Two-point Statistics Research Articles

Theoretical wavelet ℓ1-norm from one-point probability density function prediction

Context. Weak gravitational lensing, which results from the bending of light by matter along the line of sight, is a potent tool for exploring large-scale structures, particularly in quantifying non-Gaussianities. It is a pivotal objective for upcoming surveys. In the realm of current and forthcoming full-sky weak-lensing surveys, convergence maps, which represent a line-of-sight integration of the matter density field up to the source redshift, facilitate field-level inference. This provides an advantageous avenue for cosmological exploration. Traditional two-point statistics fall short of capturing non-Gaussianities, necessitating the use of higher-order statistics to extract this crucial information. Among the various available higher-order statistics, the wavelet ℓ1 -norm has proven its efficiency in inferring cosmology. However, the lack of a robust theoretical framework mandates reliance on simulations, which demand substantial resources and time. Aims. Our novel approach introduces a theoretical prediction of the wavelet ℓ1-norm for weak-lensing convergence maps that is grounded in the principles of large-deviation theory. This method builds upon recent work and offers a theoretical prescription for an aperture mass one-point probability density function. Methods. We present for the first time a theoretical prediction of the wavelet ℓ1-norm for convergence maps that is derived from the theoretical prediction of their one-point probability distribution. Additionally, we explored the cosmological dependence of this prediction and validated the results on simulations. Results. A comparison of our predicted wavelet ℓ1 -norm with simulations demonstrates a high level of accuracy in the weakly nonlinear regime. Moreover, we show its ability to capture cosmological dependence. This paves the way for a more robust and efficient parameter-inference process.

Secondary halo bias through cosmic time II. Reconstructing halo properties using clustering information

When constructing mock galaxy catalogs based on suites of dark matter halo catalogs generated with approximated, calibrated, or machine-learning approaches, assigning intrinsic properties for these tracers is a step of paramount importance, given that they can shape the abundance and spatial distribution of mock galaxies and galaxy clusters. We explore the possibility of assigning properties of dark matter halos within the context of calibrated or learning approaches, explicitly using clustering information. The goal is to retrieve the correct signal of primary and secondary large-scale effective bias as a function of properties reconstructed solely based on phase-space properties of the halo distribution and dark matter density field. The algorithm reconstructs a set of halo properties (such as virial mass, maximum circular velocity, concentration, and spin) constrained to reproduce both primary and secondary (or assembly) bias. The key ingredients of the algorithm are the implementation of individually-assigned large-scale effective bias, a multi-scale approach to account for halo exclusion, and a hierarchical assignment of halo properties. The method facilitates the assignment of halo properties, aiming to replicate the large-scale effective bias, both primary and secondary. This constitutes an improvement over previous methods in the literature, especially for the high-mass end population. We have designed a strategy for reconstructing the main properties of dark matter halos obtained using calibrated or learning algorithms, such that the one- and two-point statistics (on large scales) replicate the signal from detailed $N$-body simulations. We encourage the application of this strategy (or the implementation of our algorithm) for the generation of mock catalogs of dark matter halos based on approximated methods.

Two-point statistics in non-homogeneous rotating turbulence

Time-resolved two-dimensional two-component particle image velocimetry measurements with high spatial resolution are carried out in a water tank agitated by four blades rotating at constant speed. Different blade geometries and rotation speeds are tested for the purpose of modifying turbulent flow conditions. In all cases where no baffles are used to break the rotation, the Zeman length is an order of magnitude smaller than the Taylor length. Compared with the cases with baffles which break the rotation, in the unbaffled cases turbulence production and/or mean advection are significant and the turbulence nonlinearity is dramatically reduced for the horizontal (i.e. normal to the axis of rotation) two-point turbulence fluctuating velocities. This nonlinearity reduction is manifest not only in the interscale turbulent energy transfer but also in the interspace turbulent energy transfer, which nearly vanishes. However, the nonlinearity is not reduced for the vertical two-point turbulence fluctuating velocities: the corresponding interscale turbulent transfer rate is in fact intensified, and its dependence on the two-point separation distance, as well as that of the corresponding interspace turbulent transfer rate, which does not vanish, is significantly modified. Even though non-homogeneities are very different for different blades and rotation speeds in the unbaffled cases, the horizontal fluctuating velocity's second-order structure function collapses with scalings which resemble predictions for homogeneous turbulence subject to strong rotation. The vertical fluctuating velocity's second-order structure function does not collapse for different blade geometries by neither these nor the Kolmogorov predictions.

Reionization Parameter Inference from 3D Minkowski Functionals of the 21 cm Signals

The Minkowski functionals (MFs), a set of topological summary statistics, have emerged as a powerful tool for extracting non-Gaussian information. We investigate the prospect of constraining the reionization parameters using the MFs of the 21 cm brightness temperature field from the epoch of reionization (EOR). Realistic effects, including thermal noise, synthesized beam, and foreground avoidance, are applied to the mock observations from radio interferometric array experiments such as the Hydrogen Epoch of Reionization Array (HERA) and the Square Kilometre Array (SKA). We demonstrate that the MFs of the 21 cm signal measured with SKA-Low can be used to distinguish different reionization models, whereas the MF measurement with a HERA-like array cannot be made accurately enough. We further forecast the accuracies with which the MF measurements can place constraints on reionization parameters, using the standard Markov Chain Monte Carlo analysis for parameter inference based on forward modeling. We find that for SKA-Low observation, MFs provide unbiased estimations of the reionization parameters with accuracies comparable to the power spectrum (PS) analysis. Furthermore, joint constraints using both MFs and PS can improve the constraint accuracies by up to 30% compared to those with the PS alone. Nevertheless, the constraint accuracies can be degraded if the EOR window is shrunk with strong foreground avoidance. Our analysis demonstrates the promise of MFs as a set of summary statistics that extract complementary information from the 21 cm EOR field to the two-point statistics, which suggests a strong motivation for incorporating the MFs into the data analysis of future 21 cm observations.

Strong evidence for universality in homogeneous compressible turbulence

Quantifying the degree of universality in compressible turbulence is challenging due to the existence of different modes and their complex interactions. For a restricted family of flows, Donzis and John [Phys. Rev. Fluids 5, 084609 (2020)] showed that universal behavior is indeed observed in compressible turbulence if the ratio of dilatational to solenoidal root mean square (rms) velocities (δ=u′d/u′s) is incorporated as a scaling parameter along with the traditional turbulent Mach number (Mt=u′/〈c〉, where u′ is the rms velocity and 〈c〉 is the mean speed of sound). In this paper, we argue for the generality of those results by analyzing a wide range of compressible turbulent flows spanning a variety of flow configurations and setups to assess the degree of universal behavior. These include, among others, reacting flows, flows with solenoidal, thermal, and dilatational forcing, and flows with mean shear and bulk viscosity. We also performed new direct numerical simulations, which include turbulence in situations where vibrational modes of constitutive molecules are not in thermal equilibrium. Collectively, we offer the largest comparison across studies in the literature to date. We find that despite the wide range of forcing conditions and physical processes, universality holds across all these turbulent flows to a very satisfactory degree when both δ and Mt are considered as intrinsic compressibility parameters. The statistics investigated here—single-point statistics up to order four—are chosen such that they represent different ranges across the spectrum of dynamically relevant turbulence scales. We discuss the applicability of the purposed universal behavior for other key statistics in these turbulent flows, including two-point statistics and inhomogeneity effects, and the perspective it opens for modeling them.

A multivariate and wavelet-based analysis on the wall pressure fluctuations for propellers in ground effect

This study, conducted at the University of Bristol’s aeroacoustic facility, explores the aeroacoustic characteristics of a 10-inch APC propeller in pusher configuration within ground effect conditions. Tested at a rotational speed of 7000 r/min, the propeller’s performance was evaluated at varying distances from the ground. Wall pressure fluctuations near the propeller were measured at eight radial points, complemented by far-field acoustic measurements. Consistent with previous research, distinctive thrust and torque behaviours were observed when operating both in and out of ground effect. Notably, a significant noise increase of approximately 5 dB was observed on the ground’s reflected side, while a reduction of about 14 dB was detected on the shielded side. Higher-order statistical analysis such as skewness and kurtosis revealed substantial effects of tip vortex impingement in ground effect conditions. This study applies two-point statistics and the Corcos model to analyze propeller-induced wall pressure fluctuations. The coherence function effectively captures the intricate dynamics of pressure variations across different spatial locations and frequencies. This approach offers novel insights into the behaviour of boundary layers and flow-induced pressures in the context of propellers operating in the ground effect. Wavelet decomposition was utilized to differentiate the influences of the boundary layer and the acoustic field. This comprehensive analysis highlights the intricate dynamics of propeller operation in ground effect, contributing valuable insights to the field of aeroacoustic research.

Ray-tracing versus Born approximation in full-sky weak lensing simulations of the MillenniumTNG project

ABSTRACT Weak gravitational lensing is a powerful tool for precision tests of cosmology. As the expected deflection angles are small, predictions based on non-linear N-body simulations are commonly computed with the Born approximation. Here, we examine this assumption using DORIAN, a newly developed full-sky ray-tracing scheme applied to high-resolution mass-shell outputs of the two largest simulations in the MillenniumTNG suite, each with a 3000 Mpc box containing almost 1.1 trillion cold dark matter particles in addition to 16.7 billion particles representing massive neutrinos. We examine simple two-point statistics like the angular power spectrum of the convergence field, as well as statistics sensitive to higher order correlations such as peak and minimum statistics, void statistics, and Minkowski functionals of the convergence maps. Overall, we find only small differences between the Born approximation and a full ray-tracing treatment. While these are negligibly small at power-spectrum level, some higher order statistics show more sizeable effects; ray-tracing is necessary to achieve per cent level precision. At the resolution reached here, full-sky maps with 0.8 billion pixels and an angular resolution of 0.43 arcmin, we find that interpolation accuracy can introduce appreciable errors in ray-tracing results. We therefore implemented an interpolation method based on non-uniform fast Fourier transforms (NUFFT) along with more traditional methods. Bilinear interpolation introduces significant smoothing, while nearest grid point sampling agrees well with NUFFT, at least for our fiducial source redshift, $z_s=1.0$, and for the 1 arcmin smoothing we use for higher order statistics.

Constraining modified gravity with weak lensing peaks

Abstract It is well established that maximizing the information extracted from upcoming and ongoing stage-IV weak-lensing surveys requires higher-order summary statistics that complement the standard two-point statistics. In this work, we focus on weak-lensing peak statistics to test two popular modified gravity models, f(R) and nDGP, using the FORGE and BRIDGE weak-lensing simulations, respectively. From these simulations we measure the peak statistics as a function of both cosmological and modified gravity parameters simultaneously. Our findings indicate that the peak abundance is sensitive to the strength of modified gravity, while the peak two-point correlation function is sensitive to the nature of the screening mechanism in a modified gravity model. We combine these simulated statistics with a Gaussian Process Regression emulator and a Gaussian likelihood to generate stage-IV forecast posterior distributions for the modified gravity models. We demonstrate that, assuming small scales can be correctly modelled, peak statistics can be used to distinguish GR from f(R) and nDGP models at the two-sigma level with a stage-IV survey area of $300 \, \rm {deg}^2$ and $1000 \, \rm {deg}^2$, respectively. Finally, we show that peak statistics can constrain log10(|fR0|) = −6 to 2% precision, and log10(H0rc) = 0.5 to 25% precision.

Clustering of primordial black holes from quantum diffusion during inflation

We study how large fluctuations are spatially correlated in the presence of quantum diffusion during inflation. This is done by computing real-space correlation functions in the stochastic-δ N formalism. We first derive an exact description of physical distances as measured by a local observer at the end of inflation, improving on previous works. Our approach is based on recursive algorithmic methods that consistently include volume-weighting effects. We then propose a “large-volume” approximation under which calculations can be done using first-passage time analysis only, and from which a new formula for the power spectrum in stochastic inflation is derived. We then study the full two-point statistics of the curvature perturbation. Due to the presence of exponential tails, we find that the joint distribution of large fluctuations is of the form P(ζ R 1, ζ R 2) = F(R 1,R 2, r) P(ζ R 1)P( ζ R 2), where ζ R 1 and ζ R 2 denote the curvature perturbation coarse-grained at radii R 1 and R 2, around two spatial points distant by r. This implies that, on the tail, the reduced correlation function, defined as P(ζ R 1 > ζc, ζ R 2 > ζc)/[P(ζ R 1 > ζc) P(ζ R 2 > ζc)]-1, is independent of the threshold value ζc. This contrasts with Gaussian statistics where the same quantity strongly decays with ζc, and shows the existence of a universal clustering profile for all structures forming in the exponential tails. Structures forming in the intermediate (i.e. not yet exponential) tails may feature different, model-dependent behaviours.

Unleashing cosmic shear information with the tomographic weak lensing PDF

In this work, we demonstrate the constraining power of the tomographic weak lensing convergence PDF for StageIV-like source galaxy redshift bins and shape noise. We focus on scales of 10 to 20 arcmin in the mildly nonlinear regime, where the convergence PDF and its changes with cosmological parameters can be predicted theoretically. We model the impact of reconstructing the convergence from the shear field using the well-known Kaiser-Squires formalism. We cross-validate the predicted and the measured convergence PDF derived from convergence maps reconstructed using simulated shear catalogues. Employing a Fisher forecast, we determine the constraining power for (Ωm,S8,w0). We find that adding a 5-bin tomography improves the κ−PDF constraints by a factor of {3.8,1.3,1.6} for (Ωm,S8,w0) respectively. Additionally, we perform a joint analysis with the shear two-point correlation functions, finding an enhancement of around a factor of 1.5 on all parameters with respect to the two-point statistics alone. These improved constraints come from disentangling from w0 by extracting non-Gaussian information, in particular, including the PDF skewness at different redshift bins. We also study the effect of varying the number of parameters to forecast, in particular we add h, finding that the convergence PDF maintains its constraining power while the precision from two-point correlations degrades by a factor of {1.7,1.4,1.8} for , respectively.

On direct estimation of density parameters and Hubble constant for ΛCDM universe using Hubble measurements

The set of cosmological density parameters ( Ω0mh02 , Ω0kh02 , Ω0Λh02 ) and Hubble constant ( hˆ0 ) are useful for fundamental understanding of the Universe from many perspectives. In this article, we propose a new procedure to estimate these parameters for Λ cold dark matter (ΛCDM) universe in the Friedmann-Robertson-Walker (FRW) background. We generalize the two-point statistics first proposed by Sahni et al (2008) to the three point case and estimate the parameters using currently available Hubble parameter (H(z)) values in the redshift range 0.07 ≤ z ≤ 2.36 measured by differential age (DA) and baryon acoustic oscillation (BAO) techniques. All the parameters are estimated assuming the general case of non-flat universe. Using both DA and BAO data we obtain Ω0mh02=0.1485±0.0065 , Ω0kh02=−0.0137±0.017 , Ω0Λh02=0.3126±0.0145 and hˆ0=0.6689±0.0021 . These results are in satisfactory agreement with the Planck results. An important advantage of our method is that to estimate the value of any one of the independent cosmological parameters one does not need to use the values for the rest of them. Each parameter is obtained solely from the measured values of Hubble parameters at different redshifts without any need to use values of other parameters. Such a method is expected to be less susceptible to the undesired effects of degeneracy issues between cosmological parameters during their estimations. Moreover, there is no requirement of assuming spatial flatness in our method.

Neutrino Mass Constraint from an Implicit Likelihood Analysis of BOSS Voids

Cosmic voids identified in the spatial distribution of galaxies provide complementary information to two-point statistics. In particular, constraints on the neutrino mass sum, ∑m ν , promise to benefit from the inclusion of void statistics. We perform inference on the CMASS Northern Galactic Cap sample of the third-generation Sloan Digital Sky Survey/Baryon Oscillation Spectroscopic Survey (BOSS) with the aim of constraining ∑m ν . We utilize the void size function, the void–galaxy cross power spectrum, and the galaxy auto power spectrum. To extract constraints from these summary statistics we use a simulation-based approach, specifically implicit likelihood inference. We populate approximate gravity-only, particle-neutrino cosmological simulations with an expressive halo occupation distribution model. With a conservative scale cut of kmax=0.15hMpc−1 and a Planck-inspired Lambda cold dark matter prior, we find upper bounds on ∑m ν of 0.43 and 0.35 eV from the galaxy auto power spectrum and the full data vector, respectively (95% credible interval). Relaxing the scale cut to kmax=0.2hMpc−1 , we find 0.28 and 0.32 eV. Generally, void statistics appear to add modest but nonzero information beyond the auto power spectrum. We also substantiate the usual assumption that the void size function is Poisson distributed.

Joint inference of multiplicative and additive systematics in galaxy density fluctuations and clustering measurements

ABSTRACT Galaxy clustering measurements are a key probe of the matter density field in the Universe. With the era of precision cosmology upon us, surveys rely on precise measurements of the clustering signal for meaningful cosmological analysis. However, the presence of systematic contaminants can bias the observed galaxy number density, and thereby bias the galaxy two-point statistics. As the statistical uncertainties get smaller, correcting for these systematic contaminants becomes increasingly important for unbiased cosmological analysis. We present and validate a new method for understanding and mitigating both additive and multiplicative systematics in galaxy clustering measurements (two-point function) by joint inference of contaminants in the galaxy overdensity field (one-point function) using a maximum-likelihood estimator (MLE). We test this methodology with Kilo-Degree Survey-like mock galaxy catalogues and synthetic systematic template maps. We estimate the cosmological impact of such mitigation by quantifying uncertainties and possible biases in the inferred relationship between the observed and the true galaxy clustering signal. Our method robustly corrects the clustering signal to the sub-per cent level and reduces numerous additive and multiplicative systematics from $1.5 \sigma$ to less than $0.1\sigma$ for the scenarios we tested. In addition, we provide an empirical approach to identifying the functional form (additive, multiplicative, or other) by which specific systematics contaminate the galaxy number density. Even though this approach is tested and geared towards systematics contaminating the galaxy number density, the methods can be extended to systematics mitigation for other two-point correlation measurements.

The statistics of Rayleigh-Levy flight extrema

Rayleigh-Levy flights have played a significant role in cosmology as simplified models for understanding how matter distributes itself under gravitational influence. These models also exhibit numerous remarkable properties that enable predictions of a wide range of characteristics. Here, we derive the one- and two-point statistics for extreme points within Rayleigh-Levy flights, spanning one to three dimensions (1D-3D) and stemming directly from fundamental principles. In the context of the mean field limit, we provide straightforward closed-form expressions for Euler counts and their correlations, particularly in relation to their clustering behaviour over long distances. Additionally, quadratures allow for the computation of extreme value number densities. A comparison between theoretical predictions in 1D and Monte Carlo measurements shows remarkable agreement. Given the widespread use of Rayleigh-Levy processes, these comprehensive findings offer significant promise not only in astrophysics, but also in broader applications beyond the field.

The use of LES to analyse higher-order moments of two-point statistics

The models prescribed by the IEC standard for the generation of wind fields, as well as intermittent modifications of the original models, are used as inflows in LES. The behaviour of the wind fields is investigated with regard to higher-order moments. It is shown that all models reach an intermittent state in the LES. This intermittent state can be determined by the intermittency parameter and by higher orders of the moments. It can be seen that the Kaimal model generates significantly lower intermittency than the other models. The convergence length is significantly reduced with a correct intermittency description in the wind fields.

Fast emulation of two-point angular statistics for photometric galaxy surveys

ABSTRACT We develop a set of machine-learning-based cosmological emulators, to obtain fast model predictions for the C(ℓ) angular power spectrum coefficients, characterizing tomographic observations of galaxy clustering and weak gravitational lensing from multiband photometric surveys (and their cross-correlation). A set of neural networks are trained to map cosmological parameters into the coefficients, achieving, with respect to standard Boltzmann solvers, a speed-up of $\mathcal {O}(10^3)$ in computing the required statistics for a given set of cosmological parameters, with an accuracy better than 0.175 per cent (<0.1 per cent for the weak lensing case). This corresponds to $\lesssim 2~{{\ \rm per\ cent}}$ of the statistical error bars expected from a typical Stage IV photometric surveys. Such overall improvement in speed and accuracy is obtained through (i) a specific pre-processing optimization, ahead of the training phase, and (ii) an effective neural network architecture. Compared to previous implementations in the literature, we achieve an improvement of a factor of 5 in terms of accuracy, while training a considerably lower amount of neural networks. This results in a cheaper training procedure and a higher computational performance. Finally, we show that our emulators can recover unbiased posteriors when analysing synthetic Stage-IV galaxy survey data sets.

Distinguishing ΛCDM from Evolving Dark Energy with Om Two-point Statistics: Implications from the Space-borne Gravitational-wave Detector

The Omh 2(z i , z j ) two-point diagnostics was proposed as a litmus test of the ΛCDM model, and measurements of the cosmic expansion rate H(z) have been extensively used to perform this test. The results obtained so far suggested a tension between observations and predictions of the ΛCDM model. However, the data set of H(z) direct measurements from cosmic chronometers and baryon acoustic oscillations was quite limited. This motivated us to study the performance of this test on a larger sample obtained in an alternative way. In this paper, we propose that gravitational-wave (GW) standard sirens could provide large samples of H(z) measurements in the redshift range of 0 < z < 5, based on the measurements of the dipole anisotropy of luminosity distance arising from the matter inhomogeneities of the large-scale structure and the local motion of the observer. We discuss the effectiveness of our method in the context of the space-borne DECi-herz Interferometer Gravitational-wave Observatory, based on a comprehensive H(z) simulated data set from binary neutron star merger systems. Our results indicate that in the GW domain, the Omh 2(z i , z j ) two-point diagnostics could effectively distinguish whether ΛCDM is the best description of our Universe. We also discuss the potential of our methodology in determining possible evidence for dark energy evolution, focusing on its performance on the constant and redshift-dependent dark energy equation of state.

Mass reconstruction and noise reduction with cosmic-web environments

ABSTRACT The clustering of galaxies and their connections to their initial conditions is a major means by which we learn about cosmology. However, the stochasticity between galaxies and their underlying matter field is a major limitation for precise measurements of galaxy clustering. Efforts have been made with an optimal weighting scheme to reduce this stochasticity using the mass-dependent clustering of dark matter haloes. Here, we show that this is not optimal. We demonstrate that the cosmic-web environments (voids, sheets, filaments, and knots) of haloes, when combined linearly with the linear bias, provide extra information for reducing stochasticity in terms of two-point statistics. Using the environmental information alone can increase the signal-to-noise of clustering by a factor of 3 better than the white-noise level at the scales of the baryon acoustic oscillations. The information about the environment and halo mass are complementary. Their combination increases the signal-to-noise by another factor of 2-3. The information about the cosmic web correlates with other properties of haloes, including halo concentrations and tidal forces – all are related to the assembly bias of haloes.

Positivity from Cosmological Correlators

Effective field theories in flat space and in anti-de Sitter space are constrained by causality and unitarity, often in the form of positivity bounds. Similar bounds have been harder to demonstrate in cosmological backgrounds, where the roles of unitarity and causality are more obscure. Fortunately, the expansion of the universe ensures that late-time cosmological correlators are effectively classical and the role of unitarity is played by classical statistical inequalities. For multi-field inflation, the resulting positivity constraints have long been known in terms of the Suyama-Yamaguchi inequality. In this paper, we demonstrate that similar statistical bounds imply nontrivial constraints for massive fields in the early universe. We show that any real anomalous dimensions for principal series fields in de Sitter space must be positive. We also derive a limit on the amplitude of particular oscillatory signals from inflation, including those arising in cosmological collider physics. Finally, we demonstrate that these constraints manifest themselves directly in the two-point statistics of matter and galaxies that will be measured in upcoming surveys.

The effects of non-linearity on the growth rate constraint from velocity correlation functions

ABSTRACT The two-point statistics of the cosmic velocity field, measured from galaxy peculiar velocity (PV) surveys, can be used as a dynamical probe to constrain the growth rate of large-scale structures in the Universe. Most works use the statistics on scales down to a few tens of Megaparsecs, while using a theoretical template based on the linear theory. In addition, while the cosmic velocity is volume-weighted, the observable line-of-sight velocity two-point correlation is density-weighted, as sampled by galaxies, and therefore the density–velocity correlation term also contributes, which has often been neglected. These effects are fourth order in powers of the linear density fluctuation $\delta _{\rm L}^4$, compared to $\delta _{\rm L}^2$ of the linear velocity correlation function, and have the opposite sign. We present these terms up to $\delta _{\rm L}^4$ in real space based on the standard perturbation theory, and investigate the effect of non-linearity and the density–velocity contribution on the inferred growth rate fσ8, using N-body simulations. We find that for a next-generation PV survey of volume $\sim {\cal O}(500 \, h^{-1} \, {\rm Mpc})^3$, these effects amount to a shift of fσ8 by ∼10 per cent and is comparable to the forecasted statistical error when the minimum scale used for parameter estimation is $r_{\rm min} = 20 \, h^{-1} \, {\rm Mpc}$.