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Newtonian kinematical backreaction in cosmological $N$-body simulations with Delaunay Tesselation: "zero test" and scale dependence

The backreaction of inhomogeneities describes the effect of inhomogeneous structure on average properties of the Universe. We investigate this approach by testing the consistency of cosmological $N$-body simulations as non-linear structure evolves. Using the Delaunay Tessellation Field Estimator (DTFE), we calculate the kinematical backreaction $Q$ from simulations on different scales in order to measure how much $N$-body simulations should be corrected for this effect. This is the first step towards creating fully relativistic and inhomogeneous $N$-body simulations. In this paper we compare the interpolation techniques available in DTFE and illustrate the statistical dependence of $Q$ as a function of length scale.