Accessing the high-$\ell$ frontier under the Reduced Shear Approximation
  with $k$-cut Cosmic Shear
  The precision of Stage IV cosmic shear surveys will enable us to probe
smaller physical scales than ever before, however, model uncertainties from
baryonic physics and non-linear structure formation will become a significant
concern. The $k$-cut method -- applying a redshift-dependent $\ell$-cut after
making the Bernardeau-Nishimichi-Taruya transform -- can reduce sensitivity to
baryonic physics; allowing Stage IV surveys to include information from
increasingly higher $\ell$-modes. Here we address the question of whether it
can also mitigate the impact of making the reduced shear approximation; which
is also important in the high-$\kappa$, small-scale regime. The standard
procedure for relaxing this approximation requires the repeated evaluation of
the convergence bispectrum, and consequently can be prohibitively
computationally expensive when included in Monte Carlo analyses. We find that
the $k$-cut cosmic shear procedure suppresses the $w_0w_a$CDM cosmological
parameter biases expected from the reduced shear approximation for Stage IV
experiments, when $\ell$-modes up to $5000$ are probed. The maximum cut
required for biases from the reduced shear approximation to be below the
threshold of significance is at $k = 5.37 \, h{\rm Mpc}^{-1}$. With this cut,
the predicted $1\sigma$ constraints increase, relative to the case where the
correction is directly computed, by less than $10\%$ for all parameters. This
represents a significant improvement in constraints compared to the more
conservative case where only $\ell$-modes up to 1500 are probed, and no $k$-cut
is used. We also repeat this analysis for a hypothetical, comparable kinematic
weak lensing survey. The key parts of code used for this analysis are made
publicly available.
