# The Impact of Anisotropy on Neutron Star Properties: Insights from $I-f-C$
Universal Relations

Sailesh Ranjan Mohanty1 ID Sayantan Ghosh1 ID Pinku Routaray1 ID H. C. Das2,3
ID Bharat Kumar1 ID<EMAIL_ADDRESS>1Department of Physics and Astronomy,
National Institute of Technology, Rourkela 769008, India 2Institute of
Physics, Sachivalaya Marg, Bhubaneswar 751005, India 3Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

This study presents a universal relation for anisotropic neutron stars, called
the $I-f-C$ relation, which accounts for the local anisotropic pressure using
the Quasi-Local (QL) Model proposed by Horvat et al. Horvat  _et al._ (2010a)
to describe the anisotropy inside the neutron star. This study analyzes
approximately 60 unified tabulated EoS-ensembles, spanning from relativistic
to non-relativistic mean-field models, that comply with multimessenger
constraints and cover a broad range of stiffness. The results indicate that
the relationship between the parameters becomes more robust with positive
anisotropy, while it weakens with negative anisotropy. With the help of the
GW170817 & GW190814 tidal deformability limit, a theoretical limit for the
canonical $f$-mode frequency for both isotropic and anisotropic stars is
established. For isotropic case the canonical $f$-mode frequency for event
GW170817 & GW190814 is $f_{1.4}=2.605^{+0.487}_{-0.459}\ \mathrm{kHz}$ and
$f_{1.4}=2.093^{+0.150}_{-0.125}\ \mathrm{kHz}$ respectively. These
established relationships have the potential to serve as a reliable tool to
limit the equation of state of nuclear matter when measurements of relevant
observables are obtained.

##  I Introduction

The detection of gravitational waves (GWs) is a crucial objective in
astrophysics today, with significant efforts being devoted to this problem
globally. Several ground-based experiments and space missions have already
been devised and are poised to yield significant discoveries in the near
future Abbott  _et al._ (2009); Harry and (forthe LIGO Scientific
Collaboration); Acernese  _et al._ (2006); Accadia  _et al._ (2010); Antonucci
_et al._ (2011). Furthermore, ongoing efforts are underway to develop a third-
generation GW telescope, such as the Einstein Telescope Punturo  _et al._
(2010) and Cosmic Explorer Hall (2022), which promise even higher levels of
sensitivity. The reason why the detection of GWs is so challenging is that
they are incredibly weak, necessitating detectors with very high sensitivity,
as well as the precise waveform of the signal emitted from astrophysical
objects.

Neutron stars (NSs) exhibit oscillations that are regarded as potential
sources of gravitational waves (GWs), manifesting in various forms including
radial Chandrasekhar (1964); Chanmugam (1977); Kokkotas and Ruoff (2001);
Routaray  _et al._ (2023); Sen  _et al._ (2023) and nonradial McDermott  _et
al._ (1988); Kunjipurayil  _et al._ (2022a); Das  _et al._ (2021a) modes. When
a NS experiences external or internal disturbances, it emits GWs through
different oscillation modes known as quasi-normal modes (QNMs), each
characterized by the restoring force that brings them back to their
equilibrium state. Notable QNMs include the fundamental mode ($f$-mode) Zhao
and Lattimer (2022); Pradhan and Chatterjee (2021), pressure mode ($p$-mode)
Sotani and Kumar (2021a); Kunjipurayil  _et al._ (2022a), gravity mode
($g$-mode) Finn (1987); Reisenegger and Goldreich (1992); Zhao  _et al._
(2022a); Lozano  _et al._ (2022); Constantinou  _et al._ (2021); Jaikumar  _et
al._ (2021); Wei  _et al._ (2020); Tran  _et al._ (2022), rotational mode
($r$-mode) Haskell  _et al._ (2014); Haskell (2015); Jyothilakshmi  _et al._
(2022); Lin (2004); Rezzolla (2002); Jasiulek and Chirenti (2017), space-time
mode ($w$-mode) Benhar  _et al._ (1999); Bandyopadhyay and Chatterjee (2012),
and other modes Kokkotas and Schmidt (1999); Sotani  _et al._ (2011); Flores
and Lugones (2014); Ranea-Sandoval  _et al._ (2018). The frequencies of these
oscillations are directly linked to the internal structure and composition of
the stars Zhao  _et al._ (2022b). Theoretical studies indicate that the
$f$-mode possesses the highest likelihood of being detected initially, with
approximately 10% of the gravitational radiation attributed to $f$-mode
oscillations for $l=2$ Shibagaki  _et al._ (2020).

Different modes of oscillation exhibit distinct behaviors depending on the
type of star, offering valuable insights. For instance, the signature of the
hadron-quark phase transition can be inferred through the observation of both
$f$ and $g$-modes in hybrid stars Sotani  _et al._ (2011). A study by Flores
et al. Flores and Lugones (2014) suggests that compact objects emitting GWs
within the frequency range of 0-1 kHz may be hybrid stars, while frequencies
exceeding 7 kHz could indicate strange stars. However, the nature of compact
objects and their GW emissions still pose challenges due to the limitations of
our terrestrial detectors, which are unable to detect certain frequency
ranges. Nonetheless, constraints on the frequency of various modes can be
established by establishing relationships between the frequency and specific
properties of neutron stars (NSs). Several approaches have been proposed to
link the $f$-mode frequency with various NS properties, including compactness
Andersson and Kokkotas (1998), the moment of inertia Lau  _et al._ (2010), and
tidal deformability Chan  _et al._ (2014); Sotani and Kumar (2021b); Pradhan
_et al._ (2023). In this work, we aim to derive such relationships in a model-
independent manner, known as universal relations (URs). These URs provide
valuable insights into the properties of NSs and their associated oscillation
modes.

In literature, there are several URs have already been established between
different properties of the NS Yagi and Yunes (2013, 2015); Breu and Rezzolla
(2016); Rezzolla  _et al._ (2018); Riahi  _et al._ (2019); Gupta  _et al._
(2017); Jiang  _et al._ (2020); Yeung  _et al._ (2021); Chakrabarti  _et al._
(2014); Haskell  _et al._ (2013); Bandyopadhyay  _et al._ (2018). However, the
focus of this study primarily lies on the $I-f-C$ relations specifically for
anisotropic NSs. Previous URs proposed thus far have mainly been formulated
for isotropic NSs, assuming a matter-energy distribution characterized by an
isotropic perfect fluid. However, at extremely high-density regions, the
presence of nuclear matter can induce deviations between the tangential and
radial components of pressure, leading to the emergence of an anisotropic
fluid. To obtain more accurate results, we aim to include the effects of
anisotropy in our analysis. Anisotropy in NSs can arise from various factors,
such as the influence of a high magnetic field Yazadjiev (2012); Cardall  _et
al._ (2001); Ioka and Sasaki (2004); Ciolfi and Rezzolla (2013); Ciolfi  _et
al._ (2010); Frieben and Rezzolla (2012); Pili  _et al._ (2014); Bucciantini
_et al._ (2015), pion condensation Sawyer (1972), phase transitions Carter and
Langlois (1998), relativistic nuclear interactions Canuto (1974); Ruderman
(1972), core crystallization Nelmes and Piette (2012), and the presence of
superfluid cores Kippenhahn and Weigert (1990); Glendenning (1997); Heiselberg
and Hjorth-Jensen (2000), among others. Several models have been proposed to
incorporate anisotropy within NSs, such as the Bowers-Liang model Bowers and
Liang (1974), Horvat et al. model Horvat  _et al._ (2010b), Cosenza et al.
model Cosenza  _et al._ (1981), and others. In this study, we will primarily
focus on the Quasi-Local (QL) model, as proposed by Horvat et al. Horvat  _et
al._ (2010b), to describe anisotropy within NSs. More details about the QL-
model will be elaborated in Section III.

The presence of pressure anisotropy within NSs has a significant impact on
various macroscopic properties, including the mass-radius relation,
compactness, surface redshift, moment of inertia, tidal deformability, and
non-radial oscillations Silva  _et al._ (2015); Hillebrandt and Steinmetz
(1976); Doneva and Yazadjiev (2012); Bayin (1982); Roupas (2021); Deb  _et
al._ (2021); Estevez-Delgado and Estevez-Delgado (2018); Pattersons and
Sulaksono (2021); Rizaldy  _et al._ (2019); Rahmansyah  _et al._ (2020);
Rahmansyah and Sulaksono (2021); Herrera  _et al._ (2008); Herrera and Barreto
(2013); Biswas and Bose (2019); Das  _et al._ (2021b); Roupas and Nashed
(2020); Sulaksono (2015); Setiawan and Sulaksono (2019a); Silva  _et al._
(2015). The specific impact on these quantities varies depending on the degree
of anisotropy and the choice of the model employed. Biswas and Bose
constrained the degree of anisotropy within stars utilizing tidal
deformability data from the GW170817 event Biswas and Bose (2019). More
recently, the $I-$Love$-C$ universal relation for anisotropic NSs has been
proposed. By incorporating observational data from GW170817, constraints have
been placed on the moment of inertia and radius of canonical anisotropic stars
for different degrees of anisotropy Das (2022).

In this study, we introduce the URs for anisotropic NSs for the first time,
utilizing a range of models describing unified EOSs such as RMF models (for
npe$\mu$, hyperonic npe$\mu$Y, and strange npe$\mu$Ys matter), density-
dependent RMF models, and Skyrme-Hartree-Fock (SHF) models Fortin  _et al._
(2016); Kumar and Landry (2019); Landry and Kumar (2018); Kunjipurayil  _et
al._ (2022b); Malik  _et al._ (2018); Alam  _et al._ (2016); Parmar  _et al._
(2022a, b). The unified EOSs closely reproduce the properties of finite
nuclei, nuclear matter, and NSs and support $\geq 2.0M_{\odot}$ star. A total
of 60 EOSs (35 EOSs from RMF models and 25 EOSs from SHF models) are
considered in this paper. By employing these EOSs, we calculate various
macroscopic properties of NSs with varying degrees of anisotropy using the QL-
model. The primary objective of this work is to constrain the $f$-mode
frequency of anisotropic stars by leveraging recent observational data.
Detailed formalism, results, and discussions are presented in the subsequent
sections, shedding light on the intricate relationship between anisotropy,
macroscopic properties, and the $f$-mode frequency in NSs.

##  II Stellar Structure Equations

The metric that describes the space-time inside a spherically symmetric star
is given by Wald (1984); Schwarzschild (1916)

$ds^{2}=-e^{2\psi}dt^{2}+e^{2\lambda}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\
d\phi^{2}\right)\,,$ (1)

where $\psi$ and $\lambda$ are metric functions that depend on $r$. By
introducing anisotropy in the matter-energy distribution of the system, we
obtain the following stress-energy tensor Estevez-Delgado and Estevez-Delgado
(2018)

$T_{\mu v}=\left(\mathcal{E}+P_{t}\right)u_{\mu}u_{v}+P_{t}g_{\mu v}-\sigma
k_{\mu}k_{v}\,,$ (2)

where $u^{\mu}$ is the four-velocity of the fluid, and $k^{\mu}$ is a unit
space-like four-vector. $\mathcal{E}$ is the energy density, and $\sigma$ is
the anisotropic pressure ($\sigma=P_{t}-P_{r}$). The four-vectors $u^{\mu}$
and $k^{\mu}$ must satisfy the following conditions

$u_{\mu}u^{\mu}=-1,\quad k_{\mu}k^{\mu}=1,\quad u_{\mu}k^{\mu}=0\,.$ (3)

The Einstein field equations relate the space-time geometry and the matter-
energy distribution Einstein (1915)

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi T_{\mu\nu}\,,$ (4)

where $G_{\mu\nu}$ is the Einstein tensor, $R_{\mu\nu}$ is the Ricci tensor,
$R$ is the scalar curvature. By solving Eqs. (1–4) with the energy and
momentum conservation, we can obtain the modified Tolman-Oppenheimer-Volkoff
(TOV) equations, which describe the stellar structure of an anisotropic star
Bhar and Rej (2021); Oppenheimer and Volkoff (1939)

$\displaystyle\frac{dm}{dr}$ $\displaystyle=4\pi r^{2}\mathcal{E}\,,$
$\displaystyle\frac{dP_{r}}{dr}$
$\displaystyle=-\left(P_{r}+\mathcal{E}\right)\left(\frac{m}{r^{2}}+4\pi
rP_{r}\right)e^{2\lambda}+\frac{2}{r}\sigma\,,$
$\displaystyle\frac{d\psi}{dr}$
$\displaystyle=-\frac{1}{P_{r}+\mathcal{E}}\frac{dP_{r}}{dr}+\frac{2\sigma}{r\left(P_{r}+\mathcal{E}\right)}\,,$
(5)

where $m(r)$ is the enclosed mass corresponding to radius $r$ and $\lambda(r)$
is the metric function defined as

$e^{-2\lambda}=1-\frac{2m}{r}.$

In order to solve the above set of coupled ODEs numerically, certain boundary
conditions must be established. It is customary to set the surface of the star
at $r=R$, where the radial pressure becomes zero ($P_{r}=0$). Since the
equilibrium system is spherically symmetric, the Schwarzschild metric is used
to describe the exterior space-time. This ensures that the metric is
continuous at the surface of the anisotropic NS and imposes a boundary
condition on $\psi$. Specifically, the value of $\psi$ at $r=R$ must match the
value of the Schwarzschild metric’s $\psi$ at $r=R$.

$\psi(r=R)=\frac{1}{2}\ln\left[1-\frac{2M}{R}\right]\,.$

By selecting an EOS for the radial pressure ($P_{r}$) and employing an
anisotropic model for $\sigma$, we can numerically solve Eq. (II) for a
specified central energy density $\mathcal{E}(r=0)=\mathcal{E}_{c}$ while
imposing the initial condition of $m(r=0)=0$.

Figure 1: The radial profile of the sound speed ($c_{s,t}^{2}$) for different
$\beta_{\rm QL}$ values of the maximum mass NS corresponding to the DD2 EOS.

Figure 2: Left: Mass-radius profiles of EOS ensembles used in this work.
Right: Mass-radius profiles for anisotropic NSs with $-2<\beta_{\rm QL}<2$ for
the DD2 EOS. The black dashed line represents the isotropic case. The limits
on mass and radius from the 2019 NICER data of PSR J0030+0451 by Riley and
Miller Miller  _et al._ (2019); Riley  _et al._ (2019) are shown in the light
pink boxes, and the 2021 NICER data with X-ray MultiMirror observations of PSR
J0740+6620 Miller  _et al._ (2021) is shown in the light blue boxes. The green
horizontal bar represents the observed mass during the GW190814 merger Abbott
_et al._ (2020). The upper left region denotes the restricted causality region
where the speed of sound in the compact object surpasses the speed of light in
a vacuum.

##  III Anisotropy Model

For anisotropic NS, we use the Quasi-Local (QL) model proposed by Horvat et
al. Horvat  _et al._ (2010a) describing the quasi-local nature of anisotropy
in the following

$\sigma=P_{t}-P_{r}=\frac{\beta_{\mathrm{QL}}P_{r}\mu}{3}=\frac{\beta_{\mathrm{QL}}}{3}P_{r}(1-e^{-2\lambda})\,,$
(6)

where the factor $\beta_{\mathrm{QL}}$ depicts the measurement of the degree
of anisotropy in the fluid and ($\mu=1-e^{-2\lambda}=\frac{2m(r)}{r}$) also
known as local compactness is the quasi-local variable. Anisotropic NSs must
satisfy certain conditions to maintain their spherical symmetric configuration
Estevez-Delgado and Estevez-Delgado (2018); Setiawan and Sulaksono (2019b).
These conditions include: (i) anisotropy should not exist at the center of the
NS or $P_{r}=P_{t}$ at $r=0$, (ii) $P_{r}$, $P_{t}$, and $\mathcal{E}$ must be
positive throughout the star, (iii) the null energy $(\mathcal{E})$, the
dominant energy $(\mathcal{E}+P_{r},\mathcal{E}+P_{t})$, and the strong energy
$\left(\mathcal{E}+P_{r}+2P_{t}\right)$ must always be positive inside the
star, and (iv) the speed of sound inside the star must never be negative or
exceed the speed of light ($c=1$ taken in this study). Hence, the sound speed
in the radial and tangential directions must hold the following constraints:
$0<c_{s,r}^{2},c_{s,t}^{2}<1$.

The advantage of using the QL-model with anisotropy in this study is that it
ensures the fluid remains isotropic at the center of the star due to the
behavior of $(1-e^{-2\lambda}=2m/r)\sim r^{2}$ when $r\rightarrow 0$, while
also being applicable only to relativistic configurations where anisotropy may
arise at high densities Herrera and Barreto (2013). For the 60 EOS-ensembles
considered in this study, the QL-model with anisotropy parameter ranging from
$-2<\beta_{\rm QL}<2$ satisfies all the necessary conditions to maintain
spherical symmetry in an anisotropic NS configuration Estevez-Delgado and
Estevez-Delgado (2018); Setiawan and Sulaksono (2019b). Fig. 1 shows that the
speed of sound in the tangential direction ($c_{s,t}^{2}=\frac{\partial
P_{t}}{\partial\mathcal{E}}$) inside the NS for the maximum mass configuration
corresponding to the DD2 EOS satisfies the causality condition throughout the
star for $-2<\beta_{\rm QL}<2$.

The Mass-Radius (MR) profile for a given EOS can be obtained by solving TOV
Eq. (II) for various central densities, generating a sequence of mass and
radius data. Fig. 2 shows the MR profiles for the EOS-ensembles used in this
analysis as well as their anisotropic variation for DD2 EOS. Each MR profile
curve in the left of Fig. 2 has a color gradient that displays how the central
energy density, varies corresponding to each NS’s mass and radius. It is
observed that NSs having a radius of $9.5<R<11.5$ km and mass $M\sim
2M_{\odot}$ have central densities $\rho_{c}$ greater than $2\times 10^{18}$
kg/m3. Increasing the value of $\beta_{\rm QL}$ results in a higher maximum
mass and its corresponding radius for the NS while decreasing $\beta_{\rm QL}$
decreases NS’s mass and its corresponding radius. Observational data, such as
x-ray, NICER, and gravitational wave (GW) data, can be used to constrain the
degree of anisotropy within the NS Miller  _et al._ (2019); Riley  _et al._
(2019); Miller  _et al._ (2021); Abbott  _et al._ (2017, 2020).

##  IV Slowly Rotating NS and Moment of Inertia

Figure 3: Left: The MI is a function of NS’s mass ($M$) for 60 different
modern EOSs used in this work. Right: The MI as a function of NS’s mass for
different values of $\beta_{\rm QL}$ for DD2 EOS. The error bars were
calculated based on the results of several pulsar analyses as done in Ref.
Kumar and Landry (2019).

The Hartle-Thorne metric Hartle (1967); Hartle and Thorne (1968); Hartle
(1973) is commonly used to describe the space-time inside a slowly rotating
anisotropic NS. The metric is given by

$\displaystyle ds^{2}=$
$\displaystyle-e^{2\psi}dt^{2}+e^{2\lambda}dr+r^{2}\left(d\theta^{2}+\sin^{2}\
\theta\ d\phi^{2}\right)$ (7) $\displaystyle-2\omega(r)r^{2}\sin^{2}\theta\
dt\ d\phi\,,$

where $\omega(r)=(d\phi/dt)_{\rm ZAMO}$ is the Lenze-Thirring angular velocity
measured from a zero angular momentum observer (ZAMO) frame of reference,
which accounts for the frame-dragging effect. By solving the Einstein field
equations for the Hartle-Thorne metric (in Eq. (7)), one can obtain the MI of
the anisotropic NS.

For instance, the $t\phi$ component of the Einstein field equation can be
written as described in Ref. Rahmansyah  _et al._ (2020)

$\frac{1}{r^{4}}\frac{d}{dr}\left(r^{4}J\frac{d\bar{\omega}}{dr}\right)+\frac{4}{r}\frac{dJ}{dr}\left(1+\frac{\sigma}{\mathcal{E}+P_{r}}\right)\bar{\omega}=0,$
(8)

where

$J=e^{-\psi}\left(1-\frac{2m}{r}\right)^{1/2},\quad\text{and}\quad\bar{\omega}(r)=\Omega-\omega(r)\,.$

The MI of a slowly rotating NS can be expressed integrally using Eq. (8) with
the boundary conditions are $r\rightarrow R$, $J\rightarrow 1$ and
$\frac{d\bar{\omega}}{dr}\rightarrow\frac{6I\Omega}{R^{4}}$, where $\Omega$ is
the angular velocity and $I$ is the moment of inertia. This integral
expression can be written as Rahmansyah  _et al._ (2020); Das (2022)

$I=\frac{8\pi}{3}\int_{0}^{R}\frac{r^{5}J\bar{\omega}}{r-2m}\frac{\mathcal{E}+P_{r}}{\Omega}\left[1+\frac{\sigma}{\mathcal{E}+P_{r}}\right]dr\,.$
(9)

We display the MI of a NS as a function of its mass in Fig. 3 for EOS-
ensembles used in this study, as well as its anisotropic variability for DD2
EOS. As the mass of the NS increases, the MI also increases until a stable
configuration is achieved, beyond which it begins to decrease. Moreover, both
the mass and MI of the NS increase with an increase in $\beta_{\rm QL}$, and
vice-versa for negative values of $\beta_{\rm QL}$. The influence of
anisotropy on the MI is more pronounced for high-mass NSs than for low-mass
ones. Kumar and Landry Kumar and Landry (2019) have put constraints on MI with
the help of different NSs data, such as double NSs (DNS), millisecond pulsars
(MSP), and low-mass x-ray binaries (LMXB). The error bars in the figure
represent the possible range of values for these constraints.

##  V Tidal Deformability Parameters

Figure 4: Left: The dimensionless tidal deformability ($\Lambda$) as a
function of the NS’s mass ($M$) for the EOS ensembles considered in this work.
Right: The $\Lambda$ as a function of the NS’s mass for different values of
the $\beta_{\rm QL}$ corresponds to DD2 EOS. The error bars in both panels
represent the observational constraints from LIGO-Virgo’s detections of GWs
emitted by the NS mergers in events GW170817 Abbott  _et al._ (2017) and
GW190814 Abbott  _et al._ (2020).

When the NS is present in the external field ($\epsilon_{ij}$) created by its
companion star, it acquires a quadrupole moment ($Q_{ij}$). The magnitude of
the quadrupole moment is linearly proportional to the tidal field and is given
by Hinderer (2008, 2009)

$Q_{ij}=-\alpha\epsilon_{ij},$ (10)

where $\alpha$ is the tidal deformability of a star. $\alpha$ can be defined
in terms of tidal Love number $k_{2}$ as $\alpha=\frac{2}{3}k_{2}R^{5}$.

To determine the value of the tidal Love number $k_{2}$ for an anisotropic NS,
a linear perturbation is introduced to the background metric following the
method proposed by Thorne and Campolattaro Thorne and Campolattaro (1967).
After inserting the fluid and metric perturbation into the perturbed Einstein
equation, the following $2^{\rm nd}$ order differential equation for $H$ is
obtained from the tt-component Biswas and Bose (2019).

$H^{\prime\prime}+H^{\prime}\mathrm{U}+H\mathrm{V}=0\,.$ (11)

where

$\displaystyle\mathrm{U}$
$\displaystyle=\left[\frac{2}{r}+e^{2\lambda}\left(\frac{2m(r)}{r^{2}}+4\pi
r(P_{r}-\mathcal{E})\right)\right]$ $\displaystyle\mathrm{V}$
$\displaystyle=\left[4\pi
e^{2\lambda}\left(4\mathcal{E}+8P_{r}+\frac{\mathcal{E}+P_{r}}{dP_{t}/d\mathcal{E}}\left(1+c_{s}^{2}\right)\right)-\frac{6e^{2\lambda}}{r^{2}}-4\psi^{\prime
2}\right]$

The $k_{2}$ of a star can be obtained by ensuring the regularity of $H$ at the
center of the star and the continuity of $H(r)$ and its derivative at the
surface Hinderer (2008); Damour and Nagar (2009)

$\displaystyle k_{2}=$
$\displaystyle\frac{8}{5}C^{5}(1-2C)^{2}\left[2\left(y_{2}-1\right)C-y_{2}+2\right]$
$\displaystyle\times\left\\{\left[\left(4y_{2}+4\right)C^{4}+\left(6y_{2}-4\right)C^{3}-\left(22y_{2}-26\right)C^{2}\right.\right]$
$\displaystyle\left.+3\left(5y_{2}-8\right)C-3\left(y_{2}-2\right)\right]2C+3(1-2C)^{2}$
$\displaystyle\left.\times\left[2\left(y_{2}-1\right)C-y_{2}+2\right]\log(1-2C)\right\\}^{-1}\,,$
(12)

where we define $C$ as the compactness of the anisotropic NS given by
$C=\frac{M}{R}$. Additionally, we define $y_{2}$ as

$y_{2}=\left.\frac{rH^{\prime}}{H}\right|_{\mathrm{at}\ r=R}\,.$

Dimensionless tidal deformability ($\Lambda$) is a significant property that
can be inferred from GW data Choi  _et al._ (2019). Once $k_{2}$ and $C$ have
been obtained, we can calculate $\Lambda$ of the NS, which is defined as
follows

$\Lambda=\alpha/M^{5}=\frac{2}{3}k_{2}C^{-5}$ (13)

The tidal parameters of anisotropic NSs are shown in Figure 4. As the
anisotropy parameter $\beta_{\rm QL}$ increases, the magnitude of the Love
number $k_{2}$ and its corresponding tidal deformability $\Lambda$ decrease,
while they increase with decreasing $\beta_{\rm QL}$. The GW170817 Abbott  _et
al._ (2017) event constrains $\Lambda_{1.4}$ to be $190_{-120}^{+390}$, while
GW190814 Abbott  _et al._ (2020) sets a limit of
$\Lambda_{1.4}=616_{-158}^{+273}$ (in the NS-BH scenario). The predicted value
of $\Lambda_{1.4}$ satisfies the GW190814 limit for almost all values of
$\beta_{\rm QL}$, whereas, for DD2 EOS, the GW170817 limit is met in the range
of $0.5<\beta_{\rm QL}<2$. However, $\Lambda$ sharply decreases once the
stable configuration is exceeded. Need to change all the lines.

##  VI Non-Radial Oscillation in Cowling approximation

Figure 5: Left: The $f$-mode frequency as a function of NS’s mass for EOSs are
used in this work. Right: The $f$-mode frequency is a function of NS’s mass
for different $\beta_{\rm QL}$ corresponding to DD2 EOS. The error bars
represent the theoretical limits that we have calculated for events GW170817
Abbott  _et al._ (2017) and GW190814 Abbott  _et al._ (2020) for isotropic NS
using $f$-Love universal relation (VII.4).

The Cowling approximation, initially proposed by Cowling Cowling (1941) for
Newtonian stars and later extended to the case of NSs by McDermott McDermott
_et al._ (1988). Under this approximation, the metric perturbations are
neglected, keeping the space-time metric fixed. We will provide a brief
explanation of the derivation of the perturbation equations in the Cowling
formalism in the following, while more comprehensive details can be found in
Doneva and Yazadjiev (2012).

In order to derive the oscillation equations, we need to examine the
perturbation of the conservation equation for the energy-momentum tensor

$\nabla_{\nu}\delta T_{\mu}^{\nu}=0.$ (14)

We can then project Eq. (14) along with the four-velocity $u^{\mu}$ to obtain
the following equation

$\displaystyle
u^{\nu}\nabla_{\nu}\delta\mathcal{E}+\nabla_{\nu}\left\\{\left[(\mathcal{E}+P_{t})\delta_{\mu}^{\nu}-\sigma
k^{\nu}k_{\mu}\right]\delta u^{\mu}\right\\}$ (15)
$\displaystyle\quad+(\mathcal{E}+P_{t})a_{\nu}\delta
u^{\nu}-\nabla_{\nu}u_{\mu}\delta\left(\sigma k^{\nu}k^{\mu}\right)=0\,.$

One can obtain the following expression by projecting orthogonally to the
background 4-velocity using the operator
$\mathcal{P}{\mu}^{\nu}=\delta{\mu}^{\nu}+u^{\nu}u_{\mu}$

$\displaystyle(\delta\mathcal{E}+\delta
P_{t})a_{\mu}+(\mathcal{E}+P_{t})u^{\nu}\left(\nabla_{\nu}\delta
u_{\mu}-\nabla_{\mu}\delta u_{\nu}\right)$ (16)
$\displaystyle+\nabla_{\mu}\delta P_{t}+u_{\mu}u^{\nu}\nabla_{\nu}\delta
P_{t}-\mathcal{P}_{\mu}^{\nu}\nabla_{\alpha}\delta\left(\sigma
k^{\alpha}k_{\nu}\right)=0\,,$

where the 4-acceleration in the background metric is represented by
$a_{\mu}=u^{\nu}\nabla_{\nu}u_{\mu}$. Due to the oscillation of the star, the
fluid elements move from their equilibrium position, and these motions are
described by the displacements $\xi^{i}$, where $i=1,2,3$ ($r,\theta,\phi$).
From Eq. (16), an integrability condition can be derived to obtain the correct
expressions for $\xi_{\theta}$ and $\xi_{\phi}$.

With the help of Eq. (15) and the displacement $\xi^{i}$, one can obtain the
oscillation equations in the Cowling approximation by considering a harmonic
time dependence for the perturbation functions $W(r,t)=W(r)\mathrm{e}^{i\omega
t}$ and $V(r,t)=V(r)\mathrm{e}^{i\omega t}$, where $\omega$ represents the
oscillation frequency in the following Curi  _et al._ (2022); Doneva and
Yazadjiev (2012)

$\displaystyle W^{\prime}=$
$\displaystyle\frac{d\mathcal{E}}{dP_{r}}\left[\omega^{2}\frac{\mathcal{E}+P_{t}}{\mathcal{E}+P_{r}}\left(1+\frac{\partial\sigma}{\partial
P_{r}}\right)^{-1}\mathrm{e}^{\lambda-2\psi}r^{2}V+\psi^{\prime}W\right]$ (17)
$\displaystyle-l(l+1)\mathrm{e}^{\lambda}V-\frac{\sigma}{\mathcal{E}+P_{r}}\left[\frac{2}{r}\left(1+\frac{d\mathcal{E}}{dP_{r}}\right)W\right.$
$\displaystyle\left.+l(l+1)\mathrm{e}^{\lambda}V\right]\,,$ $\displaystyle
V^{\prime}$
$\displaystyle=2V\psi^{\prime}-\left(1+\frac{\partial\sigma}{\partial
P_{r}}\right)\frac{\mathcal{E}+P_{r}}{\mathcal{E}+P_{t}}\frac{\mathrm{e}^{\lambda}}{r^{2}}W$
(18)
$\displaystyle-\left[\frac{\sigma^{\prime}}{\mathcal{E}+P_{t}}+\left(\frac{d\mathcal{E}}{dP_{r}}+1\right)\frac{\sigma}{\mathcal{E}+P_{t}}\left(\psi^{\prime}+\frac{2}{r}\right)\right.$
$\displaystyle\left.-\frac{2}{r}\frac{\partial\sigma}{\partial
P_{r}}-\left(1+\frac{\partial\sigma}{\partial
P_{r}}\right)^{-1}\left(\frac{\partial^{2}\sigma}{\partial
P_{r}^{2}}P_{r}^{\prime}+\frac{\partial^{2}\sigma}{\partial
P_{r}\partial\mu}\mu^{\prime}\right)\right]V\,.$

To solve the equations mentioned earlier, it is necessary to consider boundary
conditions at the center and surface of the star in the following

$\displaystyle\omega^{2}\frac{\mathcal{E}+P_{t}}{\mathcal{E}+P_{r}}\left(1+\frac{\partial\sigma}{\partial
P_{r}}\right)^{-1}\mathrm{e}^{-2\psi}V$ (19)
$\displaystyle+\left(\psi^{\prime}-\frac{2}{r}\frac{\sigma}{\mathcal{E}+P_{r}}\right)\mathrm{e}^{-\lambda}\frac{W}{r^{2}}=0,$

and the boundary condition at the star center $(r=0)$ satisfies

$\tilde{W}=-l\tilde{V}$

where, the functions $\tilde{W}$ and $\tilde{V}$ are defined as
$W=\tilde{W}r^{l+1}$ and $V=\tilde{V}r^{l}$. In this work, we focus on the
quadrupolar modes, which correspond to $l=2$.

In Fig. 5 shows the $f$-mode frequency of a NS as a function of its mass for
various EOS ensembles, including the anisotropic variation for DD2 EOS. The
frequency decreases with increasing $\beta_{\rm QL}$ and increases with
decreasing $\beta_{\rm QL}$. Using the tidal deformability limit from
GW170817, we find a theoretical limit on the canonical $f$-mode frequency for
isotropic stars with a 90% confidence limit given as

  * •

GW170817: $f_{1.4}=2.605^{+0.487}_{-0.459}\ \mathrm{kHz}$ ,

  * •

GW190814: $f_{1.4}=2.093^{+0.150}_{-0.125}\ \mathrm{kHz}$

##  VII Universal Relations

In this analysis, we will explore various types of universal relations (URs)
between the moment of inertia, tidal deformability, and compactness, which
have already been defined by Yagi et al. Yagi and Yunes (2013) for the
isotropic NS. Although, the URs for the anisotropic NS have been put forward
in Ref. Yagi and Yunes (2015); Das (2022); Das  _et al._ (2023), however, our
primary focus will be on the URs between the moment of inertia, $f$-mode
frequency, and compactness of the anisotropic NS. The approximate URs for an
anisotropic NS are of great significance in astrophysical observations. This
is because, with the help of these relations, one can break the degeneracy in
data analysis and model selection for various observations, including x-ray,
radio, and GWs Yagi and Yunes (2015).

Before delving into different URs, it is necessary to normalize/dimensionless
certain key parameters of NSs that are required to obtain the URs. In this
study, we have used the unit of MI and frequency of the $f$-mode are kg/m3 and
s-1, respectively. Therefore, these quantities are needed to be normalized
which are given as

  * •

Normalised MI = $\eta=\sqrt{M^{3}/I}$

  * •

Normalised $f$-mode frequency = $\bar{\omega}=\omega M$

###  VII.1 $I-f$ relation

Figure 6: $I-f$ relation with anisotropy parameter $\beta_{\rm QL}=0$ for
selected EOSs. The black-dashed line shows the least-squares fit using Eq.
(20). The lower panel displays the residuals of the fitting obtained using the
formula in Eq. (21).

The $I-f$ universal relation for slowly rotating NS with a few EOSs was first
calculated by Lau and Leung Lau  _et al._ (2010). Breu and Rezzolla Breu and
Rezzolla (2016) studied the universal behavior of dimensionless MI, which is
defined as $\bar{I}=I/M^{3}$, and is more accurate than the dimensionless MI
defined earlier, $\bar{I}=I/MR^{2}$. Lau and Leung replaced $\bar{I}=I/M^{3}$
with $\eta=\sqrt{M^{3}/I}$, called as normalized MI/effective compactness, due
to its proportionality with compactness for stars with uniform density.
Therefore, in this study, we use $\eta=\sqrt{M^{3}/I}$ rather than
$\bar{I}=I/MR^{2}$. Here, we calculate the $I-f$ relations for anisotropic NS.
We perform the least-squares fit using the approximate formula

$\eta=\sum_{n=0}^{n=4}a_{n}(\bar{\omega})^{n}\,.$ (20)

Figure 7: Upper: Same as Fig. 6, but with $\beta_{\rm QL}=+2$. Lower: For
$\beta_{\rm QL}=-2$

The normalized MI ($\eta$) is plotted as the function of normalized $f$-mode
frequency ($\bar{\omega}$) in Figs. 6-7 with $\beta_{\rm QL}=0,+2,-2$ for
anisotropic NS. The residuals are computed with the formula,

$\Delta\eta=\frac{\eta-\eta_{\rm fit}}{\eta_{\rm fit}}\,.$ (21)

We enumerated the coefficients ($a_{n}$) with their corresponding chi-squared
($\chi^{2}$) errors in Table 1. An increase in anisotropy results in a
decrease in the value of $\chi^{2}$ error, indicating stronger EOS insensitive
relation, and vice versa.

Table 1: The fitting coefficients are listed for $I-f$, $C-f$, and $C-I$
relations with $\beta_{\mathrm{QL}}=$ -2.0, 0.0, +2.0 . The reduced chi-
squared ($\chi_{r}^{2}$) is also given for all cases.

$I-f$ | $C-f$ | $C-I$
---|---|---
$\beta_{\mathrm{QL}}=$ | -2.0 | 0.0 | +2.0 | $\beta_{\mathrm{QL}}=$ | -2.0 | 0.0 | +2.0 | $\beta_{\mathrm{QL}}=$ | -2.0 | 0.0 | +2.0
$a_{0}\left(10^{-2}\right)=$ | 2.8343 | 2.8662 | 2.9010 | $b_{0}\left(10^{-3}\right)=$ | 4.0431 | 4.1439 | 4.2493 | $c_{0}\left(10^{-3}\right)=$ | -8.6012 | -7.9976 | -7.6262
$a_{1}=$ | 3.9493 | 3.9062 | 3.8654 | $b_{1}=$ | 2.2272 | 2.2171 | 2.2101 | $c_{1}\left(10^{-1}\right)=$ | 4.4470 | 4.2716 | 4.1661
$a_{2}\left(10^{1}\right)=$ | -2.4297 | -2.2022 | -2.0197 | $b_{2}=$ | -8.5811 | -7.7939 | -7.2724 | $c_{2}=$ | 1.3717 | 1.5197 | 1.6078
$a_{3}\left(10^{2}\right)=$ | 1.3542 | 1.0996 | 0.8927 | $b_{3}\left(10^{1}\right)=$ | 2.8797 | 2.4738 | 2.1978 | $c_{3}=$ | -4.0139 | -4.4249 | -4.6549
$a_{4}\left(10^{2}\right)=$ | -2.8085 | -2.1488 | -1.6170 | $b_{4}=$ | 11.699 | 1.5765 | -6.6949 | $c_{4}=$ | 4.9121 | 5.4582 | 5.8096
$\chi_{r}^{2}\left(10^{-6}\right)=$ | 13.511 | 5.4863 | 2.5859 | $\chi_{r}^{2}\left(10^{-6}\right)=$ | 2.4094 | 2.0599 | 3.1187 | $\chi_{r}^{2}\left(10^{-6}\right)=$ | 6.7880 | 6.8402 | 6.6086

###  VII.2 $C-f$ relation

Andersson and Kokkotas Kokkotas and Schmidt (1999) first established the
correlation between $C$ and $f$-mode frequency. Here, we calculate the $C-f$
relations for anisotropic NSs, using the approximate formula obtained through
least-squares fitting

$C=\sum_{n=0}^{n=4}b_{n}(\bar{\omega})^{n}\,.$ (22)

Figure 8: $C-f$ relation with anisotropy parameter $\beta_{\rm QL}=0$ for
assumed EOSs. The black-dashed line is fitted with Eq. (22). The lower panel
shows the residuals for the fitting are calculated.

Compactness is plotted as the function of normalized $f$-mode frequency
($\bar{\omega}$) in Figs. 8-9 with $\beta_{\rm QL}=0,+2,-2$ for anisotropic
NS.

Figure 9: Upper: Same as Fig. 8, but with $\beta_{\rm QL}=+2$. Lower: For
$\beta_{\rm QL}=-2$

The coefficients ($b_{n}$) with $\chi^{2}$\- error are enumerated in Table 1.
The magnitude of $b_{n}$ increases with increasing $\beta_{\rm QL}$, implying
that the fitting is more robust for the isotropic case. Additionally,
$\chi^{2}$ also increases with the inclusion of anisotropy, and it is the
minimum for the isotropic case. Therefore, the inclusion of anisotropy
(whether positive or negative) weakens the EOS insensitive $C-I$ UR.

###  VII.3 $C-I$ relation

Figure 10: $C-I$ relation with anisotropy parameter $\beta_{\rm QL}=0$ for
assumed EOSs. The black-dashed line is fitted with Eq. (23). The lower panel
shows the residuals for the fitting are calculated.

The relationship between the dimensionless MI ($\bar{I}=I/M^{3}$) and
compactness has been established as a lower-order polynomial fit by Ravenhall
and Pethick Ravenhall and Pethick (1994). Since then, this relation has been
studied and modified by various authors, including for the double pulsar
system with higher-order polynomial fitting Lattimer and Schutz (2005),
scalar-tensor theory and $R^{2}$ gravity Staykov  _et al._ (2016); Popchev
_et al._ (2019), rotating stars Breu and Rezzolla (2016), and strange stars
Bejger, M. and Haensel, P. (2002). In this work, we investigate the $C-I$
relations for anisotropic NS using the normalized moment of inertia
($\eta=\sqrt{M^{3}/I}$) instead of the dimensionless one. We use the
approximate formula to perform a least-squares fit

$C=\sum_{n=0}^{n=4}c_{n}(\eta)^{n}.$ (23)

Figure 11: Upper: Same as Fig. 10, but with $\beta_{\rm QL}=+2$. Lower: Same
as Fig. 10, but with $\beta_{\rm QL}=-2$

We display the relationship between compactness ($C$) and normalized MI
($\eta$) for anisotropic NSs with $\beta_{\rm QL}=0,+2,-2$, respectively in
Figs. 10-11. We observe that the inclusion of anisotropy has little effect on
the ($\chi^{2}$) error, indicating that the $C-I$ UR is conserved even when
anisotropy is present, especially for NSs with low compactness.

###  VII.4 $f$-Love relation

An important tool for studying the oscillation of NSs through observational
exploration is a UR between the non-radial $f$-mode frequency (a promising
source of GWs) and the tidal deformability (a valuable property that can be
extracted from GW data). The first exploration of multipolar universal
relations between the $f$-mode frequency and tidal deformability of compact
stars was explored by Chan et al. Chan  _et al._ (2014) and further improvised
by Bikram et al. Pradhan  _et al._ (2023). Recently, Sotani and Kumar Sotani
and Kumar (2021b) introduced a UR between the quasinormal modes and tidal
deformability for isotropic NSs. In this work, we calculate the $f$-Love
relations for anisotropic NSs and perform a least-squares fit using the
approximate formula

$\bar{\omega}=\sum_{n=0}^{n=4}d_{n}(\log(\Lambda))^{n}\,.$ (24)

Table 2: The fitting coefficients are listed for $f$-Love relation with
$\beta_{\mathrm{QL}}=$ -2.0, -1.0, 0.0, 1.0, +2.0 . The reduced chi-squared
($\chi_{r}^{2}$) is also given for all cases.

$\beta_{\mathrm{QL}}=$ | -2.0 | -1.0 | 0.0 | 1.0 | +2.0
---|---|---|---|---|---
$d_{0}\left(10^{-1}\right)=$ | 1.9004 | 1.9822 | 2.0560 | 2.1321 | 2.1901
$d_{1}\left(10^{-2}\right)=$ | 0.3910 | -1.1827 | -2.3807 | -3.4981 | -4.3329
$d_{2}\left(10^{-2}\right)=$ | -2.4773 | -1.8477 | -1.3985 | -0.9821 | -0.6807
$d_{3}\left(10^{-3}\right)=$ | 4.9528 | 4.0060 | 3.3631 | 2.7581 | 2.3358
$d_{4}\left(10^{-4}\right)=$ | -2.9034 | -2.4111 | -2.0911 | -1.7825 | -1.5756
$\chi_{r}^{2}\left(10^{-6}\right)=$ | 3.9545 | 2.0994 | 1.1530 | 0.5937 | 0.3233

Figure 12: The $f$-Love relation for anisotropic NSs with $\beta_{\rm QL}=0$
for various assumed EOSs. The black-dashed line represents the best fit using
Eq. (24). The light-pink-shaded region and the orange-shaded region represent
the range of canonical tidal deformability data obtained from the GW190814
Abbott  _et al._ (2020) and GW170817 Abbott  _et al._ (2017) papers,
respectively. Table 3: The canonical normalized $f$-mode frequency
$(\bar{\omega}_{1.4})$, and f-mode frequency ($f_{1.4}$ in kHz) inferred from
GW170817 and GW190814 data.

| $\mathrm{GW170817}$ |  | $\mathrm{GW190814}$
---|---|---|---
$\beta_{\rm QL}$ | $\bar{\omega}_{1.4}$ | $f_{1.4}$ |  | $\bar{\omega}_{1.4}$ | $f_{1.4}$
-2.0 | $0.121^{+0.022}_{-0.019}$ | $2.795^{+0.504}_{-0.452}$ |  | $0.098^{+0.007}_{-0.006}$ | $2.264^{+0.159}_{-0.132}$
0.0 | $0.113^{+0.021}_{-0.019}$ | $2.605^{+0.487}_{-0.459}$ |  | $0.091^{+0.006}_{-0.005}$ | $2.093^{+0.150}_{-0.125}$
+2.0 | $0.108^{+0.021}_{-0.020}$ | $2.500^{+0.483}_{-0.471}$ |  | $0.086^{+0.006}_{-0.005}$ | $1.992^{+0.146}_{-0.122}$

Table 4: The canonical $f$-mode frequency ($f_{1.4}$ in kHz) inferred from
GW170817 and GW190814 data using $f$-Love UR obtained in different literature
for isotropic NS.

| $\mathrm{GW170817}$ |  | $\mathrm{GW190814}$
---|---|---|---
Ref. | $f_{1.4}$ |  | $f_{1.4}$
Chan et al. Chan  _et al._ (2014) | $3.469^{+0.215}_{-0.181}$ |  | $3.242^{+0.073}_{-0.058}$
Bikram et al. Pradhan  _et al._ (2023) | $3.471^{+0.216}_{-0.182}$ |  | $3.243^{+0.073}_{-0.058}$
Sotani and Kumar Sotani and Kumar (2021b) | $2.124^{+0.446}_{-0.440}$ |  | $1.656^{+0.132}_{-0.112}$
This Work | $2.605^{+0.487}_{-0.459}$ |  | $2.093^{+0.150}_{-0.125}$

Figure 13: Upper: Same as Fig. 12, but with $\beta_{\rm QL}=+2$. Lower: Same
as Fig. 12, but with $\beta_{\rm QL}=-2$

The coefficients ($d_{n}$) with $\chi^{2}$\- errors are all enumerated in
Table 2. As anisotropy increases, the errors in $\chi^{2}$ decrease,
indicating that the EOS insensitive relations become stronger with the
addition of anisotropy. On the other hand, as anisotropy decreases, the value
of $\chi^{2}$ increases, resulting in weaker EOS insensitive relations.
Therefore, including positive anisotropy strengthens the $f$-Love UR. We also
constrain the canonical $f$-mode frequency for events GW170817 Abbott  _et
al._ (2017) and GW190814 Abbott  _et al._ (2020) for different degrees of
anisotropy, which are enumerated in Table 3.Additionally, for comparison, we
also constrain the canonical $f$-mode frequency and compare them with previous
works only for isotropic NS, which are enumerated in Table 4.

##  VIII Conclusion

In conclusion, we study the $I-f-C$ relation for anisotropic NSs, taking into
account the local anisotropic pressure using the QL-Model. Our analysis
considered 60 tabulated EoS-ensembles spanning a wide range of stiffness,
complying with multimessenger constraints. The results indicate that the
anisotropy significantly impacts the macroscopic properties of the anisotropic
star, such as mass, radius, and moment of inertia. Furthermore, we found that
positive and negative anisotropy affects tidal deformability parameters and
quadrupolar non-radial $f$-mode frequency significantly, which suggests that
the star with higher anisotropy sustains more life in the inspiral-merger
phase, while the star with lower anisotropy is more likely to collapse.

Further, one can use the $I-f-C$ universal relation for anisotropic stars to
extract information about star properties that are not directly observable
with current detectors and telescopes. By varying the anisotropy value, we
calculated the $I-f$, $C-f$, and $C-I$ universal relations and fitted them
with the polynomial equation using the least-square method. Our results showed
that the reduced chi-square errors for the $I-f,C-f,$ and $C-I$ relations were
$5.4863\times 10^{-6},2.0599\times 10^{-6}$, and $6.8402\times 10^{-6}$,
respectively, for isotropic stars. In addition to the $I-f-C$ universal
relations, we also calculated the $f$-Love universal relation to constrain the
canonical $f$-mode frequency for anisotropic stars. We observed that the
sensitivity of the $C-f$ universal relation is weaker for anisotropic stars in
comparison to the isotropic case. However, the relation between $I-f$ and
$f$-Love became stronger with increasing anisotropy. The C-I relation barely
changed with the inclusion of anisotropy compared to the other universal
relations.

With the help of various observational data for dimensionless tidal
deformability, such as GW170817 and GW190814, we established a theoretical
constraint on the canonical $f$-mode frequency for both isotropic and
anisotropic stars, which is presented in Table 3. This constraint can be
refined by incorporating different anisotropy models and considering various
phenomena such as magnetic fields, quarks in the core, and dark matter in
detail in future work. Therefore, our findings provide avenues for
investigating the various mechanisms that generate anisotropy within compact
stars and for constraining its degree with observational data.

##  IX Acknowledgments

I would like to thanks P. Landry and T. Zhao for their fruitful discussions
regarding universal relations and fitting procedure. B.K. acknowledges partial
support from the Department of Science and Technology, Government of India
with grant no. CRG/2021/000101.

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# Integral closure and Hilbert series of a special monomial ideal

Amir Mafi* and Dler Naderi  Amir Mafi, Department of Mathematics, University
Of Kurdistan, P.O. Box: 416, Sanandaj, Iran<EMAIL_ADDRESS>Dler Naderi,
Department of Mathematics, University of Kurdistan, P.O. Box: 416, Sanandaj,
Iran<EMAIL_ADDRESS>
###### Abstract.

Let $R=K[x_{1},\ldots,x_{n}]$ be the polynomial ring in $n$ variables over a
field $K$ and let $M_{n,t}=(x^{e_{1}},\ldots,x^{e_{n}})$ be a monomial ideal
of $R$, where $x^{e_{i}}=x_{1}^{t}\ldots x_{i-1}^{t}x_{i+1}^{t}\ldots
x_{n}^{t}$. We study the unmixedness of its integral closure. Furthermore, we
compute the Hilbert series of this ideal and we show that this ideal is
Freiman.

###### Key words and phrases:

Hilbert Series, Monomial ideal, Integral closure, Freiman ideal.
*Corresponding author

###### 2010 Mathematics Subject Classification:

13A02, 13D40, 13F55, 13B22

## Introduction

Throughout this paper, we assume that $R=K[x_{1},\ldots,x_{n}]$ is the
polynomial ring in $n$ variables over a field $K$ with the maximal ideal
$\mathfrak{m}=(x_{1},\ldots,x_{n})$. Let $I$ be a monomial ideal of $R$ and
suppose $\operatorname{Ass}(I)$ is the set of associated prime ideals of
$R/I$. Brodmann [2] showed that there exists an integer $t_{0}$ such that
$\operatorname{Ass}(I^{t})=\operatorname{Ass}(I^{t_{0}})$ for all $t\geq
t_{0}$. The smallest such integer $t_{0}$ is called the index of Ass-stability
of $I$, and denoted by $\operatorname{astab}(I)$. Brodmann [3] also showed
that there exists an integer $t_{0}$ such that
$\operatorname{depth}R/I^{t}=\operatorname{depth}R/I^{t_{0}}$ for all $t\geq
t_{0}$. The smallest such integer $t_{0}$ is called the index of depth
stability of $I$ and denoted by $\operatorname{dstab}(I)$.

Let $d,a_{1},...,a_{n}$ be positive integers. We put
$I_{(d;a_{1},...,a_{n})}\subset R$ be the monomial ideal generated by the
monomials $u\in R$ of degree $d$ satisfying $\deg_{x_{i}}(u)\leq a_{i}$ for
all $i=1,...,n$. Monomial ideals of this type are called ideals of Veronese
type. Monomial ideals of Veronese type are polymatroidal. In addition, if
$a_{i}=1$ for all $i$, then we use $I_{d;n}$ instead of $I_{(d;1,\ldots,1)}$.
Note that the product of polymatroidal ideals is again polymatroidal (see [4,
Theorem 5.3]). In particular each power of a polymatroidal ideal is
polymatroidal. In addition, every polymatroidal ideal is a normal ideal (see
[9, Theorem 3.4]).

Katzman [13] gave an explicit formula for the Hilbert series of the Veronese
type ideals ( see also [5]). Jarrah [12] presented
$M_{n,t}=(x^{e_{1}},\ldots,x^{e_{n}})$ a monomial ideal of $R$, where
$x^{e_{i}}=x_{1}^{t}\ldots x_{i-1}^{t}x_{i+1}^{t}\ldots x_{n}^{t}$ and he
proved that these ideals are unmixed monomial ideals of $R$ such that their
integral closures have embedded primes. In [8] the equigenerated monomial
ideal $I$ of $R$ is called a Freiman ideal, if
$\mu(I^{2})=\ell(I)\mu(I)-{\ell(I)\choose 2}$, where $\ell(I)$ is the analytic
spread of $I$, that is, the dimension of
$\mathcal{R}(I)/{{\mathfrak{m}}\mathcal{R}(I)}$.

The purpose of this paper is to give a simply proof for the main theorem of
[12] and also we compute the Hilbert series of $M_{n,t}$. Furthermore, we show
that $M_{n,t}$ is a Freiman ideal. For any unexplained notion or terminology,
we refer the reader to [6] or [14].

##  1\. Integral closure of a special monomial ideal

We start this section by the following definition.

######  Definition 1.1.

Let $I$ be an ideal of $R$. Then an element $\alpha\in R$ is integral over
$I$, if there is an equation

${\alpha}^{k}+a_{1}{\alpha}^{k-1}+\ldots+a_{k-1}\alpha+a_{k}=0,$

with $a_{i}\in I^{i}$. The set of elements $\overline{I}$ in $R$ which are
integral over $I$ is the integral closure of $I$. The ideal $I$ is integrally
closed, if $I=\overline{I}$, and $I$ is normal if all powers of $I$ are
integrally closed. If $u\in\mathfrak{m}^{k-1}$ or if
$u\in\mathfrak{m}\setminus\mathfrak{m}^{2}$, then the ideal
$(u)+\mathfrak{m}^{k}$ is integrally closed (see [11, Exercises 1.18, 1.19]).
Moreover, if $I$ is monomial ideal, then $\overline{I}$ is a monomial ideal
generated by all monomials $u\in R$ for which there exists an integer $k$ such
that $u^{k}\in I^{k}$.

######  Lemma 1.2.

Let $I=(u_{1},\ldots,u_{k})$ be a monomial ideal of $R$ and
$J=(u_{1}^{t},\ldots,u_{k}^{t})$, where $t$ is a natural number. Then
$\overline{J}=\overline{I^{t}}$.

###### Proof.

Suppose that $a\in I^{t}$. Then $a=u_{1}^{e_{1}}\ldots u_{k}^{e_{k}}$, where
$\Sigma_{i=1}^{k}e_{i}=t$. It therefore follows
$a^{t}=(u_{1}^{t})^{e_{1}}\ldots(u_{k}^{t})^{e_{k}}\in J^{t}$. Consider
$x^{t}-a^{t}$ as an equation of integral dependence and so $a\in\overline{J}$.
Thus $I^{t}\subseteq\overline{J}$ and hence
$\overline{I^{t}}\subseteq\overline{J}$. For the converse it is clear
$J\subseteq I^{t}$ and so $\overline{J}\subseteq\overline{I^{t}}$. Therefore
$\overline{J}=\overline{I^{t}}$, as required.

######  Example 1.3.

Let $n=2$ and $I=(x_{1}^{k},x_{2}^{k+1})$ be an ideal of $R$. Then
$\overline{I}=((x_{1},x_{2})^{k+1},x_{1}^{k})$.

###### Proof.

It is clear $(x_{1}^{k+1},x_{2}^{k+1})\subseteq I$ and so, by Lemma, 1.2
$(x_{1},x_{2})^{k+1}=\overline{(x_{1},x_{2})^{k+1}}\subseteq\overline{I}$. It
therefore follows $((x_{1},x_{2})^{k+1},x_{1}^{k})\subseteq\overline{I}$.
Also, $I\subseteq((x_{1},x_{2})^{k+1},x_{1}^{k})$ and hance
$\overline{I}\subseteq\overline{((x_{1},x_{2})^{k+1},x_{1}^{k})}$. Since
$x_{1}^{k}\in(x_{1},x_{2})^{k}$, it follows that
$((x_{1},x_{2})^{k+1},x_{1}^{k})$ is integrally closed and so
$\overline{I}\subseteq((x_{1},x_{2})^{k+1},x_{1}^{k})$. Therefore
$\overline{I}=((x_{1},x_{2})^{k+1},x_{1}^{k})$.

######  Example 1.4.

Let $n=2$ and $I=(x_{1},x_{2}^{k})$. Then $I$ is normal.

###### Proof.

Since $x_{1}\in(x_{1},x_{2})\setminus(x_{1},x_{2})^{2}$, it follows
$\overline{I}=\overline{((x_{1},x_{2})^{k},x_{1})}=((x_{1},x_{2})^{k},x_{1})=I$.
Now, by using [11, Theorem 1.4.10] $I$ is normal, as required.

######  Lemma 1.5.

Let $I$ be a monomial ideal of $R$ such that
$\operatorname{Ass}(I)=\\{\mathfrak{p}\\}$. Then
$\operatorname{Ass}(\overline{I})=\\{\mathfrak{p}\\}$.

###### Proof.

Since $\sqrt{I}=\sqrt{\overline{I}}$, it follows that
$\mathfrak{p}\in\operatorname{Ass}(\overline{I})$. If
$\mathfrak{p}\neq\mathfrak{q}\in\operatorname{Ass}(\overline{I})$, then
$\mathfrak{q}=(\overline{I}:u)$ for some monomial element $u$ of $R$. Let
$x_{i}\in\mathfrak{q}\setminus\mathfrak{p}$. Then $x_{i}u\in\overline{I}$ and
so $(x_{i}u)^{k}\in I^{k}$ for some $k\geq 1$. Since $I$ is a
$\mathfrak{p}-$primary ideal, it follows $u^{t}\in I^{t}$ for some positive
integers $t$ and so $u\in\overline{I}$ and this is a contradiction. Therefore
$\operatorname{Ass}(\overline{I})=\\{\mathfrak{p}\\}$, as required.

######  Theorem 1.6.

Let $M_{n,t}$ and $I_{n-1;n}$ be as before. Then
$\overline{M_{n,t}}=I_{n-1;n}^{t}$.

###### Proof.

As before $M_{n,t}=(x^{e_{1}},\ldots,x^{e_{n}})$, where
$x^{e_{i}}=x_{1}^{t}\ldots x_{i-1}^{t}x_{i+1}^{t}\ldots x_{n}^{t}$. Thus, by
Lemma 1.2, $\overline{M_{n,t}}=\overline{I^{t}_{n-1,n}}$. Since Veronese type
ideals are normal, it follows that $\overline{M_{n,t}}=I_{n-1;n}^{t}$.

If you raise the $x_{i}$ uniformly to some power, say $x_{i}$ is replaced by
$x_{i}^{t}$ everywhere in the generated of square-free Veronese type ideal
$I_{n-1;n}$, then the resulting ideal $M_{n,t}$ is the image of the flat map
$R\rightarrow R$ with $x_{i}\mapsto x_{i}^{t}$ for all $i$. Thus in this case
$M_{n,t}$ will be Cohen-Macaulay, if $I_{n-1;n}$ is so ( see [10]). By [7,
Theorem 4.2] the square-free Veronese type ideal $I_{n-1;n}$ is Cohen-Macaulay
and so $M_{n,t}$ is Cohen-Macaulay.

######  Corollary 1.7.

Let $M_{n,t}$ be as before. Then $M_{n,t}$ is Cohen-Macaulay and
$\overline{M_{n,t}}$ has embedded primes.

###### Proof.

By the above arguments, since $I_{n-1;n}$ is Cohen-Macaulay by [7, Theorem
4.2], it follows that $M_{n,t}$ is Cohen-Macaulay. By using [9, Lemma 5.1],
[15, Theorem 3.4] and Theorem 1.6, $\overline{M_{n,t}}=I_{n-1;n}^{t}$ has
embedded primes, as required.

######  Theorem 1.8.

Let $M_{n,t}$ be as before. Then
$\operatorname{astab}(M_{n,t})=\operatorname{dstab}(M_{n,t})=n-1$.

###### Proof.

Consider the flat map $R\rightarrow R$ with $x_{i}\mapsto x_{i}^{t}$ for all
$i$. Therefore $I_{n-1;n}R=M_{n,t}$. Thus
$\operatorname{Ass}(M_{n,t}^{k})=\operatorname{Ass}(I^{k}_{n-1;n}R)$ and
$\operatorname{depth}(R/{M_{n,t}^{k}})=\operatorname{depth}(R/{I^{k}_{n-1;n}R})$
for all positive integer $k$. Since by using [9, Corollariy 5.7]
$\operatorname{astab}(I_{n-1;n})=\operatorname{dstab}(I_{n-1;n})=n-1$, it
follows that
$\operatorname{astab}(M_{n,t})=\operatorname{dstab}(M_{n,t})=n-1$, as
required.

##  2\. Hilbert series of a special monomial ideal

We start this section by the following theorem.

######  Theorem 2.1.

The Hilbert function of $M_{n,t}$ is given by

$H(i)=\sum\limits_{j=0}^{n-2}{{{(-1)}^{j}}}\left({\begin{array}[]{*{20}{c}}n\\\
j\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{i(n-1)-ji-j+n-1}\\\
{n-1}\end{array}}\right).$

###### Proof.

By [5, Lemma 2.1] $H(i)$ is the number of sequences
$(\beta_{1},\cdots,\beta_{n})$ satisfying
$\sum\limits_{j=1}^{n}{{\beta_{j}}=i(n-1)t}$, $0\leq\beta_{j}\leq it$ and
$\beta_{j}=t\alpha_{j}$ such that $0\leq\alpha_{j}\leq i$. So $H(i)$ is the
number of sequences $(\alpha_{1},\cdots,\alpha_{n})$ satisfying
$\sum\limits_{j=1}^{n}{{\alpha_{j}}=i(n-1)}$ and $0\leq\alpha_{j}\leq i$. In
fact $H(i)$ is the number of integer solution of the equation

$\alpha_{1}+\cdots+\alpha_{n}=i(n-1),~{}~{}~{}~{}0\leq\alpha_{j}\leq
i,~{}~{}~{}~{}1\leq j\leq n.$

For $\alpha_{j}\geq 0$, $1\leq j\leq n$, the number of integer solution of the
equation $\alpha_{1}+\cdots+\alpha_{n}=i(n-1)$ is

$\left({\begin{array}[]{*{20}{c}}{i(n-1)+n-1}\\\ {n-1}\end{array}}\right).$

For $\alpha_{1}>i$ and $\alpha_{j}\geq 0$, $2\leq j\leq n$, the number of
integer solution of the equation $\alpha_{1}+\cdots+\alpha_{n}=i(n-1)$ is the
same as the number of integer solution of the equation
$\gamma_{1}+\cdots+\gamma_{n}=i(n-1)-i-1$, where $\gamma_{1}=\alpha_{1}-i-1$
and $\gamma_{j}=\alpha_{j}$ for $2\leq j\leq n$ and equal to

$\left({\begin{array}[]{*{20}{c}}{i(n-1)-i-1+n-1}\\\
{n-1}\end{array}}\right).$

For $\alpha_{1},\alpha_{2}>i$ and $\alpha_{j}\geq 0$, $3\leq j\leq n$, the
number of integer solution of the equation
$\alpha_{1}+\cdots+\alpha_{n}=i(n-1)$ is the same as the number of integer
solution of the equation $\gamma_{1}+\cdots+\gamma_{n}=i(n-1)-2i-2$, where
$\gamma_{1}=\alpha_{1}-i-1$, $\gamma_{2}=\alpha_{2}-i-1$ and
$\gamma_{j}=\alpha_{j}$ for $3\leq j\leq n$ and equal to

$\left({\begin{array}[]{*{20}{c}}{i(n-1)-2i-2+n-1}\\\
{n-1}\end{array}}\right).$

By similar argument the number of integer solution of the equation
$\alpha_{1}+\cdots+\alpha_{n}=i(n-1)$ such that
$\alpha_{j_{1}},\alpha_{j_{2}},\cdots,\alpha_{j_{r}}>i$ for some
$\\{j_{1},\cdots,j_{r}\\}\subseteq\\{1,\cdots,n\\}$ and $\alpha_{j}\geq 0$ for
$\\{1,\cdots,n\\}\setminus\\{j_{1},\cdots,j_{r}\\}$ is equal to

$\left({\begin{array}[]{*{20}{c}}{i(n-1)-ji-j+n-1}\\\
{n-1}\end{array}}\right).$

Now, by rule of sum we have

$H(i)=\sum\limits_{j=0}^{n-2}{{{(-1)}^{j}}}\left({\begin{array}[]{*{20}{c}}n\\\
j\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{i(n-1)-ji-j+n-1}\\\
{n-1}\end{array}}\right).$

In combinatorial it is known that for positive integer $n$ and $d$, the
coefficient of $x_{1}^{n_{1}}\ldots x_{d}^{n_{d}}$ in the expansion
$(x_{1}+x_{2}+\ldots+x_{d})^{n}$ is

$\frac{n!}{n_{1}!n_{2}!n_{3}!\ldots n_{d}!}$

where each $n_{i}$ is an integer with $0\leq n_{i}\leq n$, for all $1\leq
i\leq d$, and $n_{1}+n_{2}+\ldots+n_{d}=n$. So the coefficient of $t^{i}$ in
the expansion $(1+t+t^{2}+t^{3}+\ldots+t^{d-1})^{n}$ is

$\frac{n!}{n_{1}!n_{2}!n_{3}!\ldots n_{d}!}$

where each $n_{i}$ is an integer with $0\leq n_{i}\leq n$, for all $1\leq
i\leq d$, $n_{1}+n_{2}+\ldots+n_{d}=n$ and
$n_{2}+2n_{3}+3n_{4}+\ldots+(d-1)n_{d}=i$. Katzman [13] denoted this
coefficient by $A_{i}^{n,d}$. Actually

$(1+t+t^{2}+t^{3}+\ldots+t^{d-1})^{n}=\sum\limits_{i\geq
0}{A_{i}^{n,d}}t^{i}.$

Katzman[13] proved the following lemmas in general [13, Theorem 2.5, Lemma 2.7
and Lemma 2.10].

######  Lemma 2.2.

$(1-t)^{n-r}\sum\limits_{i\geq 0}{{i(n-j-1)+n-r-1\choose
n-r-1}t^{i}}=\sum\limits_{l\geq 0}A_{l(n-j-1)}^{n-r,n-j-1}t^{l}.$

######  Lemma 2.3.

$\displaystyle\sum\limits_{i=0}^{\infty}{i(n-j-1)-j+n-1\choose n-1}t^{i}$
$\displaystyle=\sum\limits_{r=0}^{j}(-1)^{r}{j\choose
r}\sum\limits_{i=0}^{\infty}{i(n-j-1)+n-r-1\choose n-r-1}t^{i}.$

######  Lemma 2.4.

$\sum\limits_{k\geqslant 0}{A_{k(n-j-1)}^{n,n-j-1}}=(n-j-1)^{n-1}.$

Now, we are ready to prove the following result.

######  Theorem 2.5.

The Hilbert polynomial of $M_{n,t}$ is

$Q(n)=\sum\limits_{j=0}^{n-2}{{(-1)}^{j}}{n\choose
j}\sum\limits_{r=0}^{j}{{(-1)}^{r}}{j\choose
r}{(1-t)^{r}}\sum\limits_{k\geqslant 0}{A_{k(n-j-1)}^{n-1,n-j-1}{t^{k}}}.$

###### Proof.

By definition of Hilbert polynomial we have

$Q(n)={(1-t)^{n}}\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{n-2}{{{(-1)}^{j}}{n\choose
j}}}{i(n-1)-ji-j+n-1\choose n-1}{t^{i}}.$

Therefore we have

$\displaystyle{(1-t)^{n}}\sum\limits_{i=0}^{\infty}{\sum\limits_{j=0}^{n-2}{{{(-1)}^{j}}{n\choose
j}}{i(n-1)-ji-j+n-1\choose n-1}{t^{i}}}$
$\displaystyle=\sum\limits_{j=0}^{n-2}{(1-t)^{n}}{{{{(-1)}^{j}}{n\choose
j}}\sum\limits_{i=0}^{\infty}{i(n-1)-ji-j+n-1\choose n-1}{t^{i}}}$
$\displaystyle=\sum\limits_{j=0}^{n-2}(1-t)^{n}(-1)^{j}{n\choose
j}\sum\limits_{r=0}^{j}{{(-1)}^{r}}{j\choose
r}\sum\limits_{i=0}^{\infty}{i(n-j-1)+n-r-1\choose n-r-1}{t^{i}}$
$\displaystyle=\sum\limits_{j=0}^{n-2}(-1)^{j}{n\choose
j}\sum\limits_{r=0}^{j}{{(-1)}^{r}}{j\choose
r}(1-t)^{r}(1-t)^{n-r}\sum\limits_{i=0}^{\infty}{i(n-j-1)+n-r-1\choose
n-r-1}{t^{i}}$ $\displaystyle=\sum\limits_{j=0}^{n-2}(-1)^{j}{n\choose
j}\sum\limits_{r=0}^{j}{{(-1)}^{r}}{j\choose
r}(1-t)^{r}\sum\limits_{k\geqslant 0}{A_{k(n-j-1)}^{n-r,n-j-1}{t^{k}}}.$

The second equality follows from Lemma 2.3 and the last equality follows from
Lemma 2.2.

######  Corollary 2.6.

The Hilbert series of $M_{n,t}$ is

$P(n,t)=\frac{\sum\limits_{j=0}^{n-2}{{(-1)}^{j}}{n\choose
j}\sum\limits_{r=0}^{j}{{(-1)}^{r}}{j\choose
r}{(1-t)^{r}}\sum\limits_{k\geqslant
0}{A_{k(n-j-1)}^{n-1,n-j-1}{t^{k}}}}{(1-t)^{n}}.$

######  Corollary 2.7.

The multiplicity of $M_{n,t}$ is

$e(M_{n,t})=\sum\limits_{j=0}^{n-2}{(-1)^{j}{n\choose j}(n-j-1)^{n-1}}.$

If $(S,\mathfrak{m})$ is a Noetherian local ring and $J\subseteq I$ be a pair
of ideals of $S$, then $J$ is a reduction of $I$ if and only if
$\bar{J}=\bar{I}$. In particular, $\ell(I)=\ell(J)$ ( see [1, Theorem 3.2]).

Now, by Theorem 1.6 we have $\bar{M}_{n,t}$ is equal to the Veronese type
ideal $I_{(t(n-1);t,\ldots,t)}$. Therefore $\ell(M_{n,t})=n$.

######  Theorem 2.8.

$M_{n,t}$ is a Freiman ideal for all $n$ and $t$.

###### Proof.

Set $I=M_{n,t}$. Then $I$ is a Freiman ideal if and only if

$\mu(I^{2})=\ell(I)\mu(I)-{\ell(I)\choose 2}.$

$\mu(I^{2})$ is the number of sequence
$(\beta_{1},\beta_{2},\ldots,\beta_{n})$ satisfies
$\beta_{1}+\beta_{2}+\ldots+\beta_{n}=2t(n-1)$ such that $\beta_{i}=0,t$ or
$2t$. So we can say, $\mu(I^{2})$ is the number of sequence
$(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ satisfies
$\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n}=2(n-1)$ and $0\leq\alpha_{i}\leq 2$.

The number of integer solution of equation
$\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n}=2(n-1)$ such that $\alpha_{1}=0$ is
one. Therefore the number of integer solution of equation
$\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n}=2(n-1)$ such that $\alpha_{1}=0$ or
$\alpha_{2}=0$ or $\ldots$ or $\alpha_{n}=0$ is $n={n\choose 1}$.
Also the number of integer solution of equation
$\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n}=2(n-1)$ such that $\alpha_{1}=1$ is
$n-1$. Therefore the number of integer solution of equation
$\alpha_{1}+\alpha_{2}+\ldots+\alpha_{n}=2(n-1)$ such that $\alpha_{1}=1$ or
$\alpha_{2}=1$ or $\ldots$ or $\alpha_{n}=1$ is

${n-1\choose 1}+{n-2\choose 1}+\ldots+{1\choose 1}.$

So $\mu(I^{2})={n\choose 1}+{n-1\choose 1}+\ldots+{1\choose 1}={n+1\choose
2}$. Hence $\mu(I^{2})=n\mu(I)-{n\choose 2}$ and $M_{n,t}$ is Freiman ideal
for any $n$ and $t$.

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