the radius a and the average scalar field ¯
φ, but to first order in the perturbations. One gets
a series of Schrödinger equations for the rate of change of the perturbation wave functions
with respect to the time coordinate of the background metric.
Schr¨
odinger Equations
∂Ψ( dn)
1
i
=
− ∂ 2 + n 2 d 2
Ψ( dn)
etc
∂t
2 a 3
∂d 2
na 4
n
One can use the no boundary condition to obtain initial conditions for the perturbation
wave functions. One solves the field equations for a small but slightly distorted three
sphere. This gives the perturbation wave function in the exponentially expanding period.
One then can evolve it using the Schrödinger equation.
The tensor harmonics which correspond to gravitational waves are the simplest to
consider. They don’t have any gauge degrees of freedom and they don’t interact directly
with the matter perturbations. One can use the no boundary condition to solve for the
initial wave function of the coefficients dn of the tensor harmonics in the perturbed metric.
Ground State
Ψ( d
na 2 d 2
ωx 2
n)
∝ e− 12
n = e− 12
3
n
where x = a 2 dn and
ω = a
One finds that it is the ground state wave function for a harmonic oscillator at the fre-
quency of the gravitational waves. As the universe expands the frequency will fall. While
the frequency is greater than the expansion rate ˙ a/a the Schrödinger equation will allow the wave function to relax adiabatically and the mode will remain in its ground state. Eventually, however, the frequency will become less than the expansion rate which is roughly
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constant during the exponential expansion. When this happens the Schrödinger equation will no longer be able to change the wave function fast enough that it can remain in the
ground state while the frequency changes. Instead it will freeze in the shape it had when
the frequency fell below the expansion rate.
end of
inflation
wavelength/
radius
wavelength of perturbations
perturbations
on radius come back within
horiz
horizon radius
perturbation becomes greater
than the horizon radius
wave function frozen
time
adiabatic
evolution
After the end of the exponential expansion era, the expansion rate will decrease faster
than the frequency of the mode. This is equivalent to saying that an observers event
horizon, the reciprocal of the expansion rate, increases faster than the wave length of the
mode. Thus the wave length will get longer than the horizon during the inflation period
and will come back within the horizon later on. When it does, the wave function will still
be the same as when the wave function froze. The frequency, however, will be much lower.
The wave function will therefore correspond to a highly excited state rather than to the
ground state as it did when the wave function froze. These quantum excitations of the
gravitational wave modes will produce angular fluctuations in the microwave background
whose amplitude is the expansion rate (in Planck units) at the time the wave function
froze. Thus the COBE observations of fluctuations of one part in 105 in the microwave
background place an upper limit of about 10 − 10 in Planck units on the energy density when the wave function froze. This is sufficiently low that the approximations I have used
should be accurate.
However, the gravitational wave tensor harmonics give only an upper limit on the
density at the time of freezing. The reason is that it turns out that the scalar harmonics
give a larger fluctuation in the microwave background. There are two scalar harmonic
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degrees of freedom in the three metric hij and one in the scalar field. 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