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Register a new userStability Conditions in the Generalized SU(2) Proca Theory
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Yeinzon Rodriguez Garcia
Publication date
2019
fields
Physics
and research's language is
English
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Under the same spirit of the Galileon-Horndeski theories and their more modern extensions, the generalized SU(2) Proca theory was built by demanding that its action may be free of the Ostrogradskis instability. Nevertheless, the theory must also be free of other instability problems in order to ensure its viability. As a first approach to address this issue, we concentrate on a quite general variant of the theory and investigate the general conditions for the absence of ghost and gradient instabilities in the tensor sector without the need for resolving the dynamical background. The phenomenological interest of this approach as well as of the variant investigated lies on the possibility of building cosmological models driven solely by non-Abelian vector fields that may account for a successful description of both the early inflation and the late-time accelerated expansion of the universe.
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Following previous works on generalized Abelian Proca theory, also called vector Galileon, we investigate the massive extension of an SU(2) gauge theory, i.e., the generalized SU(2) Proca model, which could be dubbed non-Abelian vector Galileon. This particular symmetry group permits fruitful applications in cosmology such as inflation driven by gauge fields. Our approach consists in building, in an exhaustive way, all the Lagrangians containing up to six contracted Lorentz indices. For this purpose, and after identifying by group theoretical considerations all the independent Lagrangians which can be written at these orders, we consider the only linear combinations propagating three degrees of freedom and having healthy dynamics for their longitudinal mode, i.e., whose pure Stuckelberg contribution turns into the SU(2) multi-Galileon dynamics. Finally, and after having considered the curved space-time expansion of these Lagrangians, we discuss the form of the theory at all subsequent orders.
As a modified gravity theory that introduces new gravitational degrees of freedom, the generalized SU(2) Proca theory (GSU2P for short) is the non-Abelian version of the well-known generalized Proca theory where the action is invariant under global transformations of the SU(2) group. This theory was formulated for the first time in Phys. Rev. D 94 (2016) 084041, having implemented the required primary constraint-enforcing relation to make the Lagrangian degenerate and remove one degree of freedom from the vector field in accordance with the irreducible representations of the Poincare group. It was later shown in Phys. Rev. D 101 (2020) 045008, ibid 045009, that a secondary constraint-enforcing relation, which trivializes for the generalized Proca theory but not for the SU(2) version, was needed to close the constraint algebra. It is the purpose of this paper to implement this secondary constraint-enforcing relation in GSU2P and to make the construction of the theory more transparent. Since several terms in the Lagrangian were dismissed in Phys. Rev. D 94 (2016) 084041 via their equivalence to other terms through total derivatives, not all of the latter satisfying the secondary constraint-enforcing relation, the work was not so simple as directly applying this relation to the resultant Lagrangian pieces of the old theory. Thus, we were motivated to reconstruct the theory from scratch. In the process, we found the beyond GSU2P.
The beyond-generalized Proca theories are the extension of second-order massive vector-tensor theories (dubbed generalized Proca theories) with two transverse vector modes and one longitudinal scalar besides two tensor polarizations. Even with this extension, the propagating degrees of freedom remain unchanged on the isotropic cosmological background without an Ostrogradski instability. We study the cosmology in beyond-generalized Proca theories by paying particular attention to the dynamics of late-time cosmic acceleration and resulting observational consequences. We derive conditions for avoiding ghosts and instabilities of tensor, vector, and scalar perturbations and discuss viable parameter spaces in concrete models allowing the dark energy equation of state smaller than $-1$. The propagation speeds of those perturbations are subject to modifications beyond the domain of generalized Proca theories. There is a mixing between scalar and matter sound speeds, but such a mixing is suppressed during most of the cosmic expansion history without causing a new instability. On the other hand, we find that derivative interactions arising in beyond-generalized Proca theories give rise to important modifications to the cosmic growth history. The growth rate of matter perturbations can be compatible with the redshift-space distortion data due to the realization of gravitational interaction weaker than that in generalized Proca theories. Thus, it is possible to distinguish the dark energy model in beyond-generalized Proca theories from the counterpart in generalized Proca theories as well as from the $Lambda$CDM model.
The generalized Proca theories with second-order equations of motion can be healthily extended to a more general framework in which the number of propagating degrees of freedom remains unchanged. In the presence of a quartic-order nonminimal coupling to gravity arising in beyond-generalized Proca theories, the speed of gravitational waves $c_t$ on the Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological background can be equal to that of light $c$ under a certain condition. By using this condition alone, we show that the speed of gravitational waves in the vicinity of static and spherically symmetric black holes is also equivalent to $c$ for the propagation of odd-parity perturbations along both radial and angular directions. As a by-product, the black holes arising in our beyond-generalized Proca theories are plagued by neither ghost nor Laplacian instabilities against odd-parity perturbations. We show the existence of both exact and numerical black hole solutions endowed with vector hairs induced by the quartic-order coupling.
We derive the profile of a vector field coupled to matter on a static and spherically symmetric background in the context of generalized Proca theories. The cubic Galileon self-interaction leads to the suppression of a longitudinal vector component due to the operation of the Vainshtein mechanism. For quartic and sixth-order derivative interactions, the solutions consistent with those in the continuous limit of small derivative couplings correspond to the branch with the vanishing longitudinal mode. We compute the corrections to gravitational potentials outside a compact body induced by the vector field in the presence of cubic, quartic, and sixth-order derivative couplings, and show that the models can be consistent with local gravity constraints under mild bounds on the temporal vector component. The quintic Galileon interaction does not allow regular solutions of the longitudinal mode for a rapidly decreasing matter density outside the body.
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