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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.

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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.

Guided by some key principles, Einstein came to the important realization of a very close relationship between the curvature of spacetime and gravity.

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Taking into account further requirements, such as coordinate invariance, conservation laws, and limits that must be consistent with Newtonian gravity, he proposed his gravitational field equations Einstein Astonishingly, the same simple but powerful equations remain to date the most accurate description of gravitational physics at all scales. Shortly after that, GR gave birth to the current standard model of cosmology predicting exact solutions with expanding or contracting universes.

These models describing the background cosmological evolution were completed by the addition of cosmological perturbation theory to populate them with cosmic structures. Over the years and decades to follow, the FLRW models plus cosmological perturbations benefited from a number of theoretical developments and observational techniques that allowed us to map the whole history of cosmic evolution from very early times to the current stages of the universe as we observe it today.

However, this scientific triumph in cosmology came with two conundrums: dark matter and cosmic acceleration or dark energy. The dark matter problem motivated the introduction of modified gravity MG theories that would explain such observations by a small modification to Kepler laws such as Modified Newtonian Dynamics MOND Milgrom b , its relativistic generalization known as TeVeS tensor—vector—scalar theory Bekenstein , or other vector—tensor theories. While Dark Matter motivated proposals of some MG models, the main focus of this review is rather on models that attempt to address the question of cosmic acceleration that we describe next.

The second problem in standard cosmology is indeed that of cosmic acceleration and the dark energy associated with it. This best fit model is spatially flat. The cosmological constant can be cast into the model as an effective cosmic fluid with an equation of state of minus one. This coincides exactly with the equation of state of the vacuum energy associated with zero-level quantum fluctuations.

Namely, why is the value measured from cosmology so small compared to the one predicted from quantum field calculations? This is known as the old cosmological constant problem. If it were any larger it would have prevented cosmic structure from forming. It is worth noting that most of these dark energy models do not address the cosmological constant problem and may suffer from some form of fine-tuning as well.

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Relevant to our review, the question of cosmic acceleration motivated a number of proposals for modified gravity models that could produce such an acceleration without the need for a cosmological constant. Such models are said to be self-accelerating. In most cases, these models do not address the cosmological constant problems and it is hoped that by some mechanism, for example degravitation or some given cancellation, vacuum energy does not contribute to gravitational and cosmological dynamics.

However, in some cases, modified gravity models do provide some degravitation mechanism, although not successfully so far. We discuss these further in this review. Finally, there are also modified gravity models at high energies that have been motivated by the search for quantum gravity and other unified theories of physics which may or may not have any consequences at cosmological scales. Indeed, just a few years after GR was introduced, Weyl gravity was proposed by Weyl , and so were the theories of Eddington , Cartan b and Brans and Dicke , and many others.

Testing GR and gravity theories within the solar system and using other astrophysical objects have been the subject of intense work with a number of important results over the last five decades or so; see for a review Will Impressively, GR fits all these local tests of gravity. In fact, it fits them so well that these tests are commonly referred to as GR local tests. This is very useful to the current cosmological developments, because it has established very stringent constraints at the level of the solar system that any gravity theory must pass.

Nevertheless, to address these requirements, some MG models have some gravitational screening mechanisms that allow them to deviate from GR at cosmological scales but then become indistinguishable from it at small scales. Further motivation for testing GR at cosmological scales is the increasing quantity and quality of available cosmological data. These are indeed good times for cosmology where a plethora of complementary observational data from ongoing and planned surveys will continue to flow for the many decades to come.

These include the cosmic microwave background radiation, weak gravitational lensing, galaxy surveys, distances to supernovae, baryon acoustic oscillations, and gravitational waves.

A good piece of news is that one can not only combine these data sets to increase their constraining power, but one can also apply consistency tests between such complementary data sets. This would allow one to identify any problems with systematic effects in the data or any problems with the underlying model. Furthermore, nature has also given us a break in cosmology as we have two types of data sets. Indeed, some data sets are sensitive to the geometry and expansion of the universe and some other sets are sensitive to the growth of large-scale structure i.

These two sets of observations must be consistent with one another. Meanwhile, any departure from GR is constrained by using the growth data from large-scale structure observables. There are two general approaches that have been developed to test departures from GR at cosmological scales. The first one is where the deviation is parameterized in a phenomenological way with no necessary exact knowledge of the specific alternative theory. The growth equations are modified by the addition of MG parameters that represent the departure from GR.

It is worth noting that such an effective description may not necessarily remain valid at all scales constrained by observations and so must be used with some caution when compared to various observations. These are then implemented in cosmological analysis software and used to compare to the data.

We cover both approaches in this review.

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A related question is what one could call a modified gravity model versus a dark energy model. There are some guiding helpful prescriptions that we discuss in the review but the spectrum of models has a grey zone where such a distinction is not unambiguous. We characterize various types of deviations from general relativity and organize MG models accordingly with some illustrative examples.

In this review, we aim at providing an overall current picture of the field of testing gravity at cosmological scales including a selection of recent important results on the subject. The review is meant to provide an entry point for students and researchers interested in the field where they can find summaries and references to further readings.

This review can also serve for experienced researchers or other readers to find quickly recent developments or results in the field.

As required for the Living Review guidelines, this review is written with the depth and style of a plenary review talk on the subject. It is not meant to replace thorough comprehensive reviews on various parts of this topic and we refer the reader constantly to such specialized reviews as we discuss each sub-topic. Einstein considered some key guiding principles and well-known limits that a successful theory of gravity must obey.

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At the forefront is the principle of covariance—that is the laws of physics must be independent of any coordinate system. So the right language must be that of tensors or another coordinate independent formulation. Such a successful theory should locally be consistent with special relativity and must inherit its principles including the equivalence of local inertial frames of reference, the universal constancy of the speed of light in vacuum, and the Lorentz-invariance of the theory.

The WEP is expressed as the universality of the gravitational interaction and free-fall for all particles. WEP is satisfied if there exists some spacetime metric in the Jordan frame to which all species of matter are universally coupled. Test particles fall then along geodesics of this metric. Einstein equivalence principle EEP The EEP requires the validity of the WEP, and that in all local freely falling frames, the laws of physics reduce to those of special relativity assuming tidal gravitational forces are absent.

It is also customary to add here that the EEP contains the statement that the outcome of any local non-gravitational experiment is independent of where and when it is performed Will Strong equivalence principle SEP The SEP extends the universality of free fall of the WEP to massive gravitating objects so it is completely independent of the composition of the objects as well as their gravitational binding energy. Compact objects like Black Holes will also fall along geodesics like test particles Will , We conclude this subsection by commenting on a few other notions that guided Einstein in formulating his equations of the gravitational field.

The geometrical nature of GR and the principles it is built upon are certainly far from the Newtonian gravity based on forces and potentials, not to mention the notions of absolute space and other shortcomings that had to be abolished. However, it is interesting to remark that the notion of spacetime and its metric to explain gravity can be compared to the notion of the gravitational potential field in space created by massive objects.

However, there is a major difference, in GR there is no gravitational potential or gravity that is added on the top of spacetime, but gravity is curvature of spacetime itself. This was a major insight that Einstein got from his EEP principle. In fact, he knew well that GR must have Newtonian gravity as a limit in the weak regime and that provided to him many hints on how to formulate the field equations that we provide in the next section.

In addition to the principles above, Einstein used the fact that, in the weak field limit, the gravitational field equations must locally reduce to those of Newtonian gravity where the metric tensor components would be related to the gravitational potential and the field equations must reduce to Poisson equations. From the latter, he imposed that the curvature side of the equations must contain only up to second order derivatives of the metric and must also be of the same tensor rank as the energy-momentum tensor.

This naturally led Einstein to consider the Ricci tensor, derived from contracting twice the Riemann curvature tensor, but there was a little bit more into it. Indeed, he knew that the equations must satisfy conservation laws and thus must be divergence-free.

Cosmology: Methodological Debates in the s and s (Stanford Encyclopedia of Philosophy)

While the vanishing of the divergence of the matter-energy source side of the equations is assured by energy conservation laws and continuity equations, on the curvature side, the Ricci tensor is not divergence-free so more work was required. For that, Einstein built precisely the tensor that holds his name which, by the Bianchi identity, is divergence-free hence complies with conservation laws, as it should.

These solutions are based on symmetries of the spacetime and defined forms of the energy momentum source. Some of these solutions have found direct applications to real astrophysical systems. These include the popular Schwarzschild static spherically symmetric vacuum solution around a concentric mass Schwarzschild The solution is often used to model space around Earth, Sun, or other slowly rotating objects where it leads to more accurate predictions than Newtonian gravity, see e.

The solution is also used to represent the exterior spacetime around a static spherically symmetric black hole. A second well-know exact solution is that of Kerr representing the vacuum space around an axially symmetric rotating compact object or black hole. Next, several other static spherically symmetric non-vacuum solutions such as the Tolman family of solutions Tolman and the Buchdahl solutions Buchdahl have been used to model the interior of compact astrophysical objects such as Neutron stars Lattimer and Prakash Finally, some solutions have found applications in cosmology.

Finally, with regards to this review, it is worth clarifying that modifications to GR mean also that the above exact solutions are not anymore valid and need to be replaced by their homologous solutions in the modified theory. For example, in relation to cosmic acceleration, the cosmological constant term can be replaced by extra terms coming from the modification and that could play a similar role to it.

However, as we already mentioned in the introduction, some of these models are able to fit well the expansion and background observations so any further distinction will have to come from the growth of structure constraints and observables. This is the case for a cosmological constant. The field equations of GR have no difficulty in mathematically producing an accelerated expansion, but the real challenge is to figure out what is the physical nature of such an effective dark energy fluid.

The universe we observe at large scales is rather full of clusters and superclusters of galaxies. Sufficiently large scales are considered so linear perturbations are a valid description. This is done by adding to the metric tensor a small perturbation tensor. Then computing the Einstein tensor to the first order. At the same time, the energy momentum tensor is also linearly perturbed. The Einstein equations then give the usual background Friedmann equations plus additional equations governing the evolution of the perturbations see, e.

An insightful approach to these linear perturbations is to decompose the components of the symmetric metric tensor perturbations according to how they transform under spatial rotations. The component of the metric perturbation tensor is a scalar, the three 0i-components or equally the three i0-components constitute a vector, and the remaining nine ij components form a symmetric spatial tensor of rank two. This is known as the SVT decomposition of linear perturbations.

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The three parts transform only into components of the same type under spatial rotations. In GR, the scalar modes are, for example, associated with matter density fluctuations and used for large scale structure studies, tensor modes are associated with gravitational radiation used, for example, for primordial gravitational waves, while vector modes decay in and are usually ignored. Last, in addition to this decomposition, one needs to specify a gauge choice where the components of the perturbations can be different in the corresponding coordinate system, see e. Modification to gravity can be implemented at the level of scalar mode perturbations as we discuss further below or at the level of tensor modes as in, e.