B-Pol Science

B-Pol Science

The CMB and its Temperature Anisotropy

The photons of the fossil cosmic microwave background (CMB) radiation emanate dominantly from the epoch of recombination, some 380,000 years after the Big Bang, when the universe was only about one thousandth its present size. At this time, the free electrons and protons combined to form neutral gaseous hydrogen, causing the universe to become transparent over a relatively short interval of time. Some of the microwave photons (approximately 10% ) were subsequently re-scattered by free electrons liberated during "re-ionization" (around redshift
z ≈ 7 — 12 when the first structures in the universe were formed, allowing the first generation of massive stars and quasars to liberate numerous ionizing ultraviolet photons.

The CMB today has a blackbody spectrum, and except for small departures of approximately 1 part in 100,000, is of a uniform temperature of
T = 2.75 K After the discovery of the CMB in 1965 by Penzias and Wilson, for which they received the Nobel Prize in Physics in 1978, the importance of searching for deviations from a perfect blackbody spectrum and for variations in temperature between different parts of the sky was immediately appreciated. Nevertheless, because of the smallness of such variations, it was not until 1992 that an anisotropy in CMB temperature was first measured by the NASA COBE space mission, for which J. Mather and G. Smoot were awarded the Nobel Prize in 2006.

The temperature variations of the CMB provide a snapshot of spatial inhomogeneities of our Universe at the earliest accessible moment, before non-linear effects had a chance to intervene. Consequently, observations of the CMB offer the cleanest and most direct means of characterizing the initial conditions of our Universe, and consequently of testing theories of the very early universe, such as inflation.

Since COBE much progress has been made in characterizing the temperature anisotropies of the CMB over a broad range of angular scales, and measurements of the E-mode polarization anisotropy have also been made. The BOOMERANG balloon borne experiment established the presence of the first Doppler peak, thus ruling out topological defects as the sole source of the primordial cosmological density fluctuations. The NASA WMAP satellite carried out a full-sky survey of temperature anisotropy at moderate sensitivity and angular resolution and produced low sensitivity maps of the CMB E mode polarization. The ESA PLANCK mission, to be launched in summer 2008, will map the temperature anisotropies with even greater sensitivity and angular resolution, and will provide the first signal dominated maps of the E mode polarizations, allowing for important consistency checks between the temperature and E-mode polarization anisotropies. PLANCK will provide a sensitive probe of the scalar cosmological perturbations but is unlikely to discover tensor modes because of insufficient sensitivity and control of systematic errors.

B-Mode Polarization and Primordial Gravity Waves

The CMB photons that reach us are polarized owing to the anisotropy of the polarized Thomson scattering cross section of unpolarized electromagnetic radiation off free electrons. If we follow a microwave background photon observed today backward in time along its line of sight to its instant of last scattering, its present polarization can be related to the temperature quadrupole moment anisotropy from the point of view of the electron of last scattering. Part of this anisotropy is due to intrinsic temperature fluctuations. Another part is due to the Doppler effect caused by dilatation or shear of the plasma, and yet another part arises from gravitational redshift.

The polarization of the CMB seen by us today may be decomposed into two modes: an `electric' component, or E-mode, and a `magnetic' component or B-mode. The polarization map may best be considered as a field of double-headed vectors on the celestial sphere, mathematically represented as a traceless symmetric second-rank tensor field. The direction of the plotted lines indicates the linear polarization of highest temperature.

A polarization field on the celestial sphere may be decomposed according to its transformation properties under the rotation group using the spherical harmonics Ylm much as is done for the temperature anisotropies. For the polarization, however, there are two independent components that may be constructed. There is an `electric' or `gradient,' polarization, obtained by taking the double derivative and removing the trace:

$\displaystyle {\bf Y}_{\ell m,
}^{(E)}= \frac{1}{\ell (\ell +1)} \Bigl[ \nabla _a\nabla _b
-\frac{1}{2} \delta _{ab} \Bigr] Y_{\ell m}(\hat \Omega )

where the indices a,b indicate directions on the celestial sphere. There is also a `magnetic,' or `curl' polarization obtained by rotating the above by 45° with a certain handedness.

$\displaystyle {\bf Y}_{\ell m, ab }^{(B)}=\frac{1}{\ell (\ell +1)}
... _c\nabla _b
+\nabla _a\epsilon _{bc}\nabla _c
Y_{\ell m}(\hat \Omega )

where εab is the completely anti-symmetric unit tensor.

The figure below illustrates the two classes of polarization patterns. The pattern on the left is purely `electric', meaning that it may be represented as the double gradient of a scalar potential Φ(E)^) on the celestial sphere in the manner described above. The pattern of the right is, by contrast, purely `magnetic,' meaning that no potential representation of the kind above is possible. On the other hand, the magnetic pattern, after a rotation by 45° in a given sense, either left- or right-handed, becomes representable as a double gradient of a pseudo-scalar potential Ψ(B)^) This potential is pseudo-scalar because under a parity transformation the sense of the 45° rotation is reversed leading to a sign change.

b-mode and e-mode

E-mode (left) and B-mode (right)

Ordinary cosmological perturbations--that is, the well-known cosmological density perturbations, generated by the inflaton field in the standard inflationary scenario, which later turn into the structures now present in the universe (galaxies, clusters of galaxies, Lyman-α clouds, etc.) — are purely scalar in character. Within the framework of the linear theory for the evolution of cosmological perturbations (which is adequate for calculating the anisotropies of the CMB, except for very small and calculable nonlinear corrections), scalar cosmological perturbations can generate only E mode polarization of the CMB. Consequently, the presence of B-mode polarization not attributable to the small nonlinear corrections mentioned above is the unmistakable sign of new physics.

Inflation typically generates tensor fluctuations (i.e., gravity waves) as well as the ordinary scalar fluctuations. Tensor fluctuations from inflation are primordial gravitational waves. These gravity waves arise from quantum vacuum fluctuations of the graviton field that become frozen in during the epoch of inflationary expansion, in much the same way as vacuum fluctuations of the scalar inflaton field become frozen in to generate the ordinary scalar density fluctuations described above.

Unlike ordinary gravitational waves generated by astrophysical sources (that will presumably be detected by LIGO, VIRGO, LISA, etc.), gravitational waves from inflation are predicted to have an approximately scale-invariant primordial spectrum. This spectrum is very red in comparison, with appreciable amplitudes on as large as the size of our entire presently observable universe! After the primordial gravity waves enter the horizon, their amplitudes decay with the expansion of the universe. This is why primordial gravity waves (as opposed to those from localized sources) are detectible at the longest accessible wave lengths.

The prediction of a nearly scale-invariant spectrum of gravitational waves generated from inflation offers us a real opportunity to test inflation. The determination of the amplitude of these gravity waves measures the height of inflationary potential. The scalar perturbations by contrast also depend on the slope of the inflationary potential.

B Mode Signal Expectations

b-mode signal expectationB-mode signal expectation

Although we cannot make a precise prediction for the amplitude of the expected B mode polarization anisotropy, inflationary theory predicts that the spatial power spectrum of the tensor modes is very nearly scale invariant. Using the best fit cosmological parameters inferred from the three-year WMAP data, we plot below the several CMB anisotropies as a function of multipole number for a range of values for the tensor-to-scalar ratio (T/S). The green curves represent the anisotropies arising from the scalar perturbations which may be considered well characterized, at least for our purposes here, from WMAP. The tensor anisotropies are plotted in blue. The dot-dashed curves represent the tensor TT, TE and EE tensor anisotropies assuming a value of (T/S) = 0.1, lying modestly below present constraints. These anisotropies, however, do not provide a very sensitive means of detecting tensor modes, because the scalar modes too predict such anisotropies, albeit with a slightly different angular spectrum. The BB mode [plotted here from top to bottom for (T/S) = 10-1, 10-2 and 10-3 ], however, provides a particularly clean and robust means of detecting the presence of tensor modes when (T/S) is small, because at linear order scalar modes cannot generate B mode polarization anisotropy for reasons of symmetry. The only contaminant, plotted in red, arises from gravitational lensing of scalar E mode anisotropies by intervening gravitationally collapsed structures. This contaminant, however, can be calculated from theory, measured from higher-order correlations, and moreover has a different angular power spectrum shape. At intermediate multipole number (i.e.,12 ≲ l ≲ 60) both the BB tensor signal and the EE → BB lensed contaminant have an approximately white noise angular power spectrum. However, the tensor spectrum turns over much earlier, at around l ≈ 100. Consequently, almost all of the useful statistical information concerning the presence of primordial gravitational waves lies on angular scales greater than a degree. An examination of the plot below reveals that there are two windows for detecting a non-zero value of (T/S). l there is the "re-ionization bump" where the tensor signal rises far above its white noise extrapolation from larger angular scales. Second, there is another window, where the statistical weight is centered about
l ≈ 50. The first window, which is the more sensitive of the two, is accessible only from space by means of a full-sky survey.

B-Pol Detection Capabilities

Compared to the CMB temperature and E-mode polarization anisotropies, the expected B mode signal is very small. A successful B mode polarization experiment must improve on current and planned experiments in the following three respects:

Bpol vs PLANK

B-Pol vs PLANK

Sensitivities of B-Pol and PLANCK for measuring the primordial B mode.

The four heavy blue curves show the predicted angular spectra for the primordial gravity wave B mode signal for four values of the tensor-to-scalar ratio (T/S) = 10-1, 10-2, 10-3, 10-4 (from top to bottom) assuming the best fit cosmological parameters from the three-year WMAP analysis. The heavy red curve indicates the scalar B mode contaminant, due to the gravitational lensing of the scalar E mode by intervening structures between the surface of last scattering and us today. The lower solid black curve indicates the nominal B-Pol sensitivity (obtained by taking the combined sensitivity of 5.2 μK· arcmin of the two central channels at 100 and 143 GHz having 47' fwhm resolution). The lower dotted black curve indicates the sensitivity that would be obtained by broad binning (i.e., Δl/l ≈ 1). The lighter pair of black curves above indicate the corresponding sensitivities for the PLANCK satellite, where the three central channels at 70, 100 and 143 GHz have been combined to obtain a corresponding sensitivity of 63 μK· arcmin at 10' fwhm resolution. The lower solid and dotted green curves provide an estimate of the sensitivity after foreground removal calculated by combining all channels optimally to remove synchrotron and dust components whose frequency spectra are assumed fixed and known. The lighter pair of curves above shows the result of the same analysis for PLANCK.

B Mode Polarization and Fundamental Science

Inflationary cosmology, developed in the early 1980s, resolved a number of vexing cosmological paradoxes by combining ideas from quantum field theory with general relativity in a self-consistent way. Soon thereafter it was realized that when quantum fluctuations were taken into account, inflation also provides an elegant and predictive mechanism for generating the primordial scalar perturbations that subsequently led to the structures now seen in the Universe.

Inflation presently constitutes the most plausible and most satisfying theoretical foundation for understanding what occurred in the very early Universe, and in particular for understanding the origin of the primordial cosmological perturbations that are now being probed with ever increasing precision. Inflation, however, is not yet a complete theory. Despite many interesting proposals and ideas from the high-energy theory community, a definitive prediction for the form of the inflationary potential is not presently at hand.

Nevertheless we do have a rather clear idea of what general form the inflationary potential should take. Dimensional considerations suggest that the scalar field should traverse a distance in field space of order the Planck mass (i.e., ≈ 1019 GeV) between the epoch of inflation and the present epoch and that during inflation its height should lie a few orders of magnitude below the Planck energy — that is, around the scale of Grand Unification of the strong, weak, and electromagnetic interactions.

Knowledge of the scalar power spectrum allows in principle to reconstruct a part of the inflationary potential, up to an integration constant corresponding to the height of the potential. A measurement of the tensor amplitude at one length scale allows a direct determination of the height of the inflationary potential, breaking this degeneracy. A measurement of the tensor amplitude at more than one length scale allows one to test the consistency between the scalar and tensor spectra as predicted by inflationary theory.

The discovery of primordial gravitational waves from inflation would constitute a qualitative advance of lasting importance both for cosmology and for high-energy theory. Firstly, it would finally provide definitive proof that inflation actually took place. Second, it would provide stringent constraints on new physics around the Planck energy, where the unification of gravity with the other three fundamental interactions is expected to lie. Today the greatest challenge of fundamental physics is to understand the nature of this unification including gravity. Over the last thirty years enormous progress has beeen made on the theoretical front, most notably with the development of superstring and M theory. However, progress is being held back by the dearth of guidance from experiment. The proposed B-Pol mission offers a unique opportunity to furnish the required clues concerning the new physics that lies many orders of magnitude beyond the reach of accelerator experiments.

Other B-Pol Science

While the primary science motivation for BPOL is the search for primordial gravitational waves from inflation, the high-sensitivity polarized maps of the full microwave sky produced by BPOL will be of great value for furthering other science objectives. Below we list a few of the most important among these: