# Cosmological implications of the Weyl geometric gravity theory

Tiberiu Harko<EMAIL_ADDRESS>Department of Theoretical Physics,
National Institute of Physics and Nuclear Engineering (IFIN-HH), Bucharest,
077125 Romania,  Department of Physics, Babes-Bolyai University, 1
Kogalniceanu Street, 400084 Cluj-Napoca, Romania,  Astronomical Observatory,
19 Ciresilor Street, 400487 Cluj-Napoca, Romania,  Shahab Shahidi
<EMAIL_ADDRESS>School of Physics, Damghan University, Damghan,
41167-36716, Iran  Department of Theoretical Physics, National Institute of
Physics and Nuclear Engineering (IFIN-HH), Bucharest, 077125 Romania,
Astronomical Observatory, 19 Ciresilor Street, 400487 Cluj-Napoca, Romania,
Department of Physics, Babes-Bolyai University, 1 Kogalniceanu Street, 400084
Cluj-Napoca, Romania,  School of Physics, Sun Yat-Sen University, 510275
Guangzhou, People’s Republic of China.  School of Physics, Damghan University,
Damghan, 41167-36716, Iran

###### Abstract

We consider the cosmological implications of the Weyl geometric gravity
theory. The basic action of the model is obtained from the simplest
conformally invariant gravitational action, constructed, in Weyl geometry,
from the square of the Weyl scalar, the strength of the Weyl vector, and a
matter term, respectively. The total action is linearized in the Weyl scalar
by introducing an auxiliary scalar field. To maintain the conformal invariance
of the action the trace condition is imposed on the matter energy-momentum
tensor, thus making the matter sector of the action conformally invariant. The
field equations are derived by varying the action with respect to the metric
tensor, the Weyl vector field, and the scalar field, respectively. We
investigate the cosmological implications of the theory, and we obtain first
the cosmological evolution equations for a flat, homogeneous and isotropic
geometry, described by Friedmann-Lemaitre-Robertson-Walker metric, which
generalize the Friedmann equations of standard general relativity. In this
context we consider two cosmological models, corresponding to the vacuum
state, and to the presence of matter described by a linear barotropic equation
of state. In both cases we perform a detailed comparison of the predictions of
the theory with the cosmological observational data, and with the standard
$\Lambda$ CDM model. By assuming that the presence of the Weyl geometric
effects induce small perturbations in the homogeneous and isotropic
cosmological background, and that the anisotropy parameter is small, the
equations of the cosmological perturbations due to the presence of the Weyl
geometric effects are derived. The time evolution of the metric and matter
perturbations are explicitly obtained. Therefore, if Weyl geometric effects
are present, the Universe would acquire some anisotropic characteristics, and
its geometry will deviate from the standard FLRW one.

###### pacs:

04.50.+h,04.20.Cv, 95.35.+d

##  I Introduction

Einstein’s general relativity is a physical theory essentially based on the
geometry of Riemann [1]. The gravitational field equations, as proposed by
Einstein [2, 3], and Hilbert [4] provide a description of the gravitational
field in which the contractions of the curvature tensor determine the
gravitational interaction of massive objects. The gravitational force is thus
not due to the microscopic exchange of particles (quanta of the fields), as is
the case in electrodynamics, but it is an intrinsic property of space-time
structure. General relativity has found extensive applications in astrophysics
and cosmology, and it has been extensively tested at various scales. At the
level of the Solar System, general relativity gives an excellent description
of the of the gravitational dynamics, and fully explains the perihelion
precession of Mercury, the bending of light by the Sun, or the Shapiro time
delay effect [5, 6]. The experimental detection of the gravitational waves [7,
8] has also brilliantly confirmed the predictions of general relativity, and
it has opened a new window for the understanding of the intricate physical
processes taking black hole - black hole, or black hole - neutron star merging
processes [8]. One of the most intriguing predictions of general relativity is
related to the existence of black holes, whose existence in the static
spherically case was predicted by Schwarzschild [9], while the rotating black
hole solution was obtained by Kerr [10]. The first images of the supermassive
black hole M87* were presented recently by the Event Horizon Telescope
collaboration [11, 12, 13]. The observations point towards a Kerr-like
structure for the Sgr A* black hole. But still one cannot reject the possible
existence of differences with respect to the predictions of standard general
relativity.

The impressive success, of general relativity as a physical theory of gravity
based on geometric concepts, influenced significantly not only physics, but
also mathematics. The differential geometric approach to gravity pioneered by
Einstein led to a deeper understanding of the structure of Riemannian
geometry, and opened the ways for its generalization. The possibility of
solving physical problems by using geometrical methods suggested to study
geometries that go beyond Riemann geometry, and general relativity.
Historically, the first proposal to generalize Riemann geometry, was due to
Weyl [14, 15]. Weyl was motivated in this generalization by the intention of
solving one of the most important problems of theoretical physics, namely, the
unification of the gravitational and electromagnetic forces. For an extensive
account of the historical aspects of Weyl geometry, its evolution, and its
physical and mathematical implications see [16]. A few years later after
Weyl’s work another important generalization of Riemann geometry was
introduced by Cartan [17, 18, 19, 20], based on the concept of torsion.
Geometries with torsion represent the mathematical foundations of the
Einstein-Cartan theories of gravity [21]. In these types of theories torsion
describes the effects on the space-time geometry arising from the rotation of
compact objects.

In building his geometry, Weyl adopted two fundamental principles. The first
is the possibility of the variation of the length of a vector during its
parallel transport. The variation of the length is described by a geometric
quantity called nonmetricity $Q_{\lambda\mu\nu}$. Consequently, in Weyl
geometry, the covariant derivative of the metric tensor $g_{\mu\nu}$ satisfies
the condition $\tilde{\nabla}_{\lambda}g_{\mu\nu}=Q_{\lambda\mu\nu}$, also
called the nonmetricity condition. Secondly, Weyl postulated that the laws of
nature must be conformally invariant. In the first formulation of Weyl’s
geometry for the nonmetricity a particular form was adopted, namely,
$Q_{\lambda\mu\nu}=A_{\lambda}g_{\mu\nu}$, where $A_{\mu}$ is the Weyl vector.
Weyl also proposed the interpretation of the vector $A_{\mu}$ as the
electromagnetic field potential. However, the physical interpretation of
Weyl’s geometry was strongly disapproved by Einstein. Due to this criticism,
as well and as a result of the general development of theoretical physics, for
a long time Weyl’s unified field theory was not accepted as a possible
physical approach to the unification of the electromagnetic and gravitational
interactions. But Weyl’s geometry has many beautiful characteristics, and it
opens the way for the full implementation of the conformal invariance of
physical laws. Weyl’s geometry is also at the origin of the gauge theory [22,
23], which has become the fundamental theoretical tool in particle physics.
Thus, in the past century, Weyl’s geometry did continue to inspire the work of
both physicists and mathematicians.

One of the interesting attempts for the reconsideration of Weyl’s theory from
a physical point of view was due to Dirac [24, 25], who introduced a real
scalar field $\phi$ of weight $w(\phi)=-1$ in the basic theory. As for the
gravitational Lagrangian Dirac proposed the expression
$L=-\phi^{2}R+kD^{\mu}\phi D_{\mu}\phi+c\phi^{4}+W_{\mu\nu}W^{\mu\nu}/4,$
where $R$ is the Ricci scalar, $c$ and $k=6$ are constants, and $W_{\mu\nu}$
is the electromagnetic type tensor constructed from the Weyl vector. The Dirac
Lagrangian has the important property of conformal invariance. A slightly
modified version of the Dirac model was investigated, from a cosmological
perspective, in [26]. The cosmological implications of a Weyl-Dirac type
Lagrangian of the form
$L=W^{\lambda\rho}W_{\lambda\rho}-\phi^{2}R+\sigma\phi^{2}w^{\lambda}w_{\lambda}+2\sigma\phi
w^{\lambda}\phi_{,\lambda}+(\sigma+6)\phi_{,\rho}\phi_{,\lambda}g^{\rho\lambda}+2\Lambda\phi^{4}+L_{m},$
where $\sigma$ and $\Lambda$ are constants, were investigated in [27]. In the
cosmological model based on the action introduced in [27], the creation of
particles at the beginning of the Universe is determined by the Dirac gauge
field. During the late time evolutionary phases of the Universe, the Dirac
gauge field represents the source of the dark energy triggering the de Sitter
exponential cosmic expansion of the Universe.

Weyl’s geometry was also generalized to include torsion. The corresponding
geometry is called the Weyl-Cartan geometry, and it has also been extensively
studied from both physical and mathematical points of view [28, 29, 30, 31,
32, 33, 34, 35, 36]. For a detailed review of the geometry and of the
cosmological applications of the Riemann-Cartan and Weyl-Cartan space-times
see [37]. The role of the torsion in the Weyl-Dirac theory was investigated in
[38, 39, 40]. By including the torsion tensor in the theory one can also
obtain a conformally invariant massive electrodynamic theory, generalizing
classical electromagnetism.

Another interesting mathematical approach with important physical applications
was initiated by Weitzenböck [41], who introduced the geometrical structure
known today as the Weitzenböck space. The Weitzenböck spaces have the basic
properties $\nabla_{\mu}g_{\sigma\lambda}=0$, $T^{\mu}_{\sigma\lambda}\neq 0$,
and $R^{\mu}_{\nu\sigma\lambda}=0$, respectively, where
$T^{\mu}_{\sigma\lambda}$ is the torsion tensor. Weitzenböck spaces reduce to
a Euclidean manifold when $T^{\mu}_{\sigma\lambda}=0$. Moreover, the torsion
tensor $T^{\mu}_{\sigma\lambda}$ has distinct values in the different points
of the manifold. Since the Riemann curvature tensor vanishes identically, the
Weitzenböck spaces have the property of distant parallelism, also known as
teleparallelism, or absolute parallelism. Einstein proposed a unified
teleparallel theory of electromagnetism and gravitation based on the
Weitzenböck geometry [42]. Cosmological applications of the Wey-Cartan-
Weitzenböck geometry were considered in [43, 44].

A teleparallel formulation of the gravitational field can be obtained by
replacing the metric tensor $g_{\mu\nu}$ with a set of tetrad vectors
$e^{i}_{\mu}$. Thus, the torsion tensor, constructed from the tetrad fields,
describes completely gravity, with the curvature replaced by the torsion. This
approach represents the so-called teleparallel equivalent of General
Relativity (TEGR), which was initially developed in [45, 46, 47]. The
corresponding theory is also known as the $f(\mathbb{T})$ gravity theory, with
$\mathbb{T}$ denoting the torsion scalar. In the $f(\mathbb{T})$ type theories
the equations describing the gravitational field are of second order, while in
other modified gravity theories, like, for example, the $f(R)$ gravity theory,
the field equations in the metric formalism are of fourth order [48]. See [49]
for a review of the TEGR type theories. $f(\mathbb{T})$ theories can also
explain geometrically the recent accelerated expansion of the Universe,
without the need of introducing in the field equations a dark energy term, or
the cosmological constant [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61].

The determinations by the Planck satellite of the fluctuations in the
temperature distribution of the Cosmic Microwave Background Radiation [62,
63], as well as the investigations of the light curves of the Type Ia
supernovae [64], have provided compelling evidence that the present day
Universe is in a phase of rapid cosmological expansion. Moreover, these
observational findings have also conclusively proven that only around 5% of
the matter-energy content of the Universe consists of baryonic matter, with
95% being represented by two other components, called generically dark matter,
and dark energy, respectively. In order to interpret theoretically the
cosmological observational data, the $\Lambda$CDM ($\Lambda$ Cold Dark Matter)
model was introduced, which is essentially based on the reintroduction in the
standard gravitational field equations of the cosmological constant $\lambda$,
first proposed in 1917 by Einstein [65] to obtain a static cosmological model
of the Universe. Later on, Einstein rejected the possibility of the existence
of $\Lambda$. The $\Lambda$CDM model gives a very good description of the
observational data, especially at low redshifts, and thus it is considered the
standard cosmological paradigm of the present times. However, in its basic
formulation the $\Lambda$CDM model is confronted with an important objection
related to its theoretical foundation. Presently no satisfactory explanation
of the nature (physical or geometrical) of the cosmological constant does
exist, and thus the theoretical basis of the $\Lambda$CDM model are at least
uncertain.

Thus, it is justifiable to suppose that to obtain a realistic physical picture
of the Universe, fully consistent with the cosmological observations, one
should extend standard general relativity. It may be possible that the
Einstein field equations, describing very well the gravitational dynamics in
the Solar System, are only a first order approximation of a more general
gravity theory. Many modified gravity theories have been proposed, and for a
detailed reviews see [66, 67, 68]. Generalizations of Einstein’s relativity in
the presence of geometry-matter couplings have been introduced in [69, 70,
71]. For a unified approach to modified gravity theories see [72, 73].

The $\Lambda$CDM model is also confronted with several other problems, which
are mostly the results of the increase in accuracy of the cosmological
observations. An important cosmological challenge is the difference between
the expansion rate of the Universe as determined from the Cosmic microwave
Background Radiation satellite observations, and the numerical values obtained
from the local (low redshift) determinations [74]. This discrepancy is called
as the Hubble tension [75, 76, 77]. The Hubble constant (H0) as measured by
the Planck satellite, has the value of $66.93\pm 0.62$ km/ s/ Mpc [78], while
the value of $73.24\pm 1.74$ km/ s/ Mpc [74] is obtained by the SH0ES
collaboration. The differences between these two values is more than 3$\sigma$
[79]. The Hubble tension, if indeed it exists, points strongly towards the
need of considering new gravitational theories, and extending, or even fully
replacing, the $\Lambda$CDM model.

An important avenue for the extension of standard cosmology, and for obtaining
explanations of the present-day observations, is related to the reexamination
of Weyl’s theory as a possible description of the gravitational interaction. A
fundamental idea, first introduced by Weyl, is the consideration of the
conformal invariance of the physical laws. It turns out that the fundamental
equations of elementary particle physics are conformally invariant, but
Einstein’s general relativity is not. Hence, in order to make all the
equations describing elementary particle interactions consistent, it is
necessary to reformulate general relativity as a conformally invariant field
theory. Theories of gravity, satisfying the requirement of the conformal
invariance, as well as conformally invariant theories of elementary particle
physics were proposed and investigated in detail in [141, 81, 142, 143, 84,
85, 140].

A fully conformally invariant theory of gravity can be obtained from the
action
$S=-\alpha_{g}\int{C_{\lambda\mu\nu\kappa}C^{\lambda\mu\nu\kappa}\sqrt{-g}d^{4}x}=-2\alpha_{g}\int{\left(R_{\mu\nu}R^{\mu\nu}-R^{2}/3\right)\sqrt{-g}d^{4}x},$
where $C_{\lambda\mu\nu\kappa}$ is the (conformally invariant) Weyl tensor,
and $\alpha_{g}$ is a constant. This theory was introduced, and extensively
investigated, in [87, 88, 89, 90, 91, 92]. The theory can also provide an
explanation for the unusual motion of the massive particles orbiting around
the galactic centers, which is usually explained by assuming the existence of
a mysterious, and yet undetected, component of the Universe, called dark
matter.

Weyl geometry represents the mathematical basis of the $f(Q)$ modified gravity
theory [93, 94, 95], and of its generalizations [96, 97, 98, 99, 100]. In the
$f(Q)$ theory, the basic geometric parameter, fully describing the
gravitational interaction, is the non-metricity $Q$. The action of the $f(Q)$
theory is obtained as $S=\int{f(Q)\sqrt{-g}d^{4}x},$ where $f(Q)$ denotes an
arbitrary analytical function of $Q$. An extension of the $f(Q)$ theory, based
on the action $S=\int{f(Q,T)\sqrt{-g}d^{4}x}$ where $T$ is the trace of the
matter energy-momentum tensor, was introduced in [97]. The $f(Q)$ theory, as
well as its extensions, were intensively investigated from both theoretical
and observational points of view [101, 102, 103, 104, 105, 106, 107, 108].

A novel, and very interesting perspective on Weyl gravity, and of its
applications in elementary particle physics, cosmology and astrophysics, was
proposed, and extensively developed, in [109, 110, 111, 112, 113, 114, 115,
116, 117, 118, 119, 120, 121]. This approach to gravity heavily relies on
concepts from elementary particle physics, and in the following we will call
it the Weyl geometric gravity theory (sometimes it is also called quadratic
Weyl gravity [120]). The starting point in this approach to Weyl gravity is to
linearize, in the quadratic Weyl action [14, 15], the square of the Weyl
scalar $\tilde{R}^{2}$ via the introduction of an auxiliary scalar field
$\phi$. As a result, quadratic Weyl gravity can be reformulated as a
gravitational theory linear in the curvature scalar. This linearization has
important physical implications. In the curvature linearized Weyl action one
can introduce a spontaneous breaking of the $D(1)$ symmetry as a result of the
presence of a Stueckelberg type mechanism, having a geometric origin.
Consequently, the Weyl gauge vector field acquires a mass, originating from
the spin-zero mode of the $\tilde{R}^{2}$ term. The Stueckelberg mechanism is
implemented via the replacement of the scalar field $\phi$ with a constant
value (its vacuum expectation value), with $\phi\rightarrow<\phi>$. Hence,
through this mechanism the Weyl vector field becomes massive, and in this way
the dynamical scalar field $\phi$ is absorbed in the mass of the vector field.
Consequently, the scalar field is eliminated from the initial scalar-vector-
tensor theory. After removing $\phi$, the Einstein-Proca action is obtained
from the initial Weyl action. Therefore, a vector-tensor theory is reobtained,
which is similar to the initial Weyl geometric gravity theory.

The pathway to gravity via the linearization procedure of the quadratic Weyl
action also provides some insights on the Planck scale, and on the
cosmological constant. We have already mentioned that the Einstein-Proca
action emerges from the initial Weyl action in the broken phase. Consequently,
all mass scales, including the Planck scale, as well as the cosmological
constant $\Lambda$, have a geometric origin [116]. The Higgs field, which
plays an essential role in the standard model of elementary particles, also
originates from geometry, and it is created through the fusion of Weyl bosons
in the very early Universe, during the reheating phase.

The physical and cosmological properties of the Weyl geometric gravity have
been extensively investigated recently. The conformally invariant coupling
between geometry and matter was considered in [122, 123, 124]. The Palatini
formulation of the theory in the presence of conformally invariant matter-
geometry couplings was studied in [123]. The Palatini formulation of the
quadratic Weyl gravity $\tilde{R}^{2}+R_{\mu\nu}^{2}$ was also analyzed in
[113], by assuming that the metric and the Weyl connection are independent
quantities. In the Palatini approach all the mass scales do appear as a purely
geometric effects. Moreover, a spontaneous breaking of the gauge scale
symmetry can be implemented in the theory. An inflationary scenario can also
be constructed, and the tensor-to-scalar ratio is predicted as $0.007\leq
r\leq 0.01$, at 95 % CL, and $N=60$ efolds. The investigation of the inflation
in the Weyl geometric gravity in its metric and Palatini versions was
initiated in [114]. Since the two versions of the Weyl geometric gravity have
different non-metricities, both determined by the Weyl gauge field, the
physical predictions of the two formalisms are different. Black hole solutions
in Weyl geometric gravity were obtained, by using both numerical and
analytical methods, in [125]. The possibility that dark matter is a Weyl
geometric effect was considered in [126]. Stellar type objects in Weyl
geometric gravity were studied in [127]. The spin Hall effect for light was
generalized to the case of Weyl geometry in [128]. The thermodynamical
properties of the Weyl geometric black holes were analyzed extensively in
[129]. The behavior of the galactic rotation curves in Weyl geometric gravity
were investigated in [130].

It is the goal of the present paper to consider isotropic and anisotropic
cosmological models in the simplest model of the Weyl geometric gravity. Our
starting point is a gravitational action consisting of the sum of the square
$\tilde{R}^{2}$ of the Weyl scalar, and of the field strength $F_{\mu\nu}^{2}$
of the Weyl vector. The action can be linearized in the Weyl scalar, by
introducing an auxiliary scalar field $\phi$. Then the conformally invariant
Weyl action can be reformulated as an effective scalar-vector-tensor theory in
Riemann geometry, with the action containing effective couplings between the
scalar field and the Ricci scalar, and the Weyl vector field. These terms are
conformally invariant by construction. Moreover, a matter term is also added
to the total action in a conformally invariant way. The field equations
corresponding to this action are obtained in the metric formalism, by varying
the action respect to the metric tensor, Weyl vector and the scalar field.

An important question in conformally invariant gravitational actions is how to
implement the conformal invariance of the matter terms. In this work, we will
achieve the requirement of the conformal invariance of the matter action by
imposing the trace condition on the effective matter action $\mathcal{L}_{m}$,
constructed with the help of the ordinary matter action $L_{m}$, and the
square of the Weyl vector $\omega^{2}$, so that
$\mathcal{L}_{m}=\mathcal{L}_{m}\left(L_{m},\omega^{2}\right)$. Once this
condition is satisfied, the corresponding gravitational field equations, and
their solutions, are conformally invariant.

After obtaining the gravitational field equations of the Weyl geometric
gravity, and the consistency condition in the homogeneous and isotropic
Friedmann-Lemaitre-Robertson-Walker cosmological framework (the generalized
Friedmann equations), we consider a number of specific cosmological models.
Thus, we investigate first a vacuum cosmological model, in which the matter
Lagrangian and the baryonic matter density are assumed to vanish. The
cosmological equations are solved numerically, and the Hubble function of the
model is compared with a small observational dataset of observational values
of the Hubble function, as well as with the predictions of the standard
$\Lambda$CDM paradigm. The behaviors of the deceleration parameter, of the
Weyl vector, and of the scalar field are also obtained.

The effect of the matter is considered in the next step of our investigation.
The generalized Friedmann equations are again solved numerically, and the
comparison with the observations and $\Lambda$CDM model is performed in
detail. The variations with respect to the redshift of the matter energy
density, deceleration parameter, Weyl vector and scalar field are also
considered.

Finally, we investigate the anisotropic properties of the Weyl geometric
cosmological models. We consider the simplest extension of the isotropic and
homogeneous FLRW geometry, namely Bianchi type I spacetimes. We write down the
gravitational equations describing the Weyl geometric evolution in a Bianchi
type I geometry in an effective form, by introducing the equivalent dark
energy and dark matter terms, which also incorporate the contributions from
Weyl geometry. By assuming that the differences between the scale factor of
the isotropic FLRW model and the scale factors of the Bianchi type I spacetime
are small, it follows that the deviations from isotropy are also small, and
therefore we can assume that they represent just a small perturbation of the
homogeneous and isotropic FLRW background metric. We derive the basic
equations satisfied by the perturbations of the metric, and of the effective
(geometric) energy densities and pressures. It turns out that the cosmological
evolution of the perturbed geometric and physical quantities is determined by
the Hubble function of the background isotropic model. The perturbation
equations are solved numerically, for the case of the Bianchi type I geometry,
and the behaviors of the relevant physical and geometrical quantities, like
the anisotropy parameter, deceleration parameter or energy density and
pressure perturbations are obtained.

As a possible test of the anisotropic Weyl geometric cosmological model we
have considered the behavior of the quadrupole $Q_{2}$ of the CMBR as function
of the deviations $\delta_{1}$ and $\delta_{3}$ from the isotropic and
homogeneous FLRW geometry. Since the dynamical characteristics of the
background FLRW geometry in Weyl geometric gravity are known, one could use
the obtained expression of $Q_{2}$ to put tight constraints on the parameters
of the anisotropic Weyl geometric cosmology model. For the isotropic case we
have obtained the constraints on the model parameters from the study of the
luminosity distance by using the observational data from the type Ia
supernovae.

The present paper is organized as follows. The action and the field equations
of the Weyl geometric gravity theory are introduced in Section II, where the
conformally invariant construction of the matter action is also discussed. The
cosmological evolution of the isotropic and homogeneous Weyl geometric gravity
models is discussed in Section III. The generalized Friedmann equations are
obtained, and two cosmological models, corresponding to a vacuum Universe, and
a matter filled one, are explored in detail. A comparison with the
observational data for the Hubble function, and with the $\Lambda$CDM model is
also performed. Bianchi type I homogeneous and anisotropic cosmological models
are considered in Section IV, with the use of a perturbative approach. The
behavior of the various geometrical and cosmological quantities describing the
model are also considered in detail. Finally, we discuss and conclude our
results in Section V.

##  II Action and field equations of Weyl geometric gravity

in the present Section we write down first the action of Weyl geometric
gravity, quadratic in the Weyl scalar. Secondly, we briefly introduce the
basic concepts of Weyl geometry to be used in the sequel. The problem of the
conformally invariant coupling of ordinary matter is also considered, and the
trace condition is obtained in a general form. Then, the field equations of
the theory are obtained by varying the Weyl type action with respect to the
metric.

###  II.1 From the gravitational Lagrangian to Weyl geometry

The most general gravitational Lagrangian density that is invariant under a
gauged Weyl symmetry, and defined in Weyl geometry, is given by [114, 115,
116]

$\displaystyle\mathcal{L}_{1}=\sqrt{g}\,\,\Big{\\{}\frac{1}{4!\,\xi^{2}}\tilde{R}^{2}-\frac{1}{\eta^{2}}\,\tilde{C}_{\mu\nu\rho\sigma}^{2}-\frac{1}{4}\,F_{\mu\nu}^{2}\Big{\\}},0<\xi,\,\eta<1,$

were $F_{\mu\nu}$ is the field strength of the Weyl gauge field, and
$\tilde{C}_{\mu\nu\rho\sigma}$ is the Weyl tensor.

A local Weyl symmetry is defined as the invariance of the action under the set
of transformations

$\displaystyle\hat{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\Sigma^{q}(x)g_{\mu\nu},\quad\sqrt{-\hat{g}}=\Sigma^{2\,q}(x)\sqrt{-g},$
(2) $\displaystyle\hat{\phi}$ $\displaystyle=$
$\displaystyle\Sigma^{-q/2}(x)\phi,\quad\quad\hat{\psi}=\Sigma^{-3q/4}(x)\psi,$
(3)

where $\Sigma(x)$ is an arbitrary positive definite function of the
coordinates, $q$ is a constant called the Weyl charge of the metric, and
$\phi$ and $\psi$ are bosonic and fermionic fields, respectively.

The gauged Weyl symmetry (or Weyl gauge symmetry for short) is defined as the
invariance of the action under the transformations (2), and the transformation

$\displaystyle\hat{\omega}_{\mu}=\omega_{\mu}-\frac{1}{\alpha}\,\partial_{\mu}\ln\Sigma,$
(4)

where $\omega_{\mu}$ is the associated Weyl gauge field, while $\alpha$
denotes the Weyl gauge coupling constant.

In the following, by Weyl geometry we consider a geometry that is invariant
under the transformations (2), (3), and (4), respectively. Also, for
simplicity, and without loss of generality, we will consider only the case
$q=1$. The Weyl geometry is non-metric, and the covariant divergence of the
metric tensor satisfies the condition

$\displaystyle\tilde{\nabla}_{\mu}g_{\alpha\beta}=-\alpha\,q\,\omega_{\mu}\,g_{\alpha\beta},$
(5)

where

$\displaystyle\tilde{\nabla}_{\mu}g_{\alpha\beta}\equiv\partial_{\mu}g_{\alpha\beta}-\tilde{\Gamma}_{\alpha\mu}^{\rho}g_{\rho\beta}-\tilde{\Gamma}_{\beta\mu}^{\rho}g_{\rho\alpha},$
(6)

and $\tilde{\Gamma}_{\beta\mu}^{\rho}$ are the coefficients of the Weyl
connection.

The Weyl connection $\tilde{\Gamma}$ can be found by direct calculation from
Eq. (5), and it is given by

$\displaystyle\tilde{\Gamma}_{\mu\nu}^{\lambda}$ $\displaystyle=$
$\displaystyle\Gamma_{\mu\nu}^{\lambda}\Big{|}_{\partial_{\mu}\rightarrow\partial_{\mu}+\alpha\,\omega_{\mu}}$
(7) $\displaystyle=$
$\displaystyle\Gamma_{\mu\nu}^{\lambda}+\frac{\alpha}{2}\,\,\Big{(}\delta_{\mu}^{\lambda}\,\,\omega_{\nu}+\delta_{\nu}^{\lambda}\,\,\omega_{\mu}-g_{\mu\nu}\,\omega^{\lambda}\Big{)},$

where the Levi-Civita connection $\Gamma_{\mu\nu}^{\lambda}$ is defined by

$\displaystyle\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}\,g^{\lambda\alpha}(\partial_{\mu}g_{\alpha\nu}+\partial_{\nu}g_{\alpha\mu}-\partial_{\alpha}g_{\mu\nu}).$
(8)

The scalar curvature $\tilde{R}$ of the Weyl geometry and the square of the
Weyl tensor $\tilde{C}_{\mu\nu\rho\sigma}^{2}$ can be obtained as [118]

$\displaystyle\tilde{R}=R-3\,\alpha\,\nabla_{\mu}\omega^{\mu}-\frac{3}{2}\alpha^{2}\,\omega_{\mu}\,\omega^{\mu},$
(9)
$\displaystyle\tilde{C}_{\mu\nu\rho\sigma}^{2}=C_{\mu\nu\rho\sigma}^{2}+\frac{3}{2}\,\alpha^{2}\,F_{\mu\nu}^{2}.$
(10)

In the following the quantities without tilde represent the Riemannian
geometric counterparts of the geometric quantities defined in Weyl geometry.

Since $\tilde{\Gamma}_{\mu\nu}^{\lambda}=\tilde{\Gamma}_{\nu\mu}^{\lambda}$,
then
$F_{\mu\nu}=\tilde{\nabla}_{\mu}\omega_{\nu}-\tilde{\nabla}_{\nu}\omega_{\mu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$,
identical to $F_{\mu\nu}$ as defined in the Riemann (or flat) space-time.

We proceed now to the scalar-vector-tensor representation of the Weylian
Lagrangian density (II.1). To obtain this representation we perform the
substitution [114, 115, 116]

$\tilde{R}^{2}\rightarrow-2\phi^{2}\tilde{R}-\phi^{4},$ (11)

in the gravitational Lagrangian (II.1), where $\phi$ is an auxiliary scalar
field. Moreover, we neglect in the gravitational Lagrangian the term
containing the Weyl tensor. This is equivalent in taking the limit
$\eta\rightarrow\infty$ in our model. Hence, the action of the Weyl geometric
gravity theory can be written as

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{12\xi^{2}}\phi^{2}\left(R-3\alpha\nabla_{\mu}\omega^{\mu}-6\alpha^{2}\omega^{2}-\frac{1}{2}\phi^{2}\right)-\frac{1}{4}F^{2}+\mathcal{L}_{m}\left(L_{m},\omega^{2},\psi\right)\right],$
(12)

where $\psi$ is an external field, $L_{m}$ is the ordinary (baryonic) matter
Lagrangian, and $\mathcal{L}_{m}$ is the effective matter Lagrangian of the
Weyl geometric theory, respectively. The effective matter Lagrangian
$\mathcal{L}_{m}$ must contain minimal or non-minimal couplings between the
matter Lagrangian and the Weyl geometric quantities in order to assure the
conformal invariance of the theory. Moreover, we have denoted
$\omega^{2}\equiv\omega_{\mu}\omega^{\mu}$, and $F^{2}\equiv
F_{\mu\nu}F^{\mu\nu}$.

###  II.2 Conformal coupling of matter

It is important to note at this moment that it is not necessary for the matter
part $\mathcal{L}_{m}$ of the action (12) to be gauge invariant with respect
to the conformal transformations. However, the variation of $\mathcal{L}_{m}$
must have this important invariance property [131, 132, 133, 134]. By varying
the matter action in Eq. (12) we obtain

$\displaystyle\delta S_{m}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\int
T^{\mu\nu(tot)}\delta g_{\mu\nu}\sqrt{-g}d^{4}x+\int
G^{\mu}\delta\omega_{\mu}\sqrt{-g}d^{4}x$ (13)
$\displaystyle+\int\frac{\delta\mathcal{L}_{m}\left(L_{m},\omega^{2},\psi\right)}{\delta\psi}\delta\psi,$

where by $T^{(tot)}_{\mu\nu}$ we denote the effective total energy-momentum
tensor, defined according to

$T^{(tot)}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left[\sqrt{-g}\mathcal{L}_{m}\left(L_{m},\omega^{2},\psi\right)\right]}{\delta
g^{\mu\nu}},$ (14)

and

$G^{\mu}\left(L_{m},\omega^{2},\omega^{\mu},\psi\right)=\frac{\delta\mathcal{L}_{m}\left(L_{m},\omega^{2},\psi\right)}{\delta\omega^{\mu}},$
(15)

denotes the Weyl current [131, 132, 133, 134]. In the following we assume that
$\delta\mathcal{L}_{m}/\delta\psi=0$.

With the use of the mathematical results [131, 132, 133, 134]

$\delta
g_{\mu\nu}=-\frac{2\delta\Omega}{\Omega^{3}}\tilde{g}_{\mu\nu}=-2\frac{\delta\Omega}{\Omega}g_{\mu\nu},$
(16) $\displaystyle\delta\omega_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{4}{\alpha}\delta\frac{\partial_{\mu}\Omega}{\Omega}=-\frac{4}{\alpha}\delta\left(\partial_{\mu}\ln\Omega\right)=-\frac{4}{\alpha}\partial_{\mu}\left(\delta\ln\Omega\right)$
(17) $\displaystyle=$
$\displaystyle-\frac{4}{\alpha}\partial_{\mu}\left(\frac{\delta\Omega}{\Omega}\right)=-\frac{4}{\alpha}\nabla_{\mu}\left(\frac{\delta\Omega}{\Omega}\right),$

we obtain for the variation of the matter action the relation

$\displaystyle\delta S_{m}$ $\displaystyle=$ $\displaystyle\int
T^{\mu\nu(tot)}g_{\mu\nu}\frac{\delta\Omega}{\Omega}\sqrt{-g}d^{4}x$ (18)
$\displaystyle+\frac{4}{\alpha}\int
G^{\mu}\nabla_{\mu}\left(\frac{\delta\Omega}{\Omega}\right)\sqrt{-g}d^{4}x=0.$

After performing a partial integration of the second term in the above
equation, and with the use of the Gauss theorem, from Eq. (18) we obtain the
important consistency (trace) condition that must be satisfied by the matter
terms of the Weyl geometric gravity theory

$\displaystyle
T^{(tot)}=-\frac{8}{\alpha}\nabla_{\mu}\left[\omega^{\mu}\frac{\partial\mathcal{L}_{m}\left(L_{m},\omega^{2},\psi\right)}{\partial\omega^{2}}\right],$
(19)

where $T^{(tot)}=g^{\mu\nu}T^{(tot)}_{\mu\nu}$ is the trace of the effective
energy-momentum tensor, obtained with the help of the effective matter
Lagrangian $\mathcal{L}_{m}$.

From Eq. (19) it immediately follows that when $\mathcal{L}_{m}=L_{m}$, the
constraint (19) gives the usual trace condition $T^{(m)}=0$, where by
$T^{(m)}$ we have denoted the trace of the ordinary matter energy-momentum
tensor. This leads to the important result that the only conformally invariant
form of ordinary matter must have a traceless energy-momentum tensor, thus
corresponding to a radiation type fluid.

###  II.3 The gravitational field equations of Weyl geometric gravity

In this paper, we will assume that the total (effective) matter Lagrangian is
given by

$\displaystyle\mathcal{L}_{m}=L_{m}+\beta(-\omega^{2})^{n},$ (20)

where $L_{m}$ is the baryonic matter Lagrangian, and $\beta$ and $n$ are
constants.

Before considering the cosmological implications of the model, let us
transform the action to a form which is easier to handle in cosmology. If one
defines the new set of quantities $\left(\Phi,A_{\mu},\beta_{2}\right)$, given
by

$\displaystyle\Phi=\phi^{2},\quad
A_{\mu}=\alpha\omega_{\mu},\quad\beta_{2}=12\beta\xi^{2}\alpha^{-2n},$ (21)

one can write the action as

$\displaystyle S=\frac{1}{12\xi^{2}}\int
d^{4}x\sqrt{-g}\left[\Phi\left(R-3\nabla_{\mu}A^{\mu}-6A^{2}-\frac{1}{2}\Phi\right)-\frac{3\xi^{2}}{\alpha^{2}}F^{2}+\mathcal{L}_{m}\left(L_{m},A^{2},\psi\right)\right],$
(22)

with the matter Lagrangian given by

$\displaystyle\mathcal{L}_{m}=12\xi^{2}L_{m}+\beta_{2}(-A^{2})^{n}.$ (23)

Also the trace constraint equation is simplified as

$\displaystyle
T^{(tot)}=-8\nabla_{\mu}\left(A^{\mu}\frac{\partial\mathcal{L}_{m}\left(L_{m},A^{2},\psi\right)}{\partial
A^{2}}\right),$ (24)

Varying the action (22) with respect to $g_{\mu\nu}$, $\Phi$ and $A_{\mu}$
gives the full set of the field equations of the Weyl geometric gravity as

$\displaystyle\Phi
G_{\mu\nu}-\frac{3}{2}\Phi\left(A_{\mu}A_{\nu}-\frac{1}{2}A^{2}g_{\mu\nu}\right)+\frac{1}{4}\Phi
g_{\mu\nu}+\Box\Phi g_{\mu\nu}$
$\displaystyle-\nabla_{\mu}\nabla_{\nu}\Phi-\frac{6\xi^{2}}{\alpha^{2}}\left(F_{\mu}^{~{}\alpha}F_{\nu\alpha}-\frac{1}{4}F^{2}g_{\mu\nu}\right)-6\xi^{2}T_{\mu\nu}$
$\displaystyle-\beta_{2}nX\left(A_{\mu}A_{\nu}-\frac{1}{2n}A^{2}g_{\mu\nu}\right)$
$\displaystyle+\frac{3}{2}\left(A_{\nu}\nabla_{\mu}\Phi+A_{\mu}\nabla_{\nu}\Phi-
g_{\mu\nu}A^{\alpha}\nabla_{\alpha}\Phi\right)=0,$ (25)
$\displaystyle\frac{12\xi^{2}}{\alpha^{2}}\nabla_{\alpha}F^{\mu\alpha}+3\Phi
A^{\mu}-3\nabla^{\mu}\Phi+2\beta_{2}nXA^{\mu}=0,$ (26) $\displaystyle
R-\frac{3}{2}A^{2}-3\nabla_{\mu}A^{\mu}-\Phi=0,$ (27)

where we have defined $X\equiv(-A^{2})^{n-1}$.

The trace equation can also be written as

$\displaystyle 3\xi^{2}T$
$\displaystyle+\beta_{2}(-A^{2})^{n-4}\Bigg{(}2n(n-1)A^{\mu}\nabla A^{2}$
$\displaystyle-(n-2)(-A^{2})^{2}+2nA^{2}\nabla_{\mu}A^{\mu}\Bigg{)}=0.$ (28)

Using the trace of the metric equation to eliminate $R$ in the scalar field
equation (27), one obtains

$\displaystyle\Box\Phi-\nabla_{\rho}\left(\Phi
A^{\rho}\right)-\frac{\beta_{2}}{3}(n-2)XA^{2}-2\xi^{2}T=0,$ (29)

##  III Cosmological implications of Weyl geometric gravity - the case of the
FLRW geometry

In the present Section we consider the cosmological implications of the Weyl
geometric gravity in the presence of baryonic matter coupled in a conformally
invariant way to the Weyl geometric quantities in a homogeneous and isotropic
Universe. As a first step in our study we obtain the generalized Friedmann
equations of Weyl geometric gravity. The vacuum case and the cosmological
evolution in the presence of matter are both considered. In each case we
compare the predictions of the theory with the observational data.

###  III.1 The generalized Friedmann equations

We assume first that the geometry of the Universe can be described by a flat
FLRW ansatz of the form

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right),$ (30)

where $a$ is the scale factor. We also introduce the Hubble function
(parameter) $H$ defined as $H=\dot{a}/a$. In the following by a dot we denote
the derivative with respect to the time $t$. We will assume that the energy
momentum tensor has a perfect fluid form, and in the comoving frame it is
given by

$\displaystyle T^{\mu}_{~{}\nu}={{\rm diag}}(-\rho,p,p,p),$ (31)

where $\rho$ and $p$ denote the baryonic matter energy density and pressure,
respectively. For the matter Lagrangian we adopt the expression $L_{m}=-\rho$.
With the above assumptions we can write

$\displaystyle A_{\mu}=(A_{0},\vec{0}),$ (32)

where $A_{0}$ is a function of the cosmological time $t$ only.

The cosmological field equations of the Weyl geometric gravity theory can then
be written as

$\displaystyle\Phi(12H^{2}$
$\displaystyle-3A_{0}^{2}-\Phi)-24\xi^{2}\rho+6(2H+A_{0})\dot{\Phi}$
$\displaystyle+2\beta_{2}(1-2n)A_{0}^{2}X=0,$ (33) $\displaystyle
2\Phi\dot{H}$
$\displaystyle+\frac{3}{2}A_{0}^{2}\Phi+6\xi^{2}(\rho+p)-\left(H+3A_{0}\right)\dot{\Phi}+\ddot{\Phi}$
$\displaystyle+\beta_{2}nA_{0}^{2}X=0,$ (34)
$\displaystyle\dot{\Phi}-A_{0}\Phi-\frac{2}{3}\beta_{2}nA_{0}X=0,$ (35)
$\displaystyle 6\dot{H}+3\dot{A}_{0}-\Phi+\frac{3}{2}(A_{0}+2H)(A_{0}+4H)=0,$
(36)

and

$\displaystyle 3\xi^{2}(3p$ $\displaystyle-\rho)$
$\displaystyle=\beta_{2}X\left[(n-2)A_{0}^{2}-6nA_{0}H+2n(1-2n)\dot{A}_{0}\right],$
(37)

respectively.

Now, combining the above set of equations, one can obtain the conservation
equation of the matter energy momentum tensor as

$\displaystyle\dot{\rho}+3H(\rho+p)=0,$ (38)

which shows that the matter sector is conserved in this theory. This is
however obvious from the action (12), since there is no non-minimal coupling
between the ordinary baryonic matter, and the gravitational fields.

###  III.2 The vacuum solution

We will first investigate the cosmological evolution in Weyl geometric gravity
by neglecting the effect of the baryonic matter, that is, by assuming
$\rho=p=0$, and $L_{m}=0$, respectively. In this case the cosmological
evolution equations become

$\displaystyle
2\Phi\dot{H}+\frac{3}{2}A_{0}^{2}\Phi-(H+3A_{0})\dot{\Phi}+\ddot{\Phi}+\beta_{2}nA_{0}^{2}X=0,$
(39) $\displaystyle\dot{\Phi}-A_{0}\Phi-\frac{2}{3}\beta_{2}nA_{0}X=0,$ (40)
$\displaystyle 6\dot{H}+3\dot{A}_{0}-\Phi+\frac{3}{2}(A_{0}+2H)(A_{0}+4H)=0,$
(41)

and

$\displaystyle\beta_{2}X\left((n-2)A_{0}^{2}-6nA_{0}H+2n(1-2n)\dot{A}_{0}\right)=0,$
(42)

respectively. As can be seen from the above set of equations, we have more
equations than variables. In the following we will solve numerically the
equations (39)-(41). We will then substitute the solution into equation (42)
to check whether the extra equation is also satisfied.

We define now the following set of dimensionless variables

$\displaystyle\tau$ $\displaystyle=H_{0}t,\quad\quad H=H_{0}h,$ $\displaystyle
A_{0}$ $\displaystyle=H_{0}\bar{A}_{0},\quad\beta_{2}=\gamma
H_{0}^{4-2n},\quad\Phi=H_{0}^{2}\bar{\Phi},$ (43)

where $H_{0}$ is the present day value of the Hubble function. Moreover,
instead of the time coordinate, we introduce the redshift coordinate $z$,
defined as $1+z=1/a$.

Hence, one obtains the cosmological field equations of the Weyl geometric
gravity in the vacuum in a dimensionless form as

$\displaystyle 5(2n\gamma\bar{X}$
$\displaystyle+3\bar{\Phi})\bar{A}_{0}^{2}-4n(2n-1)(1+z)\gamma\bar{X}h\bar{A}_{0}^{\prime}$
$\displaystyle+6(1+z)\left(h^{\prime}\bar{\Phi}^{\prime}-h\bar{\Phi}(\bar{A}_{0}^{\prime}+2h^{\prime})\right)$
$\displaystyle+18(1+z)\bar{A}_{0}h\bar{\Phi}^{\prime}=0,$ (44)
$\displaystyle-\frac{2}{3}\bar{\Phi}+\bar{A}_{0}^{2}+6\bar{A}_{0}h+8h^{2}-2(1+z)h(\bar{A}_{0}^{\prime}+2h^{\prime})=0,$
(45) $\displaystyle 3(1+z)h\bar{\Phi}^{\prime}+3\bar{A}_{0}\bar{\Phi}-2\gamma
n\bar{X}\bar{A}_{0}=0,$ (46)

and

$\displaystyle\gamma\bar{X}\left(12n\bar{A}_{0}h-4n(2n-1)(1+z)h\bar{A}_{0}^{\prime}-(n-2)\bar{A}_{0}^{2}\right)=0,$
(47)

respectively. Here, we have denoted $\bar{X}=\bar{A}_{0}^{2n-2}$. Also, using
the relation $h(z=0)=1$, one can obtain the value of the parameter $\gamma$ in
terms of the present day value of the scalar and vector fields as

$\displaystyle\gamma=\frac{\bar{A}_{0}(0)^{1-2n}(\bar{\Phi}(0)^{2}-3(2+\bar{A}_{0}(0))^{2}\bar{\Phi}(0))}{2(\bar{A}_{0}(0)+4n)}.$
(48)

In order to find the best fit value of the parameters $H_{0}$,
$\bar{\Phi}_{0}\equiv\bar{\Phi}(0)$, $\bar{A}_{0}(0)$ and $n$, we use the
Likelihood analysis using the observational data on the Hubble parameter in
the redshift range $z\in(0.07,2.36)$ tabulated in [135].

In the case of independent data points, the likelihood function can be defined
as

$\displaystyle L=L_{0}e^{-\chi^{2}/2},$ (49)

where $L_{0}$ is the normalization constant, and the quantity $\chi^{2}$ is
defined as

$\displaystyle\chi^{2}=\sum_{i}\left(\frac{O_{i}-T_{i}}{\sigma_{i}}\right)^{2}.$
(50)

Here $i$ counts the data points, $O_{i}$ are the observational value, $T_{i}$
are the theoretical values, and $\sigma_{i}$ are the errors associated with
the $i$th data obtained from observations.

By maximizing the likelihood function, the best fit values of the parameters
$\bar{\Phi}_{0}\equiv\bar{\Phi}(0)$, $\bar{A}_{0}(0)$, $n$ and $H_{0}$ at
$1\sigma$ confidence level, can be obtained as

Figure 1: The corner plot for the values of the parameters $H_{0}$,
$\bar{\Phi}_{0}\equiv\bar{\Phi}(0)$ and $\bar{A}_{0}(0)$ with their $1\sigma$
and $2\sigma$ confidence levels for the vacuum solution of the FLRW Weyl
geometric gravity model.

$\displaystyle H_{0}$ $\displaystyle=71.547^{+0.779}_{-0.772},$
$\displaystyle\bar{A}_{0}(0)$ $\displaystyle=-0.040^{+0.001}_{-0.001},$
$\displaystyle\bar{\Phi}_{0}$ $\displaystyle=9.938^{+0.073}_{-0.076},$
$\displaystyle n$ $\displaystyle=1.989^{+0.101}_{-0.100}.$ (51)

The corner plot for the values of the parameters $H_{0}$, $\bar{\Phi}_{0}$ and
$\bar{A}_{0}(0)$ with their $1\sigma$ and $2\sigma$ confidence levels is shown
in Fig. 1.

The redshift evolution of the Hubble function and of the deceleration
parameter $q$ are represented, for this model, in Fig. 2. Also, we have
depicted the behavior of the cosmological fields $\bar{\Phi}$ and
$\bar{A}_{0}$ in Fig. 3.

Figure 2:  The behavior of the rescaled Hubble parameter $H/(1+z)$ (left
panel) and of the deceleration parameter $q$ (right panel) as a function of
the redshift for the vacuum FLRW Weyl geometric gravity model for the best fit
values of the parameters as given by Eqs. (III.2). The shaded area denotes the
$1\sigma$ error. The dashed line represents the $\Lambda$CDM model.

Figure 3:  The behavior of $\bar{A}_{0}$ (left panel) and of $\bar{\Phi}$
(right panel) as a function of the redshift in the vacuum FLRW Weyl geometric
gravity theory for the best fit values of the parameters as given by Eqs.
(III.2). The shaded area denotes the $1\sigma$ error.

###  III.3 Cosmological evolution in the presence of matter

We assume now that the equation of state of the baryonic matter fields is
given by $p=\omega_{m}\rho$, with a varying equation of state parameter. We
let the observations to determine $\omega_{m}$. From the matter conservation
equation, one can obtain

$\displaystyle\omega_{m}=-1-\frac{\dot{\rho}}{3H\rho}.$ (52)

After defining a set of dimensionless parameters as

$\displaystyle\tau$ $\displaystyle=H_{0}t,\quad
H=H_{0}h,\quad\bar{\rho}=\frac{2\xi^{2}}{H_{0}^{2}}\rho,$ $\displaystyle
A_{0}$ $\displaystyle=H_{0}\bar{A}_{0},\quad\beta_{2}=\gamma
H_{0}^{4-2n},\quad\Phi=H_{0}^{2}\bar{\Phi},$ (53)

and transforming to the redshift coordinate, defined again as $1+z=1/a$, one
obtains the cosmological field equations (generalized Friedmann equations) in
the presence of baryonic matter as

$\displaystyle
5(2n\gamma\bar{X}+3\bar{\Phi})\bar{A}_{0}^{2}-4n(2n-1)(1+z)\gamma\bar{X}h\bar{A}_{0}^{\prime}$
$\displaystyle+18(1+z)\bar{A}_{0}h\bar{\Phi}^{\prime}$
$\displaystyle+6(1+z)\left[h^{\prime}\bar{\Phi}^{\prime}-h\bar{\Phi}(\bar{A}_{0}^{\prime}+2h^{\prime})+\bar{\rho}^{\prime}\right]=0,$
(54)
$\displaystyle-\frac{2}{3}\bar{\Phi}+\bar{A}_{0}^{2}+6\bar{A}_{0}h+8h^{2}-2(1+z)h(\bar{A}_{0}^{\prime}+2h^{\prime})=0,$
(55)
$\displaystyle\gamma\bar{X}\left(12n\bar{A}_{0}h-4n(2n-1)(1+z)h\bar{A}_{0}^{\prime}-(n-2)\bar{A}_{0}^{2}\right)$
$\displaystyle-12\bar{\rho}+3(1+z)\bar{\rho}^{\prime}=0,$ (56)

and

$\displaystyle 3(1+z)h\bar{\Phi}^{\prime}+3\bar{A}_{0}\bar{\Phi}-2\gamma
n\bar{X}\bar{A}_{0}=0,$ (57)

respectively, where we have defined $\bar{X}=\bar{A}_{0}^{2n-2}$. Noting that
$h(z=0)=1$, one can obtain the value of the parameter $\gamma$ in terms of the
present value of the fields as

$\displaystyle\gamma=\frac{\bar{A}_{0}(0)^{1-2n}(\bar{\Phi}(0)^{2}-3(2+\bar{A}_{0}(0))^{2}\bar{\Phi}(0)+12\bar{\rho}(0))}{2(\bar{A}_{0}(0)+4n)}.$
(58)

By maximizing the likelihood function, the best fit values of the parameters
$\bar{\Phi}_{0}\equiv\bar{\Phi}(0)$, $\bar{A}_{0}(0)$, $\bar{\rho}_{0}$, $n$
and $H_{0}$ at $1\sigma$ confidence level, can be obtained as

Figure 4: The corner plot for the values of the parameters $H_{0}$,
$\bar{\rho}_{0}$,$\bar{\Phi}_{0}\equiv\bar{\Phi}(0)$ and $\bar{A}_{0}(0)$ with
their $1\sigma$ and $2\sigma$ confidence levels for the FLRW Weyl geometric
gravity model in the presence of matter.

$\displaystyle\bar{\rho}_{0}$ $\displaystyle=0.299^{+0.001}_{-0.001},$
$\displaystyle H_{0}$ $\displaystyle=71.768^{+0.784}_{-0.787},$
$\displaystyle\bar{A}_{0}(0)$ $\displaystyle=-0.040^{+0.010}_{-0.010},$
$\displaystyle\bar{\Phi}(0)$ $\displaystyle=10.210^{+0.085}_{-0.085},$
$\displaystyle n$ $\displaystyle=1.305^{+0.090}_{-0.084}.$ (59)

The corner plot for the values of the parameters $H_{0}$, $\bar{\rho}_{0}$,
$\bar{\Phi}_{0}$, $\bar{A}_{0}(0)$ and $n$ with their $1\sigma$ and $2\sigma$
confidence levels is shown in Fig. 4.

The redshift evolution of the Hubble function, of the deceleration parameter
$q$, and of the matter density parameter $\Omega=\bar{\rho}/h^{2}$ are
represented, for this model, in Figs. 5 and 6, respectively. Also, we have
depicted the behavior of the cosmological fields $\bar{\Phi}$ and
$\bar{A}_{0}$ in Fig. 7. One can compute the value of the equation of state
parameter $\omega_{m}$ as

$\displaystyle\omega_{m}=0.3216^{+0.0001}_{-0.0001}.$ (60)

Figure 5:  The behavior of the rescaled Hubble parameter $H/(1+z)$ (left
panel) and of the deceleration parameter $q$ (right panel) as a function of
the redshift for the best fit values of the parameters as given by Eqs.
(III.3) for the FLRW Weyl geometric gravity theory in the presence of matter.
The shaded area denotes the $1\sigma$ error. The dashed line represents the
$\Lambda$CDM model.

Figure 6:  The behavior of the matter density abundance $\bar{\rho}$ (left
panel) and of the equation of state parameter $\omega$ (right panel) as a
function of the redshift for the best fit values of the parameters as given by
Eqs. (III.3) for the FLRW Weyl geometric gravity theory in the presence of
matter. The shaded area denotes the $1\sigma$ error. The dashed line
represents the $\Lambda$CDM model.

Figure 7:  The behavior of $\bar{\Phi}$ (left panel) and $\bar{A}_{0}$ (right
panel) as a function of the redshift for the best fit values of the parameters
as given by Eqs. (III.3) for the FLRW Weyl geometric gravity theory in the
presence of matter. The shaded area denotes the $1\sigma$ error.

##  IV Cosmological implications of Weyl geometric gravity - the Bianchi type
I geometry

In the present Section, we consider the cosmological applications Of the Weyl
geometric gravity theory in an anisotropic cosmological setting. We begin our
analysis by writing down the gravitational field equations corresponding to an
anisotropic expansion of the Universe. We will consider only the case of flat
Bianchi type I geometries, which are the simplest anisotropic generalizations
of the flat FLRW geometry. As a first step in our analysis we analyze the
general properties of the anisotropic Bianchi type I model in Weyl geometric
gravity. In order to simplify our analysis, a perturbative approach is
developed.

###  IV.1 Gravitational field equations in the Bianchi type I geometry

In the following, we assume that on the large scale the Universe is
homogeneous, and thus all the cosmological and physical parameters (metric,
energy densities, pressures, Weyl vector, and scalar field) are functions of
the cosmological time $t$ only. In comoving Cartesian coordinates $x^{0}=t$,
$x^{1}=x$, $x^{2}=y$, and $x^{3}=z$, and in an anisotropic geometry, the
components of the ordinary matter energy-momentum tensor take the general form

$T_{0}^{0}=-\varepsilon,\qquad T_{1}^{1}=T_{2}^{2}=\Pi,\qquad T_{3}^{3}=\Psi,$
(61)

where $\varepsilon$ denotes the total (effective) energy density of the
cosmological fluid, which includes also the Weyl geometric contributions,
$\Pi=P_{x}=P_{y}$ denote the effective pressure along the $x$ and $y$
directions, while $\Psi=P_{z}$ is the pressure along the $z$ direction. All
the effective pressure terms include also the geometric contributions coming
from Weyl geometry.

Since the energy-momentum tensor (61) corresponds to an anisotropic effective
cosmological fluid, the geometry must also be anisotropic on a large
cosmological scale. From our choice of the components of the energy-momentum
tensor, as given by Eqs. (61), it follows that the effective pressure $\Psi$
along the $z-$ axis could be different with respect to the effective pressures
$\Pi$ along the $x$ and $y$ axis.

The simplest geometry presenting the symmetry (61) of the effective energy-
momentum tensor in a homogeneous Universe is the flat Bianchi type I geometry,
with the metric represented by

$ds^{2}=-dt^{2}+a_{1}^{2}(t)dx^{2}+a_{2}^{2}(t)dy^{2}+a_{3}^{2}(t)dz^{2},$
(62)

where $a_{i}$, $i=1,2,3$ are the three directional scale factors, which are
generally different. The Einstein field equations

$\displaystyle
R_{\mu\nu}=\frac{1}{2\kappa^{2}}\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right),$
(63)

take in a Bianchi type I geometry the form

$3\dot{H}+H_{1}^{2}+H_{2}^{2}+H_{3}^{2}=-\frac{1}{4\kappa^{2}}\left(\varepsilon+\Psi+2\Pi\right),$
(64)
$\frac{1}{V}\frac{d}{dt}\left(VH_{i}\right)=\frac{1}{4\kappa^{2}}\left(\varepsilon-\Psi\right),\qquad
i=1,2,$ (65)

and

$\frac{1}{V}\frac{d}{dt}\left(VH_{3}\right)=\frac{1}{4\kappa^{2}}\left(\varepsilon+\Psi-2\Pi\right),$
(66)

respectively, where, in order to simplify the mathematical formalism, we have
introduced the following notations

$V=a_{1}a_{2}a_{3},\qquad H_{i}=\frac{\dot{a}_{i}}{a_{i}},\qquad i=1,2,3,$
(67)

and

$H=\frac{1}{3}\left(\sum_{i=1}^{3}H_{i}\right)=\frac{\dot{V}}{3V},$ (68)

respectively.

$H_{i}$, $i=1,2,3$ are the three directional Hubble parameters, while by $H$
we have denoted the mean Hubble parameter. The cosmological expansion
parameter $\theta$ is related to the Hubble parameter by the relation
$\theta=3H$. The conservation of the total energy-momentum tensor of the Weyl
geometric gravity cosmological model gives the evolution equation of the
effective energy density of the cosmic fluid as

$\dot{\varepsilon}+3\left(\varepsilon+\Pi\right)H+\left(\Psi-\Pi\right)H_{3}=0.$
(69)

###  IV.2 General properties of the Bianchi type I models

By taking into account the explicit expression of the anisotropic energy-
momentum tensor as described by Eqs. (61), from Eqs. (65) it follows that we
can take $a_{1}=a_{2}$ without any loss of generality. Hence $\theta$, the
expansion parameter of the Universe, is represented as

$\theta=2\frac{\dot{a}_{1}}{a_{1}}+\frac{\dot{a}_{3}}{a_{3}}.$ (70)

The shear scalar $\sigma$ of the Universe is obtained in the form

$\sigma=\frac{1}{\sqrt{3}}\left(\frac{\dot{a}_{3}}{a_{3}}-\frac{\dot{a}_{1}}{a_{1}}\right).$
(71)

Eqs. (70) and (71) determine the relation between the directional Hubble
parameters $H_{1}$ and $H_{2}$ and the observable cosmological parameters as

$H_{1}=H_{2}=\frac{\theta}{3}-\frac{1}{\sqrt{3}}\sigma,\qquad
H_{3}=\frac{\theta}{3}+\frac{2}{\sqrt{3}}\sigma,$ (72)

respectively.

The late-time evolution of an anisotropic cosmological model can be obtained
from the study of another important cosmological quantity, the anisotropy
parameter $A$, defined according to

$A=\frac{1}{3}\sum_{i=1}^{3}\left(\frac{H_{i}-H}{H}\right)^{2}.$ (73)

If the anisotropy parameter vanishes, $A=0$, the considered cosmological model
is isotropic.

The differences of the effective pressures of the cosmological fluid are
given, as functions of the observable quantities, by the general relation

$\Psi-\Pi=2\sqrt{3}\kappa^{2}(\theta\sigma+\dot{\sigma}).$ (74)

By adding Eqs. (65) and (66) we obtain

$4\kappa^{2}\ddot{V}+\left(2\Pi+\Psi-3\varepsilon\right)V=0.$ (75)

These expressions will be considered in the next Section.

###  IV.3 Perturbative approach to the Bianchi type I cosmological models

In the present Section, we consider a simple and elementary perturbative
approach to the gravitational field equations (64)-(66), by assuming that the
anisotropic expansion due to the presence of the Weyl geometric effects
represents a small perturbation of the background flat, and isotropic FLRW
geometry.

Hence, the cosmological properties of the background geometry are described by
the isotropic scale factor $a$. Moreover, we suppose that the cosmological
expansion rates along the $x$ and $y$ axes are the same, which implies the
condition $a_{1}=a_{2}$. Furthermore, we assume that the perturbations along
these two axes are also equal. Therefore, the scale factors of the perturbed
Bianchi type I geometry can be represented as [136]

$a_{i}=a+\delta_{i},\quad\delta_{i}\ll a,\qquad i=1,2,3,$ (76)

where $\delta_{i}$, $i=1,2,3$, are small corrections terms to the scale factor
$a$ of the isotropic Universe, induced by the existence of the Weyl geometric
effects. Our previous assumption implies

$\delta_{1}=\delta_{2}.$ (77)

For the background isotropic FLRW type geometry the Hubble parameter is
$H_{0}=\dot{a}/a$. Hence, for the directional Hubble parameters $H_{i}$ of the
Bianchi type I geometry we obtain

$\displaystyle H_{i}=\frac{\dot{a}+\dot{\delta}_{i}}{a+\delta_{i}}\approx
H_{0}\left[1-\frac{\delta_{i}}{a}\right]+\frac{\dot{\delta_{i}}}{a},i=1,2,3.$

The expression of the mean Hubble parameter $H$ of the perturbed Bianchi type
I geometry in the presence of Weyl geometric effects is given by

$H=\frac{1}{3}\left[3H_{0}-H_{0}\frac{\delta}{a}+\frac{\dot{\delta}}{a}\right],$
(79)

where we have denoted

$\delta=\sum_{i=1}^{3}\delta_{i}=2\delta_{1}+\delta_{3}.$ (80)

The square of the directional Hubble parameters can be obtained as

$H_{i}^{2}\approx
H_{0}^{2}-2H_{0}^{2}\frac{\delta_{i}}{a}+2H_{0}\frac{\dot{\delta}_{i}}{a},\quad
i=1,2,3.$ (81)

For the comoving volume $V=\prod_{i=1}^{3}a_{i}$ of the Universe we obtain

$V=a^{3}\left[1+\frac{\delta}{a}\right],$ (82)

and $1/V\approx 1/a^{3}$.

To describe the perturbations of the effective energy density and pressure due
to the presence of the anisotropic effects induced by the presence of Weyl
geometric terms, we introduce the parameter $\beta$, which allows us to write
the perturbations of the effective energy density $\varepsilon$ and pressure
$\Psi=\Pi$ of the anisotropic cosmological model in Weyl geometric gravity in
the form

$\displaystyle\varepsilon=\rho_{DE}+\rho_{DM}+\beta\left(\rho_{DE}+p_{DE}+\rho_{DM}+p_{DM}\right),$
(83)

and

$\displaystyle\Psi=\Pi=p_{DE}+p_{DM}+\beta\left(\rho_{DE}+p_{DE}+\rho_{DM}+p_{DM}\right),$
(84)

respectively, where $\rho_{DE}$ and $p_{DE}$ represent the effective energy
density and pressure of the dark energy and dark matter components of the
isotropic matter distribution, constructed with the help of the Weyl geometric
quantities.

Therefore, in Eqs. (83) and (84) we have supposed that the perturbations of
the effective thermodynamic quantities of the Weyl geometric cosmological
fluid are proportional to the sum of the energy densities and pressures of the
effective quantities of the isotropic model.

Hence, in a first order approximation in the metric, the gravitational field
equations describing the slightly perturbed isotropic flat FLRW Universe due
to Weyl geometric effects become

$\displaystyle
3\dot{H}_{0}+3H_{0}^{2}-\left[\dot{H}_{0}+H_{0}^{2}\right]\frac{\delta}{a}+\frac{\ddot{\delta}}{a}$
$\displaystyle=-\frac{1}{4\kappa^{2}}\Big{[}\rho_{DE}+\rho_{DM}+3\left(p_{DE}+p_{DM}\right)$
$\displaystyle+4\beta\left(\rho_{DE}+p_{DE}+\rho_{DM}+p_{DM}\right)\Big{]},$
(85)
$\displaystyle\frac{1}{a^{3}}\frac{d}{dt}\left[a^{3}H_{0}\right]+\frac{1}{a^{3}}\frac{d}{dt}\left[a^{2}H_{0}\delta_{+}\right]+\frac{1}{a^{3}}\frac{d}{dt}\left[a^{2}\dot{\delta}_{1}\right]$
$\displaystyle-\frac{\delta}{a^{4}}\frac{d}{dt}\left[a^{3}H_{0}\right]=\frac{1}{4\kappa^{2}}(\rho_{DE}-p_{DE}+\rho_{DM}-p_{DM}),$
(86)

and

$\displaystyle\frac{1}{a^{3}}\frac{d}{dt}\left[a^{3}H_{0}\right]+\frac{2}{a^{3}}\frac{d}{dt}\left[a^{2}H_{0}\delta_{1}\right]-\frac{\delta}{a^{4}}\frac{d}{dt}\left[a^{3}H_{0}\right]$
$\displaystyle+\frac{1}{a^{3}}\frac{d}{dt}\left[a^{2}\dot{\delta}_{3}\right]=\frac{1}{4\kappa^{2}}\left(\rho_{DE}-p_{DE}+\rho_{DM}-p_{DM}\right),$

respectively, where we have denoted $\delta_{+}=\delta_{1}+\delta_{3}$.

The gravitational field equations of the background FLRW geometry are, in Weyl
geometric gravity, the generalized Friedmann equations, which give the
evolution of the scale factor $a$ as

$3\frac{\dot{a}^{2}}{a^{2}}=\frac{1}{2\kappa^{2}}(\rho_{DE}+\rho_{DM}),$ (88)
$2\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a}=-\frac{1}{2\kappa^{2}}(p_{DE}+p_{DM}),$
(89)

respectively. Eq. (75), describing the time variation of the rate of the
expansion of the volume of the Universe, becomes

$\displaystyle
4\kappa^{2}\frac{d^{2}}{dt^{2}}a^{3}+3\left[a^{3}\left(p_{DE}-\rho_{DE}+p_{DM}-\rho_{DM}\right)\right]$
$\displaystyle+4\kappa^{2}\frac{d^{2}}{dt^{2}}(a^{2}\delta)+3\left[a^{2}\left(p_{DE}-\rho_{DE}+p_{DM}-\rho_{DM}\right)\right]\delta=0.$
(90)

Figure 8: The behavior of the perturbation function $\delta$ as a function of
the redshift $z$. The shaded area corresponds to the $1\sigma$ domain for the
model parameters (III.3).

Figure 9: The behavior of the perturbation functions $\delta_{1}$ (left panel)
and $\delta_{3}$ (right panel) as a function of the redshift $z$. The shaded
area corresponds to the $1\sigma$ domain for the model parameters (III.3).

With the use of Eqs. (88) and (89) we straightforwardly find

$\frac{1}{a^{3}}\frac{d\left[a^{3}H_{0}\right]}{dt}=\frac{1}{4\kappa^{2}}\left(\rho_{DE}-p_{DE}+\rho_{DM}-p_{DM}\right),$
(91)
$3\dot{H}_{0}+3H_{0}^{2}=-\frac{1}{4\kappa^{2}}\left[\rho_{DE}+\rho_{DM}+3\left(p_{DE}+p_{DM}\right)\right],$
(92)

and

$\frac{d^{2}a^{3}}{dt^{2}}+\frac{3}{4\kappa^{2}}\left[a^{3}\left(p_{DE}-\rho_{DE}+p_{DM}-\rho_{DM}\right)\right]=0,$
(93)

respectively.

Consequently, the gravitational field equations describing the cosmological
evolution in the presence of a small anisotropy in the $z$-direction, induced
by the presence of the Weyl geometric effects, are given by

$\displaystyle\frac{\ddot{\delta}}{a}-\frac{\ddot{a}}{a}\frac{\delta}{a}=-\frac{\beta}{\kappa^{2}}\left(\rho_{DE}+p_{DE}+\rho_{DM}+p_{DM}\right),$
(94)
$\displaystyle\frac{1}{a^{3}}\frac{d}{dt}\left[a^{2}H_{0}\delta_{1}\right]+\frac{1}{a^{3}}\frac{d}{dt}\left[a^{2}\dot{\delta}_{1}\right]-\frac{\delta}{a^{4}}\frac{d}{dt}\left[a^{3}H_{0}\right]=0,$
(95)
$\frac{2}{a^{3}}\frac{d}{dt}\left[a^{2}H_{0}\delta_{1}\right]+\frac{1}{a^{3}}\frac{d}{dt}\left[a^{2}\dot{\delta}_{3}\right]-\frac{\delta}{a^{4}}\frac{d}{dt}\left[a^{3}H_{0}\right]=0.$
(96)

For a known $H_{0}$ and $a$,Eqs. (94)-(96) represent a system of three
ordinary differential equations for the three time dependent functions
$\left(\delta_{1},\delta_{3},\beta\right)$, whose solutions indicate the
effects of the presence of Weyl geometry on the cosmological evolution.

We consider now the perturbations of the average deceleration parameter $q$ of
the anisotropic Bianchi type I Universe, which is defined generally according
to

$\langle q\rangle=\frac{d}{dt}\frac{1}{H}-1.$ (97)

Assuming again that the deviations from isotropy induced by the Weyl geometric
effects are small, the average deceleration parameter becomes

$\langle
q\rangle=q_{0}+\frac{1}{3}\frac{d}{dt}\left(\frac{1}{aH_{0}}\left[\delta-\frac{\dot{\delta}}{H_{0}}\right]\right),$
(98)

where $q_{0}=d\left[1/H_{0}\right]/dt-1=-a\ddot{a}/\dot{a}^{2}$ is the
deceleration parameter as defined in the FLRW geometry.

By adding Eqs. (95) and (96), we find the second order differential equation
describing the evolution of the total perturbation $\delta$ due to the
presence of anisotropic effects induced by the Weyl geometry as

$\ddot{\delta}+4H_{0}\dot{\delta}-\left[\dot{H}_{0}+5H_{0}^{2}\right]\delta=0.$
(99)

Eq. (99) must be considered together with the two initial conditions

$\delta\left(t_{0}\right)=\delta_{0},\qquad\dot{\delta}\left(t_{0}\right)=H_{0}\left(t_{0}\right)\delta_{0}.$
(100)

Hence, we have obtained the result that the time evolution of the metric
perturbations, induced by the presence of Weyl geometric effects, is fully
determined by the Hubble parameter of the isotropic FLRW background. In Fig. 8
we have plotted the behavior of the perturbation function $\delta$ as a
function of redshift, by adopting for $H_{0}$ the isotropic Weyl geometric
form considered in the previous Section. The shaded area corresponds to the
$1\sigma$ error in the model parameters given by (III.3).

Once the total perturbation function $\delta$ is known, the behavior of the
deviations function from isotropy along the $x$ and $y$ axes, $\delta_{1}$, is
obtained as a solution of the second order differential equation

$\displaystyle
2\frac{d}{dt}(a^{2}H_{0}\delta_{1})-2\frac{d}{dt}(a^{2}\dot{\delta}_{1})+\frac{d}{dt}(a^{2}\dot{\delta})-\frac{\delta}{a}\frac{d}{dt}(a^{3}H_{0})=0.$
(101)

The behavior of $\delta_{1}$ depends on the evolution of both $a$ and $H_{0}$.
Finally, In Weyl geometric gravity the time evolution of the deviations from
isotropy along the $z$ axes can be obtained immediately from the simple
relation

$\delta_{3}=\delta-2\delta_{1}.$ (102)

In Figs. 9 we have plotted the behavior of the perturbation functions
$\delta_{1}$ and $\delta_{3}$ as a function of the redshift.

The parameter $\beta$, describing the perturbations of the effective geometric
type density and pressure can be obtained as

$\beta=\kappa^{2}\frac{(\ddot{H}_{0}+H_{0}^{2})\delta-\ddot{\delta}}{a(\rho_{DE}+p_{DE}+\rho_{DM}+p_{DM})}.$
(103)

In Figs. 10 we have plotted the behavior of the perturbation function $\beta$
as a function of the redshift.

Figure 10: The behavior of the perturbation function $\beta$ as a function of
the redshift $z$. The shaded area corresponds to the $1\sigma$ domain for the
model parameters (III.3).

In Fig. 11 we have presented the behavior of the mean anisotropy parameter
$A$, in the presence of the Weyl geometric effects, as a function of the
redshift $z$.

Figure 11: The behavior of the anisotropy parameter $A$ as a function of the
redshift $z$. The shaded area corresponds to the $1\sigma$ domain for the
model parameters (III.3).

In Fig. 12 we have presented the difference between the isotropic and
anisotropic deceleration parameter $\Delta q$, in the presence of the Weyl
geometric effects, as a function of the redshift $z$.

Figure 12: The behavior of the difference between the isotropic and
anisotropic deceleration parameter $\Delta q$ as a function of the redshift
$z$. The shaded area corresponds to the $1\sigma$ domain for the model
parameters (III.3).

###  IV.4 The quadrupole moment $Q_{2}$

The anisotropic expansion of the Universe will also have an influence on
photon trajectories, and on the CMBR spectrum. The photons emitted by
astrophysical sources located at cosmological distances travel in the Bianchi
type I geometry by following the geodesic lines, described by the geodesic
equation given by [137]

$\frac{du^{\mu}}{d\lambda}+\Gamma^{\mu}_{\alpha\beta}u^{\alpha}u^{\beta}=0,$
(104)

where $\lambda$ is the affine parameter along the trajectory. The Christoffel
symbols $\Gamma^{\mu}_{\alpha\beta}$ can be easily obtained from the metric
Eq. (62), and they are given by $\Gamma^{0}_{ii}=a_{i}^{2}H_{i}$, and
$\Gamma^{i}_{0i}=H_{i}$, $i=1,2,3$ (no summation upon $i$ in the Christoffel
symbols).

The four-velocity $u^{\mu}=dx^{\mu}/d\lambda$ of the photons is normalized
according to the relation $u^{\mu}u_{\mu}=0$, from which we obtain
$\left(u_{0}\right)^{2}=a_{i}^{2}\left(u^{i}\right)^{2}$. Let us now consider
the emission of two photons, occuring at the times $t_{0}=t_{e}$ and
$t_{1}=t_{e}+\delta\tau$, respectively, where $\delta\tau\ll t_{e}$. in the
first order of approximation in $\delta\tau$, after taking the difference of
the two normalization conditions for photons, we obtain [137]

$u^{0}\frac{d}{d\lambda}\delta\tau(\lambda)=\sum_{i=1}^{3}{a_{i}\dot{a}_{i}\left(u^{i}\right)^{2}\delta\tau(\lambda)}+O\left(\delta\tau^{2}\right),$
(105)

where $\lambda$ is the wave length of the radiation. By denoting by
$\delta\tau\left(\lambda_{r}\right)$ the time difference between the received
signals, and by introducing the redshift $z$, defined according to
$1+z\left(\lambda_{e}\right)=\delta\tau\left(\lambda_{r}\right)/\delta\tau\left(\lambda_{e}\right)$,
with the use of Eq. (105) we obtain the equation

$\frac{d}{d\lambda}\ln(1+z)=\frac{1}{u^{0}}\sum_{i=1}^{3}{a_{i}\dot{a}_{i}\left(u^{i}\right)^{2}}.$
(106)

The components $u^{i}$, $i=1,2,3$, of the photon velocity follow immediately
from the geodesic equation of motion, and they are given by

$\frac{du^{i}}{d\lambda}+2\frac{\dot{a}_{i}}{a_{i}}u^{i}u^{0}=0,\qquad
i=1,2,3,$ (107)

giving

$u^{i}(t)=\frac{u_{0i}}{a_{i}^{2}(t)},\qquad i=1,2,3,$ (108)

where $u_{0i}$, $i=1,2,3$ are arbitrary constants of integration, which can be
determined by taking into account that the present day values of the scale
factors are normalized according to $a_{i}\left(t_{0}\right)=1$.

We can reparameterize the affine parameter $\lambda$ without any impact on the
physical behavior of the radiation, and thus we normalize the present day
photon four-velocities $u^{i}\left(t_{0}\right)=\hat{u}^{i}$, $i=1,2,3$,
according to $\sum_{i=1}^{3}\left(\hat{u}^{i}\right)^{2}=1$. Moreover, we
describe the unit vector $\hat{u}$ with the help of the angles
$\left(\hat{u}_{x},\hat{u}_{y},\hat{u}_{z}\right)=\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right)$,
which give the arrival angles of the photon beams to the observer, as
estimated at the present time. After the substitution of the photon velocities
into the redshift definition, we find the relation [137]

$1+z\left(\hat{\vec{u}}\right)=\sqrt{\sum_{i=1}^{3}{\frac{\hat{u}_{i}^{2}}{a_{i}^{2}}}},$
(109)

or, equivalently,

$1+z\left(\hat{\vec{u}}\right)=\frac{1}{a_{1}}\sqrt{1+\hat{u}_{y}^{2}e_{y}^{2}+\hat{u}_{z}^{2}e_{z}^{2}}.$
(110)

In Eq. (110 we have introduced the eccentricities $e^{2}_{y}$ and $e_{z}^{2}$,
defined according to

$e_{y}^{2}=\left(\frac{a_{1}}{a_{2}}\right)^{2}-1,\qquad
e_{z}^{2}=\left(\frac{a_{1}}{a_{3}}\right)^{2}-1.$ (111)

Since in the anisotropic Weyl geometric gravity model we have assumed
$a_{1}=a_{2}$, we immediately obtain $e_{y}^{2}=0$, and

$e_{z}^{2}=\frac{2}{a}\left[\delta_{1}(t)-\delta_{3}(t)\right],$ (112)

respectively.

The multipole spectrum $Q_{l}$ of the CMBR is described in terms of the
coefficients in the spherical expansion of the temperature anisotropy field.
The observationally important quadrupole term $Q_{2}$ is obtained as [137]

$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{5\sqrt{3}}\sqrt{e_{z}^{4}+e_{y}^{4}-e_{z}^{2}e_{y}^{2}}=\frac{2}{5\sqrt{3}}e_{z}^{2}$
(113) $\displaystyle=$
$\displaystyle\frac{4}{5\sqrt{3}}\frac{1}{a}\left[\delta_{1}(t)-\delta_{3}(t)\right].$

In the case of the anisotropic Weyl geometric gravity cosmological models one
obtains

$\displaystyle Q_{2}=\frac{2}{5\sqrt{3}}(1+z)(5\delta_{1}-2\delta),$ (114)

In Fig. 13 we have plotted the quantity $Q_{2}$ as a function of the redshift
$z$.

Figure 13: The behavior of the quadrupole term $Q_{2}$ as a function of the
redshift $z$ in the anisotropic Weyl geometric gravity cosmological models.
The shaded area corresponds to the $1\sigma$ domain for the model parameter
(III.3).

The quadrupole moment measured by the Planck satellite is $Q\equiv\left(\Delta
T/T\right)_{Q}\sim 4.5\times 10^{-6}$ [138]. Hence, observational results on
the CMBR could be used to constrain the numerical values of the model
parameters, and the predictions of the Weyl geometric gravity cosmological
models.

The temperature distribution of the CMBR can be obtained from the relation

$T\left(\hat{\vec{u}}\right)=\frac{T_{*}}{1+z\left(\hat{\vec{u}}\right)},$
(115)

where by $T_{*}$ we have denoted the last scattering temperature, which is
independent on the direction. But in the presence of anisotropies in the
metric, photons travelling from distinct directions will be redshifted
differently. The spatial average $\bar{T}$ of the temperature field can be
computed from the relation
$4\pi\bar{T}=\int{T\left(\hat{\vec{u}}\right)d\Omega_{\hat{\vec{u}}}}$. The
anisotropies in the temperature field can be obtained as

$\displaystyle\delta T\left(\hat{\vec{u}}\right)$ $\displaystyle=$
$\displaystyle 1-\frac{T\left(\hat{\vec{u}}\right)}{\bar{T}}$ (116)
$\displaystyle=$ $\displaystyle
1-\frac{T_{*}}{\bar{T}}\frac{1}{\sqrt{1+2\left[\delta_{1}\left(t_{0}\right)-\delta_{3}\left(t_{0}\right)\right]\cos^{2}\theta}}.$

Hence, a detailed comparison of the predictions of the anistropic Weyl
geometric cosmological model with the CMBR data could lead to obtaining
significant constraints on the presence of the anisotropy in the Universe, and
of the Weyl geometric effects.

##  V Discussions and final remarks

In the present paper, we have investigated the theoretical possibility that
our Universe may be modeled in terms of a Weylian geometric structure, in
which the gravitational properties are described by the metric tensor, and two
other fields, namely, a scalar and a vector field. The considered theory
implements strictly the idea of the conformal invariance of the gravitational
interaction. The problem of the role of the conformal symmetry, or of the Weyl
gauge group in gravitational theories has attracted a lot of interest
recently. Gerard ’t Hooft advanced the idea that conformal symmetry is an
exact symmetry of the natural laws, which is spontaneously broken during the
evolution of the Universe [139]. Therefore, conformal symmetry may be as
important for physics as the Lorentz invariance of natural laws is. Moreover,
the breaking of the conformal symmetry may allow us to understand in geometric
terms the small scale structure of gravity, and to obtain new insights in the
physics of the Planck scale. An approach to gravity based on the assumption
that conformal symmetry is an exact local, but spontaneously broken symmetry,
was investigated in [140].

Conformal symmetry also plays a fundamental role in the Conformal Cyclic
Cosmology (CCC) model [141, 142, 143], in which the Universe is assumed to
exist as o set of eons, geometric structures corresponding to time oriented
spacetimes. Eons have interesting mathematical properties, and, for example,
they possess, as a direct result of their conformal compactification,
spacelike null infinities.

In the Weyl geometric theory considered in this work the idea of conformal
invariance is rigorously implemented at the level of both geometry and matter.
The starting point is the conformally invariant geometric quadratic Weyl
action, with gravitational Lagrangian $\tilde{R}^{2}$, supplemented by the
square of the strength of the Weyl vector field, and by a matter term. A new
perspective on the theory can be obtained by introducing an auxiliary scalar
field in the formalism, thus transforming the initial vector-tensor theory
into a scalar-vector tensor theory, linear in the Weyl (and Ricci) scalars.
The physical implications of this theory have been investigated in detail
elsewhere [118, 119, 120, 121], and it was shown that it has many attractive
features, including the possibility of representing a bridge between gravity
and elementary particle physics. We have analyzed in detail the cosmological
implications of the theory, by assuming a particular form, additive in the
baryonic matter Lagrangian, and the Weyl vector, of the effective, conformally
invariant matter Lagrangian. By using this simplified model we have considered
in detail two general classes of cosmological models.

The first class we have considered is represented by flat, homogenous and
isotropic FLRW type models. The symmetry of the problem imposes a fixed
structure of the Weyl vector, which has only a non-zero temporal component.
After obtaining the generalized Friedmann equations, we have investigated the
vacuum model, and a baryonic matter filled Universe. In both cases we have
compared the predictions of the Weyl geometric gravity theory with a small set
of observational data for the Hubble parameter, as well as with the
predictions of the $\Lambda$CDM model.

The vacuum Weyl geometric gravity model gives a relatively good description of
the observational data for the Hubble, indicating that the geometric
contributions from the Weyl geometry can successfully simulate dark energy and
dark matter, and, to some extent, even baryonic matter. The statistical
analysis predicts a value of the Hubble constant which is closer to the SHOES
value [64] than to the Planck value [62]. The deceleration parameter of the
vacuum model is relatively consistent with the $\Lambda$CDM predictions in the
redshift range $0<z<1$, but significant differences do appear at higher
redshifts. The exponent $n$, giving the contribution of the square of the Weyl
vector to the effective, conformally invariant matter energy momentum tensor,
is almost equal to 2, and hence
$\mathcal{L}_{m}=L_{m}+\beta\left(-\omega^{2}\right)^{2}$. The temporal
component of the Weyl vector takes negative values, and it is a decreasing
function of the redshift (an increasing function of time). On the other hand,
the scalar field $\Phi$ is a monotonically increasing function of the
redshift, increasing rapidly with increasing $z$.

Adding baryonic matter to the Weyl geometric FLRW model does not change
drastically the parameters of the cosmological evolution. In the presence of
matter the model also gives an acceptable description of the observational
data, as well as of the $\Lambda$CDM model, but without succeeding in
reproducing it exactly. Similar differences do appear in the case of the
deceleration parameter, with the Weyl geometric gravity model indicating much
higher values of $q$ at higher redshifts. Significant differences do appear in
the predictions of the behavior of the matter energy density, with the Weyl
geometric gravity model predicting much higher matter density values as
compared to the $\Lambda$CDM model. In the presence of the baryonic matter the
temporal component of the Weyl vector is a monotonically decreasing function
of the redshift, taking negative values, while the scalar field is a positive,
monotonically increasing function of the redshift.

There is compelling observational evidence [62] that the Universe is
isotropic, and homogeneous on large scales. However, the Planck Collaboration
experiments have not provided yet a conclusive observational proof for the
cosmological isotropy. Moreover, recently a number of observations have
questioned the nature of the geometry of cosmological spacetime itself, and
suggested the existence of deviations from the homogeneous and isotropic FLRW
geometry. In this respect we may mention the quadrupole-octupole alignment
problem, the lack of correlations on large angular scales, and the
hemispherical power asymmetry [144, 145]. All these observations seem to
suggest a violation of statistical isotropy in the Universe, and of the scale-
invariance of the primordial spectrum fluctuations. Hence, the investigation
of the anisotropic cosmological models may not be of purely theoretical
interest, but could lead to the explanation of some observed features of the
cosmological expansion. Hence, the above observations, as well as the Planck
results suggest that the possibility of the existence of a large scale
cosmological anisotropy in the Universe cannot be neglected a priori in the
theoretical models.

In the present work we have also considered an analysis of the simplest
anisotropic, Bianchi type I cosmological models, in the Weyl geometric gravity
theory. . We have assumed that the Weyl vector has two nonzero components,
$\omega_{\mu}=\left(\omega_{0},0,0,\omega_{3}\right)$. As a first step in our
analysis we have reformulated the cosmological evolution equations in terms of
an effective energy density, and of two effective pressures, which combine the
effects of the baryonic matter and of the Weyl geometry. We have also
considered that the deviations from isotropy are very small, and that the
anisotropic properties of the Universe can be described as a small
perturbation of the homogeneous and isotropic FLRW background metric. For the
case of the Bianchi type I geometry considered in the present paper we have
explicitly derived the perturbation equations, by assuming that the scale
factors of the Bianchi type I Universe are given by $a_{i}=a+\delta a_{i}$,
$i=1,2,3$, with $\delta a_{i}$ satisfying the condition $\delta_{i}\ll a$,
$i=1,2,3$. If this condition is not satisfied, non-linear effects must also be
taken into account, and included in the analysis of the cosmological
evolution.

Therefore, due to the presence of the $z$ component of the Weyl vector, and of
the anisotropic effective pressure distribution, the Weyl type Universe would
achieve some anisotropic properties, and its geometry will slightly differ
from the standard FLRW one. Our perturbative analysis has shown that the
behavior of the total perturbations of the FLRW metric $\delta$, given by Eq.
99, depend only, in the considered linear approximation, on the Hubble
function of the isotropic model. Once this function is known, by numerically
integrating Eq. (99) one can obtain a full description of the behavior of the
anisotropic perturbations in the cosmology of the Weyl geometric gravity
theory.

The total perturbation of the metric, shown in Fig. 8, is a monotonically
increasing positive function of the redshift, or a monotonically decreasing
function of the cosmological time. $\delta$ is relatively constant in the
redshift range $0<z<1$, and increases rapidly afterwards, making at higher
redshifts the Universe more and more anisotropic. $\delta_{1}$, represented in
Fig. 9, has a similar behavior as $\delta$, but $\delta_{3}$, shown in the
same Figure, takes negative values, and decreases with the redshift, thus
increasing the $z$ axis anisotropy with increasing $z$.

An important parameter describing the cosmological properties of the
anisotropic Weyl geometric gravity cosmological models is the parameter
$\beta$, defined in Eqs. (83) and (84), respectively, and which describes the
perturbations of the effective densities and pressure of the model, which
include the contributions of the Weyl vector, and of the scalar field. The
variation of $\beta$, represented in Fig. 10, indicates that for the
considered background cosmological model this quantity takes negative values,
and it decreases with the redshift.

The anisotropy parameter $A$, depicted in Fig. 11, is practically zero in the
redshift range $0<z<2$, and it increases rapidly for $z>2$. This indicates a
rapid increase, and the strong presence, of the anisotropies at higher
redshifts. Hence, in the present cosmological scenario, the initially
anisotropic Universe isotropizes in the large time limit, with the present day
Universe being isotropic on average. The difference of the isotropic and
anisotropic deceleration parameters, represented in Fig. 12, shows an increase
of the difference at higher redshifts, the differences between the two
quantities varying slowly in the redshift range $0<z<1$.

In our analysis we have also pointed out the possibilities of observationally
testing the Weyl geometric gravity cosmological model, which could be done,
for example, by using the Planck data [62]. We have explicitly obtained the
expression of the quadrupole $Q_{2}$ for an anisotropic Universe as functions
of the deviations $\delta_{1}$ and $\delta_{3}$ from the isotropic FLRW
geometry. The variation of $Q_{2}$ as a function of redshift is presented in
Fig. 13). $Q_{2}$ is practically constant in the redshift range $0<z<1$, but
it rapidly increases at higher redshifts. One can use the obtained expression
of $Q_{2}$ to obtain some constraints on the parameters of the anisotropic
Weyl geometric cosmological model. We have already obtained constraints on the
isotropic model parameters from the study of the luminosity distance by using
the observational data coming from the type Ia supernovae.

In the present paper we have investigated, in the framework of Weyl geometric
gravity, both homogeneous isotropic and anisotropic cosmological models. We
have also performed a full comparison of the predictions of the isotropic
cosmological model with the observations, and we have found that the model can
give a satisfactory description of the observational data up to a redshift of
$z=2$. We have also assumed the existence on large cosmological scales of
small deviations from isotropy, and we have obtained a full description of the
behavior of the Bianchi type I models in Weyl geometric gravity. In this study
we have obtained the basic theoretical tools necessary for the in depth
investigation of the implications of the Weyl geometric effects in cosmology.
The impact of the Weyl geometry on the cosmological evolution, the
observational implications of the presence of this geometry, as well the
possibilities of its observational testing, will be considered in future
studies.

## Acknowledgments

The work of T.H. is supported by a grant of the Romanian Ministry of Education
and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-2255 (PNCDI
III).

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