- Fermions (particles that obey Fermi statistics) come
in generations with each generation heavier than the preceding one.
- electron neutrinos interact with protons and neutrons; in
the early universe this set of reactions keeps the protons and
neutrons in balance:
anti-neutrino+proton<---->neutron+anti-electron neutrino + neutron <------> proton + electron

Note: In the very early Universe, it is possible that its so small that the separation between the quarks vanishes if quarks are arbitrarily close together the gluon force vanishes and the quarks are "free". During this free quark period, one can not have protons or neutrons. This period is called the Quark-gluon plasma. The origin of the matter - anti-matter assymmetry may be related to the end of this period. Its unclear.

Extra Material: Now let's do some funny things with numbers:

Let's assume that the Universe, for no apparent reason, knows about these three constants at arbitrarily early times:

- G
_{n}= Newton's Constant Gravity/Mass - h-bar = Planck's constant which we can maybe relate to some kind of length
- c = constant speed of light necessary to define time in a consistent manner.

1. (Gh/c^{3}) has units of length; in real units this length
is approximately 10^{-36 } meters
pretty damn small! This is called the Planck Scale

2. (hc/G)^{1/2} has units of mass; in real units this mass
is approximately 2 x 10^{-8} kg. If we multiple this mass by
c^{2} we get an Energy scale which is about 10^{19}
Gev. You might think of this as the energy per unit volume at the
"beginning of the universe". This is called the Planck Mass.

3. since we have length scale, and a speed limit, we then define
a time which is the planck scale divided by c. or
(Gh/c^{5}) or 10^{-43} seconds. One can think of
this planck time, as the time when G,h-bar and c now exist.

Back now to the neutrinoes.

When the universe is about 1 second old, or at an energy scale of 1 Mev, the neutrinos no longer interact with matter. The density of the Universe has dropped below in the interaction scale of the neutrinos and now the neutrinos "free-stream" away from the matter.

After this decoupling, the electrons and positrons annihilate (to one part in one billion) to reheat the universe and give us the photon background. However, the neutrinos are not around to participate in this reheating and so the neutrino background is colder than the photon background. We can actually calculate this exactly:

1. The entropy per volume of the universe, *S * is given by this

where N_{T} is the number of species of particles in
thermal equilbrium at temperature, T.

2. The total entropy is constant; so *S*R^{3} is constant
so N_{T}(TR)^{3} is constant.

3. The total number of species of a particle is the product of its total number of spin states, whether or not it has an anti-particle, and whether it obeys the pauli exclusion principle. So

- a photon has 2 spin states, no anti particle, and obeys
bose statistics so its species number is 2x1x1
- an electron is 2x2x7/8 = 7/2.

5. After the electron position annihilation N_{T} is 2.

6. Since N_{T}(TR)^{3} is constant we have

7. Before annihilation everything was in thermal equilibrium
so the T_{neutrino} = T_{photon}. Therefore
(T_{neutrino}R)after = (T_{neutrino}R)before = (TR)before.

So we now have
(TR)after/(TR)before = TR(after)/(T_{neutrino}R)after =
T/T_{neutrino} =
(11/4)^{1/3} = 1.4

In real numbers this means the neutrino background is 1.4 times colder than the photon background (or 2K right now).

8. Furthermore, the energy density scales as T^{4} so the
energy density in the neutrino background compared to the photon
background is:

This means that if neutrinos have a tiny mass, then their energy density (in the form of rest mass) is large. If they have no mass, then they are just like photons and their energy density is a consequence only of the neutrinos being at finite temperature.

Structure Formation:

In a completely generic way, structure formation involves the amplification of density enhancements through the actions of self gravity.

In a static case, once sufficient mass is available to overcome the internal energy in some volume, that mass will collapse and the size of the density enhancement will increase. This condition is known as the Jeans criteria - we will return to it later.

In a normal static situation, the internal energy or pressure of a gas is simply related to its temperature. Hence, cool clouds of gas can gravitational collapse while hot clouds would dissipate.

However, in the early universe there is an additional source of "pressure" and that is radiation pressure. Since the radiation and matter are coupled, then the radiation is trying to smooth out the distribution of matter (e.g. wash out any density enhancements).

At a general level, its unclear if purely baryonic density fluctuations could survive the effects of radiation pressure (also called radiation drag). To mitigate the effects of radiation pressure, it is desireable to have a new form of gravitating matter, one that can gravitationally clump but not interact with radiation. This matter, often called dark matter, could therefore provide the seed density fluctuations around which structure will later grow.

When discussion the gravitational collapse of object it is useful to refer to a timescale, known as the dynamical timescale. This is the characteristic timescale of a system after which the virial theorem holds. Its universal in all kinds of systems and I will derive this timescale under three different scenarios:

1) A pressure wave travelling through a hydrostatic fluid. Here we use the sound velocity as the characteristic velocity.

2) Gravitational free fall (or motion in 1D under constant acceleration).

3) Cluster crossing time. Here we equate the dynamical mass of the cluster with the mass obtained by considering it a spherical region of constant density.

These three cases are shown below in equation form.

We can now also do a quick estimate, based on observed large
scale structure, what we might expect the for the amplitude of
any anisotropy in the microwave background.

Structure will amplify in the matter dominated universe as
z^{3/2}. Since the redshift at decoupling was z ~ 1100,
the amplification factor is 40,000. Hence, perturbations
as small as 1/40,000 could have been amplified to produce
the factor of 2 overdensities that we observe in large scale
structure today. Hence, the expected temperature anisotropy
at the surface of last scattering is 1/40000 = 2.5 x 10^{-5}
which is consistent with the WMAP observations.

To illustrate the profound effects of radiation pressure on the suppression of the growth of structure in the radiation dominated era (e.g. the first 300,000 years), we can do a Jeans Mass analysis.

The Jeans criterion is that the gravitational potential energy of a could of gas must overcome its internal energy (IE) in order to for collapse to occur. The IE for a fluid is its pressure times its volume. This criteria can be stated in terms of density and pressure as shown below:

In the radiation dominated era the pressure is significant and is 1/3c

At recombination the density of the universe is ~ 10^{-21}
g/cc.

The Jeans mass prior to recombination is 5 x 10^{18} solar
masses and we don't observe structures this large.

After recombination the jeans mass lowers to 2 x 10^{5} solar
masses, which is the mass of a globular cluster.