> interact weakly with gravity

It would be better to rearrange this to "interact gravitationally, and maybe non-gravitationally at the scale of the weak nuclear force".

In the standard cosmology \Lambda-CDM, cold dark matter ("CDM") is allowed to interact at the scale of the weak nuclear force (wnf), which may allow participation in nuclear interactions. This motivates the search for direct detection in various track and scintillator experiments, where a WIMP (weakly interacting massive particle) could take recoil energy from an atomic nucleus, the latter leaving behind a trail of charged particles and/or emitted photons. There are of course other hypothesized interactions between WIMPs and normal matter which do not literally engage the wnf, but instead have some new interaction at no more than the same energy scale ("weak scale"). And for completeness, there are CDM models which have stronger but rarer individual interactions which average out to the weak scale (or effective collisionlessness) across volumes comparable to the size of galaxies or galaxy clusters.

(In the standard cosmology what matters is that the equation of state for dark matter is 0 or close to it so that expansion dilutes it away like cold baryons; in structure formation and galactic dynamics it's more important that most dark matter is in a halo outside the luminous structure. Both are incompatible with decays or collisions which emit relativstic particles (w > ~ 1/6 leads to early evaporation of overdensities; galactic dynamics is less tolerant of (non-radiative) clumping and other mechanisms which concentrate/gravitationally-collapse halos).

> interact ... with gravity

In a system of coordinates that absorbs linear and angular momentum, all matter -- dark or otherwise -- interacts gravitationally in proportion to its mass (or energy-density).

In General Relativity, this is the universality of free fall, which descends from Newton's Law of Universal Gravitation.

Unless we modify or abandon General Relativity, a dark matter particle and an alpha particle occupying the same starting point and having the same initial velocity will follow the same trajectory together forever barring some interaction (examples: intrinsic: one of the particles decays emitting radiation, or there is some weak scale interaction between them that imparts a recoil on one or both of the initial alpha or DM; extrinsic: scattering of a photon off the alpha, or the alpha captures electron(s) imparting a recoil -- a force that shoves the alpha onto a different trajectory, a non-gravitational acceleration). This is the weak equivalence principle (WEP), <https://en.wikipedia.org/wiki/Equivalence_principle#The_weak...> from a somewhat Fermi-Walker perspective; the "weak" in WEP is unrelated to the weak nuclear force.

Finally, dark matter and baryonic matter (and all other matter and radiation) interact the same way with the curvature of spacetime. If dark matter and baryons interact non-gravitationally at all, the (averaged) interaction is at no more than the weak scale. \Lambda-CDM does not require any non-gravitational interaction between CDM and baryons.

> so is undetected in low-density regions

Density of what?