An Open Universe from Inflation

A.S. Goldhaber (SUNYSB); M. Bucher (Princeton U.); N. Turok

REVIEW

An Open Universe from Inflation

M. Bucher (Princeton U.), A.S. Goldhaber (SUNYSB), N. Turok

High Energy Physics - Phenomenology · Submitted 2 Nov 1994 · 38 references

Pith's verdict: The founding one-bubble open-inflation paper: the Ω₀ calculation is solid and worth citing, the spectrum is a genuine first-principles result but carries a factor-of-two inconsistency and acknowledged approximation limits at large angles.

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arXiv IDhep-ph/9411206 (version 6)

Pith ID54LXMXFV, Pith's permanent record of this paper and its review

CitationA.S. Goldhaber (SUNYSB), M. Bucher (Princeton U.), N. Turok. "An Open Universe from Inflation." arXiv:hep-ph/9411206 (1994). https://pith.science/paper/54LXMXFV

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Editor's note

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This is the paper that turned open inflation from a hand-imposed initial condition into an actual scenario. Two things carry it: the Ω₀ formula in Section 2, and the spectrum in Section 8 obtained by matching Bunch-Davies modes across the bubble wall. Both are worth your time, for different reasons and with different confidence levels. Section 2 is clean. Ω₀ ≈ 1/(1 + 4e^{-8πGΔφ²}A), and the framing that you tune log(1−Ω) rather than Ω itself is the right way to say it. The criticism of Lyth-Stewart and Ratra-Peebles is also a real physics point, not a quibble: their conformal vacuum makes t=0 a genuine singularity with divergent stress-energy, and the coth factor is the honest consequence of fixing that. Credit where due — the authors are explicit that Gott and Coleman-de Luccia supplied the ingredients, and they don't oversell the novelty. Three soft spots. First, a concrete error: the text after eqn (8.10) says the difference from Lyth-Stewart is coth[πζ/2], while eqn (8.10) and concluding remark 4 both say coth[πζ]. One of those is wrong, and it's a factor of two inside an exponent. Fix before anyone quotes the number. Second, the stress-test's point about m² = 2H² and the region-II stiff approximation being least controlled at small ζ is technically correct, but the practical stake is smaller than it sounds: coth[πζ] at ζ = 0.2 is about 1.7, and the authors themselves call the factor 'very close to unity on all scales accessible to observation.' So the distinctive prediction is a modest O(1) tilt at large angles, not a dramatic signature. That said, the m² = 2H² choice is explicitly a convenience, and the generality argument lives in an unpublished companion (ref 37). That is the one place I'd wait for the follow-up before treating the spectrum as robust. Third, 'does not require significant fine tuning' is a touch generous given the quoted window 1.48 ≲ φᵢ/m_pl ≲ 1.54 — a few percent in φᵢ. The log-tuning argument is fair, but the abstract leans on it harder than the numbers do. Who should read it: anyone working on open inflation, large-angle CMB predictions, or bubble-nucleation initial conditions. The Ω₀ section stands alone and stands up. Send it out; the Ω₀ result is independent of the approximations, and the spectrum needs the factor-of-two fixed and the companion paper to land.

Referee report

Significance

If correct, this is a substantial and genuinely predictive contribution: it converts 'Ω_0 < 1' from a prima facie problem for inflation into a calculable consequence of the inflaton potential and the reheating temperature, and it does so with a definite, falsifiable perturbation spectrum rather than a phenomenological parametrization. The Ω_0 formula of Eq. (2.7) is parameter-free given the potential and depends on no observational input, which is exactly the kind of derivation that makes the scenario testable. The explicit Bogoliubov coefficients of Eq. (6.26) and the light-cone matching conditions of Appendix A are worked out in enough detail to be reproduced and checked, and the authors note independent confirmation of Eq. (6.26) by Allen–Caldwell and by Yamamoto–Tanaka–Sasaki. The distinctive coth factor in Eq. (8.10) is a concrete, in-principle-observable deviation from earlier open-inflation spectra, and the paper is appropriately explicit about which approximations it makes and why.

Major objections (1)

[§8, Eq. (8.10); §4 final paragraph; Note Added] The distinctive feature of Eq. (8.10) relative to Lyth–Stewart and Ratra–Peebles is the coth[πζ] factor, which grows as 1/(πζ) at small ζ; the novel signature is therefore concentrated at large angular scales, precisely where the approximations used are least controlled. Two approximations bear on this regime. First, the calculation is carried out for m² = 2H² in the false vacuum, a choice the authors describe in §4 as made 'simply in order to be able to pursue the calculation right through to the end in a simple way,' and which sits in tension with the m ≫ H narrow-well condition used earlier in §4 to justify the thin-wall treatment. Second, region II is treated in the stiff approximation, and concluding remark 2 argues control only for ζ ≳ 1, with the small-ζ case handled by the observation that observationally accessible scales are never far from the regime of validity. The authors acknowledge both points, and the Note Added reports that the result is 'surprisingly insensitive' to m; however, that claim is supported by ref. 37, listed as an unpublished Princeton preprint. Since the spectrum is one of the two headline results of the paper (abstract, sentence s7), the manuscript should state explicitly the range of ζ over which Eq. (8.10) is claimed to be reliable, and should either quantify the residual error at ζ ≲ 1 or qualify the claim that the coth factor is a robust prediction at large angular scales. This does not affect the Ω_0 result of §2, which is independent of both approximations.

Minor objections (9)

[§8, text following Eq. (8.10)] The text states that the result differs from Lyth–Stewart and Ratra–Peebles 'only by the factor coth[πζ/2]', whereas Eq. (8.10) as typeset contains coth[πζ]. At small ζ these differ by a factor of two, which is exactly the regime where the deviation is largest, so the two statements should be reconciled and the correct argument stated once.
[§8, Eqs. (8.9)–(8.10)] Please verify the normalization relating the two spectra. Since χ = 16πG(V²/V,φ²)Φ during inflation with a ζ-independent coefficient, P_χ/P_Φ should be ζ-independent and equal to [16πG V²/V,φ²]²; as typeset, Eqs. (8.9) and (8.10) differ by a ζ-dependent factor. This may be a transcription issue, but given that Eq. (8.10) is the paper's principal spectral result it would be worth checking the printed forms and, if the ratio is not ζ-independent, restating the convention used for P_χ.
[§2, Eq. (2.8) and the numerical example] Recomputing from Eq. (2.7) with A = 10²⁵ gives φ_i/m_pl ≈ 1.50 for Ω_0 = 0.1 and ≈ 1.56 for Ω_0 = 0.9, whereas the text quotes 1.48–1.54. The difference is small but the numbers are presented as a concrete illustration of the fine-tuning argument, so the precise inputs (the value of A, the factor of 4 in the logarithm, and whether Δφ² is identified with φ_i²) should be stated so that the range can be reproduced.
[§2, footnote on reheating] The footnote dismisses the effect of non-instantaneous reheating as 'a minor numerical detail.' Because A enters the exponent in Eq. (2.7), a short quantitative estimate of how much the allowed φ_i window shifts for, say, a matter-dominated reheating phase would strengthen the naturalness argument rather than merely asserting that the effect is small.
[§5–§6, s-wave reduction] The statement that 'all of the density perturbations may be deduced from the s-wave sector' is correct by maximal symmetry, but the text would be clearer if it said explicitly that the coincident-point two-point function determines the full two-point function at separated points, and hence the higher-multipole correlators, rather than leaving the inference implicit.
[Figures 1–3] The region boundaries in Fig. 2 are difficult to follow in the reproduction provided, and the identification of the r = t line with the surface on which φ = φ_n is central to the construction. Marking that surface and the region labels directly on the figure, rather than only in the caption, would help readers considerably.
[§9, concluding remark 1] The claim that a comparable construction for Ω > 1 is not possible is asserted rather than argued. Either a short argument or an explicit hedge would be appropriate, since the remark is a strong statement about the space of inflationary models rather than a result derived in the paper.
[Note Added and ref. 37] The insensitivity of the spectrum to the false-vacuum mass is supported by ref. 37, listed as an unpublished preprint (PUPT 95-1518). Please indicate whether that work has since appeared, or summarize the key result in enough detail that the claim can be assessed independently of an unavailable reference.
[Editorial] There are several typographical errors that should be corrected in the final version, among them 'sktec hed' (p. 5), 'in which in which' (p. 6), and the inconsistent labels '(5.15)a' and '(5.15)b' used for the two equations in that display.

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The paper

The paper claims

Inflation does not have to make the universe flat: a single bubble can leave it open, with a density today that follows from the potential.

The reading

The paper asks whether inflation is ruled out if the universe turns out to be open, with a density parameter $0<Ω_0<1$ rather than the near-unity value usually attributed to inflation. The authors argue that it is not, and they build an explicit mechanism: a first epoch of old inflation parks the inflaton in a false vacuum and solves the smoothness and horizon problems, then a single bubble nucleates, and inside that bubble the geometry is an open, negatively curved Friedmann-Robertson-Walker universe. The bubble event effectively resets $Ω$ to zero, and a second, deliberately shortened epoch of new inflation inside the bubble then drives $Ω$ back up toward one but not all the way. The payoff is that $Ω_0$ today is not a free parameter but a calculable function of how far the inflaton rolls and of the reheating temperature, and getting a value comfortably between zero and one costs only a few percent adjustment of the starting field value, not the absurd precision usually quoted for flatness. The second half of the paper computes the resulting spectrum of density perturbations by carrying the Bunch-Davies vacuum modes of the false-vacuum de Sitter phase across the bubble wall into the interior, which is what makes the scenario testable rather than merely possible.

What carries the argument

The load-bearing object is the single-bubble geometry: the Coleman-de Luccia bounce continued to Lorentzian signature, whose interior is foliated by the $SO(3,1)$-invariant hyperboloids $ds^2 = -dt^2 + a^2(t)[dξ^2 + sinh^2ξ dΩ^2]$. This coordinate system splits spacetime into three regions, and the future light cone of the nucleation point is the surface on which the field takes the value $φ_n$ and across which the quantum modes must be matched. The machinery that carries the perturbation calculation is the matching of region II mode functions (de Sitter, false vacuum) onto region I mode functions (open FRW interior) across that null cone, done in the thin-wall, small-bubble limit where the matching can be computed in flat space and then applied generally because only the leading light-cone singularities matter. On top of that sits the gauge-invariant Bardeen formalism, used to convert the field fluctuations into the conserved quantity $χ$ that tracks super-horizon density perturbations once $Ω$ is close to one.

Extended reading notes

Core claim

The central claim is that an open universe arises naturally from combining old and new inflation around a single nucleated bubble. During old inflation the inflaton sits in a false vacuum and the spacetime is essentially de Sitter, which erases pre-existing inhomogeneity; a Coleman-de Luccia bubble then nucleates, and the interior of that bubble, foliated by surfaces of constant field, is an infinite open FRW universe with negative spatial curvature. Because the interior starts with $Ω$ near zero on the light cone and rises during the subsequent slow-roll epoch according to $Ω-1 ∝ 1/(a^2 H^2)$, the present density parameter is set by the amount of new inflation: the authors give $Ω_0 ≈ 1/(1 + 4 e^{-8πG Δφ^2} A)$ with $A ≈ (T_{rh}/T_{eq})^2 T_{eq}/T_0$, so that $Ω_0$ depends on the total field excursion and the reheating temperature. For a linear potential and reheating at the electroweak scale they find that $Ω_0$ between 0.1 and 0.9 corresponds to $φ_i/m_{pl}$ between roughly 1.48 and 1.54, a range of a few percent. They then compute the perturbation spectrum, obtaining a result that is approximately scale invariant on scales small compared with the curvature scale and deviates at large scales, differing from earlier open-inflation treatments only by a factor $coth[πζ/2]$ that traces back to the choice of vacuum across the bubble wall.

Load-bearing premise

The perturbation calculation is carried out for the special case where the inflaton mass in the false vacuum is $m^2 = 2H^2$, a value the authors pick explicitly because it makes the mode equations solvable, so the predicted spectrum is only as trustworthy as the claim that a realistic false-vacuum mass gives essentially the same answer.

Editorial extensions

If this is right

A measured $Ω_0$ strictly between zero and one becomes a constraint on the inflaton potential and reheating temperature rather than evidence against inflation, since the model maps any such value onto a field excursion of order the Planck mass.
The scenario predicts a specific large-angle suppression in the cosmic microwave background: the spectrum $P_χ(ζ) ∝ coth[πζ] ζ(ζ^2+1)$ is scale invariant at small scales and turns over at scales comparable to the curvature scale.
The big-bang singularity at $t=0$ in the open FRW interior is demoted to a coordinate singularity, so a freely falling observer crossing from outside the bubble into the interior would encounter no singularity.
Because negative curvature can be introduced by a localized event whose surface propagates at nearly the speed of light, the authors argue that an open universe is achievable in a way a closed one is not, where singling out positive-curvature surfaces would require a nonlocal process.

Reading between the lines

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

If the tensor-to-scalar ratio were measured at a level implying a high inflation energy scale, the requirement that the bubble be small compared with the Hubble radius and that $ΔV ≪ V$ would translate into a joint constraint on the old-inflation false-vacuum energy and the new-inflation slope, which is a check this paper does not perform.
The factor $coth[πζ/2]$ is the entire observational signature separating this vacuum choice from the earlier conformal-vacuum treatments; a precision low-multipole CMB measurement is therefore the sharpest discriminator between the two initial-condition philosophies, sharper than any measurement of $Ω_0$ alone.
The authors' own note that the result is insensitive to the false-vacuum mass, taken together with their remark that the thin-wall and small-bubble approximations are what make the matching tractable, suggests the spectrum is protected by the light-cone structure rather than by the details of the wall, which would make the prediction more robust than the calculation's assumptions suggest.
Because the scenario converts a cosmological parameter into a logarithm of a field range, it invites a converse question the paper does not ask: whether a fundamental theory that fixes $φ_i$ in Planck units would land in the few-percent window, which is a statement about landscape statistics rather than about inflation.

Original abstract

We present a natural scenario for obtaining an open universe ($\Omega _0<1$) through inflation. In this scenario, there are two epochs of inflationary expansion---an epoch of `old inflation,' during which the inflaton field is stuck in a false vacuum, followed by an epoch of `new inflation,' during which the inflaton field slowly rolls toward its true minimum. During the first epoch, inflation solves the smoothness and horizon problems. Then an open universe (with negative spatial curvature) is created by the nucleation of a single bubble. In effect $\Omega$ is instantaneously `reset' to zero. During the subsequent `new' inflation $\Omega$ rises toward unity. The value of $\Omega$ today is calculable in terms of the parameters of the potential, and we show that obtaining values significantly different from zero or unity (though within the range $0<\Omega <1$) does not require significant fine tuning. We compute the spectrum of density perturbations by evolving the Bunch-Davies vacuum modes across the bubble wall into its interior.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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The abstract, sentence by sentence

arXiv listing abstract v6. Results and the main claim are underlined; show every role.

We present a natural scenario for obtaining an open universe ($\Omega _0<1$) through inflation. In this scenario, there are two epochs of inflationary expansion---an epoch of `old inflation,' during which the inflaton field is stuck in a false vacuum, followed by an epoch of `new inflation,' during which the inflaton field slowly rolls toward its true minimum. During the first epoch, inflation solves the smoothness and horizon problems. Then an open universe (with negative spatial curvature) is created by the nucleation of a single bubble. In effect $\Omega$ is instantaneously `reset' to zero. During the subsequent `new' inflation $\Omega$ rises toward unity. The value of $\Omega$ today is calculable in terms of the parameters of the potential, and we show that obtaining values significantly different from zero or unity (though within the range $0<\Omega <1$) does not require significant fine tuning. We compute the spectrum of density perturbations by evolving the Bunch-Davies vacuum modes across the bubble wall into its interior.

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 present a natural scenario for obtaining an open universe ($\Omega _0<1$) through inflation.	contribution 1.00	0.91	0.73	asserted 0.95
s1	In this scenario, there are two epochs of inflationary expansion---an epoch of `old inflation,' during which the inflaton field is stuck in a false vacuum, followed by an epoch of `new inflation,' during which the inflaton field slowly rolls toward its true minimum.	method 0.79	0.37	0.06	asserted 1.00
s2	During the first epoch, inflation solves the smoothness and horizon problems.	method 0.32	0.27	0.05	asserted 1.00
s3	Then an open universe (with negative spatial curvature) is created by the nucleation of a single bubble.	method 0.52	0.58	0.30	asserted 1.00
s4	In effect $\Omega$ is instantaneously `reset' to zero.	interpretation 0.84	0.52	0.11	asserted 0.99
s5	During the subsequent `new' inflation $\Omega$ rises toward unity.	result 0.50	0.46	0.08	asserted 1.00
s6	The value of $\Omega$ today is calculable in terms of the parameters of the potential, and we show that obtaining values significantly different from zero or unity (though within the range $0<\Omega <1$) does not require significant fine tuning.	result 0.93	0.93	0.41	asserted 1.00
s7	We compute the spectrum of density perturbations by evolving the Bunch-Davies vacuum modes across the bubble wall into its interior.	method 0.95	0.94	0.05	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.84.

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sources read

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not examined

full text
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Figures

**Figure 2.** The instanton and the classical expanding bubble s

Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9411206 by the authors.

References

Works cited (38)

[1]

A.Z. Dolginov and I.N. Toptygin, “Relativistic Spherica l Functions,” Zh. Eksperim. i. Teor. Fiz. 37, 1441 (1959). [Translation: Sov. Phys. JETP 10, 1022 (1960).]

View on Pith 1959
[2]

A.S. Goncharov and A.D. Linde, “ Tunneling in Expanding Un iverse: Euclidean and Hamiltonian Approaches”, Sov. J. Part. Nucl. 17 ( 1986) 3 69; A. Linde, Stochastic Approach to Tunneling an Baby Universe Formatio n” Nucl. Phys. B372 (1992) 421

View on Pith 1992
[3]

L. Abbott and R. Schaefer, “A General, Gauge-Invariant An alysis of the Cosmic Microwave Anisotropy,” Ap. J. 308, 462 (1986)

View on Pith 1986
[4]

R. Carlitz and R. Wiley, “Reﬂections on Moving Mirrors,” P hys. Rev. D36, 2327 (1987)

View on Pith 1987
[5]

N. Birrell and P. Davies, Quantum Fields in Curved Space, (Cambridge, Cam- bridge U. Press, 1982) and references therein

View on Pith 1982
[6]

J.R. Gott, III, “Creation of Open Universes from de Sitter Space,” Nature 295, 304 (1982); J.R. Gott and T. Statler, “Constraints on the For mation of Bubble Universes,” Phys. Lett. 136B, 157 (1984); J.R. Gott, “Conditions for the Formation of Bubble Universes,” in E.W. Kolb et al., Eds., Inner Space/Outer Space, (Chicago: U. of Chicago Press, 1986)

View on Pith 1982
[7]

D. Lyth and E. Stewart, “Inﬂationary Density Perturbatio ns with Ω < 1,” Phys. Lett. B252, 336 (1990)

View on Pith 1990
[8]

B. Ratra and P.J.E. Peebles, “Inﬂation in an Open Universe ,” PUPT-1444 (Feb. 1994)

View on Pith 1994
[9]

B. Ratra and P.J.E. Peebles, “CDM Cosmogony in an Open Univ erse,” Ap. J. Lett. 432, L5 (1994)

View on Pith 1994
[10]

V. Rubakov, “Particle Creation During Vacuum Decay,” Nu cl. Phys. B245, 481 (1984). 51

View on Pith 1984
[11]

T. Vachaspati and A. Vilenkin, “Quantum State of a Nuclea ting Bubble,” Phys. Rev. D43, 3846 (1991); J. Garriga and A. Vilenkin, ‘Quantum Fluctuati ons on Domain Walls, Strings, and Vacuum Bubbles,” Phys. Rev. D45, 3469 (1992)

View on Pith 1991
[12]

M. Sasaki, T. Tanaka, K. Yamamoto, and J. Yokoyama, “Quan tum State During and After Nucleation of an O(4) Symmetric Bubble,”; Prog. Theor. Phys. 90, 1019 (1993); M. Sasaki, T. Tanaka, K. Yamamoto, and J. Yokoya ma, “Quantum State Inside a Vacuum Bubble and Creation of an Open Universe,” Phys. Lett. B317, 510 (1993

View on Pith 1993
[13]

T. Tanaka and M. Sasaki, “Quantum State During and After O (4) Symmetric Bubble Nucleation with Gravitational Eﬀects”, Phys. Rev. D50, 6444 (1994)

View on Pith 1994
[14]

G. Gibbons and S. Hawking, “Cosmological Event Horizons , Thermodynamics, and Particle Creation,” Phys. Rev. D15, 2738 (1976)

View on Pith 1976
[15]

B. Allen, “Vacuum States in de Sitter Space,” Phys. Rev. D32, 3136 (1985)

View on Pith 1985
[16]

A. Vilenkin, “Did the Universe Have a Beginning?”, Phys. Rev. D46, 3255 (1992); A. Borde and A. Vilenkin, “Eternal Inﬂation and the I nitial Singularity,” Phys. Rev. Lett. 72, 3305 (1994)

View on Pith 1992
[17]

M. Bander and C. Itzykson, “Group Theory and the Hydrogen Atom (II),” Rev. Mod. Phys. 38, 346 (1966)

View on Pith 1966
[18]

S. Coleman, Phys. Rev. D15, 2929 (1977), D16, 1248(E) (1977); C. Callan and S. Coleman, Phys. Rev. D16, 1763 (1977); see also, “The Uses on Instantons,” in S. Coleman, Aspects of Symmetry, (Cambridge, Cambridge University Press, 1985)

View on Pith 1977
[19]

S. Coleman and F. De Luccia, “Gravitational Eﬀects on and of Vacuum Decay,” Phys. Rev. D21, 3305 (1980)

View on Pith 1980
[20]

A. Guth, “Inﬂationary Universe: A Possible Solution to t he Horizon and Flatness Problems,” Phys. Rev D23, 347 (1981)

View on Pith 1981
[21]

A. Guth and E. Weinberg, “Could the Universe Have Recover ed from a First- Order Phase Transition?,” Nucl. Phys. B212, 321 (1983). 52

View on Pith 1983
[22]

A. Linde, “A New Inﬂationary Universe Scenario: A Possib le Solution of the Horizon, Flatness, Homogeneity, Isotropy, and Primordial Monopole Problems,” Phys. Lett. 108B, 389 (1982)

View on Pith 1982
[23]

A. Albrecht and P. Steinhardt, “Cosmology for Grand Uniﬁ ed Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48, 1220 (1982)

View on Pith 1982
[24]

A. Linde, “Chaotic Inﬂation”, Phys. Lett. 129B, 177 (1983)

View on Pith 1983
[25]

L. Jensen and P. Steinhardt, “Bubble Nucleation and the C oleman-Weinberg Model,” Nucl. Phys. B237, 176 (1984)

View on Pith 1984
[26]

S. Hawking, “The Development of Irregularities in a Sing le Bubble Inﬂationary Universe”, Phys. Lett. 115B, 295 (1982)

View on Pith 1982
[27]

A.A. Starobinsky, “Dynamics of Phase Transition in the N ew Inﬂationary Scenario and Generation of Perturbations”, Phys. Lett. 117B, 175 (1982)

View on Pith 1982
[28]

A.H. Guth and S.-Y. Pi, “Fluctuations in the New Inﬂation ary Universe”, Phys. Rev. Lett. 49, 1110 (1982)

View on Pith 1982
[29]

J. Bardeen, P. Steinhardt, and M. Turner, “Spontaneous C reation of Almost Scale-Free Density Perturbations in an Inﬂationary Univer se,” Phys. Rev. D28, 679 (1983)

View on Pith 1983
[30]

H. Kodama and M. Sasaki, “Cosmological Perturbation The ory,” Prog. of Theor. Phys. Suppl. 78, 1 (1984)

View on Pith 1984
[31]

V. Mukhanov, H. Feldman and R. Brandenberger, “Theory of Cosmological Perturbations,” Phys. Rep. 215, 203 (1992)

View on Pith 1992
[32]

J. Bardeen, “Gauge Invariant Cosmological Perturbatio ns,” Phys. Rev. D22, 1882 (1980)

View on Pith 1980
[33]

M. Bucher and N. Turok, “Large Angle CMB Anisotropy in an I nﬂationary Ω< 1 Universe,” in preparation. 53

View on Pith
[34]

B. Allen and R. Caldwell, “Cosmic Background Radiation T emperature Fluc- tuations in a Spatially Open Inﬂationary Universe”, unpubl ished manuscript (1994)

View on Pith 1994
[35]

K. Yamamoto, T. Tanaka and M. Sasaki, “Particle Creation through Bubble Nucleation and Quantum Field Theory in the Milne Universe”, preprint in preparation (1994)

View on Pith 1994
[36]

D. Lyth, “Large Scale Energy Density Perturbations and I nﬂation,” Phys. Rev. D31, 1792 (1985)

View on Pith 1985
[37]

M. Bucher and N.Turok, “Open Inﬂation with Arbitrary Fal se Vacuum Mass,” Princeton Preprint PUPT 95-1518 (2-95)

View on Pith
[38]

K. Yamamoto, M. Sasaki and T. Tanaka, “Large Scale CMB Ani sotropy in an Open Universe in the One-Bubble Inﬂationary Scenario”, pre print KUNS 1309, astro-ph/9501109 (1995). APPENDIX A—Matching Conditions for Minkowski Space. In this appendix we present the details of the derivation of t he matching conditions for Minkowski space for the expansions in reg...

View on Pith Pith review arXiv 1995

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A first-order phase-transition exit from a displaced Starobinsky branch truncates ~12 e-folds, shifting the spectral tilt from n_s≈0.965 to ≈0.973 at r≈2×10⁻³.
Planck 2015 results. XIII. Cosmological parameters
astro-ph.CO 2015-02 accept novelty 6.0 of 10

Full-mission Planck temperature and polarization data are consistent with six-parameter flat Lambda-CDM, yielding H0 = 67.8 +/- 0.9, Omega_m = 0.308 +/- 0.012, ns = 0.968 +/- 0.006, and tau = 0.066 +/- 0.016.

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Planck 2013 results. XXII. Constraints on inflation
astro-ph.CO 2013-03 conditional novelty 6.0 of 10

Planck 2013 measures the scalar spectral index at 0.9603 plus or minus 0.0073 and bounds the tensor-to-scalar ratio below 0.11, ruling out exact scale invariance and disfavouring steep, exponential, and simple hybrid ...
What can the observation of nonzero curvature tell us?
hep-th 2012-03 conditional novelty 6.0 of 10

A future detection of negative spatial curvature above 10^-4 would imply that our universe formed by bubble nucleation, that the probability measure does not reward volume increase, and that slow-roll inflation was an...
Probabilities in the inflationary multiverse
hep-th 2005-09 conditional novelty 6.0 of 10

A comoving-horizon cutoff gives a coordinate-independent way to count pocket universes of different types, which combined with an ergodic conjecture yields a computable distribution of the constants of nature.
Inflation, Open Universes, and Dark Energy
astro-ph.CO 2026-07 conditional novelty 5.0 of 10

A slightly open universe with Ω_k ≈ 3×10^{-3} pulls n_s down to ~0.969, reconciling plateau inflation models with combined CMB and DESI data.
The galaxy power spectrum take on spatial curvature and cosmic concordance
astro-ph.CO 2020-10 conditional novelty 5.0 of 10

Planck plus the BOSS DR12 full-shape galaxy power spectrum gives a curvature parameter of 0.0023 ± 0.0028, flat to sub-percent precision, but only by combining two datasets that disagree at high significance when curv...
Exploring Cosmic Origins with CORE: Cosmological Parameters
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Simulated forecasts for the proposed CORE-M5 CMB satellite project 3 to 8 fold improvements over Planck 2015 on neutrino mass, relativistic species, curvature and dark energy, conditional on foreground residuals sitti...
Curvature constraints from Large Scale Structure
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Extending CLASS to curved geometry, the authors find that neglecting cosmic magnification in a tomographic galaxy survey can bias the inferred spatial curvature by more than one standard deviation.
Robust forecasts on fundamental physics from the foreground-obscured, gravitationally-lensed CMB polarization
astro-ph.CO 2015-09 accept novelty 5.0 of 10

An end-to-end Fisher pipeline combining parametric foreground cleaning and three delensing methods finds that a Stage-IV ground survey plus a space mission could reach sigma(r) near 1.3e-4 and sigma(M_nu) near 31 meV.
Cosmic microwave background anomalies in an open universe
astro-ph.CO 2013-06 conditional novelty 5.0 of 10

A marginally open universe whose inflation excites a super-curvature perturbation mode, plus a curvaton, can produce the observed dipole power asymmetry in the cosmic microwave background within existing Planck constraints.
Open inflation in the landscape
hep-th 2011-05 conditional novelty 5.0 of 10

In single-field open inflation, the slope of the potential right after bubble nucleation decides whether tensor CMB anisotropies remember the high tunneling energy scale or only the later, low-scale slow-roll Hubble rate.
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A satellite measuring CMB B-mode polarization could detect primordial gravitational waves down to r about 0.002 and, if r exceeds 0.01, establish that inflation occurred near 10^16 GeV with a super-Planckian inflaton ...
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Bubble collisions in eternal inflation leave a predictable, azimuthally symmetric CMB imprint (a disk with a linear temperature gradient and E-mode polarization in a ring), and a new exact metric describes the post-co...
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Two decades of supernova, baryon-acoustic-oscillation, cosmic-microwave-background, lensing and cluster measurements converge on a dark energy component with equation of state w close to -1, and the review maps which ...
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How the literature uses it

38 arXiv papers cite this in 48 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

An inflationary scenario that leads to $\Omega _0<1$ today is presented.

$\Omega_0 <1$ From Inflation · 0.98

Papers citing it · 28 sentences without a confident label

Uses its method or tool 1 paper

To determine the power spectrum of the light scalar $σ$ with respect to the Bunch–Davies vacuum [ref] , we trace the evolution of $Υ^υ_qs$ back into the parent vacuum and Klein-Gordon normalize the positive-frequency modes over a Cauchy surface, as is done in [ref] .

Observable effects of anisotropic bubble nucleation · Bunch–Davies vacuum · 0.97

Compares with it 1 paper

This is not unlike the situation in regular (isotropic) 4d open bubble inflation, where the inflationary spectrum is also modified at $q≲O(1)$ [ref] .

Observable effects of anisotropic bubble nucleation · Bunch–Davies vacuum · 1.00

Cites it as background 17 papers

This scenario is often called one-bubble open inflation [ref] .

Multi-field open inflation model and multi-field dynamics in tunneling · Introduction · 0.96
In the conventional inflationary scenario for an open universe [ref] , the spatial curvature is inflated away and a spatial flat universe is predicted.

The Cosmological Constant Problem and Inflation in the String Landscape · The open universe scenario · 0.83
This is known as the “open inflation” [ref] .

Probability of Slowroll Inflation in the Multiverse · Probability for Multifield Slowroll Inflation · 0.87

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.Z. Dolginov and I.N. Toptygin, “Relativistic Spherica l Functions,” Zh. Eksperim. i. Teor. Fiz. 37, 1441 (1959). [Translation: Sov. Phys. JETP 10, 1022 (1960).]

View on Pith 1959

[2] [2]

A.S. Goncharov and A.D. Linde, “ Tunneling in Expanding Un iverse: Euclidean and Hamiltonian Approaches”, Sov. J. Part. Nucl. 17 ( 1986) 3 69; A. Linde, Stochastic Approach to Tunneling an Baby Universe Formatio n” Nucl. Phys. B372 (1992) 421

View on Pith 1992

[3] [3]

L. Abbott and R. Schaefer, “A General, Gauge-Invariant An alysis of the Cosmic Microwave Anisotropy,” Ap. J. 308, 462 (1986)

View on Pith 1986

[4] [4]

R. Carlitz and R. Wiley, “Reﬂections on Moving Mirrors,” P hys. Rev. D36, 2327 (1987)

View on Pith 1987

[5] [5]

N. Birrell and P. Davies, Quantum Fields in Curved Space, (Cambridge, Cam- bridge U. Press, 1982) and references therein

View on Pith 1982

[6] [6]

J.R. Gott, III, “Creation of Open Universes from de Sitter Space,” Nature 295, 304 (1982); J.R. Gott and T. Statler, “Constraints on the For mation of Bubble Universes,” Phys. Lett. 136B, 157 (1984); J.R. Gott, “Conditions for the Formation of Bubble Universes,” in E.W. Kolb et al., Eds., Inner Space/Outer Space, (Chicago: U. of Chicago Press, 1986)

View on Pith 1982

[7] [7]

D. Lyth and E. Stewart, “Inﬂationary Density Perturbatio ns with Ω < 1,” Phys. Lett. B252, 336 (1990)

View on Pith 1990

[8] [8]

B. Ratra and P.J.E. Peebles, “Inﬂation in an Open Universe ,” PUPT-1444 (Feb. 1994)

View on Pith 1994

[9] [9]

B. Ratra and P.J.E. Peebles, “CDM Cosmogony in an Open Univ erse,” Ap. J. Lett. 432, L5 (1994)

View on Pith 1994

[10] [10]

V. Rubakov, “Particle Creation During Vacuum Decay,” Nu cl. Phys. B245, 481 (1984). 51

View on Pith 1984

[11] [11]

T. Vachaspati and A. Vilenkin, “Quantum State of a Nuclea ting Bubble,” Phys. Rev. D43, 3846 (1991); J. Garriga and A. Vilenkin, ‘Quantum Fluctuati ons on Domain Walls, Strings, and Vacuum Bubbles,” Phys. Rev. D45, 3469 (1992)

View on Pith 1991

[12] [12]

M. Sasaki, T. Tanaka, K. Yamamoto, and J. Yokoyama, “Quan tum State During and After Nucleation of an O(4) Symmetric Bubble,”; Prog. Theor. Phys. 90, 1019 (1993); M. Sasaki, T. Tanaka, K. Yamamoto, and J. Yokoya ma, “Quantum State Inside a Vacuum Bubble and Creation of an Open Universe,” Phys. Lett. B317, 510 (1993

View on Pith 1993

[13] [13]

T. Tanaka and M. Sasaki, “Quantum State During and After O (4) Symmetric Bubble Nucleation with Gravitational Eﬀects”, Phys. Rev. D50, 6444 (1994)

View on Pith 1994

[14] [14]

G. Gibbons and S. Hawking, “Cosmological Event Horizons , Thermodynamics, and Particle Creation,” Phys. Rev. D15, 2738 (1976)

View on Pith 1976

[15] [15]

B. Allen, “Vacuum States in de Sitter Space,” Phys. Rev. D32, 3136 (1985)

View on Pith 1985

[16] [16]

A. Vilenkin, “Did the Universe Have a Beginning?”, Phys. Rev. D46, 3255 (1992); A. Borde and A. Vilenkin, “Eternal Inﬂation and the I nitial Singularity,” Phys. Rev. Lett. 72, 3305 (1994)

View on Pith 1992

[17] [17]

M. Bander and C. Itzykson, “Group Theory and the Hydrogen Atom (II),” Rev. Mod. Phys. 38, 346 (1966)

View on Pith 1966

[18] [18]

S. Coleman, Phys. Rev. D15, 2929 (1977), D16, 1248(E) (1977); C. Callan and S. Coleman, Phys. Rev. D16, 1763 (1977); see also, “The Uses on Instantons,” in S. Coleman, Aspects of Symmetry, (Cambridge, Cambridge University Press, 1985)

View on Pith 1977

[19] [19]

S. Coleman and F. De Luccia, “Gravitational Eﬀects on and of Vacuum Decay,” Phys. Rev. D21, 3305 (1980)

View on Pith 1980

[20] [20]

A. Guth, “Inﬂationary Universe: A Possible Solution to t he Horizon and Flatness Problems,” Phys. Rev D23, 347 (1981)

View on Pith 1981

[21] [21]

A. Guth and E. Weinberg, “Could the Universe Have Recover ed from a First- Order Phase Transition?,” Nucl. Phys. B212, 321 (1983). 52

View on Pith 1983

[22] [22]

A. Linde, “A New Inﬂationary Universe Scenario: A Possib le Solution of the Horizon, Flatness, Homogeneity, Isotropy, and Primordial Monopole Problems,” Phys. Lett. 108B, 389 (1982)

View on Pith 1982

[23] [23]

A. Albrecht and P. Steinhardt, “Cosmology for Grand Uniﬁ ed Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48, 1220 (1982)

View on Pith 1982

[24] [24]

A. Linde, “Chaotic Inﬂation”, Phys. Lett. 129B, 177 (1983)

View on Pith 1983

[25] [25]

L. Jensen and P. Steinhardt, “Bubble Nucleation and the C oleman-Weinberg Model,” Nucl. Phys. B237, 176 (1984)

View on Pith 1984

[26] [26]

S. Hawking, “The Development of Irregularities in a Sing le Bubble Inﬂationary Universe”, Phys. Lett. 115B, 295 (1982)

View on Pith 1982

[27] [27]

A.A. Starobinsky, “Dynamics of Phase Transition in the N ew Inﬂationary Scenario and Generation of Perturbations”, Phys. Lett. 117B, 175 (1982)

View on Pith 1982

[28] [28]

A.H. Guth and S.-Y. Pi, “Fluctuations in the New Inﬂation ary Universe”, Phys. Rev. Lett. 49, 1110 (1982)

View on Pith 1982

[29] [29]

J. Bardeen, P. Steinhardt, and M. Turner, “Spontaneous C reation of Almost Scale-Free Density Perturbations in an Inﬂationary Univer se,” Phys. Rev. D28, 679 (1983)

View on Pith 1983

[30] [30]

H. Kodama and M. Sasaki, “Cosmological Perturbation The ory,” Prog. of Theor. Phys. Suppl. 78, 1 (1984)

View on Pith 1984

[31] [31]

V. Mukhanov, H. Feldman and R. Brandenberger, “Theory of Cosmological Perturbations,” Phys. Rep. 215, 203 (1992)

View on Pith 1992

[32] [32]

J. Bardeen, “Gauge Invariant Cosmological Perturbatio ns,” Phys. Rev. D22, 1882 (1980)

View on Pith 1980

[33] [33]

M. Bucher and N. Turok, “Large Angle CMB Anisotropy in an I nﬂationary Ω< 1 Universe,” in preparation. 53

View on Pith

[34] [34]

B. Allen and R. Caldwell, “Cosmic Background Radiation T emperature Fluc- tuations in a Spatially Open Inﬂationary Universe”, unpubl ished manuscript (1994)

View on Pith 1994

[35] [35]

K. Yamamoto, T. Tanaka and M. Sasaki, “Particle Creation through Bubble Nucleation and Quantum Field Theory in the Milne Universe”, preprint in preparation (1994)

View on Pith 1994

[36] [36]

D. Lyth, “Large Scale Energy Density Perturbations and I nﬂation,” Phys. Rev. D31, 1792 (1985)

View on Pith 1985

[37] [37]

M. Bucher and N.Turok, “Open Inﬂation with Arbitrary Fal se Vacuum Mass,” Princeton Preprint PUPT 95-1518 (2-95)

View on Pith

[38] [38]

K. Yamamoto, M. Sasaki and T. Tanaka, “Large Scale CMB Ani sotropy in an Open Universe in the One-Bubble Inﬂationary Scenario”, pre print KUNS 1309, astro-ph/9501109 (1995). APPENDIX A—Matching Conditions for Minkowski Space. In this appendix we present the details of the derivation of t he matching conditions for Minkowski space for the expansions in reg...

View on Pith Pith review arXiv 1995