The Lyman-beta forest as a cosmic thermometer

We present a comprehensive analysis of high resolution hydrodynamic simulations in terms of Lyman-alpha and Lyman-beta one dimensional flux power spectra ($P_{\alpha\alpha}$ and $P_{\beta\beta}$). In particular, we focus on the behaviour that the flux auto-power spectra and cross-power spectra ($P_{\alpha\beta}$) display when the intergalactic medium (IGM) thermal history is changed in a range of values that bracket a reference model, while cosmological parameters are kept fixed to best fit the cosmic microwave background data. We present empirical fits that describe at the sub-percent level the dependence of the power spectra on the thermal parameters. At the largest scales, the power spectra show a constant bias between each other that is set by the parameters describing the IGM thermal state. The cross-power spectrum has an oscillatory pattern and crosses zero at a scale which depends on $T_0$, the IGM temperature at the mean density, for reasonable values of the power-law index $\gamma$ of the IGM temperature-density relation ($T=T_0(1+\delta)^{\gamma-1}$). By performing a Fisher matrix analysis, we find that the power spectrum $P_{\beta\beta}$ is more sensitive to the thermal history than $P_{\alpha\alpha}$ alone, due to the fact that it probes denser regions than Lyman-alpha. When we combine the power and cross spectra the constraints on $\gamma$ can be improved by a factor $\sim 4$, while the constraints on $T_0$ improve by a factor of $\sim 2$. We address the role of signal-to-noise and resolution by mocking realistic observations and we conclude that the framework presented in this work can significantly improve the knowledge of the IGM thermal state, which will in turn guarantee better constraints on IGM-derived cosmological parameters.