Anisotropic Stress as a Signature of Nonstandard Propagation of Gravitational Waves

Abstract

We make precise the heretofore ambiguous statement that anisotropic stress is a sign of a modification of gravity. We show that in cosmological solutions of very general classes of models extending gravity -all scalar-tensor theories (Horndeski), Einstein-Aether models and bimetric massive gravity -a direct correspondence exists between perfect fluids apparently carrying anisotropic stress and a modification in the propagation of gravitational waves. Since the anisotropic stress can be measured in a model-independent manner, a comparison of the behavior of gravitational waves from cosmological sources with large-scale-structure formation could in principle lead to new constraints on the theory of gravity.Over the last decade, we have established beyond reasonable doubt that, in its recent past, the expansion of the universe has been accelerating. This has challenged our beliefs about the theory of gravity: the only possibility available in general relativity with non-exotic matter is a cosmological constant, which would suffer from severe fine-tuning issues. Alternatively, the mechanism could be dynamical, i.e. feature at least one new degree of freedom. These dynamics would modify the predictions of concordance cosmology and give us a means to carry out precision tests of gravity at extremely large scales.Frequently in extended models of gravity perfect fluids apparently carry anisotropic stress: there is gravitational slip, i.e. the values of the two scalar gravitational potentials sourced by matter are not equal. This affects structure formation and weak lensing. Recently, it was shown that the ratio of the two potentials is actually a modelindependent observable [1, 2], which Euclid should be able to measure to a precision of a few percent, depending on the precise assumptions [3]. This begs the question as to what detecting or not detecting anisotropic stress actually means.In this Letter we show that the propagation of tensor modes (gravitational waves, GWs) is also modified whenever the anisotropic stress is present at first order in perturbations sourced by perfect-fluid matter. We demonstrate this relationship in the context of three very large classes of extensions of the gravitational sector: general scalar-tensor theories (Horndeski [4, 5]), Einstein-Aether models [6][7][8] and bimetric massive gravity [9,10]. GWs are the only propagating degrees of freedom in General Relativity, and it is natural to define modified gravity models as those where the gravitational waves are modified in such a non-trivial manner. Since imperfect fluids with anisotropic stress also split the two gravitational potentials but do not modify the propagation of tensor modes, this definition allows us to separate modifications of gravity from imperfect fluids.The emphasis of this paper is not on new calculations (see e.g. the review [11]), but rather on new relations which are very general, were not noted before in the literature and could have a significant impact on tests of gravity on cosmological scales. ASSUMPTIONSWe ass...

BeyondΛCDM: Problems, solutions, and the road ahead

“…In principle, one can therefore measure the gravitational slip η (where η 1 is an unambiguous signature of modified gravity [227]) without making reference to galaxy bias [226,466]. Without assuming a value and form for the galaxy bias function, one cannot determine the underlying dark matter perturbation and so make a measurement of the effective Newton's constant µ, however.…”

“…Measuring the anisotropic stress is a promising test for deviations from ΛCDM, as it can be measured without any assumptions on the primordial matter power spectrum or the galaxy bias by combining weak-lensing measurements with peculiar velocities [226]. The presence of the anisotropic stress also appears to be connected to a modification of the propagation of gravitational waves, which can be used as a way to define what "modified gravity" means [227,228].…”

Speed of gravitational waves and the fate of scalar-tensor gravity

“…In alternative theories of gravity, additional polarizations may propagate, each with potentially different velocities, attenuations and effective masses [7]. This issue has been well studied in cosmology, and has been a topic of discussion in connection to the early [8][9][10][11] and the late Universe [12][13][14][15]. There are fairly model-independent tests for effects caused by additional polarizations [16], damping [17][18][19], mass [20], and Lorentz symmetry violations [21,22].…”

Cosmological constraints on deviations from Lorentz invariance in gravity and dark matter

“…The first two classes of effects are common to a very broad class of modified gravity models, not necessary based on the Einstein-aether [34][35][36]. A detailed study of their impact on the CMB and matter power spectrum was performed in [10] with the following outcome: (i) Whenever G cos = G N the Poisson equation for subhorizon scales in an expanding background is modified.…”