Big Bang Theory

• Figures borrowed from:  Ned Wright's Cosmology Tutorial

Outline

• Einstein's first attempt at a cosmological model
• the Standard Big Bang Model
• the Age of the Universe

Einstein's Attempt

• Assume a homogeneous and isotropic Universe
• Galaxies and clusters of galaxies are considered a small correction to the overall structure of the Universe.
• They're like little specs following the overall evolution of the Universe
• This is an approximation
• assume no spatial boundary
• Universe is infinite in space or closed like a sphere
• Using these assumptions, Einstein tried to find a solution
• He wanted a static solution - one that doesn't change with time
• but as you know, gravity sucks
• that is, with ordinary matter, gravity is always an attractive force
• In Einstein's Theory
• gravity sucks in space-time as well as ordinary matter
• a static solution is not possible
• contracting solutions are ok
• expanding solutions are ok - as long as the expansion is slowing down.
• This was Einstein's big chance to predict the expansion (or contraction) of the Universe
• but he blew it!
• he decided to fudge his equations instead!!
• He took his beautiful field equation: 
• and changed it to :
• adding the ugly "cosmological constant" term: 
• the   term describes the matter in the Universe
• It is always positive for ordinary matter
• the - sign in from of the   term means that  provides a repulsive gravitational force
• so gravity doesn't just suck any more
• You don't need to remember the equations here - just remember that the cosmological constant can provide a repulsive gravitational force.
• With the Cosmological constant term, , it is possible to balance the attractive force due to ordinary matter with the repulsive force of the cosmological constant
• Einstein found his static model
• The balance was unstable
• like a pencil balanced on its point
• an extremely tiny push will make it fall
• remember that this is only an approximate model
• the extra details will provide the push, and the model won't be balanced
• Einstein rejected the cosmological constant as an ugly fudge factor when Hubble discovered the expansion of the Universe
• In January 1998, two teams studying distant Supernovae announced possible evidence for a cosmological constant!
• But, now it's not thought of as a fudge factor
• There's thought to be a really weird kind of matter that gives a repulsive gravitational force!

The Standard Model

• Developed by Friedmann and Lemaitre
• Lemaitre was a Belgian priest motivated to find a cosmology with a beginning of time - like Genesis!
• For a mathematical description we need to a way to describe how big the Universe is
• but it could be infinite!
• define a scale factor, a(t)
• This describes the relative distance of two specs floating along with the Universal expansion (or contraction)
• t₀ = now
• for t < t₀, a(t) < a(t₀)  i.e. the Universe was smaller in the past
• our two specs were closer together
• Light waves also follow the expansion:

• the cosmological redshift formula:
• = (a(t)/a(t₀)) ₀
• You don't need to know this formula on the midterm, but we will cover it later
• Here's how the scale factor evolved with time in the Friedmann model:


• This is a plot of the scale factor as a function of time
• now is given as t = 0.
• the green line shows a low density Universe
• there's so little matter that gravity can never stop the expansion
• the Universe will expand forever
• the Universe is like a projectile launched at v > the escape velocity
• space is negatively curved - like a saddle (as shown below)
• the red line shows a high matter density Universe
• there's so much matter that gravity will stop the expansion
• the Universe will eventually collapse in the Big Crunch
• the Universe is like a projectile launched at v < the escape velocity
• space is positively curved (like a sphere) and finite
• the black line shows a critical density Universe
• there's just enough matter to stop the expansion at t = infinity
• the Universe will expand forever
• the Universe is like a projectile launched at v = the escape velocity
• space is flat, but space-time is still curved

Age of the Universe

• The age of the Universe depends on how fast the Universe is expanding now
• This is described by Hubble's law:
v = H₀ d
• where v = velocity and d = distance
• H₀ = Hubble's constant = 70±10 km/sec /Mpc
• note that km and Mpc are both units of distance
• we can also write H₀ = 1/(15 billion years)
• simple estimate for the age of the Universe
• age = 1/H₀ = 15 billion years
• this is correct if the Universe expands at the same rate
• i.e. H never changes - the empty Universe case
• but if the Universe is slowing down, then it must have expanded faster in the past
• so, it would be younger than 15 billion years
• such as 10 billion years for the critical density Universe
• or 8.5 billion years for the high density Universe shown below.
• But, we could have a cosmological constant:

• The blue curve shows the evolution of the scale factor for a Universe with a cosmological constant but no matter.
• This is the blue curve above.
• In this case the expansion accelerates for all time
• it will expand forever
• it never had a beginning
• With both a cosmological constant and matter
• This is the magenta curve above.
• It starts like a critical or high density Universe
• the cosmological constant takes over at late times
• the age is larger than it would be without the cosmological constant: 15 billion years above.

A Problem

• consider the Universe at t = 1ns
• If it has just the critical density of matter, it will expand forever
• If it has twice the critical density, it will collapse within a few ns
• If it has half the critical density, it will become essentially empty in a few tens of ns.
• It has to be very precisely tuned to be very close to the critical density in order to
• survive for 10 billion years
• and to have enough matter to make galaxies and stars

• Why is the Universe like this?
• we don't know
• but the inflationary Universe model is one possible answer.