next up previous contents
Next: Entropy and Heat Conduction Up: Photon-Baryon Fluid Previous: Compression and Vorticity

Viscosity and Polarization

  Anisotropic stress represents shear viscosity in the fluid and is generated as tight coupling breaks down on small scales where the photon diffusion length is comparable to the wavelength. For the photons, anisotropic stress is related to the quadrupole moments of the distribution $\Theta_2^{(m)}$ which is in turn coupled to the E-parity polarization E2(m). The zeroth order expansion of the polarization $(\ell=2)$ equations (Eqn. 63) gives  
E_2^{(m)} = -{\sqrt{6}\over 4} \Theta_2^{(m)}, \qquad
B_2^{(m)} = 0\end{displaymath} (70)
or $P^{(m)} = {1 \over 4} \Theta_2^{(m)}$.The quadrupole $(\ell=2)$ component of the temperature hierarchy (Eqn. 60) then becomes to lowest order in $k/\dot\tau$ 
\Theta_2^{(m)} = {4 \over 9}\sqrt{4-m^2} {k \over \dot\tau} ...
 ... {1 \over 9}\sqrt{4-m^2} {k \over \dot\tau} 
\Theta_1^{(m)}\, ,\end{displaymath} (71)
for scalars and vectors. In the tight coupling limit, the scalar and vector sources of polarization traces the structure of the photon-baryon fluid velocity. For the tensors,  
\Theta_2^{(2)} = -{4 \over 3}{ \dot H \over \dot \tau}\, ,\qquad
P^{(2)} = -{1 \over 3}{ \dot H \over \dot \tau} \, .\end{displaymath} (72)
Combining Eqns. (88) and (89), we see that polarization fluctuations are generally suppressed with respect to metric or temperature fluctuations. They are proportional to the quadrupole moments in the temperature which are suppressed by scattering. Only as the optical depth decreases can polarization be generated by scattering. Yet then the fraction of photons affected also decreases. In the standard cold dark matter model, the polarization is less than $5\%$ of the temperature anisotropy at its peak (see Fig. 6).

\epsfxsize=4in \epsfbox{} \end{center}\end{figure}

These scaling relations between the metric and anisotropic scattering sources of the temperature and polarization are important for understanding the large angle behavior of the polarization and temperature polarization cross spectrum. Here last scattering is effectively instantaneous compared with the scale of the perturbation and the tight coupling remains a good approximation through last scattering.

For the scalars, the Euler equation (80) may be used to express the scalar velocity and hence the polarization in terms of the effective temperature,
\Theta_1^{(0)}= m_{\rm eff}^{-1} \int k(\Theta_0^{(0)}+ m_{\rm eff}\Psi)
d\eta \, .\end{displaymath} (73)
Since $m_{\rm eff} \sim 1$, $\Theta_1^{(0)}$ has the same sign as $\Theta_0^{(0)}+\Psi$before $\Theta_0^{(0)}+\Psi$ itself can change signs, assuming reasonable initial conditions. It then follows that P(0) is also of the same sign and is of order  
P^{(0)} \sim (k\eta){k \over \dot\tau} [\Theta_0^{(0)}+ \Psi]\, ,\end{displaymath} (74)
which is strongly suppressed for $k\eta \ll 1$. The definite sign leads to a definite prediction for the sign of the temperature polarization cross correlation on large angles.

For the vectors  
P^{(1)} = {\sqrt{3} \over 9} {k \over \dot\tau} V \, ,\end{displaymath} (75)
and is both suppressed and has a definite sign in relation to the metric fluctuation. The tensor relation to the metric is given in Eqn. (90). In fact, in all three cases the dominant source of temperature perturbations has the same sign as the anisotropic scattering source P(m). From Eqn. (79), differences in the sense of the cross correlation between temperature and polarization thus arise only due to geometric reasons in the projection of the sources (see Fig. 5). On angles larger than the horizon at last scattering, the scalar and vector $C_\ell^{\Theta E}$is negative whereas the tensor cross power is positive [9,21].

On smaller scales, the scalar polarization follows the velocity in the tight coupling regime. It is instructive to recall the solutions for the acoustic oscillations from Eqn. (83). The velocity oscillates $\pi/2$out of phase with the temperature and hence the E-polarization acoustic peaks will be out of phase with the temperature peaks (see Fig. 6). The cross correlation oscillates as $\cos(ks +\phi)\sin(ks +\phi)$ and hence has twice the frequency. Thus between peaks of the polarization and temperature power spectra (which represents both peaks and troughs of the temperature amplitude) the cross correlation peaks. The structure of the cross correlation can be used to measure the acoustic phase $\phi$ ($\phi \approx 0$for adiabatic models) and how it changes with scale just as the temperature but like the EE power spectrum [27] has the added benefit of probing slightly larger scales than the first temperature peak. This property can help distinguish adiabatic and isocurvature models due to causal constraints on the generation of acoustic waves at the horizon at last scattering [28].

Finally, polarization also increases the viscosity of the fluid by a factor of 6/5, which has significant effects for the temperature. Even though the viscous imperfections of the fluid are small in the tight coupling region they can lead to significant dissipation of the fluctuations over time (see §IVD).

next up previous contents
Next: Entropy and Heat Conduction Up: Photon-Baryon Fluid Previous: Compression and Vorticity
Wayne Hu