Free Access
Volume 559, November 2013
Article Number A78
Number of page(s) 18
Section Cosmology (including clusters of galaxies)
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 Δ-variance 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 real-space 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 − x2/σ2]    exp [ − x2/2σ2], characterized by a positive core and negative wings);

  • 3.

    the variance Vσ of the previous cube is calculated and recast into the estimate of the power, knowing that

We refer to Arévalo et al. (2012) – Appendix A – for the technical procedure and normalization. In order to handle the non-periodicity of the cube, we use a “mask” which is 1 inside and 0 outside the domain. The big advantage of the MH is that it avoids any leakage of power linked to the non-periodicity of the data; the drawback is that it can not capture very sharp features in the power spectrum, due to the smoothening over Δk ~ k.

thumbnail 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 non-periodic 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 non-periodic box. Progressively trimming the box increases the relative normalization of the FFT spectrum, even by 20 percent, while distorting the low-frequency slope; the MH spectrum is instead unaltered.

Appendix B: β-profile in Fourier space

thumbnail Fig. B.1

Analytic 1D power spectra: β-profile (red), Kolmogorov noise (blue), β-profile perturbed by the noise (black; ). The spectrum is normalized to the value at k0 = 1/L = 0.01 (dimensionless units; 2π is dropped for clarity). The core radius is rc = 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 large-scale structures.

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We present here the analytic conversion of the β-profile to Fourier space, and its interplay with a power-law noise. Using the notation , the Fourier transform of the β-profile (Eq. (1)) results to be (B.1)where ξ ≡ 3β/2, K1/2 − 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 β-profile reduces to (B.2)The previous equation strikes for its simplicity, and can be readily used in semi-analytic 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 [ − k2]). The spectrum is dominated by the power on large scales, with the core radius playing a crucial role; a progressively rising rc 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 β-profile and a power-law Kolmogorov noise (with 1D power ∝k− 5/3), np = nβ   (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: β-profile (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 large-scale coherent structures, in order to unveil density perturbations, especially with substantial turbulence.

© ESO, 2013

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