# Emergent metric and geodesic analysis in cosmological solutions of (torsion-
free) Polynomial Affine Gravity

Oscar Castillo-Felisola<EMAIL_ADDRESS>Departamento de Física,
Universidad Técnica Federico Santa María
Casilla 110-V, Valparaíso, Chile  Centro Científico Tecnológico de Valparaíso
Casilla 110-V, Valparaíso, Chile  José Perdiguero  Departamento de Física,
Universidad Técnica Federico Santa María
Casilla 110-V, Valparaíso, Chile  Oscar Orellana  Departamento de Matemáticas,
Universidad Técnica Federico Santa María
Casilla 110-V, Valparaíso, Chile  Alfonso R. Zerwekh  Departamento de Física,
Universidad Técnica Federico Santa María
Casilla 110-V, Valparaíso, Chile  Centro Científico Tecnológico de Valparaíso
Casilla 110-V, Valparaíso, Chile

###### Abstract

Starting from an affinely connected space, we consider a model of gravity
whose fundamental field is the connection. We build up the action using as
sole premise the invariance under diffeomorphisms, and study the consequences
of a cosmological ansatz for the affine connection in the torsion-free sector.
Although the model is built without requiring a metric, we show that the
nondegenerated Ricci curvature of the affine connection can be interpreted as
an _emergent_ metric on the manifold. We show that there exists a
parametrization in which the $(r,\varphi)$-restriction of the geodesics
coincides with that of the Friedman–Robertson–Walker model. Additionally, for
connections with nondegenerated Ricci we are able to distinguish between
space-, time- and null-like self-parallel curves, providing a way to
differentiate _trajectories_ of massive and massless particles.

affine connection; affine gravity; self-parallel curve; geodesic; cosmological
models

##  I Introduction

General Relativity was proposed by A. Einstein as an attempt to compatibilise
the gravitational interactions with the postulates of special relativity
Einstein (1915a, b, c, d, 1916). The ground-breaking idea behind the proposal
was the interpretation of the gravitational interaction as the effect of
properties of the spacetime, represented by a nontrivial geometry. The
spacetime is modelled by a Riemannian manifold, whose geometric properties are
determined by the metric tensor, and therefore it is the natural field
describing the dynamics of the spacetime. The properties of the _matter_
distribution are encoded in the energy-momentum tensor. Einstein’s field
equations are the extrema of the Einstein–Hilbert action when varied with
respect to the metric.

General Relativity has been tested extensively with magnificent agreement with
the experimental data, as one can appreciate in the excellent review Ref. Will
(2014). The most recent triumph of the theory was the direct measurement of
gravitational waves by the LIGO-Virgo collaborations Abbott  _et al._ (2016,
2017).

Although General Relativity is, by far, the most successful theory of
gravitational interactions, there is an increasing interest in alternative
models of gravity, particularly driven for the lack of a complete framework of
quantum gravity DeWitt (1967a, b); Deser and van Nieuwenhuizen (1974a, b); ’t
Hooft and Veltman (1974); Ashtekar (1986, 1987), and the necessity of
hypothesising a dark sector that accounts for approximately 96% of the energy
content of the Universe Zwicky (1937); Rubin and W. Kent (1970); Sofue and
Rubin (2001); Riess  _et al._ (1998); Perlmutter  _et al._ (1999).

The existence of these problems is a signal of _new physics_ , and their
solutions require either including new fields or changing the gravitational
theory. The latter suggests that Einstein’s theory is an effective theory of
gravity and, therefore, one may consider alternative models. Among the
generalisations one encounters for example: the Einstein–Cartan theory, which
extends General Relativity by allowing a non-symmetric connection, but
considers the same action Cartan (1922, 1923, 1924, 1925); models with extra
dimensions, firstly proposed by T. Kaluza and O. Klein Kaluza (1921); Klein
(1926); Lovelock models, which are build under the same premises than General
Relativity, but in any dimension Lovelock (1971); the metric-affine models, in
which the conditions of metricity and vanishing torsion are generally dropped
Hehl  _et al._ (1995); the Lovelock–Cartan model, which are the extension of
Lovelock models with a non-symmetric connection Mardones and Zanelli (1991);
and many others.

Inspired by the fact that fundamental interactions (other than gravity) are
described by gauge theories whose dynamical field is a connection, seems
reasonable to search for a model of gravity described solely by a connection.
The first affine model of gravity was proposed by Sir A. Eddington, who
considered an action defined by the square root of the Ricci tensor Eddington
(1923) (See also Ref. Schrödinger (1950)). Moderns attempts to describe
gravity as a theory for the affine connection have been proposed in Refs.
Kijowski (1978); Krasnov (2006, 2007, 2008); Krasnov and Shtanov (2008);
Krasnov (2011); Popławski (2007a, b, 2014).

Recently, a novel model has been proposed, dubbed Polynomial Affine Gravity,
which is built out with polynomial terms of the irreducible components of the
connection Castillo-Felisola and Skirzewski (2015, 2018); Castillo-Felisola
(2018); Castillo-Felisola  _et al._ (2019), assuming invariance under the
group of diffeomorphims and no explicit use of a metric, i.e. even when the
spacetime is metric this field plays no role in the mediation of gravitational
interactions.111In General Relativity the metric plays a double role: it is
the instrument that serves to measure distances, and also it is the field that
mediates gravity. The idea behind metric-affine models is that those roles are
played by different fields, but both fields are dynamic. In our construction,
we manage to build a model of gravity without the need of an instrument that
measures distances. The action of Polynomial Affine Gravity has very
interesting features: (i) It is power-counting renormalisable;222We highlight
that this is a necessary but not sufficient condition for the model to be
renormalisable. (ii) No other term can be added, i.e. it is not possible to
add counter-terms; (iii) All the couplings are dimensionless, suggesting that
the model is a conformal theory at tree level; (iv) The torsion-free sector is
a consistent truncation compatible with General Relativity, i.e. it passes the
classical test of gravity; and (v) The structure of the model yields no three-
point graviton vertices, which might allow to overcome the no-go theorems
found in Refs. McGady and Rodina (2014); Camanho  _et al._ (2016).333Regarding
the aforementioned no-go theorem, a similar phenomenon happens in the case of
a massive spin-1 field coupled to a non-Abelian gauge field. A necessary
condition for making the theory consistent with perturbative unitarity is the
absence of a three-point vetex for the massive field Zerwekh (2013).

There are several conceptual subtleties one has to rethink about when working
with an affine model. The most recurrent question is: How do we measure
distances if the model lacks a metric? In this paper we aim to broaden current
knowledge on those subtleties. With this in mind, in Sec. II we briefly review
the polynomial affine model of gravity, and re-formulate the model in terms of
geometrical objects with simpler interpretation. In addition, we argue that
the limit of zero torsion is well defined, i.e. it is a consistent
_truncation_ of the model, and show that the field equations on this sector
are a known generalisation of the Einstein field equations. Then, in Sec. III
we proceed—by restricting ourselves to the torsion-free sector—to find
cosmological solutions of the field equations Castillo-Felisola (2018), that
extend the results reported in Ref. Castillo-Felisola  _et al._ (2019). In the
absence of a metric there is no concept of geodesic, however the concept of
self-parallel curve is still valid. Assuming that the trajectories of _free
falling_ test particles are self-parallel curves, we analyse them in Sec.IV
and show that, there exist a parametrisation in which the
$(r,\varphi)$-restriction of the equations is nothing but the expected from
the Friedman–Robertson–Walker model. At this point, the obstruction is that
without a metric it is not possible to differentiate between trajectories of
massive and massless particles. In Sec. V we show that under certain
conditions the Ricci tensor is a well-behaved (emergent) metric,444We prefer
to call this metric “emergent”, because it is a derived instead of a
fundamental geometrical object. allowing to define _space-like_ , _time-like_
or _light-like_ vectors, and providing the arena for an affine definition of
Einstein manifolds. Then, in Sec. VI we mention how some basic cosmological
quantities are defined in terms of the parametric functions of the connection.
In Sec. VII we conclude with a discussion of the results. For the sake of
completeness we include some appendixes. In Appendix A we review the dubbed
_dimensional analysis_ that allowed us to build the action. In Appendix B we
show the explicit contribution of each term of the action to the field
equations.

##  II The model of Polynomial Affine Gravity

The polynomial affine gravity is an alternative theory of gravitation, whose
sole fundamental field is the affine connection, $\hat{\Gamma}$. Notice that
without the use of a metric, one calculates the curvature and the Ricci
tensors, but not the curvature scalar. Therefore, it is not possible to write
an equivalent action to the Einstein–Hilbert action.

Our goal is to build up the most general action which is invariant under the
group of diffeomorphims. Firstly, one could try to use the connection as a
whole, however this yields no interesting models, since the obtained terms are
topological invariants,555All other possible terms are related to these, up to
boundary terms.

$S[\hat{\Gamma}]=\int\Big{(}a_{1}\hat{\mathcal{R}}_{{\mu\nu}}{}^{{\alpha}}{}_{{\beta}}\hat{\mathcal{R}}_{{\lambda\rho}}{}^{{\beta}}{}_{{\alpha}}+a_{2}\hat{\mathcal{R}}_{{\mu\nu}}{}^{{\alpha}}{}_{{\alpha}}\hat{\mathcal{R}}_{{\lambda\rho}}{}^{{\beta}}{}_{{\beta}}\Big{)}\mathop{\mathrm{d}V}\nolimits^{\mu\nu\lambda\rho},$
(1)

which are the four-dimensional Pontryagin density and the product of
(generalised) two-dimensional Pontryagin densities. In the above equation we
have introduced the _natural_ volume form, defined as the wedge product of the
coordinates, i.e.
$\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=J(x)\mathop{\mathrm{d}x}\nolimits^{\alpha}\wedge\mathop{\mathrm{d}x}\nolimits^{\beta}\wedge\mathop{\mathrm{d}x}\nolimits^{\gamma}\wedge\mathop{\mathrm{d}x}\nolimits^{\delta}$
for an arbitrary nonvanishing—within a chart—function $J(x)$.

The irreducible components of the connection, used to build the action, are
defined as666In a model with a metric, the connection can always be decomposed
into the Levi-Cività component and the distorsion. The upper index of the
later can be lowered with the metric, and since the distorsion is a tensor, it
decomposes according with the Young projection, i.e. it has a completely
symmetric and antisymmetric parts plus a component with mixed symmetries. A
detailed analysis of the irreducible components of the connection in the
metric case can be found in Ref. Iosifidis (2019a, b). The central difference
with our approach is that without the use of a metric, only the lower indices
can be decomposed through Young projection, as shown in Eq. (2).

$\displaystyle\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$
$\displaystyle=\hat{\Gamma}_{{(\mu}}{}^{{\lambda}}{}_{{\nu)}}+\hat{\Gamma}_{{[\mu}}{}^{{\lambda}}{}_{{\nu]}}$
(2)
$\displaystyle=\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+\epsilon_{\mu\nu\sigma\kappa}\mathcal{T}^{\lambda,\sigma\kappa}+{\mathcal{A}}_{[\mu}\delta^{\lambda}_{\nu]}$
$\displaystyle=\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+\delta^{\lambda}_{[\mu}{\mathcal{A}}_{\nu]},$

where we have first separated the symmetric and antisymmetric part in the
lower indices (the latter is nothing but the torsion of the connection), in
the second line we have renamed the symmetric part of the affine connection
$\hat{\Gamma}_{{(\mu}}{}^{{\lambda}}{}_{{\nu)}}\equiv\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$
and provided a reparametrisation of the torsion in terms of its trace
(${\mathcal{A}}_{\mu}$) and a _dual_ of a Curtright-like tensor,777This was
the parametrisation of the decomposition utilised in Refs. Castillo-Felisola
and Skirzewski (2015, 2018); Castillo-Felisola (2018). while in the third line
the $\mathcal{B}$ field is just the traceless part of the torsion. All these
elements transform as tensors under diffeomorphism, except for the symmetric
part of the affine connection, $\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$, which
consequently must be included in the action solely through the covariant
derivative.888Notice that under similar requirements but in odd dimensions,
there are Chern–Simons terms, in which the connection enters explicitly in the
action. See for example the three-dimensional construction in Ref. Castillo-
Felisola and Skirzewski (2015).

In order to build the most general action using the irreducible fields,
$\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$,
$\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$ and ${\mathcal{A}}_{\mu}$, the
strategy to write down the action is to define the most general scalar
density, where the dynamics is given by the covariant derivative with respect
to the symmetric part of the connection.

Using the second parametrisation in Eq. (2), the most general action is

$\begin{split}S&=\int\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\bigg{[}B_{1}\mathcal{R}_{{\mu\nu}}{}^{{\mu}}{}_{{\rho}}\mathcal{B}_{{\alpha}}{}^{{\nu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}+B_{2}\mathcal{R}_{{\alpha\beta}}{}^{{\mu}}{}_{{\rho}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\mathcal{B}_{{\mu}}{}^{{\rho}}{}_{{\nu}}+B_{3}\mathcal{R}_{{\mu\nu}}{}^{{\mu}}{}_{{\alpha}}\mathcal{B}_{{\beta}}{}^{{\nu}}{}_{{\gamma}}{\mathcal{A}}_{\delta}+B_{4}\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\rho}}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}{\mathcal{A}}_{\sigma}\\\
&\quad+B_{5}\mathcal{R}_{{\alpha\beta}}{}^{{\rho}}{}_{{\rho}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}{\mathcal{A}}_{\sigma}+C_{1}\mathcal{R}_{{\mu\alpha}}{}^{{\mu}}{}_{{\nu}}\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}+C_{2}\mathcal{R}_{{\alpha\beta}}{}^{{\rho}}{}_{{\rho}}\nabla_{\sigma}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}+D_{1}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\lambda}}\mathcal{B}_{{\mu}}{}^{{\nu}}{}_{{\alpha}}\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\lambda}}{}_{{\delta}}\\\
&\quad+D_{2}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}\nabla_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}+D_{3}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\nu}}\mathcal{B}_{{\beta}}{}^{{\lambda}}{}_{{\gamma}}\nabla_{\lambda}\mathcal{B}_{{\mu}}{}^{{\nu}}{}_{{\delta}}+D_{4}\mathcal{B}_{{\alpha}}{}^{{\lambda}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\nabla_{\lambda}{\mathcal{A}}_{\sigma}+D_{5}\mathcal{B}_{{\alpha}}{}^{{\lambda}}{}_{{\beta}}{\mathcal{A}}_{\sigma}\nabla_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\\\
&\quad+D_{6}\mathcal{B}_{{\alpha}}{}^{{\lambda}}{}_{{\beta}}{\mathcal{A}}_{\gamma}\nabla_{\lambda}{\mathcal{A}}_{\delta}+D_{7}\mathcal{B}_{{\alpha}}{}^{{\lambda}}{}_{{\beta}}{\mathcal{A}}_{\lambda}\nabla_{\gamma}{\mathcal{A}}_{\delta}+E_{1}\nabla_{\rho}\mathcal{B}_{{\alpha}}{}^{{\rho}}{}_{{\beta}}\nabla_{\sigma}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}+E_{2}\nabla_{\rho}\mathcal{B}_{{\alpha}}{}^{{\rho}}{}_{{\beta}}\nabla_{\gamma}{\mathcal{A}}_{\delta}\\\
&\quad+F_{1}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}\mathcal{B}_{{\sigma}}{}^{{\rho}}{}_{{\lambda}}+F_{2}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\lambda}}\mathcal{B}_{{\delta}}{}^{{\lambda}}{}_{{\rho}}\mathcal{B}_{{\mu}}{}^{{\rho}}{}_{{\nu}}+F_{3}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\lambda}}\mathcal{B}_{{\mu}}{}^{{\nu}}{}_{{\alpha}}\mathcal{B}_{{\beta}}{}^{{\lambda}}{}_{{\gamma}}{\mathcal{A}}_{\delta}+F_{4}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}{\mathcal{A}}_{\mu}{\mathcal{A}}_{\nu}\bigg{]}.\end{split}$
(3)

where the covariant derivation and the curvature are defined with respect to
the symmetric connection, i.e. $\nabla=\nabla^{\Gamma}$ and
$\mathcal{R}=\mathcal{R}{}^{{\Gamma}}$. The action is defined up to boundary
and topological terms. Although the dropped terms are relevant when studying
global aspects of the model, they do not contribute to the equations of
motion. In order to write down the action we used a variation of the
_dimensional analysis_ method introduced in Ref. Castillo-Felisola and
Skirzewski (2018). Details of the dimensional analysis are shown in Appendix
A.

Interestingly, all of the coupling constant are dimensionless, which from the
view point of Quantum Field Theory is desirable if one is interested in trying
to quantise the model. In addition, as shown by the dimensional analysis in
Ref. Castillo-Felisola and Skirzewski (2018), there is a finite number of
possible terms contributing to the action (once those ignored in Eq. (3) are
included), which we interpret as a _rigidity_ of the model, given than in the
hypothetical scenario of quantisation of Polynomial Affine Gravity all the
counter-terms should have the form of terms already present in the original
action.

###  II.1 Field equations

In what follows, we will obtain the field equations of the model using the
standard variational principle. It is well-known that to ensure the well-
posedness of the variational problem no second-class constraints should be
present, or higher derivative terms might appear in the field equations. In
General Relativity, one needs to add the Gibbons–Hawking–York term to solve
this problem. Although the absence of second-class constraints in Polynomial
Affine Gravity has not been proven yet, the structure of the action in Eq. (3)
suggests that the variational problem is well-posed.999Analysis of affine
analogues to the Gibbons–Hawking–York term can be found in Refs. Parattu  _et
al._ (2016); Krishnan  _et al._ (2017); Krishnan and Raju (2017); Lehner  _et
al._ (2016); Hopfmüller and Freidel (2017); Jubb  _et al._ (2017).

Under the assumption that no Gibbons–Hawking–York term is needed, and since
the action contains up to first derivatives of the fields, the field equations
are obtained through the Euler-Lagrange equations,

$\displaystyle\partial_{\mu}\left(\frac{\partial\mathscr{L}}{\partial\left(\partial_{\mu}{\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}\right)}\right)-\frac{\partial\mathscr{L}}{\partial{\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}}$
$\displaystyle=0,$ (4)
$\displaystyle\partial_{\mu}\left(\frac{\partial\mathscr{L}}{\partial\left(\partial_{\mu}{\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}\right)}\right)-\frac{\partial\mathscr{L}}{\partial{\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}}$
$\displaystyle=0,$
$\displaystyle\partial_{\mu}\left(\frac{\partial\mathscr{L}}{\partial\left(\partial_{\mu}{{\mathcal{A}}_{\nu}}\right)}\right)-\frac{\partial\mathscr{L}}{\partial{{\mathcal{A}}_{\nu}}}$
$\displaystyle=0.$

A simple way of dealing with the field equations was introduced by Kijowski in
Ref. Kijowski (1978), and we will reviewed in what follows.101010Our notation
is inspired by that of the cited work, but it might differ in both symbols and
signs. We advise to be careful if you would like to compare the results.

In the following, we present an alternative form of writing the Euler–Lagrange
equations presented above, in a way that the calculations are easier to
follow. Nonetheless, the explicit calculations are given in Appendix B.

####  II.1.1 Field equations for the symmetric connection

The canonically conjugated momenta of the connection are defined by

${{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}=\frac{\partial\mathscr{L}}{\partial\,(\partial_{\mu}\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}})}\equiv\frac{\partial\mathscr{L}}{\partial\,\Gamma_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}},$
(5)

and since the derivative of the connection appears only in the curvature
tensor, it follows that

${{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}=\frac{\partial\mathscr{L}}{\partial\mathcal{R}_{{\alpha\beta}}{}^{{\gamma}}{}_{{\delta}}}\frac{\partial\mathcal{R}_{{\alpha\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial\Gamma_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}}\equiv{{z}_{\Gamma}{}^{\alpha\beta}{}_{\gamma}{}^{\delta}}\frac{\partial\mathcal{R}_{{\alpha\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial\Gamma_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}}.$
(6)

The last term in Eq. (6) can be calculated explicitly from the definition of
the curvature tensor,

$\frac{\partial\mathcal{R}_{{\alpha\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial\Gamma_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}}=4\,\delta^{\gamma}_{\lambda}\delta^{\mu}_{[\alpha}\delta^{(\nu}_{\beta]}\delta^{\rho)}_{\delta},$
(7)

from which it follows that

${{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}=2{{z}_{\Gamma}{}^{[\mu\nu]}{}_{\lambda}{}^{\rho}}+2{{z}_{\Gamma}{}^{[\mu\rho]}{}_{\lambda}{}^{\nu}}.$
(8)

The last equation implies that the canonical momenta satisfy the
Jacobi–Bianchi identity,

${{\Pi}_{\Gamma}{}^{[\mu\nu}{}_{\lambda}{}^{\rho]}}=0.$ (9)

On the other hand, the second term in the Euler–Lagrange equations for
$\Gamma$ yields

$\frac{\partial\mathscr{L}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}=\frac{\partial\mathscr{L}}{\partial\mathcal{R}_{{\alpha\beta}}{}^{{\gamma}}{}_{{\delta}}}\frac{\partial\mathcal{R}_{{\alpha\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}.$
(10)

Once again the last term can be calculated from the definition of curvature

$\frac{\partial\mathcal{R}_{{\alpha\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}=4\Big{[}\delta^{\gamma}_{\lambda}\delta^{(\nu}_{[\alpha}\Gamma_{{\beta]}}{}^{{\rho)}}{}_{{\delta}}+\delta^{\rho}_{\delta}\delta^{(\nu}_{[\beta}\Gamma_{{\alpha]}}{}^{{\gamma)}}{}_{{\lambda}}\Big{]},$
(11)

and then,

$\displaystyle\frac{\partial\mathscr{L}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}$
$\displaystyle=2\,\Big{[}{{z}_{\Gamma}{}^{[\nu\beta]}{}_{\lambda}{}^{\delta}}\Gamma_{{\beta}}{}^{{\rho}}{}_{{\delta}}+{{z}_{\Gamma}{}^{[\rho\beta]}{}_{\lambda}{}^{\delta}}\Gamma_{{\beta}}{}^{{\nu}}{}_{{\delta}}$
$\displaystyle\quad+{{z}_{\Gamma}{}^{[\beta\nu]}{}_{\gamma}{}^{\rho}}\Gamma_{{\beta}}{}^{{\gamma}}{}_{{\lambda}}+{{z}_{\Gamma}{}^{[\beta\rho]}{}_{\gamma}{}^{\nu}}\Gamma_{{\beta}}{}^{{\gamma}}{}_{{\lambda}}\Big{]}$
$\displaystyle=\frac{1}{2}{{\Pi}_{\Gamma}{}^{\nu\beta}{}_{\lambda}{}^{\delta}}\Gamma_{{\beta}}{}^{{\rho}}{}_{{\delta}}+\frac{1}{2}{{\Pi}_{\Gamma}{}^{\rho\beta}{}_{\lambda}{}^{\delta}}\Gamma_{{\beta}}{}^{{\nu}}{}_{{\delta}}+{{\Pi}_{\Gamma}{}^{\beta\nu}{}_{\gamma}{}^{\rho}}\Gamma_{{\beta}}{}^{{\gamma}}{}_{{\lambda}}$
$\displaystyle=-{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\delta}}\Gamma_{{\mu}}{}^{{\rho}}{}_{{\delta}}-{{\Pi}_{\Gamma}{}^{\mu\rho}{}_{\lambda}{}^{\delta}}\Gamma_{{\mu}}{}^{{\nu}}{}_{{\delta}}+{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\gamma}{}^{\rho}}\Gamma_{{\mu}}{}^{{\gamma}}{}_{{\lambda}}.$
(12)

Therefore, the field equations for the symmetric part of the connection are

$\nabla_{\mu}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}=\frac{\partial^{*}\mathscr{L}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}.$
(13)

In this last equation, we have used the fact that the canonical momenta are
densities, thus there are two term (which seems to be missing above) that
cancel themselves.111111The covariant derivative of this tensor density is
given by the expression,
$\displaystyle\nabla_{\sigma}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$
$\displaystyle=\partial_{\sigma}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}+\Gamma_{{\sigma}}{}^{{\mu}}{}_{{\tau}}{{\Pi}_{\Gamma}{}^{\tau\nu}{}_{\lambda}{}^{\rho}}+\Gamma_{{\sigma}}{}^{{\nu}}{}_{{\tau}}{{\Pi}_{\Gamma}{}^{\mu\tau}{}_{\lambda}{}^{\rho}}$
$\displaystyle-\Gamma_{{\sigma}}{}^{{\tau}}{}_{{\lambda}}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\tau}{}^{\rho}}+\Gamma_{{\sigma}}{}^{{\rho}}{}_{{\tau}}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\tau}}-\Gamma_{{\sigma}}{}^{{\tau}}{}_{{\tau}}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}},$
then, when one contract the $\sigma$ and $\mu$ indices, the second and sixth
terms in the right-hand side cancel each other. The asterisk on the right-hand
side of Eq. (13) denotes the partial derivative with respect to the connection
that is not contained in the curvature tensor.

Notice that there are seven term in which the symmetric part of the connection
enters through the curvature tensor, while it enters through the covariant
derivative of the tensors $\mathcal{B}$ and ${\mathcal{A}}$ in eleven terms.
Noteworthily, there are only two terms in which the symmetric part of the
connection enters in both ways, these are the terms in the action with
coupling constants $C_{1}$ and $C_{2}$. Furthermore, the terms in the action
with couplings $C_{1}$ and $C_{2}$ are the only which are linear in either
$\mathcal{B}$ and ${\mathcal{A}}$ fields. Hence, these are the terms which
could possibly contribute to the field equations in the sector of vanishing
torsion, i.e., ${\mathcal{A}}\to 0$ and $\mathcal{B}\to 0$.

####  II.1.2 Field equations for the $\mathcal{B}$ field

Using the relations

$\frac{\partial\nabla_{\alpha}\mathcal{B}_{{\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial(\partial_{\mu}\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}})}=2\delta^{\mu}_{\alpha}\delta^{\gamma}_{\lambda}\delta^{[\nu}_{\beta}\delta^{\rho]}_{\delta}$
(14)

and

$\frac{\partial\nabla_{\alpha}\mathcal{B}_{{\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}=-2\Gamma_{{\alpha}}{}^{{[\nu}}{}_{{\beta}}\delta^{\rho]}_{\delta}\delta^{\gamma}_{\lambda}-2\Gamma_{{\alpha}}{}^{{[\rho}}{}_{{\delta}}\delta^{\nu]}_{\beta}\delta^{\gamma}_{\lambda}+2\Gamma_{{\alpha}}{}^{{\gamma}}{}_{{\lambda}}\delta^{[\nu}_{\beta}\delta^{\rho]}_{\delta}$
(15)

it is straightforward to show that the equations of motion for the
$\mathcal{B}$ field are

$\nabla_{\mu}{{\Pi}_{\mathcal{B}}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}=\frac{\partial\mathscr{L}}{\partial\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}.$
(16)

####  II.1.3 Field equations for the ${\mathcal{A}}$ field

The field equations for the ${\mathcal{A}}$ field are simpler to calculate,
but a rigorous approach as in the previous sections, yields the field
equations,

$\nabla_{\mu}{{\Pi}_{{\mathcal{A}}}{}^{\mu\nu}}=\frac{\partial\mathscr{L}}{\partial{\mathcal{A}}_{\nu}}.$
(17)

###  II.2 The torsion-less limit

An important result, obtained in Ref. Castillo-Felisola and Skirzewski (2018),
is that within the torsion-less sector of the conection, the field equations
admits all vacuum solutions of Einstein’s gravity, as solutions of the
polynomial affine gravity.

The sector of vanishing torsion is equivalent to the limit ${\mathcal{A}}\to
0$ and $\mathcal{B}\to 0$. Clearly, such limit cannot be taken at the level of
the action, but at the equation of motions. In the Appendix B, the explicit
field equations are shown from Eq. (79) to Eq. (123), and it can be checked
that the mentioned limit is well-defined. In the torsion-free sector, the only
nontrivial field equations are Eqs. (99) and (100), i.e.

$\nabla_{\mu}\left(\mathcal{R}_{{\sigma\alpha}}{}^{{\sigma}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\mu\nu\rho\alpha}+C\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\sigma}}\delta^{\mu}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}\right)=0,$
(18)

where $C$ is the ratio of the original parameters of the model,
$C=\tfrac{C_{2}}{C_{1}}$. The second term in Eq. (18) is proportional to the
trace of the curvature 2-form, which vanishes in General Relativity. Also, the
volume form is not necessarily compatible with the connection. However, if one
restricts oneself to connections that preserve a volume form, such connections
are dubbed _equiaffine_ connections Nomizu and Sasaki (1994), the trace of the
curvature 2-form is ensured and the field equations simplify further to

$\nabla_{[\mu}\mathcal{R}_{{\nu]\lambda}}=0,$ (19)

which can be written as

$\nabla_{\rho}\mathcal{R}_{{\mu\nu}}{}^{{\rho}}{}_{{\lambda}}=0,$ (20)

after using the second Bianchi identity. Equations (19) and (20) are part of a
set of well-known generalisations of Einstein’s field equations (see for
example Chap. 16 of Ref. Besse (2007)).

Particularly, Eq. (20) can be obtained as the field equation for the
connection of a gravitational Yang–Mills theory,

$S_{\textsc{sky}}=\int{\mathrm{d}}^{4}\\!{x}\;\sqrt{g}\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}\mathcal{R}{}^{{\mu\nu\rho}}{}_{{\lambda}}.$
(21)

The above model is known as Stephenson–Kilmister–Yang (or SKY for short)
Stephenson (1958); Kilmister and Newman (1961); Yang (1974), but its structure
requires the inclusion of the metric in order to build the action. Therefore,
besides Eq. (20) there is a field equation for the metric, and it spoils
desirable features of the Stephenson–Kilmister–Yang model Pavelle (1975);
Thompson (1975).121212Notice for example that in terms of the connection, Eq.
(19) is a set of second order differential equations, while if we would
interprete them as equations for the metric, become a set of third order
differential equations.

The solutions to Eq. (19) are classified in three categories: (i) Ricci-flat
connections, $R_{\mu\nu}=0$; (ii) connections with parallel Ricci,
$\nabla_{\lambda}R_{\mu\nu}=0$; and (iii) connections with harmonic curvature,
$\nabla_{\rho}\mathcal{R}_{{\mu\nu}}{}^{{\rho}}{}_{{\lambda}}=0$.
Interestingly, among the possible solutions of Eq. (19) one encounters the
vacuum solutions to the Einstein field equations, dubbed Einstein manifolds. A
key difference in Polynomial Affine Gravity is that unlike General Relativity
the cosmological constant appears as an integration constant. The same feature
occurs in other generalisations of General Relativity, e.g. unimodular gravity
Einstein (1919).

##  III Cosmological solutions on the torsion-free sector

In order to solve the equations (19) one proceeds—just as in General
Relativity—by giving an ansatz compatible with the symmetries of the problem.
Using the Lie derivative, we have found the most general torsion-free
connection compatible with the cosmological principle Castillo-Felisola
(2018). Since we shall restrict ourselves to the torsion-free sector, the
nonvanishing coefficients of the connection are

$\displaystyle\Gamma_{{t}}{}^{{t}}{}_{{t}}$ $\displaystyle=f(t),$
$\displaystyle\Gamma_{{i}}{}^{{t}}{}_{{j}}$
$\displaystyle=g\left(t\right)S_{ij},$ (22)
$\displaystyle\Gamma_{{t}}{}^{{i}}{}_{{j}}$
$\displaystyle=h\left(t\right)\delta^{i}_{j},$
$\displaystyle\Gamma_{{i}}{}^{{j}}{}_{{k}}$
$\displaystyle=\gamma_{i}{}^{j}{}_{k},$

where $f$, $g$ and $h$ are functions of time, while $S_{ij}$ and
$\gamma_{i}{}^{j}{}_{k}$ are the three-dimensional rank two symmetric tensor
and connection compatible with isotropy and homogeneity, defined by

$\displaystyle S_{ij}=\begin{pmatrix}\frac{1}{1-\kappa r^{2}}&0&0\\\
0&r^{2}&0\\\ 0&0&r^{2}\sin^{2}\theta\end{pmatrix},$ (23)

and

$\displaystyle\gamma_{r}{}^{r}{}_{r}$ $\displaystyle=\frac{\kappa r}{1-\kappa
r^{2}},$ $\displaystyle\gamma_{\theta}{}^{r}{}_{\theta}$ $\displaystyle=\kappa
r^{3}-r,$ (24) $\displaystyle\gamma_{\varphi}{}^{r}{}_{\varphi}$
$\displaystyle=\left(\kappa r^{3}-r\right)\sin^{2}\theta,$
$\displaystyle\gamma_{r}{}^{\theta}{}_{\theta}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\gamma_{\varphi}{}^{\theta}{}_{\varphi}$
$\displaystyle=-\cos\theta\sin\theta,$
$\displaystyle\gamma_{r}{}^{\varphi}{}_{\varphi}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\gamma_{\theta}{}^{\varphi}{}_{\varphi}$
$\displaystyle=\frac{\cos\theta}{\sin\theta}.$

With the connection above, one can calculate the curvature,

$\displaystyle\mathcal{R}_{{ti}}{}^{{t}}{}_{{j}}=-\mathcal{R}_{{it}}{}^{{t}}{}_{{j}}$
$\displaystyle=\left(\dot{g}+(f-h)g\right)S_{ij}$ (25)
$\displaystyle\mathcal{R}_{{mt}}{}^{{n}}{}_{{t}}=-\mathcal{R}_{{tm}}{}^{{n}}{}_{{t}}$
$\displaystyle=-\delta_{m}^{n}\left(\dot{h}+h^{2}-fh\right)$
$\displaystyle\mathcal{R}_{{mi}}{}^{{n}}{}_{{j}}=-\mathcal{R}_{{im}}{}^{{n}}{}_{{j}}$
$\displaystyle=\left(gh-\kappa\right)\delta_{m}^{n}S_{ij},$

and the Ricci tensor,

$\displaystyle\mathcal{R}_{{tt}}$
$\displaystyle=-3\left(\dot{h}+h^{2}-fh\right)$ (26)
$\displaystyle\mathcal{R}_{{ij}}$
$\displaystyle=\left(\dot{g}+(f+h)g+2\kappa\right)S_{ij}.$

The covariant derivative of the Ricci yields

$\displaystyle\nabla_{t}\mathcal{R}_{{tt}}$
$\displaystyle=\partial_{t}\mathcal{R}_{{tt}}-2\Gamma_{{t}}{}^{{t}}{}_{{t}}\mathcal{R}_{{tt}}=-3\left[\ddot{h}+2h\dot{h}-3f\dot{h}-\dot{f}h-2fh(f-h)\right],$
(27) $\displaystyle\nabla_{t}\mathcal{R}_{{ij}}$
$\displaystyle=\partial\mathcal{R}_{{ij}}-2\Gamma_{{t}}{}^{{k}}{}_{{i}}\mathcal{R}_{{kj}}=\left[\ddot{g}+\dot{f}g+\dot{h}g+(f-h)\dot{g}-2(f+h)gh-4\kappa
h\right]S_{ij},$ $\displaystyle\nabla_{i}\mathcal{R}_{{tj}}$
$\displaystyle=-\Gamma_{{i}}{}^{{k}}{}_{{t}}\mathcal{R}_{{kj}}-\Gamma_{{i}}{}^{{t}}{}_{{j}}\mathcal{R}_{{tt}}=\left[3g\dot{h}-\dot{g}h-4fg+2gh^{2}-2\kappa
h\right]S_{ij}.$

Finally, the harmonic curvature expression has a single independent component,

$\displaystyle\nabla_{[t}\mathcal{R}_{{i]j}}$
$\displaystyle=\partial_{t}\mathcal{R}_{{ij}}-\Gamma_{{t}}{}^{{k}}{}_{{i}}\mathcal{R}_{{kj}}+\Gamma_{{i}}{}^{{t}}{}_{{j}}\mathcal{R}_{{tt}}$
(28)
$\displaystyle=\left(\ddot{g}+\dot{f}g+f\dot{g}-2g\dot{h}+2fgh-4gh^{2}-2\kappa
h\right)S_{ij}.$

In the remain of this section we will solve the Eq. (19). Firstly, notice that
the Levi-Cività connection from Friedman–Robertson–Walker models is obtained
from Eq. (22) by setting $f=0$, $g=a\dot{a}$ and $h=\frac{\dot{a}}{a}$,
implying that all (vacuum) cosmological models in General Relativity are in
the space of solutions of Polynomial Affine Gravity. Moreover, it was shown in
Ref. Castillo-Felisola  _et al._ (2019), that within this space of solutions,
suitable deviations from the vacuum Friedman–Robertson–Walker connection mimic
the behaviour of matter content, even though Eq. (19) aim to describe
geometric properties of the manifold exclusively. Secondly, our classification
of solutions—into Ricci-flat, parallel Ricci and harmonic curvature—is
hierarchic, in the sense that once a condition is satisfied, the remaining are
satisfied as well. Therefore, in order to find a proper solution of the
parallel Ricci equations, we have to ensure that the connection is not Ricci-
flat; and in order to find a proper solution of the harmonic curvature
equations, neither the Ricci-flat or parallel Ricci condition should be
satisfied.

###  III.1 Cosmological solutions with vanishing Ricci

A first kind of solutions can be found by solving the system of equations
determined by vanishing Ricci. From Eq. (26) the differential equations to
solve are

$\displaystyle\dot{h}-(f-h)h$ $\displaystyle=0,$ (29)
$\displaystyle\dot{g}+(f+h)g+2\kappa$ $\displaystyle=0.$ (30)

Since $f$ is not a dynamical function, the system can be solved in terms of
$f$ (see Ref. Castillo-Felisola  _et al._ (2019)),

$\displaystyle h(t)$
$\displaystyle=\frac{\exp\left(F(t)\right)}{C_{h}+\int\mathop{\mathrm{d}t}\nolimits\,\exp(F)},$
(31) $\displaystyle g(t)$
$\displaystyle=\exp(-\Sigma(t))\left(C_{g}-2\kappa\left(\int\mathop{\mathrm{d}t}\nolimits\exp(\Sigma(t))\right)\right),$
(32)

where $F=\int\mathop{\mathrm{d}t}\nolimits\,f$ and
$\Sigma(t)=\int\mathop{\mathrm{d}t}\nolimits\,\left(f(t)+h(t)\right)$ are
integrals of the defining functions, while $C_{h}$ and $C_{g}$ are integration
constants.

####  III.1.1 Friedman–Robertson–Walker-like models

In particular, Friedman–Robertson–Walker-like models are obtained by setting
$f=0$, and besides the trivial solution—$h=g=\kappa=0$—, yielding a parametric
family of nontrivial solutions,

$\displaystyle g(t)$
$\displaystyle=\frac{1}{t+C_{h}}\left(C_{g}-\kappa(t+C_{h})^{2}\right),$
$\displaystyle h(t)$ $\displaystyle=\frac{1}{t+C_{h}}.$ (33)

Unlike in General Relativity—whose sole Ricci-flat cosmological solution is a
flat manifold—, the above solution is not flat in general, since its curvature
tensor has nonvanishing components

$\displaystyle\mathcal{R}_{{ti}}{}^{{t}}{}_{{j}}=-\mathcal{R}_{{it}}{}^{{t}}{}_{{j}}$
$\displaystyle=-2C_{g}h^{2}S_{ij},$ (34)
$\displaystyle\mathcal{R}_{{mi}}{}^{{n}}{}_{{j}}=-\mathcal{R}_{{im}}{}^{{n}}{}_{{j}}$
$\displaystyle=C_{g}h^{2}S_{ij}\delta_{m}^{n}.$

####  III.1.2 Case $h=f$

In these particular subspace, the solutions to Eqs. (29) and (30) are given by

$\displaystyle f$ $\displaystyle=C_{h}$ $\displaystyle h$
$\displaystyle=C_{h}$ $\displaystyle g$
$\displaystyle=C_{g}\exp(-2C_{h}t)-\frac{\kappa}{C_{h}}.$ (35)

####  III.1.3 Case $h=-f$

Equations (29) and (30) are solved by

$\displaystyle f$ $\displaystyle=-\frac{1}{2t+C_{h}}$ $\displaystyle h$
$\displaystyle=\frac{1}{2t+C_{h}}$ $\displaystyle g$
$\displaystyle=C_{g}-2\kappa t.$ (36)

####  III.1.4 Case $h=0$ and a given $f$

In this case, equation (29) becomes an identity, and Eq. (30) is solved by

$g(t)=\exp(-F(t))\left(C_{g}-2\kappa\left(\int\mathop{\mathrm{d}t}\nolimits\exp(F(t))\right)\right).$
(37)

####  III.1.5 Case $g=0$ and a given $f$

In this case, equation (30) requires $\kappa=0$, and $h$ can still be solved
for a given function $f$ as

$h(t)=\frac{\exp\left(F(t)\right)}{C_{h}+\int\mathop{\mathrm{d}t}\nolimits\,\exp(F)}.$
(38)

###  III.2 Cosmological solutions with parallel Ricci

In this section we solve the Eqs. (27), under the condition that the Ricci
tensor is nonzero, i.e. $\nabla_{\lambda}\mathcal{R}_{{\mu\nu}}=0$ but
$\mathcal{R}_{{\mu\nu}}\neq 0$. The strategy to solve these equations is to
propose an ansatz for the Ricci tensor, and solve for $f$, $g$ and $h$
accordingly. Nonetheless, a broaden ansatz is useless, thus we focus in the
two simpler cases.

####  III.2.1 Parallel time-independent Ricci

A time-independent Ricci has the form,

$\displaystyle\mathcal{R}_{{tt}}$ $\displaystyle=R_{1},$
$\displaystyle\mathcal{R}_{{ij}}$ $\displaystyle=R_{2}S_{ij},$ (39)

with constant $R_{1}$ and $R_{2}$. For this proposal of the Ricci tensor the
Eqs. (27) yield the constraints

$\displaystyle\nabla_{t}\mathcal{R}_{{tt}}$ $\displaystyle=0$
$\displaystyle\Rightarrow$ $\displaystyle\,f=0\lor R_{1}=0,$ (40)
$\displaystyle\nabla_{t}\mathcal{R}_{{ij}}$ $\displaystyle=0$
$\displaystyle\Rightarrow$ $\displaystyle\,h=0\lor R_{2}=0,$ (41)
$\displaystyle\nabla_{i}\mathcal{R}_{{tj}}$ $\displaystyle=0$
$\displaystyle\Rightarrow$ $\displaystyle\,h=0\lor(g=0\land\kappa=0).$ (42)

Notice that $h=0$ implies $R_{1}=0$, and $g=0\land\kappa=0$ implies $R_{2}=0$.
Therefore, there is no solution of the Eqs. (27) for a time-independent
nondegenerated Ricci.

Given the above conditions, there are solely two solutions with degenerated
nonvanishing Ricci. Firstly, for vanishing $h$, one have

$\displaystyle h$ $\displaystyle=0,$ (43) $\displaystyle f$
$\displaystyle=\text{undetermined function},$ $\displaystyle g$
$\displaystyle=\exp(-F)\left(C_{g}+(R_{2}-2\kappa)\int\mathop{\mathrm{d}t}\nolimits\,\exp(F)\right).$

Similarly, for $g=0\land\kappa=0\land f=0$, the solution is given by

$\displaystyle h$
$\displaystyle=\sqrt{\frac{R_{1}}{3}}\tanh\left(\sqrt{\frac{R_{1}}{3}}(t-t_{*})\right),$
(44) $\displaystyle f$ $\displaystyle=0,$ $\displaystyle g$ $\displaystyle=0.$

####  III.2.2 Parallel Ricci with a “scale factor”

We now consider an ansatz for the Ricci tensor with the form of a
Friedman–Robertson–Walker metric, given that this has the required symmetries,
i.e.,

$\displaystyle\mathcal{R}_{{tt}}$ $\displaystyle=-R_{1},$
$\displaystyle\mathcal{R}_{{ij}}$ $\displaystyle=A(t)S_{ij}.$ (45)

With the above ansatz, the parallel Ricci equations require,

$\displaystyle\nabla_{t}\mathcal{R}_{{tt}}$ $\displaystyle=2fR_{1}$
$\displaystyle\Rightarrow\,f=0\lor R_{1}=0,$ (46)
$\displaystyle\nabla_{t}\mathcal{R}_{{ij}}$
$\displaystyle=S_{ij}(\dot{A}-2hA)$
$\displaystyle\Rightarrow\,A=C_{A}\exp(2H),$
$\displaystyle\nabla_{i}\mathcal{R}_{{tj}}$ $\displaystyle=-S_{ij}(hA-R_{1}g)$
$\displaystyle\Rightarrow\,g=\tfrac{hA}{R_{1}};\,(R_{1}\\!\neq\\!0).$

Unlike the previous case, Eqs. (46), accept a nondegenerated solution given by

$\displaystyle f$ $\displaystyle=0,$ (47) $\displaystyle h$
$\displaystyle=\sqrt{\frac{R_{1}}{3}}\tanh\left(\sqrt{\frac{R_{1}}{3}(t-t_{*})}\right),$
$\displaystyle A$
$\displaystyle=C_{A}\cosh^{2}\left(\sqrt{\frac{R_{1}}{3}(t-t_{*})}\right),$
$\displaystyle g$
$\displaystyle=\frac{1}{\sqrt{12R_{1}}}\sinh\left(\sqrt{\frac{4R_{1}}{3}(t-t_{*})}\right).$

Notice that degenerated solutions, with $R_{1}=0$, require either vanishing
$C_{A}$ or a constant $A$ for $h=0$, which are part of previously considered
cases.

###  III.3 Cosmological solutions with harmonic curvature

####  III.3.1 Harmonic curvature with time-independent Ricci

For a time-independent Ricci tensor, see Eq. (39), the harmonic curvature
condition, Eq. (28), becomes

$\nabla_{[t}\mathcal{R}_{{i]j}}=-(gR_{1}+hR_{2})S_{ij},$ (48)

i.e. the harmonic curvature requires

$g=-\frac{R_{2}}{R_{1}}h.$ (49)

Therefore, the consistency equations for the Ricci are rewritten, after using
Eq. (49), as

$\displaystyle\dot{h}+h^{2}$ $\displaystyle=\beta_{1},$
$\displaystyle\beta_{1}$ $\displaystyle=\frac{R_{1}}{3R_{2}}(3\kappa-R_{2}),$
(50) $\displaystyle f$ $\displaystyle=\frac{\beta_{2}}{h},$
$\displaystyle\beta_{2}$ $\displaystyle=\frac{R_{1}}{3R_{2}}(3\kappa-2R_{2}).$
(51)

Notice that the general solution is parameterised by the function $h$, and for
vanishing $\beta_{2}$ the function $f$ is zero, however this solution is
degenerated for vanishing $\kappa$.

We assume that the product of the constants $R_{1}$ and $R_{2}$ is positive,
i.e. $R_{1}R_{2}>0$. Then, the solution are

$\displaystyle h$
$\displaystyle=\begin{cases}\omega\tanh\left(\omega(t-t_{*})\right)&\phantom{-}\omega^{2}=\beta_{1}>0,\\\
-\omega\tan\left(\omega(t-t_{*})\right)&-\omega^{2}=\beta_{1}<0,\\\
\frac{1}{t-C_{h}}&\phantom{-\omega^{2}=\,}\beta_{1}=0,\end{cases}$ (52)
$\displaystyle g$ $\displaystyle=-\frac{R_{2}}{R_{1}}h,$ $\displaystyle f$
$\displaystyle=\frac{\beta_{2}}{h}.$

####  III.3.2 Harmonic curvature from Ricci with a “scale factor”

Finally, the harmonic curvature condition for a Ricci with a _scale factor_ ,
obtained from (46), is

$\nabla_{[t}\mathcal{R}_{{i]j}}=(\dot{A}-gR_{1}-hA)S_{ij},$ (53)

whose solution for $g$ is

$g=\frac{\dot{A}-hA}{R_{1}}.$ (54)

Substituting Eq. (54) into the consistency equations for the Ricci, we obtain

$\displaystyle\dot{h}+h^{2}-fh$ $\displaystyle=\frac{R_{1}}{3},$ (55)
$\displaystyle\ddot{A}+f\dot{A}-\left(2fh+\frac{4R_{1}}{3}\right)A$
$\displaystyle=-2\kappa R_{1}.$ (56)

Equation (55) can be solved for a constant function $f=C_{f}$, therefore, we
shall restrict ourselves to that case. In this particular case, Eq. (55) can
be rewritten as

$\displaystyle\dot{h}+\left(h-\frac{C_{f}}{2}\right)^{2}$
$\displaystyle=\beta_{3}$ $\displaystyle\beta_{3}$
$\displaystyle=\frac{1}{12}(3C_{f}^{2}+4R_{1}),$ (57)

and its solutions are

$h=\begin{cases}\omega\tanh(\omega(t-t_{*}))+\frac{C_{f}}{2}&;\phantom{-}\omega^{2}=\beta_{3}>0,\\\
-\omega\tan(\omega(t-t_{*}))+\frac{C_{f}}{2}&;-\omega^{2}=\beta_{3}<0,\\\
\frac{1}{t-t_{*}}+\frac{C_{f}}{2}&;\phantom{-\omega^{2}=}\,\beta_{3}=0,\\\
\pm\omega+\frac{C_{f}}{2}&;\phantom{-}\omega^{2}=\beta_{3}>0.\end{cases}$ (58)

Worth to highlight, the last case is a constant solution for $h$.

For the simplest solutions of $h$—a constant function—the second consistency
condition, Eq. (56), can be integrated. Define the constant

$\alpha=2C_{f}C_{h}+\frac{4R_{1}}{3},$

where $h=C_{h}=\pm\omega+C_{f}/2$ is the constant determined from Eq. (58)
above. Therefore, the exact solutions for the _scale factor_ is

$A=\mathbb{A}_{1}\exp\Big{(}(2C_{h}-C_{f})t\Big{)}+\mathbb{A}_{2}\exp\Big{(}-2C_{h}t\Big{)}+\frac{2\kappa
R_{1}}{\alpha}.$ (59)

Additionally, second simplest solution to the Eq. (56) is obtained for
$\beta_{3}=0$, in whose case the $R_{1}$ constant is determined by the value
of $C_{f}$, and the function $h=\frac{1}{t}+\frac{C_{f}}{2}$. The solution for
$A$ is

$A=t(C_{f}t+2)\Big{[}\mathbb{A}_{1}+\mathbb{A}_{2}C_{f}\Gamma(0,C_{f}t)\Big{]}-\mathbb{A}_{2}e^{-C_{f}t}(C_{f}t+1)+\frac{3}{4}\kappa
C_{f}^{2}t^{2},$ (60)

with $\Gamma(0,C_{f}t)$ an incomplete gamma function defined by the expression

$\Gamma(0,C_{f}t)=\int_{1}^{\infty}\mathop{\mathrm{d}x}\nolimits\,\frac{e^{-xC_{f}t}}{x}.$

The _scale factor_ can be obtained for the other choices of $h$ when one sets
the $C_{f}=0$. In those cases, the solutions to Eq. (56) is

$A=\mathbb{A}_{1}\exp\left(\sqrt{\tfrac{4R_{1}}{3}}t\right)+\mathbb{A}_{2}\exp\left(-\sqrt{\tfrac{4R_{1}}{3}}t\right)+\frac{3\kappa}{2},$
(61)

where the behaviour of the exponential functions is managed by the sign of the
constant $R_{1}$.

##  IV The affine self-parallel curves

In General Relativity, given that the gravitational connection is the one of
Levi-Cività, the concepts of geodesic and self-parallel curve are equivalent,
however, for generic connections these concepts differ. Geodesics play an
important role in General Relativity, since they represent the trajectories
followed by free falling particles.

Although it is not possible to define geodesics in (purely) affine models of
gravity, we shall postulate that trajectories of free falling particles are
described by self-parallel curves

$\ddot{x}^{\mu}+\Gamma_{\nu}{}^{\mu}{}_{\lambda}\dot{x}^{\nu}\dot{x}^{\lambda}=0,$
(62)

where now $\Gamma$ is a generic connection, and the _dot_ represents—unlike in
the previous sections—derivation with respect to the affine parameter of the
curve, $\tau$.

With the coefficients of the cosmological affine connection, Eq. (22), the
self-parallel curves are given by

$\displaystyle\ddot{t}+f\dot{t}^{2}+gS_{ij}\dot{x}^{i}\dot{x}^{j}$
$\displaystyle=0,$
$\displaystyle\ddot{x}^{i}+2h\dot{t}\dot{x}^{i}+\gamma_{j}{}^{i}{}_{k}\dot{x}^{j}\dot{x}^{k}$
$\displaystyle=0.$

The term that mixes the time with the spacial coordinates can be eliminated by
a redefinition of the affine parameter as

$\frac{\mathop{\mathrm{d}}\nolimits}{\mathop{\mathrm{d}l}\nolimits}=C_{T}e^{2H}\frac{\mathop{\mathrm{d}}\nolimits}{\mathop{\mathrm{d}\tau}\nolimits}=C_{T}e^{2\int\mathop{\mathrm{d}t}\nolimits
h}\frac{\mathop{\mathrm{d}}\nolimits}{\mathop{\mathrm{d}\tau}\nolimits},$

with $C_{T}$ an arbitrary constant. In terms of the new parameter (were now
the dot derivative is with respect to the parameter $l$) the self-parallel
equations become

$\displaystyle\ddot{t}+(f-2h)\dot{t}^{2}+gS_{ij}\dot{x}^{i}\dot{x}^{j}$
$\displaystyle=0,$ (63)
$\displaystyle\ddot{x}^{i}+\gamma_{j}{}^{i}{}_{k}\dot{x}^{j}\dot{x}^{k}$
$\displaystyle=0.$

The three-dimensional restriction of Eqs. (63) are the same as the spacial
geodesic equations for a Friedman–Robertson–Walker model in General Relativity
Garfinkle  _et al._ (2018).

The above shows that there exists a parametrisation of the self-parallel
curves from Polynomial Affine Gravity, in which the restriction to the spatial
coordinates coincides with the geodesic of the Friedman–Robertson–Walker
models on General Relativity.

For the sake of completeness, we remind to the readers that due to the
isotropy, one can set $\theta=\tfrac{\pi}{2}$, and thus the spatial part of
Eq. (63) are

$\displaystyle\ddot{r}+\frac{kr}{1-kr^{2}}\dot{r}^{2}+\left(kr^{3}-r\right)\dot{\varphi}^{2}$
$\displaystyle=0$ (64)
$\displaystyle\ddot{\varphi}+2\frac{\dot{r}\dot{\varphi}}{r}$
$\displaystyle=0.$

Hence, in terms of the new affine parameter, the geometry of the
$r(\varphi)$-curves is determined by

$r(\varphi)=\begin{cases}\sin(l)&\text{for }\kappa=1\\\ l&\text{for
}\kappa=0\\\ \sinh(l)&\text{for }\kappa=-1\end{cases},$ (65)

for $\dot{\varphi}=0$, i.e. for radial self-parallels, and

$r(\varphi)=\frac{1}{\sqrt{\kappa+B^{2}\cos^{2}(\varphi+\beta)}},$ (66)

for $\dot{\varphi}\neq 0$.

Interestingly, in this context, although the equation that describes the
geometry of the orbits is the same, in Polynomial Affine Gravity there is no
way to differentiate among the orbits of massive and massless particles, since
the absence of a metric precludes the classification of vectors into time-
like, space-like or null-like.

##  V The Ricci tensor as a metric

From the last section, we understood that—generically—the geometry of the
self-parallel curves (or geodesics) is solely determined by the notion of
parallelism, however, their relation with physical notions (as trajectories,
equivalence principle and principles of relativity) requires the existence of
a metric. Nonetheless, our proposal stands on the idea that no metric is
required to formulate the model.

A coruscating fact is that, unlike in General Relativity, a metric field is
not necessarily a fundamental geometric object. In what follows, we highlight
that under certain conditions the Ricci tensor could play the role of a
_emergent_ (or derived) metric. Therefore, we shall consider the affine
connection as the fundamental field of Polynomial Affine Gravity, while the
(non-degenerated) symmetric part of the Ricci tensor serves as metric. Let us
first remind the formal definition of a metric.

######  Definition V.1 (Metric).

Let $M$ be an $m$-dimensional differential manifold. A pseudo-Riemannian
metric on $M$ is a $\binom{0}{2}$-tensor field $g$ on $M$ satisfying:

  1. 1.

$g$ is a symmetric tensor field, and

  2. 2.

$g$ is nondegenerated, i.e. $\forall\,X\in TM$ the quantity $g(X,Y)=0$ if and
only if $TM\ni Y=0$.

A pair $(M,g)$ is called a pseudo-Riemannian manifold.

Notice that the second condition in Def. V.1 ensures that as a map, $g:TM\to
T^{*}M$, the metric field has trivial kernel and thus it is invertible.

The metric is the fundamental field in General Relativity, and all other
geometrical properties of the manifold—connection, curvature, etc.—are derived
from it,

$g\to\Gamma\to\operatorname{Riem}\to\operatorname{Ric}\to\mathcal{R}.$ (67)

A peculiar, and particularly important type of manifolds are the Einstein
spaces, since the chain of derived quantities closes, see Fig. 1.

$g$$\Gamma$$\operatorname{Riem}$$\operatorname{Ric}$ Figure 1: Closed chain of
derived quantities for (Riemannian) Einstein manifolds.

The closure condition for Einstein manifolds is expressed as

$\mathcal{R}_{{\mu\nu}}=\Lambda g_{\mu\nu}.$ (68)

It is worth mentioning that in the vanishing Ricci case, despite the Eq. (68)
is satisfied, one cannot say that the chain really closes, since

$g\to\Gamma\to\operatorname{Riem}\to\operatorname{Ric}=0.$

In our model, we start with an affinely connected manifold $(M,\Gamma)$, were
the ansatz of the affine connection is determined by the symmetries of the
problem.131313A noticeable difference with the approach in General Relativity
is that (in principle) a choice of the connection does not determine the
signature of the manifold. Without the use of a metric, the chain of derided
products stops at the Ricci tensor,

$\Gamma\to\operatorname{Riem}\to\operatorname{Ric}.$ (69)

Nevertheless, the above chain could close if the Ricci tensor—as in the case
of Einstein manifolds—satisfies the conditions of a metric, i.e. conditions in
Def. V.1. Assuming such, the chain of derived quantities closes, see Fig. 2,
and therefore it is possible to define an affine analogue of Einstein
manifolds.

$\Gamma$$\operatorname{Riem}$$\operatorname{Ric}$ Figure 2: Closed chain of
derived quantities for affine Einstein manifolds.

Noticeable, along the developing of this work we restricted ourselves to
_locally equiaffine_ connections, which ensure the symmetry of the Ricci
tensor (see Proposition 3.1 in Ref. (Nomizu and Sasaki, 1994)). This is
precisely the requirement to fulfil the first condition in Def. V.1. The
second condition the Ricci tensor should satisfy in order to be a good metric
field is not to posses a zero eigenvalue.

Given the classification of solutions of the field equations (19), one
concludes that: (i) as in General Relativity, the relation between the Ricci
tensor and the metric interpretation is lost for Ricci-flat connections; (ii)
in the case of parallel Ricci—if nondegenerated—it is a _good_ metric field,
which additionally satisfies the “metricity” condition; (iii) in the case of
proper solutions of the harmonic curvature, the Ricci tensor accept
interpretation of a metric, but it is not compatible with the connection, i.e.
does not satisfy the metricity condition.

Since we started from an affinely connected space, $(M,\Gamma)$, and a derived
geometric object, the Ricci tensor, introduces a metric structure, we call it
an _emergent_ metric.

It can be shown with ease, given a locally equiaffine connection whose Ricci
tensor is parallel and nondegenerated, that

$\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=\frac{1}{2}\mathcal{R}{}^{{\lambda\sigma}}\left(\partial_{\mu}\mathcal{R}_{{\sigma\nu}}+\partial_{\nu}\mathcal{R}_{{\mu\sigma}}-\partial_{\sigma}\mathcal{R}_{{\mu\nu}}\right).$
(70)

Consequently, in the analysed case the geometry is naturally Riemannian, and
we can relate the Ricci tensor to a _canonical_ covariantly constant
$\binom{0}{2}$-tensor, $g$, by

$\mathcal{R}_{{\mu\nu}}=\Lambda g_{\mu\nu},\text{ with
}\Lambda\in\mathbb{R}^{*},$ (71)

i.e. the space is an Einstein manifold. An additional comment is that given
the relation in Eq. (71) it follows that

$\Gamma(\mathcal{R})=\Gamma(g).$

The fact that the Ricci tensor—or its symmetric part—could play the role of a
metric was (somehow) anticipated by Schrödinger Schrödinger (1950).

##  VI Cosmological quantities

In the previous section we argue that even when in principle the space does
not posses a metric structure, under certain conditions a metric structure
emerges through the symmetric component of the Ricci tensor.

A consequence of the emergence of a metric is that one can now distinguish
between null-like and time-like self-parallel curves, which could be
interpreted as the trajectories of free-falling particles—as mentioned in Sec.
IV. In particular, it is possible to define a null-like self-parallel curve by
the equations

$\displaystyle\ddot{x}^{\mu}+\Gamma_{{\lambda}}{}^{{\mu}}{}_{{\rho}}\dot{x}^{\lambda}\dot{x}^{\rho}$
$\displaystyle=0,$ (72)
$\displaystyle\mathcal{R}_{{\mu\nu}}\dot{x}^{\mu}\dot{x}^{\nu}$
$\displaystyle=0.$

The special case of a light ray coming in the radial direction yields

$\mathop{\mathrm{d}t}\nolimits=\pm\sqrt{\frac{\mathcal{R}_{{rr}}}{\mathcal{R}_{{tt}}}}\frac{\mathop{\mathrm{d}r}\nolimits}{\sqrt{1-\kappa
r^{2}}},$ (73)

which by renaming

$a(t)=\sqrt{\frac{\mathcal{R}_{{rr}}}{\mathcal{R}_{{tt}}}},$ (74)

is the same equation that serves to define the cosmological redshift, i.e.

$1+z=\frac{a(t_{0})}{a(t_{1})}=\sqrt{\frac{\mathcal{R}_{{rr}}(t_{0})\mathcal{R}_{{tt}}(t_{1})}{\mathcal{R}_{{tt}}(t_{0})\mathcal{R}_{{rr}}(t_{1})}}.$
(75)

Not surprisingly, after the emergence of the metric, the physical properties
of the space are determined by a single function, i.e. the scale factor. If we
focus on the nondegenerated Ricci tensors found in Sec. III, the standard
scale factor is defined as

$a(t)=\sqrt{A(t)},$

where $A(t)$ is the function defining the Ricci tensor in Eq. (45).

##  VII Conclusions and remarks

In this paper we have extended the set of known solutions to the parallel
Ricci and harmonic curvature equations, when these are defined in terms of a
connection instead of a metric.141414For solutions of the mentioned equations
in terms of the metric, we refer the readers to Chapter 16 of Ref. Besse
(2007). In addition, a detailed exposition of the SKY model of gravity can be
found in Chap. 7 of Ref. Mielke (2017). We also analysed the equation of self-
parallel curves and noticed that, without the aid of a metric, it is not
possible to distinguishing between the (yet hypothetical) trajectories
followed by massive or massless free-falling particles. Then, we show that the
Ricci derived from the connection would, under certain conditions, be a good
metric. In these cases, it is possible to surpass the mentioned limitation,
and distinguish trajectories followed by massive and massless free-falling
particles. Moreover, the emergent metric allows us to make contact with the
standard cosmological quantities such as the redshift, scale factor, Hubble
and des-acceleration parameters, etc.

We first would like to highlight the fact that a very important step toward
the simplification of the Polynomial Affine model of Gravity was due to the
change of the field decomposition of the connection, Eq. (2), which alleviate
both the geometric interpretation of the irreducible components of the
connection and the process of finding the complete field equations (See
Appendix B). An advantage of having the complete field equations at hand, is
that readers can convince themselves that the torsion-free sector is a
consistent truncation of the model.

Despite the field equations of Polynomial Affine Gravity are well-defined in
the torsion-free sector, the limit ${\mathcal{A}}\to 0$ and $\mathcal{B}\to 0$
is meaningless at the action level because all the terms are at least linear
in either ${\mathcal{A}}$ or $\mathcal{B}$. Hence, the Feynman rules for the
model lack vertices with only _gravitons_.151515Here we call gravitons to the
spin-2 field within the symmetric part of the connection,
$\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$. This feature suggests that the
Polynomial Affine model of Gravity could bypass the no-go theorems found in
Refs. McGady and Rodina (2014); Camanho  _et al._ (2016). Furthermore, the
effective action from which one can derive the field equations (19) contains
vertices with three and four gravitons.

In the formulation of the Polynomial Affine model of Gravity there is not
something such as a cosmological constant. Nevertheless, the integration
constant included in the process of solving Eq. (19), plays the role of
cosmological constant. This changes the paradigm on the _cosmological problem_
, similarly as in Unimodular Gravity models. The most relevant feature of
Unimodular Gravity is that vacuum fluctuations of the energy-momentum tensor
do not gravitate Weinberg (1989), removing the discrepancy between the
observed and estimated values of the vacuum energy Ng and van Dam (1991);
Smolin (2009); Ellis  _et al._ (2011).161616Very recently, it has been pointed
that from an affine point of view, the nature of the cosmological constant is
related with the volume preserving property (instead of being related to the
sectional curvature) Boskoff and Capozziello (2019).

Although cosmological solutions, in the torsion free sector of the Polynomial
Affine model of Gravity, were found in Ref. Castillo-Felisola  _et al._
(2019), in this work we were able of developing further arguments that allows
us to obtain explicit solutions in cases that were previously unexplored.

The field equations of General Relativity—without cosmological constant—in
vacuum are equivalent to Ricci-flat manifolds. In general these solutions are
curved spacetimes (i.e. Schwarzschild space), however, once one ask for
cosmological solutions the field equations require the manifold to be flat. In
the analysis of the solutions of Polynomial Affine Gravity field equations, we
notices that they accept Ricci-flat cosmological solutions which are not flat.
Let us work out the interpretation of this situation.

Remember first that a connection, $\nabla$, on a vector bundle $\pi:E\to M$
assigns to each vector field, $X$, a map $\nabla_{X}$ from the space of
sections $C^{\infty}(E)$ to itself. Therefore, for a given direction, the
connection represents an endomorphism on the space of sections, i.e.
$C^{\infty}(\operatorname{End}(E))$. Similarly, the curvature of a linear
connection, $\mathcal{R}^{\nabla}$, on a vector bundle $\pi:E\to M$ is a
2-form on $M$ with values in $C^{\infty}(\operatorname{End}(E))$.171717See
Refs. Besse (2007); Ivey and Landsberg (2003); Baez (1994). In the
gravitational case, the vector bundle is the tangent bundle, $TM$, and in
particular when one considers pseudo-Riemannian geometries the group structure
of $\operatorname{End}(TM)$ is a subgroup of the orthogonal group,
$O(p,q)$.181818We are using the standard notation that $n=p+q$ is the
dimension of the pseudo-Riemannian manifold, $p$ the number of space-like
coordinates and $q$ the number of time-like dimensions.

Now, from Eqs. (25) and (26), it follows that the Ricci-flat condition allows
the components
$\mathcal{R}_{{ti}}{}^{{t}}{}_{{j}}=-\mathcal{R}_{{it}}{}^{{t}}{}_{{j}}$ and
$\mathcal{R}_{{mi}}{}^{{n}}{}_{{j}}=-\mathcal{R}_{{im}}{}^{{n}}{}_{{j}}$ of
the curvature not to vanish. Thus, the group structure underling the
endomorphisms of the tangent bundle is the homogeneous Carroll group Lévy-
Leblond (1965).191919We thank Dr. Zurab Silagadze for help us with the
identification of this group. The homogeneous Carroll groups can be obtained
from the Lorentz group through the Inönü–Wigner contraction in the limit $c\to
0$.

Notice that the nonvanishing components of curvature tensor for the Ricci-flat
manifolds obtained in Sec. III.1 are

$\displaystyle\mathcal{R}_{{ti}}{}^{{t}}{}_{{j}}=-\mathcal{R}_{{it}}{}^{{t}}{}_{{j}}$
$\displaystyle=-2S_{ij}\big{(}gh+\kappa\big{)},$ (76)
$\displaystyle\mathcal{R}_{{mi}}{}^{{n}}{}_{{j}}=-\mathcal{R}_{{im}}{}^{{n}}{}_{{j}}$
$\displaystyle=S_{ij}\,\delta_{m}^{n}\,\big{(}gh+\kappa\big{)}.$

Thus, for example the solution in Eq. (33) is flat—irrespective of the value
of $\kappa$—if and only if $C_{g}$ vanishes.

The family of solutions in Sec. III.1.2 has vanishing curvature for $C_{g}=0$,
while the family from Sec. III.1.3 requires $C_{g}=-\kappa C_{h}$. Solutions
in Sec. III.1.4 and III.1.5 are flat solely for $\kappa=0$.

The second class of solutions of the field equations are those connections
with parallel Ricci tensor, $\nabla_{\lambda}\mathcal{R}_{{\mu\nu}}=0$. In
order to solve the field equations we required the Ricci tensor to be
compatible with the cosmological principle, i.e. to preserve the isotropic and
homogeneity symmetries. We analyse the time-independent and
Friedman–Robertson–Walker-like cases.

Noticeable, there are no connections with parallel, nondegenerated, time-
independent Ricci tensor. However, we found solutions with degenerated Ricci
[see Eqs. (43) and (44)].

There are connections with parallel, nondegenerated Friedman–Robertson–Walker-
like Ricci tensor. In this cases the Ricci represents an emergent metric, and
thus the underling structure of the manifold is Riemannian. From Eq. (47) one
notices that depending on the sign of the constant $R_{1}$ the manifold is a
sphere- or hyperbolic-like space.202020If one demand the signature to be
Lorentzian, the solutions are de Sitter and Anti de Sitter. However, as
mentioned previously the signature of the space is not fixed from the affine
structure. It can be checked that the solution in Eq. (47) satisfies—as the
standard Friedman–Robertson–Walker solutions—:

$\displaystyle f$ $\displaystyle=0,$ $\displaystyle a$
$\displaystyle=\sqrt{A},$ $\displaystyle g$ $\displaystyle=a\dot{a},$
$\displaystyle h$ $\displaystyle=\frac{\dot{a}}{a}.$

Such result, as expected from the discussion in Sec. V is not very interesting
from the view point of the Polynomial Affine model of Gravity.

The third class of solutions to the field equations, i.e. connections with
harmonic curvature, does not accept degenerated solutions for the time-
independent Ricci case, since either $f$ or $g$ diverge when $R_{1}\lor R_{2}$
vanish. The solutions shown in Eq. (52) assume that $R_{1}R_{2}>0$, which
ensures, since the Ricci tensor endows the manifold with a metric structure, a
Lorentzian signature of the metric. Worth mentioning that it is possible to
set $\beta_{2}=0$ which implies that $f$ vanishes (similar to the behaviour of
standard Friedman–Robertson–Walker models).

Even more interesting are the solutions of the harmonic curvature equations,
whose Ricci contains a scale factor. We found solution with nondegenerated
Ricci tensor, which again endows the manifold with a metric structure.
Although this metric structure has a similar form than the expected from the
Friedman–Robertson–Walker models, the cosmological scale factor, $a=\sqrt{A}$,
admits richer behaviour in the cosmological evolution. Of course, one has to
remind that that these are (so far) vacuum solutions in Polynomial Affine
Gravity.

Notice for example that for constant $f=C_{f}$ and $h=C_{h}$, Eq. (56) becomes
the equation of motion of a damped harmonic oscillator on which a constant
force is exerted. This case is the simplest case of a cosmological model in
Polynomial Affine Gravity with an emergent metric, with no equivalent in
General Relativity. These kind of solutions generalise the metric solutions to
the field equations $\nabla_{[\mu}\mathcal{R}_{{\nu]\lambda}}=0$ reported in
Ref. Castillo-Felisola  _et al._ (2019). The behaviour of the $A$ function for
certain set of values of the parameters
$(\mathbb{A}_{1},\mathbb{A}_{2},\kappa,C_{h},C_{f})$ is shown in Fig. 3.

Figure 3:  Behaviour of the function $A(t)$ for certain values of the
parameters $(\mathbb{A}_{1},\mathbb{A}_{2},\kappa,C_{h},C_{f})$

We were able to solve the field equations for the harmonic curvature in the
case $\beta_{3}=0$, which introduces interesting models of cosmologies through
the appearance of the incomplete gamma function within the scale factor [see
Eq. (60)]. In Fig. 4 the behaviour of the scale-like factor $A(t)$ is shown
for a set of values of the parameters
$(\mathbb{A}_{1},\mathbb{A}_{2},\kappa,C_{f})$. Notice the complex behaviour
for the particular case with dashed lines.

Figure 4:  Behaviour of the function $A(t)$ in Eq. (60), for certain values of
the parameters $(\mathbb{A}_{1},\mathbb{A}_{2},\kappa,C_{f})$

It is noticeable that connections with harmonic curvature could provide an
emergent metric (i.e. Ricci tensor) corresponding to the standard metric on a
Eucliean or Minkowskian space, even though their curvature do not vanish.

Further studies, which take the nature of the dark sector into account, will
need to be undertaken. Interesting proposals—in models other than the
Polynomial Affine model of Gravity—could be found in Ref. Josset  _et al._
(2017); Brensinger  _et al._ (2019); Cervantes-Cota and Liebscher (2016);
Barrow  _et al._ (2019); Kilic (2019); Roberts (1995); Chothe  _et al._
(2019); Boskoff and Capozziello (2019).

###### Acknowledgements.

We would like to thanks the following people for their support and helpful
discussions: Pietro Fré, Aureliano Skirzewski, Cristóbal Corral, Claudio Dib,
Iván Schmidt, Fernando Izaurrieta, Andrés Anabalón and Julio Oliva. We
gratefully acknowledge to Stefano Vignolo, without whose help this work would
never have been possible. We are specially thankful to the developers of the
software SageMath Stein  _et al._ (2019) and SageManifolds Gourgoulhon  _et
al._ (2019, 2015); Gourgoulhon and Mancini (2018) which we have used
extensively in our calculations. The “Centro Científico y Tecnológico de
Valparaíso” (CCTVal) is funded by the Chilean Government through the Centers
of Excellence Base Financing Program of Comisión Nacional de Investigación
Científica y Tecnológica (CONICYT), by grant number FS0821. The work of JP is
funded by USM-PIIC grant number 059/20128. AZ is funded by FONDECYT project
1160423. This research benefited from a grant from the Universidad Técnica
Federico Santa María `PI_LI_19_02`.

##  Appendix A Building the simplified polynomial affine action

Our goal is to build the most general action using the irreducible fields,
$\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$,
$\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$ and ${\mathcal{A}}_{\mu}$, where
the dynamics is given by the covariant derivative with respect to the
symmetric part of the connection, i.e., $\nabla=\nabla^{\hat{\Gamma}}$. Now,
the strategy to write down the action is to define the most general scalar
density. For that sake, inspired by the procedure described in Ref. Castillo-
Felisola and Skirzewski (2018), we introduced a two operators, $\mathscr{N}$
and $\mathscr{W}$, that count the number of free indices and the weight
density—respectively—of a given term.

The action of the operators $\mathscr{N}$and $\mathscr{W}$on the irreducible
components of the connection,

$\displaystyle\mathscr{N}({\mathcal{A}})$ $\displaystyle=-1,$
$\displaystyle\mathscr{N}(\mathcal{B})$ $\displaystyle=-1,$ (77)
$\displaystyle\mathscr{N}(\nabla)$ $\displaystyle=-1,$
$\displaystyle\mathscr{N}(\mathop{\mathrm{d}V}\nolimits)$ $\displaystyle=4,$
$\displaystyle\mathscr{W}({\mathcal{A}})$ $\displaystyle=0,$
$\displaystyle\mathscr{W}(\mathcal{B})$ $\displaystyle=0,$
$\displaystyle\mathscr{W}(\nabla)$ $\displaystyle=0,$
$\displaystyle\mathscr{W}(\mathop{\mathrm{d}V}\nolimits)$ $\displaystyle=1.$

As example of how the dimensional analysis works, we consider a general term
of the form
$\mathcal{O}={\mathcal{A}}^{m}\mathcal{B}^{n}\nabla^{p}\mathop{\mathrm{d}V}\nolimits^{q}$,
the action of the $\mathscr{N}$and $\mathscr{W}$operators on the term yield
the equations

$\displaystyle\mathscr{N}(\mathcal{O})$ $\displaystyle=4q-m-n-p,$ (78)
$\displaystyle\mathscr{W}(\mathcal{O})$ $\displaystyle=q.$

We are interested in the Lagrangian, i.e. a scalar density. Equations (78)
require $q=1$ and $m+n+p=4$. The terms contributing to this construction are
shown in Table 1.

Table 1:  Possible terms contributing to the action of Polynomial Affine Gravity $m$ |  $n$ |  $p$ |  Terms
---|---|---|---
4 |  0 |  0 |  ${\mathcal{A}}{\mathcal{A}}{\mathcal{A}}{\mathcal{A}}$
3 |  1 |  0 |  ${\mathcal{A}}{\mathcal{A}}{\mathcal{A}}\mathcal{B}$
3 |  0 |  1 |  ${\mathcal{A}}{\mathcal{A}}{\mathcal{A}}\nabla$
2 |  2 |  0 |  ${\mathcal{A}}{\mathcal{A}}\mathcal{B}\mathcal{B}$
2 |  1 |  1 |  ${\mathcal{A}}{\mathcal{A}}\mathcal{B}\nabla$
2 |  0 |  2 |  ${\mathcal{A}}{\mathcal{A}}\nabla\nabla$
1 |  3 |  0 |  ${\mathcal{A}}\mathcal{B}\mathcal{B}\mathcal{B}$
1 |  2 |  1 |  ${\mathcal{A}}\mathcal{B}\mathcal{B}\nabla$
1 |  1 |  2 |  ${\mathcal{A}}\mathcal{B}\nabla\nabla$
1 |  0 |  3 |  ${\mathcal{A}}\nabla\nabla\nabla$
0 |  4 |  0 |  $\mathcal{B}\mathcal{B}\mathcal{B}\mathcal{B}$
0 |  3 |  1 |  $\mathcal{B}\mathcal{B}\mathcal{B}\nabla$
0 |  2 |  2 |  $\mathcal{B}\mathcal{B}\nabla\nabla$
0 |  1 |  3 |  $\mathcal{B}\nabla\nabla\nabla$
0 |  0 |  4 |  $\nabla\nabla\nabla\nabla$

From Table 1 one can straightforwardly read terms that vanish, e.g. the term
with four ${\mathcal{A}}$ does not contribute to the action since its
contraction with the volume element is identically zero. Whenever two
covariant derivatives are contracted with the volume form they give a
curvature tensor, and since the curvature is defined for the symmetric
component of the connection, such curvature satisfy the torsion-free Bianchi
identities, which relate some of the several possible contractions of indices.
An additional argument that helps to drop contraction of indices is that
$\mathcal{B}$ is traceless. Finally, the terms contributing to the action come
from

$\displaystyle{\mathcal{A}}{\mathcal{A}}\mathcal{B}\mathcal{B}$
$\displaystyle\rightarrow F_{4}$
$\displaystyle{\mathcal{A}}{\mathcal{A}}\mathcal{B}\nabla$
$\displaystyle\rightarrow D_{6},D_{7}$
$\displaystyle{\mathcal{A}}\mathcal{B}\mathcal{B}\mathcal{B}$
$\displaystyle\rightarrow F_{3}$
$\displaystyle{\mathcal{A}}\mathcal{B}\mathcal{B}\nabla$
$\displaystyle\rightarrow D_{4},D_{5}$
$\displaystyle{\mathcal{A}}\mathcal{B}\nabla\nabla$ $\displaystyle\rightarrow
B_{3},B_{4},B_{5},E_{2}$
$\displaystyle\mathcal{B}\mathcal{B}\mathcal{B}\mathcal{B}$
$\displaystyle\rightarrow F_{1},F_{2}$
$\displaystyle\mathcal{B}\mathcal{B}\mathcal{B}\nabla$
$\displaystyle\rightarrow D_{1},D_{2},D_{3}$
$\displaystyle\mathcal{B}\mathcal{B}\nabla\nabla$ $\displaystyle\rightarrow
B_{1},B_{2},E_{1}$ $\displaystyle\mathcal{B}\nabla\nabla\nabla$
$\displaystyle\rightarrow C_{1},C_{2}.$

##  Appendix B Explicit calculation of the field equations

In order to obtain the field equations we proceed as proposed in Ref. Kijowski
(1978),212121Notice that the original method was proposed by Tulczyjew in
Refs. Tulczyjew (1974, 1975a, 1975b) and developed further by Kijowski and
Tulczyjew in Ref. Kijowski and Tulczyjew (1979). where an auxiliary field is
introduced to ease the process. Additionally, we calculate the contribution of
each term in the action to the field equations separately, since this allows
us to obtain manageable expressions, and to check explicitly the possible
_consistent truncations_ of the model.

###  B.1 Field equations for $\Gamma_{\mu}{}^{\lambda}{}_{\nu}$

We shown in Sec. II.1.1 that the field equations for the symmetric part of the
connection are

$\nabla_{\mu}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}=\frac{\partial^{*}\mathscr{L}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}.$

The asterisk on the right-hand side denotes the partial derivative with
respect to the connection that is not contained in the curvature tensor.

#### Calculation of the ${{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$

We start calculating the ${{z}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$ and
then use the Eq. (8), to obtain

$\displaystyle B_{1}:$
$\displaystyle\Big{[}2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\alpha}}{}^{{\nu]}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}+2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\alpha}}{}^{{\rho]}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\Big{]}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
$\displaystyle B_{2}:$ $\displaystyle
2\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathcal{B}_{{\sigma}}{}^{{(\rho}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}$
$\displaystyle B_{3}:$ $\displaystyle
2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\beta}}{}^{{\nu]}}{}_{{\gamma}}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\rho\beta\gamma\delta}+2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\beta}}{}^{{\rho]}}{}_{{\gamma}}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\nu\beta\gamma\delta}$
$\displaystyle B_{4}:$
$\displaystyle-2\mathcal{B}_{{\gamma}}{}^{{(\rho}}{}_{{\delta}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}$
$\displaystyle B_{5}:$
$\displaystyle-2\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}{\mathcal{A}}_{\sigma}\delta^{(\rho}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}$
$\displaystyle C_{1}:$ $\displaystyle
2\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}\delta^{[\mu}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu]\beta\gamma\delta}+2\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\delta^{[\mu}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\rho]\beta\gamma\delta}$
$\displaystyle C_{2}:$
$\displaystyle-2\nabla_{\sigma}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\delta^{(\rho}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}$

#### Calculation of
$\frac{\partial^{*}\mathscr{L}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}$

In Eq. (3), the connection appears explicitly (non in the curvature tensor) in
the covariant derivative. However, the covariant derivative contains different
terms depending on the field it is acting on. The two different terms in which
the connection appears are

$\nabla_{\alpha}\mathcal{B}_{{\beta}}{}^{{\gamma}}{}_{{\delta}}=\partial_{\alpha}\mathcal{B}_{{\beta}}{}^{{\gamma}}{}_{{\delta}}+\Gamma_{{\alpha}}{}^{{\lambda}}{}_{{\beta}}\mathcal{B}_{{\lambda}}{}^{{\gamma}}{}_{{\delta}}+\Gamma_{{\alpha}}{}^{{\lambda}}{}_{{\delta}}\mathcal{B}_{{\beta}}{}^{{\gamma}}{}_{{\lambda}}-\Gamma_{{\alpha}}{}^{{\gamma}}{}_{{\lambda}}\mathcal{B}_{{\beta}}{}^{{\lambda}}{}_{{\delta}}$

and

$\nabla_{\alpha}{\mathcal{A}}_{\beta}=\partial_{\alpha}{\mathcal{A}}_{\beta}-\Gamma_{{\alpha}}{}^{{\lambda}}{}_{{\beta}}{\mathcal{A}}_{\lambda}.$

Now their partial derivatives with respect to the connection are

$\frac{\partial\nabla_{\alpha}\mathcal{B}_{{\beta}}{}^{{\gamma}}{}_{{\delta}}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}=2\delta^{(\nu}_{\alpha}\delta^{\rho)}_{\beta}\mathcal{B}_{{\lambda}}{}^{{\gamma}}{}_{{\delta}}+2\delta^{(\nu}_{\alpha}\delta^{\rho)}_{\delta}\mathcal{B}_{{\beta}}{}^{{\gamma}}{}_{{\lambda}}-2\delta^{c}_{\lambda}\delta^{(\nu}_{\alpha}\mathcal{B}_{{\beta}}{}^{{\rho)}}{}_{{\delta}}$

and

$\frac{\nabla_{\alpha}{\mathcal{A}}_{\beta}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}=-2\delta^{(\nu}_{\alpha}\delta^{\rho)}_{\beta}{\mathcal{A}}_{\lambda}.$

The terms coming from these derivatives are

$\displaystyle C_{1}:$ $\displaystyle
2\mathcal{R}_{{\mu\lambda}}{}^{{\mu}}{}_{{\alpha}}\mathcal{B}_{{\gamma}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\gamma\delta}$
$\displaystyle C_{2}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\sigma}}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{(\nu}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\rho)\alpha\beta\delta}-\delta^{(\nu}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}$
$\displaystyle D_{1}:$ $\displaystyle
2\mathcal{B}_{{\tau}}{}^{{\sigma}}{}_{{\lambda}}\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\alpha}}\mathcal{B}_{{\gamma}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\gamma\delta}$
$\displaystyle D_{2}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathcal{B}_{{\sigma}}{}^{{(\nu}}{}_{{\tau}}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{|\rho)\alpha\beta\delta}-\delta^{\tau}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{|\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}$
$\displaystyle D_{3}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\tau}}\mathcal{B}_{{\beta}}{}^{{(\nu}}{}_{{\gamma}}\Big{[}\delta^{\rho)}_{\sigma}\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\delta}}+\delta^{\rho)}_{\delta}\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\lambda}}-\delta^{\tau}_{\lambda}\mathcal{B}_{{\sigma}}{}^{{|\rho)}}{}_{{\delta}}\Big{]}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
$\displaystyle D_{4}:$
$\displaystyle-2\mathcal{B}_{{\alpha}}{}^{{\nu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
$\displaystyle D_{5}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{(\nu}}{}_{{\beta}}{\mathcal{A}}_{\sigma}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{|\rho)\alpha\beta\delta}-\delta^{\sigma}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{|\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}$
$\displaystyle D_{6}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{(\nu}}{}_{{\beta}}{\mathcal{A}}_{\gamma}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\rho)\alpha\beta\gamma}$
$\displaystyle D_{7}:$ $\displaystyle 0$ $\displaystyle E_{1}:$ $\displaystyle
4\nabla_{\sigma}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\beta\delta}-\delta^{(\nu}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}$
$\displaystyle E_{2}:$ $\displaystyle
2{\mathcal{F}}_{\alpha\beta}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\beta\delta}-\delta^{(\nu}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}$

Notice that in the last set of contributions, the one coming from the term
accompanied by the coupling $D_{7}$ yields zero. This is because the
antisymmetrization of $\nabla{\mathcal{A}}$ is nothing but the field strength
of the potential ${\mathcal{A}}$ (or the curvature of an Abelian 1-form). The
strength does not depend on the symmetric connection.

### Complete contribution

$\displaystyle B_{1}:$
$\displaystyle\nabla_{\mu}\left(\Big{[}2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\alpha}}{}^{{\nu]}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}+2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\alpha}}{}^{{\rho]}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\Big{]}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\right)=0$
(79) $\displaystyle B_{2}:$
$\displaystyle\nabla_{\mu}\left(2\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathcal{B}_{{\sigma}}{}^{{(\rho}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}\right)=0$
(80) $\displaystyle B_{3}:$
$\displaystyle\nabla_{\mu}\left(2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\beta}}{}^{{\nu]}}{}_{{\gamma}}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\rho\beta\gamma\delta}+2\delta^{[\mu}_{\lambda}\mathcal{B}_{{\beta}}{}^{{\rho]}}{}_{{\gamma}}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\nu\beta\gamma\delta}\right)=0$
(81) $\displaystyle B_{4}:$
$\displaystyle\nabla_{\mu}\left(-2\mathcal{B}_{{\gamma}}{}^{{(\rho}}{}_{{\delta}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}\right)=0$
(82) $\displaystyle B_{5}:$
$\displaystyle\nabla_{\mu}\left(-2\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}{\mathcal{A}}_{\sigma}\delta^{(\rho}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}\right)=0$
(83) $\displaystyle C_{1}:$
$\displaystyle\nabla_{\mu}\left(2\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}\delta^{[\mu}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu]\beta\gamma\delta}+2\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\delta^{[\mu}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\rho]\beta\gamma\delta}\right)=2\mathcal{R}_{{\mu\lambda}}{}^{{\mu}}{}_{{\alpha}}\mathcal{B}_{{\gamma}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\gamma\delta}$
(84) $\displaystyle C_{2}:$
$\displaystyle\nabla_{\mu}\left(-2\nabla_{\sigma}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\delta^{(\rho}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu)\mu\gamma\delta}\right)=2\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\sigma}}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{(\nu}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\rho)\alpha\beta\delta}-\delta^{(\nu}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}$
(85) $\displaystyle D_{1}:$ $\displaystyle
2\mathcal{B}_{{\tau}}{}^{{\sigma}}{}_{{\lambda}}\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\alpha}}\mathcal{B}_{{\gamma}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\gamma\delta}=0$
(86) $\displaystyle D_{2}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathcal{B}_{{\sigma}}{}^{{(\nu}}{}_{{\tau}}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{|\rho)\alpha\beta\delta}-\delta^{\tau}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{|\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}=0$
(87) $\displaystyle D_{3}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\tau}}\mathcal{B}_{{\beta}}{}^{{(\nu}}{}_{{\gamma}}\Big{[}\delta^{\rho)}_{\sigma}\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\delta}}+\delta^{\rho)}_{\delta}\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\lambda}}-\delta^{\tau}_{\lambda}\mathcal{B}_{{\sigma}}{}^{{|\rho)}}{}_{{\delta}}\Big{]}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(88) $\displaystyle D_{4}:$
$\displaystyle-2\mathcal{B}_{{\alpha}}{}^{{\nu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\rho}}{}_{{\delta}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(89) $\displaystyle D_{5}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{(\nu}}{}_{{\beta}}{\mathcal{A}}_{\sigma}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{|\rho)\alpha\beta\delta}-\delta^{\sigma}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{|\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}=0$
(90) $\displaystyle D_{6}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{(\nu}}{}_{{\beta}}{\mathcal{A}}_{\gamma}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\rho)\alpha\beta\gamma}=0$
(91) $\displaystyle E_{1}:$ $\displaystyle
4\nabla_{\sigma}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\beta\delta}-\delta^{(\nu}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}=0$
(92) $\displaystyle E_{2}:$ $\displaystyle
2{\mathcal{F}}_{\alpha\beta}\Big{[}2\mathcal{B}_{{\lambda}}{}^{{(\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu)\alpha\beta\delta}-\delta^{(\nu}_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho)}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}=0$
(93)

###  B.2 Field equations for $B_{\mu}{}^{\lambda}{}_{\nu}$

In Sec. II.1.2 we show that the field equations for the $\mathcal{B}$ field
are

$\nabla_{\mu}{{\Pi}_{\mathcal{B}}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}=\frac{\partial\mathscr{L}}{\partial\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}.$

We now calculate explicitly the contribution of each term in the action to the
above field equation.

#### Calculation of ${{\Pi}_{\mathcal{B}}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$

The explicit calculation of
${{\Pi}_{\mathcal{B}}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$ yield

$\displaystyle C_{1}:$
$\displaystyle-2\mathcal{R}_{{\sigma\alpha}}{}^{{\sigma}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\mu\nu\rho\alpha}$
$\displaystyle C_{2}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\sigma}}\delta^{\mu}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle D_{1}:$
$\displaystyle-2\mathcal{B}_{{\sigma}}{}^{{\theta}}{}_{{\lambda}}\mathcal{B}_{{\theta}}{}^{{\sigma}}{}_{{\alpha}}\mathop{\mathrm{d}V}\nolimits^{\mu\nu\rho\alpha}$
$\displaystyle D_{2}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathcal{B}_{{\sigma}}{}^{{\mu}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle D_{3}:$
$\displaystyle-2\mathcal{B}_{{\alpha}}{}^{{\nu}}{}_{{\lambda}}\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}\mathop{\mathrm{d}V}\nolimits^{\rho\alpha\beta\gamma}$
$\displaystyle D_{5}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle E_{1}:$ $\displaystyle
4\delta^{\mu}_{\lambda}\nabla_{\sigma}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle E_{2}:$ $\displaystyle
2\delta^{\mu}_{\lambda}{\mathcal{F}}_{\alpha\beta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$

#### Calculation of
$\frac{\partial\mathscr{L}}{\partial\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}}$

The contributions to the right-hand side of the field equations for the
$\mathcal{B}$-field are,

$\displaystyle B_{1}:$ $\displaystyle
4\mathcal{R}_{{\mu(\sigma}}{}^{{\mu}}{}_{{\lambda)}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}$
$\displaystyle B_{2}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+2\mathcal{R}_{{\alpha\beta}}{}^{{[\nu}}{}_{{\lambda}}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
$\displaystyle B_{3}:$ $\displaystyle
2\mathcal{R}_{{\mu\lambda}}{}^{{\mu}}{}_{{\alpha}}{\mathcal{A}}_{\beta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle B_{4}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\lambda}}{\mathcal{A}}_{\sigma}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle B_{5}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\tau}}{}_{{\tau}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle D_{1}:$ $\displaystyle
2\mathcal{B}_{{\lambda}}{}^{{[\nu}}{}_{{\alpha}}\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}+2\mathcal{B}_{{\lambda}}{}^{{[\nu|}}{}_{{\sigma}}\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{|\rho]\beta\gamma\delta}$
$\displaystyle D_{2}:$ $\displaystyle
2\mathcal{B}_{{\lambda}}{}^{{\mu}}{}_{{\sigma}}\nabla_{\mu}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+2\mathcal{B}_{{\alpha}}{}^{{[\nu}}{}_{{\beta}}\nabla_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
$\displaystyle D_{3}:$ $\displaystyle
2\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}\nabla_{\mu}\mathcal{B}_{{\lambda}}{}^{{[\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu]\beta\gamma\delta}+2\mathcal{B}_{{\gamma}}{}^{{\mu}}{}_{{\sigma}}\nabla_{\lambda}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}$
$\displaystyle D_{4}:$ $\displaystyle
4\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\nabla_{(\lambda}{\mathcal{A}}_{\sigma)}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle D_{5}:$ $\displaystyle
2\nabla_{\lambda}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}{\mathcal{A}}_{\sigma}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle D_{6}:$ $\displaystyle
2{\mathcal{A}}_{\gamma}\nabla_{\lambda}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}$
$\displaystyle D_{7}:$ $\displaystyle
2{\mathcal{A}}_{\lambda}F_{\gamma\delta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}$
$\displaystyle F_{1}:$ $\displaystyle
4\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\tau}}\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+4\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{[\nu}}{}_{{\delta}}\mathcal{B}_{{\mu}}{}^{{\rho]}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
$\displaystyle F_{2}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\beta}}{}^{{\sigma}}{}_{{\tau}}\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\mu}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\lambda}}\mathcal{B}_{{\gamma}}{}^{{[\rho}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\nu]\alpha\beta\gamma}$
$\displaystyle-2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\lambda}}\mathcal{B}_{{\mu}}{}^{{[\rho}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\nu]\alpha\beta\gamma}+2\mathcal{B}_{{\alpha}}{}^{{[\nu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\sigma}}\mathcal{B}_{{\delta}}{}^{{\sigma}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
$\displaystyle F_{3}:$ $\displaystyle
2\mathcal{B}_{{\lambda}}{}^{{[\nu}}{}_{{\alpha}}\mathcal{B}_{{\beta}}{}^{{\rho]}}{}_{{\gamma}}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}+2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}{\mathcal{A}}_{\gamma}\mathcal{B}_{{\lambda}}{}^{{[\nu}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\rho]\alpha\beta\gamma}$
$\displaystyle\quad+2\mathcal{B}_{{\sigma}}{}^{{\mu}}{}_{{\lambda}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\alpha}}{\mathcal{A}}_{\beta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
$\displaystyle F_{4}:$ $\displaystyle
4\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}{\mathcal{A}}_{\mu}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$

### Complete contribution

$\displaystyle B_{1}:$ $\displaystyle
4\mathcal{R}_{{\mu(\sigma}}{}^{{\mu}}{}_{{\lambda)}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}=0$
(94) $\displaystyle B_{2}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+2\mathcal{R}_{{\alpha\beta}}{}^{{[\nu}}{}_{{\lambda}}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(95) $\displaystyle B_{3}:$ $\displaystyle
2\mathcal{R}_{{\mu\lambda}}{}^{{\mu}}{}_{{\alpha}}{\mathcal{A}}_{\beta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}=0$
(96) $\displaystyle B_{4}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\lambda}}{\mathcal{A}}_{\sigma}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}=0$
(97) $\displaystyle B_{5}:$ $\displaystyle
2\mathcal{R}_{{\alpha\beta}}{}^{{\tau}}{}_{{\tau}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}=0$
(98) $\displaystyle C_{1}:$
$\displaystyle\nabla_{\mu}\left(-2\mathcal{R}_{{\sigma\alpha}}{}^{{\sigma}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\mu\nu\rho\alpha}\right)=0$
(99) $\displaystyle C_{2}:$
$\displaystyle\nabla_{\mu}\left(2\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\sigma}}\delta^{\mu}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}\right)=0$
(100) $\displaystyle D_{1}:$
$\displaystyle\nabla_{\mu}\left(-2\mathcal{B}_{{\sigma}}{}^{{\theta}}{}_{{\lambda}}\mathcal{B}_{{\theta}}{}^{{\sigma}}{}_{{\alpha}}\mathop{\mathrm{d}V}\nolimits^{\mu\nu\rho\alpha}\right)=2\mathcal{B}_{{\lambda}}{}^{{[\nu}}{}_{{\alpha}}\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}+2\mathcal{B}_{{\lambda}}{}^{{[\nu|}}{}_{{\sigma}}\nabla_{\beta}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{|\rho]\beta\gamma\delta}$
(101) $\displaystyle D_{2}:$
$\displaystyle\nabla_{\mu}\left(2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathcal{B}_{{\sigma}}{}^{{\mu}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}\right)=2\mathcal{B}_{{\lambda}}{}^{{\mu}}{}_{{\sigma}}\nabla_{\mu}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+2\mathcal{B}_{{\alpha}}{}^{{[\nu}}{}_{{\beta}}\nabla_{\lambda}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}$
(102) $\displaystyle D_{3}:$
$\displaystyle\nabla_{\mu}\left(-2\mathcal{B}_{{\alpha}}{}^{{\nu}}{}_{{\lambda}}\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}\mathop{\mathrm{d}V}\nolimits^{\rho\alpha\beta\gamma}\right)=2\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}\nabla_{\mu}\mathcal{B}_{{\lambda}}{}^{{[\rho}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu]\beta\gamma\delta}+2\mathcal{B}_{{\gamma}}{}^{{\mu}}{}_{{\sigma}}\nabla_{\lambda}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}$
(103) $\displaystyle D_{4}:$ $\displaystyle
4\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\nabla_{(\lambda}{\mathcal{A}}_{\sigma)}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}=0$
(104) $\displaystyle D_{5}:$
$\displaystyle\nabla_{\mu}\left(2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}\right)=2\nabla_{\lambda}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}{\mathcal{A}}_{\sigma}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}$
(105) $\displaystyle D_{6}:$ $\displaystyle
2{\mathcal{A}}_{\gamma}\nabla_{\lambda}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}=0$
(106) $\displaystyle D_{7}:$ $\displaystyle
2{\mathcal{A}}_{\lambda}F_{\gamma\delta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\gamma\delta}=0$
(107) $\displaystyle E_{1}:$
$\displaystyle\nabla_{\mu}\left(4\delta^{\mu}_{\lambda}\nabla_{\sigma}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}\right)=0$
(108) $\displaystyle E_{2}:$
$\displaystyle\nabla_{\mu}\left(2\delta^{\mu}_{\lambda}{\mathcal{F}}_{\alpha\beta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}\right)=0$
(109) $\displaystyle F_{1}:$ $\displaystyle
4\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\tau}}\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+4\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{[\nu}}{}_{{\delta}}\mathcal{B}_{{\mu}}{}^{{\rho]}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(110) $\displaystyle F_{2}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\beta}}{}^{{\sigma}}{}_{{\tau}}\mathcal{B}_{{\lambda}}{}^{{\tau}}{}_{{\mu}}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}+2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\lambda}}\mathcal{B}_{{\gamma}}{}^{{[\rho}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\nu]\alpha\beta\gamma}$
$\displaystyle-2\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\lambda}}\mathcal{B}_{{\mu}}{}^{{[\rho}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\nu]\alpha\beta\gamma}+2\mathcal{B}_{{\alpha}}{}^{{[\nu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\rho]}}{}_{{\sigma}}\mathcal{B}_{{\delta}}{}^{{\sigma}}{}_{{\lambda}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(111) $\displaystyle F_{3}:$ $\displaystyle
2\mathcal{B}_{{\lambda}}{}^{{[\nu}}{}_{{\alpha}}\mathcal{B}_{{\beta}}{}^{{\rho]}}{}_{{\gamma}}{\mathcal{A}}_{\delta}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}+2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}{\mathcal{A}}_{\gamma}\mathcal{B}_{{\lambda}}{}^{{[\nu}}{}_{{\sigma}}\mathop{\mathrm{d}V}\nolimits^{\rho]\alpha\beta\gamma}+2\mathcal{B}_{{\sigma}}{}^{{\mu}}{}_{{\lambda}}\mathcal{B}_{{\mu}}{}^{{\sigma}}{}_{{\alpha}}{\mathcal{A}}_{\beta}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}=0$
(112) $\displaystyle F_{4}:$ $\displaystyle
4\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}{\mathcal{A}}_{\mu}{\mathcal{A}}_{\lambda}\mathop{\mathrm{d}V}\nolimits^{\nu\rho\alpha\beta}=0$
(113)

###  B.3 Field equations for ${\mathcal{A}}_{\mu}$

In this section, we show the _complete_ Euler–Lagrage equations for the
${\mathcal{A}}$ field.

$\displaystyle B_{3}:$
$\displaystyle\mathcal{R}_{{\sigma\tau}}{}^{{\sigma}}{}_{{\alpha}}\mathcal{B}_{{\beta}}{}^{{\tau}}{}_{{\gamma}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\nu}=0$
(114) $\displaystyle B_{4}:$
$\displaystyle\mathcal{R}_{{\alpha\beta}}{}^{{\nu}}{}_{{\sigma}}\mathcal{B}_{{\gamma}}{}^{{\sigma}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(115) $\displaystyle B_{5}:$
$\displaystyle\mathcal{R}_{{\alpha\beta}}{}^{{\sigma}}{}_{{\sigma}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(116) $\displaystyle D_{4}$
$\displaystyle\nabla_{\mu}\Big{[}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}\Big{]}=0$
(117) $\displaystyle D_{5}:$
$\displaystyle\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\nabla_{\sigma}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(118) $\displaystyle D_{6}:$
$\displaystyle\nabla_{\mu}\Big{[}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}{\mathcal{A}}_{\gamma}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\nu}\Big{]}+\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\nabla_{\mu}{\mathcal{A}}_{\gamma}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\nu}=0$
(119) $\displaystyle D_{7}:$
$\displaystyle\nabla_{\mu}\Big{[}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}{\mathcal{A}}_{\sigma}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\mu\nu}\Big{]}+\mathcal{B}_{{\alpha}}{}^{{\nu}}{}_{{\beta}}{\mathcal{F}}_{\gamma\delta}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(120) $\displaystyle E_{2}:$
$\displaystyle\nabla_{\mu}\Big{[}\nabla_{\sigma}\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\mu\nu}\Big{]}=0$
(121) $\displaystyle F_{3}:$
$\displaystyle\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\lambda}}\mathcal{B}_{{\tau}}{}^{{\sigma}}{}_{{\alpha}}\mathcal{B}_{{\beta}}{}^{{\lambda}}{}_{{\gamma}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\nu}=0$
(122) $\displaystyle F_{4}:$ $\displaystyle
2\mathcal{B}_{{\alpha}}{}^{{\sigma}}{}_{{\beta}}\mathcal{B}_{{\gamma}}{}^{{\nu}}{}_{{\delta}}{\mathcal{A}}_{\sigma}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma\delta}=0$
(123)

## References

  * Einstein (1915a) Albert Einstein, “Grundgedanken der allgemeinen relativitätstheorie und anwendung dieser theorie in der astronomie,” Preussische Akademie der Wissenschaften, Sitzungsberichte 315, 778–786 (1915a).
  * Einstein (1915b) Albert Einstein, “Zur allgemeinen relativitätstheorie,” Sitzungsber. preuss. Akad. Wiss. 1, 778 (1915b).
  * Einstein (1915c) Albert Einstein, “Erklarung der perihelionbewegung der merkur aus der allgemeinen relativitatstheorie,” Sitzungsber. preuss. Akad. Wiss. 47, 831 (1915c).
  * Einstein (1915d) Albert Einstein, “Die feldgleichungen der gravitation,” Sitzungsber. preuss. Akad. Wiss. 1, 844 (1915d).
  * Einstein (1916) Albert Einstein, “Die grundlage der allgemeinen relativitätstheorie,” Ann. Phys. 49, 284–339 (1916).
  * Will (2014) Clifford M. Will, “The Confrontation Between Gerenal Relativity and experiment,” Living Rev. Rel. 17, 4 (2014), arXiv:1403.7377 [gr-qc] .
  * Abbott  _et al._ (2016) B. P. Abbott _et al._ (Virgo, LIGO Scientific), “GW151226: Observation of Gravitational Waves From a 22-solar-Mass Binary Black Hole Coalescence,” Phys. Rev. Lett. 116, 241103 (2016), arXiv:1606.04855 [gr-qc] .
  * Abbott  _et al._ (2017) B. P. Abbott _et al._ , “Gravitational waves and gamma-rays from a binary neutron star merger: Gw170817 and grb170817a,” Astrophys. J. 848, L13 (2017).
  * DeWitt (1967a) Bryce S. DeWitt, “Quantum Theory of Gravity. 1. The Canonical Theory,” Phys. Rev. 160, 1113 (1967a).
  * DeWitt (1967b) Bryce S. DeWitt, “Quantum Theory of Gravity. 2. The Manifestly Covariant Theory,” Phys. Rev. 162, 1195 (1967b).
  * Deser and van Nieuwenhuizen (1974a) Stanley Deser and P. van Nieuwenhuizen, “One Loop Divergences of Quantized Einstein-Maxwell Fields,” Phys. Rev. D 10, 401 (1974a).
  * Deser and van Nieuwenhuizen (1974b) Stanley Deser and P. van Nieuwenhuizen, “Nonrenormalizability of the Quantized Dirac-Einstein System,” Phys. Rev. D 10, 411 (1974b).
  * ’t Hooft and Veltman (1974) Gerard ’t Hooft and M. J. G. Veltman, “One Loop Divergencies in the Theory of gravitation,” Annales Poincare Phys. Theor. A 20, 69 (1974).
  * Ashtekar (1986) A. Ashtekar, “New Variables for Classical and Quantum Gravity,” Phys. Rev. Lett. 57, 2244–2247 (1986).
  * Ashtekar (1987) A. Ashtekar, “New Hamiltonian Formulation of General Relativity,” Phys. Rev. D 36, 1587 (1987).
  * Zwicky (1937) F. Zwicky, “On the masses of nebulae and of clusters of nebulae,” Astrophys. J. 86, 217 (1937).
  * Rubin and W. Kent (1970) Vera C. Rubin and Jr. W. Kent, Ford, “Rotation of the andromeda nebula from a spectroscopic survey of emission regions,” Astrophys. J. 159, 379 (1970).
  * Sofue and Rubin (2001) Yoshiaki Sofue and Vera Rubin, “Rotation Curves of Spiral galaxies,” Ann. Rev. Astron. Astrophys. 39, 137 (2001), arXiv:astro-ph/0010594 [astro-ph] .
  * Riess  _et al._ (1998) Adam G. Riess _et al._ (Supernova Search Team), “Observational Evidence From Supernovae for an Accelerating Universe and a Cosmological constant,” Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201 [astro-ph] .
  * Perlmutter  _et al._ (1999) S. Perlmutter _et al._ (Supernova Cosmology Project), “Measurements of $\Omega$ and $\Lambda$ From 42 High-Redshift Supernovae,” Astrophys. J. 517, 565 (1999).
  * Cartan (1922) Elie Cartan, “Sur une généralisation de la notion de courbure de riemann et les espaces à torsion,” C. R. Acad. Sci. Paris 174, 593 (1922).
  * Cartan (1923) Elie Cartan, “Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie),” Ann. Ec. Norm. Super. 40, 325 (1923).
  * Cartan (1924) Elie Cartan, “Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (suite),” Ann. Ec. Norm. Super. 41, 1 (1924).
  * Cartan (1925) Elie Cartan, “Sur les variétés à connexion affine et la théorie de la relativité généralisée, (deuxième partie),” Ann. Ec. Norm. Super. 42, 17 (1925).
  * Kaluza (1921) Theodor Kaluza, “On the Problem of Unity in Physics,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1921, 966 (1921).
  * Klein (1926) O. Klein, “Quantum Theory and Five-Dimensional Theory of relativity,” Z. Phys. 37, 895 (1926).
  * Lovelock (1971) David Lovelock, “The Einstein Tensor and Its generalizations,” J. Math. Phys. 12, 498–501 (1971).
  * Hehl  _et al._ (1995) Friedrich W. Hehl, J. Dermott McCrea, Eckehard W. Mielke, and Yuval Ne’eman, “Metric Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation invariance,” Phys. Rep. 258, 1–171 (1995), arXiv:gr-qc/9402012 [gr-qc] .
  * Mardones and Zanelli (1991) Alejandro Mardones and Jorge Zanelli, “Lovelock–Cartan Theory of gravity,” Class. Quant. Grav. 8, 1545 (1991).
  * Eddington (1923) Arthur S. Eddington,  _The mathematical theory of relativity_ (Cambridge University Press, 1923).
  * Schrödinger (1950) Erwin Schrödinger,  _Space-time structure_ (Cambridge University Press, 1950).
  * Kijowski (1978) Jerzy Kijowski, “On a new variational principle in general relativity and the energy of the gravitational field,” Gen. Rel. Grav. 9, 857 (1978).
  * Krasnov (2006) Kirill Krasnov, “Renormalizable Non-Metric Quantum Gravity?” (2006), arXiv:hep-th/0611182 [hep-th] .
  * Krasnov (2007) Kirill Krasnov, “Non-Metric Gravity: A Status report,” Mod. Phys. Lett. A 22, 3013–3026 (2007), arXiv:0711.0697 [gr-qc] .
  * Krasnov (2008) Kirill Krasnov, “Non-Metric Gravity. I. Field Equations,” Class. Quant. Grav. 25, 025001 (2008), arXiv:gr-qc/0703002 [gr-qc] .
  * Krasnov and Shtanov (2008) Kirill Krasnov and Yuri Shtanov, “Non-Metric Gravity. II. Spherically Symmetric Solution, Missing Mass and Redshifts of Quasars,” Class. Quant. Grav. 25, 025002 (2008), arXiv:0705.2047 [gr-qc] .
  * Krasnov (2011) Kirill Krasnov, “Pure Connection Action Principle for General Relativity,” Phys. Rev. Lett. 106, 251103 (2011), arXiv:1103.4498 [gr-qc] .
  * Popławski (2007a) Nikodem J. Popławski, “A unified, purely affine theory of gravitation and electromagnetism,” (2007a), arXiv:0705.0351 [gr-qc] .
  * Popławski (2007b) Nikodem J. Popławski, “On the Nonsymmetric Purely Affine gravity,” Mod. Phys. Lett. A 22, 2701 (2007b), arXiv:gr-qc/0610132 [gr-qc] .
  * Popławski (2014) Nikodem J. Popławski, “Affine Theory of gravitation,” Gen. Rel. Grav. 46, 1625 (2014), arXiv:1203.0294 [gr-qc] .
  * Castillo-Felisola and Skirzewski (2015) Oscar Castillo-Felisola and Aureliano Skirzewski, “A Polynomial Model of Purely Affine Gravity,” Rev. Mex. Fis. 61, 421 (2015), arXiv:1410.6183 [gr-qc] .
  * Castillo-Felisola and Skirzewski (2018) Oscar Castillo-Felisola and Aureliano Skirzewski, “Einstein’s gravity from a polynomial affine model,” Class. Quant. Grav. 35, 055012 (2018), arXiv:1505.04634 [gr-qc] .
  * Castillo-Felisola (2018) Oscar Castillo-Felisola, “Gravity,” (IntechOpen, 2018) Chap. Beyond Einstein: A Polynomial Affine Model of Gravity, pp. 183–201, arXiv:1902.09131 [gr-qc] .
  * Castillo-Felisola  _et al._ (2019) Oscar Castillo-Felisola, José Perdiguero, and Oscar Orellana, “Redifining standard model cosmology,” (IntechOpen, 2019) Chap. Cosmological Solutions to Polynomial Affine Gravity in the Torsion-Free Sector, p. NA, arXiv:1808.05970 [gr-qc] .
  * McGady and Rodina (2014) David A. McGady and Laurentiu Rodina, “Higher-Spin Massless $S$-matrices in four-Dimensions,” Phys. Rev. D 90, 084048 (2014), arXiv:1311.2938 [hep-th] .
  * Camanho  _et al._ (2016) Xian O. Camanho, Jose D. Edelstein, Juan Maldacena, and Alexander Zhiboedov, “Causality Constraints on Corrections To the Graviton Three-Point Coupling,” J. High Energy Phys. 02, 020 (2016), arXiv:1407.5597 [hep-th] .
  * Zerwekh (2013) Alfonso R. Zerwekh, “On the quantum chromodynamics of a massive vector field in the adjoint representation,” Int. J. Mod. Phys. A 28, 1350054 (2013).
  * Iosifidis (2019a) Damianos Iosifidis,  _Metric-Affine Gravity and Cosmology: aspects of Torsion and Non-Metricity in Gravity Theories_ , Ph.D. thesis, Institute of Theoretical Physics - Physics Department of Aristotle University of Thessaloniki (2019a), arXiv:1902.09643 [gr-qc] .
  * Iosifidis (2019b) Damianos Iosifidis, “Exactly solvable connections in metric-affine gravity,” Class. Quant. Grav. 36, 085001 (2019b), arXiv:1812.04031 [gr-qc] .
  * Parattu  _et al._ (2016) Krishnamohan Parattu, Sumanta Chakraborty, Bibhas Ranjan Majhi, and T. Padmanabhan, “A boundary term for the gravitational action with null boundaries,” Gen. Rel. Grav. 48, 94 (2016).
  * Krishnan  _et al._ (2017) Chethan Krishnan, Shubham Maheshwari, and P. N. Bala Subramanian, “Robin gravity,” Journal of Physics: Conference Series 883, 012011 (2017).
  * Krishnan and Raju (2017) Chethan Krishnan and Avinash Raju, “A neumann boundary term for gravity,” Mod. Phys. Lett. A 32, 1750077 (2017).
  * Lehner  _et al._ (2016) Luis Lehner, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin, “Gravitational action with null boundaries,” Phys. Rev. D 94, 084046 (2016).
  * Hopfmüller and Freidel (2017) Florian Hopfmüller and Laurent Freidel, “Gravity degrees of freedom on a null surface,” Phys. Rev. D 95, 104006 (2017).
  * Jubb  _et al._ (2017) Ian Jubb, Joseph Samuel, Rafael D. Sorkin, and Sumati Surya, “Boundary and corner terms in the action for general relativity,” Class. Quant. Grav. 34, 065006 (2017).
  * Nomizu and Sasaki (1994) Katsumi Nomizu and Takeshi Sasaki,  _Affine differential geometry_ (Cambridge University Press, 1994).
  * Besse (2007) Arthur L. Besse,  _Einstein manifolds_ (Springer, 2007).
  * Stephenson (1958) G. Stephenson, “Quadratic lagrangians and general relativity,” Nuovo Cimento 9, 263–269 (1958).
  * Kilmister and Newman (1961) C. W. Kilmister and D. J. Newman, “The use of algebraic structures in physics,” in  _Mathematical Proceedings of the Cambridge Philosophical Society_, Vol. 57 (Cambridge University Press, 1961) p. 851.
  * Yang (1974) C. N. Yang, “Integral formalism for gauge fields,” Phys. Rev. Lett. 33, 445 (1974).
  * Pavelle (1975) Richard Pavelle, “Unphysical solutions of yang’s gravitational-field equations,” Phys. Rev. Lett. 34, 1114–1114 (1975).
  * Thompson (1975) A. H. Thompson, “Geometrically degenerate solutions of the kilmister-yang equations,” Phys. Rev. Lett. 35, 320 (1975).
  * Einstein (1919) Albert Einstein, “Spielen gravitationsfelder im auf ber der materiellen elementar-teilchen eine wesentliche rolle?” Sitzungsber. Preuss. Akad. Wiss. (Berlin) , 349–356 (1919).
  * Garfinkle  _et al._ (2018) David Garfinkle, Lawrence R. Mead, and H. I. Ringermacher, “The shape of the orbit in flrw spacetimes,”  (2018), arXiv:1808.06683 [gr-qc] .
  * Mielke (2017) Eckehard W. Mielke,  _Geometrodynamics of Gauge Fields_, Mathematical Physics Studies (Springer International Publishing, 2017) p. 373.
  * Weinberg (1989) Steven Weinberg, “The cosmological constant problem,” Rev. Mod. Phys. 61, 1–23 (1989).
  * Ng and van Dam (1991) Y. Jack Ng and H. van Dam, “Unimodular theory of gravity and the cosmological constant,” J. Math. Phys. 32, 1337–1340 (1991).
  * Smolin (2009) Lee Smolin, “Quantization of unimodular gravity and the cosmological constant problems,” Phys. Rev. D 80 (2009), 10.1103/physrevd.80.084003, arXiv:0904.4841 [hep-th] .
  * Ellis  _et al._ (2011) George F. R. Ellis, Henk van Elst, Jeff Murugan, and Jean-Philippe Uzan, “On the trace-free einstein equations as a viable alternative to general relativity,” Class. Quant. Grav. 28, 225007 (2011), arXiv:1008.1196 [gr-qc] .
  * Boskoff and Capozziello (2019) Wladimir-Georges Boskoff and Salvatore Capozziello, “Recovering the cosmological constant from affine geometry,”  (2019), arXiv:1908.02340 [gr-qc] .
  * Ivey and Landsberg (2003) Thomas A. Ivey and J. M. Landsberg,  _Cartan for Beginners_ , Graduate Studies in Mathematics, Vol. 61 (AMS, 2003).
  * Baez (1994) John Baez,  _Gauge fields, knots, and gravity_ (World Scientific, Singapore River Edge, N.J, 1994).
  * Lévy-Leblond (1965) Jean-Marc Lévy-Leblond, “Une nouvelle limite non-relativiste du groupe de poincaré,” Annales de l’I.H.P. Physique théorique 3, 1–12 (1965).
  * Josset  _et al._ (2017) Thibaut Josset, Alejandro Perez, and Daniel Sudarsky, “Dark energy from violation of energy conservation,” Phys. Rev. Lett. 118, 021102 (2017).
  * Brensinger  _et al._ (2019) Samuel Brensinger, Kenneth Heitritter, Vincent G. J. Rodgers, Kory Stiffler, and Catherine A. Whiting, “Dark energy from dynamical projective connections,”  (2019), arXiv:1907.05334 [hep-th] .
  * Cervantes-Cota and Liebscher (2016) Jorge L. Cervantes-Cota and Dierck-Ekkehard Liebscher, “On constructing purely affine theories with matter,” Gen. Rel. Grav. 48, 108 (2016).
  * Barrow  _et al._ (2019) J. D. Barrow, C. G. Tsagas, and G. Fanaras, “Friedmann-like universes with torsion: a dynamical system approach,”  (2019), arXiv:1907.07586 [gr-qc] .
  * Kilic (2019) Delalcan Kilic, “The diffeomorphism field revisited,”  (2019), arXiv:1907.08850 [hep-th] .
  * Roberts (1995) Craig W. Roberts, “The projective connections of t.y. thomas and j.h.c. whitehead applied to invariant connections,” Differential Geometry and its Applications 5, 237–255 (1995).
  * Chothe  _et al._ (2019) Hiyang Ramo Chothe, Ashim Dutta, and Sourav Sur, “Cosmological dark sector from a mimetic-metric-torsion perspective,”  (2019), arXiv:1907.12429 [gr-qc] .
  * Stein  _et al._ (2019) W. A. Stein _et al._ ,  _Sage Mathematics Software (Version 8.6)_, The Sage Development Team (2019).
  * Gourgoulhon  _et al._ (2019) Eric Gourgoulhon, Michal Bejger, _et al._ ,  _SageManifolds (Version 8.6)_, SageManifolds Development Team (2019).
  * Gourgoulhon  _et al._ (2015) Eric Gourgoulhon, Michal Bejger, and Marco Mancini, “Tensor calculus with open-source software: the sagemanifolds project,” Journal of Physics: Conference Series 600, 012002 (2015), arXiv:1412.4765 [gr-qc] .
  * Gourgoulhon and Mancini (2018) Éric Gourgoulhon and Marco Mancini, “Symbolic tensor calculus on manifolds: a sagemath implementation,” Les cours du CIRM 6, 1–54 (2018), arXiv:1804.07346 [gr-qc] .
  * Tulczyjew (1974) Wlodzimierz M. Tulczyjew, “Hamiltonian systems, lagrangian systems and the legendre tranformation,” (Academic Press, 1974) pp. 247–258.
  * Tulczyjew (1975a) Wlodzimierz M. Tulczyjew, “A Symplectic Formulation of Particle Dynamics,” in  _Conference on Differential Geometrical Methods in Mathematical Physics Bonn, Germany, July 1-4, 1975_ (1975) pp. 457–463.
  * Tulczyjew (1975b) Wlodzimierz M. Tulczyjew, “A Symplectic Formulation of Field Dynamics,” in  _Conference on Differential Geometrical Methods in Mathematical Physics Bonn, Germany, July 1-4, 1975_ (1975) pp. 464–468.
  * Kijowski and Tulczyjew (1979) Jerzy Kijowski and Wlodzimierz M. Tulczyjew,  _A symplectic framework for field theories_ (Springer-Verlag, 1979).