ner_tags: O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O B-Citation I-Citation I-Citation I-Citation O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model I-Model I-Model O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O B-Model I-Model I-Model O O O O O O O O
tokens: only by extending supernovae observations to high redshifts ( z ∼ 1) can a measurement of both Ω m, 0 and Ω Λ, 0 be obtained ( Goobar Perlmutter 1995 ). By contrast, through Eq. ( 10 ) we can estimate Ω Λ, 0 today, with no reference to Ω m, 0 , using the effect of Λ on galaxy-cluster scales rather than on the Universe as a whole, and based on low-redshift observations alone. We note that the independence of Eq. ( 10 ) from Ω m, 0 holds only at z = 0. Producing an estimate of Ω Λ, 0 using Eq. ( 10 ) requires measuring the present-day slope and intercept of Ω ta ( z ), ideally within a redshift range where Ω ta grows linearly with z . When data from higher redshifts are used, nonlinear terms will introduce some dependence between the Ω Λ, 0 and Ω m, 0 estimates, which, however, can be quantified based on Eq. ( 8 ). In practice, once measurements of Ω ta are obtained, a nonlinear fit will be performed on the predicted Ω ta ( z ) parameterized by Ω m, 0 and Ω Λ, 0 (see lower panel of Fig. 1 ), which will yield a measurement of both cosmological parameters (similar to the contours of Fig. 3 ). For the range z ≤ 0.3 considered in Sect. 3 , the deviation of Ω ta ( z ) from its linear approximation around z = 0 is 10% for any value of Ω Λ, 0 : small, but not subdominant compared to the statistical uncertainties that can be achieved using the large number of available clusters out to this distance. For this reason, the predicted shapes of the constraints from Ω ta ( z ) in Fig. 2 should be viewed as approximations. The technique discussed here depends on independent measurements of Ω ta in different structures and it does not require completeness of the sample of clusters used. For this reason, the same result can be obtained with measurements in only a fraction of the z ≤ 0.3 SDSS clusters, if the sample is optimized in terms of its redshift distribution. There is also margin for improvement in the accuracy of the measurement of Ω ta in individual clusters, for example, by applying quality cuts based on the absence of massive neighbors ( Korkidis et al. 2020 ), mass cuts ( Köhlinger et al. 2015 ), or galaxy-number cuts ( Karachentsev Nasonova 2010 ; Lee 2018 ). About 500 well-selected z 0.3 clusters, uniformly distributed over redshift, with Ω ta measured with 25% accuracy in each, would be enough to establish Ω Λ, 0 > 0 at the 5 σ level. In estimating the potential accuracy of constraints on Ω m, 0 and Ω Λ, 0 , we assumed all uncertainties to be statistical. However, before any statement on cosmology can be made based on measurements of Ω ta ( z ), systematic errors have to be carefully considered as well. For the proof-of-principle calculation of Sect. 2 we used the model of spherical collapse of a single structure in an otherwise uniform and isotropic background universe. Although Korkidis et al. (2020) have shown that the predictions of spherical collapse for Ω ta persist in N -body simulations, they have also reported a small systematic shift