General CMB and Primordial Trispectrum Estimation

Abstract

We present trispectrum estimation methods which can be applied to general non-separable primordial and CMB trispectra. We present a general optimal estimator for the connected part of the trispectrum, for which we derive a quadratic term to incorporate the effects of inhomogeneous noise and masking. We describe a general algorithm for creating simulated maps with given arbitrary (and independent) power spectra, bispectra and trispectra. We propose a universal definition of the trispectrum parameter $T_{NL}$, so that the integrated bispectrum on the observational domain can be consistently compared between theoretical models. We define a shape function for the primordial trispectrum, together with a shape correlator and a useful parametrisation for visualizing the trispectrum. We derive separable analytic CMB solutions in the large-angle limit for constant and local models. We present separable mode decompositions which can be used to describe any primordial or CMB bispectra on their respective wavenumber or multipole domains. By extracting coefficients of these separable basis functions from an observational map, we are able to present an efficient estimator for any given theoretical model with a nonseparable trispectrum. The estimator has two manifestations, comparing the theoretical and observed coefficients at either primordial or late times. These mode decomposition methods are numerically tractable with order $l^5$ operations for the CMB estimator and approximately order $l^6$ for the general primordial estimator (reducing to order $l^3$ in both cases for a special class of models). We also demonstrate how the trispectrum can be reconstructed from observational maps using these methods.

38 pages, 9 figures. In v2 Figures 4-7 are altered slightly and some extra references are included in the bibliography. v3 matches version submitted to journal. Includes discussion of special cases

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• Folded Resonant Non-Gaussianity in General Single Field Inflation
• Halo Clustering with Non-Local Non-Gaussianity
• BICEP2 / Keck Array VIII: Measurement of gravitational lensing from large-scale B-mode polarization
• The CMB Bispectrum
• On Classification of Models of Large Local-Type Non-Gaussianity
• Primordial Non-Gaussianity in the Cosmic Microwave Background
• Full covariance of CMB and lensing reconstruction power spectra
• Influence of large local and non-local bispectra on primordial black hole abundance
• The Trispectrum in the Effective Field Theory of Large Scale Structure
• On the joint analysis of CMB temperature and lensing-reconstruction power spectra
• Combining power spectrum and bispectrum measurements to detect oscillatory features
• The real shape of non-Gaussianities
• Rapid Separable Analysis of Higher Order Correlators in Large Scale Structure
• Primordial Magnetic Field Limits from CMB Trispectrum - Scalar Modes and Planck Constraints
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• Baryons do trace dark matter 380,000 years after the big bang: Search for compensated isocurvature perturbations with WMAP 9-year data
• Capturing Non-Gaussianity of the Large-Scale Structure with Weighted Skew-Spectra
• Towards optimal cosmological parameter recovery from compressed bispectrum statistics
• Fast Estimation of Gravitational and Primordial Bispectra in Large Scale Structures
• Oscillations in the Primordial Bispectrum: Mode Expansion
• Bias-Hardened CMB Lensing with Polarization
• Precise Calibration of the One-Loop Bispectrum in the Effective Field Theory of Large Scale Structure
• Seeking Inflation Fossils in the Cosmic Microwave Background
• The CMB Bispectrum, Trispectrum, non-Gaussianity, and the Cramer-Rao Bound
• Optimal constraint on g_NL from CMB
• Disorder in the Early Universe
• CMB internal delensing with general optimal estimator for higher-order correlations
• CMB lensing power spectrum estimation without instrument noise bias
• Isotropic N-Point Basis Functions and Their Properties
• Universal Non-Gaussian Initial Conditions for N-body Simulations
• Precise and Accurate Cosmology with CMBxLSS Power Spectra and Bispectra
• All sky CMB map from cosmic strings integrated Sachs-Wolfe effect
• Fundamental limits on constraining primordial non-Gaussianity
• Primordial trispectra and CMB spectral distortions
• CMB lensing and primordial squeezed non-Gaussianity
• Efficient optimal non-Gaussian CMB estimators with polarisation
• Precise Calibration of the One-Loop Trispectrum in the Effective Field Theory of Large Scale Structure
• General and consistent statistics for cosmological observations
• Non-Gaussian Structure of B-mode Polarization after Delensing
• Angular dependence of primordial trispectra and CMB spectral distortions
• Trispectrum estimation in various models of equilateral type non-Gaussianity
• Non-Gaussianity from Lifshitz Scalar
• Anatomy of bispectra in general single-field inflation -- modal expansions
• DBI Galileon in the Effective Field Theory of Inflation: Orthogonal non-Gaussianities and constraints from the Trispectrum
• Primordial Features as Evidence for Inflation
• Cosmological Angular Trispectra and Non-Gaussian Covariance
• Planck Trispectrum Constraints on Primordial Non-Gaussianity at Cubic Order
• General CMB bispectrum analysis using wavelets and separable modes
• Optimal Estimation of the Binned Mask-Free Power Spectrum, Bispectrum, and Trispectrum on the Full Sky: Scalar Edition
• Mixed Inflaton and Spectator Field Models after Planck
• Implications of Planck results for models with local type non-Gaussianity
• A Modal Approach to the Numerical Calculation of Primordial non-Gaussianities
• Constraining the WMAP9 bispectrum and trispectrum with needlets
• Statistical Anisotropic Gaussian Simulations of the CMB Temperature Field
• The probability distribution for non-Gaussianity estimators constructed from the CMB trispectrum
• An Inventory of Bispectrum Estimators for Redshift Space Distortions
• Non-Gaussian Shape Recognition
• Impact of Anisotropic Birefringence on Measuring Cosmic Microwave Background Lensing
• Multi-variate joint PDF for non-Gaussianities: exact formulation and generic approximations
• Constraints on ${\tau }_{\mathrm{NL}}$ from Planck temperature and polarization
• On primordial trispectrum from exchanging scalar modes in general multiple field inflationary models
• The bispectrum of cosmic string temperature fluctuations including recombination effects
• Weak Lensing Trispectrum and Kurt-Spectra
• The Trispectrum in the Effective Theory of Inflation with Galilean symmetry
• Accurate estimation of angular power spectra for maps with correlated masks
• Anatomy of Parity-violating Trispectra in Galaxy Surveys
• Statistical anisotropies in temperature and polarization fluctuations from a scale-dependent trispectrum
• The Trispectrum as a Diagnostic of Primordial Orthogonal non-Gaussianities
• Halo/Galaxy Bispectrum with Equilateral-type Primordial Trispectrum
• CMB lensing power spectrum without noise bias
• Principal Components of CMB non-Gaussianity
• Optimal bispectrum constraints on single-field models of inflation
• CMB temperature trispectrum from accreting primordial black holes
• Detecting CMB modulations and non-Gaussianities from power multipoles
• Observable predictions of generalised inflationary scenarios
• Constraining the early universe with primordial black holes
• Measuring CMB non-Gaussianity as a probe of Inflation and Cosmic Strings
• Running the small-correlated-against-large estimator at scale: Applications of small-scale CMB lensing estimators on realistic simulations
• Higher order non-linear parameters with PLANCK