∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: Departamento de Física, Universidad Técnica Federico Santa
María, Casilla 110-V, Valparaíso, Chile 22institutetext: Departamento de
Matemáticas, Universidad Técnica Federico Santa María, Casilla 110-V,
Valparaíso, Chile 33institutetext: Instituto de Física, Facultad de Ciencias,
Iguá 4225, esq. Mataojo, 11400 Montevideo, Uruguay.

# Aspects of the polynomial affine model of gravity in three dimensions

With focus in the cosmological solutions

Oscar Castillo-Felisolaaddr1,e1  Oscar Orellanaaddr2  José Perdigueroaddr1
Francisca Ramírezaddr1  Aureliano Skirzewskiaddr3  Alfonso R. Zerwekhaddr1

(Received: date / Accepted: date)

###### Abstract

The polynomial affine gravity is a model that is built up without the explicit
use of a metric tensor field. In this article we reformulate the three-
dimensional model and, given the decomposition of the affine connection, we
analyse the consistently truncated sectors. Using the cosmological ansatz for
the connection, we scan the cosmological solutions on the truncated sectors.
We discuss the emergence of different kinds of metrics.

###### Keywords:

Alternative models of gravity Affine gravity Cosmological models Three
dimensional

††journal: Eur. Phys. J. C

##  1 Introduction

Our current understanding of fundamental physics accounts for four different
interactions, which split into two pillars. On the one hand, the gravitational
interaction is described by General Relativity, which models the spacetime as
a Riemannian manifold $(\mathcal{M},g)$, i.e. its geometrical properties are
tied up to the metric tensor field. On the other hand, the remaining three
interactions are described by gauge theories, whose fundamental field are
connections under transformations of the gauge group, but (covariant) vectors
under the group of diffeomorphisms.

The formulation of the gauge theories requires the choice of a background
manifold, on which the theories stand, and the standard treatment does not
consider back-reaction of the gauge theories to the manifold. Nonetheless, in
General Relativity the metric tensor field defines the geometric properties of
the spacetime, and also mediates the gravitational interaction. Such dual role
of the metric tensor field originates significant differences between these
pillars of fundamental physics, being the most highlighted the quantisation
program. While the attempts to quantise General Relativity have eluded a
consistent culmination, the remaining three interactions are quantisable and
renormalisable, raising what is known as the standard model of particles.

In a quest to prevent the _inconsistent_ quantisation of the gravitational
interaction, a large number of generalisations of General Relativity have been
proposed. Inspired by the seminal work by Palatini
palatini19_deduz_invar_delle_equaz_gravit , a large amount of these
generalisations modelled the spacetime by manifolds whose connections is not
the one of Levi-Civita, dubbed _metric-affine_ models of gravity (see for
example Ref. hehl95_metric_affin_gauge_theor_gravit ). These models rely in
the metric as the field mediating the interaction. However, some have
attempted to model the gravitational interaction through affine theories,
where the mediating field is an affine (linear) connection on the spacetime.
The first proposals of affine gravity were considered by Einstein, Eddington
and Schrödinger einstein23_zur_affin_feldt ; einstein23_theor_affin_field ;
eddington23 ; schroedinger50_space , but those models were left aside because
their manipulation was significantly more complex and offered insufficient
novelty from the phenomenological point of view. More contemporaneous affine
models of gravity were proposed by Kijowski and collaborators
kijowski78_new_variat_princ_gener_relat ;
ferraris81_gener_relat_is_gauge_type_theor ;
ferraris82_equiv_relat_theor_gravit ;
kijowski07_univer_affin_formul_gener_relat , Poplawski
poplawski07_nonsy_purel_affin ;
poplawski07_unified_purel_affin_theor_gravit_elect ;
poplawski14_affin_theor_gravit , Krasnov and collaborators
krasnov07_non_metric_gravit ; krasnov08_non_metric_gravit_i ;
krasnov08_non_metric_gravit_ii ;
krasnov11_pure_connec_action_princ_gener_relat ;
delfino15_pure_connec_formal_gravit_lin ;
delfino15_pure_connec_formal_gravit_feyn , and some of us castillo-
felisola15_polyn_model_purel_affin_gravit ; castillo-
felisola18_einst_gravit_from_polyn_affin_model .

The analysis of gravitational models in lower dimensions, where the number of
parameters decreases in general, serves as a playground to test methods that
might be applied to the four-dimensional models, but also for their
applications in other branches of physics since vector and tensor gauge
theories can be interpreted as the high-temperature limit of four-dimensional
models weinberg76_under . These _simplified_ models stimulate the generation
of new ideas and insights into their four-dimensional counterparts.

The three-dimensional version of General Relativity was firstly considered by
Staruszkiewicz staruszkiewicz63_gravit_theor_three_dimen_space , where a
Schwarzschild-like solution was studied, and a relation between the presence
of massive point particles and conical singularities (which modify the
asymptotic behaviour of the spacetime) was found. The interest for three-
dimensional was left aside due to the lack of propagating degrees of freedom,
until in a series of papers Deser, Jackiw and collaborators showed that adding
a nontrivial, gauge invariant, topological term to the three-dimensional
Einstein–Hilbert action resulted in a massive model for gravity
deser82_three_dimen_massiv_gauge_theor ; deser82_topol_massiv_gauge_theor ;
deser84_three_dimen_cosmol_gravit ; deser84_three_dimen_einst_gravit . The
topological term added in this model, dubbed Deser–Jackiw–Templeton, was the
Chern–Simons term associated to the four-dimensional $\theta$-term build from
the (metric) Riemannian curvature—also known as the Pontryagin density. A
remarkable feature of the Deser–Jackiw–Templeton model is that it was able to
induce a mass for the graviton without a Higgs mechanism and preserving the
(infinitesimal) gauge invariance.

Witten showed in Ref. witten88_dimen_gravit_as_exact_solub_system , that the
three-dimensional gravity modified by the Pontryagin Chern–Simons Lagrangian
is equivalent to a Yang–Mills theory, called Chern–Simons gravity, and also
that its perturbative expansion was renormalisable.111An update on the
original ideas in this paper can be found in Ref.
witten07_three_dimen_gravit_revis . Some years later, the first black hole
solution was found by Bañados, Teitelboim and Zanelli
banados92_black_hole_three_dimen_time , disproving the _triviality_ of
classical three-dimensional gravity and revitalising the interest in the
search of exact solutions garcia-diaz17_exact_solut_three_dimen_gravit  and
further development into their quantum aspects carlip98_quant .
Contemporaneously, Mielke and Baekler generalised the topological massive
model of gravity to include torsion
mielke91_topol_gauge_model_gravit_with_torsion , and extended further to
metric-affine gravity by Tresguerres
tresguerres92_topol_gravit_three_dimen_metric_affin_time .

An interconnection between three-dimensional gravity with other branches of
physics was encountered by Dereli and Verçin in the context of the continuum
theory of lattice defects, through the identification of the dislocation and
disclination line density tensors with torsion and curvature tensors, and the
free-energy density with the Lagrangian of the Deser–Jackiw–Templeton model of
gravity dereli91_gauge_model_amorp_solid_contain_defec_ii .222For a more
recent review, see Ref. lazar10_cartan_spiral_stair_physic_and . More
recently, methods of quantum field theory in curved spaces have been applied
to the analysis of properties of graphene, viewed as a _membrane_ endowed with
an metric induced by its embedding into three-dimensional space
kerner12_inter_flexur_phonon_with_elect_graph ; chaves14_optic_conduc_curved ;
pacheco14_graph_morph_elect_proper_from ; castro-
villarreal17_pseud_field_curved ; oliveira17_signat_curved_qft_effec_optic ;
arrighi19_from_curved_spacet_to_spacet .

From a mathematical point of view, the Deser–Jackiw–Templeton model
generalises the three-dimensional Einstein–Hilbert model by taking into
account global properties of the Riemannian spacetime. The Mielke–Baekler
model relaxes the Riemannian condition, by allowing the spacetime to be
modelled by a Riemann–Cartan manifold, while in the Tresguerres model the
spacetime possesses also non-metricity. In all of these models, the metric
plays a fundamental role in their formulation. However, the existence of
contexts in which the notion of metric is not helpful, e.g. in a phase space
or a moduli space, inspires the search of _gravitational_ models which are
defined in spaces where the notion of displacement and flow is defined, but
not necessarily a notion of length. These spaces are called _affine manifolds_
, and they are the underlying structure supporting the polynomial affine model
of gravity.

The classification of the affinely connected spaces is shown schematically in
the following commutative diagram. The most general affine manifold might not
posses a metric, and thus it is characterised by a connection
$(\mathcal{M},\hat{\nabla})$. When an affine manifold is equipped with a
metric, dubbed metric-affine manifold, the connection can be decomposed into
three contributions—the Levi-Civita connection, the contorsion tensor and the
deflection tensor—, and the manifold is classified according to its curvature
$(\mathcal{R})$, torsion $(\mathcal{T})$ and non-metricity $(\mathcal{Q})$. In
the diagram, Riemannian geometries belong to the sector denoted by
$(\mathcal{R})$, Riemann–Cartan geometries to $(\mathcal{R},\mathcal{T})$,
Weizenböck geometries to $(\mathcal{T})$, et cetera.

${(\mathcal{M},\hat{\nabla})}$${(\mathcal{Q},\mathcal{R},\mathcal{T})}$${(\mathcal{Q},\mathcal{T})}$${(\mathcal{Q},\mathcal{R})}$${(\mathcal{Q})}$${(\mathcal{R},\mathcal{T})}$${(\mathcal{T})}$${(\mathcal{R})}$${(\text{Flat})}$$\scriptstyle{g}$

The use of an affine model allows to explore features that should be
attributed to the local invariance under coordinate transformations,
regardless of the metric structure on the manifold. In this context, the
polynomial affine model of gravity emerges naturally. Even though the four-
dimensional model is being analysed, the three-dimensional version is expected
to be easier to characterise, and eventually solve some issues the affine
models have encountered, such as the coupling of matter.

In this paper we re-state the three-dimensional model of polynomial affine
gravity, firstly proposed in Ref. castillo-
felisola15_polyn_model_purel_affin_gravit , analyse their field equations and
find explicit cosmological solutions. The paper is organised as follows.
Section 2 gives a brief overview of the polynomial affine model of gravity in
three dimensions. Then, in Sec. 3 the field equations are derived, and some
issues regarding their truncation—i.e. restriction to sectors where only a
subset of the fields (irreducible components of the connection) are _turned
on_ —are discussed in Sec. 4. In Sec. 5 we scan the space of solutions which
are compatible with the cosmological principle. Some conclusions are drawn in
the Sec. 6. For completeness we include some appendices, including a
discussion of our notation in A.

##  2 Building the model

The polynomial affine model of gravity was born as an attempt to build up a
theory with the affine connection as sole fundamental field castillo-
felisola15_polyn_model_purel_affin_gravit . Formally, the idea behind the
polynomial affine model of gravity is that spacetime is not a
(pseudo-)Riemannian manifold, but an affinely connected manifold
$(\mathcal{M},\hat{\nabla})$. The strategy is then to consider all possible
terms, allowed by the invariance under diffeomorphisms, as part of the
Lagrangian density. Note that a generic affine connection is a reducible
object under the group of diffeomorphisms, and therefore we can decompose it.

In the absence of a metric tensor field, the affine connection decomposes into
irreducible components as follows

$\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=\hat{\Gamma}_{{(\mu}}{}^{{\lambda}}{}_{{\nu)}}+\hat{\Gamma}_{{[\mu}}{}^{{\lambda}}{}_{{\nu]}}=\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+\mathcal{B}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+{\mathcal{A}}_{[\mu}\delta^{\lambda}_{\nu]},$
(1)

where $\mathcal{B}$ field is the traceless part of the torsion,
${\mathcal{A}}_{\mu}$ is the trace of the torsion, and
$\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}\equiv\hat{\Gamma}_{{(\mu}}{}^{{\lambda}}{}_{{\nu)}}$
is a renaming of the symmetric part of the affine connection. All these
elements transform as tensors under diffeomorphisms, with the exception of the
symmetric part of the affine connection, which must be included in the action
almost exclusively through the covariant derivative.

In order to build an action, we consider the chart-induced basis of the
tangent and cotangent spaces, i.e. $\Set{\partial_{\mu}}$ and
$\Set{\mathop{\mathrm{d}x}\nolimits^{\mu}}$, and the volume form defined as

$\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma}=\mathop{\mathrm{d}x}\nolimits^{\alpha}\wedge\mathop{\mathrm{d}x}\nolimits^{\beta}\wedge\mathop{\mathrm{d}x}\nolimits^{\gamma}.$
(2)

An interesting type of connection are those _compatible_ with the volume, i.e.
$\nabla(\mathop{\mathrm{d}V}\nolimits)=0$, which are said to be equiaffine.
Such compatibility ensures that the Ricci tensor field is symmetric, and the
trace of the curvature tensor vanishes eisenhart27_non_rieman ;
schouten13_ricci ; nomizu94_affin . Although equiaffinity is not demanded in
the following formulation, in Sec. 5 we shall encounter that symmetric
connections compatible with the cosmological principle are necessarily
equiaffine.

Now, with the aid of the irreducible components of the connection and the
volume form, we can write down the most general Lagrangian, i.e. a scalar
density in three dimensions. A _dimensional analysis_ similar to the one
presented in Refs. castillo-felisola18_einst_gravit_from_polyn_affin_model ;
castillo-felisola18_beyond_einstein ; castillo-felisola18_cosmol ; castillo-
felisola20_emerg_metric_geodes_analy_cosmol  shows that the most general
action (up to boundary terms) is given by333Note that this action is
equivalent to the one introduced in Ref. castillo-
felisola15_polyn_model_purel_affin_gravit .

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

In the action, the covariant derivative and the curvature are defined with
respect to the symmetric connection, i.e. $\nabla=\nabla^{\Gamma}$ and
$\mathcal{R}=\mathcal{R}{}^{{\Gamma}}$.

Among the features of the polynomial affine model of gravity we count: (i) The
fundamental field is a connection, like the other fundamental interactions;
(ii) The coupling constant are dimensionless, which is desirable from the view
point of Quantum Field Theory, since the superficial degree of divergence
vanishes; (iii) 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 all the counter-terms have the form of terms already
present in the original action.

##  3 Field equations

We now focus in obtaining the field equations for the fields $\Gamma$,
$\mathcal{B}$ and $\mathcal{A}$, by varying the action in Eq. (3). It is
important to highlight that although the absence of second-class constraints
in Polynomial Affine Gravity has not been proven yet, the structure of the
action suggests that the variational problem is well-posed, and therefore the
field equations below do not consider the existence of affine analogues of the
Gibbons–Hawking–York term.444Analysis of affine analogues to the
Gibbons–Hawking–York term can be found in Refs.
parattu16_bound_term_gravit_action_with_null_bound ; krishnan17_robin_gravit ;
krishnan17_neuman_bound_term_gravit ; lehner16_gravit_action_with_null_bound ;
hopfmueller17_gravit_degrees_freed_null_surfac ;
jubb17_bound_corner_terms_action_gener_relat .

Since the action contains up to first derivatives of the fields, the field
equations are obtained through the Euler-Lagrange equations,

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

We proceed utilising the formalism introduced by Kijowski in Ref.
kijowski78_new_variat_princ_gener_relat , and following the steps from Ref.
castillo-felisola20_emerg_metric_geodes_analy_cosmol .

It can be shown with ease that the Euler–Lagrange equations (4) can be
rewritten as

$\displaystyle\nabla_{\mu}{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$
$\displaystyle=\frac{\partial^{*}\mathscr{L}}{\partial\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}},$
$\displaystyle\nabla_{\mu}{{\Pi}_{\mathcal{B}}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}},$
$\displaystyle\nabla_{\mu}{{\Pi}_{{\mathcal{A}}}{}^{\mu\nu}}$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial{\mathcal{A}}_{\nu}},$ (5)

where the quantities $\Pi_{X}$ are the canonical momentum associated to the
field $X$—which are tensor densities—, defined as

$\displaystyle{{\Pi}_{\Gamma}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial\,(\partial_{\mu}\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}})},$
$\displaystyle{{\Pi}_{\mathcal{B}}{}^{\mu\nu}{}_{\lambda}{}^{\rho}}$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial\,(\partial_{\mu}\mathcal{B}_{{\nu}}{}^{{\lambda}}{}_{{\rho}})},$
$\displaystyle{{\Pi}_{{\mathcal{A}}}{}^{\mu\nu}}$
$\displaystyle=\frac{\partial\mathscr{L}}{\partial\,(\partial_{\mu}{\mathcal{A}}_{\nu})},$
(6)

and the asterisk on the right-hand side of the field equation for the
symmetric part of the connection in Eq. (5) denotes the partial derivative
with respect to the connection that is not contained in the curvature tensor.

The field equations for the fields $\mathcal{A}$, $\mathcal{B}$ and $\Gamma$
derived from the action in Eq. (3) are respectively:

$2B_{1}\mathcal{A}_{\alpha}\mathcal{B}_{{\nu}}{}^{{\alpha}}{}_{{\rho}}+2B_{2}{\mathcal{F}}_{\nu\rho}+B_{3}\nabla_{\mu}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\rho}}+B_{5}\mathcal{R}_{{\nu\rho}}{}^{{\mu}}{}_{{\mu}}=0,$
(7)
$2B_{1}\mathcal{A}_{\nu}\mathcal{A}_{\rho}-2B_{3}\nabla_{(\nu}\mathcal{A}_{\rho)}+3B_{4}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}+2B_{6}\mathcal{R}_{{\mu(\nu}}{}^{{\mu}}{}_{{\rho)}}=0,$
(8)
$B_{3}\mathcal{A}_{\mu}\mathcal{B}_{{\rho}}{}^{{\nu}}{}_{{\sigma}}+B_{5}\left(\delta_{\mu}^{\nu}{\mathcal{F}}_{\rho\sigma}+\delta_{[\rho}^{\nu}{\mathcal{F}}_{\sigma]\mu}\right)+B_{6}\left(2\delta_{\mu}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\rho}}{}^{{\tau}}{}_{{\sigma}}}+\delta_{\rho}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\mu}}}+\delta_{\sigma}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\mu}}{}^{{\tau}}{}_{{\rho}}}\right)+B_{7}\left(\delta_{\mu}^{\nu}\mathcal{R}_{{\rho\sigma}}{}^{{\lambda}}{}_{{\lambda}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\mu}}{}^{{\lambda}}{}_{{\lambda}}\right)+B_{8}\left(\mathcal{R}_{{\rho\sigma}}{}^{{\nu}}{}_{{\mu}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\lambda}}{}^{{\lambda}}{}_{{\mu}}\right)=0.$
(9)

Note that the field equation for the $\mathcal{B}$-field, i.e. Eq. (8), is
similar to the Ricci form of the Einstein field equations. Particularly, it
has been shown that the tensor
$\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}$
(when non-degenerated) might be interpreted as a torsion-descendent metric
tensor field poplawski14_affin_theor_gravit .

##  4 Truncations

Unlike the analysis of the truncations in the four-dimensional polynomial
affine model of gravity castillo-
felisola18_einst_gravit_from_polyn_affin_model ; castillo-felisola18_cosmol ;
castillo-felisola20_emerg_metric_geodes_analy_cosmol , the presence of the
Chern–Simons terms in (3) allows to switch-off certain fields at the level of
the action, and not just at the level of the field equations. In order to
distinguish these two possible _truncations_ of the model, we refer to them as
off-shell and on-shell limits. Below we list six possible truncations of the
model.

###  4.1 Torsion-free limit

On the one hand, taking the off-shell torsion-free limit yields an effective
action which is the sum of two Chern-Simons terms,

$S_{\text{off}}[\Gamma]=\int\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma}\left[B_{7}\,\Gamma_{{\alpha}}{}^{{\mu}}{}_{{\mu}}\partial_{\beta}\Gamma_{{\gamma}}{}^{{\nu}}{}_{{\nu}}+B_{8}\,\left(\Gamma_{{\alpha}}{}^{{\mu}}{}_{{\nu}}\partial_{\beta}\Gamma_{{\gamma}}{}^{{\nu}}{}_{{\mu}}+\frac{2}{3}\Gamma_{{\alpha}}{}^{{\mu}}{}_{{\nu}}\Gamma_{{\beta}}{}^{{\nu}}{}_{{\lambda}}\Gamma_{{\gamma}}{}^{{\lambda}}{}_{{\mu}}\right)\right],$
(10)

whose field equations are

$B_{7}\left(\delta_{\mu}^{\nu}\mathcal{R}_{{\rho\sigma}}{}^{{\lambda}}{}_{{\lambda}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\mu}}{}^{{\lambda}}{}_{{\lambda}}\right)+B_{8}\left(\mathcal{R}_{{\rho\sigma}}{}^{{\nu}}{}_{{\mu}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\lambda}}{}^{{\lambda}}{}_{{\mu}}\right)=0.$

Note that the contribution to the field equations of the term with coefficient
$B_{7}$ vanishes identically for equiaffine connections,555For equiaffine
connections the skew-symmetric part of the Ricci tensor field vanishes. From
the first (or algebraic) Bianchi identity, it follows that the trace of the
curvature vanishes. Therefore,
$\mathcal{R}_{{\mu\nu}}{}^{{\alpha}}{}_{{\alpha}}=\partial_{[\mu}\Gamma_{{\nu]}}{}^{{\alpha}}{}_{{\alpha}}=0$.
In the remaining of the article we shall ignore the contribution of this term
to the field equations, however, we keep the term in the action because it
might contribute to topological quantities. and thus the field equations for
the symmetric connection are

$B_{8}\left(\mathcal{R}_{{\rho\sigma}}{}^{{\nu}}{}_{{\mu}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\lambda}}{}^{{\lambda}}{}_{{\mu}}\right)=0.$
(11)

These field equations require that the connection is projectively Weyl-
flat.666Weyl introduced two different notions of curvature, both of therm are
referred as Weyl’s tensors, and sometimes they are called _projective_ and
_conformal_ Weyl tensor weyl21_zur_infin . As physicists we are custom to the
conformal Weyl tensor, which might is the curvature without traces (with
respect to the metric). The projective Weyl tensor might be defined without
requiring a metric, and it is invariant under the projective transformations
of the connection,
$\tilde{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}+\delta^{\lambda}_{\mu}V_{\nu}+\delta^{\lambda}_{\nu}V_{\mu}$.
See Refs. eisenhart27_non_rieman ; nomizu94_affin ;
luebbe13_note_coinc_projec_confor_weyl_tensor .

On the other hand, when one takes the on-shell torsion-free sector, there are
subsidiary equations,

$\mathcal{R}_{{\nu\rho}}{}^{{\mu}}{}_{{\mu}}=0\text{ and
}\mathcal{R}_{{\mu(\nu}}{}^{{\mu}}{}_{{\rho)}}=0,$

the first is satisfied when the connection is equiaffine, while the second
(Ricci-flatness) restricts the solutions of the system to be flat connections.

###  4.2 Vectorial torsion

The restriction to sole vectorial torsion in the off-shell limit yields a
Chern–Simons-like action, whose field equations are

${\mathcal{F}}_{\nu\rho}=0,$ (12)

while the on-shell truncation raises the auxiliary condition

$\displaystyle
2B_{1}\mathcal{A}_{\nu}\mathcal{A}_{\rho}-2B_{3}\partial_{(\nu}\mathcal{A}_{\rho)}$
$\displaystyle=0.$ (13)

The system of equations (12) and (13) can be solved in general. First,
equation (12) implies that locally the field $\mathcal{A}$ is an exact
$1$-form, $\mathcal{A}=\mathop{\mathrm{d}\phi}\nolimits$. Secondly, equation
(13) can be written in the form

$\partial_{\nu}\partial_{\rho}f=0,\text{ with
}f=e^{-\frac{B_{1}}{B_{3}}\phi}.$

The above equation implies that $f$ is a linear function of the coordinates,
and allows to solve for $\phi$ and therefore $\mathcal{A}$,

$\phi(x)=-\frac{B_{3}}{B_{1}}\ln\left(D+C_{\mu}x^{\mu}\right)\text{ and
}\mathcal{A}_{\nu}(x)=-\frac{B_{3}}{B_{1}}\frac{C_{\nu}}{D+C_{\mu}x^{\mu}}.$

Note that the field equations (12), implies that the $\mathcal{A}$-field
inherits a gauge redundancy. This gauge redundancy disappears with the
existence of the $\mathcal{B}$-field, even if it vanishes, i.e. $\mathcal{B}$
breaks the gauge redundancy of $\mathcal{A}$.

###  4.3 Trace-less torsion

Interestingly, the off-shell truncation to trace-less torsion leaves an
effective action whose sole term is that with coefficient $B_{4}$, which
provides no dynamics to the $\mathcal{B}$-field. The field equations are

$\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}=0.$
(14)

The on-shell truncation yields the subsidiary equations,

$\partial_{\mu}\mathcal{B}_{{\rho}}{}^{{\mu}}{}_{{\sigma}}=0\text{ and
}\partial_{[\mu}\mathcal{B}_{{\rho}}{}^{{\nu}}{}_{{\sigma]}}=0.$ (15)

Note that these equations are equivalent,777Contracting the later with
$\epsilon^{\mu\rho\sigma}$ yields the former. and their equivalency can be
extended to the covariant version of the equations.

Equation (14) can be rewritten in terms of the quasi-Hodge dual
vaz16_introd_cliff_algeb_spinor ,

$T^{\mu\alpha}=\frac{1}{2}\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}\epsilon^{\alpha\beta\gamma},$
(16)

giving a cofactor equation for the tensor $T$. Hence,
$T^{\mu\alpha}=\rho(x)V^{\mu}V^{\alpha}$ is the general solution where $\rho$
is a scalar density, and therefore

$\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\lambda}}=\rho(x)V^{\mu}V^{\sigma}\epsilon_{\sigma\nu\lambda}.$

The structure of our tensor $T^{\mu\alpha}$ is equivalent to that of the
energy-momentum tensor for cold matter (i.e. _dust_), where $\rho$ is the
energy density and $V^{\mu}$ represents the velocity of the matter
distribution. Similarly, the Eq. (15) is a generalisation of the continuity
equation and energy-momentum conservation,

$0=\partial_{\mu}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\lambda}}=\left((\partial_{\mu}\rho)V^{\mu}V^{\sigma}+\rho(\partial_{\mu}V^{\mu})V^{\sigma}+\rho
V^{\mu}(\partial_{\mu}V^{\sigma})\right)\epsilon_{\sigma\nu\lambda}=\left[\rho\nabla_{V}V+(\nabla\cdot(\rho
V))V\right]^{\sigma}\epsilon_{\sigma\nu\lambda}.$

The last line is the equation for a self-parallel vector, written in with a
non-canonical affine parameter. The expression in brackets corresponds to the
divergence of the tensor $T$, i.e. $\partial_{\mu}T^{\mu\sigma}$, which
represents—in our analogue with the cold matter energy-momentum tensor—the
momentum conservation. However, in General Relativity the each term in the
bracket vanishes independently, corresponding to the geodesic and continuity
equations respectively.

###  4.4 Symmetric connection with vectorial torsion

The _off-shell_ limit, yielding an effective action that is the sum of the
three Chern-Simons terms (whose coupling constants are $B_{2}$, $B_{7}$ and
$B_{8}$) plus an interaction coming from the term whose coupling constant is
$B_{5}$,

$0=2B_{2}{\mathcal{F}}_{\nu\rho}+B_{5}\mathcal{R}_{{\nu\rho}}{}^{{\mu}}{}_{{\mu}},$
(17)
$0=B_{5}\left(\delta_{\mu}^{\nu}{\mathcal{F}}_{\rho\sigma}+\delta_{[\rho}^{\nu}{\mathcal{F}}_{\sigma]\mu}\right)+B_{7}\left(\delta_{\mu}^{\nu}\mathcal{R}_{{\rho\sigma}}{}^{{\lambda}}{}_{{\lambda}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\mu}}{}^{{\lambda}}{}_{{\lambda}}\right)+B_{8}\left(\mathcal{R}_{{\rho\sigma}}{}^{{\nu}}{}_{{\mu}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\lambda}}{}^{{\lambda}}{}_{{\mu}}\right).$
(18)

Since Eq. (17) relates the field strength ${\mathcal{F}}$ with the trace of
the curvature tensor, Eq. (18) become an equation for just curvature objects.
In particular, if the coupling constants satisfy

$\left(\frac{2B_{2}B_{7}-B_{5}^{2}}{2B_{2}B_{8}}\right)=\frac{1}{4},$ (19)

equation (18) coincides with Weyl’s projective curvature tensor field, which
in three dimensions is

$\mathcal{W}_{\mu\nu}\,^{\lambda}\,{}_{\rho}=\mathcal{R}_{\mu\nu}\,^{\lambda}\,{}_{\rho}-\frac{1}{4}\delta^{\lambda}_{\rho}\mathcal{R}_{{\mu\nu}}{}^{{\sigma}}{}_{{\sigma}}-\frac{1}{2}\left(\mathcal{R}_{{\sigma\nu}}{}^{{\sigma}}{}_{{\rho}}\delta^{\lambda}_{\mu}-\mathcal{R}_{{\sigma\mu}}{}^{{\sigma}}{}_{{\rho}}\delta^{\lambda}_{\nu}\right)-\frac{1}{8}\left(\delta^{\lambda}_{\mu}\mathcal{R}_{{\nu\rho}}{}^{{\sigma}}{}_{{\sigma}}-\delta^{\lambda}_{\nu}\mathcal{R}_{{\mu\rho}}{}^{{\sigma}}{}_{{\sigma}}\right).$

Therefore, in this sector field equations describe a symmetric projectively-
flat connection. A projectively-flat connection is _locally_ written as

$\Gamma_{\nu}\,^{\lambda}\,{}_{\rho}=\delta_{\nu}^{\lambda}\psi_{\rho}+\delta_{\rho}^{\lambda}\psi_{\nu},$

where $\psi_{\mu}$ is a generic (differentiable) vector field. From Eq. (17),
the $\mathcal{A}$-field is proportional to $\psi_{\mu}$,

$\mathcal{A}_{\mu}=-\frac{2B_{5}}{B_{2}}\psi_{\mu}.$

Note that if the coefficients do not satisfy Eq. (19), the two equations in
(18) are incompatible unless the trace of the curvature and the field strength
vanish independently, i.e.

$\psi_{\mu}=\partial_{\mu}\alpha\text{ and
}\mathcal{A}_{\mu}=\partial_{\mu}\beta.$

In the on-shell limit the system of field equations are enriched by the
additional equation,

$B_{1}\mathcal{A}_{\nu}\mathcal{A}_{\rho}-B_{3}\nabla_{(\nu}\mathcal{A}_{\rho)}+B_{6}\mathcal{R}_{{\mu(\nu}}{}^{{\mu}}{}_{{\rho)}}=0,$
(20)

Note that the nontrivial part of the equation comes from the symmetric part,
which is an Einstein-like equation.

###  4.5 Symmetric connection with trace-less torsion

The off-shell restriction to this sector involves the terms in the action, in
Eq. (3), with coefficients from $B_{4}$, $B_{6}$, $B_{7}$ and $B_{8}$. The
field equations on this off-shell limit are,

$3B_{4}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}+2B_{6}\mathcal{R}_{{\mu(\nu}}{}^{{\mu}}{}_{{\rho)}}=0,$
(21)
$B_{6}\left(2\delta_{\mu}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\rho}}{}^{{\tau}}{}_{{\sigma}}}+\delta_{\rho}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\mu}}}+\delta_{\sigma}^{\nu}\nabla_{\tau}{\mathcal{B}_{{\mu}}{}^{{\tau}}{}_{{\rho}}}\right)+B_{7}\left(\delta_{\mu}^{\nu}\mathcal{R}_{{\rho\sigma}}{}^{{\lambda}}{}_{{\lambda}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\mu}}{}^{{\lambda}}{}_{{\lambda}}\right)+B_{8}\left(\mathcal{R}_{{\rho\sigma}}{}^{{\nu}}{}_{{\mu}}+\delta_{[\rho}^{\nu}\mathcal{R}_{{\sigma]\lambda}}{}^{{\lambda}}{}_{{\mu}}\right)=0.$
(22)

In the on-shell limit, there is an extra subsidiary condition,

$B_{3}\nabla_{\mu}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\rho}}+B_{5}\mathcal{R}_{{\nu\rho}}{}^{{\mu}}{}_{{\mu}}=0.$
(23)

Note that for volume-preserving connections, Eq. (23) is nothing but the
continuity equation of the $T$-tensor (quasi-Hodge dual of the
$\mathcal{B}$-field) which might be interpreted as a conserved “energy-
momentum tensor”, or in case of being non-degenerated it admits the
interpretation of a compatible _inverse metric_ tensor field. Similarly, the
term accompanying the coupling constant $B_{7}$ in Eq. (22) vanishes.

Equation (21) is an Einstein-like equation, where the term
$\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}$
behaves like a torsion-descendent metric.888Along the paper we might refer to
this torsion-descendent metric tensor field as the _Poplawski metric_ , due to
the analysis of such geometrical object by N. Poplawski in Ref.
poplawski14_affin_theor_gravit .

###  4.6 The torsion sector

The off-shell limit to the torsion sector of our model is given by restricting
the action (3) to the terms with coefficients from $B_{1}$ to $B_{4}$. The
field equations in this limit are

$2B_{1}\mathcal{A}_{\alpha}\mathcal{B}_{{\nu}}{}^{{\alpha}}{}_{{\rho}}+2B_{2}{\mathcal{F}}_{\nu\rho}+B_{3}\partial_{\mu}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\rho}}=0,$
(24)
$2B_{1}\mathcal{A}_{\nu}\mathcal{A}_{\rho}-2B_{3}\partial_{(\nu}\mathcal{A}_{\rho)}+3B_{4}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}=0.$
(25)

However, the on-shell restriction to the torsion sector yields an extra
condition,

$B_{3}\mathcal{A}_{\mu}\mathcal{B}_{{\rho}}{}^{{\nu}}{}_{{\sigma}}+B_{5}\left(\delta_{\mu}^{\nu}{\mathcal{F}}_{\rho\sigma}+\delta_{[\rho}^{\nu}{\mathcal{F}}_{\sigma]\mu}\right)+B_{6}\left(2\delta_{\mu}^{\nu}\partial_{\tau}{\mathcal{B}_{{\rho}}{}^{{\tau}}{}_{{\sigma}}}+\delta_{\rho}^{\nu}\partial_{\tau}{\mathcal{B}_{{\sigma}}{}^{{\tau}}{}_{{\mu}}}+\delta_{\sigma}^{\nu}\partial_{\tau}{\mathcal{B}_{{\mu}}{}^{{\tau}}{}_{{\rho}}}\right)=0.$
(26)

##  5 A scan of cosmological solutions

###  5.1 Ansatz for the connection

In order to develop further aspects of the three-dimensional polynomial affine
model of gravity, we have to provide an ansatz for the connection. The ansatz
are found by solving the equations derived from the vanishing Lie derivative
of the connection, i.e.

$\pounds_{V}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=V^{\sigma}\partial_{\sigma}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}-\hat{\Gamma}_{{\mu}}{}^{{\sigma}}{}_{{\nu}}\partial_{\sigma}V^{\lambda}+\hat{\Gamma}_{{\sigma}}{}^{{\lambda}}{}_{{\nu}}\partial_{\mu}V^{\sigma}+\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\sigma}}\partial_{\nu}V^{\sigma}+\frac{\partial^{2}V^{\lambda}}{\partial
x^{\mu}\partial
x^{\nu}}=\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}V^{\lambda}+\hat{\mathcal{R}}_{{\rho\mu}}{}^{{\lambda}}{}_{{\nu}}V^{\rho}-\hat{\nabla}_{\mu}\left(\mathcal{T}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}V^{\rho}\right)=0,$
(27)

where $V$ represents a vector associated to the generators of the symmetry
group, i.e. each $V$ defines a symmetry flow.

Two physically interesting cases are the _isotropic_ connection, which is
required to analyse spherical configurations, e.g. black hole solutions, and
the _isotropic and homogeneous_ connection, that is compatible with the
cosmological principle, and therefore required to build cosmological models.

The vector fields generating the described symmetry groups are

$\displaystyle J$ $\displaystyle=\begin{pmatrix}0&0&1\end{pmatrix},$
$\displaystyle X$ $\displaystyle=\sqrt{1-\kappa
r^{2}}\begin{pmatrix}0&\cos\varphi&-\frac{1}{r}\sin\varphi\end{pmatrix},$
$\displaystyle Y$ $\displaystyle=\sqrt{1-\kappa
r^{2}}\begin{pmatrix}0&\sin\varphi&\frac{1}{r}\cos\varphi\end{pmatrix},$ (28)

where $J$ is associated with the sole angular momentum defining the isotropy,
while $X$ and $Y$ are associated with the translations defining the
homogeneity.

####  5.1.1 Isotropic connection

Since the components of the vector field $J$ are constants, the Lie derivative
of the connection along its flow is equal to that of a $\binom{1}{2}$-tensor.
The vanishing Lie derivative along the vector field $J$ yields

$\pounds_{J}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=\partial_{\varphi}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=0,$
(29)

i.e. this symmetry does not relate different components of the connection, but
its dependence on the coordinates.

At this stage, without mentioning the field equations for the connection, if
one would try to find an analogue of the Schwarzschild solution, there are
_twenty seven_ functions to be determined.

####  5.1.2 Isotropic and homogeneous connection

Now, we require that the restricted components of the connection in Eq. (29),
are symmetric with respect to the vector fields $X$ and $Y$.999Note that due
to the isotropy, once calculating both Lie derivatives is redundant, therefore
we need to calculate and solve only one of them. The procedure to solve the
equations $\pounds_{X}\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}=0$ is tedious but
straightforward (see for example Refs. castillo-felisola18_beyond_einstein ;
castillo-felisola18_cosmol ), hence we only show the results below. This time,
we split the affine connection into its irreducible components,

$\displaystyle\Gamma_{{t}}{}^{{t}}{}_{{t}}$ $\displaystyle=j(t),$
$\displaystyle\Gamma_{{i}}{}^{{t}}{}_{{j}}$ $\displaystyle=g(t)S_{ij},$ (30)
$\displaystyle\Gamma_{{i}}{}^{{k}}{}_{{j}}$
$\displaystyle=\gamma_{i}{}^{k}{}_{j},$
$\displaystyle\Gamma_{{t}}{}^{{i}}{}_{{j}}$
$\displaystyle=\Gamma_{{j}}{}^{{i}}{}_{{t}}=h(t)\delta^{i}_{j}+f(t)S^{ik}\epsilon_{kj}\frac{r}{\sqrt{1-\kappa
r^{2}}},$

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,101010It is worth
mentioning that the general two-dimensional isotropic and homogeneous
(covariant and contravariant) tensors posses a nonvanishing skew-symmetric
component. This off-diagonal component allows us to write the three-
dimensional torsion full connection in the same form of Eq. (30) but where the
matrices $S$ and $S^{-1}$ are no longer symmetric. defined by

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

and

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

It is important to highlight the existence of an unexpected function in the
components of the affine connection, to know, the $f$-function, which might be
introduced solely in the three-dimensional case. Furthermore, as shown in B,
the function $j$ can be set to zero by a reparametrisation of the time
coordinate.

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

$\displaystyle\mathcal{B}_{{\varphi}}{}^{{t}}{}_{{r}}=-\mathcal{B}_{{r}}{}^{{t}}{}_{{\varphi}}$
$\displaystyle=\xi(t)\frac{r}{\sqrt{1-\kappa r^{2}}},$
$\displaystyle\mathcal{B}_{{t}}{}^{{r}}{}_{{\varphi}}=-\mathcal{B}_{{\varphi}}{}^{{r}}{}_{{t}}$
$\displaystyle=\psi(t)r\sqrt{1-\kappa r^{2}},$
$\displaystyle\mathcal{B}_{{r}}{}^{{\varphi}}{}_{{t}}=-\mathcal{B}_{{t}}{}^{{\varphi}}{}_{{r}}$
$\displaystyle=\frac{\psi(t)}{r\sqrt{1-\kappa r^{2}}},$ (31)

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

###  5.2 Curvature of the symmetric connection

From Eq. (30), the components of curvature are calculated, yielding

$\begin{split}\mathcal{R}_{{ti}}{}^{{t}}{}_{{j}}&=-\mathcal{R}_{{it}}{}^{{t}}{}_{{j}}=\left(\dot{g}-gh\right)S_{ij}-fg\frac{r}{\sqrt{1-\kappa
r^{2}}}\epsilon_{ij},\\\
\mathcal{R}_{{ij}}{}^{{t}}{}_{{t}}&=2fg\frac{r}{\sqrt{1-\kappa
r^{2}}}\epsilon_{ij},\\\
\mathcal{R}_{{ti}}{}^{{j}}{}_{{t}}&=-\mathcal{R}_{{it}}{}^{{j}}{}_{{t}}=\left(\dot{h}+h^{2}-f^{2}\right)\delta_{i}^{j}+\left(\dot{f}+2fh\right)\frac{r}{\sqrt{1-\kappa
r^{2}}}S^{jk}\epsilon_{ki},\\\
\mathcal{R}_{{ij}}{}^{{k}}{}_{{l}}&=2(gh+\kappa)S_{l[j}\delta_{i]}^{k}-fg\frac{r}{\sqrt{1-\kappa
r^{2}}}\epsilon_{ij}\delta_{l}^{k}.\end{split}$ (32)

It follows that the trace of the curvature vanishes,
$\mathcal{R}_{{\mu\nu}}{}^{{\sigma}}{}_{{\sigma}}=0$, and therefore our
connection is _equiaffine_. Hence, the Ricci tensor field is symmetric.

The nonvanishing components of the Ricci tensor field are,

$\displaystyle\mathcal{R}_{{tt}}$
$\displaystyle=-2\left(\dot{h}+h^{2}-f^{2}\right),$
$\displaystyle\mathcal{R}_{{ij}}$
$\displaystyle=\left(\dot{g}+\kappa\right)S_{ij}.$ (33)

The three-dimensional Weyl projective curvature for an equiaffine connection
is then,

$\mathcal{W}_{\mu\nu}{}^{\lambda}{}_{\rho}=\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}-\frac{1}{2}\left(\mathcal{R}_{{\nu\rho}}\delta_{\mu}^{\lambda}-\mathcal{R}_{{\mu\rho}}\delta_{\nu}^{\lambda}\right),$
(34)

Note that since in general three-dimensional connections are not projectively
flat, the curvature cannot be resolved in terms of the Ricci tensor, unlike
what it is expected from General Relativity.

Before turn over solving the differential equations, there are some
interesting remarks:

  * •

The $f^{2}$-function is a _parameter_ for the system of ordinary differential
equations for isotropic and homogeneous Ricci-flat spaces.

  * •

Isotropic and homogeneous flat spaces require that either $f$ or $g$ vanishes.

  * •

The dynamical equations for $f$ and $g$ are _always_ integrable in terms of
$h$ and $\kappa$.

  * •

The dynamics of $h$ is determined by a Riccati ordinary differential equation,
which is the same for both flat and Ricci-flat spaces.

Since the Riccati equation is not solvable in general, we may characterise the
solutions of the system of ordinary differential equations by the partial
solutions of the Riccati equation.

###  5.3 Space of solutions

The substitution of the cosmological ansatz for the connection into the field
equations in Eqs. (7), (8) and (9), yields the system [noalign]

$2B_{8}gf-B_{3}\xi\eta=0,$ (35) $B_{6}(2g\psi+\dot{\xi})-B_{8}fg=0,$ (36)
$B_{8}(2gh+\kappa-\dot{g})=0,$ (37) $B_{3}\eta\psi+B_{8}(2hf+\dot{f})=0,$ (38)
$B_{1}\eta^{2}-B_{3}\dot{\eta}+3B_{4}\psi^{2}-2B_{6}(\dot{h}+h^{2}-f^{2})=0,$
(39) $B_{3}g\eta-3B_{4}\psi\xi+B_{6}(\kappa+\dot{g})=0,$ (40)
$2B_{1}\eta\xi+B_{3}(2g\psi+\dot{\xi})=0.$ (41)

Note that the Eq. (35) is algebraic, from Eqs. (36) and (41) an algebraic
relation is obtained after eliminating the expression $2g\psi+\dot{\xi}$, and
similarly from Eqs. (37) and (40) after eliminating $\dot{g}$. These
expressions are [noalign]

$2B_{8}gf-B_{3}\xi\eta=0,$ (42)
$\frac{B_{8}}{B_{6}}gf+2\frac{B_{1}}{B_{3}}\xi\eta=0,$ (43)
$3\frac{B_{4}}{B_{6}}\psi\xi-2gh-\frac{B_{3}}{B_{6}}g\eta=2\kappa.$ (44)

Equations (42) and (43) can be seen as a system of equations for the variables
$f$ and $\eta$ as functions of $g$ and $\xi$, but there independence is
dictated by the determinant of the system coefficients,

$\Omega=B_{8}\left[4\frac{B_{1}}{B_{3}}+\frac{B_{3}}{B_{6}}\right]g\xi.$ (45)

We explore the space of cosmological solutions considering all possible cases,
which can be represented by _decision trees_. On the one hand, when the
determinant in Eq. (45) is nonvanishing, it follows that $f=\eta=0$, and also
that both $g$ and $\xi$ are nonvanishing. The field equations allow to
parametrise $\psi=\psi(g,h,\xi)$, and thus the final system reduces to a set
of three equations and three unknowns.

On the other hand, when the determinant $\Omega$ vanishes, following the same
approach, the decision tree has more branches.111111Note that in the decision
tree we do not include branches for the cases where the parameters of the
model are constrained, i.e. the case $B_{8}=0$ and $B_{3}^{2}+4B_{1}B_{6}=0$.

The solutions obtained after exhausting the branches of the decision trees are
presented below, but they are grouped according to the fields that are
nontrivial.

###  5.4 Torsion-free limit

Consider first the field equation (11). Although at first sight the solution
seems that spacetime is flat, when working the expression carefully, one
notice that the requirements are that,

$\displaystyle fg$ $\displaystyle=0,$ $\displaystyle\dot{g}-2gh-\kappa$
$\displaystyle=0,$ $\displaystyle\dot{f}+2fh$ $\displaystyle=0.$ (46)

However, the extra equation obtained when we take the _on-shell_ torsion-free
limit, i.e. Ricci-flatness, is satisfied simultaneously if and only if the
connection is flat.

####  5.4.1 Projectively-flat solutions

A projectively-flat connection requires that

$\displaystyle gf$ $\displaystyle=0$ $\displaystyle\dot{f}+2fh$
$\displaystyle=0$ $\displaystyle\dot{g}-2gh-\kappa$ $\displaystyle=0$ (47)

There are two _branches_ of solutions, with either $g=0$ or $f=0$. None of
these _branches_ requires the connection to be flat. Additionally, since $h$
is a non-dynamical function, it is possible to solve the field equations in
terms of an $h$-parameter function.

The branch $g=0$, requires vanishing $\kappa$. In this scenario, the
$f$-function is solved by

$f(t)=C_{f}e^{-2H(t)},$

where $H(t)=\int_{t_{0}}^{t}\mathop{\mathrm{d}\tau}\nolimits h(\tau)$.

In the branch $f=0$, again the $h$-function plays the role of a parameter
function, and the _dynamical_ function is given by

$g(t)=e^{2H(t)}\left(C_{g}+\kappa\int_{t_{0}}^{t}\mathop{\mathrm{d}\tau}\nolimits
e^{-2H(\tau)}\right),$

with $C_{g}$ the integration constant.

Note that the case $f=g=0$ leaves the function $h$ undetermined, but it is
compatible with both of the previous branches.

####  5.4.2 Flat solutions

As mentioned previously, flat connections solve the field equations obtained
from the on-shell torsion-less limit of the polynomial affine model of
gravity.

From Eqs. (32), the system of ordinary differential equations defining
isotropic and homogeneous affinely connected flat spaces are

$\displaystyle fg$ $\displaystyle=0,$ $\displaystyle\dot{g}-gh$
$\displaystyle=0,$ $\displaystyle gh+\kappa$ $\displaystyle=0,$
$\displaystyle\dot{h}+h^{2}-f^{2}$ $\displaystyle=0,$
$\displaystyle\dot{f}+2fh$ $\displaystyle=0.$

As in the previous case, there are two branches of solutions. The branch with
$f=0$, the field equations are solved by

$h(t)=\frac{1}{t+C_{h}},\quad g(t)=-\kappa(t+C_{h}),$ (48)

with $C_{h}$ the integration constant of the equation for the $h$-function.

The second branch requires $g=0\land\kappa=0$. Since the case $f=0$ is
included the other branch, we restrict to $f\neq 0$. Therefore, the
$f$-function can be integrated in terms of $h$,

$f(t)=e^{-2H(t)}.$

Then, the field equation for $h$ turns into a second order differential
equation. The solution of the system is given by

$\displaystyle h(t)$ $\displaystyle=\frac{t+c_{1}}{(t+c_{1})^{2}+1},$
$\displaystyle f(t)$ $\displaystyle=\frac{1}{(t+c_{1})^{2}+1}.$ (49)

####  5.4.3 Ricci-flat solutions

Ricci-flat connections solve the field equations from the on-shell torsion-
free limit of polynomial affine gravity solely if the coupling constant
$B_{8}$ is zero.

From Eqs. (33), the system of ordinary differential equations defining
isotropic and homogeneous affinely connected Ricci-flat spaces are

$\displaystyle\dot{h}+h^{2}-f^{2}$ $\displaystyle=0,$
$\displaystyle\dot{g}+\kappa$ $\displaystyle=0.$ (50)

The solution for the differential equation for $g$ is

$g(t)=-\kappa t+C_{g}.$

The first equation in Eq. (50) is a Riccati ordinary differential equation, in
which the $f$-function plays the role of _parameter_ function. A well-known
strategy to solve the Riccati equation is to transform it into a second order
linear differential equation. The transformation $u(t)=e^{H(t)}$, takes it
onto

$\ddot{u}-f^{2}u=0.$ (51)

Equation (51) can be immediately compared with the one-dimensional time-
independent Schrödinger equation, where $u$ would be the wave function, the
$f^{2}$-function plays the role of the quantum mechanical potential minus the
energy eigenvalue.

###  5.5 Purely vectorial sector

The off-shell limit toward the purely vectorial sector is dominated by the
term whose coefficient is $B_{2}$. The field equations are identically
satisfied, given that for $\mathcal{A}_{\mu}=\eta(t)\delta^{0}_{\mu}$ its
field strength vanishes, i.e. the field equations impose no restriction to the
function $\eta$.

On the other hand, the subsidiary condition coming from the on-shell limit is

$B_{3}\,\dot{\eta}-B_{1}\,\eta^{2}=0,$

whose solution is

$\eta(t)=-\frac{B_{3}}{B_{1}}\frac{1}{t+C_{\eta}}.$ (52)

###  5.6 Purely traceless torsion sector

In this sector, the field equations from the off-shell limit become
$\psi^{2}=\psi\xi=0$, whose solution is driven by

$\psi=0,\qquad\xi=\xi(t),$ (53)

whose solution is driven by $\psi(t)=0$, while $\xi(t)$ remains as an unknown
function.

When one considers the on-shell limit, the system of field equations is

$\displaystyle\dot{\xi}$ $\displaystyle=0,$ $\displaystyle\kappa$
$\displaystyle=0,$ $\displaystyle\psi^{2}$ $\displaystyle=0,$
$\displaystyle-3B_{4}\,\psi\xi+B_{6}\,\kappa$ $\displaystyle=0,$ (54)

which is solved by

$\displaystyle\xi(t)$ $\displaystyle=C_{\xi},$ $\displaystyle\psi(t)$
$\displaystyle=0,$ $\displaystyle\kappa$ $\displaystyle=0.$ (55)

###  5.7 Connection with vectorial torsion

The field equations, in the _off-shell_ limit, are those of the non-Abelian
Chern–Simons for the symmetric connection and the Chern–Simons for the
$\mathcal{A}$-field. They, however do not interact, and additionally in the
cosmological ansatz the field equation for $\mathcal{A}$ is automatically
satisfied. Therefore, the cosmological models with vectorial torsion in the
off-shell limit, do not differ from those of the torsion-free (off-shell)
limit mixed—but non-interacting—with an unconstrained vector field
$A_{\mu}=\eta(t)\delta^{0}_{\mu}$.

The field equations obtained in the _on-shell_ limit come with an additional
condition, which can be written as

$B_{6}\mathcal{R}_{{\mu\nu}}-B_{3}\nabla_{\mu}\mathcal{A}_{\nu}+B_{1}\mathcal{A}_{\mu}\mathcal{A}_{\nu}=0.$

The above expression yield two independent field equations,

$\displaystyle-2B_{6}\left(\dot{h}+h^{2}-f^{2}\right)-B_{3}\dot{\eta}+B_{1}\eta^{2}$
$\displaystyle=0,$ (56) $\displaystyle
B_{6}\left(\dot{g}+\kappa\right)+B_{3}\,g\eta$ $\displaystyle=0.$ (57)

Therefore the equations to solve are Eqs. (46), (56) and (57). From Eqs. (46),
the branches structure is inherited, i.e. $g=0\lor f=0$.

The branch of solutions with vanishing $g$ requires that $\kappa=0$. The
nontrivial field equations are then Eq. (56) and

$\dot{f}+2fh=0.$

Since $f(t)=\exp(-2H(t))$, Eq. (56) can be recasted as

$-2B_{6}\left(\dot{h}+h^{2}-e^{-4H(t)}\right)-B_{3}\dot{\eta}+B_{1}\eta^{2}=0.$

From the last equation it is evident that a solution is given by the flat
connection (49), together with

$\eta(t)=-\frac{B_{3}}{B_{1}}\frac{1}{t+C_{\eta}}.$

Note in addition that for a generic function $h$, unrelated to $\eta$, we can
define

$\phi(t)=\frac{2B_{6}}{B_{1}}\left(\dot{h}+h^{2}-e^{-4H(t)}\right),$

resulting in a Riccati equation for the $\eta$-function, which might be
transformed into the one-dimensional Schrödinger equation

$\ddot{u}-\frac{B_{3}}{B_{1}}\phi(t)u(t)=0,$

with the change of variable,
$u(t)=\exp\left(-\tfrac{B_{3}}{B_{1}}\mathcal{H}(t)\right)$ where
$\mathcal{H}(t)=\int_{t_{0}}^{t}\mathop{\mathrm{d}\tau}\nolimits\eta(\tau)$.
Summarising, in this branch, given a _parameter_ function $h$, it is (in
principle) possible to integrate the field equations to determine the
connection.

On the other hand, in the branch $f=0$, the nontrivial differential equations
to solve are

$\displaystyle B_{1}\eta^{2}-B_{3}\dot{\eta}-2B_{6}(\dot{h}+h^{2})$
$\displaystyle=0,$ (58) $\displaystyle B_{6}(\dot{g}+\kappa)+B_{3}\eta g$
$\displaystyle=0,$ $\displaystyle\dot{g}-2gh-\kappa$ $\displaystyle=0.$

Solving for $\eta$ and $h$ from the last two expressions and substituting into
the first, one obtains a differential equation for the $g$-function,

$\frac{B_{6}}{B_{3}}\left(B_{3}^{2}-2B_{1}B_{6}\right)\frac{\dot{g}+\kappa}{g}=0.$
(59)

There are three branches of solutions:

  1. 1.

For $\dot{g}+\kappa=0$, it follows that $g=-\kappa t+C_{g}$, $\eta=0$ and
$h=\frac{\kappa}{2(\kappa t-C_{g})}$. Note that for $\kappa=0$ this solution
is valid if $C_{g}\neq 0$, but the solution with $g=0$ requires that
$h=\frac{1}{t+C_{h}}$.

  2. 2.

For $B_{6}=0$ there are two kinds of solutions, both with arbitrary $h$
function: (i) $g=0\land\kappa=0$ and
$\eta=-\frac{B_{3}}{B_{1}}\frac{1}{t+C_{\eta}}$, or (ii) $\eta=0$ and
$g(t)=e^{2H(t)}\left(C_{g}+\kappa\int_{t_{0}}^{t}\mathop{\mathrm{d}\tau}\nolimits
e^{-2H(\tau)}\right)$.

  3. 3.

For $B_{3}^{2}=2B_{1}B_{6}$, the solutions are parametrised by the function
$g$, which is required to be nonvanishing and $C^{1}$. Hence,
$h=\frac{\dot{g}-\kappa}{2g}$ and
$\eta=-\frac{B_{3}}{B_{1}}\frac{\dot{g}+\kappa}{g}$.

###  5.8 Connection with traceless torsion

The field equations in the off-shell limit are obtained from (21) and (22),
and yield

$\displaystyle fg$ $\displaystyle=0,$ $\displaystyle
B_{6}(2g\psi+\dot{\xi})-B_{8}fg$ $\displaystyle=0,$
$\displaystyle\dot{g}-2gh-\kappa$ $\displaystyle=0,$
$\displaystyle\dot{f}+2fh$ $\displaystyle=0,$ $\displaystyle
3B_{4}\psi^{2}-2B_{6}(\dot{h}+h^{2}-f^{2})$ $\displaystyle=0,$
$\displaystyle-3B_{4}\psi\xi+B_{6}(\kappa+\dot{g})$ $\displaystyle=0.$

The _extra_ condition obtained from the on-shell limit is

$\dot{\xi}+2g\psi=0,$

was already satisfied with the off-shell equations, i.e. there is no
difference between the off-shell and on-shell $\mathcal{A}\to 0$ limit, for
the cosmological ansatz.

The solutions to the system of field equations are categorised in to classes,
those with $g=0$ or $f=0$.

The branch of solutions with $g=0$ requires that $\kappa=0$ and either
$\psi=0$ or $\xi=0$. On the one hand, for $\psi=0$, the field equations
require that $\xi=C_{\xi}$, while the nontrivial equations are solved by (49),
with the difference that the torsion tensor field is nonvanishing. on the
other hand, for $\xi=0$ the field equations to be solved are

$\dot{f}+2fh=0\quad\text{ and
}\quad\dot{h}+h^{2}-f^{2}-\frac{3B_{4}}{2B_{6}}\psi^{2}=0.$

Since there are three unknowns and only two equations, the solutions are
parametrised by one of the unknowns, e.g. $\psi$. A simple solution is
obtained for $\psi\propto f$, with results similar to those in Eq. (49). Note
that another solution is given by $f=0$, in whose case the sole nontrivial
field equation is

$\dot{h}+h^{2}-\frac{3B_{4}}{2B_{6}}\psi^{2}=0.$ (60)

This is a Riccati equation for $h$, which is equivalent to a one-dimensional
Schrödinger equation, in which the function $\psi$ is the analogous to the
quantum mechanical potential.

The system of equations for the branch with $f=0$ is

$\displaystyle\psi(t)+\frac{\dot{\xi}}{2g}$ $\displaystyle=0,$ $\displaystyle
h(t)-\frac{\dot{g}-\kappa}{2g}$ $\displaystyle=0,$ (61)
$\displaystyle-3B_{4}\dot{\xi}^{2}+2B_{6}(\kappa^{2}-\dot{g}^{2}+2g\ddot{g})$
$\displaystyle=0,$ $\displaystyle
2B_{6}(\kappa+\dot{g})+\frac{3B_{4}\xi\dot{\xi}}{g}$ $\displaystyle=0$

The system of Eqs. (61) is solved by

$\psi(t)=-\frac{\dot{\xi}}{2g},$ (62) $h(t)=\frac{\dot{g}-\kappa}{2g},$ (63)
$\xi(t)=\sqrt{C_{\xi}^{2}-\frac{2B_{6}}{3B_{4}}\bigg{(}g^{2}+2\kappa\int
g\mathop{\mathrm{d}t}\nolimits\bigg{)}},$ (64)
$0=g\ddot{g}\left(g^{2}+2\kappa\int
g\mathop{\mathrm{d}t}\nolimits\right)+\kappa
g^{2}\left(\dot{g}+1\right)+\kappa\int
g\mathop{\mathrm{d}t}\nolimits\left(1-\dot{g}^{2}\right)+\frac{3B_{4}}{4B_{6}}C_{\xi}\left(\dot{g}^{2}-2g\ddot{g}-\kappa^{2}\right).$
(65)

Particularly, for $\kappa=C_{\xi}=0$ the explicit solution is given
by121212Formally, the field equations can be solved for $C_{\xi}\neq 0$, but
the solution for the $g$-function is expressed in terms of the inverse of an
hypergeometric function. Hence, we have omitted the details of such solution.

$\displaystyle f(t)$ $\displaystyle=0,$ $\displaystyle g(t)$
$\displaystyle=C_{m}t+C_{g},$ $\displaystyle h(t)$
$\displaystyle=\frac{C_{m}}{2(C_{m}t+C_{g})},$ (66) $\displaystyle\xi(t)$
$\displaystyle=\sqrt{-\frac{2B_{6}}{3B_{4}}}\left(C_{m}t+C_{g}\right),$
$\displaystyle\psi(t)$
$\displaystyle=-\sqrt{-\frac{B_{6}}{6B_{4}}}\frac{C_{m}}{C_{m}t+C_{g}},$
$\displaystyle\eta(t)$ $\displaystyle=0,$

where
$\operatorname{sgn}\left(B_{4}\right)=-\operatorname{sgn}\left(B_{6}\right)$.

For $\kappa\neq 0$ a simplification of the field equations is to propose $g$
as a linear function of $t$, yielding

$\displaystyle f(t)$ $\displaystyle=0,$ $\displaystyle g(t)$
$\displaystyle=-\kappa t+C_{g},$ $\displaystyle h(t)$
$\displaystyle=\frac{1}{t-C_{g}\kappa},$ (67) $\displaystyle\xi(t)$
$\displaystyle=C_{\xi}$ $\displaystyle\psi(t)$ $\displaystyle=0,$
$\displaystyle\eta(t)$ $\displaystyle=0.$

###  5.9 Restriction to torsional sector

The solutions to the off-shell restriction to the torsional sector require
that either $\psi=0$ or $\xi=0$.131313Note that $\psi$ and $\xi$ cannot vanish
at the same time, since it would imply that $\mathcal{B}=0$. For $\xi=0$ the
nontrivial field equation is the Riccati-like equation,

$B_{1}\eta^{2}-B_{3}\dot{\eta}+3B_{4}\psi^{2}=0,$ (68)

which (as mentioned before) is equivalent to a one-dimensional Schrödinger
equation whose potential is related to the function $\psi^{2}$. For $\psi=0$,
the explicit solution is

$\psi(t)=0,\quad\eta(t)=-\frac{B_{3}}{B_{1}}\frac{1}{t+C_{\eta}},\quad\xi(t)=C_{\xi}\left(t+C_{\eta}\right)^{2}.$
(69)

In the on-shell limit, the subsidiary conditions are $\xi\eta=0$,
$\dot{\xi}=0$ and $\psi\eta=0$. Hence, there is no solution because the field
equations require that $\mathcal{B}=0$.

###  5.10 Exceptional solutions to the whole model

The cosmological solutions to polynomial affine model of gravity in three
dimensions with all the fields turned on are _exceptional_ , since we are
solving the seven dimensional system of field equations (41), with only six
unknowns. Generically, the consistency conditions of the system require that
either some of the functions vanish (worsening the well-being of the system)
or a relation between the parameters of the model.

A first example of these is given by the case $g=\xi=\kappa=0$, with
nontrivial field equations

$\dot{f}+2fh=-\frac{B_{3}}{B_{8}}\eta\psi,$ (70)
$\dot{h}+h^{2}-f^{2}=\frac{1}{2B_{6}}\left(B_{1}\eta^{2}-B_{3}\dot{\eta}+3B_{4}\psi^{2}\right).$
(71)

These field equations are solvable when all functions are inversely
proportional to $t$.

Another branch of solutions is found when the coupling constants are not all
independent. As an example, for $B_{3}^{2}+4B_{1}B_{6}=0$ the field equations
allow to decouple the functions $g$ and $\xi$—which are unconstrained—, from
the equations for $f$ and $\psi$, which should satisfy the differential
equations (71). We were able to find two types of solutions of this system of
equations (which do not fall into the previously presented categories).

  * •

Solution with $B_{3}=0$: With this condition, the solution is characterised by
the functions

$\displaystyle f(t)$ $\displaystyle=0,$ $\displaystyle g(t)$
$\displaystyle=-\kappa t+C_{g},$ $\displaystyle h(t)$
$\displaystyle=\frac{1}{t-\kappa C_{g}},$ (72) $\displaystyle\xi(t)$
$\displaystyle=\sqrt{-\frac{2B_{6}}{3B_{4}}}C_{g},$ $\displaystyle\psi(t)$
$\displaystyle=0,$ $\displaystyle\eta(t)$ $\displaystyle=\text{arbitrary},$

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

$\displaystyle f(t)$ $\displaystyle=0,$ $\displaystyle g(t)$
$\displaystyle=C_{m}t+C_{g},$ $\displaystyle h(t)$
$\displaystyle=\frac{C_{m}}{2(C_{m}t+C_{g})},$ (73) $\displaystyle\xi(t)$
$\displaystyle=\sqrt{-\frac{2B_{6}}{3B_{4}}}(C_{m}t+C_{g}),$
$\displaystyle\psi(t)$
$\displaystyle=\sqrt{-\frac{B_{6}}{6B_{4}}}\frac{C_{m}}{C_{m}t+C_{g}},$
$\displaystyle\eta(t)$ $\displaystyle=\text{arbitrary}.$

  * •

Solutions for $\kappa=0$: The system of field equations can be solved for the
ansatz $g(t)=t^{n}$ with
$n\in\mathbb{R}-\Set{-2,\left[\frac{1-\sqrt{33}}{4},\frac{1+\sqrt{33}}{4}\right]}$,
with the addition 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

$\displaystyle f(t)$
$\displaystyle=\frac{\sqrt{2n^{2}-n-4}\operatorname{sgn}(B_{8})}{2t},$
$\displaystyle g(t)$ $\displaystyle=t^{n},$ $\displaystyle h(t)$
$\displaystyle=\frac{n}{2t},$ (74) $\displaystyle\xi(t)$
$\displaystyle=\frac{\sqrt{2n^{2}-n-4}\operatorname{sgn}(B_{8})t^{n}}{2B_{6}},$
$\displaystyle\psi(t)$
$\displaystyle=\frac{(n-1)\sqrt{2n^{2}-n-4}\operatorname{sgn}(B_{8})}{4tB_{6}},$
$\displaystyle\eta(t)$ $\displaystyle=\frac{2B_{6}}{B_{3}t}.$

##  6 Discussion and conclusions

In an attempt to bring gravity to the same footing than gauge theories, we
have proposed a model of gravity whose sole fundamental field is the affine
connection. Such model has been named _polynomial affine gravity_. Our model
might be understood as a Schwarz topological theory, in the sense that the
metric plays no role in the model building, similar to the case of
Chern–Simons theories.

The polynomial affine model of gravity has been built in three and four
dimensions (i.e. there are no _ab initio_ restrictions on the dimension of the
space, unlike for Chern–Simons theories), and possesses attractive features.
Firstly, the model is appealing for a quantum theory of gravity, since all the
terms in the action are power-counting renormalisable, and in addition the
lack of additional invariant forms would forbid the existence of counter-
terms. Secondly, the absence of an energy scale, reflected by the fact that
all the coupling constants are dimensionless, appears as a hint (of a sort) of
conformal invariance (at least at tree level).

Customarily, the conformal transformation is understood (in metric
gravitational models) as a point-wise scaling of the metric tensor field, and
the invariant curvature under these transformations is the conformal Weyl
tensor field, i.e. the $g$-traceless part of the Riemann–Christoffel
curvature. This notion, can be generalised without evoking a metric. The idea
is that _self-parallel curves_ can be preserved under “generalised”
transformations. These are the _projective_ transformations, and the invariant
curvature under these transformations is the projective Weyl tensor field.

In this article we focus in the three-dimensional version of the polynomial
affine model of gravity, where the action and therefore the field equations
are simpler than their four-dimensional analogous, expecting the physical
interpretation to be clearer. Note that in comparison the three-dimensional
action [see Eq. (3)] is determined by eight terms (including the Chern–Simons
terms) while the four-dimensional one is composed by twenty terms
(disregarding topological terms), unrelated through boundary terms.

Interestingly, the term of the action with coefficient $B_{3}$ can be re-
written (up to boundary term) as,
$\nabla_{\mu}\mathcal{A}_{\alpha}\mathcal{B}_{{\beta}}{}^{{\mu}}{}_{{\gamma}}\mathop{\mathrm{d}V}\nolimits^{\alpha\beta\gamma}$.
In this case, the $\mathcal{B}$-field would be non-dynamical, and therefore
interpreted as an auxiliary field. Furthermore, for the case with $B_{4}=0$,
the $\mathcal{B}$-field would be a Lagrangian multiplier.

It is worth noticing that unlike the three-dimensional version of General
Relativity, in an affine model the (projective) Weyl tensor does not vanish
necessarily, and thus there is room to novel phenomenological effects. In
particular, the field equation (9) contains terms that can be related to the
projective Weyl curvature, and therefore its space of solutions might differ
from the one expected in General Relativity, even in the cases where the field
equations are alike, e.g. for flat or Ricci-flat manifolds.

The field equations derived from the action in Eq. (3), include a
generalisation of the Einstein field equations,

$\mathcal{R}_{{\mu(\nu}}{}^{{\mu}}{}_{{\rho)}}=-\frac{B_{1}}{B_{6}}\mathcal{A}_{\nu}\mathcal{A}_{\rho}+\frac{B_{3}}{B_{6}}\nabla_{(\nu}\mathcal{A}_{\rho)}-\frac{3B_{4}}{2B_{6}}\mathcal{B}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{B}_{{\rho}}{}^{{\sigma}}{}_{{\mu}}=\tilde{T}_{\nu\rho},$
(75)

obtained by varying with respect to the $\mathcal{B}$-field is the _analogous_
to the Einstein equations written in the Ricci form. Noticeable, the fact
that—even for torsion-free truncation—the field equations for the symmetric
connection appear from the variation with respect to other field, has been
interpreted as a sign of the non-uniqueness of the Lagrangian description of
the system hojman_privat .

The Eq. (75) represents a non-Riemannian generalisation of the Einstein
equations in the Ricci form, where the right-hand side geometrically encodes
what in General Relativity is attributed to the presence of matter. However,
$\tilde{T}$ does not admit a separation between material and geometrical
contributions, unlike its analogous form in General Relativity, which is
expressed in terms of the energy-momentum tensor $T_{\mu\nu}$ and its trace
$T$, as

$\tilde{T}_{\nu\rho}^{GR}\propto T_{\nu\rho}-Tg_{\nu\rho}.$

Furthermore, when $\tilde{T}$ is non-degenerated, it equips the affine
manifold with a torsion-descendent notion of metric, and hence Eq. (75)
provides a notion of _affine_ Einstein manifold. A nice feature of this
torsion-descendent metric is that, unlike the _emergent metric_ from Ref.
castillo-felisola20_emerg_metric_geodes_analy_cosmol , it might be well-
defined even when the space is Ricci-flat.

However, since the affine connection is a less intuitive geometrical object
(in comparison with the metric), we analysed the possible truncations of the
model, i.e. sectors where only a subset of the irreducible components of the
connections are nontrivial.

Turning to the solutions of the field equations, we found the ansätze of the
three-dimensional affine connection compatible with the cosmological
principle. Firstly, we found that the symmetric connection is determined by
three functions,141414When solving the differential equations obtained from
the Lie derivative of the connection, there is a fourth parametric function
($j$) characterising the affine connection, but we show in B that this
parameter can be eliminated by a reparametrisation on the _time_ coordinate.
$f$, $g$ and $h$. The $f$ function has no analogous in a cosmological Levi-
Civita connection, and therefore it is a non-Riemannian parameter. In
addition, the $g$ and $h$ functions describe a non-Riemannian cosmological
geometry unless their could be parametrised in terms of a scale factor,
$a=a(t)$, as

$g=a\dot{a}\quad\text{and}\quad h=\frac{\dot{a}}{a}.$ (76)

Hence, in the torsion-free sector of the polynomial affine model of gravity
the non-Riemannian structure percolates the Einstein-like equations if $f\neq
0$ and/or $g$ and $h$ are not parametrised as in Eq. (76).

Before discussing the cosmological structure of the torsional fields, we would
like to briefly mention the geometrical meaning of the $f$ function in the
cosmological ansatz of the symmetric connection. Consider the symmetric part
of the affine connection defining the polynomial affine model of gravity,
$\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$, and a generic metric, $g_{\mu\nu}$.
Let $\mathring{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$ the Levi-Civita
connection associated to the metric $g$. The difference between the two
connections,
$\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\nu}}-\mathring{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\nu}}$,
is a tensor defining the Weyl’s _congruent transferences_.151515These
transformations might be also called _congruent transplantation_ , which is
the translation of the original German vocable (kongruente Verpflanzung). Such
tensor does no have an analogous in Riemannian geometry, since it is related
to the non-metricity. The cosmological ansatz for the connection in three
dimensions admits certain components of the _transference_ tensor, determined
by the function $f$. Interestingly, in the cosmological ansatz in four
dimensions requires vanishing _transference_ tensor.

The (cosmological) torsion field is characterised by three functions, $\eta$,
$\xi$ and $\psi$, the first one defines the ${\mathcal{A}}$-field and the
remaining two define the $\mathcal{B}$-field. In comparison with the four-
dimensional case, which is characterised by just two functions, the torsion
tensor field is less restricted. Noticeable, the same is true about the
characterisation of the (cosmological) symmetric connection, since in the
four-dimensional scenario it is determined (after the time reparametrisation)
by solely two functions.161616We would like to stress that in Refs. castillo-
felisola18_beyond_einstein ; castillo-felisola18_cosmol ; castillo-
felisola20_emerg_metric_geodes_analy_cosmol  we were not aware of the _time_
reparametrisation, and therefore the additional function turns the
manipulation of the field equations into a more cumbersome process.

Using the cosmological ansatz, the Eq. (75) is written as

$\dot{h}+h^{2}=\Xi(\text{non-Riemannian terms in
}\hat{\Gamma}),\quad\left(\text{in GR: }\dot{H}+H^{2}\propto(\rho+3p)\text{
with }H=\frac{\dot{a}}{a}\right)$ (77)

where $h$ is one of the functions defining the symmetric connection and its
analogous in General Relativity, i.e. $H$, is the Hubble parameter. Eq. (77)
is a generalisation of the Friedmann equation, where other geometric fields
uphold what in General Relativity would be interpreted as matter effects.
Generically, the $\Xi$ function depends on all the other (i.e. non $h$)
functions characterising the affine connection, but it is peculiar that the
case where only $f$ and $h$ are nonvanishing, the Eq. (77) becomes a Riccati
ordinary differential equation

$\dot{h}+h^{2}-f^{2}=0,$

which can be expressed through the transformation $u(t)=e^{H(t)}$, where
$H(x)=\int_{x_{0}}^{x}\mathop{\mathrm{d}y}\nolimits h(y)$, onto a one-
dimensional “time-independent” Schrödinger equation, $\ddot{u}-f^{2}u=0$,
where $u$ would be the wave function, the $f^{2}$-function plays the role of
the quantum mechanical potential (with the energy eigenvalue subtracted). Note
that particularly in the flat and Ricci-flat cases (coming from the torsion-
free truncation) the Riccati equation with constant $f$ is analogous to the
(acceleration) Friedmann equation describing a Universe filled with a perfect
fluid in a dark energy dominated era, which in General Relativity is

$\dot{h}+h^{2}-\frac{8\pi G}{3}\rho_{DE}=0.$

In Sec. 5 have found cosmological solutions to the field equation of the
polynomial affine model of gravity, characterised by the system in Eq. (41).
We proceeded systematically, scanning all the possible kinds of solutions. We
noted that the solutions split into two categories, depending on whether the
function $\Omega$ from Eq. (45) vanishes or not. However, we re-classified the
solutions according to the type of truncation they belong to. It is worth
mentioning that even though in the classification of solutions we do not
consider explicitly the case with vanishing $B_{8}$, this branch of solutions
lying in the sector $\Omega=0$, yields no additional solutions to those in
Sec. 5, e.g. the solutions with $f=0$ in Sec. 5.4.2 and the solutions in Sec.
5.8. Furthermore, in the sector with $4B_{1}B_{6}+B_{3}^{2}=0$ we found
solutions to the whole system of field equations, where none of the functions
determining the components of the connection vanish.

Now, even though we have found _affine_ cosmological solutions to the
polynomial affine model of gravity in three dimensions, _real life_
applications require the existence of a metric. In purely affine models,
although the fundamental field is the connection, it is possible to define
various types of (derived) metrics. Let us mention four of these derived
metrics:

  1. 1.

The symmetric part of the Ricci tensor field, when it is non-degenerated,
serves as a metric. This was noticed very early in the development of
differential geometry (see for example section 5 of Ref.
eisenhart27_non_rieman ), and used recently in Ref. castillo-
felisola20_emerg_metric_geodes_analy_cosmol . A notable disadvantage is that
interesting cases, such as Minkowski and Schwarzschild, cannot be described
using this notion of metric.

  2. 2.

The quasi-Hodge dual of the $\mathcal{B}$-field, i.e.
$T^{\mu\nu}=\frac{1}{2}\mathcal{B}_{{\alpha}}{}^{{\mu}}{}_{{\beta}}\epsilon^{\nu\alpha\beta}$,
is a symmetric $\binom{2}{0}$-tensor density. When $T$ is non-degenerated, it
can be used as an _inverse metric density_ , similar to that used by
Eddington, Einstein, Schrödinger and others to build affine models of General
Relativity (see Ref. tonnelat14_einst ). In Ref. castillo-
felisola15_polyn_model_purel_affin_gravit  this analogue is used to
intuitively relate the three-dimensional action of polynomial affine gravity
with General Relativity nonminimally coupled to the $\mathcal{A}$-field (see
Eq. (9) of the referred article).

  3. 3.

The construction
$\mathcal{T}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}\mathcal{T}_{{\nu}}{}^{{\rho}}{}_{{\lambda}}$,
defined from the torsion, is a symmetric $\binom{0}{2}$-tensor field that
serves (when non-degenerated) as a metric. This tensor was introduced by
Poplawski in Ref. poplawski14_affin_theor_gravit , and it is related to the
symmetric part of Eq. (84) (since the $\mathcal{S}$-tensor is proportional to
the torsion).

  4. 4.

The symmetric part of $\mathcal{S}_{{\sigma\mu}}{}^{{\sigma}}{}_{{\nu}}$ in
Eq. (84) can also be interpreted as a metric, when it is non-degenerated.

Each of the examples above serves to endow the affine manifold with a metric
structure with respect to which we could measure geodesic distances and
compare with the parallel transport obtained using the symmetric connection.

Only a few of the solutions explicitly presented in the paper admit non-
degenerated metrics. From the solution to the coupled system
$\Gamma$–$\mathcal{A}$, the case with vanishing $f$ and $B_{6}$ [see the
second type of solutions after Eq. (59)] possesses a _Ricci_ metric as long as
the arbitrary function $h$ is not the reciprocal of $t$, while the case with
vanishing $f$ and $B^{2}_{3}=2B_{1}B_{6}$ [see the third type of solutions
after Eq. (59)] possesses a metric of the fourth kind, as long as
$g\neq-\kappa t+C_{g}$. The solutions of the coupled system
$\Gamma$–$\mathcal{B}$ with $\kappa=0$ [see Eq. (66)] and the exceptional
cases with vanishing $\kappa$ [see Eqs. (73) and (74)], possess the first
three types of metric described above, as long as $C_{m}\neq 0$ and $n\neq 0$.
In addition, one can endow the exceptional solutions with the fourth kind of
metric, with the constrain that $\eta$ is not a constant function in Eqs. (72)
and (73).

The above opens the discussion of the background independence of the
gravitational model, which is explicitly broken in General Relativity due to
the presence of the metric in the Einstein–Hilbert action.

##  Appendix A Notions of non-Riemannian geometry

The aim of this appendix is to provide a short summary of results in non-
Riemannian geometry, and also intends to fix the notation used through the
development of the paper. Readers interested in the subjects are recommended
to review the classical texts by L. Eisenhart eisenhart27_non_rieman  and J.
Schouten schouten13_ricci , the final chapter of the book by Synge and Schild
synge78_tensor , the book by P. Gilkey and collaborators
stana13_applic_affin_weyl_geomet , and articles like Refs.
siqueira18_topic_non_rieman_geomet ;
iosifidis19_metric_affin_gravit_cosmol_torsion ;
klemm20_einst_manif_with_torsion_nonmet .

A $d$-dimensional affine manifold $(\mathcal{M},\hat{\nabla})$ is a
$d$-dimensional differential manifold $\mathcal{M}$ equipped with a linear
connection $\hat{\nabla}$. The linear connection is determined by their
$d^{3}$-independent components
$\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}$, such that

$\hat{\nabla}_{\mu}=\partial_{\mu}+\hat{\Gamma}_{{\mu}}{}^{{\bullet}}{}_{{\bullet}}.$

Since an affine structure does not require the existence of a metric tensor
field, the affine connection admits a decomposition in their lower indices,
into their symmetric and skew-symmetric parts,

$\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}=\hat{\Gamma}_{{(\mu}}{}^{{\lambda}}{}_{{\rho)}}+\hat{\Gamma}_{{[\mu}}{}^{{\lambda}}{}_{{\rho]}}=\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\rho}}+\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}.$

The symmetric component of the connection remains as a connection, which we
denote by simply $\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\rho}}$, while the skew-
symmetric component is a tensor field proportional to the torsion tensor field
(explicitly, it is twice the tensor field).

The curvature of a connection, defined by

$\hat{\mathcal{R}}(X,Y)Z=\left(\hat{\nabla}_{X}\hat{\nabla}_{Y}-\hat{\nabla}_{Y}\hat{\nabla}_{X}-\hat{\nabla}_{[X,Y]}\right)Z,$
(78)

with $X$, $Y$ and $Z$ vector fields, can be written in components as

$\hat{\mathcal{R}}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}=\partial_{\mu}\hat{\Gamma}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}-\partial_{\nu}\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}+\hat{\Gamma}_{{\mu}}{}^{{\lambda}}{}_{{\sigma}}\hat{\Gamma}_{{\nu}}{}^{{\sigma}}{}_{{\rho}}-\hat{\Gamma}_{{\nu}}{}^{{\lambda}}{}_{{\sigma}}\hat{\Gamma}_{{\mu}}{}^{{\sigma}}{}_{{\rho}}.$
(79)

The curvature tensor field is skew-symmetric in the first couple of indices,
and therefore the contra-variant index can be contracted in two independent
ways, $\hat{\mathcal{R}}_{{\lambda\nu}}{}^{{\lambda}}{}_{{\rho}}$ and
$\hat{\mathcal{R}}_{{\mu\nu}}{}^{{\lambda}}{}_{{\lambda}}$, referred to as the
first and second Ricci curvatures. Customarily, the first Ricci curvature is
simply called Ricci tensor field, while the second Ricci curvature is also
referred to as homothetic curvature or _trace of the curvature_. Note that
when restricting oneself to Riemannian connections the Ricci tensor field is
symmetric and the trace of the curvature vanishes, but for generic affine
connections these properties do not hold.

The curvature in Eq. (79) can be decomposed in terms of the symmetric
connection $\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\rho}}$ and the tensor
$\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}$, yielding

$\hat{\mathcal{R}}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}=\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}+\hat{\nabla}_{\mu}\mathcal{S}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}-\hat{\nabla}_{\nu}\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}-\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\sigma}}\mathcal{S}_{{\nu}}{}^{{\sigma}}{}_{{\rho}}+\mathcal{S}_{{\nu}}{}^{{\lambda}}{}_{{\sigma}}\mathcal{S}_{{\mu}}{}^{{\sigma}}{}_{{\rho}}-2\mathcal{S}_{{\mu}}{}^{{\sigma}}{}_{{\nu}}\mathcal{S}_{{\sigma}}{}^{{\lambda}}{}_{{\rho}}=\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}+\nabla_{\mu}\mathcal{S}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}-\nabla_{\nu}\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}+\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\sigma}}\mathcal{S}_{{\nu}}{}^{{\sigma}}{}_{{\rho}}-\mathcal{S}_{{\nu}}{}^{{\lambda}}{}_{{\sigma}}\mathcal{S}_{{\mu}}{}^{{\sigma}}{}_{{\rho}}=\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}+\mathcal{S}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}.$
(80)

Given the split of the curvature of the affine connection in Eq. (80), we can
analyse the contractions of their components. Let us take first the curvature
of the symmetric connection,

$\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}=\partial_{\mu}\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}-\partial_{\nu}\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\rho}}+\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\sigma}}\Gamma_{{\nu}}{}^{{\sigma}}{}_{{\rho}}-\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\sigma}}\Gamma_{{\mu}}{}^{{\sigma}}{}_{{\rho}}.$

The Ricci tensor is given by

$\mathcal{R}_{{\nu\rho}}=\mathcal{R}_{{\lambda\nu}}{}^{{\lambda}}{}_{{\rho}}=\partial_{\lambda}\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}-\partial_{\nu}\Gamma_{{\rho}}{}^{{\lambda}}{}_{{\lambda}}+\Gamma_{{\sigma}}{}^{{\lambda}}{}_{{\lambda}}\Gamma_{{\nu}}{}^{{\sigma}}{}_{{\rho}}-\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\sigma}}\Gamma_{{\rho}}{}^{{\sigma}}{}_{{\lambda}},$
(81)

which splits into symmetric and skew-symmetric parts,

$\displaystyle\mathcal{R}_{{(\nu\rho)}}$
$\displaystyle=\partial_{\lambda}\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\rho}}-\frac{1}{2}\left(\partial_{\nu}\Gamma_{{\rho}}{}^{{\lambda}}{}_{{\lambda}}+\partial_{\rho}\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\lambda}}\right)+\Gamma_{{\sigma}}{}^{{\lambda}}{}_{{\lambda}}\Gamma_{{\nu}}{}^{{\sigma}}{}_{{\rho}}-\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\sigma}}\Gamma_{{\rho}}{}^{{\sigma}}{}_{{\lambda}},$
$\displaystyle\mathcal{R}_{{[\nu\rho]}}$
$\displaystyle=-\frac{1}{2}\left(\partial_{\nu}\Gamma_{{\rho}}{}^{{\lambda}}{}_{{\lambda}}-\partial_{\rho}\Gamma_{{\nu}}{}^{{\lambda}}{}_{{\lambda}}\right).$

From the transformation of the connection under diffeomorphisms, one finds
that the trace of the symmetric connection,
$\Gamma_{{\mu}}=\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\lambda}}$, transforms like

$\Gamma^{\prime}_{\mu}=\Gamma_{{\alpha}}\frac{\partial x^{\alpha}}{\partial
x^{\prime\mu}}+\partial_{\mu}\ln\mathfrak{d},$

with $\mathfrak{d}=\det\frac{\partial x^{\alpha}}{\partial x^{\prime\nu}}$ is
the scalar density defined by the determinant of the transformation.
Therefore, the skew-symmetric part of the Ricci tensor is given by the curl of
a vector

$\mathcal{R}_{{[\mu\nu]}}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu},\quad\text{
with }a_{\mu}=a_{\mu}{}^{\lambda}{}_{\lambda}.$ (82)

The tensor $a_{\mu}{}^{\lambda}{}_{\rho}$ is defined by the difference between
the connection $\Gamma_{{\mu}}{}^{{\lambda}}{}_{{\rho}}$ and a symmetric
connection of reference. In order to illustrate this point, we show two cases:
(i) if one chooses the Levi-Civita connection as reference,
$a_{\mu}{}^{\lambda}{}_{\rho}$ is the _contorsion_ tensor; and (ii) if the
reference is given by the parameters of the parallel displacement in an
Euclidean manifold, then $a_{\mu}{}^{\lambda}{}_{\rho}=\frac{\partial
x^{\lambda}}{\partial y^{\prime\alpha}}\frac{\partial
y^{\prime\alpha}}{\partial x^{(\mu}\partial x^{\rho)}}$.

The second contraction of the curvature, i.e. the trace of curvature, is given
by

$\mathcal{R}_{{\mu\nu}}{}^{{\lambda}}{}_{{\lambda}}=\partial_{\mu}\Gamma_{{\nu}}-\partial_{\nu}\Gamma_{{\mu}}=-2\mathcal{R}_{{[\mu\nu]}}.$

The last step might be obtained from the algebraic Bianchi identity for a
torsion free connection.

The second term coming from the splitting of the curvature tensor,

$\mathcal{S}_{{\mu\nu}}{}^{{\lambda}}{}_{{\rho}}=\nabla_{\mu}\mathcal{S}_{{\nu}}{}^{{\lambda}}{}_{{\rho}}-\nabla_{\nu}\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\rho}}+\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\sigma}}\mathcal{S}_{{\nu}}{}^{{\sigma}}{}_{{\rho}}-\mathcal{S}_{{\nu}}{}^{{\lambda}}{}_{{\sigma}}\mathcal{S}_{{\mu}}{}^{{\sigma}}{}_{{\rho}},$
(83)

admits two contractions

$\mathcal{S}_{{\mu\nu}}{}^{{\mu}}{}_{{\rho}}=\nabla_{\mu}\mathcal{S}_{{\nu}}{}^{{\mu}}{}_{{\rho}}+\nabla_{\nu}\mathcal{S}_{{\rho}}-\mathcal{S}_{{\sigma}}\mathcal{S}_{{\nu}}{}^{{\sigma}}{}_{{\rho}}+\mathcal{S}_{{\nu}}{}^{{\mu}}{}_{{\sigma}}\mathcal{S}_{{\rho}}{}^{{\sigma}}{}_{{\mu}},$
(84)

and

$\mathcal{S}_{{\mu\nu}}{}^{{\lambda}}{}_{{\lambda}}=\nabla_{\mu}\mathcal{S}_{{\nu}}-\nabla_{\nu}\mathcal{S}_{{\mu}}=\partial_{\mu}\mathcal{S}_{{\nu}}-\partial_{\nu}\mathcal{S}_{{\mu}}.$
(85)

In the above equations we have introduced the vector
$\mathcal{S}_{{\mu}}=\mathcal{S}_{{\mu}}{}^{{\lambda}}{}_{{\lambda}}=-\mathcal{S}_{{\lambda}}{}^{{\lambda}}{}_{{\mu}}$.

Note that both terms of the homothetic curvature of the linear connection,
$\hat{\Gamma}$, is the curl of $\mathcal{S}_{{\mu}}-a_{\mu}$ and thus
invariant under the addition of a gradient, i.e.
$\mathcal{S}_{{\mu}}-a_{\mu}\mapsto\mathcal{S}_{{\mu}}-a_{\mu}+\partial_{\mu}\phi$.
This reflects a _gauge_ redundancy in the trace of the curvature, which is
inherited from the freedom of choosing a connection of reference to define the
$a$ tensor.

Let us turn to the Bianchi identities. The torsion of the affine connection is
defined as

$\hat{\mathcal{T}}(X,Y)=\hat{\nabla}_{X}Y-\hat{\nabla}_{Y}X-[X,Y],$ (86)

and its derivative is

$\hat{\nabla}_{Z}(\hat{\mathcal{T}}(X,Y))=\hat{\nabla}_{Z}\hat{\mathcal{T}}(X,Y)+\hat{\mathcal{T}}(\hat{\nabla}_{Z}X,Y)+\hat{\mathcal{T}}(X,\hat{\nabla}_{Z}Y).$
(87)

The derivative of the vectors in last two terms of Eq. (87), are expressible
in terms of the torsion, since

$\hat{\mathcal{T}}(\hat{\mathcal{T}}(X,Y),Z)=\hat{\mathcal{T}}(\hat{\nabla}_{X}Y,Z)+\hat{\mathcal{T}}(Z,\hat{\nabla}_{Y}X)-\hat{\mathcal{T}}([X,Y],Z).$
(88)

The algebraic Bianchi identity is obtained by adding the cyclic permutation of
the vectors $X$, $Y$ and $Z$, which shall be denoted by the operator
$\mathfrak{S}_{X,Y,Z}$. Therefore, from Eq. (88) one gets

$\mathfrak{S}_{X,Y,Z}\left(\hat{\mathcal{R}}(X,Y,Z)-\hat{\mathcal{T}}(\hat{\mathcal{T}}(X,Y),Z)-\hat{\nabla}_{X}\hat{\mathcal{T}}(Y,Z)\right)=0.$
(89)

The differential Bianchi identity is obtained from the derivative of the
curvature in Eq. (78),

$\hat{\nabla}_{Z}(\hat{\mathcal{R}}(X,Y)W)=\hat{\nabla}_{Z}\hat{\mathcal{R}}(X,Y)W+\hat{\mathcal{R}}(\hat{\nabla}_{Z}X,Y)W+\hat{\mathcal{R}}(X,\hat{\nabla}_{Z}Y)W+\hat{\mathcal{R}}(X,Y)\hat{\nabla}_{Z}W,$

after expressing the derivatives of the vectors (others than $W$) in terms of
the torsion tensor, and the application of the cyclic permutation operator,

$\mathfrak{S}_{X,Y,Z}\left(\hat{\nabla}_{Z}\hat{\mathcal{R}}(X,Y)W+\hat{\mathcal{R}}(\hat{\mathcal{T}}(X,Y),Z)W\right)=0.$
(90)

In order to get the last expression one uses that

$\left[\hat{\nabla}_{Z},\hat{\mathcal{R}}(X,Y)\right]W-\hat{\mathcal{R}}([X,Y],Z)=[\hat{\nabla}_{Z},[\hat{\nabla}_{X},\hat{\nabla}_{Y}]]W+\hat{\nabla}_{[[X,Y],Z]}W,$

and the action of the cyclic permutation operator on it vanishes due to the
Jacobi identity of the Lie bracket and the commutator.

##  Appendix B Reparametrisation of time coordinate

Under a change of coordinates, $x^{\prime a}=x^{\prime a}(x^{i})$, the
component of the connection transform as

$\frac{\partial^{2}x^{i}}{\partial x^{\prime a}\partial x^{\prime
b}}+\Gamma_{{j}}{}^{{i}}{}_{{k}}\frac{\partial x^{j}}{\partial x^{\prime
a}}\frac{\partial x^{k}}{\partial x^{\prime
b}}=\Gamma^{\prime\,c}_{a\;b}\frac{\partial x^{i}}{\partial x^{\prime c}}.$
(91)

Given the generic form of the isotropic and homogeneous connection, Eq. (30),
one notices that the function $j$ has no dynamics in the curvature tensors. A
natural question is whether there is a reparametrisation of the time
coordinate that allows to set $j=0$. Since this function comes from the
component $\Gamma_{{0}}{}^{{0}}{}_{{0}}$, consider a transformation of the
form,

$\displaystyle t^{\prime}$ $\displaystyle=t^{\prime}(t),$ $\displaystyle
r^{\prime}$ $\displaystyle=r$ $\displaystyle\varphi^{\prime}$
$\displaystyle=\varphi.$

Equation (91) for $a=b=c=0$, and $\Gamma^{\prime\,0}_{0\;0}=0$, yields

$\frac{\partial^{2}t}{\partial t^{\prime 2}}+j\cdot\left(\frac{\partial
t}{\partial t^{\prime}}\right)^{2}=0,$

which can be written as a total derivative,

$\frac{1}{X}\partial_{t^{\prime}}\left(X\partial_{t^{\prime}}t\right)=0,$

for $j=\frac{1}{X}\partial_{t}X$, or equivalently
$X=e^{\int\mathop{\mathrm{d}t}\nolimits j(t)}=e^{J(t)}$. From here, one have
that

$t^{\prime}=\int\mathop{\mathrm{d}t}\nolimits e^{J(t)}.$

With the above transformation, it can be checked with ease that the effect of
the time reparametrisation is a scaling of the other three functions of time
entering in the connection, and thus $f(t)\mapsto f(t^{\prime})$, $g(t)\mapsto
g(t^{\prime})$ and $h(t)\mapsto h(t^{\prime})$.

###### 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
the grant `PI_LI_19_02` from the Universidad Técnica Federico Santa María. The
work of JP is funded by USM-PIIC grant number `059/2018`.

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