The Lyman-alpha forest is the large-scale structure probe for which we appear to have modeling control to the highest wavenumbers, which makes it of great interest for constraining the warmness/fuzziness of the dark matter and the timing of reionization processes. However, the standard statistic, the Lyman-alpha forest power spectrum, is unable to strongly constrain the IGM temperature-density relation, and this inability further limits how well other high wavenumber-sensitive parameters can be constrained. With the aim of breaking these degeneracies, we measure the power spectrum of the Lyman-beta forest and its cross correlation with the coeveal Lyman-alpha forest using the one hundred spectra of z=3.5-4.5 quasars in the VLT/X-Shooter XQ-100 Legacy Survey, motivated by the Lyman-beta transitions smaller absorption cross section that makes it sensitive to somewhat higher densities relative to the Lyman-alpha transition. Our inferences from this measurement for the IGM temperature-density relation appear to latch consistently onto the recent tight lower-redshift Lyman-alpha forest constraints of arXiv:2009.00016v1 [astro-ph.CO]. The z=3.4-4.7 trends we find using the Lyman-alpha--Lyman-beta cross correlation show a flattening of the slope of the temperature-density relation with decreasing redshift. This is the trend anticipated from ongoing HeII reionization and there being sufficient time to reach the asymptotic temperature-density slope after hydrogen reionization completes. Furthermore, our measurements provide a consistency check on IGM models that explain the Lyman-alpha forest, with the cross correlation being immune to systematics that are uncorrelated between the two forests, such as metal line contamination.
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