Summing things up
The last few posts had a lot of Mathematics in them. But, what is the bottom line?
1) The expansion of the universe is the
expansion space itself and not the motion of matter away from each other through space. It's a bit like the pictures getting magnified below,
2) Because everything is getting zoomed together, it's very convenient to express cosmology in terms of the
comoving coordinate that zooms out along with space. Thus while actual physical distance between galaxies increase with the zooming of space, in comoving coordinates, they are still in same coordinate location as shown below,
3) At the large scales involving thousands of galaxies, observation shows that the distribution of galaxies are approximately uniform. This makes it easier to calculate a first-pass analytical physical theory of the universe. The final expressions are:-
(Square of the Hubble Parameter giving the Rate of expansion at a given time)
= (Density of Matter and Radiation in the Universe at that time) - (Curvature of Space Divided by the Square of Length Scale of the Universe at that time) + Cosmological Constant of Einstein
4)Also using energy balance we get:-
(Rate of Change of Density of Matter and Radiation with time) /( Hubble Parameter) + (Pressure exerted by Matter and Radiation) + (Density of Matter and Radiation at that given time in the Universe ) = 0
5) We also saw that while matter and Radiation slows down the rate of expansion, the cosmological constant tends to speed it up. This explains the current acceleration of expansion rate that cosmologists have observed.
Next we shall look at how matter, radiation and cosmological constant individually affect the physics of the universe. For this we shall look at the energy balance equation for expansion given by point (4) above. In mathematical terminology we write this as (see
Post)
Fluid equation
ρ' + 3*(a'/a)*(ρ + p) = 0
First we look at the solution of fluid equation for matter.
For matter the pressure is given by
p= ρ*u^2 where
u is the characteristic thermal vibration velocity of the matter particle relative to the speed of light.
Now most matter particles have considerable mass and have relatively low energy such that their thermal vibrational speed is much much less than the speed of light.
Hence u«1. Hence for matter p~0. This makes the fluid equation very easy to solve for matter.
If
ρ° is the current matter density of the universe, the solution of the fluid equation for matter for a flat expanding universe with near zero curvature gives
ρ°/ρ = a^3
where a is the scale factor or zoom factor for the universe.
Then solving the Friedmann equation provides the change in the scale of the universe with time as,
a = (t/t°) ^(2/3)
where
t° is the current time of the universe where the
present length scale (or size scale) of the universe is taken to be a° = 1.
Thus for example at half the age of today, the size of the universe was
a = (0.5)^2/3 = 0.63 times the current size of the universe.
And the Hubble parameter is given by
H= a'/a = 2/(3t)
Thus the Hubble parameter continues to stay positive but decreases towards zero as age of the universe increases towards infinity.
However the above results are only for matter alone. We have to add the results from radiation as well as cosmological dark energy to get the complete picture of the behavior of the universe.
More of that in the next post.