Undergrad Accelerating Universe and spherical distribution of matter

[Ranku]

If we take a spherical distribution of matter wherein gravitational force and cosmological-constant force are equal upon an object on its surface, then does the time that it took for that volume to grow to the size wherein the two forces are equal match the time it took for the universe to start to expand acceleratingly?

 

[anuttarasammyak]

Einstein introduced cosmological constant to realize the case you say. He intended to explain eternal constant universe by it i.e. without both before it and after it. I suppose this balanced case is not stable and not on the path of progress of universe though I have no mathematical credit here now.

 

[PeterDonis]

If we take a spherical distribution of matter wherein gravitational force and cosmological-constant force are equal upon an object on its surface

Gravity and the cosmological constant aren't "forces" in GR. By "forces are equal", do you mean that an object on the surface of the matter distribution is staying at the same radial coordinate and is in free fall (i.e., feels no weight)?

the time that it took for that volume to grow to the size

Why would you expect the volume to be growing? What solution of the EFE are you talking about?

 

[Ranku]

Gravity and the cosmological constant aren't "forces" in GR. By "forces are equal", do you mean that an object on the surface of the matter distribution is staying at the same radial coordinate and is in free fall (i.e., feels no weight)?
Why would you expect the volume to be growing? What solution of the EFE are you talking about?

We start with the second Friedmann equation

a"/a = - 4πG/3 (ρ+3p/c²) + Λc²/3

which can be recast as the equation of motion of mass m on the surface of an expanding spherical distribution of matter M and radius R ≡ a

R" = - GM/R² + Λc²/3 R

We would therefore expect that there is a 'standard volume' of mass M where gravitational and cosmological-constant acceleration upon m would be equal at a certain radial distance R.

My question is if we start at the time where galaxies were first formed and consider a 'minimal' volume of M, and as the universe expanded and the 'standard volume' of M was reached, whereby gravitational and cosmological-constant acceleration were equal upon m, does the time it took for that to happen roughly correspond to the age of the universe when it started to expand acceleratingly? In other words, should not there be a correlation between when local acceleration of m starts in standard volume M due to cosmological-constant acceleration, with when the accelerating expansion of the universe started?

 

[PeterDonis]

We would therefore expect that there is a 'standard volume' of mass M where gravitational and cosmological-constant acceleration upon m would be equal at a certain radial distance R.

You are misunderstanding what this equation is telling you. This equation is telling you that the Einstein static universe is a solution of the field equation with cosmological constant. If you have the right conditions for $R'' = 0$, those conditions are true for every value of $R$ at that same instant of time. So the entire universe is static at that instant of time.

However, this static state is unstable; a small perturbation in either direction leads to expansion or collapse. That also means that, even in a universe in which there are values of the matter/energy density and $\Lambda$ that make such a static state reachable, it is astronomically unlikely to actually be reached; it would require astronomically precise fine-tuning of conditions.

Finally, if you do the math to calculate what value of the cosmological constant $\Lambda$ would be required to satisfy the $R'' = 0$ condition with the density of matter and energy in our present universe, you would find that that value is much larger than the actual value of $\Lambda$ we observe. So our universe does not even have appropriate values of density and $\Lambda$ to make the static condition reachable at all, even in principle.

 

[Ranku]

Finally, if you do the math to calculate what value of the cosmological constant Λ would be required to satisfy the R″=0 condition with the density of matter and energy in our present universe, you would find that that value is much larger than the actual value of Λ we observe. So our universe does not even have appropriate values of density and Λ to make the static condition reachable at all, even in principle.

Do you mean R"=0 is unattainable in a 'local' spherical distribution of matter, such as in a group of galaxies?

 

[PeterDonis]

Do you mean R"=0 is unattainable in a 'local' spherical distribution of matter, such as in a group of galaxies?

A "local" spherical distribution of matter, such as a group of galaxies, surrounded by vacuum, is not described by the Friedmann equation, so none of what you've said is relevant to such a case.

 

[Ranku]

A "local" spherical distribution of matter, such as a group of galaxies, surrounded by vacuum, is not described by the Friedmann equation, so none of what you've said is relevant to such a case.

So do you mean that when we consider a spherical distribution of matter, we are considering all the matter in the entire observable universe?

 

[Ibix]

If you are applying the Friedmann equations then you are considering a universe completely filled with uniform density matter. That's spherically symmetric, rather than a spherical distribution of matter.

If you tune $\Lambda$ to balance out expansion, as Einstein did and you seem to be trying to do, you will find that any region of any shape anywhere does not expand.

 

[PeterDonis]

do you mean that when we consider a spherical distribution of matter, we are considering all the matter in the entire observable universe?

Not just the observable universe, the entire universe. That is what the Friedmann equations apply to: an entire universe filled with matter whose density is the same everywhere at an instant of time.