Towards Accurate Modelling of Galaxy Clustering on Small Scales: Testing
  the Standard $\Lambda\mathrm{CDM}$ + Halo Model
  Interpreting the small-scale clustering of galaxies with halo models can
elucidate the connection between galaxies and dark matter halos. Unfortunately,
the modelling is typically not sufficiently accurate for ruling out models
statistically. It is thus difficult to use the information encoded in small
scales to test cosmological models or probe subtle features of the galaxy-halo
connection. In this paper, we attempt to push halo modelling into the
"accurate" regime with a fully numerical mock-based methodology and careful
treatment of statistical and systematic errors. With our forward-modelling
approach, we can incorporate clustering statistics beyond the traditional
two-point statistics. We use this modelling methodology to test the standard
$\Lambda\mathrm{CDM}$ + halo model against the clustering of SDSS DR7 galaxies.
Specifically, we use the projected correlation function, group multiplicity
function and galaxy number density as constraints. We find that while the model
fits each statistic separately, it struggles to fit them simultaneously. Adding
group statistics leads to a more stringent test of the model and significantly
tighter constraints on model parameters. We explore the impact of varying the
adopted halo definition and cosmological model and find that changing the
cosmology makes a significant difference. The most successful model we tried
(Planck cosmology with Mvir halos) matches the clustering of low luminosity
galaxies, but exhibits a 2.3$\sigma$ tension with the clustering of luminous
galaxies, thus providing evidence that the "standard" halo model needs to be
extended. This work opens the door to adding interesting freedom to the halo
model and including additional clustering statistics as constraints.
