Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-03T02:41:18.690Z Has data issue: false hasContentIssue false

VII - Numerical Computation of Maass Waveforms and an Application to Cosmology

Published online by Cambridge University Press:  05 January 2012

By
Ralf Aurich ,
Frank Steiner  and
Holger Then
Edited by
Jens Bolte  and
Frank Steiner
Ralf Aurich
Affiliation:
Institut für Theoretische Physik
Frank Steiner
Affiliation:
Institut für Theoretische Physik
Holger Then
Affiliation:
University of Bristol
Jens Bolte
Affiliation:
Royal Holloway, University of London
Frank Steiner
Affiliation:
Universität Ulm, Germany
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology , pp. 229 - 270
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. M., Abramowitz and I. A., Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55. U. S. Government Printing Office, Washington, D.C., 1964.Google Scholar
2. R., Aurich, E. B., Bogomolny, and F., Steiner. Periodic orbits on the regular hyperbolic octagon. Physica D, 48:91–101, 1991.Google Scholar
3. R., Aurich, S., Lustig, F., Steiner, and H., Then. Hyperbolic universes with a horned topology and the CMB anisotropy. Class. Quant. Grav. 21:4901–4925, 2004.Google Scholar
4. R., Aurich, S., Lustig, F., Steiner, and H., Then. Indications about the shape of the universe from the WMAP data. Phys. Rev. Lett. 94:021301, 2005.Google Scholar
5. R., Aurich and F., Steiner. On the periodic orbits of a strongly chaotic system. Physica D, 32:451–460, 1988.Google Scholar
6. R., Aurich and F., Steiner. Periodic-orbit sum rules for the Hadamard-Gutzwiller model. Physica D, 39:169–193, 1989.Google Scholar
7. R., Aurich and F., Steiner. Energy-level statistics of the Hadamard-Gutzwiller ensemble. Physica D, 43:155–180, 1990.Google Scholar
8. R., Aurich and F., Steiner. The cosmic microwave background for a nearly flat compact hyperbolic universe. Mon. Not. Roy. Astron. Soc., 323:1016–1024, 2001.Google Scholar
9. V. G., Avakumović. Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. (German). Math. Z., 65:327–344, 1956.Google Scholar
10. H., Avelin. On the deformation of cusp forms (Licentiate Thesis). UUDM report 2003:8, Uppsala 2003.
11. J., Bardeen. Gauge-invariant cosmological perturbations. Phys. Rev. D, 22:1882–1905, 1980.Google Scholar
12. C. L., Bennett, M., Halpern, G., Hinshaw, N., Jarosik, A., Kogut, M., Limon, S. S., Meyer, L., Page, D. N., Spergel, G. S., Tucker, E., Wollack, E. L., Wright, C., Barnes, M. R., Greason, R. S., Hill, E., Komatsu, M. R., Nolta, N., Odegard, H. V., Peirs, L., Verde, and J. L., Weiland. First year Wilkinson microwave anisotropy probe (WMAP) observations: Preliminary maps and basic results. Astroph. J. Suppl., 148:1–27, 2003.Google Scholar
13. P., de Bernardis, P. A. R., Ade, J. J., Bock, J. R., Bond, J., Borrill, A., Boscaleri, K., Coble, B. P., Crill, G., De Gasperis, P. C., Farese, P. G., Ferreira, K., Ganga, M., Giacometti, E., Hivon, V. V., Hristov, A., Iacoangeli, A. H., Jaffe, A. E., Lange, L., Martinis, S., Masi, P., Mason, P. D., Mauskopf, A., Melchiorri, L., Miglio, T., Montroy, C. B., Netterfield, E., Pascale, F., Piacentini, D., Pogosyan, S., Prunet, S., Rao, G., Romeo, J. E., Ruhl, F., Scaramuzzi, D., Sforna, and N., Vittorio. A flat universe from high-resolution maps of the cosmic microwave background radiation. Nature, 404:955–959, 2000.Google Scholar
14. M. V., Berry. Regular and irregular semiclassical wavefunctions. J. Phys. A, 10:2083–2091, 1977.Google Scholar
15. M. V., Berry and M., Tabor. Closed orbits and the regular bound spectrum. Proc. Roy. Soc. London Ser. A, 349:101–123, 1976.Google Scholar
16. E., Bogomolny, Quantum and arithmetical chaos. In P., Cartier, B., Julia, P., Moussa, and P., Vanhove, editors, Frontiers in Number Theory, Physics, and Geometry I. Springer, 2006.Google Scholar
17. E. B., Bogomolny, B., Georgeot, M.-J., Giannoni, and C., Schmit. Chaotic billiards generated by arithmetic groups. Phys. Rev. Lett., 69:1477–1480, 1992.Google Scholar
18. O., Bohigas, M.-J., Giannoni, and C., Schmit. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett., 52:1–4, 1984.Google Scholar
19. O., Bohigas, M.-J., Giannoni, and C., Schmit. Spectral fluctuations, random matrix theories and chaotic motion. Stochastic processes in classical and quantum systems. Lecture Notes in Phys., 262:118–138, 1986.Google Scholar
20. J., Bolte. Some studies on arithmetical chaos in classical and quantum mechanics. Int. J. Mod. Phys. B, 7:4451–4553, 1993.Google Scholar
21. J., Bolte, G., Steil, and F., Steiner. Arithmetical chaos and violation of universality in energy level statistics. Phys. Rev. Lett., 69:2188–2191, 1992.Google Scholar
22. A., Borel. Introduction aux groupes arithmétiques. (French). Hermann, 1969.Google Scholar
23. T., Chinburg and E., Friedman. The smallest arithmetic hyperbolic three-orbifold. Invent. Math., 86:507–527, 1986.Google Scholar
24. T., Chinburg, E., Friedman, K. N., Jones, and A. W., Reid. The arithmetic hyperbolic 3-manifold of smallest volume. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 30:1–40, 2001.Google Scholar
25. F. J., Dyson. Correlations between the eigenvalues of a random matrix. Commun. Math. Phys., 19:235–250, 1970.Google Scholar
26. A., Einstein. Zur allgemeinen Relativitätstheorie. (German). Preuß. Akad. Wiss., Sitzungsber., pp. 778–786, 1915.Google Scholar
28. A., Einstein. Die Feldgleichungen der Gravitation. (German). Preu. Akad. Wiss., Sitzungsber., pp. 844–847, 1915.Google Scholar
29. A., Einstein. Die Grundlage der allgemeinen Relativitätstheorie. (German). Ann. Phys., 49:769–822, 1916.Google Scholar
30. A., Einstein. Kosmologische Betrachtungen zur Allgemeinen Relativitätstheorie. (German). Preuß. Akad. Wiss., Sitzungsber., pp. 142–152, 1917.Google Scholar
31. J., Elstrodt, F., Grunewald, and J., Mennicke. Eisenstein series on three-dimensional hyperbolic space and imaginary quadratic number fields. J. Reine Angew. Math., 360:160–213, 1985.Google Scholar
32. J., Elstrodt, F., Grunewald, and J., Mennicke. Groups Acting on Hyperbolic Space. Springer, 1998.Google Scholar
33. A., Friedmann. Über die Krümmung des Raumes. (German). Z. Phys., 10:377–386, 1922.Google Scholar
34. A., Friedmann. Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes. (German). Z. Phys., 21:326–332, 1924.Google Scholar
35. F., Grunewald and W., Huntebrinker. A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp. Experiment. Math., 5:57–80, 1996.Google Scholar
36. S., Hanany, P., Ade, A., Balbi, J., Bock, J., Borrill, A., Boscaleri, P., de Bernardis, P. G., Ferreira, V. V., Hristov, A. H., Jaffe, A. E., Lange, A. T., Lee, P. D., Mauskopf, C. B., Netterfield, S., Oh, E., Pascale, B., Rabii, P. L., Richards, G. F., Smoot, R., Stompor, C. D., Winant, and J. H. P., Wu. MAXIMA-1: A measurement of the cosmic microwave background anisotropy on angular scales of 10 arcminutes to 5 degrees. Astroph. J., 545:L5, 2000.Google Scholar
37. D., Heitkamp. Hecke-Theorie zur SL(2; o). (German). Schriftenreihe des Mathematischen Instituts der Universität Münster, 3. Serie, 5, 1992.Google Scholar
38. D. A., Hejhal. The Selberg trace formula for PSL(2,ℝ). Lecture Notes in Math. 1001. Springer, 1983.Google Scholar
39. D. A., Hejhal. On eigenfunctions of the Laplacian for Hecke triangle groups. In D. A., Hejhal, J., Friedman, M. C., Gutzwiller, and A. M., Odlyzko, editors, Emerging applications of number theory, IMA Series No. 109, pp. 291–315. Springer, 1999.Google Scholar
40. D. A., Hejhal. On the calculation of Maass cusp forms, these proceedings.
42. E. P., Hubble. A relation between distance and radial velocity among extragalactic nebulae. Proc. Nat. Acad. Sci. (USA), 15:168–173, 1929.Google Scholar
43. G., Humbert. Sur la mesure des classes d'Hermite de discriminant donné dans un corps quadratique imaginaire, et sur certaines volumes non euclidiens. (French). C. R. Acad. Sci. Paris, 169:448–454, 1919.Google Scholar
44. W., Huntebrinker. Numerical computation of eigenvalues of the Laplace-Beltrami operator on three-dimensional hyperbolic spaces by finite-element methods. Diss. Summ. Math., 1:29–36, 1996.Google Scholar
45. K. T., Inoue, K., Tomita, and N., Sugiyama. Temperature correlations in a compact hyperbolic universe. Mon. Not. Roy. Astron. Soc., 314:L21, 2000.Google Scholar
46. T., Kubota. Elementary Theory of Eisenstein Series. Kodansha, Tokyo and Halsted Press, 1973.Google Scholar
47. C. L., Kuo, P. A. R., Ade, J. J., Bock, C., Cantalupo, M. D., Daub, J., Goldstein, W. L., Holzapfel, A. E., Lange, M., Lueker, M., Newcomb, J. B., Peterson, J., Ruhl, M. C., Runyan, and E., Torbet. High resolution observations of the CMB power spectrum with ACBAR. Astroph. J., 600:32–51, 2004.Google Scholar
48. The Legacy Archive for Microwave Background Data Analysis (LAMBDA), http://lambda.gsfc.nasa.gov/
49. G., Lemaîitre. Un univers homogène de masse constante et de rayon croissant, rendant compte de la vitesse radiale de nébuleuses extragalactiques. (French). Ann. Soc. Sci. Bruxelles, 47A:47–59, 1927.Google Scholar
50. J., Levin. Topology and the cosmic microwave background. Phys. Rep., 365:251–333, 2002.Google Scholar
51. J. J., Levin, J. D., Barrow, E. F., Bunn, and J., Silk. Flat spots: Topological signatures of an open universe in cosmic background explorer sky maps. Phys. Rev. Lett., 79:974–977, 1997.Google Scholar
52. E., Lifshitz. On the gravitational stability of the expanding universe. Zh. Eksp. Teor. Fiz., 16:587–602, 1946.Google Scholar
53. H., Maaß. Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. (German). Math. Ann., 121:141–183, 1949.Google Scholar
54. H., Maaß. Automorphe Funktionen von mehreren Veränderlichen und Dirichletsche Reihen. (German). Abh. Math. Semin. Univ. Hamb., 16:72–100, 1949.Google Scholar
55. C., Matthies. Picards Billard. Ein Modell für Arithmetisches Quantenchaos in drei Dimensionen. (German). PhD thesis, Universität Hamburg, 1995.
56. M. L., Mehta. Random matrices. Academic Press, second edition, 1991.Google Scholar
57. R., Meyerhoff. A lower bound for the covolume of hyperbolic 3-orbifolds. Duke Math. J., 57:185–203, 1988.Google Scholar
58. R., Meyerhoff. Sphere packing and volume in hyperbolic 3-space. Comment. Math. Helv., 61:271–278, 1988.Google Scholar
59. G. D., Mostow. Strong rigidity of locally symmetric spaces. Annals of Mathematics Studies, Princeton University Press, 1973.Google Scholar
60. V. F., Mukhanov, H. A., Feldman, and R. H., Brandenberger. Theory of cosmological perturbations. Phys. Reports, 215:203–333, 1992.Google Scholar
61. L., Page, M. R., Nolta, C., Barnes, C. L., Bennett, M., Halpern, G., Hinshaw, N., Jarosik, A., Kogut, M., Limon, S. S., Meyer, H. V., Peiris, D. N., Spergel, G. S., Tucker, E., Wollack, and E. L., Wright. First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Interpretation of the TT and TE angular power spectrum peaks. Astroph. J. Suppl., 148:233–241, 2003.Google Scholar
62. T. J., Pearson, B. S., Mason, A. C. S., Readhead, M. C., Shepherd, J. L., Sievers, P. S., Udomprasert, J. K., Cartwright, A. J., Farmer, S., Padin, S. T., Myers, J. R., Bond, C. R., Contaldi, U.-L., Pen, S., Prunet, D., Pogosyan, J. E., Carlstrom, J., Kovac, E. M., Leitch, C., Pryke, N. W., Halverson, W. L., Holzapfel, P., Altamirano, L., Bronfman, S., Casassus, J., May, and M., Joy. The anisotropy of the microwave background to 1 = 3500: Mosaic observations with the Cosmic Background Imager (CBI). Astroph. J., 591:556–574, 2003.Google Scholar
63. P. J. E., Peebles. The black-body radiation content of the universe and the formation of galaxies. Astroph. J., 142:1317–1326, 1965.Google Scholar
64. A. A., Penzias and R. W., Wilson. A measurement of excess antenna temperature at 4080 Mc/s. Astroph. J., 142:419–421, 1965.Google Scholar
65. E., Picard. Sur un groupe de transformations des points de l'espace situé du même coté d'un plan. Bull. Soc. Math. France 12:43–47, 1884.Google Scholar
66. G., Prasad. Strong rigidity of ℚ-rank 1 lattices. Invent. Math., 21:255–286, 1973.Google Scholar
67. A., Przeworski. Cones embedded in hyperbolic manifolds. J. Differential Geom., 58:219–232, 2001.Google Scholar
68. W., Roelcke. Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I and II. (German). Math. Ann., 167:292–337, 1966 and 168:261–324, 1967.Google Scholar
69. R. K., Sachs and A. M., Wolfe. Perturbations of a cosmological model and angular variations of the microwave background. Astroph. J., 147:73–90, 1967.Google Scholar
70. P., Sarnak. Arithmetic quantum chaos. Israel Math. Conf. Proc., 8:183–236, 1995.Google Scholar
71. B., Selander and A., Strömbergsson. Sextic coverings of genus two which are branched at three points. UUDM report 2002:16, Uppsala 2002.Google Scholar
72. A., Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc., 20:47–87, 1956.Google Scholar
73. G. F., Smoot, C. L., Bennett, A., Kogut, E. L., Wright, J., Aymon, N. W., Boggess, E. S., Cheng, G., De Amici, S., Gulkis, M. G., Hauser, G., Hinshaw, P. D., Jackson, M., Janssen, E., Kaita, T., Kelsall, P., Keegstra, C., Lineweaver, K., Loewenstein, P., Lubin, J., Mather, S. S., Meyer, S. H., Moseley, T., Murdock, L., Rokke, R. F., Silverberg, L., Tenorio, R., Weiss, and D. T., Wilkinson. Structure in the COBE Differential Microwave Radiometer first-year maps. Astroph. J., 396:L1–L5, 1992.Google Scholar
74. M. N., Smotrov and V. V., Golovčanskii. Small eigenvalues of the Laplacian on Γ\H3 for Γ = PSL2(ℤ[i]). Preprint, 91-040, Bielefeld 1991.Google Scholar
75. D. D., Sokolov and A. A., Starobinskii. Globally inhomogeneous “spliced” universes. Sov. Astron., 19:629–633, 1976.Google Scholar
76. D. N., Spergel, L., Verde, H. V., Peiris, E., Komatsu, M. R., Nolta, C. L., Bennett, M., Halpern, G., Hinshaw, N., Jarosik, A., Kogut, M., Limon, S. S., Meyer, L., Page, G. S., Tucker, J. L., Weiland, E., Wollack, and E. L., Wright. First year Wilkinson microwave anisotropy probe (WMAP) observations: Determination of cosmological parameters. Astroph. J. Suppl., 148:175–194, 2003.Google Scholar
77. H. M., Stark. Fourier coefficients of Maass waveforms. In R. A., Rankin, editor, Modular Forms, pp. 263–269. Ellis Horwood, 1984.Google Scholar
78. G., Steil. Eigenvalues of the Laplacian for Bianchi groups. In D. A., Hejhal, J., Friedman, M. C., Gutzwiller, and A. M., Odlyzko, editors, Emerging applications of number theory, IMA Series No. 109, pp. 617–641. Springer, 1999.Google Scholar
79. K., Stramm. Kleine Eigenwerte des Laplace-Operators zu Kongruenzgruppen. (German). Schriftenreihe des Mathematischen Instituts der Universität Münster, 3. Serie, 11, 1994.Google Scholar
80. F., Strömberg. Maass waveforms on (Γ0(N), χ), this volume.
81. H., Then. Maass cusp forms for large eigenvalues. Math. Comp., 74:363–381, 2005.Google Scholar
82. H., Then. Arithmetic quantum chaos of Maass waveforms. In P., Cartier, B., Julia, P., Moussa, and P., Vanhove, editors, Frontiers in Number Theory, Physics, and Geometry I, pp 183–212. Springer, 2006.Google Scholar
83. G. N., Watson. A treatise on the theory of Bessel functions. Cambridge University Press, 1944.Google Scholar
84. J., Weeks. Hyperbolic structures on 3-manifolds. PhD thesis, Princeton University, 1985.
85. H., Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. (German). Math. Ann., 71:441–479, 1912.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×