[style=theoremstyle]cdbexampleCadabra Example

# Cosmological solutions to polynomial affine gravity in the torsion-free
sector

Oscar Castillo-Felisola  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.

###### Contents

  1. 1 Introduction
  2. 2 Polynomial Affine Gravity
  3. 3 Cosmological metric solutions
    1. 3.1 … with vanishing Ricci
    2. 3.2 … with parallel Ricci
    3. 3.3 … with harmonic Riemann
  4. 4 Cosmological non-metric solutions
    1. 4.1 … with vanishing Ricci
    2. 4.2 … with parallel Ricci
    3. 4.3 … with harmonic curvature
  5. 5 Conclusions and remarks
  6. A Lie derivative and Killing vectors

##  1 Introduction

All of the _fundamental_ physics is described by four interactions:
electromagnetic, weak, strong and gravitational. The former three are bundled
into what is known as _standard model_ of particle physics, which explains
very accurately the physics at very short scales. These three interactions
share common grounds, e.g. they are modelled by connections with values in a
Lie algebra, they have been successfully quantised and renormalised, and the
simplest of them—Quantum Electrodynamics—gives the most accurate results when
compared with the experiments.

On the other hand, the model that explains gravitational interaction (General
Relativity) is a field theory for the metric, which can be thought as a
potential for the gravitational connection [1, 2]. Although General Relativity
is the most successful theory we have to explain gravity [3, 4, 5], it cannot
be formulated as a gauge theory (in four dimensions), the standard
quantisation methods lead to inconsistencies, and it is non-renormalizable,
driving the community to believe it is an effective theory of a yet unknown
fundamental one. Within the framework of cosmology, when one wants to
conciliate both _standard models_ ,111Beside the standard model of particles,
there is a _standard model of cosmology_. it was noticed that nearly the 95%
of the Universe does not fit into the picture. Therefore, a (huge) piece of
the puzzle is missing called the dark sector of the Universe, composed of dark
matter and dark energy. In order to solve this problem one needs to add _new
physics_ , by either including extra particles (say inspired in beyond
standard model physics) or changing the gravitational sector. The latter has
inspired plenty of generalisations of General Relativity.

Although it cannot be said that the mentioned troubles are due to the fact
that the model is described by the metric, given that the _physical_ quantity
associated with the gravitational interaction—the curvature—is defined for a
connection, it is worth to ask ourselves whether a _more_ fundamental model of
gravitational interactions can be built up using the affine connection as the
mediator.

The first affine model of gravity was proposed by Sir A. Eddington in Ref.
[6], were the action was defined by the square root of the determinant of the
Ricci tensor,

$S=\int\sqrt{\det(\operatorname{Ric})},$

but in Schrödinger’s words [7]

> For all that I know, no special solution has yet been found which suggests
> an application to anything that might interest us, …

However, Eddington’s idea serves as starting point to new proposals [8, 9].

In a series of seminal papers [10, 11, 12, 13], E. Cartan presented a
definition of curvature for spaces with torsion, and its relevance for General
Relativity. It is worth mentioning, that in pure gravity—described by the
Einstein–Hilbert-like action—, Cartan’s generalisation of gravity yields the
condition of vanishing torsion as an equation of motion. Therefore, it was not
seriously considered as a generalisation of General Relativity, until the
inclusion of gravitating fermionic matter [14].

Inspired by Cartan’s idea of considering an affine connection into the
modelling of gravity, new interesting proposal have been considered. Among the
interesting generalisations we mention a couple: (i) The well-known metric-
affine models of gravity [15], in which the metric and connection are not only
considered as independent, but the conditions of metricity and vanishing
torsion are in general dropped; (ii) The Lovelock–Cartan gravity [16],
includes extra terms in the action compatible with the precepts of General
Relativity, whose variation yield field equations that are second order
differential equations. Nonetheless, the metric plays a very important role in
these models.

Modern attempts to describe gravity as a theory for the affine connection have
been proposed in Refs. [17, 18, 19, 20, 21, 22, 23, 24, 25], and the
cosmological implications in a Eddington-inspired affine model were studied in
Refs. [26, 27, 28, 29].

The recently proposed Polynomial Affine Gravity [24], separates the two roles
of the metric field, as in a Palatini formulation of gravity, but does not
allow it to participate in the mediation of the interaction, by its exclusion
from the action. It turns out that the absence of the metric in the action,
results in a robust structure that—without the addition of other fields—does
not accept deformations. That robustness can be useful if one would like to
quantise the theory, because all possible counter-terms should have the form
of terms already present in the action.

In this paper, we focus in finding cosmological solutions in the context of
Polynomial Affine Gravity, restricted to torsion-free sector of equi-affine
connections, which yields a simple set of field equations generalising those
obtained in standard General Relativity [25]. This paper is divided into four
sections: In Section 2 we review briefly the polynomial affine model of
gravity. In Section 3 we use the Levì-Civita connection for a
Friedman–Robertson–Walker metric, to solve the field equations—obtained in the
torsion-free sector—of Polynomial Affine Gravity. Then, in Section 4 we solve
for, the case of (affine) Ricci flat manifold, the field equations for the
affine connection. Some remarks and conclusions are presented in Section 5. In
order for completeness, in Appendix A, we include a short exposition of the
Lie derivative applied to the connection, and show the Killing vectors
compatible with the cosmological principle.

##  2 Polynomial Affine Gravity

In the standard theory of gravity, General Relativity, the fundamental field
is the metric, $g_{\mu\nu}$, of the spacetime [1, 2]. Nevertheless, the metric
has a two fold role in this gravitational model: it measures distances, and
also define the notion of parallelism, i.e. settles the connection. Palatini,
in Ref. [30], considered a somehow separation of these roles, but at the end
of the day the metric was still the sole field of the model. It was understood
soon after that the connection, $\Gamma^{\mu}{}_{\rho\sigma}$, does not need
to be related with the metric field [10, 11, 12, 13, 31], and the therefore,
the curvature could be blind to the metric.

In this section we briefly expose the model proposed in Refs. [24, 25], which
is inspired in the aforementioned role separation. The metric is left out the
mediation of gravitational interactions by taking it out the action.

The action of the polynomial affine gravity is built up from an affine
connection, $\hat{\Gamma}^{\mu}{}_{\rho\sigma}$, which accepts a decomposition
on irreducible components as

$\hat{\Gamma}^{\mu}{}_{\rho\sigma}=\hat{\Gamma}^{\mu}{}_{(\rho\sigma)}+\hat{\Gamma}^{\mu}{}_{[\rho\sigma]}={\Gamma}^{\mu}{}_{\rho\sigma}+\epsilon_{\rho\sigma\lambda\kappa}T^{\mu,\lambda\kappa}+A_{[\rho}\delta^{\mu}_{\nu]},$
(1)

where ${\Gamma}^{\mu}{}_{\rho\sigma}=\hat{\Gamma}^{\mu}{}_{(\rho\sigma)}$ is
symmetric in the lower indices, $A_{\rho}$ is a vector field corresponding to
the trace of torsion, and $T^{\mu,\lambda\kappa}$ is a Curtright field [32],
which satisfy the properties $T^{\kappa,\mu\nu}=-T^{\kappa,\nu\mu}$ and
$\epsilon_{\lambda\kappa\mu\nu}T^{\kappa,\mu\nu}=0$. The metric field, which
might or might not exists, cannot be used for contracting nor lowering or
raising indices. The relation between the epsilons with lower and upper
indices is given by
$\epsilon^{\delta\eta\lambda\kappa}\epsilon_{\mu\nu\rho\sigma}=4!\delta^{\delta}{}_{[\mu}\delta^{\eta}{}_{\nu}\delta^{\lambda}{}_{\rho}\delta^{\kappa}{}_{\sigma]}$.

The most general action preserving diffeomorphisms invariance, written in
terms of the fields in Eq. (1), is

$S[{\Gamma},T,A]=\int{\mathrm{d}}^{4}\\!{x}\;\bigg{[}B_{1}\,R_{\mu\nu}{}^{\mu}{}_{\rho}T^{\nu,\alpha\beta}T^{\rho,\gamma\delta}\epsilon_{\alpha\beta\gamma\delta}+B_{2}\,R_{\mu\nu}{}^{\sigma}{}_{\rho}T^{\beta,\mu\nu}T^{\rho,\gamma\delta}\epsilon_{\sigma\beta\gamma\delta}+B_{3}\,R_{\mu\nu}{}^{\mu}{}_{\rho}T^{\nu,\rho\sigma}A_{\sigma}+B_{4}\,R_{\mu\nu}{}^{\sigma}{}_{\rho}T^{\rho,\mu\nu}A_{\sigma}+B_{5}\,R_{\mu\nu}{}^{\rho}{}_{\rho}T^{\sigma,\mu\nu}A_{\sigma}+C_{1}\,R_{\mu\rho}{}^{\mu}{}_{\nu}\nabla_{\sigma}T^{\nu,\rho\sigma}+C_{2}\,R_{\mu\nu}{}^{\rho}{}_{\rho}\nabla_{\sigma}T^{\sigma,\mu\nu}+D_{1}\,T^{\alpha,\mu\nu}T^{\beta,\rho\sigma}\nabla_{\gamma}T^{(\lambda,\kappa)\gamma}\epsilon_{\beta\mu\nu\lambda}\epsilon_{\alpha\rho\sigma\kappa}+D_{2}\,T^{\alpha,\mu\nu}T^{\lambda,\beta\gamma}\nabla_{\lambda}T^{\delta,\rho\sigma}\epsilon_{\alpha\beta\gamma\delta}\epsilon_{\mu\nu\rho\sigma}+D_{3}\,T^{\mu,\alpha\beta}T^{\lambda,\nu\gamma}\nabla_{\lambda}T^{\delta,\rho\sigma}\epsilon_{\alpha\beta\gamma\delta}\epsilon_{\mu\nu\rho\sigma}+D_{4}\,T^{\lambda,\mu\nu}T^{\kappa,\rho\sigma}\nabla_{(\lambda}A_{\kappa)}\epsilon_{\mu\nu\rho\sigma}+D_{5}\,T^{\lambda,\mu\nu}\nabla_{[\lambda}T^{\kappa,\rho\sigma}A_{\kappa]}\epsilon_{\mu\nu\rho\sigma}+D_{6}\,T^{\lambda,\mu\nu}A_{\nu}\nabla_{(\lambda}A_{\mu)}+D_{7}\,T^{\lambda,\mu\nu}A_{\lambda}\nabla_{[\mu}A_{\nu]}+E_{1}\,\nabla_{(\rho}T^{\rho,\mu\nu}\nabla_{\sigma)}T^{\sigma,\lambda\kappa}\epsilon_{\mu\nu\lambda\kappa}+E_{2}\,\nabla_{(\lambda}T^{\lambda,\mu\nu}\nabla_{\mu)}A_{\nu}+T^{\alpha,\beta\gamma}T^{\delta,\eta\kappa}T^{\lambda,\mu\nu}T^{\rho,\sigma\tau}\Big{(}F_{1}\,\epsilon_{\beta\gamma\eta\kappa}\epsilon_{\alpha\rho\mu\nu}\epsilon_{\delta\lambda\sigma\tau}+F_{2}\,\epsilon_{\beta\lambda\eta\kappa}\epsilon_{\gamma\rho\mu\nu}\epsilon_{\alpha\delta\sigma\tau}\Big{)}+F_{3}\,T^{\rho,\alpha\beta}T^{\gamma,\mu\nu}T^{\lambda,\sigma\tau}A_{\tau}\epsilon_{\alpha\beta\gamma\lambda}\epsilon_{\mu\nu\rho\sigma}+F_{4}\,T^{\eta,\alpha\beta}T^{\kappa,\gamma\delta}A_{\eta}A_{\kappa}\epsilon_{\alpha\beta\gamma\delta}\bigg{]},$
(2)

where terms related through partial integration, and topological invariant
have been dropped.222An example of four-dimensional topological term is the
Euler density. One can prove via a dimensional analysis, the _uniqueness_ of
the above action (see Ref. [25]).

The action in Eq. (2) shows up very interesting features: (i) it is power-
counting renormalizable,333power-counting renormalizability does not guarantee
renormalizability. (ii) all coupling constants are dimensionless which hints
the conformal invariance of the model [33], (iii) yields no three-point
graviton vertices, which might allow to overcome the _no-go_ theorems found in
Refs. [34, camanho16_causal_const_correc_to_gravit], (iv) its non-relativistic
geodesic deviation agrees with that produced by a Keplerian potential [24],
and (v) the effective equations of motion in the torsion-free limit are a
generalization of the Einstein’s equations [25]. In the remaining of this
section we will sketch how to find the relativistic limit of this model, when
the torsion vanishes.

First, notice that the vanishing torsion condition is equivalent to setting
both $T^{\lambda,\mu\nu}$ and $A_{\mu}$ equal to zero. Although this limit is
not well-defined at the action level, it is well-defined at the level of
equation of motion.444The field equations can be consistently truncated under
the requirement of vanishing torsion. It is worth noticing that this condition
does not yield the Riemannian theory, since we are not yet asking for a
metricity condition. In order to simplify the task of finding the equations of
motion to take the limit, we restrict ourselves to the terms in the action
which are linear in either $T^{\lambda,\mu\nu}$ or $A_{\mu}$, since these are
the only terms which, after the extremisation, will survive the torsion-free
limit. Therefore, after the described considerations, the effective torsion-
free action is

$S_{\text{eff}}=\int{\mathrm{d}}^{4}\\!{x}\;\Big{(}C_{1}\,R_{\lambda\mu}{}^{\lambda}{}_{\nu}\nabla_{\rho}+C_{2}\,R_{\mu\rho}{}^{\lambda}{}_{\lambda}\nabla_{\nu}\Big{)}T^{\nu,\mu\rho}.$
(3)

The nontrivial equations of motion for this action are those for the Curtright
field, $T^{\nu,\mu\rho}$,

$\nabla_{[\rho}R_{\mu]\nu}+\kappa\nabla_{\nu}R_{\mu\rho}{}^{\lambda}{}_{\lambda}=0,$
(4)

where $\kappa$ is a constant related with the original couplings of the model.

In the Riemannian formulation of differential geometry, since the curvature
tersor is anti-symmetric in the last couple of indices, the second term in Eq.
(4) vanishes identically. However, for non-Riemannian connections, such term
still vanishes if the connection is compatible with a volume form. These
connections are known as equi-affine connections [35, 36]. In addition, the
Ricci tensor for equi-affine connections is symmetric. For these connections,
the gravitational equations are simply

$\nabla_{[\rho}R_{\mu]\nu}=0.$ (5)

The Eq. (5) is a generalisation of the parallel Ricci curvature condition,
$\nabla_{\rho}R_{\mu\nu}=0$, which is a known extension of the Einstein’s
equations [37, 38]. Moreover, these field equations are also obtained as part
of a à la Palatini approach to a Yang–Mills formulation of gravity, known as
the Stephenson–Kilmister–Yang (or SKY) model, proposed in Refs. [39,
kilmister61, 40]. Such Yang–Mills-like gravity is described by the action

$S_{\textsc{sky}}=\int{\mathrm{d}}^{4}\\!{x}\;\sqrt{g}\,g^{\mu\nu}g^{\sigma\tau}\,R_{\mu\sigma}{}^{\lambda}{}_{\rho}R_{\nu\tau}{}^{\rho}{}_{\lambda},$
(6)

which can be written using the curvature two-form as

$S_{\textsc{sky}}=\int\operatorname{Tr}\left(\bm{\mathcal{R}}\operatorname{\star}\bm{\mathcal{R}}\right)=\int\left(\bm{\mathcal{R}}^{{a}}{}_{{b}}\operatorname{\star}\bm{\mathcal{R}}^{{b}}{}_{{a}}\right).$
(7)

Although the field equations of the connection obtained from Eq. (6) are the
harmonic curvature condition [41],

$\nabla_{\lambda}R_{\mu\nu}{}^{\lambda}{}_{\rho}=0,$ (8)

these are equivalent to Eq. (5) through the second Bianchi identity [42, 38].

The Stephenson–Kilmister–Yang model is a field theory for the metric—not for
the connection—, and thus there is an extra field equation for the metric. The
field equation for the metric is very restrictive, and it does not accept
Schwarzschild-like solutions [43]. However, in the Polynomial Affine Gravity,
since the metric does not participate in the mediation of gravitational
interaction, that problem is solved trivially. Meanwhile, the physical field
associated with the gravitational interaction is the connection. This
difference makes a huge distinction in the phenomenological interpretation of
these models.

In the following sections we shall present solutions to the field equations
(5), in the cases where the connection is metric or not. To this end, in
appendix A we show how to propose an ansatz compatible with the desired
symmetries. Moreover, equation (5) can be solved in three ways, yielding to a
sub-classification of the solutions: (i) Ricci flat solutions, $R_{\mu\nu}=0$;
(ii) Parallel Ricci solutions, $\nabla_{\lambda}R_{\mu\nu}=0$; and (iii)
Harmonic Riemann solutions,
$\nabla_{\lambda}R_{\mu\nu}{}^{\lambda}{}_{\rho}=0$.

##  3 Cosmological metric solutions

The conditions of isotropy and homogeneity are very stringent, when imposed on
a symmetric rank-two tensor, and the possible ansatz is just the
Friedmann–Robertson–Walker metric,

$g=G_{00}\left(t\right)\mathrm{d}t\otimes\mathrm{d}t+G_{11}\left(t\right)\left(\frac{1}{1-{\kappa}r^{2}}\mathrm{d}r\otimes\mathrm{d}r+r^{2}\mathrm{d}{\theta}\otimes\mathrm{d}{\theta}+r^{2}\sin\left({\theta}\right)^{2}\mathrm{d}{\phi}\otimes\mathrm{d}{\phi}\right).$
(9)

In the remaining of this section, we shall use the standard parametrisation of
a Friedmann–Robertson–Walker metric, i.e.,

$g=-\mathrm{d}t\otimes\mathrm{d}t+a^{2}\left(t\right)\left(\frac{1}{1-{\kappa}r^{2}}\mathrm{d}r\otimes\mathrm{d}r+r^{2}\mathrm{d}{\theta}\otimes\mathrm{d}{\theta}+r^{2}\sin\left({\theta}\right)^{2}\mathrm{d}{\phi}\otimes\mathrm{d}{\phi}\right).$
(10)

The nonvanishing component of the Levi-Cività connection for the metric in Eq.
(10) are

$\displaystyle\Gamma_{\phantom{\,t}\,r\,r}^{\,t\phantom{\,r}\phantom{\,r}}$
$\displaystyle=-\frac{a\dot{a}}{kr^{2}-1}$
$\displaystyle\Gamma_{\phantom{\,t}\,{\theta}\,{\theta}}^{\,t\phantom{\,{\theta}}\phantom{\,{\theta}}}$
$\displaystyle=r^{2}a\dot{a}$
$\displaystyle\Gamma_{\phantom{\,t}\,{\phi}\,{\phi}}^{\,t\phantom{\,{\phi}}\phantom{\,{\phi}}}$
$\displaystyle=r^{2}a\sin^{2}\left({\theta}\right)\dot{a}$ (11)
$\displaystyle\Gamma_{\phantom{\,r}\,t\,r}^{\,r\phantom{\,t}\phantom{\,r}}$
$\displaystyle=\frac{\dot{a}}{a}$
$\displaystyle\Gamma_{\phantom{\,r}\,r\,t}^{\,r\phantom{\,r}\phantom{\,t}}$
$\displaystyle=\frac{\dot{a}}{a}$
$\displaystyle\Gamma_{\phantom{\,r}\,r\,r}^{\,r\phantom{\,r}\phantom{\,r}}$
$\displaystyle=-\frac{kr}{kr^{2}-1}$
$\displaystyle\Gamma_{\phantom{\,r}\,{\theta}\,{\theta}}^{\,r\phantom{\,{\theta}}\phantom{\,{\theta}}}$
$\displaystyle=kr^{3}-r$
$\displaystyle\Gamma_{\phantom{\,r}\,{\phi}\,{\phi}}^{\,r\phantom{\,{\phi}}\phantom{\,{\phi}}}$
$\displaystyle={\left(kr^{3}-r\right)}\sin^{2}\left({\theta}\right)$
$\displaystyle\Gamma_{\phantom{\,{\theta}}\,t\,{\theta}}^{\,{\theta}\phantom{\,t}\phantom{\,{\theta}}}$
$\displaystyle=\frac{\dot{a}}{a}$
$\displaystyle\Gamma_{\phantom{\,{\theta}}\,r\,{\theta}}^{\,{\theta}\phantom{\,r}\phantom{\,{\theta}}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{\,{\theta}}\,{\theta}\,t}^{\,{\theta}\phantom{\,{\theta}}\phantom{\,t}}$
$\displaystyle=\frac{\dot{a}}{a}$
$\displaystyle\Gamma_{\phantom{\,{\theta}}\,{\theta}\,r}^{\,{\theta}\phantom{\,{\theta}}\phantom{\,r}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{\,{\theta}}\,{\phi}\,{\phi}}^{\,{\theta}\phantom{\,{\phi}}\phantom{\,{\phi}}}$
$\displaystyle=-\cos\left({\theta}\right)\sin\left({\theta}\right)$
$\displaystyle\Gamma_{\phantom{\,{\phi}}\,t\,{\phi}}^{\,{\phi}\phantom{\,t}\phantom{\,{\phi}}}$
$\displaystyle=\frac{\dot{a}}{a}$
$\displaystyle\Gamma_{\phantom{\,{\phi}}\,r\,{\phi}}^{\,{\phi}\phantom{\,r}\phantom{\,{\phi}}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{\,{\phi}}\,{\theta}\,{\phi}}^{\,{\phi}\phantom{\,{\theta}}\phantom{\,{\phi}}}$
$\displaystyle=\frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)}$
$\displaystyle\Gamma_{\phantom{\,{\phi}}\,{\phi}\,t}^{\,{\phi}\phantom{\,{\phi}}\phantom{\,t}}$
$\displaystyle=\frac{\dot{a}}{a}$
$\displaystyle\Gamma_{\phantom{\,{\phi}}\,{\phi}\,r}^{\,{\phi}\phantom{\,{\phi}}\phantom{\,r}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{\,{\phi}}\,{\phi}\,{\theta}}^{\,{\phi}\phantom{\,{\phi}}\phantom{\,{\theta}}}$
$\displaystyle=\frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)}$

###  3.1 … with vanishing Ricci

This particular case is a metric model of gravity, whose field equations are
vanishing Ricci. It is expected to obtain the cosmological vacuum solution of
General Relativity (without cosmological constant), i.e. Minkowski spacetime.

From the connection in Eq. (11), it is straightforward to calculate the Ricci
tensor, and the field equations are then

$\displaystyle R_{tt}$ $\displaystyle=-\frac{3\ddot{a}}{a}=0,$ $\displaystyle
R_{ii}$
$\displaystyle=f_{i}(r,\theta)\left(\dot{a}^{2}+a\ddot{a}+2\kappa\right)=0,$
(12)

where the functions $f_{i}$ are $f_{r}=(1-\kappa r^{2})^{-1}$,
$f_{\theta}=r^{2}$ and $f_{\varphi}=r^{2}\sin^{2}(\theta)$.

The solutions to Eqs. (12) are shown in Table 1, and (as expected) are two
parametrisations of Minkowski spacetime, see for example Ref. [44].

Scale factor for the metric vanishing Ricci case $\kappa=-1$ |  $\kappa=0$ |  $\kappa=1$
---|---|---
$\sqrt{2}t+B$ |  $B\in\mathbb{R}^{+}$ |  $\not\exists$
Table 1: Scale factor solving the vanishing Ricci condition, for a
cosmological metric connection

###  3.2 … with parallel Ricci

Secondly, we shall analyse the possible solutions to the parallel Ricci
equations,

$\nabla_{\lambda}R_{\mu\nu}=0.$ (13)

Notice that in the case of Riemannian geometry, there is a _natural_ parallel
symmetric $\binom{0}{2}$-type tensor, i.e. the metric. Therefore, a simple
solution to Eq. (13) is that the Ricci is proportional to the metric—the
spacetime is an Einstein manifold—, and the proportionality factor is related
with the cosmological constant.

The independent components of Eq. (13) for the ansätze in Eq. (10) are,

$\displaystyle\nabla_{t}R_{tt}$
$\displaystyle\simeq\dot{a}\ddot{a}-a\dddot{a}=0,$ (14)
$\displaystyle\nabla_{i}R_{ti}$
$\displaystyle\simeq\left(\dot{a}^{2}-{a}\ddot{a}+\kappa\right)\dot{a}=0.$
(15)

Additionally, Eq. (14) can be rewritten as

$\frac{d}{dt}\left(\frac{\ddot{a}}{a}\right)=0\quad\Rightarrow\quad\ddot{a}+Ca=0.$
(16)

According to the value of the integration constant $C$, we parametrise it as

$C=\begin{cases}\omega^{2}&\text{ for }C>0\\\ \omega=0&\text{ for }C=0\\\
-\omega^{2}&\text{ for }C<0\end{cases}$

Using Eq. (16) to eliminate the $\ddot{a}$ dependence from Eq. (15) yields

$\dot{a}^{2}+Ca^{2}+\kappa=0.$ (17)

Scale factor for the metric parallel Ricci case |  $\kappa=-1$ |  $\kappa=0$ |  $\kappa=1$
---|---|---|---
$C=-\omega^{2}<0$ |  $\pm\frac{\sinh(\omega t)}{2\omega}$ |  $A\exp(\pm\omega t)$ |  $\pm\frac{\cosh(\omega t)}{2\omega}$
$C=\omega=0$ |  $\pm t+B$ |  $B\in\mathbb{R}^{+}$ |  $\not\exists$
$C=\omega^{2}>0$ |  $\frac{\sin(\omega t+\varphi)}{\omega}$ |  $\not\exists$ |  $\not\exists$
Table 2: Scale factor solving the parallel Ricci condition, for a cosmological
metric connection

The solutions to Eq. (16) are presented in Table 2, and they are known from
General Relativity, see for example Ref. [44]. Interestingly our integration
constant, $C$, could be identified as $C=-\frac{\Lambda}{3}$ from the vacuum
Friedmann’s equations. However, our equations are compatible with Friedmann’s
equations, interacting with a vacuum energy perfect fluid, if the integration
constant is identified with

$C=\frac{4\pi G_{N}}{3}(\rho+3p)-\frac{\Lambda}{3}.$ (18)

###  3.3 … with harmonic Riemann

Now that we showed that the solutions of the parallel Ricci equations are
equivalent to those of General Relativity, we turn our attention to the Eq.
(5). For the metric ansatz in Eq. (10), interestingly, only an independent
equation is obtained,

$\frac{2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a}+2\kappa\dot{a}}{a}=0,$

that should determine the scale factor. It can be rewritten as

$-\frac{d}{dt}\left(\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{\kappa}{a^{2}}\right)=0,$
(19)

i.e.

$\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{\kappa}{a^{2}}=-C.$ (20)

After a change of variable, $f=a^{2}$, Eq. (20) becomes

$\ddot{f}+2Cf+2\kappa=0,$ (21)

whose solutions are

$f(t)=\begin{cases}-\kappa t^{2}+At+B&C=0\\\ A\sin(\omega t)+B\cos(\omega
t)-\frac{2\kappa}{\omega^{2}}&2C=\omega^{2}>0\\\ A\sinh(\omega
t)+B\cosh(\omega t)+\frac{2\kappa}{\omega^{2}}&2C=-\omega^{2}<0\end{cases}$
(22)

Therefore, the scale factors are those presented in Table 3. Notice, however,
that in this case we are not separating the cases according to the value of
$\kappa$, but the existence of a solution for a given $\kappa$ is determined
by the domain of time, and also by the values of the integration constants $A$
and $B$.

Scale factor for the metric harmonic curvature case |  $\kappa={-1,0,1}$
---|---
$C=-\omega^{2}<0$ |  $\sqrt{A\sinh(\omega t)+B\cosh(\omega t)+\frac{2\kappa}{\omega^{2}}}$
$C=\omega=0$ |  $\sqrt{-\kappa t^{2}+At+B}$
$C=\omega^{2}>0$ |  $\sqrt{\frac{\sin(\omega t+\varphi)}{\omega}}$
Table 3: Scale factor solving the harmonic curvature condition, for a
cosmological metric connection

##  4 Cosmological non-metric solutions

In order to solve the set of coupled, non-linear, partial differential
equations for the connection, 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 [45].555See appendix A for a brief comment
about the Lie derivative of a connection. The nonvanishing components of the
connection are,

$\displaystyle\Gamma_{\phantom{t}tt}^{t\phantom{t}\phantom{t}}$
$\displaystyle=f\left(t\right)$
$\displaystyle\Gamma_{\phantom{t}rr}^{t\phantom{r}\phantom{r}}$
$\displaystyle=\frac{g\left(t\right)}{1-{\kappa}r^{2}}$
$\displaystyle\Gamma_{\phantom{t}{\theta}{\theta}}^{t\phantom{{\theta}}\phantom{{\theta}}}$
$\displaystyle=r^{2}g\left(t\right)$ (23)
$\displaystyle\Gamma_{\phantom{t}{\phi}{\phi}}^{t\phantom{{\phi}}\phantom{{\phi}}}$
$\displaystyle=r^{2}g\left(t\right)\sin^{2}\left({\theta}\right)$
$\displaystyle\Gamma_{\phantom{r}tr}^{r\phantom{t}\phantom{r}}$
$\displaystyle=h\left(t\right)$
$\displaystyle\Gamma_{\phantom{r}rt}^{r\phantom{r}\phantom{t}}$
$\displaystyle=h\left(t\right)$
$\displaystyle\Gamma_{\phantom{r}rr}^{r\phantom{r}\phantom{r}}$
$\displaystyle=\frac{{\kappa}r}{1-{\kappa}r^{2}}$
$\displaystyle\Gamma_{\phantom{r}{\theta}{\theta}}^{r\phantom{{\theta}}\phantom{{\theta}}}$
$\displaystyle={\kappa}r^{3}-r$
$\displaystyle\Gamma_{\phantom{r}{\phi}{\phi}}^{r\phantom{{\phi}}\phantom{{\phi}}}$
$\displaystyle={\left({\kappa}r^{3}-r\right)}\sin^{2}\left({\theta}\right)$
$\displaystyle\Gamma_{\phantom{{\theta}}t{\theta}}^{{\theta}\phantom{t}\phantom{{\theta}}}$
$\displaystyle=h\left(t\right)$
$\displaystyle\Gamma_{\phantom{{\theta}}r{\theta}}^{{\theta}\phantom{r}\phantom{{\theta}}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{{\theta}}{\theta}t}^{{\theta}\phantom{{\theta}}\phantom{t}}$
$\displaystyle=h\left(t\right)$
$\displaystyle\Gamma_{\phantom{{\theta}}{\theta}r}^{{\theta}\phantom{{\theta}}\phantom{r}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{{\theta}}{\phi}{\phi}}^{{\theta}\phantom{{\phi}}\phantom{{\phi}}}$
$\displaystyle=-\cos\left({\theta}\right)\sin\left({\theta}\right)$
$\displaystyle\Gamma_{\phantom{{\phi}}t{\phi}}^{{\phi}\phantom{t}\phantom{{\phi}}}$
$\displaystyle=h\left(t\right)$
$\displaystyle\Gamma_{\phantom{{\phi}}r{\phi}}^{{\phi}\phantom{r}\phantom{{\phi}}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{{\phi}}{\theta}{\phi}}^{{\phi}\phantom{{\theta}}\phantom{{\phi}}}$
$\displaystyle=\frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)}$
$\displaystyle\Gamma_{\phantom{{\phi}}{\phi}t}^{{\phi}\phantom{{\phi}}\phantom{t}}$
$\displaystyle=h\left(t\right)$
$\displaystyle\Gamma_{\phantom{{\phi}}{\phi}r}^{{\phi}\phantom{{\phi}}\phantom{r}}$
$\displaystyle=\frac{1}{r}$
$\displaystyle\Gamma_{\phantom{{\phi}}{\phi}{\theta}}^{{\phi}\phantom{{\phi}}\phantom{{\theta}}}$
$\displaystyle=\frac{\cos\left({\theta}\right)}{\sin\left({\theta}\right)}$

with $f$, $g$ and $h$ the unknown functions of time to be determined. The
Levi-Cività connection compatible with the Friedman–Robertson–Walker metric is
obtained from Eq. (23) by setting $f=0$, $g=a\dot{a}$ and
$h=\frac{\dot{a}}{a}$—Compare with Eq. (11).

The Ricci tensor calculated for the connection in Eq. (23) has only two
independent components

$\displaystyle R_{tt}$ $\displaystyle=3fh-3h^{2}-3\frac{\partial h}{\partial
t},$ (24) $\displaystyle R_{ii}$
$\displaystyle\simeq{fg+gh+2{\kappa}+\frac{\partial g}{\partial t}}.$ (25)

We now proceed to find solutions to Eq. (5). As in the previous section, we
present the three possibilities of solutions, but we will restrict ourselves
to finding solutions to the (affine) Ricci flat case.

###  4.1 … with vanishing Ricci

A first kind of solutions can be found by solving the system of equations
determined by vanishing Ricci. However, this strategy requires the fixing of
one of the unknown functions. The equations to solve are written as

$\displaystyle\dot{h}-(f-h)h$ $\displaystyle=0,$ (26)
$\displaystyle\dot{g}+(f+h)g+2\kappa$ $\displaystyle=0.$ (27)

Noticing that in the above equations $f$ is not a dynamical function, from Eq.
(26) we can solve $h$ as a function of $f$,

$h(t)=\frac{\exp\left(F(t)\right)}{C_{h}+\int\mathop{\mathrm{d}t}\nolimits\,\exp(F)},$
(28)

where we have defined $F=\int\mathop{\mathrm{d}t}\nolimits\,f(t)$ and $C_{h}$
is an integration constant. Then, Eq. (27) can be solved for $g$,

$g(t)=\exp(-\Sigma(t))\left(C_{g}-2\kappa\left(\int\mathop{\mathrm{d}t}\nolimits\exp(\Sigma(t))\right)\right),$
(29)

where $\Sigma(t)=\int\mathop{\mathrm{d}t}\nolimits\,\left(f(t)+h(t)\right)$,
and $C_{g}$ is another integration constant.

A particular solution inspired in the components of the connection for
Friedmann–Robertson–Walker, in whose case $f=0$, gives

$\displaystyle f(t)$ $\displaystyle=0,$ $\displaystyle g(t)$
$\displaystyle=\frac{1}{t+C_{h}}\left(C_{g}-\kappa(t+C_{h})^{2}\right),$
$\displaystyle h(t)$ $\displaystyle=\frac{1}{t+C_{h}},$ (30)

which for $C_{h}=C_{g}=0$ and $\kappa=-1$ yields the expected solution from
Table 1.666The standard parametrisation of Minkowski spacetime is achived by
the trivial solution of Eqs. (26) and (27), i.e., $f=g=h=\kappa=0$. However,
in Eq. (30) there are Ricci flat solutions which cannot be associated with the
sole existence of a metric, i.e. non-Riemannian manifolds; as for example
solutions with $\kappa>0$.

There are special solutions that cannot be obtained from Eqs. (28) and (29),
since they represent degenerated point in the moduli space.

Case $f=h$:

In these particular subspace on the Moduli, the first equation is linear, and
therefore the solution above is not valid. However, the solutions to Eqs. (26)
and (27) are given by

$\displaystyle f$ $\displaystyle=C_{h}$ $\displaystyle h$
$\displaystyle=C_{h}$ $\displaystyle g$
$\displaystyle=C_{g}\exp(-2Ct)-\frac{\kappa}{C_{h}}.$ (31)

Case $h=-f$:

In this case again Eq. (26) decouples from Eq. (27), and there solutions are
given by

$\displaystyle f$ $\displaystyle=-\frac{1}{2t+C_{h}}$ $\displaystyle h$
$\displaystyle=\frac{1}{2t+C_{h}}$ $\displaystyle g$
$\displaystyle=C_{g}-2\kappa t$ (32)

Case $h=0$ and $f$ given:

In this case, Eq. (26) becomes an identity, and $g$ can still be solved for a
given function $f$ as

$g(t)=\exp(-F(t))\left(C_{g}-2\kappa\left(\int\mathop{\mathrm{d}t}\nolimits\exp(F(t))\right)\right).$
(33)

Case $g=0$ and $f$ given:

In this case, Eq. (27) requires $\kappa=0$, and $h$ can still be solved for a
given function $f$ as in Eq. (28).

At this point, we have shown that a spacetime described by a Ricci flat,
torsion-free, equi-affine connection with the form presented in Eq. (23)
reproduces the cosmological Ricci flat solutions to General Relativity,
presented in Table 1, and there exist generalisations to these solutions which
are not possibly obtained in the Riemannian case. However, one can go even
further, and ask oneself whether the affine Ricci flat condition yield
more—real life—useful solutions, such as those solutions of General Relativity
presented in Table 2.

Therefore, we would like to obtain the Einstein equations from the affine
Ricci flat equation, i.e.,

$R^{\text{Aff}}_{\mu\nu}=M^{\text{GR}}_{\mu\nu}=0,$ (34)

where

$M^{\text{GR}}_{\mu\nu}=R^{\text{GR}}_{\mu\nu}-\Lambda g_{\mu\nu}-8\pi
G\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right).$

In the following, we are considering that the stress-energy tensor describes a
perfect fluid, i.e.,

$T_{\mu\nu}=\operatorname{diag}\begin{pmatrix}\rho&\frac{pa^{2}}{1-\kappa
r^{2}}&pa^{2}r^{2}&pa^{2}r^{2}\sin^{2}\theta\end{pmatrix}.$

In General Relativity, the Einstein equations in the form of Ricci, for the
cosmological ansatz yields

$\displaystyle M^{\text{GR}}_{tt}$ $\displaystyle\simeq 3\ddot{a}-\Lambda
a+4\pi Ga(\rho+3p),$ $\displaystyle M^{\text{GR}}_{tt}$ $\displaystyle\simeq
a\ddot{a}+2\dot{a}^{2}-\Lambda a^{2}-4\pi Ga^{2}(\rho-p)+2\kappa,$ (35)

Now, comparing Eq. (35) with Eqs. (26) and (27), a parametrisation for $f$,
$g$ and $h$ can be found such that once one compute the Ricci tensor for the
affine connection, the compatibility in Eq. (34) is satisfied. The
parametrisation is given by

$\displaystyle h$ $\displaystyle=\dot{a}+x,$ $\displaystyle f$
$\displaystyle=x,$ $\displaystyle g$ $\displaystyle=a\dot{a}+y,$ (36)

where the functions $x$ and $y$ satisfy the equations,

$\displaystyle\dot{x}+x\dot{a}-F_{1}$ $\displaystyle=0,$ (37)
$\displaystyle\dot{y}+yF_{2}-F_{3}$ $\displaystyle=0,$ (38)

with

$\displaystyle F_{1}$ $\displaystyle=\frac{a}{3}\Big{(}4\pi
G\left(3p+\rho\right)-\Lambda\Big{)}-(\dot{a})^{2},$ $\displaystyle F_{2}$
$\displaystyle=\dot{a}+2x,$ $\displaystyle F_{3}$
$\displaystyle=a^{2}\left(4\pi
G(p-\rho)-\Lambda\right)-2a\dot{a}x-a(\dot{a})^{2}+(\dot{a})^{2}.$ (39)

The Eqs. (37) and (38) can be formally integrated in terms of functions $a$,
$\rho$ and $p$, yielding

$\displaystyle x$
$\displaystyle=e^{-a}\left(C_{x}+\int\mathop{\mathrm{d}t}\nolimits
F_{1}\,e^{a}\right),$ (40) $\displaystyle y$
$\displaystyle=e^{-\int\mathop{\mathrm{d}t}\nolimits
F_{2}}\left(C_{y}+\int\mathop{\mathrm{d}t}\nolimits
F_{3}\,e^{\int\mathop{\mathrm{d}t}\nolimits F_{2}}\right).$ (41)

Therefore, a subspace of the possible solutions of the affine Ricci flat
geometries, describes the cosmological scenarios from General Relativity
coupled with perfect fluids. However, the explicit expressions for Eqs. (40)
and (41) for obtaining specific solutions to
Friedman–Lemaître–Robertson–Walker models are very complicated.

###  4.2 … with parallel Ricci

A second class of solutions can be found by solving the parallel Ricci
equation, $\nabla_{\lambda}R_{\mu\nu}=0$, which yield three independent field
equations,

$\displaystyle\nabla_{t}R_{tt}$
$\displaystyle\simeq\ddot{h}-(3f-2h)\dot{h}-h\dot{f}+2fh(f-h),$ (42)
$\displaystyle\nabla_{i}R_{ti}=\nabla_{i}R_{it}$ $\displaystyle\simeq
3g\dot{h}-h\dot{g}-2gh(2f-h)-2\kappa h,$ (43) $\displaystyle\nabla_{t}R_{ii}$
$\displaystyle\simeq\ddot{g}+g(\dot{f}+\dot{h})+(f-h)\dot{g}-gh(f+h)-4\kappa
h.$ (44)

However, the system of equations is complicated enough to avoid an analytic
solution.

Despite the complication, we can try a couple of assumptions that simplify the
system of equations, for example, if one consider the parametrisation inspired
in the Friedmann–Robertson–Walker results, i.e. setting $f=0$, and can solve
$h$ from Eq. (42), which is a total derivative in this particular case.
Nonetheless, despite the value of the first integration constant, the system
of equations imposes that both $\kappa$ and $g$ vanish.

###  4.3 … with harmonic curvature

Finally, the third class of solutions are those of Eq. (5). The set of
equations degenerate and yield a single independent field equation,

$\nabla_{\lambda}R_{ti}{}^{\lambda}{}_{i}\simeq\ddot{g}+g\dot{f}+f\dot{g}-2g\dot{h}+2gh(f-2h)-2\kappa
h$ (45)

Therefore, we need to set two out of the three unknown functions to be able of
solving for the connection.

##  5 Conclusions and remarks

In this article we have shortly reviewed the Polynomial Affine Gravity, which
is an alternative model for gravitational interactions described solely by the
connection, i.e., the metric does not play role in the mediation of the
interactions. Among the features of the model one encounters that despite the
numerous possible terms in the action, see Eq. (2), the absence of a metric
tensor gives a sort of rigidity to the action, in the sense that no other type
of terms can be added. Such rigidity suggests that if one attempts to quantise
the model, it could be renormalisable. Additionally, all of the coupling
constants, in the pure gravity regime, are dimensionless, pointing to a
possible conformal invariance of the (pure) gravitational interactions.

Restricting ourselves to equi-affine, torsion-free connections, the field
equations are a generalisation of those from General Relativity, Eq. (5). We
solved the field equations for a isotropic and homogeneous connection, either
compatible with a metric or not.

When the affine connection is the Levi-Cività connection for a
Friedman–Robertson–Walker metric, we showed that the sole solution for a Ricci
flat spacetime was described by the connection of Minkowski’s space, see Table
1. In the parallel Ricci case, we shown that—as intuitively expected—one
recovers the vacuum cosmological models, see Table 2, where the cosmological
constant enters as an integration constant, but such constant could be
interpreted as (partially) coming from the stress-energy tensor of a vacuum
energy perfect fluid, as mentioned—in the context of General Relativity—in
Ref. [46]. Finally, the (formal) solutions to the harmonic curvature are
presented in Table 3, but yet some work remains to be done to extract physical
phenomenology from these solutions.

In the case of the cosmological affine connection, we found that the Ricci
flat condition yield only two independent equations, which are not enough to
find the three unknown functions that parametrise the homogeneous and
isotropic connection. Nonetheless, since $f$ is not a dynamical function, the
remaining two, $h$ and $g$, can be solved in terms of the third one, $f$.
Interestingly, the three functions can be chosen in a way that Ricci flatness
condition for the affine connection, yields the Friedmann-Lemaître equations
from General Relativity coupled with a perfect fluid. In this sense, the pure
Polynomial Affine Gravity supersize General Relativity, since geometrically it
can mimic effects that are usually interpreted as _matter_ effects. However,
among the possible solutions for the Ricci flat condition there are countless
(yet) nonphysical solutions, and what is more, there is nothing that favours
the specific choice in Eq. (36) over others. Such landscape, drives us to
think that another type of condition should be use to restrict even further
the possible solutions for the affine connection.

The conditions of affine parallel Ricci could be the cornerstone in solving
the aforementioned degeneracy, since these conditions raise three independent
equations, that would serve to determine the three unknown functions. However,
at the moment we have not achieve any interesting result in pursuing this
goal.

On the other hand, the harmonic curvature condition yields a sole
(independent) field equation, and therefore the solutions are even more
degenerated than those from the Ricci flat condition, leaving even more space
for nonphysical solutions.

We would like to finish our discussion highlighting that, the geometric
emulation of matter content can serve as a starting point to a change of
paradigm related with the interpretation of the matter content of the
Universe, in particular the _dark_ sector.

### Acknowledgements

We would like to thank the following people for their support, without whose
help this work would never have been possible: Aureliano Skirzewski, Cristóbal
Corral, Iván Schmidt, Alfonso R. Zerwekh, Claudio Dib, Stefano Vignolo and
Jorge Zanelli. We are particularly grateful to the developers of the software
SageMath [47], SageManifolds [48, 49, 50] and Cadabra2 [51, 52, 53], which
were used extensively along the development of this work. The “Centro
Científico y Tecnológico de Valpara’iso” (CCTVal) is funded by the Chilean
Government through the Centers of Excellence Base Financing Program of
CONICYT, under project No. FS0821.

##  Appendix A Lie derivative and Killing vectors

The usual procedure for solving the Einstein’s equation is to propose an
ansatz for the metric. That ansatz must be compatible with the symmetries we
would like to respect in the problem. A first application is seen in the
Schwarzschild’s metric [54], which is the _most general_ symmetric rank-two
tensor compatible with the rotation group in three dimensions, an thus is
_spherically symmetric_.

The formal study of the symmetries of the fields is accomplish via the Lie
derivative (for reviews, see Refs. [55, 56, 57, 58]). Below, we briefly
explain the use of the Lie derivative for obtaining ansatzes for either the
metric or the connection.

The Lie derivative of a connection possesses an inhomogeneous part, in
comparison with the one of a rank three tensor. This can be written
schematically as

$\mathcal{L}_{\xi}\Gamma^{a}{}_{bc}=\mathcal{L}_{\xi}T^{a}{}_{bc}+\frac{\partial^{2}\xi^{a}}{\partial
x^{b}\partial x^{c}},$

or explicitly,

$\mathcal{L}_{\xi}\Gamma^{a}{}_{bc}=\xi^{m}\partial_{m}\Gamma^{a}{}_{bc}-\Gamma^{m}{}_{bc}\partial_{m}\xi^{a}+\Gamma^{a}{}_{mc}\partial_{b}\xi^{m}+\Gamma^{a}{}_{bm}\partial_{c}\xi^{m}+\frac{\partial^{2}\xi^{a}}{\partial
x^{b}\partial x^{c}},$ (46)

where $\xi$ is the vector defining the symmetry flow.

In particular, for cosmological applications, one asks for _isotropy_ and
_homogeneity_ , which in four dimensions restricts the isometry group to
either $SO(4)$, $SO(3,1)$ or $ISO(3)$. The algebra of these groups can be
obtained from the algebra $\mathfrak{so}(4)$ through a $3+1$ decomposition,
i.e. $J_{AB}=\Set{J_{ab},J_{a*}}$, where the extra dimension has been denoted
by an asterisk. In term of these new generators, the algebra reads

$\begin{split}\left[J_{ab},J_{cd}\right]&=\delta_{bc}J_{ad}-\delta_{ac}J_{bd}+\delta_{ad}J_{bc}-\delta_{bd}J_{ac},\\\
\left[J_{ab},J_{c*}\right]&=\delta_{bc}J_{a*}-\delta_{ac}J_{b*},\\\
\left[J_{a*},J_{c*}\right]&=-\kappa J_{ac},\end{split}$ (47)

with777The inhomogeneous algebra of $ISO(n)$ can be obtained from those of
$SO(n+1)$ or $SO(n,1)$ through the Inönü–Wigner contraction [59].

$\kappa=\begin{cases}1&SO(4)\\\ 0&ISO(3)\\\ -1&SO(3,1)\end{cases}.$ (48)

The six Killing vectors of these algebras, expressed in spherical coordinates
are,

$\displaystyle J_{1}=J_{23}$
$\displaystyle=\begin{pmatrix}0&0&-\cos\left({\varphi}\right)&\cot\left({\theta}\right)\sin\left({\varphi}\right)\end{pmatrix},$
(49) $\displaystyle J_{2}=J_{31}$
$\displaystyle=\begin{pmatrix}0&0&\sin\left({\varphi}\right)&\cos\left({\varphi}\right)\cot\left({\theta}\right)\end{pmatrix},$
$\displaystyle J_{3}=J_{12}$
$\displaystyle=\begin{pmatrix}0&0&0&1\end{pmatrix},$ $\displaystyle
P_{1}=J_{1*}$
$\displaystyle=\sqrt{1-{\kappa}r^{2}}\begin{pmatrix}0&\cos\left({\varphi}\right)\sin\left({\theta}\right)&\frac{\cos\left({\varphi}\right)\cos\left({\theta}\right)}{r}&-\frac{\sin\left({\varphi}\right)}{r\sin\left({\theta}\right)}\end{pmatrix},$
$\displaystyle P_{2}=J_{2*}$
$\displaystyle=\sqrt{1-{\kappa}r^{2}}\begin{pmatrix}0&\sin\left({\varphi}\right)\sin\left({\theta}\right)&\frac{\cos\left({\theta}\right)\sin\left({\varphi}\right)}{r}&\frac{\cos\left({\varphi}\right)}{r\sin\left({\theta}\right)}\end{pmatrix},$
$\displaystyle P_{3}=J_{3*}$
$\displaystyle=\sqrt{1-{\kappa}r^{2}}\begin{pmatrix}0&\cos\left({\theta}\right)&-\frac{\sin\left({\theta}\right)}{r}&0\end{pmatrix}.$

Using the Eq. (46), for the above Killing vectors, the most general connection
compatible with the desired symmetries can be obtained [45], giving the
components structure shown in Eq. (23).

## References

  * [1] A. Einstein, “Zur allgemeinen relativitätstheorie,” Sitzungsber. preuss. Akad. Wiss. 1 (1915) 778.
  * [2] D. Hilbert, “Die grundlagen der physik.(erste mitteilung.),” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1915 (1915) 395.
  * [3] C. M. Will, “The Confrontation Between Gerenal Relativity and experiment,” Living Rev. Rel. 17 (2014) 4, arXiv:1403.7377 [gr-qc].
  * [4] Virgo and LIGO scientific Collaboration, B. P. Abbott et al., “Observation of Gravitational Waves From a Binary Black Hole merger,” Phys. Rev. Lett. 116 no. 6, (2016) 061102, arXiv:1602.03837 [gr-qc].
  * [5] B. P. Abbott et al., “Gravitational waves and gamma-rays from a binary neutron star merger: Gw170817 and grb170817a,” Astrophys. J. 848 no. 2, (2017) L13.
  * [6] A. S. Eddington, The mathematical theory of relativity.  Cambridge University Press, 1923.
  * [7] E. Schrödinger, Space-time structure.  Cambridge University Press, 1950.
  * [8] M. Bañados and P. G. Ferreira, “Eddington’s theory of gravity and its progeny,” Phys. Rev. Lett. 105 no. 1, (2010) 011101.
  * [9] M. Bañados and P. G. Ferreira, “Erratum: Eddington’s theory of gravity and its progeny [phys. rev. lett.105, 011101 (2010)],” Phys. Rev. Lett. 113 no. 11, (2014) 119901.
  * [10] E. Cartan, “Sur une généralisation de la notion de courbure de riemann et les espaces à torsion,” C. R. Acad. Sci. Paris 174 (1922) 593. http://gallica.bnf.fr/ark:/12148/bpt6k3127j.image.langFR.
  * [11] E. Cartan, “Sur les variétés à connexion affine et la théorie de la relativité généralisée (première partie),” Ann. Ec. Norm. Super. 40 (1923) 325. http://archive.numdam.org/article/ASENS_1923_3_40__325_0.pdf.
  * [12] E. Cartan, “Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (suite),” Ann. Ec. Norm. Super. 41 (1924) 1. http://www.numdam.org/numdam-bin/item?id=ASENS_1924_3_41__1_0.
  * [13] E. Cartan, “Sur les variétés à connexion affine et la théorie de la relativité généralisée, (deuxième partie),” Ann. Ec. Norm. Super. 42 (1925) 17. http://www.numdam.org/numdam-bin/item?id=ASENS_1925_3_42__17_0.
  * [14] T. W. B. Kibble, “Lorentz Invariance and the Gravitational field,” J. Math. Phys. 2 (1961) 212–221.
  * [15] F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, “Metric Affine Gauge Theory of Gravity: Field Equations, Noether Identities, World Spinors, and Breaking of Dilation invariance,” Phys. Rep. 258 (1995) 1–171, arXiv:gr-qc/9402012 [gr-qc].
  * [16] A. Mardones and J. Zanelli, “Lovelock–Cartan Theory of gravity,” Class. Quant. Grav. 8 (1991) 1545.
  * [17] J. Kijowski, “On a new variational principle in general relativity and the energy of the gravitational field,” Gen. Rel. Grav. 9 no. 10, (1978) 857.
  * [18] K. Krasnov, “Non-Metric Gravity: A Status report,” Mod. Phys. Lett. A 22 (2007) 3013–3026, arXiv:0711.0697 [gr-qc].
  * [19] K. Krasnov, “Non-Metric Gravity. I. Field Equations,” Class. Quant. Grav. 25 (2008) 025001, arXiv:gr-qc/0703002 [gr-qc].
  * [20] K. Krasnov, “Pure Connection Action Principle for General Relativity,” Phys. Rev. Lett. 106 (2011) 251103, arXiv:1103.4498 [gr-qc].
  * [21] N. J. Popławski, “A unified, purely affine theory of gravitation and electromagnetism,” arXiv:0705.0351 [gr-qc].
  * [22] N. J. Popławski, “On the Nonsymmetric Purely Affine gravity,” Mod. Phys. Lett. A 22 no. 36, (2007) 2701, arXiv:gr-qc/0610132 [gr-qc].
  * [23] N. J. Popławski, “Affine Theory of gravitation,” Gen. Rel. Grav. 46 (2014) 1625, arXiv:1203.0294 [gr-qc].
  * [24] O. Castillo-Felisola and A. Skirzewski, “A Polynomial Model of Purely Affine Gravity,” Rev. Mex. Fis. 61 (2015) 421, arXiv:1410.6183 [gr-qc].
  * [25] O. Castillo-Felisola and A. Skirzewski, “Einstein’s gravity from a polynomial affine model,” Class. Quant. Grav. 35 no. 5, (2018) 055012, arXiv:1505.04634 [gr-qc].
  * [26] H. Azri and D. Demir, “Affine inflation,” Phys. Rev. D 95 no. 12, (2017) 124007.
  * [27] H. Azri and D. Demir, “Induced affine inflation,” Phys. Rev. D 97 no. 4, (2018) 044025.
  * [28] H. Azri, “Are there really conformal frames? uniqueness of affine inflation,” Int. J. Mod. Phys. D 27 no. 09, (2018) 1830006.
  * [29] H. Azri, Cosmological Implications of Affine Gravity.  PhD thesis, İzmi̇r Institute of Technology, 2018.  arXiv:1805.03936 [gr-qc].
  * [30] A. Palatini, “Deduzione Invariantiva Delle Equazioni Gravitazionali Dal Principio Di Hamilton,” Rend. Circ. Mat. (Palermo) 43 no. 1, (1919) 203.
  * [31] R. Debever, ed., Elie Cartan - Albert Einstein Letters on Absolute Parallelism 1929-1932.  Princeton University Press, 1979.
  * [32] T. Curtright, “Generalized Gauge Fields,” Phys. Lett. B 165 (1985) 304.
  * [33] D. Buchholz and K. Fredenhagen, “Dilations and Interaction,” J. Math. Phys. 18 (1977) 1107.
  * [34] D. A. McGady and L. Rodina, “Higher-Spin Massless $S$-matrices in four-Dimensions,” Phys. Rev. D 90 no. 8, (2014) 084048, arXiv:1311.2938 [hep-th].
  * [35] K. Nomizu and T. Sasaki, Affine differential geometry.  Cambridge University Press, 1994.
  * [36] R. L. Bryant, “Symmetries of non-Riemannian curvature tensor (answer).” MathOverflow.  URL: http://mathoverflow.net/a/212794/25356 (visited on 2015-08-03).
  * [37] A. Derdziński, “Classification of Certain Compact Riemannian Manifolds With Harmonic Curvature and Non-Parallel Ricci tensor,” Math. Z. 172 no. 3, (1980) 273.
  * [38] A. L. Besse, Einstein manifolds.  Springer, 2007.
  * [39] G. Stephenson, “Quadratic lagrangians and general relativity,” Nuovo Cimento 9 no. 2, (1958) 263–269.
  * [40] C. N. Yang, “Integral formalism for gauge fields,” Phys. Rev. Lett. 33 (Aug, 1974) 445.
  * [41] J.-P. Bourguignon, “Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’einstein,” Inventiones mathematicae 63 no. 2, (1981) 263.
  * [42] A. Derdziński, “Riemannian manifolds with harmonic curvature,” in Global Differential Geometry and Global Analysis 1984, D. Ferus, R. B. Gardner, S. Helgason, and U. Simon, eds., vol. 1156 of Lecture Notes in Mathematics, p. 74.  Springer Berlin Heidelberg, 1985.
  * [43] J. Zanelli, “Private communication.”. We thank J. Zanelli for sharing his personal communications with C. N. Yang, and clarify this aspect of the effective model.
  * [44] T. Dray, Differential forms and the geometry of general relativity.  CRC Press, 2014.
  * [45] O. Castillo-Felisola, Gravity, ch. Beyond Einstein: A Polynomial Affine Model of Gravity, pp. 183–201.  IntechOpen, 2018.
  * [46] S. Weinberg, “The cosmological constant problem,” Rev. Mod. Phys. 61 no. 1, (1989) 1–23.
  * [47] W. A. Stein et al., Sage Mathematics Software (Version 8.2).  The Sage Development Team, 2018.  http://www.sagemath.org.
  * [48] E. Gourgoulhon, M. Bejger, et al., SageManifolds (Version 1.2).  SageManifolds Development Team, 2018.  http://sagemanifolds.obspm.fr/index.html.
  * [49] E. Gourgoulhon, M. Bejger, and M. Mancini, “Tensor calculus with open-source software: the sagemanifolds project,” Journal of Physics: Conference Series 600 (2015) 012002, arXiv:1412.4765 [gr-qc].
  * [50] E. Gourgoulhon and M. Mancini, “Symbolic tensor calculus on manifolds: a sagemath implementation,” CoRR (2018) , arXiv:1804.07346 [gr-qc]. http://arxiv.org/abs/1804.07346v1.
  * [51] K. Peeters, “Symbolic field theory with cadabra,” Computeralgebra Rundbrief 41 (2007) 16. http://www.fachgruppe-computeralgebra.de/CA-Rundbrief/car41.pdf.
  * [52] K. Peeters, “Introducing Cadabra: A Symbolic Computer Algebra System for Field Theory problems,” arXiv:hep-th/0701238 [hep-th].
  * [53] K. Peeters, “Cadabra: a field-theory motivated symbolic computer algebra system,” Comput. Phys. Commun. 176 no. 8, (2007) 550.
  * [54] K. Schwarzschild, “On the gravitational field of a mass point according to Einstein’s theory,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916 (1916) 189–196, arXiv:physics/9905030 [physics].
  * [55] K. Yano, The theory of Lie derivatives and its applications, vol. 3.  North-Holland, 1957.
  * [56] Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, manifolds and physics, vol. 1 & 2\.  North-Holland, 1989.
  * [57] M. Nakahara, Geometry, Topology and Physics.  Institute Of Physics, 2005.
  * [58] A. McInerney, First Steps in Differential Geometry.  Springer, 2013.
  * [59] R. Gilmore, Lie Groups, Lie Algebras, and some of their Applications.  Dover, 2005.