OtherGround Forums Atheism debunked

6/5/18 9:02 PM
3/12/15
Posts: 835

Mendel's laws of Genetics (scientific facts) contradict evolution. Second, if the universe came into existence BY random chance then there would be no reason for science because then ANY scientific laws could randomly change because of "random chance." For example, you atheists think that there was nothing and that the universe "just appeared" out of nowhere because of random chance. According to that logic then some magical unicorn could randomly appear or gravity might stop working because of "random chance" which is BS but it's still much more likely than our entire universe (if you've actually taken any science classes you'd know how complex our universe is) coming into existence by random chance. Also go ahead and tell me how you would prove that nothing created the universe. Go ahead, I DARE you. And remember, the Appeal to Ignorance fallacy is where you're saying something is true because there's no evidence against it, so you can't try to disprove any gods or anything. Also, all the events mentioned in the Bible have been shown to be historically accurate and many of the very specific prophecies that it made which were ridiculous at the time actually came true in the exact ways it said it would happen. Evolution has been proven to not work and there is 0% evidence that it actually happened. Ever wonder why they call them MISSING links? Go ahead, show me your evidence for evolution. I DARE you, but I know you won't because you can't.?

Edited: 6/5/18 9:07 PM
3/16/11
Posts: 4454
Wtf GIF - Find & Share on GIPHY

 

6/5/18 9:06 PM
9/18/02
Posts: 1822

Trouble with your post is that it has nothing to do with your thread title. 

6/5/18 9:07 PM
1/30/10
Posts: 4109

Agree with OP

6/5/18 9:08 PM
3/17/14
Posts: 12598
The Lion King - 

Mendel's laws of Genetics (scientific facts) contradict evolution. Second, if the universe came into existence BY random chance then there would be no reason for science because then ANY scientific laws could randomly change because of "random chance." For example, you atheists think that there was nothing and that the universe "just appeared" out of nowhere because of random chance. According to that logic then some magical unicorn could randomly appear or gravity might stop working because of "random chance" which is BS but it's still much more likely than our entire universe (if you've actually taken any science classes you'd know how complex our universe is) coming into existence by random chance. Also go ahead and tell me how you would prove that nothing created the universe. Go ahead, I DARE you. And remember, the Appeal to Ignorance fallacy is where you're saying something is true because there's no evidence against it, so you can't try to disprove any gods or anything. Also, all the events mentioned in the Bible have been shown to be historically accurate and many of the very specific prophecies that it made which were ridiculous at the time actually came true in the exact ways it said it would happen. Evolution has been proven to not work and there is 0% evidence that it actually happened. Ever wonder why they call them MISSING links? Go ahead, show me your evidence for evolution. I DARE you, but I know you won't because you can't.?


Dogs.
6/5/18 9:08 PM
5/31/16
Posts: 5125

 

6/5/18 9:08 PM
12/13/09
Posts: 1935
For later
6/5/18 9:11 PM
5/31/16
Posts: 5126

Who created God?

6/5/18 9:14 PM
3/12/15
Posts: 836
Poleeko -

So if your God created the universe then where did he/she come from?

God is an eternal being who transcends mortal concepts such as time and space. He has always existed. 

Educate yourself, unwashed barbarian. 

*smug French laugh*

6/5/18 9:15 PM
3/12/15
Posts: 837
Glowman -

Who created God?

Read above post, ignorant caveman.

6/5/18 9:16 PM
9/8/10
Posts: 13515

I tried to make it through your wall of text.  I really, really did.  But by the fourth sentence I could already feel brain cells dying so I decided to take the L.

6/5/18 9:16 PM
6/3/09
Posts: 10743

" Also go ahead and tell me how you would prove that nothing created the universe. Go ahead, I DARE you."

 

Mathematical proof, in the form of the Wheeler-DeWitt equation, can be found here:

https://arxiv.org/abs/1404.1207

6/5/18 9:19 PM
3/12/15
Posts: 838
Poleeko -
The Lion King -
Poleeko -

So if your God created the universe then where did he/she come from?

God is an eternal being who transcends mortal concepts such as time and space. He has always existed. 

Educate yourself, unwashed barbarian. 

*smug French laugh*

So he was created out of nothing one day?

God transcends the concept of nothingness 

6/5/18 9:19 PM
5/31/16
Posts: 5127

God could not have come from nothing. So who created him?

6/5/18 9:20 PM
3/12/15
Posts: 839
Curtis_E_Bare -

" Also go ahead and tell me how you would prove that nothing created the universe. Go ahead, I DARE you."

 

Mathematical proof, in the form of the Wheeler-DeWitt equation, can be found here:

https://arxiv.org/abs/1404.1207

I'm not clicking a link from an evil atheist who is likely trying to hack me. Post your proof in this thread. 

6/5/18 9:21 PM
3/12/15
Posts: 840
Glowman -

God could not have come from nothing. So who created him?

Again, read the above post, brute. 

*smug French laufh* 

6/5/18 9:21 PM
3/17/14
Posts: 12601
The Lion King - 
Poleeko -
The Lion King -
Poleeko -

So if your God created the universe then where did he/she come from?

God is an eternal being who transcends mortal concepts such as time and space. He has always existed. 

Educate yourself, unwashed barbarian. 

*smug French laugh*

So he was created out of nothing one day?

God transcends the concept of nothingness 


I answered your question but you missed it.

Also there is the Twany Owl.
6/5/18 9:22 PM
8/15/07
Posts: 12839
Glowman - 

God could not have come from nothing. So who created him?


If God cannot come from nothing, how can a universe?
6/5/18 9:22 PM
3/12/15
Posts: 841

Logging off for the night. Any atheists who disagree with me post your arguments in this thread and I will expose you for the uneducated double digit IQ brutes that you are. 

6/5/18 9:23 PM
6/3/09
Posts: 10744
The Lion King -
Curtis_E_Bare -

" Also go ahead and tell me how you would prove that nothing created the universe. Go ahead, I DARE you."

 

Mathematical proof, in the form of the Wheeler-DeWitt equation, can be found here:

https://arxiv.org/abs/1404.1207

I'm not clicking a link from an evil atheist who is likely trying to hack me. Post your proof in this thread. 

Spontaneous creation of the universe from nothing Dongshan He,1, 2 Dongfeng Gao, 1 and Qing-yu Cai1, ∗ 1State Key Laboratory of Magnetic Resonances and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China 2Graduate University of the Chinese Academy of Sciences, Beijing 100049, China An interesting idea is that the universe could be spontaneously created from nothing, but no rigorous proof has been given. In this paper, we present such a proof based on the analytic solutions of the Wheeler-DeWitt equation (WDWE). Explicit solutions of the WDWE for the special operator ordering factor p = −2 (or 4) show that, once a small true vacuum bubble is created by quantum fluctuations of the metastable false vacuum, it can expand exponentially no matter whether the bubble is closed, flat or open. The exponential expansion will end when the bubble becomes large and thus the early universe appears. With the de Broglie-Bohm quantum trajectory theory, we show explicitly that it is the quantum potential that plays the role of the cosmological constant and provides the power for the exponential expansion of the true vacuum bubble. So it is clear that the birth of the early universe completely depends on the quantum nature of the theory. PACS numbers: 98.80.Cq, 98.80.Qc I. INTRODUCTION With the lambda-cold dark matter (ΛCDM) model and all available observations (cosmic microwave background, abundance of light elements), it has been widely accepted that the universe was created in a big bang. However, there are still some puzzles, such as problems of the flatness, the horizon, the monopole, and the singularity [1]. Quantum mechanics has been applied to cosmology to study the formation of the universe and its early evolution. In particular, inflation theories, which suggest that the universe experienced an exponential expansion period, were proposed to solve puzzles of the early universe [2–4]. In quantum cosmology theory, the universe is described by a wave function rather than the classical spacetime. The wave function of the universe should satisfy the Wheeler-DeWitt equation (WDWE) [5]. With the development of quantum cosmology theory, it has been suggested that the universe can be created spontaneously from nothing, where “nothing” means there is neither matter nor space or time [6], and the problem of singularity can be avoided naturally. Although the picture of the universe created spontaneously from nothing has emerged for a long time, a rigorous mathematical foundation for such a picture is still missing. According to Heisenberg’s uncertainty principle, a small empty space, also called a small true vacuum bubble, can be created probabilistically by quantum fluctuations of the metastable false vacuum. But if the small bubble cannot expand rapidly, it will disappear soon due to quantum fluctuations. In this case, the early universe would disappear before it grows up. On the other side, if the small bubble expands rapidly to a large enough size, the universe can then be created irreversibly. In this paper, we obtain analytic solutions of the ∗Corresponding author. Electronic address: qycai@wipm.ac.cn WDWE of the true vacuum bubble. With the de BroglieBohm quantum trajectory theory, we prove that once a small true vacuum bubble is created, it has the chance to expand exponentially when it is very small, i.e. , a ? 1. The exponential expansion will end when the true vacuum bubble becomes very large, i.e., a ? 1. It is the quantum potential of the small true vacuum bubble that plays the role of the cosmological constant and provides the power for its exponential expansion. This explicitly shows that the universe can be created spontaneously by virtue of a quantum mechanism. II. WDWE FOR THE SIMPLEST MINISUPERSPACE MODEL Heisenberg’s uncertainty principle indicates that a small true vacuum bubble can be created probabilistically in a metastable false vacuum. The small bubble has 1 degree of freedom, the bubble radius. We can assume that the bubble is nearly spherical, isotropic and homogeneous, since it is a small true vacuum bubble. As we will show below, the small bubble will expand exponentially after its birth and all asymmetries will be erased by the inflation. Since the small true vacuum bubble is nearly spherical, it can be described by a minisuperspace model [7–9] with one single parameter of the scalar factor a. The action of the minisuperspace can be written as S = 1 16πG Z R √ −gd 4x. (1) Since the bubble is homogeneous and isotropic, the metric in the minisuperspace model is given by ds 2 = σ 2 [ N 2 ( t )dt 2 − a 2 ( t ) d ? 23 ] . (2) Here, N ( t) is an arbitrary lapse function, d ? 23 is the metric on a unit three-sphere, and σ2 = 2G/3π is a normalizing factor chosen for later convenience. Substituting Eq.

6/5/18 9:23 PM
6/3/09
Posts: 10745

(2) into Eq. (1), we obtain the Lagrangian L = N 2 a(k − a? 2 N2 ), (3) where the dot denotes the derivative with respect to the time, t, and the momentum pa = −aa/N. ? The Lagrangian (3) can be expressed in the canonical form, L =paa? − NH, where H = − 1 2 ( p 2 a a + ka). In quantum cosmology theory, the evolution of the universe is completely determined by its quantum state that should satisfy the WDWE. With HΨ = 0 and p 2 a = −a −p ∂ ∂a (a p ∂ ∂a ), we get the WDWE [6, 10] ( 1 a p ∂ ∂aa p ∂ ∂a − ka2 )ψ(a) = 0. (4) Here, k = 1, 0, −1 are for spatially closed, flat, and open bubbles, respectively. The factor p represents the uncertainty in the choice of operator ordering. For simplicity, we have set ~ = c = G = 1. III. QUANTUM TRAJECTORY FROM WDWE The complex function ψ(a) can be rewritten as ψ(a) = R(a) exp(iS(a)), (5) where R and S are real functions [11, 12]. Inserting ψ(a) into Eq. (4) and separating the equation into real and imaginary parts, we get two equations [11, 12]: S ′′ + 2 R′S ′ R + p a S ′ = 0, (6) (S ′ ) 2 + V + Q = 0. (7) Here V (a) = ka2 is the classical potential of the minisuperspace, the prime denotes derivatives with respect to a, and Q(a) is the quantum potential, Q(a) = −( R′′ R + p a R′ R ). (8) In the minisuperspace model, the current is [13] j a = i 2 a p (ψ ∗ ∂aψ − ψ∂aψ ∗ ) = −a pR 2S ′ From Eq. (6), we derive the following equations step by step: p a RS′ + 1 R (R 2S ′ ) ′ = 0, d(R2S ′ ) R2S′ + p da a = 0, a pR 2S ′ = const. Then we have ∂aj a = 0. This implies that Eq. (6) is the continuity equation. It should be pointed out that Eq. (7) is similar to the classical Hamilton-Jacobi equation, supplemented by an extra term called quantum potential Q(a). R and S in Eq. (7) can be obtained conveniently from ψ(a) by solving Eq. (4) with relations, ψ(a) = U + iW = R(a) exp(iS(a)), (9) R 2 = U 2 + W2 , S = tan−1 (W/U). (10) Generally speaking, the wave function of the bubble should be complex. Specifically, if the wave function of the universe is pure real or pure imaginary (W = 0 or U = 0), we have S ′ = 0. That means the quantum potential Q will counteract the ordinary potential V at all times. Thus, the vacuum bubble would evolve at a constant speed, and the small bubble cannot grow up rapidly. In the following, we consider the general case for the vacuum bubble, i.e., both U and W are nonzero functions. By analogy with cases of nonrelativistic particle physics and quantum field theory in flat space-time, quantum trajectories can be obtained from the guidance relation [7, 14], ∂L ∂ · a = −aa? = ∂S ∂a , (11) a? = − 1 a ∂S ∂a . (12) Equation (12) is a first order differential equation, so the three-metric for all values of the parameter t can be obtained by integration. IV. INFLATION OF THE TRUE VACUUM BUBBLE (p 6= 1) In the following, we solve the WDWE of the bubble with k = 1, −1, 0, respectively. When the ordering factor takes a special value p = −2 (or 4 for equivalence), exponential expansion of the small true vacuum bubble induced by quantum potential can be obtained no matter whether the bubble is closed, open, or flat.

6/5/18 9:23 PM
6/3/09
Posts: 10746

A. The closed bubble In this case, the analytic solution of Eq. (4) is ψ(a) = a (1−p)/2 [ic1Iν ( a 2 2 ) − c2Kν( a 2 2 )], (13) where Iν ’s are modified Bessel functions of the first kind, Kν’s are the modified Bessel function of the second kind, the coefficients c1 and c2 are arbitrary constants that should be determined by the state of the bubble, and ν = |1−p|/4. As discussed previously, the wave function of the bubble should be complex. For simplicity, we set c1 and c2 as real numbers to find the inflation solution. Using Eqs. (9) and (10), we can get S = tan−1 [− c1 c2 Iν ( a 2 2 ) Kν( a2 2 ) ], and R = a (1−p)/2 r [c1Iν ( a 2 2 )]2 + [c2Kν( a 2 2 )]2 . Here, we omit the sign “±” in front of R, since it does not affect the value of Q(a) in Eq.(8). For small arguments 0 < x ? √ ν + 1, Bessel functions take the following asymptotic forms: Iν (x) ∼ 1 Γ(ν + 1) x 2 ν and Kν(x) ∼ Γ(ν) 2  2 x ν , ν 6= 0. where Γ(z) is the Gamma function. It is easy to get S(a ? 1) ≈ − 2c1 c2Γ(ν)Γ(ν+1) ( a 2 4 ) 2ν , ν 6= 0. Using the guidance relation (12), we can get the general Bohmian trajectories for any small scale factor a(t) = ? ??? ???  (3 − 4ν)λ(ν) 3 (t + t0)  1 3−4ν , ν 6= 0, 3 4 e λ(3/4)(t+t0) , ν = 3 4 , where λ(ν) = 6c1/(42ν c2Γ(ν)Γ(ν + 1)). For the case of ν = 0 (i.e., p = 1), we will discuss it later. It is clear that only the ordering factor takes the value p = −2 (or p = 4 for equivalence), i.e., ν = 3/4, has the scale factor a(t) an exponential behavior. λ(3/4) > 0 corresponds to an expansionary bubble, and λ(3/4) < 0 implies a contractive bubble that does not satisfy the evolution of the early universe. Therefore, with the condition λ(3/4) > 0, we draw the conclusion that, for a closed true vacuum bubble, it can expand exponentially, and then the early universe appears irreversibly. The quantum mechanism of spontaneous creation of the early universe can be seen from the quantum potential of the bubble. For the case of p = −2 (or 4), the quantum potential of the small true vacuum bubble is Q(a → 0) = −a 2 − λ(3/4)2 a 4 . (14) We find that the first term in quantum potential Q(a → 0) exactly cancels the classical potential V (a) = a 2 . The effect of the second term −λ(3/4)2a 4 is quite similar to that of the scalar field potential in [15] or the cosmological constant in [16] for inflation. For the small true vacuum bubble, we have H ≡ a/a ? and Λ = 3H2 . Then we can get the effective “cosmological constant” Λ for the vacuum bubble as Λ ≈ 3λ(3/4)2 . In this way, we can see that the quantum potential of the small true vacuum bubble plays the role of the cosmological constant and provides the power for the exponential expansion. It is the quantum mechanism (i.e., the quantum potential) that dominates the exponential expansion of the vacuum bubble. B. The open bubble For this case, the analytic solution of Eq. (4) is found to be ψ(a) = a (1−p)/2  ic1Jν  a 2 2  + c2Yν  a 2 2  , (15) where Jν’s are Bessel functions of the first kind, and Yν’s are Bessel function of the second kind and ν = |1 − p|/4. Likewise, we get S = tan−1 " c1 c2 Jν( a 2 2 ) Yν ( a2 2 ) # , and R = a (1−p)/2 s c1Jν  a 2 2 2 +  c2Yν  a 2 2 2 . For small arguments 0 < x ? √ ν + 1, Bessel functions take the following asymptotic forms, Jν(x) ∼ (x/2)ν/Γ(ν + 1), and Yν(x) ∼ −Γ(ν)2ν−1/xν for (ν 6= 0). Then we find S(a ? 1) ≈ − πc1 c2Γ(ν)Γ(ν+1)  a 2 4 2ν , v 6= 0. and a(t) = ? ??? ???  (3 − 4ν)λ¯(ν) 3 (t + t0)  1 3−4ν , ν 6= 0, 3 4 e λ¯(3/4)(t+t0) , ν = 3 4 ,

6/5/18 9:23 PM
11/25/09
Posts: 1335

This is why I became a Scientologist. 

6/5/18 9:23 PM
6/3/09
Posts: 10747

where λ¯(ν) = 3πc1/(42ν c2Γ(ν)Γ(ν + 1)). It is interesting that the scale factor for the open bubble (k = −1) is quite similar to that of the closed one (k = 1). For the special case of p = −2 (or 4), the scale factor a(t) has an exponential behavior like before. In this case, the quantum potential for the open bubble can be obtained as Q(a → 0) = a 2 − λ¯(3/4)2 a 4 . (16) Comparing with the case of the closed bubble, we find that the terms a 2 in quantum potential Q(a → 0) and classical potential V (a) change sign simultaneously, so they can still cancel each other exactly. Thus, it is the term −λ¯(3/4)2a 4 in quantum potential Q(a → 0) that causes the exponential expansion of the vacuum bubble. Likewise, we can get the effective cosmological constant for the small true vacuum bubble, Λ ≈ 3λ¯(3/4)2 . C. The flat bubble The analytic solution of Eq. (4) is ψ(a) = ic1 a 1−p 1 − p − c2, (17) where p 6= 1, and hence S = tan−1 [− c1 c2 a 1−p 1 − p ], p 6= 1, R = s c 2 2 + (c1 a 1−p 1 − p ) 2, p 6= 1. Using the guidance relation (12), we can get the general form of the Bohmian trajectories as a(t) = ? ?? ??  c1 c2 (3 − |1 − p|)(t + t0)  1 3−|1−p| , |1 − p| 6= 0, 3, e c1(t+t0)/c2 , |1 − p| = 3. Likewise, only conditions p = −2 (or 4) and c1/c2 > 0 are satisfied, will the small true vacuum bubble expand exponentially. For the case of exponential expansion, the quantum potential for the vacuum bubble can be obtained as Q(a → 0) = −(c1/c2) 2a 4 , while the classical potential is V (a) = 0. This definitely indicates that quantum potential Q(a) is the origin of exponential expansion for the small true vacuum bubble. Similarly, we can get the effective cosmological constant for the small true vacuum bubble as Λ ≈ 3(c1/c2) 2 . V. THE BOHMIAN TRAJECTORIES FOR p = 1 Solutions of Eq. (4) for the p = 1 case are still Eq. (13) and Eq. (15) for the closed and open bubbles, respectively. For the flat bubble, the solution of Eq. (4) for p = 1 is ψ(a) = ic1 − c2 ln a. It is clear that the quantum potential Q(a) of the bubble approaches infinity when the bubble is very small a → 0, no matter whether the small bubble is closed, open or flat. The requirement of a finite value of Q(a → 0) will result in a(t) = constant for k = 0, ±1. VI. THE BEHAVIOR OF LARGE VACUUM BUBBLES Let us look at behaviors of our solutions for large vacuum bubbles [19]. For the closed bubble, we get S(a ? 1) = − tan−1 (c1e a 2 /c2), and hence ?a = 2c1e −a 2 /c2 → 0. The quantum potential of the bubble is Q(a ? 1) ∼ −a 2 . It is obvious that there is no classical limit for the closed bubble. For the open bubble, we have S(a ? 1) ∼ − tan−1 [c1 tan(a 2/2+π/4−νπ/2)/c2]. When |c1/c2| = 1, we can get its classical limit ?a 2 = 1 as Q(a ? 1) → 0. For the case of the flat bubble, we get ?a = c1a −|1−p|−2/c2. When the bubble becomes large enough, it can reach the classical limit, ?a 2 → 0 with Q(a ? 1) → 0. When the vacuum bubble becomes very large, it will stop expanding for k = 0, 1, or it will expand with a constant velocity for k = −1. In one word, it turns out that the vacuum bubble will stop accelerating when it becomes very large, no matter whether it is closed, flat, or open. VII. THE OPERATOR ORDERING FACTOR Generally speaking, the factor p in Eq. (4) represents the uncertainty of the operator ordering. Different p gives different rule of quantization for the classical system. From Eqs. (6) and (8), we get a general form of the quantum potential, Q(a) = −[ −p 2 + 2p 4a 2 + 3(S ′′) 2 4(S′) 2 − S ′′′ 2S′ ]. (18) It is clear that the effect of the ordering factor p is important only to small bubbles, and different p will result in different quantum potential. In other words, for small bubbles (i.e., a ? 1), the first term is significant to Q(a), while for large bubbles (i.e., a ? 1), it is negligible. So, the factor p represents quantum effects of the system described by the WDWE in Eq. (4). It is interesting that only when the ordering factor p takes value −2 (or 4) can one get the exponential expansion for the small true vacuum bubble, no matter whether the bubble is closed, flat, or open. It is generally believed that the operator ordering factor p can be restricted by the quantum to classical transition of the system [17]. Maybe a more elegant treatment of the quantum to classical transition is needed to restrict the interesting values of p, since the classical limit is independent of p in the

6/5/18 9:24 PM
6/3/09
Posts: 10748

present treatment. A hint from loop quantum gravity (LQG) theory is that when one wants to remove the ambiguities from LQG, the ordering factor should take the value p = −2 [18]. VIII. DISCUSSION AND CONCLUSION In summary, we have presented a mathematical proof that the universe can be created spontaneously from nothing. When a small true vacuum bubble is created by quantum fluctuations of the metastable false vacuum, it can expand exponentially if the ordering factor takes the value p = −2 (or 4). In this way, the early universe appears irreversibly. We have shown that it is the quantum potential that provides the power for the exponential expansion of the bubble. Thus, we can conclude that the birth of the early universe is completely determined by quantum mechanism. One may ask the question when and how space, time and matter appear in the early universe from nothing. With the exponential expansion of the bubble, it is doubtless that space and time will emerge. Due to Heisenberg’s uncertainty principle, there should be virtual particle pairs created by quantum fluctuations. Generally speaking, a virtual particle pair will annihilate soon after its birth. But, two virtual particles from a pair can be separated immediately before annihilation due to the exponential expansion of the bubble. Therefore, there would be a large amount of real particles created as vacuum bubble expands exponentially. A rigorous mathematical calculation for the rate of particle creation with the exponential expansion of the bubble will be studied in our future work. IX. ACKNOWLEDGMENT We thank the referees for their helpful comments and suggestions that significantly polish this work. Financial support from NSFC under Grant No. 11074283, and NBRPC under Grant Nos. 2013CB922003 is appreciated. [1] B. A. Bassett, Rev. Mod. Phys. 78, 537(2006). [2] A. A. Starobinsky, JETP Lett. 30, 682 (1979) [Pisma Zh. Eksp. Teor. Fiz. 30,719 (1979)]. [3] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980). [4] A. H. Guth, Phys. Rev. D 23, 347 (1981). [5] B. S. DeWitt, Phys. Rev. 160, 1113 (1967). [6] A. Vilenkin, Phys. Rev. D 50, 2581 (1994). [7] N. Pinto-Neto and J. C. Fabris, Classical Quantum Gravity 30, 143001 (2013). [8] N. Pinto-Neto, F. T. Falciano, R. Pereira, and E. S. Santini, Phys. Rev. D 86, 063504 (2012). [9] S. P. Kim, Phys. Lett. A 236, 11 (1997). [10] S. W. Hawking, Nucl. Phys. B 239, 257 (1984). [11] D. Bohm, Phys. Rev. 85, 166 (1952). [12] P. R. Holland, The quantum Theory of Motion. Cambridge University Press, Cambridge (1993). [13] A. Vilenkin, Phys. Rev. D 37, 888 (1988). [14] L. P. Grishchuk, Classical Quantum Gravity 10, 2449 (1993). [15] J. B. Hartle, S. W. Hawking, and T. Hertog, J. Cosmol. Astropart. Phys. 01 (2014) 015. [16] D. H. Coule, Classical Quantum Gravity 22, R125 (2005). [17] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983). [18] W. Nelson and M. Sakellariadou, Phys. Rev. D 78, 024006 (2008). [19] When x ? |ν 2 − 1/4|, Bessel functions take the asymptotic forms, Iν(x) ∼ e x / √ 2πx, Kν(x) ∼ e −x / √ 2πx, Jν (x ? |ν 2 − 1/4|) ∼ p 2/πx cos(x − νπ/2 − π/4), and Yν (x ? |ν 2−1/4|) ∼ p 2/πx sin(x−νπ/2−π/4). We can use these asymptotic forms to calculate S(a) and a(t) for large bubbles