# QCD, Gravitational Waves, and Pulsars

Partha Bagchi<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, Jatni, Odisha -752050, India
Oindrila Ganguly<EMAIL_ADDRESS>The Institute of Mathematical Sciences,
Chennai -600113, India  Biswanath Layek<EMAIL_ADDRESS>Department of Physics, Birla Institute of Technology and Science, Pilani
-333031, India  Anjishnu Sarkar<EMAIL_ADDRESS>Physics Department, The
LNM Institute of Information Technology, Jaipur -302031, India  Ajit M.
Srivastava<EMAIL_ADDRESS>Institute of Physics, Bhubaneswar-751005, India

###### Abstract

Investigations of the phase diagram of quantum chromodynamics (QCD) have
revealed exotic new possibilities. Experimental investigations with
relativistic heavy ion collision experiments (RHICE) have been able to probe
relatively low baryon chemical potential regime of the QCD phase diagram,
namely the quark-gluon plasma (QGP) phase. An entire new spectrum of phases is
expected to arise at very high baryon chemical potential, which may remain out
of reach at these terrestrial experiments. These phases are collectively
referred to as color superconducting phases. Attention is naturally directed
towards astrophysics where gravity assisted ultra high baryon density objects
naturally occur. The densest such object, which can be directly observed, is a
neutron star. It is thus speculated that cores of neutron stars provide a
natural laboratory where such exotic phases of QCD may exist. In fact, the
theoretically well established neutron superfluidity phase is also believed to
be found only inside neutron stars. Focus on neutrons stars has tremendously
intensified in recent years from a completely different angle. Even though the
first indirect detection of gravitational waves had come long ago from a
binary neutron star (BNS) system, it is the direct detection of gravitational
waves by LIGO/Virgo from BNS merger events which has allowed the possibility
of directly probing the properties of the interior of a neutron star. A truly
remarkable phenomenon manifested by rapidly rotating neutron stars is in their
avatar as Pulsars. The accuracy of pulsar timing can reach the level of one
part in 1015, comparable to that of atomic clocks. Indeed, it was such a great
accuracy which had allowed the first indirect detection of gravitational waves
from a BNS system. Such an incredible accuracy of pulse timings points to a
very interesting possibility. Any deformation of the pulsar, even if it is
extremely tiny, has the potential of leaving its imprints on the pulses
through introduction of tiny perturbations in the entire moment of inertial
(MI) tensor. While, the diagonal components of perturbed MI tensor affect the
pulse timings, the off-diagonal components lead to wobbling of pulsar,
directly affecting the pulse profile. This opens up a new window of
opportunity for exploring various phase transitions occurring inside a pulsar
core, through induced density fluctuations, which may be observable as
perturbations in the pulse timing as well as its profile. Such perturbations
also naturally induce a rapidly changing quadrupole moment of the star,
thereby providing a new source of gravitational wave emission. Another
remarkable possibility arises when we consider the effect of an external
gravitational wave on neutron star. With the possibility of detecting any
minute changes in its configuration through pulse observations, the neutron
star has the potential of performing as a Weber detector of gravitational
wave. This brief review will focus on these specific aspects of a pulsar.
Specifically, the focus will be on the type of physics which can be probed by
utilizing the effect of changes in the MI tensor of the pulsar on pulse
properties.

###### pacs:

12.38.Mh,97.60.Gb,95.55.Ym,04.80.Nn,26.60.+c

##  I Introduction

Cosmos has always proved to be the ultimate laboratory where physical systems
may exist in extreme environments, even those which are beyond the reach of
any terrestrial experiments. Early hot and dense stages of the Universe is one
such case where extremely high temperatures are achieved. Fortunately, some of
those stages, with temperatures reaching 1012 K (few hundred MeV) are possible
to be partially probed in terrestrial experiments now in relativistic heavy-
ion collision experiments (RHICE) [1, 2]. This possibility has put the physics
of quark-gluon plasma (QGP) phase of QCD matter at a central stage, the phase
which is believed to exist in the Universe when it was few tens of
microseconds old. Experimental observations at RHICE have already shown
completely unexpected results, for example, a near-perfect fluid nature of QGP
with a value of shear viscosity to entropy ratio which is close to the
proposed lowest bound on this number [3]. With continued efforts in RHICE with
varying collision energy, it has been possible to extensively investigate
certain part of the phase diagram of quantum chromodynamics (QCD) which
corresponds to relatively low baron number density. As the early Universe was
filled with a matter with extremely low baryon number density, it allows one
to claim that conditions like early Universe have been recreated in laboratory
(at least for the strongly interacting matter part).

At the same time, theoretical investigations have revealed the possibility of
an entire new spectrum of phases of strongly interacting matter which are
expected to arise at very high baryon chemical potential [4]. It is reasonably
clear now that much of this extremely rich part of the QCD phase diagram may
remain out of reach at these terrestrial experiments. These phases are
collectively referred to as the color superconducting phases [5, 6]. Attention
is thus naturally directed towards astrophysics where gravity assisted ultra
high baryon density objects routinely occur. Extreme conditions of high baryon
density are expected to be reached in supernova explosions, in neutrons stars,
and in matter undergoing collapse to a black hole. The densest such object,
which can be directly observed at present, is a neutron star. It is speculated
that cores of neutron stars provide a natural laboratory where various exotic
phases of QCD may occur [7]. Even exotic forms of matter, stable only under
extreme conditions of density and pressure, may form in these objects, such as
strange stars [8, 9, 10, 11, 12]. Interestingly, the theoretically very well
established neutron superfluidity phase has never been seen in any terrestrial
experiment. At the same time it is expected to routinely occur inside neutron
stars [13]. In fact, superfluid vortices in such a phase provide the most
convincing explanation of the phenomena of pulsar glitches [14].

Neutron stars have long been investigated theoretically and experimentally,
especially with pulsar observations. Pulsars are rapidly rotating neutron
stars with pulse timings which are observed on earth with incredible accuracy,
reaching one part in 1015 for certain pulsars, comparable to that of atomic
clocks. This extreme accuracy of pulse timings had allowed the first indirect
detection of gravitational waves from a binary neutron star (BNS) system [15,
16]. Neutron star physics has acquired a centre stage recently with the advent
of gravitational wave detectors. After the first direct detection of
gravitational waves (GWs) by LIGO coming from a binary black hole merger event
[17], the stage was set for the detection of GWs from spiral-in of other
compact dense objects. The first such event of binary neutron star merger was
detected by LIGO/Virgo in 2017 [18] and that opened the remarkable possibility
of directly probing the properties of the interior of neutron stars.

Neutron stars thus acquire a unique status of providing a laboratory for
probing microphysics of exotic phases of QCD on one hand, while providing a
window to probe the physics of its interior using GW detectors on earth in BNS
merger events on the other. This remarkable story of neutron stars still
allows for one more chapter, that relating to its avatar as a pulsar with
extreme accuracy of pulsar timing observations. Such an incredible accuracy of
pulse timings points to a very interesting possibility. Any deformation of the
neutron star, even if it is extremely tiny, has the potential of leaving its
imprints on the pulses through introduction of tiny perturbations in the
entire moment of inertial (MI) tensor. Clearly, it will directly affect the
pulse timing. However, a general deformation of NS will change the entire MI
tensor, including its off-diagonal components. The diagonal components of
perturbed MI tensor will affect the pulse timings, at the same time, the
perturbed off-diagonal components will induce wobbling of pulsar. Wobbling of
pulsar (on top of any previously present) will directly affect the profile of
pulses as observed on earth. Thus observations of changes in pulse timings,
along with any accompanying changes in the pulse profile will contain
information about details of minute changes in the configuration of NS, e.g.
density perturbations inside the NS, or its overall deformations. This opens
up a new window of opportunity for exploring various phase transitions
occurring inside a pulsar core, through induced density fluctuations, which
may be observable as perturbations in the pulse timing as well as its profile
[19, 20, 21]. Such perturbations also naturally induce a rapidly changing
quadrupole moment of the star, thereby providing a new source of gravitational
wave emission [19]. Another remarkable possibility arises when we consider the
effect of an external gravitational wave on neutron star. With the possibility
of detecting any minute changes in its configuration through pulse
observations, the neutron star has the potential of performing as a Weber
detector of gravitational wave [22]. The possibility of such Weber detectors,
spread out in space, with their signals (carrying imprints of any GWs)
monitored at earth, has tremendous potential, especially in allowing us to re-
visit GW events whose signal may have passed through earth in past [23].

In this brief review we will focus on this particular aspect of NS physics,
namely the range of phenomena which can be probed utilizing the effect of
changes in the configuration of NS using high precision measurements of pulses
coming from a pulsar. We will begin in Section II with a discussion of salient
features of the QCD phase diagram, specially the high baryon density regime.
We will briefly discuss theoretical expectations of different phases in this
regime. We will also connect with the experimental situation and discuss what
parts of QCD phase diagram can be probed by the present and future
relativistic heavy-ion collision experiments, and which regimes may remain out
of reach of these terrestrial experiments, leading us towards cosmos, in
particular to neutron stars. Section III will then be devoted to basic physics
of neutron stars, including the superfluid phase in its interior as well as
the possibility of exotic QCD phases in the inner core. There are excellent
reviews on this subject (as well as on the subject matter of Section II, i.e.
QCD phase diagram). We will only recollect essential parts of these
discussions for self-completeness of the discussion here. Thus, we will also
briefly discuss how recent gravitational wave detections have allowed the
probe of NS interior properties. Section IV will be devoted to pulsars
recalling the extreme accuracy of pulsar timing observations. We will also
recall here the first (indirect) detection of gravitational waves (GWs) by
pulsar observations, as well as ongoing attempts of pulsar timing arrays for
detection of ultra low frequency GWs. In Section V we will discuss various
proposals from the literature for possible observational signatures of various
phases in NS interiors. Among these, glitches take a prominent role as well
established signals for the existence of superfluid phase in NS interior. We
will discuss here difficulty of this explanation in accounting for relatively
recent observations of anti-glitches. We will also discuss various proposals
for detection of the exotic color superconducting phases of QCD in NS
interior. Here we will also point to a new possibility where possibly the
highest observable baryon density phases could arise, that is in the matter
undergoing collapse to a black hole. For example, for a stellar mass black
hole, the Schwarzschild radius is about one third of the typical neutron star
radius. Thus it is possible that baryon densities in matter collapsing to
black hole may become about 20 times larger than that in a neutron star
(depending on the density profile, also the density contrast will be smaller
for a more massive black hole). Certainly, it will only be transient, lasting
only for tiny fractions of seconds, but still may allow observational
signatures of novel QCD phase transition which can occur with typical local
time scale of QCD, i.e. fm/c.

In Section VI we will discuss the implications of phase transitions occurring
inside a pulsar on its pulses. The consequences of phase transition (for
example from nuclear matter to QGP) occurring in the core of a neutron star in
terms of its effect on the moment of inertia have been discussed in the
literature with its observational implications on the spin rate of the neutron
star [24, 25]. These discussions primarily focused on the change in the
equation of state during the phase transition, and hence the main implications
related to changes in the diagonal components of the moment of inertia tensor
affecting the spin rate of the neutron star. However, phase transitions
necessarily produce density fluctuations, which perturb entire moment of
inertia tensor, including its off-diagonal components. This was discussed by
some of us in [19, 20], pointing out that phase transitions induced density
fluctuations modify the entire moment of inertial tensor of the pulsar which
affects pulse timings, but also induces modulations of the pulse profile. The
detailed modification of the pulses carries the information of statistical
nature of density fluctuations, and hence the precise nature of phase
transition occurring inside the NS interior. In Section VII we will discuss a
special case in detail when the density fluctuations are modelled in terms of
random components of MI tensor added to the unperturbed diagonal MI tensor of
the neutron star [21]. We will see here the effect on pulse timings as well as
the nature of modulations expected in the pulse profile. We will discuss that
even for very tiny density fluctuations, even if pulsar timings remain
extremely small, pulse profile modification may become relatively large. This
is because while pulse timing changes will be proportional to typical density
fluctuation magnitude $\epsilon$, the pulse profile modification will be
proportional to $\epsilon/\eta^{2}$ where $\eta$ is the NS deformation
parameter. ($\eta$ is typically very small $\sim 10^{-8}-10^{-4}$. This
observation will play an important role in later discussion when we discuss
possible detection of external GWs using NS deformations.) We will also
briefly discuss how the same technique can be used to detect other
perturbations occurring in neutron stars, e.g. collision with an asteroid.

In Section VIII we will discuss how phase transitions occurring inside NS may
provide a new high frequency source of GWs through density fluctuation induced
rapidly changing quadrupole moment. Section IX will change the direction of
discussion towards the effects of an external gravitational wave on the
neutron star configuration. Clearly, expected deformations in NS will be
extremely tiny. However, we will discuss how it may be possible to detect even
such tiny deformations utilizing the impressive accuracy of pulsar timing
observations, and in particular possible changes in the pulse profile from
induced wobbling of pulsar (recalling discussion from the results of Section
VII that even if pulse timing changes remain very small, pulse profile
modulations may become relatively large). We will therefore conclude in this
section that pulsars will effectively act as remotely stationed Weber
detectors whose GW perturbed signals may be observable on earth [22]. A very
important part of discussion here will involve the so called ringing of the
pulsar which will allow folding of very large number of pulses, thereby
tremendously increasing the signal to noise ratio, exactly what is done in a
Weber detector. We will make a point here on the importance of the quantity
the Quality factor Q of the NS matter which will directly determine the
strength of the ringing effect. While much focus has been there in the
literature in calculating the shear viscosity to entropy ratio of the QCD
matter (in the QGP phase as well as the hadronic phase), there is no such
discussion for the Quality factor Q.

This possibility of pulsars acting as Weber detectors, spread in cosmos will
lead to a truly remarkable possibility of re-vising past GW events. We will
discuss this in Section X, how some GW event (collisions of black holes,
neutron stars, supernova explosions etc.) from far away, whose signal may have
passed through earth in past, may become observable on earth again via
observations of certain pulsars which also get affected by the same GW source,
and transmit the perturbed pulses which are then later detected on earth [23].
Knowing the location of GW source and various pulsar coordinates, it is
possible to predict at what time in future such GW events may become
observable again through specific pulsars. Even the GWs from the earliest
detected supernova SN185, observed in AD 185, may be observable through
observations of perturbed pulses of specific pulsars, (in this case, via
pulsars J0900-3144 and pulsar J1858-2216 with perturbed pulsar signal arrival
date reaching earth during 2016-2049). The final Section XI will present
conclusions and future directions where we will discuss various limitations of
these proposals, as well as what specific efforts are needed to make these
proposed techniques more effective.

##  II QCD Phase diagram

This section will discuss salient features of the QCD phase diagram, and
possible experimental probes of different regimes of the phase diagram. The
earliest known part of the QCD phase diagram relates to the liquid/gas
transition of nucleonic matter. Indeed, it was the liquid drop model of the
nucleus which led Lise Meitner to propose theory of fission [26, 27]. Further
structure in this phase diagram emerged from astrophysics, with the notion of
neutron superfluidity in the cores of neutron stars [13]. Up to this stage, it
would be more appropriate to call it the phase diagram of nucleonic (hadronic)
matter. With the discovery of asymptotic freedom, Quantum Chromo Dynamics
(QCD) was established as the fundamental theory of strong interactions, with
quarks and gluons being the fundamental constituents, and gluons being the
mediators of the color force among these constituents. Hadrons were composed
of quarks and the nuclear (hadronic) interactions arise as residual
interaction from this fundamental color interaction. With the color gauge
group SU(3), the QCD Lagrangian is

$\displaystyle
L_{QCD}=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}+\sum_{\alpha}\overline{\psi}_{\alpha}\left(i\gamma^{\mu}D_{\mu}-m_{\alpha}\right)\psi_{\alpha}$

where $\alpha=u,d,c,s,t,b$ is the flavor index for quarks.

The covariant derivative is

$\displaystyle
D_{\mu}\psi_{\alpha}=\left(\partial_{\mu}-ig_{s}T^{a}A_{\mu}^{a}\right)\psi_{\alpha}$

$g_{s}$ is the strong interaction coupling constant. $\psi_{\alpha}$ is in the
3-dimensional fundamental representation of color $SU(3)$, with the generators
$T^{a}$ = $\frac{\lambda^{a}}{2}$, $a=1,..8$. $A_{\mu}^{a}$ are the 8 gluon
fields and $\lambda^{a}$ are the Gell-Mann matrices. The field strength tensor
is

$\displaystyle F_{\mu
v}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g_{s}f^{abc}A_{\mu}^{b}A_{\nu}^{c}$

where $f^{abc}$ are the antisymmetric structure constants for the Lie algebra
of SU(3). As isolated quarks are not seen, confinement is an essential part of
color interaction. Isolated objects can only be color singlets. Running of the
strong interaction coupling constant $g_{s}$ with energy exhibits the famous
asymptotic freedom. Writing $\frac{g_{s}^{2}}{4\pi}\equiv\alpha_{s}$,

$\displaystyle\alpha_{s}\left(Q^{2}\right)=\frac{4\pi}{\left(11-\frac{2n_{F}}{3}\right)\,ln\,Q^{2}/\Lambda^{2}}$

$\Lambda$ is the QCD scale typically taken to be of order 200 MeV. Thus color
interaction becomes weaker with large momentum transfer, equivalently at short
distances. On the other hand, at large distances, interaction becomes very
strong, which is consistent with the notion of color confinement.

With asymptotic freedom, it is but natural to expect that the behavior of
strongly interacting matter (system of hadrons) at ultra high temperatures, or
ultra-high densities, may show qualitatively different behavior compared to
the standard confining hadronic phase. One would expect in such extreme
regimes, typical interaction between quarks will be at very large
energies/short distances, hence will become much weaker. In the limiting case
quarks and gluons should become almost free, hence the notion of an almost
ideal gas of quarks and gluons, the quark-gluon plasma (QGP) phase of strongly
interacting matter. Extreme limits of temperature are only found in the very
early stages of the Universe, while very large baryon densities are expected
to arise in cores of neutron stars. These two systems thus provide natural
laboratories for the QGP phase of the QCD phase diagram. However, the QGP
phase in the early Universe is primarily probed theoretically, with no
reasonably direct observable signals expected for the present stage of the
Universe. (This is with the present understanding that for very small net
baryon densities, the quark-hadron transition is a cross-over. Some time back
when the possibility of a first order transition was also there, there were
tantalizing possibilities of forming quark nuggets which could be candidates
for dark matter [8].) Situation is different with neutron stars which are
directly accessible to observations. Observational data available for masses
and sizes of neutron stars put strong constraints on the equation of state of
the matter inside the neutron star. Detailed information about phases in the
interior of a pulsar is provided by observations of pulses, in particular from
the pulsar glitches [28]. A remarkable new probe of the neutron star interior
in terms of tidal deformability became available from the direct detection of
gravitational waves by LIGO/Virgo coming from binary neutron star (BNS) merger
events [18].

Study of QGP phase, and exploration of QCD phase diagram in general, has
acquired new dimensions with the arrival of relativistic heavy-ion collider
experiments (RHICE). Heavy ions (e.g. Lead, Gold, etc.) are accelerated to
ultra-relativistic speeds and made to collide, thereby creating a transient
ultra hot and dense system of strongly interacting matter. For high enough
center of mass energy, the temperatures are expected to be high enough for the
creation of a thermalized system of quarks and gluons, the so called quark-
gluon plasma (QGP) phase of QCD. These experiments have allowed unprecedented
control over the properties of strongly interacting matter created reaching
temperatures and densities which were only available so far in cosmos.
Controlled experiments have given wealth of knowledge about the system, from
thermodynamic properties to transport coefficients, and even allow to study
QCD matter in strong external electromagnetic field. The initial goal of these
experiments was to find the QGP phase, hence the drive for larger and larger
centre of mass energy. Indeed, from Super Proton Synchrotron (SPS) at CERN to
Relativistic Heavy-Ion Collider (RHIC) at BNL, USA, and then to Large Hadron
Collider (LHC) at CERN, center of mass energy per nucleon-nucleon pair has
increased from few tens of GeV at SPS to 200 GeV at RHIC, and 5.36 TeV at LHC
for nucleus-nucleus collisions. The temperatures reached at such energies have
certainly been high enough to convincingly demonstrate creation of the QGP
phase. However, increasing collision energy at the same time leads to lower
baryon density for the produced system. This is because of asymptotic freedom
of QCD, larger momentum transfer leads to weaker interaction, thereby reducing
possibilities of baryon stopping in the produced system. The QGP system
produced at ultra high energies, therefore, resembles more closely the QGP
phase of the early Universe which also had very small baryon number density.

With these experiments at ultra high energy, creating a QGP system with very
small baryon densities, reaching a level of maturity, the attention is now
being focused on the other extreme condition, namely very high baryon density
regime of QCD matter. QCD matter is also expected to go to QGP phase in this
regime, but with different properties. While high temperature low baryon
density QGP teaches us about the early Universe, the regime of very high
baryon density QGP directly relates to the interior of neutron stars. In fact,
it is now realized that the QCD phase diagram has extremely rich structure
precisely in this ultra-high baryon density regime. The situation is quite
like the phase diagram of water. While at high temperatures we have liquid and
gas phases, at ultra low temperatures and densities there are numerous phases
of ice which appear. Possibilities of various phases of QCD are illustrated in
the QCD phase diagram in Fig.1. Here we show the phase diagram in terms of two
most important variables, temperature $T$ and baryon chemical potential $\mu$
(representing baryon density). In different situations, one can use other
variables such as strangeness chemical potential etc. For theoretical
discussions it is also useful to invoke another axis where quark masses can be
varied. This allows for the discussion of various possible phases, and
critical points in the phase diagram. We will confine our discussion to the
standard 2-d phase diagram as shown in Fig.1, in the $T,\mu$ plane.

Figure 1: QCD phase diagram in the $T,\mu$ plane (from ref.[29])

The phase diagram in Fig.1 shows possible phases in different regions of
temperature $T$ and baryon chemical potential $\mu_{B}$. The figure also shows
where different regions of the phase diagram are expected to arise. It is
useful to consider two different regimes in the entire phase diagram. One is
for relatively low baryon density with baryon chemical potential values less
than about 1 GeV, or ultra high temperatures, and the other regime is for much
higher values of $\mu_{B}$ and relatively low temperatures.

###  II.1 Low baryon density/ultra high temperature regime

There are several regions for separating this regime. First, there are only
two phases to discuss here. The hadron gas phase appears at low values of
$\mu_{B}$ and relatively low temperatures. Boundary of this phase is denoted
by the white curve, part solid, joined to dashed white curve, the joining
point denoted as the critical point. The solid white curve denotes line of
first order transition, which ends at the critical end point where the
transition is second order (a continuous phase transition). The dashed line
denotes a cross-over transition, which is not a proper phase transition (in
the sense that the partition function remains analytic across this boundary).
The entire region of the phase diagram outside this boundary, and above the
solid yellow line is the second phase, the quark-gluon phase. Importantly, all
the present and future planned heavy-ion collision experiments probe this part
of the phase diagram only. First system to discuss here will be the early
Universe denoted by the solid white arrow (almost) on the $\mu_{B}=0$ line.
The temperatures reached in the Universe in its earliest stages can reach
close to the Planck temperature ($\sim 10^{19}$ GeV), depending on the
inflationary models. The Universe, expanding and cooling from these earliest
stages, reaches the quark-hadron transition temperature (where the solid
dashed line intersects the $T$ axis). Lattice calculations are under good
control for $\mu_{B}=0$ case, and give the value of quark-hadron transition
temperature to be $T_{c}=156$ MeV. This value of temperature is expected to be
reached in the early Universe when its age is few tens microseconds. Upto that
time, the Universe was filled with a plasma of quarks, gluons, as well as
other elementary particles, e.g. leptons, photons. After this time, the
Universe undergoes quark-hadron transition, (which is expected to be a smooth
cross over from lattice calculations) when quarks and gluons get confined into
system of hadrons. As we mentioned, with smooth cross-over the earlier much
discussed quark nugget scenario [8] is no more possible. Though, there have
been alternate proposals for the formation of quark nuggets which do not
depend on the nature of quark-hadron phase transition [30, 31].

The ultra-relativistic collisions at RHIC and LHC, with centre of mass
energies ranging from 200 GeV to 5.36 TeV (per nucleon-nucleon collision) lead
to a fireball with temperatures well above the quark-hadron transition
temperature with relatively low values of $\mu_{B}$ ($\sim 50$ MeV for 200 GeV
collision energy). As we mentioned, increasing collision energy leads to
smaller values of $\mu_{B}$ due to asymptotic freedom of QCD. For these large
values of collision energies, confining forces become irrelevant, so basically
the collision is between quarks and gluons (partons) contained in each
colliding nuclei. For very large collision energies, partons scatter little,
basically the colliding nuclei go through each other. Even with smaller
scattering at large energies, the energy available for secondary parton
production monotonically increases with collision energies, leading to higher
temperatures. But with smaller scattering, proton stopping is less, hence net
baryon number of the produced QGP system is monotonically decreasing function
of collision energy. (Strong interactions conserve baryon number, so baryon
density in the central fireball can only come from the stopping of initial
valance quarks of the colliding nuclei.)

It is believed that at these collision energies, the expanding QGP system
undergoes quark-hadron transition through the dashed white line in Fig.1, that
is the cross-over line. This is with several estimates of the location of the
critical point. Though it has not been possible to have a good control over
lattice calculations for non-zero $\mu_{B}$ (due to the so called fermion sign
problem), several techniques have been devised to extend lattice calculations
for non-zero $\mu_{B}$. Recent estimates suggest the location of the critical
point to be about ($T\sim 156$ MeV, $\mu_{B}\sim 65$ MeV) [32]. It should be
mentioned that there are strong theoretical arguments for the existence of a
first order transition line at non-zero values of $\mu_{B}$. These are
typically based on various effective field theory models. Combined with the
solid knowledge of cross-over transition at $\mu_{B}=0$, it automatically
follows that the first order line has to end somewhere at a critical end
point.

Physics near the critical point (in the critical regime) is dominated by
critical fluctuations showing universal properties. Experimental evidence for
such fluctuations will give deep insight into this very important part of the
QCD phase diagram. With this aim, the beam energy scan program was designed at
RHIC, with collision energies as low as 7.7 GeV (thereby creating the QGP
system with much higher values of $\mu_{B}$, hopefully evolving through the
critical regime).

One more region deserves mention in this part of the phase diagram, denoted as
Nuclear matter near $\mu_{B}=900$ MeV. There is a short line of first order
transition (not shown in Fig.1) which corresponds to liquid-gas transition of
nuclear matter. What is more interesting is that at further larger values of
$\mu_{B}$ but still within the hadronic phase (so not crossing solid white
curve), there is the possibility of nucleonic superfluid phase. It is
precisely this superfluid phase, and associated superfluid vortices which are
supposed to play crucial role in neutrons stars, especially in relation to the
phenomenon of pulsar glitches. We will have details discussion of this phase
in the next section.

###  II.2 Very high baryon density, low temperature regime, color
superconductivity

Discussions relating to this part of the phase diagram have intensified
relatively recently. There have been insightful exchanges of ideas between the
usual condensed matter physics and this area, which is appropriately being
called as condensed matter physics of strongly interacting matter. The phases
in the part of the phase diagram are typically termed as color superconducting
phases. Note here that for this part of the QCD phase diagram, we are not
including the so called high baryon density QGP phase which will be the part
above the solid yellow line with very large values of $\mu_{B}$. Though there
will be important differences in the physical properties of QGP with high
baryon density from QGP with very low baryon density (and high temperature),
they are the same thermodynamic phase, with no phase boundary expected between
these two regimes. That is why we included high $\mu_{B}$ QGP phase in the
preceding subsection. Discussion below is primary taken from refs.[4, 5, 6, 7]
which can be consulted for details.

High $\mu_{B}$ with low temperature changes the physics qualitatively giving
rise to new thermodynamic phases. The basic physics of these phases lies in
the realization that at very high values of $\mu_{B}$, and relatively low
temperatures, the physics will be governed by low energy excitations near the
Fermi level. Standard BCS Superconductivity has taught us that any attractive
interaction between fermions at the Fermi level destabilizes the Fermi level,
forming Cooper pairs of fermions. First we note that for the relevant values
of chemical potential, only relevant quark flavors are $u,d,$ and $s$ quarks.
(We do not discuss here heavier quarks as there is no physical system known
where one could reach quark chemical potential high enough so that heavier
quarks can play any important role for these condensed phases of QCD.) At
ultra high chemical potential, of order 500 MeV for quarks, asymptotic freedom
will make perturbative calculations reliable, especially for $u$ and $d$
quarks. Consider scattering of light quarks with one gluon exchange as shown
in Fig.2,

Figure 2: quark-quark scattering with one-gluon exchange

Here, $i,j,k,l=1,2,3$ (or r,g,b) refer to colors of u and d quarks. Now we
note that the amplitude for this process is the same as that for the QED
process $e^{-}\mu^{-}\rightarrow e^{-}\mu^{-}$ with the replacement of the
electromagnetic coupling $e$ by the strong coupling $g_{s}$ (i.e. replacing
$\alpha$ by $\alpha_{s}$, and inclusion of the color factor),

$C_{F}(ik\rightarrow jl)={1\over 4}\lambda^{a}_{ij}\lambda^{a}_{kl}$ (1)

quark-quark can combine in two ways, with $3\times 3=3^{*}+6$. The color
factors for the two representations are, $C_{F}(3^{*})=-2/3$ and
$C_{F}(6)=1/3$.

As the EM potential between $e^{-}$ and $\mu^{-}$ is repulsive, the qq
potential is attractive in the 3* channel, and repulsive in the 6 channel. BCS
pairing of quarks in the $3^{*}$ channel leads to the color superconducting
phase. Depending on the relative masses of the quarks forming the condensate,
different phases are possible.

Color-Flavor Locked (CFL) phase:

With color antisymmetric ($3^{*}$ channel), spin antisymmetric (for
${}^{1}S_{0}$ pairing), we conclude that the condensate should have flavor
antisymmetric. For very high chemical potential it may be reasonable to treat
all three quarks ($u,d,s$) as massless. Condensate for this most symmetric
case, (with ${}^{1}S_{0}$ spin pairing) will have the following structure

$<q_{i}^{\alpha}q_{j}^{\beta}>\sim\Delta_{CFL}(\delta^{\alpha}_{i}\delta^{\beta}_{j}-\delta^{\alpha}_{j}\delta^{\beta}_{i})=\Delta_{CFL}\epsilon^{\alpha\beta
n}\epsilon_{ijn}$ (2)

where $\alpha\beta$ are flavor indices and $ij$ are color indices. Note that
the condensate is invariant under equal and opposite rotations of color and
(vector) flavor. Hence, it leads to the following spontaneous symmetry
breaking

$SU(3)_{Color}\times SU(3)_{L}\times SU(3)_{R}\times U(1)_{B}\rightarrow
SU(3)_{C+L+R}\times Z_{2}$ (3)

Thus color symmetry of QCD $SU(3)_{C}$ is spontaneously broken. Three flavor
Chiral symmetry is also spontaneously broken. Interestingly, QGP phase
restores the chiral symmetry (at least for $\mu_{B}=0$ case). Here, at large
$\mu_{B}$ also, we expect chiral symmetry restoration in the high density QGP
phase which occurs at higher temperatures. But at low temperatures, high
$\mu_{B}$ quark-gluon phase breaks the chiral symmetry spontaneously. We will
discuss the situation with chiral symmetry further below.

Spontaneous Breaking of SU(3) color symmetry implies that all gluons become
massive, (so there are no long range color forces even in this deconfined
phase), hence the name Color superconductor. We mention here that the role of
photon in this broken phase is played by the usual photon with a small
admixture of gluon, see [4] for details. Also, note that spontaneous breaking
of chiral symmetry, here is not by a color neutral condensate (as for the
$\mu_{B}=0$ case, but by a colored condensate. This implies multiplets of
colored hadrons, (and also colored Skyrmions),

2SC Pairing:

For Cooper pairing of 2 light quarks, we get the so called $2SC$ phase. This
breaks color group $SU(3)_{C}$ to color $SU(2)$. In this phase, 5 out of 8
gluons become massive.

Crystalline Color Superconducting phase:

This phase is similar to the so called Larkin–Ovchinnikov–Fulde–Ferrell (LOFF)
phase in condensed matter systems: When chemical potential is not very large
compared to the strange quark mass, then difference between $u,d$ quark masses
and $s$ quark mass becomes significant. In that case Cooper pairing of
different quarks (having different Fermi momenta), leads to spatial modulation
of the order parameter. This implies spontaneous breaking of spatial
translations (and rotation) symmetries.

We are only giving brief details of various phases here, just to give an idea
of the richness of the QCD phase diagram in different regimes. There are
excellent references available in literature, we have listed some here in
refs.[4, 5, 6, 7]. We end this section with brief discussion the important
subject of chiral symmetry in QCD.

###  II.3 Chiral Symmetry in QCD

We here briefly recall the notion of chiral symmetry in QCD. For details,
literature can be consulted, e.g. ref. [33]. Note that we have been using the
term quark-hadron transition in the above discussion. This term can have two
meanings. One is the confinement-deconfinement (C-D) transition where a system
of hadrons, (with quarks/gluons being confined inside hadrons) makes a
transition to the deconfined phase of a plasma of quarks and gluons. The other
meaning can refer to the chiral transition. For 2 massless flavors, QCD
Lagrangian has exact $SU(2)_{L}\times SU(2)_{R}$ symmetry, called as 2-flavor
chiral symmetry, which corresponds to independent transformations of left and
right components of $u$ and $d$ quarks. Hadron spectrum does not show any such
doubling of mass spectrum, but it does show multiplet structure of
$SU(2)_{isospin}$. This leads to the conclusion that the chiral symmetry
$SU(2)_{L}\times SU(2)_{R}$ is spontaneously broken to the diagonal subgroup
$SU(2)_{isospin}$, with pions as the Goldstone bosons. For three massless
flavors, all $SU(2)$ groups should be replaced by $SU(3)$ groups, leading to
3-flavor chiral symmetry breaking. Of course, quarks are not massless, this
leads to explicit breaking of chiral symmetry, leading to small masses for the
Goldstone bosons, and contributing to mass differences within the multiplets.
The explicit symmetry breaking being relatively small, especially for the
2-flavor case, the notion of chiral symmetry in QCD has been immensely useful,
especially in constructing effective field theory models which capture physics
at low energy scale. Chiral sigma models, Nambu-Jona-Lasinio (NJL) models etc.
are all bases on the notion of chiral symmetry breaking and have been the only
tools for discussions of QCD phase diagram at high baryon densities.

It is believed that chiral symmetry transition and confinement-deconfinement
(C-D) transition are the same. Lattice results support this idea (though at
times there are differing results also). Conceptually, these two transitions
are completely different. Indeed, different order parameters characterize
these two transitions, with expectation value of the Polyakov loop
characterizing the C-D phase, and the ${\bar{\psi}}\psi$ condensate
charactering the chiral transition ($\psi$ being the quark field). We will
have discussion of some of these (especially the Polyakov loop condensate)
later in Section VI. The conceptual difference between these two transitions
becomes clear when we consider high $\mu_{B}$ phases, in particular the color
superconducting phases. These phases appear in the regime where inter-quark
separation is so small that confining forces are irrelevant. This is like a
plasma of deconfined quarks and gluons, but not the usual QGP phase as here
thermal effects are insignificant compared to the quantum statistics effects.
Thus, we have chiral symmetry breaking here (as discussed above for the $CFL$
phase), even though we have a system of deconfined quarks and gluons. Note
that although we have massive gluons here, there is no relevance of color
singlet objects here. Indeed, as mentioned above, we expect colored hadrons
here arising from the chiral symmetry breaking via a colored di-quark
condensate.

##  III Neutron stars

This section will discuss basic physics of neutron stars, including the
superfluid phase in its interior as well as the possibility of exotic QCD
phases in the inner core. We will also briefly discuss how the recent
gravitational wave detections have allowed the probe on NS interior
properties.

Basics of a neutron star : The existence of neutron stars as one of the
possible end states of massive stars was predicted by Walter Baade and Fritz
Zwicky in 1933 [34], a long time before the discovery of the pulsating neutron
star (little green man) in late 1967 [35]. Neutron stars are the remnants of
the supernova explosion of supergiant stars of mass in the range $10~{}\text{
M}_{\odot}-20~{}\text{M}_{\odot}$. During its formation, gravity squeezes the
matter to achieve extremely high baryon density. Gravitational collapse is
counterbalanced by the neutron degeneracy pressure (along with repulsive
nuclear forces) leading to the formation of a stable neutron star. The radius
(R) of a neutron star lies in the range (10 - 14) km with mass in the range
$~{}\text{1.1 M}_{\odot}-2.1~{}\text{M}_{\odot}$. Thus the average mass
density is of the order of nuclear matter’s saturation density
$\rho_{0}=2.8\times 10^{14}~{}\text{g cm}^{-3}$. The discovery of the neutron
star put the discussions of such compact astrophysical objects on deep
footing, with the realization that it provides the opportunity for testing
exciting physics at a high baryon density regime that cannot be tested
otherwise by terrestrial experiments. However, with the opportunity, new
theoretical challenges arise, of understanding the properties of dense
interior materials, the inner core in particular, and relating these
properties to various testable observables.

There have been continuous efforts towards a theoretical understanding of the
internal structure of neutron stars keeping in view its astrophysical
implications and observational constraints. These include the mass-radius
relations, the moments of inertia (MI) of the star (including the fractional
contribution of the solid crust and superfluid/superconductor components), the
extent of rigidity of the outer crust, and the deformation from the sphericity
etc. The above quantities, in turn, determine the frequency of pulsar’s free
precession, glitch size in the context of various pulsar glitch models
(crustquake, superfluid-vortex model etc.), or the feasibility of GW emission
etc.

The standard approach of constructing a model of neutron star’s internal
structure is to implement hydrostatic equilibrium of a gravitating fluid
system, resulting in the well known Tolman-Oppenheimer-Volkoff (TOV) equations
[36, 37] ($c=1=G$),

$\displaystyle\frac{dP(r)}{dr}$ $\displaystyle=$
$\displaystyle-\frac{(P+\rho)(M+4\pi r^{3}P)}{r(r-2M)};$ (4)
$\displaystyle\frac{dM(r)}{dr}$ $\displaystyle=$ $\displaystyle
4\pi\rho(r)r^{2}.$ (5)

The solutions of the above set of equations provide the density profile
$\rho(r)$, including the mass and the radius of the neutron star, provided the
equation of state (EOS) $P=P(\rho)$ and the central density $\rho(0)$ are
supplied. The above procedure thus produces a neutron star structure model for
a fixed value of central density, which finally determines various physical
parameters, such as the mass, radius, moment of inertia, etc., of the star.
However, the main challenge lies in providing the EOS, the most critical input
for solving the Eqs. (4,5). Although the EOS of the outer crust and, to some
extent, the inner crust are known [38], the first principle calculations of
the many-body QCD interaction relevant to the inner core are unknown. Thus,
one has to resort to phenomenological models for extracting the EOS for the
core region. One can then use the experimental measured values of various
physical parameters to test the predictions based on constructed stellar
models. For most practical calculations, one uses a set of canonical values
$M=1.4~{}\text{M}_{\odot}$, R = 10 km, and MI $=10^{45}~{}\text{g~{}cm}^{2}$.
For the above set of values, and from the observations of various pulsar
events, such as glitches (to be discussed later), one broadly takes the
neutron star’s internal structure as shown in Fig. 3.

Figure 3: The expected internal structure of a neutron star. Credit : Dany P
Page (https://phys.org/news/2015-09-neutron-star.html).

Observation of glitches in the rotation of young pulsars indicates a solid
crust containing $\geq 1.4\%$ of total MI [39], the outer layer of which
consists of crystalline solid iron nuclei and a sea of degenerate electrons
with mass density $\rho\simeq 10^{6}~{}\text{g cm}^{-3}$. Going deeper into
the inner crust region, as the density increases from $\rho\simeq
10^{11}~{}\text{g cm}^{-3}$ onward, protons and electrons start combining to
form neutrons for creating neutron-rich nuclei. Eventually, it becomes
energetically favorable for the neutrons dripping out of the nuclei and
forming a sea of free (unbounded) neutrons. A few hundred meters thick inner
crust region with the density ranging from approximately $10^{11}~{}\text{g
cm}^{-3}$ to $10^{14}~{}\text{g cm}^{-3}$ plays a crucial role in the theory
of glitches. There is a strong belief, supported by glitch observations, that
a significant fraction of free neutrons in this region (see Fig. 3) are in a
${}^{1}S_{0}$ neutron superfluid state. Further deeper, below the inner crust
region, a more favorable ${}^{3}P_{2}$ neutron superfluid state is believed to
co-exist with a ${}^{1}S_{0}$ proton superconductor.

The composition of the inner core is highly speculative. The states of matter
at the high pressures in the deep interior can form various hadronic states,
such as hyperons, condensed-pions etc. The central region of the neutron star
can also accommodate different QCD phases, such as QGP, CFL, 2SC, etc. The
core centre can become quite mysterious for massive neutron stars. One needs
well defined signatures that can be tested to confirm possibilities of any
such phases which remains an important theoretical challenge.

We will be discussing later various proposals and observational techniques for
probing physics of pulsar interior. But, it seems most appropriate to close
this section with a discussion of a truly exciting recent development towards
probing neutron star interior. This is the direct detection of gravitational
waves by LIGO/Virgo which, for GWs originating from a BNS merger allows to put
observational constraints on deformation of neutron stars during last stages
of coalescence, thereby directly probing the stage of matter inside the
neutron star.

Probing NS interior with direct detection of gravitational waves from BNS
merger

Possibly the most important advancement in experimental General Relativity in
recent years has been the direct detection of gravitational waves by LIGO and
Virgo coming from distant events of binary black hole mergers and
subsequently, binary neutron star mergers. Black hole merger events allow the
possibility of directly probing the dynamics of intense gravity regime of near
horizon regions of black holes by analysis of the GW waveform corresponding to
the near coalescence regime of black holes. Similarly, for binary neutron star
merger events, GW waveform contains information about strong tidal
deformations towards the end of spiral-in of the neutron stars, when
separation between the two neutron stars approaches neutron star size. In such
regime, the coalescence is accelerated by a quadrupolar deformation of NS by
the tidal field of the companion NS. Indeed, GW waveform observations have
been used [18] to put constraints on the dimensionless tidal deformability
$\Lambda={2\over 3}K_{2}R^{5}/M^{5}$ (in gravitational units with G = c = 1)
where $k_{2}$ is called the second Love number. ($\Lambda$ is related to the
tidal deformability $\lambda$ discussed later in Sec. IX, by
$\Lambda=\lambda/M^{5}$.) value of $k_{2}$ and hence the tidal deformability
depends on the equation of state of NS matter. GW observations thus provide a
completely independent probe of the equation of state of NS matter, which is
usually constrained by mass-radius relationship of neutron stars. Direct
observation of source identification for BNS mergers by resulting
electromagnetic radiation (which would be absent for black hole mergers) has
started the new chapter of multi-messenger astronomy in our exploration of
cosmos. With a range of observations, from gravitational waves, to
electromagnetic radiation in a range of energies, along with the possibility
of neutrino bursts from such BNS merger events will jointly give a powerful
probe of the structure and property of neutron stars.

##  IV Pulsars

This section will be devoted to pulsars recalling the extreme accuracy of
pulsar timing observations. We will also briefly discuss here the first
(indirect) detection of gravitational waves (GWs) by pulsar observations, as
well as ongoing attempts of pulsar timing arrays for detection of ultra low
frequency GWs.

Pulsars Timing :  The atomic clock-like stability of pulsar rotation period
allows one, through monitoring of the pulsar rotation, to study a rich variety
of phenomenon affecting the propagation of pulses while reaching earth. Most
applications of pulsars involve a powerful technique known as the pulsar
timing. Measurement of a sequence of time of arrival (ToA) of pulses over
intervals ranging from hours to decades is the basis of pulsar timing [40].
These ToAs are first transferred, normally to the solar-system barycentre, to
remove the effects of rotation and orbital motion of the Earth. The amount of
useful information that can be extracted critically depends on the precision
at which the pulse arrival times are measured. In order to understand the
pulsar timing, we will take the example of isolated pulsars and describe the
rotation in the pulsar’s comoving frame.
For an isolated pulsar, one can express the spin frequency $\nu$ (or time
period $P$) in a Taylor series about some reference epoch $t_{0}$ [40]

$\nu(t)=\nu_{0}+\dot{\nu}(t-t_{0})+\ddot{\nu}\frac{(t-t_{0})^{2}}{2}+...$ (6)

Where $\nu_{0}=\nu(t_{0})$ is the pulsar’s spin frequency at $t_{0}$ and
$\dot{\nu},\ddot{\nu},...$ are the higher order time derivatives of $\nu$ to
be evaluated at $t_{0}$. These parameters are associated with some physical
process, knowledge of which provides valuable information about the underline
process. For a normal (i.e., rotation-powered) pulsar, the period $P$
($\sim\text{0.3 s - 3 s}$) and its first derivative $\dot{P}$ ($\sim
10^{-15}~{}\text{s~{}s}^{-1}$) are observed with high accuracy through timing
measurements. These parameters capture the spin-down history of the isolated
pulsar. On the other hand, the millisecond pulsars (MSP) have the most exotic
applications, including the detection of low-frequency gravitational waves
(see below) because of their extreme stability in their periodicity ($P\sim 3$
ms with $\dot{P}\sim 10^{-20}~{}\text{s~{}s}^{-1}$) compared to the normal
pulsar.

Indirect detection of GWs :  The celebrated discovery of the pulsar PSR
B1913+16 in binary star system by Russell Hulse and Joseph Taylor in 1974 [15]
using the data from the Arecibo radio telescope opened up the possibilities
for the study of relativistic gravity in moderately strong field regime. The
above discovery provided the first indirect quantitative confirmation test in
favor of the existence of gravitational waves within the framework of
Einstein’s theory of gravity [41, 16, 42]. The Hulse–Taylor pulsar (PSR
B1913+16) is a binary star system composed of a pulsar of mass $\simeq
1.44~{}\text{M}_{\odot}$ and the silent companion neutron star of mass $\simeq
1.39~{}\text{M}_{\odot}$ [43], moving around in elliptical orbits about their
center of mass. As per Einstein’s theory of gravity, the orbital period of
this binary system is expected to decay with time. The heartening agreement
between the observed data with the theoretical prediction (see Fig. 4) not
only provided conclusive evidence for the existence of gravitational waves; it
laid the foundation for the belief of the possibility of direct detection of
GWs. The remarkable first-ever direct detection of GWs in 2015 by LIGO [17],
arising from a binary black hole merger, fulfills that belief and opens a new
era in gravitational wave astronomy. Since then, there have been quite a few
significant detections of GWs. The peak strain amplitude ($h_{0}$) for all
these detections has been in the range $10^{-21}~{}-~{}10^{-22}$.

There were several theoretical studies [44, 45, 19, 46], where the authors
suggested that isolated pulsars can be a potential source of GWs. A few other
neutron star activities, such as the neutron star flaring, the formation of
hyper-massive NS following binary coalescence, etc., are capable of exciting
quasi-normal modes of a pulsar, resulting in the GWs emission. The primary
purpose of such studies is to probe the internal structure of the pulsars.
Hopefully, the more sensitive ground-based advanced detectors, namely, aLIGO,
VIRGO, Einstein Telescope (ET) etc., will be able to measure the gravitational
waves produced by isolated pulsars shortly and answer a few questions on
pulsar physics. In fact, with the above purpose, even before the direct
detection of GWs, there were a few attempts for GWs searches from isolated
pulsars. The GWs associated with the timing glitch in the Vela pulsar in
August 2006 [47] was one among those searches worth mentioning. The observed
timing noise, i.e., glitch, can be one such activity that can excite quasi-
normal modes in the pulsar and cause GWs. Although the searches during the
August 2006 Vela pulsar timing glitch produced no detectable GWs [17], with
the improving sensitivity of advanced detectors, continuous attempts in this
direction may produce more conclusive results shortly.

Figure 4: The shift in the periastron passage of the binary pulsar PSR
B1913+16 with time, caused by gravitational radiation. Figure taken from Ref.
[43]

Pulsar Timing Array (PTA) : Pulsar timing arrays are the GW detector setups
provided by nature itself, in the form of a population of highly stable
millisecond pulsars, with timing accuracies of $\sim 10$ ns over several
years, with arm lengths of galactic scale. They are sensitive to much lower
frequencies than ground-based instruments. With precise pulsar timing, an
array of pulsars could detect extremely low-frequency sources with typical
frequencies less than $10^{-6}$ Hz. The origins of GWs with such low frequency
can be supermassive black hole mergers with masses in the range
$(10^{9}-10^{10})~{}\text{M}_{\odot}$, the stochastic background of GWs
arising from cosmological phase transition during the early stages of the
Universe, or otherwise, or more exotic objects such as cosmic strings. The
basic principle is that the set of stable MSPs serves as an array of clocks
whose time , as observed on earth, would be modulated by gravitational waves
passing through the space between MSP and Earth. With observations of many
pulsars, phenomena which affect all pulsar periods in a correlated way can be
separated from phenomena which affect different pulsars differently. For
example, a stochastic gravitational wave background can be separated from
errors in the time standard because of their different dependence on pulsar
sky position. For example, clock errors will lead to all pulsars having the
same Time of Arrival (TOA) variations (monopole signature), solar-system
ephemeris errors will lead to a dipole signature. In contrast, gravitational
waves will lead to a quadrupole signature. These effects can be separated if
one has sufficient number of widely distributed pulsars. With this purpose,
there is a worldwide effort to search for and observe the set of stable
millisecond pulsars for detecting gravitational waves through pulse
modulation. There are four current efforts in this direction, operating under
the joint umbrella of the International Pulsar Timing Array (IPTA [48]); North
American Nanohertz Observatory for Gravitational Waves (NANOGrav) in the USA,
the Parkes Pulsar Timing Array (PPTA) in Australia, the European Pulsar Timing
Array (EPTA), and the Indian Pulsar Timing Array Project (InPTA). IPTA aims to
construct the most sensitive low-frequency gravitational wave detector, which
can be achieved through sharing resources among the stakeholders and creating
combined pulsar timing data sets. The current sensitivity of the experiments
is exciting from the perspective of the potential detection of GWs through
PTA.

##  V Observational aspects of neutron star interiors

Here we will discuss various proposals from the literature for possible
observational signatures of various phases in NS interiors. Among these,
glitches take a prominent role as well established signals for the existence
of the superfluid phase in NS interior. We will discuss here difficulty of
this explanation in accounting for relatively recent observations of anti-
glitches. We will also discuss various proposals for detection of the exotic
color superconducting phases of QCD in NS interior.

Pulsar glitches : Pulsars are magnetized rotating neutron stars that emits
periodic short pulses of electromagnetic radiation with periods between 1.4 ms
- 0.3 s. The non-alignment of the rotation axis with the magnetic axis causes
the light-house-like appearance of the pulses to the observer at Earth. Amid
the pulsars’ extraordinarily stable rotational frequency, many pulsars show
sudden spin-up events (glitches) followed by a period of slow recovery.

Figure 5: A schematic representation of a typical glitch pattern of a pulsar.
Here Q is the recovery fraction, which measures the part of $\Delta\Omega$
which decays. The post-glitch recovery time scale typically ranges from a few
days to weeks. (Taken from: http://hdl.handle.net/11343/36537.)

Since the discovery of the first glitch in the Vela pulsar [49], many glitches
have been observed and reported to date [50]. A typical glitch pattern is
shown in Fig. 5. The fractional change of rotational frequency, i.e., the
glitch size $(\Delta\Omega/\Omega)$ lies in the range $10^{-11}-10^{-5}$ with
an average inter-glitch time of a few months to a few years. The pulsar
recovers from weeks to months to a period close to the pre-glitch value. The
oldest theoretical model for pulsar glitches [51], namely, the crustquake
model, assumes the existence of a deformed solid crust of a pulsar. The
oblateness can be characterized by the parameter
$\epsilon=\frac{I_{zz}-I_{xx}}{I_{0}}$. Where $I_{zz}$, $I_{xx}$ and $I_{0}$
are the moment of inertia about the z-axis (rotation axis), x-axis, and the
spherical star, respectively [52]. The sudden decrease of oblateness
$\Delta\epsilon$ decreases the MI, resulting in a spin-up event (following
angular momentum conservation), i.e.,
$\frac{\Delta\Omega}{\Omega}=-\frac{\Delta I}{I_{0}}=\Delta\epsilon$. It was
immediately realized [52] that inter-glitch time being proportional to
$\Delta\epsilon$, successive large-size glitches need longer waiting periods.
Thus, crustquake might be responsible for producing smaller Crab-like glitches
$\Delta\Omega/\Omega\simeq 10^{-8}$; the model is incompatible with Vela-like
large-size glitches $\Delta\Omega/\Omega\simeq 10^{-6}$.

Anderson & Itoh proposed the most popular superfluid-vortex model in 1975
[28]. The model assumes the existence of neutron superfluidity in the inner
crust, with the core being in a superfluidity/superconducting state.
Interestingly, superfluidity in a neutron star was hypothesized [53] a long
time before the above-proposed glitch model. The basic idea of the vortex
model is that the vortices (spinning neutron star causes an array of quantized
vortices for stability) act as an angular momentum reservoir while being
pinned to the nuclear sites [54, 46, 55, 56]. As the neutron star slows down
due to radiation loss, differential angular velocity $\delta\Omega$ between
the inner crust superfluid and the rest (crust+core) is built until it reaches
a critical value $(\delta\Omega)_{c}$. Once the Magnus force overcomes the
pinning force, just above $(\delta\Omega)_{c}$, many vortices ($\sim 10^{18}$)
are released from the pinning site and share the excess angular momentum to
the rigid crust-core co-rotating system, causing the pulsar to spin-up. The
vortex model not only can account for large-size Vela-like glitches ($\sim
10^{-6}$), the long-relaxation time scale (weeks to months) associated with
the post-glitch recovery phase also arises naturally in this picture, thus
providing indirect evidence of superfluidity in the interior [57]. In this
context we mention that an interesting spin-down event was observed a decade
earlier in the magnetar 1E 2259+586. As per the report [58], X-ray timing
observation [59] of the magnetar clearly shows an anti-glitch. Evidence for
anti-glitches in the X-ray pulsar NGC 300 ULX-1 is also reported recently
[60]. Such events challenge the standard glitch theory and suggest the need
for rethinking this issue [19].

Mesonic condensate phases

It has been suggested that at high baryon densities in the core of neutron
stars, apart from the neutrons and protons, various mesonic condensates may
form. Formation of pion condensate has been suggested, (see, e.g. [61, 62,
63]) and it has been argued that it may lead to significant modification of
equation of state. Kaon condensates have also been considered and its effects
on softening the equation of state, hence constraining the maximum mass of
neutron stars has been discussed [64, 65]. These condensates may also affect
the cooling rate of the neutron star. Along with such condensates, dominant
presence of hyperons in the NS interior has been discussed, and constraints on
maximum mass of NS have been derived [66]. Effect of hyperons on the
possibility of pion condensates has been discussed in [67] and it has been
argued that with hyperons, it is unlikely that s-wave pion condensate could
form in the NS interior.

Quark matter core:

At high baryon densities, the core of the neutron star is likely to convert to
quark matter. With the outer regions being in the hadronic phase, there will
be a phase boundary separating the quark core from the hadronic region. With
quark matter equation of state for the core, the mass and size relationship of
the neutron star is affected [68] which can be probed by observations, e.g.
using thermonuclear X-ray bursts from matter accretion on NS surface in binary
systems [69]. The compactness of the neutron star ($M/R$) is directly probed
by the tidal deformability from gravitational wave detection from binary
neutron star mergers. Possibility of strange quark matter in the high density
core of neutron star has been discussed where three-flavor quark matter is
assumed more stable than nuclear matter at low density [8, 9, 10, 11, 12]. The
composition of the core also affects the cooling of neutron star by neutrino
emission.

Color superconducting phases : The quark matter core itself can have very rich
phase structure at very high baryon densities, as we discussed earlier in the
Section II. In the QCD phase diagram, we have observed that the QCD theory
allows the appearance of various exotic phases at the low-temperature ($T$)
and very high baryon chemical potential ($\mu_{B}$) regime. The neutron star’s
core can provide such conditions to accommodate those phases. One such phase,
the color superconducting phase, may arise due to spontaneously broken
$SU(3)_{C}$ color and chiral symmetry for three light quark flavors (u, d &
s). For two light quarks, color group $SU(3)_{C}$ is broken down to color
$SU(2)_{C}$ causing the so-called $2SC$ phase. Observational signatures of
these phases have been discussed in the literature [5, 6]. For example,
central core with CFL phase leads to suppressed cooling by neutrino emission,
and also has smaller specific heat. Thus the total heat capacity and neutrino
emission of NS with CFL core will be dominated by outer layer which are in the
standard nucleonic phase. For somewhat lower baryon densities, crystalline
color superconductivity may arise in the NS core. It has been suggested that
the rigidity of such a phase (possibly with suitably misaligned magnetic field
form the rotation axis) will lead to non-zero quadrupole moment [5]. This will
be a significant source of gravitational waves, which can be observationally
constrained with the present generation of gravitational wave detectors.

With the realization that many of these possible signatures of the exotic QCD
phases are subject to model uncertainties, an attempt was made by some of us
in [19] that the above symmetric breaking phase transitions may cause density
fluctuations in the core through the formation of topological defects, leading
to a transient change of MI tensors components. It was shown in Ref. [19] that
the change of diagonal components of the MI tensor may lead to the change of
the spin frequency of the pulsars and may be responsible for glitches and/or
anti-glitches. As the fluctuations are random, there is a possibility of the
generation of quadrupole moment leading to the emission of GWs. Development of
the non-zero MI tensor’s diagonal components may also lead to modulations of
pulse profile [21]. Contrary to the above proposal, there was also a view in
Ref. [70] that the color superconducting phase associated with the three light
flavors might not exist. The authors also claimed that two-flavor color
superconductivity (2SC) might be marginally inconsistent with pulsar data.

###  V.1 Possibility of ultra-high baryon densities in matter collapsing to a
black hole

Neutron star core is generally believed to be the place where the highest
baryon density can be achieved in nature. Thus the possibility of quark matter
and other exotic QCD phases are usually discussed in that context (here we
include quark star, strange star etc. in the same category). These are
supposed to be equilibrium configurations achieved by using suitable equation
of state for the interior. What happens when such compact objects accrete
matter? Beyond a limit, the equilibrium is broken and the collapse to black
hole occurs. Typically, the collapse is very rapid, occurring in milliseconds
for a typical stellar mass object. During this collapse, density keeps
increasing, while an event horizon starts forming near the centre, which grows
outwards. It is reasonable to expect that during this dynamical evolution,
density will be significantly larger than the initial density of the
equilibrium configuration. Even before the event horizon forms at the centre,
the central density will keep growing, and after the formation of horizon, the
region outside it will also have large density before it gets engulfed by the
growing horizon. For a naive estimate, consider a stellar-mass black hole of
mass, say, $2~{}\text{M}_{\odot}$ (approximately the upper mass limit of a
neutron star, though there is still a theoretical uncertainty in estimates of
this limiting mass). The Schwarzschild radius $r_{s}=2GM/c^{2}$ of this type
of black hole is about 6 km. Thus it is possible that baryon densities in
matter $\rho_{bh}$ collapsing to a black hole may become about
$(2/1.4)(10/6)^{3}\simeq 5$ times larger than the average mass density of a
canonical neutron star of mass $1.4~{}\text{M}_{\odot}$ and radius 10 km.
Thus, such collapsing proto black holes may provide possibly the highest
possible densities available in nature, larger than any compact equilibrium
stars. This should be most optimistic place for looking for extreme exotic QCD
phases, such as CFL phase which require very high baryon densities. Even
though such high densities will last for a very short times, less than a
millisecond, it is a very large time for the QCD processes which occur at time
scale of fm/c. Thus observational signals from phase transitions to different
high density QCD phases may be detectable when such collapse occurs.
Possibility of transition to quark matter core in core-collapse supernova
simulations was discussed in [71]. What we are proposing is that even more
exotic QCD phases requiring extreme baryon densities may show up transiently
in matter collapsing to black hole, for suitable masses of collapsing object.
Numerical simulations of neutron stars accreting matter and collapsing to
black hole show that central density can significantly increase even before
event horizon forms at the centre [72, 73, 74]. The signatures from such
collapse should be very unique, as there will be a succession of phase
transitions to various high baryon density QCD phases, all occurring within a
time span of order milliseconds. Apart from signaling exotic QCD phases, it
can provide unique signature of such a gravitational collapse to black holes.

##  VI Detecting phase transition occurring inside a pulsar

Here we will discuss observational implications of phase transitions occurring
inside a pulsar, in particular, on the nature of its pulses. The consequences
of phase transition (for example from nuclear matter to QGP) occurring in the
core of a neutron star on its rotational dynamics has been discussed in [24,
25]. Basic physics of their model is that as pulsar rotation slows down due to
radiation breaking, the density of the core steadily increases (with reduction
of centrifugal force). If the density was initially below the critical density
of the pulsar core (corresponding to a phase transition), then at some stage
central density crosses the critical value leading to phase transition. It is
assumed that as the high density core grows in size (slowly, over the time
scale of millions of years), it continuously converts to the high density QGP
phase (even when the transition is is of first order). The dependence of
pressure on density will determine the manner in which the phase transition
will affect the moment of inertia of the neutron star, hence the angular
velocity of the star. It was emphasized in these works that the quantity of
special interest is the breaking index defined as
$n(\Omega)\equiv\frac{\ddot{\Omega}\Omega}{\dot{\Omega}^{2}}$. It is argued
that while usually $n(\Omega)$ will be equal to the intrinsic index n of the
energy-loss mechanism (with the energy loss $dE/dt={d\over dt}({1\over
2}I\Omega^{2})=-C\Omega^{{\rm n}+1}$), during phase transitions, it can differ
markedly from this value, possibly even by orders of magnitude. Thus, even if
the changes in moment of inertia, and hence the spin rate changes are not
directly observable (due to very large time scale), breaking index may provide
a more promising signal for phase transitions occurring inside pulsars.

The discussions in refs. [24, 25] primarily focused on the change in the
equation of state during the phase transition. Main consequence of the phase
transition was thus related to change in the diagonal components of the moment
of inertia tensor affecting the spin rate of the neutron star. However, all
phase transitions necessarily produce density fluctuations. In fact, a rich
spectrum of physics is encoded in the distribution of density fluctuations
relating to the nature of phase transition (first order, second order), and in
particular the symmetry breaking pattern (if any) associated with the phase
transition. Density fluctuations perturb entire moment of inertia tensor,
including its off-diagonal components. This was discussed by some of us in
[19, 20] for the situation when phase transition occurs rapidly in a large
core of neutron star. It was pointed out that as phase transitions induced
density fluctuations modify the entire moment of inertial tensor of the
pulsar, the resulting off-diagonal components will lead to wobbling of star
(in addition to any previously present) which will induces modulations of the
pulse profile. Thus, it was argued that the detailed modification of the
pulses carries the information of statistical nature of density fluctuations,
and hence the precise nature of phase transition occurring inside the NS
interior. An important aspect of this model is that it predicts that off-
diagonal components of the MI tensor components necessarily become non-zero
along with its diagonal components, with all perturbations being of similar
order, due to statistical nature of the phase transition induced density
fluctuations. Thus, in this model, spin rate changes will be necessarily
associated with the modulations of pulse profile. This can be used to test
this model for any observed pulse modifications, as if spin rate changes occur
due to de-pinning of vortices, then dominant changes only occur in the
diagonal components of the MI tensor (as all vortices are aligned along the
rotation axis).

###  VI.1 Effects of density fluctuations

The earlier discussions in refs. [24, 25] related to a scenario of slow
transition as applicable for slowly evolving star (e.g. by accretion), with a
transition which is either a weak ﬁrst order, or a second order, (or a
crossover). For a strongly ﬁrst order transition a different possibility
arises as discussed in [19, 20]. Strong supercooling can lead to a highly
suppressed nucleation rate, so that no bubble nucleation occurs for a very
long time. The transition can then occur suddenly, possibly due to some
inhomogeneity, after the supercritical core becomes macroscopic in size. The
phase transition thus occurs rapidly over a macroscopically large core. This
scenario would be quite similar to the one discussed by Witten [8] where a
very low nucleation rate could lead to macroscopic length scales, of order of
meters, for bubble nucleations for the quark-hadron transition where the
typical length scale would be of order fm. (The discussion in [8] assumed a
first order transition. With lattice results, now one knows that for very low
baryonic chemical potential the quark-hadron transition is a cross-over).
Rapid phase transition occurring over a large core can occur for in other
situations also, for example during early hot stages of the neutron star
undergoing rapid cooling.) The discussion in ref. [19, 20] related to a
general such situation and argued that, along with the expected change in the
moment of inertia, and hence the spin rate, (which could be directly
observable), density fluctuations will be produced in the entire large core
region undergoing this rapid phase transition. This will then affect off-
diagonal MI components and hence induce wobbling of the neutron star.

The situation considered in ref. [19, 20] related to the case when phase
transition induced spin rate remains small, (say, within the range set by
observations of glitches), and determined the effects of density fluctuations
on the MI tensor. In a simplified, two density picture of the phase transition
occurring in the NS, it is assumed that the phase transition converts the core
of radius $R_{0}$ to the new phase with density $\rho_{2}$, while the rest of
NS remains in the old phase with density $\rho_{1}$. The resulting fractional
change in the moment of inertia [25] is

$\frac{\Delta{I}}{I}\approx\frac{5}{3}(\frac{\rho_{2}}{\rho_{1}}-1)\frac{R_{0}^{3}}{R^{3}}$
(7)

Where $R$ is the radius of the star (taken to be spherical) in absence of the
dense core. If we consider the possibility that phase transitions could have
occurred in the pulsars which have been regularly monitored, then observations
of glitches can be used to constrain the size of the core undergoing phase
transition. One would then constrain the largest fractional change of moment
of inertia to be less than $10^{-5}$, relating to strongest glitches observed
so far. Various phase transition cases can then be considered. For example, a
sample value of change in density due to phase transition can be taken to be
about $30\%$ (which could be appropriate for QCD transition where density
change can be of order one). This will constrain the value of $R_{0}$ to be
less than about $300m$ (taking $R=10$ km). Another important case is that of
superfluid transition where the density change can be taken to be of order of
the superfluid condensation energy density ($\approx 0.1MeV/fm^{3}$) [75, 13].
In this case, glitch constraint will implying that $R_{0}$ can be as large as
$5$ km.

With the size of the core undergoing phase transition being constrained, one
can then discuss effects of density fluctuations occurring in this core [19,
20]. First it is useful to get generic estimates which depend simply on the
nature of phase transition. First, we focus on density fluctuations due to the
nucleation of bubbles. As discussed above, for a strong first-order case, a
core of size a few hundred meters (or larger) can undergo rapid phase
transition. However, in general, the core region will be expected to have
minute nonuniformities, even of purely statistical origin, so that one can
consider a situation where many bubbles may nucleate in different parts of the
supercritical core. The bubbles nucleated with a critical size of the order of
tens of $fm$, will expand and coalesce. At the time of coalescence, the
supercritical core region will consist of close packing of bubbles of the new
phase, embedded in the old phase. We carried out simulation of random
spherical bubble nucleation of radius $r_{0}$ (at the coalescence stage)
filling up a spherical core of radius $R_{0}=300m$. The density change in
bubble nucleation is taken to $\sim 160$ MeV/fm3, as appropriate for the QCD
transition. We find fractional change in moment of inertia $\Delta{I}/I\approx
4\times 10^{-8}$ for $r_{0}=20$ meters and remains of same order when $r_{0}$
varies from $5$ meters to $20$ meters. Due to random nucleation of the bubble,
the off-diagonal component of components of the moment of inertia, as well as
the quadrupole moment become nonzero. The ratio of both to the initial moment
of inertia are found to be of same order, ${Q\over I_{0}}\simeq{I_{xy}\over
I_{0}}\simeq 10^{-11}-10^{-10}$.

###  VI.2 Density Fluctuation From Topological Defects

The formation of topological defects routinely occurs in symmetry-breaking
transitions and has been extensively discussed in the literature, from the
early Universe to condensed matter systems. Depending on the relevant energy
scales these defects can be a source of large density fluctuations. The
underlying dominant mechanism for their formation in a phase transition is the
so-called Kibble mechanism [76, 77] which predicts a defect density
proportional to the number density of correlation domains, with
proportionality constant determined using universal arguments depending only
on the specific symmetry breaking pattern and dimensionality of space under
consideration. Thus, the random network of defects arising in any phase
transition and resulting defect distribution can be determined entirely using
the symmetry-breaking pattern. For example, a random network of vortices
arises from superfluid transition. A network of domain walls and global
strings arises from the spontaneous breaking of $Z(3)$ center symmetry for
confinement–deconfinement QCD transition [78]. Some QCD transitions ( e.g. the
color flavor locked (CFL) phase, expected to arise at very large values of
baryonic chemical potential) may give rise to only global strings, with
$SU(3)_{c}\times SU(3)_{L}\times SU(3)_{R}\times U(1)_{B}$ symmetry broken
down to the diagonal subgroup $SU(3)_{c+L+R}\times Z_{2}$ [5]. The evolution
of such a defect network shows universal characteristics. Starting with
initial defect densities (basically determined using correlation length and
topological probability calculations), the later evolution of string defects
and domain wall defects shows scaling behavior. This has important
implications for predictions of changes in the moment of inertia (hence
glitches/anti-glitches), quadrupole moment, and subsequent relaxation to the
original state of rotation in a reasonably model-independent way. An important
aspect of topological defect sourced density fluctuations relates to the
manner in which density fluctuations evolve in time. Eventually density
fluctuations decay away, leaving uniform new phase (in the core region
undergoing phase transition). But the manner of this decay, and the time
duration, depend crucially on the specific nature of the defect, and hence on
the symmetry breaking pattern. For example, while bubble-generated density
fluctuations decay away quickly in the time scale of coalescence of bubbles,
the domain wall network and the string network coarsen on much larger time
scales (with specific scaling exponents) which are characteristic of specific
type of defect. Thus the precise measurement of the pulsar spinning rate and
modification of pulse profile, and its time evolution, can also provide
important information about the specific transition occurring inside the
neutron star.

####  VI.2.1 Lattice simulation of string network

Due to topological nature of these defects, generic features of the defect
network can be determined using simple lattice picture with lattice size
representing the correlation length. An estimate of the change in MI due to
string and wall formation can be made by producing a network of defects inside
the core of the pulsar by modelling the correlation domain formation in a
cubic lattice, with lattice spacing $\xi$ representing the correlation length
[79]. To model U(1) global string formation each lattice site is associated
with an angle $\theta$ (randomly varying between $0$ and $2\pi$), or two
discrete values $0,1$ while modelling $Z_{2}$ domain wall formation. ($Z_{2}$
domain wall is considered instead of $Z_{3}$ walls of QCD just for simplicity
). For string case geodesic rule [76, 77, 79] is used to determine the winding
of $\theta$ on each face of the cube.

Starting with correlation length of order of $fm$ to simulate the network of
order few hundreds meter is numerically not possible. So the estimates were
made in [19, 20] by considering spherical star of size $R$ and a spherical
core of radius $R_{c}=\frac{0.3}{10}R$. Then, considering $\xi=10fm$, $R_{c}$
is increased from $5\xi$ to $400\xi$, and it was found that
$\frac{\delta{I}}{I_{i}}\simeq$ appears to stabilize at $10^{-13}-10^{-14}$.
Using this numerical results, it was suggested that the same fractional change
in the MI may also be possible for realistic value of $R=10$ km, especially
when one accounts for statistical fluctuations in the core. For the case of
domain wall formation, one finds fractional change in the off-diagonal
components of MI (as well as quadrupole moments) to be larger by a factor of
$40$. For the case of superfluid transition, a rapid superfluid transition can
take place due to transient heating, and subsequent cooling of star. This may
occur either due to another transition releasing latent heat, or due to
accretion, etc.. Taking vortex energy per unit length to be $100MeV/fm$ and
correlation length for vortex formation of order $10fm$ ([75, 13]) it is found
that the superfluid vortices induced transient fractional change in MI is of
order $10^{-10}$ (compared to net fractional change in MI of order $10^{-5}$
as discussed in Section VI.1). Also it was found that the ratios of the
quadrupole moment and off-diagonal components of MI to the net MI of the
pulsar are of order $10^{-10}$.

####  VI.2.2 field theory simulation of strings and domain walls

The technique proposed in [19, 20] has potential of using detailed
observations of pulse modification to learn about precise nature of phase
transition occurring inside the pulsar. For this, one would need to know
specific nature of density fluctuations for different cases and their precise
time evolution. As the relevant cases here refer to quantum field theory phase
transitions, one has to resort to field theory simulations to get such
details. Unfortunately, with typical length scales of such QFT phase
transitions being microscopic (e.g. of order fm for QCD transition), one can
only hope to do these simulations in very small spatial regions. (This is
different from the case of bubble nucleation where generic arguments of strong
supercooling were invoked to determine density fluctuations over macroscopic
regions even for QCD transition. even for lattice modelling of defect networks
as discussed above, one could consider relatively large lattice sizes.)

This was achieved in [19, 20] by studying string and wall formation in a field
theory simulation for confinement-deconfinement (C-D) QCD transition using
effective field theory Polyakov loop model. The expectation value of the
Polyakov loop, $l(x)$, is the order parameter for the C-D transition [80, 81].
$l(x)$ vanishes in the confined phase and is non-zero in the deconfined phase
where $Z(3)$ center symmetry (for the SU(3) color group) is spontaneously
broken as $l(x)$ transforms non-trivially under $Z(3)$. This gives rise to
three different vacua for different value of $l(x)$ in QGP phase, leads to
topological domain wall defects (which interpolate between different $Z(3)$
vacua) [82], and also string defects (QGP strings) forming at the junction of
these $Z(3)$ walls [83, 78, 84]. (Note, we have used the notations $Z(3)$ and
$Z_{3}$ interchangeably.) Defect formation is studied using field theory
simulation of the evolution of $l(x)$ from an initial value of zero
(appropriate for the confining phase) as the system is assumed to undergo a
rapid transition (quench) to the deconfined phase (as in [85]). Use of quench
is not an important point here as the formation of defects only requires
formation of uncorrelated domains, and the size of the domains in this model
has to be treated as a parameter as it is not possible to cover length scales
of km (for star) to fm (QCD scale). Hence these simulations are necessarily
restricted to system sizes of tens of $fm$ only. The physical size of the
lattice is taken as $(7.5fm)^{3}$ and $(15fm)^{3}$. Another possible phase
transition is to the so called color flavor locked (CFL) phase inside the core
of a pulsar where the QCD symmetry for three flavors (for very large baryon
chemical potential such mass differences between these quarks become
unimportant)), $SU(3)_{c}\times SU(3)_{L}\times SU(3)_{R}\times U(1)_{B}$ is
broken down to the diagonal subgroup $SU(3)_{c+L+R}\times Z_{2}$ [5, 86]. This
transition will give rise to global strings. To roughly estimate resulting
change in MI, a simplified case is considered by removing the cubic term from
effective potential. This modification in the potential gives rise to string
defects only without any domain walls, as appropriate for the transition from,
say, QGP phase to the CFL phase, while ensuring that one has the correct
energy scale for these string defects.

Figure 6:  Time evolution of fractional changes in the moment of inertia as
well as the quadrupole moment during the process of phase transition. (a,b),
and (c,d) correspond to lattice sizes $(7.5fm)^{3}$ and $(15fm)^{3}$
respectively. Plots in (a) and (c) correspond to the confinement-deconfinement
phase transition which results in the production of Z(3) walls and associated
strings, while plots in (b) and (d) correspond to the transition where only
string defects form which will be the case for the CFL phase. (Fig. taken from
[19].)

Fig. 6 shows how the induced fractional off-diagonal MI components and ratio
of quadrupole moment to MI change in time. Even though the exact values of
these quantities remain to be determined for the truly macroscopic core sizes
(as for the lattice simulation of defect network discussed above), it is clear
that in principle, exact temporal profile of these quantities can be
determined. Thus resulting pulse modifications can be predicted in complete
detail, in principle. One important, completely robust feature of these
results is that density fluctuation induced contributions to even the diagonal
components of the MI tensor can be positive as well as negative (with equal
probability). These changes in diagonal components can then account for
glitches as well as anti-glithces in the same unified framework. For this it
is important to note that density fluctuations arise during phase transition
to a new phase which by itself leads to change in the spin rate. Thus, when
density fluctuations die away, the original spin rate is only partially
restored (depending on the density difference between the two phases). This is
precisely the qualitative behavior observed for glitches.

##  VII Pulse Modification for Random Density Fluctuations

As we discussed above, in principle, a detailed analysis of perturbed pulses
should be able to reveal a wealth of information about underlying density
fluctuations occurring inside a pulsar. The order of phase transition, the
symmetry breaking pattern, the time scale of phase transition, etc. will leave
distinctive characteristic perturbations on the pulses. For this, one should
consider specific source of density fluctuation, with well defined
distribution of density fluctuations, and calculate in detail the resulting
pulse modification. However, certain aspects of generic density fluctuations,
in particular those resulting from any phase transition, will have qualitative
implications for pulse modifications. Such density fluctuations will be
statistical in nature, and will lead to random contributions to the components
of the MI tensor. A general study of such fluctuations was carried out by some
of us in [21] where the density fluctuations were modeled in terms of Gaussian
distributed random components of MI tensor added to the unperturbed diagonal
MI tensor of the neutron star. With the form of perturbed MI tensor
prescribed, the resulting effects on pulse timings as well as specific nature
of pulse modulations was then calculated. An important finding in [21] was
that even for very tiny density fluctuations, with resulting changes in pulsar
timings being extremely small, pulse profile modification were found to become
relatively large due to modulations resulting from wobbling of NS. Main reason
for this was that while pulse timing changes remain proportional to typical
density fluctuation magnitude $\epsilon$, the pulse profile modification were
found to be proportional to $\epsilon/\eta^{2}$ where $\eta$ is the NS
deformation parameter. ($\eta$ is typically very small $\sim 10^{-8}-10^{-4}$.
This observation will play an important role in later discussion when we
discuss possible detection of external GWs using NS deformations.) We first
discuss this case of random MI components from [21]. Subsequently, we will
also briefly mention how this technique can be applied to cases where where
modified MI tensor is precisely known, for example, the case of specific
density fluctuations occurring in neutron stars during collision with an
asteroid.

Basic physics of calculations in [21] is to consider a reasonably symmetric
initial configuration of neutron star with homogeneous density, and then
incorporate effects of random density fluctuations on its rotational dynamics.
The shape of the unperturbed pulsar was taken to be an oblate spheroid,
rotating about the $z$-axis with angular frequency $\omega$ and angular
momentum $L_{z}=L$ ($L_{x}=L_{y}=0$). The principal moment of inertia
components were taken as $I^{0}_{11}\equiv I^{0}_{1}=I^{0}_{22}\equiv
I^{0}_{2}$ and $I^{0}_{33}\equiv I^{0}_{3}=I_{0}$ with
$I_{0}>I^{0}_{1},I^{0}_{2}$. The oblateness is parameterized through
$\eta=(I_{0}-I^{0}_{1})/I_{0}$, the value of which depends on the neutron
star’s mass, the rigidity of the crust and magnetic field etc. Theoretical
studies by [87, 88] put the upper bound of $\eta$ as $\simeq 10^{-6}$, whereas
work of [89] puts the upper bound limit as $10^{-4}$. Observational studies
put the upper bound close to $10^{-5}-10^{-4}$ [90] with some pulsars having
$\eta\sim 10^{-2}-10^{-3}$ [91].

Immediately after a phase transition (at $t=0$), density fluctuation will
alter the MI tensor of the pulsar. The $S_{0}$ frame in Fig. 7 shows the set
of principal axes immediately after the phase transition. The subsequent
dynamics of the perturbed pulsar are governed by the set of Euler equations,
[92, 93]

$\displaystyle I_{1}\dot{\omega}_{1}-(I_{2}-I_{3})\omega_{2}\omega_{3}$
$\displaystyle=0$ (8) $\displaystyle
I_{2}\dot{\omega}_{2}-(I_{3}-I_{1})\omega_{1}\omega_{3}$ $\displaystyle=0$ (9)
$\displaystyle I_{3}\dot{\omega}_{3}-(I_{1}-I_{2})\omega_{1}\omega_{2}$
$\displaystyle=0.$ (10)

Where $I_{i}~{}(i=1,2,3)$ denotes the principal MI tensor components relative
to the body fixed frame ($S^{\prime}$ in Fig. 7) and
$\omega_{1}(t),\omega_{2}(t)$, and $\omega_{3}(t)$ are the angular frequencies
of the star with respect to space fixed frame (which momentarily coincides
with $S^{\prime}$). As $\omega_{1}$, and $\omega_{2}$ are expected to be very
small compared to $\omega_{3}\simeq\omega$, one can write the equation of
motion for $\omega_{1}$ as

$\displaystyle\ddot{\omega_{1}}$ $\displaystyle+\Omega^{2}~{}\omega_{1}=0\,,$
(11) $\displaystyle\textrm{where,}\qquad\Omega$
$\displaystyle=\omega_{3}\left[\frac{(I_{3}-I_{1})(I_{3}-I_{2})}{(I_{1}I_{2})}\right]^{1/2}\,.$
(12)

Here, $\Omega$ is the precession frequency due to the perturbation. We
consider the situation when perturbations are tiny in comparison to the
oblateness parameter ($\epsilon<<\eta$), hence the condition that
$I_{3}>I_{1},I_{2}$ is still valid (as we will see below). The solution of the
above equation is then given by

$\omega_{1}(t)=A~{}\cos(\Omega~{}t)+B~{}\sin(\Omega~{}t).$ (13)

$A$ and $B$ are two arbitrary constants determined from the initial
conditions. Similarly using Eq. (13)) and Eq. (9), one obtains the time
evolution of $\omega_{2}$

$\omega_{2}(t)=k[A\sin(\Omega~{}t)-B\cos(\Omega~{}t)].$ (14)

Where the factor $k$ is given by

$k=\left[\frac{I_{1}(I_{3}-I_{1})}{(I_{2}(I_{3}-I_{2}))}\right]^{1/2}.$ (15)

The time $t=0$ is assumed to be the onset of the pulsar’s precession
immediately after the completion of phase transition. Denoting the respective
angular velocities at $t=0$ by $\omega^{0}_{1}$ and $\omega^{0}_{2}$
respectively, Eqs. (13) and (14) can be rewritten as

$\displaystyle\omega_{1}(t)$
$\displaystyle=\omega_{1}^{0}\cos(\Omega~{}t)-\frac{\omega_{2}^{0}}{k}\sin(\Omega~{}t)$
(16) $\displaystyle\omega_{2}(t)$
$\displaystyle=k\omega_{1}^{0}\sin(\Omega~{}t)+\omega_{2}^{0}\cos(\Omega~{}t).$
(17)

Here, the arbitrary constants $\omega^{0}_{1}$ and $\omega^{0}_{2}$ can be
fixed by initial conditions.

Figure 7:  An oblate shape unperturbed pulsar is initially (i.e. before any
phase transition) rotating about the z-axis relative to a space-fixed frame S
(solid black lines). The red dotted lines show the principal axes ($x_{0}$,
$y_{0}$, $z_{0}$) of $S_{0}$ frame immediately after the phase transition (at
$t=0$). The body-fixed $S^{\prime}-$frame at any arbitrary time $t>0$ is shown
with blue dashed lines (taken from [21]).

###  VII.1 Initial conditions and analytical estimates of various parameters

Immediately after the phase transition, the orientation of the principal axes
($S_{0}$ frame) is shown in Fig. 7. The standard polar angle and the azimuthal
angle of $z_{0}$ axis with respect to the $S$ frame are denoted by
$\theta_{0}$ and $\phi_{0}$. These angles are obtained by diagonalizing the
perturbed moment of inertia tensor and finding the eigenvalues and the
corresponding set of eigenvectors (see [21] for details). In the absence of
external torque, the angular momentum of the pulsar must be conserved. Thus,
after the phase transition, the components of the angular momentum along
$x_{0},y_{0}$, and $z_{0}$ are given by

$\displaystyle L_{x_{0}}(t=0)$
$\displaystyle=I_{1}\omega_{1}^{0}=-L\theta_{0}\cos\phi_{0}$ (18)
$\displaystyle L_{y_{0}}(t=0)$
$\displaystyle=I_{2}\omega_{2}^{0}=-L\theta_{0}\sin\phi_{0}$ (19)
$\displaystyle L_{z_{0}}(t=0)$ $\displaystyle=I_{3}\omega_{3}^{0}=L.$ (20)

Because $\omega_{3}$ is approximately constant and the angle $\theta_{0}$ is
small for tiny density fluctuations, one can use the approximation
$L/I_{1}\simeq L/I_{3}\simeq\omega$. The above set of equations can be used
now to express Eqs. (13) and (14) in terms of $\theta_{0}$ and $\phi_{0}$.
Since the phase transition fluctuations are small, the perturbed MI tensor can
be written as

$\displaystyle I_{1,2}$ $\displaystyle=I_{0}(1-\eta+\epsilon_{1,2})\,,$ (21)
$\displaystyle\textrm{and}\qquad I_{3}$
$\displaystyle=I_{0}(1+\epsilon_{3})\,,$ (22)

with $\mathcal{O}(\epsilon_{i})\simeq\mathcal{O}(\epsilon)$ for $i=1,2,3$.
Therefore the precession frequency (Eq. (12)) can be written as

$\Omega\simeq\left(\frac{\eta+\epsilon}{1-\eta+\epsilon}\right)\omega\simeq\eta~{}\omega.$
(23)

Here it is assumed that $\epsilon,~{}\eta\ll 1$ and $\epsilon\ll\eta$. The
initial angle $\theta_{0}$ is obtained by diagonalizing the perturbed MI
tensor $I_{ij}$. For an order of magnitude estimate, the perturbation
$\epsilon_{ij}\equiv\delta I_{ij}/I_{0}$ can be taken as
$\epsilon_{ij}=\epsilon/I_{0}$. This results in

$\displaystyle\cos\theta_{0}$ $\displaystyle=\left(1+2\left(\frac{\epsilon
I_{0}}{I_{3}-I_{1}-\epsilon I_{0}}\right)^{2}\right)^{-1/2}.$ (24)

Assuming $\epsilon<<\eta$, one gets
$\theta_{0}\simeq\sqrt{2}\left(\frac{\epsilon}{\eta}\right)$. On similar
argument, one obtains the factor $k=1+\epsilon/2\eta$. Thus, the Eq.(16) and
Eq.(17)can be expressed as

$\displaystyle\omega_{1}(t)$ $\displaystyle=-\omega\theta_{0}\left[\cos(\Omega
t+\phi_{0})+\frac{\epsilon}{2\eta}\sin\phi_{0}\sin(\Omega t)\right]$ (25)
$\displaystyle\omega_{2}(t)$ $\displaystyle=-\omega\theta_{0}\left[\sin(\Omega
t+\phi_{0})+\frac{\epsilon}{2\eta}\cos\phi_{0}\sin(\Omega t)\right].$ (26)

The corresponding rotational angles $\theta_{1}(t)$ and $\theta_{2}(t)$ can
also be re-expressed as

$\displaystyle\theta_{1}(t)$
$\displaystyle=-\frac{\omega\theta_{0}}{\Omega}\left[\sin(\Omega
t+\phi_{0})-\frac{\epsilon}{2\eta}\sin\phi_{0}\cos(\Omega t)\right]$ (27)
$\displaystyle\theta_{2}(t)$
$\displaystyle=\frac{\omega\theta_{0}}{\Omega}\left[\cos(\Omega
t+\phi_{0})+\frac{\epsilon}{2\eta}\cos\phi_{0}\cos(\Omega t)\right].$ (28)

Putting the values of $\Omega$ and $\theta_{0}$, the amplitude $\omega_{m}$
(From Eqs. (25) and (26)), and $\theta_{m}$ (From Eqs. (27) and (28)) are
determined as

$\displaystyle\omega_{m}$
$\displaystyle=\omega\theta_{0}\simeq\sqrt{2}\left(\frac{\epsilon}{\eta}\right)\omega\,,$
(29) $\displaystyle\theta_{m}$
$\displaystyle=\left(\frac{\omega}{\Omega}\right)\theta_{0}\simeq\sqrt{2}\left(\frac{\epsilon}{\eta^{2}}\right)\,.$
(30)

Thus for $\eta=10^{-3}$, the oscillation amplitude $\theta_{m}$ is of order
$10^{6}~{}\epsilon\simeq 1^{\circ}$ for $\epsilon=10^{-8}$. It was shown in
ref. [21] that the above analytical estimates approximately match the results
obtained from the simulation.

###  VII.2 Numerical Results

Here we briefly review the simulation results of Ref. [21]. There, two sets of
values $(\eta,\epsilon)=(10^{-2},10^{-5})$ and $(10^{-3},10^{-8})$ were chosen
to observe the impact of perturbation on pulse modulations (See Table 1 in
[21] for the values of other parameters). Note that Eqs. (25) and (26) can be
approximately written as $\omega_{1,2}\sim\omega_{m}\cos(\Omega t)$. Since
$\Omega\sim\eta\omega$ (Eq. (23)), the time period
$T_{\Omega}=T_{\omega}/\eta$. Thus $T_{\Omega}=0.1$ for $\eta=10^{-2}$ and $1$
sec for $\eta=10^{-3}$. This matches the numerical simulation results in Ref.
[21]. The results are shown in Figs. 8, and 9.

Other than the above (first) modulation, another (second) modulation is also
expected since $\omega_{1}$ and $\omega_{2}$ also oscillates about the $x$ and
$y$ axis, respectively. From the frequency oscillation amplitude
$\omega_{m}\sim(\epsilon/\eta)\omega=(2\pi\epsilon/\eta)1000$ /sec., the
second modulation time scale can be approximately determined as $T_{m}\simeq
10^{-3}~{}(\eta/\epsilon)$ sec. Thus, the time scale $T_{m}$ varies from a few
seconds [for $(\eta,\epsilon)=(10^{-2},10^{-5}$)] to a few hundred seconds
[for $(\eta,\epsilon)=(10^{-3},10^{-8})$]. The figures 8, and 9 indeed show
that there is a second modulation, though the numerical results show time
scale is larger compared to the analytical estimate $T_{m}$. (For clarity, in
Fig. 9, only the top part of the pulses is shown compared to Fig. 8.) Of
course, considering the complexity of the rigid body dynamics, and the order
of magnitude estimates used, one can expect uncertainty in the analytical
estimates of the concerned quantities.

An important feature of these results can be termed as the memory effect. This
relates to the fact that, even after the density fluctuations fade away
leading to vanishing off-diagonal components, thereby restoring the original
rotation axes, and hence the original pulse profile, there will in general be
a net shift of the angular position of the pulse. (This is apart from the
effect of any possible net change in the spin rate. So, if net spin rate
change remains unobservable, this net shift of the angular position may still
be observable as it originates from the transient pulse modification from
intermediate stage of wobbling of star.) Therefore, a net, residual shift of
the angular position of the pulse could signal a missed phase transition.

Figure 8: The figure shows the time evolution of the pulse profile
$I(\theta_{p})/I_{0}$ of a millisecond pulsar for
$(\eta,\epsilon)=(10^{-2},10^{-5})$. The top plot shows the two different
modulations at different time scales. Bottom-left shows the same plot with
better resolution. Bottom-right shows a few pulses for a typical millisecond
pulsar. The figures are taken from [21]. Figure 9: (taken from [21]) Same plot
as Fig. 8, however for $(\eta,\epsilon)=(10^{-3},10^{-8})$. An apparent ‘kink’
at the top figure appears just below 500 s smoothed out with improved
resolution (bottom-left). For clarity, here only the top part of the pulses is
shown compared to Fig. 8.

###  VII.3 Pulse modification due to asteroid impact on neutron star

Impact of asteroids, comets etc. on astrophysical bodies frequently occur and
with intense gravity of neutron stars, such impacts have dramatic effects. For
example, it has been proposed that certain specific gamma ray burst event may
have origin in the impact of a solid body (comet or asteroid) with mass of
order 1018 g colliding with NS surface [94]. The tidal distortion of the body
during last stages of impact, with intense magnetic field of neutron star
leads to strong compression between magnetic longitudes. The interaction of
this with NS surface material, with exploding material falling back at
magnetic conjugate points was studied in detail in [94] and it was proposed
that it could explain specific gamma ray burst events. In view of discussions
in previous sections, it becomes natural to expect that such a collision
occurring on the surface of a pulsar should lead to perturbation in the MI
tensor of the pulsar, and hence should leave imprints on the pulses. This
interesting possibility is explored in [95]. Special feature of this case is
that, in this case, one can determine exact modification of the MI tensor of
the pulsar. This is done by following detailed impact dynamics of the body on
the NS surface as calculated in [94]. With the perturbed MI tensor known, it
is then straightforward to apply the technique discussed above, and determine
the detailed nature of perturbed pulses. One important unknown in this case is
the NS deformation parameter $\eta$ which determines the pulse modulations (as
discussed above). As resulting change in MI tensor will have very small
magnitude in this case (with $\epsilon$ of order 10-15 for a 1018 g body
impacting on a solar mass NS), observation of pulse modulations will only be
possible for NS with very small values of $\eta$. Still, this suggests an
interesting technique to probe impact of bodies on pulsar surfaces by
observing pulse modifications. In particular, for any proposed explanations on
gamma ray bursts etc., one will expect an accompanying observation of pulse
modification (depending on value of $\eta$).

##  VIII Gravitational waves due to phase transition induced density
fluctuations

Here we discuss how phase transitions occurring inside neutron stars may
provide a new high frequency source of gravitational waves through density
fluctuation induced rapidly changing quadrupole moment of the star.

Neutron stars are considered to be one of the most sought after sources of
gravitational waves. As the emission mechanism from such a source is governed
by its internal structure, searches for gravitational waves from such a source
can provide valuable information. For example, through a series of works,
Thorne et al. [96, 97, 98, 99] developed a non-radial pulsation theory for a
general relativistic static neutron star model. The author suggested that
various mechanisms can excite quasinormal modes of the parent neutron star and
may result in gravitational wave emission. The mechanisms capable of exciting
quasinormal modes include flaring activity, the formation of hypermassive
neutron stars following the coalescence of binary neutron stars etc. The
mechanism associated with the timing glitch can excite quasinormal modes in
the parent pulsar also. With this motivation, the searches for the emission of
GWs from isolated pulsars began even before the first ever direct detection of
GWs from the binary black hole merger. In this context, an attempt for GW
search associated with the timing glitch in the Vela pulsar in August 2006
[47] is worth mentioning. Although the above search produced no detectable
GWs, with improving sensitivity of the detectors, continuous attempts in this
direction are believed to yield fruitful results soon.

In the literature, there are a few other theoretical studies where the authors
[100, 44, 45, 87, 46] have explored the feasibility of emission of
gravitational waves from isolated pulsars (see the review [101] for more
details). These studies considered various reasons for GW emissions. These
include the existence of deformity of neutron stars in the form of crustal
mountains on the surface [44], or permanent triaxiality [45] of the star. This
mechanism, as mentioned above, produces monochromatic GWs, the frequency of
which is determined by the spin frequency of the star. There was also a
suggestion that crustquake (see V) may have a role in generating GWs [102,
46]. In reference [102], the authors considered the crustquake as a trigger
which can excite various oscillatory modes of the pulsar [102], causing
emission of GWs. The role of crustquake as a possible source of GWs was also
suggested in [46], where the authors noticed that the sudden change of the
oblateness corresponded to a change of the quadrupole moment within a very
small time scale and crustquake could indeed cause the bursts of GWs [46]. In
the context of bursts of GWs from an isolated pulsar, we will discuss the case
discussed in [19] where density fluctuations induced during phase transitions
were discussed. These density fluctuations were found to perturb the entire MI
tensor of the neutron star. Along with that, it also necessarily produced non-
zero quadrupole moment. With a very short time scale associated with the phase
transition, e.g. for a QCD phase transition, in [19], it was suggested the
rapidly evolving quadrupole moment can provide a new source of gravitational
waves.

In [19], the authors have discussed how density perturbations due to bubble
nucleation or formation of topological defects during phase transitions change
the moment of inertia (MI) tensor of the star, in general with the addition of
off diagonal components in the MI tensor. In other words, the rotation axis of
the star no longer remains aligned with one of its principal axes. As the
shape of the star diverges from sphericity during phase transition, it
develops a quadrupole moment $Q$ also. The quadrupole moment tensor $Q_{ij}$
($i,j=1,2,3$) varies rapidly during phase transition, thereby giving rise to
quadrupolar gravitational radiation. In the weak field limit, the power
emitted by the star through gravitational radiation is given by [103]

$\displaystyle\frac{dE}{dt}$ $\displaystyle\,=-\frac{32G}{5c^{5}}(\Delta
Q)^{2}\omega^{6}$ $\displaystyle\,\approx-(10^{33}J/s)\left(\frac{\Delta
Q/I_{0}}{10^{-6}}\right)^{2}\left(\frac{10^{-3}s}{\Delta t}\right)^{6}.$ (31)

Here, we use $\Delta Q$ to represent generic value of $\Delta Q_{ij}$ which is
the change in the $ij^{\textrm{th}}$ component of the quadrupole moment tensor
during a time interval $\Delta t$ and $I_{0}$ is the initial moment of inertia
of the spherical NS. (The above expressions were derived in [103] for a
periodically varying quadrupole moment with angular frequency $\omega$. As our
aim here is only to provide the possibility of phase transition induced
density fluctuations being a new source of GWs, we use these expressions with
the typical time scale of the phase transition as playing the role of the
period associated with $\omega$. The important factors to be noted above are
$(\Delta Q)^{2}$ and $\omega^{6}$ leading to $(\Delta t)^{-6}$ dependence for
the GW power with $\Delta t$ being the time scale of the phase transition.)
According to [19], $\frac{\Delta Q_{ij}}{I_{0}}$ lies within the range
$10^{-11}-10^{-10}$ for phase transition through bubble nucleation and within
the range $10^{-14}-10^{-10}$ when inhomogeneities in density of the star
arise due to formation of topological defects (see for example Table 1 of
[19]). Though this value is much lower than the typical value of $10^{-6}$
used for deformed neutron stars, the fact that the change in $Q$ occurs over a
time scale of microseconds (as a conservative estimate) makes the power
radiated by gravitational waves in this situation significant. An estimate of
the strain amplitude $h$ arising from a pulsar at a distance $r$ from the
field point is given by

$\displaystyle h$ $\displaystyle\,=\frac{4\pi^{2}G\Delta Qf^{2}}{c^{4}r}$
$\displaystyle\,\approx 10^{-24}\left(\frac{\Delta
Q/I_{0}}{10^{-6}}\right)^{2}\left(\frac{10^{-3}s}{\Delta
t}\right)^{6}\frac{1kpc}{r}.$ (32)

Taking $\frac{\Delta Q}{I_{0}}\approx 10^{-10}$, $\Delta t\approx
10^{-6}-10^{-5}$ s, and $r=1kpc$, they get $h\approx 10^{-24}-10^{-22}$. In
reality, $\Delta t$ could be smaller, thereby enhancing $h$ and $dE/dt$. Note
that as the gravitational wave emission is extremely short lived, net energy
lost by the star is negligible compared to its mass.

In conclusion, transient changes in MI of a pulsar due to phase transition
induced density perturbations give rise to non-vanishing rapidly varying
quadrupole moment and hence quadrupolar gravitational radiation. In general,
the MI tensor in this case gets off diagonal contributions that cause wobbling
of the pulsar and consequent modification of the peak pulse intensity. This is
a special feature of the model of [19]. Through observation of modulation of
peak pulse intensity and gravitational waves, it is possible to identify the
specific nature of phase transition occurring within the core of the star.

##  IX Pulsar as a Weber detector of gravitational waves

So far we discussed the effects of internal dynamics of neutron star leading
to density fluctuations and changes in the nature of pulses from the pulsar.
In this section, and next, we will discuss the effects of an external
gravitational wave on the neutron star configuration. It is natural to be
sceptic, as expected deformations in NS will be extremely tiny. However, at
the same time we also recall the impressive accuracy of pulsar timing
observations, better than 1 part in $10^{15}$. Further, as discussed above in
Section VII, possible changes in the pulse profile from induced wobbling of
pulsar may be large, even if pulse timing changes remain very small. We will
put forward arguments here that pulsars can effectively act as remotely
stationed resonant Weber detectors whose GW perturbed signals may be
observable on earth. Such a possibility is worth exploring, even if it
requires challenging observations. Many gravitational wave detectors (like
LIGO/Virgo) are being set up around the globe, in order to be able to detect
gravitational waves with good localization of the source in the sky. This will
be complemented by future space-based detectors for the search for
gravitational wave sources with very wide range of wavelengths and strengths.
However, even all these near earth detectors will be limited in their scope as
most of the powerful gravitational wave sources occur very far and also,
triangulating the location of the sources will also be limited. Clearly, if
one could have a family of detectors placed far away in space, so that their
signals could be monitored on earth, that will immensely boost our ability for
detecting and identifying GW sources.

The discussion in this section is taken primarily from ref.[22]. Basic physics
proposed here is, to put it simply, taking the entire neutron star as a
resonant Weber detector. While for the conventional Weber detectors [104, 105,
106, 107], GW induced deformations are detected and converted into electrical
signals, for neutron stars (pulsars), deformations induced by external GWs
will be imprinted on the detailed nature of pulses which can be monitored on
earth. As for the conventional Weber detector, resonance will play a crucial
role for pulsar Weber detectors also which will lead to amplitude enhancement,
and more importantly, the ringing effect [107] which will allow folding of a
large number of pulses to tremendously improve the signal to noise ratio.

It is important to make a distinction between the proposal here for using
pulsar itself as a Weber detector, and the conventional technique of pulsar
timing array (PTA) for GW detection [108]. For PTA, one monitors the pulse
arrival times from a network of pulsars, and looks for gradual changes due to
passage of very low frequency gravitational waves in the intervening region.
By its very nature, PTA technique is limited to extremely low frequency
sources with frequencies of order $10^{-6}$ Hz or less. Such GWs are expected
to arise from supermassive black hole mergers or exotic objects such as cosmic
strings. In contrast, for the technique discussed here, a single pulsar acts
as a GW detector. In principle, even a single pulsar Weber detector can give
some information about the GW source direction through changes in its spin
rate and pulse profile. In the context of the proposal made here, we note that
it was proposed long ago [109] that with an array of seismometers, the whole
earth may be used as gravitational wave detector for low frequency GWs.
Proposals have also been made that gravitational waves may be detected due to
their effects on nearby stars, in particular on the solar acoustic modes
(helioseismology and astro-seismology) [110, 111, 112]. There is some
similarity in spirit of these proposals with the pulsar Weber detector
proposed here. However, to our knowledge, detection of GW using its effects on
pulse modifications of a pulsar [22], had not been been discussed earlier.

Consider a monochromatic gravitational wave with wavelength $\lambda$ from a
far away GW source, reaching a neutron star. Passing of GW means periodic
changes in the Riemann curvature tensor $R_{\mu\nu\lambda\rho}$ which will
induce deformations of any body in its path. For a neutron star, one can
calculate changes in its quadrupole moment $Q_{ij}$ induced by a GW in the
static limit (essentially, large wavelength limit for the GW compared to the
neutron star size). It can be written in the following form [113]:

$Q_{ij}=-\lambda_{d}E_{ij}~{}.$ (33)

Here, $E_{ij}=R_{i0j0}$ is the external tidal field. $\lambda_{d}$ is called
the tidal deformability,

$\lambda_{d}=\frac{2}{3}k_{2}\frac{R^{5}}{G}.$ (34)

Here R denotes the equilibrium (undisturbed) radius of the neutron star and
$k_{2}$ is called the second Love number. There have been theoretical
estimates of this. For polytropic pressure-density relation
$P=K\rho^{1+{1\over n}}$, where $K$ is a constant and $n$ is the polytropic
index, numerical results (for $0.5\leq n\leq 1.0$, and $0.1\leq(M/R)\leq
0.24$) can be fitted by the formula [113].

$k_{2}\simeq{3\over 2}(-0.41+{0.56\over n^{0.33}})({M\over R})^{-0.003}$ (35)

Remarkably, the BNS merger event detected by LIGO/Virgo [18] has been used to
put a direct observational constraint on the value of $k_{2}$ to lie within
the range, $k_{2}\simeq 0.05-0.15$, and we will be using values of $k_{2}$
within this allowed range.

For a GW travelling along z direction, the tidal field $E_{ij}$ can be
calculated for the two polarizations (${}^{\prime}+^{\prime}$, and
${}^{\prime}\times^{\prime}$ polarizations) in the transverse traceless (TT)
gauge. We consider the ${}^{\prime}+^{\prime}$ polarization and denote the
gravitational wave strain amplitude by $h$ for this polarization. The
resulting tidal field amplitude is given by [114],

$E_{xx}=-E_{yy}=\frac{2\pi^{2}hc^{2}}{\lambda^{2}},$ (36)

We take the neutron star to have a spherical shape and mass $M$. This tidal
field of the gravitational wave will then induce a quadrupole moment tensor as
given above in Eq.(33). Taking this deformation to be of an ellipsoidal shape,
one can estimate the resulting change in the moment of inertia tensor of the
neutron star.

$\frac{\Delta I_{xx}}{I}=-\frac{\Delta
I_{yy}}{I}\simeq\frac{k_{2}}{3}\frac{R^{3}c^{2}}{GM\lambda^{2}}20h$ (37)

For sample values, $M=1.0M_{\odot}$ and $R=10$ km and $\lambda$ for a
gravitational wave with 1 kHz frequency (for GW from a typical astrophysical
source, as detected by LIGO/Virgo).

$\frac{\Delta I_{xx}}{I}\simeq 10^{-2}h~{}.$ (38)

Here we have used a sample value $k_{2}$ = 0.1 within the allowed value from
BNS merger event [18]. This event had the GW source about 130 million light
years away from earth, and had the peak strength of the signal $h\simeq
10^{-19}$. The advantage of the proposed pulsar Weber detector is that it
could be anywhere in space. Most optimistically, one can even imagine that a
pulsar was about 1 light years away from the BNS undergoing merger. (Such a
possibility can be considered as most neutron stars and pulsars arise in
globular clusters in our galaxy, so one could imagine it to be also true for
other galaxies, i.e. for this particular BNS event, though extragalactic
pulsar observations are not very frequent). The value of $h$ at the location
of that pulsar will be about $h\simeq 10^{-11}$. As the spin rate change for
the pulsar is directly related to change in its MI, change in the spin rate of
the pulsar will be

$\frac{\Delta\nu}{\nu}=\frac{\Delta I}{I}\simeq 10^{-13}~{}.$ (39)

where $\Delta I$ represent the change in the relevant component of MI. Given
the extreme levels of accuracy of measurements of pulsar signals, this
magnitude of spin rate change is well within observations. It is important to
note that for a generic direction of propagation of the GW, the MI tensor will
develop non-zero off-diagonal components also. This will induce wobbling of
the pulsar leading to the modulation of the pulse intensity profile. Exactly
this type of modulation was discussed above in Section VII. One important
result from that section is that even for very tiny changes in the spin rate
due to changes in the diagonal components of MI tensor, induced wobbling (and
hence changes in the pulse profile) may be much larger, especially for neutron
stars with small deformation parameter $\eta$. Thus profile changes may become
more important as a signal of GW passing through a pulsar. However, in this
section we will keep focusing on the spin rate change, as profile change
discussion is much more involved. Discussion of Section VII can be
straightforwardly applied to the present case of GW induced changes in the MI
tensor.

A crucial requirement for the Weber detector is that it needs to work at
resonance. The resulting increase in amplitude is not too large as GW signals
are short pulses, limiting resonant enhancement of the amplitude. However, at
resonance, the solid detector exhibits the so called ringing effect. Ringing
effect refers to continued vibration of the detector in the resonant mode for
a long time even after complete passing of the GW pulse through the detector.
This happens because at resonance energy absorption from the wave is highly
efficient. That energy has to be dissipated in sound and heat, until then the
detector will keep vibrating. For example, for a GW pulse of duration a few
ms, at resonance, the vibrations of the Weber bar can continue for time of
order 10 min [106, 107]. With the particular template for the pulse, a large
number of pulses, getting repeated during this ringing, can be folded. This
leads to tremendous increase in the signal to noise ratio. In the same way,
for the pulsar Weber detector working at resonance, one would expect the
pulsar to continue ringing for a long time after the passing of the GW. This
should allow folding of many pulses to separate this ringing signal. In fact,
this is the standard way in which a large number of regular pulses from the
pulsar are folded, leading to impressive accuracy of pulse timings. The
difference is that regular pulses from the pulsar are periodic, and hardly
change in relevant time scales (apart from occasional glitches), so pulse
folding is standard. The ringing of pulsar Weber detector shows that even for
a short GW pulse, similar folding of large number of pulses may be possible.

There is a wide range of resonant frequencies for neutron stars with
frequencies of few Hz all the way up to 20 kHz [115]. For specific modes, the
resonant frequencies of NS can be in the Range of 100 Hz to 1 kHz, which is
precisely the range relevant for a typical BNS merger GW source, also for
typical black hole mergers (with masses within tens of solar masses).

###  IX.1 The quality factor Q for the neutron star matter

Effectiveness of Weber detector crucially depends on the quality factor Q of
the detector material. Very high values of Q will lead to better enhancement
in the vibration amplitude. More importantly, the energy dissipated per
vibration cycle will be low, prolonging the ringing effect, thereby allowing
folding of much larger number of pulses for better signal to noise ratio. It
has been argued that various viscous effects may not be very important even on
time scales of the orbital decay time [116]. Thus it is entirely possible that
the quality factor Q for neutron star interior may be very large, possibly
much larger than the material available on earth for conventional Weber
detectors. Thus, one needs to know the Q factor for NS interior. This is a new
challenge for QCD calculations, apart from the well known problem of
determining the equation of state, and transport coefficients like shear
viscosity, the quality factor Q for different phases of QCD (at least for the
hadronic and the QGP phases) needs to be calculated. It is illuminating to
quote here from the review article on Detection of gravitational waves [117].
In the section on the Antenna materials for the resonant-mass (Weber)
detectors, it is stated,

”An ideal resonant bar would consist of a piece of nuclear matter, with high
density and a velocity of sound comparable to the velocity of light! Since
this is not available except in neutron stars, we must find a form of
molecular matter which, to maximize coupling to gravitational waves, combines
high velocity of sound $v_{s}$ , and high density $\rho$. To reduce the
thermal noise we require a low acoustic loss $Q^{-1}$”.

The arguments presented in this section propose that, going with the spirit of
above quotation, we realize that neutron star material can possibly be
available for use in a Weber detector, in fact by using the entire neutron
star itself as a pulsar Weber detector.

##  X Re-visiting past gravitational waves via pulsars as Webber detectors

An interesting outcome of the possibility of using pulsars as Webber
gravitational wave detectors spread out in the cosmos is that they can be used
to revisit past GW events, like collisions of black holes, neutron stars,
supernova explosions etc., again and again. These past events are taken to be
such that their signals have already passed through Earth. They might have
been detected by LIGO/Virgo or their signals might have been missed.
Supernovae within our galaxy whose existence have been deduced from
astronomical data collected around the world fall within the latter category.
This novel idea has been proposed in [23].

Figure 10: Schematic diagram showing the relative positions of Earth, a
distant GW source and a pulsar whose change in pulse profile and spin rate we
can measure on Earth. (Fig. taken from [23].)

Figure 10 schematically represents the situation we are talking about.
Gravitational waves from the source directly travel to Earth via path C but it
also travels to a pulsar via path B, transiently inducing a quadrupole moment
in the pulsar and changing its moment of inertia and hence its pulse frequency
and profile. The modified pulses reach earth via path A. If $r_{A,B,C}$ are
the distances along paths A, B, C respectively then the path length difference
between the direct path C and the path A+B via a pulsar is,

$\displaystyle\Delta r$ $\displaystyle\,=(r_{A}+r_{B})-r_{C},$ (40)
$\displaystyle\,=r_{A}+(r_{A}^{2}+r_{C}^{2}-2r_{A}r_{C}\cos\alpha)^{1/2}-r_{C}.$
(41)