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 ${\bar{\mathcal{M}}}_{g,n}$. 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 ${\bar{\mathcal{M}}}_{g,n}$ comes from the boundary $\mathrm{\partial }{\mathcal{M}}_{g,n}$.