# Multiscale analysis of the CMB temperature derivatives

## TL;DR

This paper analyzes the Planck CMB temperature derivatives across multiple scales to examine local shape, isotropy, and anomalies, confirming low variance at large scales and identifying specific features like the Cold Spot and anisotropic directions.

## Contribution

It introduces a multiscale derivative analysis of the CMB temperature field, addressing incomplete sky coverage and proposing a new estimator for directional anisotropy in spinorial quantities.

## Key findings

- Confirmed low variance in large-scale CMB derivatives.
- Detected the Cold Spot as a prominent curvature peak.
- Identified an anisotropic direction with less than 1% probability.

## Abstract

We study the Planck CMB temperature at different scales through its derivatives up to second order, which allows one to characterize the local shape and isotropy of the field. The problem of having an incomplete sky in the calculation and statistical characterization of the derivatives is addressed in the paper. The analysis confirms the existence of a low variance in the CMB at large scales, which is also noticeable in the derivatives. Moreover, deviations from the standard model in the gradient, curvature and the eccentricity tensor are studied in terms of extreme values on the data. As it is expected, the Cold Spot is detected as one of the most prominent peaks in terms of curvature, but additionally, when the information of the temperature and its Laplacian are combined, another feature with similar probability at the scale of $10^\circ$ is also observed. However, the $p$-value of these two deviations increase above the $6\%$ when they are referred to the variance calculated from the theoretical fiducial model, indicating that these deviations can be associated to the low variance anomaly. Finally, an estimator of the directional anisotropy for spinorial quantities is introduced, which is applied to the spinors derived from the field derivatives. An anisotropic direction whose probability is $<1\%$ is detected in the eccentricity tensor.