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Is asymptotically safe inflation eternal?
Open accessAbstract
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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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.
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Claims the paper makes
25 No basis given · 24 Own interpretation · 22 Own result · 16 Sources not found · 3 Own method
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In the paper
“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 .”
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
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“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 .”
§ Is asymptotically safe inflation eternal? > 3.2 Starobinsky model
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“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 .”
§ Is asymptotically safe inflation eternal? > 3.1 Alpha-attractor models
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“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].”
§ Is asymptotically safe inflation eternal? > 1 Introduction
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“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.”
§ Is asymptotically safe inflation eternal? > 3.1 Alpha-attractor models
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“Fokker-Planck equation is analogously solved by the Gaussian distribution (2.10) with:”
- Also stated in: § Case 2. Linear potential. · plain
“µ (t) = α 3H t, σ 2 (t) = H 3 4π 2 t.”
§ Is asymptotically safe inflation eternal? > Case 2. Linear potential.
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“For every α, φ 0 necessary to produce 60 e-folds is safely below φ EI .”
§ Is asymptotically safe inflation eternal? > 3.1 Alpha-attractor models
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“Similarly, the second condition for the eternal inflation may be derived from the quadratic hilltop potential: - V V < 3 M 2 Pl .”
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
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“then in order for the Universe to inflate eternally, the positive exponential factor 3H in eq.(2.18) and the negative exponential factor -Γ in (2.15) must satisfy: 3H > Γ.”
§ Is asymptotically safe inflation eternal? > 2.3 Eternal inflation conditions
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“It has been found, that decay rate Γ decreases approximately exponentially with φ 0 .”
§ Is asymptotically safe inflation eternal? > 3.2 Starobinsky model
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“In this regime, probability of descending into each of the minima φ -and φ + is similar, giving |1 -R| 1.”
§ Is asymptotically safe inflation eternal? > 2.4 Tunneling and eternal inflation
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“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.”
§ Is asymptotically safe inflation eternal? > 2.2 Analytic solutions
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“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.”
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
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“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.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“Starobinsky inflation stemming from R 2 gravity gives around 60 e-folds for φ 0 = 5.5 M Pl .”
§ Is asymptotically safe inflation eternal? > 5 Conclusions
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“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.”
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
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“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 -.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“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.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“For large values of φ the potential flattens out and the slow-roll conditions are not violated.”
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
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“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.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“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.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“The above figures are qualitatively similar because for the small values of φ max , the effective potential is highly asymmetric (”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“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 .”
§ Is asymptotically safe inflation eternal? > 4.2 RG-improved Starobinsky inflation
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“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.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“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].”
§ Is asymptotically safe inflation eternal? > 1 Introduction
What the sources say
- Not matchedcites [3]
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In the paper
“As pointed out by Donoghue [27] below the Planck scale, for quantum gravity one can safely take the effective field theory perspective.”
§ Is asymptotically safe inflation eternal? > 1 Introduction
What the sources say
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In the paper
“In line of these swampland criteria the no eternal inflation principle has been put forward [14]”
§ Is asymptotically safe inflation eternal? > 1 Introduction
What the sources say
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“our findings suggest no tension between eternal inflation and asymptotic safety, both in the matter and gravitational sector in contradistinction to string theory”
§ Is asymptotically safe inflation eternal? > Abstract
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“We verified, that the change of H caused by the field fluctuation does not affect the conclusions for eternal inflation.”
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
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In the paper
“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).”
§ Is asymptotically safe inflation eternal? > 4.2 RG-improved Starobinsky inflation
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“This shows that the model is on a verge of being eternally inflating, which may point out to the interesting phenomenology.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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“The creation of scattered, causally disconnected regions of spacetime -the multiverse is not confirmed observationally and raises question about the inflationary predictions [77].”
§ Is asymptotically safe inflation eternal? > 5 Conclusions
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“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].”
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
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“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.”
§ Is asymptotically safe inflation eternal? > 4.4 Tunneling through the potential barrier
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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”
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.
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In the paper
§ Is asymptotically safe inflation eternal? > 2.3 Eternal inflation conditions
“P r[φ > φ c , t] = 1 2 erfc α 3H t -φ c H 2π √ 2Ht . (2.20)”
§ Is asymptotically safe inflation eternal? > Case 2. Linear potential.
“µ (t) = α 3H t, σ 2 (t) = H 3 4π 2 t.”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Is asymptotically safe inflation eternal? > JCAP04(2021)076
“|V | V 3 2 < √ 2 2π 1 M 2 Pl . (2.23)”
§ Is asymptotically safe inflation eternal? > 2.1 Fokker-Planck equation
“3H φ + ∂V ∂φ ≈ 0, H 2 M 2 Pl ≈ 1 3 V (φ) . (2.3)”
Citations
Cites
Inflationary universe: A possible solution to the horizon and flatness problems
Guth, A. H. (1981) · Physical Review D · doi:10.1103/physrevd.23.347 · Cited by 9,983
A new type of isotropic cosmological models without singularity
Starobinsky, A. A. (1980) · Physics Letters B · doi:10.1016/0370-2693(80)90670-x · Cited by 7,677
Ade, P. A. R., Aghanim, N., Arnaud, M., Arroja, F. M. A., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., et al. (2016) · ORCA Online Research @Cardiff (Cardiff University) · doi:10.17863/cam.52 · Cited by 2,497
Akrami, Y., Arroja, F. M. A., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., Bartolo, N., Basak, S., et al. (2019) · Astronomy and Astrophysics · doi:10.1051/0004-6361/201833887 · Cited by 2,435 · cited 1×
“The time of inflation larger than 60 Planck-times is unlikely according to the Planck Collaboration data [88].”
MethodGeneral relativity as an effective field theory: The leading quantum corrections
Donoghue, J. F. (1994) · Physical Review D · doi:10.1103/physrevd.50.3874 · Cited by 1,294
On the geometry of the string landscape and the swampland
Ooguri, H., & Vafa, C. (2006) · Nuclear Physics B · doi:10.1016/j.nuclphysb.2006.10.033 · Cited by 1,185 · cited 1×
“In string theory this goes under the name of swampland conjectures [2, 3].”
BackgroundNonperturbative evolution equation for quantum gravity
Reuter, M. (1998) · Physical Review D · doi:10.1103/physrevd.57.971 · Cited by 1,160
Birth of inflationary universes
Vilenkin, A. (1983) · Physical Review D · doi:10.1103/physrevd.27.2848 · Cited by 998 · cited 1×
“Eternal inflation can be understood as a random walk of a field and a diffusion process on top of the classical motion [15, 84, 85].”
BackgroundReconstructing the inflaton potential—an overview
Lidsey, J. E., Liddle, A. R., Kolb, E. W., Copeland, E. J., Barreiro, T., & Abney, M. (1997) · Reviews of Modern Physics · doi:10.1103/revmodphys.69.373 · Cited by 878 · cited 1×
“It is possible to perturbatively recover information about the shape of the potential from the CMB, for details see [90].”
BackgroundRenormalizable asymptotically free quantum theory of gravity
Fradkin, E. S., & Tseytlin, A. A. (1982) · Nuclear Physics B · doi:10.1016/0550-3213(82)90444-8 · Cited by 665
Cited by
Sorted by citations
Dimensional reduction and (Anti) de Sitter bounds
Rudelius, T. (2021) · Journal of High Energy Physics · doi:10.1007/jhep08(2021)041 · Cited by 68
Analytic extensions of Starobinsky model of inflation
Ivanov, V. R., Ketov, S. V., Pozdeeva, E. O., & Vernov, S. Y. (2022) · Journal of Cosmology and Astroparticle Physics · doi:10.1088/1475-7516/2022/03/058 · Cited by 67
“, in the framework of asymptotically-safe quantum gravity [39, 40].”
BackgroundAsymptotic Safety: Swampland or Wonderland?
Basile, I., & Platania, A. (2021) · Universe · doi:10.3390/universe7100389 · Cited by 39
Rodrigues-da-Silva, G., Bezerra-Sobrinho, J., & Medeiros, L. G. (2022) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.105.063504 · Cited by 34
From quantum gravity to gravitational waves through cosmic strings
Eichhorn, A., dos Santos, R. R. L., & Miqueleto, J. L. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.109.026013 · Cited by 40
Finite action principle and Hořava-Lifshitz gravity: Early universe, black holes, and wormholes
Chojnacki, J., & Kwapisz, J. H. (2021) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.104.103504 · Cited by 14
Self-citationConstraining the quintessential α -attractor inflation through dynamical horizon exit method
Sarkar, A., & Ghosh, B. (2023) · Physics of the Dark Universe · doi:10.1016/j.dark.2023.101239 · Cited by 5
“advantage is in fact a generic feature of α-attractors [130, 131].”
BackgroundA novel way of constraining the $\alpha$-attractor chaotic inflation through Planck dataPreprint
Sarkar, A., Sarkar, C., & Ghosh, B. (2021) · arXiv (Cornell University) · doi:10.1088/1475-7516/2021/11/029
(No) Eternal Inflation in the Starobinsky Inflation Corrected by Higher Curvature Invariants
Chojnacki, J., & Kwapisz, J. (2023) · Astronomy · doi:10.3390/astronomy2010003
“In our previous work [14], we verified the applicability of Condition (2) (and further conditions stemming from the no eternal inflation principle) for a handful of asymptotically safe (AS) inflationary models [15–18].”
MethodSelf-citation
Metadata
- 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.
- 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
- 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
- bronze
- doi.org
- 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 › Nuclear and High Energy 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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