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The post-reionization thermal state of the intergalactic medium ischaracterized by a power-law relationship between temperature and density, witha slope determined by the parameter $\gamma$. We describe a new method tomeasure $\gamma$ using the ratio of flux curvature in the Lyman-$\alpha$ and$\beta$ forests. At a given redshift, this curvature ratio incorporatesinformation from the different gas densities traced by Lyman-$\alpha$ and$\beta$ absorption. It is relatively simple and fast to compute and appearsrobust against several observational uncertainties. We apply this technique toa sample of 27 high-resolution quasar spectra from the Very Large Telescope.While promising statistical errors on $\gamma$ appear to be achievable withthese spectra, to reach its full potential, the dependence of the curvatureratio on the thermal state of the gas in the foreground Lyman-$\alpha$ forestwill require further, detailed forward modelling.

Constraining the temperature-density relation of the intergalactic medium with the Lyman-$\alpha$ and $\beta$ forests