Abstract. We present a new N-body code, gevolution, for the evolution of large scale structure in the Universe. Our code is based on a weak field expansion of General Relativity and calculates all six metric degrees of freedom in Poisson gauge. N-body particles are evolved by solving the geodesic equation which we write in terms of a canonical momentum such that it remains valid also for relativistic particles. We validate the code by considering the Schwarzschild solution and, in the Newtonian limit, by comparing with the Newtonian N-body codes Gadget-2 and RAMSES. We then proceed with a simulation of large scale structure in a Universe with massive neutrinos where we study the gravitational slip induced by the neutrino shear stress. The code can be extended to include different kinds of dark energy or modified gravity models and going beyond the usually adopted quasi-static approximation. Our code is publicly available.

Numerical simulations are a versatile tool providing insight into the complicated process of structure formation in cosmology [1]. This process is mainly governed by gravity, which is the dominant force on large scales. To date, a century after the formulation of general relativity [2], numerical codes for structure formation still employ Newton's law of gravitation. This approximation relies on the two assumptions that gravitational fields are weak and that they are only sourced by non-relativistic matter. While the former appears well justified on cosmological scales, the latter imposes restrictions on the nature of the "dark" components of the Universe (dark matter and dark energy) which are, however, poorly understood. Here we present the first simulations of cosmic structure formation using equations consistently derived from general relativity. We study in detail the small relativistic effects for a standard ΛCDM cosmology which cannot be obtained within a purely Newtonian framework. Our particle-mesh N-body code computes all six degrees of freedom of the metric and consistently solves the geodesic equation for particles, taking into account the relativistic potentials and the frame-dragging force. This conceptually clean approach is very general and can be applied to various settings where the Newtonian approximation fails or becomes inaccurate, ranging from simulations of models with dynamical dark energy [3] or warm/hot dark matter [4] to core collapse supernova explosions [5].The applicability of Newton's law of gravitation in the context of cosmic structure formation has been discussed extensively in the recent literature [6,7,8]. In particular, it is now well understood that this simplified description is quite accurate when applied within standard ΛCDM cosmology where perturbations come entirely from non-relativistic matter. However, the situation is not satisfactory for two reasons. Firstly, the quality of observational data is rapidly increasing, and upcoming galaxy surveys will eventually reach a precision where a naive treatment of the effects of spacetime geometry becomes insufficient [9,10]. Secondly, the true nature of dark matter and dark energy is not yet established. In order to study models beyond ΛCDM, some of which may feature relativistic sources of stress-energy (e.g. warm dark matter or dynamical dark energy), employing the Newtonian approximation is not always justified. A number of numerical codes has been developed for particular models [11,12,13,14], yet a general framework would be desirable. Furthermore, Newtonian gravity is acausal and not sensitive to the presence of a cosmological horizon. Even if a judicious interpretation of the output of Newtonian simulations can cure this problem at the linear level, it comes back when one goes beyond linear perturbation theory and it would be preferable to use the correct physics from the outset.Moving from the absolute space and time of the Newtonian picture towards a general relativistic view where geometry is dynamical poses ...

We investigate cosmic string networks in the Abelian Higgs model using data from a campaign of large-scale numerical simulations on lattices of up to 4096 3 grid points. We observe scaling or selfsimilarity of the networks over a wide range of scales, and estimate the asymptotic values of the mean string separation in horizon length unitsξ and of the mean square string velocityv 2 in the continuum and large time limits. The scaling occurs because the strings lose energy into classical radiation of the scalar and gauge fields of the Abelian Higgs model. We quantify the energy loss with a dimensionless radiative efficiency parameter, and show that it does not vary significantly with lattice spacing or string separation. This implies that the radiative energy loss underlying the scaling behaviour is not a lattice artefact, and justifies the extrapolation of measured network properties to large times for computations of cosmological perturbations. We also show that the core growth method, which increases the defect core width with time to extend the dynamic range of simulations, does not introduce significant systematic error. We compareξ andv 2 to values measured in simulations using the Nambu-Goto approximation, finding that the latter underestimate the mean string separation by about 25%, and overestimatev 2 by about 10%. The scaling of the string separation implies that string loops decay by the emission of massive radiation within a Hubble time in field theory simulations, in contrast to the Nambu-Goto scenario which neglects this energy loss mechanism. String loops surviving for only one Hubble time emit much less gravitational radiation than in the Nambu-Goto scenario, and are consequently subject to much weaker gravitational wave constraints on their tension.