Capozziello / Faraoni Beyond Einstein Gravity

A Survey of Gravitational Theories for Cosmology and Astrophysics
1. Auflage 2010
ISBN: 978-94-007-0165-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 428 Seiten

Beyond Einstein’s Gravity is a graduate level introduction to extended theories of gravity and cosmology, including variational principles, the weak-field limit, gravitational waves, mathematical tools, exact solutions, as well as cosmological and astrophysical applications. The book provides a critical overview of the research in this area and unifies the existing literature using a consistent notation. Although the results apply in principle to all alternative gravities, a special emphasis is on scalar-tensor and f(R) theories. They were studied by theoretical physicists from early on, and in the 1980s they appeared in attempts to renormalize General Relativity and in models of the early universe. Recently, these theories have seen a new lease of life, in both their metric and metric-affine versions, as models of the present acceleration of the universe without introducing the mysterious and exotic dark energy. The dark matter problem can also be addressed in extended gravity. These applications are contributing to a deeper understanding of the gravitational interaction from both the theoretical and the experimental point of view. An extensive bibliography guides the reader into more detailed literature on particular topics.

Autoren/Hrsg.

Capozziello, Salvatore

Faraoni, Valerio

Inhaltsverzeichnis

1;Preface;8
2;Acknowledgements;12
3;Contents;14
4;Acronyms;20
5;Chapter 1:Extended gravity: a primer;22
5.1;1.1 Why extending gravity?;22
5.2;1.2 Cosmological and astrophysical motivation;24
5.3;1.3 Mathematical motivation;27
5.4;1.4 Quantum gravity motivation;28
5.4.1;1.4.1 Emergent gravity and thermodynamics of spacetime;33
5.5;1.5 What a good theory of gravity should do: General Relativity and its extensions;34
5.6;1.6 Quantum field theory in curved space;39
5.7;1.7 Mach's principle and other fundamental issues;44
5.7.1;1.7.1 Higher order corrections to Einstein's theory;46
5.7.2;1.7.2 Minimal and non-minimal coupling and the Equivalence Principle;48
5.7.3;1.7.3 Mach's principle and the variation of G;53
5.8;1.8 Extended gravity from higher dimensions and area metric approach;56
5.9;1.9 Conclusions;61
6;Chapter 2:Mathematical tools;62
6.1;2.1 Conformal transformations;62
6.2;2.2 Variational principles in General Relativity;68
6.2.1;2.2.1 Geodesics;68
6.2.2;2.2.2 Field equations;70
6.3;2.3 Adding torsion;72
6.4;2.4 Noether symmetries;75
6.5;2.5 Conclusions;78
7;Chapter 3:The landscape beyond Einstein gravity;79
7.1;3.1 The variational principle and the field equations of Brans-Dicke gravity;79
7.2;3.2 The variational principle and the field equations of metric f(R) gravity;82
7.2.1;3.2.1 f(R)=R +R2 theory;82
7.2.2;3.2.2 Metric f(R) gravity in general;84
7.3;3.3 A more general class of ETGs;87
7.4;3.4 The Palatini formalism;87
7.4.1;3.4.1 The Palatini approach and the conformal structure of the theory;88
7.4.2;3.4.2 Problems with the Palatini formalism;93
7.5;3.5 Equivalence between f(R) and scalar-tensor gravity;97
7.5.1;3.5.1 Equivalence between scalar-tensor and metric f(R) gravity;97
7.5.2;3.5.2 Equivalence between scalar-tensor and Palatinif(R) gravity;98
7.6;3.6 Conformal transformations applied to extended gravity;99
7.6.1;3.6.1 Brans-Dicke gravity;99
7.6.2;3.6.2 Scalar-tensor theories;103
7.6.3;3.6.3 Mixed f(R)/scalar-tensor gravity;105
7.6.4;3.6.4 The issue of the conformal frame ;106
7.7;3.7 The initial value problem;110
7.7.1;3.7.1 The Cauchy problem of scalar-tensor gravity;112
7.7.2;3.7.2 The initial value problem of f(R) gravity in the ADM formulation;117
7.7.3;3.7.3 The Gaussian normal coordinates approach;118
7.7.3.1;3.7.3.1 The Cauchy problem of GR;119
7.7.3.2;3.7.3.2 The Cauchy problem of vacuum f(bold0mu mumu RR*RRRR) gravityin the metric-affine formalism;121
7.7.3.3;3.7.3.3 The Cauchy problem in the metric-affine formalism with matter;121
7.8;3.8 Conclusions;126
8;Chapter 4:Spherical symmetry;127
8.1;4.1 Spherically symmetric solutions of GR and metric f(R) gravity;127
8.1.1;4.1.1 Spherical symmetry;128
8.1.2;4.1.2 The Ricci scalar in spherical symmetry;129
8.1.3;4.1.3 Spherical symmetry in metric f(R) gravity;130
8.1.4;4.1.4 Solutions with constant Ricci scalar;132
8.1.5;4.1.5 Solutions with R=R(r);135
8.1.6;4.1.6 Perturbations;137
8.1.7;4.1.7 Spherical symmetry in f(R) gravity and the Noether approach;139
8.1.7.1;4.1.7.1 The point-like f(R) Lagrangian in spherical symmetry;139
8.1.8;4.1.8 Noether solutions of spherically symmetric f(R) gravity;144
8.1.9;4.1.9 Non-asymptotically flat and non-static spherical solutions of metric f(R) gravity;148
8.1.9.1;4.1.9.1 Clifton and Barrow's static solution in f(R)=R1+ gravity;149
8.1.9.2;4.1.9.2 A dynamical solution in f(R)=R1+ gravity;149
8.2;4.2 Spherical symmetry in scalar-tensor gravity;154
8.2.1;4.2.1 Static solutions of Brans-Dicke theory;154
8.2.2;4.2.2 Dynamical and asymptotically FLRW solutions;156
8.2.3;4.2.3 Collapse to black holes in scalar-tensor theory;157
8.3;4.3 The Jebsen-Birkhoff theorem;159
8.3.1;4.3.1 The Jebsen-Birkhoff theorem of GR;159
8.3.2;4.3.2 The non-vacuum case;160
8.3.3;4.3.3 The vacuum case;162
8.3.4;4.3.4 The Jebsen-Birkhoff theorem in scalar-tensor gravity;163
8.3.5;4.3.5 The trivial case = constant;164
8.3.6;4.3.6 Static non-constant Brans-Dicke-like field;165
8.3.7;4.3.7 The Jebsen-Birkhoff theorem in Einstein frame scalar-tensor gravity;166
8.3.8;4.3.8 Hawking's theorem and Jebsen-Birkhoff in Brans-Dicke gravity;168
8.3.9;4.3.9 The Jebsen-Birkhoff theorem in f(R) gravity;170
8.3.9.1;4.3.9.1 Palatini f(R) gravity;170
8.3.9.2;4.3.9.2 Metric f(R) gravity;171
8.4;4.4 Black hole thermodynamics in extended gravity;171
8.4.1;4.4.1 Scalar-tensor gravity;173
8.4.2;4.4.2 Metric modified gravity;175
8.4.3;4.4.3 Palatini modified gravity;176
8.4.4;4.4.4 Dilaton gravity;177
8.5;4.5 From spherical to axial symmetry: an application to f(R) gravity;178
8.6;4.6 Conclusions;183
9;Chapter 5: Weak-field limit;185
9.1;5.1 The weak-field limit of extended gravity;185
9.2;5.2 The Newtonian and post-Newtonian approximations:general remarks;187
9.2.1;5.2.1 The Newtonian and post-Newtonian limits of metric f(R) gravity with spherical symmetry;191
9.2.2;5.2.2 Comparison with the standard formalism and the chameleon effect;200
9.3;5.3 The Post-Minkowskian approximation;205
9.3.1;5.3.1 The energy-momentum pseudotensor in f(R) gravity and gravitational radiation;207
9.4;5.4 Gravitational waves;210
9.4.1;5.4.1 Gravitational waves in scalar-tensor gravity;212
9.4.2;5.4.2 Gravitational waves in higher order gravity;215
9.4.2.1;5.4.2.1 Polarization states of gravitational waves;221
9.4.2.2;5.4.2.2 Detector response;224
9.5;5.5 Conclusions;228
10;Chapter 6:Qualitative analysis and exact solutions in cosmology;229
10.1;6.1 The Ehlers-Geren-Sachs theorem;229
10.2;6.2 The phase space of FLRW cosmology in scalar-tensor and f(R) gravity;230
10.2.1;6.2.1 The dynamical system;232
10.2.1.1;6.2.1.1 The phase space with vacuum, free scalar field, and any three-geometry;233
10.2.1.2;6.2.1.2 The phase space for vacuum, V=m22/2, and flat three-sections;235
10.2.1.3;6.2.1.3 The phase space in vacuo with V0 and spatially flat three-geometry;236
10.2.1.4;6.2.1.4 The phase space with P=-/3 and a free scalar;238
10.2.1.5;6.2.1.5 The phase space of f(R) gravity;240
10.3;6.3 Analytical solutions of Brans-Dicke and scalar-tensor cosmology;240
10.3.1;6.3.1 Analytical solutions of Brans-Dicke cosmology;241
10.3.1.1;6.3.1.1 Spatially flat FLRW solutions of Brans-Dicke theory;243
10.3.1.2;6.3.1.2 Spatially curved FLRW solutions with V =0 and Bianchi models;248
10.3.1.3;6.3.1.3 Phase space for V =m22/2 and any three-geometry;251
10.3.2;6.3.2 Exact scalar-tensor cosmologies;252
10.4;6.4 Analytical solutions of metric f(R) cosmology by the Noether approach;253
10.4.1;6.4.1 Point-like f(R) cosmology;253
10.4.2;6.4.2 Noether symmetries in metric f(R) cosmology;255
10.4.3;6.4.3 Exact cosmologies;258
10.4.3.1;6.4.3.1 c1=0;258
10.4.3.2;6.4.3.2 c2=0;261
10.4.4;6.4.4 c1,c20;263
10.4.4.1;6.4.4.1 Cosmological constant and dust;264
10.4.4.2;6.4.4.2 Non-Noether solutions;268
10.5;6.5 Analytical cosmological solutions of f ( R, R, , k R ) gravity;273
10.5.1;6.5.1 Higher order point-like Lagrangians for cosmology;273
10.5.2;6.5.2 The Noether symmetry approach for higher order gravities;276
10.6;6.6 Conclusions;280
11;Chapter 7:Cosmology;281
11.1;7.1 Big Bang, inflationary, and late-time cosmology in GR;282
11.1.1;7.1.1 The standard Big Bang model;283
11.1.2;7.1.2 Inflation in the early universe;283
11.1.3;7.1.3 The present-day acceleration;285
11.2;7.2 Using cosmography to map the structure of the universe;293
11.2.1;7.2.1 The cosmographic apparatus;294
11.2.1.1;7.2.1.1 The scale factor series;295
11.2.1.2;7.2.1.2 Cosmography and extended gravity;301
11.2.1.3;7.2.1.3 Cosmography and the derivatives of f(R);302
11.2.1.4;7.2.1.4 f(R) gravity and the CPL model;307
11.2.1.5;7.2.1.5 The CDM model;308
11.2.1.6;7.2.1.6 The constant EoS model;310
11.2.1.7;7.2.1.7 The general case;312
11.2.1.8;7.2.1.8 Constraining the f(R) parameters;314
11.2.1.9;7.2.1.9 A double power-law action;314
11.2.1.10;7.2.1.10 The Hu and Sawicki model;316
11.2.1.11;7.2.1.11 Observational constraints on the derivatives of f(R);318
11.2.1.12;7.2.1.12 What does cosmography teach us after all?;322
11.3;7.3 Large scale structure and galaxy clusters;324
11.3.1;7.3.1 The weak-field limit of f(R) gravity and galaxy clusters;325
11.3.2;7.3.2 Extended systems;326
11.3.3;7.3.3 The cluster mass profiles;327
11.3.4;7.3.4 The galaxy clusters sample;330
11.3.5;7.3.5 The gas density model;330
11.3.6;7.3.6 Temperature profiles;331
11.3.7;7.3.7 The galaxy distribution model;331
11.3.8;7.3.8 Uncertainties in the mass profiles;334
11.3.9;7.3.9 Fitting the mass profiles;334
11.3.10;7.3.10 Results;336
11.3.11;7.3.11 Outlooks;341
11.4;7.4 Testing cosmological models with observations;346
11.4.1;7.4.1 Toward a new cosmological standard model;347
11.4.1.1;7.4.1.1 The CDM model and its generalizations;347
11.4.1.2;7.4.1.2 Generalizing the EoS: parametric density models;349
11.4.1.3;7.4.1.3 Curvature quintessence;350
11.4.2;7.4.2 Methods to constrain models;351
11.4.3;7.4.3 Data samples for constraining models: large scale structure;356
11.4.4;7.4.4 Testing cosmological models: an example;357
11.5;7.5 Conclusions;365
12;Chapter 8: From the early to the present universe;367
12.1;8.1 Quantum cosmology;367
12.1.1;8.1.1 Noether symmetries in quantum cosmology;370
12.1.2;8.1.2 Scalar-tensor quantum cosmology;372
12.1.3;8.1.3 The quantum cosmology of fourth order gravity;375
12.1.3.1;8.1.3.1 The case s=0;377
12.1.3.2;8.1.3.2 The case s=-2;378
12.1.4;8.1.4 Quantum cosmology with gravity of order higher than fourth;379
12.2;8.2 Inflation in ETGs;382
12.2.1;8.2.1 Scalar-tensor gravity: extended and hyperextended inflation;382
12.2.2;8.2.2 Inflation with quadratic corrections;385
12.3;8.3 Cosmological perturbations;386
12.3.1;8.3.1 Scalar perturbations;387
12.3.2;8.3.2 Gravitational wave perturbations;396
12.4;8.4 Constraints on ETGs from primordial nucleosynthesis;401
12.5;8.5 The present universe: f(R) gravity as an alternative to dark energy;404
12.5.1;8.5.1 Background universe;405
12.5.2;8.5.2 Perturbations;408
12.6;8.6 Conclusions;409
13;Appendix A Physical constants and astrophysical and cosmological parameters;411
13.1;A.1 Physical constants;411
13.2;A.2 Conversion factors;412
13.3;A.3 Astrophysical and cosmological quantities;412
13.4;A.4 Planck scale quantities;413
14;Appendix B:The Noether symmetry approach to f(R) gravity;414
14.1;B.1 The field equations and the Noether vector for spherically symmetric f(R) gravity;414
14.2;B.2 Noether symmetries in metric f(R) cosmology;415
15;Appendix C:The weak-field limit of metric f(R) gravity;418
16;References;420
17;Index;444