Relativistic corrections to the evolution of structure can be used to test general relativity on cosmological scales. They are also a well-known systematic contamination in the search for a primordial non-Gaussian signal. We present a numerical framework to generate RELativistic second-order Initial Conditions (RELIC) based on a generic (not necessarily separable) second-order kernel for the density perturbations. In order to keep the time complexity manageable we introduce a scale cut that separates long and short scales, and neglect the “short-short” coupling that will eventually be swamped by uncontrollable higher-order effects. To test our approach, we use the second-order Einstein-Boltzmann code SONG to provide the numerical second-order kernel in a ΛCDM model, and we demonstrate that the realisations generated by RELIC reproduce the bispectra well whenever at least one of the scales is a “long” mode. We then present a generic algorithm that takes a perturbed density field as an input and provides particle initial data that matches this input to arbitrary order in perturbations for a given particle-mesh scheme. We implement this algorithm in the relativistic N-body code gevolution to demonstrate how our framework can be used to set precise initial conditions for cosmological simulations of large-scale structure.

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We study the information content of summary statistics built from the multi-scale topology of large-scale structures on primordial non-Gaussianity of the local and equilateral type. We use halo catalogs generated from numerical N-body simulations of the Universe on large scales as a proxy for observed galaxies. Besides calculating the Fisher matrix for halos in real space, we also check more realistic scenarios in redshift space. Without needing to take a distant observer approximation, we place the observer on a corner of the box. We also add redshift errors mimicking spectroscopic and photometric samples. We perform several tests to assess the reliability of our Fisher matrix, including the Gaussianity of our summary statistics and convergence. We find that the marginalized 1-σ uncertainties in redshift space are ∆f loc NL ∼ 16 and ∆f equi NL ∼ 41 on a survey volume of 1 (Gpc/h) 3 . These constraints are weakly affected by redshift errors. We close by speculating as to how this approach can be made robust against small-scale uncertainties by exploiting (non)locality.

We study the information content of summary statistics built from the multi-scale topology of large-scale structures on primordial non-Gaussianity of the local and equilateral type. We use halo catalogs generated from numerical N-body simulations of the Universe on large scales as a proxy for observed galaxies. Besides calculating the Fisher matrix for halos in real space, we also check more realistic scenarios in redshift space. Without needing to take a distant observer approximation, we place the observer on a corner of the box. We also add redshift errors mimicking spectroscopic and photometric samples. We perform several tests to assess the reliability of our Fisher matrix, including the Gaussianity of our summary statistics and convergence. We find that the marginalized 1-σ uncertainties in redshift space are Δf NL loc ∼ 16 and Δf NL equi ∼ 41 on a survey volume of 1 (Gpc/h)3. These constraints are weakly affected by redshift errors. We close by speculating as to how this approach can be made robust against small-scale uncertainties by exploiting (non)locality.

We write down the Lagrangian bias expansion in general relativity up to 4th order in terms of operators describing the curvature of an early-time hypersurface for comoving observers. They can be easily expanded in synchronous or comoving gauges. This is necessary for the computation of the one-loop halo bispectrum, where relativistic effects can be degenerate with a primordial non-Gaussian signal. Since the bispectrum couples scales, an accurate prediction of the squeezed limit behavior needs to be both non-linear and relativistic. We then evolve the Lagrangian bias operators in time in comoving gauge, obtaining non-local operators analogous to what is known in the Newtonian limit. Finally, we show how to renormalize the bias expansion at an arbitrary time and find that this is crucial in order to cancel unphysical 1/k2 divergences in the large-scale power spectrum and bispectrum that could be mistaken for a contamination to the non-Gaussian signal.