Neyrinck, Szapudi & Szalay (2011)

The big idea¶

Paper II carries the log-density Gaussianization of Neyrinck, Szapudi & Szalay (2009) from the continuous matter field to the discrete, shot-noise-limited galaxy field. A Gaussianizing transform (one that makes the one-point distribution Gaussian) again reduces power-spectrum non-linearity and increases the Fisher information, in real and redshift space — with the redshift-space gain smaller because peculiar velocities partly Gaussianize the field already.

Two results progenax uses¶

1-point + 2-point sufficiency. The paper states the assumption cleanly: “in the approximation that a non-Gaussian field is a non-linear transformation of a Gaussian field, PDF Gaussianization will produce a Gaussian field, vanishing all higher-order correlations.” That is exactly the gravoturb generative model — a Gaussian random field passed through a monotone marginal map (the copula). For such a field the one-point PDF plus the two-point function are sufficient statistics; the higher-order correlations carry no independent information. This is the formal licence for restricting the Differentiable inference — natal cloud parameters from cluster substructure likelihood to 1pt + 2pt: it is not a lossy shortcut but the complete description of the model, and therefore the most honest use of it.

The Fisher gain peaks at a finite resolution. The cumulative Fisher information recovered by Gaussianizing “peaks at a particular grid resolution [that] depends on the sampling level.” Finer than that, shot noise dominates and adds nothing; coarser, the small-scale information is averaged away. This is the cosmology precedent for the resolution / $N_{\rm eff}$ forecast in Differentiable inference — natal cloud parameters from cluster substructure: there is an optimal map resolution for a given tracer density, and pushing past it does not help.

Use in progenax¶

• Justifies the 1pt+2pt inference design — the GRF-through-monotone-map field is fully captured by its one- and two-point statistics, so the phase-randomness of the model is not a misspecification for this likelihood.

• Frames the resolution trade-off behind the survey-design forecast (how many independent resolution elements a map needs).

Notes¶

• The genuine filamentary phase coherence that a Gaussian-mapped field cannot reproduce is the signal reserved for the (deferred) 3-point null test — the one place the 1pt+2pt sufficiency argument is expected to break for real clouds.

• Companion to Neyrinck, Szapudi & Szalay (2009); optimal-transform theory in Carron & Szapudi (2013); discrete sufficiency in Carron & Szapudi (2014).