Page 56 - Nature Of Space And Time

degrees of freedom in the three metric h ij and one in the scalar eld. However two of these
scalar degrees correspond to coordinate freedom. Thus there is only one physical scalar
degree of freedom and it corresponds to density perturbations.
The analysis for the scalar perturbations is very similar to that for the tensor harmon-
ics if one uses one coordinate choice for the period up to the wave function freezing and

another after that. In converting from one coordinate system to the other, the amplitudes
get multiplied by a factor of the expansion rate divided by the average rate of change
of phi. This factor will depend on the slope of the potential, but will be at least 10 for

reasonable potentials. This means the
uctuations in the microwave background that the
density perturbations produce will be at least 10 times bigger than from the gravitational
waves. Thus the upper limit on the energy density at the time of wave function freezing
is only 10 −12 of the Planck density. This is well within the range of the validity of the
approximations I have been using. Thus it seems we don't need string theory even for the

beginning of the universe.
The spectrum of the
uctuations with angular scale agrees within the accuracy of the
present observations with the prediction that it should be almost scale free. And the size
of the density perturbations is just that required to explain the formation of galaxies and

stars. Thus it seems the no boundary proposal can explain all the structure of the universe
including little inhomogeneities like ourselves.
One can think of the perturbations in the microwave background as arising from
thermal
uctuations in the scalar eld . The in
ationary period has a temperature

of the expansion rate over 2 because it is approximately periodic in imaginary time.
Thus, in a sense, we don't need to nd a little primordial black hole: we have already
observed an intrinsic gravitational temperature of about 10 26 degrees, or 10 −6 of the Planck
temperature.

COBE predictions plus upper limit on energy density
⇒
gravitational wave perturbations 10 −10 Planck density
upper limit on energy density
plus density perturbations ⇒
10 −12 Planck density
intrinsic gravitational 10 −6 Planck temperature
≈
temperature of early universe =10 26 degrees

What about the intrinsic entropy associated with the cosmological event horizon. Can

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