Table of 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 θ-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: νMSM 31
1.3.3 The moduli space of Dirac operators 34
1.3.4 Bosons and inner fluctuations 36
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 RGE flows 62
2.1.1 RGE flow from the Minimal Standard Model 62
2.1.2 RGE flow from the νMSM 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
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 spacetimcs 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 Non commutative mixmaster cosmologies 139
5.3.1 Mixmaster universes with torus sections 139
5.3.2 Noncommutative θ-deformations 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 a2 term period 175
7.3 The periods of the higher order terms a2n 177
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 Bethe 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
9.4.4 Continuum limit and the Wilson action 249
9.5 Random finite noncommutative geometries 250
Bibliography 255
Index 271
