Self-acceleration and matter content in bicosmology from Noether Symmetries

Self-acceleration and matter content in bicosmology from Noether Symmetries

Mariam Bouhmadi-López M. Bouhmadi-López Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain.
IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain.
22email: mariam.bouhmadi@ehu.eusS. Capozziello Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II, Compl. Univ. di Monte Sant’Angelo, Via Cinthia 9, I-80126 Napoli, Italy.
Istituto Nazionale di Fisica Nucleare (INFN) Sez. di Napoli, Compl. Univ. di Monte Sant’Angelo, Via Cinthia 9, I-80126 Napoli, Italy.
Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L’Aquila, Italy.
44email: capozziello@na.infn.itP. Martín-Moruno Departamento de Física Teórica I, Ciudad Universitaria, Universidad Complutense de Madrid, E-28040 Madrid, Spain.
66email: pradomm@ucm.es
   Salvatore Capozziello M. Bouhmadi-López Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain.
IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain.
22email: mariam.bouhmadi@ehu.eusS. Capozziello Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II, Compl. Univ. di Monte Sant’Angelo, Via Cinthia 9, I-80126 Napoli, Italy.
Istituto Nazionale di Fisica Nucleare (INFN) Sez. di Napoli, Compl. Univ. di Monte Sant’Angelo, Via Cinthia 9, I-80126 Napoli, Italy.
Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L’Aquila, Italy.
44email: capozziello@na.infn.itP. Martín-Moruno Departamento de Física Teórica I, Ciudad Universitaria, Universidad Complutense de Madrid, E-28040 Madrid, Spain.
66email: pradomm@ucm.es
   Prado Martín-Moruno M. Bouhmadi-López Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain.
IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain.
22email: mariam.bouhmadi@ehu.eusS. Capozziello Dipartimento di Fisica “E. Pancini”, Università di Napoli “Federico II, Compl. Univ. di Monte Sant’Angelo, Via Cinthia 9, I-80126 Napoli, Italy.
Istituto Nazionale di Fisica Nucleare (INFN) Sez. di Napoli, Compl. Univ. di Monte Sant’Angelo, Via Cinthia 9, I-80126 Napoli, Italy.
Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L’Aquila, Italy.
44email: capozziello@na.infn.itP. Martín-Moruno Departamento de Física Teórica I, Ciudad Universitaria, Universidad Complutense de Madrid, E-28040 Madrid, Spain.
66email: pradomm@ucm.es
Received: date / Accepted: date
Abstract

In bigravity, when taking into account the potential existence of matter fields minimally coupled to the second gravitation sector, the dynamics of our Universe depends on some matter that cannot be observed in a direct way. In this paper, we assume the existence of a Noether symmetry in bigravity cosmologies in order to constrain the dynamics of that matter. By imposing this assumption we obtain cosmological models with interesting phenomenology. In fact, considering that our universe is filled with standard matter and radiation, we show that the existence of a Noether symmetry implies that either the dynamics of the second sector decouples, being the model equivalent to General Relativity, or the cosmological evolution of our universe tends to a de Sitter state with the vacuum energy in it given by the conserved quantity associated with the symmetry. The physical consequences of the genuine bigravity models obtained are briefly discussed. We also point out that the first model, which is equivalent to General Relativity, may be favored due to the potential appearance of instabilities in the second model.

Keywords:
Noether symmetries cosmology bigravity
pacs:
04.50.Kd 95.36.+x 98.80.Jk
journal: General Relativity and Gravitation

1 Introduction

Bigravity theories, which are models of two mutually interacting dynamical metrics, were initially introduced by Isham, Salam, and Strathdee in the seventies Isham (). These theories can be interpreted as describing two universes interacting in a classical way through their gravitational effects Damour:2002ws (). Recently, they have attracted considerable attention since Hassan and Rosen formulated a bigravity theory Hassan:2011zd () that is free of the Boulware–Deser ghost BD (). The formulation of this ghost-free bigravity theory was possible thanks to the development of a theory of massive gravity by de Rham, Gabadadze and Tolley that was also potentially stable deRham:2010ik (); deRham:2010kj () (see references Hinterbichler:2011tt (); deRham:2014zqa (); Schmidt-May:2015vnx () for a reviews). This massive gravity theory has 5 propagating modes, however, some solutions may still present stability issues related to the scalar polarization Fasiello (); Deser (); Chamseddine:2013lid (); Guarato:2013gba (). In addition, there could be nontrivial gravitational effects in vacuum for massive gravity theories with a Friedmann-Lemaître-Robertson-Walker (FLRW) background metric nonvacuum (). Nevertheless, these potential shortcomings would not necessarily affect the ghost-free bigravity theory nonvacuum (); Fasiello:2013woa (), which could also be considered to be conceptually favoured against massive gravity since it is a background independent theory. In bigravity the two gravitational sectors may be interpreted as weakly coupled worlds in the nomenclature introduced in reference Damour:2002ws (). Unlike other modified gravity theories, the Solar System constraint on this type of theories are so tight that they automatically fulfils the constraint coming from the recently observed GW170817 and GRB 170817A events Baker:2017hug ().

Cosmological scenarios of the ghost-free bigravity theory have been studied assuming that there is only matter minimally coupled to one gravitational sector vonStrauss:2011mq (); Volkov:2011an (); Comelli:2011zm (). These models can describe accelerated solutions in absence of a cosmological constant Akrami (); Konnig:2013gxa (). Nevertheless, as assuming the absence of matter coupled to one sector is a strong assumption Capozziello:2012re (), the possible presence of two sets of matter fields minimally coupled to each gravitational sector has also been considered Capozziello:2012re (); Aoki:2013joa (). The future cosmological evolution could approach a de Sitter state, a matter dominated universe, or even a spacetime future singularity when assuming ordinary matter in both sectors Aoki:2013joa (). Moreover, perturbations in bimetric cosmologies for both cases have been investigated Comelli:2012db (); Comelli:2014bqa (); DeFelice:2014nja (); Konnig:2014dna (); Solomon:2014dua (); Konnig:2014xva (); Akrami:2015qga (). It should be noted, however, that if the matter content minimally coupled to the second gravitational sector is completely arbitrary, then one could re-construct any desired cosmological evolution for our universe by simply fine-tuning that material content which is not directly observable Capozziello:2012re (). Therefore, it would be desirable to find an argument which would favor a particular kind of material content to be coupled to the second gravitational sector. On the other hand, more general couplings of matter to the gravitational sectors have also been considered Akrami:2013ffa (); deRham:2014naa (); Gumrukcuoglu:2015nua (); Heisenberg:2015iqa (). Although some of these couplings could be free of the Boulware–Deser ghost below a cut-off scale deRham:2014naa (), we focus our attention on the ghost-free minimally coupled case.

Ghost-free multigravity theories Hinterbichler:2012cn (); Luben:2016lku (); Tamanini:2013xia (), general kinetic terms for the metrics Nojiri:2012zu (), and Lanczos–Lovelock terms Hassan:2012rq () in higher dimensional generalizations Do:2016uef () have also been investigated. Nevertheless, in this paper, we consider a theory of the form

(1)

where we assume that the interaction term is independent of the derivatives of the metrics (and it is of the particular form introduced in Hassan:2011zd ()), that , , and are different constants. The interaction term leading to a ghost-free bigravity theory Hassan:2011zd () is the same as the Lagrangian terms giving mass to the graviton in the ghost-free massive gravity theory deRham:2010ik (); deRham:2010kj (); Hassan:2011tf (). However, the characteristics and underlying philosophy of those theories is completely different. As it has been pointed out in reference Baccetti:2012bk (), if one insists in considering massive gravity as a particular limit of bimetric gravity, that limit would corresponds to setting . Such a limit must be taken carefully since there is not complete continuity in the parameter space of the theory Baccetti:2012bk ().

With these considerations in mind, in this paper we explore the possibility of fixing the role and dynamics of the minimally coupled matter in bigravity. The existence of symmetries has been a guidance principle in the construction of different models in physics. Usually one assumes a particular symmetry which is physically well motivated to restrict the forms of the possible Lagrangian models of the theory. The Noether Symmetry Approach consists in giving a class of Lagrangians, restricting to those presenting Noether symmetries, and integrating the related dynamics Noether (). Specifically, in the context of theories of gravity extended1 (); extended2 () the Noether Symmetry Approach has been proven to be a useful method for obtaining exact solutions in cosmological scenarios Noether1 (); Noether2 (). The method consists in assuming the existence of an arbitrary Noether symmetry for a family of Lagrangians in cosmological scenarios. Since a symmetry does not always exist, the method select both the form of the Lagrangians that have a symmetry (fixing, in particular, the form of couplings and potentials) and the particular symmetry (i.e. the associated conserved quantity). Moreover, once the symmetry is known, the equations of motion can be easily integrated. The Noether Symmetry Approach has allowed to select particular theories of interest and to obtain exact cosmological solutions in several cases (see for example felicia (); basilakos (); defelice (); nesseris ()). It is worth noticing that symmetries have always a physical meaning and their existence discriminates between physical and unphysical models besides allowing integrability of related dynamical systems. In the present case, the existence of symmetries determines the matter content of the bigravity system. In particular, the existence of a Noether symmetry implies that either the dynamics of the second cosmological sector decouples, being the model equivalent to general relativity, or the cosmological evolution of the observed universe evolves towards a de Sitter state with the vacuum energy (the cosmological constant) determined by the conserved quantity associated to the symmetry. In other words, the matter content (and the source) of the cosmological dynamical model is determined by the symmetry. On the contrary, the absence of symmetry does not allow any conclusion on matter content and cosmological constant. Another important example is discussed in Noether2 () where minisuperspace models, related to quantum cosmology, are taken into account. There, the presence of Noether symmetries determines oscillating behaviors in the wave function of the universe, solution of the Wheeler-de Witt equation. In this case, according to the Hartle criterion, correlated quantities can be identified and conserved momenta imply that observable universes are selected. If the symmetries do not exist, it is not possible to state if observable models can be found or not. In this sense, symmetries are a sort of criterion to select physical models.

In bigravity the situation is different than in the case of extended theories of gravity. Here, the Lagrangian is already fixed and belongs to the family of ghost-free Lagrangians, which includes only four free parameters, but the kind of matter fields, minimally coupled to the second gravitational sector, is completely arbitrary, entailing a great freedom Capozziello:2012re (). Hence, we consider that the material content, minimally coupled to the second sector, should be such that there is a Noether symmetry for the resulting cosmological model. Therefore, in this paper, we will apply the Noether Symmetry Approach to bigravity cosmology not only to select the material content minimally coupled to the second gravitational sector (when assuming the presence of a suitable amount of ordinary matter minimally coupled to our sector) but also to fix some of the free parameters appearing in the interaction Lagrangian.

Specifically, searching for Noether symmetries in bigravity cosmology has some advantages that can be listed as follows: it allows to restrict the range of free parameters according to the criterion of existence or not of symmetry. This seems an arbitrary assumption, but it is motivated by the fact that the presence of conserved quantities allows the reduction of dynamical systems, and, in principle, their exact integration, if the number of first integrals is the same as the number of configuration space variables; if the observed universe is filled with standard matter and radiation, the presence of Noether symmetries allows the decoupling with respect to the second gravitational sector, being the model equivalent to general relativity. This decoupling is not guaranteed, if a symmetry is not present: in fact, the decoupling strictly relies on the possibility to integrate the dynamical system; finally, if the observed universe evolves towards a de Sitter phase, the conserved quantity related to the Noether symmetry allows to fix the vacuum energy (i.e. the cosmological constant) of the model.

Despite the advantages of the existence of a Noether symmetry that we have just discussed, we have to comment also a potential difficulty of some bigravity models that we obtain in this paper. We start by considering a particular bigravity Lagrangian because it propagates the correct number of modes and is, therefore, free of the Boulware–Deser ghost. However, we obtain that the purely bigravity models, which are not equivalent to General Relativity (GR) due to the decoupling of the gravitational sectors, with a Noether symmetry, present gravitational couplings in both sectors with different signs. So these models could still lead to other instabilities. This conclusion would point out that GR is the only stable bigravity model with a Noether symmetry. This statement needs further detailed analysis.

The layout of the paper is as follows: In section 2, we present the basic formalism of cosmological solutions in bigravity. We summarize how to obtain the Friedmann equations in subsection 2.1 and we construct the point-like Lagrangian in subsection 2.2. In section 3, we consider the Noether Symmetry Approach in the context of cosmological bigravity solutions in order to fix the material content minimally coupled to the second gravitational sector. We review the Noether Symmetry Approach in subsection 3.1, apply it to bigravity cosmological models in subsection 3.2, and discuss the particular solutions that we obtain in subsection 3.3. In section 4, we discuss the potential issues of some particular cosmological solutions with a Noether symmetry. We summarize the results in section 5 and relegate some considerations about the definition of the point-like Lagrangian to appendix A.

2 Bicosmology

In this section, we include some results about cosmological scenarios of the ghost-free bimetric theory of gravity Hassan:2011zd (). In the first place, in the subsection 2.1, we summarize the equations of motion retrieved from the action of the theory and, in particular, the modified Friedmann equations, obtained by substituting the cosmological ansatz in the modified Einstein equations. In subsection 2.2, we re-obtain a point-like Lagrangian by considering the cosmological ansatz in the general action of the theory. As we will show, the modified Friedmann equations can also be extracted by varying the point-like Lagrangian, therefore, being both procedures compatible. Finally, in subsection 2.3, we present an equivalent but non-degenerate point-like Lagrangian that will be suitable for the subsequent analysis.

2.1 Bimetric cosmology

The interaction Lagrangian of the ghost-free bigravity theory formulated by Hassan and Rosen is a function of the matrix , implicitely defined as

(2)

The action reads Hassan:2011zd ()

(3)

where the parameter appearing in equation (1) has been absorved in . On the above expression, the elementary symmetric polynomials are Hassan:2011vm ()

(4)
(5)
(6)
(7)
(8)

and the fields and are minimally coupled to and , respectively. In addition, is the Planck mass111It must be noted that is related with the gravitational coupling appearing in the modified Einstein equations (9). However, it could be considered that the physical Planck mass is the one of the massless spin-2 mode mediating the gravitational force, that is ., a dimensionless constant, and and are (free) constants of the model with dimensions of mass and inverse mass squared, respectively. It must be noted that although we have not explicitly written a cosmological constant for each kinetic term in the action (3), those cosmological constants have been absorbed in the and terms of the interaction Lagrangian since they are equivalent to cosmological constant contributions for the -space and -space Hassan:2011vm (); Baccetti:2012re (). The variation of the action (3) with respect to the two metrics leads to two sets of modified Einstein equations. These are vonStrauss:2011mq (); Baccetti:2012bk ()

(9)

and

(10)

where

(11)

and

(12)

The indexes of equations (9) and (10) must be raised and lowered using and , respectively. In addition, and correspond to the matter energy-momentum tensors of the -space and -space, respectively,

Now, let us consider a cosmological scenario. We assume that the metrics can be written as follows:

(13)

and

(14)

where and is the spatial curvature parameter. More general cosmological ansatzs have been also considered in the literature (cf. references Nersisyan:2015oha () and Garcia-Garcia:2016dcw ()). The modified Friedman equations for each space can be obtained substituting the ansatzs (13) and (14) in equations (9) and (10). That can be done either by brute force vonStrauss:2011mq () or noting that (13) and (14) are related through the generalized Gordon ansatz Baccetti:2012ge (). Following any of the two procedures, one obtains

(15)

The modified Friedmann equations are

(16)

and

(17)

where the effective energy density due to the interaction in each space has been defined as Baccetti:2012ge ()

(18)
(19)

and the Hubble functions are and , respectively, with .

On the other hand, due to the diffeomorphism invariance of the matter actions appearing in the action (3), the stress-energy tensors of the matter components of both spaces are conserved. Therefore one has and . Moreover, taking into account the Bianchi identities together with these conservations, the effective stress energy tensors (11) and (12) must be also conserved. This leads to the Bianchi-inspired constraint222The use of the generalized Gordon ansatz implies that we only consider the branch of the Bianchi-inspired constraint given by equation (20). However, it should be noted that there is a second branch that leads to solutions whose scale factors are proportional to and (with fixed constant of proportionality) vonStrauss:2011mq (). As pointed out in reference nonvacuum (), restriction to the second branch implies the elimination of all nontrivial interaction terms, i.e., those which are not equivalent to a cosmological constant. Moreover, cosmological solutions of this branch have a nonlinear instability DeFelice:2012mx (). Volkov:2011an (); vonStrauss:2011mq (); Baccetti:2012ge ()

(20)

As is already well known, this implies that the Hubble function of the -space can be expressed as . Therefore, the Friedmann equations of both spaces, equations (16) and (17), are coupled Volkov:2011an (); vonStrauss:2011mq (). This allows us to write the Friedmann equation of the second universe (17) as

(21)

which can be combined with equation (16) to remove the term . Considering a general material content in both universes, this procedure leads to Capozziello:2012re ()

(22)

where , , , , , , , . This equation can be seen as an algebraic equation in and, therefore, it can be solved to obtain at least in principle. Solutions for the case in which the second universe is empty are generically easier to obtain, as one can define the quantity as in reference vonStrauss:2011mq () and then equation (22) becomes a quartic equation in (it could even be simpler for models in which some parameters vanish vonStrauss:2011mq ()). Once one obtains from equation (22), it can be substituted in the Friedmann equation to study the dynamics of the universe by integrating .

2.2 The point-like Lagrangian

In the previous subsection, we have written the general equations for the dynamics of the metrics, that is the modified Einstein equations. Then we have restricted our attention to solutions described by two FLRW geometries. Nevertheless, one could have also studied the problem considering a point-like Lagrangian, which leads to dynamics in a minisuperspace. To obtain such a point-like Lagrangian, one has to substitute the cosmological metrics given by equations (13) and (14) in action (3). Therefore, by replacing the expressions (15) in the interaction term of the Lagrangian (3) and integrating by parts the Einstein-Hilbert Lagrangian of action (3), one gets the following point-like Lagrangian:

(23)

with and have been defined in equations (18) and (19), respectively. This is the same Lagrangian as that presented in reference Fasiello:2013woa (), but we have split the interaction term in and for convenience. It must be noted that the Lagrangian (23) is defined in the tangent space, , coming from the configuration space .

Varying the Lagrangian (23) with respect to and , we obtain the modified Friedmann equations which are equations (16) and (17), respectively. Now, the variation with respect to the scale factor leads to

(24)

with and . The conservation equation can be written as . Therefore, if the interaction term vanishes (for ), the acceleration equation (24) will be equivalent to considering the derivative of Friedmann equation and the conservation of the fluid, as in usual GR; i.e. the standard Raychaudhuri equation. By similarity to what happens in GR, we can define an effective pressure, , as follows

(25)

Combining the derivative of the Friedmann equation (16) with equation (24), one obtains

(26)

which is exactly the Bianchi-inspired constraint (20). Therefore, this procedure is equivalent to that presented in the previous subsection.

Nevertheless, it can be noted that the determinant of the Hessian333The Hessian of the Lagrangian is defined as . of the Lagrangian (23) vanishes and, therefore, the Lagrangian is degenerate Noether () in the tangent space . This fact can be problematic when applying certain procedures, as the Noether Symmetry Approach or the quantization of the corresponding Hamiltonian. Therefore, one can interpret the fact that the Bianchi inspired constraint (20) is obtained by deriving the equations of motion and requiring their compatibility, as being a consequence of the large number of equations of motion provided by an initial degenerate Lagrangian. The information is there, but it is redundant. As we will see in the next section, one can define a non-degenerate Lagrangian containing the information encoded in equation (20) from the beginning.

2.3 Non-degenerate point-like Lagrangian

The Lagrangian (23) is degenerate because it is independent of and . Therefore, to find a non-degenerate point-like Lagrangian we need to remove the dependence on the non-dynamical variables and in . In the first place, we note that we can use the temporal gauge of freedom to fix without loss of generality; then the cosmic time of the -universe will be . As in this theory we have only one global invariance under changes of coordinates, this choice already fixes the temporal gauge freedom and, therefore, we cannot freely choose . Indeed, we already know that has to be given by equation (20). Therefore, we take

(27)

in the Lagrangian (23) to obtain444Although, as we have discussed in the previous section, the information of the Bianchi-inspired constraint was already included in the Lagrangian (23), by considering the equations (27), we remove redundant information and with it the degeneracy of the Lagrangian.

(28)

It has to be emphasized that this Lagrangian is not symmetric under inter-change of the -universe and -universe, as it has been the case until fixing the gauge of freedom. The reason is that the gauge fixing (27) breaks the symmetry between both gravitational sectors, so is not invariant under the inter-change . In particular, we are choosing to express the physics in terms of our cosmic time. As it is suggested in the appendix A, a symmetry breaking gauge fixing seems to be necessary to obtain a non-degenerate Lagrangian.

In order to check that the Lagrangian (28) describes the same physical situation as (23), one should obtain the same information; i.e. dynamical evolution, as in the previous section. In fact, varying the Lagrangian with respect to and after some simplifications, one obtains

(29)

This is just the Friedmann equation of the second universe, equation (17), with written in terms of through equation (27). Moreover, varying with respect to , simplifying, substituting equation (29) when needed, and simplifying again, one gets

(30)

with given by equation (25), which is equation (24). Thus, the system of equations is equivalent to that obtained from the Lagrangian (23).

On the other hand, as the non-degenerate point-like Lagrangian has no explicit dependence on , the energy is conserved. That is

(31)

where we know that this function vanishes for compatibility with the Friedmann equation. Moreover, the Hamiltonian function of the system can be obtained from the energy (31) when re-expressing the “velocities” in terms of the momenta; therefore, it has to vanish due to the diffeomorphism invariance. Furthermore, although equation (31) could suggest that the energy of each universe is conserved separately, this is not the case because writing , cannot be obtained using only (it has contributions from ), and vice versa. Anyway, it is important to emphasize that the consideration of the equation implied by is equivalent to one of the equations of motion of the system, and it would be easier to calculate in general.

3 Bigravity cosmologies with a Noether symmetry

As we have discussed in the introduction, the dynamics of our Universe in bigravity is determined not only by the material content minimally coupled to our gravitational sector but also by the material content minimally coupled to the other sector. As that set of matter fields cannot be measured in a direct way, it seems that any cosmological dynamics could be described by fine-tuning it. Therefore, the theory would loss some predictive power if the nature of the “hidden” matter content cannot be fixed by fundamental principles. We assume that such a fundamental principle is the existence of an additional Noether symmetry for cosmological solutions. We consider that this assumption is more robust than assuming an empty second gravitational sector from the beginning. The existence of symmetries commonly simplifies the treatment of physical problems. As we will show, however, our universe cannot have a Noether symmetry during its whole evolution, but it can tend to a state where this symmetry is present and the kind of material content is, therefore, fixed. The phenomenological interest of the solutions obtained are, at the end of the day, the stronger indication in favor of the considered assumption.

In this section, we review the Noether Symmetry Approach formalism in subsection 3.1, we apply this formalism to general bigravity cosmological solutions in subsection 3.2, and, we particularize the study considering a specific kind of material content minimally coupled to our sector in subsection 3.3 extracting the particular solutions.

3.1 The Noether Symmetry Approach

The motivation behind the use of the Noether Symmetry Approach in cosmology is to select a particular modified gravity model from the general class of theories considered and to find exact cosmological solutions for those model. From a classical point of view, the selected model may be interpreted as being favored against other Lagrangians since it has an additional symmetry Noether (). This interpretation entails, of course, the underlying assumption that the most suitable theory of the family of theories is that implying the existence of more symmetries. On the other hand, from a quantum point of view, the presence of Noether symmetries allows to select observable universes by the Hartle criterion, since the absence of Noether symmetries gives rise to non-correlated variables and then to non-observable universes Noether2 () .

In order to apply the Noether Symmetry Approach, one needs a non-degenerate point-like Lagrangian which is independent of time. Then, one supposes the existence of a Noether symmetry, which implies that Noether ()

(32)

where is a vector field defined on the tangent space , that is

(33)

and is the Lie derivative along the direction . Equation (32) gives rise to a system of partial differential equations whose solution is not unique. The solutions of equation (32) fix the vector components (and consequently and the symmetry) and the functional form of the Lagrangian , that is the couplings and potentials (see Noether () for details). Besides, the Lagrangian, and then the dynamics, are associated to a conserved quantity that can be used to integrate the equations of motion, being

(34)

and then

(35)

as a consequence of . It should be emphasized that in this approach one starts considering the existence of an arbitrary Noether symmetry (37), and the particular form of this symmetry is obtained once one imposes for a particular class of Lagrangians. Furthermore, the energy function

(36)

is conserved as well. The right hand side of equation (36) corresponds to the Hamiltonian of the system, , and given that -, it follows immediately that is conserved, whenever does not depend on time. Therefore, the use of this equation further simplifies the problem and analytic solutions can be easily found. This approach has been used several times in cosmology giving exact solutions of physical interest (see for example felicia (); basilakos (); defelice (); nesseris ()). Although in most cases the particular symmetry has not a clear physical interpretation, it can select a model from the corresponding family of theories.

3.2 Noether symmetries for bigravity cosmology

Now that we have a point-like Lagrangian that is non-degenerate, we can look for the potential existence of Noether symmetries. Taking into account equation (33), a general vector field on can be expressed as

(37)

The point-like Lagrangian (28) has a Noether symmetry if the condition (32) is satisfied for the vector field (37). The Noether symmetry is at this point arbitrary, it will be fixed when and will be specified.

It can be shown that555We refer to the appendix B for the mathematical details. the two components of the vector reads

(38)

where , , are constants. The functions and must as well fullfill the conditions (95) and (97) given in the Appendix B. Those two equations will fix the matter content for a given matter content .

3.3 Solutions compatible with matter in our universe

Let us now take into account that in our universe there is only matter (dust) and radiation. That is

(39)

Therefore, we want to find which material content should be present in the other universe, , to have a Noether symmetry for the biuniversal dynamics, given by equations (37) and (B) (cf. Appendix B). Hence, we look for functions that are solutions of both equations (95) and (97), once expressions (39) and (B) have been introduced in these equations. The solutions that we present in the following discussion also restrict some of the parameters and the function given by equation (39). Restriction of the parameters means that only some of the ghost-free theories will be able to have a Noether symmetry for cosmological solutions, whereas further restriction of implies that not all the matter and radiation contents are compatible with the existence of this symmetry. In particular, as we will show, compatibility of the system of differential equations requires ; therefore, a universe with a non-vanishing radiation contribution cannot have a Noether symmetry. However, as we will discuss in detail, a universe like ours that contains radiation can tend to a state with a Noether symmetry when the radiation component is sufficiently diluted. As we will show, this final state is of particular phenomenological interest.

3.3.1 General Relativity with a cosmological constant

The first solution is given by

(40)

So the only non-vanishing parameter in the interaction Lagrangian is , implying that interaction between both sectors reduces to a cosmological constant in the second universe. That is

(41)

In this case, the dynamics of both universes is decoupled and have the same spatial geometry, i.e. the same value of (which is arbitrary). The second universe is empty, since equations (95) and (97) (cf. Appendix B) imply that the cosmological constant appearing in the geometric term cancel the contribution coming form as

(42)

Moreover, one can substitute the parameters in equations (40) and (42) into equation (22) to study the solution of this system. This leads to

(43)

As , this implies that if . Assuming that this condition is preserved in time, one has that implies and, therefore, there is no second gravitational sector. In this case, there is only one gravitational sector filled with dust matter and, therefore, there is no bigravity solutions for this model. If one had instead of , then the cosmological model with a Noether symmetry would correspond to a general relativistic model that is empty. Our Universe could tend to that state when the matter is infinitely diluted by the expansion.

3.3.2 Putative bigravity solution

The second solution for the system (95) and (97) (please cf. Appendix B) is given by

(44)

In this case, the interaction between both gravitational sectors is not equivalent to a cosmological constant, since . Indeed one has a non-trivial interaction affecting the dynamics of our universe, which is given by

(45)

That is, a cosmological constant component plus a purely bimetric term. It must be noted that the solution requires , that is, there is no radiation in this universe. Our Universe contains, of course, relativistic matter, so this result could seem incompatible with the existence of a Noether symmetry in physical interesting situations. Nevertheless, it can be noted that the state with a Noether symmetry can be approached when the radiation component is diluted. So the universe could be in a state with a Noether symmetry in its asymptotic past or future; i.e. during a “pre-inflationary” phase to be followed by a standard inflationary era or at the very late-time dark energy era. In both situations, the radiation contribution can be negligible as compared with the material content. Moreover, this solution is a solution valid for any .

In the other universe, the material content compatible with the existence of a Noether symmetry is

(46)

Therefore, the other universe only contains vacuum energy when it is in the state with a Noether symmetry666One could be surprised about this non-symmetric result for the matter content given the symmetry of the original interaction term. Nevertheless, in order to obtain a non-degenerate and nontrivial Lagrangian we have shown that the original symmetry should be broken in order to obtain a dynamical evolution., this corresponds to in action (1) but leading to a cosmological constant contribution. Hence, the solution for this particular model corresponds to the “minimal model” discussed in vonStrauss:2011mq () plus an explicitly cosmological constant contribution. This model was called minimal in reference vonStrauss:2011mq () because, as , the only nonlinear interaction term in both universes is the quadratic term (it should not be confused with the minimal model introduced in reference Konnig:2013gxa ()). In this case one has that some coefficients appearing in equation (22) are fixed, . Furthermore, in our model and are not independent and, therefore, one additionally has .

It is particularly simple to obtain the analytic solution using equation (22) and we do not need to use the Noether symmetry to find out the exact solution, as in other alternative theories of gravity. Noether (); Noether1 (); Noether2 (). Substituting these values of the parameters and equation (46) in equation (22), it can be seen that the solutions for this model are given by

(47)

As in the previous case, if we take , then we will have . Thus, assuming , one can conclude that this model is not compatible with having a second gravitational sector. On the other hand, if one insists in considering to have a bigravity model, then our Universe will restore a state with a Noether symmetry when the matter component is infinitely diluted, that is . In this case, however, is left undetermined by the Friedmann equations; so will depend on an undetermined function and we will not be able to conclude anything about the dynamics of our Universe (at least without assuming a form for ). Therefore, requiring the existence of a Noether symmetry leads either to a general relativistic world with a cosmological constant term (fixed by ), which is suitable to describe the current acceleration of our universe, or to an empty model whose dynamics cannot be determined.

3.3.3 Bigravity solution I

The last set of solutions of the system (95) and (97) (cf. Appendix B) is

(48)

which leads to a non-trivial interaction given by

(49)

The content of the other universe is expressed as

(50)

therefore, we have again in action (1) but is equivalent to a cosmological constant contribution. As in the previous case, given the simplicity of the model corresponding to the parameters (48) and (50), we can easily solve equation (22) even before calculating the Noether symmetry and conserved quantity. Taking into account (48) and (50) in equation (22), we obtain

(51)

Hence, on the one hand, in order to have a real function , one needs to consider and/or . On the other hand, considering equation (51) in the Friedmann equation (16), or equivalently in equation (21), with the definitions (18) and (19), one obtains

(52)

This implies that and should have the same sign. Therefore, both parameters should be negative. From equations (51) and (52) it can be seen that, in this case, both and can be well defined. (Some comments regarding the physics of the other universe are included in section 4.) Defining and integrating equation (52), we obtain

(53)

So, even if we have considered the presence of matter in this universe (), the evolution given by equation (53) is exactly de Sitter. The scale factor (53) can be re-written as

(54)

Therefore, the effective cosmological constant which appears in equation (54), is different from the cosmological constant induced by the interaction term (18), which is . Indeed, one has

(55)

where in the case that one considers an explicit cosmological constant in the action, the term has to be replaced by .

On the other hand, the components of the Noether vector field (37) assume the explicit forms

(56)

Following the considerations in reference Noether (), the Noether vector field can be written in a simplified form as

(57)

by defining a new par of variables such that is cyclic; i.e. the Lagrangian can be recast in a form such that . This change of variables is always possible since there is a Noether symmetry and the condition (35) is satisfied. The change of variables can be realized assuming a regular transformation and where the Jacobian of the transformation is different from zero. Thus, we have

(58)

and we can obtain

(59)

where and are integration constants. Reverting (59) and assuming, without loss of generality, , we obtain

(60)

The Lagrangian, in terms of these new variables, assume the form where the new variable is cyclic. From equation (57), the constant of motion can be expressed as

(61)

which is the same as that defined in equation (35) but expressed in the new variables. This quantity can be rewritten in terms of the scale factor of both universes as

(62)

The fact that this quantity is conserved can be used to integrate the equations of motion. In our case, however, they were easy to integrate so one could check that is indeed conserved using the solutions. Taking into account the equations (53) and (60), equation (62) simplifies to

(63)

Therefore, the conserved quantity associated to the Noether symmetry fixes the value of the effective cosmological constant . The existence of a Noether symmetry for this solutions implies that the model evolves exactly as a de Sitter space even when matter has not been completely diluted. It has to be emphasized that, of course, our Universe contains both matter and radiation. However, we consider that it could approach the state with a Noether symmetry when the radiation component is completely negligible as compared with the matter component. In particular, it tends to a state where the effect of matter in the dynamics is negligible even before is small enough to assume that the material content is diluted.

3.3.4 Bigravity solution II

The last set of solutions of the system (95) and (97) is given by

(64)

For this case, the interaction term in our universe has the same form as the one given in equation (49), that is

(65)

but now the material content of the other universe is

(66)

which is equivalent to consider a cosmological constant contribution plus a spatial curvature contribution. Therefore, from equation (22), we get (considering positive, see Eq. (3.22))

(67)

which can be introduced in the Friedmann equation (16) to obtain

(68)

Assuming that the state with a Noether symmetry is approached at later times in our universe, that is for large , and taking into account equation (67) and (68) one needs to consider negative values777In this case one could also have well-defined solutions for positive values of and by restricting attention to small values of the scale factor . We are interested, however, in the situation in which the solution having a Noether symmetry is approached at late-time. Notice that a positive value of would imply a minimum value for the scale factor which would lead to an avoidance of the Big Bang singularity in our universe. Nevertheless, as we said before, we are mainly interested on the late-time behavior of the universe. of and , and one can define again . As in the previous case, we obtain

(69)

although in this case this accelerating universe could have a non-vanishing spatial curvature as shown in equation (64), its contribution to the dynamics is cancelled by the interaction term.

In this case, the vector field associated with the Noether symmetry, is given by

(70)

For the the cyclic variable, one can again perform the transformation obtaining

(71)