The background Friedmannian Hubble constant in relativistic inhomogeneous cosmology and the age of the Universe

In relativistic inhomogeneous cosmology, structure formation couples to average cosmological expansion. A conservative approach to modelling this assumes an Einstein–de Sitter model (EdS) at early times and extrapolates this forward in cosmological time as a “background model” against which average properties of today’s Universe can be measured. This requires adopting an early-epoch–normalised background Hubble constant $H_1^{bg}$. Here, we show that the $\mathrm{\Lambda }$CDM model can be used as an observational proxy to estimate $H_1^{bg}$ rather than choose it arbitrarily. We assume (i) an EdS model at early times; (ii) a zero dark energy parameter; (iii) bi-domain scalar averaging—division of the spatial sections into over- and underdense regions; and (iv) virialisation (stable clustering) of collapsed regions. We find $H_1^{bg}=37.7\pm 0.4$ km/s/Mpc (random error only) based on a Planck $\mathrm{\Lambda }$CDM observational proxy. Moreover, since the scalar-averaged expansion rate is expected to exceed the (extrapolated) background expansion rate, the expected age of the Universe should be much less than $2\mathrm{/}(3H_1^{bg})=17.3$ Gyr. The maximum stellar age of Galactic Bulge microlensed low-mass stars (most likely: 14.7 Gyr; 68% confidence: 14.0–15.0 Gyr) suggests an age about a Gyr older than the (no-backreaction) $\mathrm{\Lambda }$CDM estimate.