Fourth order gravity: equations, history, and applications to cosmology

Abstract

The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e., the details how within fourth order gravity with L= R + R^2 the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented.

51 pages, LaTeX, no figure, lecture for 42nd Karpacz Winter School 6.-11.2.06, references 99-109 and related comments are added

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Cited by in corpus (116)

• f(R) Theories Of Gravity
• Modified Gravity and Cosmology
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• Testing General Relativity with Present and Future Astrophysical Observations
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• Constraining f(R) Gravity as a Scalar Tensor Theory
• The Newtonian Limit of F(R) gravity
• Modified theories of Gravity: Why, How and What?
• The Lagrangian description of perfect fluids and modified gravity with an extra force
• The Lovelock Black Holes
• Junction Conditions in f(R) Theories of Gravity
• The Influence of Modification of Gravity on the Dynamics of Radiating Spherical Fluids
• Massive 3D Supergravity
• Notes on $f(T)$ Theories
• Some remarks on the dynamical systems approach to fourth order gravity
• Beyond Concordance Cosmology with Magnification of Gravitational-Wave Standard Sirens
• Static Solutions for 4th order gravity
• The Cauchy problem of f(R) gravity
• Higher Derivative Gravity and Conformal Gravity From Bimetric and Partially Massless Bimetric Theory
• Black hole entropy in scalar-tensor and f(R) gravity: an overview
• Covariant equations of motion for test bodies in gravitational theories with general nonminimal coupling
• Gödel-type universes in f(R) gravity
• The Jebsen-Birkhoff theorem in alternative gravity
• Motion of test bodies in theories with nonminimal coupling
• Reconstructing f(R) modified gravity from ordinary and entropy-corrected versions of the holographic and new agegraphic dark energy models
• 6+1 lessons from f(R) gravity
• On dynamical systems approaches and methods in $f(R)$ cosmology
• Conservation laws in gravitational theories with general nonminimal coupling
• f(R) gravity: successes and challenges
• On the absence of the usual weak-field limit, and the impossibility of embedding some known solutions for isolated masses in cosmologies with f(R) dark energy
• Effective field description of the Anton-Schmidt cosmic fluid
• On cosmological-type solutions in multi-dimensional model with Gauss-Bonnet term
• Conservation laws in gravity: A unified framework
• Self-interaction in the Bopp-Podolsky electrodynamics: Can the observable mass of a charged particle depend on its acceleration?
• Multi-Scalar-Tensor Equivalents for Modified Gravitational Actions
• Gauge invariant formulation of gravitational waves in metric $f(R)$ gravity
• On The Existence Of Anisotropic Cosmological Models In Higher-Order Theories Of Gravity
• The generalized second law of gravitational thermodynamics on the apparent horizon in f(R)-gravity
• Massive Gravity with Mass Term in Three Dimensions
• Perihelion precession for modified Newtonian gravity
• Scalar field mass in generalized gravity
• Inflation with a Weyl term, or ghosts at work
• Gravitoelectromagnetism and other decompositions of the Riemann tensor
• Some exact solutions with torsion in 5-D Einstein-Gauss-Bonnet gravity
• Gauss-Bonnet lagrangian G ln G and cosmological exact solutions
• Cosmic Acceleration From Matter-Curvature Coupling
• Position and frequency shifts induced by massive modes of the gravitational wave background in alternative gravity
• Neutron stars in generalized f(R) gravity
• A Kinematical Approach to Conformal Cosmology
• Universal thermodynamics in different gravity theories: Modified entropy on the horizons
• Generality of Starobinsky and Higgs inflation in the Jordan frame
• An analysis of the Sultana-Dyer cosmological black hole solution of the Einstein equations
• Scalar modes in extended hybrid metric-Palatini gravity: weak field phenomenology
• Naked Singularity Formation In f(R) Gravity
• No-go theorem for inflation in an extended Ricci-inverse gravity model
• Spherical Structures in Conformal Gravity and its Scalar-Tensor Extension
• Higher Dimensional Inhomogeneous Perfect Fluid Collapse in \emph{f(R)} Gravity
• Dynamical Systems analysis of an interacting dark energy model in the Brane Scenario
• How a projectively flat geometry regulates $F(R)$-gravity theory?
• Testing generalized logotropic models with cosmic growth
• Kerr Metric, Geodesic Motion, and Flyby Anomaly in Fourth-Order Conformal Gravity
• Weak field limit and gravitational waves in higher-order gravity
• The tetralogy of Birkhoff theorems
• Clifton's spherical solution in f(R) vacuo harbours a naked singularity
• Spatial Curvature in $f(R)$ Gravity
• Multipolar test body equations of motion in generalized gravity theories
• Implications of non-analytical f(R) gravity at Solar-System scales
• Polytropic and Chaplygin f(R)-gravity models
• Tomography from the Next Generation of Cosmic Shear Experiments for Viable f(R) Models
• New rotating AdS/dS black holes in $\mathrm{f}(\mathrm{R})$ gravity
• Weakly Charged Compact Stars in $f\left(R\right)$ gravity
• Dynamical apparent horizons in inhomogeneous Brans-Dicke universes
• An anisotropic bouncing universe in non-local gravity
• Gravitons in Flatland
• On stability of the Kasner solution in quadratic gravity
• An updated analysis of two classes of f(R) theories of gravity
• Implications of the Holst term in a $f(R)$ theory with torsion
• Universal thermodynamics in different gravity theories: Conditions for generalized second law of thermodynamics and thermodynamical equilibrium on the horizons
• Characteristic Formulation for Metric $f(R)$ Gravity
• Power-law solutions and accelerated expansion in scalar-tensor theories
• Spontaneous breaking of conformal invariance, solitons and gravitational waves in theories of conformally invariant gravitation
• Dynamic properties of cyclic cosmologies
• Higher-order gravity in higher dimensions: Geometrical origins of four-dimensional cosmology?
• Spherical Non-Abelian Solutions in Conformal Gravity
• Perihelion advance for orbits with large eccentricities in the Schwarzschild black hole
• Fundamental Physics and Cosmology with TianQin
• Modified Bekenstein-Hawking system in $f(R)$ gravity
• Isotropic universe with almost scale-invariant fourth-order gravity
• Bianchi $VI{I}_A$ solutions of quadratic gravity
• Pseudo generalized Ricci-recurrent spacetimes with certain applications to modified gravity
• Interacting dark energy model in the brane scenario: A Dynamical System Analysis
• Critical Cosmology in Higher Order Gravity
• Investigation of Bouncing Universe and Phantom Crossing in Modified Gravity Coupled with Weyl Tensor and its Reconstruction
• Interacting Induced Dark Energy Model
• A Study of Universal Thermodynamics in Massive Gravity: Modified Entropy on the Horizons
• An overview of f(R) theories
• Vacuum $f(R)$ thick brane solution with a Gaussian warp function
• Bianchi I solutions of effective quadratic gravity
• Horizons and singularity in Clifton's spherical solution of f(R) vacuum
• Quadratic Curvature Gravity with Second Order Trace and Massive Gravity Models in Three Dimensions
• A study on causality in $f(R,\varphi ,X)$ theory
• Homogeneous Solutions of Quadratic Gravity
• Double gauge invariance and covariantly-constant vector fields in Weyl geometry
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• Hubble Diagrams in the Jordan and Einstein Frames
• An exact solution of the higher-order gravity in standard radiation-dominated era
• Third Order Lagrangians, Weyl Invariants & Classical Trace Anomaly in Six Dimensions
• Notes on f(R) Theories of Gravity
• An ostentatious model of cosmological scalar-tensor theory
• Static cosmological solutions in quadratic gravity
• Non-analytical power law correction to the Einstein-Hilbert action: gravitational wave propagation
• Is the use of Christoffel connection in gravity theories conceptually correct?
• Cosmological Asymptotics in Higher-Order Gravity Theories
• Tensor Potential Description of Matter and Space, III. Gravitation
• The Bach equations in Spin-Coefficient form