Page 54 - Nature Of Space And Time

Tensor harmonics - Gravitational waves
Vector harmonics - Gauge
Scalar harmonics - Density perturbations

One can then expand the Wheeler-DeWitt equation for the wave function to all orders in

the radius a and the average scalar eld , but to rst order in the perturbations. One gets
aseries of Schrodinger equations for the rate of change of the perturbation wave functions

with respect to the time coordinate of the background metric.

Schrodinger Equations

2
@ (d n) 1 @
2 2 4
i = − + n d a (d n )etc
@t 2a 3 @d 2 n
n

One can use the no boundary condition to obtain initial conditions for the perturbation

wave functions. One solves the eld 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 Schrodinger 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 coe cients d n of the tensor harmonics in the perturbed metric.

Ground State

2 2
1
1
(d n) ∝ e − na d n = e − !x 2
2
2
3 n
where x = a d n and ! =
2
a

One nds 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 Schrodinger equation will allow the
wave function to relax adiabatically and the mode will remain in its ground state. Even-
tually, however, the frequency will become less than the expansion rate which is roughly

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