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3.2 A parameter space for tests of gravity

The two parameters, ε and ξ, define a parameter space on which we can quantify the strengths of the gravitational fields probed by different tests of gravity (see Figure 1View Image). Only a fraction of this parameter space is accessible to experiments. Regions of the parameter space with potential ε > 1 correspond to distances from a gravitating object that are smaller than the horizon radius and are, therefore, inaccessible to observers. (I neglect here, for simplicity, the small numerical factor in the horizon radius that depends on the spin of the black hole.) In Figure 1View Image, this region is outlined by the vertical red line.
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Figure 1: A parameter space for quantifying the strength of a gravitational field. The x-axis measures the potential ε ≡ GM ∕rc2 and the y-axis measures the spacetime curvature ξ ≡ GM ∕r3c2 of the gravitational field at a radius r away from a central object of mass M. These two parameters provide two different quantitative measures of the strength of the gravitational fields. The various curves, points, and legends are described in the text.

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 [7View Equation])

2 ( ) ( )2 ξ ≤ 3G-ΩΛH-0-≃ 5 × 10−58 Ω-Λ- -----H0------- cm −2 , (11 ) 8πc2 0.73 73 km ∕s∕Mpc
where ΩΛ is the current density of dark energy in units of the critical density and H0 is the current value of the Hubble constant. Phenomena that probe such low values of curvature (i.e., below the horizontal green line in Figure 1View Image) can lead to quantitative tests of general relativity only if a specific model of dark energy (e.g., a cosmological constant) is assumed.

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 −8 −2 a0 ≃ 10 cm s [9714014]. (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 a0 ≃ cH0. Systems for which dark matter is necessary to account for their gravitational fields are characterized by

( a0)2 1 (H0 )2 1 ξ ≤ -2- --≃ --- --. (12 ) c ε c ε
This region of the parameter space is outlined by the purple line in Figure 1View Image. The fact that the three lines that correspond to the Schwarzschild horizon, the MOND acceleration scale, and the dark energy all seem to intersect roughly in one point in the parameter space is directly related to the cosmic-coincidence problem, i.e., that the universe is flat, with comparable amounts of (mostly dark) matter and dark energy.

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 M that is comparable to the Compton wavelength λC ≡ h∕M c, where h is Planck’s constant. Quantum effects are, therefore, expected to dominate when

ξ ≥ -1-ε2 , (13 ) L2P
where 3 1∕2 −33 LP ≡ (Gh ∕c ) ≃ 4 × 10 cm is the Planck length. This part of the parameter space is not shown in Figure 1View Image, as it is many orders of magnitude away from the values of the parameters that correspond to astrophysical systems.

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 M will follow curves of the form

c4 3 ξ = --2--2 ε , (14 ) G M
whereas probes at a constant distance r away from the central object will follow curves of the form
ξ = 1-ε . (15 ) r2
Figure 1View Image shows a number of representative contours of constant mass and distance.

The strongest gravitational fields around astrophysical systems can be found in the vicinities of neutron stars (NS in Figure 1View Image) 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 (2 4 ∼ 10 – 10 M ⊙; IMBHs) and in active galactic nuclei (106– 1010M ⊙; 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 1View Image, I have used a typical mass-radius relation for neutron stars and white dwarfs [147], the calculated mass-radius relation of main-sequence stars [34], and the inferred mass-radius profile of the inner region around Sgr A* [143], which smoothly approaches the mass profile inferred from the rotation curve of the Milky Way [46].

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Figure 2: Tests of general relativity placed on an appropriate parameter space. The long-dashed line represents the event horizon of Schwarzschild black holes.

Current tests of general relativity with astrophysical objects probe a wide range of gravitational potentials and curvatures (see Figure 2View Image). 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

-6-( 2) R = α2 αα¨+ ˙α , (16 )
where α is the scale factor. Using the Friedmann equation, the scalar curvature becomes
( 0 ) ( )2 R = 3 Ωm-+ 4Ω0 H0- , (17 ) α3 d c
where Ω0 m and Ω0 d are the (non-relativistic) matter and dark energy densities in the present universe, respectively, in units of the critical density. Equation (17View Equation) shows that, at late times, the radius of curvature of the universe is comparable to the Hubble distance.

The evolution of the scalar curvature with redshift for a flat universe and for the best-fit cosmological parameters obtained by the WMAP mission [156Jump To The Next Citation Point] is shown in Figure 3View Image. Identified on this figure are several characteristic epochs that have been used in testing general relativistic predictions: the z ≃ 1 epoch of type I supernovae that are used to measure the value of the cosmological constant [122136], the z ≃ 1000 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 [14131]. 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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Figure 3: The scalar curvature of our universe, as a function of redshift. The curve corresponds to a flat universe with the best-fit values of the cosmological parameters obtained by the WMAP mission [156]. The arrows point to the curvature and redshift of the universe during various epochs.

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