Coupled multi-Proca vector dark energy

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Abstract

We study a new class of vector dark energy models where multi-Proca fields Aμa are coupled to cold dark matter by the term f(X)L̃m where f(X) is a general function of X12AaμAμa and L̃m is the cold dark matter Lagrangian. From here, we derive the general covariant form of the novel interaction term sourcing the field equations. This result is quite general in the sense that encompasses Abelian and non-Abelian vector fields. In particular, we investigate the effects of this type of coupling in a simple dark energy model based on three copies of canonical Maxwell fields to realize isotropic expansion. The cosmological background dynamics of the model is examined by means of a dynamical system analysis to determine the stability of the emergent cosmological solutions. As an interesting result, we find that the coupling function leads to the existence of a novel scaling solution during the dark matter dominance. Furthermore, the critical points show an early contribution of the vector field in the form of dark radiation and a stable de Sitter-type attractor at late times mimicking dark energy. The cosmological evolution of the system as well as the aforementioned features are verified by numerical computations. Observational constraints are also discussed to put the model in a more phenomenological context in the light of future observations.

Introduction

Our current understanding of the universe is based on the standard model of cosmology, in short ΛCDM (Λ Cold Dark Matter) [1]. Under this realization, the universe is experiencing today an accelerated expansion due to a constant energy density characterized by some repulsive pressure and attributed to the cosmological constant [2], [3], [4]. Despite its success, the model has been the subject of intense debate due to some controversies and problems when confronted with observations, such as the cosmological constant problem [5], [6] and the inference of some spatial curvature suggested by the lensing amplitude in the cosmic microwave background power spectra [7]. Likewise, recent observations of the redshift-space distorsion [8] and cluster counts [9], [10], [11] have pointed out a lower rate for the cosmic growth than that predicted by the ΛCDM model.

However, perhaps one of the major concerns in the scientific community about the standard model of cosmology, among the aforementioned internal inconsistencies, is the lack of conciliation between early and late time universe measurements which has been referred to as the Hubble tension [12]. The Planck experiment infers a value H0=67.4±0.5 km/s/Mpc at 68% C.L., assuming the standard ΛCDM model [1] while the recent value determined from the observation of long-period Cepheids in the Large Magellanic Cloud [13] is H0=74.03±1.42 km/s/Mpc at 68% C.L., thereby putting the standard model in tension.

Three major routes have been established, among the numerous theoretical proposals (see e.g Refs. [6], [14], [15] and references therein), to reconcile such discrepancies and other issues in the ΛCDM model. These are, modifying the geometric sector of Einstein gravity by breaking its fundamental assumptions [16], [17], [18], [19], including extra fields minimally coupled to gravity which leads to dynamical dark energy (DE) such as quintessence [20], [21] and k-essence [22], [23], [24], or extending gravity by building non-minimal interactions between matter and gravity1 [19], [25], [26], [27], [28], [29]. Yet, within the second possibility, interactions between dark mater and DE are allowed. This idea has been explored intensively through phenomenological interactions introduced into the conservation equations. We are not going to discuss, however, the vast amount of possible choices studied in the literature; instead, we refer to Refs. [30], [31] and references therein for the cosmological implications of non-trivial functional forms of couplings. It has been shown, however, that this artificial description may introduce both inconsistencies with the covariant stress energy conservation and instabilities [32]. Hence, one must appeal for a more fundamental and robust approach to account for such a coupling within the dark sector based, for instance, on a field theoretical description. Under this perspective, several consistent approaches have been developed to build coupled dark energy models at the level of the action such as conformal/disformal transformations [33], [34], [35], scalar-fluid theories [36], [37] and the post-Friedmannian formalism [38]. Yet another possibility is to couple directly the DE field to the matter Lagrangian, i.e., at the Lagrangian level, through a coupling function as has been proposed in a pioneering work called coupled quintessence [39]. This type of coupling, and more general non-trivial ones, can arise naturally after an appropriate conformal transformation which relates the metric in the Jordan frame, where matter lives, with the Einstein frame, where gravity is described by general relativity [34]. No matter the underlying physical origin of the coupling, that is, whether it comes from high energy physics arguments or from another yet unknown physical reason, one can motivate the interaction with the aim of addressing some inconsistencies in the standard cosmological model with the phenomenological interest of testing their observational signatures.

It is important to mention that most of the existing (coupled) dark energy models are based on scalar fields with little or no presence of space-like vector fields. This is because, unlike time-like vector fields, space-like vector fields can generate large amounts of anisotropy which is inconsistent with observations [1]. Nevertheless, one may avoid this undesirable result, for instance, by taking a large number of random vector fields so that, on average, they lead to an isotropic universe [40] or by considering a set of three space-like vector fields of the same norm and pointing towards mutually orthogonal spatial directions, shaping what is called the cosmic triad [41]. We highlight, up to the best of our knowledge, some significant progress in the construction of coupled vector dark energy models based on space-like vector fields such as the cosmic triad cosmology [42], three-form dark energy models [43], [44], [45], extensions of the pioneering vector-like dark energy model [41] through phenomenological interactions [46], [47] and a direct coupling driving anisotropic expansion [48]. It is worth mentioning that a coupled vector dark energy model has been proposed recently [49] in the context of the generalized Proca theory [50], [51], [52], [53], [54] with a trivial time-like vector configuration to comply with the background symmetry.

We study in this work a novel coupling between multiple vector fields and matter of the form f(X)L̃m, where f(X) is a general function of X12AaμAμa and L̃m is the matter Lagrangian, following closely the spirit of the kinetically coupled scalar dark energy scenario [55]. Our main purpose is to extend the kinetic scalar coupling to an analogous vector coupling which is fully independent of the assumed model and on the Abelian/non-Abelian nature of the vector fields. To investigate its cosmological implications, we frame this coupling into a model consisting of three copies of the standard Maxwell-theory, consistent with the isotropic background, with a specific power law form for the coupling and an exponential potential.

The content of this paper is structured as follows. We derive, in Section 2, the general interaction term for the proposed coupling function. In Section 3, we study its cosmological behaviour by means of a dynamical system analysis in a specific setup consisting of the cosmic triad compatible with a Friedmann–Lemaıˆtre–Robertson–Walker (FLRW) universe. We confirm the obtained qualitative features by numerically solving the cosmological evolution of the system in Section 4. Finally, the discussion and perspectives of this work are presented in Section 5.

Section snippets

The model

We start by writing the general L2 piece of the Lagrangian for the generalized multi-Proca theory [56], [57], [58] that involves only gauge-invariant and spontaneous symmetry breaking quantities which are present in the Abelian and non-Abelian cases. Indeed, any function of a set of vector fields Aμa, their associated gauge field strength tensors Fμνa, and their Hodge duals F̃μνa belongs to L22

Dynamical system

In order to study the background evolution of the model given by Eqs. (14)–(16) and Eqs. (19)–(20) by means of the dynamical system analysis, we transform the system of equations into an autonomous system of first-order differential equations. To do so, we introduce the following dimensionless variables that will define the phase space portrait: x(Ȧ+HA)22Mp2H2;yV3Mp2H2;zρm3Mp2H2;uAMp. These variables satisfy, in turn, the Friedmann constraint x2+y2+z2=1.At this point, it is necessary to

Numerical results: Cosmological background evolution

We employ numerical methods in this section to solve the cosmological evolution of the model in order to verify the dynamical behaviour found in the previous section. We start by exploring the viable region of the parameter space in accordance with the analysis already performed. After an exhaustive exploration of the numerical solutions in terms of the model parameters, we notice that the behaviour of the density parameters associated to the vector field ΩA and the matter component Ωm is

Discussion and conclusions

We proposed a novel coupling between multi-Proca vector fields and cold dark matter at the level of the action through a vector mass-type term. The general interacting term in a FLRW universe, with the cosmic triad configuration for the space-like vector fields, is of the form Q=3f,Xfρm, it being fully independent of the vector field nature (i.e., the specific structure of the associated group is not manifest, see Eq. (11)).

The dynamical system analysis revealed the existence of three physical

CRediT authorship contribution statement

L. Gabriel Gómez: Conception and design of study, Writing - original draft. Yeinzon Rodríguez: Conception and design of study, Writing - original draft.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

L. G. G is supported by the Postdoctoral Fellowship Programme N 2020000102 of the Vicerrectoría de Investigación y Extensión - Universidad Industrial de Santander. This work was supported by the following grants: Patrimonio Autónomo - Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas (MINCIENCIAS - COLOMBIA) Grant No. 110685269447 RC-80740-465-2020, project 69553, and Dirección de Investigación y Extensión de la Facultad de

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