Adhesion a sticky way of understanding Large Scale Structure Johan Hidding Supervisor: Rien van de Weijgaert July 26, 2010 Contents 1 Introduction 7 1.1 Goals of this project . 9 1.2 Outline . 10 2 Theory of modern cosmology and large scale structure 11 2.1 Standard cosmology . 11 2.2 Concordance model . 13 2.3 Co-moving coordinates and density perturbation . 13 2.4 Initial conditions . 15 2.5 Equations of motion . 17 2.6 Linear theory . 17 2.7 Non-linear structure formation . 20 3 Zel’dovich approximation 21 3.1 Eulerian vs. Lagrangian . 22 3.2 Expansion, vorticity and shear . 23 3.3 Lagrangian perturbation . 23 3.4 Generalised Zel’dovich formalism . 25 3.5 Anisotropic collapse . 26 3.6 Caustics . 29 4 Burgers’ dynamics 31 4.1 Burgers’ equation . 32 4.2 Inviscid limit . 32 4.3 Dynamics of shocks . 34 4.4 Convex hull interpretation . 35 4.4.1 Legendre-Fenchel conjugate and convex hulls . 37 4.4.2 Maxwell’s rule . 38 4.4.3 Hierarchical collapse . 39 2 5 Numerical Algorithms 41 5.1 Periodic boundary conditions . 41 5.2 Convolution method . 41 5.3 Discrete scale-space . 42 5.4 Fast Legendre Transform . 44 5.4.1 Sub-pixel post minimization . 44 5.5 Density . 47 6 Hierarchical evolution of the Cosmic Web 49 6.1 From Voronoi to Adhesion . 49 6.1.1 Voronoi model . 49 6.1.2 EL-Duality . 51 6.1.3 Smooth potential models . 51 6.2 Geometric analysis . 54 6.2.1 Mathematical definitions . 54 6.2.2 Numerical implementation . 55 6.3 Voids . 59 6.3.1 Void definitions . 59 6.3.2 Percolation theory . 60 6.3.3 Void sociology . 60 6.3.4 Alignment . 62 6.4 Wall dynamics . 62 6.5 Merging and fragmentation of clusters . 62 7 Results 65 7.1 Simple statistics . 65 7.1.1 Void volume . 65 7.2 Adhesion as a geometric tool . 66 7.3 Results on 2-D scale-free models . 68 7.3.1 Filament strength . 69 7.3.2 Quadrilateral deformation . 74 7.3.3 Tidal fields . 79 8 Discussion 85 8.1 Large scale evolution of the cosmic web . 85 8.2 Validity of adhesion . 86 8.3 Future work . 86 8.3.1 Caustics . 86 8.3.2 Legendre transforms . 87 8.3.3 Dynamics within nodes . 87 8.3.4 Forced Burgers dynamics . 87 8.4 Conclusion . 87 8.5 Acknowledgements . 88 3 A Conventions 97 B Newtonian Friedman equations and the Einstein-de Sitter universe 98 C Comoving equations of motion 100 C.1 Euler equation . 100 C.2 Continuity equation . 100 C.3 Poisson equation . 101 D Cubic splines 102 E About the code 105 4 List of Figures 1.1 WMAP 5 year CMB temperature map . 8 1.2 Slice from SDSS . 9 1.3 Slice from the Millennium simulation . 10 2.1 Scale factor . 13 2.2 CDM power spectrum . 16 2.3 Growing mode solution . 19 3.1 Eulerian vs. Lagrangian coordinates . 22 3.2 Ellipsoidals . 27 3.3 Integrals of the density . 28 3.4 Multistream region . 29 3.5 Cusp singularity . 30 3.6 Caustics in a swimming pool . 30 4.1 Parabolic interpretation . 33 4.2 Convex hull interpretation . 36 4.3 Construction of the convex hull . 38 4.4 Maxwell’s rule . 38 4.5 Merger cascade . 39 5.1 Discrete Gaussian kernel . 43 5.2 Enhanced convex hull . 45 5.3 Area of quadrilateral . 47 5.4 Smooth 2-D density maps . 48 6.1 the Cosmic Web in Lagrangian view . 50 6.2 Comparison with Voronoi model . 50 6.3 From convex hull to cosmic spine . 53 6.4 Tessellating E ............................. 55 5 6.5 Grid deformation of E ........................ 56 6.6 Pixel connectivity . 57 6.7 Adhesion tessellation . 58 6.8 Number of voids as a function of time . 60 6.9 The life of a void . 61 6.10 Wall dynamics . 63 7.1 Cluster mass function . 66 7.2 Void volume function . 67 7.3 Adhesion shape finder . 67 7.4 2-D Mass density for scale-free models . 70 7.5 2-D Volume density for scale-free models . 71 7.6 Correlation of filament density with cluster mass . 72 7.7 Non-correlation of filament density with filament length . 73 7.8 Quadrilateral deformation . 74 7.9 Quadrilateral deformation for n = 0.5 . 76 7.10 Quadrilateral deformation for n = 0.0 . 77 7.11 Quadrilateral deformation for n = −0.5 . 78 7.12 Tidal field strength . 80 7.13 Tidal field for n = 0.5 . 81 7.14 Tidal field for n = 0.0 . 82 7.15 Tidal field for n = −0.5 . 83 7.16 Correlation between tidal strength and density . 84 6 Chapter 1 Introduction The story of the Big Bang has many sides. It is not only the Story of the cre- ation of Life, the Universe and Everything; it is also one of the greatest success stories of science in the twentieth century and it sets the stage on which the formation of structure in the Universe plays its part. Questions on the origin of structure have haunted many a philosopher, but since developments in quantum mechanics have shaken our understanding of causality, this theory serves us well in finding answers to these questions. Tiny quantum fluctuations were blown out of proportion by cosmic inflation during the first moments after the Big Bang. By gravitational interaction these fluctuations then grew to become the structures we now find our selves em- bedded in. It is our mission to describe this process. Current observations show us both ends of the pipeline, but relatively little of what is in between. Moreover, the largest agent of gravitational collapse, dark matter, is also the largest unknown. At the one end we see the cosmic microwave background radiation (fig. 1.1), which shows the conditions of the Universe only 340000 years after the Big Bang. At the other end we can look at the distribution of galaxies as it is now (or up-to a few billion years in the past). In figure 1.2 we see a slice of the Sloan survey. In this image we see a web-like structure consisting of clusters inter- woven by filaments and walls, leaving vast volumes of empty space: the voids. This structure we call the Cosmic Web. (van de Weygaert and Bond, 2005a,b) Knowing the statistics of the conditions in the Universe at the time of recom- bination, given the right physics, we should be able to derive the statistics of the cosmic web. However, even if we knew the right physics, this problem is mathematically very hard. One way to treat this problem is to run N-body simulations of which we see a famous example in figure (1.3). The results can then be compared to obser- 7 Figure 1.1: WMAP 5 year CMB temperature anisotropy map: The fluctuations are of the order of a ten thousandth of a degree. This is the fingerprint of the pristine condition the Universe was in when it was only 340000 years old. (Komatsu et al., 2009) Source: http://lambda.gsfc.nasa.gov/ vations. Even though the structures found largely agree with what we see in surveys, the simulations don’t get it quite right. Despite the immense difficul- ties in either measuring dark matter halo properties in real galaxies, or putting virtual galaxies inside simulated haloes, there are numerous inconsistencies. There is to much substructure in the simulations. Galaxies should be popu- lated by numerous dark matter sub-haloes, and should be accompanied with a large body of satellite dwarfs. These dwarfs are not seen in the numbers predicted by the N-body simulations (Klypin et al., 1999). A possibly related problem is that voids seem to be more empty than seen in simulations (Pee- bles, 2001). Dark matter halo profiles have a sharp cusp in simulations, while observations suggest that the profile should show a cut-off (Navarro et al., 1996; de Blok, 2010). Furthermore it is unknown how galaxies get there angu- lar momentum. (Perivolaropoulos 2008 gives a review of several problems in.