cold dark matter protohalo structure around collapse

YITP-21-93

Astronomy & Astrophysics manuscript no. ms_collapse ©ESO 2021November 18, 2021

Cold dark matter protohalo structure around collapse: Lagrangiancosmological perturbation theory versus Vlasov simulations

Shohei Saga1, Atsushi Taruya2, 3, and Stéphane Colombi4

1 Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, Université de Paris, CNRS, F-92190 Meudon, Francee-mail: [email protected]

2 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan3 Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai institute for Advanced Study, University of Tokyo,

Kashiwa, Chiba 277-8568, Japan4 Sorbonne Universtié, CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis boulevard Arago, F-75014 Paris, France

Received –; accepted –

ABSTRACT

We explore the structure around shell-crossing time of cold dark matter protohaloes seeded by two or three crossed sine waves of var-ious relative initial amplitudes, by comparing Lagrangian perturbation theory (LPT) up to 10th order to high-resolution cosmologicalsimulations performed with the public Vlasov code ColDICE. Accurate analyses of the density, the velocity, and related quantitiessuch as the vorticity are performed by exploiting the fact that ColDICE can follow locally the phase-space sheet at the quadratic level.To test LPT predictions beyond shell-crossing, we employ a ballistic approximation, which assumes that the velocity field is frozenjust after shell-crossing.In the generic case, where the amplitudes of the sine waves are all different, high-order LPT predictions match very well the exact so-lution, even beyond collapse. As expected, convergence slows down when going from quasi-1D dynamics where one wave dominatesover the two others, to the axial-symmetric configuration, where all the amplitudes of the waves are equal. It is also noticed that LPTconvergence is slower when considering velocity related quantities. Additionally, the structure of the system at and beyond collapsegiven by LPT and the simulations agrees very well with singularity theory predictions, in particular with respect to the caustic andvorticity patterns that develop beyond collapse. Again, this does not apply to axial-symmetric configurations, that are still correctfrom the qualitative point of view, but where multiple foldings of the phase-space sheet produce very high density contrasts, hence astrong backreaction of the gravitational force.

Key words. gravitation - methods: numerical - galaxies: kinematics and dynamics - dark matter

1. Introduction

In the concordant model of large-scale structure formation, thematter content of the Universe is dominated by collisionlesscold dark matter (CDM) following Vlasov-Poisson equations(Peebles 1982, 1984; Blumenthal et al. 1984). The cold na-ture of initial conditions implies that the dark matter distribu-tion is concentrated on a three-dimensional sheet evolving insix-dimensional phase-space. At shell crossing, that is in placeswhere the phase-space sheet first self-intersects, the seeds of firstdark matter haloes are created. In these regions, the fluid enters amulti-stream regime during which violent relaxation takes place(Lynden-Bell 1967) to form primordial CDM haloes. Numericalinvestigations suggest that first dark matter haloes formed dur-ing this process have a power-law density profile ρ ∝ rα withα ≈ −1.5 (Moutarde et al. 1991; Diemand et al. 2005; Ishiyama2014; Angulo et al. 2017; Ogiya & Hahn 2018; Delos et al. 2018;Colombi 2021). During their subsequent evolution, which in-cludes successive mergers, dark matter haloes relax to the wellknown universal Navarro-Frenk-White profile (hereafter NFW,Navarro et al. 1996, 1997).

From an analytical point of view, many approaches havebeen proposed to describe the results of simulations and thedifferent steps of dark matter haloes history, relying, for ex-ample, on entropy maximization (Lynden-Bell 1967; Hjorth &Williams 2010; Carron & Szapudi 2013; Pontzen & Governato

2013), self-similarity (Fillmore & Goldreich 1984; Bertschinger1985; Henriksen & Widrow 1995; Sikivie et al. 1997; Zukin& Bertschinger 2010b,a; Alard 2013), or more recently, apost-collapse perturbative treatment (Colombi 2015; Taruya &Colombi 2017; Rampf et al. 2021). However, because of thehighly nonlinear complex processes taking place during the post-collapse phase, the formation and evolution of dark matter haloesremain a subject of debate, and a consistent framework explain-ing the different phases of halo history remains yet to be pro-posed.

The early growth of large scale structures up to first shell-crossing, on the other hand, is well understood thanks to per-turbation theory. Indeed, restricting to the early phase of struc-ture formation, i.e., to the single-stream regime before shell-crossing, we can employ perturbation theory as long as fluctua-tions in the density field remain small (see e.g., Bernardeau et al.2002, for a review and references therein). Lagrangian perturba-tion theory (LPT) (e.g., Shandarin & Zeldovich 1989; Bouchetet al. 1992; Buchert 1992; Buchert & Ehlers 1993; Bouchetet al. 1995; Bernardeau 1994) uses the displacement field asa small quantity in the expansion of the equations of motion.First-order LPT corresponds to the classic Zel’dovich approx-imation (Zel’dovich 1970), and in one-dimensional space, it isan exact solution until shell-crossing (Novikov 1969). Becausethey follow elements of fluid along the motion, Zel’dovich ap-

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proximation and higher-order LPT provide us with a rather ac-curate description of the large scale matter distribution, even inthe nonlinear regime, shortly after shell crossing. The families ofsingularities that form at shell-crossing and after have been ex-amined in detail in the Lagrangian dynamics framework (Arnoldet al. 1982; Hidding et al. 2014; Feldbrugge et al. 2018), and thestructure of cosmological systems at shell-crossing has been in-vestigated for specific initial conditions (Novikov 1969; Rampf& Frisch 2017; Saga et al. 2018; Rampf 2019) and for randominitial conditions (Rampf & Hahn 2021).

As described above, investigations into the early stages oflarge-scale structure formation represent a key to understand thebottom-up scenario underlying the CDM model. Practically, per-turbation theory under the single-stream assumption, which isstrictly valid only during the early stages of the dynamical evolu-tion, provides the basis for predicting statistics of the large-scalestructure distribution, and has been successfully confronted withN-body simulations and observations (see e.g., Bernardeau et al.2002, for a review). Recently, in order to incorporate the effectsof multi-streaming at small scales, it has been proposed to intro-duce effective fluid equations with a non-vanishing stress tensorin the dark matter fluids, the so-called effective field theory oflarge-scale structure (Baumann et al. 2012; Carrasco et al. 2012;Hertzberg 2014; Baldauf et al. 2015). Although this approachneeds parameters in the stress tensor to be calibrated with N-body simulations, it has attracted much attention, and has beenrecently applied to real observational datasets to derive cosmo-logical parameter constraints (Ivanov et al. 2020). Accordingly,the understanding of multi-streaming effects from first princi-ples, even at an early stage, would provide significant insightsinto the precision theoretical modelling of large-scale structure.

The aim of this paper is to extend the investigations of Sagaet al. (2018, hereafter STC18). By relying on comparisons be-tween LPT predictions up to 10th order to numerical simula-tions performed with the state-of-the-art Vlasov-Poisson solverColDICE (Sousbie & Colombi 2016), we aim to thoroughly ex-amine the structure both at and shortly after shell-crossing ofprotohaloes with simplified initial conditions composed of twoor three crossed sine waves, following the footsteps of Moutardeet al. (1991, 1995) and STC18. Although these initial configu-rations are restrictive, they are to a large extent representative ofhigh peaks of a smooth random Gaussian field (see e.g., Bardeenet al. 1986), and therefore still quite generic. The high level ofsymmetry of these initial conditions considerably simplifies an-alytical calculations while still allowing one a thorough and in-sightful exploration of the pre- and post-collapse dynamics.

The prominent features shortly after shell-crossing are,for instance, the appearance of caustics, and in the multi-stream region delimited by these latter, non-trivial vorticity pat-terns (Doroshkevich 1973; Chernin 1993; Pichon & Bernardeau1999; Hahn et al. 2015). Thanks to a description of the phase-space sheet at the quadratic level in ColDICE, we can measurethese quantities in high-resolution Vlasov simulations, in par-ticular the vorticity, with unprecedented accuracy. To performtheoretical predictions shortly after shell-crossing, we use a sim-ple dynamical approximation based on ballistic motion appliedto the state of the system described by high-order LPT solutionsat shell-crossing. While convergence with perturbation order ofLPT remains a complex subject of investigation (Zheligovsky &Frisch 2014; Rampf et al. 2015; Michaux et al. 2021), it seems totake place at least up to shell crossing, not only for the three sinewaves configurations we aim to examine (STC18), but also formore general, random initial conditions (Rampf & Hahn 2021),although the effects of cut-offs on power-spectra remain to be

investigated furthermore in the latter case. Therefore, as long asthe backreaction from multi-stream flows on post-collapse dy-namics remains negligible, shortly after shell-crossing, the ap-proximation of the dynamics we propose here should work. Thisis the first step toward a proper analytical description of post-collapse dynamics in 6D phase-space.

This paper is organized as follows. In Sec. 2, we begin by in-troducing the basics of LPT and its recurrence relations, as wellas its applications to initial conditions given by trigonometricpolynomials, such as our sine waves initial conditions. In Sec. 3,the important features of the Vlasov solver ColDICE are brieflysummarized. In Sec. 4, we examine the phase-space structureand radial profiles at shell-crossing with a comparison of ana-lytic predictions to simulations, in the framework of singular-ity theory. In Sec. 5, the structure shortly after shell-crossing isexplored using the ballistic approximation by examining phase-space diagrams, caustics, density and vorticity fields. Again, an-alytical predictions are compared to the Vlasov runs. Finally,Sec. 6 summarizes the main results of this article. To comple-ment the main text, Appendices A, B, C and D provide detailson the explicit form of high-order LPT solutions for the sinewaves initial conditions up to 5th order, the measurements in theVlasov simulations, the predictions from quasi-1D LPT, and theexpected structure of singularities at collapse time, respectively.

Throughout the paper, we will use the following units: a boxsize L = 1 and an inverse of the Hubble parameter at presenttime H−1

0 = 1 for the dimensions of length and time, respectively.In particular, the Lagrangian/Eulerian coordinates, velocity, andvorticity will be explicitly expressed as q/L, x/L, v/(L H0), andω/H0 = (L∇) × (v/(L H0)), respectively.

2. Lagrangian Perturbation Theory

The Lagrangian equation of motion of a fluid element is givenby (e.g., Peebles 1980)

d2xdt2 + 2H

dxdt

= −1a2∇xφ(x), (1)

where the quantities x, a, and H(t) = a−1da/dt are the Euleriancomoving position, the scale factor of the Universe, and Hubbleparameter, respectively. The derivative operator ∇x = ∂/∂x isa spatial derivative in Eulerian space. The Newton gravitationalpotential φ(x) is related to the matter density contrast δ(x) =ρ(x)/ρ − 1 with ρ being the background mass density, throughthe Poisson equation:

∇2xφ(x) = 4πGρ a2 δ(x). (2)

In this framework, the velocity of each mass element is given byv = a dx/dt.

Taking the divergence and curl of Eq. (1) with respect toEulerian coordinates, Eqs. (1) and (2) can be expressed by theequivalent set of equations:

∇x · (x + 2H x) = 4πGρ δ(x), (3)∇x × (x + 2H x) = 0, (4)

where the dot represents the Lagrangian derivative of time, d/dt.In the Lagrangian approach, for each mass element, the Eu-

lerian position x at the time of interest t is related to the La-grangian coordinate (initial position) q through the displacementfield Ψ(q, t) by

x(q, t) = q +Ψ(q, t). (5)


S. Saga et al.: Cold dark matter structure around collapse

The velocity field is expressed as v(q, t) = a dΨ/dt. In the singleflow regime, i.e., before first shell-crossing time tsc, mass con-servation implies ρ d3q = ρ(x) d3x, which leads to

1 + δ(x) =ρ(x)ρ

=1J, (6)

where the quantity J = det Ji j is the Jacobian of the matrix Ji jdefined by

Ji j(q, t) =∂xi(q, t)∂q j

= δi j + Ψi, j(q, t), (7)

and its inverse is given as

J−1i j =

∂qi

∂x j. (8)

While equation (6) is well defined only until the first shell-crossing time tsc, that is the first occurrence of J = 0, Eqs. (7)and (8) can be formally used beyond tsc, as long as they are ex-pressed in terms of Lagrangian coordinates (except that Eq. (8)might become singular).

2.1. Recursion relation

In Lagrangian perturbation theory, the displacement field, Ψ, isthe fundamental building block which is considered as a smallquantity. It can be systematically formally expanded as follows,

Ψ(q, t) =

∞∑n=1

Ψ(n)(q, t). (9)

In what follows, we shall assume that the fastest growing modesdominate. They are well known to be given to a very good ap-proximation by (see, e.g. Bernardeau et al. 2002, and referencestherein)

Ψ(n)(q, t) = Dn+(t)Ψ(n)(q), (10)

where the purely time-dependent function D+(t) corresponds tothe linear growth factor. The velocity field is then given by

v(q, t) = a H f∞∑

n=1

n Dn+(t)Ψ(n)(q), (11)

where function f (t) ≡ d ln D+/d ln a corresponds to the loga-rithmic derivative of the growth factor. Note that the analysesperformed in subsequent sections will consider the Einstein-deSitter cosmology, that is a total matter density parameter Ωm = 1and a cosmological constant density parameter ΩΛ = 0. In thiscase, one simply has D+ ∝ a and f = 1.

During early phases of structure formation, the Universe ap-proaches Einstein-de Sitter cosmology and f is approximatelygiven by f ≈ Ω

3/5m . With the further approximation f ≈ Ω

1/2m , as

implicitly assumed in all the subsequent calculations (see, e.g,Peebles 1980; Bernardeau et al. 2002; Matsubara 2015), by sub-stituting Eqs. (5), (6), (7), and (9) into Eqs. (3) and (4), one ob-tains simple recurrence formulae for the longitudinal and trans-verse parts of the motion (Rampf 2012; Zheligovsky & Frisch2014; Rampf et al. 2015; Matsubara 2015):(T −

32

)Ψ

(n)k,k = −εim ε jk

∑n1+n2=n

Ψ(n1)m,k

(T −

34

)Ψ

(n2)i, j , (12)

εi j TΨ(n)j,i = εi j

∑n1+n2=n

Ψ(n1)k, j TΨ

(n2)k,i , (13)

in the two-dimensional case, and(T −

32

)Ψ

(n)k,k = −εi jk εipq

∑n1+n2=n

Ψ(n1)j,p

(T −

34

)Ψ

(n2)k,q

−12εi jk εpqr

×∑

n1+n2+n3=n

Ψ(n1)i,p Ψ

(n2)j,q

(T −

12

)Ψ

(n3)k,r , (14)

εi jk TΨ(n)j,k = −εi jk

∑n1+n2=n

Ψ(n1)p, j TΨ

(n2)p,k , (15)

in the three-dimensional case. Here, we defined Ψi, j ≡ ∂Ψi/∂q j,and the tensors εi j and εi jk are, respectively, the two-dimensionaland three-dimensional Levi-Civita symbols. The symbol Tstands for a differential operator:

T ≡∂2

∂ ln D+2 +

12

∂

∂ ln D+

. (16)

In obtaining the recursion relations, we used the following for-mulae of linear algebra:

J =12εi j εkr Jik J jr, J−1

i j =1Jεik ε jr Jrk, (17)

for the two-dimensional case, and

J =16εi jk εpqr Jip J jq Jkr, J−1

i j =1

2Jε jkp εiqr Jkq Jpr, (18)

for the three-dimensional case.In the fastest-growing mode regime (10), which will be as-

sumed in the LPT calculations of this work, the time dependencein the recursion relations simplifies and one obtains

Ψ(n)k,k = −εim ε jk

∑n1+n2=n

4n22 + 2n2 − 3

2(n − 1)(2n + 3)Ψ

(n1)m,k Ψ

(n2)i, j , (19)

εi jΨ(n)j,i = εi j

∑n1+n2=n

n2(2n2 + 1)n(2n + 1)

Ψ(n1)k, j Ψ

(n2)k,i , (20)

for the two-dimensional case, and

Ψ(n)k,k = −εi jk εipq

∑n1+n2=n

4n22 + 2n2 − 3

2(n − 1)(2n + 3)Ψ

(n1)j,p Ψ

(n2)k,q

− εi jk εpqr

×∑

n1+n2+n3=n

(n3 + 1)(2n3 − 1)2(n − 1)(2n + 3)

Ψ(n1)i,p Ψ

(n2)j,q Ψ

(n3)k,r , (21)

εi jk Ψ(n)j,k = −εi jk

∑n1+n2=n

n2(2n2 + 1)n(2n + 1)

Ψ(n1)p, j Ψ

(n2)p,k , (22)

for the three-dimensional case.

2.2. Two sine and three sine waves initial conditions

Throughout this paper, we focus on specific initial conditions:two or three sine waves in a periodic box covering the interval[−L/2, L/2[:

Ψinii (q, tini) =

L2π

D+(tini) εi sin(

2πL

qi

). (23)



The parameters εi < 0 with |εx| ≥ |εy| ≥ |εz| quantify the lin-ear amplitudes of the sine waves in each direction. The ini-tial time, tini, is chosen such that D+(tini)|εi| ≤ 0.012 1,which makes the fastest growing mode approximation very ac-curate, as shown by STC18. In this framework, the dependenceon εi of the dynamics is reduced to a function of the ratiosε2D = εy/εx and ε3D = (εy/εx, εz/εx), respectively for two andthree sine waves initial conditions. These ratios will thereforebe the quantities of relevance to define our initial conditions. Inthis setting, the initial density field, given by δini ' −∇q ·Ψ

ini =D+(tini)

∑i |εi| cos (2π/L qi), presents a small peak at the origin,

and mass elements subsequently infall toward the central over-dense region. With the proper choice of εi, this initial overdensitycan asymptotically match any peak of a smooth random Gaus-sian field (see, e.g., Bardeen et al. 1986), which actually makesthe three sine waves initial conditions quite generic, hence pro-viding many insights into the dynamics during the early stagesof protohalo formation.

Naturally, this very symmetrical set-up remains unrealistic,with a tidal environment restricted to periodic replica, but has theadvantage of being a simple trigonometric polynomial, whichconsiderably simplifies LPT calculations, as described below.Initial conditions expressed as trigonometric polynomials in-clude fast Fourier transforms, so are in principle very general.Furthermore, with only a few Fourier modes, one can theoret-ically account for more realistic initial conditions with propertidal environments and mergers, while keeping the analyticaldescription still very tractable. However, as far as we are con-cerned, applications beyond two or three sine waves are left forfuture work.

For initial conditions given by trigonometric polynomials, itis trivial to see that all the terms of the perturbation series are alsogiven by trigonometric polynomials, which can be schematicallywritten as

∇q ·Ψ(n)L =

∑mα(n)

m ei m·q, (24)

∇q ×Ψ(n)T =

∑mβ(n)

m ei m·q, (25)

where the nth-order scalar coefficients α(n)m and the nth-order vec-

tor coefficients β(n)m are obtained recursively by calculating the

right-hand-side of Eqs. (21) and (22), which depends on lowerorder terms, starting from the n = 1 coefficients determined byequation (23).

By imposing the conditions ∇q ×Ψ(n)L = 0 and ∇q ·Ψ

(n)T = 0,

one can build up the perturbative solutions Ψ(n)L and Ψ(n)

T usingsimple algebraic manipulations involving coefficients α(n)

m andβ(n)

m :

Ψ(n)L =

∑m

(−i)α(n)m

m|m|2

ei m·q, (26)

Ψ(n)T =

∑m

im× β(n)

m

|m|2ei m·q. (27)

These solutions lead to the nth-order displacement field givenby Ψ(n) = Ψ

(n)L + Ψ

(n)T . Using this solution for Ψ(n) as well as

Eqs. (21) and (22), we subsequently construct the source of the(n + 1)th-order derivatives ∇q ·Ψ

(n+1)L and ∇q ×Ψ

(n+1)T . By repeat-

ing the above operation together with the recursive relations, onecan derive, in principle, arbitrary high-order LPT solutions. InAppendix A, we present the explicit forms of the LPT solutionsup to 5th order derived in this way.

The above prescription is valid in three-dimensional space,and can be applied to the 2D case by cancelling all fluctuationsalong z-axis, that is by performing 3D calculations with εz =0, i.e., ε3D = (εy/εx, 0) for the three sine waves case. However,one can also realize that in two dimensions, vector coefficientsβ(n)

m become scalars β(n)m , and that the solutions take the following

form:

Ψ(n)L =

∑m

(−i)α(n)

m

|m|2

(mxmy

)ei m·q, (28)

Ψ(n)T =

∑m

iβ(n)

m

|m|2

(my−mx

)ei m·q. (29)

Our analytical investigations can easily cover a large range ofvalues of ε2D and ε3D, while the simulations, much more costly,will only focus on three configurations, as detailed in Table 1, re-flecting various regimes in the dynamics: quasi one-dimensionalwith |εx| |εy,z|, anisotropic with |εx| > |εy| > |εz|, and what wedesign by axial-symmetric, with |εx| = |εy|(= |εz|), denoted byQ1D, ANI and SYM, respectively1.

3. Vlasov-Poisson simulations

To perform the numerical experiments, we use the public paral-lel Vlasov solver ColDICE (Sousbie & Colombi 2016). ColDICEfollows the phase-space sheet with an adaptive tessellation ofsimplices, composed, in 2 and 3 dimensions, of connected tri-angles and connected tetrahedra, respectively. The phase-spacecoordinates of the vertices of the tessellation, [X(t),V(t)], fol-low the standard Lagrangian equations of motion, similarly asin an N-body code, but matter is distributed linearly inside eachsimplex instead of being transported by the vertices.

The Lagrangian coordinates defined in Sec. 2.1 correspondto the following unperturbed initial set-up,

X(Q, t0) ≡ Q, (30)V(Q, t0) ≡ 0, (31)

where t0 is a virtual time corresponding to a = 0, Q is the La-grangian coordinate of each vertex. Note that we use capital let-ters to distinguish between vertices coordinates and actual coor-dinates of fluid elements of the phase-space sheet that they aresupposed to trace. These notations are used in Appendix B.

Vertices phase-space coordinates are then perturbed usingZel’dovich motion to set actual initial conditions defined inSec. 2.2:

X(Q, tini) = Q +Ψini(Q, tini), (32)

V(Q, tini) = a H f Ψini(Q, tini), (33)

with Ψini given by equation (23).To update the position and the velocity of each vertex, a

standard second order predictor-corrector scheme with slowlyvarying time step is employed. Constraints on the value of thetime step combine bounds on the relative variations of the ex-pansion factor, classical Courant-Friedrichs-Lewy criterion anda harmonic condition related to the local projected density. Moredetails about the parameters used for the time step constraints inthe 3D simulations are given in Colombi (2021, hereafter C21),

1 The designations named here, Q1D-3SIN, ANI-3SIN, and SYM-3SIN, are the same as Q1D-S, ASY-Sb, and SYM-S, used in Saga et al.(2018), respectively.



Designation ε2D or ε3D ng ns a∞sc asc asc asc + ∆aQuasi 1D

Q1D-2SIN 1/9 2048 2048 0.05279 0.05285 0.05281 0.05402Q1D-3SIN (1/6, 1/8) 512 256 0.03815 0.03832 0.03819 0.03907AnisotropicANI-2SIN 2/3 2048 2048 0.04531 0.04545 0.04534 0.04601ANI-3SIN (3/4, 1/2) 512 512 0.02911 0.02919 0.02915 0.03003

Axial-symmetricSYM-2SIN 1 2048 2048 0.04076 0.04090 0.04078 0.04101SYM-3SIN (1, 1) 512 512 0.03190 0.03155 0.03190 0.03201

Table 1. Parameters of the runs performed with ColDICE (Sousbie & Colombi 2016). The first column indicates the designation of the run. Thesecond column corresponds to the relative amplitudes of the initial sine waves, namely, ε2D = εy/εx and ε3D = (εy/εx, εz/εx) for two and three sinewaves, respectively. The third and fourth columns, respectively, indicate the spatial resolution ng of the grid used to solve the Poisson equation, andthe spatial resolution ns of the mesh of vertices used to construct the initial tessellation (see Sec. 3 for details). The fifth column indicates the scalefactor a∞sc at shell-crossing estimated by LPT extrapolated to infinite order (Saga et al. 2018), while the sixth one provides the value asc measuredin the Vlasov runs (see Appendix B.1) and which is actually used in Sec. 4.2. The seventh column indicates the value asc of the expansion factor ofthe closest available simulation snapshot to collapse time for the comparisons performed in section 4. Finally, the last column indicates the valueof the expansion factor used for the analyses performed beyond collapse time in Sec 5.

so we do not repeat these details here. Additionally, we chose thesame constraints on the time step for the 2D runs as for the 3Dsimulations with ng = ns = 512 (see below for the definitions ofng and ns).

Poisson equation is solved using the Fast-Fourier-Techniquein a mesh of fixed resolution ng. To estimate the projected den-sity ρ(x) on this mesh, the intersection of each simplex of thephase-space sheet with each voxel/pixel of the mesh is computedexactly up to linear order with a special ray-tracing technique.Once the gravitational potential is obtained on the mesh, thegravitational force is computed from the gradient of the potentialusing a standard 4 points stencil, and then it is interpolated to thevertices using second-order triangular shape cloud interpolation(see, e.g., Hockney & Eastwood 1988), in order to update theirvelocities.

Initially, the tessellation is constructed from a regular net-work of nD

s vertices corresponding, respectively to 2n2s and 6n3

ssimplices in 2 and 3 dimensions. ColDICE allows for localrefinement of the simplices following criteria based on localPoincaré invariant conservation as explained more in detail inSousbie & Colombi (2016). Values of the refinement criterionparameter I used for our 3D runs are listed in detail in C21. Forcompleteness, following the notations of C21, we used I = 10−8

for the 2D runs.To perform local refinement, the phase-space sheet is locally

described at quadratic order inside each simplex with the helpof 3 and 6 additional tracers per simplex in 2D and 3D, respec-tively. At the dynamical times considered in this work, whichcorrespond at most to short periods after collapse, refinement isnot triggered, except for a small number of simplices in the 3Daxial-symmetric simulation, SYM-3SIN, so we do not deem itnecessary to discuss more about refinement. However, the abilityto describe the phase-space sheet at the quadratic level is impor-tant to have correct estimates of derivatives of the velocity field,in particular of the local vorticity of the mean flow.

The parameters used for all the simulations are listed in Ta-ble 1, in particular the resolution ng of the mesh used to solvePoisson equation and the initial number of vertices of the tes-sellation, n3

s . In Appendix B, we explain how measurements ofvarious quantities are performed, such as the set of curves cor-responding to the intersection of the phase-space sheet with thehyperplane y(= z) = 0 used in Secs. 4.2 and 5.2, the collapsetime tsc shown in Table 1, the radial profiles used in Sec. 4.3, as

well as the caustic network, the projected density and the vortic-ity analysed in Sec. 5.3.

4. Shell crossing structure

We are now in a position to study the structure of our proto-haloes at collapse time, tsc, and concentrate our investigations onphase-space diagrams and radial profiles, with comparisons ofLPT pushed up to 10th order to the Vlasov runs. This section isorganized as follows. First, Sec. 4.1 presents the calculation ofcollapse time itself. Indeed, this quantity depends on initial con-ditions and perturbation order, and the ability of LPT to providean accurate determination of tsc is of prime importance. We dis-cuss the extrapolation to infinite order of the LPT series proposedby STC18 and generalize it to the 2D case. Second, Sec. 4.2 ex-amines the convergence of LPT at collapse with phase-space di-agrams, extending the earlier investigation of STC18 to the 2Dcase. For comparison, we also test the formal extension of LPTto infinite order and the predictions of the quasi one-dimensionalapproach proposed by Rampf & Frisch (2017, hereafter RF17).Finally, in Sec. 4.3, LPT predictions and their convergence arestudied in terms of radial profiles of the density, the velocity aswell as the pseudo phase-space density, and put into perspectivein relation to singularity theory.

4.1. Shell-crossing time

This subsection presents estimates of the expansion factor at firstshell-crossing to which we refer to as collapse time. In the fol-lowing, the expansion factor will be formally identified to a timevariable, still denoted by a to contrast with actual physical timet. We compare the value a(n)

sc of collapse time obtained from nth-order LPT and its extrapolation to infinite order a∞sc, as describedin STC18, to the value asc measured in the Vlasov runs as ex-plained in Appendix B.2. Our approach follows closely that ofSTC18. It is mainly meant to repeat its main steps and to supple-ment it with additional discussions and comparisons to Vlasovruns in the 2D case.

Using nth-order LPT predictions, we explore the sequence ofshell-crossing times, a(n)

sc , as a function of order n up to n = 10,by solving J(n) = 0 at the origin, where J(n) is the Jacobian ofthe nth-order LPT solution. Generally, except in the pure one-dimensional case where first-order LPT is exact before collapse,



a(n)sc becomes smaller with increasing perturbation order n (see

e.g., RF17, Rampf & Hahn 2021, hereafter RH21, for recentworks). As illustrated by Fig. 1, the shell-crossing time calcu-lated at nth-order with LPT is very accurately described by thefollowing fitting form (STC18):

a(n)sc = a∞sc +

(b + c exp

[d ne])−1 . (34)

This fitting form, also used for each coordinate of the positionsand velocities in Fig. 2 below, does not necessarily represent thesole choice for approximating the n dependence of collapse time,but using the exponential of a power-law might be the only wayto match the convergence speed of LPT at large n, when consid-ering quantities computed at collapse time a(n)

sc of each respec-tive order. Note importantly that this property is not incompati-ble with the findings of RH21, who examined LPT convergenceterm by term in the slightly different context of a Gaussian ran-dom field in a periodic box. In RH21, convergence is studiedin terms of the L2 norm of the coefficients of the displacementfield at each order in the perturbation series, for a given timea, while we consider the sequence of shell-crossing times a(n)

sc ,which is fundamentally different. Yet it would be interesting torelate equation (34) to the findings of RH21 which suggest theexistence of a power-law pole in the perturbation series.

One important thing to note is that convergence of collapsetime with order is rather slow, except in the quasi-1D case, whichstill requires at least third order for reaching a ∼percent level ofaccuracy for approximate convergence, while much higher or-der is required for other configurations, especially the 3D axial-symmetric case, ε3D = (1, 1), for which convergence does notseem to be achieved even at 10th order. Fig. 1 thus demonstratesthat low-order perturbation theory cannot be used to estimate ac-curately collapse times (see also STC18, in particular, Fig. 2 ofthis article, and RH21).

Despite these convergence issues, the extrapolated values ofa∞sc obtained with equation (34) are very accurate, as illustratedby Table 1. Indeed, the relative difference between a∞sc and thevalue asc measured in the Vlasov simulations remains of the or-der of the percent or below, which suggests that the error on theextrapolated estimate of collapse time remains small and shouldnot affect significantly the analytical predictions of the phase-space structure and the radial profiles at shell-crossing. There-fore, we will use a∞sc for the analytical predictions of “exact”collapse time in the remaining section 4. As for the simulationsdata, we will use the snapshot with the closest possible availablevalue asc of the expansion factor to asc, as indicated in Table 1.

4.2. Phase-space structure

Fig. 2 shows the (x, vx) subspace of the phase-space structureat shell-crossing by considering the intersection of the phase-space sheet with the hyperplane y = 0 for two sine waves andy = z = 0 for three sine waves. For comparison, in additionto standard LPT results, we also present the formal extensionof LPT to infinite order as developed by STC18 and describedin Sec. 4.1 for the shell-crossing time, as well as the analyticalprediction obtained in the context of the quasi one-dimensionalapproach developed by RF17, that we extended to second orderin the transverse direction (see Appendix C for details).

For quasi-1D (Q1D-2SIN and Q1D-3SIN) and anisotropic(ANI-2SIN and ANI-3SIN) initial conditions, the phase-spacesheet first self-intersects along x-axis, and, until then, the dynam-ics is similar to the pure one-dimensional case, in which 1LPT

1 2 3 4 5 6 7 8 9 10

0.030

0.035

0.040

0.045

0.050

0.055

0.060

a sc

2SIN-Q1D2SIN-ANI2SIN-SYM

3SIN-Q1D3SIN-ANI3SIN-SYM

1 2 3 4 5 6 7 8 9 10order

0

2 × 10 4

4 × 10 4

|a sc

a sc|

Fig. 1. Top: the shell-crossing time calculated at nth-order LPT (dots),together with the fitting function given by equation (34) (solid lines),as a function of the perturbation order, for various initial conditions asindicated on the panel. The dashed horizontal lines presents the valuesestimated from the simulations (see Appendix B.1), corresponding tothe sixth column of Table 1. Bottom: The relative error between theshell-crossing time calculated with nth-order LPT and the one obtainedwith the fitting formula.

(the Zel’dovich solution) is exact until shell-crossing. As illus-trated by Fig. 2, the LPT prediction quickly converges with per-turbation order n and describes the simulation results well evenat shell-crossing, especially in the Q1D case. Note that the keypoint is to match phase-space diagrams in the vicinity of the lo-cal extrema of the velocity, where LPT predictions perform theleast.

In the axial-symmetric cases, SYM-2SIN and SYM-3SIN,the phase-space sheet self-intersects simultaneously along allthe axes of the dynamics. Interestingly, the phase-space struc-ture for ε2D = 1 (SYM-2SIN) is still qualitatively similar toone-dimensional collapse and LPT again quickly converges withperturbation order to the exact solution. This is clearly not thecase for the three-dimensional axial-symmetric configuration,ε3D = (1, 1), where LPT convergence is very slow. This setup isindeed qualitatively different from other initial conditions, withthe appearance of a spiky structure in phase-space (SCT18), sim-ilarly as in spherical collapse. The exact properties of this spike,in particular, whether the velocity diverges or remains finite, and,whether if finite, the velocity is actually smooth at the fine level,remain unknown. However, while this spike is not present in theLPT predictions at finite order, it is well reproduced by the for-mal extrapolation to infinite order.

A question might arise whether, because of such a spikeand because of their highly contrasted nature, close to axial-symmetric 3D configurations correspond to a potentially differ-ent population of protohaloes. A partial answer can be foundin C21, who followed numerically the evolution of the threesine-wave configurations further in the non-linear regime. As de-scribed in C21, collapse of our protohaloes is followed by a vio-lent relaxation phase leading to a power-law profile ρ(r) ∝ r−α,



0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5v x

Q1D-2SIN ( 2D = 1/9)

1LPT2LPT3LPT4LPT5LPT6LPT7LPT

8LPT9LPT10LPTQ1D(2nd)EXTSimulation

0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

ANI-2SIN ( 2D = 2/3)

0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

SYM-2SIN ( 2D = 1)

0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

Q1D-3SIN ( 3D = (1/6, 1/8))

0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

ANI-3SIN ( 3D = (3/4, 1/2))

0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

SYM-3SIN ( 3D = (1, 1))

Fig. 2. Phase-space structure for two and three sine waves initial conditions at collapse time: Q1D-2SIN (top left), ANI-2SIN (center left), SYM-2SIN (bottom left), Q1D-3SIN (top right), ANI-3SIN (center right), and SYM-3SIN (bottom right). The intersection of the phase-space sheet withthe y = 0 plane for two sine waves and y = z = 0 hyper-plane for three sine waves is displayed in (x, vx) subspace. Simulation results are comparedto standard LPT predictions, that are supplemented with the blue line, denoted by “EXT”, which corresponds to the formal extrapolation to infiniteorder proposed by STC18 and sketched in Sec. 4.1 for the collapse time. For completeness, the quasi one-dimensional approach (Rampf & Frisch2017), denoted by Q1D, is also presented (see Appendix C for details).

with α ∈ [1.5, 1.7] and then by relaxation to an NFW like uni-versal profile. After violent relaxation, C21 did not find specificsignatures in the density profile nor the pseudo phase-space den-sity for the axial-symmetric case compared to the non-axial-symmetric ones, except that α tends to augment when goingfrom Q1D to axial-symmetric, and that the axial-symmetric con-figuration is subject to significant (and expected) radial orbit in-stabilities.

To complete the analyses, note that the formal extrapolationof LPT to infinite order matches well (but not perfectly) the sim-ulation results for all the configurations, as already found bySCT18 in the three-dimensional case. We also notice that thequasi one-dimensional approach of RF17 can describe very wellthe quasi-1D configurations. Interestingly, at the second order inthe transverse fluctuations considered here, the predictions are

rather similar to those of standard 4th-order LPT, irrespective ofinitial conditions. Thus, as expected, the quasi one-dimensionalapproach becomes less accurate when the ratio ε2D or ε3D ap-proaches unity, particularly in 3D as already shown by STC18.

4.3. Radial profiles

We now focus on radial profiles, and define Eulerian polarand spherical coordinates for two and three sine waves ini-tial conditions by (x, y) = (r cos θ, r sin θ) and (x, y, z) =(r sin θ cos φ, r sin θ sin φ, r cos θ), respectively. Then the angularaveraged radial profiles of the density ρ/ρ, velocity dispersionv2, radial velocity dispersion v2

r , and infall velocity −vr are given



by

ρ(r, t)/ρ =⟨J−1(q, t)

⟩Ω, (35)

v2(r, t) =

⟨J−1(q, t) v2(q, t)

⟩Ω⟨

J−1(q, t)⟩

Ω

, (36)

v2r (r, t) =

⟨J−1(q, t) (v(q, t) · x)2

⟩Ω⟨

J−1(q, t)⟩

Ω

, (37)

vr(r, t) =

⟨J−1(q, t) (v(q, t) · x)

⟩⟨J−1(q, t)

⟩Ω

, (38)

where x = x/r with r = |x| being the radial coordinate. In theseequations, we used the angle average defined by

〈 f (q)〉Ω =

∫dθ2π

f (q)|x=x(q,t) , (39)

for two sine waves initial conditions, and

〈 f (q)〉Ω =

∫sin θdθdφ

4πf (q)|x=x(q,t) , (40)

for three sine waves initial conditions. It is important to note thatthe angular coordinates θ and φ in the integrands are the Euleriancoordinates, and the integrands should be evaluated in terms ofthe Eulerian coordinate by solving the equation x = x(q, t).

To complete the analyses, we also study the pseudo phase-space density Q(r, t), defined by

Q(r, t) =ρ(r, t)/ρv3(r, t)

. (41)

Note in addition that we present not only the radial profiles butalso their logarithmic slopes defined by

n ≡d ln Xd ln r

, (42)

where the quantity X stands for the radial profile under consid-eration.

Radial profiles at shell-crossing are presented in Figs. 3–6.In these figures, measurements in the Vlasov code are performedby replacing each simplex of the phase-space sheet with a largeensemble of particles as explained in detail in C21 (see alsoAppendix B.4). Two approaches of collapse are considered. InFigs. 3 and 4, which correspond respectively to two and threesine waves initial conditions, calculations are performed at “ex-act” shell-crossing time, that is the extrapolated value a∞sc forLPT and the approximate value asc for the simulations. Figs. 5and 6 are analogous, but examine LPT predictions of nth-ordersynchronized to their own shell-crossing time a(n)

sc , which allowsus to examine in detail the structure of the singularity at col-lapse produced at each perturbation order. It is important to notethat from now on, we do not examine further the quasi one-dimensional approach proposed by RF17. We also do not per-form the formal extension to infinite order (Eq. 34), because cal-culating radial profiles in this framework was found to be toocostly with the computational methods we are currently using.

The examination of these figures confirms the conclusions ofthe phase-space diagram analysis in Sec. 4.2. In particular, quasi-1D profiles (Q1D-2SIN and Q1D-3SIN) are well described byLPT predictions even at low order. At “exact” collapse, con-sidered in Figs. 3 and 4, LPT predictions generally accuratelydescribe the outer part of the halo, where density contrasts are

10 3 10 2 10 1

100

101

102

103

(r)/

Q1D-2SIN ( 2D = 1/9)1LPT2LPT3LPT4LPT5LPT6LPT

7LPT8LPT9LPT10LPTSimulation

10 3 10 2 10 1

r

2

1

0

slop

e 2/3

10 3 10 2 10 110 3

10 2

10 1

100

v2 (r)

10 3 10 2 10 1

r

0

2

slop

e 2/3

10 3 10 2 10 110 3

10 2

10 1

100

v2 r(r)

10 3 10 2 10 1

r

0

2sl

ope 2/3

10 3 10 2 10 110 2

10 1

100

v r(r

)

10 3 10 2 10 1

r

1

0

1

slop

e

1/3

10 3 10 2 10 1101102103104105

Q(r

)

10 3 10 2 10 1

r

3210

slop

e 5/3

10 3 10 2 10 1

100

101

102

103

ANI-2SIN ( 2D = 2/3)

10 3 10 2 10 1

r

2

1

0 2/3

10 3 10 2 10 110 3

10 2

10 1

100

10 3 10 2 10 1

r

0

2 2/3

10 3 10 2 10 110 3

10 2

10 1

100

10 3 10 2 10 1

r

0

2 2/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

1/3

10 3 10 2 10 1101102103104105

10 3 10 2 10 1

r

3210

5/3

10 3 10 2 10 1

100

101

102

103

SYM-2SIN ( 2D = 1)

10 3 10 2 10 1

r

2

1

0 4/3

10 3 10 2 10 110 3

10 2

10 1

100

10 3 10 2 10 1

r

0

2 2/3

10 3 10 2 10 110 3

10 2

10 1

100

10 3 10 2 10 1

r

0

2 2/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

1/3

10 3 10 2 10 1101102103104105

10 3 10 2 10 1

r

3210

7/3

Fig. 3. Radial profiles and their logarithmic slopes for the two sinewaves initial conditions, at shell-crossing time. As indicated on topleft panel, LPT predictions are given by solid lines of various colourswhile Vlasov simulations results are represented by red dots. From leftto right, the initial conditions are given by Q1D-2SIN (ε2D = 1/9), ANI-2SIN (ε2D = 2/3), and SYM-2SIN (ε2D = 1), respectively. From top tobottom, we present the radial profiles of the normalized density ρ/ρ, thevelocity dispersion v2, the radial velocity dispersion v2

r , the infall veloc-ity −vr, and the pseudo pseudo phase-space density Q(r) respectively.Note that when plotting the logarithmic slopes in Vlasov simulations,we used the Savitzky-Golay filter implemented in savgol_filter ofSciPy (Virtanen et al. 2020) to smooth the data, for presentation pur-poses. In the logarithmic slopes panels, the horizontal dashed lines cor-respond to the theoretical predictions at small radii of Appendix D aslisted in Table 2.

lower, and then deviate from the exact solution when the ra-dius becomes smaller than some value rmin(n) decreasing with



10 3 10 2 10 1

100

101

102

103

(r)/

Q1D-3SIN ( 3D = (1/6, 1/8))1LPT2LPT3LPT4LPT5LPT6LPT


10 3 10 2 10 1

r

2

1

0

slop

e 2/3

10 3 10 2 10 110 3

10 2

10 1

100

v2 (r)

10 3 10 2 10 1

r

0

2

slop

e 2/3

10 3 10 2 10 110 3

10 2

10 1

100

v2 r(r)

10 3 10 2 10 1

r

0

2

slop

e 2/3

10 3 10 2 10 110 2

10 1

100

v r(r

)

10 3 10 2 10 1

r

1

0

1

slop

e

1/3

10 3 10 2 10 1101102103104105

Q(r

)

10 3 10 2 10 1

r

3210

slop

e 5/3

10 3 10 2 10 1

100

101

102

103

ANI-3SIN ( 3D = (3/4, 1/2))

10 3 10 2 10 1

r

2

1

0 2/3

10 3 10 2 10 110 3

10 2

10 1

100

10 3 10 2 10 1

r

0

2 2/3

10 3 10 2 10 110 3

10 2

10 1

100

10 3 10 2 10 1

r

0

2 2/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

1/3

10 3 10 2 10 1101102103104105

10 3 10 2 10 1

r

3210

5/3

10 2 10 1

100

101

102

103

SYM-3SIN ( 3D = (1, 1))

10 2 10 1

r

2

1

0 2

10 2 10 110 3

10 2

10 1

100

10 2 10 1

r

0

2 2/3

10 2 10 110 3

10 2

10 1

100

10 2 10 1

r

0

2 2/3

10 2 10 110 2

10 1

100

10 2 10 1

r

1

0

1

1/3

10 2 10 1101102103104105

10 2 10 1

r

3210

3

Fig. 4. Same as Fig. 3 but for the three sine waves initial condi-tions, from left to right, Q1D-3SIN [ε3D = (1/6, 1/8)], ANI-3SIN[ε3D = (3/4, 1/2)], and SYM-3SIN [ε3D = (1, 1)], respectively. Notethat in SYM-3SIN, the closest snapshot from collapse we had at dis-posal from our Vlasov runs, asc = 0.03190, is significantly beyond ac-tual shell-crossing time estimated by the method described in Sec. B.2,asc = 0.03155, which explains some discrepancies between the theoryand the simulation at small radii.

increasing n, where n is the perturbation order. LPT can describearbitrarily large densities, as long as n is large enough. In the re-sults presented here, density contrasts as large as 100 or largercan be probed accurately by LPT, depending on the order consid-ered and on the nature of initial conditions. Again, convergenceworsens when approaching three-dimensional axial-symmetry,the worse case being, as expected, SYM-3SIN with ε3D = (1, 1).It is worth noticing that velocity profiles require significantly

10 3 10 2 10 1

100

101

102

103

(r)/

Q1D-2SIN ( 2D = 1/9)1LPT2LPT3LPT4LPT5LPT6LPT


10 3 10 2 10 1

r

2

1

slop

e

2/3

10 3 10 2 10 110 2

10 1

100

v2 (r)

10 3 10 2 10 1

r

1

0

1

slop

e

2/3

10 3 10 2 10 110 2

10 1

100

v2 r(r)

10 3 10 2 10 1

r

1

0

1sl

ope

2/3

10 3 10 2 10 1

10 1

100

v r(r

)

10 3 10 2 10 1

r

1

0

1

slop

e

1/3

10 3 10 2 10 1100101102103104

Q(r

)

10 3 10 2 10 1

r

321

slop

e 5/3

10 3 10 2 10 1

100

101

102

103

ANI-2SIN ( 2D = 2/3)

10 3 10 2 10 1

r

2

12/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

2/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

2/3

10 3 10 2 10 1

10 1

100

10 3 10 2 10 1

r

1

0

1

1/3

10 3 10 2 10 1100101102103104

10 3 10 2 10 1

r

321 5/3

10 3 10 2 10 1

100

101

102

103

SYM-2SIN ( 2D = 1)

10 3 10 2 10 1

r

2

14/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

2/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

2/3

10 3 10 2 10 1

10 1

100

10 3 10 2 10 1

r

1

0

1

1/3

10 3 10 2 10 1100101102103104

10 3 10 2 10 1

r

321 7/3

Fig. 5. Same as Fig. 3 but all analytical predictions are evaluated atindividual shell-crossing times computed at each perturbation order.

higher LPT order than density to achieve a comparable visualmatch between theory and measurements. In particular, devi-ations between LPT predictions and simulations arise at quitelarger radii rmin(n) for the velocities than for the density, withdifferences as large as an order of magnitude. With synchro-nization, performances of LPT predictions greatly improve, asexpected and as illustrated by Figs. 5 and 6. All perturbation or-ders predict strikingly similar density profiles, in close to perfectmatch with the simulations, except for SYM-3SIN discussed fur-ther below, while the velocity profiles still diverge slightly fromeach other except for quasi one-dimensional initial conditions,Q1D-2SIN and Q1D-3SIN.

We now turn to the logarithmic slope of various profilesshown in the bottom insert of each panel of Figs. 3–6. The na-



10 3 10 2 10 1

100

101

102

103

(r)/

Q1D-3SIN ( 3D = (1/6, 1/8))1LPT2LPT3LPT4LPT5LPT6LPT


10 3 10 2 10 1

r

2

1

slop

e

2/3

10 3 10 2 10 110 2

10 1

100

v2 (r)

10 3 10 2 10 1

r

1

0

1

slop

e

2/3

10 3 10 2 10 110 2

10 1

100

v2 r(r)

10 3 10 2 10 1

r

1

0

1

slop

e

2/3

10 3 10 2 10 1

10 1

100

v r(r

)

10 3 10 2 10 1

r

1

0

1

slop

e

1/3

10 3 10 2 10 1100101102103104

Q(r

)

10 3 10 2 10 1

r

321

slop

e 5/3

10 3 10 2 10 1

100

101

102

103

ANI-3SIN ( 3D = (3/4, 1/2))

10 3 10 2 10 1

r

2

12/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

2/3

10 3 10 2 10 110 2

10 1

100

10 3 10 2 10 1

r

1

0

1

2/3

10 3 10 2 10 1

10 1

100

10 3 10 2 10 1

r

1

0

1

1/3

10 3 10 2 10 1100101102103104

10 3 10 2 10 1

r

321 5/3

10 2 10 1

100

101

102

103

SYM-3SIN ( 3D = (1, 1))

10 2 10 1

r

2

12

10 2 10 110 2

10 1

100

10 2 10 1

r

1

0

1

2/3

10 2 10 110 2

10 1

100

10 2 10 1

r

1

0

1

2/3

10 2 10 1

10 1

100

10 2 10 1

r

1

0

1

1/3

10 2 10 1100101102103104

10 2 10 1

r

321 3

Fig. 6. Same as Fig. 4 but all analytical predictions are evaluated atindividual shell-crossing times computed at each perturbation order.

ture of the singularities of cold dark matter structures at collapsetime and beyond has been widely studied in the literature, partic-ularly in the framework of Zel’dovich motion (e.g., Zel’dovich1970; Arnold et al. 1982; Hidding et al. 2014; Feldbrugge et al.2018). In Appendix D, we rederive the asymptotic properties ofour protohaloes at small radii by Taylor expanding the equationsof motion up to third order in the Lagrangian coordinate q. Forour sine-wave configurations, whatever the order n of the pertur-bation order,2 three kinds of singularities are expected at shell-crossing, as summarized in Table 2: the classic one-dimensionalpancake with a power-law profile at small radii of the form (S1)

2 Strictly speaking, our calculations are performed for n ≤ 10, but it isreasonable to expect that the result applies to arbitrary order.

ρ(r) ∝ r−2/3, valid for all the configurations expect SYM-2SINand SYM-3SIN; then (S2) ρ(r) ∝ r−4/3 and (S3) ρ(r) ∝ r−2, forSYM-2SIN and SYM-3SIN, respectively. On the other hand, ve-locities are expected to follow the same power-law pattern what-ever initial conditions or dimensionality, with logarithmic slopesequal to 2/3, 2/3, and 1/3 respectively, for v2, v2

r , and −vr, whichin turn implies Q(r) ∝ r−7/3 for SYM-2SIN, r−3 for SYM-3SIN,and r−5/3 for other configurations.

Figs. 3 and 4 consider LPT predictions for perturbation or-der n calculated at “exact” theoretical collapse time a∞sc and not atindividual collapse time a(n)

sc at this perturbation order, so shell-crossing is not reached exactly, but gets nearer as n increases.Consequently, the asymptotic slope is only approached approx-imately, and better so with larger n. Note that convergence isslower for velocities than for the density, especially for axial-symmetric configurations, in particular SYM-3SIN. With syn-chronization, as illustrated by Figs. 5 and 6, the convergence ata small radius to the prediction of singularity theory becomesclear.

Except for SYM-3SIN for which the best available snapshotis still too far off collapse with asc > asc, one notices that allthe simulated data show a close to perfect agreement betweenthe measured slope at small radii and the predicted ones for allthe radial profiles. This result is non-trivial in the sense that theTaylor expansion at small |q| underlying singularity theory is notnecessarily valid in the actual fully nonlinear framework. Indeed,while singularity theory seems to apply to each order n for an ar-bitrarily large value of n in LPT framework, this does not meanthat the limit to n → ∞ should converge to the same singular-ity. The fact that the simulation agrees with singularity theorypredictions beyond the well known, but nonetheless somewhattrivial, 1D case, is remarkable even though naturally expected.

Several additional remarks are in order. First, if crossed sinewaves are considered as generic approximations of local peaksof a smooth random Gaussian field, we also point out in thisframework that the probability of having exactly ε2D = εSYM

2D ≡ 1or ε3D = εSYM

3D ≡ (1, 1) is null, hence one expects the classicone-dimensional pancake to be exclusively dominant, from thepure mathematical view. However, this can be unrealistic at thecoarse level, where high-density peaks can be close to spheri-cal or filaments locally close to cylindrically symmetric. Thisis illustrated by Fig. 7, which examines the transition betweenvarious regimes in the vicinity of εSYM

2D (top panels) and εSYM3D

(bottom panels) for 10th-order LPT. Clearly, at sufficiently largeradii, values of ε2D or ε3D close to εSYM

2D and εSYM3D give similar

results for the profiles, even for the logarithmic slope which canapproach that of the axial-symmetric case. The exact mathemat-ical asymptotic behaviour is indeed reached only at very smallradii. In practice, the axial-symmetric configurations cannot beignored, which shows the limits of singularity theory if appliedblindly.

Second, we note that the slope of the density profile pre-dicted by LPT at collapse in the axial-symmetric 3D case, SYM-3SIN, is different from the prediction of pure spherical collapse,ρ(r) ∝ r−12/7, for an initial profile with the same asymptoticbehaviour at a small radius of the form ρini ∝ 1 − αr2 withα > 0 (see, e.g. Nakamura 1985; Moutarde et al. 1995). Hencethe anisotropic nature of the axial-symmetric case (containedin terms beyond leading order r2 in the Taylor expansion oftrigonometric functions) provides significantly different resultsfrom pure spherical collapse. Note, though, that this state of factis not fully proved by our simulations measurements, which asmentioned above, are slightly beyond actual collapse time.



Sine wave amplitudes Designation d ln ρ/ρd ln r

d ln v2

d ln rd ln v2

rd ln r

d ln (−vr)d ln r

d ln Qd ln r

0 ≤ −εz ≤ −εy < −εx Q1D-2SIN, ANI-2SIN, Q1D-3SIN, ANI-3SIN -2/3 2/3 2/3 1/3 -5/3εz = 0, 0 < −εy = −εx SYM-2SIN -4/3 2/3 2/3 1/3 -7/30 < −εz = −εy = −εx SYM-3SIN -2 2/3 2/3 1/3 -3

Table 2. Summary of the logarithmic slopes at shell-crossing obtained by singularity theory applied to LPT predictions at various orders (seeAppendix D for details). The first and second columns indicate the relative amplitudes of the initial sine waves and the designation of the runs,respectively. The third through seventh columns show the logarithmic slopes of the normalized density, velocity dispersion, radial velocity dis-persion, infall velocity, and pseudo phase-space density, respectively. Note thus that the logarithmic slopes do not depend, as expected, on theperturbation order of LPT.

10 5 10 4 10 3 10 2 10 1100101102103104105

(r)/

2D = 0.90

2D = 1

10 5 10 4 10 3 10 2 10 1

r1.61.41.21.00.80.6

slop

e

2/3 4/3

10 5 10 4 10 3 10 2 10 1

10 2

10 1

100

v2 (r)

2D = 0.90

2D = 1

10 5 10 4 10 3 10 2 10 1

r0.2

0.4

0.6

0.8

1.0

slop

e

2/3 10 5 10 4 10 3 10 2 10 1

10 2

10 1

100

v2 r(r)

2D = 0.90

2D = 1

10 5 10 4 10 3 10 2 10 1

r0.20.40.60.81.01.2

slop

e2/3 10 5 10 4 10 3 10 2 10 1

10 1

100

v r(r

)

2D = 0.90

2D = 1

10 5 10 4 10 3 10 2 10 1

r0.2

0.4

0.6

slop

e

1/3 10 5 10 4 10 3 10 2 10 1

102

105

108

Q(r

)

2D = 0.90

2D = 1

10 5 10 4 10 3 10 2 10 1

r

2.5

2.0

1.5

slop

e

5/3 7/3

10 5 10 4 10 3 10 2 10 1

101

103

105

107

109

(r)/

3D = (0.97, 0.97)

3D = (1, 0.97)3D = (1, 1)

10 5 10 4 10 3 10 2 10 1

r

2.0

1.5

1.0

0.5

slop

e

2/3 4/3 2 10 5 10 4 10 3 10 2 10 1

10 2

10 1

100

v2 (r)

3D = (0.97, 0.97)

3D = (1, 0.97)

3D = (1, 1)

10 5 10 4 10 3 10 2 10 1

r0.4

0.6

0.8

1.0

slop

e

2/3 10 5 10 4 10 3 10 2 10 1

10 2

10 1

100

v2 r(r)

3D = (0.97, 0.97)

3D = (1, 0.97)

3D = (1, 1)

10 5 10 4 10 3 10 2 10 1

r0.40.60.81.01.21.4

slop

e

2/3 10 5 10 4 10 3 10 2 10 1

10 1

100v r

(r)

3D = (0.97, 0.97)

3D = (1, 0.97)

3D = (1, 1)

10 5 10 4 10 3 10 2 10 1

r

0.2

0.4

0.6

0.8

slop

e

1/3 10 5 10 4 10 3 10 2 10 1

102

105

108

1011

Q(r

)

3D = (0.97, 0.97)

3D = (1, 0.97)3D = (1, 1)

10 5 10 4 10 3 10 2 10 1

r

3.5

3.0

2.5

2.0

slop

e5/3 7/3 3

2D = 0.90 2D = 1, (10LPT)

3D = (0.97, 0.97) 3D = (1, 0.97) 3D = (1, 1), (10LPT)

Fig. 7. Radial profiles at shell-crossing using 10th-order LPT in the vicinity of ε2D ' εSYM2D ≡ 1 (top panels) and ε3D ' εSYM

3D ≡ (1, 1) (bottompanels). On top panels, ε2D is progressively varying in the range [0.9, 1] with linear intervals of ∆ε = 0.1/64. On Bottom panels, ε3D progressivelychanges from (0.97, 0.97) to (1, 1) with linear intervals of ∆ε = 0.03/64. From left to right, we present radial profiles of the normalized densityρ/ρ, the velocity dispersion v2, the radial velocity dispersion v2

r , the infall velocity −vr, and the pseudo phase-space density Q, respectively. Inthe lower inserts corresponding to logarithmic slopes, the horizontal dashed lines correspond to the values expected from singularity theory ascomputed in Appendix D and listed in Table 2.

Third, we can compare, in the three-dimensional case, thelogarithmic slope of the pseudo phase-space density to that seenafter violent relaxation and at late stages of the evolution ofdark matter haloes, Q(r) ∼ r−β (see, e.g. Taylor & Navarro2001; Navarro et al. 2010; Ludlow et al. 2010), which has beenfound to be close to the prediction of secondary infall model,βsph = 1.875 (Bertschinger 1985). This is also the case of ourthree sine waves haloes, as analysed in detail by C21, includingSYM-3SIN, which shows again that this particularly symmet-ric set-up, while being significantly more singular at collapse,with β = 3, relaxes to a state similar to more generic three-dimensional configurations. The slope prior to collapse in the“generic” cases, β = −5/3, although of the same order of βsph,remains still different. Concerning the 2D case, which is differ-ent from the dynamical point of view, further analyses of theevolution of the haloes beyond collapse need to be done.

5. Structure of the system shortly after shellcrossing

Once shell-crossing time is passed, the system enters into ahighly non-linear phase of violent relaxation. In this regime,standard LPT is not applicable anymore, but it is still possi-ble, shortly after collapse, to take into account the effects of themulti-streaming flow on the force field using an LPT approach(see, e.g., Colombi 2015; Taruya & Colombi 2017, for the intro-duction of “post-collapse” perturbation theory). While the mo-tion can still be described again for a short while in a perturba-tive way from shell-crossing, it is not, overall, a smooth functionof time anymore due to the formation of the singularity at col-lapse time (Rampf et al. 2021). Still, it is reasonable to assumethat very shortly after shell crossing, the backreaction correc-tions due to the multi-streaming flow can be neglected, as a firstapproximation. However, it is important to notice that the con-vergence radius in time of the LPT series is finite (e.g., Zheligov-



sky & Frisch 2014; Rampf et al. 2015, RH21). As illustratedby the measurement of the previous section, the convergence ofthe perturbation series is expected (although not proven) at leastup to collapse time, but not necessarily far beyond collapse (seealso RH21). Therefore, in what follows, we shall use the LPTsolution of nth-order at collapse as a starting point, and fromthere, the standard ballistic approximation, where velocity fieldis frozen, to study the structure of the system shortly after col-lapse.

In practice, one would like to use sufficiently large pertur-bation order so that convergence of LPT is achieved, and ideallythe formal extrapolation to infinite order proposed by STC18 andreintroduced in Sec. 4.1. However, this extrapolation is insuffi-ciently accurate for the purpose of the calculations performednext, where very small time intervals after collapse are consid-ered. Also, it is very costly from the computational point of viewto exploit this extrapolation when solving the multivalued prob-lem intrinsic to the multi-streaming solutions. So from now on,we shall consider only higher-order LPT calculations and nottheir extrapolation to infinite order (nor the quasi-1D approxi-mation proposed by RF17).

This section is organized as follows. In Sec. 5.1, we intro-duce the ballistic approximation. We also relate, in the multi-stream regime, the Eulerian density and velocity fields to theirLagrangian counterparts, and introduce, in this framework, thevorticity field. Indeed, after shell-crossing, caustics form andnon-zero vorticity is generated in the multi-stream region delim-ited by the caustics. In this section, we also discuss the causticnetwork created up to second order by our systems seeded bysine waves. Sec. 5.2 turns to actual comparisons of analyticalpredictions to Vlasov simulations using, similarly as in Sec. 4.2,phase-space diagrams, but shortly after collapse. In particular,we explore the limits of the ballistic approximation. Finally,Sec. 5.3 focuses on the overall structure of the multi-stream re-gion in configuration space, by successively testing LPT predic-tions against simulations for the caustic pattern, the projecteddensity and the vorticity fields.

5.1. Multi-stream regime and ballistic approximation

Because collapse time a(n)sc depends on perturbation order, it

only makes sense to test various perturbation orders shortly af-ter shell-crossing and compare them to simulations only if col-lapse times are synchronized. In other words, in what follows,we consider the time a(n)

pc = a(n)sc + ∆a for LPT of nth-order and

aSIMpc = asc + ∆a for the simulation, as listed in last column of

Table 1. The value of ∆a used for each sine waves configurationis given by the difference between the values of the last and 6thcolumns of the table. Remind that the quantity asc correspondsto the “true” collapse time measured in the Vlasov runs as ex-plained in Appendix B.2.

From the Eulerian coordinate and velocity fields given bynth-order LPT solutions at shell-crossing time a(n)

sc of each per-turbation order, we model the Eulerian coordinate after collapseas follows:

x(q, a) = x(n)(q, a(n)sc ) +

∂x(n)

∂a

∣∣∣∣∣∣a=a(n)

sc

∆a, (43)

= x(n)(q, a(n)sc ) +

v(n)

a2H(a)

∣∣∣∣∣∣a=a(n)

sc

∆a, (44)

while the nth-order velocity field is frozen to its value at shell-crossing time of each perturbation order, v(q, a) = v(n)(q, a(n)

sc ).

In the multi-stream region, given the Eulerian coordinate x,the solution of the equation x = x(q) has an odd number of solu-tions for the Lagrangian coordinate, that we denote qF labelledby the subscript F = [1, · · · , nF]. Shortly after collapse, if εi , ε jfor i , j, there are at most three streams, nF ≤ 3, in a given pointof space. For symmetric cases, e.g. ε2D = 1 in 2D or εx = εy in3D, due to simultaneous shell-crossing along several axes, thenumber of streams can reach 9 and 27, respectively in two andthree dimensions, as discussed further in Sec. 5.2.

In what follows, we omit the time dependence for brevity.After defining the Lagrangian density and velocity fields:

ρL(q) = | det J(q)|−1, (45)

vL(q) = adΨ(q)

dt, (46)

the Eulerian density and velocity fields are expressed by the su-perpositions of each flow:

ρ(x) =∑

F

ρL(qF), (47)

v(x) =

∑F ρL(qF) vL(qF)∑

F ρL(qF). (48)

where the summation∑

F is performed over all the solutions qFof the equation x = x(qF).

The generation of vorticity is one of the prominent featuresafter shell-crossing (e.g., Pichon & Bernardeau 1999; Pueblas& Scoccimarro 2009). By taking the curl of the velocity fieldwith respect to the Eulerian coordinate, the vorticity is given,respectively in 2 and 3 dimensions, by

ω2D(x) = εi j∂v j(x)∂xi

= εi j1ρ(x)

∑F

J−1mi (qF)

∂ρL(qF)∂qF,m

[vL, j(qF) − v j(x)

]+ εi j

1ρ(x)

∑F

J−1mi (qF)

∂vL, j(qF)∂qF,m

ρL(qF), (49)

ω3Di (x) = εi jk

∂vk(x)∂x j

,

= εi jk1ρ(x)

∑F

J−1m j(qF)

∂ρL(qF)∂qF,m

[vL,k(qF) − vk(x)

]+ εi jk

1ρ(x)

∑F

J−1m j(qF)

∂vL,k(qF)∂qF,m

ρL(qF). (50)

The vorticity fields in two- and three-dimensional space arescalar and vector quantities, respectively. Because the accelera-tion derives from a gravitational potential, local vorticity on eachindividual fold of the phase-space sheet cancels. Only the nonlin-ear superposition of folds in the first term of equations (49) and(50) induces non-zero vorticity, while the second term should notcontribute. Using LPT in the fastest growing mode approxima-tion however generates spurious vorticity in Eulerian space dueto the finite truncation of the perturbation order (see e.g., Buchert& Ehlers 1993; Buchert 1994). This spurious vorticity is mainlycoming from the second term in Eqs. (49) and (50), and can beespecially noticeable in the single-stream region. We neglectedit in our predictions by simply forcing this term to zero.

Before studying the structure of the system beyond collapse,we take the time to analytically investigate the properties of the



ballistic solution for the sine waves case in the first- and second-order LPT, in order to understand better the measurements per-formed in the next paragraphs. In particular, it is important toknow how the solution behaves in the multi-stream region, ofwhich the boundaries are given by the caustics, where J(q) = 0.

First, we start with 1LPT (Zel’dovich approximation). In theballistic approximation, the Eulerian coordinate can be easilycalculated shortly after shell-crossing:

A(1LPT)(q, a) = qA +L2π

D(1LPT)+,sc εA

(1 + f

∆aasc

)sin

(2πL

qA

), (51)

with A = x, y, or z and D(1LPT)+,sc = −1/εx stands for the growth

factor at shell-crossing time evaluated by using 1LPT. Note thatin equation (51), the growth rate f and the expansion factor asc

are evaluated at the same time as D(1LPT)+,sc (remind that f = 1 in

the Einstein-de Sitter cosmology considered in this work). Thevalue of D(1LPT)

+,sc is obtained in practice by solving the equation∂x(1LPT)/∂qx = 0 at the origin, keeping in mind that |εx| ≥ |εy,z|.Remind that equation (51) is exact in the pure 1D case, that iswhen εy = εz = 0. The condition for the caustics, J(1LPT)(q, a) =0, is reduced to the relation:

cos(

2πL

qi

)=

(1 + f

∆aasc

)−1

. (52)

Eq. (52) implies that the equations of the caustics are givenby q = q0 with q0 being a constant vector depending ontime, i.e. on ∆a. Therefore the 1LPT caustics consist of one-dimensional lines in 2D and two-dimensional planes in 3D, evenat collapse time. This configuration is actually degenerate andis not expected in realistic cases, for instance in the frame-work of a smooth Gaussian random field (see e.g., Pichon &Bernardeau 1999). Indeed, the regions where first shell-crossingoccurs should be composed, in the non-degenerate case (that isnon-vanishing εi), of a set of points. In the vicinity of such pointsand shortly after shell-crossing, even in 1LPT, the equation ofthe caustics should correspond, at leading order and in the non-axial-symmetric case, εi , ε j, i , j, to the equation of an ellipsein 2D and an ellipsoid in 3D (see, e.g., Hidding et al. 2014; Feld-brugge et al. 2018). The reason for this not being the case hereis due to the extremely restrictive class of initial conditions wehave chosen.

Due to the degenerate nature of the sine waves initial con-ditions, it is therefore needed to go beyond the first order of theperturbative development of the dynamical equations to obtain amore realistic shape for the caustics. Indeed, for instance, 2LPTbrings non-linear couplings between various axes of the dynam-ics:

A(2LPT)(q, a) = qA +L2π

D(2LPT)+,sc εA

(1 + f

∆aasc

)sin

(2πL

qA

)−

314

L2π

(D(2LPT)

+,sc

)2 ∑B

(1 − δAB) εA εB

×

(1 + 2 f

∆aasc

)sin

(2πL

qA

)cos

(2πL

qB

), (53)

where D(2LPT)+,sc stands for the growth factor at the shell-crossing

time evaluated by using 2LPT, which can be obtained as a func-tion of ε2D or ε3D by solving the following second order polyno-mial:

1 + D(2LPT)+,sc εx −

314

(D(2LPT)

+,sc

)2εx(εy + εz) = 0. (54)

qx

0.040.02

0.000.02

0.04

q y

0.100.05

0.000.05

0.10

q z

0.1

0.0

0.1

ANI-3SIN ( 3D = (3/4, 1/2))

x

0.0005

0.0000

0.0005

y0.02

0.00

0.02

z

0.05

0.00

0.05

qx

0.060.03

0.000.03

0.06q y

0.060.03

0.000.03

0.06

q z

0.06

0.03

0.00

0.03

0.06

SYM-3SIN ( 3D = (1, 1))

x

×10 31.0

0.50.0

0.51.0

y×10

3

1.00.5

0.00.5

1.0

z

×10

3

1.0

0.5

0.0

0.5

1.0

Lagrangian caustics

Eulerian caustics

Lagrangian caustics

Eulerian caustics

Fig. 8. Three-dimensional view of the expected caustic pattern shortlyafter shell-crossing for three sine waves initial conditions. The lefts pan-els correspond to the most typical pancake, here embodied by ANI-3SIN, but Q1D-3SIN would look alike, while the right ones showthe axial-symmetric case, that is SYM-3SIN. The calculations are per-formed using 2LPT along with ballistic approximation (Eq. 53), buthigher-order LPT would provide the same topology from the qualita-tive point of view, except that the position of the caustic surfaces wouldchange, especially in the axial-symmetric case.

In Eq. (53) , the growth rate f and the scale factor asc are eval-uated at the same time as D(2LPT)

+,sc . The additional terms com-pared to Eq. (51) imply, shortly after collapse, non-vanishingcontributions from all axes at the quadratic level (and above) inthe Lagrangian coordinate q in the equation J(2LPT)(q, a) = 0(as long as none of the εi cancels). This leads to, in the non-axial-symmetric case (i.e., Q1D-2SIN, Q1D-3SIN, ANI-2SINand ANI-3SIN), elliptic and ellipsoid shapes for the causticcurve/surface in Lagrangian space, respectively in 2D and 3D,as expected and as illustrated in the three-dimensional case bythe top left panel of Fig 8, while the bottom left one shows theexpected typical pancake shape in Eulerian space. Accordingly,in the analyses performed below, except for the phase-space di-agrams that we examine first, LPT is examined from 2nd or-der and above. Note that, as will be studied below, the axial-symmetric cases ε2D = 1 and ε3D = (1, 1) are a bit more complex,as they require Taylor expansion of the motion beyond quadraticorder in q to obtain a full description of the correct and more in-tricate topology of the caustic surfaces (e.g. 4th order in q insteadof second for SYM-2SIN), as illustrated in 3D by right panels ofFig. 8, due to simultaneous collapses along several axes. It wouldbe beyond the scope of this article to go further in analytic detailsof catastrophe theory for these very particular configurations, sowe shall be content with a descriptive analysis of the LPT resultsin this latter case.

5.2. Phase-space structure

Figs. 9 and 10 display phase-space diagrams for our varioussine-wave systems shortly after collapse, namely the intersec-tion of the phase-space sheet with the hyperplanes y = 0 and



0.004 0.002 0.000 0.002 0.004x

0.4

0.2

0.0

0.2

0.4v x

Q1D-2SIN ( 2D = 1/9)

a = asc + aa = asc

Ballistic

0.0010 0.0005 0.0000 0.0005 0.0010x

0.6

0.4

0.2

0.0

0.2

0.4

0.6

v x

ANI-2SIN ( 2D = 2/3)

0.0001 0.0000 0.0001x

0.2

0.0

0.2

v x

SYM-2SIN ( 2D = 1)

0.001 0.000 0.001x

0.4

0.2

0.0

0.2

0.4

v x

Q1D-3SIN ( 3D = (1/6, 1/8))

0.004 0.002 0.000 0.002 0.004x

1.0

0.5

0.0

0.5

1.0

v x

ANI-3SIN ( 3D = (3/4, 1/2))

0.001 0.000 0.001x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

SYM-3SIN ( 3D = (1, 1))

Fig. 9. Tests of the ballistic approximation in Vlasov simulations: measured phase-space structure for two and three sine waves initial conditions.This figure is analogous to Fig. 2, except that it shows a zoom on the central part of the system shortly after collapse. We test the validity of theballistic approximation, by using two simulation snapshots slightly before (red curves, with a = asc as shown in Table 1 except for ε3D = (1, 1),for which a = 0.03103) and slightly after collapse time (black, with a = asc + ∆a). The ballistic approximation, as described in the main text, isapplied to the red curves to obtain the magenta ones, to be compared directly to the “exact” solution, in black, given by the simulation.

y = z = 0, for 2D and 3D configurations, respectively. Thefirst figure examines the validity of the ballistic approximationdirectly from simulations data, while the second one comparespredictions from LPT at various orders to the simulations.

Interestingly, Fig. 9 shows that the ballistic approximationcan deviate significantly from the exact solution even quiteshortly after collapse. While it remains precise for all config-urations in 2D, with a slight worsening of the accuracy whengoing from the quasi-1D to the axial-symmetric set-up, as ex-pected, significant or even dramatic deviations from the exact so-lution can be seen in 3D for the anisotropic configuration (ANI-3SIN) and the axial-symmetric configuration (SYM-3SIN). Inthe last case (bottom right panel), the intersection of the phase-space sheet with the hyperplane y = z = 0 is composed, in themulti-stream region, of three curves with a “S” shape instead ofa single one, as a result of simultaneous shell-crossing along all

coordinates axes. The prediction from the ballistic approxima-tion has then to be compared to the “S” with the largest ampli-tude along x-axis. Similar arguments apply to the 2D case (bot-tom left panel), but in this case, the intersection with the y = 0hyperplane is composed of two curves instead of three.3

3 Note that when examining the bottom panels of Figs. 9 (black curves)and 10, solving the equation x(q) = x0 in the multi-stream region doesnot seem to have the expected number of solution, that is either 3, 9 or27 (only in the bottom right panel for the latter case). But this is becausewe are sitting exactly at the centre of the system and symmetries implysuperposition of solutions in the subspace y = 0 in 2D and y = z = 0 in3D. In particular, as discussed in Sec. 5.1, first-order LPT has each co-ordinate axis totally decoupled from the others, which means that onlyone “S” shape, which corresponds respectively, in the (x, vx) subspace,to 3 and 9 superposed “S” shapes, is visible as the orange curve on bot-tom panels of Fig. 10. On the other hand, higher-order LPT induces



0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5v x

Q1D-2SIN ( 2D = 1/9)

0.004 0.002 0.000 0.002 0.004x

0.50

0.25

0.00

0.25

0.50

v x

1LPT2LPT3LPT4LPT5LPT6LPT


0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

ANI-2SIN ( 2D = 2/3)

0.001 0.000 0.001x

0.5

0.0

0.5

v x

0.4 0.2 0.0 0.2 0.4x

1.5

1.0

0.5

0.0

0.5

1.0

1.5

v x

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Fig. 10. Phase-space structure for two and three sine waves initial conditions shortly after collapse. This figure is analogous to Fig. 2, expect thatwe compare predictions of LPT at various orders in the ballistic approximation framework to measurements in Vlasov simulations shortly aftercollapse, that is for a = asc + ∆a as listed in Table 1 and discussed in the main text.

The results shown in Fig. 9 must however be interpreted withcaution. Indeed, strictly speaking, we aim to apply the ballisticapproximation from collapse time, which is not exactly the casein this figure. The red curves, which correspond to the snapshotsused to originate the ballistic motion, all correspond to a time abcslightly before actual collapse (namely abc = 0.03103 for SYM-3SIN and abc = asc as given in Table 1 for other cases), becausewe do not have exactly access to this latter. The level of prox-imity to collapse is, at least from the visual point of view, thelowest for the axial-symmetric cases, particularly SYM-3SIN. Itis also for this latter case that the ratio ∆aeff/abc ' 0.03 is thelargest, where ∆aeff is the difference between apc ≡ asc + ∆a

nonlinear couplings between various axes of the dynamics, which im-plies that we have, respectively, only 2 and 3 visible “S” shapes on thebottom left and right panels of Figs. 9 (black curves) and 10, both forthe simulations and LPT predictions at order n ≥ 2 among those theremaining 3 − 2 = 1 and 9 − 3 = 6 curves overlap with their visible “S”curves in (x, vx) subspace.

and abc and is used to test the ballistic approximation. Note alsothat ∆aeff/abc ' 0.03 is also of the same order for ANI-3SIN.The greater the value of ∆aeff/abc, the greater the deviation dueto nonlinear corrections of the motions to be expected. Further-more, we also expect the ballistic approximation to worsen fromquasi-1D cases to fully axial-symmetric.

When examining Fig. 9 more in detail, we also note that forANI-3SIN, the central part of the “S” shape is well reproducedby the ballistic approximation, while the tails underestimate ve-locities. This last defect is in fact present to some extent in allconfigurations except for Q1D cases, given the uncertainties onthe measurements. In the 3D axial-symmetric case, not only thevelocities in the tails are strongly underestimated, but also themagnitude of the position of the caustics along x-axis (local ex-tremum of x coordinate). This suggests that the effects of accel-eration in the vicinity of collapse are strong, which implies thatthe ballistic approximation can be applied only for a very short



0.025 0.000 0.025qx

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×10 4 Eulerian caustics2LPT3LPT4LPT5LPT6LPT7LPT8LPT9LPT10LPTSimulation

Fig. 11. The caustic pattern shortly after shell crossing in the two-dimensional case: comparison of LPT predictions using ballistic approximationto Vlasov runs. Left, middle and right panel correspond respectively to Q1D-2SIN, ANI-2SIN and SYM-2SIN configurations, while top andbottom panels show the caustics in Lagrangian and Eulerian spaces, respectively.

time (or it could be that it is not applicable, mathematically, dueto the strength of the singularity).

Now we turn to the comparison of LPT predictions of vari-ous orders to the simulations, as shown in Fig. 10. Remind thatthe ballistic approximation is applied from the respective col-lapse times of each perturbation order, which obviously makespredictions of LPT artificially more realistic than it should be.The conclusions of Sec. 4.2, where we compared LPT to simu-lations at collapse, still stand at the coarse level, obviously, sincethe time considered in Fig. 10 is nearly equal to collapse time.When zooming on the central part of the system, we notice thatthe ballistic approximation applied to LPT works increasinglywell with the order, as expected, especially when approachingquasi-1D dynamics. It can fail in the tails of the “S” shape of thephase-space sheet, as a combination of limits of LPT to describethe system in the vicinity of the velocity extrema and the limits ofthe ballistic approximation itself just discussed above. The worseperformances are, as expected, for the axial-symmetric configu-rations. In this case, the topology of the phase-space structurein the multi-stream region (two or three curves according to thenumber of dimensions) is correct and the very central part of theshell corresponding to x-axis motion remains synchronized withthe simulation, but except for this, the structure of the systemis reproduced only qualitatively. This will be confirmed by theanalyses that follow next.

5.3. Configuration space: caustics, density, vorticity

We now enter into the heart of this work, which consists of exam-ining in detail the structure of the system shortly after collapsein configuration space, both in Lagrangian and Eulerian coordi-nates. The multi-stream region is delimited by caustics, wheredensity and vorticity are singular, so an accurate description ofthe caustic pattern represents the first test of LPT predictions.In most cases, the singularity corresponds to a simple fold ofthe phase-space sheet, with ρ(r) ∼ r−1/2 (see, e.g., Feldbruggeet al. 2018) and likewise for the vorticity along the direction or-

thogonal to the caustic (see, e.g., Pichon & Bernardeau 1999).This means that a large part of the vorticity signal is generatedin the vicinity of caustics. Having the correct shape of causticsis therefore fundamental because this information mostly deter-mines the subsequent internal structure of the multi-stream re-gion both for the density and the vorticity fields. Hence, in whatfollows, we first comment on Figs. 11 and 12, which compare,respectively in the 2D and 3D cases, the caustic pattern at variousorders of LPT to the Vlasov code for our various sine waves ini-tial conditions. Then, we turn to the normalized density and thevorticity in Figs. 13 to 18, and compare LPT predictions at 2ndand 10th order to the simulations. We remind the reader againthat LPT predictions with the ballistic approximation are com-puted by synchronizing the respective collapse times obtainedat each perturbation order, which obviously artificially improvesthe performances of LPT. Note cautiously that in plotting the re-sults below in 3D cases, the caustics outputs (Figs. 11 and 12)are shown at the intersections with the x = 0, y = 0 and z = 0planes, while the density and vorticity outputs (Figs. 14 and 16–18) are shown in slices slightly away from the origin because thevorticity vanishes in the x = 0, y = 0, and z = 0 planes due tothe symmetries in the chosen setting.

The examination of left four panels of Fig. 11 and top twelvepanels of Fig. 12 provides us insights on the caustic pattern inthe most typical configurations, εi , ε j, i , j, that is the non-axial-symmetric case, both in 2D and in 3D. Note that in the3D case, to have a more clear view of the caustic pattern, weconsider the intersection of the caustic surfaces with the planesx = 0, y = 0 and z = 0 (remind however that a 3D view ofthe caustic surfaces was given for 2LPT in Fig. 8). In the caseof the simulations, as explained in Appendix B.3, the causticpattern corresponds to a tessellation, that is a set of ensembles ofconnected segments in 2D and of connected triangles in 3D. Theintersection of a tessellation of triangles with a plane also gives aset of ensembles of connected segments. On the figures, only thevertices of these segments are represented, and the initial regular



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2LPT3LPT4LPT5LPT6LPT7LPT8LPT9LPT10LPTSimulation

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Fig. 12. The structure of caustics shortly after shell crossing for three sine waves initial conditions: comparison of LPT predictions using ballisticapproximation to Vlasov runs. Top six, middle six, and bottom two panels respectively correspond to Q1D-3SIN, ANI-3SIN and SYM-3SIN. Oneach group of 6 panels, the top and bottom lines correspond to Lagrangian and Eulerian spaces, and the intersection of the caustic surfaces withthe plane qx = 0, qy = 0 and qz = 0 respectively for the first, second and third column of each group. In the bottom group of two panels, the leftand right panels correspond respectively to Lagrangian and Eulerian space and only show the intersection of the caustic surface network with theplane qz = 0 due to the symmetry of the system.



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Fig. 13. Two-dimensional density shortly after shell-crossing: compar-ison of LPT predictions using ballistic approximation to Vlasov runs.From top to bottom, Q1D-2SIN, ANI-2SIN and SYM 2SIN. From leftto right, 2LPT, 10LPT and measurements in Vlasov simulations.

pattern of the tessellation used to represent the phase-space sheetis clearly visible in Lagrangian space.

Figs. 11 and 12 confirm the results of the previous section,namely that high order LPT provides a rather accurate descrip-tion of the caustic pattern in the Q1D case, with a clear conver-gence to the simulation when the perturbation order increases. Inthe anisotropic cases, ANI-2SIN and ANI-3SIN, the match be-tween LPT and simulations is not perfect, even at 10th order, butremains still reasonably good, especially in 2D. For the axial-symmetric configurations, the convergence of LPT with orderseems much slower, particularly in Eulerian space and in 3D.

While the shape of the caustic pattern is very simple inthe generic cases, it becomes significantly more intricate in theaxial-symmetric cases, particularly in Eulerian space. Indeed,the caustics are composed of two connected curves instead ofone for ε2D = 1, and three connected surfaces instead of one forε3D = (1, 1). For ε2D = 1, the shapes of simulation caustics areapproximately reproduced by LPT, but we already see that con-vergence to the exact solution is slow. In particular, the extensionof the outer caustic in Eulerian space is quite underestimated byLPT and it seems that increasing perturbation order to arbitraryvalues is not going to improve the agreement with the simula-tions. The situation is much worse in 3D, particularly in Eule-rian space, although the simulation measurements are very noisy,which makes the comparison between theory and measurementsdifficult. We know from the previous paragraph that these mis-matches are at least partly attributable to limits of the ballistic

approximation, along with limits of LPT to be able to describethe velocity field at collapse in the vicinity of its extrema, partic-ularly the spike observed on the phase-space diagram in the 3Dcase.

Not surprisingly, the projected density measurements ofFigs. 13 and 14 fully confirm the analyses of the caustic pat-tern. Putting aside the axial-symmetric cases, we can see that theagreement between simulations and LPT of high order is quitegood, inside and outside the caustics. Note that the very highdensity contrasts observed in the axial-symmetric cases are dueto the multiple foldings of the phase-space sheet, even in 2D,which explains the limits of the ballistic approximation, sincefeedback effects from the gravitational force field after collapseare expected to be orders of magnitude larger than in the genericcase.

Turning to the vorticity, as shown in Figs. 15 to 18, we obtainagain a good agreement between theory and measurements in thenon-axial-symmetric configurations, except that, at the exact lo-cation of the caustics, the simulation measurements are expectedto be spurious. This is discussed in Appendix B.5, which pro-vides details on the measurements of various fields, in particularhow we exploit a quadratic description of the phase-space sheetto achieve vorticity measurements with unprecedented accuracy,but yet still limited by strong variations of the fields in the vicin-ity of the caustics, in particular the strong discontinuous transi-tion between the inner part and the outer part of the multi-streamregions at the precise locations of the caustics.

From the qualitative point of view, the topology of the vor-ticity field for the generic configurations, εi , ε j, i , j, agreesperfectly with the predictions of Pichon & Bernardeau (1999)based on Zel’dovich dynamics.4 In 2D, the vorticity field is ascalar which can be decomposed into the four sectors inside thecaustic, two of positive sign and two of negative one. In 3D,the vorticity field is a vector. Because shell crossing takes placealong x-axis, each component of this vector field has specificproperties, which are related to variations of the velocity anddensity field along with the phase-space sheet. In particular, itis easy to convince oneself, that if collapse happens along x-axis, the strongest variations of all the fields are expected alongthis direction. That means, since each coordinate of the vorticityvector depends on variations of the velocity field in other coor-dinates, that the magnitude of ωx is expected to be small. Due tosymmetries, we also expect that ωy and ωz are, respectively, anodd function of z and y, which means that ωy approaches zerowhen approaching the z = 0 plane, and similarly for ωz whenapproaching the y = 0 plane, which explains the pattern of thevorticity field on Figs. 16 and 17. These symmetries impose us toperform measurements on slices slightly shifted from the centreof the system, as detailed in the caption of Fig. 14. Note as well,in these figures, that the pattern is analogous to 2D for the y andz coordinate of the vorticity fields in the x− z and x−y subspace,respectively.

Turning to the axial-symmetric case, the agreement betweentheory and measurements is again only partial. Yet the high mag-nitude of vorticity is of the same order for theory and measure-ments. The structure of the vorticity field, significantly morecomplex than in the generic case due to multiple foldings of thephase-space sheet, is qualitatively in an agreement between LPTand Vlasov code in 2D, while simulations measurements are too

4 Remind however that the crossed sine waves configurations we con-sider are degenerate with respect to 1LPT, as discussed at the end ofSec. 5.1, so the reasoning of Pichon & Bernardeau (1999) requires 2LPTto be applicable in this particular case.



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Fig. 14. Slices of the projected density shortly after shell-crossing: comparison of LPT using ballistic approximation to Vlasov runs. The left andright groups of nine panels correspond respectively to Q1D-3SIN (top, middle and bottom line of panels: x = −1.55 × 10−4, y = −1.17 × 10−3 andz = −1.56 × 10−3 slice) and ANI-3SIN (x = −5.16 × 10−4, y = −2.15 × 10−4 and z = −5.47 × 10−4 slice), while the bottom group of three panelscorresponds to SYM-3SIN (z = −1.17 × 10−5 slice). On each group of panels, left, middle and right columns give respectively the 2nd-order LPTprediction, the 10th-order LPT prediction and the Vlasov code measurements. Due to the symmetry of the system for SYM-3SIN, only one sliceis shown for the bottom panels.

noisy in the 3D case to make definitive conclusions. What isclear, however, is that the outer part of the multi-stream regionseems to be qualitatively reproduced by the theory, but its size istotally wrong.

6. Summary

In this paper, following the footsteps of Saga et al. (2018), wehave investigated the structure of primordial cold dark matter(CDM) haloes seeded by two or three crossed sine waves ofvarious amplitudes at and shortly after shell-crossing, by com-paring thoroughly up to 10th-order Lagrangian perturbation the-ory (LPT) to high-resolution Vlasov-Poisson simulations per-formed with the public Vlasov solver ColDICE. We devoted ourattention first to the phase-space structure and radial profilesof the density and velocities at shell-crossing, and, second, tothe phase-space structure, caustics, density and vorticity fields

shortly after shell-crossing. In particular, measurements of un-precedented accuracy of the vorticity in the simulations weremade possible by exploiting the fact that ColDICE is able to fol-low locally the phase-space sheet structure at the quadratic level.

We studied three qualitatively different initial conditionscharacterized by the amplitude of three crossed sine waves,as summarized in Table 1: quasi one-dimensional (Q1D-2SIN,Q1D-3SIN), where one amplitude of the sine waves dominatesover the other one(s), anisotropic (ANI-2SIN, ANI-3SIN), wherethe amplitude of each wave is different but remains of the sameorder, and axial-symmetric (SYM-2SIN, SYM-3SIN), where allamplitudes are the same. In order to predict the protohalo struc-ture shortly after shell-crossing in an analytical way, we used theballistic approximation, where the acceleration is neglected afterthe shell-crossing time.

Our main findings can be summarized as follows:



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Fig. 15. Two-dimensional vorticity fields: comparison of LPT predic-tions to Vlasov runs.

– Phase-space diagrams at collapse: except for SYM-3SIN,one expects the system to build up a classic pancake singu-larity at shell-crossing, with a phase-space structure alongx-axis analogous to what is obtained in one dimension. Thispancake is well reproduced by LPT which converges increas-ingly well to the exact solution when approaching quasi one-dimensional dynamics, as expected. The local extrema of thevelocity field around the singularity are the locations whereLPT differs most from the exact solution, by underestimat-ing the magnitude of the velocity. Convergence with per-turbation order becomes very slow when approaching three-dimensional axial-symmetry, where spikes appear on the ve-locity field on each side of the singularity.

– Radial profiles at collapse: with a sufficiently high order ofthe perturbation, LPT can reproduce arbitrarily high densitycontrasts at collapse, but we note a slower convergence whenturning to velocity profiles. Still, the convergence of LPT issufficiently good to probe the asymptotic logarithmic slopeat small radii expected at collapse for various profiles fromsingularity theory, as summarized in Table 2, and confirmedby simulation measurements. These profiles are ρ(r) ∝ r−2/3

for generic initial conditions, that is Q1D and ANI, andρ(r) ∝ r−4/3 for SYM-2SIN, ρ(r) ∝ r−2 for SYM-3SIN,while velocities profiles always present the same power-lawbehaviour, e.g. v2(r) ∝ r2/3. Confirming predictions of singu-larity theory at collapse and explicit convergence to asymp-totic profiles at small radii is expected but non-trivial.

– Synchronization: agreement with singularity theory predic-tions is made even more explicit by computing LPT profilesat the respective collapse times computed at each perturba-

tion order. In this case, convergence to the power-law profilepredicted at small radii by singularity theory is explicit. Infact, the radial profiles obtained from LPT with such syn-chronization are strikingly alike, particularly for the density,which motivates the use of a ballistic approximation to studythe motion slightly beyond collapse.

– Ballistic approximation: measurements in simulations showthat the ballistic approximation provides an accurate descrip-tion of the phase-space structure of the system slightly be-yond collapse, except in the axial-symmetric case SYM-3SIN, where the very highly contrasted nature of the systemdue to multiple superpositions of phase-space sheet folds in-troduces strong force feedback effects. We notice howeverthat even in the generic pancake case, the ballistic approxi-mation can still slightly underestimate velocities in the vicin-ity of the singularity. Obviously, these conclusions depend onhow long this approximation is used. In this work, we con-sidered maximum relative variations of the expansion factorof the order of three percent. For these values, the deviationsfrom pure ballistic motion were the most significant, as ex-pected.

– Structure beyond collapse: given the limits discussed above,we find, when considering generic, non-axial-symmetricconfigurations, that LPT of sufficient order combined withthe ballistic approximation provides a very good descriptionof the structure of the system beyond collapse time, with ex-cellent agreement between the predicted density and the vor-ticity fields inside the multi-stream region and those mea-sured in the Vlasov simulations. Turning to axial-symmetriccases, this agreement is only obtained at the qualitative level,even for 10th-order LPT, particularly for SYM-3SIN, wherethe size of the outer caustics is strongly underestimated.

Obviously, the ballistic approximation is only the first stepfor a more complete calculation taking into account the feedbackdue to the force field, as first proposed in the one dimensionalcase by Colombi (2015); Taruya & Colombi (2017) (see alsothe recent work of Rampf et al. 2021), and formulated in termsof “post-collapse perturbation theory”. The next step is indeedto implement a Lagrangian perturbation theory approach wherethe small parameter is the interval of time ∆a = a − asc fromcollapse time and where a Taylor expansion of the phase-spacesheet is performed around the singularity in terms of Lagrangiancoordinates. Shortly after collapse, in the generic, non-axial-symmetric case, the system presents a “S” shape in x − vx sub-space, similar to the 1D case where position and velocities canbe approximated as third-order polynomials of the Lagrangiancoordinate q. The calculation of the force field requires then tosolve a 3 value problem in the multi-stream region, similarlyas in the 1D case. While technically challenging, generalizationof post-collapse perturbation theory to 2D and 3D seems possi-ble. It might bring strong insights on nonlinear corrections dueto multi-stream dynamics on large scale structure statistics, e.g.predictions of the power spectrum of the large-scale matter dis-tribution from higher order perturbation theory.

Acknowledgements

This work was supported in part by JSPS Research FellowGrant-in-Aid Number 17J10553 (SS), MEXT/JSPS KAKENHIGrants Numbers JP17H06359, JP20H05861, and JP21H01081,JST AIP Acceleration Research grant JP20317829 (AT), ANRgrant ANR-13-MONU-0003 (SC), as well as Programme Na-tional Cosmology et Galaxies (PNCG) of CNRS/INSU with



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Fig. 16. Vorticity components shortly after shell-crossing: comparison of LPT with the ballistic approximation to Vlasov runs for Q1D-3SIN. Thetwo-dimensional slices are the same as those shown in the top left group of nine panels of Fig. 14, except that the slice considered changes fromleft to right, while the vorticity component changes from top to bottom. Again, on each line of three panels, 2LPT (left panel) and 10LPT (middlepanel) are compared to simulation measurements (right panel).

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Fig. 17. Same as Fig. 16, but for ANI-3SIN. The two-dimensional slices considered are the same as in the top right group of nine panels of Fig. 14.

INP and IN2P3, co-funded by CEA and CNES (SC). Nu-merical computation with ColDICE was carried out usingthe HPC resources of CINES (Occigen supercomputer) underthe GENCI allocations c2016047568, 2017-A0040407568 and2018-A0040407568. Post-treatment of ColDICE data were per-formed on HORIZON cluster of Institut d’Astrophysique de

Paris. Calculations of theoretical predictions have made use ofthe Yukawa Institute Computer Facility.

ReferencesAlard, C. 2013, MNRAS, 428, 340Angulo, R. E., Hahn, O., Ludlow, A. D., & Bonoli, S. 2017, MNRAS, 471, 4687



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2500

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Fig. 18. Same as Fig. 16, but for SYM-3SIN. Due to the symmetry ofthe initial conditions, only a x-y slice is shown, which is the same as inbottom three panels of Fig. 14.

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Appendix A: Expressions of the LPT solutions

In this appendix, we present the LPT solutions up to 5th order,which are obtained by solving the recursion relations given inEqs. (21) and (22). Since higher-order solutions are straightfor-wardly derived in the same way, we do not explicitly show themhere.

The results are partly presented in Moutarde et al. (1991)up to 3rd order, taking into account the contributions other thanthe fastest growth mode (see also Buchert et al. 1997, for thesolutions including decaying modes). Here, we first explicitly

show the analytical solutions of the sine waves initial conditionsup to 5th order.

As shown in Sec. 2.1, the fastest growing mode can be ex-panded as follows:

Ψ(q, t) =

∞∑n=1

Dn+(t)Ψ(n)(q) . (A.1)

For presentation purposes, we only show the x-components ofthe displacement field, i.e., Ψ

(n)x (q) with the definition (ε1, ε2) ≡

(εy/εx, εz/εx). Given the x-components, the y- and z-componentscan be derived by permutating all x, y, and z.

Ψ(1)x =

εx

2πsin(2π qx) , (A.2)

Ψ(2)x = −

3ε2x

28π

[ε1 cos(2π qy) + ε2 cos(2π qz)

]sin(2π qx) , (A.3)

Ψ(3)x =

ε3x

2520π

[78 cos(2πqx)(ε1 cos(2πqy) + ε2 cos(2πqz))

+ 160ε1ε2 cos(2πqy) cos(2πqz) − 3ε21 cos(4πqy) − 3ε2

2 cos(4πqz) + 75(ε2

1 + ε22

)]sin(2π qx) , (A.4)

Ψ(4)x = −

ε4x

7761600π

[ε2 cos(2πqz)

(4242 cos(4πqx) + 28550ε2

1 cos(4πqy) + 208850ε21 + 57015ε2

2 + 89166)

+ 60 cos(2πqx)(6010ε1ε2 cos(2πqy) cos(2πqz) + 1274ε2

1 cos(4πqy) + 1274ε22 cos(4πqz) + 2039

(ε2

1 + ε22

))+ 2ε1 cos(2πqy)

(2121 cos(4πqx) − 9303ε2

1 cos(4πqy) + 14275ε22 cos(4πqz) + 33159ε2

1 + 104425ε22 + 44583

)− 9303ε3

2 cos(6πqz)]

sin(2π qx) , (A.5)

Ψ(5)x =

ε5x

36793476720000π

[895050 cos(4πqx)

(181296ε1ε2 cos(2πqy) cos(2πqz) + 41657

(ε2

1 + ε22

))+ 560226137900ε2

1ε2 cos(2πqx) cos(4πqy) cos(2πqz) + 560226137900ε1ε22 cos(2πqx) cos(2πqy) cos(4πqz)

+ 594423768510ε31 cos(2πqx) cos(2πqy) − 18642271230ε3

1 cos(2πqx) cos(6πqy)

+ 57401651835ε21 cos(4π(qx − qy)) + 57401651835ε2

1 cos(4π(qx + qy)) + 1006179766020ε1ε22 cos(2πqx) cos(2πqy)

+ 106526974554ε1 cos(2πqx) cos(2πqy) − 6828912090ε1 cos(6πqx) cos(2πqy) + 1006179766020ε21ε2 cos(2πqx) cos(2πqz)

+ 57401651835ε22 cos(4π(qx − qz)) + 594423768510ε3

2 cos(2πqx) cos(2πqz) − 18642271230ε32 cos(2πqx) cos(6πqz)

+ 57401651835ε22 cos(4π(qx + qz)) + 106526974554ε2 cos(2πqx) cos(2πqz) − 6828912090ε2 cos(6πqx) cos(2πqz)

+ 46659033850ε21ε

22 cos(4π(qy − qz)) + 46659033850ε2

1ε22 cos(4π(qy + qz)) + 503168866200ε3

1ε2 cos(2πqy) cos(2πqz)

− 54358176600ε31ε2 cos(6πqy) cos(2πqz) + 503168866200ε1ε

32 cos(2πqy) cos(2πqz) − 54358176600ε1ε

32 cos(2πqy) cos(6πqz)

+ 877320054000ε1ε2 cos(2πqy) cos(2πqz) + 115709716980ε21ε

22 cos(4πqy) − 35064073044ε4

1 cos(4πqy)

− 1607701095ε41 cos(8πqy) + 241408089450ε2

1 cos(4πqy) + 115709716980ε21ε

22 cos(4πqz) − 35064073044ε4

2 cos(4πqz)

+ 241408089450ε22 cos(4πqz) − 1607701095ε4

2 cos(8πqz)

+ 546975(ε2

1

(940700ε2

2 + 443058)

+ 171045ε41 + 21ε2

2

(8145ε2

2 + 21098))]

sin(2π qx) . (A.6)

Note that because of the underlying symmetry of the initialconditions, the x-components of the LPT solutions are symmet-ric under the exchange of y↔ z and ε1 ↔ ε2.

Appendix B: Measurements on the tessellation ofVlasov simulations

This appendix explains in detail how measurements of variousquantities studied in this article are performed on the output of

ColDICE, which consists of a tessellation of the phase-spacesheet with simplices, respectively triangles and tetrahedra in 4Dand 6D phase-space.

B.1. Phase-space diagrams

The intersection of a hypersurface of dimension D1 = D with ahyperplane P of dimension D2 = D + 1 inside a phase-space ofdimension D3 = 2D is of dimension D1 + D2 − D3 = 1. In other



words, this intersection corresponds in the non-trivial case to aset of curves. Therefore, in 2D, the intersection of the phase-space sheet with the hyperplane x = 0 is a set of curves, andlikewise in 3D, the intersection of the phase-space sheet withthe hyperplane x = y = 0. Furthermore, as discussed more indetail in C21, because the phase-space sheet remains at all timesa fully connected hypersurface, this set of curves should also befully connected, with no hanging point.

To produce a phase-space diagram, we employ an approachvalid at linear order, consisting in simply computing, for eachsimplex, its geometric intersection with the hyperplane P, whichis either empty, a point or a segment. As a result, phase-spacediagrams extracted from the Vlasov runs consist of sets of con-nected segments of which the ends are plotted in Figs. 2 and 10.

B.2. Expansion factor at collapse: asc

An accurate estimate of the value of the expansion factor corre-sponding to collapse time is essential when studying the proper-ties of the system at shell crossing and shortly after. The curvesgenerated in Appendix B.1 can be used for this purpose, at twoexpansion times ai, i = 1, 2, supposed to be just before andjust after collapse. At both these times, we consider the uniquephase-space diagram segment portion S of which one of the endsis as close as possible to the origin, while the other one, with ab-scissa xi, has vx > 0 and the magnitude of other coordinate(s) ofthe velocity, in theory null, as small as possible. The latter con-dition excludes, in the axial-symmetric case, the component(s)of the flow corresponding to simultaneous collapse along the yor/and z direction, which has/have non-zero value of vy or/andvz. Once this segment is identified at both times correspondingto a1 and a2, a simple linear interpolation is used to estimateexpansion factor at collapse:

asc 'a1x2 − a2x1

x2 − x1. (B.1)

This formula does not require to be just before and after collapsetime to estimate asc, but, to provide sufficiently accurate results,it is obviously needed to consider times sufficiently close to ac-tual collapse time. A first guess of collapse time is estimated byusing perturbation theory predictions extrapolated to infinite or-der (STC18), as listed in 5th column of Table 1. Examination ofTable 1 shows that the value a∞sc predicted this way agrees ex-tremely well with the measurements in the Vlasov runs providedby equation (B.1), and shown in 6th column of the table.

Note however that, beyond the approximate nature of the lin-ear approximation underlying equation (B.1), collapse time esti-mate can be significantly influenced by force resolution, that isthe value of ng. As discussed in detail in C21, decreasing forceresolution delays collapse time, hence, to have an accurate es-timate of this latter, it is required to have a sufficiently largevalue of ng. We performed extensive force resolution tests forthe 3D simulation with ε3D = (3/4, 1/2). Our measurements ofasc with the above method provide asc = 0.02912 (in excellentagreement with the predicted value a∞sc = 0.02911), 0.0292 and0.0293 respectively for ng = 1024, 512 and 256. Our ng = 512simulation thus provides, for this value of ε3D, an estimate of col-lapse expansion factor accurate at approximately the 10−4 leveland we expect this to apply as well to the quasi one-dimensionalcase ε3D = (1/6, 1/8). However, for the axial-symmetric case,ε3D = (1, 1), the measured value might still overestimate the ac-tual one by an amount larger than ∼ 10−4 due to the high strengthof the singularity building up at the center of the system. On thecontrary, in the 2D case, given the high value of ng = 2048 used

to perform the simulations, we expect our estimates of collapsetime to be very accurate, at an order better than the 10−4 level.

B.3. Caustics

Caustics are regions where the determinant J of the Jacobian ma-trix changes sign. At linear order in the local description of thephase-space sheet, geometrically this means that the orientationof the simplex changes in configuration space, which allows oneto define unambiguously regions corresponding to the intersec-tions of simplices with J ≥ 0 and simplices with J < 0, wherethe sign of J is directly estimated from the current orientation ofthe simplex with respect to the original one. This is actually per-formed during runtime by ColDICE which can output causticsdirectly when needed. In 2D, the phase-space sheet is composedof a tessellation of triangles, hence the caustics estimated thisway are given by sets of segments of which the ends are shownin Fig. 11. In 3D, the caustics are given by sets of triangles, ofwhich we compute the intersection with the y = z = 0 plane toget again a set of segments of which the extremities are shownin Fig. 12. Because we are using a leading order approach, thecaustic lines or surfaces are not necessarily smooth but shouldtrace accurately enough the actual caustics for the purpose ofthis work.

B.4. Radial profiles

To measure radial profiles in logarithmic bins, each simplex isreplaced with a large number of particles as explained for the3D case in Appendix A2 of C21. We refer to this work for thereader interested in the details of this procedure, that we straight-forwardly generalized to the 2D case.

B.5. Density field and vorticity: from linear to quadratic order

In this section, we aim to compute the following quantities:

– the jacobian of the transformation between initial and finalpositions, J(q) as defined in equation (7);

– the Lagrangian projected density ρL(q) defined in equation(45);

– the total Eulerian projected density, which stems from thesuperposition of one or more folds of the phase-space sheet,as described by equation (47);

– the Eulerian velocity field, given by equation (48);– the vorticity, ω2D and ω3D, defined in equations (49) and

(50). Again, because the acceleration derives from a poten-tial, local vorticity on each phase-space sheet fold cancels.Only the nonlinear superposition of phase-space sheet foldscontained in the first term of these equations induces non-zero vorticity, while the second term should not contribute,although we shall take it into account in our numerical cal-culations.

In what follows, we explain how to compute the various quanti-ties defined above by exploiting the decomposition in simplicesof the phase-space sheet by ColDICE. In § B.5.1, we introducebarycentric coordinates, which are useful to define the positionof any point inside each simplex. In § B.5.2, we show how thebarycentric coordinates can be used to perform calculations atthe linear level inside each simplex, in particular partial deriva-tives of a function, as already discussed for instance by Hahnet al. (2015). In § B.5.3, we generalize the procedure to the casewhen a quadratic description of the phase-space sheet is avail-able. Finally, in § B.5.4, we describe the way we sample various



quantities described above on an high resolution cartesian grid,by using proper sets of sampling particles associated to each sim-plex.

B.5.1. Barycentric coordinates

Since the phase-space sheet is sampled with simplices, it is use-ful to define a well known local system of coordinates on eachsimplex. Given the positions Xk, k = 1, · · · ,Ns of the simplexvertices, with Ns = D + 1 where D is dimension of configurationspace, and a function g(X), one can define the following linearinterpolation

glinear(X) =

Ns∑k=1

ξkgk, (B.2)

for

X = Xlinear ≡

Ns∑k=1

ξk Xk, (B.3)

where the barycentric coordinates ξk are positive quantities ver-ifying ξk = 1, ξk′,k = 0, for X = Xk and

∑k ξk = 1. When

working in Lagrangian space, the space of initial positions, thephase-space sheet is flat and the linear interpolation

Qlinear =

Ns∑k=1

ξkQk (B.4)

is exact. In what follows, we shall therefore use equation (B.4)to define barycentric coordinates. With this definition of ξk, thismeans that at given time t, equation (B.3) remains valid, but onlyat the linear level, since dynamical evolution of the phase-spacesheet produces curvature.

B.5.2. Calculations of various quantities at the linear level

With the barycentric coordinate representation, any function de-fined at vertices positions can be estimated locally at the linearlevel using equation (B.2). The derivative of the function insideeach simplex can thus be written

∂g∂xβ

=

Ns∑k=1

gk∂ξk

∂xβ=

Ns∑k=1

gk M−1k,β+1, (B.5)

where M is the matrix of the partial derivatives of the vector P ≡(∑

k ξk, X1, X2, X3) with respect to ξk, with X given by equation(B.3). In other words,

M1,k = 1, (B.6)

Mα+1,k =∂Xα

∂ξk= Xk,α, (B.7)

with Xk,α the αth coordinate of kth vertex with position Xk. At thelinear level, the derivative defined by equation (B.5) is thereforesimply constant inside each simplex.

This calculation can be performed in Lagrangian space toobtain directly the Jacobian of the transformation between initialand present position, equation (7), from the jacobian matrix

Jα,β ≡∂Xα

∂Qβ=

Ns∑k=1

Mα+1,kW−1k,β+1, (B.8)

where W is defined similarly as M but in Lagrangian space,

W1,k = 1, (B.9)

Wα+1,k =∂Qα

∂ξk= Qk,α, (B.10)

with Qk,α the αth Lagrangian coordinate of kth vertex with La-grangian position Qk.

This means that the linear description allows one to ob-tain the Jacobian J(q) and the corresponding Lagrangian densityρL(q) = 1/|J(q)| as constant quantities inside each simplex. Un-fortunately, this is not sufficient, since equations (49) and (50)involve partial derivatives of the projected density. Rigorouslyspeaking, this means that a representation of the simplices at thequadratic level is required.

B.5.3. Calculations of various quantities at the quadratic level

Fortunately, ColDICE employs a quadratic description of thephase-space sheet for performing local refinement of the tessel-lation, by using tracers defined as mid-points of edges of thesimplices in Lagrangian space. For instance, the tracer associ-ated to vertices (k, k′) corresponds to barycentric coordinatesξk = ξk′ = 1/2, ξ j = 0 for j , k and j , k′.

This allows one to define in a unique way a quadratic hyper-surface inside each simplex coinciding with vertices and trac-ers belonging to it. Note that the second order representationpreserves the continuity of the phase-space sheet but does notwarrant its smoothness at the differential level. Exploiting thisrepresentation will nevertheless provide, in practice, sufficientlyaccurate measurements of the vorticity.

In the quadratic representation, on defines the following con-ventional shape functions Nk,

Nk(ξ) ≡ ξk(2ξk − 1), k ≤ Ns, (B.11)≡ 4ξK(k)ξL(k), Ns < k ≤ Ns + Nt, (B.12)

where functions K(k) and L(k) are appropriately chosen to coverall the combinations 1 ≤ K(k) < L(k) ≤ Ns, and where Ns =D + 1 is the number of simplices and Nt = D(D + 1)/2 is thenumber of tracers, given D the dimension of space.

Then equation (B.2) becomes

gquad(X) =

Ns+Nt∑k=1

Nk(ξ)gk, (B.13)

where gk is now also defined over tracers. This is true in particu-lar for quadratically interpolated positions

X = Xquad =

Ns+Nt∑k=1

Nk(ξ)Xk, (B.14)

while corresponding Lagrangian coordinates Q are still given bythe linear representation (B.4) since the phase-space sheet is ini-tially flat, which means that matrix W does not change.

From equation (B.14), we obtain the matrix Mquad of partialderivatives of the vector P ≡ (

∑k ξk, X1, X2, X3) with respect to

ξk, which replaces equations (B.6) and (B.7):

Mquad1,k = 1, (B.15)

Mquadα+1,k =

∂Xα

∂ξk=

Ns+Nt∑k′=1

∂Nk′

∂ξkXk′,α, (B.16)



with Xk′,α the αth coordinate of k′th vertex/tracer with positionXk′ . While, at the linear level, this matrix was constant insideeach simplex, it now depends linearly on barycentric coordi-nates, given the quadratic nature of the shape functions Nk.

The calculation of the jacobian matrix exploiting thequadratic representation then just consists in replacing matrix Mby Mquad in equation (B.8), since matrix W remains unchanged,which allows us to compute J(q) in the quadratic representa-tion. In particular, we can compute Jk = J(Qk) at each vertexand tracer position to ready it for the quadratic interpolation pro-cedure given by equation (B.13) (but staying aware of the factthat this calculation is accurate only up to linear order), which,in addition to phase-space coordinates themselves, is enough tocompute all the quantities we are interested in.

Once we have a scalar function g(x) defined at vertices andtracers positions, its derivative is given from equation (B.13) by[∂g∂xβ

]quad

=

Ns+Nt∑k=1

gk

Ns∑k′=1

∂Nk

∂ξk′

∂ξk′

∂xβ(B.17)

=

Ns+Nt∑k=1

gk

Ns∑k′=1

∂Nk

∂ξk′[Mquad(ξ)]−1

k′,β+1. (B.18)

This quantity depends in a non-linear way on barycentric posi-tion and requires matrix Mquad to be invertible, so to be able tohave a finite estimate of the partial derivative, it is necessary toavoid the subspace occupied by the caustics (or, more exactly theelement of surface/curve approximating the actual caustics).

In particular, we estimate the partial derivatives of the pro-jected density as follows

∂ρ

∂xβ= −

sign(J)J2

∂J∂xβ

, (B.19)

using equation (B.18) to estimate ∂J/∂xβ. Indeed, the jacobian isa smooth function of the Lagrangian coordinate. This is not thecase of function ρL(q) = 1/|J(q)|, which presents strong varia-tions in the vicinity of caustics. Using the Jacobian instead of thedensity to estimate derivatives allows us to capture better asymp-totic behaviours in the vicinity of the caustics.

B.5.4. Practical estimate: particle representation and localcoarse graining

In order to compute any quantity at position x0, one has to com-pute the intersection of the phase space sheet with the positionx = x0, or to resolve inside each simplex the equation X(ξ) = x0.While this is a straightforward linear problem in the linear rep-resentation of the phase-space sheet (equation B.3), it becomesnon-trivial in the quadratic case (equation B.14), for which someiterative procedure seems necessary. Here, to simplify the ap-proach (and also to reduce effects of aliasing and divergencesnear caustics), we adopt a forward point of view consisting inprojecting the tessellation on a cartesian mesh of resolution nanain configuration space, similarly as it is performed by ColDICEto compute the projected density. This means that we apply ad-ditional coarse graining over each pixel/voxel of the cartesianmesh: Integrals of the form∫

dDv g(x, v) f (x, v) (B.20)

become1

vpixel/voxel

∫pixel/voxel

dDx∫

dDv g(x, v) f (x, v), (B.21)

with vpixel/voxel the area/volume of the pixel/voxel under consid-eration.

While Sousbie & Colombi (2016) use actual ray-tracing ex-act to linear order to compute such an integral, we replace eachsimplex with a large number of particles, which allows us to givean account of the quadratic shape of the simplices in a very sim-ple way. However, employing particles introduces some discretenoise. To reduce the noise, we use cloud-in-cell (CIC) interpo-lation procedure on the target cartesian mesh (Hockney & East-wood 1988), that is each particle is replaced by a voxel/pixelof the same (small) size as the voxels/pixels of the cartesianmesh, and associate a weight Winter to the particle proportionalto the volume of intersection between the voxel/pixel associ-ated with the particle and the voxel/pixel of the mesh. In ad-dition, we do not place particles at random within the simplex,but instead refine the simplex hierarchically in an homogeneousfashion `max times, and then replace each sub-simplex obtainedthis way with a particle corresponding to the centre of the sim-plex in the barycentric representation. The calculation of `max issuch that discreteness effects due to the projection inside eachvoxel/pixel of the cartesian mesh are kept under control:

`max = maxblog2(S/∆) + 4 − Dc, 0

(B.22)

where D is the dimension of space, ∆ is the step of the cartesianmesh and

S = maxk,k′,α|Xk,α − Xk′,α|, (B.23)

with Xk,α the αth coordinate of kth vertex with position Xk andlikewise for Xk′,α.

In practice, prior to CIC interpolation, each particle p sam-ples a small element of volume δVp in the spatial part of theintegral of right hand side of equation (B.21),

δVp = VL2−D`max Jp, (B.24)

where VL is the Lagrangian volume of the simplex and Jp the ja-cobian (quadratically) interpolated at position of the particle. So,at the end, the contribution of each particle in right hand integralof equation (B.21) is equal to δVp Winter g(Xp,Vp), where we re-mind that Winter is the CIC weight defined previously and wherefunction g is estimated at particle phase-space position (Xp,Vp)using the methods described in previous paragraphs.

Note that coarse graining due to the CIC interpolation is ex-pected to introduce defects/biases in regions where large varia-tions are expected, that is in the vicinity of caustics. Obviouslythese effects become more dramatic when differenciating quan-tities, so one expects the vorticity to be affected numericallynearby the caustics, as can be noted in Figs. 15, 16, 17 and 18.

Appendix C: Perturbative treatment of quasione-dimensional flow

In Sec. 4.2, we test theoretical predictions relying on the pertur-bative treatment of a quasi one-dimensional flow as proposed byRF17. In this appendix, we briefly present this approach, focus-ing mainly on three sine waves initial conditions configurations.Generalization to the two sine waves case is straightforward.

When the system is exactly one dimensional, the first-orderLPT solution is exact before shell-crossing. This fact leads to anapproximate treatment in three-dimensional space. We exploitthe exact solution of the one-dimensional problem along x-axisgiven by Zel’dovich approximation as the unperturbed zeroth-order state:

Ψ(0)i (q, t) = D+(t) Ψini

x (qx) δi,x, (C.1)



with the subscript i = x, y for two sine waves and i = x, y, z forthree sine waves. The transverse fluctuations are considered assmall first-order perturbations to this set-up:

Ψ(1)(q, t) =

∞∑n=1

Dn+(t)Ψ(1,n)(q). (C.2)

In the case of the three sine waves, we assume the explicit formof the functions Ψini

x and Ψ(1,1) to be

Ψinix =

L2π

εx sin(

2πL

qx

), (C.3)

Ψ(1,1) =

0

L2π εy sin

(2πL qy

)L2π εz sin

(2πL qz

) . (C.4)

Then, substituting these initial conditions into the recursion re-lations in Eqs. (21) and (22), we obtain

∇q ·Ψ(1,n) = −Ψini

x,x2n − 12n + 3

(Ψ(1,n−1)

y,y + Ψ(1,n−1)z,z

), (C.5)

∇q ×Ψ(1,n) = −Ψini

x,xn − 2

n

0

Ψ(1,n−1)x,z

−Ψ(1,n−1)x,y

. (C.6)

By solving Eqs. (C.5) and (C.6), one can construct the Q1D first-order solutions.

We further extend the Q1D treatment proposed by RF17 upto second order in the transverse fluctuations as described below.On top of the zeroth and first order perturbations, we define thesecond order perturbation by

Ψ(2)(q, t) =

∞∑n=2

Dn+(t)Ψ(2,n). (C.7)

Again, using Eqs. (21) and (22), we obtain the following recur-sion relation:

∇q ·Ψ(2,n) = −Ψini

x,x2n2 − 3n + 12n2 + n − 3

(Ψ(2,n−1)

y,y + Ψ(2,n−1)z,z

)− εi jk εipq

n−1∑m=1

2m2 + m − 3/22n2 + n − 3

Ψ(1,n−m)j,p Ψ

(1,m)k,q

− Ψinix,x εx jk εxqr

×

n−2∑m=1

2m2 + m2n2 + n − 3

Ψ(1,n−m−1)j,q Ψ

(1,m)k,r , (C.8)

[∇q ×Ψ

(2,n)]i= Ψini

x,x2n2 − 3n − 2

2n2 + nεix j Ψ

(2,n−1)x, j

+

n−1∑m=1

2m2 + m2n2 + n

εi jk Ψ(1,n−m)l, j Ψ

(1,m)l,k . (C.9)

The recursion is initialized by

∇q ·Ψ(2,2) = −

314

εi jk εipq Ψ(1,1)j,p Ψ

(1,1)k,q , (C.10)

∇q ×Ψ(2,2) = 0. (C.11)

Using the above recursion relations, the perturbative expansionis then performed by keeping terms proportional to εn

y and εmz up

to second order, n + m = 2, while keeping terms proportional toεk

x up to tenth order, k = 10, as shown in Fig. 2. This approachprovides a very accurate description of the dynamics when thesystem is initially quasi one-dimensional, i.e., |εx| |εy,z|.

The results of the Q1D solution at first order in the transversefluctuations are presented in RF17. Here, we show (for the firsttime) the Q1D solution at second second-order in the transversefluctuations and up to 5th-order in the longitudinal direction interms of the growth factor, i.e.,Ψ(2,5) as follows. We use the samenotations as in Appendix A.

For the x-components of the Q1D solutions, we have

Ψ(2,2)x = −

3ε2x

28πsin(2πqx)

[ε1 cos(2πqy) + ε2 cos(2πqz)

], (C.12)

Ψ(2,3)x =

ε3x

2520πsin(2πqx)

[39ε1 cos(2π(qx − qy)) + 39ε1 cos(2π(qx + qy)) + 39ε2 cos(2π(qx − qz)) + 39ε2 cos(2π(qx + qz))

+ 80ε1ε2 cos(2π(qy − qz)) + 80ε1ε2 cos(2π(qy + qz)) − 3ε21 cos(4πqy) − 3ε2

2 cos(4πqz) + 75(ε2

1 + ε22

)], (C.13)

Ψ(2,4)x =

ε4x

2587200π

[−10 sin(4πqx)

(6010ε1ε2 cos(2πqy) cos(2πqz) + 1274ε2

1 cos(4πqy) + 1274ε22 cos(4πqz) + 2039

(ε2

1 + ε22

))− 29015 sin(2πqx)(ε1 cos(2πqy) + ε2 cos(2πqz)) − 707 sin(6πqx)(ε1 cos(2πqy) + ε2 cos(2πqz))

], (C.14)

Ψ(2,5)x =

ε5x

4088164080000π

[4507471800ε1ε2 cos(2π(2qx − qy − qz)) + 4507471800ε1ε2 cos(2π(2qx + qy − qz))

+ 4507471800ε1ε2 cos(2π(2qx − qy + qz)) + 4507471800ε1ε2 cos(2π(2qx + qy + qz)) + 6377961315ε21 cos(4π(qx − qy))

+ 6377961315ε21 cos(4π(qx + qy)) + 5918165253ε1 cos(2π(qx − qy)) + 5918165253ε1 cos(2π(qx + qy))

− 379384005ε1 cos(2π(3qx + qy)) − 379384005ε1 cos(6πqx − 2πqy)

+ 33(193271555ε2

2 cos(4π(qx − qz)) + 179338341ε2 cos(2π(qx − qz)) + 815965150(ε2

1 + ε22

))+ 6377961315ε2

2 cos(4π(qx + qz)) + 5918165253ε2 cos(2π(qx + qz)) − 379384005ε2 cos(2π(3qx + qz))

− 379384005ε2 cos(6πqx − 2πqz) + 4142788650 cos(4πqx)(ε2

1 + ε22

)+ 48740003000ε1ε2 cos(2π(qy − qz))

+ 48740003000ε1ε2 cos(2π(qy + qz)) + 26823121050ε21 cos(4πqy) + 26823121050ε2

2 cos(4πqz)]

sin (2πqx) . (C.15)



For the y-components of the Q1D solutions, we derive

Ψ(2,2)y = −

3ε1ε2x

28π

[sin(2πqy)(cos(2πqx) + ε2 cos(2πqz))

], (C.16)

Ψ(2,3)y =

ε1ε3x

2520π

[sin(2πqy)(2 cos(2πqx)(39ε1 cos(2πqy) + 80ε2 cos(2πqz)) − 3 cos(4πqx) + 75)

], (C.17)

Ψ(2,4)y = −

ε1ε4x

7761600π

[sin(2πqy)(60ε1(1274 cos(4πqx) + 2039) cos(2πqy)

+ 50ε2(571 cos(4πqx) + 4177) cos(2πqz) + 57015 cos(2πqx) − 9303 cos(6πqx))], (C.18)

Ψ(2,5)y =

ε1ε5x

4088164080000π

[sin(2πqy)(33(−340ε1 cos(2πqx)(369227 cos(4πqx) − 6071163) cos(2πqy)

− 91(1297372 cos(4πqx) + 59485 cos(8πqx) − 3461625)) − 618800ε2 cos(2πqx)(19521 cos(4πqx) − 100109) cos(2πqz))].

(C.19)

Note that the y-components and z-components are symmet-ric under the exchange of qy ↔ qz and εy ↔ εz in the aboveexpressions.

Appendix D: Asymptotic structure of the singularityat collapse

In this appendix, we analytically investigate the dependence oninitial conditions of the logarithmic slopes of various profiles ex-pected at shell-crossing and small radii, as discussed in Sec. 4.3.The main results of this investigation are summarized in Table 2.Since the extension to the two-dimensional case is obvious bysimply setting εz = 0, we present here the structure of the singu-larity obtained from LPT to some order for the three sine wavescase. Note that the calculations presented in this appendix areconceptually not new since singularity theory applied to cosmol-ogy is already well known (see, e.g. Arnold et al. 1982; Shan-darin & Zeldovich 1989; Hidding et al. 2014; Feldbrugge et al.2018).

D.1. Taylor expansion around the singularity

To facilitate the analysis, we focus on the the relation betweenEulerian and Lagrangian coordinates around the origin. Expand-ing the relation (5) between the Eulerian coordinate x and theLagrangian coordinate q around the origin, we obtain, after ne-glecting O(q4) and higher order terms,

xi(q, t) = Aia(t)qa + Ciabc(t) qa qb qc, (D.1)

with Ai j(t) ≡ δi j +Ai j(t), where Ai j(t) is some function of time. Inthese equations, we have exploited the symmetric nature of thethree sine waves initial conditions, which imply that Euleriancoordinates around the shell-crossing point can be expanded interms of odd third-order polynomials of the Lagrangian coor-dinates, that is polynomial forms P verifying P(−q) = −P(q)(see e.g., Colombi 2015; Taruya & Colombi 2017, for the one-dimensional case).

The coefficients Ai j and Ci jkl are expressed in terms of par-tial derivatives of the displacement field with respect to the La-grangian coordinates at the origin as follows,

Ai j(t) = Ψi, j(0, t), Ci jkl(t) =13!

Ψi, jkl(0, t). (D.2)

Using Eq. (D.1), the Jacobi matrix is given by

Ji j(q, t) = Ai j(t) + 3Ci jab(t) qa qb. (D.3)

The shell-crossing time tsc is obtained by solving the equationJ(0, tsc) = 0, that is

det A(tsc) = 0. (D.4)

Hereafter, the dependence on collapse time tsc will be omitted inthe notations.

Using Eq. (D.3), the Jacobian is given by

J(q) =32εi jk εabc

[Aia A jb Ckcde qd qe

+ 3Aia C jbde Ckc f g qd qe q f qg

+ 3Ciade C jb f g Ckcmn qd qe q f qg qm qn

]. (D.5)

To simplify furthermore the calculations, we noticed, up to10th order of the LPT development, that for an initial displace-ment field given by three orthogonal sine waves aligned witheach coordinate axis, the matrix Ai j is diagonal and the matrixCi jkl verifies

Ciiii , 0,Cii j j = Ci ji j = Ci j ji , 0 , i , j,Ci jkl = 0 , otherwise. (D.6)

We did not find any simple way to demonstrate that these prop-erties stand at any order, but the fact that they are verified up to10th-order LPT is a really convincing clue. In this case, the shell-crossing condition, Eq. (D.4), is reduced to Axx Ayy Azz = 0. Priorto shell-crossing, the Eulerian coordinates read

xi(q) =(Axx qx δix + Ayy qy δiy + Azz qz δiz

)+ Ciiiiq3

i + 3(1 − δi j

)Cii j j qi q2

j , (D.7)

and the Jacobian

J(q) = 3(Axx Ayy Czzde + Ayy Azz Cxxde + Azz Axx Cyyde

)× qd qe

+92

(εx jk εxbc Axx + εy jk εybc Ayy + εz jk εzbc Azz

)×C jbde Ckc f g qd qe q f qg

+92εi jk εabc Ciade C jb f g Ckcmn qd qe q f qg qm qn. (D.8)



In this last expression, we did not exploit yet explicitly the sym-metries on Ci jkl matrix. On the basis of these equations, we cannow investigate the slope of the density profile for the threetypes of singularities we consider, depending on the values ofthe eigenvalues of matrix Ai j.

D.2. Asymptotic behaviour of the profiles at collapse

We now analyse the asymptotic behaviour of the profiles aroundthe origin. To this end, when we consider the following scaling,x→ s x, implying r → s r, we examine how the Lagrangian co-ordinate changes. By doing so, we can understand the behaviourof the scaling of the Jacobian in terms of the Lagrangian coordi-nates, and thus reveal the behaviour of the density profile at theorigin.

It is important to note that this proof is somewhat simplifiedand ignores details on the angular dependence when perform-ing integrals over spherical shells to obtain the radial profiles.Therefore, the proof given in this appendix is not mathematicallyrigorous, but it leads to the same conclusions as the exact calcu-lations in which proper form factors are estimated. The purposeof this section is indeed to provide a simplified rephrasing of sin-gularity theory already presented in a more rigorous fashion inother works (see, e.g. Arnold et al. 1982; Shandarin & Zeldovich1989; Hidding et al. 2014; Feldbrugge et al. 2018).

First, we examine the case 0 = Axx , Ayy , Azz, whichcorresponds to the following sine waves initial conditions: Q1D-2SIN, Q1D-3SIN, ANI-2SIN, and ANI-3SIN, i.e., 0 ≤ ε2D < 1or 0 ≤ ε3D,i < 1 for i = 1, 2, as summarized in first line ofTable 2. This configuration corresponds to shell-crossing alongx-axis.

When computing the radial profiles close to the center of thesystem, we consider spherical shells of radius r with r2 = x2 +y2 + z2 1. At leading order in q we have, at shell-crossing,

x ' Cxxxx q3x + 3Cxxyy qx q2

y + 3Cxxzz qx q2z , (D.9)

y ' Ayy qy, (D.10)

z ' Azz qz. (D.11)

Applying the scaling x→ sx implies

qy → s qy, qz → s qz, (D.12)

then, applying the same scaling in Eq. (D.9) and taking the limits 1, we simply obtain

qx → s1/3qx. (D.13)

After exploiting the symmetries of matrix Ci jkl (Eq. D.6), theJacobian up at leading order in q is given by

J(q) ' 3Ayy Azz

(Cxxxx q2

x + Cxxyy q2y + Cxxzz q2

z

). (D.14)

Using the scalings in Eqs. (D.12) and (D.13), we have

J(q)→ 3Ayy Azz

(Cxxxx s2/3q2

x + Cxxyy s2q2y + Cxxzz s2q2

z

). (D.15)

Then, we have J ∝ s2/3 when s 1, resulting in ρ ∝ r−2/3.Second, we consider the case, 0 = Axx = Ayy , Azz which

corresponds to the initial condition SYM-2SIN, i.e., ε2D = 1 orε3D,1 = 1 and ε3D,2 < 1, as summarized in second line of Table 2.This configuration corresponds to simultaneous shell-crossings

along x and y axes. In this case, the main contribution to Euleriancoordinates reads, at shell-crossing,

x ' C1q3x + 3C2 qx q2

y + 3C3 qx q2z , (D.16)

y ' C1q3y + 3C2 qy q2

x + 3C3 qy q2z , (D.17)

z ' Azz qz, (D.18)

where we have exploited symmetry between x and y which im-plies Cxxxx = Cyyyy ≡ C1, Cxxyy = Cyyxx ≡ C2 and Cxxzz = Cyyzz ≡

C3. Applying the same reasoning as above, the scaling x → sximplies

qz → s qz, (D.19)

which leads us, when applying the scaling to equations (D.16)and (D.17) and assuming s 1 to neglect the last term in theseequations, hence

s x ' C1q3x + 3C2 qx q2

y , (D.20)

s y ' C1q3y + 3C2 qy q2

x. (D.21)

which leads to the scalings

qx → s1/3qx, qy → s1/3qy. (D.22)

With Axx = Ayy = 0, the first term in Eq. (D.8) vanishes, and thesecond term becomes the leading contribution:

J(q) =92εz jk εzbc AzzC jbde Ckc f g qd qe q f qg. (D.23)

Exploiting equation (D.6) then allows us to write this term as

J(q) '∑

i+ j+k=2

bi jk q2ix q2 j

y q2kz , (D.24)

where bi jk are constant coefficients. Using the scalings given byEqs. (D.19) and (D.22), we have

J(q)→∑

i+ j+k=2

s2(i+ j)/3+k bi jk q2ix q2 j

y q2kz . (D.25)

The lowest power of s in Eq. (D.25) is s4/3 with i = j = 1 andk = 0. Thus we have ρ ∝ r−4/3.

Third, we consider the case Axx = Ayy = Azz = 0, whichcorresponds in the three sine waves case to the SYM-3SIN, i.e.,ε3D = (1, 1), as summarized in third line of Table 2. This con-figuration corresponds to simultaneous shell-crossings along allthe axes. The approach is analogous to the previous case. TheEulerian coordinates can be written as follows:

x ' C1q3x + 3C2 qx q2

y + 3C3 qx q2z , (D.26)

y ' C1q3y + 3C2 qy q2

z + 3C3 qy q2x, (D.27)

z ' C1q3z + 3C2 qz q2

x + 3C3 qz q2y , (D.28)

with, again C1 = Cxxxx = Cyyyy = Czzzz, C2 = Cii j j for i , j.Eqs. (D.26)–(D.28) directly lead to the scaling of q:

qx → s1/3qx, qy → s1/3qy, qz → s1/3qz, (D.29)

Now, the first and second terms in Eq. (D.8) vanish, and we have

J(q) =92εi jk εabc Ciade C jb f g Ckcmn qd qe q f qg qm qn, (D.30)



which reads, after exploitation of the symmetries (D.6),

J(q) '∑

i+ j+k=3

bi jk q2ix q2 j

y q2kz , (D.31)

Using the scaling given by Eq. (D.29), we have

J(q)→∑

i+ j+k=3

s2(i+ j+k)/3bi jkq2ix q2 j

y q2kz . (D.32)

The lowest power of s in Eq. (D.32) is s2. Thus we have ρ ∝ r−2.Finally, we focus on the velocity and pseudo phase-space

density profiles. Using Eq. (D.1), the velocity field up to lead-ing order in the Lagrangian coordinate is given by

vi(q, tsc) = a ˙Ai j(tsc) q j, (D.33)

where the dot denotes derivative with respect to the cosmic timet. As can be seen from this relation, the velocity field is propor-tional to the Lagrangian coordinate, irrespective of initial con-ditions, because the shell-crossing condition Ai j(tsc) = 0 doesnot imply ˙Ai j(tsc) = 0. Since, according to the calculations per-formed above, the leading contribution from the Lagrangian vec-tor always come from the scaling qi → s1/3qi when s 1,it is fairly easy to convince oneself that the velocity profiles aregiven, at small radii, by v2 ∝ r2/3, v2

r ∝ r2/3, and −vr ∝ r1/3, fromwhich we can infer as well the logarithmic slope of the pseudophase-space density Q(r) through Eq. (41).


cold dark matter protohalo structure around collapse

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