Backreaction in Numerical Relativity: Averaging on Newtonian gauge-like hypersurfaces in Einstein Toolkit cosmological simulations

We introduce a spatial averaging scheme and use it to study the evolution of spatial averages in large-scale simulations of cosmological structure formation performed with the Einstein Toolkit. The averages are performed on the spatial hypersurfaces of the simulation setup which, at least initially, represent the hypersurfaces of statistical homogeneity and isotropy. We find only negligible cosmic backreaction on these hypersurfaces even on very small scales, but find significant curvature fluctuations of up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>10</mml:mn> <mml:mi>%</mml:mi> </mml:mrow> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>R</mml:mi> </mml:msub> </mml:math> for sub-volumes with radius <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>200</mml:mn> </mml:mrow> </mml:math> Mpc and even larger fluctuations in smaller sub-volumes. In addition, we quantify fluid flow in and out of these sub-volumes. We find this to be significant, up to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>5</mml:mn> <mml:mi>%</mml:mi> </mml:mrow> </mml:math> change in the density between redshift <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> of a single sphere of radius <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:mn>200</mml:mn> </mml:mrow> </mml:math> Mpc (and larger for smaller spheres). We suggest this may be important for studies basing averages on volumes co-moving with the simulation hypersurfaces.