×

Alpha, Betti and the megaparsec universe: on the topology of the cosmic web. (English) Zbl 1250.85007

Gavrilova, Marina L. (ed.) et al., Transactions on Computational Science XIV. Special issue on Voronoi diagrams and Delaunay triangulation. Berlin: Springer (ISBN 978-3-642-25248-8/pbk). Lecture Notes in Computer Science 6970. Journal Subline, 60-101 (2011).
Summary: We study the topology of the megaparsec cosmic web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them.
For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, \(\alpha \). As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of \(\alpha \), and their relation to the morphological patterns in the cosmic web, we first study them within the context of simple heuristic Voronoi clustering models. These can be tuned to consist of specific morphological elements of the cosmic web, i.e. clusters, filaments, or sheets. To elucidate the relative prominence of the various Betti numbers in different stages of morphological evolution, we introduce the concept of alpha tracks.
Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution of the Betti numbers is shown to reflect the hierarchical evolution of the cosmic web. We also demonstrate that the scale-dependence of the Betti numbers yields a promising measure of cosmological parameters, with a potential to help in determining the nature of dark energy and to probe primordial non-Gaussianities. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field.
Finally, we introduce the concept of persistent homology. It measures scale levels of the mass distribution and allows us to separate small from large scale features. Within the context of the hierarchical cosmic structure formation, persistence provides a natural formalism for a multiscale topology study of the cosmic web.
For the entire collection see [Zbl 1232.68009].

MSC:

85A40 Astrophysical cosmology
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)

Software:

DTFE

References:

[1] Adler, R.J., Bobrowski, O., Borman, M.S., Subag, E., Weinberger, S.: Persistent homology for random fields and complexes. Collections, 1–6 (2010), http://arxiv.org/abs/1003.1001
[2] Adler, R.J., Taylor, J.E.: Random fields and geometry (2007) · Zbl 1149.60003
[3] Bardeen, J.M., Bond, J.R., Kaiser, N., Szalay, A.S.: The statistics of peaks of Gaussian random fields. Astrophys. J. 304, 15–61 (1986) · doi:10.1086/164143
[4] Bendich, P., Edelsbrunner, H., Kerber, M.: Computing robustness and persistence for images. IEEE Trans. Vis. Comput. Graph. 16(6), 1251–1260 (2010) · doi:10.1109/TVCG.2010.139
[5] Bennett, C.L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S.S., Page, L., Spergel, D.N., Tucker, G.S., Wollack, E., Wright, E.L., Barnes, C., Greason, M.R., Hill, R.S., Komatsu, E., Nolta, M.R., Odegard, N., Peiris, H.V., Verde, L., Weiland, J.L.: First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results. Astrophys. J. Suppl. 148, 1–27 (2003) · doi:10.1086/377253
[6] Bond, J., Kofman, L., Pogosyan, D.: How filaments are woven into the cosmic web. Nature 380, 603–606 (1996) · doi:10.1038/380603a0
[7] de Boni, C., Dolag, K., Ettori, S., Moscardini, L., Pettorino, V., Baccigalupi, C.: Hydrodynamical simulations of galaxy clusters in dark energy cosmologies. Mon. Not. R. Astron. Soc. 414, 780 (2011)
[8] Bos, P.C. , van de Weygaert, R., Dolag, K., Pettorino, V.: Void shapes as probes of the nature of dark energy. Mon. Not. R. Astron. Soc. (2011)
[9] Calvo, M.A.A., Shandarin, S.F., Szalay, A.: Geometry of the cosmic web: Minkowski functionals from the delaunay tessellation. In: International Symposium on Voronoi Diagrams in Science and Engineering, pp. 235–243 (2010) · doi:10.1109/ISVD.2010.33
[10] Cautun, M.C., van de Weygaert, R.: The DTFE public software - The Delaunay Tessellation Field Estimator code. ArXiv e-prints (May 2011)
[11] Choi, Y.Y., Park, C., Kim, J., Gott, J.R., Weinberg, D.H., Vogeley, M.S., Kim, S.S.: For the SDSS Collaboration: Galaxy Clustering Topology in the Sloan Digital Sky Survey Main Galaxy Sample: A Test for Galaxy Formation Models. Astrophys. J. Suppl. 190, 181–202 (2010) · doi:10.1088/0067-0049/190/1/181
[12] Colless, M.: 2dF consortium: The 2df galaxy redshift survey: Final data release pp. 1–32 (2003); astroph/0306581
[13] Delfinado, C.J.A., Edelsbrunner, H.: An incremental algorithm for betti numbers of simplicial complexes. In: Symposium on Computational Geometry, pp. 232–239 (1993) · doi:10.1145/160985.161140
[14] Dey, T., Edelsbrunner, H., Guha, S.: Computational topology. In: Chazelle, B., Goodman, J.E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry, pp. 109–143. American Mathematical Society (1999) · Zbl 0916.68202 · doi:10.1090/conm/223/03135
[15] Edelsbrunner, H.: Alpha shapes - a survey. In: van de Weygaert, R., Vegter, G., Ritzerveld, J., Icke, V. (eds.) Tessellations in the Sciences; Virtues, Techniques and Applications of Geometric Tilings. Springer, Heidelberg (2010)
[16] Edelsbrunner, H., Facello, M., Liang, J.: On the definition and the construction of pockets in macromolecules. Discrete Appl. Math. 88, 83–102 (1998) · Zbl 0928.68113 · doi:10.1016/S0166-218X(98)00067-5
[17] Edelsbrunner, H., Harer, J.: Computational Topology, An Introduction. American Mathematical Society (2010) · Zbl 1193.55001
[18] Edelsbrunner, H., Kirkpatrick, D., Seidel, R.: On the shape of a set of points in the plane. IEEE Trans. Inform. Theory 29, 551–559 (1983) · Zbl 0512.52001 · doi:10.1109/TIT.1983.1056714
[19] Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistance and simplification. Discrete and Computational Geometry 28, 511–533 (2002) · Zbl 1011.68152 · doi:10.1007/s00454-002-2885-2
[20] Edelsbrunner, H., Muecke, E.: Three-dimensional alpha shapes. ACM Trans. Graphics 13, 43–72 (1994) · Zbl 0806.68107 · doi:10.1145/174462.156635
[21] Eisenstein, D.J., Hu, W.: Baryonic Features in the Matter Transfer Function. Astrophys.J. 496, 605–614 (1998) · doi:10.1086/305424
[22] Eldering, B.: Topology of Galaxy Models, MSc thesis, Univ. Groningen (2006)
[23] Gott, J.R., Choi, Y.Y., Park, C., Kim, J.: Three-Dimensional Genus Topology of Luminous Red Galaxies. Astrophys. J. Lett. 695, L45–L48 (2009) · doi:10.1088/0004-637X/695/1/L45
[24] Gott III, J.R., Miller, J., Thuan, T.X., Schneider, S.E., Weinberg, D.H., Gammie, C., Polk, K., Vogeley, M., Jeffrey, S., Bhavsar, S.P., Melott, A.L., Giovanelli, R., Hayes, M.P., Tully, R.B., Hamilton, A.J.S.: The topology of large-scale structure. III - Analysis of observations. Astrophys. J. 340, 625–646 (1989) · doi:10.1086/167425
[25] Gott, J., Dickinson, M., Melott, A.: The sponge-like topology of large-scale structure in the universe. Astrophys. J. 306, 341–357 (1986) · doi:10.1086/164347
[26] Hamilton, A.J.S., Gott III, J.R., Weinberg, D.: The topology of the large-scale structure of the universe. Astrophys. J. 309, 1–12 (1986) · doi:10.1086/164571
[27] Hoyle, F., Vogeley, M.S., Gott III, J.R., Blanton, M., Tegmark, M., Weinberg, D.H., Bahcall, N., Brinkmann, J., York, D.: Two-dimensional Topology of the Sloan Digital Sky Survey. Astrophys. J. 580, 663–671 (2002) · doi:10.1086/343734
[28] Huchra, J., et al.: The 2mass redshift survey and low galactic latitude large-scale structure. In: Fairall, A.P., Woudt, P.A. (eds.) Nearby Large-Scale Structures and the Zone of Avoidance. ASP Conf. Ser., vol. 239, pp. 135–146. Astron. Soc. Pacific, San Francisco (2005)
[29] Icke, V.: Voids and filaments. Mon. Not. R. Astron. Soc. 206, 1P–3P (1984) · doi:10.1093/mnras/206.1.1P
[30] Kauffmann, G., Colberg, J.M., Diaferio, A., White, S.D.M.: Clustering of galaxies in a hierarchical universe - I. Methods and results at z=0. Mon. Not. R. Astron. Soc. 303, 188–206 (1999) · doi:10.1046/j.1365-8711.1999.02202.x
[31] Komatsu, E., Smith, K.M., Dunkley, J., et al.: Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation. eprint arXiv:1001.4538 (January 2010)
[32] Liang, J., Edelsbrunner, H., Fu, P., Sudhakar, P., Subramaniam, S.: Analytical shape computation of macromolecules: I. molecular area and volume through alpha shape. Proteins: Structure, Function, and Genetics 33, 1–17 (1998) · doi:10.1002/(SICI)1097-0134(19981001)33:1<1::AID-PROT1>3.0.CO;2-O
[33] Liang, J., Edelsbrunner, H., Fu, P., Sudhakar, P., Subramaniam, S.: Analytical shape computation of macromolecules: Ii. inaccessible cavities in proteins. Proteins: Structure, Function, and Genetics 33, 18–29 (1998) · doi:10.1002/(SICI)1097-0134(19981001)33:1<18::AID-PROT2>3.0.CO;2-H
[34] Liang, J., Woodward, C., Edelsbrunner, H.: Anatomy of protein pockets and cavities: measurement of binding site geometry and implications for ligand design. Protein Science 7, 1884–1897 (1998) · doi:10.1002/pro.5560070905
[35] Mecke, K., Buchert, T., Wagner, H.: Robust morphological measures for large-scale structure in the universe. Astron. Astrophys. 288, 697–704 (1994)
[36] Moore, B., Frenk, C.S., Weinberg, D.H., Saunders, W., Lawrence, A., Ellis, R.S., Kaiser, N., Efstathiou, G., Rowan-Robinson, M.: The topology of the QDOT IRAS redshift survey. Mon. Not. R. Astron. Soc. 256, 477–499 (1992) · doi:10.1093/mnras/256.3.477
[37] Muecke, E.: Shapes and Implementations in three-dimensional geometry, PhD thesis, Univ. Illinois Urbana-Champaign (1993)
[38] Park, C., Chingangbam, P., van de Weygaert, R., Vegter, G., Kim, I., Hidding, J., Hellwing, W., Pranav, P.: Betti numbers of gaussian random fields. Astrophys. J. (2011) (to be subm.)
[39] Park, C., Gott III, J.R., Melott, A.L., Karachentsev, I.D.: The topology of large-scale structure. VI - Slices of the universe. Astrophys. J. 387, 1–8 (1992)
[40] Park, C., Kim, J., Gott III, J.R.: Effects of Gravitational Evolution, Biasing, and Redshift Space Distortion on Topology. Astrophys. J. 633, 1–10 (2005) · doi:10.1086/452621
[41] Park, C., Kim, Y.R.: Large-scale Structure of the Universe as a Cosmic Standard Ruler. Astrophys. J. Lett. 715, L185–L188 (2010) · doi:10.1088/2041-8205/715/2/L185
[42] Peebles, P.: The Large Scale Structure of the Universe. Princeton Univ. Press (1980)
[43] Pranav, P., Edelsbrunner, H., van de Weygaert, R., Vegter, G.: On the alpha and betti of the universe: Multiscale persistence of the cosmic web. Mon. Not. R. Astron. Soc. (2011) (to be subm.)
[44] Ratra, B., Peebles, P.J.E.: Cosmological consequences of a rolling homogeneous scalar field. Phys. Rev. D. 37, 3406–3427 (1988) · doi:10.1103/PhysRevD.37.3406
[45] Sahni, V., Sathyprakash, B.S., Shandarin, S.: Shapefinders: A new shape diagnostic for large-scale structure. Astrophys. J. 507, L109–L112 (1998) · doi:10.1086/311689
[46] Schaap, W.E., van de Weygaert, R.: Continuous fields and discrete samples: reconstruction through Delaunay tessellations. Astron. Astrophys. 32, L29–L32 (2000)
[47] Schmalzing, J., Buchert, T.: Beyond Genus Statistics: A Unifying Approach to the Morphology of Cosmic Structure. Astrophys. J. Lett. 482, L1–L4 (1997) · doi:10.1086/310680
[48] Schmalzing, J., Buchert, T., Melott, A., Sahni, V., Sathyaprakash, B., Shandarin, S.: Disentangling the cosmic web. i. morphology of isodensity contours. Astrophys. J. 526, 568–578 (1999) · doi:10.1086/308039
[49] Sheth, R.K., van de Weygaert, R.: A hierarchy of voids: much ado about nothing. Mon. Not. R. Astron. Soc. 350, 517–538 (2004) · doi:10.1111/j.1365-2966.2004.07661.x
[50] Smoot, G.F., Bennett, C.L., Kogut, A., Wright, E.L., Aymon, J., Boggess, N.W., Cheng, E.S., de Amici, G., Gulkis, S., Hauser, M.G., Hinshaw, G., Jackson, P.D., Janssen, M., Kaita, E., Kelsall, T., Keegstra, P., Lineweaver, C., Loewenstein, K., Lubin, P., Mather, J., Meyer, S.S., Moseley, S.H., Murdock, T., Rokke, L., Silverberg, R.F., Tenorio, L., Weiss, R., Wilkinson, D.T.: Structure in the COBE differential microwave radiometer first-year maps. Astrophys. J. Lett. 396, L1–L5 (1992) · doi:10.1086/186504
[51] Sousbie, T.: The persistent cosmic web and its filamentary structure - I. Theory and implementation. Mon. Not. R. Astron. Soc., pp. 511–+ (April 2011)
[52] Sousbie, T., Colombi, S., Pichon, C.: The fully connected N-dimensional skeleton: probing the evolution of the cosmic web. Mon. Not. R. Astron. Soc. 393, 457–477 (2009) · doi:10.1111/j.1365-2966.2008.14244.x
[53] Sousbie, T., Pichon, C., Colombi, S., Novikov, D., Pogosyan, D.: The 3D skeleton: tracing the filamentary structure of the Universe. Mon. Not. R. Astron. Soc. 383, 1655–1670 (2008) · doi:10.1111/j.1365-2966.2007.12685.x
[54] Sousbie, T., Pichon, C., Courtois, H., Colombi, S., Novikov, D.: The Three-dimensional Skeleton of the SDSS. Astrophys. J. Lett. 4, L1–L4 (2008) · doi:10.1086/523669
[55] Sousbie, T., Pichon, C., Kawahara, H.: The persistent cosmic web and its filamentary structure - II. Illustrations. Mon. Not. R. Astron. Soc., pp. 530–+ (2011)
[56] Spergel, D.N., Bean, R., Doré, O., Nolta, M.R., Bennett, C.L., Dunkley, J., Hinshaw, G., Jarosik, N., Komatsu, E., Page, L., Peiris, H.V., Verde, L., Halpern, M., Hill, R.S., Kogut, A., Limon, M., Meyer, S.S., Odegard, N., Tucker, G.S., Weiland, J.L., Wollack, E., Wright, E.L.: Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Cosmology. Astrophys. J. Suppl. 170, 377–408 (2007) · doi:10.1086/513700
[57] Springel, V., et al.: Simulations of the formation, evolution and clustering of galaxies and quasars. Nature 435, 629–636 (2005) · doi:10.1038/nature03597
[58] Tegmark, M., et al.: The three-dimensional power spectrum of galaxies from the sloan digital sky survey. Astrophys. J. 606, 702–740 (2004) · doi:10.1086/382125
[59] van de Weygaert, R.: Voids and the geometry of large scale structure, PhD thesis, University of Leiden (1991)
[60] van de Weygaert, R., Bertschinger, E.: Peak and gravity constraints in Gaussian primordial density fields: An application of the Hoffman-Ribak method. Mon. Not. R. Astron. Soc. 281, 84–118 (1996) · doi:10.1093/mnras/281.1.84
[61] van de Weygaert, R., Bond, J.R.: Clusters and the Theory of the Cosmic Web. In: Plionis, M., López-Cruz, O., Hughes, D. (eds.) A Pan-Chromatic View of Clusters of Galaxies and the Large-Scale Structure. Lecture Notes in Physics, vol. 740, pp. 335–407. Springer, Heidelberg (2008) · doi:10.1007/978-1-4020-6941-3_10
[62] van de Weygaert, R., Bond, J.R.: Observations and Morphology of the Cosmic Web. In: Plionis, M., López-Cruz, O., Hughes, D. (eds.) A Pan-Chromatic View of Clusters of Galaxies and the Large-Scale Structure. Lecture Notes in Physics, vol. 740, pp. 409–467. Springer, Berlin (2008) · doi:10.1007/978-1-4020-6941-3_11
[63] van de Weygaert, R., Icke, V.: Fragmenting the universe. II - Voronoi vertices as Abell clusters. Astron. Astrophys. 213, 1–9 (1989)
[64] van de Weygaert, R., Schaap, W.: The Cosmic Web: Geometric Analysis. In: Martínez, V.J., Saar, E., Martínez-González, E., Pons-Bordería, M.-J. (eds.) Data Analysis in Cosmology. Lecture Notes in Physics, vol. 665, pp. 291–413. Springer, Berlin (2009) · Zbl 1205.85048 · doi:10.1007/978-3-540-44767-2_11
[65] Vegter, G.: Computational topology. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., ch. 32, pp. 719–742. CRC Press LLC, Boca Raton (2004)
[66] Vegter, G., van de Weygaert, R., Platen, E., Kruithof, N., Eldering, B.: Alpha shapes and the topology of cosmic large scale structure. Mon. Not. R. Astron. Soc. (2010) (in prep.)
[67] Vogeley, M.S., Park, C., Geller, M.J., Huchra, J.P., Gott III, J.R.: Topological analysis of the CfA redshift survey. Astrophys. J. 420, 525–544 (1994) · doi:10.1086/173583
[68] van de Weygaert, R., Pranav, P., Jones, B., Vegter, G., Bos, P., Park, C., Hellwing, W.: Probing dark energy with betti-analysis of simulations. Astrophys. J. Lett. (2011) (to be subm.)
[69] van de Weygaert, R., Platen, E., Vegter, G., Eldering, B., Kruithof, N.: Alpha shape topology of the cosmic web. In: International Symposium on Voronoi Diagrams in Science and Engineering, pp. 224–234 (2010) · doi:10.1109/ISVD.2010.24
[70] van de Weygaert, R.: Voronoi tessellations and the cosmic web: Spatial patterns and clustering across the universe. In: ISVD 2007: Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering, pp. 230–239. IEEE Computer Society, Washington, DC (2007)
[71] Zhang, Y., Springel, V., Yang, X.: Genus Statistics Using the Delaunay Tessellation Field Estimation Method. I. Tests with the Millennium Simulation and the SDSS DR7. Astrophys. J. 722, 812–824 (2010) · doi:10.1088/0004-637X/722/1/812
[72] Zomorodian, A.: Topology for Computing. Cambr. Mon. Appl. Comp. Math., Cambr. Univ. Press (2005) · Zbl 1065.68001 · doi:10.1017/CBO9780511546945
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.