# Constraining Weyl type $f(Q,T)$ gravity with Big Bang Nucleosynthesis

Jian Ge<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen University,
Guangzhou 510275, People’s Republic of China,  Lei Ming<EMAIL_ADDRESS>Key Laboratory of Atomic and Subatomic Structure and Quantum Control(Ministry
of Education), Guangdong Basic Research Center of Excellence for Structure and
Fundamental Interactons of Matter, School of Physics, South China Normal
University, Guangzhou 510006, China  Guangdong Provincial Key Laboratory of
Quantum Engineering and Quantum Materials, Guangdong-Hong Kong Joint
Laboratory of Quantum Matter, South China Normal University, Guangzhou 510006,
China  School of Physics, Sun Yat-Sen University, Guangzhou 510275, People’s
Republic of China,  Shi-Dong Liang<EMAIL_ADDRESS>School of Physics,
Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China,  Hong-
Hao Zhang<EMAIL_ADDRESS>School of Physics, Sun Yat-Sen University,
Guangzhou 510275, People’s Republic of China,  Tiberiu Harko
<EMAIL_ADDRESS>Department of Physics, Babes-Bolyai University,
Kogalniceanu Street, Cluj-Napoca, 400084, Romania,  Astronomical Observatory,
19 Ciresilor Street, Cluj-Napoca 400487, Romania

###### Abstract

The Weyl type $f(Q,T)$ modified gravity theory is an extension of the $f(Q)$
and $f(Q,T)$ type theories, where $T$ is the trace of the matter energy-
momentum tensor, and the scalar non-metricity $Q$ is represented in its
standard Weyl form, and it is fully determined by a vector field
$\omega_{\mu}$. The theory can give a good description of the observational
data, and of the evolution of the late-time Universe, including a geometric
explanation of the dark energy. In this work we investigate the Big Bang
Nucleosynthesis (BBN) constraints on several Weyl type $f(Q,T)$ gravity
models. In particular, we consider the corrections that Weyl type $f(Q,T)$
terms induce on the freeze-out temperature $\mathcal{T}_{f}$, as compared to
the standard $\Lambda$CDM results. We analyze in detail three distinct
cosmological models, corresponding to specific choices of the functional form
of $f(Q,T)$. The first model has a simple linear additive structure in $Q$ and
$T$, the second model is multiplicative in $Q$ and $T$, while the third is
additive in $T$ and the exponential of $Q$. For each $f(Q,T)$ we consider
first the cosmological evolution in the radiation dominated era, and then we
impose the observational bound on
$\left|\delta\mathcal{T}_{f}/\mathcal{T}_{f}\right|$ to obtain constraints on
the model parameters from the primordial abundances of the light elements such
as helium-4, deuterium and lithium-7. The abundances of helium-4 and deuterium
agree with theoretical predictions, however, the lithium problem, even
slightly alleviated, still persists for the considered Weyl type $f(Q,T)$
models. Generally, these models satisfy the BBN constraints, and thus they
represent viable cosmologies describing the entire dynamical time scale of the
evolution of the Universe.

###### Contents

  1. I Introduction
  2. II Weyl type $f(Q,T)$ gravity
    1. II.1 The gravitational field equations of the $f(Q,T)$ theory
      1. II.1.1 Action and geometry
      2. II.1.2 Field equations
    2. II.2 The generalized Friedmann equations
  3. III Standard $\Lambda$CDM Big Bang Nucleosynthesis
    1. III.1 The radiation dominated Universe
    2. III.2 The freeze-out constraint
    3. III.3 ${}^{2}H$, ${}^{4}He$ and ${}^{7}Li$ abundances
  4. IV Nucleosynthesis constraints on Weyl type $f(Q,T)$ gravity models
    1. IV.1 Freeze-out temperature in Weyl type $f(Q,T)$ gravity
    2. IV.2 Model I: $f(Q,T)=\alpha Q+\frac{\beta}{6\kappa^{2}}T$
      1. IV.2.1 The freeze-out constraint.
      2. IV.2.2 Deuterium and helium constraints.
      3. IV.2.3 The ${}^{7}Li/H$ ratio.
      4. IV.2.4 Model II: $f(Q,T)=\frac{\gamma}{6H_{0}^{2}\kappa^{2}}QT$
      5. IV.2.5 The freeze-out constraint
      6. IV.2.6 The combined ${}^{7}Li/H$ ratio
    3. IV.3 Model III: $f(Q,T)=\eta H_{0}^{2}e^{\frac{\mu}{6H_{0}^{2}}Q}+\frac{\nu}{6\kappa^{2}}T$
      1. IV.3.1 The freeze-out constraint
      2. IV.3.2 The ${}^{7}Li/H$ ratio
  5. V Discussions and final remarks

##  I Introduction

The theory of general relativity plays a vital role in modern physics and
cosmology. Einstein Ein  and Hilbert Hil  did extensively use the Riemannian
geometry, by endowing the space-time with a metric, and an affine structure.
The geometric and gravitational properties of the space-time can thus be
described by a metric, and by the Riemann curvature tensor, and its
contractions.

Shortly after the theory of general relativity was proposed, in 1918 Weyl
introduced an extension of Riemannian geometry Weyl1 ; Weyl2 , with the goal
to develop the first unified theory of gravity and electromagnetism. In Weyl’s
geometry, the presence of electromagnetic field can be attributed to the
existence of the nonmetricity of the space-time. A few years later, Cartan
Cart1 ; Cart2  introduced another generalization of the Riemann geometry, by
considering another important geometric concept, the torsion tensor,
describing the properties of a space-time with non-symmetric Christoffel
symbols.

In its first 50 years of existence, Weyl’s geometry was largely ignored by
physicists, mostly due to Einstein’s strong criticism of the physical
interpretation of the nonmetricity. But gradually the interest of physicists
in this geometry begin to increase, and interesting physical and mathematical
effects on both microscopic and macroscopic level have been intensively
investigated Scholz .

A number of recent theoretical investigations have arrived to the important
result that general relativity can be described in three mathematically
equivalent geometric formalisms, by using the three basic geometric concepts:
curvature, torsion, and non-metricity Jim . Hence, one can consider, as a
viable alternative to the curvature formulation, the teleparallel formulation,
with the torsion vanishing for the first approach, and the curvature vanishing
for the second alternative, called the teleparallel equivalent of General
Relativity (TEGR) TEGR ; TEGR1 . The TEGR theory is also called the
$f(\mathcal{T})$ gravity theory, and in it the torsion $\mathcal{T}$ exactly
compensates curvature, with the space-time becoming flat.

The possibility of formulating general relativity by using only non-metricity
was investigated in Nester , leading to the development of the symmetric
teleparallel formulation of gravity. This approach was further generalized
into the $f(Q)$ gravity theory Jim1 , which is also called coincident general
relativity, and in which $Q$ represents quadratic nonmetricity scalar. The
coupling of matter with nonmetricity was considered in fQ1 , and this initial
study led to the extension of the symmetric teleparallel gravity to the
$f(Q,T$) theory ref1 , in which the gravitational Lagrangian is given by an
arbitrary function $f$ of the non-metricity $Q$ and of the trace $T$ of the
matter-energy-momentum tensor $T$.

The role of the modified theories of gravity has recently become central in
the understanding of the gravitational dynamics. This situation was determined
by the recent theoretical challenges faces by general relativity. These
challenges come from a plethora of observational data, which cannot be
explained using standard general relativity without adding at least one extra
parameter to the theory. The first major challenge general relativity is
encountering is the explanation of the observations of the accelerated
expansion of the Universe P1 ; P2 ; P3 , which is confirmed by data of the
Planck satellite 42 , and studies of Baryon Acoustic Oscillations 43 ; 44 ; 45
. For a recent review of the cosmic acceleration problem see 46 .

The problem of the cosmic acceleration was addressed within the framework of
the $f(Q,T)$ gravity ref1 , by obtaining first the cosmological evolution
equations for a flat, homogeneous and isotropic geometry, which generalize the
Friedmann equations of general relativity. Several cosmological models were
investigated by adopting some simple functional forms for the function
$f(Q,T)$: $f(Q,T)=\alpha Q+\beta T$, $f(Q,T)=\alpha Q^{n+1}+\beta T$, and
$f(Q,T)=-\alpha Q-\beta T^{2}$, respectively. The main cosmological parameters
(Hubble function, the deceleration parameter, and the matter-energy density)
have been obtained as a function of the redshift by using analytical and
numerical methods.

A particular class of the $f(Q,T)$ gravity theory was investigated in ref2 ,
in which the scalar non-metricity $Q_{\alpha\mu\nu}$ is represented in a
standard Weyl form, and it is fully determined by a vector field
$\omega_{\mu}$. The field equations of the theory have been obtained by
assuming the vanishing of the total scalar curvature, with this condition
added into the gravitational action via a Lagrange multiplier. The
gravitational field equations have been obtained from a variational principle,
and they depend on the scalar nonmetricity, as well as on the Lagrange
multiplier. The cosmological implications of the theory have been also
investigated, and the cosmological evolution equations, which generalize the
Friedmann equations of standard general relativity, have been obtained for a
flat, homogeneous and isotropic geometry. Several cosmological models have
been considered for some simple functional forms of $f(Q,T)$. The predictions
of the theory have been compared with the standard $\Lambda$CDM model, and the
observational data. It turns out that the Weyl type $f(Q,T)$ gravity theory
can give a good description of the observational data, and hence it can be
considered as an attractive theoretical alternative to the standard
$\Lambda$CDM model. Various physical and cosmological aspects of the $f(Q,T)$
type theory have been investigated in W1 ; W2 ; W3 ; W4 ; W5 ; W6 ; W7 ; W8 ;
W9 ; W10 ; W11 ; W12 ; W13 ; W14 ; W15 ; W16 ; W17 ; W18 ; W19 .

Big Bang Nucleosynthesis happened within the first three minutes of the
Universe’s existence. After the first $180\;\rm s$ from the Big Bang, the
temperature of the Universe (defined as the radiation temperature) drops from
beyond $10^{16}\;\rm GeV$ $=1.2\times 10^{29}\;\rm K$ to $1\;\rm MeV$
$=1.2\times 10^{10}\;\rm K$, which represents the temperature at which the
density of the nucleon component allowed for nuclear reactions to form stable
nuclei Copi ; Fields . The presence of a small nucleon content allowed for the
production of light nuclide, such as hydrogen, the most abundant element in
the Universe, helium, the second most common element, and an isotope of
lithium, respectively. The BBN processes took place during the rapid expansion
of the Universe, and they lasted until the nuclear reactions ceased due to the
low temperatures and matter densities, a result of the expansion of the
Universe .

The standard model of cosmology heavily relies on the standard model of
particle physics, thus incorporating many results of the elementary particle
physics, like, for example, the existence of three families of neutrinos. The
formation of the light nuclei in the early Universe depend on several physical
factors like temperature, nucleon density, expansion rate, neutrino content,
and neutrino-antineutrino asymmetry, respectively. Therefore, BBN provides
powerful methods to investigate and to test cosmological scenarios and
physical conditions that might have influenced the formation of the primordial
elements Copi ; Fields ; Steigman-2004 ; Sepico ; Steigman-2007 ;
Steigman-2008 ; Li1 ; Steigman-2012 ; Li2 ; Cybrut ; 2016On ; F ; De1 ; N1 .

BBN data were used in Barrow  to impose constraints on the exponent of the
Barrow entropy, an extended entropy relation arising from the incorporation of
quantum-gravitational effects on the black-hole structure. Constraints on the
parameters of the bimetric theory of gravity, a ghost-free and observationally
viable extension of general relativity, exhibiting both a massless and a
massive graviton, were obtained in N2 . Constraints obtained from BBN on a
number of higher-order modified gravity theories, including the $f(G)$ Gauss-
Bonnet gravity, the $f(P)$ cubic gravities, obtained through the use of the
quadratic-curvature Gauss-Bonnet $G$ term, and the cubic-curvature
combination, string-inspired quadratic Gauss-Bonnet gravity coupled to the
dilaton field, models with string-inspired quartic curvature corrections, and
running vacuum models were investigated in N3 . It was found that all the
above mentioned models can satisfy the BBN constraints, and hence they
represent viable cosmological scenarios.
$f\left(\mathcal{T},\mathcal{T}_{G}\right)$ gravity, were $\mathcal{T}$ is the
torsion scalar, and $\mathcal{T}_{G}$ is the teleparallel equivalent of the
Gauss–Bonnet term, was analyzed with respect to the BBN constraints in N4 .
The deviations of the freeze-out temperature determined by the presence of the
extra torsion terms were calculated, and a comparison to the standard
$\lambda$CDM model was performed. It turns out that the BBN constraints are
satisfied by a specific logarithmic model for large regions of the model
parameters. The implications of Einstein-Aether and modified Horava-Lifshitz
theories of gravity on the formation of light elements in the early Universe
were considered in N5 . Some models in the framework of the considered
theories can satisfy the Big Bang nucleosynthesis constraints. The BBN
formalism and observations were used in Anag  to obtain constraints on several
classes of $f(Q)$ gravity models. It was shown that models based on the $f(Q)$
gravity theory can easily satisfy the BBN constraints. This result indicates
that $f(Q)$ modified gravity theories have an important advantage with respect
to other modified gravity approaches. The BBN constraints on the cosmology of
the bumblebee model, a vector-tensor theory of gravitation, were considered in
N6 . In this model a single Lorentz-violating timelike vector field with a
nonzero vacuum expectation value couples to the Ricci tensor and scalar. By
using the observational data coming from the BBN, and the matter-antimatter
asymmetry in the Baryogenesis era, some constraints on the vacuum expectation
value of the bumblebee timelike vector field can be imposed. The BBN was used
in N7  to constrain the free parameters of the extended Starobinsky model. The
primordial abundances of the light elements, such as He, D, He, or Li, have
been compared with the most recent observational data. Most non-standard
cosmic evolutions can not easily satisfy these BBN constraints, but a free
parameter of the viable models has an upper limit imposed by the constraints.
BBN bounds on the model parameters have been obtained for the $f(R)$ gravity
scalarons N8 , on the $f(T,\mathcal{T}$ gravity N9 , on the variation of the
gravitational constant N10 , and on the quadratic energy–momentum squared
gravity N11 , respectively.

It is the goal of this study to study the BIg Bang Nucleosynthesis (BBN)
process within the framework of the Weyl type $f(Q,T)$ modified gravity
theory. We use the Big Bang Nucleosynthesis (BBN) formalism and observed light
element abundance to extract constraints on several $f(Q,T)$ models. The three
models we are considering are given by $f(Q,T)=\alpha
Q+\left(\beta/6\kappa^{2}\right)T$,
$f(Q,T)=\left(\alpha/6H_{0}^{2}\kappa^{2}\right)QT$, and $f(Q,T)=\eta
H_{0}^{2}\exp\left[\left(\mu/6H_{0}^{2}\right)\right]+\left(\nu/6\kappa^{2}\right)T$,
respectively, where $\alpha$, $\beta$, $H_{0}$, $\nu$, and $\eta$ are
constants, while $\kappa^{2}$ denotes the gravitational coupling constant. As
a first step in our approach we obtain a general, although approximate,
expression for the ratio of the variation of the freeze-out temperature
$\delta\mathcal{T}_{f}/\mathcal{T}_{f}$, by considering the corrections to the
Hubble function due to the presence of the Weyl geometry effects. Moreover, a
second approach is also used, in which we directly obtain the freeze-out
temperature from the equality of the Hubble function and of the particle decay
rate. For each model we begin our analysis of the BBN constraints by
considering the full numerical evolution of the models during the radiation
dominated period. We simultaneously analyze the behaviors of the Hubble
functions of the models, of the particle decay rates, and of the Weyl vector.
A comparison with the Hubble function behavior of standard general relativity
is also performed. Then we explicitly obtain the BBN constraints on the model
parameters from the observational values of the 4He mass fraction and 7Li
abundance. The combined constraints coming from the observed numerical values
of the 4He 7Li are also considered. Based on the obtained results we can
conclude that the Weyl type $f(Q,T)$ cosmological models can not only pass the
cosmological tests in the early Universe, but they can also represent some
viable and attractive alternatives for the physical description of the very
early Universe.

The present paper is organized as follows. The theoretical formalism of the
Weyl type $f(Q,T)$ gravity is briefly reviewed in Section II. The standard BBN
theory, the theoretical freeze-out constraints, as well as the observational
data and results are presented in Section III. The BBN constraints on three
Weyl type $f(Q,T)$ are analyzed in detail in Section IV. We discuss and
conclude our results in Section V.

##  II Weyl type $f(Q,T)$ gravity

In the present Section we introduce the action of the Weyl type $f(Q,T)$
gravity, and we briefly review the basic geometric concepts necessary for the
development of the basic theory. The field equations of the theory are
obtained by varying the action with respect to the metric tensor. The
generalized Friedmann equations, obtained for a flat FLRW geometry, are also
presented.

###  II.1 The gravitational field equations of the $f(Q,T)$ theory

In the present Subsection we briefly present the action of the Weyl type
$f(Q,T)$ gravity theory, as well the geometrical foundations of the theory.

####  II.1.1 Action and geometry

The action of the Weyl type $f(Q,T)$ gravity theory is given by ref2

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\int\left[\kappa^{2}f(Q,T)-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}-\frac{1}{2}m^{2}\omega^{2}+\mathcal{L}_{m}\right]$
(1) $\displaystyle\times\sqrt{-g}d^{4}x,$

where $\kappa^{2}=1/16\pi G$, $Q$ is the non-metricity scalar,
$\Omega_{\mu\nu}$ is the strength of the Weyl vector, while $\mathcal{L}_{m}$
is the action of the ordinary (baryonic) matter. This action is defined in a
Weyl geometric framework, in which the metric condition is not satisfied, so
that

$Q_{\alpha\mu\nu}\equiv\bar{\nabla}_{\alpha}g_{\mu\nu}=2g_{\mu\nu}\omega_{\alpha},$
(2)

where $Q_{\alpha\mu\nu}$ is the general non-metricity tensor, and
$\omega_{\alpha}$ is the Weyl vector, with square given by
$\omega^{2}=\omega_{\alpha}\omega^{\alpha}$. The deformation tensor
$L^{\lambda}_{\ \mu\nu}$ can be defined by using the non-metricity tensor as

$\displaystyle L^{\lambda}_{\ \mu\nu}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{2}g_{\lambda\gamma}(Q_{\mu\gamma\nu}+Q_{\nu\gamma\mu}-Q_{\gamma\mu\nu})$
(3) $\displaystyle=$
$\displaystyle-\delta^{\lambda}_{\nu}\omega_{\mu}-\delta^{\lambda}_{\mu}\omega_{\nu}+g_{\mu\nu}\omega^{\lambda}.$

The non-metricity scalar $Q$ is introduced according to the definition

$\begin{split}Q&\equiv-g^{\mu\nu}(L^{\alpha}_{\ \beta\mu}L^{\beta}_{\
\nu\alpha}-L^{\alpha}_{\ \beta\alpha}L^{\beta}_{\
\mu\nu})=-6\omega^{2}.\end{split}$ (4)

The strength of the Weyl vector is defined as

$\Omega_{\mu\nu}\equiv\nabla_{\nu}\omega_{\mu}-\nabla_{\mu}\omega_{\nu},$ (5)

where $\nabla_{\mu}$ denotes the covariant derivative with respect to Levi-
Civita connection of the Riemannian geometry.

The Weyl scalar, the second contraction of the Weyl curvature tensor, is
obtained as

$\bar{R}=R+6(\nabla_{\nu}\omega^{\nu}-\omega^{2}),$ (6)

where $R$ is the Ricci scalar of the Riemannian geometry.

In the following we impose the flatness condition, which requires $\bar{R}=0$.
This condition is implemented in the action with the use of a Lagrangian
multiplier $\lambda$. Hence, for the action of the Weyl type $f(Q,T)$ gravity
we obtain the final expression

$\begin{split}\mathcal{S}&=\int
d^{4}x\sqrt{-g}[\kappa^{2}f(Q,T)-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}-\frac{1}{2}m^{2}\omega^{2}\\\
&+\lambda(R+6\nabla_{\alpha}\omega^{\alpha}-6\omega^{2})+\mathcal{L}_{m}].\end{split}$
(7)

We also introduce the matter energy-momentum tensor, defined according to

$T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}}=g_{\mu\nu}-2\frac{\delta\mathcal{L}_{m}}{\delta g^{\mu\nu}},$ (8)

The trace of the energy-momentum tensor is defined as $T=T_{\mu}^{\mu}$.

####  II.1.2 Field equations

By considering the variation of the action with respect to the Weyl vector
$\omega^{\mu}$, and by taking into account that $\delta
S/\delta\omega^{\mu}=0$, after an integration by parts, we obtain the equation
describing the variation of the Weyl vector as

$\nabla^{\nu}\Omega_{\mu\nu}-m_{eff}^{2}\omega_{\mu}=6\nabla_{\mu}\lambda,$
(9)

where we have denoted

$m_{eff}^{2}=m^{2}+12\kappa^{2}f_{Q}+12\lambda.$ (10)

Thus, in the present theory, the Weyl vector satisfies a Proca-type field
equation. We also introduce the notations $f_{T}=\partial f(Q,T)/\partial T$,
and $f_{q}=\partial f(Q,T)/\partial Q$, respectively. The variation of the
action with respect to the metric $g^{\mu\nu}$ yields the field equations ref2

$\displaystyle\frac{1}{2}(T_{\mu\nu}+S_{\mu\nu})-\kappa^{2}f_{T}(T_{\mu\nu}+\Theta_{\mu\nu})+\frac{\kappa^{2}}{2}g_{\mu\nu}f$
$\displaystyle=-6\kappa^{2}f_{Q}\omega_{\mu}\omega_{\nu}+\lambda(R_{\mu\nu}-6\omega_{\mu}\omega_{\nu}+3g_{\mu\nu}\nabla_{\rho}\omega^{\rho})$
$\displaystyle+3g_{\mu\nu}\omega^{\rho}\nabla_{\rho}\lambda-6\omega_{(\mu}\nabla_{\nu)}\lambda+g_{\mu\nu}\Box\lambda-\nabla_{\mu}\nabla_{\nu}\lambda,$
(11)

where we have introduced the tensor $\Theta_{\mu\nu}$, defined as

$\Theta_{\mu\nu}\equiv g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta
g^{\mu\nu}}=g_{\mu\nu}\mathcal{L}_{m}-2T_{\mu\nu}.$ (12)

The tensor $S_{\mu\nu}$ represents the energy-momentum tensor of the Proca
field

$\displaystyle S_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{4}g_{\mu\nu}\Omega_{\rho\sigma}\Omega^{\rho\sigma}+\Omega_{\mu\rho}\Omega_{\nu}^{\rho}-\frac{1}{2}m^{2}g_{\mu\nu}\omega^{2}+m^{2}\omega_{\mu}\omega_{\nu}.$

The gravitational equations of the Weyl type $f(Q,T)$ theory can be
reformulated in a form similar to the standard Einstein field equations as

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{1}{2\lambda}(T_{\mu\nu}+S_{\mu\nu})-\frac{\kappa^{2}}{\lambda}f_{T}(T_{\mu\nu}+\Theta_{\mu\nu})$
$\displaystyle+\frac{\kappa^{2}}{2\lambda}g_{\mu\nu}f+\frac{6\kappa^{2}}{\lambda}f_{Q}\omega_{\mu}\omega_{\nu}-\frac{3}{\lambda}g_{\mu\nu}\omega^{\rho}\nabla_{\rho}\lambda+\frac{6}{\lambda}\omega_{(\mu}\nabla_{\nu)}\lambda$
$\displaystyle-\frac{1}{\lambda}(g_{\mu\nu}\Box\lambda-\nabla_{\mu}\nabla_{\nu}\lambda)+6\omega_{\mu}\omega_{\nu}-3g_{\mu\nu}\omega^{2}.$
(14)

From the trace of the field equations we find

$\displaystyle\frac{1}{2}(T+S)-\kappa^{2}f_{T}(T+\Theta)$ $\displaystyle=$
$\displaystyle-2\kappa^{2}f-6\kappa^{2}f_{Q}\omega^{2}$ $\displaystyle-$
$\displaystyle\frac{m^{2}}{2}\nabla^{\mu}\omega_{\mu}-6\kappa^{2}\nabla^{\mu}(f_{Q}\omega_{\mu}),$

where $S=S_{\mu}^{\mu}=-m^{2}\omega^{2}$

The covariant derivative of both sides of Eq. (II.1.2) gives

$\displaystyle\left(\frac{1}{2}+\kappa^{2}f_{T}\right)\nabla^{\mu}T_{\mu\nu}$
$\displaystyle=$
$\displaystyle\kappa^{2}g_{\mu\nu}\nabla^{\mu}(f_{T}\mathcal{L}_{m})-\kappa^{2}T_{\mu\nu}\nabla^{\mu}f_{T}$
(16) $\displaystyle-\frac{\kappa^{2}}{2}f_{T}\nabla_{\nu}T.$

From Eq. (16) it follows that due to the geometry-matter interaction, in the
Weyl type $f(Q,T)$ theory an extra-force, acting on the massive test particle,
is generated. The extra force is given by ref2

$\displaystyle\mathcal{F}^{\rho}$ $\displaystyle=$
$\displaystyle-\frac{h^{\nu\rho}\nabla_{\nu}p}{\rho+p}+\frac{h^{\nu\rho}\nabla^{\mu}T_{\mu\nu}}{\rho+p}$
$\displaystyle=$
$\displaystyle\frac{\kappa^{2}h^{\nu\rho}}{(\rho+p)(1+2\kappa^{2}f_{T})}\Big{[}-\frac{1}{\kappa^{2}}\nabla_{\nu}p+2\nabla_{\nu}\left(f_{T}\mathcal{L}_{m}\right)$
(17) $\displaystyle-f_{T}\nabla_{\nu}T-2T_{\mu\nu}\nabla^{\mu}f_{T}\Big{]},$

where by $h^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu}$ we have denoted the projection
operator.

###  II.2 The generalized Friedmann equations

To investigate Big Bang Nucleosynthesis in the Weyl type $f(Q,T)$ gravity
theory we assume first that the early Universe can be described by the flat,
homogeneous and isotropic Friedmann-Lemaitre-Robertson-Walker (FLRW) metric,
given by

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j},$ (18)

where $a(t)$ is the scale factor. We also define the Hubble function as
$H=\dot{a}/a$. In the following by a dot we denote the derivative with respect
to the cosmological time $t$. For the Weyl vector we adopt the particular form

$\omega_{\mu}=(\psi(t),0,0,0)$ (19)

which follows from the conditions of the homogeneity and isotropy of the
Universe. With this choice we easily obtain the relations
$\omega^{2}=-\psi^{2}$, and $Q=6\psi^{2}$, respectively.

Hence, the energy balance equation and the generalized Friedmann equations of
the Weyl type $f(Q,T)$ gravity can be written down as

$\dot{\rho}+3H(\rho+p)=\frac{-\kappa^{2}}{1+2\kappa^{2}f_{T}}[2(\rho+p)\dot{f}_{T}+f_{T}(\dot{\rho}-\dot{p})],$
(20) $\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda}[(1+2\kappa^{2}f_{T})\rho+2\kappa^{2}f_{T}p-\kappa^{2}f-\frac{m^{2}\psi^{2}}{2}$
(21)
$\displaystyle-6\lambda\psi^{2}+m_{eff}^{2}H\psi]=\frac{1}{2\lambda}(\rho+\rho_{eff}),$

and

$\displaystyle 2\dot{H}$ $\displaystyle=$
$\displaystyle\frac{-1}{2\lambda}[(1+2\kappa^{2}f_{T})(\rho+p)-\frac{m_{eff}^{2}}{3}(\dot{\psi}+\psi^{2}-H\psi)$
(22)
$\displaystyle-4\kappa^{2}\dot{f}_{Q}\psi]=\frac{-1}{2\lambda}(\rho+\rho_{eff}+p+p_{eff}),$

respectively, where we have introduced the notations

$\rho_{eff}\equiv
m_{eff}^{2}H\psi+2\kappa^{2}f_{T}(\rho+p)-\kappa^{2}f-\frac{m^{2}\psi^{2}}{2}-6\lambda\psi^{2},$
(23)

and

$\displaystyle p_{eff}$ $\displaystyle\equiv$
$\displaystyle-\frac{m_{eff}^{2}}{3}(\dot{\psi}+\psi^{2}-H\psi)-4\kappa^{2}\dot{f}_{Q}\psi$
(24)
$\displaystyle-\left(m_{eff}^{2}H\psi-\kappa^{2}f-\frac{m^{2}\psi^{2}}{2}-6\lambda\psi^{2}\right)$
$\displaystyle=$
$\displaystyle-\frac{m_{eff}^{2}}{3}(\dot{\psi}+\psi^{2}+2H\psi)-4\kappa^{2}\dot{f}_{Q}\psi+\kappa^{2}f$
$\displaystyle+\frac{m^{2}\psi^{2}}{2}+6\lambda\psi^{2},$

respectively.

##  III Standard $\Lambda$CDM Big Bang Nucleosynthesis

We begin our analysis of the Big Bang Nucleosynthesis constraints on the Weyl
type $f(Q,T)$ gravity theory by briefly presenting the standard model of Big
Bang Nucleosynthesis, developed in the framework of standard general
relativity, and of the $\Lambda$CDM cosmological model. The standard Big Bang
Nucleosynthesis model makes some definite predictions on the abundances of
primordial elements, which we will review below.

###  III.1 The radiation dominated Universe

In the standard $\Lambda$CDM model, the two Friedmann equations describing the
cosmological evolution are given by ref4

$3H_{GR}^{2}=\frac{\rho}{2\kappa^{2}},$ (25)

and

$2\dot{H}_{GR}=-\frac{1}{2\kappa^{2}}(\rho+p),$ (26)

respectively, where by $H_{GR}$ we have denoted the standard general
relativistic expression of the Hubble function. The matter content of the very
early Universe is assumed to consist mainly of radiation, satisfying the
equation of state $p_{r}=\rho_{r}/3$, and with energy density given by

$\rho\simeq\rho_{r}=\frac{\pi^{2}g_{\ast}}{30}\mathcal{T}^{4}$ (27)

where $\mathcal{T}$ is the temperature of the photon gas, and the effective
number of degrees of freedom $g_{\ast}=10.6$. The evolution of the Hubble
function is thus fully determined by the temperature of the photon gas

$H_{GR}=\sqrt{\frac{\pi^{2}g_{\ast}}{180\kappa^{2}}}\mathcal{T}^{2}.$ (28)

In the radiation-dominated epoch, from the energy conservation equation we can
easily find the relation

$\rho=\frac{\rho_{0}a_{0}^{4}}{a^{4}},$ (29)

where $\rho_{0}$ and $a_{0}$ are constants of integration. Combining this
relation with Eq. (25), after integrating on both sides, we obtain

$a=\sqrt{\frac{\rho_{0}a_{0}^{4}}{6\kappa^{2}}t},$ (30)

and

$\frac{1}{t}=\sqrt{\frac{2\rho_{r}}{3\kappa^{2}}}=\sqrt{\frac{\pi^{2}g_{\ast}}{45\kappa^{2}}}\mathcal{T}^{2},$
(31)

respectively.

##### The photon mass correction.

In the radiation-dominated epoch, we can consider the photon mass correction
in the expression of the pressure of the radiation fluid. In the de Broglie-
Proca cosmology 2016de , the pressure of the photons is given by

$p_{r}=\frac{\rho_{r}}{3}\left(1-\frac{\sqrt{15g_{\ast}}m_{\gamma}^{2}}{6\pi\sqrt{2\rho_{r}}}\right)=\frac{\rho_{r}}{3}-\frac{\sqrt{15g_{\ast}}m_{\gamma}^{2}}{18\sqrt{2}\pi}\sqrt{\rho_{r}},$
(32)

with $m_{\gamma}\leq 10^{-18}\;{\rm eV}$. Hence, in the radiation-dominated
epoch, the radiation pressure can be expressed as

$p_{r}=\frac{\rho_{r}}{3}-\frac{g_{\ast}}{2}\dfrac{m_{\gamma}^{2}\mathcal{T}^{2}}{18}.$
(33)

For the trace of the energy-momentum tensor of the Proca type radiation field
we obtain
$T_{r}=-\rho_{r}+3p_{r}=-\left(g_{\ast}/2\right)m_{\gamma}^{2}\mathcal{T}^{2}/6$.
As the photon mass is small as compared with the Plank mass, in some physical
situations we can directly ignore the last term in the expression of the
pressure, and approximate it as $p_{r}=\rho_{r}/3$.

###  III.2 The freeze-out constraint

An important observational quantity in the study of the Big Bang
Nucleosynthesis is the freeze-out temperature $\mathcal{T}_{f}$. A fundamental
physical process in the early Universe is the conversion of protons and
neutrons taking place through the weak interactions. If the temperature of the
Universe is greater than $1\;{\rm MeV}$, the neutron-proton conversion
reactions are in equilibrium. Once the temperature of the Universe becomes
lower than $1\;{\rm MeV}$, the neutron-to-proton ratio freezes out at a value
of around $1/6$, and then it slowly decrease due to free neutron decay ref4 .
Consequently, the neutron abundance can be estimated once the neutron to
proton $\Gamma_{pn}(\mathcal{T})$ and the proton to neutron
$\Gamma_{np}(\mathcal{T})$ conversion rates are known.

Neutrons decay into protons through three distinct channels Cybrut ; ref4

$n\longrightarrow p+e^{-}+\bar{\nu}_{e},$ $n+\nu_{e}\longrightarrow p+e^{-}$

and

$n+e^{+}\longrightarrow p+\bar{\nu}_{e},$

respectively. Hence

$\displaystyle\Gamma_{pn}(T)$ $\displaystyle=$
$\displaystyle\Gamma_{(n\longrightarrow
p+e^{-}+\bar{\nu}_{e})}+\Gamma_{(n+\nu_{e}\longrightarrow p+e^{-})}$ (34)
$\displaystyle+\Gamma_{(n+e^{+}\longrightarrow p+\bar{\nu}_{e})}.$

Once the inverse decay rate $\Gamma_{np}(\mathcal{T})$ is known, the total
conversion rate
$\Gamma_{tot}(\mathcal{T})=\Gamma_{pn}(\mathcal{T})+\Gamma_{np}(\mathcal{T})$
can be obtained as Kolb ; Bern

$\Gamma_{tot}(\mathcal{T})\equiv\Gamma(\mathcal{T})=4A\mathcal{T}^{3}(4!\mathcal{T}^{2}+2\times
3!\mathcal{Q}\mathcal{T}+2!\mathcal{Q}^{2})$ (35)

here $\mathcal{Q}=m_{n}-m_{p}=1.29\times 10^{-3}\;{\rm GeV}$ and $A=1.02\times
10^{-11}\;{\rm GeV}^{-4}$.

When the temperature is large, the total conversion rate is much larger than
the Hubble rate, and thus the protons and the neutrons are in the equilibrium.
This equilibrium situation is maintained until a freeze-out temperature is
reached, when the two particles decouple from each other.

The freeze-out temperature is defined in terms of the Hubble function
according to the relation Kolb ; Bern

$H(\mathcal{T}_{f})=\Gamma_{tot}(\mathcal{T}_{f})\approx
c_{q}\mathcal{T}_{f}^{5},$ (36)

where $c_{q}=4A4!\approx 9.8\times 10^{-10}\;{\rm GeV}^{-4}$. Using Eq. (28)
at $\mathcal{T}=\mathcal{T}_{f}$, we find

$\mathcal{T}_{f}=\bigg{(}\frac{\pi^{2}g_{*}}{180\kappa^{2}c_{q}^{2}}\bigg{)}^{1/6}\approx
0.6786\;{\rm MeV}.$ (37)

At the freeze-out time, the neutron-proton ratio is given as

$\left(\frac{n}{p}\right)_{f}\simeq e^{-\frac{\mathcal{Q}}{\mathcal{T}}}$ (38)

We define the neutron mass fraction as

$X\equiv\frac{n}{n+p}.$ (39)

The evolution equation of $X$ is given by ref4

$\frac{dX}{dx}=\frac{x\Gamma_{np}}{\sqrt{\frac{4\pi^{3}G\mathcal{Q}^{4}g_{\ast}}{45}}}[e^{-x}-X(1+e^{-x})].$
(40)

By substituting $X$ with e-μ(x)X’, where

$\displaystyle\mu(x)$ $\displaystyle\equiv$
$\displaystyle\int_{x_{i}}^{x}\frac{x^{\prime}\Gamma_{np}(x^{\prime})}{\sqrt{\frac{4\pi^{3}G\mathcal{Q}^{4}g_{\ast}}{45}}}(1+e^{-x^{\prime}})dx^{\prime}=-\frac{255}{\tau_{n}\sqrt{\frac{4\pi^{3}G\mathcal{Q}^{4}g_{\ast}}{45}}}$
(41) $\displaystyle\times$
$\displaystyle\left.\left[[\left(\frac{4}{x^{3}}+\frac{3}{x^{2}}+\frac{1}{x}\right)+\left(\frac{4}{x^{3}}+\frac{1}{x^{2}}\right)e^{-x}\right]\right|_{x_{i}}^{x},$

with $x_{i}$ denoting the value of $x$ corresponding to a very early time,
$X\left(x_{i}\right)=X^{\prime}\left(x_{i}\right)=0.5$, we can obtain the
solution of Eq. (40) as

$X(x)=0.5e^{-\mu(x)}+\int_{x_{i}}^{x}\frac{x^{\prime}\Gamma_{np}(x^{\prime})e^{-x^{\prime}}}{\sqrt{\frac{4\pi^{3}G\mathcal{Q}^{4}g_{\ast}}{45}}}e^{\mu(x^{\prime})-\mu(x)}dx^{\prime}.$
(42)

Then we obtain $X(\infty)=0.148$. The value of $X$ in the standard BBN is also
represented as

$X_{f}=\frac{e^{-\frac{\mathcal{Q}}{\mathcal{T}_{f}}}}{1+e^{-\frac{\mathcal{Q}}{\mathcal{T}_{f}}}}=X(\infty),$
(43)

from which we obtain $(n/p)_{f}=0.1737$, and $\mathcal{T}_{f}\approx 0.7387$
MeV,

Between the freeze-out and the nucleosynthesis phases, a fraction of neutrons
decay into protons

$X_{n}(t_{N})=X_{f}e^{-\frac{t_{N}-t_{f}}{\tau_{n}}}=\chi\frac{e^{-\frac{\mathcal{Q}}{\mathcal{T}_{f}}}}{1+e^{-\frac{\mathcal{Q}}{\mathcal{T}_{f}}}},$
(44)

where $\chi\equiv e^{-\frac{t_{N}-t_{f}}{\tau_{n}}}$, thus reducing the
neutron mass fraction further. Supposing all neutrons have transformed into
${}^{4}He$ after the BBN, the ${}^{4}He$ mass fraction is given as ref4

$Y_{P}=2X_{n}(t_{N})=\chi\frac{2e^{-\frac{\mathcal{Q}}{\mathcal{T}_{f}}}}{1+e^{-\frac{\mathcal{Q}}{\mathcal{T}_{f}}}}.$
(45)

##### The freeze-out temperature constraint.

Let’s assume now that the freeze-out temperature can vary due to the presence
of modified gravity, or other physical effects. Since
$H(\mathcal{T}_{f})\approx c_{q}\mathcal{T}_{f}^{5}$, we obtain immediately
the relation $\delta
H\left(\mathcal{T}_{f}\right)=5c_{q}\mathcal{T}_{f}^{4}\delta\mathcal{T}_{f}$,
which gives the relation

$\frac{\delta
H\left(\mathcal{T}_{f}\right)}{H\left(\mathcal{T}_{f}\right)}=5\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}.$
(46)

The magnitude of the freeze-out constraint can be obtained as follows. From
Eq. (31), we obtain first

$\frac{dt_{f}}{d\mathcal{T}_{f}}=-\frac{2t_{f}}{\mathcal{T}_{f}}.$ (47)

We take now the derivative of Eq. (45) to find

$\displaystyle\frac{dY_{P}}{d\mathcal{T}_{f}}$ $\displaystyle=$
$\displaystyle-\frac{2t_{f}}{\tau_{n}\mathcal{T}_{f}}Y_{P}+\frac{\mathcal{Q}}{\mathcal{T}_{f}^{2}(1+x)}Y_{P}$
(48) $\displaystyle=$
$\displaystyle-\frac{2t_{f}}{\tau_{n}\mathcal{T}_{f}}Y_{P}+\left(1-\frac{Y_{P}}{2\chi}\right)\ln(\frac{2\chi}{Y_{P}}-1)\frac{Y_{P}}{\mathcal{T}_{f}}.$

Hence the deviation of the ${}^{4}He$ abundance can be represented as the
deviation of the freeze-out temperature

$\delta
Y_{P}=Y_{P}\left[\left(1-\frac{Y_{P}}{2\chi}\right)\ln\left(\frac{2\chi}{Y_{P}}-1\right)-\frac{2t_{f}}{\tau_{n}}\right]\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}.$
(49)

From the observational data (see Anag  and references therein), we obtain for
the deviation of the freeze-out temperature the upper limit

$\left|\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}\right|<4.7\times
10^{-4}.$ (50)

As one can see, the deviation of the 4He abundance has a strong dependence on
the freeze-out temperature, whose form does not change significantly in the
non standard models. Hence, we can reformulate the observations of the
${}^{4}He$ abundance into the constraints on the freeze-out temperature
$\mathcal{T}_{f}$. Thus, we do not need to use explicitly the value $Y_{P}$.
We just need to make the freeze-out temperature not to be too large or too
small to obtain the correct expression of the 4He abundance, whatever is the
specific form of the freeze-out temperature.

With the degrees of freedom in the freeze-out temperature taken as
$g_{\ast}=10.6$ 2016On , we obtain the freeze-out time as $t_{f}=1.36$ s. From
the observational data, we have for the $He^{4}$mass fraction the value
$Y_{P}=0.245$, and $\left|\delta Y_{P}\right|<2\times 10^{-3}$. Combining this
relation with the freeze-out time, we have $t_{N}=169\;{\rm s}=2.57\times
10^{26}\;{\rm GeV}^{-1}$.

By assuming for the number of the effective degree of freedom the value
$g_{\ast}=3.37$, we obtain the nucleosynthesis temperature of the deuterium as
$\mathcal{T}_{N}=8.83\times 10^{-2}$ MeV.

In the following we introduce the notation

$Z\equiv\frac{H}{H_{GR}}.$ (51)

###  III.3 ${}^{2}H$, ${}^{4}He$ and ${}^{7}Li$ abundances

#####  ${}^{2}H$.

Deuterium ${}^{2}H$ is produced in the early Universe through the reaction
$n+p\rightarrow^{2}H+\gamma$ ref4 . The best fit of the deuterium abundance is
obtained as Steigman-2012

$Y_{D_{p}}=2.6\times(1\pm
0.06)\times\left[\frac{6}{\eta_{10}-6(Z-1)}\right]^{1.6}$ (52)

For $Z=1$ and $\eta_{10}=6$, we obtain $y_{D_{p}}=2.6\pm 0.16$. The
observational constraint on the deuterium abundance is $y_{Dp}=2.55\pm 0.03$
De1 . Hence $Z$ is constrained in the range $Z=1.062\pm 0.444$ N9 .

#####  ${}^{4}He$.

The production of ${}^{4}He$ begins with the production of ${}^{2}H$ from the
reaction of a neutron $n$ and of a proton $p$. Then deuterium is converted
into ${}^{3}He$ and tritium $T$ via the reactions
$n+p\rightarrow^{2}H+\gamma$, ${}^{2}H+^{2}H\rightarrow^{3}He+n$, and
${}^{2}H+^{2}H\rightarrow^{3}H+p$, respectively. Finally, ${}^{4}He$ is
produced from the reactions of ${}^{3}H$ with ${}^{2}H$ and ${}^{3}He$, given
by ${}^{2}H+^{3}H\rightarrow^{4}He+n$ and ${}^{2}H+^{3}He\rightarrow^{4}He+p$
Kolb ; ref4 .

The cosmological ${}^{4}He$ abundance is obtained from the numerical best fit
of the observational data as Steigman-2004 ; Steigman-2007

$Y_{{}^{4}He}=0.2485\pm
0.0006+0.0016\left[\left(\eta_{10}-6\right)+100\left(Z-1\right)\right].$ (53)

For $Z=1$, corresponding to standard Big Bang Nucleosynthesis, we obtain
$Y_{{}^{4}He}=0.2485\pm 0.0006$. The cosmological observations give for
${}^{4}He$ abundance the value $0.2449\pm 0.0040$ De1 . This constrains $Z$ in
the range $1.0475\pm 0.105$ N9 .

#####  ${}^{7}Li$.

The lithium abundance is also an important cosmological indicator, and the
study of the abundance of these nuclei could provide a strong constraint on
the theoretical model. The lithium abundance is given by the following
numerical best fit Li1 ; Li2 ; Steigman-2012

$\frac{{}^{7}Li}{H}=4.82\times 10^{-10}\times(1\pm
0.1)\times\left[\frac{10^{10}\eta-3(Z-1)}{6}\right]^{2}.$ (54)

In the standard BBN case, $Z=1$, and we obtain

$\frac{{}^{7}Li}{H}=4.82\times 10^{-10}\times(1\pm
0.1)\times\left(\frac{10^{10}\eta}{6}\right)^{2}.$ (55)

After using the observed value of baryon-photon ratio $\eta=6\times 10^{-10}$,
we can obtain the ratio

$\frac{{}^{7}Li}{H}=4.82\times 10^{-10}\times\left(1\pm 0.1\right),$ (56)

which is much larger than the observed value

$\left.\frac{{}^{7}Li}{H}\right|_{obs}=(1.6\pm 0.3)\times 10^{-10}.$ (57)

The lithium abundance is important since the $\eta_{10}$ parameter, which fits
well the abundances of hydrogen and helium, cannot describe the observations
of ${}^{7}Li$. The ratio of the value of ${}^{7}Li$ abundance predicted by the
standard Big Bang Nucleosynthesis model and the observations is of the order
of $2.4-4.3$ Li1 ; Li2 . Hence, the standard BBN cannot explain the observed
abundance of ${}^{7}Li$. This problem is known as the Lithium problem Li1 .
The observational value of the lithium abundance is $Y_{p}^{(Li)}=1.6\pm 0.3$
De1 . Thus, the constraint on $Z$ obtained from the observed ${}^{7}Li$
abundance is $Z=1.960025\pm 0.076675$. It is important to note that this
constraint does not overlap with the constraints on Deuterium-2 and Helium-4
N9 .

##  IV Nucleosynthesis constraints on Weyl type $f(Q,T)$ gravity models

In the present Section we consider the BBN constraints on several Weyl type
$f(Q,T)$ gravity cosmological models. We begin our analysis by imposing the
flatness condition, which allows to express the time component Weyl vector as
a function of $H$. Then, we will obtain the general expression of the freeze-
out temperature in Weyl type $f(Q,T)$ gravity, and we will apply these general
equations to constrain some particular cosmological models.

###  IV.1 Freeze-out temperature in Weyl type $f(Q,T)$ gravity

The flatness condition (zero total Weyl curvature $\tilde{R}$), introduced in
the action via a Lagrange multiplier, can be formulated in the cosmological
case as

$\dot{H}+2H^{2}-\dot{\psi}-3H\psi+\psi^{2}=0,$ (58)

or, equivalently

$\left(\dot{\psi}-\dot{H}\right)-(\psi-2H)(\psi-H)=0.$ (59)

The flatness condition is identically satisfied by the choice

$\psi=H,$ (60)

and in the following we will adopt, in the approximate analysis of the
cosmological implications of our models, this expression for $\psi$. We will
also adopt for the Lagrange multiplier $\lambda$ the standard general
relativistic approximation $\lambda\approx\kappa^{2}={\rm constant}$.
Moreover, in these approximations we obtain for the non-metricity scalar $Q$
the expression

$Q=6H^{2}.$ (61)

From Eq. (21) we obtain for the Hubble function the general expression

$H=\sqrt{\frac{1}{6\lambda}}\sqrt{\rho_{r}}\sqrt{1+\frac{\rho_{eff}}{\rho_{r}}}.$
(62)

By performing a first order series expansion of the square root we obtain

$\delta H=H-H_{GR}=\frac{1}{2}\frac{\rho_{eff}}{\rho_{r}}H_{GR}.$ (63)

Hence, for the variation of the freeze-out temperature we obtain the general
expression

$\left|\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}\right|=\frac{1}{10c_{q}}\frac{1}{\mathcal{T}_{f}^{5}}\left.\left[\frac{\rho_{eff}\left(\mathcal{T}\right)}{\rho_{r}\left(\mathcal{T}\right)}H_{GR}\left(\mathcal{T}\right)\right]\right|_{\mathcal{T}=\mathcal{T}_{f}}.$
(64)

Under the considered approximations we obtain successively

$m_{eff}^{2}=m^{2}+12\kappa^{2}\left(1+f_{Q}\right),$ (65)

and

$\displaystyle\rho_{eff}$ $\displaystyle=$
$\displaystyle-\kappa^{2}f+2\kappa^{2}f_{T}\left(\rho_{r}+p_{r}\right)$ (66)
$\displaystyle+\frac{1}{2}H^{2}\left[m^{2}+12\kappa^{2}\left(1+2f_{Q}\right)\right],$

respectively. In Eq. (66) we can approximate $H$ by its general relativistic
expression $H_{GR}$, $H\approx H_{GR}$, and $\rho_{r}$ and $p_{r}$ by their
standard expressions, by ignoring the corrections due to the photon mass.
Moreover, we have $H_{GR}^{2}/\rho_{r}=1/6\kappa^{2}$. Hence we obtain

$\frac{\rho_{eff}}{\rho_{r}}=1-\kappa^{2}\frac{f}{\rho_{r}}+2f_{Q}+\frac{8}{3}\kappa^{2}f_{T}+\frac{m^{2}}{12\kappa^{2}}.$
(67)

Thus, for the freeze-out constraint in the Weyl type $f(Q,T)$ gravity we
obtain the general expression

$\displaystyle\left|\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{10c_{q}}\sqrt{\frac{\pi^{2}g_{\ast}}{180\kappa^{2}}}\frac{1}{\mathcal{T}_{f}^{3}}\Bigg{(}1-\frac{30\kappa^{2}}{\pi^{2}g_{\ast}}\frac{f}{\mathcal{T}_{f}^{4}}+2f_{Q}$
(68)
$\displaystyle+\frac{8}{3}\kappa^{2}f_{T}+\frac{m^{2}}{12\kappa^{2}}\Bigg{)}.$

As for the quantity $Z=H/H_{GR}$, from Eq. (63) we obtain the expressions

$\displaystyle Z$ $\displaystyle=$ $\displaystyle
1+\frac{1}{2}\frac{\rho_{eff}}{\rho_{r}}$ (69) $\displaystyle=$ $\displaystyle
1+\frac{1}{2}\left(1-\kappa^{2}\frac{f}{\rho_{r}}+2f_{Q}+\frac{8}{3}\kappa^{2}f_{T}+\frac{m^{2}}{12\kappa^{2}}\right).$

###  IV.2 Model I: $f(Q,T)=\alpha Q+\frac{\beta}{6\kappa^{2}}T$

As a first example of a specific cosmological model in $f(Q,T)$ gravity, we
consider the case when the function $f$ is given by

$f(Q,T)=\alpha Q+\frac{\beta}{6\kappa^{2}}T,$ (70)

where $\alpha$ and $\beta$ are constants. In this case $f_{Q}=\alpha$, and
$f_{T}=\beta/6\kappa^{2}$.

$H^{2}=\frac{1}{108\lambda}[(18+8\beta)\rho+18m_{eff}^{2}H\psi-9m_{eff}^{2}\psi^{2}],$
(71)

and

$\dot{H}=\frac{-1}{12\lambda}\Bigg{[}\frac{4}{3}(3+\beta)\rho+m_{eff}^{2}(-\dot{\psi}-\psi^{2}+H\psi)\Bigg{]},$
(72)

respectively. In the following we assume that $\lambda$ is a constant,
$\lambda\equiv\kappa^{2}$. Moreover, we approximate $Q=6H^{2}$ as $Q\approx
6H_{GR}^{2}=\rho_{r}/\kappa^{2}$. For $T$ we adopt the expression
$T=T_{r}=-\left(g_{\ast}/2\right)m_{\gamma}^{2}\mathcal{T}^{2}/6$.

##### The cosmological evolution in the radiation dominated era.

The comparative dynamical behavior of $H$, $H_{GR}$, $\psi$, $\Gamma$, and $Z$
is represented in Fig. 1. As one can see from Fig. 1, at hight temperatures
there are significant differences between the Hubble functions of the $f(Q,T)$
model, and of the standard general relativity. These differences do decrease
once the temperature of the Universe becomes smaller. The decay rate of the
particles $\Gamma$ also decreases with the temperature, and at the freeze-out
temperature it intersects the Hubble function. The Weyl vector $\psi$ also
monotonically decreases with the temperature decrease.

Figure 1: Temperature evolution of $H_{GR}$, $\Gamma$, $H$ and $\psi$ for the
first Weyl type $f(Q,T)$ gravity model (left panel), and of the ratio
$Z=H/H_{GR}$ (right panel). For the left panel plots the initial value of
$\psi$, $\psi_{ini}$, is $\psi_{ini}=0$, while the parameter values are given
by $\alpha+\frac{m^{2}}{12\kappa^{2}}=-1$, and $\beta=1$, respectively. For
the right panel plot $M=m/\sqrt{12}\kappa=0.08$, and the values of $\alpha$
are $\alpha=-0.10$ (solid line), $\alpha=-0.50$ (dotted line), $\alpha=-0.90$
(short dashed line), $\alpha=-1.30$ (dashed line), and $\alpha=-1.60$ (long
dashed line), respectively.

####  IV.2.1 The freeze-out constraint.

##### Numerical constraints.

As we have already mentioned, the freeze-out temperature is obtained as a
solution of the equation
$H\left(\mathcal{T}_{f}\right)=\Gamma\left(\mathcal{T}_{f}\right)$. The field
equations can be integrated numerically for different values of the model
parameter to obtain the variation of the Weyl vector $\psi$, and the freeze-
out temperature $\mathcal{T}_{f}$. The numerical solution is obtained by
adopting some initial values for $\psi$, which correspond to an initial
temperature of $\mathcal{T}=2.7\times 10^{10}$ K. The constraints on the model
parameters $\left(\alpha+m^{2}/12\kappa^{2},\beta\right)$ and $(\alpha,m)$,
respectively, are presented in Fig. 2.

Figure 2: Numerical freeze-out temperature constraint on the parameters
$\left(\alpha+m^{2}/12\kappa^{2},\beta\right)$ (left panel) and $(\alpha,m)$
(right panel) of the first Weyl type $f(Q,T)$ cosmological model. For the left
panel plots $\psi_{ini}=-1\times 10^{-24}$ GeV, while for the right panel
plots $\psi_{ini}=2\times 10^{-25}$ GeV. The permitted region of the
parameters is the area between the two curves.

##### The analytical freeze-out constraint.

The freeze-out constraint (68) becomes in the first considered model of the
Weyl type $f(Q,T)$ gravity

$\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}=\frac{1}{10c_{q}}\sqrt{\frac{\pi^{2}g_{\ast}}{180\kappa^{2}}}\frac{1}{\mathcal{T}_{f}^{3}}\Bigg{(}1+\alpha+\frac{4}{9}\beta+\frac{5\beta}{12\pi^{2}}\frac{m_{\gamma}^{2}}{\mathcal{T}_{f}^{2}}+\frac{m^{2}}{12\kappa^{2}}\Bigg{)}.$
(73)

By considering that the parameters $\alpha$ and $m^{2}/12\kappa^{2}$ satisfy
the condition $\alpha+m^{2}/12\kappa^{2}\approx-1$, and that the mass of the
photon is negligible in the present context, for the parameter $\beta$ we
obtain the constraint

$\beta\approx\frac{45c_{q}}{2}\sqrt{\frac{180\kappa^{2}}{\pi^{2}g_{\star}}}\mathcal{T}^{3}_{f}\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}.$
(74)

For the numerical estimations we will consider the following numerical values
of the parameters: $g_{\ast}=10$, $\mathcal{T}_{f}=0.0006$ GeV,
$c_{q}=9.8\times 10^{-10}$ GeV-4, and $\lambda=\kappa^{2}=M_{p}^{2}/16\pi$,
where $M_{p}$, the Planck mass, is given by $M_{p}=1.22\times 10^{19}$ GeV.
Due to its smallness we will neglect the term related to the photon mass.

By using the constraint (50) it follows that $\beta<0.0005$. On the other
hand, assuming $\alpha\approx-1$, it follows that the parameters $\beta$ and
$m$ are given by

$\frac{4}{9}\beta+\frac{m^{2}}{12\kappa^{2}}\approx
10c_{q}\sqrt{\frac{180\kappa^{2}}{\pi^{2}g_{\star}}}\mathcal{T}^{3}_{f}\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}.$
(75)

Hence we obtain the constraint $4\beta/9+m^{2}/12\kappa^{2}<0.002$. The
variations of the ratio $\delta\mathcal{T}_{f}/\mathcal{T}_{f}$ are
represented, for a fixed $\alpha$, and for different values of $\beta$, in the
left panel of Fig. 3, while the variation of
$\delta\mathcal{T}_{f}/\mathcal{T}_{f}$ as a function of both $\alpha$ and
$M=m/\sqrt{12}\kappa$ is represented in the right panel of the Figure.

Figure 3:  Variation of the ratio $\delta\mathcal{T}_{f}/\mathcal{T}_{f}$ as
function of $\beta$ in the first Weyl type $f(Q,T)$ cosmological model (left
panel), for $m^{2}/12\kappa^{2}=10^{-8}$, and for different values of
$\alpha$: $\alpha=-0.997$ (solid line), $\alpha=-1.000$ (dotted line),
$\alpha=-1.001$ (short dashed line), $\alpha=-1.003$ (dashed line), and
$\alpha=-1.005$ (long dashed line), respectively. In the right panel the
variation of the ratio $\delta\mathcal{T}_{f}/\mathcal{T}_{f}$ is presented as
function of $\alpha$ and $M=m/\sqrt{12}\kappa$ for $\beta=0.05$.

####  IV.2.2 Deuterium and helium constraints.

Both the deuterium and helium constraints of the BBN can be satisfied if $Z$
is in the (approximate) range $Z\in(0.5,1.6)$. The variation of $Z$ as a
function of $\beta$, for fixed values of $\alpha$ and $M$, presented in the
left panel of Fig. 1, indicates that the present model satisfies both the
deuterium and lithium constraints.

####  IV.2.3 The ${}^{7}Li/H$ ratio.

##### Numerical results.

In this Subsubsection, we will combine the ${}^{4}He$ constraints and the
${}^{7}Li$ constrains to obtain the range of the model parameters. The results
obtained through the numerical integration of the generalized Friedmann
equations are presented in Fig. 4.

Figure 4: Numerical Lithium abundance constraints on the parameters
$\left(\alpha+m^{2}/12\kappa^{2},\beta\right)$ (left panel) and $(\alpha,m)$
(right panel) of the first Weyl type $f(Q,T)$ cosmological model for different
initial values of the Weyl vector $\psi$.

From the numerical analysis one obtains the constraints on the model
parameters as

$-1.35<\alpha+\frac{m^{2}}{12\kappa^{2}}<-1.16,\;0.43<\beta<0.69.$ (76)

and

$-1.83<\alpha+\frac{m^{2}}{12\kappa^{2}}<-1.53,\;1.22<\beta<1.70,$ (77)

respectively.

##### Analytical results.

In the first considered cosmological model in the Weyl type $f(Q,T)$ gravity,
the Lithium abundance is obtained as

$\displaystyle\frac{{}^{7}Li}{H}$ $\displaystyle=$ $\displaystyle 4.82\times
10^{-10}\times(1\pm 0.1)$
$\displaystyle\times\left[\frac{10^{10}\eta-(3/2)\left(1+\alpha+4\beta/9+m^{2}/12\kappa^{2}\right)}{6}\right]^{2}.$

The variation of the ${}^{7}Li/H$ ratio as a function of the model parameter
$\beta$, for fixed $m^{2}/12\kappa^{2}$ values, different values of $\alpha$,
is represented in Fig. 5.

Figure 5:  Variation of the ratio ${}^{7}Li/H$ as function of $\beta$ for
$M=m/\sqrt{12}\kappa=0.10$, and for different values of $\alpha$:
$\alpha=0.72$ (solid line), $\alpha=0.70$ (dotted line), $\alpha=0.68$ (short
dashed line), $\alpha=0.66$ (dashed line), and $\alpha=0.64$ (long dashed
line), respectively.

##### The combined 7Li and freeze-out constraint.

The combined freeze-out temperature and 7Li constraints are represented in
Fig. 6.

Figure 6: The combined freeze-out temperature and Lithium abundance constraint
on the first model parameters $\left(\alpha+m^{2}/12\kappa^{2}\right)$ and
$\beta$. For the initial values of $\psi$ we have considered the values
$-1\times 10^{-24}$ GeV, $0$, and $1\times 10^{-24}$ GeV. The allowed range of
parameters lies in the shadowed area.

####  IV.2.4 Model II: $f(Q,T)=\frac{\gamma}{6H_{0}^{2}\kappa^{2}}QT$

As a second example of an $f(Q,T)$ type cosmological model we consider the
case in which the function $f$ is given by

$f(Q,T)=\frac{\gamma}{6H_{0}^{2}\kappa^{2}}QT,$ (79)

where $\gamma$ is a constant, and $H_{0}$, the Hubble rate at the freeze-out
point, is given by $H_{0}=H_{GR}\left(\mathcal{T}_{f}\right)=2.42\times
10^{-25}$ GeV. For this model, we have two modified Friedmann equations

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6\lambda}\Bigg{[}\rho+(m^{2}+12\lambda)(H\psi-\frac{1}{2}\psi^{2})+\frac{\gamma}{H_{0}^{2}}(6p-2\rho)H\psi$
(80) $\displaystyle+\frac{\gamma}{H_{0}^{2}}(3\rho-p)\psi^{2}\Bigg{]},$
$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle\frac{-1}{4\lambda}\Bigg{[}(1+\frac{2\gamma\psi^{2}}{H_{0}^{2}})(\rho+p)-\frac{1}{3}(m^{2}+\frac{6\gamma}{H_{0}^{2}}p-\frac{2\gamma}{H_{0}^{2}}\rho$
(81)
$\displaystyle+12\lambda)(\dot{\psi}+\psi^{2}-H\psi)-\frac{2\gamma}{3H_{0}^{2}}(3\dot{p}-\dot{\rho})\psi\Bigg{]}.$

In the following we adopt again the approximation $\lambda=\kappa^{2}$.

##### Cosmological evolution in the radiation dominated epoch.

The variations of the functions $H$, $H_{GR}$, $\psi$, $\Gamma$, and of $Z$
are represented, for the second model, in Fig. 7. In this case there are
significant differences between the evolution of the general relativistic
Hubble function, and the Hubble function of the model, with the differences
increasing with decreasing temperature. This behavior can be also observed
from the variation of $Z$, indicating significant differences, of two orders
of magnitude, at low temperatures. However, for enough high temperatures of
the order of 0.025 GeV, and higher, the two Hubble functions basically
coincide. For temperatures lower than 0.0005 GeV, the Hubble function $H$
becomes approximately a constant, indicating the transition to an accelerated,
de Sitter type evolution. During the entire period of the cosmological
evolution in the radiation dominated phase, the Weyl vector is a constant, and
thus it plays the role of an effective cosmological constant. The freeze-out
temperature, obtained from the intersection of the decay rate $\Gamma$ and the
Hubble functions, takes very different values in the two models.

Figure 7: Temperature evolution of $H_{GR}$, $\Gamma$, $H$ and $\psi$ for the
second Weyl type $f(Q,T)$ cosmological model (left panel), and of the ratio
$Z=H/H_{GR}$ (right panel). For the left panel plots the initial value of
$\psi$, $\psi_{ini}$, is $\psi_{ini}=1\times 10^{-25}$ GeV, while $\gamma=-1$.
For the right panel plot $\psi_{ini}=10^{-25}$ GeV.

For the function $\rho_{eff}/\rho_{r}$ we obtain

$\frac{\rho_{eff}}{\rho_{r}}=1-\frac{\gamma
g_{\ast}m_{\gamma}^{2}}{72H_{0}^{2}\kappa^{2}}\mathcal{T}^{2}+\frac{2\pi^{2}\gamma
g_{\ast}}{135H_{0}^{2}\kappa^{2}}\mathcal{T}^{4}+\frac{m^{2}}{12\kappa^{2}}.$
(82)

####  IV.2.5 The freeze-out constraint

##### The numerical freeze-out constraint.

The freeze-out constraint on the parameters of the second model, obtained by
numerically solving the gravitational field equations for different initial
values of the Weyl vector $\psi$, are represented in Fig. 8, for both positive
and negative values of $\gamma$. As one can see from the Figure, the behavior
of the solution of the constraint equation
$H\left(\mathcal{T}_{f}\right)=\Gamma\left(\mathcal{T}_{f}\right)$ is strongly
dependent on the sign, and on the numerical value of $\gamma$ and $m$.
However, for both signs of $]\gamma$ one can obtain a range of numerical
values of $(\gamma,m)$ satisfying the observational freeze-out constraint.

Figure 8: The freeze-out constraint on the parameters $\gamma$ and $m$ of the
second Weyl type cosmological model. For the left panel curves the initial
values of the Weyl vector are $1\times 10^{-25}$ GeV, and $2\times 10^{-25}$
GeV, respectively, while the right panel curves have been obtained for the
initial values of $\psi$ given by $5\times 10^{-25}$ GeV, and $1\times
10^{-24}$ GeV, respectively.

##### Analytical results.

For the second Weyl type $f(Q,T)$ theory cosmological model the freeze-out
constraint (68) takes the form

$\displaystyle\left|\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{10c_{q}}\sqrt{\frac{\pi^{2}g_{\ast}}{180\kappa^{2}}}\frac{1}{\mathcal{T}_{f}^{3}}\Bigg{(}1+\frac{\gamma}{6H_{0}^{2}\kappa^{2}}T$
(83)
$\displaystyle+\frac{4\pi^{2}g_{\ast}\mathcal{T}_{f}^{4}}{270H_{0}^{2}\kappa^{2}}\gamma+\frac{m^{2}}{12\kappa^{2}}\Bigg{)}.$

By neglecting the trace of the matter energy-momentum tensor, we obtain the
freeze-out temperature BBN constraint as

$1+\frac{4\pi^{2}g_{\ast}\mathcal{T}_{f}^{4}}{270H_{0}^{2}\kappa^{2}}\gamma+\frac{m^{2}}{12\kappa^{2}}<0.002.$
(84)

By taking into account that
$\left(4\pi^{2}g_{\ast}\mathcal{T}_{f}^{4}/270H_{0}^{2}\kappa^{2}\right)=1.09$,
we obtain the upper limit of the model parameters as given by

$1+1.09\gamma+\frac{m^{2}}{12\kappa^{2}}<0.002.$ (85)

This constraint is satisfied only for negative values of $\gamma$.

####  IV.2.6 The combined ${}^{7}Li/H$ ratio

For the second Weyl type $f(Q,T)$ modified gravity theory the combined Lithium
and freeze-out temperature constraints on the model parameters $\gamma$ and
$m$ are depicted in Fig. 9.

Figure 9: The combined freeze-out temperature and Lithium constraints on the
parameters $\gamma$ and $m$ of the second Weyl type cosmological model. The
initial values of the Weyl vector are $1\times 10^{-25}$ GeV, $2\times
10^{-25}$ GeV, $5\times 10^{-25}$ GeV, and $1\times 10^{-24}$ GeV,
respectively.

###  IV.3 Model III: $f(Q,T)=\eta
H_{0}^{2}e^{\frac{\mu}{6H_{0}^{2}}Q}+\frac{\nu}{6\kappa^{2}}T$

As a third example of Weyl type $f(Q,T)$ models we assume that the function
$f$ can be written as

$f(Q,T)=\eta H_{0}^{2}e^{\frac{\mu}{6H_{0}^{2}}Q}+\frac{\nu}{6\kappa^{2}}T,$
(86)

where $\mu$, $\nu$ and $\eta$ are constants. The two modified Friedmann
equations of this Weyl type $f(Q,T)$ model can be written as

$\displaystyle H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{6\lambda}\Bigg{[}\rho+\frac{\nu}{6}(3\rho-p)-\kappa^{2}\eta
H_{0}^{2}e^{\frac{\mu}{H_{0}^{2}}\psi^{2}}+\Big{(}m^{2}$ (87)
$\displaystyle+2\kappa^{2}\eta\mu
e^{\frac{\mu}{H_{0}^{2}}\psi^{2}}+12\lambda\Big{)}H\psi-\frac{m^{2}\psi^{2}}{2}-6\lambda\psi^{2}\Bigg{]},$
$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle\frac{-1}{4\lambda}\Bigg{[}\left(1+\frac{\nu}{3}\right)(\rho+p)-\frac{1}{3}\Big{(}m^{2}+2\kappa^{2}\eta\mu
e^{\frac{\mu}{H_{0}^{2}}\psi^{2}}$
$\displaystyle+12\lambda\Big{)}\left(\dot{\psi}+\psi^{2}-H\psi\right)-\frac{4\kappa^{2}\eta\mu^{2}\psi^{2}\dot{\psi}}{3H_{0}^{2}}e^{\frac{\mu}{H_{0}^{2}}\psi^{2}}\Bigg{]}.$

We again adopt the approximation $\lambda=\kappa^{2}$.

##### The cosmological evolution.

For the third Weyl type $f(Q,T)$ cosmological model the comparative variations
of the functions $H$, $H_{GR}$, $\psi$, $\Gamma$, and $Z$ are represented in
Fig. 10. At enough high temperatures the behaviors of the two Hubble functions
become similar, but significant differences do exist in the lower temperature
regime, during the BBN phase. However, at the freeze-out time, $Z$ takes
values of the order of two, or smaller. The Weyl vector decreases from the
high temperature regime up to around the BBN epoch, and then it becomes a
constant, indicating again the possible transition to an accelerated phase in
the evolution of the Universe. But once the transition to the matter dominated
era takes place, the Universe may enter into a decelerating phase.

Figure 10: Temperature evolution of $H_{GR}$, $\Gamma$, $H$, and $\psi$ for
the third Weyl type $f(Q,T)$ cosmological model (left panel), and of the ratio
$Z=H/H_{GR}$ (right panel). For the left panel plots the initial value of
$\psi$, $\psi_{ini}$, is $\psi_{ini}=1\times 10^{-25}$ GeV, while $\gamma=-1$.
For the right panel plot $\psi_{ini}=10^{-25}$ GeV.

####  IV.3.1 The freeze-out constraint

##### The numerical freeze-out constraint.

The dependence of the freeze-out temperature on the parameters of the third
Weyl type cosmological model and the freeze-out constraint on the model
parameters are presented in Fig 11.

Figure 11: The freeze-out temperature of the third Weyl type cosmological
model as a function of the model parameter $\eta$ (left panel), and the
freeze-out constraint (right panel), obtained from the numerical solution of
the field equation. For the initial value of the Weyl vector we have adopted
the value $\psi(0)=0$.

##### The analytical freeze-out constrain.

By taking into account that for this model $f_{Q}=\left(\eta\nu/6\right)e^{\mu
Q/\left(6H_{0}^{2}\right)}$, and $f_{T}=\nu/6\kappa^{2}$, we obtain for the
freeze-out constraint the general expression

$\displaystyle\left|\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{10c_{q}}\sqrt{\frac{\pi^{2}g_{\ast}}{180\kappa^{2}}}\frac{1}{\mathcal{T}_{f}^{3}}\Bigg{(}1-\frac{30\kappa^{2}}{\pi^{2}g_{\ast}}\frac{\eta
H_{0}^{2}e^{\frac{\mu
Q}{6H_{0}^{2}}}+\frac{\nu}{6\kappa^{2}}T}{\mathcal{T}_{f}^{4}}$ (89)
$\displaystyle+\frac{\eta\mu}{3}e^{\frac{\mu
Q}{6H_{0}^{2}}}+\frac{4}{9}\nu+\frac{m^{2}}{12\kappa^{2}}\Bigg{)}.$

Since $Q|_{\mathcal{T}_{f}}=6H^{2}|_{\mathcal{T}_{f}}\approx
6H_{GR}^{2}|_{\mathcal{T}_{f}}=\left(\pi^{2}g_{\ast}/30\kappa^{2}\right)\mathcal{T}_{f}^{4}$,
by performing a first order expansion of the exponential, and by neglecting
the trace of the energy-momentum tensor, we obtain

$\displaystyle\left|\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}\right|$
$\displaystyle=$
$\displaystyle\frac{1}{10c_{q}}\sqrt{\frac{\pi^{2}g_{\ast}}{180\kappa^{2}}}\frac{1}{\mathcal{T}_{f}^{3}}\Bigg{(}1-\frac{30\kappa^{2}\eta
H_{0}^{2}}{\pi^{2}g_{\ast}\mathcal{T}_{f}^{4}}-\frac{2}{3}\eta\mu$ (90)
$\displaystyle+\frac{\pi^{2}\eta\mu^{2}g_{\ast}}{30H_{0}^{2}\kappa^{2}}\mathcal{T}_{f}^{4}+\frac{4}{9}\nu+\frac{m^{2}}{12\kappa^{2}}\Bigg{)}.$

By neglecting the terms in the bracket containing the freeze-out temperature
$\mathcal{T}_{f}^{4}$, we obtain the following expression on the model
parameters

$1-\frac{2}{3}\eta\mu+\frac{4}{9}\nu+\frac{m^{2}}{12\kappa^{2}}\approx
10c_{q}\sqrt{\frac{180\kappa^{2}}{\pi^{2}g_{\ast}}}\mathcal{T}_{f}^{3}\left|\frac{\delta\mathcal{T}_{f}}{\mathcal{T}_{f}}\right|,$
(91)

or

$1-\frac{2}{3}\eta\mu+\frac{4}{9}\nu+\frac{m^{2}}{12\kappa^{2}}<0.002.$ (92)

####  IV.3.2 The ${}^{7}Li/H$ ratio

The dependence of the ${}^{7}Li/H$ ratio on the third Weyl type $f(Q,T)$
theory cosmological model on the parameters $\eta$ and $\nu$ is presented in
Fig. 12.

Figure 12: The Lithium abundance constraint on the parameters $\eta$ and $\nu$
of the third Weyl type $f(Q,T)$ theory cosmological model. For the parameter
$\mu$ w we have adopted the value $\mu=0.5$, while the initial conditions for
the Weyl vector $\psi$ are $1\times 10^{-25}$ GeV, $2\times 10^{-25}$ GeV,
$5\times 10^{-25}$ GeV, and $1\times 10^{-24}$ GeV, respectively.

##  V Discussions and final remarks

General relativity is presently facing several observational challenges,
mostly coming from the late time Universe observations. There is a plethora of
such observational data, obtained from the intensive investigation of the
Cosmic Microwave Background, Supernovae Type Ia, Baryonic Acoustic
Oscillations, Gamma-ray bursts, large scale structure growth data, Cosmic
Chronometers etc. Even that some of these observations can be described by the
standard $\Lambda$CDM model, several open problems confronting this model have
become more and more important recently, like the Hubble or the $\sigma_{8}$
tensions Val . This raises some new questions about the validity of standard
cosmology, and of general relativity. And, of course, the problems of the
cosmological constant and of the renormalizability of a possible quantum
theory of gravity are still unsolved.

Hence, modified gravity theories can help in solving these problems, and they
can open a new window on the gravitational dynamics. There are a large number
of such theories, whose predictions sometimes significantly differ from those
of standard general relativity. Among these theories those based on Weyl
geometry are particularly interesting, since they could represent a bridge
between gravity and elementary particle physics, due to their properties of
conformal invariance. The $f(Q)$ type theories are completely equivalent to GR
for the choice $f(Q)=Q$, and for an arbitrary $f(Q)$ they can also describe
the late-time properties of the cosmological expansion. An extension of the
$f(Q)$ theory, the $f(Q,T)$ gravity, allows the existence of a coupling
between matter and geometry, thus adding a new degree of freedom to the basic
gravitational theory. A simpler formulation of the $f(Q,T)$ theory, which is
closer to the spirit of the initial Weyl geometry, is represented by the Weyl
type $f(Q,T)$ theory, in which the Weyl vector appears as an integral part of
the gravitational dynamics. The applications of this theory in cosmology and
astrophysics have pointed out its potential as an alternative to general
relativity.

Modified gravity theories usually introduce a number of numerical parameters,
which must be fine-tuned in order to explain the observational data. The most
common method to constrain these parameters is by comparison with the late-
time cosmological observations. However, many modified gravity theories that
describe well the recent acceleration of the Universe do introduce some
supplementary terms in the field equations that could significantly change the
early time evolution. Therefore, it is also necessary to test modified gravity
theories in the cosmological background of the early Universe, no matter how
successful they are in explaining the recent dynamics of the Universe. late-
universe successes that a modified gravity model may have, one should always
examine whether the model can pass the BBN constraint

One of such phases in the evolution of the Universe, which can be successfully
used as a testing ground for modified gravity theories, is the BBN phase. The
concordance cosmology model is extremely successful in the description of the
BBN phase, and thus it also offers the possibility of a detailed quantitative
and qualitative comparison between general relativity and its alternatives. In
the present work we have compared the predictions of the Weyl type $f(Q,T)$
gravity with respect to the observational data of the BBN. We have adopted
three specific functional forms of the function $f$, involving additive and
multiplicative algebraic structures of the function $f$, as well as an
exponential dependence on the nonmetricity scalar. We have investigated the
range of the model parameters that allow to explain the abundances of the
light elements in the early Universe, as well as the freeze-out temperature.
For all our models we have also investigated numerically the early Universe
evolution in the radiation dominated epoch, and we have obtained the
temperature dependence of the Hubble function.

The first cosmological model we have considered has the Lagrangian density
given by $f(Q,T)=\alpha Q+\left(\beta/6\kappa^{2}\right)$, where $\alpha$ and
$\beta$ are the model parameters. Recently, the cosmological implications of
the Weyl type $f(Q,T)$ cosmological models have been investigated in W19 . For
the first model comparison with the observational data the values
$\alpha=-1.07$ and $\beta=0.5$ have been adopted. These numerical values
guarantee the positivity of the energy density, and a dark energy equation of
state parameter that satisfies the observational constraints. These values
also satisfy the freeze-out constrain, as one can see from Fig. 2. The same
range of values of $\alpha$ and $\beta$ also satisfies the Lithium constraint,
as one can see from Fig. 4. On the other hand, for the parameter $\beta$
taking values in the range $0.5<\beta<3$, the Strong Energy Condition (SEC) is
violated by this model. However, for the first Weyl type $f(Q,T)$ gravity
cosmological model, there is a range of the parameters $\alpha$ and $\beta$
satisfying both the SEC and the BBN constraints.

The late time behavior of the second cosmological model we have analyzed, with
$f(Q,T)$ given by $f(Q,T)=\left(\gamma/6H_{0}^{2}\kappa^{2}\right)QT$, was
also investigated in W19 , for $\gamma=0.315$. For the model the parameter of
the dark energy equation of state $\omega$ lies in the range $-1<\omega<-1/3$,
and hence the dark energy is of quintessence type. On the other hand, the
Strong Energy Condition is violated at the present epoch for small variations
of $\gamma$. Even for a small change in $\gamma$, the SEC doe nit hold
anymore. On the other hand it is important to note that the NEC and the DEC
are satisfied in this model. From the point of view of the BBN, all the
observational constraints (freeze-out temperature, and light element
abundances) can be satisfied by a range of the parameter $\gamma$, which is
also consistent with the late time behavior observations.

In the present study, in order to constrain our theoretical models we have
used the most stringent cosmological limits from BBN. In particular, we have
used we utilized the primordial Deuterium, Helium, and Lithium abundances to
constrain the parameters of the three cosmological models in the Weyl type
$f(Q,T)$ gravity. The constraints obtained can be further improved once the
new results of the future observations, like, for example, the CMB-S4
measurements S4  will become known. At temperatures smaller than 0.1 MeV, an
important reaction taking place in the early Universe is the production of
deuterium via the reaction $n+p\leftrightarrow D+\gamma$. The abundance of
Deuterium decreases in the post-BBN Universe due to its annihilation during
stellar evolution. Therefore BBN is the unique significant source of this
element. By using the primordial abundances of deuterium, we have obtained a
set of constraints on the free parameters of the Weyl type $f(Q,T)$ gravity
cosmological models. Deuterium is the most constrained light nucleus among all
light elements, with its observational determination reaching 1% accuracy.
Recently, the EMPRESS collaboration has presented a measurement of the
primordial 4He abundance that is over 3$\sigma$ from the $\Lambda$CDM
theoretical value. A small 2$\sigma$ discrepancy may also exist in the
deuterium abundance. These observational results strongly suggest that the
$\Lambda$CDM paradigm may face some problems not only in the late Universe,
but also during the BBN phase. Hence, modified gravity theories could play an
important role in elucidating not only the cosmological behavior of the late
Universe, but also in the description of the very early phases of cosmological
evolution. In the present study we have considered the effects of such a
modified gravity theory on the BBN phase, which could open a new perspective
on the physics and cosmology of the early Universe.

## Acknowledgments

HHZhang is supported by the National Natural Science Foundation of China under
Grant No. 12275367. LMing acknowledges the Project funded by China
Postdoctoral Science Foundation under Grant No. 2022M723677. This work is also
supported by the Fundamental Research Funds for the Central Universities, the
Natural Science Foundation of Guangdong Province, and the Sun Yat-Sen
University Science Foundation.

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