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...

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The aim of this paper is to answer the following two questions: (1) Given cosmological observations of the expansion history and linear perturbations in a range of redshifts and scales as precise as is required, which of the properties of dark energy could actually be reconstructed without imposing any parameterization? (2) Are these observables sufficient to rule out not just a particular dark energy model, but the entire general class of viable models comprising a single scalar field?This paper bears both good and bad news. On one hand, we find that the goal of reconstructing dark energy models is fundamentally limited by the unobservability of the present values of the matter density Ωm0, the perturbation normalization σ8 as well as the present matter power spectrum. On the other, we find that, under certain conditions, cosmological observations can nonetheless rule out the entire class of the most general single scalar-field models, i.e. those based on the Horndeski Lagrangian.

Despite its continued observational successes, there is a persistent (and growing) interest in extending cosmology beyond the standard model, ΛCDM. This is motivated by a range of apparently serious theoretical issues, involving such questions as the cosmological constant problem, the particle nature of dark matter, the validity of general relativity on large scales, the existence of anomalies in the CMB and on small scales, and the predictivity and testability of the inflationary paradigm. In this paper, we summarize the current status of ΛCDM as a physical theory, and review investigations into possible alternatives along a number of different lines, with a particular focus on highlighting the most promising directions. While the fundamental problems are proving reluctant to yield, the study of alternative cosmologies has led to considerable progress, with much more to come if hopes about forthcoming high-precision observations and new theoretical ideas are fulfilled.Keywords: cosmology -dark energy -cosmological constant problem -modified gravitydark matter -early universe Cosmology has been both blessed and cursed by the establishment of a standard model: ΛCDM. On the one hand, the model has turned out to be extremely predictive, explanatory, and observationally robust, providing us with a substantial understanding of the formation of large-scale structure, the state of the early Universe, and the cosmic abundance of different types of matter and energy. It has also survived an impressive battery of precision observational tests -anomalies are few and far between, and their significance is contentious where they do arise -and its predictions are continually being vindicated through the discovery of new effects (B-mode polarization [1] and lensing [2,3] of the cosmic microwave background (CMB), and the kinetic Sunyaev-Zel'dovich effect [4] being some recent examples). These are the hallmarks of a good and valuable physical theory.On the other hand, the model suffers from profound theoretical difficulties. The two largest contributions to the energy content at late times -cold dark matter (CDM) and the cosmological constant (Λ) -have entirely mysterious physical origins. CDM has so far evaded direct detection by laboratory experiments, and so the particle field responsible for it -presumably a manifestation of "beyond the standard model" particle physics -is unknown. Curious discrepancies also appear to exist between the predicted clustering properties of CDM on small scales and observations. The cosmological constant is even more puzzling, giving rise to quite simply the biggest problem in all of fundamental physics: the question of why Λ appears to take such an unnatural value [5,6,7]. Inflation, the theory of the very early Universe, has also been criticized for being fine-tuned and under-predictive [8], and appears to leave many problems either unsolved or fundamentally unresolvable. These problems are indicative of a crisis.From January 14th-17th 2015, we held a conference in Oslo, Norway to surve...

General Relativity and the ΛCDM framework are currently the standard lore and constitute the concordance paradigm. Nevertheless, long-standing open theoretical issues, as well as possible new observational ones arising from the explosive development of cosmology the last two decades, offer the motivation and lead a large amount of research to be devoted in constructing various extensions and modifications.All extended theories and scenarios are first examined under the light of theoretical consistency, and then are applied to various geometrical backgrounds, such as the cosmological and the spherical symmetric ones. Their predictions at both the background and perturbation levels, and concerning cosmology at early, intermediate and late times, are then confronted with the huge amount of observational data that astrophysics and cosmology are able to offer recently. Theories, scenarios and models that successfully and efficiently pass the above steps are classified as viable and are candidates for the description of Nature.We list the recent developments in the fields of gravity and cosmology, presenting the state of the art, high-lighting the open problems, and outlining the directions of future research.