REVIEW 3 major objections 4 minor 24 references
Field-level likelihood for projected fields: Evolved projected fields from initial projected fields
T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read The paper derives a closed-form prediction for the mean evolved projected matter field at fixed initial projected modes: it is a 2D Zel'dovich-evolved field multiplied by a Gaussian damping factor whose width is set by the variance of the u
desk verdict A useful and honest paper on the conditional mean of evolved projected fields; the abstract oversells an HMC implementation that isn't in the body, but the core derivation and simulation test are worth referee time. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is Eq. (2.24), which packages the whole argument as a 2D Zel'dovich-evolved field times a Gaussian damping factor. It is obtained by writing initial conditions conditioned on fixed projected modes, expanding the evolved field in Zeldovich displacements, and Gaussian-averaging over the unconstrained line-of-sight modes. The three variance terms in Sigma^2 arise, respectively, from the full 3D displacement variance, the variance of the projected component of the residual modes, and the cross-term that depends on the line-of-sight window; an expansion of that window term in q_parallel turns the result into a pure Gaussian damping.
What would settle it
Evaluate Eq. (2.23) numerically without dropping the q_parallel integral, and compare it with Eq. (2.24) for a given window function; if the projection produces any k_perp-dependent correction beyond a constant normalization, or if the measured P_me/P_ee in a fixed-projected-modes N-body suite deviates from exp(-k_perp^2 Sigma^2) on scales where the expansion was assumed convergent, the central relation is wrong.
Extended reading notes
Core claim
The central claim is Equation (2.24): the average over all initial conditions sharing a fixed projected density field d(k_perp) gives the mean evolved projected field Delta(Delta0,t) = exp(-1/2 D^2 k_perp^2 Sigma^2) Z(Delta0,t), where Z(Delta0,t) is the 2D Zel'dovich evolution of the initial projected modes and Sigma^2 = Sigma^2_Z - Sigma^2_W + Sigma^2_W2. In words, the deterministic part of the evolution is a 2D Zel'dovich map, while the unknown bulk modes act as a Gaussian random smearing that exponentially suppresses memory of the initial projected state on small scales. The paper tests this against 100 small N-body simulations with fixed projected initial conditions and finds the predict
Load-bearing premise
The step from Eq. (2.23) to Eq. (2.24) is asserted rather than demonstrated: it assumes the line-of-sight integral in Eq. (2.23) only changes the normalization and that, after projection, the damping is exactly the Gaussian exp(-1/2 D^2 k_perp^2 Sigma^2) with amplitude fixed by linear theory.
Editorial extensions
If this is right
- On scales where k_perp^2 Sigma^2 becomes large, the mean evolved projected field loses almost all memory of the initial projected modes; information is exponentially suppressed, so a field-level likelihood for 2D surveys mainly constrains large scales.
- The paper proposes a hybrid likelihood: evolve the initial projected field deterministically via Eq. (2.24), then model the residual from bulk modes statistically (e.g. Gaussian with covariance P - P_mm), and use summary statistics of residuals to capture the statistical information in bulk modes.
- The suppression has the same physical origin as BAO damping in 3D but applies to the one-point field, with a factor of 1/2; the window-correction terms reduce the damping because some modes are evolved explicitly.
- At z=0 the measured suppression departs from growth-factor-squared scaling by about 20%, indicating higher-order Lagrangian corrections matter at low redshift.
Reading between the lines
- Inference: Because the derivation expands cos(k_parallel q_parallel) only to leading order, the exact Gaussian shape is an approximation; for narrow window functions one should see the next term as a k^4 correction to the log-ratio, a measurable prediction.
- Inference: For broad weak-lensing kernels that integrate over cosmic time, Sigma^2 cannot be a single number; a tomographic generalisation of Eq. (2.24) would need a redshift-dependent damping applied bin-by-bin.
- Inference: The formula suggests a natural upgrade path: replace the Zel'dovich 2D evolution with a fast dedicated 2D non-linear solver and test whether the residual damping stays Gaussian with the same Sigma^2, which would make the hybrid likelihood fully non-linear.
- Inference: The information-loss curve makes a concrete survey-design prediction: the recoverable information at scale k_perp is controlled by the combination Sigma^2(L_window), so photometric bin widths and window shapes can be optimised to maximise the retained initial-projected information.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper addresses what can be predicted for the two-dimensional projected matter density field from the initial projected matter density field alone. At linear order the relation is deterministic; at nonlinear order the unconstrained 'bulk' modes contribute through mode coupling. Using Lagrangian perturbation theory in the Zeldovich approximation, the authors derive an expression for the ensemble mean of the evolved projected field conditioned on fixed initial projected modes: Eq. (2.24), a 2D Zeldovich-evolved field multiplied by a Gaussian damping exp(-1/2 D^2 k⊥^2 Σ^2). This prediction is tested with 100 small-box N-body simulations that share the same k_z=0 initial modes but differ in all other modes, by comparing the power spectrum of the mean field to that of a simulation evolved only from the projected modes. The reported agreement is good in shape, with the predicted damping amplitude matching the measured one to 7% at z=3, 20% at z=1, and 30% at z=0. The paper concludes by sketching a hybrid field-level likelihood in which the mean field is modeled deterministically and residual bulk-mode effects are described statistically.
Significance. If the central formula is correct, it gives a parameter-free, falsifiable prediction for how much information about initial projected modes survives into the nonlinear projected field, and it provides a natural functional form for a projected-field likelihood. The simulation protocol is a clean and honest test: the suppression exponent is not fit in the theory curves, and the 'best fit' in Fig. 3 is used only as a comparison diagnostic. The degrading agreement with redshift is reported transparently. The main issues are an asserted rather than demonstrated step in the LPT derivation, an apparent inversion of the growth-factor dependence in the quoted Σ² values, and a mismatch between the abstract and the actual content regarding the HMC implementation.
major comments (3)
- [§2.2, Eqs. (2.23)–(2.24)] The step from Eq. (2.23) to Eq. (2.24) is not demonstrated and, as written, is not correct. Evaluating the q∥ integral in Eq. (2.23) as a Gaussian gives a prefactor proportional to 1/|k⊥|, not a constant. Even after correcting the coefficient from Eqs. (2.14)–(2.20) — the characteristic-function exponent should contain −¼ D² k⊥² q∥² Σ_W^(2), not −½ — the |k⊥|⁻¹ factor survives because the projection operator P̂ acts only on k∥ and leaves the k⊥-dependent prefactor untouched. The exact q∥ integral before the quadratic expansion is not the Gaussian shown; over a finite window it approaches a constant at low k⊥ and crosses to a 1/k⊥ scaling only at large k⊥, with corrections that are not computed. Thus Eq. (2.24) is not a proven consequence of Eq. (2.23). This is the central derivation of the paper and needs to be fixed or replaced with a controlled computation of the q∥ integral.
- [§3.2, Fig. 3 and text] The quoted suppression values are internally inconsistent with Eq. (2.24). Equation (2.24) contains the combination D² Σ², so the effective Gaussian width at redshift z should scale as D(z)² Σ²(z=0), decreasing toward high redshift. Yet the text reports predicted/measured Σ² = 264/246, 103/84, 25.1/17.13 (Mpc/h)² at z=3, 1, 0 — the opposite trend. These numbers are close to Σ²(z=0)/D(z)², suggesting an inversion of the growth factor. Because this comparison is the main quantitative test of the model, the definition of Σ² in Eq. (3.1) and the role of D(t) must be clarified or corrected.
- [Abstract vs. §4] The abstract states that the approach is implemented in a likelihood code and that HMC sampling reconstructs initial fields in the presence of non-trivial masks. The main text contains no such implementation, results, or code description; §4 only sketches the likelihood in Eq. (4.1) and lists it as future work. Either the implementation should be added or the abstract claim removed. This is a substantial mismatch between the advertised scope and the actual content.
minor comments (4)
- [Eq. (2.23)] The exponent of the q∥ Gaussian appears to be off by a factor of 2 relative to the derivation in Eqs. (2.20)–(2.22); please check the sign and coefficient.
- [Table 1 and §3.2] Table 1 is labeled as z=0, but the text then quotes values at z=3 and z=1 without an explicit formula relating them. State the growth-factor convention explicitly so the reader can reproduce the quoted numbers.
- [§2.2, 'common swindle'] The replacement of the Zeldovich operator by a fully nonlinear field is acknowledged as a 'common swindle.' It would be helpful to state explicitly that Eq. (2.24) is therefore a resummed/phenomenological model rather than a strict LPT result.
- [General] Typographical issues: 'preform' should be 'perform' (§3.1); 'Efstathion' should be 'Efstathiou'; the notation 'Mpc/ℎ2' throughout is unformatted.
Circularity Check
No significant circularity; the damping-factor prediction is derived from LPT and tested against simulations, not fitted.
full rationale
The paper's central result Eq. (2.24) is obtained by taking the ensemble mean over non-projected modes in Zeldovich LPT. The variance in Eq. (2.14) is computed from the Gaussian initial conditions and the projection window; Sigma^2_Z, Sigma^2_W, Sigma^2_W2 are evaluated numerically from the assumed power spectrum and W(k_parallel)=sinc(k_parallel L/2) rather than fit to the final projected-field statistics. The measured P_me/P_ee ratio in Fig. 3 is compared with this theory (red dashed) and with Sigma^2_Z (red dotted); the 'best fit' curve is labeled as a fit and is not used as the theoretical prediction (Eq. 3.1 uses the computed Sigma^2). Therefore no fitted parameter is renamed as a prediction. There are no load-bearing self-citations: the paper uses standard references (Zeldovich approximation, GADGET-4, LPT) as external, reproducible methods, not to justify its own central claim. The step from Eq. (2.23) to Eq. (2.24), in which the q_parallel integral is asserted to only renormalize the amplitude, is under-demonstrated and likely k_perp-dependent; however, this is a gap or potential error in the derivation, not a circular reduction of the output to the input. The paper also openly labels the replacement of Z with a fully non-linear field as 'a common swindle.' Thus the derivation chain is self-contained enough that no circularity score above 0 is warranted.
Assumptions & free parameters
assumptions (5)
- domain assumption The evolved density field in the Zeldovich approximation is given by Eq. 2.9.
- domain assumption The initial density field is Gaussian, so the conditional field with fixed projected modes is Gaussian.
- domain assumption The window function W has support only at low k∥, justifying the expansion of cos(k∥ q∥) and the convergence of the Σ²_W series.
- ad hoc to paper The q∥ integral in Eq. 2.23 only changes overall normalization.
- ad hoc to paper Replacing the Zeldovich-evolved field Z with a fully non-linear evolved field (the 'common swindle') is valid.
Cite this review
Pith. "Pith review of Field-level likelihood for projected fields: Evolved projected fields from initial projected fields." pith.science (2026). https://pith.science/paper/FHP7WGSE
@misc{pith2026251014089,
author = {Pith},
title = {Pith review of: Field-level likelihood for projected fields: Evolved projected fields from initial projected fields},
year = {2026},
howpublished = {\url{https://pith.science/paper/FHP7WGSE}},
note = {Machine review of arXiv:2510.14089}
}
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 suppressed on non-linear scales. We implement this approach in a likelihood code and use Hamiltonian Monte-Carlo sampling to show that initial fields can be reconstructed even in the presence of non-trival mask features.
Reference graph
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Reviewed August 4, 2026 · model on record in the stance chip above.
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