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.
