The Hot Big Bang Theory

Figures borrowed from: Ned Wright's Cosmology Tutorial
and from: Smoot Group Education/Outreach Page

Big Bang Review

Expanding Universe -> Big Bang Theory
The scale factor: a(t)

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

For an expanding Universe like ours, a(t) grows with time

But, the gravity of ordinary matter slows it down
The age and fate of the Universe depend on the matter density:

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 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 black line shows a critical density Universe

there's just enough matter to stop the expansion at t = infinity
he Universe will expand forever

Here's what the plots look like in the distant future:

Cosmological Red Shift

In the Big Bang Theory, the Doppler formula we learned before does not describe large redshifts correctly.
We can view the redshifts as being caused indirectly by the expansion of the Universe:

If the Universe was contracting, we'd see blue shifts.

Each observer sees a redshift because the Universe has expanded while the light was traveling to her:
Here's Doppler's redshift formula that you already learned:

unless

But, in cosmology, we often observe sources with z > 1, so this formula doesn't work.
The correct formula is:

This just means that the wavelength grows with the expansion of the Universe.

Let's consider an example of this:

Say a galaxy is observed at a redshift of z = 5 (this is the current record holder)
At what wavelength would we observe the Lyman-alpha line of Hydrogen?

which has a wavelength of = 121.6 nm in the lab

= 121.6 nm and 1 + z = 6,

= ( 1 + z )

= 6 ( 121.6 nm ) = 729.6 nm

This line is redshifted from the far ultra-violet to the far red part of the spectrum.

The expansion of the Universe also changes the energy of the radiation since

In the z = 5 example above, the energy of the radiation drops by a factor of 1 + z = 6 between the time it is emitted from the distant galaxy and now.

Since temperature is just a measure of average energy, , and we have

So, if the average stellar surface temperature of the galaxy at z = 5 is 6000 K, we see the average temperature as:

T_now = T_then / (1 + z ) = T_then / 6 = 1000 K

The Universe is cooling as it expands
So, it was hotter in the past:

The Hot Big Bang Theory

The early Universe was very hot
High Energy Physics experiments probe the physics of the early Universe

the Fermilab collider (1 hour west of Chicago)

Here's a schematic view of the early Universe:

How was the Universe different early on?

Before t = 10,000 years or so, radiation was the dominant form of matter in the Universe

there was more total mass in radiation than in ordinary matter
also more mass in radiation than in dark matter

Why?

Energy density of ordinary (slow moving) matter (or dark matter) is dominated by Einstein's

E = mc² term

as long as v << c, gravity is determined just by the mass, m.
For radiation, we saw that
So the radiation energy was higher in the past while the energy of massive particles doesn't change

(unless v approaches c)

Primordial Nucleosynthesis

How the elements were made in the first few minutes of the Big Bang
Original calculation by Gamow - wanted to produced all the observed elements
At t = 1 sec, collisions are frequent so we have thermal equilibrium

radiation and particles are at the same temperature, T
T drops slowly compared to the pace of the particle interactions
The distribution of radiation energy is given by the black-body curve:

For the ordinary particles, this distribution is slightly different, but similar:

At t = 1 sec, the energy is high enough for the following interactions to occur:

photons collide and produce electron-positron pairs

electron-positron pairs annihilate into a pair of photons

protons and neutrons convert into each other

these interactions include neutrinos and electrons

We can also have protons and Neutrons bounce off of each other:

Or, if their energy is low enough the neutron and proton can stick:

to form a deuterium nucleus or deuteron
But, they will get knocked apart again if it is too hot
There are lots more photons than protons and neutrons

Eventually at t = 100 sec or so, the deterium nuclei last long enough to collide
Heavier elements can form:

Helium-4 is very tightly bound, and most neutrons end up as Helium-4 or they decay
Very few heavier elements can form - no Carbon, Oxygen, or Iron

higher temperatures are required to initiate these reactions
but deuterium only forms at low temperatures

deuterium "bottleneck" effectively prevents the formation of heavy elements in the Big Bang

A little bit of Lithium, Berillium, and Boron form
heavy elements can form in stars because

once Helium forms, stellar core can contract and heat up
gravity provides energy source for this heat
at higher temperatures, heavy elements can form

Here's the time history: