The Scale Factor Equation

The equation for the scale factor $a$ is derived from the Friedmann equations

$$\ddot{a} = -{4 \pi a \over 3} (\rho + 3 p)$$ (6.19)

$$\left({\dot{a} \over a}\right)^2 = {8 \pi \over 3} \rho$$ (6.20)

For a set of scalar fields $f_i$ in an FRW universe

$$\rho = T + G + V;\;p = T - {1 \over 3} G - V$$ (6.21)

where $T$, $G$, and $V$ are kinetic (time derivative), gradient, and potential energy respectively with

$$T = {1 \over 2} \dot{f}_i^2;\;G = {1 \over 2 a^2} \vert \nabla f_i\vert^2.$$ (6.22)

Equations (6.19) and (6.20) and the field evolution equations form an overdetermined system. In principle either scale factor equation could be used but in practice it is easiest to combine them so as to eliminate the time derivative term $T$ because in the staggered leapfrog algorithm $f$ and $\dot{f}$ are known at different times. Eliminating $T$ we get

$$T = {3 \over 8 \pi}\left({\dot{a} \over a}\right)^2 - G - V$$ (6.23)

$$\ddot{a} = -{4 \pi a \over 3} (4 T - 2 V) = -2 {\dot{a}^2 \o... ...er a}\left({1 \over 3} \vert \nabla f_i\vert^2 + a^2 V\right).$$ (6.24)

To convert to program variables note that

$$\dot{a} = B a^s a';\;\ddot{a} = B^2\left(a^{2s} a'' + s a^{2s-1} a'^2\right),$$ (6.25)

so the scale factor equation becomes

$$a'' = (-s-2){a'^2 \over a} + {8 \pi \over A^2} a^{-2s-2r-1} ... ... 3} \vert\nabla_{pr} f_{i,pr}\vert^2 + a^{2s+2} V_{pr}\right).$$ (6.26)