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RegPT: Direct and fast calculation of regularized cosmological power spectrum at two-loop order

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Atsushi Taruya, Francis Bernardeau, Takahiro Nishimichi, Sandrine Codis

Cosmology and Nongalactic Astrophysics · Submitted 6 Aug 2012 · 58 references

Pith's verdict: A solid, honest methods paper: two-loop RegPT assembled and validated, with a fast fiducial-expansion trick and released code — the one soft spot is that the damping scale sigma_d(k) is calibrated, not derived, and the calibration is never stress-tested across redshift or cosmology.

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arXiv ID1208.1191 (version 1)
Pith IDIDVFGIL6, Pith's permanent record of this paper and its review
CitationAtsushi Taruya, Francis Bernardeau, Sandrine Codis, Takahiro Nishimichi. "RegPT: Direct and fast calculation of regularized cosmological power spectrum at two-loop order." arXiv:1208.1191 (2012). https://pith.science/paper/IDVFGIL6
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The paper

The paper claims

RegPT computes the two-loop cosmological matter power spectrum and correlation function directly, and then again in seconds by expanding around precomputed fiducial models, matching N-body simulations at the percent level in the weakly non-

The reading

The paper asks a practical question: can perturbation theory, pushed to next-to-next-to-leading (two-loop) order, predict the matter power spectrum and the two-point correlation function well enough to be used for real survey analysis, and can it be made fast enough to run over many cosmological models? The authors answer both by building the power spectrum out of multi-point propagators whose loop expansions are regularized so that they keep the standard perturbative result at low wavenumber while picking up the expected exponential damping at high wavenumber. Compared against a large set of N-body simulations, this RegPT construction stays within about one percent of the simulated spectrum and correlation function over the weakly nonlinear regime, including the baryon acoustic peak and its smearing. The second half of the paper shows that the whole two-loop result, which otherwise needs multi-dimensional Monte Carlo integrals, can be Taylor expanded around a few precomputed fiducial models so that every remaining integral is one-dimensional, cutting the runtime from minutes to seconds. The payoff is a publicly released Fortran code that returns percent-level spectra for a broad range of flat wCDM models on a laptop.

What carries the argument

The load-bearing object is the regularized multi-point propagator, $Gamma^{(p)}_{a,reg}$. In the standard $Gamma$-expansion, the power spectrum is a sum over products of multi-point propagators glued together by linear power spectra; each propagator is the ensemble-averaged response of the final field to an infinitesimal change in the initial conditions, and each is known to decay as $exp(-k^2 sigma_d^2 e^{2 eta}/2)$ in the high-$k$ limit while reducing to the standard PT kernels $F^{(p)}_{a,sym}$ at low $k$. The regularization scheme of the paper's earlier reference splices these two behaviors by multiplying the standard PT series (tree, one-loop, two-loop pieces plus compensating counter-terms in $k^2 sigma_d^2 e^{2 eta}/2$) by the exponential damping factor, so that no order is double-counted and both limits are respected. The second piece of machinery is the fiducial-model Taylor expansion: the nonlinear spectrum is written as an unperturbed part evaluated with a fiducial $P_{0,fid}$ plus a correction linear in $delta P_0$, and the kernels $L^{(1)}, M^{(1)}, X^{(2)}, Y^{(2)}, Z^{(2)}, Q^{(2)}, R^{(2)}, S^{(3)}$ are precomputed once per fiducial model on $301 times 301$ logarithmic grids, so that a target model costs only 1D quadratures. A free rescaling $P_{0,fid} to c P_{0,fid}$ lets the code slide the fiducial model continuously toward the target before expanding.

Extended reading notes

Core claim

The central claim is that the regularized $Gamma$-expansion, taken to two-loop order, is an accurate and usable theory of the weakly nonlinear matter clustering. The regularization replaces each multi-point propagator by an expression that reproduces standard perturbation theory order by order at low $k$ and the resummed $e^{-k^2 sigma_d^2 e^{2 eta}/2}$ damping at high $k$; inserting these into the $Gamma$-expansion gives a power spectrum built from three terms involving $Gamma^{(1)}_{reg}$, $Gamma^{(2)}_{reg}$ and $Gamma^{(3)}_{reg}$, each of which contributes a localized bump at a different scale. Against N-body simulations with $2048 h^{-1}$Mpc boxes and $1024^3$ particles at $z = 3, 2, 1, 0.35$, the authors report percent-level agreement in $[P_{N body} - P_{RegPT}]/P_{no wiggle}$ up to a redshift-dependent wavenumber, and near-perfect agreement for the correlation function around the baryon acoustic peak and down to $r sim 30 h^{-1}$Mpc at $z = 0.35$. The second claim is that the same result can be obtained in seconds: because the nonlinear spectrum is a functional of the linear spectrum, expanding in $delta P_0 = P_{0,target} - P_{0,fid}$ to first order turns every diagram into a one-dimensional integral against precomputed kernels, and with three fiducial models spanning the shape parameter the fast method reproduces the direct calculation to better than $0.4$ percent at $k lesssim 0.3 h$Mpc$^{-1}$ across 38 cosmologies.

Load-bearing premise

Everything rests on the scale-dependent displacement variance $sigma_d(k)$ built from a running UV cutoff $k_Lambda(k) = k/2$; the paper states in Sec. III B that cutoffs in the range $k/2$ to $k/5$ are favored by comparison with simulation 'although there is no clear physical reason why this is so', so the percent-level agreement is calibrated against a choice that is phenomenological rather than derived.

Editorial extensions

If this is right

  • Cosmological parameter inference from galaxy surveys can use a two-loop perturbation theory prediction for the matter power spectrum and correlation function at percent-level accuracy over the baryon acoustic scale, at a computational cost compatible with sampling a likelihood.
  • The speedup makes it feasible to run the full two-loop calculation across large model grids: the paper demonstrates this on 38 flat wCDM models spanning $0.120 < Omega_m h^2 < 0.155$, $0.85 < n_s < 1.05$, $-1.30 < w < -0.70$ and $0.616 < sigma_8 < 0.9$.
  • The empirical criterion $k^2_{crit}/(6pi^2) int_0^{k_{crit}} dq, P_{lin}(q;z) = 0.7$ gives users a redshift- and model-dependent wavenumber below which the two-loop result can be trusted, and can be shipped with the code as an automatic validity flag.
  • Because the same regularization applies to any multi-point propagator, the construction extends to velocity and density-velocity spectra and provides a starting point for redshift-space power spectrum modeling, where multi-point spectra enter.

Reading between the lines

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

  • The fast method is a general trick for any diagrammatic expansion whose result is a functional of a single input spectrum; it would apply equally to one-loop closure theory or to Lagrangian resummation, provided the relevant kernels are precomputed and the fiducial set spans the shape-parameter range.
  • The paper's own comparison suggests the residual disagreement at $z=0.35$ and low $k$ traces back to the convergence of $Gamma^{(1)}_{reg}$ rather than to the $Gamma$-expansion itself, which makes the two-loop two-point propagator the natural next target for improvement.
  • The appendix note that the emulator and RegPT disagree for model M015 even well below $k_{crit}$, with a dedicated N-body run favoring RegPT, points to a systematic in how emulator training sets estimate the smooth spectrum; independent simulation suites would settle it.
  • The dependence on a running cutoff with no derived physical justification suggests a testable extension: derive $k_Lambda(k,z)$ from the same soft/hard separation used in the resummation and check whether $k/2$ emerges rather than being fitted.
Original abstract

We present a specific prescription for the calculation of cosmological power spectra, exploited here at two-loop order in perturbation theory (PT), based on the multi-point propagator expansion. In this approach power spectra are constructed from the regularized expressions of the propagators that reproduce both the resummed behavior in the high-k limit and the standard PT results at low-k. With the help of N-body simulations, we show that such a construction gives robust and accurate predictions for both the density power spectrum and the correlation function at percent-level in the weakly non-linear regime. We then present an algorithm that allows accelerated evaluations of all the required diagrams by reducing the computational tasks to one-dimensional integrals. This is achieved by means of pre-computed kernel sets defined for appropriately chosen fiducial models. The computational time for two-loop results is then reduced from a few minutes, with the direct method, to a few seconds with the fast one. The robustness and applicability of this method are tested against the power spectrum cosmic emulator from which a wide variety of cosmological models can be explored. The fortran program with which direct and fast calculations of power spectra can be done, RegPT, is publicly released as part of this paper.

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-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,302 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.24 · randomized assignment 0.01 · states quantitative result 0.93 · compares with prior results 0.91 · mentions code or data availability 0.98.

The abstract, sentence by sentence

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

We present a specific prescription for the calculation of cosmological power spectra, exploited here at two-loop order in perturbation theory (PT), based on the multi-point propagator expansion. In this approach power spectra are constructed from the regularized expressions of the propagators that reproduce both the resummed behavior in the high-k limit and the standard PT results at low-k. With the help of N-body simulations, we show that such a construction gives robust and accurate predictions for both the density power spectrum and the correlation function at percent-level in the weakly non-linear regime. We then present an algorithm that allows accelerated evaluations of all the required diagrams by reducing the computational tasks to one-dimensional integrals. This is achieved by means of pre-computed kernel sets defined for appropriately chosen fiducial models. The computational time for two-loop results is then reduced from a few minutes, with the direct method, to a few seconds with the fast one. The robustness and applicability of this method are tested against the power spectrum cosmic emulator from which a wide variety of cosmological models can be explored. The fortran program with which direct and fast calculations of power spectra can be done, RegPT, is publicly released as part of this paper.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We present a specific prescription for the calculation of cosmological power spectra, exploited here at two-loop order in perturbation theory (PT), based on the multi-point propagator expansion. contribution 0.97 0.94 0.16 asserted 1.00
s1 In this approach power spectra are constructed from the regularized expressions of the propagators that reproduce both the resummed behavior in the high-k limit and the standard PT results at low-k. method 0.97 0.60 0.12 asserted 1.00
s2 With the help of N-body simulations, we show that such a construction gives robust and accurate predictions for both the density power spectrum and the correlation function at percent-level in the weakly non-linear regime. result 1.00 0.96 0.40 asserted 1.00
s3 We then present an algorithm that allows accelerated evaluations of all the required diagrams by reducing the computational tasks to one-dimensional integrals. method 0.82 0.94 0.10 asserted 0.97
s4 This is achieved by means of pre-computed kernel sets defined for appropriately chosen fiducial models. method 1.00 0.84 0.04 asserted 1.00
s5 The computational time for two-loop results is then reduced from a few minutes, with the direct method, to a few seconds with the fast one. result 0.93 0.91 0.14 asserted 1.00
s6 The robustness and applicability of this method are tested against the power spectrum cosmic emulator from which a wide variety of cosmological models can be explored. method 0.63 0.86 0.09 asserted 0.99
s7 The fortran program with which direct and fast calculations of power spectra can be done, RegPT, is publicly released as part of this paper. contribution 0.98 0.89 0.12 asserted 1.00

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 1.00.

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
not run
  • 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 1. FIG. 1: Diagrammatic representation of the standard PT ex
Figure 2. FIG. 2: Example of the multi-point propagator, Γ
Figure 3. FIG. 3: Diagrammatic representation of the power spectrum b
Figure 4. FIG. 4: Diagrammatic representation of the standard PT expa
Figure 5. FIG. 5: Diagrammatic representation of the regularized thr
Figure 6. FIG. 6: Contribution of multi-point propagators to the powe
Figure 7. FIG. 7: Sensitivity of the power spectrum prediction at
Figure 8. FIG. 8: Sensitivity of the power spectrum prediction to the U
Figure 9. FIG. 9: Comparison of power spectrum results between
Figure 10. FIG. 10: Comparison of two-point correlation function betw
Figure 11. FIG. 11: Example of the performances of the
Figure 12. FIG. 12: Linear power spectra
Figure 13. FIG. 13: Convergence between the
Figure 14. FIG. 14: Ratio of power spectra,
Figure 15. FIG. 15: Same as Fig. 14, but for the models M019–M037.

Figures reproduced from arXiv:1208.1191 by the authors.

References
How later papers used it

99 later papers cite it

Evidence: substantial. 77 of the citing papers share no author with it, and 34 of those carry a confident label from Pith's citation classifier.

Pith read 28 of the 170 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 13 of 35 papers with this label · 3 by the same authors

By the same authors

Uses its data or values 1 of 3 papers with this label · 1 by the same authors

By the same authors

  • Used the dark matter distributions created by the simulations in this paper for subsequent analysis.

    Chasing Unbiased Spectra of the Universe (2013) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

66 arXiv papers cite this in 127 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 74 sentences without a confident label

Contests it 1 paper
  • This is not the case for the former resummation techniques [ref] , while it is the case for ours, as it can be seen clearly in Fig. [ref] [The reader who wants to check this claim more explicitly can compare our eq.

    The IR-resummed Effective Field Theory of Large Scale Structures · Matter Power Spectrum · contest 0.83

Uses its method or tool 18 papers
Uses its data or values 3 papers
Compares with it 7 papers
Cites it as background 16 papers
  • There are several models that attempt to do this task: the halo model (see [ref] for a review), HALOFIT [ref] , cosmological standard perturbation theory [ref] , and other perturbation theories approaches based on Lagrangian perturbation theory [ref] , time renormalization [ref] , Eulerian renormalized (or resummed) perturbation theories [ref] and closure theory [ref] (see [ref] for comparison of some of these theories).

    Perturbation theory approach for the power spectrum: from dark matter in real space to massive haloes in redshift space · Perturbation theory in real space · 0.97

  • Many different approaches have been developed over the past decade for understanding this evolution: standard perturbation theory (SPT; [ref] ), Lagrangian resummation theory (LRT; [ref] ), renormalized perturbation theory (RPT; [ref] ), closure theory [ref] , multi-point propagator method (the $Γ$-expansion method; [ref] ), regularized multi-point propagator method (Reg PT; [ref] ), the Wiener Hermite expansion method [ref] , as well as other techniques [ref] .

    How does non-linear dynamics affect the baryon acoustic oscillation? · Introduction · 0.99

  • A diversity of modeling methods have been developed ranging from simulation-based methods such as emulator [ref] , fast simulations [ref] , and machine learning [ref] to analytical methods such as Standard Perturbation Theory (SPT) [ref] , Lagrangian Perturbation Theory (LPT) [ref] , Effective-Field Theory of Large Scale Structure (EFTofLSS) [ref] , and various Renormalized Perturbation Theory (RPT) [ref] , including RegPT [ref] .

    Perturbation Theory Remixed: Improved Nonlinearity Modeling beyond Standard Perturbation Theory · Introduction · 0.98

What it cites

Cites as background 3 papers
  • Precious information regarding the growth of structure are and will also be obtained from redshift-space distortions (e.g., [ref] ) and weak lensing measurements (see [ref] and review papers [ref] ) at scales ranging to the linear or quasi-linear to the non-linear regimes.

    Detection of correlated galaxy ellipticities on CFHT data: first evidence for gravitational lensing by large-scale structures · Introduction · 0.96

  • Precious information regarding the growth of structure are and will also be obtained from redshift-space distortions (e.g., [ref] ) and weak lensing measurements (see [ref] and review papers [ref] ) at scales ranging to the linear or quasi-linear to the non-linear regimes.

    Gravitational Lensing by Large Scale Structures: A Review · Introduction · 0.96

  • Precious information regarding the growth of structure are and will also be obtained from redshift-space distortions (e.g., [ref] ) and weak lensing measurements (see [ref] and review papers [ref] ) at scales ranging to the linear or quasi-linear to the non-linear regimes.

    Weak Gravitational Lensing · Introduction · 0.96

Discussion

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