Threshold Clustering Functions and Thermal Inhomogeneities in the Lyman-Alpha Forest

We introduce to astrophysics the threshold probability functions S_2, C_2, and D_2 first derived by \citet{torq+88}, which effectively samples the flux probability distribution (PDF) of the Lya forest at different spatial scales. These statistics are tested on mock Lya forest spectra based on various toy models for HeII reionization, with homogeneous models with various temperature-density relations as well as models with temperature inhomogeneities. These mock samples have systematics and noise added to simulate the latest Sloan Digital Sky Survey Data Release 7 (SDSS DR7) data. We find that the flux PDF from SDSS DR7 can be used to constrain the temperature-density relation $\gamma$ (where $T \propto (1 + \Delta)^{\gamma-1}$) of the intergalactic medium (IGM) at z=2.5 to a precision of $\Delta \gamma = 0.2$ at $\sim 4\sigma$ confidence. The flux PDF is degenerate to temperature inhomogeneities in the IGM arising from HeII reionization, but we find S_2 can detect these inhomogeneities at $\sim 3 \sigma$, with the assumption that the flux continuum of the Lya forest can be determined to 9% accuracy, approximately the error from current fitting methods. If the quasar continuum can be determined to 3% accuracy, then S_2 is capable of constraining the characteristic scale of temperature inhomogeneities, with $\sim 4 \sigma$ differentiation between toy models with hot bubble radii of 50 Mpc/h and 25 Mpc/h, comoving.