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5.7.2. Multi-field theories

A recent trend in inflationary model building has been the exploration of models with more than one scalar field. The classic example is the hybrid inflation model, [13] which seems particularly promising for particle physics model building. The simplest version has a potential with two fields phi and psi of the form

Equation 53 (53)

which is illustrated in Figure 4. When phi2 is large, the minimum of the potential in the psi-direction is at psi = 0. The field rolls down this `channel' until it reaches phiinst2 = lambdaM2 / lambda', at which point psi = 0 becomes unstable and the field rolls into one of the true minima at phi = 0 and psi = ±M.

Figure 4

Figure 4. The potential for the hybrid inflation model. The field rolls down the channel at psi = 0 until it reaches the critical phi value, then falls off the side to the true minimum at phi = 0 and psi = ±M.

While in the `channel', which is where all the interesting behaviour takes place, this is just like a single field model with an effective potential for phi of the form

Equation 54 (54)

This is a fairly standard form, the unusual thing being the constant term, which would not normally be allowed as it would give a present-day cosmological constant. The most interesting regime is where that constant dominates, and it gives quite an unusual phenomenology. In particular, the energy density during inflation can be much lower than normal while still giving suitably large density perturbations, and secondly the field phi can be rolling extremely slowly which is of benefit to particle physics model building.

Within the more general class of two and multi-field inflation models, it is quite common for only one field to be dynamically important, as in the hybrid inflation model - this effectively reduces the situation back to the single field case of the previous subsection. However, it may also be possible to have more than one important dynamical degree of freedom. In that case there is no attractor behaviour giving a unique route into the potential minimum, as in the single field case; for example, if the potential is of the form of an asymmetric bowl one could roll into the base down any direction. In that situation, the model loses some of its predictive power, because the late-time behaviour is not independent of the initial conditions. (5)

5 Of course, there is no requirement that the `true' physical theory does have predictive power, but it would be unfortunate for us if it does not.

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