# Tuning Goodness-of-Fit Tests

## TL;DR

This paper introduces a Karhunen-Loève based method to enhance goodness-of-fit tests for detecting subtle deviations from the standard $\Lambda$CDM model in cosmological data, demonstrated on Planck CMB spectra.

## Contribution

It presents a novel data compression technique that improves sensitivity of goodness-of-fit tests to small signals beyond the standard cosmological model.

## Key findings

- Weak evidence of extra-$\\Lambda$CDM physics in Planck data
- Method detects features partially mimicking the tuning model
- Sensitive to specific types of deviations

## Abstract

As modern precision cosmological measurements continue to show agreement with the broad features of the standard $\Lambda$-Cold Dark Matter ($\Lambda$CDM) cosmological model, we are increasingly motivated to look for small departures from the standard model's predictions which might not be detected with standard approaches. While searches for extensions and modifications of $\Lambda$CDM have to date turned up no convincing evidence of beyond-the-standard-model cosmology, the list of models compared against $\Lambda$CDM is by no means complete and is often governed by readily-coded modifications to standard Boltzmann codes. Also, standard goodness-of-fit methods such as a naive $\chi^2$ test fail to put strong pressure on the null $\Lambda$CDM hypothesis, since modern datasets have orders of magnitudes more degrees of freedom than $\Lambda$CDM. Here we present a method of tuning goodness-of-fit tests to detect potential sub-dominant extra-$\Lambda$CDM signals present in the data through compressing observations in a way that maximizes extra-$\Lambda$CDM signal variation over noise and $\Lambda$CDM variation. This method, based on a Karhunen-Lo\`{e}ve transformation of the data, is tuned to be maximally sensitive to particular types of variations characteristic of the tuning model; but, unlike direct model comparison, the test is also sensitive to features that only partially mimic the tuning model. As an example of its use, we apply this method in the context of a nonstandard primordial power spectrum compared against the $2015$ $Planck$ CMB temperature and polarization power spectrum. We find weak evidence of extra-$\Lambda$CDM physics, conceivably due to known systematics in the 2015 Planck polarization release.