Maximizing S/N in Angular Power Spectrum Signals

[SherLOCKed]

The signal-to-noise ratio for angular power spectrum signal Cl under theoretical noise Nl, where Cl and Nl are functions of multipole l, is given as

(S/N)^2= \sum (2l+1) (Cl/Nl)^2To increase the S/N we bin the power spectrum signal, if bin width \Delta l, this in principle decreases Nl by a factor of 1/sqrt(\Delta l).

Now, in (S/N)^2 should we replace the sum over multipoles with the sum over bin centers?

 

[Hyperfine]

Consult Section 3.2.5. Binning of Planck 2018 results V. CMB power spectra and likelihoods.

 

[SherLOCKed]

Thanks for the response. I checked the paper, it talks about the power spectrum binning. Suppose I bin the power spectrum as described in the paper.
The confusion I had is, if I just sum over the binned multipoles, I will end with the similar cummulative signal-to-noise ratio as before I started binning. So, binning is not necessarily helping to increase the signal-to-noise ratio.

 

[LastScattered1090]

@SherLOCKed I guess one thing that confuses me is why there is a summation over 2ℓ + 1 in this case. That would make sense whenever averaging over all the m modes within a given multipole ℓ. But C_ℓ is the same for every m mode at a given ℓ by assumption of statistical isotropy. So summing over m modes doesn't make sense to me. What does the summation do, and why isn't S/N just quantified as C_ℓ/N_ℓ at every multipole?

I agree that because we're considering power, not just amplitude, random noise produces an N_ℓ that enters your spectrum as a bias, not just as variance. You can't get rid of it by binning multipoles. But for any actual measurement, the noise also causes multipole-to-multipole variance in the estimation of C_ℓ that would average down through binning.