Can Integrating out Dark Matter give Modified Gravity? I'm sure I misunderstood something obvious, but reading this question, I wondered what philosophically is the difference between modified gravity (like TeVeS, f(R), etc.) and dark matter, if/since we don't have direct evidence (e.g. from a lab experient) for the latter.
My question is: Given an action for dark matter, can't we just integrate it out (assuming energy scales smaller than its mass)? At the cost of generating an infinite number of higher derivative terms for the metric field, hence going beyond Einstein gravity. But we expect those to be present (from an EFT point of view) anyway, since gravity is not renormalizable.
How can we tell the difference?
EDIT: Justin Khoury’s superfluid dark matter seems to go in this direction:

Superfluid phonons, in particular, are assumed to be governed by a MOND-like effective action and mediate a MONDian acceleration between baryonic matter particles.

Does anyone know the current status of this now six years old proposal?
 A: There are observations where modified gravity and dark matter predict different outcomes. These can be used to disentangle the two models.
I leave it to somebody else to comment on your picture of integrating out the action and instead comment on one observation that can be used to disentangle the two scenarios. Indeed modified gravity works well on galactic scales but it fails on cosmological scales. In addition to a potential future detection of dark matter quanta in the lab, cosmological observations thus provide means to disentangle the two scenarios. For example, the absence of strong Baryon-Acoustic Oscillations provide evidence for dark matter and against modified gravity. Most famously this is true for the power spectrum in the Cosmic Microwave Background. Personally I find this other puzzle piece convincing, it comes from Scott Dodelson in his paper The Real Problem with MOND:

Shown is the matter power spectrum. Values larger than 1 are required for structure formation to become non-linear, and thus, for structure (such as galaxies and stars and planets) to form. Red is data. Black is the Lambda-CDM model which fits nicely. Blue dashed is what you would expect if baryons were all there was: nowhere near enough matter in the universe to form structure, and from the strong coupling of baryons to photons, strong Baryon Acoustic Oscillations. Blue solid is the way out by modifying gravity: you scale those oscillations up until the spectrum goes above 1 to form structure, but then that blue solid power spectrum is significantly different from the smooth-ish red power spectrum that we observe.
