The universe

In the 1920's, V. M. Slipher measured velocities of nearby galaxies. Hubble estimated their distances. Hubble (Hubble, E., 1929, Proc. Nat. Acad. Sci. 15, 168) found velocities v are proportional to distances D.

This is called Hubble's Law.
	v=H D

Hubble expansion is special

ListPlot[{{{0,0},{0,1},{2,0}},1.5{{0,1},{2,0}}},Epilog->{Text["MW",{0,0},{-1.5,0}],Text["A",{0,1},{-1.5,0}],Text["B",{2,0},{1.5,0}],Text["(Subscript[A, later])",1.5{0,1},{-1.5,0}],Text["(Subscript[B, later])",1.5{2,0},{1.5,0}]},AspectRatio->Automatic,Axes->False,ImageSize->300,BaseStyle->{FontFamily->"Helvetica",FontSize->Medium}]

Expansion by Hubble's Law is very special. Consider Milky Way, galaxy A at distance 1, and galaxy B at distance 2.
1. Expansion is by a scale factor. (Subscript[v, A])=1, and (Subscript[v, B])=2. Suppose some time passed, and A has moved by 0.5 to 1.5. Then B has moved by 1 to 3. B remains twice as far as A.
2. Centerless expansion.
A is (5^(1/2)) from B. MW is 2 from B.
After time passed, A is ((((3^2)+(1.5^2)))^(1/2))=(3/2)(5^(1/2)) from B, and MW is (3/2)2 from B.
Galaxy B is the center of the expansion too.
3. There exists a beginning, when the scale factor is 0. In this example, let time be -1.
4. Hubble did not find special directions. The universe is isotropic.

Puzzle

Can galaxies go faster than light? H=70km/s/Mpc. (A parsec is 180×3600/πAU.) A galaxy is at 6,000Mpc. Its speed v=HD=420,000km/s is faster than the speed of light. How is that possible? What could happen if I could go faster than the speed of light?

Isotropic & homogeneous spaces

The universe is the same everywhere (homogeneous) and the same in all directions (isotropic).

Simplicio: We see light from distant galaxies that were forming stars for the first time. There is not the same as here.

Can you think of a 2-d homogeneous & isotropic space?

Friedman-Robertson-Walker spaces

A 3-d space that is homogeneous and isotropic has a special choice of time. A choice of coordinates is
	(t,r,θ,ϕ)
and the metric is
	(ds^2)=-(dt^2)+((a(t))^2) [ d (r^2)/(1-(((r/(Subscript[r, 0])))^2))+(r^2)(((dθ)^2)+(sin^2) θ ((dϕ)^2))].
(r,θ,ϕ) is called the comoving coordinate. A galaxy stays at the same position; time changes.

(Subscript[r, 0]^2) can have any value, positive or negative.

a(t) is called the expansion parameter. Descibe the effect of the expansion parameter.

fig[α_:π/6]:=Module[{c={0,0}},ListPlot[c,AspectRatio->Automatic,PlotRange->{1.05{-1,1},1.05{-1,1}},Epilog->{Circle[c,1],Circle[c,.3,{π/2-α,π/2}],Text["ξ",.3{Sin@α/2,Cos@α},{0,-2}],Dashed,Line[{c,{0,1}}],Line[{c,{Sin@α,Cos@α},{0,Cos@α}}]},ImageSize->200,Axes->False]]

What is (Subscript[r, 0]^2)?

(ds^2)=-(dt^2)+((a(t))^2) [ d (r^2)/(1-(((r/(Subscript[r, 0])))^2))+(r^2)(((dθ)^2)+(sin^2) θ ((dϕ)^2))]

Let time be fixed. Consider the spatial part of the metric. 
	 d (r^2)/(1-(((r/(Subscript[r, 0])))^2))+(r^2)(((dθ)^2)+(sin^2) θ ((dϕ)^2))

Suppose (Subscript[r, 0]^2)->∞. What is the space?

Suppose (Subscript[r, 0]^2)>0. What is the space?
Map 3-d space into 2-d by supressing ϕ. The space is the 2-d surface of a sphere. I draw a slice.

fig[]

Define sinξ=r/(Subscript[r, 0]). (Subscript[r, 0])d ξ=d r/cosξ=d r/[1-(((r/(Subscript[r, 0])))^2)]. Therefore
	(ds^2)=(Subscript[r, 0]^2)d (latitude^2)+(r^2)d (longitude^2).

Lesson: If (Subscript[r, 0]^2)>0, then (Subscript[r, 0]) is the radius of curvature of the 3-d space. If ((Subscript[r, 0])^2)<0, the space is like a saddle, and the space is infinite.

Hubble's Law

Consider the case (r^2)≪|((Subscript[r, 0])^2)|. The radial distance of a galaxy at r is
	D(t)=a(t)r.
The galaxy moves away at speed 
	v=(d/(d t)) D(t)=(1/a)((d a(t))/(d t))a r=(1/a)((d a(t))/(d t))D.
Define Hubble's constant H=(1/a)((d a(t))/(d t)). Then
	v=H D
This is called Hubble's Law.