The Linear Response of the CMB-Inferred Hubble Constant to Perturbations in the Expansion History

We develop a model-independent response-window formalism that describes how small perturbations of the cosmological expansion history around that of the standard ΛCDM model affect the Hubble constant, H0, inferred from the cosmic microwave background (CMB) acoustic-scale constraint. The formalism is independent of the physical origin of the perturbation, which may, for example, arise from a modification of the dark-energy contribution to the expansion history. By considering perturbations at fixed physical matter and radiation densities, we derive a first-order functional relation in which the response is determined by a single projection of the expansion-history perturbation onto a signed response kernel. The kernel is normalized by the difference between the pre-recombination sound-horizon and post-recombination distance contributions, with the latter dominating for the fiducial cosmology. We show that the resulting response window is strongly localized at late times, with an effective redshift zeff≃0.46 and a dominant contribution concentrated near the present epoch. Numerical tests demonstrate substantial sensitivity to the temporal profile of smooth perturbations, while perturbations with identical kernel projections are linearly equivalent in their effect on the inferred H0. The formalism provides a model-independent framework for quantifying and comparing the CMB response to modifications of the late-time expansion history.