Constraints on anisotropic properties of the universe in f ( Q, T ) gravity theory

A. Zhadyranova ,1, ∗ M. Koussour ,2, † V. Zhumabekova ,3, ‡

O. Donmez ,4, § S. Muminov ,5, ¶ and J. Rayimbaev 6, 7, 8, 9, ∗∗
1 Generaland Theoretical Physics Department, L. N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan.
2 Department of Physics, University of Hassan II Casablanca, Morocco.
3 Theoretical and Nuclear Physics Department, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan.
4 College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait.
5 Mamun University, Bolkhovuz Street 2, Khiva 220900, Uzbekistan.
6 Institute of Fundamental and Applied Research, National Research University TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan.
7 University of Tashkent for Applied Sciences, Str. Gavhar 1, Tashkent 100149, Uzbekistan.
8 Urgench State University, Kh. Alimjan Str. 14, Urgench 221100, Uzbekistan
arXiv:2409.15562v1 [astro-ph.CO] 23 Sep 2024

9 Shahrisabz State Pedagogical Institute, Shahrisabz Str. 10, Shahrisabz 181301, Uzbekistan.

Motivated by anomalies in cosmic microwave background observations, we investigate the impli-

cations of f ( Q, T ) gravity in Bianchi type-I spacetime, aiming to characterize the universe’s spatially
homogeneous and anisotropic properties. By using a linear combination of non-metricity Q and the
energy-momentum tensor trace T, we parametrize the deceleration parameter and derive the Hub-
ble solution, which we then impose in the Friedmann equations of f ( Q, T ) gravity. Bayesian analy-
sis is employed to find the best-fit values of model parameters, with 1 − σ and 2 − σ contour plots
illustrating the constraints from observational data, including H (z) data and the Pantheon+ sam-
ple. Our analysis reveals a transition from a decelerated to an accelerated expansion phase, with the
present deceleration parameter indicating an accelerating universe. The energy density gradually
decreases over time, approaching zero for the present and future, indicating continuous expansion.
The anisotropic pressure, initially notably negative, transitions to slightly negative values, suggesting
the presence of dark energy. The evolving equation of state parameter ω exhibits behavior akin to
phantom energy, influenced by spacetime anisotropy. Violations of the null energy condition and the
strong energy condition imply phantom-like behavior and accelerated expansion.

I. INTRODUCTION corresponds to ω0 = −1 in the standard ΛCDM model.

DE is hypothesized to constitute approximately 68% of
Cosmology underwent a dramatic transformation the total energy density of the universe, exerting a nega-
when observational evidence from Type Ia supernovae tive pressure that drives the accelerated expansion. This
(SNe Ia) searches [1, 2] confirmed the accelerating ex- concept aligns well with observational data from vari-
pansion of the universe. In the past two decades, ous cosmological probes, such as SNe Ia, CMB, and LSS
numerous observational results, including those from surveys. The exact nature of DE remains one of the most
the WMAP experiment [3, 4], cosmic microwave back- profound mysteries in modern cosmology, with numer-
ground (CMB) radiation [5, 6], large-scale structure ous theoretical models proposed to elucidate its prop-
(LSS) studies [7, 8], and baryon acoustic oscillation erties and behavior. Among these models, the standard
(BAO) [9, 10], have consistently supported the observed ΛCDM model is the most widely accepted by theoretical
cosmic acceleration. The leading explanation for this ac- physicists for explaining DE and the late-time accelera-
celerating scenario is the presence of a dark energy (DE) tion of the universe. However, it faces two fundamen-
component. This mysterious form of energy is com- tal challenges: fine-tuning and the cosmic coincidence
monly described by an equation of state (EoS) param- problem [12–14]. An alternative method for explaining
eter ω0 = −1.03 ± 0.03 in a flat universe [11]. While this the accelerating expansion of the universe, without re-
value suggests a dynamical form of DE, the leading ex- lying on undetected DE, involves considering a more
planation remains the cosmological constant (Λ), which general description of the gravitational field. This ap-
proach modifies the Einstein–Hilbert action of general
relativity (GR) by introducing a generic function f ( R),
∗ Email: a.a.zhadyranova@gmail.com where R is the Ricci scalar curvature, as initially pro-
† Email: pr.mouhssine@gmail.com posed in [15–17]. The f ( R) gravity model can account
‡ Email: zh.venera@mail.ru
§ for the expansion mechanism without requiring any ex-
Email: orhan.donmez@aum.edu.kw
¶ Email: sokhibjan.muminov@gmail.com otic DE component [18, 19]. Studies have examined the
∗∗ Email: javlon@astrin.uz observational implications of f ( R) gravity models, in-
 2

cluding constraints from the solar system and the equiv- accelerated expansion of the universe using observa-
alence principle [20–22]. Furthermore, viable cosmolog- tional constraints. Additionally, significant research has
ical models of f ( R) gravity that pass solar system tests been conducted on topics such as baryogenesis [57], cos-
have been proposed [23–25]. mological inflation [58], and cosmological perturbations
In the literature, several modified theories of grav- [59]. However, the astrophysical implications of f ( Q, T )
ity have been proposed, including f (T ) theory [26–29], gravity have not been extensively explored. Tayde et
f ( R, T ) theory [30–32], f ( G ) theory [33], f ( R, G ) theory al. [60] investigated static spherically symmetric worm-
[34, 35], and f ( R, Lm ) theory [36, 37]. Recently, f ( Q) hole solutions in f ( Q, T ) gravity, considering both linear
theories of gravity have garnered significant attention. and non-linear models under various equations of state.
Symmetric teleparallel gravity, or f ( Q) gravity, was in- Furthermore, Pradhan et al. [61] examined the thin-shell
troduced by Jiménez et al. [38], where gravitational in- gravastar model within the framework of f ( Q, T ) grav-
teractions are described by the non-metricity Q, which ity.. Recently, Bourakadi et al. [62] explored constant-
geometrically represents the variation in the length of roll inflation and the formation of primordial black holes
a vector during parallel transport. This is distinct from within the framework of f ( Q, T ) gravity.
teleparallel f (T ) gravity, where gravity is described us- Contrary to popular belief, which holds that the uni-
µ
ing torsion generated by tetrad fields ei , replacing the verse is homogeneous and isotropic, there is evidence
metric tensor gµν as the primary geometric variable. In that anisotropies may have existed in the early past and
this approach, the torsion replaces curvature as the de- may reappear in the future. Small-scale anisotropies in
scriptor of gravitational effects. In symmetric teleparal- the CMB were discovered by the cosmic background
lel gravity, the covariant divergence of the metric ten- explorer in 1996 [63]. This idea of anisotropic space-
sor is non-zero, similar to Weyl’s theory. Numerous time geometry is substantially supported by observa-
studies have investigated f ( Q) gravity [39–46], includ- tional data from experiments like the cosmic back-
ing the first cosmological solutions [47, 48], geodesic de- ground imager [64] and the WMAP [65]. Moreover, re-
viation equations derived from its covariant formula- cent developments indicate that the cosmos is expand-
tion [49], and quantum cosmology for power-law mod- ing anisotropically, as indicated by differences in the
els [50]. Cosmological solutions and the growth index intensities of microwaves received from various direc-
of matter perturbations for polynomial functional forms tions [66]. An efficient framework for characterizing the
of f ( Q) have also been examined [51]. Furthermore, homogeneous and anisotropic properties of spacetime
Refs. [52, 53] introduced the f ( Q, Lm ) theory, which ex- is presented by Bianchi-type cosmology. In the litera-
tends modified gravity by incorporating a non-minimal ture, numerous models inspired by Bianchi cosmology
coupling between the non-metricity scalar Q and the have been explored [67–71]. Recent studies have fo-
matter Lagrangian Lm within the framework of the Ein- cused on the isotropization mechanism in anisotropic
stein–Hilbert action. Bianchi type-I cosmology, particularly through a poly-
In a recent work, Xu et al. [54] introduced an exten- nomial f ( Q) model [72]. Loo et al. [73] investigated
sion of f ( Q) gravity that involves a non-minimal cou- the dynamics of an anisotropic universe in the context
pling between the non-metricity Q and the trace T of the of f ( Q, T ) gravity, while Narawade et al. [74] exam-
matter-energy-momentum tensor. They formulated the ined observational constraints on the hybrid scale fac-
Lagrangian density of the gravitational field as a general tor in anisotropic spacetimes under f ( Q, T ) gravity. In
function of both Q and T, denoted as L = f ( Q, T ). This this work, we use the locally rotationally symmetric
theory resembles the f ( R, T ) theory [30], but instead of (LRS) Bianchi type-I metric, which is essentially an ex-
using the geometric sector of the Einstein–Hilbert ac- tension of the isotropic case and has a strong resem-
tion, it utilizes the symmetric teleparallel formulation. blance to the Friedmann–Lemaı̂tre–Robertson–Walker
Like the standard couplings between the curvature and (FLRW) metric. This work differs from previous stud-
the trace of the energy-momentum tensor, the coupling ies by examining the anisotropic dynamics of the uni-
between Q and T in the f ( Q, T ) theory also causes the verse in f ( Q, T ) gravity within a Bianchi type-I space-
energy-momentum tensor to be non-conserved. This time. Specifically, we consider a linear combination of
non-conservation has important physical consequences, Q and T in the form f ( Q, T ) = Q + bT [75], moti-
including substantial changes in the thermodynamics of vated by its simplicity and ability to capture deviations
the universe, similar to those seen in the f ( R, T ) the- from GR while still allowing for analytic solutions in an
ory [30]. Further, the non-geodesic motion of test par- anisotropic spacetime.
ticles leads to the emergence of an extra force. Kous- The present study is organized as follows: Sec. II out-
sour et al. [55, 56] specifically examined the late-time lines the basic formalism of f ( Q, T ) gravity. In Sec. III,
 3

our focus is on the anisotropic Bianchi type-I cosmolog- Hence, the non-metricity scalar is defined as [38, 47],
ical model and the equations of motion in the context of
1
f ( Q, T ) gravity. Sec. IV is dedicated to exploring a par- Q = − Q βµν P βµν = − − Q βνρ Q βνρ + 2Q βνρ Qρβν
ticular functional form of f ( Q, T ) and its cosmological  4
solutions for f ( Q, T ) gravity, using parameterization of −2Qρ Q̃ρ + Qρ Qρ . (7)
the deceleration parameter. In Sec. V, we employ obser-
vational data from H (z) data and Pantheon+ sample to The field equation of f ( Q, T ) gravity is derived by
determine the model parameters. In Sec. VI, we discuss varying the action (1) with respect to the metric com-
the cosmological implications of the model and validate ponent gµν ,
its consistency with the energy conditions. Lastly, in Sec.
2 1
∇ β ( f Q − gP µν ) − f gµν + f T ( Tµν + Θµν )
p β
VII, we review and summarize our results. √
−g 2
βα βα
+ f Q ( Pµβα Qν − 2Q µ Pβαν ) = 8πTµν . (8)
II. BASIC FORMALISM OF f ( Q, T ) GRAVITY
∂f ∂f
where f ≡ f ( Q, T ), f Q ≡ ∂Q , f T ≡ ∂T , and Tµν is the
The action principle for the f ( Q, T ) gravity model, energy-momentum tensor derived from the matter La-
as proposed by Xu et al. [54], is defined using the grangian and is defined as
non-metricity scalar Q and the trace of the energy- √
momentum tensor T as 2 δ( − gLm )
Tµν = − √ . (9)
  −g δgµν
1
Z p
S= −g f ( Q, T ) + Lm d4 x, (1)
16π Furthermore, we have

where f ( Q, T ) is an arbitrary function of Q and T, and g δ g µν T µν

= T αβ + Θ α β , (10)
is the determinant of the metric tensor gµν . δ gαβ
The non-metricity scalar Q is defined in terms of the δT
β
metric tensor gµν and the disformation tensor Lαγ as fol- where Θµν = gαβ δgµν αβ
. In addition, it is crucial to note
lows: that in f ( Q, T ) gravity, the divergence of the energy-
µ
momentum tensor can be expressed as Dµ T ν = Bν ̸= 0,
β β
Q ≡ − gµν ( L α
αµ L νβ −L α
αβ L µν ), (2) where Dµ denotes the covariant derivative with respect
to the connection associated with the non-metricity Q.
where The term Bν is the non-conservation vector, which de-
β 1 βη   pends on Q, T, and the thermodynamic properties of
Lαγ = g Qγαη + Qαηγ − Qηαγ = L β γα . (3) the system. Therefore, in this framework, the energy-
2
momentum tensor is not conserved.
and

Qγµν = −∇γ gµν = −∂γ gµν + gνσ Γ

eσ µγ + gσµ Γ
eσ νγ . (4) III. BIANCHI TYPE-I METRIC AND MOTION
EQUATIONS IN f ( Q, T ) GRAVITY
Here, Γeγ µν = Lγ µν + Γγ µν represents the Weyl con-
nection, where Γγ µν denotes the well-known Levi-Civita In the context of cosmology, the Bianchi type-I metric
connection associated with the metric. In addition, the plays a significant role in describing an anisotropic uni-
trace of the non-metricity tensor is given by verse. Unlike the isotropic models, where isotropy im-
plies a uniform expansion in all directions, the Bianchi
Q β = gµν Q βµν , e β = gµν Qµβν .
Q (5)
type-I metric allows for different scale factors along each
Further, we introduce the superpotential tensor, also spatial direction. This anisotropy introduces complexi-
referred to as the non-metricity conjugate. It is defined ties into the dynamics of the universe, leading to distinct
as follows: observational signatures compared to isotropic models.
" Understanding the behavior of cosmological models un-
β 1 β β der this metric is crucial for accurately modeling the
P µν ≡ − Q µν + 2Q + Q β gµν − Q
e β gµν
4 (µ ν) evolution of our universe and interpreting observational
# data. The metric is diagonal and can be written as
β 1 β 1 β e β gµν − 1 δ β Qν) . (6)

−δ (µ Qν) = − L µν + Q −Q
2 4 4 (µ ds2 = −dt2 + A2 (t)dx2 + B2 (t)(dy2 + dz2 ), (11)
 4

where A (t) and B (t) are the scale factors correspond- are expressed as [73, 74],
ing to each spatial dimension, and t is the cosmic time.
f
Here, we assume the symmetry between y and z to sim- + 6 f Q (2H − Hy ) Hy ,
(8π + f T )ρ + f T p = (17)
plify the model, as this assumption is often sufficient to 2
f ∂ h i
capture the effects of anisotropy while preserving ana- 8π p = − − 2 f Q Hy − 6 f Q Hy H, (18)
2 ∂t
lytical traceability [42]. In particular, the standard flat
f ∂ h i
FLRW cosmology is derived when the scale factors A(t) 8π p = − − f (3H − Hy ) − 3 f Q (3H − Hy ) H .
2 ∂t Q
and B(t) are equal to the scale factor a(t). (19)
The non-metricity scalar associated with the Bianchi
type-I metric can then be expressed as [73]: Through algebraic manipulations of Eqs. (17) to (19),
we can simplify the field equations to
Q = −6(2H − Hy ) Hy , (12)
f 6 fQ h i
8πρ = + 8π (2H − Hy ) Hy + f T H 2 (20)
where Hx = Ȧ Ḃ
A , Hy = B , and Hz = Hy denote direc-
2 8π + f T
tional Hubble parameters. Moreover, the average Hub- 2 fT ∂  
+ fQ H ,
ble parameter, which characterizes the rate of expansion 8π + f T ∂t
of the universe, can be computed as f ∂ 
f H − 6 f Q H2 .

8π p = − − 2 (21)
2 ∂t Q
ȧ 1 
H= = Hx + 2Hy . (13)
a 3
IV. COSMOLOGICAL SOLUTIONS
By examining Eq. (12), we observe that when Hx =
Hy = H, it simplifies to Q = −6H 2 , which corresponds
Recently, Xu et al. [54] considered a linear form for
to the isotropic FLRW case.
the function f ( Q, T ), in which the cosmological evo-
The anisotropy parameter quantifies the degree of de-
lution follows a de Sitter type, resulting in the uni-
viation from isotropy in a cosmological model. It mea-
verse expanding exponentially. Here, to investigate the
sures how much the expansion rate of the universe dif-
anisotropic dynamics of the universe, we consider the
fers along different spatial directions. It is defined as
simple linear functional form of f ( Q, T ) gravity given
1 3

Hi − H
2
2  2 by [74]
∆= ∑ = H x − Hy . (14)
3 i =1 H 9H 2 f ( Q, T ) = Q + bT, (22)
In addition, the expansion scalar θ and the shear where b is a free parameter, and the scenario equivalent
scalar σ of the fluid are defined as follows: to GR is retained for b = 0. In this scenario, we have
1   f Q = 1 and f T = b. Therefore, Eqs. (20) and (21) can be
θ = Hx + 2Hy = 3H, σ = √ Hx − Hy . (15) rewritten as
3
. .
In cosmology, a perfect fluid is a theoretical model of- 3H 2 + H 3( H − Hy )2 + H
ρ = − , (23)
ten used to describe a continuous distribution of mat- 2b + 8π b + 8π
. .
ter exhibiting certain idealized properties. Its simplic- 3H 2 + H 3( H − Hy )2 + H
ity and ability to describe large-scale properties of mat- p = − − , (24)
2b + 8π b + 8π
ter make it a valuable tool in studying the dynamics of
where the dot represents the derivative with respect to
the universe. The energy-momentum tensor associated
cosmic time t. Further, we assume a physical condition
with the metric (11) is described by the standard form
where the shear scalar is proportional to the expansion
for a perfect fluid:
scalar (σ2 ∝ θ 2 ), leading to the relation A = Bn , where
Tµν = (ρ + p)uµ uν + pgµν . (16) n is a real number. This physical law is based on obser-
vations of the velocity-redshift relation for extragalac-
Here ρ is the energy density, p is the anisotropic pres- tic sources, suggesting that the Hubble expansion of the
sure, and uµ = (1, 0, 0, 0) is the four-velocity of the fluid universe may achieve isotropy when σθ = constant [76].
element. In addition, we assume the matter Lagrangian This condition has been used in several instances in the
to be Lm = p, which results in Θµν = pgµν − 2Tµν . literature [73, 74, 76–78]. In terms of the directional
The generalized Friedmann equations, which de- Hubble parameter, this condition can be expressed as
scribe the dynamics of the universe in f ( Q, T ) gravity, Hx = nHy , n ̸= (0, 1). For n = 1, the isotropic flat FLRW
 5

cosmology is recovered. Thus, the average Hubble pa- − (1 + z) H (z) dHdz(z) . Using Eq. (28), we have
rameter is expressed as ! 2α −1
. αH02 (1 + z)3 (1 + z )3 + β 3

( n + 2) H=− . (29)
H= Hy . (25) 1+β 1+β
3
By substituting Eq. (28) into Eqs. (14) and (15), we de-
Now, we have a system of two equations as described
rive the expressions for the anisotropy parameter, scalar
in Eqs. (23) and (24), which involve three unknowns: H,
expansion, and shear scalar as follows:
p, and ρ. Therefore, to obtain a complete solution for
the system, an additional plausible condition is neces- 2( n − 1)2
∆ = , (30)
sary to determine the unique solution for this system of ( n + 2)2
equations. !α/3
(1 + z )3 + β
In this study, the focus is on a specific form of θ = 3H0 , (31)
parametrization of the deceleration parameter q as de- 1+β
fined by Koussour et al. [79], which is represented by √ !α/3
3H0 (n − 1) (1 + z )3 + β
..
σ = . (32)
a α n+2 1+β
q = − 2 = −1 + (26)
aH 1 + βa3 respectively. From the above expressions, it is evident
that the anisotropy parameter remains constant, indi-
Where α and β are positive constants, and a is the scale
cating uniform anisotropy throughout the evolution of
factor of the universe, defined in terms of redshift z as
the universe. The scalar expansion and shear scalar di-
a = (1 + z)−1 . The sign of the deceleration parameter
verge during the early stages (z >> 1) of the universe
indicates the direction of the universe’s expansion. For
and approach zero as time progresses toward infinity
q > 0, the universe experiences deceleration, meaning
(z → −1). This indicates that the universe initially
its expansion rate decreases over time. In contrast, for
undergoes a phase of infinite expansion rate, eventu-
q < 0, the universe undergoes acceleration, indicating
ally transitioning to a constant expansion rate in later
an increasing expansion rate. It is important to note that
epochs.
current observations, such as SNe Ia and CMB [1, 2, 5,
6], tend to favor accelerating models characterized by
q < 0. From the above expression, we can see that at V. OBSERVATIONAL DATA
present (z = 0), the current value of the deceleration
parameter q0 is given by q0 = −1 + 1+α β . This implies In this section, a statistical analysis is performed to
that the present-day acceleration or deceleration of the compare the predictions of the theoretical model with
Universe depends on the values of α and β. If α < 1 + β, observational data. The goal is to establish constraints
the Universe is currently accelerating; if α > 1 + β, the on the free parameters of the model, namely H0 , α, and
Universe is currently decelerating; and if α = 1 + β, the β. The analysis employs a sample of Cosmic Chronome-
universe is currently coasting. ters (CC), consisting of 31 measurements, along with the
The following equation establishes the relationship Pantheon+ sample, which includes 1701 data points. We
between the deceleration parameter and the Hubble pa- use the emcee Python package [80], which implements
rameter: the affine-invariant ensemble sampler for Markov Chain
Z z
! Monte Carlo (MCMC) simulations, a widely used tool
1 + q (z) in Bayesian methods in cosmology [81]. The MCMC
H (z) = H0 exp dz . (27)
0 (1 + z ) sampler is used to estimate the posterior distribution
of the model parameters. The likelihood function is
By substituting Eq. (26) into Eq. (27), we derive the constructed using observational data, and the posterior
expression for H (z) as follows: distribution is computed through multiple iterations of
!α/3 MCMC sampling. Our MCMC analysis uses 100 walk-
(1 + z )3 + β ers and 1000 steps to obtain the fitting results. The like-
H (z) = H0 , (28)
1+β lihood function is given by [82, 83]

L ∝ exp(−χ2 /2). (33)

where H0 denotes the present value of the Hubble
parameter (at z = 0). The time derivative of the
.
where χ2 represents the pseudo chi-squared function
Hubble parameter can be expressed as follows: H = [81]. Further details about the construction of the χ2
 6

function for different data samples are provided in the model, and it can be expressed as:
following subsections.
" #
th D L ( zi )
A. Observational H (z) data
µ (zi ) = 5log10 + 25, (36)
1Mpc

Cosmic chronometers are a method for estimating the

Hubble parameter by comparing the relative ages of where d L (z) represents the luminosity distance in the
passively evolving galaxies. They are identifiable by assumed theoretical model, and it can be expressed as
specific features in their spectra and color profiles [84]. follows:
Cosmic chronometers use the estimated ages of galaxies
at various redshifts to obtain their data. For our analysis, Z z
we incorporate a collection of 31 independent measure- dx
d L ( z ) = c (1 + z ) (37)
ments of H (z) spanning the redshift range 0.07 ≤ z ≤ 0 H ( x, θ )
2.41 [85]. These H (z) measurements are obtained using
1 dz dz
the relationship H (z) = − 1+ z dt . In this context, dt is
∆z where θ represents the parameter space of the assumed
estimated by ∆t , where ∆z and ∆t denote the change in
model. In contrast to the Pantheon dataset, the Pan-
redshift and the corresponding change in age between
theon+ compilation effectively resolves the degeneracy
two galaxies. The corresponding χ2 function is given
between the parameters H0 and M by redefining the
by:
vector D as follows:
31 [ Hth (zk ) − Hobs,k ]2
χ2Hz = ∑ 2
σH,k
. (34) 
k =1 m − M − µCeph i ∈ Cepheid hosts
Bi i
D̄ = (38)
Here, Hobs,k represents the observed Hubble value, m Bi − M − µth (zi ) otherwise
while Hth (zk ) denotes the theoretical value of H (z) at
redshift zk . The standard error is given by σH,k .
Ceph
Here, µi is independently estimated using
Cepheid calibrators. Consequently, the equation
B. Pantheon+ SNe Ia sample −1
χ2SNe = D̄ T CSN D̄ is obtained.

The Pantheon+ sample spans a wide range of red-

shifts from 0.001 to 2.3, incorporating the latest observa- In addition, we use the total χ2total to obtain joint con-
tional data and surpassing previous collections of SNe straints for the parameters H0 , α, and β from the H (z)
Ia. SNe Ia are known for their consistent brightness, and Pantheon+ samples. Thus, the relevant chi-square
making them reliable standard candles for measuring functions are defined as
relative distances using the distance modulus technique.
Over the past two decades, several compilations of SNe
Ia data have been introduced, including Union [86], χ2total = χ2Hz + χ2SNe (39)
Union2 [87], Union2.1 [88], JLA [89], Pantheon [90], and
the latest addition, Pantheon+ [91]. The corresponding
χ2 function is given by:
In our analysis, we use Gaussian priors as follows:
−1 [50, 100] for H0 , [0, 10] for α, and [0, 10] for β. The cor-
χ2SNe = D T CSN D, (35)
responding 1 − σ and 2 − σ contour plots for the H (z)
In this context, CSN [91] denotes the covariance ma- data, Pantheon+ sample, and the combined observa-
trix linked with the Pantheon+ samples, which includes tional data are displayed in Figs. 1, 2, and 3, respectively.
both statistical and systematic uncertainties. Also, the The results with a 68% confidence limit are as follows:
vector D is defined as D = m Bi − M − µth (zi ), where H0 = 67.8+ 1.3 +0.43 +1.5
−1.3 , α = 1.53−0.40 , and β = 2.2−1.5 for H ( z )
m Bi and M represent the apparent magnitude and abso- data; H0 = 72.0+ 1.4 +0.14 +1.0
−1.4 , α = 1.56−0.13 , and β = 3.55−0.90
+1.2 +0.059
lute magnitude, respectively. Furthermore, µth (zi ) rep- for Pantheon+ data; H0 = 68.1−1.2 , α = 1.508−0.054 , and
resents the distance modulus of the assumed theoretical β = 2.57+ 0.58
−0.53 for the combined data.
 7

H0 = 67.8+1.3
1.3 H0 = 68.1+1.2
1.2
H(z) dataset H(z)+Pantheon+ dataset

= 1.53+0.43
0.40 = 1.508+0.059
0.054

1.6
2.0

1.5 1.5

1.0 1.4
= 2.2+1.5
1.5 = 2.57+0.58
0.53

3.5
4
3.0

2 2.5
2.0
0
66 67 68 69 70 1.0 1.5 2.0 0 1 2 3 4 5 66 67 68 69 70 1.4 1.5 1.6 2.0 2.5 3.0 3.5
H0 H0

FIG. 1: Confidence intervals for model parameters using FIG. 3: Confidence intervals for model parameters using
the H (z) dataset: 1-σ and 2-σ levels. the combined dataset: 1-σ and 2-σ levels.

VI. COSMOLOGICAL IMPLICATIONS OF THE MODEL

A. Cosmological parameters

The deceleration parameter is a crucial metric for un-

H0 = 72.0+1.4
1.4 derstanding the universe’s expansion phases. Fig. 4
Pantheon+ dataset shows that the model transitions from a decelerated
epoch to a de-sitter-type accelerated expansion phase.
Specifically, the transition redshift is zt = 0.61 for the
H (z) data, zt = 0.85 for the Pantheon+ samples, and
= 1.56+0.14
0.13
zt = 0.72 for the combined data [92–95]. The present
value of the deceleration parameter is q(z = 0) = q0 =
1.7 −0.52, q0 = −0.65, and q0 = −0.57, respectively, at a
1.6 68% confidence limit, aligning well with observed val-
1.5 ues [96–101]. This consistency with observational data
1.4 supports the model’s validity and its effectiveness in de-
= 3.55+1.0
0.90

5
scribing the universe’s expansion history. The uniform
transition redshift across different datasets further high-
4 lights the robustness of the model’s predictions.
3 From Fig. 5, the energy density demonstrates a com-
pelling trend across all constrained values of the model
70 71 72 73 74 1.4 1.5 1.6 1.7 3 4 5 parameters. It starts with a significant initial magnitude
H0 but gradually diminishes over time, ultimately converg-
ing towards zero for the present (z = 0) and future
(z → −1). This striking behavior strongly suggests the
FIG. 2: Confidence intervals for model parameters using
continuous expansion of the universe. As illustrated in
the Pantheon+ SNe Ia sample: 1-σ and 2-σ levels.
Fig. 6, the pressure initially shows significantly nega-
 8

0.5
- 0.2

0.0 - 0.4

p/H02
q

- 0.6
- 0.5 Hz Hz
- 0.8 Pantheon+
Pantheon+
Joint Joint
- 1.0 - 1.0
-1 0 1 2 3 4 -1 0 1 2 3 4
z z

FIG. 4: Variation of the deceleration parameter as a FIG. 6: Variation of the pressure as a function of
function of redshift z. redshift z with b = −0.1 and n = 0.087 [75].

- 0.2
tive values, which gradually evolve toward less nega-
tive values over time. This gradual shift in the pressure - 0.4

profile reflects the influence of modified gravity effects - 0.6

within the model, without requiring an explicit DE com-
ponent. This behavior aligns with the modified grav- - 0.8
ω

ity framework, where late-time cosmic acceleration is - 1.0

driven by geometric contributions, rather than the in- Hz
troduction of a separate DE term. - 1.2
Pantheon+
- 1.4 Joint
1.4
Hz
1.2 -1 0 1 2 3 4
Pantheon+
z
1.0 Joint
0.8 FIG. 7: Variation of the EoS parameter as a function of
2
ρ/3H0

redshift z with b = −0.1 and n = 0.087 [75].

0.6

0.4
as DE, dark matter, and radiation, as they evolve over
0.2
time. The value of the EoS parameter can vary depend-
0.0 ing on the substance: For ordinary matter (baryonic
-1 0 1 2 3 4 matter and non-relativistic matter), the EoS parameter
z is approximately ω = 0, indicating that the pressure is
negligible compared to the energy density. For radia-
FIG. 5: Variation of the energy density as a function of tion, including photons and relativistic particles, the EoS
redshift z with b = −0.1 and n = 0.087 [75]. parameter is ω = 13 , indicating that the pressure is one-
third of the energy density. For DE, which is thought
The equation of state (EoS) parameter in cosmology to be driving the accelerated expansion of the universe,
describes the relationship between pressure and energy the EoS parameter is typically represented by ω ≈ −1.
density for a given substance or component of the uni- This value is consistent with a cosmological constant (Λ)
verse. It is denoted by the symbol ω and is defined as or vacuum energy, leading to a constant energy density
the ratio of anisotropic pressure p to energy density ρ: and negative pressure. In addition to the values men-
p tioned earlier, there are two other important categories
ω= . (40) in cosmology for the EoS parameter:
ρ

This parameter plays a crucial role in determining the • Quintessence [102–104]: is a hypothetical form of
behavior of different components of the universe, such DE postulated to explain the accelerating expan-
 9

sion of the universe. It is characterized by a dy- B. Energy conditions

namic EoS parameter ω that evolves over time.
Typically, quintessence models have −1 < ω < In GR, energy conditions are a set of constraints im-
− 31 , allowing for a range of behaviors that can posed on the stress-energy tensor Tµν to ensure physi-
mimic both cosmological constant-like behavior cally reasonable matter and energy distributions. These
(ω ≈ −1) and more exotic forms of DE. conditions help in understanding the behavior of mat-
ter and energy under various circumstances, especially
• Phantom [105–107]: is another theoretical form of
in the context of gravitational collapse, black holes, and
DE with an EoS parameter ω < −1. This im-
cosmological models. Here, the primary role of these en-
plies that the energy density of phantom energy
ergy conditions is to verify the accelerated expansion of
increases as the universe expands, leading to a
the universe [112–114]. These conditions originate from
”big rip” scenario where the universe is ultimately
the well-established Raychaudhuri equations, which de-
torn apart. The concept of phantom DE is highly
scribe the behavior of geodesic congruences (families) in
speculative and is not supported by current obser-
a given spacetime, illustrating the focusing of geodesic
vational data.
flows due to gravity [117]. The Raychaudhuri equation
The EoS parameter is often dynamic, particularly for has different forms depending on the type of geodesics
DE, allowing for transitions between different epochs. considered, whether timelike or null [115, 116]:
A notable example is a fluid with ω > −1 at certain 1. Timelike geodesics:
times and ω < −1 at others, combining quintessence- dθ 1
like and phantom-like behaviors. Such a transition can = − θ 2 − σµν σµν + ωµν ω µν − Rµν uµ uν (41)
dτ 3
be seen in Fig. 7, where the EoS parameter evolves
where θ is the expansion scalar, σµν is the shear tensor,
from the quintessence region (ω > −1) to the phantom
ωµν is the vorticity tensor, Rµν is the Ricci tensor, and uµ
regime (ω < −1) as the universe evolves. This dynamic
is the timelike vector tangent to the geodesics.
evolution indicates that the EoS parameter is not static,
2. Null geodesics:
but rather evolves with time, a behavior characteristic
of models that allow for EoS crossing the phantom di- dθ 1
vide line (ω = −1). It is important to clarify that while
= − θ 2 − σµν σµν + ωµν ω µν − Rµν kµ kν (42)
dλ 2
quintessence and phantom DE are commonly explored
where θ is the expansion scalar for null geodesics and kµ
within cosmological models, the results discussed here
is the null vector tangent to the geodesics.
pertain specifically to our model within the framework
The Raychaudhuri equations are fundamental in un-
of f ( Q, T ) modified gravity. Therefore, the crossing of
derstanding gravitational focusing and the formation of
the phantom divide line in our context does not im-
singularities in spacetime. They show how the expan-
ply the existence of a traditional DE fluid, but rather
sion, shear, and rotation of geodesic congruences evolve
emerges from the geometric contributions of the mod-
and are influenced by the curvature of spacetime. The
ified gravity model itself. Observational data, such as
main energy conditions are:
from SNe Ia and the CMB, are used to constrain the pos-
1. Null energy condition (NEC):
sible values of ω and understand the nature of DE. The
WMAP9 data [65], combining measurements from the Tµν kµ kν ≥ 0 (43)
Hubble parameter (H0 ), SNe Ia, CMB, and BAO, sug-
gests ω0 = −1.084 ± 0.063. In contrast, the Planck col- for any null vector kµ . This condition implies that the
laboration’s findings in 2015 indicated ω0 = −1.006 ± energy density as seen by a light-like observer is non-
0.045 [111], and in 2018, it reported ω0 = −1.028 ± 0.032 negative. This results in the form ρ + p ≥ 0. When
[11]. In addition, for the constrained values of the model ω < −1, it violates the NEC, which in turn leads to the
parameters, we determine the present value of the EoS violation of the second law of thermodynamics.
parameter as ω0 = −1.07, ω0 = −1.19, and ω0 = −1.12, 2. Weak energy condition (WEC):
respectively [108–110]. The observational data refer- Tµν uµ uν ≥ 0 (44)
enced serve as constraints on the EoS parameter, provid-
ing useful benchmarks. However, these constraints do for any timelike vector uµ . This condition ensures that
not directly imply the presence of quintessence or phan- the energy density measured by any observer is non-
tom DE in our model, but instead highlight how mod- negative. In addition, the NEC must be satisfied. This
ified gravity effects can lead to an evolving EoS similar condition for energy density simplifies to ρ ≥ 0 and
to that of DE models. ρ + p ≥ 0.
 10

3. Dominant energy condition (DEC): 5 Hz

Pantheon+
Tµν uµ uν ≥ 0 T µν uν is a non-spacelike vector
and 4
Joint
(45)

2
3

(ρ-p)/H0
µ
for any timelike vector u . This condition ensures that
energy density is non-negative and energy flux is non-
2
spacelike, meaning energy cannot flow faster than light.
This condition leads to ρ ≥ 0 and ρ ± p ≥ 0.
1
4. Strong energy condition (SEC):
0
  -1 0 1 2 3 4
1
Tµν − Tgµν uµ uν ≥ 0 (46) z
2
FIG. 9: Variation of the DEC as a function of redshift z
for any timelike vector uµ , where T = Tαα is the trace with b = −0.1 and n = 0.087 [75].
of the stress-energy tensor. This condition implies that
gravity is always attractive, which leads to ρ + p ≥ 0
Hz
and ρ + 3p ≥ 0.
1.0 Pantheon+
The above energy conditions indicate that the viola-
Joint
tion of the NEC leads to the violation of the other energy 2
(ρ+3p)/H0
conditions. This signifies a depletion of energy density
0.5
as the universe expands. Furthermore, the violation of
the SEC represents the acceleration of the universe. Figs.
8, 9, and 10 show the evolution of the energy conditions
with respect to redshift. From these figures, we observe 0.0
that ρ + p ≤ 0 and ρ + 3p ≤ 0, indicating the violation
of both the NEC and the SEC at present (z = 0) and
-1 0 1 2 3 4
in the future (z → −1). In addition, the figures show
z
that ρ − p ≥ 0, demonstrating that the DEC is satis-
fied. Therefore, the violation of the NEC and SEC, along FIG. 10: Variation of the SEC as a function of redshift z
with ω < −1, suggests the phantom-like behavior of the with b = −0.1 and n = 0.087 [75].
universe, which is associated with its accelerated expan-
sion.
VII. CONCLUDING REMARKS

3.0 Hz
The accelerating expansion of the universe remains
Pantheon+ one of the most compelling phenomena in modern cos-
2.5
Joint mology. Taking into account anomalies observed in
2.0 0.15 the CMB [63], we have explored the implications of
2
(ρ+p)/H0

0.10
1.5 f ( Q, T ) gravity in a Bianchi type-I spacetime, which is
0.05
0.00
characterized by spatial homogeneity and anisotropy, to
1.0
study cosmic acceleration. Our approach, using a lin-
-1.0 -0.5 0.0 0.5 1.0
0.5 ear combination of the non-metricity Q and the trace of
the energy-momentum tensor T in the form f ( Q, T ) =
0.0
Q + bT, has allowed us to investigate the evolution of
-1 0 1 2 3 4
the universe in a modified gravity framework.
z
Through the parametrization of the deceleration pa-
FIG. 8: Variation of the NEC as a function of redshift z rameter and the derivation of the Hubble solution, we
with b = −0.1 and n = 0.087 [75]. imposed these conditions in the Friedmann equations
for f ( Q, T ) gravity. Employing a Bayesian approach,
we estimated the best-fit values of the model parame-
 11

ters using MCMC sampling, constrained by H (z), Pan- of the EoS parameter is found to be ω is ω0 = −1.07,
theon+ samples, and combined observational data. The ω0 = −1.19, and ω0 = −1.12 for the different datasets,
corresponding 1 − σ and 2 − σ contour plots for the respectively. Lastly, we analyzed the evolution of the en-
H (z) data, Pantheon+ sample, and the combined obser- ergy conditions. We observed the violation of both the
vational data are displayed in Figs. 1, 2, and 3, respec- NEC and the SEC at present and in the future. How-
tively. The obtained best fit values are: H0 = 67.8+ 1.3
−1.3 , ever, the DEC is satisfied. Therefore, the violation of
α = 1.53+ 0.43 +1.5
−0.40 , and β = 2.2−1.5 for H ( z ) data; H0 =
these energy conditions implies phantom-like behavior
72.0−1.4 , α = 1.56−0.13 , and β = 3.55+
+1.4 +0.14 1.0 and the accelerated expansion of the universe. Lym-
−0.90 for Pantheon+
data; H0 = 68.1+ 1.2 +0.059 +0.58 peris [118] explored phantom DE within f ( Q) gravity,
−1.2 , α = 1.508−0.054 , and β = 2.57−0.53
for the combined data. Our results indicate that the showing how modified gravity can drive cosmic accel-
model successfully describes the transition from a de- eration. Our findings, particularly the transition of the
celerated epoch to an accelerated de-Sitter-like phase, EoS into the phantom regime, align with this work, fur-
with the transition redshift zt varying depending on the ther supporting the role of f ( Q, T ) gravity in explaining
dataset. Specifically, the transition redshift is found to late-time acceleration without invoking an explicit DE
be zt = 0.61 for the H (z) data, zt = 0.85 for the Pan- component.
theon+ samples, and zt = 0.72 for the combined data. Future research directions in f ( Q, T ) gravity could
The present value of the deceleration parameter is de- involve exploring more complex functional forms of
termined to be q0 = −0.52, q0 = −0.65, and q0 = −0.57, f ( Q, T ), such as f ( Q, T ) = Qn + bT [73], to account for a
respectively. broader range of cosmological phenomena. In addition,
In addition, our analysis shows that the energy den- addressing standing issues such as the detailed stabil-
sity starts with a high initial value and decreases over ity of solutions, the investigation of perturbation the-
time, approaching zero for the present and future, indi- ory [59], and the connection to quantum gravity could
cating the continuous expansion of the universe. More- deepen our understanding of the model’s predictions.
over, the pressure initially exhibits large negative val- In anisotropic cosmologies, extending this framework
ues, gradually transitioning to smaller negative values, to include interactions between different cosmic compo-
driven by the modified gravity effects rather than a DE nents or exploring the impact of inhomogeneities could
component. One of the key results of our analysis is the provide new insights into the early and late universe’s
dynamic nature of the equation of state (EoS) param- structure.
eter ω, which evolves from quintessence-like behavior
(ω > −1) to the phantom regime (ω < −1), a hallmark
of models that cross the phantom divide. This behav-
ior is driven entirely by the geometric contributions of DATA AVAILABILITY STATEMENT
the modified gravity model, without invoking a sep-
arate DE component. Furthermore, the present value There are no new data associated with this article.

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