In any case, after my previous post I began to feel guilty about my brief characterization of CMB photons at a single frequency equivalent to 2.725 K temperature. The CMB photons are described by a spectrum, a curve describing about how much power (equivalent to about how many photons there are) at each of a range of frequencies, so it is a little misleading to speak about a single frequency. I accordingly dropped my other work (a physics study I have spent two years on at this point) to add a little more to my discussion.

I was surprised (again, and why is it that most surprises are undesirable? must be my life's experience or the deterioration of this world during my lifetime) to find that the equations and presentations of the data related to blackbody radiation (the CMB radiation is almost exactly that of a blackbody spectrum equivalent to a perfect radiator at 2.725 degrees Kelvin) are ambiguous at best, and possibly flatly wrong on occasion, depending on the author (there are many different forms for the equations and units involved, e.g., intensity at unit area vs integrated over a hemisphere). It is not that the mathematics and physics involved is new, Max Planck got the ball rolling back in 1901 or so, when he invented what became known as the Planck constant, h, in order to avoid the ultraviolet catastrophe. Scientists up to that time had made very accurate measurements of the frequencies of radiation emitted from hot objects, in particular those constructed to be the equivalent of a blackbody. By blackbody I mean an object that does not reflect any electromagnetic energy, like a mirror reflects, instead absorbing everything that comes its way and in turn emitting a very precise spectrum of energy related to its own temperature. At the time they created hollow cavities, termed "hohlraum", with a tiny hole out of which they could measure radiation intensity and character inside. Almost all the radiation of the hohlraum is trapped inside, bouncing around internally, which has the effect that its internal heat is solely related to its own temperature rather than through exchange processes with the environment by which it might otherwise come to equilibrium.

A few had offered equations to characterize the blackbody radiation, but they had the undesired quality of blowing up a short wavelength, i.e., creating an ultraviolet catastrophe. Planck laboriously (I have discovered through the accounts of many scientists that scientific work requires a lot of effort, which makes me feel a bit better about my own self-inflicted pain in that area) came up with the idea that the radiation could not simply be any number on the real number line, but rather would have to come in integer multiples of this odd constant, 6.62607004e-34 Joules seconds (I have one of his 1901 papers and see he got very close to this present value of h, he giving 6.55e-27 erg second; an erg is 1e-7 Joule). He hoped for some time that someone would find a way to explain this in some other way, but quantum mechanics took off within a few years, changing everything Planck and every other scientist had known in the way of a world view (along with special and general relativity, most people are aware Einstein created that new area, but less aware that he only received the Nobel Prize for his work in creating quantum mechanics, he having found immediate use for Planck's constant in describing the photoelectric effect).

I'll see if I can somehow type the Planck law equation of interest to me (it is a real pain in the neck to work without LaTex math or its online version MathJax):

๐(ฮฝ,T) = (2โ๐³ / c²) (1 / exp(โ๐/kT) - 1)

Whew that is ugly. Well anyway. That equation gives you the radiation intensity as a function of frequency (that is the ฮฝ, a Greek letter nu) and time (per second) in, depending on what constant units you use, J or W per m^2 (meter squared) per steradian (a chunk out of the surface of a sphere, picture the radiation emerging from the eyes of angry Superman stuck in a big ball, say a hohlraum, and cones of death ray hit the surface). The famous Planck's constant is the โ, which is privileged to have its own Unicode symbol up here apparently. exp means "exponential", i.e., Euler's number raised to the power of the stuff in parentheses next to it. I would have liked to put a "B" subscript on the k to signal that this is Boltzman's constant, put that was perversely impossible (every letter other than b had a subscript when I looked at the menu here). T is temperature in Kelvin absolute degrees. The frequency ฮฝ is in Hz (cycles per second). I wrote some computer code (I work in a Jupyter notebook environment, where I can type perfectly typeset mathematics in one place and active Python computer code in another, with all the scientific software available for that environment, SciPy, NumPy, SymPy, Matplotlib, etc.---first time in years I have felt like computers were actually fulfilling some of their promise, despite Gates' constant attempts to sabotage anyone's efforts to accomplish that...the Borg operating system, as it is known in the world-wide computing community) to graph the result of a spectrum of frequencies from 100 MHz to 1000 GHz input to that equation above:

And there you have it (hoot, there it is?). I encountered so many errors and misuse of the Planck equations in the literature that I intentionally used the same spectrum covered in a figure in a 2011 paper by P.J.E. Peebles (

*The natural science of cosmology,*2011) as a cross check on my methods. Peebles, by the way, was one of the scientists who Penzias and Wilson shared their discovery with in 1965; they published a letter in 1965

*Astrophysical Journal*, vol. 142, p.419-421 and Peebles, along with his collaborators, published "a possible explanation for the observed excess noise temperature [of the antenna]" You have to consider that this was an age when men in science conducted themselves like gentlemen, with integrity and understatement. It is probably difficult to understand for anyone of the current era. These folks knew they had probably found the simmering heat of the Creation, but described it in clear, non-hyperbolic terms as the Bell Telephone scientists that they were (they had used a re-purposed piece of satellite communication equipment, a giant horn, I kid you not, used to communicate with a couple of early telecommunications satellites). Bell Telephone Labs, that is another loss of that age. Amazing discoveries that came out of their work, most of it freely shared with the world. I was just reading Nyquist's 1924 paper (published in a Bell technical journal) on

*Certain Factors Affecting Telegraph Speed*the other day. That was the work that in many ways started the field of communications theory, his name still used today as the term for the minimum sample frequency to prevent aliasing (the Nyquist criterion). I should add that "the horn" was a

*microwave*horn, not something you might find in the Alps. At high frequencies electromagnetic fields can be guided through suitably configured plumbing as it were.

In any case, I wanted to make it clear that there were more frequencies involved in the CMB than my June 17, 2019 post discussed. I had mentioned 57 GHz. You can see that the peak of the spectrum is around 160 GHz. Penzias and Wilson's 1965 article specifically dealt with the 7.35 cm flux, 4080 MHz or 4.080 GHz (their horn was tuned to that frequency, like trumpets with air, there are favored notes in electromagnetic radiation for the particular construction), which is pretty far out on the left tail of the distribution I graphed above.