Conformal Coordinates

The equation of motion for a scalar field in an expanding universe is

$$\ddot{f} + 3 {\dot{a} \over a} \dot{f} - {1 \over a^2} \nabla^2 f + {\partial V \over \partial f} = 0.$$ (5.10)

To define an adiabatic invariant occupation number for this field we need to switch to conformal variables in which this equation becomes a more standard oscillator equation. These variables are

$$f_c \equiv a f;\;dt_c \equiv {1 \over a} dt.$$ (5.11)

Using conformal time and noting that

$$\dot{f} = {1 \over a} f';\;\ddot{f} = {1 \over a^2} f'' - {a' \over a^3} f'$$ (5.12)

the equation of motion becomes

$$f'' + 2 {a' \over a} f' - \nabla^2 f + a^2 {\partial V \over \partial f} = 0.$$ (5.13)

Then switching to conformal field values note that

$$f' = {1 \over a} f_c' - {a' \over a^2} f_c;\;f'' = {1 \over ... ...\over a^2} f_c' + 2 {a'^2 \over a^3} f_c - {a'' \over a^2} f_c$$ (5.14)

and thus

$$f_c'' - {a'' \over a} f_c - \nabla^2 f_c + a^4 {\partial V \over \partial f_c} = 0.$$ (5.15)

This equation can be approximated in Fourier space by

$$\tilde{F}_{k,c}'' + \omega_k \tilde{F}_{k,c} = 0$$ (5.16)

where $\tilde{F}_{k,c} = a \tilde{F}_k$ and

$$\omega_k^2 \equiv k^2 + a^4 \left<{\partial^2 V \over \parti... ...left<{\partial^2 V \over \partial f^2}\right> - {a'' \over a}.$$ (5.17)