Exact Cosmology in Myrzakulov Gravity
Dinesh Chandra Maurya1, Ratbay Myrzakulov2

1 Centre for Cosmology, Astrophysics and Space Science, GLA University,
Mathura-281 406, Uttar Pradesh, India.

2 Eurasian International Centre for Theoretical Physics and Department of
General & Theoretical Physics, Eurasian National University, Astana 010008,
Kazakhstan.

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###### Abstract

In this paper, we have investigated some exact cosmological models in
Myrzakulov gravity using a flat Friedmann-Lematre-Robertson-Walker (FLRW)
spacetime metric. We have considered the modified Lagrangian function as
$F(R,T)=R+\lambda T$, where $R,T$ are respectively the Ricci curvature scalar
and the torsion scalar with respect to non-special connection, and $\lambda$
is a model parameter. We have obtained two exact solutions in two different
situations for the scale factor $a(t)$. Using this scale factor, we have
obtained various geometrical parameters to investigate cosmological properties
of the universe. We have obtained the best fit values of model parameters
through the MCMC analysis of two types latest observational datasets like
$H(z)$ and Pantheon SNe Ia samples, with $1-\sigma,2-\sigma$ & $3-\sigma$
regions. We have performed a comparative and relativistic study of geometrical
and cosmological parameters. In model-I, we have found that the effective
equation of state (EoS) parameter $\omega_{eff}$ varies in the range
$-1\leq\omega_{eff}\leq 0$ while in the model-II, it varies as
$-1.031\leq\omega_{eff}\leq 0$. We have found that both models are transit
phase (decelerating to accelerating) universe with transition redshift in the
range $0.6<z_{t}<0.8$ and present age of the universe $t_{0}\approx 13.5$
Gyrs.

Keywords: Myrzakulov Gravity; FLRW flat universe; Exact Cosmology; Transit
phase expansion; Observational constraints.

PACS number: 98.80-k, 98.80.Jk, 04.50.Kd

###### Contents

  1. 1 Introduction
  2. 2 Myrzakulov Gravity
  3. 3 Field Equations and Solutions
    1. 3.1 Model-I
    2. 3.2 Model-II
  4. 4 Observational Constraints
    1. 4.1 Hubble function $H(z)$
    2. 4.2 Apparent Magnitude $m(z)$
  5. 5 Result Discussion
    1. 5.1 Model-I
    2. 5.2 Model-II
  6. 6 Conclusions
  7. 7 Acknowledgments

##  1 Introduction

Recent cosmological studies [1]-[7] suggest that the Universe evolves with two
accelerating phases of expansion, one at early times and one at late times.
Although the latter can be explained by the presence of a cosmological
constant $\Lambda$, the inclusion of some modification was necessary due to
the associated theoretical issue, the potential for a dynamical behavior, and
particularly the cosmological constant’s incapacity to explain the early
accelerated phase. One possible approach is to stick with general relativity
as the basic theory and change the universe’s matter content by adding new
fields, like the dark energy sector at later times [8, 9] and/or the inflaton
at earlier times [10, 11]. A second option involves altering the gravitational
sector itself, creating a theory that, although having general relativity as a
specific limit, generally shows more degrees of freedom [12, 13].

Gravitational modifications are constructed in a variety of methods, each of
which modifies a different aspect of general relativity. The braneworld
theories [14] comes from changing the dimensionality; the $F(R)$ gravity [15,
16], $F(G)$ gravity [17, 18], Lovelock theories [19, 20], etc. come from
changing the Einstein-Hilbert Lagrangian; and the Horndeski/Galileon theories
[21, 22, 23] come from adding a scalar field coupled in different ways with
curvature. Furthermore, one can create modifications utilizing torsional
invariants, as in $F(T)$ gravity [24, 25], in $F(T,T_{G})$ gravity [26], or in
scalar-torsion theories [27, 28], beginning with the analogous, teleparallel
formulation of gravity [29, 30]. Furthermore, a broad class of metric-affine
theories [31, 32, 33] can be built, which includes a generic linear connection
structure; alternatively, nonlinear connections can be introduced, as in the
case of Finsler and Finsler-like theories [34]-[39].

Motivated by these, a theory utilizing a specific but non-special connection
could be built from such affinely connected metric theories, and specifically
from their Riemann-Cartan subclass [40]. This would result in both non-zero
curvature and non-zero torsion at the same time, providing the additional
degrees of freedom usually required in any gravitational modification [41]. As
a result, Myrzakulov gravity can produce a good phenomenology that can explain
both the early and late universe evolution [42]–[46]. A recent analysis of the
resulting cosmology of such a framework and the computation of the evolution
of observable quantities such as the density parameters and the effective dark
energy equation-of-state parameter have been published in [42]. Through the
use of the mini-super-space technique, they have studied the cosmological
behavior with a focus on the effect of the connection, expressing the theory
as a deformation from both general relativity and its teleparallel
counterpart. [47] has investigated the observational restrictions on
Myrzakulov $F(R,T)$-gravity. In [48]–[54], several Metric-Affine Myrzakulov
Gravity Theories and their applications are covered.

Recently [55] has investigated observational constraints on some Myrzakulov
gravity models using a wide range of observational datasets. In [56], we have
investigated some exact solutions in Myrzakulov $F(R,T)$ gravity with
observational constraints. Recently we have investigated some exact
cosmological models in different scenarios in various modified gravity
theories [57]-[63]. Motivated by above discussions, in this paper, we find
some new exact cosmological models in Myrzakulov gravity.

This paper is organized as follows. Section 1 is introductory. A brief
introduction of Myrzakulov gravity is given in Section 2. Field equations and
their some exact solutions are given in Section 3. We perform some
observational constraints on different parameters using latest datasets in
Section 4. Result discussions are given in Section 5. Finally, conclusions are
given in last Section 6.

##  2 Myrzakulov Gravity

In the present section, we organize a brief review of Myrzakulov gravity, also
called as $F(R,T)$ gravity with $R$ as curvature and $T$ known as torsion [41,
42]. The key factor of this gravity is the choice of non-special connection.
Specifically, it is established that enforcing a generic connection
${\omega^{\alpha}}_{\beta\gamma}$ one defines the torsion tensor and the
curvature as [26]

${R^{\alpha}}_{\beta ij}={\omega^{\alpha}}_{\beta
j,i}-{\omega^{\alpha}}_{\beta i,j}+{\omega^{\alpha}}_{\gamma
i}{\omega^{\gamma}}_{\beta j}-{\omega^{\alpha}}_{\gamma
j}{\omega^{\gamma}}_{\beta i}$ (1)
${T^{\alpha}}_{ij}={e^{\alpha}}_{j,i}-{e^{\alpha}}_{i,j}+{\omega^{\alpha}}_{\beta
i}{e^{\beta}}_{j}-{\omega^{\alpha}}_{\beta j}{e^{\beta}}_{i}$ (2)

where ${e_{\alpha}}^{i}\partial_{i}$ is the tetrad field associated with the
metric through $g_{ij}=\eta_{\alpha\beta}{e^{\alpha}}_{i}{e^{\beta}}_{j}$,
where $\eta_{\alpha\beta}=diag(1,-1,-1,-1)$, where the Latin, Greek indices
signify the coordinate and tangent space, respectively, and where the comma
indicates differentiation.
There are numerous possibilities for connections. The only connection that
provides vanishing torsion is the Levi-Civita $\Gamma_{\alpha\beta\gamma}$.
Henceforth, we will refer to the curvature (Riemann) tensor as
${R^{(LC)\alpha}}_{\beta ij}={\Gamma^{\alpha}}_{\beta
j,i}-{\Gamma^{\alpha}}_{\beta i,j}+{\Gamma^{\alpha}}_{\gamma
i}{\Gamma^{\gamma}}_{\beta j}-{\Gamma^{\alpha}}_{\gamma
j}{\Gamma^{\gamma}}_{\beta i}$. The Weitzenböck connection, on the other hand,
can be used to obtain the following:
${T^{(W)k}}_{ij}={W^{k}}_{ji}-{W^{k}}_{ij}$. This connection is curvatureless
and yields only torsion. Quantities corresponding to ${W^{k}}_{ij}$ are
denoted by the label “W”. It is inferred from the preceding that the Levi-
Civita connection’s associated Ricci scalar is

$R^{(LC)}=\eta^{\alpha\beta}{e_{\alpha}}^{i}{e_{\beta}}^{j}\left[{\Gamma^{k}}_{ij,k}-{\Gamma^{k}}_{ik,j}+{\Gamma^{l}}_{ij}{\Gamma^{k}}_{kl}-{\Gamma^{l}}_{ik}{\Gamma^{k}}_{jl}\right]$
(3)

while the torsion scalar corresponding to the Weitzenböck connection is

$T^{(W)}=\frac{1}{4}\left(W^{ikj}-W^{ijk}\right)\left(W_{ikj}-W_{ijk}\right)+\frac{1}{2}\left(W^{ikj}-W^{ijk}\right)\left(W_{kij}-W_{kji}\right)-\left({W_{j}}^{ij}-{W_{j}}^{ji}\right)\left({W^{k}}_{ik}-{W^{k}}_{ki}\right)$
(4)

The first is utilized in the Lagrangian of General Relativity and in all
curvature-based modified gravities, such as in $F(R)$ gravity [15], whereas
the second is utilized in the Lagrangian of the teleparallel equivalent of
General Relativity and in all torsion-modified gravities, such as in $F(T)$
gravity [13].
A non-special connection with simultaneous non-zero curvature and torsion is
used in Myrzakulov gravity [42]. Since general relativity and its teleparallel
counterpart, which both have two degrees of freedom corresponding to the
massless graviton, do not have extra degrees of freedom, the resulting theory
will generally have them, even if the imposed Lagrangian is simple. Here, we
consider the action as

$I=\int{d^{4}x~{}e\left[\frac{1}{2\kappa^{2}}F(R,T)+L_{m}\right]}$ (5)

where the gravitational constant is $\kappa^{2}=8\pi G$, the matter Lagrangian
$L_{m}$ has also been introduced for completeness, and
$e=\det(e^{\alpha}_{i})=\sqrt{-g}$. Take note that the arbitrary function
$F(R,T)$ has two variables, $R$ and $T$, which represent the curvature and
torsion scalars, respectively, corresponding to the non-special connection
employed [26].

$T=\frac{1}{4}T^{ijk}T_{ijk}+\frac{1}{2}T^{ijk}T_{kji}-{T_{j}}^{ji}{T^{k}}_{ki},$
(6) $R=R^{(LC)}+T-2{{T_{j}}^{ji}}_{;i}$ (7)

where the covariant differentiation with regard to the Levi-Civita connection
is indicated by the symbol (;). So, while $R$ depends on the tetrad and its
first derivative as well as the connection and its first derivative, it also
depends on the second tetrad derivative because of the final term in (7). In
contrast, $T$ depends on the tetrad field, its first derivative, and the
connection. Therefore, we can finally write using Eqs. (4), (6), and (7).

$R=R^{(LC)}+u,$ (8) $T=T^{(W)}+v,$ (9)

The scalar $u$ is contingent upon the tetrad, its first and second
derivatives, the connection, and its first derivative, whereas the scalar $v$
is contingent upon the tetrad, its first derivative, and the connection.
The effect of the specific but non-special imposed connection is quantified by
the numbers $u$ and $v$. If this link turns into the Levi-Civita connection,
then $u=0$ and $v=-T^{(W)}$. This makes the above theory the standard $F(R)$
gravity, which under $F(R)=R$ coincides with general relativity. Conversely,
if the connection is the Weitzenböck one, then $v=0$ and $u=-R^{(LC)}$, and so
the theory corresponds with $F(T)$ gravity. For $F(T)=T$, this gravity thus
becomes the teleparallel counterpart of general relativity.

##  3 Field Equations and Solutions

We put all of this to a cosmological framework and get the corresponding field
equations that govern the evolution of the universe [42]. We consider the
Friedmann-Robertson-Walker (FRW) flat Friedmann homogeneous and isotropic
geometry spacetime metric as

$ds^{2}=dt^{2}-a^{2}(t)\delta_{\mu\nu}dx^{\mu}dx^{\nu},$ (10)

which is equivalent to the tetrad $e^{\alpha}_{i}=diag[1,a(t),a(t),a(t)]$, in
which the scale factor is denoted by $a(t)$. The torsion and curvature scalar
$T^{(W)}=-6\left(\frac{\dot{a}}{a}\right)^{2}$ and
$R^{(LC)}=6\left(\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}\right)$
are easily found in this situation, respectively. We further consider the
conventional substitution $L_{m}=-\rho_{m}(a)$ [64, 65, 66]. Ultimately, in
light of the previous conversation regarding the interdependence of $u$ and
$v$, we consistently enforce that $u=u(a,\dot{a},\ddot{a})$ and
$v=v(a,\dot{a})$.
We aim to investigate the cosmic behavior resulting just from the non-special
connection in Myrzakulov gravity in this work. Since the coupling coefficient
of $R$ can be absorbed into $\kappa^{2}$, we remove it. Instead, we
concentrate on the simplest case, where the involved arbitrary function is
trivial, which is $F(R,T)=R+\lambda T$. Be aware that the Lagrangian does not
explicitly take into account a cosmological constant term. By incorporating
the aforementioned spacetime metric (10) into the action (5), we obtain
$I=\int{Ldt}$ for this Lagrangian selection [55] as

$L=\frac{3}{\kappa^{2}}[\lambda+1]a\dot{a}^{2}-\frac{a^{3}}{2\kappa^{2}}[u(a,\dot{a},\ddot{a})+\lambda
v(a,\dot{a})]+a^{3}\rho_{m}(a).$ (11)

Taking the Hamiltonian constraint $H=\dot{a}\left[\frac{\partial
L}{\partial\dot{a}}-\frac{\partial}{\partial t}\frac{\partial
L}{\partial\ddot{a}}\right]+\ddot{a}\frac{\partial L}{\partial\ddot{a}}-L=0$
and extracting the equations of motion for $a(t)$, we have the Friedmann
equations as [55]

$3(1+\lambda)H^{2}-\frac{1}{2}[Ha(u_{\dot{a}}+\lambda v_{\dot{a}})-(u+\lambda
v)+au_{\ddot{a}}(\dot{H}-2H^{2})]=\kappa^{2}\rho_{m}$ (12)
$(1+\lambda)(2\dot{H}+3H^{2})-\frac{1}{6}\left[3Ha(u_{\dot{a}}+\lambda
v_{\dot{a}})-3(u+\lambda v)\right.\\\ \left.-a(u_{a}+\lambda
v_{a}-\dot{u}_{\dot{a}}-\lambda\dot{v}_{\dot{a}})-3a(\dot{H}+3H^{2})u_{\ddot{a}}-6Ha\dot{u}_{\ddot{a}}-a\ddot{u}_{\ddot{a}}\right]=-\kappa^{2}p_{m}$
(13) $\dot{\rho}_{m}+3H(\rho_{m}+p_{m})=0$ (14)

where $H$ is the Hubble parameter defined by $H=\frac{\dot{a}}{a}$, $p_{m}$
denotes the matter pressure, and the subscripts $a,\dot{a},\ddot{a}$ indicate
the partial derivatives with respect to this arguments.
Now, we find the solution of above field equations for two different choices
of $u$ and $v$, and we investigate these in next two sub-sections. As per
definitions of $u$ and $v$, in first model, we choice $u=k_{1}H-k_{2}$,
$v=k_{3}aH-k_{4}$, and in second model, we consider
$u=k_{1}(\dot{H}+H^{2})-k_{2}$, $v=k_{3}aH-k_{4}$ with $k_{i}$’s $i=1,2,3,4$
constants,.

###  3.1 Model-I

In this model, we choose $u=k_{1}H-k_{2}$ and $v=k_{3}aH-k_{4}$ with $k_{i}$’s
$i=1,2,3,4$ constants and $H=\frac{\dot{a}}{a}$, then the above field
equations (12) & (13) become

$3(1+\lambda)H^{2}-\frac{1}{2}(k_{2}+\lambda k_{4})=\kappa^{2}\rho_{m}$ (15)
$(1+\lambda)(2\dot{H}+3H^{2})-\frac{1}{2}(k_{2}+\lambda
k_{4})=-\kappa^{2}p_{m}$ (16)

Taking the non-relativistic matter pressure $p_{m}\approx 0$ in (16) and
rewrite the Eq. (16) as

$2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}-\frac{k_{2}+\lambda
k_{4}}{2(1+\lambda)}=0,~{}~{}~{}~{}\lambda\neq-1.$ (17)

Solving the Eq. (17) for the scale factor $a(t)$, we get

$a(t)=\left[\frac{3\sqrt{3}c_{2}e^{\frac{\sqrt{3n_{1}}}{2}t}-\sqrt{3}c_{1}e^{-\frac{\sqrt{3n_{1}}}{2}t}}{6\sqrt{n_{1}}}\right]^{\frac{2}{3}},~{}~{}~{}~{}n_{1}>0,$
(18)

where $c_{1},c_{2}$ are arbitrary constants and $n_{1}=\frac{k_{2}+\lambda
k_{4}}{2(1+\lambda)}$, $\lambda\neq-1$. Without loss of generality, we choose
$c_{1}=k\sqrt{3},c_{2}=\frac{k}{\sqrt{3}}$ so that we can put the scale factor
in the following simplified form (hyperbolic expansion law cosmology)

$a(t)=\left[\frac{k}{\sqrt{n_{1}}}\sinh{\left(\frac{\sqrt{3n_{1}}}{2}t\right)}\right]^{\frac{2}{3}},~{}~{}~{}~{}n_{1}>0$
(19)

Alternatively, if we choose $c_{1}=0$, then we find the exponential expansion
law cosmology as
$a(t)=\frac{\sqrt{3}c_{2}}{2\sqrt{n_{1}}}e^{\sqrt{\frac{n_{1}}{3}}t}$ which
gives a constant deceleration parameter $q$ that reveals either decelerating
or accelerating expanding universe, but we seeking a transit phase
(decelerating-accelerating) expanding universe model. Therefore, we consider
the first choice of $c_{1},c_{2}$ to investigate the model. Thus using the
scale factor mentioned in (19), we derived the Hubble parameter
$H=\frac{\dot{a}}{a}$ and deceleration parameter
$q=-\frac{a\ddot{a}}{\dot{a}^{2}}$, respectively as

$H(t)=\sqrt{\dfrac{n_{1}}{3}}\coth{\left(\frac{\sqrt{3n_{1}}}{2}t\right)}$
(20) $q(t)=-1+\frac{3}{2}sech^{2}{\left(\frac{\sqrt{3n_{1}}}{2}t\right)}$ (21)

Now we use the relationship $a(t)=a_{0}(1+z)^{-1}$ [8], in (20), (21), with
present value of scale factor $a_{0}=1$ in standard convention, and $z$ as the
redshift whose positive values show the early evolution of the universe while
the negative values of $z$ reveals the future predictions, and $z=0$ represent
the present stage of the universe, we obtain the Hubble parameter $H(z)$ and
deceleration parameter $q(z)$ as

$H(z)=\frac{1}{\sqrt{3}}\sqrt{k^{2}(1+z)^{3}+\frac{k_{2}+\lambda
k_{4}}{2(1+\lambda)}}$ (22)
$q(z)=-1+\frac{3}{2}\frac{k^{2}(1+z)^{3}}{k^{2}(1+z)^{3}+\frac{k_{2}+\lambda
k_{4}}{2(1+\lambda)}}$ (23)

Now we define two more geometrical parameters, proposed in [67], called as
statefinder parameters $r,s$, which reveals the geometrical evolution of
universe and different stages of dark energy models [67, 68, 69]. These
parameters are defined in terms of scale factor as

$r=\frac{\dddot{a}}{aH^{3}},~{}~{}~{}~{}~{}~{}s=\frac{r-1}{3(q-\frac{1}{2})}$
(24)

Using the scale factor (19) in (24), we obtain the statefinder diagnostic
parameters $r(t)$ and $s(t)$ as below

$r(t)=1-3~{}sech^{2}{\left(\frac{\sqrt{3n_{1}}}{2}t\right)}$ (25)
$s(t)=\frac{2~{}sech^{2}{\left(\frac{\sqrt{3n_{1}}}{2}t\right)}}{3\left[1-sech^{2}{\left(\frac{\sqrt{3n_{1}}}{2}t\right)}\right]}$
(26)

Now we define the effective equation of state (EoS) parameter $\omega_{eff}$
by comparing the Eqs. (15) & (16) to the standard Friedmann equations in a
flat spacetime universe, as

$\omega_{eff}=\frac{p_{eff}}{\rho_{eff}}$ (27)

or

$\omega_{eff}=-1+\frac{2n_{1}(\kappa^{2}\rho_{m0}-\lambda
k^{2})}{2n_{1}(\kappa^{2}\rho_{m0}-\lambda k^{2})+(k_{2}+\lambda
k_{4}-2n_{1}\lambda)k^{2}\sinh^{2}{\left(\frac{\sqrt{3n_{1}}}{2}t\right)}}$
(28)

Also, from Eq. (15), we can derive total energy density parameter as

$\Omega_{m}+\Omega_{F}=1,$ (29)

where

$\Omega_{m}=\frac{\kappa^{2}\rho_{m}}{3H^{2}},~{}~{}~{}~{}\Omega_{F}=\frac{k_{2}+\lambda
k_{4}}{6H^{2}}-\lambda.$ (30)

called respectively as matter energy density parameter $\Omega_{m}$ and dark
energy density parameter $\Omega_{F}$ due to $F(R,T)$ gravity function.

In the field equations (15) & (16), for $\lambda=0$, one can obtain the
Einstein’s field equations in GR with cosmological constant as
$\Lambda=\frac{k_{2}}{2}$. And in this case,
$\Omega_{F}=\frac{k_{2}}{6H^{2}}=\frac{\Lambda}{3H^{2}}=\Omega_{\Lambda}$.
Also, for $\lambda\neq 0$, we can obtain varying $\Lambda$-term as a function
of Hubble parameter $H$, as $\Lambda(H)=\frac{1}{2}(k_{2}-6\lambda H^{2})$.

###  3.2 Model-II

In this model, we choose $u=k_{1}(\dot{H}+H^{2})-k_{2}$ and $v=k_{3}aH-k_{4}$
with $k_{i}$’s $i=1,2,3,4$ constants, and $\dot{H}+H^{2}=\frac{\ddot{a}}{a}$,
in Eqs. (12) & (13), we obtain the following simplified field equations

$\frac{3}{2}(2+2\lambda+k_{1})H^{2}-\frac{1}{2}(k_{2}+\lambda
k_{4})=\kappa^{2}\rho_{m}$ (31)
$\frac{1}{3}(3+k_{1}+3\lambda)(2\dot{H}+3H^{2})-\frac{1}{2}(k_{2}+\lambda
k_{4})=-\kappa^{2}p_{m}$ (32)

Taking the non-relativistic matter pressure $p_{m}\approx 0$ in Eq. (32), we
rewrite it as

$2\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}-\frac{3(k_{2}+\lambda
k_{4})}{2(3+k_{1}+3\lambda)}=0$ (33)

Solving Eq. (33) for scale factor $a(t)$, we get

$a(t)=\left[\frac{3\sqrt{3}c_{4}e^{\frac{\sqrt{3n_{2}}}{2}t}-\sqrt{3}c_{3}e^{-\frac{\sqrt{3n_{2}}}{2}t}}{6\sqrt{n_{2}}}\right]^{\frac{2}{3}},~{}~{}~{}~{}n_{2}>0$
(34)

where $c_{3},c_{4}$ are arbitrary constants and $n_{2}=\frac{3(k_{2}+\lambda
k_{4})}{2(3+k_{1}+3\lambda)}$. Without loss of generality, we choose
$c_{3}=k\sqrt{3},c_{4}=\frac{k}{\sqrt{3}}$ so that we can write the scale
factor as the hyperbolic expansion law

$a(t)=\left[\frac{k}{\sqrt{n_{2}}}\sinh{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}\right]^{\frac{2}{3}}$
(35)

Using this scale factor as in Eq. (35), we derive the Hubble parameter $H(t)$
and deceleration parameter $q(t)$ as given below:

$H(t)=\sqrt{\dfrac{n_{2}}{3}}\coth{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}$
(36) $q(t)=-1+\frac{3}{2}sech^{2}{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}$ (37)

Again, we can express these in terms of redshift $z$ as

$H(z)=\frac{1}{\sqrt{3}}\sqrt{k^{2}(1+z)^{3}+\frac{3(k_{2}+\lambda
k_{4})}{2(3+k_{1}+3\lambda)}}$ (38)
$q(z)=-1+\frac{3}{2}\frac{k^{2}(1+z)^{3}}{k^{2}(1+z)^{3}+\frac{3(k_{2}+\lambda
k_{4})}{2(3+k_{1}+3\lambda)}}$ (39)

For model-II, the statefinder diagnostic parameters $r(t)$ and $s(t)$ are
derived as

$r(t)=1-3~{}sech^{2}{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}$ (40)
$s(t)=\frac{2~{}sech^{2}{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}}{3\left[1-sech^{2}{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}\right]}$
(41)

The effective EoS parameter for model-II is derived as

$\omega_{eff}=-1+\frac{3n_{2}(2\kappa^{2}\rho_{m0}-k_{1}k^{2}-2\lambda
k^{2})-n_{2}k_{1}k^{2}\sinh^{2}{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}}{3n_{2}(2\kappa^{2}\rho_{m0}-k_{1}k^{2}-2\lambda
k^{2})-3k^{2}(2n_{2}\lambda+n_{2}k_{1}-k_{2}-k_{4}\lambda)\sinh^{2}{\left(\frac{\sqrt{3n_{2}}}{2}t\right)}}$
(42)

Also, from Eq. (31), we can derive total energy density parameter for model-
II, as

$\Omega_{m}+\Omega_{F}=1,$ (43)

where

$\Omega_{m}=\frac{\kappa^{2}\rho_{m}}{3H^{2}},~{}~{}~{}~{}\Omega_{F}=\frac{k_{2}+\lambda
k_{4}}{6H^{2}}-\lambda-\frac{1}{2}k_{1}.$ (44)

Now, we can find the Original Einstein’s field equations with cosmological
constant $\Lambda$-term in GR, by substituting $\lambda=0,k_{1}=0$, with
$\Lambda=\frac{k_{2}}{2}$ otherwise
$\Lambda(H)=\frac{1}{2}[k_{2}-(3k_{1}+6\lambda)H^{2}]$.

##  4 Observational Constraints

In this section, we make observational constraints on the model parameters
with observational datasets in our derived model. For this, we use the freely
available emcee program at [70], to perform an MCMC (Monte Carlo Markov Chain)
analysis so that we can compare our derived model with observational datasets.
By varying the parameter values in a probable range of priors and analysis of
the parameter space posteriors, the MCMC sampler constraints the cosmological
and model parameters.

###  4.1 Hubble function $H(z)$

The Hubble parameter is one of the most important cosmological parameter for
the study of evolution of the universe for both theoretical and observational
cosmologists. Due to the availability of observed values of Hubble datasets
$H(z)$ with redshift $z$, we first compare our derived Hubble function from
the field equations with observed values of $H(z)$ through MCMC analysis, to
find the best fit values of model parameters with error bars. For this, we
consider $32$ observed Hubble $H(z)$ datasets from [71]-[72] with errors. We
use the following $\chi^{2}$-test formula in our analysis

$\chi^{2}(\phi)=\sum_{i=1}^{i=N}\frac{[(H_{ob})_{i}-(H_{th})_{i}]^{2}}{\sigma_{i}^{2}}$

Where $N$ denotes the total amount of data, $H_{ob},~{}H_{th}$, respectively,
the observed and hypothesized datasets of $H(z)$ and standard deviations are
displayed by $\sigma_{i}$. Here for the Model-I $\phi=(k,k_{2},k_{4},\lambda)$
and for the Model-II $\phi=(k,k_{1},k_{2},k_{4},\lambda)$.

Figure 1: The contour plots of $k,k_{2},k_{4},\lambda$ at $1-\sigma,2-\sigma$ and $3-\sigma$ confidence level in MCMC analysis of $H(z)$ datasets. Parameter | Prior | Model-I | Model-II
---|---|---|---
$k$ | $(50,100)$ | $66.751_{-0.47347}^{+0.46}$ | $66.746_{-0.24954}^{+0.25992}$
$k_{1}$ | $(-1.0,2.5)$ | $-$ | $0.19416_{-0.22047}^{+0.21862}$
$k_{2}$ | $(10^{4},2\times 10^{4})$ | $18730_{-0.46756}^{+0.48868}$ | $18930_{-0.24383}^{+0.23751}$
$k_{4}$ | $(10^{4},3\times 10^{4})$ | $29654_{-0.48363}^{+0.50787}$ | $29843_{-0.2494}^{+0.25406}$
$\lambda$ | $(-0.7,3.0)$ | $0.34563_{-0.25994}^{+0.27994}$ | $0.20855_{-0.15423}^{+0.21128}$
Table 1: The MCMC Results in $H(z)$ datasets analysis. Figure 2: The contour
plots of $k,k_{1},k_{2},k_{4},\lambda$ at $1-\sigma,2-\sigma$ and $3-\sigma$
confidence level in MCMC analysis of $H(z)$ datasets.

a. b.

Figure 3: The Hubble error bar plots of $H(z)$ datasets over $z$, respectively
for Model-I, and Model-II.

Figure 1 & 2 represent the contour plots of model parameters involved in
Model-I & Model-II, respectively, with observed Hubble datasets $H(z)$, using
MCMC analysis. We have obtained the best fit values of various model
parameters $k,k_{1},k_{2},k_{4},\lambda$, by using the different values of
parameters in the range $50<k<100$, $-1<k_{1}<2.5$, $10000<k_{2}<20000$,
$10000<k_{4}<30000$ and $-0.7<\lambda<3.0$ with $1-\sigma,2-\sigma$ and
$3-\sigma$ confidence level. The best fit values are mentioned in Table 1, for
both models, respectively. We have estimated the best fit values of model
parameter
$\lambda=0.34563_{-0.25994}^{+0.27994},0.20855_{-0.15423}^{+0.21128}$ for
Model-I, Model-II, respectively. Recently [55] has estimated the value of
$\lambda$ as $\lambda=0.491_{-0.533}^{+0.387},0.537_{-0.550}^{+0.403}$,
respectively, in two different models. Figure 3a & 3b depicts the Hubble error
bar plots for Model-I & Model-II, respectively.

###  4.2 Apparent Magnitude $m(z)$

The relationship between luminosity distance and redshift is one of the main
observational techniques used to track the universe’s evolution. The expansion
of the cosmos and the redshift of the light from distant brilliant objects are
taken into consideration when calculating the luminosity distance ($D_{L}$) in
terms of the cosmic redshift ($z$). It is provided as

$D_{L}=a_{0}r(1+z),$ (45)

where the radial coordinate of the source $r$, is established by

$r=\int^{r}_{0}dr=\int^{t}_{0}\frac{cdt}{a(t)}=\frac{1}{a_{0}}\int^{z}_{0}\frac{cdz}{H(z)},$
(46)

where we have used $dt=dz/\dot{z},\dot{z}=-H(1+z)$.
As a result, the luminosity distance is calculated as follows:

$D_{L}=c(1+z)\int^{z}_{0}\frac{dz}{H(z)}.$ (47)

Hence, the apparent magnitude $m(z)$ of a supernova is defined as:

$m(z)=16.08+5~{}log_{10}\left[\frac{(1+z)H_{0}}{0.026}\int^{z}_{0}\frac{dz}{H(z)}\right].$
(48)

We use the most recent collection of $1048$ datasets of the Pantheon SNe Ia
samples in the ($0.01\leq z\leq 1.7$) range [79] in our MCMC analysis. We have
used the following $\chi^{2}$ formula to constrain different model parameters:

$\chi^{2}(\phi)=\sum_{i=1}^{i=N}\frac{[(m_{ob})_{i}-(m_{th})_{i}]^{2}}{\sigma_{i}^{2}}.$

The entire amount of data is denoted by $N$, the observed and theoretical
datasets of $m(z)$ are represented by $m_{ob}$ and $m_{th}$, respectively, and
standard deviations are denoted by $\sigma_{i}$. Here for the Model-I
$\phi=(H_{0},k,k_{2},k_{4},\lambda)$ and for the Model-II
$\phi=(H_{0},k,k_{1},k_{2},k_{4},\lambda)$.

Figure 4: The contour plots of $H_{0},k,k_{2},k_{4},\lambda$ at $1-\sigma,2-\sigma$ and $3-\sigma$ confidence level in MCMC analysis of Pantheon SNe Ia datasets. Figure 5: The contour plots of $H_{0},k,k_{1},k_{2},k_{4},\lambda$ at $1-\sigma,2-\sigma$ and $3-\sigma$ confidence level in MCMC analysis of Pantheon SNe Ia datasets. Parameter | Prior | Model-I | Model-II
---|---|---|---
$H_{0}$ | $(50,100)$ | $67.967_{-0.078202}^{+0.063326}$ | $68.193_{-0.12074}^{+0.10342}$
$k$ | $(50,100)$ | $65.336_{-0.072302}^{+0.077101}$ | $67.749_{-0.092949}^{+0.11758}$
$k_{1}$ | $(-1.0,2.5)$ | $-$ | $0.17981_{-0.13219}^{+0.062515}$
$k_{2}$ | $(10^{4},2\times 10^{4})$ | $18610_{-0.068945}^{+0.07286}$ | $18966_{-0.11619}^{+0.10197}$
$k_{4}$ | $(10^{4},3\times 10^{4})$ | $29668_{-0.069035}^{+0.068959}$ | $29745_{-0.10137}^{+0.11386}$
$\lambda$ | $(-0.7,3.0)$ | $0.43317_{-0.030127}^{+0.084908}$ | $0.22418_{-0.040733}^{+0.10975}$
Table 2: The MCMC Results in Pantheon SNe Ia datasets analysis.

Figure 4 & 5 represent the contour plots of model parameters involved in
Model-I & Model-II, respectively, with apparent magnitude $m(z)$ from observed
Pantheon SNe Ia datasets, using MCMC analysis. We have obtained the best fit
values of various model parameters $H_{0},k,k_{1},k_{2},k_{4},\lambda$, by
using the different values of parameters in the range $40<H_{0}<100$,
$50<k<100$, $-1<k_{1}<2.5$, $10000<k_{2}<20000$, $10000<k_{4}<30000$ and
$-0.7<\lambda<3.0$ with $1-\sigma,2-\sigma$ and $3-\sigma$ confidence level.
The best fit values of various model parameters are mentioned in Table 2, for
both models, respectively. We have estimated the best fit values of model
parameter
$\lambda=0.43317_{-0.030127}^{+0.084908},0.22418_{-0.040733}^{+0.10975}$ for
Model-I, Model-II, respectively. Recently [55] has estimated the value of
$\lambda$ as $\lambda=0.491_{-0.533}^{+0.387},0.537_{-0.550}^{+0.403}$,
respectively, in two different models. We have estimated the value of Hubble
constant as
$H_{0}=67.967_{-0.078202}^{+0.063326},68.193_{-0.12074}^{+0.10342}$ Km/s/Mpc,
respectively for two models, Model-I & Model-II along Pantheon SNe Ia
datasets. Recently, the value of Hubble constant is measured as $H_{0}=69.8\pm
1.3~{}Kms^{-1}Mpc^{-1}$ in [80], in [81], it is measured as $H_{0}=69.7\pm
1.2~{}Kms^{-1}Mpc^{-1}$. Recently, we have measured the value of $H_{0}$ as
$H_{0}=68.3721\pm 1.7205$ Km/s/Mpc in [59], $H_{0}=68.3721\pm 1.65678$
Km/s/Mpc in [57] and $H_{0}=71.66123\pm 0.33061$ in [58].

##  5 Result Discussion

First, we discuss each model in details as follows:

###  5.1 Model-I

The scale factor $a(t)$ is the most powerful parameter in cosmology that
governs every geometrical parameters behaviour. From the solution of the field
equation (16), we obtain the scale factor $a(t)$ as in (19), and its
geometrical evolution over cosmic time $t$ is given in figure 6a. From the
figure 6a, we can observe that the scale factor $a(t)$ is an increasing
function of cosmic time $t$ that shows that our universe model is expanding.
One can see that as $t\to t_{0}$ (present time) then the value of $a(t)\to 1$
(i.e. $a_{0}=a(t_{0})=1$), and as $t\to\infty$ then $a(t)\to\infty$. The
present age of the universe is estimated as
$t_{0}=0.01363_{-0.00020}^{+0.00027},0.01379_{-0.00006}^{+0.00003}$ or
$t_{0}=13.33_{-0.19}^{+0.26},13.48_{-0.05}^{+0.02}$ Gyrs, respectively along
two datasets $H(z)$ and Pantheon SNe Ia. In general, the present age of the
universe can be expressed as
$t_{0}=978\times\frac{2}{\sqrt{3}}\sqrt{\frac{2(1+\lambda)}{k_{2}+\lambda
k_{4}}}\sinh^{-1}\left(k\sqrt{\frac{2(1+\lambda)}{k_{2}+\lambda
k_{4}}}\right)$ Gyrs.

a. b.

Figure 6: Variation of scale factor over cosmic time $t$, and deceleration
parameter $q(z)$ over $z$, respectively.

The geometrical parameter $q(t)$ is defined in terms of scale factor as
$q(t)=-\frac{a\ddot{a}}{\dot{a}^{2}}$ that reveals the expansion phase of the
universe (decelerating or accelerating). In our model-I, it is derived as in
Eq. (21) from the scale factor (19), and expressed in terms of redshift $z$ as
in Eq. (23). The geometrical behaviour of deceleration parameter $q(z)$ is
shown in figure 6b, and one can see that $q(z)$ is an increasing function of
redshift $z$ and shows a signature-flipping point (decelerating-accelerating)
called as transition redshift $z$. The present value of $q(z)$ is obtained as
$q_{0}=-0.5610_{-0.0141}^{+0.0254},-0.58_{-0.0051}^{+0.0016}$, respectively
for two observational datasets $H(z)$ and Pantheon SNe Ia, that reveals that
our universe model is accelerating phase of expansion at present (at $z=0$).
We observe that as $z\to-1$, then $q\to-1$ (late-time acceleration) while as
$z\to\infty$ then $q\to 0.5$ early decelerating universe. This model can not
explain the early accelerating (inflation) scenario of the evolution of the
universe. Also, one can obtain as $z\to 0$, then the value of deceleration
parameter $q\to-1+\frac{3}{2}\left[1+\frac{k_{2}+\lambda
k_{4}}{2k^{2}(1+\lambda)}\right]^{-1}$. The estimated transition redshift is
obtained as $z_{t}=0.69_{-0.056}^{+0.028},0.726_{-0.004}^{+0.009}$,
respectively along two datasets, and the general expression for transition
redshift is obtained as

$z_{t}=\left[\frac{k_{2}+\lambda
k_{4}}{k^{2}(1+\lambda)}\right]^{\frac{1}{3}}-1,~{}~{}~{}~{}\lambda\neq-1.$
(49)

This transition value shows that the universe model is in decelerating phase
of expansion for $z>z_{t}$ and the model is in accelerating phase of expansion
for $z<z_{t}$. Recently, this transition redshift $z_{t}=0.74\pm 0.05$ is
obtained in [82] and in [83] it is measured as $z_{t}=0.74\pm 0.04$. In [84],
this transition redshift is obtained as $z_{t}=0.72\pm 0.05$ and in 2018, [85]
has suggested that the transition redshift varies over $0.33<z_{t}<1.0$.
Recently, we have found this transition redshift $z_{t}\approx 0.7$ in [86,
87]. Thus, the transition redshift
$z_{t}=0.69_{-0.056}^{+0.028},0.726_{-0.004}^{+0.009}$ obtained in our derived
model is in good agreement with recent observed values in [82]-[87].

The cosmological parameters $\Omega_{m}$ and $\Omega_{F}$ are derived from the
field Eqs. (15) as in Eqs. (29) & (30) for the model-I. Here, $\Omega_{m}$ is
the non-relativistic matter energy density parameter and $\Omega_{F}$ is the
dark energy density parameter from geometrical modifications. The geometrical
behaviour of $\Omega_{m}$ & $\Omega_{F}$ are shown in figure 7a & 7b,
respectively. The estimated present values of total energy density parameters
are $\Omega_{m0}=0.3938_{-0.0577}^{+0.0656},0.4013_{-0.0069}^{+0.0186}$ and
$\Omega_{F0}=0.6062_{-0.0656}^{+0.0577},0.5987_{-0.0186}^{+0.0069}$,
respectively along two observational datasets $H(z)$ and Pantheon SNe Ia. From
figure 7a & 7b, one can see that $\Omega_{m}\to 0$ & $\Omega_{F}\to 1$ as
$z\to-1$ (at late-time) which reveals that our model tends to $\Lambda$CDM
model at late-time universe, and in early universe $\Omega_{m}\to 1$ and
$\Omega_{F}\to-\lambda$ as $z\to\infty$ that shows the matter dominated early
universe.

a. b.

Figure 7: The plot of total energy density parameters $\Omega_{m}$ and
$\Omega_{F}$ over redshift $z$, respectively.

a. b.

Figure 8: The plot of effective EoS parameter $\omega_{eff}$ versus $z$ &
$\Omega_{m}$, respectively.

The effective EoS parameter $\omega_{eff}$ for model-I is derived in Eq. (28),
and its geometrical evolution over $z$ is shown in figure 8a. From the figure
8a, one can see that $\omega_{eff}\to-1$ as $z\to-1$ (at late-time) and
$\omega_{eff}\to 0$ as $z\to\infty$ (at early universe time), that support
early matter dominated universe and late-time dark energy dominated universe.
The present value of effective EoS parameter for model-I is estimated as
$\omega_{eff}=-0.9946_{-0.0020}^{+0.0024},-0.9950_{-0.0006}^{+0.0002}$,
respectively along two observational datasets. Figure 8b reveals the variation
of $\omega_{eff}$ over matter energy density parameter $\Omega_{m}$ and
mathematically we can find this relationship as
$(A-B)k^{2}\Omega_{m}\omega_{eff}+B\rho_{m0}\kappa^{2}\omega_{eff}-Bk^{2}\Omega_{m}+B\rho_{m0}\kappa^{2}=0$,
where $A=2n_{1}(\kappa^{2}\rho_{m0}-\lambda k^{2})$ and $B=(k_{2}+\lambda
k_{4}-2n_{1}\lambda)k^{2}$. Also, it can be expressed as

$\omega_{eff}=\frac{Bk^{2}\Omega_{m}-B\rho_{m0}\kappa^{2}}{(A-B)k^{2}\Omega_{m}+B\rho_{m0}\kappa^{2}}$
(50)

From Eq. (50) and figure 7b, it is clear that the EoS parameter
$\omega_{eff}\to-1$ for $\Omega_{m}\to 0$, and $\omega_{eff}\to 0$ for
$\Omega_{m}\to\frac{\rho_{m0}\kappa^{2}}{k^{2}}$. From the above relationship,
it is predicted that one can obtain radiation dominated universe for
$\Omega_{m}>\frac{\rho_{m0}\kappa^{2}}{k^{2}}$.

Now, we discuss two more geometrical parameters derived from the scale factor
$a(t)$ as the statefinder diagnostic parameters $r(t)$, $s(t)$ which are
defined by the Eq. (24). For the model-I, we have derived these parameters
$r(t)$ & $s(t)$ as in Eqs. (25) & (26), respectively. The geometrical
evolution of these parameters $r,s$ are shown in figure 9a & 9b, over redshift
$z$, respectively. From the figure 9a, 9b, one can see that $r(z)\to 1$,
$s(z)\to 0$ as $z\to-1$, that reveals the late-time $\Lambda$CDM tendency of
the model. The estimated present values of $r,s$ are as
$r_{0}=0.122_{-0.0508}^{+0.0302},0.16_{-0.0033}^{+0.0103}$ and
$s_{0}=0.2759_{-0.0133}^{+0.0231},0.2593_{-0.0044}^{+0.0014}$, respectively
along two datasets $H(z)$ and Pantheon SNe Ia.

a. b.

Figure 9: The plot of statefinder diagnostic parameter $r(z)$ & $s(z)$ versus
redshift $z$, respectively.

a. b.

Figure 10: The variation of statefinder diagnostic parameter $s(z)$ versus
$r(z)$, and $r(z)$ versus $q(z)$, respectively.

Figure 10a represents the variation of statefinder parameter $s(z)$ with the
variation of statefinder parameter $r(z)$ that reveals that these parameters
are related to each other. Also, we can find the relation between $r$ & $s$
from Eqs. (25) & (26) as $3rs+6s+2r-2=0$ for the model-I. It can be expressed
as $s=\frac{2(1-r)}{3(2+r)}$ which shows that $s\to\infty$ for $r\to-2$
(singular point) and $s\to 0$ for $r\to 1$. Figure 10b shows the relationship
between statefinder parameter $r$ and deceleration parameter $q$ that reveals
that $r$ & $q$ are linearly related to each other. From the Eqs. (21) & (25),
we can find the relationship between $r$ & $q$ as $r+2q+1=0$ for the model-I.
Also, we can rewrite as $q=-\frac{1+r}{2}$ which gives $q>0$ for $r<-1$ and
$q<0$ for $r>-1$, and the model undergoes a transition point at $r=-1$.

a. b.

Figure 11: The plots of effective EoS parameter $\omega_{eff}$ versus
statefinder parameters $r(z)$ and $s(z)$, respectively.

Figure 11a depicts the evolution of effective EoS parameter $\omega_{eff}$
over the variation of statefinder parameter $r(z)$, that depicts
$\omega_{eff}$ & $r$ are related each other. From the Eqs. (25) & (28), one
can obtain the following relationship between $\omega_{eff}$ & $r$ as
$(B-A)r\omega_{eff}+(A+2B)\omega_{eff}+Br+2B=0$ for the model-I, where
$A=2n_{1}(\kappa^{2}\rho_{m0}-\lambda k^{2})$ and $B=(k_{2}+\lambda
k_{4}-2n_{1}\lambda)k^{2}$. It can be expressed as
$\omega_{eff}=\frac{B(r+2)}{A(r-1)-B(r+2)}$ that shows as $r\to 1$ then
$\omega_{eff}\to-1$. Figure 11b depicts the variation of effective EoS
parameter over statefinder $s(z)$, that reveals that the EoS $\omega_{eff}$ is
directly proportional to $r(z)$. From Eqs. (26) & (28), we can obtain the
relationship between $\omega_{eff}$ & $s$ as
$3As\omega_{eff}+2B\omega_{eff}+2B=0$ for model-I. Also, we can express it as
$\omega_{eff}=-\frac{2B}{3As+2B}$ that represents as $s\to 0$ then
$\omega_{eff}\to-1$ and as $s\to\infty$ then $\omega_{eff}\to 0$.

a. b.

Figure 12: The plots of statefinder parameters $r(z)$ & $s(z)$ versus matter
energy density parameter $\Omega_{m}$, respectively.

Figure 12a depicts the variation of statefinder parameter $r(z)$ over
$\Omega_{m}$ that reveals the relationship between them. From Eqs. (25) &
(30), we can derive the relationship between $r$ & $\Omega_{m}$ as
$r=1-C\Omega_{m}$, $C=\frac{3k^{2}}{\rho_{m0}\kappa^{2}}$. Figure 12b shows
the variation of statefinder $s$ over matter energy density parameter
$\Omega_{m}$ that reveals the relationship between them. The relationship
between $s$ & $\Omega_{m}$ is derived from Eqs. (26) & (30) as
$3k^{2}s\Omega_{m}+2k^{2}\Omega_{m}-3\rho_{m0}\kappa^{2}s=0$ for model-I.

###  5.2 Model-II

In the second model, we have obtained the scale factor by solving the field
equation (32) and obtained the scale factor $a(t)$ as in Eq. (35). The
geometrical evolution of $a(t)$ for model-II is shown in figure 13a with
cosmic time $t$ for two observational datasets $H(z)$ and Pantheon SNe Ia. At
present, we have assumed the value of scale factor as $a_{0}=a_{t_{0}}=1$,
where $t_{0}$ denotes the present age of the universe. We have estimated the
present age of the universe as
$t_{0}=0.01381_{-0.00008}^{+0.00005},0.01365_{-0.00006}^{+0.00002}$ or
$t_{0}=13.50_{-0.07824}^{+0.04890},13.29_{-0.05868}^{+0.01956}$ Gyrs,
respectively for two observational datasets. At $t=0$, $a(0)=0$ and as
$t\to\infty$, then $a(t)\to\infty$, this shows the expansion of the universe.

a. b.

Figure 13: The plot of scale factor $a(t)$ versus cosmic time $t$, and the
deceleration parameter $q(z)$ versus redshift $z$, respectively.

The equation (37) represents the mathematical expression for the deceleration
parameter derived from the scale factor (35). We have obtained the expression
of $q$ in terms of redshift $z$ as in Eq. (39). The geometrical behaviour of
$q(z)$ is represents in figure 12b, and from the figure 13b, one can see that
$q(z)$ is an increasing function of $z$. The evolution of $q(z)$ shows a
signature-flipping point (transition) denoted by $z_{t}$. We have estimated
the present values of transition redshift as
$z_{t}=0.642_{-0.001}^{+0.003},0.633_{-0.001}^{+0.012}$, respectively for two
observational datasets $H(z)$ and Pantheon SNe Ia, which are in good agreement
with recent observed values of $z_{t}$ in [82]-[87]. At $z=0$, the value of
deceleration parameter is called present value and it is denoted by $q_{0}$.
We have estimated the present value of deceleration parameter $q$ as
$q_{0}=-0.5338\pm 0.0047,-0.5275\pm 0.0063$, respectively along two considered
datasets. Thus the value $q_{0}<0$ reveals that our universe model is
accelerating at present. From figure 13b, we can see that $q\to-1$ as $z\to-1$
and $q\to 0.5$ for $z\to\infty$ that depicts the existence of such universe
which is early decelerating and late-time accelerating expanding in nature.
The transition redshift is generally obtained as

$z_{t}=\left[\frac{3(k_{2}+\lambda
k_{4})}{k^{2}(3+k_{1}+3\lambda)}\right]^{\frac{1}{3}}-1.$ (51)

For the model-II, the mathematical expression for matter energy density
parameter $\Omega_{m}$ and dark energy density parameter $\Omega_{F}$ are
shown by the Eqs. (43) & (44), respectively. The geometrical evolution of
$\Omega_{m}$ & $\Omega_{F}$ over $z$ are shown in figure 14a & 14b,
respectively. From figure 14a & 14b, one can see that as $z\to-1$, then
$\Omega_{m}\to 0$ & $\Omega_{F}\to 1$ that represents the $\Lambda$CDM
tendency of the model, on the other hand, as $z\to\infty$, then $\Omega_{m}$
tends to $1.4$ and $\Omega_{F}\to-\lambda-0.5k_{1}$ that reveals the early
matter dominated universe. We have estimated the present values of these
parameters as
$\Omega_{m0}=0.4167_{-0.0073}^{+0.0024},0.4160_{-0.0130}^{+0.0130}$ and
$\Omega_{F0}=0.5719_{-0.1198}^{+0.1080},0.5655_{-0.0456}^{+0.0546}$,
respectively for two observational datasets $H(z)$ and Pantheon SNe Ia.

a. b.

Figure 14: The plots of total energy density parameters $\Omega_{m}$ and
$\Omega_{F}$ versus redshift $z$, respectively.

a. b.

Figure 15: The variation of effective EoS parameter $\omega_{eff}$ versus $z$
& $\Omega_{m}$, respectively.

For model-II, the mathematical expression of effective EoS parameter
$\omega_{eff}$ is obtained as in Eq. (42), and its geometrical evolution over
$z$ is shown in figure 15a. From figure 15a, one can see that
$\omega_{eff}\to-1.031$ as $z\to-1$, and it crosses the $\Lambda$CDM value
$\omega_{eff}=-1$ at $z=-0.599,-0.6$, respectively for two datasets. On the
other hand, $\omega_{eff}\to 0$ as $z\to\infty$ that represents that early
universe was matter dominated. The present value of effective EoS parameter is
measured as
$\omega_{eff}=-0.6971_{-0.1066}^{+0.0696},-0.6979_{-0.0440}^{+0.0287}$,
respectively along two observational datasets $H(z)$ and Pantheon SNe Ia.
Figure 15b represents the variation of $\omega_{eff}$ versus $\Omega_{m}$, and
one can see that they have relationship which is obtained from Eqs. (42) &
(44) as
$(A+B)k^{2}\Omega_{m}\omega_{eff}-B\rho_{m0}\kappa^{2}\omega_{eff}+(B-n_{2}k_{1}k^{2})k^{2}\Omega_{m}-B\rho_{m0}\kappa^{2}+n_{2}\rho_{m0}k_{1}k^{2}\kappa^{2}=0$
where $A=3n_{2}(2\kappa^{2}\rho_{m0}-k_{1}k^{2}-2\lambda k^{2})$ and
$B=3k^{2}(2n_{2}\lambda+n_{2}k_{1}-k_{2}-k_{4}\lambda)$. It can be also
expressed as

$\omega_{eff}=\frac{B\rho_{m0}\kappa^{2}-n_{2}\rho_{m0}k_{1}k^{2}\kappa^{2}-(B-n_{2}k_{1}k^{2})k^{2}\Omega_{m}}{(A+B)k^{2}\Omega_{m}-B\rho_{m0}\kappa^{2}}$
(52)

From Eq. (52), we can obtained for $\Omega_{m}\to 0$,
$\omega_{eff}\to-1-\frac{k_{1}}{6-k_{1}},k_{1}<6$ that gives cosmological
constant value $\omega_{eff}=-1$ for $k_{1}=0$, phantom and super-phantom
value for $k_{1}<6$.
The mathematical expressions of statefinder parameters for model-II are
represented by the Eqs. (40) & (41), respectively, and their geometrical
evolution over $z$ are shown in figure 16a & 16b. From figure 16a & 16b, one
can observe that $(s,r)\to(0,1)$ as $z\to-1$ that reveals that model obtained
$\Lambda$CDM stage at late-time universe. We have measured the present values
of these parameters as
$r_{0}=0.0676_{-0.0094}^{+0.0094},0.0550_{-0.0126}^{+0.0126}$ and
$s_{0}=0.3007_{-0.0055}^{+0.0055},0.3066_{-0.006}^{+0.006}$, respectively for
two datasets.

a. b.

Figure 16: The plots of statefinder diagnostic parameters $r(z)$ & $s(z)$
versus $z$, respectively.

a. b.

Figure 17: The plots of statefinder $s(z)$ versus $r(z)$, and statefinder
$r(z)$ versus deceleration parameter $q(z)$, respectively.

a. b.

Figure 18: The plots of effective EoS parameter versus statefinder parameters
$r(z)$ & $s(z)$, respectively.

Figure 17a represents the relationship between statefinder parameters $r,s$
for the model-II, and figure 17b represents the relationship between
statefinder $r(z)$ and deceleration parameter $q(z)$. From figure 17a & 17b,
one can observe that as $r\to 1$, then $s\to 0$, while $r\to 1$ when $q\to-1$.

Figure 18a & 18b represent the relationship between effective EoS parameters
$\omega_{eff}$ and statefinder parameters $r,s$. From Eqs. (40), (41) & (42),
we obtain these relationship as
$(A+C)r\omega_{eff}+(2C-A)\omega_{eff}+(C-B)r+2C-2B=0$, and
$3As\omega_{eff}-2C\omega_{eff}-2C+2B=0$ where
$A=3n_{2}(2\kappa^{2}\rho_{m0}-k_{1}k^{2}-2\lambda k^{2})$,
$B=n_{2}k_{1}k^{2}$ and
$C=3k^{2}(2n_{2}\lambda+n_{2}k_{1}-k_{2}-k_{4}\lambda)$. These relations can
be expressed as

$\omega_{eff}=\frac{(B-C)(r+2)}{(A+B)r+(2C-A)},~{}~{}~{}~{}\omega_{eff}=\frac{2(C-B)}{3As-2C}$
(53)

For $r\to 1$ & $s\to 0$, we find respectively the effective EoS parameter as
$\omega_{eff}\to\frac{3(B-C)}{B+2C}$ and $\omega_{eff}\to-1+\frac{B}{C}$.

a. b.

Figure 19: The plots of statefinder parameters $r(z)$ & $s(z)$ versus
$\Omega_{m}$, respectively.

Figure 19a & 19b represent the relationship between statefinder parameters
$r,s$ and matter energy density parameter $\Omega_{m}$. We can derive these
relationship from the Eqs. (40), (41) & (44) as
$r=1-\frac{3k^{2}}{\rho_{m0}\kappa^{2}}\Omega_{m}$ and
$3k^{2}s\Omega_{m}-3\rho_{m0}\kappa^{2}s+2k^{2}\Omega_{m}=0$.

Thus, both the model-I & model-II are able to explain the late-time
accelerating as well as early decelerating scenario of the universe but it
does not able to explain the early time accelerating (inflation) scenario of
the universe. In the model-II, the effective EoS parameter crosses the
cosmological constant value $\omega_{eff}=-1$ while model-I does not so and it
tends to cosmological constant value $-1$.

##  6 Conclusions

Using a flat Friedmann-Lematre-Robertson-Walker (FLRW) spacetime metric, we
have studied various exact cosmological models in Myrzakulov gravity in this
study. We have taken into consideration the modified Lagrangian function
$F(R,T)=R+\lambda T$, where $\lambda$ is a model parameter and $R,T$ are the
Ricci curvature scalar and the torsion scalar with regard to non-special
connection, respectively. We have found two exact solutions for the factor of
scale $a(t)$ in two distinct scenarios. We have derived a number of
geometrical parameters to study the universe’s cosmological properties by
using this scale factor. With $1-\sigma,2-\sigma$, and $3-\sigma$ areas, we
have determined the best fit values of model parameters using MCMC analysis on
two types of recent observational datasets, such as $H(z)$ and Pantheon SNe Ia
samples. We have studied geometrical and cosmological parameters in a
relativistic comparison manner. The effective equation of state (EoS)
parameter, $\omega_{eff}$, varies as $-1.031\leq\omega_{eff}\leq 0$ in model-
II, whereas it varies in the range $-1\leq\omega_{eff}\leq 0$ in model-I. The
main features of the derived models are as follows:

  * •

Both model are transit phase expanding universe with transition redshift in
the range $0.6<z_{t}<0.8$ which is in good agreement with recent measurement
[82]-[87].

  * •

Both models are decelerating to accelerating phase of expansion and the
present values of $q(z)$ varies over $-0.6<q_{0}<-0.5$ which are in good
agreement with recent observations [1]-[7].

  * •

Both models are able to explain the late-time accelerating as well as early
decelerating scenario of the universe but it does not able to explain the
early time accelerating (inflation) scenario of the universe.

  * •

The effective EoS parameter in Model-I evolves as $-1\leq\omega_{eff}\leq 0$
over $-1\leq z\leq\infty$.

  * •

In Model-II, the effective EoS parameter varies as $-1.031\leq\omega_{eff}\leq
0$ over $-1\leq z\leq\infty$.

  * •

We have found that the effective EoS parameter $\omega_{eff}$ is related to
statefinder parameters $r,s$ and their behaviour is in acceptable range.

  * •

Model-II evolves from matter dominated stage, passes through quintessence,
cosmological constant and tends to phantom and super-phantom stages of the
expanding universe.

  * •

While Model-I evolves from matter dominated, passes to quintessence and tends
to $\Lambda$CDM model.

  * •

We have found good relationship among the parameters $\omega_{eff}$,
$\Omega_{m}$, $q,r$ & $s$.

  * •

We have found the value of Hubble constant
$H_{0}=67.967_{-0.078202}^{+0.063326},68.193_{-0.12074}^{+0.10342}$ Km/s/Mpc,
respectively for two models.

  * •

We have found the present age of the universe as
$t_{0}=13.33_{-0.19}^{+0.26},13.48_{-0.05}^{+0.02}$ Gyrs, respectively along
two datasets $H(z)$ and Pantheon SNe Ia for model-I, while for model-II, it is
found as $t_{0}=13.50_{-0.07824}^{+0.04890},13.29_{-0.05868}^{+0.01956}$ Gyrs,
respectively for two observational datasets

Thus, we can conclude that the choice of non-special connection (i.e. in
resultant the choice of $u$ & $v$) has an important role in the dynamical dark
energy evolution history of the universe. Therefore, it needs further
investigation and attracts cosmologists to review it.

##  7 Acknowledgments

This work was supported by the Ministry of Science and Higher Education of the
Republic of Kaza- khstan, GrantAP14870191.

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