N -body methods for relativistic cosmology

Abstract

Abstract. We present a framework for general relativistic N -body simulations in the regime of weak gravitational fields. In this approach, Einstein's equations are expanded in terms of metric perturbations about a Friedmann-Lemaître background, which are assumed to remain small. The metric perturbations themselves are only kept to linear order, but we keep their first spatial derivatives to second order and treat their second spatial derivatives as well as sources of stress-energy fully nonperturbatively. The evolution of matter is modelled by an N -body ensemble which can consist of free-streaming nonrelativistic (e.g. cold dark matter) or relativistic particle species (e.g. cosmic neutrinos), but the framework is fully general and also allows for other sources of stress-energy, in particular additional relativistic sources like modifiedgravity models or topological defects. We compare our method with the traditional Newtonian approach and argue that relativistic methods are conceptually more robust and flexible, at the cost of a moderate increase of numerical difficulty. However, for a ΛCDM cosmology, where nonrelativistic matter is the only source of perturbations, the relativistic corrections are expected to be small. We quantify this statement by extracting post-Newtonian estimates from Newtonian N -body simulations.PACS numbers: 98.80. Jk, 04.25.Nx

First-Order Cosmological Perturbations Engendered by Point-Like Masses

“…Here attention is concentrated solely on the perturbations with non-negligible sources, and cosmological gravitational waves are not investigated. Similarly to, e.g., Adamek et al (2013Adamek et al ( , 2014, it is demanded that…”

“…Both approaches make great progress in describing the inhomogeneous world within the limits of their applicability. Nevertheless, the linear cosmological perturbations theory certainly fails in describing nonlinear dynamics at small distances, while Newtonian simulations do not take into account relativistic effects becoming non-negligible at large distances (Green & Wald 2012;Adamek et al 2013Adamek et al , 2014Milillo et al 2015). In this connection, in the latter case certain effort is required for extracting relativistic features from the large-scale Newtonian description (Chisari & Zaldarriaga 2011;Fidler et al 2015;Hahn & Paranjape 2016).…”

“…First, the generalization of the wellknown nonrelativistic post-Minkowski formalism (Landau & Lifshitz 2000) to the cosmological case in the form of the relativistic post-Friedmann formalism, which would be valid on all scales and include the full nonlinearity of Newtonian gravity at small distances, has been made in the recent paper by Milillo et al (2015), but the authors resorted to expansion of the metric in powers of the parameter 1/c (the inverse speed of light). Second, the formalism for relativistic N -body simulations in the weak field regime, suitable for cosmological applications, has been developed by Adamek et al (2013), but the authors gave different orders of smallness to the metric corrections and to their spatial derivatives (a similar "dictionary" can be found in (Green & Wald 2012;Adamek et al 2014)). …”

Estimating the weak-lensing rotation signal in radio cosmic shear surveys

“…In a ΛCDM cosmology, on linear scales, the vectors and tensors are not generated at first order. Furthermore, even on nonlinear scales in ΛCDM, the vector and tensor perturbations are negligible (Bruni et al 2014;Adamek et al 2014). At second order however, the scalar perturbations can generate rotation, through the term in the deflection angle that corresponds to the Born correction and lens-lens coupling (see Dodelson et al 2005;Krause & Hirata 2010;Thomas et al 2015).…”

Interactive mixture of inhomogeneous dark fluids driven by dark energy: a dynamical system analysis

“…Ideally, fully relativistic inhomogeneous DE and CDM interacting sources should be examined through high power numerical relativistic codes (whether assuming a continuous modelling or N-body simulations). However, since the latter codes are in their early development stages [22][23][24][25], we can resort to more idealised (yet still relativistic and non-perturbative) description by means of inhomogeneous exact solutions of Einstein's equations.…”