skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Inflation in a closed universe

Abstract

To derive a power spectrum for energy density inhomogeneities in a closed universe, we investigate a spatially-closed inflation-modified hot big bang model whose evolutionary history is divided into three epochs: an early slowly-rolling scalar field inflation epoch and the usual radiation and nonrelativistic matter epochs. (For our usage it is not necessary to consider a final dark energy dominated epoch.) We derive general solutions of the relativistic linear perturbation equations in each epoch. The constants of integration in the inflation epoch solutions are determined from de Sitter invariant quantum-mechanical initial conditions in the Lorentzian section of the inflating closed de Sitter space derived from Hawking’s prescription that the quantum state of the universe only include field configurations that are regular on the Euclidean (de Sitter) sphere section. The constants of integration in the radiation and matter epoch solutions are determined from joining conditions derived by requiring that the linear perturbation equations remain nonsingular at the transitions between epochs. The matter epoch power spectrum of gauge-invariant energy density inhomogeneities is not a power law, and depends on spatial wave number in the way expected for a generalization to the closed model of the standard flat-space scale-invariant power spectrum. The power spectrummore » we derive appears to differ from a number of other closed inflation model power spectra derived assuming different (presumably non de Sitter invariant) initial conditions.« less

Authors:
 [1]
  1. Kansas State Univ., Manhattan, KS (United States)
Publication Date:
Research Org.:
Kansas State Univ., Manhattan, KS (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP)
OSTI Identifier:
1511831
Alternate Identifier(s):
OSTI ID: 1409869
Grant/Contract Number:  
SC0011840; SC001184
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Physical Review. D.
Additional Journal Information:
Journal Volume: 96; Journal Issue: 10; Journal ID: ISSN 2470-0010
Publisher:
American Physical Society (APS)
Country of Publication:
United States
Language:
English
Subject:
79 ASTRONOMY AND ASTROPHYSICS

Citation Formats

Ratra, Bharat. Inflation in a closed universe. United States: N. p., 2017. Web. doi:10.1103/physrevd.96.103534.
Ratra, Bharat. Inflation in a closed universe. United States. https://doi.org/10.1103/physrevd.96.103534
Ratra, Bharat. 2017. "Inflation in a closed universe". United States. https://doi.org/10.1103/physrevd.96.103534. https://www.osti.gov/servlets/purl/1511831.
@article{osti_1511831,
title = {Inflation in a closed universe},
author = {Ratra, Bharat},
abstractNote = {To derive a power spectrum for energy density inhomogeneities in a closed universe, we investigate a spatially-closed inflation-modified hot big bang model whose evolutionary history is divided into three epochs: an early slowly-rolling scalar field inflation epoch and the usual radiation and nonrelativistic matter epochs. (For our usage it is not necessary to consider a final dark energy dominated epoch.) We derive general solutions of the relativistic linear perturbation equations in each epoch. The constants of integration in the inflation epoch solutions are determined from de Sitter invariant quantum-mechanical initial conditions in the Lorentzian section of the inflating closed de Sitter space derived from Hawking’s prescription that the quantum state of the universe only include field configurations that are regular on the Euclidean (de Sitter) sphere section. The constants of integration in the radiation and matter epoch solutions are determined from joining conditions derived by requiring that the linear perturbation equations remain nonsingular at the transitions between epochs. The matter epoch power spectrum of gauge-invariant energy density inhomogeneities is not a power law, and depends on spatial wave number in the way expected for a generalization to the closed model of the standard flat-space scale-invariant power spectrum. The power spectrum we derive appears to differ from a number of other closed inflation model power spectra derived assuming different (presumably non de Sitter invariant) initial conditions.},
doi = {10.1103/physrevd.96.103534},
url = {https://www.osti.gov/biblio/1511831}, journal = {Physical Review. D.},
issn = {2470-0010},
number = 10,
volume = 96,
place = {United States},
year = {Wed Nov 22 00:00:00 EST 2017},
month = {Wed Nov 22 00:00:00 EST 2017}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 40 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Large-scale structure in COBE-normalized cold dark matter cosmogonies
journal, July 1997


Cosmographic bounds on the cosmological deceleration-acceleration transition redshift in f ( R ) gravity
journal, August 2014


New Model-Independent Method to test the Curvature of the Universe
journal, September 2016


Encyclopædia Inflationaris
journal, December 2014


Quantum mechanics of inflation
journal, September 1985


Constraints on the redshift dependence of the dark energy potential
journal, June 2005


An Improved Method to Measure the Cosmic Curvature
journal, April 2017


Cosmological Constraints from Hubble Parameter versus Redshift Data
journal, September 2006


Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann‐Robertson‐Walker Models
journal, August 2000


Dark energy degeneracies in the background dynamics
journal, January 2008


COBE DMR-normalized open inflation cold dark matter cosmogony
journal, May 1995


A Hypothesis, Unifying the Structure and the Entropy of the Universe
journal, October 1972


Causality in Inflationary Universes with Positive Spatial Curvature
journal, September 2002


Closed universes from cosmological instantons
journal, January 2002


Dark energy in a hyperbolic universe
journal, August 2002


The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample
journal, March 2017


Inflation with variable Ω
journal, May 1995


Planck 2015 results : XIII. Cosmological parameters
journal, September 2016


Cosmic chronometers: constraining the equation of state of dark energy. I: H ( z ) measurements
journal, February 2010


Dynamical dark energy or simply cosmic curvature?
journal, August 2007


Primeval Adiabatic Perturbation in an Expanding Universe
journal, November 1970


Binned Hubble parameter measurements and the cosmological deceleration–acceleration transition
journal, October 2013


Hints of Dynamical Vacuum Energy in the Expanding Universe
journal, September 2015


Growth rate in the dynamical dark energy models
journal, November 2014


The Atacama Cosmology Telescope: cosmological parameters from three seasons of data
journal, October 2013


Dark energy parametrizations and the curvature of the universe
journal, February 2007


COBE ‐DMR–normalized Open Cold Dark Matter Cosmogonies
journal, January 1998

  • Gorski, Krzysztof M.; Ratra, Bharat; Stompor, Radosław
  • The Astrophysical Journal Supplement Series, Vol. 114, Issue 1
  • https://doi.org/10.1086/313062

Large-Angle Cosmic Microwave Background Anisotropy in an Open Universe in the One-Bubble Inflationary Scenario
journal, December 1995


Wilkinson Microwave Anisotropy Probe data and the curvature of space
journal, October 2003


Galaxy cluster number count data constraints on cosmological parameters
journal, November 2012


Constraining cosmic curvature by using age of galaxies and gravitational lenses
journal, March 2017


First Evidence of Running Cosmic Vacuum: Challenging the Concordance Model
journal, February 2017


Cosmology with a time-variable cosmological 'constant'
journal, February 1988


Stochastic approach to tunneling and baby universe formation
journal, March 1992


On the determination of curvature and dynamical dark energy
journal, December 2008


Median Statistics and the Hubble Constant
journal, September 2011


Inflation in an exponential-potential scalar field model
journal, March 1992


Is There Evidence for dark Energy Evolution?
journal, April 2015


Quintessence with a constant equation of state in hyperbolic universes
journal, June 2003


The Hubble constant and dark energy from cosmological distance measures
journal, April 2008


MODEL-INDEPENDENT DETERMINATION OF CURVATURE PARAMETER USING H ( z ) AND D A ( z ) DATA PAIRS FROM BAO MEASUREMENTS
journal, June 2014


The Beginning and Evolution of the Universe
journal, March 2008


NINE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE ( WMAP ) OBSERVATIONS: COSMOLOGICAL PARAMETER RESULTS
journal, September 2013


WHAT ARE THE Omh 2 ( z 1 , z 2 ) AND Om ( z 1 , z 2 ) DIAGNOSTICS TELLING US IN LIGHT OF H ( z ) DATA?
journal, June 2016


Spatial curvature endgame: Reaching the limit of curvature determination
journal, July 2016


Investigations in relativistic cosmology
journal, April 1963


Why Not Consider Closed Universes?
journal, March 1996


Determining the Hubble Constant from Hubble Parameter Measurements
journal, January 2017


Cosmological implications of baryon acoustic oscillation measurements
journal, December 2015


Imprint of spatial curvature on inflation power spectrum
journal, August 2008


Constraints on Scalar-Field Dark Energy from Galaxy Cluster Gas Mass Fraction versus Redshift Data
journal, July 2004


Dynamical dark energy: Scalar fields and running vacuum
journal, March 2017


Inflation in an open universe
journal, August 1995


Cosmology and astrophysics from relaxed galaxy clusters – II. Cosmological constraints
journal, April 2014


Constraints on dark energy from the lookback time versus redshift test
journal, October 2010


Cosmological Constraints from Compact Radio Source Angular Size versus Redshift Data
journal, January 2003


A 2.4% Determination of the Local Value of the Hubble Constant
journal, July 2016


Carnegie Hubble Program: a Mid-Infrared Calibration of the Hubble Constant
journal, September 2012


Fast and accurate CMB computations in non-flat FLRW universes
journal, September 2014


Cosmological constraints on the DGP braneworld model with gamma-ray bursts
journal, April 2011


A 6% measurement of the Hubble parameter at z ∼0.45: direct evidence of the epoch of cosmic re-acceleration
journal, May 2016


Dynamics of Inflationary Universes with Positive Spatial Curvature
journal, September 2002


Reconstructing the properties of dark energy using standard sirens
journal, April 2013


Testing flatness of the universe with probes of cosmic distances and growth
journal, December 2009


Constraining thawing and freezing models with cluster number counts
journal, June 2014


Large scale perturbations in the open universe
journal, September 1995


Forecasting Cosmological Parameter Constraints from Near-Future Space-Based Galaxy Surveys
journal, October 2012


Hubble Parameter Measurement Constraints on dark Energy
journal, February 2013


Median Statistics, H 0 , and the Accelerating Universe
journal, March 2001


Time and distance constraints on accelerating cosmological models
journal, May 2011


Inflation in the closed FLRW model and the CMB
journal, October 2016


Cosmology with gamma-ray bursts: II. Cosmography challenges and cosmological scenarios for the accelerated Universe
journal, February 2017


Fluctuations at the Threshold of Classical Cosmology
journal, May 1970


Cosmological constraints from large-scale structure growth rate measurements
journal, July 2014


Polar Network Index as a Magnetic Proxy for the Solar Cycle Studies
journal, September 2014


Creation of open universes from de Sitter space
journal, January 1982


Probing dynamics of dark energy with latest observations
journal, May 2017


Model-Independent Estimations for the Curvature from Standard Candles and Clocks
journal, December 2016


Integral Solution for the Microwave Background Anisotropies in Nonflat Universes
journal, February 1998


Supernova Ia Constraints on a Time‐variable Cosmological “Constant”
journal, March 2000


CONSTRAINTS ON NON-FLAT COSMOLOGIES WITH MASSIVE NEUTRINOS AFTER PLANCK 2015
journal, September 2016


Hubble parameter data constraints on dark energy
journal, September 2011


Four new observational H ( z ) data from luminous red galaxies in the Sloan Digital Sky Survey data release seven
journal, September 2014


Nonflat time-variable dark energy cosmology
journal, December 2013


Inflationary density perturbations with
journal, December 1990


Age Problem in Holographic dark Energy
journal, June 2010


Cold dark matter cosmogony in an open universe
journal, September 1994


Constraining thawing dark energy using galaxy cluster number counts
journal, May 2013


Dark Energy and Cosmic Curvature: Monte Carlo Markov Chain Approach
journal, July 2008


The quantum state of the universe
journal, June 1984


Quintessence and the Curvature of the Universe After wmap
journal, January 2004


Observational constraints on dark energy and cosmic curvature
journal, November 2007


First-order phase transition of a vacuum and the expansion of the Universe
journal, July 1981


Dark energy evolution and the curvature of the universe from recent observations
journal, April 2006


Implications of dark energy parametrizations for the determination of the curvature of the universe
journal, December 2006


Dark Energy Versus Modified Gravity
journal, October 2016


Constraining the dark energy equation of state with H ii galaxies
journal, July 2016


Quantum mechanics of exponential-potential inflation
journal, December 1989


Cosmological consequences of a rolling homogeneous scalar field
journal, June 1988


Dynamics of the universe and spontaneous symmetry breaking
journal, October 1980


The trouble with H 0
journal, October 2016


Model-independent test of the FLRW metric, the flatness of the Universe, and non-local estimation of H 0 r d
journal, January 2017


Observational constraints on non-flat dynamical dark energy cosmological models
journal, April 2015


Curvature versus distances: Testing the FLRW cosmology
journal, July 2014


Is the low cosmic microwave background quadrupole a signature of spatial curvature?
journal, August 2003


Impact of H 0 prior on the evidence for dark radiation
journal, August 2012


Probing for dynamics of dark energy and curvature of universe with latest cosmological observations
journal, April 2007


Open universe from inflation
journal, September 1995


Inflation with Ω≠1
journal, December 1995


Can we have inflation with   > 1?*
journal, May 2003


Constraining dark Energy with Gamma-Ray Bursts
journal, April 2010


Null test of the cosmic curvature using H ( z ) and supernovae data
journal, February 2016


Tests of cosmological models constrained by inflation
journal, September 1984


Non‐Gaussian Error Distribution of Hubble Constant Measurements
journal, November 2003

  • Chen, Gang; Gott III, J. Richard; Ratra, Bharat
  • Publications of the Astronomical Society of the Pacific, Vol. 115, Issue 813
  • https://doi.org/10.1086/379219

Galaxy cluster angular-size data constraints on dark energy
journal, July 2012


Normal Modes of Vibrations of the Universe
journal, October 1967


Towards a precision cosmology from starburst galaxies at z > 2
journal, January 2005


Is there a concordance value for H 0 ?
journal, November 2016


Closed universes, de Sitter space, and inflation
journal, March 2005


Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples
journal, August 2014


Global Cosmological Parameters Determined Using Classical Double Radio Galaxies
journal, December 2000


Cosmic background radiation anisotropy in an open inflation, cold dark matter cosmogony
journal, October 1994


The H ii galaxy Hubble diagram strongly favours R h = ct over ΛCDM
journal, August 2016


Cosmic degeneracy with dark energy equation of state
journal, December 2003


Cosmology with gamma-ray bursts: I. The Hubble diagram through the calibrated E p,iE iso correlation
journal, February 2017


Planck 2015 results : X. Diffuse component separation: Foreground maps
journal, September 2016


Planck 2015 results : XXVI. The Second
journal, September 2016


Constraining dark energy with gamma-ray bursts
text, January 2009


Galaxy cluster number count data constraints on cosmological parameters
text, January 2011


The Atacama Cosmology Telescope: Cosmological parameters from three seasons of data
text, January 2013


Constraining thawing and freezing models with cluster number counts
text, January 2014


Polar Network Index as a magnetic proxy for the solar cycle studies
text, January 2014


Planck 2015 results. XIII. Cosmological parameters
text, January 2015


Is there evidence for dark energy evolution?
text, January 2015


Hints of dynamical vacuum energy in the expanding Universe
text, January 2015


Spatial curvature endgame: Reaching the limit of curvature determination
text, January 2016


A 2.4% Determination of the Local Value of the Hubble Constant
text, January 2016


Inflation in the closed FLRW model and the CMB
text, January 2016


Cosmic Degeneracy with Dark Energy Equation of State
text, January 2003


Imprint of spatial curvature on inflation power spectrum
text, January 2006


Why Not Consider Closed Universes?
text, January 1995


An Open Universe from Inflation
text, January 1994


Works referencing / citing this record:

Theoretical cosmology
journal, November 2019


Constraints on dark energy dynamics and spatial curvature from Hubble parameter and baryon acoustic oscillation data
journal, July 2018


Baryon acoustic oscillation, Hubble parameter, and angular size measurement constraints on the Hubble constant, dark energy dynamics, and spatial curvature
journal, July 2019


General cosmography model with spatial curvature
journal, December 2019


General theory of cosmological perturbations in open and closed universes from the Horndeski action
journal, February 2019


Planck 2015 Constraints on the Nonflat ϕ CDM Inflation Model
journal, October 2018


Flatness without CMB: The Entanglement of Spatial Curvature and Dark Energy Equation of State
journal, May 2019


Planck 2015 constraints on spatially-flat dynamical dark energy models
journal, October 2019


Primordial power spectra for curved inflating universes
journal, December 2019


Planck 2015 Constraints on the Non-flat XCDM Inflation Model
journal, December 2018


First study of reionization in tilted flat and untilted non-flat dynamical dark energy inflation models
journal, June 2019


Planck 2015 Constraints on the Non-flat ΛCDM Inflation Model
journal, August 2018


Planck 2015 constraints on the non-flat XCDM inflation model
text, January 2017


Theoretical Cosmology
text, January 2019


Lorentzian vacuum transitions: Open or closed universes?
journal, July 2021


General theory of cosmological perturbations in open and closed universes from the Horndeski action
journal, February 2019


Observable relics of the simple harmonic universe
preprint, January 2019


The galaxy power spectrum take on spatial curvature and cosmic concordance
text, January 2020