Lecture 20: Cosmology

Reading Assignment: Arny: Chapter 16 & 17, Cosmos: Chapter 10.

Olbers' Paradox

Last week we discussed the problem of why the sky is dark? If we assume the Universe is
  1. infinite in extent
  2. homogenously filled with matter
  3. inifinitely old
Then we are faced with the paradox that the night sky should be as bright at the surface of the Sun and we should be fried by the intense radiation.

Assuming there is obscuring dust doesn't help. It would just heat up and re-emit all of the radiation. In such a Universe it must reach thermodynamic equillibrium: the same temperature everywhere.

So, why is the sky dark at night? We think that it is because the Universe is not infinitely old (c). The Universe has an age somewhere between 10-20 billion years.

The Big Bang Theory

We observe the Universe to be expanding. Galaxies that are farther away from us are moving away from us faster than those nearby.

Hubble's Law: v = Hod

There is no center of the expansion (infinite raisin bread) and all observers see the same thing.

The density of the Universe decreases as the Universe expands.

We define a scale factor, R, which can be thought of as the distance between any two clusters of galaxies. This scale factor is getting larger with time (the Universe is expanding).

Run the expansion backward in time. Density and Temperature increase as R decreases. When R = 0, density = infinity, temperature = infinity (a singularity!!!).

We call this the Hot Big Bang. Which is a misnomer. It was actually a name given the theory by a Fred Hoyle who was trying to discredit the theory. Unfortunately the name has stuck.

The Big Bang was not an explosion. An explosion implies the sudden release of energy and expansion into space. The expansion of the Universe is not an expansion of matter into space, but rather an expansion of space itself. Therefore the Big Bang did not happen at some point in space, but rather everywhere in space at the same time.

The Age of the Universe

If the expansion happens at a constant rate (i.e., a galaxy never slows down) then we can estimate the age of the Universe from the current expansion rate (Hubble's constant).

We know that d = vt. And v = Hod. Therefore, the age of the Universe is to = 1/Ho.

If Ho = 50 km/s/Mpc then,

to = (1/50)·(Mpc/km)·s = 2x10-2s·(1 Mpc·3.086x1013 km/pc· 106pc/Mpc /km)

= 6.17x1017 s ·(1 yr/3.16x107) = 1.96x1010 yrs = 19.6 billion years.

If Ho = 80 km/s/Mpc then,

to = (1/80)·(Mpc/km)·s = 1.25x10-2s·(1 Mpc·3.086x1013 km/pc· 106pc/Mpc /km)

= 3.86x1017 s ·(1 yr/3.16x107) = 1.22x1010 yrs = 12.2 billion years.

This is called the Hubble Time.

(if we ignore the effects of a possible Cosmological Constant). Then we must conclude that the expansion was faster in the past: gravity is slowing it down. So the Universe must be in fact younger than the Hubble Time.

Globular stars clusters are thought to be between 11-15 billion years old: Possibly inconsistent with the expansion age. The discrepancy was worse in the past, but new data is causing us to date the globular clusters a bit younger and find the Universe a bit older. So things are starting to work out. STAY TUNED!!

Geometry of the Universe

We attempt to understand the expansion of the Universe using Einstein's Theory of General Relativity.. To do this we use a concept called the Cosmological Principle.

The Cosmological Principle states that on the largest scales the Universe must look the same for any observer in space. The Universe must therefore be Homogeneous (same average density) and Isotropic (looks the same in all directions).

Evidence to support these assumptions: (1) Largest structures are superclusters. Beyond that things seem to smooth out. (2) Cosmic "Black body radiation" the same everywhere.

NOTE: density can change with time. These assumptions of for all space at any given time.

The results are that there are 3 types of possible homogeneous and isotropic Universes and the global geometry of the Universe can be 1 of 3 possible geometries:

  1. Flat (Euclidean Geometry)
    2-D example: plane
  2. Positive Curvature
    2-D example: sphere
  3. Negative Curvature
    2-D example: infinite saddle or Pringles potato chip
The geometry of the Universe is determined by the matter/energy content of the Universe. Recall that we discovered that mass warps spacetime, and that since energy is equivalent to matter (E = mc2) that it too can warp spacetime. Locally the Universe can have any of the three geometries. On the surface of the Earth we have mountains and valleys, but its global geometry is that of a sphere. In an analogous way the Universe has some global spacetime curvature.

There is a critical average density (matter + energy) that makes the Universe globally flat. It is calculated from Einstein's equations:

crit = 3H2/8G
Where H is the Hubble parameter which evolves with time. At the current epoch we call it Ho. So clearly the critical density evolves with time as well.

If we choose Ho = 65 km/s/Mpc, then the current critical density is

crit,o = 8 x 10-30 g/cm3

> crit Positive Curvature: Closed. Volume is finite but unbounded.
= crit Zero Curvature: Flat. Volume is infinite.
< crit Negative Curvature: Open. Volume is infinite.

Deceleration and Acceleration

Gravity causes all 3 models of the Universe to decelerate. Whether or not the expansion ever stops and reverses depends the average density of the Universe (if there is no Cosmological Constant).

If < crit Universe Expands Forever.
If = crit Universe Expands Forever, but stops at time equal to infinity.
If > crit Universe Eventually Stops Expanding and Collapses Back to a Hot, Dense State. (Big Crunch).

If the Universe Expands Forever, eventually all galaxies will redshift away from our cosmic horizon. Eventually, star formation comes to an end as all matter is locked up in white dwarfs, brown dwarfs, planets, and other such debris. After 1045 years the protons and neutrons that make up atomic nuclei decay into neutrinos, electrons, and photons. All that remains with any significant gravity are black holes. After 10100 years black holes will have evaporated via Hawking radiation. The Universe is very cold and empty, containing only photons and leptons (electrons, neutrinos, etc.).

If the Universe Stops Expanding and Collapses, eventually we will see all of the galaxies go from all being redshifted to blueshifted. The Universe will get hotter and denser as space gets smaller and smaller. Eventually all life will be extinguished in the overwhelming temperatures (matter becomes completely ionized and then broken up into its fundamental particles) and the Universe will collapse to some incredibly small size of unknown proportions. Some speculate that from this dense and hot state a new Universe will born from the ashes of the former. Some call this the Phoenix Universe.

Return to Class Notes Page