non-Gaussian Covariance Research Articles

CSST WL preparation I: forecast the impact from non-Gaussian covariances and requirements on systematics control

ABSTRACT The precise estimation of the statistical errors and accurate removal of the systematical errors are the two major challenges for the stage IV cosmic shear surveys. We explore their impact for the China Space Station Telescope (CSST) with survey area ${\sim} 17\,500\deg ^2$ up to redshift ∼4. We consider statistical error contributed from Gaussian covariance, connected non-Gaussian covariance, and super-sample covariance. We find the non-Gaussian covariances, which is dominated by the super-sample covariance, can largely reduce the signal-to-noise ratio of the two-point statistics for CSST, leading to an ∼1/3 loss in the figure of merit for the matter clustering properties (σ8–Ωm plane) and 1/6 in the dark energy equation of state (w0–wa plane). We further put requirements of systematics mitigation on intrinsic alignment of galaxies, baryonic feedback, shear multiplicative bias, and bias in the redshift distribution, for an unbiased cosmology. The 10−2–10−3 level requirements emphasize strong needs in related studies, to support future model selections and the associated priors for the nuisance parameters.

The impact of cross-covariances between the CMB and reconstructed lensing power

Weak gravitational lensing of the Cosmic Microwave Background (CMB) changes CMB statistics in a nontrivial way, allowing for reconstruction of the lensing potential and the use of these reconstructed maps in determining cosmological parameters that affect the formation of intervening large-scale structures. Although in principle there are correlations between the primary CMB and the reconstructed lensing potential due to the lensing procedure itself, in practice CMB analyses treat these as negligible when combining these band powers in likelihoods. In this paper we quantify explicitly the impact on parameter constraints due to these cross-covariances between the lensed CMB and reconstructed lensing power, and we compare to the effect of including all lensing-induced non-Gaussian covariances, which have previously shown to impact parameter constraints on the order of 10%. We perform our analysis for a range of experimental setups, scanning over instrumental noise levels of 0.5 to 10.0 μK-arcmin in temperature assuming fully polarized detectors, and using a fixed beam size of 1.4 arcmin. When the correlations between the lensed CMB and lensing power are neglected, we find that forecasted constraints shift by at most 3% of the error bar for a 6-parameter ΛCDM model, and for the noise levels considered in this paper. For some of the ΛCDM extensions considered here, however, these correlations have a nontrivial impact, in some cases more than 10% of the error bar, even for current experimental noise levels.

CSST forecast: impact from non-Gaussian covariances and requirements on systematics-control

CSST forecast: impact from non-Gaussian covariances and requirements on systematics-control

Primordial non-Gaussianity and non-Gaussian covariance

In the pursuit of primordial non-Gaussianities, we hope to access smaller scales across larger comoving volumes. At low redshift, the search for primordial non-Gaussianities is hindered by gravitational collapse, to which we often associate a scale $k_{\rm NL}$. Beyond these scales, it will be hard to reconstruct the modes sensitive to the primordial distribution. When forecasting future constraints on the amplitude of primordial non-Gaussianity, $f_{\rm NL}$, off-diagonal components are usually neglected in the covariance because these are small compared to the diagonal. We show that the induced non-Gaussian off-diagonal components in the covariance degrade forecast constraints on primordial non-Gaussianity, even when all modes are well within what is usually considered the linear regime. As a testing ground, we examine the effects of these off-diagonal components on the constraining power of the matter bispectrum on $f_{\rm NL}$ as a function of $k_{\rm max}$ and redshift, confirming our results against N-body simulations out to redshift $z=10$. We then consider these effects on the hydrogen bispectrum as observed from a PUMA-like 21-cm intensity mapping survey at redshifts $2<z<6$ and show that not including off-diagonal covariance over-predicts the constraining power on $f_{\rm NL}$ by up to a factor of $5$. For future surveys targeting even higher redshifts, such as Cosmic Dawn and the Dark Ages, which are considered ultimate surveys for primordial non-Gaussianity, we predict that non-Gaussian covariance would severely limit prospects to constrain $f_{\rm NL}$ from the bispectrum.

The covariance of squeezed bispectrum configurations

We measure the halo bispectrum covariance in a large set of N-body simulations and compare it with theoretical expectations. We find a large correlation among (even mildly) squeezed halo bispectrum configurations. A similarly large correlation can be found between squeezed triangles and the long-wavelength halo power spectrum. This shows that the diagonal Gaussian contribution fails to describe, even approximately, the full covariance in these cases. We compare our numerical estimate with a model that includes, in addition to the Gaussian one, only the non-Gaussian terms that are large for squeezed configurations. We find that accounting for these large terms in the modeling greatly improves the agreement of the full covariance with simulations. We apply these results to a simple Fisher matrix forecast, and find that constraints on primordial non-Gaussianity are degraded by a factor of ∼ 2 when a non-Gaussian covariance is assumed instead of the diagonal, Gaussian approximation.

Adaptive Masreliez–Martin fractional interpolatory cubature Kalman filter with recursive noise estimation

In this study, we propose a Masreliez–Martin fractional interpolatory cubature Kalman filter and an adaptive Masreliez–Martin fractional interpolatory cubature Kalman filter to design the estimators for fractional-order discrete-time nonlinear systems under non-Gaussian measurement noise and unknown process noise covariance. Ther Masreliez–Martin fractional interpolatory cubature Kalman filter is developed by extending the interpolatory cubature Kalman filter to the fractional nonlinear discrete system to increase the robustness of estimation under noisy environment by applying the Masreliez–Martin method, including both time update and measurement update. Meanwhile, the adaptive Masreliez–Martin fractional interpolatory cubature Kalman filter is put forward to enhance the state estimation adaptive to the fractional system with uncertain process noise through recursive estimation. Simulation results on the re-entry target tracking system have demonstrated the adaptiveness, robustness and effectiveness of the propose filters.

Measurement of the Relativistic Sunyaev–Zeldovich Correction in RX J1347.5-1145

We present a measurement of the relativistic corrections to the thermal Sunyaev–Zel’dovich (SZ) effect spectrum, the rSZ effect, toward the massive galaxy cluster RX J1347.5-1145 by combining submillimeter images from Herschel-SPIRE with millimeter wavelength Bolocam maps. Our analysis simultaneously models the SZ effect signal, the population of cosmic infrared background galaxies, and the galactic cirrus dust emission in a manner that fully accounts for their spatial and frequency-dependent correlations. Gravitational lensing of background galaxies by RX J1347.5-1145 is included in our methodology based on a mass model derived from the Hubble Space Telescope observations. Utilizing a set of realistic mock observations, we employ a forward modeling approach that accounts for the non-Gaussian covariances between the observed astrophysical components to determine the posterior distribution of SZ effect brightness values consistent with the observed data. We determine a maximum a posteriori (MAP) value of the average Comptonization parameter of the intracluster medium (ICM) within R 2500 to be 〈y〉2500 = 1.56 × 10−4, with corresponding 68% credible interval [1.42, 1.63] × 10−4, and a MAP ICM electron temperature of 〈T sz〉2500 = 22.4 keV with 68% credible interval spanning [10.4, 33.0] keV. This is in good agreement with the pressure-weighted temperature obtained from Chandra X-ray observations, 〈T x,pw〉2500 = 17.4 ± 2.3 keV. We aim to apply this methodology to comparable existing data for a sample of 39 galaxy clusters, with an estimated uncertainty on the ensemble mean 〈T sz〉2500 at the ≃ 1 keV level, sufficiently precise to probe ICM physics and to inform X-ray temperature calibration.

Cosmology in the non-linear regime: the small scale miracle

Interest is rising in exploiting the full shape information of the galaxy power spectrum, and in pushing analyses to smaller non-linear scales. Here I use the halo model to quantify the information content in the tomographic angular power spectrum of galaxies Cℓgal(iz) for the future high-resolution surveys Euclid and SKA2. I study how this information varies as a function of the scale cut applied, either with angular cut ℓmax or physical cut kmax. For this, I use analytical covariances with the most complete census of non-Gaussian terms, which proves to be critical. I find that the Fisher information on most cosmological and astrophysical parameters shows a striking behaviour. Beyond the perturbative regime, we first get decreasing returns: the information continues to rise but the slope slows down until reaching saturation. The location of this plateau, at k ∼ 2 Mpc−1, is slightly beyond the reach of current modelling methods and depends to some extent on the parameter and redshift bin considered. I explain the origin of this plateau, which is due to non-linear effects both on the power spectrum, and more importantly on non-Gaussian covariance terms. Then, pushing further, we see the information rising again in the highly non-linear regime, with a steep slope. This is the small-scale miracle, for which I give my interpretation and discuss the properties. There are suggestions that it may be possible to disentangle this information from the astrophysical content, and improve dark energy constraints. Finally, more hints are shown that high-order statistics may yield significant improvements over the power spectrum in this regime, with the improvements increasing with kmax.

Euclid: Covariance of weak lensing pseudo-Cℓ estimates

An accurate covariance matrix is essential for obtaining reliable cosmological results when using a Gaussian likelihood. In this paper we study the covariance of pseudo-Cℓ estimates of tomographic cosmic shear power spectra. Using two existing publicly available codes in combination, we calculate the full covariance matrix, including mode-coupling contributions arising from both partial sky coverage and non-linear structure growth. For three different sky masks, we compare the theoretical covariance matrix to that estimated from publicly available N-body weak lensing simulations, finding good agreement. We find that as a more extreme sky cut is applied, a corresponding increase in both Gaussian off-diagonal covariance and non-Gaussian super-sample covariance is observed in both theory and simulations, in accordance with expectations. Studying the different contributions to the covariance in detail, we find that the Gaussian covariance dominates along the main diagonal and the closest off-diagonals, but farther away from the main diagonal the super-sample covariance is dominant. Forming mock constraints in parameters that describe matter clustering and dark energy, we find that neglecting non-Gaussian contributions to the covariance can lead to underestimating the true size of confidence regions by up to 70 per cent. The dominant non-Gaussian covariance component is the super-sample covariance, but neglecting the smaller connected non-Gaussian covariance can still lead to the underestimation of uncertainties by 10–20 per cent. A real cosmological analysis will require marginalisation over many nuisance parameters, which will decrease the relative importance of all cosmological contributions to the covariance, so these values should be taken as upper limits on the importance of each component.

Impact of survey geometry and super-sample covariance on future photometric galaxy surveys

Photometric galaxy surveys probe the late-time Universe where the density field is highly non-Gaussian. A consequence is the emergence of the super-sample covariance (SSC), a non-Gaussian covariance term that is sensitive to fluctuations on scales larger than the survey window. In this work, we study the impact of the survey geometry on the SSC and, subsequently, on cosmological parameter inference. We devise a fast SSC approximation that accounts for the survey geometry and compare its performance to the common approximation of rescaling the results by the fraction of the sky covered by the survey, fSKY, dubbed ‘full-sky approximation’. To gauge the impact of our new SSC recipe, that we call ‘partial-sky’, we perform Fisher forecasts on the parameters of the (w0, wa)-CDM model in a 3 × 2 point analysis, varying the survey area, the geometry of the mask, and the galaxy distribution inside our redshift bins. The differences in the marginalised forecast errors –with the full-sky approximation performing poorly for small survey areas but excellently for stage-IV-like areas– are found to be absorbed by the marginalisation on galaxy bias nuisance parameters. For large survey areas, the unmarginalised errors are underestimated by about 10% for all probes considered. This is a hint that, even for stage-IV-like surveys, the partial-sky method introduced in this work will be necessary if tight priors are applied on these nuisance parameters. We make the partial-sky method public with a new release of the public code PySSC.

Super sample covariance of the thermal Sunyaev-Zel’dovich effect

The thermal Sunyaev-Zel'dovich (tSZ) effect is a powerful probe of cosmology. The statistical errors in the tSZ power spectrum measurements are dominated by the presence of massive clusters in a survey volume that are easy to identify on individual cluster basis. First, we study the impact of super sample covariance (SSC) on the tSZ power spectrum measurements, and find that the sample variance is dominated by the connected non-Gaussian (cNG) covariance arising mainly from Poisson number fluctuations of massive clusters in the survey volume. Second, we find that removing such individually-detected, massive clusters from the analysis significantly reduces the cNG contribution, thereby leading the SSC to be a leading source of the sample variance. We then show, based on Fisher analysis, that the power spectrum measured from the remaining diffuse tSZ effects can be used to obtain tight constraints on cosmological parameters as well as the hydrostatic mass bias parameter. Our method offers complementary use of individual tSZ cluster counts and the power spectrum measurements of diffuse tSZ signals for cosmology and intracluster gas physics.

Fast lightcones for combined cosmological probes

The combination of different cosmological probes offers stringent tests of the ΛCDM model and enhanced control of systematics. For this purpose, we present an extension of the lightcone generator UFALCON first introduced in Sgier et al. [1], enabling the simulation of a self-consistent set of maps for different cosmological probes. Each realization is generated from the same underlying simulated density field, and contains full-sky maps of different probes, namely weak lensing shear, galaxy overdensity including RSD, CMB lensing, and CMB temperature anisotropies from the ISW effect. The lightcone generation performed by UFALCON is parallelized and based on the replication of a large periodic volume simulated with the GPU-accelerated N-Body code PKDGRAV3. The post-processing to construct the lightcones requires only a runtime of about 1 walltime-hour corresponding to about 100 CPU-hours. We use a randomization procedure to increase the number of quasi-independent full-sky UFALCON map-realizations, which enables us to compute an accurate multi-probe covariance matrix. Using this framework, we forecast cosmological parameter constraints by performing a multi-probe likelihood analysis for a combination of simulated future stage-IV-like surveys. We find that the inclusion of the cross-correlations between the probes significantly increases the information gain in the parameter constraints. We also find that the use of a non-Gaussian covariance matrix is increasingly important, as more probes and cross-correlation power spectra are included. A version of the UFALCON package currently including weak gravitational lensing is publicly available.

Covariance of the matter power spectrum including the survey window function effect: N -body simulations versus fifth-order perturbation theory on grids

We present a Next-to-next-to-leading (fifth or NNLO) order calculation for the covariance matrix of the matter power spectrum, taking into account the effect of survey window functions. Using the grid-based calculation scheme for the standard perturbation theory, GridSPT, we quickly generate multiple realizations of the nonlinear density fields to fifth order in perturbation theory, then estimate the power spectrum and the covariance matrix from the sample. To the end, we have obtained the non-Gaussian covariance originated from the one-loop trispectrum without explicitly computing the trispectrum. By comparing the GridSPT calculations with the N-body results, we show that NNLO GridSPT result reproduces the N-body results on quasi-linear scales where SPT accurately models nonlinear matter power spectrum. Incorporating the survey window function effect to GridSPT is rather straightforward, and the resulting NNLO covariance matrix also matches well with the N-body results.

Joint cosmology and mass calibration from thermal Sunyaev-Zel’dovich cluster counts and cosmic shear

We present a new method for joint cosmological parameter inference and cluster mass calibration from a combination of weak lensing measurements and the abundance of thermal Sunyaev-Zel'dovich (tSZ) selected galaxy clusters. We combine cluster counts with the spherical harmonic cosmic shear power spectrum and the cross-correlation between cluster overdensity and cosmic shear. These correlations constrain the cluster mass-observable relation. We model the observables using a halo model framework, including their full non-Gaussian covariance. Forecasting constraints on cosmological and mass calibration parameters for a combination of LSST cosmic shear and Simons Observatory tSZ cluster counts, we find competitive constraints for cluster cosmology, with a factor two improvement in the dark energy figure of merit compared to LSST cosmic shear alone. We find most of the mass calibration information will be in the large and intermediate scales of the cross-correlation between cluster overdensity and cosmic shear. Finally, we find broadly comparable constraints to traditional analyses based on calibrating masses using stacked cluster lensing measurements, with the benefit of consistently accounting for the correlations with cosmic shear.

Elucidating ecological complexity: Unsupervised learning determines global marine eco-provinces.

An unsupervised learning method is presented for determining global marine ecological provinces (eco-provinces) from plankton community structure and nutrient flux data. The systematic aggregated eco-province (SAGE) method identifies eco-provinces within a highly nonlinear ecosystem model. To accommodate the non-Gaussian covariance of the data, SAGE uses t-stochastic neighbor embedding (t-SNE) to reduce dimensionality. Over a hundred eco-provinces are identified with the density-based spatial clustering of applications with noise (DBSCAN) algorithm. Using a connectivity graph with ecological dissimilarity as the distance metric, robust aggregated eco-provinces (AEPs) are objectively defined by nesting the eco-provinces. Using the AEPs, the control of nutrient supply rates on community structure is explored. Eco-provinces and AEPs are unique and aid model interpretation. They could facilitate model intercomparison and potentially improve understanding and monitoring of marine ecosystems.

Cosmological angular trispectra and non-Gaussian covariance

Angular cosmological correlators are infamously difficult to compute due to the highly oscillatory nature of the projection integrals. Motivated by recent development on analytic approaches to cosmological perturbation theory, in this paper we present an efficient method for computing cosmological four-point correlations in angular space, generalizing previous works on lower-point functions. This builds on the FFTLog algorithm that approximates the matter power spectrum as a sum over power-law functions, which makes certain momentum integrals analytically solvable. The computational complexity is drastically reduced for correlators in a “separable” form—we define a suitable notion of separability for cosmological trispectra, and derive formulas for angular correlators of different separability classes. As an application of our formalism, we compute the angular galaxy trispectrum at tree level, with and without primordial non-Gaussianity. This includes effects of redshift space distortion and bias parameters up to cubic order. We also compute the non-Gaussian covariance of the angular matter power spectrum due to the connected four-point function, beyond the Limber approximation. We demonstrate that, in contrast to the standard lore, the Limber approximation can fail for the non-Gaussian covariance computation even for large multipoles.

Super-CMB fluctuations and the Hubble tension

We study the covariance in the angular power spectrum estimates of CMB fluctuations when the primordial fluctuations are non-Gaussian. The non-Gaussian covariance comes from a nonzero connected four-point correlation function – or the trispectrum in Fourier space – and can be large when long-wavelength (super-CMB) modes are strongly coupled to short-wavelength modes. The effect of such non-Gaussian covariance can be modeled through additional freedom in the theoretical CMB angular power spectrum and can lead to different inferred values of the standard cosmological parameters relative to those in ΛCDM. Taking the collapsed limit of the primordial trispectrum in the quasi-single field inflation model as an example, we study how the six standard ΛCDM parameters shift when two additional parameters describing the trispectrum are allowed. The reduced statistical significance of the Hubble tension in the extended model allows us to combine the Planck temperature data and the type Ia supernovae data from Panstarrs with the distance-ladder measurement of the Hubble constant. This combination of data shows strong evidence for a primordial trispectrum-induced non-Gaussian covariance, with a likelihood improvement of Δχ2≈−15 (with two additional parameters) relative to ΛCDM.

The impact of braiding covariance and in-survey covariance on next-generation galaxy surveys

As galaxy surveys improve their precision thanks to lower levels of noise and the push toward small, non-linear scales, the need for accurate covariances beyond the classical Gaussian formula becomes more acute. Here I investigate the analytical implementation and impact of non-Gaussian covariance terms that I had previously derived for the galaxy angular power spectrum. Braiding covariance is such an interesting class of such terms and it gets contributions both from in-survey and super-survey modes, the latter proving difficult to calibrate through simulations. I present an approximation for braiding covariance which speeds up the process of numerical computation. I show that including braiding covariance is a necessary condition for including other non-Gaussian terms, namely the in-survey 2-, 3-, and 4-halo covariance. Indeed these terms yield incorrect covariance matrices with negative eigenvalues if considered on their own. I then move to quantify the impact on parameter constraints, with forecasts for a survey with Euclid-like galaxy density and angular scales. Compared with the Gaussian case, braiding and in-survey covariances significantly increase the error bars on cosmological parameters, in particular by 50% for the dark energy equation of state w. The error bars on the halo occupation distribution (HOD) parameters are also affected between 12% and 39%. Accounting for super-sample covariance (SSC) also increases parameter errors, by 90% for w and between 7% and 64% for HOD. In total, non-Gaussianity increases the error bar on w by 120% (between 15% and 80% for other cosmological parameters) and the error bars on HOD parameters between 17% and 85%. Accounting for the 1-halo trispectrum term on top of SSC, as has been done in some current analyses, is not sufficient for capturing the full non-Gaussian impact: braiding and the rest of in-survey covariance have to be accounted for. Finally, I discuss why the inclusion of non-Gaussianity generally eases up parameter degeneracies, making cosmological constraints more robust for astrophysical uncertainties. I released publicly the data and a Python notebook reproducing the results and plots of the article.

Tomographic weak lensing bispectrum: a thorough analysis towards the next generation of galaxy surveys

ABSTRACT We address key points for an efficient implementation of likelihood codes for modern weak lensing large-scale structure surveys. Specifically, we focus on the joint weak lensing convergence power spectrum–bispectrum probe and we tackle the numerical challenges required by a realistic analysis. Under the assumption of (multivariate) Gaussian likelihoods, we have developed a high performance code that allows highly parallelized prediction of the binned tomographic observables and of their joint non-Gaussian covariance matrix accounting for terms up to the six-point correlation function and supersample effects. This performance allows us to qualitatively address several interesting scientific questions. We find that the bispectrum provides an improvement in terms of signal-to-noise ratio (S/N) of about 10 per cent on top of the power spectrum, making it a non-negligible source of information for future surveys. Furthermore, we are capable to test the impact of theoretical uncertainties in the halo model used to build our observables; with presently allowed variations we conclude that the impact is negligible on the S/N. Finally, we consider data compression possibilities to optimize future analyses of the weak lensing bispectrum. We find that, ignoring systematics, five equipopulated redshift bins are enough to recover the information content of a Euclid-like survey, with negligible improvement when increasing to 10 bins. We also explore principal component analysis and dependence on the triangle shapes as ways to reduce the numerical complexity of the problem.

Statistical anisotropies in temperature and polarization fluctuations from a scale-dependent trispectrum

We study statistical anisotropies generated in the observed two-point function of the cosmic microwave background (CMB) fluctuations if the primordial statistics are non-Gaussian. Focusing on the dipole modulations of the anisotropies, we find that the hemispherical power asymmetry observed in the CMB temperature fluctuations can be modeled by a local-type trispectrum with amplitude $\tau_{\rm NL}(k_p=0.05~{\rm Mpc}^{-1}) \approx 2 \times 10^4$ and a large red tilt $n\approx -0.68$. We numerically evaluate the non-Gaussian covariance of the modulation estimators for both temperature and E-mode polarization fluctuations and discuss the prospects of constraining the model using Planck satellite data. We then discuss other effects of the scale-dependent trispectrum that could be used to distinguish this scenario from other explanations of the power asymmetry: higher-order modulations of the two-point function and the non-Gaussian angular power spectrum covariance. As an important consequence of the non-Gaussian power spectrum covariance, we discuss how the CMB-inferred spectral index of primordial scalar fluctuations can be significantly biased in the presence of a scale-dependent local-type trispectrum.