Quantum Fluctuation in the Inflating Universe

With the RFW metric ds = dt − R(t)d~x, the scale factor grows exponentially R(t) = R(0)e. We now switch to the conformally flat metric ds = R(η)(dη − d~x). Therefore, dt = R(η)dη, or equivalently, dη = ±R(t)−1dt = ±R(0)−1e−Htdt. Note that η is nothing but the size of the particle horizon, namely the comoving distance for which the light could propagate since the initial time. This differential equation is easy to integrate, and we find