Constrain the linear scalar perturbation theory of Cotton gravity

We perform a cosmological test of Cotton gravity, which describes gravity by cotton tensor. The model we consider allows for the same background evolution as the ΛCDM model. We derive the cosmological perturbation theory of the scalar mode at the linear level, where the difference from the ΛCDM model is characterized by the parameter β. 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 β=-0.00008^+0.00080_-0.00104 at the 1-σ confidence level. We conclude that currently, there is no obvious deviation between Cotton gravity and the ΛCDM model in the linear cosmological perturbation level for observations.