Beyond dust in local cosmography: perturbative Vlasov predictions

The dust approximation used in cosmography neglects the velocity dispersion generated during structure formation. This dispersion produces isotropic and anisotropic pressure, whose gradients accelerate the mean matter flow even though individual particles follow geodesics. The acceleration introduces an intrinsic dipole into the local distance-redshift relation. We calculate this dipole and the accompanying changes in expansion and shear using linear Vlasov perturbation theory about an FLRW background, with a mean dispersion history motivated by halo estimates. The assumed FLRW background prevents an independent test of the cosmological principle; the numerical predictions also depend on the adopted LCDM cosmology. Pressure also changes the growth of structure, modifying the shear and the spatial gradients that govern how the distance anisotropy varies with redshift. The largest correction at the lowest order in redshift is to the shear, while changes in the expansion gradient can produce a larger dipole at the next order. The resulting intrinsic dipole is far smaller than the observed low-redshift dipole, whose behaviour nearest the observer is dominated by motion between the matter and CMB frames.