Histograms showing the distribution of best-fit ℓ < 800 parameters from simulations performed using our low-ℓ approximation. The vertical line is the input fiducial model and the contours show the posteriors from an ℓ < 800 chain using the actual Commander likelihood at low ℓ. The unbiased recovery of the fiducial parameters and agreement with the posteriors is a stringent test of the validity of our low-ℓ approximation. We note that these simulations, unlike the ones used in the main body of the paper to determine significance levels, have the prior on τ handled slightly differently, so as to allow us to use them as a test of the low-ℓ approximation (see Appendix B for discussion).