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Table 2

Review of advantages and disadvantages of measuring spectra vs. fields to infer an ISW detection (top). Review of statistical methods in the literature and their respective advantages and disadvantages (bottom).

Measured Advantage Disadvantage
quantity

Methods exist for calculation with missing data. Assumes C()’s or estimator are Gaussian.
Spectra Can easily introduce tomography. Most methods requires
to estimate of the covariance matrix.

Fields No assumption about galaxy/matter density field Missing data is an ill-posed problem

Statistical Advantage Disadvantage
method

Simple correlation Independent of cosmology. Measure of significance
(spectra/fields) assumes test distribution to be Gaussian

Amplitude estimation Detection depends on cosmology/model
(spectra/fields) Validates signal and model simultaneously Measure of significance assumes estimator is Gaussian
Assumes underlying theory (e.g., ΛCDM) is correct

χ2 (spectra/fields) Assumes C()’s are Gaussian (spectra)
Validates signal and model simultaneously. Only gives confidence of rejecting null hypothesis
Assumes underlying theory (e.g., ΛCDM) is correct.

Model comparison Asks a different question than other tests. ISW signal usually too weak to be detected this way
(spectra) Assumes underlying theory (e.g., ΛCDM) is correct

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