Mach's principle and dark matter

In this paper we entertain a Machian setting where local physics is non-locally affected by the whole Universe, taking the liberty to identify the local (``Newton's bucket'') with our visible Universe, and the whole Universe (Mach's ``fixed stars'') with the global Universe beyond our horizon. Crucially, we allow for the two to have different properties, so that we are beyond the traditional FRW setting. For definiteness we focus on theories where non-locality arises from evolution in the laws of physics in terms of spatially global time variables dual to the constants of Nature. Since non-local theories are foliation-dependent, the {\it local} (but not the global) Hamiltonian constraint is lost. This is true not only while non-locality is taking place, but also after it ceases: the local Hamiltonian constraint is only recovered up to a constant in time, keeping a memory of the integrated past non-locality. We show that this integration constant is equivalent to preserving the local Hamiltonian constraint and adding an extra fluid with the same cosmological properties as conventional pressureless dark matter. The equivalence breaks down in terms of clustering properties, with the new component attracting other matter, but not budging from its location. This is the ultimate ``painted-on'' dark matter, attracting but not being attracted, and nailing down a preferred frame.
Comment: Revised version corrects small technical issue at Eqs. 52-53. This does not change anything in the discussion that follows