Topological Data Analysis of the Large-Scale Structure
Tsutomu T. Takeuchi · 2025
The distribution of matter and galaxies at each cosmic age contains fundamental information about the formation and evolution of the structure of the Universe (e.g., Bernardeau et al., 2002 ; Efstathiou & Silk, 1983 ; Peebles, 1980 ). One key fact that must be emphasized is that cosmology theoretically predicts the statistical properties of matter or galaxy distributions, but it does not have the predictive power to determine where exactly in space-time any specific galaxy will form. Therefore, comparisons between cosmological theory and observations can only be made through statistical descriptions. For this reason, numerous sophisticated methods have been proposed to characterize the statistical properties of fluctuations in the matter distribution (e.g., Bernardeau et al., 2002 ; Martinez & Saar, 2001 ; Peebles, 1980 ). Among these methods, the n -point correlation function is the most commonly studied and widely used in the analysis of observed galaxy distributions (e.g., Bernardeau et al., 2002 ; Martinez & Saar, 2001 ; Peebles, 1980 , see Chapter 6 .). To understand the new method we adopt in this chapter, we first recall the method of correlation function, introduced in Chapter 6 . We measure the number density of galaxies at a position https://www.w3.org/1998/Math/MathML" display="inline"> x → https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003104315/27ee123e-46ef-4238-8126-364fd8f26fd4/content/math15_1.tif "/> in the Universe, denoted as https://www.w3.org/1998/Math/MathML" display="inline"> n ( x → ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003104315/27ee123e-46ef-4238-8126-364fd8f26fd4/content/math15_2.tif "/> , and compare it to the statistical average https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ n ⟩ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003104315/27ee123e-46ef-4238-8126-364fd8f26fd4/content/math15_3.tif "/> . While we can only observe one Universe, we can conceptually assume an ensemble of many Universes and consider the average of a quantity Q across many Universes. This average is referred to as the statistical or ensemble average, denoted by https://www.w3.org/1998/Math/MathML" display="inline"> ⟨ Q ⟩ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003104315/27ee123e-46ef-4238-8126-364fd8f26fd4/content/math15_4.tif "/> . On the other hand, the average taken over space is called the volume average, denoted by https://www.w3.org/1998/Math/MathML" display="inline"> Q ¯ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003104315/27ee123e-46ef-4238-8126-364fd8f26fd4/content/math15_5.tif "/> In cosmology, we often base discussions on the assumption that these two averages are equal (the cosmological ergodic hypothesis).