#  Isotropic compact stars in 4D Einstein-Gauss-Bonnet gravity

Sudan Hansraj1, Ayan Banerjee 1, Lushen Moodly1 and M. K. Jasim2 1
Astrophysics and Cosmology Research Unit, University of KwaZulu Natal, Private
Bag X54001, Durban 4000, South Africa,
4Department of Mathematical and Physical Sciences, College of Arts and
Science, University of Nizwa, Nizwa, Sultanate of Oman

###### Abstract

Recently it has been proposed that the Gauss-Bonnet coupling parameter of
Lovelock gravity may suitably be rescaled in order to admit physically viable
models of celestial phenomena such that higher curvature effects are active in
standard four dimensions as opposed to the usual higher dimensions. We
investigate the consequences of this modification in the context of stellar
modelling. The evolution of perfect fluid distributions is governed by the
pressure isotropy condition and through stipulation of one of the metric
potentials complete models emerge from solutions of the master differential
equation. New classes of exact solution with this approach have been reported.
One particular model is analysed in detail and shown to comport with
elementary physical requirements demanded of realistic compact stars
suggesting that the modified theory is not inconsistent with observations.

4$D$-EGB gravity; Compact Star; Stability

##  I Introduction

In a recent contribution Glavan and Lin Glavan:2019inb  proposed that the
Einstein–Gauss–Bonnet (EGB) gravity theory, which is well known to be
quintessentially a higher dimensional theory with extra curvature terms that
are dynamic in dimensions 5 and 6, may in fact be studied in the four
dimensional context by a scaling of the Gauss-Bonnet coupling $\alpha$ by a
factor of $\alpha/(D-4)$, and taking the limit $D\to 4$. A major advantage of
this procedure is that the Lovelock theorem demanding up to second order
equations of motion is satisfied and Ostrogradsky instability is avoided. The
mathematical consequences to this approach are mitigated by the dimensional
regularization process proposed by Tomozawa:2011gp . One application of this
proposal was made by Glavan and Lin in generating a static spherically
symmetric singularity–free black hole solution. It turns out that this
interesting scenario is different from those of black hole solutions in higher
dimensional EGB gravity.

Notwithstanding the ostensible positive features of the 4$D$ EGB theory there
are several criticisms against the proposal. At the outset, subjecting a
discrete element such as spacetime dimension to a limiting process has evoked
concern but the dimensional regularization technique endeavours to deal with
this aspect Tomozawa:2011gp . Gurses et al Gurses:2020ofy ; Ai:2020peo ;
Mahapatra:2020rds ; Shu:2020cjw  have discussed two independent lines of
reasoning to show why such a theory is not palatable in a four dimensional
setting in general. These authors do however agree that in spherically
symmetric spacetimes there may be no objection and so in this article we
endeavour to analyse the behaviour of such a theory in the context of static
stellar modelling. In particular, our interest lies in determining whether
this theory admits physically viable compact star models.

The literature is replete with investigations of other configurations of
matter fields in 4D EGB. For example, rotating and non-rotating black hole
solutions and their physical properties have received attention in 4D EGB
theory Ghosh:2020syx ; Konoplya:2020juj ; Kumar:2020uyz ; Kumar:2020xvu ;
Zhang:2020sjh ; Liu:2020vkh . In addition, the strong/ weak gravitational
lensing problem by black holes was considered by Islam:2020xmy ; Kumar:2020sag
; Heydari-Fard:2020sib ; Jin:2020emq . The question of geodesic motion and the
shadow of black holes was investigated by Zeng:2020dco  while Hawking
radiation was considered by Zhang:2020qam ; Konoplya:2020cbv . Quasinormal
modes were discussed by Churilova:2020aca ; Mishra:2020gce  and wormhole and
thin-shell wormhole solutions generated by Jusufi:2020yus ; Liu:2020yhu . For
a number of other matter configurations that have been treated in the
literature see the recent works of Easson:2020mpq ; Jusufi:2020qyw ;
Yang:2020jno ; Ma:2020ufk ; Dadhich:2020ukj ; Samart:2020sxj .

Note also that there are other attempts at studying the effects of higher
curvature EGB terms in a four dimensional setting by the inclusion of a scalar
field such as a dilaton. This is known as Einstein dilaton Gauss-Bonnet (EdGB)
gravity and has been extensively considered by Boulware:1986dr ; Kanti:1995vq
; Kanti:1997br ; Torii:1996yi ; Nojiri:2005vv ; Nojiri:2006je ; Leith:2007bu ;
Bamba:2007ef ; Niz:2007gq ; Capozziello:2008gu ; Ohta:2009pe ; Maeda:2009uy ;
Guo:2009uk ; Guo:2010jr ; Chen:2009rv ; Nojiri:2010wj . The major drawback of
this proposal is that it involves a distortion of the Einstein frame
necessitating a conformal transformation of the frame Iihoshi:2011ew . This
procedure then generates extra couplings with matter and these are generally
conveniently ignored bouldes2  when models are developed. Nevertheless
numerous explorations of this proposal have been undertaken. Early studies on
EdGB gravity focussed on the low-energy effective action of the bosonic sector
of heterotic string theory Kanti:1995vq ; Pani:2009wy . Recently, BH solutions
have been found for more general coupling functions Antoniou:2017acq . A study
of neutron stars in the EdGB gravity was conducted in Ref. Pani:2011xm
including both the static and the slowly rotating cases. It is worth
mentioning that for certain fixed values of the coupling parameter no neutron
star solutions exist above some critical central energy density as discussed
in Pani:2011xm . In the context of these theories, the axial quasinormal modes
of neutron stars were examined in Blazquez-Salcedo:2015ets , and additional
neutron star solutions were constructed by Doneva:2017duq  (see Ref. Blazquez-
Salcedo:2018pxo  for further reviews). In addition traversable wormholes were
numerically derived with a coupling function of exponential type with no need
to resort to exotic matter in (3 + 1)-dimensions. In a similar vein some
authors Cuyubamba:2018jdl  have shown that the well known ‘Kanti-Kleihaus-
Kunz’ wormhole is unstable against small perturbations for any values of its
parameters.

In this article we probe the novel 4$D$ approach through rescaling of the GB
parameter. Before elucidating the proposal we consider briefly the motivation
to pursue extensions of the Einstein theory of gravity which, despite its
numerous experimental successes, leave many questions unanswered. The current
accelerated expansion of the universe has been confirmed by several
independent observations, including supernovae Ia data Perlmutter , Baryon
Acoustic Oscillations Eisenstein  and cosmic microwave background radiation
(CMB) Spergel  as studied by the WMAP. This raises many intriguing questions
amongst them is the missing mass problem which was noted for the first time by
Fritz Zwicky Zwicky . The puzzle can be addressed either introducing some form
of dark energy or assuming some kinds of modifications to the classical GR.
Several such ideas have been explored including higher derivative gravity
(HDG) theories such as $f(R)$ gravity Starobinsky . In particular, HDG was
motivated by the search for singularity free black holes solutions, avoiding
causality problems at the classical level and so on. An alternative
modification of GR is to add higher curvature effects through polynomial
invariants of the Riemann tensor, Ricci tensor and Ricci scalar without
compromising diffeomorphism invariance or the Bianchi identies. In this
regard, Lovelock gravity (LG) Lovelock ; Lovelock:1972vz  is a natural
generalization of Einstein’s general relativity. In this higher curvature
effects are only dynamic for dimensions higher than 4. Additionally Lovelock
gravity preserves conservation of energy momentum and are free from unique
ghosts when expanded on a flat space, avoiding problems with unitarity
Zumino:1985dp ; Zwiebach:1985uq . At the quadratic level the Lovelock action
is called Gauss–Bonnet (GB) action or the Lanczos action Lanczos:1938sf . This
is the simplest nontrivial case to study higher curvature effects.
Interestingly, The GB action appears in the low energy effective action of
heterotic string theory Wiltshire:1985us ; Boulware:1985wk ; Wheeler:1986 .

Since GB terms are known to affect physics in dimensions higher than four it
is interesting to study the viability of the 4D EGB version in the context of
stellar modelling. Compact stars and their properties have been an active area
of research for many decades. The existence of a mass limit for white dwarfs
by Chandrasekhar Chandrasekhar  advanced the way to understanding the nature
and structure of compact stars. In his work, it was shown that white dwarfs
are compact stars in which the pressure support against collapse comes from
the quantum degeneracy of the electrons. In the context of GR there exists a
large number of works studying the behaviour of static spherically symmetric
objects composed of a perfect fluid. A comprehensive listing of such exact
solutions for isotropic fluids may be found in Delgaty:1998uy . Exact
solutions in five dimensional EGB theory were generated in Maharaj:2015gsd ;
Chilambwe:2015rra ; Dadhich:2015rea ; Hansraj:2016gmk  and references therein.
The objective of this paper is to find new exact interior models in 4$D$ EGB
theory with a spherical distribution of perfect fluid. Some attempts in this
direction have been made by (see refs. Doneva:2020ped ; Banerjee:2020stc ;
Banerjee:2020yhu ) where authors aimed to address several shortcomings coming
out in the study of compact objects at high densities. Recently, Banerjee _et
al_ Banerjee:2020dad  studied spherically symmetric static solutions in this
scenario with interacting quark EoS. The effects of the coupling constant
$\alpha$ on the physical properties of the strange stars were analysed.

A notable body of work has emerged in the area of analysing the properties of
ultracompact objects in general relativity. Horizonless spacetimes with closed
light rings (null circular geodesics) have been proposed as possible spatially
regular exotic alternatives to canonical black-hole spacetimes. Comprehensive
treatments may be found in cunha ; cardoso ; hod2 ; hod3 ; hod4 ; hod5 . In
the work of Novotný, Hladík, and Stuchlík novotny  spherically symmetric self-
gravitating isotropic ultra-compact objects were studied numerically. It was
demonstrated in hod1  that the horizonless curved spacetimes of spatially
regular compact matter generally possess two light rings satisfying the
relationship that the mass-radius ratio of the inner light ring being bounded
above by the mass-radius ratio of the outer ring. Hod hod1  succeeded in
generating an analytical proof for the phenomenon. The calculations were
possible in the context of asymptotically flat horizonless spacetimes.In
particular self-gravitating ultra-compact trapping polytropic spheres were
first considered in stuchlik . It would be interesting to investigate the
contribution of the higher curvature Gauss-Bonnet terms with regards to the
bounds established in Einstein gravity.

Realistic stellar models are expected to satisfy some elementary physical
requirements knutsen . These conditions on 4D EGB are extrapolated from the
Einstein version which are as follows. The energy density ($\rho$) and
pressure ($P$) profiles of a compact object are positive definite. The fluid’s
pressure and energy density are monotonically decreasing outward from the
centre and the boundary defined by $P(R)=0$ should have a real valued
solution. This is used in matching of the interior metric with the exterior
vacuum solution. The energy conditions should also be satisfied. Here, we
focused on null, weak, dominant and strong energy conditions that can be
identified as $\rho-P>0$, $\rho+P>0$ and $\rho+3P>0$. Another important
feature worth mentioning, is that velocity of sound should obey
$0<\frac{dP}{d\rho}<1$ to be subluminal. Chandrasekhar also established a
lower bound for a certain quantity in order to guarantee the adiabatic
stability of the model. All of these physical constraints will be examined
once an exact model is developed.

The plan of this paper is as follows: After the introduction in Sec. I, we
briefly review the field equations in the 4$D$ EGB gravity and show that it
makes a nontrivial contribution to gravitational dynamics in 4$D$ in Sec. II.
In Sec. III we note some known solutions namely the exterior metric which will
be used in the matching procedure as well as the interior Schwarzschild
spacetime. We choose particular forms for one of the gravitational potentials
which enables us to obtain the condition of pressure isotropy in the remaining
gravitational potential in Sec. IV and analyze the physical properties of the
model such as their energy density, pressure, speed of sound, energy
conditions and adiabatic stability. In Sec. V, we give some other examples of
the gravitational potentials that yield exact solutions. Finally, in Sec. VI,
we end with a discussion of our results and a conclusion.

##  II Basic equations of EGB gravity

The Einstein-Gauss-Bonnet gravity in the $D$-dimensions spacetime can be
described by the following action

$\mathcal{I}_{G}=\frac{1}{16\pi}\int
d^{D}x\sqrt{-g}\left[R+\frac{\alpha}{D-4}\mathcal{L}_{\text{GB}}\right]+\mathcal{S}_{\text{matter}},$
(1)

where $g$ denotes the determinant of the metric $g_{\mu\nu}$ and $\alpha$ is
the Gauss-Bonnet (GB) coupling constant. The Ricci scalar $R$ provides the GR
part of the action and $\mathcal{L}_{\text{GB}}$ is the Gauss-Bonnet
Lagrangian defined as

$\mathcal{L}_{\text{GB}}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^{2}.$
(2)

The $\mathcal{S}_{\text{matter}}$ is the action of the matter fields. The
equation of motion can be obtained by varying the action with respect to
$g_{\mu\nu}$, as

$G_{\mu\nu}+\frac{\alpha}{D-4}H_{\mu\nu}=8\pi
T_{\mu\nu}~{},~{}~{}~{}\mbox{where}~{}~{}~{}~{}~{}~{}T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\mathcal{S}_{m}\right)}{\delta
g^{\mu\nu}},$ (3)

Since, $\mathcal{L}_{m}$ represents the Lagrangian density of matter that
depends only on the metric tensor components $g_{\mu\nu}$, and not on its
derivatives. $G_{\mu\nu}$ is the Einstein tensor with $H_{\mu\nu}$ is the
contribution of the GB term with the following expression

$\displaystyle G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}R~{}g_{\mu\nu},$
$\displaystyle
H_{\mu\nu}=2\Bigr{(}RR_{\mu\nu}-2R_{\mu\sigma}{R}{{}^{\sigma}}_{\nu}-2R_{\mu\sigma\nu\rho}{R}^{\sigma\rho}-R_{\mu\sigma\rho\delta}{R}^{\sigma\rho\delta}{{}_{\nu}}\Bigl{)}-\frac{1}{2}~{}g_{\mu\nu}~{}\mathcal{L}_{\text{GB}},$
(4)

In the above expression, $R$ is the Ricci scalar, $R_{\mu\nu}$ the Ricci
tensor, $H_{\mu\nu}$ the Lanczos tensor and $R_{\mu\sigma\nu\rho}$ is the
Riemann tensor, respectively. In general, $H_{\mu\nu}$ vanishes in 4$D$ space-
time, and hence does not contribute to the field equations. However, by taking
the limit $D\to 4$ in (3) and by rescaling the Gauss-Bonnet coupling constant
by $\alpha/(D-4)$, one can prevent the vanishing of $H_{\mu\nu}$ in 4$D$
space-time Glavan:2019inb .

Here, we consider the energy-momentum tensor $T_{\mu\nu}$ as a perfect fluid
source, that is $T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$, where
$\rho=\rho(r)$ is the energy density of matter, $p=p(r)$ is the pressure, and
$u_{\nu}$ is the fluid’s $D$-velocity. Now, we start by writing down the
static, spherically symmetric $D$-dimensional metric ansatz Mehdizadeh:2015jra
in the usual form:

$\displaystyle
ds^{2}_{D}=-e^{2\Phi(r)}c^{2}dt^{2}+e^{2\lambda(r)}dr^{2}+r^{2}d\Omega_{D-2}^{2},$
(5)

where $d\Omega_{D-2}^{2}$ is the metric on the unit $(D-2)$-dimensional
sphere. As the metric is time independent and spherically symmetric, the
$\Phi(r)$ and $\lambda(r)$ are functions of the radial coordinate $r$ only.
Now, we are going to obtain the Einstein field equations in the limit $D\to 4$
for 4$D$ EGB gravity, the $tt$, $rr$ and $\theta\theta$ components are given
by the following differential equations:

$\displaystyle\frac{\alpha(1-e^{-2\lambda})}{r^{3}}\left[4\lambda^{\prime}e^{-2\lambda}-\frac{(1-e^{-2\lambda})}{r}\right]+e^{-2\lambda}\left({2\lambda^{\prime}\over
r}-{1\over r^{2}}\right)+{1\over r^{2}}=8\pi\rho(r),$ (8)
$\displaystyle\frac{\alpha(1-e^{-2\lambda})}{r^{3}}\left[4\Phi^{\prime}e^{-2\lambda}+\frac{(1-e^{-2\lambda})}{r}\right]+e^{-2\lambda}\left({2\Phi^{\prime}\over
r}+{1\over r^{2}}\right)-{1\over r^{2}}=8\pi p(r),$ $\displaystyle
e^{-2\lambda}\left[\Phi^{\prime\prime}+\Phi^{\prime
2}+\frac{1}{r}\left(\Phi^{\prime}-\lambda^{\prime}\right)+\Phi^{\prime}\lambda^{\prime}\left(\frac{8\alpha
e^{-2\lambda}}{r^{2}}-1\right)-\frac{2\alpha(1-e^{-2\lambda})}{r^{2}}\right.$
$\displaystyle\left.\times\Bigg{\\{}\frac{1}{r}\left(\Phi^{\prime}-\lambda^{\prime}\right)-2\left(\Phi^{\prime\prime}+\Phi^{\prime
2}-\Phi^{\prime}\lambda^{\prime}\right)+\frac{1}{r^{2}}(e^{2\lambda}-1)\Bigg{\\}}\right]=8\pi
p(r).$

It must be emphasized that we use the regularization process (see Ref.
Glavan:2019inb ; Cognola:2013fva ), that leads to exactly the same black hole
solutions Lu:2020iav ; Hennigar:2020lsl ; Casalino:2020kbt ; Ma:2020ufk  at
least for the case of 4$D$ spherically symmetric spacetimes. Now, we can deal
with the stellar structure equations under the standard of dynamical systems.
Now, combining Eq. (8) and (8), one can get the equation of pressure isotropy,
which yield

$\displaystyle
2\alpha\left(2e^{-2\lambda}r^{2}\Phi^{\prime}\lambda^{\prime}+(1-e^{2\lambda})+(1-e^{-2\lambda})(r^{2}W+r(\lambda^{\prime}-3\Phi^{\prime})+1)\right)$
$\displaystyle+r^{2}\left(r^{2}W-r(\phi^{\prime}+\lambda^{\prime})+(e^{2\lambda}-1)\right)=0,$
(9)

where we have defined
$W=\Phi^{\prime\prime}+\Phi^{2}-\Phi^{\prime}\lambda^{\prime}$. Of course, the
conservation equation of the energy-momentum tensor, $T^{\mu\nu}_{;\mu}=0$
that for a perfect fluid is

$\displaystyle\frac{dp}{dr}=-(\rho+p)\frac{d\Phi}{dr},$ (10)

At this stage we introduce a coordinate transformation

$\displaystyle
x=Cr^{2},~{}~{}e^{-2\lambda}=Z(x)~{}~{}~{}\mathrm{and}~{}~{}~{}e^{2\Phi}=y^{2}(x),$
(11)

where $C$ is an arbitrary constant. The benefit of this transformation is that
the master nonlinear isotropy field equation is transformed to a linear
differential equation and we can profit from the vast knowledge on such
equations. With this choice of (11), we are able to express the components of
the field Eqs. (8), (8) and (9) can be written as

$\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle\frac{C}{2x^{2}}\bigg{[}4x(\beta(Z-1)-x)\dot{Z}-(Z-1)(2x+\beta(Z-1))\bigg{]},$
(12) $\displaystyle 8\pi p$ $\displaystyle=$
$\displaystyle\frac{C}{2x^{2}y}\bigg{[}8xZ\dot{y}(x-\beta(Z-1))+y(Z-1)(2x+\beta(Z-1))\bigg{]},$
(13) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\beta\left(4x^{2}Z(1-Z)\ddot{y}+2x\left[2Z(Z-1)+x\dot{Z}(1-3Z)\right]\dot{y}-(1-Z)(\dot{Z}x-Z+1)y\right)+$
(14) $\displaystyle x(4x^{2}Z\ddot{y}+2x^{2}\dot{Z}\dot{y}+(\dot{Z}x-Z+1)y),$

where dots denote differentiation with respect to the variable $x$, which is
quasilinear. Here we define $\beta=2\alpha C$. It is a second order linear
differential equation in the variable $y$ and first order Abel equation of the
second kind when written in terms of $Z$. The field equation bears a striking
resemblance to the structure of the 6 dimensional Einstein–Gauss–Bonnet
equation of pressure isotropy but not the five dimensional version. When the
Gauss-Bonnet coupling $\beta$ vanishes the four dimensional equation of
pressure isotropy in Einstein gravity

$\displaystyle
4x^{2}Z\ddot{y}+2x^{2}\dot{Z}\dot{y}+\left(x\dot{Z}-Z+1\right)y=0,$ (15)

is regained. It is also interesting to draw a comparison with the standard EGB
equations. For five dimensional EGB gravity using the metric (5) the pressure
isotropy equation has the form

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\beta\left(2xZ(1-Z)\ddot{y}+(2Z(Z-1)+x\dot{Z}(1-3Z))\dot{y}\right)+\left(2x^{2}Z\ddot{y}+x^{2}\dot{Z}\dot{y}+(\dot{Z}x-Z+1)y\right),$
(16)

for an unscaled coupling parameter $\alpha$ while in six dimensional EGB
gravity the master isotropy equation is given by

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
6\beta\left(4x^{2}Z(1-Z)\ddot{y}+2x(2Z(Z-1)+\dot{Z}x(1-3Z))\dot{y}+(1-Z)(\dot{Z}x-Z+1)y\right)$
(17)
$\displaystyle+x\left(4x^{2}Z\ddot{y}+2x^{2}\dot{Z}\dot{y}+3(\dot{Z}x-Z+1)y\right),$

after inserting a suitable extra angular dependence in (5).

##  III Known solutions

In order to establish the applicable exterior metric we may set $\rho=0$ in
(12). The solution of the differential equation is then given by

$\displaystyle Z=1+\frac{x}{\beta}\left(1\pm\sqrt{1+\frac{\beta
C_{1}}{x^{\frac{3}{2}}}}\right),$ (18)

where $C_{1}$ is an integration constant. This corroborates the result of
Doneva and Yazadjiev Doneva:2020ped , (see Ghosh:2020syx  for black hole
solution). Note that in the limit of $\beta$ approaching zero, the
Schwarzschild exterior solution is regained from (18) but only for the
negative branch of the solution. The limit otherwise is indeterminate. This
can be seen by a series expansion of the square root term in powers of
$\beta$. Following Boulware and Deser Boulware:1986dr  we may write (18) in
the form

$e^{-2\lambda}=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8M\alpha}{r^{3}}}\right),$
(19)

which will be useful during the matching of the interior and exterior
spacetime in order to settle the values of the constant $C$ and any other
integration constant. Note that the coefficient
$\frac{C_{1}}{C^{{\frac{3}{2}}}}$ is identified with the mass $M$ of the star.
Observe also that the solution (18) may also be obtained by setting
$y=\sqrt{Z}$ in isotropy equation (14) since it is well known that vacuum
solutions have the form

$ds^{2}=-F(r)dt^{2}+\frac{1}{F(r)}dr^{2}+r^{2}d\Omega^{2},$ (20)

for some function $F(r)$ and this serves as a useful check on the veracity of
the field equations.

Another known solution is that of the constant density or incompressible fluid
sphere. This was studied in Doneva:2020ped , however the solution obtained is
precisely the form of the Schwarschild interior metric of classical GR. This
supports the conclusion of Dadhich et al who proved that the Schwarzschild
interior solution is universal in EGB in all dimensions and evidently this is
true for the 4$D$ scenario. We can easily verify this by putting $Z=1+x$ into
field equation (14) and obtaining the potential

$y=A+B\sqrt{1+x},$ (21)

where $A$ and $B$ are constants of integration. Essentially for the constant
density value $\rho=\rho_{0}$ the spatial potential may be written as

$e^{-2\lambda}=1+\frac{r^{2}}{2\alpha}\left(1-\sqrt{1+\frac{8\rho_{0}\alpha}{3}}\right),$
(22)

as reported in Doneva:2020ped ; Banerjee:2020stc  and which is clearly of
Schwarzschild form $Z=e^{-2\lambda}=1+Cr^{2}$.

##  IV A physically reasonable exact model

Generating exact solutions to the equations of motion comprising the Einstein
equations supplemented by the Gauss–Bonnet contributions is a nontrivial
matter. Historically, the Einstein equations in the case of isotropic perfect
fluids, have admitted some 120 exact solutions most of which were found
through mathematical motivations. This is because the system of field
equations is underdetermined and requires a further stipulation to close the
system. Implementing physically relevant conditions such as specifying an
equation of state such as a linear barotropic or polytropic one has drawn a
blank as respects exact solutions. Each of these cases were solved numerically
by Nillson and Uggla nilsson1 ; nilsson2 . In general, the freedom of choice
has often been exercised with the selection of a suitable potential ansatz and
by integration determining the remaining potential and all remaining physical
quantities. Post-facto it is then checked to see whether an equation of state
$p=p(\rho)$ could be explicitly determined. A notable exception to this
approach was that of Tolman tolman . By a cunning rearrangement of the master
isotropy equation, he was able to determine eight classes of solutions by
considering the vanishing of certain parts of the equation. Five of these
turned out to be new and some such as Tolman IV and Tolman VII have been shown
to be physically viable. Implementing the same scheme in our case does not
yield palatable results. In fact inserting the Tolman IV and VII prescriptions
into the isotropy equation (23) leads to an intractable differential equation.
The approach we follow has been successful in finding exact solutions to the
five dimensional EGB system and constitute the only ones detectable to the
present time.

The equation (14) may be converted to the form

$\displaystyle\beta\left(x(2x(1-3Z)\dot{y}+y(Z-1))\dot{Z}-(4x^{2}\ddot{y}-4x^{2}\dot{y}+y)Z^{2}+(4x^{2}\ddot{y}-4x\dot{y}+2y)Z-y\right)$
$\displaystyle+x\left(x(2x\dot{y}+y)\dot{Z}+(4x^{2}\ddot{y}-y)Z+y\right)$
$\displaystyle=$ $\displaystyle 0,$ (23)

by rearrangement. Viewed as a differential equation in $Z(x)$ equation (23) is
an Abel equation of the second kind with very few classes of known solutions
expressible as elementary functions. In order to determine an exact solution
we prescribe the temporal potential to have the form $y=\sqrt{x}$. This choice
is made on mathematical grounds and corresponds to a case of the Tolman VI
ansatz. This approach has been followed repeatedly historically in finding
exact solutions of the much simpler Einstein equations as opposed to the
method of imposing a physical prescription such as an equation of state. This
latter approach generally ends in resorting to the use of numerical techniques
with their accompanying approximations. Our objective is to detect an exact
solution and thereafter to deduce the physical characteristics of the model.
The one drawback of this choice of metric potential is that at the stellar
centre $x=0$ it is not defined. This suggests that if all other physical
requirements are met, then our model may model a shell of perfect fluid
surrounding a core with well behaved physical properties at the centre. This
phenomenon is bound to show up in what comes next in the form of singularities
in the physical quantities which are not removable.

Inserting $y=\sqrt{x}$ in (23) the real valued function

$\displaystyle
Z=\frac{1}{2}\left(\frac{\sqrt[3]{-2k{}^{6}x^{3}+2\sqrt{k{}^{15}x^{3}+k{}^{12}x^{6}}-k{}^{9}}}{k{}^{3}}+\frac{k{}^{3}}{\sqrt[3]{-2k{}^{6}x^{3}+2\sqrt{k{}^{15}x^{3}+k{}^{12}x^{6}}-k{}^{9}}}+2x+1\right),$
(24)

where $k$ is a constant of integration and two complex valued potentials for
$\beta=1$ emerge.The complex roots are discarded as they have no physical
meaning in the context of stellar structure. At this stage of computational
capability the general solution for all $\beta$ is not available, but it is
pleasing that we have a nontrivial value of the coupling constant present so
that we can study the influence of the higher curvature terms in the
gravitational behaviour of the four dimensional spherically symmetric
spacetime.

The density $\rho$ and isotropic particle pressure $p$ evaluate to

$\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle\frac{C}{2x^{2}}\left(x\left(k^{3}x^{2}\left(k^{6}-v_{2}^{2/3}\right)+2v_{1}\sqrt[3]{v_{2}}\right)\left(\beta
k^{6}+k^{3}\sqrt[3]{v_{2}}\left(2v_{3}x-\beta\right)+\beta
v_{2}^{2/3}\right)\right./$ (25)
$\displaystyle\left.k^{3}v_{1}v_{2}^{2/3}-\frac{1}{2}\left(\frac{\sqrt[3]{v_{2}}}{k^{3}}+\frac{k^{3}}{\sqrt[3]{v_{2}}}+2x-1\right)\left(\frac{1}{2}\beta\left(\frac{\sqrt[3]{v_{2}}}{k^{3}}+\frac{k^{3}}{\sqrt[3]{v_{2}}}-1\right)+(\beta+2)x\right)\right),$
$\displaystyle 8\pi p$ $\displaystyle=$
$\displaystyle\frac{C}{2x^{2}}\left(\frac{1}{4}\beta\left(-\frac{3v_{2}^{2/3}}{k^{6}}-\frac{3k^{6}}{v_{2}^{2/3}}-\frac{2\sqrt[3]{v_{2}}}{k^{3}}-\frac{2k^{3}}{\sqrt[3]{v_{2}}}-1\right)-\frac{v_{3}x\left(3k^{6}+k^{3}\sqrt[3]{v_{2}}+3v_{2}^{2/3}\right)}{k^{3}\sqrt[3]{v_{2}}}-3(\beta-2)x^{2}\right),$
(26)

where we have applied the substitutions
$v_{1}=\sqrt{k^{12}x^{3}\left(k^{3}+x^{3}\right)}$ ,
$v_{2}=-k^{9}-2k^{6}x^{3}+2v_{1}$ and $v_{3}=\beta-1$.In light of the fact
that both density and pressure have very complicated forms, the determining of
a barotropic equation of state $p=p(\rho)$ becomes intractable. Consequently
we will rely upon the quantity $\frac{p}{\rho}$ to convey an idea of the
variation of the equation of state. The square of the sound speed has the
complicated form

$\displaystyle\frac{dp}{d\rho}$ $\displaystyle=$
$\displaystyle-\left(\left(k^{3}+x^{3}\right)\left(2k^{15}x^{3}\left(9v_{3}x-\beta\right)+2v_{1}v_{2}^{2/3}x^{3}\left(\beta+2v_{3}x\right)+2k^{12}x^{3}\left(2\beta\sqrt[3]{v_{2}}+12v_{3}x^{4}-9v_{3}\sqrt[3]{v_{2}}x+2\beta
x^{3}\right)\right.\right.$ $\displaystyle+k^{3}v_{1}\left(\beta
v_{2}^{2/3}+24v_{3}\sqrt[3]{v_{2}}x^{4}-6\beta\sqrt[3]{v_{2}}x^{3}+2v_{3}v_{2}^{2/3}x-8\beta
x^{6}\right)$ $\displaystyle+k^{9}\left(\beta
v_{1}-24v_{3}\sqrt[3]{v_{2}}x^{7}+6\beta\sqrt[3]{v_{2}}x^{6}-4v_{3}v_{2}^{2/3}x^{4}-2\beta
v_{2}^{2/3}x^{3}-6v_{1}v_{3}x+8\beta x^{9}\right)$
$\displaystyle\left.\left.-k^{6}\left(\beta
v_{1}\sqrt[3]{v_{2}}+4v_{3}v_{2}^{2/3}x^{7}+2\beta
v_{2}^{2/3}x^{6}+24v_{1}v_{3}x^{4}-6v_{1}v_{3}\sqrt[3]{v_{2}}x\right)\right)\right)/$
$\displaystyle\left(2k^{18}x^{3}\left(5\beta+2v_{3}x\right)-2k^{15}x^{3}\left(4\beta\sqrt[3]{v_{2}}+7v_{3}x^{4}+4v_{3}\sqrt[3]{v_{2}}x-20\beta
x^{3}\right)-2v_{1}v_{2}^{2/3}x^{6}\left(3\beta+2v_{3}x\right)\right.$
$\displaystyle+k^{3}v_{1}x^{3}\left(-9\beta
v_{2}^{2/3}-8v_{3}\sqrt[3]{v_{2}}x^{4}+30\beta\sqrt[3]{v_{2}}x^{3}-6v_{3}v_{2}^{2/3}x+8\beta
x^{6}\right)$ $\displaystyle+k^{12}\left(-3\beta
v_{1}-24v_{3}x^{10}-6v_{3}\sqrt[3]{v_{2}}x^{7}-32\beta\sqrt[3]{v_{2}}x^{6}+4v_{3}v_{2}^{2/3}x^{4}+6\beta
v_{2}^{2/3}x^{3}-2v_{1}v_{3}x+28\beta x^{9}\right)$
$\displaystyle+k^{9}\left(3\beta
v_{1}\sqrt[3]{v_{2}}+8v_{3}\sqrt[3]{v_{2}}x^{10}-30\beta\sqrt[3]{v_{2}}x^{9}+8v_{3}v_{2}^{2/3}x^{7}+12\beta
v_{2}^{2/3}x^{6}+2v_{1}v_{3}x^{4}-23\beta
v_{1}x^{3}+2v_{1}v_{3}\sqrt[3]{v_{2}}x-8\beta x^{12}\right)$
$\displaystyle\left.+k^{6}\left(-3\beta
v_{1}v_{2}^{2/3}+4v_{3}v_{2}^{2/3}x^{10}+6\beta
v_{2}^{2/3}x^{9}+24v_{1}v_{3}x^{7}-32\beta
v_{1}x^{6}+10v_{1}v_{3}\sqrt[3]{v_{2}}x^{4}+17\beta
v_{1}\sqrt[3]{v_{2}}x^{3}-2v_{1}v_{3}v_{2}^{2/3}x\right)\right),$

while the expressions

$\displaystyle\rho-p$ $\displaystyle=$ $\displaystyle\frac{C}{32\pi
v_{1}v_{2}^{2/3}x^{2}}\left(4k^{12}x^{3}\left(\beta+5v_{3}x\right)+2k^{9}x^{3}\left(\beta\sqrt[3]{v_{2}}+12v_{3}x^{4}+2v_{3}\sqrt[3]{v_{2}}x\right)+k^{6}\left(-\beta
v_{1}+6\beta\sqrt[3]{v_{2}}x^{6}-8v_{1}v_{3}x\right)\right.$ (28)
$\displaystyle\left.+k^{3}v_{1}\left(\beta\sqrt[3]{v_{2}}-24v_{3}x^{4}+8v_{3}\sqrt[3]{v_{2}}x\right)-v_{1}\sqrt[3]{v_{2}}\left(\beta\sqrt[3]{v_{2}}-12(\beta-2)\sqrt[3]{v_{2}}x^{2}+6\beta
x^{3}\right)\right),$ $\displaystyle\rho+p$ $\displaystyle=$
$\displaystyle\frac{C}{16\pi
v_{1}^{5/3}x^{2}}\left(-2k^{12}v_{1}x^{4}+k^{6}v_{1}\left(2v_{1}x-\beta\right)-v_{1}^{2/3}v_{1}\left(\beta+2v_{1}x\right)\right.$
(29)
$\displaystyle\left.\left.k^{3}v_{1}\left(\beta\sqrt[3]{v_{1}}-2\sqrt[3]{v_{1}}v_{1}x+2\beta
x^{3}\right)-2k^{9}x^{3}\left(\beta\sqrt[3]{v_{1}}-\sqrt[3]{v_{1}}v_{1}x+\beta
x^{3}\right)\right)\right),$ $\displaystyle\rho+3p$ $\displaystyle=$
$\displaystyle\frac{C}{32\pi
v_{1}^{5/3}x^{2}}\left(v_{1}\sqrt[3]{v_{1}}\left(-3\beta\sqrt[3]{v_{1}}-12(\beta-2)\sqrt[3]{v_{1}}x^{2}-8v_{1}\sqrt[3]{v_{1}}x+6\beta
x^{3}\right)-4k^{12}x^{3}\left(\beta+7v_{1}x\right)\right.$ (30)
$\displaystyle-2k^{9}x^{3}\left(5\beta\sqrt[3]{v_{1}}+12v_{1}x^{4}-2v_{1}\sqrt[3]{v_{1}}x+4\beta
x^{3}\right)+k^{6}\left(-3\beta
v_{1}-6\beta\sqrt[3]{v_{1}}x^{6}+16v_{1}v_{1}x\right)$
$\displaystyle\left.\left.+k^{3}v_{1}\left(3\beta\sqrt[3]{v_{1}}+24v_{1}x^{4}-16v_{1}\sqrt[3]{v_{1}}x+8\beta
x^{3}\right)\right)\right),$

will be useful in studying the energy conditions. Finally we exhibit the
Chandrasekhar adiabatic stability index

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle-\left(\left(4v_{1}\left(2k^{12}v_{3}x^{4}+2k^{9}x^{3}\left(\beta\sqrt[3]{v_{2}}-v_{3}\sqrt[3]{v_{2}}x+\beta
x^{3}\right)-k^{6}v_{1}\left(2v_{3}x-\beta\right)+v_{1}v_{2}^{2/3}\left(\beta+2v_{3}x\right)\right.\right.\right.$
(31)
$\displaystyle\left.-k^{3}v_{1}\left(\beta\sqrt[3]{v_{2}}-2v_{3}\sqrt[3]{v_{2}}x+2\beta
x^{3}\right)\right)\left(2k^{15}x^{3}\left(9v_{3}x-\beta\right)+2v_{1}v_{2}^{2/3}x^{3}\left(\beta+2v_{3}x\right)\right.$
$\displaystyle+2k^{12}x^{3}\left(2\beta\sqrt[3]{v_{2}}+12v_{3}x^{4}-9v_{3}\sqrt[3]{v_{2}}x+2\beta
x^{3}\right)$ $\displaystyle-k^{3}v_{1}\left(-\beta
v_{2}^{2/3}-24v_{3}\sqrt[3]{v_{2}}x^{4}+6\beta\sqrt[3]{v_{2}}x^{3}-2v_{3}v_{2}^{2/3}x+8\beta
x^{6}\right)$ $\displaystyle-k^{9}\left(-\beta
v_{1}+24v_{3}\sqrt[3]{v_{2}}x^{7}-6\beta\sqrt[3]{v_{2}}x^{6}+4v_{3}v_{2}^{2/3}x^{4}+2\beta
v_{2}^{2/3}x^{3}+6v_{1}v_{3}x-8\beta x^{9}\right)$
$\displaystyle\left.\left.-k^{6}\left(\beta
v_{1}\sqrt[3]{v_{2}}+4v_{3}v_{2}^{2/3}x^{7}+2\beta
v_{2}^{2/3}x^{6}+24v_{1}v_{3}x^{4}-6v_{1}v_{3}\sqrt[3]{v_{2}}x\right)\right)\right)/$
$\displaystyle\left(k^{6}x^{3}\left(6\beta
v_{1}\sqrt[3]{v_{2}}+k^{12}\left(\beta-12v_{3}x\right)+4k^{3}v_{1}\left(\beta+6v_{3}x\right)+k^{9}\left(-\beta\sqrt[3]{v_{2}}-24v_{3}x^{4}+12v_{3}\sqrt[3]{v_{2}}x-4\beta
x^{3}\right)\right.\right.$ $\displaystyle\left.+k^{6}\left(\beta
v_{2}^{2/3}-6\beta\sqrt[3]{v_{2}}x^{3}+12(\beta-2)v_{2}^{2/3}x^{2}+4v_{3}v_{2}^{2/3}x\right)\right)$
$\displaystyle\times\left(2k^{18}x^{3}\left(5\beta+2v_{3}x\right)-2k^{15}x^{3}\left(4\beta\sqrt[3]{v_{2}}+7v_{3}x^{4}+4v_{3}\sqrt[3]{v_{2}}x-20\beta
x^{3}\right)-2v_{1}v_{2}^{2/3}x^{6}\left(3\beta+2v_{3}x\right)\right.$
$\displaystyle+k^{3}v_{1}x^{3}\left(-9\beta
v_{2}^{2/3}-8v_{3}\sqrt[3]{v_{2}}x^{4}+30\beta\sqrt[3]{v_{2}}x^{3}-6v_{3}v_{2}^{2/3}x+8\beta
x^{6}\right)$ $\displaystyle-k^{12}\left(3\beta
v_{1}+24v_{3}x^{10}+6v_{3}\sqrt[3]{v_{2}}x^{7}+32\beta\sqrt[3]{v_{2}}x^{6}-4v_{3}v_{2}^{2/3}x^{4}-6\beta
v_{2}^{2/3}x^{3}+2v_{1}v_{3}x-28\beta x^{9}\right)$
$\displaystyle+k^{9}\left(3\beta
v_{1}\sqrt[3]{v_{2}}+8v_{3}\sqrt[3]{v_{2}}x^{10}-30\beta\sqrt[3]{v_{2}}x^{9}+8v_{3}v_{2}^{2/3}x^{7}+12\beta
v_{2}^{2/3}x^{6}\right)$ $\displaystyle+k^{9}\left(2v_{1}v_{3}x^{4}-23\beta
v_{1}x^{3}+2v_{1}v_{3}\sqrt[3]{v_{2}}x-8\beta x^{12}\right)$
$\displaystyle+k^{6}\left(4v_{3}v_{2}^{2/3}x^{10}+24v_{1}v_{3}x^{7}+10v_{1}v_{3}\sqrt[3]{v_{2}}x^{4}-2v_{1}v_{3}v_{2}^{2/3}x\right)$
$\displaystyle\left.\left.\left.+k^{6}\left(-3\beta v_{1}v_{2}^{2/3}+6\beta
v_{2}^{2/3}x^{9}-32\beta v_{1}x^{6}+17\beta
v_{1}\sqrt[3]{v_{2}}x^{3}\right)\right)\right)\right),$

is established with the help of the formula
$\Gamma=\left(\frac{\rho+p}{p}\right)\frac{dp}{d\rho}$. These lengthy and
complicated expressions are common in the study of higher curvature effects
induced by the Gauss–Bonnet contribution. Clearly an analytic investigation of
these physical quantities lies outside the realm of possibility and so we
endeavour to analyse the effect of the higher curvature terms through
graphical means.

###  IV.1 Qualitative physical analysis

We now consider the dynamical quantities graphically using the parameter
values $k=-1$, $\beta=-1$ and $C=1$. These parameter values have been so
selected through a careful and tedious process of empirical fine tuning until
physically reasonable behaviour of the dynamical variables emerged. Observe
there is no reason to demand a positive coupling constant $\beta=2\alpha C$.
It is known that $\alpha$ is connected to the string tension in string theory
and so is expected to be positive there, however, it has a different meaning
as a coupling parameter in the present gravitational theory. Additionally, for
the same parameter values, we have plotted the Einstein only ($\beta=0$) on
the same system of axes to facilitate a comparison and to examine the
influence of the GB terms.

  * •

Density and pressure profiles

At the very least it is demanded that both density and pressure be positive
definite functions that are monotonically decreasing outwards from the stellar
centre. This is clearly in evidence in both figures 1 and 2. Moreover from
figure 2, it is clear that the pressure vanishes for some finite radius at
roughly $x=0.085$ in the GB case and at a higher value in the Einstein case.
This suggests that the higher curvature terms have the effect of decreasing
the radius of the sphere. The density plot in figure 1 also indicates that the
Einstein counterpart is not physically viable being negative everywhere. The
additional GB terms appear to correct this undesirable behaviour.

Figure 1: Variation of density $\rho$ against the radial parameter $x$.

Figure 2: Variation of pressure $p$ against the radial parameter $x$.

  * •

Energy conditions

In order to investigate further the structure and property of our model, we
focus on the energy conditions (ECs). It is reasonably expected that every
type of classical matter satisfies the energy conditions whether in modified
gravity or standard general relativity. There are several energy conditions
that are sets of inequalities depending on the energy-momentum tensor of
matter. The most fundamental of these is the weak energy condition (WEC), i.e.
$T_{\mu\nu}U^{\mu}U^{\nu}$ $\geq 0$, where $U^{\mu}$ is a timelike vector. For
the given diagonal energy momentum tensor, the WEC requires that,

$\rho(r)\geq 0~{}~{}\text{and}~{}~{}\rho(r)+p(r)\geq 0.$ (32)

Indeed, continuity of the WEC implies the null energy condition which requires
for any null vector $k^{\mu}$ that $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$. The
strong energy condition (SEC) is given by,
$\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)U^{\mu}U^{\nu}\geq 0$ for any
timelike vector $U^{\mu}$, where $T$ is the trace of the stress energy tensor.
In terms of the diagonal stress energy tensor the SEC reads

$\rho(r)+\sum p_{i}(r)\geq 0~{}~{}\implies~{}~{}\rho(r)+3p(r)\geq 0.$ (33)

Finally the dominant energy condition (DEC) demands that for any timelike
vector $U^{\mu}$ that $T_{\mu\nu}U^{\mu}U^{\nu}$ $\geq 0$ and
$T_{\mu\nu}U^{\nu}$ is not spacelike. Accordingly the constraints

$\rho(r)\geq 0~{}~{}\text{and}~{}~{}\rho(r)-|p(r)|\geq 0.$ (34)

arise. The results obtained from our requirements (32 – 34) are illustrated in
Fig. 3, for $\beta=-1$ and $\beta=0$. These plots imply that all energy
conditions are satisfied in the 4D EGB case however the DEC is violated in the
Einstein model when $\beta=0$. The null energy condition is verified in Fig 1
for both standard and modified models. 1.

Figure 3: Variation of energy conditions versus the radial variable $x$.

  * •

Sound speed and causality

In order for the causality to be preserved, it is natural to require that the
sound speed does not exceed the speed of light, i.e. $0\leq
v^{2}_{s}=dp/d\rho<1$ (in geometrical units). Here, we investigate the speed
of sound propagation using the expression (LABEL:sp). From figure 3 it is
observed that the sound speed is always subluminal ($0<\frac{dp}{d\rho}<1$)
for the GB ($\beta=1$) case within the radial value $x=0.08$ but fails totally
for the Einstein case ($\beta=0$). We may infer that the Gauss–Bonnet higher
curvature terms were responsible for the correction in behaviour of the
Einstein model.

Figure 4: Square of sound speed $\frac{dp}{d\rho}$ versus the radial variable
$x$.

  * •

Chandrasekhar adiabatic stability criterion

The dynamical stability of a compact is computed through the adiabatic index
($\gamma$) which is an important thermodynamical quantity. It has been
discussed by Chandrasekhar Chandrasekhar  who considered the instability
problem based on the variational method. The usual representations of the
high-density EoS of perfect fluid are based on parameterizing the adiabatic
index $\gamma$ defined by

$\gamma\equiv\left(1+\frac{\rho}{p}\right)\left(\frac{dp}{d\rho}\right)_{S},$
(35)

where the derivation is performed at constant entropy $S$ and $dp/d\rho$ is
the speed of sound in units of the speed of light. Thus, the sound speed is an
important quantity closely related to the adiabatic index. For instance, for
the Schwarzschild star with constant-density, the $\gamma$ for the fluid is
infinite (incompressible fluid). However, in Ref. Glass  the authors have
shown that the adiabatic index should exceed $4/3$ for a stable polytropic
star by an amount that depends on that ratio $\rho/p$ at the centre of the
star. But from recent observations it is suggested that the range of $\gamma$
should be 2 to 4 in most neutron stars Haensel . In Fig. 5 we plot $\gamma$ as
a function of radial distance. These results are once again consistent with
expectations i.e. $\gamma>4/3$.

Figure 5: Chandrasekhar adiabatic stability index versus the radial variable
$x$.

  * •

Matching

The line element for the star at the boundary $(r=R)$ has the form

$\displaystyle{ds}^{2}$ $\displaystyle=$
$\displaystyle-\left(CR^{2}\right){dt}^{2}+\frac{2}{\left(\frac{1}{\eta}+\eta+2{CR}^{2}+1\right)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (36)

where we have set
$\eta=\frac{k^{3}}{\sqrt[3]{-2\left({CR}^{2}\right)^{3}k^{6}+2\sqrt{\left({CR}^{2}\right)^{3}k^{15}+\left({CR}^{2}\right)^{6}k^{12}}-k^{9}}}$.
The metric (36) should be matched with the exterior line element (19).
Accordingly the arbitrary constant $C$ may be expressed as

$\displaystyle C=\frac{1}{R^{2}}\left(1-\frac{2M}{R}\right)$ (37)

in terms of the radius $R$ and mass $M$ of the sphere. From the matching of
the $g_{11}$ component the integration constant $k$ may be established as

$\displaystyle k$ $\displaystyle=$ $\displaystyle
CR^{2}\left(\zeta_{2}+\zeta_{3}\right)\bigg{/}\sqrt[3]{2}\zeta_{1}\left(-\left(1+\frac{1}{8\zeta_{1}^{3}}\left(\zeta_{2}+\zeta^{2}_{3}\right)^{3}\right)^{2}\right)^{\frac{1}{3}},$
(38)

where we have set $\zeta_{1}=\left(\alpha-2MR+R^{2}\right),$
$\zeta_{2}=\alpha+4CMR^{3}-2CR^{4}-2\alpha CR^{2}+R^{2}\sqrt{\frac{8\alpha
M}{R^{3}}+1}+2MR,$

$\zeta_{3}^{2}=\left(\alpha+2M\left(2CR^{3}+R\right)-2CR^{2}\left(\alpha+R^{2}\right)+R^{2}\sqrt{\frac{8\alpha
M}{R^{3}}+1}\right)^{2}-4\left(\alpha-2MR+R^{2}\right)^{2}$.

Thus, we conclude the matching of the interior and exterior spacetime across
the boundary interface $r=R$. Note we have reverted to the canonical
coordinates in the above calculations.

###  IV.2 Hydrostatic equilibrium

Given the significance of the above results in the astrophysical context, it
would be interesting if some comment regarding stability of the solution can
be made. For this purpose we discuss the equilibrium stage that can be
achieved by formulating the modified TOV equation (10). Along these lines, a
wide variety of astrophysical solutions (including wormholes/compact stars)
have been studied (for a review, see, e.g., Refs. Rahaman:2013xoa ;
Rahaman:2014dpa ; Jawad:2015uea ; Maurya:2018kxg ). For compact star, the
equilibrium picture can be discussed via,

$\displaystyle-\frac{dp}{dr}-\Phi^{\prime}(\rho+p)=0,~{}~{}~{}\text{or}~{}~{}~{}F_{h}+F_{g}=0,$
(39)

where the first term represents the hydrodynamic force ($F_{h}$) and the
second term is gravitational force ($F_{g}$), respectively. The Eq. (39)
predicts that the system is in equilibrium if the sum of different forces must
be zero. As is evident in the plot of Fig. 9, that the equilibrium of the
forces is achieved for our chosen parametric values and confirms stability of
the system.

Figure 6: Variation of the hydrodynamic and gravitational forces with respect
to the radial coordinate $x$.

###  IV.3 Stability under radial pulsations

Models of perfect fluids are expected to be stable under radial pulsations.
Historically, the first comprehensive treatment of stability was by
Chandrasekhar chandra  who developed a test for stability under perturbations
of the pressure and energy density for a linearised system of Einstein’s
equations. We emphasise that his scheme is only applicable to the Einstein’s
equations and not directly useful in our work with the injection of more
complex higher curvature terms. In fact, it would be an interesting programme
to emulate his work in this context but this is bound to be very complicated.
The analysis boiled down to examining a boundary value problem of Sturm-
Liouville type with the use of some trial functions moustakidis . Bardeen et
al soon thereafter devised a catalogue of methods to study the normal modes of
radial pulsations of stellar models. A normal mode of radial pulsation for an
equilibrium solution has the form $\delta r=\xi(r)\exp(i\sigma t)$ where $\xi$
is a trial function and the eigenvalue $\sigma$ is the frequency. We note that
even though we start with a static distribution, if it undergoes some
disturbance from the equilibrium position then it is expected that the
configuration will oscillate and consequently be time dependent. Additionally
it is assumed, for this process that, the Lagrangian change of the pressure
vanishes at the boundary and that the trial function is finite at the centre.

It is worthwhile noting that studies of stability can be treacherous
territory. Due warnings are contained in the work of Bardeen et al bardeen .
For example, Knutsen knutsen , applying method 2D of the Bardeen et al bardeen
scheme, came to the conclusion that the well known Buchdahl exact solution is
unstable with respect to radial oscillations. Contradictory results were
obtained by Negi negi  and Moustakidis moustakidis . Kokkotas and Ruoff
kokkotas  utilise a numerical technique and give two approaches to studying
stability: the shooting method and the method of finite differences. In the
context of EGB gravity the stability of scalarized black holes has been
considered by Silva et al silva . In our case, since there is no worked out
equivalent of the Chandrasekhar integral condition for EGB gravity, we shall
analyse our exact solution under radial pulsations by perturbing the radial
coordinate $x$ as $x+\epsilon x$, where $\epsilon$ is a very small quantity in
relation to the stellar radius, in the energy density, pressure and sound
speed. It will be immediately apparent that for small values of $\epsilon$ the
dynamical quantities converge to the original shapes. We provide plots where
$\epsilon$ is of the order $0.1,0.01,0.001$ in order to confirm that our
solution is indeed stable under radial pulsations. This is in the same spirit
as the well known Lyapunov stability of curves and follows the approach of
Herrera herrera ; abreu  in discussing the concept of cracking in anisotropic
spheres. Of course, cracking does not apply to perfect fluids of the type we
are studying here.

Fig. 7, 8 and 9 depicts the energy density, pressure and sound speed profiles
under pulsations of the type $x+\epsilon x$ for three different $\epsilon$
values namely $0.1,0.01,0.001$ and each plot showing increments up to seven
curves. It is clear that in all cases as $\epsilon$ decreases the perturbed
curves tend toward the equilibrium curve $\epsilon=0$. This indicates that
these dynamical quantities are stable under radial pulsations. Moreover, in
the case of the isotropic particle pressure, it is evident that even when the
$\epsilon$ values are large $(0.1)$ the curves all tend to converge at the
same stellar radius where the pressure vanishes.

Figure 7: Energy density versus radial pulsation $x+\epsilon x$ for $\epsilon$
of the order $0.1,0.01$ and $0.001$, respectively. The parameters are the same
as given in Fig. 1.  Figure 8: Pressure versus radial pulsation $x+\epsilon x$
for $\epsilon$ of the order $0.1,0.01$ and $0.001$, respectively.  Figure 9:
Sound speed squared versus radial pulsation $x+\epsilon x$ for $\epsilon$ of
the order $0.1,0.01$ and $0.001$, respectively.

##  V Additional exact Solutions

It is extremely difficult finding exact solutions to the field equations when
higher curvature terms are present. Nevertheless, some success was achieved in
locating a few classes of exact solutions which are listed below. However, we
do not study these in detail. Despite several efforts we were unable to
determine a suitable parameter space for these solutions such that all the
physical requirements are satisfied. Accordingly we merely list these as exact
solutions and in some cases we write out the dynamical quantities in detail
where this is simple.

###  V.1 Constant space potential $Z=k$

It is interesting to examine the case of a constant spatial potential $Z=k$,
for some constant $k$ since it is known that this is a necessary and
sufficient condition for isothermal behaviour in standard EGB gravity. By
isothermal behaviour we mean that the density and pressure vary as
$\frac{1}{r^{d-2}}$ where $d$ is the dimension of the spacetime. In standard
Einstein gravity the isothermal property demands that the energy density and
pressure go as $\frac{1}{r^{2}}$ as in the Newtonian gravity theory. In the
current context the choice $Z=k$ results in the form

$4kx^{2}(\beta(1-k)+x)\ddot{y}+4kx\beta(k-1)\dot{y}-(k-1)(\beta(k-1)+x)y=0,$
(40)

for the isotropy equation (14). This selection gives

$\displaystyle y$ $\displaystyle=$ $\displaystyle
C_{12}F_{1}\left(\left[-\frac{-3\sqrt{2}\sqrt{4k-2}+2\sqrt{3}\sqrt{9k+3}-6\sqrt{k}}{12\sqrt{k}},-\frac{\sqrt{2}\sqrt{4k-2}+\frac{2}{3}\sqrt{3}\sqrt{9k+3}-2\sqrt{k}}{4\sqrt{k}}\right],\right.$
(41)
$\displaystyle\left.\left[-\frac{\sqrt{3}\sqrt{9k+3}-3\sqrt{k}}{3\sqrt{k}}\right],\frac{x}{\beta(k-1)}\right)x^{-\frac{\sqrt{3}\sqrt{9k+3}-6\sqrt{k}}{6\sqrt{k}}}$
$\displaystyle+C_{22}F1\left(\left[\frac{3\sqrt{2}\sqrt{4k-2}+2\sqrt{3}\sqrt{9k+3}+6\sqrt{k}}{12\sqrt{k}},-\frac{\sqrt{2}\sqrt{4k-2}-2\sqrt{k}-\frac{2}{3}\sqrt{3}\sqrt{9k+3}}{4\sqrt{k}}\right],\right.$
$\displaystyle\left.\left[\frac{\sqrt{3}\sqrt{9k+3}+3\sqrt{k}}{3\sqrt{k}}\right],\frac{x}{\beta(k-1)}\right)x^{\frac{\sqrt{3}\sqrt{9k+3}+6\sqrt{k}}{6\sqrt{k}}}$

in terms of hypergeometric functions and where $C_{11}$ and $C_{22}$ are
integration constants. As it was not possible to find constant values $k$ for
which $y$ could reduce to elementary functions it cannot be checked if
isothermal behaviour results in 4D EGB as it does in standard EGB theory.

###  V.2 $Z=x$

The prescription $Z=x$ is a case of Kuchowicz’s kuch  exact solution in
Einstein theory where the radial potential was chosen in the form $x^{(b-1)}$
for some natural number $b$. The general solution was obtained for general $b$
but it was not observed that the $b=2$ solution could be resolved as
trigonometric functions. In our more complicated case, the $b=2$ case gives

$\displaystyle y$ $\displaystyle=$ $\displaystyle
c_{1}e^{-\frac{1}{\sqrt{x}}}x-c_{2}e^{\frac{1}{\sqrt{x}}}x,$ (42)
$\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{2x^{2}}-\frac{3}{2},$ (43) $\displaystyle 8\pi p$
$\displaystyle=$
$\displaystyle\frac{3}{2}+\frac{1}{2x^{2}}\left(1+4x+\sqrt{x}\left(8c_{1}\left(c_{1}-c_{2}e^{\frac{2}{\sqrt{x}}}\right)^{-1}-4\right)\right),$
(44)

for $\beta=1$. It was not possible to determine the solution for all $\beta$.
Given the simplicity of the present solution we list the sound speed squared
index and energy conditions

$\displaystyle\frac{dp}{d\rho}$ $\displaystyle=$
$\displaystyle-\frac{c_{1}^{2}\left(2x+3\sqrt{x}+1\right)-2c_{2}c_{1}e^{\frac{2}{\sqrt{x}}}(2x-1)+c_{2}^{2}e^{\frac{4}{\sqrt{x}}}\left(2x-3\sqrt{x}+1\right)}{\left(c_{1}-c_{2}e^{\frac{2}{\sqrt{x}}}\right){}^{2}},$
(45) $\displaystyle 8\pi(\rho-p)$ $\displaystyle=$
$\displaystyle-\frac{4c_{1}}{x^{3/2}\left(c_{1}-c_{2}e^{\frac{2}{\sqrt{x}}}\right)}+\frac{2}{x^{3/2}}-\frac{1}{x^{2}}-\frac{2}{x}-3,$
(46) $\displaystyle 8\pi(\rho+p)$ $\displaystyle=$
$\displaystyle\frac{2\left(c_{1}\left(\sqrt{x}+1\right)-c_{2}e^{\frac{2}{\sqrt{x}}}\left(\sqrt{x}-1\right)\right)}{x^{3/2}\left(c_{1}-c_{2}e^{\frac{2}{\sqrt{x}}}\right)},$
(47) $\displaystyle 8\pi(\rho+3p)$ $\displaystyle=$
$\displaystyle\frac{12c_{1}}{x^{3/2}\left(c_{1}-c_{2}e^{\frac{2}{\sqrt{x}}}\right)}-\frac{6}{x^{3/2}}+\frac{1}{x^{2}}+\frac{6}{x}+3.$
(48)

Finally the Chandrasekhar adiabatic stability index is given by

$\displaystyle\left(\frac{\rho+p}{p}\right)\frac{dp}{d\rho}$ $\displaystyle=$
$\displaystyle-\frac{4\sqrt{x}\left(c_{1}\left(\sqrt{x}+1\right)-c_{2}e^{\frac{2}{\sqrt{x}}}\left(\sqrt{x}-1\right)\right){}^{2}\left(c_{1}\left(2\sqrt{x}+1\right)-c_{2}e^{\frac{2}{\sqrt{x}}}\left(2\sqrt{x}-1\right)\right)}{\left(c_{1}-c_{2}e^{\frac{2}{\sqrt{x}}}\right){}^{2}\left(c_{1}\left(3x^{2}+4x+4\sqrt{x}+1\right)-c_{2}e^{\frac{2}{\sqrt{x}}}\left(3x^{2}+4x-4\sqrt{x}+1\right)\right)}.$

Despite its simplicity, it was not possible to find a solution which displayed
pleasing physical properties.

###  V.3 $y=\sqrt{1+x}$

The stipulation $y=\sqrt{1+x}$ corresponds to the Schwarzschild interior
temporal potential as can be seen from (21). It remains to see whether the the
unique radial potential is indeed $Z=1+x$. A major advantage of this choice of
$y$ is that it is singularity free everywhere. The isotropy equation is solved
to give gives the general solution

$\displaystyle Z$ $\displaystyle=$
$\displaystyle\frac{1}{2(2x-1)}\left(\frac{xv}{k\beta}+\frac{4(\beta-1)^{2}xk}{\beta
v}+\frac{4x^{2}}{\beta}+2(x-1)\right),$

where we have defined

$v=\sqrt[3]{4\left(k^{2}(x+1)(1-2x)^{2}-2k^{3}(\beta-1)^{3}+k^{2}(1-2x)\sqrt{(x+1)\left(4x^{3}-3x+1-4(\beta-1)^{3}k\right)}\right)}.$

which is not exactly the radial potential of the constant density fluid.
However, it is interesting to note that in the special case $\beta=1$ reduces
to a much simpler exact solution that is more tractable. Setting $\beta=1$
gives

$\displaystyle Z=1+x\hskip 28.45274pt{\mbox{or}}\hskip
28.45274ptZ=1+x+\frac{c_{1}x\sqrt[3]{x+1}}{\sqrt[3]{1-2x}},$ (51)

the first being the expected Schwarzschild potential and the second being a
totally new solution. We briefly investigate the properties of the second
solution. For the density and pressure we have

$\displaystyle 8\pi\rho$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{c_{1}^{2}(x(6x-1)-3)}{(2x-1)\sqrt[3]{1-2x}^{2}\sqrt[3]{x+1}}-3\right),$
(52) $\displaystyle 8\pi p$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{c_{1}^{2}(1-3x)\sqrt[3]{x+1}^{2}}{(x+1)\sqrt[3]{1-2x}^{2}}+3\right),$
(53)

and it is clear that the density is not constant but varies with the radius.
It can also be shown that the pressure does vanish for a finite radius thus
the model may be used to model compact stars. In addition we list the sound
speed squared index and energy conditions

$\displaystyle\frac{dp}{d\rho}$ $\displaystyle=$
$\displaystyle\frac{(2x-1)(x-1)}{2x^{2}-x-5},$ (54) $\displaystyle
8\pi(\rho-p)$ $\displaystyle=$
$\displaystyle\frac{c_{1}^{2}\left(6x^{2}-3x-1\right)}{(2x-1)\sqrt[3]{1-2x}^{2}\sqrt[3]{x+1}}-3,$
(55) $\displaystyle 8\pi(\rho+p)$ $\displaystyle=$
$\displaystyle\frac{2cc_{1}^{2}(x-1)}{(2x-1)\sqrt[3]{1-2x}^{2}\sqrt[3]{x+1}},$
(56) $\displaystyle 8\pi(\rho+3p)$ $\displaystyle=$
$\displaystyle\frac{c_{1}^{2}((7-6x)x-3)}{(2x-1)\sqrt[3]{1-2x}^{2}\sqrt[3]{x+1}}+3.$
(57)

Finally the Chandrasekhar adiabatic stability index is evaluates to

$\displaystyle\left(\frac{\rho+p}{p}\right)\frac{dp}{d\rho}$ $\displaystyle=$
$\displaystyle-\frac{4c_{1}^{2}(x-1)^{2}}{(x(2x-1)-5)\left(c_{1}^{2}(3x-1)-3\sqrt[3]{1-2x}^{2}\sqrt[3]{x+1}\right)}.$
(58)

The presence of a singularity at $x=\frac{1}{2}$ is not necessarily a
debilitating factor in this model since the pressure could vanish before this
value of the radius. However, it can be seen that the central density is zero
at the stellar centre which is not usually palatable since we expect a density
profile that decreases monotonically from the centre of the distribution.
Moreover, following extensive empirical testing, it was not possible to obtain
suitable parameter values such that all the elementary conditions are
satisfied.

##  VI Summary and Discussion

We have considered the impact of the Gauss–Bonnet higher curvature invariants
on stellar structure in the 4D framework with a rescaled coupling constant.
Our interest was in this proposal’s suitability in generating models of
compact stars. After converting the master pressure isotropy equation to a
form of an Abel equation we obtained an exact solution by prescribing the
temporal potential to be proportional to the stellar radius. This prescription
permitted the solving of the Abel equation and the complete model could then
be constructed. The physical properties of the model were studied and shown to
satisfy the usual requirements of positivity of density and pressure,
existence of a surface of vanishing pressure, all the energy conditions being
met, a subluminal sound speed and adiabatic stability in the sense of
Chandrasekhar. It turns out that the higher curvature terms had the effect of
correcting the deficiencies of the associated Einstein model. Finally, some
more exact solutions were discovered and reported on and which merit closer
scrutiny in the future. This study has demonstrated that compact star models
satisfying basic physical requirements are admissible in the 4D EGB framework.

## Acknowledgments

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# A Study of Acoustic Features in Arabic Speaker Identification under Noisy
Environmental Conditions

Zhor Benhafid, Kawthar Yasmine Zergat, Abderrahmane Amrouche  Speech Com &
Signal Proc. Lab.-LCPTS
Faculty of Electronics and Computer Sciences
USTHB, Bab Ezzouar, 16 111, ALGERIA
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

One of the major parts of the voice recognition field is the choice of
acoustic features which have to be robust against the variability of the
speech signal, mismatched conditions, and noisy environments. Thus, different
speech feature extraction techniques have been developed. In this paper, we
investigate the robustness of several front-end techniques in Arabic speaker
identification. We evaluate five different features in babble, factory and
subway conditions at the various signal to noise ratios (SNR). The obtained
results showed that two of the auditory feature i.e. gammatone frequency
cepstral coefficient (GFCC) and power normalization cepstral coefficients
(PNCC), unlike their combination performs substantially better than a
conventional speaker features i.e. Mel-frequency cepstral coefficients (MFCC).

###### Index Terms:

Arabic speaker identification; noisy environment; varied SNR ; Gaussian
mixture models; auditory features.

##  I Introduction

Voice biometry is to identify the speaker using specific information included
in speaker speech waves. It can be split into two types: speaker verification
(SV) and speaker identification (SI), which can both be text-dependent or
text-independent. SV aims to determine if the speaker is who claims to be or
not, while speaker identification decides which speaker is talking among a set
of speakers.

Speaker identification system (SIS) as every pattern recognition system has
two parts shown in Fig.1: (1) front-end process block where pre-processing and
features extraction parameters are done, and (2) back-end process block where
pattern classification using speaker modelling, and decision are calculated.
However, SIS decreases performance in noisy environments, and mismatched
conditions [1]. Thus, the improvement of robustness of speaker identification
system has become a focus question, where much research has been conducted.
For example, Missing-data approaches using Gaussian mixture models (GMM) and
universal background model (UBM) are used to optimize recognition performance
under noisy conditions [[2], [3]. Speech pre-processor such as voice activity
detection (VAD) in [4] has shown robustness against noise. Also, speaker
features like gammatone frequency cepstral coefficient, based on an auditory
periphery model (CASA) show that this feature captures speaker characteristics
and performs substantially better than conventional speaker features in noisy
environment [5].

Figure 1:  Speaker identification diagram.

Other studies using GFCC exhibits superior noise robustness to commonly used
Mel-frequency cepstral coefficients (e.g. [6],[7], [8], [9], and [10]). On the
other hand, power normalization cepstral coefficients for speech recognition
have shown a substantial improvement in recognition accuracy in the presence
of various types of additive noise and reverberant environments [11] that why
paper [12], and [13]proposed a speaker verification system using PNCC combined
with other methods which gave good results.

This paper investigates the problem of speaker identification using speech
samples distorted by additive environmental noises. We review five speech
characteristics including MFCC, GFCC, PNCC, perceptual linear predictive (PLP)
and line spectral frequency (LSF). The experiments were carried out using the
GMM algorithm and Arabic spoken digit (ARADIGIT) database in clean, babble,
factory and subway noises conditions.

The rest of the paper is organized as follows: Section II presents the front-
end process, the back-end process is given in Section III. Section IV presents
the experimental process; results and discussion are given in section V;
finally, conclusions are drawn in Section VI.

##  II FRONT-END PROCESS

###  II-A Mel-frequency cepstral coefficients

The MFCCs are heuristic representations of acoustic properties, that simulate
the human ear. More precisely, they mimic the perceptual representations of
acoustic information conveyed by the human auditory system [14]. The
extraction of MFCC features is listed as follow:

  * •

Pre-emphasize input speech signal.

  * •

Calculate short-time Fourier transform (STFT) to get magnitude spectrum.

  * •

Wrap the magnitude spectrum into Mel-spectrum using tri-angular overlapping
windows where centre frequencies of the windows are equally distributed on the
Mel scale.

  * •

Take the log operation on the power spectrum

  * •

Apply the discrete cosine transform (DCT) on the log-Mel-power- spectrum to
derive cepstral features. The mel-scale calculation is given by the following
equation:

$f_{mel}=2595log_{10}(1+\dfrac{f_{hz}}{700})$ (1)

###  II-B Gammatone frequency cepstral coefficients

GFCCs use technique based on the gammatone filter bank, which attempts to
model the human auditory system as a series of overlapping bandpass filters
[8]. GFCCs are calculated as follow:

  * •

Get gammatone filter bank which impulse response is shown in 2:

$g(t)=at^{n-1}e^{-2\pi b_{i}t}cos(2\pi f_{i}t+\varphi)U(t)$ (2)

Where , $f_{i}$ is the center frequency,$\varphi$ is the phase of carrier, $a$
is the gain of filter, $n$ is the order of filter, $N$ is the number of filter
and $b_{i}$ is the attenuation factor that decide the decay rate. The relation
between attenuation factor and bandwidth is: $b_{i}=1.019ERE(f_{i})$ where
$ERB(f_{i})$ is the equivalent rectangular bandwidth [10], it’s definition is
shown in 3:

$ERB(f_{i}=24.7(\frac{4.37f_{i}}{1000}+1)$ (3)

  * •

Obtain the filter response and decimate it to 100 Hz as a way of time
windowing.

  * •

Apply the cubic root operation on the magnitudes of the decimated outputs. The
result provides a T-F decomposition of input which represent a variant of
cochleagram.

  * •

Compute the DCT to have GFCCs. The coefficients are not cepstral coefficients
because a cepstral analysis requires a log operation between the first and the
second frequency analysis for the deconvolution purpose. Here, we call them
cepstral coefficients because of the functional similarities between the above
transformation and that of a typical cepstral analysis in the derivation of
MFCC [5].

###  II-C Perceptual linear predictive

In PLP technique, several well-known properties of hearing are simulated by
practical engineering. The principal of PLP method is illustrated by the
abouve block diagram at Fig.2.

Figure 2:  The computation step of PLP [15].

###  II-D Power Normalized Cepstral Coefficients

The major innovations of PNCC processing include the redesigned nonlinear
rate-intensity function, along with the series of processing elements to
suppress the effects of background acoustical activity based on medium-time
analysis [11]. The PNCC process is done in three stages: initial processing,
environment processing, and final processing.

  * •

Apply pre-emphasis followed by STFT on the speech signal and get the short-
time spectral power by weighting the magnitude-squared STFT outputs for
positive frequencies by the frequency response associated with gammatone
filter bank as initial processing.

  * •

In environmental processing, the main purpose is estimating the noise floor of
the speech by temporal asymmetric low pass filter and temporal masking filter,
then estimate the “transfer function” by smoothing the ratio of noise floor
and mid-time spectral with temporal average, followed by smoothing the short-
time spectral using the “transfer function”. These operations called time-
frequency normalization processing [13].

  * •

The PNCC vector is obtained in the final process by power-law non-linearity
function with exponent 1/15 followed by DCT.

###  II-E Line spectral frequency

Line spectral frequencies are an alternative representation to the linear
predictive coefficient (LPC) which are the parameters of the vocal tract
filter that transformed the excitation signal into a speech signal. The
frequency-domain representation of LSFs makes easier the incorporation of
human perception system properties. The overall process of the LSF algorithm
is shown in the block diagram in Fig.3.

Figure 3:  LSF step estimation.

##  III BACK-END PROCESS

In this study, we used the state of the art back-end process for speaker
recognition i.e. GMM. This feature modelling is a statistical unsupervised
machine learning algorithm that uses the statistical variations of the
features extracted from the input speech signal to model the speaker. Thus, it
provides us with a statistical representation of how a speaker produces
sounds; it is a representation of a speaker identity for text-independent
speaker recognition[16]. The d-dimensional Gaussian Probability Density
Function (PDF)is given by 4:

$N(x|\mu,\Sigma)=\dfrac{1}{(2\pi)^{d}/2\sqrt{|}\Sigma|}e^{(-\frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu))}$
(4)

Where $\mu$ is the mean and $\Sigma$ is the covariance matrix of the Gaussian.
The probability given in a mixture of $K$ Gaussians is:

$p(x)=\sum\limits_{j=1}^{K}w_{j}N(x\mu_{j},\Sigma_{j})$ (5)

Where $w_{j}$ is the probability (weight) of the jth Gaussian. All the weights
are positive and sum to one. The Expectation-Maximization (EM) algorithm is
used to find GMM parameters i.e. $w_{j}$ , $\mu_{j}$ , $\Sigma_{j}$.

##  IV SPEAKER IDENTIFICATION EXPERIMENT

###  IV-A Description of the database

The database used in this work is an Arabic spoken digit database named
ARADIGIT with 62 speakers (31 males and 31 females), recorded in a quiet
environment with an ambient noise level below 35 dB, in WAV format, at 22.050
kHz and down-sampled at 16 kHz. It consists of a set of ten digits of the
Arabic language (0 to 9), where each speaker repeats the same list 3 times.
The speakers are Algerian aged between 18 and 50 years from different regions
of Algeria.

In our experiments, we used the concatenated sequences of eight numbers (0 to
7) for training and sequences of two numbers (8 to 9) for testing used in
[17].

###  IV-B Expirimental process

In this study, we reconsidered an SIS in a noisy environment using GMM. First,
we modelled speakers at 16 mixtures of GMMs using the training concatenated
ARADIGT database. In the testing phase, utterances that are different from the
training ones, were digitally mixed at various SNR levels from -6 dB to 18 dB
at 6dB intervals using filtering and noise adding tool (FaNT) [18] and
NOISEX’92 database (NATO: AC 243/RSG 10). Three different noise types were
used for evaluation: factory noise, speech babble, and subway noise.

In the front-end process, we extracted 39-dimensional auditory features. For
each speech frame, we pulled 13 coefficients of PLP, GFCC, and PNCC combined
with their first and second derivatives as dynamic features. For the state of
the art of speaker, recognition features i.e. MFCC, we took 12 coefficients
per frame and its energy which gave a vector of 13 parameters which is
concatenated with its delta and Delta delta derivatives. For LSF we extracted
the commonly used number of parameters i.e. 10.

As the second step we evaluated the combination of the more robustness
parameter. For that, we combined 26-dimensional features of GFCC with PNCC,
PLP with GFCC, and PLP with PNCC.

For evaluating performances of the proposed Arabic speaker identification
system, we used the identification rate accuracy (IR) given by:

$IR=\dfrac{numberofcorrectidentifications}{totalnumberoftrails}*100$ (6)

##  V RESULTS AND DISCUSSION

TABLE I: speaker identification accuracy ($\%$) in three environmental noisy conditions Noise | SNR | Acoustic Features
---|---|---
MFCC | LSF | PLP | PNCC | GFCC
Babble | -6 | 4,83 | 9,67 | 6,45 | 6,45 | 12,90
0 | 12,90 | 19,35 | 16,12 | 24,19 | 24,19
6 | 20,96 | 45,16 | 45,16 | 70,96 | 72,58
12 | 38,70 | 64,51 | 91,93 | 91,93 | 95,16
18 | 72,58 | 77,41 | 98,38 | 95,16 | 100
Factory | -6 | 1,61 | 1,61 | 1,61 | 4,83 | 4,83
0 | 4,83 | 4,83 | 4,83 | 11,29 | 20,96
6 | 9,67 | 19,35 | 16,12 | 48,38 | 62,9032
12 | 19,35 | 30,64 | 45,16 | 75,80 | 93,54
18 | 53,22 | 53,22 | 87,09 | 88,70 | 100
Subway | -6 | 4,83 | 1,61 | 1,61 | 4,83 | 1,61
0 | 6,45 | 1,61 | 1,61 | 6,45 | 3,22
6 | 16,13 | 1,61 | 1,61 | 29,03 | 17,74
12 | 27,41 | 1,61 | 19,35 | 53,22 | 62,90
18 | 53,22 | 8,06 | 40,32 | 77,41 | 88,70

Table I shows the Arabic SI performances in babble, factory and subway noisy
environments conditions using MFCC, GFCC, PLP, and PNCC as auditory features
and LSF. From the obtained results, it can be said that the GFCCs combined
with their dynamic information give the best performances in the three
different noisy environments, followed by PNCCs in the second place. It
depends mainly on the use of gammatone filter bank for both GFCC and PNCC.
Also, the implementation details such as the use of the cubic root of the
decimated filter response in the GFCC process and estimating the noise by
medium temporary analysis then reducing the effect of noise by the short time
spectral in PNCC computation, improve the SI performances. The experiments
show also that:

  * •

Each noisy environment influences the robustness of speaker identification
differently; subway noise is the one that most decreases the speaker
identification rate accuracy.

  * •

LSF and MFCC have less performance in the three environmental conditions.
However, MFCC is better than LSF in subway noise.

  * •

PLPs start to give good performances at 12 dB in babble and factory noise,
contrary to MFCC and LSF. This is because PLP like GECC take into account that
the perceived loudness is a cubic root of voice intensity.

  * •

It is noticed that the Arabic speaker identification using the GMM gives the
best performances using the GFCC followed by PNCC and PLP at third position.
However, their combination doesn’t provide more robustness for identification
as it is shown in Fig.4.

Figure 4: SID accuracy ($\%$) comparisons of the proposed combined system.

##  VI Conclusion

This work describes the evaluation of speaker identification system using
different speech features in babble, factory, and subway noisy environmental
conditions. We investigated the MFCCs, PLPs, GFCCs and PNCCs combined with
their dynamic characteristics as auditory features and LSF with an Arabic
database using GMM. The obtained results showed that GFCC and PNCC are the
most robust features against environmental noise, unlike their combination. As
future work, we recommend evaluating the robustness of the GFCC and PNCC in
environmental noisy conditions to discriminate Arabic speakers using a deep
neural network classifier in the verification tasks.

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