$. The critical temperature as a function of NS mass has been
plotted in Fig.-9. For the calculation of $T_{c}$, the timescale of shear
viscous drag arising from electron-electron scattering has been used. It is
observed that the critical temperature $T_{c}$ decreases rapidly with NS mass.

It is observed from Fig.-8 and Fig.-9 that $\nu_{c}$ and $T_{c}$ decrease with
mass causing the r-mode instability window to increase with NS mass. This
implies that for the same temperature and EoS, compared to the less massive
NSs, more massive configurations are prone to a great extent to the r-mode
instability and hence to the emission of GWs. The fact that
$\widetilde{\tau}_{GR}$ is much less than $\widetilde{\tau}_{ee}$ and
$\widetilde{\tau}_{nn}$ for massive NSs as in Fig.-6, the preceding
observation can also be substantiated indirectly. The isolated young massive
NSs thus are more prone to GW emissions because of r-mode instability.

It is worthwhile to mention here that $\Omega_{c}$ is sensitive to the density
dependence of the symmetry energy and its slope $L$. Moreover, full radius
$R$, core radius $R_{c}$, $I(R_{c})$ and $\rho_{t}$ also depend upon the
values of $L$. Therefore, for fixed mass and temperature, the quantity
$\Omega_{c}$ depends upon the above parameters through the relation,

$\Omega_{c}\sim\frac{R_{c}^{12/11}}{[I(R_{c})]^{4/11}}\varrho_{t}^{3/11}$ (39)

Figure 13: Plots displaying the amplitude of the strain tensor ($h_{0}$) in
relation to the distance for two distinct rotational frequencies of neutron
stars with varying masses, while the r-mode amplitude estimated from the
‘thermal equilibrium’.

In our case, for a fixed temperature and NS mass $L$, $\rho_{t}$ and $R_{c}$
are constants. The amplitude $\alpha_{r}$ of the r-mode grows till it attains
a saturation value when a NS enters into the region of instability because of
mass accretion from its binary companion. At this point a NS emits GWs
releasing its energy and angular momentum leading to spin down to reach the
stability region. The time evolution of angular velocity of spinning and the
rate of spin down can be calculated for a NS from Eq.(22) and Eq.(24),
respectively, assuming the ideal condition that the decrease in temperature
due to emission of GWs is compensated by the heat produced due to viscous
effects, provided T, M, $\alpha_{r}$ and $\Omega_{in}$ of the star are known.
The spin evolution is calculated for different NS masses using the
representative values of $\nu_{in}=\frac{\Omega_{in}}{2\pi}=700$ Hz and
$\alpha_{r}=2\times 10^{-7}$ as used by Moustakidis Moustakidis2015  and shown
in Fig.-10. The rates of spin down are shown for those masses in Fig.-11. The
rates of spin down as a function of frequency of spinning are shown in
Fig.-12.

In a previous study Vidana2012  using different EoS models, the critical
frequency $\Omega_{c}$ of the pulsar 4U 1608-52 as a function of the symmetry
energy slope parameter $L$ was plotted using an estimated temperature $\sim
4.55\times 10^{8}$ K of the core. Like Fig.-6 of Vidana2012 , using the
measured frequency of spinning and the estimated temperature of core, if the
mass of 4U 1608-52 is $1.4M_{\odot}$ then it should be only slightly unstable
($\Omega_{c}$ is lower than its frequency of spinning), since the radius
corresponding to this mass is $\sim 11.87$ kms from the present mass-radius
relationship (Fig.-5) and is greater than 11.5 kms. As $L<60$ MeV for NRAPR
EoS, the highest mass configuration of 1.9445 M⊙ with a radius of $\sim$9.93
kms is also likely to be in the region of instability. In agreement with We12
, if one considers the dissipation to arise at the crust-core interface, it
would imply that the instability window of the r-mode would be broadened for
an isolated NS with a rigid crust.

Figure 14: Plots displaying the amplitude of the strain
tensor ($h_{0}$) in relation to the angular frequency of the
neutron stars with varying masses, while the r-mode
amplitude estimated from the ‘thermal equilibrium’.

It is worthwhile to acknowledge the observations made by Demorest et al. De10
of the binary millisecond pulsar J1614-2230 that rotates with 3.1 ms period
which suggests that its mass lies in the range $1.97\pm 0.04$ M⊙. The
measurements of radio timing for pulsar PSR J0348+0432 and its companion
(white dwarf) have confirmed the mass of the pulsar to be in the range
1.97$-$2.05 $M_{\odot}$ at the confidence level of 68.27$\%(1\sigma)$ or
1.90$-$2.18 $M_{\odot}$ at the confidence level of 99.73$\%(3\sigma)$ An13 .
Very recently, the studies for PSR J0740+6620 Fon21  and for PSR J0952-0607
Rom22  find masses of 2.08 $\pm$ 0.07 $M_{\odot}$ and 2.35 $\pm$ 0.17
$M_{\odot}$, respectively, but only with a $1\sigma$ credibility. Some recent
works Leg21 ; Ril21  constrain the equatorial radius and mass of PSR
J0740+6620 to be 12.39${}^{+1.30}_{-0.98}$ km and 2.072${}^{+0.067}_{-0.066}$
$M_{\odot}$ respectively, each reported as the posterior credible interval
bounded by the 16$\%$ and 84$\%$ quantiles. All these imply that NRAPR
somewhat underestimates the observed NS masses at the $1\sigma$ level
measurement credibility but is reasonably good at predicting observations at
the $3\sigma$ level.

Figure 15: Plots displaying the amplitude of the strain tensor ($h_{0}$) in
relation to the angular frequency of neutron stars with varying masses, while
the r-mode amplitude estimated from the ‘spin equilibrium’.

The plots of strain tensor amplitude $h_{0}$ as a function of distance for two
different NS masses for two spin frequencies of 200 Hz and 600 Hz are shown in
Fig.-13. The quantity $\alpha_{r}$ is estimated from ’thermal equilibrium’
using the red-shifted effective surface temperature $T_{eff}^{\infty}=100$ eV.
For same $T_{eff}^{\infty}$, effective surface temperature $T_{eff}$ for
higher mass would be higher which explains the trend resulting from higher
$\alpha_{r}$ and correspondingly higher strain tensor amplitude $h_{0}$. In
Fig.-14, the strain tensor amplitude $h_{0}$ is plotted as a function of
evolving angular velocity $\Omega$ of the star in units of rad s-1 for three
different NS masses which arise from GW emissions. In Fig.-15, $\alpha_{r}$ is
estimated from ’spin equilibrium’ using Eq.(25) where the experimental data
used were from the source IGR J00291 of Table-II of Ref.Mahmoodifar2013 .
Entire calculations have been performed presuming the range of angular
momentum of the star such that the CFS two-stream instability
Chandrasekhar1970  applies enhancing the r-mode perturbation as the GW
emission pumps into the perturbation instead of damping it. Thus the NS,
spinning at a particular angular velocity emits GW because of the perturbation
and causing loss of its angular momentum. As a consequence while the the
star’s angular velocity continues to reduce till it crosses the critical limit
($\Omega_{c}$) of CFS two-stream instability, the intensity of GW emission
increases.

##  VII Summary and conclusions

The current study explores the instability due to the r-mode oscillations for
slowly rotating neutron stars having a solid crust, by utilizing an equation
of state derived from the Skyrme effective nucleon-nucleon interaction with
the NRAPR parametrization. The fiducial gravitational radiation timescale and
the timescales for shear viscosity drag have been calculated in the framework
of Skyrme-NRAPR interaction for a wide range of NS masses. Using the present
EoS, it is seen that while the viscous damping timescales increase more or
less linearly with increasing NS mass, the GW emission timescales decrease
rapidly. Further studies have been conducted on the variation of the critical
angular frequency with temperature for diverse NS masses. As observed in
Fig.-8, the core temperatures and spin frequencies of the observed Millisecond
Radio Pulsars Ha12  and Low Mass X-ray Binaries always lie below the region of
the r-mode instability. This fact implies that for the NSs spinning faster
than their corresponding critical frequencies have unstable r-modes leading to
the emission of GWs. In the present study it is observed that with increasing
mass the temperature and critical frequency decrease. It may therefore be
concluded that hot massive NSs are more vulnerable to instability caused by
r-mode through GW radiation. The rates of spin down and the time evolution of
angular frequency of NSs due to the r-mode instability have been calculated as
well. We highlight the dependence of the critical angular frequency on the EoS
by examining the radius and the slope parameter $L$ of symmetry energy. The
region of the r-mode instability extends to the lower values of $L$ if the
r-mode dissipation arising out of shear viscous drag proceeds along the layer
separating the core from the crust at the core-crust interface.

###### Acknowledgements.

One of the authors (DNB) acknowledges support from Science and Engineering
Research Board, Department of Science and Technology, Government of India,
through Grant No. CRG/2021/007333.

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as $\alpha=\mathbb{E}_{\alpha}[N]/\mathbb{E}_{\alpha}[T_{1}]$ (this can be
obtained from the previous considerations by choosing $w(t,x)=g(t)$ for
$t\;\geqslant\;0$ and $x\in\mathbb{R}^{d}$). For a constant $c\;\geqslant\;0$
we add $c\cdot g$ to $w$ (i.e. we add $c$ to $v$) and obtain

$\mathbb{E}_{\alpha}\bigg{[}e^{-(\psi(\alpha)+\alpha
c-\alpha)T_{1}}\mathbb{E}_{\mathcal{W}}\Big{[}\prod_{i=1}^{N}v(t_{i}-s_{i},X_{s_{i},t_{i}})+c\Big{]}\bigg{]}=1.$

If we again assume that we may differentiate under the integral we obtain

$\alpha\mathbb{E}_{\alpha}\big{[}T_{1}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})\big{]}=\mathbb{E}_{\alpha}\big{[}Ne^{-(\psi(\alpha)-\alpha)T_{1}}\overline{F}(\xi_{1})]$

where

$\overline{F}(\xi_{1})=\frac{1}{N}\sum_{j=1}^{N}\mathbb{E}_{\mathcal{W}}\Big{[}\prod_{i\neq
j}v(t_{i}-s_{i},X_{s_{i},t_{i}})\Big{]}.$

Combining both equalities yields

$\psi^{\prime}(\alpha)=\frac{\mathbb{E}_{\alpha}\big{[}Ne^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})]}{\mathbb{E}_{\alpha}\big{[}Ne^{-(\psi(\alpha)-\alpha)T_{1}}\overline{F}(\xi_{1})]}.$
(7.2)

In other words, $\psi^{\prime}(\alpha)$ is a measure for how much $F$ changes
on average if we randomly delete a point of $\xi_{1}$. If there exists a
measurable function $f:[0,\infty)\to(0,\infty)$ satisfying
$\int_{0}^{\infty}(1+t)f(t)\,\mathrm{d}t<\infty$ such that
$w(t,x)\;\leqslant\;f(t)$ for all $x\in\mathbb{R}^{d}$ and $t\;\geqslant\;0$
(i.e. if the assumptions of Proposition 2.4 a) are satisfied) then we may
choose $g=f/\|f\|_{L^{1}}$ and obtain $v\;\leqslant\;C$ with
$C\coloneqq\|f\|_{L^{1}}$. This means that
$F(\xi_{1})\;\leqslant\;C\bar{F}(\xi_{1})$, and then (7.2) implies a linear
upper bound for the growth of $\psi(\alpha)$ with $\alpha$. It should be
noted, however, that this can be obtained by an elementary calculation, using
$e^{C\cdot c_{\alpha,T}}\sim e^{2\alpha CT-\alpha C\mathbb{E}[\tau_{1}]}$.
Similarly, lower linear bounds can be derived. For the Fröhlich polaron,
$\psi(\alpha)\sim g_{0}\alpha^{2}$ and thus $\psi^{\prime}(\alpha)\to\infty$
as $\alpha\to\infty$ (by convexity of $\psi$). Combining this with Equation
(7.1), we see that, as a consequence of the singularity of the potential,
reweighting the point process leads to a relative increase of the number of
customers per unit of time that diverges to $+\infty$ as $\alpha\to\infty$.

By Theorem 3.2 we have

$\hat{\mathbb{E}}_{\alpha}[\widehat{T}_{1}]=\mathbb{E}_{\alpha}\big{[}T_{1}e^{-(\psi(\alpha)-\alpha)T_{1}}F(\xi_{1})\big{]}=\frac{1}{\psi(\alpha)\lim_{T\to\infty}Z_{\alpha,T}e^{-\psi(\alpha)T}}.$

As seen in the proof of Proposition 2.4 b) and Corollary 2.5, for the Fröhlich
polaron this yields

$\hat{\mathbb{E}}_{\alpha}[\widehat{T}_{1}]=-\frac{1}{E_{\alpha}(0)|\langle\Omega,\Psi_{\alpha}\rangle|^{2}}$

where $\Omega$ is the Fock-vacuum and $\Psi_{\alpha}$ and $E_{\alpha}(0)$ are
the ground state and the ground state energy of the Hamiltonian of the
Fröhlich polaron at total momentum zero. Notice that

$|\langle\Omega,\Psi_{\alpha}\rangle|^{2}=\hat{\mathbb{E}}_{\alpha}[\hat{d}_{1}]/\hat{\mathbb{E}}_{\alpha}[\widehat{T}_{1}]$

is the probability that we are dormant at a given point in time under the
stationary measure $\widehat{\Gamma}_{\alpha,\operatorname{st}}$. We can also
put the coupling parameter into the potential, to obtain

$\mathbb{E}_{1}\Big{[}\alpha^{N}e^{-(\psi(\alpha)-1)T_{1}}F(\xi_{1})\Big{]}=1.$

This identity might potentially be used in order to show analyticity of $\psi$
using the implicit function theorem.

## Appendix: Polaron models and Polaron path measures

Here we give an overview over polaron models and their connection to Polaron
path measures. For details and proofs we refer to [DS20] and [Møl06], see also
Chapters 5 and 6 of [LHB11].

The polaron describes a $d$-dimensional quantum particle coupled to a scalar
Bosonic field, e.g. the lattice vibrations of a polar crystal. Its Hamiltonian
acts in the space $L^{2}({\mathbb{R}}^{d})\otimes{\mathcal{F}}$, where
${\mathcal{F}}$ the Hilbert space for the Bose field, i.e. the symmetric Fock
space over $L^{2}({\mathbb{R}}^{d})$. The Hamiltonian is given by

$H=\frac{1}{2}p^{2}+\int_{{\mathbb{R}}^{d}}\omega(k)a^{\ast}(k)a(k)\,{\rm
d}k+\sqrt{\alpha}\int_{{\mathbb{R}}^{d}}\frac{\hat{\varrho}(k)}{\sqrt{2\omega(k)}}\big{(}\,{\rm
e}^{{\rm i}k\cdot x}\,a(k)+\,{\rm e}^{-{\rm i}k\cdot
x}\,a^{\ast}(k)\big{)}\,{\rm d}k.$

Here $a^{\ast}(k)$, $a(k)$ are the creation and annihilation operators of the
free Bose field, respectively, satisfying the canonical commutation relations
$[a^{\ast}(k),a(k^{\prime})]=\delta(k-k^{\prime})$. The first term represents
the momentum operator of the free particle, and the second term is the energy
of the free field, which is the differential second quantization of the
operator of multiplication with $\omega$. The energy-momentum relation
$\omega$ of the Bose field is assumed to be nonnegative, strictly positive
almost everywhere, continuous, and invariant under rotations. The third term
implements the coupling between particle and field, $x$ being the position
operator of the particle. The function $\varrho$ is used to ’smear out’ the
coupling of the particle to the field. $\hat{\varrho}$ is assumed to be
rotation invariant and real-valued. The function

$g(k)=\hat{\varrho}(k)/\sqrt{2\omega(k)}$

is usually called the coupling function, and $\alpha$ is the coupling
constant. Important special cases are the Fröhlich polaron where

$\omega(k)=1,\qquad g(k)=(\sqrt{2}\pi|k|)^{-1},$

and the Nelson model where

$\omega(k)=\sqrt{k^{2}+m^{2}}\text{ with }m\;\geqslant\;0,\quad\text{and
}\quad g(k)={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{[\kappa,\Lambda]}(|k|)\omega(k)^{-1/2}\text{ with
}0\;\leqslant\;\kappa<\Lambda<\infty.$

$m$ is the mass of the Bosons, and $\kappa,\Lambda$ are the infrared and
ultraviolet cutoffs, respectively. They restrict the interaction of the
particle with the field modes to those modes with energy between $\kappa$ and
$\Lambda$.

Under suitable assumptions on $\omega$ and $g$ (which are fulfilled for the
two examples above), the operator $H$ is self-adjoint and bounded below. Since
the coupling of the particle to the field is invariant under translations, $H$
commutes with the total momentum operator

$P_{\rm tot}=p+P_{\rm f}=p+\int_{{\mathbb{R}}^{d}}k\,a^{\ast}(k)a(k)\,{\rm
d}k.$

Therefore, $H$ admits a fiber decomposition
$H=U^{\ast}\int_{{\mathbb{R}}^{d}}^{\oplus}H(P)\,{\rm d}P\,\,U$ with a
suitable unitary operator $U$, where for each $P\in{\mathbb{R}}^{d}$ the
operator

$H(P)=\frac{1}{2}(P-P_{\rm
f})^{2}+\int_{{\mathbb{R}}^{d}}\omega(k)a^{\ast}(k)a(k)\,{\rm
d}k+\sqrt{\alpha}\int_{{\mathbb{R}}^{d}}g(k)(a(k)+a^{\ast}(k))\,{\rm d}k.$

now acts only on Fock space. $H(P)$ is bounded below and self-adjoint whenever
$H$ is, and the map that takes $P\in{\mathbb{R}}^{d}$ to the bottom $E(P)$ of
the spectrum of $H(P)$ is the energy-momentum relation for the particle. By
the rotation invariance of $\omega$ and $g$, $E(P)=E_{\rm r}(|P|)$ only
depends on $|P|$, and $E^{\prime\prime}_{\rm r}(0)$ is the inverse of the
effective mass of the particle interacting with the Bose field.

Polaron path measures are related to polaron models by a Feynman-Kac formula:
the particle Hamiltonian $\frac{1}{2}p^{2}$ is the generator of Brownian
motion ${\mathcal{W}}$, and the field Hamiltonian
$\int\omega(k)a^{\ast}(k)a(k)\,{\rm d}k$ is unitarily equivalent to the
generator of an infinite dimensional Ornstein-Uhlenbeck process
${\mathcal{G}}$. Its probability distribution is supported on distribution-
valued functions $s\mapsto\phi_{s}$, and its covariance function is given by

${\mathbb{E}}_{\mathcal{G}}(\phi_{s}(u)\phi_{t}(v))=\int\hat{u}(k)\frac{1}{2\omega(k)}\,{\rm
e}^{-|t-s|\omega(k)}\,\overline{\hat{v}(k)}\,{\rm d}k$

for suitable test functions $u,v$. Much like in the ordinary Feynman-Kac
formula, this allows to write matrix elements $\langle\Psi,\,{\rm
e}^{-tH}\,\Phi\rangle$ as integrals with respect to the measure
${\mathcal{G}}\otimes{\mathcal{W}}$. When $\Omega$ is the Fock vacuum and
$f_{1},f_{2}\in L^{2}({\mathbb{R}}^{d})$, this leads to the equality

$\langle f_{1}\otimes\Omega,\,{\rm
e}^{-tH}\,f_{2}\otimes\Omega\rangle=\int{\rm
d}xf_{1}(x)\int\mathcal{W}^{x}({\rm d}\mathbf{x})\int{\mathcal{G}}({\rm
d}\phi)\exp\Big{(}-\sqrt{\alpha}\int_{0}^{t}\phi_{s}(\rho(\cdot-\mathbf{x}_{s}))\,{\rm
d}s\Big{)}f_{2}(\mathbf{x}_{t}),$

where ${\mathcal{W}}^{x}$ is Brownian motion started at $x$. Since the
exponent is linear in the field variable $\phi$, the Gaussian integral can be
carried out explicitly, with the result

$\langle f_{1}\otimes\Omega,\,{\rm
e}^{-tH}\,f_{2}\otimes\Omega\rangle=\int{\rm d}xf_{1}(x)\int W^{x}({\rm
d}\mathbf{x})\,{\rm e}^{\frac{\alpha}{2}\int_{0}^{t}{\rm d}r\int_{0}^{t}{\rm
d}s\,w(r-s,\mathbf{x}_{r}-\mathbf{x}_{s})}\,f_{2}(\mathbf{x}_{t}),$ (7.3)

where

$w(r-s,\mathbf{x}_{r}-\mathbf{x}_{s})={\mathbb{E}}_{{\mathcal{G}}}\big{(}\phi_{r}\big{(}\varrho(\cdot-\mathbf{x}_{r})\big{)}\phi_{s}\big{(}\varrho(\cdot-\mathbf{x}_{s})\big{)}\big{)}=\int|g(k)|^{2}\,{\rm
e}^{{\rm i}k\cdot(\mathbf{x}_{r}-\mathbf{x}_{s})}\,\,{\rm
e}^{-\omega(k)|r-s|}\,.$

This establishes the connection between the Polaron models and Polaron path
measures. The formal choice $f_{1}=\delta(0)$ and $f_{2}=1$ in (7.3)
corresponds to the matrix element $\langle\Omega,e^{-tH(0)}\Omega\rangle$, see
e.g. [DS20], where also expressions for
$\langle\Omega,e^{-tH(P)}\Omega\rangle$ are derived for arbitrary $P$.

For the Fröhlich polaron in three dimensions, an explicit computation leads to
$w(t,x)=\frac{1}{|x|}\,{\rm e}^{-|t|}\,$. For the massless Nelson model (i.e.
$m=0$) in three dimensions, one obtains

$w(t,x)=4\pi\int_{\kappa}^{\Lambda}\,{\rm
e}^{-r|t|}\,\frac{\sin(r|x|)}{|x|}{\rm d}r,$

When $\kappa=0$, i.e. when the particle is allowed to interact with low energy
field modes, this $w$ decays like $|t|^{-2}$ for large $t$.

Acknowledgement: We thank Herbert Spohn for useful comments on an earlier
version of this paper.

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