Minkowski Tensors in Two Dimensions: Probing the Morphology and Isotropy of the Matter and Galaxy Density Fields

Abstract

We apply the Minkowski Tensor statistics to two dimensional slices of the three dimensional matter density field. The Minkowski Tensors are a set of functions that are sensitive to directionally dependent signals in the data, and furthermore can be used to quantify the mean shape of density fields. We begin by reviewing the definition of Minkowski Tensors and introducing a method of calculating them from a discretely sampled field. Focusing on the statistic W 1,1 2 -a 2 × 2 matrix -we calculate its value for both the entire excursion set and for individual connected regions and holes within the set. To study the morphology of structures within the excursion set, we calculate the eigenvalues λ 1 , λ 2 for the matrix W 1,1 2 of each distinct connected region and hole and measure their mean shape using the ratio β ≡ λ 2 /λ 1 . We compare both W 1,1 2 and β for a Gaussian field and a smoothed density field generated from the latest Horizon Run 4 cosmological simulation, to study the effect of gravitational collapse on these functions. The global statistic W 1,1 2 is essentially independent of gravitational collapse, as the process maintains statistical isotropy. However, β is modified significantly, with overdensities becoming relatively more circular compared to underdensities at low redshifts. When applying the statistics to a redshift-space distorted density field, the matrix W 1,1 2 is no longer proportional to the identity matrix and measurements of its diagonal elements can be used to probe the large-scale velocity field.

Filaments and striations: anisotropies in observed, supersonic, highly magnetized turbulent clouds

“…This is similar to what has been done in Appleby et al (2018), where they use Minkowski tensors to approximate the anisotropy of dark matter fields. Our anisotropy parameter is analogous to taking the average of the ratio between the two eigenvalues of the (W 1,1 2 )ij tensor (equation 15 in Appleby et al 2018) over all excursion sets on the 2D power spectrum.…”

“…The power spectrum allows us to reveal order out of complicated and stochastic structures that are mostly intangible in real space. There are, however, also techniques for directly analysing real-space structures, such as structure functions, topological (Appleby et al 2018;Henderson et al 2019), and fractal methods (Scalo 1990;Elmegreen & Falgarone 1996;Stutzki et al 1998;Kowal et al 2007;Federrath et al 2009;Roman-Duval et al 2010;Donovan Meyer et al 2013;Konstandin et al 2016;Beattie et al 2019b). The power spectrum has a special role in the study of turbulence, since turbulence models rely heavily upon understanding flow characteristics on different length scales in real space, e.g., on the driving scale, in the inertial (for incompressible flows) scaling range (cascade) of turbulence, and on the dissipation scale (Kolmogorov 1941;Burgers 1948).…”

Morphology of CMB fields—effect of weak gravitational lensing

“…Critical points of a field are intrinsically higher order quantities, and hence the method will fail to accurately represent structures of size ∼ O(∆ 2 ), where ∆ is the resolution of the pixel grid. This issue is ameliorated by smoothing the field with scale R G , and it was shown in [42] that smoothing over five pixel lengths R G > 5∆ is sufficient to reduce the numerical error on the W 1 statistic to below 1% for threshold values in the range −4 < ν < 4.…”

Morphology of 21cm brightness temperature during the Epoch of Reionization using Contour Minkowski Tensor

“…and ds is the infinitesimal arc length. Our notation follows [37] 1 where W 1 is referred to as W 1,1 2 in [34,37,38]. Tr (W 1 ) is two times the second scalar MF i.e.…”

“…However, closed form expressions for n con,hole ,r con,hole andβ con,hole , are not known. n con,hole has been calculated numerically in [19], whileβ con,hole has been studied extensively using numerical computation in [38].r con,hole has not been studied before. We can infer their behaviour at very high and positive and very low and negative thresholds.…”