  The Lyman-alpha forest is the large-scale structure probe for which we appear
to have modeling control to the highest wavenumbers. This makes the Lyman-alpha
forest of great interest for constraining the warmness/fuzziness of 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 coeval 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 transition's 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. 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.
