The dependence of galaxy clustering on local density provides an effective method for extracting non-Gaussian information from galaxy surveys. The two-point correlation function (2PCF) provides a complete statistical description of a Gaussian density field. However, the late-time density field becomes non-Gaussian due to non-linear gravitational evolution and higher-order summary statistics are required to capture all of its cosmological information. Using a Fisher formalism based on halo catalogues from the Quijote simulations, we explore the possibility of retrieving this information using the density-split clustering (DS) method, which combines clustering statistics from regions of different environmental density. We show that DS provides more precise constraints on the parameters of the $νΛ$CDM model compared to the 2PCF, and we provide suggestions for where the extra information may come from. DS improves the constraints on the sum of neutrino masses by a factor of $7$ and by factors of 4, 3, 3, 6, and 5 for $Ω_{\rm m}$, $Ω_{\rm b}$, $h$, $n_s$, and $σ_8$, respectively. We compare DS statistics when the local density environment is estimated from the real or redshift-space positions of haloes. The inclusion of DS autocorrelation functions, in addition to the cross-correlation functions between DS environments and haloes, recovers most of the information that is lost when using the redshift-space halo positions to estimate the environment. We discuss the possibility of constructing simulation-based methods to model DS clustering statistics in different scenarios. Author's accepted manuscript
