Noncommutative cosmology:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hongkong ; Taipei ; Chennai ; Tokyo
World Scientific
[2018]
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references |
| Physical Description: | xv, 275 Seiten Illustrationen, Diagramme |
| ISBN: | 9789813202832 9789813202849 |
Staff View
MARC
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| 100 | 1 | |a Marcolli, Matilde |d 1969- |0 (DE-588)141750545 |4 aut | |
| 245 | 1 | 0 | |a Noncommutative cosmology |c Matilde Marcolli (California Institute of Technology, USA, Perimeter Institute for Theoretical Physics, Canada, University of Toronto, Canada) |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hongkong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2018] | |
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| 650 | 4 | |a Cosmology | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface v
Acknowledgments ix
1. Gravity and Matter in Noncommutative Geometry 1
1.1 Spectral triples........................................... 4
1.1.1 Dimension in noncommutative geometry............... 6
1.1.2 Manifolds.......................................... 7
1.1.3 Almost-commutative geometries..................... 10
1.1.4 Cartesian products ............................... 11
1.1.5 Finite spectral triples........................... 12
1.1.6 0-deformations.................................... 15
1.1.7 Fractals.......................................... 17
1.2 The Spectral Action functional . ....................... 19
1.2.1 Asymptotic expansion.............................. 20
1.2.2 The spectral action as modified gravity........... 22
1.2.3 Riemannian versus Lorentzian geometries........... 23
1.2.4 Einstein-Hilbert action with cosmological term . . 24
1.2.5 Conformal gravity................................. 24
1.2.6 Gauss-Bonnet gravity.............................. 25
1.2.7 Poisson summation................................. 26
1.2.8 Spectral action and expansion on fractals......... 28
1.3 Particle Physics models................................... 30
1.3.1 Gravity coupled to matter......................... 30
1.3.2 The Standard Model: i/MSM......................... 31
1.3.3 The moduli space of Dirac operators .............. 34
1.3.4 Bosons and inner fluctuations..................... 36
XI
Noncommutative Cosmology
xii
1.3.5 Asymptotic expansion of the spectral action ... 39
1.3.6 Coefficients of the gravitational terms............ 42
1.3.7 Fused algebra approach............................. 43
1.3.8 Supersymmetric theories............................ 45
1.3.9 Adinkras, SUSY algebras, and the spectral action 51
1.3.10 Grand Unified theories............................. 57
2. Renormalization Group Flows and Early Universe Models 61
2.1 RGB flows .............................................. 62
2.1.1 RGE flow from the Minimal Standard Model ... 62
2.1.2 RGE flow from the i^MSM............................ 65
2.1.3 Geometric constraints at unification . ............ 67
2.1.4 Maximal mixing and initial condition at unification 68
2.1.5 Sensitive dependence and fine tuning............... 69
2.2 Gravitational terms ....................................... 69
2.3 Early universe models...................................... 71
2.3.1 Effective gravitational constant .................. 73
2.3.2 Effective cosmological constant.................... 74
2.3.3 Antigravity in the early universe.................. 75
2.3.4 Gravity balls...................................... 76
2.3.5 Primordial black holes with gravitational memory 77
2.3.6 Emergent Hoyle—Narlikar cosmologies................ 78
2.3.7 Slow-roll inflation................................ 79
2.4 Higgs mass estimates....................................... 80
2.4.1 Scalar fields and the Higgs mass problem........... 81
2.4.2 Asymptotic safety and anomalous dimensions . . 84
3. Cosmic Topology 89
3.1 The problem of cosmic topology ............................ 90
3.2 The spectral action and cosmic topology.................... 92
3.2.1 Slow-roll potential and slow-roll parameters ... 93
3.2.2 Spherical space forms.............................. 96
3.2.3 Flat tori and Bieberbach manifolds................ 102
3.2.4 A heat kernel view................................ 106
3.2.5 Gravity coupled to matter and slow-roll potential 109
3.2.6 Engineering inflation via Dirac spectra........... 113
Contents xiii
4. Algebro-geometric models in Cosmology 115
4.1 Spacetimes and complex geometry............................ 115
4.1.1 Complexified spacetimes and Grassmannians ... 116
4.1.2 Twistor spaces................................... 116
4.2 Blowup models.............................................. 117
4.2.1 Gluing spacetimes................................ 117
4.2.2 Conformally cyclic cosmological models........... 119
4.2.3 Eternal inflation via trees of projective spaces . . 120
4.3 Time and elliptic curves................................... 121
4.4 Noncommutativity and gluing of spacetimes ........ 122
5. Mixmaster Cosmologies 125
5.1 Kasner metrics and mixmaster universe models............... 125
5.1.1 The shift of the continued fraction expansion . . . 126
5.1.2 Continued fractions and the mixmaster universe . 127
5.1.3 Continued fractions and modular curves ............. 129
5.1.4 Kasner times and geodesic lengths................ 132
5.2 Modular curves, C*-algebras, and mixmaster models . . . 134
5.3 Noncommutative mixmaster cosmologies....................... 139
5.3.1 Mixmaster universes with torus sections.......... 139
5.3.2 Noncommutative deformat ions........................ 141
5.3.3 Spectral action and inflation scenario........... 142
5.4 Bianchi IX SU(2)-gravitational instantons .......... 146
5.4.1 Painlevé VI equation................................ 147
5.4.2 Gravitational instantons and Painlevé............ 148
5.5 Noncommutativity in the early universe .................... 150
6. The Spectral Action on Bianchi IX Cosmologies 155
6.1 Pseudodifferential calculus and parametrix method .... 156
6.2 Wodzicki residues method................................... 160
6.3 Rationality result......................................... 162
6.4 Gravitational instantons and the spectral action .......... 163
6.5 The spectral action and modular forms...................... 165
7. Motives and Periods in Cosmology 171
7.1 Robertson-Walker metrics................................... 173
7.2 The û2 term period......................................... 175
7.3 The periods of the higher order terms a2n.................. 177
XIV
Noncommutative Cosmology
7.4 The mixed motives of Robertson-Walker gravity......... 179
7.4.1 Triangulated category of motives............... 180
7.4.2 Grothendieck classes of RW spacetimes.............. 181
7.4.3 Mixed Tate motives of RW spacetimes ........... 182
8. Fractal and Multifractal Structures in Cosmology 187
8.1 Packed Swiss Cheese Cosmology and the spectral action . 187
8.1.1 Apollonian sphere packings......................... 188
8.1.2 Length spectrum and zeta functions................. 191
8.1.3 Models of fractal spacetimes....................... 195
8.1.4 The spectral action functional on fractal
cosmologies........................................ 197
8.1.5 Slow-roll inflation potentials with fractality .... 201
8.2 A p-adic model of eternal inflation...................201
8.2.1 Bruhat-Tits tree and Bet he tree....................202
8.2.2 Multifractals, symbolic dynamics, and stochastics 203
9. Noncommutative Quantum Cosmology 209
9.1 Hartle-Hawking quantum cosmology.......................... 209
9.1.1 Path integral and wave function of the universe . 210
9.1.2 Hamiltonian constraint, Wheeler-DeWitt equation 212
9.1.3 Minisuperspace models.............................. 214
9.1.4 Exotic smoothness.................................. 216
9.2 Categories and algebras of geometries..................... 218
9.2.1 Cobordisms: equivalences and 2-categories .... 222
9.2.2 Vertical composition and Hartle-Hawking gravity 227
9.2.3 Horizontal composition and Connes-Chern
character.......................................... 228
9.2.4 Almost-commutative cobordisms.......................230
9.3 Topological spin networks and foams........................233
9.3.1 Spin networks and monodromies...................... 233
9.3.2 Spin foams and monodromies .........................237
9.3.3 2-categories and convolution algebras...............239
9.3.4 Quantized area operator and dynamics............... 240
9.4 Discretized almost-commutative geometries..................243
9.4.1 Categorical data and finite spectral triples .... 243
9.4.2 Gauge networks..................................... 245
9.4.3 Spectral action on a lattice....................... 247
Contents
xv
9.4.4 Continuum limit and the Wilson action..........249
9.5 Random finite noneommutative geometries ............ 250
Bibliography 255
Index 271
Noncommutative
Cosmology
Modified gravity models play an important role in contemporary
theoretical cosmology. The present book proposes a novel
approach to the topic based on techniques from noncommutative
geometry, especially the spectral action functional as a gravity
model. The book discusses applications to early universe models
and slow-roll inflation models, to the problem of cosmic topology,
to non-isotropic cosmologies like mixmaster universes and Bianchi
IX gravitational instantons, and to multifractal structures in
cosmology.
Relations between noncommutative and algebro-geometric
methods in cosmology are also discussed, including the
occurrence of motives, periods, and modular forms in spectral
models of gravity.
World Scientific
www.worldscientific.com
10335 he
ISBN 978-981 3202-83-2
789
B13
20
2832
|
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| id | DE-604.BV044749694 |
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| indexdate | 2024-07-10T08:01:09Z |
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| physical | xv, 275 Seiten Illustrationen, Diagramme |
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| spelling | Marcolli, Matilde 1969- (DE-588)141750545 aut Noncommutative cosmology Matilde Marcolli (California Institute of Technology, USA, Perimeter Institute for Theoretical Physics, Canada, University of Toronto, Canada) New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hongkong ; Taipei ; Chennai ; Tokyo World Scientific [2018] © 2018 xv, 275 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references Noncommutative differential geometry Mathematical physics Cosmology Quantenkosmologie (DE-588)4302168-2 gnd rswk-swf Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf Gravitation (DE-588)4021908-2 gnd rswk-swf Nichtkommutative Geometrie (DE-588)4171742-9 s Gravitation (DE-588)4021908-2 s Quantenkosmologie (DE-588)4302168-2 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030145373&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030145373&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Marcolli, Matilde 1969- Noncommutative cosmology Noncommutative differential geometry Mathematical physics Cosmology Quantenkosmologie (DE-588)4302168-2 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd Gravitation (DE-588)4021908-2 gnd |
| subject_GND | (DE-588)4302168-2 (DE-588)4171742-9 (DE-588)4021908-2 |
| title | Noncommutative cosmology |
| title_auth | Noncommutative cosmology |
| title_exact_search | Noncommutative cosmology |
| title_full | Noncommutative cosmology Matilde Marcolli (California Institute of Technology, USA, Perimeter Institute for Theoretical Physics, Canada, University of Toronto, Canada) |
| title_fullStr | Noncommutative cosmology Matilde Marcolli (California Institute of Technology, USA, Perimeter Institute for Theoretical Physics, Canada, University of Toronto, Canada) |
| title_full_unstemmed | Noncommutative cosmology Matilde Marcolli (California Institute of Technology, USA, Perimeter Institute for Theoretical Physics, Canada, University of Toronto, Canada) |
| title_short | Noncommutative cosmology |
| title_sort | noncommutative cosmology |
| topic | Noncommutative differential geometry Mathematical physics Cosmology Quantenkosmologie (DE-588)4302168-2 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd Gravitation (DE-588)4021908-2 gnd |
| topic_facet | Noncommutative differential geometry Mathematical physics Cosmology Quantenkosmologie Nichtkommutative Geometrie Gravitation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030145373&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030145373&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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