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Scenario of THDE model in Bianchi type VI
0
Universe
Dr. Nitin Sarma
Associate Professor
Department of Mathematics, ADP College, Nagaon-782002, Assam, India
DOI: https://doi.org/10.51583/IJLTEMAS.2026.150600235
Received: 13 July 2026; Accepted: 18 July 2026; Published: 28 July 2026
ABSTRACT
In this paper, we study the anisotropic Bianchi type VI
0
metric that consists of Tsallis holographic dark energy
(THDE) and dark matter (DM). The hybrid expansion law for the average scale factor is used to calculate some
cosmological parameters and achieve exact solutions to Einstein's field equations. These parameters geometrical
and physical behaviours demonstrate how the universe eventually becomes homogeneous, flat, and isotropic.
Analyzing the equation of state (EoS) parameter also reveals that our model acts like a lambda cold dark matter
(Λ) scenario in the late universe.
Keywords: Bianchi-VI
0
type, Hybrid expansion laws, Tsallis holographic dark energy model.
INTRODUCTION
The universe is undergoing an accelerated growth period as indicated by type Ia supernovae [1], CMB radiation
[2], and galaxy redshift surveys [3]. This has accelerated the expansion of a new kind of energy known as dark
energy (DE) [4]. To investigate the correct nature of DE, numerous researchers have talked about various DE
theories. The cosmological constant lambda [5], quintessence [6], tachyon [7], phantom [8], k-essence model
[9], and chaplygin gas model [10] are just a few of the models.
Among other DE models, the holographic dark energy (HDE) model holds an important position. The
holographic theory, which states that the number of degrees of freedom in a bounded system should be
constrained and related to the area of the boundary, is the foundation of the model HDE. The energy density of
the HDE is
[11] (
Planck mass, L infrared (IR) cut-off radius and c numerical constant).
Recently, a type of HDE model known as "Tsallis holographic dark energy (THDE)" [12] has been proposed to
use Tsallis generalized entropy ‘
′ where γ is a constant and δ is the parameter (non-additive) to explain
the cosmological phenomena.
According to the holographic theory, which posits that a physical system’s degrees of freedom should correspond
with the boundary area rather than its volume [13] and must adhere to an infrared cut-off, Cohen, Kaplan, and
Nelson [14] established an inequality
Λ
. By integrating this with
, they derived Λ
where Λ
represents the vacuum's energy density. The energy density of THDE can be
determined using this inequality as
where D is an unknown variable [15]. When δ = 1, THDE
simplifies to the HDE model. By applying the Hubble horizon as the system's IR cut-off, we express the energy
density of THDE as
.
Recently several scholars like Santhi and Sobhanbabu [16], Pandey et al. [17], Sadeghi et al. [18], Sharif et al.
[19], Sarma [20, 21] explored THDE in different prospect. These works encouraged us to use the hybrid
expansion approach to explore Bianchi type VI
0
metric stuffed with DM and THDE.
The following is a breakdown of the paper’s structure: Section 2 contains the metric and field equation. Section
3 describes the THDE model’s cosmological solutions and physical properties. In Section4, we sum up our
findings.
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2. Bianchi type-VI
0
metric and field equations
For the metric functions
,
and
, the Bianchi type VI
0
metric is defined as
(1)
We consider that the universe is stuffed with DM and THDE components.
Here the Einstein’s field equations are
(2)
where
and
are the tensors of matter energy and the THDE energy momentum respectively.
For the matter energy density (
), the tensor for matter energy momentum (
) is
(3)
In addition, the tensor for THDE energy momentum is taken as
(4)
where
is the energy density of THDE, pressures and EoS parameters of THDE along x axis
and
; along y axis
and
; along z axis
and
respectively.
Using (1), (3), (4) in (2), we get the following equations
(5)
(6)
(7)
(8)
(9)
For the metric (1) scale factor, spatial volume, the mean generalised Hubble’s parameters are expressed as
(10)
(11)
(12)
where
,
,
are the Hubble’s parameters along three axes respectively.
Moreover, the parameter for deceleration (q), growth scalar
, shear scalar
, average anisotropy
parameter
are described as
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(13)
(14)
(15)
(16)
where
.
3. Cosmological solutions and physical properties of THDE model
From (9), we obtain
() (17)
Using (17), the equations (5-8) now can be written as
(18)
(19)
(20)
Solving equations (18) and (19), after using equation (11) we get
(21)
where
is constant.
We use the condition
(22)
which is presented by K. S. Adhav [22] to solve equation (21).
Using (22) in (21), we get
(23)
Also conservation law of energy
gives the continuity equation as
(24)
Since the two fluids under consideration are non-mixing, the continuity equation (24) can be used differently for
matter and THDE, therefore the continuity equation for matter becomes
(25)
and the continuity equation for THDE is
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(26)
The barotropic EoS is
(27)
To find the parameters
,
,
,
,
we consider two additional conditions which are given below.
(i) Density of THDE as
(28)
(ii) The scale factor ‘a’ as
(29)
where and j are constants[23].
From (29) and (11), we obtain V of this model as
(30)
Again Eq. (23) gives
(31)
Eq. (17), (30) and (31) yield
(32)
(33)
(34)
The directional Hubble’s parameter, deceleration parameter, average anisotropy parameter, expansion scalar,
shear scalar become
(35)
(36)
(37)
(38)
(39)
(40)
(41)
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From Eqs. (30), (37), (40) and (41), it is found that for , V (spatial volume) is zero and H, , are diverges
and H, , are approaches to zero and when . As a result, the universe starts with null volume
and evolves at a rate of infinity.
From Eq. (38), we notice that for
and for . It is observed that for
,
. In this case, and according to or , respectively. It can be seen from
(39), that
when t , indicating that universe’s evolution is anisotropic.
From Eq. (27), we get THDE EoS parameter as
(42)
It shows that
when. As a result, it acts like CDM model. The astrophysical data, particularly
SNe Ia data [24], the three years WMAP data [25], the SDSS data [26], all show that the CDM is the standard
model for describing the universe’s evolution.
Now, by using equations (18), (32), (33), (34) in (25), (26) we get
(43)
(44)
From these equations, we observe that energy densities of THDE, matter reduce with cosmic time. For this
model, THDE density parameter
and matter density parameter
are expressed as
(45)
and
(46)
The total energy density
parameter is
(47)
The (total energy density) approaches to 1, as shown by (47).
CONCLUSIONS
Here, using the hybrid expansion rule in the framework of general relativity, we investigated the THDE in a
Here, Bianchi type VI
0
universe. We looked at a number of cosmological factors in our model to explain the
accelerated evolution of the universe. The q (deceleration parameter) is seen to represent universe’s accelerated
phase under certain condition, which is in strong agreement with observations.
It is observed that the energy densities of matter and THDE decrease with time (t). Furthermore, the total density
tends to 1 as the universe gets older. Later on, our model becomes flat, isotropic, and spatially uniform.
Additionally, THDE's EoS parameter
tends to -1 in the future. Consequently, our model and the Λ CDM
model are identical.
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