Euclid: Forecasts for $k$-cut $3 \times 2$ Point Statistics
  Modelling uncertainties at small scales, i.e. high $k$ in the power spectrum
$P(k)$, due to baryonic feedback, nonlinear structure growth and the fact that
galaxies are biased tracers poses a significant obstacle to fully leverage the
constraining power of the {\it Euclid} wide-field survey. $k$-cut cosmic shear
has recently been proposed as a method to optimally remove sensitivity to these
scales while preserving usable information. In this paper we generalise the
$k$-cut cosmic shear formalism to $3 \times 2$ point statistics and estimate
the loss of information for different $k$-cuts in a $3 \times 2$ point analysis
of the {\it Euclid} data. Extending the Fisher matrix analysis
of~\citet{blanchard2019euclid}, we assess the degradation in constraining power
for different $k$-cuts. We work in the idealised case and assume the galaxy
bias is linear, the covariance is Gaussian, while neglecting uncertainties due
to photo-z errors and baryonic feedback. We find that taking a $k$-cut at $2.6
\ h \ {\rm Mpc} ^{-1}$ yields a dark energy Figure of Merit (FOM) of 1018. This
is comparable to taking a weak lensing cut at $\ell = 5000$ and a galaxy
clustering and galaxy-galaxy lensing cut at $\ell = 3000$ in a traditional $3
\times 2$ point analysis. We also find that the fraction of the observed
galaxies used in the photometric clustering part of the analysis is one of the
main drivers of the FOM. Removing $50 \% \ (90 \%)$ of the clustering galaxies
decreases the FOM by $19 \% \ (62 \%)$. Given that the FOM depends so heavily
on the fraction of galaxies used in the clustering analysis, extensive efforts
should be made to handle the real-world systematics present when extending the
analysis beyond the luminous red galaxy (LRG) sample.
