text:
\section{Introduction}
The standard cosmology has assumed that the universe is exactly homogeneous
and isotropic at large scales \cite{Tedesco_2018}. The spacetime of
this kind of universe is described by the Friedmann-Lema\^itre-Robertson-Walker
 (FLRW) metric, and it is assumed to be occupied by some ideal fluids with a
diagonal energy momentum tensor \cite{Pebbles_1994,J.Colin_2019}. Various
cosmological observational data like, data of Type Ia Supernovae
(SNe Ia) \cite{Hossienkhani_2021,Campanelli_2011,Akarsu_2019,Ma_2011,
Tedesco_2018,Maurya_2018, Bamba_2012}, Cosmic Microwave Background (CMB) \cite{Akarsu_2019,Bunn_1996,Costa_2004,
Hossienkhani_2021,Ma_2011,Tedesco_2018,Campanelli_2011,Copi_2004, Eriksen_2004_a,Eriksen_2004_b}, Baryon Acoustic
Oscillation (BAO) \cite{Akarsu_2019,Hossienkhani_2021,Ma_2011,Tedesco_2018,
Campanelli_2011}, Hubble parameter \cite{Hossienkhani_2021,Campanelli_2011,
Akarsu_2019,Ma_2011,Tedesco_2018,Maurya_2018,Frieman_2008,
Perivolaropoulos_2014,Cea_2022}, the Wilkinson Microwave Anisotropy Probe (WMAP)
\cite{Hossienkhani_2021,Frieman_2008,Perivolaropoulos_2014,Buniy_2006, Campanelli_2006, Campanelli_2007,Berera_2004, Vielva_2004, Hansen_2004} and the Large
Scale Structure (LSS) \cite{Hossienkhani_2021,Frieman_2008} etc.\ have
suggested that the current expansion of the  universe is accelerating
\cite{Hossienkhani_2021,Ma_2011}. This current accelerated expansion of the
universe is considered to be due to the presence of a large negative pressure
component in the universe, which is referred as the Dark Energy (DE). On the
other hand, an unknown form of matter known as the Dark Matter (DM) is thought
to be responsible for the formation of LSS in the universe \cite{J.Colin_2019,
Hossienkhani_2018}. However, at this point it would be appropriate to mention
that according to the modified gravity theories scenario, the current
accelerated expansion of the universe and DM are the manifestations of the
modifications of spacetime behaviours at large scales, which could not be
accounted for by the general theory of relativity \cite{Nojiri_2017,Gogoi_2021,
Parbin_2021}. Although the distribution of galaxies under the background of
dominated DM is observed to be inhomogeneous \cite{J.Colin_2019}, other
observational sources like different galaxy redshift catalogues have suggested
a statistical transition from the inhomogeneity to homogeneity on scale
exceeding $\sim$ 100 Mpc \cite{J.Colin_2019, Hogg_2005,Scrimgeour_2012}. Thus,
the assumption of isotropy and homogeneity of the universe is applicable only
for the large scales \cite{Tedesco_2018}. However, the standard cosmological
models are challenged by few other puzzling cosmological observations
\cite{Perivolaropoulos_2014,Wang_2018}, such as power asymmetry of the CMB
perturbation maps \citep{Eriksen H. K_2007,Hoftuft_2009,Paci_2010,
Mariano_2013,Zhao_2016}, anisotropy in accelerating
expansion rate \cite{Antoniou_2010, Mariano_2012, Wang_2014}, the large scale
velocity flow \cite{Kashlinsky_2010,Kashlinsky_2008,Watkins_2009,Feldman_2010,
Lavaux_2010}, spatial dependence of the value of the fine structure constant
\cite{Perivolaropoulos_2014, Moss_2011, Webb_2011, King_2012, Pinho_2016},
etc. Based on the observations of these cosmic anomalies
\cite{Perivolaropoulos_2014}, the Planck collaboration had rightly stated
that \textit{``The Universe is still weird and interesting"} \cite{Zhao_2016}.
The origin of these anomalies are still unknown. These may either be arised due
to the large statistical fluctuations or may have some physical origin like
geometry or energy of the universe \cite{Perivolaropoulos_2014} or may favour
of a preferred cosmological direction, and perhaps depict the anisotropic nature
of the universe.

Testing the large-scale geometry of the universe by using cosmological scale
is one of the major challenges of modern cosmology \cite{Campanelli_2011}.
Reliable observational data from the various sources like WMAP, BAO, Planck
\cite{Planck_2018} etc.\ have given new insight on the study of the basic
assumptions (i.e.\ isotropy and homogeneity) of the standard cosmological
models and also shown the anisotropic characteristics of the universe at least
at the local scale. In our real universe, there exist some kind of strange
motions due to the local
inhomogeneity and anisotropy of surrounding structures, which can not be
neglected \cite{J.Colin_2019}. To explain the anisotropic character of the
universe, spatially homogeneous and anisotropic cosmological models play a
significant role to describe the large-scale behaviour of the universe. These
models have been widely studied under the regime of general relativity to
understand the relativistic picture of the universe at its early stage
\cite{Maurya_2018}. Thus, we need a simple and convincing cosmological model
with a relativistic background metric \cite{Akarsu_2019}, which allows
directional scale factors \cite{Akarsu_2019,Tedesco_2018,Campanelli_2011,
Maurya_2018,Paul_2008} while maintaining the spatial homogeneity and flatness.
The Bianchi Type I model \cite{Akarsu_2019,Tedesco_2018,Maurya_2018,
Campanelli_2011, Hossienkhani_2018,Paul_2008,Perivolaropoulos_2014} and its
metric fulfill the above criteria and hence it is suitable for study the
anisotropic nature of the universe in its early stages as well as in the
current scenario. Bianchi Type I model describes the anisotropic nature of the
universe by considering the anisotropic and homogeneous background
\cite{Tedesco_2018}. It gives a very small deviation from the exact isotropy.
Hence, use of Bianchi Type I geometry corresponds to replacing the spatially
flat FLRW background by Bianchi Type I background metric \cite{Tedesco_2018}.
Therefore, the Bianchi Type I geometry can be considered as an alternative
option to the FLRW metric to study the various cosmic anomalies as mentioned
above.

Keeping in view of the above points, in this study, we have used the Bianchi
Type I metric \citep{Akarsu_2019,Maurya_2018,Hossienkhani_2018,Paul_2008,
Perivolaropoulos_2014} with the ``customized scale factors" in different
directions. Here we have considered two cases. For case I, we have modified
directional scale factors in the Bianchi type I metric by considering the
second and third scale factors in terms of the first one with some
multiplicative constants. For case II, we have again considered the second and
third scale factors in terms of the first one, but here we have added constant
terms with them rather than multiplying. With these two cases of scale
factors in the metric, we have derived various cosmological parameters
\cite{Tedesco_2018,Akarsu_2019,Maurya_2018,Perivolaropoulos_2014,Frieman_2008,
Hossienkhani_2018,Gogoi_2021,Paul_2008,Barrow_1997} including the directional
Hubble parameters, average or mean Hubble parameter, deceleration parameter,
shear scalar, relative shear anisotropy parameter or Hubble normalized shear
parameter, average anisotropic expansion, density parameters (matter,
radiation, anisotropy and dark energy), distance modulus, Equation of State
(EoS) etc. We have also studied the evolution of various cosmological
parameters with respect to cosmological redshift ($z$) \cite{Frieman_2008,
Gogoi_2021,Hossienkhani_2018} by plotting those parameters against $z$ or
($1+z$) and compared them with the standard cosmological model ($\Lambda$CDM
model) and also with the available observational data of various sources for
both the cases.

This paper is organized as follows. In Section \ref{2}, the basics of the
Bianchi Type I model have been discussed. Here, we have mainly included the
various mathematical equations and expressions of the Bianchi type I universe
from various literatures. In Section \ref{Model}, we have derived all the
mathematical equations and expressions using the ``customized scale factors
approach" for the Bianchi Type I model of the universe. This section is
subdivided into two subsections \ref{s1} and \ref{s2}, in which we have
derived all the expressions and equations for two separate cases respectively.
In Section \ref{4}, we have made graphical analysis of the various cosmological
parameters with respect to $z$ and tried to give the explanations of those
results obtained from the plots for both the cases. The paper has been
summarized in Section \ref{5} with conclusions. Throughout this work we use the
geometrized unit system, where $c = G = 1$ with $(-,+,+,+)$ metric convention.

\section{Basic equations of Bianchi type I model}\label{2}
The general form of the metric for the Bianchi Type I universe \cite{Perivolaropoulos_2014,Akarsu_2019,Paul_2008,Maurya_2018,Hossienkhani_2018} can be
written as
\begin{equation}\label{Bianchi_Metric}
ds^2= -\,dt^2+\sum_{i=1}^{3}a_{i}(dx^i)^{2},
\end{equation}
where $a_{i}$ is the directional scale factor along the $i$th direction,
which is the function of time $t$ only. The average expansion scale factor
$(a_{1}a_{2}a_{3})^{\frac{1}{3}}$ arises from the average Hubble parameter
defined as $$H=\frac{1}{3} {\sum\limits_{i=1}^{3}H_{i}}$$ with
$H_{i}=\frac{\dot{a_{i}}}{a_{i}}$ is the directional Hubble parameter along
the $i$th direction
The most general form of the energy-momentum tensor $T_{\mu\nu}$ for the given
metric can be considered in the form:
\begin{equation}\label{Stress_energy_tensor}
 T_{\mu}^{\nu}= diag\big[-\rho,P,P,P\big],
\end{equation}
where $\rho$ is the energy density and $P$ is the isotropic pressure of a
perfect fluid. Now, in view of the present scenario of our accelerating
universe, we consider the Einstein field equations with the cosmological
constant $\Lambda$ as
\begin{equation}\label{field_equation}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu},
\end{equation}
where $R_{\mu\nu}$ is the Ricci tensor, R is the Ricci scalar and $g_{\mu\nu}$
is the metric tensor. For the above metric (\ref{Bianchi_Metric}) and the
energy-momentum tensor \eqref{Stress_energy_tensor}, the following set of
field equations can be obtained from the Einstein field equations:
\begin{align}\label{temporal_component}
H_{1}H_{2}+H_{1}H_{3}+H_{2}H_{3}&=8\pi \rho + \Lambda,\\[5pt]
\label{spatial_component_I}
\frac{\ddot{a_{2}}}{a_{2}}+\frac{\ddot{a_{3}}}{a_{3}}+H_{2}H_{3}&=-\,8\pi  P+\Lambda,\\[5pt]
\label{spatial_component_II}
\frac{\ddot{a_{3}}}{a_{3}}+\frac{\ddot{a_{1}}}{a_{1}}+H_{3}H_{1}&=-\,8\pi  P+\Lambda,\\[5pt]
\label{spatial_component_III}
\frac{\ddot{a_{1}}}{a_{1}}+\frac{\ddot{a_{2}}}{a_{2}}+H_{1}H_{2}&=-\,8\pi  P+\Lambda.
\end{align}
Combining Eqs.\ (\ref{spatial_component_I}), (\ref{spatial_component_II}) and
(\ref{spatial_component_III}) and then subtracting Eq.\
(\ref{temporal_component}) from the combined one, we may write the resulting
Friedmann equation as
\begin{equation}\label{Friedmann_equation}
\sum_{i=1}^{3}\frac{\ddot{a_{i}}}{a_{i}}=-\,4\pi \big(3P+\rho\big)+\Lambda.
\end{equation}
The continuity equation in this model of the universe is given by
\cite{Paul_2008},
\begin{equation}\label{continuity_equation}
\dot{\rho}+ \big(\rho+P\big) \sum_{i=1}^{3}\frac{\dot a_{i}}{a_{i}}=0.
\end{equation}
Considering the EoS $P=\omega \rho$ in Eq.\ (\ref{continuity_equation}), in
which the parameter $\omega$ tells us about the nature of energy or matter
density in the universe, for example, $\omega=0$ for the non-relativistic
matter, $\omega=\frac{1}{3}$ for the radiation and $\omega=-1$ for the
vacuum energy \cite{Frieman_2008, Maurya_2018,Akarsu_2019}, the energy
density of the universe can be found as
\begin{equation}\label{density}
\rho=\rho_{0}\prod_{i=1}^{3} a_{i}^{-(1+\omega)}.
\end{equation}
Here $\rho_0$ is the present energy density of the universe. From this
equation it is seen that the future energy density of this model of the
universe is the EoS parameter $\omega$ dependent, i.e.\ on the nature of
the matter or energy content of the universe. It is clear that $\rho \propto
(a_1 a_2 a_3)^{-1}$ for the matter, $\rho \propto
(a_1 a_2 a_3)^{-\frac{4}{3}}$ for the radiation and $\rho = \rho_0$ (constant)
for the vacuum energy dominated cases respectively.

Using Eq.\ (\ref{temporal_component}) the Hubble parameter of this universe
can be expressed in terms of the present density parameters of the universe as
\begin{equation}\label{Hubble_parameter}
H= H_{0}{\sqrt{\Omega_{m0}(a_{1}a_{2}a_{3})^{-1}+\Omega_{r0}(a_{1}a_{2}a_{3})^{-\frac{4}{3}}+\Omega_{\Lambda 0}+\Omega_{\sigma 0}(a_{1}a_{2}a_{3})^{-2}}},
\end{equation}
where $H_0$ is the current Hubble parameter, and $\Omega_{m0} =
8\pi \rho_{m0}/3H_{0}^{2}$ is the density parameters for the matter content,
$\Omega_{r0} = 8\pi  \rho_{r0}/3H_{0}^{2}$ is the density parameter for the
radiation content, $\Omega_{\Lambda 0} = \Lambda/3H_{0}^{2}$ is the density
parameter for the vacuum energy and $\Omega_{\sigma 0} =
\sigma_{0}^{2}/3H_{0}^{2}$ is the density parameter for the anisotropy
\cite{Akarsu_2019,Maurya_2018,Hossienkhani_2021} of the present universe.
Here, $\rho_{m0}$ and $\rho_{r0}$ are current values of matter density
$\rho_{m}$ and radiation density $\rho_{r}$ of the universe respectively.
{The term $\sigma_{0}^{2}$ in $\Omega_{\sigma 0}$ is the current value of
shear scalar. The shear scalar is the parameter through which the contribution
of the expansion anisotropy in Bianchi type I model of universe can be
quantified. It arises due to the different scale factor taken for different
directions in the Bianchi model of universe. For the isotropic case,
$\sigma_{0}^{2} = 0$.} The current value  { of shear scalar,
i.e.~$\sigma_{0}^{2}$} is related with the shear scalar  {$\sigma^{2}$} by
the relation $\sigma^{2}= \sigma_0^2(a_1 a_2 a_3 )^{-2}$. {In general} the shear scalar {$\sigma^{2}$} in terms of the average and directional Hubble parameters can be written as
\cite{Maurya_2018,Akarsu_2019,Paul_2008}
\begin{equation}\label{shear_scalar_II}
\sigma^{2}= \frac{1}{2}\Big[\sum_{i=1}^{3}H_{i}^{2}-3H^2\Big].
\end{equation}

Again, the deceleration parameter for the Bianchi Type I universe can be
derived as
\begin{equation}\label{dec_par_I}
q=-\frac{1}{3H^2}\bigg(\frac{\ddot a_1}{a_1}+\frac{\ddot a_2}{a_2}+\frac{\ddot a_3}{a_3}\bigg)+\frac{2\sigma^2}{3H^2}.
\end{equation}
Using Eq.\ \eqref{Friedmann_equation} and the EoS mentioned above, this equation
for the deceleration parameter $q$ can expressed in terms of density parameters
of the universe as
\begin{equation}\label{dec_par_II}
q=\frac{1}{2}\Omega_m + \Omega_r -\Omega_{\Lambda}+2\,\Omega_{\sigma},
\end{equation}
where $\Omega_m = 8\pi \rho_{m}/3H^{2}$, $\Omega_r = 8\pi \rho_r/3H^{2}$,
$\Omega_{\Lambda} = \Lambda/3H^{2}$ and $\Omega_{\sigma}=\frac{\sigma^2}{3H^2}$ \cite{Akarsu_2019} are the matter density parameter, radiation density
parameter, vacuum energy density parameter and anisotropy or shear density
parameter respectively of the universe at any given instant. Similarly, the
average anisotropic expansion \cite{Maurya_2018, Paul_2008} for Bianchi type I
universe is
\begin{equation}\label{Avg_exp}
A=\frac{1}{3}\sum_{i=1}^{3}\Big(\,\frac{\Delta H_i}{H}\Big)^2 =
2\,\Omega_{\sigma}, \;\; \Delta H_i= (H_i-H).
\end{equation}

The cosmological redshift ($z$) is another cosmological parameter, which is a
very important parameter to understand the evolution and history of the
universe in the sense that it is a directly observable parameter, not the
scale factor. Hence to quantify the
effect of anisotropy in the observable domain of the universe, it is necessary
to express all other cosmological parameters mentioned above as the functions
of $z$. In this context, it is to be noted that in the case of the Bianchi
type I universe, there exist three redshifts in three spatial directions due
to the directional dependence of scale factor. If $z_1, z_2, z_3$ are these
three redshift parameters along three spatial directions, then they can be
defined as
\begin{equation}
\frac{a_1(t)}{a_{10}}=\frac{1}{1+z_{1}}, \;\;
\frac{a_2(t)}{a_{20}}=\frac{1}{1+z_{2}} \;\;\mbox{and}\;\;
\frac{a_3(t)}{a_{30}}=\frac{1}{1+z_{3}}.
\end{equation}
Here, $a_{10}, a_{20}, a_{30}$ are the scale factors of present time along the
three spatial directions. Now, from the current observations, we have taken
$a_{10} = a_{20}= a_{30} = 1$ \cite{Maurya_2018} and thus the scale factors
can be rewritten in terms of $z_1, z_2, z_3$ as

\begin{equation}\label{redshift_I}
a_1(t)=\frac{1}{1+z_{1}},\;\; {a_2(t)}=\frac{1}{1+z_{2}}\;\; \mbox{and}\;\;
{a_3(t)}=\frac{1}{1+z_{3}}.
\end{equation}
With this form of the scale factors, the directional Hubble parameters ($H_i)$
can be written in terms of $z_i$ as
\begin{equation}\label{dir_hubble}
H_i=-\,\frac{\dot{z}_i}{1+z_i}
\end{equation}
and hence the average Hubble parameter takes the form:
\begin{equation}\label{avg_hubble}
H= -\frac{1}{3}\sum_{i=1}^{3}\frac{\dot{z}_i}{1+z_i}.
\end{equation}
Thus, Eq.\ (\ref{Hubble_parameter}) for the average Hubble parameter can be
rewritten in terms of redshift parameters $(z_1, z_2, z_3)$ as
\begin{equation}\label{H_E}
H=H_0 \sqrt{E(z_1,z_2,z_3)},
\end{equation}
\begin{multline}\label{E_z}
\!\!\!\!\!\!\!\mbox{where}\;\;
E(z_1,z_2,z_3) = \Omega_{m0}\big[(1+z_1)(1+z_2)(1+z_3)\big]\\+\Omega_{r0}\big[(1+z_1)(1+z_2)(1+z_3)\big]^{\frac{4}{3}}+\Omega_{\Lambda 0}+\Omega_{\sigma 0}\big[(1+z_1)(1+z_2)(1+z_3)\big]^{2}.
\end{multline}
With the help of Eqs.\ (\ref{dir_hubble}), (\ref{avg_hubble}) and (\ref{H_E}),
the age of the universe can be expressed in terms of the directional redshift
parameters $(z_1,z_2,z_3)$ as follows:
\begin{multline}\label{age_univ}
t_{age}=\int_{0}^{t}dt=\frac{1}{3H_0}\Bigg[\int_{0}^{\infty}\frac{dz_1}{(1+z_1)\sqrt{E(z_1,z_2,z_3)}}+\int_{0}^{\infty}\frac{dz_2}{(1+z_2)\sqrt{E(z_1,z_2,z_3)}}+\int_{0}^{\infty}\frac{dz_3}{(1+z_3)\sqrt{E(z_1,z_2,z_3)}}\Bigg].
\end{multline}
Other cosmological parameters in terms of redshift can be derived for this model
of the universe as follows:\\

(i) Deceleration parameter:
\begin{multline}\label{dec_final}
q(z_1,z_2,z_3)=\frac{1}{E(z_1,z_2,z_3)}\bigg[\frac{1}{2}\,\Omega_{m0}\,(1+z_1)(1+z_2)(1+z_3) +\Omega_{r0}\Big\{(1+z_1)(1+z_2)(1+z_3)\Big\}^{\frac{4}{3}}\\-\,\Omega_{\Lambda 0}+2\,\Omega_{\sigma 0}\Big\{(1+z_1)(1+z_2)(1+z_3)\Big\}^{2}\bigg].
\end{multline}

 (ii) Equation of State:
\begin{equation}\label{eos_final}
\omega(z_1,z_2,z_3) = \frac{\frac{1}{3}\,\Omega_{r0}\big[(1+z_1)(1+z_2)(1+z_3)\big]^{\frac{4}{3}}-\Omega_{\Lambda 0}}{\Omega_{m0}\big[(1+z_1)(1+z_2)(1+z_3)\big]+\Omega_{r0}\big[(1+z_1)(1+z_2)(1+z_3)\big]^{\frac{4}{3}}+\Omega_{\Lambda 0}}.
\end{equation}

(iii) Ricci Scalar:
\begin{equation}\label{Ricci_tensor_final}
R(z_1,z_2,z_3) = 3H_0^2\Big[\Omega_{m0}(1+z_1)(1+z_2)(1+z_3)+4\,\Omega_{\Lambda 0}\Big].
\end{equation}

(iv) Luminosity distance:
\begin{multline}\label{lumin_final}
d_L(z_1,z_2,z_3) =\frac{\big[(1+z_1)(1+z_2)(1+z_3)\big]^{\frac{1}{3}}}{3H_0}\int_{0}^{\infty} \Big\{\frac{\big[(1+z_2)(1+z_3)\big]^{\frac{1}{3}}}{(1+z_1)^{\frac{2}{3}}\sqrt{E(z_1,z_2,z_3)}}\,dz_1 + \frac{\big[(1+z_1)(1+z_3)\big]^{\frac{1}{3}}}{(1+z_2)^{\frac{2}{3}}\sqrt{E(z_1,z_2,z_3)}}\,dz_2\\+ \frac{\big[(1+z_1)(1+z_2)\big]^{\frac{1}{3}}}{(1+z_3)^{\frac{2}{3}}\sqrt{E(z_1,z_2,z_3)}}\,dz_3\Big\}.
\end{multline}

(v) Distance Modulus:
\begin{equation}\label{dist_mod}
D_m = 5 \log{d_L} + 25.
\end{equation}
These are some important expressions in general Bianchi Type I cosmology in
generic form, which will be useful to study the properties of the universe
in terms of the observable parameter $z$ for our considered cases as discussed
in the following sections.

\section{Customized scale factor approach for Bianchi Type I model}
\label{Model}
In this section we implement two different cases to the Bianchi type I model
for which we use two different types of directional scale factors to calculate
the various cosmological parameters as discussed below.
\subsection{\underline{Case I}}\label{s1}
In this case we consider the following set of directional scale factors:
\begin{equation}\label{scale_1}
a_1=a(t), \;\; a_2=\alpha\, a(t), \;\; \mbox{and}\;\; a_3=\beta\, a(t),
\end{equation}
where $\alpha$ and $\beta$ are two multiplicative constants. Since $\alpha$ and
$\beta$ are two time independent constants, the directional dependence of scale
factors or the anisotropy in the universe in this case should be considerable
only for very small values of the scale factor $a(t)$, i.e.\ for the
cosmological redshift $z\gg 1$. {The major advantage of these forms of
scale factors is that we can express every equation and cosmological parameter
in terms of a single scale factor multiplied by some constants. It helps to
simplify the problem and hence reduces the complexity.} The directional Hubble parameters for the above considered
set of scale factors take the same form as in the case of the isotropic
situation, i.e.
\begin{equation}\label{dir_hub_case_I}
H_1=H_2=H_3=\frac{\dot a}{a},
\end{equation}
and consequently the average Hubble parameter is
\begin{equation}\label{avg_hubble_case_I}
H=\frac{\dot a}{a}.
\end{equation}
Also it can be easily seen that
\begin{equation}
\frac{\ddot a_1}{a_1}=\frac{\ddot a_2}{a_2}=\frac{\ddot a_3}{a_3}=\frac{\ddot a}{a}.
\end{equation}
Thus the temporal component of the Einstein field equations, i.e.\ Eq.\
(\ref{temporal_component}) is transformed for this case as
\begin{equation}\label{temporal_case_I}
H^2=\frac{8\pi  \rho}{3}+\frac{\Lambda}{3}.
\end{equation}
Whereas all the three spatial components of the Einstein field equations, given
in Eqs.\ (\ref{spatial_component_I}), (\ref{spatial_component_II}),
(\ref{spatial_component_III}) take the same form for this case as
\begin{equation}\label{spatial_case_I}
2\,\frac{\ddot a}{a}+H^2=-\,8\pi P+\Lambda,
\end{equation}
and hence the combination of all the three spatial field equations leads to the
equation:
\begin{equation}\label{combined_comp_case_I}
6\,\frac{\ddot a}{a}+3H^2=-\,8\pi\, (3P)+3\Lambda.
\end{equation}
Now multiplication of Eq.\ (\ref{temporal_case_I}) by a factor $3$ and then
subtracting it from Eq.\ (\ref{combined_comp_case_I}) give rise a new equation:
\begin{equation}\label{friedmann_case_I}
\frac{\ddot a}{a}=-\frac{4\pi\,  (\rho+3P)}{3}+\frac{\Lambda}{3}.
\end{equation}
It should be pointed out that this Eq.\ (\ref{friedmann_case_I}) can also be
derived directly from Eq.\ (\ref{Friedmann_equation}).
Eqs.\ (\ref{temporal_case_I}) and
(\ref{friedmann_case_I}) are the two independent form of field equations.
Similarly the Eq.\ (\ref{continuity_equation}) i.e.\ the continuity equation
can be rewritten in this case as
\begin{equation}\label{continuity_case_I}
\dot{\rho}+3\,{\frac{\dot a}{a}}\,(\rho+P)=0.
\end{equation}
Again Eq.\ (\ref{density}), i.e.\ the expression of energy density for this case
becomes,
\begin{equation}\label{density_case_I}
\rho=(\alpha\beta)^{-(1+\omega)}{\rho_{0}}\,{a^{-3(1+\omega)}}.
\end{equation}
It is clear that this expression is different from the isotropic situation as
it depends on the parameters $\alpha$ and $\beta$. Hence, although the
expressions for directional and average Hubble parameters, and some other
expressions derived above look identical to the isotropic situation, they
would behave differently because of the different form of the energy density in
this case. This can be seen clearly from the expression
(\ref{Hubble_parameter}) for the Hubble parameter, which can rewritten for
this case as
\begin{equation}\label{hub_dens_case_I}
H= H_{0}{\sqrt{(\alpha\beta)^{-1}\Omega_{m0}\,a^{-3}+(\alpha\beta)^{-\frac{4}{3}}\Omega_{r0}\,a^{-4}+\Omega_{\Lambda 0}}}.
\end{equation}
Here $\Omega_{\sigma}=0$ as $\sigma^{2}=0$ from Eq.\ (\ref{shear_scalar_II})
for this case. We see that the matter and radiation parts of energy density
contributed differently in the Hubble parameter than the isotropic condition
because of the expression \eqref{density_case_I} for the energy density. This
situation will be similar to all other cosmological parameters discussed here.
The deceleration parameter of Eq.\ (\ref{dec_par_I}) in this case reduces to
\begin{equation}
q=-\frac{\ddot a}{aH^2}.
\end{equation}
And in terms of density parameter, $q$ can be rewritten as
\begin{equation}\label{q_dens_case_I_}
q=\Omega_r+\frac{1}{2}\,\Omega_m-\Omega_{\Lambda}.
\end{equation}

If we denote $z_1 = z$, the relations between the cosmological redshift
parameters and the directional scale factors given in Eq.\ \eqref{redshift_I}
will be transformed in this case as
\begin{equation}\label{a_Z_caseI}
a_1(t)=\frac{1}{1+z},\;\; {a_2(t)}=\frac{\alpha}{1+z},\;\;\mbox{and}\;\;
{a_3(t)}=\frac{\beta}{1+z}.
\end{equation}
Here, the other two redshift parameters $z_2$ and $z_3$ can be found as
$$z_2 = \frac{1+z}{\alpha} - 1,\;\; \mbox{and}\;\;
z_3 = \frac{1+z}{\beta} - 1.$$
Accordingly, the directional Hubble parameters can be written as
\begin{equation}
H_i= -\frac{\dot{z}}{1+z}.
\end{equation}
and hence the form of the expression of the average Hubble parameter becomes
exactly the same as the directional Hubble parameters' expression given above.
Thus, in terms of the cosmological redshift parameters
Eq.\ (\ref{hub_dens_case_I}) can be rewritten as
\begin{equation}\label{H_Z_caseI}
H= H_{0}{\sqrt{(\alpha \beta)^{-1} \Omega_{m0}(1+z)^{3}+(\alpha \beta)^{-\frac{4}{3}}\Omega_{r0}(1+z)^{4}+\Omega_{\Lambda 0}}}\;\; = H_{0}\sqrt{E(z)},
\end{equation}
where
\begin{equation}\label{E_caseI}
E(z)=(\alpha \beta)^{-1} \Omega_{m0}(1+z)^{3}+(\alpha \beta)^{-\frac{4}{3}}\Omega_{r0}(1+z)^{4}+\Omega_{\Lambda 0}.
\end{equation}
Consequently, the age of the universe given by Eq.\ (\ref{age_univ}) can be
rewritten for this case as
\begin{equation}\label{age_caseI}
t_{age}=\int_{0}^{t}dt=\frac{1}{H_0}\int_{0}^{\infty}\frac{dz}{(1+z)\sqrt{E(z)}}.
\end{equation}
Similarly, with the same approach the other cosmological parameters for the
present case can be obtained as a function of redshift parameters as\\

(i) Deceleration parameter:
\begin{equation}\label{q_Z_caseI}
q(z)=\frac{1}{\sqrt{E(z)}}\Big[\frac{1}{2}(\alpha \beta)^{-1}\Omega_{m0}(1+z)^{3}+(\alpha \beta)^{-\frac{4}{3}}\Omega_{r0}(1+z)^{4}-\Omega_{\Lambda 0}\Big].
\end{equation}

(ii) Equation of State:
\begin{equation}\label{eos_Z_caseI}
\omega(z) = \frac{\frac{1}{3}\Omega_{r0}(\alpha \beta)^{-\frac{4}{3}}(1+z)^{4}-\Omega_{\Lambda 0}}{(\alpha \beta)^{-1}\Omega_{m0}(1+z)^{3}+\Omega_{r0}(\alpha \beta)^{-\frac{4}{3}}(1+z)^{4}+\Omega_{\Lambda 0}}.
\end{equation}

(iii) Ricci Scalar:
\begin{equation}\label{R_Z_caseI}
R(z)= 3H_0^2\Big[(\alpha \beta)^{-1}\Omega_{m0}(1+z)^{3}+4\Omega_{\Lambda 0}\Big].
\end{equation}

(iv) Luminosity distance:
\begin{equation}\label{Lumin_Z_caseI}
d_L=\frac{(\alpha \beta)^{-\frac{2}{3}}(1+z)}{H_0}\int_{0}^{\infty} \frac{dz}{\sqrt{E(z)}}.
\end{equation}

(v) Density parameter of matter:
\begin{equation}\label{dens_mat_I}
\Omega_{m}(z) =  \frac{\Omega_{m0}(\alpha\beta)^{-1}(1+z)^{3}}{E(z)}.
\end{equation}

{These are the forms of expressions of the cosmological parameters we have
obtained for the given set of scale factors in Eq.~({\ref{scale_1}}). The
detailed analysis of these expressions are done graphically in
section {\ref{4}}}.

\subsection{\underline{Case II}}\label{s2}

In this case we consider that scale factors of the three dimensional universe
differ from each other by some constant amounts and hence here we consider the
following set of anisotropic scale factors:
\begin{equation}\label{scale1_caseII}
a_1=a(t),\;\; a_2=a(t)+\delta, \;\; \mbox{and}\;\; a_3=a(t)+\gamma,
\end{equation}
where $\delta$ and $\gamma$ are two constants.  {As in the previous case,
these form of scale factors help us to reduce the complexity and hence easier
to deal with the problem.} For convenience, this set of scale factors can also
be written as
\begin{equation}\label{scale1_caseII_ad}
a_1=a(t),\;\; a_2=a(t)f_{a\delta}, \;\; \mbox{and}\;\; a_3=a(t)f_{a\gamma},
\end{equation}
where $f_{a\delta} = 1+\delta/a(t)$ and $f_{a\gamma} = 1+\gamma/a(t)$. It is
clear that $f_{a\delta}, f_{a\gamma}\rightarrow 1$, when the scale factor
$a(t)\rightarrow \infty$ or the cosmological redshift $z\rightarrow -1$. On
the other hand when $a(t)\rightarrow 0$ or $z\gg1$, $f_{a\delta}$, $f_{a\gamma} \gg 1$. Thus, according to this case the universe will be almost in an
isotropic state in the distant future and the anisotropy in the universe was
significant in the distant past as in the case I. With this set of scale
factors \eqref{scale1_caseII_ad} the directional Hubble parameters of the
anistropic universe becomes:
\begin{equation}\label{dir_hub1_caseII}
H_1=\frac{\dot a}{a},\;\;
H_2=\frac{H_1}{f_{a\delta}},\;\;
H_3=\frac{H_1}{f_{a\gamma}}
\end{equation}
and hence the average Hubble parameter can be written as
\begin{equation}\label{avg_hub_caseII}
H=\frac{H_1}{3}\Big[1 + f_{a\delta}^{-1} + f_{a\gamma}^{-1} \Big].
\end{equation}
Again,
\begin{equation}
\frac{\ddot a_1}{a_1}=\frac{\ddot a}{a},\;\;
\frac{\ddot a_2}{a_2}=\frac{\ddot a}{a}\,f_{a\delta}^{-1},\;\;
\frac{\ddot a_3}{a_3}=\frac{\ddot a}{a}\,f_{a\gamma}^{-1}.
\end{equation}
The Einstein field equations as mentioned in Eqs.\ (\ref{temporal_component}),
(\ref{spatial_component_I}), (\ref{spatial_component_II}) and
(\ref{spatial_component_III}) are transformed for the present case as
\begin{align}\label{temp_caseII}
H_1^2\Big[f_{a\delta}^{-1} + f_{a\gamma}^{-1} + \big(f_{a\delta}f_{a\gamma}\big)^{-1}\Big]& = 8\pi \rho+\Lambda,\\[5pt]
\label{spatial1_caseII}
\frac{\ddot{a}}{a}\Big[f_{a\delta}^{-1} + f_{a\gamma}^{-1}\Big] + H_1^2\big(f_{a\delta}f_{a\gamma}\big)^{-1} & = -\,8\pi P+\Lambda,\\[5pt]
\label{spatial2_caseII}
\frac{\ddot{a}}{a}\Big[1+ f_{a\gamma}^{-1}\Big] + H_1^2\,f_{a\gamma}^{-1} & = -\,8\pi P+\Lambda,\\[5pt]
\label{spatial3_caseII}
\frac{\ddot{a}}{a}\Big[1+ f_{a\delta}^{-1}\Big] + H_1^2\,f_{a\delta}^{-1} & = -\,8\pi P+\Lambda.
\end{align}
And Eq.\ (\ref{Friedmann_equation}) can be rewritten as
\begin{equation}\label{friedmann_caseII}
\frac{\ddot{a}}{a}\Big[1 + f_{a\delta}^{-1} + f_{a\gamma}^{-1} \Big] =-\,4\pi (3P+\rho)+\Lambda.
\end{equation}
The continuity equation for this case becomes,
\begin{equation}\label{cont_caseII}
\dot{\rho}+ \big(\rho+P\big)H_1\Big[1 + f_{a\delta}^{-1} + f_{a\gamma}^{-1}\Big] = 0
\end{equation}
and the energy density Eq.\ (\ref{density}) can be rewritten as
\begin{equation}\label{dens_caseII}
\rho=\rho_0 \big(f_{a\delta}f_{a\gamma})^{-(1+\omega)}a^{-3(1+\omega)}.
\end{equation}
Similarly, the expression of shear scalar takes the form:
\begin{equation}\label{shear_scalar_caseII}
\sigma^2=\frac{H_1^2}{3}\bigg[1 + f_{a\delta}^{-2} + f_{a\gamma}^{-2} -\Big(f_{a\delta}^{-1} + f_{a\gamma}^{-1} + \big(f_{a\delta}f_{a\gamma}\big)^{-1}\Big)\bigg].
\end{equation}
This expression of shear scalar can be used to calculate the density parameter
for anisotropy, $\Omega_{\sigma}=\sigma^2/3H^2$.
The Hubble parameter expression (\ref{Hubble_parameter}) for this case can be
found as
\begin{equation}\label{hubble_density_caseII}
H=H_0\sqrt{\Omega_{mo}\big(f_{a\delta}f_{a\gamma}\big)^{-1}a^{-3} +\Omega_{ro}\big(f_{a\delta}f_{a\gamma}\big)^{-\frac{4}{3}} a^{-4}+\Omega_{\Lambda 0}+\Omega_{\sigma 0}\big(f_{a\delta}f_{a\gamma}\big)^{-2}a^{-6}}
\end{equation}
The deceleration parameter of Eq.\ (\ref{dec_par_I}) for the present case
takes the form:
\begin{equation}\label{q_caseII}
q=-\frac{1}{3H^2}\frac{\ddot a}{a}\Big[1+f_{a\delta}^{-1} + f_{a\gamma}^{-1}\Big]+2\,\Omega_{\sigma}.
\end{equation}
However, the expression of $q$ in terms of density parameters is same as
Eq.\ (\ref{dec_par_II}).

As in the case I, in this case also all cosmological parameters need to be
expressed in terms of the cosmological redshift parameter, which is an
observable quantity. The relation between cosmological redshift parameters and
directional scale factors in this present case will take the form:
\begin{equation}\label{a_Z_caseII}
a_1(t)=\frac{1}{1+z},\;\; {a_2(t)}=\frac{1}{1+z}+\delta,\;\; \mbox{and}\;\;
{a_3(t)}={\frac{1}{1+z}}+\gamma.
\end{equation}
As earlier, for the convenient notational purpose this set of scale factors can
 also be re-expressed as
\begin{equation}\label{a_Z_caseIIad}
a_1(t) = \frac{1}{1+z},\;\; {a_2(t)}=\frac{f_{z\delta}}{1+z},\;\; \mbox{and}\;\;
{a_3(t)}={\frac{f_{z\gamma}}{1+z}},
\end{equation}
where $f_{z\delta} = 1 + (1+z)\delta$ and $f_{z\gamma} = 1 + (1+z)\gamma$.
Indeed, the behaviours of $f_{z\delta}$ and $f_{z\gamma}$ are exactly same
as $f_{a\delta}$ and $f_{a\gamma}$ for different values of $z$ as discussed
above. Here the directional cosmological redshift parameters $z_2$ and $z_3$
can be obtained as
$$z_2 = \frac{1+z}{f_{z\delta}} - 1, \;\; \mbox{and}\;\;
z_3 = \frac{1+z}{f_{z\gamma}} - 1.$$
For this set of scale factors the directional Hubble parameters can be written
as
\begin{equation}\label{dir_hub1_Z_caseII}
H_1= -\frac{\dot{z}}{1+z},\;\;H_2= -\,\frac{\dot{z}}{(1+z)f_{z\delta}},\;\;
H_3= -\,\frac{\dot{z}}{(1+z)f_{z\gamma}}.
\end{equation}
Thus the expression of average Hubble parameter takes the form:
\begin{equation}\label{avg_hub_caseII}
H=-\frac{\dot{z}}{3(1+z)}\Big[1+f_{z\delta}^{-1}+ f_{z\gamma}^{-1}\Big].
\end{equation}
In view of the above set of scale factors, Eq.\ (\ref{hubble_density_caseII})
can now be rewritten as
\begin{equation}\label{H_Z_caseII}
H= H_{0}\sqrt{E(z)},
\end{equation}
where
\begin{equation}
E(z) = \Omega_{m0}\big(f_{z\delta}f_{z\gamma}\big)^{-1}(1+z)^3 +\Omega_{r0}\big(f_{z\delta}f_{z\gamma}\big)^{-\frac{4}{3}}(1+z)^4 + \Omega_{\Lambda 0}+
\Omega_{\sigma 0}\big(f_{z\delta}f_{z\gamma}\big)^{-2} (1+z)^{6}.
\end{equation}
Correspondingly, the age of the universe can be written as
\begin{equation}\label{age_univ_caseII}
t_{age}=\frac{1}{H_0}\int_{0}^{\infty}\frac{dz}{(1+z)\sqrt{E(z)}}-\frac{1}{3H_0}\int_{0}^{\infty}\Big[\frac{\delta}{f_{z\delta}\sqrt{E(z)}}+ \frac{\gamma}{f_{z\gamma}\sqrt{E(z)}}\Big]dz.
\end{equation}
As before, other cosmological parameters in terms of redshift are as follows:\\

(i) Deceleration parameter:
\begin{equation}\label{dec_Z_caseII}
q(z)= \frac{1}{\sqrt{E(z)}}\Big[\frac{1}{2}\,\Omega_{m0}\big(f_{z\delta}f_{z\gamma}\big)^{-1}(1+z)^{3}+\Omega_{r0}\big(f_{z\delta}f_{z\gamma}\big)^{-\frac{4}{3}}(1+z)^{4}+\Omega_{\Lambda 0}+2\Omega_{\sigma 0}\big(f_{z\delta}f_{z\gamma}\big)^{-2}(1+z)^{6}\Big].
\end{equation}

(ii) Equation of State:
\begin{equation}\label{eos_Z_caseII}
\omega(z) = \frac{\frac{1}{3}\Omega_{r0}\big(f_{z\delta}f_{z\gamma}\big)^{-\frac{4}{3}}(1+z)^{4}-\Omega_{\Lambda 0}}{\Omega_{m0}\big(f_{z\delta}f_{z\gamma}\big)^{-1}(1+z)^{3}+\Omega_{r0}\big(f_{z\delta}f_{z\gamma}\big)^{-\frac{4}{3}}(1+z)^{4}+\Omega_{\Lambda 0}}.
\end{equation}

(iii) Ricci Scalar:
\begin{equation}\label{R_Z_caseII}
R(z)= 3H_0^2\Big[\Omega_{m0}\big(f_{z\delta}f_{z\gamma}\big)^{-1}(1+z)^{3}+4\,\Omega_{\Lambda 0}\Big].
\end{equation}

(iv) Luminosity distance:
\begin{equation}\label{lumin_Z_caseII}
d_L(z)=\frac{\big(f_{z\delta}f_{z\gamma}\big)^{-\frac{1}{3}}(1+z)}{H_0}\int_{0}^{\infty}\bigg[\frac{1}{\big(f_{z\delta}f_{z\gamma}\big)^{\frac{1}{3}}\sqrt{E(z)}}-\frac{\delta (1+z)}{3\,f_{z\delta}^{\frac{4}{3}}f_{z\gamma}^{\frac{1}{3}}\sqrt{E(z)}}-\frac{\gamma (1+z)}{3\,f_{z\delta}^{\frac{1}{3}}f_{z\gamma}^{\frac{4}{3}}\sqrt{E(z)}}\bigg]dz.
\end{equation}

(vi) Density parameter for anisotropy:
\begin{equation}\label{density_sigma_Z_caseII}
\Omega_{\sigma} = \frac{AB}{C E(z)},
\end{equation}
where $A = \Omega_{m0}\big(f_{z\delta}f_{z\gamma}\big)^{-1}(1+z)^{3}+\Omega_{r0}\big(f_{z\delta}f_{z\gamma}\big)^{-\frac{4}{3}} (1+z)^{4} + \Omega_{\Lambda 0}$,
$B = 1+f_{z\delta}^{-2}+f_{z\gamma}^{-2}-\Big[f_{z\delta}^{-1}+f_{z\gamma}^{-1}+\big(f_{z\delta}f_{z\gamma}\big)^{-1}\Big]$ and
$ C = f_{z\delta}^{-1}+f_{z\gamma}^{-1}+\big(f_{z\delta}f_{z\gamma}\big)^{-1}$.
\\

(vii) Density parameter for matter:
\begin{equation}\label{dens_mat_II}
\Omega_m(z) = \frac{\Omega_{m0}\big(f_{z\delta}f_{z\gamma}\big)^{-1}(1+z)^{3}}{E(z)}.
\end{equation}

{As in the previous case, we have obtained these forms of expressions of
various cosmological parameters by considering the scale factors as
given in Eq.~{\eqref{scale1_caseII_ad}}. The detailed graphical analysis
of these expressions are presented in the next section.}

\section{Graphical analysis and results}\label{4}

In this section we focus on the graphical analysis of various cosmological
parameters discussed above and try to give possible explanations of the
results obtained from the emerging graphs. Here, in the numerical calculations, we have taken
the Planck 2018 results on cosmological parameters \citep{Planck_2018}.

\subsubsection{\textbf{Hubble parameter}}
To understand the behaviour of Hubble parameter $H(z)$ with respect to $z$, we
first plot the values of $H(z)$ with $1+z$ over a large range of its values for three different sets of values of $\alpha$ and $\beta$ for the case I and
two sets of values of $\gamma$ and $\delta$ for the case II as shown in the
Fig.\ \ref{fig1a} by using Eqs.\ \eqref{H_Z_caseI} and \eqref{H_Z_caseII}
respectively. As mentioned earlier, it is seen from the figure that both the
cases deviate significantly from the isotropic universe ($\Lambda$CDM model) in
the remote past, whereas they agree with the isotropic universe at present and
at distant future epochs. However, there is a difference of variation
patterns of $H(z)$ in between these two cases in the past. In the case I, the
values of $H(z)$ can be greater or smaller than its corresponding values in
the $\Lambda$CDM model in the past depending on the values of parameters
$\alpha$ and $\beta$. On the other hand in the case II, the values of $H(z)$
never greater than its corresponding values in the $\Lambda$CDM model in the
past for any set of values of parameters $\gamma$ and $\delta$. Also we have
noticed that, more the values of $\alpha$ and $\beta$ moves towards one, higher
the tendency of the Hubble parameter plot in case I to move towards the
$\Lambda$CDM model. Whereas, smaller values of $\delta$ and $\gamma$ show
excellent agreement with the $\Lambda$CDM plot.
\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{H_Z_case_I_revised.eps}\hspace{1cm}
  \includegraphics[scale = 0.35]{hubble_cs_II_large_z_mod_revised.eps}}
\caption{Behaviour of Hubble parameter $H(z)$ with respect to $1+z$ for the
case I (left) and case II (right) with different associated model parameters.}
\label{fig1a}
\end{figure}

Moreover, to choose a reliable range of values of model parameters, i.e.\
$\alpha, \beta$ for the case I and $\gamma, \delta$ for the case II, we have
used four sets of available $H(z)$ data, viz., HKP and SVJ05 data
\cite{Simon_2005}, SJVKS10 data \cite{Stern_2010} and GCH09 data
\cite{Gaztanaga_2009} in the $H(z)$ versus $z$ plots of Fig.\ \ref{fig1b} to
constrained the model parameters within the observable range of $z$ values.
Since the SVJ05 data set has been replaced already by SJVKS10 data, we have
taken this data set only as a reference \cite{Ma_2011}. From this figure, we
have found a good range of values of the product of the two free parameters
(i.e.\ $\alpha\beta$) from 0.6 to 1.21 for the case I. For different sets of
values of $\alpha$ and $\beta$, whose products lie in between the above
mentioned range of values will give us consistent results with the
observational data. In this paper, we have taken three different sets of values
of $\alpha$ and $\beta$: (0.75, 0.8), (0.905, 0.954) and (1.1, 1.1) as shown
in the left plot of Fig.\ \ref{fig1b}. For the case II, we have found a best
possible range of values of $\gamma$ and $\delta$ as (0.001, 0.0095) to
(0.1, 0.05), within which any set of values of $\gamma$ and $\delta$ gives
results that fit to data within range of error bars. It is also seen that the
set $\gamma = 0.001$ and $\delta = 0.0095$ produces the results that almost
agree with the results of the $\Lambda$CDM model upto the values of $z\sim 1.2$.\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{Hubble_case_I.eps}\hspace{1cm}
  \includegraphics[scale = 0.35]{Hubble_caseII_mod_small.eps}}
\caption{Behaviour of Hubble parameter $H(z)$ for the small $z$ values with
different values of model parameters in conformity with four sets of observed
data within the available ranges of errors for the case I (left) and case II
(right).}
\label{fig1b}
\end{figure}
\begin{figure}[!htb]
 \centerline{
   \includegraphics[scale = 0.35]{fitting_plot_mod.eps}}
\caption{Least square fitting curve and the best fitted curve of $H(z)$
obtained from Eq.\ \eqref{H_Z_caseII} for the values of the parameters
$\delta$ and $\gamma$ as 0.001 and 0.0095 respectively to the observed Hubble
parameter data set shown in Table \ref{tableI}.}
\label{fig2}
\end{figure}

Further, to get the precise values of above free parameters for both the cases,
which should be consistent with the $\Lambda$CDM model at least around the
present epoch, we have considered the compilation of 43 observational Hubble
parameter data against 43 different values of redshift \cite{Akarsu_2019,
Gogoi_2021} as shown in Table \ref{tableI}. We have plotted the best fitted
curve to this set of Hubble parameter data against the redshift by using a
non-linear curve fitting (least square fitting) technique as shown in
Fig.\ \ref{fig2}. From the curve we have calculated the value of
$\alpha\beta = 0.86275$ for the case I at $z=0$. It is important to mention
that we may be able to consider a set of values of $\alpha$ and $\beta$, which
satisfy the condition that $\alpha\beta = 0.86275$. Thus, we can use different
sets of $\alpha$ and $\beta$ with this condition. Here, we have not consider
any special set of values of $\alpha$ and $\beta$, but just considered the
condition $\alpha\beta = 0.86275$ in the rest of the paper. However, it is not
easy to get the values of the parameters $\delta$ and $\gamma$ for the case II
using this method. So, for the case II we have used a suitable set of values of
$\delta$ and $\gamma$, for which the expression of $H(z)$ for this case
(Eq.\ \eqref{H_Z_caseII}) gives a curve very close to the least square fitting
curve to the observational data as shown in Fig.\ \ref{fig2}. This set of
values of $\delta$ and $\gamma$ are found to be 0.001 and 0.0095 respectively,
which is the same as the lower limit values of these parameters found from the
analysis with Fig.\ \ref{fig1b}. Thus the graphical plots of $H(z)$ for both the
cases obtained from these derived free parameters values as mentioned above
should be consistent with the $\Lambda$CDM plot shown in Fig.\ \ref{fig1b},
especially for the range of small $z$.
\begin{center}
\begin{table}[!hbt]
\caption{Presently available observational Hubble parameter ($H^{obs}(z)$)
data set. $H^{obs}(z)$ is in unit of km/s/Mpc.}
\vspace{2mm}
\begin{tabular}{ccc|ccc}
\hline
\rule[-1ex]{0pt}{2.5ex} \hspace{0.5cm} $z$ \hspace{0.5cm}  & \hspace{0.5cm} $ H^{obs}(z)$ \hspace{0.5cm} & \hspace{0.5cm} Reference \hspace{0.5cm} &
\hspace{0.5cm} $z$ \hspace{0.5cm}  & \hspace{0.5cm} $ H^{obs}(z)$ \hspace{0.5cm} & \hspace{0.5cm} Reference \hspace{0.5cm} \\
\hline
\rule[-1ex]{0pt}{2.5ex} 0.0708 & 69.0 $\pm$ 19.68 & \cite{Zhang_2014} &
0.48 & 97.0 $\pm$ 62.0 & \cite{Ratsimbazafy_2017}\\
\rule[-1ex]{0pt}{2.5ex} 0.09 & 69.0 $\pm$ 12.0 & \cite{Simon_2005} &
0.51 & 90.8 $\pm$ 1.9 & \cite{Alam_2017}\\
\rule[-1ex]{0pt}{2.5ex} 0.12 & 68.6 $\pm$ 26.2 & \cite{Zhang_2014} &
0.57 & 92.4 $\pm$ 4.5 & \cite{Samushia_2013} \\
\rule[-1ex]{0pt}{2.5ex} 0.17 & 83.0 $\pm$ 8.0 & \cite{Simon_2005} &
0.593 & 104.0 $\pm$ 13.0 & \cite{Moresco_2012}\\
\rule[-1ex]{0pt}{2.5ex} 0.179 & 75.0 $\pm$ 4.0 & \cite{Moresco_2012} &
0.60 & 87.9 $\pm$ 6.1 & \cite{Blake_2012}\\
\rule[-1ex]{0pt}{2.5ex} 0.199 & 75.0 $\pm$ 5.0 & \cite{Moresco_2012} &
0.61 & 97.8 $\pm$ 2.1 & \cite{Alam_2017}\\
\rule[-1ex]{0pt}{2.5ex} 0.20 & 72.9 $\pm$ 29.6 & \cite{Zhang_2014} &
0.68 & 92.0 $\pm$ 8.0 & \cite{Moresco_2012}\\
\rule[-1ex]{0pt}{2.5ex} 0.24 & 79.69 $\pm$ 2.65 & \cite{Gaztanaga_2009} &
0.73 & 97.3 $\pm$ 7.0 & \cite{Blake_2012}\\
\rule[-1ex]{0pt}{2.5ex} 0.27 & 77.0 $\pm$ 14.0 & \cite{Simon_2005} &
0.781 & 105.0 $\pm$ 12.0 & \cite{Moresco_2012}\\
\rule[-1ex]{0pt}{2.5ex} 0.28 & 88.8 $\pm$ 36.6 & \cite{Zhang_2014} &
0.875 & 125.0 $\pm$ 17.0 & \cite{Moresco_2012}\\
\rule[-1ex]{0pt}{2.5ex} 0.35 & 84.4 $\pm$ 7.0  & \cite{Xu_2013} &
0.88 & 90.0 $\pm$ 40.0 & \cite{Ratsimbazafy_2017}\\
\rule[-1ex]{0pt}{2.5ex} 0.352 & 83.0 $\pm$ 14.0 & \cite{Moresco_2012}&
0.90 & 117.0 $\pm$ 23.0 & \cite{Simon_2005}\\
\rule[-1ex]{0pt}{2.5ex} 0.38 & 81.9 $\pm$ 1.9 & \cite{Alam_2017} &
1.037 & 154.0 $\pm$ 20.0 & \cite{Moresco_2012}\\
\rule[-1ex]{0pt}{2.5ex} 0.3802 & 83.0 $\pm$ 13.5 & \cite{Moresco_2016} &
1.30 & 168.0 $\pm$ 17.0 & \cite{Simon_2005}\\
\rule[-1ex]{0pt}{2.5ex} 0.40 & 95.0 $\pm$ 17.0 & \cite{Simon_2005} &
1.363 & 160.0 $\pm$ 33.6 & \cite{Moresco_2015}\\
\rule[-1ex]{0pt}{2.5ex} 0.4004 & 77.0 $\pm$ 10.2 & \cite{Moresco_2016} &
1.43 & 177.0 $\pm$ 18.0 & \cite{Simon_2005}\\
\rule[-1ex]{0pt}{2.5ex} 0.4247 & 87.1 $\pm$ 11.2 & \cite{Moresco_2016} &
1.53 & 140.0 $\pm$ 14.0 & \cite{Simon_2005}\\
\rule[-1ex]{0pt}{2.5ex} 0.43 & 86.45 $\pm$ 3.68 & \cite{Gaztanaga_2009} &
1.75 & 202.0 $\pm$ 40.0 & \cite{Simon_2005}\\
\rule[-1ex]{0pt}{2.5ex} 0.44 & 82.6 $\pm$ 7.8 & \cite{Blake_2012} &
1.965 & 186.5 $\pm$ 50.4 & \cite{Moresco_2015}\\
\rule[-1ex]{0pt}{2.5ex} 0.4497 & 92.8 $\pm$ 12.9 & \cite{Moresco_2016} &
2.34 & 223.0 $\pm$ 7.0 & \cite{Delubac_2015}\\
\rule[-1ex]{0pt}{2.5ex} 0.47 & 89.0 $\pm$ 50.0 & \cite{Ratsimbazafy_2017} &
2.36 & 227.0 $\pm$ 8.0 & \cite{Ribera_2014}\\
\rule[-1ex]{0pt}{2.5ex} 0.4783 & 80.9 $\pm$ 9.0 & \cite{Moresco_2016} &&&\\
\hline
\end{tabular}
\label{tableI}
\end{table}
\end{center}

\subsubsection{\textbf{Ricci scalar}}
Ricci scalar is an important parameter to specify the geometric property of
spacetime of the universe. {Some mathematical and graphical analysis of
the evolution of Ricci scalar for various cosmological models in modified
cosmology are also found in Ref.} \cite{Gogoi_2021}. So, to understand the
evolution of Ricci scalar in
our anisotropic model of the universe, we have plotted Ricci scalar $R(z)$
against the redshift ($z$) for the case I and case II by using Eqs.\
(\ref{R_Z_caseI}) and (\ref{R_Z_caseII}) respectively as shown in
Fig.\ \ref{fig3}. In this plot we have used the observationally constrained
value of $\alpha\beta = 0.86275$ for the case I, and $\gamma = 0.001$ and
$\delta = 0.0095$ for the case II as mentioned in the previous subsection.
Here, we have also shown the plot of this cosmological parameter for the
$\Lambda$CDM model. Figure shows that both the cases of our model are
consistent with the $\Lambda$CDM model from the present epoch to the near past.
However, the deviation of both the cases increases with higher curvature of
spacetime for the case I and with lower curvature for the case II, from the
$\Lambda$CDM model as we move towards the distant past.

\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{R_Z_cs_I.eps}\hspace{1cm}
  \includegraphics[scale = 0.35]{R_Z_cs_II_mod.eps}}
\caption{Ricci scalar versus redshift plots for the case I (left) and case II
(right) of the proposed anisotropic model of the universe with the constrained
values of free model parameters along with the plot for the $\Lambda$CDM model.}
\label{fig3}
\end{figure}

\subsubsection{\textbf{Deceleration Parameter and Equation of State}}
As the deceleration parameter $q(z)$ is an important parameter to understand
the pattern of evolution of the universe, we have plotted it with respect to
$1+z$ for the both the cases of our model for the constrained values of
the free parameters as mentioned above by using Eqs.\ (\ref{q_Z_caseI}) and
(\ref{dec_Z_caseII}) along with the plot for the $\Lambda$CDM model as shown in
Fig.\ \ref{fig4}. It is found that for both the cases, at $z$ = 0, $q(0)$
tends to $-0.55$, which is a good agreement with the $\Lambda$CDM model's
prediction for the current value of $q(z)$. In reality for any value of $z$,
$q(z)$ for both the cases shows a good agreement with the $\Lambda$CDM model,
especially the case II has the excellent agreement. This implies that the
pattern of evolution of our anisotropic model of universe is same as that of
the isotropic model.

\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{q_z_cs_I.eps}\hspace{1cm}
  \includegraphics[scale = 0.35]{q_z_cs_II_mod.eps}}
\caption{Deceleration parameter versus redshift plots for the case I (left) and
case II (right) of anisotropic model of the universe with the constrained
values free model parameters along with the plot for the $\Lambda$CDM model.}
\label{fig4}
\end{figure}
Similarly, the Equation of State (EoS) is an important cosmological parameter
to understand the composition of the universe as well as its characteristics.
We have plotted the EoS ($\omega$) against $1+z$ for both the cases by
using Eq.\ (\ref{eos_Z_caseI}) and (\ref{eos_Z_caseII}) with the constrained
values of free model parameters as shown in Fig.\ \ref{fig5}. We found that
at $z=0$, EoS $\omega$ is close to $-0.7$ for both the cases, which is also
consistent with the $\Lambda$CDM model and agrees with the current era of the
universe as the dark energy era. The curves show good agreement with the
$\Lambda$CDM model (in fact it is excellent for the case II) and hence support
the three phases of the universe, viz., the radiation dominated phase
($\omega= 1/3$), the matter dominated phase ($\omega =0$) and the dark energy
phase ($\omega=-1$). Thus this anisotropic model of the universe has the same
evolution characteristics as the isotropic model of the universe.

\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{eos_cs_I.eps}\hspace{1cm}
  \includegraphics[scale = 0.35]{Eos_cs_II_mod.eps}}
\caption{Equation of State parameter with respect to redshift for the
case I (left) and case II (right) of anisotropic model of the universe with
the constrained values free model parameters along with the plot for the
$\Lambda$CDM model.}
\label{fig5}
\end{figure}
\subsubsection{\textbf{Distance modulus}}
The distance modulus is a reliable observational parameter to understand the
cosmic evolution with redshift. Here, we have plotted the distance modulus
($D_m$) against redshift for both the cases with the free model parameters as
mention above by using Eqs.\ (\ref{Lumin_Z_caseI}) and (\ref{lumin_Z_caseII})
in Eq.\ (\ref{dist_mod}) along with the Union2.1 observational data
\cite{Suzuki_2012} and $D_m$ for the $\Lambda$CDM model as shown in Fig.\
\ref{fig6}. It is clear from the figure that for both the cases, for the
constrained free parameter values, the distance modulus plot shows excellent
agreement with observational data along with the $\Lambda$CDM plot. This
implies that the anisotropic model of the universe agrees very well with
the observed data of the distance modulus of galaxies.

\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{Dm_Z_cs_I.eps}\hspace{1cm}
  \includegraphics[scale = 0.35]{Dm_Z_cs_II_mod.eps}}
\caption{Distance modulus versus redshift plots for the case I (left) and
case II (right) of anisotropic model of the universe with the constrained
values free model parameters in comparison with Union2.1 data along with the
plot for the $\Lambda$CDM model.}
\label{fig6}
\end{figure}

\subsubsection{\textbf{Anisotropy density parameter}}
The anisotropy density parameter is an important parameter to understand
the possible anisotropy that exists in the universe. For the case I it's value
is zero, but for the case II it has a definite expression as mentioned in
Eq.\ (\ref{density_sigma_Z_caseII}). Fig.\ \ref{fig7} shows the plot of this
$\Omega_\sigma$ against $1+z$ for the constrained values of $\delta$ and
$\gamma$. This figure suggests that a considerable amount of
anisotropy exists for a much higher value of redshift, which indicates that the
universe had prominent anisotropic character in its early stages and which is
now almost reduced to  zero. {The existence of anisotropy after inflation
period can be explained with the idea like breaking the Lorentz invariance by
introducing a condensation of vector field} \cite{Kanno_2008}{, slow roll
phase of vector fields like as inflation field in chaotic inflationary
scenario,} \cite{Golovnev_2008,Kanno_2008,Watanabe_2009} {or considering
anisotropic inflation model with vector impurity}\cite{Kanno_2008} {etc.
However, this is a broad area of study and hence to be considered latter.} In
Ref.\ \cite{Akarsu_2019}, the upper limits
of the present value of the anisotropy density parameter ($\Omega_{\sigma 0}$)
has been calculated by using the BAO and CMB data for the matter dominated
recombination era as $\Omega_{\sigma 0}\le 10^{-15}$, and by demanding that
the standard big bang nucleosynthesis (BBN) is not significantly affected by the
expansion anisotropy as $\Omega_{\sigma 0}\le 10^{-23}$. From these two upper
limits of $\Omega_{\sigma 0}$ we have calculated the corresponding values of
the free parameters $\delta$ and $\gamma$ of the case II, and found that their
values may lie in between $10^{-3}$ and $10^{-8}$. It should be noted that this
calculated upper limit of values of $\delta$ and $\gamma$ is in agreement with
their constrained values obtained from the Hubble parameter data as mentioned
above.
\begin{figure}[!htb]
\centerline{
  \includegraphics[scale=0.35]{sigma_modified.eps}}
\caption{Anisotropy density parameter versus redshift plot for the case II
with the constrained values of free model parameters $\gamma = 0.001$ and
$\delta = 0.0095$.}
\label{fig7}
\end{figure}

\subsubsection{\textbf{Relative shear anisotropy parameter}}
The relative shear anisotropy parameters $r$ and $s$ can be defined as \cite{Barrow_1997}
$$r = \frac{H_1-H_2}{H},\;\; s = \frac{H_1-H_3}{H},$$

where as mentioned above $H_1, H_2, H_3$ are the directional Hubble parameters
and $H$ is the average Hubble parameter. The parameters $r$ and $s$ provide
the information about if the anisotropy exists in different directions and
their plot against redshift gives the picture about the evolution of
directional anisotropy. Fig.\ \ref{fig8} shows the plots of both the
parameters for case II. The figure shows that there was a considerable amount
of relative shear anisotropy present for the higher value of redshift $(z)$.
Thus the model predicts the existence of higher value of relative shear
anisotropy in the early stage of the universe. However, for the case I both
these parameters vanish. Thus for case I, the universe has no relative shear
anisotropy.
\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{rel_anisotropy_mod.eps}}
\caption{Relative shear anisotropy versus redshift plot for the case II.}
\label{fig8}
\end{figure}
\subsubsection{\textbf{Density parameter of matter}}
To visualize how the matter density parameter varies from early stages to
present epoch according to our anisotropic model of the universe, we have
plotted the $\Omega_{m}/\Omega_{mo}$ against the redshift in Fig.\ \ref{fig9}
by using Eq.\ (\ref{dens_mat_I}) and (\ref{dens_mat_II}) for the case I and
case II respectively for the constrained values of free parameters for each
cases. The left panel of Fig.\ \ref{fig9} is for case I and the right
panel is for case II. We have plotted this parameter for both the cases
against the value of $z$ up to $10^{6}$. From the figure, we have seen that
there is a matter dominated era up to $z$ $\sim 10^{3}$ for both the cases and
it supports the matter dominated region $0.5 \leq z \leq 3000$ as mentioned in
Ref.\ \cite{Frieman_2008}. For very high values of $z$, the density parameter
for matter drops down considerably close to zero. It suggests that the universe
was radiation dominated at it's early stage. Thus this anisotropic model of the
universe agrees well with the observed or predicted evolution nature of the
standard cosmology. However, there are differences between the matter density
parameter evolution in case I and case II. Firstly, falling of this
parameter for very high $z$ values is not smooth in the case II in comparison
to in the case I. Secondly, in the case I the ratio $\Omega_{m}/\Omega_{mo}$
moves gradually towards one as $z$ tends to zero as expected, but in the case
II this ratio reached to the value one before $z$ reached its zero value.

\begin{figure}[!htb]
\centerline{
  \includegraphics[scale = 0.35]{Om_enlrg_cs_I.eps}\hspace{1cm}
  \includegraphics[scale = 0.35]{Om_z_cs_II_L_mod.eps}}
\caption{Variation of matter density parameter with respect to redshift for
case I (left) and case II (right) with the constrained set of free model
parameters.}
\label{fig9}
\end{figure}

\subsubsection{\textbf{Age of the universe}}
According to the standard model of cosmology ($\Lambda$CDM) the age of the
universe is calculated to be $\sim$ 13.786 Gyr. To find the age of the universe
in our model we have calculated it by using Eqs.\ \eqref{age_caseI} and
\eqref{age_univ_caseII} for the case I and case II respectively with the
constrained sets of corresponding free parameters. We found that according to
case I the age of the universe is $\sim$ 13.084 Gyr and according to case II
it is $\sim$ 13.741 Gyr. Thus we see that case II result agrees very well with
the results of the standard model of cosmology with a deviation of only 0.33\%.
Whereas the deviation of the result in case I is 5.09\%.

\section{Conclusions}\label{5}
In this study we have considered the simplest Bianchi type I metric with slight
modifications in directional scale factors for two different cases to understand
the possible anisotropy that may exist in the universe. Here, we have used the
generalized equations of Bianchi type I cosmological model and implemented them
in both the considered cases. Starting from the field equations, we have
construct all the cosmological parameters including average Hubble parameter,
luminosity distance, distance modulus, deceleration parameter, equation of
state, shear scalar and various density parameters etc.\ for both the cases.
After the mathematical constructions, we have tried to plot various parameters
with respect to cosmological redshift and compare the results with
$\Lambda$CDM model which is the best fitted model to cosmological data till
date. Also we compare the results with various observational data like HKP
data, SVJ05 data, SJVKS10 data and GCH09 data for the Hubble parameter and with
Union 2.1 data for the distance modulus. Further, we have used observational
data of Hubble parameter given in Table \ref{tableI} that are obtained
from various sources as mentioned in the table to constrained the free
parameters of both the cases i.e. $\alpha$ and $\beta$ for the case I, and
$\delta$ and $\gamma$ for the case II as shown in Fig.\ref{fig2}.
{In this context, explicitly we should state that we have used the
observational Hubble parameter data to find out the best possible values of
model parameters as mentioned above. For the case I, we have equated
Eq.}~\eqref{H_Z_caseI} {with $H_0$ for $z=0$ using the Planck $2018$ data}
\cite{Planck_2018} {for the same and found that $\alpha \beta = 0.86275$.
Thus any set of values of $\alpha$
and $\beta$ which satisfies this condition are eligible for consideration in
the case I. In our work, we have not considered any special set of values of
$\alpha$ and $\beta$ but taken the condition as a whole for the graphical
analysis. However, for case II this method is not suitable as the
expression for the Hubble parameter in this case is complicated as compared to
the case I. Therefore we have fitted the 43 observational Hubble parameter
data in Table} \ref{tableI} {using the least square fitting technique.
Then we have changes our model parameters $\delta$ and $\gamma$ in
Eq.}~\eqref{H_Z_caseII} {for $H(z)$ of the case II to find out which
values of these parameters give the suitable or close result with the least
square fitting curve. In our analysis we have found that for $\delta = 0.001$
and $\gamma = 0.0095$, the plot of the Hubble parameter fairly agrees with the
fitting curve as shown in Fig.} \ref{fig2}. {Thus from our analysis we have
taken  $\alpha\beta = 0.86275$ for the case I, and  $\delta = 0.001$,
$\gamma = 0.0095$ for the case II as the constrained values of the model
parameters in all other graphical analysis.}

After {constraining the model parameters} we have plotted various
cosmological parameters against cosmological redshift for both the cases with
the constrained values of free parameters and tried to explain those plots.
From the results of this analysis we have found that for the case I, where the
directional scale factors are modified by multiplicative constants, the model
unable to show any kind of anisotropy as the shear scalars and density
parameter of anisotropy are zero in this case. It is to be noted that for
$\alpha = \beta = 1$, the case I is reduced to the $\Lambda$CDM model.
For the case II, we found  {relatively higher amount of anisotropy  as compared to case I} for higher values
of redshift and it becomes very very small at $z \le$ $20$ (see
Fig.\ \ref{fig7}). The case II also shows the presence of relative shear
anisotropy, which is
obviously very small for small value of redshift. The case II also reduces to
the $\Lambda$CDM model when $\delta = \gamma = 0$. Moreover, in terms of age
of the universe the case II is found to be in very good agreement with the
$\Lambda$CDM model. {Further, we have found that the current value of the
Hubble parameter ($H_0$) for the case I is $69.31$ and for the case II is
$67.56$. The result obtained for case II shows more consistency with the
Planck 2018 result} \cite{Planck_2018}{. Similarly, the current value
of deceleration parameter for case I is $-0.47$ and for case II is $-0.54$.
Here also the result obtained from case II is more consistent with the Planck
$2018$ data} \cite{Planck_2018}{. Furthermore, the constrained values of
model parameters for both the cases give the matter dominated region in the
density parameter of matter vs redshift ($z$) plots of Fig.}~\ref{fig9}{,
which show good agreement within the range $0.5 \leq z \leq 3000$ as
mentioned in Ref.} \cite{Frieman_2008}{. For the range of values of
density parameters
of anisotropy $\Omega_{\sigma 0}$ as suggested in Ref.}~\cite{Akarsu_2019}
 {i.e.~$10^{-23} \leq \Omega_{\sigma 0} \leq 10^{-15}$, the values of
$\delta$ and $\gamma$ lies in between $10^{-8}$ and $10^{-3}$ for the case II.}

Thus from our work, we have shown that even there exist variations in
directional scale factors, the Bianchi type I model sometime gives purely
isotropic universe just like the case I. Also we have found that although
very small variations of directional scale factors unable to change
the isotropic nature of the universe significantly in the current state, but it
may possess {relatively higher} contribution of anisotropy in early stage of the
universe {compared to the current state}.

{Finally, it needs to be mentioned that in most of the past studies on
various anisotropic cosmological models authors tried to constraint the
cosmological parameters related to the possible anisotropy of the universe
using the available cosmological data with various statistical methods. For
example, in Refs.}~\cite{Akarsu_2019, Andrade_2018} {Bayesian
inference
technique is used, goodness of fit is used in Refs.}~\cite{Bunn_1996} {and
simulation techniques like Monte Carlo is used in Refs.}~\cite{Hansen_2004,
Bunn_1996}{. Whereas in our work we constrain our model parameters using
the available cosmological data as well as the constrained anisotropic density
parameter to predict the possible anisotropy present in the universe as
mentioned above. Apart from this difference, our work is mainly based on
analytical techniques in contrast to statistical methods of previous studies
as mentioned. We have derived analytical expressions of all the cosmological
parameters based on our model, that contain the anisotropic parameters of the
model. So these cosmological parameters and hence our findings can hopefully
be tested with the early universe cosmological data that may be available in
future from the future advanced telescopes, such as the Thirty Meter
Telescope} \cite{TMT}{, Extremely Large Telescope} \cite{ELT}{, CTA}
\cite{CTA}
{etc.}

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Our money is a blessing from God that is used by God to change lives. Imagine the potential impact if everyone equally sacrificed through generous giving. Let's unite and give with a thankful and cheerful heart.
Our giving is an act of worship. So give cheerfully and sacrificially 2 Corinthians 9:6-7.
Our giving is an act of faith in the Lord. In the end we can't out give God; He has given everything Malachi 3:10.
Our giving is an indicator of our heart. It reveals our motives, both good and bad Matthew 6:21.
Our giving is tied to Jesus' sacrifice. It's Living the Gospel, Serving the City, and Being the Church 1 John 3:16-18.


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text:
By duckingbeamers Andrew Symonds, Assimilation, Australia, Racism, West Indies
A White West Indian Cricketer?
Peter English had a wonderful article about Brendan Nash, a West Indian cricketer of mixed heritage and often "accused" of being white. (Over at the Guardian, Paul Weaver also provides some interesting quotes from the man in question.) He's something of a sensation in the West Indies: racially mixed, conceived in Jamaica but born in Australia, where he grew up, then a migrant back to Jamaica, which he now calls home, and, because of his fair(er) skin, called "white."
A couple of things pop out for me: first, there is another cricketer of mixed heritage — Andrew Symonds — who hasn't had the best of times lately (and has gone downhill ever since he got into that racially tinged war-of-words with Harbhajan in 2007). Symonds's biography mirrors Nash: of West Indian origin, Symonds grew up in Australia, where he was adopted and raised, I think, by white parents. Hopefully, things will turn out better for Nash than they have for Symonds, who apparently still feels bitter and angry at Cricket Australia for abandoning him.
And the mirror is complete with recent history: Singh calls Symonds a "monkey," even though British colonialists would have wasted no time in calling Singh one as well. Meanwhile, Nash, a brown man in a black society, gets called 'white,' even though he looks just the same shade as Symonds. It's strange how these things work, but it shows how different our local racial constructions can be: white in the West Indies obviously isn't white in Australia, where black is clearly not the same black in India.
The New Yorker had an article about African businessmen in China this week, and included the results of a survey:
"Sims conducted an informal sidewalk experiment in Fuzhou, a large southern city, in which he asked people to identify photographs of famous black Americans by race. Oprah was indentified as Indian; Lena Horne didn't 'have the lips of black people'; Beyonce was white, as was Benjamin Davis, the commander of the Tuskegee Airmen."
It's funny, because in the United States, usually the one-drop rule applies: if you have even one drop of black blood in you, you cannot be considered white (as is the case with President Obama now, who is almost never considered 'white'). White women can have black babies, but black women can almost never have white young'uns. But in the West Indies, a different rule works, at least with Nash: a touch of white, and you're as white as the moon.
Tagged Andrew Symonds, brendon nash, obama, white man's burden, whiteness
5 thoughts on "A White West Indian Cricketer?"
I have a touch of white in me and I am not called white.
That's not how it works. You get called a name based on the ethnicity which comes off the most.
For example, my friend is mixed with indian and they call her miss singh and they called my other friend who is part chinese mis chin.
They are not derrogatory terms. As to Nash, he should be fine in the Caribbean.
Thanks for the comment, Candy. If it isn't how it works, though, why is Nash called white?
heners says:
bit mistruthful to say symnods and nash look exactly the same shade since they don't, i think its not either or, hes mixed race but people categorise by what they see, some mixed race kids look more like one supposed race than another thats all
This is a very weird discussion to have but.. same shade? you cannot be serious.
http://www.cricinfo.com/ausvwi09/content/current/player/6950.html
http://www.cricinfo.com/ci/content/player/7702.html
zast23 says:
Other than color of skin, features of the face is another identification criteria by people of various cultures. Indians can be dark with caucasian features. A lot of people with negroid features are also very fair.
http://en.wikipedia.org/wiki/Caucasian_race
http://en.wikipedia.org/wiki/Negroid
At the end of the day, neither facial feature nor color should matter.

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Individuals with disabilities receiving Supplemental Security Income or other Social Security benefits will see a small increase next year.
Payments will rise 0.3 percent in 2017, the Social Security Administration said Tuesday.
The bump up is due to an automatic cost-of-living adjustment, known as COLA, which is mandated by law and based on inflation.
The increase, which takes effect in January, will mean more money each month for 60 million Americans on Social Security and 8 million receiving SSI, officials said.
With the change, the maximum federal SSI benefit for individuals will be $735 per month, up from $733 in 2016. For couples, the highest federal payment will grow to $1,103 per month from $1,100.
Many states provide additional funding to SSI beneficiaries so actual payments could be higher.
Despite the modest increase, the adjustment is greater than what was announced at this time last year. Benefits for 2016 were unchanged compared to the year prior.


text:
Gunman Who Ambushed, Murdered Selma Officer Arrested
Holly MatkinJuly 29, 2021
Tuscaloosa, AL – The gunman accused of murdering Selma Police Officer Marquis Moorer was apprehended by police near Tuscaloosa on Wednesday.
Javonte Cornius Stubbs, 18, was identified as a suspect after the Alabama Law Enforcement Agency (ALEA) followed up on numerous tips and leads and conducted multiple interviews, WSFA reported.
The motive for the ambush attack remains unclear.
The U.S. Marshal's Gulf Coast Regional Fugitive Task Force and Alabama State Bureau of Investigation (SBI) officers located Stubbs at an apartment complex in Tuscaloosa, The Birmingham News reported.
He was booked into the Chilton County Jail at 4:12 a.m. on Wednesday on charges of attempted murder and two counts of capital murder.
One of the capital murder counts is for allegedly firing into an occupied building, while the second count is for allegedly murdering Officer Moorer.
The charge of attempted murder is for allegedly shooting Officer Moorer's "significant other," The Birmingham News reported.
"Agents and officers with the Alabama Attorney General's Office, The Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF), U.S. Marshal's Service, 4th Judicial Task Force, 17th Judicial Task Force, and the Dallas County Sheriff's Office were instrumental in assisting and supporting the investigation, along with locating and apprehending the suspect," the ALEA said in a statement to WAFF.
Selma Mayor James Perkins also credited the community for helping law enforcement identify and apprehend Stubbs, according to WFSA.
"People – the community – stepped forward and shared information," Perkins told reporters after the accused killer's arrest. "If you see it, say it and if you know it, tell it."
Officer Moorer was murdered in the line of duty as he was stopping by his apartment for his meal break early Tuesday morning, police said.
The 25-year-old officer was heading into his residence at the Selma Square Apartments between 3:30 a.m. and 4 a.m. when a gunman approached him and opened fire, Dallas County District Attorney Michael Jackson told The Birmingham News.
Jackson described the attack as an "ambush," according to the Associated Press.
A woman at the apartment, identified only as Officer Moorer's "significant other," was also shot in the attack, Jackson said.
She was rushed to a local hospital in unknown condition.
Early this morning, Officer Marquis Moorer was on duty with the Selma AL Police Department and had stopped by his home for a lunch break when he was shot and killed. #EndofWatch #Hero #EnoughIsEnough pic.twitter.com/VUPL3d5n45
— Blake Williams (@blwilliamsmedia) July 27, 2021
Jackson described Officer Moorer as an "outstanding officer" who "took his job very seriously."
Perkins said that he and Selma Police Chief Kenta Fulford met with the slain officer's family earlier Tuesday morning.
"We had prayer and expressed our condolences to the family,'' Perkins said.
He said Officer Moorer's murder is a "tragic moment" for the city, The Birmingham News reported.
"The Selma Police Department is devastated," Chief Fulford said during a news conference, according to the Associated Press. "We're asking the community for your prayers."
Alabama Governor Kay Ivey has ordered flags be lowered to half-staff in honor of Officer Moorer as soon as a date for his memorial service is announced.
Alabama Attorney General Steve Marshall released a statement urging citizens to recognize the risks law enforcement officers face on a daily basis, WBMA reported.
"We must honor them and never take for granted the sacrifices they make on our behalf," Marshall said. "I ask you to join me in prayerful gratitude for Officer Moorer as we hold his loved ones in our hearts."
NBA's Jaxson Hayes Tased Twice During Fight That Sent Cop To Hospital
New Laws Prevent Deputies From Pursuing Man In Yellow Dress Who Stole Bus, Demolished House


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text:
GUAA Adopts a New Visual Identity
Since its founding in 1881, the Georgetown University Alumni Association's (GUAA) mission has been to connect alumni around the world. Until this year, GUAA used the university seal as a logo in...
Club Spotlight: The Alumni Club of Los Angeles
By Na'Tasha Jones, Managing Editor, Georgetown Alumni Georgetown's alumni clubs serve as beacons for alumni around the world, offering a way to connect, learn from and grow the Hoya community in each...
Welcome to the Neighborhood 2014
On September 30, local alumni clubs around the globe hosted Welcome to the Neighborhood events to greet both new and seasoned generations of Hoyas. Whether they were new to the area or simply...
Healey Family Student Center Virtual Tour
By Na'Tasha Jones Managing Editor, Georgetown Alumni Georgetown's new Healey Family Student Center, located in the New South residence hall building, opened to students on September 5. The innovative...


text:
Jonathan Kenneth Corupe (born January 18, 1980) is a Canadian retired professional ice hockey player.

During the 2013–14 GET-ligaen season, Corupe scored 66 points (30 goals and 36 assists) to tie with Morten Ask for the most points scored in the GET-ligaen.

Career statistics

References

External links

1980 births
Living people
Birmingham Bulls players
Bolzano HC players
Canadian ice hockey centres
Columbus Cottonmouths (ECHL) players
Ice hockey people from Ontario
Johnstown Chiefs players
Lørenskog IK players
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Le Verger du diable est un roman d'Alain Gerber paru le  aux éditions Grasset et ayant reçu le Prix Interallié la même année.

Résumé

Éditions
Le Verger du diable, Éditions Grasset, 1989 .

Roman français paru en 1989
Roman récompensé par le prix Interallié


Have you ever noticed these fascinating things on some French doors?
September 17, 2018 Oliver Gee1 Comment on Have you ever noticed these fascinating things on some French doors?
I have a new doorway fascination and it can't be cured. You may remember my borderline obsession with chasse-roues, the doorway protectors in Paris.
But what I've found in the western city of Cognac may beat even this.
Yes, as I was walking along the main pedestrianized street in Cognac, the Rue Aristade Briand (pictured below) I noticed a little cubby-hole to the bottom right of a doorway. In fact, you can see it in the picture above.
On closer inspection I saw there was a thin metal strip in front of this hole in the wall, and I realized that it must have been an in-built shoe-scraper!
Here's a better look at it so you can understand me.
Now, I've seen shoe-scrapers before in Paris and elsewhere – and while they're sometimes ornate and interesting, I've never seen them built into the wall. Below is an example of a typical and rather boring shoe scraper, also found in Cognac. This will give you an idea of how excited I was to see the ornate variety.
So, imagine my surprise when I continued to walk along the street and notice that many, many of the door were equipped with these intriguing little shoe scrapers, seemingly unused for decades.
Some even had TWO scrapers – his and hers?
So what did I do? I walked the street from end to end and took pictures of them all. Now you don't need to visit Cognac! (Although I highly recommend it).
Anyway, here are the rest of the pictures, you can click to make them bigger. I included a full shot of the door each time for perspective.
As for me, if you want to hear me get out of breath just talking about these shoe scrapers, you should listen to the latest episode of the podcast, where I also talk about Cognac, a vintage car race in Angouleme, and the three things you need to know before living in a French village.
Click play to hear it.
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So, for the time being, the heavy woolens, shawls & socks are the saviors while fashion (which calls for something not quite so practical) has to almost take a back seat! Though every night, before drifting off to sweet & warm sleep (under layers of blankets) I try to think up of practical, warm, comfortable, yet stylish outfits for the next day (sometimes I succeed, sometimes I give up after half an hour of restless tossing & turning).............most days, I end up wearing whatever will keep me covered & very, very warm, while slipping on my thickest socks & covered shoes!!! Sigh!
Anyways, all my above ranting doesn't mean at all that the fashionista in me has gone into hibernation!! In fact, I'm still bringing out old sweaters, cardigans, jackets & jumpers from the dark recesses of my closet, so that I can do more of mix & match as well as layering!!
Grey is one of my latest favorite color (I have this phases of liking a certain color for a certain period of time) & anything in basic grey appeals to me in a way I just can't ignore! I was actually eyeing 2 other grey cardis at Mango, but once I spotted this one, all the others went back to the rack........this was a clear winner because of the length, neck, sleeves & the loose, comfy silhouette!!
Nicely done! :) I love the blue, grey combo a LOT! :) And hey, congrats on your first Etsy purchase!
It is been exceptionally cold in the whole Northern Hemisphere this year.
Princess Sweetie, I LOVE LOVE LOVE all the extra details, the bow, the scarf and of course I LOVE your grey mango tunic too, I can imagine you wear it in Spring too, you ROCK babe!!!!
@Vish: Thank you......happy winters to you too!
What a cute little bow darling ... You look adorbss!!!
I love pieces in grey, and a good wool cardi is one of them. I also love grey jersey knit dresses. They seem so classic and the perfect foil to a bold print or bright color. Both of which we both seem very partial to. Looking good, Princess!
Nice dress,, Excellent for this freezing winter..
@Z: I love it too!!
@Fashion Tales: Thank you dear!!
Love your outfit!!!! Thanks so much for stopping by my blog!


A number of Des Moines companies are sponsoring Tech Camp 2013 at Des Moines public schools this week. The camp is designed to bring information technology officials together to work with eighth-grade students, and is the first of a series of camps to be hosted by a new nonprofit group called Tech Journey. A final presentation from students will be held at 2:15 p.m. on Thursday at Central Campus. Sponsors for the event are Source Allies Inc., Principal Financial Group Inc., Bankers Trust Co., Visual Logic, FBL Financial Group Inc., Signature Consultants, Mumo Systems LLC, Edward D. Jones and Co. LP and Verizon Wireless.


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Q: How close together can an antiparallel magnetic moments free electron pair get? See illustration below:

Is there a calculation-prediction of the minimum distance separation of an antiparallel electron pair due its like charge repulsion?
There must be an equilibrium between the magnetic attraction and the charge repulsion?
(the two electrons in the illustration, antiparallel magnetic moments N-S and S-N will attract the two electrons together forming a stable pair but at the same time the Coulomb charge repulsion force will keep them separated at a distance d).
Any analytical solution to this problem?
I believe I heard or read somewhere once I cannot recall, that this distance d described in the question is equal to the Reduced Compton Wavelength of the electron ƛ=ħ/mc =3.861 592 6796(12) x 10E-13 m = 386 fm, which is also the uncertainty Δχ in this case but I cannot find any analytical solution?
Specifically, the minimum Δχ uncertainty in the position of a free electron at rest is Δχ= 0.5 (ħ/mc) = 0.5ƛ = 193 fm thus half of its reduced Compton wavelength. Notice here that the Reduced Compton Wavelength for the electron also written as ƛe=λe/2π where λe is the Compton wavelength of the electron, represents the radius r of the electron's rest mass field shown in the above illustration (blue sphere).
Therefore, any analytical prediction presented as an answer in this question page must have a minimum uncertainty in the separation distance d for an antiparallel pair of electrons set at d(uncertainty)=2Δχ=ƛe=386 fm.
Thus any value prediction must be written as:
d=................ (193 fm), where the value in the parenthesis indicates the ± , uncertainty in the prediction.
Note: Definition of the term rest mass field of the electron used in this question, is (or equals) a sphere volume of radius the total minimum uncertainty in position Δχ of an electron at rest, ±193fm = 386fm. Of course the electron particle by known theory is a massive dimensionless point particle.

A: According to Alexander A. Mikhailichenko [1] (eq. 8), reference suggested to me by Prof. Behnam Farid (https://www.researchgate.net/profile/Behnam-Farid-2) this equilibrium linear distance d where the two forces equalize is:
d=√6ƛe~2.5ƛe
where ƛe is the Reduced Compton Wavelength of the electron thus ƛe=386 fm =0.386 picometers.
Therefore resulting in a final value of linear separation between the two electron's magnetic moments in an electron pair at rest of d~2.5x386=965 fm.
That is x1.25 the diameter cross-section of the single electron's rest mass field (blue sphere in illustration of anti parallel electron pair). Meaning in the separated antiparallel electron pair a whole third electron can fit in between!
Final value adding uncertainty is,
d=965 (193) fm.
[1] A.A. Mikhailichenko, TO THE POSSIBILITY OF BOUND STATES BETWEEN TWO ELECTRONS, https://accelconf.web.cern.ch/ipac2012/papers/weppp031.pdf  (accessed September 26, 2021).

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UN SG launches #Youth2030 strategy
Youth 2030: The United Nations Youth Strategy envisions a world in which the human rights of every young person are realized so they can achieve their full potential to advance sustainable development and peace.
New York, 24 September 2018 – United Nations Member States, private sector and civil society today joined Secretary-General António Guterres' call to enhance the role of the world's 1.8 billion young people in driving global efforts to promote a peaceful, just and sustainable world.
Youth 2030: The United Nations Youth Strategy, launched today by Mr. Guterres at a high-level event of the 73rd session of the General Assembly, seeks to strengthen and increase commitments at the global, regional and national level to meet young people's needs, help them realize their rights, and recognize their positive contributions as agents of change.
"All of our hopes for a better world rest on young people," Mr. Guterres said. "Sustainable development, human rights, and peace and security can only be achieved if we empower these young people as leaders and enable them to unleash their full potential."
Commitments to the Youth Strategy were voiced throughout the event, with Denmark being the first partner to make a direct financial contribution towards its implementation. "If we want real change for the better, we need to ensure development not just for young people – but with and by young people" said H.E. Lars Løkke Rasmussen, Prime Minister of Denmark, as he spoke alongside Denmark's Youth Delegate. Similarly, Mr. Sultan AlMusallam of the Misk Foundation spoke out in support of the Strategy. Tomorrow, the Foundation will sign a partnership with the Office of the United Nations Secretary-General's Envoy on Youth making it the first non-government organization to make a concrete pledge to Youth 2030.
While this is the largest and most connected, tech-savvy generation the world has ever known, young people face tremendous challenges. Too many do not have access to a quality education and struggle to secure decent work. Many lack adequate health care, including for sexual and reproductive health. There are few opportunities to participate in decisions that can determine the course of their lives.
Youth 2030: The United Nations Youth Strategy, which is based on lessons learned from youth programmes around the world, stresses the need to engage young people so they can more meaningfully contribute to the work of the United Nations, and seeks to ensure the UN can benefit from their views, insights and ideas.
"Today marks the beginning of a reset and reorientation of the UN system's commitment to ensure that young people are not only heard, but understood; not only engaged, but empowered; and not only supporting, but leading global, regional and national efforts to achieve the Sustainable Development Goals and build a better world for generations to come," said Jayathma Wickramanayake, the Secretary-General's Envoy on Youth.
Jayathma Wickramanayake, the Secretary-General's Envoy on Youth, in the SDG Media Zone just prior to the launch.
The majority of the world's 1.8 billion young people live in developing countries, and their numbers are expected to continue growing.
"Being young means being ambitious, creative, determined, innovative and courageous. What else do we need to solve all the problems around us?" asked youth leader Batool Alwahdani, Vice-President of the International Federation of Medical Students' Associations during the event. "The new UN Youth Strategy is an investment in a better future for me and my generation. It establishes all possible means to support, enable and strengthen meaningful youth participation all over the world." Ms. Alwahdani was selected to speak at this event by her peers through a key youth platform representing thousands of youth organizations and millions of young people around the world.
The strategy has five priority areas: youth engagement, participation and advocacy; ensuring informed and healthy foundations through education and health care; economic empowerment through decent work; human rights; and peace and resilience building.
Delivering on Youth 2030 will require bold new approaches and partnerships. A multi-stakeholder initiative, Generation Unlimited, was also launched today to address the desperate lack of quality education, skills training and employment opportunities for young people. This partnership will bring young people together with a wide range of partners, including governments, the private sector, academia, international and civil society organizations to create, fund and scale up innovative solutions and expand opportunities.
Other key speakers at the launch of Youth2030: The United Nations Youth Strategy included:
H.E. Ms. Maria Fernanda Espinosa President of the General Assembly
H.E. Paul Kagame, President of the Republic of Rwanda
H.E. Nana Addo Dankwa Akufo-Addo, President of Ghana;
H.E. Uhuru Muigai Kenyatta, President of the Republic of Kenya
Note to Editors:
The ownership of the UN Youth Strategy is equally shared by all members of the UN system at the global, regional and national levels. The Secretary-General's Envoy on Youth, Jayathma Wickramanayake, will be the public face of the UN Youth Strategy and will lead global advocacy efforts and its High-Level Steering Committee in support of its successful implementation. In addition to the Envoy as the Chair of the High Level Steering Committee, it will consist of Senior Leaders (ASG level) from across the UN-System whose entities' organizational mandates relate to any of the priority areas of the Strategy, along with two youth organizations on a biennial rotational basis. The High-Level Steering Committee will seek to ensure alignment with other UN processes and the UN reform, guide effective linkages and synergies at all levels, and guide the efforts to mobilize resources and leverage partnerships in support of the strategy.
www.un.org/youthenvoy/
Twitter at: @UNYouthEnvoy
Facebook: www.facebook.com/UNYouthEnvoy
Press Contact Information for UN Office of the Secretary-General's Envoy on Youth:
Esther Agbarakwe ; E-mail: esther.agbarakwe@un.org; Tel: +1 786 603 3643
Tolu Olubunmi; E-mail: tolu.olubunmi@un.org; Tel: +1 917 742 0537
Martin2019-01-17T21:59:42-05:0024 Sep 2018|Featured, News, Press material, Youth|
The 17 Goals


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Kelly Clarkson On How Blake Shelton Got Her To Join 'The Voice' Over 'American Idol'
Emily Hutchinson
Kelly Clarkson is revealing why she signed up as a coach on The Voice despite being offered a spot on the judging panel of the revamped American Idol, the talent search she won back in 2002.
It was confirmed earlier this year that Kelly would be heading to the NBC show alongside Adam Levine and Blake Shelton for Season 14, and now she's revealing how the latter influenced her decision.
Speaking to Entertainment Weekly in a new cover feature, the star revealed that her decision was actually influenced by her relationship with Blake and what would be best for her family, particularly her two children, three-year-old River Rose and one-year-old Remington Alexander.
"I want it to be a great experience for my whole family, and that's what I have to think about at this point in my life," Kelly said of why she seemingly snubbed the show that made her a household name to join its rival show.
She explained that her husband Brandon Blackstock, who also serves as her manager, also manages the career of Blake and is often backstage at the NBC show with the country star. That means she and her two children are already familiar with The Voice and what goes on behind the scenes.
[Image by Todd Williamson/Getty Images for InStyle]
"We're used to The Voice schedule, we already have to work around it because of our family," Kelly said of why she ultimately chose a seat on the NBC series over her former home American Idol, adding that she'd actually been in talks to sit alongside Blake and Adam for years but always had to turn down producers' offers because she was pregnant.
"We have four children — that can be very taxing with a schedule," she continued, referring to her two step-children, Seth and Savannah Blackstock. "It was just a no-brainer for me with The Voice because of that alone."
But just because she chose The Voice, Kelly explained that there's certainly no bad blood between herself and American Idol – which she became the first ever winner of back in 2002 – because she turned down their offer to judge.
[Image by Kevin Winter/Getty Images]
"I was kind of bummed too when [Idol] came to me," the "Since U Been Gone" singer explained, admitting that she hopes the revamped version of the series is "super successful" when it debuts on ABC next year.
"Would it have been awesome to come back to the show that started me and help give someone that start? Yes!" she added. "But that doesn't mean I can't do the same thing on The Voice. That's my goal. I want to have someone on my team, whether they win or not, to have a successful career after being on the show."
Sources confirmed earlier this year that she was in the running to become a judge on the re-vamped ABC version of the talent search and it was largely assumed that Clarkson would be the first officially announced American Idol judge.
However, just days after the speculation began, Blake shockingly announced in a Facebook Live video that Kelly would actually be heading to The Voice in early 2018.
Blake confirmed that Kelly would be joining the show as a coach on Season 14 in May.
He described her as being "a very near and dear friend" in the video and continued to gush over Clarkson by claiming that he already considers her to be family because of their connection via his manager.
"We have felt as a family at The Voice that this person would fit in perfectly," he added, to which Kelly joked that the country star needs to "watch out" when she joins the show next year.
And it sounds like there's going to be some rivalry despite their friendship, as Clarkson teased earlier this year that she's going to "annihilate" Gwen Stefani's man when they go head-to-head.
The Voice Season 13 is set to debut on September 18 on NBC, while Kelly Clarkson's season is expected to debut in February 2018.
[Featured Image by Michael Buckner/Getty Images for Celebrity Fight Night]


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Chedraui in Debut Equity Offering
Cleary Gottlieb represented Grupo Comercial Chedraui and its principal shareholder in Chedraui's recent $425 million debut global equity offering, which was comprised of an initial public offering on the Mexican Stock Exchange and a Rule 144A/Reg S offering in the United States and outside Mexico and the United States. The offering was Mexico's first initial public offering since June 2008. The base transaction closed on May 5. The underwriters subsequently exercised their over-allotment option in full.
Chedraui is a leading Mexican food and general merchandise retailer with operations under the Chedraui and Super Che formats in Mexico and under the El Super format in the United States. Chedraui intends to use the proceeds of the offering for general corporate purposes and to fund its ongoing expansion plan.