**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,
~
*M*_{p}^{4}.

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

(204) |

where we have defined
*m* /
*M*_{p} << 1. We also have

(205) |

where

(206) |

so that the scalar field equation of motion becomes

(207) |

This is solved by

(208) |

where
*m*^{2}
/ 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
*t*_{c} = (3
)^{-1}.
Since *t*_{c} <<
*t*, all
solutions rapidly decay to (208) - it is an attractor.