Probing cosmology via the clustering of critical points

Exclusion zones in the cross-correlations between critical points (peak-void, peak-wall, filament-wall, filament-void) of the density field define quasi-standard rulers that can be used to constrain dark matter and dark energy cosmological parameters. The average size of the exclusion zone is found to scale linearly with the typical distance between extrema. The latter changes as a function of the matter content of the universe in a predictable manner, but its comoving size remains essentially constant in the linear regime of structure growth on large scales, unless the incorrect cosmology is assumed in the redshift-distance relation. This can be used to constrain the dark energy parameters when considering a survey that scans a range of redshifts. The precision of the parameter estimation is assessed using a set of cosmological simulations, and is found to be a 4$\sigma$ detection of a change in matter content of 5%, or about 3.8$\sigma$ detection of 50% shift in the dark energy parameter using a full sky survey up to redshift 0.5.


INTRODUCTION
The large-scale matter distribution is a valuable source of information because its clustering properties are sensitive to cosmology.Indeed, stringent constraints have been set on cosmological parameters thanks to measurements of the baryonic acoustic oscillation (e.g.Eisenstein et al. 2005;Percival et al. 2007;Okumura et al. 2008;Beutler et al. 2011;Dawson et al. 2013;Alam et al. 2021;Xu et al. 2023), redshift-space distortions (e.g.Da Ângela et al. 2005;Marinoni & Buzzi 2010;Okumura et al. 2016;Neveux et al. 2020), and Alcock-Paczynski effect (e.g.Blake et al. 2011;Li et al. 2016;Beutler et al. 2017;Li et al. 2018;Zhang et al. 2019;Dong et al. 2023) in the two-point correlation functions of galaxies.However, as the evolution of the density field becomes increasingly non-linear, it departs from its Gaussian initial state.This causes an increasing amount of information to be contained in statistics beyond the two-point functions, which can be captured by measuring three-point correlation functions (e.g.Peebles & Groth 1975;Hinshaw et al. 1995;Nichol et al. 2006;Marín et al. 2013;Slepian et al. 2017;Sugiyama et al. 2023) or higher-order moments (e.g.Fry 1985;Bouchet et al. 1993; ★ E-mail:jshim@asiaa.sinica.edu.twBernardeau 1994;Croton et al. 2004;Cappi et al. 2015;Sabiu et al. 2019;Philcox 2022).While the hierarchy typically converges, each extra order becomes increasingly more difficult to measure robustly.This has fostered the development of alternative probes to obtain information beyond the simple two-point functions.
Recently, it has been shown that the relative clustering of critical points of a density field is maintained nearly-constant throughout the gravitational evolution (Shim et al. 2021, Appendix B), and can be measured accurately.Critical points are topological elements of a given field, and their attributes including position, height, curvature, and relative orientation encode the topological characteristics of the underlying field (Bond et al. 1996;Pogosyan et al. 2009;Sousbie et al. 2009;Gay et al. 2012) and its evolution (Cadiou et al. 2020).One of their characteristic clustering features is that a pair of critical points with different curvatures and heights cannot be arbitrarily close.This exclusion zone, or the strong anti-clustering region (Lumsden et al. 1989;Mo & White 1996;Sheth & Lemson 1999;Baldauf et al. 2016;Shim et al. 2021), appears more evidently in the crosscorrelations between critical points with the opposite-sign biases (Shim et al. 2021).Interestingly, Shim et al. (2021) showed that the sizes of the exclusion zones in the initial Gaussian field are fairly consistent with those measured at late time, suggesting that we have a theoretical handle on their cosmology dependence since the Gaussian expectation value can be derived from first principles.On the other hand, the amplitude of this exclusion zone is shown to depend on how smooth the underlying field is (Baldauf et al. 2016), indicating a cosmological dependence which we set forth to establish in this paper.
In expanding such exploration into cosmological tests, the redshift invariance of topology statistics emerges as a pivotal metric, as first introduced in Park & Kim (2010).Leveraging the conserved nature of the genus amplitude in density fields smoothed on large scales, these authors present a method to constrain the cosmological model.This involves identifying the correct expansion history of the Universe that minimizes the evolution of the genus amplitude with redshift.Aligning with this strategy, we introduce an approach utilizing the exclusion radius as a standard ruler to probe cosmology without delving into specific practicalities.We measure the size of the exclusion zone in critical point correlation functions and show how it can perform as a cosmological probe.Specifically, we focus on cross-correlations involving peak-wall, peak-void, filament-wall, and filament-void pairs -tracers that are oppositely biased to matter density fields.We rely on the suite of cosmological -body simulations to measure the critical point statistics.
The outline of the paper is as follows: §2 describes briefly the multiverse simulation; §3 introduces estimators for the exclusion zone; §4 presents our results on the Ω m -dependent variations of the exclusion zone radius, and on estimating  de when different redshifts fields are considered, while §5 concludes.Appendix §A shows the apparent evolution of the exclusion radius for a larger smoothing scale, while §B discusses the theoretical expectation of the exclusion radius.

SIMULATION SET
In this paper, we rely on the multiverse simulations introduced in Park et al. ( 2019 1. The power spectra are normalized such that the root mean square of the linearly evolved matter fluctuation at  = 0 yields a value of  8 = 0.794 when smoothed with a spherical top hat with  s = 8 ℎ −1 Mpc.The number of particles in each simulation is 2048 3 and the comoving size of the simulation box is 1024 ℎ −1 Mpc.The initial power spectrum at a redshift  init = 99 was computed with the CAMB package.The N-body integrator is an extension of the original GOTPM code (Dubinski et al. 2004) which evolve particles according to the modified Poisson equation where  de and  m are the linear growth factors of the dark energy and matter, respectively (see Sefusatti & Vernizzi 2011 for details).

EXCLUSION ZONE ESTIMATORS
Let us first briefly describe how we define and measure the size of the exclusion zones.We refer to Shim et al. (2021) for more details.The first step is to compute a smooth density field in real space at the relevant redshifts.Density fields are calculated on 512 3 grids by applying the Cloud-In-Cell method to the dark matter particle distribution.We then smooth these density fields with Gaussian kernel over 4 different smoothing scales  s = 16, 14, 10 and 6 ℎ −1 Mpc.Note that the smallest smoothing scale corresponds to a typical size of an average density region of mass-scale around 10 15  ⊙ .
0.5 Correlation functions for the F W, F V, P V, and P W are shown clockwise from the upper-left panel.The adopted Gaussian smoothing scale  s is 6 ℎ −1 Mpc.Vertical lines represent the exclusion zone radii and shaded regions show the standard errors around the fiducial cosmology.Note that the correlation function for the fiducial cosmology (red-solid) is nearly identical to those with the different equations of state dark energy models (blue).However, the two-point correlation function depends on Ω m , see Fig. 2.

Critical points
The critical points are defined as positions where the gradient of the dark matter density field vanishes (Milnor 1963;Bardeen et al. 1986;Shim et al. 2021).Based on the typical shape of isosurfaces in that neighborhood, the four types are labeled according to the sign of the Hessian's eigenvalues: peaks (P) with signature − − −, voids (V) with signature +++, filament-type saddles (F ) with signature − −+, and wall-type saddles (W) with signature −++.They are proxies for the geometry of the cosmic web (Bond et al. 1996) of the underlying density field, tracing respectively clusters, voids, filaments, and walls (Pogosyan et al. 2009;Sousbie et al. 2009) on a mass scale relevant to the adopted smoothing scale.

Finding critical points
The detection of critical points in a smoothed dark matter density field relies on a second-order Taylor-expansion of the density field near a critical point, x c : where  is the density field, ∇ the local gradient and ∇∇ the local hessian matrix.The detection algorithm proceeds as follows: a) For each cell in the grid, compute ∇ and ∇∇, b) solve equation ( 2) discarding solutions beyond a distance larger than one pixel and c) loop over cells that contain multiple critical points of the same kind, retaining for each only the critical point closest to the center of the cell (the technique was originally introduced by one of the authors for Colombi et al. 2000).

Computing clustering correlation functions
We count the pairs of critical points with rarity above or below a certain threshold to quantify their clustering characteristics.We define the rarity of critical points as  ≡ /, with  ≡ / ρ − 1, and where  is the over-density contrast of the smoothed density field, ρ the average density and  is the root mean square fluctuation of the field.
For peak and filament (respectively void and wall) critical points, we identify points with rarity higher (respectively lower) than a given threshold.In this analysis, the rarities  + type and  − type are chosen to trace the highest and lowest 20%-rarity critical points, so that respectively for peaks and filaments, and for voids and walls.Here  type ( ≥  thresh ) represents the number of critical points of a given type above the threshold  thresh (here 20% rarest) while  type is the total number of critical points of this type.As discussed in Shim et al. (2021), this choice is driven by our requirement to sample populations that represent the same abundance for a given type of critical point.We then measure cross-correlation functions with the estimator given by Davis & Peebles (1983) 1 where  shown in red and blue, respectively.We normalized the exclusion zone radius by the smoothing scale  s adopted since  ex linearly depends on the smoothing scale.Error bars represent the standard errors of the mean.The grey horizontal line marks the exclusion zone radius for the fiducial cosmology with Ω m = 0.26 and  de = −1.For this plot, we adopt the Gaussian smoothing scale  s = 6 ℎ −1 Mpc.The trend of the exclusion radius with Ω m is consistent with the scaling involving   in equation ( 11) if one associates  ex with  * , see the text for details.

Defining exclusion radii
The exclusion zone size, denoted as  ex , is measured from the crosscorrelation functions.Specifically, this is achieved by identifying the minimum distance at which the cross-correlation deviates from  = −1.In practice, we identify the radius at which with the deviation from perfect anti-correlation set at  = 0.01.We note that the exclusion radius for a particular density field depends on the choice of rarity levels of critical points and on the choice of  that defines departure from perfect anti-correlation.Decreasing rarity levels tends to yield smaller exclusion radii, as the difference in height between critical points will decrease.Conversely, it should increase the number of critical point pairs, thereby providing more reliable estimates for the exclusion radius.We estimate measurement uncertainties to evaluate the statistical significance of the impact of cosmological parameters on the exclusion radius.We divide the simulation box into eight separate, nonintersecting regions.For each sub-volume, we compute the cross-correlation function using critical points within that sub-volume.While doing this, we accounted for edge effects by creating a random sample within the same sub-volume.We then measure the exclusion radii from those cross-correlation functions.The measurement uncertainties are represented by the standard error of the mean, which is the standard deviation divided by the square root of the number of sub-volumes.

RESULTS
Let us now turn to our main results on matter density and dark energy equation of state estimation using exclusion zone measurements.

Exclusion radius and matter density parameter
We first explore how the exclusion radius varies with the matter density parameter.As illustrated in Fig. 1, we measure the crosscorrelation functions between peak/filament and void/wall critical points across five different cosmologies.The cross-correlations reveal varying exclusion zones based on the combination of critical points.Notably, the filament-wall combination presents the smallest exclusion zone, while the peak-void combination showcases the largest.This observation aligns with the understanding that the exclusion zone expands with increasing differences in height and curvature, as shown in (Baldauf et al. 2016;Shim et al. 2021).After the emergence of the exclusion zone, cross-correlations start to deviate from  = −1 and become less anti-clustered with increasing separation.Because peak/filament and void/wall points are oppositely biased tracers of the underlying matter density field, their crosscorrelations are always negative, eventually approaching  = 0 as expected (Kaiser 1984) at separations larger than  ≈ 10 s .
Comparing cases for different matter density parameters reveals that the cross-correlations show a smaller exclusion radius for a larger matter density parameter.This implies that the exclusion radius is impacted by the matter density.In contrast, when focusing on dark energy models with non-standard dark energy EOS parameters  de , their exclusion radii are very similar to the fiducial model.This suggests that changes to the dark energy parameter do not significantly affect the exclusion radius.Thus, the size of the exclusion zone is mainly dictated by the amount of matter and remains largely unaffected by variations in the dark energy EOS.
We now quantitatively compare the mean exclusion radii across five different cosmologies at lower redshifts, as illustrated in Fig. 2. The cross-correlations involving filaments and peaks are presented in the top and lower panels, respectively.Different shaded bars represent different redshift snapshot boxes for each cosmological parameter set.We observe a clear Ω m -dependence of the exclusion radius.For instance, the exclusion zone shrinks as the matter density increases.However, adjusting the dark energy parameter shows a minimal impact on the exclusion radius.This is a consequence of the power spectrum slope at the scales of interest becoming shallower as Ω m increases but remaining nearly invariant with  de .We discuss the cosmological parameter dependence of the exclusion zone in detail below while relating it to the theoretical expectation for the distance between extrema.Finally, when focusing on the time evolution, the exclusion zone remains remarkably stable in the redshift ranges investigated.
In Fig. 3, we quantify how significant the difference in the exclusion radius of each cosmology is from the fiducial case.The significance S is calculated as where  x ex and Σ x represent the exclusion radius and standard error for a particular cosmology, while  fid ex and Σ fid are for the fiducial model.We find that the significance values tend to be larger for the filament-wall correlation compared to other critical point combinations.Conversely, the peak-void correlation shows the smallest significance.This trend of larger (smaller) significance values when involving only saddles (extrema) arises from the differing numbers of critical points of each type.In a Gaussian random field, the ratio of saddle-to-extrema is approximately 3 (see e.g.Gay et al. 2012;Shim et al. 2021).Consequently, the number of pairs for the filament-wall correlation is roughly an order of magnitude larger than that for the peak-void case.Therefore, the standard error is typically smaller for the case only involving saddles than extrema leading to a larger significance value for cross-correlations involving saddles.While the significance for one type of cross-correlation might not be sufficient to distinguish between different cosmologies, combining results from all four distinct measurements will enhance the overall constraining power.
We identify critical points on a particular scale determined by the smoothing length adopted, so next we examine how the exclusion radius changes with the smoothing scale, as this allows us to probe the clustering of critical points corresponding to a different mass scale.Fig. 4 illustrates the variation of the exclusion radius measured from the matter density field smoothed on different smoothing scales.We observe a common trend that the rescaled exclusion radius decreases with the smoothing scale.This is evident across different cosmologies and types of cross-correlations considered.We consistently confirm that the exclusion radius shows a distinct dependence on the matter density on different smoothing scales.For instance, the rescaled  ex is always larger at all smoothing scales investigated for a cosmology with a smaller matter density.On the other hand, when varying the dark energy EOS parameters, the exclusion radii are remarkably consistent with the fiducial case.
Interestingly, we find that the behavior of  ex / s seen in Fig. 4 is similar to the prediction for  * / s as depicted in Fig. B2, which leads us to examine the relation between them.Here,  * represents the typical separation between extrema points and is defined by The moments of the smoothed power spectrum are calculated as where we adopt the Gaussian smoothing kernel, which serves as a low-pass filter suppressing power on scales below 1/ s for the power spectrum () weighted by varying powers of  2 .As depicted in Fig. B3, we find a linear scaling relation between  ex and  * .Associating these two provides a way to connect the measurements of exclusion radius to theoretical predictions based on Gaussian random fields.Indeed, for a Gaussian random field with (locally) scale-invariant power-spectra, () ∝   s , the rescaled typical distance between extrema, is solely determined by the effective power-law index of the power spectrum,  s [ = 1/ s ].The effective slope is itself sensitive to the chosen smoothing scale,  s , and to the matter density, Ω m , as displayed in Fig. B1, bottom panel.Now, we can understand how the rescaled  * varies with respective matter density, dark energy EOS, and smoothing scale.Since the power spectrum slope either increases when the smoothing scale or matter density parameter becomes larger (see Fig. B1), and hence, according to the equation ( 11) the rescaled  * will decrease with the smoothing scale and matter density.On the other hand,  * / s does not vary with the dark energy EOS since it has no impact on the power spectrum slope.Consequently, when relying on the linear relation between  ex and  * (see Fig. B3) the observed behaviors of  ex depicted in Figures 2 and 4 can be qualitatively explained with the prediction made for  * based on Gaussian random fields.

Exclusion radius and dark energy EOS
Let us now describe how we can extract information on the dark energy parameter using the exclusion radius, even though the effect of dark energy on the measured exclusion radius is shown to be negligible.Our strategy is based on the redshift invariance of the exclusion radius in comoving space.
We follow the approach presented in Park & Kim (2010); Blake et al. ( 2014); Appleby et al. (2018a), where matter density and dark energy parameters were constrained to minimize the apparent redshift evolution of the genus of the underlying matter density field.
Let us recall the strategy.It relies on the fact that the redshift invariance of the measured genus amplitude is achieved only when the adopted cosmology for the redshift-distance relation matches the underlying true cosmology.Indeed, the genus amplitude, A, obeys where  0 ≡  0 / 1 characterizes the typical separation between zero crossing of a density field.Hence A is expressed in terms of the ratio of moments defined by equation ( 9).Again, for scaleinvariant power-spectra, the genus amplitude A only depends on the effective power-index,  s , as √︁ 2/( s + 3).Since the slope of a power spectrum changes with  s (because the power spectrum is not scale invariant on the scales we are considering, the lower panel of Fig. B1), if an incorrect cosmology is assumed, the adopted smoothing scale effectively corresponds to a different length scale, hence A is altered1 .
Similarly and importantly, if the wrong cosmology is adopted, we also expect to observe an apparent redshift evolution of  ex because it is linearly related to  * (as described in Appendix B).We highlight that the measurements of exclusion radii in the evolved matter density field, which has deviated from its initial Gaussian nature, well align with the predictions made in the Gaussian random field for  * .Therefore, in theory, it is indeed possible to estimate the deviation of the exclusion radius from the reference point as a function of redshift by calculating the effective smoothing scale for an adopted cosmological model.
In Fig. 5, we show how the exclusion radius for a trial cosmology deviates from the fiducial case as a function of redshift.We observe a redshift evolution of the exclusion radius when the trial cosmology is inconsistent with the true underlying cosmology.For example, the exclusion radius monotonically grows with redshift when the matter density and dark energy parameters are larger than the true values, whereas it becomes smaller at higher redshifts in cosmologies with smaller matter density and dark energy parameters.We observe a larger departure from the reference for non-fiducial dark energy models.This reflects the fact that, at low , the distance estimates are more strongly impacted by dark energy than matter density, in the redshift range of interest.Thus, the evolution of the apparent exclusion radius is more sensitive to the properties of dark energy.
We then calculate the potential measurement errors for an all-sky survey up to various redshifts to assess if it is possible to detect such redshift evolution of the exclusion radius given those survey volumes.We base our error estimates on the standard error achieved from the simulation volume  sim = 1 ℎ −3 Gpc 3 .Let us assume that these errors reduce by √︁  sim / sur , similar to the behavior of shot-noise, where  sur is survey volume.For a survey with  sur ≈ 2.5 ℎ −3 Gpc 3 , scanning up to  = 0.3, the departure of  de -shifted dark energy models from the fiducial cosmology can be detected approximately at 1.5 significance level.When a survey extends to  = 0.5 reaching  sur ≈ 10 ℎ −3 Gpc 3 , the detection significances for the non-fiducial dark energy models can increase approximately to 3.8, while those for Ω m -shifted models eventually become marginal.We expect that one can more significantly detect the redshift variation of the exclusion radius when measuring it at higher redshifts.This is partly because the predicted amplitude of redshift variation gradually grows with redshift.In addition, even with a relatively narrow redshift span, the volume scanned through a non-full-sky survey targeting higher redshifts can be larger than the largest comoving volume considered here, for instance,  sur = 10 ℎ −3 Gpc 3 up to  ≈ 0.5.Therefore, a higher significance detection should be available from surveys aiming at high redshifts.Note that we can also combine the exclusion radius measurements using four distinct types of cross-correlations, which should yield more robust measurements (although these measurements from the same density field are not strictly independent).For completeness, we also show the expected redshift evolution of the exclusion radius for  s = 16 ℎ −1 Mpc with the same mock surveys in Fig. A1.As expected, the error bars are larger.

CONCLUSIONS & PERSPECTIVES
This paper advocates that the exclusion radii,  ex , define standard rulers on multiple scales: on linear scales, they can be computed using Lagrangian theory (Bardeen et al. 1986;Shim et al. 2021), while on smaller non-linear scales they can be extracted from cosmological simulations, as was done in this paper.The value of  ex mainly constrains Ω m (Fig. 3) and may also be potentially used for measuring  s as can be done with the genus amplitude analysis (Appleby et al. 2020).Our results support that, at a given physical scale, the exclusion radius is redshift-invariant at first order, which can be leveraged to constrain the equation of state of dark energy  de (Fig. 5).More precise results could be obtained by calibrating the scale-dependence and redshift evolution of the relation on -body simulations.We find that the redshift evolution and cosmology dependence of  ex can be remarkably captured using the locally-scale-invariant quantity  * , which we can predict from linear theory.This implies that the topology we measure at late time is already imprinted in the Gaussian initial state, and is not destroyed by gravitational collapse.Thus, the non-Gaussian final state mostly preserves its Gaussian initial topology.We carried out a naive error budget and found that the accuracy of the parameter estimation corresponds to a 4 detection of a change in matter content of 5% in 1 ℎ −3 Gpc 3 , or roughly at 3.8 detection of 50% change in the dark energy parameter using a full sky survey up to redshift 0.5.These numbers could be improved by performing a joint analysis of all exclusion radii, accounting for the fact that they are not independent, or by exploring jointly different rarity thresholds.At this stage, we assumed that the linear relation between  ex and  * given in Appendix B holds for the range of redshifts considered; investigating its redshift dependence is of interest as it would allow us to mitigate its residual effect (beyond the linear scaling shown in Fig. B3) on smaller scales, but is beyond the scope of the current analysis.Conversely, extending this investigation to higher redshift would clearly be of interest given the depth of upcoming surveys2 , and is also postponed to later work.
The accuracy of these standard rulers was assessed here using the multiverse set of simulations in a fairly idealized setting.Our goal was to demonstrate the potential to build such rulers out of the two-point correlation functions of critical points, setting aside complications from completeness issues, tracer biases, SNR, and redshift space distortions.The companion paper, Kraljic et al. (2022), investigated how Lyman− tomography can, in the context of the Weave mission (Dalton et al. 2012;Pieri et al. 2016), open access to the clustering of critical points and the possibility of exclusion zone estimation.While the focus of this paper was on exclusion zones, Kraljic et al. (2022) showed that the position of the maxima of the crosscorrelations of critical points of the same sign could also be estimated from tomography.Combining both approaches could potentially allow even tighter constraints to be set on cosmological parameters.While more realistic, Kraljic et al. (2022) still only involved mock data, and should of course be revisited with other tomographic 3D surveys such as PFS (Takada et al. 2014), MOSAIC on ELT (Puech et al. 2018), MSE (The MSE Science Team et al. 2019), but also with spectroscopic or photometric redshifts galactic surveys, such as Euclid (Laureĳs et al. 2011), DESI (DESI Collaboration et al. 2016), PFS, WFIRST (Roman Space Telescope; Spergel et al. 2013), or LSST (Rubin Observatory;Ivezić et al. 2019).
The cosmology dependence of the clustering of critical point should also be investigated in 2D maps such as weak lensing maps (Laureĳs et al. 2011;Hildebrandt et al. 2017), line intensity maps (CHIME Collaboration et al. 2022) and photometric redshift surveys (Doré et al. 2014).
It would finally be of interest to compute the auto-correlation function of saddle points in redshift space, which are less subject to finger of god effects given the density they probe.This would allow for a more robust Alcock-Paczyński test (Alcock & Paczynski 1979), as one would expect that the overall motion of the local cosmic web does not impact such correlations.
); Tonegawa et al. (2020) which are a set of cosmological

Figure 1 .
Figure1.Two-point cross-correlation functions for pairs of positively and negatively biased critical points for the five different cosmologies as labeled.Correlation functions for the F W, F V, P V, and P W are shown clockwise from the upper-left panel.The adopted Gaussian smoothing scale  s is 6 ℎ −1 Mpc.Vertical lines represent the exclusion zone radii and shaded regions show the standard errors around the fiducial cosmology.Note that the correlation function for the fiducial cosmology (red-solid) is nearly identical to those with the different equations of state dark energy models (blue).However, the two-point correlation function depends on Ω m , see Fig.2.

Figure 2 .
Figure2.Exclusion zone radius averaged over 8 measurements from the simulations for the five different cosmologies.Different Ω m -and  de -cosmologies are shown in red and blue, respectively.We normalized the exclusion zone radius by the smoothing scale  s adopted since  ex linearly depends on the smoothing scale.Error bars represent the standard errors of the mean.The grey horizontal line marks the exclusion zone radius for the fiducial cosmology with Ω m = 0.26 and  de = −1.For this plot, we adopt the Gaussian smoothing scale  s = 6 ℎ −1 Mpc.The trend of the exclusion radius with Ω m is consistent with the scaling involving   in equation (11) if one associates  ex with  * , see the text for details.

Figure 3 .
Figure3.Significance, S, of the difference in the exclusion radius from the fiducial cosmology as a function of redshift.Non-fiducial Ω m and  de models are shown in red and blue, respectively.For this plot, we adopt the Gaussian smoothing scale  s = 6 ℎ −1 Mpc.Note that we allow the significance to have both signs as they can indicate the direction of the difference from the fiducial case.

Figure 4 .
Figure 4. Measured exclusion zone radius as a function of smoothing scale for four different types of cross-correlations.Correlation functions for F W, F V, P V, and P W are shown clockwise from the upper-left panel.The symbols and lines represent the measured exclusion radius and best fit to these measurements, respectively.Error bars are measured as the standard error of the mean.Again the trend is consistent with theoretical expectation if one associates  ex to  * , as confirmed to the first order in Fig. B3.

Figure 5 .
Figure 5. Expected redshift evolution of  ex relative to the true value when adopting trial cosmologies with different Ω m (red) or  de (blue) for the redshift-distance relation.The exclusion radius for the true underlying cosmology is shown in a grey dotted line and vertical error bars at redshifts  = 0.3, 0.4, and 0.5 represent the estimated measurement uncertainties about the fiducial values, given the volumes surveyed up to those respective redshifts.Note that the true exclusion radius does not evolve with redshift.

Table 1 .
Summary of cosmological parameter values of five cosmologies considered.The second-row model is the fiducial cosmology.