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 Pg(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
Tl 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 ~
3
M / 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.