[astro-ph/0412581] Cosmic Microwave Background Fluctuations from Gravitational Waves: An Analytic Approach

Authors:  Jonathan R. Pritchard, Marc Kamionkowski (Caltech)
Abstract:  We develop an analytic approach to calculation of the temperature and polarisation power spectra of the cosmic microwave background due to inflationary gravitational waves. This approach complements the more precise numerical results by providing insight into the physical origins of the features in the power spectra. We explore the use of analytic approximations for the gravitational-wave evolution, making use of the WKB approach to handle the radiation-matter transition. In the process, we describe scaling relations for the temperature and polarisation power spectra. We illustrate the dependence of the amplitude, shape, and peak locations on the details of recombination, the gravitational-wave power spectrum, and the cosmological parameters, and explain the origin of the peak locations in the temperature and polarisation power spectra. The decline in power on small scales in the polarisation power spectra is discussed in terms of phase-damping. In an appendix we detail numerical techniques for integrating the gravitational-wave evolution in the presence of anisotropic stress from free-streaming neutrinos.
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Antony Lewis
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[astro-ph/0412581] Cosmic Microwave Background Fluctuations

Post by Antony Lewis » January 04 2005

In Appendix A the authors consider the effect of neutrino anisotropic stress on the tensor CMB power spectra.

The authors say that the damping is only significant at high [tex]\ell[/tex] (modes entering the horizon during radiation domination). However as discussed in (version 2 of) astro-ph/0306304 there is also a significant (if smaller) damping for modes which enter the horizon during matter domination, so the effect is not only at large [tex]\ell[/tex] - you can see a numerical comparison plot on page 14 of the CAMB notes showing a nearly 10% effect on BB for [tex]\ell < 100[/tex].

However since reionization dominates the signal on very large scales, it is certainly true that the effect of neutrino damping is negligible there. The large scale effect on the temperature is also negligible. Interestingly the phase shift means that at around [tex]\ell \sim 150[/tex] there is actually a slight increase in BB power there due to the damping.

On another note, I'm not sure why Fig. 3 has an EE spectrum that seems to have reionization bump at low [tex]\ell[/tex] but BB doesn't?

Jonathan Pritchard
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[astro-ph/0412581] Cosmic Microwave Background Fluctuations

Post by Jonathan Pritchard » January 07 2005

Thanks for your comments on the paper. I agree with your comments on neutrino damping being relevant to the polarisation at l < 100 and should have been more careful to draw attention to that point. The numerical comparison in the CAMB notes is a nice way of illustrating the damping. I'd calculated something very similar, but didn't think to include it in the paper.

The "reionization" bump in the EE spectrum at low l seems to be a numerical artifact from CMBFAST. That said, a similar feature is visible in the early papers of Zaldarriaga and Seljak e.g. PRD 55 1830 (1997), fig. 1. Redoing the calculations in CAMB seems to remove most of the effect. There is still a slight break in the spectrum and so a small excess of power in the EE over the BB spectrum at l=2. Do you think that's a real feature or simply the limits of precision? Its not intuitive to me why you'd expect a loss of precision in these low l.

My thinking on why you get that feature in the EE and not the BB is the appearance of second deriviatives of the visibility and \Psi terms when you calculate the EE source term. Exactly how that leads to a loss of precision and why that would appear in CMBFAST but not CAMB isn't clear to me. What do you think?

Antony Lewis
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Re: [astro-ph/0412581] Cosmic Microwave Background Fluctuati

Post by Antony Lewis » January 09 2005

I do remember seeing some numerical problems at low [tex]\ell[/tex], though I can't remember their origin off hand. Certainly without reionization I don't think there should be a feature there. Not a big deal, since if we know [tex]z_{re} > 6[/tex] the very large scales will be dominated by the reionization signal anyway.

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