C$^3$-Cluster Clustering Cosmology II. First detection of the BAO peak
  in the three-point correlation function of galaxy clusters
  Third-order statistics of the cosmic density field provides a powerful
cosmological probe containing synergistic information to the more commonly
explored second-order statistics. Here, we exploit a spectroscopic catalog of
72,563 clusters of galaxies extracted from the Sloan Digital Sky Survey,
providing the first detection of the baryon acoustic oscillations (BAO) peak in
the three-point correlation function (3PCF) of galaxy clusters. We measure and
analyze both the connected and the reduced 3PCF of SDSS clusters from
intermediate ($r\sim10$ Mpc/h) up to large ($r\sim140$ Mpc/h) scales, exploring
a variety of different configurations. From the analysis of reduced 3PCF at
intermediate scales, in combination with the analysis of the two-point
correlation function, we constrain both the cluster linear and non-linear bias
parameters, $b_1=2.75\pm0.03$ and $b_2=1.2\pm0.5$. We analyze the measurements
of the 3PCF at larger scales, comparing them with theoretical models. The data
show clear evidence of the BAO peak in different configurations, which appears
more visible in the reduced 3PCF rather than in the connected one. From the
comparison between theoretical models considering or not the BAO peak, we
obtain a quantitative estimate of this evidence, with a $\Delta \chi^2$ between
2 and 75, depending on the considered configuration. Finally, we set up a
generic framework to estimate the expected signal-to-noise ratio of the BAO
peak in the 3PCF exploring different possible definitions, that can be used to
forecast the most favorable configurations to be explored also in different
future surveys, and applied it to the case of the Euclid mission.
