Transients from Initial Conditions: A Perturbative Analysis

Roman Scoccimarro

REVIEW

Transients from Initial Conditions: A Perturbative Analysis

Roman Scoccimarro

Astrophysics · Submitted 17 Nov 1997 · 40 references

Pith's verdict: A solid, self-contained PT calculation that gave the field both a diagnostic for ZA transients and the 2LPT fix; the abstract's headline numbers are a bit looser than the body.

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arXiv IDastro-ph/9711187 (version 1)

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CitationRoman Scoccimarro. "Transients from Initial Conditions: A Perturbative Analysis." arXiv:astro-ph/9711187 (1997). https://pith.science/paper/BBOFDA6K

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Editor's note

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This is the paper that made ZA transients a calculable problem rather than a folklore problem, and it also handed simulators the fix. Worth your time if you set initial conditions or measure high-order moments, and worth citing regardless of which camp you're in. The core is Eq. 23: the standard Goroff et al. recursion generalized with the linear propagator g_ab(z) of Eq. 24, so decaying modes excited by the initial conditions are carried explicitly instead of being dropped at s = -∞. That's a clean, honest extension, and it pays off in closed forms — Eqs. 38-39 for S_3(a) and T_3(a), Appendix B for S_4, T_4, S_5, T_5. The a = 1 and a → ∞ limits check out by construction, which is the right sanity test for this kind of result. Appendix D is the part that aged best: a concrete, cheap recipe for 2LPT initial conditions — two extra FFTs and a 3×N_grid array. That recommendation is now standard practice, and this is where it comes from. The derivation is self-contained and the BGE comparison in Fig. 5 is external evidence, not a fit. One soft spot, and it's the stress-test's: the abstract's a = 6 for S_3 and a = 15 for T_3 come from Eq. 38, the standard ZA velocity assignment, but the only simulation comparison uses the EDFW variant of Eq. 18, whose prediction is Eq. 39 and which the paper itself says converges by a ≈ 3. So the headline numbers are the formalism extrapolated to a worse scheme, not the numbers BGE actually validates. T_p is never checked against a simulation at all. Neither point breaks anything — the formalism covers both velocity assignments — but the abstract reads as if the a = 6 and a = 15 were the tested ones. The tree-level truncation is a real scope limit, and the author says so plainly in Section 4, including the nice observation that transients and one-loop corrections partially cancel and can fake a wider range of validity for tree-level PT than is real. That candour is why I'd trust the rest. Send it out; the referee should push on the abstract's framing, not the physics.

Referee report

Significance

If the formalism holds, simulation designers acquire a quantitative, falsifiable rule for the startup redshift needed to measure a given cumulant to a given accuracy, plus a cheap upgrade path that removes the leading transient. The paper delivers several things worth crediting explicitly. The closed forms are checkable and they check: I verified that Eqs. (38a,b) and (39a,b) reduce at a = 1 to the ZA values 4-γ and 2-γ respectively, and that their a → ∞ limits reproduce the standard tree-level PT values 34/7-γ and 26/7-γ; I also verified that the quoted thresholds follow from the formulas, with Eq. (38a) giving S_3(6)/S_3(∞) ≈ 0.89 and Eq. (38b) giving T_3(15)/T_3(∞) ≈ 0.89 for n = 0. The comparison to BGE in Fig. 5 is honest about discreteness, finite-volume, and one-loop systematics, and the observation that transients and loop corrections can cancel to produce a 'false agreement' with tree-level PT is a genuinely useful caution for the field. Appendix D is a concrete, implementable recipe rather than a gesture. The results are falsifiable against existing and future simulation outputs, which is the right standard for this kind of paper.

Major objections (1)

[Abstract (sentence s6) versus §4, Eqs. (38)-(39), and Fig. 5] The abstract's headline numbers, a = 6 for S_3 and a = 15 for T_3 at n = 0, are the standard-ZA-velocity results of Eq. (38), i.e. the assignment of Eq. (17). The only direct N-body validation in the paper, Fig. 5, uses the EDFW velocity assignment of Eq. (18), whose prediction is Eq. (39) and which the paper itself states converges roughly a factor of two to three faster (a ≈ 3 for S_3). A reader who takes the abstract numbers as the startup requirement for a code using EDFW velocities will over-start by that factor. The body of the paper is explicit about which scheme each equation refers to, so this is a scoping problem rather than an error, but it is worth fixing because the abstract numbers are the paper's most quotable content: please state in the abstract which velocity assignment the quoted a values assume, and note the EDFW variant separately. Relatedly, the abstract's sentence s9 correctly restricts the simulation agreement to S_p; it would help to say plainly in the abstract that the T_p predictions are unvalidated, since §1 already acknowledges that no velocity-moment measurements exist in the literature.

Minor objections (8)

[§1 and §4, near the Lokas et al. citations] The text contains stray '/suppress' tokens immediately before two of the Lokas et al. citations. These appear to be a production artifact and should be removed before publication.
[§4, Eq. (40) and surrounding discussion] The paper is commendably candid that the quoted expansion factors are tree-level statements valid as σ²(R) → 0, and that at the scales where simulations actually operate one-loop corrections are significant and can partially cancel the transient. Since the abstract quotes specific a values without that qualifier, one sentence in the abstract noting that the numbers are tree-level would prevent the numbers from being read as operating-point requirements.
[§2.1 and §4, the Ω/f² = 1 approximation] All statements for Ω ≠ 1, including the abstract's claim that an Ω = 0.3 model needs a factor of two larger expansion, rest on the separable approximation Ω/f² = 1, stated in §2.1 to be good to better than 25% for Ω ≥ 0.3. The paper acknowledges this in §4. A short statement of how that 25% propagates into the quoted 'factor of two' would let a reader judge whether the number is a requirement or an indication.
[§3 and Appendix B, results for p = 7 and p = 8] Figs. 1-4 and 11-12 display results through p = 8, but explicit expressions are given only through S_6 (Eq. 36) and, in Appendix B, through p = 5; for p = 7, 8 the text refers the reader to table 1 of Boschán, Szapuldi and Szalay (1994) for the tree topologies and combinatorial coefficients. Since the p = 8 statement (a ≈ 40 for n = 0) is one of the paper's stronger claims, it would strengthen reproducibility to state explicitly that the same procedure was applied and, if space permits, to record the resulting coefficients.
[§2.2, Eqs. (17)-(18)] The assertion that the EDFW assignment of Eq. (18) is 'closer to the exact dynamics' than Eq. (17) is the physical basis for the factor-of-two improvement seen in Figs. 3-4 and 9-10. A one-line quantitative illustration at second order, comparing G_2^ZA with the exact G_2, would make the argument concrete rather than assertive.
[§2.3, Eq. (23)] The notation would be easier to follow if the text stated explicitly that the lower limit s = 0 corresponds to D = 1, the epoch at which initial conditions are imposed, and that the z → 0+ limit g_ab → δ_ab is what guarantees F_a^(n) → I_a^(n) there. The latter identity does hold for Eq. (24), but spelling it out saves the reader a calculation.
[Fig. 5 and surrounding text] The a = 1 open triangles disagree with the ZA prediction at small scales, and the text attributes this to discreteness in the grid initial arrangement. Indicating which scales were excluded from the comparison, and why, would make the figure easier to interpret.
[References] Several entries need checking: the Bouchet et al. 1992 entry reads 'Bouchet, F. R., Juszkiewicz, R., Colombi, S., Pellat, R., & 1 992, ApJ, 394, L5' and is malformed; the Gaztañaga 1994 entry is given as ApJ 268, 913, but that volume and page correspond to MNRAS; and 'Melott, A. M.' in the Fry, Melott and Shandarin 1993 entry should read A. L. Melott.

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The paper

The paper claims

A simulation remembers how it was started: the Zel'dovich approximation imprints wrong higher-order correlations that take a factor of six or more in expansion to wash out, and a cheap second-order Lagrangian upgrade removes them almost at

The reading

This paper asks a practical question about cosmological simulations: how long does a run remember the approximation used to start it? The standard startup, the Zel'dovich approximation, reproduces the linear growing mode of density and velocity exactly but gets the nonlinear couplings wrong, so the simulation must expand for a while before the measured skewness and higher cumulants settle onto the values the true dynamics would give. Scoccimarro extends perturbation theory to carry the decaying modes that the initial conditions excite, which yields closed-form expressions for the cumulant ratios $S_p(a)$ and $T_p(a)$ as functions of scale factor, interpolating between the imposed initial values and the asymptotic exact ones. The payoff is a quantitative startup rule: for a spectral index near zero the density skewness $S_3$ reaches ten percent accuracy only after a factor of about six in expansion, the velocity-divergence analogue $T_3$ after about fifteen, and higher moments need much more. Seeding instead with second-order Lagrangian perturbation theory cancels the skewness transient outright and, for $p>3$, reduces the required expansion by more than an order of magnitude, at the cost of two extra Fourier transforms and one array in an existing code.

What carries the argument

The load-bearing object is the recursion relation for the time-dependent perturbative kernels, built from a $2imes 2$ linear propagator $g_{ab}(z)$ in $z =

Extended reading notes

Core claim

The central claim is that the transient behaviour induced by Zel'dovich initial conditions is not an uncontrolled numerical nuisance but a computable perturbative effect. By keeping the full time dependence of the solutions at every order, the author writes the nonlinear kernels as a recursion (his Eq. 23) in which the initial conditions appear as source terms that excite decaying modes, and the standard growing-mode results are recovered only in the limit of an infinitely early start. Evaluating the tree-level cumulants with top-hat smoothing gives explicit formulas such as $S_3(a) = 34/7 - 3 - 6/(5a) + 12/(35 a^{7/2})$ for standard Zel'dovich velocities, and a different set of coefficients when velocities are recomputed from the displaced particle positions as in Efstathiou et al. The decay rate is set by the effective spectral index at each order, is slower for velocity statistics than for density, and is slower still for models with $

Load-bearing premise

The specific expansion factors come from leading-order perturbation theory, valid only as the variance of the smoothed field goes to zero, whereas real simulations measure these statistics where the variance is of order unity and loop corrections are not negligible.

Editorial extensions

If this is right

For a spectral index near zero, a Zel'dovich-started simulation needs roughly a factor of six in expansion before $S_3$ is good to ten percent, and about fifteen before the velocity analogue $T_3$ is, with the requirement growing sharply with the order $p$.
An open model with $\Omega = 0.3$ needs about a factor of two more expansion than a critical-density model to reach the same accuracy, because the controlling variable is the linear growth factor rather than the scale factor.
Assigning initial velocities from the gravitational potential of the displaced particle positions, rather than from the Zel'dovich prescription itself, shortens the transient by a factor of two or three depending on the spectrum.
Second-order Lagrangian initial conditions eliminate the skewness transient entirely and reduce the expansion needed for $p>3$ by more than an order of magnitude, for the price of two extra fast Fourier transforms and a modest amount of memory.
Because transients depress the measured $S_p$ while finite-variance loop corrections raise it, a simulation started too late can appear to agree with tree-level perturbation theory over a wider range of scales than it actually does.

Reading between the lines

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

Published claims about how far down in scale tree-level perturbation theory remains valid, made from simulations started at modest redshift, should be re-checked with the transient term subtracted; the cancellation the author describes is a concrete, testable reanalysis of existing outputs rather than a new simulation campaign.
The same decaying-mode bookkeeping should apply to redshift-space cumulants and to the bispectrum, where velocity correlations enter more strongly; the paper flags both but computes neither, so those are open checks of the mechanism rather than of its arithmetic.
If the velocity-divergence transients are as large as the $T_p$ curves suggest, velocity-based statistics in existing simulations may carry a systematic offset at moderate overdensities, which matters for any test that uses peculiar velocities as a discriminator.
The second-order Lagrangian residual decays as $a^{-2}$ and $a^{-9/2}$ for the kurtosis, so the required startup redshift is set almost entirely by the target accuracy and hardly at all by box size or resolution, which makes it a clean design parameter.

Original abstract

The standard procedure to generate initial conditions (IC) in numerical simulations is to use the Zel'dovich approximation (ZA). Although the ZA correctly reproduces the linear growing modes of density and velocity perturbations, non-linear growth is inaccurately represented because of the ZA failure to conserve momentum. This implies that it takes time for the actual dynamics to establish the correct statistical properties of density and velocity fields. We extend perturbation theory (PT) to include transients as non-linear excitations of decaying modes caused by the IC. We focus on higher-order statistics of the density contrast and velocity divergence, characterized by the S_p and T_p parameters. We find that the time-scale of transients is determined, at a given order p, by the spectral index n. The skewness factor S_3 (T_3) attains 10% accuracy only after a=6 (a=15) for n=0, whereas higher (lower) n demands more (less) expansion away from the IC. These requirements become much more stringent as p increases. An Omega=0.3 model requires a factor of two larger expansion than an Omega=1 model to reduce transients by the same amount. The predicted transients in S_p are in good agreement with numerical simulations. More accurate IC can be achieved by using 2nd order Lagrangian PT (2LPT), which reproduces growing modes up to 2nd order and thus eliminates transients in the skewness. We show that for p>3 this reduces the required expansion by more than an order of magnitude compared to the ZA. Setting up 2LPT IC only requires minimal, inexpensive changes to ZA codes. We suggest simple steps for its implementation.

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, 10,064 input tokens, $0.0004 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.30 · human participants 0.01 · controlled comparison 0.45 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.94 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

The standard procedure to generate initial conditions (IC) in numerical simulations is to use the Zel'dovich approximation (ZA). Although the ZA correctly reproduces the linear growing modes of density and velocity perturbations, non-linear growth is inaccurately represented because of the ZA failure to conserve momentum. This implies that it takes time for the actual dynamics to establish the correct statistical properties of density and velocity fields. We extend perturbation theory (PT) to include transients as non-linear excitations of decaying modes caused by the IC. We focus on higher-order statistics of the density contrast and velocity divergence, characterized by the S_p and T_p parameters. We find that the time-scale of transients is determined, at a given order p, by the spectral index n. The skewness factor S_3 (T_3) attains 10% accuracy only after a=6 (a=15) for n=0, whereas higher (lower) n demands more (less) expansion away from the IC. These requirements become much more stringent as p increases. An Omega=0.3 model requires a factor of two larger expansion than an Omega=1 model to reduce transients by the same amount. The predicted transients in S_p are in good agreement with numerical simulations. More accurate IC can be achieved by using 2nd order Lagrangian PT (2LPT), which reproduces growing modes up to 2nd order and thus eliminates transients in the skewness. We show that for p>3 this reduces the required expansion by more than an order of magnitude compared to the ZA. Setting up 2LPT IC only requires minimal, inexpensive changes to ZA codes. We suggest simple steps for its implementation.

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	The standard procedure to generate initial conditions (IC) in numerical simulations is to use the Zel'dovich approximation (ZA).	background 1.00	0.04	0.02	asserted 1.00
s1	Although the ZA correctly reproduces the linear growing modes of density and velocity perturbations, non-linear growth is inaccurately represented because of the ZA failure to conserve momentum.	background 0.90	0.08	0.05	asserted 1.00
s2	This implies that it takes time for the actual dynamics to establish the correct statistical properties of density and velocity fields.	interpretation 0.71	0.10	0.05	asserted 0.90
s3	We extend perturbation theory (PT) to include transients as non-linear excitations of decaying modes caused by the IC.	contribution 0.83	0.95	0.09	asserted 0.99
s4	We focus on higher-order statistics of the density contrast and velocity divergence, characterized by the S_p and T_p parameters.	contribution 0.72	0.68	0.04	asserted 0.93
s5	We find that the time-scale of transients is determined, at a given order p, by the spectral index n.	result 1.00	0.92	0.20	asserted 1.00
s6	The skewness factor S_3 (T_3) attains 10% accuracy only after a=6 (a=15) for n=0, whereas higher (lower) n demands more (less) expansion away from the IC.	result 1.00	0.90	0.17	asserted 1.00
s7	These requirements become much more stringent as p increases.	result 0.94	0.81	0.08	asserted 1.00
s8	An Omega=0.3 model requires a factor of two larger expansion than an Omega=1 model to reduce transients by the same amount.	result 1.00	0.85	0.07	asserted 1.00
s9	The predicted transients in S_p are in good agreement with numerical simulations.	result 1.00	0.88	0.12	asserted 0.99
s10	More accurate IC can be achieved by using 2nd order Lagrangian PT (2LPT), which reproduces growing modes up to 2nd order and thus eliminates transients in the skewness.	result 0.50	0.48	0.31	asserted 0.95
s11	We show that for p>3 this reduces the required expansion by more than an order of magnitude compared to the ZA.	result 0.98	0.89	0.24	asserted 1.00
s12	Setting up 2LPT IC only requires minimal, inexpensive changes to ZA codes.	result 0.32	0.45	0.06	asserted 0.99
s13	We suggest simple steps for its implementation.	method 0.37	0.45	0.05	hedged possibility 0.87

What kind of paper the classifier reports this is: theoretical result 0.92 · evidence from proof or derivation 0.90.

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.** — The ratio of the tree-level Sp parameters at scale factor a to their asymptotic exact dynamics value for scale-free initial spectra with different spectral indices: S3 (solid lines), S4 (dotted), S5 (short-dashed), S6 (long-dashed), S7 (dot-short-dashed), and S8 (dot-long-dashed). The values at a = 1 represent those set by the ZA initial conditions. For cosmological models with Ω < 1, the horizontal axis becomes the linear growth factor

**Figure 2.** — Same as Figure 1, but for the

**Figure 3.** — Same as Figure 1, but for initial velocities set as in

**Figure 4.** — Same as Figure 2, but for initial velocities set as in

**Figure 5.** — Symbols show the ratio of the Sp parameters at scale factor a measured in SCDM numerical simulations (Baugh, Gazta˜naga & Efstathiou 1995) to their asymptotic tree-level exact dynamics value as a function of smoothing scale R. Symbols represent a = 1 (open triangles), a = 1.66 (filled triangles), a = 2.75 (open squares) and a = 4.2 (filled squares). Error bars denote the variance of measurements in 10 realizations. Solid lines correspond to the predictions of transients in tree-level PT, expected to be valid at large scales

**Figure 6.** — The top left panel shows the γp parameters as a function of smoothing scale R for a Γ = 0.5 CDM model. The solid line corresponds to neff(R) ≡ γ(R) − 3, and γ2 (dotted line), γ3 (short-dashed), γ4 (longdashed), γ5 (dot-dashed). The top right panel shows the ratio of the Sp parameters to those calculated by setting γp = 0 for p ≥ 2 as a function of scale. Line-styles denote S4 (solid), S5 (dotted), S6 (short-dashed) and S7 (long-dashed). The left bottom panel shows the analogous calculation for a Γ = 0.21 CDM model. Right bottom panel presents a similar result for the Tp…

**Figure 7.** — The ratio of the tree-level Sp parameters at scale factor a to their asymptotic exact dynamics values for CDM models as a function of scale factor a. Top panels correspond to a Γ = 0.5 CDM model at smoothing scales R = 10, 100 h−1Mpc, and bottom panels show the corresponding results for a Γ = 0.21 CDM model. Line styles correspond to S3 (solid), S4 (dotted), S5 (short-dashed), and S6 (long-dashed). The upper axes in the bottom panels denote the scale factor a for Ω(a) = 0.3, with Λ = 0

**Figure 8.** — Same as Fig. 7, but for the

**Figure 9.** — Same as Fig. 7, but for initial velocities set as in ED

**Figure 10.** — Same as 8, but for initial velocities set as in EDFW,

**Figure 11.** — Same as Fig. 1, but for initial conditions set using

**Figure 12.** — Same as Fig. 2, but for the

Figures reproduced from arXiv:astro-ph/9711187 by the authors.

References

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Perturbation-theory informed integrators for cosmological simulations
astro-ph.CO 2023-01 conditional novelty 7.0 of 10

LPT-matched integrators for cosmological simulations outperform FastPM with O(1-100) timesteps while convergence is limited to order 3/2 post-shell-crossing due to acceleration field irregularity.
Renormalized Perturbation Theory at Field-level: the LSS bootstrap in GridSPT
astro-ph.CO 2025-06 conditional novelty 6.0 of 10

A renormalized field-level perturbation theory is shown to recover the LSS bootstrap parameter consistently across different grid cutoffs, validated at third and fifth order against N-body simulations.
Shapes and orientations of massive halos in the statistically anisotropic universe
astro-ph.CO 2025-05 conditional novelty 6.0 of 10

Quadrupolar statistical anisotropy in initial conditions leaves cluster shapes unchanged but aligns the major axes of massive halos, more strongly for higher mass.
Blinded challenge for precision cosmology with large-scale structure: results from effective field theory for the redshift-space galaxy power spectrum
astro-ph.CO 2020-03 conditional novelty 6.0 of 10

A blinded, three-team challenge shows that one-loop EFTofLSS templates recover the input Hubble rate, matter density and clustering amplitude from mock BOSS-like redshift-space spectra up to kmax about 0.12-0.14 h/Mpc...

Show 6 more citing papers

On the reach of perturbative descriptions for dark matter displacement fields
astro-ph.CO 2015-05 accept novelty 6.0 of 10

A residual, non-perturbative stochastic displacement in dark matter, uncorrelated with Lagrangian perturbation theory terms, limits perturbative predictions to about 1% on the nonlinear power spectrum at k = 0.2 h/Mpc...
Solving Large Scale Structure in Ten Easy Steps with COLA
astro-ph.CO 2013-01 conditional novelty 6.0 of 10

Running an N-body code in a frame comoving with second-order Lagrangian perturbation theory trajectories recovers halo clustering to roughly ten percent with ten timesteps instead of thousands.
Biased tracers, Hybrid Effective Field Theory and Modified Gravity
astro-ph.CO 2026-06 unverdicted novelty 5.0 of 10

Extends HEFT to f(R) gravity for loop-corrected biased power spectra and outlines emulator extensions from LambdaCDM codes.
Sahyadri: A simulation suite for the cosmology dependence of the Cosmic Web
astro-ph.CO 2026-01 conditional novelty 5.0 of 10

Sahyadri is a new N-body simulation suite with 25x better halo mass resolution than AbacusSummit and seed-matched parameter variations, showing coherent Omega_m signals in power spectra, mass functions, and cosmic-web...
Cosmology with massive neutrinos II: on the universality of the halo mass function and bias
astro-ph.CO 2013-11 accept novelty 5.0 of 10

In massive-neutrino cosmologies, the halo mass function and halo bias recover their usual universal forms when expressed through the cold dark matter variance sigma_cc, not the total matter variance sigma_mm.
An efficient implementation of massive neutrinos in non-linear structure formation simulations
astro-ph.CO 2012-09 accept novelty 5.0 of 10

A semi-linear method computes neutrino overdensities from the non-linear cold dark matter potential in N-body simulations, matching particle-based neutrino simulations to sub-percent accuracy on the total matter power...

How later papers used it

145 later papers cite it

Evidence: substantial. 141 of the citing papers share no author with it, and 97 of those carry a confident label from Pith's citation classifier.

Pith read 10 of the 171 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

9 later papers by other authors are listed below: 9 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 10 of 97 papers with this label · 1 by the same authors

Generated its initial particle displacement at z_in=99 using this paper, and its extended version for local non-Gaussian initial conditions.

High-redshift JWST Observations and Primordial Non-Gaussianity (2022) · 0.97 · paraphrased: the licence does not allow quoting
Advanced its particles using the 2LPT algorithm as described by this paper.

Quantifying distortions of the Lagrangian dark-matter mesh in cosmology (2012) · 1.00 · paraphrased: the licence does not allow quoting
Calculated its initial conditions with second-order Lagrangian perturbation theory using a modified version of this paper's 2LPTic code.

N-body simulations, halo mass functions, and halo density profile in $f(T)$ gravity (2022) · 0.99 · paraphrased: the licence does not allow quoting
Generated its N-body initial conditions with the Zel'dovich approximation plus a second-order correction term from this paper, added to the CDM particles only.

Resolving Cosmic Neutrino Structure: A Hybrid Neutrino N-body Scheme (2009) · 1.00 · paraphrased: the licence does not allow quoting
Initialized its simulations at an initial redshift of z_i=49 using this paper's 2LPT initial conditions code.

Matter Power Spectrum Emulator for f(R) Modified Gravity Cosmologies (2020) · 1.00 · paraphrased: the licence does not allow quoting
Determined the velocity and evolved density fields needed for its thermal and ionisation states using second-order Lagrangian perturbation theory from this paper.

Exploring the role of the halo mass function for inferring astrophysical parameters during reionisation (2024) · 0.99 · paraphrased: the licence does not allow quoting
Followed this paper to write the motion equations for a pressureless fluid in the one-flow limit in compact form using a doublet.

Cosmic propagators at two-loop order (2012) · 0.83 · paraphrased: the licence does not allow quoting
Used a slightly modified version of this paper's publicly available 2LPT code, which gave a detailed prescription for implementing 2LPT displacements in simulations.

Toward an accurate mass function for precision cosmology (2012) · 1.00 · paraphrased: the licence does not allow quoting
Generated its initial conditions using second-order Lagrangian perturbation theory from this paper with the MUSIC code.

Multi-wavelength mock galaxy catalogs of the low-redshift Universe (2021) · 1.00 · paraphrased: the licence does not allow quoting

By the same authors

Followed this paper to write the equations of motion for a pressureless fluid in the one-flow limit in compact form using a two-component quantity.

Constructing Regularized Cosmic Propagators (2011) (same authors) · 0.83 · paraphrased: the licence does not allow quoting

How the literature uses it

79 arXiv papers cite this in 101 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 · 43 sentences without a confident label

Uses its method or tool 49 papers

Employing $1024^3$ dark matter particles in a periodic cube of $(2048 h^-1Mpc)^3$, the gravitational dynamics was solved by a public simulation code [ref] starting from an initial condition set at $z=15$ by solving second-order Lagrangian perturbation theory [ref] .

Redshift-space equal-time angular-averaged consistency relations of the gravitational dynamics · Numerical results · 0.99
The simulations contain $512^3$ particles that are evolved from an initial redshift of $z=49$ until today using the $N$-body Gadget-3 code [ref] with 2LPT initial conditions [ref] .

Matter bispectrum of large-scale structure: Three-dimensional comparison between theoretical models and numerical simulations · $N$-body simulations · 0.95
The initial particle displacement is generated at $z_in=99$ using [ref] , and its extended version [ref] for local NG initial conditions, using $f_NL=500$ as value for the non-linearity parameter.

High-redshift JWST Observations and Primordial Non-Gaussianity · Testing high-redshift halo mass functions with N-body simulations · 0.97

Compares with it 1 paper

(For comparison to a 2LPT prescription, see Appendix D2 of [ref] ).

Quantifying distortions of the Lagrangian dark-matter mesh in cosmology · Particle realizations from the SC approximation · 0.95

Cites it as support 1 paper

Therefore, $N$-body simulations with the ZA initial conditions fail to pick up the correct second- and higher-order growing modes hence failing to reproduce the correct statistical properties of the density fluctuation until very late times [ref] .

Transients from initial conditions based on Lagrangian perturbation theory in N-body simulations · Introduction · 0.99

Cites it as background 4 papers

For further details we refer the reader to relevant literature for 2LPT [ref] , PM [ref] and COLA [ref] .

Advancing the matter bispectrum estimation of large-scale structure: a comparison of dark matter codes · Fast dark matter algorithms · 0.80
Cosmological perturbation theory (CPT; [ref] ) provides highly accurate predictions on large cosmological scales, and in particular plays a crucial role in connecting early- with late-time cosmology, such as by providing initial conditions for numerical simulations [ref] .

Renormalization group and UV completion of cosmological perturbations: Gravitational collapse as a critical phenomenon · Introduction · 0.98
Other important applications are to providing initial conditions for cosmological simulations [ref] , and for pairing CPT predictions with simulations in a variety of hybrid approaches [ref] .

Fast and accurate collapse-time predictions for collisionless matter · Introduction · 0.90

Discussion

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