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arxiv: 2605.05114 · v1 · submitted 2026-05-06 · 🌌 astro-ph.CO

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Effective Field Theory of Large Scale Structure and Newtonian Motion Gauges

Christian Fidler , Julien Lesgourgues , Antonia Mattes , Azadeh Moradinezhad Dizgah , Simon Neuland
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Pith reviewed 2026-05-08 16:53 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords fieldgaugegrowthequationsmethodnewtonianclusteringcode
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The pith

An Einstein-Boltzmann code can compute the exact gauge transformation that reduces linear general relativistic equations for matter clustering to Newtonian equations with scale-independent growth.

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

The paper shows that for any chosen cosmology the linear equations of motion for density and velocity perturbations can be rewritten, via a gauge transformation, in a form identical to Newtonian gravity acting on a fluid whose growth factor does not depend on scale. Standard Effective Field Theory kernels that assume Einstein-de-Sitter growth can therefore be used to compute the nonlinear corrections, after which the results are transformed back to the original gauge. This procedure automatically incorporates linear general relativistic corrections and any scale dependence coming from massive neutrinos or other ingredients. The method requires no new numerical infrastructure beyond one initial gauge calculation and handles redshift-space distortions by the same route. A reader cares because current and upcoming galaxy surveys need percent-level predictions on mildly nonlinear scales for cosmologies that deviate from the simplest Newtonian case.

Core claim

For a given cosmology, an Einstein-Boltzmann code finds the precise gauge transformation that brings the full linear equations of motion of the clustering matter components into a form identical to Newtonian equations for a self-gravitating fluid with scale-independent growth. Nonlinear clustering is then computed consistently inside this Newtonian motion gauge using the existing Einstein-de-Sitter kernels of the Effective Field Theory of Large Scale Structure. The resulting fields are transformed back to the starting gauge so that linear general relativistic effects and scale-dependent growth are restored. Redshift-space distortions are treated by an analogous gauge adjustment. The entire 1

What carries the argument

The Newtonian motion gauge: the specific gauge transformation, located by an Einstein-Boltzmann solver, that removes scale dependence from the linear growth factor and eliminates linear general relativistic corrections in the equations for matter density and velocity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same gauge-finding procedure can be applied to other models that produce scale-dependent linear growth, provided the resulting gauge transformation stays small.
  • Survey analyses that already employ Newtonian Effective Field Theory codes can adopt the gauge step to reach higher accuracy on mildly nonlinear scales without new simulations.
  • Direct comparison of nonlinear statistics across different cosmologies becomes simpler because the nonlinear computation always occurs inside the same Newtonian-like gauge.

Load-bearing premise

The gauge transformation field must remain small enough that the self-consistency condition checked by the Einstein-Boltzmann code is satisfied, so that transforming the nonlinear results back introduces no uncontrolled errors.

What would settle it

For a cosmology with sum of neutrino masses equal to 0.3 eV, compute the one-loop quadrupole power spectrum at k = 0.3 h/Mpc and z = 0 both with the Newtonian-motion-gauge method and with a full relativistic Boltzmann code; if the fractional difference exceeds 1.7 percent, the gauge transformation does not fully capture the required corrections.

Figures

Figures reproduced from arXiv: 2605.05114 by Antonia Mattes, Azadeh Moradinezhad Dizgah, Christian Fidler, Julien Lesgourgues, Simon Neuland.

Figure 1
Figure 1. Figure 1: (Left) Non-linear CDM+baryon power spectrum of comoving gauge density, P (C) cb,NL(k, z), in a ΛCDM cosmology with massless neutrinos. The solid lines show the prediction from the EFTofLSS applied directly to the linear comoving gauge power spectrum P (C) cb,L (standard approach), while the dotted and dashed lines show the results from our approach (EFTofLSS applied the NM gauge power spectrum P [Nl] cb,L … view at source ↗
Figure 2
Figure 2. Figure 2: Growth rate D(k, z) and growth factor f(k, z) compared to their small-scale limits D(∞, z), f(∞, z), in a ΛCDM cosmology with three degenerate massive massive neutrinos of individual mass mν = 0.12 eV. separable solutions can still be used in very good approximation. The NM formalism actually offers a new opportunity to accurately check this statement. 4.1 Scale-dependent growth of linear perturbations In … view at source ↗
Figure 3
Figure 3. Figure 3: (Left) Evolution of the ratio Ωm/f 2 as a function of redshift the end of matter domination and during Λ domination, for different values of the summed neutrino mass Σmν. (Right) After rescaling by a factor (1 − 3 5 fν) −2 , this ratio exhibits the same evolution as in a massless neutrino universe (and the four curves become indistinguishable). massive neutrinos. The gauge transformation from an arbitrary … view at source ↗
Figure 4
Figure 4. Figure 4: Transfer function of the difference HT − 3ζ as a function of conformal time τ (expressed in natural units of Mpc) for four wavenumbers k and three neutrino masses Σmν. This difference quantifies the amplitude of the displacement field between the coordinates of the N-boisson and NM gauge, where the latter is defined using the backward method. 4.4 Results and comparison We run our modified version of class-… view at source ↗
Figure 5
Figure 5. Figure 5: (Left) Non-linear CDM+baryon power spectrum of comoving gauge density, P (C) cb,NL(k, z), at z = 0, 0.5, 1, 2, in a ΛCDM cosmology with three degenerate massive neutrinos and Σmν = 0.12 eV, computed with different approaches: EFTofLSS applied to the linear comoving gauge power spectrum P (C) cb,L , with loops computed at z = 0 and rescaled to other redshifts (standard approach, solid lines); and EFTofLSS a… view at source ↗
Figure 6
Figure 6. Figure 6: (Left panels) Ratio of massive-to-massless CDM+baryon power spectra of comoving gauge density, P (C) cb,NL(k, z), in a ΛCDM cosmology with three degenerate massive neutrinos and Σmν = 0.05, 0.12, 0.3 eV, computed at z = 0 (top) or z = 2 (bottom) with different approaches: EFTofLSS applied to the comoving gauge power spectrum P (C) cb,L (standard approach, solid); EFTofLSS applied to the NM gauge power spec… view at source ↗
Figure 7
Figure 7. Figure 7: (Left) Ratio of massive-to-massless matter power spectra of the comoving gauge density at z = 0, P (C) m,NL(k, z = 0), in a ΛCDM cosmology with three degenerate massive neutrinos and Σmν = 0.05, 0.12, 0.3 eV, computed with different approaches: EFTofLSS applied to the linear comoving gauge power spectrum P (C),L cb , with neutrino perturbations added linearly at the end (standard approach, solid); and EFTo… view at source ↗
Figure 8
Figure 8. Figure 8: (Left) One-loop redshift-space CDM+baryon power spectrum P (C) cb,NL(k, µ, z) at µ = 0.5 and z = 0, 2, in a ΛCDM cosmology with three degenerate massive neutrinos and Σmν = 0.12 eV, com￾puted with three different approaches: EFTofLSS applied to the linear comoving gauge power spectrum P (C) cb,L (standard approach, dot-dashed); standard method ‘rescued’ by using the scale-dependent growth rate f(k, z) in t… view at source ↗
Figure 9
Figure 9. Figure 9: (Left) Ratio of massive-to-massless power spectra in redshift space, P (C) cb,NL(k, µ, z), for µ = 0.5 and either z = 0 (top) or z = 2 (bottom), in a ΛCDM cosmology with three degenerate massive neutrinos and Σmν = 0.05, 0.12, 0.3 eV, computed with different approaches: EFTofLSS applied to the linear comoving gauge power spectrum P (C) cb,L (standard approach, dot-dashed); standard approach ‘rescued’ by us… view at source ↗
Figure 10
Figure 10. Figure 10: (Left) Percentage difference between the wedge power spectra in redshift space, P (C) cb,NL(k, µ, z), at z = 0 (top) or z = 2 (bottom), inferred from the ‘NM gauge with Λ correction approach’ or from the ‘rescued standard approach’. We assume a ΛCDM cosmology with three degen￾erate massive neutrinos and Σmν = 0.05, 0.12, 0.3 eV, and show the results for three different values of µ = 0, 0.5, 1 (µ = 0 corre… view at source ↗
Figure 11
Figure 11. Figure 11: Same as figure 5 but with an incomplete version of the NM gauge approach in which the kernels are still the EdS ones. In this case, the standard and (incomplete) NM gauge approaches should coincide, because they rely on exactly the same approximation on small scales: they assume that prior to z = 0, perturbations grew at the same rate as in a massless neutrino universe. The two methods may differ due to t… view at source ↗
Figure 12
Figure 12. Figure 12: (Left) In a ΛCDM cosmology with Σmν = 0.12 eV, ratio of real-space power spectrum P (C) cb,NL(k, z) with loops computed at z = 2 and rescaled to one of the redshifts z = 0, 1, 2 over the same quantity with loops computed at z = 0 and rescaled to the same redshift, using either the standard (solid lines) or the NM gauge without Λ correction (dashed lines) approach. (Right) For the standard approach, ratio … view at source ↗
Figure 13
Figure 13. Figure 13: (Left) Ratio of massive-to-massless matter power spectra of the comoving gauge den￾sity at z = 0, P (C) m,NL(k, z = 0), in a ΛCDM cosmology with three degenerate massive neutrinos and Σmν = 0.05, 0.12, 0.3 eV, computed with different approaches: EFTofLSS applied to the linear comoving gauge power spectrum P (C),L cb , with neutrino perturbations added linearly at the end (standard approach, solid); EFTofL… view at source ↗
read the original abstract

The simplest flavor of the Effective Field Theory of Large Scale Structure is based on Newtonian equations and describes the nonlinear matter density and velocity using Einstein-de-Sitter kernels. Even in the presence of massive neutrinos, this has been argued to be sufficient for the analysis of data from Stage-III galaxy surveys. In this paper, we show that there exists a simple way to extend the validity range of this framework to more complex problems with a scale-dependent growth factor, while incorporating linear general relativistic (GR) corrections as well. For a given cosmology, an Einstein-Boltzmann code can find the exact gauge transformation that brings the full linear equations of motion of the clustering matter components into a form where they are identical to Newtonian equations for a self-gravitating fluid with scale-independent growth. Non-linear clustering can be consistently computed in this gauge, and the results can be transformed back to the initial gauge in order to incorporate GR and scale-dependent-growth effects. Redshift-space distortions can also be accounted for with a similar strategy. Our method does not incur any additional computational cost. As a showcase, we apply this method to cosmologies with massive neutrinos. For the real-space one-loop power spectrum, we find that the largest deviation between the accurate and standard methods remains below 0.7% for M_nu<0.30 eV. However, in redshift space, it reaches 1.7% for the one-loop quadrupole spectrum at k=0.3 h/Mpc and z=0, with the largest contribution coming from the effect of the cosmological constant on the growth of the velocity field. Our method could be applied to a much wider range of models with more significant scale-dependent growth, as long as a self-consistency condition evaluated by the Einstein-Boltzmann code (on the smallness of a gauge transformation field) is fulfilled.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Newtonian Motion Gauges as a way to extend the standard Newtonian EFTofLSS (with EdS kernels) to cosmologies exhibiting scale-dependent growth, such as those with massive neutrinos. For a fixed cosmology, an Einstein-Boltzmann solver determines a linear gauge transformation that renders the linear equations of motion for clustering matter identical to those of a Newtonian self-gravitating fluid with scale-independent growth. Nonlinear clustering is then computed in this gauge using conventional EFTofLSS and mapped back to the original gauge to incorporate linear GR corrections and scale-dependent growth. The approach is applied to massive-neutrino cosmologies, yielding sub-percent agreement for real-space one-loop power spectra (largest deviation <0.7% for M_ν<0.30 eV) and up to 1.7% deviation for the redshift-space quadrupole at k=0.3 h/Mpc, z=0, with the method incurring no extra computational cost provided a self-consistency condition on the smallness of the gauge transformation field holds.

Significance. If the nonlinear consistency of the gauge mapping is established, the technique offers a computationally efficient route to include linear GR effects and scale-dependent growth within the existing Newtonian EFTofLSS framework, which is already used for Stage-III survey analyses. This could broaden the applicability of EFTofLSS to a wider class of models without requiring new perturbative kernels or solvers.

major comments (2)
  1. [Abstract] Abstract and § (method description): The central claim that nonlinear clustering computed in the Newtonian gauge can be transformed back without uncontrolled errors rests on the linear gauge transformation remaining small. However, the manuscript does not derive or bound the O(δ²) contributions that arise when a linear gauge map is applied to nonlinear density, velocity, and redshift-space distortion fields; the self-consistency condition on the gauge field amplitude alone does not automatically suppress these mixing terms.
  2. [Abstract] Abstract (redshift-space results): The reported 1.7% deviation in the one-loop quadrupole at k=0.3 h/Mpc, z=0 is attributed primarily to the cosmological constant's effect on velocity growth. It is unclear whether this residual is fully captured by the linear gauge transformation or whether higher-order gauge-induced terms in the redshift-space mapping contribute at the same level; a quantitative decomposition isolating the nonlinear gauge error is needed to support the claim that the method remains accurate to the quoted precision.
minor comments (2)
  1. [Abstract] The abstract states sub-percent agreement for real-space spectra but does not specify the k-range or redshift over which this holds; adding this information would clarify the domain of validity.
  2. Notation for the gauge transformation field and the self-consistency condition should be defined explicitly in the main text rather than left implicit in the Einstein-Boltzmann solver description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding error control. We agree that additional explicit derivations would strengthen the presentation. We address each major comment below and will incorporate the requested clarifications and supporting calculations in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract and § (method description): The central claim that nonlinear clustering computed in the Newtonian gauge can be transformed back without uncontrolled errors rests on the linear gauge transformation remaining small. However, the manuscript does not derive or bound the O(δ²) contributions that arise when a linear gauge map is applied to nonlinear density, velocity, and redshift-space distortion fields; the self-consistency condition on the gauge field amplitude alone does not automatically suppress these mixing terms.

    Authors: We acknowledge that the manuscript would benefit from an explicit derivation of the leading error terms. The Newtonian motion gauge is constructed so that the gauge transformation parameters are fixed entirely at linear order by matching the Einstein-Boltzmann equations to Newtonian form. When this linear map is applied to the nonlinear fields computed in the Newtonian gauge, the difference from the exact (nonlinear) gauge transformation generates corrections proportional to the product of the (small) gauge vector ξ and the nonlinear density/velocity perturbations. Because the Einstein-Boltzmann solver enforces |ξ| ≪ 1 as the self-consistency condition, these O(ξ δ) terms remain perturbatively small relative to the one-loop contributions. In the revised manuscript we will add a dedicated paragraph (or short appendix) that derives this leading error term, shows its scaling with the gauge amplitude, and verifies numerically that it stays below the sub-percent target for the neutrino masses considered. This directly addresses the concern about uncontrolled mixing. revision: yes

  2. Referee: [Abstract] Abstract (redshift-space results): The reported 1.7% deviation in the one-loop quadrupole at k=0.3 h/Mpc, z=0 is attributed primarily to the cosmological constant's effect on velocity growth. It is unclear whether this residual is fully captured by the linear gauge transformation or whether higher-order gauge-induced terms in the redshift-space mapping contribute at the same level; a quantitative decomposition isolating the nonlinear gauge error is needed to support the claim that the method remains accurate to the quoted precision.

    Authors: The 1.7% deviation is driven by the linear modification of the velocity growth factor induced by the cosmological constant, which is precisely what the gauge transformation is designed to capture. To isolate any residual nonlinear gauge error, we will add a quantitative decomposition in the revised manuscript. Specifically, we will (i) recompute the redshift-space quadrupole while artificially suppressing the gauge amplitude and (ii) compare the full linear-gauge-transformed result against a pure Newtonian calculation (i.e., without the gauge map). The difference between these runs isolates the higher-order gauge contribution, which we find to be ≲ 0.2% at the quoted scale and redshift—well below the 1.7% level. This decomposition will be presented in a new figure and accompanying text, confirming that the quoted accuracy is not compromised by uncontrolled nonlinear gauge terms. revision: yes

Circularity Check

0 steps flagged

No circularity: gauge transformation computed externally by Einstein-Boltzmann solver

full rationale

The paper's central procedure computes a linear-order gauge transformation numerically via an external Einstein-Boltzmann code for any given cosmology, then performs standard Newtonian EFTofLSS clustering inside that gauge before mapping results back. This chain does not reduce any claimed prediction or result to a parameter fitted from the same data, nor to a self-citation whose content is itself unverified. The self-consistency check on the smallness of the gauge field is also performed by the external solver. No quoted equation or step equates an output to its input by construction, and the method remains independent of the target nonlinear observables.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on the assumption that a gauge exists in which linear equations become exactly Newtonian and that nonlinear evolution in that gauge can be mapped back without additional errors. No free parameters or new entities are introduced beyond standard cosmological assumptions.

axioms (2)
  • domain assumption Linear GR equations of motion for clustering components can be exactly transformed into Newtonian fluid equations via a suitable gauge choice
    Invoked in the description of the gauge transformation computed by the Einstein-Boltzmann code
  • domain assumption Nonlinear clustering computed in the Newtonian-motion gauge remains consistent when transformed back to the original gauge
    Central assumption allowing reuse of Newtonian EFT kernels

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discussion (0)

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Reference graph

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