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Cited by 73Cosmology and Gravitation TheoriesNoncommutative and Quantum Gravity TheoriesMathematical Dynamics and Fractals

Attractor solutions in scalar-field cosmology

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Remmen, Grant N., Carroll, Sean M.·2013·Physical Review D
DOI 10.1103/physrevd.88.083518
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Abstract

Models of cosmological scalar fields often feature ``attractor solutions'' to which the system evolves for a wide range of initial conditions. There is some tension between this well-known fact and another well-known fact: Liouville's theorem forbids true attractor behavior in a Hamiltonian system. In universes with vanishing spatial curvature, the field variables $\ensuremath{\phi}$ and $\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}$ specify the system completely, defining an effective phase space. We investigate whether one can define a unique conserved measure on this effective phase space, showing that it exists for ${m}^{2}{\ensuremath{\phi}}^{2}$ potentials and deriving conditions for its existence in more general theories. We show that apparent attractors are places where this conserved measure diverges in the $\ensuremath{\phi}\mathrm{\text{\ensuremath{-}}}\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}$ variables and suggest a physical understanding of attractor behavior that is compatible with Liouville's theorem.

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  • The problem of obtaining a measure on flat universes was later solved in [68], which derived a measure on the phase space for flat universes by demanding Liouville's theorem, and [69], which used this to derive a measure on the space of trajectories.

    “This problem was later solved in [68], which derived a measure on the phase space for flat universes by demanding that it obey Liouville's theorem, and [69], which used this to derive a measure on the space of trajectories.”

    From In What Sense Is the Early Universe Fine-Tuned? (2014) · cites this paper as [68]

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Research works printed but not linked: 6

  1. [6]A. Liddle and D. Lyth, Cosmological Inflation and LargeScale Structure (Cambridge University Press, 2000).
  2. [8]V. Kiselev and S. Timofeev (2008), arXiv:0801.2453 [grqc]. · arXiv
  3. [18]C.-Y. Tseng, Ph.D. thesis, California Institute of Technology (2013).
  4. [21]S. M. Carroll and H. Tam (2010), arXiv:1007.1417 [hepth]. · arXiv
  5. [23]P. Ade et al. (Planck Collaboration) (2013), arXiv:1303.5076 [astro-ph.CO]. · arXiv
  6. [26]J. Douglas, Trans. Amer. Math. Soc. 50 , 71 (1941). http://www.jstor.org/stable/1989912

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Authors
Remmen, G. N., & Carroll, S. M.
Affiliations
California Institute of Technology (US)
ORCID
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2013-10-18
Volume
88
Issue
8
ISSN
  • 0556-2821 (ISSN-L)
  • 1550-7998
  • 1550-2368
DOI
10.1103/physrevd.88.083518
OpenAlex
W2028443183
Cited by
73
References
26
Citations by year
2026: 3 · 2025: 10 · 2024: 6 · 2023: 6 · 2022: 2 · 2021: 5 · 2020: 9 · 2019: 3 · 2018: 6 · 2017: 4 · 2016: 9 · 2015: 5 · 2014: 5
FWCI
3.42
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CaltechAUTHORS (California Institute of Technology) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
Keywords
attractor solutions, scalar field cosmology, Liouville's theorem, phase space, Hamiltonian systems, quadratic potential, flat universe, initial conditions

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