Measuring the thermal and ionization state of the low-$z$ IGM using likelihood free inference

We present a new approach to measure the power-law temperature density relationship $T=T_0 (\rho / \bar{\rho})^{\gamma -1}$ and the UV background photoionization rate $\Gamma_{\rm HI}$ of the IGM based on the Voigt profile decomposition of the Ly$\alpha$ forest into a set of discrete absorption lines with Doppler parameter $b$ and the neutral hydrogen column density $N_{\rm HI}$. Previous work demonstrated that the shape of the $b$-$N_{\rm HI}$ distribution is sensitive to the IGM thermal parameters $T_0$ and $\gamma$, whereas our new inference algorithm also takes into account the normalization of the distribution, i.e. the line-density d$N$/d$z$, and we demonstrate that precise constraints can also be obtained on $\Gamma_{\rm HI}$. We use density-estimation likelihood-free inference (DELFI) to emulate the dependence of the $b$-$N_{\rm HI}$ distribution on IGM parameters trained on an ensemble of 624 Nyx hydrodynamical simulations at $z = 0.1$, which we combine with a Gaussian process emulator of the normalization. To demonstrate the efficacy of this approach, we generate hundreds of realizations of realistic mock HST/COS datasets, each comprising 34 quasar sightlines, and forward model the noise and resolution to match the real data. We use this large ensemble of mocks to extensively test our inference and empirically demonstrate that our posterior distributions are robust. Our analysis shows that by applying our new approach to existing Ly$\alpha$ forest spectra at $z\simeq 0.1$, one can measure the thermal and ionization state of the IGM with very high precision ($\sigma_{\log T_0} \sim 0.08$ dex, $\sigma_\gamma \sim 0.06$, and $\sigma_{\log \Gamma_{\rm HI}} \sim 0.07$ dex).