Why are we still using 3D masses for cluster cosmology?
  The abundance of clusters of galaxies is highly sensitive to the late-time
evolution of the matter distribution, since clusters form at the highest
density peaks. However, the 3D cluster mass cannot be inferred without
deprojecting the observations, introducing model-dependent biases and
uncertainties due to the mismatch between the assumed and the true cluster
density profile and the neglected matter along the sightline. Since projected
aperture masses can be measured directly in simulations and observationally
through weak lensing, we argue that they are better suited for cluster
cosmology. Using the Mira-Titan suite of gravity-only simulations, we show that
aperture masses correlate strongly with 3D halo masses, albeit with large
intrinsic scatter due to the varying matter distribution along the sightline.
Nonetheless, aperture masses can be measured $\approx 2-3$ times more precisely
from observations, since they do not require assumptions about the density
profile and are only affected by the shape noise in the weak lensing
measurements. We emulate the cosmology dependence of the aperture mass function
directly with a Gaussian process. Comparing the cosmology sensitivity of the
aperture mass function and the 3D halo mass function for a fixed survey solid
angle and redshift interval, we find the aperture mass sensitivity is higher
for $\Omega_\mathrm{m}$ and $w_a$, similar for $\sigma_8$, $n_\mathrm{s}$, and
$w_0$, and slightly lower for $h$. With a carefully calibrated aperture mass
function emulator, cluster cosmology analyses can use cluster aperture masses
directly, reducing the sensitivity to model-dependent mass calibration biases
and uncertainties.
