Extended scalar-tensor theories of gravity

Abstract

We study new consistent scalar-tensor theories of gravity recently introduced by Langlois and Noui with potentially interesting cosmological applications. We derive the conditions for the existence of a primary constraint that prevents the propagation of an additional dangerous mode associated with higher order equations of motion. We then classify the most general, consistent scalar-tensor theories that are at most quadratic in the second derivatives of the scalar field. In addition, we investigate the possible connection between these theories and (beyond) Horndeski through conformal and disformal transformations. Finally, we point out that these theories can be associated with new operators in the effective field theory of dark energy, which might open up new possibilities to test dark energy models in future surveys. I. INTRODUCTIONScalar tensor theories of gravity play an essential role for building models of inflation and dark energy. The most general scalar-tensor theory leading to second order equations of motion, the theory of Horndeski [1, 2], propagates three degrees of freedom (dof) in the vacuum. When coupled with matter, this set-up provides a convenient framework for parametrising linear cosmological perturbations and for developing tests of General Relativity using cosmological observations (see e.g. [3] for a review). On the other hand, it has been recently pointed out [4-6] that there are scalar-tensor theories more general than Horndeski, which are characterised by higher order equations of motion, but which nevertheless propagate three degrees of freedom in the vacuum. This property is ensured by the existence of constraints that forbid the propagation of additional, dangerous modes. Such theories are dubbed 'beyond Horndeski' [4,5].The existence of these theories raises various broad questions that we consider in this paper:• What is the most general, consistent theory of gravity coupled with a scalar field? Beyond Horndeski is an explicit example of a consistent generalisation of Horndenski, but it is not necessarily the most general one. In fact, [7,8] provided explicit examples of theories more general than beyond Horndeski. They study degeneracy conditions for the kinetic matrix associated with Lagrangians that are at most quadratic in the scalar second derivatives. In the present work, we re-derive the results of [7] using a method that provides the conditions for the existence of a primary constraint starting from the conjugate momenta of the scalar and tensor fields. Our method is equivalent to the one developed in [7] to impose the degeneracy conditions for the kinetic matrix. In the present work we limit our attention to theories quadratic in second derivatives of the scalar and we dub them extended scalar-tensor theories of gravity, in brief EST theories, and classify them carefully.• Is there any connection between EST theories and the original Horndeski, or beyond Horndeski actions? It is known that generalised disformal transformations allow to generate beyond Horndeski Lagrangia...

Cosmological self-tuning and local solutions in generalized Horndeski theories

“…Under the simplifying but reasonable hypothesis that the spin-2 sector does not hide any subtlety, Refs. [14,15,31,32] then showed that this is also the case for most of these generalized Horndeski models, but curiously enough, not all of them -confirming thereby that previously published arguments were incomplete. It was shown in [14,31,32] …”

“…(15) that we obtained in cosmology, which is not a surprise since the asymptotic behavior of the present solution at large radii should match with this cosmological solution. But we find here some restrictions with respect to (15): The f 3 function needs to be very precisely tuned at the cosmological value X = X c (in order not to contribute to any background equation at this precise value), while f 5 and s 5 should be related in a specific way at this value of X (again so that their sum Xf 5 + 2s 5 does not contribute to any background equation). It should be underlined that this set of equations (28) only needs to be satisfied at the cosmological value X = X c , and notably that (28c) and (28d) should not be imposed for all X.…”

“…IV and V B that the subclass of models L (2,0) + L (4,0) + L (4,1) , that we may call the "Three Graces", is the most interesting. Let us therefore focus on this subclass for the present illustration, and to simplify, let us choose monomials f 2 = k 2 X α , f 4 = k 4 X β and s 4 = κ 4 X γ , where k 2 , k 4 and κ 4 are O(1) dimensionless constants, whose signs are imposed by the two cosmological equations (15). As underlined above, we must choose α = −1 otherwise the L (2,0) Lagrangian, Eq.…”

“…8 In conclusion, in this simple L (2,0) + L (3,0) model, the metric cannot be close to the Schwarzschild solution, and solar-system tests are not passed. The only ways out are either that the contribution of L (3,0) is negligible in the cosmological equations (15), so thatφ is actually unrelated to f 3 and the backreaction ∝φ 3 f 3 can be small enough, or that other Lagrangians than L (3,0) dominate at small distances, which depends on the functions f n (X) entering them.…”

“…III below Eqs. (15)], although this would also give a vanishing backreaction. Indeed, this would correspond to ϕ ′ = 0 in Eq.…”