Cosmological Parameter Estimation from the Two-dimensional Genus Topology: Measuring the Shape of the Matter Power Spectrum

Appleby, Stephen; Park, Changbom; Hong, Sungwook E.; Hwang, Ho Seong; Kim, Juhan

doi:10.3847/1538-4357/ab952e

Cited by 16 publications

(19 citation statements)

References 97 publications

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“…While itself a purely topological quantity, the Euler characteristic is also the alternating sum of the Betti numbers of all ambient dimensions of a manifold, as denoted by the Euler-Poincaré formula (Adler 1981;Pranav et al 2017Pranav et al , 2019b. The Euler characteristic has a long history in the analysis of cosmological fields (Gott et al 1986;Pogosyan et al 2009;Park et al 2013;Appleby et al 2020).…”

Section: Introductionmentioning

confidence: 99%

Unexpected topology of the temperature fluctuations in the cosmic microwave background

Pranav

Adler

Buchert

et al. 2019

A&A

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We study the topology generated by the temperature fluctuations of the Cosmic Microwave Background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the LCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33 degrees. The survey of the CMB over S 2 is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of "masks" is of importance, we introduce the concept of relative homology. The parametric χ 2 -test shows differences between observations and simulations, yielding p-values at per-cent to less than per-mil levels roughly between 2 to 7 degrees, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b 0 and b 1 is approximately between 3σ-4σ at scales of 3 to 7 degrees. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66 degrees in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck's predecessor WMAP (Wilkinson Microwave Anisotropy Probe) satellite. The mildly anomalous behaviour of Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and the first Betti numbers, respectively. Further, since these topological descriptors show anomalous behaviour consistently over independent measurements of Planck and WMAP, instrumental errors and systematics may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations, based on power spectrum matching the characteristics of the observed dipped power spectrum, are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB -whether cosmological in nature, or arising due to late-time effects is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate to look at primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology.

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Section: Introductionmentioning

confidence: 99%

Unexpected topology of the temperature fluctuations in the cosmic microwave background

Pranav

Adler

Buchert

et al. 2019

A&A

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“…The geometrical aspects involve the notion of volume, area and so on, via the Minkowski functionals, or the Lifshitz-Killing curvatures (Crofton 1868;Hadwiger 1957;Adler 1981), while the topological characterization involves the notion of critical points (Milnor 1963;Edelsbrunner & Harer 2010), and topological properties associated with them, such as topological cycles or equivalently the topological holes, finding their basis in homology theory (Munkres 1984;Edelsbrunner & Harer 2010;Pranav 2015;Pranav et al 2017). The notion of Euler characteristic (Euler 1758;Gott et al 1986Gott et al , 1989Park et al 2013;Appleby et al 2018Appleby et al , 2020 provides a bridge between purely topological and purely geometrical concepts, as while being a purely topological quantity, it can be expressed in a purely integral geometric setting, as established by the Theorema Egrerium due to Gauss (Gauss 1900).…”

Section: Characterization Of the Properties Of Random Fieldsmentioning

confidence: 99%

Topology and geometry of Gaussian random fields II: on critical points, excursion sets, and persistent homology

Pranav¹

2021

Preprint

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This paper is second in the series, following , focused on the characterization of geometric and topological properties of 3D Gaussian random fields. We focus on the formalism of persistent homology, the mainstay of Topological Data Analysis (TDA), in the context of excursion set formalism. We also focus on the structure of critical points of stochastic fields, and their relationship with formation and evolution of structures in the universe.The topological background is accompanied by an investigation of Gaussian field simulations based on the LCDM spectrum, as well as power-law spectra with varying spectral indices. We present the statistical properties in terms of the intensity and difference maps constructed from the persistence diagrams, as well as their distribution functions. We demonstrate that the intensity maps encapsulate information about the distribution of power across the hierarchies of structures in more detailed than the Betti numbers or the Euler characteristic. In particular, the white noise (n = 0) case with flat spectrum stands out as the divide between models with positive and negative spectral index. It has the highest proportion of low significance features. This level of information is not available from the geometric Minkowski functionals or the topological Euler characteristic, or even the Betti numbers, and demonstrates the usefulness of hierarchical topological methods. Another important result is the observation that topological characteristics of Gaussian fields depend on the power spectrum, as opposed to the geometric measures that are insensitive to the power spectrum characteristics.

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“…In a recent series of works, we have measured the genus of two-dimensional slices of the SDSS-III BOSS data, extracting cosmological information from the genus amplitude, and placing constraints on the scalar spectral index n s and dark matter fraction Ω c h 2 , assuming a flat ΛCDM cosmology (Appleby et al 2020). Given that the BOSS data is spectroscopic, we have precise redshift information.…”

Section: Introductionmentioning

confidence: 99%

“…After generating smoothed number density fields from the point galaxy distribution, we measure the statistics, ensuring that the complex survey geometry does not impact our results. We then extract information from the amplitudes of these functions, using similar methodology to Appleby et al (2020). We also measure the bispectrum components from the MFs, although we do not convert this information into cosmological parameter constraints in this work.…”

Section: Introductionmentioning

confidence: 99%

Minkowski Functionals of SDSS-III BOSS : Hints of Possible Anisotropy in the Density Field?

Appleby,

Park,

Pranav

et al. 2021

Preprint

Self Cite

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We present measurements of the Minkowski functionals extracted from the SDSS-III BOSS catalogs. After defining the Minkowski functionals, we describe how an unbiased reconstruction of these statistics can be obtained from a field with masked regions and survey boundaries, validating our methodology with Gaussian random fields and mock galaxy snapshot data. From the BOSS galaxy data we generate a set of four density fields in three dimensions corresponding to the northern and southern skies of LOWZ and CMASS catalogs, smoothing over large scales such that the field is perturbatively non-Gaussian. We extract the Minkowski functionals from each data set separately, and measure their shapes and amplitudes by fitting a Hermite polynomial expansion. For the shape parameter of the Minkowski functional curves a 0 , that is related to the bispectrum of the field, we find that the LOWZ-South data presents a systematically lower value of a 0 = −0.080 ± 0.040 than its northern sky counterpart a 0 = 0.032 ± 0.024. Although the significance of this discrepancy is low, it potentially indicates some systematics in the data or that the matter density field exhibits anisotropy at low redshift. By assuming a standard isotropic flat ΛCDM cosmology, the amplitudes of Minkowski functionals from the combination of northern and southern sky data give the constraints Ω c h 2 n s = 0.110 ± 0.006 and 0.111 ± 0.008 for CMASS and LOWZ, respectively, which is in agreement with the Planck ΛCDM best-fit Ω c h 2 n s = 0.116 ± 0.001.

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Cosmological Parameter Estimation from the Two-dimensional Genus Topology: Measuring the Shape of the Matter Power Spectrum

Cited by 16 publications

References 97 publications

Unexpected topology of the temperature fluctuations in the cosmic microwave background

Unexpected topology of the temperature fluctuations in the cosmic microwave background

Topology and geometry of Gaussian random fields II: on critical points, excursion sets, and persistent homology

Minkowski Functionals of SDSS-III BOSS : Hints of Possible Anisotropy in the Density Field?

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