Occupation Number and Energy Spectrum

In terms of the conformal variables of the previous section it makes sense to define occupation number as

$$n_k \equiv {1 \over 2}\left(\omega_k \vert\tilde{F}_{k,c}\vert^2 + {1 \over \omega_k} \vert\tilde{F}_{k,c}'\vert^2\right).$$ (5.18)

Note that this quantity is adiabatically invariant, meaning it is conserved in the limit ${\omega_k' \over \omega_k^2} \ll 1$. Note also that because it is defined in terms of $\tilde{F}$ instead of $F_k$, $n_k$ is unitless.

The energy density $\rho_k$ is defined as

$$\rho_k \equiv \omega_k n_k = {1 \over 2}\left(\omega_k^2 \vert\tilde{F}_{k,c}\vert^2 + \vert\tilde{F}_{k,c}'\vert^2\right).$$ (5.19)

To convert these definitions back to physical coordinates note that

$$a' = a \dot{a};\;a'' = a^2 \ddot{a} + a \dot{a}^2;\;\tilde{F... ...a' \tilde{F}_k = a^2 \dot{\tilde{F}}_k + a \dot{a} \tilde{F}_k$$ (5.20)

so

$$n_k ={1 \over 2}\left( a^2 \omega_k \vert\tilde{F}_k\vert^2 ... ...dot{\tilde{F}}_k + {\dot{a} \over a} \tilde{F}_k\vert^2\right)$$ (5.21)

$$\rho_k ={1 \over 2}\left( a^2 \omega_k^2 \vert\tilde{F}_k\ve... ...dot{\tilde{F}}_k + {\dot{a} \over a} \tilde{F}_k\vert^2\right)$$ (5.22)

$$\omega_k^2 = k^2 + a^2 \left<{\partial^2 V \over \partial f^2}\right> - a \ddot{a} - \dot{a}^2.$$ (5.23)

Finally, in terms of the discrete Fourier transform $f_k$

$$n_k = {a^2 dx^6 \over 2 L^3} \left[\omega_k \vert f_k\vert^2... ...omega_k} \vert\dot{f}_k + {\dot{a} \over a} f_k\vert^2\right].$$ (5.24)

$$\rho_k = {a^2 dx^6 \over 2 L^3} \left[\omega_k^2 \vert f_k\vert^2 + a^2 \vert\dot{f}_k + {\dot{a} \over a} f_k\vert^2\right].$$ (5.25)