Beyond Einstein gravity : a survey of gravitational theories for cosmology and astrophysics

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.

Author Notes

Valerio Faraoni received a PhD in Astrophysics at the International School for Advanced Studies in Trieste, Italy. He is known for his research on alternative theories of gravity, cosmology, and gravitational waves. He is currently Associate Professor at Bishop's University in Sherbrooke, Canada. Salvatore Capozziello graduated in Physics at University of Rome "La Sapienza" and received a PhD in Theoretical Physics at University of Naples "Federico II", Italy. He is the author of almost 300 hundred papers and monographs including theory of gravity, gravitational waves, theoretical and observational cosmology. He is currently Associate Professor at the University of Naples "Federico II", Italy.

l1 Extended gravity: a primer	p. 1
1.1 Why extending gravity?	p. 1
1.2 Cosmological and astrophysical motivation	p. 3
1.3 Mathematical motivation	p. 6
1.4 Quantum gravity motivation	p. 7
1.4.1 Emergent gravity and thermodynamics of spacetime	p. 12
1.5 What a good theory of gravity should do: General Relativity and its extensions	p. 13
1.6 Quantum field theory in curved space	p. 18
1.7 Mach's principle and other fundamental issues	p. 23
1.7.1 Higher order corrections to Einstein's theory	p. 25
1.7.2 Minimal and non-minimal coupling and the Equivalence Principle	p. 27
1.7.3 Mach's principle and the variation of G	p. 32
1.8 Extended gravity from higher dimensions and area metric approach	p. 35
1.9 Conclusions	p. 40
2 Mathematical tools	p. 41
2.1 Conformal transformations	p. 41
2.2 Variational principles in General Relativity	p. 47
2.2.1 Geodesies	p. 47
2.2.2 Field equations	p. 49
2.3 Adding torsion	p. 51
2.4 Noether symmetries	p. 54
2.5 Conclusions	p. 57
3 The landscape beyond Einstein gravity	p. 59
3.1 The variational principle and the field equations of Brans-Dicke gravity	p. 59
3.2 The variational principle and the field equations of metric f(R) gravity	p. 62
3.2.1 f(R) = R + ¿R 2 theory	p. 62
3.2.2 Metric f(R) gravity in general	p. 64
3.3 A more general class of ETGs	p. 67
3.4 The Palatini formalism	p. 67
3.4.1 The Palatini approach and the conformal structure of the theory	p. 68
3.4.2 Problems with the Palatini formalism	p. 73
3.5 Equivalence between f(R) and scalar-tensor gravity	p. 77
3.5.1 Equivalence between scalar-tensor and metric f(R) gravity	p. 77
3.5.2 Equivalence between scalar-tensor and Palatini f(R) gravity	p. 78
3.6 Conformal transformations applied to extended gravity	p. 79
3.6.1 Brans-Dicke gravity	p. 79
3.6.2 Scalar-tensor theories	p. 83
3.6.3 Mixed f(R)/scalar-tensor gravity	p. 85
3.6.4 The issue of the conformal frame	p. 86
3.7 The initial value problem	p. 90
3.7.1 The Cauchy problem of scalar-tensor gravity	p. 92
3.7.2 The initial value problem of f(R) gravity in the ADM formulation	p. 97
3.7.3 The Gaussian normal coordinates approach	p. 98
3.8 Conclusions	p. 106
4 Spherical symmetry	p. 107
4.1 Spherically symmetric solutions of GR and metric f(R) gravity	p. 107
4.1.1 Spherical symmetry	p. 108
4.1.2 The Ricci scalar in spherical symmetry	p. 109
4.1.3 Spherical symmetry in metric f(R) gravity	p. 110
4.1.4 Solutions with constant Ricci scalar	p. 112
4.1.5 Solutions with R = R(r)	p. 115
4.1.6 Perturbations	p. 117
4.1.7 Spherical symmetry in f(R) gravity and the Noether approach	p. 119
4.1.8 Noether solutions of spherically symmetric f(R) gravity	p. 124
4.1.9 Non-asymptotically flat and non-static spherical solutions of metric f(R) gravity	p. 128
4.2 Spherical symmetry in scalar-tensor gravity	p. 134
4.2.1 Static solutions of Brans-Dicke theory	p. 134
4.2.2 Dynamical and asymptotically FLRW solutions	p. 136
4.2.3 Collapse to black holes in scalar-tensor theory	p. 137
4.3 The Jebsen-Birkhoff theorem	p. 139
4.3.1 The Jebsen-Birkhoff theorem of GR	p. 139
4.3.2 The non-vacuum case	p. 140
4.3.3 The vacuum case	p. 142
4.3.4 The Jebsen-Birkhoff theorem in scalar-tensor gravity	p. 143
4.3.5 The trivial case ¿ = constant	p. 144
4.3.6 Static non-constant Brans-Dicke-like field	p. 145
4.3.7 The Jebsen-Birkhoff theorem in Einstein frame scalar-tensor gravity	p. 146
4.3.8 Hawking's theorem and Jebsen-Birkhoff in Brans-Dicke gravity	p. 148
4.3.9 The Jebsen-Birkhoff theorem in f(R) gravity	p. 150
4.4 Black hole thermodynamics in extended gravity	p. 151
4.4.1 Scalar-tensor gravity	p. 153
4.4.2 Metric modified gravity	p. 155
4.4.3 Palatini modified gravity	p. 156
4.4.4 Dilaton gravity	p. 157
4.5 From spherical to axial symmetry: an application to f(R) gravity	p. 158
4.6 Conclusions	p. 163
5 Weak-field limit	p. 165
5.1 The weak-field limit of extended gravity	p. 165
5.2 The Newtonian and post-Newtonian approximations: general remarks	p. 167
5.2.1 The Newtonian and post-Newtonian limits of metric f(R) gravity with spherical symmetry	p. 171
5.2.2 Comparison with the standard formalism and the chameleon effect	p. 180
5.3 The Post-Minkowskian approximation	p. 185
5.3.1 The energy-momentum pseudotensor in f(R) gravity and gravitational radiation	p. 187
5.4 Gravitational waves	p. 190
5.4.1 Gravitational waves in scalar-tensor gravity	p. 192
5.4.2 Gravitational waves in higher order gravity	p. 195
5.5 Conclusions	p. 208
6 Qualitative analysis and exact solutions in cosmology	p. 209
6.1 The Ehlers-Geren-Sachs theorem	p. 209
6.2 The phase space of FLRW cosmology in scalar-tensor and f(R) gravity	p. 210
6.2.1 The dynamical system	p. 212
6.3 Analytical solutions of Brans-Dicke and scalar-tensor cosmology	p. 220
6.3.1 Analytical solutions of Brans-Dicke cosmology	p. 221
6.3.2 Exact scalar-tensor cosmologies	p. 232
6.4 Analytical solutions of metric f(R) cosmology by the Noether approach	p. 233
6.4.1 Point-like f(R) cosmology	p. 233
6.4.2 Noether symmetries in metric f(R) cosmology	p. 235
6.4.3 Exact cosmologies	p. 238
6.4.4 c 1 , c 2 ≠0	p. 243
6.5 Analytical cosmological solutions of f(R, R, ..., k R) gravity	p. 253
6.5.1 Higher order point-like Lagrangians for cosmology	p. 253
6.5.2 The Noether symmetry approach for higher order gravities	p. 256
6.6 Conclusions	p. 260
7 Cosmology	p. 261
7.1 Big Bang, inflationary, and late-time cosmology in GR	p. 262
7.1.1 The standard Big Bang model	p. 263
7.1.2 Inflation in the early universe	p. 263
7.1.3 The present-day acceleration	p. 265
7.2 Using cosmography to map the structure of the universe	p. 273
7.2.1 The cosmographic apparatus	p. 274
7.3 Large scale structure and galaxy clusters	p. 304
7.3.1 The weak-field limit of f(R) gravity and galaxy clusters	p. 305
7.3.2 Extended systems	p. 306
7.3.3 The cluster mass profiles	p. 307
7.3.4 The galaxy clusters sample	p. 310
7.3.5 The gas density model	p. 310
7.3.6 Temperature profiles	p. 311
7.3.7 The galaxy distribution model	p. 311
7.3.8 Uncertainties in the mass profiles	p. 314
7.3.9 Fitting the mass profiles	p. 314
7.3.10 Results	p. 316
7.3.11 Outlooks	p. 321
7.4 Testing cosmological models with observations	p. 326
7.4.1 Toward a new cosmological standard model	p. 327
7.4.2 Methods to constrain models	p. 331
7.4.3 Data samples for constraining models: large scale structure	p. 336
7.4.4 Testing cosmological models: an example	p. 337
7.5 Conclusions	p. 345
8 From the early to the present universe	p. 347
8.1 Quantum cosmology	p. 347
8.1.1 Noether symmetries in quantum cosmology	p. 350
8.1.2 Scalar-tensor quantum cosmology	p. 352
8.1.3 The quantum cosmology of fourth order gravity	p. 355
8.1.4 Quantum cosmology with gravity of order higher than fourth	p. 359
8.2 Inflation in ETGs	p. 362
8.2.1 Scalar-tensor gravity: extended and hyperextended inflation	p. 362
8.2.2 Inflation with quadratic corrections	p. 365
8.3 Cosmological perturbations	p. 366
8.3.1 Scalar perturbations	p. 367
8.3.2 Gravitational wave perturbations	p. 376
8.4 Constraints on ETGs from primordial nucleosynthesis	p. 381
8.5 The present universe: f(R) gravity as an alternative to dark energy	p. 384
8.5.1 Background universe	p. 385
8.5.2 Perturbations	p. 388
8.6 Conclusions	p. 389
A Physical constants and astrophysical and cosmological parameters	p. 391
A.1 Physical constants	p. 391
A.2 Conversion factors	p. 392
A.3 Astrophysical and cosmological.quantities	p. 392
A.4 Planck scale quantities	p. 393
B The Noether symmetry approach to f(R) gravity	p. 395
B.1 The field equations and the Noether vector for spherically symmetric f(R) gravity	p. 395
B.2 Noether symmetries in metric f(R) cosmology	p. 396
C The weak-field limit of metric f(R) gravity	p. 399
References	p. 401
Index	p. 425

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