© CAMBRIDGE UNIVERSITY PRESS 1999

1.2.2 Is the Gravitational Force r^-1 at Large r?

Back to the question whether our conventional theory of gravity is trustworthy on large scales. The reason for raising this question is that interpreting modern observations within the context of the standard theory leads to the conclusion that at least 90% of the matter in the universe is dark. Moreover, there is no observational confirmation that the gravitational force falls as r^-2 on large scales.

Tohline (1983) pointed out that a modified gravitational force law, with the gravitational acceleration given by a' = (G M_lum r²) (1 + r/d), could be an alternative to dark matter galactic halos as an explanation of the constant-velocity rotation curves of spiral galaxies. The mass is written as M_lum to emphasize that there is not supposed to be any dark matter.) Indeed, this equation implies v² = G M_lum / d = constant for r >> d. However, with the distance scale d where the force shifts from r^-2 to r^-1 taken to be a physical constant, the same for all galaxies, this implies that M_lum v², whereas observationally M_lum L_B v with ~ 4 (``Tully-Fisher law''), where L_B is the galaxy luminosity in the blue band.

Milgrom (1983, 1994, 1995; cf. Mannheim & Kazanas 1994) proposed an alternative idea, that the separation between the classical and modified regimes is determined by the value of the gravitational acceleration a' rather than the distance scale r. Specifically, Milgrom proposed that

(1.3)

where the value of the critical acceleration a'₀ 8 x 10^-8 h² cm s^-2 is determined for large spiral galaxies with M_lum ~ 10¹¹ M. (This value for a'₀ happens to be numerically approximately equal to c H₀.) This equation implies that v⁴ = a'₀ G M_lum for a' << a'₀, which is now consistent with the Tully-Fisher law.

Although Milgrom's proposed modifications of gravity are consistent with a large amount of data, they are entirely ad hoc. Also, not only do they not predict the gravitational lensing by galaxies and clusters that is observed, it has recently been shown (Bekenstein & Sandars 1994) that in any theory that describes gravity by the usual tensor field of GR plus one or more scalar fields, the bending of light cannot exceed that predicted by GR just including the actual matter. Thus, if Milgrom's nonrelativistic theory, in which by assumption there is no dark matter, were extended to any scalar-tensor gravity theory, the light bending could only be due to the visible mass. However, the evidence is becoming increasingly convincing that the mass indicated by gravitational lensing in clusters of galaxies is at least as large as that implied by the velocities of the galaxies and the temperature of the gas in the clusters (see e.g. Wu & Fang 1996).

Moreover, it is difficult to fit an r^-1 force law into the larger framework of either cosmology or theoretical physics. The cosmological difficulty is that an r^-1 force never saturates: distant masses are more important than nearby masses. Regarding theoretical physics, all one needs to assume in order to get the weak-field limit of general relativity is that gravitation is carried by a massless spin-two particle (the graviton): masslessness implies the standard r^-2 force, and then spin two implies coupling to the energy-momentum tensor (Weinberg 1965). In the absence of an intrinsically attractive and plausible theory of gravity which leads to a r^-1 force law at large distances, it seems to be preferable by far to assume GR and take dark matter seriously, as done below. But until the nature of the dark matter is determined - e.g., by discovering dark matter particles in laboratory experiments - it is good to remember that there may be alternative explanations for the data.