Towards field-level likelihood for projected fields: Evolved projected fields from initial projected fields

TL;DR

The paper addresses how the 2D cosmological projected density field evolves when conditioned on fixed initial projected modes, revealing that non-linear evolution couples to bulk modes that erase information about those initial projection modes. It derives a mean evolution using Lagrangian perturbation theory, showing the evolved projected field equals a 2D Zeldovich-evolved field multiplied by a Gaussian damping factor $e^{-\frac{1}{2}D^2 k_\perp^2 \Sigma^2}$, with a total suppression $\Sigma^2 = \Sigma^2_\mathrm{Z}-\Sigma^2_{W}+\Sigma^2_{W^{2}}$. The authors validate the theory with 100 small fixed-projection N-body simulations and find good agreement, demonstrating exponential suppression of projection-mode information on non-linear scales and quantifying the residual discrepancy at low redshift. They propose a practical hybrid field-level likelihood for projected fields, combining deterministic linear information with stochastic residuals from bulk modes, and discuss implications for weak lensing and photometric clustering analyses. This work clarifies the limits and potential of 2D field-level inference and motivates efficient 2D forward-models that can leverage partial field-level information while accommodating non-linear mode coupling.

Abstract

The evolved cosmological matter density field is fully determined by the initial matter density field at fixed cosmological parameters. However, the two-dimensional cosmological projected matter density field, relevant for weak-lensing and photometric galaxy studies, is fully determined by the initial projected matter density field only at the linear order. At non-linear order, the entire volume of initial matter contributes. We study a model for the evolved projected density field that is deterministic in the initial projected density fields and probabilistic in the effects of the remaining modes in the initial conditions. We write down predictions for the mean evolved projected field model using Lagrangian perturbation theory. We run a suite of small N-body simulations with fixed projected initial conditions and measure the statistical properties of the ensemble of evolved projected fields. Measurements and theory are in good agreement and show that the information on the initial projected fields is exponentially suppresses on non-linear scales. Our model offers a potential approach to a field-level likelihood of projected fields.

Figures (4)

• Figure 1: This figure shows the result of our simulations. In each panel we show the three-dimensional density field projected along the $z$ axis on the $x-y$ plane. Top three rows show three randomly chosen realization out of the 100 we have run, the last by one row corresponds the field-level mean of realizations and the final plot is for the projected-mode only. Columns from left to right correspond to decreasing redshifts as labeled on top. Note that the plotted dynamic range is adapted at every redshift, but is uniform across the plots (see the bottom color-scales). See text for the discussion.
• Figure 2: Two-dimensional power spectra for 100 simulations. Black line corresponds to the mean power spectra of 100 realizations (at fixed initial projected modes; top 3 rows in Figure \ref{['fig:projplots']}). The red line shows the only-projected mode realization (bottom row in Figure \ref{['fig:projplots']}). The purple line shows power spectra of the mean field (last by one row in Figure \ref{['fig:projplots']}).
• Figure 3: The quantity of Equation \ref{['eq:plot']} measured at three different redshifts. We plot a theoretical suppression $\Sigma^2$ (red dashed) as well as suppression expected from Zeldovich only $\Sigma_Z^2$ (red dotted) and a theoretical fit $\Sigma_{\rm fit}^2$ (green) against the measured values (black solid line).
• Figure 4: Same as Figure \ref{['fig:projplots']} but for a projection along the $x$-axis.