Coincidental Cosmic Forces

Continuing our inadvertent theme of "alternative physics", we contemplate whether the Force really is with us, and find the fundamental mass scale for MOND...

Earlier, we had a nice little discussion about the persistent fringe phenomenon of one of the alternative models for gravity that persistently bob up in astrophysics...

Specifically MOND.

Now, the details of MOND are kinda irrelevant, what is a key point is that MOND provides a phenomenological argument for a fundamental acceleration scale, a₀ ~ 10^-10 m s^-2.

Whether this is really "fundamental" is disputable, and at some level semantics, the basic point is that looking at some structure of the scale of galaxies or so, there is an indication of either massive coincidence or unmodeled physics at this scale.

So, accelerations are kinda awkward, since they are unit dependent so it is interesting to find what happens when you throw other fundamental constants at a₀.

And, the first thing you discover is t_H ~ c/a₀
where c is the speed of light and is of course equal to 1
and t_H is the Hubble time, or the approximate current age of the universe.
Now, after the second beer, you contemplate that this must either be a coincidence or indicate dynamics.
As Ben pointed out in the comments, the natural thing to do then is to fudge it into dark energy dynamics.

So... well, Ω_Λ ~ 0.7 is the approximate current energy density of the universe due to the dark energy, as a fraction of the critical density.
The critical density, at the level of blog precision, is about ρ_c ~ 10^-26 kg m^-3 - yes, this requires real units.

But... an energy density is really a force per unit area, at least classically, and since we are playing in a blog with post-Newtonian modifications, that is an acceptable level of conceptual approximation.

What force, we ask: well, if I did me 'rithmetic rite (and if I did not you will be sure to let me know), then this is about 10^-8 (erg/c²) cm^-3, per unit area.

What area - well the natural scale is L ~ c t_H ~ c²/a₀

So, there is a natural mass scale, M_M ~ ρ_c c² L²/a₀

Or M_M ~ 10⁵⁴ gm

Which is approximately the mass of observable universe, up to factors of π etc which I neglected.
Which it should be, because of the coincidence that t_H ~ c/a₀

But this is a little bit strange, because a₀ is a measured parameter, it is the characteristic acceleration at a 2-3 scale lengths for disk galaxies, which naively is not something tightly coupled to the mass of the universe as a whole.

In other words, the MOND acceleration is roughly the acceleration of the horizon of the observable universe, due to the dark energy.
Of course the universe is not Newtonian and a unique well defined relativistic MOND is not at hand, but now see why people contemplate f(R) extensions of general relativity etc as possible alternative to CDM and Λ
There are about 500 f(R) papers on ADS - comparable to the MOND literature.
2% of the former mention the latter in the abstract ;-)

Or not.