Cluster-Void Degeneracy Breaking: Dark Energy, Planck and the Largest Cluster & Void

Combining galaxy cluster and void abundances breaks the degeneracy between mean matter density $Ω_{\rm m}$ and power spectrum normalization $σ_8$. In a first for voids, we constrain $Ω_{\rm m} = 0.21 \pm 0.10$ and $σ_8 = 0.95 \pm 0.21$ for a flat $Λ$CDM universe, using extreme-value statistics on the claimed largest cluster and void. The Planck-consistent results detect dark energy with two objects, independently of other dark energy probes. Cluster-void studies also offer complementarity in scale, density, and non-linearity - of particular interest for testing modified-gravity models.

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