  It is well-known that stars have the potential to be excellent dark matter
detectors. Infalling dark matter that scatters within stars could lead to a
range of observational signatures, including stellar heating, black hole
formation, and modified heat transport. To make robust predictions for such
phenomena, it is necessary to calculate the scattering rate for dark matter
inside the star. As we show in this paper, for small enough momentum transfers,
this requires taking into account collective effects within the dense stellar
medium. These effects have been neglected in many previous treatments; we
demonstrate how to incorporate them systematically, and show that they can
parametrically enhance or suppress dark matter scattering rates depending on
how dark matter couples to the Standard Model. We show that, as a result,
collective effects can significantly modify the potential discovery or
exclusion reach for observations of compact objects such as white dwarfs and
neutron stars. While the effects are more pronounced for dark matter coupling
through a light mediator, we show that even for dark matter coupling via a
heavy mediator, scattering rates can differ by orders of magnitude from their
naive values for dark matter masses <~ 100 MeV. We also illustrate how
collective effects can be important for dark matter scattering in more dilute
media, such as the Solar core. Our results demonstrate the need to
systematically incorporate collective effects in a wide range of astroparticle
contexts; to facilitate this, we provide expressions for in-medium
self-energies for a variety of different media, which are applicable to many
other processes of interest (such as particle production).
