The above statement is quite simply wrong! 13.77 billion years is the light travel time.
The observable universe, also known as the Hubble volume, is the region of space that it is theoretically possible for us to observe, small enough that light from the furthest regions has had sufficient time to reach Earth since the Big Bang. This region of space has a diameter of approximately 92.94 billion light-years, centered on the planet Earth. Each different portion of space has its own visible universe, some overlapping, some not.
I gave you the Wiki link with the size of the observable universe: According to calculations, the current comoving distance—proper distance, which takes into account that the universe has expanded since the light was emitted—to particles from which the cosmic microwave background radiation (CMBR) was emitted, which represent the radius of the visible universe, is about 14.0 billion parsecs(about 45.7 billion light-years), while the comoving distance to the edge of the observable universe is about 14.3 billion parsecs (about 46.6 billion light-years), about 2% larger. The radius of the observable universe is therefore estimated to be about 46.5 billion light-years and its diameter about 28.5 gigaparsecs (93 billion light-years, 8.8×1023 kilometres or 5.5×1023 miles)
There are countless other links saying the same thing.
As I said, when you are calculating the total energy, or the energy density, it makes no sense at all to use the radius of the light travel time, and I don't know of any cosmologists who would do that.
I tell you what; go ahead and do your calculations for energy density and total energy, using the 13.77 Blyr radius as your size, then work out the total energy of the CMB, then work that back to the time of last scattering and show what the size of the universe was then and that the total energy of the CMB is the same, conserved.
Show the numerical answers and show the energy is the same then and now, if you can.
Maybe you can convince me you are right?
When you are done I will post my numbers for comparison.
This is nothing personal, but we should both want to get to the bottom of this for a better understanding.
Of course it's not personal! I agree with you!
My problem is this:
1) When I read data from NASA or academic sources (heavy stuff, like the link I provided), they talk about
the Hubble's Radius, at which galaxies start to recede faster than light.
They call this radius as the one for VISIBLE Universe. So, COBE, WMAP and PLANCK maps of anisotropies
are a 2D projection of a sphere with the HUBBLE RADIUS.
2) When relativity and inflation enter into the picture, it appears a OBSERVABLE RADIUS. But, at the same time,
cosmologists agree that TODAY it's impossible to get signals beyond VISIBLE RADIUS.
I believe that even them don't know what they are saying.
Another thing: Co-moving and proper distances appear when they talk about the OBSERVABLE universe.
And both distances are ALMOST IDENTICAL (they tell why, but I don't get it).
Finally, I know when to stop. I think that I'll never will be able to understand the "observable" radius, specially
in the way it's defined: It exist, but nothing can be measure at its edge with TODAY's TECHNOLOGY.
And that excuse just kill any desire I can have to understand it, because I feel that it's completely non-logical.
I have NO FORMATION in astronomy, astrophysics or cosmology. I only know things that I read since childhood
with books like Isaac Assimov's "The Universe", some Atlas books on stars, constellations and galaxies, and my
own inference using my technical basis on math and physics, plus analytical thinking.
Only recently, when I wrote a 30 pages document about the history of the Black Body Radiation (1859-1900), I
became aware of Stefan, Wien, Planck and others whose works are currently applied at astrophysics and cosmology.
Also, don't forget that I'm a non-believer on relativity (none of them, but specially GTR), black holes and the BBT plus inflation.
The fact that I'm fooling around these theories, RESPECTING THEM, is because I need to crunch some numbers to avoide being bored.
Also, I can develop my own CONCEPTS (not theories) about the Universe and the building of physics in general. I find interesting to
criticize what seems unlogical for me.
No other agenda than what I expressed above. Just a curious but renegade mind fooling around.
But one thing is certain: If something is ILL DEFINED, even when I dislike the original idea, I express myself
against it, showing the fails, paradoxes or fallacies.
And I do this just for fun, as a "food for thought".
Why don't you try to enter at NASA COBE's and WMAP's sites or at ESA's PLANCK satellite mission
and try to find out which is the radius they used for the different anisotropies maps?
I think that you'll be surprised with the answer. One thing is astrophysics (the engineering branch of
astronomy) and a very different thing is cosmology (the philosophical branch of astronomy).
I hope you may understand me (I dropped Wiki as a serious source of information almost when it started,
and because of that I seek data at academic place or gov. institutions).
Are we OK?
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.