Dissipation

  The generation of viscosity and heat conduction in the fluid dissipates fluctuations through the Euler equations with (89) and (94),  

$$\begin{array} {rcl}\displaystyle{}(1+R)\dot\Theta_1^{(0)}&=&... ...k \over \dot \tau} {4 \over 15}k\Theta_1^{(1)}\, ,\end{array}$$

where we have dropped the $\dot a/a$ factors under the assumption that the expansion can be neglected during the dissipation period. We have also employed Eqn. (85) to eliminate higher order terms in the vector equation. With the continuity equation for the scalars $\dot \Theta_0^{(0)} = -k \Theta_1^{(0)}/3 - \dot\Phi$(see Eqn. 60 $\ell=0$, m=0 ), we obtain 

$$\begin{array} {rcl}\displaystyle{}\ddot \Theta_0^{(0)}+ {1 \... ...heta_0^{(0)}= -{k^3 \over 3}\Psi - \ddot \Phi \, ,\end{array}$$ (76)

which is a damped forced oscillator equation.

An interesting case to consider is the behavior in the absence of metric fluctuations $\Psi$, $\Phi$, and V. The result, immediately apparent from Eqn. (96) and (97), is that the acoustic amplitude (m=0) and vorticity (m=1) damp as $\exp[-(k/k_D^{(m)})^2]$ where 

$$\begin{array} {rcl}\displaystyle{}\left[{1 \over k_D^{(0)}} ... ...int d\eta{ 1\over \dot \tau} {1 \over {1 + R} } \, .\end{array}$$

Notice that dissipation is less rapid for the vectors compared with the scalars once the fluid becomes baryon dominated $R\gg 1$ because of the absence of heat conduction damping. In principle, this allows vectors to contribute more CMB anisotropies at small scales due to fluid contributions. In practice, the dissipative cut off scales are not very far apart since $R \alt1$ at recombination.

Vectors may also dominate if there is a continual metric source. There is a competition between the metric source and dissipational sinks in Eqns. (96) and (97). Scalars retain contributions to $\Theta^{(0)}_0 + \Psi$ of ${\cal O}(R\Psi,(\ddot\Psi-\ddot\Phi)/k^2)$ (see Eqn. (84) and [29]). The vector solution becomes

$$\Theta_1^{(1)}(\eta) = e^{-[k/k_D^{(1)}(\eta)]^2} \int_0^\eta d\eta' \dot V e^{[k/{k_D^{(1)}}(\eta')]^2} ,$$ (77)

which says that if variations in the metric are rapid compared with the damping then $\Theta_1^{(1)}= V$ and damping does not occur.