| Extended gravity: a primer | p. 1 |
| Why extending gravity? | p. 1 |
| Cosmological and astrophysical motivation | p. 3 |
| Mathematical motivation | p. 6 |
| Quantum gravity motivation | p. 7 |
| Emergent gravity and thermodynamics of spacetime | p. 12 |
| What a good theory of gravity should do: General Relativity and its extensions | p. 13 |
| Quantum field theory in curved space | p. 18 |
| Mach's principle and other fundamental issues | p. 23 |
| Higher order corrections to Einstein's theory | p. 25 |
| Minimal and non-minimal coupling and the Equivalence Principle | p. 27 |
| Mach's principle and the variation of G | p. 32 |
| Extended gravity from higher dimensions and area metric approach | p. 35 |
| Conclusions | p. 40 |
| Mathematical tools | p. 41 |
| Conformal transformations | p. 41 |
| Variational principles in General Relativity | p. 47 |
| Geodesies | p. 47 |
| Field equations | p. 49 |
| Adding torsion | p. 51 |
| Noether symmetries | p. 54 |
| Conclusions | p. 57 |
| The landscape beyond Einstein gravity | p. 59 |
| The variational principle and the field equations of Brans-Dicke gravity | p. 59 |
| The variational principle and the field equations of metric f(R) gravity | p. 62 |
| f(R) = R + R2 theory | p. 62 |
| Metric f(R) gravity in general | p. 64 |
| A more general class of ETGs | p. 67 |
| The Palatini formalism | p. 67 |
| The Palatini approach and the conformal structure of the theory | p. 68 |
| Problems with the Palatini formalism | p. 73 |
| Equivalence between f(R) and scalar-tensor gravity | p. 77 |
| Equivalence between scalar-tensor and metric f(R) gravity | p. 77 |
| Equivalence between scalar-tensor and Palatini f(R) gravity | p. 78 |
| Conformal transformations applied to extended gravity | p. 79 |
| Brans-Dicke gravity | p. 79 |
| Scalar-tensor theories | p. 83 |
| Mixed f(R)/scalar-tensor gravity | p. 85 |
| The issue of the conformal frame | p. 86 |
| The initial value problem | p. 90 |
| The Cauchy problem of scalar-tensor gravity | p. 92 |
| The initial value problem of f(R) gravity in the ADM formulation | p. 97 |
| The Gaussian normal coordinates approach | p. 98 |
| Conclusions | p. 106 |
| Spherical symmetry | p. 107 |
| Spherically symmetric solutions of GR and metric f(R) gravity | p. 107 |
| Spherical symmetry | p. 108 |
| The Ricci scalar in spherical symmetry | p. 109 |
| Spherical symmetry in metric f(R) gravity | p. 110 |
| Solutions with constant Ricci scalar | p. 112 |
| Solutions with R = R(r) | p. 115 |
| Perturbations | p. 117 |
| Spherical symmetry in f(R) gravity and the Noether approach | p. 119 |
| Noether solutions of spherically symmetric f(R) gravity | p. 124 |
| Non-asymptotically flat and non-static spherical solutions of metric f(R) gravity | p. 128 |
| Spherical symmetry in scalar-tensor gravity | p. 134 |
| Static solutions of Brans-Dicke theory | p. 134 |
| Dynamical and asymptotically FLRW solutions | p. 136 |
| Collapse to black holes in scalar-tensor theory | p. 137 |
| The Jebsen-Birkhoff theorem | p. 139 |
| The Jebsen-Birkhoff theorem of GR | p. 139 |
| The non-vacuum case | p. 140 |
| The vacuum case | p. 142 |
| The Jebsen-Birkhoff theorem in scalar-tensor gravity | p. 143 |
| The trivial case = constant | p. 144 |
| Static non-constant Brans-Dicke-like field | p. 145 |
| The Jebsen-Birkhoff theorem in Einstein frame scalar-tensor gravity | p. 146 |
| Hawking's theorem and Jebsen-Birkhoff in Brans-Dicke gravity | p. 148 |
| The Jebsen-Birkhoff theorem in f(R) gravity | p. 150 |
| Black hole thermodynamics in extended gravity | p. 151 |
| Scalar-tensor gravity | p. 153 |
| Metric modified gravity | p. 155 |
| Palatini modified gravity | p. 156 |
| Dilaton gravity | p. 157 |
| From spherical to axial symmetry: an application to f(R) gravity | p. 158 |
| Conclusions | p. 163 |
| Weak-field limit | p. 165 |
| The weak-field limit of extended gravity | p. 165 |
| The Newtonian and post-Newtonian approximations: general remarks | p. 167 |
| The Newtonian and post-Newtonian limits of metric f(R) gravity with spherical symmetry | p. 171 |
| Comparison with the standard formalism and the chameleon effect | p. 180 |
| The Post-Minkowskian approximation | p. 185 |
| The energy-momentum pseudotensor in f(R) gravity and gravitational radiation | p. 187 |
| Gravitational waves | p. 190 |
| Gravitational waves in scalar-tensor gravity | p. 192 |
| Gravitational waves in higher order gravity | p. 195 |
| Conclusions | p. 208 |
| Qualitative analysis and exact solutions in cosmology | p. 209 |
| The Ehlers-Geren-Sachs theorem | p. 209 |
| The phase space of FLRW cosmology in scalar-tensor and f(R) gravity | p. 210 |
| The dynamical system | p. 212 |
| Analytical solutions of Brans-Dicke and scalar-tensor cosmology | p. 220 |
| Analytical solutions of Brans-Dicke cosmology | p. 221 |
| Exact scalar-tensor cosmologies | p. 232 |
| Analytical solutions of metric f(R) cosmology by the Noether approach | p. 233 |
| Point-like f(R) cosmology | p. 233 |
| Noether symmetries in metric f(R) cosmology | p. 235 |
| Exact cosmologies | p. 238 |
| c1, c2 ≠0 | p. 243 |
| Analytical cosmological solutions of f(R, R, …, k R) gravity | p. 253 |
| Higher order point-like Lagrangians for cosmology | p. 253 |
| The Noether symmetry approach for higher order gravities | p. 256 |
| Conclusions | p. 260 |
| Cosmology | p. 261 |
| Big Bang, inflationary, and late-time cosmology in GR | p. 262 |
| The standard Big Bang model | p. 263 |
| Inflation in the early universe | p. 263 |
| The present-day acceleration | p. 265 |
| Using cosmography to map the structure of the universe | p. 273 |
| The cosmographic apparatus | p. 274 |
| Large scale structure and galaxy clusters | p. 304 |
| The weak-field limit of f(R) gravity and galaxy clusters | p. 305 |
| Extended systems | p. 306 |
| The cluster mass profiles | p. 307 |
| The galaxy clusters sample | p. 310 |
| The gas density model | p. 310 |
| Temperature profiles | p. 311 |
| The galaxy distribution model | p. 311 |
| Uncertainties in the mass profiles | p. 314 |
| Fitting the mass profiles | p. 314 |
| Results | p. 316 |
| Outlooks | p. 321 |
| Testing cosmological models with observations | p. 326 |
| Toward a new cosmological standard model | p. 327 |
| Methods to constrain models | p. 331 |
| Data samples for constraining models: large scale structure | p. 336 |
| Testing cosmological models: an example | p. 337 |
| Conclusions | p. 345 |
| From the early to the present universe | p. 347 |
| Quantum cosmology | p. 347 |
| Noether symmetries in quantum cosmology | p. 350 |
| Scalar-tensor quantum cosmology | p. 352 |
| The quantum cosmology of fourth order gravity | p. 355 |
| Quantum cosmology with gravity of order higher than fourth | p. 359 |
| Inflation in ETGs | p. 362 |
| Scalar-tensor gravity: extended and hyperextended inflation | p. 362 |
| Inflation with quadratic corrections | p. 365 |
| Cosmological perturbations | p. 366 |
| Scalar perturbations | p. 367 |
| Gravitational wave perturbations | p. 376 |
| Constraints on ETGs from primordial nucleosynthesis | p. 381 |
| The present universe: f(R) gravity as an alternative to dark energy | p. 384 |
| Background universe | p. 385 |
| Perturbations | p. 388 |
| Conclusions | p. 389 |
| Physical constants and astrophysical and cosmological parameters | p. 391 |
| Physical constants | p. 391 |
| Conversion factors | p. 392 |
| Astrophysical and cosmological.quantities | p. 392 |
| Planck scale quantities | p. 393 |
| The Noether symmetry approach to f(R) gravity | p. 395 |
| The field equations and the Noether vector for spherically symmetric f(R) gravity | p. 395 |
| Noether symmetries in metric f(R) cosmology | p. 396 |
| The weak-field limit of metric f(R) gravity | p. 399 |
| References | p. 401 |
| Index | p. 425 |
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