Creation of a compact topologically nontrivial inflationary universe

It may state two separate claims as one.

Particle production in the very early universe can make the universe locally isotropic, but cannot make it globally isotropic in the hot big bang scenario; such effects may play an important role in this scenario by making the initial small patch isotropic, and then this isotropy will be further enhanced by inflation.

“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”

• cites [23] de Oliveira-Costa A, 2004, Phys. Rev. sourceCould not verify

  Split into 2 atoms — 0 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 2 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

It may state two separate claims as one.

The probability of a universe being created with kinetic energy K > ~ V^{-2} 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).

“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].”

• cites [14] sourceCould not verify

  Split into 2 atoms — 0 confirmed, 0 partially supported, 0 disputed, 0 misattributed, 2 unverifiable. Expand the claim to see per-atom verdicts.

  Experimental — accuracy not yet measured

• AssumptionThe authors state this as an assumption: The potential is flat, V(φ)=V.

It may state two separate claims as one.

Ref. [26] also found that the probability of quantum creation of compact flat and open dS universes (for K=0) is not exponentially suppressed, which agrees with [3] and with our results.

“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.”

• cites [26] Barrow J, 2001, Class. Quantum Grav. sourceCould not verify

  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)

  Experimental — accuracy not yet measured
• cites [3] Schwarzschild K, 1900, Vierteljahrschrift der Ast. Ges.Not checked yet

Probably not a substantial claim. The paper’s conclusions may not depend on it.

Inflation in the string theory landscape scenario is eternal.

“This is an important consideration, because inflation in the string theory landscape scenario is eternal [37,38].”

• cites [37] Linde A D, 1998, Phys. Rev. sourcePartly supported

  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)

  Experimental — accuracy not yet measured
• cites [38] Bucher M, 1995, Phys. Rev. sourceCould not verify

  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)

  Experimental — accuracy not yet measured

It may state two separate claims as one.

For a ≫ K V^{1/6}, the term K a^{-2} is much smaller than a^4 V, so the solutions of the Wheeler-DeWitt equation look the same as those of Eq. (5).

“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.”

• AssumptionThe authors state this as an assumption: The potential is flat, V(φ)=V.

It may state two separate claims as one.

When studying creation of a closed universe due to tunneling from a=0 to a=3V through the potential barrier V(a)=6π^2 a^2 -2π^2 a^4 V, the potential is positive at small a, so the description of tunneling begins under the barrier, where the wave function does not oscillate, and there are no incident and reflected waves.

“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.”

• AssumptionThe authors state this as an assumption: The potential is flat, V(φ)=V.

It may state two separate claims as one. The quote only partly states this claim.

For compact flat or open universes, for K ≫ 1/48, incident waves coming from α=log a=-∞ are adiabatically transformed into waves moving towards α=log a=+∞; this result was verified numerically and agrees with the conclusion that there is no exponential suppression of the probability of creation of such universes.

“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.”

It may state two separate claims as one.

For large a, the solution of Eq. (5) can be obtained in the WKB semiclassical approximation, Ψ ∼ a^-1 exp[±i 2a√V/√3]; the positive sign corresponds to an expanding universe.

“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.”

• AssumptionThe authors state this as an assumption: The inflaton potential V(φ) is constant during the derivation of the Wheeler-DeWitt equation.
• AssumptionThe authors state this as an assumption: The scalar field φ is constant for the derivation.

It may state two separate claims as one.

If one adds the kinetic energy term -2π^2 K/a^2 to V(a), the effective potential becomes negative at small a, but for K < 1/(192π^4) the solutions of the Wheeler-DeWitt equation remain non-oscillatory at small a, and the standard description of tunneling breaks down.

“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.”

• AssumptionThe authors state this as an assumption: The potential is flat, V(φ)=V.

It may state two separate claims as one.

For the compact topologically nontrivial inflationary universe solution, the action during the first quantum-mechanical part of evolution was O(1), so for any potential V(φ) there was no exponential suppression of the probability of quantum creation associated with tunneling.

“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.”

It may state two separate claims as one.

For K > 1/(192π^4), the WKB approximation is applicable at small a, leading to solutions oscillating at a < ~ K^{1/4}; in terms of α=log a, these can be interpreted as waves coming from -∞ and almost completely reflected back from the barrier at a=(K/3)^{1/4}, so the standard quantum mechanical description of tunneling can be used.

“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.”

• AssumptionThe authors state this as an assumption: The potential is flat, V(φ)=V.

It may state two separate claims as one.

Once the universe grows larger than a ~ V^{-1/6}, its action rapidly becomes exponentially large and its classical description becomes possible.

“However, once the universe grows larger than a ∼ V -1/6 , its action rapidly becomes exponentially large and its classical description becomes possible.”

It may state two separate claims as one.

The universe appears from nothing without tunneling and without exponential suppression of the probability of its creation.

“The universe appears “from nothing” without tunneling and without exponential suppression of the probability of its creation.”

It may state two separate claims as one.

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 and curvature scalar remain small, R=4V << 1.

“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.”

It may state two separate claims as one.

Without inflation, the effects considered cannot solve the major cosmological problems; however, they can keep the universe sufficiently homogeneous until the onset of inflation.

“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.”

It may state two separate claims as one.

If a closed universe is born small, with total kinetic energy of the scalar field smaller than V^{-2} at the Planck time, then at the classical level it will collapse; it may survive due to tunneling, but the probability of this event is suppressed by exp(-24π^2/V(φ)).

“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) .”

• AssumptionThe authors state this as an assumption: The potential is flat, V(φ)=V.

It may state two separate claims as one.

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.

“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.”

Probably not a substantial claim. The paper’s conclusions may not depend on it.

In the eternal inflation scenario, the problem of initial conditions may become irrelevant.

“One should note, however, that in the eternal inflation scenario [34,35] the problem of initial conditions may become irrelevant.”

• cites [34] Vilenkin A, 1984, Phys. Rev. sourceNot checked — no full text of the source
• cites [35] Hartle J B, 1983, Phys. Rev. sourceNot checked — no full text of the source

It may state two separate claims as one.

If the initial size of the universe was O(1), then at t >> 1 each particle within one cosmological time would run all over the torus many times, appearing in all parts with nearly equal probability; this chaotic mixing leads to rapid homogenization.

“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].”

• cites [5] sourceNot checked — no full text of the source
• cites [6] sourceNot checked — no full text of the source

It may state two separate claims as one.

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 Eq. (1).

“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).”

It may state two separate claims as one.

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 the inflaton 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 decreases as radiation until inflation begins.

“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 )”

It may state two separate claims as one.

With a specific fine-tuning of inflationary potential [12], the inflationary stage can be made sufficiently short, in which case remnants of the original anisotropy of the universe may become observable [6].

“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].”

• cites [12] Lockhart C N, 1982, Phys. Rev. sourceNot checked — no full text of the source
• cites [6] sourceNot checked — no full text of the source

It may state two separate claims as one.

At a <~ V^-1/6, the effective action becomes small, the classical description breaks down, and quantum uncertainty becomes large.

“Meanwhile at a < ∼ V -1/6 , the effective action becomes small, the classical description breaks down, and quantum uncertainty becomes large.”

It may state two separate claims as one.

In the open universe case, the effective energy associated with curvature decreases as a^-2, becomes dominant soon after expansion, making the relative role of normal matter and its density perturbations insignificant; this is an additional reason density perturbations in a compact hyperbolic universe will rapidly decrease.

“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”

• cites [22] sourceNot checked yet

  Checked before — its earlier verdict is held back until a new check runs.

Probably not a substantial claim.

Flat and open universes may be compact if they have nontrivial topology.

“However, in his papers Friedmann noted that even flat and open universes may be compact if they have nontrivial topology [1].”

• cites [1] Friedmann A, 1923, Mir Kak Prostranstvo i Vremya, The Universe as Space and TimeNot checked yet

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

A similar result is valid for the open universe as well.

“A similar result is valid for the open universe as well.”

It may state two separate claims as one.

The Casimir effect may give an important quantum contribution O(a^-4) to the vacuum energy density; under certain conditions it may be negative, forbidding classical evolution at a < O(H^-1); however, it is suppressed by supersymmetry, disappears in some anisotropic versions of toroidal compactification, and becomes unimportant with usual matter or 3D curvature in the open universe case.

“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”

• cites [3] Schwarzschild K, 1900, Vierteljahrschrift der Ast. Ges.Not checked yet
• cites [4] Zeldovich Y B, 1984, Sov. Astron. Lett.Not checked — no full text of the source

It may state two separate claims as one.

After creation, 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.

“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.”

It may state two separate claims as one.

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 Wheeler-DeWitt equation remains semiclassical and oscillatory for all a.

“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.”

• AssumptionThe authors state this as an assumption: The potential is flat, V(φ)=V.

It may state two separate claims as one.

Casimir energy is suppressed by supersymmetry; in the absence of supersymmetry it is O(a^-4), so if a is somewhat greater than 1 at the Planck density, it remains smaller than the thermal energy.

“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.”

• cites [3] Schwarzschild K, 1900, Vierteljahrschrift der Ast. Ges.Not checked yet
• cites [4] Zeldovich Y B, 1984, Sov. Astron. Lett.Not checked — no full text of the source

It may state two separate claims as one.

The kinetic energy of the scalar field gives the leading contribution in the limit of going back in time, because it scales as a^{-6}, faster than the energy density of relativistic and nonrelativistic matter.

“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.”

It may state two separate claims as one.

The Wheeler-DeWitt equation for the compact flat toroidal universe differs from that for the closed universe not only by numerical coefficients, but mainly by the absence of the term 6π^2 a^2, which led to the existence of the barrier in the effective potential for the closed universe; in the flat toroidal universe case there is no such barrier.

“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.”

It may state two separate claims as one.

The Hartle-Hawking wave function yields 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).

“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).”

• cites [17] Sokolov D D, 1976, Sov. Astron.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It may state two separate claims as one.

If inflation is not eternal, then our investigation suggests that most probably we live in a flat or open compact universe with nontrivial topology.

“If inflation is not eternal, then our investigation suggests that most probably we live in a flat or open compact universe with nontrivial topology.”