Generalized Proca Theories Research Articles

GRBoondi: A code for evolving Generalized Proca theories on arbitrary backgrounds

GRBoondi: A code for evolving Generalized Proca theories on arbitrary backgrounds

Scrutinizing black hole stability in cubic vector Galileon theories

In a subclass of generalized Proca theories where a cubic vector Galileon term breaks the U(1) gauge invariance, it is known that there are static and spherically symmetric black hole (BH) solutions endowed with nonvanishing temporal and longitudinal vector components. Such hairy BHs are present for a vanishing vector-field mass (m=0) with a non-zero cubic Galileon coupling β 3. We study the linear stability of those hairy BHs by considering even-parity perturbations in the eikonal limit. In the angular direction, we show that one of the three dynamical perturbations has a nontrivial squared propagation speed c Ω,1 2, while the other two dynamical modes are luminal.We could detect two different unstable behaviors of perturbations in all the parameter spaces of hairy asymptotically flat BH solutions we searched for. In the first case, an angular Laplacian instability on the horizon is induced by negative c Ω,1 2. For the second case, it is possible to avoid this horizon instability, but in such cases, the positivity of c Ω,1 2 is violated at large distances. Hence these hairy BHs are generally prone to Laplacian instabilities along the angular direction in some regions outside the horizon. Moreover, we also encounter a pathological behavior of the radial propagation speeds c r possessing two different values of c r 2 for one of the dynamical perturbations. Introducing the vector-field mass m to cubic vector Galileons, however, we show that the resulting no-hair Schwarzschild BH solution satisfies all the linear stability conditions in the small-scale limit, with luminal propagation speeds of three dynamical even-parity perturbations.

Constraint structure of the generalized Proca model in the Lagrangian formalism

We introduce a novel Lagrangian approach to analyze the dynamical structure of the generalized Proca theory (GP). This method incorporates the A-Z constraint structure of the model within the Lagrangian formalism, providing a precise determination of the degrees of freedom. Furthermore, we present a comprehensive analysis of the model's Hamiltonian constraint structure.

Black hole thermodynamics in generalized Proca theories

Black hole thermodynamics in generalized Proca theories

Polarization modes of gravitational waves in generalized Proca theory

Polarization modes of gravitational waves in generalized Proca theory

Loss of hyperbolicity and tachyons in generalized Proca theories

Loss of hyperbolicity and tachyons in generalized Proca theories

Kinetic field theory: generic effects of alternative gravity theories on non-linear cosmic density-fluctuations

Non-linear cosmic structures contain valuable information on the expansion history of the background space-time, the nature of dark matter, and the gravitational interaction. The recently developed kinetic field theory of cosmic structure formation (KFT) allows to accurately calculate the non-linear power spectrum of cosmic density fluctuations up to wave numbers of k ≲ 10 h Mpc-1 at redshift zero. Cosmology and gravity enter this calculation via two functions, viz. the background expansion function and possibly a time-dependent modification of the gravitational coupling strength.The success of the cosmological standard model based on general relativity suggests that cosmological models in generalized theories of gravity should have observable effects differing only weakly from those in standard cosmology. Based on this assumption, we derive the functional, first-order Taylor expansion of the non-linear power spectrum of cosmic density fluctuations obtained from the mean-field approximation in KFT in terms of the expansion function and the gravitational coupling strength. This allows us to study non-linear power spectra expected in large classes of generalized gravity theories. To give one example, we apply our formalism to generalized Proca theories.

Coupled vector Gauss-Bonnet theories and hairy black holes

We study vector-tensor theories in which a 4-dimensional vector field Aμ is coupled to a vector quantity Jμ, which is expressed in terms of Aμ and a metric tensor gμν. The divergence of Jμ is equivalent to a Gauss-Bonnet (GB) term. We show that an interacting Lagrangian of the form f(X)AμJμ, where f is an arbitrary function of X=−(1/2)AμAμ, belongs to a scheme of beyond generalized Proca theories. For f(X)=α=constant, this interacting Lagrangian reduces to a particular class of generalized Proca theories. We apply the latter coupling to a static and spherically symmetric vacuum configuration by incorporating the Einstein-Hilbert term, Maxwell scalar, and vector mass term ηX (η is a constant). Under an expansion of the small coupling constant α with η≠0, we derive hairy black hole solutions endowed with nonvanishing temporal and radial vector field profiles. The asymptotic properties of solutions around the horizon and at spatial infinity are different from those of hairy black holes present in scalar-GB theories. We also show that black hole solutions without the vector mass term, i.e., η=0, are prone to ghost instability of odd-parity perturbations.

Vector-tensor gravity from a broken gauge symmetry

In this paper we present a Yang-Mills type gauge theory of vector-tensor gravity, where the tetrad, the spin connection and vector field are identified with components of the gauge field. This setup leads to a theory that in flat spacetime is contained in Generalized Proca theories, while in curved spacetime is closely related to beyond Generalized Proca. We solve for static and spherically symmetric space-time and show that there are two branches of solutions, one where the metric is asymptotically Schwarzschild even though there is a cosmological constant in the action, and another one where the metric is asymptotically (anti-)de Sitter. Also, we study the effect of the vector field on homogeneous and isotropic spacetimes, finding that it contributes to the accelerated expansion of the spacetime.

Gravitational Wave Propagation for The Generalized Proca Theories

In general relativity, a gravitational wave propagates with the speed of light, but inthe alternative theories of gravity, propagation speed could deviate from the speed of lightdue to the modification of gravity. Gravitational waves are influenced by modified gravityduring propagation at the cosmological distance. In this paper, we investigate thepropagation of a gravitational wave of the generalized Proca theories by consideringgravitational wave as the gravitational field propagates in spacetime as a wave perturbing flatspacetime. We show that the arbitrary functions G3, G4, and G5 can be the sources ofdeviation of the speed of the gravitational wave.

Constant‐Roll Inflation in the Generalized SU(2) Proca Theory

AbstractThe generalized SU(2) Proca theory (GSU2P) is a variant of the well known generalized Proca theory where the vector field belongs to the Lie algebra of the SU(2) group of global transformations under which the action is made invariant. New interesting possibilities arise in this framework because of the existence of new interactions of purely non‐Abelian character and new configurations of the vector field resulting in spatial spherical symmetry and the cosmological dynamics being driven by the propagating degrees of freedom. We study the 2D phase space of the system that results when the cosmic triad configuration is employed in the Friedmann–Lemaitre–Robertson–Walker background and find an attractor curve whose attraction basin both covers almost all the allowed region and does not include a Big‐Bang singularity. Such an attractor curve corresponds to a primordial inflationary solution that has the following characteristic properties: 1) it is a de Sitter solution whose Hubble parameter is regulated by a generalized version of the SU(2) group coupling constant, 2) it is constant‐roll including, as a limiting case, the slow‐roll variety, 3) a number of e‐folds is easily reached, 4) it has a graceful exit into a radiation dominated period powered by the canonical kinetic term of the vector field and the Einstein–Hilbert term. The free parameters of the action are chosen such that the tensor sector of the theory is the same as that of general relativity at least up to second‐order perturbations, thereby avoiding the presence of ghost and Laplacian instabilities in the tensor sector as well as making the gravity waves propagate at light speed. This is a proof of concept of the interesting properties that could be found in this scenario when the coupling constants be replaced by general coupling functions and more terms be discovered in the GSU2P.

Covariant effective action for generalized Proca theories

We investigate the quantum stability of generalised Proca theories in curved spacetime treating gravity as a dynamical field. To compute the quantum gravitational corrections, we evaluate the divergent part of the effective action at one-loop level using Vilkovisky-DeWitt formalism, which gives us a gauge invariant and gauge condition independent effective action. It is shown that the quantum corrections are suppressed by a UV cutoff parametrically higher than the Proca mass, if the coupling constants are restricted to lie in a certain range. Furthermore, it is shown that the quantum corrections remain suppressed even at scales where classical non-linearities dominate over kinetic terms, allowing Vainshtein mechanism to work.

Black holes in the quadratic-order extended vector–tensor theories

We investigate the static and spherically black hole solutions in the quadratic-order extended vector–tensor theories without suffering from the Ostrogradsky instabilities, which include the quartic-order (beyond-)generalized Proca theories as the subclass. We start from the most general action of the vector–tensor theories constructed with up to the quadratic-order terms of the first-order covariant derivatives of the vector field, and derive the Euler–Lagrange equations for the metric and vector field variables in the static and spherically symmetric backgrounds. We then substitute the spacetime metric functions of the Schwarzschild, Schwarzschild–de Sitter/anti-de Sitter, Reissner–Nordström-type, and Reissner–Nordström–de Sitter/anti-de Sitter-type solutions and the vector field with the constant spacetime norm into the Euler–Lagrange equations, and obtain the conditions for the existence of these black hole solutions. These solutions are classified into the two cases 1) the solutions with the vanishing vector field strength; the stealth Schwarzschild and the Schwarzschild–de Sitter/anti-de Sitter solutions, and 2) those with the nonvanishing vector field strength; the charged stealth Schwarzschild and the charged Schwarzschild–de Sitter/anti-de Sitter solutions, in the case that the tuning relation among the coupling functions is satisfied. In the latter case, if this tuning relation is violated, the solution becomes the Reissner–Nordström-type solution. We show that the conditions for the existence of these solutions are compatible with the degeneracy conditions for the class-A theories, and recover the black hole solutions in the generalized Proca theories as the particular cases.

A dark energy model from generalized Proca theory

We consider a specific dark energy model, which only includes the Lagrangian up to the cubic order in terms of the vector field self-interactions in the generalized Proca theory. We examine the cosmological parameters in the model by using the data sets of CMB and CMB+HST, respectively. In particular, the Hubble constant is found to be $H_0=71.80^{+1.07}_{-0.72}$ ($72.48^{+0.72}_{-0.60}$) $\rm kms^{-1}Mpc^{-1}$ at $68\%$~C.L. with CMB (CMB+HST), which would alleviate the Hubble constant tension. We also obtain that the reduced $\chi^2$ values in our model are close to unity when fitting with CMB and CMB+HST, illustrating that our model is a good candidate to describe the cosmological evolutions of the universe.

Quantum stability of generalized Proca theories

We establish radiative stability of generalized Proca effective field theories. While standard power-counting arguments would conclude otherwise, we find non-trivial cancellations of leading order corrections by explicit computation of divergent one-loop diagrams up to four-point. These results are crosschecked against an effective action based generalized Schwinger–DeWitt method. Further, the cancellations are understood as coming from the specific structure of the theory through a decoupling limit analysis which at the same time allows for an extension of the results to higher orders.

Disformal transformation of stationary and axisymmetric solutions in modified gravity

The extended scalar-tensor and vector-tensor theories admit black hole solutions with the nontrivial profiles of the scalar and vector fields, respectively. The disformal transformation maps a solution in a class of the scalar-tensor or vector-tensor theories to that in another class, and hence it can be a useful tool to construct a new nontrivial solution from the known one. First, we investigate how the stationary and axisymmetric solutions in the vector-tensor theories without and with the $U(1)$ gauge symmetry are disformally transformed. We start from a stationary and axisymmetric solution satisfying the circularity conditions, and show that in both the cases the metric of the disformed solution in general does not satisfy the circularity conditions. Using the fact that a solution in a class of the vector-tensor theories with the vanishing field strength is mapped to that in a class of the shift-symmetric scalar-tensor theories, we derive the disformed stationary and axisymmetric solutions in a class of these theories, and show that the metric of the disformed solutions does not satisfy the circularity conditions if the scalar field depends on the time or azimuthal coordinate. We also confirm that in the scalar-tensor theories without the shift symmetry, the disformed stationary and axisymmetric solutions satisfy the circularity conditions. Second, we investigate the disformal transformations of the stationary and axisymmetric black hole solutions in the generalized Proca theory with the nonminimal coupling to the Einstein tensor, the shift-symmetric scalar-tensor theory with the nonminimal derivative coupling to the Einstein tensor, the Einstein-Maxwell theory, and the Einstein-conformally coupled scalar field theory. We show that the disformal transformations modify the causal properties of the spacetime.

Generalized SU(2) Proca theory reconstructed and beyond

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 Poincar\'e 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.

Suppressed cosmic growth in coupled vector-tensor theories

We study a coupled dark energy scenario in which a massive vector field $A_{\mu}$ with broken $U(1)$ gauge symmetry interacts with the four-velocity $u_c^{\mu}$ of cold dark matter (CDM) through the scalar product $Z=-u_c^{\mu} A_{\mu}$. This new coupling corresponds to the momentum transfer, so that the background vector and CDM continuity equations do not have explicit interacting terms analogous to the energy exchange. Hence the observational preference of uncoupled generalized Proca theories over the $\Lambda$CDM model can be still maintained at the background level. Meanwhile, the same coupling strongly affects the evolution of cosmological perturbations. While the effective sound speed of CDM vanishes, the propagation speed and no-ghost condition of a longitudinal scalar of $A_{\mu}$ and the CDM no-ghost condition are subject to nontrivial modifications by the $Z$ dependence in the Lagrangian. We propose a concrete dark energy model and show that the gravitational interaction on scales relevant to the linear growth of large-scale structures can be smaller than the Newton constant at low redshifts. This leads to the suppression of growth rates of both CDM and total matter density perturbations, so our model allows an interesting possibility for reducing the tension of matter density contrast $\sigma_8$ between high- and low-redshift measurements.

Rotating black holes in valid vector-tensor theories after GW170817

Vector-tensor theories beyond General Relativity have widely been studied in the context of ultraviolet completion of gravity, endowing a mass to the graviton and explaining dark energy phenomena. We here construct rotating black hole solutions in vector-tensor theories valid after the binary neutron star merger event GW170817 that placed very stringent bound on the propagation speed of gravitational waves away from the speed of light. Such valid vector-tensor theories are constructed by performing a generic conformal transformation to Einstein-Maxwell theory, and the new rotating black hole solutions are constructed by applying the same conformal transformation to the Kerr-Newman solution. These theories fall outside of beyond generalized Proca theories but are within an extended class of vector-tensor theories that satisfy a degenerate condition to eliminate instability modes and are thus healthy. We find that such conformal Kerr-Newman solutions preserve the location of the singularities, event horizons and ergoregion boundary from Kerr-Newman, as well as the multipole moments and the Petrov type. On the other hand, the Hamilton-Jacobi equation is no longer separable, suggesting that the Carter-like constant does not exist in this solution. The standard Newman-Janis algorithm also does not work to construct the new solutions. We also compute the epicyclic frequencies, the location of the innermost stable circular orbits, and the Schwarzschild precession and apply the latter to the recent GRAVITY measurement to place bounds on the deviations away from Kerr-Newman for Sgr A$^*$.

Reducing the H0 tension with generalized Proca theory

We investigate the cosmological viability of the generalized proca theory. We first implement the background and linear perturbation equations of motion in the Boltzmann code and then study the constraints on the parameters of the generalized proca theory after running MCMC against the cosmological data set. With Planck + HST data, we obtain the constraint h=0.7334−0.0269+0.0246, which indicates that the tension between early universe and late time universe within this theory is removed. By adding other late-time data sets (BAO, RSD, etc.) we show that the tension is reduced, as the 2σ allowed region for h in Proca, h=0.7041−0.0087+0.0094, overlaps with the 2σ region of the HST data.