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Optimal Trispectrum Estimators and WMAP Constraints

J. R. Fergusson, D. M. Regan, E. P. S. Shellard

TL;DR

The paper develops and validates an optimal CMB trispectrum estimator that accounts for anisotropic noise and incomplete sky coverage using a separable mode expansion. It demonstrates the approach on WMAP5 data, constraining non-diagonal trispectra such as the local cubic $g_{\rm NL}$ term, equilateral, and constant shapes, and derives a conservative bound on cosmic strings; all results are consistent with a Gaussian Universe. The methodology reduces computational complexity via a diagonal-free, separable representation and a late-time transformation, enabling efficient estimation with realistic noise and masks. These results strengthen the Gaussian paradigm for primordial fluctuations and lay the groundwork for Planck-era trispectrum constraints and broad testing of non-Gaussian models beyond diagonal shapes.

Abstract

We present an implementation of an optimal CMB trispectrum estimator which accounts for anisotropic noise and incomplete sky coverage. We use a general separable mode expansion which can and has been applied to constrain both primordial and late-time models. We validate our methods on large angular scales using known analytic results in the Sachs-Wolfe limit. We present the first near-optimal trispectrum constraints from WMAP data on the cubic term of local model inflation $ g_{\rm NL} = (1.6 \pm 7.0)\times 10^5$, for the equilateral model $t_{\rm NL}^{\rm{equil}}=(-3.11\pm 7.5)\times 10^6 $ and for the constant model $t_{\rm NL}^{\rm{const}}=(-1.33\pm 3.62)$. These results, particularly the equilateral constraint, are relevant to a number of well-motivated models (such as DBI and K-inflation) with closely correlated trispectrum shapes. We also use the trispectrum signal predicted for cosmic strings to provide a conservative upper limit on the string tension $Gμ\le 1.1\times 10^{-6}$ (at 95% confidence), which is largely background and model independent. All these new trispectrum results are consistent with a Gaussian Universe. We discuss the importance of constraining general classes of trispectra using these methods and the prospects for higher precision with the Planck satellite.

Optimal Trispectrum Estimators and WMAP Constraints

TL;DR

The paper develops and validates an optimal CMB trispectrum estimator that accounts for anisotropic noise and incomplete sky coverage using a separable mode expansion. It demonstrates the approach on WMAP5 data, constraining non-diagonal trispectra such as the local cubic term, equilateral, and constant shapes, and derives a conservative bound on cosmic strings; all results are consistent with a Gaussian Universe. The methodology reduces computational complexity via a diagonal-free, separable representation and a late-time transformation, enabling efficient estimation with realistic noise and masks. These results strengthen the Gaussian paradigm for primordial fluctuations and lay the groundwork for Planck-era trispectrum constraints and broad testing of non-Gaussian models beyond diagonal shapes.

Abstract

We present an implementation of an optimal CMB trispectrum estimator which accounts for anisotropic noise and incomplete sky coverage. We use a general separable mode expansion which can and has been applied to constrain both primordial and late-time models. We validate our methods on large angular scales using known analytic results in the Sachs-Wolfe limit. We present the first near-optimal trispectrum constraints from WMAP data on the cubic term of local model inflation , for the equilateral model and for the constant model . These results, particularly the equilateral constraint, are relevant to a number of well-motivated models (such as DBI and K-inflation) with closely correlated trispectrum shapes. We also use the trispectrum signal predicted for cosmic strings to provide a conservative upper limit on the string tension (at 95% confidence), which is largely background and model independent. All these new trispectrum results are consistent with a Gaussian Universe. We discuss the importance of constraining general classes of trispectra using these methods and the prospects for higher precision with the Planck satellite.

Paper Structure

This paper contains 9 sections, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Optimal Trispectrum Estimation
  3. Separable Mode Expansion
  4. WMAP constraints on non-diagonal models
  5. Local Model (cubic $g_{\rm NL}$ term)
  6. Constant Model
  7. Equilateral Model
  8. Cosmic strings
  9. Discussion

Figures (1)

  • Figure 1: Plot of the ratio of the full model decomposition trispectrum with the analytic Sachs-Wolfe result \ref{['eq:SachsWolfegnl']} for the local $g_{NL}$ model. The excellent agreement for $l\lesssim 60$ demonstrates the accuracy of the modal formalism adopted in this paper.