# 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 Y_{lm}
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:

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.

where ε

_{ab }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)}(Ω^{^})
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)}(Ω^{^})
This potential is
pseudo-scalar because under a parity transformation the sense of the
45° rotation is reversed leading to a sign change.

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

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:

**A dramatic increase in raw sensitivity.**With current detector technology, one is within a factor of two of the intrinsic quantum photon shot noise limit. Consequently, the only way to obtain a significant improvement in sensitivity is to increase the number of detectors. Photon shot noise produces a white noise spectrum on the sky maps. For the combined central channels, for Planck this noise will be approximately at the effective level of 65 μK over an arcminute square pixel as opposed to 5 μK for B-Pol. That means that PLANCK would have to integrate for approximately 169 times as long, or three centuries to achieve the same nominal sensitivity!**Full sky coverage and exquisite control of systematic errors, especially on large angular scales.**Given the importance of the polarization anisotropy on large angular scales and the much larger E mode polarization signal arising from scalar modes, special care must be taken to minimize E to B leakage and to adopt a scanning strategy where widely separated points on the sky are rapidly compared before (1/ƒ) noise is allowed to intervene.**Broad frequency coverage in order to allow effective subtraction of galactic foreground contaminants.**In addition to the primordial polarized signal, there is also a contaminant signal from polarized galactic synchrotron emission and from dust that must be removed. Because each of the underlying components has a different spectrum, foreground removal can be achieved by combining maps at different frequencies.

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.,
≈ 10^{19} 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:

**Gravitational lensing.**From the point of view of discovering primordial tensor mode, the lensing of E polarization mode into the B mode is simply a nuisance, just like instrument noise and foregrounds. However the characterization of the underlying spectrum contains invaluable information that can be exploited to constrain neutrino masses and dark energy. While the lensing potential can in principle be reconstructed from quadratic estimators constructed from TT correlations, such estimators are extremely noisy due to cosmic variance present even in the absence of lensing. Since the B mode (in the absence of tensor modes) is predicted to be zero, correlations involving the observed B mode are much cleaner.**Re-ionization history of the Universe.**Perhaps the most striking contribution of the WMAP space mission was its measurement of the reionization optical depth of the microwave photons through its characterization of the E mode polarization on very large angular scales, summarized by the single number τ = 0.093 +/- 0.027 from their 3yr analysis. Such reionization results from ionizing radiation emitted by very first structures formed in the high-redshift Universe--either by very massive stars or by quasars--and a better characterization of the full reionization history as a function of redshift would provide important clues for understanding structure formation. The precise polarization data that BPol will provide at very low multipoles will be invaluable to this endeavor.**Magnetic fields in the early Universe.**Large-scale magnetic field are ubiquitous in our present Universe; however, their origin is poorly understood. It is unknown to what extent these fields are primordial. If present very early on at sufficient amplitude, such magnetic fields through their Faraday rotation would generate B mode polarization detectable with BPOL.