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We show for each $k$, any critical point for the $C^2$-Morse function $\syst$or the systole function that is topologically Morse on $\mathcal M_{g,n}$ hasindex greater than $k$ when $g$ or $n$ is sufficiently large. In other words,there are no critical points of index $\le k$ in those moduli spaces, and allcritical points for $\syst$ of index $\le k$ live in the Deligne-Mumfordboundary. In the Morse handle decomposition given by $\syst$, all $k'$-handleslive in the boundary of such $\overline{\mathcal M}_{g,n}$ for $k'\le k$.

No low index critical points for the systole function and sys_T functions in large M_{g,n}