Covariance of cross-correlations: towards efficient measures for large-scale structure

We study the covariance of the cross-power spectrum of different tracers for the large-scale structure. We develop the counts-in-cells framework for the multi-tracer approach, and use this to derive expressions for the full non-Gaussian covariance matrix. We show, that for the usual auto-power statistic, besides the off-diagonal covariance generated through gravitational mode-coupling, the discreteness of the tracers and their associated sampling distribution can generate strong off-diagonal covariance, and that this becomes the dominant source of covariance as k>>k_f=2 pi/L. On comparison with the derived expressions for the cross-power covariance, we show that the off-diagonal terms can be suppressed, if one cross-correlates a high tracer-density sample with a low one. Taking the effective estimator efficiency to be proportional to the signal-to-noise ratio (SN), we show that, to probe clustering as a function of physical properties of the sample, i.e. cluster mass or galaxy luminosity, then the cross-power approach can out perform the auto-power one by factors of a few. We confront the theory with measurements of the mass-mass, halo-mass, and halo-halo power spectra from a large ensemble of N-body simulations. We show that there is a significant SN advantage to be gained from using the cross-power approach when studying the bias of rare haloes. The analysis is repeated in configuration space and again SN improvement is found. We estimate the covariance matrix for these samples, and find strong off-diagonal contributions. The covariance depends on halo mass, with higher mass samples having stronger covariance. In agreement with theory, we show that the covariance is suppressed for the cross-power. This work points the way towards improved estimators for clustering studies.Comment: Several significant improvements to the earlier version: for instance it is shown more clearly how shot noise corrections may generate off-diagonal covariance in the power spectrum. Original version submitted to MNRAS on 18th September 2008. This version 18 pages, 7 figure

SKA Weak Lensing III: Added Value of Multi-Wavelength Synergies for the Mitigation of Systematics

In this third paper of a series on radio weak lensing for cosmology with the Square Kilometre Array, we scrutinise synergies between cosmic shear measurements in the radio and optical/near-IR bands for mitigating systematic effects. We focus on three main classes of systematics: (i) experimental systematic errors in the observed shear; (ii) signal contamination by intrinsic alignments; and (iii) systematic effects due to an incorrect modelling of non-linear scales. First, we show that a comprehensive, multi-wavelength analysis provides a self-calibration method for experimental systematic effects, only implying <50% increment on the errors on cosmological parameters. We also illustrate how the cross-correlation between radio and optical/near-IR surveys alone is able to remove residual systematics with variance as large as 0.00001, i.e. the same order of magnitude of the cosmological signal. This also opens the possibility of using such a cross-correlation as a means to detect unknown experimental systematics. Secondly, we demonstrate that, thanks to polarisation information, radio weak lensing surveys will be able to mitigate contamination by intrinsic alignments, in a way similar but fully complementary to available self-calibration methods based on position-shear correlations. Lastly, we illustrate how radio weak lensing experiments, reaching higher redshifts than those accessible to optical surveys, will probe dark energy and the growth of cosmic structures in regimes less contaminated by non-linearities in the matter perturbations. For instance, the higher-redshift bins of radio catalogues peak at z~0.8-1, whereas their optical/near-IR counterparts are limited to z<0.5-0.7. This translates into having a cosmological signal 2 to 5 times less contaminated by non-linear perturbations.Comment: 16 pages, 10 figures, 2 tables; improved discussion of experimental systematics in Sec. 2; updated to match published versio

Evolution of the Clustering of Photometrically Selected SDSS Galaxies

We measure the angular auto-correlation functions (w) of SDSS galaxies selected to have photometric redshifts 0.1 < z < 0.4 and absolute r-band magnitudes Mr < -21.2. We split these galaxies into five overlapping redshift shells of width 0.1 and measure w in each subsample in order to investigate the evolution of SDSS galaxies. We find that the bias increases substantially with redshift - much more so than one would expect for a passively evolving sample. We use halo-model analysis to determine the best-fit halo-occupation-distribution (HOD) for each subsample, and the best-fit models allow us to interpret the change in bias physically. In order to properly interpret our best-fit HODs, we convert each halo mass to its z = 0 passively evolved bias (bo), enabling a direct comparison of the best-fit HODs at different redshifts. We find that the minimum halo bo required to host a galaxy decreases as the redshift decreases, suggesting that galaxies with Mr < -21.2 are forming in halos at the low-mass end of the HODs over our redshift range. We use the best-fit HODs to determine the change in occupation number divided by the change in mass of halos with constant bo and we find a sharp peak at bo ~ 0.9 - corresponding to an average halo mass of ~ 10^12Msol/h. We thus present the following scenario: the bias of galaxies with Mr < -21.2 decreases as the Universe evolves because these galaxies form in halos of mass ~ 10^12Msol/h (independent of redshift), and the bias of these halos naturally decreases as the Universe evolves.Comment: 17 pages, 14 figures, matches version accepted for publication in MNRA

A new statistical test based on the wavelet cross-spectrum to detect time–frequency dependence between non-stationary signals: Application to the analysis of cortico-muscular interactions

The study of the correlations that may exist between neurophysiological signals is at the heart of modern techniques for data analysis in neuroscience. Wavelet coherence is a popular method to construct a time-frequency map that can be used to analyze the time-frequency correlations be- tween two time series. Coherence is a normalized measure of dependence, for which it is possible to construct confidence intervals, and that is commonly considered as being more interpretable than the wavelet cross-spectrum (WCS). In this paper, we provide empirical and theoretical arguments to show that a significant level of wavelet coherence does not necessarily correspond to a significant level of dependence between random signals, especially when the number of trials is small. In such cases, we demonstrate that the WCS is a much better measure of statistical dependence, and a new statistical test to detect significant values of the cross-spectrum is proposed. This test clearly outperforms the limitations of coherence analysis while still allowing a consistent estimation of the time-frequency correlations between two non-stationary stochastic processes. Simulated data are used to investigate the advantages of this new approach over coherence analysis. The method is also applied to experimental data sets to analyze the time-frequency correlations that may exist between electroencephalogram (EEG) and surface electromyogram (EMG)