Simulated responses from our ΛCDM simulation with two different L window sizes at z = 0. The small gray markers represent the measured sub-volumes. This data have been re-binned, and the black circles with error bars represent the average response in each bin and the 1σ deviation. The linear (R = 1 + R₁δ_w) and second-order () responses from SPT are plotted with the green and blue curves, respectively. The red curves represent the new, σ-dependent power-law response.