We begin with computing the power spectrum of density fluctuations from the Millen- nium Run, to check whether we reproduce the KFT predictions with the same precision as in Bartelmann et al. (2016).
7Unlike the rest of the thesis, in (4.4) and (4.5) I explicitly indicated the fields in Fourier space with
Figure 4.1: Dimensionless power spectrum of matter density fluctuations at redshift
z = 0, obtained from the Millennium Run in this work (orange solid line) and in
Springel et al. 2005b (magenta dashed line). The shaded areas around the solid orange and dashed magenta lines represent the respective scatter due to cosmic variance. Our code reproduces Springel (2005) for k < 2 h cMpc. The differences at larger k are due to the different algorithms implemented (see text for details).
First of all, we build the density fluctuation field from the Millennium Run by binning the dark matter particles into a regular Cartesian grid, through a CIC scheme. For each simulation, we set the size of the grid cell equal to the mean interparticle separation, i.e. 231 h−1kpc. This ensures converge of the power spectra up to k ∼ 3 h cMpc−1. The detailed convergence tests are presented in the Appendix G.
To compute the power spectrum, we start by fast-Fourier transforming the density fluc- tuation field. We then consider a set of spherical surfaces in Fourier space, the radii of which are multiples of 2π/L, where L is the box size.8 The radii are the values of the wavenumbers ksat which the power spectrum will be computed. The amplitude of each
Fourier mode δ(k) is split between the two closest Fourier-space spheres, weighted by the Fourier-space distance from such spheres. The power spectrum P (ks) corresponding
to the Fourier mode ksis then given by the average of the square modulus of all weighted
amplitudes assigned to the sphere with radius ks.
We show the dimensionless power spectrum of density fluctuations k3P (k)/(2π2) at
z = 0 that we obtain from the Millennium Run in Figure 4.1 (orange solid line). The
shot noise Psn = (L/N )3, where N is the number of grid points along each side of the
box, is subtracted from the power spectrum P (k). The orange shaded area indicates the scatter due to cosmic variance. The scatter ΔP (k) of the power spectrum at each mode
8The radius of the largest sphere in Fourier space is set to be equal to the Nyquist frequency of the
Figure 4.2: Left panel : Power spectrum of matter density fluctuations at redshift
z = 0, predicted by KFT (dashed blue line) and by the Millennium Run (solid orange
line). The orange shaded areas represents the error due to cosmic variance in the Millennium Run. The vertical dotted orange line is the Nyquist mode in the grid used to compute the power spectrum (see text for details). Right panel : relative difference between the power spectrum of matter density fluctuations predicted by KFT and by the Millennium Run. The differences are consistent with the findings in Bartelmann et al. 2016 (see text for details).
k is computed as (Schneider et al., 2016)
ΔP (k) = 2 ΔNm 1 2 [P (k) + Psn(k)] , (4.9)
where ΔNm = L3k2Δk/(2π2) is the number of modes within the bin centered in k,
with width Δk. To check whether our code is reliable, we overplot in Fig. 4.1 the power spectrum of the Millennium Run at z = 0 as published in Springel (2005), which is shown with a magenta dashed line. The magenta shaded area indicates the cosmic variance as estimated in Springel (2005).
Our code perfectly resembles Springel (2005) results for 0.06h−1cMpc < k < 2h−1cMpc, and is consistent with them for k < 0.06h−1cMpc. The differences at k > 2h−1cMpc arise from the different algorithms used to compute the power spectrum. In our case we computed it on a regular Cartesian grid, treating all Fourier modes in the same way. Instead, Springel (2005) measured the power spectrum in the Millennium Run on the fly, distinguishing between “large-scale modes” and “small-scale modes”. For the large-scale modes, the power spectrum was computed through a Fourier transform of the entire box. The small-scale modes were obtained by self-folding the density field, and assuming periodicity in a fraction of the box, following Jenkins et al. (1998). The algorithm implemented in Springel (2005) is equivalent to computing the power spectrum on a regular 819203 mesh. In conclusion, our code accurately resembles the power spectrum of matter density fluctuations in the Millennium Run up to k ∼ 2h−1cMpc. The differences at larger k are due to the different algorithms implemented.
Having verified that our code for computing the power spectrum is reliable, we now compare it with the power spectrum of density fluctuations predicted by KFT, choosing the same cosmological model as the Millennium Run. In the left panel of Figure 4.2 we plot the power spectrum predicted by the Millennium Run and KFT with the solid orange and dashed blue lines, respectively. The shaded orange area indicates the scatter due to cosmic variance in the Millennium Run, estimated with (4.9). In the right panel of Figure 4.2 we show the relative di↵erence between the power spectra predicted by KFT and the Millennium Run (solid red line). The red shaded area is the scatter in the relative di↵erence resulting from propagating the error due to cosmic variance of the Millennium Run. We investigate only modes up to 3 h cMpc 1, because at larger k the convergence of our computation of the power spectrum of the Millennium Run is worse than 5%. This is due to the resolution of the grid onto which we CIC-binned the dark matter density upon computing the power spectrum (see also the Appendix G).
The predictions of KFT are overall in good agreement with the Millennium Run, in fact even better than what found by Bartelmann et al. (2016) in comparison with the Coyote emulator (Heitmann et al., 2009, 2014, 2010, Lawrence et al., 2010). In the range 0.1 h cMpc 1 < k < 3 h cMpc 1, the relative di↵erence remains below 10%. In particular, for 0.2 h cMpc 1 < k < 0.5 h cMpc 1, well beyond the breaking point of perturbation theory (Taruya et al., 2012), the accuracy is ⇠ 3%. On the other hand, for k < 0.1 h cMpc 1 the di↵erence can be as high as 30%. However, in this regime, the scatter due to cosmic variance is too large to draw robust conclusions. Indeed, the lower bound of the red shaded area in the right panel of Figure 4.2 suggests that the di↵erence may be as small as 10%. To get more insight in this regime, it would be necessary to consider a larger simulation (e.g., the Millennium XXL Run Angulo et al., 2012), in order to reduce the uncertainty due to cosmic variance on the scales probed by the Millennium Run. This would also enable us to probe modes at larger scales (3 h 1cGpc).
At this point, we compute the power spectrum of the momentum density fluctuations in the Millennium Run, using the same codes adopted for the power spectrum of matter density fluctuations. Specifically, in Figure 4.3 we show, from top to bottom, the power spectra of the kinetic energy density fluctuations, and of the divergence and curl of momentum density fluctuations, respectively, obtained from the Millennium Run. All power spectra are computed at redshift z = 0. In all panels, the shaded orange area is again the cosmic variance computed as in (4.9). Unfortunately, the corresponding predictions of KFT are not available yet, as the implementation of the theory into a symbolic code to generate the relevant power spectra is still in progress.
Figure 4.3: Power spectrum of kinetic energy density fluctuations, divergence and curl of momentum density fluctuations (top, middle, and bottom panels, respectively), predicted by the Millennium Run (solid orange line). All power spectra at computed at redshift z = 0. In all panels, the orange shaded area represents the error due to cosmic variance in the Millennium Run, while the vertical dotted orange line marks the Nyquist mode in the grid used to compute the power spectra (see text for details).