Bianchi type-V Transitioning model in Brans-Dicke theory with Observational
Constraint
Vinod Kumar Bhardwaj1, Archana Dixit2, Anirudh Pradhan3

1,2Department of Mathematics, Institute of Applied Sciences and Humanities,
GLA University, Mathura-281 406, Uttar Pradesh, India

3Centre for Cosmology, Astrophysics and Space Science (CCASS), GLA University,
Mathura-281 406, Uttar Pradesh, India

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###### Abstract

In this paper, we have examined the viability of the Bianchi type-V universe
in Brans-Dicke (BD) theory of gravitation. We have discussed the interacting
and non-interacting scenarios between dark matter (DM) and dark energy (DE) of
the derived universe within the framework of BD theory. CCA technique has been
applied to constrain the model parameters using 46 values of observational
Hubble data (OHD), Pantheon data (the latest compilation of SNIa with 40
binned in the redshift range $0.014\leq z\leq 1.62)$ and their combined
datasets. We establish an exact solution of the field equations to derive the
dynamics of the derived universe and the obtained results are found to agree
with the observations We also noted a distinctive change in the sign of the
deceleration parameter from positive to negative, as well as the presence of a
transition red-shift exists. Using various observational data points, the
evolution trajectories for $(r~{}-s)$ diagnostic planes are shown to
understand the geometrical behavior of the Bianchi-V model. Some physical
properties of the universe are also discussed. It’s also worth noting that the
conclusions of the cosmological parameter are consistent with modern
observational data.

Keywords : LRS Bianchi-V, Brans-Dicke theory, $\Lambda$CDM Model,
Statefinders.

PACS: 98.80.-k

##  1 Introduction

According to recent studies of type Ia supernovae [1, 2] our current universe
is undergoing an accelerated phase of expansion proceeded by a period of
deceleration. To examine the current phase of acceleration, a new form of
substance with negative pressure, called dark energy, has been suggested. In
literature several models have been examined, to predict cosmic acceleration
by assuming dark energy with repulsive gravity as the major content of the
cosmos. According to the Planck observations, Universe’s total mass-energy
budget is composed of 68.3 % DE, 26.8 % dark matter and 4.9 % ordinary matter.
This exotic form of energy is known as dark energy which is to be less
effective in early times but dominates at the present epoch. The attracting
force of DM is involved in the creation of structure and clustering of
galaxies.

Although several observational experiments support DE as a general
clarification for the expansion mystery, but there are many questions which
remain unanswered. The cosmological constant ($\Lambda$) has a repulsive
nature, it is a usual candidate for DE. It posseses a low magnitutde in
comparison with prediction of particle physics. Apart from dark energy models,
a variety of gravity theories such as f (R) gravity [3, 4],$f(T)$ gravity [5],
and $f(G)$ gravity [6, 7], $f(Q)$ gravity [8, 7b], $f(Q,T)$ [10], Einstein-
Gauss-Bonnet theory [11, 12, 13, 14, 15] were discussed. In the same
direction, $f(R,T)$ theory is the another perspective of modified version of
theory of gravity, which has been proposed by Harko et al.[16]. Similaly, many
theories such as “scalartensor theories, scalar field theories, bimetric
theories and vector-tensor theories [17, 18], Weyl theory [19], Lyra theory
[20] Ylmaz theory [21] and Brans-Dicke theory [22, 23]” are some examples of
these alternative gravitation theories [24]. For the previous coupled of
decades, cosmologists have been interested in alternative theories of gravity,
particularly ”scalar-tensor” theories. Among all these alternative theories,
the “Brans-Dicke theory (BDT)” is the one of the most successful alternate
theory. Alonso et al. [25] have shown that the scalar tensor theory of Brans-
Dicke type provides a relativistic generalization of Newtonian gravity in 2+1
dimensions. The theory is metric and test particles follow the space-time
geodesics.

BD theory, on the other hand, is often considered as the prototype of a wide
category of gravitational theories known as scalar-tensor theories. These
theories are derived through a non-minimal coupling between the curvature
scalar and a scalar field. These hypotheses have gotten a lot of interest in
the cosmology community because they can accurately represent the early
expansion of the universe [26]. The BD hypothesis predicts an extended period
of inflation to address the universe’s graceful exit dilemma, as outlined by
Mathiazhagan and Johri [27], La and Steinhardt [28]. Furthermore, with
suitable parameter values, the theory can successfully construct late-time
accelerated expansion without the contribution of DE ([29]) Several authors
have been explored BD cosmology in different prospectives. The work of Singh
and Rai [30] provides a detailed discussion of Brans-Dick cosmological models.
Remarkable studies of BD theory with the application of a non-vanishing
cosmological constant can be seen in [31, 32, 33, 34, 35, 36, 37]. In BD
theory, authors [38, 39, 40] studied a model of accelerating universe with
changeable deceleration parameter. In 1984 Brans-Dike cosmological model was
investigated by taking $\Lambda$. The same idea was further extended
inhomogeneous and anisotropic Bianchi type I spacetime [41, 42]. Akarsu, et
al. [43] studied the anisotropic $\Lambda$CDM model with Brans–Dicke gravity.
Recently, exact solutions of accelerating cosmological models in modified
Brans-Dicke theory are proposed [44, 45].

In cosmology, Bianchi-type space-times play a very important role in
comprehending and describing the early phases of the universe’s evolution. The
study of Bianchi types II, VIII, and IX universe is particularly relevant
since they correlate to well-known solutions such as the FRW universe with
positive curvature, the de Sitter universe, the Taub-NUT solutions, and so on.
The Bianchi type-V frame of reference is very important for developing the
models for measuring the expansion of the Universe in its early stages [46,
47]. Several authors [48, 49, 50, 51, 52, 53] have been interested in “Bianchi
type-V space-time type over the last decade. The dynamical behavior of the
Bianchi-V model is significantly more general than the simple FLRW model. The
study of Bianchi type-V cosmological models creates more interest as these
models contain isotropic special cases and permit arbitrarily small
anisotropic levels at certain stages [54]. However, we know that, in recent
years, Bianchi cosmological models have become extremely important in
observational cosmology, as evidenced by the WMAP observational data. It has
validated an addition to the standard “$\Lambda CDM$” model that resembles the
Bianchi morphology. The WMAP data analysis supports the fact that the universe
has a preferred direction and it should reach a slightly anisotropic geometry.
Therefore, in view of the background anisotropy, the models with anisotropic
backgrounds are more suitable to describe early stages of the universe [55,
56].

Cosmology is now one of the fastest-developing branches of science,
particularly in terms of observations. In the last decade, cosmological data
from various sources has been rapidly updated, not only in terms of quantity
but also in terms of quality [57]. The latest “CMB anisotropies” results by
“Planck Collaboration” provide the precise observations on temperature and
polarization of the photons from the last scattering surface near the redshift
about $z=1090$ [58]. Another observable, which is worthy of mentioning, is the
“redshift-space distortion (RSD)”. It has been measured more and more
precisely in the past few years. This observable is a key signature to
disclose the large-scale structure. In the present study, we proposed
transitioning model of the universe Bianchi-V space-time in the framework of
BD theory. To achieve the explicit solution for the model, we assumed the
scalar field $\phi=\phi_{0}(t)[(k_{1}e^{n\beta t}-1)^{1/n}]^{\eta}$; where
$\phi_{0}$, $k_{1}$, $n$, $\beta$ and $\eta$ are constants. In the present
study, the recent OHD and Pantheon data sets are used to decide the model
parameters“ $n$ and $\beta$ [59]. We investigate a transitioning model of the
universe for interacting and non-interacting scenarios within the framework of
Brans-Dicke theory.

In our derived model, the best fit values of parameters are determined by
using $\chi^{2}$ statistic. Two observational data sets, i) A data set of 46
OHD and ii) Pantheon data (the latest compilation of SNIa with 40 binned in
the redshift range $0.014\leq z\leq 1.62)$ are considered to discuss the
dynamics of the derived universe. This paper is structured as follows: In
section 2, we introduce the basic field equation of the Bianchi- V universe.
In Section 3, we discussed the observational constraints of the model
parameter. In Section 4, we have discussed the non-interacting and interacting
models. The physical behavior of the model is analyzed in Section 5. In
Section 6, we apply the Statefinder diagnostic. Conclusions are mentioned in
section 7.

##  2 Field Equations of Bianchi type-V model

The action for Brans-Dicke theory is given by

$S=\frac{1}{16\pi}\int{\sqrt{-g}\left(\phi
R-\omega\frac{\phi_{,i}\phi^{,i}}{\phi}\right)d^{4}x}+\frac{1}{16\pi}\int{\sqrt{-g}L_{m}d^{4}x},$
(1)

where $L_{m}$ stands for Lagrangian matter field, $\phi$ denotes the Brans-
Dicke scalar field and $\omega$ is the Brans-Dicke coupling constant.

The field equations in Brans-Dicke theory [60, 61] by assuming dark matter
(DM) and dark energy (DE) are read as:

$R_{ij}-\frac{1}{2}Rg_{ij}=-\frac{8\pi}{\phi}\left(T^{m}_{ij}+T^{de}_{ij}\right)-\frac{\omega}{\phi^{2}}\left(\phi_{i}\phi_{j}-\frac{1}{2}g_{ij}\phi_{k}\phi^{k}\right)-\frac{1}{\phi}\left(\phi_{ij}-g_{ij}\Box\phi\right),$
(2)

and

$\Box\phi=\frac{8\pi}{(3+2\omega)}\left(T^{m}+T^{de}\right),$ (3)

where $T^{m}_{ij}=dia\left[-1,\omega_{m},\omega_{m},\omega_{m}\right]\rho_{m}$
and
$T^{de}_{ij}=dia\left[-1,\omega_{de},\omega_{de},\omega_{de}\right]\rho_{de}$
are respectively the energy momentum tensors for DM and DE.

The energy conservation equation is read as

$\left(T^{m}_{ij}+T^{de}_{ij}\right)_{;j}=0,$ (4)

which leads to

$\dot{\rho_{de}}+3H(1+\rho_{m}+\omega_{de})\rho_{de}+\dot{\rho_{m}}=0$ (5)

We consider the Bianchi type-V metric of the form

$ds^{2}=dt^{2}-A(t)^{2}dx^{2}-e^{-2\alpha
x}\left[B(t)^{2}dy^{2}+C(t)^{2}dz^{2}\right],$ (6)

where $\alpha$ is a constant and the functions $A(t)$, $B(t)$ and $C(t)$ are
the three anisotropic directions of expansion in normal three dimensional
space. Those three functions are equal in FRW models due to the radial
symmetry and so we have only one function $a(t)$ there. The average scale
factor $a$, the spatial volume $V$ and the average Hubble’s parameter $H$ are
defined as $a=(ABC)^{1/3}$, $V=a^{3}=ABC$, and
$H=\frac{1}{3}\left(\frac{{\dot{A}}}{A}+\frac{{\dot{B}}}{B}+\frac{{\dot{C}}}{C}\right)$
respectively.

In a comoving coordinate system, the field equations given by Eqs. (2) and
(3), for Bianchi type-V spacetime mentioned in Eq. (6) with energy-momentum
tensors defined previously, are read as

$\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}+\frac{\dot{B}\dot{C}}{BC}-\frac{\alpha^{2}}{A^{2}}+\frac{\omega}{2}\frac{\dot{\phi}^{2}}{\phi^{2}}+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)+\frac{\ddot{\phi}}{\phi}=-\phi^{-1}p_{de}$
(7)
$\frac{\ddot{C}}{C}+\frac{\ddot{A}}{A}+\frac{\dot{C}\dot{A}}{CA}-\frac{\alpha^{2}}{A^{2}}+\frac{\omega}{2}\frac{\dot{\phi}^{2}}{\phi^{2}}+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{C}}{C}+\frac{\dot{A}}{A}\right)+\frac{\ddot{\phi}}{\phi}=-\phi^{-1}p_{de}$
(8)
$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}-\frac{\alpha^{2}}{A^{2}}+\frac{\omega}{2}\frac{\dot{\phi}^{2}}{\phi^{2}}+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}\right)+\frac{\ddot{\phi}}{\phi}=-\phi^{-1}p_{de}$
(9)
$\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}\dot{C}}{BC}+\frac{\dot{C}\dot{A}}{CA}-\frac{3\alpha^{2}}{A^{2}}-\frac{\omega}{2}\frac{\dot{\phi}^{2}}{\phi^{2}}+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)=\phi^{-1}\left(\rho_{m}+\rho_{de}\right)$
(10) $2\frac{\dot{A}}{A}-\frac{\dot{B}}{B}-\frac{\dot{C}}{C}=0$ (11)
$\frac{\ddot{\phi}}{\phi}+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)=\frac{\rho_{de}(1-3\omega_{de})+\rho_{m}}{(3+2\omega)}$
(12)

From Eqs. (7)- (9), we have

$\frac{\ddot{A}}{A}+\frac{\dot{A}\dot{C}}{AC}-\left(\frac{\ddot{B}}{B}+\frac{\dot{B}\dot{C}}{BC}\right)+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{A}}{A}-\frac{\dot{B}}{B}\right)=0$
(13)

$\frac{\ddot{B}}{B}-\frac{\ddot{C}}{C}+\frac{\dot{A}\dot{B}}{AB}-\frac{\dot{A}\dot{C}}{AC}+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{B}}{B}-\frac{\dot{C}}{C}\right)=0$
(14)

$\frac{\ddot{C}}{C}-\frac{\ddot{A}}{A}+\frac{\dot{B}\dot{C}}{BC}-\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{\phi}}{\phi}\left(\frac{\dot{C}}{C}-\frac{\dot{A}}{A}\right)=0$
(15)

Integrating Eq. (11) and omitting constant of integration, we get

$\displaystyle BC=A^{2}$ $\displaystyle\Rightarrow C=AD,\ \&\ B=A/D,$ (16)

here $D=D(t)$ denotes the measures of anisotropy in the Bianchi-V model.

From Eqs. (14) and (2), we obtain

$\left(\frac{\ddot{D}}{D}-\frac{\dot{D}^{2}}{D^{2}}\right)+\frac{\dot{D}}{D}\left(3\frac{\dot{A}}{A}+\frac{\phi}{\phi}\right)=0,$
(17)

which on integration reduces to

$D=exp\left(\int c_{1}A^{-3}\phi^{-1}dt\right)$ (18)

where $c_{1}$ is the integration constant.

Usng Eq. (2), we can determine the scale factor as $a^{3}=A^{3}\Rightarrow
a=A$.

Thus, we get the following equations finally,

$2\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}-\frac{\alpha^{2}}{a^{2}}+\frac{c_{1}^{2}}{a^{6}\phi^{2}}+\frac{\omega}{2}\frac{\dot{\phi}^{2}}{\phi^{2}}+2\frac{\dot{\phi}}{\phi}\frac{\dot{a}}{a}+\frac{\ddot{\phi}}{\phi}=-\phi^{-1}p_{de},$
(19)
$3\frac{\dot{a}^{2}}{a^{2}}-3\frac{\alpha^{2}}{a^{2}}-\frac{c_{1}^{2}}{a^{6}\phi^{2}}-\frac{\omega}{2}\frac{\dot{\phi}^{2}}{\phi^{2}}+3\frac{\dot{\phi}}{\phi}\frac{\dot{a}}{a}=\phi^{-1}\left(\rho_{m}+\rho_{de}\right),$
(20)
$\frac{\ddot{\phi}}{\phi}+3\frac{\dot{\phi}}{\phi}\frac{\dot{a}}{a}=\frac{\rho_{de}(1-3\omega_{de})+\rho_{m}}{(3+2\omega)}.$
(21)

On solving these equation we get dyanamic features of the proposed model.

Authors[62, 63] have considered the funtional relation of Brans-Dicke scalar
field $\phi$ and scale factor $a$ as

$\phi=\phi_{0}a^{\eta},$ (22)

where $\eta$ and $\phi_{0}$ are constants. Choice of $\phi=\phi_{0}a^{\eta}$
leads to be consistent with results [64].

The observations of type Ia supernovae [65, 66, 67], “CMB anisotropies” [68],
and the “Planck Collaborations” [69] have verified that the universe is
currently expanding speedly, which was decelerating in past. As a result, the
universe must have a “signature flipping” from past deceleration to current
acceleration [70, 71]. To determine the explicit solutions of transitioning
universe, we have considered a special parameterization of the Hubble
parameter [72, 73, 74] $H=\frac{\dot{a}}{a}=\beta(1+a^{-n})$, where $\beta>0$
, $n>1$ are constants. On integration of this parametrization, we get an
explicit form of the scale factor as $a(t)=\left(k_{1}e^{n\beta
t}-1\right)^{1/n}$; where $k_{1}>0$ is the constant of integration [59]. The
present explicit form of scale factor is an exponential function containing
two model parameters $n$ and $\beta$ which describe the dynamics of the
universe. As $t\to 0$, we can have $a(0)=(k_{1}-1)^{1/n}$, which provides a
non-zero initial value of scale factor for $k_{1}\neq 1$ (or a cold initiation
of universe with finite volume).
Now, the Brans–Dicke scalar field $\phi$ reads as

$\phi=\phi_{0}\left[\left(k_{1}e^{n\beta
t}-1\right)^{\frac{1}{n}}\right]^{\eta}$ (23)

The deceleration parameter (DP) $q$ is obtained as

$q=-\frac{a\ddot{a}}{\dot{a}^{2}}=-\left(1+\frac{\dot{H}}{H^{2}}\right)=-1+\frac{n}{k_{1}e^{n\beta
t}}$ (24)

Eq. (24) shows that the DP is time-dependent, which can take both positive and
negative values representing early decelerating phase and later accelerating
phase. From Eq. (24) we can see that $q=-1+\frac{n}{k_{1}}$ as $t\to 0$, which
is constant and positive for $n>1$ and $k_{1}<n$ and for $k_{1}>n$, DP
possesses negative value. This indicates that DP has a signature flipping
nature from positive to negative era with the evaluation.
By using the relation $(1+z)=\frac{a_{0}}{a}$, we express the cosmological
parameters in terms of redshift. Thus, by applying the above transformation,
$t$ in terms of $z$ can be found as
$\frac{1}{n\beta}log\left[\frac{1+(1+z)^{n}}{k_{1}(1+z)^{n}}\right]$. To
describe the dynamics of the Universe, the Hubble parameter $H$ in terms of
redshift can be read as

$H(z)=\beta[(1+z)^{n}+1].$ (25)

or

$H(z)=\frac{H_{0}}{2}(1+(1+z)^{n}).$ (26)

In order to describe the dynamics properties of the model and the bahaviour of
physical parameters, we discuss two observational data sets in following
section.

##  3 Observational constraints of the model

###  3.1 Observational Hubble data set (OHD)

In this section, we have estimate the model parameters $n$ and $H_{0}$ using
recent 46 points of the $H(z)$ data set (OHD) in the red-shift range $0\leq
z\leq 2.36$ and their corresponding standard deviation $\sigma_{i}$. These 46
OHD points are summarized in the Table-I [61]-[83]. The best fit value of
model parameters $n$ and $H_{0}$ obtained using $\chi^{2}$ statistic, which is
equivalent to the maximum likelihood analysis given as

$\chi^{2}_{OHD}\left(n,H_{0}\right)=\sum_{i=1}^{46}{\frac{\left(H_{th}(n,H_{0},z_{i})-H_{ob}(z_{i})\right)^{2}}{\sigma(i)^{2}}},$
(27)

where $`H_{th}^{\prime}$ and $`H_{obs}^{\prime}$ denote respectively the
theoretical and observed value of Hubble parameter $H$. By analysis, we found
the values of parameters $n=1.457\pm 0.042$ and$H_{0}=68.46\pm 1.97$
$km/s/Mpc$, which are comparable with the current Hubble parameter from Planck
2014 results [85].

Table 1: The OHD data set of Hubble parameter $H(z)$ $S.No$ | $Z$ | $H(Obs)$ | $\sigma_{i}$ | References | $S.No$ | $Z$ | $H(Obs)$ | $\sigma_{i}$ | References
---|---|---|---|---|---|---|---|---|---
1 | 0 | 67.77 | 1.30 | [75] | 24 | 0.4783 | 80.9 | 9 | [69]
2 | 0.07 | 69 | 19.6 | [76] | 25 | 0.48 | 97 | 60 | [64]
3 | 0.09 | 69 | 12 | [77] | 26 | 0.51 | 90.4 | 1.9 | [68]
4 | 0.01 | 69 | 12 | [64] | 27 | 0.57 | 96.8 | 3.4 | [79]
5 | 0.12 | 68.6 | 26.2 | [76] | 28 | 0.593 | 104 | 13 | [66]
6 | 0.17 | 83 | 8 | [64] | 29 | 0.60 | 87.9 | 6.1 | [70]
7 | 0.179 | 75 | 4 | [66] | 30 | 0.61 | 97.3 | 2.1 | [68]
8 | 0.1993 | 75 | 5 | [66] | 31 | 0.68 | 92 | 8 | [66]
9 | 0.2 | 72.9 | 29.6 | [76] | 32 | 0.73 | 97.3 | 7 | [61]
10 | 0.24 | 79.7 | 2.7 | [78] | 33 | 0.781 | 105 | 12 | [66]
11 | 0.27 | 77 | 14 | [64] | 34 | 0.875 | 125 | 17 | [66]
12 | 0.28 | 88.8 | 36.6 | [76] | 35 | 0.88 | 90 | 40 | [64]
13 | 0.35 | 82.7 | 8.4 | [67] | 36 | 0.9 | 117 | 23 | [64]
14 | 0.352 | 83 | 14 | [66] | 37 | 1.037 | 154 | 20 | [66]
15 | 0.38 | 81.5 | 1.9 | [68] | 38 | 1.3 | 168 | 17 | [64]
16 | 0.3802 | 83 | 13.5 | [69] | 39 | 1.363 | 160 | 33.6 | [80]
17 | 0.4 | 95 | 17 | [77] | 40 | 1.43 | 177 | 18 | [64]
18 | 0.4004 | 77 | 10.2 | [69] | 41 | 1.53 | 140 | 14 | [64]
19 | 0.4247 | 87.1 | 11.2 | [69] | 42 | 1.75 | 202 | 40 | [64]
20 | 0.43 | 86.5 | 3.7 | [78] | 43 | 1.965 | 186.5 | 50.4 | [80]
21 | 0.44 | 82.6 | 7.8 | [70] | 44 | 2.3 | 224 | 8 | [81]
22 | 0.44497 | 92.8 | 12.9 | [69] | 45 | 2.34 | 222 | 7 | [82]
23 | 0.47 | 89 | 49.6 | [71] | 46 | 2.36 | 226 | 8 | [83]

###  3.2 Pantheon data

In this part, we fit the Pantheon data (the latest compilation of SNIa with 40
binned in the redshift range $0.014\leq z\leq 1.62)$ to get the best fit
values of model parameters $n$ and $H_{0}$, by minimizing
$\chi^{2}(\mu_{0},n,H_{0})$ statistic:

$\chi^{2}(\mu_{0},n)=\sum_{i=1}\frac{[\text{$\mu$th}(\text{z}(\mu_{0},n,z_{i}))-\text{$\mu$obs}(z_{i})]^{2}}{\text{$\sigma$}(i)^{2}}$
(28)

“$\mu_{th}$ and $\mu_{obs}$” are represents as the theoretical and observed
distance modulus. $\sigma(i)$ denotes the standard error of the observed
values.
The distance modulus $\mu(z)$ is read by

$\mu=m-M=\mu_{0}+5\log_{10}D_{L}(z),$ (29)

where $m$ and $M$ are “apparent magnitude and absolute magnitude” of any
distant luminous object respectively. The parameter $\mu_{0}$ and the
luminosity distance $D_{L}(z)$ are defined as

$\mu_{0}=5\log_{10}\left[\frac{H^{-1}_{0}}{Mpc}\right]+25,$ (30)

and

$D_{L}(z)=(1+z)\int_{0}^{z}\frac{dz}{H(z)}.$ (31)

The absolute magnitude $M$ and apparent magnitude $m$ are read as

$M=16.08-25+5\log_{10}\left(\frac{H_{0}}{0.026}\right),$ (32)
$m=16.08+5\log_{10}\left(\frac{(1+z)}{0.026}\int_{0}^{z}\frac{dz}{H(z)}\right).$
(33)

The best fit contour plots for the $H(z)$ data set and Pantheon data set and
their combined data set are shown in this figures $1$a, $1$b $\&$ $1$c.

(a) Contour plot for OHD

(b) Contour plot for Pantheon data

(c) Contour plot for OHD+Pantheon

Figure 1: In this figure, the contour plots for likelihood values of model
parameters $H_{0}$ $\&$ $n$ with samples of OHD and Pantheon data and combined
datasets are displayed at $1-\sigma$, $2-\sigma$, and $3-\sigma$ levels.

(a) (b)

Figure 2: Fig.$2$a shows the comparison of theoretical model with error bar
plots of the OHD data and the $\Lambda$CDM model. Table 1 summarises the
experimental data in the points with bars. When we perform joint analysis of
two datasets, it is evident that our generated model is best-fitted to data.
The distance modulus error plot of the Pantheon data and the $\Lambda$CDM are
shown in Figure $2$b, respectively.

Using a differential age technique and galaxy clustering method, many
cosmologists [87, 88] have computed the values of the Hubble constant at
various red-shifts. They described several observed Hubble constant Hob values
as well as corrections in the range $0\leq z\leq 2.36$ [61, 89]. In our
derived model the observed and theoretical values are found to agree quite
well, indicating that our model is correct. The dots indicate the 46 observed
Hubble constant (Hob) values with corrections, while the linear curves show
the theoretical Hubble constant H(z) graphs with marginal corrections.

##  4 Non-Interacting Model

We suppose that DM and DE only interact gravitationally in the non-interacting
model, therefore each source must satisfy the continuity equation
independently. Thus, from Eq. (4), we have

$\dot{\rho_{m}}+3H\rho_{m}=0,$ (34)
$\dot{\rho_{de}}+3H\rho_{de}(1+\omega_{de})=0.$ (35)

The energy density of dark matter is calculated by using Eq. (34)

$\rho_{m}=\rho_{0}a^{-3}=\rho_{0}\left(k_{1}e^{n\beta
t}-1\right)^{\frac{-3}{n}}$ (36)

The energy density of dark energy is calculated by using Eqs. (23) and (36) in
Eq. (20).

$\displaystyle\rho_{de}=\phi_{0}\left(k_{1}e^{\beta
nt}-1\right)^{\eta/n}\bigg{[}-\frac{c_{1}^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-6/n}\left(\left(k_{1}e^{\beta
nt}-1\right)^{1/n}\right)^{-2\eta}}{\phi_{0}^{2}}-3\alpha^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-2/n}$
$\displaystyle-\frac{k^{2}_{1}\beta^{2}\left(\eta^{2}\omega-6\eta-6\right)e^{2\beta
nt}}{2\left(k_{1}e^{\beta
nt}-1\right)^{2}}\bigg{]}-\rho_{0}\left(k_{1}e^{\beta nt}-1\right)^{-3/n}$
(37)

Using the values of $\phi$ and $\rho_{de}$, the value of EoS parameter is
obtained as

$\displaystyle\omega_{de}=\frac{-\bigg{[}\frac{c_{1}^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-6/n}\left(\left(k_{1}e^{\beta
nt}-1\right)^{1/n}\right)^{-2\eta}}{\phi_{0}^{2}}-\alpha^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-2/n}+\frac{k_{1}\beta^{2}e^{\beta
nt}\left(\left(\eta^{2}(\omega+2)+4\eta+6\right)k_{1}e^{\beta
nt}-2(\eta+2)n\right)}{2\left(k_{1}e^{\beta
nt}-1\right)^{2}}\bigg{]}}{\bigg{[}-\frac{c_{1}^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-6/n}\left(\left(k_{1}e^{\beta
nt}-1\right)^{1/n}\right)^{-2\eta}}{\phi_{0}^{2}}-3\alpha^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-2/n}-\frac{k^{2}_{1}\beta^{2}\left(\eta^{2}\omega-6\eta-6\right)e^{2\beta
nt}}{2\left(k_{1}e^{\beta
nt}-1\right)^{2}}\bigg{]}-\frac{\rho_{0}}{\phi_{0}}\left(k_{1}e^{\beta
nt}-1\right)^{-(3+\eta)/n}}.$ (38)

##  5 Interacting Model

We discuss the interacting cosmological models of the universe by considering
the interaction between DE and DM components. As a result, the DM and DE the
continuity equations are written as

$3H\rho_{m}+\dot{\rho_{m}}-Q=0,$ (39)
$3H\rho_{de}(1+\omega_{de})+\dot{\rho_{de}}+Q=0.$ (40)

The coupling between DM and DE is denoted by $Q$. We estimate the coupling
between DE and DM as a function of $\rho_{m}$ and $H$, $Q\propto\rho_{m}H$. As
a result, for the interaction model, we use “$Q=3b^{2}\rho_{m}H$”; $b^{2}$ is
a constant.
Now, the energy densities of DM and DE are determined as follows:

$\rho_{m}=k\left(k_{1}e^{\beta
nt}-1\right)^{\frac{3\left(b^{2}-1\right)}{n}}.$ (41)

Here, $k$ is an integration constant.

$\displaystyle\rho_{de}=\phi_{0}\left(k_{1}e^{\beta
nt}-1\right)^{\eta/n}\bigg{[}-\frac{c_{1}^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-6/n}\left(\left(k_{1}e^{\beta
nt}-1\right)^{1/n}\right)^{-2\eta}}{\phi_{0}^{2}}-3\alpha^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-2/n}$
$\displaystyle-\frac{k^{2}_{1}\beta^{2}\left(\eta^{2}\omega-6\eta-6\right)e^{2\beta
nt}}{2\left(k_{1}e^{\beta nt}-1\right)^{2}}\bigg{]}-k\left(k_{1}e^{\beta
nt}-1\right)^{\frac{3\left(b^{2}-1\right)}{n}}$ (42)

For interacting model, the EoS parameter is obtained as:

$\displaystyle\omega_{de}=\frac{-\bigg{[}\frac{c_{1}^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-6/n}\left(\left(k_{1}e^{\beta
nt}-1\right)^{1/n}\right)^{-2\eta}}{\phi_{0}^{2}}-\alpha^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-2/n}+\frac{k_{1}\beta^{2}e^{\beta
nt}\left(\left(\eta^{2}(\omega+2)+4\eta+6\right)k_{1}e^{\beta
nt}-2(\eta+2)n\right)}{2\left(k_{1}e^{\beta
nt}-1\right)^{2}}\bigg{]}}{\bigg{[}-\frac{c_{1}^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-6/n}\left(\left(k_{1}e^{\beta
nt}-1\right)^{1/n}\right)^{-2\eta}}{\phi_{0}^{2}}-3\alpha^{2}\left(k_{1}e^{\beta
nt}-1\right)^{-2/n}-\frac{k^{2}_{1}\beta^{2}\left(\eta^{2}\omega-6\eta-6\right)e^{2\beta
nt}}{2\left(k_{1}e^{\beta
nt}-1\right)^{2}}\bigg{]}-\frac{k}{\phi_{0}}\left(k_{1}e^{\beta
nt}-1\right)^{\frac{3\left(b^{2}-1-\eta\right)}{n}}}.$ (43)

##  6 Physical properties and dynamical behaviour of model

The evolution of density parameter for DM and DE are shown in Fig. 3

(a) (b) (c)

Figure 3:  Plots of DM and DE energy densities versus redshift $z$ for non-
interacting (left) and interacting (right) models $\&$ EoS parameter

Figure 3(a) shows the energy density for dark matter for the interacting and
non-interacting cases with respect to redshift. Figure (3b) shows the
variation of energy density for DE versus redshift. In both figures, we
conclude that as $\rho\to 0$ for (or $z\to-1$) which is consistent with a
well-established scenario. From figure 3c, it is clear that the EoS of DE is
varying in quintessence region and crossing the Phantom Divide line (PDL)
$\omega_{de}=-1$ and finally enters in the phantom region in both interacting
and non-interacting cases. The curve exhibits a negative tendency before
evolving through several stages of acceleration, deceleration, and finally
reaching to phantom phase.

The possibility of universe evolution is indicated by the shift from one phase
to the next.

###  6.1 Age of universe

The age of the universe is obtained as

$dt=-\frac{dz}{(1+z)H(z)}\implies\int_{t}^{t_{0}}dt=-\int_{z}^{0}\frac{1}{(1+z)H(z)}dz.$
(44)

Using Eq. (26), we get

$t_{0}-t=\int_{0}^{z}\frac{2}{H_{0}(z+1)\left[(z+1)^{n}+1\right]}\,dz,$ (45)

where $t_{0}$ is the “present age of universe” and it is given by

$t_{0}=\lim_{x\to\infty}\int_{0}^{x}\frac{2}{H_{0}(z+1)\left[(z+1)^{n}+1\right]}\,dz.$
(46)

Integrating Eq.(43), we get

$H_{0}\ t_{0}=0.9425$ (47)

In this study, we have calculated the numerical value of $H_{0}$ as $0.07173$
$Gyr^{-1}$ $\sim$ $70.29$ $kms^{-1}Mpc^{-1}$. Therefore, the present age of
the universe for the derived model is obtained as
$t_{0}=\frac{0.9425}{H_{0}}=13.14$ Gyrs. Figure 7 shows the variation of
“$H_{0}(t_{0}-t)$” with redshift $z$. According to WMAP data [55], the
empirical value of current age of the universe is $t_{0}=13.73^{+.13}_{-.17}$.
It’s worth noting that $t_{0}=13.81\pm 0.038$ Gyrs is the current age of the
universe in Planck collaboration results [90]. The universe’s present has been
calculated as $14.46\pm 0.8$ Gyrs [105], $14.3\pm 0.6$ Gyrs [106] and $14.5\pm
1.5$ Gyrs [107] in various cosmological studies.

Figure 4: The plot of $H_{0}(t_{0}-t)$ versus $z$

###  6.2 Deceleartion parameter

Figure 5 depicts the dynamics of the decelerating parameter $(q)$ with respect
to $z$. The models for OHD and (OHD+Pantheon) coincide with $\Lambda$CDM with
the signature flipping at $z_{t}=0.809$ while for Pantheon data model shows
signature flipping at $z_{t}=4.44123$ due to the dominance of DE in the
universe. As a result, the current universe evolves with a negative sign of
$q$, causing the accelerated expansion of the universe. Moreover the
reconstruction of $q(z)$ is done by the joined $(SNIa+CC+H_{0})$, which have
obtained the transition redshift
$z_{t}={0.69}^{+0.09}_{-0.06},{0.65}^{+0.10}_{-0.07}$ and
${0.61}^{+0.12}_{-0.08}$ within $(1\sigma)$ [91]. which are seen as well
consistent with past outcomes [92, 93, 94, 95, 96] including the $\Lambda CDM$
expectation $z_{t}\approx 0.7$. $0.6\leq z_{t}\leq 1.18$ ($2\sigma$, joint
examination ) [97] is the other limit of transition redshift.

The recent 36 observational Hubble data (OHD) provides the redshift range
$0.07\leq z\leq 2.36$ [98]. The joint light curves (JLA) sample, comprised of
740 type-Ia supernovae (SN Ia) indicates the redshift range $0.01\leq z\leq
1.30$. In this way, our developed model depicts a transition from the early
deceleration phase to the current speeding phase for OHD, Pantheon,
(OHD+Pantheon) data. Furthermore, we find that the deceleration parameter will
remain negative in the future, $z\to-1$, $q\to-1$.

The DP in terms of the redshift $z$ can be written as

$q(z)=\frac{(n-1)(1+z)^{n}-1}{1+(1+z)^{n}}$ (48)

Figure 5: Plot of deceleration parameter $q$ versus $z$.

###  6.3 Luminosity Distance

The redshift-luminosity distance connection [99] is a significant
observational tool for studying the universe’s evolution. The luminosity
distance $(D_{L})$ is calculated in terms of the redshift, which occurs when
light coming out of a distant luminous body is redshifted due to the expansion
of the cosmos. The luminosity distance is used to calculate a source’s flux.
It’s written as $D_{L}=a_{0}r(1+z)$, where $r$ denotes the radial coordinate
of the source. For the model, the radial coordinate $r$ is obtained as
$r=\int_{0}^{dr}=\int_{0}^{t}\frac{cdt}{a(t)}=\frac{1}{a_{0}H_{0}}\int_{0}^{z}\frac{cdz}{h(z)}$,
where $h(z)=\frac{H(z)}{H_{0}}$.

Therefore, we get the luminosity distance as

$\frac{H_{0}D_{L}}{c}=(1+z)\int_{0}^{0}\frac{dz}{h(z)}$ (49)

###  6.4 Particle horizon

The particle horizon is a measurement of the size of the observable cosmos
[100]. The particle horizon is reads as

$R_{P}=\lim_{t_{P}\to
0}a_{0}\int_{t_{0}}^{t_{p}}\frac{dt}{a(t)}=\lim_{z\to\infty}\int_{0}^{z}\frac{dz}{H(z)}$
(50)

where $t_{p}$ denotes the ”time in past” at which the light signal is sent
from the source. When we integrate Eq. (29) for a large value of red-shift, we
get $R_{P}=2.7845$ $H_{0}^{-1}$ as the particle horizon. Figure 6 depicts the
dynamics of correct distance vs red-shift. We can see from Figure 7 that when
$z=0$), $a_{0}H_{0}x$ is null, implying that when $z=0$, the appropriate
distance x becomes infinite. Thus we are at very large distance ( $\sim$ at
infinite distance) from an event occurred in the beginning of the universe.

Figure 6: Distance of particle horizon versus $z$

##  7 Statefinder diagnostics

The statefinder pairs $\\{r,s\\}$ are the geometrical quantities that are
directly obtained from the metric. This diagnostic is used to distinguish
different dark energy models and hence becomes an important tool in modern
cosmology. The statefinder parameters $r$ and $s$ have been defined as follows
[101, 102, 103]

$r=\frac{\dddot{a}}{aH^{3}},\ s=\frac{r-1}{3(q-\frac{1}{2})}$ (51)

$r=\frac{n^{2}\left(2(z+1)^{n}+1\right)(z+1)^{n}}{\left((z+1)^{n}+1\right)^{2}}-\frac{3n(z+1)^{n}}{(z+1)^{n}+1}+1$
(52)
$s=\frac{2n(z+1)^{n}\left(2n(z+1)^{n}-3\left((z+1)^{n}+1\right)+n\right)}{3\left((z+1)^{n}+1\right)\left(2n(z+1)^{n}-3\left((z+1)^{n}+1\right)\right)}$
(53)

(a) (b) (c)

Figure 7:  Plots of statefinders $(r~{}-s)$ $(r~{}-q)$ and $(s~{}-q)$ versus
redshift $z$.

The statefinder diagnostic is a valuable tool in current cosmology used to
differentiate various dark energy models [101, 102, 103, 104]. Different
trajectories in the $(r-s)$ $(r-q)$ and $(s-q)$ planes express the time
evolution for various dark energy models. The present model initially lies in
quintessence region ($r<1,s>0$), passes through the $\Lambda$CDM $(r,s)=(1,0)$
and finally approaches to Chaplygin gas region $r>1$ and $s<0$ as clearly seen
in Fig.7a. The statefinder pair for $\Lambda$CDM and standard cold dark
matter(SCDM) are respectively $(r,s)=(1,0)$ and $(r,s)=(1,1)$ depicts in the
framwork of Bianchi-V background. To get more information on the
parametrization, we analyzed the temporal growth of our model by the $(r-q)$
plane. In Fig. 7(b) The solid line in the diagnostic plane’s shows the
evolution of the $\Lambda$CDM cosmological model. Initially our model start
with the SCDM point $(r,q)=(1,-0.5)$ passing the $\Lambda$CDM $(r,q)=(1,0)$
point and finally reaches to SS point $(r,q)=(1,-1)$. The SCDM universe is
clearly represented by the $q>0$ and $r>1$ area of the profile. The $s-q$
trajectory begins with a decelerating period and progress to an accelerating
era (see Fig.7c).

Table 2: Constrained values of the model parameters with minimum $\chi^{2}$ values  Datasets | $n$ | $H_{0}$ | $q(z=0)$ | $z_{t}$ | $\chi^{2}$
---|---|---|---|---|---
$H(z)$ | 1.457$\pm 0.042$ | 68.46$\pm 1.97$ | -0.2715 | 0.711645 | 31.8169
Pantheon | 1.144$\pm 0.085$ | 69 $\pm 0.49$ | -0.428 | 4.44123 | 569.787
$H(z)+$Pantheon | 1.426$\pm 0.030$ | 68.39 $\pm 0.40$ | -0.287 | 0.8192 | 628.552

##  8 Conclusion

In this paper, first, we observed the cosmological parameters for the
observable universe in Bianchi type V space time in Branc Dicke theory.
Second, we used statistical $\chi^{2}$ tests to limit the various model
parameters of the universe in the resultant model. Table-2 summaries the major
findings of the statistical analysis.

The main objective of this article is to use independent observables such as
Pantheon dataset, OHD, and their joint combination (OHD+Pantheon) to constrain
the free parameters of the theoretical models, which potentially increase the
sensitivity of our estimations. Out of these measurements, we specifically
discuss the $H_{0}$ determination. The main features of the derived model are
discussed in the following manner:

  * •

First, we have found a perfect solution to Einstein’s field equations for both
interacting and non-interacting cases in Bianchi type V space-time.

  * •

We have calculated best-fitted values of the model parameters from the
marginal 1$\sigma$, 2$\sigma$, and 3$\sigma$ contour plots of OHD, Pantheon,
and (OHD+Pantheon) as shown in Fig.1. The constraints on $H_{0}$ and $n$ for
the derived model by fitting the OHD points and the estimate of Hubble’s
constant agree closely with Riess et al.[1].

  * •

In Fig. $2$a, we have shown the error bar plots for OHD data set, theoretical
model plot for OHD data, and the $\Lambda$CDM model. The points with bars
indicate the experimental data summarized in Table-1. The 46 points of the
$H(z)$ and Union 2.1 compilation data are used to constrain the model
parameter $n$ and $H_{0}$. The derived model agrees well with $H(z)$ and
Pantheon data and closely resembles the behavior of the $\Lambda$CDM. The
distance modulus error plot of the Pantheon data alongwith the $\Lambda$CDM
are shown in Figure $2$b.

  * •

Figures 3a and 3b show the graphical behavior of the energy densities for DM
and DE. We noticed that $\rho_{m}$ and $\rho_{d}$ of the model under
discussion are positive decreases with time and vary individually with
redshift in both interacting and non-interacting scenarios. Figure 3c shows
the variation of the DE ($\omega_{de})$ equation of state parameter versus
redshift for interacting and non-interacting models.

It has been observed that $(\omega_{de})$ begins in the positive and ends in
the negative. Note that $\omega_{de}>-1$ and $\omega_{de}<-1$ respectively
indicate quintessence and phantom regimes. While $\omega_{de}=-1$ represents a
universe dominated by cosmological constant. This nature of EoS is ruled out
by SN Ia findings. As a result, the evolving range of $\omega_{de}$ of our
derived model favors the phantom universe in the present epoch.

  * •

The age of the universe in the derived model is $13.14$ Gyrs, as determined by
OHD observations (see Fig. 4). The current age of the universe is believed to
be $t_{0}=13:65$ Gyrs. This age of the cosmos is very similar to the results
found through Plank’s observation[90] and several other observations[105, 106,
107].

  * •

In our derived model, there is a smooth transition from decelerating to
accelerating phase, which is consistent with the present scenario of the
modern cosmology (see Fig. 5). We have plotted the deceleration parameter $q$
by getting its numerical solution, which exhibits a transition at
$z_{t}=0.711$, from the early decelerated phase to a late time accelerated
phase. This is in good agreement with the current cosmological observations.
We have depicted that the deceleration parameter for OHD, (OHD+Pantheon)
coincide with the $\Lambda$CDM model and have the same transition point(see
Fig. 5).

  * •

Particle horizon exists in the derived model, and its value differs from the
$\Lambda$CDM universe model (see Fig.6). The age of the universe in the
developed model matches the empirical value obtained from WMAP measurements
and Plank collaborations quite well.

  * •

The variation of the trajectories $r-s$, $r-q$ and $s-q$ planes are depicted
in Fig. 7. In the diagram, the paths of the $r-s$ trajectories are depicted by
arrows, which represent various dark energy models, and eventually approach to
$\Lambda$CDM (see Fig. 7a). The evolution of the trajectories begins at SCDM
in the $r-q$ plane diagram, and as time passes, the trajectories of the $r-q$
approach the steady-state model SS (see Fig. 7b). The $s-q$ trajectories begin
with a decelerating period and progress to an accelerating era (see Fig. 7c).

We noticed the range of the deceleration parameter from the equation of
$q(z)$, which clearly demonstrates the signature flipping behavior. As a
result, we find that the current OHD and Pantheon data give well-constrained
$H_{0}$ values, and our model is consistent with recent observations.
Furthermore, we describe the dynamics of the cosmos for both interacting and
non-interacting scenarios.

## Acknowledgments

A. Pradhan is thankful to IUCAA, Pune, India for providing support and
facility under Visting Associateship prpgram.

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