View source for Bandpass Convention - What does flat mean

Nov 6 2017, Clem Pryke

For CMB-S4 sims we have a table of bandpasses in [[CMB-S4 frequency bands v1.99]]
which specifies the lower/upper edges of assumed tophat bandpasses.
In [[Tophat bands for Data Challenge]] (Nov 2 2016) Colin specified how to
define "flat" tophat - he states "I define this tophat to be such that a single-moded antenna
(AΩ scales as λ2) would have uniform response as a function of frequency to a beam-filling Rayleigh-Jeans source."

The Planck bandpass files are specified in units of (relative) spectral radiance (SR) although the
WMAP ones are specified in RJ units.

PySM works in spectral radiance units (Jy/Sr) and it makes a lot of sense to do
all astrophysics type modelling in these units.

When working in SR units the correct shape for a "flat" bandpass is not
flat - it is nu^-2 because real detectors absorb more photons to lower
frequencies.
Sky models 00 and 05 did this in the creation step.
However, apparently in the re-analysis presented in
[http://bicep.rc.fas.harvard.edu/CMB-S4/analysis_logbook/20170927_dc02/ Maximum likelihood search results for Data Challenge 02] flat in SR
was assumed and this may be all or part of the cause of the biases
found in the results.
(Raphael's re-analysis does not need to know the bandpasses.)

As part of the PySM documentation Ben Thorne offers
[http://bicep.rc.fas.harvard.edu/pryke/PySM_docs/unit_conversion.pdf this document] - 
I am not fully able to tell but I think flat for him means flat in SR units?
Can we please get confirmation what was assumed when generating the PySM files
on NSERSC?

Flavien Vansyngel said: "The monochromatic templates of both the dust and the synchrotron are produced in MJy/sr.
They are integrated through bandpasses that are flat in MJy/sr (using their respective SED in those units).
After integration, the maps are converted in K_CMB by dividing them by the following factor

[[File:20171106_f1.png|300px]]

where \nu_c is the central frequency of the band, f is the fractional bandpass and (K_CMB -> MJy/sr)(\nu) is the
monochromatic conversation factor from K_CMB to MJy/sr."

Bottom line: The conclusions that I would like to get agreement on are:

1) When working with models in spectral radiance units (Jy/Sr) the most sensible thing to use for a "flat bandpass"
is actually nu^-2 - because that is what an ideal physical detector would do.

2) Assuming one has taken the product of the source response and the bandpass as above the SR to T_CMB
conversion factor then also needs to be a weighted mean over the bandpass.

Here is Colin's freq_scaling.m Matlab code which I believe does the right thing - and which
should be a fed a nu^-2 bandpass [http://bicep.rc.fas.harvard.edu/pryke/freq_scaling.m]