Well, your understanding of the relevance of spherical harmonics in the CBR analysis is just plain wrong.
I think that you like to argue just for the sake of it, and not for a valid scientific doubt. Also, I note that your
way to addressing to me is a little bit tilted to the downplaying game, to prove that you're the smartest child
in the block (you're far from being a child, of course).
So, this is mi final attempt to give you insights about the complex mechanisms behind the measurement of
the Cosmic Backgroud Radiation. I'll base this simple explanation on the final analysis of the data of COBE,
which was written by 18 scientists involved in the posterior evaluation of the data recopiled by COBE FIRAS.
As the final paper is heavily protected to readings by non-members, as it is in custody of the American
Astronomical Society, I've managed to get a link to a photographic copy at this link:
Measurements of the Cosmic Background Spectrum by the COBE FIRAS Instrument
But also, I searched a copy of a derivate paper at this link, which is written by the same 18 scientists, and is
less protected (if you select PRINT to a virtual PDF printer will obtain the document):
Cosmic Background Dipole Spectrum Measured by the COBE FIRAS Instrument
Some explanations to make it easier to understand:
1) The Universe was measured as a black body spherical cavity by COBE, 30 years ago,
As the capabilities of FIRAS to make accurate measurements over small areas was low,
scientists invented (specifically for COBE) the reduction of a sphere to a "cube), which
was called: quadriteralized spherical cube.
This cube was decomposed into 6144 macropixels, to make average measurements over each one of them (they only could
measure 90% of the total).
2) The measurements covered the band between 60 and 660 Ghz (THIS has to make your eyebrows to raise). It was
EQUIVALENT to measure wavelengths between 0.2 mm and 2 mm.
What was measure was a "fluctuating thermal radiation" using a wideband bolometer. The measurements were a total
TEMPORAL AVERAGE per macro-pixel, and were stored (thousands of them) to further and heavy post-processing
here, at Earth,
And this is a KEY FACTOR. The Planck's formula is an SPECTRAL formula, not a TEMPORAL formula, and is based
on statistical averages (Boltzmann, Planck).
To obtain the SPECTRAL data of a macro-pixel, you HAVE TO make a spectrum transform of bidimensional order,
But each macropixel is not a cartesian area, but a 2D representation of a 3D surface, so you HAVE TO apply
spherical harmonics transforms to EACH macropixel (either to analyze power or to its derivate temperature, which is
obtained FROM Planck's formula. Or Rayleigh's formula, as frequency is very low).
3) Having your thousands of samples per each of the 6144 pixels, you need to ELIMINATE distortions (galactic noise,
cosmic noise - difficult to define, instrument's missbehaviors, glitches, etc).
To do that, you use the spectral spherical decomposition in monopoles, dipoles, quadrupoles, etc., and HEAVILY
DOCTOR the data to eliminate what you don't like.
4) Finally, you have to SHOW to the audience HOW GOOD is the job COBE did, by proving that CBR is fully compliant
with Planck's law for a black body cavity,
But you have A PROBLEM:
Planck's formula is about DENSITY of thermal radiation that fills the cavity, wich is a volumetric measurement.
But you only have 6000+ pieces of spectral data (heavily doctored) and you need to MAKE PUBLIC a single
chart (measured radiation intensity that escapes from a small hole at the Planck's cavity versus wavelength).
Then, what you do is A WEIGHTED SPECTRAL AVERAGE of 6000+ spectral data (with a wide range of wavelengths)
and PRESENT IT as a SINGLE spectral formula, like Planck did.
Then, when you fitted data to your target, you present it to the general audience, which cheers you and the mission
and recover their faith on science and its reconciliation with history: Planck tell us that CBR is exactly as he predicted,
He's the father of quantum physics, so the man knows and, also, the COBE results HAVE TO be true,
I end it here, as I'm sicked of this, But remember that I claimed for "petitio pricipii" sophistic mechanism behind COBE, WMAP
or Planck satellite; The calibrations of the antenna and the measurement instrument use not one, but two Black Body radiation
So, going back to this topic and my OP: "Why Planck's formula for black body radiation is used to measure the CBR?"