Constrain the linear scalar perturbation theory of Cotton gravity

We perform a cosmological test of Cotton gravity, which describes gravity by cotton tensor. We assume in Cotton gravity the background evolution is the same as the flat FLRW background. We derive the cosmological perturbation theory of the scalar mode at the linear level, where the difference from the $\Lambda$CDM model is characterized by the parameter $\beta$. We incorporate Cotton gravity with a neutrino model and perform a Monte Carlo Markov Chain (MCMC) analysis using data from the Cosmic Microwave Background (CMB) and Sloan Digital Sky Survey (SDSS). The analysis constrains parameter $\beta=-0.00008^{+0.00080}_{-0.00104}$ at the 1-$\sigma$ confidence level. We conclude that currently, there is no obvious deviation between Cotton gravity and the $\Lambda$CDM model in the linear cosmological perturbation level for observations.