Why does mass of observable universe match distance squared in early universe model?

[dedocta]

TL;DR

Playing around with a dark matter theory, noticed (mass of obs Universe / distance squared) approaches unity

I was playing around with a theory that dark matter was behind the opaque wall of the early universe, as gravity would not be opaque.

Not sure if the numbers fit yet, but one odd thing I calculated was the distance squared of 46 billion light years, as the mass of the early universe, by time, is interacting with us via a thin shell beyond the darkness due to the decreased volume of the Universe ~1 million years in.

Anyways, the number comes out to 1.89*10^53 m^2, while the mass of the observable Universe is estimated to be ~1.5*10^53 kg. m/d^2 would approach unity based off of our mass estimates of the Universe.

Is there any reason those numbers come out like that, or is it mere coincidence?

 

[PeroK]

What precisely is this calculation you've done?

 

[dedocta]

Well the first calculation I was doing is not appropriate due to the shell theorem, however, basically if you had a point source of mass at the edge of the Universe, I was wondering what you would get out for force and acceleration at that distance.

F=Gmm/r^2, so the force would be proportional to 5x missing mass of the Universe at that distance. I converted 46 billion light years to meters, which is ~4.35*`10^26 m, square that and you get
~1.89*10^53 m^2. Since that's about 1:1 observable mass:radius of obs Universe squared, a = Gm/r^2 so acceleration would be ~ 5G which again seemed funky!

It's not appropriate, as there's no net force in the shell theorem. However, there's still some kind of affect on the gravitational potential, no? A clock would tick slower in this scenario of a shell of mass surrounding us at 46 billion ly vs a scenario without that (i.e. assumed homogeneity.)

 

[dedocta]

Oh! I left something out, the radius to the edge of the observable Universe is pretty much the radius including what would be beyond the dark opaque wall, as radius of the first seconds of the Universe temporally, would lie maybe @ most 4-10 million lightyears of comoving distance beyond observable radius? I'm not sure how to even calculate that, but I think the radius of the observable Universe >>> distance of the earliest epoch (dark) minus distance of observable Universe. Sorry for edits, first time I'm trying to put these thoughts together...

 

[Nugatory]

Is there any reason those numbers come out like that, or is it mere coincidence?

It is a coincidence and has no physical significance - the pejorative term for stuff like this is “numerology”.

One way of seeing this is to consider that the ratio only approaches one (to the extent that it does) if we choose to measure distances in meters and masses in kilograms - we can make the ratio come out to be anything we please just by choosing different units. And clearly the specific number cannot have much cosmic significance if it’s based on minutiae like the orbital period of one planet that matters only to us.

Generally the real physics is in numbers that don’t change with our arbitrary choices of units. For example, the mass of the electron is $9.10\times 10^{-31}$ kilograms; that tells us more about the kilogram than the electron. On the other hand, the mass of the electron is $5.45\times 10^{-4}$ times the mass of the proton; this will be true no matter what units we use, and has a lot to do with why atoms behave the way they do.

 

[Nugatory]

As this thread is based on a misunderstanding we have closed it.