Gravitational waves and geometrical optics in scalar-tensor theories

Abstract

The detection of gravitational waves (GWs) propagating through cosmic structures can provide invaluable information on the geometry and content of our Universe as well as on the fundamental theory of gravity. In order to test possible departures from General Relativity, it is essential to analyze, in a modified gravity setting, how GWs propagate through a perturbed cosmological space-time. Working within the framework of geometrical optics, we develop tools to address this topic for a broad class of scalar-tensor theories, including scenarios with non-minimal, derivative couplings between scalar and tensor modes.

Gravitational-wave cosmological distances in scalar-tensor theories of gravity

“…One of the delicate issues in studying GW propagation in modified gravity is to distinguish tensor from scalar fluctuations, and correctly identify their roles in the evolution equations of high-frequency fields. This topic started with the classic papers [52,53], and has been recently reconsidered in [54][55][56][57][58] using a variety of methods. The issue can be subtle in theories where scalar and metric fluctuations propagate with different speed, a phenomenon associated with spontaneous breaking of global Lorentz invariance by means of a non-vanishing time-like gradient for the dark energy field.…”

“…Recently, various scenarios have been analyzed[55][56][57][58] where, at the price of tunings, tensor and scalar propagate with the same speed. We do not consider this case in this work.…”

Unifying gravitational waves and dark energy

“…The presence of φ changes the dynamics of the gravitational potentials, possibly introducing scale dependencies in the growth of cosmic structures, and the EFT formalism is able to produce predictions for the modified observables [9][10][11][12]. On the other hand, GWs in General Relativity and ST theories are usually addressed in the high energy limit [13][14][15][16]. By introducing an expansion in derivatives of high-frequency (HF) perturbations, it is possible to study the propagation of GWs over slow-varying, but otherwise unknown, backgrounds.…”

“…The separation of variation scales between perturbation and background can be regarded as another SSB and helps to identify the true degrees of freedom (dofs) of the theory [17,18]. For example, ST theories predict the presence of an additional scalar wave (SW) [11,19,20] and possibly introduce extra damping [15,[21][22][23] or modifications to the propagation speed of the modes [24]. Moreover, the HF expansion is well suited to describe the GWs observed by LVK and third-generation interferometers [25,26] because their frequency can be as high as the EFT energy cutoff, and their sources may be located close enough for the spacetime geometry not to be FRW.…”

“…High-frequency limit.-We study the ǫ ≪ 1 limit of the EoMs above to study HF GWs and SWs. Following the considerations illustrated when discussing the gauge transformations, we keep the first non-null orders of the ǫ expansions ( 14)- (15). Since all of the dofs satisfy a wave equation, we make the following WKB ansatz γ µν = Υ µν e iθ/ǫ , δφ 0 = Φ e iθ/ǫ , ψ = Ψ e iθ/ǫ , (31) where Υ µν , Φ, Ψ are complex and of order O(ǫ 0 ), θ is real and they are all slow-varying functions of the spacetime coordinates.…”