Vector modes in $\Lambda$CDM: the gravitomagnetic potential in dark
  matter haloes from relativistic $N$-body simulations
  We investigate the transverse modes of the gravitational and velocity fields
in $\Lambda$CDM, based on a high-resolution simulation performed using the
adaptive-mesh refinement general-relativistic $N$-body code GRAMSES. We study
the generation of vorticity in the dark matter velocity field at low redshift,
providing fits to the shape and evolution of its power spectrum over a range of
scales. By analysing the gravitomagnetic vector potential, which is absent in
Newtonian simulations, in dark matter haloes with masses ranging from
$\sim10^{12.5}~h^{-1}{M}_{\odot}$ to $\sim10^{15}~h^{-1}{M}_{\odot}$, we find
that its magnitude correlates with the halo mass, peaking in the inner regions.
Nevertheless, on average, its ratio against the scalar gravitational potential
remains fairly constant, below percent level, decreasing roughly linearly with
redshift and showing a weak dependence on halo mass. Furthermore, we show that
the gravitomagnetic acceleration in haloes peaks towards the core and reaches
almost $10^{-10}$ $h$ cm/s$^2$ in the most massive halo of the simulation.
However, regardless of the halo mass, the ratio between the magnitudes of the
gravitomagnetic force and the standard gravitational force is typically at
around the $10^{-5}$ level inside the haloes, again without significant radius
dependence. This result confirms that the gravitomagnetic effects have a
negligible impact on structure formation, even for the most massive structures,
although its behaviour in low density regions remains to be explored. Likewise,
the impact on observations remains to be understood in the future.
