# Gravitational Perturbation in Nonlocal Modified Tolman VII Model

Byon N. Jayawiguna<EMAIL_ADDRESS>Piyabut Burikham
<EMAIL_ADDRESS>High Energy Physics Theory Group, Department of Physics,
Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

###### Abstract

In comparison to the original Tolman VII model, Exact Modified Tolman VII
(EMTVII) with one additional parameter can increase the compactness of compact
object. When the compactness is in the ultracompact regime, the quasinormal
modes (QNMs) of the trapped mode as well as the gravitational echoes become
more viable. Starting with the EMTVII model, we introduce nonlocality into the
matter sector and analyze the effective potential, the QNMs, and the
gravitational echoes of the compact and ultracompact object in the nonlocal
model. The nonlocal gravity version of EMTVII (NEMTVII) is parametrized by the
nonlocal parameter ($\beta$), modified Tolman VII parameter ($\alpha$), and
the compactness ($\mathcal{C}$). It is found that the nonlocal profile
produces the smeared surface and consequently reduce the compactness. The
maximum compactness $\mathcal{C}_{max}=0.4$ occurs when $\alpha=0=\beta$,
i.e., EMTVII with no smearing. For relatively small value of $\beta=0.01$ and
the compactness $\mathcal{C}\lesssim 0.2667$ (with $M=2.14$ solar masses,
$R=11.835$ km at $\alpha=1.4$), the causality condition and the dominant
energy condition (DEC) are satisfied. The quasinormal modes of the
gravitational perturbation are calculated using Bohr-Sommerfeld (BS) fitting
and we find that the nonlocality produces less trapped modes than the original
(EMTVII) counterpart. At high compactness, gravitational echoes are simulated
numerically. Echoes are found to exist in the parameter space where the
dominant energy condition and the causality condition are violated.

##  I Introduction

A long journey from the first prediction up until the first direct detection
of the black hole merger (GW150914) LIGOScientific:2016aoc , the binary
neutron stars (NS) inspiral (GW170817) LIGOScientific:2017vwq  and the two
compact object coalesences (GW200105 and GW200115) recently in
LIGOScientific:2021qlt  proves the existence of gravitational waves (GW). At
the dawn of the gravitational waves era, we still have rich information to
obtain from observations which can be extensively studied both experimentally
and theoretically. From the phenomenological point of view, NS and black holes
could be just two specific objects of a larger family of the astrophysical
objects with high compactness. There are numerous theoretical investigations
to justify the existence of the horizonless compact object whose mass are
large and the radius are relatively small: the black hole mimickers. These
compact object are usually described by the compactness, $\mathcal{C}\equiv
GM/Rc^{2},$ where $G$ is the gravitational constant, $M$ and $R$ are the mass
and the radius of the object and $c$ is the speed of light (Here and
henceforth, we will work in geometric unit where $G=c=1$). They can be
classified as the following CP2019 :

  1. 1.

Compact object: $\mathcal{C}>0.167,$

  2. 2.

Ultracompact Object: $\mathcal{C}>0.33,$

  3. 3.

Object above Buchdahl Limit: $\mathcal{C}>0.444,$

  4. 4.

Object with clean photon sphere: $\mathcal{C}>0.49,$

  5. 5.

Object with near-horizon scale quantum effects: $\mathcal{C}\approx 0.5,$

  6. 6.

Black hole: $\mathcal{C}=0.5.$

NS with realistic equation of state (EoS) has a total mass between 1.4 and
roughly two solar masses ($M_{\odot}$) and its radius is roughly within the
range of $10-20$ km Clark:2002db ; Antoniadis:2013pzd ; Miller:2021qha .
Therefore, NS lies in the compact object category. If the object from the
merger state is compact enough to have a photon sphere, then the post merger
could have a sequence ringdown called the echoes that only emerge in the
ultracompact region. However, some of the object exists, as a black hole
mimicker, with the compactness larger than the Buchdahl limit. The object
which satisfies the surface condition $r_{0}<2.038M$ allows the stable photon
sphere to be formed CP2019 . Examples of objects with clean photon sphere are
the fuzzball Hertog:2017vod ; Bianchi:2020bxa ; Bianchi:2020miz ; Bah:2021jno
; Bianchi:2021xpr ; Bena:2022rna ; Bianchi:2022qph  and gravastar Mazur:2001fv
; Chirenti:2007mk ; Pani:2009hk ; Pani:2009ss ; Pani:2010em ; Chirenti:2016hzd
; Uchikata:2016qku ; Volkel:2017ofl ; Volkel:2017kfj . If the quantum
corrections are taken into account, the gravitational-wave signatures in the
train-echo form could exist Cardoso:2016oxy  for exotic compact object with
$\mathcal{C}\approx 0.5$. Finally, the black hole’s compactness is exactly at
$\mathcal{C}=0.5$.

Gravitational echoes provide information of the effective potential, the
quasinormal modes (QNMs) and the eigen solutions of the time dependent Regge-
Wheeler equation (RW). The production of gravitational echoes (waves) requires
the existence of photon sphere around the object, i.e., when the compactness
reaches $\mathcal{C}>1/3$. It is intriguing to investigate the building blocks
of matter which support this kind of compactness. Among the solutions to the
Einstein field equations in general relativity, Tolman VII (TVII)
Tolman:1939jz ; Lattimer:2000nx ; Jiang:2019vmf  model has a number of
interesting physical properties. It has an analytic solution and represents a
realistic NS which brings us the important information about the Equation of
State (EoS) including nuclear physics Raghoonundun:2015wga ;
Raghoonundun:2016cun ; Hensh:2019rtb ; Stuchlik:2021coc ; Stuchlik:2021vdd ;
Posada:2021zxk ; Posada:2023bnm . For TVII, the scaling axial and polar GW
modes are studied in Ref. Tsui:2004qd  whereas the quasinormal modes are
studied in Ref. Neary:2001ai ; Neary:2001yg . Subsequently, Jiang and Yagi
Jiang:2019vmf  introduce a modified Tolman-VII (MTVII): an additional quartic
term to energy density profile with a free parameter, $\alpha,$ in order to
describe more realistic NS. The numerical solution of this model is studied by
Ref. Posada:2022lij  which we will call the exact numerical solution of the
MTVII (EMTVII) in order to distinguish it from the original MTVII approximate
solution studied in Ref. Jiang:2019vmf . It is found that the relevant range
of the free parameter is $\alpha\in[0,2]$, otherwise the EoS becomes
unphysical due to negative density near the surface. The maximum compactness
of the EMTVII model is $\mathcal{C}_{max}=0.4$ for $\alpha=0$ and the causal
limit is $\mathcal{C}=0.276$ for $\alpha=1.3$.

The post-merger state of the exotic compact object with $\mathcal{C}\approx
0.5$ is almost identical to a black hole. The ringdown waveform from the
coalescence of the binary objects provides a tool for testing the existence of
the compact object (horizonless) and BH dominated by QNMs. The quantum effects
on the near-horizon scale could appear in the late-time ringdown waveform
Cardoso:2016oxy ; Cardoso:2016rao ; Conklin:2017lwb ; Oshita:2018fqu ;
Wang:2019rcf ; Cardoso:2019apo ; Buoninfante:2020tfb . In Ref. Maggio:2020jml
, the pathology in the classical black hole is replaced by the horizonless,
singularity-free solution, by extending the black hole concepts into the
membrane paradigm. Recent observations can be used to put constraint on the
compactness in the order of $\mathcal{C}>0.49$ (99% of the black hole
compactness).

Another black hole mimicker is the nonlocal star which can smear out the
curvature singularity via the nonlocal interaction that could potentially
regulate the UV behavior of nonrenormalizable gravitational theory
Nicolini:2005vd ; Koshelev:2017bxd ; Buoninfante:2018rlq ; Buoninfante:2018xif
; Biswas:2011ar ; Nicolini:2012eu ; Frolov:2015bta ; Buoninfante:2019swn ;
Buoninfante:2019zws . Interestingly, nonlocality could amplify the
gravitational echoes in certain system Buoninfante:2019teo . The other types
of nonlocal theory such as the generalized uncertainty principle (GUP)
Battisti:2007jd ; Shibusa:2007ju ; Battisti:2007zg ; Nozari:2009cs ;
Bojowald:2011jd ; Huang:2012hia ; Sprenger:2012uc ; Isi:2013cxa ; Ali:2013ma ;
Tawfik:2014zca ; Bruneton:2016yws ; Li:2020vqg  and noncommutative geometry
Nicolini:2005zi ; Nicolini:2005de ; Rizzo:2006zb ; Ansoldi:2006vg ;
Casadio:2008qy ; Spallucci:2008ez ; Banerjee:2009xx ; Gingrich:2010ed ;
Modesto:2010rv ; Mann:2011mm ; Mureika:2011py ; Nicolini:2005vd ;
Nicolini:2009gw ; Nicolini:2011dp ; Nicolini:2008aj ; Banerjee:2008gc  are
also discussed.

In this work, we investigate the impact of nonlocal gravity to the MTVII
density model by analytical and numerical analyses. We focus on the
ultracompact object in nonlocal gravity with ideal and isotropic fluid. It is
found that the ultracompact nonlocal object could exist in the range
$\beta\in[0,~{}0.15].$ The effective potential, quasinormal modes, and the
eigen solutions of Regge-Wheeler (RW) equation are subsequently obtained and
explored.

This work is organized as the following. In Section II, we review the ordinary
and modified Tolman VII model and their limitation. In Section III, nonlocal
gravity coupled with the MTVII model is considered and the equations of motion
are solved numerically. Since the maximum compactness and the energy condition
depend on the choices of MTVII parameter $\alpha$, nonlocal parameter $\beta$,
and the compactness $\mathcal{C}$, the results are discussed with respect to
those parameters in Section IV. In Section V, we employ axial perturbations to
the metric describing nonlocal gravity coupled with the EMTVII. The effective
potential, quasinormal modes, and the gravitational echoes are examined.
Section VI concludes our work.

##  II (Exact) Modified Tolman VII Model

In this section, we will briefly review the ordinary TVII model and the
(modified) MTVII model. Generically, the spherically symmetric metric is in
the form

$ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}d\Omega^{2},$ (1)

where $tt$ and $rr$ components are a function of $r,$. The matter sector for
this case is the perfect fluid tensor energy momentum which can be written as

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (2)

where $\rho,p,u_{\mu}$ are energy density, pressure, and 4-velocity
respectively. The Einstein field equation, $G_{\mu\nu}=8\pi T_{\mu\nu},$ gives

$\displaystyle m^{\prime}=4\pi r^{2}\rho,$ (3)
$\displaystyle\left(\frac{e^{-\lambda}-1}{r^{2}}\right)^{\prime}+\left(\frac{e^{-\lambda}\nu^{\prime}}{2r}\right)^{\prime}+e^{-(\lambda+\nu)}\left(\frac{e^{\nu}\nu^{\prime}}{2r}\right)^{\prime}=0,$
(4) $\displaystyle
e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=8\pi
p,$ (5)

Note that the prime symbol ′ denotes $d/dr$ and $e^{-\lambda}\equiv
1-2m(r)/r$. The first eqn. (3) represents the equation for the mass of the
star. The second eqn. (4) represents the equation for solving $e^{\nu}$, and
the pressure can be obtained by substituting the $\nu(r)$ solution to the
third eqn. (5). In the Tolman VII model, the corresponding energy density
profile is defined in a simple quadratic form which can be written as

$\rho(r)\equiv\rho_{c}\left[1-(r/R)^{2}\right],$ (6)

where $\rho_{c}$ is the energy density at the center and $R$ is a stellar
radius. From these equations, we can obtain a complete solution of the metric
inside the star within the TVII model. Substitute the density profile into
Eqn. (3) and integrate with the appropriate choices of boundary conditions at
$r=R$ gives the mass profile of this model

$\displaystyle m_{\rm
TVII}(r)=4\pi\rho_{c}\left(\frac{r^{3}}{3}-\frac{r^{5}}{5R^{2}}\right),$ (7)

hence, the explicit value of $\rho_{c}$ can be obtained by using condition
$M\equiv m(R)$. The result is $\rho_{c}=15M/8\pi R^{3}$. Then, we can obtain
the metric in the $rr$-component as

$\displaystyle e^{-\lambda_{\rm
TVII}}(r)=1-\frac{M}{R}\left(\frac{5r^{2}}{R^{2}}-\frac{3r^{4}}{R^{4}}\right).$
(8)

This result can be used to obtain the $\nu$-function by substituting the
latter equation into eqn. (4). The result reads

$e^{\nu_{\rm TVII}}(r)=C_{1,~{}{\rm TVII}}\cos^{2}[\phi_{\rm TVII}(r)],$ (9)

where

$\phi_{\rm TVII}(r)=C_{2,~{}{\rm
TVII}}-\frac{1}{2}\log\bigg{|}\frac{r^{2}}{R^{2}}-\frac{5}{6}+\sqrt{\frac{e^{-\lambda_{\rm
TVII}}}{3\mathcal{C}}}\bigg{|}.$ (10)

Moreover, the pressure reads

$p=\frac{1}{4\pi R^{2}}\left[\sqrt{3\mathcal{C}e^{-\lambda_{\rm
TVII}}}\tan\phi_{\rm
TVII}-\frac{\mathcal{C}}{2}\left(5-\frac{3r^{2}}{R^{2}}\right)\right].$ (11)

The $C_{1,~{}{\rm TVII}}$ and $C_{2,~{}{\rm TVII}}$ are defined as

$\displaystyle C_{1,~{}{\rm TVII}}$ $\displaystyle\equiv$ $\displaystyle
1-\frac{5\mathcal{C}}{3},$ (12) $\displaystyle C_{2,~{}{\rm TVII}}$
$\displaystyle\equiv$
$\displaystyle\tan^{-1}\left[\sqrt{\frac{\mathcal{C}}{3(1-2\mathcal{C})}}~{}~{}\right]+\frac{1}{2}\log\bigg{|}\frac{1}{6}+\sqrt{\frac{1-2\mathcal{C}}{3\mathcal{C}}}\bigg{|},$
(13)

where the compactness $\mathcal{C}\equiv M/R.$ These integration constants are
determined from the boundary condition at $r=R$; by matching the exterior
solution and use zero pressure value at the object’s surface. The allowed
compactness based on the finiteness of the pressure and the metric for the
TVII model Jiang:2019vmf ; Posada:2021zxk  is $\mathcal{C}\lesssim 0.386$
whereas the allowed compactness with the causality condition (subluminal sound
speed, $dp/d\rho<1$) is $\mathcal{C}\lesssim 0.26$. However, Jiang and Yagi
Jiang:2019vmf  proposed a new model by adding quartic term to the density
model in order to describe more realistic NS. The density model reads

$\rho_{\rm
mod}(r)=\rho_{c}\left[1-\alpha\frac{r^{2}}{R^{2}}+(\alpha-1)\frac{r^{4}}{R^{4}}\right]$
(15)

with a new parameter $\alpha$. The central density, $\rho_{c},$ can be written
as

$\rho_{c}=\frac{105M}{8\pi R^{3}(10-3\alpha),}$ (16)

The mass and the $g^{rr}$ metric component are then given by

$\displaystyle m_{\rm mod}(r)$ $\displaystyle=$
$\displaystyle\frac{M}{(20-6\alpha)}\left[35\frac{r^{3}}{R^{3}}-21\alpha\frac{r^{5}}{R^{5}}+15(\alpha-1)\frac{r^{7}}{R^{7}}\right],$
$\displaystyle e^{-\lambda_{\rm mod}}$ $\displaystyle=$ $\displaystyle
1-\frac{M}{R(10-3\alpha)}\left[35\frac{r^{2}}{R^{2}}-21\alpha\frac{r^{4}}{R^{4}}+15(\alpha-1)\frac{r^{6}}{R^{6}}\right],$
(17)

where “mod” denotes the modified solution corresponds to the density in (15).
In MTVII model with additional term in the density presented in (15), the
$\nu(r)$ equation is no longer analytic. Jiang and Yagi make use of
approximate solution by using $g_{rr}$ metric from the original Tolman VII
with the modified boundary condition Jiang:2019vmf . Later the equations are
numerically solved by Camilo Posada, et al Posada:2022lij  and the physically
allowed range is in $\alpha\in[0,2]$. The allowed (maximum) compactness for
compact object in EMTVII, as presented in Posada:2022lij , is $\mathcal{C}\leq
0.4$ for $\alpha=0$, whereas in causality domain we have $\mathcal{C}\leq
0.276$ for $\alpha=1.3$. Notably, both upper bounds on the compactness are
greater than the TVII model. As a further extension, we will include the
nonlocality effect to the matter sector with EMTVII model and consider how
nonlocal parameter, $\beta,$ affect the range of compactness $\mathcal{C}.$ We
will also compare our solutions with the EMTVII.

##  III Nonlocal Gravity with (Exact) Modified Tolman VII Model (NEMTVII)

The nonlocality in the gravity sector can be mapped into the nonlocality in
the matter sector and has been used recently to study the impact of
nonlocality in a black hole Nicolini:2012eu ; Gaete:2010sp ; Mureika:2010je ;
Moffat:2010bh ; Modesto:2010uh . The action is

$S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}~{}\mathcal{R}(x)+S_{\textrm{matter}},$
(18)

where the ordinary Ricci tensor, $R,$ is embedded through the following form

$\mathcal{R}(x)=\int d^{4}y\sqrt{-g}\mathcal{A}^{2}(x-y)R(y),$ (19)

with the bi-local distribution defined by

$\mathcal{A}^{2}(x-y)\equiv\mathcal{A}^{2}(\Box_{x})\delta^{(4)}(x-y).$ (20)

The operator represents nonlocal interaction between 2 points in spacetime and
could potentially correlate points in the inner and outer region of the black
hole horizon.

Here, the operator $\Box_{x}=\beta g_{\mu\nu}\nabla^{\mu}\nabla^{\nu},$ is
dimensionless D’Alembertian operator. We can obtained the Einstein field
equation (EFE) by varying the action in Eqn. (18) with respect to the metric
tensor $g_{\mu\nu}$ and neglecting the surface term related to variation of
the D’Alembertian. The nonlocal EFE reads

$\mathcal{A}^{2}(\Box)\left(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right)=8\pi
T_{\mu\nu}.$ (21)

The bi-local operator works on the gravity sector. However, the EFE can be
written in the following form

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi\mathcal{T}_{\mu\nu},$
(22)

where $\mathcal{T}_{\mu\nu}\equiv\mathcal{A}^{-2}(\Box)T_{\mu\nu}$. The
operator $\mathcal{A}$ has a unique and well-defined inverse Nicolini:2012eu .
The role of the nonlocal operator now is to correlate energy-matter between 2
points in spacetime. The action (18) can be represented in two possible ways;
Eqn. (21) denotes the nonlocal geometry coupled to the ordinary matter,
whereas Eqn. (22) denotes the ordinary local gravity coupled to the
generalized matter. The two forms are equivalent Nicolini:2012eu . In the
entire calculation in this work, we use Eqn. (22) as the representation of
nonlocal EFE to be solved. The nonlocal effect appears in the matter sector,
and hence, the tensor energy-momentum presented in Eqn. (2) is modified to

$\mathcal{T}_{\mu\nu}=(\tilde{\rho}+\tilde{p})u_{\mu}u_{\nu}+\tilde{p}~{}g_{\mu\nu},$
(23)

and the component of the Einstein field equation reads

$\displaystyle\tilde{m}^{\prime}=4\pi r^{2}\tilde{\rho},$ (24)
$\displaystyle\left(\frac{e^{-\lambda}-1}{r^{2}}\right)^{\prime}+\left(\frac{e^{-\lambda}\nu^{\prime}}{2r}\right)^{\prime}+e^{-(\lambda+\nu)}\left(\frac{e^{\nu}\nu^{\prime}}{2r}\right)^{\prime}=0,$
(25) $\displaystyle
e^{-\lambda}\left(\frac{\nu^{\prime}}{r}+\frac{1}{r^{2}}\right)-\frac{1}{r^{2}}=8\pi\tilde{p},$
(26)

The EFEs for the isotropic perfect fluid lead to three equations for the
matter variables, $\tilde{\rho}~{}\textrm{and}~{}\tilde{p}$, and the two
metric variables, $\nu(r)~{}\textrm{and}~{}\lambda(r)$. Our next task is to
obtain the modified nonlocal density profile. In order to find $\tilde{\rho},$
it is necessary to choose a particular $\mathcal{A}(\Box)$. To the best of our
knowledge, there is no experimental information about the quantum gravity
hence there is no restrictions for choosing a suitable form of
$\mathcal{A}(\Box)$ Isi:2013cxa . We choose the function so that the higher
momenta can be taken into account

$\mathcal{A}(\Box)=(1-\Box)^{1/2}.$ (27)

By using the Schwinger representation111The definition can be written as
Isi:2013cxa
$\hat{\Delta}^{\gamma}=\frac{1}{\Gamma(-\gamma)}\int_{0}^{\infty}ds~{}s^{-\gamma-1}e^{-s\hat{\Delta}}.$
Since the operator acting on the density is $\mathcal{A}^{-2}$, hence
$\hat{\Delta}=(1-\Box)$ and $\gamma=-1.$, we obtain

$\mathcal{A}^{-2}=\int_{0}^{\infty}ds~{}e^{-s[1+(-\Box)]}.$ (28)

Hence the density can be written as

$\displaystyle\tilde{\rho}=\mathcal{A}^{-2}\rho$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}ds~{}e^{-s(1-\beta\nabla^{2})}\int
d^{3}p~{}\rho(p)~{}e^{i\vec{x}.\vec{p}},$ (29) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int_{0}^{\infty}ds~{}e^{-s(1+\beta
p^{2})}\int d^{3}p~{}\rho(p)~{}e^{i\vec{x}.\vec{p}},$ (30) $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\beta}x}\int_{0}^{R}dx^{\prime}~{}x^{\prime}~{}\rho(x^{\prime})\left[e^{-\frac{|x-x^{\prime}|}{\sqrt{\beta}}}-e^{-\frac{|x+x^{\prime}|}{\sqrt{\beta}}}\right],$
(31)

where $\beta$ is the nonlocal parameter that has a dimension
$\textit{length}^{2}$, $\beta\sim\ell^{2}$. After some calculation, we obtain
the expression for the nonlocal density. Since we have absolute signs, this
expression can be split into interior and exterior region and do the
integration with appropriate interval. In general, we can put any density
$\rho(x^{\prime})$ in the integrand. In this work we only use the (E)MTVII
density. Changing variables and inserting the model into Eqn. (31) gives

$\displaystyle\tilde{\rho}_{\textrm{int}}(r)$ $\displaystyle=$
$\displaystyle\frac{\rho_{c}~{}e^{-(r+R)/\sqrt{\beta}}}{r/\sqrt{\beta}}\bigg{\\{}\left(1-e^{2r/\beta}\right)\bigg{[}(\alpha-2)+(7\alpha-10)\frac{\sqrt{\beta}}{R}+3(9\alpha-10)\frac{\beta}{R^{2}}$
(32)
$\displaystyle+60(\alpha-1)\frac{\beta^{3/2}}{R^{3}}+60(\alpha-1)\frac{\beta^{2}}{R^{4}}\bigg{]}+\frac{e^{(r+R)/\sqrt{\beta}}~{}r}{\sqrt{\beta}}\bigg{[}1-\alpha\frac{r^{2}}{R^{2}}+(\alpha-1)\frac{r^{4}}{R^{4}}$
$\displaystyle-6\alpha\frac{\beta}{R^{2}}+120(\alpha-1)\frac{\beta^{2}}{R^{4}}+20(\alpha-1)\frac{\beta}{R^{2}}\bigg{]}\bigg{\\}},$

and

$\displaystyle\tilde{\rho}_{\textrm{ext}}(r)$ $\displaystyle=$
$\displaystyle\frac{\rho_{c}~{}e^{-\frac{(r+R)}{\sqrt{\beta}}}}{r/\sqrt{\beta}}\bigg{\\{}\left(1-e^{2R/\sqrt{\beta}}\right)\bigg{[}(\alpha-2)+3(9\alpha-10)\frac{\beta}{R^{2}}+60(\alpha-1)\frac{\beta^{2}}{R^{4}}\bigg{]}$
(33)
$\displaystyle+\left(1+e^{2R/\sqrt{\beta}}\right)\bigg{[}(7\alpha-10)\frac{\sqrt{\beta}}{R}+60(\alpha-1)\frac{\beta^{3/2}}{R^{3}}\bigg{]}\bigg{\\}}.$

The modified nonlocal density profile is characterized by its star radius $R,$
nonlocal parameter $\beta$ and the energy density value at the center. The
mass can be written as

$\displaystyle\tilde{m}_{\textrm{int}}(r)$ $\displaystyle=$ $\displaystyle
4\pi\rho_{c}\bigg{\\{}\frac{(\alpha-1)}{7}\frac{r^{7}}{R^{4}}+\frac{r^{5}}{R^{2}}\left[\frac{4\beta}{R^{2}}(\alpha-1)-\frac{\alpha}{5}\right]+\frac{r^{3}}{3}\left[1-\frac{6\alpha\beta}{R^{2}}+\frac{120(\alpha-1)\beta^{2}}{R^{4}}\right]$
(34) $\displaystyle-\left[e^{\frac{r-R}{\sqrt{\beta}}}(\beta
r-\beta^{3/2})+e^{-\frac{r+R}{\sqrt{\beta}}}(\beta
r+\beta^{3/2})\right]\times\bigg{[}(\alpha-2)+(7\alpha-10)\frac{\sqrt{\beta}}{R}$
$\displaystyle+\frac{3\beta}{R^{2}}(9\alpha-10)+\frac{60\beta^{3/2}(\alpha-1)}{R^{3}}+\frac{60\beta^{2}(\alpha-1)}{R^{4}}\bigg{]}\bigg{\\}},$

and

$\displaystyle\tilde{m}_{\textrm{ext}}(r)$ $\displaystyle=$ $\displaystyle
4\pi\rho_{c}\left[e^{-\frac{(r+R)}{\sqrt{\beta}}}(\beta
r+\beta^{3/2})-e^{-\frac{2R}{\sqrt{\beta}}}(\beta
R+\beta^{3/2})\right]\times\bigg{\\{}\left(e^{2R/\sqrt{\beta}}-1\right)\bigg{[}(\alpha-2)$
$\displaystyle+3(9\alpha-10)\frac{\beta}{R^{2}}+60(\alpha-1)\frac{\beta^{2}}{R^{4}}\bigg{]}-\left(1+e^{2R/\sqrt{\beta}}\right)\bigg{[}(7\alpha-10)\frac{\sqrt{\beta}}{R}+60(\alpha-1)\frac{\beta^{3/2}}{R^{3}}\bigg{]}\bigg{\\}}.$

We can subsequently solve the second-order differential equation for $\nu(r)$
and obtain the nonlocal pressure. In the next section, we will perform the
numerical calculation and analyze the parametric plot between the maximum
compactness and nonlocal parameter for each value of $\alpha$.

##  IV Validity of the NEMTVII Solution

In Section II and Ref. Posada:2022lij 222Orange solid line in FIG.6, it was
observed that the allowed maximum compactness predicted by EMTVII model for
$0\leq\alpha<1~{}(1<\alpha\leq 2)$ is higher (lower) than the TVII model
$(\alpha=1)$. When the nonlocality is included in the matter sector, the
nonlocal parameter should affect compactness as well as radius of the star. In
this section, the Einstein’s equations are solved numerically under suitable
boundary conditions that match the Schwarzschild solution. From the numerical
solutions, we examine the parameter space of the physically allowed solution
as shown in Fig. 1.

Figure 1: [Left] Maximum compactness versus $\alpha$ for $\alpha\in[0,2]$.
[Right] Parametric plot between compactness $\mathcal{C}$ within the causal
limit ($c_{s}\big{|}_{r=0}\leq 1$) and $\alpha$. Several exact values are
presented in VII.1 and VII.2.

In the left panel of Fig. 1, $\mathcal{C}_{max}$ represents the maximum
compactness for each model. The right panel of Fig. 1 represents the
parametric plot between compactness $\mathcal{C}$ and $\alpha$. The red dotted
line in the left (right) figures represent the maximum compactness without
(within) the causal limit, $c_{s}\big{|}_{r=0}\leq 1$, in EMTVII model
Posada:2022lij . Incorporating nonlocality makes the compactness decreases due
to larger radius of the smeared surface of the star. For each $\beta,$ the
maximum compactness only occurs when $\alpha=0$. Nonlocal parameter, $\beta,$
covers the ultracompact object only in the region $\beta\in[0,~{}0.15].$ The
exact values for the maximum compactness in the nonlocal gravity model that we
will use in the subsequent section are $\mathcal{C}_{max}=0.382$ for
$\beta=0.01$ and $\mathcal{C}_{max}=0.368$ for $\beta=0.03$, both at
$\alpha=0$. On the other hand within the causal limit, the nonlocality can
increase the compactness only for certain values of $\alpha$. The maximum
compactness at causal limit is $\mathcal{C}=0.2667$ for $\beta=0.01$ and
$\alpha=1.4$. Note that the maximum compactness in red dotted line (EMTVII) in
both panel of Fig. 1 is the same as reported in Ref. Posada:2022lij  (solid
orange and blue line in Fig. 6) for the exact numerical solution of MTVII
whereas Ref. Posada:2021zxk  uses approximate solution of MTVII.

Figure 2: Profiles of the energy condition at maximum compactness with and
without the causality condition. The null energy condition is automatically
satisfied since it has non-negative density.

The physically acceptable density and the pressure profile should satisfy the
energy conditions. Hence, it is crucial to check whether the equation of state
of ultracompact stars within the NEMTVII model satisfies the energy condition.
The energy conditions can be categorized into four types, i.e., Strong Energy
Condition (SEC), Weak Energy Condition (WEC), Dominant Energy Condition (DEC),
and Null Energy Condition (NEC). These conditions can be mathematically
summarized as

SEC $\displaystyle:$ $\displaystyle~{}~{}\tilde{\rho}+\tilde{p}\geq
0,~{}~{}~{}\tilde{\rho}+3\tilde{p}\geq 0,$ (36) DEC $\displaystyle:$
$\displaystyle~{}~{}\tilde{\rho}\geq|\tilde{p}|,$ (37) WEC $\displaystyle:$
$\displaystyle~{}~{}\tilde{\rho}\geq 0,~{}~{}~{}\tilde{\rho}+\tilde{p}\geq 0,$
(38) NEC $\displaystyle:$ $\displaystyle~{}~{}\tilde{\rho}\geq 0.$ (39)

We also assume that matter is isotropic. In Fig. 2, we present corresponding
combination between nonlocal density and nonlocal pressure relevant to the
energy conditions. The black and red solid line denote the maximum compactness
the star can have as predicted by EMTVII and the nonlocal extension model
whereas the green line denotes the compactness that obeys causal condition. It
is shown that the green line (i.e., causal NEMTVII model at maximum
compactness) satisfies all of the energy conditions while other causality-
violating profiles at larger compactness only violate DEC.

Figure 3: [Left] Nonlocal density profile with respect to radial coordinate
$r$. The red and green solid line represent the density with nonlocal gravity.
Nonlocality notably smears out the surface of the star.
$\\{\beta,R_{N}/\textrm{Km}\\}=~{}\\{0.01,~{}11.93\\},~{}\\{0.01,~{}12.4\\},~{}\\{10^{-12}~{}(\textrm{EMTVII}),11.4\\}$.
[Right] The nonlocal pressure as a function of radius $r$. $\tilde{p}_{c}$
denotes the critical pressure and the surface is defined by the condition
$\tilde{p}(R_{N})=0.$

With the choices of parameter that we obtain in Fig. 1, we can proceed the
analysis further by considering the profile of the nonlocal density as well as
the pressure which are shown in Fig. 3. We also compare the nonlocal solution
with the original EMTVII profile in order to see the effects of the nonlocal
parameter. From Fig. 3, it is shown that the nonlocality allows the star to
have a larger radius. The blue (red) and pink (green) lines are the
$\tilde{\rho}_{\textrm{ext}}$ ($\tilde{\rho}_{\textrm{int}}$) profile. The
more we incorporate the nonlocality $\beta$, the larger the exterior part and
thus a larger radius. The new radius of the smeared object, in the presence of
$\beta,$ is defined when the density approaches very small value, $\sim
10^{-7}$. For $\beta=0.01,$ we have $R_{N}=11.93~{}\textrm{Km}.$ For
$\beta=0.03$ we have $R_{N}=12.4~{}\textrm{Km}$ and no smeared surface appear
in the EMTVII ($\beta\approx 0$) model. The nonlocal pressure with respect to
radial coordinate is also plotted by solving the TOV equation in (26).
However, if the compactness is larger than a certain maximum value
($\mathcal{C}>\mathcal{C}_{max}$), the function $\nu(r)$ and $\tilde{p}(r)$
become divergent. From Fig. 3, it can be inferred that nonlocality extends
radius of the star.

###  IV.1 Mass-Radius (MR) diagram in NEMTVII model

To understand the range of physical mass of star the NEMTVII mass model can
provide, we present the MR diagram in Fig. 4. In EMTVII model at fixed radius
without the causality condition, increasing $\alpha$ reduces mass of the star
and the maximum mass $M_{max}=3.1M_{\odot}$ occurs at $\alpha=0$. Under
causality condition, however, the maximum mass $M_{max}\simeq 2.14M_{\odot}$
occurs at $\alpha=1.3-1.4$ and it is almost zero at $\alpha=0$, see also Table
2.

Figure 4: [Left] MR-diagram of NEMTVII star at maximum compactness, DEC and
causality condition are not imposed. [Right] MR-diagram of NEMTVII star at
maximum compactness under causality condition. Nonlocality smears the surface
and enlarge the star radius for the massive branch.

When nonlocality is turned on, the star surface is smeared to larger radii as
depicted in Fig. 4. Interestingly, around $R\simeq 11.8$ km, NEMTVII mass
model satisfying causality condition can cover stars with mass around
$0.86-2.1~{}M_{\odot}$ for $\alpha=0.4-1.6$.

##  V Gravitational Perturbation of the Ultracompact Star

One of the interesting aspect of ultracompact object with high compactness
($0.33\leq\mathcal{C}\leq 0.44$) is the existence of light ring and the well
of the effective potential inside the star CP2019 . The incoming waves can be
trapped in the interior resulting in the gravitational waves in the
quasinormal modes. We only focus on the axial gravitational perturbation
considered in Chandrasekhar:1985kt ; Chandrasekhar:1991fi . The nonzero Ricci
tensors from a perturbed metric are given in the following differential
equations Chandrasekhar:1985kt ; Chandrasekhar:1991fi

$\displaystyle(e^{3\psi-\nu-\mu_{2}-\mu_{3}}Q_{23}),_{3}+e^{3\psi-\nu-\mu_{2}+\mu_{3}}Q_{02,0}$
$\displaystyle=$ $\displaystyle
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(40)
$\displaystyle(e^{3\psi-\nu-\mu_{2}-\mu_{3}}Q_{23}),_{2}-e^{3\psi-\nu+\mu_{2}-\mu_{3}}Q_{03,0}$
$\displaystyle=$ $\displaystyle
0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(41)

where $Q_{AB}=q_{A,B}-q_{B,A}$ and
$Q_{A0}=q_{A,0}-\omega_{,A}~{}~{}(\textrm{A}=2,3).$ The variable
$\nu,~{}\mu_{2},~{}\mu_{3},~{}\textrm{and }\psi$ are the unperturbed metric
functions and $\omega,q_{2},q_{3}$ are the perturbations defined in the
following metric Chandrasekhar:1985kt ; Chandrasekhar:1991fi ,

$ds^{2}=e^{2\nu}dt^{2}-e^{2\psi}(d\phi-\omega~{}dt-
q_{2}dx^{2}-q_{3}dx^{3})^{2}-e^{2\mu_{2}}(dx^{2})^{2}-e^{2\mu_{3}}(dx^{3})^{2},$
(42)

where $x^{2},x^{3}=r,\theta$ respectively. From the integrability of (40) and
(41), we obtain

$[e^{-3\psi+\nu+\mu_{2}-\mu_{3}}(e^{3\psi+\nu-\mu_{2}-\mu_{3}}Q_{23}),_{3}],_{3}+[e^{-3\psi+\nu+\mu_{2}+\mu_{3}}(e^{3\psi+\nu-\mu_{2}-\mu_{3}}Q_{23}),_{2}],_{2}=Q_{23,0,0}.$
(43)

Eqn. (43) can be separated by expanding in terms of Gegenbauer function
$\mathcal{C}_{n}^{\alpha}(\theta),$ which obeys

$\left[\frac{d}{d\theta}\sin^{2\alpha}\theta\frac{d}{d\theta}+n(n+2\alpha)\sin^{2\alpha}\theta\right]\mathcal{C}_{n}^{\alpha}(\theta)=0.$
(44)

By separation of variables

$\displaystyle
e^{3\psi+\nu-\mu_{2}-\mu_{3}}Q_{23}=r~{}\Psi(r)~{}\mathcal{C}_{l+2}^{-3/2}(\theta)~{}~{}~{}~{}\textrm{and
introducing}~{}~{}~{}~{}\frac{dr_{*}}{dr}=e^{-\nu+\mu_{2}},$ (45)

the tortoise coordinate, Eqn. (43) can be reduced to the Schrodinger-like wave
equation in $r_{*}$ as

$\left[\frac{\partial^{2}}{\partial t^{2}}-\frac{\partial^{2}}{\partial
r_{*}^{2}}+V(r)\right]\Psi(r_{*},t)=0,$ (46)

where the effective potential is

$V(r)=\frac{e^{\nu}}{r^{3}}\left[l(l+1)r+4\pi
r^{3}\left(\tilde{\rho}-\tilde{p}\right)-6\tilde{m}(r)\right].$ (47)

This master equation is often called time-dependent Regge-Wheeler (RW)
equation which can be analyzed in terms of the tortoise coordinate ($r_{*}$)

$r_{*}=\int_{0}^{r}\sqrt{-\frac{g_{rr}}{g_{tt}}}~{}dr,$ (48)

where $g_{tt}$ and $g_{rr}$ are the metrics of spacetime both inside and
outside of the star.

Figure 5: Tortoise coordinate vs radial coordinate

Fig. 5 represents the relation between $r_{*}$ and $r$ for the cases
$\mathcal{C}_{max}=0.368$ with $\beta=0.03$, $\mathcal{C}_{max}=0.382$ with
$\beta=0.01$, and EMTVII parameter. The value of $r_{*}(0)$ is finite and
becomes more negative as compactness increases. From this aspect, it can be
inferred that compactness plays an important role on the “tortoise-width” of
the interior of the star. Furthermore, the exterior region of the star
exhibits linear relation between $r_{*}$ and $r$.

Figure 6: Effective potential ($V_{eff}$) versus radial coordinate, $r,$ and
the tortoise coordinate, $r_{*}$. The black and red solid line represents the
EMTVII for $l=2$ and $l=3$ respectively.

In Fig 6, we show the effective potential as the function of the radius, $r,$
as well as the tortoise coordinate, $r_{*}.$ Here, we consider first the
EMTVII model in the compactness range of ultracompact objects i.e.,
$0.33\leq\mathcal{C}\leq 0.44$. The star surface is located at $R<3M$, and the
exterior solution has a barrier at $R\approx 3M.$ A perturbed star’s effective
potential also has the infinite potential value at the center due to the
centrifugal contribution in the first term on the right-hand side (RHS) of
Eqn. (47). In the tortoise coordinate, the light ring occurs almost at the
peak of the potential. The effective potential also possess a second light
ring at the minimum. This light ring should be stable since
$V_{eff}^{\prime\prime}>0.$ Moreover, $l$ in the potential determines the
height of the barrier. Later, we will see that the quasinormal mode also
depends on $l$.

Figure 7: Effective potential ($V_{eff}$) with respect to radial $r,$ and the
tortoise coordinate, $r_{*}$ saturating the causal limit. The figures
represent the profile within the causal limit for $l=2$ and $l=3$.

Figure 8: Effective potential ($V_{eff}$) with respect to radial $r,$ and the
tortoise coordinate, $r_{*}$ saturating the DEC.

On the other hand, Fig. 7 shows effective potential in the presence of
nonlocal gravity at $\mathcal{C}_{max}$ with causality condition imposed (see
Right plot in Fig. 1). The upper (lower) part represents the nonlocal
potential with $\beta=0.01,~{}\alpha=1.4,~{}\mathcal{C}=0.2667$ for
$l=3~{}(2)$ respectively. Since the compactness is below the ultracompact
region, the light ring does not exist for both $l.$ The potential well rapidly
disappears and does not have sufficient width and depth to produce quasinormal
modes and gravitational echoes. Similarly for DEC saturating star in NEMTVII
model shown in Fig. 8, the effective potential well is too shallow to sustain
echoes.

Figure 9: Effective potential ($V_{eff}$) versus radial coordinate $r,$ and
the tortoise coordinate, $r_{*}$. The upper (lower) curve represents the
$V_{eff}$ profile with $\beta=0.01,~{}\alpha=0,~{}\mathcal{C}_{max}=0.382$
($\beta=0.03,~{}\alpha=0,~{}\mathcal{C}_{max}=0.368$) for $l=3~{}(2)$
respectively.

However, if causality condition is not imposed, generically the nonlocal EoS
of NEMTVII with $\mathcal{C}_{max}$ provides a well potential with surface at
$R_{N}<3M$ as presented in Fig. 9. Notably, the larger the nonlocal parameter,
the wider the smeared surface (green colour) becomes. The whole interior of
the star is within $0\leq r\leq R_{N}$. For $r>R_{N}$ we match it with the
exterior Schwarzschild $V_{\textrm{eff}}$ profile. The existence of potential
well is crucial for the production of gravitational echoes. The deeper the
well, the more pronounced echoes could be produced. The wider the well, the
longer it will take for the echo to propagate through the interior of the
ultracompact star.

###  V.1 Quasinormal Modes

The literatures from the early stage Kojima:1995nc ; Andersson:1995ez
categorize the QNM spectrum related to the effective potential into three
kinds of spacetime modes, i.e., the trapped modes Kokkotas:1994an  produced by
potential well of ultracompact stars, $w$-modes Kokkotas:1992xak  produced
from scattering on the top of the potential barrier, and $w_{II}$ mode
Leins:1993zz  that corresponds to the extremely fast damped mode. In this
section, we will discuss only the crucial mode for ultracompact stars (axial
part), i.e., the trapped modes.

For the waves with definite energy and possible damping,
$\Psi(r_{*},t)=\psi(r)e^{-i\omega_{n}t}$, Eqn. (46) can be cast into

$\frac{d^{2}\psi}{dr_{*}^{2}}+[\omega^{2}_{n}-V_{\textrm{eff}}(r)]\psi=0,$
(49)

where $\omega_{n}$ is the QNM and $\psi(r)$ is the eigen solution of the time
independent RW equation. QNMs, in general, encode dissipative property of
spacetime and play a crucial role in the formation of gravitational wave
echoes when stars reach the final stage during ringdown phase. When the star
are in these states, the unstable circular orbit traps the primary signal. In
order to obtain the QNMs, the complex eigenvalues on the time-independent
equation shown in (49) must be solved with the appropriate boundary
conditions. From the black hole point of view Urbano:2018nrs , since nothing
can escape from it, we must employ first, an incoming spherical boundary
condition at the black hole horizon and second, an outgoing waves at spatial
infinity Urbano:2018nrs . However, when we consider the ultracompact or
compact stars the boundary conditions will be different since they lack event
horizon. The first boundary condition should be replaced by the regularity
condition at the center, whereas the second boundary condition is unaltered.

Here, we employ Volkel and Kokkotas’s Volkel:2017ofl  procedure to obtain the
QNMs using Wentzel-Kramers-Brillouin (WKB) approximation. In quantum
mechanics, Bohr-Sommerfeld (BS) rule is a well-known method to approximate the
energy spectrum, $E_{n},$ of bound states in a potential. With the WKB theory,
it is possible to include higher-order correction of the BS rule popov  as
follows atomicwkb

$\int_{x_{0}}^{x_{1}}\sqrt{E_{n}-V(x)}dx=\pi\left(n+\frac{1}{2}\right)-\frac{i}{4}e^{2i\int_{x_{1}}^{x_{2}}\sqrt{E_{n}-V(x)}~{}dx},$
(50)

where $x_{0}$ and $x_{1}$ are the classical turning point(s) determined by the
roots of the integrand and also depend on the energy spectrum. The second term
in Eqn. (50) is the additional term in the generalized BS rule, where $x_{2}$
is the third classical turning point on the right side of the potential
barrier333For a detailed discussion of this procedure, see Ref. Volkel:2017ofl
.. This additional term denotes an exponentially small imaginary part of the
energy spectrum, which measures the barrier penetrability. We can simplify
further by writing the energy spectrum into $E_{n}=E_{0n}+iE_{1n}$, where
$E_{0n}$ is the real part of the energy whereas $E_{1n}$ is small imaginary
energy. Insert these energy spectra to the left-hand side of Eqn. (50) and
match with the real and imaginary parts on the right-hand side of Eqn. (50).
The results can be written as

$\int_{x_{0}(E_{0n})}^{x_{1}(E_{0n})}\sqrt{E_{0n}-V(x)}dx=\pi\left(n+\frac{1}{2}\right),$
(51)

and

$E_{1n}=-\frac{1}{2}\left(\int_{x_{0}(E_{0n})}^{x_{1}(E_{0n})}\frac{1}{\sqrt{E_{0n}-V(x)}}dx\right)^{-1}e^{2i\int_{x_{1}(E_{0n})}^{x_{2}(E_{0n})}\sqrt{E_{0n}-V(x)}~{}dx}.$
(52)

Furthermore, we use the well known analytic fitting potential described below
Volkel:2017ofl

$\displaystyle
U_{Q}=U_{0}+\lambda_{0}^{2}(x-x_{min})^{4}~{}~{}~{}\textrm{and}~{}~{}~{}U_{BW}=\frac{U_{1}}{1+\lambda_{1}(x-x_{max})^{2}}.$
(53)

Both functions above are called quartic oscillator potential and Breit-Wigner
potential respectively. These two functions depend on $U_{0},$ $U_{1},$
$\lambda_{0},$ and $\lambda_{1}$ parameters. The value of these parameters is
then matched with the true effective potential (in tortoise coordinate).
$U_{0}$ is the minimum of the true potential, whereas $U_{1}$ is the maximum
of the true potential. Next, $\lambda_{1}$ can be obtained by identifying
$V_{max}^{\prime\prime}$ on the $U_{BW}$ function, which can be written as
$\lambda_{1}=-V_{max}^{\prime\prime}/2V_{max}.$ The last parameter,
$\lambda_{0},$ we demand that the quartic oscillator must be equal to the
Regge-Wheeler equation at the surface Volkel:2017ofl . We can evaluate the
energy spectrum by substituting both fitting functions into Eqn.(51) and
Eqn.(52). The detailed calculation of the integral can be found in Ref.
Volkel:2017ofl . The final solution reads

$\displaystyle E_{0n}$ $\displaystyle=$ $\displaystyle
U_{0}+\lambda_{0}^{2/3}\left[\frac{3\pi}{4~{}\mathcal{K}(-1)}\left(n+\frac{1}{2}\right)\right]^{4/3},$
(54) $\displaystyle E_{1n}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{\lambda_{0}}(E_{0n}-U_{0})^{1/4}}{4~{}\mathcal{K}(-1)}~{}\textrm{Exp}\left[4i\sqrt{\frac{U_{1}-E_{0n}}{\lambda_{1}}}~{}\mathcal{E}\left(i\sinh^{-1}\left(\sqrt{\frac{U_{1}}{E_{0n}}-1}\right),\frac{E_{0n}}{E_{0n}-U_{1}}\right)\right],$
(55)

where $\mathcal{K}(a)$ is the complete elliptic integral of the first kind and
$\mathcal{E}(a,b)$ denotes the elliptic integral of the second kind Abraham .
The variable $n$ shown in the Eqn. (54) is the overtone number. The numerical
results from Eqn. (54) and Eqn. (55) are the same to those obtained from the
formula given in Ref. Volkel:2017ofl .

Figure 10: Quasinormal modes $|\omega_{i}|$ vs $\omega_{r}$. The exact values
are presented in VII.3.

We can match the energy with the complex eigenvalue shown in Eqn.(49) which
can be written as,
$\omega_{n}=\sqrt{E_{n}}=\sqrt{E_{0n}+iE_{1n}}=\omega_{r}+i\omega_{i}$, where
the real part describes the frequency of the oscillation and the imaginary
part describes the inverse of the damping time, $\tau_{d}$. This brings us to
the approximate analytic results for all trapped modes in ultracompact stars.

The result of the QNMs is shown in Fig. 10. We plot the QNM, $|\omega_{i}|$ vs
$\omega_{r}$, with different $l$ and maximum compactness. The number of the
dots in the plot represents the overtone number. For the original EMTVII, the
overtone number for $l=2$ exceed $n\sim 35$ and for $l=3$ we have $n\sim 55$.
For NEMTVII ($\beta=0.03$), the overtone number for the lowest mode is $n\sim
27$ and for $l=3$ is $n\sim 43$. From these overtone numbers, we can conclude
that both $l=2$ and $l=3$ modes allow the ultracompact star EMTVII to have
more trapped modes than the NEMTVII counterpart. On the other hand, for both
$l$, the imaginary parts increase as the compactness decreases (from EMTVII to
NEMTVII). The horizonless ultracompact object with EMTVII matter allows the
axial perturbations to persist with damping time longer than the NEMTVII
model. This is the consequence of the fact that the well of the EMTVII is
wider than the NEMTVII.

###  V.2 Gravitational Echoes and Echo Time

In this section, we numerically analyze the gravitational echoes from the
perturbed ultracompact star by solving the time-dependent partial differential
equation in Eqn. (46). Since the highest differential operator in the equation
is of the second order in $t$ and $r_{*},$ the solution requires four
conditions.

I. $\displaystyle~{}\Psi(r_{*},0)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=~{}0,$ II.
$\displaystyle~{}\Psi(r_{*}^{c},t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=~{}0,$ III.
$\displaystyle~{}\frac{\partial\Psi(t,r_{*})}{\partial
t}\bigg{|}_{t=0}~{}~{}~{}=~{}f(r_{*}),$ IV.
$\displaystyle~{}\frac{\partial\Psi(r_{*},t)}{\partial
r}\bigg{|}_{r_{*}\rightarrow\infty}=-\frac{\partial\Psi(r_{*},t)}{\partial
t}\bigg{|}_{r_{*}\rightarrow\infty}.$ (56)

Condition I and III are the initial data and II and IV are the boundary
conditions. Both initial data denotes the post-merger phase with an initial
Gaussian pulse centered at $r_{*}=r_{g}$ and with spread $\sigma$;
$f(r_{*})=e^{-\frac{(r_{*}-r_{g})^{2}}{\sigma^{2}}}$ Urbano:2018nrs ;
Cardoso:2016oxy . III is the condition for the star to be regular at the
center, where $r_{*}^{c}(r)=r_{*}(0).$ Lastly, condition IV relates outgoing
waves at spatial infinity Urbano:2018nrs . After evaluating numerically, the
solution of the differential equation depends on the time and tortoise radius,
$\Psi(r_{*},t).$ We will focus on the time evolution of the signal from the
EMTVII and NEMTVII model and present the lowest mode only. The signal can be
numerically obtained by inserting the effective potential (as a function of
tortoise coordinate $r_{*}$) into the Eqn. (46). Since $V_{\textrm{eff}}$ in
the EMTVII model has wider interior, the signal needs more time to traverse
the interior as shown in Fig. 11. The effective potential of compact object
with compactness consistent with the causal condition does not exhibit any
well in the interior, and thus such a star does not produce the gravitational
echoes even in the NEMTVII model.

Figure 11: Eigen solution of the time-dependent Regge-Wheeler equation with
respect to time for EMTVII (upper) and NEMTVII (middle and lower) model for
$\alpha=0,1$. The horizonless object in the ultracompact region produces the
train echoes in both models only when DEC and causality condition are
violated. Figure 12: The echo time $\tau$ versus compactness $\mathcal{C}$ of
the EMTVII and NEMTVII star. When the compactness approaches maximum, the echo
time saturates to a certain bound.

Echo time is defined as the time needed for the waves to propagate from center
of the star to the photon sphere radius Mannarelli:2018pjb ; Pani:2018flj .
The echo time can be written as

$\tau_{echo}=\int_{0}^{3M}\sqrt{-\frac{g_{rr}}{g_{tt}}}~{}dr.$ (57)

In our case, the solution is obtained numerically. The echo time is plotted by
varying the compactness as shown in Fig. 12. In the EMTVII model, as denoted
by the black solid line, the limiting compactness where $\tau\to\infty$ is far
below the Buchdahl limit since its compactness is bounded in order to obtain
finite pressure profile. The maximum compactness a star can reach is
$\mathcal{C}_{max}=0.4$ for $\alpha=0$. When we include the nonlocal effect,
the star surface is smeared and the compactness decreases. Thus, the echo time
curve shifts to lower values. The echo time corresponds to the propagation
time from the star center to the light ring as depicted in Fig. 11. The
“$2\tau_{\textrm{echo}}$” denotes the time wave travels from light ring to the
center and then back to the light ring.

On the other hand, the maximum compactness at which DEC is still minimally
satisfied appears at $\alpha=0$, $\beta=0.01$, and $\mathcal{C}=0.324$ (but
causality condition, $c_{s}\leq 1$, is violated) for NEMTVII model. As shown
in Fig. 8, the effective potential well almost disappeared and there is no
echoes. Notably, this compactness is smaller than the corresponding value from
EMTVII model where $\alpha=0,~{}\mathcal{C}=0.34$, just above the lower limit
of the ultracompact object regime.

##  VI Conclusions and Discussions

Exact Modified Tolman VII (EMTVII), an extension of Tolman VII (TVII) with the
additional free parameter, appears as a solution of realistic candidate EoS of
NS. The numerical solution Posada:2022lij  shows that the additional free
parameter, $\alpha$ is adequate to increase both the maximum compactness and
the causal limit. If the compactness of the compact stars takes the value in
the ultracompact regime, the gravitational perturbation can be more viable.

We introduce the nonlocal gravity contribution to the EMTVII solution
(NEMTVII), which could provide the interior structure of the compact object in
the ultracompact regime ($0.33\leq\mathcal{C}\leq 0.44$). In the Einstein
field equation, the nonlocality only couples to the matter sector and
therefore affects only the density and pressure. Notably, nonlocality smears
out surface of the star and effectively reduce its compactness.

It is found that the maximum compactness for NEMTVII model is reduced to
$\mathcal{C}_{max}=0.382$ for $\beta=0.01$ and $\alpha=0$ whether for EMTVII
model the maximum compactness is $\mathcal{C}_{max}=0.4$ and $\alpha=0$. We
further explore two important aspects of the gravitational perturbations: the
quasinormal modes and the time-dependent gravitational wave echoes. By
considering the effective potential in the tortoise coordinate, we found that
the interior of the star in NEMTVII model has smaller well. As a consequence,
the EMTVII model can excite more trapped modes than the nonlocal one. To study
the echoes, gravitational waves solution of the time dependent Regge-Wheeler
equation can be evaluated with appropriate boundary conditions. Each of the
model exhibits train echoes from the incoming waves that enter the light ring
barrier, propagate to the center and come back again to the light ring. The
NEMTVII object has shorter echo time than the EMTVII counterpart due to the
smaller potential well in the interior of the object.

For compact object in NEMTVII mass model with maximum compactness saturating
causality condition such as NS with
$\mathcal{C}=0.2667,~{}\beta=0.01,~{}\alpha=1.4$ (see Table 2), the potential
well and the light ring do not exist since the compactness is below the
ultracompact regime. Notably this star has $M=2.14M_{\odot},R=11.835$ km, the
star profile very close to the NS with multiquark core and stiff nuclear crust
studied in Ref. Pinkanjanarod:2020mgi ; Pinkanjanarod:2021qto ;
Burikham:2021xpn  and the multiquark star studied in Ref.
Ponglertsakul:2022vni . Our results in Fig. 4 explore certain classes of
physical EoS that the EMTVII and NEMTVII parameter space could cover at low
compactness where causality condition is s͡aturating within the star.

It is also interesting to investigate how the NEMTVII model fit into
observational constraints by considering the tidal deformability parametrized
by Love number $k_{2}$ Hinderer:2007mb . There are literatures discussing the
tidal deformation in Tolman VII and EMTVII Posada:2022lij ; Yagi:2013bca ;
Yagi:2013awa ; Doneva:2017jop ; Jiang:2020uvb , we leave this for our further
investigations.

###### Acknowledgements.

We thank to anonymous reviewer for the technical comment. B.N. Jayawiguna is
supported by the Second Century Fund (C2F), Chulalongkorn University,
Thailand. P.B. is supported in part by National Research Council of Thailand
(NRCT) and Chulalongkorn University under Grant N42A660500. This research has
received funding support from the NSRF via the Program Management Unit for
Human Resources & Institutional Development, Research and Innovation [grant
number B39G660025].

##  VII Appendix

###  VII.1 Maximum Compactness ($\mathcal{C}_{max}$)

The maximum compactness is presented in Table 1.

Table 1: Nonlocal parameter $\beta$, new star radius and maximum compactness
for $\beta=10^{-12},~{}0.01,~{}0.03.$

$\begin{array}[]{|c | c | c | c |}\hline\cr&\textrm{EMTVII}&\\\ \hline\cr\hline\cr\alpha&R_{N}~{}({\rm km})&\mathcal{C}_{\textrm{max}}\\\ \hline\cr 0&11.4&0.400\\\ 0.1&11.4&0.399\\\ 0.2&11.4&0.398\\\ 0.3&11.4&0.397\\\ 0.4&11.4&0.395\\\ 0.5&11.4&0.394\\\ 0.6&11.4&0.393\\\ 0.7&11.4&0.391\\\ 0.8&11.4&0.390\\\ 0.9&11.4&0.388\\\ 1&11.4&0.386\\\ 1.1&11.4&0.383\\\ 1.2&11.4&0.381\\\ 1.3&11.4&0.378\\\ 1.4&11.4&0.374\\\ 1.5&11.4&0.371\\\ 1.6&11.4&0.366\\\ 1.7&11.4&0.362\\\ 1.8&11.4&0.356\\\ 1.9&11.4&0.349\\\ 2&11.4&0.341\\\ \hline\cr\end{array}\begin{array}[]{|c | c | c | c |}\hline\cr&\beta=0.01&\\\ \hline\cr\hline\cr\alpha&R_{N}~{}({\rm km})&\mathcal{C}_{\textrm{max}}\\\ \hline\cr 0&11.933&0.382\\\ 0.1&11.930&0.381\\\ 0.2&11.928&0.380\\\ 0.3&11.925&0.379\\\ 0.4&11.922&0.378\\\ 0.5&11.919&0.377\\\ 0.6&11.916&0.376\\\ 0.7&11.912&0.375\\\ 0.8&11.908&0.373\\\ 0.9&11.90&0.371\\\ 1&11.897&0.370\\\ 1.1&11.891&0.368\\\ 1.2&11.88&0.365\\\ 1.3&11.87&0.363\\\ 1.4&11.865&0.360\\\ 1.5&11.852&0.357\\\ 1.6&11.836&0.353\\\ 1.7&11.815&0.349\\\ 1.8&11.784&0.344\\\ 1.9&11.734&0.339\\\ 2&11.60&0.335\\\ \hline\cr\end{array}\begin{array}[]{|c | c | c | c |}\hline\cr&\beta=0.03&\\\ \hline\cr\hline\cr\alpha&R_{N}({\rm km})&\mathcal{C}_{\textrm{max}}\\\ \hline\cr 0&12.406&0.368\\\ 0.1&12.402&0.367\\\ 0.2&12.398&0.366\\\ 0.3&12.394&0.365\\\ 0.4&12.389&0.364\\\ 0.5&12.384&0.363\\\ 0.6&12.378&0.362\\\ 0.7&12.372&0.361\\\ 0.8&12.364&0.360\\\ 0.9&12.356&0.358\\\ 1&12.347&0.357\\\ 1.1&12.337&0.355\\\ 1.2&12.325&0.353\\\ 1.3&12.311&0.350\\\ 1.4&12.294&0.348\\\ 1.5&12.273&0.345\\\ 1.6&12.247&0.342\\\ 1.7&12.213&0.338\\\ 1.8&12.166&0.334\\\ 1.9&12.092&0.330\\\ 2&1.936&0.326\\\ \hline\cr\end{array}$

###  VII.2 Compactness within the Causal Condition, $c_{s}\leq 1$

The compactness within causality condition is presented in Table 2.

Table 2: Compactness within the causal condition ($c_{s}\leq 1$).

$\begin{array}[]{|c | c | c | c |}\hline\cr&\textrm{EMTVII}&\\\ \hline\cr\hline\cr\alpha&R_{N}~{}({\rm km})&\mathcal{C}_{\textrm{max}}\\\ \hline\cr 0&11.4&10^{-5}\\\ 0.1&11.4&0.084\\\ 0.2&11.4&0.144\\\ 0.3&11.4&0.179\\\ 0.4&11.4&0.208\\\ 0.5&11.4&0.226\\\ 0.6&11.4&0.240\\\ 0.7&11.4&0.250\\\ 0.8&11.4&0.259\\\ 0.9&11.4&0.265\\\ 1&11.4&0.269\\\ 1.1&11.4&0.273\\\ 1.2&11.4&0.275\\\ 1.3&11.4&0.2765\\\ 1.4&11.4&0.276\\\ 1.5&11.4&0.275\\\ 1.6&11.4&0.274\\\ 1.7&11.4&0.272\\\ 1.8&11.4&0.268\\\ 1.9&11.4&0.263\\\ 2&11.4&0.258\\\ \hline\cr\end{array}\begin{array}[]{|c | c | c | c |}\hline\cr&\beta=0.01&\\\ \hline\cr\hline\cr\alpha&R_{N}~{}({\rm km})&\mathcal{C}_{\textrm{max}}\\\ \hline\cr 0&11.438&0.001\\\ 0.1&11.782&0.086\\\ 0.2&11.828&0.139\\\ 0.3&11.848&0.175\\\ 0.4&11.858&0.199\\\ 0.5&11.864&0.217\\\ 0.6&11.867&0.230\\\ 0.7&11.868&0.240\\\ 0.8&11.867&0.249\\\ 0.9&11.865&0.254\\\ 1&11.862&0.259\\\ 1.1&11.857&0.262\\\ 1.2&11.851&0.265\\\ 1.3&11.844&0.2664\\\ 1.4&11.835&0.2667\\\ 1.5&11.823&0.2664\\\ 1.6&11.807&0.265\\\ 1.7&11.786&0.263\\\ 1.8&11.756&0.260\\\ 1.9&11.706&0.257\\\ 2&11.578&0.254\\\ \hline\cr\end{array}\begin{array}[]{|c | c | c | c |}\hline\cr&\beta=0.2&\\\ \hline\cr\hline\cr\alpha&R_{N}({\rm km})&\mathcal{C}_{\textrm{max}}\\\ \hline\cr 0&13.218&0.028\\\ 0.1&13.733&0.090\\\ 0.2&13.885&0.128\\\ 0.3&13.957&0.154\\\ 0.4&14&0.173\\\ 0.5&14.024&0.188\\\ 0.6&14.038&0.198\\\ 0.7&14.044&0.207\\\ 0.8&14.041&0.213\\\ 0.9&14.03&0.218\\\ 1&14.024&0.221\\\ 1.1&14.012&0.225\\\ 1.2&13.991&0.227\\\ 1.3&13.968&0.2291\\\ 1.4&13.935&0.2290\\\ 1.5&13.896&0.2294\\\ 1.6&13.847&0.2290\\\ 1.7&13.784&0.227\\\ 1.8&13.700&0.226\\\ 1.9&13.580&0.224\\\ 2&13.395&0.222\\\ \hline\cr\end{array}$

###  VII.3 Table for QNM

The QNMs are presented in Table 3, 4.

Table 3: The lowest-mode QNM for EMTVII ($\alpha=0~{}\textrm{and}~{}\mathcal{C}_{max}=0.4$) and NEMTVII ($\beta=0.03,~{}\alpha=0,~{}\mathcal{C}_{max}=0.368$). $l=2,$ |  EMTVII |  |  NEMTVII |  |
---|---|---|---|---|---
$n$ | Re$(\omega_{n})$ | $\big{|}\textrm{Im}(\omega_{n})\big{|}$ |  |  | Re$(\omega_{n})$ | $\big{|}\textrm{Im}(\omega_{n})\big{|}$ |  |  |  |
$0$ | 0.0067 | $5.16\times 10^{-15}$ |  |  | 0.0081 | $2.91\times 10^{-14}$ |  |  |  |
$1$ | 0.0112 | $3.69\times 10^{-13}$ |  |  | 0.0134 | $2.14\times 10^{-12}$ |  |  |  |
$2$ | 0.0151 | $4.49\times 10^{-12}$ |  |  | 0.0180 | $2.68\times 10^{-10}$ |  |  |  |
$3$ | 0.0185 | $2.63\times 10^{-11}$ |  |  | 0.0221 | $1.61\times 10^{-10}$ |  |  |  |
$4$ | 0.0217 | $1.04\times 10^{-10}$ |  |  | 0.0259 | $6.60\times 10^{-10}$ |  |  |  |
$5$ | 0.0247 | $3.23\times 10^{-10}$ |  |  | 0.0295 | $2.11\times 10^{-9}$ |  |  |  |
$6$ | 0.0275 | $8.51\times 10^{-10}$ |  |  | 0.0328 | $5.71\times 10^{-9}$ |  |  |  |
$7$ | 0.0302 | $1.98\times 10^{-9}$ |  |  | 0.0360 | $1.37\times 10^{-8}$ |  |  |  |
$8$ | 0.0328 | $4.22\times 10^{-9}$ |  |  | 0.0391 | $2.00\times 10^{-8}$ |  |  |  |
$9$ | 0.0352 | $8.35\times 10^{-9}$ |  |  | 0.0421 | $6.12\times 10^{-8}$ |  |  |  |
$10$ | 0.0376 | $1.56\times 10^{-8}$ |  |  | 0.0449 | $1.17\times 10^{-7}$ |  |  |  |
$11$ | 0.0399 | $2.78\times 10^{-8}$ |  |  | 0.0477 | $2.16\times 10^{-7}$ |  |  |  |
$12$ | 0.0422 | $4.77\times 10^{-8}$ |  |  | 0.0504 | $3.81\times 10^{-7}$ |  |  |  |
$13$ | 0.0444 | $7.90\times 10^{-8}$ |  |  | 0.0530 | $6.49\times 10^{-7}$ |  |  |  |
$14$ | 0.0465 | $1.27\times 10^{-7}$ |  |  | 0.0556 | $1.07\times 10^{-6}$ |  |  |  |
$15$ | 0.0486 | $1.99\times 10^{-7}$ |  |  | 0.0581 | $1.73\times 10^{-6}$ |  |  |  |
$16$ | 0.0507 | $3.06\times 10^{-7}$ |  |  | 0.0605 | $2.74\times 10^{-6}$ |  |  |  |
$17$ | 0.0527 | $4.61\times 10^{-7}$ |  |  | 0.0629 | $4.24\times 10^{-6}$ |  |  |  |
$18$ | 0.0547 | $6.28\times 10^{-7}$ |  |  | 0.0653 | $6.44\times 10^{-6}$ |  |  |  |
$19$ | 0.0566 | $9.93\times 10^{-7}$ |  |  | 0.0676 | $9.65\times 10^{-6}$ |  |  |  |
$20$ | 0.0585 | $1.42\times 10^{-6}$ |  |  | 0.0699 | $1.42\times 10^{-5}$ |  |  |  |
$21$ | 0.0604 | $2.02\times 10^{-6}$ |  |  | 0.0721 | $2.07\times 10^{-5}$ |  |  |  |
$22$ | 0.0623 | $2.83\times 10^{-6}$ |  |  | 0.0743 | $2.98\times 10^{-5}$ |  |  |  |
$23$ | 0.0641 | $3.92\times 10^{-6}$ |  |  | 0.0765 | $4.24\times 10^{-5}$ |  |  |  |
$24$ | 0.0659 | $5.38\times 10^{-6}$ |  |  | 0.0786 | $5.98\times 10^{-5}$ |  |  |  |
$25$ | 0.0677 | $7.32\times 10^{-7}$ |  |  | 0.0808 | $8.35\times 10^{-5}$ |  |  |  |
$26$ | 0.0694 | $9.88\times 10^{-6}$ |  |  | 0.0828 | $1.15\times 10^{-4}$ |  |  |  |
$27$ | 0.0711 | $1.32\times 10^{-5}$ |  |  | 0.0849 | $1.58\times 10^{-4}$ |  |  |  |
$28$ | 0.0728 | $1.75\times 10^{-5}$ |  |  |  |  |  |  |  |
$29$ | 0.0745 | $2.31\times 10^{-5}$ |  |  |  |  |  |  |  |
$30$ | 0.0762 | $3.03\times 10^{-5}$ |  |  |  |  |  |  |  |
$31$ | 0.0778 | $3.95\times 10^{-5}$ |  |  |  |  |  |  |  |
$32$ | 0.0795 | $5.12\times 10^{-5}$ |  |  |  |  |  |  |  |
$33$ | 0.0811 | $6.61\times 10^{-5}$ |  |  |  |  |  |  |  |
$34$ | 0.0827 | $8.48\times 10^{-5}$ |  |  |  |  |  |  |  |
$35$ | 0.0843 | $1.08\times 10^{-4}$ |  |  |  |  |  |  |  |
Table 4: QNMs with $l=3$ for EMTVII ($\alpha=0~{}\textrm{and}~{}\mathcal{C}_{max}=0.40$) and NEMTVII ($\beta=0.03,~{}\alpha=0,~{}\mathcal{C}_{max}=0.368$). $l=3,$ |  EMTVII |  |  NEMTVII |  |
---|---|---|---|---|---
$n$ | Re$(\omega_{n})$ | $\big{|}\textrm{Im}(\omega_{n})\big{|}$ |  |  | Re$(\omega_{n})$ | $\big{|}\textrm{Im}(\omega_{n})\big{|}$ |  |  |  |
$0$ | 0.0090 | $1.36\times 10^{-21}$ |  |  | 0.0107 | $1.63\times 10^{-20}$ |  |  |  |
$1$ | 0.0138 | $3.56\times 10^{-19}$ |  |  | 0.0163 | $4.19\times 10^{-18}$ |  |  |  |
$2$ | 0.0181 | $1.23\times 10^{-17}$ |  |  | 0.0215 | $1.46\times 10^{-16}$ |  |  |  |
$3$ | 0.0220 | $1.61\times 10^{-16}$ |  |  | 0.0261 | $1.94\times 10^{-15}$ |  |  |  |
$4$ | 0.0257 | $1.20\times 10^{-15}$ |  |  | 0.0304 | $1.49\times 10^{-14}$ |  |  |  |
$5$ | 0.0291 | $6.36\times 10^{-15}$ |  |  | 0.0345 | $8.08\times 10^{-14}$ |  |  |  |
$6$ | 0.0324 | $2.62\times 10^{-14}$ |  |  | 0.0384 | $3.42\times 10^{-13}$ |  |  |  |
$7$ | 0.0355 | $9.06\times 10^{-14}$ |  |  | 0.0420 | $1.21\times 10^{-12}$ |  |  |  |
$8$ | 0.0385 | $2.72\times 10^{-13}$ |  |  | 0.0455 | $3.74\times 10^{-12}$ |  |  |  |
$9$ | 0.0413 | $7.36\times 10^{-13}$ |  |  | 0.0489 | $1.03\times 10^{-11}$ |  |  |  |
$10$ | 0.0441 | $1.82\times 10^{-12}$ |  |  | 0.0522 | $2.64\times 10^{-11}$ |  |  |  |
$11$ | 0.0468 | $4.20\times 10^{-12}$ |  |  | 0.0554 | $6.26\times 10^{-11}$ |  |  |  |
$12$ | 0.0494 | $9.13\times 10^{-12}$ |  |  | 0.0585 | $1.39\times 10^{-10}$ |  |  |  |
$13$ | 0.0520 | $1.88\times 10^{-11}$ |  |  | 0.0615 | $2.95\times 10^{-10}$ |  |  |  |
$14$ | 0.0545 | $3.71\times 10^{-11}$ |  |  | 0.0645 | $6.00\times 10^{-10}$ |  |  |  |
$15$ | 0.0569 | $7.06\times 10^{-11}$ |  |  | 0.0674 | $1.17\times 10^{-9}$ |  |  |  |
$16$ | 0.0593 | $1.29\times 10^{-10}$ |  |  | 0.0702 | $2.20\times 10^{-9}$ |  |  |  |
$17$ | 0.0617 | $2.31\times 10^{-10}$ |  |  | 0.0730 | $4.04\times 10^{-9}$ |  |  |  |
$18$ | 0.0640 | $4.01\times 10^{-10}$ |  |  | 0.0757 | $7.20\times 10^{-9}$ |  |  |  |
$19$ | 0.0662 | $6.81\times 10^{-10}$ |  |  | 0.0784 | $1.25\times 10^{-8}$ |  |  |  |
$20$ | 0.0685 | $1.13\times 10^{-9}$ |  |  | 0.0810 | $2.13\times 10^{-8}$ |  |  |  |
$21$ | 0.0706 | $1.83\times 10^{-9}$ |  |  | 0.0836 | $3.56\times 10^{-8}$ |  |  |  |
$22$ | 0.0728 | $2.94\times 10^{-9}$ |  |  | 0.0861 | $5.85\times 10^{-8}$ |  |  |  |
$23$ | 0.0749 | $4.62\times 10^{-9}$ |  |  | 0.0887 | $9.45\times 10^{-8}$ |  |  |  |
$24$ | 0.0770 | $7.16\times 10^{-9}$ |  |  | 0.0911 | $1.50\times 10^{-7}$ |  |  |  |
$25$ | 0.0791 | $1.09\times 10^{-8}$ |  |  | 0.0936 | $2.35\times 10^{-7}$ |  |  |  |
$26$ | 0.0811 | $1.65\times 10^{-8}$ |  |  | 0.0960 | $3.64\times 10^{-7}$ |  |  |  |
$27$ | 0.0831 | $2.46\times 10^{-8}$ |  |  | 0.0984 | $5.58\times 10^{-7}$ |  |  |  |
$28$ | 0.0851 | $3.63\times 10^{-8}$ |  |  | 0.1007 | $8.44\times 10^{-7}$ |  |  |  |
$29$ | 0.0871 | $5.30\times 10^{-8}$ |  |  | 0.1031 | $1.26\times 10^{-6}$ |  |  |  |
$30$ | 0.0890 | $7.67\times 10^{-8}$ |  |  | 0.1054 | $1.87\times 10^{-6}$ |  |  |  |
|  EMTVII |  |  NEMTVII |  |
---|---|---|---|---|---
$n$ | Re$(\omega_{n})$ | $\big{|}\textrm{Im}(\omega_{n})\big{|}$ |  |  | Re$(\omega_{n})$ | $\big{|}\textrm{Im}(\omega_{n})\big{|}$ |  |  |  |
$31$ | 0.0910 | $1.10\times 10^{-7}$ |  |  | 0.1076 | $2.75\times 10^{-6}$ |  |  |  |
$32$ | 0.0929 | $1.56\times 10^{-7}$ |  |  | 0.1099 | $4.01\times 10^{-6}$ |  |  |  |
$33$ | 0.0947 | $2.20\times 10^{-7}$ |  |  | 0.1121 | $5.80\times 10^{-6}$ |  |  |  |
$34$ | 0.0966 | $3.08\times 10^{-7}$ |  |  | 0.1143 | $8.31\times 10^{-6}$ |  |  |  |
$35$ | 0.0985 | $4.28\times 10^{-7}$ |  |  | 0.1165 | $1.18\times 10^{-5}$ |  |  |  |
$36$ | 0.1003 | $5.91\times 10^{-7}$ |  |  | 0.1187 | $1.67\times 10^{-5}$ |  |  |  |
$37$ | 0.1021 | $8.11\times 10^{-7}$ |  |  | 0.1208 | $2.35\times 10^{-5}$ |  |  |  |
$38$ | 0.1039 | $1.10\times 10^{-6}$ |  |  | 0.1230 | $3.29\times 10^{-5}$ |  |  |  |
$39$ | 0.1057 | $1.49\times 10^{-6}$ |  |  | 0.1251 | $4.56\times 10^{-5}$ |  |  |  |
$40$ | 0.1075 | $2.02\times 10^{-6}$ |  |  | 0.1272 | $6.31\times 10^{-5}$ |  |  |  |
$41$ | 0.1092 | $2.70\times 10^{-6}$ |  |  | 0.1292 | $8.67\times 10^{-5}$ |  |  |  |
$42$ | 0.1110 | $3.61\times 10^{-6}$ |  |  | 0.1313 | $1.18\times 10^{-4}$ |  |  |  |
$43$ | 0.1127 | $4.80\times 10^{-6}$ |  |  | 0.1333 | $1.61\times 10^{-4}$ |  |  |  |
$44$ | 0.1144 | $6.35\times 10^{-6}$ |  |  |  |  |  |  |  |
$45$ | 0.1161 | $8.36\times 10^{-6}$ |  |  |  |  |  |  |  |
$46$ | 0.1178 | $1.09\times 10^{-5}$ |  |  |  |  |  |  |  |
$47$ | 0.1195 | $1.43\times 10^{-5}$ |  |  |  |  |  |  |  |
$48$ | 0.1211 | $1.86\times 10^{-5}$ |  |  |  |  |  |  |  |
$49$ | 0.1228 | $2.41\times 10^{-5}$ |  |  |  |  |  |  |  |
$50$ | 0.1244 | $3.11\times 10^{-5}$ |  |  |  |  |  |  |  |
$51$ | 0.1261 | $4.01\times 10^{-5}$ |  |  |  |  |  |  |  |
$52$ | 0.1277 | $5.14\times 10^{-5}$ |  |  |  |  |  |  |  |
$53$ | 0.1293 | $6.58\times 10^{-5}$ |  |  |  |  |  |  |  |
$54$ | 0.1309 | $8.39\times 10^{-5}$ |  |  |  |  |  |  |  |
$55$ | 0.1325 | $1.06\times 10^{-4}$ |  |  |  |  |  |  |  |

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