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’.

We develop a model of eternal topological inflation using a racetrack potential within the context of type IIB string theory with KKLT volume stabilization. The inflaton field is the imaginary part of the Kähler structure modulus, which is an axion-like field in the 4D effective field theory. This model does not require moving branes, and in this sense it is simpler than other models of string theory inflation. Contrary to single-exponential models, the structure of the potential in this example allows for the existence of saddle points between two degenerate local minima for which the slow-roll conditions can be satisfied in a particular range of parameter space. We conjecture that this type of inflation should be present in more general realizations of the modular landscape. We also consider 'irrational' models having a dense set of minima, and discuss their possible relevance for the cosmological constant problem.

Using recent simulation results, we provide the mass and speed spectrum of cosmic string loops. This is the quantity of primary interest for many phenomenological signatures of cosmic strings, and it can be accurately predicted using recently acquired detailed knowledge of the loop production function. We emphasize that gravitational smoothing of long strings does not play any role in determining the total number of existing loops. We derive a bound on the string tension imposed by recent constraints on the stochastic gravitational wave background from pulsar timing arrays, finding Gµ ≤ 2.8 × 10 −9 . We also provide a derivation of the Boltzmann equation for cosmic string loops in the language of differential forms.