very small [@Robertson13; @Noh14; @Yong14].

![The inverse capture time ($\tau^{-1}_{ec}$) for the electron of an exciton in monolayer MoS$_{2}$ on quartz is plotted as a function of the exciton density. The plotted capture time has been normalized by multiplying it by $|b_{c}|^{2}$. The defect density $n_{d}$ is $2\times 10^{11}$ cm$^{-2}$ and the defect energy $E_{d}$ is assumed to be 1.5 eV above the valence band edge. The inverse capture time increases with the exciton density $n_{ex}$ roughly as, $\tau^{-1}_{ec} \sim A + Bn_{ex}$ ($A$ and $B$ are constants).[]{data-label="fig:capture2"}](./capture2){width=".45\textwidth"}

Carrier Capture Times at High Exciton Densities {#sec:num2}
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At large exciton densities (typically larger than $10^{12}$ cm$^{-2}$, but smaller than $10^{13}$ cm$^{-2}$, for 2D-TMDs [@Changjian14]), phase-space filling effects cannot be ignored in the description of the exciton states. We use the formalism developed by Kira and Koch [@Kira12; @Kira06]. When phase-space filling is taken into account, exciton eigenvalue equation in the relative co-ordinates becomes non-Hermitian (see the Appendix) and its solutions are expressed in terms of the [*left*]{} and the [*right*]{} eigenfunctions, $\phi^{L}_{\alpha,s,s'}(\vec{k},\vec{Q})$ and $\phi^{R}_{\alpha,s,s'}(\vec{k},\vec{Q})$, respectively. These eigenfunctions are a also a function of the center of mass momentum $\vec{Q}$, and are related as follows [@Kira12; @Kira06], $$\begin{aligned}
& & \phi^{R}_{s,s',\alpha}(\vec{k},\vec{Q}) 