13. The gravitational inverse square law

The inverse square law for gravity determines the relation between the mass distribution and the gravitationally-driven peculiar velocities that enter estimates of the matter density parameter _M0. The peculiar velocities also figure in the evolution of the mass distribution, and hence the relation between the present mass fluctuation spectrum and the spectrum of cosmic microwave background temperature fluctuations imprinted at redshift z ~ 1000. We are starting to see demanding tests of both aspects of the inverse square law.

We have a reasonably well checked set of measurements of the apparent value of _M0 on scales ranging from 100 kpc to 10 Mpc (as reviewed under test [7]). Most agree with a constant value of the apparent _M0, within a factor of three or so. This is not the precision one would like, but the subject has been under discussion for a long time, and, we believe, is now pretty reliably understood, within the factor of three or so. If galaxies were biased tracers of mass one might have expected to have seen that _M0 increases with increasing length scale, as the increasing scale includes the outer parts of extended massive halos. Maybe that is masked by a gravitational force law that decreases more rapidly than the inverse square law at large distance. But the much more straightforward reading is that the slow variation of _M0 sampled over two orders of magnitude in length scale agrees with the evidence from tests (7) to (10) that galaxies are useful mass tracers, and that the inverse square law therefore is a useful approximation on these scales.

The toy model in Eq. (57) illustrates how a failure of the inverse square law would affect the evolution of the shape of the mass fluctuation power spectrum P(k, t) as a function of the comoving wavenumber k, in linear perturbation theory. This is tested by the measurements of the mass and cosmic microwave background temperature fluctuation power spectra. The galaxy power spectrum P_g(k) varies with wavenumber at k ~ 0.1h Mpc^-1 about as expected under the assumptions that the mass distribution grew by gravity out of adiabatic scale-invariant initial conditions, the mass is dominated by dark matter that does not suffer radiation drag at high redshift, the galaxies are useful tracers of the present mass distribution, the matter density parameter is _M0 ~ 0.3, and, of course, the evolution is adequately described by conventional physics (Hamilton and Tegmark, 2002, and references therein). If the inverse square law were significantly wrong at k ~ 0.1h Mpc^-1, the near scale-invariant form would have to be an accidental effect of some failure in this rather long list of assumptions. This seems unlikely, but a check certainly is desirable. We have one, from the cosmic microwave background anisotropy measurements. They also are consistent with near scale-invariant initial conditions applied at redshift z ~ 1000. This preliminary check on the effect of the gravitational inverse square law applied on cosmological length scales and back to redshift z ~ 1000 will be improved by better understanding of the effect on T_l of primeval tensor perturbations to spacetime, and of the dynamical response of the dark energy distribution to the large-scale mass distribution.

Another aspect of this check is the comparison of values of the large-scale rms fluctuations in the present distributions of mass and the cosmic microwave background radiation. The latter is largely set at decoupling, after which the former grows by a factor of about 10³ to the present epoch, in the standard relativistic cosmological model. If space curvature is negligible the growth factor agrees with the observations to about 30%, assuming galaxies trace mass. In a low density universe with = 0 the standard model requires that mass is more smoothly distributed than galaxies, N / N ~ 3M / M, or that the gravitational growth factor since decoupling is a factor of three off the predicted factor ~ 1000; this factor of three is about as large a deviation from unity as is viable. We are not proposing this interpretation of the data, rather we are impressed by the modest size of the allowed adjustment to the inverse square law.