A Phase Space Diagram for Gravity

In modified theories of gravity including a critical acceleration scale, $a_{0}$, a critical length scale, $r_{M}=(GM/a_{0})^{1/2}$, will naturally arise, with the transition from the Newtonian to the dark matter mimicking regime occurring for systems larger than $r_{M}$. This adds a second critical scale to gravity, in addition to the one introduced by the criterion $v<c$ of the Shwarzschild radius, $r_{S}=2GM/c^{2}$. The distinct dependencies of the two above length scales give rise to non-trivial phenomenology in the (mass, length) plane for astrophysical structures, which we explore here. Surprisingly, extrapolation to atomic scales suggests gravity should be at the dark matter mimicking regime there.

From a cosmological perspective, the qualitative similarity between the early inflationary phase and 20 the current late accelerated expansion phase, has been interpreted as evidence for a common physical ori-21 gin for both, in terms of modified gravity, [7]. This approach has been extensively explored over the past 22 years by several authors, who have now showed the consistency of the proposal with all global expansion 23 history observations, for a variety of extensions to general relativity e.g. [ [14]. The connection between such approaches and dark matter inferences at galactic dynamics level has 25 also been explored for the case of F (R) modifications to general relativity by e.g. [15], [16], [17]. 26 Very recently, independent observations for three distinct types of astrophysical systems have severely below a 0 . Finally, [22] showed that the inferred infall velocity of the bullet cluster is inconsistent with tations or not. A general consistency check for the gravitational interpretation of astrophysical dynamics is found in that not a single high acceleration system (a > a 0 ) is known where dark matter is required, 62 and conversely, not a single low acceleration system (a < a 0 ) is known where dark matter is not required, 63 when interpreting observations under Newtonian gravity. An exception to either of the two above rules 64 would seriously challenge many of the modified theories of gravity currently under consideration.
Going back to the usual hierarchy r S << r < r M or r S << r M < r for astrophysical objects in the 66 Newtonian or dark matter mimicking regimes, we note that the distinct mass scalings of r S and r M imply 67 that at sufficiently large masses the situation r M < r S could arise. This leads to non-trivial structure in 68 the (mass, radius) plane for astrophysical objects, which we explore across 25 orders in magnitude in   74 We begin by examining the distinct dependencies of the two critical length scales which will appear 75 in any covariant theory of gravity aiming at explaining the observed astrophysical phenomenology at 76 galactic scales, without invoking dark matter:

A Gravitational Phase Space Diagram
and It is now obvious that a critical dimensionless parameter of the problem will be the ratio of the above 79 two radii, b = r S /r M . This parameter will be very small for most astrophysical objects. Whilst r S scales 80 with M, r M scales only with M 1/2 . This implies a reversal of the accustomed hierarchy r S << r M 81 into r M < r S at sufficiently large masses, when b will transit from b < 1 to b > 1, with a critical point 82 appearing at b = 1.

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To better appreciate the distinct regions which will appear in the (mass, radius) plane, we plot figure   84 easily explained by the appearance of non Newtonian effects outside their core regions, in consistency 95 with the approach of the a = a 0 threshold.

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Within this Newtonian wedge region, increasing the mass or reducing the radius drives a system into 97 the relativistic region, and then into the black hole regime. Conversely, reducing the mass or increas-98 ing the radius shifts an object from the Newtonian regime into the dark matter mimicking region, for 99 example, in going from globular clusters to dwarf spheroidal galaxies (dSphs), objects with comparable  . The approach to the former from below, signals the relativistic region, whilst the approach to latter from the left, denotes the transition from the Newtonian to the dark matter mimicking regime. The labels identify the regions occupied by different astrophysical objects; the solar system, SS, stars, S, wide binaries, WB, globular clusters, GC, dwarf spheroidal galaxies, dSph, elliptical galaxies, E, spiral galaxies, S Gal and galaxy clusters, GaC. Distinct regions of the diagram are labelled; black holes, BH, appearance of relativistic effects, GR, the Newtonian region, N, the modified gravity regime, M, and the critical density of the universe, or the dark energy density, coinciding with the critical point b=1 where r S = r M .
horizontal dashed line. We see that galaxy clusters lie very close to this line, in fact, dispersion velocities 131 in clusters of galaxies often exceed 1000km/s, much more than the values of around 50km/s of the 132 orbit of Mercury, where relativistic effects begin to become apparent. This alerts to the fact that galaxy 133 clusters probably present non-negligible relativistic effects, and can not be treated under non-relativistic 134 modified gravity schemes. This appears obvious from the region occupied by galaxy clusters in figure   135 (1), only slightly below the horizontal dotted line mentioned.

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In view of the above, it is probably more correct to think of the relativistic regime, which within the  gravity ideas, implies gravity at atomic scales will be at the dark matter mimicking regime.

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One should therefore expect that at atomic scales, a test mass in the presence of a much larger mass 150 M, will experience a gravitational attraction several thousand times larger than the Newtonian prediction.

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This remains many orders of magnitude below the electromagnetic effects, which obviously still largely 152 dominate. However, if such effects can be accounted for, a residual force per unit mass should appear 153 given by: with a corresponding potential Φ = (c 2 b/2)ln(r/r S ), where r S has been introduced for dimensional 155 consistency. We see again the critical parameter b appearing. This force will be several orders of mag-156 nitude larger than the Newtonian value. It is important to notice that this prediction is generic to many The disappearance of the Newtonian region for masses slightly above galactic cluster scales identifies 166 a limit above which low velocity dark matter mimicking phenomenology can transit into its relativistic 167 regime, without an intermediary Newtonian region.

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The coincidence of the critical mass and radius at this point with the critical density of the universe 169 could be interpreted as a clue towards understanding the recent appearance of the accelerated expansion 170 of the Universe, within the framework of modified theories of gravity in general.

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In going to the smallest scales available to direct experimentation, we see that a prediction appears, in 172 the form of gravity at atomic level being decidedly at the dark matter mimicking regime. This constitutes 173 an exciting prediction for future micro-gravity experiments. 174