One of the key datasets of the recent BICEP2 results is the B mode power spectrum shown below. The existence of these B modes implies the existence of gravitational waves prior to inflation.

My question is: what is the relationship between the multipole number of the B mode and the wavelength of the gravitational waves which generated it? My naive expectation is that it is one to one with some sort of scaling factor which includes information about the expansion of the universe and the conversion between spherical and linear coordinates.

B mode power spectrum from ref. 1


1 Answer 1


Wayne Hu (professor, Univ. Chicago) has a CMB website that seems to answer this question.

It is stated that "The one-to-one mapping between wavenumber and multipole moment described in [a previous section] is only approximately true and comes from the fact that the spherical Bessel function is strongly peaked at $kD \approx l$"

where $l$ is the multipole moment (x-axis of the graph in the question); $D$ is the distance light can travel between recombination and present, and $k$ is the wavenumber.


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