The DOI matches its Crossref record; 91 of the 91 references it cites were found in a registry or an index; 11 of the 40 sources its claims cite were checked for support.
Creation of a compact topologically nontrivial inflationary universe
Open accessAbstract
If inflation can occur only at the energy density V much smaller than the Planck density, which is the case for many inflationary models based on string theory, then the probability of quantum creation of a closed or an infinitely large open inflationary universe is exponentially suppressed for all known choices of the wavefunction of the universe. Meanwhile under certain conditions there is no exponential suppression for creation of topologically nontrivial compact flat or open inflationary universes. This suggests, contrary to the standard textbook lore, that compact flat or open universes with nontrivial topology should be considered a rule rather than an exception.
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TL;DR
For low-scale inflation with \(V \ll 1\) in Planck units, quantum creation of a closed or infinite open inflationary universe is exponentially suppressed. Linde argues that compact flat or open universes with nontrivial topology can instead be created without such exponential suppression, so they may be the rule rather than the exception.
Key findings
- Standard closed or infinite open creation is suppressed by \(P \sim \exp(-24\pi^2/V)\); for \(V \sim 10^{-12}\), the suppression is \(10^{-10^{14}}\). A radiation-dominated closed universe must have initial entropy \(S \sim V^{-3/4}\), e.g. \(S > 10^9\) for \(V < 10^{-12}\), with probability \(\exp(-S) \sim \exp(-V^{-3/4}) < e^{-10^9}\).
- In a compact flat toroidal universe, an initial Planckian size \(O(1)\) with radiation domination gives total entropy \(O(1)\); chaotic mixing homogenizes the universe, and inflation begins at \(a \sim V^{-1/4}\), which is smaller than the de Sitter horizon \(H^{-1} \sim V^{-1/2}\). The probability of creating the initial \(O(1)\) patch is not exponentially suppressed and practically does not depend on \(V(\phi)\).
- The Wheeler-DeWitt equation for the flat toroidal de Sitter universe has no potential barrier, unlike the closed case. The WKB approximation breaks down at \(a \lesssim V^{-1/6}\), and classical space-time description begins at \(a \gtrsim V^{-1/6}\); the action in the quantum regime is \(O(1)\), so there is no tunneling suppression.
- Including scalar kinetic energy \(K\), for \(48K < 1\) the wave function is non-oscillatory and vanishes as \(a \to 0\); for \(48K > 1\) it is oscillatory. For compact flat or open universes with \(K \gg 1/48\), incident waves are adiabatically transformed without reflection, confirming no exponential suppression.
Methods
- Theoretical quantum cosmology analysis using the Wheeler-DeWitt equation for closed, flat toroidal, and compact open universes, comparing tunneling barriers and wave-function behavior.
- Uses chaotic mixing arguments for compact universes and estimates of entropy, radiation density, and scale factor at inflation onset; includes the scalar-field kinetic energy term \(K\) and reports numerical verification for \(K \gg 1/48\).
- No observational data are used; the paper is an analytic and numerical theoretical study.
Limitations
- The paper states that there is still no compelling observational evidence for nontrivial topology, and inflation typically makes the universe too large to observe its topology unless the number of e-folds and scalar potential are fine-tuned.
- The paper states that it ignores the Casimir effect, which is suppressed by supersymmetry but may be negative under certain conditions and could forbid classical evolution at \(a < O(H^{-1})\); adding matter leads to a curvature singularity, so kinetic energy of the scalar field must be included.
- The paper states that eternal inflation may make initial conditions irrelevant, but probability measures in eternal inflation are delicate, and many inflationary models do not lead to eternal inflation. Evident from the text: the conclusions rely on assumptions such as Planckian initial size, radiation domination, a flat inflationary potential, and specific compact topology, with no observational test proposed.
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Claims the paper makes
29 Own interpretation · 19 No basis given · 10 Not checked yet · 6 No full text of the sources · 4 Couldn’t confirm · 2 Partly supported · 2 Sources not found · 2 Own result · accuracy not yet measured
None of the sources cited for this claim have been checked yet.
In the paper
“If inflation can occur only at the energy density V much smaller than the Planck density, which is the case for many inflationary models based on string theory, then the probability of quantum creation of a closed or an infinitely large open inflationary universe is exponentially suppressed for all known choices of the wave function of the universe.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > Abstract
What the sources say
- Not checked yetcites [15] Sokolov D D, 1975, Sov. Phys. JETP
- Not checked — no full text of the sourcecites [16] Paal G, 1971, Acta. Phys. Acad. Sci. Hung. source
- Could not verifycites [19] Levin J, 1998, Class. Quantum Grav. sourceExperimental — accuracy not yet measured
Split into 2 atoms — 0 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 2 unverifiable. Expand the claim to see per-atom verdicts.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In our opinion, however, the last 20 years of the debates on the probability of quantum creation a closed universe or an infinite open universe [15,16,17,19,21] demonstrated that in all known cases the probability of creation of a low energy density inflationary universe is exponentially suppressed.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
What the sources say
- Not checked yetcites [15] Sokolov D D, 1975, Sov. Phys. JETP
- Not checked — no full text of the sourcecites [16] Paal G, 1971, Acta. Phys. Acad. Sci. Hung. source
- Not matchedcites [17] Sokolov D D, 1976, Sov. Astron.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Could not verifycites [19] Levin J, 1998, Class. Quantum Grav. sourceExperimental — accuracy not yet measured
Section III (page unavailable) asks whether quantum creation favors compact hyperbolic universes but gives no probability for creating a low-energy-density inflationary universe. A further search for exponential suppression and quantum creation found no passage addressing that claim; the paper's discussion of inflation concerns observable topology instead. (LLM, conf=0.73)
- Could not verifycites [21] Bond J R, 2000, Phys. Rev. sourceExperimental — accuracy not yet measured
The Conclusion discusses whether small compact hyperbolic spaces might be more probable products of early-universe quantum processes, but gives no probability for creating a low-energy-density inflationary universe. A search for the claimed exponential suppression and for quantum-creation probabilities found no passage addressing that point. (LLM, conf=0.75)
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In the paper
“An estimate of the probability of such event for the universe initially dominated by relativistic matter suggests that it should be suppressed by a factor of exp(-S) ∼ exp(-V -3/4 ) < e -10 9 , and the probability that not only the energy density but also the scalar field in such a universe will be sufficiently homogeneous may be even smaller [14].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > II. THE PROBLEM OF INITIAL CONDITIONS FOR THE LOW-SCALE INFLATION
What the sources say
- Not checked — no full text of the sourcecites [13] source
- Could not verifycites [14] sourceExperimental — accuracy not yet measured
The paper’s discussion of the initial universe (section “Does Chaotic Mixing Facilitate Ω < 1 Inflation?”, page unavailable) says that small-volume universes may be more likely to be created and considers radiation-dominated beginnings. It does not give the claimed probability estimate for homogeneous initial conditions in a closed universe, the exp(-S) scaling, or the e^-10^9 bound; searches for those terms found no supporting passage. (LLM, conf=0.76)
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In the paper
“Both cases involve tunneling and exponential suppression of the probability of creation of an inflationary universe with V ≪ 1 [19].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > II. THE PROBLEM OF INITIAL CONDITIONS FOR THE LOW-SCALE INFLATION
What the sources say
- Could not verifycites [19] Levin J, 1998, Class. Quantum Grav. sourceExperimental — accuracy not yet measured
The discussion of compact hyperbolic universes (section III; page unavailable) raises the question of how quantum cosmology might assess their creation, but does not describe two ways to create an infinite homogeneous open universe. Searches for tunneling, exponential suppression, and the inflationary potential V found no passage addressing those claims. (LLM, conf=0.76)
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In the paper
“For 48K < 1 the solution is non-oscillatory, Ψ(a) = √ a C 1 a -1 2 √ 1-48K + C 2 a 1 2 √ 1-48K . (10”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plaindirect- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: The potential is flat, V(φ)=V.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The general solution for Eq. ( 5) can be represented as a sum of two Bessel functions:”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“As we see from this figure, quantum mechanical description of the universe near a = 0 is quite regular.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“Note that for K ≪ V -2 an addition of the term -2π 2 K/a 2 practically does not affect the shape of the potential barrier at a ∼ H -1 ∼ V -1/2 , so the final result for the exponential suppression of tunneling (1) remains intact.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plain- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: The potential is flat, V(φ)=V.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“As we see, for a < V -1/6 the action becomes smaller than 1, which means that one must describe evolution of the universe quantum mechanically.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“Meanwhile, for 48K > 1 at a ≪ K V 1/6 one has an oscillatory solution Ψ(a) = √ a C 1 sin 1 2 √ 48K -1 log a + C 2 cos 1 2 √ 48K -1 log a .(11)”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- It may state two separate claims as one. 30
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The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Similarly, it was shown long ago that particle production in the very early universe can make the universe locally isotropic [23].However, this process could not make the universe globally isotropic because in the hot big bang scenario the universe consisted of many causally disconnected domains.But such effects may play important role in our scenario by making the initial small patch of the”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
What the sources say
- Could not verifycites [23] de Oliveira-Costa A, 2004, Phys. Rev. sourceExperimental — accuracy not yet measured
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In the paper
“As we already discussed in Section II, the probability of such event for V ≪ 1 is expected to be exponentially small, even though under certain conditions it might be greater than the probability given by the square of the tunneling wave function (1); see Eqs. (18) and (19) in [14].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
What the sources say
- Could not verifycites [14] sourceExperimental — accuracy not yet measured
Split into 2 atoms — 0 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 2 unverifiable. Expand the claim to see per-atom verdicts.
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- Could not verify
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In the paper
“Moreover, the effective energy associated with the curvature of 3D space in the open universe case decreases as a -2 , i.e. much more slowly than the energy of ordinary matter.Soon after the beginning of expansion, the curvature energy density becomes dominant, and the relative role of normal matter and of its density perturbations become insignificant.This is an additional reason to expect that”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
What the sources say
- Partly supportedcites [22] sourceExperimental — accuracy not yet measured
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plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“After this work was completed we became aware of Ref.[26], where it was also found that the probability of quantum creation of compact flat and open dS universes (in the case K = 0, see below) is not exponentially suppressed, which agrees with[3] and with our results.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
What the sources say
- Could not verifycites [26] Barrow J, 2001, Class. Quantum Grav. sourceExperimental — accuracy not yet measured
Section 2 discusses compact flat and open universes and mentions the naturalness of finite spatial sections in quantum cosmologies, but it does not address their probability of quantum creation. Searches for exponential suppression and quantum nucleation found no passage establishing the claimed result for K = 0. (LLM, conf=0.78)
- Not checked yetcites [3] Schwarzschild K, 1900, Vierteljahrschrift der Ast. Ges.
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“This is an important consideration, because inflation in the string theory landscape scenario is eternal [37,38].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
What the sources say
- Partly supportedcites [37] Linde A D, 1998, Phys. Rev. sourceExperimental — accuracy not yet measured
In “Open universes and the tunneling wave function” (page not supplied), Linde says that most versions of chaotic inflation enter a regime of eternal self-reproduction. Our copy does not connect that finding to a string theory landscape scenario, so it supports eternal inflation in the models discussed, but not the claim's specified setting. (LLM, conf=0.87)
- Could not verifycites [38] Bucher M, 1995, Phys. Rev. sourceExperimental — accuracy not yet measured
The Introduction and Concluding Remarks describe an open-universe model with old and new inflation, not a string-theory landscape scenario. The Concluding Remarks say that eternal inflation backward in time seems inconsistent with a finite bubble-nucleation rate, but do not establish whether inflation in a landscape scenario is eternal. Searched for string-theory landscape, eternal inflation, and future-directed inflation. (LLM, conf=0.76)
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
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In the paper
“At a ≫ K V 1/6 the term Ka -2 is much smaller than a 4 V , and therefore the solutions of Eqs.(9) look the same as the solutions of Eq. ( 5), see Fig. 2.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Meanwhile when one studies creation of a closed universe due to the tunneling from a = 0 to a = 3 V through the potential barrier V(a) = 6π 2 a 2 -2π 2 a 4 V , one finds that the potential is positive at small a. Therefore the description of tunneling begins under the barrier, where the wave function does not oscillate, so there are no incident and reflected waves.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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- The authors state this as an assumption: The potential is flat, V(φ)=V.
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In the paper
“Meanwhile, as we have seen, there is no exponential suppression of creation of the low energy density inflationary universe if one considers compact open or flat universes.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
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In the paper
“For large a, the solution of Eq. ( 5) can be easily obtained in the WKB (semiclassical) approximation, Ψ ∼ a -1 exp[±i 2afoot_2 √ V √ 3 ]; positive sign corresponds to an expanding universe.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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- The authors state this as an assumption: The inflaton potential V(φ) is constant during the derivation of the Wheeler-DeWitt equation.
- The authors state this as an assumption: The scalar field φ is constant for the derivation.
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In the paper
“If one adds to V(a) the kinetic energy term -2π 2 K/a 2 , the effective potential V(a) becomes negative at small a, but one can show that for K < 1/192π 4 the solutions of the WDW equation remain non-oscillatory at small a, and the standard description of tunneling breaks down.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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- The authors state this as an assumption: The potential is flat, V(φ)=V.
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In the paper
“What is most important to us here is that the action at the first (quantum mechanical) part of the evolution of the universe was O(1), i.e. for any V (φ) there was no exponential suppression of the probability of quantum creation of the universe associated with tunneling.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“However, for K > 1/192π 4 , the WKB approximation is applicable at small a, which leads to the existence of solutions oscillating at a < ∼ K 1/4 .In terms of the variable α = log a, one can interpret these solutions as waves coming from -∞ and almost completely reflected back from the barrier at a = (K/3) 1/4 .In this case one can use the standard quantum mechanical description of tunneling.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“However, once the universe grows larger than a ∼ V -1/6 , its action rapidly becomes exponentially large and its classical description becomes possible.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“The universe appears “from nothing” without tunneling and without exponential suppression of the probability of its creation.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“In other words, contrary to the usual expectations, at a < ∼ V -1/6 one cannot describe the universe in terms of a classical space-time even though the size of the universe at a ∼ V -1/6 is much greater than the Planck size, and the density of matter as well as the curvature scalar in this regime remains small, R = 4V ≪ 1.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“Thus without the help of inflation the effects considered above cannot solve the major cosmological problems.However, they can keep the universe sufficiently homogeneous until the onset of inflation.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
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In the paper
“If a closed universe is born small, with the total kinetic energy of the scalar field being smaller than V -2 at the Planck time, then, at the classical level, it is going to collapse.It may survive due to tunneling, but the probability of this event is suppressed by exp -24π 2 V (φ) .”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“The meaning of this result, to be discussed below in a more detailed way, is that at a ≫ V -1/6 the effective action corresponding to the expanding universe is very large, and the universe can be described in terms of classical space and time.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
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In the paper
“One should note, however, that in the eternal inflation scenario [34,35] the problem of initial conditions may become irrelevant.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
What the sources say
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
A claim is checked only against the cited paper’s full text, never its abstract, and none is available for its sources.
In the paper
“If the initial size of the universe was O(1), then at the time t ≫ 1 each particle (or a gravitational perturbation of metric) within one cosmological time would run all over the torus many times, appearing in all of its parts with nearly equal probability.This effect, called “chaotic mixing,” should lead to a rapid homogenization of the universe [5,6].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
What the sources say
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Of course, there will be some suppression of the probability of inflationary initial conditions if an inflationary regime occurs only for a narrow range of initial values of the inflaton field.However, this phase space suppression is much milder than the exponential suppression of the type described by Eq. ( 1).”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
plainconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since the mass squared of the inflaton field is much smaller than V , and the size of the universe prior to inflation is smaller than H -1 , perturbations of this field behave as ultrarelativistic particles, the distribution of density of scalar perturbations becomes nearly homogeneous, and the energy density of the gradients of the scalar field will decrease as the energy of radiation (as a -4 )”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A claim is checked only against the cited paper’s full text, never its abstract, and none is available for its sources.
In the paper
“However, with a specific fine-tuning of inflationary potential [12], the inflationary stage can be made sufficiently short, in which case the remnants of the original anisotropy of the universe may become observable [6].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
What the sources say
hedgedconditional- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Meanwhile at a < ∼ V -1/6 , the effective action becomes small, the classical description breaks down, and quantum uncertainty becomes large.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“However, in his papers Friedmann noted that even flat and open universes may be compact if they have nontrivial topology [1].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > I. INTRODUCTION
What the sources say
- Not checked yetcites [1] Friedmann A, 1923, Mir Kak Prostranstvo i Vremya, The Universe as Space and Time
hedgedconditional- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“A similar result is valid for the open universe as well.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plain- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“One should note that here we ignored the Casimir effect, which may give an important quantum contribution O(a -4 ) to the vacuum energy density [3,4].Under certain conditions, this contribution may be negative, which will again forbid the classical evolution at a < O(H -1 ).However, this effect is suppressed by supersymmetry [3,4], it disappears in some anisotropic versions of toroidal compactifi”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
What the sources say
- Not checked yetcites [3] Schwarzschild K, 1900, Vierteljahrschrift der Ast. Ges.
- Not checked — no full text of the sourcecites [4] Zeldovich Y B, 1984, Sov. Astron. Lett.
hedgedconditional- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“After that, inflation occurs in an expanding and cooling universe automatically, if the inflaton field did not strongly interact with matter and if from the very beginning it was staying at the flat (inflationary) region of the effective potential.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Meanwhile, for 48K ≫ 1 the kinetic energy becomes Planckian when the universe is large in Planckian units, at a = K 1/6 ≫ 1, and the solution of the WDW equation remains semiclassical and oscillatory for all a.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- The authors state this as an assumption: The potential is flat, V(φ)=V.
None of the sources cited for this claim have been checked yet.
In the paper
“However, Casimir energy is suppressed by supersymmetry.In the absence of supersymmetry, this energy is O(a -4 ), so if a is somewhat greater than 1 at the Planck density, this energy always remains smaller than the thermal energy.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > III. LOW-SCALE INFLATION IN A COMPACT OPEN OR FLAT UNIVERSE
What the sources say
- Not checked yetcites [3] Schwarzschild K, 1900, Vierteljahrschrift der Ast. Ges.
- Not checked — no full text of the sourcecites [4] Zeldovich Y B, 1984, Sov. Astron. Lett.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Usually, kinetic energy of the scalar field φ gives the leading contribution in this limit, since it scales as a -6 , i.e. much faster than the energy density of relativistic and nonrelativistic matter.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Eqs. (5) differs from Eq. ( 3) not only by numerical coefficients, but mainly by the absence of the term 6π 2 a 2 , which was leading to the existence of the barrier in the effective potential V(a) for the closed universe.In the flat toroidal universe case there is no such barrier.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > B. Quantum creation of a compact inflationary universe
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
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It may state two separate claims as one.
In the paper
“The Hartle-Hawking wave function[17] yields a different result, P ∼ exp(+ 24π 2 V (φ)).In this case, the probability of inflation is also exponentially suppressed, as compared with the probability of living in the deepest minimum of V (φ).”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
What the sources say
- Not matchedcites [17] Sokolov D D, 1976, Sov. Astron.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“If inflation is not eternal, then our investigation suggests that most probably we live in a flat or open compact universe with nontrivial topology.”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
hedgedconditional- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
The authors state this as an assumption.
At the Planck time the universe was radiation dominated, with V << T^4 = O(1).
“Let us assume, as we did in the beginning of the previous section, that at the Planck time t p ∼ M -1 p = 1 the universe was radiation dominated, V ≪ T 4 = O(1).”
The authors state this as an assumption.
At the Planck time the total size of the box was Planckian, a(tp)=O(1).
“Let us also assume that at the Planck time the total size of the box was Planckian, a(t p ) = O(1).”
The authors state this as an assumption.
The inflaton potential V(φ) is constant during the derivation of the Wheeler-DeWitt equation.
“we temporarily ignored the evolution of the scalar fields (we will discuss it later, see Eqs. (8), ( 9)), and assumed that V (φ) = const, which is a good approximation in inflationary cosmology if the potential V is sufficiently flat.”
The authors state this as an assumption.
The scalar field φ is constant for the derivation.
“Let us assume for a moment that φ is constant (see below).”
The authors state this as an assumption.
The standard description of the universe in terms of classical space-time becomes possible only below the Planck density M^4_p=1.
“If we assume that the standard description of the universe in terms of the classical space-time becomes possible only below the Planck density M 4 p = 1”
The authors state this as an assumption.
The potential is flat, V(φ)=V.
“Assuming, as before, that the potential is flat, V (φ) = V , one can separate the variables and reduce this equation to the equation for the wave function of the scale factor Ψ(a):”
The paper relies on this without support.
Adding any matter contribution to the energy-momentum tensor makes going back in time result in a curvature singularity.
“It is clear, however, that at least one of the properties of our solution is not generic: If one adds any matter contribution to the energy-momentum tensor, then going back in time will result in the curvature singularity.”
The authors state this as an assumption.
Initially all space dimensions were compact; later some continued expansion while others became stabilized.
“Therefore it seems natural to assume that initially all space dimensions were compact; later some of them continued expansion, whereas some others become stabilized, e.g. by the KKLT mechanism [28].”
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 cited source doesn’t support the claim as written.
In the citing paper
“The authors of Refs.[7][8][9] nonetheless interpret the wave function for this model as a tunneling wave function, which amounts to dropping the condition that the probability flux must enter superspace through non-singular configurations.”
From On quantum creation of a toroidal universe (2026) · cites this paper as [9]
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredThe cited source doesn’t support the claim as written.
In the citing paper
“Thus, the tunneling boundary condition does not allow quantum nucleation of a toroidal universe, at least not in the minisuperspace model considered in Refs.[7][8][9].”
From On quantum creation of a toroidal universe (2026) · cites this paper as [9]
Checked by a reasoning model — may be wrong.
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Coule & Martin [8] and Linde [9] note that the wave function (67) does not oscillate; hence the semiclassical description breaks down at small a.”
From On quantum creation of a toroidal universe (2026) · cites this paper as [9]
The cited source says what the claim attributes to it.
In the citing paper
“The general solution of this equation is a linear superposition of Hankel functions of the first and second kind [9],”
From On quantum creation of a toroidal universe (2026) · cites this paper as [9]
The cited source says what the claim attributes to it.
In the citing paper
“Here we argued that this conclusion is not true.”
From On quantum creation of a toroidal universe (2026) · cites this paper as [9]
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.
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
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > IV. DISCUSSION AND SUMMARY
“some others become stabilized, e.g. by the KKLT mechanism [28].”
§ Creation of a Compact Topologically Nontrivial Inflationary Universe > References
“28. A. D. Linde and D. A. Linde, "Topological defects as seeds for eternal inflation," Phys. Rev. D 50, 2456 (1994) [arXiv:hep-th/9402115].”
Citations
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Metadata
- Authors
- Linde, A.
- Affiliations
- Stanford University (US)
- Corresponding author
- Linde, Andrei
- ORCID
- Linde, Andrei
- Venue
- Journal of Cosmology and Astroparticle Physics
- Publisher
- IOP Publishing
- Published
- 2004-10-05
- Volume
- 2004
- Issue
- 10
- Pages
- 004-004
- ISSN
- 1475-7508 (ISSN-L)
- 1475-7516
- OpenAlex
- W2094697662
- Cited by
- 158
- References
- 59
- Citations by year
- 2026: 11 · 2025: 4 · 2024: 13 · 2023: 2 · 2022: 3 · 2021: 3 · 2020: 2 · 2019: 6 · 2018: 3 · 2017: 6 · 2016: 5 · 2015: 10 · 2014: 8 · 2013: 7 · 2012: 8
- FWCI
- 5.29
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Cosmology and Astroparticle Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (0244728)
- 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
- quantum creation of the universe, inflationary universe, wave function of the universe, exponential suppression
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Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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