Generalized Ghost Dark Energy Model in Extended Modified Theory of Symmetric Telleparallel Gravity
M. Sharif Iqra Ibrar





Generalized Ghost Dark Energy Model in Extended Modified Theory of Symmetric
Telleparallel Gravity
    M. Sharif msharif.math@pu.edu.pk  and Iqra Ibrar
iqraibrar26@gmail.com

Department of Mathematics and Statistics, The University of Lahore,

1-KM Defence Road Lahore-54000, Pakistan.
    
==========================================================================================================================================================================================

§ ABSTRACT 
This paper focuses on developing a generalized ghost dark energy
model f(𝒬,𝒯) through the correspondence scheme for a
non-interacting scenario involving pressureless matter and a
power-law scale factor. We analyze the cosmological implications of
our model using the equation of state parameter,
ω_GGDE-ω'_GGDE and the r-s planes. We also examine
stability of the model by considering squared speed of sound
parameter. The equation of state parameter shows the phantom era,
while the squared speed of sound indicates a stable generalized
ghost dark energy model throughout the cosmic evolution. The
ω_GGDE-ω'_GGDE plane depicts a freezing region,
whereas the r-s plane corresponds to the Chaplygin gas model. We
conclude that our results are consistent with the current
observational data.

Keywords: f(𝒬,𝒯) gravity; Generalized ghost
dark energy; Cosmic  diagnostic parameters.

PACS: 04.50.kd; 95.36.+x; 64.30.+t.



§ INTRODUCTION


In the early 20th century, Albert Einstein introduced the general
theory of relativity (GR), which revolutionized our comprehension of
the cosmos. It is expanding through a significant accumulation of
accurate observations and delving into the concealed realms of the
universe in current cosmology. The current concept of cosmic
expansion has been reinforced by several observable findings,
including large-scale structures, cosmic microwave background
radiations and supernovae type Ia <cit.>. The universe in which we
live is experiencing a unique epoch, in which it expands at an
accelerated rate. Cosmologists examined this expansion by conducting
numerous attempts on distant galaxies. It is assumed that this
universe expansion is due to an extraordinary force called dark
energy (DE), which works through immense negative pressure. Several
models have been proposed to explain phenomena related to DE and the
evolution of the universe. There are two main approaches to studying
DE, i.e., dynamical DE models and modified theories of gravity
<cit.>.

In recent years, scholars have proposed various approaches to
address these issues, yet they remain enigmatic to this day. To
characterize DE, researchers found the equation of state (EoS)
parameter beneficial for the model being examined. The Veneziano
ghost DE (GDE) model possesses noteworthy physical properties within
the cosmos <cit.>. In flat spacetime, the GDE does not contribute
to vacuum energy density but in curved spacetime, it enhances small
vacuum energy density in proportion to Λ_QCD^3 H, where
Λ_QCD represents the quantum chromodynamics (QCD) mass
scale and H denotes the Hubble parameter. With Λ_QCD∼
100 MeV and H∼ 10^-33eV, Λ^3_QCDH gives the
order of observed GDE density. This small value efficiently provides
the essential exotic force driving the accelerating universe,
thereby resolving the fine-tuning issue. <cit.>. The energy
density of GDE is given as

    ρ_GDE=α H,

where α is an arbitrary constant.

The GDE model solves various issues effectively, but it encounters
stability problems <cit.>. It suggests that the energy density
relies not only on H explicitly but also on higher-order terms of
H, leading to what is known as the generalized GDE (GGDE) model.
In this model, the vacuum energy associated with the ghost field can
be regarded as a dynamic cosmological constant. In reference
<cit.>, the author explained that the Veneziano QCD ghost field's
contribution to vacuum energy does not precisely follow the
H-order. The additional term H^2 is particularly important
during the initial phase of the universe evolution, serving as early
DE <cit.>. Instead, a subleading term H^2 arises because the
conservation of the energy-momentum tensor holds independently
<cit.>. The GGDE density is given as follows

    ρ_GGDE=α H+β H^2,

where β is another arbitrary constant.

Numerous DE models can be explored to study the ongoing accelerated
expansion of the universe. Khodam et al. <cit.> investigated the
reconstruction of f(ℛ,𝒯) gravity (here ℛ and
𝒯 are the Ricci scalar and trace of the energy-momentum
tensor, respectively) within the GGDE model, delving into cosmic
evolution through the analysis of cosmological parameters. Malekjani
<cit.> explored the cosmographic aspects of GGDE and concluded
that this model demonstrates strong consistency with observational
data. Ebrahimi et al. <cit.> studied the interacting GGDE within
a non-flat cosmos, revealing that the EoS parameter aligns with the
phantom era of the universe. Chattopadhyay <cit.> reconstructed
the QCD ghost f(T) model (T represents the torsion scalar) to
examine the cosmic evolution and found both the phantom and
quintessence phases of the cosmos. Sharif and Saba <cit.>
explored the cosmography of GGDE in f(𝒢, 𝒯)
(𝒢 is the Gauss-Bonnet invariant) gravity. Biswas et al.
<cit.> investigated the evolution trajectories of the EoS
parameter, ω_D-ω'_D and the state finder planes in
GGDE DGP model. Recently, Sharif and Ajmal <cit.> studied the
cosmography of GGDE f(𝒬) theory.

Several gravitational theories, including GR, are acquiring
importance because of their ability to explain the accelerating
expansion of the universe. The Levi-Civita connection in GR explains
gravitational interactions in Riemannian space based on the
assumption of geometry free from torsion and non-metricity.
Moreover, it is important to consider that the general affine
connection has a broader formulation <cit.>, and GR can be
derived within alternative spacetime geometries beyond Riemannian.
Teleparallel gravity presents an alternative framework to GR, where
gravitational interaction is characterized by torsion, denoted as
T <cit.>. Teleparallel equivalent of GR utilizes the
Weitzenböck connection, which entails zero curvature and
non-metricity <cit.>. In the study of a cosmological model
within Weyl-Carten (WC) spacetime, the Weitzenböck
connection is investigated with vanishing of the combined curvature
and scalar torsion <cit.>.

In a torsion-free space, gravity is modified by non-metricity,
defined as
𝒬_λζξ=∇_λg_ζξ. This
theory is called symmetric teleparallel gravity (STG) <cit.>.
Jimenez et al. <cit.> recently introduced f(𝒬) theory
also named as non-metric gravity, the scalar function 𝒬
represents the non-metricity. Lazkoz et al. <cit.> described the
constraints on f(𝒬) gravity by using polynomial
functions of the red-shift. The energy conditions for two distinct
f(𝒬) gravity models have also been discussed <cit.>.
The behavior of cosmic parameters in the same context was examined
by Koussour et al. <cit.>. Chanda and Paul <cit.> studied the
formation of primordial black holes in this theory.

Xu et al. <cit.> extended f(𝒬) gravity to
f(𝒬,𝒯) gravity by adding trace of the energy-momentum
tensor into f(𝒬) theory. The Lagrangian governing the
gravitational field is considered to be a gravitational function of
both 𝒬 and 𝒯. The field equations of the
corresponding theory are derived by varying the gravitational action
with respect to both the metric and the connection. The goal of
introducing this theory is to analyze its theoretical implications,
its consistency with real-world experimental evidence, and its
applicability to cosmological scenarios. Nájera and
Fajardo <cit.> studied cosmic perturbation theory in
f(𝒬,𝒯) gravity and found that non-minimal coupling
between matter and curvature perturbations have a major influence on
the universe evolution. Pati et al. <cit.> investigated the
dynamical features and cosmic evolution in the corresponding
gravity. Mandal et al. <cit.> studied the cosmography of
holographic DE f(𝒬,𝒯) gravity.

In this paper, we use a correspondence technique to recreate the
GGDE f(𝒬,𝒯) model in a non-interacting scenario. We
investigate cosmic evolution using the EoS parameter and phase
planes. The paper is structured as follows. In section 2,
we introduce f(𝒬,𝒯) gravity and its corresponding field
equations. In section 3, we explore the reconstruction
procedure to formulate the GGDE f(𝒬,𝒯) paradigm. In
section 4, we analyze cosmic behavior using the EoS
parameter and phase planes. We also examine the stability of the
resulting model. Finally, we discuss our findings in the last
section.



§ THE F(𝒬,𝒯) THEORY


This section presents the basic structure of modified
f(𝒬,𝒯) theory. In this theory, the framework of
spacetime is the torsion-free teleprallel geometry, i.e.,
R^ρ_ ηγν=0 and T^ρ_γν=0. The
connection in the WC geometry is expressed as <cit.>

    Γ̂^λ_αβ=Γ^λ_αβ
    +ℂ^λ_αβ+𝕃^λ_αβ,

where Γ^ζ_ γα is the Levi-Civita
connection, ℂ^λ_αβ is the contortion
tensor and 𝕃^λ_αβ is the deformation
tensor, given by

    Γ^ζ_μα   =1/2g^ζσ
    (g_σα,μ+g_σμ,α-g_σμ,α),


    ℂ^λ_αβ=Γ̂^λ_[αβ]
    +g^λϕg_ακΓ̂^κ_[βϕ]
    +g^λϕg_βκΓ̂^κ_[αϕ],

and

    𝕃^λ_αβ=1/2g^λϕ(𝒬_βαϕ
    +𝒬_αβϕ-𝒬_λαβ),

where

    𝒬_λαβ=∇_λg_αβ=-g_αβ,_λ
    +g_βϕΓ̂^ϕ_αλ
    +g_ϕαΓ̂^ϕ_βλ.

The gravitational action can be written as <cit.>

    S=1/2k∫ g^αβ(Γ^ρ_ϕαΓ^ϕ_βρ -Γ^ρ_ϕρΓ^ϕ_αβ)√(-g)d^4x.

Using the anti-symmetric relation
(Γ^μ_ςα=-𝕃^μ_ ςα)
in Eq.(<ref>), the integral action takes the form

    S=-1/2k∫ g^αβ(𝕃^ρ_ϕα𝕃^ϕ_βρ - 𝕃^ρ_ϕρ𝕃^ϕ_αβ) √(-g) d^4x.

This action is known as the action of STG which is equivalent to the
Einstein-Hilbert action. There are some significant differences
between two gravitational paradigms. One of them is that the
vanishing of curvature tensor in STG causes the system to appear as
flat structure throughout. Furthermore, the gravitational effects
occur due to variations in the length of vector itself, rather than
rotation of an angle formed by two vectors in parallel transport.

Now, we look at an extension of STG and take an arbitrary function
of 𝒬, the above action takes the form

    S=∫[1/2kf(𝒬)+L_m]√(-g)d^4x,

where

    𝒬=-g^αβ(𝕃^ρ_μα𝕃^μ_βρ
    -𝕃^ρ_μρ𝕃^μ_αβ),

which leads to f(𝒬) theory. If we couple this theory
with the trace of the energy-momentum tensor, we can obtain
f(𝒬,𝒯) theory whose action is given as

    𝒮=1/2k∫ f(𝒬,𝒯) √(-g)d^4x+ ∫
    L_m√(-g)d^4x,

here L_m represents the matter lagrangian. The traces of
non-metricity tensor are given by

    𝒬_ρ= 𝒬^ α_ρ α,   𝒬̃_ρ= 𝒬^α_ρα.

The superpotential in view of 𝒬 is written as

    P^ρ_αβ   =-1/2ℒ^ρ_αβ
    +1/4(𝒬^ρ-𝒬̃^ρ)
    g_αβ- 1/4δ ^ρ_ [α𝒬_β].

Furthermore, the expression of 𝒬 obtained using the
superpotential becomes

    𝒬=-𝒬_ραβP^ραβ
    =-1/4(-𝒬^ρβζ𝒬_ρβζ
    +2𝒬^ρβζ𝒬_ζρβ
    -2𝒬^ζ𝒬̃_ζ+𝒬^ζ𝒬_ζ).


Taking the variation of S with respect to the metric tensor as
zero yields the field equations

    δ S   =0=∫1/2δ
    [f(𝒬,𝒯)√(-g)]+δ[L_m√(-g)]d^4x 
    
    0   =∫1/2( -1/2 f g_αβ√(-g)δ g^αβ + f_𝒬√(-g)δ𝒬 + f_𝒯√(-g)δ𝒯)
       -1/2𝒯_αβ√(-g)δ g^αβd^ 4x.

Furthermore, we define

    𝒯_αβ   = -2/√(-g)δ
    (√(-g) L_m)/δ g^αβ,   Θ_αβ= g^ρμδ𝒯_ρμ/δ g^αβ,

which implies that δ𝒯=δ
(𝒯_αβg^αβ)=(𝒯_αβ+
Θ_αβ)δ g^αβ. Inserting the
aforementioned factors in Eq.(<ref>), we have

    δ S    =   0=∫1/2{-1/2f
    g_αβ√(-g)δ g^αβ +
    f_𝒯(𝒯_αβ+
    Θ_αβ)√(-g)δ g^αβ
       -   f_𝒬√(-g)(P_αρμ𝒬_β ^ρμ- 2𝒬^ρβ _α
    P_ρμβ) δ g^αβ+2f_𝒬√(-g)
    P_ραβ∇^ρδ g^αβ
       +   2f_𝒬√(-g)P_ραβ∇^ρδ
    g^αβ}- 1/2𝒯_αβ√(-g)δ g^αβd^ 4x.

Integration of the term 2 f_𝒬√(-g)
P_ραβ∇^ρδ g^αβ along with
the boundary conditions takes the form -2 ∇^ρ
(f_𝒬√(-g) P_ραβ)δ
g^αβ. The terms f_𝒬 and f_𝒯
represent partial derivatives with respect to 𝒬 and
𝒯, respectively. Finally, we obtain the field equations
as

    T_αβ   =-2/√(-g)∇_ρ
    (f_𝒬√(-g) P^ρ_αβ)- 1/2 f
    g_αβ + f_𝒯 (𝒯_αβ +
    Θ_αβ) -f_𝒬 (P_αρμ𝒬_β ^ρμ
       -2𝒬^ρμ _α P_ρμβ).

Its covariant derivative yields the non-conservation equation as

    ∇_αT^α_ β   =1/f_𝒯-1[∇_α(1/√(-g)∇_ρH^ ρα_β)-∇_α
    (f_𝒯Θ^α_ β)
    -1/√(-g)∇_ρ∇_αH^ ρα_β
       -2∇_αA^α_ β
    +1/2f_𝒯∂_β𝒯],

where hyper-momentum tensor density is defined as

    H^ ρα_β=√(-g)/2f_𝒯δ𝒯/δΓ̂^β _ρα+δ√(-g)L_m/δΓ̂^β _ρα.




§ RECONSTRUCTION OF GGDE F(𝒬,𝒯) MODEL


In this section, we use a correspondence technique to recreate the
GGDE f(𝒬,𝒯) model. The line element of the isotropic and
spatially homogeneous universe model is given by

    ds^2 = -dt^2+ a^2(t)[dx^2+ dy^2+dz^2],

where the scale factor is represented by a(t). The
configuration of isotropic matter with four-velocity fluid
u_μ, pressure (P_M), and normal matter density
(ρ_M) is given as

    𝒯̃_μν=(ρ_M+P_M)u_μu_ν+P_Mg_μν.

The modified Friedmann equations in the context of
f(𝒬,𝒯) gravity are expressed as

    3H^2    =ρ_eff=ρ_M+ρ_GGDE,
    
    
    2Ḣ+3H^2   =P_eff=P_M+P_GGDE,

where H =ȧ/a. The dot demonstrates
derivative with respect to cosmic time t. The non-metricity
𝒬 in terms of the Hubble parameter is

    𝒬=-1/4[-𝒬_ρξη𝒬^ρξη
    +2𝒬_ρξη𝒬^ξρη-2𝒬̃^ρ𝒬_ρ+𝒬^ρ𝒬_ρ].

Simplifying this equation leads to

    𝒬=6H^2.

Furthermore, ρ_GGDE and P_GGDE denote the density and
pressure of DE, respectively, given as

    ρ_GGDE   =1/2f(𝒬,𝒯)-𝒬f_𝒬
    -f_𝒯(ρ_M+P_M),
    
    P_GGDE   =-1/2f(𝒬,𝒯)+2f_𝒬𝒬H+2f_𝒬Ḣ
    +𝒬f_𝒬.

The conservation equation (<ref>) takes the following form for an
ideal fluid

    ρ̇_M+3H(ρ_M+P_M)=1/(f_𝒯-1)[2∇_β(P_Mμ^βf_𝒯)
    +f_𝒯∇_βμ^β𝒯
    +2μ^β𝒯_αβ∇^α
    f_𝒯μ^β].

The first field equation (<ref>) leads to

    Ω_M+Ω_GGDE=1,

where Ω_M=ρ_M/3H^2 and
Ω_GGDE=ρ_GGDE/3H^2 represent the fractional
energy densities of normal matter and dark source, respectively.
Dynamic DE models with energy density proportional to Hubble
parameter are crucial to explaining the accelerated expansion of the
universe.

Next, we will use a correspondence approach in an ideal fluid
configuration to create the GGDE f(𝒬,𝒯) model with
emphasis on the dust case (P_M=0).



 §.§ Non-interacting GGDE f(𝒬,𝒯) Model


Here, we consider the standard f(𝒬,𝒯) function in the
following form <cit.>

    f(𝒬,𝒯)=f_1(𝒬)+f_2(𝒯),

where f_1 and f_2 depend upon 𝒬 and
𝒯, respectively. In this scenario, it is evident that
there is a minimal coupling between curvature and matter
constituents. This version of the generic function reveals that the
interaction is purely gravitational and hence easy to handle. This
can effectively explain the ongoing expansion of the universe.
Moreover, the reconstruction methodology demonstrates that such
generated models are physically viable <cit.>. Using dust fluid
and Eq.(<ref>), the field equations (<ref>) and (<ref>)
yield

    3H^2=ρ_eff=ρ_M+ρ_GGDE,   
    2Ḣ+3H^2=P_eff=P_GGDE,

where

    ρ_GGDE   =1/2f_1(𝒬)+1/2f_2(𝒯)-𝒬
    f_1𝒬+ f_2𝒯ρ_M, 
    
    P_GGDE   =-1/2f_1(𝒬)-1/2f_2(𝒯)
    +2f_1𝒬Ḣ+2
    f_1𝒬𝒬H+𝒬f_1𝒬.

The associated conservation equation (<ref>) reduces to

    ρ̇_M+3Hρ_M=1/f_2𝒯-1[2𝒯
    f_2𝒯𝒯+f_2𝒯𝒯̇].

This equation is consistent with the standard continuity equation
when the right-hand side is assumed to be zero, implying

    ρ̇_M+3Hρ_M=0  ⟹  ρ_M=ρ_0a(t)^-3,

with the constraint

    f_2𝒯+2𝒯f_2𝒯𝒯= 0,

whose solution provides

    f_2(𝒯)=a𝒯^1/2+b,

where a and b represent the integration constants.

We utilize Eqs.(<ref>) and (<ref>), in conjunction with the
constraint on f_2(𝒯) as given in (<ref>), to
formulate a reconstruction framework employing the correspondence
approach. The differential equation for f_1(𝒬) is
expressed as

    f_1(𝒬)/2-𝒬
    f_1𝒬+a𝒯^1/2+1/2b=α
    H+β H^2.

We use the power-law solution for the scale factor given as follows

    a(t)=a_0t^m,      m > 0.

In this context, a_0 denotes the current value of the
scale factor. Employing this relation, the expressions for H, its
derivative, and the non-metricity scalar in terms of cosmic time t
are given as follows

    H=m/t,   Ḣ=-m/t^2,   𝒬=6
    m^2/t^2.

Substituting (<ref>) in (<ref>), it follows that

    ρ_M=d(a_0t^m)^-3,

where ρ_0=d for the sake of similicity. Using Eq.(<ref>) in
(<ref>), we can find the function f_1(𝒬) as

    f_1(𝒬)   =√(𝒬)[-a √(d)
    2^3 m/8+1 3^3 m/8(𝒬/m^2)^-3 m/8/(-3
    m/8-1/2) √(𝒬)+2
    b/√(𝒬)+β m 𝒬/√(6)+22^3/4α𝒬/3 √(3)] 
       +c_1
    √(𝒬),

where c_1 is the integration constant. Consequently, the
reconstructed f(𝒬,𝒯) model is obtained by substituting
Eqs.(<ref>) and (<ref>) into (<ref>) as follows

    f(𝒬,𝒯)   =1/18[-a √(d) 6^3
    m/8+2(𝒬/m^2)^-3
    m/8/-3 m/8-1/2+18 √(ad)+54 b+18 c_1
    √(𝒬)
       +3 √(6)β  m 𝒬^3/2+4 6^3/4α𝒬^3/2].


Now we write down this function in terms of redshift parameter z.
The expression for the deceleration parameter is given by

    q=-aä/ȧ^2=-1+1/m.

In relation to the deceleration parameter, the evolution of the
cosmic scale factor can be characterized as

    a(t)=t^1/(1+q),

Here, we assume a_0 to be equal to unity. It is worth
noting that the power-law model corresponds to an expanding universe
when q > -1. The description of both the expanding phase and the
present cosmic evolution is given by

    H=(1+q)^-1t^-1,    H_0=(1+q)^-1t^-1_0.

In power-law cosmology, the evolution of the universe expansion is
determined by two fundamental parameters, the Hubble constant
(H_0) and the deceleration parameter (q). By examining the
correlation between the scale factor a and redshift z, we can
elucidate

    H=H_0Γ^1+q,

where Γ=1+z. Using Eq.(<ref>) in (<ref>), 𝒬
appears as

    𝒬=6H_0^2Γ^2+2q.

The reconstructed model for the GGDE f(𝒬,𝒯) in terms of
the redshift parameter is derived by substituting this value in
Eq.(<ref>), resulting in

    f(𝒬,𝒯)   =16 a √(d)(H_0^2 Γ^2
    q+2/m^2)^-3 m/8/3 m+4+√(ad)+3
    b+√(6) c_1 √(H_0^2 Γ^2 q+2)
       +(H_0^2 Γ^2 q+2)^3/26 β  m +8 √(6)α(H_0^2 Γ^2 q+2)^3/2.



For the graphical analysis, we set a = 1, b = -4, and c_1 =
-15. Figure 1 indicates that the reconstructed GGDE model
remains positive and increases with respect to z. The GGDE model
indicates a rapid expansion, according to this analysis. Figure
2 illustrates the behavior of ρ_GGDE and P_GGDE
with the redshift parameter. The energy density ρ_GGDE is
positive and increasing, whereas P_GGDE is negative and
consistent with the DE behavior. We then investigate the properties
of ρ_GGDE and P_GGDE in the context of the reconstructed
GGDE f(𝒬,𝒯) gravity model. Putting (<ref>) into
(<ref>) and (<ref>), we have


    ρ_GGDE   =1/18(H_0^2 Γ^2 q+2/m^2)^-3 m/8[36 a √(d)-18 (H_0^2 Γ^2
    q+2/m^2)^3 m/8{√(ad) (d-1)
    
       +b (d-2)+2 (H_0^2 Γ^2 q+2)^3/2(4
    √(6)α +3 β m)}],
    
    
    P_GGDE   =1/36[1/H_0{(3 H_0-1)
    {6 √(6)√(H_0^2 Γ^2 q+2){c_1+3
    √(6)β  H_0^2 m Γ^2 q+2
       +4 6^3/4α H_0^2 Γ^2 q+2}-72 a
    √(d) m (H_0^2 Γ^2 q+2/m^2)^-3
    m/8/3 m+4}}
       +1/12 H_0^3[Γ^-3 (q+1){{18 a
    √(d) m(3 m+8)(H_0^2 Γ^2
    q+2/m^2)^-3 m/8}(3 m+4)
    
       +√(6)√(H_0^2 Γ^2 q+2){-6 c_1+18 √(6)β H_0^2 m Γ^2 q+2+24 6^3/4α  H_0^2 Γ^2
    q+2}}]
       -288 a √(d)(H_0^2 Γ^2 q+2/m^2)^-3 m/8/3
    m+4 -18 √(ad)-54 b-18 √(6) c_1 √(H_0^2
    Γ^2 q+2)
       -108 β  m (H_0^2 Γ^2 q+2)^3/2-144
    √(6)α_0a (H_0^2 Γ^2
    q+2)^3/2].




§ COSMOLOGICAL ANALYSIS


In this section, we explore the evolutionary stages of the universe
through different phases. To achieve this, we utilize the
reconstructed GGDE f(𝒬,𝒯) model under non-interacting
conditions, as defined in Eq.(<ref>). Furthermore, we depict the
dynamics of various cosmological parameters, including the EoS
parameter, ω_GGDE-ω'_GGDE and state finder planes.
The stability of this model is also examined.



 §.§ Equation of State Parameter


The EoS parameter (ω=P/ρ) of DE plays a crucial
role in describing both the cosmic inflationary phase and the
subsequent expansion of the universe. We analyze the criterion for
an accelerating universe, which occurs when the EoS parameter
ω<-1/3. When ω=-1, it corresponds to the
cosmological constant. However, when ω=1/3 and
ω=0, it signifies the radiation-dominated and
matter-dominated universe, respectively. Moreover, the phantom
scenario manifests when we assume ω<-1. The expression for
the EoS parameter is given by

    ω_GGDE=
    P_eff/ρ_eff=P_GGDE/ρ_GGDE+ρ_M.

Equations (<ref>), (<ref>) and (<ref>) are employed in
the aforementioned expression to compute the respective parameter as

    ω_GGDE   =-[f^3 Γ^-2 q-2{a √(d)
    2^3 m/8+1 3^3 m/8+2(-432 h^4 m Γ^4
    q+4 -48 h^2 Γ^2 q+2(12 h^2
    
       ×Γ^2 q+2-3 h m Γ^2 q+2) +3 h m (3 m+8)
    Γ^q+1) -6^3 mx/8 +1/2 (3 m+4) √(h^2
    Γ^2 q+2)
       ×(h^2 Γ^2 q+2/m^2)^3 m/8{108 √(6)√(ad)(H_0^2 Γ^2
    q+2)^3/2+324 √(6) b (H_0^2 Γ^2
    q+2)^3/2+216
    
       × c_1 H_0^3 Γ^4 q+4+18 c_1 H_0Γ^q+1-1296
    √(6)β  H_0^6 m Γ^6 q+6 -1728 6^3/4α
    H_0^6 Γ^6 q+6
       +648 √(6)β H_0^5 m Γ^6 q+6 +864 6^3/4α
    H_0^5 Γ^6 q+6-54 √(6)β H_0^3 m Γ^3
    q+3-72 6^3/4
       ×α  H_0^3 Γ^3 q+3}}] [18
    H_0^2 (3 m+4) {a √(d) f^3 2^3 m/8 3^3
    m/8+2 m +H_0^2 6^3 m/8+1Γ^2 q+2
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8(6 f^3 √(H_0^2 Γ^2 q+2)(4 √(6)α +3 β m)-12 d)}]^-1.

Figure 3 illustrates the behavior of EoS parameter against
z from which one can find that phantom epoch for current as well
as late time cosmic evolution.




 §.§ ω_GGDE-ω'_GGDE Plane


Here, we utilize the ω_GGDE-ω'_GGDE phase plane,
where ω'_GGDE indicates the evolutionary mode of
ω_GGDE and prime denotes the derivative with respect to
𝒬. This cosmic plane was established by Caldwell and
Linder <cit.> to investigate the quintessence DE paradigm which
can be split into freezing (ω_GGDE <0, ω'_GGDE<0 )
and thawing (ω_GGDE<0, ω'_GGDE>0 ) regions. To
depict the prevailing cosmic expansion paradigm, the freezing region
signifies a more accelerated phase compared to thawing. The cosmic
trajectories of ω_GGDE-ω'_GGDE plane for specific
choices of m are shown in Figure 4 which provides the
freezing region of the cosmos. The expression of ω'_GGDE is
given in Appendix A.




 §.§ State Finder Analysis


One of the techniques to examine the dynamics of the cosmos using a
cosmological perspective is state finder analysis. It is an
essential approach for understanding numerous DE models. As a
combination of the Hubble parameter and its temporal derivatives,
Sahni et al. <cit.> established two dimensionless parameters
(r,s) given as

    r=⃛a/a H^3,    s=r-1/3(q-1/2).

The acceleration of cosmic expansion is determined by the parameter
s, while the deviation from pure power-law behavior is precisely
described by the parameter r. This is a geometric diagnostic that
does not favor any particular cosmological paradigm. It is such an
approach that does not depend upon any specific model to distinguish
between numerous DE scenarios, i.e., CG (Chaplygin gas), HDE
(Holographic DE), SCDM (standard CDM) and quintessence.

Several DE scenarios for the appropriate choices of r and s
parametric values are given below.


  * When r=1, s=0, it indicates the CDM model.

  * If r=1, s=1, then it denotes SCDM epoch.

  * When r=1, s=2/3, this epoch demonstrates the HDE
model.

  * When we have r>1, s<0, we get CG scenario.

  * Lastly, r<1, s>0 corresponds to quintessence paradigm.

For our considered setup, the parameters r and s in terms of
required factors are given in Appendix B. The graphical
analysis of r-s phase plane in Figure 5 gives r>1 and
s<0, indicating the CG model.




 §.§ Squared Speed of Sound Parameter


Perturbation theory provides a direct analysis for evaluating the
stability of the DE model through examination of the sign of
(ν_s^2). In this context, we investigate the squared speed
of sound parameter to analyze the stability of the GGDE
f(𝒬,𝒯) model, represented by

    ν_s^2=P_GGDE/ρ'_GGDEω'_GGDE
    +ω_GGDE.

A positive value signifies a stable configuration, whereas a
negative value indicates unstable behavior for the associated model.
Substituting the necessary expressions on the right-hand side of the
equation above for the reconstructed model, we derive the squared
speed of sound parameter as provided in Appendix C. Figure
6 illustrates that the speed component remains positive for
all assumed values of m, indicating the stability of the
reconstructed GGDE f(𝒬,𝒯) model throughout the cosmic
evolution.




§ FINAL REMARKS


In this study, we have explored various features of the GGDE model
within the framework of the recently developed f(𝒬,𝒯)
theory of gravity. To comprehend the influence of 𝒬 and
𝒯 in the GGDE ansatz, we have adopted a specific
f(𝒬,𝒯)=f_1(𝒬)+f_2(𝒯) model. We
have formulated a reconstruction framework involving a power-law
scale factor and a corresponding scenario for the flat FRW universe.
Subsequently, we have analyzed the EoS parameter,
ω_GGDE-ω'_GGDE and state finder planes, and
conducted a squared sound speed analysis for the derived model. We
summarize the main results as follows.


  * In the non-interacting case, the reconstructed GGDE f(𝒬,𝒯)
model shows an increasing trend for z, indicating the realistic
nature of the reconstructed model (Figure 1).

  * The energy density demonstrates positive behavior, showing an
increase, while the pressure exhibits negative behavior. These
patterns align with the characteristics of DE (Figure 2).

  * The early and late time universe are characterized by the EoS
parameter, which involves phantom field and DE. Additionally, we
have noted a trend where this parameter assumes increasingly
negative values below -1 (Figure 3). These observations
align with the current understanding of accelerated cosmic behavior.

  * The evolutionary pattern in the ω_GGDE-ω'_GGDE plane
indicates a freezing region for all values of m (Figure
4). This affirms that the non-interacting GGDE
f(𝒬,𝒯) gravity model implies a more accelerated
expanding universe.

  * The r-s plane depicts the CG model as both r
and s satisfy the respective model criteria (Figure 5).

  * We have found that the squared speed of sound parameter is
positive and hence GGDE f(𝒬,𝒯) gravity model is stable
for all values of m (Figure 6).


Finally, we conclude that our results align with the current
observational data <cit.>, as provided below

    ω_GGDE   =   -1.023^+0.091_-0.096  (Planck
    TT+LowP+ext),
    ω_GGDE   =   -1.006^+0.085_-0.091  (Planck
    TT+LowP+lensing+ext),
    ω_GGDE   =   -1.019^+0.075_-0.080   (Planck TT, TE,
    EE+LowP+ext).

These values have been established using diverse observational
methodologies, ensuring a confidence level of 95%. Moreover, we
have verified that the cosmic diagnostic state finder parameters for
our derived model align with the constraints and limitations on the
kinematics of the universe <cit.>. Sharif and Saba <cit.>
established the correspondence of modified Gauss-Bonnet theory with
GGDE paradigm and found phantom phase as well as the stable
reconstructed model in non-interacting case. Our findings align with
these results. Our results are also consistent with the recent work
in f(𝒬) theory <cit.>.



§ APPENDIX A: CALCULATION OF Ω'_GGDE





    ω'_GGDE   =[f^3 Γ^-4 q-4{2^3 m/8+1
    3^3 m/8+2 a √(d)(3 H_0 m (3 m+8)
    Γ^q+1-48 H_0^2 (12 H_0^2 Γ^2 q+2
       
    -3 H_0 m Γ^2 q+2) Γ^2 q+2-432 H_0^4 m
    Γ^4 q+4) -6^3 m/8+1/2 (3 m+4)
    √(H_0^2 Γ^2 q+2)
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8{324 √(6) b (H_0^2 Γ^2
    q+2)^3/2+108 √(6)√(ad)(H_0^2 Γ^2
    q+2)^3/2
       -726^3/4 H_0^3 Γ^3 q+3α -1728 6^3/4 H_0^6
    Γ^6 q+6α+864 6^3/4 H_0^5 Γ^6 q+6α
       -54 √(6) H_0^3m Γ^3 q+3β -1296 √(6) H_0^6m
    Γ^6 q+6β+648 √(6) H_0^5 m Γ^6 q+6β
       +18 H_0Γ^q+1 c_1+216 H_0^3 Γ^4 q+4
    c_1}}][108 H_0^4 (3 m+4) {6^3
    m/8+1 H_0^2 Γ^2 q+2
       ×(6 f^3 √(H_0^2 Γ^2 q+2)(4 √(6)α +3 m β) -12 d) (H_0^2 Γ^2
    q+2/m^2)^3 m/8 +2^3 m/8 3^3
    m/8+2
       × a √(d) f^3 m}]^-1+[f^3 Γ^-2 q-2{2^3 m/8 3^3 m/8+1 f^3 √(H_0^2
    Γ^2 q+2)(4 √(6)α+3 m β)
    
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8 +6^3 m/8(6 f^3 √(H_0^2 Γ^2 q+2)(4 √(6)α +3 m β) (H_0^2
    Γ^2 q+2/m^2)^3 m/8
       +[2^3 m/8-3 3^3 m/8+1H_0^2 Γ^2
    q+2(6 f^3 √(H_0^2 Γ^2 q+2)(4 √(6)α+3 m β)(H_0^2 Γ^2 q+2/m^2)^3 m/8-1]
    
       ×1/m}(2^3 m/8+1 3^3
    m/8+2a √(d){3 H_0 m (3 m+8) Γ^q+1-48 H_0^2
    (12 H_0^2 Γ^2 q+2
       -3 H_0 m Γ^2 q+2) Γ^2 q+2-432 H_0^4 m
    Γ^4 q+4} -6^3 m/8+1/2 (3
    m+4)(H_0^2 Γ^2 q+2/m^2)^3 m/8
       ×√(H_0^2 Γ^2 q+2)(324 √(6) b
    ×(H_0^2 Γ^2 q+2)^3/2+108 √(6)√(ad)(H_0^2 Γ^2 q+2)^3/2
        -726^3/4 H_0^3 Γ^3 q+3α -1728  6^3/4 H_0^6
    Γ^6 q+6α +864 6^3/4 H_0^5 Γ^6 q+6α
       -54 √(6) H_0^3 m Γ^3 q+3β -1296 √(6) H_0^6
    m Γ^6 q+6β +648 √(6) H_0^5mΓ^6 q+6β
       +18 H_0Γ^q+1 c_1+216 H_0^3 Γ^4 q+4
    c_1))] [18 H_0^2 (3 m+4) (6^3
    m/8 +1 H_0^2 Γ^2 q+2
       ×(6 f^3 √(H_0^2 Γ^2 q+2)(4√(6)α +3 m β)-12 d) (H_0^2 Γ^2
    q+2/m^2)^3 m/8 +2^3 m/8 3^3
    m/8+2
       × a √(d) f^3 m)^2]^-1-[f^3Γ^-2
    q-2(-6^3 m/8+1/2 (3 m+4)√(H_0^2
    Γ^2 q+2){-12 6^3/4
       × H_0αΓ^q+1-9 √(6) H_0 m βΓ^q+1 +108 H_0^2 c_1 Γ^2 q+2+36 H_0 c_1 Γ^2
    q+2
       -432 6^3/4 H_0^4 αΓ^4 q+4+2886^3/4 H_0^3
    αΓ^4 q+4 -324 ×√(6) H_0^4 m βΓ^4
    q+4
       +216 √(6) H_0^3 m βΓ^4 q+4+81 √(6) b
    √(H_0^2 Γ^2 q+2) +27 √(6)√(ad)√(H_0^2 Γ^2 q+2)}
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8 +2^3 m/8+1 3^3 m/8+2 a √(d){-96 H_0^2 Γ^2 q+2 -72 H_0^2 m Γ^2 q+2
       -8 (12 H_0^2 Γ^2 q+2-3 H_0 m Γ^2
    q+2)}-[2^3 m/8-3/2 3^3
    m/8-1/2 (3 m+4)(H_0^2 Γ^2
    q+2/m^2)^3 m/8
       ×{324 √(6) b (H_0^2 Γ^2 q+2)^3/2
    +108√(6)√(ad)(H_0^2 Γ^2 q+2)^3/2
    -6^3/4 H_0^3 Γ^3 q+3α
       -1728 6^3/4 H_0^6 Γ^6 q+6α+864 6^3/4
    H_0^5Γ^6 q+6α -54 √(6) H_0^3 m Γ^3 q+3β
       -1296 √(6) H_0^6 m Γ^6 q+6β +√(6) H_0^5
    mΓ^6 q+6β +18 H_0Γ^q+1 c_1 +H_0^3 Γ^4
    q+4 c_1}]
    
       ×[√(H_0^2 Γ^2 q+2)]^-1
    -[2^3 m/8-7/2 3^3 m/8+1/2
    (3 m+4) √(H_0^2 Γ^2 q+2)(H_0^2 Γ^2
    q+2/m^2)^3 m/8-1
       ×(324 √(6) b (H_0^2 Γ^2
    q+2)^3/2+108 √(6)(H_0^2 Γ^2 q+2)^3/2
    -72 6^3/4 H_0^3 Γ^3 q+3α
       -1728 6^3/4 H_0^6 Γ^6 q+6α +864 6^3/4
    H_0^5Γ^6 q+6α -54 √(6) H_0^3 m Γ^3 q+3β
       -1296 √(6) H_0^6 m Γ^6 q+6β +648 √(6) H_0^5
    m Γ^6 q+6β+18 H_0Γ^q+1 c_1
    
       +216 H_0^3 Γ^4 q+4 c_1)][18 H_0^2 (3 m+4)
    (6^3 m/8+1 H_0^2 Γ^2 q+2(6 f^3
    √(H_0^2 Γ^2 q+2)
       ×(4 √(6)α +3 m β)-12 d)
    (H_0^2 Γ^2 q+2/m^2)^3
    m/8+2^3 m/8 3^3 m/8+2 a √(d) f^3
    m)].




§ APPENDIX B: EVALUATION OF R AND S PARAMETERS





    r   =2 (3/2[f^3 Γ^-4 q-4(2^3
    m/8+1 3^3 m/8+2 a √(d){3 H_0 m (3 m+8)
    Γ^q+1-48 H_0^2 (12 H_0^2 Γ^2 q+2
        - Γ^2 q+2) Γ^2 q+2 -432 H_0^4 m Γ^4
    q+4}-6^3 m/8 +1/2 (3 m+4) √(^H_02
    Γ^2 q+2)(h^2 Γ^2 q+2/m^2)^3
    m/8
       ×(324 √(6) b (H_0^2 Γ ^2 q+2)^3/2
    +108 √(6)√(ad)(H_0^2 Γ^2
    q+2)^3/2-72 6^3/4 h^3 Γ^3 q+3α
       × 6^3/4 H_0^6 Γ^6 q+6α +864 6^3/4 h^5
    Γ^6 q+6×α -54 √(6) H_0^3 m Γ^3 q+3β -1296 √(6) H_0^6m
    
       ×Γ^6 q+6β +648 √(6) H_0^5 m Γ^6 q+6β +18 H_0Γ^q+1 c_1+216 h^3 Γ^4 q+4
    c_1))] [108 h^4
    
       ×(6^3 m/8+1H_0^2 Γ^2 q+2(6 f^3
    √(H_0^2 Γ^2 q+2){4 √(6)α +3 m β}(H_0^2 Γ^2 q+2/m^2)^3
    m/8
       × 3^3 m/8+2 a √(d) f^3 m)]^-1
    +[f^3 Γ^-2 q-2(2^3 m/8 3^3
    m/8+1 f^3 √(H_0^2 Γ^2 q+2)(4 √(6)α
    r
    
       +3 m β)(H_0^2 Γ^2
    q+2/m^2)^3 m/8 +6^3 m/8(6 f^3
    √(H_0^2 Γ^2 q+2)(4√(6)α +3 m β)
    
       -12 d) (H_0^2 Γ^2 q+2/m^2)^3
    m/8+1/m[2^3 m/8-3 3^3 m/8+1 h^2
    Γ^2 q+2(6 f^3 √(h^2 Γ^2 q+2)(4
    √(6)α
       +3 m β)- (h^2 Γ^2
    q+2/m^2)^3 m/8-1](2^3 m/8+1
    3^3 m/8+2 a √(d)(3 h m (3 m+8) Γ^q+1
        -48 h^2 (12 H_0^2 Γ^2 q+2 -3 H_0 m Γ^2
    q+2)-432 h^4 m Γ^4 q+4) - (h^2
    Γ^2 q+2/m^2)^3 m/8
       ×√(h^2 Γ^2 q+2)(324 √(6) b (h^2
    Γ^2 q+2)^3/2 108 √(6)√(ad) + (h^2
    Γ^2 q+2)^3/2-72 6^3/4
       × h^3 Γ^3 q+3α -1728 6^3/4 h^6 Γ^6 q+6α +864 6^3/4 h^5 Γ^6 q+6α-54 √(6) h^3 m
    Γ^3 q+3β
       -1296 √(6) H_0^6 m Γ^6 q+6β+216 H_0^3 Γ^4
    q+4 c_1][18 h^2 (3 m+4) (6^3 m/8+1 H_0^2
    Γ^2 q+2
       × 6 f^3 √(H_0^2 Γ^2 q+2)(4 √(6)α 3 m β)-12 d) (H_0^2 Γ^2
    q+2/m^2)^3 m/8+2^3 m/8 3^3
    m/8+2f^3 m)^2]^-1
       -[f^3 Γ^-2 q-2(-6^3 m/8+1/2(3
    m+4) √(H_0^2 Γ^2 q+2)(-12 6^3/4 H_0αΓ^q+1
       +36 H_0 c_1 Γ^2 q+2-432 6^3/4 h^4 αΓ^4 q+4
    +288 6^3/4 h^3 αΓ^4 q+4 -324 √(6) H_0^4 m
    
       ×βΓ^4 q+4+216 √(6) H_0^3 m βΓ^4
    q+4 + √(H_0^2 Γ^2 q+2) +27 √(6)√(ad)√(H_0^2 Γ^2 q+2))
    
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8+2^3 m/8+1 3^3 m/8+2 a √(d){-
    H_0^2 Γ^2 q+2- H_0^2 m Γ^2 q+2
       -8 (12 h^2 Γ^2 q+2-3 H_0 m Γ^2
    q+2)}-[2^3 m/8-3/2 3^3
    m/8-1/2 (3 m+4) (H_0^2 Γ^2
    q+2/m^2)^3 m/8
       ×(324 √(6) b (H_0^2 Γ^2 q+2)^3/2
    +108 √(6)√(ad)(H_0^2 Γ^2
    q+2)^3/2 -72 6^3/4
       × H_0^3 Γ^3 q+3α -1728 6^3/4 H_0^6
    Γ^6 q+6α +864 6^3/4 H_0^5 Γ^6 q+6α
    -54 √(6)
       × H_0^3 m Γ^3 q+3β - √(6) h^6 m Γ^6
    q+6β + √(6) h^5 m Γ^6 q+6β +18 H_0Γ^q+1 c_1
    
       +216 h^3 Γ^4 q+4 c_1)] ×[√(H_0^2
    Γ^2 q+2)]^-1-1/m[2^3 m/8
    -7/2 3^3 m/8+1/2(H_0^2
    Γ^2 q+2/m^2)^3 m/8-1
       ×√(h^2 Γ^2 q+2)(324 √(6) b (H_0^2
    Γ^2 q+2)^3/2 +√(6)√(ad)(h^2
    Γ^2 q+2)^3/2
       -72 6^3/4 H_0^3 Γ^3 q+3α -1728 6^3/4 H_0^6
    Γ^6 q+6α + 6^3/4 H_0^5 Γ^6 q+6α -54
    √(6) h^3
    
       × m Γ^3 q+3β -1296 √(6) H_0^6 m Γ^6
    q+6β +648 √(6) h^5 m Γ^6 q+6β +18 h
    Γ^q+1 c_1
    
       +216 h^3 Γ^4 q+4 c_1)]][18 h^2 (3 m+4)
    (6^3 m/8+1 h^2 Γ^2 q+2(6 f^3 √(h^2
    Γ^2 q+2)(4 √(6)α
       +3 m β)-) (h^2 Γ^2
    q+2/m^2)^3 m/8+2^3 m/8 3^3
    m/8+2 a √(d) f^3 m)]^-1)
    +1/2)+3/2[f^3 Γ^-4 q-4
       ×(2^3 m/8+1 3^3 m/8+2 a
    √(d)(3 H_0 m (3 m+8) Γ^q+1 -48 H_0^2 (12
    H_0^2 Γ^2 q+2 -3 H_0 m
    
       ×Γ^2 q+2) Γ^2 q+2 -H_0^4 m Γ^4
    q+4)-6^3 m/8 +1/2 (3 m+4) √(H_0^2
    Γ^2 q+2)(H_0^2 Γ^2
    q+2/m^2)^3 m/8
       ×(324 √(6) b (H_0^2 Γ^2 q+2)^3/2
    +108 √(6)√(ad)(H_0^2 Γ^2
    q+2)^3/2 -72 6^3/4 H_0^3 Γ^3 q+3α
        -1728 6^3/4 H_0^6 Γ^6 q+6α +864 6^3/4 H_0^5
    Γ^6 q+6α -54 √(6) H_0^3 m Γ^3 q+3β
    -1296 √(6) H_0^6 m
    
       ×Γ^6 q+6β +648 √(6) H_0^5 m Γ^6 q+6βΓ^q+1 c_1 +216 H_0^3 Γ^4 q+4
    c_1))] [108 H_0^4 (3 m+4)
    
       ×(6^3 m/8+1 H_0^2 Γ^2 q+2(√(H_0^2 Γ^2 q+2)(4 √(6)α +3 m
    β) -12 d) (H_0^2 Γ^2
    q+2/m^2)^3 m/8
       ×{2^3 m/8 3^3 m/8+1 f^3
    √(H_0^2 Γ^2 q+2)(4 √(6)α +3 m β) (H_0^2 Γ^2 q+2/m^2)^3
    m/8 + (√(H_0^2 Γ^2 q+2)
       ×(4 √(6)α +3 m β)-12 d)
    (H_0^2 Γ^2 q+2/m^2)^3 m/8
    +[2^3 m/8-3 3^3 m/8+1 H_0^2 Γ^2
    q+2√(H_0^2 Γ^2 q+2)
       ×(4 √(6)α +3 m β) -12 d
    (H_0^2 Γ^2 q+2/m^2)^3
    m/8-1][m]^-1}(2^3 m/8+1
    3^3 m/8+2 a √(d)(3 H_0
       × m (3 m+8) Γ^q+1 -48 H_0^2 (12 H_0^2
    Γ^2 q+2-3 H_0 m Γ^2 q+2) Γ^2 q+2 -432
    H_0^4 m Γ^4 q+4)
    
       -6^3 m/8 +1/2 (3 m+4)√(H_0^2 Γ^2
    q+2)(H_0^2 Γ^2 q+2/m^2)^3 m/8(324 √(6) b (H_0^2 Γ^2 q+2)^3/2+108
    √(6)
       ×(H_0^2 Γ^2 q+2)^3/2 -72 6^3/4 H_0^3
    Γ^3 q+3α-1728 6^3/4 H_0^6 Γ^6 q+6α
    +864 6^3/4 H_0^5 Γ^6 q+6
       ×α -54 √(6) H_0^3 m Γ^3 q+3β -√(6)
    H_0^6 m Γ^6 q+6β +√(6) H_0^5 m Γ^6 q+6β +H_0Γ^q+1
       +H_0^3 Γ^4 q+4 c_1))] [18 H_0^2 (3
    m+4) (6^3 m/8+1 H_0^2 Γ^2 q+2{6 f^3
    √(H_0^2 Γ^2 q+2)(4 √(6)α +β)
    
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8 +2^3 m/83^3 m/8+2 a √(d) f^3
    m)^2] -[f^3 Γ^-2 q-2(-6^3
    m/8+1/2 (3 m+4)
    
       ×√(H_0^2 Γ^2 q+2)(-12 6^3/4 H_0αΓ^q+1 - √(6) H_0 m βΓ^q+1+ H_0^2 c_1
    Γ^2 q+2+ H_0 c_1 Γ^2 q+2
       - 6^3/4 H_0^4 αΓ^4 q+4+ 6^3/4× H_0^3
    αΓ^4 q+4-√(6) H_0^4 m βΓ^4 q+4
    +√(6) H_0^3 m βΓ^4 q+4
       + √(6) b √(H_0^2 Γ^2 q+2)+ √(6)√(ad)√(H_0^2 Γ^2 q+2))
    (H_0^2 Γ^2 q+2/m^2)^3 m/8[3 m/8 +2^3 m/8+1 3^3 m/8+2
       ×(-96 H_0^2 Γ^2 q+2-72 H_0^2 m Γ^2 q+2
    -8 (12 H_0^2 Γ^2 q+2-3 H_0 m Γ^2
    q+2)) -(3 m+4)
    
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8 b (H_0^2 Γ^2 q+2)^3/2( √(6)+
    √(6)√(ad)(H_0^2 Γ^2 q+2)^3/2
    -72 6^3/4 H_0^3
    
       ×Γ^3 q+3α -6^3/4 H_0^6 Γ^6 q+6α
    +(324 √(6)(324 √(6) 6^3/4 H_0^5 Γ^6
    q+6α -54 √(6) H_0^3 m
    
       ×Γ^3 q+3β -√(6) H_0^6 m Γ^6 q+6β
    +√(6) H_0^5 m Γ^6 q+6β +18 H_0Γ^q+1
    c_1+H_0^3 Γ^4 q+4 c_1)]
    
       ×[√(H_0^2 Γ^2 q+2)] -[2^3
    m/8 -7/2 3^3 m/8+1/2√(H_0^2
    Γ^2 q+2)(H_0^2 Γ^2
    q+2/m^2)^3 m/8-1(H_0^2 Γ^2
    q+2)^3/2
       +√(6)√(ad)(H_0^2 Γ^2 q+2)^3/2
    - 6^3/4 H_0^3 Γ^3 q+3α -1728 6^3/4 H_0^6
    Γ^6 q+6α +864 6^3/4
       × H_0^5 Γ^6 q+6α -54 √(6) H_0^3 m
    Γ^3 q+3β -1296 √(6) H_0^6 m Γ^6 q+6β
    +648 √(6) H_0^5 m Γ^6 q+6β
       +18 H_0Γ^q+1 c_1+216 H_0^3 Γ^4 q+4
    c_1)][m]^-1] [18 H_0^2 (3 m+4)
    (6^3 m/8+1 H_0^2 Γ^2 q+2(6 f^3
    
       ×√(H_0^2 Γ^2 q+2)(4 √(6)α +3
    m β)-12 d ) (H_0^2 Γ^2
    q+2/m^2)^3 m/8 +2^3 m/8 3^3
    m/8+2 a √(d) f^3 m)]^-1
        -3 Γ^-q-1(f^3 Γ^-2 q-2([2^3
    m/8-2 3^3 m/8+1α +3 m β)
    (H_0^2 Γ^2 q+2/m^2)^3
    m/8][√(H_0^2 Γ^2 q+2)]^-1
       +[2^3 m/8-4 3^3 m/8(3
    m/8-1) H_0^2 Γ^2 q+2(6 f^3 √(H_0^2
    Γ^2 q+2)(4 √(6)α +3 m β)-
    (H_0^2 /m^2)^3 m/8-2]
    
       ×[m^3]^-1 +2^3 m/8-3 3^3 m/8+1
    f^3 √(H_0^2 Γ^2 q+2)(4 √(6)α +3 m
    β)  (H_0^2 Γ^2 q+2/m^2)^3
    m/8-1 +[2^3 m/8-3
       × 3^3 m/8(6 f^3 √(H_0^2 Γ^2 q+2)
    (4 √(6)α +3 m β) -12 d)
    (H_0^2 Γ^2 q+2/m^2)^3
    m/8-1][m]^-1(2^3 m/8+1
       × 3^3 m/8+2 a √(d)(3 H_0 m
    Γ^q+1-48 H_0^2 (12H_0^2 Γ^2 q+2 -3 H_0 m
    Γ^2 q+2) Γ^2 q+2 432 H_0^4 m
    
       ×Γ^4 q+4)-6^3 m/8 +1/2 -(3 m+4)
    √(H_0^2 Γ^2 q+2)(H_0^2 Γ^2
    q+2/m^2)^3 m/8(324 √(6) b (H_0^2
    Γ^2 q+2)^3/2
       +108 √(6)√(ad)(H_0^2 Γ^2
    q+2)^3/2-72 6^3/4 H_0^3 Γ^3 q+3α -17286^3/4H_0^6 Γ^6 q+6α +864 6^3/4
       × h^5 Γ^6 q+6α -54 √(6) h^3 m Γ^3 q+3β -1296 √(6) h^6 m Γ^6 q+6β +648 √(6) h^5
    × m Γ^6 q+6β
        +18 h Γ^q+1 c_1 +216 h^3 Γ^4 q+4 c_1))18
    h^2 (3 m+4) (6^3 m/8+1 h^2 Γ^2 q+2(6 f^3
    √(h^2 Γ^2 q+2)
       ×(4 √(6)α +3 m β) -12 d)
    (h^2 Γ^2 q+2/m^2)^3
    m/8+2^3 m/8 3^3 m/8+2 a √(d) f^3 m)^2
    +f^3 Γ^-4 q-4
       ×(-6^3 m8 +1/2 (3 m+4) √(h^2 Γ^2
    q+2)(-6^3/4 H_0αΓ^q+1-9 √(6) H_0 m
    βΓ^q+1 +108
    
       × H_0^2 c_1 Γ^2 q+2 + H_0 c_1 Γ^2 q+2 -6^3/4 H_0^4 αΓ^4 q+4 + 6^3/4 H_0^3 αΓ^4 q+4 -√(6) H_0^4 m βΓ^4 q+4
       +√(6) H_0^3 m βΓ^4 q+4 +81 √(6) b
    √(H_0^2 Γ^2 q+2)+27 √(6)√(ad)√(H_0^2 Γ^2 q+2) ) (H_0^2 Γ^2
    q+2/m^2)^3 m/8
       ×+2^3 m/8+1 3^3 m/8+2 a √(d)(-96
    H_0^2 Γ^2 q+2-72H_0^2 m Γ^2 q+2 -8 (12
    H_0^2 Γ^2 q+2 -3 H_0 m
    
       ×Γ^2q+2-2^3 m/8-3/2 3^3
    m/8-1/2 (3 m+4) (H_0^2 Γ^2
    q+2/m^2)^3 m/8(324 √(6) b (H_0^2
    Γ^2 q+2)^3/2 +108
    
       ×√(6)√(ad)(H_0^2 Γ^2
    q+2)^3/2 -72 6^3/4 H_0^3 Γ^3 q+3α -
    6^3/4 H_0^6 Γ^6 q+6α +6^3/4 H_0^5
    
       ×Γ^6 q+6α -54 √(6) H_0^3 m Γ^3 q+3β -1296 √(6) H_0^6 m Γ^6 q+6β +648 √(6)
    H_0^5 m Γ^6 q+6β
        +18 H_0Γ^q+1 c_1].
    
    
    s   =1/2+[H_0αΓ^q+1/72
    (H_0^2 Γ^2 q+2)^7/2-[5 H_0(3 √(6)log(6 H_0^2 Γ^2 q+2) α -2 √(6)α
    +36 c_1)Γ^q+1]
    
       ×[√(6)(H_0^2 Γ^2
    q+2)^7/2]^-1 +[ (3 H_0^2 Γ^2 q+2
    -H_0Γ^2 q+2)(αΓ^-2
    q-2/√(6) H_0^2 +2 √(2/3)β/√(H_0^2 Γ^2 q+2))]
    
       ×{√(H_0^2 Γ^2 q+2)}^-1-8 β
    -2 α/√(H_0^2 Γ^2 q+2) -2 √(6)
    c_1/√(H_0^2 Γ^2 q+2)-αlog(6 H_0^2
    Γ^2q+2)/√(H_0^2 Γ^2 q+2)
        -[(3 H_0^2 Γ^2 q+2 -H_0Γ^2 q+2)
    (√(6)log(6 H_0^2 Γ^2 q+2) α +
    α +√(H_0^2 Γ^2 q+2)β +12 c_1)]
    
       ×[(H_0^2 Γ^2 q+2)^3/2]^-1]
    (-H_0^2 βΓ^2 q+2+(d+1)+b
    (d+3/2) - √(H_0^2 Γ^2 q+2)α
       ×1/2+a√(d)+d/f^3)]^-1
    -[(-α/24 √(H_0^2 Γ^2 q+2)
    -β/6) ([H_0(3 √(6)log
    (6 H_0^2 Γ^2 q+2) α
        -2 √(6)α +36 c_1) Γ^q+1] [36
    √(6)(H_0^2 Γ^2 q+2)^5/2]-48 H_0^2
    βΓ^2 q+2-√(H_0^2 Γ^2 q+2) c_1
    
        -12 √(H_0^2 Γ^2 q+2)α +8 √(6)√(H_0^2
    Γ^2 q+2)β +12 c_1)][√(H_0^2
    Γ^2 q+2)]^-1-48 a √(d))]
    
       ×[-H_0^2 βΓ^2 q+2 +√(ad) (d+1)
    +b (d+3/2)-1/2√(H_0^2 Γ^2
    q+2)α+a√(d)+d/f^3)^2]
    
       +36 c_1(H_0^2 Γ^2 q+2)^9/2-H_0αΓ^q+1/72 (H_0^2 Γ^2 q+2)^9/2+
    √(2/3)(3 H_0^2 Γ^2 q+2 -H_0Γ^2
    q+2)
    
       -(αΓ^-4 q-4)(6 √(6) H_0^4)^-1q+2)^3/2 +[αlog(6 H_0^2
    Γ^2q+2)][12 (H_0^2 Γ^2 q+2
    )^3/2]^-1
       +(3 H_0^2 Γ^2 q+2 -H_0Γ^2 q+2)
    (√(6)log(6H_0^2 Γ^2 q+2] ) α +2
    √(6)α + √(h^2 Γ^2 q+2)β + 12 c_1
    
       ×12 √(6)(h^2 Γ^2 q+2)^5/2 - (3 h^2
    Γ^2 q+2-h Γ^2 q+2) (αΓ^-2
    q-2/√(6) h^2 +2 √(2/3)β/√(h^2
    Γ^2 q+2))3 (h^2 Γ^2 q+2)^3/2
       ×32 (-h^2 βΓ^2 q+2 +√(ad) (d+1)+b
    (d+3/2)-1/2√(h^2 Γ^2 q+2)α +a √(d)+d/f^3 )
    
       ×-5 (-α/24 √(h^2 Γ^2 q+2)
    -β/6 ) (h αΓ^q+1/72
    (h^2 Γ^2 q+2)^7/2 -(5 H_0(3 √(6)log(6 H_0^2 Γ^2 q+2)
    
       ×α -2 √(6)α +36 c_1)
    Γ^q+1)(432 √(6)(H_0^2 Γ^2
    q+2)^7/2)^-1 +2 √(2/3)(3 H_0^2
    Γ^2 q+2
       -[αlog(6 H_0^2 Γ^2
    q+2)][√(H_0^2 Γ^2 q+2)]
    -[(3 H_0^2 Γ^2 q+2 -H_0Γ^2 q+2)
    (6 H_0^2Γ^2 q+2) α
       +2 √(6)α +8 √(6)√(H_0^2 Γ^2 q+2)β
    +12 c_1][3 √(6)(H_0^2 Γ^2
    q+2)^3/2]].




§ APPENDIX C: DETERMINATION OF Ν_S^2 





    ν_s^2   =[(h^2 Γ ^2
    q+2/m^2)^-3 m/836 a √(d)-18 (h^2
    Γ^2 q+2/m^2)^3 m/82 (4 √(6)α +3 m β) (h^2 Γ^2 q+2)^3/2
       +b (d-2)+√(ad) (d-1)[f^3 Γ^-4 q-4 2^3
    m/8+1 3^3 m/8+2 a √(d)(3 h m (3 m+8)
    Γ^q+1
       -48 h^2 (12 h^2 Γ^2 q+2-3 h m Γ^2 q+2)
    Γ^2 q+2-432 h^4 m Γ^4 q+4)-6^3 m/8
    +1/2√(h^2 Γ^2 q+2)
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8 324 √(6) b (H_0^2 Γ^2 q+2)^3/2+108
    √(6)√(ad)(H_0^2 Γ^2 q+2)^3/2
       -72 6^3/4 H_0^3 Γ^3 q+3α -1728 6^3/4 H_0^6
    Γ^6 q+6α +864 6^3/4 H_0^5 Γ^6 q+6α
    -54 √(6) H_0^3
    
       × m Γ^3 q+3β -1296 √(6) H_0^6 m Γ^6
    q+6β +648 √(6) H_0^5 m Γ^6 q+6β +18 H_0Γ^q+1 c_1
    
       +216 H_0^3 Γ^4 q+4 c_1] [108 H_0^4 (3 m+4)
    (6^3 m/8+1 H_0^2 Γ^2 q+2(6 f^3
    √(H_0^2 Γ^2 q+2)(4 √(6)α
       +3 m β) -12 d) (H_0^2 Γ^2
    q+2/m^2)^3 m/8 +2^3 m/8 3^3
    m/8+2 a √(d) f^3 m)]^-1+[f^3 Γ^-2 q-2
       ×2^3 m/8 3^3 m/8+1 f^3 √(H_0^2
    Γ^2 q+2)(4 √(6)α +3 m β)
    (H_0^2 Γ^2 q+2/m^2)^3
    m/8+(√(H_0^2 Γ^2 q+2)
       ×(4 √(6)α +3 m β) -12 d)
    (H_0^2 Γ^2 q+2/m^2)^3 m/8
    +[2^3 m/8-3 3^3 m/8+1 H_0^2 (z+1) ^2
    q+2(6 f^3
    
       ×√(H_0^2 Γ^2 q+2)(4 √(6)α-432
    H_0^4 m Γ^4 q+4) +3 m β)(H_0^2
    (z+1) ^2 q+2/m^2)^3 m/8-1][m]^-1
       ×2^3 m/8+1 3^3 m/8+2 a √(d)3 H_0 m
    (3 m+8) (z+1) ^q+1-48 H_0^2 (12 H_0^2 Γ^2 q+2-3
    H_0 m
    
       ×(z+1) ^2 q+2) Γ^2 q+2-6^3 m/8
    +1/2 (3 m+4) √(H_0^2 (z+1) ^2 q+2)(H_0^2 Γ^2 q+2/m^2)^3 m/8
       - (H_0^2 (z+1) ^2 q+2)^3/2 +108 √(6)√(ad) (H_0^2 Γ^2 q+2)^3/2-726^3/4 H_0^3 (z+1)^3 q+3α
       -1728 6^3/4 H_0^6 Γ^6 q+6α +864 6^3/4 -54
    √(6) H_0^3 m (z+1) ^3 q+3β -1296 √(6) H_0^6
    
       ×  m (z+1) ^6 q+6β +648 √(6) H_0^5 m Γ^6
    q+6β +H_0Γ^q+1 c_1 +H_0^3 (z+1) ^4 q+4 c_1]
    
       ×[18 H_0^2 (3 m+4) (6^3 m/8+1 H_0^2
    Γ^2 q+2(6 f^3 √(H_0^2 Γ^2 q+2) (4
    √(6)α +3 m β)-12 d)
    
       ×(H_0^2 Γ^2 q+2/m^2)^3
    m/8+2^3 m/8 3^3 m/8+2 a √(d) f^3
    m)^2]^-1 -[f^3 (z+1) ^-2 q-2(-6^3
    m/8 +1/2
       ×(3 m+4) √(H_0^2 Γ^2 q+2)-12 6^3/4 H_0α  (z+1) ^q+1-9 √(6) H_0 m βΓ^q+1 +108
    H_0^2
    
       × c_1 (z+1) ^2 q+2 +36 H_0 c_1 Γ^2 q+2-432 6^3/4
    H_0^4 α (z+1) ^4 q+4 +6^3/4 H_0^3 αΓ^4 q+4
       -324 √(6) H_0^4 m β (z+1) ^4 q+4+216 √(6) H_0^3 m
    βΓ^4 q+4]+b √(H_0^2 (z+1) ^2 q+2)
       +27 √(6)√(ad)√(H_0^2 Γ^2 q+2)(H_0^2 (z+1) ^2 q+2/m^2)^3 m/8
    +2^3 m/8+1 3^3 m/8+2 a √(d) [-96
    H_0^2
    
       ×Γ^2 q+2-72 H_0^2 m (z+1) ^2 q+2 -8 (12
    H_0^2 (z+1) ^2 q+2-3 H_0 m Γ^2 q+2)].



Data availability: No new data were generated or analyzed
in support of this research.

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