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[1003.0299] The local B-polarization of the CMB: a very sen

Posted: March 05 2010
by Antony Lewis
Topological defects can source scalar, vector and tensor modes in the early universe. The vector modes have power on small scales and can generate E and B polarization; the B signal can be quite distinctive, and used to constrain defect models with future data.

This paper appears to take some previous results for the B-mode power spectrum and multiply them by l^4, so e.g. in Fig 1 the power is very blue. Of course to be consistent you also have to multiply the noise and the any other spectrum of interest by l^4 as well, so you seem to gain nothing by doing this. Is there some point I have missed?

The paper also defines a 'local' scalar [tex]\tilde{B}[/tex] by taking two derivatives of the polarization tensor. However you gain nothing by doing this; with noisy or non-band-limited data you cannot calculate derivatives on a scale L without having data available over a scale L - the non-locality just hits you in a different form (see astro-ph/0305545 and refs).

[1003.0299] The local B-polarization of the CMB: a very sen

Posted: March 05 2010
by Juan Garcia-Bellido
The main point is that vector components of defects' contribution to CMB polarization anisotropies peak at scales smaller than those from inflation.

On the other hand, the ordinary E- and B-modes depend non-locally on the Stokes parameters, so they cannot be used to put constraints on causal sources like defects using the angular correlation function of E- and B-modes on small scales. That is the reason why Baumann and Zaldarriaga [0901.0958] suggested using instead the local modes. Those are the true causal modes, written in terms of derivatives of the Stokes parameters.

These local B-modes then have power spectra that are much bluer than the non-local ones, and hence enhance the small scale (high-l) end of the spectrum. It is by looking at the angular correlation functions at small separations (tens of arcmin) that one has a chance to measure the defect's contribution to the local B-modes, and distinguish it from the one of inflation.

Of course, the usual white noise power spectrum for polarization will also be modified by this [tex]\ell^4[/tex] factor, but by a suitable gaussian smoothing of the data (following Baumann&Zaldarriaga), we can indeed obtain large signal to noise ratios for binned data at small angular scales.

Baumann&Zaldarriaga looked at the model-independent signature of inflation at angles [tex]\theta>2[/tex] degrees. What we have realiazed is that, although model-dependent, the signal at angles [tex]\theta < 1[/tex] degrees can be much more significant. In fact, the feature at small angles is rather universal. The differences between defect models (and we considered four different ones) is just in the height and width of the first and second oscillations in the angular correlation functions (related to the heigth and position of the angular power spectrum). Therefore, with sufficient angular resolution one could not only detect defects (if they are there) but also differentiate between different models.

Re: [1003.0299] The local B-polarization of the CMB: a very

Posted: March 06 2010
by Antony Lewis
I think it is clear from the normal power spectra that the sourced vector mode B-polarization peaks at much smaller scales than the gravitational wave spectrum: mostly scales sub-horizon at recombination as opposed to tensor modes which decay on sub-horizon scales. I agree that with low enough noise this is an interesting signal (and has been calculated many times before), though it needs to be distinguished from other possible vector mode sources like magnetic fields.

I thought the point of the Baumann paper was to make a nice picture showing visually the structure of the correlations. The E and B modes contain exactly the same information as the tilde versions; in the same way the WMAP7 papers make some nice plots of the polarization-temperature correlation to visually show a physical effect, but these constrain the same information as the usual power spectra. In the Gaussian limit the usual E/B spectra contain all the information on the defect power spectrum.

Only Q and U can actually be measured locally on the sky (in one pixel you cannot calculate any spatial derivatives). The two-point Q/U correlations can be calculated from the usual E and B spectra.