Peak theory is one of ΛCDM's oldest quantitative successes: treat the initial density field as Gaussian, identify the rare high peaks that collapse into massive halos and clusters, and the linear power spectrum plus growth history predicts their abundance, their height distribution, and the high-mass end of the halo mass function (Press and Schechter 1974; Sheth and Tormen 1999). Rare peaks are exponentially sensitive to the field's statistics, which makes them powerful tests and dangerous liabilities.
The measured extremes keep straining the framework. Surveys and simulations report excesses or deficits of very massive clusters relative to the calibrated baselines, environment-dependent peak statistics the universal mass function does not anticipate, and peak-height distributions whose shape resists the Gaussian-plus-growth prescription (Jenkins et al. 2001; Bhattacharya et al. 2011). Each discrepancy is individually arguable, but they share a structure: the deepest potential wells in the real universe are not distributed quite the way rare-peak statistics of a Gaussian field demand, and the model's correction tools, recalibrated mass functions and baryonic adjustments, are post-hoc fits rather than predictions.
The standing is an accumulation of strain at the statistical extremes, sharpened by the same lensing-sector results that drive the S8 and peak-count discussions. Euclid, LSST, and Roman will measure the cluster mass function and peak-height statistics across unprecedented volume, turning the rare-peak tail from an argument into a measurement.