Cyclic cosmology and geodesic completeness
Open accessAbstract
We consider recently proposed bouncing cosmological models for which the Hubble parameter is periodic in time, but the scale factor grows from one cycle to the next as a mechanism for shedding entropy. Since the scale factor for a flat universe is equivalent to an overall conformal factor, it has been argued that this growth corresponds to a physically irrelevant rescaling, and such bouncing universes can be made perfectly cyclic, extending infinitely into the past and future. We show that any bouncing universe which uses growth of the scale factor to dissipate entropy must necessarily be geodesically past-incomplete, and therefore cannot be truly cyclic in time.
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Written from the paper’s full text.
TL;DR
The authors apply the Borde–Guth–Vilenkin (BGV) theorem to the Ijjas–Steinhardt cyclic cosmology, in which the Hubble parameter is periodic but the scale factor grows by a factor e^N each cycle. They show that any bouncing spacetime satisfying the model's entropy-dissipation condition and the Null Energy Condition away from the bounce is geodesically past-incomplete, so it is not truly cyclic.
Key findings
- The Ijjas–Steinhardt cyclic model, which solves Tolman's entropy-growth problem by having periodic H(t) with a(t+T) = e^N a(t), is geodesically past-incomplete for the same reason inflationary spacetimes are.
- The result is stated as general: any bouncing spacetime obeying the entropy-dissipation condition ∫₀^τ H₊ dt > ∫_τ^T H₋ dt (eq. 1.3) and the NEC outside the bounce must be geodesically incomplete.
- For de Sitter expansion the past proper time of a geodesic is finite, Δs = (1/2H) ln[(√(1+v₀²)+1)/(√(1+v₀²)−1)], for any nonzero v₀.
- For the cyclic case, the e-fold number is bounded between two de Sitter solutions, e^{H̄t} < exp(∫₀^t H dt) < e^{H̄t+N_max}, so the proper time Δs is squeezed between finite bounds Δs_min < Δs < Δs_max on −∞ < t < 0.
Methods
- Analytic study: a geodesic test particle in a flat Friedmann–Robertson–Walker metric ds² = dt² − a²(t)dx², with the geodesic equation giving a²(t) dx/ds ≡ v₀ = const and dt/ds = √(1 + v₀² a⁻²(t)).
- The BGV theorem is applied to a cosmology with H(t+T) = H(t), a(t+T) = e^N a(t), and a rapid bounce treated as a discontinuity in H(t) with continuous a(t); the NEC (dH/dt ≤ 0) is assumed to hold everywhere except at the bounce.
- Proper time along past-directed geodesics is integrated from t = 0 back to t = −∞ and bounded above and below by the corresponding de Sitter integrals, which are shown to be finite.
Limitations
- Stated: the analysis assumes the bounce is rapid enough that H(t) can be treated as discontinuous across it, with a(t) continuous; the bounce itself necessarily violates the Null Energy Condition and requires modification of General Relativity.
- Stated: the result is claimed to be independent of the details of the dynamics, but it is derived under the specific conditions of periodic H(t), exponential scale-factor growth, and the entropy-dissipation inequality (1.3).
- Evident from the text: the argument is restricted to flat FRW backgrounds with a single test-particle geodesic treatment, and the paper does not address black-hole formation or inhomogeneities, which it notes make a bounce strongly inhomogeneous.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
8 of 8 cited sources checked · 7 supported · 0 flagged · accuracy not yet measured
- Premisecitation
Cyclic cosmology suffers from the problem of entropy growth, as known since Tolman's 1930s work.
“It has been known since work by Tolman in the 1930s that cyclic cosmology suffers from the problem of entropy growth [2].”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [2] 2. R.C. Tolman, On the theoretical requirements for a periodic behaviour of the universe, Phys. Rev. 38 (1931) 1758 [INSPIRE]. source
Section 8 says entropy growth would prevent successive expansions and contractions from being identical, so Tolman identifies the problem. But §12 concludes that idealized cyclic models can expand and contract without increasing entropy; the claim overstates the problem as applying to cyclic cosmology generally. (LLM, conf=0.88)
Experimental — accuracy not yet measured
- Premisecitation
In the IS model, the scale factor grows exponentially from one cycle to the next, a(t+T)=e^N a(t).
“The scale factor, however, grows exponentially from one cycle to the next, as a mechanism for shedding the entropy produced in each cycle, a (t + T ) = e N a (t) , (1.2)”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [3] 3. A. Ijjas and P.J. Steinhardt, A new kind of cyclic universe, Phys. Lett. B 795 (2019) 666 [arXiv:1904.08022] [INSPIRE]. source
- Premisecitation
Inflationary spacetimes are geodesically past-incomplete, as shown by Borde, Guth, and Vilenkin.
“inflation, which was shown by Borde, Guth, and Vilenkin (BGV) to be geodesically past incomplete [1].”
- Also stated in: § JCAP06(2022)011 · boosted
“Inflationary spacetimes were shown to be generically geodesically incomplete by BGV in ref. [1].”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [1] 1. A. Borde, A.H. Guth and A. Vilenkin, Inflationary space-times are incompletein past directions, Phys. Rev. Lett. 90 (2003) 151301 [gr-qc/0110012] [INSPIRE]. source
- Also stated in: § JCAP06(2022)011 · boosted
- Findingreasoning
de Sitter expansion spacetime is geodesically incomplete.
“Since this integral is finite for any nonzero v 0 , the past proper time along the geodesic is finite, and the space is therefore geodesically incomplete.”
- Also stated in: § JCAP06(2022)011 · boosted
“it is straightforward to show that the spacetime is geodesically incomplete by integrating the proper time backward from time t = 0 to t = -∞ [6]”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [6] 6. A. Aguirre and S. Gratton, Steady state eternal inflation, Phys. Rev. D 65 (2002) 083507 [astro-ph/0111191] [INSPIRE]. source
- Also stated in: § JCAP06(2022)011 · boosted
- Premisecitation
The BGV theorem shows that any spacetime for which the average Hubble parameter is positive must be geodesically incomplete.
“This is consistent with the BGV theorem, which shows that any spacetime for which the average Hubble parameter is positive must be similarly geodesically incomplete.”
Quote in the paperSubstantial claim1 of 1 sources checked- cites [1] 1. A. Borde, A.H. Guth and A. Vilenkin, Inflationary space-times are incompletein past directions, Phys. Rev. Lett. 90 (2003) 151301 [gr-qc/0110012] [INSPIRE]. source
- Findingreasoning
Proper time ∆s satisfies ∆s_min < ∆s < ∆s_max.
“From eq. (2.16), proper time satisfies a bound ∆s min < ∆s < ∆s max (figure 3), where ∆s min = 0 -t i dt 1 + v 2 0 e -2Ht .(2.19)”
Quote in the paperSubstantial claim - Interpretationreasoning
Any bouncing spacetime which satisfies the condition (1.3) for entropy dissipation must be geodesically past-incomplete.
“In this section, we show that any bouncing spacetime which satisfies the condition (1.3) for entropy dissipation must be geodesically past-incomplete.”
- Also stated in: § JCAP06(2022)011 · plain
“Any spacetime periodic in H (t) and satisfying the condition (1.3) must therefore be geodesically incomplete.”
- Also stated in: § 3 Conclusions · boosted
“This result is completely general: any bouncing spacetime which obeys the condition (1.3) for entropy dissipation and the Null Energy Condition outside the bounce must be geodesically incomplete.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The bounce is rapid enough that H(t) can be taken discontinuous across the bounce, with the scale factor continuous.
- Also stated in: § JCAP06(2022)011 · plain
- Interpretationreasoning
In solving the problem of entropy growth, the IS model introduces the problem of geodesic incompleteness in its place.
“In solving the problem of entropy growth, the model introduces the problem of geodesic incompleteness in its place.”
Quote in the paperSubstantial claim - Interpretationcitation
The IS model is geodesically past-incomplete.
“In the next section, we apply the BGV theorem to the IS model, and show that it, like the inflationary spacetime, is geodesically incomplete, and for the same reason.”
- Also stated in: § 3 Conclusions · boosted
“In this paper, we use the BGV theorem to demonstrate that growth in the scale factor inevitably means that the spacetime is geodesically past-incomplete.”
Quote in the paperSubstantial claim - Also stated in: § 3 Conclusions · boosted
- Interpretationnone given
In the limit that the Hubble parameter undergoes a discontinuity at the bounce, the scale factor is well-behaved and continuous.
“In the limit that the Hubble parameter undergoes JCAP06(2022)011 a discontinuity at the bounce, the scale factor is nonetheless well-behaved and continuous.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The bounce is rapid enough that H(t) can be taken discontinuous across the bounce, with the scale factor continuous.
- Interpretationreasoning
Despite periodic H(t), a difference in average equation of state between expanding and contracting phases results in growth of the scale factor.
“Despite periodic H (t), a difference in average equation of state between expanding and contracting phases results in growth of the scale factor.”
Quote in the paperSubstantial claim - Interpretationreasoning
Geodesic spacetime paths can be traced through multiple cycles of expansion, contraction, and bounce.
“This means that geodesic spacetime paths can be traced through multiple cycles of expansion, contraction, and bounce.”
Quote in the paperSubstantial claim - Interpretationreasoning
For a universe with H(t) periodic satisfying condition (1.3), there must exist a mean expansion rate H > 0.
“The condition for entropy dissipation (1.3) means there must exist a mean expansion rate H on the interval 0 < t < T such that H ≡ 1 T T 0 H t dt > 0. (2.10)”
Quote in the paperSubstantial claim - Interpretationreasoning
The IS cosmology cannot be past eternal, independent of the details of the dynamics.
“The IS cosmology satisfies this condition and therefore cannot be past eternal, independent of the details of the dynamics.”
Quote in the paperSubstantial claim - Interpretationreasoning
The number of e-folds of expansion is bounded from above and below by two de Sitter solutions.
“The number of e-folds of expansion is then bounded from above and below by two de Sitter solutions (figure 2), e Ht < exp t 0 H t dt < e Ht+Nmax .(2.16)”
Quote in the paperSubstantial claim - Interpretationreasoning
Any such spacetime is not truly cyclic, in the sense of being perfectly periodic in time.
“Therefore any such spacetime is not truly cyclic, in the sense of being perfectly periodic in time.”
Quote in the paperSubstantial claim
- Classified as probably not a substantial claim. 3
- May combine two claims. 4
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationThe Ijjas-Steinhardt (IS) model suggests a solution to the entropy problem in cyclic cosmology.
“A model recently proposed by Ijjas and Steinhardt (IS) [3] suggests a solution to the entropy problem in cyclic cosmology [4].”
Quote in the paperNot a substantial claim2 of 2 sources checked- cites [3] 3. A. Ijjas and P.J. Steinhardt, A new kind of cyclic universe, Phys. Lett. B 795 (2019) 666 [arXiv:1904.08022] [INSPIRE]. source
- cites [4] 4. A. Ijjas and P.J. Steinhardt, Entropy, black holes, and the new cyclic universe, Phys. Lett. B 824 (2022) 136823 [arXiv:2108.07101] [INSPIRE]. source
Probably not a substantial claim: may not pass S3 (load-bearing).
PremisecitationIn the IS model, the Hubble parameter is exactly periodic in time, H(t+T)=H(t).
“The IS model postulates a cosmology with alternating expanding and contracting phases with period T , such that the Hubble parameter is exactly periodic, H (t + T ) = H (t) .”
Quote in the paperNot a substantial claim1 of 1 sources checked- cites [3] 3. A. Ijjas and P.J. Steinhardt, A new kind of cyclic universe, Phys. Lett. B 795 (2019) 666 [arXiv:1904.08022] [INSPIRE]. source
May combine two claims.
Findingreasoning∆s_min and ∆s_max are finite on the interval -∞ < t < 0, so ∆s is finite on the same interval.
“From eq. (2.9), ∆s min and ∆s max are finite on the interval -∞ < t < 0, which means that ∆s min < ∆s < ∆s max must also be finite on the same interval.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.
InterpretationreasoningThe spacetime is trivially both future- and past-infinite for comoving observers.
“The spacetime is trivially both future-and past-infinite for comoving observers.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationnone givenAt the bounce, the Hubble parameter undergoes a sudden transition from small and negative to large and positive, which necessarily violates the Null Energy Condition, requiring modification to General Relativity.
“At the bounce itself, the Hubble parameter undergoes a sudden transition from small and negative to large and positive, which necessarily violates the Null Energy Condition, requiring modification to General Relativity.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningThe spacetime is nonsingular over the range t = [-∞, ∞], with a(t→-∞) ∝ e^N → 0 as N → -∞.
“Because the scale factor varies exponentially over the number of cycles, the spacetime is nonsingular over the range t = [-∞, ∞], with a (t → -∞) ∝ e N → 0 (1.13) as N → -∞.”
Quote in the paperSubstantial claim
The authors state this as an assumption.
The bounce is rapid enough that H(t) can be taken discontinuous across the bounce, with the scale factor continuous.
“We assume that the bounce is rapid enough that we can take H (t) to be discontinuous across the bounce at the boundary of the interval, with scale factor a (t) continuous, as shown in figure 1.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The bounding-space argument applies to cyclic universes whose scale factor is oscillatory but grows from one cycle to the next.
“This applies not only to spaces where the scale factor a grows monotonically, but also to cyclic universes for which the scale factor a ( t ) is oscillatory, but grows from one cycle to the next [2].”
From Borde-Guth-Vilenkin theorem in extended de Sitter spaces (2024) · cites this paper as [2]
Geodesic incompleteness of the bounding de Sitter space implies geodesic incompleteness of the target space with expansion history H(s).
“Then geodesic incompleteness of the bounding de Sitter space implies geodesic incompleteness of the space with expansion history H ( s ), which we will refer to in what follows as the target space.”
From Borde-Guth-Vilenkin theorem in extended de Sitter spaces (2024) · cites this paper as [2]
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.
Contracting-branch sign conflicts with the stated Hubble rate
MinorDuring contraction, ȧ/a is negative. Equation (2.2) assigns the minus sign to that branch, which would make H positive, contrary to the paper's earlier definition of the contracting rate as negative.
“H ≡ ± ȧ a , (2.2) with the (+) branch corresponding to an expanding phase, and the (-) branch corresponding to a contracting phase.”
“Calling H + (t) > 0 the Hubble parameter during expansion, and H -(t) < 0 the corresponding rate of contraction”
Cites
Inflationary Spacetimes Are Incomplete in Past Directions
Borde, A., Guth, A. H., & Vilenkin, A. (2003) · Physical Review Letters · doi:10.1103/physrevlett.90.151301 · Cited by 582 · see citing works
Ekpyrotic and Cyclic Cosmology
Lehners, J.-L. (2008) · Physics Reports · doi:10.1016/j.physrep.2008.06.001 · Cited by 381 · see citing works
Ijjas, A., & Steinhardt, P. J. (2019) · Physics Letters B · doi:10.1016/j.physletb.2019.06.056 · Cited by 145 · see citing works
On the Theoretical Requirements for a Periodic Behaviour of the Universe
Tolman, R. C. (1931) · Physical Review · doi:10.1103/physrev.38.1758 · Cited by 145 · see citing works
Steady state eternal inflation
Aguirre, A., & Gratton, S. (2002) · Physical Review D · doi:10.1103/physrevd.65.083507 · Cited by 68 · see citing works
Entropy, black holes, and the new cyclic universe
Ijjas, A., & Steinhardt, P. J. (2022) · Physics Letters B · doi:10.1016/j.physletb.2021.136823
Steady-State Eternal Inflation
Aguirre, A., & Gratton, S. (2001) · arXiv · doi:10.48550/arxiv.astro-ph/0111191
Ijjas, A., & Steinhardt, P. J. (2019) · arXiv · doi:10.48550/arxiv.1904.08022
Entropy, Black holes, and the New Cyclic Universe
Ijjas, A., & Steinhardt, P. J. (2021) · arXiv · doi:10.48550/arxiv.2108.07101
Ekpyrotic and Cyclic Cosmology
Lehners, J.-L. (2008) · arXiv · doi:10.48550/arxiv.0806.1245
Flagged first, then by citations
Cited by
Sorted by citations
On the initial singularity and extendibility of flat quasi-de Sitter spacetimes
Geshnizjani, G., Ling, E., & Quintin, J. (2023) · Journal of High Energy Physics · doi:10.1007/jhep10(2023)182 · Cited by 20 · see citing works
Inflation and late-time accelerated expansion driven by k -essence degenerate dynamics
(2024) · doi:10.1103/physrevd.109.023515 · Cited by 12 · see citing works
“[36], in which there is a net growth of the scale factor, have, however, been shown to be geodesic past– incomplete [37, 38] and, therefore, not truly cyclical, as the geodesics meet somewhere in the past.”
BackgroundBorde-Guth-Vilenkin theorem in extended de Sitter spaces
Kinney, W. H., Maity, S., & Sriramkumar, L. (2024) · Physical Review D · doi:10.1103/physrevd.109.043519 · Cited by 10 · see citing works
Self-citationWaiting for inflation: A new initial state for the Universe
Melcher, B., Pradhan, A., & Watson, S. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.110.063517 · Cited by 4 · see citing works
Geodesic completeness, cosmological bounces, and inflation
García-Sáenz, S., Hua, J., & Zhao, Y. (2024) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.110.l061304 · Cited by 4 · see citing works
On Efforts to Decouple Early Universe Cosmology and Quantum Gravity Phenomenology
Schneider, M. D. (2023) · Foundations of Physics · doi:10.1007/s10701-023-00720-y · Cited by 2 · see citing works
“4 A modification of the BVG result [8] is used in [26] to argue that the new cyclic cosmology must be past-incomplete with respect to at least some non-comoving geodesic.”
MethodGeodesically complete cyclic cosmologies and entropy
Pavlović, P., & Sossich, M. (2024) · The European Physical Journal C · doi:10.1140/epjc/s10052-024-12621-z · Cited by 3 · see citing works
An Infinitely Old Universe with Planck Fields Before and After the Big Bang
Pilipović, D. (2024) · Universe · doi:10.3390/universe10100400
“In search of the answer, some research considers bouncing cosmological models to show that a growing scale factor allowing entropy to dissipate cannot be cyclic in time [1,2].”
Kambouris, M. E., Ludovici, G. M., Karageorgou, K., Καντζάνου, Μ., & Patrinos, G. P. (2026) · One Health Microbiology & Infection · doi:10.66585/ohmi.2026.2.0024
Bouncing completion of eternal inflation
Ferreira, A. L., Pinto-Neto, N., & Xavier, V. N. (2025) · Physical review. D/Physical review. D. · doi:10.1103/rl3f-v3p3
“Recently, the theorem gained renewed attention as cos-mologies with cyclic properties, such as periodic Hubble functions and/or conformal cycles [5–7], were argued to be incomplete [8, 9].”
Is a quantum gravity era necessary?
Veklych, B. (2025) · Reviews in Physics · doi:10.1016/j.revip.2025.100116
- Authors
- Kinney, W. H., & Stein, N. K.
- Affiliations
- University at Buffalo, State University of New York (US)
- Corresponding author
- Kinney, William H.
- Stein, Nina K.
- ORCID
- Kinney, William H.
- Venue
- Journal of Cosmology and Astroparticle Physics
- Publisher
- IOP Publishing
- Published
- 2022-06-01
- Volume
- 2022
- Issue
- 06
- Pages
- 011
- ISSN
- 1475-7508 (ISSN-L)
- 1475-7516
- OpenAlex
- W3209243388
- Cited by
- 13
- References
- 7
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 7 · 2023: 2 · 2022: 1
- FWCI
- 1.20
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Journal of Cosmology and Astroparticle Physics (publishedVersion)
- arXiv (Cornell University) (acceptedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (2014021)
- 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 › Astronomy and Astrophysics › Advanced Differential Geometry Research
- Keywords
- cyclic cosmology, bounce cosmology, geodesic completeness, entropy dissipation, conformal rescaling, flat universe
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