Super-sample Covariance Research Articles

The squeezed matter bispectrum covariance with responses

We present a calculation of the angle-averaged squeezed matter bispectrum covariance Cov(Bm(k1, k1′, s1), Bm(k2, k2′, s2)), si ≪ ki,ki′ (i=1,2), that uses matter power spectrum responses to describe the coupling of large- to short-scale modes in the nonlinear regime. The covariance is given by a certain configuration of the 6-point function, which we show is dominated by response-type mode-coupling terms in the squeezed bispectrum limit. The terms that are not captured by responses are small, effectively rendering our calculation complete and predictive for linear s1,s2 values and any nonlinear values of k1,k1′,k2,k2′. Our numerical results show that the squeezed bispectrum super-sample covariance is only a negligible contribution. We also compute the power spectrum-bispectrum cross-covariance using responses. Our derivation for the squeezed matter bispectrum is the starting point to calculate analytical covariances for more realistic galaxy clustering and weak-lensing applications. It can also be used in cross-checks of numerical ensemble estimates of the general bispectrum covariance, given that it is effectively noise-free and complete in the squeezed limit.

Covariances for cosmic shear and galaxy–galaxy lensing in the response approach

In this study, we measure the response of matter and halo projected power spectra $P^{\rm 2D}_{\rm XY}(k)$ (X, Y are matter and/or halos), to a large-scale density contrast, $\delta_{\rm b}$, using separate universe simulations. We show that the fractional response functions, i.e., $\mathrm{d}\ln P^{\rm 2D}_{\rm XY}(k)/\mathrm{d}\delta_{\rm b}$, are identical to their respective three-dimensional power spectra within simulation measurement errors. Then, using various $N$-body simulation combinations (small-box simulations with periodic boundary conditions and sub-volumes of large-box simulations) to construct {mock observations of projected fields}, we study how super-survey modes, in both parallel and perpendicular directions to the projection direction, affect the covariance matrix of $P^{\rm 2D}_{\rm XY}(k)$, known as super-sample covariance (SSC). Our results indicate that the SSC term provides dominant contributions to the covariances of matter-matter and matter-halo spectra at small scales but does not provide significant contributions in the halo-halo spectrum. We observe that the large-scale density contrast in each redshift shell causes most of the SSC effect, and we did not observe a SSC signature arising from large-scale tidal field within the levels of measurement accuracy. We also develop a response approach to calibrate the SSC term for cosmic shear correlation function and galaxy--galaxy weak lensing, and validate the method by comparison with the light-cone ray-tracing simulations. Our method provides a reasonably accurate, albeit computationally inexpensive, way to calibrate the covariance matrix for clustering observables available from wide-area galaxy surveys.

Accurate cosmic shear errors: do we need ensembles of simulations?

Accurate inference of cosmology from weak lensing shear requires an accurate shear power spectrum covariance matrix. Here, we investigate this accuracy requirement and quantify the relative importance of the Gaussian (G), super-sample covariance (SSC) and connected non-Gaussian (cNG) contributions to the covariance. Specifically, we forecast cosmological parameter constraints for future wide-field surveys and study how different covariance matrix components affect parameter bounds. Our main result is that the cNG term represents only a small and potentially negligible contribution to statistical parameter errors: the errors obtained using the G+SSC subset are within ≲ 5% of those obtained with the full G+SSC+cNG matrix for a Euclid-like survey. This result also holds for the shear two-point correlation function, variations in survey specifications and for different analytical prescriptions of the cNG term. The cNG term is that which is often tackled using numerically expensive ensembles of survey realizations. Our results suggest however that the accuracy of analytical or approximate numerical methods to compute the cNG term is likely to be sufficient for cosmic shear inference from the next generation of surveys.

Covariance of the galaxy angular power spectrum with the halo model

As the determination of density fluctuations becomes more precise with larger surveys, it becomes more important to account for the increased covariance due to the non-linearity of the field. Here I have focussed on the galaxy density, with analytical prediction of the non-Gaussianity using the halo model coupled with standard perturbation theory in real space. I carried out an exact and exhaustive derivation of all tree-level terms of the non-Gaussian covariance of the galaxyCℓ, with the computation developed up to the third order in perturbation theory and local halo bias, including the non-local tidal tensor effect. A diagrammatic method was used to derive the involved galaxy 3D trispectra, including shot-noise contributions. The projection to the angular covariance was derived in all trispectra cases with and without Limber’s approximation, with the formulae being of potential interest for other observables than galaxies. The effect of subtracting shot-noise from the measured spectrum is also discussed, and does simplify the covariance, though some non-Gaussian shot-noise terms still remain. I make the link between this complete derivation and partial terms which have been used previously in the literature, including super-sample covariance (SSC). I uncover a wealth of additional terms which were not previously considered, including a whole new class which I dub braiding terms as it contains multipole-mixing kernels. The importance of all these new terms is discussed with analytical arguments. I find that they become comparable to, if not bigger than, SSC if the survey is large or deep enough to probe scales comparable with the matter-radiation equalitykeq. A short self-contained summary of the equations is provided in Sect. 9 for the busy reader, ready to be implemented numerically for analysis of current and future galaxy surveys.

Power spectrum in the presence of large-scale overdensity and tidal fields: breaking azimuthal symmetry

We consider the power spectrum of a biased tracer observed in a finite volume in the presence of a large-scale overdensity and tidal fields. Expanding both the observed power spectrum and the source fields (linear power spectrum, scalar overdensity and tidal field tensor) in spherical harmonics, we explicitly confirm that each (ℓ,m) source generates just the corresponding (ℓ,m) modes of power spectrum in real space. In redshift space, each (ℓ,m) source additionally couples only to (ℓ+2n,m) modes of tracer power spectra. This generalizes the Kaiser formula for monopole, quadrupole and hexadecapole of the power spectrum to all (ℓ,m) modes generated to the second order in perturbation theory. This formalism can find applications in constraining the super-sample covariance and in the local power spectrum based bispectrum estimators. As an example application, we forecast the ability to measure these modes a survey with BOSS-like galaxy number densities.

Complete super-sample lensing covariance in the response approach

We derive the complete super-sample covariance (SSC) of the matter and weak lensing convergence power spectra using the power spectrum response formalism to accurately describe the coupling of super- to sub-survey modes. The SSC term is completely characterized by the survey window function, the nonlinear matter power spectrum and the full first-order nonlinear power spectrum response function, which describes the response to super-survey density and tidal field perturbations. Generalized separate universe simulations can efficiently measure these responses in the nonlinear regime of structure formation, which is necessary for lensing applications. We derive the lensing SSC formulae for two cases: one under the Limber and flat-sky approximations, and a more general one that goes beyond the Limber approximation in the super-survey mode and is valid for curved sky applications. Quantitatively, we find that for sky fractions fsky ≈ 0.3 and a single source redshift at zS=1, the use of the flat-sky and Limber approximation underestimates the total SSC contribution by ≈ 10%. The contribution from super-survey tidal fields to the lensing SSC, which has not been included in cosmological analyses so far, is shown to represent about 5% of the total lensing covariance on multipoles ℓ1,ℓ2 ≳ 300. The SSC is the dominant off-diagonal contribution to the total lensing covariance, making it appropriate to include these tidal terms and beyond flat-sky/Limber corrections in cosmic shear analyses.

Super-sample covariance approximations and partial sky coverage

Super-sample covariance (SSC) is the dominant source of statistical error on large scale structure (LSS) observables for both current and future galaxy surveys. In this work, we concentrate on the SSC of cluster counts, also known as sample variance, which is particularly useful for the self-calibration of the cluster observable-mass relation; our approach can similarly be applied to other observables, such as galaxy clustering and lensing shear. We first examined the accuracy of two analytical approximations proposed in the literature for the flat sky limit, finding that they are accurate at the 15% and 30–35% level, respectively, for covariances of counts in the same redshift bin. We then developed a harmonic expansion formalism that allows for the prediction of SSC in an arbitrary survey mask geometry, such as large sky areas of current and future surveys. We show analytically and numerically that this formalism recovers the full sky and flat sky limits present in the literature. We then present an efficient numerical implementation of the formalism, which allows fast and easy runs of covariance predictions when the survey mask is modified. We applied our method to a mask that is broadly similar to the Dark Energy Survey footprint, finding a non-negligible negative cross-z covariance, i.e. redshift bins are anti-correlated. We also examined the case of data removal from holes due to, for example bright stars, quality cuts, or systematic removals, and find that this does not have noticeable effects on the structure of the SSC matrix, only rescaling its amplitude by the effective survey area. These advances enable analytical covariances of LSS observables to be computed for current and future galaxy surveys, which cover large areas of the sky where the flat sky approximation fails.

Bispectrum supersample covariance

Modes with wavelengths larger than the survey window can have significant impact on the covariance within the survey window. The supersample covariance has been recognized as an important source of covariance for the power spectrum on small scales, and it can potentially be important for the bispectrum covariance as well. In this paper, using the response function formalism, we model the supersample covariance contributions to the bispectrum covariance and the cross covariance between the power spectrum and the bispectrum. The supersample covariances due to the long wavelength density and tidal perturbations are investigated, and the tidal contribution is a few orders of magnitude smaller than the density one because in configuration space the bispectrum estimator involves angular averaging and the tidal response function is anisotropic. The impact of the super-survey modes is quantified using numerical measurements with periodic box and subbox setups. For the matter bispectrum, the ratio between the supersample covariance correction and the small scale covariance, which can be computed using a periodic box, is roughly an order of magnitude smaller than that for the matter power spectrum. This is because for the bispectrum, the small scale non-Gaussian covariance is significantly larger than that for the power spectrum. For the cross covariance, the supersample covariance is as important as for the power spectrum covariance. The supersample covariance prediction with the halo model response function is in good agreement with numerical results.

Galaxy power-spectrum responses and redshift-space super-sample effect

As a major source of cosmological information, galaxy clustering is susceptible to long-wavelength density and tidal fluctuations. These long modes modulate the growth and expansion rate of local structures, shifting them in both amplitude and scale. These effects are often named the growth and dilation effects, respectively. In particular the dilation shifts the baryon acoustic oscillation (BAO) peak and breaks the assumption of the Alcock-Paczynski (AP) test. This cannot be removed with reconstruction techniques because the effect originates from long modes outside the survey. In redshift space, the long modes generate a large-scale radial peculiar velocity that affects the redshift-space distortion (RSD) signal. We compute the redshift-space response functions of the galaxy power spectrum to long density and tidal modes at leading order in perturbation theory, including both the growth and dilation terms. We validate these response functions against measurements from simulated galaxy mock catalogs. As one application, long density and tidal modes beyond the scale of a survey correlate various observables leading to an excess error known as the super-sample covariance, and thus weaken their constraining power. We quantify the super-sample effect on BAO, AP, and RSD measurements, and study its impact on current and future surveys.

Galaxy two-point covariance matrix estimation for next generation surveys

We perform a detailed analysis of the covariance matrix of the spherically averaged galaxy power spectrum and present a new, practical method for estimating this within an arbitrary survey without the need for running mock galaxy simulations that cover the full survey volume. The method uses theoretical arguments to modify the covariance matrix measured from a set of small-volume cubic galaxy simulations, which are computationally cheap to produce compared to larger simulations and match the measured small-scale galaxy clustering more accurately than is possible using theoretical modelling. We include prescriptions to analytically account for the window function of the survey, which convolves the measured covariance matrix in a non-trivial way. We also present a new method to include the effects of supersample covariance and modes outside the small simulation volume which requires no additional simulations and still allows us to scale the covariance matrix. As validation, we compare the covariance matrix estimated using our new method to that from a brute force calculation using 500 simulations originally created for analysis of the Sloan Digital Sky Survey Main Galaxy Sample (SDSS-MGS). We find excellent agreement on all scales of interest for large scale structure analysis, including those dominated by the effects of the survey window, and on scales where theoretical models of the clustering normally break-down, but the new method produces a covariance matrix with significantly better signal-to-noise. Although only formally correct in real-space, we also discuss how our method can be extended to incorporate the effects of Redshift Space Distortions.

Inadequacy of internal covariance estimation for super-sample covariance

We give an analytical interpretation of how subsample-based internal covariance estimators lead to biased estimates of the covariance, due to underestimating the super-sample covariance (SSC). This includes the jackknife and bootstrap methods as estimators for the full survey area, and subsampling as an estimator of the covariance of subsamples. The limitations of the jackknife covariance have been previously presented in the literature because it is effectively a rescaling of the covariance of the subsample area. However we point out that subsampling is also biased, but for a different reason: the subsamples are not independent, and the corresponding lack of power results in SSC underprediction. We develop the formalism in the case of cluster counts that allows the bias of each covariance estimator to be exactly predicted. We find significant effects for a small-scale area or when a low number of subsamples is used, with auto-redshift biases ranging from 0.4% to 15% for subsampling and from 5% to 75% for jackknife covariance estimates. The cross-redshift covariance is even more affected; biases range from 8% to 25% for subsampling and from 50% to 90% for jackknife. Owing to the redshift evolution of the probe, the covariances cannot be debiased by a simple rescaling factor, and an exact debiasing has the same requirements as the full SSC prediction. These results thus disfavour the use of internal covariance estimators on data itself or a single simulation, leaving analytical prediction and simulations suites as possible SSC predictors.

Galaxy–galaxy lensing estimators and their covariance properties

We study the covariance properties of real space correlation function estimators -- primarily galaxy-shear correlations, or galaxy-galaxy lensing -- using SDSS data for both shear catalogs and lenses (specifically the BOSS LOWZ sample). Using mock catalogs of lenses and sources, we disentangle the various contributions to the covariance matrix and compare them with a simple analytical model. We show that not subtracting the lensing measurement around random points from the measurement around the lens sample is equivalent to performing the measurement using the lens density field instead of the lens over-density field. While the measurement using the lens density field is unbiased (in the absence of systematics), its error is significantly larger due to an additional term in the covariance. Therefore, this subtraction should be performed regardless of its beneficial effects on systematics. Comparing the error estimates from data and mocks for estimators that involve the over-density, we find that the errors are dominated by the shape noise and lens clustering, that empirically estimated covariances (jackknife and standard deviation across mocks) are consistent with theoretical estimates, and that both the connected parts of the 4-point function and the super-sample covariance can be neglected for the current levels of noise. While the trade-off between different terms in the covariance depends on the survey configuration (area, source number density), the diagnostics that we use in this work should be useful for future works to test their empirically-determined covariances.

Combining cluster number counts and galaxy clustering

The abundance of clusters and the clustering of galaxies are two of the important cosmological probes for current and future large scale surveys of galaxies, such as the Dark Energy Survey. In order to combine them one has to account for the fact that they are not independent quantities, since they probe the same density field. It is important to develop a good understanding of their correlation in order to extract parameter constraints. We present a detailed modelling of the joint covariance matrix between cluster number counts and the galaxy angular power spectrum. We employ the framework of the halo model complemented by a Halo Occupation Distribution model (HOD). We demonstrate the importance of accounting for non-Gaussianity to produce accurate covariance predictions. Indeed, we show that the non-Gaussian covariance becomes dominant at small scales, low redshifts or high cluster masses. We discuss in particular the case of the super-sample covariance (SSC), including the effects of galaxy shot-noise, halo second order bias and non-local bias. We demonstrate that the SSC obeys mathematical inequalities and positivity. Using the joint covariance matrix and a Fisher matrix methodology, we examine the prospects of combining these two probes to constrain cosmological and HOD parameters. We find that the combination indeed results in noticeably better constraints, with improvements of order 20% on cosmological parameters compared to the best single probe, and even greater improvement on HOD parameters, with reduction of error bars by a factor 1.4-4.8. This happens in particular because the cross-covariance introduces a synergy between the probes on small scales. We conclude that accounting for non-Gaussian effects is required for the joint analysis of these observables in galaxy surveys.

The impact of supersurvey modes on cosmological constraints from cosmic shear fields

Owing to the mass-sheet degeneracy, cosmic shear maps do not probe directly the Fourier modes of the underlying mass distribution on scales comparable to the survey size and larger. To assess the corresponding effect on attainable cosmological parameter constraints, we quantify the information on super-survey modes in a lognormal model and, when interpreted as nuisance parameters, their degeneracies to cosmological parameters. Our analytical and numerical calculations clarify the central role of super-sample covariance (SSC) in shaping the statistical power of cosmological observables. Reconstructing the background modes from their non-Gaussian statistical dependence to small scales modes yields the renormalized convergence. This diagonalizes the spectrum covariance matrix, and the information content of the corresponding power spectrum is increased by a factor of two over standard methods. Unfortunately, careful calculation of the Cramer-Rao bound shows that the information recovery can never be made complete, any observable built from shear fields, including optimal sufficient statistics, are subject to severe information loss, typically $80\%$ to $90\%$ below $\ell \sim 3000$ for generic cosmological parameters. The lost information can only be recovered from additional, non-shear based data. Our predictions hold just as well for a tomographic analysis, and/or full sky surveys.

An optimal survey geometry of weak lensing survey: minimizing supersample covariance

Upcoming wide-area weak lensing surveys are expensive both in time and cost and require an optimal survey design in order to attain maximum scientific returns from a fixed amount of available telescope time. The supersample covariance (SSC), which arises from unobservable modes that are larger than the survey size, significantly degrades the statistical precision of weak lensing power spectrum measurement even for a wide-area survey. Using the 1000 mock realizations of the lognormal model, which approximates the weak lensing field for a Λ-dominated cold dark matter model, we study an optimal survey geometry to minimize the impact of SSC contamination. For a continuous survey geometry with a fixed survey area, a more elongated geometry such as a rectangular shape of 1:400 side length ratio reduces the SSC effect and allows for a factor of 2 improvement in the cumulative signal-to-noise ratio (S/N) of power spectrum measurement up to ℓmax ≃ a few 103, compared to compact geometries such as squares or circles. When we allow the survey geometry to be disconnected but with a fixed total area, assuming 1 × 1 deg2 patches as the fundamental building blocks of survey footprints, the best strategy is to locate the patches with ∼15 deg separation. This separation angle corresponds to the scale at which the two-point correlation function has a negative minimum. The best configuration allows for a factor of 100 gain in the effective area coverage as well as a factor of 2.5 improvement in the S/N at high multipoles, yielding a much wider coverage of multipoles than in the compact geometry.

Joint analysis of cluster number counts and weak lensing power spectrum to correct for the super-sample covariance

A coherent over- or under-density contrast across a finite survey volume causes an upward- or downward-fluctuation in the observed number of halos. This fluctuation in halo number adds a significant co-variant scatter in the observed amplitudes of weak lensing power spectrum at nonlinear, small scales -- the so-called super-sample variance or the halo sample variance. In this paper, we show that by measuring both the number counts of clusters and the power spectrum in the same survey region, we can mitigate this loss of information and significantly enhance the scientific return from the upcoming surveys. First, using the halo model approach, we derive the cross-correlation between the halo number counts and the weak lensing power spectrum, taking into account the super-sample covariance effect, which well matches the distributions measured from 1000 realizations for a Lambda-dominated cold dark matter model. Then we show that adding the observed number counts of massive halos with M> 10^14 Ms/h can significantly improve the information content of weak lensing power spectrum, almost recovering the Gaussian information up to lmax~1000, if the average mass profiles of the massive halos are known, which can be estimated from stacked lensing. When combined with the halo number counts for M > 3 or 1 x 10^14 Ms/h, the improvement is up to a factor of 1.4 or 2 at lmax~1000 --2000, equivalent to a factor 2 or 4 times larger survey volume, compared to the power spectrum measurement alone.

Super-sample covariance in simulations

Using separate universe simulations, we accurately quantify super-sample covariance (SSC), the typically dominant sampling error for matter power spectrum estimators in a finite volume, which arises from the presence of super survey modes. By quantifying the power spectrum response to a background mode, this approach automatically captures the separate effects of beat coupling in the quasilinear regime, halo sample variance in the nonlinear regime and a new dilation effect which changes scales in the power spectrum coherently across the survey volume, including the baryon acoustic oscillation scale. It models these effects at typically the few percent level or better with a handful of small volume simulations for any survey geometry compared with directly using many thousands of survey volumes in a suite of large volume simulations. The stochasticity of the response is sufficiently small that in the quasilinear regime, SSC can be alternately included by fitting the mean density in the volume with these fixed templates in parameter estimation. We also test the halo model prescription and find agreement typically at better than the 10% level for the response.

Power spectrum super-sample covariance

We provide a simple, unified approach to describing the impact of super-sample covariance, or beat coupling, on power spectrum estimation in a finite-volume survey. For a wide range of survey volumes, the sample variance that arises from modes that are larger than the survey dominates the covariance of power spectrum estimators for modes much smaller than the survey. The deeply nonlinear version of this effect is known as halo sample variance. We show that all variants are unified by the matter trispectrum of squeezed configurations and that such configurations obey a consistency relation which relates them to the response of the power spectrum to a change in the background density. Our method also applies to statistics that are based on radial projections of the density field such as weak lensing shear. While we use the halo model for an analytic description to expose the nature of the effect, the consistency description enables an accurate calibration of the full effect directly from simulations. It also suggests that super-sample covariance may be viewed as an additional interesting signal rather than excess noise.