Program Variables

In general the program uses neither physical nor conformal coordinates but a run-specific set of rescalings defined as

$$f_{pr} \equiv A a^r f;\;dt_{pr} \equiv B a^s dt;\;x_{pr} \equiv B x.$$ (5.26)

(See section 6.1.) Noting that $k$ scales like $1/x$ and

$$\dot{a} = B a^s a';\;\ddot{a} = B^2 \left(a^{2 s} a'' + s a^{2 s - 1} a'^2\right)$$ (5.27)

$$f_k = {a^{-r} \over A} f_{k,pr};\;\dot{f}_k = B a^s f_k' = {... ...r A} \left(a^{s-r} f_{k,pr}' - r a^{s-r-1} a' f_{k,pr}\right),$$ (5.28)

the equations for $n_k$, $\rho_k$, and $\omega_k$ become

$$n_k = {a^{2 - 2 r} \over A^2 B^3} {dx_{pr}^6 \over 2 L_{pr}^... ...k} \vert f_{k,pr}' + (1-r) {a' \over a} f_{k,pr}\vert^2\right]$$ (5.29)

$$\rho_k = {a^{2 - 2 r} \over A^2 B^3} {dx_{pr}^6 \over 2 L_{p... ...^2 \vert f_{k,pr}' + (1-r) {a' \over a} f_{k,pr}\vert^2\right]$$ (5.30)

$$\omega_k^2 = B^2\left(k_{pr}^2 + a^{2 + 2 r} {A^2 \over B^2}... ..._{pr}^2}\right> - a^{2 s + 1} a'' - (1+s) a^{2 s} a'^2\right).$$ (5.31)

Using the additional definitions (sections 6.1.1 and 6.3.2)

$$V_{pr} \equiv {A^2 \over B^2} a^{-2 s + 2 r} V;\;m_{pr}^2 \e... ...al f_{pr}^2}\right>;\;\omega_{k,pr} \equiv {\omega_k \over B},$$ (5.32)

we bring the equations to their final forms

$$n_k = {a^{2 - 2 r} \over A^2 B^2} {dx_{pr}^6 \over 2 L_{pr}^... ...}} \vert f_{k,pr}' + (1-r) {a' \over a} f_{k,pr}\vert^2\right]$$ (5.33)

$$\rho_k = {a^{2 - 2 r} \over A^2 B} {dx_{pr}^6 \over 2 L_{pr}... ...s} \vert f_{k,pr}' + (1-r) {a' \over a} f_{k,pr}\vert^2\right]$$ (5.34)

$$\omega_{k,pr}^2 = k_{pr}^2 + m_{pr}^2 - a^{2 s + 1} a'' - (1+s) a^{2 s} a'^2.$$ (5.35)