It can be boiled down to a simple assertion:
"We're nothing special."
Earth's location is not special.
This principle is generalized under Relativity theories to apply to the whole universe, as the cosmological principle:
"There are no special places".
Mathematical physicists have applied the (currently) most-widely employed solutions to the Einstein equations, the Friedman Lemaitre Walker Robertson (FLRW) solutions, in order to derive the so-called "balloon universe" we have all heard advanced as an explanation for why our cosmological observations have always seemed to put us in the center of it all.
In this "balloon universe" explanation, the two dimensional surface of the balloon serves as an analogy for the three dimensional space of an expanding universe.
If we take a balloon, put a bunch of dots on its surface, and start to blow it up, we will notice that all the dots expand away from each other as the surface area expands.
If we were to imagine the dots as galaxies, and ourselves as observers looking out from one of them, we would think we were in the center, since we would observe all the other dots expanding away from us in all directions.
This "Friedman" universe is the basis upon which rests a recent demand of a correspondent that I essentially, "shut up and calculate"; that is, that I stop dealing with the metaphysical absurdities which have sprouted like mushrooms in the standard model of cosmology, and instead show what something like the CMB Axis would mean mathematically for FLRW equations.
It was a clever move, since of course the arcane mathematics of Relativity has always been the final thicket that mere mortals- you know, the Tom, Dick, and Harrys who actually pay for all of this stuff- can never hope to penetrate.
But, thankfully, someone else has recently been thinking along these precise lines.
His name is Yukio Tomazawa, and he is a professor of physics at the University of Michigan ( Michigan University has really been the center of Axis research).
In August of last year, he posted a paper which explicitly addresses the implications of the CMB dipole for FLRW mathematics.
The assertions are quite remarkable.
"1) In a B-type universe, nobody observes a cmb dipole. Not even a peculiar velocity yields a cmb dipole."
The math Tomazawa presents in his paper leads to the astonishing conclusion that in a B (balloon) type universe, nobody would see a CMB dipole!
Since we do see a CMB dipole.........yup.
Tomazawa's math says that the balloon-type FLRW universe exists not in reality, but instead only in FLRW math.
It gets much more interesting.
Tomozawa matter-of-factly delivers this bombshell a little further in:
"II) If the observed cmb dipole and the peculiar velocity of the solar system coincide, as is assumed among some physicists, the solar system must reside at the center of the universe."
It appears that the geocentric hypothesis is getting quite a bit more attention these days!
Now Tomozawa goes on to advance other observational evidence he thinks shows we are not exactly at, but only relatively close to, the center, but his point above is remarkable nonetheless because the consensus today among physicists is that the observed CMB dipole and the peculiar velocity of the solar system *do* coincide!
My interlocutor in the ongoing geocentrism debate says exactly that, for example.
When I brought Paul's demand that I essentially ought to "shut up and calculate" to another physicist, whose doctoral dissertation was on General Relativity, I received the following, very relevant response: