Lecture03 The Thermal History of the universe

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Lecture-03 The Thermal History of the universe
Ping He ITP.CAS.CN 2006.1.13
http//power.itp.ac.cn/hep/cosmology.htm
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3.1 Thermal history to study
(1) T t temperature time
(2)
(3) Degree of freedom T, t
(4) Decouple relic background
(5) Nucleosynthesis
(6) Baryogenesis
They are typical events in the early Universe
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3.2. Equilibrium Thermodynamics
Piston La(t)r
Quasi static
?Thermal equilibrium
Reaction rate
Expansion rate
(Eq-3.1)
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This analysis can be applied to cosmology
Eq-2.1 is the kernel of this lecture
Thermal equilibrium
Coupling (1) (2)
A, C equilibrium
A, B equilibrium
(1) AC
Coupling mode
(2) AB
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3.3 Distribution Function in Thermal Equilibrium
g spin-degeneracy factor (inner degree of
freedom)
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If equilibrium
(relativistic)
If chemical equilibrium
From Eq-2.2 Eq-2.3
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From Eq-2.6 , Eq-2.7 , Eq-2.8 we have
The above is the general form for relativistic
quantum cases. In kinetic equilibrium
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3.4 Distributions as a function of E
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Specifically
(1) relativistic limit, and non-degeneracy
(2) non-relativistic
In above calculation , we used the fact that
Maxwell-Boltzmann
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(3) For non-degenerate ,relativistic species
average energy /particle
For a non-relativistic species
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3.5 The excess of fermions over its antiparticle
From thermodynamics and statistical dynamics
for photon
From Eq-2.19, we have
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The net (the excess of ) Fermion number density
(relativistic)
(non-relativistic)
For proton
Most of the particle species have
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3.6 Degrees of freedom
From eq-2.15
From eq-2.16
At the early epoch of the Universe, T is very
high. All are in Relativistic. Non-relativistic
exponentially decrease ? negligible
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Here, the effective degrees of freedom
(1) Tlt1 MeV
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From eq-2.24
(2) 1 MeVltTlt100 MeV
(3) Tgt300 GeV
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3.7 Time Temperature
when radiation dominated
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3.8 Evolution of Entropy
In the expanding Universe, 2nd law of
thermodynamics
(unit coordinate volume)
More,
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From Eq-2.26 with Eq-2.27
Up to an additive constant , the entropy per
commoving volume
From
And with Eq-2.26, we have
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The entropy per commoving volume is consented
during the expansion of the Universe.
density of physical volume
Entropy density s
dominated by relativistic particles
For most of the history of the Universe
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For photon number-density
Commoving entropy density is conserved
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From eq-2.32
Boson - relativistic
Fermions - relativistic
non - relativistic
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Define
(relativistic)
(non-relativistic)
If the number of a given species in a commoving
volume is not changing, ie, particles of that
species are not being created or destroyed, then
If no baryon non-conserving mechanism, then
So, with eq-2.31, we have
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More over
So, the temperature of the Universe evolves as
Explanation
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3.9 Decoupling
for massless
When A is decoupled
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In addition
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(2) Massive particle decoupling ,
So
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Summary
In both cases
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(3) general cases
The phase-space distribution does not maintain an
equilibrium distribution. In the absence of
interactions
You cannot find a simple relation , for
So the equilibrium distribution cannot be
maintained
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3.10 Brief Thermal History of the Universe
Some famous events
Key the interaction rate per particle
The correct way to evolve particlen
distributions is to integrate the Boltzmann
equation
3.10.1 Neutrino decoupling
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when
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Relic neutrino background
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And
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3.10.2 Matter-Radiation Equality
In above calculation, we have used
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3.10.3 Photon Decoupling and Recombination
Thomson cross-station
Radiation-Matter decoupling
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B binding energy of hydrogen ,
Define the fractional ionization (ionization
degree)
neutral. Ionization0
ionization total
From eq-2.52 the equilibrium ionization fraction
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When recombination , matterdominated
Decoupling
Summary
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3.10.4 The baryon number of the Universe
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entropy per baryon
To compare with, in a star, the entropy is
So the entropy of the Universe is enormous !!!
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A human age
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Key points
phase-space distribution function
(1)
(2)
entropy is concerned
(3)
interaction rate per particle
thermal equilibrium
decouple
Qualitative and semi-quantitative
Full-quantitative treatment solve collisional
Boltzmann Equation
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References

• E.W. Kolb M.S. Turner, The Early Universe,
  Addison-Wesley Publishing Company, 1993
• T. Padmanabhan, Theoretical Astrophysics III
  Galaxies and Cosmology, Cambridge, 2002