# Polynomial affine model of gravity in three-dimensions

Oscar Castillo-Felisola,1,∗ Bastian Grez,1 Oscar Orellana,2 Jose Perdiguero,1
Francisca Ramirez,1 Aureliano Skirzewski,3 Alfonso R. Zerwekh.1

(1 Departamento de Física, Universidad Técnica Federico Santa María
Casilla 110-V, Valparaíso, Chile
2 Departamento de Matemáticas, Universidad Técnica Federico Santa María
Casilla 110-V, Valparaíso, Chile
3 Instituto de Física, Facultad de Ciencias
Iguá 4225, esq. Mataojo, 11400 Montevideo, Uruguay.
∗ Email<EMAIL_ADDRESS>
###### Abstract

In this work, we explore a three-dimensional formulation of the polynomial
affine model of gravity, which is a model that extends general relativity by
relaxing the equivalence principle through the exclusion of the metric from
the set of fundamental fields. In particular, in an attempt to gain insight of
the role of the torsion and nonmetricity in the gravitational models, we
consider homogeneous and isotropic cosmological models, for which their
solutions are classified in a _decisions tree_. We also show a few of these
explicit solutions that allow the definition of (alternative/emergent) metrics
derived from the connection.

##  1 Introduction

During the first quarter of the twentieth century, Einstein built the general
theory of relativity, attempting to render gravitational interactions and the
special theory of relativity compatible. The resulting model, known as General
Relativity, turned to provide a geometrical interpretation of the
gravitational interactions, where gravitational forces are a manifestation of
the curvature of spacetime.

General Relativity is so far the most successful model of gravitational
interactions at our disposal, explaining a large amount of experimental
observations [1]. Nonetheless, the completion of General Relativity with the
standard model of particles is insufficient to account for effects such as the
rotation of the Halos of Galaxies [2, 3, 4], and the value of the cosmological
constant [5, 6]. We should add to that the difficulty to consistently quantise
the model [7, 8, 9, 10, 11, 12, 13, 14].

In order to bypass these vicissitudes, it is necessary either to modify the
matter content of the standard model of particles or to find an alternative
model to General Relativity. In this work, we focus on the second type of
modifications, particularly to enhance the gravitational sector by allowing a
general connection and restricting the role of the metric.111In General
Relativity the role of the metric is two-folded, it is the geometric object
that defines distances, but also is the field responsible for the mediation of
the gravitational interactions.

Interestingly, a general notion of (continuous and connected) manifolds is
determined by the _affine connection_ , which is a geometrical object more
fundamental than the metric. A manifold ($M$) equipped with an affine
connection ($\hat{\Gamma}$) is called an affine manifold, and it admits the
notion of parallel transports. Hence, the Riemannian geometry behind General
Relativity is a particular case of the above when the connection is symmetric
in their lower indices and compatible with a metric tensor field, i.e.,
$\hat{\Gamma}_{{[\mu}}{}^{{\lambda}}{}_{{\nu]}}=0$ and
$\nabla^{(\hat{\Gamma})}g=0$. In affinely connected manifolds, it is possible
to define $\binom{0}{2}$ tensors derived from the connection, e.g., the Ricci
tensor field, for which its symmetric part might serve as a metric field, if
it is non-degenerated [15]. Sometimes, these tensors are referred to as
connection descendent or induced or emergent metric tensor fields.

From a modern perspective, the idea of formulating gravity as an affine theory
is attractive since other fundamental interactions are gauge theories, for
which their fundamental fields are gauge connections, and such theories are
renormalisable. Therefore, an affine formulation of gravity would bring the
theory to the same footing with gauge theories.

The first proposals of affine gravity were considered by Einstein, Eddington
and Schrödinger [16, 17, 18, 19], but those models did not offer sufficient
phenomenological novelty, while their treatments were significantly more
complex. More recent affine models of gravity were proposed by Kijowski and
collaborators [20, 21, 22, 23], Popławski [24, 25, 26, 27], Krasnov and
collaborators [28, 29, 30, 31, 32, 33], and ourselves [34, 35].

Albeit the fact that affine formulations of gravity conceptually differ from
general relativity, the predictions obtained from earlier models do not
provide significant differences from those of general relativity, while their
manipulation was harder.222In the words of Schrödinger, “For all that I know,
no special solution has yet been found which suggests an application to
anything that might interest us …” [19]. It has been understood that even if
two models are equivalent at the dynamical level, their generalisations might
be inequivalent [36]. However, subsequent models predict novel effects.

In the four-dimensional version of the polynomial affine model of gravity,
even if we restrict ourselves to the torsion-free sector of the model, the
vacuum solutions of Polynomial Affine Gravity include the vacuum solutions of
general relativity as a subset [35]. Moreover, some vacuum solutions of our
affine model account for effects that in general relativity are induced by the
presence of matter. This is interpreted as a mimicking of matter effects by
the non-Riemannian structure of geometry [37]. Although the underlying
geometry of the polynomial affine model of gravity does not require the
existence of a metric ab~initio, it is possible to define connection-
descendent metrics,333We also refer to these metrics as _emergent_ metrics.
allowing to distinguish between null and non-null _geodesics_ (self-parallel),
which would describe the free-fall trajectories for mass-less and massive
particles [38]. What is worth noticing is the existence of a connection-
descendent metric permits making contact with other quantities of interest in
cosmological and astronomical/astrophysical applications, such as the red
shift.

Moved by the moral from the early attempts to build affine models of gravity,
it is convenient to analyse the model in lower dimensions, where the number of
parameter is in general diminished. These models are a playground to test
methods that would be applied in four dimensions, but also might be applied in
other branches of physics since some gauge theories in three dimensions can be
interpreted as the high-temperature limit of four-dimensional models [39].

Historically, the three-dimensional version of General Relativity was firstly
studied by Staruszkiewicz [40], who considered Schwarzschild-like solutions
and noticed a relation between the presence of massive point particles and
conical singularities. It was soon understood that in three-dimensional
General Relativity there were no propagating degrees of freedom as a
consequence of a relation between the Riemann and Ricci tensors (they are not
independent since the conformal Weyl tensor in three dimensions is trivial).
Due to this, the interest for three-dimensional gravity decreased. It was only
after a series of papers by Deser, Jackiw and Templeton [41, 42, 43, 44], who
added a nontrivial topological term to the action, that three-dimensional
gravity re-emerged. In those works, the authors showed that the additional
term induced a massive model of gravity dubbed Deser–Jackiw–Templeton gravity
(or DJT for short).

Later, it was shown that the three-dimensional gravity modified by the
Pontryagin–Chern–Simons Lagrangian is equivalent to a Yang–Mills theory
(called Chern–Simons gravity), and its perturbative expansion was
renormalisable [45].444An update on the original ideas in this paper can be
found in Ref. [46]. Later, Bañados, Teitelboim and Zanelli found the first
black hole solution [47], disproving the _triviality_ of the classical three-
dimensional gravity. After these there was a boost in the search of exact
solutions [48] and analysis of their quantum aspects [49].

From all of the above, it is possible to conclude that the non-triviality of
three-dimensional gravity comes from topological aspects. However, when one
considers affinely connected manifolds, it is possible to define the
_projective Weyl tensor field_ :

$W_{\mu\nu}{}^{\lambda}{}_{\rho}=\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}-\frac{1}{D-1}\left(\mathcal{R}_{{\nu\rho}}\delta^{\lambda}_{\mu}-\mathcal{R}_{{\mu\rho}}\delta^{\lambda}_{\nu}\right),$

without the use of a metric tensor. The projective Weyl tensor, unlike the
conformal Weyl tensor (which requires a metric and has terms, with the scalar
curvature), does not vanish necessarily. Therefore, gravitational models
involving the affine connection might express dynamical effects unaccounted
for by General Relativity. An example of the modifications to the dynamics
comes from the existence of nonmetricity which, unlike the torsion, introduces
a term that is responsible of the non-coincidence of geodesics and self-
parallel curves. Moreover, such a quantity cannot be set to zero through a
coordinate transformation due to its tensor character, i.e., it is only zero
in a frame if it vanishes in any other frame.

The aim of this study is to review some of the aspects of the polynomial
affine model of gravity introduced in Ref. [34], focusing our attention on the
three-dimensional scenario. In this context, we find ansätze compatible with
the cosmological principle, which allows us to solve the field equations.
Interestingly, some of the solutions admit non-degenerated derived metrics.

##  2 What is the polynomial affine model of gravity?

Our purpose is to build an alternative model of gravity where the degrees of
freedom mediating the interaction come from a fundamental affine connection,
instead of coming from the metric tensor field. In order to achieve our goals,
we suppressed the use of the metric in the formulation of the action and
focused on terms that are polynomials in the fields (and their derivatives).
As a consequence, the number of possible terms involved in the action is
finite, as we shall show by an analysis of the index structure later in this
section.

Starting from the affine connection,
$\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}$, we can decompose it into its
irreducible components by separating the symmetric and skew-symmetric parts
(in the lower indices),

$\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=\hat{\Gamma}_{{(\mu}}{}^{{\lambda}}{}_{{\nu)}}+\hat{\Gamma}_{{[\mu}}{}^{{\lambda}}{}_{{\nu]}}=\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+\mathcal{A}_{[\mu}\delta^{\lambda}_{\nu]}.$

In the last equality, we split the skew-symmetric component of the connection
further into its trace, $\mathcal{A}$, and its traceless part, $\mathcal{B}$.
Hereafter, we denote the symmetric component of the connection by $\Gamma$
(without the hat). It is worth mentioning that the skew-symmetric part of the
connection is related to the torsion tensor field; therefore, both
$\mathcal{A}$ and $\mathcal{B}$ are the building blocks of torsion.

Our guideline to build the action functional would be the invariance under
diffeomorphisms, and the field content of the model would be the symmetric
connection ($\Gamma$), the trace of the torsion ($\mathcal{A}$), and the
traceless torsion ($\mathcal{B}$). Note that the torsion-descendent fields are
tensor fields, while $\Gamma$ is (still) a connection. Consequently, the
symmetric connection cannot (in principle) enter directly to the action, but
instead it should enter through the covariant derivative, $\nabla$.

In order to build an action functional successfully, we require a volume form
in the affine manifold. Such volume form is a generalisation of the well-known
Riemannian volume form, which is defined by the metric tensor field,
$\mathop{\mathrm{d}V}\nolimits_{g}={\mathrm{d}}^{3}\\!{x}\;\sqrt{g}$, where
the symbol $g$ represents the (absolute value of the) determinant of the
metric tensor field. We denote the affine volume by
$\mathop{\mathrm{d}V}\nolimits^{\mu\nu\lambda}=\mathop{\mathrm{d}x}\nolimits^{\mu}\mathop{\scriptstyle\wedge}\mathop{\mathrm{d}x}\nolimits^{\nu}\mathop{\scriptstyle\wedge}\mathop{\mathrm{d}x}\nolimits^{\lambda}$.

How do we build the action with these ingredients? Since the action is the
integral of the Lagrangian, which is a scalar density, we should write down
all possible terms that are polynomial in the fields, which can be contracted
to form scalar densities. Since $\mathcal{A}$, $\mathcal{B}$, and $\nabla$
have tensor character and the volume form is the only element with _weight_ ,
we can classify the terms in the action by an analysis of the indices
structure.555In previous articles we refer to this as _dimensional analysis_.

Note that all “tensor” fields have a simple net lower index (e.g., the
$\mathcal{B}$ field has one upper and two lower indices; thus, it would
contribute to a net single index), while the volume form has three upper
indices. Let us define the _index balance operator_ as $\mathscr{N}$, which
returns the value $+1$ for each upper index and $-1$ for lower indices, and
the _weight operator_ as $\mathscr{W}$, which returns the weight of the tensor
density. Then, an operator of the following form is obtained:

$\mathcal{O}=\mathcal{A}^{m}\mathcal{B}^{n}\nabla^{p}\mathop{\mathrm{d}V}\nolimits^{q},$

where the superindices denote the power of the element (number of times it
might appear), and it can be part of the Lagrangian when the net index balance
is zero and its weight is one; thus, thus it should satisfy the following:

$\mathscr{N}(\mathcal{O})=3q-m-n-p=0,\qquad\mathscr{W}(\mathcal{O})=q=1,$

if the operator could take part in the action.

Now that we know the restrictions in the power of the fields, we write down
all possible contractions, and use the symmetries of the fields to eliminate
redundant terms. The most general action (up to boundary terms) is given by
the following (see Ref. [34]):

$S=\int\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma}\bigg{(}B_{1}\,\mathcal{A}_{\alpha}\mathcal{A}_{\mu}\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}+B_{2}\,\mathcal{A}_{\alpha}\mathcal{F}_{\beta\gamma}+B_{3}\,\mathcal{A}_{\alpha}\nabla_{\mu}\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}+B_{4}\,\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\nu}}\mathcal{B}_{{\beta}}{}^{{\nu}}{}_{{\lambda}}\mathcal{B}_{{\gamma}}{}^{{\lambda}}{}_{{\mu}}+B_{5}\mathcal{R}_{{\alpha\beta}}{}^{{\mu}}{}_{{\mu}}\mathcal{A}_{\gamma}+B_{6}\,\mathcal{R}_{{\mu\alpha}}{}^{{\mu}}{}_{{\nu}}\mathcal{B}_{{\beta}}{}^{{\nu}}{}_{{\gamma}}+B_{7}\,\Gamma_{{\alpha}}{}^{{\mu}}{}_{{\mu}}\partial_{\beta}\Gamma_{{\gamma}}{}^{{\nu}}{}_{{\nu}}+B_{8}\,\left(\Gamma_{{\alpha}}{}^{{\mu}}{}_{{\nu}}\partial_{\beta}\Gamma_{{\gamma}}{}^{{\nu}}{}_{{\mu}}+\frac{2}{3}\Gamma_{{\alpha}}{}^{{\mu}}{}_{{\nu}}\Gamma_{{\beta}}{}^{{\nu}}{}_{{\lambda}}\Gamma_{{\gamma}}{}^{{\lambda}}{}_{{\mu}}\right)\bigg{)},$
(1)

where the topological Chern–Simons terms have been included explicitly. In the
above, we use the symbols $\mathcal{F}$ to denote the _field strength_ of the
$\mathcal{A}$-field, i.e.,
$\mathcal{F}_{\beta\gamma}=\partial_{\beta}\mathcal{A}_{\gamma}-\partial_{\gamma}\mathcal{A}_{\beta}$,
and we use $\mathcal{R}=\mathcal{R}{}^{{\Gamma}}$ to denote the curvature of
the symmetric connection.

In comparison with the action of General Relativity, the Eq. (1) seems
unpleasant at first. However, we should have in mind that the Hilbert–Einstein
action is far from being the _most general_ one compatible with the
symmetries. For this reason it is useful to highlight some nice features of
the polynomial affine model of gravity, which can be read from its action: (i)
Similarly for the other fundamental interactions, the fundamental field is a
connection; (ii) All coupling constants are dimensionless, which is desirable
from the point of view of Quantum Field Theory, since the superficial degree
of divergence vanishes; (iii) At least at a classical level, the model seems
to exhibit scale invariance; (iv) The number of possible terms in the action
is finite (we usually refer to this property as the _rigidity_ of the model),
giving the impression that in the hypothetical scenario of quantisation, the
model would be renormalisable, since all counter-terms should have the form of
terms already present in the original action.

The field equations derived from the action in Eq. (1) are as follows:

$2B_{1}\mathcal{A}_{\alpha}\mathcal{B}_{{\nu}}{}^{{\alpha}}{}_{{\rho}}+2B_{2}\mathcal{F}_{\nu\rho}+B_{3}\nabla_{\mu}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\rho}}+B_{5}\mathcal{R}_{{\nu\rho}}{}^{{\mu}}{}_{{\mu}}=0,$
(2)
$2B_{1}\mathcal{A}_{\nu}\mathcal{A}_{\rho}-2B_{3}\nabla_{(\nu}\mathcal{A}_{\rho)}+3B_{4}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}+2B_{6}\mathcal{R}_{{\mu(\nu}}{}^{{\mu}}{}_{{\rho)}}=0,$
(3)
$B_{3}\mathcal{A}_{\mu}\mathcal{B}_{{\rho}}{}^{{\nu}}{}_{{\sigma}}+B_{5}\left(\delta_{\mu}^{\nu}\mathcal{F}_{\rho\sigma}+\delta_{[\rho}^{\nu}\mathcal{F}_{\sigma]\mu}\right)+B_{6}\left(2\delta_{\mu}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\rho}}{}^{{\tau}}{}_{{\sigma}}}+\delta_{\rho}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\mu}}}+\delta_{\sigma}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\mu}}{}^{{\tau}}{}_{{\rho}}}\right)+B_{7}\left(\delta_{\mu}^{\nu}\mathcal{R}_{{\rho\sigma}}{}^{{\lambda}}{}_{{\lambda}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\mu}}{}^{{\lambda}}{}_{{\lambda}}\right)+B_{8}\left(\mathcal{R}_{{\rho\sigma}}{}^{{\nu}}{}_{{\mu}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\lambda}}{}^{{\lambda}}{}_{{\mu}}\right)=0.$
(4)

We have to mention that the field equations presented above are obtained by a
naive optimization of the action functional, without taking into account the
possible existence of second class constraints that could result in the
necessity of adding boundary contributions to the action.

In order to solve the field equations, we use the same strategy as in General
Relativity to provide an ansatz that must be compatible with the requirements
of our problem.

##  3 Building the ansätze

Riemannian geometry is the arena where General Relativity lays its grounds,
and the _fundamental_ geometrical object in Riemannian geometry is the metric
tensor field. Since the metric tensor field is related to the notion of
distance, in the easiest examples it is straightforward to propose ansätze.
However, for less trivial examples one utilises the notion of the Lie
derivative and its relation with the concept of symmetries.

Although the action of the Lie derivative on tensors (and even tensor
densities) is commonly known [50], it is less known that it is possible to
define the Lie derivative of an affine connection [51, 52, 53], given by the
following relation ,

$\pounds_{V}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=V^{\sigma}\partial_{\sigma}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}-\hat{\Gamma}_{{\mu}}{}^{{\sigma}}{}_{{\nu}}\partial_{\sigma}V^{\lambda}+\hat{\Gamma}_{{\sigma}}{}^{{\lambda}}{}_{{\nu}}\partial_{\mu}V^{\sigma}+\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\sigma}}\partial_{\nu}V^{\sigma}+\frac{\partial^{2}V^{\lambda}}{\partial
x^{\mu}\partial
x^{\nu}}=\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}V^{\lambda}+\hat{\mathcal{R}}_{{\rho\mu}}{}^{{\lambda}}{}_{{\nu}}V^{\rho}-2\hat{\nabla}_{\mu}\left(\Gamma_{{[\nu}}{}^{{\lambda}}{}_{{\rho]}}V^{\rho}\right).$
(5)

Above, the vector field $V$ represents a generator of the symmetry group,
i.e., in mathematical terminology, $V$ defines the symmetry flow. Note that
the second line is a covariant expression, since the skew-symmetric part of
the connection (i.e., the torsion) is a tensor field.

Therefore, the strategy to find ansätze would be the following: (i) Define the
desired symmetries in concordance with the physical problem we would like to
solve; (ii) Obtain the vector fields associated to the generators of the
symmetry group and express them in the coordinate system of preference; (iii)
Restrict the coefficients of the affine connection by solving the set of
equations defined by
$\pounds_{V}\hat{\Gamma}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}=0$. Note that this
represents a set of $27$ partial differential equations for each vector $V$.

In order to illustrate the procedure, we use _spherical_ coordinates,
$(t,r,\varphi)$, and show that the general affine connection is compatible
with isotropy and the cosmological principle (i.e., isotropic and
homogeneous).

In three-dimensions, the vector fields generating the rotation and
translations along a two-dimensional subspace of constant curvature are
represented by the following:

$\displaystyle J$ $\displaystyle=\begin{pmatrix}0&0&1\end{pmatrix},$ (6)
$\displaystyle X$ $\displaystyle=\sqrt{1-\kappa
r^{2}}\begin{pmatrix}0&\cos\varphi&-\frac{1}{r}\sin\varphi\end{pmatrix},$
$\displaystyle Y$ $\displaystyle=\sqrt{1-\kappa
r^{2}}\begin{pmatrix}0&\sin\varphi&\frac{1}{r}\cos\varphi\end{pmatrix},$

where $\kappa=-1,0,+1$ is the normalised Gaußian curvature of the two-
dimensional subspace.

###  3.1 Isotropic connection

Straightforwardly, one can observe from Eq. (5) that, since the vector field
$J$ is constant, the non-homogeneous contribution to the Lie derivative
vanishes; thus, it coincides with the Lie derivative of a
$\binom{1}{2}$-tensor.

Moreover, the explicit relations derived from the condition of the vanishing
Lie derivative of the connection along the vector $J$ are as follows:

$\pounds_{J}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=\partial_{\varphi}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=0.$
(7)

The solution is that none of the components of the affine connection depend on
the angular coordinate $\varphi$. Unfortunately, this symmetry argument does
not kill any of the components of the connection! Consequently, using only an
argument of isotropy, it would be almost impossible to find an analogous case
of the Schwarzschild black hole—which in three-dimensions it is called
Bañados–Teitelboim–Zanelli (BTZ) black hole [47, 54, 55, 48].

###  3.2 Isotropic and homogeneous connection

In order to find the affine connection compatible with the cosmological
principle, we have to solve the conditions of the vanishing Lie derivative of
the connection along the vector fields $X$ and $Y$. This procedure is long and
tedious, but straightforward, and we refer to the interested reader to check
Refs. [56, 37] to see more details.

After solving the conditions, the irreducible components of the affine
connection are given by the following:

$\displaystyle\Gamma_{{t}}{}^{{t}}{}_{{t}}$ $\displaystyle=j(t),$
$\displaystyle\Gamma_{{i}}{}^{{t}}{}_{{j}}$ $\displaystyle=g(t)S_{ij},$ (8)
$\displaystyle\Gamma_{{i}}{}^{{k}}{}_{{j}}$
$\displaystyle=\gamma_{i}{}^{k}{}_{j},$
$\displaystyle\Gamma_{{t}}{}^{{i}}{}_{{j}}=\Gamma_{{j}}{}^{{i}}{}_{{t}}$
$\displaystyle=h(t)\delta^{i}_{j}+f(t)S^{ik}\epsilon_{kj}\frac{r}{\sqrt{1-\kappa
r^{2}}},$

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

$S_{ij}=\begin{pmatrix}\frac{1}{1-\kappa r^{2}}&0\\\ 0&r^{2}\end{pmatrix},$

and

$\displaystyle\gamma_{r}{}^{r}{}_{r}$ $\displaystyle=\frac{\kappa r}{1-\kappa
r^{2}},$ $\displaystyle\gamma_{\varphi}{}^{r}{}_{\varphi}$
$\displaystyle=-r(1-\kappa r^{2}),$
$\displaystyle\gamma_{r}{}^{\varphi}{}_{\varphi}$ $\displaystyle=\frac{1}{r},$
$\displaystyle\gamma_{\varphi}{}^{\varphi}{}_{r}$ $\displaystyle=\frac{1}{r}.$

If one compares with the Levi-Civita connection obtained from a
Friedman–Robertson–Walker metric, one notices that the function $j$ is zero,
the function $f$ is something entirely new (and it exists solely in three-
dimensions), and the functions $g$ and $h$ are defined in terms of the scale
factor $a$, as $g=a\dot{a}$ and $h=\dot{a}/a$.

It is possible to show that in the affine case, a reparametrisation of the
time allows us to set the function $j=0$ [57].

The nonvanishing components of the $\mathcal{B}$-field are as follows:

$\displaystyle\mathcal{B}_{{\varphi}}{}^{{t}}{}_{{r}}=-\mathcal{B}_{{r}}{}^{{t}}{}_{{\varphi}}$
$\displaystyle=\xi(t)\frac{r}{\sqrt{1-\kappa r^{2}}},$ (9)
$\displaystyle\mathcal{B}_{{t}}{}^{{r}}{}_{{\varphi}}=-\mathcal{B}_{{\varphi}}{}^{{r}}{}_{{t}}$
$\displaystyle=\psi(t)r\sqrt{1-\kappa r^{2}},$
$\displaystyle\mathcal{B}_{{r}}{}^{{\varphi}}{}_{{t}}=-\mathcal{B}_{{t}}{}^{{\varphi}}{}_{{r}}$
$\displaystyle=\frac{\psi(t)}{r\sqrt{1-\kappa r^{2}}},$

while the nonvanishing component of the $\mathcal{A}$-field is
$\mathcal{A}_{t}=\eta(t)$.

##  4 Cosmological solutions

Now that we have the general form of the fields, we substitute such ansatz
into the field equations in Eq. (2), (3) and (4). The resulting system of
equations is [noalign]

$2B_{8}gf-B_{3}\xi\eta=0,$ (10) $B_{6}(2g\psi+\dot{\xi})-B_{8}fg=0,$ (11)
$B_{8}(2gh+\kappa-\dot{g})=0,$ (12) $B_{3}\eta\psi+B_{8}(2hf+\dot{f})=0,$ (13)
$B_{1}\eta^{2}-B_{3}\dot{\eta}+3B_{4}\psi^{2}-2B_{6}(\dot{h}+h^{2}-f^{2})=0,$
(14) $B_{3}g\eta-3B_{4}\psi\xi+B_{6}(\kappa+\dot{g})=0,$ (15)
$2B_{1}\eta\xi+B_{3}(2g\psi+\dot{\xi})=0.$ (16)

Notably, the Eq. (10) is algebraic. However, we can obtain other algebraic
relations from Eqs. (11) and (16), and from Eqs. (12) and (15). These
expressions are [noalign]

$2B_{8}gf-B_{3}\xi\eta=0,$ (17)
$\frac{B_{8}}{B_{6}}gf+2\frac{B_{1}}{B_{3}}\xi\eta=0,$ (18)
$3\frac{B_{4}}{B_{6}}\psi\xi-2gh-\frac{B_{3}}{B_{6}}g\eta=2\kappa.$ (19)

Equations (17) and (18) can be seen as a system of equations for variables $f$
and $\eta$ as functions of $g$ and $\xi$, but their independence is dictated
by the determinant of the following system coefficients,

$\Omega=B_{8}\left[4\frac{B_{1}}{B_{3}}+\frac{B_{3}}{B_{6}}\right]g\xi.$ (20)

A thorough search for solutions requires taking into account all possible
cases, which can be represented by _decision trees_. We shall sketch the types
of solutions in two cases: (i) when the determinant in Eq. (20) is
nonvanishing, which requires $f=\eta=0$; and (ii) when the determinant
$\Omega$ vanishes, for which the decision tree has many branches.

Interestingly, from the solutions obtained in the scan, just a few of them
have all the fields turned on [57]. Below we present some of these
_exceptional_ solutions.

Firstly, the case with $g=\kappa=\xi=0$ has as nontrivial field equations the
set

$\dot{f}+2fh=-\frac{B_{3}}{B_{8}}\eta\psi,$ (21)
$\dot{h}+h^{2}-f^{2}=\frac{1}{2B_{6}}\left(B_{1}\eta^{2}-B_{3}\dot{\eta}+3B_{4}\psi^{2}\right),$
(22)

which are solved when all the functions are inversely proportional to $t$.
Note that with this configuration, $\Gamma$, $\mathcal{A}$, and $\mathcal{B}$
are nonvanishing.

Another branch of solutions is found when the coupling constants are not all
independent, e.g., for $B_{3}^{2}+4B_{1}B_{6}=0$, condition $\Omega=0$ is
satisfied without fixing $g$ or $\xi$. Under these conditions, we observed a
couple of exceptional solutions.

First, for $B_{1}=B_{3}=0$, the solution is characterised by the functions

$\displaystyle f(t)$ $\displaystyle=0,$ $\displaystyle g(t)$
$\displaystyle=-\kappa t+C_{g},$ $\displaystyle h(t)$
$\displaystyle=\frac{1}{t-\kappa C_{g}},$ (23) $\displaystyle\xi(t)$
$\displaystyle=\sqrt{-\frac{2B_{6}}{3B_{4}}}C_{g},$ $\displaystyle\psi(t)$
$\displaystyle=0,$ $\displaystyle\eta(t)$ $\displaystyle=\text{arbitrary},$

for $\kappa\neq 0$. While for $\kappa=0$, the functions defining the
connection are as follows:

$\displaystyle f(t)$ $\displaystyle=0,$ $\displaystyle g(t)$
$\displaystyle=C_{m}t+C_{g},$ $\displaystyle h(t)$
$\displaystyle=\frac{C_{m}}{2(C_{m}t+C_{g})},$ (24) $\displaystyle\xi(t)$
$\displaystyle=\sqrt{-\frac{2B_{6}}{3B_{4}}}(C_{m}t+C_{g}),$
$\displaystyle\psi(t)$
$\displaystyle=\sqrt{-\frac{B_{6}}{6B_{4}}}\frac{C_{m}}{C_{m}t+C_{g}},$
$\displaystyle\eta(t)$ $\displaystyle=\text{arbitrary}.$

Secondly, for $\kappa=0$ we proposed the ansatz $g(t)=t^{n}$ with
$n\in\mathbb{R}-\Set{-2,\left[\tfrac{1-\sqrt{33}}{4},\tfrac{1+\sqrt{33}}{4}\right]}$,
with the condition

$B_{4}=-\frac{8B_{6}^{3}(n+2)}{3B_{8}^{2}\left(2n^{3}-3n^{2}-3n+4\right)}.$

The functions defining the connection are as follows,

$\displaystyle f(t)$
$\displaystyle=\frac{\sqrt{2n^{2}-n-4}\operatorname{sgn}(B_{8})}{2t},$
$\displaystyle g(t)$ $\displaystyle=t^{n},$ $\displaystyle h(t)$
$\displaystyle=\frac{n}{2t},$ (25) $\displaystyle\xi(t)$
$\displaystyle=\frac{\sqrt{2n^{2}-n-4}\operatorname{sgn}(B_{8})t^{n}}{2B_{6}},$
$\displaystyle\psi(t)$
$\displaystyle=\frac{(n-1)\sqrt{2n^{2}-n-4}\operatorname{sgn}(B_{8})}{4tB_{6}},$
$\displaystyle\eta(t)$ $\displaystyle=\frac{2B_{6}}{B_{3}t}.$

The solutions shown above are interesting because they admit some sort of
connection-descendent metric. In particular, the solution from Eq. (25) admits
several _emergent metrics_ , for example, (i) The Ricci tensor field, and (ii)
The Popławski metric [27],666We shall denote the Popławski metric by
$\operatorname{Popl}$ or $g^{(P)}$. defined from the torsion tensor field as
$g_{\mu\nu}=\mathcal{T}{}_{\mu}{}^{\lambda}{}_{\rho}\mathcal{T}{}_{\nu}{}^{\rho}{}_{\lambda}$,
which are in general non-degenerated. Without further details, for this
solution the metrics are of the following form:

$\operatorname{Ric}=-\frac{A(n)}{t^{2}}\mathop{\mathrm{d}t}\nolimits\otimes\mathop{\mathrm{d}t}\nolimits+nt^{n-1}\left(\frac{\mathop{\mathrm{d}r}\nolimits\otimes\mathop{\mathrm{d}r}\nolimits}{1-\kappa
r^{2}}+r^{2}\mathop{\mathrm{d}\varphi}\nolimits\otimes\mathop{\mathrm{d}\varphi}\nolimits\right),$

and

$\operatorname{Popl}=-\frac{F(n,B_{i})}{t^{2}}\mathop{\mathrm{d}t}\nolimits\otimes\mathop{\mathrm{d}t}\nolimits+{G(n,B_{i})t^{n-1}}\left(\frac{\mathop{\mathrm{d}r}\nolimits\otimes\mathop{\mathrm{d}r}\nolimits}{1-\kappa
r^{2}}+r^{2}\mathop{\mathrm{d}\varphi}\nolimits\otimes\mathop{\mathrm{d}\varphi}\nolimits\right),$

where $A$, $F$ and $G$ are numerical factors depending on coefficients $B_{i}$
and the value of $n$ [57].

##  5 Discussion and concluding remarks

In this article, we have briefly reviewed the polynomial affine model of
gravity, which attempts to model the gravitational interaction as mediated by
the affine connection. Its action functional is built up without the use of a
metric tensor field by explicitly excluding it from the action.

Despite the fact that an affine model of gravity brings gravitation a step
closer to the formulation of gauge theories, it is interesting that the number
of possible terms in the action [see Eq. (1)] is finite.777Note that once the
metric is included, as in General Relativity, the number of possible terms is
naturally unbound, since we allow the contraction of the indices that are
initially on the same footing. As an example of these terms we have the scalar
curvature, $\mathcal{R}$, and its powers, $\mathcal{R}{}^{{n}}$, opening the
window to the $f(\mathcal{R})$-models.

Remarkably, even if we start with an affinely connected manifold, without
referring to a fundamental metric, there are symmetric $\binom{0}{2}$ tensors
derived from the connection that might play the role of metric tensor fields
when non-degenerated.

The first connection-descendent object that might be a metric is the symmetric
Ricci tensor field [15]. In fact, Einstein and Eddington proposed
independently their affine model of gravity by using the square-root of the
determinant of the Ricci, i.e., $\sqrt{\operatorname{Ric}}$, as Lagrangian
density. Another emergent metric is given by a quadratic combination of the
torsion tensor,
$g^{(P)}_{\mu\nu}=\mathcal{T}{}_{\mu}{}^{\lambda}{}_{\rho}\mathcal{T}{}_{\nu}{}^{\rho}{}_{\lambda}$,
which was considered by Popławski in Ref. [27]. We can define a Popławski-like
metric by using the $\mathcal{B}$-field, as
$g^{(\mathcal{B})}_{\mu\nu}=\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}\mathcal{B}_{{\nu}}{}^{{\rho}}{}_{{\lambda}}$.

In addition, in three-dimensions, the dual of the $\mathcal{B}$-field,
$\mathfrak{B}^{\lambda\rho}=\frac{1}{2}\epsilon^{\mu\nu\lambda}\mathcal{B}_{{\mu}}{}^{{\rho}}{}_{{\nu}}$,
is a symmetric $\binom{2}{0}$-tensor density. When this tensor density is non-
degenerated, it might be interpreted as an analogous case of the Riemannian
$\sqrt{g}g^{\mu\nu}$. Hence, such quantity induces a notion of distance, i.e.,
introduces a metric.

The general field equations are a clear generalisation of the Einstein
equations; in particular, the field equation for the $\mathcal{B}$-field [see
Eq. (3)] is the analogous case for the standard equations in General
Relativity, where the following term is provided:

$2B_{1}\mathcal{A}_{\nu}\mathcal{A}_{\rho}-2B_{3}\nabla_{(\nu}\mathcal{A}_{\rho)}+3B_{4}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}$

and it might be interpreted as the energy-momentum tensor field of a
geometrically induced matter content. It is worth mentioning that the Ricci
tensor field in Eq. (3) in general encodes information about nonmetricity,
since the symmetric connection, $\Gamma$, is not necessarily compatible with
emergent metrics [38, 57].

In the sense of the last paragraph, the polynomial affine model of gravity
provides nontrivial solutions in the absence of matter, i.e., there are
matter-like effects induced by non-Riemannian geometrical quantities.

Lastly, although~the Hamiltonian analysis of the polynomial affine model of
gravity has not been completed, we have reasons to think that in this model
there are propagating degrees of freedom, unlike the three-dimensional version
of general~relativity.

Before finishing, we would like to remark some of the features that, in~four
dimensions, differentiate the polynomial affine model of gravity from
general~relativity.

We know that the four-dimensional version of Polynomial Affine Gravity yields
to gravitational interactions (forces) and vacuum solutions that differ from
those of general relativity. Nevertheless, our vacuum solutions contain
effects that in general relativity can be obtained solely in presence of
matter. Ergo, the~previously mentioned non-Riemannian matter-like effects
might (at least partially) account in the four-dimensional context, for~the
dark sector of the~Universe.

#### Acknowledgements

The work OCF and ARZ is sponsored by the “Centro Científico y Tecnológico de
Valparaíso” (CCTVal), funded by the Chilean Government through the Centers of
Excellence Base Financing Program of Agencia Nacional de Investigación y
Desarrollo (ANID), by grant ANID PIA/APOYO AFB180002. This work was funded by
ANID Millennium Science Initiative Program ICN2019_044, and benefited from
grant PI_LI_19_02 from the Universidad Técnica Federico Santa María. The work
of FR is funded by ANID BECAS/DOCTORADO NACIONAL 21211990. BG acknowledges
financial support from UTFSM master scholarship No. 034/2021.

We gratefully acknowledge the constructive comments of anonymous referees,
which helped to improve the quality of the final version of the manuscript.

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