The grand challenges of contemporary fundamental physics—dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem—all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on ‘Black holes, Gravitational waves and Fundamental Physics’.

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We study black holes in the infrared sector of three-dimensional Hořava gravity. It is shown that black hole solutions with anti-de Sitter asymptotics are admissible only in the sector of the theory in which the scalar degree of freedom propagates infinitely fast. We derive the most general class of stationary, circularly symmetric, asymptotically anti-de Sitter black hole solutions. We also show that the theory admits black hole solutions with de Sitter and flat asymptotics, unlike three-dimensional general relativity. For all these cases, universal horizons may or may not exist depending on the choice of parameters. Solutions with de Sitter asymptotics can have universal horizons that lie beyond the de Sitter horizon.

Modifications of general relativity usually include extra dynamical degrees of freedom, which to date remain undetected. Here we explore the possibility of modifying Einstein's theory by adding solely nondynamical fields. With the minimal requirement that the theory satisfies the weak equivalence principle and admits a covariant Lagrangian formulation, we show that the field equations generically have to include higher-order derivatives of the matter fields. This has profound consequences for the viability of these theories. We develop a parametrization based on a derivative expansion and show that -to next-to-leading order -all theories are described by just two parameters. Our approach can be used to put stringent, theory-independent constraints on such theories, as we demonstrate using the Newtonian limit as an example.In four dimensions, the Lovelock theorem [1,2] states that the only divergence-free rank-2 tensor which is constructed solely from the metric g ab and its derivatives up to second differential order is the Einstein tensor, G ab ≡ R ab − 1 2 g ab R, plus a cosmological constant term (see also some previous, more restrictive proofs by Weyl [3] and Cartan [4]). This suggests a natural choice for the lefthand side of Einstein's equations (we work in units where c = 8π G = 1):(1)The right-hand side, T ab , is the matter stress-energy tensor and the contracted Bianchi identity then implies that T ab is divergence free, ∇ a T ab = 0. This property is necessary for geodesic motion, which guarantees that the weak equivalence principle (universality of free fall) is satisfied.With the mild requirement that the field equations for the gravitational field and the matter fields be derived by an action, the arguments above single out the action

We consider f(Q) extended symmetric teleparallel cosmologies, where Q is the non-metricity scalar, and constrain its functional form through the order reduction method. By using this technique, we are able to reduce and integrate the field equations and thus to select the corresponding models giving rise to bouncing cosmology. The selected Lagrangian is then used to develop the Hamiltonian formalism and to obtain the Wave Function of the Universe which suggests that classical observable universes can be recovered according to the Hartle Criterion.

We consider Hořava gravity within the framework of the effective field theory (EFT) of dark energy and modified gravity. We work out a complete mapping of the theory into the EFT language for an action including all the operators which are relevant for linear perturbations with up to sixth order spatial derivatives. We then employ an updated version of the EFTCAMB/EFTCosmoMC package to study the cosmology of the low-energy limit of Hořava gravity and place constraints on its parameters using several cosmological data sets. In particular we use cosmic microwave background (CMB) temperature-temperature and lensing power spectra by Planck 2013, WMAP low-polarization spectra, WiggleZ galaxy power spectrum, local Hubble measurements, Supernovae data from SNLS, SDSS and HST and the baryon acoustic oscillations measurements from BOSS, SDSS and 6dFGS. We get improved upper bounds, with respect to those from Big Bang Nucleosynthesis, on the deviation of the cosmological gravitational constant from the local Newtonian one. At the level of the background phenomenology, we find a relevant rescaling of the Hubble rate at all epoch, which has a strong impact on the cosmological observables; at the level of perturbations, we discuss in details all the relevant effects on the observables and find that in general the quasi-static approximation is not safe to describe the evolution of perturbations. Overall we find that the effects of the modifications induced by the low-energy Hořava gravity action are quite dramatic and current data place tight bounds on the theory parameters. *

Spherical symmetry for f (R)-gravity is discussed by searching for Noether symmetries. The method consists in selecting conserved quantities in form of currents that reduce dynamics of f (R)-models compatible with symmetries. In this way we get a general method to obtain constants of motion without setting a priori the form of f (R). In this sense, the Noether symmetry results a physical criterium. Relevant cases are discussed.