Szapudi, Pan, Prunet & Budavári (2005)

The big idea¶

The two-point correlation function $\xi(r)$ and the power spectrum $P(k)$ are a Fourier-transform pair (Wiener–Khinchin), yet in practice they are measured by different, complementary methods — pair counting on small scales, FFT band-powers on large scales — with different edge-correction problems. Szapudi, Pan, Prunet & Budavári present a unified, fast pair: an FFT algorithm that produces an edge-corrected $\xi(r)$ matching a direct pair-count, plus a numerical Hankel transform (with a Gauss–Bessel quadrature) that turns the measured $\xi(r)$ into an edge-corrected $P(k)$, both at $N\log N$ cost. The key practical point is that $\xi(r)$ and $P(k)$ are equivalent — one can use whichever is convenient and convert between them robustly.

Use in progenax¶

• The measurement oracle. progenax’s measure.py uses exactly this Wiener–Khinchin route: autocovariance_3d computes $\xi(r)$ as the inverse FFT of the squared field transform, and the inference covariance layer measures power-spectrum band-powers from the same field. These are the ground-truth measurements against which the analytic gaussianized_xi prediction and the Limber-projected two-point are validated.

• The $\xi \leftrightarrow P$ duality is what lets the Differentiable inference — natal cloud parameters from cluster substructure 2-point block be expressed either as the configuration-space $\xi_s(r)$ (Gaussianization series) or as Fourier band-powers (the likelihood data vector), interchangeably.

Notes¶

• Same lead author and CIC lineage as Szapudi & Pan (2004); here the focus is the estimator rather than the density model.

• Edge correction matters for real, bounded survey footprints; the FDF works on periodic simulation boxes, where the estimator is exact and the edge term is absent — but the same duality and FFT machinery apply.