6.3 Projected power spectrum covariance
6.3.3 Comparing the covariances
In order to determine a more realistic standard error for the covariance mean of the Gems simulation, we produce 50 bootstrap subsamples of size Nboot = 219 which correspond to the size of the original sample. The resulting standard deviation is presented in the lower plot of Fig. 6.9 against the wave-numbers (li, lj). In comparison to the former method where we found covariance estimates from subsets of the available κ-maps (upper plot) the error as determined with the bootstrap method decreased to approximately one-third of the previous values. Along the diagonal entries the mean standard deviation is around 10%, the overall mean standard deviation is 20%. Varying the number of bootstrap subsamples shows that this result is stable.
6.3 Projected power spectrum covariance 155
following, we analyze the figures for the different simulations in detail. The scales considered for each simulation are listed in Tab. 6.3.
Table 6.3: Binning scheme and wave-number ranges of the simulations.
Simulation lmin lmax ∆l
Virgo 720 14400 720
Gems 90 9000 90
Borgani 84 9884 140
VLS 109 10441 211
Grossi 52 5244 88
Millennium Run 144 3672 72
Virgo
Fig. 6.10 compares the halo model prediction of the lensing power spectrum covariance with that from the Virgo simulation (zs = 1) by plotting the relative covariance difference ∆Cij against the wave-numbers (li, lj). The binning scheme is linear with
∆l = 720 going from l0 = 720 to l19 = 14400. The upper plots illustrate the relative error for a deterministic relation between halo concentration and mass as described in Sect. 5.4. The left panel shows the full covariance, whereas the right panel shows a zoom of the lower right corner up to l5 = 4320. On small scales, for wave-numbers larger than l10 = 7920, the halo model underestimates the results from the simulation up to 70%−80%, whereas on large scales, for wave-numbers smaller thanl3 = 2160, it overestimates the simulation results by about 50%. The best agreement is achieved on intermediate scales for l between 2200 and 4500 with a relative difference smaller or equal to 30%. The lower plots of Fig. 6.10 show that this can be improved significantly by considering a dispersion in the concentration parameter as defined in Eq. (5.49).
Since we showed in Sect. 6.2.2 that the effect of a concentration dispersion is rather small for the projected power spectrum on the scales considered, we neglect this effect here and in the following in the Gaussian contribution of the covariance. For the halo model prediction of the covariance in the left panel we chose σlnc = 0.2; for the right panel we usedσlnc = 0.3. In both cases the relative difference between the covariances is lowered significantly, in particular along the borders. For σlnc = 0.3 the region with a deviation smaller than 30% extends up to l = 104. For wave-numbers smaller than l ' 5000 the error becomes 20% or less, which is within the uncertainty of the simulation. We obtain very similar results for sources at a redshift zs = 2, as shown in Fig. 6.11.
Gems
In Fig. 6.12 we illustrate the relative covariance difference between the halo model prediction and the Gems simulation (zs = 1). In contrast to the Virgo simulation the binning is much finer with a bin-width ∆l = 90. The upper plots display the halo model prediction for the deterministic concentration-mass relation. The left panel shows the complete range of bins, whereas the right panel shows a zoom of the region withl < 1800. Again the simulation covariance is underestimated by the halo model up to 80% for wave-numbers larger than l '6300. The best agreement with ∆Cij <30%
is found on large scales, where l < 1000, and along the diagonal, where ∆Cij <50%.
Including a scatter in the concentration of σlnc = 0.2 (lower left panel) and σlnc = 0.3 (lower right panel) decreases the relative difference significantly. In case of a concentration dispersionσlnc = 0.3 the region up tol '3600 differs about 50% or less from the simulation results. In contrast to the Virgo simulation the halo model never overestimates the simulation.
Borgani
Fig. 6.13 displays the relative covariance difference between the halo model prediction and the Borgani simulation (zs = 1). The comparison covers a range from l0 = 84 to l70 = 8840 using a bin-width ∆l = 140. As one can see, this simulation is strongly affected by noise around the covariance border. The reason for this is twofold: all small wave-numbers have a large sampling variance and this simulation has only Nmap = 60 κ-maps to average over. The foregoing two simulations had both around Nmap ≈ 200 and thus much fewer problems with noise. However, the overall impression of the quantitative analysis is very similar. The best agreement between theoretical prediction and simulation is along the diagonal entries and for wave-numbers smaller than l ≈ 3000. In this regions the deviation is approximately 50%. Considering an additional scatter in the concentration parameter improves the result only slightly.
Millennium Run
In Fig. 6.14 which depicts the quantitative difference of halo model to Millennium Run simulation covariance the effect of noise is even more severe: The first eight bins are too noisy to make a quantitative analysis. This is due to the finer binning with
∆l = 72 and even more importantly due to the very small number ofκ-maps available which amounts to Nmap = 20. As a result the resemblance from theory to simulation is seldom better than 50%.
6.3 Projected power spectrum covariance 157
lj bin
0 2 4 6 8 10 12 14 16 18 0
2 4 6 8 10 12 14 16 18
0 1 2 3 4 5
li bin lj bin
0 2 4 6 8 10 12 14 16 18 0
2 4 6 8 10 12 14 16 18
-1 -0.5 0 0.5 1
li bin
0 2 4 6 8 10 12 14 16 18
Figure 6.10: Relative error ∆Cij of the theoretical halo model prediction for the projected power spectrum covariance in comparison with the results from the Virgo simulation (zs= 1) against wave-numbers (li, lj). The binning scheme is linear with ∆l= 720 going from l0 = 720 to l19 = 14400. The upper plots illustrate the relative error as found when considering the 1-halo, 2-halo and Gaussian term with σlnc = 0 for the halo model. Left and right plot differ only in the number of considered bins. In the lower left panel, we consider σlnc = 0.2 and the lower right panel displays a varianceσlnc= 0.3.
lj bin
0 2 4 6 8 10 12 14 16 18 0
2 4 6 8 10 12 14 16 18
0 1 2 3 4 5
li bin lj bin
0 2 4 6 8 10 12 14 16 18 0
2 4 6 8 10 12 14 16 18
-1 -0.5 0 0.5 1
li bin
0 2 4 6 8 10 12 14 16 18
Figure 6.11: Relative error ∆Cij of the theoretical halo model prediction for the projected power spectrum covariance in comparison with the results from the Virgo simulation (zs= 2) against wave-numbers (li, lj). The binning scheme is linear with ∆l= 720 going from l0 = 720 to l19 = 14400. The upper plots illustrate the relative error as found when considering the 1-halo, 2-halo and Gaussian term with σlnc = 0 for the halo model. The right plot is a zoom into the left plot. In the lower left panel, we consider σlnc = 0.2 and the lower right panel displays a variance σlnc = 0.3.
6.3 Projected power spectrum covariance 159
lj bin
0 10 20 30 40 50 60 70 80 90 0
10 20 30 40 50 60 70 80 90
0 5 10 15 20
li bin lj bin
0 10 20 30 40 50 60 70 80 90 0
10 20 30 40 50 60 70 80 90
-1 -0.5 0 0.5 1
li bin
0 10 20 30 40 50 60 70 80 90
Figure 6.12: Relative error ∆Cij of the theoretical halo model prediction for the projected power spectrum covariance in comparison with the results from the Gems simulation (zs= 1) against wave-numbers (li, lj). The binning scheme is linear with ∆l= 90going froml0 = 90 to l99 = 9000. The upper plots illustrate the relative error as found when considering the 1-halo, 2-halo and Gaussian term withσlnc = 0 for the halo model. The right plot is a zoom into the left plot. In the lower left panel, we consider σlnc = 0.2 and the lower right panel displays a variance σlnc = 0.4.
lj bin
0 10 20 30 40 50 60 70 0
10 20 30 40 50 60 70
5 10 15 20 25
li bin lj bin
0 10 20 30 40 50 60 70 0
10 20 30 40 50 60 70
-1 -0.5 0 0.5 1
li bin
0 10 20 30 40 50 60 70
Figure 6.13: Relative error ∆Cij of the theoretical halo model prediction for the projected power spectrum covariance in comparison with the results from the Borgani simulation (zs= 1) against wave-numbers (li, lj). The binning scheme is linear with ∆l = 140 going from l0 = 84tol70= 9884. The upper plots illustrate the relative error as found when considering the 1-halo, 2-halo and Gaussian term with σlnc = 0 for the halo model. The right plot is a zoom into the left plot. In the lower left panel, we consider σlnc = 0.2 and the lower right panel displays a variance σlnc = 0.3. The simulation shows a lot of noise close to the covariance borders.
6.3 Projected power spectrum covariance 161
lj bin
0 5 10 15 20 25 30 35 40 45 0
5 10 15 20 25 30 35 40 45
10 15 20 25 30
li bin lj bin
0 5 10 15 20 25 30 35 40 45 0
5 10 15 20 25 30 35 40 45
-1 -0.5 0 0.5 1
li bin
0 5 10 15 20 25 30 35 40 45
Figure 6.14: Relative error ∆Cij of the theoretical halo model prediction for the projected power spectrum covariance in comparison with the results from the Millennium Run simula-tion (zs = 1) against wave-numbers (li, lj). The binning scheme is linear with∆l= 72going from l0 = 144 to l49 = 3672. The upper plots illustrate the relative error as found when considering the 1-halo, 2-halo and Gaussian term with σlnc = 0 for the halo model. The right plot is a zoom into the left plot. In the lower left panel, we considerσlnc = 0.2 and the lower right panel displays a variance σlnc = 0.4. The simulation shows a lot of noise close to the covariance borders.