The Slope Tax: How Lens Mass-Profile Slope Uncertainty Propagates into the Hubble Constant

Lens models commonly describe the deflector as a power-law ellipsoid whose logarithmic slope is near isothermal. The inferred Hubble constant responds to that slope with a coupling of approximately minus 0.9 in logarithmic terms, so a slope uncertainty of 0.05 becomes a 0.045 fractional Hubble uncertainty per lens. Tabulating the propagation: slope control to 0.02 reaches two per cent cosmography with 1 lenses, while slope control of only 0.10 needs 21 lenses for the same target and 81 for one per cent. Imaging that constrains the slope is therefore worth sample size at a computable exchange rate, and the rate is steep. Computed from the adopted coupling; not measured.