Nonlinear evolution of cosmological power spectra

Edinburgh); J.A. Peacock; S.J. Dodds (Royal Observatory

REVIEW

Nonlinear evolution of cosmological power spectra

J.A. Peacock, S.J. Dodds (Royal Observatory, Edinburgh)

Astrophysics · Submitted 8 Mar 1996 · 1 reference

Pith's verdict: The spectrum-dependent HKLM fit is the real deliverable and it holds up; the one untested claim is that the factor-2 index shift in Eq. (22) doesn't matter, which is asserted exactly where the (1+n/3) exponents bite hardest.

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arXiv IDastro-ph/9603031 (version 1)

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CitationEdinburgh), J.A. Peacock, S.J. Dodds (Royal Observatory. "Nonlinear evolution of cosmological power spectra." arXiv:astro-ph/9603031 (1996). https://pith.science/paper/7GKDVI7B

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Editor's note

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Short version: this is the paper that made the HKLM scaling usable, and it holds up. The deliverable is Eq. (21) with coefficients (23)-(27): a closed-form map from linear to nonlinear Delta^2(k) covering power-law and CDM spectra, open and flat-Lambda, with cosmology entering only through g(Omega). That last point is the actual result. It says quasilinear evolution doesn't care about the cosmology, only about the displacement amplitude, which is what the Zeldovich picture predicts. The simulation work behind it is careful: they modified AP3M's equations of motion for Omega != 1 (Eqs. 11-16), handled shot noise and FFT binning explicitly, and used sub-cube analysis to stretch the k range. Eighteen simulations, 48 f_NL determinations. That is a real library for 1996, and the code modifications are described well enough to reproduce. The soft spots are the usual ones for a fitting paper, plus one specific one. The 14 per cent rms in f_NL and 7 per cent in Delta^2_NL are in-sample residuals from the same 48 points used to fix ten parameters. No held-out set, no per-point errors. Normal for the genre, but it means the quoted accuracy cannot adjudicate the modelling choices. The choice that matters is Eq. (22), the factor-2 shift in where you evaluate the tangent index. The paper says the shift size 'is not critical' and that the unshifted index already gives 'very good predictions', but shows no scan. That is not a neutral omission. Every coefficient is a power of (1+n/3) with exponents up to -1.78, and CDM's tangent index reaches -2.5, outside the -2 < n < 0 calibration range. A delta-n of 0.3 near n = -2.5 moves A by roughly 50 per cent. The fit evidently absorbs this, since the CDM points sit inside the 14 per cent, but the specific claim that shift size is immaterial is untested precisely where the exponents bite. That deserves a paragraph and a sensitivity plot, not a rejection. One more thing: the abstract sells the wrong result. abstract:s3, the steeper nonlinearity for more negative n and lower Omega, is a side observation. The formula is the product. Who should read it: anyone linearizing survey data or generating nonlinear P(k) for lensing or velocity work. It is infrastructure and has been used as such. Recommendation: accept, with a request for a shift-sensitivity check and an explicit note that the quoted accuracies are in-sample. Send it to a serious referee.

Referee report

Significance

If the calibration holds, this is a genuinely useful result: a single closed-form recipe, with coefficients that are explicit functions of the local spectral index and one cosmological factor g(Ω), replaces per-model simulation campaigns for a wide range of hierarchical spectra. The paper is commendably explicit about what it does not deliver — no fitting formula for ξ̄ and no inverse nonlinear function (§4) — and about the numerical limits of the suite, including the shot-noise and box-power constraints that make n < -2 infeasible at N=80³ (§3.2). The comparison against the earlier PD formula in Fig. 5 is a fair and informative calibration of the improvement, and the statement that errors in the deduced Δ²_L stay near 20 per cent in the regime PD actually used is the right way to frame continuity with prior work. The physical argument that quasilinear evolution is Zeldovich-like, so that model dependence should collapse into a single displacement-scaling factor, is a falsifiable organizing principle rather than a post-hoc rationalization, and it is the part of the paper most likely to be tested by later work.

Major objections (4)

[§3.3, Eqs. (21)–(27)] The quoted 'rms accuracy of about 14 per cent in f_NL(x)' and 'about 7 per cent' transverse scatter are residuals of the same 48 f_NL determinations used to fix the ten coefficients of Eqs. (23)–(27); no held-out simulation set, per-point error bar, or covariance is given. This matters because the abstract and §4 present these numbers as the accuracy of the method, not as a goodness-of-fit statistic. Please relabel them explicitly as in-sample fit residuals, and, if feasible, add a leave-one-simulation-out or jackknife estimate so the reader can see how much the residual grows when a run is not in the fit. A short table of per-simulation residuals would settle the question.
[§3.3, Eq. (22)] The factor-of-two shift in the scale at which the tangent index is evaluated is load-bearing for the CDM branch: the text states that without it the CDM points 'generally lay below those predicted from the power-law fits using n_eff(k_L)', and that a shift of a factor 2 'gives outstandingly good results'. The accompanying claim that 'the exact size of the shift is not critical' is asserted rather than demonstrated. This is not a cosmetic point, because every coefficient is a steep power of u = 1+n/3, so at the CDM tangent index n_eff ≈ -2.5 the logarithmic sensitivity ∂lnA/∂n = -0.947/(3u) is about 1.9 per unit n, i.e. roughly a 19 per cent change in A for δn = 0.1, before the other coefficients compensate. Please show a scan over shift factors (e.g. 1, 1.5, 2, 3) with the resulting rms residual for the CDM subset, so the insensitivity claim is quantified rather than asserted.
[§3.3, calibration range versus CDM extrapolation] The power-law calibration covers -2 < n < 0, while §3.3 and the Fig. 4 caption note that the CDM tangent index 'can be as negative as n = -2.5 for the models studied here'. The (1+n/3) scalings of Eqs. (23)–(27) are therefore extrapolated precisely where their exponents are largest, and the paper does not separate the CDM points from the power-law points when quoting the global 14 per cent figure. Please report the residual split by spectral class (power law versus CDM) and, if possible, by Ω, so a reader can judge whether the CDM extrapolation is carrying disproportionate error.
[§3.3 and §4, scope of the g(Ω)-only claim] The statement that 'the cosmological model does not enter anywhere in these parameters' and is present 'only through the growth factor g' is tested over 0.1 < Ω_m < 1 with Ω_v = 0 or Ω_v + Ω_m = 1, and with no dark-energy equation of state other than w = -1 and no massive neutrinos. That is a real and useful test, but the claim as phrased is broader than the tested domain. Please state the tested range explicitly in the sentence that makes the claim, so later users do not apply the formula outside it without checking.

Minor objections (6)

[§3.2] The 48 f_NL determinations come from 18 simulations, with several output times taken from the same run; they are therefore not independent, and the effective sample size is closer to 18 than to 48. This should be stated where the fit sample is described.
[§3.2] The paper notes that 'improper' realizations (exact mean power, no random phases) were used. This removes realization-to-realization scatter from the quoted residuals, which is worth saying explicitly, since a user of the formula will care about the scatter between independent realizations as much as about the fit residual.
[§4] The remark that an observed nonlinear index β ≈ 1.8 would require n ≈ -2.2 via Eq. (30) is an interesting aside but is presented without the caveats that the virialized limit may not apply and that the observed slope is itself uncertain. Please label it as speculative or soften it.
[§4] The statement that no fitting formula is provided for ξ̄ or for the inverse nonlinear function is appropriately candid; it would help to add one sentence telling the reader how inversion should be performed in practice (the text says 'iterative' but does not outline the iteration).
[Figs. 1–4 captions] The figure captions would be clearer if each stated the box size, particle number, and output epoch for the plotted runs, or if a single table of simulation parameters were added; at present the reader cannot tell which curves come from which configuration.
[Figs. 2–4 captions and §4] There are typographical slips in the captions and text, for example 'a a flat model' in the Fig. 4 caption and 'the the power spectrum' in §4. These should be corrected in proof.

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The paper

The paper claims

A single closed-form recipe, with one slope-dependent correction and one growth factor, converts a linear density power spectrum into its nonlinear counterpart across open, flat, and cosmological-constant universes.

The reading

The paper asks whether one closed-form rule can carry a linear density power spectrum into the nonlinear regime across the cosmologies people actually consider — open low-density universes and flat universes with a cosmological constant, not just the Einstein–de Sitter case. The earlier scaling ansatz of Hamilton and collaborators worked well near spectral index $-1$ but was known to fail for steeper, more negative indices, which is exactly where cold dark matter spectra spend their small scales. The authors run a library of N-body simulations covering $-2

What carries the argument

The carrying object is the generalized Hamilton–Kumar–Lu–Matthews mapping: nonlinear power at wavenumber $k_{NL}$ is read off linear power at a shifted, smaller wavenumber $k_L=[1+2_{NL}(k_{NL})]^{-1/3}k_{NL}$, and the two are related by a fitting function with a linear-growth term, a quasilinear power law, and a virialized $x^{3/2}$ asymptote, softened by a parameter $1$. The new ingredient is that the quasilinear coefficients $A$, $B$, $1$, $1$, $V$ are not constants but scale with $(1+n/3)$ raised to fitted powers, with $n$ taken from the tangent slope of the linear spectrum at $k_L/2$ — a factor-of-two shift the authors adopt because using the unshifted slope left the CDM results systematically below the power-law fits. The $g^3$ denominator on the virialized term is the only place the cosmological model appears.

Extended reading notes

Core claim

The central claim is that the nonlinear-to-linear mapping $f_{NL}$ is not universal, but is well described by a five-parameter family whose coefficients are simple power laws in $(1+n/3)$, where $n$ is the tangent spectral index of the linear spectrum evaluated at half the linear wavenumber: $n_L(k_L) 1 d1$ at $k=k_L/2$. Fitted globally to power-law and CDM simulations, this gives about 14 per cent rms in $f_{NL}(x)$ over $0.1<2.5$ in $1$, and about 7 per cent scatter in the direction that governs reconstructing a spectrum from nonlinear data. The cosmological dependence — open versus flat-with-$1$, and the value of $1$ — is carried entirely by the $g^3(1)$ factor that sets the amplitude of the stable-clustering asymptote $f_{NL}(x)1$. The qualitative result is that nonlinear effects boost small-scale power more strongly for more negative spectral indices and for lower densities, but that a cosmological constant softens the boost relative to an open universe of the same $1$.

Load-bearing premise

The load-bearing premise is that the tangent spectral index should be read at half the linear wavenumber being predicted; the paper asserts that the exact size of this shift is not critical, but never shows a sensitivity scan, and the whole cold-dark-matter branch of the fit depends on it.

Editorial extensions

If this is right

Linearizing an observed nonlinear clustering spectrum to recover the underlying linear spectrum becomes a numerical iteration using a single formula, rather than a separate simulation campaign for each cosmology.
The earlier conclusion that small-scale clustering data imply a linear spectral index near $-1.5$ survives: in the regime those data actually covered ($2_{NL}1$), the deduced linear power shifts by at most about 20 per cent.
Because the quasilinear slope $f_{NL}1$ with $1$–$4.5$ is steeper than the stable-clustering answer, the predicted growth of nonlinear power with time is faster than linear growth, which bears on the weak angular clustering of faint galaxies.
An observed nonlinear slope of about 1.8 would imply a linear index near $-2.2$ through the paper's relation between $1$ and $n$, rather than the $n=0$ that stable clustering alone would suggest.
The method is proposed as general for any convex hierarchical spectrum, with hot or warm dark matter cutoffs explicitly excluded as needing separate treatment.

Reading between the lines

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

The finding that cosmology enters only through $g(1)$ is close to a statement that quasilinear evolution is Zeldovich-like: if final positions are initial displacements scaled by a universal factor, then matching displacement amplitude should erase the cosmology. That suggests the residual cosmological dependence could be computed analytically from the known Zeldovich power spectrum rather than fi
Reading the tangent index at a smaller scale than the one being predicted makes the mapping implicitly non-local in $k$. A spectrum with a sharp feature — a baryon oscillation, a mild cutoff — would then be mispredicted in a way a smooth CDM spectrum conceals, which is a testable weakness rather than a defect of the present fit.
The steep quasilinear slope offers a growth-rate probe: measuring the nonlinear slope at a fixed nonlinear amplitude across two epochs constrains the growth factor more sharply than linear growth alone, and this is checkable against existing simulations at earlier output times.
The calibration used boxes of $100\,h^{-1}$ Mpc with final $1$ in the range 0.4–1, so applying the formula to very high normalizations or to much smaller boxes is an extrapolation the paper does not test.

Original abstract

Hamilton et al. have suggested an invaluable scaling formula which describes how the power spectra of density fluctuations evolve into the nonlinear regime of hierarchical clustering. This paper presents an extension of their method to low-density universes and universes with nonzero cosmological constant. We pay particular attention to models with large negative spectral indices, and give a spectrum-dependent fitting formula which is of significantly improved accuracy by comparison with an earlier version of this work. The tendency of nonlinear effects to increase power on small scales is stronger for spectra with more negative spectral indices, and for lower densities. However, for low-density models with a cosmological constant, the nonlinear effects are less strong than for an open universe of the same $\Omega$.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.26 · human participants 0.01 · controlled comparison 0.32 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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Hamilton et al. have suggested an invaluable scaling formula which describes how the power spectra of density fluctuations evolve into the nonlinear regime of hierarchical clustering. This paper presents an extension of their method to low-density universes and universes with nonzero cosmological constant. We pay particular attention to models with large negative spectral indices, and give a spectrum-dependent fitting formula which is of significantly improved accuracy by comparison with an earlier version of this work. The tendency of nonlinear effects to increase power on small scales is stronger for spectra with more negative spectral indices, and for lower densities. However, for low-density models with a cosmological constant, the nonlinear effects are less strong than for an open universe of the same $\Omega$.

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	Hamilton et al. have suggested an invaluable scaling formula which describes how the power spectra of density fluctuations evolve into the nonlinear regime of hierarchical clustering.	background 1.00	0.03	0.03	asserted 0.67
s1	This paper presents an extension of their method to low-density universes and universes with nonzero cosmological constant.	contribution 1.00	0.86	0.07	asserted 0.98
s2	We pay particular attention to models with large negative spectral indices, and give a spectrum-dependent fitting formula which is of significantly improved accuracy by comparison with an earlier version of this work.	contribution 0.95	0.94	0.15	asserted 1.00
s3	The tendency of nonlinear effects to increase power on small scales is stronger for spectra with more negative spectral indices, and for lower densities.	result 0.99	0.83	0.35	asserted 1.00
s4	However, for low-density models with a cosmological constant, the nonlinear effects are less strong than for an open universe of the same $\Omega$.	result 1.00	0.84	0.14	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.72 · evidence from simulation or computation 0.82.

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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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.** The generalized HKLM function relating nonlinear power to linear power, for an Ω = 1 Einstein-de Sitter universe and power-law spectra with n = 0, −1, −1.5, −2. The solid lines show the fitting formula of Section 3.3. For spectra with n >∼ −1, the function shows very little spectral dependence. For flatter spectra, the nonlinear power at a given linear power is higher; in particular the slope of the quasilinear portion around ∆2 NL ∼ 10 increases as n decreases, from approximately fNL(x) ∝ x 3 for n = 0 to fNL(x) ∝ x 4 for n = −2. This contains 5 free parameters, each of which i

**Figure 2.** The generalized HKLM function relating nonlinear power to linear power, for Ωm = 1 & 0.2 and zero vacuum energy. (a) an n = 0 spectrum; (b) n = −1.5. The solid lines show the fitting formula of Section 3.3. As Ω decreases, the nonlinear power increases and the quasilinear portion steepens. However, these changes are largely associated with the increase of fNL in the virialized regime with ∆2 NL >∼ 100. via the power-law model corresponding to neff at the nonlinear scale. However, it seemed to us that the whole philosophy of the HKLM method is that the nonlinear power at kNL…

**Figure 5.** The present fitting formula for the nonlinear function (solid lines) compared with the spectrum-independent form suggested by PD (dashed lines), for the cases n = −1 & −1.5 and open models with Ωm = 1, 0.5, 0.2. The PD formula was an approximation to the average effect of these different spectra, but the detailed agreement is often poor, particularly at low Ω for flatter spectra. However, these deviations are small in the regime of the data used by PD (∆2 NL <∼ 3). than the power spectrum. It is probably best to proceed numerically with the forward nonlinear function as a…

Showing 3 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9603031 by the authors.

References

Forward citations

Cited by 13 Pith papers

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The 6dF Galaxy Survey: z \approx 0 measurement of the growth rate and sigma_8
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Redshift-space distortions in 82,000 galaxies at z = 0.067 give a growth rate f-sigma-8 = 0.423 +/- 0.055, and combined with WMAP7 constrain the growth index to gamma = 0.547 +/- 0.088.
Massive Neutrinos and the Non-linear Matter Power Spectrum
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Particle-based N-body simulations show HALOFIT over-predicts the neutrino free-streaming suppression of the non-linear matter power spectrum by up to 10 percent of the suppression, and a refitted HALOFIT variant repro...

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First cosmological simulations with the chameleon mechanism show it suppresses f(R) power spectrum enhancements at small scales by up to a factor of four, while leaving measurable signals at intermediate scales.
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Including non-linear density perturbations in the Q-ball decay scenario yields a second-order gravitational wave spectrum that can match the nanohertz pulsar timing array signal for a tuned amplitude of small-scale cu...
Searching for modified growth patterns with tomographic surveys
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Fisher-matrix forecasts show that DES can constrain five modified-growth parameters at the 20-40 percent level and LSST at better than 10 percent, using tomographic galaxy clustering, weak lensing, CMB and supernova data.
Large Scale Structure as a Probe of Gravitational Slip
astro-ph 2008-02 conditional novelty 5.0 of 10

A corrected one-parameter gravitational-slip model is confronted with CMB, weak-lensing, and ISW-galaxy data, which mildly favor nonzero slip but show inter-probe tension.
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Marginalising over intrinsic galaxy alignments roughly triples the photometric redshift precision and at least doubles the number of tomographic bins a cosmic shear survey needs to recover most of its dark energy information.

How later papers used it

46 later papers cite it

Evidence: substantial. 46 of the citing papers share no author with it, and 12 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 56 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

4 later papers by other authors are listed below: 3 used its method or tool and 1 compared its own results with it.

Uses its method or tool 3 of 7 papers with this label

Used the method of this paper to determine the non-linear matter power spectrum because of its simplicity and speed.

Building a Predictive Model of Galaxy Formation - I: Phenomenological Model Constrained to the $z=0$ Stellar Mass Function (2014) · 1.00 · paraphrased: the licence does not allow quoting
Used the fitting function from this paper to get the nonlinear density power spectrum from the linear power spectrum calculated with its modified CMBfast code.

Large Scale Structure as a Probe of Gravitational Slip (2008) · 0.99 · paraphrased: the licence does not allow quoting
Used the cosmology- and simulation-dependent fitting formula of this paper to make its plot of expansion coefficients versus nonlinear wavenumber.

Fitting functions on the cheap: the relative nonlinear matter power spectrum (2019) · 0.99 · paraphrased: the licence does not allow quoting

Compares with it 1 of 3 papers with this label

Compared its own MA99 and HALOFIT with this paper's formula to identify the origin of a discrepancy, using a smooth linear power spectrum without baryonic oscillations.

Cosmic Microwave Background Bispectrum from the Lensing--Rees-Sciama Correlation Reexamined: Effects of Non-linear Matter Clustering (2012) · 0.91 · paraphrased: the licence does not allow quoting

How the literature uses it

37 arXiv papers cite this in 47 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 · 28 sentences without a confident label

Contests it 2 papers

[ref] ) are uncapable to reach the required level of accuracy [ref] .

Flowing with Time: a New Approach to Nonlinear Cosmological Perturbations · Introduction · contest 0.87
Although there are well-known methods [ref] to analytically transform the linear power spectrum into the non-linear power spectrum, we find that they do not yield reasonable results in this application.

Primordial Black Hole Reformation in the Early Universe · Power Spectrum · contest 0.77

Uses its method or tool 7 papers

Here we take into account the effect of uncorrelated large scale structure projected along the line of sight on the error covariance matrix $C^lss_ij$ as $C=C^stat+C^lss$, where $C^lss$ is given as [ref] [equation] Here $C^κκ(l)$ is the weak-lensing power spectrum as a function of angular multipole $l$ evaluated for a given source population and a cosmology, and $Ĵ_0(lθ_i)$ is the Bessel function of the first kind and order zero averaged over the $i$th annulus between $θ_i,1$ and $θ_i,2 (>θ_i,1)$, given as [equation] We will assume the concordance $Λ$CDM cosmological model of [ref] and use the fitting formula of [ref] to compute the nonlinear mass power spectrum that enters in equation ([ref]).

A Precise Cluster Mass Profile Averaged from the Highest-Quality Lensing Data · Combining Weak and Strong Lensing · 0.98
There are two approaches to compute the $P_δ^NL(k,z)$ that have been extensively tested and used in the literature: the more recent Halofit [ref] model, which is included in CAMB [ref] and the Peacock and Dodds (PD) [ref] method (generalized for dark energy cosmologies by [ref] ).

Non-Gaussianity and the CMB Bispectrum: confusion between Primordial and Lensing-Rees Sciama contribution? · Bispectrum calculation and expected signal-to-noise · 0.97
We demonstrate that by integrating the mapping approach recommended in [ref] , it is feasible to derive a concise expression for the nonlinear power spectrum based on its linear counterpart and several critical cosmological parameters.

$\texttt{SPINE}$: Symbolic Models to Predict the Evolution of the $\Lambda$CDM Nonlinear Power Spectrum · Introduction · 0.92

Compares with it 3 papers

Fig. [ref] shows the sum of the linear and non-linear contributions compared to the non-linear power spectrum derived by [ref] from numerical simulations.

Non-equilibrium statistical field theory for classical particles: Non-linear structure evolution with first-order interaction · Results · 0.99
In order to identify the origin of this discrepancy, we have also compared MA99 and HALOFIT with the formula by Peacock and Dodds (1996; PD96) [ref] (not shown in Figure 1; in order to use their formula, it is necessary to use the smooth linear power spectrum without baryonic oscillations; thus, we used the smooth power spectrum given in [ref] ).

Cosmic Microwave Background Bispectrum from the Lensing--Rees-Sciama Correlation Reexamined: Effects of Non-linear Matter Clustering · Comparing $P(k,z)$ · 0.91
In $δ$, the migration of power is qualitatively as in [ref] and [ref] : power moves from large to small scales.

Ringing the initial Universe: the response of overdensity and transformed-density power spectra to initial spikes · Results · 0.94

Cites it as background 6 papers

A different idea to address the issue is expressing the non-linear power spectrum (or correlation function) as a function of its linear counterpart, evaluated at a different scale [ref] .

An Optimally Weighted Estimator of the Linear Power Spectrum Disentangling the Growth of Density Perturbations Across Galaxy Surveys · Growth of Perturbations · 0.90
Cosmological inference has traditionally focused on analyzing 2-point correlations, typically the power spectrum $P(k)$ [ref] , on large, weakly non-linear scales, where it is amenable to analytical modeling using perturbation theory (PT) [ref] .

PatchNet: A hierarchical approach for neural field-level inference from Quijote Simulations · Introduction · 0.97
For flat cosmological models in particular, fitting formulae or emulators based on the predictions of $N$-body simulations have been developed [ref] .

Flat to nonflat: Calculating nonlinear power spectra of biased tracers for nonflat $\Lambda$CDM model · Introduction · 0.92

Discussion

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[1] [1]

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