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tokens: and neutrinos), which is observationally indistinguishable at that redshift from the EdS value of . 2.2. Key equations Since the spherical collapse overdensity threshold is several hundred, volume-weighted averaging, together with assumptions (iii) and (iv), implies that the average expansion rate is close to that of the underdense regions, especially at late times, and can be rewritten as where is the peculiar expansion rate of voids, or in other words, the expansion rate above that of the extrapolated high-redshift background EdS model (cf. Eq. (32) of Buchert Carfora 2008 ; Eq. (2.27) of Roukema et al. 2013 ). At early epochs, before the main virialisation epoch, the expansion is dominated by the EdS background model, that is, while at the present, the effective local expansion (measured by local estimates of the Hubble constant) is the sum of the background expansion rate and the peculiar expansion rate of voids, which reads Since we have an early-epoch EdS model that we extrapolate to later epochs (assumption (i)), matter conservation gives the effective present-day matter density parameter (e.g. Eq. (6), Buchert Carfora 2008 ) where are the current values of , respectively. Equation ( 6 ) has solutions A high-redshift () model that contracts would not be physically realistic, so that we have positive . We have a void-dominated model, therefore we also have positive . Thus, the solution of physical interest is We can now estimate the effective scalar curvature. The Hamiltonian constraint (e.g. Eq. (7), Buchert Carfora 2008 ) at the current epoch gives where is the effective (averaged) present-day scalar (3-Ricci) curvature parameter and is the effective (averaged) present-day kinematical backreaction parameter. See for example Eq. (2.9), Roukema et al. (2013) . The sum (Sect. 2.4.1, Buchert 2008 ) is not only a relativistic alternative to dark energy on large scales, it may also provide a relativistic contribution to dark matter on small scales. As summarised in Sect. 1 , observational constraints on recently evolved average spatial hyberbolicity remain weak, and we comment on this further in Sect. 5 . The FLRW equivalents of two of the parameters in Eq. ( 7 )- and -have been the subject of low-redshift observational work for many decades. In Eqs. ( 9 ) and ( 10 ) below, we show that adding a third long-studied observational parameter, , lets us observationally estimate both and when we use the CDM model as a proxy, in the sense that it provides a phenomenological fit to many observations. 3. <inline-formula specific-use="simple-math">Λ</inline-formula>CDM as an observational proxy With the aim of using CDM as an extragalactic observational proxy, we can use Eq. ( 1 ) and assumption (i) to write in alternative form to that in Eq. ( 7 ), and using Eqs ( 1 ) and ( 7 ), we can now write as Thus, Eqs. ( 9 ) and ( 10 ) show that and are constrained by the values of , and , estimated either by methods that minimise model dependence, or