  We show that in the Dirac-Milne universe (a matter-antimatter symmetric
universe where the two components repel each other), rotation curves are
generically flat beyond the characteristic distance of about 3 virial radii,
and that a Tully-Fisher relation with exponent $\approx 3$ is satisfied. Using
3D simulations with a modified version of the RAMSES code, we show that the
Dirac-Milne cosmology presents a Faber-Jackson relation with a very small
scatter and an exponent equal to $\approx 3$ between the mass and the velocity
dispersion. We also show that the mass derived from the rotation curves
assuming Newtonian gravity is systematically overestimated compared to the mass
really present. We also show that the Dirac-Milne universe, featuring a
polarization between its matter and antimatter components, presents a behavior
similar to that of MOND (Modified Newtonian Dynamics), characterized by an
additional surface gravity compared to the Newtonian case. We show that in the
Dirac-Milne universe, at the present epoch, the intensity of the additional
gravitational field $g_{am}$ due to the presence of clouds of antimatter is of
the order of a few $10^{-11}$ m/s$^2$, similar to the characteristic
acceleration of MOND. We study the evolution of this additional acceleration
$g_{am}$ and show that it depends on the redshift, and is therefore not a
fundamental constant. Combined with its known concordance properties on SNIa
luminosity distance, age, nucleosynthesis and structure formation, the
Dirac-Milne cosmology may then represent an interesting alternative to the
$\Lambda$CDM, MOND, and other scenarios for explaining the Dark Matter and Dark
Energy conundrum.
