Link between Universe mass density and Cosmic Microwave Background energy

[johne1618]

Present critical density of Universe

rho_crit = 3 H^2 / 8 Pi G

H = Hubble Constant = 2.2E-18 sec^-1

rho_crit = 8.6E-27 kg / m^3

What particle mass does this represent?

The length scale associated with a quantum particle of mass m is
the Compton wavelength lambda where

lambda = h / m c

This translates to the following density:

rho_particle = mass / volume

rho_particle = mass / lambda^3

rho_particle = m / (h / m c)^3

rho_particle = m^4 c^3 / h^3

Let us equate the two densities:

rho_particle = rho_crit

m^4 = h^3 * 8.6E-27 / c^3

m = 1.7E-38 kg

m = 1 meV / c^2

The Cosmic Microwave Background has a photon energy of 1 meV.

Surprising coincidence or not?

I suppose this is not surprising if the mass and radiation energy at
the recombination time were similar.

Or maybe it is surprising as mass density is supposed to scale as R^-3 whereas radiation density scales as R^-4.

 

[zhermes]

What's the coincidence?
What are you asking here?
What is the purpose of this?

 

[johne1618]

I'm not sure what I'm getting at!

 

[Naty1]

johne: why not try to explain to yourself, that is write down and maybe post, what each of your results means to you...make an interpretation of the math...

When I was in college I always found I learned the most when I tried to explain to myself in my own words what I was trying to learn...