International Journal of Modern Physics D

We introduce the standard model of cosmology based on general relativity and discuss its successes and problems. We then discuss motivations to consider gravitational theories beyond general relativity and summarize observational and theoretical constraints that these theories need to satisfy. A special focus is laid on screening mechanisms, which hide deviations from general relativity in the Solar System and enable large modifications to general relativity on astrophysical and cosmological scales. Finally, several modified gravity models are introduced, which satisfy the Solar System constrains as well as the constraint on the speed of gravitational waves obtained from almost simultaneous detections of gravitational waves and gamma ray bursts from a neutron star merger (GW170817/GRB 170817A).

With the increasing wealth of high-quality astronomical and cosmological data and the manifold departures from General Relativity in principle conceivable, the development of generalized parametrization frameworks that unify gravitational models and cover a wide range of length scales and a variety of observational probes to enable systematic high-precision tests of gravity has been a stimulus for intensive research. A review is presented here for some of the formalisms devised for this purpose, covering the cosmological large- and small-scale structures, the astronomical static weak-field regime as well as emission and propagation effects for gravitational waves. This includes linear and nonlinear parametrized post-Friedmannian frameworks, effective field theory approaches, the parametrized post-Newtonian expansion, the parametrized post-Einsteinian formalism as well as an inspiral-merger-ringdown waveform model among others. Connections between the different formalisms are highlighted where they have been established and a brief outlook is provided for general steps towards a unified global framework for tests of gravity and dark sector models.

The standard paradigm of cosmology assumes General Relativity (GR) is a valid theory for gravity at scales in which it has not been properly tested. Developing novel tests of GR and its alternatives is crucial if we want to give strength to the model or find departures from GR in the data. Since alternatives to GR are usually defined through nonlinear equations, designing new tests for these theories implies a jump in complexity and thus, a need for refining the simulation techniques. We summarize existing techniques for dealing with modified gravity (MG) in the context of cosmological simulations. $N$-body codes for MG are usually based on standard gravity codes. We describe the required extensions, classifying the models not according to their original motivation, but by the numerical challenges that must be faced by numericists. MG models usually give rise to elliptic equations, for which multigrid techniques are well suited. Thus, we devote a large fraction of this review to describing this particular technique. Contrary to other reviews on multigrid methods, we focus on the specific techniques that are required to solve MG equations and describe useful tricks. Finally, we describe extensions for going beyond the static approximation and dealing with baryons.

We review some of the commonly used approximation methods to predict large-scale structure formation in modified gravity (MG) models for the cosmic acceleration. These methods are developed to speed up the often slow $N$-body simulations in these models, or directly make approximate predictions of relevant physical quantities. In both cases, they are orders of magnitude more efficient than full simulations, making it possible to explore and delineate the large cosmological parameter space. On the other hand, there is a wide variation of their accuracies and ranges of validity, and these are usually not known a priori and must be validated against simulations. Therefore, a combination of full simulations and approximation methods will offer both efficiency and reliability. The approximation methods are also important from a theoretical point of view, since they can often offer useful insight into the nonlinear physics in MG models and inspire new algorithms for simulations.

Observations of the evolution of large-scale structures in the Universe provides unique tools to confront Einstein’s theory of General Relativity on cosmological scales. We review weak gravitational lensing and galaxy clustering studies, discussing how these can be used in combination in order to constrain a range of different modified gravity theories. We argue that in order to maximise the future information gain from these probes, theoretical effort will be required in order to model the impact of beyond-Einstein gravity in the nonlinear regime of structure formation.

Changes in the law of gravity have far-reaching implications for the formation and evolution of galaxy clusters, and appear as peculiar signatures in their mass-observable relations, structural properties, internal dynamics, and abundance. We review the outstanding progress made in recent years towards constraining deviations from General Relativity with galaxy clusters, and give an overview of the yet untapped information becoming accessible with forthcoming surveys that will map large portions of the sky in great detail and unprecedented depth.

Cosmic void as a cosmological probe is reviewed. We introduce the theoretical background and recent development in observations of cosmic voids, focusing on its potential to test theorists of gravity. Four observables about cosmic voids are introduced: void number counts, weak gravitational lensing, redshift-space distortions and ISW around voids. We discuss opportunities on their application to current and future galaxy surveys and highlight challenges for this subject.

Screened modified gravity models evade solar system tests of relativistic gravitation but exhibit novel and interesting effects on scales between the solar system and the Hubble flow: astrophysical scales. In this paper, we review how astrophysical tests using stars, galaxies, and clusters can be used to constrain these theories. We classify screening into three categories: thin-shell (chameleon, symmetron, and dilaton models), Vainshtein screening (e.g. Galileons and Horndeski), and Vainshtein breaking (e.g. beyond Horndeski and DHOST) and discuss the optimal strategy for probing each. In many cases, this is driven by whether a specific category violates the equivalence principles (strong or weak). We summarize the general model-independent bounds on each screening category that have been derived in the literature.

We review laboratory constraints on theories of modified gravity and show that they are complementary to cosmological and astrophysical tests. We particularly focus on the environmentally dependent dilaton, as a worked example to show how such constraints are derived. Finally we discuss precision photons experiments, and why these may also give us information about possible modifications of gravity.