The Parameterized Post-Friedmann Framework for Theories of Modified Gravity: Concepts, Formalism and Examples

Abstract

A unified framework for theories of modified gravity will be an essential tool for interpreting the forthcoming deluge of cosmological data. We present such a formalism, the Parameterized Post-Friedmann framework (PPF), which parameterizes the cosmological perturbation theory of a wide variety of modified gravity models. PPF is able to handle spin-0 degrees of freedom from new scalar, vector and tensor fields, meaning that it is not restricted to simple models based solely on cosmological scalar fields. A direct correspondence is maintained between the parameterization and the underlying space of theories, which allows us to build up a `dictionary' of modified gravity theories and their PPF correspondences. In this paper we describe the construction of the parameterization and demonstrate its use through a number of worked examples relevant to the current literature. We indicate how the formalism will be implemented numerically, so that the dictionary of modified gravity can be pitted against forthcoming observations.

24 pages, updated to match version published in PRD. Discussion of section 4 extended. Suggestions for the busy reader are given at the end of section 1

References in corpus (86)

• Efficient Computation of CMB anisotropies in closed FRW models
• f(R) Theories Of Gravity
• Modified Gravity and Cosmology
• f(R) theories
• Cosmological Imprint of an Energy Component with General Equation of State
• 4D Gravity on a Brane in 5D Minkowski Space
• Quantum Gravity at a Lifshitz Point
• k-Inflation
• The galileon as a local modification of gravity
• Essentials of k-essence
• Coupled Quintessence
• Chameleon Fields: Awaiting Surprises for Tests of Gravity in Space
• Chameleon Cosmology
• The Confrontation between General Relativity and Experiment
• Relativistic gravitation theory for the MOND paradigm
• Generalized G-inflation: Inflation with the most general second-order field equations
• From k-essence to generalised Galileons
• Theoretical Aspects of Massive Gravity
• Brane-World Gravity
• Gauss-Bonnet dark energy
• Symmetron Fields: Screening Long-Range Forces Through Local Symmetry Restoration
• Imperfect Dark Energy from Kinetic Gravity Braiding
• A healthy extension of Horava gravity
• Fundamental Theoretical Bias in Gravitational Wave Astrophysics and the Parameterized Post-Einsteinian Framework
• General second order scalar-tensor theory, self tuning, and the Fab Four
• Structure Formation with Generalized Dark Matter
• A Parameterized Post-Friedmann Framework for Modified Gravity
• Scalar-tensor gravity in an accelerating universe
• Einstein-Aether Waves
• Eddington's theory of gravity and its progeny
• Massive Cosmologies
• Strong coupling in Horava gravity
• The pattern of growth in viable f(R) cosmologies
• Modifying gravity with the Aether: an alternative to Dark Matter
• Distinguishing Modified Gravity from Dark Energy
• On the Extra Mode and Inconsistency of Horava Gravity
• Symmetron Cosmology
• Cosmology and Astrophysical Constraints of Gauss-Bonnet Dark Energy
• Gravitational Wave Tests of General Relativity with the Parameterized Post-Einsteinian Framework
• Horava-Lifshitz gravity: a status report
• Effective gravitational couplings for cosmological perturbations in the most general scalar-tensor theories with second-order field equations
• The Dilaton and Modified Gravity
• More on ghosts in DGP model
• A Unified Description of Screened Modified Gravity
• Current constraints on the cosmic growth history
• DGP Specteroscopy
• Coupled and Extended Quintessence: theoretical differences and structure formation
• Self-tuning and the derivation of the Fab Four
• Gauge-invariant gravitational perturbations of maximally symmetric spacetimes
• How to optimally parametrize deviations from General Relativity in the evolution of cosmological perturbations
• Are there ghosts in the self-accelerating brane universe?
• Challenges to the DGP Model from Horizon-Scale Growth and Geometry
• Inflation and primordial non-Gaussianities of "generalized Galileons"
• Bouncing Eddington-inspired Born-Infeld cosmologies: an alternative to Inflation ?
• Examining the evidence for dynamical dark energy
• Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum: I. Properties of a Quasi-Kerr Spacetime
• Generalizing TeVeS Cosmology
• Testing alternative theories of gravity using the Sun
• Cosmological Perturbations in Hořava-Lifshitz Gravity
• Constraints on Gauss-Bonnet Gravity in Dark Energy Cosmologies
• Cosmological Tests of General Relativity with Future Tomographic Surveys
• Effective action approach to cosmological perturbations in dark energy and modified gravity
• de Sitter Galileon
• Towards a fully consistent parameterization of modified gravity
• Exact self-accelerating cosmologies in the ghost-free bigravity and massive gravity
• Measuring the metric: a parametrized post-Friedmanian approach to the cosmic dark energy problem
• Lessons from the decoupling limit of Horava gravity
• A tensor instability in the Eddington inspired Born-Infeld Theory of Gravity
• Cosmological tests of General Relativity: a principal component analysis
• Testing General Relativity at Cosmological Scales: Implementation and Parameter Correlations
• Eddington-Born-Infeld gravity and the large scale structure of the Universe
• On the growth of structure in theories with a dynamical preferred frame
• TeVeS Cosmology : Covariant formalism for the background evolution and linear perturbation theory
• Eddington-Born-Infeld action for dark energy and dark matter
• Radial oscillations and stability of compact stars in Eddington-inspired Born-Infeld gravity
• Confronting General Relativity with Further Cosmological Data
• Consistent cosmological modifications to the Einstein equations
• A Class of Effective Field Theory Models of Cosmic Acceleration
• Conserved cosmological perturbation in Galileon models
• Observational Constraints on Kinetic Gravity Braiding from the Integrated Sachs-Wolfe Effect
• Linear Growth of Structure in the Symmetron Model
• Ambiguous Tests of General Relativity on Cosmological Scales
• Structure Formation in Modified Gravity Scenarios
• Parametrised modified gravity and the CMB Bispectrum
• Testing model independent modified gravity with future large scale surveys
• Cosmological post-Newtonian approximation compared with perturbation theory

Cited by in corpus (131)

• The Confrontation between General Relativity and Experiment
• Testing General Relativity with Present and Future Astrophysical Observations
• Beyond the Cosmological Standard Model
• Cosmology and Fundamental Physics with the Euclid Satellite
• Cosmology and fundamental physics with the Euclid satellite
• Cosmological Tests of Modified Gravity
• Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian
• Beyond $\Lambda$CDM: Problems, solutions, and the road ahead
• The Confrontation between General Relativity and Experiment
• The Effective Field Theory of Dark Energy
• Holographic Dark Energy
• Dark Energy vs. Modified Gravity
• Testing General Relativity in Cosmology
• Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity
• Essential Building Blocks of Dark Energy
• Dark Energy Survey Year 1 Results: Constraints on Extended Cosmological Models from Galaxy Clustering and Weak Lensing
• KiDS-450 + 2dFLenS: Cosmological parameter constraints from weak gravitational lensing tomography and overlapping redshift-space galaxy clustering
• Dark Energy and Modified Gravity in Degenerate Higher-Order Scalar-Tensor (DHOST) theories: a review
• The Clustering of Galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS): measuring growth rate and geometry with anisotropic clustering
• Effective Field Theory of Cosmic Acceleration: an implementation in CAMB
• A unifying description of dark energy
• Black Hole Based Tests of General Relativity
• Effective Field Theory of Cosmic Acceleration: constraining dark energy with CMB data
• Effective Description of Higher-Order Scalar-Tensor Theories
• Cosmological Tests of Gravity
• Testing General Relativity with 21 cm intensity mapping
• Models of coupled dark matter to dark energy
• Observables and unobservables in dark energy cosmologies
• A practical approach to cosmological perturbations in modified gravity
• Effective Field Theory of Cosmological Perturbations
• Limits of Quasi-Static Approximation in Modified-Gravity Cosmologies
• The Observational Future of Cosmological Scalar-Tensor Theories
• Nonlocal Models of Cosmic Acceleration
• A Simplified Approach to General Scalar-Tensor Theories
• Effective Theory of Interacting Dark Energy
• The Parameterized Post-Friedmannian Framework for Interacting Dark Energy Theories
• What can Cosmology tell us about Gravity? Constraining Horndeski with Sigma and Mu
• Constraining chameleon models with cosmology
• The Novel Probes Project -- Tests of Gravity on Astrophysical Scales
• The parameter space in Galileon gravity models
• Fixing extensions to General Relativity in the non-linear regime
• Effective Theory of Dark Energy at Redshift Survey Scales
• Strong gravitational lensing by an electrically charged black hole in Eddington-inspired Born-Infeld gravity
• Phenomenology of dark energy: exploring the space of theories with future redshift surveys
• Nonlinear Effective Theory of Dark Energy
• Void Lensing as a Test of Gravity
• A general theory of linear cosmological perturbations: scalar-tensor and vector-tensor theories
• Nonlinear/non-iterative treatment of EFT-motivated gravity
• Relativistic effects in galaxy clustering in a parametrized post-Friedmann universe
• Properties of an electrically charged black hole in Eddington-inspired Born-Infeld gravity
• Nonlocal Gravity and Structure in the Universe
• Towards the nonlinear regime in extensions to GR: assessing possible options
• Designing Horndeski and the effective fluid approach
• Probing Dark Energy through Scale Dependence
• A general theory of linear cosmological perturbations: stability conditions, the quasistatic limit and dynamics
• New Gravitational Scales in Cosmological Surveys
• An extensive investigation of the Generalised Dark Matter model
• Regularized cosmological power spectrum and correlation function in modified gravity models
• Principal Component Analysis of Modified Gravity using Weak Lensing and Peculiar Velocity Measurements
• Growth of Cosmic Structure
• Weakening Gravity on Redshift-Survey Scales with Kinetic Matter Mixing
• $k$-evolution: a relativistic N-body code for clustering dark energy
• Reconstructing the gravitational field of the local universe
• Parameterized Post-Newtonian Cosmology
• Effective Field Theory of Dark Energy: a Dynamical Analysis
• A parametrisation of modified gravity on nonlinear cosmological scales
• Parametrized modified gravity constraints after Planck
• A Fast Route to Modified Gravitational Growth
• Probing non-standard gravity with the growth index: a background independent analysis
• Parameterizing dark sector perturbations via equations of state
• Einstein-aether theory with a Maxwell field: General formalism
• Observational discrimination of Eddington-inspired Born-Infeld gravity from general relativity
• Semi-dynamical perturbations of unified dark energy
• Exploring degeneracies in modified gravity with weak lensing
• Stochastic dark energy from inflationary quantum fluctuations
• Cosmological Implications of the Effective Field Theory of Cosmic Acceleration
• The Parametrized Post-Newtonian-Vainshteinian Formalism
• A Rosetta Stone for Parameterized Tests of Gravity
• Probing Modified Gravity Theories with ISW and CMB Lensing
• f(R) as a dark energy fluid
• Revival of the Deser-Woodard nonlocal gravity model: Comparison of the original nonlocal form and a localized formulation
• Cosmological perturbation theory in Generalized Einstein-Aether models
• Computing model independent perturbations in dark energy and modified gravity
• Testing gravity theories using tensor perturbations
• $\text{Hi-COLA}$: Fast, approximate simulations of structure formation in Horndeski gravity
• Gravitational Redshift Constraints on the Effective Theory of Interacting Dark Energy
• Do cosmological data rule out $f(\mathcal{R})$ with $w\ne -1$?
• Forecasted constraints on modified gravity from Sunyaev Zel'dovich tomography
• 50 Years of Horndeski Gravity: Past, Present and Future
• Parametrizing theories of gravity on large and small scales in cosmology
• Observable physical modes of modified gravity
• A null test of the equivalence principle using relativistic effects in galaxy surveys
• The traces of anisotropic dark energy in light of Planck
• A general theory of linear cosmological perturbations: bimetric theories
• How can we tell whether dark energy is composed by multiple fields?
• A null test to probe the scale-dependence of the growth of structure as a test of General Relativity
• The effective field theory of K-mouflage
• Future CMB ISW-Lensing bispectrum constraints on modified gravity in the Parameterized Post-Friedmann formalism
• Combined constraints on deviations of dark energy from an ideal fluid from Euclid and Planck
• Parametrizations for tests of gravity
• Growth of perturbations in nonlocal gravity with non-$\Lambda$CDM background
• Cosmologically viable generalized Einstein-Aether theories
• Constraining dark sector perturbations I: cosmic shear and CMB lensing
• Post-Newtonian $\gamma$-like parameters and the gravitational slip in scalar-tensor and $f(R)$ theories
• Constraining dark sector perturbations II: ISW and CMB lensing tomography
• Testing General Relativity with the Doppler magnification effect
• Magnetized relativistic stellar models in Eddington-inspired Born-Infeld gravity
• The Parameterized Post-Newtonian-Vainshteinian formalism for the Galileon field
• On the phenomenology of extended Brans-Dicke Gravity
• Cosmological tests of gravity with latest observations
• Shift-symmetric Horndeski gravity in the asymptotic-safety paradigm
• Searching for dark energy in the matter-dominated era
• A first-order approach to conformal gravity
• Simple numerical implementation of general dark energy models
• A covariant approach to parameterised cosmological perturbations
• Cosmic Growth Signatures of Modified Gravitational Strength
• The momentum constraint equation in Parameterised Post-Newtonian Cosmology
• Fluctuating Dark Energy and the Luminosity Distance
• Static topological black hole with nonminimal derivative coupling and nonlinear electromagnetic field of Born-Infeld type
• Post-Newtonian Gravity in Cosmology
• K-mouflage at high $k$: extending the reach of $\text{Hi-COLA}$
• Minimal parameterizations for modified gravity
• Analytical $poly\Lambda$CDM dynamics
• On phenomenological parametrizations for the luminosity distance of gravitational waves
• The Possibility of a Non-Lagrangian Theory of Gravity
• PPN expansion and FRW scalar perturbations in n-DBI gravity
• Does General Relativity Highlight Necessary Connections in Nature?
• DUCA: Dynamic Universe Cosmological Analysis. II. The impact of clustering dark energy on the halo mass function
• Extended datasamples under the lens of Brane World Theory
• Cosmological perturbations in generalised dark Lagrangians
• Matter growth in extended $\Lambda$CDM cosmology