6.4 Non-Gaussian to Gaussian ratio
6.4.2 Results
Figs. 6.15 and 6.16 show the estimates obtained from different N-body simulations and compare them to the halo model prediction. From the plots one can see that both estimates are of similar magnitude, but differ in the behavior on small scales.
For the Virgo and the Gems simulations the constructed estimator from Eq. (6.17) is too noisy for a reliable comparison. One factor for this might be that the estimator is constructed from two other estimators which suffer from noise effects as well. The VLS and Millennium Run simulations show an increasing amount of non-Gaussianity on small scales contrary to the decreasing slope predicted by the halo model. This behavior is most likely the result of discreteness effects as the amount of increase is very steep. This is presumedly due to three effects: shot noise and force softening (see Sect. 6.1.3) become important for wave-numbers larger than l ' 104. Additionally, aliasing might occur on small scales causing a decrease of the power spectrum estimate [39]. As the power spectrum is inversely proportional to the ratio this can explain the rise of the ratio on small scales. The overall impression is that the halo model prediction and the simulation estimates do not coincide well. Currently, one cannot say whether this is because the halo model misinterprets the non-Gaussian contribution or because the present simulations are not reliable enough on small scales.
10-1 100 101
103 104
l CNG /CG (l)
Virgo HM
10-1 100 101
103 104
l CNG /CG (l)
Gems HM
Figure 6.15: Ratio of non-Gaussian to Gaussian contribution of the convergence power spectrum covariance Rl against wave-number l. The halo model (HM, black line) predicts a reduction of non-Gaussianity on small scales. The Virgo and Gems simulation estimates (red points) are very noisy and do not show a clear behavior.
6.4 Non-Gaussian to Gaussian ratio 165
10-1 100 101 102
103 104
l CNG /CG (l)
VLS HM
10-1 100 101
103 104
l CNG /CG (l)
Millennium Run HM
Figure 6.16: Ratio of non-Gaussian to Gaussian contribution of the convergence power spectrum covariance Rl against wave-number l. The halo model (HM, black line) predicts a reduction of non-Gaussianity on small scales. The VLS and Millennium Run simulation estimates (red points) predict the opposite behavior.
Summary and conclusions
The motivation behind this thesis was to provide an analytical treatment of higher-order correlation functions in the cosmic matter density field and to compare the results obtained with numerical N-body simulations. As the perturbative description holds only for densities of the order|δ| '1, the tool of choice was a semi-analytic halo model which combines results from both perturbation theory and numerical simula-tions. In this work, we focused on the fourth-order correlation function and its Fourier counterpart, the trispectrum, since it allows us to study the non-Gaussianities of the matter field and to calculate the full non-Gaussian covariance of the power spectrum.
This provides a way to estimate the error and mode coupling in the matter power spectrum to higher accuracy than has been previously.
To calculate the trispectrum and, subsequently, the full non-Gaussian covariance of the convergence power spectrum in the halo model approach, we had to combine results from different areas in mathematics, physics and cosmology. We have summarized the most important ones in the first chapters of this thesis. This includes a detailed overview of the standard model of cosmology, perturbation theory and the properties of cosmological random fields. Additionally, Chapter 4 gives a comprehensive overview of the halo model description of dark matter and its ingredients such as the halo mass abundance, halo profile and clustering of halos.
With the halo model at hand, one has a recipe to calculate matter correlation functions of arbitrary order. We used the halo model to find the expectation value both for the three-dimensional and the convergence power spectrum covariance and confirmed the analytical results of Cooray and Hu [16] and Scoccimarro et al. [72]. In order to find a fast way to calculate the full non-Gaussian covariance, we studied the accuracy of different approximations to the complete trispectrum in the halo model approach.
As a result, we found that the combination of 1-halo and 2-halo contributions of the trispectrum yields an error smaller than 10% on intermediate and large scales.
More precisely, in the three-dimensional case this approximation is accurate for k &
0.3hMpc−1 and in the projected case for l & 300 and zs = 1 for our fiducial ΛCDM model. Furthermore, we extended this result to the non-Gaussian contribution of the covariance. The aforementioned approximation yields for the same wave-numbers a comparable error to the total trispectrum. Thus a combination of 1-halo and 2-halo terms of the bin-averaged trispectrum together with the Gaussian contribution of the covariance provides an accurate and efficient approximation to the full non-Gaussian covariance of the matter power spectrum.
Since recent results from numericalN-body simulations suggest that the concentration-mass relation is probabilistic [38], i.e. there is scatter in the concentration parameter for a fixed mass m, we investigated the impact of this on the 1- and 2-halo term contributions to the power spectrum and trispectrum. In the three-dimensional case, we found a significant deviation from a deterministic concentration relation on scales smaller than k '5hMpc−1, which is less pronounced for the power spectrum. Addi-tionally, we were able to show that this effect is also present in the projected versions of the spectra on scales smaller than l ' 2000. In contrast to the three-dimensional case, the increase in the small scale tails of the spectra is diminished, but alters the covariance on small scales up to 12% for reasonable concentration dispersions.
In order to understand how different wave-number modes in the power spectrum co-variance couple, we analyzed the angular dependence of the normalized trispectrum for each halo term. We found the strongest mode coupling in the 4-halo term for colinear configurations of the trispectrum wave-vectors. Towards orthogonal configurations the angular dependence vanishes and there is almost no coupling visible. We discovered a similar but less pronounced behavior for the 3-halo term. The 2-halo and 1-halo terms show no visible angular dependence of the modes. Furthermore, we studied the mode coupling for trispectrum configurations with different wave-vector length. This revealed a dependence on both the length of the wave-vectors and their ratio. The minimum wave-length, in particular, affects the amplitude of the trispectrum.
Studying the non-Gaussian-to-Gaussian ratio of the covariance, we found an increase of non-Gaussianity towards small scales for the three-dimensional case. In the projected case, the non-Gaussian contribution reaches a maximum amplitude around l ' 800 and decreases on small scales.
With the results from the halo model consideration, we developed a fitting formula for the non-Gaussian contribution of the convergence power spectrum covariance. The fitting formula is valid in a range 1000 < l < 5000 and provides on average a 10%
accuracy to the corresponding halo model prediction. For the diagonal of the non-Gaussian covariance we achieve in this way a 15% accuracy.
An additional aspect of this thesis was the comparison of the halo model to results from simulations. For this we calculated the power spectrum and covariance from six different numericalN-body simulations that spanned a wide range of possible designs.
Our comparison with the halo model revealed a good correspondence with the power spectra. However, the convergence power spectrum covariance with a determinis-tic concentration parameter consistently underestimates the covariance estimate from simulations on small scales. This effect is mitigated if we include in the halo model prediction a probabilistic concentration relation with a dispersion σlnc = 0.2−0.3 in the logarithmic halo concentration. Nevertheless, the halo model prediction of the covariance underestimates still the simulation estimate on small scales. This discrep-ancy hints that the halo model description does not reflect well enough the underlying physics of higher-order correlations on small scales. One also has to be aware that the limited size and resolution of simulations can lead to high noise and unreliable
predic-Summary and conclusions 169
tions as well. This was particularly visible in our analysis of the ratio of non-Gaussian-to-Gaussian contributions to the convergence power spectrum covariance. While the halo model predicted a decrease towards small scales, the simulations showed an in-crease. A final conclusion whether the halo model has to be corrected for this will require future simulations of improved quality.
Appendix A
Halo model trispectrum
In Sect. 5 we showed that for the calculation of the power spectrum covariance in the halo model description, we only have to consider parallelogram configurations of the trispectrum wave-vectors. The restriction to these configurations allows us to simplify the expressions for the terms of the trispectrum. In the following, we perform this calculation in detail after summarizing the most important properties of the second order coupling functions.
A.1 Second-order coupling functions
From the recursion relations (2.26) and (2.27) for the n-th order density contrast δn and divergence velocity field θn that solve in a perturbative approach the collision-less Boltzmann equation for an ideal dark matter fluid, we find for the second-order coupling functions the following expressions:
F2(q1,q2) = 5
7α(k1,k2) + 2
7β(k1,k2), (A.1) G2(q1,q2) = 5
7α(k1,k2) + 2
7β(k1,k2), (A.2) where
α(k1,k2)≡ (k1+k2)·k1
k12 and β(k1,k2)≡ k122 (k1·k2)
2k21k22 , (A.3) denote the mode coupling functions (see Sect. 2.4.3). The symmetrized versions of the second-order coupling functions are:
F2(s)(q1,q2) = 5 7 +2
7
(q1·q2)2 q12q22 + 1
2 q1·q2
q1q2 q1
q2 + q2 q1
, (A.4)
G(s)2 (q1,q2) = 3 7 +4
7
(q1·q2)2 q12q22 + 1
2 q1·q2
q1q2 q1
q2 + q2 q1
. (A.5)
Since these expressions play an important role for the subsequent calculation of the trispectrum in the halo model description, we summarize their properties which follow immediately from their definitions [26]:
• F2(s)(q1,−q1) = G(s)2 (q1,−q1) = 0 ,
• F2(s)(q1,q1) = G(s)2 (q1,q1) = 2 ,
• F2(s)(q1,q2) = F2(s)(−q1,−q2) ,
• G(s)2 (q1,q2) = G(s)2 (−q1,−q2) ,
• lim
||→0F2(s)(q1,) = lim
||→0G(s)2 (q1,) = lim
||→0/||2 → ∞.