Scalar Einstein Equations

  With the form of the scalar metric and stress energy tensor given in Eqs. (A4) and (15), the ``Poisson'' equations become in the Newtonian gauge  

$$\begin{array} {rcl} (k^2-3K) \Phi &=& 4\pi G a^2 \left[ (\rh... ...0)}\right), \vphantom{\displaystyle{\dot a \over a}}\end{array}$$ (38)

and in the synchronous gauge

$$\begin{array} {rcl}(k^2 - 3K)(h_L + {1 \over 3} h_T) + 3{\do... ...T) &=& -8\pi Ga^2 [p_f \pi_f^{(0)}+ \pi_s^{(0)}]\,.\end{array}$$

Two out of four of the synchronous gauge equations are redundant.

The corresponding evolution of the matter is given by covariant conservation of the stress energy tensor $T_{\mu\nu}$:  

$$\begin{array} {rcl}\dot \delta_f & = & -(1+w_f)kv_f^{(0)}- 3... ...over 3}(1-3K/k^2)\pi_f^{(0)}\right] + S^{(0)}_v \, ,\end{array}$$

for the fluid part. The gravitational sources are

$$\begin{array} {lll} S_\delta = -3(1+w_f)\dot \Phi\,, \quad& ... ... = 0\,,\qquad & (\rm synchronous) \vphantom{\Big[}.\end{array}$$ (39)

These equations remain true for each fluid individually in the absence of momentum exchange, e.g. for the cold dark matter. The baryons have an additional term to the Euler equation due to momentum exchange from Compton scattering with the photons. For a given velocity perturbation the momentum density ratio between the two fluids is  

$$R \equiv {\rho_B+p_B \over \rho_\gamma + p_\gamma} \approx {3\rho_B \over 4\rho_\gamma} \, .$$ (40)

A comparison with photon Euler equation (33; $\ell=1$) gives the source modification for the baryon Euler equation  

$$S_v^{(0)} \rightarrow S_v^{(0)} + {\dot\tau \over R} (\Theta_1^{(0)}- v_B^{(0)})\, .$$ (41)

For a seed source, the conservation equations become

$$\begin{array} {rcl}\dot \rho_s &=& -3{\dot a \over a} (\rho... ...[ p_s -{2 \over 3} (1-3K/k^2)\pi_s^{(0)}\right]\, ,\end{array}$$

independent of gauge since the metric fluctuations produce higher order terms.

Finally for a scalar field, $\varphi = \phi+ \delta\phi$, the conservation equations become

$$\ddot{\delta\phi} + 2{\dot a \over a} \dot{\delta\phi} + (k^2 + a^2 {\cal V}_{,\phi\phi})\delta\phi = S_\phi \,,$$ (42)

where

$$S_\phi = \cases{ (\dot\Psi-3\dot\Phi)\dot\phi - 2 a^2 {\cal... ... -3\dot h_L \dot\phi\,, & (synchronous), \vphantom{\Big[} \cr }$$ (43)

are the gravitational sources.