we're fine. it is the rest of the world that has the problem

P.S.: I promise you that, when I find the mood, I'll try to understand co-moving and proper

distances, and inflation & GTR and its impact on the observable universe of 45.6 Gyl radius.

I'm going to hold you to that, starting now!

First, the radiation constant, [math] \alpha \quad =\frac { 8{ \pi }^{ 5 }{ k }^{ 4 } }{ 15{ c }^{ 3 }{ h }^{ 3 } }[/math]

Where c is the speed of light, k is Boltzmann's constant, and h is Planck's constant.

Numerically [math]\alpha \quad =\quad 7.5657\quad E-16\quad J\quad { m }^{ -3 }\quad { K }^{ -4 }[/math]

Photon energy density = [math]\alpha { T }^{ 4 }[/math]

[math]{ T }_{ now }=\quad 2.728\quad K[/math]

So, energy density of the CMB photons is [math]4.19\quad E-14\quad J/{ m }^{ 3 }[/math]

All we need now to find the total energy of the CMB photons is the volume of the observable universe.

The Radius is 46 E9 lyrs, = 4.35 E26 m

[math]Volume\quad =\quad 4/3\quad \pi \quad { r }^{ 3 }[/math] = [math]3.453\quad E80\quad { m }^{ 3 }[/math]

**Total energy of CMB photons = 1.45 E67 Joules**

The object now is to show that this value is conserved; that it is the same now as at the time of last scattering.

To do that, I use the linear scaling factor of 1100. That is, at last scattering the universe’s linear dimensions were 1100 times smaller than today.

That scaling factor also applied to the Temperature, so that 2.728 K today corresponds to 3000 K at decoupling and last scattering.

So, [math]\alpha { T }^{ 4 }[/math] at that time was [math] 0.0613\quad J/{ m }^{ 3 }[/math]

Now, here is where it is easy to go wrong . . .

If you calculate the volume of the universe at last scattering by taking the cube of the scaling factor, and dividing that into the present-day volume, you get:

[math]\frac { 3.453\quad E80\quad { m }^{ 3 } }{ { 1100 }^{ 3 } } =\quad 2.594\quad E71\quad { m }^{ 3 }[/math]

That is of course the correct volume at last scattering, BUT when multiplied by the energy density of [math] 0.0613\quad J/{ m }^{ 3 }[/math], that will **not** get back to the total value of energy that must be conserved, that was calculated to be 1.45 E67 Joules.

So what gives?

Well, here is a paper that explains why, but the **only** part that you need to read is the first few sentences, which I quote here:

*“According to present cosmological views the energy density of CMB (Cosmic Microwave Background) photons, freely propagating through the expanding cosmos, varies proportional to 1/S^4 with S being the scale factor of the universe. This behavior is expected, because General Theory of Relativity, in application to FLRW- (Friedmann-Lemaitre-RobertsonWalker) cosmological universes, leads to the conclusion that the photon wavelengths increase during their free passage through the spacetime metrics of the universe by the same factor as does the scale factor S . This appears to be a reasonable explanation for the presently observed Planckian CMB spectrum with its actual temperature of about 2.7 K, while at the time of its origin after the last scattering during the recombination phase its temperature should have been about 3000 K, at an epoch of about 380 ky after the Big Bang, when the scale of the universe S r was smaller by roughly a factor of S/S r = 1+zr = 1100 compared to the present scale S = S 0 of the universe”*

**(I strongly advise that you do NOT read the rest of this paper)**

So, you see, while the volume varies according to the cube of the scaling factor, the energy density varies according to the fourth power!

The simplest way to apply this is to calculate an “effective volume” of the early universe:

[math]\frac { 3.453\quad E80\quad { m }^{ 3 } }{ { 1100 }^{ 4 } } =\quad 2.358\quad E68\quad { m }^{ 3 }[/math]

Now, when this is multiplied by the energy density of [math] 0.0613\quad J/{ m }^{ 3 }[/math],

We find that the total energy of the CMB in the early universe was **1.45 E67 Joules same as today.**

I hope that clears up all of your questions while opening up even deeper doubts and confusions!