question about average/specific SNR

[dbkeitel]

Hi @TomWagg @katiebreivik here's a final question about a point in the documentation that confuses my physics intuitoon. At https://legwork.readthedocs.io/en/latest/notebooks/Source.html#Position-inclination-polarisation-specfic-sources there is the finding

As you can see, the SNR for a specific source is always lower than for the average source. This is because the modulation reduces the strain amplitude as the smearing in frequency reduces the amount of signal build up at the true source frequency.

• Doesn't this mean that by calling the SNR functions in the simple, position/orientation-independent way, one will overestimate the actual detectability of a population?
• Also doesn't it imply that in some sense the analytic averaging might have been done in the wrong order at some step - when I read "average SNR", I'd naively expect this to include the average effect over the actual modulations for each source, not an averaged-out modulation. In other words, the brute-force numerical computation of an "average SNR" would be to calculate lots of source-dependent SNRs with their actual modulations and average those, but according to that plot it wouldn't agree with the defined analytic "average SNR"?

Maybe this is just nomenclature confusion, but I'd be happy for some enlightenment.

[katiebreivik]

@dbkeitel this is indeed pretty tricky and ends up revealing some of the complexities of LISA data analysis. The Cornish & Larson paper which we are drawing the position/orientation dependence from also includes the effect of frequency spreading due to the doppler modulation arising from the detector motion. In principle, this modulation can be calculated and "undone" to remove the signal degradation if you know the signal is there. However, the actual data analysis the LISA team eventually does is very unlikely to be the calculation done in Cornish & Larson, and may be closer to the recent work of Cornish/Littenberg/others (i.e. ldasoft).

In the case of large populations with random inclinations, the average SNR will over predict and under predict the true value if the frequency modulation isn't accounted for. If we assume that the frequency spreading from the doppler motion of the detector is not able to be accounted for accurately, then the average will always be a bit over predicted. In the end, the uncertainties from this are roughly factors of ~1-5 (at the very most), while the uncertainties in the progenitor models which produce the populations that LEGWORK is applied to can be several orders of magnitude, so a factor of ~1-5 in SNR is a bit less difficult to stomach for the community.

Very happy to keep this conversation open if you see any red flags in my description!