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5.6. Attractor Solutions in Inflation

For simplicity, let us consider a particularly simple potential

Equation 203 (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, rho ~ Mp4.

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

Equation 204 (204)

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

Equation 205 (205)


Equation 206 (206)

so that the scalar field equation of motion becomes

Equation 207 (207)

This is solved by

Equation 208 (208)

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

Equation 209 (209)

and therefore, (208) is valid for a time

Equation 210 (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

Equation 211 (211)

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

Equation 212 (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 = (3alpha phi)-1. Since tc << Deltat, all solutions rapidly decay to (208) - it is an attractor.

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