Boltzmann codes are used extensively by several groups for constraining cosmological parameters with Cosmic Microwave Background and Large Scale Structure data. This activity is computationally expensive, since a typical project requires from 10 4 to 10 5 Boltzmann code executions. The newly released code CLASS (Cosmic Linear Anisotropy Solving System) incorporates improved approximation schemes leading to a simultaneous gain in speed and precision. We describe here the three approximations used by CLASS for basic ΛCDM models, namely: a baryon-photon tight-coupling approximation which can be set to first order, second order or to a compromise between the two; an ultra-relativistic fluid approximation which had not been implemented in public distributions before; and finally a radiation streaming approximation taking reionisation into account.

We propose a natural extension of Hořava's model for quantum gravity, which is free from the notorious pathologies of the original proposal. The new model endows the scalar graviton mode with a regular quadratic action and remains power-counting renormalizable. At low energies, it reduces to a Lorentzviolating scalar-tensor gravity theory. The deviations with respect to general relativity can be made weak by an appropriate choice of parameters. Introduction.-Recently, Hořava has proposed a new approach to quantum gravity [1]. The key idea is to abandon local Lorentz invariance as fundamental and to assume instead that it appears at low energies as an approximate symmetry. The breaking of Lorentz invariance is achieved by equipping the space-time with a preferred foliation by three-dimensional spacelike surfaces, which defines the splitting of coordinates into space and time. This allows us to complete the action of general relativity (GR) with higher spatial derivatives of the metric, improving the UV behavior of the graviton propagator and making the theory power-counting renormalizable. Besides, the action remains second order in time derivatives, avoiding the ghosts of covariant theories of higher-derivative gravity [2].The concrete realization of this idea as developed in [1] unfolds as follows. One considers the 3 þ 1 decomposition of the space-time metric in the preferred foliation,

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 address the consistency of Hořava's proposal for a theory of quantum gravity from the low-energy perspective. We uncover the additional scalar degree of freedom arising from the explicit breaking of the general covariance and study its properties.The analysis is performed both in the original formulation of the theory and in the Stückelberg picture. A peculiarity of the new mode is that it satisfies an equation of motion that is of first order in time derivatives. At linear level the mode is manifest only around spatially inhomogeneous and time-dependent backgrounds. We find two serious problems associated with this mode. First, the mode develops very fast exponential instabilities at short distances. Second, it becomes strongly coupled at an extremely low cutoff scale. We also discuss the "projectable" version of Hořava's proposal and argue that this version can be understood as a certain limit of the ghost condensate model.The theory is still problematic since the additional field generically forms caustics and, again, has a very low strong coupling scale. We clarify some subtleties that arise in the application of the Stückelberg formalism to Hořava's model due to its non-relativistic nature. 1Recently, Hořava has proposed a new approach to the theory of quantum gravity [1]. The key idea of the proposal is to equip space-time with a new structure: a foliation by spacelike surfaces. This foliation defines the splitting of the coordinates into "space" and "time" and breaks the general covariance of general relativity (GR). Then one can improve the UV behavior of the graviton propagator and ultimately make the theory power-counting renormalizable by adding to the GR action terms with higher spatial derivatives. At the same time the action in the ADM formalism contains only first order time derivatives, which allows to circumvent the problems with the ghosts appearing in covariant higher order gravity theories [2]. The higher derivative terms naively become irrelevant in the infrared and it was argued in [1] that the theory reduces to GR at large distances.However, the consistency of the above proposal is far from being clear. The main concern comes from the fact that the introduction of a preferred foliation explicitly breaks the gauge group of GR down to the group of space-time diffeomorphisms preserving this foliation. As already pointed out in [1] this breaking is expected to introduce extra degrees of freedom compared to GR. The new degrees of freedom can persist down to the infrared and lead to various pathologies (instabilities, strong coupling) that may invalidate the theory. An illustration of this phenomenon is provided by theories of massive gravity where special care is needed to make the additional degrees of freedom well-behaved [3,4,5].

Hořava's proposal for non-relativistic quantum gravity introduces a preferred time foliation of space-time which violates the local Lorentz invariance. The foliation is encoded in a dynamical scalar field which we call 'khronon'. The dynamics of the khronon field is sensitive to the symmetries and other details of the particular implementations of the proposal. In this paper we examine several consistency issues present in three non-relativistic gravity theories: Hořava's projectable theory, the healthy non-projectable extension, and a new extension related to ghost condensation. We find that the only model which is free from instabilities and strong coupling is the non-projectable one. We elaborate on the phenomenology of the latter model including a discussion of the couplings of the khronon to matter. In particular, we obtain the parameters of the post-Newtonian expansion in this model and show that they are compatible with current observations.

Under broad assumptions breaking of Lorentz invariance in gravitational theories leads to tension with unitarity because it allows for processes that apparently violate the second law of thermodynamics. The crucial ingredient of this argument is the existence of black hole solutions with the interior shielded from infinity by a causal horizon. We study how the paradox can be resolved in the healthy extension of Hořava gravity. To this aim we analyze classical solutions describing large black holes in this theory with the emphasis on their causal structure. The notion of causality is subtle in this theory due to the presence of instantaneous interactions. Despite this fact, we find that within exact spherical symmetry a black hole solution contains a space-time region causally disconnected from infinity by a surface of finite area -the 'universal horizon'.We then consider small perturbations of arbitrary angular dependence in the black hole background. We argue that aspherical perturbations destabilize the universal horizon and, at non-linear level, turn it into a finite-area singularity. The causal structure of the region outside the singularity is trivial. If the higher-derivative terms in the equations of motion smear the singularity while preserving the trivial causal structure of the solutions, the thermodynamics paradox would be obviated. As a byproduct of our analysis we prove that the black holes do not have any non-standard long-range hair. We also comment on the relation with Einstein-aether theory, where the solutions with universal horizon appear to be stable.

We propose in this White Paper a concept for a space experiment using cold atoms to search for ultra-light dark matter, and to detect gravitational waves in the frequency range between the most sensitive ranges of LISA and the terrestrial LIGO/Virgo/KAGRA/INDIGO experiments. This interdisciplinary experiment, called Atomic Experiment for Dark Matter and Gravity Exploration (AEDGE), will also complement other planned searches for dark matter, and exploit synergies with other gravitational wave detectors. We give examples of the extended range of sensitivity to ultra-light dark matter offered by AEDGE, and how its gravitational-wave measurements could explore the assembly of super-massive black holes, first-order phase transitions in the early universe and cosmic strings. AEDGE will be based upon technologies now being developed for terrestrial experiments using cold atoms, and will benefit from the space experience obtained with, e.g., LISA and cold atom experiments in microgravity.KCL-PH-TH/2019-65, CERN-TH-2019-126

We consider some flat space theories for spin 2 gravitons, with less invariance than full diffeomorphisms. For the massless case, classical stability and absence of ghosts require invariance under transverse diffeomorphisms (TDiff), h µν → h µν + 2∂ (ν ξ µ) , with ∂ µ ξ µ = 0. Generic TDiff invariant theories contain a propagating scalar, which disappears if the symmetry is enhanced in one of two ways. One possibility is to consider full diffeomorphisms (Diff). The other (which we denote WTDiff) adds a Weyl symmetry, by which the Lagrangian becomes independent of the trace. The first possibility corresponds to General Relativity, whereas the second corresponds to "unimodular" gravity (in a certain gauge). Phenomenologically, both options are equally acceptable. For massive gravitons, the situation is more restrictive. Up to field redefinitions, classical stability and absence of ghosts lead directly to the standard Fierz-Pauli Lagrangian. In this sense, the WTDiff theory is more rigid against deformations than linearized GR, since a mass term cannot be added without provoking the appearance of ghosts.