Vector Einstein Equations

  The vector metric source evolution is similarly constructed from a ``Poisson'' equation: in the generalized Newtonian gauge

$$\dot V + 2 {\dot a \over a} V = -8\pi G a^2 (p_f \pi^{(1)}_f + \pi_s^{(1)})/k \, ,$$ (44)

and for the synchronous gauge,

$$\ddot h_V + 2 {\dot a \over a} \dot h_V = -8\pi G a^2 (p_f \pi^{(1)}_f + \pi_s^{(1)})/k^2 \, .$$ (45)

Likewise momentum conservation implies the Euler equation  

$$\dot v_f^{(1)}= - (1 - 3c_f^2 ) {\dot a \over a} v^{(1)}_f -... ...er 2 } k {w_f \over 1+w_f} (1 - 2K/k^2)\pi^{(1)}_f + S_v^{(1)};$$ (46)

where recall $c_f^2=\dot{p}_f/\dot{\rho}_f$ is the sound speed and the gravitational sources are

$$S_v^{(1)} = \cases { \dot V + (1-3c_f^2) {\displaystyle{\dot a \over a}} V\,, & (Newtonian), \cr 0\,, & (synchronous). \cr}$$ (47)

The seed Euler equation is given by

$$\dot v_s^{(1)}= - 4{\dot a \over a} v_s^{(1)}- {1 \over 2}k(1-2K/k^2)\pi_s^{(1)},$$ (48)

Again, the first of these equations remains true for each fluid individually save for momentum exchange terms. The baryon Euler equation has an additional term in the source of the same form as Eq. (A12) with $m=0 \rightarrow m=1$.