The Redshift-Space Galaxy Power Spectrum at Two-Loop Order

Oliver H. E. Philcox

REVIEW

The Redshift-Space Galaxy Power Spectrum at Two-Loop Order

Oliver H. E. Philcox

Cosmology and Nongalactic Astrophysics · Submitted 9 Oct 2026

Pith's verdict: First complete two-loop redshift-space galaxy power spectrum, carefully derived and honestly validated; the load-bearing renormalization closure check is asserted rather than shipped.

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arXiv ID2610.13867 (version 2)

Pith IDHKYZNNBH, Pith's permanent record of this paper and its review

CitationOliver H. E. Philcox. "The Redshift-Space Galaxy Power Spectrum at Two-Loop Order." arXiv:2610.13867 (2026). https://pith.science/paper/HKYZNNBH

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Editor's note

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Two things you should know. First, this is the first complete two-loop redshift-space galaxy power spectrum in EFT, and it is a serious piece of work. Second, the one load-bearing step—closure of the fifth-order operator basis under renormalization—is verified internally but not shipped in any checkable form. My verdict tracks the reader's CONDITIONAL.

What is new is real. The real-space two-loop galaxy model existed (Bakx et al., Ivanov) and redshift-space two-loop dark matter existed (Taule and Garny), but nobody had the full redshift-space galaxy case with fifth-order bias, counterterms, stochasticity, and IR resummation together. The same 49-parameter model carries the deterministic one-loop bispectrum and trispectrum, which is a genuine bonus for joint analyses.

Credit where earned. The one-loop pipeline is anchored by a class-pt comparison agreeing to about 0.2% once IR-resummation conventions are matched. Dark-matter hard limits are checked against published results, including a corrected factor of two in an earlier double-hard limit. Validation uses 25 AbacusSummit realizations, three separate failure metrics, and both individual-box and combined analyses. The kmax numbers (0.34 h/Mpc for LRGs at z=0.5, 0.4 for ELGs) are credible as stated, and the paper is honest that sigma8 constraints do not improve.

Soft spots, in proportion. The closure check is the real one. The paper states that the single-, nested, and double-hard limits of Z3–Z5 are exactly expressible in lower-order kernels, verified explicitly "at every power of f," but the notebooks and templates are not released and the algebra is too large to check from the text. The stress-test note is right that a missed structure would be absorbed by the 49 coefficients and invisible to the fits. I do not read that as evidence of an error—the two-language implementation, rank-saturation enumeration, and dark-matter anchors all point the right way—but it limits what the manuscript can certify. Code release would settle it. Second, the kmax cuts are selected on the same simulations used for validation; standard practice, but the headline reach is a fit claim, not a prediction. Third, the sigma8 recovery test is much stronger for dark matter (0.3% precision) than galaxies (2%), so galaxy-sector validation mostly tests flexibility. The AI-computation disclosure is transparent, and only adds to the case for shipping the kernels.

Bottom line: bring it to the reading group, cite it if you work on EFT power spectra, and send it to referees. The completeness claim deserves careful refereeing, and the code should be required to ship.

Referee report

Significance

If the closure claim holds, this is a landmark calculation: the first complete two-loop redshift-space galaxy power spectrum model, with operator enumeration, hard-limit renormalization, and an efficient implementation presented together. The paper's strengths are concrete: cross-validation of the one-loop pipeline against class-pt at better than 0.2% agreement without IR resummation (Appendix B); dark-matter hard limits matching published results [22,28,37,93]; dual Mathematica/Python kernel implementations verified against the LaTeX definitions (§III A); explicitly stated scale cuts determined by three distinct metrics; and an honest conclusion, repeated in the abstract, that the additional small-scale information mostly calibrates fifth-order nuisance parameters rather than improving sigma_8. The main gap is that the completeness of the operator basis, which is what distinguishes a complete two-loop model from an empirically adequate parameterization, rests on an assertion about unpublished hard-limit algebra; the simulation fits cannot certify this step because the fitted coefficients would absorb a missed UV-sensitive structure.

Major objections (2)

[§II C-D and §III B, Eqs. (25)-(26), (61)-(64)] The closure of the fifth-order operator basis under renormalization is the single load-bearing step for the paper's central 'complete model' claim, and it is currently certified only by assertion. For the two-loop integrals to be UV-finite, the angle-averaged single-hard limits of Z3-Z5 and the nested and double-hard limits of Z5 must be exactly expressible in terms of lower-order kernels (Z3; Z1 and ZNLG1) at every power of f. The paper states this was 'verified explicitly' (§II D) and that the dark-matter limits match [22,28,37,93] (§III B), but the galaxy-sector, fifth-order, redshift-space verification itself — the Mathematica/Python kernel computations, the rank-saturation enumeration of §III A starting from 1982 candidates, and the hard-limit reduction templates — is not shipped or documented in a checkable form. Because a missed UV-sensitive structure would be absorbed by the 49 fitted coefficients at the fitting scale without changing chi^2 or sigma_8, the AbacusSummit fits cannot test this step. I request that the verification artifacts be provided as ancillary material, or at minimum that a fully worked non-trivial example of a fifth-order redshift-space hard-limit reduction (including nested and double-hard limits at general f) be written out, and that the text state explicitly which claims are certified by the closure check versus by the simulation fits.
[§IV C-D and Tab. IV] The simulation validation, while convincing as a test of the model's predictive performance at the chosen scale cuts, does not test the completeness of the operator basis, and the paper does not draw this distinction sharply. The three metrics of §IV C are evaluated with all parameters free (49 for galaxies, 28 for dark matter in redshift space, with most entering analytically after marginalization); since the two-loop corrections are largely absorbed by the fifth-order coefficients — as the paper itself notes in the abstract and §V — a chi^2/nu_eff near unity at k_max = 0.34 h/Mpc demonstrates that the model is flexible enough to describe the data, not that the operator list is closed under renormalization. I recommend that the paper separate these two claims explicitly: an effective description valid on the tested scales (supported by Figs. 2, 3, 5, and 7) versus theoretical completeness (certified only by the closure check of §III B). The caveat of footnote 16, that some operator degeneracies hold only after angular averaging and formally fail under anisotropic IR resummation at three-loop order, should be connected to this distinction.

Minor objections (5)

[§II D, opening paragraph] In the previous section, we discussed the renormalize of the real-space density field' — 'renormalize' should read 'renormalization'.
[§IV D, numerical convergence paragraph] Such biases are are negligible' contains a duplicated 'are'; please correct.
[Fig. 6 caption] The caption contains a subject-verb agreement error ('the quadrupoles receives') and a stray double period after 'hexadecapole moment.' before 'While the monopoles'.
[Appendix C and Appendix D] The symbol G is overloaded: it denotes both the SPT velocity kernels G_n in Appendix D and the Green's-function integrals G^{(p)}_delta, G^{(p)}_S, G^{(p)}_V in Appendix C. In a paper of this length, renaming one set would materially reduce reader confusion.
[§II G, Eq. (50)] The reduction of the stochastic sector to a Taylor expansion in k^2 and k^2_parallel is asserted in a single sentence, with the mixed deterministic-stochastic loop integrals dismissed by citation to [40,42,69]. For the power spectrum the statement may well be true, but one clarifying sentence on why those integrals reduce to the displayed k^2 and k^4 terms would help the reader.

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The paper

The paper claims

This paper presents the first complete two-loop effective field theory model of the galaxy power spectrum in redshift space, with 49 free coefficients, and shows it fits simulated LRG-like galaxies without bias out to $k_{\rm max} =…

The reading

Contemporary galaxy-survey cosmology leans on the one-loop effective field theory (EFT) power spectrum, which is trusted only down to wavenumbers near $k \approx 0.2\,h\,{\rm Mpc}^{-1}$. This paper aims to build the complete two-loop model in redshift space: every galaxy-bias, counterterm, redshift-space, stochastic, infrared-resummation, and coordinate-distortion contribution through fifth order in the density field, packaged into 49 free coefficients. If the construction is right, full-shape analyses can trust the model to $k_{\rm max} = 0.34\,h\,{\rm Mpc}^{-1}$ for LRG-like galaxies, roughly 70 percent more wavenumber range, and, because the same fifth-order kernels generate the one-loop bispectrum and trispectrum, joint analyses of several statistics become possible with a shared parameter set. The model is validated against 25 AbacusSummit simulations, and the authors report that the added small-scale information mostly calibrates the new coefficients rather than sharpening $\sigma_8$ constraints.

What carries the argument

The load-bearing object is the renormalized kernel tower $Z_1$ through $Z_5$: the Fourier-space response of the galaxy field through fifth order in the linear density field. These kernels are assembled from bias operators built from the convective tidal tensors $\Pi^{[n]}$ — fields advected along fluid trajectories, which fold time non-locality into local operators — mapped to redshift space through the exponential distortion $(1+\delta_g)\exp(-i f k_\parallel u_\parallel)$, and supplemented by higher-rank counterterms $C^{i_1\cdots i_m}_{u^m}$ that renormalize the velocity moments. The argument is carried by the renormalization scheme: the single-hard, nested, and double-hard momentum limits are subtracted from $Z_3$ through $Z_5$, and the operator basis is declared closed only if every angle-averaged subtracted limit is exactly re-expressible in terms of lower-order kernels. That closure converts a long operator list into a complete model. The computational backbone is a Vegas Monte Carlo evaluation of the loop integrals over precomputed bias templates, which brings each likelihood evaluation down to milliseconds.

Extended reading notes

Core claim

The central claim is that the redshift-space galaxy power spectrum can be modeled completely, within the EFT of large-scale structure, through two-loop order. The model assembles a fifth-order Eulerian bias expansion built from convective tidal tensors $\Pi^{[n]}$, redshift-space counterterms of tensor rank zero through four that renormalize velocity moments, two non-local-in-space operators forced by dark-matter renormalization, an anisotropic wiggle-no-wiggle infrared resummation, six stochastic amplitudes, and Alcock-Paczynski rescaling, for a total of 49 independent coefficients (29 in real space). Against 25 AbacusSummit realizations, the model fits the monopole, quadrupole, and hexadecapole of an LRG-like sample at $z=0.5$ without $\sigma_8$ bias, parameter running, or excess $\chi^2$ up to $k_{\rm max} = 0.34\,h\,{\rm Mpc}^{-1}$, where the one-loop model breaks down near $0.2\,h\,{\rm Mpc}^{-1}$; an ELG-like sample at $z=1.1$ is fit to at least $0.40\,h\,{\rm Mpc}^{-1}$. The same kernels fix the deterministic one-loop bispectrum and trispectrum and the tree-level quadspectrum and pentaspectrum, so the work is simultaneously a joint-statistics model. The authors are explicit that two-loop terms do not dramatically improve $\sigma_8$ constraints from the power spectrum alone, since the new small-scale information is largely absorbed by the biases and counterterms; only $\sigma_8$ and the Wilson coefficients are varied in the validation, with full cosmological inference left to future work.

Load-bearing premise

The load-bearing premise is that the fifth-order operator basis is closed under renormalization: every UV-divergent structure produced by the two-loop integrals — the single-hard, nested, and double-hard limits of the $Z_5$ kernels — is exactly expressible in terms of the tabulated lower-order operators, so that no missing operator is needed to absorb the cutoff dependence. The paper states that this was verified explicitly, but the verification is not shipped as code or certificates, so the completeness of the 49-parameter model rests on a check a reader cannot rerun without re-deriving it.

Editorial extensions

If this is right

Full-shape survey analyses can extend the usable wavenumber range from about $0.2$ to $0.34\,h\,{\rm Mpc}^{-1}$ for LRG-like samples (and beyond $0.4\,h\,{\rm Mpc}^{-1}$ for higher-redshift ELG-like samples) before the model shows bias or parameter running.
The deterministic one-loop bispectrum and trispectrum, and the tree-level quadspectrum and pentaspectrum, are fixed by the same fifth-order kernels, so power-spectrum and higher-point analyses can share one parameter set instead of being calibrated separately.
Two-loop terms alone will not dramatically tighten $\sigma_8$ constraints from the power spectrum, because the new small-scale information mostly calibrates the additional bias and counterterm coefficients rather than the broadband shape.
The millisecond likelihood evaluation makes a 49-parameter two-loop fit computationally practical for survey-scale data, provided the loop templates are precomputed once per cosmology.
The one-loop limit of the pipeline agrees with existing public codes once infrared-resummation conventions are aligned, which is the consistency anchor for trusting the two-loop results.

Reading between the lines

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

If the basis is genuinely closed at fifth order, the same enumeration-and-hard-limit machinery should extend to the three-loop power spectrum without a new derivation of the operator set, making the next bottleneck computational rather than conceptual.
The reported $\sigma_8$ saturation suggests the two-loop information is most likely to surface in parameters that shift the broadband shape differently from galaxy bias — growth, the Hubble constant, or primordial non-Gaussianity — or in joint power-spectrum-bispectrum-trispectrum fits, which the paper leaves as future work.
A sharper validation than fitting would be a prediction test: fix the coefficients from $k<0.15\,h\,{\rm Mpc}^{-1}$ data and check whether the model then predicts the $0.2$–$0.34\,h\,{\rm Mpc}^{-1}$ band; a systematic residual would reveal operators missing from the tabulated basis even if the fit passes.
Because the non-local-in-space counterterms are inherited from dark-matter renormalization, multi-tracer surveys at one redshift should see shared amplitudes for those operators, and measuring that sharing would test the locality-in-space assumption for galaxy formation that the paper adopts.

Original abstract

Contemporary large-scale structure analyses center around the galaxy power spectrum, modeled with one-loop perturbation theory. This description naturally breaks down at large wavenumbers, prohibiting the extraction of cosmological information from small-scale clustering. In this work, we present a complete model for the redshift-space galaxy power spectrum up to two-loop order, derived in the framework of the Effective Field Theory of Large Scale Structure. This accounts for a wide variety of effects, including fifth-order galaxy bias, redshift-space distortions, small-scale non-linearities, spatial non-locality, bulk flows, coordinate distortions, and stochasticity, involving 49 free coefficients. Our model additionally encodes all deterministic and counterterm contributions to the one-loop bispectrum and trispectrum (as well as the tree-level quadspectrum and pentaspectrum), which depend on many of the same parameters. We implement the model numerically, using the VEGAS algorithm to precompute the loop integrals; this allows the full likelihood to be evaluated in milliseconds. The approach is validated using power spectra measured from high-resolution simulations, considering both dark matter and realistic galaxy samples. The extension to two-loop order greatly increases the range of scales over which the power spectrum can be modeled, with $k_{\rm max}$ rising from $0.20h\,\mathrm{Mpc}^{-1}$ to $0.34h\,\mathrm{Mpc}^{-1}$ for a luminous red galaxy sample at $z=0.5$, or $0.27h\,\mathrm{Mpc}^{-1}$ to $0.42h\,\mathrm{Mpc}^{-1}$ for an emission line galaxy sample at $z=1.1$. Much of the additional information is absorbed by the numerous fifth-order biases and counterterms; for this reason, constraints on $\sigma_8$ are not significantly enhanced by the inclusion of two-loop terms. Future work will assess the implications of the two-loop model for cosmological parameter inference.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-12: 1 request, 7,898 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.95 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.89 · mentions code or data availability 0.04.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Contemporary large-scale structure analyses center around the galaxy power spectrum, modeled with one-loop perturbation theory. This description naturally breaks down at large wavenumbers, prohibiting the extraction of cosmological information from small-scale clustering. In this work, we present a complete model for the redshift-space galaxy power spectrum up to two-loop order, derived in the framework of the Effective Field Theory of Large Scale Structure. This accounts for a wide variety of effects, including fifth-order galaxy bias, redshift-space distortions, small-scale non-linearities, spatial non-locality, long-wavelength bulk flows, coordinate distortions, and stochasticity, involving a total of 49 free coefficients. Our model additionally encodes all deterministic contributions to the one-loop bispectrum and trispectrum (as well as the tree-level quadspectrum and pentaspectrum), which depend on many of the same parameters. We implement the model numerically, using the \textsc{vegas} algorithm to precompute the loop integrals; this allows the full likelihood to be evaluated in milliseconds. The approach is validated using power spectra measured from high-resolution simulations, considering both dark matter and realistic galaxy samples. The extension to two-loop order greatly increases the range of scales over which the power spectrum can be modeled, with $k_{\rm max}$ rising from $0.2h\,\mathrm{Mpc}^{-1}$ to $0.34h\,\mathrm{Mpc}^{-1}$ for a luminous red galaxy sample at $z=0.5$. Much of the additional information is absorbed by the numerous fifth-order biases and counterterms; for this reason, constraints on $\sigma_8$ are not significantly enhanced by the inclusion of two-loop terms. Future work will assess the full implications of the two-loop model for cosmological parameter inference, both alone and in combination with higher-order statistics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Contemporary large-scale structure analyses center around the galaxy power spectrum, modeled with one-loop perturbation theory.	background 1.00	0.04	0.02	asserted 1.00
s1	This description naturally breaks down at large wavenumbers, prohibiting the extraction of cosmological information from small-scale clustering.	background 0.57	0.07	0.02	asserted 1.00
s2	In this work, we present a complete model for the redshift-space galaxy power spectrum up to two-loop order, derived in the framework of the Effective Field Theory of Large Scale Structure.	contribution 1.00	0.97	0.32	asserted 1.00
s3	This accounts for a wide variety of effects, including fifth-order galaxy bias, redshift-space distortions, small-scale non-linearities, spatial non-locality, long-wavelength bulk flows, coordinate distortions, and stochasticity, involving a total of 49 free coefficients.	contribution 0.56	0.93	0.12	asserted 1.00
s4	Our model additionally encodes all deterministic contributions to the one-loop bispectrum and trispectrum (as well as the tree-level quadspectrum and pentaspectrum), which depend on many of the same parameters.	contribution 0.97	0.90	0.07	asserted 1.00
s5	We implement the model numerically, using the \textsc{vegas} algorithm to precompute the loop integrals; this allows the full likelihood to be evaluated in milliseconds.	method 0.99	0.96	0.04	asserted 1.00
s6	The approach is validated using power spectra measured from high-resolution simulations, considering both dark matter and realistic galaxy samples.	method 0.88	0.93	0.04	asserted 1.00
s7	The extension to two-loop order greatly increases the range of scales over which the power spectrum can be modeled, with $k_{\rm max}$ rising from $0.2h\,\mathrm{Mpc}^{-1}$ to $0.34h\,\mathrm{Mpc}^{-1}$ for a luminous red galaxy sample at $z=0.5$.	result 1.00	0.92	0.30	asserted 1.00
s8	Much of the additional information is absorbed by the numerous fifth-order biases and counterterms; for this reason, constraints on $\sigma_8$ are not significantly enhanced by the inclusion of two-loop terms.	interpretation 0.80	0.88	0.27	asserted 0.99
s9	Future work will assess the full implications of the two-loop model for cosmological parameter inference, both alone and in combination with higher-order statistics.	future work 1.00	0.07	0.05	unclear 0.56

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from simulation or computation 0.81.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 8.** FIG. 8. Comparison of the one-loop galaxy power spectra computed with

Showing 1 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.13867 by the authors.

References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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astro-ph.CO 2026-10 conditional novelty 7.0 of 10

A precomputed tensor method using FFTLog and Mellin-Barnes kite integrals evaluates the two-loop EFTofLSS matter power spectrum for a new cosmology in milliseconds, agreeing with direct integration to better than 10^-...
Toward the one- and two-loop EFTofLSS for Simons Observatory CMB lensing
astro-ph.CO 2026-11 conditional novelty 6.0 of 10

A two-loop EFT of large-scale structure with one PCA-compressed counterterm parameter fits Simons Observatory CMB lensing forecasts across baryonic feedback scenarios and outperforms the dark-matter-only model at unbi...

Discussion

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