Background:

CMB polarization is induced via Thomson scattering, which occurs either at decoupling or during a later epoch of reionization. The level of polarization is linked to the local quadrupole anisotropy of the incident radiation on the scattering eletrons, and it is expected to be 1-10% of the amplitude of the temperature anisotropies depending on the angular scale.

CMB polarization is important for two reasons: first, polarization measurements can substantially improve the accuracy with which parameters are measured by breaking the degeneracy between certain parameter combinations; second, it also offers an independent test of the basic assumptions that underly the standard cosmological model.

Contrary to most of the Astronomy field that uses the Stokes parameters Q and U to describe their polarization measurements, in the CMB field we make use of two scalar fields E and B that are independent of how the coordinate system is oriented, and are related to the tensor field (Q,U) by a non-local transformation. Scalar CMB fluctuations have been shown to generate only E-fluctuations, whereas gravity waves, CMB lensing and foregrounds generate both E and B.

See also:

Polarization Movies
Other Polarization Resorces

Polarization from the 2003 Flight of BOOMERanG
The Cosmic Microwave Background and Its Polarization
The Large-Scale Polarization of the Microwave Background and Foreground
Data Analysis of the POLAR Experiment
E/B Decomposition of CMB Maps
First attempt at measuring the CMB cross-polarization
A Limit on the Large Angular Scale Polarization of the CMB
How to measure CMB polarization power spectra without losing information

Polarization Movies:

Omega_Baryon: Like in the temperature case, increasing the fraction of ordinary matter (baryons) raises the "acoustic peaks". There is no polarization at low l: it happens because the Sachs-Wolfe effect (which is a pure gravity effect) doesn't involve Thomson scattering.



Omega_CDM: Increasing the density of dark matter reduces not only the temperature, but also the polarization spectra.



Neutrino Fraction: The CMB power spectrum (temperature and polarization) changes only very weakly as you replace cold dark matter by hot. This is because the neutrinos were already quite cold (nonrelativistic) the time the CMB fluctuations are formed.



Omega_Lambda: The cosmological constant was completely irrelevant at z>1000, when the acoustic oscillations were created. It therefore doesn't change the shape of the peaks at all - it merely shifts them sideways, since it affects the conversion from the physical scale of the wiggles (in meters) into the angular scale (in degrees, or multipole l).



Omega_Curvature: Spatial curvature was completely irrelevant at z>1000, when the acoustic oscillations were created. Omega_k therefore doesn't change the shape of the peaks at all - it merely shifts them sideways, since the conversion from the physical scale of the wiggles (in meters) into the angular scale (in degrees, or multipole l) depends on whether space is curved. If space has negative curvature (positive Omega_k), like a Pringles potato chip or a saddle, then the angle subtended on the sky decreases, shifting the peaks to the right. If space has positive curvature, like a balloon, then the peaks shift to the left. In addition to shifting the peaks sideways, curvature also causes fluctuations in the gravitational field to decrease over time. This also has no effect in the polarization since it is a pure gravity effect that doesn't involve Thomson scattering.



Scalar Normalization: The amplitude of density fluctuations (a.k.a. "scalar" fluctuations) simply determines the overall normalization of the two power spectra.



Tensor Normalization: As you can see, gravity waves (so-called tensor fluctuations) contribute only to fairly large angular scales in the CMB power spectrum, i.e., boost the left part. These fluctuations have a unique signature in the polarized case, since they produce B-polarization. It's important to point out that B-polarization cannot be generated by density modes, thus its detection is a signature of gravity waves (which can be used to constrain inflation).



Scalar Tilt: Changing the spectral index of scalar fluctuations simply tilts the power spectra, altering the ratio of small-scale and large-scale power in all curves.



Reionization Optical Depth: If the Universe was reionized long ago by quasars or early stars, then there is a non-zero optical depth tau. This means that there is a non-zero probability that a CMB photon arriving in our detector once Thomson scattered off of a free electron, and in fact originated from a somewhat different direction than we thought. This effect smears out small-scale features in the CMB power spectrum (temperature and polarization), suppressing all acoustic peaks by a constant factor exp(-tau), while leaving the power on the largest scales unaffected. The new last-scattering surface at low redshift also creates new power at low l, but this is dwarfed by the Sachs-Wolfe effect in the unpolarized case. Since the Sachs-Wolfe effect is unpolarized, this new power is much easier to see and quantify with E-polarization. Specifically, the location of the first new peak gives the redshift of reionization, since it reflects the horizon size at that time.

Other Polarization Resources:

Theory:

Wayne Hu
Max Tegmark
Matias Zaldarriaga


Polarization Experiments:

Old CMB Polarization Measurements
COMPASS
PIQUE
POLAR

Ongoing Ground-based:
CAPMAP
CBI
DASI
KuPID
Polatron

Upcoming Ground-based:
AMiBA
BICEP
PolarBear
QUEST

Balloon:
Archeops
BOOMERanG
MAXIPOL

Space:
WMAP
Planck
SPOrt

Polarization from the 2003 Flight of BOOMERanG:

We present the BOOMERanG-03 experiment and maps of the Stokes parameters I, Q, U of the microwave sky obtained during a 14 day balloon flight in 2003. Three regions of the southern sky were surveyed: a deep survey (~ 90 square degrees) and a shallow survey (~ 750 square degrees) at high Galactic latitudes (both centered at RA ~ 5.5 h, dec ~ -45 deg) and a survey of ~ 300 square degrees across the Galactic plane at RA ~ 9.1 h, dec ~ -47 deg. All three surveys were carried out in three wide frequency bands centered at 145, 245 and 345 GHz, with an angular resolution of ~ 10'. The 145 GHz maps of Stokes I are dominated by Cosmic Microwave Background (CMB) temperature anisotropy, which is mapped with high signal to noise ratio. The measured anisotropy pattern is consistent with the pattern measured in the same region by BOOMERanG-98 and by WMAP. The 145 GHz maps of Stokes Q and U provide a robust statistical detection of polarization of the CMB when subjected to a power spectrum analysis. This amplitude of the polarization is consistent with that of the CMB in the $\Lambda$CDM cosmological scenario. At 145 GHz, in the CMB surveys, the intensity and polarization of the astrophysical foregrounds are found to be negligible with respect to the cosmological signal. At 245 and 345 GHz we detect ISD emission correlated to the 3000 GHz IRAS/DIRBE maps, and give upper limits for any other non-CMB component. We also present intensity maps of the surveyed section of the Galactic plane. These are compared to monitors of different interstellar components, showing that a variety of emission mechanisms is present in that region. Details about this analysis is found in Masi et al.(2005), Jones et al.(2005), Montroy et al.(2005), Piacentini et al.(2005) and MacTavish et al.(2005).

Top figure: EE power spectra. Bottom Figure: TE (or X) power spectra.

The DASI discovery of CMB polarization, confirmed by WMAP, has opened a new chapter in cosmology. Most of the useful information about inflationary gravitational waves and reionization is on large angular scales where Galactic foreground contamination is the worst. The goal of the present review is to provide the state-of-the-art of the CMB polarization from a practical point of view, connecting real-world data to physical models. We present the physics of this polarized phenomena and illustrate how it depends of various cosmological parameters for standard adiabatic models. We also present all observational constraints to date and discuss how much we have learned about polarized foregrounds so far from the CMB studies. Finally, we comment on future prospects for the measurement of CMB polarization. de Oliveira-Costa (2004).

The DASI discovery of CMB polarization has opened a new chapter in cosmology. Most of the useful information about inflationary gravitational waves and reionization is on large angular scales where Galactic foreground contamination is the worst, so a key challenge is to model, quantify and remove polarized foregrounds. We use the POLAR experiment, COBE/DMR and radio surveys to provide the strongest limits to date on the TE cross power spectrum of the CMB on large angular scales and to quantify the polarized synchrotron radiation, which is likely to be the most challenging polarized contaminant for the MAP satellite. We find that the synchrotron E- and B-contributions are equal to within 10% from 408-820 MHz with a hint of E-domination at higher frequencies. We quantify Faraday Rotation and Depolarization effects in the two-dimensional (l,nu)-plane and show that they cause the synchrotron polarization percentage to drop both towards lower frequencies and towards lower multipoles. Details about this analysis may be found in de Oliveira-Costa et al.(2003).

The coming flood of CMB polarization experiments, spurred by the recent detection of CMB polarization by DASI and WMAP, will be confronted by many new analysis tasks specific to polarization. For the analysis of CMB polarization data sets, the devil is truly in the details. With this in mind, we present the details of the data analysis for the POLAR experiment, which recently led to the tightest upper limits on the polarization of the Cosmic Microwave Background Radiation at large angular scales. We discuss the data selection process, mapmaking algorithms, offset removal, and the likelihood analysis which were used to find upper limits on the polarization. Stated using the modern convention for reporting CMB Stokes parameters, these limits are 5.0 \uK on both E-type and B-type polarization at 95% confidence. Finally, we discuss simulations used to test our analysis techniques and to probe the fundamental limitations of the experiment. Details about this analysis may be found in O'Dell et al.(2002).

Separation of the E and B components of a microwave background polarization map or a weak lensing map is an essential step in extracting science from it, but when the map covers only part of the sky and/or is pixelized, this decomposition cannot be done perfectly. We present a method for decomposing an arbitrary sky map into a sum of three orthogonal components that we term ``pure E'', ``pure B'' and ``ambiguous''. This method is useful both for providing intuition for experimental design and for analyzing data sets in practice. We show how to find orthonormal bases for all three components in terms of bilaplacian eigenfunctions. The number of ambiguous modes is proportional to the length of the map boundary so fairly round maps are preferred. For real-world data analysis, we present a simple matrix eigenvalue method for calculating nearly pure E and B modes in pixelized maps. We find that the dominant source of leakage between E and B is aliasing of small-scale power caused by the pixelization. This problem can be eliminated by heavily oversampling the map, but is exacerbated by the fact that the E power spectrum is expected to be much larger than the B power spectrum and extremely blue. We found that a factor of 2 to 3 more pixels are needed in a polarization map to achieve the same level of contamination by aliased power than in a temperature map. Details about this analysis can be found in Bunn et al. (2002).

We compute upper limits on CMB cross-polarization by cross-correlating the PIQUE and Saskatoon experiments. We also discuss theoretical and practical issues relevant to measuring cross-polarization and illustrate them with simulations of the upcoming BOOMERanG 2002 experiment. We present a method that separates all six polarization power spectra (TT, EE, BB, TE, TB, EB) without any other "leakage" than the familiar EE-BB mixing caused by incomplete sky coverage. Since E and B get mixed, one might expect leakage between TE and TB, between EE and EB and between BB and EB - our method eliminates this by preserving the parity symmetry under which TB and EB are odd and the other four power spectra are even. Details about this analysis can be found in de Oliveira-Costa et al. (2002).

We present an upper limit on the polarization of the Cosmic Microwave Background at 7 degree angular scales in the frequency band between 26 and 36 GHz, produced by the POLAR experiment. The campaign produced a map of linear polarization over the RA range 112-275 degrees at declination 43 degrees. The model-independent upper limit on the E-mode polarization component of the CMB at angular scales l = 2-20 is 10 uK (95% confidence). The corresponding limit for the B-mode is also 10 uK. Constraining the B-mode power to be zero, the 95% confidence limit on E-mode power alone is 8 uK. Details about this analysis can be found in Keating et al. (2001).

We present a method for measuring CMB polarization power spectra given incomplete sky coverage and test it with simulated examples such as Boomerang 2001 and MAP. By augmenting the quadratic estimator method with an additional step, we find that the E and B power spectra can be effectively disentangled on angular scales substantially smaller than the width of the sky patch in the narrowest direction. We find that the basic quadratic and maximum-likelihood methods display a unneccesary sensitivity to systematic errors when T-E cross-correlation is involved, and show how this problem can be eliminated at negligible cost in increased error bars. We also test numerically the widely used approximation that sample variance scales inversely with sky coverage, and find it to be an excellent approximation on scales substantially smaller than the sky patch. Details about this analysis can be found in Tegmark & de Oliveira-Costa (2000).

1. "Instrument, Method, Brightness and Polarization Maps from the 2003 Flight of BOOMERanG" Masi & the BOOMERanG Collaboration 2005, ApJ,submitted.
2. "A Measurement of the Angular Power Spectrum of the CMB Temperature Anisotropy from the 2003 Flight of BOOMERanG" Jones & the BOOMERanG Collaboration 2005, ApJ,submitted.
3. "A Measurement of the CMB EE Spectrum from the 2003 Flight of BOOMERanG" Montroy & the BOOMERanG Collaboration 2005, ApJ,submitted.
4. "A Measurement of the Polarization-Temperature Angular Cross Power Spectrum of the CMB from the 2003 Flight of BOOMERanG" Piacentini & the BOOMERanG Collaboration 2005, ApJ,submitted.
5. "Cosmological Parameters from the 2003 Flight of BOOMERanG" MacTavish & the BOOMERanG Collaboration 2005, ApJ,submitted.
6. "BOOMERANG Results" Polenta & the BOOMERanG Collaboration 2005, AdSpR, 36(6):1064.
7. "The Cosmic Microwave Background and Its Polarization" de Oliveira-Costa 2004, Review to be published in the proceedings of "Astronomical Polarimetry - Current Status and Future Directions'', Hawaii, March 15-19.
8. "Measuring CMB Polarization with BOOMERANG" Montroy & BOOMERanG Collaboration 2003, Proceedings from "The Cosmic Microwave Background and its Polarization", New Astronomy Reviews, Eds. S. Hanany and K.A. Olive.
9. "The Large-Scale Polarization of the Microwave Foreground" de Oliveira-Costa, Tegmark, O'Dell, Keating, Timbie, Efstathiou & Smoot 2003, Proceedings from "The Cosmic Microwave Background and its Polarization", New Astronomy Reviews, Eds. S. Hanany and K.A. Olive.
10. "The Large-Scale Polarization of the Microwave Background and Foreground" de Oliveira-Costa, Tegmark, O'Dell, Keating, Timbie, Efstathiou & Smoot 2002, Phys. Rev. D. 68:083003.
11. "CMB Polarization at Large Angular Scales: Data Analysis of the POLAR Experiment" O'Dell, Keating, de Oliveira-Costa, Tegmark & Timbie 2002, Phys. Rev. D. 68:042002.
12. "E/B decomposition of finite pixelized CMB maps" Bunn, Zaldarriaga, Tegmark & de Oliveira-Costa 2002, Phys. Rev. D. 67:023501.
13. "First attempt at measuring the CMB cross-polarization" de Oliveira-Costa, Tegmark, Zaldarriaga, Barkats, Gundersen, Hedman, Staggs & Winstein 2002, Phys. Rev. D. 67:023003.
14. "A Limit on the Large Angular Scale Polarization of the Cosmic Microwave Background" Keating, O'Dell, de Oliveira-Costa, Klawikowski, Stebor, Piccirillo, Tegmark & Timbie 2001, ApJL, 560:L1.
15. "How to Measure CMB Polarization Power Spectra without Losing Information" Tegmark & de Oliveira-Costa, 2001, Phys.Rev. D, 64:063001.