Consider a sphere with $n$ punctures. If you pick a holomorphic cotangent vector at each puncture, you can canonically construct a holomorphic top form in the corresponding moduli space. (The specific construction I'm thinking of comes from string theory, but I believe that it's essentially the only thing you can write down once you fix a normalization for the Haar measure on $SL(2, \mathbb{C})$: just fix three points and contract the three extra cotangent vectors with the three Killing vector fields.)

Now consider the hyperbolic metric on the above sphere: since it is possible to assign coordinates $z_i$ near a cusp in a way well defined up to rotating $z_i \mapsto e^{i \theta} z_i$, one can make a volume form $\omega_S$ on the moduli space $\mathcal{M}_{0, n}$ coming from putting $dz_i \wedge d \overline{z}_i$ at each puncture. There is also the Weil-Petersson volume form $\omega_{WP}$. Let $f = \omega_S / \omega_{WP}$.

From physical considerations, $f$ is expected to diverge near the boundary of $\mathcal{M}_{0, n}$, so let the region $R_{n, \epsilon}$ be the sub-region of $\mathcal{M}_{0, n}$ where a punctured sphere $\Sigma \in \mathcal{M}_{0, n}$ is in $R$ if $\Sigma$ has no closed geodesics of length less than $\epsilon$. Does anybody have a bound on the behavior of $\sup_{R_{n, \epsilon}} f$ as $n$ approaches infinity?

(If it helps, the actual question that lead to this is to determine the asymptotics of $\int_{R_{n, \epsilon}} \omega_S$, where the punctures are taken to be unordered, and in particular to determine if they grow exponentially or factorially.)

Consider a sphere with $n$ punctures. If you pick a holomorphic cotangent vector at each puncture, you can canonically construct a holomorphic top form in the corresponding moduli space. (The specific construction I'm thinking of comes from string theory, but I believe that it's essentially the only thing you can write down once you fix a normalization for the Haar measure on $\operatorname{SL}(2, \mathbb{C})$: just fix three points and contract the three extra cotangent vectors with the three Killing vector fields.)

Now consider the hyperbolic metric on the above sphere: since it is possible to assign coordinates $z_i$ near a cusp in a way well defined up to rotating $z_i \mapsto e^{i \theta} z_i$, one can make a volume form $\omega_S$ on the moduli space $\mathcal{M}_{0, n}$ coming from putting $dz_i \wedge d \overline{z}_i$ at each puncture. There is also the Weil–Petersson volume form $\omega_\text{WP}$. Let $f = \omega_S / \omega_\text{WP}$.

From physical considerations, $f$ is expected to diverge near the boundary of $\mathcal{M}_{0, n}$, so let the region $R_{n, \epsilon}$ be the sub-region of $\mathcal{M}_{0, n}$ where a punctured sphere $\Sigma \in \mathcal{M}_{0, n}$ is in $R$ if $\Sigma$ has no closed geodesics of length less than $\epsilon$. Does anybody have a bound on the behavior of $\sup_{R_{n, \epsilon}} f$ as $n$ approaches infinity?

(If it helps, the actual question that led to this is to determine the asymptotics of $\int_{R_{n, \epsilon}} \omega_S$, where the punctures are taken to be unordered, and in particular to determine if they grow exponentially or factorially.)