3.7.1. Multi-field theories

While much of early investigation of inflation featured potentials such as the massive field discussed above, such models are widely regarded by inflation model builders as unsatisfactory. The reason is that current thinking in particle physics is dominated by supersymmetry, implying that in a cosmological context we should be operating within the framework of supergravity. Once supergravity is brought into play, the numerical value of the scalar field acquires a well-defined meaning, and it is believed that its value must be less than the (reduced) Planck mass m_Pl / (8 ) if the potential is not to be vulnerable to large nonrenormalizable corrections, which typically will destroy slow-roll and anyway will render theoretical calculations unreliable. As we have just seen, one cannot obtain sufficient inflation (or indeed any at all) with the polynomial potential under this restriction, and this conclusion is fairly generic for models where there is only a single scalar field.

An attractive way of circumventing this problem is the hybrid inflation model [10], where a second field provides an additional energy density which dominates over that from the inflaton itself. A typical potential takes the form

(38)

where g² is the coupling constant governing the interaction between the two fields. This is shown in Fig. 1. For large inflaton values the coupling stabilizes at zero, where it contributes a potential energy M⁴ / 4 but otherwise does not participate in the dynamics, so that the inflaton sees a potential

(39)

The interesting case is where the constant term dominates, as it provides extra friction to the equation of motion which makes it roll much more slowly, enabling sufficient inflation without violating the condition << m_Pl / (8 ). Inflation ends when the field drops below a critical value, destabilizing the field and allowing the system to rapidly evolve into its true minimum at = 0, = ±M.

Figure 1. The potential for the hybrid inflation model. The field rolls down the channel at = 0 until it reaches the critical value, then falls off the side to the true minimum at = 0 and = ±M.

The original models [10] assumed that the inflaton potential V() was just that of a massive field, but unfortunately this choice is vulnerable to large loop corrections which dominate over the mass term. However many other possible models have been derived within the hybrid framework; for an extensive discussion of this and other model-building issues see the review of Lyth and Riotto [11].