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The number of e-folds $N=\log a$ satisfiesWhen $\varphi$ is monotone, the chain rule gives $dt=d\varphi /\dot{\varphi }$, and henceUsing the two slow-roll equations, $3H\dot{\varphi }\simeq -V^{\prime }(\varphi )$ and $H^2\simeq 8\pi GV/3$, eliminates both $H$ and time:This is the differential form of the slow-roll e-fold count.

In the decelerating hot Big Bang model, the particle horizon at recombination encloses only a finite comoving region. Widely separated parts of the Cosmic microwave background therefore had no overlapping past light cones in that model, although they have almost the same temperature. This is the Horizon problem. During cosmic inflation, accelerated expansion makes the comoving Hubble radius $(aH{)}^{-1}$ shrink. A small region that was initially in causal contact can then expand to contain the whole presently observable universe.

The Flatness problem follows fromDuring ordinary radiation- or matter-dominated decelerating expansion this quantity grows, so the observed closeness of the cosmological density parameter $\Omega$ to one requires extreme tuning at early times. Inflation makes $aH$ grow nearly exponentially and therefore drives $|\Omega -1|$ rapidly toward zero. A sufficiently large number of e-folds makes spatial flatness a dynamical outcome rather than a finely tuned initial condition.