Perturbative Lagrangian Approach to Gravitational Instability

E. Hivon; F.R. Bouchet; R. Juszkiewicz; S. Colombi

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Perturbative Lagrangian Approach to Gravitational Instability

F.R. Bouchet, S. Colombi, E. Hivon, R. Juszkiewicz

Astrophysics · Submitted 6 Jun 1994 · 1 reference

Pith's verdict: The derivations are clean and the third-order growth rates are genuinely new; the 'powerful' claim rests on one N-body box with a smoothing scale chosen after the fact.

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CitationE. Hivon, F.R. Bouchet, R. Juszkiewicz, S. Colombi. "Perturbative Lagrangian Approach to Gravitational Instability." arXiv:astro-ph/9406013 (1994). https://pith.science/paper/75HBCUWE

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Editor's note

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This is the paper people cite for Lagrangian perturbation theory growth rates, and it earns that. The core is a clean separable expansion of the Jacobian equation (1.13) order by order, and the payoff is real: third-order growth rates for Omega not equal to 1 (Eqs. 3.36-3.38) and for flat Lambda not equal to 0 (4.43-4.47) were not in the literature, and the S3 = 4 - 2 g2/g1^2 result (5.57) recovers Peebles' 34/7 and shows it barely moves with Omega. The l_a = -3/2 step below Eq. 3.32, fixing the second-order growing mode by requiring g2/g1^2 to vanish as Omega goes to 1, is exactly the kind of argument that could be hand-waving and isn't. The spherical-model check and the PM run are external validation, not self-confirmation, so circularity is low here despite the paper being partly a synthesis of the authors' own earlier letter (1992). That earlier result is published, so building on it is fine; the new material is the third order and the statistics section, and both are substantial. Where I would push back on the reader: the extrapolation past Eq. (7.3) is not a hidden assumption, it is the hypothesis under test, and the authors say so in section 4.2.2. They are also upfront that L/2 fails in voids (Eq. 7.6) and that C1 and C2 are dominated by the densest regions. That candour counts. The stress-test's complaint still lands, though. 'Powerful' rests on one realization of one n = -2 spectrum, 64^3 particles on a 128^3 grid, with no realization-to-realization scatter quoted for C1, C2, the CPDF or Table 1, and a smoothing scale picked because the smoothed variance peaks at the comparison epoch, which is a choice made with the answer in hand. The spherical test is also the friendliest geometry available: Birkhoff kills the tidal field and vorticity identically, so agreement there does not transfer for free to generic three-dimensional fields. The demonstrated claim is narrower than the headline: L/2 beats L/1 in dense regions, in one box, under a tuned filter. One soft spot the reader did not flag: Eq. (5.59) leaves E undetermined between 0 and 1, so the redshift-space skewness is a one-parameter family rather than a prediction. They point to Hivon et al. for the full calculation, but it is worth knowing before quoting a number. None of this is load-bearing against the derivations. Send it out. Anyone working on second-order biasing, redshift-space moments, or reconstruction should read it, and I would cite the growth-rate fits.

Referee report

Significance

If the second-order solution is as accurate beyond the perturbative regime as the paper argues, it provides a cheap, essentially parameter-free tool for the weakly nonlinear density and velocity fields, and the S3 = 4 - 2 g2/g1^2 result is a clean, nearly Omega-independent prediction that can be confronted with galaxy surveys. The strengths are real and should be named: the growth rates contain no fitted parameters; the Omega = 1 limit reproduces the known 34/7 result; the spherical-model comparison in Section 4 is a genuine external check against an exactly solvable case; and the redshift-space generalization is a concrete, falsifiable extension. The paper is also commendably candid about its failure modes, stating in Section 5.3 that the cross-correlation statistics are dominated by the densest regions and hide the fact that L2 is worse than L1 for the lowest negative density contrasts, and in Section 4.2.2 that L2 fails in voids. The assessment below concerns the scope and statistical weight of the empirical claim, not the derivational core.

Major objections (4)

[Section 5 and Table 1] The entire N-body validation rests on a single realization of a single spectrum (64^3 particles on a 128^3 PM grid, n = -2). No realization-to-realization scatter is quoted for the cross-correlation coefficients C1 and C2, for the CPDF panels of Figs. 14 and 16, or for the 'largest number of standard deviations' entries of Table 1. Because abstract:s11 ('both simple and powerful') is an empirical claim, the reader cannot judge whether the quoted improvements of L2 over L1 exceed the sampling noise of one 64^3 realization. Please either quote the scatter from an ensemble of realizations or, at minimum, state explicitly that the comparisons are single-realization and estimate the expected cosmic variance at the quoted scales.
[Section 5.1 and Fig. 11 (bottom-left panel)] The smoothing scale that yields the best L1S/L2S agreement at a = 16 is selected by the criterion that the smoothed variance be maximal at the comparison epoch, a criterion that uses knowledge of the answer. The gains attributed to smoothing in Table 1 and in Figs. 12, 14 and 16 therefore depend on a choice tuned at the comparison time. Please demonstrate that the conclusions are robust to the smoothing criterion, for example by fixing the filter scale from the linear variance at the initial epoch or by varying it by a factor of two, and report how much the quoted gains move.
[Section 5, abstract:s11] The simulation uses n = -2, the spectral index with the least small-scale power and hence the most favourable case for Zel'dovich-type approximations; the paper itself notes in Section 5.1 that small-scale power must be truncated for L1 and L2 to remain usable at late times. A general claim that second-order theory is 'powerful' is not established from the single most favourable index. Please add at least one steeper spectrum (n = -1 or a CDM-like spectrum), or restrict the claim explicitly to red spectra with little small-scale power.
[Section 4.2.2, Eq. (73), and Section 6] The authors state that for the larger-amplitude case epsilon g1/g1,i is approximately 1.45, so the second validity condition, Eq. (73), fails, and they proceed to use L2 as an approximation to the real dynamics. The resulting statements, that L2 reproduces density contrasts as large as ten and is 'powerful', are therefore extrapolations rather than controlled expansions. The authors are explicit about this in Section 4.2.2 and about the void failure in Eq. (7.6) and the C1/C2 bias in Section 5.3, which is to their credit; what remains unresolved is that the abstract and Section 6 do not carry the qualification. Please state the demonstrated domain (halos up to delta of order ten, failure in voids and underdense regions) in the abstract and conclusions, so that 'powerful' is read as an empirically established range rather than a general claim.

Minor objections (9)

[Section 2.2, Eqs. (3.32)-(3.33)] The constant l_a is fixed by the statement that one must have g2/g1 -> 0 when tau -> infinity, i.e. when Omega -> 1, and this is asserted rather than shown. As printed the condition reads as conflicting with the Omega = 1 limit g2 = -(3/7) g1^2 of Eq. (3.34), since g2/g1 does not vanish there. Please spell out the argument, presumably that the unwanted decaying-mode contamination must vanish, so that the reader can follow the selection of the physically relevant solution.
[Section 3.2, text below Eq. (5.58)] The redshift-space variance is found to be boosted by 30/15 for Omega = 1, and the text says this 'slightly differs from Kaiser's calculation (who did a calculation in spherical coordinates instead of our rectangular ones)'. The actual origin of the difference is that Kaiser averages over the angle between the line of sight and the wavevector, giving the 1/5 coefficient on f^2, whereas this calculation fixes the line of sight along the x3 axis, giving 1/3. Please state this explicitly so that the passage is not read as an unresolved discrepancy with a published result.
[Section 3.2, Eq. (5.59)] The quantity E is bounded between 0 and 1 but is not evaluated, so the redshift-space skewness for Gaussian initial conditions is quoted only as the range (35 + E)/7. Please give the resulting numerical range of S3^z explicitly and state how it compares with the real-space 34/7, since the claim that S3 is barely affected by redshift-space distortions rests on this bound.
[Section 2.3, Eq. (3.38), and Section 2.4, Eq. (4.47)] The third-order fits are described as providing 'very good fits' on the basis of Figs. 2 and 4 alone. Since the abstract advertises third-order results as a deliverable, please quote the maximum fractional error of the fits over the stated range 0.1 < Omega < 1, rather than leaving the accuracy to visual inspection.
[Section 4.2.1 and Figs. 6-7] Bernardeau's model is cited in the text as 'Bernardeau (1992)', while the reference list gives 'Bernardeau, Juszkiewicz, Dekel, Bouchet 1993, in preparation'. The year and publication status should be reconciled, and because an unpublished source is load-bearing for the comparison in Figs. 6 and 7, the text should say so. The text already notes that the model is statistical rather than applicable to a single object, which is appropriate.
[Section 3.2 and reference list] The spherical-coordinate redshift-space calculation is supported only by 'Hivon et al. 1994, in preparation'. Please flag this as unpublished in the text, or supply the result, since it is the stated basis for extending Eq. (5.59) beyond the infinite-observer limit.
[Table 1 caption] The caption ends with a garbled fragment, 'These measurements were made at a=16. second and third one give', which should be removed or completed.
[Fig. 1 caption and Section 1.2] The Fig. 1 caption contains the typo 'sho es' for 'shows', and Section 1.2 reads 'Matarese and its colleagues' where 'his colleagues' is intended. Please also check the author list and year of the Matarese et al. reference against the in-text citation.
[Section 4.2.2, Eq. (7.7)] The initial profile uses the symbol 'th', presumably tanh, and the width parameter xi; please define both explicitly at first use.

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The paper

The paper claims

Expanding particle paths, not density fields, extends gravitational perturbation theory into the mildly nonlinear regime.

The reading

The paper asks how far a perturbative expansion of particle trajectories, rather than of the density field, can carry the standard theory of gravitational instability. The authors' answer is that the Lagrangian route buys real reach: its first-order term is exactly the Zel'dovich approximation, its second-order term is solvable in closed form for any density parameter $/Omega$ (with $/Lambda = 0$, and also for flat models with $/Lambda /neq 0$), and the third-order growth rates can be given analytically or fitted numerically. Because mass conservation is built in through the Jacobian of the coordinate map, the approximation keeps describing strong density contrasts long after the Eulerian series has stopped being usable. The paper then does two things with that machinery: it computes statistical consequences (the skewness of the density distribution, including its redshift-space version) and it tests the second-order solution against exactly solvable spherical collapse and against an $n = -2$ N-body simulation. The claim a sympathetic reader should take away is that the second-order Lagrangian solution is a cheap, closed-form stand-in for real dynamics that beats the Zel'dovich approximation in the weakly nonlinear regime, especially for dense regions and for velocity fields.

What carries the argument

The carrying object is the displacement field $/boldsymbol{/Psi}$ in the mapping from Lagrangian label $/boldsymbol{q}$ to Eulerian position $/boldsymbol{x}$, together with the Jacobian determinant $J = /det( /delta_{ij} + /partial /Psi_i / /partial q_j )$ that enforces mass conservation through $/delta = J^{-1} - 1$. Expanding $J$ in the invariant scalars $K = /nabla /cdot /boldsymbol{/Psi}$, $L$ (the two-index combination of first-order gradients), and $M = /det(/partial /Psi_i / /partial q_j)$ turns the divergence of the equation of motion into a hierarchy of ordinary differential equations for the growth factors. This is what makes the scheme work where the Eulerian one fails: the density contrast is generated by the geometry of the map rather than by a truncated series in $/delta$, so a first-order displacement (Zel'dovich) already produces an arbitrarily large $/delta$ at caustics. The same structure makes statistics tractable, because ensemble averages over the density become averages over the initial displacement gradients.

Extended reading notes

Core claim

The central claim is that a perturbative expansion of the displacement field $/boldsymbol{x} = /boldsymbol{q} + /boldsymbol{/Psi}^{(1)} + /boldsymbol{/Psi}^{(2)} + /dots$ gives a practical approximation to gravitational clustering well outside its formal domain. The authors derive the growth factors order by order: for $/Omega = 1$ the second-order rate behaves as $g_2 = -(3/7) g_1^2$, and for general $/Omega$ they give the exact integral form plus the accurate fit $g_2 /approx -(3/7) g_1^2 /Omega^{-2/63}$. They extend the calculation to third order, where the equation no longer separates for $/Omega /neq 1$, providing $g_{3a}$ and $g_{3b}$ numerically and as power-law fits near $/Omega = 1$; a parallel treatment covers flat models with a cosmological constant. From this they obtain the skewness-to-variance ratio $S_3 = 4 - 2 g_2/g_1^2$, which equals the known $34/7$ for $/Omega = 1$ and varies only weakly with $/Omega$ (the fit $S_3 /approx [28 + 6(1 - /Omega^{-2/63})]/7$), and they show how to redo the same calculation in redshift space, where the answer becomes $(35 + E)/7$ with $0 /le E /le 1$ for Gaussian initial conditions. The final claim is empirical: in spherical collapse the second-order solution tracks density contrasts up to about ten and outperforms Eulerian second-order theory, and in the $n = -2$ simulation it corrects the Zel'dovich approximation for high-density regions, for the velocity modulus, and for the third through fifth normalized moments of the count probability distribution.

Load-bearing premise

The paper's practical claims rest on using the second-order solution as an approximation to real dynamics even when its own validity condition fails, as the authors openly do in their larger-amplitude spherical test where $/varepsilon g_1/g_{1,i} /approx 1.45$.

Editorial extensions

If this is right

Any weakly nonlinear prediction that depends on the second-order growth rate, such as the skewness of the density distribution, can be written in closed form for arbitrary $/Omega$ rather than computed model by model.
The ratio $S_3 = 4 - 2 g_2/g_1^2 /approx 34/7$ is nearly independent of $/Omega$ and nearly unaffected by redshift-space mapping for Gaussian initial conditions, so a measured skewness close to $34/7$ is not by itself evidence for a critical-density universe.
Second-order Lagrangian displacements give a cheap way to generate mock density and velocity fields that outperform Zel'dovich mocks in dense regions and in the velocity modulus, without running a simulation.
For non-Gaussian initial conditions the skewness does respond to initial skewness and kurtosis, and the redshift-space distortion of $S_3$ can be large, so the same formalism gives a route to constraining primordial non-Gaussianity from low-order moments.

Reading between the lines

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

The paper effectively proposes a division of labour: use the Lagrangian series where mass conservation matters (density, high-density peaks) and switch to Eulerian or smoothed initial conditions where the local flow is smooth, since the authors' own scans show second-order losing to Zel'dovich in underdense regions.
The finding that second-order theory gives reasonable values for $S_5$ — a quantity that formally needs third order — suggests the series converges faster in the cumulative statistical averages than in the pointwise field, which would be worth testing directly by computing $S_5$ at third order and comparing.
The deliberate extrapolation past the formal validity condition invites a systematic study of where the second-order map breaks: the authors note structures are washed out by $a /approx 80$ in their toy model, which is a concrete, checkable failure mode rather than a vague caveat.
Because the second-order correction is a local functional of the initial field, it could in principle be applied as a fast reconstruction step on observed galaxy displacements; whether that improves reconstruction over Zel'dovich-based methods is an open, testable question.

Original abstract

This paper deals with the time evolution in the matter era of perturbations in Friedman-Lemaitre models with arbitrary density parameter $\Omega$, with either a zero cosmological constant, $\Lambda = 0$, or with a non-zero cosmological constant in a spatially flat Universe. Unlike the classical Eulerian approach where the density contrast is expanded in a perturbative series, this analysis relies instead on a perturbative expansion of particles trajectories in Lagrangian coordinates. This brings a number of advantages over the classical analysis. In particular, it enables the description of stronger density contrasts. Indeed the linear term is the famous Zel'dovich approximate solution (1970). We present here a systematic and detailed account of this approach. We give analytical results (or fits to numerical results) up to the third order. We then proceed to explore the link between the lagrangian description and statistical measures. We show in particular that Lagrangian perturbation theory provides a natural framework to compute the effect of redshift distortions, using the skewness of the density distribution function as an example. Finally, we show how well the second order theory does as compared to other approximat- ions in the case of spherically symmetric perturbations. We also compare this second order approximation and Zel'dovich solution to N-body simulations in the description of large-scale structure formation starting from a power law (n=-2) power spectrum of Gaussian perturbation. We find that second order theory is both simple and powerful.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

This paper deals with the time evolution in the matter era of perturbations in Friedman-Lemaitre models with arbitrary density parameter $\Omega$, with either a zero cosmological constant, $\Lambda = 0$, or with a non-zero cosmological constant in a spatially flat Universe. Unlike the classical Eulerian approach where the density contrast is expanded in a perturbative series, this analysis relies instead on a perturbative expansion of particles trajectories in Lagrangian coordinates. This brings a number of advantages over the classical analysis. In particular, it enables the description of stronger density contrasts. Indeed the linear term is the famous Zel'dovich approximate solution (1970). We present here a systematic and detailed account of this approach. We give analytical results (or fits to numerical results) up to the third order. We then proceed to explore the link between the lagrangian description and statistical measures. We show in particular that Lagrangian perturbation theory provides a natural framework to compute the effect of redshift distortions, using the skewness of the density distribution function as an example. Finally, we show how well the second order theory does as compared to other approximat- ions in the case of spherically symmetric perturbations. We also compare this second order approximation and Zel'dovich solution to N-body simulations in the description of large-scale structure formation starting from a power law (n=-2) power spectrum of Gaussian perturbation. We find that second order theory is both simple and powerful.

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	This paper deals with the time evolution in the matter era of perturbations in Friedman-Lemaitre models with arbitrary density parameter $\Omega$, with either a zero cosmological constant, $\Lambda = 0$, or with a non-zero cosmological constant in a spatially flat Universe.	background 0.61	0.39	0.03	asserted 0.91
s1	Unlike the classical Eulerian approach where the density contrast is expanded in a perturbative series, this analysis relies instead on a perturbative expansion of particles trajectories in Lagrangian coordinates.	method 0.82	0.69	0.06	asserted 1.00
s2	This brings a number of advantages over the classical analysis.	interpretation 0.52	0.20	0.06	asserted 0.99
s3	In particular, it enables the description of stronger density contrasts.	contribution 0.55	0.52	0.11	asserted 0.96
s4	Indeed the linear term is the famous Zel'dovich approximate solution (1970).	background 0.77	0.06	0.05	asserted 1.00
s5	We present here a systematic and detailed account of this approach.	contribution 1.00	0.83	0.04	asserted 0.81
s6	We give analytical results (or fits to numerical results) up to the third order.	contribution 0.91	0.95	0.05	asserted 0.99
s7	We then proceed to explore the link between the lagrangian description and statistical measures.	contribution 0.92	0.87	0.04	asserted 0.58
s8	We show in particular that Lagrangian perturbation theory provides a natural framework to compute the effect of redshift distortions, using the skewness of the density distribution function as an example.	contribution 0.61	0.94	0.11	asserted 1.00
s9	Finally, we show how well the second order theory does as compared to other approximat- ions in the case of spherically symmetric perturbations.	contribution 0.88	0.91	0.09	asserted 0.94
s10	We also compare this second order approximation and Zel'dovich solution to N-body simulations in the description of large-scale structure formation starting from a power law (n=-2) power spectrum of Gaussian perturbation.	contribution 0.55	0.94	0.11	asserted 0.98
s11	We find that second order theory is both simple and powerful.	interpretation 0.64	0.95	0.56	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.67 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

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

levels run

P abstract

not run

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T body sentences
C citations
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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.** Trajectories of particles for 2 in-phase, perpendicular, sine waves. The exact trajectories computed by direct numerical 12 12 simulation are shown at the top left. The top right panel shoes the trajectories in Zeldovich approximation. The bottom panel shows the trajectories obtained by using the frozen ow approximation (left), and the second-order Lagrangian approximation (right). well in which they will oscillate. The gure shows x(a) for the particles initially closest to y = 1=2. According to Zeldovich approximation a particle with coordinate q before the perturbation has an Euler

**Figure 2.** Growth factors for Friedman-Lema^tre models with arbitrary and = 0. The upper panels display the rst order growth rate divided by the expansion factor (left), and the second order one (right), while the lower panels show the two third order growth factors. The solid curves show the exact solutions and the dashed curves correspond to the asymptotic behaviors near = 1. while g2=g2 1 ! 1=2 when approaches 0. If > 1, the solutions are obtained by the usual transformation, and in particular, the solution corresponding to (33) is g2 = 1 2 + 9 2 ( 2 + 1) 1 S 1 2

**Figure 3.** Logarithmic derivatives of the growth factors of

**Figure 5.** Logarithmic derivatives of the growth factors for a at Universe with 6= 0. The upper panels correspond to the rst order (left) and second order (right) rates. The lower panels correspond to the two third order factors. The solid curves show the exact solutions, the dotted lines display their slope in = 1 [see Eq. (49)], and the dashed ones a better t for 0:1 < < 1 [see Eq. (51)]. We also show for comparison in the upper left plot the values for f1 provided by Peebles (triangles), Lahav et al. (long dashes), and Martel (dashes-dots). 3.1. Skewness factor in real space Now that large…

**Figure 6.** Computed density contrast as a function of the exact solution E for various approximations for a spherical top hat perturbation ( = 0:1). The left panel and the right panel respectively correspond to E 0 and E 0. The abbreviated labels in the right panel correspond to the following models: \Ex." stands for the exact solution, \E1" for Eulerian linear theory, \E2" for Eulerian second order perturbation theory, \L1" for Zeldovich approximation, \L2" for Lagrangian second order perturbation theory, and \B" for Bernardeau's model. is rather useful when one wants to use it as a…

**Figure 7.** The quantity rx:v=(aH ) as a function of , in logarithmic coordinates, for various approximations in the spherical top hat model ( = 0:1). Notations are the same as those used in

**Figure 8.** The density contrast (left panels) and the velocity eld (right panels) versus the distance to the center of the halo for two initial values of (see text). The top and bottom panels respectively correspond to = 0:002 and = 0:005. For a better legibility, the bottom left graph is in logarithmic coordinates. order approximation (hereafter L2). Note that at second order, arbitrarily underdense regions cannot be obtained. In the Eulerian case, the largest underdensity that can be reached is min;E2 ' 3 28(14 + 3 2=63) ' 0:3: (75) For Lagrangian second order perturbation theory,

**Figure 10.** Slices Lbox=50 thick of a model evolved from n = 2 gaussian initial conditions with respectively a PM numerical simulation (left column), Zeldovich approximation (middle column) and the Lagrangian second order approximation (right column). The rows from top to bottom correspond to a = 4, 8, and 16

**Figure 11.** The tree top rows show the density contours for the same slices than in the previous gure (i.e. PM, L1, and L2 are from left to right, and a =4, 8, and 16, from top to the third row). The left gure of the bottom row show the computed evolution of the variance (in cubic cells of size 1/64), when the initial conditions are smoothed with Gaussian lters of various scale `s. The adjacent gures show the corresponding a = 16 density contours for the L1 and L2 case, when an initial smoothing has applied at a scale `s = 1=64, to be compared with the unsmoothed case directly above

**Figure 12.** Density measured along a one-dimensional scan in a box evolved with a PM simulation (solid), and by using the L1 and L2 approximations (dots and dashes respectively) at a=16. In both cases, the density has been obtained by smoothing the data with a gaussian lter with `s = Lbox=64 (see text). The top panel compares the results for identical initial conditions, while on the bottom the initial conditions for the L1 and L2 approximations have been smoothed with the same `s = Lbox=64 gaussian lter used at a=16

**Figure 13.** Approximate density contrasts, 1 1 2, obtained by L1 (left) and L2 (right) versus the PM one, 1. The top panel is for a = 8. . The middle and bottom panels are at a = 16 and correspond respectively to unsmoothed and smoothed IC. See text for details

**Figure 14.** The CPDF PN (`) at a =16, for a PM, L1, and L2 \mover" (solid, dots, and dashes respectively). From left to right, the curves are for the scale log10 `=-1.6, -1.4, -1.2, -1.0, -0.8. The corresponding variances (see eq.(85)) measured in the PM simulation are

**Figure 15.** The third, forth and fth normalized moments of the density distribution for a = 8 (left) and a = 16 with smoothed initial conditions for approximations (right). The squares correspond to the simulation results, while triangles and circles denote the results obtained using respectively Zeldovich approximation and L2

**Figure 16.** Distribution function P () at a =16, for a PM, L1, and L2 \mover" (solid, dots, and dashes respectively). The top pannel shows a comparison for unsmoothed intial conditions, and the bottom one for smoothed ones (with a gaussian lter of size `s = Lbox=64).In each panel, the curves from top to bottom, are for the following 2 PM of the PM simulation : 3.3, 1.6, 1/1.26, 1/2.5, 1/5. For a greater legibility, each curve has been o set as compared to its neighbor above by -1 on the y-axis

**Figure 17.** 1 1 Same as gure 13, but for the modulus of the velocity. L1 is on the left, L2 on the right. The top row is at a=8 and is not smoothed. The bottom rows corresponds to a latter time, a = 16, when the measurements are performed after smoothing of the nal velocity eld, in the case of unsmoothed (middle) and smoothed (bottom) initial conditions

**Figure 18.** Distribution of angles between the true velocity, and the one computed using L1 (dots), or L2 (solid). The top plot shows the result (for unsmoothed IC) at a = 8. The other plots show the result after smoothing of the nal velocity eld, at a = 16, for unsmoothed (middle) and smoothed IC (bottom). the comparison is made, improves dramatically the predictions of Zeldovich approximation. One may think of several ways of choosing the exact value of the scale at which the initial data is ltered. One would like to choose the smallest scale that still prevents, at the time the comparison

Showing 16 of the paper's 18 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9406013 by the authors.

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HETDEX [OII] galaxies give a linear growth index gamma = 0.60 +/- 0.07 (or 0.55 +/- 0.05 with Alcock-Paczynski parameters fixed), consistent with general relativity and with Planck flat LambdaCDM.

Show 19 more citing papers

A First Post-Friedmann Extension of the Schr\"odinger Approach to Cosmic Structure Formation
astro-ph.CO 2026-06 unverdicted novelty 6.0 of 10

Derives a 1PF relativistic extension of the Schrödinger approach to cold matter dynamics that requires an effective vector potential for the transverse velocity component in the cosmological frame.
DISCO-DJ II: a differentiable particle-mesh code for cosmology
astro-ph.CO 2025-10 conditional novelty 6.0 of 10

A GPU-accelerated, differentiable particle-mesh N-body code achieves per-cent-level power-spectrum accuracy with few time steps and recovers sigma_8 plus initial conditions from a noisy mock field.
$S_8$ from peculiar velocities: agreement with Planck for Tully--Fisher and supernovae, tension for the fundamental plane
astro-ph.CO 2025-09 conditional novelty 6.0 of 10

A unified Bayesian analysis of Tully-Fisher, fundamental plane, and supernova distances gives S8 = 0.819±0.030, consistent with the Planck CMB value of 0.832±0.013.
Using Neural Emulators and Hamiltonian Monte Carlo to constrain the Epoch of Reionization's History with the Ly$\alpha$ Forest Power Spectrum
astro-ph.CO 2025-09 conditional novelty 6.0 of 10

A JAX pipeline that emulates both the Lyα forest power spectrum and its model-dependent covariance with neural networks, then runs Hamiltonian Monte Carlo, recovers reionization-history parameters from mock observations.
$\texttt{GENGARS}$: Accurate non-Gaussian initial conditions with arbitrary bispectrum for N-body simulations
astro-ph.CO 2025-08 conditional novelty 6.0 of 10

GENGARS uses a Schwinger-parameterized reduced bispectrum kernel to generate N-body initial conditions for arbitrary separable PNG shapes, reducing spurious power-spectrum contributions relative to 2LPT-PNG.
The Manticore Project I: a digital twin of our cosmic neighbourhood from Bayesian field-level analysis
astro-ph.CO 2025-05 conditional novelty 6.0 of 10

Manticore-Local delivers Bayesian constrained simulations that reproduce major clusters and beat prior velocity-field reconstructions, but its LCDM-consistency and no-underdensity findings are partly enforced by priors.
Measuring the redshift-space distortions by cross-correlating the density fields before and after reconstruction
astro-ph.CO 2025-02 conditional novelty 6.0 of 10

A one-loop perturbation theory model for the cross-power spectrum of pre- and post-reconstructed density fields recovers the cosmic growth rate from N-body simulations up to k = 0.29 h/Mpc for smoothing scales Rs = 15...
Diagnosing Systematic Effects Using the Inferred Initial Power Spectrum
astro-ph.CO 2024-12 conditional novelty 6.0 of 10

The inferred initial matter power spectrum from the SELFI algorithm reveals misspecified galaxy bias, selection, mask, redshift, and gravity models, exposing a >2σ cosmological bias before parameter inference.
Fast Large Scale Structure Perturbation Theory using 1D FFTs
astro-ph.CO 2016-03 conditional novelty 6.0 of 10

The 1-loop SPT power spectrum corrections P22 and P13 reduce exactly to sums of products of 1D Hankel transforms of the linear power spectrum, each computable with a single FFTLog call.
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...
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.
Large-Scale Galaxy Bias
astro-ph.CO 2016-11 accept novelty 5.0 of 10

The paper reviews and extends the perturbative theory of galaxy bias, showing that quasi-linear galaxy clustering is captured by a finite set of bias parameters built from local gravitational observables.
On the reach of perturbative methods for dark matter density fields
astro-ph.CO 2015-07 conditional novelty 5.0 of 10

The exponentiated Lagrangian-to-Eulerian mapping, combined with fitted transfer functions, reproduces the nonlinear matter power spectrum to 1% at k = 0.25 h/Mpc at z = 0, with the residual attributed to a stochastic ...
Coupled dark matter-dark energy in light of near Universe observations
astro-ph.CO 2010-06 conditional novelty 5.0 of 10

Low-redshift probes, chiefly the dark matter deficit in voids, bound the dark matter-dark energy coupling to |xi| < 0.2, and a class of coupled models is shown to modify structure growth while leaving General Relativi...
A critical look at cosmological perturbation theory techniques
astro-ph.CO 2009-05 accept novelty 5.0 of 10

A side-by-side test against N-body simulations shows every current analytic perturbation theory for the matter power spectrum breaks down near k about 0.1 per Mpc at redshift zero, and the newly computed 2-loop standa...
Variability in Cosmological Hydrodynamical Simulations: how Stochastic Processes, Numerical Effects, and Reproducibility Limits impact Predictability
astro-ph.GA 2026-06 unverdicted novelty 4.0 of 10

Four controlled identical realizations of a galaxy-cluster zoom-in simulation reveal 10-25% variability in galaxy masses driven by stochastic processes and modulated by feedback, establishing a noise-dominated but sta...
Cosmic Structure Formation in the Non-linear Regime: Beyond Gaussian Statistics and Standard Cosmologies
astro-ph.CO 2024-11 conditional novelty 4.0 of 10

The matter density PDF can be predicted in extended cosmologies with large deviations theory, and wave-based forward models capture phase-space dynamics beyond the fluid approximation.
From quantum fluctuations to galaxy power spectrum multipoles
astro-ph.CO 2026-09 conditional novelty 3.0 of 10

A self-contained lecture-note derivation of galaxy power-spectrum multipoles from primordial curvature perturbations, with a closed-form FFTLog master-integral basis and illustrative validation against AbacusPNG halo ...
Bridging perturbation theory and simulations: initial conditions and fast integrators for cosmological simulations
astro-ph.CO 2026-08 unverdicted

A lecture-note review of Gaussian random field initial conditions, Lagrangian perturbation theory, and perturbation-theory-informed integrators for N-body simulations.

How later papers used it

86 later papers cite it

Evidence: substantial. 83 of the citing papers share no author with it, and 16 of those carry a confident label from Pith's citation classifier.

Pith read 14 of the 137 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

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 13 papers with this label

We expand the smoothed HI density contrast given in <ref> in Fourier space as [equation] where we have introduced the following Fourier space kernels [equation] Here, we made use of the Fourier space kernels for the dark matter density field in an Einstein de Sitter universe [ref] [equation] and $F_3$ is given in [ref] .

The effect of finite halo size on the clustering of neutral hydrogen (2021) · HI power spectrum at one-loop and stochasticity · 0.90

How the literature uses it

83 arXiv papers cite this in 136 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 · 90 sentences without a confident label

Uses its method or tool 13 papers

In this study we capture these effects closely following the recent work of [ref] , that employed the Gaussian Streaming Model (GSM) approach [ref] with Lagrangian Perturbation Theory [ref] in the context of MG [ref] , in order to successfully model the multipoles of the anisotropic correlation function of halos in theories beyond GR.

Constraints on $f(R)$ and nDGP Modified Gravity Model Parameters with Cluster Abundances and Galaxy Clustering · Introduction · 0.99
These particles were evolved via 2lpt [ref] up to redshift $z_LPT=19$, followed by $N$-body evolution with cola [ref] from $z_LPT$ to $z_1=0.1500$, both implemented within Simbelmynë.

Diagnosing Systematic Effects Using the Inferred Initial Power Spectrum · Gravitational evolution · 1.00
These particles are evolved to the redshift of $z=19$ via second order Lagrangian Perturbation theory (2LPT) [ref] , then with an efficient implementation of cola (COmoving Lagrangian Acceleration, [ref] ) from $z=19$ to $z=0$.

Primordial power spectrum and cosmology from black-box galaxy surveys · Galaxy surveys · 1.00

Compares with it 1 paper

Given the $Λ$CDM Universe with general relativity, i.e. $w^(0)=-1$ and $c_i=β=0$, we can reproduce the standard result $ξ_λ=3/572$ [ref] , which corresponds to $ξ_t=15/1144$.

Second-order peculiar velocity field as a novel probe of scalar-tensor theories · Second-order · 0.80

Cites it as support 1 paper

Note that unobserved and observed regions in the inferred final conditionsfinal density fielddensity fields do not appear visually distinct, a consequence of the excellent approximation of 2LPT not just to the first but also high-order correlation functionhigher-order moments [ref] .

Bayesian large-scale structure inference and cosmic web analysis · Inferred 3D density fields · 0.96

Cites it as background 28 papers

As seen in section [ref], in the Lagrangian frame, the quantity of interest is not the position, but the displacement field $Ψ(q)$ which maps the initial comoving coordinatescomoving particle position $q$ to its final comoving coordinatescomoving Eulerian position x (see e.g. [ref] or [ref] for overviews), [equation] It is important to note that, though $Ψ(q)$ is a priori a full three-dimensional vector field, it is curl-free up to second order in LPT (appendix D in [ref] or [ref] for a review).

Bayesian large-scale structure inference and cosmic web analysis · Lagrangian $ψ$ versus Eulerian $δ$: one-point statistics · 0.96
In the absence of $a⃗_ct$ and $a⃗_stoch$, i.e. in the standard Lagrangian perturbation theory (LPT), the scalar part of the displacement field can be solved for recursively in terms of the underlying linear density field ($δ_0$): s⃗^(n)(k⃗)=-/n!k⃗/k^2∏_i=1^n{∫_p⃗_i δ_0(p⃗_i)}(p⃗_1+…+p⃗_n)l_n(p⃗_1,…,p⃗_n) , where $k=p⃗_1+…+p⃗_n$ and $l_n$ are kernels than can be found in the literature, e.g. [ref] .

On the reach of perturbative methods for dark matter density fields · Dynamics in the EFT framework · 0.91
Normally, in Lagrangian perturbation theory [ref] , the SIGF approximation is taken, namely treating the baryon and CDM density fields the same and solving for the total matter density $δ_m$.

Does it matter? A more careful treatment of density fluctuations in 21-cm simulations · Scale-dependent Lagrangian perturbation theory · 0.87

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