Observables in single field slow-roll inflation (lets assume a monomial potential for simplicity) depend only on the number of e-folds at the pivot-scale, $n_s=1-2/N_*$, $r=8/N_*$ etc.
On the other hand the field equation together with the Friedmann equation form an initial value problem. If I were to solve it directly I would need to choose initial values for the field and the derivative (later possibly from slow-roll conditions).
How come that the results are independent of the initial conditions? I think to know how this comes about technically, but I do not understand the physical reason and implications. Also, I know that one distinguishes small-field inflation and large-field inflation. How can the size of the initial field value play a role if the results are approximately independent of the initial values?