30
Phase transitions in the early universe
example, topologicaUy stable objects such as domain walls, cosmic strings and
magnetic monopoles can be formed when the 'alignment' of the spontaneous
symmetry breaking expectation value is different in adjacent causal domains.
These can make substantial contributions to the energy density of the universe.
Moreover, if supercooling occurs before the phase transition is completed, the
reheating that takes place when the phase transition occurs can greatly modify
pre-existing particle densities. In addition, if the universe spends some time with
positive vacuum energy (cosmological constant) before relaxing to a minimum
with zero vacuum energy, then rapid expansion can occur. Such an 'inflationary'
stage in the history of the universe, to be discussed in later chapters, may explain
the extreme isotropy, homogeneity and flatness of the present day observed
universe. For all of these reasons it is important to understand any phase
transitions that may have occurred as the universe cooled.
In this chapter we shall begin by developing the partition function and
the effective potential for the gauge field theories at finite temperature [1-5)
before applying these methods to the Higgs model, as a warm-up, and then to
electroweak theory and grand unified theory. In each case, the nature of the phase
transitions that occur as the temperature of the universe drops will be studied.
We shall then extend the discussion to gauge theories with global supersymmetry
and local supersymmetry (supergravity). Finally, the nucleation of (stable) 'true'
vacuum from (metastable) 'false' vacuum in first-order phase transitions will be
considered. This nucleation rate wi\1 control the extent of any supercooling that
occurs before the phase transition is complete.
2.2 Partition functions
One of the fundamental objects in the statistical thermodynamics of a finite
temperature system is the partition function Z defined by
Z = Tre- fJH
(2.1)
where iI is the Hamiltonian operator and
{J = (kBT)-1 = T- 1
(2.2)
in units where the Boltzmann constant kB is set equal to l. The trace in (2.1)
means that we are to sum over the (diagonal) matrix elements of e- fJH for all
independent states of the system. Once the partition function has been evaluated,
the (Helmholtz) free energy F is given by
Z = e- fJF
(2.3)
where, as usual in thermodynamics, F is related to the internal energy E and the
entropy S by
F=E-TS.
(2.4)
Phase transitions in the early universe
example, topologicaUy stable objects such as domain walls, cosmic strings and
magnetic monopoles can be formed when the 'alignment' of the spontaneous
symmetry breaking expectation value is different in adjacent causal domains.
These can make substantial contributions to the energy density of the universe.
Moreover, if supercooling occurs before the phase transition is completed, the
reheating that takes place when the phase transition occurs can greatly modify
pre-existing particle densities. In addition, if the universe spends some time with
positive vacuum energy (cosmological constant) before relaxing to a minimum
with zero vacuum energy, then rapid expansion can occur. Such an 'inflationary'
stage in the history of the universe, to be discussed in later chapters, may explain
the extreme isotropy, homogeneity and flatness of the present day observed
universe. For all of these reasons it is important to understand any phase
transitions that may have occurred as the universe cooled.
In this chapter we shall begin by developing the partition function and
the effective potential for the gauge field theories at finite temperature [1-5)
before applying these methods to the Higgs model, as a warm-up, and then to
electroweak theory and grand unified theory. In each case, the nature of the phase
transitions that occur as the temperature of the universe drops will be studied.
We shall then extend the discussion to gauge theories with global supersymmetry
and local supersymmetry (supergravity). Finally, the nucleation of (stable) 'true'
vacuum from (metastable) 'false' vacuum in first-order phase transitions will be
considered. This nucleation rate wi\1 control the extent of any supercooling that
occurs before the phase transition is complete.
2.2 Partition functions
One of the fundamental objects in the statistical thermodynamics of a finite
temperature system is the partition function Z defined by
Z = Tre- fJH
(2.1)
where iI is the Hamiltonian operator and
{J = (kBT)-1 = T- 1
(2.2)
in units where the Boltzmann constant kB is set equal to l. The trace in (2.1)
means that we are to sum over the (diagonal) matrix elements of e- fJH for all
independent states of the system. Once the partition function has been evaluated,
the (Helmholtz) free energy F is given by
Z = e- fJF
(2.3)
where, as usual in thermodynamics, F is related to the internal energy E and the
entropy S by
F=E-TS.
(2.4)