Re-cap, ...

Force Law

Newton's Law ===> R(t)**2 d[dR(t)/dt]/dt = - 4 pi G rho(0) / 3

Energy Conservation

• -k/2 = 1/2 [dR(t)/dt]**2 - 4 pi / 3 G rho(t) R(t)**2

• This expression sound be familiar. Recall that the energy of a particle in a gravitational field is E = K.E. + P.E. = 1/2 mv**2 -GMm/r ===> E/m = 1/2 v**2 - G [4 pi/3 density r**2]

  This is quite suggestive. The constant k in the Newtonian theory is a measure of the total energy (kinetic plus potential) of the mass m. A GR calculation leads to the same relationship given above, however, the constant -k/2 defines the curvature of the universe (and not the energy) and a new term can appear (which contains the Cosmological constant)

• -k/2 = 1/2 [dR(t)/dt]**2 - 4 pi / 3 G rho(t) R(t)**2 - CC/3 R(t)**2

Here CC is the Cosmological constant (usually represented by the capital Greek letter Lambda). Solutions for the Universe without Lambda (the CC).

Fate of the Universe

This notion that an unbounded Universe has positive energy (negative curvature), leads us to a natural test to determine the ultimate fate of the Universe. Recall:

• - k = [dR(t)/dt]**2 - [ 8 pi / 3 ] G rho(t) R(t)**2

• v = H(now) distance (Hubble's Law) ===> R(0) dR(t)/dt = H(now) R(0) R(t) ===> dR(t)/dt = H(now) R(t) ===> -k = [H(now) R(t)]**2 - [ 8 pi / 3 ] G rho(0) / R(t)

• So, since the critical universe has k = 0

  ===> rho(0) / R(t)**3 less than H(now)**2 / [8 pi G / 3 ] ===> the Universe is open (negative curvature).

• The critical density of the Universe is then given by
  • rho(critical) = 3 H(now)**2 / [8 pi G]
• For H(now) = 25 (km per second) per Million light years,
  • rho(critical) = 1.3 x 10**(-29) grams per cubic centimeter
  or 1 hydrogen atom per 130,000 cubic centimeters. There are 10**18 - 10**19 molecules per cubic centimeter in this room.

• Define Omega = rho(t)/rho(critical), that is, Omega (as used before) is the ratio of the current density of the Universe to the current critical density.

• The current best measurements for rho(Universe) are
  • visible matter, Omega ~ 0.007
  • matter in galaxies, Omega ~ 0.007
  • Clusters of Galaxies, Omega ~ 0.1 - 0.3

  The preceding is a nice test because all we need to do is measure the current density and expansion rate of the Universe. Much of our current observational efforts are directed toward this seemingly simple task. I will have much more to say about this exercise later.

So, our simple test to determine the fate of the Universe is to:

• Measure H(now) ===> expansion rate of Universe
• Measure Rho(now) ===> how large the escape speed of the Universe is

(Recall that we can also try to measure the topology of Universe: Hubble Relation; Density; and Angular Size

Evolution of the Universe