Constraints on the density fluctuation parameters in the state, mass, and redshift sub-samples used in this analysis. (a) Distribution of sub-samples in mass-redshift plane. The shaded areas indicate the mass and redshift sub-samples extracted from Tiers 1 and 2, respectively. The colour of the dots reflects the dynamical state of the clusters as measured by the centroid shift w. (b) Joint posterior distributions of the standard deviation, σ_δ, the injection scale, ℓ_inj, and the spectral index α of the density fluctuation power spectrum parameters, evaluated on the full sample including the core < 0.15 R₅₀₀ region and (c) excluding the core region. The dashed black line represents the expected 11/3 index from Kolmogorov theory. (d), (e), and (f) are the same plot using the sub-samples associated with the state, mass, and redshift, respectively, as defined in Table 1, including the core region. (g), (h), and (i) use the same sub-samples without the core regions.