An Axially Symmetric Transitioning models with Observational Constraints
Vinod Kumar Bhardwaj1, Archana Dixit2, Rita Rani3, G. K. Goswami4, Anirudh
Pradhan5

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

3,4Department of Mathematics, Netaji Subhas University of Technology, Delhi,
India

5Centre for Cosmology, Astrophysics and Space Science (CCASS), GLA University,
Mathura-281 406, Uttar Pradesh, India

1E-mail<EMAIL_ADDRESS>
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3E-mail<EMAIL_ADDRESS>
4E-mail<EMAIL_ADDRESS>
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###### Abstract

In this study, we have demonstrated the expansion history of an axially
symmetric Bianchi type-I model of the universe. Our model as of now presents
an accelerating universe, which had been in the decelerating phase in the
past. Roles of the two crucial Hubble $H(z)$ and deceleration $q(z)$
parameters are examined. The energy parameters of the universe are estimated
with the help of the latest observational Hubble data (46-data points) and
Pantheon data (the latest compilation of SNIa with 40 binned in the redshift
range $0.014\leq z\leq 1.62)$. We also discuss the stability analysis of the
model by state finder diagnosis. The analysis reveals that in late time, the
model is a quintessence type and points towards the $\Lambda$CDM model. Our
developed model agrees with observational findings in a proper way. We have
discussed some of the physical aspects of the model.

Keywords : LRS Bianchi-I, $\Lambda$CDM Model, Statefinder diagnosis.

PACS: 98.80.-k

##  1 Introduction

Type Ia Supernova observations, announced in 1998, indicate that the physical
universe is accelerating which had been in the decelerating phase in the
past[1, 2]. The background behind these observations are the studies of
supernova exploration and explosions. SN Ia supernovae are standard candles
(populations II and III stars). At shorter redshift, nearby SN Ia help in the
determination of Hubble constant H whereas large red-shifted long distant
objects provide an estimation of deceleration constant q [3, 4, 5]. The
Supernova Cosmology Project was launched in 1988, in which magnitude-redshift
relations were used to estimate the cosmological parameters. In this project,
cosmological parameters were estimated with the help of more than 75 SN Ia
($0.18\leq z\leq 0.86$) magnitude-redshift relation. Authors[6] discussed that
the cosmological constant-based dark energy is dominant over baryon matter-
energy. Authors [7] worked with the 33 additional high-red shifted supernovae
and developed a confidence region of an accelerating universe. They have also
searched for a possibility of a low-mass $\Lambda=0$ cosmology. In this
context, Riess et al. [8] had also done an independent study of 10 high-
redshift supernovae and found that the universe is accelerating. In this
direction, CMB anisotropy [9], SN Ia magnitude-redshift relation [10, 11],
baryonic acoustic oscillation (BAO), peak length scale [12, 13, 14, 15], and
Hubble parameter versus redshift measurements [16, 17, 18, 19] point to the
accelerating universe.

Most recent, Hubble parameters for various red-shifts have been determined
with the help of cosmic chronometric (CC) and BAO techniques [16, 19, 20].
They provided cosmic “deceleration-acceleration transition” when the universe
is thought to have the presence of dark energy in abundance along with non-
relativistic baryon matter. This redshift transition were discussed in [19,
21, 22]. Hubble H(z) data sets are also used to derive energy parameters
$\Omega_{m}$ and $\Omega_{de}$ [23, 24, 25, 26, 27, 28, 29, 30] and are
comparable with the results with SNIa data, BAO and CMB observations. They
estimated the present value of Hubble constant $H_{0}$ [31, 32]. The derived
values of $H_{0}$ are compatible with Huchra’s $H_{0}$ compilation statistical
analysis [17]. All of the above findings were discovered by a simple analysis
of “55 supernova data points”.

46 Hubble parameters H(z) data set in the redshifts range $0\leq z\leq 2.36$
were used to estimate present value of Hubble constant along with baryon and
dark energy parameters [33, 34, 35, 36]. The authors of these references have
also analyzed the cumulative effect of H(z) data, SN Ia pantheon compilation
data, and BAO data in the estimation of model parameters. They have used the
$\chi^{2}$ minimization technique. SN Ia pantheon compilation data set is
given in Scolnic et al. [37].

The late-time acceleration of the universe is described in General relativity
(GR) by dark energy density along with matter density in Einstein’s field
equation [38, 39, 40, 41]. It is suggested that dark energy of repulsive anti
gravitating origin present in abundance in our universe, which is responsible
for the change in the mode of the universe. In some modified theories of
gravity accelerated expansion of the universe has been explained without
applying the DE component in a different way of thinking. A variety of gravity
theories such as f(R) [42, 43, 44], f(T) [45], and f(G) [46], f(Q,T) [47, 48],
f(Q) [49, 50, 51], f(R,G) [52], f(T,B) gravity [53], and Einstein-Gauss-Bonnet
theory [54, 55, 56, 57, 58, 59] were proposed. In the same direction, Harko et
al. [60] developed f(R,T)gravity theory. In these modified theories, non-
linear curvature scalar and energy traces describe acceleration in the
universe. Their origin lies in the corrections towards curvature and energy
trace.

Off late considerable number of Bianchi type-I cosmological models that fit
well with the latest observations and exhibit acceleration in the universe
were surfaced in the literature [61, 62, 63, 64, 65, 66]. It is a well-known
fact that neutrino viscosity develops anisotropy in the universe and was in
abundance during the primordial fireball(P.F.) [67, 68]. As of now the
contribution of neutrinos and CMBR(Cosmic Microwave Background Radiations) in
the total content of the universe are 0.1% $-$ 0.5% [69]. They are leftover
remains of the P.F. . So just after the investigation of the CMBR phenomenon,
Spatially homogeneous and anisotropic cosmological models attracted the
scientific community [70, 71]. It is 1962, when Bianchi type-I cosmological
models were introduced in the book “An Introduction to Current Research” [70].
Sometimes they were called Heckmann and Schucking models in the author’s name.
The Bianchi-type models are the best and simplest anisotropic models, which
completely describe the anisotropic effects. The advantages of these
anisotropic models are that they play an important role in describing the
history of the early Universe and that they contribute to the formation of
more generalized cosmological models than isotropic FRW models [72, 73, 74].

Locally rotational symmetric(LRS) spacetimes are a sub-class of Bianchi type I
models in the sense that they are spatially homogeneous and describe symmetry
along a particular direction(axis). They may also be called axially symmetric
spacetimes. A considerable number of present days accelerating universe models
on LRS spacetimes have appeared in the literatures [75, 76, 77, 78, 79, 80,
81].

In this study, we have demonstrated the expansion history of an axially
symmetric Bianchi type-I model of the universe. Our model as of now presents
an accelerating universe, which had been in the decelerating phase in the
past. Roles of the two crucial Hubble $H(z)$ and deceleration $q(z)$
parameters are examined. The energy parameters of the universe are estimated
with the help of the latest observational Hubble data (46-data points) and
Pantheon data (the latest compilation of SNIa with 40 binned in the redshift
range $0.014\leq z\leq 1.62)$. We also discuss the stability analysis of the
model by state finder diagnosis. The analysis reveals that in late time, the
model is a quintessence type and points towards the $\Lambda$ CDM model. Our
developed model agrees with observational findings in a proper way. We have
discussed some of the physical aspects of the model.

The paper is structured as follows: The present cosmological scenario is
discussed in section I. In section 2 we present a model and field equation. In
section 3, we constrain the model parameters using Hubble data set H(z) and
Pantheon data set. We have also discussed the Luminosity Distance, Apparent
Magnitude, and Distance Modulus of the model. Section 4 contains the age of
the universe. Section 5 shows the behavior of the deceleration parameter.
Evolutionary trajectories are discussed in section 6. Conclusions are
mentioned in section 7.

##  2  Metric and the Field Equations

We consider the LRS Bianchi type I metric of the form

$ds^{2}=-A(dx)^{2}-B^{2}\left(dy^{2}+dz^{2}\right)+dt^{2}$ (1)

Here, “A and B” are the time-dependent metric functions. In general
relativity, Einstein’s field equation with the cosmological constant is:

$\Lambda g_{ij}+R_{ij}-\frac{1}{2}g_{ij}R=-T_{ij}$ (2)

We consider the energy–momentum tensor in the form
$T_{ij}=(p_{m}+\rho_{m})u_{i}u_{j}+p_{m}g_{ij}$, where $p_{m}$ and $\rho_{m}$
“represents the matter pressure and matter energy density”. For the LRS
Bianchi type-I universe, the field equations are as follows:

$2\frac{\ddot{B}}{B}+\frac{\dot{B}^{2}}{B^{2}}=-p_{m}+\Lambda$ (3)
$\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}=-p_{m}+\Lambda$
(4) $2\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}^{2}}{B^{2}}=\rho_{m}+\Lambda$
(5)

The volume for the model is given by $V=AB^{2}$, the scale factor is assumed
as $a=\left(AB^{2}\right)^{1/3}$ and the Hubble’s parameter can be written as
$H=\frac{1}{3}\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)=\frac{\dot{a}}{a}$
in order to obtained the solutions of the field equations.
From Eq.(3)-Eq.(4), we have

$\frac{d}{dt}\left(\frac{\dot{A}}{A}-\frac{\dot{B}}{B}\right)+\left(\frac{\dot{A}}{A}-\frac{\dot{B}}{B}\right)\left(\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}\right)=0$
(6)

On integration, we get

$\frac{\dot{A}}{A}-\frac{\dot{B}}{B}=\frac{c_{1}}{a^{3}}$ (7)

Also, we have

$\frac{\dot{A}}{A}+2\frac{\dot{B}}{B}=3\frac{\dot{a}}{a}$ (8)

Solving Eq.(7) and Eq.(8), we get

$\frac{\dot{A}}{A}=\frac{\dot{a}}{a}+\frac{2}{3}\frac{c_{1}}{a^{3}}$ (9)
$\frac{\dot{B}}{B}=\frac{\dot{a}}{a}-\frac{1}{3}\frac{c_{1}}{a^{3}}$ (10)

Using Eq.(9), Eq.(10) in Eq.(5), we get

$H^{2}=\frac{1}{3}\left(\rho_{m}+\Lambda+\frac{1}{3}\frac{c_{1}^{2}}{a^{6}}\right)$
(11)

For barotropic matter, the energy conservation law holds i.e.,
$\dot{\rho_{m}}+3H(p_{m}+\rho_{m})=0$. The present universe is “dust filled
for which $p_{m}=0$” and $\rho_{m}\propto a^{-3}$. Using the relation
$\frac{a_{0}}{a}=1+z$ between scale factor $a$ and redshift $z$, we get
$\rho_{m}=\rho_{m0}(1+z)^{3}$. The term
$\frac{1}{3}\frac{c_{1}^{2}}{a^{6}}=\rho_{\sigma 0}(1+z)^{6}=\rho_{\sigma}$
represents the anisotropy energy density. For the dust filled universe
($p_{m}=0$), the density parameters are defined as
$\Omega_{m}=\frac{\rho_{m}}{\rho_{c}}$ and
$\Omega_{\sigma}=\frac{\rho_{\sigma}}{\rho_{c}}$, where
$\rho_{c}=\frac{3H^{2}}{8\pi G}$ & $8\pi G\approx 1$. Thus, Eq.(11) can also
be witten as

$H^{2}=H^{2}_{0}\left[(1+z)^{3}\Omega_{m0}+\Omega_{\Lambda
0}+(1+z)^{6}\Omega_{\sigma 0}\right]$ (12)

For $z=0$, the relationship between energy parameters is obtained from Eq.(12)
as:

$1=\Omega_{m0}+\Omega_{\Lambda 0}+\Omega_{\sigma 0}$ (13)

The deceleration parameter (DP) for the model can be expressed as:

$q=\frac{(1+z)^{3}\Omega_{m0}-2\Omega_{\Lambda 0}+4(1+z)^{6}\Omega_{\sigma
0}}{2\left[(1+z)^{3}\Omega_{m0}+\Omega_{\Lambda 0}+(1+z)^{6}\Omega_{\sigma
0}\right]}$ (14)

The Hubble parameter $H$ and deceleration parameter $q$ is the important
physical quantities for describing the evolution of the universe. For
explaining the expansion of the Universe, $H$ plays a vital role and is also
very useful in the estimation of the age of the universe. On the other hand,
the deceleration parameter describes the phase transition (acceleration or
deceleration) during the evolution of the universe.

##  3 Observasional Constraints

###  3.1 Observational Hubble Data of 46 Data Set of $H(z)$

. In this segment, we present the observational data as well as the
statistical methodological analysis that was used to constrain the model
parameters of the derived Universe see in (Fig.1, 2). Here we applied 46 H(z)
observational data points in the ranges $0\leq z\leq 2.36$, which were
obtained by using the “cosmic chronometric approach (CCA)” [18, 21].

To determine the best-fitting values and limits for a fitted model, we use the
$\chi^{2}$ statistic as

$\chi^{2}\left(p\right)=\sum_{i=1}^{46}{\frac{\left(H_{th}(i)-H_{ob}(i)\right)^{2}}{\sigma(i)^{2}}}.$
(15)

Our estimated values for various cosmological parameters for minimum
$\chi^{2}$ have been computed as $H_{0}=69.48$, $\Omega_{m0}=0.2548$,
$\Omega_{\Lambda 0}=0.7411$, $\Omega_{\sigma 0}=0.0005$, $\chi^{2}=29.1136$.

Here $p$ is the set of model parameters, where ($p=H_{0},\Omega_{m0}$). The
$\chi^{2}$ expression in Eq. (15) holds for the H(z) measurements listed in
Table 1. This table contains a data set of the observed values of the Hubble
parameters H(z) versus redshift $z$ with a possible error obtained using the
different age approach by various cosmologists.

Table 1: “The behaviour of Hubble parameter $H(z)$ with redshift $S.No$ | $Z$ | $H(Obs)$ | $\sigma_{i}$ | References | $S.No$ | $Z$ | $H(Obs)$ | $\sigma_{i}$ | References
---|---|---|---|---|---|---|---|---|---
1 | 0 | 67.77 | 1.30 | [82] | 24 | 0.4783 | 80.9 | 9 | [22]
2 | 0.07 | 69 | 19.6 | [83] | 25 | 0.48 | 97 | 60 | [84]
3 | 0.09 | 69 | 12 | [20] | 26 | 0.51 | 90.4 | 1.9 | [86]
4 | 0.01 | 69 | 12 | [84] | 27 | 0.57 | 96.8 | 3.4 | [73]
5 | 0.12 | 68.6 | 26.2 | [83] | 28 | 0.593 | 104 | 13 | [19]
6 | 0.17 | 83 | 8 | [84] | 29 | 0.60 | 87.9 | 6.1 | [88]
7 | 0.179 | 75 | 4 | [19] | 30 | 0.61 | 97.3 | 2.1 | [86]
8 | 0.1993 | 75 | 5 | [19] | 31 | 0.68 | 92 | 8 | [19]
9 | 0.2 | 72.9 | 29.6 | [83] | 32 | 0.73 | 97.3 | 7 | [88]
10 | 0.24 | 79.7 | 2.7 | [85] | 33 | 0.781 | 105 | 12 | [19]
11 | 0.27 | 77 | 14 | [84] | 34 | 0.875 | 125 | 17 | [19]
12 | 0.28 | 88.8 | 36.6 | [83] | 35 | 0.88 | 90 | 40 | [84]
13 | 0.35 | 82.7 | 8.4 | [87] | 36 | 0.9 | 117 | 23 | [84]
14 | 0.352 | 83 | 14 | [19] | 37 | 1.037 | 154 | 20 | [85]
15 | 0.38 | 81.5 | 1.9 | [86] | 38 | 1.3 | 168 | 17 | [84]
16 | 0.3802 | 83 | 13.5 | [87] | 39 | 1.363 | 160 | 33.6 | [91]
17 | 0.4 | 95 | 17 | [20] | 40 | 1.43 | 177 | 18 | [84]
18 | 0.4004 | 77 | 10.2 | [22] | 41 | 1.53 | 140 | 14 | [84]
19 | 0.4247 | 87.1 | 11.2 | [22] | 42 | 1.75 | 202 | 40 | [91]
20 | 0.43 | 86.5 | 3.7 | [85] | 43 | 1.965 | 186.5 | 50.4 | [85]
21 | 0.44 | 82.6 | 7.8 | [88] | 44 | 2.3 | 224 | 8 | [16]
22 | 0.44497 | 92.8 | 12.9 | [22] | 45 | 2.34 | 222 | 7 | [92]
23 | 0.47 | 89 | 49.6 | [89] | 46 | 2.36 | 226 | 8 |  [93]”

###  3.2 Pantheon Data

In this investigation, we use “Pantheon data (the latest compilation of SNIa
with 40 binned in the redshift range $0.014\leq z\leq 1.62)$” [37]. In this
scenario, $\chi^{2}$ for Pantheon data was calculated by using the following
formula.

$\chi^{2}(H_{0},\Omega_{m0})\text{=}\sum_{i=1}\frac{(\text{$\mu$th}(\text{z
}(i),H_{0},\Omega_{m0})-\text{$\mu$obs}(i))^{2}}{\text{$\sigma$}(i)^{2}},$
(16)

where $\mu_{th}$ denoted the theoretical distance modulus and $\mu_{obs}$
represents the observed “distance modulus” with the standard error
$\sigma{(i)^{2}}$. In earlier work [38, 40] researchers presented a technique
for estimating the present values of the energy parameters $\Omega_{m0}$
$\Omega_{\Lambda_{0}}$, and $\Omega_{\sigma 0}$ by “comparing the theoretical
and observed results” with the help of the $\chi^{2}$ statistic.

Our estimated values for various cosmological parameters for minimum
$\chi^{2}$ have been computed as $H_{0}=70.02$, $\Omega_{m0}=0.2728$,
$\Omega_{\Lambda 0}=0.7262$, $\Omega_{\sigma 0}=0.001$, $\chi^{2}=562.242$.

We have formed a new data set by joining OHD and Pantheom datas and estimated
an other set of values for these parameters with minimum $\chi^{2}$. In this
case we obtaine $H_{0}=70.17$, $\Omega_{m0}=0.2542$, $\Omega_{\Lambda
0}=0.745731$, $\Omega_{\sigma 0}=0.000069$, $\chi^{2}=593.356$.

(a) (b) (c)

Figure 1:  Two-dimensional contours for (OHD, Pantheon, OHD+Pantheon data set
) at $1\sigma$, $2\sigma$ and $3\sigma$ confidence regions, bounded with
latest 46 OHD and Pantheon data.

(a) (b) .

Figure 2: Hubble parameter versus redshift error bar plot. The plot of Hubble
rate H(z)/(1+z) versus $z$

The Hubble parameter is clearly not a constant as demonstrated in the figures.
Over the redshift, it varies slowly. Using the differential age technique and
galaxy clustering method, many physicists have computed the values of the
Hubble constant at various redshifts. They have described a variety of
observed Hubble constant $H_{ob}$ values as well as the corrections in the
range $0\leq z\leq 2.36$. The observed and theoretical values are found to
agree quite well, indicating that our model is stable. We can see in Fig 2a
that H grows as the redshift increases. The dots in the figures represents 46
observed Hubble constant $H_{ob}$ values with corrections, whereas the linear
curve represents the theoretical graphs of the Hubble constant. Fig2b
represents, the error bar plot of Hubble rate $H(z)/(1+z)$ versus the redshift
$z$. It is clear that using a joint dataset gives raise to a better fit of the
data. The dots with bars indicate the experimental observations of H(z) data,
$H_{0}$ is the present value of the Hubble constant.

####  3.2.1 Luminosity Distance

The luminosity distance relation [94, 95] is a useful method for examining the
evolution of the universe. The equation for luminosity distance $(D_{L})$ is
defined in terms of redshift, as the light coming out of a distant luminous
body gets red-shifted as the cosmos expands. The luminosity distance is used
to calculate a source’s flux. It is written as

$D_{L}=a_{0}(1+z)r$ (17)

Here $r$ is denoted as the radial co-ordinate of the source. The luminosity
distance can be expressed as [26].

$D_{L}=(1+z)a_{0}c\int_{0}^{z}\frac{dz}{H(z)}$ (18)

We noticed that the luminosity distance is an increasing function of redshift.

####  3.2.2 Apparent Magnitude and Distance Modulus in the model

The luminosity distance is related to the distance modulus by the formula
below.

$\mu=M-m_{b}=5log_{10}\frac{D_{L}}{Mpc}+25$ (19)

where “$m_{b}$ and M are the apparent and absolute magnitude of the source”,
respectively.

We use the equation to calculate the $D_{L}$ for a supernova at very small
redshift.

$D_{L}=\frac{cz}{H_{0}}$ (20)

There are many supernovae with low red shift whose “apparent magnitudes” are
known in the literature. The common “absolute magnitude (M)” for Pantheon data
is determined by this. Using Eqs. (19) and (20) in our previous study [96,
97], we derived (M) as follows:

$M=5log_{10}\frac{H_{0}}{.026c}-8.92$ (21)

We obtained the expression for the apparent magnitude by using Eqs. (18) and
(19) as:

$m_{b}=16.08+5log_{10}[\frac{1+z}{.026c}\int_{0}^{z}\frac{dz}{h(z)}]$ (22)

We are numerically solving Eqs.(12) & (21) and by using Pantheon data (the
latest compilation of SN Ia with 40 binned in the redshift range $0.014\leq
z\leq 1.62$. According to our model, we derive the corresponding theoretical
values. Figures 3 show the closeness of observational and theoretical results,
demonstrating the validity of our model.

Figure 3: Plot of Distance modulus $\mu$ versus $z$

.

Figure 3 shows the progression of $\mu(z)$ for the best-fit values of model
parameters Here, $\mu(z)$ represents the distance modulus, which is the
difference between the “apparent magnitude and the absolute magnitude” of the
observed supernova, is given by [89, 90] $\mu(z)=25+5log10(dL/Mpc)$, where
$d_{L}$ is the luminosity distance. The blue dots in this plot correspond to
the Pantheon data (the latest compilation of SNIa with 40 binned in the
redshift range $0.014\leq z\leq 1.62)$ for error bar plot [98].

##  4 Age of the Universe

The age of the cosmos is calculated as

$dt=-\frac{dz}{(1+z)H(z)}\implies\int_{t}^{t_{0}}dt=-\int_{z}^{0}\frac{1}{(1+z)H(z)}dz$
(23)

Using Eq.(12) and Eq.(23), we get

$t_{0}-t=\int_{0}^{z}\frac{1}{H_{0}(z+1)\sqrt{\Omega_{\sigma
0}(z+1)^{6}+\Omega_{m0}(z+1)^{3}+\Omega_{\Lambda 0}}}\,dz$ (24)

The present age of universe is represents as $t_{0}$ and it is written by

$t_{0}=\lim_{x\to\infty}\int_{0}^{x}\frac{1}{H_{0}(z+1)\sqrt{\Omega_{\sigma
0}(z+1)^{6}+\Omega_{m0}(z+1)^{3}+\Omega_{\Lambda 0}}}\,dz$ (25)

Integrating Eq.(25), we get

$H_{0}\ t_{0}=0.97786$ (26)

In this paper, we have estimated the numerical value of $H_{0}$ as $0.6948$
$Gyr^{-1}$ $\sim$ $69.48$ $kms^{-1}Mpc^{-1}$. Therefore, the present age of
universe for the derived model is estimated as
$t_{0}=\frac{0.97786}{H_{0}}=13.79$ $Gyrs$.

Figure 4 depicts the fluctuation of $H_{0}(t_{0}-t)$ as a function of redshift
$z$. According to WMAP data, the empirical value of the universe’s current age
is $t_{0}=13.73^{+.13}_{-.17}$.

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

Figure 4 shows, the plot of the age of the universe versus redshift in our
derived model. In several cosmological research, the age of the universe is
estimated as $14.46\pm 0.8$ Gyrs [99], $14.3\pm 0.6$ Gyrs [100] and $14.5\pm
1.5$ Gyrs [101].

##  5 Deceleration Parameter

Figure 5 depicts the dynamics of the decelerating parameter $(q)$ with respect
to $z$ for “OHD, Pantheon, and OHD + Pantheon”. As we know the early universe
was in decelerated era $q$, with the signature flipping at $z_{t}=0.809$ for
(OHD+Pantheon), at $z_{t}=0.781$ for (OHD), at $z_{t}=0.705$ for (Pantheon),
due to the dominance of dark energy in the universe. As a result, the current
universe evolves with a negative sign of $q$, causing the accelerated
expansion of the universe. In this way, our developed model depicts a
transition from the early deceleration phase to the current speeding phase.
Here $q_{0}=-0.42$ is the current value of the deceleration parameter.
Furthermore, we find that the deceleration parameter will remain negative in
the future, $z\to-1$, $q\to-1$.

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

##  6 Statefinders

In this section, we focus on the diagnosis of the statefinder. The Hubble
parameter H, which represents the universe’s expansion rate, and the
deceleration parameter $q$, which represents the rate of
acceleration/deceleration of the expanding cosmos, are two well-known
geometrical variables that characterize the universe’s expansion history. They
only depend on the scaling factor $a$. However, with the enhancing amount of
cosmological models and the remarkable increase in the accuracies of
cosmological observational data, these two parameters are no longer sensitive
enough to discriminate between different models. As a consequence, the
statefinder diagnosis was developed to distinguish between an increasing
number of cosmological models containing dark energy. Since different
cosmological models have different evolutionary paths in the $(r-~{}s)$ plane,
the statefinder diagnostic is likely a good way distinguish cosmological
models.The remarkable property is that “$(r-~{}~{}s)=(1~{}~{}0)$” corresponds
to the $\Lambda CDM$ model see in figure 6a.

There are so many dark energy models, for example, the quintessence, the
phantom, the Chaplygin gas, the holographic dark energy models, and the
interacting DE models, which have been studied in the literature [102, 103,
104, 105, 106], one can clearly identify the distance from a given
cosmological model to a $\Lambda CDM$ model in the $(r-~{}s)$ plane.

$r=\frac{\ddot{H}}{H^{3}}+3\frac{\dot{H}}{H^{2}}+1=\frac{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda
0}+10\Omega_{\sigma 0}(1+z)^{6}}{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda
0}+\Omega_{\sigma 0}(1+z)^{6}}$ (27)
$s=\frac{r-1}{3(q-\frac{1}{2})}=\frac{2\Omega_{\sigma
0}(1+z)^{6}}{-\Omega_{\Lambda 0}+\Omega_{\sigma 0}(1+z)^{6}}$ (28)

It’s worth noting that different combinations of $r$ and $s$ indicate
different DE models, as referenced in [102, 107, 108]. Figure 6(a) dipicts
that our model lies in quintessence region $(r<1,s>0)$ and CG region
$(r>1,s<0)$ and also meets SCDM $(r=1,s=1)$ point and LCDM $(r=1,s=0)$ point.

(a)  (b)

Figure 6: (a) Plot of $r$ vs $s$. (b) Plot of $r$ vs $q$.

Figure 6(b) shows the evolutionary trajectories in the $(r-q)$ plane. In the
figure the fixed point $(q=-1,r=1)$ shows the SS model (de sitter expansion)
and the horizontal line represents the evolution of the trajectory
corresponding to $\Lambda CDM$ $(r=1,q=0)$. The trajectories corresponding to
our model start from the fixed point $\Lambda CDM$ and passes through the SS
point.

##  7 Conclusion

With the currently available data points, we have analyzed the Pantheon and
OHD data and constrain several cosmological parameters. We would like to point
out that our research does not take the effects of correlation. The goal of
this study was to concentrate on a few key theoretical issues that have
received insufficient attention in the literature. The 46 OHD data points and
Pantheon data (the latest compilation of SNIa with 40 binned in the redshift
range $0.014\leq z\leq 1.62)$ were used in this analysis. In particular, we
have examined the estimated values of cosmological parameters from the
OHD+Pantheon data sets. While the parameter estimation errors decreased
dramatically for all the data sets.

We have summarise the main findings once more:

  * •

Fig.1 shows the 2-dimensional joint contours (OHD+Pantheon), at 68$\%$and
95$\%$ confidence regions, bounded with latest .

  * •

Fig.2 represents the error bar plots for OHD and Pantheon data sets. In both
graphs, the dots represent observed values with corrections, whereas red lines
are the present model compared with the $\Lambda CDM$ model shown in black
dashed lines.

  * •

Figure 3 shows the progression of $\mu(z)$ for the best-fit values of model
parameters

  * •

Our model is based on latest observational findings of 46 OHD data and
Pantheon data (the latest compilation of SNIa with 40 binned in the redshift
range $0.014\leq z\leq 1.62)$. Figure 4 depicts the age- redshift plot of the
universe in our derived model. In this paper, we have estimated the numerical
value of $H_{0}$ as $0.6948$ $Gyr^{-1}$ $\sim$ $69.48$ $kms^{-1}Mpc^{-1}$.
Therefore, the present age of universe for the derived model is estimated as
$t_{0}=\frac{0.97786}{H_{0}}=13.79$ $Gyrs$.

  * •

The deceleration parameter $q$ fluctuates with $z$ from positive to negative,
as seen in the graph. This shows a transition from early slowdown to the
current acceleration of the universe (see fig 5).

  * •

We also use the $(r,s)$ and $r,q$ planes parameters to diagnose the DE model
geometrically, as shown in fig.6.Our derived model initially shows a Chaplygin
gas (CG) type DE model, which later on evolves into a quintessence DE model at
a few points. The model, later on, reverts back again in CG. Interestingly,
the model deviates significantly from the point $(r,s)=(1,0)$ and it does not
coincide with $\Lambda CDM$ (see Fig. 6).

  * •

The estimated results on the basis of $\chi^{2}$ statistic using OHD,
Pantheom, and OHD+ Pantheom datas are shown in the following Table-2:

Table 2: Numerical estimations for the derived models Datasets | $H_{0}$ | $\Omega_{m0}$ | $\Omega_{\Lambda 0}$ | $\Omega_{\sigma 0}$ | $\chi^{2}$ | $z_{t}$
---|---|---|---|---|---|---
$OHD$ | $69.48$ | $0.2584$ | $0.7411$ | $0.0005$ | $29.1136$ | $0.781$
$Pantheon$ | $70.02$ | $0.2728$ | $0.7262$ | $0.001$ | $562.242$ | $0.705$
$OHD+Pantheon$ | $70.17$ | $0.2542$ | $0.745731$ | $0.000069$ | $593.356$ | $0.809$

Finally, we can state that all of the preceding conclusions in LRS Bianchi-I
are good and consistent with recent cosmological observations.

## CRediT authorship contribution statement

Vinod Kumar Bhardwaj : Conceptualization, Ideas, Formulation, Writing –
original draft. Archana Dixit: Writing-review & editing, Conceptualization.
Rita Rani: Analysis of figures and review of literatures. G. K. Goswami:
Formal analysis, Writing – review & editing. Anirudh Pradhan: Methodology,
Supervision, Final writing & editing.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgement

A. Pradhan thanks to the IUCAA, Pune, India for providing support and facility
under associateship program.

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