Computing general-relativistic effects from Newtonian N-body simulations: Frame dragging in the post-Friedmann approach

We present the first calculation of an intrinsically relativistic quantity, the leading-order correction to Newtonian theory, in fully nonlinear cosmological large-scale structure studies. Traditionally, nonlinear structure formation in standard $\mathrm{\Lambda }\text{CDM}$ cosmology is studied using N-body simulations, based on Newtonian gravitational dynamics on an expanding background. When one derives the Newtonian regime in a way that is a consistent approximation to the Einstein equations, the first relativistic correction to the usual Newtonian scalar potential is a gravitomagnetic vector potential, giving rise to frame dragging. At leading order, this vector potential does not affect the matter dynamics, thus it can be computed from Newtonian N-body simulations. We explain how we compute the vector potential from simulations in $\mathrm{\Lambda }\text{CDM}$ and examine its magnitude relative to the scalar potential, finding that the power spectrum of the vector potential is of the order ${10}^{-5}$ times the scalar power spectrum over the range of nonlinear scales we consider. On these scales the vector potential is up to two orders of magnitudes larger than the value predicted by second-order perturbation theory extrapolated to the same scales. We also discuss some possible observable effects and future developments.

Figure 1

The power spectra of the three source terms of the vector potential, as extracted from N-body simulations. The solid (red) line is for $\delta \nabla \times \mathbf{v}$, the dot-dashed (blue) line is for $\nabla \delta \times \mathbf{v}$, the dashed (black) line is for $\nabla \times \mathbf{v}$ and the dotted (magenta) lines are the linear and nonlinear matter power spectra for comparison. The power spectra plotted here are given by $P(k)/(f^2{\mathcal{H}}^2{(2\pi )}^3)$, where $\mathcal{H}$ is the conformal time Hubble constant and $f=d\ln D/d\ln a$ is the logarithmic derivative of the linear growth factor $D$, see text and [25].

Figure 2

The power spectra ${\mathcal{P}}_{\mathrm{\Phi }}$ and ${\mathcal{P}}_{{\mathbf{B}}^N}$ of the Newtonian scalar potential (dashed red line) and the frame-dragging vector potential (solid blue line) as a function of scale, as extracted from N-body simulations. The dotted (black) line is the power spectrum of the linear theory scalar potential for comparison.

Figure 3

The ratio of the power spectra ${\mathcal{P}}_{{\mathbf{B}}^N}$ and ${\mathcal{P}}_{\mathrm{\Phi }}$ of the gravitomagnetic vector potential and the Newtonian scalar potential as a function of scale, as extracted from N-body simulations.