Bispectrum Covariance Research Articles

Bispectrum non-Gaussian covariance in redshift space

We provide an analytical description of the galaxy bispectrum covariance and the power spectrum-bispectrum cross-covariance in redshift space that captures the dominant non-Gaussian contributions at mildly non-linear scales, particularly those arising from the squeezed configurations.The Gaussian prediction for the variance of the halo bispectrum monopole significantly underestimates numerical estimates particularly for squeezed triangles, that is bispectrum triangular configurations where one side is much smaller than the other two, whereas the effect is relatively less important when considering the quadrupole.We propose an expression for the missing non-Gaussian contribution valid at mildly non-linear scales in the squeezed limit that requires an accurate modeling of the bispectrum alone. We validate our model against the numerical covariance estimated from a large suite of mock catalogs and find that it accurately predicts the variance as well as the dominant off-diagonal terms. We also present an expression for the cross-covariance between power spectrum and bispectrum multipoles and likewise find it to provide a good description of the numerical results.

A roadmap to cosmological parameter analysis with third-order shear statistics

Context. Third-order weak lensing statistics are a promising tool for cosmological analyses since they extract cosmological information in the non-Gaussianity of the cosmic large-scale structure. However, such analyses require precise and accurate models for the covariance of the statistics. Aims. In this second paper of a series on third-order weak lensing statistics, we derive and validate an analytic model for the covariance of the third-order aperture statistics 〈Map3〉. Methods. We derive the covariance model from a real-space estimator for 〈Map3〉, including the Gaussian and non-Gaussian parts. We validate the model by comparing it to estimates from simulated Gaussian random fields (GRFs) and two sets of N-body simulations. Finally, we perform mock cosmological analyses with the model covariance and the simulation estimate to compare the resulting parameter constraints. Results. We find good agreement between the analytic model and the simulations, both for the GRFs and the N-body simulations. The figure of merit in the S8 − Ωm plane from our covariance model is within 3% of the one obtained from the simulated covariances. We also show that our model, which is based on an estimator using convergence maps, can be used to obtain upper and lower bounds for the covariance of an estimator based on three-point shear correlation functions. This second estimator is required to measure 〈Map3〉 in realistic survey data. In our derivation, we find that the covariance of 〈Map3〉 cannot be obtained from the bispectrum covariance and that it includes several ‘finite-field terms’ that do not scale with the inverse survey area. Conclusions. Our covariance model is sufficiently accurate for analysing stage III surveys. Covariances for statistics in Fourier space cannot always be straightforwardly converted into covariance for real-space statistics.

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.

Breaking degeneracies with the Sunyaev-Zeldovich full bispectrum

Non-Gaussian (NG) statistics of the thermal Sunyaev-Zeldovich (tSZ) effect carry significant information which is not contained in the power spectrum. Here, we perform a joint Fisher analysis of the tSZ power spectrum and bispectrum to verify how much the full bispectrum can contribute to improve parameter constraints. We go beyond similar studies of this kind in several respects: first of all, we include the complete power spectrum and bispectrum (auto- and cross-) covariance in the analysis, computing all NG contributions; furthermore we consider a multi-component foreground scenario and model the effects of component separation in the forecasts; finally, we consider an extended set of both cosmological and intra-cluster medium parameters. We show that the tSZ bispectrum is very efficient at breaking parameter degeneracies, making it able to produce even stronger cosmological constraints than the tSZ power spectrum: e.g. the standard deviation on σ8 shrinks from σPS(σ8)=0.35 to σBS(σ8)=0.065 when we consider a multi-parameter analysis. We find that this is mostly due to the different response of separate triangle types (e.g. equilateral and squeezed) to changes in model parameters. While weak, this shape dependence is clearly non-negligible for cosmological parameters, and it is even stronger, as expected, for intra-cluster medium parameters.

Constraining primordial non-Gaussianity with postreconstructed galaxy bispectrum in redshift space

Galaxy bispectrum is a promising probe of inflationary physics in the early Universe as a measure of primordial non-Gaussianity (PNG), whereas its signal-to-noise ratio is significantly affected by the mode coupling due to nonlinear gravitational growth. In this paper, we examine the standard reconstruction method of linear cosmic mass density fields from nonlinear galaxy density fields to decorrelate the covariance in redshift-space galaxy bispectra. In particular, we evaluate the covariance of the bispectrum for massive-galaxy-sized dark matter halos with reconstruction by using 4000 independent $N$-body simulations. Our results show that the bispectrum covariance for the postreconstructed field approaches the Gaussian prediction at scale of $k<0.2\text{ }\text{ }h\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}$. We also verify the leading-order PNG-induced bispectrum is not affected by details of the reconstruction with perturbative theory. We then demonstrate the constraining power of the postreconstructed bispectrum for PNG at redshift of approximately 0.5. Further, we perform a Fisher analysis to make a forecast of PNG constraints by galaxy bispectra including anisotropic signals. Assuming a massive galaxy sample in the Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey, we find that the postreconstructed bispectrum can constrain the local, equilateral, and orthogonal types of PNG with $\mathrm{\ensuremath{\Delta}}{f}_{\mathrm{NL}}\ensuremath{\sim}13$, 90, and 42, respectively, improving the constraints with the prereconstructed bispectrum by a factor of 1.3--3.2. In conclusion, the reconstruction plays an essential role in constraining various types of PNG signatures with a level of $\mathrm{\ensuremath{\Delta}}{f}_{\mathrm{NL}}\ensuremath{\lesssim}1$ from the galaxy bispectrum based on upcoming galaxy surveys.

Nonlinear 3D cosmic web simulation with heavy-tailed generative adversarial networks

Fast and accurate simulations of the non-linear evolution of the cosmic density field are a major component of many cosmological analyses, but the computational time and storage required to run them can be exceedingly large. For this reason, we use generative adversarial networks (GANs) to learn a compressed representation of the 3D matter density field that is fast and easy to sample, and for the first time show that GANs are capable of generating samples at the level of accuracy of other conventional methods. Using sub-volumes from a suite of GADGET-2 N-body simulations, we demonstrate that a deep-convolutional GAN can generate samples that capture both large- and small-scale features of the matter density field, as validated through a variety of n-point statistics. The use of a data scaling that preserves high-density features and a heavy-tailed latent space prior allow us to obtain state of the art results for fast 3D cosmic web generation. In particular, the mean power spectra from generated samples agree to within 5% up to k=3 and within 10% for k<5 when compared with N-body simulations, and similar accuracy is obtained for a variety of bispectra. By modeling the latent space with a heavy-tailed prior rather than a standard Gaussian, we better capture sample variance in the high-density voxel PDF and reduce errors in power spectrum and bispectrum covariance on all scales. Furthermore, we show that a conditional GAN can smoothly interpolate between samples conditioned on redshift. Deep generative models, such as the ones described in this work, provide great promise as fast, low-memory, high-fidelity forward models of large-scale structure.

Minimizing gravitational lensing contributions to the primordial bispectrum covariance

The next generation of ground-based CMB experiments aim to measure temperature and polarization fluctuations up to $\ell_{\rm max} \approx 5000$ over half of the sky. Combined with Planck data on large scales, this will provide improved constraints on primordial non-Gaussianity. However, the impressive resolution of these experiments will come at a price. Besides signal confusion from galactic foregrounds, extra-galactic foregrounds and late-time gravitational effects, gravitational lensing will introduce large non-Gaussianity that can become the leading contribution to the bispectrum covariance through the connected 4-point function. Here, we compute this effect analytically for the first time on the full sky for both temperature and polarization. We compare our analytical results with those obtained directly from map-based simulations of the CMB sky for several levels of instrumental noise. Of the standard shapes considered in the literature, the local shape is most affected, resulting in a 35\% increase of the estimator standard deviation for an experiment like the Simons Observatory (SO) and a 110\% increase for a cosmic-variance limited experiment, including both temperature and polarization modes up to $\ell_{\rm max} = 3800$. Because of the nature of the lensing 4-point function, the impact on other shapes is reduced while still non negligible for the orthogonal shape. Two possible avenues to reduce the non-Gaussian contribution to the covariance are proposed. First by marginalizing over lensing contributions, such as the ISW-lensing 3pt function in temperature, and second by delensing the CMB. We show the latter method can remove almost all extra covariance, reducing the effect to below $<$5\% for local bispectra. At the same time, delensing would remove signal biases from secondaries induced by lensing, such as ISW-lensing.

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.

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.

Assessment of the information content of the power spectrum and bispectrum

The covariance matrix of the matter and halo power spectrum and bispectrum are studied. Using a large suite of simulations, we find that the non-Gaussianity in the covariance is significant already at mildly nonlinear scales. We compute the leading disconnected non-Gaussian correction to the matter bispectrum covariance using perturbation theory, and find that the corrections result in good agreement in the mildly nonlinear regime. The shot noise contribution to the halo power spectrum and bispectrum covariance is computed using the Poisson model, and the model yields decent agreement with simulation results. However, when the shot noise is estimated from the individual realization, which is usually done in reality, we find that the halo covariance is substantially reduced and gets close to the Gaussian covariance. This is because most of the non-Gaussianity in the covariance arises from the fluctuations in the Poisson shot noise. We use the measured non-Gaussian covariance to access the information content of the power spectrum and bispectrum. The signal-to-noise ratio, S/N, of the matter and halo power spectrum levels off in the mildly nonlinear regime, $k \sim 0.1 - 0.2 \,\mathrm{Mpc} h^{-1}$. In the nonlinear regime the S/N of the matter and halo bispectrum increases but much slower than the Gaussian results suggest. We find that both the S/N for power spectrum and bispectrum are overestimated by the Gaussian covariances, but the problem being much more serious for the bispectrum. Because the bispectrum is affected strongly by nonlinearity and shot noise, inclusion of the bispectrum only adds modest amount of S/N compared to that of the power spectrum.

The bispectrum covariance beyond Gaussianity

To investigate and specify the statistical properties of cosmological fields with particular attention to possible non-Gaussian features, accurate formulae for the bispectrum and the bispectrum covariance are required. The bispectrum is the lowest-order statistic providing an estimate for non-Gaussianities of a distribution, and the bispectrum covariance depicts the errors of the bispectrum measurement and their correlation on different scales. Currently, there do exist fitting formulae for the bispectrum and an analytical expression for the bispectrum covariance, but the former is not very accurate and the latter contains several intricate terms and only one of them can be readily evaluated from the power spectrum of the studied field. Neglecting all higher-order terms results in the Gaussian approximation of the bispectrum covariance. We study the range of validity of this Gaussian approximation for two-dimensional non-Gaussian random fields. For this purpose, we simulate Gaussian and non-Gaussian random fields, the latter represented by log-normal fields and obtained directly from the former by a simple transformation. From the simulated fields, we calculate the power spectra, the bispectra, and the covariance from the sample variance of the bispectra, for different degrees of non-Gaussianity \alpha, which is equivalent to the skewness on a given angular scale \theta g. We find that the Gaussian approximation provides a good approximation for \alpha<0.6 and a reasonably accurate approximation for \alpha< 1, both on scales >8\theta g. Using results from cosmic shear simulations, we estimate that the cosmic shear convergence fields are described by \alpha<0.7 at \theta g~4". We therefore conclude that the Gaussian approximation for the bispectrum covariance is likely to be applicable in ongoing and future cosmic shear studies.

Bispectrum covariance in the flat-sky limit

<i>Aims. <i/>To probe cosmological fields beyond the Gaussian level, three-point statistics can be used, all of which are related to the bispectrum. Hence, measurements of CMB anisotropies, galaxy clustering, and weak gravitational lensing alike have to rely upon an accurate theoretical background concerning the bispectrum and its noise properties. If only small portions of the sky are considered, it is often desirable to perform the analysis in the flat-sky limit. We aim at a formal, detailed derivation of the bispectrum covariance in the flat-sky approximation, focusing on a pure two-dimensional Fourier-plane approach.<i>Methods. <i/>We define an unbiased estimator of the bispectrum, which takes the average over the overlap of annuli in Fourier space, and compute its full covariance. The outcome of our formalism is compared to the flat-sky spherical harmonic approximation in terms of the covariance, the behavior under parity transformations, and the information content. We introduce a geometrical interpretation of the averaging process in the estimator, thus providing an intuitive understanding.<i>Results. <i/>Contrary to foregoing work, we find a difference by a factor of two between the covariances of the Fourier-plane and the spherical harmonic approach. We argue that this discrepancy can be explained by the differing behavior with respect to parity. However, in an exemplary analysis it is demonstrated that the Fisher information of both formalisms agrees to high accuracy. Via the geometrical interpretation we are able to link the normalization in the bispectrum estimator to the area enclosed by the triangle configuration at consideration as well as to the Wigner symbol, which leads to convenient approximation formulae for the covariances of both approaches.