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An important development in our understanding of the `vacuum energy' was associated with the phenomenon of symmetry breaking in the electroweak Weinberg-Salam model. Consider the scalar field action

Equation 69 (69)

where curlyL is the Lagrangian density

Equation 70 (70)

and the scalar field potential has the form

Equation 71 (71)

This particular form of the potential (illustrated in Fig. 12) endows the system with some interesting properties. For instance since the symmetric state phi = 0 is unstable (V''(phi) < 0) the system settles in the ground state phi = +sigma or phi = -sigma, where sigma = [µ2 / lambda]1/2 thus breaking the reflection symmetry phi leftrightarrow -phi present in the Lagrangian. The energy momentum tensor Tik of a scalar field with Lagrangian density curlyL is given by

Equation 72 (72)

Assuming phi to be homogeneous and time-independent one finds the ground state energy-momentum tensor to be

Equation 73 (73)

the vacuum state therefore has precisely the form of an effective cosmological constant Tik = gik Lambdaeff where Lambdaeff = V(phi = sigma) = V0 - µ4 / 4lambda. Setting V0 = 0, results in a negative cosmological term Lambdaeff = -µ4 / 4lambda. Substituting parameters arising in the electroweak theory results in a lower limit on the value of the vacuum energy density [205] rhovac = |Lambdaeff| / 8pi G = 106 GeV4, which is almost 1053 times larger than current observational upper limits on the cosmological constant rhovac,0 = Lambda0 / 8pi G ~ 10-29 g/cm3 appeq 10-47 GeV4. Clearly in order not to violate observational bounds today, one must set V0 appeq µ4 / 4lambda so that Lambdaeff ~ Lambda0. An interesting feature of this `regularization' of the cosmological constant is that, while drastically reducing the value of the cosmological constant today it simultaneously generates a large cosmological constant ~ V0 during an early epoch before symmetry breaking, thereby giving rise to the possibility of Inflation ! The cosmological constant problem therefore presents us with a dilemma: it is certainly good to have a large cosmological constant during an early epoch so as to resolve - via Inflation - the horizon and flatness problems and possibly generate seed fluctuations for galaxy formation. However one must simultaneously ensure that the value of Lambda today is small so as not to conflict with observations. As we have seen, in models with SSB this dual requirement of `large Lambda in the past + small Lambda at present' results in an enormous fine tuning of initial conditions.

Figure 12

Figure 12. The `Mexican top-hat' potential describing spontaneous symmetry breaking shown: before (dashed) and after (solid) the cosmological constant has been `renormalized'.

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