Issue 
A&A
Volume 638, June 2020



Article Number  L8  
Number of page(s)  5  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/201937358  
Published online  17 June 2020 
Letter to the Editor
Turnaround density as a probe of the cosmological constant
^{1}
Department of Physics and Institute for Theoretical and Computational Physics, University of Crete, 70013 Heraklio, Greece
email: pavlidou@physics.uoc.gr
^{2}
Institute of Astrophysics, Foundation for Research and Technology – Hellas, Vassilika Vouton, 70013 Heraklio, Greece
^{3}
Department of Astronomy and Astrophysics and KICP, University of Chicago, Chicago, IL 60637, USA
Received:
19
December
2019
Accepted:
20
April
2020
Spherical collapse predicts that a single value of the turnaround density, meaning the average matter density within the scale on which a structure detaches from the Hubble flow, characterizes all cosmic structures at the same redshift. It was recently shown by Korkidis et al. that this feature persists in complex nonspherical galaxy clusters that have been identified in Nbody simulations. Here we show that the lowredshift evolution of the turnaround density constrains the cosmological parameters and it can be used to derive a local constraint on Ω_{Λ, 0} alone, independent of Ω_{m, 0}. The turnaround density thus offers a promising new method for exploiting upcoming large cosmological datasets.
Key words: cosmological parameters / largescale structure of Universe / galaxies: clusters: general
© ESO 2020
1. Introduction
Despite the many successes of concordance ΛCDM cosmology, increasingly accurate cosmological datasets are starting to reveal tensions (see e.g., Bernal et al. 2016; Zhao et al. 2017; Hildebrandt et al. 2017; Joudaki et al. 2017; Charnock et al. 2017; Motloch & Hu 2018; Planck Collaboration VI 2020; Riess et al. 2018, 2019; Raveri & Hu 2019; Adhikari & Huterer 2019; Handley 2019; Di Valentino et al. 2020a,b). Additionally, our evidence for the existence of vacuum energy, whether it is in the form of a cosmological constant Λ or not, remains indirect, with different datasets constraining primarily the relation between the presentday values of the matter and Λ density parameters (Ω_{m, 0} and Ω_{Λ, 0}, respectively), rather than Ω_{Λ, 0} alone. This is in contrast to Ω_{m, 0}, to which certain datasets (e.g., cluster abundances, baryon acoustic oscillations) are almost exclusively sensitive, independently of the value or sheer existence of Ω_{Λ, 0}. This periodically leads to a critical revisiting of the strength of the evidence that Ω_{Λ, 0} ≠ 0 (Kolb et al. 2006; Buchert & Räsänen 2012; Nielsen et al. 2016; Dam et al. 2017; Colin et al. 2019; Kang et al. 2020), which is also fuelled by the lack of generally accepted fundamentalphysicsdriven candidates for the nature of vacuum energy. In this context, unexplored probes of the cosmological parameters can provide new insights for the cosmological model.
In recent years, considerable attention has been paid to the properties of cosmic structures on the largest scales as a means of locally probing cosmology and alternative theories of gravity (e.g., Cuesta et al. 2008; Tavio et al. 2008; Tiret & Combes 2008; Pavlidou & Tomaras 2014; Pavlidou et al. 2014; Diemer & Kravtsov 2014; Stark et al. 2017; Baxter et al. 2017; Adhikari et al. 2018; Walker et al. 2019). The turnaround radius (the scale on which a cosmic structure detaches from the Hubble flow) has been the focus of many such studies (Pavlidou et al. 2014; Tanoglidis et al. 2015, 2016; Lee & Li 2017; Bhattacharya et al. 2017; Nojiri et al. 2018; Capozziello et al. 2019; Santa & Enea Romano 2019; Lopes et al. 2019; Wong 2019). The turnaround radius can be measured kinematically in any galaxy cluster as the boundary between the cluster and the expanding Universe. Spherical collapse predicts that all structures turning around at some cosmic epoch share a characteristic average density within the turnaround radius, that is, the turnaround density, ρ_{ta}. Korkidis et al. (2020) have used Nbody simulations to show that a single turnaround radius also meaningfully describes simulated galaxy clusters with realistic shapes and that the average matter density within that turnaround radius has a narrow distribution around a characteristic value for clusters of all masses, which is consistent with the predictions of spherical collapse.
The turnaround density is sensitive to the presence of a cosmological constant Λ. Once the effect of Λ becomes dominant over the gravitational selfattraction of matter, it halts structure growth (Busha et al. 2003, 2005; Pavlidou & Tomaras 2014; Tanoglidis et al. 2015). As a consequence, in an everexpanding Universe with Λ, not all overdensities are destined to eventually detach from the Hubble flow (e.g., Pavlidou & Fields 2005), and ρ_{ta} has a hard lower bound of 2ρ_{Λ} = 2(Λc^{2}/8πG) (Pavlidou & Tomaras 2014). The evolution of ρ_{ta} thus changes between early and late cosmic times in a cosmologyrevealing manner. In ΛCDM, at early times, when matter dominates, ρ_{ta} falls as a^{−3}, where a = (1 + z)^{−1} is the scale factor of the Universe (see e.g., Peebles 1980; Padmanabhan 1993; and Sect. 2). At late times, when Λ dominates, ρ_{ta} → 2ρ_{Λ} ∝ a^{0} (Fig. 1, upper panel, black solid line). In contrast, in an open matteronly Universe, ρ_{ta} decreases without bound, as a^{−3} while matter dominates, and as a^{−2} when curvature takes over (Fig. 1, upper panel, red dashed line). For the present cosmic epoch, concordance ΛCDM predicts that ρ_{ta} ∝ a^{−1.5} (Fig. 1, upper panel, dotted black line), which is already shallower than the asymptotic latetime behavior of a matteronly Universe. It is therefore reasonable to expect that a measurement of the evolution of ρ_{ta} with redshift could provide evidence for the existence of Λ. This result is independent of the earlytimes behavior of ρ_{ta} (universal for all cosmologies), so observations at low redshifts would be sufficient to establish it. The turnaround density could thus provide a ‘local’ probe of the cosmological parameters, demonstrating the existence of dark energy by using its effect on scales that are much smaller than the observable universe and by using structures located at low redshifts.
Fig. 1. Upper panel: evolution with scale factor a of ρ_{ta} in units of the presentday critical density, ρ_{c, 0}. Black solid line: flat ΛCDM cosmology with Ω_{m, 0} = 0.315 (Planck Collaboration VI 2020). Red dashed line: matteronly open cosmology, with Ω_{m, 0} = 0.315, Ω_{Λ, 0} = 0. Black dotted line: presentday (a = 1) tangent to the black solid line, with slope ρ_{ta} ∼ a^{−1.5}, shallower than the asymptotic behavior of an Ω_{Λ, 0} = 0 Universe. Lower panel: evolution of ρ_{ta} with redshift z, for Ω_{m, 0} = 0.315 and different values of Ω_{Λ, 0}. The magenta shaded box shows the accuracy that can be achieved by measuring ρ_{ta} in 100 clusters at z = 0.3 with fractional uncertainty of 50% in each, and is indicative of the discriminating power of such an experiment. 

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In this Letter, we explore the type of constraints that could be placed on the cosmological parameters, Ω_{m, 0} and Ω_{Λ, 0}, by using measurements of the presentday value of the turnaround density, ρ_{ta, 0}, and of its presentday rate of change with redshift, dρ_{ta}/dz_{0}. For the predictions of the concordance ΛCDM model, we use the 2018 Planck cosmological parameters (Planck Collaboration VI 2020): Ω_{m, 0} = 0.315, Ω_{Λ, 0} = 1 − Ω_{m, 0}, H_{0} = 67.4 km s^{−1} Mpc^{−1}. This choice has no qualitative effect on our conclusions.
2. Evolution of ρ_{ta}
We consider a spherical shell of evolving radius R_{s} that is destined to eventually turn around, surrounding a single spherical perturbation in an otherwise homogeneous and isotropic Universe. We consider the background Universe on a scale R large enough so that the perturbation negligibly alters its expansion properties.
The evolution of both the background Universe and the shell is described by a Friedmann equation, each with a different curvature constant κ. We consider shells that turn around late enough so that their turnaround radius is much larger than their size at matterradiation equality and, therefore, we only consider the matter, curvature, and cosmological constant contributions to the Friedmann equation. For the background Universe we write:
From Eq. (1), we can obtain κ_{U} in terms of the presentday value of the Hubble parameter, H_{0} = Ṙ_{0}/Ṙ_{0} (quantities with subscript 0 refer to z = 0):
We define the presentday critical density, , and the backgroundUniverse scale factor, a = R/R_{0} = 1/(1 + z) (so that a = 1 today and ρ_{m} = ρ_{m, 0}a^{−3}). Then, substituting in (1) yields the Friedmann equation in its most frequently encountered form,
where . For the spherical shell around the perturbation, we write:
In this case, we obtain κ_{s} by considering the state of the shell at its time of turnaround, a_{ta}^{1}. Then, Ṙ_{s} is equal to zero; R_{s} is equal to the turnaround radius, R_{s, ta}; and the mass density enclosed by R_{s} is equal to the turnaround density ρ_{ta} at a_{ta}:
Substituting in (4), defining the shell scale factor a_{s} = R_{s}/R_{s, ta} (so that a_{s} = 1 at the time of turnaround and ), and measuring densities in units of the backgroundUniverse critical density ρ_{c, 0} yields:
where Ω_{ta} = ρ_{ta}/ρ_{c, 0}. Ω_{ta} is a function of a_{ta}, which, in turn, depends on the initial overdensity within the shell: initially denser perturbations turn around earlier. Dividing Eq. (6) by Eq. (3) and taking the positive square root (since for a_{s} ≤ 1 both Universe and perturbation expand) we obtain:
The turnaround density Ω_{ta} as a function of turnaround time a_{ta} can be obtained by integrating the perturbation scale factor, a_{s}, from 0 to 1 and the Universe scale factor, a, from 0 to a_{ta}:
The result of this integration is plotted in the upper panel of Fig. 1 for a flat ΛCDM cosmology, with Ω_{m, 0} = 0.315 (black solid line) and an open CDM universe with Ω_{m, 0} = 0.315 and Ω_{Λ, 0} = 0 (red dashed line). The presentday slope of the scaling is shown with the dotted line. Remarkably, the present cosmic epoch coincides with the era of transition between asymptotic behaviors. For this reason, Ω_{ta}(z) curves for different values of Ω_{Λ, 0} deviate from each other very quickly, already at low z, as shown in the lower panel of Fig. 1.
In principle, two exact measurements of Ω_{ta} at two different redshifts would uniquely determine Ω_{m, 0} and Ω_{Λ, 0}. Alternatively, if Ω_{ta} is measured at some redshift (e.g., z = 0, a_{ta} = 1), then Eq. (8) yields a constraint on the relative values of Ω_{m, 0} and Ω_{Λ, 0}. A measurement of the presentday value of Ω_{ta} is most sensitive to the value of Ω_{m, 0} (see Fig. 2, red contours; we also note in the lower panel of Fig. 1 that different values for Ω_{Λ, 0} yield very similar presentday Ω_{ta} when Ω_{m, 0} remains the same).
Fig. 2. Constraints on Ω_{m, 0} and Ω_{Λ, 0} from the cosmic microwave background (CMB, orange contours, WMAP, Komatsu et al. 2011), supernovae (blue contours, Union 2 SN Ia compilation, Amanullah et al. 2010), and baryon acoustic oscillations (green contours, SDSS, Eisenstein et al. 2005; from Amanullah et al. 2010, Fig. 10). The red and purple contours correspond to projected 1, 2, and 3σ constraints implied by Eqs. (8) and (10), respectively, using a presumed highaccuracy measurement of the evolution of Ω_{ta} at the lowredshift Universe (∼42 000 galaxy clusters at z ≤ 0.3, with an individualcluster Ω_{ta} uncertainty of 50%, see Sect. 3), yielding a 1.5% accuracy estimate of Ω_{ta}_{0} and a 3.5% estimate of dΩ_{ta}/dz_{0}. We have assumed that Ω_{ta} evolves with z as predicted by flat Ω_{m, 0} = 0.315, ΛCDM cosmology. At this level of accuracy, Ω_{Λ, 0} > 0 could be established at a 14σ confidence level from the turnaround density data alone. 

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The lower panel of Fig. 1 shows that the presentday value of the rate of change of Ω_{ta} with z for a given value of Ω_{m, 0} is very sensitive to Ω_{Λ, 0}. This implies that dΩ_{ta}/dz_{0} would be a useful cosmological observable. We can obtain a prediction for dΩ_{ta}/da_{ta}_{0} = −dΩ_{ta}/dz_{0} by differentiating Eq. (8) with respect to a_{ta}:
For the present cosmic epoch (a_{ta} = 1) Eq. (9) becomes
independent of Ω_{m, 0}.
If a measurement is then made of Ω_{ta} in a sufficient number of clusters at a range of (low) redshifts, then both the slope dΩ_{ta}/dz_{0} and the intercept Ω_{ta}_{0} of the scaling can be derived. A constraint on Ω_{Λ, 0} alone can thus be obtained from Eq. (10) (see Fig. 2, purple contours).
3. Possible constraints on cosmological parameters
Here we present a simplified estimate of the potential accuracy of a measurement of Ω_{ta}_{0} and dΩ_{ta}/da_{ta}_{0}, and of the associated inference of the cosmological parameters Ω_{m, 0} and Ω_{Λ, 0}, under the following three assumptions. 1. Errors are dominated by statistical uncertainties. 2. Ω_{ta}(z) behaves approximately linearly with z in the low redshift range we will consider. 3. The redshift distribution of measured clusters follows^{2} dn/dz ∝ z^{2}.
We can obtain an estimate of the uncertainties in dΩ_{ta}/dz_{0} and Ω_{ta}_{0} by considering the errors in the slope and intercept of a linear regression fit to Ω_{ta}(z)≈Ω_{ta}_{0} + (dΩ_{ta}/dz_{0})z of a sample of n measurements of (z_{i}, Ω_{ta}_{i}) in individual galaxy clusters with z ≤ z_{max}. In each cluster, Ω_{ta}_{i} is measured with some uncertainty σ_{Ωta, i}. In linear regression, the standard error of the slope, σ_{dΩta/dz0}, is , where ϵ_{i} are the regression residuals and Var(z) the variance of the independent variable. is equal to . If Ω_{ta} in all clusters can be measured with the same fractional accuracy f, then . To calculate both the latter average and Var(z), we use dn/dz ∼ z^{2}. Finally, dividing by dΩ_{ta}/dz_{0}, we obtain the fractional uncertainty of the slope:
The error of the intercept is related to that of the slope through . Calculating the latter average and using Eq. (11), we obtain:
Measurements of Ω_{ta} in individual clusters will likely have a relatively poor accuracy (f ∼ 0.5)^{3}. If concordance ΛCDM cosmology holds, then Ω_{ta}_{0} = 3.71 (Eq. (8)) and dΩ_{ta}/dz_{0} = −dΩ_{ta}/da_{ta}_{0} = 5.67 (Eq. (10)). If then Ω_{ta} were measured in the ∼42 000 clusters with z_{max} = 0.3 and M_{200} > 0.6 × 10^{14} M_{⊙} in the 14 000 square degrees of the Sloan Digital Sky Survey (SDSS) III (Wen et al. 2012) with f = 0.5, we could obtain an estimate of Ω_{ta}_{0} with an uncertainty of ∼1.5%, and an estimate of dΩ_{ta}/dz_{0} with an uncertainty of ∼3.5%. These would yield the constraints on Ω_{m, 0} and Ω_{Λ, 0} shown in Fig. 2 with the red (Eq. (8)) and purple (Eq. (10)) contours. The projected measurement of Ω_{Λ, 0} shown features a 7% accuracy, corresponding to a 14σ confidence level that Ω_{Λ, 0} > 0. In Fig. 3, we have combined the constraints from Eqs. (8) and (10) to obtain confidence levels on the values of Ω_{m, 0} and Ω_{Λ, 0}, assuming all uncertainties to be Gaussian and ignoring nonlinear corrections to the lowredshift behavior of Ω_{ta}(z).
Fig. 3. Red and pink contours: projected 1, 2, and 3σ confidence intervals on Ω_{m, 0} and Ω_{Λ, 0} from turnaround density data only, obtained by combining the constraints from Eqs. (8) and (10) shown in Fig. 2. As in Fig. 2, the contours are overplotted on Fig. 10 of Amanullah et al. (2010) showing constraints from supernovae (blue), the CMB (orange), and baryon acoustic oscillations (green). 

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4. Discussion
Equation (10) being independent of Ω_{m, 0} is a nontrivial feature, not shared by any of the currently used probes of the cosmological parameters, which makes the evolution of ρ_{ta} especially attractive as a direct, local probe of Ω_{Λ, 0}. The CMB primarily probes the geometry of the Universe, so the confidence region on the Ω_{m, 0} − Ω_{Λ, 0} plane that can be directly derived from it resembles a negativeslope strip (Fig. 2, orange), and it is only in combination with other probes (e.g., weak lensing, or baryon acoustic oscillations) that the CMB reveals a clear preference for Ω_{Λ, 0} ∼ 0.7 (e.g., Spergel et al. 2003; Planck Collaboration VI 2020). Cluster abundances and baryon acoustic oscillations on the other hand are primarily sensitive to Ω_{m, 0} (as is the presentday value of the turnaround density) and by themselves reveal very little about the existence of a cosmological constant (Fig. 2, green and red). This is why observational evidence in support of the currently accepted Ω_{m, 0} ∼ 0.3 predated the general adoption of a nonzero cosmological constant (e.g., Davis et al. 1985; Bahcall & Cen 1993; Cole et al. 1995). Finally, type Ia supernovae, which, as standard candles, probe the cosmological parameters by mapping the redshift dependence of the luminosity distance, only constrain the relative values of Ω_{m, 0} and Ω_{Λ, 0} if they are observed in the lowredshift Universe. This constraint is a diagonal positiveslope strip on the Ω_{m, 0} − Ω_{Λ, 0} plane (Perlmutter et al. 1997). The slope of this strip changes with increasing redshift and, thus, 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 galaxycluster scales rather than on the Universe as a whole, and based on lowredshift 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 presentday 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 galaxynumber cuts (Karachentsev & Nasonova 2010; Lee 2018). About 500 wellselected 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 5σ 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 proofofprinciple 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 Nbody simulations, they have also reported a small systematic shift towards higher values of Ω_{ta} due to effects that oppose gravity in realistic cosmic structures (primarily tidal forces from massive neighbors and, to a much smaller extent, rotation; see also Bhattacharya & Tomaras 2019). Whether this shift evolves with redshift, and how, remains to be explored through simulations.
Measuring ρ_{ta} in a single cluster requires separate measurements of the cluster turnaround radius, R_{ta}, and of the cluster mass, M_{ta}, within R_{ta}; then the turnaround density is simply obtained from . The quantities R_{ta} and M_{ta} might also suffer from systematic biases in their measurement that also have to be quantified and accounted for, using both mock observations of simulated structures and crosscalibration of measurements using different techniques. The uncertainty of a single measurement of ρ_{ta} is dominated by that of R_{ta}, which, in turn, can be obtained from observations of peculiar velocities with respect to the Hubble flow of cluster member galaxies, provided these galaxies have distances measured by some indicator other than redshift. Estimates of the turnaround radius have been attempted in several nearby structures (Karachentsev & Kashibadze 2005; Lee 2018), including the Virgo cluster (Karachentsev & Nasonova 2010) and the FornaxEridanus complex (Nasonova et al. 2011). Although deriving cosmological parameters from these measurements requires a careful consideration of the uncertainties involved, which is beyond of scope of this work, it is worth noting that these measurements are consistent with Planckparameter concordance ΛCDM.
Deriving Ω_{ta} from ρ_{ta} in general requires an assumption on the value of H_{0} = h × 100 km s^{−1} Mpc^{−1}. However, many techniques for the measurement of cluster masses and cluster radii are themselves calibrated on the local expansion of the Universe (i.e., they yield masses in h^{−1} M_{⊙} and radii in h^{−1} Mpc). Estimates of Ω_{ta} obtained from such measurements are independent of h.
Extending such analyses to a large number of structures will require the measurement of a large number of cluster masses, redshifts, and independent distance estimates of member galaxies. However, these measurements are already considered of high cosmological significance and largescale campaigns to obtain them are planned or underway. In this context, the turnaround density provides a promising new way to analyze and exploit upcoming large cosmological datasets.
This is equivalent to assuming distances ∝z and a constant number density of clusters in the nearby Universe. Such a distribution approximates well the redshift distribution of clusters in e.g., Wen et al. (2012) out to z ∼ 0.2. For higher redshifts the number of clusters in Wen et al. (2012) grows more slowly with z, which for a fixed number of clusters results in tighter constraints.
This comes from a typical uncertainty of ∼30% in the cluster mass measurement (e.g., Köhlinger et al. 2015), an uncertainty of ∼10% in the turnaround radius (comparable to what has been claimed for nearby clusters, e.g., Karachentsev & Kashibadze 2005; Karachentsev & Nasonova 2010), and a ∼25% halotohalo scatter seen in Nbody simulations (Korkidis et al. 2020).
Acknowledgments
We thank A. Zezas, K. Tassis, I. Papadakis, V. Charmandaris, N. Kylafis, and I. Papamastorakis for valuable discussions, and an anonymous referee for their careful review and constructive comments. GK acknowledges support from the European Research Council under the European Union’s Horizon 2020 research and innovation program, under grant agreement no 771282.
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All Figures
Fig. 1. Upper panel: evolution with scale factor a of ρ_{ta} in units of the presentday critical density, ρ_{c, 0}. Black solid line: flat ΛCDM cosmology with Ω_{m, 0} = 0.315 (Planck Collaboration VI 2020). Red dashed line: matteronly open cosmology, with Ω_{m, 0} = 0.315, Ω_{Λ, 0} = 0. Black dotted line: presentday (a = 1) tangent to the black solid line, with slope ρ_{ta} ∼ a^{−1.5}, shallower than the asymptotic behavior of an Ω_{Λ, 0} = 0 Universe. Lower panel: evolution of ρ_{ta} with redshift z, for Ω_{m, 0} = 0.315 and different values of Ω_{Λ, 0}. The magenta shaded box shows the accuracy that can be achieved by measuring ρ_{ta} in 100 clusters at z = 0.3 with fractional uncertainty of 50% in each, and is indicative of the discriminating power of such an experiment. 

Open with DEXTER  
In the text 
Fig. 2. Constraints on Ω_{m, 0} and Ω_{Λ, 0} from the cosmic microwave background (CMB, orange contours, WMAP, Komatsu et al. 2011), supernovae (blue contours, Union 2 SN Ia compilation, Amanullah et al. 2010), and baryon acoustic oscillations (green contours, SDSS, Eisenstein et al. 2005; from Amanullah et al. 2010, Fig. 10). The red and purple contours correspond to projected 1, 2, and 3σ constraints implied by Eqs. (8) and (10), respectively, using a presumed highaccuracy measurement of the evolution of Ω_{ta} at the lowredshift Universe (∼42 000 galaxy clusters at z ≤ 0.3, with an individualcluster Ω_{ta} uncertainty of 50%, see Sect. 3), yielding a 1.5% accuracy estimate of Ω_{ta}_{0} and a 3.5% estimate of dΩ_{ta}/dz_{0}. We have assumed that Ω_{ta} evolves with z as predicted by flat Ω_{m, 0} = 0.315, ΛCDM cosmology. At this level of accuracy, Ω_{Λ, 0} > 0 could be established at a 14σ confidence level from the turnaround density data alone. 

Open with DEXTER  
In the text 
Fig. 3. Red and pink contours: projected 1, 2, and 3σ confidence intervals on Ω_{m, 0} and Ω_{Λ, 0} from turnaround density data only, obtained by combining the constraints from Eqs. (8) and (10) shown in Fig. 2. As in Fig. 2, the contours are overplotted on Fig. 10 of Amanullah et al. (2010) showing constraints from supernovae (blue), the CMB (orange), and baryon acoustic oscillations (green). 

Open with DEXTER  
In the text 
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