Eugeny Babichev

An introduction to the Vainshtein mechanism

We introduce the Vainshtein mechanism which plays a crucial role in massive gravities, as well as in related theories such as Galileons and their extensions. This mechanism, also known as k-mouflage, allows to hide via nonlinear effects—typically for source distances smaller than a so-called Vainshtein radius which depends on the source and on the theory considered—some degrees of freedom whose effects are then only left important at large distances, e.g. for cosmology. It is introduced here in nonlinear Fierz–Pauli theories (massive gravities), including the dRGT theories, in their decoupling limits, as well as in other models such as the DGP model or generalized Galileons. This presentation is self-contained, and before discussing the Vainshtein mechanism we introduce some useful results and concepts concerning massive gravity, such as the vDVZ discontinuity, the decoupling limits or the Boulware–Deser ghost.

Instability of black holes in massive gravity

We show that linear perturbations around the simplest black hole solution of massive bi-gravity theories, the bi-Schwarzschild solution, exhibit an unstable mode featuring the Gregory–Laflamme instability of higher dimensional black strings. The result is obtained for the massive gravity theory which is free from the Boulware–Deser ghost, as well as for its extension with two dynamical metrics. These results may indicate that static black holes in massive gravity do not exist. For the graviton mass of the order of the Hubble scale, however, the instability timescale is of order of the Hubble time.

Constraints on shift-symmetric scalar-tensor theories with a Vainshtein mechanism from bounds on the time variation of G.

We show that the current bounds on the time variation of the Newton constant G can put severe constraints on many interesting scalar-tensor theories which possess a shift symmetry and a nonminimal matter-scalar coupling. This includes, in particular, Galileon-like models with a Vainshtein screening mechanism. We underline that this mechanism, if efficient to hide the effects of the scalar field at short distance and in the static approximation, can in general not alter the cosmological time evolution of the scalar field. This results in a locally measured time variation of G which is too large when the matter-scalar coupling is of order one.

Black holes and stars in Horndeski theory

We review black hole and star solutions for Horndeski theory. For non-shift symmetric theories, black holes involve a Kaluza-Klein reduction of higher dimensional Lovelock solutions. On the other hand, for shift symmetric theories of Horndeski and beyond Horndeski, black holes involve two classes of solutions: those that include, at the level of the action, a linear coupling to the Gauss-Bonnet term and those that involve time dependence in the galileon field. We analyze the latter class in detail for a specific subclass of Horndeski theory, discussing the general solution of a static and spherically symmetric spacetime. We then discuss stability issues, slowly rotating solutions as well as black holes coupled to matter. The latter case involves a conformally coupled scalar field as well as an electromagnetic field and the (primary) hair black holes thus obtained. We review and discuss the recent results on neutron stars in Horndeski theories.

Restoring general relativity in massive bigravity theory

We study static spherically symmetric solutions of massive bi-gravity theory, free from the Boulware-Deser ghost. We show the recovery of General Relativity via the Vainshtein mechanism, in the weak limit of the physical metric. We find a single polynomial equation determining the behavior of the solution for distances smaller than the inverse graviton mass. This equation is generically of the seventh order, while for a specific choice of the parameters of the theory it can be reduced to lower orders. The solution is analytic in different regimes: for distances below the Vainshtein radius (where General Relativity is recovered), and in the opposite regime, beyond the Vainshtein radius, where the solution approaches the flat metric.

Time-Dependent Spherically Symmetric Covariant Galileons

We study spherically symmetric solutions of the cubic covariant Galileon model in curved spacetime in presence of a matter source, in the test scalar field approximation. We show that a cosmological time evolution of the Galileon field gives rise to an induced matter-scalar coupling, due to the Galileon-graviton kinetic braiding, therefore the solution for the Galileon field is non trivial even if the bare matter-scalar coupling constant is set to zero. The local solution crucially depends on the asymptotic boundary conditions, and in particular, Minkowski and de Sitter asymptotics correspond to different branches of the solution. We study the stability of these solutions, namely, the well-posedness of the Cauchy problem and the positivity of energy for scalar and tensor perturbations, by diagonalizing the kinetic terms of the spin-2 and spin-0 degrees of freedom. In addition, we find that in presence of a cosmological time evolution of the Galileon field, its kinetic mixing with the graviton leads to a friction force, resulting to efficient damping of scalar perturbations within matter.

Black holes in massive gravity

We review the black hole solutions of the ghost-free massive gravity theory and its bimetric extension and outline the main results on the stability of these solutions against small perturbations. Massive (bi)-gravity accommodates exact black hole solutions, analogous to those of General Relativity. In addition to these solutions, hairy black holes -- solutions with no correspondent in General Relativity -- have been found numerically, whose existence is a natural consequence of the absence of Birkhoffs theorem in these theories. The existence of extra propagating degrees of freedom, makes the stability properties of these black holes richer and more complex than those of General Relativity. In particular, the bi-Schwarzschild black hole exhibits an unstable spherically symmetric mode, while the bi-Kerr geometry is also generically unstable, both against the spherical mode and against superradiant instabilities. If astrophysical black holes are described by these solutions, the superradiant instability of the Kerr solution imposes stringent bounds on the graviton mass.

Stability analysis of black holes in massive gravity: a unified treatment

We consider the analytic solutions of massive (bi)gravity which can be written in a simple form using advanced Eddington-Finkelstein coordinates. We analyse the stability of these solutions against radial perturbations. First we recover the previously obtained result on the instability of the bidiagonal bi-Schwarzschild solutions. In the non-bidiagonal case (which contains, in particular, the Schwarzschild solution with Minkowski fiducial metric) we show that generically there are physical spherically symmetric perturbations, but no unstable modes.

Asymptotically flat black holes in Horndeski theory and beyond

We find spherically symmetric and static black holes in shift-symmetric Horndeski and beyond Horndeski theories. They are asymptotically flat and sourced by a non trivial static scalar field. The first class of solutions is constructed in such a way that the Noether current associated with shift symmetry vanishes, while the scalar field cannot be trivial. This in certain cases leads to hairy black hole solutions (for the quartic Horndeski Lagrangian), and in others to singular solutions (for a Gauss-Bonnet term). Additionally, we find the general spherically symmetric and static solutions for a pure quartic Lagrangian, the metric of which is Schwarzschild. We show that under two requirements on the theory in question, any vacuum GR solution is also solution to the quartic theory. As an example, we show that a Kerr black hole with a non-trivial scalar field is an exact solution to these theories.

Rotating black holes in massive gravity

We present a solution for rotating black holes in massive gravity. We first give a solution of massive gravity with one dynamical metric. Both metrics of this solution are expressed in the advanced Eddington-Finkelstein-like coordinates: the physical metric has the original Kerr line element, while the fiducial metric is flat, but written in a rotating Eddington-Finkelstein form. For the bi-gravity theory we give an analogue of this solution: the two metrics have the original Kerr form, but, in general, different black hole masses. The generalisation of the solution to include the electric charge is also given, it is an analogue of the Kerr-Newman solution in General Relativity. We also discuss further possible ways to generalise the solutions.