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.