A Novel Statistical Method for Measuring the Temperature-Density
  Relation in the IGM Using the $b$-$N_{\text{HI}}$ Distribution of Absorbers
  in the Ly$\alpha$ Forest
  We present a new method for determining the thermal state of the
intergalactic medium based on Voigt profile decomposition of the Ly$\alpha$
forest. The distribution of Doppler parameter and column density
($b$-$N_{\text{HI}}$ distribution) is sensitive to the temperature density
relation $T=T_0 (\rho/\rho_0)^{\gamma-1}$, and previous work has inferred $T_0$
and $\gamma$ by fitting its low-$b$ cutoff. This approach discards the majority
of available data, and is susceptible to systematics related to cutoff
determination. We present a method that exploits all information encoded in the
$b$-$N_{\text{HI}}$ distribution by modeling its entire shape. We apply kernel
density estimation to discrete absorption lines to generate model probability
density functions, then use principal component decomposition to create an
emulator which can be evaluated anywhere in thermal parameter space. We
introduce a Bayesian likelihood based on these models enabling parameter
inference via Markov chain Monte Carlo. The method's robustness is tested by
applying it to a large grid of thermal history simulations. By conducting 160
mock measurements we establish that our approach delivers unbiased estimates
and valid uncertainties for a 2D $(T_0, \gamma)$ measurement. Furthermore, we
conduct a pilot study applying this methodology to real observational data at
$z=2$. Using 200 absorbers, equivalent in pathlength to a single Ly$\alpha$
forest spectrum, we measure $\log T_0 =4.092^{+0.050}_{-0.055}$ and
$\gamma=1.49^{+0.073}_{-0.074}$ in excellent agreement with cutoff fitting
determinations using the same data. Our method is far more sensitive than
cutoff fitting, enabling measurements of $\log T_0$ and $\gamma$ with precision
on $\log T_0$ ($\gamma$) nearly two (three) times higher for current dataset
sizes.
