Issue 
A&A
Volume 556, August 2013



Article Number  A70  
Number of page(s)  16  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201321718  
Published online  30 July 2013 
Online material
Appendix A: Equivalence of the simulation methods
In this appendix, we want to prove analytically that the two simulation methods mentioned in are in fact equivalent.
The established method calculates the correlation function components ξ_{m}, m = 0,..., N − 1 directly from the field components g_{n}, n = 0,..., N − 1. Since we impose periodic boundary conditions on the field, this can be done using the estimator (A.1)The realspace field components are calculated from the Fourier components by (A.2)Here, we set , which is equivalent to the field having zero mean in real space. Discretization and periodicity already imply – still, in order not to give double weight to this mode, we set it to zero as well. Of course, we then have to do the same in our new method, i.e. P_{N/2} ≡ 0. Note that for the sake of readability, we still include this term in the formulae of this work.
Our new method draws a realization of the power spectrum and Fourier transforms it to obtain the correlation function: (A.3)In both methods, the variance of the field components in Fourier space is determined by the power spectrum, namely, for one realization, (A.4)To prove the equivalence of the two methods, we insert the Fourier transforms as given in Eq. (A.2) into the estimator, Eq. (A.1): (A.5)The sum over n simply gives a Kronecker δ, since (A.6)Thus, we obtain (A.7)where in the last step, we used the fact that the g_{i} are real, implying . In order to show that this is equivalent to Eq. (A.3), we now split the sum into two parts, omitting the zero term (since , as we explained before): (A.8)Inserting Eq. (A.4) we end up with (A.9)This is exactly the way we calculate ξ_{m} in our new method – thus, we proved analytically that the two methods are indeed equivalent. As mentioned before, we also confirmed this fact numerically.
Appendix B: Analytical calculation of the ξ_{0}dependence of mean and covariance matrix
As mentioned in , our analytical calculation of the ξ_{0}dependence of the mean and the covariance matrix does not produce practically usable results – nonetheless, it is interesting from a theoretical point of view and is thus presented in this appendix.
We are ultimately interested in the mean y and the covariance matrix C_{y}, however, we will first show calculations in ξspace before addressing the problem of how to transform the results to yspace.
Fig. B.1
Mean of ξ_{n} for different n as function of ξ_{0}, determined from simulations (black points with error bars) and analytically to zeroth (red crosses), first (blue circles), second (green filled triangles; left panel only), third (purple empty triangles; left panel only), and tenth (brown squares) order. 

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Appendix B.1: Calculation in ξspace
The ξ_{0}dependence of the mean ⟨ ξ_{1} ⟩ (where the index is purely a numbering and does not denote the lag) can be computed as (B.1)with the conditional probability (B.2)We define the corresponding characteristic function as Fourier transform of the probability distribution (Φ ↔ p in shorthand notation; for details on characteristic functions see Keitel & Schneider 2011, hereafter KS2011, and references therein): Making use of the characteristic function Ψ(s_{0},s_{1}) (where Ψ(s_{0},s_{1}) ↔ p(ξ_{0},ξ_{1})) computed in KS2011, we can also write (B.5)Comparison with Eq. (B.3) yields (B.6)Now, we can calculate the mean (i.e. the first moment) from the characteristic function (equivalent to Eq. (B.1) – again, see KS2011 and references there): Using the result from KS2011 for the bivariate characteristic function, (B.9)where , we can calculate the derivative as (B.10)where we inserted the univariate characteristic function computed in KS2011, (B.11)To calculate the integral in Eq. (B.7), we use a Taylor expansion of Y_{n}(s_{0}) from Eq. (B.10): (B.12)We insert the derivative into Eq. (B.7) and thus obtain (B.13)According to the definition of Ψ(s_{0}) ↔ p(ξ_{0}), (B.14)and thus, after changing the order of summation and integration, Eq. (B.13) can finally be written as (B.15)Inserting the known result for p(ξ_{0}) and calculating its derivatives allows us to compare the analytical result to simulations. The results can be seen in ; here, the black points with error bars show the mean of ξ_{n} for different lags n as determined from simulations (100 000 realizations, Gaussian power spectrum with Lk_{0} = 100), and the colored symbols show the analytical results to different order (see figure caption). It seems that, although the Taylor series in Eq. (B.15) does not converge, a truncation at order 10 yields sufficient accuracy, barring some numerical issues for very low ξ_{0}values.
Fig. B.2
Different elements of the covariance matrix C({ ξ_{n} }), determined from simulations (black points with error bars) and analytically to zeroth (red crosses), first (blue circles), fifth (purple triangles), and tenth (brown squares) order. 

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The ξ_{0}dependence of the covariance matrix C_{ξ} can be calculated in a similar way. We start from the general definition of covariance, The integral A can again be expressed in terms of the characteristic function Φ(s_{1},s_{2};ξ_{0}) ↔ p(ξ_{1},ξ_{2}  ξ_{0}): (B.16)Similar to the previous calculations, (B.17)with the trivariate characteristic function (B.18)Calculating the second derivative of B.18 yields (B.19)The Taylor expansions of Z_{n}(s_{0}) and Z_{k,n}(s_{0}) read Using it as well as the expansion (B.12), we finally obtain (B.22)We show a comparison of the results (for different elements of the covariance matrix) from simulations and the analytical formula in . Again, the black dots are obtained from simulations and the colored symbols represent the results from Eq. (B.22), where the last term (i.e. the one containing the mean values ⟨ ξ_{n} ⟩) was calculated up to tenth order, thus providing sufficient accuracy, as previously shown. As before, there are some numerical problems for very small values of ξ_{0}. Additionally, the analytical results do not agree with the simulations for small lags, as can be seen from the leftmost panel (the same holds for other covariance matrix elements involving small lags). However, for the the higherlag examples (i.e. the right two panels), a truncation of the Taylor series at tenth order seems to be accurate enough.
Appendix B.2: Transformation of mean and covariance matrix to yspace
In the previous section, we showed how to calculate the (ξ_{0}dependent) mean and covariance matrix in ξspace. The computation of the quasiGaussian approximation, however, requires the mean y and the covariance matrix C_{y} in yspace, which cannot be obtained from those in ξspace in a trivial way due to the highly nonlinear nature of the transformation ξ → y.
Thus, instead of settling for a linear approximation, we have to choose a more computationally expensive approach. Namely, we calculate the first and second moments (in ξ) of the quasiGaussian distribution as functions of the mean and (inverse) covariance matrix in yspace and equate the result to the analytical results, i.e. we solve a set of equation of the form where we did not write down the ξ_{0}dependence explicitly for the sake of readability.
Note that this is a complicated procedure, since the integration on the equations’ lefthand sides can only be performed
numerically (we make use of a MonteCarlo code from Press et al. 2007). In order to solve the equation set (consisting of equation for an Nvariate distribution) we use a multidimensional rootfinding algorithm (as provided within the GSL, Galassi et al. 2009). However, due to the high dimensionality of the problem, this procedure does not seem practical, since it is computationally very expensive – in addition to that, any possible gain in accuracy is averted by the required heavy use of purely numerical methods.
Thus, as described in , we refrain from using our analytical results for the mean and covariance matrix and simply determine them (as well as their ξ_{0}dependence) from simulations, which we have shown to be sufficiently accurate.
© ESO, 2013
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