Minimising the impact of scale-dependent galaxy bias on the joint
  cosmological analysis of large scale structures
  We present a mitigation strategy to reduce the impact of non-linear galaxy
bias on the joint `$3 \times 2 $pt' cosmological analysis of weak lensing and
galaxy surveys. The $\Psi$-statistics that we adopt are based on Complete
Orthogonal Sets of E/B Integrals (COSEBIs). As such they are designed to
minimise the contributions to the observable from the smallest physical scales
where models are highly uncertain. We demonstrate that $\Psi$-statistics carry
the same constraining power as the standard two-point galaxy clustering and
galaxy-galaxy lensing statistics, but are significantly less sensitive to
scale-dependent galaxy bias. Using two galaxy bias models, motivated by
halo-model fits to data and simulations, we quantify the error in a standard $3
\times 2$pt analysis where constant galaxy bias is assumed. Even when adopting
conservative angular scale cuts, that degrade the overall cosmological
parameter constraints, we find of order $1 \sigma$ biases for Stage III surveys
on the cosmological parameter $S_8 = \sigma_8(\Omega_{\rm m}/0.3)^{\alpha}$.
This arises from a leakage of the smallest physical scales to all angular
scales in the standard two-point correlation functions. In contrast, when
analysing $\Psi$-statistics under the same approximation of constant galaxy
bias, we show that the bias on the recovered value for $S_8$ can be decreased
by a factor of $\sim 2$, with less conservative scale cuts. Given the
challenges in determining accurate galaxy bias models in the highly non-linear
regime, we argue that $3 \times 2$pt analyses should move towards new
statistics that are less sensitive to the smallest physical scales.
