Index gap of the systole function

It is known that the systole function is topologically Morse on the moduli space $\mathcal M_{g,n}$ and the $\text{sys}_T$ functions are $C^2$-Morse on the Deligne-Mumford compactification $\overline{\mathcal M}_{g,n}$. In this paper, We show that these Morse functions admit an index gap on $\mathcal M_{g,n}$. Specifically, there exists a universal constant $C>0$ such that any critical point in $\mathcal M_{g,n}$ has Morse index at least $C\log\log(g+n)$. This implies by Morse theory that the low degree homology of the Deligne-Mumford compactification $\overline{\mathcal M}_{g,n}$ comes from the boundary $\partial\mathcal M_{g,n}$.