Living Reviews in Relativity (2019) 22:1 https://doi.org/10.1007/s41114-018-0017-4 REVIEW ARTICLE Testing general relativity in cosmology Mustapha Ishak1 Received: 28 May 2018 / Accepted: 6 November 2018 © The Author(s) 2018 Abstract We review recent developments and results in testing general relativity (GR) at cosmo- logical scales. The subject has witnessed rapid growth during the last two decades with the aim of addressing the question of cosmic acceleration and the dark energy asso- ciated with it. However, with the advent of precision cosmology, it has also become a well-motivated endeavor by itself to test gravitational physics at cosmic scales. We overview cosmological probes of gravity, formalisms and parameterizations for testing deviations from GR at cosmological scales, selected modified gravity (MG) theories, gravitational screening mechanisms, and computer codes developed for these tests. We then provide summaries of recent cosmological constraints on MG parameters and selected MG models. We supplement these cosmological constraints with a summary of implications from the recent binary neutron star merger event. Next, we summarize some results on MG parameter forecasts with and without astrophysical systematics that will dominate the uncertainties. The review aims at providing an overall picture of the subject and an entry point to students and researchers interested in joining the field. It can also serve as a quick reference to recent results and constraints on testing gravity at cosmological scales. Keywords Tests of relativistic gravity · Theories of gravity · Modified gravity · Cosmological tests · Post-Friedmann limit · Gravitational waves Contents 1 Introduction ............................................... 2 General relativity (GR) ......................................... 2.1 Basic principles .......................................... 2.2 Einstein field equations (EFEs) and their exact solutions ..................... 3 The standard model of cosmology ................................... 3.1 The homogeneous cosmological background ........................... 3.1.1 FLRW metric and Friedmann’s equations ......................... B Mustapha Ishak [email protected] 1 Department of Physics, The University of Texas at Dallas, Richardson, TX 75080, USA 0123456789().: V,-vol 123 1 Page 2 of 204 M. Ishak 3.1.2 Cosmic mass-energy budget, dark energy and cosmic acceleration ............ 3.1.3 Cosmological distances .................................. 3.2 The inhomogeneous lumpy universe and the growth of large-scale structure .......... 3.2.1 Large-scale structure and cosmological perturbations .................. 3.2.2 Growth factor and growth rate of large-scale structure .................. 3.2.3 Correlation function and matter power spectrum ..................... 4 Cosmological probes of gravity theory ................................. 4.1 Probes of cosmic geometry and expansion ............................ 4.1.1 Standard candles: type Ia supernova ........................... 4.1.2 Standard rulers: angular distance to CMB last scattering surface and baryon acoustic oscillations ......................................... 4.1.3 Local measurements of the Hubble constant or measurements of H(z) .......... 4.2 Weak gravitational lensing .................................... 4.3 Galaxy surveys: clustering and redshift space distortions (RSD) ................. 4.4 Cosmic microwave background radiation ............................. 4.4.1 Integrated Sachs–Wolfe (ISW) effect ........................... 4.4.2 CMB lensing ........................................ 5 Formalisms and approaches to testing GR at cosmological scales ................... 5.1 Effective field theory (EFT) approach to dark energy and modified gravity ........... 5.2 Modified growth parameters ................................... 5.3 Evolution of MG parameters in time and scale .......................... 5.4 The growth index parameter γ .................................. 5.5 The EG -parameter test ...................................... 5.6 Parameterized post-Friedmann formalism ............................ 5.7 Remarks on transition to nonlinear scales ............................ 6 Constraints and results on MG parameters (i.e., deviations from GR) from current cosmological data sets ................................................. 6.1 Constraints on modified growth parameters ........................... 6.1.1 Constraints from Planck CMB, ISW, CMB lensing, and other data sets ......... 6.1.2 Constraints on MG parameters from mainly weak lensing data ............. 6.1.3 Constraints on MG parameters from various probes and analyses ............ 6.2 Constraints on f σ8 from galaxy surveys and RSD measurements ............... 6.3 Constraints on EG ........................................ 7 Types of modifications to GR at cosmological scales and corresponding MG models ........ 7.1 Cartan–Weyl–Lovelock theorem ................................. 7.2 Modified gravity versus dark energy ............................... 7.3 Modified gravity theories with extra fields ............................ 7.3.1 Theories with extra scalar field .............................. 7.3.2 Extra vector field(s) .................................... 7.3.3 Extra vector and scalar fields ............................... 7.3.4 Extra tensor fields ..................................... 7.4 Modified gravity theories with higher-order derivatives ..................... 7.4.1 Illustrative example 1: f (R) theories ........................... 7.4.2 Illustrative example 2: Hoˇrava–Lifshitz .......................... 7.4.3 Other higher order derivative theories ........................... 7.5 Modified gravity theories with higher-dimensions ........................ 7.5.1 Theories with compact dimensions versus braneworld models .............. 7.5.2 Illustrative example : Dvali–Gabadadze–Porrati gravity (DGP) ............. 7.6 Non-local modified gravity theories ............................... 7.6.1 Illustrative example: RR model .............................. 7.6.2 Other non-local gravity theories .............................. 8 Screening mechanisms ......................................... 8.1 Large-mass based screening .................................... 8.2 Weak-coupling based screening .................................. 8.3 Large kinetic terms based screening ............................... 9 Constraints on MG models from current cosmological data sets .................... 9.1 Constraints on Horndeski and beyond Horndeski models .................... 123 Testing general relativity in cosmology Page 3 of 204 1 9.2 Constraints on Brans–Dicke theory ................................ 9.3 Constraints on vector–tensor and generalized Einstein aether theories .............. 9.4 Constraints on massive gravity and bigravity ........................... 9.5 Constraints on f (R) models ................................... 9.6 Constraints on DGP models .................................... 9.7 Constraints on Galileon models .................................. 9.8 Constraints on TeVeS ....................................... 9.9 Constraints on non-local gravity models ............................. 10 Constraints on deviations from GR and MG models from neutron star merger event GW170817/GRB170817A 10.1 Implications for scalar–ensor theories .............................. 10.1.1 Implications for Horndeski models ............................ 10.1.2 Implications for Beyond Horndeski models ........................ 10.2 Implications for vector–tensor theories .............................. 10.3 Implications for massive gravity and bigravity theories ..................... 10.4 Implications for ghost condensates and Horˇava–Lifshitz Gravity ................ 10.5 Implications for higher dimension models ............................ 10.6 Implications for results on MG parameters and large-scale-structure from GW170817 and GRB170817A ........................................... 10.7 Implications for Vainshtein screening mechanism after GW170810 and GRB170817A ..... 10.8 Further notes or caveats on the implications of GW170817 and GRB170817A ......... 11 Computer codes and packages for testing gravity at cosmological scales ............... 11.1 Integrated Software in Testing General Relativity (ISiTGR) .................. 11.2 Modification of growth with CAMB (MGCAMB)andMGCosmoMC ................ 11.3 Horndeski in CLASS (hi_class) ............................... 11.4 Effective field theory with CAMB (EFTCAMB)and(EFTCosmoMC) ............... 11.5 Recent comparison of Einstein–Boltzmann solver codes for testing general relativity ...... 12 Systematic effects in cosmological probes and degeneracies with modifications to GR ........ 13 Future cosmological constraints on GR and MG parameter forecasts ................. 14 Concluding remarks and outlook .................................... References .................................................. 1 Introduction For over a century, Einstein’s general relativity (GR) has continued to be an impressive theory of gravity that fits observations from our solar system to the entire cosmological model of the universe. Guided by some key principles, Einstein came to the important realization of a very close relationship between the curvature of spacetime and gravity. Taking into account further requirements, such as coordinate invariance, conservation laws, and limits that must be consistent with Newtonian gravity, he proposed his grav- itational field equations (Einstein 1915). Astonishingly, the same simple