A Cyclic Model of the Universe

Neil Turok; Paul J. Steinhardt

REVIEW

A Cyclic Model of the Universe

Paul J. Steinhardt, Neil Turok

High Energy Physics - Theory · Submitted 4 Nov 2001 · 29 references

Pith's verdict: A genuinely new cosmological model with a sharp falsifiable prediction, built on one uncomputed step the authors are upfront about.

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arXiv IDhep-th/0111030 (version 2)

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CitationNeil Turok, Paul J. Steinhardt. "A Cyclic Model of the Universe." arXiv:hep-th/0111030 (2001). https://pith.science/paper/MCP2MPOA

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Editor's note

read the letter

The thing to know: this is not a revival of the 1930s oscillatory universe. Steinhardt and Turok build a flat universe that cycles forever, and they make the bounce finite by coupling matter and radiation to a conformal factor β(φ) so that the brane scale factor â = aβ stays nonzero as a → 0. That single move is what lets them claim finite temperature and density at the crunch-to-bang transition (abstract:s0), and it is the real content of the paper. The equations of motion (2)–(5) drop out of the action cleanly, and the stage-by-stage tour of the cycle is internally consistent. The best part is the prediction: no long-wavelength gravitational waves, because the perturbations come from φ rolling down a potential rather than from gravity stretching modes. That is a real bet, it does not depend on any fitted number, and it is the kind of thing that gets settled by data. Giving the observed dark energy a structural job — diluting entropy and debris so the next cycle starts clean — is also a nice piece of model building, and it is a better story than inflation's unexplained coincidence.

Referee report

Significance

If the construction holds together, it is a significant reframing of cosmic history: it dispenses with an inflationary epoch, treats the crunch as a transition rather than a singularity, gives the recently observed acceleration a functional rather than incidental role, and yields a clean observational discriminant from inflation (no long-wavelength tensor modes, testable via CMB polarization and direct gravitational-wave searches). The brane-world realization ties the ingredients to M-theory rather than introducing them ad hoc. Credit where due: the equations of motion (2)–(5) follow cleanly from the action (1); the same V0 does multiple duty (acceleration, dilution, and the reversal from expansion to contraction), which is an economical use of one tuned quantity; and the prediction of no long-wavelength gravitational waves is a genuine, falsifiable statement that distinguishes the model from inflation.

Major objections (4)

[Cyclic solution as Cosmic Attractor] The attractor argument is heuristic and covers only the amplitude of φ on the positive plateau: critical damping erases memory of where the field started, so it arrives at V = 0 with a universal rolling rate. The Tolman objection, however, concerns the full return map, which must include the radiation injected with fraction χ at the bounce (Eq. 7) and the subsequent radiation-dominated damping. No such map is written down, and no bound on χ is given, so the statement that 'the cyclic solution is a stable attractor' is asserted for a one-dimensional projection of a multi-variable map. A concrete demonstration would be to exhibit the map (φ_max, ρ_R) → (φ_max′, ρ_R′) across one full cycle and show that it contracts, together with an argument that entropy per cycle is bounded rather than accumulating.
[Obtaining scale-invariant perturbations] The step that turns the scale-invariant spectrum of φ generated during slow contraction into a scale-invariant spectrum of density fluctuations is attributed to ref. 25, listed as 'Phys. Rev. D, in press (2002).' Because the bounce is described in the same manuscript as non-adiabatic and particle-producing, it is not obvious that the spectral index or gaussianity survives the matching rule of Eq. (7), and the letter does not indicate the mechanism by which they do. At minimum the manuscript should state the result it relies on and the conditions under which the conversion preserves the index. Separately, the amplitude of the fluctuations (the COBE normalization) is not discussed at all; a nearly scale-invariant index does not by itself establish that the model reproduces the observed fluctuation amplitude.
[Essential Ingredients and Stringy motivation] The quantitative backbone of the 'realistic cosmology' claim rests on ref. 14 (hep-th/0111098), which supplies the choice c ≥ 10, the identification of V0 with today's dark energy density, the fifth-force bound (ln β),φ ≪ 10⁻³, and the attractor analysis. That reference is an unpublished preprint at the time of writing, so the reader cannot check the load-bearing numbers. The letter should summarize the key results it depends on — the allowed range of c, the derivation of the fifth-force bound, and the sense in which the solution is an attractor — so that the central claim can be assessed from the manuscript itself.
[Comparing cyclic and inflationary model] The manuscript states that 'the cyclic model leads naturally to the prediction of quintessence and cosmic acceleration,' but V0 in Eq. (6) is set equal to the observed dark energy density, as the text following Eq. (6) makes explicit. The model therefore does not predict the magnitude of the acceleration; what it predicts is that, given a positive plateau, acceleration occurs and performs the required dilution. The authors are candid that 'this choice entails tuning V0,' but the word 'prediction' in the comparison section is stronger than that candor supports and should be qualified. Relatedly, the attractor argument addresses insensitivity to φ's starting amplitude, not the timing question of why acceleration is beginning now; that coincidence should either be addressed or explicitly set aside.

Minor objections (7)

[A brief tour of the cyclic universe] The stage numbering jumps from stage (3) to stage (5); stage (4) is never described in the text or identified in the Fig. 1 caption. Either define stage (4) or renumber the sequence.
[Eq. (5)] The symbol ∂ρ_i/∂φ_i carries a stray index i on φ; it should be ∂ρ_i/∂φ. The notation β′ should be defined explicitly as β,φ to match the notation used in Eq. (4).
[Eq. (6) and Fig. 1] The role of F(φ) on the positive plateau deserves one explicit sentence: the text says it is 'nearly unity for φ to the right of potential minimum,' but the reader must reconstruct why the plateau value V0 is unaffected while the φ → −∞ limit is driven to zero.
[Fig. 1 caption] The caption's stage markers (solid, grey, open, broken circles) should be cross-referenced to the numbered stages used in the text, particularly since radiation domination occurs at both stage (1) and stage (7) and the broken circle appears to denote both.
[Eq. (7)] The matching rule would be clearer if the text stated that both sides are evaluated at the bounce and explained that the exponential factor e^{√(3/2)φ} is what renders both sides finite as φ → −∞.
[Dark energy and the cyclic model] The statements that the acceleration is 'nearly 100 orders of magnitude smaller' than in inflation and that the particle number 'may be suppressed to less than one per Hubble volume' are order-of-magnitude claims; a short footnote giving the estimate would let the reader check them.
[References] Ref. 26 contains a stray 'i' before 'Phys. Rev. D64'; please correct. Ref. 25 is listed as 'in press,' and its status should be updated if a volume and page are available.

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The paper

The paper claims

The authors propose a universe that expands, accelerates, contracts and bounces forever, with finite temperature and density at every crunch, and argue this single loop reproduces what inflation explains while giving the observed cosmic加速 a

The reading

The paper proposes that the universe has neither a first moment nor a last one: it repeats a cycle of hot expansion, a long stretch of slow accelerated expansion, a slow contraction, and a bounce back to a hot state. The authors want to show this loop can deliver everything inflation is usually invoked for — a flat, homogeneous universe seeded with a nearly scale-invariant spectrum of density fluctuations — without an inflationary epoch, and that the recently observed cosmic acceleration is a working part of the mechanism rather than an unexplained extra. The claim matters because it removes the need to specify initial conditions, gives dark energy a job, and draws a sharp observational line against inflation: no long-wavelength gravitational waves. The load-bearing step is a postulated rule for what happens when the branes collide at the crunch, which the authors say must eventually be derived from a fuller calculation.

What carries the argument

Three pieces carry the argument. First, a scalar potential $V(φ)$ with a specific shape: it rises to a shallow positive plateau $V_0$ at large $φ$ (the dark energy), passes through zero, goes negative at intermediate $φ$, and falls to zero as $φ → -∞$ where the branes collide. Second, the conformal coupling $β(φ)$ in the action, which rescales the scale factor to $â = aβ(φ)$ and is what keeps energy density and curvature finite at the crunch — the paper's answer to the singularity. Third, a matching rule at the bounce, $φ̇ e^{√{3/2}φ} → -(1+χ) φ̇ e^{√{3/2}φ}$, which converts a small fraction of the field's kinetic energy into radiation, with the efficiency $χ$ left as a free parameter. The perturbation spectrum is generated not by gravity stretching sub-Planckian modes but by the rolling field during quasi-static contraction, so only $φ$ gets a scale-invariant spectrum and gravitational waves do not.

Extended reading notes

Core claim

The central claim is that a flat universe can cycle indefinitely through bang, radiation- and matter-dominated expansion, a plateau of dark-energy-driven acceleration lasting trillions of years, slow contraction, and a bounce, with temperature, density and brane curvature finite at the transition. Finiteness comes from coupling the scalar field $V(φ)$ to matter and radiation through a function $β(φ)$ so that the physically measured scale factor is $â = aβ(φ)$ rather than $a$; as $a → 0$ and $φ → -∞$, $β$ falls as $e^{-φ/√6}$ and $â$ stays finite. The long accelerating phase is not decoration: it dilutes entropy, black holes and other debris from the previous cycle, flattens the universe, and makes the cyclic trajectory an attractor, answering the old Tolman objection that entropy should pile up cycle after cycle. During the slow contraction that follows, quantum fluctuations in $φ$ are amplified as the potential steepens, producing a nearly scale-invariant, adiabatic, gaussian spectrum of density perturbations that mimics inflation's without invoking it.

Load-bearing premise

The crunch-to-bang matching rule, equation (7), which converts a small fraction of the scalar field's kinetic energy into radiation with an efficiency $χ$ that the paper postulates by analogy with string theory and explicitly leaves for a future calculation to determine.

Editorial extensions

If this is right

A flat universe needs no beginning: the cyclic solution is claimed to be an attractor, so a wide range of initial conditions on the potential plateau converge to the same sequence of epochs.
The observed cosmic acceleration is predicted rather than surprising, since hundreds of e-folds of dark energy domination are required to reset the universe before each contraction.
The primordial perturbation spectrum is nearly scale-invariant, adiabatic and gaussian, matching current cosmic microwave background and large-scale structure data as well as inflation does.
There should be no significant background of long-wavelength gravitational waves, in contrast to standard inflation.
The true ground state has large negative energy density, but the universe never settles into it; it hovers above it, spending most of its time on the positive-energy side.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the brane-collision matching rule can be computed from first principles, $χ$ stops being a free dial and the model turns into a parameter-free statement about the equation of state at the bounce; the same calculation would settle whether entropy really dilutes between cycles.
Because the coupling $β(φ)$ that regularizes the crunch also mediates a force between matter particles, laboratory equivalence-principle and varying-constant measurements constrain the very parameter that controls whether the bounce is finite — a link the paper notes but does not quantify.
A recycling universe turns the low-entropy past from a one-time initial condition into a per-cycle requirement: the accelerating phase must be efficient enough to erase the previous cycle's entropy, which is a quantitative constraint on how many e-folds of dark energy domination actually occur.
The picture reframes the cosmological constant problem as a question about why the universe never reaches its ground state, rather than why the ground-state energy is zero; any proposal that ties vacuum energy to observable quantities has to explain the hovering.

Original abstract

We propose a cosmological model in which the universe undergoes an endless sequence of cosmic epochs each beginning with a `bang' and ending in a `crunch.' The temperature and density are finite at each transition from crunch to bang. Instead of having an inflationary epoch, each cycle includes a period of slow accelerated expansion (as recently observed) followed by slow contraction. The combination produces the homogeneity, flatness, density fluctuations and energy needed to begin the next cycle.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,950 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.33 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We propose a cosmological model in which the universe undergoes an endless sequence of cosmic epochs each beginning with a `bang' and ending in a `crunch.' The temperature and density are finite at each transition from crunch to bang. Instead of having an inflationary epoch, each cycle includes a period of slow accelerated expansion (as recently observed) followed by slow contraction. The combination produces the homogeneity, flatness, density fluctuations and energy needed to begin the next cycle.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a cosmological model in which the universe undergoes an endless sequence of cosmic epochs each beginning with a `bang' and ending in a `crunch.' The temperature and density are finite at each transition from crunch to bang.	contribution 0.99	0.71	0.63	asserted 0.69
s1	Instead of having an inflationary epoch, each cycle includes a period of slow accelerated expansion (as recently observed) followed by slow contraction.	contribution 0.58	0.37	0.14	asserted 1.00
s2	The combination produces the homogeneity, flatness, density fluctuations and energy needed to begin the next cycle.	interpretation 0.71	0.42	0.29	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from argument 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Schematic plot of the potential V (φ) versus field φ. In M theory, φ determines the distance between branes, and φ → −∞ as the branes collide. We define φ to be zero where V (φ) crosses zero and, therefore, φ is positive when the branes are at their maximal separation. Far to the right, the potential asymptotes to ρ 0 Q, the current value of the quintessence (dark energy) density. The solid circles represent the dark energy dominated stage; the grey represent the contracting phase during which density fluctuations are generated; the open circles represent the phase when the scalar…

Figures reproduced from arXiv:hep-th/0111030 by the authors.

References

Works cited (29)

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A. H. Guth, Phys. Rev. D 23 (1981) 347

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A. D. Linde, Phys. Lett. B 108 (1982) 389

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A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220

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J. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28, 679 (1983)

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N. Bahcall, J.P. Ostriker. S. Perlmutter, and P.J. Stein hardt, Science 284 (1999) 1481

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R.C. Tolman, Relativity, Thermodynamics and Cosmology , (Oxford U. Press, Clarendon Press, 1934)

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See, for example, R.H. Dicke and J.P.E. Peebles, in General Relativity: An Ein- stein Centenary Survey , ed. by S.H. Hawking and W. Israel, (Cambridge U. Press, Cambridge, 1979)

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T. Damour, gr-qc/0109063

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H. Liu, G. Moore, and N. Seiberg, hep-th/0204618

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J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Phys. Rev. D64, 123522, (2001)

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R.R. Caldwell, R. Dave and P.J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998); see also, J. Ostriker and P.J. Steinhardt, Sci. Am. , p. 46, January, 2001

View on Pith 1998
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J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, hep- th/0109050; Phys. Rev. D, in press (2002)

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R. Kallosh, L. Kofman, and A. Linde, i Phys. Rev. D64 , 123523 (2001)

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J. Khoury, B.A. Ovrut, P.J.Steinhardt and N. Turok, hep- th/0105212

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We thank M. Bucher, R. Durrer, S. Gratton, J. Khoury, B.A. Ovrut, J. Ostriker, P.J.E. Peebles, A. Polyakov, M. Rees, N. Seiberg, D. Spergel , A. Tolley, T. Wiseman and E. Witten for useful conversations. We thank L. Rocher fo r pointing out historical references. This work was supported in part by US Department of Energy grant DE- FG02-91ER40671 (PJS) and...

View on Pith

Forward citations

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How the literature uses it

56 arXiv papers cite this in 76 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 23 sentences without a confident label

Compares with it 1 paper

In our case, different from the scenario discussed in Ref. [ref] , the vacuum energy density would decrease after each cycle due to dissipation.

Relaxation of vacuum energy in q-theory · Conclusion · 0.98

Cites it as support 2 papers

But we can think it over that the cyclic models of the universe must have some initial cycle, otherwise we would be at odds with the results in thermodynamics, [ref] .

New results concerning the curvature of the Universe · Conclusion · 0.97
This assumption seems well justified in light of recent works analysing the beginning stages of inflation [ref] , and it is well founded for ekpyrosis, in particular in the context of its cyclic realisations [ref] .

Quantum Circuit Complexity of Primordial Perturbations · Quantisation of cosmological perturbations and complexity thereof · 0.97

Cites it as background 41 papers

In the cyclic universe scenario [ref] an evolution of the universe consists of the sequence of classical and quantum phases.

Dirac quantization of membrane in time dependent orbifold · Introduction · 0.98
An alternative model has been proposed by Steinhardt and Turok (ST) [ref] .

Modelling cosmological singularity with compactified Milne space · Introduction · 0.90
Steinhardt and Turok worked with some sort of phantom field in their cyclic model of the universe from the year 2001, [ref] .

New results concerning the curvature of the Universe · Theoretical evidence for positive curvature · 0.88

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

A. H. Guth, Phys. Rev. D 23 (1981) 347

View on Pith 1981

[2] [2]

A. D. Linde, Phys. Lett. B 108 (1982) 389

View on Pith 1982

[3] [3]

A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220

View on Pith 1982

[4] [4]

J. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28, 679 (1983)

View on Pith 1983

[5] [5]

A. H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110

View on Pith 1982

[6] [6]

A. A. Starobinskii, Phys. Lett. B 117 (1982) 175

View on Pith 1982

[7] [7]

S. W. Hawking, Phys. Lett. B 115 (1982) 295

View on Pith 1982

[8] [8]

S. Perlmutter, et al , Ap. J. 517 (1999) 565

View on Pith 1999

[9] [9]

A. G. Riess, et al , Astron.J. 116 (1998) 1009. 14

View on Pith 1998

[10] [10]

P. M. Garnavich et al , Ap. J. 509 (1998) 74

View on Pith 1998

[11] [11]

N. Bahcall, J.P. Ostriker. S. Perlmutter, and P.J. Stein hardt, Science 284 (1999) 1481

View on Pith 1999

[12] [12]

R.C. Tolman, Relativity, Thermodynamics and Cosmology , (Oxford U. Press, Clarendon Press, 1934)

View on Pith 1934

[13] [13]

See, for example, R.H. Dicke and J.P.E. Peebles, in General Relativity: An Ein- stein Centenary Survey , ed. by S.H. Hawking and W. Israel, (Cambridge U. Press, Cambridge, 1979)

View on Pith 1979

[14] [14]

P.J. Steinhardt and N. Turok, hep-th/0111098

View on Pith Pith review arXiv

[15] [15]

T. Damour and A. Polyakov, Nucl. Phys. B 423 (1994) 532

View on Pith 1994

[16] [16]

T. Damour, gr-qc/0109063

View on Pith Pith review arXiv

[17] [17]

H. Liu, G. Moore, and N. Seiberg, hep-th/0204618

View on Pith arXiv

[18] [18]

A.J. Tolley and N. Turok, hep-th/0204091

View on Pith Pith review arXiv

[19] [19]

J. Dai, R.G. Leigh and J. Polchinski, Mod. Phys. Lett. A 4 (1989) 2073

View on Pith 1989

[20] [20]

P. Hoˇ rava and E. Witten, Nucl. Phys. B 460 (1996) 506; B475 (1996) 94

View on Pith 1996

[21] [21]

J. Polchinski, String Theory, Vols. I and II , (Cambridge University Press, Cam- bridge, 1998)

View on Pith 1998

[22] [22]

J. Khoury, B.A. Ovrut, N. Seiberg, P.J. Steinhardt and N. Turok, Phys. Rev. D65, 086007 (2002). 15

View on Pith 2002

[23] [23]

J. Khoury, B.A. Ovrut, P.J. Steinhardt and N. Turok, Phys. Rev. D64, 123522, (2001)

View on Pith 2001

[24] [24]

R.R. Caldwell, R. Dave and P.J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998); see also, J. Ostriker and P.J. Steinhardt, Sci. Am. , p. 46, January, 2001

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We thank M. Bucher, R. Durrer, S. Gratton, J. Khoury, B.A. Ovrut, J. Ostriker, P.J.E. Peebles, A. Polyakov, M. Rees, N. Seiberg, D. Spergel , A. Tolley, T. Wiseman and E. Witten for useful conversations. We thank L. Rocher fo r pointing out historical references. This work was supported in part by US Department of Energy grant DE- FG02-91ER40671 (PJS) and...

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