Professor, Department of Astronomy and Astrophysics
University of Chicago

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The dissipation of the acoustic oscillations leaves a signature in the polarization of CMB in its wake (see e.g. [Hu & White, 1997a] and references therein for a more complete treatment). Much like reflection off of a surface, Thomson scattering induces a linear polarization in the scattered radiation. Consider incoming radiation in the $-{\bf x}$ direction scattered at right angles into the ${\bf z}$ direction (see Plate 2, left panel). Heuristically, incoming radiation shakes an electron in the direction of its electric field vector or polarization $\hat{\bf\epsilon}'$ causing it to radiate with an outgoing polarization parallel to that direction. However since the outgoing polarization $\hat{\bf\epsilon}$ must be orthogonal to the outgoing direction, incoming radiation that is polarized parallel to the outgoing direction cannot scatter leaving only one polarization state. More generally, the Thomson differential cross section $d\sigma_T/d\Omega \propto
\vert\hat{\bf\epsilon'} \cdot \hat{\bf\epsilon}\vert^2.$

Unlike the reflection of sunlight off of a surface, the incoming radiation comes from all angles. If it were completely isotropic in intensity, radiation coming along the $\hat {\bf y}$ would provide the polarization state that is missing from that coming along $\hat{\bf x}$ leaving the net outgoing radiation unpolarized. Only a quadrupole temperature anisotropy in the radiation generates a net linear polarization from Thomson scattering. As we have seen, a quadrupole can only be generated causally by the motion of photons and then only if the Universe is optically thin to Thomson scattering across this scale (i.e. it is inversely proportional to $\dot\tau$). Polarization generation suffers from a Catch-22: the scattering which generates polarization also suppresses its quadrupole source.

% latex2html id marker 529
\centerline{ \epsfxsize = 5.0in \ep...
$E$-mode and the one at $45^\circ$\ angles is called the $B$-mode.}\end{plate}


The fact that the polarization strength is of order the quadrupole explains the shape and height of the polarization spectra in Plate 1b. The monopole and dipole $\Theta$ and $v_\gamma$ are of the same order of magnitude at recombination, but their oscillations are $\pi/2$ out of phase as follows from Equation (9) and Equation (10). Since the quadrupole is of order $kv_\gamma/\dot\tau$ (see Figure 3), the polarization spectrum should be smaller than the temperature spectrum by a factor of order $k/\dot \tau$ at recombination. As in the case of the damping, the precise value requires numerical work [Bond & Efstathiou, 1987] since $\dot\tau$ changes so rapidly near recombination. Calculations show a steady rise in the polarized fraction with increasing $l$ or $k$ to a maximum of about ten percent before damping destroys the oscillations and hence the dipole source. Since $v_\gamma$ is out of phase with the monopole, the polarization peaks should also be out of phase with the temperature peaks. Indeed, Plate 1b shows that this is the case. Furthermore, the phase relation also tells us that the polarization is correlated with the temperature perturbations. The correlation power $C_\ell^{\Theta E}$ being the product of the two, exhibits oscillations at twice the acoustic frequency.

Until now, we have focused on the polarization strength without regard to its orientation. The orientation, like a 2 dimensional vector, is described by two components $E$ and $B$. The $E$ and $B$ decomposition is simplest to visualize in the small scale limit, where spherical harmonic analysis coincides with Fourier analysis [Seljak, 1997]. Then the wavevector ${\bf k}$ picks out a preferred direction against which the polarization direction is measured (see Plate 2, right panel). Since the linear polarization is a ``headless vector'' that remains unchanged upon a $180^\circ$ rotation, the two numbers $E$ and $B$ that define it represent polarization aligned or orthogonal with the wavevector (positive and negative $E$) and crossed at $\pm 45^\circ$ (positive and negative $B$).

In linear theory, scalar perturbations like the gravitational potential and temperature perturbations have only one intrinsic direction associated with them, that provided by ${\bf k}$, and the orientation of the polarization inevitably takes it cue from that one direction, thereby producing an $E-$mode. The generalization to an all-sky characterization of the polarization changes none of these qualitative features. The $E-$mode and the $B-$mode are formally distinguished by the orientation of the Hessian of the Stokes parameters which define the direction of the polarization itself. This geometric distinction is preserved under summation of all Fourier modes as well as the generalization of Fourier analysis to spherical harmonic analysis.

The acoustic peaks in the polarization appear exclusively in the $EE$ power spectrum of Equation (5). This distinction is very useful as it allows a clean separation of this effect from those occuring beyond the scope of the linear perturbation theory of scalar fluctuations: in particular, gravitational waves (see §4.2.3) and gravitational lensing (see §4.2.4). Moreover, in the working cosmological model, the polarization peaks and correlation are precise predictions of the temperature peaks as they depend on the same physics. As such their detection would represent a sharp test on the implicit assumptions of the working model, especially its initial conditions and ionization history.

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Next: Integral Approach Up: ACOUSTIC PEAKS Previous: Damping
Wayne Hu 2001-10-15