Scale factor solving the vanishing Ricci condition, for a cosmological metric connection.
Abstract
We find possible cosmological models of the polynomial affine gravity described by connections that are either compatible or not with a metric. When possible, we compare them with those of general relativity. We show that the set of cosmological vacuum solutions in general relativity are a subset of the solutions of polynomial affine gravity. In our model, the cosmological constant appears as an integration constant, and, additionally, we show that some forms of matter can be emulated by the affine structure—even in the metric compatible case. In the case of connections not compatible with a metric, we obtain formal families of solutions, which should be constrained by physical arguments. We show that for a certain parametrisation of the connection, the affine Ricciflat condition yields the cosmological field equations of general relativity coupled with a perfect fluid, pointing towards a geometrical emulation of—what is interpreted in general relativity as—matter effects.
Keywords
 affine gravity
 exact solutions
 cosmological models
1. Introduction
All of the
On the other hand, the model that explains gravitational interaction (general relativity) is a field theory for the metric, which can be thought as a potential for the gravitational connection [1, 2]. Although general relativity is the most successful theory, we have to explain gravity [3, 4, 5], it cannot be formulated as a gauge theory (in four dimensions), the standard quantisation methods lead to inconsistencies, and it is nonrenormalisable, driving the community to believe it is an effective theory of a yet unknown fundamental one. Within the framework of cosmology, when one wants to conciliate both
Although it cannot be said that the mentioned troubles are due to the fact that the model is described by the metric, given that the
The first affine model of gravity was proposed by Eddington in Ref. [6], where the action was defined by the square root of the determinant of the Ricci tensor:
but in Schrödinger’s words [7]:
However, Eddington’s idea serves as a starting point to new proposals [8, 9].
In a series of seminal papers [10, 11, 12, 13], Cartan presented a definition of curvature for spaces with torsion and its relevance for general relativity. It is worth mentioning that in pure gravity—described by the EinsteinHilbertlike action, Cartan’s generalisation of gravity yields the condition of vanishing torsion as an equation of motion. Therefore, it was not seriously considered as a generalisation of general relativity, until the inclusion of gravitating fermionic matter [14].
Inspired by Cartan’s idea of considering an affine connection into modelling of gravity, a new interesting proposal has been considered. Among the interesting generalisations, we mention a couple: (i) the wellknown metricaffine models of gravity [15], in which the metric and connection are not only considered as independent, but the conditions of metricity and vanishing torsion are in general dropped and (ii) the LovelockCartan gravity [16], includes extra terms in the action compatible with the precepts of general relativity, whose variation yields field equations that are secondorder differential equations. Nonetheless, the metric plays a very important role in these models.
Modern attempts to describe gravity as a theory for the affine connection have been proposed in Refs. [17, 18, 19, 20, 21, 22, 23, 24, 25], and the cosmological implications in an Eddingtoninspired affine model were studied in Refs. [26, 27, 28, 29].
The recently proposed polynomial affine gravity [24] separates the two roles of the metric field, as in a Palatini formulation of gravity, but does not allow it to participate in the mediation of the interaction, by its exclusion from the action. It turns out that the absence of the metric in the action results in a robust structure that—without the addition of other fields—does not accept deformations. That robustness can be useful if one would like to quantise the theory, because all possible counterterms should have the form of terms already present in the action.
In this chapter, we focus in finding cosmological solutions in the context of polynomial affine gravity, restricted to torsionfree sector of equiaffine connections, which yields a simple set of field equations generalising those obtained in standard general relativity [25]. This chapter is divided into four sections: In Section 2, we review briefly the polynomial affine model of gravity. In Section 3, we use the LevìCivita connection for a FriedmanRobertsonWalker metric, to solve the field equations—obtained in the torsionfree sector—of polynomial affine gravity. Then, in Section 4, we solve for, the case of (affine) Ricciflat manifold, the field equations for the affine connection. Some remarks and conclusions are presented in Section 5. For completeness, in Appendix A, we include a short exposition of the Lie derivative applied to the connection and show the Killing vectors compatible with the cosmological principle.
2. Polynomial affine gravity
In the standard theory of gravity, general relativity, the fundamental field is the metric,
In this section, we briefly expose the model proposed in Refs. [24, 25], which is inspired in the aforementioned role separation. The metric is left out the mediation of gravitational interactions by taking it out the action.
The action of the polynomial affine gravity is built up from an affine connection,
where
The most general action preserving diffeomorphism invariance, written in terms of the fields in Eq. (2), is
where terms related through partial integration and topological invariant have been dropped.^{2} One can prove via a dimensional analysis, the
The action in Eq. (3) shows up very interesting features: (i) it is powercounting renormalisable;^{3} (ii) all coupling constants are dimensionless which hints the conformal invariance of the model [33]; (iii) yields no threepoint graviton vertices, which might allow to overcome the
First, notice that the vanishing torsion condition is equivalent to setting both
The nontrivial equations of motion for this action are those for the Curtright field,
where
In the Riemannian formulation of differential geometry, since the curvature tensor is antisymmetric in the last couple of indices, the second term in Eq. (5) vanishes identically. However, for nonRiemannian connections, such term still vanishes if the connection is compatible with a volume form. These connections are known as equiaffine connections [36, 37]. In addition, the Ricci tensor for equiaffine connections is symmetric. For these connections, the gravitational equations are simply
Eq. (6) is a generalisation of the parallel Ricci curvature condition,
which can be written using the curvature twoform as
Although the field equations of the connection obtained from Eq. (7) are the harmonic curvature condition [43],
these are equivalent to Eq. (6) through the second Bianchi identity [39, 44].
The StephensonKilmisterYang model is a field theory for the metric—not for the connection, and thus there is an extra field equation for the metric. The field equation for the metric is very restrictive, and it does not accept Schwarzschildlike solutions [45]. However, in the polynomial affine gravity, since the metric does not participate in the mediation of gravitational interaction, that problem is solved trivially. Meanwhile, the physical field associated with the gravitational interaction is the connection. This difference makes a huge distinction in the phenomenological interpretation of these models.
In the following sections, we shall present solutions to the field Eqs. (6), in the cases where the connection is metric or not. To this end, in Appendix A we show how to propose an ansatz compatible with the desired symmetries. Moreover, Eq. (6) can be solved in three ways, yielding to a subclassification of the solutions: (i) Ricci flat solutions,
3. Cosmological metric solutions
The conditions of isotropy and homogeneity are very stringent, when imposed on a symmetric ranktwo tensor, and the possible ansatz is just the FriedmannRobertsonWalker metric:
In the remaining of this section, we shall use the standard parametrisation of a FriedmannRobertsonWalker metric, that is
The nonvanishing components of the LeviCività connection for the metric in Eq. (11) are
3.1. … with vanishing Ricci
This particular case is a metric model of gravity, whose field equations are vanishing Ricci. It is expected to obtain the cosmological vacuum solution of general relativity (without cosmological constant), that is, Minkowski space–time.
From the connection in Eq. (12), it is straightforward to calculate the Ricci tensor, and the field equations are then
where the functions
The solutions to Eq. (13) are shown in Table 1 and (as expected) are two parametrisations of Minkowski space–time (see, for example, Ref. [46]).
Scale factor for the metric vanishing Ricci case  







3.2. … with parallel Ricci
Secondly, we shall analyse the possible solutions to the parallel Ricci equations:
Notice that in the case of Riemannian geometry, there is a
The independent components of Eq. (14) for the ansätze in Eq. (11) are
Additionally, Eq. (15) can be rewritten as
According to the value of the integration constant
Using Eq. (17) to eliminate the
The solutions to Eq. (17) are presented in Table 2, and they are known from general relativity (see, for example, Ref. [46]). Interestingly, our integration constant,
Scale factor for the metric parallel Ricci case  

















3.3. … with harmonic Riemann
Now that we showed that the solutions of the parallel Ricci equations are equivalent to those of general relativity, we turn our attention to Eq. (6). For the metric ansatz in Eq. (11), interestingly, only an independent equation is obtained:
that should determine the scale factor. It can be rewritten as
that is,
After a change of variable,
whose solutions are
Therefore, the scale factors are those presented in Table 3. Notice, however, that in this case we are not separating the cases according to the value of
Scale factor for the metric harmonic curvature case  









4. Cosmological nonmetric solutions
In order to solve the set of coupled, nonlinear, partial differential equations for the connection, one proceeds—just as in general relativity—by giving an ansatz compatible with the symmetries of the problem. Using the Lie derivative, we have found the most general torsionfree connection compatible with the cosmological principle [47].^{5} The nonvanishing components of the connection are
with
The Ricci tensor calculated for the connection in Eq. (24) has only two independent components:
We now proceed to find solutions to Eq. (6). As in the previous section, we present the three possibilities of solutions, but we will restrict ourselves to finding solutions to the (affine) Ricciflat case.
4.1. … with vanishing Ricci
The first kind of solutions can be found by solving the system of equations determined by vanishing Ricci. However, this strategy requires the fixing of one of the unknown functions. The equations to solve are written as
Noticing that in the above equations
where we have defined
where
A particular solution inspired in the components of the connection for FriedmannRobertsonWalker, in whose case
which for
There are special solutions that cannot be obtained from Eqs. (29) and (30), since they represent degenerated point in the moduli space.
At this point, we have shown that a space–time described by a Ricci flat, torsionfree, equiaffine connection with the form presented in Eq. (24) reproduces the cosmological Ricciflat solutions to general relativity, presented in Table 1, and there exist generalisations to these solutions which are not possibly obtained in the Riemannian case. However, one can go even further and ask oneself whether the affine Ricciflat condition yields more—real life—useful solutions, such as those solutions of general relativity presented in Table 2.
Therefore, we would like to obtain the Einstein equations from the affine Ricciflat equation, that is,
where
In the following, we are considering that the stressenergy tensor describes a perfect fluid, that is,
In general relativity, the Einstein equations in the form of Ricci, for the cosmological ansatz, yield
Now, comparing Eq. (36) with Eqs. (27) and (28), a parametrisation for
where the functions
with
Eqs. (38) and (39) can be formally integrated in terms of functions
Therefore, a subspace of the possible solutions of the affine Ricciflat geometries describes the cosmological scenarios from general relativity coupled with perfect fluids. However, the explicit expressions for Eqs. (41) and (42) for obtaining specific solutions to FriedmannLemaîtreRobertsonWalker models are very complicated.
4.2. … with parallel Ricci
The second class of solutions can be found by solving the parallel Ricci equation,
However, the system of equations is complicated enough to avoid an analytic solution.
Despite the complication, we can try a couple of assumptions that simplify the system of equations, for example, if one considers the parametrisation inspired in the FriedmannRobertsonWalker results, that is, setting
4.3. … with harmonic curvature
Finally, the third class of solutions are those of Eq. (6). The set of equations degenerate and yield a single independent field equation:
Therefore, we need to set two out of the three unknown functions to be able to solve for the connection.
5. Conclusions and remarks
In this chapter, we have shortly reviewed the recently proposed model of polynomial affine gravity, which is an alternative model for gravitational interactions described solely by the connection, that is, the metric does not play any role in the mediation of the interaction. Among the features of the model, one encounters that despite the numerous possible terms in the action (see Eq. (3)), the absence of a metric tensor gives a sort of rigidity to the action, in the sense that only a very restricted set of terms can be added. Such rigidity suggests that if one attempts to quantise the model, it could be renormalisable. Additionally, all of the coupling constants, in the pure gravity regime, are dimensionless, pointing to a possible conformal invariance of the (pure) gravitational interaction.^{7}
Restricting ourselves to equiaffine, torsionfree connections, the field equations are a generalisation of those from general relativity (Eq. (6)). We solved the field equations for an isotropic and homogeneous connection, either compatible with a metric or not. These solutions are classified under three conditions: Ricci flat, parallel Ricci and harmonic curvature.
When the affine connection is the LeviCività connection for a FriedmanRobertsonWalker metric, we show that the sole solution for a Ricciflat space–time is described by the connection of Minkowski’s space (see Table 1). In the parallel Ricci case, we show that—as intuitively expected—one recovers the vacuum cosmological models from general relativity (see Table 2), where the cosmological constant enters as an integration constant, but such constant could be interpreted as (partially) coming from the stressenergy tensor of a vacuum energy perfect fluid, as mentioned—in the context of general relativity—in Ref. [48]. Finally, the (formal) solutions to the harmonic curvature are presented in Table 3, but yet some work remains to be done to extract the phenomenology from these solutions.
In the case of the cosmological affine connection, we found that the Ricciflat condition yields only two independent equations, which are not enough to find the three unknown functions that parametrise the homogeneous and isotropic connection. Nonetheless, since
The conditions of affine parallel Ricci could be the cornerstone in solving the aforementioned degeneracy, since these conditions raise three independent equations that would serve to determine the three unknown functions. However, at the moment we have not yet achieved any interesting result in pursuing this goal.
On the other hand, the harmonic curvature condition yields a sole (independent) field equation, and therefore the solutions are even more degenerated than those from the Ricciflat condition, leaving even more space for nonphysical solutions.
Our research has stressed the importance of considering the connection as the mediator of the gravitational interactions. We have confirmed that in the framework of polynomial affine gravity, the cosmological solutions described by a connection compatible with a FriedmannRobertsonWalker metric are compatible with those of general relativity, with the possible exception of the case of harmonic curvature. The impact of our contribution lies in the fact that for a generic affine connection, even the simplest condition—Ricci flatness—allows solutions which are (dynamically) equivalent to the system of FriedmannLamaîtreRobertsonWalker for a perfect fluid (in general relativity), despite the absence of matter in the affine model.
We would like to finish our discussion highlighting that the geometric emulation of matter content can serve as a starting point to a change of paradigm related with the interpretation of the matter content of the Universe, in particular the
Acknowledgments
We would like to thank the following people for their support; without those help this work would never have been possible: Aureliano Skirzewski, Cristóbal Corral, Iván Schmidt, Alfonso R. Zerwekh, Claudio Dib, Stefano Vignolo and Jorge Zanelli. We are particularly grateful to the developers of the software SageMath [53], SageManifolds [54, 55, 56] and Cadabra2 [57, 58, 59], which were used extensively along the development of this work. We gratefully acknowledge the constructive comments of the anonymous referee. The “Centro Científico y Tecnológico de Valparaíso” (CCTVal) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of CONICYT, under project No. FS0821.
The usual procedure for solving Einstein’s equation is to propose an ansatz for the metric. That ansatz must be compatible with the symmetries we would like to respect in the problem. A first application is seen in Schwarzschild’s metric [60], which is the
The formal study of the symmetries of the fields is accomplished via the Lie derivative (for reviews, see Refs. [61, 62, 63, 64]). Below, we briefly explain the use of the Lie derivative for obtaining ansatzes for either the metric or the connection.
The Lie derivative of a connection possesses an inhomogeneous part, in comparison with the one of a rankthree tensor. This can be written schematically as
or explicitly
where
In particular, for cosmological applications, one asks for
with^{8}
The six Killing vectors of these algebras, expressed in spherical coordinates, are,
Using Eq. (47), for the above Killing vectors, the most general connection compatible with the desired symmetries can be obtained [47], giving the components structure shown in Eq. (24).
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Notes
 Besides the standard model of particles, there is a standard model of cosmology.
 An example of fourdimensional topological term is the Euler density.
 Powercounting renormalisability does not guarantee renormalisability.
 The field equations can be consistently truncated under the requirement of vanishing torsion. It is worth noticing that this condition does not yield the Riemannian theory, since we are not yet asking for a metricity condition.
 See Appendix A for a brief comment about the Lie derivative of a connection.
 The standard parametrisation of Minkowski space–time is achieved by the trivial solution of Eqs. (27) and (28), i.e. f = g = h = κ = 0 .
 At least at classical level.
 The inhomogeneous algebra of ISO n can be obtained from those of SO n + 1 or SO n 1 through the InönüWigner contraction [65].