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What is microwave background polarization good for? One basic and model-independent answer to this question was outlined above: polarization can provide a clean demonstration of the existence of acoustic oscillations in the early universe. The fact that three of the six polarization-temperature power spectra are zero for linear scalar perturbations gives several other interesting and model-independent probes of physics.

The most important is that the ``curl'' polarization power spectrum directly reflects the existence of any vector (vorticity) or tensor (gravitational wave) metric perturbations. Inflation models generically predict a nearly-scale invariant spectrum of tensor perturbations, while defects or other active sources produce significant amounts of both vector and tensor perturbations. If the measured temperature power spectrum of the microwave background turns out to look different than what is expected in the broad class of inflation-like cosmological models, polarization will tell what part of the temperature anisotropies arise from vector and tensor perturbations. More intriguingly, in inflation models, the amplitude of the tensor perturbations is directly proportional to the energy scale at which inflation occurred, so characterizing the gravitational wave background becomes a probe of GUT-scale physics at 1016 GeV! Inflation also predicts potentially measurable relationships between the amplitudes and power law indices of the primordial density and gravitational wave perturbations (see (Lidsey et al., 1997) a comprehensive overview), and measuring a ClC power spectrum appears to be the only way to obtain precise enough measurements of the tensor perturbations to test these predictions. A microwave background map with forseeable sensitivity could measure gravitational wave perturbations with amplitudes smaller than 10-3 times the amplitude of density perturbations (Kamionkowski and Kosowsky, 1998), thanks to the fact that the density perturbations don't contribute to ClC. The tensor perturbations generally contribute significantly to the temperature perturbations at angular scales larger than two degrees (l ltapprox 100) in a flat universe but have a much broader range of scales in polarization (50 ltapprox l ltapprox 500). For tensor and vector perturbations, the amplitude of the C-polarization is generally about the same as that of the G-polarization; if the perturbations inducing the COBE temperature anisotropies are 10% tensors, then we expect the peak of l ClC appeq 10-15 at angular scales around l = 80. An experimental challenge not for the faint of heart!

A second source of C-type polarization is gravitational lensing. The mass distribution in the universe between us and the surface of last scatter will bend the geodesics of the microwave background photons. This lensing can be described by an effective displacement field, in which the temperature and polarization at each point of the sky in an unlensed universe is mapped to a nearby but different point on the sky when lensing is accounted for. The displacement alters the shape of temperature contours in the microwave background, and likewise distorts the polarization pattern, inducing some curl component to the polarization field. Detailed calculations of this effect and the induced ClC have been made by Zaldarriaga and Seljak (1998). The amplitude of this effect is expected to be around l2 ClC appeq 10-14 on a broad range of subdegree angular scales (200 ltapprox l ltapprox 3000) with the power spectrum peaking around l = 1000 in a flat universe. This lensing polarization signal is just at the limit of detectability for the upcoming Planck satellite; future polarization satellites with better sensitivity could make detailed lensing maps based on the curl component of microwave background polarization. It is interesting to note that tensor perturbations and gravitational lensing are substantially distinguishable by their different angular scales. Note that the most recent version of the publicly available CMBFAST code by Seljak and Zaldarriaga (Seljak and Zaldarriaga, 1996) computes polarization from both tensor modes and from gravitational lensing.

A third source of C-type polarization is a primordial magnetic field. If a magnetic field was present at recombination, the linear polarization of electromagnetic radiation would undergo a Faraday rotation as it propagated through the surface of last scatter while significant numbers of free electrons were still present. (Such rotation could also occur after reionization, but both the electron density and the field strength would be much smaller and the resulting rotation is small compared to the primordial signal). This effect rotates an initial G-type polarization field into a C-type polarization field. A detailed estimate of the magnitude of this effect (Kosowsky and Loeb, 1997) shows that a primordial field with present strength 10-9 gauss induces a measurable one-degree rotation in the polarization at a frequency of 30 GHz. Faraday rotation depends quadratically on wavelength of the radiation, so down at 3 GHz, the rotation would be a huge 100 degrees (although the polarized emission from synchrotron radiation would also be correspondingly larger). Such a rotation will induce l2CC at a level of between 10-15 for one degree of rotation and 10-11 for large rotations. Additionally, it has been pointed out that Faraday rotation will contribute also to the ClTC cross-correlation at corresponding levels (Scannapieco and Ferreira, 1997) as well as to ClGC. Investigation of the angular dependence and detectability of such a signal is ongoing (Mack and Kosowsky, 1999). The best current constraints on a homogeneous component of a primordial magnetic field come from COBE constraints on anisotropic Bianchi spacetimes (Barrow, Ferreira and Silk, 1997), because a universe which contains a homogeneous magnetic field cannot be statistically isotropic. Detection of a significant primordial magnetic field would both provide the seed field needed to generate current galactic and subgalactic-scale magnetic fields via the dynamo mechanism, and also provide a very interesting constraint on fundamental particle physics, particularly if a field on large scales is detected (see, e.g., Turner and Widrow (1988) or Gasperini et al. (1995)).

Faraday rotation from magnetic fields is a special case of cosmological birefringence: rotation of polarization by differing amounts depending on direction of observation. Such rotation could arise from interactions between photons and other unknown fields. Constraints on the C-polarization of the microwave background could strongly constrain new pseudoscalar particles (see, e.g., Carroll and Field (1997)). More generally, non-zero cosmological contributions to the ClTC and ClGC cross correlations, which must be zero if parity is a valid symmetry of the cosmological perturbations, would indicate some intrinsic parity to either the primordial perturbations (Lue, Wang, and Kamionkowski, 1998) or to some interaction of the microwave background photons (Carroll, 1998). These types of effects are generally independent of photon frequency, so they can be distinguished from Faraday rotation through microwave background frequency dependence.

The above signals are all model-independent probes of new physics using microwave background polarization. An additional less daring but initially more useful and important use of polarization is in determining and constraining the basic background cosmology of the universe. It has been appreciated for several years now that the microwave background offers the cleanest and most powerful constraint on the gross features of the universe (Jungman et al., 1996). If the universe is described by an inflation-type model, with nearly scale-invariant initial adiabatic perturbations which evolve via gravitational instability, then the power spectrum of microwave background temperature fluctuations can strongly constrain nearly all cosmological parameters describing the universe: densities of various matter and energy components, amplitudes and power laws of initial density and gravitational wave perturbations, the Hubble parameter, and the redshift of reionization. More recent work (Eisenstein, Hu and Tegmark, 1999; Zaldarriaga, Seljak, and Spergel, 1997) has shown that the addition of polarization information can help tighten these constraints considerably, mainly because the new information now gives four theoretical power spectra to match instead of just one. Polarization particularly helps constrain the reionization redshift and the baryon density (Zaldarriaga and Harari, 1995). Polarization will also be important for deciphering the universe if measurements of the temperature anisotropies reveal that the universe is not described by the simple class of inflation-like cosmological models: it is a strong discriminator between vector and tensor perturbations and scalar perturbations (Kamionkowski, Kosowsky, and Stebbins, 1997).

Finally, no discussion of this sort would be completely honest without mentioning the thorny issue of foreground emission. We are gradually concluding that foregrounds have some non-negligible effect on temperature anisotropies, but that the amplitudes of various foregrounds are small enough that they will not substantially hinder our ability to draw cosmological conclusions from microwave background temperature maps (see, e.g., Tegmark (1998) for a recent estimate). Whether the same will prove true for polarization is unknown at present. Free-free emission is likely to have only negligible polarization, but synchrotron emission will be strongly polarized, and the polarization of dust emission is difficult to estimate reliably (Draine and Lazarian, 1998). Polarized emission from radio point sources is another potential problem. No present measurements have had sufficient sensitivity to detect polarized emission from any of these foreground sources, so it is difficult to predict the foreground impact. My own guess is that the G-polarization component, from which acoustic oscillations can be confirmed and from which parameter estimation can be significantly improved, will face foreground contamination comparable to the temperature anisotropies. If so, and if the polarization foregrounds are divided evenly between C and G polarization components, then control of foregrounds will become crucial for the very interesting physics probed by the cosmological C polarization. But I fully expect that through a combination of techniques, including carefully tailored sky cuts, measurements at many frequencies, improved theoretical understanding, foreground nongaussianity, and foreground template matching, we will separate out the small cosmological polarization signals from whatever polarized foregrounds are out there.

The next five years will bring us microwave background temperature maps of vastly improved sensitivity and resolution, and almost certainly the first detection of microwave background polarization. These observations will provide us with very tight constraints on our cosmological model, or else will reveal some new and unexpected aspect of our universe. Either way, the microwave background will be the cornerstone of a mature cosmology. What is left to do after Planck? One good answer to this question, I believe, is very high sensitivity measurements of microwave background polarization. Such observations hold the promise of probing the potential driving inflation, detecting primordial magnetic fields, mapping the matter distribution in the universe, and likely a variety of other interesting physics yet to be explored.

This work has been supported by the NASA Theory Program. Portions of this work were done at the Institute for Advanced Study.

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