III. The Restoration of Spontaneously Broken Symmetries in the Early Universe and its Consequences

The conventional results (eq. [2.9]) on spontaneous symmetry breakdown are all obtained in the approximation that fluctuations in the Higgs field are important in determining the symmetry of the ''vacuum.'' However, when the ambient temperature () is sufficiently high, this approximation will inevitably break down, as thermal fluctuations in the Higgs field strength become comparable to the difference between its zero temperature value () in the ''ordered phase'' (in which symmetry breakdown occurs) and in the ''disordered phase,'' or ordinary vacuum state () (Kirzhnits and Linde 1972; Kirzhnits 1972). At zero temperature the ''order parameter is given simply by equation (2.9): . However, at finite temperature a simple argument suffices to show that fluctuations in the field about lead to (Kirzhnits and Linde 1972; Weinberg 1974; Dolan and Jaciw 1974):

It is therefore clear that a phase transition to the disordered phase occurs when

If now we include the coupling of the Higgs field to the gauge bosons (which is essential in order to generate the gauge boson masses), then the order parameter becomes (Kirzhnits and Linde 1972; Weinberg 1974; Dolan and Jaciw 1974):

where the are the couplings of the gauge bosons ) to . The difference in the coefficients of the terms in equation (3.3) corresponding to field and gauge boson fluctuations is a consequence simply of the different numbers of spin states for the fields. In the Weinberg-Salam model, therefore, the critical temperature is given by (1)

For , the term associated with gauge boson fluctuations dominates, so that using ,

If the gauge boson term dominates, is the natural scale for , rather than as assumed by Bludman and Ruderman (1977). (For , ) High fermion densities can influence only through the strong gauge boson fields which result from them. If is the expectation value of the fermion four-current , then typically in the ordered phase

so that a large fermion charge density can serve to prevent symmetry restoration (Linde 1976b).

In the standard Friedmann model for the evolution of the universe, the expansion scale factor is determined from the equations ( denotes time derivative):

where account has been taken of a possible cosmological term in Einstein's equations by using the ''generalized'' density and pressure defined in equation (2.7). Since we shall consider the very early universe, the terms proportional to the intrinsic curvature in equation (3.7) may be neglected.

We must now assume an equation of state for and in order to compute the evolution of the universe using equation (3.7). Quantum chromodynamics (QCD) has been found to provide an excellent theory for strong interactions, and it agrees with experiment in all cases where a comparison has been made (see, e.g., Field 1978). In particular, its property of asymptotic freedom has been tested in deep-inelastic lepton-nucleon interactions. This property leads to the prediction that at temperatures exceeding about and/or densities above about or , assemblies of hadrons should behave like an ideal gas of quarks and gluons (Collins and Perry 1975). (Note that models [Hagedorn 1965; Harrison 1972] predicting a maximum temperature for hadronic matter are disfavored by recent experimental results indicating the presence of pointlike weakly interacting constituents within hadrons at short distances, in agreement with QCD.) In this case the energy density of the universe at high temperatures will be

where is the effective number of particle species in thermal equilibrium at a temperature . Ultrarelativistic boson (fermion) spin states contribute to . For , due to contributions from , , , , while above , it receives a contribution 1.75 from . According to QCD, above it should be the quarks and gluons rather than the hadrons into which they are seen to be combined at lower temperatures, which contribute to , giving . In deriving we have also included a massless and , since above the symmetry is restored and they are massless. The pressure should be related to the energy density by the ideal gas law for ultrarelativistic particles:

Using the result (3.9), equations (3.7) become

where was defined in equation (2.15) as . Below , the symmetry is spontaneously broken, and one can arrange . Above , the symmetry should be restored, so that , leaving uncanceled the large cosmological term , which contributes a constant energy density, independent of temperature. At early times, the radiation energy density, which grows like . should have been entirely dominant. However, as decreased, may have fallen below . At ,

where the second equality follows as long as . On the other hand, the effective vacuum energy density contributed by the cosmological term is (minus the Higgs condensate energy density for ):

The condition for to be greater than prior to symmetry breaking (when ) is

The limit in inequality (3.13) includes , which is favored from other considerations (Coleman and Weinberg 1973; Weinberg 1979).

If satisfies the inequality (3.13) the following scenario can be imagined.

1. . In this region . The generalized energy density is dominated by the contribution of relativistic particles, and equation (3.10a) becomes :

Since , the solution to equation (3.14) is . This is the standard hot big-bang expansion rate.

2. . If inequality (3.13) is satisfied, then . The generalized energy density is dominated by , and equation (3.10a) becomes

The solution to equation (3.15) is . This expansion rate differs from the standard expansion rate.

3. . Below , the symmetry is broken, , and the expansion rate is once again the standard form (3.14).

Our conclusion is that the vacuum energy density of the Higgs scalar field could have had important dynamical effects in the early universe; it could have for a brief period dominated the expansion rate. The requirement for this to happen is that the Higgs be much lighter than the vector bosons (but as shown above, still allowed). In this case the Higgs mass sets the scale for the critical temperature, rather than the vector mass.

A glitch in the expansion rate at may not have direct observational consequences. However, there are several interesting phenomena that may occur. If , there is likely to be a first order phase transition, generating entropy and perhaps causing inhomogeneities to develop. If for any period , then the effective pressure of the universe (2.7a) is negative.

Although we have considered only the Weinberg-Salam model, our results could apply in any spontaneously broken gauge theory, in which the Higgs mass is much less than the gauge field mass. If the Lagrangian for the scalar field is not incorporated into a gauge theory, the results of Bludman and Ruderman (1977) obtain, and the vacuum energy can never be important in determining the expansion rate.

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