COLEMAN–DE LUCCIA TUNNELING AND THE GIBBONS–HAWKING TEMPERATURE

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Case (IV): the inside does not reach V-, and the outside does not reach the false vacuum in the Euclidean action due to the de Sitter horizon; HM tunneling is a special limit.

“(IV) Not only does the inside of the bubble not reach V -, the outside of the bubble does not reach the false vacuum in the Euclidean action, due to the presence of the de Sitter horizon.HM tunneling is a special limit in this case.”

It may state two separate claims as one.

For large V_T and ǫ > 0, case (IV) is generic, with HM tunneling as a special limit.

“For large V T and ǫ > 0, case (IV) is generic, where HM tunneling is a special limit.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Case (I): the center of the nucleation bubble is in the true vacuum V- while the outside reaches the false vacuum V+; the thin-wall approximation is a special limit (ǫ → 0).

“(I) The center of the nucleation bubble is in the true vacuum V -while the outside of the bubble reaches the false vacuum V + .The thin-wall approximation is a special limit (ǫ → 0) in this case.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

For a general smooth potential with varying parameters, the four cases may reduce to case (IV) only; within case (IV), it contains regions resembling the various cases.

“Strictly speaking, for a”

It may state two separate claims as one.

In Eq. (4.14), the two terms are comparable; the back-reaction is important.

“So we see explicitly that the two terms in Eq. (4.14) are comparable.That is, the back-reaction is important.”

It may state two separate claims as one.

For the particular potential used, the slope of case (IV) (∼ -3.74) never quite reaches the HM value due to corrections, and case (III) does not appear.

“For this particular potential, the slope case (IV) (∼ -3.74) never quite reaches the HM value due to corrections.The specific potential used here is described in Sec. 6. Case (III) does not appear for this potential.”

It may state two separate claims as one.

The HM formula for B is an over-estimate by as much as a factor of three; the actual decay is a combination of thermal and quantum tunneling, whose rate can be much faster.

“We see that the HM formula for B is an over-estimate, by as much as a factor of three.That is, the actual decay is a combination of thermal tunneling and quantum tunneling, whose rate can be much faster.”

• AssumptionRests on an assumption: In the semiclassical approximation, the tunneling rate per unit volume is given by Γ ≃ A e^{-B}, with B = S_E(φ) - S_E(φ+).

It may state two separate claims as one.

Regions (I) and (II) are divided by β: case (I) if 0 ≤ β < 1 and case (II) if β > 1; for β = 1, only the center of the bubble reaches the true vacuum.

“In summary, the regions (I) and (II) are divided by the value of β (3.13): case (I) if 0 ≤ β < 1 and case (II) (φ(0) = φ 0 < φ -) if β > 1.For β = 1, only the center of the bubble reaches the true vacuum.”

It may state two separate claims as one.

In case (II), tunneling is always downward, and ∆V+/∆V0 ≤ 1/4.

“which implies that tunneling is always downward, and ∆V + /∆V 0 ≤ 1/4.”

It may state two separate claims as one.

Case (II): φ(ξ) reaches φ+ but not φ- when both αλ/H^2 Δφ+ ≥ 1 and (λ/H^2)I(H^2 Δφ+/λ) < Δφ-.

“• Case (II) φ(ξ) reaches φ + but not φ -.That is, φ(ξ) reaches φ + and φ t-, where φ T ≤ φ t-< φ -.This happens when both αλ/H 2 ∆φ + ≥ 1 and (λ/H 2 )I H 2 ∆φ + /λ < ∆φ -are satisfied.”

It may state two separate claims as one.

When ε is small, the thin-wall approximation coincides with our computation, verifying that the thin-wall approximation is good when ε is small.

“It means that when ǫ is small the thin-wall approximation coincides with our computation and so verifies that the thin-wall appoximation is good when ǫ is small.”

Probably not a substantial claim. It may state two separate claims as one.

To emphasize the effect of gravity, consider V_T ≫ ∆V- = V_T - V-, where V_T/M_P^4 ≪ 1 so gravitational effects can be important while the semiclassical approximation is still valid.

“To emphasize the effect of gravity, let us consider the situation where V T ≫ ∆V -= V T -V -(4.1) where V T /M 4 P ≪ 1 so gravitational effects can be important for the tunneling while the semi-classical approximation is still valid.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

Tunneling in the absence of gravity is always downwards, while gravitational effects allow tunneling upwards.

“Another interesting point is that tunneling in the absence of gravity is always downwards, while gravitational effects allow tunneling upwards.”

It may state two separate claims as one.

For case (II), φ(ξ) reaches the false vacuum V+ at φ+ outside the bubble within the horizon but the bubble inside never reaches the true vacuum V- at φ- at the moment of creation.

“In this case, φ(ξ) reaches the false vacuum V + at φ + outside the bubble within the horizon but the bubble inside never reaches the true vacuum V -at φ -at the moment of creation.”

It may state two separate claims as one.

When ε is fixed and V_T increases, the difference between B_tw and B gets larger; this is because the solution moves toward case (II), where the solution does not reach φ- in contrast to the thin-wall approximation.

“It means when ǫ is fixed and V T is increasing, the difference is getting larger and larger.An immediate explanation is: in this situation the solution is moving toward case (II) and finally when tunneling becomes case (II) the difference is very large, since the solution does not reach φ -in contrast to the thin-wall approximation.”

Probably not a substantial claim. The paper may give no reason why it holds.

The absolute height of the potential, corresponding to the vacuum energy density, is important for tunneling with gravity.

“Here the absolute height of the potential, which corresponds to the vacuum energy density, is important for tunneling with gravity.”

It may state two separate claims as one.

It is crucial to include the back-reaction effect due to δr; if the δr contribution is dropped, B(δr-excluded) ∼ -2992, which is negative and has no physical meaning.

“We see that it is crucial to include the back-reaction effect due to δr here.In fact, if we drop the contribution from δr in Eq.(4.14), then B(δr-excluded) ∼ S E,0 + S E,1,φ -S(φ + ) ∼ -2992, which is negative and has no physical meaning.”

It may state two separate claims as one.

In the thin-wall approximation, η → 1 as ǫ/∆V- becomes small, and B reduces to the usual thin-wall formula B → 27π2τ4/(2ǫ3).

“In the thin-wall approximation, η → 1 as ǫ/∆V -becomes small, and B reduces to the usual thin-wall formula for B, B → 27π 2 2ǫ 3 τ 4 (3.20)”

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

Case (III): the inside reaches V-, but the outside does not reach the false vacuum in the Euclidean action due to the de Sitter horizon.

“(III) The inside of the bubble reaches V -, but the outside of the bubble does not reach the false vacuum in the Euclidean action, due to the presence of the de Sitter horizon.”

It may state two separate claims as one.

For case (IV), B = S_E(φ)-S(φ+) = 24π^2 M_p^4(1/V+ - 1/V_T) - C λ^2 M_p^6/V_T^3 + O(...). The first term is the Hawking-Moss bounce B_HM, and the second term lowers B, so the actual tunneling rate is larger than that given by the HM formula.

“B = S E (φ) -S(φ + ) = 24π 2 M 4 p 1 V + - 1 V T -C λ 2 M 6 p V 3 T + O λ 4 M 8 p V 5 T .”

• Also stated in: § 5.1 Case (IV) · plain

  “The first term is just the Hawking-Moss bounce B HM (1.4).while the second term, correction from this bounce solution, lowers B and therefore the actual tunneling rate is larger than that given by the HM formula.”

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

The value of τ as defined should be very close to the actual domain wall tension.

“The value of τ as defined should be very close to the actual domain wall tension.”

It may state two separate claims as one.

Case (II) only happens if φ0 is to the right side of the barrier and φ0 ≤ φ-, and V+ ≥ V(φ0) ≥ V-.

“This case only happens if φ 0 is to the right side of the barrier (i.e., near φ -) and φ 0 ≤ φ -. Also, we expect V + ≥ V (φ 0 ) ≥ V -.”

It may state two separate claims as one.

This contribution is perturbative in the couplings while its effect in tunneling is non-perturbative.

“This contribution is perturbative in the couplings while its effect in tunneling is non-perturbative.”

It may state two separate claims as one.

For case (IV), the integral (4.12) is carried out analytically, and the leading order correction has two terms of the same form: S_E(φ)=24π^2 M_p^4/V_T - C λ^2 M_p^6/V_T^3 + O(...), where C=C1+C2∼65.49, C1∼-196.47 from δr and C2∼261.96 from φ.

“S E (φ) = 24π 2 M 4 p V T -C λ 2 M 6 p V 3 T + O λ 4 M 8 p V 5 T .”

• Also stated in: § 5.1 Case (IV) · plain

  “where C = C 1 + C 2 ∼ 65.49 is a positive constant.C 1 ∼ -196.47 corresponds to the δr(ξ) contribution and C 2 ∼ 261.96 is from φ(ξ).”

It may state two separate claims as one.

Case (I): φ(ξ) reaches both φ+ and φ- when αλ/H^2 Δφ+ ≥ 1 and (λ/H^2)I(H^2 Δφ+/λ) ≥ Δφ-.

“• Case (I) φ(ξ) reaches both φ + and φ -.This happens when αλ/H 2 ∆φ + ≥ 1 and (λ/H 2 )I H 2 ∆φ + /λ ≥ ∆φ -where α ∼ 0.4 and I(x) is a monotonic function shown in Figure 6.”

Probably not a substantial claim. The paper may give no reason why it holds.

For a general potential, the coupled equations of φ and the cosmic scale factor are too complicated to solve except in the thin-wall approximation.

“For a general potential V (φ), the coupled equations of φ(ξ) and the cosmic scale factor r(ξ) are too complicated to solve even in the absence of the gravity except in the thin-wall approximation.”

It may state two separate claims as one.

If λ- = λ+, β simplifies to (1 - η)^2/√η, and the dividing point is η_c = (√2 - 1)^2 = 0.17.

“If λ -= λ + , we see that β simplifies somewhat, β = (1 -η) 2 √ η (3.21) where 0 < η < 1 is given in Eq.(2.8); so the dividing point is at η c = ( √ 2 -1) 2 = 0.17.”

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

For more general potentials, the coefficient 2/3 in Eq. (3.18) will have order one variations.

“For more general potentials, the coefficient 2/3 in Eq. ( 3.18) will have order one variations.”

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

In the absence of gravity, the overall height of the potential is not important, so the triangular potential is parameterized by four parameters (∆V±, ∆λ±).

“In the absence of gravity, the overall height of the potential is not important, so the triangular potential is parameterized by a set of four parameters, namely (∆V ± , ∆λ ± ).”

It may state two separate claims as one.

The tunneling rate for the case where the inside does not reach the true vacuum and the outside is cut off by the horizon can be estimated, and the corrections to the Hawking-Moss formula can be large.

“We estimate the tunneling rate for this case and find that the corrections to the Hawking-Moss formula can be large.”

• AssumptionRests on an assumption: In the semiclassical approximation, the tunneling rate per unit volume is given by Γ ≃ A e^{-B}, with B = S_E(φ) - S_E(φ+).

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

Case (II): the outside reaches the false vacuum V+ but the inside never reaches the true vacuum V- in the Euclidean solution; after creation, the inside falls towards the true vacuum as the bubble grows.

“(II) The outside of the bubble reaches the false vacuum V + but the inside of the nucleation bubble never reaches the true vacuum V -in the Euclidean solution.After the creation of the bubble, its inside falls towards the true vacuum V -as the bubble grows.”

It may state two separate claims as one.

For c < 1 (gradient towards false vacuum steeper than towards true vacuum), the Euclidean solution has ΔV_R < ΔV_L, meaning tunneling is from V_L = V_T - ΔV_L going up to V_R = V_T - ΔV_R.

“On the other hand, for c < 1 (that is, when the gradient towards the false vacuum is steeper than that towards the true vacuum), the Euclidean solution has ∆V R < ∆V L .This means the tunneling is from V L = V T -∆V L going up to V R = V T -∆V R .”

It may state two separate claims as one.

This happens if κ = 0, or as a very good approximation when the overall height of the potential is much smaller than the Planck scale.

“This happens if we set κ = 0.Alternatively, this is a very good approximation when the overall height of the potential is much smaller than the Planck scale.”

It may state two separate claims as one.

The bounce solution φ(ξ) will modify the geometric background r(ξ) in (4.5), and it is important to include this back-reaction.

“The bounce solution φ(ξ) will modify the geometric background used, i.e., r(ξ) in (4.5).As we shall see, it is important to include this back-reaction.”

It may state two separate claims as one.

In all cases, the bounce of the Euclidean action decreases rapidly as the vacuum energy density increases, signaling that the tunneling is not exponentially suppressed.

“In all cases, we see that the bounce of the Euclidean action decreases rapidly as the vacuum energy density increases, signaling that the tunneling is not exponentially suppressed.”

• AssumptionRests on an assumption: In the semiclassical approximation, the tunneling rate per unit volume is given by Γ ≃ A e^{-B}, with B = S_E(φ) - S_E(φ+).

It may state two separate claims as one.

For general c = λ-/λ+ ≠ 1, the leading order correction is at most about 64%; the HM formula is an overestimate of B by as much as a factor of three.

“which means that the leading order correction is at most about 64%.We see that the HM formula is an over estimate of the value of B, by as much as a factor of three.”

The quote only partly states this claim.

For a potential with small ε = V+ - V-, λ = 1×10^-7, V_T = 0.2×10^-6, Δφ+ = 0.005, and Δφ- = 0.0055, B = 1794.32, which is close to the thin-wall approximation result B_tw = 1848.76.

“B = S E -S(φ + ) ∼ S E,0 + S E,1,δr + S E,1,φ -S(φ + ) = 1794.32 which is quite close to the thin-wall approximation result B tw = 1848.76by Eq.(5.50).”

The quote only partly states this claim.

For a specific background temperature, S3/T_H = 8π∆V+ / 3H^4 + O(λ^2 M_p^6 / V_T^3).

“Evaluating (7.2) using (7.8) we find that, for a specific background temperature,”

It may state two separate claims as one.

As H ∼ √V_T increases, the tunneling behavior goes from case I to case IV via case II; that is, from the CDL thin-wall approximation (slope ∼ -3) to HM tunneling (slope ∼ -4).

“As H ∼ √ V T increases, the tunneling behavior goes from case (I) to case (IV) via case (II); that is, it goes from the CDL thin-wall approximation (to the left in case (I), with slope ∼ -3 as given by Eq.(1.3)), to the HM tunneling (to the right of case (IV), with slope ∼ -4 as given by Eq.(1.4)).”

It may state two separate claims as one.

In case I thermal tunneling, the solutions have the same qualitative features as in case I Euclidean tunneling, but the Euclidean bounce B is smaller than S3/T_H.

“In general the solutions have the same qualitative features as in case I Euclidean tunneling, but the Euclidean bounce B is smaller than S 3 /T H .”

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

In the thin-wall approximation, maximizing S3 gives 1/R^2 = 1/(R0^(3))^2 + (H+^2 + H-^2)/2 + κ^2 τ^2 / 36, where R0^(3) = 2τ/ǫ.

“Maximizing S 3 , we find R given by”

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

There are four types of thermal tunneling solutions: Case I reaches both φ+ and φ-; Case II reaches φ+ but not φ-; Case III reaches φ- but not φ+; Case IV reaches neither φ+ nor φ-.

“Like the Euclidean case discussed above, there are also four types of thermal tunneling solutions:”

The quote only partly states this claim.

B varies by 25 orders of magnitude in the comparison of Euclidean action and free energy divided by GH temperature.

“We see that B varies by 25 orders of magnitude.”

It may state two separate claims as one.

For the case II solution to be consistent, φ1(0) - φT < ∆φ- or equivalently I S3 (H^2 ∆φ+ / λ) = 1/2 - HξT/2 cot(HξT) < H^2 ∆φ- / λ.

“For our solution to be consistent, we need φ 1 (0) -φ T < ∆φ -or equivalently”

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

Using Eq. (5.47), τ_i = π/2 α_i^2 f_i.

“τ i = φ + φ - dφ 2(2α i ) cos( φ i f i ) = π 2 α i 2 f i (8”

It may state two separate claims as one.

For cases (I) and (II), φ_f+ = φ+, so the first term in (6.2) vanishes; for cases (III) and (IV), φ_f+ > φ+, so both terms contribute.

“For cases (I) and (II), we have φ f + = φ + , so the first term vanishes.For cases (III) and (IV), φ f + > φ + , so both terms contribute.”

It may state two separate claims as one.

A crude estimate of the prefactor gives Γ ≃ (B/2π)^2 |V''(φ_t)|^2 e^{-B}, where B = 2π^2 τ / H^3 and |V''(φ_t)| = α/f^2, yielding Γ = π^4/(8H^9) Σ_{i=1}^N α_i^3 f_i^2 e^{-B_i}.

“and a crude estimate of the prefactor gives Γ ≃ B 2π 2 |V ′′ (φ t )| 2 e -B (8.4) where B = 2π 2 τ /H 3 (8.5)and |V ′′ (φ t )| = α/f 2 , yielding Γ = π 4 8H 9 N i=1 α 3 i f 2 i e -B i (8.6)”

It may state two separate claims as one.

For the specific potential used in Sec. 6, the slope in case IV (∼ -3.74) never quite reaches the HM value due to corrections, and case III does not appear.

“For this particular potential, the slope case (IV) (∼ -3.74) never quite reaches the HM value due to corrections.The specific potential used here is described in Sec. 6. Case (III) does not appear for this potential.”

It may state two separate claims as one.

Numerically, I S3 (H^2 ∆φ+ / λ) > H^2 ∆φ+ / λ, so case II thermal tunneling is always from V+ to some VF where V+ > VF > V-, as in case II Euclidean tunneling.

“Numerically we find that I S 3 ( H 2 ∆φ + λ ) > H 2 ∆φ + λ so case II thermal tunneling is always from V + to some V F where V + > V F > V -as we saw above in case (II) Euclidean tunneling.”

It may state two separate claims as one.

The approximation works for V_T ≫ ∆V-, which reads V_T ≫ 7 × 10^-10.

“Notice that our approximation works for VT ≫ ∆V-, which reads VT ≫ 7 × 10 -10 .”

The quote only partly states this claim.

In the large VT limit, the general solution to (7.4) is φ = λ H^2 A + Hξ/2 cot(Hξ) + B cot(Hξ).

“In the large V T limit considered above, the general solution to (7.4) is”

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

Case III thermal tunneling only occurs when λ+ = λ-, as in the Euclidean case.

“Case III tunneling only occurs when λ + = λ -as in the Euclidean case.”

It may state two separate claims as one.

As B decreases, multi-nucleation bubbles will form and the phase transition involves bubble collisions.

“As B decreases, multi-nucleation bubbles will form and the phase transition involves bubble collisions.”

It may state two separate claims as one.

This tunneling up phenomenon does not happen in the absence of gravity; it is a gravitational effect.

“As noted earlier, this tunneling up phenomenon does not happen in the absence of gravity.So this possibility is a gravitational effect.”

It may state two separate claims as one.

The familiar CDL thin-wall approximation occurs in case (I), but a solution of case (I) is not necessarily the thin-wall solution; B_tw may not be very accurate.

“The familiar CDL thin-wall approximation occurs in case (I), however, a solution of the case (I) is not necessarily the”

It may state two separate claims as one.

In case (I), φ(ξ) reaches both φ+ and φ-, and the solution looks like the thin-wall tunneling solution in [1]; the thin-wall approximation belongs to case (I).

“The solution looks like the thin-wall tunneling solution in [1].”

• Also stated in: § 5.4 Case (I) · boosted

  “It is clear that the thin-wall approximation belongs to case (I), since the inside of the bubble reaches φ -while the outside reaches φ + (see Figure 4).”

• cites [1]Not matched

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

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

This can be interpreted as a finite volume effect due to Gauss's law, H^2 φ^2, where H^-1 is the horizon size.

“One an also interpret this as a finite volume effect, due to Gauss's law, H 2 φ 2 , where H -1 is the horizon size.”

It may state two separate claims as one.

The picture of case (IV) is consistent with the interpretation proposed in [12], [13], [14]: HM tunneling should be interpreted as a quantum fluctuation up the potential barrier; the field will thermally fluctuate from V+ at φ+ up to V(φ_L(π/H)) < V_T at φ_L(π/H) > φ+, tunnel to φ_R(0) < φ-, and then classically roll down to φ-.

“The picture of case (IV) is consistent with the interpretation proposed in [12], [13], [14].HM tunneling should be interpreted as a quantum fluctuation up the potential barrier.Here we see that the field will thermally fluctuate from V + at φ + up to V (φ L (π/H)) < V T at φ L (π/H) > φ + , tunnel to φ R (0) < φ -and then classically roll down to φ -.”

• cites [12]Not matched

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

• cites [13]Not matched

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

• cites [14]Not matched

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

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

The thin-wall approximation is a special limit of case (I), where inside the nucleation bubble is the true vacuum while the outside reaches the false vacuum.

“The thin-wall approximation is a special limit of the case (of only quantum tunneling) where inside the nucleation bubble is the true vacuum while the outside reaches the false vacuum.”

It may state two separate claims as one.

Because φ ≡ φ_T is the Hawking-Moss bounce solution, case (IV) tunneling is like a fluctuation around HM tunneling and approaches the Hawking-Moss solution when λ/H^2 is small.

“Because φ ≡ φ T is the Hawking-Moss bounce solution, case (IV) tunneling is like a fluctuation around HM tunneling and will approach the Hawking-Moss solution when λ/H 2 is small.”

Probably not a substantial claim. It may state two separate claims as one.

Unlike Ref. [9], the paper cannot get an exact analytic formula for B in the presence of gravity, but the resulting analytic study gives a very good approximation and simplifies enough to see the overall picture.

“Unlike Ref. [9], we cannot get an exact analytic formula for B in the presence of gravity, but the resulting analytic study gives a very good approximation and does simplify enough for us to see the overall picture.”

• cites [9]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.

By condition (4.1) and (5.11), λ^2/(H^2 V_T) ≪ 1, so L/V_T ≪ 1 and the expansion (4.12) and (4.14) works for case (IV).

“By the condition (4.1) and (5.11), the ratio λ 2 /(H 2 V T ) ≪ 1 and hence L/V T ≪ 1 and the expansion (4.12) and (4.14) works for case (IV).”

It may state two separate claims as one.

For case (IV), the small expansion factor is λ^2/(H^2 V_T), not ΔV-/V_T, because the bounce solution (5.6) does not feel φ-.

“The only new issue is that for case (IV), the small expansion factor is λ 2 /H 2 V T not ∆V -/V T , simply because the bounce solution (5.6) does not”

It may state two separate claims as one.

In the Planck region (V ∼ 1), B ≪ 1 and Γ ∼ 1. At low energy scale, the background curvature radius is quite large and the bubble size is relatively small, and then the tunneling rate is insensitive to the vacuum energy of the false vacuum.

“In Planck region (V ∼ 1), B ≪ 1 and Γ ∼ 1.At low energy scale, the background curvature radius is quite large and the bubble size is relatively small, and then the tunneling rate is insensitive to the vacuum energy of the false vacuum.”

It may state two separate claims as one.

For case (II), after quantum tunneling, the field will continue to roll down until reaching the true vacuum; this also happens in de Sitter space.

“For [9], we can verify that, after the quantum tunneling, the field will continue to roll down until reaching the true vacuum.This also happens in de Sitter space.”

• cites [9]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.

Once B is small, Eq. (1.1) is no longer valid; the prefactor and sub-leading terms become important.

“Once B is small (say B 1), Eq.(1.1) is no longer valid.The prefactor and sub-leading terms will become important.”

• AssumptionRests on an assumption: In the semiclassical approximation, the tunneling rate per unit volume is given by Γ ≃ A e^{-B}, with B = S_E(φ) - S_E(φ+).

It may state two separate claims as one.

As γ → 1/α = 2.51, φ_L(π/H) → φ+; the outside of the bubble can reach the false vacuum within the horizon. When this happens, the HM formula is no longer accurate.

“On the other hand, as γ → 1/α = 2.51, φ L (π/H) → φ + ; the outside of the bubble can reach the false vacuum within the horizon.When this happens, the HM formula is no longer accurate.”

It may state two separate claims as one.

When B is small, O(4) nucleation bubbles no longer dominate; bubbles of other shapes and bubble collisions become important.

“We do expect that O(4) nucleation bubbles no longer dominate.That is, bubbles of other shapes as well as bubble collisions become important.”

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

Because of fast tunneling when the wavefunction of the universe is high up in the landscape, it is likely that the universe is quite mobile there.

“Because of fast tunneling when the wavefunction of the universe is high up in the landscape, it is likely that the universe is quite mobile there.”

It may state two separate claims as one.

The correction is typically not very big and the HM formula is quite good in general.

“Here, we see that the correction is typically not very big and the HM formula is quite good in general.”

Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. It may state two separate claims as one.

The picture is in agreement with the qualitative understanding one has already, but the details allow a quantitative understanding of the validity of each approximation and the sizes of the corrections.

“The picture is in agreement with the qualitative understanding one has already, but the details allow us to obtain a quantitative understanding of the validity of each approximation and the sizes of the corrections.”

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

The role of resonance tunneling in the development of the Bloch wave is well known.

“In particular, its role in the development of the Bloch wave is well known.”

It may state two separate claims as one.

The temperature T in the thermal tunneling formula is the Gibbons-Hawking temperature T_H, which strictly speaking is a gravitational, not thermal, effect.

“The difference is that here, the temperature T is the Gibbons-Hawking temperature T H , which strictly speaking is a gravitational, not thermal, effect.”

It may state two separate claims as one.

This coherence is particularly likely along the axionic directions since the effective potential is periodic or close to periodic and the barrier heights can be very low.

“This is particularly likely along the axionic directions since the effective potential is periodic or close to being periodic and the heights of the barriers can be very low.”

It may state two separate claims as one.

Compactification and moduli stabilization usually introduce a term like H^2 φ^2, which tends to remove barriers; this term is a finite volume effect arising from the equivalent of Gauss's Law and causes the η problem for slow-roll inflation.

“In fact, compactification and moduli stabilization usually introduces a term like H 2 φ 2 , which will tend to remove a lot of barriers.This term, a finite volume effect, arises from the equivalent of the Gauss's Law.(Actually, this is the term that causes the η problem for slow-roll inflation [25])”

• cites [25]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 in any direction 24 f_i^2 < T_H^2, then there is no barrier in that direction, so the wavefunction is coherent along that direction.

“we see that, if in any direction, 24f 2 i < T 2 H (8.10)”

It may state two separate claims as one.

The derivation of Ref. [6] assumes a fixed de Sitter background, whereas our derivation includes back-reaction, which can be large.

“Note that the derivation of Ref. [6] assumes a fixed de Sitter background., while the back-reaction (which can be large) is included in our derivation.”

• cites [6]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.

As one goes up in the cosmic landscape (e.g., turning on D-terms), tunneling will be faster and the wavefunction will be Bloch wave-like in more axionic directions.

“As we go up in the cosmic landscape (say, turning on D-terms), tunneling will be faster and so the wavefunction will be Bloch wave-like in more of the axionic directions.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

Above the GUT scale, quantum gravity effects become important and the paper has nothing to say.

“Above the GUT scale, quantum gravity effects become important and we have nothing to say in this case.”

It may state two separate claims as one.

The enhancement can be interpreted as a GH temperature effect, but the actual enhancement is exponentially larger than a naive estimate based on the usual temperature effect, due to the enhanced O(4) symmetry of the Euclidean action versus the SO(3) × U(1) symmetry in the finite temperature case.

“Qualitatively, one may interpret this as a GH temperature effect, but the actually enhancement is exponentially larger than a naive estimate based on the usual temperature effect, due to the enhanced O(4) symmetry of the Euclidean action (versus the SO(3) × U (1) symmetry in the finite temperature case).”

It may state two separate claims as one.

Since R ≪ R+, the Euclidean S4 bounce B is smaller, so it dominates.

“Since R ≪ R + , we see that the Euclidean S 4 bounce B is smaller so it dominates.”

It may state two separate claims as one.

Resonance tunneling also enhances tunneling among the vacua in the landscape, and such enhancement can be substantial.

“This also enhances the tunneling among the vacua in the landscape.Such enhancement can be substantial.”

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

The finite temperature effect on the effective potential tends to decrease the barrier, allowing faster tunneling.

“It is again well known that the finite temperature effect on the effective potential tends to decrease the barrier, allowing faster tunneling.”

It may state two separate claims as one.

Using the saddle-point approximation, J(E*) = S_E(φ_bounce) - S_E(φ_f+), and the tunneling takes a Euclidean time Δτ = 1/2T_H.

“This yields J(E * ) = S E (φ bounce ) -S E (φ f + ). This result also implies that the tunneling takes a Euclidean time ∆τ = 1/2T H and”

Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

Presumably, the QCD θ vacuum we live in today is described by such a Bloch wave.

“Presumably, the QCD θ vacuum we live in today is described by such a Bloch wave.”

It may state two separate claims as one.

Our result (6.2) agrees with the result of Ref. [6]: its first term corresponds to a thermal (Gibbons-Hawking temperature) fluctuation from the false vacuum at φ+ part way up the barrier to φf+, and its second term corresponds to quantum tunneling.

“So our result (6.2) agrees with this result of Ref. [6]: the first term in (6.2) corresponds to a thermal (i.e., Gibbons-Hawking temperature) fluctuation from the false vacuum (at φ + ) part way up the barrier to φ f + and the second term in (6.2) corresponds to a quantum tunneling.”

• cites [6]Not matched

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

Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.

This result is expected and consistent with [18].

“Based on the discussion in the last section, this result is expected and consistent with [18].”

• cites [18]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.

The perturbation series (4.14)-(4.15) start from the de-Sitter space Euclidean action, not the HM tunneling exponential factor B_HM; so the correction, though large compared to B_HM, is still much smaller than the zero-order de-Sitter space Euclidean action, and the leading correction yields a very good approximation.

“Notice that the perturbation series (4.14), (4.14) and (4.15) start from the de-Sitter space Euclidean action, not the HM tunneling exponential factor BHM .So the correction, though large compared to BHM , is still much smaller than the zero order, de-Sitter space Euclidean action.Hence the leading correction yields a very good approximation here.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

In the multi-field (d > 1) case, as expected in the cosmic landscape, there are typically many tunneling directions for any false vacuum, so one expects fast tunneling.

“In the multi-field (d > 1) case, as we expect to be the situation in the cosmic landscape, there are typically many tunneling directions for any false vacuum, so one expects fast tunneling to take place.”

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

The effective potential in de Sitter space has been calculated for some simple potentials in [7], [8].

“The effective potential in de Sitter space has been calculated for some simple potentials in [7], [8].”

• cites [7]Not matched

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

• cites [8]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.

In this thermal system, tunneling does not need to occur from the bottom of the false potential well; instead, tunneling proceeds by a combination of thermal excitation part way up the barrier followed by quantum tunneling through the barrier.

“In this thermal system, tunneling does not need to occur from the bottom of the false potential well.Instead, tunneling proceeds by a combination of thermal excitation part way up the barrier followed by quantum tunneling through the barrier.”

• cites [6]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.

In quantum mechanics, tunneling through a barrier becomes less suppressed when the energy of the particle increases, as happens in a thermal bath with rising temperature.

“In quantum mechanics, tunneling through a barrier becomes less suppressed when the energy of the particle increases.This happens when the particle is in a thermal bath with rising temperature.”

Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.

It is reasonable to expect that the wavefunction of the universe tends to spread along some of the axionic directions.

“So it is reasonable to expect that the wavefunction of the universe tends to spread along some of the axionic directions.”

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

The QCD vacuum is expected to have a wavefunction that is a Bloch wave with angle θ_QCD.

“This what we expect for the QCD vacuum; that is, the wavefunction is a Bloch wave with angle θ QCD .”

It may state two separate claims as one.

A similar term arises due to the finite temperature effect on the potential, and one may reinterpret this effect as a finite volume effect due to the Gibbons-Hawking horizon 1/H.

“Here, a similar term arises due to the finite temperature effect on the potential.One may reinterpret this effect as a finite volume effect, due to the presence of the Gibbons-Hawking horizon 1/H.”

It may state two separate claims as one.

If the wavefunction starts localized as a classically stable vacuum site, it will take time to spread; this time is shorter higher up in the landscape and with more axionic directions.

“If we start with the wavefeunction localized as a classically stable vacuum site, it will take time for the wavefunction to spread.This time will be shorter when we are higher up in the landscape and when there are more axionic directions present.”

It may state two separate claims as one.

The origin of the factor of four is that in CDL one has S3 with size 2π^2 versus S2 × S1 with size 4π × 2π = 8π^2 in the thermal case; this shows that O(4) symmetry lowers the Euclidean action B and yields the correct answer.

“It is easy to understand the origin of this factor of four.In CDL, we have S 3 with size 2π 2 versus S 2 × S 1 with size 4π × 2π = 8π 2 in the thermal case.This just shows that the O(4) symmetry lowers the Euclidean action B and so yields the correct answer.”

It may state two separate claims as one.

The existence of a case IV Euclidean tunneling solution implies the existence of a case IV thermal tunneling solution, and some potentials admit a case IV thermal tunneling solution and a case II Euclidean tunneling solution.

“Comparing (7.9) to (5.11) we see that the existence of a case (IV) Euclidean tunneling solution implies the existence of a case IV thermal tunneling solution and that some potentials will admit a case IV thermal tunneling solution and a case (II) Euclidean tunneling solution.”

Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it.

Without detailed knowledge of the structure of the landscape, it is difficult to make definitive statements.

“However, without a detailed knowledge of the structure of the landscape (vacuum sites and their nearby neighbors as well as the barriers between them), it is difficult to make definitive statements.”

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

Linde [21] argued that Γ ∼ T^4 (S3/2πT)^{3/2} e^{-S3/T}.

“Linde [21] argued that”

• cites [21]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.

Case (IV), close to HM tunneling, happens for a flatter barrier (small λ), while case (I), of which thin-wall tunneling is a special case, happens for a sharper barrier (large λ).

“Therefore case (IV), which is close to HM tunneling, happens for a”

It may state two separate claims as one.

In the limit φ_f+ → φ_T, φ(ξ) → φ_T, so the second term in (6.2) vanishes and the resulting formula reduces to the HM limit.

“In the limit φ f + → φ T , φ(ξ) → φ T , so the second term vanishes and the resulting formula reduces to the HM limit.”

Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds. It may state two separate claims as one.

This argument is based on the large dimensionality of the landscape and does not use the gravitational effects discussed in this paper; so it should apply all the way to very low vacuum energy density sites.

“This argument is based on the large dimensionality of the landscape and does not use the gravitational effects discussed in this paper; so it should apply all the way to very low vacuum energy density sites.”

It may state two separate claims as one.

Here, a similar term arises due to the finite temperature effect on the potential, and one may reinterpret this effect as a finite volume effect due to the presence of the Gibbons-Hawking horizon 1/H.

“Here, a similar term arises due to the finite temperature effect on the potential.One may reinterpret this effect as a finite volume effect, due to the presence of the Gibbons-Hawking horizon 1/H.”

It may state two separate claims as one.

Even if the wavefunction is trapped, it is more likely to be weakly trapped, so the wavefunction has a long tail outside the classically allowed region, rendering tunneling less suppressed.

“Even if the wavefunction is trapped, it is more likely to be weakly trapped, so the wavefunction has a long tail outside the classically allowed region, rendering less suppressed tunneling.”

It may state two separate claims as one.

In quantum field theory, thermal effects typically lift the potential such that tunneling becomes faster, until the barrier disappears and the tunneling probability approaches unity.

“In quantum field theory, thermal effects typically lift the potential in a way such that the tunneling becomes faster, until the barrier disappears (i.e., the tunneling probability approaches unity).”

Probably not a substantial claim. It may be too vague to check against evidence. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

In a single dimension at low scales, these effects are probably negligible without a large degree of fine-tuning.

“In a single dimension at low scales these effects are probably negligible without a large degree of fine-tuning.”

It may state two separate claims as one.

Cosmological upper bounds on axionic dark matter constrain f below 10^12 GeV, and astrophysical measurements of red-giant cooling constrain f to be at least 10^9 GeV.

“Cosmological upper bounds on the amount of axionic dark matter constrain f to be below 10 12 GeV, and astrophysical measurements of the cooling of red giants constrain f to be at least 10 9 GeV [24].”

• cites [24]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.

Increasing D both increases the false vacuum and decreases the barrier; the de Sitter vacuum disappears classically if the D-term uplifting is too large.

“Increasing D both increases the false vacuum and decreases the barrier.The de Sitter vacuum disappears classically if the D-term uplifting is too large.”