The Ermakov-Pinney Equation in Scalar Field Cosmologies

Abstract

It is shown that the dynamics of cosmologies sourced by a mixture of perfect fluids and self-interacting scalar fields are described by the non-linear, Ermakov-Pinney equation. The general solution of this equation can be expressed in terms of particular solutions to a related, linear differential equation. This characteristic is employed to derive exact cosmologies in the inflationary and quintessential scenarios. The relevance of the Ermakov-Pinney equation to the braneworld scenario is discussed.

16 pages, 2 figures, references added, extended discussion emphasising importance of scale factor in derivation

References in corpus (50)

• Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
• Measurements of Omega and Lambda from 42 High-Redshift Supernovae
• A Large Mass Hierarchy from a Small Extra Dimension
• An Alternative to Compactification
• The Hierarchy Problem and New Dimensions at a Millimeter
• Cosmological Imprint of an Energy Component with General Equation of State
• Cosmology and the Fate of Dilatation Symmetry
• The Ekpyrotic Universe: Colliding Branes and the Origin of the Hot Big Bang
• The High-Z Supernova Search: Measuring Cosmic Deceleration and Global Cur vature of the Universe Using Type Ia Supernovae
• Exponential potentials and cosmological scaling solutions
• MAXIMA-1: A Measurement of the Cosmic Microwave Background Anisotropy on angular scales of 10 arcminutes to 5 degrees
• Brane cosmological evolution in a bulk with cosmological constant
• Reconstructing the Inflaton Potential --- an Overview
• Further Evidence for Cosmological Evolution of the Fine Structure Constant
• Cosmological Expansion in the Presence of an Extra Dimension
• Cosmology of One Extra Dimension with Localized Gravity
• Cosmology from Maxima-1, Boomerang and COBE/DMR CMB Observations
• Superstring Cosmology
• Prospects for probing the dark energy via supernova distance measurements
• Cosmic Concordance and Quintessence
• Dynamics of Anti-de Sitter Domain Walls
• A Measurement of the Angular Power Spectrum of the CMB from l = 100 to 400
• Reconstructing the Cosmic Equation of State from Supernova distances
• Multiple Peaks in the Angular Power Spectrum of the Cosmic Microwave Background: Significance and Consequences for Cosmology
• Brane-world cosmology
• Dynamic Dilatonic Domain Walls
• A Simple Varying-alpha Cosmology
• How to determine an effective potential for a variable cosmological term
• Inflationary Models with Logarithmic Potentials
• Cosmological Expansion in the Randall-Sundrum Brane World Scenario
• Limitations in using luminosity distance to determine the equation of state of the universe
• Gravitational Lorentz Violations and Adjustment of the Cosmological Constant in Asymmetrically Warped Spacetimes
• Brane-world solutions, standard cosmology, and dark radiation
• Our World as an Expanding Shell
• Opportunities for future supernova studies of cosmic acceleration
• Is cosmology consistent?
• Covariant and locally Lorentz-invariant varying speed of light theories
• Determining the Equation of State of the Expanding Universe: Inverse Problem in Cosmology
• Living on the edge: cosmology on the boundary of anti-de Sitter space
• Scaling Solutions and reconstruction of Scalar Field Potentials
• Adiabatic Invariants and Scalar Fields in a de Sitter Space-Time
• Inflation on a single brane - exact solutions
• Non-BPS D8-branes and Dynamic Domain Walls in Massive IIA Supergravities
• Chaotic inflation with a scalar field in non-classical states
• Quantum Fields in an Expanding Universe
• Solving for the dynamics of the universe
• Quintessence in a Brane World
• Brane Quintessence
• Darboux class of cosmological fluids with time-dependent adiabatic indices
• The Classical Schrodinger's Equation

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• Cosmic Dynamics of Bose-Einstein Condensates
• Recent Applications of the Theory of Lie Systems in Ermakov Systems
• Non-oscillating power spectra in Loop Quantum Cosmology
• Decoupling of the general scalar field mode and the solution space for Bianchi type I and V cosmologies coupled to perfect fluid sources
• Dynamical Invariants for Variable Quadratic Hamiltonians
• New integrable models and analytical solutions in $f(R)$~cosmology with an ideal gas
• The Quantum Arnold Transformation and the Ermakov-Pinney equation
• Barotropic FRW cosmologies with Chiellini damping
• From Feshbach-Resonance Managed Bose-Einstein Condensates to Anisotropic Universes: Some Applications of the Ermakov-Pinney equation with Time-Dependent Nonlinearity
• Schrödinger Symmetry in Gravitational Mini-Superspaces
• Metric versus observable operator representation, higher spin models
• Evolution Operators for Linearly Polarized Two-Killing Cosmological Models
• Integrable equations with Ermakov-Pinney nonlinearities and Chiellini damping
• Multi-fluid cosmology in Einstein gravity: analytical solutions
• Inflation as an attractor in scalar cosmology
• A dynamic correspondence between Bose-Einstein condensates and Friedmann-Lemaître-Robertson-Walker and Bianchi I cosmology with a cosmological constant
• Scalar field exact solutions for non-flat FLRW cosmology: A technique from non-linear Schrödinger-type formulation
• Integral representation for scattering phase shifts via the phase-amplitude approach
• Lie Point Symmetries for Reduced Ermakov Systems
• Power-law expansion cosmology in Schrödinger-type formulation
• On the Time Dependent Oscillator and the Nonlinear Realizations of the Virasoro Group
• Ermakov-Pinney and Emden-Fowler equations: new solutions from novel Bäcklund transformations
• Phantom expansion with non-linear Schrödinger-type formulation of scalar field cosmology
• Quantum potentiality in Inhomogeneous Cosmology
• The Geometry of Isochrone Orbits: from Archimedes' parabolae to Kepler's third law
• Quantum Operations: technical or fundamental challenge?
• Integrability cases for the anharmonic oscillator equation
• Ladder operators and squeezed coherent states of a 3-dimensional generalized isotonic nonlinear oscillator
• Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order
• Applications of Elliptic and Theta Functions to Friedmann-Robertson-Lemaitre-Walker Cosmology with Cosmological Constant
• Slow-roll, acceleration, the Big Rip and WKB approximation in NLS-type formulation of scalar field cosmology
• Relativistic Ermakov-Milne-Pinney Systems and First Integrals
• A non-linear Schrodinger type formulation of FLRW scalar field cosmology
• Non-linear Schrödinger-type formulation of scalar field cosmology: two barotropic fluids and exact solutions
• A Schrodinger formulation of Bianchi I scalar field cosmology
• Schwarzian derivative, Painlevé XXV-Ermakov equation and Bäcklund transformations
• The virial relation for the Q-balls in the thermal logarithmic potential revisited analytically
• Generalized EMP and nonlinear Schrodinger-type reformulations of some scalar field cosmological models
• The Gross-Pitaevskii equation: Bäcklund transformations and admitted solutions
• On Negative Order KdV Equations
• On the $\mathcal{P}\mathcal{T}$-symmetric parametric amplifier
• Revisiting solving procedure for Ermakov-Pinney equation (with applications in the field of cosmology)
• Emergence of the Gambier equation in cosmology