Lecture 7: the very early universe

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Lecture 7: the very early universe
Lecture 7: the very early universe
Using well-established physics we can described the universe with some confidence back to the
first trillionth of a second of its existence. To probe further back to the very earliest times
requires some speculation!
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Overview
One of the great achievements of the Big-Bang theory is that it provides us with a detailed
description of the physics of the early universe, and a link between cosmology and particle physics.
Relics of those early stages of the universe can be used to test the Big-Bang scenario.
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Dynamics of the early universe
Recall the Friedmann equation:
8πG
Λc 2 kc 2
H =
ρ+
− 2
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3
R
2
The energy (or mass)€density can, in fact, be split into two components: relativistic and
non-relativistic. The non-relativistic component comprises slow-moving (cold) massive
particles. We normally think of the relativistic component being photons, although any fast
moving (hot) massive particles will actually behave like radiation
The non-relativistic mass density is simply proportional to the number density of particles,
which goes with 1/R3 . However radiation requires an additional 1/R which is a redshift
effect. Thus ρ R ∝1/ R 4 .
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€
Dynamics of the early universe
If R becomes small, then the last two terms will become increasingly irrelevant in the
above equation, so in studying the early universe we normally do not have to worry about a
cosmological constant or curvature. This leaves:
H2 =
8πG
( ρ R + ρ NR )
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And at some point the relativistic density will dominate over the non-relativistic density.
This is usually referred to as the transition between radiation dominated evolution and
matter dominated evolution.
Remember (chapter 4), in the matter dominated phase:
€
R 2 ∝1 / R ⇒ R ∝ τ 2/3
But this is modified in the radiation dominated phase, since now the energy density
reduces by an extra power of R due to the redshift i.e.
ρ∝
1
R4
⇒
R 2 ∝1 / R 2
⇒
∫ R dR = ∫ dτ
⇒ R2 ∝ τ
€
⇒ R = R0 (τ / τ 0 )1/2
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Thermodynamics of the early universe
The microphysics of the early universe depends mainly on the interaction rates between
particles, which in turn depends on their typical energies (set by the temperature),
densities and the strengths of the forces between different species.
(Approximate) thermodynamic equilibrium occurs when the time scale for interactions or
exchange of energy is short compared to the time scales for any global evolution of the
system. Providing this is the case, all the components of the system are at roughly the
same temperature. In the early universe, we can use the Hubble time at any particular
moment to represent the time-scale for evolution of the system (in this case the system is
the universe!).
In the context of a system of particles, for them to be in thermodynamic equilibrium there
must be rapid interactions which distribute the energy between the particles. If the
forces between certain particles are very weak or short range, then they will only be in
equilibrium if the interactions are very frequent and high energy.
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Thermodynamics of the early universe
When different species of particles are interacting frequently with each other, and hence
in TE, we say that the particles are coupled. Once particles are decoupled they are free to
evolve differently, possibly following different temperature evolution.
Clearly, as the scale factor reduces in the past, the density of matter and radiation per
unit volume increase. In addition the temperature rises as the radiation blueshifts in the
past. Since the scale-factor correlates directly with the wavelength of the photons, which
in turn is inversely proportional to the black-body temperature by Wien's law, we have:
T
€
~
1
λmax
~
1
R
For this reason, the model is often called the Hot Big Bang model. Understanding the
physics of the early universe to a large extent depends on following the changes in
temperature. The consequence of the rise in temperature and pressure is that as we trace
further back in time, the matter is broken down into its component subatomic particles. 6
Chemical equilibrium
In the early universe, the particle decays and interactions will often end up producing new
particles. Again, if the interactions are common enough, the numbers of different types of
particles will tend to an equilibrium value. In other words, the rate of production will
become the same as the rate of annihilation (possibly simply decay) for each type of
particle. By analogy, this is often referred to as chemical equilibrium.
The deciding factors in determining the abundance are the mass of the particle in question
and the strength (or likelihood) of its interactions. Only if the original interacting
particles have enough energy to make the rest-mass energy of the new particle(s), can it
form. If the average energy in the plasma is barely sufficient to form a certain species of
particle, then it will not be present in large numbers. The rest-mass energy of each species
of particle acts like a latent heat, since the energy required to make the particles does not
go into increasing the temperature. Equally, if the particle species is very weakly
interacting, it will tend to drop out of equilibrium earlier than if it is strongly interacting.
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Chemical equilibrium
When the temperature and density of the plasma is very high (so that the KE is typically
much larger than the rest mass energy of even the most massive particles) then there will
be very frequent interactions and the energy will be effectively distributed evenly
between all the particles. Each species will have the same number density of particles
(strictly speaking, we have to account for density of states for Fermions). Since we don't
necessarily know all the particles which exist, let alone all their properties, we can only say
approximately how temperature varies with scale factor back to the earliest times.
As the temperature drops, heavier particles will reduce slowly (so they remain
approximately in equilibrium) in number as fewer interactions have enough energy to
produce them. At some point the universe will be sufficiently dilute that interactions
become infrequent (when this occurs depends on the cross-sections for the reactions) and
they drop out of CE: we say the species freezes out. After they freeze-out some particle
species (e.g. free neutrons) may continue to decline in number if they are unstable and
hence decay.
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Some practice problems
1.
Using dimensional analysis, work out how the Planck mass and length are
constructed from the fundamental constants c, G and ћ. Hence evaluate them
numerically.
2.
Assuming the average kinetic energy of particles in thermodynamic equilibrium is
given by E = kT (where k is the Boltzmann constant), calculate the temperature
at which random collisions are able to produce proton-antiproton pairs.
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