Critical density of the Universe

Critical density of the Universe
The Hubble constant (H) is also important in
predicting the ultimate fate of our universe,
related to the concept of its critical density.
One galaxy
escaping from all
the rest
The velocity of recession of a galaxy can be
considered as its escape velocity from the rest
of the Universe
By Hubble’s formula:
Velocity of recession (v) = HR where R is the
distance of the galaxy.
But the escape velocity is given by the formula:
Escape velocity (v) = [2GM/R] so v2 = 2GM/R
Mass of the “rest” of the
Universe (M)
where G is the gravitational constant (see
Gravitation section)
v2 = 2GM/R = [2G4/3]R3]/R = 8/3[GR2 ]
But from the first equation v2= H2R2 and so:
Critical density = 3H2/8G
[We have assumed both constant v and a constant value of the Hubble constant (H) in this
simplified calculation.]
If the density of the Universe is greater than this the universe will contract, if it is less it will
expand for ever!
Take H = 70 kms-1 Mpc-1 then
Critical density ( = 3H2/8G = [2.27x10-18]2x3/[8x3.14x6.67x10-11] = 9.22x10-27 kgm-3
The mass of a proton is 1.66x10-27 kg so this density is equivalent to just over five protons in
every cubic metre of space! (actually 5.56 protons).