  In previous work [L. Blanchet and A. Le Tiec, Phys. Rev. D 80, 023524
(2009)], motivated by the phenomenology of dark matter at galactic scales, a
model of dipolar dark matter (DDM) was introduced. At linear order in
cosmological perturbations, the dynamics of the DDM was shown to be identical
to that of standard cold dark matter (CDM). In this paper, the DDM model is
investigated at second order in cosmological perturbation theory. We find that
the internal energy of the DDM fluid modifies the curvature perturbation
generated by CDM with a term quadratic in the dipole field. This correction
induces a new type of non-Gaussianity in the bispectrum of the curvature
perturbation with respect to standard CDM. Leaving unspecified the primordial
amplitude of the dipole field, which could in principle be determined by a more
fundamental description of DDM, we find that, in contrast with usual models of
primordial non-Gaussianities, the non-Gaussianity induced by DDM increases with
time after the radiation-matter equality on super-Hubble scales. This
distinctive feature of the DDM model, as compared with standard CDM, could thus
provide a specific signature in the CMB and large-scale structure probes of
non-Gaussianity.
