# Testing the isotropy of the log N - log S slope for the NVSS radio catalogue

## TL;DR

This study tests the isotropy of the log N - log S slope in the NVSS radio catalog by analyzing spectral index variations across the sky, finding only weak, statistically insignificant deviations from isotropy.

## Contribution

The paper introduces a rigorous likelihood maximization algorithm to assess isotropy of the spectral index in radio source counts, providing a novel method for such analyses.

## Key findings

- Maximum difference in spectral index is 1.3% of the full sky value.
- Deviation from isotropy is only at 2σ significance, indicating weak departure.
- Dipole fit results are consistent with isotropy.

## Abstract

The cumulative number count $N$, of sources above a threshold is known to approximately follow a power law behaviour $N\propto S^{-x}$. We study the variation of spectral index $x$ across the sky in order to look for possible signals of violation of isotropy. We develop a rigorous algorithm of likelihood maximisation to accurately fit for the spectral index. We divide the sky into upper and lower hemispheres for a particular choice of $z$-axis and determine the difference $\Delta x$ between the best fit values of the spectral indices between the two hemispheres. The maximum value of this difference obtained by varying over the $z$-axis provides us with a measure of departure from isotropy. We find that the data support isotropy of the spectral index. The maximum difference is found to be 1.3\% of the full sky best fit value of $x$. The deviation is found to be significant only at 2$\sigma$ level which indicates a weak departure from isotropy. We also perform a dipole fit to the spectral index as a function of the angular coordinates. The result is found to be consistent with isotropy.