The Origami Cosmic WebMark Neyrinck, Physics & Astronomy Dept, Johns Hopkins University

Galaxies observed in the See http://vipers.inaf.it/

z=0.5

z=0.8

6 billion light-years 9 billion light-yearssurvey, several billion light-years away/ago

What the galaxies might have done since then (schematically) !Designed with help from Robert J. Lang’s Tessellatica Mathematica package

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Figure 1. A dark-matter sheet in a 2D universe, that distorts and folds through an ap-proximation to gravity, called the Zel’dovich approximation. The darkness of the color ateach position gives the number of streams there. Initially, all vertices were nearly on aregular lattice. Since then, gravity has distorted the mesh, causing regions with a bit morematter than average to accumulate more matter around them. The patch shown is > 108

light-years on a side; nodes correspond to galaxies or clusters of galaxies.

Lagrangian space, the ‘crease pattern’ where q lives, as L, and Eulerian space, the post-folding space wherex(q) lives, as E . All coordinates here are comoving, meaning that the homogeneous, isotropic expansionof the universe is scaled out. Initially, particles were arranged almost uniformly, with vanishing velocity ast ! 0. Thus, in position-velocity phase space (3D position, plus 3D velocity), they were initially arrangedon a flat sheet, flat as viewed in the velocity dimensions. The displacement field (q, t) ⌘ x(q, t) � q, andthe velocity v = @x/@t. A stream is a region of L delimited by caustics, not crossing any of them [Ney12].A void is a single-stream region, i.e. a region of E or L in which x(q) is one-to-one [Sha11, FNS12].

The origami approximation imposes the following assumptions on the functions (q) or x(q). ‘Reality’means ‘the current cosmological structure-formation paradigm,’ which fits many observations quite well.

(1) (q, t) ⌘ | (q, t)| is bounded over all q 2 R3. This property holds in reality; each component i has a roughly Gaussian distribution of dispersion ⇠ 5⇥ 107 light-years, small compared to theradius of the observable universe, ⇠ 5⇥ 1010 light-years.

(2) is irrotational in a void. This holds in reality, since any initial vorticity decays with the expansionof the Universe, and since gravity is a potential force. However, in multi-stream regions (wherex(q) is many-to-one), the flow, averaged among streams, often carries vorticity.

(3) x(q) is continuous and piecewise-isometric, i.e. rLagrangian · = 0, except at creases, where it isundefined. This is the only assumption that is manifestly broken in reality, but we explore theconsequences which it helps to establish, which we hope will apply more generally.

Cosmological Origami: Folding up the Dark-Matter Sheet into the Cosmic Web

Mark C. NeyrinckDepartment of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA

neyrinck@pha.jhu.edu

An origami analogy helps to understand the formation of galaxies, and their spatial arrangement,known as the cosmic web. In paper origami, a non-stretchy two-dimensional manifold (paper)is folded in three dimensions to build objects. In cosmological origami, an initially flat three-dimensional manifold that pervades space (the ‘dark-matter sheet’) is folded up to build galaxiesand the cosmic web of filaments between them. This folding occurs in six-dimensional phase space,consisting of three usual position dimensions, and three dimensions of velocity. A particle with agiven position and velocity occupies a point in 6D phase space. The sheet can never cross itself,or tear, just as in 2D paper origami. The creases, or caustics, as they are called in cosmology,are physically important because they mark the edges of structures like galaxies, and filaments ofmatter and galaxies. Going from paper to cosmological origami, there are two major di↵erences:the dimensionality increases, and the sheet is allowed to stretch. Some mathematical constraints areknown about the types of caustics that can form (Arnold et al. 1982), but these are very local, andorigami mathematics may help to constrain global properties.

Here, I explore properties of the cosmological origami model in a 2D universe, mentioning a 3Dcase as well. Some patterns emerge that origami practitioners will know as origami tessellations(Gjerde 2008). The model imposes extra constraints on origami from one side, and on cosmologicalstructure formation from the other. The origami folds flat, i.e. folding of the 2D sheet is allowedin 4D, but the result is constrained to 2D position space. Also, single-layer regions, or voids, areconstrained not to undergo rotation from the initial state.

A folded-up example of two galaxies and the web around them is shown below. The origamiterm for a cosmological-origami galaxy is a twist fold. In this model, whenever a galaxy formswithout stretching the sheet, it must have some rotation, and must form filaments (the thick bands)radiating from it. This helps to explain the ubiquity of galaxy spin and filaments in the Universe. Ialso show some properties of ‘galaxies’ in a 3D universe.

referencesArnold V. I., Shandarin S. F., Zeldovich I. B., 1982, “The large scale structure of the universe. I - General

properties One- and two-dimensional models,” Geophys. and Astrophys. Fluid Dynamics, 20, 111Gjerde E., 2008, Origami Tessellations: Awe-inspiring Geometric Designs, A. K. Peters, London

FIGURE 2. Triangular collapse of a pair of galaxies, on smaller scales than at right.

Submitted, Proceedings of the 6th International Meeting on Origami in Science, Mathematics and Education

Cosmological Origami: Properties of Cosmic-WebComponents When a Non-Stretchy Dark-Matter Sheet Folds

Mark C. Neyrinck

Abstract. In the current cosmological paradigm, an initially flat three-dim-ensional manifold that pervades space (the ‘dark-matter sheet’) folds up tobuild concentrations of mass (galaxies), and a cosmic web between them.Galaxies are nodes, connected by a network of filaments and walls. The foldingis in six-dimensional (3D position, plus 3D velocity) phase space. The posi-tions of creases, or caustics, mark the edges of structures.

Here, I introduce an origami approximation to cosmological structure forma-tion, in which the dark-matter sheet is not allowed to stretch. But it stillproduces an idealized cosmic web, with nodes, filaments, walls and voids. In2D, nodes form in ‘polygonal collapse’ (a twist-fold in origami), necessarily gen-erating filaments simultaneously. In 3D, nodes form in ‘polyhedral collapse,’simultaneously generating filaments and walls. The masses, spatial arrange-ment, and angular momenta of nodes and filaments are related in the model.I describe some ‘tetrahedral collapse’, or tetrahedral twist-fold, models.

1. Introduction

The formation of structure in the Universe proceeds somewhat like the origami-folding of a sheet. This concept is a Lagrangian fluid-dynamics framework (follow-ing the mass elements, not staying in a fixed spatial coordinate system). Thisapproach in cosmology started with the Zel’dovich approximation [Zel70]. Catas-trophe theory has given some further understanding into the types of singulari-ties that can occur when this sheet begins to fold [ASZ82, HSv14]. Recently,many have realized the power of Lagrangian dynamics in general, and of explic-itly following the dynamics of the sheet in a cosmological simulation, insteadof considering the particles within it to be just fuzzy, isotropic blobs of matter[SHH12, AHK12, FNS12]. Fig. 1 shows an example cosmic web [BKP96]folded from a collisionless dark-matter sheet that has distorted and moved aroundaccording to the Zel’dovich approximation.

I am grateful for support from a New Frontiers in Astronomy and Cosmology grant from theJohn Templeton Foundation, and from a grant in Data-Intensive Science from the Gordon andBetty Moore and Alfred P. Sloan Foundations.

c�0000 (copyright holder)

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References:!Falck, B., Neyrinck, M., & Szalay, A., 2012, ORIGAMI: Delineating Halos Using Phase-space Folds,! ApJ, 754, 126, arXiv:1201.2353!Guzzo, L. et al., 2014, The VIMOS Public Extragalactic Redshift Survey (VIPERS). An unprecedented! view of galaxies and large-scale structure at 0.5 < z < 1.2, A&A, 566, 108!Neyrinck, M., Cosmological Origami: Properties of Cosmic-Web Components when a Non-Stretchy ! Dark-Matter Sheet Folds, to appear in Origami6, arXiv:1408.2219!

See links at https://2014.spaceappschallenge.org/project/fold-your-own-universe/ to fold a “universe” from an arbitrary photo.

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Figure 6. Tetrahedral-collapse models. Filament creases (green) are indicated by trian-gular tubes, intersecting at the central node. Wall creases (blue), extend from filamentedges through the thin lines drawn between filaments. Node creases are in red. Left:Pre-folding/collapse (Lagrangian). Right: Post-folding/collapse (Eulerian). Top: An ir-rotational model (↵1 = ⇡/2). Each filament vector f̂i ? a face of the central tetrahedron.Walls, filaments, and the node invert along their central planes, axes, and point, but re-main connected as before. Void regions simply move inward. All 15 initial regions over-lap at the center. Bottom: A rotational model (↵1 = ⇡/6). The top filament rotatescounter-clockwise by ⇡/3, while the smaller, bottom filaments rotate clockwise by 2⇡/3. Seehttp://skysrv.pha.jhu.edu/

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neyrinck/TetCollapse for an interactive model.

Many of the results given here, particularly in 3D, were numerical. This is fine for comparison tocosmological simulations, as we plan to do. But there is much room for further rigorous mathematical studyof polyhedral collapse, both of isolated nodes, and of how networks of collapsed polyhedra behave together.