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Introduction
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The argument for an extended period of reionization is as follows. The WMAP has detected the correlation between temperature and polarization on large angular scales [@kogut03] that has an amplitude proportional to the total optical depth of CMB photons to Thomson scattering, $\tau$ [@sunyaev80; @zaldarriaga97a; @kaplinghat02]. Modeling reionization with a single sharp transition at $z_{ri}$, a multi–parameter fit to the WMAP data gives $z_{ri} = 17 \pm 5$ [@spergel03]. On the other hand, the evolution of quasar spectra from $z=6.3$ and $z=6.4$ to $z = 6$ shows a rapid decrease in the amount of neutral Hydrogen, indicating the end of reionization [@fan03]. A simple interpretation to explain these two very different datasets is that reionization started early, $z_{ri} \sim 20$, but did not conclude until much later ($z \sim 6$).

An extended period of reionization has its effect at low $\ell$ in the polarization (as detected by WMAP), and also at higher $\ell$ in temperature. In Section II we disucss the low $\ell$ effect and in Section III the high $\ell$ effect. In section IV we discuss them in combination. All the work I present here is described in more detail in @haiman03 (reionization models), @kaplinghat03a [@holder03] (low $\ell$ signal) and @santos03 (high $\ell$).

The Reionization Bumps
======================

Quadrupole radiation incident upon an electron leads to linear polarization. For a review of CMB polarization, see [@hu97a]. As drawn in Fig. \[fig:bump\] a free electron at high redshift sees a quadrupole from free–streaming of the monopole on its last–scattering surface. A monopole with wavenumber $k$ free streams primarily into $l=k(\eta_{ri} - \eta_{lss})$ so the quadrupole is given primarily by $k=2/(\eta_{ri} - \eta_{lss})$. The linear polarization thus generated at $\eta_{ri}$ projects to $l = k(\eta_0-\eta_{ri}) = 2(\eta_0-\eta_{ri})/(\eta_{ri} - \eta_{lss})$ today. Thus, the feature appears at low $\ell$. The polarization signature is proportional to the amount of scattering and hence the optical depth $\tau$. Therefore $C_l^{EE} \propto \tau^2$ and $C_l^{TE} \propto \tau$. See @zaldarriaga97a and @kaplinghat03a. The same considerations lead to a reionization bump in the tensor B mode with $C_l^{BB} \propto \tau^2$ so higher $\tau$ can improve the detectability of gravitational waves as quantified in @knox02 [@kaplinghat03b]. The polarization angular power spectra are shown in Fig. \[fig:foregrounds\].

The low l polarization has already been detected by WMAP through the correlation of this effect with the temperature map [@kogut03]. From the WMAP measurements only one number can be inferred: a joint fit of the TT and TE power spectra to a six-parameter model results in $\tau = 0.17 \pm 0.06$ [@spergel03]. If the EE power spectrum is measured with cosmic variance precision, there are 5 uncorrelated numbers (the amplitudes of 5 eigenmodes of the ionization history), that can be measured with signal-to-noise greater than 1 [@hu03]. These 5 numbers will provide strong constraints on models of the first generation of stars, since these are presumably what cause the reionization of the inter-galactic medium.

Models of foreground polarization indicate that this low l signal can, in principle, be measured with near cosmic variance precision, as seen in Fig. \[fig:foregrounds\]. Although these models are highly uncertain. We will know more soon from further releases of WMAP data.

To quantify how much improvement is possible beyond WMAP we show forecasted constraints on $\tau$ and the initial reionization redshift $z_e$ assuming statistical errors only for WMAP, Planck/LFI and Planck/HFI. Model B has sudden and complete reionization at $z_e=15$. Model A has a two–stage reionization, first increasing $X_e$ suddently to 0.42 at $z_e=25$ and then suddenly completely ionizing at $z=6.3$. The ionization fraction, $X_e\equiv n_e/n_p$ cannot be greater than 1.08 if Helium does not doubly ionize. Thus for a given $z_e$ there is a minimum value of $\tau$ that gives rise to the straight lines in the lower portions of the contours around model A.

The kSZ power spectrum
======================

An extended period of reionization (as suggested by combination of WMAP data and high-redshift quasar spectra) is likely to be a period with a highly inhomogeneous reionization fraction. Prior to percolation of the ionized regions, the ionization fraction will be near unity in the vicinity of the sources of the reioinizing radiation and zero far away from any sources. If the sources are in high-mass halos, as semi-analytic models suggest, then the patches of high ionization fraction will be highly correlated. These correlated patches can lead to a kinetic SZ power spectrum with an amplitude of about $10^{-12}$ at $l$ values of 1500 and higher. At $l > 3000$ this kinetic SZ power spectrum may be the dominant source of flucutation power on the sky, for components with the same spectral signature as thermal fluctuations about a 2.7 K black body. Here we reproduce a figure from @santos03 showing a range of the kSZ power spectra that come from the reionization models of @haiman03.

The error bars on one of the curves are for an observation of 1% of the sky with 0.9’ resolution. They are due to the “noise” from primary + lensing CMB and residual point sources. Instrument noise is assumed to be subdominant. We assume thermal SZ has been removed by taking advantage of its unique spectral dependence.

We see that the models do not show much variation in shape, but only in amplitude. There is a degeneracy between the clustering bias of the dark matter halos hosting the reionizing sources and the optical depth. The low $\ell$ measurements can break this degeneracy, allowing us to place some constraint on the clustering bias, and therefore the halo masses.

Although predicting the amplitude of the signal is difficult, the shape appears to be robust, particularly at $l < 3000$ where the kSZ power spectrum might contaminate attempts to measure cosmological parameters and reconstruct the gravitational lensing potential. We can therefore model this contaminant with one free parameter: an amplitude. Doing parameter estimation without such modeling (i.e., ignoring the kSZ power spectrum) can lead to significant biases for parameter estimation from Planck and higher–resolution observations [@knox98; @santos03]. But @santos03 find that modeling the kSZ power spectrum as this robust shape times a floating amplitude removes all significant parameter estimation biases even for an experiment that is cosmic–variance limited out to $l=3,000$.

The simple prescription for removing the bias works because there is so little variation in the shape of the kSZ power spectrum in our reionization models. This shape is merely a projection of the matter power spectrum from high redshift. The small dependence on shape that there is comes from the different mean angular diameter distances in the different models. If necessary, the modeling could be extended to include this as a free parameter also.

If the real kSZ power spectrum shape (at $l < 3,000$) were significantly different from the shapes we calculate then our simple modeling (with one or maybe two parameters) may not be sufficient. However, this could only be the case if the density of free electrons does not trace the density of matter on scales larger than about 3 Mpc. Note that even if $X_e$ were completely homogeneous, the density of free electrons traces the density of matter. Even in this homogeneous $X_e$ case the shape would still be a projection of the matter power spectrum, although this time from a smaller mean angular-diameter distance.

Our calculations have relied on semi–analytic models of reionization. For a more numerical approach, see the contribution from Naoshi Sugiyama. For more on patchy reionization, see @aghanim96 [@gruzinov98; @knox98; @valageas01] and other papers already mentioned above.
