Compression and Vorticity

  For the (m=0) scalars, the well-known result of expanding the Boltzmann equations (60) for $\ell=0,1$ and the baryon Euler equation (67) is  

$$\begin{array} {rcl}\displaystyle{}\dot \Theta_0^{(0)}&=& -... ...tom{A}} &=& k(\Theta_0^{(0)}+ m_{\rm eff}\Psi) \, ,\end{array}$$

which represent the photon fluid continuity and Euler equations and gives the baryon fluid quantities directly as

$$\dot \delta_B = {1 \over 3}\dot\Theta_0^{(0)}, \qquad v_B^{(0)}= \Theta_1^{(0)},$$ (64)

to lowest order. Here $m_{\rm eff}= 1+R$ where recall that R is the baryon-photon momentum density ratio. We have dropped the viscosity term $\Theta_2^{(0)}= {\cal O}(k/\dot\tau)\Theta_1^{(0)}$ (see § IVB). The effect of the baryons is to introduce a Compton drag term that slows the oscillation and enhances infall into gravitational potential wells $\Psi$. That these equations describe forced acoustic oscillations in the fluid is clear when we rewrite the equations as  

$$(m_{\rm eff} \dot\Theta_0^{(0)})\dot{\vphantom{A}} + {k^2 \o... ... m_{\rm eff}\Psi - (m_{\rm eff}\dot\Phi)\dot{\vphantom{A}} \, ,$$ (65)

whose solution in the absence of metric fluctuations is 

$$\begin{array} {rcl}\displaystyle{}\Theta_0^{(0)}&=& A m_{\rm... ...=& \sqrt{3} A m_{\rm eff}^{-3/4}\sin(ks + \phi) \, ,\end{array}$$

where $s=\int c_{\gamma B} d\eta = \int (3m_{\rm eff})^{-1/2} d\eta$ is the sound horizon, A is a constant amplitude and $\phi$ is a constant phase shift. In the presence of potential perturbations, the redshift a photon experiences climbing out of a potential well makes the effective temperature $\Theta_0^{(0)}+\Psi$ (see Eqn. 78), which satisfies  

$$[m_{\rm eff} (\dot\Theta_0^{(0)}+\dot\Psi)] \,\dot{\vphantom{... ...si + [m_{\rm eff}(\dot\Psi - \dot\Phi)]\,\dot{\vphantom{A}} ,$$ (66)

and shows that the effective force on the oscillator is due to baryon drag $R\Psi$ and differential gravitational redshifts from the time dependence of the metric. As seen in Eqn. (78) and (83), the effective temperature at last scattering forms the main contribution at last scattering with the Doppler effect $v_B^{(0)}= \Theta_1^{(0)}$ playing a secondary role for $m_{\rm eff} \gt 1$. Furthermore, because of the nature of the monopole versus dipole projection, features in $\ell$space are mainly created by the effective temperature (see Fig. 3 and §IIID).

If $R \ll 1$, then one expects contributions of ${\cal O}(\ddot \Psi - \ddot \Phi)/k^2$ to the oscillations in $\Theta_0^{(0)}+\Psi$ in addition to the initial fluctuations. These acoustic contributions should be compared with the ${\cal O}(\Delta\Psi-\Delta\Phi)$ contributions from gravitational redshifts in a time dependent metric after last scattering. The stimulation of oscillations at $k\eta \gg 1$ thus either requires large or rapidly varying metric fluctuations. In the case of the former, acoustic oscillations would be small compared to gravitational redshift contributions.

Vector perturbations on the other hand lack pressure support and cannot generate acoustic or compressional waves. The tight coupling expansion of the photon $(\ell=1,m=1)$ and baryon Euler equations (60) and (71) leads to  

$$[ m_{\rm eff} (\Theta^{(1)}_1 - V) ] \dot{\hphantom{A}} = 0 \, ,$$ (67)

and $v^{(1)}_B = \Theta^{(1)}_1$. Thus the vorticity in the photon baryon fluid is of equal amplitude to the vector metric perturbation. In the absence of sources, it is constant in a photon-dominated fluid and decays as a^-1 with the expansion in a baryon-dominated fluid. In the presence of sources, the solution is  

$$\Theta^{(1)}_1(\eta,k) = V(\eta,k) + {1 \over m_{\rm eff}} [\Theta^{(1)}(0,k)-V(0,k)] \, ,$$ (68)

so that the photon dipole tracks the evolution of the metric fluctuation. With $v_B^{(1)}=\Theta^{(1)}_1$ in Eqn. (61), vorticity leads to a Doppler effect in the CMB of magnitude on order the vector metric fluctuation at last scattering V in contrast to scalar acoustic effects which depend on the time rate of change of the metric.

In turn the vector metric depends on the vector anisotropic stress of the matter as

$$V(\eta_*,k) = - 8\pi G a_*^{-2} \int_0^{\eta_*} d\eta a^4 (p_f \pi_f^{(1)}+ \pi_s^{(1)})/k .$$ (69)

In the absence of sources $V \propto a^{-2}$ and decays with the expansion. They are thus generally negligible if the universe contains only the usual fluids. Only seeded models such as cosmological defects may have their contributions to the CMB anisotropy dominated by vector modes. However even though the vector to scalar fluid contribution to the anisotropy for seeded models is of order $k^2 V/(\ddot\Psi -\ddot\Phi)$ and may be large, the vector to scalar gravitational redshift contributions, of order $V/(\Psi-\Phi)$ is not necessarily large. Furthermore from the integral solution for the vectors Eqn. (78) and the tight coupling approximation Eqn. (86), the fluid effects tend to cancel part of the gravitational effect.