Why and When to Expect Gaussian Error Distributions in Epoch of Reionization 21-cm Power Spectrum Measurements

TL;DR

This paper analyzes error distributions in Epoch of Reionization 21-cm power spectrum measurements, providing mathematical solutions and conditions for Gaussianity, which impact how results are interpreted and reported.

Contribution

It offers closed-form solutions for error distributions of two estimators and clarifies when their binned spectra follow Gaussian errors, informing better analysis practices.

Findings

Gaussian errors depend on estimator type and binning method

Nonstandard central limit theorem conditions are necessary for Gaussianity

Assuming Gaussian errors can mislead upper limit estimates

Abstract

We explore error distributions in Epoch of Reionization 21-cm power spectrum estimators using a combination of mathematical analysis and numerical simulations. We provide closed form solutions for the error distributions of individual bins in 3d-power spectra for two estimators currently in use in the field, which we designate as ``straight-square" and ``cross-multiply" estimators. We then demonstrate when the corresponding spherically binned power spectra should (and should not) have Gaussian error distributions, which requires appealing to nonstandard statements of the central limit theorem. This has important implications for how upper limits are reported, as well as how cosmological inferences are performed based on power spectrum measurements. Specifically, assuming a Gaussian error distribution can over or underestimate the upper limit depending on the type of estimator, and produces overly compact likelihood functions for the power spectrum.