An inevitable consequence of quantized field theories is the existence of zero-point quantum fluctuations in the fields, usually leading to a nonzero energy density for the ''vacuum.'' These effects may be removed by considering only normal-ordered products of field operators. In unbroken supersymmetric theories they cancel between fluctuations in boson and fermion fields and so are exactly zero. It therefore appears possible that these processes should have no gravitational effects, at least when gravitation is treated classically. In the Higgs mechanism, however, there must exist a field whose vacuum expectation value is purely classical. The resulting vacuum energy density is of a somewhat different character than that associated with zero-point fluctuations, and it appears likely that its gravitational effects cannot be neglected. (To determine whether this is indeed correct, one would have to investigate the Higgs mechanism in a quantized theory of gravitation.)
We begin from Einstein's field equations with a cosmological term (we choose units such that ):
For a perfect relativistic fluid the energy-momentum is given by (our metric has signature ):
where is the pressure, the energy density, and the velocity of the fluid in a comoving frame. We may absorb the cosmological term in equation (2.1) by defining a generalized energy-momentum tensor (see, e.g., Zel'dovich and Novikov 1971):
so that Einstein's equations (2.1) become
Introducing a ''vacuum energy density,''
the generalized energy-momentum tensor for a perfect fluid may be written in the form
For a field theory with spontaneous symmetry breaking (SSB), extra terms must be added to the energy-momentum tensor (2.2). We consider the complex scalar field () theory with Lagrangian density
The field then has a classical vacuum expectation when , so that
A form of equation (2.8) is necessary in the Lagrangian for the Weinberg-Salam model. In that model
where is the Fermi coupling constant, is the mass of the charged vector boson, and is the SU(2) coupling constant:
The classical expectation value of the energy-momentum tensor for the field is simply
For spontaneous symmetry breaking to occur, must correspond to the absolute minimum of , so that . One then finds ( represents the effect of higher order terms in the effective potential):
In equation (2.13) is the mass of the Higgs particle fluctuations of the field about its classical vacuum expectation value, :
To the one-loop level, ignoring contributions from Higgs scalar loops (Coleman and Weinberg 1973),
where the are the couplings of the fermions (gauge bosons) to the Higgs. Note that because of Fermi statistics (closed fermion loops have a relative minus sign) the fermion contribution to is negative. >From equations (2.12) and (2.13) it is clear that the spontaneous symmetry breaking vacuum energy is
For numerical estimates of see, e.g., Politzer and Wolfram 1979; here we ignore the fermion contribution. In this model, (Linde 1976a; Weinberg 1976), since for spontaneous symmetry breakdown to occur. There is some theoretical incentive for the guess (Coleman and Weinberg 1973).
Observations of the present rate of expansion of the universe show that its average mass density is less than . According to equation (2.16) the vacuum energy density contributed by the Higgs field is typically more than a factor of too large. (The possibility that is experimentally excluded.) This discrepancy could be taken to show that the Higgs mechanism is not operative and would suggest that the symmetry breakdown is dynamical in origin. We shall, however, not to take this point of view, but rather assume the Higgs mechanism and investigate a possible cosmological resolution of the discrepancy. Addition of further Higgs fields cannot remove the vacuum energy, since each contributes a negative amount to , as they must all have for spontaneous symmetry breakdown to have occurred.
One possible mechanism (Linde 1974) which may remove the effects of the vacuum energy arising from the Higgs field is the presence of a large compensating cosmological term in Einstein's equations (2.1) which leads to a canceling effective vacuum energy density (see eq. [2.5]). To avoid contradictions with observation we must then demand that in the present universe
Of course, it would be very surprising if these two contributions to the effective energy density of the universe, having such different origins, should cancel to the required accuracy of better than about one part in . Our purpose is to investigate whether such a delicate cancellation could be maintained throughout the history of the universe. We find that, while it is probable that the cancellation failed under certain conditions, its failure does not appear to result in observable consequences for the present universe. However, we do find that it is quite possible that the expansion rate of the universe, i.e., (where is the Robertson-Walker scale parameter) may for a period have been determined by rather than the temperature of the relativistic particles.