# Friedmann-like universes with non-metricity

Damianos Iosifidis<EMAIL_ADDRESS>Institute of Theoretical Physics,
Department of Physics, Aristotle University of Thessaloniki, Thessaloniki
54124, Greece  Ioannis Georgios Vogiatzis<EMAIL_ADDRESS>Department
of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
Christos G. Tsagas<EMAIL_ADDRESS>Section of Astrophysics, Astronomy
and Mechanics, Department of Physics, Aristotle University of Thessaloniki,
Thessaloniki 54124, Greece
and
Clare Hall, University of Cambridge, Herschel Road, Cambridge, CB3 9AL, UK

###### Abstract

We study the potential effects of spacetime non-metricity in cosmology. In the
spirit of Einstein-Cartan gravity, but with non-metricity replacing torsion,
we consider the Einstein-Hilbert action and assume zero torsion. Adopting
certain hyperfluid models, with non-vanishing hypermomentum that can source
spacetime non-metricity, we add a matter component into the action and derive
the field equations, along with the conservation laws. Applying our formulae
to cosmology, we generalize the Friedmann and the Raychaudhuri equations in
the presence of non-metricity. Our results show that, in a number of cases,
non-metricity can mimic the effects of matter with unconventional equation of
state. For instance, specific types of hypermomentum are found to act as an
effective stiff fluid, thus opening the possibility that non-metricity could
have played a significant role in the early stages of the universe’s
evolution. Alternative forms of hypermomentum can dominate the universal
dynamics at late times. In either case, the equilibrium moment depends on the
initial conditions and it is determined by a simple relation between the
matter component and the hyperfluid.

##  I Introduction

Non-Riemannian geometries eisenhart2012non  offer a natural playground for
developing gravitational theories beyond the Riemannian framework of general
relativity. The additional geometric structures of such theories come from
both torsion and non-metricity, as is the case of the Metric-Affine Gravity
(MAG) framework hehl1995metric ; iosifidis2019metric . Naturally, such
fundamental modifications have a strong impact on the cosmic evolution and
alter the Friedmann equations in a non-trivial manner puetzfeld2005prospects .
For instance, in kranas2019friedmann ; pereira2019acceleration ;
Guimaraes:2020drj  and also in Iosifidis:2021iuw , it was shown that torsion
can drastically alter the cosmological evolution and provide intriguing new
results. From the geometrical point of view, torsion reflects the fact that
the parallel transport of two vectors along each other does not form a closed
parallelogram. As for non-metricity, it manifests the inability of the metric
to be covariantly conserved, which in turn implies that lengths and angles are
not preserved during the parallel transport of vectors iosifidis2019metric .
Possible experimental effects/manifestations of torsion and of non-metricity
have been studied by Kostelecky kostelecky2008constraints , while other
potential signatures of non-metricity have been investigated in
delhom2018observable ; bahamonde2021observational .

These non-Riemannian degrees of freedom, namely torsion and non-metricity, are
sourced by the hypermomentum tensor of the matter hehl1976hypermomentum . The
latter is formally defined as the variational derivative of the matter action
with respect to the connection and encapsulates the microstructure properties
of the matter fields, that is dilation, shear and spin. Continua with non-
vanishing hypermomentum tensor have been constructed in obukhov1993hyperfluid
; obukhov1996model ; babourova1995variational  and the development of the
Perfect Hyperfluid structure has been formulated in Iosifidis:2020gth , with
its generalization given in iosifidis2021perfect . The latter extends the
classical perfect fluid notion of general relativity (GR) to account for the
microspcopic properties of the matter. This model can be used to study the
cosmological aspects of torsion and non-metricity, since it excites all their
degrees of freedom in a homogeneous and isotropic Friedmann-like universes
Iosifidis:2020gth . The role of torsion in Friedmann-like universes has been
studied by several authors (e.g. see kranas2019friedmann ; Iosifidis:2021iuw ;
barrow2019friedmann ; pereira2019acceleration ). Here, instead, we focus on
non-metricity and study its role in homogeneous and isotropic Friedmann-like
universes.111The cosmological aspects of the general quadratic metric-affine
gravity with torsion and non-metricity have been recently studied in
iosifidis2022cosmology . We find that, depending on the form of the
hypermomentum tensor, the effects of spacetime non-metricty can mimic those of
matter with different equations of state. When the hyperfluid has only a
dilation component, for example, it behaves like stiff matter and it can
dominate the early evolution of the universe. When dealing with purely shear
hypermomentum, on the other hand, the effects of non-metricity are identical
to those of a perfect fluid with an arbitrary barotropic index and can
dominate at late times as well. Whether the moment of equilibrium (between the
matter component and the hyperfluid) lies in the past or in the future, is
determined by a simple relation that depends on the initial conditions.

The paper is organised as follows. In section II we develop the framework,
including the necessary definitions and the basic geometric notions of non-
metricity, the matter conservation laws and the field equations of the theory.
Then, in section III, we outline the key cosmological aspects of Friedmann-
like universes with non-metricity and the sources that produce them. Section
IV is primarily devoted to the investigation of the hyperfluid model proposed
in obukhov1996model . In section V we adopt the most general perfect
cosmological (hypermomentum-preserving) hyperfluid and demonstrate how it
excites all the non-metricity degrees of freedom. There, we also derive the
modified Friedmann and Raychaudhuri equations, together with the associated
conservation laws. These are accompanied by a number of exact cosmological
solutions and their physical interpretation. Finally, we conclude with a
discussion of our results and an outline of possible future extensions.

##  II The theoretical framework

The fundamental novelty of non-metricity is that, in direct contrast to
Riemannian geometry, the metric tensor is no longer covariantly constant. Put
another way, the covariant derivative of the metric is not necessarily zero.
This introduces a number of unconventional degrees of freedom to any
theoretical model of the universe.

###  II.1 The non-metricity tensor

Relaxing the metricity constraint allows magnitudes and angles to change
during the parallel transport of vectors and tensors. These changes reflect
the fact that the metric tensor is not covariantly conserved, and its failure
to remain so is measured by the non-metricity tensor

$Q_{\lambda\mu\nu}=-\nabla_{\lambda}g_{\mu\nu}\,,$ (1)

with $Q_{\lambda\mu\nu}=Q_{\lambda(\mu\nu)}$ by construction. The symmetries
of $Q_{\lambda\mu\nu}$ ensure that one can construct a pair of independent
non-metricity vectors, namely

$Q_{\lambda}=Q_{\lambda\mu\nu}g^{\mu\nu}\hskip 42.67912pt{\rm and}\hskip
42.67912ptq_{\nu}=Q_{\lambda\mu\nu}g^{\lambda\mu}\,,$ (2)

the former of which is usually referred to as the Weyl vector.

In addition to non-metricity, spacetime torsion is also present in a general
non-Riemannian space. The torsion tensor is defined as the antisymmetric part
of the connection

$S_{\kappa\lambda}^{\;\;\;\;\mu}=\Gamma^{\mu}_{\;\;\;[\kappa\lambda]}\;.$ (3)

out of which one can construct the torsion vector

$S_{\mu}=S_{\mu\nu}^{\;\;\;\;\nu}\,.$ (4)

Let us note that, with the exception of this section, in the rest of this
manuscript the connection will be treated as symmetric and therefore the space
will be assumed torsion free.

###  II.2 Connection and curvature

In the general metric-affine framework, the connection and the metric are
regarded as independent variables. Thus, contrary to GR, the connection is no
longer expressed as a function of the metric (and of its derivatives) alone.
More specifically, in a torsion-free space, the symmetric connection is given
by

$\Gamma^{\lambda}_{\;\;\;\mu\nu}=\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}+\frac{1}{2}\,g^{\rho\lambda}\left(Q_{\mu\nu\rho}+Q_{\nu\rho\mu}-Q_{\rho\mu\nu}\right)\,.$
(5)

The first term on the right-hand side of the above is the Levi-Civita
connection, with

$\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}=\frac{1}{2}\,g^{\rho\lambda}\left(\partial_{\mu}g_{\nu\rho}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\right)\,,$
(6)

while the rest form the so-called “distortion tensor”222In spaces where only
non-metricity is present, the distortion tensor is sometimes referred to as
“deflection”.

$N^{\lambda}_{\;\;\;\mu\nu}=\Gamma^{\lambda}_{\;\;\;\mu\nu}-\tilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}=\frac{1}{2}\,g^{\rho\lambda}\left(Q_{\mu\nu\rho}+Q_{\nu\rho\mu}-Q_{\rho\mu\nu}\right)\,.$
(7)

Therefore, a general affine connection decomposes into its Levi-Civita
component and a non-metricity contribution. As a result, any quantity that
depends on the connection splits into a Riemannian and a non-Riemannian part.

In the presence of non-metricity, spacetime curvature is determined by the
associated generalised Riemann tensor. The latter satisfies an expression
formally identical to that of its Riemannian analogue, namely

$R^{\mu}_{\;\;\;\nu\alpha\beta}=2\partial_{[\alpha}\Gamma^{\mu}_{\;\;\;|\nu|\beta]}+2\Gamma^{\mu}_{\;\;\;\rho[\alpha}\Gamma^{\rho}_{\;\;\;|\nu|\beta]}\,.$
(8)

Nevertheless, $R^{\mu}_{\;\;\;\nu\alpha\beta}$ lacks some of the symmetries we
are accustomed with from standard GR. As a result, in non-Riemannian
manifolds, there are three independent contractions of the Riemann tensor. The
first corresponds to the familiar Ricci tensor defined as

$R_{\nu\beta}=R^{\mu}_{\;\;\;\nu\mu\beta}=2\partial_{[\mu}\Gamma^{\mu}_{\;\;\;|\nu|\beta]}+2\Gamma^{\mu}_{\;\;\;\rho[\mu}\Gamma^{\rho}_{\;\;\;|\nu|\beta]}\,,$
(9)

which (in contrast to GR) is not necessarily symmetric. The other two
contractions are known as the “homothetic curvature tensor” and the “co-Ricci
tensor”. These are given by333The symmetries of the conventional curvature
tensor ensure that the homothetic curvature tensor vanishes in Riemannian
spaces, while the co-Ricci tensor is the opposite of the Ricci tensor.

$\hat{R}_{\alpha\beta}=R^{\mu}_{\;\;\;\mu\alpha\beta}\hskip 42.67912pt{\rm
and}\hskip
42.67912pt\check{R}^{\mu}_{\;\;\;\beta}=g^{\nu\alpha}R^{\mu}_{\;\;\;\nu\alpha\beta}\,,$
(10)

respectively. Finally, the trace of the above leads to the uniquely defined
Ricci scalar

$R=R^{\beta}_{\;\;\;\beta}=g^{\nu\beta}R_{\nu\beta}\,,\hskip 28.45274pt{\rm
with}\hskip 28.45274pt\hat{R}_{\mu\nu}g^{\mu\nu}\equiv 0\,.$ (11)

It goes without saying that all these tensors and scalars also decompose into
their Riemannian and non-metricity parts (see section IV below).

###  II.3 Sources and conservation laws

In metric affine gravity the action is a functional of the metric, of the
independent affine connection and of the various matter fields. In other
words,

$S[g,\Gamma,\phi]=S_{G}[g,\Gamma]+S_{M}[g,\Gamma,\phi]\,,$ (12)

where $S_{G}$ is the gravitational and $S_{M}$ is the matter component of the
action.444Gravitational theories where the matter part of the action does not
explicitly depend on the connection are often called Palatini theories. The
tensors associated to the matter are the usual (metrical) energy-momentum
tensor

$T^{\alpha\beta}=\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\,\mathcal{L}_{M})}{\delta
g_{\alpha\beta}}$ (13)

and (since matter couples to the connection) the hypermomentum tensor. The
latter encapsulates the microscopic characteristics of the matter (i.e. spin,
dilation and shear) and it is given by hehl1976hypermomentum

$\Delta_{\lambda}^{\;\;\;\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\,\mathcal{L}_{M})}{\delta\Gamma^{\lambda}_{\;\;\;\mu\nu}}\,.$
(14)

Working in the equivalent exterior calculus formalism and using the vielbien
$e_{\mu}^{\;\;\;c}$, we can also define the canonical energy-momentum tensor
by means of

$t^{\mu}_{\;\;c}=\frac{1}{\sqrt{-g}}\,\frac{\delta S_{M}}{\delta
e_{\mu}^{\;\;c}}\,.$ (15)

The metrical and the canonical energy-momentum tensors are not independent,
but they are related and obey the conservation laws obukhov2013conservation ;
Iosifidis:2020gth

$t^{\mu}_{\;\;\lambda}=T^{\mu}_{\;\;\lambda}-\frac{1}{2\sqrt{-g}}\,\hat{\nabla}_{\nu}(\sqrt{-g}\,\Delta_{\lambda}^{\;\;\mu\nu})$
(16)

and

$\frac{1}{\sqrt{-g}}\,\hat{\nabla}_{\mu}(\sqrt{-g}\,t^{\mu}_{\;\;\alpha})=-\frac{1}{2}\,\Delta^{\lambda\mu\nu}R_{\lambda\mu\nu\alpha}+\frac{1}{2}\,Q_{\alpha\mu\nu}T^{\mu\nu}+2S_{\alpha\mu\nu}t^{\mu\nu}\,,$
(17)

with $S_{\alpha\mu\nu}$ and $S_{\nu}$ representing the torsion tensor and
torsion vector respectively. Also, the operator $\hat{\nabla}_{\nu}$ seen in
both of the above is defined as

$\hat{\nabla}_{\nu}:=2S_{\nu}-\nabla_{\nu}\,.$ (18)

Note that our conservation laws (16) and (17) follow directly from the
diffeomorphic invariance of the matter action and also from the GL invariance
that the theory exhibits when written in the language of differential forms.

Let us now switch the torsion off and also adopt the scheme of
Iosifidis:2020gth , where the metrical and the canonical energy-momentum
tensors coincide. This assumption does not always hold, since the metrical
energy-momentum tensor is symmetric and the canonical is generally not.
Nevertheless, in homogeneous and isotropic spacetimes (like those considered
here), both tensors are symmetric and the above assumption holds. In addition,
allowing the metrical and the canonical energy-momentum tensors to differ,
leads to a hypermomentum tensor that is not conserved and to considerably more
involved equations. We shall therefore confine to hypermomentum-conserving
hyperfluids, with

$\nabla_{\nu}(\sqrt{-g}\,\Delta_{\lambda}^{\;\;\mu\nu})=0\ $ (19)

and

$\frac{1}{\sqrt{-g}}\,\nabla_{\mu}(\sqrt{-g}\,T^{\mu}_{\;\;\alpha})=\frac{1}{2}\,\Delta^{\lambda\mu\nu}R_{\lambda\mu\nu\alpha}-\frac{1}{2}\,Q_{\alpha\mu\nu}T^{\mu\nu}\,,$
(20)

throughout the rest of this manuscript.

###  II.4 The field equations

Our aim is to study the cosmological implications of a gravitational theory
that is analogous to the classical Einstein-Cartan gravity, but with non-
metricity replacing torsion. To this end, we shall consider the action

$S[g,\Gamma,\varphi]=\frac{1}{2\kappa}\int d^{n}x\sqrt{-g}R+S_{M}\,,$ (21)

where the first term on the right-hand side is the usual Einstein-Hilbert
action, $S_{M}$ denotes the matter sector and $\varphi$ represents the matter
fields that produce spacetime non-metricity. Since the latter is at the centre
of our study, we will assume a symmetric (i.e. torsion-free) connection from
now on. Then, variation with respect to the metric and the independent
connection, yields the field equations

$R_{(\mu\nu)}-\frac{1}{2}\,Rg_{\mu\nu}=\kappa T_{\mu\nu}\hskip 42.67912pt{\rm
and}\hskip
42.67912ptP_{\lambda}^{\;\;(\mu\nu)}=\kappa\Delta_{\lambda}{}^{(\mu\nu)}\,,$
(22)

with

$P_{\lambda}{}^{(\mu\nu)}=\frac{1}{2}\,Q_{\lambda}g^{\mu\nu}-Q_{\lambda}{}^{\mu\nu}+\left(q^{(\mu}-\frac{1}{2}\,Q^{(\mu}\right)\delta_{\lambda}{}^{\nu)}\,,$
(23)

giving the symmetrised (torsion-free) Palatini tensor. The first set of
formulae represents the modified Einstein equations in the presence of non-
metricity, while the second shows how the hypermomentum sources non-metricity.
In what follows, we will apply the above to a homogeneous and isotropic
Friedmann-like universe.

##  III Cosmology with non-metricity

The Friedmann models are characterised by their spatial homogeneity and
isotropy. Non-metricity introduces new degrees of freedom to all cosmological
scenarios and broadens their phenomenology. This is also true for the
Friedmann universes, although their high symmetry demands that the matter
content has the perfect-fluid form.

###  III.1 Geometrical aspects of Friedmann-like universes non-metricity

Let us consider a spatially flat (i.e. with $K=0$, where $K$ is the
3-curvature index) Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime,
satisfying the familiar Robertson-Walker line element

$ds^{2}=-dt^{2}+a^{2}\delta_{ij}dx^{i}dx^{j}\,,$ (24)

where $a=a(t)$ is the cosmological scale factor and $i,j=1,2,3$. Introducing
the normalized $4$-velocity field $u_{\mu}$, with co-moving coordinates
$u^{\mu}=\delta^{\mu}{}_{0}=(1,0,0,0)$ and $u_{\mu}u^{\mu}=-1$, the associated
temporal-derivative operator is555Evaluating the temporal derivative with
respect to the Levi-Civita connection for a scalar yields the same result.

$\dot{}=u^{\alpha}\nabla_{\alpha}\,.$ (25)

In addition, the symmetric tensor

$h_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}\,,$ (26)

acts as the metric of the spatial hypersurfaces. By construction, the above
projects into the 3-space orthogonal to $u^{\mu}$ (since
$h_{\mu\nu}u^{\nu}=0$) and also satisfies the constraints $h_{\mu}{}^{\mu}=3$
and $h_{\mu\alpha}h^{\alpha}{}_{\nu}=h_{\mu\nu}$. On using the $u_{\mu}$ and
the $h_{\mu\nu}$ fields, one achieves an 1+3 decomposition of the spacetime
into time and 3-D space.

In non-Riemannian Friedmann-like spacetimes, torsion is described by two and
non-metricity by three degrees of freedom, as it was respectively shown in
tsamparlis1979cosmological ; minkevich1998isotropic . Covariantly written, the
non-metricity tensor reads Iosifidis:2020gth

$Q_{\alpha\mu\nu}=Au_{\alpha}h_{\mu\nu}+Bh_{\alpha(\mu}u_{\nu)}+Cu_{\alpha}u_{\mu}u_{\nu}\,,$
(27)

the contractions of which lead to the non-metricity vectors

$Q_{\mu}=(3A-C)u_{\mu}\hskip 42.67912pt{\rm and}\hskip
42.67912ptq_{\mu}=\left(\frac{3}{2}\,B-C\right)u_{\mu}\,.$ (28)

The functions $A=A(t)$, $B=B(t)$ and $C=C(t)$ monitor the cosmological effects
of non-metricity in Friedmann-like universes. Also, together with the scale
factor, the above describe non-metric FLRW-like spacetimes. For instance,
along with the FLRW metric, expressions (27) and (28) can be used to calculate
the Riemann tensor and its contractions (by means of Eq. (5) – see Appendix
A).

###  III.2 The perfect cosmological hyperfluid

In order to probe the implications of non-metricity for cosmology, we will
consider an FLRW-type universe filled with a perfect hyperfluid
Iosifidis:2020gth . The latter generalises the familiar perfec fluid by
accounting for the non-Riemannian degrees of freedom. Then, relative to a
family of observers with 4-velocity $u_{\mu}$, the energy-momentum tensor of
the matter reads

$T_{\mu\nu}=\rho u_{\mu}u_{\nu}+ph_{\mu\nu}\,,$ (29)

with $\rho$ and $p$ representing the energy density and the isotropic pressure
of the matter respectively. As we have already discussed, the canonical
energy-momentum tensor also takes the above form. In addition, demanding that
the hypermomentum tensor obeys the Cosmological Principle, we have
Iosifidis:2020gth

$\Delta_{\alpha\mu\nu}=\phi h_{\mu\alpha}u_{\nu}+\chi
h_{\nu\alpha}u_{\mu}+\psi u_{\alpha}h_{\mu\nu}+\omega
u_{\alpha}u_{\mu}u_{\nu}+\epsilon_{\alpha\mu\nu\rho}u^{\rho}\zeta\,.$ (30)

The above sources both torsion and non-metricity, with the former related to
the spin part ($\Delta_{[\alpha\mu]\nu}$). Therefore, in order to isolate the
non-metricity effects, we set $\Delta_{[\alpha\mu]\nu}=0$. This ensures that
$\psi=\chi$ and subsequently leads to

$\Delta_{\alpha\mu\nu}=\phi h_{\mu\alpha}u_{\nu}+2\psi
u_{(\alpha}h_{\mu)\nu}+\omega u_{\alpha}u_{\mu}u_{\nu}\,.$ (31)

We have now developed all the necessary formalism needed for the rest of our
analysis. Next, in section IV, we will start with the cosmological
implications of the Obukhov hyperfluid obukhov1996model . Then, in section V,
we turn our attention to the generalised non-metricity tensor (27) developed
in Iosifidis:2020gth .

##  IV Cosmology with Obukhov’s hyperfluid

In Obukhov’s model the canonical energy-momentum tensor coincides with the
metrical, while the hypermomentum tensor has the special (density)x(current)
form obukhov1996model . Here, we will further constrain this hypermomentum
form, in order to comply with the Cosmological Principle.

###  IV.1 Energy-momentum and hypermomentum

As mentioned above, the energy-momentum tensor has the familiar perfect fluid
form, with $T_{\mu\nu}=\rho u_{\mu}u_{\nu}+ph_{\mu\nu}$. At the same time, the
hypermomentum tensor reads

$\Delta_{\lambda}^{\;\;\;\mu\nu}=J_{\lambda}^{\;\;\mu}u^{\nu}\,,$ (32)

where $J_{\mu\nu}$ is the hypermomentum density of the hyperfluid. Given that
the antisymmetric part ($J_{[\mu\nu]}$) is related to spin and torsion
hehl1995metric , hereafter $J_{\mu\nu}$ will be assumed symmetric. The
simplest assumption is that the hypermomentum density is proportional to the
projection tensor $h_{\mu\nu}$.666This is not the only possibility and we will
discuss the most general case in the next section. Then, due to the high
symmetry of the FLRW-type host, there exists a time-dependent function
$\phi=\phi(t)$ so that

$J_{\mu\nu}=\phi h_{\mu\nu}$ (33)

and $\Delta_{\alpha\mu\nu}=\phi h_{\alpha\mu}u_{\nu}\,$. As we will see next,
the scalar field $\phi$ completely determines the non-metricity efects.

###  IV.2 Non-metricity and curvature

Non-metricity and hypermomentum are related through the hypermomentum field
equations. Substituting the hypermomentum form seen in (IV.1) above into Eq.
(22), leads to the non-metricity tensor

$Q_{\alpha\mu\nu}=-\frac{1}{2}\,\kappa\phi
u_{\alpha}(h_{\mu\nu}+u_{\mu}u_{\nu})\,,$ (34)

the contractions of which provide the associated non-metricity vectors

$Q_{\mu}=-\kappa\phi u_{\mu}\hskip 42.67912pt{\rm and}\hskip
42.67912ptq_{\mu}=\frac{1}{2}\,\kappa\phi u_{\mu}\,.$ (35)

Using the above, one can calculate the corresponding curvature tensors. The
Ricci scalar, in particular, decomposes into its Riemannian and non-Riemannian
parts as

$R=\tilde{R}+\frac{1}{4}\,Q_{\alpha\mu\nu}Q^{\alpha\mu\nu}-\frac{1}{2}\,Q_{\alpha\mu\nu}Q^{\mu\nu\alpha}-\frac{1}{4}\,Q_{\mu}Q^{\mu}+\frac{1}{2}\,Q_{\mu}q^{\mu}+\tilde{\nabla}_{\mu}(q^{\mu}-Q^{\mu})\,.$
(36)

Note that the tilded variables are evaluated with respect to the Christoffel
connection, which makes $\tilde{R}$ the Riemannian Ricci scalar. Then,
employing (34) and (35), one arrives at

$R=\tilde{R}+\frac{3}{8}\,\kappa^{2}\phi^{2}+\frac{3}{2}\,\kappa\frac{1}{a^{3}}\,\partial_{t}(a^{3}\phi)\,,$
(37)

where

$\tilde{R}=6\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}\right]\,,$
(38)

is the Riemannian Ricci scalar. Proceeding in a similar manner one can compute
the Ricci tensor in the presence of spacetime non-metricity. In particular,
the timelike component of the latter is found to be

$R_{00}=\tilde{R}_{00}-\frac{3}{2}\,A^{2}-\frac{3}{4}\kappa\frac{1}{a^{3}}\partial_{t}\Big{(}a^{3}\phi\Big{)}\,,$
(39)

with

$\tilde{R}_{00}=-3\frac{\ddot{a}}{a}\,,$ (40)

representing the Riemannian analogue. Likewise, for the spacelike part of the
Ricci tensor, we obtain

$R_{ij}=\tilde{R}_{ij}+\frac{1}{4}\,\kappa
g_{ij}\left[\frac{1}{a^{3}}\,\partial_{t}\left(a^{3}\phi\right)\right]\,,$
(41)

the trace of which ($R=g^{00}R_{00}+g^{ij}R_{ij}$) leads to the Ricci scalar
given in Eq. (37) previously. Substituting the above post-Riemannian
expansions into the metric field equations, one arrives at the generalized
Friedmann equations in the presence of non-metric degrees of freedom described
by $(\ref{Qh})$.

###  IV.3 Modified Friedmann equations and conservation laws

Taking the timelike ($00$) and the spatial ($ij$, with $i,j=1,2,3$) components
of the metric field equations and using the curvature relations of the last
section, we obtain the modified Friedmann equations. When spscetime non-
metricity is sourced by the Obukhov hyperfluid, these take the form

$H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{3}\,\kappa\rho+\frac{1}{16}\,\kappa^{2}\phi^{2}\hskip
42.67912pt{\rm and}\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(\rho+3p)-\frac{1}{8}\,\kappa^{2}\phi^{2}-\frac{1}{4}\,\kappa\frac{\partial_{t}(a^{3}\phi)}{a^{3}}\,,$
(42)

with $H=\dot{a}/a$ representing the Hubble parameter (e.g. see
tsagas2008relativistic ). The first of the above, which measures the
contribution of matter and non-metricity to the Hubble expansion, also recasts
as

$1=\Omega_{\rho}+\Omega_{\phi}\,,$ (43)

where $\Omega_{\rho}=\kappa\rho/3H^{2}$ is the familiar density parameter
(e.g. see tsagas2008relativistic ) and
$\Omega_{\phi}=\kappa^{2}\phi^{2}/16H^{2}$ is its non-metricity analogue.
Accordingly, matter dominates the energy density of the universe and drives
its expansion, when $\Omega_{\rho}\gg\Omega_{\phi}$. In the opposite case, the
universe is dominated by spacetime non-metricity.

Turning to the conservation laws, we point out that, in FLRW-type spacetimes
with hypermomentum of the form introduced in section IV.1, one can easily show
that

$\Delta^{\alpha\beta\gamma}R_{\alpha\beta\gamma\nu}\equiv 0\,.$ (44)

Then, the conservation law of the associated energy-momentum tensor (see Eq.
$(\ref{QlawEM})$ earlier) reduces to the familiar continuity equation

$\dot{\rho}+3H(\rho+p)=0\,,$ (45)

of a standard perfect fluid tsagas2008relativistic . The corresponding
conservation law of the hypermomentum (see expression (19)) provides the
evolution formula of $\phi$. By employing the identity

$\nabla_{\lambda}(\sqrt{-g}\phi u^{\lambda})=\partial_{\lambda}(\sqrt{-g}\phi
u^{\lambda}),$ (46)

which holds for vanishing torsion, the hypermomentum conservation law reads

$\dot{\phi}+3H\phi=0\,,$ (47)

ensuring the $\phi\propto a^{-3}$. In other words, the non-metricity scalar
decays with the expansion at the rate of pressure free “dust”. Finally,
substituting the evolution law of $\phi$ back into Eq. (42), we arrive at

$\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(\rho+3p)-\frac{1}{8}\,\kappa^{2}\phi_{0}^{2}\left(\frac{a_{0}}{a}\right)^{6}\,,$
(48)

where the zero suffix corresponds to a chosen moment in time. The above is the
modified Raychaudhuri equation, monitoring the acceleration/deceleration of
the expansion of a Friedmann-like universe in the presence of Obukhov-type
non-metricity. In particular, negative terms on the right-hand side of the
above decelerate the expansion, while positive ones tend to accelerate.
Alternatively, one can say that the negative terms assist gravitational
collapse, whereas the positive ones tend to resist contraction.

Before closing this section, we should note that the non-metricity
contribution to both Eqs. (42a) and (48) is proportional to $a^{-6}$, which
means that its effects resemble those of a “stiff” fluid with $p=\rho$ (in
agreement with PhysRevD.56.7769 ). Consequently, the Obukhov non-metricity can
play a significant role only at early times. This is not unexpected, since the
post-Riemannian effects typically dominate at the early stages of the universe
2005 . Also, following the modified Friedmann equation – see (42a), non-
metricity contributes positively to the total energy density of the universe
and it cannot act as an effective “ghost” fluid. Finally, according to the
modified Raychaudhuri equation – see (48), the Obukhov non-metricity always
tends to decelerate the expansion (or alternatively accelerate the collapse).

###  IV.4 Generalized unconstrained hyperfluid

In an FLRW-like universe with non-metrictiy, the hypermmentum density of the
hyperfluid ($J_{\mu\nu}$) is generally spanned by two time functions. More
specifically, expression (33) generalises to

$J_{\mu\nu}=\phi h_{\mu\nu}+\psi u_{\mu}u_{\nu}\,,$ (49)

with $\psi=\psi(t)$ as well. Then, the corresponding hypermomentum tensor
acquires the general form

$\Delta_{\alpha\mu\nu}=(\phi h_{\alpha\mu}+\psi u_{\alpha}u_{\mu})u_{\nu}\,.$
(50)

In this scenario, the dynamics of the hypermomentum and the effects of non-
metricity are determined by $\psi$ as well as $\phi,\psi$. The question then
is whether the inclusion of $\psi$ could radically change the results of the
previous section. In order to investigate this, we will employ the “first
theorem” formulated in Iosifidis:2018jwu  and solve the connection in terms of
the Palatini tensor. The result reads

$\Gamma^{\lambda}{}_{\mu\nu}=\tilde{\Gamma}^{\lambda}{}_{\mu\nu}+\frac{1}{2}\,g^{\lambda\alpha}\left(P_{\alpha\mu\nu}-P_{\nu\alpha\mu}-P_{\mu\nu\alpha}\right)+\frac{1}{2}\,g^{\alpha\lambda}g_{\nu[\mu}\left(P_{\alpha]}-\tilde{P}_{\alpha]}\right)+\frac{1}{2}\,\delta_{\mu}{}^{\lambda}q_{\nu}\,,$
(51)

which means that the general distortion tensor (including both Q and S) is
given by

$N^{\lambda}{}_{\mu\nu}=\frac{1}{2}\,g^{\lambda\alpha}\left(P_{\alpha\mu\nu}-P_{\nu\alpha\mu}-P_{\mu\nu\alpha}\right)+\frac{1}{2}\,g^{\alpha\lambda}g_{\nu[\mu}\left(P_{\alpha]}-\tilde{P}_{\alpha]}\right)+\frac{1}{2}\,\delta_{\mu}{}^{\lambda}q_{\nu}\,.$
(52)

In the case of vanishing torsion, we set $S_{\mu\nu\lambda}=0$ and also
symmetrize the above over the indices $\mu$ and $\nu$, since the connection is
now symmetric. Therefore, for zero torsion, the distortion tensor simplifies
to

$N_{\alpha\mu\nu}=N_{\alpha(\mu\nu)}=\frac{1}{2}\Big{(}P_{\alpha(\mu\nu)}-P_{(\nu|\alpha|\mu)}-P_{(\mu\nu)\alpha}\Big{)}+\frac{1}{2}\,g_{\alpha(\mu}Q_{\nu)}+\frac{1}{2}\left(q_{\alpha}-Q_{\alpha}\right)g_{\mu\nu}\,.$
(53)

Finally, upon using the connection field equations (see relation (22b)) and
the fact that $\Delta_{\alpha\mu\nu}=\Delta_{(\alpha\mu)\nu}$, the above
expression reduces further to

$N_{\alpha\mu\nu}=-\frac{1}{2}\,\kappa\Delta_{\mu\nu\alpha}+\frac{1}{2}\,g_{\alpha(\mu}Q_{\nu)}+\frac{1}{2}\left(q_{\alpha}-Q_{\alpha}\right)g_{\mu\nu}\,.$
(54)

With this in hand and using the post-Riemannian expansions of the Ricci tensor
and the Ricci scalar, a lengthy calculation leads to the associated Friedmann
equations

$H^{2}=\frac{1}{3}\,\kappa\left[\rho+\frac{1}{2a^{3}}\,\partial_{t}\left(a^{3}\psi\right)\right]+Y^{2}\hskip
42.67912pt{\rm and}\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(\rho+3p)+\frac{1}{a^{3}}\,\partial_{t}\left(a^{3}Y\right)-2Y^{2}\,,$
(55)

where

$Y=-\frac{1}{12}\,\kappa(3\phi+\psi)\,.$ (56)

These relations are supplement by the conservation laws of the matter fields
and the non-metricity sources. It can be easily seen that the continuity
equation of the matter remains the same, namely

$\dot{\rho}+3H(\rho+p)=0\,.$ (57)

On the other hand, the evolution of the non-metricity fields $\phi$ and
$\psi$, follows form the conservation law of the hypermomentumd. More
specifically, taking the trace of Eq. (19), we obtain

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

which implies that $3\phi-\psi\propto a^{-3}$. Furthermore, expanding Eq. (19)
gives

$\delta_{\lambda}{}^{\mu}\partial_{\nu}\left(\sqrt{-g}\phi
u^{\nu}\right)+u_{\lambda}u^{\mu}\partial_{\nu}\left(\sqrt{-g}\,(\phi+\psi)u^{\nu}\right)+\sqrt{-g}\,(\phi+\psi)(\dot{u}_{\lambda}u^{\mu}+u_{\lambda}\dot{u}^{\mu})=0\,,$
(59)

the timelike and the spacelike components of which integrate to

$\phi=\phi_{0}\left(\frac{a_{0}}{a}\right)^{3}\hskip 42.67912pt{\rm and}\hskip
42.67912pt\psi=\psi_{0}\left(\frac{a_{0}}{a}\right)^{3}\,,$ (60)

respectively.777Taking the time derivative of Eq. (55a) and using the
conservation laws (57) and (60) leads to (55b) as expected. Our last step is
to plug these results back into the Friedmann equations (see expressions (55a)
and (55b) above) and in so doing arrive at

$H^{2}=\frac{1}{3}\,\kappa\rho+\frac{1}{16}\,\kappa^{2}\left(\phi+\frac{1}{3}\,\psi\right)^{2}\hskip
42.67912pt{\rm and}\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(\rho+3p)-\frac{1}{8}\,\kappa^{2}\left(\phi+\frac{1}{3}\,\psi\right)^{2}\,.$
(61)

Comparing these results with Eqsd. (42a) and (42b), we realise that there is
no essential differences between the two sets. In fact, one can recover the
latter set of formulae from the former by simply introducing the “shift”
$\phi\rightarrow\phi+\psi/3$ and vice versa. It is therefore not surprising
that this type of non-metricity also decelerates the expansion (or accelerates
gravitational collapse) and it cannot act as an effective ghost fluid.
Nevertheless, this happens for the model of the unconstrained hyperfluid,
where the hypermomentum is either of purely dilatonic type, or it is related
to it by means of a disformal transformation of the metric.888Following
$(\ref{D2})$, we may write
$\Delta_{\alpha\mu\nu}=\tilde{g}_{\alpha\mu}u_{\nu}$, with
$\tilde{g}_{\alpha\mu}=\phi g_{\alpha\mu}+(\phi+\psi)u_{\alpha}u_{\mu}$. It is
possible for non-metric degrees of freedom, associated with more general forms
for hypermomentum, to accelerate the expansion, or resist gravitational
collpase (see next section and also section V below).

###  IV.5 Weyl non-metricity

In order to demonstrate the versatility of the non-Riemannian degrees of
freedom, let us consider the simple case of Weyl non-metricity, with

$Q_{\alpha\mu\nu}=Au_{\alpha}g_{\mu\nu}\,,$ (62)

with $A=A(t)$. Following a procedure identical to the one outlined above, one
arrives at the modified Friedmann equations

$H^{2}=\frac{1}{\,}\kappa\rho+HA-\frac{1}{2}\,A^{2}\hskip 42.67912pt{\rm
and}\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(\rho+3p)+\frac{1}{2}\,\dot{A}+\frac{1}{2}\,HA\,.$
(63)

Accordingly, depending on the signs of $A$ and $\dot{A}$, the hypefluid and
act as conventional or ghost-like matter, while at the same time it can either
decelerate or accelerate the expansion of the host universe.

Finally, it is worth noting the duality between the above set of Friedmann
equations and their purely torsional counterparts obtained in
kranas2019friedmann ; Iosifidis:2018jwu . Indeed, the two sets are mapped to
each other upon the duality exchange $A\leftrightarrow-4\phi$.

##  V General non-metricity sourced by the perfect Ccosmological hyperfluid

The most general form on non-metricity allowed by the high symmetry of the
FLRW models, is sourced by the “perfect cosmological hyperfluid”, recently
formulated and developed in Iosifidis:2020gth . As a result, this model opens
up a plethora of new phenomenological possibilities when studying the
implications of non-metricity for cosmology.

###  V.1 The perfect hyperfluid

Allowing for the maximum number of non-metric degrees of freedom compatible
with the Cosmological Principle, translates into a non-metricity tensor of the
form

$Q_{\alpha\mu\nu}=Au_{\alpha}h_{\mu\nu}+Bh_{\alpha(\mu}u_{\nu)}+Cu_{\alpha}u_{\mu}u_{\nu}\,,$
(64)

spanned by three temporal functions $A$, $B$ and $C$. Recall that the
unconstrained hyperfluid discussed earlier contained only two such parameters.
Following Iosifidis:2020gth , the hypermomentum tensor of the perfect
hyperfluid reads

$\Delta_{\alpha\mu\nu}=\phi h_{\mu\alpha}u_{\nu}+\chi
h_{\nu\alpha}u_{\mu}+\psi u_{\alpha}h_{\mu\nu}+\omega
u_{\alpha}u_{\mu}u_{\nu}+\epsilon_{\alpha\mu\nu\rho}u^{\rho}\zeta\,.$ (65)

The spin part of the above is known to source torsion hehl1995metric . The
latter vanishes by default in our study, which transalets into the constraints
$\psi=\chi$ and $\zeta=0$. As a result, when dealing with pure non-metricity,
the hypermomentum tensor reduces to

$\Delta_{\alpha\mu\nu}=\phi h_{\mu\alpha}u_{\nu}+\psi
h_{\nu\alpha}u_{\mu}+\psi u_{\alpha}h_{\mu\nu}+\omega
u_{\alpha}u_{\mu}u_{\nu}\,,$ (66)

which is the form we will be using for the rest of this study.

###  V.2 Relating non-metricity and hypermomentum

Substituting expression (64), together with the associated non-metricity
vectors $Q_{\mu}=(3A-C)u_{\mu}$ and $q_{\mu}=[(3B/2)-C]u_{\mu}$, back into Eq.
(23) leads to the (symmetrized) Palatini tensor

$P_{\alpha\mu\nu}=P_{\alpha(\mu\nu)}=\frac{1}{2}\,(A-C)u_{\alpha}h_{\mu\nu}+\frac{1}{2}\,(B-3A-C)h_{\alpha(\mu}u_{\nu)}-\frac{3}{2}\,Bu_{\alpha}u_{\mu}u_{\nu}\,.$
(67)

The above and Eq. (66) combine in the connection field equation (22b) – notice
the symmetrization in the hypermomentum in the latter expression – to provide
the relations between the non-metricity functions and the sources (i.e. the
hypermomentum variables). To be precise, we obtain

$A=-\frac{1}{6}\,\kappa(3\phi+\omega)\,,\hskip
42.67912ptB=-\frac{2}{3}\,\kappa\omega\hskip 42.67912pt{\rm and}\hskip
42.67912ptC=-\frac{1}{6}\,\kappa(3\phi+\omega+12\psi)\,.$ (68)

Alternatively, one arrives at the same result by contracting the connection
field equations to express the non-metricity vectors in terms of their
hypermomentum counterparts (see Appendix B).

We finally note that the case of the unconstrained hyperfluid (see section IV)
is characterised by $\chi=0=\psi=\omega$, which substituted back into Eqs (68)
give

$C=A=-\frac{1}{2}\,\kappa\phi\hskip 42.67912pt{\rm and}\hskip
42.67912ptB=0\,,$ (69)

in agreement with our earlier results. Therefore, as expected, the
unconstrained hyperfluid is a special case of the perfect cosmological
pyperfluid.

###  V.3 Generalised conservation laws

The conservation laws of the energy-momentum and the hypermomentum tensors
that describe the perfect cosmological hyperfluid follow from Eqs. (19) and
(20) by means of (64) and (66). Recalling that we have assumed zero torsion
($S_{\mu}=0$) and we have set $\chi=\psi$, we find

$\dot{\rho}+3H(\rho+p)=-\frac{1}{2}\,\psi
u^{\mu}u^{\nu}(R_{\mu\nu}+\check{R}_{\mu\nu})$ (70)

and

$\displaystyle-\delta^{\mu}{}_{\lambda}\,\frac{\partial_{\nu}\left(\sqrt{-g}\,\phi
u^{\nu}\right)}{\sqrt{-g}}-u^{\mu}u_{\lambda}\frac{\partial_{\nu}\left[\sqrt{-g}\,(\phi+2\psi+\omega)u^{\nu}\right]}{\sqrt{-g}}+\bigg{[}\frac{1}{2}\,Q_{\lambda}u^{\mu}-\nabla_{\lambda}u^{\mu}$
$\displaystyle+\left(\frac{1}{2}\,Q^{\mu}-q^{\mu}\right)u_{\lambda}-g^{\mu\nu}\nabla_{\nu}u_{\lambda}\bigg{]}\psi+2\dot{\psi}u^{\mu}u_{\lambda}-(\phi+2\psi+\omega)(\dot{u}^{\mu}u_{\lambda}+u^{\mu}\dot{u}_{\lambda})=0\,,$
(71)

for the matter and the hyperfluid respectively. The former is the modified
continuity equation, with non-Riemannian effects (hypermomentum contributions)
on the right-hand side, while the latter determines the evolution of the
hyperfluid itself.

The conservation law (71) provides us with two evolution equations for the
hypermomentum variables. More specifically, the trace of the above expression
yields

$\partial_{\mu}\Big{[}\Big{(}3\phi-\omega\Big{)}u^{\mu}\Big{]}=0$ (72)

and integrates immediately to give

$3\phi-\omega=\frac{C_{0}}{a^{3}}$ (73)

where $C_{0}=a^{3}_{0}(3\phi_{0}-\omega_{0})$ is the integration constant.
Moreover, taking the trace of the specelike component of (71), namely setting
$\mu=i$, $\lambda=j$ and then contracting along $i$ and $j$, gives

$\dot{\phi}+3H\phi+2H\psi+\frac{1}{6}\,\kappa\psi\Big{(}3\phi-\omega\Big{)}=0\,.$
(74)

Note that the unconstrained hyperfluid model is recovered as the special case
when $\psi=0$, as expected. In addition, after computing the right-hand side
of the generalized continuity equation given in (70), we arrive at999See
Appendix A for details.

$\dot{\rho}+3H(\rho+p)=-\frac{3}{2}\,\psi\left[\frac{1}{2}\,\dot{B}+\left(A+\frac{1}{2}\,B+C\right)H+{1\over
2}\left(\frac{1}{2}\,B-A\right)(C+A)\right]\,.$ (75)

Before closing we should point out that, in order to completely describe the
hypermomentum degrees of freedom, we need to introduce “equations of state”
between the variables $\phi$, $\psi$ and $\omega$, analogous to that between
the pressure ($p$) and the density ($\rho$) of the conventional matter (see
section V.5 below).

###  V.4 Generalised Friedmann equations

Starting with the generalised symmetric connection, which differs from its
Levi-Civita counterpart due to the non-metricity, one generalises the Ricci
tensor and the Ricci scalar. Then, taking the timelike and the spacelike
components of the metric field equations, leads to (see Appendix A)

$H^{2}=\frac{1}{3}\,\kappa\rho-\frac{1}{2}\,H\left(\frac{3}{2}\,B-A+C\right)-\frac{1}{4}\,\dot{B}+\frac{1}{8}\,B(A-C)+\frac{1}{4}\,AC$
(76)

and

$\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa\left(\rho+3p\right)+H\left(A+\frac{1}{2}\,C\right)+\frac{1}{2}\,\dot{A}-\frac{1}{4}\,A(A+C)\,.$
(77)

These are the modified Friedmann and Raychaudhuri equations respectively.
Together with the conservation laws (73)-(75), the above monitor the evolution
of an FLRW-like universe with generalised non-metricity. In what follows, we
will attempt to explore the potential implications of our model for cosmology.

###  V.5 Closing the system

We begin our investigation by employing the relations between the non-
metricity and the hypermomentum variables (see Eqs. (68) in section V.2
earlier) to express (75), (76) and (77) in terms of $\phi$, $\psi$ and
$\omega$. This results into the following expression for the continuity
equation

$\dot{\rho}+3H(\rho+p)=\frac{1}{2}\,\kappa\left[\dot{\omega}+H\left(3\phi+2\omega+6\psi\right)+\frac{1}{12}\,\kappa\left(3\phi-\omega\right)\left(3\phi+\omega+6\psi\right)\right]\psi\,,$
(78)

supplemented by the generalised Friedmann

$H^{2}=\frac{1}{3}\,\kappa\rho+\frac{1}{2}\,\kappa
H\left(2\psi+\omega\right)+\frac{1}{6}\,\kappa\left[\dot{\omega}-\kappa\omega\psi+\frac{1}{24}\,\kappa\left(3\phi+\omega\right)\left(\omega+3\phi+24\psi\right)\right]$
(79)

and Raychaudhuri

$\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa\left(\rho+3p\right)-\frac{1}{4}\,\kappa
H\left(\omega+3\phi+4\psi\right)-\frac{1}{12}\,\kappa\left(3\dot{\phi}+\dot{\omega}\right)-\frac{1}{72}\,\kappa^{2}\left(3\phi+\omega\right)\left(\omega+3\phi+6\psi\right)\,,$
(80)

formulae. The above relations simplify considerably after replacing $\omega$,
$\dot{\omega}$ and $\dot{\phi}$ with $\phi$, by means of (73) and (74).
Indeed, a straightforward calculation leads to the more transparent
expressions101010The system of (81) and (82) is self-consistent, since (81b)
can be obtained by taking the time derivative of (81a) and then utilizing the
remaining equations.

$H^{2}=\frac{1}{3}\,\kappa\rho+\frac{1}{4}\,{\kappa}^{2}\left(\phi-\frac{\mathcal{C}_{0}}{6a^{3}}\right)^{2}\,,\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa\left(\rho+3p\right)-\frac{1}{2}\,{\kappa}^{2}\left(\phi-\frac{\mathcal{C}_{0}}{6a^{3}}\right)^{2}-\frac{1}{2}\,{\kappa}^{2}\psi\left(\phi-\frac{\mathcal{C}_{0}}{3a^{3}}\right)\,,$
(81)
$\dot{\rho}+3H(\rho+p)=\frac{1}{2}\,\kappa\left[H+\frac{1}{2}\,\kappa\left(\phi-\frac{\mathcal{C}_{0}}{6a^{3}}\right)\right]\psi\frac{\mathcal{C}_{0}}{a^{3}}\hskip
42.67912pt\text{and}\hskip
42.67912pt\dot{\phi}+H(3\phi+2\psi)+\kappa\psi\,\frac{\mathcal{C}_{0}}{6a^{3}}=0\,.$
(82)

In (81a) one can now see that the extra term due to non-metricity is always
positive, which ensures that the hypermomentum acts as a non-ghost fluid with
a positive effective energy density. In the second of the Friedmann equations
(namely in Raychaudhuri’s formula – see (81b)), the first of the two extra
terms always decelerates the expansion. The second non-metricity term, on the
other hand, has no definite sign and safe conclusions can be drawn only after
solving the full system of equations. However, in order to do so, we need to
take additional steps.

Let us turn or attention to the modified continuity equation (82a), which has
nonzero right-hand side due to the non-metricity effects. The latter propagate
via $\phi$ and $\psi$, of which only $\phi$ has an evolution law (given by
(82b) above). Therefore, to proceed, we need to introduce an effective
“equation of state” of the form $\psi=\psi(\phi)$. Such a step is physically
motivated by our desire to represent the hypermomentum as an effective fluid.
Then, given that only $\phi$ appears in Eq. (81a) and $\psi$ appears only in
(81b), we may associate the former with the effective energy density and the
latter with the effective pressure. On these grounds, the simplest equation of
state to assume is a “barotropic” one with $\psi=w_{1}\phi$.111111This is one
of the many possibilities, since hyperfluids could generally obey more
“exotic” equations of state. As we will see, the simple cases presented below
will naturally emerge as subcases of such a “barotropic” hyperfluid. In our
case, it is convenient to introduce the relation

$\psi=w_{1}\Phi\,,$ (83)

with

$\Phi=\phi-\frac{C_{0}}{6a^{3}}\,,$ (84)

is a combination that appears in all three of (81a), (81b) and (82a). On using
the above equation of state, the system (81a)-(82b) recasts into

$H^{2}=\frac{1}{3}\,\kappa\rho+\frac{1}{4}\,\kappa^{2}\Phi^{2}\,,\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(\rho+3p)-\frac{1}{2}\,\kappa^{2}\Phi^{2}-\frac{1}{2}\,\kappa^{2}w_{1}\Phi\left(\Phi-\frac{C_{0}}{6a^{3}}\right)\,,$
(85) $\dot{\rho}+3H(\rho+p)=\frac{1}{2}\,\kappa
w_{1}\Phi\left(H+\frac{1}{2}\,\kappa\Phi\right)\frac{C_{0}}{a^{3}}\hskip
42.67912pt{\rm and}\hskip
42.67912pt\dot{\Phi}+(3+2w_{1})H\Phi+\frac{1}{6}\,\kappa
w_{1}\Phi\,\frac{C_{0}}{a^{3}}=0\,,$ (86)

where now $C_{0}=2a_{0}^{3}(3\Phi_{0}-\omega_{0})$. Given that the last term
in (85b) has no definitive sign, one may in principle select an equation of
state that could drive cosmic acceleration. Setting $w_{1}=-1$ and demanding
that $C_{0}\Phi>0$, for example, ensures that the last term in the modified
Raychaudhuri equation tends to accelerate the expansion.

###  V.6 Solutions

Let us now turn our attention to specific scenarios that also represent
physically motivated simplifications of the general case. The hypermomentum
tensor splits into its irreducible parts, namely spin, dilation and shear as
follows hehl1995metric

$\Delta_{\alpha\mu\nu}=\Delta_{[\alpha\mu]\nu}+\frac{1}{4}\,g_{\alpha\mu}\mathit{D}_{\nu}+\breve{\Delta}_{\alpha\mu\nu}\,.$
(87)

Given that the spin part (i.e. first term) is due to torsion, we will
hereafter exclude it from our analysis. Of the remaining terms, the dilation
part is equal to

$\mathit{D}_{\nu}:=\Delta_{\alpha\mu\nu}g^{\alpha\mu}=(3\phi-\omega)u_{\nu}\,,$
(88)

while the symmetric traceless shear part reads

$\breve{\Delta}_{\alpha\mu\nu}=2\psi
u_{[\mu}h_{\alpha]\nu}+\frac{1}{4}\,(\phi+\omega)h_{\alpha\mu}u_{\nu}+\frac{3}{4}\,(\phi+\omega)u_{\nu}u_{\mu}u_{\alpha}\,.$
(89)

In what follows, we isolate the effects of these two sources and investigate
their potential cosmological implications separately.

####  V.6.1 Pure dilation

Let us assume that the shear component vanishes and that only the dilation
part of the hypermomentum survives. As seen from Eq. ($\ref{shear})$, this
leads to the constraints

$\breve{\Delta}_{\alpha\mu\nu}=0\Leftrightarrow\left\\{\begin{array}[]{rl}\psi&=0\\\
\phi+\omega&=0\end{array}\right.\,.$ (90)

Setting $\psi=0$ simplifies our system of equations considerably, while
applying the $\phi+\omega=0$ condition to Eq. (73), the latter gives

$\Phi=\frac{C_{0}}{12a^{3}}\,,$ (91)

which follows form (86) as well. When $\psi=0$, the continuity equation does
not explicitly depend on the hypermomentum. Then, supplementing a barotropic
equation of state ($p=w\rho$) for the matter leads to $\rho\propto
a^{-3(1+w)}$. On these grounds, the Friedmann and the Raychaudhuri equations
read

$H^{2}=\frac{1}{3}\,\kappa\rho+\kappa^{2}\frac{C_{0}^{2}}{24^{2}a^{6}}\hskip
42.67912pt{\rm and}\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(1+3w)\rho-\frac{1}{2}\,\kappa^{2}\frac{C_{0}^{2}}{12^{2}a^{6}}\,,$
(92)

respectively. Therefore, qualitatively speaking, the effective energy density
contribution of the pure-dilation case is always positive, which means that
the extra non-metricity term in the Raychaudhuri equation is negative and
decelerates the expansion at all times (just like conventional matter with
$1+3w>0$). To facilitate the presentation, let us define the effective energy
density for the hyperfluid as

$\rho_{e}=\frac{1}{3}\,\kappa\frac{C_{0}^{2}}{4^{3}a^{6}}$ (93)

and recast the above into

$H^{2}=\frac{1}{3}\,\kappa(\rho+\rho_{e})\hskip 42.67912pt{\rm and}\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(1+3w)\rho-\frac{2}{3}\,\kappa\rho_{e}\,.$
(94)

Accordingly, in dynamical terms, non-metricity mimics a classical stiff fluid,
namely like a perfect medium with barotropic index $w=1$. This behaviour is
essentially identical to the one found in sections III and IV earlier.
Overall, generalising the host spacetime to include non-metricity, provides an
alternative (geometrical) way of producing an effective stiff component.

From the stiff matter interpretation of pure-dilation non-metricity it becomes
evident that, even if non-metricity is dynamically negligible today, it could
have dominated at sufficiently early times and thus it might have dictated the
kinematical evolution of the universe in the past. Indeed, Eq. (94a) accepts
the solutions

$a\propto t^{1/3}\hskip 42.67912pt{\rm and}\hskip 42.67912pta\propto
t^{2/3(1+w)}\,,$ (95)

at early and late times respectively. Note that the first result corresponds
to $\rho_{e}\gg\rho$ and the latter to $\rho_{e}\ll\rho$. Also, the two
solutions coincide when $w=1$, namely when the conventional matter also has a
stiff equation of state. The possibility raised by solution (95), is
intriguing, since it provides theoretical support to speculations that matter
with hyperfluid characteristics could have dominated the very early stages of
the expansion. Here, we see that this can happen naturally within the
framework of MAG, by allowing for non-Riemannian geometry (i.e. for spacetime
non-metricity). Having said that, assuming that non-metricity dominates, the
timing is crucial because it could interfere with fundamental physical
processes, such as nucleosynthesis for example (see next section).

Let us now consider analytical solutions for specific values of the barotropic
indices. Setting $w=1=w_{e}$, for example, the two fluids have identical
behaviour. The solution is the identical to its standard GR counterpart, with
the matter density replaced by the sum of sum of the densities of the fluids

$a=a_{0}\left(\sqrt{\frac{3}{4}\,\kappa\left(\rho_{0}+\rho_{e_{0}}\right)}(1+w)t\right)^{1/3}\,,$
(96)

where the zero suffix denotes the present time. Assuming that matter is in the
form of pressure-free dust (i.e. setting $w=0$) leads to

$a=a_{0}\left(\sqrt{\frac{3}{4}\,\kappa\rho_{e_{0}}}t+\frac{3}{4}\,\kappa\rho_{0}t^{2}\right)^{1/3}\,.$
(97)

The first mode in the above grows as $\propto t^{1/3}$ and dictates the early
evolution of the model, while the second is proportional to $t^{2}/3$ and
drives the late-time expansion (both in agreement with solutions (95)).

Let us also consider the effects of pure-dilation non-metricity on standard
inflation. Assuming matter with a de Sitter inflationary equation of state, we
may set $w=-1$. Then, we obtain the solution

$a=a_{0}\left(\frac{\rho_{e}}{\rho}\right)_{0}^{1/6}\left[\sinh{\left(\sqrt{3\kappa\rho_{0}}t\right)}\right]^{1/3}\,,$
(98)

which also depends on the ratio $\rho_{e}/\rho$ today. At early times, that is
for small values of $t$, the above approaches solution (95a) and therefore
inflation is suppressed. In contrast, at late times, we recover the
exponential expansion of standard de Sitter-type inflation.

####  V.6.2 Pure shear

Turning our attention to pure shear hypermomentum, we set the dilation
component of $\Delta_{\alpha\mu\nu}$ (see Eq. (88)) to zero. Combining the
latter with (73), gives

$\mathit{D}_{\nu}=0\Leftrightarrow C_{0}=0$ (99)

which in turn recast the system of (85) and (86) into

$H^{2}=\frac{1}{3}\,\kappa\rho+\frac{1}{4}\,\kappa^{2}\Phi^{2}\,,\hskip
71.13188pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(\rho+3p)-{1\over
2}\,\kappa(1+w_{1})\Phi^{2}\,,$ (100) $\dot{\rho}+3H(\rho+p)=0\hskip
42.67912pt{\rm and}\hskip 42.67912pt\dot{\Phi}+(3+2w_{1})\Phi H=0\,,$ (101)

respectively. As it turns out, the above relations are the same with the ones
derived in Iosifidis:2020upr , were both torsion and non-metricity were
included, though there the hypermomentum tensor had only shear component. For
completeness, we will briefly recover some of the solutions given in
Iosifidis:2020upr , before providing additional results. Following (100), the
non-metricity input to the generalized Friedmann equation is always positive,
while its contribution to the Raychaudhuri equation depends on $w_{1}$-index.
In addition, according to (101), the two continuity equations have decoupled,
which guarantees that matter evolves independently of the $\Phi$-field. The
physical interpretation becomes clearer if we introduce the effective energy
density

$\rho_{e}=\frac{3}{4}\,\kappa\Phi^{2}\,.$ (102)

On using the above, the associated onservation law (see Eq. (101b)) becomes
identical to the continuity equation of a barotropic perfect fluid, namely

$\dot{\rho_{e}}+3H(1+w_{e})\rho_{e}=0\,,$ (103)

with $w_{e}=[1+(4w_{1}/3)]$. At the same time, the Friedmann and the
Raychaudhuri equations given in (100) read

$H^{2}=\frac{1}{3}\,\kappa\rho+\frac{1}{3}\,\kappa\rho_{e}\hskip
42.67912pt{\rm and}\hskip
42.67912pt\frac{\ddot{a}}{a}=-\frac{1}{6}\,\kappa(1+3w)\rho-\frac{1}{6}\,\kappa(1+3w_{e})\rho_{e}\,,$
(104)

respectively. This cosmology is described by usual matter plus an additional
effective perfect fluid that has emerged after generalizing the Riemannian
geometry of the host spacetime by including non-metricity effects. Following
(104a) conventional matter dominates when $\rho\gg\rho_{e}$. Otherwise, it is
the non-metricity that dictates the universal kinematics. According to Eq.
(104b), matter that satisfies the strong energy condition (i.e. with $1+3w>0$)
decelerates the expansion (as expected). Similarly, the (hypermomentum) second
term on the right-hand side of (104b) slows the expansion down when
$1+3w_{e}>0$, or equivalently when $w_{1}>-1$. In the opposite case the
expansion is accelerated, whereas setting $w=-1/3=w_{e}$ leads to $q=0$ and to
the familiar “coasting universe”. We should also point out that, at the
moment, there are no known constraints imposed on $w_{1}$ and therefore on
$w_{e}$.

The continuity equations of the conventional and the effective fluids (see
expressions (101a) and (103) previously) are integrated immediately,
respectively giving

$\rho=\rho_{0}\Big{(}\frac{a_{0}}{a}\Big{)}^{3(1+w)}\hskip 42.67912pt{\rm
and}\hskip
42.67912pt\rho_{e}=\rho_{e_{0}}\Big{(}\frac{a_{0}}{a}\Big{)}^{3(1+w_{e})}\,,$
(105)

where $\rho_{0}$ and $(\rho_{e})_{0}$ are the matter density and the effective
density of the hypermomentum component today. On using the above, the
generalized Friedmann equation (see expression (104a)) yields

$H^{2}=\frac{1}{3}\,\kappa\rho_{0}\Big{(}\frac{a_{0}}{a}\Big{)}^{3(1+w)}\bigg{[}1+\Big{(}\frac{\rho_{e}}{\rho}\Big{)}_{0}\Big{(}\frac{a_{0}}{a}\Big{)}^{3(w_{e}-w)}\bigg{]}\,.$
(106)

The above implies that, although the hypermomentum component may be
subdominant today (i.e. even if $(\rho_{e})_{0}\ll\rho_{0}$), non-metricity
could have dominated in the distant past, or it may take over in the far
future. Which of these alternative scenarios occurs depends on the sign of the
difference $w_{e}-w$ between the two barotropic indices. More specifically,
when $w_{e}-w>0$, the shear hypermomentum dominates the early evolution of the
universe. In the opposite case, that is for $w_{e}-w<0$, this happens in the
late future. Clearly, when $w_{e}=w$, a universe that starts matter-dominated
remains so at all times and vice versa.

Provided that $w_{e}-w\neq 0$, the two constituents reach equipartition,
analogous to that between radiation and dust, at a specific moment in the
lifetime of the universe, which can lie either in the past or in the future.
Following (106), the scale factor at the equilibrium point is

$a_{\star}=a_{0}\left(\frac{\rho_{e}}{\rho}\right)_{0}^{[1/3(w_{e}-w)]}\,.$
(107)

Making the plausible assumption that $\rho_{e}/\rho\ll 1$ at present, we may
set $(\rho_{e}/\rho)_{0}=10^{-\beta}$, with $\beta$ being a small positive
number. Then, the scale factor at equilibrium will be

$a_{\star}=a_{0}\times 10^{-[\beta/3(w_{e}-w)]}\,.$ (108)

The above ensures that $a_{\star}<a_{0}$ when $w_{e}-w>0$, which puts the time
of equipartition in the past. In the opposite case we find that
$a_{\star}>a_{0}$ and non-metricity takes over in the future. Finally,
recalling that the temperature of the universe evolves inversely proportional
to the scale factor, namely that $T\propto 1/a$, the temperature at the moment
of equipartition is given by

$T_{\star}=T_{0}\times 10^{[\beta/3(w_{e}-w)]}\,,$ (109)

with $\beta>0$. Clearly, $T_{\star}>T_{0}$ when $w_{e}-w>0$ and
$T_{\star}<T_{0}$ for $w_{e}-w<0$, putting the equilibrium time in the past
and in the future respectively.

##  VI Conclusions

In this work we have examined the role of spacetime non-metricity in
homogeneous and isotropic (Friedmann-like) cosmologies. Non-metricity reflects
the failure of the metric to be covariantly conserved, which changes the
magnitude of vectors and tensors when they are parallelly transported. Many
familiar geometric notions change in the presence of non-metricity, such as
the symmetries of the Riemann tensor, its contractions, the conservation laws,
etc. It is therefore not surprising that including non-metricity changes the
cosmic evolution in non-trivial ways.

We begun our investigation by considering the usual Einstein-Hilbert action
and a matter sector that also depends on the connection, namely with non-
vanishing hypermomentum. The spacetime was assumed torsionless to begin with,
that is the connection was symmetric by default, in order to isolate the
effects of non-metricity. Within this framework, we provided the connection,
the field equations and the associated conservation laws. Turning our
attention to cosmology, we considered a spatially flat Fridemann-like universe
along with the allowed form for the distortion and non-metricity tensors.
Clearly, the symmetries of the FRLW metric restrict the allowed forms of the
various energy tensors. More specifically, the energy-momentum tensor can only
have the perfect fluid form and the hypermomentum must be compatible;e with
the Cosmological Principle Iosifidis:2020gth .

With this in hand, we examined the unconstrained hyperfluid developed in
obukhov1996model . This model can source two of the three non-metricity
degrees of freedom allowed in a cosmological setting. Nevertheless, having
extracted the modified Friedmann equations and after employing the associated
cosmological laws, we obtained a number of results. The conservation laws of
the energy-momentum and the hypermomentum tensors, in particular, decouple and
can be trivially integrated to give the evolution laws (i.e. the continuity
equations) of $\rho$ , as well as those of $\phi$ and $\psi$. The latter
showed that both $\psi$ and $\phi$ evolve like pressureless matter, with
$\phi,\,\psi\propto 1/a^{3}$, where $a=a(t)$ is the cosmological scale
factor). Moreover, the fact that the aforementioned two fields contribute
quadratically to the modified Friedmann equations implies that their input to
the cosmological dynamics is proportional to $1/a^{6}$. Consequently, the
effect of $\phi$ and $\psi$ is phenomenologically identical to that of a
perfect fluid with a stiff equation of state. Put another way, an effective
stiff-matter component emerges naturally as a cosmological consequence of the
underlying non-Riemannian geometry of the universe and without any further
assumptions.

We have also explored in depth the most general form of non-metricity sourced
by a perfect hypermomentum-preserving hyperfluid Iosifidis:2020gth . This
generalized model is described by three additional degrees of freedom, both
for the non-metricity and the hypermomentum that sources it. The complete
system of equations immediately led to some qualitative conclusions. For
example, the additional terms introduced to the Friedmann equation were found
to be positive always, indicating that non-metricity mimics matter with
positive (i.e. non-ghost-like) effective energy density. The extra terms in
the modified Raychaudhuri equation, on the other hand, suggested that non-
metricity can accelerate or decelerate the universal expansion, depending on
the (effective) equation of state one assumes for the hypermomentum variables.

Looking for analytic solutions, we examined two physically distinct subcases,
that is for hypermomentum with a dilation part only and for shear
hypermomentum. In the former case, we recover the (effective) stiff-matter
scenario mentioned above and also provided exact cosmological solutions,
corresponding to specific values of the barotropic index ($w$) of the matter.
For shear hypermomentum, we found that the effects of non-metricity mimic
those of a perfect fluid with a “free” barotropic index ($w_{e}$). This
freedom provides a rich phenomenology, since different values of $w_{e}$ can
lead to universal acceleration, or deceleration, to exponential expansion,
etc. In this latter scenario, there is also a distinct moment in time, where
the conventional matter and the non-metricity induced (effective) fluid have
equal energy densities (i.e. $\rho_{e}=\rho$). Such a “moment of
equipartition” is dynamically similar to the “equilibrium point”, marking the
transition from the radiation to the dust era of the universe. In fact,
assuming that the current value of $\rho_{e}$ can be constrained, one could
determine the transition time from a non-metricity dominated epoch of the
universe to the era of conventional matter domination (and vice versa).
Depending on the sign of the difference $w-w_{e}$, this moment can lie in our
past or in our future.

Before closing, we should point out that our initial assumption that the
canonical and the metrical energy momentum tensors coincide, restricts the
allowed forms of non-metricity. For example, “fixed length non-metricity” is
incompatible with our equations.121212Where vector lengths are preserved and
the totally symmetric part of the non-metricity tensor is identically zero In
order to include more general types of non-metricity, one needs to ease the
restrictions imposed in Eq. (71) and thus break away from the hypermomentum-
conserving hyperfluids Iosifidis:2021fnq . We plan consider such generalized
non-metricity models in the near future.

##  Appendix A Curvature in the presence of general non-metricity

Here we provide the curvature tensors and scalars necessary for the derivation
of the modified Friedman equations and the associated conservation laws.
Following section II.2, the general affine connection splits into its
Riemannian and Non-Riemannian components as
$\Gamma^{\lambda}{}_{\mu\nu}=\tilde{\Gamma}^{\lambda}{}_{\mu\nu}+N^{\lambda}{}_{\mu\nu}$,
with

$N^{\lambda}{}_{\mu\nu}=\frac{1}{2}\,g^{\rho\lambda}\left(Q_{\mu\nu\rho}+Q_{\nu\rho\mu}-Q_{\rho\mu\nu}\right)\,.$
(110)

being the distortion tensor in the absence of torsion. Splitting the
connection leads to the decomposition of the Riemann tensor and its
contractions. In particular,

$R^{\mu}{}_{\nu\alpha\beta}=\tilde{R}^{\mu}{}_{\nu\alpha\beta}+2\tilde{\nabla}_{[\alpha}N^{\mu}{}_{|\nu|\beta]}+2N^{\mu}{}_{\rho[\alpha}N^{\rho}{}_{|\nu|\beta]}\,,$
(111)

with the tilded quantities evaluated with respect to the Levi-Civita
connection. For the most general form of non-metricity compatible with the
cosmological principle, namely when
$Q_{\alpha\mu\nu}=Au_{\alpha}h_{\mu\nu}+Bh_{\alpha(\mu}u_{\nu)}+Cu_{\alpha}u_{\mu}u_{\nu}$,
the distortion tensor becomes

$N^{\lambda}{}_{\mu\nu}=\frac{1}{2}\,(B-A)h_{\mu\nu}u^{\lambda}+A\delta^{\lambda}{}_{(\mu}u_{\nu)}+\left(A+\frac{1}{2}\,C\right)u^{\lambda}u_{\mu}u_{\nu}\,.$
(112)

Contracting (111) and using the above distortion tensor, as well as its
contractions, leads to the curvature variables. The Ricci tensor, in
particular, reads

$\displaystyle R_{\nu\beta}$ $\displaystyle=$
$\displaystyle\tilde{R}_{\nu\beta}+\frac{1}{2}\,g_{\nu\beta}\tilde{\nabla}_{\mu}\left[(B-A)u^{\mu}\right]+\frac{1}{2}\,\tilde{\nabla}_{\beta}\left[\left(C-2A\right)u_{\nu}\right]+\frac{1}{2}\,\tilde{\nabla}_{\nu}\left(Au_{\beta}\right)+\frac{1}{2}\,\tilde{\nabla}_{\mu}\left[(A+B+C)u_{\nu}u_{\beta}u^{\mu}\right]$
(113)
$\displaystyle+\frac{1}{4}\,(A-B)(A-C)h_{\nu\beta}-\frac{3}{4}\,A(A+C)u_{\nu}u_{\beta}\,.$

Then, after taking into account the FLRW metric and the associated the Levi-
Civita connection, we obtain

$R_{00}=\tilde{R}_{00}+\frac{3}{2}\,\dot{A}+\frac{3}{2}\,(2A+C)H-\frac{3}{4}\,A(A+C)$
(114)

and

$R_{ii}=\tilde{R}_{ii}+\frac{1}{2}\,(C+3B-4A)a\dot{a}+\frac{1}{2}\,a^{2}(\dot{B}-\dot{A})+\frac{1}{4}\,a^{2}(B-A)(C-A)\,,$
(115)

for the timelike and spacelike compontents respectively. Contracting again,
leads to the Ricci scalar

$R=\tilde{R}+\frac{3}{2}\,\dot{B}-3\dot{A}+9H\left(\frac{1}{2}\,B-A\right)+\frac{3}{2}\,A^{2}+\frac{3}{4}\,B(C-A)\,.$
(116)

These results combine with the Einstein field equations to give the
generalised Friedmann equations presented in section V.4 of the main document.

We also note that the Co-Ricci tensor involved in the conservation law of the
hypermomentum reads

$\displaystyle\check{R}_{\lambda\beta}$ $\displaystyle=$
$\displaystyle-\tilde{R}_{\lambda\beta}+\frac{1}{2}\,\tilde{\nabla}_{\beta}\left[(2A-2B+C)u_{\lambda}\right]+\frac{1}{2}\,g_{\lambda\beta}\tilde{\nabla}_{\alpha}\left(Au^{\alpha}\right)+\frac{1}{2}\,\tilde{\nabla}_{\lambda}\left(Au_{\beta}\right)+\frac{1}{2}\,\tilde{\nabla}_{\mu}\left[(A+B+C)u_{\lambda}u_{\beta}u^{\mu}\right]$
(117)
$\displaystyle-\frac{1}{4}\,A(3A-2B+C)h_{\lambda\beta}+\frac{3}{4}\,(B-A)(A+C)u_{\lambda}u_{\beta}\,.$

Finally, the homothetic curvature is antisymmetric and thus always zero in
isotropic spaces.

##  Appendix B Alternative derivation of the non-metricity tensor and vectors

One can arrive at Eqs. (68) given in section V.2 by taking contractions of the
connection field equations to express the non-metricity vectors in terms of
those of the hypermomentum. The results read

$Q^{\lambda}=\kappa\left[\Delta^{\lambda\alpha}{}_{\alpha}-\frac{1}{3}\,(\Delta_{\alpha}{}^{\alpha\lambda}+\Delta_{\alpha}{}^{\lambda\alpha})\right]\hskip
42.67912pt{\rm and}\hskip
42.67912ptq^{\lambda}=\frac{1}{2}\,\kappa\left[\Delta^{\lambda\alpha}{}_{\alpha}+\frac{1}{3}\,\left(\Delta_{\alpha}{}^{\alpha\lambda}+\Delta_{\alpha}{}^{\lambda\alpha}\right)\right]\,.$
(118)

Plugging the above back into the connection field equations and solving for
the non-metricity tensor, we arrive at

$Q_{\lambda}{}^{\mu\nu}=\kappa\left(-\Delta_{\lambda}{}^{(\mu\nu)}+\frac{2}{3}\,\Delta_{\alpha}{}^{\alpha(\mu}\delta^{\nu)}_{\lambda}+D_{\lambda}g^{\mu\nu}\right)\,,$
(119)

where

$D_{\lambda}=\frac{1}{2}\left(\Delta_{\lambda\alpha}{}^{\alpha}-\frac{2}{3}\,\Delta^{\alpha}{}_{\alpha\lambda}\right)\,.$
(120)

This procedure also leads to relations (68) between the two sets of variables.

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