# Riemann Tensor and Gauss-Bonnet density in Metric-Affine Cosmology

Damianos Iosifidis  Institute of Theoretical Physics, Department of Physics
Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
<EMAIL_ADDRESS>
###### Abstract

We analytically derive the covariant form of the Riemann (curvature) tensor
for homogeneous Metric-Affine Cosmologies. That is, we present, in a
Cosmological setting, the most general covariant form of the full Riemann
tensor including also its non-Riemannian pieces which are associated to
spacetime torsion and non-metricity. Having done so we also compute a list of
the curvature tensor by-products such as Ricci tensor, homothetic curvature,
Ricci scalar, Einstein tensor etc. Finally we derive the generalized Metric-
Affine version of the usual Gauss-Bonnet density in this background and
demonstrate how under certain circumstances the latter represents a total
derivative term.

###### Contents

  1. I Introduction
  2. II The setup
  3. III Cosmology with Torsion/Non-metricity
  4. IV Riemann Tensor for Metric-Affine Cosmologies
    1. IV.1  Riemann-Cartan Space (vanishing non-metricity)
    2. IV.2  Symmetric Space (vanishing torsion)
  5. V Gauss-Bonnet Density
  6. VI On Applications: Gauss-Bonnet Gravity in Metric-Affine Formalism
  7. VII Conclusions
  8. VIII Acknowledgments
  9. A Supplementary material for the proofs
    1. A.1 Gauss-Bonnet density for $L_{n}$ Cosmology
    2. A.2 Case $n=4$, $B\neq 0$

##  I Introduction

Non-Riemannian effects (i.e. torsion and non-metricity) are believed to play a
crucial role in the early time Cosmological evolution puetzfeld2005prospects .
These non-Riemannian contributions which are associated with the
microstructure of matter hehl1976hypermomentum ; hehl1997ahadronic ;
obukhov1993hyperfluid ; Iosifidis:2020gth  appear naturally in the so-called
Metric-Affine Gravity (MAG) formulation hehl1995metric ; iosifidis2019metric
where the metric and the affine connection are regarded as independent
variables with the latter possessing both torsion and non-metricity. There are
many characteristics of MAG that make it an attractive and promising Theory of
Gravity. For instance, when written in the language of exterior differential
forms, it can be seen to be a gauge Theory of Gravity hehl1995metric . In
addition, it takes into account the microscopic characteristics of matter such
as spin, dilation and shear and also offers appealing features of quantization
percacci2020towards . Possible experimental effects of these non-Riemannian
effects were discussed in kostelecky2008constraints ; foster2017constraints .

This Metric-Affine framework has received a renewed attention over the last
years iosifidis2018raychaudhuri ; iosifidis2019exactly ; iosifidis2019scale ;
vitagliano2011dynamics ; sotiriou2007metric ; capozziello2010metric ;
percacci2020new ; Jimenez:2020dpn ; BeltranJimenez:2019acz ; aoki2019scalar ;
Cabral:2020fax ; Ariwahjoedi:2020wmo ; Yang:2021fjy ; helpin2020metric ;
bahamonde2020new  especially in regards to its cosmological applications
iosifidis2020cosmic ; Iosifidis:2020gth ; Iosifidis:2021iuw ;
jimenez2016spacetimes ; beltran2017modified ; kranas2019friedmann ;
barragan2009bouncing ; shimada2019metric ; kubota2021cosmological ;
Mikura:2020qhc ; mikura2021minimal . It therefore becomes apparent that it is
of great importance to have a complete setup for both the geometrical and the
matter structures on this non-Riemannian Cosmology. Regarding the latter
point, we had earlier developed in Iosifidis:2020gth  the generalized fluid
notion that sources the non-Riemannian degrees of freedom in a Cosmological
setup. We named this perfect fluid generalization (where the microscopic
characteristics of matter are taken into account) the Perfect Cosmological
Hyperfluid Iosifidis:2020gth 111Quite recently we have generalized this
hyperfluid notion even further, even outside of the realm of Cosmology
iosifidis2021perfect .. There it was shown that this kind of fluid is the
appropriate one for the description of the matter sources in Metric-Affine
Cosmology, since it sources all the non-Riemannian degrees of freedom that are
allowed by the Cosmological Principle. Having the complete set for the sources
one also needs the associated compatible set of geometrical objects in this
Metric-Affine Cosmological setting. These are torsion, non-metricity and
curvature, the three fundamental geometrical objects of the non-Riemannian
geometry eisenhart2012non . The covariant expressions for both torsion and
non-metricity, compatible with the Cosmological Principle, have also been
presented in Iosifidis:2020gth . Therefore, the only remaining missing link is
the covariant expression for the Riemann (curvature) tensor that respects both
isotropy and homogeneity. It is this gap that a part of this work here intends
to fill. It goes without saying that having the exact covariant form of the
Riemann tensor serves as a very helpful tool for studying the Cosmological
aspects of Metric-Affine Gravity and especially Theories that are constructed
by the latter, such as $f(R)$ sotiriou2007metric ; olmo2011palatini  or
certain generalizations of it delhom2019ricci . Of course, having the
covariant form of the Riemann tensor one could then go on and compute several
important by-products of the latter such as Ricci tensor, Ricci scalar,
Riemann dual etc. We list several of these quantities after the derivation of
the Riemann tensor. In addition, one highly interesting quantity one can
construct, when having this covariant form of the Riemann tensor, is the well-
known Gauss-Bonnet density. Using a certain generalization of the
aforementioned density for non-Riemannian spaces, we derive its form for a
homogeneous torsionful and non-metric Cosmology. As expected we find that when
the full non-metricity is present the Gauss-Bonnet density fails to be an
exact differential babourova1997pontryagin ; janssen2019topological . It is
also known that for the restricted class of Cartan-Weyl spaces the generalized
Gauss-Bonnet density is indeed topological. In our Cosmological setting we
extend this result here for a slightly more general class of non-metricity,
that is for geometries that generalize the Cartan-Weyl structure.

The paper is organized as follows. Firstly, we fix notation and definitions
and go through the basic geometric and Cosmological setup. We then derive the
covariant form of the Riemann tensor for homogeneous and isotropic Cosmologies
with the inclusion of both torsion and non-metricity. Continuing, we present a
list of the forms of some important quantities that appear in actual
applications (Ricci tensor and scalar, Einstein tensor, totally antisymmetric
Riemann etc). Finally with the use of the previously derived Riemann tensor we
obtain the form of the generalized Gauss-Bonnet density for non-Riemannian
Cosmologies. We confirm that the latter is not a total divergence in general
and then prove that for a certain geometric subsector of the full non-
metricity tensor the latter is a total derivative.

##  II The setup

Let us consider a generalized $n-dim$ non-Riemannian Geometry, where the space
apart from curvature possesses also torsion and non-metricity. Following
Schouten schouten1954ricci  we shall refer to this general $n-dim$ manifold
endowed with a metric and an independent affine connection as an $L_{n}$
space. We will use the definitions and notations of Iosifidis:2020gth  so we
will go through the basic setup rather briefly here and refer the reader to
Iosifidis:2020gth  for more details. We consider a non-Riemannian manifold
endowed with a metric $g_{\mu\nu}$ and an independent affine connection
$\Gamma^{\lambda}_{\;\;\;\mu\nu}$. We define the curvature, torsion and non-
metricity tensors according to

$R^{\mu}_{\;\;\;\nu\alpha\beta}:=2\partial_{[\alpha}\Gamma^{\mu}_{\;\;\;|\nu|\beta]}+2\Gamma^{\mu}_{\;\;\;\rho[\alpha}\Gamma^{\rho}_{\;\;\;|\nu|\beta]}$
(1) $S_{\mu\nu}^{\;\;\;\lambda}:=\Gamma^{\lambda}_{\;\;\;[\mu\nu]}$ (2)
$Q_{\alpha\mu\nu}:=-\nabla_{\alpha}g_{\mu\nu}$ (3)

and the deviation of the affine connection $\Gamma^{\lambda}_{\;\;\;\mu\nu}$
from the Levi-Civita one defines the distortion tensor schouten1954ricci

$\displaystyle
N^{\lambda}_{\;\;\;\;\mu\nu}:=\Gamma^{\lambda}_{\;\;\;\mu\nu}-\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}=\frac{1}{2}g^{\alpha\lambda}(Q_{\mu\nu\alpha}+Q_{\nu\alpha\mu}-Q_{\alpha\mu\nu})-g^{\alpha\lambda}(S_{\alpha\mu\nu}+S_{\alpha\nu\mu}-S_{\mu\nu\alpha})$
(4)

where $\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}$ is the usual Levi-Civita
connection calculated only by the metric and its first derivatives. Once the
distortion is given, torsion and non-metricity can be quite easily computed
through (see for instance iosifidis2019metric )

$S_{\mu\nu\alpha}=N_{\alpha[\mu\nu]}\;\;,\;\;\;Q_{\nu\alpha\mu}=2N_{(\alpha\mu)\nu}$
(5)

Out of torsion we can construct a vector as well as a pseudo-vector. Our
definitions for the torsion vector and pseudo-vector are

$S_{\mu}:=S_{\mu\lambda}^{\;\;\;\;\lambda}\;\;,\;\;\;\tilde{S}_{\mu}:=\epsilon_{\mu\alpha\beta\gamma}S^{\alpha\beta\gamma}$
(6)

respectively. Note that the former is defined for any dimension and the latter
only for $n=4$. Continuing with non-metricity, we define the Weyl and the
second non-metricity vector according to

$Q_{\alpha}:=Q_{\alpha\mu\nu}g^{\mu\nu}\;,\;\;\tilde{Q}_{\nu}=Q_{\alpha\mu\nu}g^{\alpha\mu}$
(7)

Finally, from the Riemann tensor we can construct the three contractions

$R_{\nu\beta}:=R^{\mu}_{\;\;\nu\mu\beta}$ (8)
$\hat{R}_{\alpha\beta}:=R^{\mu}_{\;\;\mu\alpha\beta}$ (9)
$\breve{R}^{\mu}_{\;\;\beta}:=R^{\mu}_{\;\;\nu\alpha\beta}g^{\nu\alpha}$ (10)

The first one is the usual Ricci tensor (which is not symmetric now), the
second one is the homothetic curvature and the last one is oftentimes referred
to as the co-Ricci tensor. Note that the first two aforementioned tensors can
be formed without the use of any metric while for the latter a metric is
needed. Finally, let us mention that by virtue of (4) each quantity can be
split into its Riemannian part (i.e. computed with respect to the Levi-Civita
connection) plus non-Riemannian contributions. For instance inserting the
connection decomposition (4) into the definition (1) we obtain for the Riemann
tensor 222Quantities with $\widetilde{}$ will always denote Riemannian parts
unless otherwise stated.

${R^{\mu}}_{\nu\alpha\beta}=\widetilde{R}^{\mu}_{\phantom{\mu}\nu\alpha\beta}+2\widetilde{\nabla}_{[\alpha}{N^{\mu}}_{|\nu|\beta]}+2{N^{\mu}}_{\lambda|\alpha}{N^{\lambda}}_{|\nu|\beta]}\,,$
(11)

The last decomposition will be essential for the proof of the first Theorem
that we give in the sequel.

##  III Cosmology with Torsion/Non-metricity

Let us consider an FLRW Cosmology, with the usual Robertson-Walker line
element

$ds^{2}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}$ (12)

where $i,j=1,2,...,n-1$ and $a(t)$ is as usual the scale factor of the
Universe. As usual we will denote the Hubble parameter as $H:=\dot{a}/a$. In
addition let $u^{\mu}$ represent the normalized $n$-velocity field of a given
fluid which in co-moving coordinates is expressed as
$u^{\mu}=\delta^{\mu}_{0}=(1,0,0,...,0)$. Accordingly we define the projection
tensor in the usual manner

$h_{\mu\nu}:=g_{\mu\nu}+u_{\mu}u_{\nu}$ (13)

projecting objects on the space orthogonal to $u^{\mu}$. In such a highly
symmetric spacetime, torsion has only $2$ (or $1$ for $n\neq 4$) degrees of
freedom tsamparlis1979cosmological  and non-metricity contributes $3$
minkevich1998isotropic . In Iosifidis:2020gth  we showed that the most general
covariant forms of the torsion and non-metricity tensors, in such a spacetime,
are given by

$S_{\mu\nu\alpha}^{(n)}=2u_{[\mu}h_{\nu]\alpha}\Phi(t)+\epsilon_{\mu\nu\alpha\rho}u^{\rho}P(t)\delta_{n,4}$
(14)
$Q_{\alpha\mu\nu}=A(t)u_{\alpha}h_{\mu\nu}+B(t)h_{\alpha(\mu}u_{\nu)}+C(t)u_{\alpha}u_{\mu}u_{\nu}$
(15)

with the five functions $\Phi,P,A,B,C$ representing the non-Riemannian
Cosmological effects. Let us note that in the above expression for non-
metricity the Weyl non-metricity333Recall that the Weyl non-metricity is given
by $Q_{\alpha\mu\nu}=A_{\alpha}g_{\mu\nu}$. corresponds to the special choice
$B=0$ and $A=-C$. As a result the above form of non-metricity is more general
than Weyl.

Going back to the geometrical notions, in this case the distortion tensor is
given by Iosifidis:2020gth

$\displaystyle
N_{\alpha\mu\nu}^{(n)}=X(t)u_{\alpha}h_{\mu\nu}+Y(t)u_{\mu}h_{\alpha\nu}+Z(t)u_{\nu}h_{\alpha\mu}+V(t)u_{\alpha}u_{\mu}u_{\nu}+\epsilon_{\alpha\mu\nu\lambda}u^{\lambda}W(t)\delta_{n,4}$
(16)

where the functions $X(t),Y(t),Z(t),V(t),W(t)$ are linearly related to
$\Phi(t),P(t),A(t),B(t),C(t)$ according to

$2(X+Y)=B\;,\;\;2Z=A\;,\;\;2V=C\;,\;\;2\Phi=Y-Z\;,\;\;P=W$ (17)

or inverting them

$W=P\;,\;\;V=C/2\;,\;\;Z=A/2$ (18)
$Y=2\Phi+\frac{A}{2}\;\;,\;\;\;X=\frac{B}{2}-2\Phi-\frac{A}{2}$ (19)

as can be trivially checked by employing (5). As a result, the affine
connection of the non-Riemannian FLRW spacetime reads

$\Gamma^{\lambda}_{\;\;\;\mu\nu}=\widetilde{\Gamma}^{\lambda}_{\;\;\;\mu\nu}+X(t)u^{\lambda}h_{\mu\nu}+Y(t)u_{\mu}h^{\lambda}_{\;\;\nu}+Z(t)u_{\nu}h^{\lambda}_{\;\;\mu}+V(t)u^{\lambda}u_{\mu}u_{\nu}+\epsilon^{\lambda}_{\;\;\mu\nu\rho}u^{\rho}W(t)\delta_{n,4}$
(20)

and recall that the non-vanishing components of the Levi-Civita connection are
in this case

$\widetilde{\Gamma}^{0}_{\;\;ij}=\widetilde{\Gamma}^{0}_{\;\;ji}=\dot{a}a\delta_{ij}=Hg_{ij}\;,\;\;\widetilde{\Gamma}^{i}_{\;\;j0}=\widetilde{\Gamma}^{i}_{\;\;0j}=\frac{\dot{a}}{a}\delta^{i}_{\;\;j}=H\delta^{i}_{\;j}$
(21)

Now, as it is clear from the above discussion the pseudo-scalar mode $W$ (or P
equivalently) exists only in $4$ dimensions and this is why the Kronecker
delta appears. However, in order to save some space and time we will omit the
inclusion of $\delta_{n,4}$, keeping in mind that $W$ vanishes for $n\neq 4$.

Finally, let us mention that in $4$ dimensions one usually defines the three
dimensional Levi-Civita form (see for instance ellis2012relativistic ;
tsagas2008relativistic )

$\varepsilon_{\mu\nu\lambda}:=\varepsilon_{\mu\nu\lambda\rho}u^{\rho}$ (22)

living on the orthogonal tangent planes. Note that by construction
$\varepsilon_{\mu\nu\lambda}u^{\mu}=0$ confirming the fact that the latter is
purely spatial. The above expression will be useful in our subsequent
discussion. Let us close this section with one definition on which the rest of
our analysis will lie upon.

######  Definition 1.

The metric (12) along with the connection (20) will constitute a Cosmological
$L_{n}$ space.

##  IV Riemann Tensor for Metric-Affine Cosmologies

In this section we analytically derive the covariant expression of the Riemann
tensor for a Cosmological $L_{n}$ and subsequently apply this result and
compute the form of various objects constructed out of the Riemann tensor such
as Ricci tensor, homothetic curvature, Ricci scalar etc.

Let us start with the the covariant expression of the Riemann tensor which we
state and prove below.

######  Theorem 1.

Consider a Cosmological $L_{n}$ as given by the metric (12) and the connection
(20). Then, for this geometry the _Riemann (curvature) tensor_ takes the
covariant form $\displaystyle
R_{\mu\nu\alpha\beta}=\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}2h_{\nu[\alpha}u_{\beta]}u_{\mu}$
$\displaystyle+\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}2h_{\mu[\alpha}u_{\beta]}u_{\nu}$
$\displaystyle+\Big{[}H(X-Y)-XY-W^{2}+H^{2}\Big{]}2h_{\mu[\alpha}h_{\beta]\nu}$
$\displaystyle+\Big{[}\dot{W}-W(H+2X)\Big{]}\varepsilon_{\alpha\beta\nu}u_{\mu}-\Big{[}\dot{W}-W(H-2Y)\Big{]}\varepsilon_{\alpha\beta\mu}u_{\nu}-(\dot{W}+HW)\varepsilon_{\mu\nu\alpha\beta}$
(23) This is the most general form of the Riemann tensor that respects spatial
isotropy as well as homogeneity.

###### Proof.

We start by the Riemann tensor decomposition $(\ref{decomp})$ and compute its
parts piece by piece. For the metric $(\ref{metric})$ the Riemannian part is
trivially calculated to be

$\widetilde{R}_{\mu\nu\alpha\beta}=2H^{2}h_{\mu[\alpha}h_{\beta]\nu}+4(\dot{H}+H^{2})u_{[\mu}h_{\nu][\alpha}u_{\beta]}$
(24)

Next, using also the connection form (20), we compute

$\displaystyle\widetilde{\nabla}_{\alpha}N^{\mu}_{\;\;\;\nu\beta}=h_{\nu\beta}\Big{(}-\dot{X}u^{\mu}u_{\alpha}+HXh^{\mu}_{\;\;\alpha}\Big{)}+HXu^{\mu}\Big{(}h_{\alpha\nu}u_{\beta}+h_{\alpha\beta}u_{\nu}\Big{)}$
$\displaystyle+h^{\mu}_{\;\;\beta}\Big{(}-\dot{Y}u_{\alpha}u_{\nu}+HYh_{\alpha\nu}\Big{)}+HYu_{\nu}\Big{(}h_{\alpha}^{\;\;\mu}u_{\beta}+h_{\alpha\beta}u^{\mu}\Big{)}$
$\displaystyle+h^{\mu}_{\;\;\nu}\Big{(}-\dot{Z}u_{\alpha}u_{\beta}+HZh_{\alpha\beta}\Big{)}+HZu_{\beta}\Big{(}h_{\alpha}^{\;\;\mu}u_{\nu}+h_{\alpha\nu}u^{\mu}\Big{)}$
$\displaystyle+u^{\mu}u_{\nu}\Big{(}-\dot{V}u_{\alpha}u_{\beta}+HVh_{\alpha\beta}\Big{)}+HVu_{\beta}\Big{(}h_{\alpha}^{\;\;\mu}u_{\nu}+h_{\alpha\nu}u^{\mu}\Big{)}$
$\displaystyle+\varepsilon^{\mu}_{\;\;\nu\rho\beta}\Big{(}\dot{W}u^{\rho}u_{\alpha}-HWh_{\alpha}^{\;\;\rho}\Big{)}$
(25)

and antisymmetrizing in $\alpha,\beta$ it follows that

$\displaystyle\widetilde{\nabla}_{[\alpha}N^{\mu}_{\;\;\;|\nu|\beta]}=\Big{(}\dot{X}+H(X+Z+V)\Big{)}h_{\nu[\alpha}u_{\beta]}u^{\mu}+H(Y-X)h_{\nu[\alpha}h_{\beta]}^{\;\;\mu}+\Big{(}\dot{Y}+H(Y+Z+V)\Big{)}h^{\mu}_{\;\;[\alpha}u_{\beta]}u_{\nu}$
$\displaystyle+\varepsilon^{\mu}_{\;\;\nu\rho[\beta}(\dot{W}u_{\alpha]}u^{\rho}-HWh_{\alpha]}^{\;\;\rho})$
(26)

Continuing, we compute

$\displaystyle
N^{\mu}_{\;\;\rho[\alpha}N^{\rho}_{\;\;\;|\nu|\beta]}=X(Z+V)u^{\mu}h_{\nu[\alpha}u_{\beta]}+XWu^{\mu}h^{\rho}_{\;\;[\alpha}\varepsilon_{\beta]\nu\rho\lambda}u^{\lambda}-XYh^{\mu}_{\;\;[\alpha}h_{\beta]\nu}-Y(Z+V)h^{\mu}_{\;\;[\alpha}u_{\beta]}u_{\nu}$
$\displaystyle+ZWh^{\mu\rho}u_{[\alpha}\varepsilon_{\beta]\rho\nu\lambda}u^{\lambda}+YWh^{\rho}_{\;\;[\beta}\varepsilon_{\alpha]\rho}^{\;\;\;\;\;\mu\kappa}u_{\kappa}u_{\nu}-WZ\varepsilon^{\mu}_{\;\;\rho\kappa[\alpha}u_{\beta]}u^{\kappa}h^{\rho}_{\;\;\nu}+W^{2}\varepsilon^{\mu}_{\;\;\rho\kappa[\alpha}\varepsilon^{\rho}_{\;\;\beta]\nu\lambda}u^{\kappa}u^{\lambda}$
(27)

Substituting all the above into (11) after some algebra we find

$\displaystyle
R^{\mu}_{\;\;\nu\alpha\beta}=\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}2h_{\nu[\alpha}u_{\beta]}u^{\mu}$
$\displaystyle+\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}2h^{\mu}_{\;\;[\alpha}u_{\beta]}u_{\nu}$
$\displaystyle+\Big{[}H(X-Y)-XY-W^{2}+H^{2}\Big{]}2h^{\mu}_{\;\;[\alpha}h_{\beta]\nu}-2XWu^{\mu}\varepsilon_{\alpha\beta\nu}-2YW\epsilon_{\alpha\beta}^{\;\;\;\;\mu}u_{\nu}+2(\dot{W}-HW)\varepsilon^{\mu}_{\;\;\nu[\alpha}u_{\beta]}-2HW\varepsilon^{\mu}_{\;\;\nu\alpha\beta}$
(28)

Finally, using the identity

$2\varepsilon_{\mu\nu[\beta}u_{\alpha]}=\varepsilon_{\mu\nu\alpha\beta}+2\varepsilon_{\alpha\beta[\mu}u_{\nu]}$
(29)

we obtain the stated result $(\ref{Riemann})$. ∎

Some comments are now in order.

Comment 1. From the above expression of the Riemann tensor we see that in a
Cosmological $L_{n}$ the latter is completely specified by $6$ time functions
for $n=4$ and by $3$ functions for $n\neq 4$. Indeed, this can be easily
verified by constructing all the possible independent invariants with the
projection and the $n$-velocity and we trivially find

$u^{\alpha}u^{\mu}h^{\nu\beta}R_{\mu\nu\alpha\beta}=-(n-1)\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}$
(30)
$u^{\alpha}u^{\nu}h^{\mu\beta}R_{\mu\nu\alpha\beta}=-(n-1)\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}$
(31)
$h^{\mu\alpha}h^{\nu\beta}R_{\mu\nu\alpha\beta}=(n-1)(n-2)\Big{[}H(X-Y)-XY-W^{2}+H^{2}\Big{]}$
(32)
$h^{\mu\beta}h^{\nu\alpha}R_{\mu\nu\alpha\beta}=-(n-1)(n-2)\Big{[}H(X-Y)-XY-W^{2}+H^{2}\Big{]}$
(33) $u^{\mu}\varepsilon^{\nu\alpha\beta}R_{\mu\nu\alpha\beta}=2\cdot
3!(H+X)W$ (34)
$u^{\nu}\varepsilon^{\mu\alpha\beta}R_{\mu\nu\alpha\beta}=2\cdot 3!(Y-H)W$
(35) $u^{\alpha}\varepsilon^{\mu\nu\beta}R_{\mu\nu\alpha\beta}=(\dot{W}+HW)3!$
(36) $\varepsilon^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}=2\cdot
3!\Big{[}(\dot{W}+3HW)+W(X-Y)\Big{]}$ (37)

from which we conclude that out of the above $8$ invariant quantities, the
third and fourth are the same (with a sign difference) and the last term is a
linear combination of the three invariants appearing above it. As a result in
a Cosmological $L_{n}$ the Riemann tensor is specified by exactly $6$
functions for $n=4$. This result seems to be in contradiction with
minkevich1998isotropic  where the description of the Riemann tensor in
$4$-dimensions was attributed only to $3$ time functions. Probably this
contradiction arises because in minkevich1998isotropic  the pseudo-scalar mode
$W$ was constrained to vanish. This is certainly so for $n\neq 4$ but not for
$n=4$. For $n\neq 4$ the pseudo-scalar mode disappears and then and only then
the Riemann tensor is specified by three functions alone444Obviously this is
so because for $W=0$ the last $4$ terms of the list of invariants vanish
identically.. Come to think about it the appearance of these $6$ degrees of
freedom does make sense. The reasoning goes as follows. In $4$ dimensions the
Cosmological $L_{n}$ as given by the metric (12) and the connection (20) is
completely specified by $6$ time functions, one coming from the metric (the
scale factor) and $5$ from the connection (the distortion variables). These
$6$ degrees of freedom are then rearranged in a certain way and produce the
$6$ degrees of freedom that appear in the expression for the Riemann tensor
given above.

Comment 2. The last $3$ terms in (23) disappear when $W=0$. That is, these
terms arise due to torsion and more precisely due to the totally antisymmetric
part of the latter which exists only for $n=4$. Subsequently, for the
vanishing torsion subsector, the curvature is specified by three functions
only.

Let us also give a list of some essential geometrical objects that are derived
from the above expression of the Riemann tensor.

  * •

Ricci Tensor

$\displaystyle
R_{\mu\nu}=\Big{[}\dot{X}+H\Big{(}(n-1)X-(n-2)Y+(Z+V)\Big{)}-(n-2)XY+X(Z+V)-2W^{2}+\dot{H}+(n-1)H^{2}\Big{]}h_{\mu\nu}$
$\displaystyle+\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}(n-1)u_{\mu}u_{\nu}$
(38)

  * •

Homothetic Curvature

$\displaystyle\hat{R}_{\mu\nu}\equiv 0$ (39)

  * •

Co-Ricci

$\displaystyle\breve{R}_{\mu\nu}=\Big{[}\dot{Y}+H\Big{(}-(n-2)X+(n-1)Y+(Z+V)\Big{)}+(n-2)XY-Y(Z+V)+2W^{2}-\dot{H}-(n-1)\dot{H}^{2}\Big{]}h_{\mu\nu}$
$\displaystyle+\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}(n-1)u_{\mu}u_{\nu}$
(40)

  * •

Ricci Scalar

$\displaystyle
R=(n-1)\Big{[}(\dot{X}-\dot{Y})+(n-1)H(X-Y)+(X+Y)(Z+V)-(n-2)XY-2W^{2}+2\dot{H}+nH^{2}\Big{]}$
(41)

  * •

Ricci+co-Ricci

$\displaystyle(R_{\mu\nu}+\breve{R}_{\mu\nu})=\Big{[}(\dot{X}+\dot{Y})+H(X+Y)+2H(Z+V)+(X-Y)(Z+V)\Big{]}\Big{(}h_{\mu\nu}+(n-1)u_{\mu}u_{\nu}\Big{)}$
(42)

  * •

Ricci-co-Ricci

$\displaystyle(R_{\mu\nu}-\breve{R}_{\mu\nu})=\Big{[}(\dot{X}-\dot{Y})+(2n-3)H(X-Y)+(X+Y)(Z+V)-2(n-2)XY-4W^{2}+2\dot{H}+2(n-1)H^{2}\Big{]}h_{\mu\nu}$
$\displaystyle-\Big{[}(\dot{X}-\dot{Y})+H(X-Y)+(X+Y)(Z+V)+2(\dot{H}+H^{2})\Big{]}(n-1)u_{\mu}u_{\nu}$
(43)

  * •

Generalized Einstein Tensor555The definition we adopt for the Einstein tensor
comes from the contraction of the generalized Riemann double dual. Our
definition agrees with the one used in aoki2019scalar . We should note however
that this generalized version has neither properties of the usual Einstein
tensor, that is the generalized Einstein tensor is, in general, neither
symmetric nor covariantly conserved.

$\displaystyle
G_{\mu\nu}:=\frac{(n-3)}{2}\Big{(}R_{\mu\nu}-\breve{R}_{\mu\nu}-Rg_{\mu\nu}\Big{)}=$
$\displaystyle=\frac{(n-2)(n-3)}{2}\Big{[}-\Big{(}(\dot{X}-\dot{Y})+(n-2)H(X-Y)-(X+Y)(Z+V)+(n-3)XY-W^{2}+2\dot{H}-H^{2}\Big{)}h_{\mu\nu}$
$\displaystyle+\Big{(}H(X-Y)-XY-W^{2}+H^{2}\Big{)}(n-1)u_{\mu}u_{\nu}\Big{]}$
(44)

  * •

Right dual

$\displaystyle
R_{\mu\nu\alpha\beta}\varepsilon^{\alpha\beta\kappa\lambda}=\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}2u_{\mu}\varepsilon_{\nu}^{\;\;\kappa\lambda}$
$\displaystyle+\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}2u_{\nu}\varepsilon_{\mu}^{\;\;\kappa\lambda}$
$\displaystyle+\Big{[}H^{2}+H(X-Y)-XY-W^{2}\Big{]}2\Big{(}\varepsilon_{\mu\nu}^{\;\;\;\;\kappa\lambda}+u_{\nu}\varepsilon_{\mu}^{\;\;\kappa\lambda}-u_{\mu}\varepsilon_{\nu}^{\;\;\kappa\lambda}\Big{)}+8XWu_{\mu}\delta_{\nu}^{[\kappa}u^{\lambda]}+8YWu_{\nu}\delta_{\mu}^{[\kappa}u^{\lambda]}$
$\displaystyle+4(\dot{W}-HW)h^{\kappa}_{\;\;[\mu}h_{\nu]}^{\;\;\;\lambda}+8HW\delta^{[\kappa}_{\mu}\delta^{\lambda]}_{\nu}$
(45)

  * •

Symmetric Part of Riemann

$\displaystyle
R_{(\mu\nu)\alpha\beta}=\Big{[}(\dot{X}+\dot{Y})+H(X+Y)+2H(Z+V)+(X-Y)(Z+V)\Big{]}2u_{(\mu}h_{\nu)[\alpha}u_{\beta]}-2W(X+Y)\varepsilon_{\alpha\beta(\mu}u_{\nu)}$
(46)

  * •

Totally Antisymmetric Riemann666In the literature this term is sometimes
called the Hojman or more frequently the Holst term.

$\varepsilon^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}=12\Big{[}(\dot{W}+3HW)+W(X-Y)\Big{]}$
(47)

Of course the above list could go on and on. Here our intention was to present
the quantities that play an actual role in applications. For completeness let
us also present the form of the Riemann tensor of the characteristic cases of
vanishing non-metricity (Riemann-Cartan Space) and vanishing torsion
(Symmetric Space). For transparency in each case below we express the
distortion variables in terms of torsion and non-metric degrees of freedom
respectively. We have:

###  IV.1  Riemann-Cartan Space (vanishing non-metricity)

$\displaystyle
R_{\mu\nu\alpha\beta}=\Big{[}(\dot{H}-2\dot{\Phi})+H(H-2\Phi)\Big{]}4u_{[\mu}h_{\nu][\alpha}u_{\beta]}+\Big{[}(H-2\Phi)^{2}-P^{2}\Big{]}2h_{\mu[\alpha}h_{\beta]\nu}$
$\displaystyle+\Big{[}\dot{P}-P(H-4\Phi)\Big{]}2\varepsilon_{\alpha\beta[\nu}u_{\mu]}-(\dot{P}+HP)\varepsilon_{\mu\nu\alpha\beta}$
(48)

###  IV.2  Symmetric Space (vanishing torsion)

$\displaystyle
R_{\mu\nu\alpha\beta}=\left[\Big{(}\dot{H}-\frac{\dot{A}}{2}\Big{)}+H\Big{(}H-\frac{A}{2}\Big{)}\right]4u_{[\mu}h_{\nu][\alpha}u_{\beta]}+\left[\dot{B}+HB+\Big{(}H-\frac{A}{2}\Big{)}(A+C)\right]2u_{(\mu}h_{\nu)[\alpha}u_{\beta]}$
$\displaystyle+\Big{(}H-\frac{A}{2}\Big{)}\left[\frac{B}{2}+\Big{(}H-\frac{A}{2}\Big{)}\right]2h_{\mu[\alpha}h_{\beta]\nu}$
(49)

Note: If $P=0$ in the former case and $B=0$, $A=-C$ in the latter case the two
resulting Riemann tensors are related through the duality relation
$2\Phi\leftrightarrow A/2$. This is the known duality which was reported
earlier in the literature (see iosifidis2019torsion ) and holds true for
specific subspaces of torsion/non-metricity.

##  V Gauss-Bonnet Density

Having the form of the Riemann tensor, a highly interesting object one can
compute is the so-called Gauss-Bonnet term. Of course in metric Theories its
form is well-known to be given by

$\mathcal{G}=\widetilde{R}_{\mu\nu\alpha\beta}\widetilde{R}^{\alpha\beta\mu\nu}-4\widetilde{R}_{\mu\nu}\widetilde{R}^{\mu\nu}+\widetilde{R}^{2}$
(50)

and subsequently the Gauss-Bonnet density $\sqrt{-g}\mathcal{\hat{G}}$ is
known to be a topological invariant (i.e. independent of the metric) in $4$
dimensions and as a total divergence does not contribute to the field
equations when included in a Gravity Theory. It is interesting now, to explore
what the generalized Gauss-Bonnet term looks like in this non-Riemannian
Cosmological background we are considering here, given the form
$(\ref{Riemann})$. As a generalized version of the above Riemannian form, we
adopt the definition for the generalized Gauss-Bonnet term as given by

$\mathcal{G}:=R_{\mu\nu\alpha\beta}R^{\alpha\beta\mu\nu}-(R_{\mu\nu}-\breve{R}_{\mu\nu})(R^{\nu\mu}-\breve{R}^{\nu\mu})+R^{2}$
(51)

Our definition agrees with the one considered in both babourova1996gauss  and
janssen2019topological 777Note that in jimenez2016spacetimes ;
beltran2017modified  a different definition of the Gauss-Bonnet term was
considered. There, a linear combination of quadratic curvature invariants was
considered with the demand that in the Riemannian limit this combination will
yield the Riemannian Gauss-Bonnet term.  and follows from the contraction of
the Riemann tensor with the Riemann double dual after expanding the
generalized Kronecker deltas. Let us also stress out that the inclusion of the
aforementioned term in Metric-Affine Gravitational Theories is interesting for
various reasons. For instance, this term is projectively invariant888This
aspect is crucial for constructing healthy Theories aoki2019scalar ;
BeltranJimenez:2019acz .  and also this exact linear combination of Riemann
squared terms is free of ghosts jimenez2016spacetimes , a feature that is not
true for general quadratic curvature Theories (see BeltranJimenez:2019acz ).
In addition, this definition is valid for any dimension and $(\ref{GB})$
becomes topological as soon as the connection is reduced to be Levi-Civita.

From the generalized Gauss-Bonnet term we can then define the Gauss-Bonnet
density

$\mathcal{\hat{G}}:=\sqrt{-g}\mathcal{G}$ (52)

It is also known that the latter density is topological for Cartan-Weyl spaces
babourova1996gauss  but it fails to be so when the full non-metricity is
included janssen2019topological . Let us now compute the generalized Gauss-
Bonnet term for a Cosmological $L_{n}$. We have the following result.

######  Theorem 2.

Consider the generalized Gauss-Bonnet term as given in (51). In a Cosmological
$L_{n}$ the latter assumes the form
$\displaystyle\mathcal{G}=(n-1)(n-2)(n-3)\Big{[}(n-4)(H^{2}+HX-HY-
XY-W^{2})^{2}$
$\displaystyle+2\Big{(}(\dot{X}-\dot{Y})+H(X-Y)+(X+Y)(Z+V)+2(\dot{H}+H^{2})\Big{)}(H^{2}+HX-
HY-XY-W^{2})$ $\displaystyle-4(\dot{W}+HW)(2HW+WX-WY)\Big{]}$ (53)

###### Proof.

Starting with the covariant form (23) we carefully compute each term appearing
in (51). We find

$\displaystyle
R_{\mu\nu\alpha\beta}R^{\alpha\beta\mu\nu}=(n-1)\Big{(}(\dot{X}-\dot{Y})+H(X-Y)+(X+Y)(Z+V)+2(\dot{H}+H^{2})\Big{)}^{2}$
$\displaystyle+2(n-1)(n-2)\Big{(}H^{2}+H(X-Y)-XY-W^{2}\Big{)}^{2}$
$\displaystyle-4!(\dot{W}W+HW^{2})\Big{(}2H+(X-Y)\Big{)}$ (54)
$\displaystyle(R_{\mu\nu}-\breve{R}_{\mu\nu})(R^{\nu\mu}-\breve{R}^{\nu\mu})=(n-1)\Big{[}(n-1)\Big{(}(\dot{X}-\dot{Y})+H(X-Y)+(X+Y)(Z+V)+2(\dot{H}+H^{2})\Big{)}^{2}$
$\displaystyle+\Big{(}(\dot{X}-\dot{Y})+H(X-Y)+(X+Y)(Z+V)+2(\dot{H}+H^{2})+2(n-2)(H^{2}+HX-
HY-XY-W^{2})\Big{)}^{2}\Big{]}$ (55) $\displaystyle
R^{2}=(n-1)^{2}\Big{(}(\dot{X}-\dot{Y})+H(X-Y)+(X+Y)(Z+V)+2(\dot{H}+H^{2})$
$\displaystyle+(n-2)(H^{2}+H(X-Y)-XY-W^{2})\Big{)}^{2}$ (56)

Of course the next step is to now substitute all of these back to equation
(51). Even though the situation is seemingly complicated it can be greatly
simplified by observing that there are certain combinations appearing. To be
more specific, defining

$\lambda:=(\dot{X}-\dot{Y})+H(X-Y)+(X+Y)(Z+V)+2(\dot{H}+H^{2})$ (57)

and

$\xi:=H^{2}+H(X-Y)-XY$ (58)

the above terms simplify considerably (see Appendix). With this observation,
and plugging everything into (51) after some algebra we arrive at

$\mathcal{\hat{G}}=(n-1)(n-2)(n-3)\Big{[}(n-4)(\xi-W^{2})^{2}+2\lambda(\xi-W^{2})-4(\dot{W}+HW)(2HW+WX-
WY)\Big{]}$ (59)

Note that the $\lambda^{2}$ terms have cancelled out in the final expression.
Then by substituting in the latter equation the definitions of $\lambda$ and
$\xi$ we arrive at the stated result (53).

∎

An immediate comment follows from the above result.

Comment 3. In a Cosmological $L_{n}$ the generalized Gauss-Bonnet term (51)
identically vanishes when the number of dimensions is $n=1,2$ or $3$.

Now, one may ask what happens for $n=4$ which is the case of our physical
spacetime and compare it with the standard metric case. Of course in the
latter case, the Gauss-Bonnet density is a total derivative and as such it
does not contribute to the field equations of the Theory when $n=4$. It is
interesting to see what happens with the generalized Gauss-Bonnet density
here. It is also known that the latter density continues to be a total
derivative in Weyl Cartan spaces (see ref. babourova1996gauss ) but it fails
to be so when the full non-metricity is included babourova1997pontryagin  (see
also janssen2019topological ). It is then natural to ask, are there any
extended geometries more general then Cartan-Weyl, for which the Gauss-Bonnet
density (52) continues to be a total derivative. As we show below the answer
to this question is affirmative, for the Cosmological setting we are
considering here. We have the following result.

######  Theorem 3.

For the restricted $B=0$ subsector of a Cosmological $L_{n}$ the generalized
Gauss-Bonnet density is a total derivative in $4$ dimensions. In particular,
for $B=0$ and $n=4$ it holds that
$\displaystyle\mathcal{\hat{G}}=\sqrt{-g}\mathcal{G}=\frac{d}{dt}\Big{[}8a^{3}x\Big{(}x^{2}-3W^{2}\Big{)}\Big{]}$
(60) where $x=H-Y$ (61)

###### Proof.

We start by a making a trivial rearrangement in the expressions we have for
$\lambda$ and $\xi$ which with the help of (19) recast to

$\lambda=2(\dot{H}-\dot{Y})+2H(H-Y)+\frac{1}{2}(\dot{B}+HB)+\frac{B}{2}(Z+V)$
(62) $\xi=(H-Y)^{2}+\frac{B}{2}(H-Y)$ (63)

Defining

$x:=H-Y$ (64)

and setting $B=0$ they read

$\lambda=2(\dot{x}+Hx)\;\;,\;\;\;\xi=x^{2}$ (65)

Then, if we take these expressions and substitute them back in (53) for $n=4$,
after some algebra we recognise that these terms form the derivative of a
product and finally we get (see Appendix)

$\displaystyle\mathcal{\hat{G}}=\sqrt{-g}\mathcal{G}=\frac{d}{dt}\Big{[}8a^{3}x\Big{(}x^{2}-3W^{2}\Big{)}\Big{]}$
(66)

as stated. ∎

Comment 4. Let us note that the geometry with $B=0$ we considered above is a
generalization of the Weyl-Cartan geometry where the non-metricity apart from
the usual Weyl part has another degree of freedom999This can be easily seen
from (15). Recall that for Weyl non-metricity one has $B=0$ and $A=-C$. Here
we have only $B=0$ without the latter constraint (i.e no relation between $A$
and $C$), which represents a generalization of the Weyl spacetime.. Therefore
we conclude that the generalized Gauss-Bonnet density is a total derivative
term for this extended Cartan-Weyl geometry in a Cosmological setting.

We shall now prove that for $B\neq 0$ the density (53) is not a total
derivative in $4$ dimensions. Leaving $B\neq 0$ and carrying out a similar
analysis as above, after spotting the various appearances of total derivatives
we finally find (see Appendix for details)

$\mathcal{\hat{G}}=\frac{d}{dt}\Big{[}a^{3}\Big{(}8x(x^{2}-3W^{2})+6B(x^{2}-W^{2})\Big{)}\Big{]}+\frac{a^{3}xB}{4}(\dot{B}+HB)+\frac{a^{3}}{2}B(Z+V)\Big{(}x^{2}+\frac{B}{2}x-W^{2}\Big{)}$
(67)

From which we see that the first term of the latter expression, is obviously a
total derivative derivative but it is impossible to express the last two
combinations as total differentials. We see therefore that the presence of $B$
has spoiled the total derivative nature of the generalized Gauss-Bonnet
density. It is also interesting to note that for higher dimensions ($n>4$) the
Gauss-Bonnet density cannot (in general) be written as a total differential
even in the restricted $B=0$ geometry because of the very presence of the
term101010Note that for $n\neq 4$ the $W$ term disappears as we have already
discussed in the introduction. $(n-4)\xi^{2}$. Collecting all the
aforementioned results we have the following corollary.

######  Corollary 1.

In a Cosmological $L_{n}$ the generalized Gauss-Bonnet density (52)

  * •

Identically vanishes for $n=1,2$ and $3$, where $n$ denotes the number of
dimensions.

  * •

For $n=4$ it is not in a general a total derivative. For the reduced space
$B=0$, which serves as a certain generalization of Weyl-Cartan space, the
latter density becomes indeed a total differential.

  * •

For $n>4$ the Gauss-Bonnet density is not, in general, a total derivative term
even for the restricted case $B=0$.

##  VI On Applications: Gauss-Bonnet Gravity in Metric-Affine Formalism

As an application of our derived results, let us start consider the usual
Einstein-Hilbert action and add to it the above defined Gauss-Bonnet term. As
we have already proved since our affine connection is a priori generic, the
Gauss-Bonnet term will not be a total derivative even in $4$ dimensions. It is
interesting then, to study the Cosmology of this Theory. So our starting point
will be the action

$S[g,\Gamma]=\frac{1}{2\kappa}\int
d^{n}x\sqrt{-g}\Big{(}R+\lambda\hat{G}\Big{)}$ (68)

where $\lambda$ is a parameter. We leave the spacetime dimension $n$
unspecified at this point but we are going to specialize to $n=4$ shortly.
Note that the Gauss-Bonnet term is projectively invariant in any dimension and
is also conformally invariant only in $4$ dimensions. So in $4$ dimensions the
Gauss-Bonnet term is both projectively and conformally invariant. Let us now
find the field equations of the Theory. Variation with respect to the metric
gives

$R_{(\mu\nu)}-\frac{R}{2}g_{\mu\nu}+\lambda\Big{(}4G^{\alpha\beta}_{\;\;\;\;(\mu|\lambda}R_{\alpha\beta|\nu)}^{\;\;\;\;\;\;\;\;\lambda}+2G^{\lambda}_{\;\;(\nu|\alpha\beta|}R_{\mu)\lambda}^{\;\;\;\;\;\alpha\beta}-\frac{1}{2}g_{\mu\nu}\hat{G}\Big{)}=0$
(69)

Contracting the latter with the metric, we obtain

$(2-n)R+\lambda(4-n)\hat{G}=0$ (70)

from which we conclude that in $4-dim$ the contribution from the Gauss-Bonnet
term drops out. Of course this is to be expected, since as we pointed out, in
this dimensionality the Gauss-Bonnet term is conformally invariant, and
therefore its variation with respect to the metric produces a traceless
tensor. To close the system of field equations we also vary with respect to
the connection, to get

$P_{\lambda}^{\;\;\;\mu\nu}-\lambda\left(2\frac{\nabla_{\alpha}(\sqrt{-g}G_{\lambda}^{\;\;\mu\nu\alpha})}{\sqrt{-g}}+4G_{\lambda}^{\;\;\mu\nu\alpha}S_{\alpha}+2G_{\lambda}^{\;\;\mu\alpha\beta}S_{\alpha\beta}^{\;\;\;\;\nu}\right)=0$
(71)

where $P_{\lambda}^{\;\;\;\mu\nu}$ is the so-called Palatini tensor and is
explicitly given by

$P_{\lambda}^{\;\;\;\mu\nu}=-\frac{\nabla_{\lambda}(\sqrt{-g}g^{\mu\nu})}{\sqrt{-g}}+\frac{\nabla_{\sigma}(\sqrt{-g}g^{\mu\sigma})\delta^{\nu}_{\lambda}}{\sqrt{-g}}\\\
+2(S_{\lambda}g^{\mu\nu}-S^{\mu}\delta_{\lambda}^{\nu}+g^{\mu\sigma}S_{\sigma\lambda}^{\;\;\;\;\nu})$
(72)

Now, due to the fact that both the Ricci scalar and the Gauss-Bonnet term are
projectively invariant, equation $(\ref{P})$ is identically traceless when
contracted in $\mu=\lambda$. Setting now $n=4$ in $(\ref{RG})$ we conclude
that $R=0$. Expanding the latter equation and considering a homogeneous FLRW
background we arrive at

$\displaystyle\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}+4\Phi^{2}-P^{2}+\frac{1}{8}\Big{(}2A^{2}+B(C-A)\Big{)}+\Phi(2A-B)+\dot{f}+3Hf=0\;\;,\;\;f:=\frac{1}{2}\left[\frac{B}{2}-A-4\Phi\right]$
(73)

which is a variant of the modified Friedmann equation. As for the acceleration
equation, the form of the latter for non-Riemannian Universes has been derived
in iosifidis2020cosmic  and reads

$\displaystyle\frac{\ddot{a}}{a}=-\frac{1}{3}R_{\mu\nu}u^{\mu}u^{\nu}+2\left(\frac{\dot{a}}{a}\right)\Phi+2\dot{\Phi}+\left(\frac{\dot{a}}{a}\right)\left(A+\frac{C}{2}\right)+\frac{\dot{A}}{2}-\frac{A^{2}}{2}-\frac{1}{2}AC-2A\Phi-2C\Phi$
(74)

One could then proceed by contracting the metric field equations with
$u^{\mu}u^{\nu}$ in order to eliminate the first term
($R_{\mu\nu}u^{\mu}u^{\nu}$) and express everything in terms of the scale
factor and the torsion and non-metricity variables. This results in a fairly
complicated expression which we refrain from presenting here since it goes
beyond the scope of the present treatment. From the above equations we see
that both torsion and non-metricity have interesting and highly non-trivial
effects on the behaviour of the expansion.

##  VII Conclusions

We have derived the covariant form of the generalized Riemann tensor (i.e.
including also torsion and non-metricity) for non-Riemannian Cosmological
spacetimes. This is the most general covariant expression for the Riemann
tensor that is compatible with the Cosmological Principle. In deriving this
expression we used the earlier reported covariant forms of torsion and non-
metricity for homogeneous Cosmological settings. Our result shows that in a
Friedman-like Cosmological setting, the curvature is completely specified by
$6$ time functions when the spacetime dimension is equal to $4$ while for
$n\neq 4$ the latter possesses only $3$ degrees of freedom. It goes without
saying, that the aforementioned covariant form is an indispensable ingredient
for the study of Cosmological aspects of Metric-Affine Gravity. Therefore, our
result is directly applicable and serves as a necessary ingredient in
exploring the role of non-Riemannian degrees of freedom in Cosmology.

Having obtained the covariant form of the Riemann tensor in a Cosmological
$L_{n}$ we then went on and computed some of the basic objects that appear in
applications, such as the Ricci tensor, Co-Ricci tensor, Ricci scalar etc.
With the help of these relations we then proceeded in computing the
generalized Gauss-Bonnet density. As expected our results show that the
presence of the full non-metricity makes it impossible to express the latter
density as a total differential in general. However, we proved that by
constraining slightly the geometry by setting one of the non-metricity degrees
of freedom to zero, the generalized version of the Gauss-Bonnet density
becomes indeed a total derivative. Note that it was known in the literature
that for Cartan-Weyl geometries (i.e. arbitrary torsion but Weyl non-
metricity) the latter is indeed always a total differential babourova1996gauss
. Here we have shown that in a Cosmological setup, the Gauss-Bonnet density is
a total differential for geometries for which the non-metricity is more
general from Weyl. Of course our result holds true for a Cosmological setup
but this should not lead us to the conclusion that this would be so for any
background. Here we proved that this is so only for highly symmetric FLRW
backgrounds. It would be interesting, however, to see under what circumstances
the Gauss-Bonnet term we presented here is a total derivative for the more
general family of spherically symmetric Metric-Affine geometries as derived in
hohmann2020metric 111111Possibly, it would also be interesting to find the
Gauss-Bonnet density analogues for the symmetric teleparallel
nester1998symmetric ; jimenez2018teleparallel  and general teleparallel
jimenez2019general  Theories as was found for metric teleparallelism in
kofinas2014teleparallel . . Finally, it would be interesting to find the most
general structure of the non-Riemannian geometries for which (51) is a total
derivative irrespective of the background.

##  VIII Acknowledgments

I would like to thank Christos Charmousis, Alejandro Jimenez Cano and
Konstantinos Palikaris for useful discussions. This research is co-financed by
Greece and the European Union (European Social Fund- ESF) through the
Operational Programme ”Human Resources Development, Education and Lifelong
Learning” in the context of the project ”Reinforcement of Postdoctoral
Researchers - 2 nd Cycle” (MIS-5033021), implemented by the State Scholarships
Foundation (IKY).

##  Appendix A Supplementary material for the proofs

###  A.1 Gauss-Bonnet density for $L_{n}$ Cosmology

Below we give some more details regarding the derivation of the Gauss-Bonnet
term $(\ref{GB})$. In computing the invariants entering the Gauss-Bonnet term
the following relations were quite helpful

$u^{\mu}R_{\mu\nu\alpha\beta}=-\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}2h_{\nu[\alpha}u_{\beta]}+2W(X+H)\varepsilon_{\alpha\beta\nu}$
(75)
$u^{\nu}R_{\mu\nu\alpha\beta}=-\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}2h_{\mu[\alpha}u_{\beta]}+2W(Y-H)\varepsilon_{\mu\alpha\beta}$
(76) $\displaystyle
u^{\alpha}R_{\mu\nu\alpha\beta}=\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}u_{\mu}h_{\nu\beta}+(\dot{W}+HW)\varepsilon_{\mu\nu\beta}$
$\displaystyle+\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}u_{\nu}h_{\mu\beta}$
(77)
$h^{\mu\alpha}R_{\mu\nu\alpha\beta}=(n-1)\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}u_{\nu}u_{\beta}+(n-2)\Big{[}H(X-Y)-XY-W^{2}+H^{2}\Big{]}h_{\nu\beta}$
(78)
$h^{\nu\alpha}R_{\mu\nu\alpha\beta}=(n-1)\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}u_{\mu}u_{\beta}-(n-2)\Big{[}H(X-Y)-XY-W^{2}+H^{2}\Big{]}h_{\mu\beta}$
(79)
$u^{\mu}u^{\alpha}R_{\mu\nu\alpha\beta}=-\Big{[}\dot{X}+H(X+Z+V)+X(Z+V)+(\dot{H}+H^{2})\Big{]}h_{\nu\beta}$
(80)
$u^{\nu}u^{\alpha}R_{\mu\nu\alpha\beta}=-\Big{[}\dot{Y}+H(Y+Z+V)-Y(Z+V)-(\dot{H}+H^{2})\Big{]}h_{\mu\beta}$
(81) $h^{\mu\nu}R_{\mu\nu\alpha\beta}=0=u^{\mu}u^{\nu}R_{\mu\nu\alpha\beta}$
(82)

Then with the abbreviations (57) and $(\ref{xi})$ we find for each piece

$R_{\mu\nu\alpha\beta}R^{\alpha\beta\mu\nu}=(n-1)\lambda^{2}+2(n-1)(n-2)\Big{(}\xi-W^{2}\Big{)}^{2}-4!(\dot{W}W+HW^{2})\Big{(}2H+(X-Y)\Big{)}$
(83)
$(R_{\mu\nu}-\breve{R}_{\mu\nu})(R^{\nu\mu}-\breve{R}^{\nu\mu})=(n-1)\Big{[}\Big{(}\lambda+2(n-2)(\xi-W^{2})\Big{)}^{2}+(n-1)\lambda^{2}\Big{]}$
(84) $R^{2}=(n-1)^{2}\Big{(}\lambda+(n-2)(\xi-W^{2})\Big{)}^{2}$ (85)

and by substituting the latter back in (51) we see that the $\lambda^{2}$
cancel out and we straightforwardly arrive at (53).

###  A.2 Case $n=4$, $B\neq 0$

Let us now derive the stated form (67) for $n=4$ and the general case $B\neq
0$. For $n=4$ the Gauss-Bonnet density becomes

$\mathcal{\hat{G}}=12\Big{[}a^{3}\lambda(\xi-W^{2})-2(\dot{W}W+HW^{2})\Big{(}2x+\frac{B}{2}\Big{)}\Big{]}$
(86)

Now it is convenient to set

$y=2x+\frac{B}{2}$ (87)

and thus definitions (57) and (58) become respectively

$\lambda=(\dot{y}+Hy)+\frac{1}{2}(\dot{B}+HB)+\frac{B}{2}(Z+V)$ (88)
$\xi=\frac{1}{4}y^{2}-\frac{1}{16}B^{2}$ (89)

With these the above expression recasts to

$\displaystyle\frac{1}{12}\mathcal{\hat{G}}=a^{3}(\dot{y}+Hy)\xi-a^{3}(\dot{y}+Hy)W^{2}+a^{3}\frac{B}{2}(Z+V)(\xi-W^{2})-2ya^{3}(W\dot{W}+HW^{2})$
(90)

We observe now that the second and the last term of the above can be added to
give the total derivative of a product yielding

$-a^{3}(\dot{y}+Hy)W^{2}-2ya^{3}(W\dot{W}+HW^{2})=-a^{2}W^{2}\frac{d}{dt}(ay)-ay\frac{d}{dt}(W^{2}a^{2})=-\frac{d}{dt}\Big{(}a^{3}yW^{2}\Big{)}$
(91)

Substituting also $\xi$ and performing the same technique, after pairing all
the terms that form derivatives of products we find

$\mathcal{\hat{G}}=\frac{d}{dt}\Big{[}a^{3}\Big{(}8x(x^{2}-3W^{2})+6B(x^{2}-W^{2})\Big{)}\Big{]}+\frac{a^{3}xB}{4}(\dot{B}+HB)+\frac{a^{3}}{2}B(Z+V)\Big{(}x^{2}+\frac{B}{2}x-W^{2}\Big{)}$
(92)

where $x=H-Y$. Note that in the Riemannian limit ($i.e.$ $B=0=Z=V=W=Y$) the
latter reduces to the metric Gauss-Bonnet density

$\mathcal{\hat{G}}=\frac{d}{dt}(8\dot{a}^{3})$ (93)

as expected.

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