Euclid preparation. The impact of redshift interlopers on the two-point correlation function analysis

In fiber-fed galaxy redshift surveys, the finite size of the fiber plugs prevents two fibers from being placed too close to one another, limiting the ability of studying galaxy clustering on all scales. We present a new method for correcting such fiber collision effects in galaxy clustering statistics based on spectroscopic observations. Our method makes use of observations in tile overlap regions to measure the contributions from the collided population, and to therefore recover the full clustering statistics. The method is rooted in solid theoretical ground and is tested extensively on mock galaxy catalogs. We demonstrate that our method can well recover the projected and the full three-dimensional redshift-space two-point correlation functions on scales both below and above the fiber collision scale, superior to the commonly used nearest neighbor and angular correction methods. We discuss potential systematic effects in our method. The statistical correction accuracy of our method is only limited by sample variance, which scales down with (the square root of) the volume probed. For a sample similar to the final SDSS-III BOSS galaxy sample, the statistical correction error is expected to be at the level of 1% on scales 0.1--30Mpc/h for the two-point correlation functions. The systematic error only occurs on small scales, caused by non-perfect correction of collision multiplets, and its magnitude is expected to be smaller than 5%. Our correction method, which can be generalized to other clustering statistics as well, enables more accurate measurements of full three-dimensional galaxy clustering on all scales with galaxy redshift surveys. (abridged)

Embedded random matrix ensembles with k-body interactions are well established to be appropriate for many quantum systems. For these ensembles the two point correlation function is not yet derived, though these ensembles are introduced 50 years back. Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues, say E and E^{'}. Fluctuation measures such as the number variance and Dyson-Mehta Δ_{3} statistic are defined by the two-point function and so also the variance of the level motion in the ensemble. Recently, it is recognized that for the embedded ensembles with k-body interactions the one-point function (ensemble averaged density of eigenvalues) follows the so called q-normal distribution. With this, the eigenvalue density can be expanded by starting with the q-normal form and using the associated q-Hermite polynomials He_{ζ}(x|q). Covariances S_{ζ}S_{ζ^{'}}[over ¯] (overline representing ensemble average) of the expansion coefficients S_{ζ} with ζ≥1 here determine the two-point function as they are a linear combination of the bivariate moments Σ_{PQ} of the two-point function. Besides describing all these, in this paper formulas are derived for the bivariate moments Σ_{PQ} with P+Q≤8, of the two-point correlation function, for the embedded Gaussian unitary ensembles with k-body interactions [EGUE(k)] as appropriate for systems with m fermions in N single particle states. Used for obtaining the formulas is the SU(N) Wigner-Racah algebra. These formulas with finite N corrections are used to derive formulas for the covariances S_{ζ}S_{ζ^{'}}[over ¯] in the asymptotic limit. These show that the present work extends to all k values, the results known in the past in the two extreme limits with k/m→0 (same as q→1) and k=m (same as q=0).

We study the sensitivity of weak lensing surveys to the effects of catastrophic redshift errors - cases where the true redshift is misestimated by a significant amount. To compute the biases in cosmological parameters, we adopt an efficient linearized analysis where the redshift errors are directly related to shifts in the weak lensing convergence power spectra. We estimate the number Nspec of unbiased spectroscopic redshifts needed to determine the catastrophic error rate well enough that biases in cosmological parameters are below statistical errors of weak lensing tomography. While the straightforward estimate of Nspec is ~10^6 we find that using only the photometric redshifts with z<=2.5 leads to a drastic reduction in Nspec to ~30,000 while negligibly increasing statistical errors in dark energy parameters. Therefore, the size of spectroscopic survey needed to control catastrophic errors is similar to that previously deemed necessary to constrain the core of the z_s-z_p distribution. We also study the efficacy of the recent proposal to measure redshift errors by cross-correlation between the photo-z and spectroscopic samples. We find that this method requires ~10% a priori knowledge of the bias and stochasticity of the outlier population, and is also easily confounded by lensing magnification bias. The cross-correlation method is therefore unlikely to supplant the need for a complete spectroscopic redshift survey of the source population.

We present measurements of the redshift-space three-point correlation function of 50 967 luminous red galaxies (LRGs) from Data Release 3 (DR3) of the Sloan Digital Sky Survey (SDSS). We have studied the shape dependence of the reduced three-point correlation function (Qz(s, q, θ)) on three different scales, s= 4, 7 and 10 h−1 Mpc, and over the range of 1 < q < 3 and 0° < θ < 180°. On small scales (s= 4 h−1 Mpc), Qz is nearly constant, with little change as a function of q and θ. However, there is evidence for a shallow U-shaped behaviour (with θ) which is expected from theoretical modelling of Qz(s, q, θ). On larger scales (s= 7 and 10 h−1 Mpc), the U-shaped anisotropy in Qz (with θ) is more clearly detected. We compare this shape dependence in Qz(s, q, θ) with that seen in mock galaxy catalogues which were generated by populating the dark matter haloes in large N-body simulations with mock galaxies using various halo occupation distributions (HOD). We find that the combination of the observed number density of LRGs, the (redshift-space) two-point correlation function and Qz(s, q, θ) provides a strong constraint on the allowed HOD parameters (Mmin, M1, α) and breaks key degeneracies between these parameters. For example, our observed Qz(s, q, θ) disfavours mock catalogues that overpopulate massive dark matter haloes with many LRG satellites. We also estimate the linear bias of LRGs to be b= 1.87 ± 0.07 in excellent agreement with other measurements.

Two-point correlation functions are used to statistically characterize the microstructure of computationally generated CNT-polymer nanocomposite RVEs. Since the electro-mechanical properties of nanocomposite are controlled by the formation and disruption of networks between CNTs and the tunnelling of electrons between CNTs, the two-point correlation functions are calculated for the conductivity contours corresponding to the generated RVEs. The two-point correlation functions are represented in a lower dimensional subspace by performing Principal Component Analysis (PCA) on them. The principal scores of the two-point correlation functions are used as predictors in linear regression models to predict the effective stiffness, electrical conductivity, and piezoresistivity of the RVEs. The regression models predict the stiffness coefficients accurately. The models are less accurate in predicting the conductivity and piezoresistivity coefficients due to the large variance of these coefficients at certain CNT volume fractions.

Context.The LOFAR Two-metre Sky Survey (LoTSS) will eventually map the complete Northern sky and provide an excellent opportunity to study the distribution and evolution of the large-scale structure of the Universe.Aims.We test the quality of LoTSS observations through a statistical comparison of the LoTSS first data release (DR1) catalogues to expectations from the established cosmological model of a statistically isotropic and homogeneous Universe.Methods.We study the point-source completeness and define several quality cuts, in order to determine the count-in-cell statistics and differential source count statistics, and measure the angular two-point correlation function. We use the photometric redshift estimates, which are available for about half of the LoTSS-DR1 radio sources, to compare the clustering throughout the history of the Universe.Results.For the masked LoTSS-DR1 value-added source catalogue, we find a point-source completeness of 99% above flux densities of 0.8 mJy. The counts-in-cell statistic reveals that the distribution of radio sources cannot be described by a spatial Poisson process. Instead, a good fit is provided by a compound Poisson distribution. The differential source counts are in good agreement with previous findings in deep fields at low radio frequencies and with simulated catalogues from the SKA Design Study and the Tiered Radio Extragalactic Continuum Simulation. Restricting the value added source catalogue to low-noise regions and applying a flux density threshold of 2 mJy provides our most reliable estimate of the angular two-point correlation. Based on the distribution of photometric redshifts and thePlanck2018 best-fit cosmological model, the theoretically predicted angular two-point correlation between 0.1 deg and 6 deg agrees reasonably well with the measured clustering for the sub-sample of radio sources with redshift information.Conclusions.The deviation from a Poissonian distribution might be a consequence of the multi-component nature of a large number of resolved radio sources and/or of uncertainties on the flux density calibration. The angular two-point correlation function is &lt; 10−2at angular scales &gt; 1 deg and up to the largest scales probed. At a 2 mJy flux density threshold and at a pivot angle of 1 deg, we find a clustering amplitude ofA = (5.1 ± 0.6) × 10−3with a slope parameter ofγ = 0.74 ± 0.16. For smaller flux density thresholds, systematic issues are identified, which are most likely related to the flux density calibration of the individual pointings. We conclude that we find agreement with the expectation of large-scale statistical isotropy of the radio sky at the per cent level. The angular two-point correlation agrees well with the expectation of the cosmological standard model.

In the first part of this series of two papers, we proposed a theoretical formalism that enables one to model and categorize heterogeneous materials (media) via two-point correlation functions S(2) and introduced an efficient heterogeneous-medium (re)construction algorithm called the "lattice-point" algorithm. Here we discuss the algorithmic details of the lattice-point procedure and an algorithm modification using surface optimization to further speed up the (re)construction process. The importance of the error tolerance, which indicates to what accuracy the media are (re)constructed, is also emphasized and discussed. We apply the algorithm to generate three-dimensional digitized realizations of a Fontainebleau sandstone and a boron-carbide/aluminum composite from the two-dimensional tomographic images of their slices through the materials. To ascertain whether the information contained in S(2) is sufficient to capture the salient structural features, we compute the two-point cluster functions of the media, which are superior signatures of the microstructure because they incorporate topological connectedness information. We also study the reconstruction of a binary laser-speckle pattern in two dimensions, in which the algorithm fails to reproduce the pattern accurately. We conclude that in general reconstructions using S(2) only work well for heterogeneous materials with single-scale structures. However, two-point information via S(2) is not sufficient to accurately model multiscale random media. Moreover, we construct realizations of hypothetical materials with desired structural characteristics obtained by manipulating their two-point correlation functions.

Extraction of the Baryon Acoustic Oscillations (BAO) to per cent level accuracy is challenging and demands an understanding of many potential systematics to an accuracy well below 1 per cent, in order to ensure that they do not combine significantly when compared to statistical error of the BAO measurement. Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 11 (DR11) reaches a distance measurement with ?1 per cent statistical error and this prompts an extensive search for all possible sub-per cent level systematic errors which could previously be safely ignored. In this paper, we analyse the potential systematics in BAO fitting methodology using mocks and data from BOSS DR10 and DR11. We demonstrate the robustness of the fiducial multipole fitting methodology to be at 0.1-0.2 per cent level with a wide range of tests in mock galaxy catalogues pre- and post-reconstruction. We also find the DR10 and DR11 data from BOSS to be robust against changes in methodology at a similar level. This systematic error budget is incorporated into the BOSS DR10 and DR11 BAO measurements. Of the wide range of changes we have investigated, we find that when fitting post-reconstructed data or mocks, the only change which has an effect >0.1 per cent on the best-fitting values of distance measurements is varying the order of the polynomials to describe the broad-band terms (?0.2 per cent). Finally, we compare an alternative methodology denoted as Clustering Wedges with Multipoles, and find that it is consistent with the standard approach.

Three-dimensional reconstruction and homogenization of heterogeneous materials using statistical correlation functions and FEM

Reconstruction of three-dimensional heterogeneous media from a single two-dimensional section via co-occurrence correlation function

Padmanabhan has suggested a model to relate the non-linear two-point correlation function to the linear two-point correlation function. In this paper, we extend this model in two directions: (1) by averaging over the initial Gaussian distribution of density contrasts, we estimate the spectral dependence of the scaling between nonlinear and linear correlation functions; (2) by using a physically motivated Ansatz, we generalize the model to N-ppoint correlation functions and relate the non-linear, volume averaged N-point correlation function with the linearly extrapolated volume averaged two-point correlation function evaluated at a different scale. We compare the point of transition between the different regimes obtained from our model with that from numerical simulations, and show that the spectral dependence of the scaling relations seen in the simulations can be easily understood. A comparison of the calculated form of with the simulations shows reasonable agreement. We discuss several implications of the results.

The locations of most gamma-ray bursts are known only to several degrees. As a consequence, to detect repeated outbursts from gamma-ray burst sources in the largest gamma-ray burst catalogs, one must apply statistical tests for clustering. I show that the two-point correlation function is superior to the nearest neighbor test for detecting repetition whenever the average angle of separation between bursts in the sample is larger than the location error. The two-point correlation function is particularly sensitive to repeating sources that each produce a large number of observed gamma-ray bursts. The effects of Earth blockage and the disabling of the burst trigger are examined, and the ability of different repetition models to produce an observable repetition signal is calculated. I show that only a large number of repetitions per source can produce an observable signal, which underscores the strength of the two-point correlation function as a test of burst repetition. From these results, one must conclude that the deviation from isotropy found in the BATSE 1B catalog with the nearest neighbor test is a statistical fluctuation, and not a manifestation of burst repetition.

We perform theoretical and numerical studies of the full relativistic two-point galaxy correlation function, considering the linear-order scalar and tensor perturbation contributions and the wide-angle effects. Using the gauge-invariant relativistic description of galaxy clustering and accounting for the contributions at the observer position, we demonstrate that the complete theoretical expression is devoid of any long-mode contributions from scalar or tensor perturbations and it lacks the infrared divergences in agreement with the equivalence principle. By showing that the gravitational potential contribution to the correlation function converges in the infrared, our study justifies an IR cut-off (kIR ⩽ H0) in computing the gravitational potential contribution. Using the full gauge-invariant expression, we numerically compute the galaxy two-point correlation function and study the individual contributions in the conformal Newtonian gauge. We find that the terms at the observer position such as the coordinate lapses and the observer velocity (missing in the standard formalism) dominate over the other relativistic contributions in the conformal Newtonian gauge such as the source velocity, the gravitational potential, the integrated Sachs-Wolf effect, the Shapiro time-delay and the lensing convergence. Compared to the standard Newtonian theoretical predictions that consider only the density fluctuation and redshift-space distortions, the relativistic effects in galaxy clustering result in a few percent-level systematic errors beyond the scale of the baryonic acoustic oscillation (∼ 2% at 150 Mpc/h and redshift one). Our theoretical and numerical study provides a comprehensive understanding of the relativistic effects in the galaxy two-point correlation function, as it proves the validity of the theoretical prediction and accounts for effects that are often neglected in its numerical evaluation.

The theory of diffraction tomography is generalized to random media. A relationship between the two-point spatial correlation function of the dielectric susceptibility and the cross-spectral density of the scattered field is derived, and the reconstruction of the two-point spatial correlation function from measurements of the cross-spectral density in two planes, on different sides and at finite distances from the medium, is investigated. It is shown that the two-point spatial correlation function cannot be determined uniquely, in general, from scattered field measurements, except for a class of random media known as quasi-homogeneous random media.

We calculate the effective temperature ($T_{\rm eff}$) of ionizing star(s), oxygen abundance of the gas phase $(\rm O/H)$, and the ionization parameter $U$ for a sample of H\,{\sc ii} regions located in the disks of 59 spiral galaxies in the 0.005 < z < 0.03 redshift range. We use spectroscopic data taken from the CALIFA data release 3 (DR3) and theoretical (for $T_{\rm eff}$ and $U$) and empirical (for O/H) calibrations based on strong emission-lines. We consider spatial distribution and radial gradients of those parameters in each galactic disk for the objects in our sample. Most of the galaxies in our sample ($\sim70$ \%) shows positive $T_{\rm eff}$ radial gradients even though some them exhibit negative or flat ones. The median value of the $T_{\rm eff}$ radial gradient is 0.762 kK/$R_{25}$. We find that radial gradients of both $\log U$ and $T_{\rm eff}$ depend on the oxygen abundance gradient, in the sense that the gradient of $\log U$ increases as $\log(\rm O/H)$ gradient increases while there is an anti-correlation between the gradient of $T_{\rm eff}$ and the oxygen abundance gradient. Moreover, galaxies with flat oxygen abundance gradients tend to have flat $\log U$ and $T_{\rm eff}$ gradients as well. Although our results are in agreement with the idea of the existence of positive $T_{\rm eff}$ gradients along the disk of the majority of spiral galaxies, this seems not to be an universal property for these objects.