#  $f(Q,L_{m})$ gravity, and its cosmological implications

Ayush Hazarika 0009-0004-5255-0730<EMAIL_ADDRESS>Department of
Physics, Tezpur University, Napaam, Tezpur, 784028, Assam, India Simran Arora
0000-0003-0326-8945<EMAIL_ADDRESS>Department of Mathematics, CDOE,
Chandigarh University, Punjab, 140413, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Tiberiu Harko 0000-0002-1990-9172
<EMAIL_ADDRESS>Department of Physics, Babes-Bolyai University, 1
Kogalniceanu Street, 400084 Cluj-Napoca, Romania Astronomical Observatory, 19
Ciresilor Street, 400487, Cluj-Napoca, Romania

###### Abstract

In the present work, we extend the $f(Q)$ symmetric teleparallel gravity by
introducing an arbitrary coupling between the non-metricity $Q$ and matter
Lagrangian $L_{m}$ in the Lagrangian density $f$ of the theory, which thus
leads to the $f\left(Q,L_{m}\right)$ theory. This generalisation encompasses
Coincident General Relativity (CGR), and the Symmetric Teleparallel Equivalent
to GR (STEGR). Using the metric formalism, we derive the field equation of the
theory, which generalizes the field equations of $f(Q)$ gravity. From the
study of the covariant divergence of the field equations, it follows that the
presence of the geometry-matter coupling leads to the non-conservation of the
matter energy-momentum tensor. The cosmological implications of the theory are
investigated in the case of a flat, homogeneous, and isotropic Friedmann-
Lemaitre-Robertson-Walker geometry. As a first step in this direction, we
obtain the modified Friedmann equations for the $f(Q,L_{m})$ gravity in a
general form. Specific cosmological models are investigated for several
choices of $f(Q,L_{m})$, including $f(Q,L_{m})=-\alpha Q+2L_{m}+\beta$, and
$f(Q,L_{m})=-\alpha Q+(2L_{m})^{2}+\beta$, respectively. Comparative analyses
with the standard $\Lambda$ CDM paradigm are carried out, and the
observational implications of the models are investigated in detail.

Keywords: $f(Q,L_{m})$ gravity, cosmology, observational constraints, dark
energy

###### Contents

  1. I Introduction
  2. II Field equations of $f(Q,L_{m})$ gravity
    1. II.1 Geometric Preliminaries
    2. II.2 The variational principle and the field equation
    3. II.3 Energy-momentum tensor balance equations
  3. III Cosmological evolution of FLRW universe in $f(Q,L_{m})$ gravity
    1. III.1 The Friedmann equations
    2. III.2 The de Sitter solution
  4. IV Cosmological models
    1. IV.1 Data and methodology of MCMC analyses
      1. IV.1.1 Data sets
      2. IV.1.2 Statistical analysis
    2. IV.2 Model A: $f=-\alpha\,Q+2L_{m}+\beta$ with $L_{m}=p$
      1. IV.2.1 Statistical analysis of Model A
      2. IV.2.2 Cosmological evolution: Model A
    3. IV.3 Model B: $f=-\alpha\,Q+(2L_{m})^{2}+\beta$ with $L_{m}=p$
      1. IV.3.1 Statistical analysis
      2. IV.3.2 Cosmological evolution: Model B
    4. IV.4 Model comparisons, and correlation results
      1. IV.4.1 Correlation results
  5. V Conclusions
  6. A Derivation of the Friedmann equations

##  I Introduction

The theory of General Relativity (GR) [1, 2, 3] revolutionized our
understanding of gravity by conceptualizing it not as a conventional force but
as an inherent property of spacetime, rooted in Riemannian geometry [4]. Thus,
General Relativity developed Contrary to Einstein’s initial intent of not
“geometrizing gravity” [5]. In GR, the metric and matter interact minimally,
as defined by the Einstein-Hilbert action ($S_{EH}$), given as
$S_{EH}=(1/2\kappa)\int\sqrt{-g}R\,d^{4}x+S_{m}$, where $\kappa$ denotes the
gravitational coupling constant term. $g$ is the determinant of the metric
tensor, and $R$ represents the Ricci scalar. By $S_{m}$, we have denoted the
matter action. This linearity in $R$ gives a second-order field equation
$G_{\mu\nu}=R_{\mu\nu}-(1/2)R\,g_{\mu\nu}=\kappa T_{\mu\nu}$, which governs
the dynamics of matter in the curved spacetime and relates geometry, described
by the Einstein tensor $G_{\mu\nu}$ to the matter energy-momentum tensor
$T_{\mu\nu}$.

A large number of observations have established GR as a very successful theory
of gravity by confirming many of its predictions, such as the deflection of
light by the Sun’s gravitational field [6], the perihelion motion of Mercury
[7], the existence of gravitational waves [8], gravitational redshift [9],
orbital decay of the Hulse-Taylor binary pulsar [10], and the radar echo delay
[11, 12], respectively. An in-depth analysis of all the experimental and
observational tests of GR can be found in [13, 14].

Despite its remarkable success, which spanned almost one hundred years, the
theory of General Relativity currently faces numerous challenges. At a quantum
level, it cannot explain the quantum properties of the gravitational
interaction. Also, gravitational collapse can result in geodesic
incompleteness under specific assumptions regarding the energy-momentum
tensor. This implies that certain types of geodesics are constrained by an
upper limit on an affine parameter, indicating a singular structure in
spacetime. One notable consequence of this phenomenon is the appearance of
cosmological singularities during the Big Bang and the existence of the black
holes.

A significant challenge for GR did appear when it was faced with the problem
of explaining the late-time cosmic accelerated expansion. Evidence from
observations of type Ia supernovae [15, 16, 17], large-scale structure
observations, and measurements of the cosmic microwave background (CMB)
anisotropies from the Wilkinson Microwave Anisotropy Probe (WMAP) [18], and of
the Planck satellite [Planck] highlighted a limitation in GR’s ability to
fully describe and comprehend the dynamics of the Universe at cosmic scales
during its later stages. This failure of GR prompted the exploration of
alternative theories of gravity and the consideration of additional factors or
components in the gravitational field equations, such as dark energy or the
addition of a cosmological constant ($\Lambda$) in action ($S_{EH}$), to
reconcile observations with theoretical predictions [4]. Hence, a more general
gravitational framework is required to explain the gravitational dynamics
across various scales, ranging from the Solar System to galaxies and the large
scale Universe.

In pursuit of a more comprehensive understanding of gravity that aligns with
the observational evidence, a plethora of modified gravity theories have been
proposed, such as Scalar-tensor theories [19, 20, 21, 22, 23, 24], Tensor-
Vector-Scalar (TeVeS) [25], Dvali-Gabadadze-Porrati (DGP) gravity [26],
Einstein-Gauss-Bonnet gravity [27], brane-world gravity [28], Einstein-Aether
theory [29], Eddington-inspired Born-Infeld (EiBI) gravity [30, 31] etc.

A class of gravitational theories, known as $f(R)$ gravity, arises through a
straightforward extension of the Einsetin-Hilbert action $S_{EH}$ by replacing
$R$ with an arbitrary functions of the Ricci scalar $R$ [32]. The geometrical
structures of the $f(R)$ gravity were able to explain the accelerated cosmic
expansion [33], and also the flat rotation curves of galaxies, without
introducing dark matter [34]. Even though it failed when subjected to Solar-
System tests [35, 36, 37], $f(R)$ gravity could still be a valuable approach
to the foundational framework for a “parameterized post-Friedmann” description
of linear phenomena and could draw parallels with the parameterized post-
Newtonian framework for small-scale tests of gravity.

An alternative method for extending the Einstein-Hilbert action involves
postulating the presence of a non-minimal coupling between geometry and
matter, and it leads to the $f(R,L_{m})$ gravity [38]. For the various
astrophysical and cosmological implications of this theory see [39, 40, 41,
42, 43, 44, 45, 46, 47, 48]. Another similar approach is based on the
inclusion of a non-minimal coupling between geometry, described by the Ricci
scalar $R$, and the trace of the energy-momentum tensor $T$, giving rise to
$f(R,T)$ gravity [49]. A more comprehensive exploration of this theory is
available in the detailed investigations presented in [50, 51, 52, 53, 54, 55,
56, 57, 58, 59]. In all these extended theories, the gravitational dynamics is
described by more general functions of the curvature scalar, matter
Lagrangian, and the trace of momentum-energy tensor, respectively, which
allows for obtaining a broader range of gravitational behaviours going beyond
the predictions of GR. For a detailed review of modified gravity and its
implications see [60, 61, 62, 63, 32, 64, 65, 66, 67, 68, 69, 70].

GR is based solely on the metric and on the Riemannian curvature tensor to
define gravity. However, within the broader context of metric-affine geometry,
gravity is not limited to curvature alone; it can also be mediated by two
additional geometric quantities, torsion and non-metricity, respectively.

In the context of the Riemannian geometry, the torsion tensor faces a severe
limitation. Specifically, due to the symmetry of the Christoffel symbols, the
torsion tensor is restricted to zero, that is, $T^{\mu}_{\rho\lambda}=0$. In
an interesting extension of Riemann geometry, in the Weitzenböck space [71],
the torsion tensor is non-zero ($T^{\mu}_{\rho\lambda}\neq 0$), and the
Riemann curvature tensor is zero, leading to a spacetime characterized by flat
geometry, endowed with a significant property known as absolute parallelism,
or teleparallelism. The applications of Weitzenböck-type spacetimes in physics
were pioneered by Einstein to introduce a unified teleparallel theory,
unifying electromagnetism and gravity [72]. In the teleparallel approach, the
fundamental characteristic is the replacement of the metric $g_{\mu\nu}$,
which serves as the primary physical variable that describes gravitational
properties, with a set of tetrad vectors $e^{i}_{\mu}$. Torsion, originating
from the tetrad fields, can be employed as a comprehensive descriptor of the
gravitational effects, thus replacing curvature with torsion. This leads to
the theory known as the teleparallel equivalent of general relativity (TEGR)
[73, 74, 75], which was extended to the $f(T)$ gravity theory.

In teleparallel or $f(T)$ type gravity theories, torsion exactly compensates
curvature, resulting in a flat spacetime. A notable advantage of $f(T)$
gravity theory lies in its second-order field equations, which differentiates
it from the $f(R)$ gravity, which, within the metric approach, is described by
fourth-order field equations. [76]. The applications of $f(T)$ gravity
theories have been extensively explored in the study of astrophysical
processes, and in cosmology. Significantly, these theories are extensively
used to provide an alternative explanation for large-scale structure, the
late-time accelerating expansion of the Universe, thus eliminating the need to
introduce dark energy [77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,
91, 92, 93, 94, 95, 96, 97, 98].

The third geometric formulation of gravitational theories is based on the non-
metricity $Q$ of the metric [99]. Geometrically, this quantity captures the
variation in the length of a vector during parallel transport. Moreover, it
offers the advantage of covariantizing conventional coordinate calculations in
general relativity. In the framework of symmetric teleparallel gravity, the
associated energy-momentum density is fundamentally the Einstein pseudotensor,
transformed into a true tensor. In the context of gravitational actions
containing non-metricity, the action
$S_{STEGR}=(-1/2k)\int\sqrt{-g}Qd^{4}x+S_{m}$, which substitutes the curvature
scalar with the non-metricity, is at the basic of a theory called the
Symmetric Teleparallel Equivalent of General Relativity (STEGR) [100]. The
extension of the symmetric teleparallel gravity led to the formulation of the
$f(Q)$ gravity theory, also known as Coincident General Relativity [101] or
nonmetric gravity. In this theory, the connection is flat and torsionless.
These conditions lead to a connection that is purely inertial, differing from
the Levi-Civita connection through a general linear gauge transformation.
Furthermore, the torsionless condition simplifies the connection to
$Y^{\alpha}_{\;\;\mu\beta}=(\partial
x^{\alpha}/\partial\xi^{\lambda})\partial_{\mu}\partial_{\beta}\xi^{\lambda}$
for some arbitrary $\xi^{\lambda}$. This crucial outcome indicates that the
connection can be entirely removed through a diffeomorphism. Consequently, the
$\xi^{\lambda}$ fields emerge as Stückelberg fields, restoring this gauge
symmetry [101].

In exploring extensions of symmetric teleparallel gravity, recent studies have
considered the characteristics of gravitational wave propagation. An analysis
particularly of the speed and polarization of gravitational waves [102] has
remarkably extended the results obtained in general relativity, unveiling
consistent speeds and polarizations.

In another line of research, in [103], a derivation of the exact propagator
for the most general infinite-derivative, even-parity, and generally covariant
theory within symmetric teleparallel spacetimes was presented. This approach
involves decomposing the action, containing the non-metricity tensor and its
contractions, into terms involving the metric and a gauge vector field.

Further insights emerged from the study of new general relativistic type
solutions in symmetric teleparallel gravity theories [104]. The investigation
of the gravitational wave propagation in a Minkowski spacetime revealed that
all gravitational waves propagate with the speed of light. The Noether
symmetry approach played a key role in classifying possible quadratic, first-
order derivative terms of the non-metricity tensor in the framework of
symmetric teleparallel geometry [105]. The cosmology of the $f(Q)$ theory and
its observational constraints were considered in [106] and [107], where it was
shown that the accelerating expansion is an intrinsic property of the
Universe’s geometry, thus eliminating the need for exotic dark energy or
additional fields. and also used a dynamical system approach. For more work,
check the Refs. [108, 109, 110, 111]

Investigation of cosmological perturbations in $f(Q)$ gravity [112] revealed
intriguing findings, such as the re-scaling of the Newton constant in tensor
perturbations and the absence of vector contributions without vector sources
being present. Notably, the scalar sector introduced two additional
propagating modes, suggesting that $f(Q)$ theories add at least two extra
degrees of freedom. Moreover, extending non-metric gravity by incorporating
the trace of the matter-energy-momentum tensor $T$ into a general function
$f(Q,T)$ has been investigated in [113, 114]. These $f(Q,T)$ gravity models
have been observationally constrained as noted in [115], and some models have
successfully described the accelerated expansion of the Universe [116].
Additionally, a spherically symmetric stellar system in $f(Q,T)$ gravity has
been shown to satisfy all the physical conditions [117]. For more works in
$f(Q,T)$ gravity, see Refs. [118, 119, 120, 121, 122, 123, 124]. Over the past
two decades, numerous studies have been devoted to the geometrical and
physical aspects of symmetric teleparallel gravity, with a surge in interest
in recent years [125, 126, 127, 128, 129, 130, 131, 132, 102, 103, 133, 134,
135, 104, 105, 106, 107, 136, 112, 137].

Riemannian geometry represents a specific case within the broader framework of
metric-affine geometry, offering a restricted perspective on gravitational
dynamics. However, there exist no definitive physical principles that
exclusively favor Riemannian geometry as the sole representation of gravity.
Instead, metric-affine geometry presents three distinct yet physically
equivalent avenues for describing gravitational phenomena. These approaches
attribute the gravitational effects to the presence of non-zero curvature,
non-zero torsion, or non-zero non-metricity within a given geometric
framework. Together, these descriptions constitute the geometric trinity of
General Relativity [138, 137]. It is essential to investigate all three
approaches equally to gain a comprehensive understanding of gravity.

The coupling between the gravitational field and matter fields defines the
dynamics in spacetime. In GR, the minimal coupling principle dictates that
matter theories formulated in flat Minkowski space are seamlessly extended to
incorporate gravitational interactions by replacing the flat metric and
partial derivatives with the curved metric and covariant derivatives. This
principle holds as long as the matter fields are coupled solely to the metric
and its determinant without involving derivatives of the metric. In
teleparallel gravity, for the electromagnetic potential, the presence of
torsion introduces additional terms in the Maxwell action, violating the
expected equivalence with GR [139]. Similarly, fermionic fields are affected
by torsion, further challenging the minimal coupling principle. However, in
symmetric teleparallel gravity, the scenario shifts. The minimal coupling
principle remains intact even in the presence of non-metricity [139]. For
electromagnetic fields and fermions alike, non-metricity does not interfere
with the standard coupling prescriptions, ensuring compatibility with GR. In
essence, while the symmetric teleparallel theory maintains equivalence with GR
in the presence of matter fields, teleparallel theories diverge from this
equivalence, underscoring the nuanced interplay between gravity and matter
within these distinct frameworks.

The flat $\Lambda$CDM model generally aligns well with observations, but
recent data indicate possible discrepancies. These include variations in the
measured values of the Hubble constant $H_{0}$, and the amplitude of matter
fluctuations $\sigma_{8}$, when different methods are used. Additionally, some
anomalies arise when comparing the model’s theoretical predictions, based on
the best-fit cosmological parameters, with actual observations. These
potential inconsistencies encourage the investigation of extensions of the
$\Lambda$CDM model. The well-known discrepancy between $H_{0}$, measured by
the SH0ES collaboration using local distance ladder measurements from Type Ia
supernovae $(H_{0}=73\pm 1\,{\rm km/s/Mpc})$ [140, 141], and the value
inferred by the Planck collaboration from observations of temperature and
polarization anisotropies in the Cosmic Microwave Background (CMB) radiation,
assuming a $\Lambda$CDM cosmology $(H_{0}=67.4\pm 0.5\,{\rm km/s/Mpc})$ [142],
has reached a statistical significance exceeding $5\sigma$. Unless this
discrepancy is due to systematic errors, an intriguing possibility is that the
Hubble tension could indicate new physics beyond the standard $\Lambda$CDM
model of cosmology.

The main objective of our study is to generalize symmetric teleparallel
gravity by incorporating the matter Lagrangian in the Lagrangian density of
the $f(Q)$ theory, thus obtaining the $f(Q,L_{m})$ theory. The corresponding
action could describe minimal and non-minimal couplings between geometry and
matter. After introducing the basic action of the model, we obtain the general
system of field equations by varying the action with respect to the metric. We
also investigate the conservation problem of the matter energy-momentum tensor
and show that, in the present theory, it is not conserved. Furthermore, we
investigate the cosmic evolution for the case of a flat Friedmann-Lemaitre-
Robertson-Walker metric by first obtaining the generalized Friedmann
equations. Furthermore, we consider two particular gravitational models
corresponding to two distinct forms of the function $f\left(Q,L_{m}\right)$.
The predictions of the two gravitational models are compared with two distinct
observational datasets. Our results unravel the intricate dynamics of the
Universe within this extended gravitational framework.

The paper is structured as follows. Section II.1 provides the geometrical
background for building the general theory, which is followed by the
derivation of the field equation using the metric formalism (Section II.2).
Furthermore, in Section II.3, we obtain the energy-momentum tensor balance
equation. In Section III, the cosmic evolution of the Universe governed by
$f(Q,L_{m})$ gravity is investigated. Two specific expressions for the
arbitrary function $f\left(Q,L_{m}\right)$ are considered in Section IV.2 and
Section IV.3, respectively. The model parameters are constrained using MCMC in
Section IV.4. A concise summary of the work is presented in Section V. The
detailed calculations for obtaining the Friedmann equations are provided in
Appendix V.

##  II Field equations of $f(Q,L_{m})$ gravity

This section first provides a concise overview of the geometric foundations
underlying gravitational theories, which are based on the existence of a
general line element in spacetime. Then we introduce the action of the
$f(Q,L_{m})$ gravitational theory, and, with the help of the variational
principle, we obtain the corresponding gravitational field equation, offering
new insights into the understanding of gravitational phenomena within this
geometric framework. Furthermore, we explore the non-conservation of the
matter-energy momentum tensor, highlighting the impact of the coupling between
matter Lagrangian and geometry.

###  II.1 Geometric Preliminaries

Once the definition of a metric is provided, the geometric interpretation of
gravity is given by the Riemann tensor

$R^{\alpha}_{\;\;\beta\mu\nu}=\partial_{\mu}Y^{\alpha}_{\;\;\nu\beta}-\partial_{\nu}Y^{\alpha}_{\;\;\mu\beta}+Y^{\alpha}_{\;\;\mu\lambda}Y^{\lambda}_{\;\;\nu\beta}-Y^{\alpha}_{\;\;\nu\lambda}Y^{\lambda}_{\;\;\mu\beta},$
(1)

and of its contractions. The Riemann tensor is constructed with the help of an
affine connection. The general form of the affine connection
$Y^{\alpha}_{\;\;\mu\nu}$ consists of three parts: a symmetric part known as
the Levi-Civita connection $\Gamma^{\alpha}_{\;\;\mu\nu}$, a contortion tensor
$K^{\alpha}_{\;\;\mu\nu}$ describing the anti-symmetric part, and the
disformation tensor $L^{\alpha}_{\;\;\mu\nu}$, accounting for the presence of
non-metricity,

$Y^{\alpha}_{\;\;\mu\nu}=\Gamma^{\alpha}_{\;\;\mu\nu}+K^{\alpha}_{\;\;\mu\nu}+L^{\alpha}_{\;\;\mu\nu}.$
(2)

The torsion-free Levi-Civita connection $\Gamma^{\alpha}_{\;\;\mu\nu}$ is
equivalent to the 2nd order Christoffel symbol in terms of the metric; it
preserves the inner product of the various tangent vectors when a vector is
parallelly transported, and it is defined according to

$\Gamma^{\alpha}_{\;\;\mu\nu}=\frac{1}{2}g^{\alpha\lambda}(\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu}).$
(3)

The contortion tensor $K^{\alpha}_{\;\;\mu\nu}$ is represented in terms of
torsion tensor $T^{\alpha}_{\;\;\mu\nu}$ as

$K^{\alpha}_{\;\;\mu\nu}=\frac{1}{2}(T^{\alpha}_{\;\;\mu\nu}+T_{\mu\;\;\nu}^{\;\;\alpha}+T_{\nu\;\;\mu}^{\;\;\alpha}).$
(4)

The torsion tensor characterizes the deviation of a connection from symmetry,
which indicates that parallel transport around a closed loop does not
necessarily bring a vector back to its original position.

The disformation tensor $L^{\alpha}_{\;\;\mu\nu}$ describes the overall
expansion or contraction of spacetime. When a vector is parallelly
transported, its magnitude changes along its path. The variation of the length
is measured by the non-metricity tensor,

$L^{\alpha}_{\;\;\mu\nu}=\frac{1}{2}(Q^{\alpha}_{\;\;\mu\nu}-Q^{\;\;\alpha}_{\mu\;\;\nu}-Q^{\;\;\alpha}_{\nu\;\;\mu}).$
(5)

The non-metricity tensor $Q_{\alpha\mu\nu}$ is defined according to

$Q_{\alpha\mu\nu}=\nabla_{\alpha}g_{\mu\nu}=\partial_{\alpha}g_{\mu\nu}-Y^{\beta}_{\;\;\alpha\mu}g_{\beta\nu}-Y^{\beta}_{\;\;\alpha\nu}g_{\mu\beta}.$
(6)

To construct a boundary term in the action of the metric-affine gravity
theories, we need a non-metricity conjugate, known as the superpotential
$P^{\alpha}_{\;\;\mu\nu}$, defined as [113]

$P^{\alpha}_{\;\;\mu\nu}=-\frac{1}{2}L^{\alpha}_{\;\;\mu\nu}+\frac{1}{4}(Q^{\alpha}-\tilde{Q}^{\alpha})g_{\mu\nu}-\frac{1}{4}\delta^{\alpha}_{\;\;(\mu}Q_{\nu)}.$
(7)

Here, $Q^{\alpha}=Q^{\alpha\;\;\mu}_{\;\;\mu}$ and
$\tilde{Q}^{\alpha}=Q_{\mu}^{\;\;\alpha\mu}$ are the non-metricity vectors.
The non-metricity scalar can be obtained by contracting the superpotential
tensor with the non-metricity tensor,

$Q=-Q_{\lambda\mu\nu}P^{\lambda\mu\nu}.$ (8)

The non-metricity scalar $Q$ describes the deviation of the manifold geometry
from isotropy and can be thought of as a measure of how much the volume of a
parallelly transported object changes as it moves through spacetime.

###  II.2 The variational principle and the field equation

The dynamics of a physical system are studied by using the action principle.
The action for the $f(Q,L_{m})$ modified gravity takes the following form

$S=\int f(Q,L_{m})\sqrt{-g}d^{4}x,$ (9)

where $\sqrt{-g}$ is the determinant of the metric, and $f(Q,L_{m})$ is an
arbitrary function of non-metricity scalar $Q$ and of the matter Lagrangian
$L_{m}$.
By varying the action with respect to the metric tensor, we obtain the
gravitational field equation, which describes how spacetime geometry responds
to the presence of matter and energy. Hence, we first obtain

$\delta S=\int\bigr{[}(f_{Q}\delta Q+f_{L_{m}}\delta
L_{m})\sqrt{-g}+f\delta\sqrt{-g}\bigl{]}d^{4}x.$ (10)

Here, $f_{Q}=\partial f(Q,L_{m})/\partial Q$ and $f_{L_{m}}=\partial
f(Q,L_{m})/\partial L_{m}$.
The variation of $Q$ is given by [113]

$\delta Q=2P_{\alpha\nu\rho}\nabla^{\alpha}\delta
g^{\nu\rho}-(P_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q^{\alpha\beta}_{\;\;\;\mu}P_{\alpha\beta\nu})\delta
g^{\mu\nu}.$ (11)

The energy-momentum tensor $T_{\mu\nu}$ of the matter is defined as [9]

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}=g_{\mu\nu}L_{m}-2\frac{\partial L_{m}}{\partial g^{\mu\nu}},$ (12)

The variation of the determinant of the metric tensor is

$\delta\sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}.$ (13)

From Eqs. (11), (12), and (13) it follows that Eq. (10) can be written as

$\displaystyle\delta
S=\int\bigr{[}\bigr{(}f_{Q}(2P_{\alpha\nu\rho}\nabla^{\alpha}\delta
g^{\nu\rho}-(P_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q^{\alpha\beta}_{\;\;\;\mu}P_{\alpha\beta\nu})\delta
g^{\mu\nu})$ (14)
$\displaystyle+\frac{1}{2}f_{L_{m}}(g_{\mu\nu}L_{m}-T_{\mu\nu})\delta
g^{\mu\nu}\bigl{)}-\frac{1}{2}fg_{\mu\nu}\delta
g^{\mu\nu}\bigl{]}\sqrt{-g}d^{4}x.$

After applying the boundary conditions and integrating the first term in Eq.
(14) becomes $-2\nabla^{\alpha}(f_{Q}\sqrt{-g}P_{\alpha\mu\nu})\delta
g^{\mu\nu}$. Equating the metric variation of the action to zero, we obtain
the field equation of the $f(Q,L_{m})$ gravity

$\frac{2}{\sqrt{-g}}\nabla_{\alpha}(f_{Q}\sqrt{-g}P^{\alpha}_{\;\;\mu\nu})+f_{Q}(P_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q^{\alpha\beta}_{\;\;\;\mu}P_{\alpha\beta\nu})\\\
+\frac{1}{2}fg_{\mu\nu}=\frac{1}{2}f_{L_{m}}(g_{\mu\nu}L_{m}-T_{\mu\nu}).$
(15)

For $f(Q,L_{m})=f(Q)+2\,L_{m}$, it reduces to the field equation of $f(Q)$
gravity (as seen in [143])

$\frac{2}{\sqrt{-g}}\nabla_{\alpha}(\sqrt{-g}f_{Q}\,P^{\alpha}_{\;\;\mu\nu})+f_{Q}\,q_{\mu\nu}+\frac{1}{2}f(Q)g_{\mu\nu}=-T_{\mu\nu},$
(16)

where
$q_{\mu\nu}=P_{\mu\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}-2Q^{\alpha\beta}_{\;\;\;\mu}P_{\alpha\beta\nu}$.
Furthermore, the field equation (15) can also be reduced to the STEGR.
In the mixed tensor representation, the field equation (15) is given by

$\displaystyle\frac{2}{\sqrt{-g}}\nabla_{\alpha}(f_{Q}\sqrt{-g}P^{\alpha\mu}_{\;\;\;\;\nu})+f_{Q}\,P^{\mu}_{\;\;\alpha\beta}Q_{\nu}^{\;\;\alpha\beta}+\frac{1}{2}\delta^{\mu}_{\;\;\nu}\,f$
(17)
$\displaystyle=\frac{1}{2}f_{L_{m}}(\delta^{\mu}_{\;\;\nu}L_{m}-T^{\mu}_{\;\;\nu}).$

Using the Lagrange multiplier method with constraints
$T^{\alpha}_{\;\;\beta\gamma}=0$ and $R^{\alpha}_{\;\;\beta\mu\nu}=0$, the
action (9) reads as

$\displaystyle
S=\int\bigr{[}f(Q,L_{m})\sqrt{-g}+\lambda_{\alpha}^{\;\;\beta\gamma}\,T^{\alpha}_{\;\;\beta\gamma}+\xi_{\alpha}^{\;\;\beta\mu\nu}\,R^{\alpha}_{\;\;\beta\mu\nu}\bigl{]}d^{4}x.$
(18)

The variation of the Lagrange multipliers is given as

$\delta(\lambda_{\alpha}^{\;\;\beta\gamma}\,T^{\alpha}_{\;\;\beta\gamma})=2\,\lambda_{\alpha}^{\;\;\beta\gamma}\,\delta
Y^{\alpha}_{\;\;\beta\gamma},$ (19)
$\displaystyle\delta(\xi_{\alpha}^{\;\;\beta\mu\nu}\,R^{\alpha}_{\;\;\beta\mu\nu})$
$\displaystyle=\xi_{\alpha}^{\;\;\beta\mu\nu}\big{[}\nabla_{\mu}(\delta\,Y^{\alpha}_{\;\;\nu\beta})-\nabla_{\nu}(\delta\,Y^{\alpha}_{\;\;\mu\beta})\big{]}$
(20) $\displaystyle=2\,\xi_{\alpha}^{\;\;\nu\beta\mu}\nabla_{\beta}(\delta
Y^{\alpha}_{\;\;\mu\nu})\simeq
2(\nabla_{\beta}\xi_{\alpha}^{\;\;\nu\beta\mu})\delta
Y^{\alpha}_{\;\;\mu\nu}.$ (21)

Varying now the action (18) with respect to the connection gives

$\delta\,S=\scalebox{1.7}{$\displaystyle\int$}\biggl{(}4\sqrt{-g}\,f_{Q}\,P^{\mu\nu}_{\;\;\;\;\alpha}+H_{\alpha}^{\;\;\mu\nu}+2\,\nabla_{\beta}\xi_{\alpha}^{\;\;\nu\beta\mu}\\\
+2\,\lambda_{\alpha}^{\;\;\mu\nu}\biggr{)}d^{4}x\,\delta
Y^{\alpha}_{\;\;\mu\nu}.$ (22)

Here $H_{\alpha}^{\;\;\mu\nu}$ is the hypermomentum density defined as

$H_{\alpha}^{\;\;\mu\nu}=\sqrt{-g}f_{L_{m}}\frac{\delta L_{m}}{\delta
Y^{\alpha}_{\;\;\mu\nu}}.$ (23)

In the action variation, we introduce two covariant derivatives
$\nabla_{\mu}\nabla_{\nu}$ to eliminate the Lagrange multiplier coefficients
with the anti-symmetry property of $\mu$ and $\nu$. Then, the field equation
becomes

$\nabla_{\mu}\nabla_{\nu}\Bigl{(}4\sqrt{-g}\,f_{Q}\,P^{\mu\nu}_{\;\;\;\;\alpha}+H_{\alpha}^{\;\;\mu\nu}\Bigl{)}=0.$
(24)

###  II.3 Energy-momentum tensor balance equations

The covariant derivative of $\omega^{\mu}_{\;\;\nu}$ is given by

$\nabla_{\mu}\,\omega^{\mu}_{\;\;\nu}=D_{\mu}\,\omega^{\mu}_{\;\;\nu}-\frac{1}{2}Q_{\rho}\,\omega^{\rho}_{\;\;\nu}-L^{\lambda}_{\;\;\mu\nu}\,\omega^{\mu}_{\;\;\lambda},$
(25)

where $D_{\mu}$ is the covariant derivative with respect to the Levi-Civita
connection. The covariant derivative of the field equation (17) is

$D_{\mu}\Bigr{[}\frac{1}{2}f_{L_{m}}(\delta^{\mu}_{\nu}\,L_{m}-T^{\mu}_{\;\;\nu})\Bigl{]}=\frac{1}{2}\partial_{\nu}f+D_{\mu}(f_{Q}P^{\mu}_{\;\;\alpha\beta}Q_{\nu}^{\;\;\alpha\beta})\\\
+D_{\mu}\Bigr{[}\frac{2}{\sqrt{-g}}\nabla_{\alpha}(f_{Q}\sqrt{-g}P^{\alpha\mu}_{\;\;\;\;\nu})\Bigr{]},$
(26)

From Eq. (25) $D_{\mu}$ can be expressed as
$D_{\mu}=\nabla_{\mu}+\frac{1}{2}Q_{\mu}+L^{\rho}_{\;\;\mu\nu}$. Thus, Eq.
(26) becomes

$\frac{1}{2\sqrt{-g}}\nabla_{\alpha}\nabla_{\mu}H_{\nu}^{\;\;\alpha\mu}-\frac{1}{2}f_{L_{m}}D_{\mu}\,T^{\mu}_{\;\;\nu}=\frac{1}{2}f_{Q}\,\partial_{\nu}Q\
+\\\ \nabla_{\mu}(f_{Q}P^{\mu}_{\;\;\alpha\beta}Q_{\nu}^{\;\;\alpha\beta})\
+\frac{1}{2}Q_{\mu}(f_{Q}P^{\mu}_{\;\;\alpha\beta}Q_{\nu}^{\;\;\alpha\beta})\
+\\\
L^{\rho}_{\;\;\mu\nu}(f_{Q}P^{\mu}_{\;\;\alpha\beta}Q_{\rho}^{\;\;\alpha\beta})+\frac{2}{\sqrt{-g}}L^{\rho}_{\;\;\mu\nu}\nabla_{\alpha}(\sqrt{-g}f_{Q}\,P^{\alpha\mu}_{\;\;\;\;\rho})\\\
+\frac{1}{\sqrt{-g}}Q_{\mu}\nabla_{\alpha}(\sqrt{-g}f_{Q}\,P^{\alpha\mu}_{\;\;\;\;\nu}).$
(27)

The detailed calculations are shown in [113], and they lead to

$\displaystyle
D_{\mu}\,T^{\mu}_{\;\;\nu}=\frac{1}{f_{L_{m}}\sqrt{-g}}\Bigl{[}\nabla_{\alpha}\nabla_{\mu}H_{\nu}^{\;\;\alpha\mu}-2\,Q_{\mu}\,\nabla_{\alpha}(f_{Q}\sqrt{-g}P^{\alpha\mu}_{\;\;\;\;\nu})\Bigr{]}.$

To simplify the above equation, we introduce the tensor $A^{\mu}_{\;\;\alpha}$
and define Eq. (24) such that

$\nabla_{\mu}\Bigl{(}4\sqrt{-g}\,f_{Q}\,P^{\mu\nu}_{\;\;\;\;\alpha}+H_{\alpha}^{\;\;\mu\nu}\Bigl{)}=\sqrt{-g}A^{\nu}_{\;\;\alpha}.$
(29)

Then the covariant derivative of the RHS of Eq. (29) is

$\nabla_{\nu}(\sqrt{-g}A^{\nu}_{\;\;\alpha})=\sqrt{-g}\nabla_{\nu}\,A^{\nu}_{\;\;\alpha}+\frac{\sqrt{-g}}{2}Q_{\nu}\,A^{\nu}_{\;\;\alpha}=0.$
(30)

Eq. (II.3) simplifies by the combination of Eqs. (29) and (30) as

$D_{\mu}\,T^{\mu}_{\;\;\nu}=\frac{1}{f_{L_{m}}}\Bigl{[}\frac{2}{\sqrt{-g}}\nabla_{\alpha}\nabla_{\mu}H_{\nu}^{\;\;\alpha\mu}+\nabla_{\mu}\,A^{\mu}_{\;\;\nu}\\\
-\nabla_{\mu}\bigr{(}\frac{1}{\sqrt{-g}}\nabla_{\alpha}H_{\nu}^{\;\;\alpha\mu}\bigr{)}\Bigr{]}=B_{\nu}\neq
0.$ (31)

From Eq. (31), it follows that the matter energy-momentum tensor is not
conserved in the $f(Q,L_{m})$ gravity theory. The non-conservation tensor
$B_{\nu}$ is a function of dynamical variables like $Q$, $L_{m}$, and the
thermodynamic parameters.

In a broader context, dissipative processes pose significant challenges when
reconciling cosmic microwave background radiation (CMBR) and large-scale
structure (LSS). In [54], the cosmological and solar system consequences of a
class of models with geometry-matter coupling were investigated. The findings
of this work suggest that these models often exhibit inconsistent behaviour as
compared to observational data. This inconsistency may potentially manifest
and amplify when extended to cosmological scales at both galactic and
extragalactic levels. In particular, incompatibility with CMBR or LSS appears
to be a model-dependent phenomenon. However, the study in [54] reveals that
some or all of these inconsistencies can be mitigated through meticulous fine-
tuning of model parameters.

At larger scales, specifically galactic and extra-galactic levels, the non-
minimal matter coupling with geometry introduces intriguing implications. The
observed flattening of galaxy rotation curves considered a dynamically
generated effect, is attributed to the non-minimal coupling [144, 145]. The
non-conservation of the energy-momentum tensor leads to a deviation from
geodesic motion, explaining the observed deviation between measured rotation
velocity and classical predictions. Moreover, a specific type of non-minimal
matter coupling with geometry is shown to mimic the presence of dark matter in
galaxy clusters. In [146], they explore this phenomenon in the context of the
Abell cluster A586, demonstrating its potential extension to a larger sample
of galaxy clusters. Adding to the complexity of the physical behaviour,
dissipative processes play a distinctive role in the evolution of radio
galaxies, as discussed in [147].

If we consider matter as a perfect fluid described by its pressure $p$ and
energy density $\rho$, the energy-momentum tensor can be defined as

$T^{\mu}_{\;\;\nu}=(\rho+p)u_{\nu}\,u^{\mu}+p\,\delta^{\mu}_{\nu},$ (32)

where $u^{\mu}$ denotes the four-velocity of the fluid. Following [135] we
have

$\dot{\rho}+3\,H(\rho+p)=B_{\mu}\,u^{\mu}.$ (33)

The continuity equation presented above deviates noticeably from the standard
form, incorporating additional terms on the right-hand side (RHS) that account
for the deviations from the geodesic motion. In this context, the source term,
denoted by $B_{\mu}\,u^{\mu}$, is associated with the generation or
dissipation of energy. When $B_{\mu}\,u^{\mu}=0$, the system obeys the energy
conservation law of standard gravity. On the contrary, if $B_{\mu}\,u^{\mu}$
takes nonzero values, the energy transfer processes become dominant.

The momentum conservation equation, which describes the movement of massive
particles [113, 135], is expressed as

$\frac{d^{2}x^{\mu}}{ds^{2}}+\Gamma^{\mu}_{\alpha\beta}u^{\alpha}u^{\beta}=\frac{h^{\mu\nu}}{\rho+p}(B_{\nu}-D_{\nu}\,p)=F^{\mu},$
(34)

where $h^{\mu\nu}$ represents the projection operator, defined as
$h^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$. The equation of motion exhibits a
notable departure from the geodesic motion of the massive particles. An
additional force, $F^{\mu}$, emerges as a consequence of the coupling between
$Q$ and $L_{m}$. This coupling introduces a non-gravitational influence,
leading to deviations from the trajectories determined by the standard
geodesic motion of general relativity thus influencing the dynamical evolution
of massive particles.

##  III Cosmological evolution of FLRW universe in $f(Q,L_{m})$ gravity

In the present Section, we will investigate, in a general framework, the
cosmological implications of the $f(Q,L_{m})$ gravity theory. By considering a
flat Friedmann-Lemaitre-Robertson-Walker geometry, the generalized Friedmann
equations are derived. The cosmological evolution equations do contain some
extra-terms, coming from the presence of the nonmetricity and geometry matter
coupling, which generate an effective density of pressure, which can be
interpreted as representing geometric dark energy. The general form of the
energy balance equation is also obtained. The de Sitter limiting behaviour of
the cosmological models is also investigated.

###  III.1 The Friedmann equations

To study the cosmological evolution in $f(Q,L_{m})$ gravity, we assume that
the Universe is described by a flat Friedmann-Lemaître-Robertson-Walker (FLRW)
geometry, with the spacetime interval of the form

$ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (35)

where $a(t)$ is the scale factor111Here, we assume the Lapse function as
$N(t)=1$.. We define the rate of expansion of the universe as
$H=\frac{\dot{a}}{a}$. Let us also assume that the Universe is filled with a
perfect fluid. We adopt the expressions $L_{m}=-\rho$, or $L_{m}=p$ for the
Lagrangian density of the cosmic matter. Hence, in the comoving frame, the
non-zero components of the energy-momentum tensor are given by
$T^{\mu}_{\nu}=(-\rho,p,p,p)$.

Using the FLRW metric, the field equations Eq. (15) give the two generalized
Friedmann equations (the detailed calculations are presented in the Appendix
A)

$\displaystyle
3H^{2}=\frac{1}{4f_{Q}}\bigr{[}f-f_{L_{m}}(\rho+L_{m})\bigl{]},$ (36)
$\displaystyle\dot{H}+3H^{2}+\frac{\dot{f_{Q}}}{f_{Q}}H=\frac{1}{4f_{Q}}\bigr{[}f+f_{L_{m}}(p-L_{m})\bigl{]}.$
(37)

For $f(Q,L_{m})=f(Q)+2\,L_{m}$ the Friedmann equations reduces to $f(Q)$ [148,
149], it can further be simplified to STEGR. By subtracting Eqs. (36) and (37)
we obtain

$\frac{d}{dt}(f_{Q}\,H)=\frac{f_{L_{m}}}{4}(p+\rho).$ (38)

The generalized expression of the deceleration parameter is obtained as

$\displaystyle q$ $\displaystyle=-1-\frac{\dot{H}}{H^{2}}$
$\displaystyle=\frac{1}{4f_{Q}H^{2}}\left(2Qf_{Q}+4\dot{f_{Q}}H-f-
f_{L_{m}}(p-L_{m})\right)-1.$ (39)

With the help of Eq. (36) and Eq. (37) can be rewritten as

$2\dot{H}+3H^{2}=\frac{1}{4f_{Q}}\left[f+f_{L_{m}}\left(\rho+2p-L_{m}\right)\right]-2\frac{\dot{f}_{Q}}{f_{Q}}H.$
(40)

Thus, we can reformulate the generalized Friedmann equations of the
$f\left(Q,L_{m}\right)$ gravity theory in the form

$3H^{2}=\rho_{eff},\ 2\dot{H}+3H^{2}=-p_{eff},$ (41)

where we have introduced the effective energy density and pressure, defined as

$\rho_{eff}=\frac{1}{4f_{Q}}\bigr{[}f-f_{L_{m}}(\rho+L_{m})\bigl{]},$ (42)

and

$p_{eff}=2\frac{\dot{f}_{Q}}{f_{Q}}H-\frac{1}{4f_{Q}}\left[f+f_{L_{m}}\left(\rho+2p-L_{m}\right)\right],$
(43)

respectively. Eq. (41) allow to formulate the generalized effective
conservation equation of the $f\left(Q,L_{m}\right)$ gravity theory as

$\dot{\rho}_{eff}+3H\left(\rho_{eff}+p_{eff}\right)=0.$ (44)

By introducing the notations

$\Delta=\frac{f_{L_{m}}}{f_{Q}},\delta=\frac{f}{f_{L_{m}}},$ (45)

we can represent the effective energy density and pressure as

$\rho_{eff}=\frac{1}{4}\Delta\left[\delta-\left(\rho+L_{m}\right)\right],$
(46)

and

$p_{eff}=2\frac{\dot{f}_{Q}}{f_{Q}}H-\frac{1}{4}\Delta\left[\delta+\left(\rho+2p-L_{m}\right)\right],$
(47)

respectively. Then the conservation equation (44) can be reformulated as

$\displaystyle\dot{\rho}$ $\displaystyle+$ $\displaystyle
3H\left(\rho+p\right)=\frac{1}{\Delta}\frac{d}{dt}\left[\Delta\left(\delta-
L_{m}\right)\right]$ (48)
$\displaystyle+3H\left\\{8\frac{\dot{f}_{Q}}{f_{Q}}\frac{H}{\Delta}-\left[\left(1+\frac{\dot{\Delta}}{\Delta}\right)\rho+p\right]\right\\}=\Gamma.$

The function $\Gamma$ describes the non-conservation level of the present
modified gravity theory. If $\Gamma>0$, the energy of the particles increases
due to the energy transfer of matter to the gravitational field. The case
$\Gamma<0$ can be interpreted as describing particle decay due to the matter-
geometry coupling.

From Eqs. (41) we also obtain the expression of the deceleration parameter as

$\displaystyle q$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{3}{2}\frac{p_{eff}}{\rho_{eff}}$ (49)
$\displaystyle=$
$\displaystyle\frac{1}{2}+6\frac{2\dot{f}_{Q}H-(1/4)\left[f+f_{L_{m}}\left(\rho+2p-L_{m}\right)\right]}{f-f_{L_{m}}\left(\rho+L_{m}\right)}.$

The Universe enters into an accelerating phase when $q<0$, or
$p_{eff}<-\rho_{eff}/3$. This gives the condition that must be satisfied by
the function $f$ and its derivatives to describe an accelerated expansion

$12\dot{H}f_{Q}+f-f_{L_{m}}\left(\rho+L_{m}\right)>0.$ (50)

To compare the theoretical results with the cosmological observations, we
introduce an independent variable redshift $z$ instead of the usual time
variable $t$, defined as $a=\frac{1}{1+z}$, where we have used a normalization
of the scale factor by imposing $a(0)=1$. Thus, we can replace the derivatives
with respect to the time with the derivatives with respect to the redshift
using the relation

$\displaystyle\frac{d}{dt}=-(1+z)H(z)\frac{d}{dz}.$ (51)

Moreover, the redshift dependence of the deceleration parameter is given by

$\displaystyle q(z)=-1+(1+z)\frac{H^{\prime}(z)}{H(z)}.$ (52)

###  III.2 The de Sitter solution

As a first step in considering explicit theoretical models, we consider the
problem of the existence of a de-sitter-type vacuum solution of the
cosmological field equations. The de Sitter solution corresponds to $p=0$,
$\rho=0$ and $H=H_{0}={\rm constant}$, respectively. For a vacuum de Sitter
type Universe, Eq. (38) gives $\dot{f_{Q}}=0$, and further results in
$f_{Q}=F_{0}$, where $F_{0}$ is a constant.

The condition $f_{Q}=F_{0}$ is satisfied for any $Q$, when we have [113, 135]

$\displaystyle f(Q)=F_{0}Q+2\Lambda,$ (53)

where $\Lambda$ is an integration constant. In the vacuum de Sitter phase, the
first field equation (36) reduces to the form

$\displaystyle 3H_{0}^{2}=\frac{6F_{0}H_{0}^{2}+2\Lambda}{4F_{0}}.$ (54)

One can also write the above equation as

$\displaystyle H_{0}=\sqrt{\frac{\Lambda}{3F_{0}}}.$ (55)

Hence, the $f(Q,L_{m})$ theory admits the de Sitter type evolution in the
limiting case of a vacuum Universe. As can be easily calculated, for the de-
Sitter solution, we have $q=-1$ and $\omega=-1$, respectively.

##  IV Cosmological models

In this Section, we will explore various cosmological models based on the
$f(Q,L_{m})$ gravity theory. The models are determined by specific choices for
the functional form of $f\left(Q,L_{m}\right)$. To keep our analysis as
general as possible, we will assume that the matter in the Universe obeys an
equation of state given by $p=(\gamma-1)\rho$ where $p$ is the pressure and
$\rho$ is the energy density, $1\leq\gamma\leq 2$. For $\gamma=4/3$, this
linear relationship between pressure and energy density describes the
behaviour of the radiation in the early Universe, characterized by high
density, as well as, for $\gamma=1$, in the present Universe, when the matter
density is low. However, we will begin our presentation by describing the data
sets and the statistical methods used to compare the models with observations.

###  IV.1 Data and methodology of MCMC analyses

The present Section presents the observational datasets that constrain the
$f(Q,L_{m})$ modified gravity theory with the MCMC methodology. We perform a
Bayesian statistical analysis based on MCMC tools, using the Emcee module
under the Python environment, to establish bounds over the model parameters.
To find the maximum of the likelihood function for each data set, we use the
following priors: $H_{0}\mathrel{\mathop{\mathchar 58\relax}}[60,80]$,
$\alpha\mathrel{\mathop{\mathchar 58\relax}}[0,0.5]$,
$\beta\mathrel{\mathop{\mathchar 58\relax}}[-3000,-2000]$, and
$\gamma\mathrel{\mathop{\mathchar 58\relax}}[1,2]$. We also perform our
analysis by combining the samples $OHD+SN+BAO$.

####  IV.1.1 Data sets

  * •

Hubble parameter: The values of the Hubble parameter $H(z)$ are usually
derived from the so-called differential age of galaxies (DAH) methodology. As
the Hubble parameter can be estimated at a redshift $z$ from
$H(z)=\frac{-1}{1+z}\frac{dz}{dt}$, $dz/dt$ can be obtained from measurements
of massive and very slowly evolving galaxies, dubbed Cosmic Chronometers (CC).
We use 31 points compiled in [150], where we take them as uncorrelated with
each other.

  * •

Type Ia supernovae (SN): We use measurements of the most recent Pantheon$+$
dataset [151, 152], consisting of 1701 light curves from 1550 Type Ia
supernovae. This is an improvement from the previous Pantheon data set, where,
in particular, there is an increase in the low redshift range. This data set
includes measurements of Cepheid hosts from the SH0ES collaboration, composed
of 77 points. The theoretical estimate for the distance modulus
$\mu(z)=m_{b}-M_{B}$ is

$\mu_{th}=5\log_{10}D_{L}(z)/\text{Mpc}+25,$ (56)

with $D_{L}(z)$ being the luminosity distance.

  * •

Baryonic Acoustic Oscillations (BAO): Finally, we constrain our model using
BAO [153, 154, 155]. To obtain the BAO constraints, we use the acoustic scale
$l_{A}=\pi\frac{d_{A}(z_{d})}{r_{s}(z_{*})}$, where
$d_{A}(z)=\int^{z}_{0}\frac{dz^{\prime}}{H(z^{\prime})}$ is the angular
diameter distance in the comoving coordinates and $r_{s}$ is the sound horizon
determined as
$r_{s}(z_{d})=\int_{z_{d}}^{\infty}\frac{c_{s}(z^{\prime})}{H(z)}$ at the drag
epoch $z_{d}$ with sound speed $c_{s}(z)$. Here, $D_{V}(z_{BAO})$ is the
dilation scale $D_{V}(z)=\left[\frac{d_{A}(z)^{2}cz}{H(z)}\right]^{1/3}$.
Finally, we obtain the BAO constraints $\frac{d_{A}(z_{d})}{D_{V}(z_{BAO})}$.

####  IV.1.2 Statistical analysis

To perform the MCMC sampling, we consider the chi-squared functions
$\chi^{2}_{SN}$ and $\chi^{2}_{OHD}+\chi^{2}_{SN}+\chi^{2}_{BAO}$ obtained by
minimizing the corresponding log-likelihood function
$\mathcal{L}=\exp(-\chi^{2}/2)$. The best-fit values and their uncertainties
at 68% confidence level (CL) for the datasets are reported in Table I.

Table 1: Confidence-level constraints on the investigated models; we have considered $SN$ and $OHD+SN+BAO$. Model | Data | $H_{0}$ | $\alpha$ | $\beta$ | $\gamma$ | $q_{0}$ | $z_{t}$
---|---|---|---|---|---|---|---
A | SN | $72.454\pm 0.093$ | $0.1231\pm 0.0042$ | $-2484.96\pm 0.10$ | $1.0313^{+0.0085}_{-0.031}$ | $-0.44^{+0.040}_{-0.053}$ | $0.46^{+0.11}_{-0.05}$
OHD+SN+BAO | $72.527\pm 0.095$ | $0.1191\pm 0.0024$ | $-2484.952\pm 0.097$ | $1.0046^{+0.0013}_{-0.0045}$ | $-0.48^{+0.022}_{-0.025}$ | $0.56^{+0.04}_{-0.03}$
B | SN | $72.456\pm 0.094$ | $0.1228^{+0.0036}_{-0.0032}$ | $-2457.873\pm 0.094$ | $1.046^{+0.011}_{-0.047}$ | $-0.44^{+0.03}_{-0.04}$ | $0.47^{+0.08}_{-0.04}$
OHD+SN+BAO | $72.546\pm 0.093$ | $0.1167\pm 0.0024$ | $-2457.858\pm 0.099$ | $1.0188^{+0.046}_{-0.018}$ | $-0.49^{+0.03}_{-0.02}$ | $0.56^{+0.05}_{-0.07}$

It is useful to estimate how preferred the proposed models are in comparison
with the standard $\Lambda$CDM one. We then incorporate statistical criteria,
the Akaike Information Criterion (AIC) and Bayesian Information Criterion
(BIC), defined as [156]

$\mathrm{AIC}=\chi^{2}_{\mathrm{min}}+2d,\quad\mathrm{BIC}=\chi^{2}_{\mathrm{min}}+d\ln
N,$ (57)

where $d$ is the number of free parameters, and $N$ is the total size of the
data.

In this criterion, if the difference in AIC value between a given model and
the best one ($\Delta$AIC) is less than 4, both models are equally supported
by the data. For $\Delta$AIC values in the interval $4<\Delta AIC<10$, the
data still support the given model but less than the preferred one. For
$\Delta AIC>10$, the observations do not support the given model.

Similarly, BIC discriminates between models as follows: For $\Delta BIC<2$,
there is no appreciable evidence against the model. If $2<\Delta BIC<6$, there
is modest evidence against the considered model. For the interval $6<\Delta
BIC<10$, the evidence against the candidate model is strong, and even stronger
evidence against it exists in the data when $\Delta BIC>10$.

###  IV.2 Model A: $f=-\alpha\,Q+2L_{m}+\beta$ with $L_{m}=p$

As a first example of a cosmological model, we consider the functional form of
$f$ as represented by $f=-\alpha\,Q+2L_{m}+\beta$ with $L_{m}=p$, where
$\alpha$ and $\beta$ are constants. Hence, for this particular $f(Q,L_{m})$
model with $L_{m}=p=(\gamma-1)\rho$, the Friedmann equations reduces to

$\displaystyle 3H^{2}=-\frac{\beta}{2\alpha}+\frac{\rho}{\alpha},$ (58)
$\displaystyle
2\,\dot{H}+3H^{2}=-3H^{2}(\gamma-1)-\frac{\beta\gamma}{2\alpha}.$ (59)

The system of Eqs. (63) and (59) does admit a de Sitter type solution for a
vacuum Universe, with $H=H_{0}={\rm constant}$, corresponding to
$3H_{0}^{2}=-\beta/2\alpha$, which requires that either $\alpha$ or $\beta$
are negative. The effective densities and pressures are given by

$\rho_{eff}=\frac{\rho}{\alpha}-\frac{\beta}{2\alpha},\
p_{eff}=3H^{2}(\gamma-1)+\frac{\beta\gamma}{2\alpha},$ (60)

leading to the energy balance equation

$\dot{\rho}+3H\left(\rho+p\right)=0,$ (61)

Hence, in this model, the energy-momentum tensor of the matter is conserved.

Further, using the relation $\frac{1}{H}\frac{dH}{dt}=\frac{dH}{d\,lna}$, the
above equations have an exact solution for $H$, which takes the form

$\displaystyle
H(z)=\left[\frac{(6H_{0}^{2}\alpha+\beta)(1+z)^{3\gamma}-\beta}{6\alpha}\right]^{\frac{1}{2}},$
(62)

where $H(0)=H_{0}$ is the present Hubble parameter. It should be noted that
the positivity condition on $H(z)$ in Eq. (62) gives us the constraints on
model parameters for the priors to be set in MCMC analysis.

####  IV.2.1 Statistical analysis of Model A

The 1D posterior distributions and 2D confidence level contours at $1\sigma$
CL and $2\sigma$ CL are presented in Model A in Figs. 1.

Figure 1: The corner plot for the parameter space
$(H_{0},\alpha,\beta,\gamma)$ with their $1\sigma$ and $2\sigma$ confidence
levels for model A in $f(Q,L_{m})$ gravity.

According to our results, we obtain the $\chi^{2}$ for $SN$ and $OHD+SN+BAO$
in the case of model A. We have $\Delta AIC=2.13$ & $3.4$, $\Delta BIC=1.57$
and $8.87$ for $SN$ and $OHD+SN+BAO$. We find that our model A is well
supported by the observational data $SN$ and, on average, supported by the
combined set $OHD+SN+BAO$.

####  IV.2.2 Cosmological evolution: Model A

The variations as a function of the redshift of the energy density,
deceleration parameter, and the effective equation of state for Model A of
$f(Q,L_{m})$ gravity are represented in Figs. 2, 3, and 4, respectively. The
deceleration parameter $q(z)$ shows a significant dependence on the numerical
values of the model parameters. In the redshift range $z\in(-1,0.6)$, the
model can reproduce well the results of the standard $\Lambda$CDM model. The
deceleration parameter for model A at higher redshifts takes much larger
positive values than for the $\Lambda$CDM case, indicating a decelerating
evolution followed by a quicker transition to the accelerating phase. However,
the model we consider enters the accelerating phase with $q<0$ at the redshift
$0.46$ and $0.56$ for the $SN$ and $OHD+SN+BAO$ [157, 158, 159]. The
comparison of the deceleration parameter variation of Model A with the
$\Lambda$CDM paradigm shows a qualitative similarity between the two models.
Cosmological phases with a de Sitter-type expansion with $q=-1$ can also be
obtained at late times.

In addition, the energy density is an increasing function of redshift $z$. The
increase is almost linear for small redshifts, and it is almost independent of
the numerical values of model parameters. However, a significant dependence on
the model parameter can be seen at higher redshifts. The $f(Q,L_{m})$ Model A
can provide a viable alternative explanation for matter dynamics, as the plot
shows.

The plots in Fig. 4 depict the behaviour of the effective EoS parameter versus
the redshift $z$ of Model A for different sets of observational data.
According to the observations, $\gamma=4/3$ represents the radiation phase,
and $\gamma=1$ corresponds to the matter-dominated (non-relativistic) phase.
The first model here depicts the quintessence era of the universe at present,
$-1<\omega<0$ and approaches towards $\Lambda$CDM, i.e. $\omega=-1$ at late
times. However, the current function $f(Q,L_{m})$ cannot fully mimic the
standard cosmic evolution in a quantitative manner. However, it produces
identical qualitative results.

Figure 2: The behaviour of the energy density
$\overline{\rho}=\rho/3H_{0}^{2}$ as a function of redshift using the best-fit
values of model parameters for model A with $1\sigma$ levels. Figure 3: The
behaviour of the deceleration parameter as a function of redshift using the
best-fit values of model parameters for model A with $1\sigma$ levels. The
dashed black curve represents the evolution of the $q$ in the standard
$\Lambda$CDM cosmological model with $\Omega_{m}=0.3089$ and
$\Omega_{\Lambda}=0.6911$, obtained from the Planck data. Figure 4: The
behaviour of the effective equation of state parameter using the best-fit
values of model parameters for model A with $1\sigma$ levels.

###  IV.3 Model B: $f=-\alpha\,Q+(2L_{m})^{2}+\beta$ with $L_{m}=p$

As a second cosmological model, we consider the functional form as
$f(Q,L_{m})=-\alpha\,Q+(2L_{m})^{2}+\beta$ with $L_{m}=p$, where $\alpha>0$
and $\beta$ are constants. For the specific functional form, the Friedmann
equations are reduced to

$\displaystyle
3H^{2}=-\frac{2}{\alpha}(1-\gamma^{2})\rho^{2}-\frac{\beta}{2\alpha},$ (63)
$\displaystyle
2\dot{H}+3H^{2}=-\frac{\beta}{2\alpha}-\frac{(\gamma-1)(\beta+6\alpha
H^{2})}{2\alpha(\gamma+1)}.$ (64)

Hence, for this model, the effective energy density and pressure can be
defined as

$\rho_{eff}=-\frac{2}{\alpha}(1-\gamma^{2})\rho^{2}-\frac{\beta}{2\alpha},$
(65)

and

$p_{eff}=\frac{\beta}{2\alpha}+\frac{(\gamma-1)(\beta+6\alpha
H^{2})}{2\alpha(\gamma+1)}.$ (66)

For the energy balance equation, we obtain

$\dot{\rho}+\frac{3\gamma}{\gamma+1}H\rho=0,$ (67)

which can be integrated to give the density-scale factor dependence in the
form

$\rho(a)=\rho_{0}a^{-3\gamma/(\gamma+1)},$ (68)

where $\rho_{0}$ is an arbitrary integration constant. Eq. (67) can be
reformulated as

$\dot{\rho}+3\gamma H\rho=\frac{3\gamma^{2}}{\gamma+1}H\rho=\Gamma,$ (69)

which shows that $\Gamma$, the matter non-conservation rate, is proportional
to the factor $H\rho$. In the present model $\Gamma>0$, and thus it describes
energy transfer from the gravitational field to matter. Still, in the limit of
very low densities matter creation processes can be safely ignored.

The expression of the function $H(z)$ for this model is obtained as

$\displaystyle
H(z)=\left[\frac{(6H_{0}^{2}\alpha+\beta)(1+z)^{\frac{6\gamma}{1+\gamma}}-\beta}{6\alpha}\right]^{\frac{1}{2}}.$
(70)

Figure 5: The corner plot for the parameter space
$(H_{0},\alpha,\beta,\gamma)$ with their $1\sigma$ and $2\sigma$ confidence
levels for Model B in $f(Q,L_{m})$ gravity.

####  IV.3.1 Statistical analysis

The 2D confidence level contours at $1\sigma$ CL and $2\sigma$ CL for Model B
are presented in Fig. 5.

Similarly, we obtain the following results for Model B: $\chi^{2}=1609.95$,
and $\chi^{2}=1632.15$. Hence, using the above information, we obtain $\Delta
AIC=2.05$ & $2.9$, $\Delta BIC=1.49$ & $8.4$ for $SN$ and $OHD+SN+BAO$.
Consequently, Model B is also similarly supported by the observational data.

####  IV.3.2 Cosmological evolution: Model B

The energy density in Fig. 6 is a monotonically increasing function of the
redshift for all adopted numerical values of the model parameters. The
increase is almost linear for small redshifts. However, a dependence on the
model parameter can be seen at higher redshifts. This particular $f(Q,L_{m})$
model predicts a lower energy density at high and low redshifts compared to
the previous model.

Figure 6: The behaviour of the energy density
$\overline{\rho}=\rho/3H_{0}^{2}$ as a function of redshift using the best-fit
values of model parameters for model B with $1\sigma$ levels.

In this model, the deceleration parameter provides a good qualitative
concordance with the predictions of the standard cosmology. Moreover, the
behaviour of $q$ overlaps at $1\sigma$ level for $SN$ and $OHD+SN+BAO$ sets.
Up to redshifts of around $z\sim 1$, the deceleration parameter is roughly a
constant in the range $q\in(0.2,0.5)$, and the Universe is decelerating.
Furthermore, the Universe began to accelerate, and after a short cosmological
time interval, it entered an accelerating phase at the redshift $0.47$ and
$0.56$ for $SN$ and $OHD+SN+BAO$, for the considered $f(Q,L_{m})$ gravity
model. The second model in Fig. 8 depicts the quintessence era of the universe
at present, $-1<\omega<0$ and approaches towards $\Lambda$CDM, i.e.
$\omega=-1$ at late times. Thus, in the present mode, we get the overlapping
and tighter constraints for both datasets, $SN$ and $OHD+SN+BAO$.

Figure 7: The behaviour of the deceleration parameter as a function of redshift using the best-fit values of model parameters for model B with $1\sigma$ levels. The dashed black curve represents the evolution of the $q$ in the standard $\Lambda$CDM cosmological model with $\Omega_{m}=0.3089$ and $\Omega_{\Lambda}=0.6911$, obtained from the Planck data. Table 2: The corresponding $\chi^{2}$ of the models for each sample and the information criteria AIC and BIC for the examined cosmological models. Model |  Data |  $\chi^{2}$ |  $AIC$ |  $\Delta\,AIC$ |  $BIC$ |  $\Delta\,BIC$
---|---|---|---|---|---|---
$\Lambda$CDM |  SN |  $1609.9$ |  $1615.90$ |  $0$ |  $1638.21$ |  $0$
OHD+SN+BAO |  $1631.217$ |  $1637.217$ |  $0$ |  $1653.59$ |  $0$
A |  SN |  $1610.03$ |  $1618.03$ |  $2.13$ |  $1639.78$ |  $1.57$
OHD+SN+BAO |  $1632.62$ |  $1640.62$ |  $3.4$ |  $1662.46$ |  $8.87$
B |  SN |  $1609.95$ |  $1617.95$ |  $2.05$ |  $1639.70$ |  $1.49$
OHD+SN+BAO |  $1632.15$ |  $1640.15$ |  $2.9$ |  $1661.99$ |  $8.4$
Figure 8: The behavior of the effective equation of state parameter using the
best-fit values of model parameters for model B with $1\sigma$ levels. Figure
9: The behavior of the Hubble parameter $H(z)$ as a function of the redshift
for model A and B in $f(Q,L_{m})$ gravity.

###  IV.4 Model comparisons, and correlation results

Here, Fig. 9 illustrates the redshift evolution of the Hubble parameter,
providing a clear comparison of the fit for both models and the $\Lambda$CDM
at both low and high redshifts. The black line depicts the $\Lambda$CDM model.
The error-bar plot of the distance modulus $\mu(z)$ and $\mu(z)-\mu_{\Lambda
CDM}$ for the Pantheon+ data is shown in Fig. 10, with the standard
$\Lambda$CDM model represented by a black solid line. Notably, both models
correspond closely with the $\Lambda$CDM at low redshifts, but they start to
diverge slightly after $z\approx 1$.

Figure 10: The behavior of $\mu(z)$ and $\mu(z)-\mu_{\Lambda CDM}(z)$ as a
function of the redshift for Models A (blue dashed line) and B (red dotted
line) in $f(Q,L_{m})$ gravity.

####  IV.4.1 Correlation results

We further investigate the correlations among key physical parameters within
the datasets. From the results presented in Fig. 11, we infer the following
conclusions:

  * •

Fig. 11(a) shows that the Hubble parameter $H_{0}$ exhibits a strong positive
linear relationship with redshift $z$, as indicated by the correlation
coefficient of $0.95$. The standard deviation $\sigma$ shows a moderate
positive correlation with both $H_{0}$ (0.37) and $z$ (0.39).

  * •

Fig. 11(b) illustrates that $z_{BAO}$ has a strong negative relation with
$\sigma$ and $D(=d_{A}/D_{V})$, as indicated by the correlation coefficient of
-0.75 and -0.87. The standard deviation $\sigma$ shows a strong positive
correlation with $D$ (0.93).

  * •

Fig. 11(c) shows that $\mu$ has strong positive correlation with $z_{CMB}$
(0.83) and moderate negative correlation with $\sigma_{\mu}$ (-0.31). The
correlation coefficient of -0.011 indicates a very weak negative linear
relationship between $\sigma_{\mu}$ and $z_{CMB}$.

##  V Conclusions

In this paper, we have investigated the theoretical aspects of the third
geometric description of gravity, known as the symmetric teleparallel gravity,
or the $f(Q)$ gravity. From a geometric and mathematical perspective, $f(Q)$
gravity uses the Weylian extension of Riemann geometry, where the fundamental
metricity condition no longer holds. The violation of the metricity condition
thus becomes the source of gravitational phenomena, with the non-metricity
scalar $Q$ playing a similar role to that of the Ricci scalar in general
relativity.

In the present study we have introduced a novel class of theories,
representing an extension of the $f(Q)$ gravity, where the non-metricity $Q$
is coupled non-minimally with the matter Lagrangian $L_{m}$. Mathematically,
our analysis was conducted within the framework of the metric-affine
formalism. Our theory is constructed similarly to the $f(Q,T)$ theory, but
with the trace of the matter energy-momentum tensor replaced by the matter
Lagrangian. Similarly to the energy-momentum tensor trace-curvature couplings,
in $f(Q,L_{m})$ theory, the coupling between $Q$ and $L_{m}$ leads to the non-
conservation of the matter energy-momentum tensor.

By applying a variational principle, we have derived the gravitational field
equations for the $f(Q,L_{m})$ gravity theory. For particular choices of
$f(Q,L_{m})$, it reduces to both $f(Q)$ and STEGR. This theory provides the
freedom to explore different sets of coupling between $Q$ and $L_{m}$, and
thus, the theory sheds light on the coupling mechanisms between the third,
non-metric geometric description of gravity and matter, representing new
avenues for further theoretical exploration. Consequently, the fundamental
equations describing the cosmological evolution in $f(Q,L_{m})$ gravity are
expressed in terms of an effective energy density and pressure of a purely
geometric origin. But they also depend on the ordinary matter-energy and
pressure components of the energy-momentum tensor, as well as on the functions
$f(Q,L_{m})$, $f_{Q}(Q,L_{m})$, and $f_{L_{m}}(Q,L_{m})$.

Additionally, we have obtained the general relationship describing the non-
conservation of the matter-energy-momentum tensor. The equation of motion of
the particles reveals a notable departure from the geodesic motion for massive
particles, specific to standard general relativity. An additional force
emerges as a consequence of the coupling between $Q$ and $L_{m}$. This
coupling introduces a non-gravitational effect, leading to deviations from the
paths followed in the standard geodesic motion and influencing the dynamical
evolution of massive particles. The investigations presented may also
contribute to a better understanding of the geometrical formulation of gravity
theories, particularly regarding the aspects related to the geometry-matter
coupling.

For the description of the dynamics of the Universe, we have adopted the
homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker type metric,
describing the cosmological evolution in a flat geometry. In this study, we
have examined two specific classes of cosmological models by adopting some
simple functional forms of $f(Q,L_{m})$. For the first case, we have assumed
the simple additive Lagrangian, $f(Q,L_{m})=-\alpha Q+2L_{m}+\beta$. For this
model, we have obtained a wide range of cosmological scenarios and evolution
corresponding to the specific numerical values of the model parameters. These
scenarios may include cosmological evolution describing both the decelerating
and the accelerating expansion phases of the Universe and de Sitter-type
dynamics at late times. The model $f(Q,L_{m})=-\alpha Q+2L_{m}+\beta$ can
provide an effective description of cosmological data up to redshifts of
around $z\approx 1$. Specifically, in this model, the Universe undergoes a
rapid transition from a decelerating phase, characterized by a positive value
of $q$, to an accelerating state where $q<0$. This transition can result, in
its final stages, in a de Sitter-type expansion. The second model with
$f(Q,L_{m})=-\alpha\,Q+(2L_{m})^{2}+\beta$ also evolves from a decelerating to
an accelerating state. The nature of the cosmological evolution is heavily
influenced by the numerical values of the model parameters and the specific
functional form of $f$. Our fundamental finding for the specific models and
for the range of the cosmological parameters we have examined indicates that
the Universe initially underwent a decelerating phase in its recent evolution,
followed by an accelerating phase with $q<0$, in which the Universe entered at
$z=z_{crit}$, and which continued for $0<z<z_{crit}$. In the future, the
Universe enter into an accelerating de Sitter-type phase, with $q=-1$. From
our analysis, it follows that $H_{0}$ in the second model has higher values as
compared to the first model, while the values of other model parameters
decrease under the same scenario. Additionally, we find consistent DE EoS
behavior with the assumption of quintessence dynamics within $1\sigma$.

(a) OHD

(b) BAO

(c) Pantheon+

Figure 11: Correlation matrix heatmap for (a) Observed Hubble Data (OHD), (b)
Baryonic Acoustic Oscillations (BAO) and (c) SN (Pantheon+SHOES)

We have also compared the theoretical predictions of the $f(Q,L_{m})$ theory
with the corresponding results in the standard $\Lambda$CDM cosmology. Both
our considered $f(Q,L_{m})$ models align well with $\Lambda$CDM at lower
redshifts. However, at higher redshifts, significant differences emerge in the
behavior of the $\mu(z)$ function and the deceleration parameter, as compared
to the $\Lambda$CDM model. Henceforth, in the presence of matter, the models
give an acceptable description of the observational data, as well as of the
$\Lambda$CDM model, but without reproducing it exactly at the present time.

Another potential application of the $f(Q,L_{m})$ theory would be to consider
inflation in the presence of scalar fields, which might offer a completely new
perspective on the geometrical, gravitational, and cosmological processes that
significantly influenced the early dynamics of the Universe. Consequently, the
predictions of the present model could lead to major differences compared to
those of standard general relativity or its extensions that ignore the role of
matter. These differences could impact several current areas of interest, such
as cosmology, gravitational collapse, and the generation of gravitational
waves. To conclude, in the present investigation, we have introduced a new
version of the symmetric teleparallel theory, and we have demonstrated its
theoretical consistency. This approach also motivates and encourages the
exploration of further extensions within the $f(Q,L_{m})$ family of theories.

##  Appendix A Derivation of the Friedmann equations

The metric tensor components are given by
$g_{\mu\nu}=diag(-1,a^{2},a^{2},a^{2})$,
$g^{\mu\nu}=diag(-1,a^{-2},a^{-2},a^{-2})$, and its determinant
$\sqrt{-g}=a^{3}$. For the nonmetricity tensor, have the following non-zero
terms,

$\displaystyle Q_{011}=Q_{022}=Q_{033}=2a\dot{a},$ (A1) $\displaystyle
Q_{0}^{\;\;11}=Q_{0}^{\;\;22}=Q_{0}^{\;\;33}=\frac{2\dot{a}}{a^{3}},$ (A2)
$\displaystyle
Q^{01}_{\;\;\;1}=Q^{02}_{\;\;\;2}=Q^{03}_{\;\;\;3}=-\frac{2\dot{a}}{a},$ (A3)
$\displaystyle L^{0}_{\;\;11}=L^{0}_{\;\;22}=L^{0}_{\;\;33}=-a\dot{a},$ (A4)
$\displaystyle
L^{1}_{\;\;01}=L^{1}_{\;\;10}=L^{2}_{\;\;02}=L^{2}_{\;\;20}=L^{3}_{\;\;03}=L^{3}_{\;\;30}=-\frac{\dot{a}}{a},$
(A5) $\displaystyle P^{0}_{\;\;11}=P^{0}_{\;\;22}=P^{0}_{\;\;33}=-a\dot{a},$
(A6) $\displaystyle P^{011}=P^{022}=P^{033}=-\frac{\dot{a}}{a^{3}},$ (A7)
$\displaystyle P_{011}=P_{022}=P_{033}=a\dot{a},$ (A8) $\displaystyle
P^{1}_{\;\;01}=P^{1}_{\;\;10}=P^{2}_{\;\;02}=P^{2}_{\;\;20}=P^{3}_{\;\;03}=P^{3}_{\;\;30}=-\frac{\dot{a}}{4a},$
(A9) $\displaystyle
P_{110}=P_{101}=P_{220}=P_{202}=P_{330}=P_{303}=-\frac{a\dot{a}}{4},$ (A10)
$\displaystyle
P^{110}=P^{101}=P^{220}=P^{202}=P^{330}=P^{303}=-\frac{\dot{a}}{4a^{3}}.$
(A11)

The non-metricity scalar $Q$ is calculated using Eq.(8) as

$Q=-(Q_{011}P^{011}+Q_{022}P^{022}+Q_{033}P^{033}).$ (A12)

We obtain thus $Q=6H^{2}$, where $H=\dot{a}/a$. The energy-momentum tensor
$T_{\mu\nu}$ for a perfect fluid has the components
$T_{\mu\nu}=diag(\rho,pa^{2},pa^{2},pa^{2})$. Evaluating the field equation
(15) for the tt-component

$\displaystyle\frac{2}{a^{3}}\nabla_{\alpha}(f_{Q}\sqrt{-g}P^{\alpha}_{\;\;00})+f_{Q}(P_{0\alpha\beta}Q_{0}^{\;\;\alpha\beta}-2Q^{\alpha\beta}_{\;\;\;0}P_{\alpha\beta
0})$ (A13)
$\displaystyle+\frac{1}{2}fg_{00}=\frac{1}{2}f_{L_{m}}(g_{00}L_{m}-T_{00}),$
$\displaystyle
f_{Q}(P_{011}Q_{0}^{\;\;11}+P_{022}Q_{0}^{\;\;22}+P_{033}Q_{0}^{\;\;33})-\frac{1}{2}f$
(A14) $\displaystyle=-\frac{1}{2}f_{L_{m}}(\rho+L_{m}).$

gives the first generalized Friedmann equation

$3H^{2}=\frac{1}{4f_{Q}}\bigr{[}f-f_{L_{m}}(\rho+L_{m})\bigl{]}.$ (A15)

By evaluating the field equation (15) for the xx-component

$\displaystyle\frac{2}{a^{3}}\nabla_{\alpha}(f_{Q}\sqrt{-g}P^{\alpha}_{\;\;11})+f_{Q}(P_{1\alpha\beta}Q_{1}^{\;\;\alpha\beta}-2Q^{\alpha\beta}_{\;\;\;\;1}P_{\alpha\beta
1})$ (A16)
$\displaystyle+\frac{1}{2}fg_{11}=\frac{1}{2}f_{L_{m}}(g_{11}L_{m}-T_{11}),$
$\displaystyle\frac{2}{a^{3}}\frac{\partial}{\partial
t}(f_{Q}a^{3}(-a\dot{a}))-2f_{Q}\bigr{(}\frac{2\dot{a}}{a}\bigl{)}(a\dot{a})+\frac{a^{2}}{2}f$
(A17) $\displaystyle=\frac{a^{2}}{2}f_{L_{m}}(L_{m}-p).$

leads to the second generalized Friedmann equation

$\dot{H}+3H^{2}+\frac{\dot{f_{Q}}}{f_{Q}}H=\frac{1}{4f_{Q}}\bigr{[}f+f_{L_{m}}(p-L_{m})\bigl{]}.$
(A18)

## Data Availability

Data sharing is not applicable to this article as no datasets were generated
or analyzed during the current study.

## Acknowledgments

AH thanks Mr. Sayantan Ghosh (BITS-Pilani, Hyderabad) for fruitful discussion
on modified gravity. PKS acknowledges Science and Engineering Research Board,
Department of Science and Technology, Government of India for financial
support to carry out Research project No.: CRG/2022/001847 and IUCAA, Pune,
India, for providing support through the visiting Associateship program.

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