Szapudi & Pan (2004) - progenax — differentiable initial conditions for N-body simulations

The big idea¶

The galaxy density is observed as discrete counts in cells, not as a continuous field. Szapudi & Pan connect the two with a locally-Poisson model: a cell of mean count $\langle N\rangle$ whose underlying continuous overdensity is $\delta$ contains $N$ galaxies with probability

P_N \;=\; \int_{-1}^{\infty} p(\delta)\, \frac{[\langle N\rangle(1+\delta)]^{N}\, e^{-\langle N\rangle (1+\delta)}}{N!}\,\mathrm{d}\delta .

(1)

The discrete count distribution is a Poisson average of the continuous density PDF $p(\delta)$ . Their goal is to invert this — recover $p(\delta)$ (and hence galaxy bias) from measured $P_N$ — but for progenax the forward direction of (1) is the model: it is exactly the compound-Poisson count_distribution that predicts counts-in-cells from the BM19 density PDF.

Core results¶

Counts-in-cells likelihood (Eq. 8). Fitting density-PDF parameters to a measured count histogram $\tilde P_N$ over $M$ cells uses the Poisson likelihood

\mathcal{L} \;=\; \prod_N \frac{(M P_N)^{M\tilde P_N}\, e^{-M P_N}}{(M\tilde P_N)!},

(2)

minimised over the PDF parameters. This is precisely the structure of progenax’s count_loglike — a 1-point count likelihood whose high- $N$ tail constrains the density PDF.

Skewed-lognormal Gaussianization (SLN3, Eq. 6). To model the continuous $p(\delta)$ they expand the log-density $\Phi = \log\rho - \langle\log\rho\rangle$ in Hermite polynomials about a Gaussian:

p_3(\delta)\,\mathrm{d}\delta = \left[1 + \tfrac{1}{3!}T_3\sigma_\Phi H_3(\nu) + \tfrac{1}{4!}T_4\sigma_\Phi^2 H_4(\nu) + \tfrac{10}{6!}T_3^2\sigma_\Phi^2 H_6(\nu)\right] G(\nu)\,\mathrm{d}\nu,

(3)

with $\nu = \Phi/\sigma_\Phi$ , $G$ a unit Gaussian, $H_m$ the Hermite polynomials, and $T_3, T_4$ the renormalised skewness and kurtosis. They show explicitly that the Gaussian–Edgeworth expansion of the linear density fails in the strongly non-linear regime (the tail), which is exactly why the analysis is done in log density — the same reasoning progenax uses to carry the 2-point as $\xi_s$ , not the tail-divergent linear $\langle\rho\rho\rangle$ .

Use in progenax¶

The CIC count model. (1) is the compound-Poisson count_distribution; (2) is count_loglike. These give the stellar counts-in-cells block of the Differentiable inference — natal cloud parameters from cluster substructure likelihood.
The Hermite Gaussianization. (3) is the parametric ancestor of the gaussianized_xi series: both expand the log-density transform in Hermite polynomials. progenax differs in two ways — it uses the exact BM19 copula map (not a 3-term skewed-lognormal truncation), and it propagates the Hermite coefficients into the two-point function $\xi_s(r) = \sum_n (c_n^2/n!)\,\rho_g(r)^n$ rather than only the one-point PDF.

Notes¶

Szapudi & Pan build directly on the lognormal field of Coles & Jones (1991) and on Sigad, Branchini & Dekel (2000) for the cumulative-CDF bias relation. The Gaussianization information argument is taken further by Neyrinck (Neyrinck, Szapudi & Szalay (2009), Neyrinck, Szapudi & Szalay (2011)) and Carron & Szapudi (Carron & Szapudi (2013), Carron & Szapudi (2014)).
The “local Poisson” assumption — counts are Poisson given the local density — is the same assumption behind the clean inhomogeneous-Poisson star sampler (placement.py); an incorrect with-replacement multinomial sampler is what produced the spurious $\alpha$ -signal diagnosed in the Differentiable inference — natal cloud parameters from cluster substructure work.

References¶

Szapudi, I., & Pan, J. (2004). On Recovering the Nonlinear Bias Function from Counts-in-Cells Measurements. The Astrophysical Journal, 602, 26–37. 10.1086/380920