The Expansion Rate Inside Voids: A Local Hubble Diagram for Underdense Regions

The interior of a void expands faster than the universe as a whole, and this paper computes by how much, for the interior contrasts that catalogues actually report. For a mean interior contrast of minus 0.65 the interior expansion rate at the present day is enhanced by a factor of 1.115, corresponding to a local rate of 75.1 kilometres per second per megaparsec against the global 67.4. A shallower interior at contrast minus 0.4 gives an enhancement of 1.071. Applied to a local underdensity of the kind reported on 300 megaparsec scales, with contrast minus 0.30, the apparent expansion rate inferred by an observer who ignored the underdensity would be 71.0 in the same units, a factor of 1.053. Explaining the full difference between local and early-universe expansion determinations in this way would require a local contrast of -0.47, far deeper than any catalogue reports on that scale. Interior expansion is therefore a real, computable correction to local measurements, and at the same time an insufficient explanation, by a wide margin, of the present discrepancy in the expansion rate. All values are computed from stated assumptions, not measured.