Index Divisibility in Dynamical Sequences and Cyclic Orbits Modulo $p$

Let $\varphi (x)=x^d+c$ be an integral polynomial of degree at least 2, and consider the sequence $({\varphi }^n(0){)}_{n=0}^{\mathrm{\infty }}$, which is the orbit of $0$ under iteration by $\varphi$. Let $D_{d,c}$ denote the set of positive integers $n$ for which $n\mid {\varphi }^n(0)$. We give a characterization of $D_{d,c}$ in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes $p$ have the property that the orbit of $0$ is a single $p$-cycle modulo $p$. We show that the set of such primes is finite when $d$ is even, and conjecture that it is infinite when $d$ is odd.