5.6. Attractor Solutions in Inflation

For simplicity, let us consider a particularly simple potential

 (203)

where m has dimensions of mass. We shall also assume initial conditions that at the end of the quantum epoch, which we label as t = 0, ~ Mp4.

The slow-roll conditions give that, at t = 0,

 (204)

where we have defined m / Mp << 1. We also have

 (205)

where

 (206)

so that the scalar field equation of motion becomes

 (207)

This is solved by

 (208)

where m2 / 3. Now, the slow-roll conditions remain satisfied provided that

 (209)

and therefore, (208) is valid for a time

 (210)

Why is this important? This is important because (208) is an attractor solution! Let's see how this arises. Consider perturbing (208) by writing

 (211)

where (t) is a "small" perturbation, substituting in to the equation of motion and linearizing in . One obtains

 (212)

This equation exhibits two solutions. The first is just a constant, which just serves to move along the trajectory. The second solution is a decaying mode with time constant tc = (3 )-1. Since tc << t, all solutions rapidly decay to (208) - it is an attractor.