Quantifying deviations from general relativity for part of the parameter space requires a detailed understanding of the properties of dark matter and dark energy, which is beyond current capabilities. In the limit of very small values of the curvature, the presence of a non-zero cosmological constant affects the outcome of gravitational experiments when (see Equation )
The ability to perform a quantitative test of a gravity theory also relies on an independent measurement of the mass that generates the gravitational field. This is not always possible, especially in various cosmological settings, where gravitational phenomena are used mostly to infer the presence of dark matter and not to test general relativistic predictions. Dark matter is typically required in systems for which the acceleration drops below the so-called MOND acceleration scale [97, 140, 14]. (This is an observed fact, independent of whether the inability of Newtonian gravity to account for observations is due to the presence of dark matter or to the breakdown of the theory itself.) This acceleration scale is also comparable to . Systems for which dark matter is necessary to account for their gravitational fields are characterized by
In the opposite limit of very strong gravitational fields, general relativity is expected to break down when quantum effects become impossible to neglect. This is expected to happen if a gravitational test probes a distance from an object of mass that is comparable to the Compton wavelength , where is Planck’s constant. Quantum effects are, therefore, expected to dominate when
Having defined the parameter space and outlined the various limiting cases, I can now identify the astrophysical systems that probe its various regimes. In general, systems of constant central mass will follow curves of the form
The strongest gravitational fields around astrophysical systems can be found in the vicinities of neutron stars (NS in Figure 1) and black holes in X-ray binaries (XRB). Large gravitational potentials but smaller curvatures can be found around the horizons of intermediate-mass black holes (; IMBHs) and in active galactic nuclei (; AGN). Weaker gravitational fields exist near the surfaces of white dwarfs (WD), main-sequence stars (MS), or at the distances of the various planets in our solar system (SS). Finally, even weaker gravitational fields are probed by observations of the motions of stars in the vicinity of the black hole in the center of the Milky Way (Sgr A*), and by studies of the rotational curve of the Milky Way (MW) and other galaxies. In placing the various systems on the parameter space shown in Figure 1, I have used a typical mass-radius relation for neutron stars and white dwarfs , the calculated mass-radius relation of main-sequence stars , and the inferred mass-radius profile of the inner region around Sgr A* , which smoothly approaches the mass profile inferred from the rotation curve of the Milky Way .
Current tests of general relativity with astrophysical objects probe a wide range of gravitational potentials and curvatures (see Figure 2). However, they fall short of probing the most extreme phenomena that are predicted by the theory to occur in the vicinities of compact objects, for example, tests during solar eclipses, with double neutron stars (such as the Hulse–Taylor pulsar), or with Grav Prob B probe curvatures that are the same as those found near the horizons of supermassive black holes, but potentials that are smaller by six to ten orders of magnitude. Moreover, all these tests probe curvatures that are smaller by thirteen or more orders of magnitude from those found near the surfaces of neutron stars and the horizons of stellar-mass black holes. Future experiments, such as the gravitational wave detectors and the Beyond Einstein missions, will offer for the first time the opportunity to probe directly such strong gravitational fields.
The whole range of gravitational fields, from the weakest to the strongest, can also be found during various epochs of the evolution of the universe. As a result, observations of cosmological phenomena may also probe very strong gravitational fields. The scalar curvature of a flat universe is given by
The evolution of the scalar curvature with redshift for a flat universe and for the best-fit cosmological parameters obtained by the WMAP mission  is shown in Figure 3. Identified on this figure are several characteristic epochs that have been used in testing general relativistic predictions: the epoch of type I supernovae that are used to measure the value of the cosmological constant [122, 136], the epoch at which the acoustic peaks of the cosmic microwave background observed by WMAP are produced, and the period of nucleosynthesis during which the temperature of the universe was in the range 60 keV – 1 MeV [141, 31]. The period of Big-Bang nucleosynthesis is the earliest epoch for which quantitative tests have been performed. The corresponding scalar curvature of the universe at that time, however, is still small and comparable to the curvatures of gravitational fields probed by current tests of general relativity in the solar system. It was only when the temperature of the universe was 100 GeV that its curvature was 10–12 cm–2, i.e., comparable to that found around a neutron star or stellar-mass black hole. This is the period of electroweak baryogenesis, for which no detailed theoretical models or data exist to date.
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