In a series of papers, Blanchet & Le Tiec [55, 56, 57, 58, 59, 60] have pushed further the idea that the MOND phenomenology could arise from the fundamental properties of a form of dark matter itself, by suggesting that dark matter could carry a space-like53 four-vector gravitational dipole moment , following the analogy between Milgrom’s law and Coulomb’s law in a dieletric medium proposed by  (see Eq. 9) or between the Bekenstein–Milgrom modified Poisson equation and Gauss’ law in terms of free charge density (see Eq. 17). The dark matter medium is described as a fluid with mass current (where is the equivalent of the mass density of the atoms in a dielectric medium, i.e., it is the ordinary mass density of a pressureless perfect fluid, and is the four-velocity of the fluid54.) endowed with the dipole moment vector (which will affect the total density in addition to the above mass density ), with the following action :perpendicular to the four-velocity (not the norm of the polarization field55) of the polarization field , and where the dot denotes the covariant proper time derivative. The specific dynamics of this dark matter fluid will thus arise from the coupling between the current and the dipolar field (analogue to the coupling to an external polarization field in electromagnetism), as well as from the internal non-gravitational force acting on the dipolar dark particles and characterized by the potential . Let us note that the normal matter action and the gravitational Einstein–Hilbert action are just the same as in GR.
The equations of motion of the dark matter fluid are then gotten by varying the action w.r.t. the dipole moment variable and w.r.t. to the current , boiling down in the non-relativistic limit to:
This model has many advantages. The monopolar density of the dipolar atoms will play the role of CDM in the early universe, while the minimum of the potential naturally adds a cosmological constant term, thus making the theory precisely equivalent to the CDM model for expansion and large scale structure formation. The dark matter fluid behaves like a perfect fluid with zero pressure at first-order cosmological perturbation around a FLRW background and thus reproduces CMB anisotropies. Let us also note that, if the potential defining the internal force of the dipolar medium is to come from a fundamental theory at the microscopic level, one expects that the dimensionless coefficients in the expansion all be of order unity after rescaling by , thus naturally leading to the coincidence .
However, while the weak clustering hypothesis and stationarity of the dark matter fluid in galaxies are suppported by an exact and stable solution in spherical symmetry , it remains to be seen whether such a configuration would be a natural outcome of structure formation within this model. The presence of this stationary DM fluid being necessary to reproduce Milgrom’s law in stellar systems, this theory loses a bit of the initial predictability of MOND, and inherits a bit of the flexibility of CDM, inherent to invoking the presence of a DM fluid. This DM fluid could, e.g., be absent from some systems such as the globular clusters Pal 14 or NGC 2419 (see Section 6.6.3), thereby naturally explaining their apparent Newtonian behavior. However, the weak clustering hypothesis in itself might be problematic for explaining the missing mass in galaxy clusters, due to the fact that the MOND missing mass is essentially concentrated in the central parts of these objects (see Section 6.6.4).
Living Rev. Relativity 15, (2012), 10
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