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On large scales, galaxy clustering is fixed by symmetries plus a finite set of EFT parameters that absorb small-scale chaos.

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2026-07-31 11:30 UTC pith:TNC32UP3

load-bearing objection Solid, self-contained lecture notes that package the standard one-loop EFT-of-LSS toolkit for students; useful training material, not a new result. the 1 major comments →

arxiv 2607.28314 v1 pith:TNC32UP3 submitted 2026-07-30 astro-ph.CO gr-qchep-phhep-th

GGI Lectures on Large-Scale Structure Perturbation Theory (Effective Field Theory)

Mikhail M. Ivanov This is my paper
classification astro-ph.CO gr-qchep-phhep-th
keywords effective field theorylarge-scale structureperturbation theorygalaxy biasredshift-space distortionsbaryon acoustic oscillationsIR resummationLagrangian perturbation theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

These lecture notes build, from scratch, the effective-field-theory description of cosmological large-scale structure. The claim is practical and structural: once scales are long compared with the nonlinear and galaxy-formation scales, the statistics of dark matter and galaxies in real and redshift space are controlled by symmetry (rotations, Galilean boosts, the equivalence principle) and a short list of free coefficients—sound-speed-like counterterms, bias operators, and stochastic noise—rather than by the microscopic details of halos or baryons. Standard pressureless fluid perturbation theory is shown to fail both in the ultraviolet (uncontrolled loop contributions from small scales) and for baryon acoustic oscillations (large bulk flows that must be kept unexpanded). The notes assemble the one-loop toolkit that repairs those failures: renormalized loops, IR resummation of the BAO, the Eulerian and Lagrangian bias expansions, redshift-space kernels and contact-operator renormalization, and the matching between Lagrangian and Eulerian formulations. A sympathetic reader cares because this is the template used to interpret ongoing and future galaxy surveys without pretending to solve nonlinear collapse from first principles.

Core claim

The notes establish that Standard Perturbation Theory is incomplete: its loops are UV-sensitive and its treatment of BAO wiggles breaks down, while an effective stress tensor, stochastic terms, and a symmetry-based bias and redshift-space expansion supply exactly the counterterms and operators needed. With IR resummation (equivalently organized in Lagrangian perturbation theory), the resulting one-loop EFT is claimed to be sufficient to model observed galaxy clustering in real and redshift space up to the quasi-linear scales relevant for surveys.

What carries the argument

The effective stress tensor in the Euler equation (plus its stochastic piece), constrained by Galilean invariance and the equivalence principle: it generates the k²P₁₁ counterterm and k⁴ stochastic noise that renormalize SPT loops, while the same symmetry principles fix the galaxy bias operators and the structure of redshift-space contact-operator counterterms.

Load-bearing premise

Nonlinear mode-coupling kernels are computed in the Einstein–de Sitter shortcut while only linear growth is taken from the true cosmology, on the claim that the error stays at the percent level or below for practical survey analyses.

What would settle it

Field-level and power-spectrum comparisons to N-body and hydro simulations: if one-loop EFT with the stated operators systematically fails to match phases and multipoles up to the advertised k_max once counterterms are fit, or if the EdS-kernel error exceeds the claimed sub-percent level in precision mocks, the completeness claim is wrong.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Survey full-shape analyses can treat small-scale and baryonic physics as a handful of fit nuisance parameters rather than as a barrier to theory.
  • BAO peak shape and damping are predicted once bulk flows are resummed; coherent long-wavelength displacements drop out by the equivalence principle.
  • Redshift-space multipoles need only a truncated set of µ-dependent counterterms and stochastic terms fixed by symmetries, not an arbitrary angular expansion.
  • Lagrangian and Eulerian one-loop EFT are the same expansion organized differently; LPT makes IR resummation automatic.
  • Stochastic power for galaxies can be white (constant) at low k, unlike matter’s k⁴ noise, because number is not conserved.

Where Pith is reading between the lines

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

  • If the scale hierarchy k_RSD ≪ k_NL is sample-dependent as the notes stress, survey strategy may favor emission-line or high-z samples where the EFT cutoff is higher.
  • The same symmetry skeleton should extend cleanly to the bispectrum and to mixed stochastic operators, which the outlook flags but does not fully develop here.
  • Field-level forward modeling is the natural stress test of whether the finite operator set is truly complete, beyond smooth power-spectrum fits.
  • Primordial non-Gaussianity enters as extra bias operators evaluated on Lagrangian coordinates—another direct use of the equivalence-principle logic taught for BAO.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 7 minor

Summary. These are pedagogical lecture notes developing the effective field theory of cosmological large-scale structure from symmetry principles, aimed at undergraduates and beginning graduate students with no assumed cosmology or QFT background. The notes build Newtonian fluid EFT, diagnose IR/UV failures of Standard Perturbation Theory (including BAO damping and equivalence-principle cancellations), introduce the effective stress tensor, counterterms, and stochastic terms, then extend the framework to galaxy bias, redshift-space distortions (including contact-operator renormalization and RSD stochasticity), and Lagrangian perturbation theory with IR resummation. The central claim is that a finite, symmetry-fixed set of one-loop operators and free EFT coefficients is sufficient to model observed galaxy clustering in real and redshift space on large scales.

Significance. The manuscript is a coherent, self-contained training resource for a framework that is now standard in full-shape analyses of BOSS/DESI and related surveys. Strengths include: symmetry-first derivations of the Navier–Stokes/EFT stress tensor and bias/RSD operators; explicit IR cancellation from the boost/equivalence principle with soft limits of F2/F3; the Peebles k^4 stochastic argument; clear treatment of redundant bias operators and RSD contact-term renormalization (including the structural truncation at μ^4); and LPT–SPT matching with a position-space BAO picture. Free parameters (γ, bi, αi, C̃i) are treated honestly as fit quantities rather than first-principles outputs. For students and new practitioners this is a high-value synthesis; it does not claim new precision results beyond the established toolkit.

major comments (1)
  1. [Chapter 3 (after Eq. 71)] Chapter 3 (after Eq. 71) and the EdS/Ω_m=f² kernel approximation: the notes assert that using EdS Fn,Gn with exact D+,f mismatches the fully time-dependent solution by ≲1% and is negligible for current/future surveys, citing prior work. For lecture notes whose claim is completeness of one-loop templates, a short explicit pointer (which reference, at which order, and for which observables—Pgg multipoles vs. bispectrum) would better let students judge when the shortcut fails. This is not a correctness error in the derivation, but it is the main modeling assumption load-bearing for the “sufficient for survey modeling” claim and should be documented more tightly in the text itself rather than only by citation.
minor comments (7)
  1. [§3.1] Section 3.1: “histrionically” should be “historically.”
  2. [§5.1] Section 5.1 title: “Phenomeological” → “Phenomenological.”
  3. [Figures 8–19] Several figures (e.g. Figs. 8–13, 16–19) are central to the pedagogy; ensure captions state cosmology, redshift, and whether IR resummation/counterterms are included so the plots are readable standalone.
  4. [§6.8] Chapter 6 summary table of free parameters is very useful; a parallel one-line power-counting table (which operators enter at tree / one-loop / stochastic for P0,P2,P4) would help students design analyses.
  5. [§4.4, §5.4, §6.6] Cross-references: the Peebles argument is invoked in both §4.4 and later bias/RSD stochastic sections; a forward/back pointer would reduce duplication confusion for first-time readers.
  6. [Appendices B–C] Appendix B/C are valuable; ensure the published version has complete, non-truncated derivations (the review copy cut off mid-heading in B.3) and that homework prompts have enough hints for the stated audience.
  7. [Chapter 7; global] Minor notation: Θ vs. θ, and q as loop momentum vs. Lagrangian coordinate, are flagged in Ch. 7 but could be restated once in a notation table early on.

Circularity Check

0 steps flagged

No significant circularity: pedagogical EFT notes with free counterterms/bias parameters, not fitted inputs sold as predictions.

full rationale

These are teaching notes that build the one-loop EFT of LSS from symmetries (continuity/Euler/Poisson, Galilean invariance, equivalence principle) and standard SPT/LPT bookkeeping. Loop UV pieces are absorbed into free Wilson coefficients (γ, A0, b_i, α_i, C̃_i) that the text explicitly treats as fit-to-data/simulation inputs, not first-principles outputs. IR BAO damping and k^4 stochasticity are derived from the boost/EP argument and Peebles mass–momentum conservation, not from renaming a fit. The EdS/Ω_m=f² kernel shortcut is flagged as an approximation with external accuracy claims, not smuggled uniqueness. Self-citations point to technical implementations in the same program but do not force the central pedagogical claim. No step reduces a claimed prediction to its own definition or fitted input by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The notes inherit the standard Newtonian sub-horizon cosmology, Gaussian adiabatic initial conditions, and the EFT power-counting ideology. Dynamical calculations lean on the EdS-style factorization of time and the single-stream fluid description before shell crossing. Free EFT coefficients are part of the framework being taught, not new inventions of this text; no new particles or forces are postulated.

free parameters (5)
  • Effective sound speed / counterterm γ = ≈1.8 (Mpc/h)^2 at z=0 (illustrative)
    Absorbs UV sensitivity of P13-type loops; fit to N-body (notes quote γ≈1.8 (Mpc/h)^2 at z=0 from Horizon Run).
  • Stochastic amplitude A0 (matter) / α0,α1,α2 (galaxies, RSD)
    Normalize short-mode noise; galaxies allow constant shot noise unlike matter’s k^4.
  • Galaxy bias coefficients b1, b2, b_G2, b_Γ3, b_∇²δ
    Independent deterministic bias parameters retained at one loop after removing redundant operators.
  • RSD counterterms C0, C̃1, C̃2 (optional c̃ k^4 μ^4)
    Renormalize contact operators in the real-to-redshift map; FoG may enhance line-of-sight coefficients.
  • IR cutoff k_IR in BAO damping Σ²
    Separates modes resummed into the BAO damping exponent from perturbative loops; scheme choice in IR resummation.
axioms (7)
  • domain assumption Separation of scales L_long ≫ L_short with analytic derivative expansion of long-wavelength observables
    Stated as the EFT ideology in Chapter 1; controls all power counting.
  • domain assumption Newtonian sub-horizon cosmology with matter+Λ background; other components smooth at late times
    Chapter 3 equations of motion; Poisson source 4πGρ−Λ.
  • domain assumption Initial conditions are Gaussian, adiabatic, and statistically homogeneous and isotropic
    Chapter 2; Wick theorem used for all loop diagrams.
  • domain assumption Equivalence principle / time-dependent uniform boost symmetry of the fluid+gravity system
    Chapter 3.3; enforces IR cancellation and structure of BAO damping.
  • domain assumption EdS approximation Ω_m=f² for nonlinear kernels while using exact linear D_+, f
    Chapter 3; claimed ≲1% error versus exact time dependence.
  • domain assumption Single-stream perfect-fluid SPT as the leading “monopole,” with τ_ij encoding short-scale physics
    Chapters 3–4; breaks down at shell crossing, motivating EFT.
  • standard math Standard calculus, Fourier analysis, and Gaussian functional integrals (Wick theorem)
    Appendix A and loop computations throughout.

pith-pipeline@v1.2.0-daily-grok45 · 69349 in / 3121 out tokens · 60684 ms · 2026-07-31T11:30:51.344054+00:00 · methodology

0 comments
read the original abstract

These notes are an introduction to non-linear perturbation theory for cosmological large-scale structure. They are aimed at undergraduate and beginning graduate students and do not require any cosmology or quantum field theory background. All necessary concepts are developed from scratch. The lectures are intended to explain all key ingredients needed to model the observed clustering of galaxies in real and redshift spaces. After a brief pedagogical introduction to the ideas of effective field theory (EFT), we develop large-scale structure EFT in the context of Newtonian cosmology using symmetry principles. We discuss in detail the shortcomings of Standard Perturbation Theory, the non-linear evolution of baryon acoustic oscillations and its relation to the equivalence principle, counterterms and renormalization of the loop diagrams, and stochastic effects. Then we develop EFT for galaxy bias and redshift space distortions. We also highlight some important facts about redshift-space stochasticity, relevant for ongoing and future galaxy surveys. Finally, we introduce Lagrangian Perturbation Theory.

Figures

Figures reproduced from arXiv: 2607.28314 by Mikhail M. Ivanov.

Figure 1
Figure 1. Figure 1: Left: the linear matter power spectrum as a function of wavenumber k for several values of redshift, computed with the class code [3]. Right: the z = 0 linear matter power spectrum and the asymptotic behaviours ∝ k as k → 0 and ∝ k −3 ln(k/keq), keq ≈ 0.01 h/Mpc. We ignore the power spectrum tilt ns [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Linear correlation function ξlin(r) of the matter overdensity in position space. The bump at r ≃ 100 Mpc/h is the peak of baryon acoustic oscillations (BAO). The right panel of fig. 1 demonstrates the above high-k and low-k asymptotics. This funny shape is the consequence of different dynamics of the dark matter field during the radiation and matter domination. This is why it has a peak at the equality sca… view at source ↗
Figure 3
Figure 3. Figure 3: Left: the decomposition of the linear matter power spectrum into the “wiggly” and “smooth” components. The results are multiplied by k to enhance the visibility of the wiggles. Right: the ratio of the wiggly to the smooth power spectrum, which is about ∼ 5% [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: the growth factor as a function of redshift z. We show the results for the fiducial ΛCDM cosmology and for a matter dominated Universe, where D+ = 1/(1 + z). Right: the logarithmic growth factor f(z) along with the approximate function [Ωm(z)]0.5 . linear matter density extrapolated to redshift zero, in which case the growth factor now can be normalized to unity, D+(z = 0) = 1. In a matter-dominated … view at source ↗
Figure 5
Figure 5. Figure 5: Left: The dimensionless power spectrum ∆2 (k, z) = P11(k) k 3/(2π 2 ) at z = 0 (blue) and z = 1 (orange). The nonlinear scale kNL(z) is defined by ∆2 (kNL) = 1 (black horizontal line); one finds kNL ≈ 0.25 h Mpc−1 (at z = 0) and kNL ≈ 0.75 h Mpc−1 (at z = 0). For k ≪ kNL the density field is perturbative (∆2 ≪ 1). Right: The linear power spectrum P11(k, z = 0) computed with class (blue) compared to the pow… view at source ↗
Figure 6
Figure 6. Figure 6: Left: The root-mean-square density fluctuation σ(R) = hR d 3q (2π) 3 P11(q) W˜ 2 R (q) i1/2 as a function of the smoothing scale R at z = 0, using a top-hat window function. The familiar nor￾malisation convention σ8 ≡ σ(R = 8 h −1Mpc) ≈ 0.8 is visible at R = 8 h −1Mpc. As R → 0, σ(R) diverges, reflecting the UV sensitivity of the unsmoothed variance — precisely the divergence that the EFT counterterms are … view at source ↗
Figure 7
Figure 7. Figure 7: Top: the SPT vertex Fn(q1, . . . , qn) with outgoing momentum k = q1 + · · · + qn (dashed lines stand for the remaining legs). Bottom: the tree-level and one-loop power spectrum diagrams. Each fat dot is a Wick contraction of two linear fields, i.e. an insertion of Plin, with the momentum of both lines flowing into it; the open squares are the kernels Fn, with F1 = 1 the trivial vertex. 3. Multiplicity: mu… view at source ↗
Figure 8
Figure 8. Figure 8: The one-loop matter power spectrum ∆P1−loop, which is a sum of the 2P13 and P22 diagrams. Dot-dashed lines show the leading infrared (IR) asymptotics of the loop integrals proportional to the displacement variance σ 2 d . The computation is carried out numerically with class-pt code [15] [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Matter power spectrum at z = 0: SPT loop corrections versus Horizon Run N-body simulation data [16]. The three-loop SPT power spectrum is adapted from Ref. [17] [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: One-loop power spectrum and its asymptotic IR and UV limits at [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: BAO in the position space correlation function at [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Left: effective sound speed estimator γˆ, showing the prediction for this parameter from individual (independent) k bins from the N-body data of [16]. The sound speed is constant γ ∼ 1.8 [ hMpc−1 ] 2 on large scales. The apparent scale-dependence at k ≳ 0.13 hMpc−1 is the effect of two-loop corrections not included in the estimator. Right: the stochastic power spectrum from the Quijote HR simulation suite… view at source ↗
Figure 13
Figure 13. Figure 13: Dark-matter power spectrum at z = 0, divided by the linear power spectrum Plin(k). Left: Standard Perturbation Theory (SPT). The one-loop correction (light blue) overshoots the N-body data (black points) already at k ≳ 0.05 h Mpc−1 , and the two-loop result (red) develops unphysical oscillations without converging toward the data. Right: The one-loop EFT result, which includes the counterterm −2γ k2Plin(k… view at source ↗
Figure 14
Figure 14. Figure 14: 1D cartoon explaining the mechanism of correlating proto-halo positions with long [PITH_FULL_IMAGE:figures/full_fig_p042_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: One-loop galaxy power spectrum basis shapes for the fiducial cosmology at [PITH_FULL_IMAGE:figures/full_fig_p047_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: EFT fits to galaxy power spectrum and galaxy-matter cross-spectrum of the mock [PITH_FULL_IMAGE:figures/full_fig_p048_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: One loop EFT predictions for redshift-space power spectrum multipoles of the lumi [PITH_FULL_IMAGE:figures/full_fig_p057_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The galaxy density field of DESI-like luminous red galaxies from the [PITH_FULL_IMAGE:figures/full_fig_p058_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The stochastic power spectra Pstoch of DESI-like Luminous Red Galaxies (LRG) from field-level EFT fits to MTNG hydrodynamical simulation (upper left panel) and Abacus halo occupation distribution (HOD) models (upper right panel). n¯ is the number density of the simulated galaxies. Pstochn¯ is different from unity on large scales due to halo exclusion effects. It is scale and orientation-independent (white… view at source ↗

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