Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs

It is not known whether or not the stable rational cohomology groups $\tilde H^*(\operatorname{Aut}(F_\infty);\mathbb{Q})$ always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and $H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and $X_m$ of &quote;outer space&quote: and &quote;auter space&quote;, respectively, introduced in [3] by Culler and Vogtmann and [6] by Hatcher and Vogtmann.