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tokens: the strongest deviations growing at the particle Nyquist wavenumber k Ny (indicated by a dotted vertical line) – and linear perturbation theory is not recovered even on the largest scales. While we show results only for 1LPT and at starting time z start = 49, this result is almost independent of the used LPT order. Since the error is driven by a spurious growing mode, its amplitude is to first order simply determined by the starting time – with earlier starts leading to a larger error. It is also insensitive to the specific choice of softening length, as long as it is appreciably smaller than the mean particle separations. Once the small scales have shell-crossed and collapsed, at z ≲ 2.5, the spurious growth is slowed. Figure 6. Evolution of the baryon–CDM density difference power spectrum P bc in a gravity-only N -body simulations, where both baryon and CDM are simulated with N -body particles with small softening. Simulation results in solid lines (faint dashed lines) with 2 × 512 3 (2 × 1024 3 ) particles initialized at z start = 49 ( z start = 24) using 1LPT (3LPT) ICs, by suitably backscaling the amplitudes from Class at the reference redshift z ref = 2.125. Those backscaled power spectra are shown as black dashed lines, before applying the particle realization. The vertical dotted line indicates the particle Nyquist wavenumber. Top panel: Simulation results where is absorbed into particle displacements (using equation 14 ). We observe the well-known spurious growing mode due to discreteness errors in the particle discretization of the fluids (the discrete δ bc is not compensated at the discretization scale, if not sufficiently smoothed by gravitational softening, and therefore grows). Bottom panel: Same as above but now is absorbed into perturbed particle masses (using equation 15 ). Here, discretization errors are strongly suppressed, and the late-time power asymptotes to a constant noise spectrum beyond k Ny . These errors are furthermore vastly reduced when 2 × 1024 3 particles and 3LPT ICs are used. In stark contrast, the situation improves dramatically when the initial density perturbation is incorporated by perturbing the initial particle masses (using equation 15 ), instead of adding initial displacements. This result is shown in the lower panel of Fig. 6 in solid lines with the same particle load and resolution as above, while the results from the higher resolution simulation (2 × 1024 3 particles, 3LPT with z start = 24) are shown as faint dashed lines. In this case, P bc is exactly constant on large scales and exhibits non-linear growth at intermediate scales, as expected. On the smallest scales, we observe that in all cases the solution asymptotes to a constant power spectrum for k ≳ k Ny with a scaling inversely proportional to the particle number (as is expected for shot noise in the power spectrum, e.g. Colombi et al. 2009 ). We expect that this behaviour close to the particle Nyquist wavenumber is thus a combination of