Galactic Compton Wavelengths in $f(R)$ Screening Theories

$f(R)$ theories of modified gravity may be compatible with current observations if the deviations from general relativity are sufficiently well screened in dense environments. In recent work [arXiv:2310.19955] we have shown that approximations commonly used to assess whether galaxies are screened, or unscreened, fail to hold in observationally interesting parts of parameter space. One of the assumptions commonly made in these approximations, and more broadly in the study of $f(R)$ models, is that the mass of the scalar mode can be neglected inside a galaxy. In this work we demonstrate that this approximation may fail spectacularly and discuss the implications of this for tests of the theory.