Issue 
A&A
Volume 559, November 2013



Article Number  A78  
Number of page(s)  18  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201322295  
Published online  18 November 2013 
Online material
Appendix A: Mexican Hat versus FFT spectrum
Since the data cube is non periodic, computing the power spectrum via FFTs is in principle inconsistent. As a consequence, the unresolved large scale power can leak into the available frequency range, distorting the spectrum. We use thus a modified Δvariance method, known as “Mexican Hat” filtering (MH; cf. Arévalo et al. 2012). For each spatial scale σ, the method consists of three steps:
 1.
the realspace cube C is convolved with two Gaussian filters having slightly different smoothing lengths: and , where ϵ ≪ 1;
 2.
the difference of the two cubes is computed, resulting in a cube dominated by the fluctuations at scales ≈σ (the difference of two Gaussian filters is simply the Mexican Hat filter, F(x) ∝ ϵ [1 − x^{2}/σ^{2}] exp [ − x^{2}/2σ^{2}], characterized by a positive core and negative wings);
 3.
the variance V_{σ} of the previous cube is calculated and recast into the estimate of the power, knowing that
Fig. A.1
Comparison of the characteristic amplitude spectra (for the run with M ~ 0.5 and f = 10^{2}), computed with two different methods: Mexican Hat filtering (black) and fast Fourier transforms (blue). The retrieved spectrum is consistent in both cases, without major differences. 

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In Fig. A.1, we show the comparison between the MH and FFT spectrum, for the run with M ~ 0.5 and f ~ 10^{2}. In our study, there is no dramatic difference between the two methods. The slope in the inertial range is almost identical. At very small scales, the FFT spectrum produces a characteristic hook, in part due to the numerical noise near the maximum resolution, but also due to the contamination of jumps at the nonperiodic boundaries. The MH spectrum shows instead a gentle decline. In the opposite regime, the MH filter tends to smooth the scales greater than the injection scale, while the FFT spectrum shows a steeper decrease. The FFT peak is slightly higher, typically by 2−3 percent, likely affected by the nonperiodic box. Progressively trimming the box increases the relative normalization of the FFT spectrum, even by 20 percent, while distorting the lowfrequency slope; the MH spectrum is instead unaltered.
Appendix B: βprofile in Fourier space
Fig. B.1
Analytic 1D power spectra: βprofile (red), Kolmogorov noise (blue), βprofile perturbed by the noise (black; ). The spectrum is normalized to the value at k_{0} = 1/L = 0.01 (dimensionless units; 2π is dropped for clarity). The core radius is r_{c} = 20, i.e. L/5. The relative amplitude of the noise is ~10 percent. The noise clearly emerges beyond the core radius (k > 0.05), regardless of largescale structures. 

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We present here the analytic conversion of the βprofile to Fourier space, and its interplay with a powerlaw noise. Using the notation , the Fourier transform of the βprofile (Eq. (1)) results to be (B.1)where ξ ≡ 3β/2, K_{1/2 − ξ} is a modified Bessel function of the second kind, and Γ is the Gamma function. The (1D) power spectrum is as usual retrieved as . Assuming β = 2/3 ≃ 0.66 (a typical value for galaxy clusters), the power spectrum of the βprofile reduces to (B.2)The previous equation strikes for its simplicity, and can be readily used in semianalytic studies. Changing β in the range 0.5−1 does not significantly alter P(k), hence Eq. (B.2) is an excellent approximation for the majority of clusters (Fig. B.1, red line). A remarkable feature is that the transition from real to Fourier space does not dramatically deform the profile, in tight analogy with Gaussian functions (∝exp [ − k^{2}]). The spectrum is dominated by the power on large scales, with the core radius playing a crucial role; a progressively rising r_{c} leads to an increase in both the normalization and steepness of the spectrum.
For our study, it is useful to analyze the superposition of the βprofile and a powerlaw Kolmogorov noise (with 1D power ∝k^{− 5/3}), n_{p} = n_{β} (1 + δ). Using the convolution theorem, the power spectrum of the perturbed density profile is given by . The cross terms cancel out since the
δ field is random and the phases are uncorrelated. In Fig. B.1, we show three power spectra: βprofile (red), noise with ~10 percent relative amplitude (blue), and the superposition of both (black). Beyond the core radius (k ≳ 0.05), the noise clearly starts to dominate. It is thus not essential to remove the underlying profile or largescale coherent structures, in order to unveil density perturbations, especially with substantial turbulence.
© ESO, 2013
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