Cosmology for quadratic gravity in generalized Weyl geometry
Abstract
A class of vectortensor theories arises naturally in the framework of quadratic gravity in spacetimes with linear vector distortion. Requiring the absence of ghosts for the vector field imposes an interesting condition on the allowed connections with vector distortion: the resulting oneparameter family of connections generalises the usual Weyl geometry with polar torsion. The cosmology of this class of theories is studied, focusing on isotropic solutions wherein the vector field is dominated by the temporal component. De Sitter attractors are found and inhomogeneous perturbations around such backgrounds are analysed. In particular, further constraints on the models are imposed by excluding pathologies in the scalar, vector and tensor fluctuations. Various exact background solutions are presented, describing a constant and an evolving dark energy, a bounce and a selftuning de Sitter phase. However, the latter two scenarios are not viable under a closer scrutiny.
pacs:
04.50.Kd,98.80.Cq,04.20.Fy,02.40.HwI Introduction
General Relativity (GR) is the most widely accepted theory of gravity. It has been subjected to intense experimental confrontation, but yet there is no
direct evidence that would signal a need for modifications of the theory Will (2014). Nevertheless, research of different gravity theories is flourishing. This is partially because GR is not found to be theoretically satisfactory: it is not a quantum theory, and it predicts singularities, for example. Also, coupled with the Standard Model of elementary particles, it predicts a catastrophically large cosmological constant, if one does not resort to technically highly unnatural finetuning that is unstable against quantum corrections Weinberg (1989); Martin (2012). Furthermore, if we accept that GR is valid at all scales, we then need to invoke new ingredients in the cosmological matter sector: the notorious dark energy and the dark matter that has escaped all our attempts at direct or indirect detection Adamek et al. (2015); Durrer and Maartens (2008); Blanchet and
Heisenberg (2015a, b).
A defining property of GR is the unity of the metric and the affine structures of spacetime. In GR, the latter is tied to be fully determined as the Christoffel symbols of the metric tensor which is thus the only independent field. Most modifications of GR have been introduced in this context, by considering then more general actions for the metric and by introducing additional fields that couple nonminimally to the metric. The cosmological constant problem, for example, might be addressed by devising an action for the metric that would degravitate Dvali et al. (2002); ArkaniHamed et al. (2002); Dvali et al. (2007); de Rham et al. (2011), adjust suitably the underlying symmetries such as by introducing unimodularity Unruh (1989); Shaposhnikov and Zenhausern (2009a) or feature selftuning with or without extra fields Kehagias and Tamvakis (2002); Charmousis et al. (2012a); de Rham and Heisenberg (2011). On the other hand, the ultraviolet singularities and the obstacles to renormalisation could be smoothened by quantum effects or by higherderivative terms for the metric in the gravitational action Stelle (1977); Bonanno and Reuter (2000); Biswas et al. (2012); Bambi et al. (2013).
In the more general context of nonRiemannian geometries, richer possilibilities emerge as the connection may carry also torsion Hammond (2002) and nonmetricity Krasnov (2008); Sobreiro and Vasquez Otoya (2010); Vitagliano (2014). As also Einstein himself argued, the connection appears to have a more fundamental ontological status than the metric field Einstein (2001), which has become apparent especially in the developments of gauge theories of gravity Hehl et al. (1976); Blagojevic (2002). In the Palatini approach, wherein one regards the spacetime connection as an independent field besides the metric, it is well known that GR is dynamically recovered for the EinsteinHilbert action (and all the subsequent Lovelock invariants Exirifard and SheikhJabbari (2008); Borunda et al. (2008)), but more general actions become different theories when subjected either to the metric or to the Palatini variation Olmo (2011); Amendola et al. (2011); Capozziello et al. (2015); Jiménez et al. (2014). A perhaps more conservative than the Palatini approach is however to consider that only some of the degrees of freedom residing in the nonmetricity or the torsion sector are physically relevant. A prototype example is the Weyl geometry, wherein the only additional field is the trace of the nonmetricity, the so called Weyl vector.
Since the first conception of the idea of gauge symmetry, Weyl geometry has provided an important framework for the development of fundamental theories Scholz (2011). In Weyl’s unified theory of electromagnetism and gravity, the principle of relativity applied not only to the choice of reference frames, but also to the choice of local units of length. The corresponding spacetime geometry can accommodate scaleinvariance, meaning that physics becomes insensitive to for example the difference of masses of various particles. Though the original formulations of Weyl’s unified theory can be abandoned as not viable, the idea that physics at some fundamental level is scale invariant remains alive and appealing. Obviously, such a symmetry should be somehow broken at our low energy world, but it still could provide a key to a solution of the most fundamental problems of gravitational physics, if it is accepted that the elimination of the physical propagation of the conformal mode of GR could redeem gravity from both the ultraviolet singularities and the infrared vacuum’s weight Shaposhnikov and Zenhausern (2009b); ’t Hooft (2011). It was recently argued that local conformal invariance has to be an exact symmetry and further, broken in a spontaneous manner ’t Hooft (2015).
In this paper we study cosmologies in an extended Weyl geometry. It is not our aim to formulate a scaleinvariant theory (for such, see e.g. Mannheim (2012); Varieschi (2010)) but simply explore the cosmological implications of a fundamental gravitational vector mode. It is perhaps surprising that the leading order, quadratic curvature corrections to the gravitational action were elucidated in the context of Weyl geometry only very recently Beltrán Jiménez and Koivisto (2014); Haghani et al. (2015). The result is interesting as it introduces a novel fourparameter class of potentially viable (at least, ghostfree) vectortensor theories. We considered an extension of the Weyl geometry wherein the distortion (i.e. the nonLeviCivita part of the connection) is linear in a vector field in the most general nonderivative way, featuring then two extra terms besides the pure Weyltype nonmetricity Beltrán Jiménez and Koivisto (2015). As one of the special cases, this geometry includes the oneparameter nonmetric spaces of Ref. Aringazin and Mikhailov (1991). Here we will systematically derive the quadratic curvature theory in the geometry with linear vector distortion (for studies of other quadratic nonmetric theories, see e.g. Heinicke et al. (2005); Baekler et al. (2006)), and apply the results to study various new cosmological scenarios. Vector field cosmologies have been already studied extensively in the literature, with interest in e.g. dark energy ArmendarizPicon (2004); Koivisto and Mota (2008a); Beltran Jimenez and Maroto (2008), inflation Golovnev and Vanchurin (2009); Golovnev (2011); Solomon and Barrow (2014), anisotropies Koivisto and Mota (2008b); Thorsrud et al. (2012); Akarsu et al. (2014); Koivisto and Urban (2015), dual fields Koivisto and Nunes (2010); Koivisto and Urban (2012), GaussBonnet couplings Oliveros et al. (2015); Geng and Lu (2016), screening Beltran Jimenez et al. (2013); De Felice et al. (2016), a cosmological constant Beltran Jimenez and Maroto (2009); Beltran Jimenez and Maroto (2010a); Beltran Jimenez et al. (2009) and stability EspositoFarese et al. (2010); Fleury et al. (2014). Our framework however suggests a fundamental geometric origin for the possible existence of a cosmic vector, furthermore predicting a rather wellspecified vectortensor action.
We begin in the next Section II by deriving this action from the most general quadraticincurvature theory, on which we then impose restrictions by various consistency requirements (we also take a brief look at possible higher order curvature invariants). Remarkably, it turns out that we are then restricted to the class of generalised Weyl geometry, whose special status was already recognised from another aspect Beltrán Jiménez and Koivisto (2015). In Section III we then study the existence of de Sitter solutions, and find out that there are 04 such solutions depending on the theory parameters. We go further and analyse the propagation of scalar, vector and tensor fluctuations for generic models in de Sitter background. In Section IV we then study in more detail some specific analytical solutions that are found for some given parameter combinations. We present both interesting late time (dark energy, cosmological constant) and early time (selftuned de Sitter with first order phase transition, bouncing solution) scenarios. Finally, we conclude in Section VI and complete some derivations with details in the Appendix.
Ii Generalizing Weyl geometry: Spacetimes with linear vector distortion
In this section we will start by briefly reviewing the basic properties of Weyl geometry and how it can be generalized to include the most general connection linearly determined by a vector field as introduced in Beltrán Jiménez and Koivisto (2015). Then, we will proceed to the construction of gravitational actions based on these geometries.
ii.1 Geometrical framework
The defining property of Weyl geometry is the breaking of the metricity condition by introducing a vector field as follows
(1) 
This relation is preserved by the transformation when simultaneously we transform and, thus, Weyl geometry is a natural arena to formulate conformally invariant theories, although we will not pursue this here. The above expression can be easily solved for the connection (assuming vanishing torsion) to obtain
(2) 
We see that in Weyl geometry, the connection acquires a distortion linearly depending on .
The class of geometries introduced in Beltrán Jiménez and Koivisto (2015) extends the Weyl geometry by allowing for the most general connection with a distortion tensor linearly determined by a vector field. The form of the desired connection is thus
(3) 
where are the usual Christoffel symbols of the metric and are arbitrary coefficients. This connection gives rise to nonmetricity, but also contains the trace vector part of the torsion. Interestingly, although the conformal invariance of the metric compatibility condition is lost by the general connection (3), the extra torsion component allows to recuperate it for more general connections. To see it explicitly, we can compute the covariant derivative of the metric, which is given by
(4) 
Thus, we recover the Weyl condition (1) provided we impose . In the absence of torsion , we exactly recover the Weyl condition, but the torsion term allows to maintain the gauge invariance of (4) for more general connections. It is not difficult to see that for , the equation (4) remains invariant under while . We can now introduce the Riemann tensor of our connection as usual
(5) 
It is important to keep in mind that, since we have torsion and nonmetricity, this Riemann tensor does not have the usual symmetries of LeviCivita connections, but only the antisymmetry in the first two indices inherited from its definition as a commutator: . In particular, this means that we can construct 3 independent traces, namely: the usual Ricci tensor , the coRicci tensor and the homothetic tensor^{1}^{1}1Sometimes referred to as the segmental curvature tensor. . For the connection given in (3), these tensors can be expressed as
(6)  
(7)  
(8)  
where and are the Ricci tensor and covariant derivative of the LeviCivita connection of the spacetime metric, is the strength tensor of the vector field and is the spacetime dimension. Notice that the Ricci tensor is not symmetric, not even in the torsion free case with , since nonmetricity can also induce an antisymmetric part for the Ricci tensor. It is also convenient to keep in mind that the homothetic tensor is always antisymmetric and for symmetric connections it is proportional to the antisymmetric part of the Ricci tensor. Finally, the Ricci scalar is unambiguously defined as and, for our connection (4), it is given by
(9) 
where is the Ricci scalar of the LeviCivita connection.
To end this section we will give some important geometrical objects. A defining property of geometries with nonmetricity is that the length of vectors is not preserved under parallel transport. If we have a vector , its length under a parallel displacement is
(10) 
Obviously, for a pure metric geometry with , the length is conserved. However, it is remarkable that the presence of torsion () also allows to have a wider class of geometries that preserve the length of vectors given by . This actually allows to avoid one of the main problems of the pure Weyl geometries, where the length of a vector changes as it is parallel transported and, consequently, the properties of a physical object may depend on its history^{2}^{2}2It is important to emphasize that this crucially depends on the connection seen by matter fields. For instance, if bosonic fields are minimally coupled to the geometry, then they will be sensitive only to the LeviCivita piece of the connection.. In our more general geometrical setup, this can be avoided even for nontrivial connections with nonmetricity. One could relax the condition of invariance of the length of vectors only when they are transported along closed loops. In that case, the variation is determined by the homothetic or segmental curvature tensor introduced above. Again, we have a special family of geometries with in which the length of a vector remains invariant when transported around a closed path. Of course, these geometries contain the aforementioned case with , but more general cases are possible in which the length of a vector may vary under a parallel transport while remaining the same when the trajectory closes.
Besides the variation of the length of vectors, the presence of nonmetricity will also affect the properties of the geodesics. Since only the symmetric part of the connection enters the geodesic equation, the torsion term determined by will not enter here. A crucial feature of the geodesics is their projective similarity, i.e., two families of geodesics related by a change of affine parameter will describe the same class of paths. More precisely, it can be shown that two connections differing by give rise to the same geodesics up to a redefinition of the affine parameter Thomas (1925). The projective invariant object determining the class of paths is the Thomas projective parameter, which for our connection is given by:
(11) 
where is the piece corresponding to the LeviCivita part of the connection. In the above expression, only the symmetric (torsionfree) part of the connection should be considered. As expected, this expression does not depend on , since this term precisely corresponds to a projective transformation and, consequently, the geodesic trajectories will not be affected by it.
ii.2 Gravitational actions
Theories based on the Ricci scalar given in (9) lead to interesting phenomenologies, including the Starobinsky inflationary model and its socalled attractor generalisation Ozkan et al. (2015); Ozkan and Roest (2015); Beltrán Jiménez and Koivisto (2015). This is possible because in type of theories with (9) the vector field is dynamically constrained to be the gradient of a scalar and, thus, similarly to usual theories, they are equivalent to a scalartensor theory.
In order to have a fully dynamical vector field (and not imposed to be a pure gradient) we need to consider more general actions. The natural step is then to include quadratic curvature terms. For LeviCivita connections, the requirement of having second order equations of motions (so that Ostrogradski instabilities are avoided) leads to the wellknown Lovelock invariants. The quadratic term of such invariants is the GaussBonnet term, which in 4 dimensions is a total derivative and, thus, it does not contribute to the gravitational field equations. Considering the corresponding GaussBonnet term with our vector connection will also give a total derivative in 4 dimensions. However, as found in Beltrán Jiménez and Koivisto (2014), more general quadratic curvature terms can give rise to a new interesting class of vectortensor theories in the context of Weyl geometry, see also Haghani et al. (2015). Following the same procedure, we will write down the most general action quadratic in curvature invariants in the more general linear vector geometry defined by the connection (4). Such terms are given by
(12) 
where and are dimensionless constants and has dimension . In order not to have Ostrogradski instabilities associated to higher order equations of motion for the metric we will restrict the parameters in order to recover the GaussBonnet term for the LeviCivita part of the connection (i.e, when ) so that
(13) 
Since identically vanishes for , the coefficients remain fully free. Now we can rewrite (12) as a vectortensor theory in a Riemannian geometry. To that end, we will use the decomposition of the connection (4) in (12) and express everything in terms of and the LeviCivita connection . After some straightforward algebra and a few integrations by parts as done in Beltrán Jiménez and Koivisto (2014), the action can finally be expressed as
(14) 
where , , , and are dimensionless parameters that depend on , , and the spacetime dimension . The combination in the brackets in the first line is nothing but the GaussBonnet term for the LeviCivita connection, as a consequence of having imposed (13). The remaining terms in the first line are the same as were obtained in Beltrán Jiménez and Koivisto (2014) for the case of pure Weyl geometry. Despite the derivative interaction and the nonminimal coupling , those terms only propagate 3 degrees of freedom, very much like the simpler case of a Proca field. However, the terms in the second line were not present in the pure Weyl case and only arise for our general connection (4). These are however undesirable because they will propagate one additional degree of freedom besides the 3 polarizations corresponding to a massive vector field and this extra mode will generically suffer from the Ostrogradski instability, i.e., it will be a ghost. Thus, in order to have a stable theory we need to impose the conditions . The details are given in the Appendix A. Remarkably, the only solution for is , which exactly coincides with the generalised Weyl geometry discussed above that preserves the local conformal invariance of the metric compatibility condition. In that case, we can canonically normalize the field by means of the rescaling . We can include the standard EinsteinHilbert term for completeness, which will simply give a mass term for the vector field, so the final action for the vector reads
(15) 
with
(16)  
(17)  
(18)  
(19) 
As we see, in 4 spacetime dimensions only the mass term remains while all the interactions vanish. To obtain a nontrivial theory in 4 dimensions we could consider the limit simultaneously with while keeping and fixed. Then there remains only two free parameters, in particular the action could be reduced to (when choose the free parameters as the constants and )
(20) 
Throughout this work, we will focus on and for generality consider all 4 parameters as independent. It can be useful to note that our general vectortensor action can be rewritten in an alternative way by using that together with so that we finally obtain
(21) 
where we can recognize the typical Horndeski form for the vector field derivative selfinteractions.
So far we have only considered actions up to quadratic order in curvature invariants. The same program can be straightforwardly applied to higher order curvature invariants by imposing the terms at a given order to reduce to the corresponding Lovelock invariant. Instead of discussing the general framework for this construction, we will simply comment on a class of terms that give rise to additional nontrivial derivative interactions for the vector field. For simplicity, we will consider the case . At cubic order we know that a healthy coupling is given by the Horndeski vectortensor interaction Horndeski (1976) , with the double dual Riemann tensor, so we can use it to construct new terms in our class of geometries. For that, we notice that the antisymmetric part of the 3 independent traces of the Riemann tensor given in (8) are all proportional to . Thus, we can consider an interaction of the form
(22) 
that will give the Horndeski interaction for the nonminimal derivative coupling of the vector. Although we have written down explicitly all the possible terms, they all will contribute exactly the same interactions for the vector field. Explicitly, the above cubic interaction leads to
(23) 
where is the dual of the vector field strength tensor and some energy scale. The first term is the advertised Horndeski vectortensor interaction (which has been studied in, e.g., Barrow et al. (2013); Beltrán Jiménez et al. (2013)), while the second term in the first line is a derivative selfinteraction of the vector field. Although this term does not respect gauge invariance and contains derivatives of the vector field, its structure does not spoil the constraint making nondynamical. This can be seen by noticing that the time derivative of couples to , which is proportional to the magnetic part of . Since this magnetic component only contains spatial gradients, will not acquire second derivatives in the field equations and, consequently, it will not propagate. These interactions are in fact within the class of derivative selfinteractions for a massive vector field discussed in Beltran Jimenez and Heisenberg (2016); Allys et al. (2016); Tasinato (2014); Heisenberg (2014). Moreover, it is expected that by considering higher order curvature terms within our framework, the higher order terms interactions introduced in these references will be generated.
Iii de Sitter solutions
In this section we will show the existence of isotropic de Sitter solutions for the vectortensor theory given by (15). In order to comply with the given symmetries, we will consider a purely temporal and homogeneous vector field configuration as well as a homogeneous and isotropic metric described by the FriedmannLemaîtreRobertsonWalker (FLRW) line element
(24) 
where is the scale factor. The only nontrivial vector field equation for this configuration is given by
(25) 
with the Hubble expansion rate. As expected, this is an algebraic equation showing that is not dynamical and it is fully determined by . This equation leads to 3 branches, namely: the trivial one with and 2 nontrivial ones with
(26) 
From this expression we can see that de Sitter solutions are not guaranteed for any values of the parameters. For the singular case , the degree of the equation is reduced and only one nontrivial solution remains. This particular case will be studied separately below. In order to obtain the value of the Hubble parameter for the de Sitter solution we need to look at the corresponding Friedmann equation
(27) 
For the trivial branch with we obtain that , i.e., we recover the Minkowski solution as expected. On the other hand, for the nontrivial branch given by (26), the Friedmann equation gives an algebraic equation for . Such an equation forces to be constant and, therefore, these branches actually correspond to a de Sitter universe. However, the existence of such solutions will be subject to the existence of real solutions for the corresponding system of algebraic equations, which imposes restrictions on the parameters as already mentioned above. In fact, we can see that the equations reduce to a system of two polynomial (one quadratic and one quartic) equations so that we will have in general up to 8 different branches. They have some properties that can simplify the analysis of the solutions. First, one can easily see that there is a symmetry in the equations . Moreover, if is a solution for , then will be a solution for . Thus, without loss of generality we can absorb into a rescaling of and focus on solutions with (i.e., expanding solutions), keeping in mind that a corresponding contracting solution will be guaranteed to exist as well. Further, one can get rid of another parameter (up to its sign) by absorbing it into the normalization of . The most convenient one is to absorb so that .
In order to characterize the models for which de Sitter solutions exist, we will introduce the following rescalings and a dimensionless variable
(28) 
Obviously, this can only be done if both and are nonvanishing. From now on, we will assume this and treat separately below the case of vanishing or . Then, from the vector field equation we can solve for in terms of as
(29) 
This already allows us to put some conditions on the parameters to have de Sitter solutions, since we need to have . Now we can plug this solution into the Friedman equation to obtain the following quartic equation for :
(30) 
This is the crucial equation that will determine the existence of de Sitter solutions. From here we see that if is negative, there will be at least a couple of real solutions. It is important to notice that, since this equation does not depend on , we can take its real solutions, plug them into (29) and choose the sign of in order to guarantee that is positive. In Fig. 1 we show the number of solutions in the parameter space spanned by . For the singular case it is not difficult to see that there is 1 solution for and 3 for .
We can see that, quite generally, the vectortensor theories considered here give rise to de Sitter solutions. The next natural question is then if such solutions are stable. This is the subject of the next subsection.
Stability
After showing the existence of de Sitter solutions, we will now proceed to check the stability of such solutions. For that, we will study the behaviour of the inhomogeneous perturbations around the de Sitter solutions found above. Thus, the background will be given by a constant temporal component for the vector field and a constant Hubble expansion rate . The perturbations for the metric will be decomposed into irreducible representations of the symmetry of the background in the usual manner Mukhanov et al. (1992)
(31) 
where it is understood that all the metric perturbations depend on time and space and we have the constraints . On the other hand, the vector field will be analogously perturbed as
(32) 
with .
Naïvely counted, we encounter 14 dof’s in this decomposition, namely: in the traceless symmetric tensor (), in the divergencefree vectors (, , ) and in the scalars (, , , , , ). However, we have 4 diffeomorphism gauge symmetries that remove 2 dof’s each, so we have dof’s. In addition, as discussed above, the temporal component of the vector is not dynamical, but an auxiliary field, so we should substract yet another dof and, thus, the number of physical propagating modes will be , i.e., the 2 polarizations corresponding to the massless graviton plus the 3 polarizations of the massive vector. In the following we will explicitly see this by studying the tensor, vector and scalar perturbations individually, and further establish the stability conditions for the 2 tensor, 2 vector and 1 scalar dof’s in the theory. Due to the homogeneity of the background, it will be convenient to decompose the perturbations in Fourier modes with respect to the spatial coordinates. Hence, all perturbations will be expanded as
(33) 
where represents a given perturbation.
As a final remark, let us stress that we will perform the analysis without a priori fixing the gauge at the level of the action. We have also carried out the analysis in the Newtonian gauge where as a consistency check and found the same results.
Tensor perturbations
Let us start by studying the tensor perturbations. This is the simplest case because there are no dynamical dof’s to be integrated out. After inserting the metric perturbations (III) into our vectortensor action with the fields decomposed in Fourier modes and using the dynamical background equations from the previous section, the action quadratic in the tensor perturbations reads
(34) 
We see that the quadratic action for tensor perturbations is modified in the presence of a background and, as one would expect, only the nonminimal coupling to the Einstein tensor (term) contributes. However, there is a dependence on the remaining parameters through the background value , which is determined by all the parameters.
From the above action, we easily conclude that tensor perturbations around the de Sitter solutions will avoid ghostlike instabilities, i.e., they have the right sign for the kinetic term, if we impose
(35) 
which is trivially satisfied if . On the other hand, the propagation speed of the perturbations is
(36) 
that also must be positive to avoid gradient instabilities. These results are in agreement with those found in Beltran Jimenez and Maroto (2010b). We can avoid both ghosts and gradient instabilities for the tensor perturbations if
(37) 
In any case, this nontrivial effect on the propagation speed of gravitational waves will be tightly constrained at present time since binary pulsar observations put stringent limits, at the level of deviations from the speed of light Beltrán Jiménez et al. (2016).
Vector perturbations
Let us now turn to the slightly more involved case of vector modes. As we mentioned previously, two of the vector perturbations can be integrated out and only one vector will propagate (two dof’s, as a transverse 3vector). We expand our Lagrangian to second order in the vector perturbations and immediately observe that the vector field does not have any kinetic terms. Therefore we can simply compute the equation of motion with respect to and integrate them out. This yields
(38) 
After plugging this expression back into the action and adding total derivatives, the dependence on drops. Thus, as advertised, we end up with the quadratic action for only one vector,
(39) 
The propagation speed is given by
(40) 
again in agreement with the findings in Beltran Jimenez and Maroto (2010b). As in the case of tensor perturbations, we see that only the coupling to the Einstein tensor (term) modifies the quadratic action of vector perturbations, and the remaining parameters only enter through the background value . We see that the vector perturbations are never ghostly and the only instability that can appear is a gradient one. This was expected because the kinetic term for our vector field is nothing but the usual Maxwell term. In order to avoid the Laplacian instability we have to require . Interestingly, it is trivially satisfied for models with .
Scalar perturbations
As usual, the scalar sector is the most involved one. We have six scalars (, , , , , ), but only one propagates, corresponding to the longitudinal mode of . Again, we expand the action to quadratic order in the scalar perturbations. The first thing to be noticed is the fact that the corresponding kinetic matrix (or alternatively the Hessian matrix) contains already at this stage three vanishing eigenvalues, imposing three constraint equations that makes three out of the six scalar fields not propagating. The kinetic matrix is
(41) 
As one can see, the quadratic action does not evolve any kinetic term for the scalar fields and and . Thus we can simply replace them by using their equations of motion. For instance, the equation of motion for the scalar field gives
(42) 
Similarly, the expressions for and are obtained using their equations of motion, which we omit here. After inserting the solutions for , and back into the quadratic action, the resulting expression depends only on the remaining three scalar fields (, , ). On a closer inspection, one realizes that the kinetic matrix of the three scalar fields still has a vanishing determinant, pointing to the presence of more constraints that can be used to integrate out some of the scalar fields. To be precise, the kinetic matrix contains two zero eigenvalues and one nonvanishing eigenvalue. We can diagonalize the kinetic matrix by performing the following field redefinitions, which will make the only propagating scalar field manifest:
(43) 
After adding total derivatives, the resulting action depends only on and reads
(44) 
where and are some functions of the theory parameters and the background solution. Their exact expressions are very cumbersome in the general case. However, their UV limits with can be expressed as
(45)  
(46) 
For the stability of the perturbations we have to impose that both and be positive. Unlike in the previous cases, we see that for the scalar sector all the terms contribute to the perturbations and not only through the background evolution.
Iv Cosmological solutions
After showing the existence of de Sitter solutions and studying the corresponding perturbations around them, we will now consider a more general case in the presence of a matter component. Again, we will consider homogeneous and isotropic universes with the FLRW line element. In such a background metric and for a homogeneous vector field configuration with , the vector field equations read
(47)  
(48) 
From these equations we see that it is a consistent Ansatz to consider purely isotropic solutions with so that our isotropic solutions will be supported by the temporal component of the vector field. Having isotropic solutions based on the spatial part of the vector usually requires to have a set of vector fields with a global symmetry Bento et al. (1993); ArmendarizPicon (2004), sometimes called triad, or rapidly oscillating fields Cembranos et al. (2013, 2012).
Let us now consider a universe filled with the vector field plus a matter component with energy density . We will still restrict to purely isotropic solutions. Then, the vector field and Friedmann equations are
(49)  
(50) 
respectively. For we have the trivial branch that recovers standard gravity so we will assume that . In such a case, we can simply integrate out by (algebraically) solving its own equation of motion, whose solution is still given by (26) and, in general, it yields . Then, we can use this solution again in Friedmann equation to obtain a generalised version of it as
(51) 
Then, we can invert this equation to obtain a Cardassianlike model where the Hubble expansion rate is given by a nonlinear function of the energy density. This is nothing but a particular example of the general result that gravity with auxiliary fields leads to a modified matter coupling. The explicit expression is
(52) 
where parameterizes the two nontrivial branches. If we expand this expression for small we obtain
(53) 
recovering the usual GR Friedmann equation with an effective cosmological constant and a rescaled Planck mass. Notice that the case with is singular, as a consequence of reducing the degree of the vector field equation (which becomes linear in in that case). This singular case will be studied in more detail separately below. Also notice that the small regime does not need to correspond to a low density regime.
A very interesting property of the cosmological evolution in the presence of the vector field is the possibility of having a maximum and/or minimum value for the Hubble expansion rate. This is determined by the discriminant of the vector field equation, i.e., the behavior of
(54) 
which must be positive in order to have real solutions. If both combinations and are positive, we always have real solutions, while if they are negative, real solutions do not exist. However, if they have different signs, we encounter two possibilities:

If and , there is an upper limit for given by
(55) 
If and , then we have a lower bound for the Hubble expansion rate so that .
The above conditions guarantee that the solution for the vector field is real. However, we need additional conditions to guarantee the existence of physical solutions because, once the vector field has been solved for, we will obtain an equation for which must also have real solutions. The overall effect will be the presence of bounds for the energy density of the matter component. The corresponding analysis for the general theory is very cumbersome, so we will focus on some particular cases instead below.
The models with an upper bound for could be useful to resolve the Big Bang singularities. This is expected to be a more general feature (not only for cosmology) that could help regularising other types of singularities, e.g. black hole singularities. Notice that this stems from the quadratic equation for the vector field.
iv.1 Bouncing solutions
An interesting question that arises is whether it is possible to obtain viable bouncing solutions. As we have shown, the presence of the auxiliary field gives rise to a modified Friedmann equation and, therefore, it would be plausible to encounter some bouncing solutions. Such solutions are characterized by the existence of a finite (nonvanishing) energy density for which the Hubble expansion rate vanishes, i.e.,