Abstract
The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincaré dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincaré homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aurich R., Lustig S., Steiner F., Then H.: Cosmic microwave background alignment in multi-connected universes. Class. Quantum Grav. 24, 1879–1894 (2007)
Bär C.: The Dirac operator on space forms of positive curvature. J. Math. Soc. Japan 48(1), 69–83 (1996)
Bär C.: The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces. Arch. Math. 59, 65–79 (1992)
Bär, C.: Dependence of Dirac Spectrum on the Spin Structure. In: Séminaires & Congrès, 4. Bouoguignon J.P., Bânson, T., Hija-â, O. (eds.) Global Anal. and Harmonic Anal. (Luming, 2000), Paris: French Math. Soc., 2000, pp. 17–33
de Bernardis P., Ade P.A.R., Bock J.J., Bond J.R., Borrill J., Boscaleri A., Coble K., Crill B.P., De Gasperis G., Farese P.C., Ferreira P.G., Ganga K., Giacometti M., Hivon E., Hristov V.V., Iacoangeli A., Jaffe A.H., Lange A.E., Martinis L., Masi S., Mason P.V., Mauskopf P.D., Melchiorri A., Miglio L., Montroy T., Netterfield C.B., Pascale E., Piacentini F., Pogosyan D., Prunet S., Rao S., Romeo G., Ruhl J.E., Scaramuzzi F., Sforna D., Vittorio N.: A flat Universe from high-resolution maps of the cosmic microwave background radiation. Nature 404, 955–959 (2000)
van den Broek, T., van Suijlekom, W.D.: Supersymmetric QCD and noncommutative geometry. http://arXiv.org/abs/1003.3788v1 [hepth], 2010
Caillerie S., Lachièze-Rey M., Luminet J.P., Lehoucq R., Riazuelo A., Weeks J.: A new analysis of the Poincaré dodecahedral space model. Astron. and Astrophys. 476(2), 691–696 (2007)
Chamseddine A., Connes A.: The spectral action principle. Commun. Math. Phys. 186(3), 731–750 (1997)
Chamseddine A., Connes A.: The uncanny precision of the spectral action. Commun. Math. Phys. 293, 867–897 (2010)
Chamseddine A., Connes A., Marcolli M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11(6), 991–1089 (2007)
Connes A.: Gravity coupled with matter and foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996)
Cornish N.J., Spergel D.N., Starkman G.D., Komatsu E.: Constraining the topology of the universe. Phys. Rev. Lett. 92, 201302 (2004)
Dahl M.: Prescribing eigenvalues of the Dirac operator. Manus. Math. 118, 191–199 (2005)
Dahl M.: Dirac eigenvalues for generic metrics on three-manifolds. Ann. Global Anal. Geom. 24, 95–100 (2003)
De Simone A., Hertzberg M.P., Wilczek F.: Running inflation in the Standard Model. Phys. Lett. B 678, 1–8 (2009)
Dowker J.S.: Spherical universe topology and the Casimir effect. Class. Quant. Grav. 21, 4247–4271 (2004)
Gausmann E., Lehoucq R., Luminet J.P., Uzan J.P., Weeks J.: Topological lensing in spherical spaces. Class. Quant. Grav. 18, 5155–5186 (2001)
Ginoux N.: The spectrum of the Dirac operator on SU 2/Q 8. Manus. Math. 125(3), 383–409 (2008)
Gomero G.I., Reboucas M.J., Tavakol R.: Detectability of cosmic topology in almost flat universes. Class. Quant. Grav. 18, 4461–4476 (2001)
Gomero G.I., Reboucas M.J., Teixeira A.F.F.: Spikes in cosmic crystallography II: topological signature of compact flat universes. Phys. Lett. A 275, 355–367 (2000)
Hitchin N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)
Kamionkowski M., Spergel D.N., Sugiyama N.: Small-scale cosmic microwave background anisotropies as a probe of the geometry of the universe. Astrophys. J. 426, L57–60 (1994)
Lachièze-Rey M., Luminet J.P.: Cosmic topology. Phys. Rep. 254, 135–214 (1995)
Lehoucq R., Weeks J., Uzan J.P., Gausmann E., Luminet J.P.: Eigenmodes of three-dimensional spherical spaces and their applications to cosmology. Class. Quant. Grav. 19, 4683–4708 (2002)
Luminet J.P., Weeks J., Riazuelo A., Lehoucq R.: Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background. Nature 425, 593–595 (2003)
Marcolli, M., Pierpaoli, E.: Early universe models from noncommutative geometry. http://arXiv.org/abs/0908.3683v1 [hepth], 2009
McInnes B.: APS instability and the topology of the brane-world. Phys. Lett. B 593(1-4), 10–16 (2004)
Nelson W., Sakellariadou M.: Natural inflation mechanism in asymptotic noncommutative geometry. Phys. Lett. B 680, 263–266 (2009)
Niarchou A., Jaffe A.: Imprints of spherical nontrivial topologies on the cosmic microwave background. Phys. Rev. Lett. 99, 081302 (2007)
de Oliveira-Costa A., Tegmark M., Zaldarriaga M., Hamilton A.: Significance of the largest scale CMB fluctuations in WMAP. Phys. Rev. D 69, 063516 (2004)
Pfäffle F.: The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35, 367–385 (2000)
Riazuelo A., Uzan J.P., Lehoucq R., Weeks J.: Simulating Cosmic Microwave Background maps in multi-connected spaces. Phys. Rev. D 69, 103514 (2004)
Riazuelo A., Weeks J., Uzan J.P., Lehoucq R., Luminet J.P.: Cosmic microwave background anisotropies in multiconnected flat spaces. Phys. Rev. D 69, 103518 (2004)
Roukema B.F., Rózański P.T.: The residual gravity acceleration effect in the Poincaré dodecahedral space. Astron. and Astrophy. 502, 27 (2009)
Souradeep, T., Hajian, A.: Statistical isotropy of CMB anisotropy from WMAP. http://arXiv.org/abs/astro-ph/0502248v1, 2005
Spergel D.N., Verde L., Peiris H.V., Komatsu E., Nolta M.R., Bennett C.L., Halpern M., Hinshaw G., Jarosik N., Kogut A., Limon M., Meyer S.S., Page L., Tucker G.S., Weiland J.L., Wollack E., Wright E.L.: First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parameters. Astrophys. J. Suppl. 148, 175–194 (2003)
Tegmark M., de Oliveira-Costa A., Hamilton A.: A high resolution foreground cleaned CMB map from WMAP. Phys. Rev. D. 68, 123523 (2003)
Uzan J.P., Kirchner U., Ellis G.F.R.: WMAP data and the curvature of space. Mon. Not. Roy. Astron. Soc. 344, L65 (2003)
Uzan J.P., Riazuelo A., Lehoucq R., Weeks J.: Cosmic microwave background constraints on lens spaces. Phys. Rev. D 69, 043003 (2004)
Weeks J., Gundermann J.: Dodecahedral topology fails to explain quadrupole-octupole alignment. Class. Quant. Grav. 24, 1863–1866 (2007)
Weeks J., Lehoucq R., Uzan J.P.: Detecting topology in a nearly flat spherical universe. Class. Quant. Grav. 20, 1529–1542 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Connes
In memory of Andrew Lange
Rights and permissions
About this article
Cite this article
Marcolli, M., Pierpaoli, E. & Teh, K. The Spectral Action and Cosmic Topology. Commun. Math. Phys. 304, 125–174 (2011). https://doi.org/10.1007/s00220-011-1211-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1211-3