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Cited by 10Cosmology and Gravitation TheoriesNoncommutative and Quantum Gravity TheoriesBlack Holes and Theoretical Physics

Is asymptotically safe inflation eternal?

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Chojnacki, J., Krajecka, J., Kwapisz, J.H., Slowik, O., Strag, A.·2021·Journal of Cosmology and Astroparticle Physics
DOI 10.1088/1475-7516/2021/04/076
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Abstract

Recently, based on swampland considerations in string theory, the (no) eternal inflation principle has been put forward. The natural question arises whether similar conditions hold in other approaches to quantum gravity. In this article, the asymptotic safety hypothesis is considered in the context of eternal inflation. As exemplary inflationary models the SU(N) Yang-Mills in the Veneziano limit and various RG-improvements of the gravitational action are studied. The existence of UV fixed point generically flattens the potential and our findings suggest no tension between eternal inflation and asymptotic safety, both in the matter and gravitational sector in contradistinction to string theory. Moreover, the eternal inflation cannot take place in the range of applicability of effective field quantum gravity theory. We employ the analytical relations for eternal inflation to some of the models with single minima, such as Starobinsky inflation, alpha-attractors, or the RG-improved models and verify them with the massive numerical simulations. The validity of these constraints is also discussed for a multi-minima model.

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Written from the paper’s full text.

TL;DR

The paper studies whether eternal inflation can occur in asymptotically safe quantum gravity, using stochastic inflation, Fokker-Planck conditions, and Langevin simulations for several inflationary models. It finds no generic tension between eternal inflation and asymptotic safety, unlike string-theory swampland expectations, though eternal inflation lies outside the applicability of effective field quantum gravity in some cases.

Key findings

  • Eternal inflation is present for asymptotically flat effective potentials at large field values in asymptotically safe models, both in gravitational and matter sectors; the authors conclude there is no obstruction to the multiverse scenario in asymptotically safe models.
  • For Starobinsky inflation, the eternal-inflation threshold is estimated as \(\phi_0 = 16.7\,M_{\rm Pl}\), while 60 e-folds require \(\phi_0 = 5.5\,M_{\rm Pl}\); numerical simulations with 10,000 samples confirm the analytical prediction, and for \(\phi_0 = 3\,M_{\rm Pl}\) the decay rate is \(\Gamma \approx 0.15\) versus \(3H \sim 10^{-5}\), so inflation is not eternal.
  • In RG-improved Starobinsky inflation, for \(\Lambda=1,\alpha=1.6\) the analytical eternal-inflation threshold is \(\phi_{\rm EI}=22.6\,M_{\rm Pl}\), while the numerical simulation gives \(\Gamma=0.0001\,M_{\rm Pl}\) and \(3H=0.0003\,M_{\rm Pl}\), so eternal inflation begins slightly below the analytical value.
  • In the SU(N) Yang-Mills Veneziano-limit model, tunneling through a potential barrier provides a new eternal-inflation mechanism: for \(N_F=10,\mu=10^{-3}M_{\rm Pl},\xi=1/6,\delta=0.1\), the steepest-descent approximation gives \(R=0.92\) and the numerical simulation gives \(R=0.97\); above \(9.2\,M_{\rm Pl}\) there is non-zero probability of eternal inflation, while at \(9.2\,M_{\rm Pl}\) inflation classically produces roughly 54 e-folds.

Methods

  • The authors use stochastic inflation, the Langevin equation, and the Fokker-Planck equation to derive analytical eternal-inflation conditions, including the linear and quadratic hilltop conditions and higher-derivative conditions from Rudelius.
  • They numerically simulate the discretized Langevin equation with Gaussian noise of variance \(H^3/(4\pi^2)\delta t\), assuming constant \(H\) in slow roll, and build histograms of inflationary time; tunneling probabilities are also computed via first-passage formalism and checked with 10,000 simulations.
  • Models analyzed include \(\alpha\)-attractor T-models, Starobinsky inflation, leading-log quantum-corrected Starobinsky inflation, RG-improved quadratic gravity, and an SU(\(N_C\)) gauge theory with \(N_F\) Dirac fermions in the Veneziano limit.

Limitations

  • The paper states that eternal inflation cannot take place in the range of applicability of effective field quantum gravity theory; for the leading-log corrected Starobinsky model, eternal inflation requires \(\Phi \approx 1000\,M_{\rm Pl}\), far beyond the model’s applicability.
  • The authors state that the analytical conditions (2.23) and (2.24) are not well suited for multiple-minima models and cannot capture the full global influence of quantum fluctuations; fixed-point values and the potential shape may be obscured by quantum gravity effects, to be investigated elsewhere.
  • The numerical method assumes constant \(H\) and uses a large timeout to end long evolutions as an approximation; the authors checked that changing \(H\) from field fluctuations does not affect the eternal-inflation conclusions.

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

Integrity signals

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Crossref · 10.1088/1475-7516/2021/04/076

Cites

210 of 210 references verified0 citation errors · 0 not checkable

Statistics

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Claims the paper makes

0 of 24 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    For a linear potential, using slow-roll, the eternal inflation condition can be rewritten as |V'|/V^{3/2} < sqrt(2)/(2π) 1/M_Pl^2.

    “For linear potential α = V (φ) using the slow-roll equations equation (2.3), above condition can be rewritten: |V | V 3 2 < √ 2 2π 1 M 2 Pl .”
    Quote in the paperSubstantial claim
  • Findingreasoning

    Applying the analytical eternal inflation conditions to the Starobinsky potential, the initial value of the field above which eternal inflation occurs is φ0=16.7 M_Pl.

    “Applying the analytical eternal inflation conditions (2.23), (2.24) to the Starobinsky potential, the initial value of the field, above which the eternal inflation occurs, has been estimated to be φ 0 = 16.7 M Pl .”
    Quote in the paperSubstantial claim
  • Findingown data

    For alpha-attractor models, using the first eternal inflation condition, the space is eternally inflating for all initial values of φ above a certain φ_EI.

    “Using the first condition (2.23) we have verified, that generally the space is eternally inflating for all initial values of the φ, above certain φ EI .”
    Quote in the paperSubstantial claim
  • Findingown data

    In the Yang-Mills Veneziano limit model, the authors uncovered a new type of eternal inflation scenario relying on tunneling to a false vacuum, in the opposite direction to the old-inflation proposal.

    “In this model, we have uncovered a new type of eternal inflation scenario relying on tunneling to a false vacuum -in the opposite direction as it was considered in the old-inflation proposal [76].”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [70]

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

    • cites [71]

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

    • cites [76]

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

  • Findingown data

    For alpha-attractor models, the second eternal condition and higher order conditions are satisfied for almost all values of φ0 above 0, providing no new information.

    “The second eternal condition (2.24) as well as the higher order conditions are satisfied for almost all values of φ 0 above 0, providing no new information.”
    Quote in the paperNot a substantial claim
  • May combine two claims. 20
  • Classified as probably not a substantial claim. 5

  • May combine two claims.

    Findingreasoning

    In the regime where the amplitude of v(φ) is much smaller than 1, the probability of descending into each minimum is similar, giving |1-R|<<1.

    “In this regime, probability of descending into each of the minima φ -and φ + is similar, giving |1 -R| 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For a constant potential, the Fokker-Planck equation reduces to the heat equation and has a Gaussian solution with mean 0 and variance H^3 t/(4π^2).

    “Then the Fokker Planck equation reduces to the standard heat equation, which has a solution given by a Gaussian distribution:”
    • Also stated in: § 2.2 Analytic solutions · plain
      “with µ (t) = 0, σ 2 (t) = H 3 4π 2 t.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    Numerical solutions to the FP equation show there is no possibility of eternal inflation in that region, because it is an unstable maximum and any quantum fluctuation will drop the field from that position.

    “Numerical solutions to the FP equation shows that the is no possibility of eternal inflation in that region, since it is an unstable maximum and hence any quantum fluctuation will drop it from that position.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For N_F = 10, μ = 10^-3 M_Pl, ξ = 1/6, δ = 0.1, the steepest descent approximation yields R = 0.92 and numerical analysis yields R = 0.97, demonstrating good accuracy of the analytical formula.

    “For example it was found, that for the set of the parameters N F = 10, µ = 10 -3 M Pl , ξ = 1 6 , δ = 0.1 the steepest descent approximation yields R = 0.92, and the numerical analysis results in R = 0.97, which proves good accuracy of the analytical formula.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Findingnone given

    Starobinsky inflation stemming from R^2 gravity gives around 60 e-folds for φ_0 = 5.5 M_Pl.

    “Starobinsky inflation stemming from R 2 gravity gives around 60 e-folds for φ 0 = 5.5 M Pl .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    Eternal inflation for the model described by (4.3) takes place at Φ≈1000 M_Pl, which is far beyond the applicability of the model.

    “To be concrete, we have checked that eternal inflation for model described by (4.3) takes place at Φ ≈ 1000 M Pl , which is far beyond the applicability of the model.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The initial condition for which p_+ = 1/2 is shifted to the right of φ_max; starting from the maximum, it is slightly more probable to land in φ_-.

    “Notice, that the initial condition, for which p + = 1 2 is shifted to the right of φ max .This means, that starting from the maximum, it is slightly more probable to land in φ -.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The analytical conditions (2.23) and (2.24) did not allow for eternal inflation, even though the tunneling process may evolve any initial point above 9.2 Planck Masses to φ_+ = ∞ without violating the slow-roll conditions.

    “To sum up the critical point of our analysis is that the analytical conditions (2.23), (2.24) did not allow for eternal inflation, even though the tunneling process may evolve any initial point above 9.2 Planck Masses to φ + = ∞, and not violate the slow-roll conditions.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For large values of φ the potential flattens out and the slow-roll conditions are not violated.

    “For large values of φ the potential flattens out and the slow-roll conditions are not violated.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The ratio R(α, ξ) of probabilities of right-side to left-side descent from the maximum is close to 1 and favors the right side (left side) of the potential for large (small) values of the parameters.

    “As expected, the ratio R(α, ξ) is close to 1 and favors the right-side (left-side) of the potential for large (small) values of the parameters.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For small values of the parameters it is more probable to fall from the maximum towards φ_- = 0 with non-eternal inflation, while for large values of parameters the minimum at φ_+ = ∞ is favored, resulting in an eternally inflating universe.

    “For the small values of the parameters it is more probable to fall from the maximum towards φ -= 0 with noneternal inflation, while for the large values of parameters, minimum at φ + = ∞ is favored, resulting in eternally inflating universe.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    The effective potential is highly asymmetric (step-like) for small values of φ_max, which breaks the symmetry between the right and left descent probabilities.

    “The above figures are qualitatively similar because for the small values of φ max , the effective potential is highly asymmetric (”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Direct numerical simulation for Λ=1 and α=1.6 yields Γ=0.0001 M_Pl and 3H=0.0003 M_Pl, meaning that eternal inflation begins slightly below the expected value φ_EI.

    “The direct numerical simulation for this set of parameters yields Γ = 0.0001 M Pl , and 3H = 0.0003 M Pl , meaning that the eternal inflation begins slightly below the expected value φ EI .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For the parameter set (4.18), the green ball cannot tunnel (p_+ = 0) below 9.2 M_Pl, and the red ball cannot tunnel (p_- = 0) at 24.8 M_Pl.

    “There is a point, below which the green ball cannot tunnel p + = 0 (for the set of parameters given by (4.18), at 9.2 M Pl ), and the other limiting case (at 24.8 M P ), when the red ball cannot tunnel p -= 0.”
    Quote in the paperSubstantial claim

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

    Premisecitation

    The de-Sitter conjecture states that string theory cannot have de-Sitter vacua and is in tension with single field inflation.

    “Recently widely discussed is so-called de-Sitter conjecture [3,4] which states that string theory cannot have de-Sitter vacua and is in tension with single field inflation [5,6].”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [3]

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

    • cites [4]

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

    • cites [5]

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

    • cites [6]

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

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

    Premisecitation

    Below the Planck scale, for quantum gravity one can safely take the effective field theory perspective.

    “As pointed out by Donoghue [27] below the Planck scale, for quantum gravity one can safely take the effective field theory perspective.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [27]

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

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

    Premisecitation

    The no eternal inflation principle has been put forward in line with swampland criteria.

    “In line of these swampland criteria the no eternal inflation principle has been put forward [14]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [14]

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

  • May combine two claims.

    Interpretationreasoning

    The findings suggest no tension between eternal inflation and asymptotic safety, both in the matter and gravitational sector, in contradistinction to string theory.

    “our findings suggest no tension between eternal inflation and asymptotic safety, both in the matter and gravitational sector in contradistinction to string theory”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Gravity admits a non-trivial UV fixed point.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Study design and conductnone given

    The change of H caused by the field fluctuation does not affect the conclusions for eternal inflation.

    “We verified, that the change of H caused by the field fluctuation does not affect the conclusions for eternal inflation.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    In the large field expansion, the potential reduces to the linear hilltop model, which justifies the usage of formulae (2.23), (2.24) and the functional form of φ0(α).

    “and by the substitution φ = e -1 2 3 2 φ the potential reduces to the linear hilltop model, which justifies the usage of formulae (2.23), (2.24) and the functional form of φ 0 (α).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The model is on the verge of being eternally inflating, which may point to interesting phenomenology.

    “This shows that the model is on a verge of being eternally inflating, which may point out to the interesting phenomenology.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    The creation of scattered, causally disconnected regions of spacetime-the multiverse-is not confirmed observationally and raises questions about inflationary predictions.

    “The creation of scattered, causally disconnected regions of spacetime -the multiverse is not confirmed observationally and raises question about the inflationary predictions [77].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [77]

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

  • May combine two claims.

    Premisecitation

    The region to the left of the maximum is where inflation can be brought to an end and reheating takes place.

    “The region to the left of the maximum is the region, where the inflation can be brought to an end and the reheating takes place [104].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [104]

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

  • May combine two claims.

    Interpretationreasoning

    Conditions (2.23) and (2.24) are not well suited for multiple minima models and cannot contain the full information of the global influence of quantum fluctuations in the early universe.

    “This shows that conditions (2.23), (2.24) are not well suited for the multiple minima models and cannot contain the full information of the global influence of quantum fluctuations in the early universe.”
    Quote in the paperSubstantial claim
24 of 24

  • The authors state this as an assumption.

    Both the background and the fluctuations are homogeneous.

    “Hence from now on we shall assume that both background and fluctuations are homogeneous, which is the standard treatment of eternal inflation (if not otherwise specified).”

  • The authors state this as an assumption.

    The Hubble parameter is constant and respects the slow-roll regime H = 1 M Pl V (φ 0 ) 3, where V(φ0) is the value of the potential at the start of the simulation.

    “We further assume the Hubble parameter to be constant and respecting the slow-roll regime H = 1 M Pl V (φ 0 ) 3 , where V (φ 0 ) is a value of the potential at the start of the simulation.”

  • The authors state this as an assumption.

    θ3 = 1.

    “Following [93] we shall assume θ 3 = 1, then the transformation from the Jordan to the Einstein frame yields an effective potential [93,95]:”

  • The authors state this as an assumption.

    Gravity admits a non-trivial UV fixed point.

    “The first type relies on the RG-improvement of the gravitational actions and is based on the asymptotic safety hypothesis that gravity admits a non-trivial UV fixed point.”

  • The authors state this as an assumption.

    M = M_Pl.

    “We assume, that M = M Pl .”

  • The authors state this as an assumption.

    The system is ergodic.

    “assuming the ergodicity of the system.”

  • The authors state this as an assumption.

    Inflation is driven by a single field with slow-roll.

    “Assuming a single field slow-roll inflation, we examine inflationary predictions of the potential”
7 of 7

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.

  • Gaussian upper-tail probability has the wrong sign

    Minor

    For the stated Gaussian with positive drift α, the probability that φ exceeds a fixed φ_c approaches 1 as time increases. Equation (2.20) instead approaches 0: its erfc argument gives the lower tail, not the upper tail defined in (2.14).

  • Eternal-inflation bound has the wrong Planck-mass power

    Minor

    Substituting the stated slow-roll relation H2M_Pl2 ≈ V/3 into (2.22) gives |V′|/V^(3/2) < 1/(√2 π M_Pl3). Equation (2.23) prints M_Pl2 instead of M_Pl3, changing the bound used to identify eternal inflation.

2 of 2

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Authors
Chojnacki, J., Krajecka, J., Kwapisz, J. H., Slowik, O., & Strag, A.
Affiliations
University of Warsaw (PL)
Corresponding author
  • Chojnacki, J.
  • Krajecka, J.
  • Strag, A.
ORCID
Venue
Journal of Cosmology and Astroparticle Physics
Publisher
IOP Publishing
Published
2021-04-01
Volume
2021
Issue
04
Pages
076
ISSN
  • 1475-7508 (ISSN-L)
  • 1475-7516
DOI
10.1088/1475-7516/2021/04/076
OpenAlex
W3120559008
Semantic Scholar
a677528ab158bbe8dc3025197813bb4ce6260f73
Cited by
10
References
124
Citations by year
2024: 1 · 2023: 2 · 2022: 2 · 2021: 5
FWCI
1.34
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of Cosmology and Astroparticle Physics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University) (cc-by)
Funders
Infrastruktura PL-Grid
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
asymptotic safety, eternal inflation, swampland conjectures, quantum gravity, ultraviolet fixed point, Starobinsky inflation, alpha-attractors, inflationary potential

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