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Cited by 30Cosmology and Gravitation TheoriesNoncommutative and Quantum Gravity TheoriesBlack Holes and Theoretical Physics

Quantum Instability of the Emergent Universe

Open access
Aguirre, Anthony, Kehayias, John·2013·Physical Review D
DOI 10.1103/physrevd.88.103504arXiv 1306.3232
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Abstract

We perform a semiclassical analysis of the emergent universe scenario for inflation. Fixing the background, and taking the inflaton to be homogenous, we cast the inflaton's evolution as a one-dimensional quantum mechanics problem. The potential is taken to be flat or linear, as an approximation to the monotonic and slowly varying asymptotic behavior of the emergent universe potential. We find that the tuning required over a long time scale for this inflationary scenario is unstable quantum mechanically. Considering the inflaton field value as a wave packet, the spreading of the wave packet destroys any chance of both starting and ending with a well-formed state. Thus, one cannot have an Einstein static universe to begin with that evolves into a well-defined beginning to inflation a long time later.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper argues that the Emergent Universe scenario is quantum mechanically unstable: a homogeneous inflaton wavepacket cannot remain well-defined over an infinite past and still evolve into a well-defined pre-inflationary state. For flat and linear potentials, the overlap probability falls as \(1/t_0\) for long times, so it vanishes in the past-eternal limit.

Key findings

  • Treating the inflaton as a Gaussian wavepacket in a fixed classical background, the probability of a well-formed initial state evolving into another well-formed state after time \(t_0\) is proportional to \(1/t_0\) for large \(t_0\) in both flat and linear potentials.
  • For the linear potential, if the final wavepacket velocity does not match the evolved wavepacket velocity, there is additional exponential suppression depending on the slope \(-b\); at best, matching velocities reduces the result to the flat-potential \(1/t_0\) behavior.
  • Because the Emergent Universe assumes an infinite past, this probability goes to zero: the wavepacket spreads and destroys any chance of both starting and ending with a well-defined state.
  • The authors conclude it is inconsistent to have both a well-defined semi-classical field description and an infinite nonsingular past in this scenario.

Methods

  • Semi-classical analysis: the metric is treated classically and fixed, while the homogeneous inflaton field is quantized and cast as a one-dimensional quantum mechanics problem with \(\phi\) as the coordinate.
  • The potential is modeled as flat or linear/constant-slope to approximate the asymptotic behavior of the Emergent Universe potential.
  • A Gaussian wavepacket ansatz is evolved via the Schrödinger equation, and the probability is computed for the initial wavepacket to overlap with a well-defined final wavepacket after time \(t_0\).

Limitations

  • The authors state they analyze only one simplified version of the Emergent Universe, with a fixed classical background and homogeneous scalar field; they do not treat the metric quantum mechanically or include anisotropic perturbations.
  • They explicitly do not address classical stability concerns about maintaining a mathematically perfect balance in the zero-mode.
  • The text makes evident that alternative gravitational theories and models with different field dynamics or material content would require separate analysis, though the authors expect the basic problem may be generic.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevd.88.103504

Cites

52 of 52 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

3 of 9 cited sources checked · 2 supported · 0 flagged · accuracy not yet measured

  • Interpretationcitation
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    A tunneling scenario for the Emergent Universe would necessarily not be eternal.

    “Another alternative is perhaps a tunneling scenario (for instance, [13]). However, then the universe is necessarily not eternal.”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [13] [13] Pedro Labrana, 'Emergent Universe by Tunneling,' Phys.Rev. D86 , 083524 (2012), arXiv:1111.5360 [gr-qc]. - source

      The Introduction (p. 4) says the initial metastable state in this paper’s tunneling scheme could exist only for a finite time. But the claim’s “necessarily” extends that conclusion to tunneling scenarios generally; the Conclusions (p. 17) propose applying the scheme to models with stable past-eternal static regimes. (LLM, conf=0.88)

      Experimental — accuracy not yet measured
  • Findingreasoning
    § A. In a Flat or Linear Potential · p. 4

    For a linear potential with slope -b, if the final state velocity exactly matches the velocity of the wavepacket evolved in the potential, then the probability is the same as for the flat potential.

    “If this velocity exactly matches the velocity of the wavepacket which was evolved in the potential, then the probability is the same as above, eq. (23).”
    Quote in the paperNot a substantial claim
  • Findingreasoning
    § V. DISCUSSION AND CONCLUSIONS · p. 4

    In both flat and linear potential cases, the probability of a well-defined initial wavepacket evolving into a well-defined state after a time t0 is proportional to 1/t0 for large t0.

    “In both cases the probability is proportional to 1 /t 0 for large t 0 .”
    Quote in the paperSubstantial claim
  • Findingown data
    § I. INTRODUCTION · p. 1

    The probability of the initial state evolving into a well-defined pre-inflationary state is severely suppressed, and effectively zero in the past eternal limit.

    “Computing the probability of the initial state evolving into a well-defined pre-inflationary state, we see that it is severely suppressed (and effectively zero in the 'past eternal' limit).”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The perfect balance of kinetic energy and vacuum energy in the zero-mode can be maintained classically.
  • Findingreasoning
    § Appendix A: Linear Potential Calculation · p. 6

    At best the linear potential can have the same probability, proportional to 1/t0, as the flat potential.

    “Therefore, at best the linear potential can have the same probability, proportional to 1 /t 0 , as the flat potential.”
    Quote in the paperNot a substantial claim
  • Classified as probably not a substantial claim. 5
  • May combine two claims. 15

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    Graham et al. [14] construct static and oscillating universes with a specific non-perfect-fluid energy component that are stable against small perturbations.

    “For example, Graham et al. [14] construct static and oscillating universes with a specific non-perfect-fluid energy component that are stable against small perturbations.”
    Quote in the paperNot a substantial claim1 of 1 sources checked
    • cites [14] [14] Peter W. Graham, Bart Horn, Shamit Kachru, Surjeet Rajendran, and Gonzalo Torroba, 'A Simple Harmonic Universe,' (2011), arXiv:1109.0282 [hep-th]. - source

  • May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    Classically, the Einstein static universe is unstable to homogeneous perturbations, but stable to inhomogeneous perturbations if the fluid sound speed is sufficiently high.

    “Classically, the Einstein static universe is unstable to homogeneous perturbations, but stable to inhomogeneous perturbations if the fluid sound speed is sufficiently high [4].”
    Quote in the paperSubstantial claim1 of 1 sources checked
    • cites [4] [4] A.S. Eddington, 'On the Instability of Einstein's Spherical World,' Mon.Not.Roy.Astron.Soc. 90 , 668-678 (1930); E.R. Harrison, 'Normal Modes of Vibrations of the Universe,' Rev.Mod.Phys. 39 , 862-882 (1967); 8 As a check, setting b = 0, so the potential is a constant, also reproduces the result of the previous section. G.W. Gibbons, 'The entropy and stability of the universe,' Nucl.Phys. B292 , 784 (1987); 'Sobolev's inequality, Jensen's theorem and the mass and entropy of the universe,' Nucl.Phys. B310 , 636 (1988); John D. Barrow, George F.R. Ellis, Roy Maartens, and Christos G. Tsagas, source

  • May combine two claims.

    Findingreasoning
    § A. In a Flat or Linear Potential · p. 4

    The probability can be maximized to go like σ0/√T0, but only if σ1 ∝ √T0, which does not correspond to a well-defined classical configuration for large T0.

    “This can be maximized to go like σ 0 / √ T 0 , but only if σ 1 ∝ √ T 0 , which does not correspond to a well-defined classical configuration for large T 0”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § IV. INSTABILITY OF THE EMERGENT UNIVERSE SCENARIO · p. 3

    In momentum-space, the wavepacket has nonzero width ∆ ∝ 1/σ0, so at a given time, unless ∆ → 0, there is an infinitesimal probability of a measurement yielding the value of ̇φ which gives stability.

    “In momentum-space, the wavepacket has nonzero width ∆ ∝ 1 /σ 0 , so at a given time, unless ∆ → 0, there is an infinitesimal probability of a measurement yielding the value of ˙ φ which gives stability.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Appendix A: Linear Potential Calculation · p. 6

    For large t0 in the linear potential, the probability falls exponentially fast, unless x = 1.

    “which falls exponentially fast, unless x = 1 (maximizing the probability with respect to x ).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § II. THE EMERGENT UNIVERSE · p. 2

    Although there is an infinite time for inflation, the amount of inflation is finite and can be made large.

    “Although there is an infinite time for inflation, the amount of inflation is finite (and can be made large).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [3] [3] George F.R. Ellis and Roy Maartens, 'The emergent universe: Inflationary cosmology with no singularity,' Class.Quant.Grav. 21 , 223-232 (2004), arXiv:gr-qc/0211082 [gr-qc]; George F.R. Ellis, Jeff Murugan, and Christos G. Tsagas, 'The Emergent universe: An Explicit construction,' Class.Quant.Grav. 21 , 233-250 (2004), arXiv:gr-qc/0307112 [gr-qc]. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The first e-folding of inflation takes an unbounded amount of time, but the second and subsequent e-foldings proceed essentially as usual, ending in reheating and ordinary cosmological evolution.

    “That is, the first e-folding of inflation take an unbounded amount of time, but the second and subsequent e-foldings proceed essentially as usual, ending in reheating and ordinary cosmological evolution.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [3] [3] George F.R. Ellis and Roy Maartens, 'The emergent universe: Inflationary cosmology with no singularity,' Class.Quant.Grav. 21 , 223-232 (2004), arXiv:gr-qc/0211082 [gr-qc]; George F.R. Ellis, Jeff Murugan, and Christos G. Tsagas, 'The Emergent universe: An Explicit construction,' Class.Quant.Grav. 21 , 233-250 (2004), arXiv:gr-qc/0307112 [gr-qc]. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisecitation
    § II. THE EMERGENT UNIVERSE · p. 2

    The Emergent Universe is closed, K = +1, and has H = 0 initially.

    “The Emergent Universe is closed, K = +1, and has H = 0 initially.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [3] [3] George F.R. Ellis and Roy Maartens, 'The emergent universe: Inflationary cosmology with no singularity,' Class.Quant.Grav. 21 , 223-232 (2004), arXiv:gr-qc/0211082 [gr-qc]; George F.R. Ellis, Jeff Murugan, and Christos G. Tsagas, 'The Emergent universe: An Explicit construction,' Class.Quant.Grav. 21 , 233-250 (2004), arXiv:gr-qc/0307112 [gr-qc]. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    The tendency of quantum effects to destabilize even classically stable configurations suggests that even if an emergent model were possible, it would have to be posed at the quantum (and quantum-gravitational) level, largely undermining the motivation to provide an early state in which quantum gravitational effects are not crucial.

    “Moreover, the tendency of quantum effects to destabilize even classically stable configurations suggests that even if an emergent model were possible, it would have to be posed at the quantum (and quantum-gravitational) level, largely undermining the motivation to provide an early state in which quantum gravitational effects are not crucial.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    A truly stationary or periodic quantum state, which would last forever, would never evolve, whereas one with any instability will not endure for an indefinite time.

    “A truly stationary or periodic quantum state, which would last forever, would never evolve, whereas one with any instability will not endure for an indefinite time.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § IV. INSTABILITY OF THE EMERGENT UNIVERSE SCENARIO · p. 3

    For a given value of V(φ), only one precise value of ̇φ yields stability, yet φ and p ∝ ̇φ are subject to an uncertainty relation.

    “For a given value of V ( φ ), only one precise value of ˙ φ yields stability, yet φ and p ∝ ˙ φ are subject to an uncertainty relation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    Even in alternative theories of gravity in which the Einstein static universe is more stable than in standard General Relativity, once the wavefunctional has spread enough, the geometry must follow, and the spacetime becomes classically ill-defined as well as containing portions corresponding to singularities.

    “Even in alternative theories in which the Einstein static universe is more stable than in standard General Relativity, we anticipate that once the wavefunctional has spread enough, the geometry must follow, and the spacetime becomes classically ill-defined as well as containing portions corresponding to singularities.”
    Quote in the paperSubstantial claim
    • Rests on an assumption: Treating the metric quantum mechanically or including anisotropic perturbations is very unlikely to increase the cosmology's stability.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § Appendix A: Linear Potential Calculation · p. 6

    If one wants the field velocity to remain below some critical value (e.g. a slow roll condition), then this constrains both the potential and the amount of time the field evolves in the potential.

    “If one wants the field velocity to remain below some critical value (e.g. a slow roll condition), then this constrains both the potential and the amount of time the field evolves in the potential.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § IV. INSTABILITY OF THE EMERGENT UNIVERSE SCENARIO · p. 3

    In the ∆ → 0 limit, the value of φ is completely uncertain and one cannot describe the situation as a single classical universe.

    “Yet in the ∆ → 0 limit, the value of φ is completely uncertain and one cannot describe the situation as a single classical universe.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § V. DISCUSSION AND CONCLUSIONS · p. 4

    The Emergent Universe is built on an infinite past, and thus this probability goes to zero.

    “The Emergent Universe is built on an infinite past, and thus this probability goes to zero.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § II. THE EMERGENT UNIVERSE · p. 2

    The balancing of kinetic energy and vacuum energy must persist if the universe is to be considered static, and thus it will be sensitive to quantum fluctuations in the scalar field.

    “This balancing must persist if the universe is to be considered static and thus will be sensitive to quantum fluctuations in the scalar field.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Interpretationreasoning
    § V. DISCUSSION AND CONCLUSIONS · p. 5

    The effect that this analysis points to may be rather generic.

    “Nonetheless, we believe that the effect that this analysis points to may be rather generic.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: Treating the metric quantum mechanically or including anisotropic perturbations is very unlikely to increase the cosmology's stability.

  • May combine two claims.

    Premisecitation
    § II. THE EMERGENT UNIVERSE · p. 2

    The Emergent Universe starts as an Einstein static universe with finite size in the infinite past, inflates, and then reheats in the usual manner.

    “This scenario does not bounce, but starts as an Einstein static universe with finite size in the infinite past, inflates, and then reheats in the usual manner.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [3] [3] George F.R. Ellis and Roy Maartens, 'The emergent universe: Inflationary cosmology with no singularity,' Class.Quant.Grav. 21 , 223-232 (2004), arXiv:gr-qc/0211082 [gr-qc]; George F.R. Ellis, Jeff Murugan, and Christos G. Tsagas, 'The Emergent universe: An Explicit construction,' Class.Quant.Grav. 21 , 233-250 (2004), arXiv:gr-qc/0307112 [gr-qc]. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § II. THE EMERGENT UNIVERSE · p. 2

    Homogeneous perturbations will break the balance of the curvature to vacuum energy, leading to inflation.

    “Homogeneous perturbations will break the balance of the curvature to vacuum energy, leading to inflation”
    Quote in the paperSubstantial claim
19 of 19

  • The authors state this as an assumption.

    The perfect balance of kinetic energy and vacuum energy in the zero-mode can be maintained classically.

    “Rather, we assume that such a perfect balance is maintainable classically, and investigate the same problem when quantized.”

  • The paper relies on this without support.

    Treating the metric quantum mechanically or including anisotropic perturbations is very unlikely to increase the cosmology's stability.

    “Treating the metric quantum mechanically (e.g. using the Wheeler-deWitt formalism) or including anisotropic perturbations seems very unlikely to increase the cosmology's stability.”

  • The authors state this as an assumption.

    The universe can always be treated in a quasi-classical way and should have compact support in both field and field-velocity space.

    “Similarly, if we assume that the universe can always be treated in a quasi-classical way (as implicitly assumed by [3]), it should have compact support in both field and field-velocity space.”
3 of 3

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • The stated volume factor disagrees with the conjugate momentum

    Minor

    The note states a volume factor of 2πa3, but the conjugate momentum uses 2π2a3. Those factors differ by π.

  • The two overlap expressions use different time shifts

    Minor

    The first expression shifts the comparison wavepacket by T0v0/(2π2), while the expression stated equal to it uses Tv0/(2π2). The comparison is at T = 0, so the second shift vanishes rather than matching the first.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. A. Vilenkin, 2002, Workshop on Conference on the Future of Theoretical Physics and Cosmology in Honor of Steven Hawking’s 60th Birthday
  2. A. Banerjee, 2007, Gravitation Cosmol., 13, 290

Cited by

Sorted by citations

Related

Authors
Aguirre, A., & Kehayias, J.
Affiliations
University of California, Santa Cruz (US)
ORCID
Venue
Physical Review D
Publisher
American Physical Society (APS)
Published
2013-11-07
Volume
88
Issue
10
ISSN
  • 0556-2821 (ISSN-L)
  • 1550-7998
  • 1550-2368
DOI
10.1103/physrevd.88.103504
arXiv
1306.3232
OpenAlex
W2162682791
Semantic Scholar
cac97721636b04e60f8c978811245a58e4a681a1
Cited by
30
References
44
Citations by year
2025: 1 · 2024: 2 · 2023: 2 · 2022: 4 · 2021: 4 · 2020: 1 · 2019: 1 · 2018: 3 · 2017: 3 · 2016: 2 · 2015: 4 · 2014: 3
FWCI
1.62
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) (publishedVersion)
  • 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 › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
emergent universe, quantum instability, inflation (cosmology), semiclassical analysis, inflaton, wave packet spreading, Einstein static universe, flat potential, linear potential, quantum tunneling, quantum cosmology

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