So far, we demanded that only one scalar field varies during inflation. Now we allow two or more fields to vary. In the first section we generalize the slow-roll paradigm, and in the second section we give a more general discussion which applies even if the fields are not responsible for inflation.

Multi-field slow-roll inflation

Let us suppose that more than one field varies during inflation, corresponding to an inflationary trajectory φ1(t), φ2(t), …. We continue to assume Einstein gravity, with the energy density dominated by the scalar field potential of canonically normalized fields.

We recover the single-field case if the inflationary trajectory is a straight line in field space, because we can then perform a rotation of the field basis so that just a single field φ has significant variation. Even if the trajectory is not straight, we recover the single-field case provided that the trajectory lies in a steep-sided valley of the potential. The inflaton field then just corresponds to the distance in field space along the valley bottom.

To get something non-trivial, we suppose that the inflationary trajectory is not straight, and that it is one of a continuous family of possible slow-roll trajectories. The family is obtained by displacing the original trajectory sideways in field space. We then say that there is multi-field slow-roll inflation.