Flatness Problem
Updated: October 21, 2020
Overview #
In the Big Bang model there are multiple parameters which appear to be “fine tuned” in that, small changes to a given parameter have massive effects on the evolution of the universe as a whole.
Wrt the flatness problem itself, the questionable parameter is the Density Parameter (Ω), which is defined as
Ω≡ρρc
where the subscript c is in reference to the “critical value” of density (ρ). Which, today, is measured to be–roughly–five atoms of Hyrodgen per cubic meter. ρc is defined:
ρc=3H28πG
Remembering, of course, that density (ρ) is defined as:
ρ=mV
which implies
Ω≡ρρc=8πGm3H2V
This value Ω determines the shape (curvature) of the universe, i.e.:
Ω<1
Ω=1
Ω>1
For each case:
- the universe will be open (hyperbolic), will expand forever
- the universe will be flat
- the universe will be closed (spherical), will eventually stop expanding then collapse
What’s the problem? #
If Ω is not equivalent/near unity, the universe will undergo expansion or contraction. This is problematic because taking eq. 3, above, we have a variable volume (due to the contraction or expansion), which shows that any deviation from unity in the density parameter will have acclerating changes in the shape of the universe over the course of the universe’s timeline. Working backwards given the universe’s age, we find that the original density must have only deviated from ρc by less than one part in 1062. This, naturally, leads one to ask,
Why is the universe’s density sitting so close to this critical value?