A Novel Estimator for the Equation of State of the IGM by Ly$\alpha$ Forest Tomography.

We present a novel procedure to estimate the Equation of State of the intergalactic medium in the quasi-linear regime of structure formation based on Ly$\alpha$ forest tomography and apply it to 31 high quality quasar spectra from the UVES_SQUAD survey at redshift $z=2.5$. Our estimation is based on a full tomographic inversion of the line of sight. We invert the data with two different inversion algorithms, the iterative Gauss-Newton method and the regularized probability conservation approach, which depend on different priors and compare the inversion results in flux space and in density space. In this way our method combines fitting of absorption profiles in flux space with an analysis of the recovered density distributions featuring prior knowledge of the matter distribution. Our estimates are more precise than existing estimates, in particular on small redshift bins. In particular, we model the temperature-density relation with a power law and observe for the temperature at mean density $T_0 = 13000^{+1300}_{-1200}\,\mathrm{K}$ and for the slope of the power-law (polytropic index) $\gamma = 1.44 \pm 0.09$ for the power-law parameters describing the temperature-density relation. Moreover, we measure an photoionization rate $\Gamma_{-12} = 1.12 \pm 0.17$. An implementation of the inversion techniques used will be made publicly available.