Title: Tuning Goodness-of-Fit Tests,
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-Loève 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.