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courses:ast403:two-point-correlation-function [2026/04/04 01:01] – [The BAO Peak in the Correlation Function] shuvocourses:ast403:two-point-correlation-function [2026/04/04 01:17] (current) – shuvo
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 ===== Mathematical Definition ====== ===== Mathematical Definition ======
-Imagine a universe with a mean galaxy number density of $\bar{n}$. If we randomly drop two infinitesimal volume elements, $dV_1$ and $dV_2$, separated by a distance $r$, the probability $dP$ of finding one galaxy in $dV_1$ and another in $dV_2$ is given by:+Imagine a Universe with a mean galaxy number density of $\bar{n}$. If we randomly drop two infinitesimal volume elements, $dV_1$ and $dV_2$, separated by a distance $r$, the probability $dP$ of finding one galaxy in $dV_1$ and another in $dV_2$ is given by:
  
 $$dP = \bar{n}^2 [1 + \xi(r)] dV_1 dV_2$$ $$dP = \bar{n}^2 [1 + \xi(r)] dV_1 dV_2$$
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 ===== Practical Estimation: The Landy-Szalay Estimator ===== ===== Practical Estimation: The Landy-Szalay Estimator =====
-In practice, cosmologists do not have an infinite universe to measure; they have a finite survey with complex boundaries, varying observation depths, and instrumental artifacts. +In practice, cosmologists do not have an infinite Universe to measure; they have a finite survey with complex boundaries, varying observation depths, and instrumental artifacts. 
  
 To calculate $\xi(r)$ from real data, observers generate a "Random" catalog—a simulated dataset of points distributed completely randomly, but matching the exact 3D geometry and selection effects of the actual "Data" survey. They then count the number of pairs separated by a distance $r$. To calculate $\xi(r)$ from real data, observers generate a "Random" catalog—a simulated dataset of points distributed completely randomly, but matching the exact 3D geometry and selection effects of the actual "Data" survey. They then count the number of pairs separated by a distance $r$.
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 In cosmology, it is often mathematically convenient to work in Fourier space rather than configuration (real) space. The Fourier transform equivalent of the two-point correlation function is the **Matter Power Spectrum**, $P(k)$, where $k$ is the wavenumber ($k \sim 2\pi/r$). In cosmology, it is often mathematically convenient to work in Fourier space rather than configuration (real) space. The Fourier transform equivalent of the two-point correlation function is the **Matter Power Spectrum**, $P(k)$, where $k$ is the wavenumber ($k \sim 2\pi/r$).
  
-For a statistically isotropic universe (where clustering depends only on the magnitude of the distance $r$, not the direction), the two-point correlation function is the Fourier transform of the power spectrum:+For a statistically isotropic Universe (where clustering depends only on the magnitude of the distance $r$, not the direction), the two-point correlation function is the Fourier transform of the power spectrum:
  
 $$\xi(r) = \frac{1}{2\pi^2} \int_0^{\infty} P(k) \frac{\sin(kr)}{kr} k^2 dk$$ $$\xi(r) = \frac{1}{2\pi^2} \int_0^{\infty} P(k) \frac{\sin(kr)}{kr} k^2 dk$$
courses/ast403/two-point-correlation-function.1775286080.txt.gz · Last modified: by shuvo
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