colorlinks,citecolor=blue colorlinks=true,linkcolor=red,filecolor=magenta,
urlcolor=blue

Grav. Cosmol. No. issue, year

Cosmological Model with Cosmic transit behaviour in Brans-Dicke Theory

Sunil K. Tripathya,1, Alaka Priyadarsini Sendhaa,2, Sasmita Kumari Pradhanb,3,
Zashmir Naikc,4, B. Mishrad,5

a

Department of Physics, Indira Gandhi Institute of Technology, Sarang,
Dhenkanal, Odisha-759146,
India.

b

Department of Physics, Centurion University of Technology and Management,
Odisha, India and School of Physics, Sambalpur University, Jyotivihar,
Sambalpur, Odisha-768019, India.

c

School of Physics, Sambalpur University, Jyotivihar, Sambalpur, Odisha, India,
768019.

d

Department of Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

We have constructed a dark energy cosmological model in the framework of
generalized Brans Dicke theory (GBD) with a self-interacting potential. The
source of dark energy is considered through a unified dark fluid (UDF)
characterized by a linear equation of state (EoS). A time varying deceleration
parameter simulating the cosmic transit behaviour has been introduced to get
the dynamical behaviour of the model. The $H(z)$ data have been explored to
constrain the model parameters and to study the dynamical aspects of the
Brans-Dicke parameter and the scalar field.

Keywords: Cosmic transit; Brans-Dicke theory; Deceleration parameter;
Accelerating model.

11footnotetext: e-mail<EMAIL_ADDRESS>e-mail:
<EMAIL_ADDRESS>e-mail:
<EMAIL_ADDRESS>e-mail:
<EMAIL_ADDRESS>e-mail<EMAIL_ADDRESS>
(Corresponding author)

##  1 Introduction

Cosmological observations have shown that the late time cosmic acceleration
issue is due to the presence of an unknown form of energy, known as dark
energy (DE) which occupies $68.3$ percent of the total mass energy budget of
the Universe. Within the framework of GR, the explanation of DE requires some
additional dynamical degrees of freedom in the form of scalar field. Even
though, there have been geometrical modification routes suggested in
literature to understand the late time cosmic speed up issue[1, 2, 3, 4, 5],
the scalar field mediated gravity theories have gained much importance. The
scalar fields provide dynamical variables in the gravitational field equations
and also be useful in studying the early evolutionary or inflationary era.
Among the scalar-tensor theories proposed, Brans-Dicke (BD) theory [6] is
considered to be most successful. In the BD theory, the scalar field ($\phi$)
is coupled to the geometry with a coupling constant known as BD parameter
($\omega$). In this process of coupling, the gravity is mediated through the
scalar field and we naturally get a time varying Newtonian Gravitational
constant ($G$)as an inverse of the scalar field. The BD theory has passed the
experimental test from solar system [7] and is also tested against the Cosmic
Microwave Background (CMB) data and large scale structure [8, 9]. Scalar-
tensor theories of gravity including the BD theory have been used to model the
variations of constants of nature including $G$ [10, 11]. Also they provide
the archetypal class of theories used to quantify deviations from Einstein’s
theory[12, 13].

The study of DE cosmological models is the emerging research in modern
cosmology. Eingorn and Kiefer [14] have investigated the scalar cosmological
perturbations in the framework of a model with interacting DE and dark matter.
Xu and Zhang [15] have compared ten popular DE models according to their
capabilities of fitting the current observational data. Zhai et al. [16]
compared several theoretical cosmological models with the observational data
from the Type Ia supernova measurements of expansion, cosmic microwave
background, baryon acoustic oscillation measurements of expansion and the
local Hubble constant. Bruck and Mifsud [17] have considered cosmological
models that allow non-gravitational direct couplings between dark matter and
DE and distinguished the cosmological features of these couplings by the
cosmological observations. Mishra et al. [18] and Ray et al. [19] studied the
two fluid cosmological models in anisotropic space-time with the energy
momentum tensor as the combination of DE fluid and ordinary matter fluid.
Without any field theoretic interpretation, Paul and Sengupta [20] have
explained the recent observations with a dynamical cosmological constant.
Akarsu et al. [21] have studied the cosmological constant in the standard
$\Lambda$CDM model by introducing the graduated DE. Several research work have
been performed in recent times on the issue of late time cosmic speed up
phenomena and the associated exotic dark energy concept with the scalar-tensor
BD theory that provide some ample insight into the accelerated expansion.
Banerjee and Pavon [22] have argued that BD theory can explain the late time
cosmic expansion of the Universe without resorting the cosmological constant
or quintessence matter. Kim [23] suggested that BD theory as unified model for
dark matter and DE. Acquaviva and Verde [24] have framed the Jordan BD model
of gravity and with some analysis shown the inconsistencies in the expansion
history studied through the usual approach. In the framework of BD theory,
Chattopadhyay et al. [25] have used some reconstruction scheme for new
holographic DE model and derived the modified field equation of BD theory to
study the expansion history of the universe. Sharif and Shah [26] have
discussed the stability of the pilgrim DE model in BD theory. Zadeh and
Sheykhi [27] have studied the agegraphic dark energy in the background of BD
theory with pressureless dark matter. Tripathy et al. [28] have studied the
cosmology with a hybrid scale factor in BD theory. Verma has analysed the data
of gravitational waves from spinning neutron stars within Brans-Dicke theory
[29]. Du has studied the scalar stochastic gravitational wave background from
compact binary mergers and stellar collapses in BD theory [30].

The generalization of BD theory was proposed long time back to incorporate the
various requirements for the value of the BD parameter, where $\omega$ is
considered as a function of the scalar field $\phi$ [31, 32, 33]. The reason
behind the generalization is that for $2\omega+3=\frac{1}{\alpha\phi}$ and
$2\omega+3=\frac{1}{\phi-1}$, GBD reduces to Schwinger’s theory[34] and to
Barker’s scalar tensor theory [35] respectively. However, the controversy in
estimating the value of the BD parameter $\omega$ continues. Several
experiments have suggested the range for the BD parameter, prominent among
them are solar system experiment, $\omega>40000$ [7]; WMAP and SDSS,
$\omega<-120$ [36, 37]; $417<\omega<714$ [38]. In a recent work, within the
framework of GBD theory, Tripathy et al. [39] have shown the linkage between
the rate of anisotropy and the non-evolving part of the BD parameter. In
another work, Tripathy et al.[28] have obtained low negative values for the BD
parameter in the range $\omega>-1.5$. Low negative value of the BD parameter
is required to explain the structure formation, coincidence problem and late
time cosmic acceleration [44, 40, 41, 42, 43]. Pradhan et al. have discussed
the possibility of a Big Rip scenario within the generalized BD theory and
have shown that the Universe within the specified model may witness a finite
time doomsday at a time $t_{BR}\approx 16.14GYr$ requiring a large negative
value of the BD parameter [45]. Because of the success of the GBD theory in
explaining the key issues in cosmology, recently there have an increasing
interest in this scalar-tensor theory either in the original form or with
geometrical modification of the Einstein-Hilbert action [46, 47, 48, 49, 50].

In the present work, we wish to construct some dark energy cosmological models
within the framework of the generalized Brans-Dicke theory. Since, the present
observations regarding the late time cosmic speed up issue hint for a cosmic
transit from a decelerated phase of expansion to an accelerated phase, we will
construct such a cosmic transit model which may be simulated by a signature
flipping deceleration parameter. In fact, the redshift $z_{t}$ at which the
transition occurs may be considered as an important parameter and is order of
unity i.e $z_{t}\sim 1$. There have been different constraints on the cosmic
transit redshift from observational data. Some of recent constraints on the
transition redshift parameter may include Farooq and Ratra: $z_{t}=0.74\pm
0.05$ [51], Capozziello et al.: $z_{t}=0.7679^{+0.1831}_{-0.1829}$ [52], Reiss
et al.: $z_{t}=0.426^{+0.27}_{-0.089}$ [53], Lu et al. :
$z_{t}=0.69^{+0.23}_{-0.12}$ [54], Moresco et al.:$z_{t}=0.4\pm 0.1$ [55],
Yang and Gong: $z_{t}=0.61^{+0.231}_{-0.12}$ [56], Jesus et al.:
$z_{t}=0.806\pm 0.094$ [57]. Cosmological models showing cosmic transit
behaviour with proper prediction of the transition redshift may be considered
to be more viable and realistic. We use the observational $H(z)$ data to the
constrain the model parameters and to study the cosmic transit behaviour. The
paper is organized as: In Sec. II, the basic field equations with a time
varying BD parameter and self-interacting potential are derived. The DE EoS is
presented in Sec. III. In Sec. IV, the cosmic transit behaviour along with the
variable deceleration parameter are analysed. The behaviour of the time
varying BD parameter and the BD scalar field are discussed. The final remarks
along with the conclusions of the present work are listed in Sec. V.

##  2 Basic Formalism

The action for generalized Brans-Dicke (GBD) theory in a Jordan frame is given
by [31, 32, 33]

$S=\int d^{4}x\sqrt{-g}\left[\varphi
R-\frac{\omega(\varphi)}{\varphi}\varphi^{,\alpha}\varphi_{,\alpha}-V(\varphi)+\mathcal{L}_{m}\right],$
(1)

where $\omega(\varphi)$ is the modified Brans-Dicke parameter that varies with
the Brans-Dicke scalar field $\varphi$. $V(\varphi)$ is the self-interacting
potential, $R$ is the scalar curvature and $\mathcal{L}_{m}$ is the matter
Lagrangian. The field equations of GBD theory can be obtained as

$\displaystyle
G_{\mu\nu}-\dfrac{\omega(\varphi)}{\varphi^{2}}\left[\varphi_{,\mu}\varphi_{,\nu}-\frac{1}{2}g_{\mu\nu}\varphi_{,\alpha}\varphi^{,\alpha}\right]$
$\displaystyle-\frac{1}{\varphi}[\varphi_{,\mu;\nu}-g_{\mu\nu}\Box\varphi]+\frac{V(\varphi)}{2\varphi}g_{\mu\nu}$
$\displaystyle=$ $\displaystyle\frac{T_{\mu\nu}}{\varphi},$ (2)
$\displaystyle\Box\varphi$ $\displaystyle=$
$\displaystyle\frac{1}{2\omega(\varphi)+3}\Box[T-\left(2V(\varphi)-\varphi\frac{\partial
V(\varphi)}{\partial\varphi}\right)$ (3)
$\displaystyle-\frac{\partial\omega(\varphi)}{\partial\varphi}\varphi_{,\mu}\varphi^{,\mu}\Box].$

Here we chose $8\pi G_{0}=c=1$; $G_{0}$ and $c$ respectively denote the
Newtonian Gravitational Constant and the speed of light in vacuum.
$T=g^{\mu\nu}T_{\mu\nu}$ is the trace of the energy momentum tensor
$T_{\mu\nu}$, $\Box$ is the d’Alembert operator. In the present work, we
assume the energy momentum tensor as
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}$ where $\rho$ is the dark
energy density and $p$ is the dark energy pressure. We consider a plane
symmetric LRS Bianchi I (LRSBI) metric in the form

$ds^{2}=-dt^{2}+a_{1}^{2}dx^{2}+a_{2}^{2}(dy^{2}+dz^{2}),$ (4)

where $a_{1}$ and $a_{2}$ are functions of cosmic time only. The GBD field
equations for the anisotropic LRSBI Universe can be obtained as,

$\displaystyle
2\frac{\dot{a}_{1}\dot{a}_{2}}{a_{1}a_{2}}+\left(\frac{\dot{a}_{2}}{a_{2}}\right)^{2}-\frac{\omega(\varphi)}{2}\left(\frac{\dot{\varphi}}{\varphi}\right)^{2}+\left(\frac{\dot{a}_{1}}{a_{1}}+2\frac{\dot{a}_{2}}{a_{2}}\right)\frac{\dot{\varphi}}{\varphi}-\frac{V(\varphi)}{2\varphi}$
$\displaystyle=$ $\displaystyle\frac{\rho}{\varphi},$ (5) $\displaystyle
2\frac{\ddot{a}_{2}}{a_{2}}+\left(\frac{\dot{a}_{2}}{a_{2}}\right)^{2}+\frac{\omega(\varphi)}{2}\left(\frac{\dot{\varphi}}{\varphi}\right)^{2}+2\left(\frac{\dot{a}_{2}}{a_{2}}\right)\frac{\dot{\varphi}}{\varphi}+\dfrac{\ddot{\varphi}}{\varphi}-\dfrac{V(\varphi)}{2\varphi}$
$\displaystyle=$ $\displaystyle-\frac{p}{\varphi},$ (6)
$\displaystyle\frac{\ddot{a}_{1}}{a_{1}}+\frac{\ddot{a}_{2}}{a_{2}}+\frac{\dot{a}_{1}\dot{a}_{2}}{a_{1}a_{2}}+\frac{\omega(\varphi)}{2}\left(\frac{\dot{\varphi}}{\varphi}\right)^{2}+\left(\frac{\dot{a}_{1}}{a_{1}}+\frac{\dot{a}_{2}}{a_{2}}\right)\frac{\dot{\varphi}}{\varphi}+\dfrac{\ddot{\varphi}}{\varphi}-\dfrac{V(\varphi)}{2\varphi}$
$\displaystyle=$ $\displaystyle-\frac{p}{\varphi}.$ (7)

The directional Hubble parameters can be expressed as
$H_{x}=\frac{\dot{a_{1}}}{a_{2}}$ and $H_{y}=\frac{\dot{a_{2}}}{a_{2}}$.
Incorporating an anisotropic relation among the directional Hubble parameters
as, $H_{x}=kH_{y},k\neq 1$, the mean Hubble parameter may be expressed as
$H=\frac{1}{\xi}H_{y}$, where the anisotropic parameter $\xi$ is related to
the parameter $k$ as $\xi=\frac{3}{k+2}$. It is obvious that, for $\xi=1$, the
model reduces to that of a flat isotropic model. In terms of the Hubble
parameter and the anisotropic parameter $\xi$, the GBD field equations are
expressed as [28]

$\displaystyle(2k+1)\xi^{2}H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\varphi}\left(\rho+\rho_{\varphi}\right),$ (8)
$\displaystyle 2\xi\dot{H}+3\xi^{2}H^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\varphi}\left(p+p^{1}_{\varphi}\right),$ (9)
$\displaystyle(k+1)\xi\dot{H}+(k^{2}+k+1)\xi^{2}H^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\varphi}\left(p+p^{2}_{\varphi}\right),$

where we have

$\displaystyle\rho_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\omega(\varphi)\varphi\left(\frac{\dot{\varphi}}{\varphi}\right)^{2}-3H\dot{\varphi}+\frac{1}{2}V(\varphi),$
(11) $\displaystyle p^{1}_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\omega(\varphi)\varphi\left(\frac{\dot{\varphi}}{\varphi}\right)^{2}+2\xi
H\dot{\varphi}+\ddot{\varphi}-\dfrac{1}{2}V(\varphi),$ (12) $\displaystyle
p^{2}_{\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\omega(\varphi)\varphi\left(\frac{\dot{\varphi}}{\varphi}\right)^{2}+(k+1)\xi
H\dot{\varphi}+\ddot{\varphi}-\frac{1}{2}V(\varphi).$

The pressure anisotropy for the DE pressure may now be expressed as

$\triangle p_{\varphi}=p^{2}_{\varphi}-p^{1}_{\varphi}=(k-1)\xi
H\dot{\varphi}.$ (14)

It is obvious that, the pressure anisotropy vanishes for an isotropic model.
Also, we do not have a pressure anisotropy for a constant BD scalar field. The
wave equation for the scalar field becomes

$\displaystyle\dfrac{\ddot{\varphi}}{\varphi}+3H\frac{\dot{\varphi}}{\varphi}=\frac{1}{{2\omega(\varphi)+3}}\Box[\rho-3{p}-\frac{\partial\omega(\varphi)}{\partial\varphi}\dot{\varphi}^{2}$
$\displaystyle-2V(\varphi)+\varphi\frac{\partial
V(\varphi)}{\partial\varphi}\Box].$ (15)

The scalar field dependence of the Brans-Dicke parameter $\omega(\varphi)$ and
self-interacting potential $V(\varphi)$ may be obtained from the Eqs. (8)-(2)
as [28, 45],

$\displaystyle\omega(\varphi)$ $\displaystyle=$
$\displaystyle\left(\frac{\dot{\varphi}}{\varphi}\right)^{-2}\Box[-\dfrac{\rho+p}{\varphi}-\frac{\ddot{\varphi}}{\varphi}+k\xi
H\dfrac{\dot{\varphi}}{\varphi}$ (16)
$\displaystyle-2\xi\dot{H}-2(1-k)\xi^{2}H^{2}\Box],$ $\displaystyle
V(\varphi)$ $\displaystyle=$ $\displaystyle
2\xi^{2}H^{2}\varphi-2\rho-\omega(\varphi)\varphi\left(\frac{\dot{\varphi}}{\varphi}\right)^{2}+6H\dot{\varphi}.$

The dynamical behaviour of the BD parameter and the self interacting potential
requires the knowledge of the equation of state in the form of $p=p(\rho)$ and
the dynamical evolution of the scale factor. In the present work, we are
interested in a cosmic transit model that simulates a signature flipping
behaviour of the deceleration parameter which in turn fixes the Hubble rate.
From a tabulated observational $H(z)$ data we will fix the scale factor.
Another aspect is the fixation of the anisotropic parameter $\xi$. In fact,
recent works on cosmic anisotropy show a small value for the cosmic shear
which provides a little deviation of the anisotropic parameter from its
isotropic value $1$. The cosmic anisotropy issue has not yet resolved
substantially and therefore, we intend to keep this factor considering a
little departure from unity.

##  3 Equation of state

In order to handle the dark energy issue, we consider a linear relationship
between the pressure and energy density in the form

$p=\zeta(\rho-\rho_{0}),$ (18)

where $\zeta$ and $\rho_{0}$ are positive constants. A hydrodynamically stable
fluid description may be obtained for a positive value of $\zeta$ [58]. The
choice of the linear equation of state speculates an approach to unify the
dark matter and dark energy into a single dark fluid known as unified dark
fluid (UDF) [39]. In fact, the UDF EoS (18) signifies the degeneracy of the
dark sector as its first part behaving like a usual cosmic fluid and the other
as a cosmological constant. Moreover, $\zeta=0$ refers to the case of dark
matter and $\zeta=1$ implies a stiff fluid dominated with dark energy. In
literature, there have been many suggestions on the generalized EoSs where the
dark energy and dark matter are considered to be two different aspects of the
same fluid [61, 60, 59]. In some of these unified dark sector models, the EoS
may depend on the energy density, the Hubble parameter and its derivatives
[61].

The conservation equation for the given UDF may be expressed as

$\dot{\rho}+3H\left[\kappa\left(\rho-\rho_{0}\right)+\rho_{0}\right]=0,$ (19)

which may be integrated to obtain the energy density as

${\rho}={\rho_{\Lambda}}+\rho_{\kappa}\left(\frac{\mathcal{R}}{\mathcal{R}_{0}}\right)^{-3\kappa},$
(20)

where, $\rho_{\Lambda}=\left(1-\frac{1}{\kappa}\right)\rho_{0}$,
$\rho_{\kappa}=\rho_{1}-\rho_{\Lambda}$ and $\kappa=\zeta+1$. $\rho_{1}$ is
related to the dark energy density at the present epoch and $\mathcal{R}$ is
the scale factor. $\mathcal{R}_{0}$ is the scale factor at the present epoch.
The dark energy pressure becomes

$p={-\rho_{\Lambda}}+\left(\kappa-1\right)\rho_{\kappa}\left(\frac{\mathcal{R}}{\mathcal{R}_{0}}\right)^{-3\kappa}.$
(21)

The energy density and pressure for the given UDF equation of state may be
expressed in terms of the redshift
$z=-1+\left(\frac{\mathcal{R}}{\mathcal{R}_{0}}\right)^{-1}$ as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle{\rho_{\Lambda}}+{\rho_{\kappa}}(1+z)^{3\kappa},$ (22)
$\displaystyle p$ $\displaystyle=$
$\displaystyle{-\rho_{\Lambda}}+\left(\kappa-1\right)\rho_{\kappa}(1+z)^{3\kappa}.$
(23)

At an early epoch, we have $z\rightarrow\infty$ and consequently, the
magnitudes of the energy density and pressure become infinitely large. On the
other hand, with the cosmic evolution, as we move to late phase,
$z\rightarrow-1$, the energy density overlaps with $\rho_{\Lambda}$ and the
pressure becomes $p=-\rho_{\Lambda}$. In view of this, we may identify
$\rho_{\Lambda}$ as the vacuum energy density. At the present epoch, ($z=0$),
these quantities respectively assume the values
$\rho=\rho_{\Lambda}+\rho_{\kappa}$ and
$p=-\rho_{\Lambda}+(\kappa-1)\rho_{\kappa}$. As has been supported by
observational facts, in the present model, the late phase of the cosmic
evolution witnesses an accelerated expansion mostly dominated by the dark
energy.

Since, for the present UDF model, we have
$\rho+p=\kappa\rho_{\kappa}(1+z)^{3\kappa}$, the dark energy EoS,
$\omega_{D}=-1+\frac{\rho+p}{\rho}$ may be obtained as

$\omega_{D}=-1+\dfrac{\kappa}{1+\left(\dfrac{\rho_{\Lambda}}{\rho_{\kappa}}\right)(1+z)^{-3\kappa}},$
(24)

which may also be expressed as

$\omega_{D}=-1+\dfrac{\kappa}{1+\frac{\chi}{1-\chi}(1+z)^{-3\kappa}},$ (25)

where $\chi=\left(\dfrac{\rho_{\Lambda}}{\rho_{1}}\right)$. In fact the dark
energy density parameter $\Omega_{D}=\frac{\rho_{\Lambda}}{3H^{2}}\simeq 0.7$
and it dominates the energy density from usual matter at the present epoch. In
view of this, we may infer that, $\frac{\rho_{\Lambda}}{\rho_{\kappa}}>>1$. In
some of the earlier work, attempts have been made to constrain the UDF model
parameters from reasonable basis [60, 39, 58, 28, 62]. One may note from the
above expression (21) that, at an early epoch ($z\rightarrow\infty$),
$\omega_{D}\rightarrow-1+\kappa$ and at a late epoch ($z\rightarrow-1$),
$\omega_{D}\rightarrow-1$. In otherwords, the present UDF model overlaps with
the concordant $\Lambda$CDM model at late times and is in conformity with
recent observations. At the present epoch, we may have approximate value
$\omega_{D}\simeq-1-\kappa\chi$. This approximate relation may be used to
constrain the values of the parameter $\kappa$ from the known observational
constraints on the dark energy EoS. For an assumed value of $\chi$, we may
obtain $\kappa$ from known $\omega_{D}$ through the relation

$\kappa=-\frac{1+\omega_{D}}{\chi}.$ (26)

Since $\chi$ is assumed to be a positive quantity, a positive $\kappa$
requires that $\omega_{D}<-1$ which shows that, the present UDF model may
favour a phantom field dominated universe. In an earlier work [28], similar
conclusion has been derived for the UDF model within a GBD theory using
experimental $H(z)$ data.

##  4 Variable Deceleration Parameter And Cosmic Transit Behaviour

It is confirmed from a lot of observations in recent times that, the universe
is speeding up in the present epoch [63, 64]. It is believed that the Universe
may have undergone a transition from a decelerated phase to an accelerated
one. The cosmic redshift at which this transition occurs is called transition
redshift $z_{t}$. This implies a signature flipping of the deceleration
parameter $q$ from a positive value at some early time to negative value at
late time of cosmic evolution. Here, we will construct some cosmological
models that are consistent with this behaviour of the Universe and can provide
a signature flipping of the deceleration parameter. We consider a scale factor
that simulates a cosmic transit behaviour from an early decelerating phase to
a late time acceleration as,

$\mathcal{R}=\left(\frac{e^{\delta\gamma{t}}-\beta}{\delta}\right)^{\frac{1}{\gamma}},$
(27)

where $\delta,\beta$ and $\gamma$ are the model parameters. The Hubble
parameter for this model becomes
$H=\delta+\frac{\beta}{{\mathcal{R}}^{\gamma}}$ and can be expressed in terms
of redshift $z$ as

$H(z)=\delta+\beta\left(1+z\right)^{\gamma}.$ (28)

Here we have defined the redshift as $1+z=\frac{1}{\mathcal{R}}$ by assuming
the scale factor at the present epoch to be unity i.e $\mathcal{R}_{0}=1$. the
scale factor at the present epoch to be $1$. For the present model, the
deceleration parameter is obtained as,

$q=-1-\frac{\dot{H}}{H^{2}}=-1+\frac{\beta\gamma}{\delta{\mathcal{R}}^{\gamma}+\beta}.$
(29)

One may infer that, at an initial epoch with $\mathcal{R}\rightarrow 0$, the
deceleration parameter is $q=\gamma-1$. At a late phase of cosmic evolution
with $\mathcal{R}\rightarrow\infty$, we have $q=-1$. At the present epoch, it
becomes $q_{0}=-1+\frac{\beta\gamma}{\delta+\beta}$. It is clear that, for
$\gamma>1$, the deceleration parameter evolves from a positive value at an
early epoch to overlap with the concordance $\Lambda$CDM value and/or with the
de Sitter kind of expansion at late phase.

###  4.1 Transition redshift

On the basis of a lot of recent observational data, the Universe is believed
to have undergone a transition from deceleration to acceleration phases at a
transition redshift $z_{t}$ whose value primarily depends on the cosmology and
the assumed gravitational theory. $z_{t}$ is of the order of unity i.e.
$z_{t}\sim 1$. In the present work, we considered the available observational
$H(z)$ data to constrain the model parameters through a $\chi^{2}$ fitting
procedure

${\chi^{2}_{H}}(p)={\sum^{N}_{i=1}}\frac{(H^{t}-H^{o})^{2}}{\sigma^{2}_{H}}$
(30)

for $N$ measured values of $H^{o}$ with variance $\sigma^{2}_{H}$ at red shift
${z}$ where $H^{t}$ is the predicated value of $H(z)$ from the present
cosmological model. In the fitting procedure, the constrained value of the
parameters are obtained as $\delta=36.67\pm 9.58$, $\beta=27.90\pm 7.09$,
$\gamma=1.58\pm 0.17$. The $\chi^{2}$ is found to be $19.62$. In FIG. 1, the
Hubble parameter $H(z)$ for the present model with the constrained parameter
space is shown as a function of the redshift. The observational data for the
Hubble parameter are also shown in the figure as unconnected solid dots. In
FIG.2, $\frac{H(z)}{1+z}=\dot{\mathcal{R}}$ is plotted as a function of the
redshift along with the observational values. This function $H(z)$ directly
maps the accelerating behaviour of the Universe. For accelerating Universe, it
has a positive slope and for a decelerating Universe, it should have a
negative slope. Since the plot shows a positive slope for $H(z)$, the
accelerating nature of the present Universe is quite certain which is also
supported by the behaviour of the acceleration plot for
$\frac{H(z)}{1+z}=\dot{\mathcal{R}}$ in FIG.2.

Figure 1:  Hubble parameter $H(z)$ as function of redshift $z$ with unit
$Kms^{-1}Mpc^{-1}$. Figure 2:  $\frac{H(z)}{1+z}$ as a function of redshift
$z$ with unit $Kms^{-1}Mpc^{-1}$ for the constructed models. Figure 3:
Deceleration parameter $q$ as a function of redshift $z$ for the constructed
models.

The so called $H_{0}$ tension arises out of the discrepancy between the local
distance ladder measurement and those inferred from Cosmic Microwave
Background Radiation (CMBR) poses question on the the success of $\Lambda$CDM
model[65, 66, 67, 68]. While the measurement of Riess et al. provides
$H_{0}=74.03\pm 1.42km~{}s^{-1}Mpc^{-1}$ [66, 67, 69], the CMB temperature
from Planck collaboration suggests $H_{0}=67.36\pm 0.54km~{}s^{-1}Mpc^{-1}$
[68]. On the other hand, the results from the cosmological ladder and
classical Cepheids and supernovae type Ia of the S$H0$ES collaboration
estimates $H_{0}=73.3^{+1.7}_{-1.8}km~{}s^{-1}Mpc^{-1}$ [70] and the H0LiCOW
collaboration tells us that $H_{0}=73.3^{+1.7}_{-1.8}km~{}s^{-1}Mpc^{-1}$[71].
Eventhough the methods of determination of the value of the Hubble parameter
at the present epoch have substantial accuracy, within the standard
$\Lambda$CDM model, both the approaches do not reconcile and there emerges a
tension of $5.0\sigma$. Till date there is no satisfactory explanation for the
tension arising on the systematic errors. There are many attempts in recent
times to reduce the $H_{0}$ tension. Recently, it has been shown that, the BD
theory coupled with a $\Lambda$-term or a modified gravitational action may be
helpful in reducing the above mentioned tensions [72, 48]. In a recent work,
Capozziello et al. have considered the look back time and inferred that the
Hubble tension can be removed if the look-back time is correctly referred to
the redshift where the measurement is performed [73]. The discrepancy in the
measurements from different observations leading to the $H_{0}$ tension may
hint for a new Physics involving the dark energy and/or dark matter components
either beyond the standard $\Lambda$CDM model, or beyond the standard model of
particle physics [74, 75]. As reported in [73], different estimates of the
Hubble parameter from different measurements include $H_{0}=69.8\pm
0.8km~{}s^{-1}Mpc^{-1}$ [76], $H_{0}=67.8\pm 1.3km~{}s^{-1}Mpc^{-1}$ [77] and
$H_{0}=69.5^{+3.0}_{-3.5}km~{}s^{-1}Mpc^{-1}$ [78]. Goswami et al. have
constrained the Hubble parameter from the analysis of joint $H(z)$ and
Pantheon data as $H_{0}=69.36\pm 1.42km~{}s^{-1}Mpc^{-1}$ [79]. In the present
work, through the procedure of constraining the values of the model
parameters, we obtain the Hubble parameter at the present epoch as
$H_{0}=64.57\pm 16.67km~{}s^{-1}Mpc^{-1}$ which is well within the recent
estimates.

The signature flipping of the deceleration parameter $q$ for the constructed
model is shown in In FIG-3. As we have discussed earlier, the deceleration
parameter smoothly evolves from positive values in initial epoch to become
$q=-1$ at late times showing a transition from a decelerated phase of
expansion to an accelerated phase. The transition redshift corresponding to
$q=0$ is obtained as $z_{t}=0.68$. This value of transition redshift is in
conformity with that constrained from different analysis [52, 53, 54, 56, 79].
The deceleration parameter at the present epoch is obtained in the present
analysis as $q_{0}(z=0)=-0.32\pm 0.42$ signifying an accelerating phase of
expansion at the present time.

###  4.2 Brans-Dicke scalar field

From the field equations (6) and (7), one may get a relation between the BD
scalar field and the Hubble parameter as
$\frac{\dot{\varphi}}{\varphi}=-\frac{\dot{H}}{H}-3H$ and on integration, we
obtain

$\varphi=\varphi_{0}\left[\frac{H_{0}}{H{\mathcal{R}^{3}}}\right],$ (31)

where, $\varphi_{0}$ is the value of the scalar field at the present epoch.
The scalar field depend on the scale factor and decreases with the growth of
scale factor. The evolution of the BD scalar field is presented graphically in
FIG. 4 as a function of redshift. The scalar field, for the constructed model,
decreases from some large values at early phase to small values at the present
epoch.

Figure 4:  Evolution of the Brans-Dicke scalar field $\varphi$ with red shift
$z$. Figure 5:  Evaluation of BD parameter $\omega(\varphi)$ as function of
red shift with z . Here, the unified dark fluid parameter $\zeta$ is taken to
be $0.00172$.

###  4.3 Brans-Dicke parameter

In the GBD theory, the parameter $\omega$ is considered as a function of the
BD scalar field. We have also derived the general expression for $\omega$
which depends on an assumed relationship between the rest energy density and
proper pressure, the Hubble rate and the scalar field. It is to note that, in
order to simulate the presence of a dark sector of the Universe, we have
considered a unified dark fluid EoS which takes into account of both the dark
energy and dark matter. With the assumed scale factor in the model, the
expression of the BD parameter can be written as,

$\displaystyle\omega(H)=\left(\frac{\dot{H}}{H}+3{H}\right)^{-2}\Box[-\dfrac{\rho+p}{\varphi}-\frac{\ddot{H}}{H}+6\dot{H}$
$\displaystyle+\dfrac{54}{(k+2)^{2}}H^{2}\Box]$ (32)

Tripathy et al. [39], have shown the behavior of the BD parameter with the
power law and exponential expansion and observed that the anisotropic
parameter only affects the non-evolving part of the BD parameter. In some
other works, Tripathy and collaborators [45, 80] have shown that, the
anisotropic parameter affects the non-evolving part of the BD parameter when
the deceleration parameter comes out to be constant quantity. Of course, this
deceleration parameter dependence of the BD parameter is quite model dependent
[80]. In the present case, one may observe that, it is not possible to
separate out the evolving and non-evolving parts. In fact, the evolution of
the BD parameter is more involved in the present model. Here we have presented
graphically the evolutionary behaviour of the BD parameter for some
representative values of the anisotropic parameter in FIG. 5. In order to
obtain the plot, we fix the values of the UDF parameter as $\zeta=0.00172$.
The effect of the anisotropic parameter on the evolutionary aspect of the BD
parameter is clearly visible. Moreover, the BD parameter is more sensitive to
the choice of the anisotropic parameter. It is seen that, for a choice of
$k=0.9$, the BD parameter $\omega(\varphi)$ decreases gradually from an early
epoch to the late epoch. However, for higher values of the anisotropic
parameter such as $k=1.1$ and $k=1.3$, $\omega(\varphi)$ increases slowly from
an early epoch to late epoch. At a given epoch, $\omega(\varphi)$ has a lower
value for higher cosmic anisotropy. It is also interesting that, the values of
$\omega(\varphi)$ at an initial epoch and at an final epoch vary widely with
the choices of the anisotropic parameter. The predicted values of the BD
parameter at the present epoch for different choices of the anisotropic
parameter are given in Table-1.

###  4.4 Self interacting potential

The self interacting potential $V(\varphi)$, for the space-time (4) and scale
factor (21) can be calculated as,

$\displaystyle
V(\varphi)=2\varphi\left[\frac{9(2k+1)H^{2}}{(k+2)^{2}}-\frac{\rho}{\varphi}-\frac{\omega(\varphi)}{2}\left(-\frac{\dot{H}}{H}-3{H}\right)^{-2}+3H\left(\frac{\dot{H}}{H}-3{H}\right)\right]$
(33)

Graphically the behaviour of $V(\varphi)$ for the representative values of
cosmic anisotropy is represented in FIG. 6 using the UDF parameter
$\zeta=0.00172$. In general, the self interacting potential is negative for
all the choices of the anisotropic parameter. The evolutionary behaviour shows
that the self interacting potential increases from some large negative value
to almost vanish at the late time of the evolution. The self interacting
potential is also sensitive to the choice of $k$. At an initial epoch, for
higher $k$, the curve of $V(\varphi)$ remains atop of those with lower values
of $k$. However, at the present epoch and at late epochs, all the curves of
$V(\varphi)$ overlap and vanish at late times. In Table-1, we have presented
the predicted values of the self interacting potential required for the
viability of the model for different values of the anisotropic parameter.

Figure 6: Evolution of the self interacting potential $V(\varphi)$ is shown as
a function of red shift $z$ Figure 7:  $\frac{\dot{G}}{G}$ as function of
redshift $z$. The unified dark fluid parameter $\zeta$ is taken to be
$0.00172$.

###  4.5 Variation of Newtonian Gravitational Constant $G$

In generalized BD theory, the Newtonian gravitational constant $G(\varphi)$ is
expressed as [32, 81, 82, 28]

$G(\varphi)=\frac{4+2\omega(\varphi)}{\varphi(3+2\omega(\varphi))}.$ (34)

The time variation of the Newtonian gravitational constant is obtained as

$\frac{\dot{G}}{G}=-\frac{2\dot{\omega}(\varphi)}{(4+2\omega(\varphi))(3+2\omega(\varphi))}-\frac{\dot{\varphi}}{\varphi},$
(35)

which may be reduced to $\frac{\dot{G}}{G}=-\frac{\dot{\varphi}}{\varphi}$ for
a non-evolving BD parameter $\omega$. In other words, the time evolution of
the Newtonian Gravitational constant comes from the time evolution of the BD
scalar field. However, in the present case with a dynamical BD parameter, we
have some extra contribution coming from the dynamical nature of the BD
parameter. It is worth to mention here that the Newtonian gravitational
constant $G$ witnesses a very small variation whose measurement from different
observations usually involve large errors. Different estimates for the
variation of the $G$ include that of $\frac{\dot{G}}{G}(z=0)\simeq(2\pm
4)\times 10^{-12}~{}yr^{-1}$ from the Viking Lander Ranging [83],
$\frac{\dot{G}}{G}(z=0)<4\times 10^{-10}~{}yr^{-1}$ from the gravochemical
heating process [84] and $\frac{\dot{G}}{G}(z=0)\simeq(1.10\pm 1.07)\times
10^{-11}~{}yr^{-1}$ from the observations of double Neutron star binary [85].
In FIG.7, we have plotted $\frac{\dot{G}}{G}$ as a function of redshift $z$
for $\zeta=0.00172$. It may be observed that, the variation of $G$ is
prominent at an early epoch and becomes negligible at late times. The
variation of $G$ is also affected by the choice of the anisotropic parameter.
At least at late times, the variation in $G$ is substantial as compared to
that for lower values of the anisotropic parameter.

Table 1:  The predicted values of Hubble rate ($H_{0}$), Deceleration parameter ($q_{0}$), Brans-Dicke parameter ($\omega_{0}$), Self interacting potential ($\omega_{0}$), Transition red shift ($z_{0}$), at the present epoch. Parameters | Predicted value
---|---
$H_{0}$ | 64.57
$q_{0}$ | -0.32
$\omega_{0}(k=0.9)$ | 0.72
$\omega_{0}(k=1.1)$ | 0.571
$\omega_{0}(k=1.3)$ | 0.448
$V_{0}(k=0.9)$ | -343982.6
$V_{0}(k=1.1)$ | -320553.2
$V_{0}(k=1.9)$ | -302539.8
$z_{t}$ | 0.68

##  5 Final Remarks

In this paper, we have presented the dark energy cosmological model in
generalized BD theory. We have assumed an anisotropic metric in the form of
Bianchi type I space-time. Since our intention is to study some more details
on the late time cosmic speed up phenomena, we have used the unknown form of
energy, the concept of the dark energy as the unified dark fluid EoS. In order
to frame the cosmological model, we have considered a scale factor that allows
the deceleration parameter to vary with time showing a signature flipping
behaviour. Basing upon a tabulated observational $H(z)$ data we constrained
the parameters of the scale factor and that of the deceleration parameter.
Some of the salient features of the model are:

  * •

The observational constraints obtained on the model parameters are,
$\delta=36.67\pm 9.58$, $\beta=27.90\pm 7.09$, $\gamma=1.58\pm 0.17$. With
these constraints, the Hubble parameter shows an increasing trend with an
increase in redshift signifying an accelerating Universe at the present epoch.
The present value of the Hubble parameter as obtained from the present model
is $H_{0}=64.57\pm 16.67km~{}s^{-1}Mpc^{-1}$ which is in close conformity with
other recent estimates.

  * •

The deceleration parameter for the present model shows a cosmic transit
behaviour with an early deceleration to late time acceleration. The cosmic
transit occurs at a redshift $z_{t}=0.68$. At the present epoch, the
deceleration parameter is found to be $q_{0}=-0.32\pm 0.42$. These estimates
shows an accelerating expansion of the Universe at the present epoch.

  * •

A linear EoS is considered in the present work in the form of the unified dark
fluid that has a combined effect of dark energy and dark matter sector of the
Universe commonly known as dark degeneracy. The parameters of the UDF EoS are
considered on the basis of providing suitable mechanical stability in the
cosmic fluid given through the speed of sound within the fluid. The effective
equation of state parameter is observed to depend on the choice of the UDF
parameters and shows a gradual evolution from an early positive value to late
time negative values. From a simple analysis, we inferred that, the present
model may favour a phantom like phase.

  * •

Using the constrained model along with the concept of unified dark fluid,
different aspects of the generalized BD theory have been determined. The
evolution of the BD scalar field is shown to have a decreasing trend with
cosmic time. At the present time, it decreases to a vanishingly small value.

  * •

The BD parameter is assumed to vary with the scalar field and have a dynamical
trajectory. The evolution of the BD parameter is found to be more sensitive to
the choice of the anisotropic parameter. Usually, for constant deceleration
parameter, it is possible to separate out the evolving part of the BD
parameter from its non-evolving part and the anisotropic parameter can affect
only its non-evolving part. However in the present cosmic transit model, the
evolutionary behaviour of BD parameter is more involved and is controlled by
the anisotropic parameter as can be seen from FIG.5. The interesting aspect
is, while for low values of the anisotropic parameter i.e. $k=0.9$, it
decreases gradually with cosmic time, for higher values of the anisotropic
parameter, $k>1$, it shows an increasing trend. Interestingly, the trajectory
of the BD parameter for $k=1.3$ appears to be a mirror image of that for
$k=0.9$. At a given epoch, the BD parameter assumes a lower value for larger
cosmic anisotropy.

  * •

The evolutionary behaviour of the self interacting potential and that for
variation of the Newtonian gravitational constant are alike and are mostly
anisotropy dependent. The self interacting potential increase from a higher
negative value; however from different time zone for the representative value
of the anisotropic parameter. It is worthy to note here that at late times the
self interacting potential vanishes irrespective of the value of the
anisotropic parameter. $\frac{\dot{G}}{G}$ also shows the similar behaviour
and vanishes at late time of the evolution. At least at late times, the
variation in $G$ is substantial as compared to that for lower values of the
anisotropic parameter.

## Acknowledgement

SKT and BM thank IUCAA, Pune (India) for providing support during an academic
visit where a part of this work is carried out.

## Conflict of interest:

The authors have stated explicitly that there are no conflicts of interest in
connection with this article.

## References

  * [1] S. M. Carroll, et al., Phys. Rev. D 70, 043528, (2004).
  * [2] S. Nojiri, S. D.Odintsov, Phys. Lett. B 631, 1 (2005).
  * [3] S. Nojiri S and S. D. Odintsov, Int. J. Geom. Methods. Mod. Phys. 04,115 (2007).
  * [4] E. V. Linder EV, Phys. Rev. D 81, 127301 (2010).
  * [5] T. Harko, et al., Phys. Rev. D 84, 024020 (2011).
  * [6] C. Brans, R. H. Dicke, Phys. Rev. D 124,925 (1961).
  * [7] B. Bertotti, L. Iess, P. Tortora, Nature 425, 374 (2003).
  * [8] F. -Q. Xu, X. Chen, Phys. Rev. D. 82, 083003 (2010).
  * [9] Y. -C. Li, F. -Q. Xu, X. Chen, Phys. Rev. D. 88, 084053 ( 2013).
  * [10] J.-P. Uzan, Rev. Mod. Phys. 75, 403 (2003).
  * [11] J.-P., Uzan, Living Rev. Relativ. 14, 2 (2011).
  * [12] C. M. Will, Theory and Experiment in Gravitational Physics, Cambridge University Press (1993).
  * [13] J. Durk and T. Clifton, Class. Quantum Grav. 36, 185011 (2019). arXiv:1903.03043
  * [14] M. Eingorn, C. Kiefer, J. Cosmol. Astropart. Phys. 07, 036 (2015).
  * [15] Y. Y. Xu, X. Zhang, Eur. Phys. J. C 76, 588 (2016).
  * [16] Z. Zhai, et al, Astrophys. J 850, 183 (2017).
  * [17] C. Bruck, J. Mifsud, Phys. Rev. D 97, 023506 (2018).
  * [18] B. Mishra, P. P. Ray, R. Myrzakulov, Eur. Phys. J. C 79, 34 (2019).
  * [19] P. P. Ray, B. Mishra, S. K. Tripathy, Int. J. Mod. Phys. D 28, 1950093 (2019).
  * [20] P. Paul, R. Sengupta, Advance in High Energy Physics 2020, 5249839 (2020).
  * [21] O. Akarsu, et al., Phys. Rev. D 101, 063528 (2020).
  * [22] N. Banerjee, D. Pavon, Phys. Rev. D 63, 043504 (2001).
  * [23] H. Kim, Mon. Not. Roy. Astron. Soc. 364, 813 (2005).
  * [24] V. Acquaviva, L. Verde, J. Cosmol. Astropart. Phys. 0712, 001 (2007).
  * [25] S. Chattopadhyay, A. Pasqua, M. Khurshudyan, Eur. Phy. J. C 74, 3080 (2014).
  * [26] N. Sharif M, S. A. A. Shah, Mod. Phys. Lett A 34, 1950083 (2019).
  * [27] M. A. Zadeh, A. Sheykhi, Can. J. Phys. 98, 643 (2020).
  * [28] S. K. Tripathy, S. K. Pradhan, Z. Naik, D. Behera and B. Mishra, Phys. Dark Universe 30, 100722 (2020).
  * [29] P. Verma, Universe 7, 235 (2021).
  * [30] S. M. Du, Phys. Rev. D 99, 044057 (2019).
  * [31] P. G. Bergmann, Int. J. Theor. Phys. 1, 25 (1968).
  * [32] K. J. Nordtvedt, Astrophys. J. 161, 1059 (1970).
  * [33] R. V. Wagoner, Phys. Rev. D 1, 3209 (1970).
  * [34] J. Schwinger, Particles, Sources and Fields, Addison Wesley; (1970).
  * [35] B. M. Barker, Astrophys. J. 219, 5 (1978).
  * [36] F. Wu, X. Chen, Phys. Rev. D 82, 083003 (2010).
  * [37] Y. Li, F. Wu, X. Chen, Phys. Rev. D 88, 084053 (2013).
  * [38] H. Alavirad, A. Sheykhi, Phys. Lett. B 734, 148 (2014).
  * [39] S.K. Tripathy, D. Behera, B. Mishra. Unified dark fluid in Brans–Dicke theory. Eur. Phys. J. C 75, 149 (2015).
  * [40] O. Bertolami, P. J. Martins, Phys. Rev. D., 61, 064007 (2000).
  * [41] N. Banerjee, D. Pavon, Phys. Rev. D, 63, 043504 (2001).
  * [42] A. A. Sen, S. Sen and S. Sethi, Phys. Rev. D, 63, 107501 (2001).
  * [43] S. Sen and T. R. Seshadri, Int. J. Mod. Phys. D, 12, 445 (2003).
  * [44] B. K. Sahoo and L. P. Singh, Mod. Phys. Lett. A, 17, 2409 (2002).
  * [45] S. K. Pradhan, S. K. Tripathy, Z. Naik, D. Behera, M. Bhuyan, Foundations, 2, 128 (2022).
  * [46] N. Roy, N. Banerjee, Phys. Rev. D 95, 064048 (2017).
  * [47] A. Zucca, et al., Phys. Rev. D 101, 043518 (2020).
  * [48] J. Sola Peracaula, A. G. Valent, J. De C. Perez, C. M. Pulido, Class. Quantum Grav. 37, 245003(2020).
  * [49] A. Bonino, S.Camera, L. Fatibene, A. Orizzonte, Eur. Phys. J. Plus. 135, 951 (2020).
  * [50] G. Leon, A. Paliathanansis, L. V. Abab, Gen. Relativ. and Gravit. 52, 71 (2020).
  * [51] O. Farooq and B. Ratra, Astrophys J. Lett. 766, L7(2013).
  * [52] S. Capozziello, O. Farooq, O. Luongo and B. Ratra, Phys. Rev. D 90, 044016 (2014).
  * [53] A. G. Riess, et al., Astrophys. J. 659, 98 (2007).
  * [54] J. Lu, L. Xu and M. Liu, Phys. Lett. B 699, 246(2011).
  * [55] M. Moresco, et al., J. Cosmol. Astropart. Phys. 2016, 1605, 014(2016).
  * [56] Y. Yang and Y. Gong, J. Cosmol. Astropart. Phys. 2016, 1606,059(2016).arXiv:1912.07375
  * [57] J. F. Jesus, R. F. L. Holanda and S. H. Pereira, J. Cosmol. Astropart. Phys. 2018, 1805, 073(2018).
  * [58] E. Babichev, V. Dockuchaev, Y. Eroshenko, Phys. Rev. Lett. 93, 021102 ( 2004).
  * [59] L. Xu, Y. Wang, H. Noh, Phys. Rev.D 85, 043003 (2012).
  * [60] K. Liao, et al., Phys Lett. B 710, 17 (2012).
  * [61] S. Capozziello, et al., Phys. Rev. D 73,043512 (2006).
  * [62] S. K. Tripathy, S. Pandey, A. P. Sendha, D. Behera, Int. J. Geom. Method. Mod. Phys. 17, 2050056 (2020).
  * [63] A. G. Riess, et al.,Astron. Journal 116, 1009 (1998).
  * [64] S. Perlmutter, et al., Astrophys. J. 517, 565(1999).
  * [65] E. Abdalla, G. F. Abellan, A. Aboubrahim, A. Agnello, Ozgur Akarsu, Y. Akrami, G. Alestas, D. Aloni, L. Amendola, L. A. Anchordoqui, et al., Journal of High Energy Astrophysics 34, 49 (2022).
  * [66] A. G. Riess et al., Astrophys. J. 826(1), 56 (2016). arXiv:1604.01424
  * [67] A. G. Riess et al., Astrophys. J. 861(2), 126 (2018). arXiv:1804.10655
  * [68] N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak,et al. (Planck Collaboration), Astronomy & Astrophysics 641, A6 (2020).
  * [69] A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, G. S. Anand, L. Breuval, et al., The Astrophysical Journal Letters 934, L7 (2022),
  * [70] M. J. Reid, D. W. Pesce and A. G. Riess, Astrophys. J. 886(2), L27 (2019). arXiv:1908.05625
  * [71] K. C. Wong et al., (H0LiCOW Collaboration), arXiv:1907.04869
  * [72] G. Lambiase, S. Mohanty, A. Narang and P. Parashari, Eur. Phys. J. C 79, 141 (2019).
  * [73] S. Capozziello, G. Sarracino and A. D. A. M. Spallicci, arxiv:2302.13671.
  * [74] J. L. Bernal, L. Verde, A. G. Riess, J. Cosmol. Astropart. Phys. 1610, 019 (2016).
  * [75] E. Mortsell, S. Dhawan, J. Cosmol. Astropart. Phys 1809, 025 (2018).
  * [76] W. L. Freedman, B. F. Madore, D. Hatt, T. J. Hoyt, I. S. Jang, R. L. Beaton, C. R. Burns, M. G. Lee, A. J. Monson, J. R. Neeley, et al., The Astrophysical Journal 882, 34 (2019).
  * [77] E. Macaulay et al., Mon. Not. Royal Astron. Soc., 486, 2184 (2019).
  * [78] G. S. Farren, O. H. E. Philcox, and B. D. Sherwin, Phys. Rev. D 105, 063503 (2022).
  * [79] G. K. Goswami, A. K. Yadav, B. Mishra and S. K. Tripathy, Forschr. Phys. 2021, 2100007 (2021).
  * [80] S. K. Tripathy, S. K. Pradhan, B. Barik and Z. Naik, Phys.Sci. Forum 3, (2023).
  * [81] J. D. Benkenstein and A. Meisels, Phys. Rev. D 18, 4378 (1978).
  * [82] A. Serna and J. M. Alimi, Phys. Rev. D 53, 3074 (1996).
  * [83] R. W. Hellings et al., Phys. Rev. Lett. 51, 1609 (1983).
  * [84] P. Jofre, A. Reisenegger and R. Fernandez, Phys. Rev. Lett. 97, 131102 (2006).
  * [85] T. Damour and J. H. Taylor, Astrophys. J. 366, 501 (1991).


hence we also need to transform between $\dot{f}$ and $\partial
f(x,t)/\partial t$ (recall our notation $\dot{f}\equiv\partial f(r,t)/\partial
t$, $f^{\prime}\equiv\partial f(r,t)/\partial r$).

With all that, our first order accurate finite volume form of (89), which we
repeat here for reference:

$\dot{f}(t,r)-g^{\prime}(t,r)+h(t,r)+\delta(r-R)S(t,r)=0,$ (142)

can be written as

$\displaystyle\frac{d}{dt}f_{i_{0}}=$ $\displaystyle-$
$\displaystyle\frac{1}{4}[g^{\prime}+f^{\prime}\cdot
r_{t}]_{i_{0}+1}+\frac{1}{4}[g^{\prime}+f^{\prime}\cdot r_{t}]_{i_{0}-1}$
(143) $\displaystyle-$ $\displaystyle\frac{1}{4}[f\cdot r_{tr}+(f\cdot
r_{t}+g)x^{\prime\prime}\cdot r_{x}]_{i_{0}+1}$ $\displaystyle+$
$\displaystyle\frac{1}{4}[f\cdot r_{tr}+(f\cdot r_{t}+g)x^{\prime\prime}\cdot
r_{x}]_{i_{0}-1}$ $\displaystyle-$
$\displaystyle\frac{1}{2}(h_{L}+h_{R})-\frac{1}{2}(f_{L}+f_{R})\cdot r_{tr}$
$\displaystyle-$ $\displaystyle\frac{1}{2}[(f_{L}\cdot r_{t,i_{0}}+g_{L})\cdot
x^{\prime\prime}_{L}$ $\displaystyle\ \ \ +(f_{R}\cdot r_{t,i_{0}}+g_{R})\cdot
x^{\prime\prime}_{R}]\cdot r_{x,i0}$ $\displaystyle+$
$\displaystyle\frac{3}{4\Delta x}\bigg{[}(g_{R}-g_{L}+(f_{R}-f_{L})\cdot
r_{t})\cdot x^{\prime}_{i_{0}}$ $\displaystyle\ \ \ \ \ \ \ +[(g+f\cdot
r_{t})\cdot x^{\prime}]_{i_{0}+1}$ $\displaystyle\ \ \ \ \ \ \ -[(g+f\cdot
r_{t})\cdot x^{\prime}]_{i_{0}-1}$ $\displaystyle\ \ \ \ \ \ \ -2S_{i0}\cdot
x^{\prime}_{i0}\bigg{]},$

where $r_{t}\equiv\partial r(x,t)/\partial t$, $r_{x}\equiv\partial
r(x,t)/\partial x$, $r_{tr}\equiv[\partial^{2}r(x,t)/(\partial t\partial
x)]\cdot\partial x(r,t)/\partial r$. This elevates to a second order accurate
scheme when $S_{i0}\rightarrow 0$, and all “L” values equal their “R” value
neighbours.

##  Appendix E Rate of Energy Loss from AdS interior

In light-like coordinates, the line element for AdS is:

$ds^{2}=\cos^{-2}(\ell r/\sqrt{3})(-dt^{2}+dr^{2})+3/\ell^{2}\tan^{2}(\ell
r/\sqrt{3})d\Omega^{2}$ (144)

which we distinguish from the “standard coordinates” ($r,R$) which give

$ds^{2}=-(1+R^{2}\ell^{2}/3)dt^{2}+(1+R^{2}\ell^{2}/3)^{-1}dr^{2}+R^{2}d\Omega^{2}$
(145)

General solutions to scalar field propagation in AdS in light-like coordinates
can be expressed as a superposition of modes given by

$\displaystyle\Phi_{j}(t,x)$ $\displaystyle=$ $\displaystyle
d_{j}\cos(\omega_{j}\ell t/\sqrt{3})\cos^{3}(\ell r/\sqrt{3})$ (146)
$\displaystyle\,{}_{2}F_{1}(-j,3+j,3/2,\sin^{2}(\ell r/\sqrt{3}))\,;$

with $\omega_{j}^{2}=(3+2j)^{2}\ell^{2}/3$ and $d_{j}=4\sqrt{(j+1)(j+2)/\pi}$
($j\in 0,1,..$). Such modes are orthonormal and complete as
$\ell\rightarrow\infty$. For our regime of interest then we can use this
(quasi) basis, for large $\ell$, to describe the exterior pulse once it enters
the AdS region. In particular, we are interested in the reduction of (the
interior) energy ($E$) within the AdS. To this end, we can make use of the
analysis presented in Calabrese_2003  to reach the intuitive result of

$E_{,t}=4\pi(V_{+}^{2}-V_{-}^{2})R_{o}^{2}$ (147)

with $V_{+,-}=\pm\alpha u^{a}\partial_{a}\Phi+\alpha D$ the incoming
(outgoing) modes of the solution at the outer boundary $r=R_{o}$; $u^{a}$ is
the unit timelike normal at $R_{o}$, $D=\gamma^{ij}n_{i}\partial_{j}\Phi$, and
$\alpha=1/\cos(\ell r/\sqrt{3})$. At such a boundary, the AdS region looses
energy through $V_{-}$ but does not gain energy through $V_{+}$ as little
“comes back” from the exterior region. We can thus take it to zero, so energy
is lost at a rate $E_{,t}=-4\pi V_{-}^{2}R_{o}^{2}$. We can then replace
$\Phi$ in terms of its normal modes; it is clear higher modes will reduce the
energy more effectively than the lowest one $\omega_{0}$. Said differently,
energy supported by higher frequency modes leaks out at a faster rate out of
the AdS region. The long-term behavior is given by the lowest mode, and the
energy loss within 1 period of oscillation is, to leading order in $1/\ell$,

$\Delta E\propto-A_{0}^{2}\pi^{2}\frac{1}{m^{2}\ell^{3}}$ (148)

(with $A_{0}$ the amplitude of the mode). We can then use a “quasi-adiabatic”
argument to say an amount of energy

$\Delta E\propto-A_{0}^{2}\pi^{2}\frac{1}{m^{2}\ell^{2}}\Delta T$ (149)

is lost in the interior region over the period $\Delta
T=(2\pi)/(\ell\sqrt{3})$. Now, the energy within the AdS region is $\propto
A_{0}^{2}/\ell$, so $dE/dt\propto 2A_{0}/\ell dA_{0}/dt$ and we can use
$\Delta E/\Delta T$ to approximate the left hand side to arrive at,

$\frac{dA_{0}}{dt}\propto-\frac{\pi A_{0}}{m^{2}\ell^{2}}\ell$ (150)

and so, $A_{0}(t)\approx\exp(-pt\ell)$ with

$p\approx-\frac{\pi}{m^{2}\ell^{2}}.$ (151)

Thus the energy decays as $E(t)\propto\exp(-2pt\ell)/\ell$, and so $\log
E\approx-\frac{2\pi}{m^{2}\ell}\,t$. Notice the above expression is with
respect to time measured at the origin of AdS, which is related to the
asymptotic time $t_{a}$ by $t\approx t_{a}/(m\ell)$. Consequently, $\log
E\approx-\frac{2\pi}{m^{3}\ell^{2}}\,t_{a}$. This behavior is consistent with
the results shown in figures 6 and 7. For reference, we can now explore the
associated timescale for this energy to leak out of AdS and become an
“observable signature” in the asymptotically flat (AF) region. The timescale
is given by $\tau_{D}\simeq m^{3}\ell^{2}$; taking $\ell=1/L$ with $L$ a
lengthscale and assuming $m=10^{q}M_{\odot}$, one has

$\tau_{D}=10^{3q}(\mbox{m}/L)^{2}\mbox{s}$ (152)

For instance, for $L=L_{\rm{Planck}}\simeq 10^{-35}\mbox{m}$ and $q=1$,
$\tau_{D}=10^{53}\mbox{s}\approx 10^{56}t_{\rm{Hubble}}$. Requiring instead
that $\tau_{D}\simeq t_{\rm{Hubble}}$ or $\tau_{D}\simeq 1\mbox{yr}$, $L$
should be $\simeq 10^{-7},10^{-4}$m respectively.

As a last remark, we can employ a similar argument to explore what takes place
at early times. When a pulse with a given frequency $\omega_{\rm{AF}}$ in the
AF region, begins to fall in the AdS region, its frequency would be
blueshifted to $\omega_{i}\simeq\omega_{\rm{AF}}m\ell$, and would be
supported, in terms of the AdS modes, by a spectra of (almost) normal
frequencies given by $\omega_{j}=\pm(3+2j)\ell/\sqrt{3}$. Thus, the pulse
would be described by the same modes in a way that is largely insensitive to
the scale determined by $\ell$. For a mode with index $j$ the above timescale
estimate results in

$\tau_{D}=10^{3q}j^{-2}(\mbox{m}/L)^{2}\mbox{s}$ (153)

indicating the AdS could help potentially render microscopic $j$ scales into
significantly longer ones for higher values of $j$. Of course, this depends on
the content of the pulse in the AF region. Rough estimates however imply not
very high $j$’s are encountered with significant strength for the relatively
simple frequency content of waves driven by a quasi-circular merger.

## References

  * (1) O. Lunin and S. D. Mathur, “AdS / CFT duality and the black hole information paradox,” Nucl. Phys. B, vol. 623, pp. 342–394, 2002.
  * (2) P. O. Mazur and E. Mottola, “Gravitational condensate stars: An alternative to black holes,” 9 2001.
  * (3) U. H. Danielsson, G. Dibitetto, and S. Giri, “Black holes as bubbles of AdS,” JHEP, vol. 10, p. 171, 2017.
  * (4) V. Cardoso and P. Pani, “Testing the nature of dark compact objects: a status report,” Living Rev. Rel., vol. 22, no. 1, p. 4, 2019.
  * (5) B. Abbott, R. Abbott, T. Abbott, S. Abraham, F. Acernese, K. Ackley, C. Adams, R. Adhikari, V. Adya, C. Affeldt, and et al., “Gwtc-1: A gravitational-wave transient catalog of compact binary mergers observed by ligo and virgo during the first and second observing runs,” Physical Review X, vol. 9, Sep 2019\.
  * (6) T. L. S. Collaboration and the Virgo Collaboration, “Gwtc-2.1: Deep extended catalog of compact binary coalescences observed by ligo and virgo during the first half of the third observing run,” 2021.
  * (7) E. H. T. Collaboration, “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. Letters, vol. 875, p. L1, Apr. 2019.
  * (8) U. Danielsson and S. Giri, “Observational signatures from horizonless black shells imitating rotating black holes,” JHEP, vol. 07, p. 070, 2018.
  * (9) W. Israel, “Singular hypersurfaces and thin shells in general relativity,” Nuovo Cimento B Serie, vol. 44, pp. 1–14, July 1966.
  * (10) B. P. Abbott et al., “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett., vol. 116, no. 6, p. 061102, 2016\.
  * (11) R. Abbott et al., “GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run,” Phys. Rev. X, vol. 11, p. 021053, 2021.
  * (12) K. Akiyama et al., “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. Lett., vol. 875, p. L1, 2019.
  * (13) F. S. Bemfica, M. M. Disconzi, and J. Noronha, “General-relativistic viscous fluid dynamics,” 2020.
  * (14) F. S. Bemfica, M. M. Disconzi, and J. Noronha, “Causality and existence of solutions of relativistic viscous fluid dynamics with gravity,” Physical Review D, vol. 98, Nov 2018.
  * (15) P. Kovtun, “First-order relativistic hydrodynamics is stable,” Journal of High Energy Physics, vol. 2019, Oct 2019.
  * (16) A. Pandya and F. Pretorius, “Numerical exploration of first-order relativistic hydrodynamics,” Phys. Rev. D, vol. 104, no. 2, p. 023015, 2021.
  * (17) G. Calabrese, L. Lehner, O. Reula, O. Sarbach, and M. Tiglio, “Summation by parts and dissipation for domains with excised regions,” Class. Quant. Grav., vol. 21, pp. 5735–5758, 2004.
  * (18) S. Deser and O. Levin, “Accelerated detectors and temperature in (anti)-de Sitter spaces,” Class. Quant. Grav., vol. 14, pp. L163–L168, 1997.
  * (19) H. Kreiss and J. Oliger, Methods for the Approximate Solution of Time Dependent Problems, vol. 10 of Global Atmospheric Research Programme (GARP): GARP Publication Series.  GARP Publication, 1973.
  * (20) M. A. Scheel, H. P. Pfeiffer, L. Lindblom, L. E. Kidder, O. Rinne, and S. A. Teukolsky, “Solving Einstein’s equations with dual coordinate frames,” Phys. Rev. D, vol. 74, p. 104006, 2006.
  * (21) D. Hilditch, “Dual Foliation Formulations of General Relativity,” 9 2015.
  * (22) C. W. Misner and D. H. Sharp, “Relativistic equations for adiabatic, spherically symmetric gravitational collapse,” Phys. Rev., vol. 136, pp. B571–B576, 1964.
  * (23) G. Calabrese, L. Lehner, D. Neilsen, J. Pullin, O. Reula, O. Sarbach, and M. Tiglio, “Novel finite-differencing techniques for numerical relativity: application to black-hole excision,” Classical and Quantum Gravity, vol. 20, p. L245–L251, Sep 2003.
  * (24) J. Westerweck, Y. Sherf, C. D. Capano, and R. Brustein, “Sub-atomic constraints on the Kerr geometry of GW150914,” 8 2021.
  * (25) J. Keir, “Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars,” Class. Quant. Grav., vol. 33, no. 13, p. 135009, 2016.
  * (26) V. Cardoso, L. C. B. Crispino, C. F. B. Macedo, H. Okawa, and P. Pani, “Light rings as observational evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects,” Phys. Rev. D, vol. 90, no. 4, p. 044069, 2014.
  * (27) P. V. P. Cunha, E. Berti, and C. A. R. Herdeiro, “Light-Ring Stability for Ultracompact Objects,” Phys. Rev. Lett., vol. 119, no. 25, p. 251102, 2017\.
  * (28) P. Bizon and A. Rostworowski, “On weakly turbulent instability of anti-de Sitter space,” Phys. Rev. Lett., vol. 107, p. 031102, 2011.
  * (29) J. L. Friedman, “Ergosphere instability,” Communications in Mathematical Physics, vol. 63, pp. 243–255, Oct. 1978.
  * (30) G. Moschidis, “A Proof of Friedman’s Ergosphere Instability for Scalar Waves,” Commun. Math. Phys., vol. 358, no. 2, pp. 437–520, 2018.
  * (31) J. Keir, “Evanescent ergosurface instability,” Anal. Part. Diff. Eq., vol. 13, no. 6, pp. 1833–1896, 2020.
  * (32) R. Brito, V. Cardoso, and P. Pani, “Superradiance: New Frontiers in Black Hole Physics,” Lect. Notes Phys., vol. 906, pp. pp.1–237, 2015.
  * (33) S. Chandrasekhar, “The Effect of Gravitational Radiation on the Secular Stability of the Maclaurin Spheroid,” Astrophys. J. , vol. 161, p. 561, Aug. 1970\.
  * (34) J. L. Friedman and B. F. Schutz, “Secular instability of rotating Newtonian stars,” Astrophys. J., vol. 222, p. 281, 1978.