Persistent Holes in the Universe
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1 Persistent Holes in the Universe Pratyush Pranav Kapteyn Astronomical Institute Institute for Maths and its Applications, Minneapolis 14 th Oct, 2013
2 Rien van de Weygaert Kapteyn Astronomical Institute Herbert Edelsbrunner Institute of Science and Technology, Austria Gert Vegter Johan Bernoulli Institute of CS and Math Bernard Jones Kapteyn Astronomical Institute
3 Rien van de Weygaert Kapteyn Astronomical Institute Herbert Edelsbrunner Institute of Science and Technology, Austria Gert Vegter Johan Bernoulli Institute of CS and Math Bernard Jones Kapteyn Astronomical Institute
4 Outline Cosmic web & CMB : description and challenges Why (multi-scale) topology? Persistence : multi-scale topology Gaussian random fields & LSS: results Morse geometry and filament detection Conclusion
5 Cosmic Web & Cosmic Microwave Background
6 Cosmic Web : complications Discretely sampled : galaxies (observation) & particles (Nbody simulatons) Complex structural connectivity Lack of structural symmetry Intrinsic multi-scale nature Wide range of densities
7 Cosmic Microwave Bacground : Edge of the visible universe Progenitor of LSS: quantum fluctuations in infant Universe (CMB) Earliest View of our Cosmos: the Universe 379,000 years after the Big Bang Described as near-perfect (?) gaussian random field P N = exp 1 2 N N i= 1 j= 1 ( 1 M ) [( ) ( )] N 2π det M f i 1/ 2 ij f j N k = 1 df k
8 Correlation functions: Structural Insensitivity ξ () r r = r0 γ γ 1.8 r h Mpc 2-pt correlation function is highly insensitive to the geometry & morphology of weblike patterns: compare 2 distributions with same ξ(r), cq. P(k), but totally different phase distribution
9 Topology, Manifolds & topological holes
10 Why Topology? Cosmic web : a complex pattern of connected structures Topology studies connectivity Intrinsic topology insensitive to trivial change in the shape of structure (stretching, compression) or trivial coordinate transformation (expansion/contraction, distortion, rotation) In the cosmic context : topology insensitive to gravitational amplification and redshift distortion
11 Manifolds» Euclidean space» Locally Euclidean R,R 2,R 3 M 1,M 2,M 3
12 Functions on Manifolds Height of depth of the surface of earth with respect to sea level Primoridal fluctuation field, CMB Grid-densities of N-body simulations
13 Topological holes Topology in terms of k-dimensional holes Betti numbers β k (k=0, d) : count the no. of k-dimensional holes 0 dimensional holes : connected objects 1 dimensional holes : loops/tunnels 2 dimensional holes : voids
14 Are all holes important? Need to distinguish structures at different scales (cosmic web is multi-scale) Some holes are noise : discrete sampling, instrumental errors etc A need to assign a relative weight
15 Persistence : Multi-scale topology
16 Critical points : Change in topology Topological Structure of continuous field determined by singularities: - maxima - minima - saddle points Topology changes only while crossing a critical point
17 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
18 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
19 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
20 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
21 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
22 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
23 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
24 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
25 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
26 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
27 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
28 Filtration : Study the change in topology as we sweep from highest to lowest function values Persistence : Multi-scale topology Topology changes only at critical points (maxima, minima, saddle points): birth and death process
29 Persistence Diagrams 0-dimensional diagram: representation of merger of isolated objects (merger trees?) Representation of multi-scale topology Dots in the diagram record birth and death A diagram for each ambient dimension of the manifold 1-dimensional diagrams: formation and filling up of loops 2-dimensional diagrams: formation and destruction of topological voids
30 Persistence Diagrams : our preferred representation Rotation of co-ordinates DD + BB 22 = MMMMMMMM dddddddddddddd DD BB = pppppppppppppppppppppp
31 A typical persistence diagram Islands (0-dimension) Loops (1-dimension) Voids (2-dimension) Gaussian random field, power-law power spectrum, n=0 (all dimensions)
32 Statistics of persistence topology Topology connected to distribution of critical points Critical point distribution well defined for stochatic processes (GRF,CMB, LSS) : probabilistic description Are the diagrams well defined and stable/convergent over many realization? Statistical description?
33 Persistence landscape : emperical probability description Discretize the diagram into regular grids Empirical intensity function : I ij = N N ij tot Are the diagrams well defined and stable/convergent over many realization? Statistical description?
34 Persistence intensity maps : empirical probability description Discretize the diagram into regular grids Empirical intensity function : I ij = N N ij tot Are the diagrams well defined and stable/convergent over many realization? Statistical description?
35 Gaussian random field : Models Pranav et. al 2013
36 Gaussian random field : Persistence intensity maps Pranav et. al 2013
37 Model comparison : Ratio intensity maps Local difference in topology Excess/deficit of topological objects in the neighborhood of the point defined by (mean density:pers) Ratio function: ij ( f 1, f 2) = N N ij tot f 1 f 1 / N N ij tot f 2 f 2 = N N ij ij f 1 f 2 * Ω ( 1 f 1, f 2) Pranav et. al 2013
38 Ratio intensity maps : Gaussian Random fields Novel and extremely detailed descrimination of topological differences between models Pranav et. al 2013
39 Non-gaussianity: The untamable beast
40 Non-gaussianity: The untamable beast Constraining inflationary models Described as higher order terms in the expansion around a gaussian (fnl, gnl) Any function is potentially nongaussian in nature! Cosmological principle: isotropy and homogeneity as constraints imply gaussian fields described strictly only by 2-point correlation function ( a good constraint on separating gaussian from non-gaussian)
41 Ratio intensity maps: non-gaussian case Pranav et. al 2013
42 Model comparison : Intensity ratio maps Local difference in topology Excess/deficit of topological objects in the local neighborhood of the point defined by (mean density:pers) Ratio function: ij ( f 1, f 2) N = N ij tot f 1 f 1 N / N ij tot f 2 f 2 Gaussian Random field, wrt n=0 Non-gaussian field, wrt n=0 Non-gaussian simulations, courtesy Licia Verde Pranav et. al 2013b
43 Voronoi Kinematic Models: Heuristic description of structural evolution in the Cosmos d=0 d=1 d=2 Persistence diagrams segregate structures remarkably well Pranav et. al 2013a
44 Soneira-Peebles Models: Heuristic description Galaxy clustering Parameters of the model : Number of levels (n) Number of children (η) Ratio between the radius of parent and children spheres(λ) Pranav et. al 2013a Randomly place η spheres inside the top-level and continue for all levels
45 Betti Numbers Topology of excursion sets in terms of k- dimensional holes β k = # k-dimensional holes β 0 = connected components β 1 = independent loops β 2 = independent voids β 0 =1, β 1 = 2, β 2 =1
46 Betti numbers : Euler Characteristic and genus extended Euler characteristic is a compression of topological information i` i χ = ( 1) βi = ( 1) c i χ = β β + β...( 1) 0 1 Genus defined only for 2D surfaces (number of independent ways of cutting a 2D surface without leaving it disconnected) : no satisfactory generalization in lower and higher dimensions (!) 2 i d β i d Pranav et. al 2013
47 Gaussian random fields : Betti numbers and genus g 3/ k 2 υ (1 υ ) 2 = e 8π 3 / 2 Pranav et. al 2013 Park, Pranav et. al (2012)
48 Morse geometry & Filament detection
49 Morse Functions and critical points» Domain» Gradient» Critical points» Smooth functions 2, R, R R 3 f f = 0 2 f
50 Morse geometry : Critical points, gradient lines & descending/ascending manifolds Gradient lines
51 Morse geometry : Filaments as ascending manifolds of 2-saddles Gradient lines
52 Shivshankar, Pranav et. al 2013 Morse geometry : Filaments as ascending manifolds of 2-saddles
53 Pranav et. al 2013c Spin alignment of DM haloes along filaments
54 Interactive filament detection : New software
55 Universe in the Brain? I think therefore I am - Descartes
56 Universe in the Brain? I think therefore I am, therefore the Universe is - Pratyush
57 Conclusions Topology ideal tool for studying complex spatial patterns manifested in the universe Persistence topology: powerful tool for hierarchical topological description of LSS and CMB (Persistence) Homology offers new insights into structure, complexity and connectivity of the Cosmic Web Persistence provides rich language for description of multi-scale (hierarchical) topology of cosmic structure Persistence & ratio intensity maps: unprecedented detailed topological description (model discrimination) Ratio maps highly sensitive to deviations from gaussianity : Powerful probes for non-gaussianity Geometric connections : pattern/shape identification and description (cosmic filaments)
58 Holes and Homology Homology : algebraic formalization of topological holes in terms of chain, cycle and boundary groups p th homology group H p : collection of independent p- dimensional cycles Homology groups : analogous to vector spaces β p : p-dimensional Betti number (rank of the p th homology group H p )
59 Morse Functions and critical points Topology changes oly when crossing critical points» Domain» Gradient» Critical points» Smooth functions R,R 2,R 3 f f = 0 2 f
60 Morse Functions and critical points Topology changes oly when crossing critical points» Domain» Gradient» Critical points» Smooth functions R,R 2,R 3 f f = 0 2 f
61 Morse Functions and critical points Topology changes oly when crossing critical points!» Domain» Gradient» Critical points» Smooth functions R,R 2,R 3 f f = 0 2 f
62 Morse Functions and critical points Topology changes oly when crossing critical points» Domain» Gradient» Critical points» Smooth functions R,R 2,R 3 f f = 0 2 f
63 Morse Functions and critical points Topology changes oly when crossing critical points» Domain» Gradient» Critical points» Smooth functions R,R 2,R 3 f f = 0 2 f
64 Morse Functions and critical points Topology changes oly when crossing critical points» Domain» Gradient» Critical points» Smooth functions R,R 2,R 3 f f = 0 2 f
65 Morse Functions and critical points Topology changes oly when crossing critical points» Domain» Gradient» Critical points» Smooth functions R,R 2,R 3 f f = 0 2 f
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