  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.
