Dipolar dark matter and dark energy

In previous work [L. Blanchet and A. Le Tiec, Phys. Rev. D 78, 024031 (2008)], a model of dark matter and dark energy based on the concept of gravitational polarization was investigated. This model was shown to recover the concordance cosmological scenario ($\Lambda$-cold dark matter) at cosmological scales, and the phenomenology of the modified Newtonian dynamics at galactic scales. In this article we prove that the model can be formulated with a simple and physically meaningful matter action in general relativity. We also provide alternative derivations of the main results of the model, and some details on the variation of the action.

Figure 1

The dipolar fluid is entirely characterized by (i) its four-velocity $u^{\mu }$ tangent to the worldline $x^{\mu }(\tau )$, (ii) its rest mass density $\sigma$, and (iii) the dipole moment ${\xi }^{\mu }$. The polarization ${\Pi }_{\perp }=\sigma {\xi }_{\perp }$ is build from the norm ${\xi }_{\perp }$ of the projection ${\xi }_{\perp }^{\mu }$ of the dipole moment ${\xi }^{\mu }$ onto the spacelike hypersurface ${\Sigma }_t$ orthogonal to $u^{\mu }$.

Figure 2

The potential $\mathcal{W}$ is function of the polarization field ${\Pi }_{\perp }$. It has an anharmonic behavior in the MOND regime ${\Pi }_{\perp }\ll a_0$, and its minimum is given by the cosmological constant $\Lambda$. The leading order nonharmonicity is parametrized by the MOND acceleration scale $a_0$ [cf. Eq. (3.1)]. The Newtonian regime ${\Pi }_{\perp }\gg a_0$ is discussed in details in paper I.