Double-well instantons in finite volume

The result of Refs. [13, 21] can be obtained from Eq. (4.29) by neglecting the contributions from n = 0 fluctuations.

“This can be obtained from the result in Eq. (4.29) by neglecting the contributions from n = 0 fl uctuations, namely”

• cites [13] J. Alexandre and D. Backhouse, Null energy condition violation: Tunneling versus the Casimir effect, Phys. Rev. D 107 (2023) 085022 [arXiv:2301.02455] [INSPIRE]. sourceCould not verify

  Appendix A separates an n=0 temperature-independent term from a sum over n≥1, but it does not say that the cited result is obtained by neglecting n=0 fluctuations. I searched for Eq. (4.29) and for a derivation that drops those fluctuations; neither appears in our copy of this paper. (LLM, conf=0.72)

  Experimental — accuracy not yet measured
• cites [21] J. Alexandre and S. Pla, Cosmic bounce and phantom-like equation of state from tunnelling, JHEP 05 (2023) 145 [arXiv:2301.08652] [INSPIRE]. sourceCould not verify

  Section 2.2, Eqs. (2.5)–(2.6), discusses fluctuation factors for saddle points, but does not say that neglecting n = 0 fluctuations recovers an earlier result. Searches for that condition and Eq. (4.29) found no matching statement in our copy. (LLM, conf=0.72)

  Experimental — accuracy not yet measured

Ignoring the Casimir energy and the space-dependence of quantum fluctuations about the instanton, tunnelling can dynamically generate an effective equation of state that supports a cosmological bounce when coupled to gravity.

“Ignoring the Casimir energy and the space-dependence of quantum fluctuations about the instanton, it is found that tunnelling can indeed dynamically generate an effective equation of state that supports a cosmological bounce when coupled to gravity.”

• cites [21] J. Alexandre and S. Pla, Cosmic bounce and phantom-like equation of state from tunnelling, JHEP 05 (2023) 145 [arXiv:2301.08652] [INSPIRE]. sourceSupported

• AssumptionThe authors state this as an assumption: The tunnelling rate is large compared to the expansion rate of the universe, allowing calculation of quantum fluctuations for a constant scale factor.

The modified Green's function for the fluctuation operator around the static saddle points can be solved in terms of associated Legendre functions of the first and second kind P_{μ_n/0}(u) and Q_{μ_n/0}(u).

“which can be solved in terms of associated Legendre functions of the first and second kind P µ n 0 ( u ) and Q µ n 0 ( u ). See for example [31]. We have [17]”

• cites [31] B. Garbrecht and P. Millington, Fluctuations about the Fubini-Lipatov instanton for false vacuum decay in classically scale invariant models, Phys. Rev. D 98 (2018) 016001 [arXiv:1804.04944] [INSPIRE]. sourceSupported
• cites [17] W.-Y. Ai, B. Garbrecht and C. Tamarit, Functional methods for false vacuum decay in real time, JHEP 12 (2019) 095 [arXiv:1905.04236] [INSPIRE]. sourcePartly supported

  Section 5 states that the fluctuation equation has associated Legendre-function solutions and displays solutions denoted P. Appendix B.3 treats a Green’s-function resolvent in a kink background, but the passages do not give the claimed solution of a modified Green’s function around static saddle points in terms of both P and Q with the stated indices. (LLM, conf=0.83)

  Experimental — accuracy not yet measured

The solution for the τ-dependent part of the modified Green's function around the instanton can be obtained in terms of associated Legendre functions P_{μ_n/2}(u) and Q_{μ_n/2}(u).

“The solution to the equation above can be obtained in terms of associated Legendre functions P µ n 2 ( u ) and Q µ n 2 ( u ) [17, 31]. It reads [28]”

• cites [17] W.-Y. Ai, B. Garbrecht and C. Tamarit, Functional methods for false vacuum decay in real time, JHEP 12 (2019) 095 [arXiv:1905.04236] [INSPIRE]. sourcePartly supported

  Section 5.2 (eqs. 5.10–5.12; page unavailable) gives τ-dependent fluctuation solutions around a kink in terms of associated Legendre P functions. Appendix B.3 (eq. B.34; page unavailable) discusses the corresponding Green’s-function equation, but the cited passages do not give its solution as P_{μ_n/2}(u) and Q_{μ_n/2}(u) or identify it as a modified Green’s function around an instanton. (LLM, conf=0.83)

  Experimental — accuracy not yet measured
• cites [31] B. Garbrecht and P. Millington, Fluctuations about the Fubini-Lipatov instanton for false vacuum decay in classically scale invariant models, Phys. Rev. D 98 (2018) 016001 [arXiv:1804.04944] [INSPIRE]. sourcePartly supported

  Appendix B (Eqs. B1–B2) identifies the Green-function equation in the Fubini–Lipatov background as an associated Legendre equation and writes its general solution using P and Q functions. The paper formulates this as a hyperradial Green function with indices 2 and j+1; it does not identify the solution as a τ-dependent part of a modified Green function or use the claimed μ_n/2 index. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [28] B. Garbrecht and P. Millington, Green’s function method for handling radiative effects on false vacuum decay, Phys. Rev. D 91 (2015) 105021 [arXiv:1501.07466] [INSPIRE]. sourcePartly supported

  Section III (p. 5) says the Green’s-function equation has solutions expressible in associated Legendre functions. Appendix B, §3 (p. 13) identifies first- and second-kind functions P and Q, but the available text does not establish the specific index μ_n/2 attributed to this source. (LLM, conf=0.82)

  Experimental — accuracy not yet measured

The resolvent method states that the ratio of the determinants of two operators can be written as an integral of some modified Green's function.

“The method states that the ratio of the determinants of two operators can be written as an integral of some modified Green's function.”

• cites [24] J. Baacke and S. Junker, Quantum corrections to the electroweak sphaleron transition, Mod. Phys. Lett. A 8 (1993) 2869 [hep-ph/9306307] [INSPIRE]. sourcePartly supported

  On p. 2, the paper says that the logarithm of a fluctuation-determinant ratio can be written as an integral whose integrand is related to Euclidean Green functions. The claim captures the Green-function method but omits the logarithm: the paper does not say the ratio itself equals the integral. (LLM, conf=0.87)

  Experimental — accuracy not yet measured
• cites [25] J. Baacke and S. Junker, Quantum fluctuations around the electroweak sphaleron, Phys. Rev. D 49 (1994) 2055 [hep-ph/9308310] [INSPIRE]. sourcePartly supported

  Section 6 (pp. 11–12) defines Green functions for the fluctuation and vacuum operators and says that integrating the resulting expression gives a regulated version of the logarithm of the fluctuation determinant. This supports the resolvent-method connection, but the integral gives the logarithm of the determinant ratio, not the ratio directly as the claim states. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [26] J. Baacke and S. Junker, Quantum fluctuations of the electroweak sphaleron: Erratum and addendum, Phys. Rev. D 50 (1994) 4227 [hep-th/9402078] [INSPIRE]. sourceCould not verify

  The erratum discusses a corrected electroweak-sphaleron fluctuation determinant and tadpole contributions (pp. 1–3), but does not describe a resolvent method. Searches for a determinant ratio expressed as an integral of a modified Green’s function found no supporting passage in our copy; several formulas are undecoded. (LLM, conf=0.72)

  Experimental — accuracy not yet measured
• cites [27] J. Baacke, One-loop corrections to the instanton transition in the Abelian Higgs model: Gel’fand-Yaglom and Green’s function methods, Phys. Rev. D 78 (2008) 065039 [arXiv:0803.4333] [INSPIRE]. sourceSupported

In Refs. [13, 21], the instanton contribution to the ground state energy is calculated by only keeping homogeneous fluctuations.

“In Refs. [13, 21], the instanton contribution to the ground state energy is calculated by only keeping homogeneous fluctuations.”

• cites [13] J. Alexandre and D. Backhouse, Null energy condition violation: Tunneling versus the Casimir effect, Phys. Rev. D 107 (2023) 085022 [arXiv:2301.02455] [INSPIRE]. sourcePartly supported

  Section IV (page unavailable) says the one-loop corrections from instanton jumps do not depend on momentum, supporting the claim about the jump contribution. But Appendix A (page unavailable) calculates fluctuations around the static saddle points using a three-dimensional momentum integral, and Appendix B includes those fluctuation factors in the instanton-pair contribution. Thus the full calculation does not keep only homogeneous fluctuations. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites [21] J. Alexandre and S. Pla, Cosmic bounce and phantom-like equation of state from tunnelling, JHEP 05 (2023) 145 [arXiv:2301.08652] [INSPIRE]. sourcePartly supported

  Section 2.2 says the calculation focuses on homogeneous saddle points, and Appendix B calculates fluctuation factors for the instanton contribution. But the paper does not say it keeps only homogeneous fluctuations: its discussion of quantum corrections describes using continuous momentum. The claim conflates a restriction on saddle points with a restriction on fluctuations. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

The prior result on the convex effective potential was obtained by neglecting the spatially inhomogeneous fluctuations around the instanton configuration, as well as the Casimir effect.

“This result was obtained by neglecting the spatially inhomogeneous fluctuations around the instanton configuration, as well as the Casimir effect.”

• cites [12] J. Alexandre and J. Polonyi, Symmetry restoration, tunneling, and the null energy condition, Phys. Rev. D 106 (2022) 065008 [arXiv:2205.00768] [INSPIRE]. sourceNot checked — no full text of the source
• cites [13] J. Alexandre and D. Backhouse, Null energy condition violation: Tunneling versus the Casimir effect, Phys. Rev. D 107 (2023) 085022 [arXiv:2301.02455] [INSPIRE]. sourcePartly supported

  Section IV (page unavailable) says the earlier calculation ignored momentum quantisation and therefore omitted the Casimir contribution, which was added later. The paper does not clearly say that the convex-potential result was obtained by neglecting spatially inhomogeneous fluctuations around instantons; it says one-loop corrections from instanton jumps do not depend on momentum. (LLM, conf=0.78)

  Experimental — accuracy not yet measured

In addition to the known Casimir effect, the tunnelling effect also contributes to NEC violation, extending the results in Ref. [21].

“In addition to the known Casimir effect, we show that the tunnelling effect also contributes to NEC violation, extending the results in Ref. [21].”

• cites [21] J. Alexandre and S. Pla, Cosmic bounce and phantom-like equation of state from tunnelling, JHEP 05 (2023) 145 [arXiv:2301.08652] [INSPIRE]. sourceSupported

In the simplified model analysed in Refs. [21, 22], where E_cas^(3)=0 and E_inst=E_inst^(hom), it was found L_bounce ≃ 10^-5 metres for U0=κ^-1Λ=10^-122 l_p^-4, m=10^-3 eV and λ∼1.

“For example, in the simplified model analysed in Refs. [21, 22], where E (3) cas = 0 and E inst = E (hom) inst with E (hom) inst given in Eq. (4.31), it was found L bounce ≃ 10 -5 metres for U 0 = κ -1 Λ = 10 -122 ℓ -4 p , m = 10 -3 eV and λ ∼ 1.”

• cites [21] J. Alexandre and S. Pla, Cosmic bounce and phantom-like equation of state from tunnelling, JHEP 05 (2023) 145 [arXiv:2301.08652] [INSPIRE]. sourcePartly supported

  Section 2.1 describes a finite-volume tunnelling model that focuses on tunnelling rather than the Casimir effect. The discussion around eqs. (5.1)–(5.2) estimates the bounce volume, but our copy does not report L_bounce ≃ 10⁻⁵ metres for the stated values of U₀, m and λ. Searches for the length and numerical values did not find that result. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites [22] J. Alexandre, K. Clough and S. Pla, Tunneling-induced cosmic bounce in the presence of anisotropies, Phys. Rev. D 108 (2023) 103515 [arXiv:2308.00765] [INSPIRE]. sourcePartly supported

  Section C, “Size of the Universe at the bounce” (page unavailable), relates bounce size to field parameters and assumes a self-coupling of order one. But its numerical example is a bounce at about 1 metre with a mass of about 10^-7 eV and Λ of about 10^-140 ℓp^-2; it does not report the claimed 10^-5-metre result for the specified parameters. The Discussion distinguishes tunnelling from the Casimir effect, but does not state the cited sentence’s exact energy-term assumptions. (LLM, conf=0.84)

  Experimental — accuracy not yet measured

For mL ∼ 1, the instanton energy can become of the same order of magnitude as the Casimir energy.

“However, for mL ∼ 1 the latter can become of the same order of magnitude as the former, as can be seen in the bottom panel of Figure 3.”

The instanton effects result in a steeper curve of ρ+p for λ = 1.

“The instanton effects result in a steeper curve, thus making the NEC violation stronger.”

For large enough coupling λ, the effect of instantons becomes of the same order as the Casimir effect.

“The top panel clearly shows that for large enough coupling λ the effect of instantons becomes of the same order as the Casimir effect.”

The calculation of the full one-loop functional determinant for fluctuations about the instanton is done for the first time.

“This calculation is done for the first time, and all the details are therefore given in this section.”

The energy E_stat is the vacuum energy E0^(3) of a massive scalar field on a three-dimensional torus.

“The energy E stat is nothing but the vacuum energy E (3) 0 of a massive scalar field on a three-dimensional torus.”

The UV divergences extracted from the continuum limit are cancelled by the counter-term contribution in Eq. (4.29).

“which are indeed cancelled by the counter-term contribution in Eq. (4.29).”

Setting equal sizes a1 = a2 = a3 = L gives the Casimir energy formula Eq. (3.12).

“Setting equal sizes a 1 = a 2 = a 3 = L we finally find 4”

The Casimir energy is computed for an arbitrary number of dimensions using the Abel-Plana formula.

“The Casimir energy is computed for an arbitrary number of dimensions using the Abel-Plana formula”

• Also stated in: § 1 Introduction · p. 3 · plain

  “In Section 3 we derive the Casimir energy for a massive scalar field in a three-torus, using the Abel-Plana formula. This is done first in one space dimension and then in arbitrary dimensions, based on a recursion formula which is derived in Appendix B”

E_Casimir^(3) is finite.

“E (3) Casimir is finite.”

There are two trivial solutions in this model, φ identically equal to ±v.

“There are two trivial solutions in this model, ϕ ≡ ± v .”

The Casimir energy is generally larger than the instanton one.

“At first glance, we see that the Casimir energy is generally larger than the instanton one.”

The expression for the ratio of determinants contains a divergent integral.

“The expression above still contains a divergent integral”

Tunnelling can be of the same order of magnitude as the Casimir effect for mL less than or of order 1.

“The true ground state energy is then discussed, and we find that tunnelling can be of the same order of magnitude as the Casimir effect for mL ≲ 1.”

The counter-terms correctly renormalise the series, as checked by considering the continuum limit and extracting its UV divergences.

“We can check that these counter-terms correctly renormalise the series by considering the continuum limit of the latter, and extracting its UV divergences.”

The Euclidean equations of motion admit Euclidean-time-dependent but spatially homogeneous solutions, the instanton and the anti-instanton, i.e., the kink and the anti-kink in Euclidean time.

“Equation (2.6) also admits Euclidean-time-dependent but spatially homogeneous solutions, the instanton and the anti-instanton, i.e., the kink ( K ) and the anti-kink ( K ) in Euclidean time”

Removing the zero mode from G_A would lead to a divergent contribution log(λ_{B,0}+S) ∼ log S for S to infinity from the second term on the right-hand side.

“Removing the zero mode from G A would lead to a divergent contribution log( λ B, 0 + S ) ∼ log S for S → ∞ from the second term on the RHS.”

For λ ≪ 1, the QM result approximates the full QFT result really well.

“In particular, for λ ≪ 1, the QM result approximates the full QFT result really well.”

When taking S to infinity, the second term on the right-hand side of Eq. (4.6) vanishes.

“When taking S →∞ , the second term on the RHS vanishes.”

The function has a pole for μ_n = 2, namely for n = 0 and s = 0.

“Note that the function above has a pole for µ n = 2, namely for n = 0 and s = 0.”

The result in Eq. (4.29) includes the full one-loop contribution of the fluctuations around the jump.

“The result in Eq. (4.29) includes the full one-loop contribution of the fluctuations around the jump.”

The full one-loop vacuum energy for a scalar field tunnelling between two degenerate vacua in a toroidal finite volume is derived analytically, taking into account discrete momentum.

“Assuming a toroidal space with finite volume, we derive analytically the full one-loop vacuum energy for a scalar field tunnelling between two degenerate vacua, taking into account discrete momentum.”

• Also stated in: § 1 Introduction · p. 3 · plain

  “In this article we derive these effects precisely, taking into account the full one-loop quantum fluctuations about the instanton, with discrete momenta.”

Repeating the computation with the one-loop ground state energy E0 as obtained in this work gives L_bounce ≃ 10^-2 metres instead.

“Repeating the computation with the one-loop ground state energy E 0 as obtained in this work, we obtain L bounce ≃ 10 -2 metres instead.”

• AssumptionThe authors state this as an assumption: The space is a flat three-torus (K = 0) for simplicity.

The Casimir energy for the real massive scalar field on the D-torus of sides {a1,...,a_D} is given by formula (3.10).

“Ansatz: Let D > 1 be the number of spatial dimensions. Then, the Casimir energy for the real massive scalar field on the D -torus of sides { a 1 , . . . , a D } is given by”

The matching between the analytical and numerical renormalisation procedures for the Casimir energy is pretty good but for the asymptotic regions mL ≪ 1 and mL ≫ 1.

“The matching is pretty good but for the asymptotic regions mL ≪ 1 and mL ≫ 1.”

The effect of the instantons becomes relevant for λ = 1 and mL ∼ 1.

“The effect of the instantons becomes relevant for λ = 1 and mL ∼ 1.”

There is no boundary term left in the Casimir energy on a circle because a circle has no boundary.

“First, we observe that there is no boundary term left, namely a term independent of L . This is because we are computing the energy on a circle, which has no boundary at all.”

Depending on the coupling constant, the tunnelling effect can be of a similar order of magnitude as the Casimir effect.

“Depending on the coupling constant, the tunnelling effect can be of a similar order of magnitude as the Casimir effect.”

The singularity theorems assume restrictions on the energy-momentum tensor, in particular that it satisfies the NEC Tμν lμ lν ≥ 0.

“In particular, it is assumed to satisfy the NEC T µν ℓ µ ℓ ν ≥ 0 where ℓ µ is a null vector.”

• cites [33] S.W. Hawking and R. Penrose, The Singularities of gravitational collapse and cosmology, Proc. Roy. Soc. Lond. A 314 (1970) 529 [INSPIRE]. sourceNot checked — no full text of the source

For a saddle point to have a finite Euclidean action for β tending to infinity, it must approach asymptotically either of the two minima of the classical potential ±v at τ = ±β/2.

“In order for the saddle point to have a finite Euclidean action for β → ∞ , it must approach asymptotically either of the two minima of the classical potential ± v at τ = ± β 2 .”

The UV divergences are sensitive to neither the background nor the topology of space.

“The UV divergences are sensitive to neither the background nor the topology of space.”

This quantity can be identified as the one-loop correction to the 1PI effective action around the kink configuration, having subtracted the correction around the static one.

“We can identify this quantity as the one-loop correction to the 1PI effective action [32] around the kink configuration, having subtracted the correction around the static one.”

• cites [32] R. Jackiw, Functional evaluation of the effective potential, Phys. Rev. D 9 (1974) 1686 [INSPIRE]. sourceNot checked — no full text of the source

The divergences from the last term in Eq. (3.3) and in (4.20) can only be removed via renormalisation of the one-loop effective action Γ^(1)[φ].

“The divergences from the last term in Eq. (3.3) and in (4.20) can only be removed via renormalisation of the one-loop effective action Γ (1) [ ϕ ].”

In the mapping between source and vev, a vanishing source corresponds to a vanishing vev.

“In this mapping a vanishing source corresponds to a vanishing vev.”

O(4)-symmetric saddle points, representing bubbles of true vacuum developing in a false vacuum, are allowed only in the situation of non-degenerate vacua.

“We note here that O (4)-symmetric saddle points, representing bubbles of true vacuum developing in a false vacuum, are allowed in the situation of non-degenerate vacua only [14, 15]”

• cites [14] S.R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D 15 (1977) 2929 [Erratum ibid. 16 (1977) 1248] [INSPIRE]. sourceNot checked — no full text of the source
• cites [15] C.G. Callan Jr. and S.R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D 16 (1977) 1762 [INSPIRE]. sourceNot checked — no full text of the source

The contribution E_stat of the static saddle points to the ground state energy can be identified with the Casimir energy once it is renormalised.

“As explained in Section 3, the contribution E stat of the static saddle points to the ground state energy can be identified with the Casimir energy once we renormalise it.”

For mL much greater than 1, tunnelling is negligible compared to the Casimir effect.

“As a result, tunnelling is negligible compared to the Casimir effect if we consider the situation mL ≫ 1.”

This divergence is a mere artefact of removing a zero mode in the resolvent-formula for the functional determinants Eq. (4.2) and should be dropped out.

“Therefore, this divergence is a mere artefact of removing a zero mode in the resolvent-formula for the functional determinants Eq. (4.2) and should be dropped out.”

Therefore, δm^2 and δλ can be found by simply renormalising the one-loop effective potential in the infinite volume limit obtained from the effective action for constant field backgrounds.

“Therefore, we can find δm 2 and δλ by simply renormalising the one-loop effective potential in the infinite volume limit that is obtained from the effective action for constant field backgrounds”

The instanton effects make the NEC violation stronger.

“The instanton effects result in a steeper curve, thus making the NEC violation stronger.”

The inhomogeneous fluctuations yield additional one-loop corrections that are expected to be perturbatively small compared to the QM result, at least for small coupling λ.

“The inhomogeneous fluctuations yield additional one-loop corrections, which we expect to be perturbatively small compared to the QM result, at least for small coupling λ .”

The result for the Casimir energy is completely independent of the regulator.

“As explained in [1], we never need to state our regulator explicitly, as we will find a result that is completely independent of it.”

• cites [1] M. Bordag, G.L. Klimchitskaya, U. Mohideen and V.M. Mostepanenko, Advances in the Casimir Effect, Oxford University Press (2009) [https://doi.org/10.1093/acprof:oso/9780199238743.001.0001]. sourceNot checked — no full text of the source

The divergent integral arises because of the mismatch in the number of eigenmodes between the two operators in the ratio, a consequence of having subtracted the zero mode from the fluctuation operator around the kink.

“This arises because of the mismatch in the number of eigenmodes between the two operators in the ratio, a consequence of having subtracted the zero mode from the fluctuation operator around the kink.”

Tunnelling is typically suppressed as the exponential of the volume, approximately exp[-(mL)^3], where (mL)^3 is the action of a Euclidean-time-dependent and spatially homogeneous instanton connecting the two degenerate vacua.

“Tunnelling, on the other hand, is typically suppressed as the exponential of the volume ≈ exp[ -( mL ) 3 ], where ( mL ) 3 is the action of a Euclidean-time-dependent and spatially homogeneous instanton connecting the two degenerate vacua.”

In the situation where mL is less than or of order 1, tunnelling and Casimir effects could be of the same order, potentially leading to new phenomena.

“In the situation where mL ≲ 1, tunnelling and Casimir effects could be of the same order, potentially leading to new phenomena.”

With the above renormalisation procedure, the divergent term in Eq. (3.3) has essentially been regularised by the counter-term δU0, leaving the renormalised, physical U0 through the condition (4.26a).

“With the above renormalisation procedure, the divergent term in Eq. (3.3) has essentially been regularised by the counter-term δU 0 , leaving the renormalised, physical U 0 through the condition (4.26a).”

When considering a field potential with degenerate minima, the finite volume allows tunnelling between those degenerate minima, which would otherwise be prohibited in the infinite-volume limit.

“When considering a field potential with degenerate minima, the finite volume allows tunnelling between those degenerate minima, which would otherwise be prohibited in the infinite-volume limit.”

Finite-volume mechanisms intrinsically favour expansion over contraction since they are exponentially suppressed after a period of expansion.

“An advantage of these finite-volume mechanisms is that they intrinsically favour expansion over contraction since they are exponentially suppressed after a period of expansion (i.e., for large volumes).”

The dilute instanton gas approximation is valid if consecutive jumps are widely separated, namely if the average separation is much larger than the jump width.

“The dilute instanton gas approximation is valid if consecutive jumps are widely separated, namely if the average separation is much larger than the jump width”

The function can be written in terms of modified Bessel functions of the second type.

“The function above can also be written in terms of modified Bessel functions of the second type [1]”

• cites [1] M. Bordag, G.L. Klimchitskaya, U. Mohideen and V.M. Mostepanenko, Advances in the Casimir Effect, Oxford University Press (2009) [https://doi.org/10.1093/acprof:oso/9780199238743.001.0001]. sourceNot checked — no full text of the source

Splitting the numerical function into two regions with independent j_max and Λ is allowed since the result is cut-off independent.

“This is allowed since the result is cut-off independent.”

Better agreement in the asymptotic regions can be achieved by splitting the numerical function into two regions and defining appropriate j_max and Λ independently in either region.

“Better agreement in the asymptotic regions can be achieved by splitting the numerical function into two regions and defining appropriate j max and Λ independently in either region.”

The finite volume constraint has remarkable energetic effects on the confined system, allowing the violation of the NEC.

“The finite volume constraint has remarkable energetic effects on the confined system, allowing the violation of the NEC.”

We expect to find a bouncing solution to the Friedmann equations if we start from a contracting phase.

“which means that we expect to find a bouncing solution to the Friedmann equations if we start from a contracting phase.”