  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 $\nu \Lambda$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 $\Omega_{\rm m}$, $\Omega_{\rm b}$,
$h$, $n_s$, and $\sigma_8$, 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.
