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REVIEW 3 major objections 4 minor 1 cited by

Perturbative Likelihoods for Large-Scale Structure of the Universe

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that the likelihood of the observed density field can be derived from perturbation theory, and that the derivation automatically specifies the correct summary statistics and theoretical predictions at each order.

desk verdict A clean formal derivation with a genuinely new second-order result, but the fragile stochastic-sector assumption and thin numerical validation mean the paper needs revision before its central claim is fully supported. read the letter →

arxiv 2505.23750 v1 pith:CFGNXCJ3 submitted 2025-05-29 astro-ph.CO

Rodrigo Voivodic This is my paper
classification astro-ph.CO
keywords cosmologicalperturbationtheorylarge-scalestructurelikelihoodgalaxybiasstochasticfieldpowerspectrumbispectrumfield-levelinference
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the likelihood of the observed cosmological density field can be derived from first principles in perturbation theory, rather than assembled from ad-hoc choices of summary statistics, theoretical terms, and covariances. The derivation starts from a Gaussian primordial density field and a Gaussian stochastic field, writes the observed tracer field as a perturbative expansion in both, and marginalizes over the unobserved fields analytically. This yields a full likelihood that depends on the data only through specific summary statistics—at second order, the tree-level power spectrum, the tree-level bispectrum, and the (2,2) one-loop power-spectrum correction—and a field-level likelihood that retains the primordial density field. At leading order the likelihood is a gamma distribution with the same Fisher information as the standard Gaussian likelihood, while at second order it is a correction factor around that gamma form. If correct, the method makes the theoretical prediction and its covariance consistent by construction, which would matter for galaxy-survey parameter constraints.

What carries the argument

The machinery is a path-integral Gaussian marginalization. The full likelihood is written as a product of Dirac delta functions in Fourier space, the Gaussian priors on the primordial and stochastic fields are inserted, and the unobserved fields are integrated out exactly. The key non-trivial step is inverting the induced non-diagonal covariance at next-to-leading order, which brings in the tree-level bispectrum $\mathcal{B}_t$ and the (2,2) one-loop power-spectrum correction $P_{t,(22)}$ automatically. The field duplet $\phi=(\delta,\epsilon)$ is doubled into a quartet field $\Phi=(\phi_r,\phi_i)$ to handle the coupling between real and imaginary parts at second order.

What would settle it

Simulate a tracer field from an N-body or high-resolution forward model with known initial conditions in many realizations; subtract the perturbation-theory prediction conditioned on those initial conditions, and measure the residual's bispectrum and its cross-correlation with the initial density field. If either is significantly nonzero on the scales of interest, the Gaussian, independent stochastic field assumed in Eq. (2.2) is wrong and the likelihood is misspecified.

Watch

Extended reading notes

Core claim

The central claim is that a likelihood for the observed tracer density field that is self-consistently matched to perturbation theory is obtained by writing the observed field as a perturbative expansion in a Gaussian primordial density field and a Gaussian stochastic field, then exactly marginalizing over those fields. The marginalization produces a full likelihood—Eq. (4.20) at second order—in which the data enter only through specific summary statistics: the power spectrum at tree level, the tree-level bispectrum, and the (2,2) one-loop correction to the power spectrum. It also produces field-level likelihoods in which the primordial density field is kept fixed and only the stochastic field is integrated out. The method automatically determines the kernel–covariance contractions needed at each order, so the theoretical prediction and its covariance are mathematically consistent rather than assembled by hand.

Load-bearing premise

The load-bearing premise is that the stochastic sector is exactly one Gaussian field with diagonal covariance, independent of the initial density field, and that its perturbative expansion is suppressed by $(P/Q)^{n-1}$ on the scales used; if real small-scale stochasticity is non-Gaussian or correlated with initial conditions, the derived likelihood is misspecified at every order.

Editorial extensions

If this is right

  • At first order, the exact likelihood is a gamma distribution rather than a Gaussian, and its Fisher information coincides with the Gaussian power-spectrum likelihood; parameter constraints agree in the many-modes limit but can differ where the central-limit theorem fails.
  • At second order, the likelihood automatically combines the tree-level bispectrum and the (2,2) one-loop power-spectrum correction with the same theoretical power spectrum in the leading exponential, so the mean prediction and the covariance are evaluated at consistent perturbative orders.
  • The likelihood's positivity sets an effective $k_{\rm max}$: where the perturbative correction to the gamma form becomes order one, the PDF would go negative, which acts as a parameter-dependent prior on the analysis scale.
  • Field-level likelihoods derived by marginalizing only the stochastic field reproduce the standard effective-field-theory field-level likelihood at leading order and extend it to include the stochastic bispectrum term from the $(1,1)$ and $(0,2)$ operators.
  • The framework is general enough to be applied to any quantity described by perturbation theory, including multiple tracers and different cutoff scales for data and theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The formalism suggests a data-driven way to choose $k_{\rm max}$ for galaxy-survey analyses, since the point where the perturbative correction to the likelihood becomes order one is where the PDF would lose positivity.
  • One testable extension is to apply the derived field-level likelihood to high-resolution simulations with fixed initial conditions and check whether the residual stochastic field is actually Gaussian and independent of the initial density field.
  • The same order-by-order contraction rules should extend to redshift-space distortions and survey selection functions, which the paper leaves for future work, opening a route to practical field-level analyses of real surveys.
  • Because the formalism forces the summary statistics to appear in specific combinations, it could be used to derive optimal, survey-specific estimators for higher-point functions without a separate covariance-estimation step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper develops a formalism to derive likelihoods for the observed tracer density field directly from cosmological perturbation theory. The author assumes a Gaussian primordial density field δ and a Gaussian, δ-independent stochastic field ε with diagonal covariance Q(k|θ), and writes the observed field as a deterministic perturbative expansion in powers of δ and ε. Marginalizing over the underlying fields yields two likelihoods: a first-order likelihood that reduces to a gamma distribution for the band-averaged power spectrum and whose Fisher information is identical to the standard Gaussian approximation, and a second-order likelihood that automatically incorporates the tree-level bispectrum and the (2,2) one-loop power spectrum as summary statistics. The paper also derives a field-level likelihood by marginalizing only over ε, including terms that couple the deterministic and stochastic sectors. Numerical MCMC comparisons for the first-order likelihood show nearly identical posteriors to the Gaussian likelihood for standard parameters, but visibly different posteriors for a scale-dependent PNG parameter. The central advertised strength is that the formalism automatically specifies the precise combinations of covariances and PT kernels required at each order, avoiding ad-hoc choices in the statistical model.

Significance. If correct, the paper provides a principled and largely self-contained route from PT to likelihoods, and its automatic identification of summary statistics together with consistent covariance order would be a useful contribution to LSS inference. The first-order derivation is explicit and the Fisher-matrix result is a clean, checkable statement. The paper is honest about its assumptions and explicitly acknowledges the lack of second-order numerical tests. However, the central claim that the likelihood is 'fully consistent with PT' rests on a strong generative assumption about the stochastic sector, and the paper's own numerical validation does not exercise the second-order likelihood or test the assumed stochastic model. Despite these caveats, the analytic machinery and the first-order numerical comparison are valuable, and the paper could be a solid methods contribution after revision.

major comments (3)
  1. [§3.2, Eq. (3.16)] The statement that the likelihood in Eq. (3.16) is a gamma distribution with shape parameter α=1 is incorrect. For a bin containing N(ki) independent modes, the band-averaged spectrum Po(ki) has a gamma distribution with shape N(ki) and scale Pt(ki)/N(ki); Eq. (3.16) is the θ-dependent part of that density, with the data-dependent factor Po^{N-1} omitted. This error does not affect the Fisher matrix in Eq. (3.18) or the numerical posteriors, since the omitted factor is independent of θ, but it is a specific technical mistake in the interpretation of a central equation and should be corrected.
  2. [§5, Conclusions] The second-order likelihood of Eq. (4.20) is never numerically validated. The MCMC tests in Sec. 5 compare only the first-order likelihood (Eq. 3.16) with the Gaussian likelihood; there is no check of the bispectrum and one-loop corrections of Eq. (4.20) against synthetic data, nor of the positivity of the approximate PDF. Since the paper itself concludes that "Exact numerical comparisons between the two likelihoods are an interesting topic and will be addressed in future works," the status of Eq. (4.20) as a validated likelihood should be stated more explicitly, and ideally the paper should include at least a consistency test of the second-order expression.
  3. [App. A, Sec. 5] The justification for expanding the stochastic sector around a Gaussian field rests on the suppression factor [P/Q]^(n-1) being small on large scales and for dense samples. In the numerical setup of Sec. 5 (n̄ = 3×10^-4 (h/Mpc)^3, Q = 10^3–10^4), P/Q is of order unity at k ≈ 0.1 h/Mpc, so the stated suppression is not present in the tested regime. Furthermore, the mock data are generated from exactly the same Gaussian prior assumed in the likelihood, so the numerical tests cannot detect misspecification of the stochastic sector. The central claim that the likelihood is "fully consistent with PT" therefore rests on an assumed, not demonstrated, property of tracer stochasticity; this should be reframed as an explicit model assumption and its validity discussed.
minor comments (4)
  1. [§3.2] The sentence "This likelihood was also pointed out to be the correct one for the power spectrum of the temperatures fluctuations in the cosmic microwave background in [23]" cites Ref. [23], which is an LSS EFT paper and does not discuss CMB power spectrum likelihoods; a correct reference (e.g., Bond, Jaffe & Knox 1998) should be provided.
  2. [Fig. 1 caption] The caption says "The blue lines are the results for the Gaussian likelihood ... and the blue lines are the results for the perturbative likelihood," apparently duplicating "blue". The second should presumably be the perturbative color (red in Fig. 2). The row labels inside the figure (n̄ = 3×10^-4, 3×10^-3) should be reconciled with the caption values Q = 10^4, 10^3.
  3. [§4.1, Eq. (4.13)] The approximate second-order likelihood is not manifestly nonnegative; the text argues that it is positive when the perturbative expansion is valid (Eq. 4.18). This is a regime assumption, not a property of the expression itself, and the sentence "this fact is significant because it enforces the likelihood to be positive" should be qualified accordingly.
  4. [§6.2, Eq. (6.15)] The phrase "despite the fields cubed in the expression" appears garbled; it should be e.g., "despite the presence of cubic field combinations in the expression". Please clarify.

Circularity Check

1 steps flagged · score 3.0 of 10

The likelihood derivation is algebraically self-contained, but the numerical 'validation' is by construction: the mocks are generated from the same Gaussian prior whose covariance defines the likelihood's summary statistics.

  1. fitted input called prediction [Sec. 5 'Numerical posteriors'; Eqs. (3.15)-(3.16)]
    "In this section, we compare the perturbative likelihood of Eq. (3.16) with the usual Gaussian likelihood. To accomplish this, we generate a Gaussian field for δ(k) and for ϵ(k) and compute the numerical posterior of some parameters using the emcee code. ... we can also identify the theoretical prediction for the power spectrum at tree-level Pt(k|θ) := K a1(k|θ) C ab(k|θ) K b1(k|θ) = K2(1,0)(k|θ) P(k|θ) + K2(0,1)(k|θ) Q(k|θ)."

    The mock data are generated from the same diagonal Gaussian prior (Eq. 2.2) whose covariances P and Q enter the likelihood exclusively through Pt in Eq. (3.15). If the observed field is the linear model K(1,0)δ + K(0,1)ε, then by construction E[Po] = Pt. The MCMC recovery of the fiducial parameters is therefore guaranteed up to sampling noise; it is an internal consistency check, not an external validation of the PT or stochastic-Gaussian assumptions. The paper presents this as 'to validate our framework,' which overstates the evidential weight of a test whose null hypothesis is the model itself.

full rationale

The central derivation is not circular. Eqs. (2.1)-(2.2) state explicit assumptions, and the path-integral manipulations leading to Eqs. (3.16) and (4.20) are closed algebraic computations. The identification of Pt and Bt in Eqs. (3.15) and (4.14) follows directly from the covariance and third-moment structure of the assumed Gaussian generative model; this is a derivation, not a renaming of a fitted result. Self-citations (e.g., [13,20,26,27,47]) are used for context, literature comparisons, or mock generation, and none carries the mathematical argument, so they are not load-bearing circularity. The main circular element is the numerical validation: the mocks in Sec. 5 are drawn from the same Gaussian prior that defines the likelihood's summary statistics, so the numerical agreement is by construction. This does not invalidate the analytic derivation, but it means the numerical tests cannot confirm the stochastic-sector assumptions. App. A's [P/Q]^(n-1) suppression argument is an unverified assumption and a correctness risk, not a circularity. Because the analytic likelihood derivation would stand even without the numerical checks, the score is 3 rather than higher.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central derivation rests on the Gaussianity and mutual independence of the initial density field and the stochastic field, plus the exact delta-function relation between observed and modeled density. These are model assumptions, not derived facts. The PT kernels and stochastic power spectrum are free inputs, so the paper contributes a structural likelihood formula rather than a parameter-free prediction.

free parameters (3)
  • PT kernel functions K_a(n,m)(theta)
    The likelihood is conditioned on these kernels, including bias coefficients such as b1, b2, c1, and stochastic coefficients. Their values are not derived from first principles and must be supplied for any data application.
  • Stochastic sector power spectrum Q(k|theta) = constant 10^3 or 10^4 (h/Mpc)^3 in numerical tests
    The diagonal covariance of the Gaussian stochastic field is an input. The paper uses constant Q in simulations, but for real data it would be a free function or set of parameters.
  • Cutoff scales Lambda and k_NL
    The theoretical cutoff Lambda and the non-linear scale k_NL in kernel parametrizations are chosen by hand. The likelihood depends on them, though the paper notes renormalizability is not required.
assumptions (6)
  • domain assumption Primordial density field delta is exactly Gaussian with diagonal covariance P(k|theta).
    Used in the prior Eq. (2.2) and throughout; motivates the Gaussian path integrals in Secs. 3 and 4.
  • domain assumption There is one primordial Gaussian stochastic field epsilon per tracer, independent of delta, with diagonal covariance Q(k|theta).
    Invoked in Eq. (2.2) and App. A. The expansion of the stochastic sector around a Gaussian field is not proven and relies on a suppression factor [P/Q]^(n-1).
  • domain assumption The observed density field is exactly equal to the perturbative expansion of Eq. (2.1), enforced by the delta-function full likelihood Eq. (2.3).
    No observational noise or smoothing model beyond the cutoff is included; the deterministic mapping is the backbone of the derivation.
  • domain assumption The perturbative series Eq. (2.1) converges over the scales used, with lambda set to 1.
    The validity of the expansion and the positivity of the likelihood are assumed; the paper gives only a heuristic scaling argument in Sec. 4.1.
  • standard math Standard Gaussian integral identities and the Fisher information identity are valid in the continuum limit with V(k) cells.
    Used to derive Eqs. (3.9), (3.18), and App. C.
  • domain assumption SO(3) symmetry and constant kernels per radial bin in the discrete approximation.
    Used to convert continuous likelihoods to binned summary statistics in Secs. 3.2 and 4.2.
invented entities (1)
  • Primordial Gaussian stochastic field epsilon(k)
    purpose: Latent Gaussian field used as the expansion basis for the stochastic sector of the bias expansion; carries unresolved small-scale physics.
    The field itself is not observable and its statistics Q(k|theta) are free inputs. Although the field is standard in the EFT-of-LSS literature, the paper provides no new falsifiable handle beyond the assumed power spectrum.

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Pith. "Pith review of Perturbative Likelihoods for Large-Scale Structure of the Universe." pith.science (2026). https://pith.science/paper/CFGNXCJ3

@misc{pith2026250523750,
  author       = {Pith},
  title        = {Pith review of: Perturbative Likelihoods for Large-Scale Structure of the Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CFGNXCJ3}},
  note         = {Machine review of arXiv:2505.23750}
}
read the original abstract

This work presents a formalism for deriving likelihoods of the cosmological density field directly from first principles within Perturbation Theory (PT). By assuming a perturbative expansion around the Gaussian initial density field and additional stochastic components, we analytically compute two forms of the likelihood. Full marginalization over all underlying fields yields the likelihood of the observed density field, expressed in terms of its summary statistics (such as the power spectrum and bispectrum), which are naturally given by the formalism, and conditioned on model parameters. Marginalizing only over the stochastic fields results in the field-level likelihood. A key strength of this method is its ability to automatically specify the precise combinations of initial field covariances and PT expansion kernels required at each perturbative order (e.g., tree-level power spectrum and bispectrum, and the 1-loop power spectrum). This guarantees that the resulting likelihoods are fully consistent with PT at the chosen order of accuracy, avoiding ad-hoc choices in constructing the statistical model.

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Forward citations

Cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.