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Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background

Amit P. S. Yadav, Eiichiro Komatsu, Benjamin D. Wandelt

TL;DR

This work presents a fast cubic estimator for the CMB bispectrum to constrain primordial non-Gaussianity characterized by $f_{NL}$. By tomographically reconstructing the primordial perturbations from both temperature and polarization maps and forming a radial integral of a cubic combination, the authors derive an unbiased estimator $oxed{oxed{f_{NL}}}$ whose computation scales as $O(N^{3/2})$, vastly faster than full bispectrum methods. They prove the estimator is optimal in the homogeneous-noise limit by showing equivalence to the Babich–Zaldarriaga Fisher-optimal estimator, and they validate its performance with Monte Carlo simulations under realistic instrument effects and sky cuts. Planck-like surveys can therefore robustly constrain $f_{NL}$ with feasible computation, and the framework accommodates extensions to inhomogeneous noise and higher-order statistics such as the trispectrum.

Abstract

Measurements of primordial non-Gaussianity ($f_{NL}$) open a new window onto the physics of inflation. We describe a fast cubic (bispectrum) estimator of $f_{NL}$, using a combined analysis of temperature and polarization observations. The speed of our estimator allows us to use a sufficient number of Monte Carlo simulations to characterize its statistical properties in the presence of real world issues such as instrumental effects, partial sky coverage, and foreground contamination. We find that our estimator is optimal, where optimality is defined by saturation of the Cramer Rao bound, if noise is homogeneous. Our estimator is also computationally efficient, scaling as $O(N^{3/2})$ compared to the $O(N^{5/2})$ scaling of the brute force bispectrum calculation for sky maps with $N$ pixels. For Planck this translates into a speed-up by factors of millions, reducing the required computing time from thousands of years to just hours and thus making $f_{NL}$ estimation feasible for future surveys. Our estimator in its current form is optimal if noise is homogeneous. In future work our fast polarized bispectrum estimator should be extended to deal with inhomogeneous noise in an analogous way to how the existing fast temperature estimator was generalized.

Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background

TL;DR

This work presents a fast cubic estimator for the CMB bispectrum to constrain primordial non-Gaussianity characterized by . By tomographically reconstructing the primordial perturbations from both temperature and polarization maps and forming a radial integral of a cubic combination, the authors derive an unbiased estimator whose computation scales as , vastly faster than full bispectrum methods. They prove the estimator is optimal in the homogeneous-noise limit by showing equivalence to the Babich–Zaldarriaga Fisher-optimal estimator, and they validate its performance with Monte Carlo simulations under realistic instrument effects and sky cuts. Planck-like surveys can therefore robustly constrain with feasible computation, and the framework accommodates extensions to inhomogeneous noise and higher-order statistics such as the trispectrum.

Abstract

Measurements of primordial non-Gaussianity () open a new window onto the physics of inflation. We describe a fast cubic (bispectrum) estimator of , using a combined analysis of temperature and polarization observations. The speed of our estimator allows us to use a sufficient number of Monte Carlo simulations to characterize its statistical properties in the presence of real world issues such as instrumental effects, partial sky coverage, and foreground contamination. We find that our estimator is optimal, where optimality is defined by saturation of the Cramer Rao bound, if noise is homogeneous. Our estimator is also computationally efficient, scaling as compared to the scaling of the brute force bispectrum calculation for sky maps with pixels. For Planck this translates into a speed-up by factors of millions, reducing the required computing time from thousands of years to just hours and thus making estimation feasible for future surveys. Our estimator in its current form is optimal if noise is homogeneous. In future work our fast polarized bispectrum estimator should be extended to deal with inhomogeneous noise in an analogous way to how the existing fast temperature estimator was generalized.

Paper Structure

This paper contains 11 sections, 30 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Primordial Non-Gaussianity
  3. A Model
  4. Reconstructing Primordial Perturbations using Temperature and Polarization Anisotropy
  5. Fast Cubic Statistics
  6. Optimality of the fast estimator
  7. Results
  8. Computational Speed
  9. Conclusions
  10. Derivation of Fast Cubic Estimator
  11. Proof of Covariance matrix

Figures (3)

  • Figure 1: Reconstructed primordial perturbation map using only temperature information (left), and, using both temperature and polarization information (right). Perturbations are reconstructed at the surface of last scattering.
  • Figure 2: Fisher predictions for minimum detectable $f_{NL}$ at the 1-$\sigma$ level. Left panel: ideal experiment. Right panel: Planck satellite. Solid lines: temperature and polarization information combined. Dashed lines: temperature information only. Dot-dashed line: polarization information only.
  • Figure 3: Testing optimality of our $f_{NL}$ estimator. In all panels lines show the optimal Cramer Rao bounds given by $\left( F^{-1}/f_{sky} \right) ^{1/2}$. Symbols show the 1-$\sigma$ errors on $f_{NL}$ derived from Monte Carlo simulations. Upper panels: Monte Carlo errors as a function of $f_{sky}$. The star shows the WMAP Kp2 mask. Triangles: straight sky cuts excluding regions of low galactic latitudes. The left panel shows only the effect of the mask, while the right panel includes homogeneous white noise and beam smoothing at the level of the Planck satellite. Lower left: Monte Carlo errors as a function of $\ell_{max}$. Lower right: Incomplete sky coverage causes excess variancde of the low $\ell$ modes of the polarization bispectra. We show results as a function of $\ell_{min}$, below which multipoles have been removed from the analysis of polarization bispectra in two cases: 1) stars show the simluated errors in the noiseless case; 2) the dashed line and triangles show a case with homogeneous noise similar to the Planck satellite. The variance excess due to overweighting of the polarization modes is cleary visible in the noiseless case. This panel demonstrates that excluding the lowest $\ell$ polarization modes from the analysis avoids this variance excess without significant loss of information.