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

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