Wave propagation and shock formation in the most general scalar–tensor theories

Abstract

This work studies wave propagation in the most general scalar-tensor theories, particularly focusing on the causal structure realized in these theories and also the shock formation process induced by nonlinear effects. For these studies we use the Horndeski theory and its generalization to the two scalar field case. We show that propagation speeds of gravitational wave and scalar field wave in these theories may differ from the light speed depending on background field configuration, and find that a Killing horizon becomes a boundary of causal domain if the scalar fields share the symmetry of the background spacetime. About the shock formation, we focus on transport of discontinuity in second derivatives of the metric and scalar field in the shift-symmetric Horndeski theory. We find that amplitude of the discontinuity generically diverges within finite time, which corresponds to shock formation. It turns out that the canonical scalar field and the scalar DBI model, among other theories described by the Horndeski theory, are free from such shock formation even when the background geometry and scalar field configuration are nontrivial. We also observe that gravitational wave is protected against shock formation when the background has some symmetries at least. This fact may indicate that the gravitational wave in this theory is more well-behaved compared to the scalar field, which typically suffers from shock formation. CONTENTS

Causal structure of black holes in shift-symmetric Horndeski theories

“…Our results agree with previous results in the literature that used the method of characteristics [28] when there is overlap, but they go a bit further to prove that H is actually a Killing horizon for f −1 ab . They also help interpret previous result in the intuitive physical picture of effective metrics and pin down certain subtleties of the characteristics approach.…”

“…Condition I also identifies the characteristics for scalar perturbations, as is discussed in more detail in Appendix C. As shown there, if one applies the approach of Ref. [28] to the decoupling limit and considers perturbation around a stationary background configuration, one does indeed obtain condition I. The characteristics can be thought of as maximal speed of propagation surfaces and the boundary of the future of a given subset of the manifold is a characteristic.…”

“…This last statement entails some implicit assumptions not spelled out in Ref. [28]. Our approach follows a different path and it provides a rigorous proof about when Killing horizons for different excitations coincide.…”

Recovering P(X) from a canonical complex field

“…10 Expansion of the Universe only leads to actual damping for c 2 s > 1/3, otherwise it works as an antidamping. 11 Alternatively, one can consider a damped anisotropic oscillator moving in a rotating frame, so that the gyroscopic force on the r.h.s. of Eqs.…”

Healthy imperfect dark matter from effective theory of mimetic cosmological perturbations

“…To what extent such gradient instabilities are dangerous depends on the time scale on which perturbations grow, but it is better if one could remove this potential danger from the theory in the first place. Another problem is an appearance of a caustic singularity [37], though it is a generic nature of scalartensor theories [38,39] rather than a problem specific to mimetic/Hořava-Lifshitz gravity, and there exist some mechanisms to avoid formation of the caustic [40,41].…”