Aspects of False Vacuum Decay

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Aspects of False Vacuum Decay

60 ECs Master Project

Annalisa Traverso - 11804483

Supervisor:

Second supervisor:

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1 Introduction

Consider a classical theory with a potential that has two minima, an absolute one and a relative one, then two different ground states exist.

Figure 1: Potential U (φ): φ− is the absolute minimum, while φ+ is the relative one.

However, if this one was a quantum field theory, the ground state of higher energy would be unstable and it would be possible for it to decay via barrier penetration to the real ground state. In other words, there is a certain probability that the false vacuum state φ+ would

decay into the true vacuum state φ−. This problem has been widely studied in a cosmological

context, especially as an explanation to the beginning and the ending of inflation. If we consider the minima of the potential depicted in figure 1, the one with energy zero would correspond to a Minkowski space, while the one with negative energy would correspond to an anti-de Sitter space. If there was a minimum with positive energy that would correspond to a de Sitter space, and therefore we can interpret tunnelling between different minima as tunnelling between different spacetimes. This problem has also received renewed attention in recent times in the context of the string theory landscape containing a vast number of metastable vacua.

One of the first descriptions of false vacuum decay was presented between 1977 and 1980 by Sidney Coleman and Frank de Luccia, both in absence [1] and in presence of gravity [2]. They claim that the decay of false vacuum is like the nucleation processes associated to first order phase transitions in statistical mechanics: the tunnelling is represented by the

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materialization of a bubble of true vacuum within the false vacuum. Once it materializes, the bubble begins to expand, with a speed asymptotic to the speed of light, converting false vacuum into true vacuum as it grows. This theory can be applied to cosmology: at the time of the Big Bang, the energy per unit volume of the universe was very high, far from any vacuum and as the universe cooled down it might have settled into a false vacuum state. In the semiclassical limit, we can expand the probability per unit time per unit volume of the decay Γ/V to find

Γ V = Ae

−B

~[1 + o(~)] , (1.1)

where the coefficients A and B depend on the form of the potential. Our focus will be mainly on the coefficient B, that can be derived in a closed form, while a derivation of A can be found in a review that Coleman wrote, together with Curtis Callan, in 1977 [3]. To derive B, Coleman and de Luccia elaborate a formalism that makes use of Euclidean path integrals, writing at first the Euclidean action and then minimizing it. The solution to the equations of motion will be called ”bounce” and it will have a Euclidean spherical symmetry O(4).

The Coleman-de Luccia (CdL) result relies heavily on the assumption that the gravity would not change the Euclidean spherical symmetry of the bounce, but without ever proving it ex-plicitly. In 2007, Adam Brown and Erick Weinberg tried to overcome this issue by re-deriving the CdL coefficient B using a thermal approach [4], by restricting the field theory to the static patch of de Sitter spacetime and treating it as a thermal system. In this case the Euclidean O(4) symmetry of the bubble is found as a result, which confirms the validity of the assumption made by Coleman and de Luccia. However, though they reproduce the same result, they assign a different interpretation to the bounce, which is seen as a sequence interpolating all possible configurations of the field between the two sides of the barrier.

The purpose of the following pages is to introduce the reader to the problem of false vacuum decay and to show some of its interesting features. The mathematical steps have been re-produced in detail, although the reader is advised to check the appendices for more context.

After reviewing the CdL formalism, we will introduce an alternative to the Euclidean path integrals: the Hamiltonian formalism, based on the ADM1 decomposition of General Rela-tivity, presented in 1990 by Willy Fischler, Daniel Morgan and Joseph Polchinski [5, 6]. The

1_{The ADM formalism of General Relativity is named after its authors Richard Arnowitt, Stanley Deser}

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main reason behind this, was to avoid the confusion about the definition and interpretation of the Euclidean path integrals for quantum gravity. The coefficient B will be the same as the one in (1.1), but it will be derived in a way that recalls the tunnelling amplitude of a wave through a barrier. In this case we will assume once again spherical symmetry of the bounce, so we will be able to restrict ourselves to spherically symmetric metrics. In partic-ular, we will focus on the boundary conditions that we need to impose when computing the gravitational action. We will write the theory in an entirely covariant formalism, following the work of Thomas Bachlechner [7] and we will try to apply it to the problem of false vac-uum decay. Although the physical meaning of the result might appear enigmatic, the whole procedure will be reproduced in great mathematical detail and will constitute an interesting application of the methods derived so far.

Lastly, we will use the Hamiltonian formalism to compute quantum transitions between dif-ferent spacetimes [8] and we will apply these results to some models proposed in the context of early universe cosmology. We will briefly review the theory, proposed by Alexander Vilenkin in 1982 [9], of a universe nucleating from nothing, a scenario that depicts the beginning of inflation without the need for its initial conditions at the big bang. This will be reprised in 1987 by Blau, Guendelman and Guth [10], who tried to make the concept of tunnelling from nothing more physical, by suggesting that an inflationary universe can be created starting with a Minkowski space. For this purpose, we will compute the tunnelling probability in this case starting with a transition from Schwarzschild to Schwarzschild-de Sitter spacetimes. This case is particularly interesting, not only because it describes the scenario depicted in [10], as Schwarzschild is an asymptotically flat spacetime, but also because it can be used to derive the probability of a transition from a Minkowski spacetime to a de Sitter one. In fact, it will be just enough to take the result of the first case in the limit of a black hole with no mass. We will see that the outcome of this calculation will be in disagreement with the result we would get for the same problem using the Euclidean path integrals, although the reason behind this is still unknown.

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2 Tunnelling in Quantum Mechanics

We are interested in computing tunnelling probabilities in Quantum Field Theory. We will use this section to introduce some different examples of tunnelling in Quantum Mechanics and different approaches. In particular, after quickly reviewing the WKB method, we will focus on the method of Euclidean path integrals and we will show that they both lead to the same results. We will consider a particle of unit mass (m = 1) moving in one dimension, so that the system will be described by the following Lagrangian

L = 1 2˙x

2_{− V (x) .} _(2.1)

The WKB method requires that we rewrite the wavefunctions as sum of exponentials and to semiclassically expand them. The tunnelling probability, given by the ratio of the amplitudes of the outgoing and incoming wavefunctions, will turn out to be exponentially suppressed. Let’s start with the following potential barrier

Figure 2: Potential V(x)

with the particle in a state with energy E = 0 (false vacuum) in x0. We can find the

tunnelling amplitude T through the barrier using the WKB method E = 1 2m ˙x 2 + V (x) = p 2 2m + V (x) → p(x) = p 2m(E − V ) , T = exp − 2 ~ Z x1 x0 |p(x)|dx . (2.2)

In particular, for the problem described in (2.1) the particle has zero energy, thus we have p(x) =p2| − V (x)| → T = exp − 2 ~ Z x1 x0 p 2| − V (x)|dx . (2.3)

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If we compared this result with (1.1), we could identify B as B = Z x1 x0 p 2| − V (x)|dx and, more generally, for a particle with energy E

B = Z x1

x0

p

2(E − V (x))dx . (2.4) Now we want to find the tunnelling amplitude for the same problem, but we are going to do so using the Euclidean path integral formalism. To introduce it, let’s at first consider the quantum mechanical example of the double well potential [11] (see Fig. 3).

Figure 3: Double well potential

The Euclidean path integrals are obtained from ordinary path integrals by performing a Wick rotation to imaginary time τ = −it, in the following way:

Z

e−~iS(x)Dx −t→iτ−−→

Z

e−1~S(x)Dx .

We want to prove that with this method we can retrieve the same result of equation (2.2), therefore we need to compute the transmission amplitude

hf | e−Hτ~ | ii ∼ Z e−S/~Dx with S = Z τ /2 −τ /2 Ldτ , (2.5) where | f i ≡ x(τ /2) and | ii ≡ x(−τ /2). In our case the Lagrangian in (2.1) becomes

L = 1 2 dx dt 2 − V (x) = 1 2 dτ dt dx dτ 2 − V (x) = −1 2 dx dτ 2 − V (x) ≡ −LE, (2.6)

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Figure 4: The potential of the classical problem. Define ω2 ≡ V00_(±a).

where LE is the Euclidean Lagrangian. The quantum problem of the double well has thus

become a classical problem of maximum and minimum, where the potential is given by −V (x). The path integral is evaluated over all the functions x(τ ) but it is dominated by the path ¯x(τ ) that minimizes the action, that is the one which satisfies the equations of motion

δS = Z δLEdτ = Z ¨ x + V0(x) δxdτ = 0 → _{x(τ ) − V}¨_¯ 0 (¯x) = 0 e.o.m. . Integrating the equations of motion we find the energy E, which is a constant of the motion and that, in our case, we choose to be E = 0

d2x¯ dτ2 − dV (¯x) d¯x = 0 → 1 2 d¯x dτ 2 − V (¯x) = E = 0 . (2.7) There are different function ¯x(τ ) that satisfy the equations of motion:

1. trivial solution ¯x = ±a: in this solution the particle stays fixed in its position on top of one of the two hills of the potential;

2. less trivial solution: the particle starts off from the top of one hill (say at −a) and reaches the top of the other.

We are more interested in the second case. Let’s say that the particle starts off −a at −τ /2 and arrives in a at τ /2. We drew the solution in Fig.5. This solution is called instanton and we can find the analytic expression for the it by solving equation (2.7)

1 2 dx dτ 2 − V (x) = 0 → dx dτ = p 2V (x) → Z x 0 dx0 p2V (x0₎ = τ + τ1, (2.8)

where ¯x(τ1) = 0. Moreover we can scale the time so that τ1 = 0 to obtain Fig. 5. If we

compute the action for the instanton S0 we get

S0 = Z +∞ −∞ 1 2 dx dτ 2 + V (x) dτ = Z +∞ −∞ dx dτ 2 dτ = Z +a −a dx dτ dx = Z +a −a p 2V (x)dx , (2.9)

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Figure 5: Instanton. Note that we can get the anti-instanton simply by replacing τ with −τ .

where we integrated over time from −∞ to +∞: since the potential does not depend on τ we can decide to take the initial state back in time as much as we want, for example back to −∞. Same for the finale state, where we can simply take τ → +∞.

The most important remark about equation (2.9) is that we found the same result of equation (2.2). Consider once again the transmission amplitude

hf | e−Hτ /~ | ii ∼ Z

e−S[x]/~Dx ∼ e−S[¯x]/~,

where S[¯x] = S0 is the on-shell action [12]. In our case the on-shell action is the one found

in equation (2.9), so the argument of the exponential becomes −S[¯x] ~ = − S0 ~ = − 1 ~ Z +a −a p 2V (x)dx .

If we compare this result with equation (2.4) we find that the coefficient B in this case is just the on-shell action for the instanton.

We can see how the method of the Euclidean path integral allowed us to treat a quantum mechanical problem as a classical one. We were able to find a solution that minimized the action, the instanton, and with that we were able to compute the transition amplitude to find the same result as the one found with WKB. Let’s apply this procedure to the problem of the potential in Fig.2.

We start by writing the Lagrangian, which will be the same of (2.1) and from that we will construct the Euclidean Lagrangian

L = 1 2˙x 2_{− V (x)} ₋t→iτ₋₋_→ _L E = 1 2 dx dτ 2 + V (x) .

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Figure 6: V (x) and −V (x)

We now have to solve the equations of motion of the classical problem. In particular we are interested in the classical solution with E = 0 in which the particle starts in x0, moves

towards x1 and after reaching the turning point bounces back to x0. If we consider the initial

and final states to be both x0 at τ → ±∞ and x1 ≡ x(τ = 0), then we can represent x(τ ) in

the following way:

Figure 7: In this picture we scaled the time such that x1 ≡ x(τ = 0) and we set x0 = 0

From the equations of motion we find that E = 1 2 dx dτ 2 − V (x) = 0 → dx dτ = p 2V (x) .

This solution is called the bounce. The transmission amplitude will then be given by hf | e−Hτ /~ | ii ∼

Z

e−S[x]/~Dx ∼ e−S[¯x]/~_,

where S[¯x] is the action computed for the bounce S[¯x] =

Z x1

x0

p

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In conclusion, to compute the coefficient B using the method of Euclidean path integrals, we need to follow these steps:

1. we write the Euclidean lagrangian, making the quantum mechanical problem into a classical one;

2. we derive the equations of motion and we find the solution that minimizes the action; 3. the coefficient B is given by the on-shell action for that solution.

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3 Quantum Field Theory

In this section we will extend the procedure of the Euclidean path integrals to Quantum Field Theory [1]. Consider a 3+1 dimensional theory with a scalar field φ(t, ~x)

L = 1 2∂µφ∂

µ_{φ − U (φ)} _η

µν = diag(+1, −1, −1, −1) . (3.1)

Now perform the Wick rotation to obtain the Euclidean Lagrangian LE =

1 2∂µφ∂

µ

φ + U (φ) η(E)_µν = diag(+1, +1, +1, +1) . (3.2)

Figure 8: U (φ) and −U (φ)

As it has been said in the introduction, φ+ represents a false vacuum state, while φ− is the

true vacuum. We want to compute the coefficient B, which is nothing else but the on-shell action, introduced in equation (1.1)

Γ V = Ae

−B

~[1 + o(~)] .

The reason why we are considering the probability per unit time per unit volume is that, this time, the non trivial solutions to the equations of motion are not invariant under spatial translations.

Let’s proceed exactly as we did before: we start by finding the equations of motion LE = 1 2∂µφ∂ µ_{φ + U (φ)} _→ _∂ µ∂µφ − dU dφ = 0 → E = 1 2∂µφ∂ µ_{φ − U (φ) .} (3.3)

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We want to find a field ¯φ that solves the equations of motion: a non trivial solution is given by the bounce, for E = 0. In this case the initial and final state would be φ+. We can write

the coefficient B B = Z ₁ 2∂µφ∂ µ_{φ + U (φ)} dτ d3~x . (3.4) We need to impose the following boundary conditions:

(1) lim

τ →±∞φ(τ, ~x) = φ+ (2) |~x|→∞lim φ(τ, ~x) = φ+ (3)

dφ

dτ |τ =0= 0 . (3.5) With the boundary condition (3) we know that the bounce has a stationary point at τ = 0, that we choose using time-translation invariance. The condition (2), instead, is consistent with the fact that when a bubble of true vacuum nucleates at some position in space, very far away from it the universe remains in its false vacuum state; moreover ensures that the integral B is not divergent. Finally, condition (1) it’s just the request that the initial and final state are both φ+.

We want to construct the bounce ¯φ. Since the equations of motion and the boundary conditions for φ are invariant under transformations of the group O(4)2_{, we can assume that}

¯

φ is O(4) invariant. This does not mean that all the solutions to the equations of motion in (3.3) are O(4) invariant, but that we assume that those that are have always lower action than the non invariant ones3. In this case we can rewrite φ introducing the 4-dimensional distance ρ

ρ ≡pτ2_{+ |~}_x|2 _(3.6)

and then rewrite the equations of motion as d2φ dρ2 + 3 ρ dφ dρ | {z } viscous term −dU dφ = 0 . (3.7)

The boundary conditions introduced before would become: (1) & (2) lim

ρ→∞φ(ρ) = φ+ (3)

dφ

dρ |ρ=0= 0 , (3.8) and the coefficient B

B = 2π2 Z ∞ 0 1 2 dφ dρ 2 + U (φ) ρ3dρ , (3.9)

2_{When we refer to spherical symmetry of these solutions we do imply a Euclidean spherical symmetry,}

not to be confused with a spatial spherical symmetry.

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where this time the integral is taken over the volume of the 3-sphere S3. If we interpret ¯φ as the position of a particle at a time ρ, then the equations of motion (3.7) can be seen as the equations of motion of a particle moving in a potential −U (φ) and subject to a viscous damping. We need therefore to choose the initial position ¯φ(ρ) carefully.

We can see from the picture that we have two possibilities for the choice of ¯φ(ρ = 0) ≡ ¯φ0:

1. if we take ¯φ0 to the right of σ, then the particle doesn’t have enough energy to reach

φ+, especially if we consider that the viscous damping would additionally reduce the

energy. In this case the particle will undershoot;

2. if we take ¯φ0 to the left of σ the risk is that the particle might overshoot, even with

the viscous damping to reduce the energy. Consider the equations of motion (3.7): if we choose ¯φ0 to be arbitrarily close to φ− for arbitrarily large ρ, then for large enough

ρ we can neglect the viscous damping (∼ 1_ρ). Without viscous damping to reduce its energy, the particle overshoots.

However we can argue that, by continuity, it must exist an initial condition ¯φ0 at which the

particle can start and reach φ+ at ρ → ∞.

3.1 Thin wall approximation

At this point we can find an explicit form for both the bounce ¯φ(ρ) and for B using the so called thin wall approximation. We start by expressing the potential U (φ) in the following way:

U (φ) = U0(φ) +

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where U0(φ) is the double well potential of Fig. 3 and is the energy density difference

between false and true vacuum: the thin wall approximation is the approximation for small . We can write an expression for U0 in the following way:

U0(φ) = λ 8 φ2− µ 2 λ 2 , λ ≡ µ 2 a2, µ 2 _{≡ U}00 0(φ±) . (3.11)

Therefore the potential in (8) becomes U (φ) = λ 8 φ2−µ 2 λ 2 + 2a(φ − φ+) . (3.12) If we continue with our mechanic analogy, we need to choose a starting point for the particle

¯

φ0, then we assume that the particle will stay in that position for a very long time ρ ' R,

at time R it will then cross quickly the minimum of −U (φ) and it will slowly reach φ+ at

ρ → ∞. In QFT the bounce is a four-dimensional bubble with a very big radius R and a thin separation wall between the false vacuum (outside) and the true one (inside). In the light of this fact we can finally compute the coefficient B by splitting the integral in three parts:

• 0 < ρ < R : inside the bubble we know that φ = φ−. Let’s compute U (φ−)

U (φ−) = λ 8 φ−2− µ2 λ 2 + 2a(φ−− a) = 0 + 2a(−2a) = − . The coefficient B becomes

B = 2π2 Z R 0 ρ3 1 2 dφ dρ 2 − dρ = −1 2R 4_π2_. _(3.13)

• ρ ' R : this integral is carried out over the wall itself and the potential is given by U (φ) = λ 8 φ2 −µ 2 λ 2 + 2a(φ − a) ' λ 8 φ2− µ 2 λ 2

We can use the thin wall approximation to get rid of the term proportional to . Moreover, since R is very big, we can neglect the viscous damping. We are left with the double well potential that we saw in paragraph 2.2: we know that the solution for this potential is given by the instanton. We can derive the result analogous to equation (2.9)

d2φ dρ2 + 3 R dφ dρ | {z }

neglect this term

−dU dφ ' d2φ dρ2 − dU dφ = 0 → 1 2 dφ dρ 2 − U (φ) = 0 (3.14)

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since E = 0. The coefficient B becomes B = 2π2 Z ₁ 2 dφ dρ 2 + U (φ) ρ3dρ ' 2π2R3 Z φ+ φ− p 2U (φ)dφ ≡ 2π2R3S0 = 2π2R3 Z φ+ φ− √ λ 2 φ2− µ 2 λ dφ = 2π2R3 √ λa3 3 = 2π 2_R3µ3 3λ. (3.15)

• R < ρ < ∞ : this is the region outside the bubble, where we know φ = φ+. The potential

is U (φ+) = λ 8 φ+2− µ2 λ 2 + 2a(φ+− a) = 0 , therefore the coefficient in this region is B = 0.

The final value for B is given by the sum of the previous results B = 2π2R3µ 3 3λ− 1 2R 4 π2 (3.16)

and we have also found a description for the bounce

¯ φ(ρ) =          φ− for 0 < ρ < R

the instanton for ρ ' R φ+ for R < ρ < ∞

. (3.17)

We want to find an expression for R that extremizes the action: we must differentiate B and see for which value of R is stationary

dB dR = 2π 2_R2µ3 λ − 2π 2_R3 _{= 0} _→ _{R =} µ3 λ → B = π2_µ12 6λ43 .

Moreover we can give an estimate of how small has to be for the thin wall approximation to hold R = µ 3 λ 1 → µ3 λ .

3.2 Conclusions

Using the Euclidean path integral approach we derived the coefficient B in the limit of the thin wall approximation

B = π

2_µ12

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In this case we can also find an explicit expression for the instanton U (φ) = λ 8 φ2− µ 2 λ 2 → dφ dρ = √ λ 2 φ2 −µ 2 λ , 2 µ Z dx x2_{− 1} ∼ ρ → φ(ρ) ∼ µ √ λtanh µ 2ρ .

Figure 9: Plot of y = tanh(x). We can see that has the behaviour that we expect from the instanton as in Fig. 5

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4 Inclusion of Gravity

In this section we will include the effect of gravity in the computation of the coefficient B [2]. At first sight this might seem pointless, as false vacuum decays takes place at scales at which gravitational effects are negligible. Yet, as the bubble grows, its volume radius will become comparable to its Schwarzschild radius, therefore we must take gravity into account.

In the thin wall approximation, the energy released by the conversion of false vacuum into true vacuum goes into the growth of the bubble. Therefore the energy is proportional to the volume of the bubble, as well as to its Schwarzschild radius. Consider a sphere with energy density and radius R, we can find its Schwarzschild radius in the following way:

E = 4 3πR 3_, RS = 2GM = 2G 4 3πR 3 _→ _R S = r 3 8πG. For ' (1Gev)4 _{we have R}

S ' 0.8km. For the energy scales we are considering, in a

cosmological contest, the Schwarzschild radius is too small to be neglected. It is clear then, that gravity affects vacuum decay.

4.1 The action

In the previous section, without gravity, we used this action S = Z d4x 1 2∂µφ∂ µ_{φ − U (φ)} (4.1) and, adding a constant, we shifted the potential in order to make the energy zero. On the other hand, if we include gravity in the action, this is no longer possible.

S = Z d4x√−g 1 2g µν_∂ µφ∂νφ − U (φ) − R 16πG . (4.2)

This time, adding a constant to the potential results in adding a term proportional to √−g to the action: this is equivalent to adding a cosmological constant. We will see later what are the implications for the calculation of B.

In the meantime we will proceed as we did for the non-gravitational case: 1. we will construct the Euclidean lagrangian;

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3. finally we will compute the coefficient B imposing the right initial conditions.

Once again we will assume the bounce to be O(4) invariant: although we cannot prove it explicitly, there is no apparent reason why this should not hold.

4.1.1 Construct the Euclidean Lagrangian

We can construct the Euclidean Lagrangian starting from (4.2) as we did for the non-gravitational case SE = Z d4x√−gE 1 2g µν_∂ µφ∂νφ + U (φ) − R 16πG , (4.3)

where gE is a 4-dimensional rotationally invariant Euclidean metric. To construct it, in the

most general case, we can foliate the 4-dimensional space with 3-spheres S3_{. We will indicate}

the position on the radial curve, a curve that has fixed angular coordinates, with ξ and, since it can be redefined by addition of a constant, we can take it to be the distance from the origin of the coordinate system. The metric gE is then given by

ds2 = dxi2+ ρ(ξ)dΩ2, (4.4)

where dΩ = dψ2_{+ sin}2_ψ(dθ2_{+ sin}2_θdφ2_{) is the line element on S}3_{. The function ρ(ξ) is the}

radius of curvature of S3 _{and it’s the only one-variable function that we need to determine}

to write the metric. We can do it by solving the Einstein equations of motion

Gµν = κTµν, (4.5)

where κ ≡ 8πG. In particular, the ξξ component

Gξξ= κTξξ (4.6)

is the only one we will need, as all the others are identically zero or follow from this one. The stress energy tensor is given by

Tξξ = gE ξξ ∂LE ∂(∂ξφ) − δξξLE = 1 2 dφ dξ 2 − U (φ) , (4.7) while the ξξ-component of the Einstein’s tensor is

Gξξ = 3 (ρ0₎2 ρ2 − 1 ρ2 , (4.8)

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where the derivative with respect to ξ has been indicated by the prime. Thus equation (4.6) becomes 3 (ρ 0₎2 ρ2 − 1 ρ2 = κ 1 2 dφ dξ 2 − U (φ) → (ρ0)2 = 1 +κρ 2 3 1 2(φ 0 )2− U (φ) . (4.9) The final term that we need to compute the action, is the Ricci scalar

R = − 6 ρ2(ρρ

00

+ (ρ0)2− 1) . (4.10) We are finally ready to compute the action

SE = Z dx4√−gE 1 2g µν_∂ µφ∂νφ + U (φ) − R 16πG = 2π2ρ3 Z dξρ3 1 2(φ 0 )2+ U (φ) + 1 2κ 6 ρ2(ρρ 00 + (ρ0)2− 1) = 2π2 Z ρ3 1 2(φ 0 )2 + U (φ) + 3 κ(ρ 2_ρ00 + ρ(ρ0)2− ρ) dξ . (4.11)

Note that, in the non gravitational case, the independent variable was ρ, while this time it is ξ. In principle, we could find an expression ρ(ξ), but since we are not going to need it, we will not compute it. Before applying the thin-wall approximation we can rearrange this integral, using integration by parts to eliminate the term proportional to ρ00

ρ2ρ00 = d dξ(ρ 2 ρ0) − 2ρ(ρ0)2 → ρ2ρ00+ ρ(ρ0)2− ρ = d dξ(ρ 2 ρ0) | {z } =0 −2ρ(ρ0)2+ ρ(ρ0)2− ρ ,

where the underlined term vanishes because we are interested in the difference between two solutions vanishing at infinity. The integral then becomes

SE = 2π2 Z ρ3 1 2(φ 0 )2+ U (φ) − 3 κ(ρ(ρ 0 )2+ ρ) dξ . We can simplify more, using the ξξ-component of the Einstein equations ρ3 1 2(φ 0 )2 + U (φ) − 3 κ(ρ(ρ 0 )2+ ρ) = ρ3 1 2(φ 0 )2+ U (φ) − 3 κρ − 3ρ κ 1 + κρ 2 3 1 2(φ 0 )2− U (φ) = ρ3 1 2(φ 0 )2+ U (φ) − 6 κρ − ρ 3 1 2(φ 0 )2− U (φ) = 2ρ3U (φ) − 6 κρ .

Finally, keeping in mind that so far we didn’t make any approximation, we can write the action as SE = 4π2 Z ρ3U (φ) − 3 κρ dξ . (4.12)

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4.1.2 Finding the equation of motions For the action (4.12), the equations of motion are

φ00+ 3ρ 0 ρ φ 0 = dU dξ . (4.13)

We can compare them to the non-gravitational case No Gravity: φ00+ 3 ρφ 0 = dU dρ Gravity: φ 00 + 3ρ 0 ρ φ 0 = dU dξ . (4.14) As we mentioned earlier, the independent variable in the gravitational case is ξ and not ρ. In both cases we have a term proportional to φ0, which can be neglected using the thin-wall approximation.

As we did in the non-gravitational case, we could integrate the equations of motion to find an expression for φ φ00+3ρ 0 ρ φ 0₋ dU dξ = 0 → Z φ00+3ρ 0 ρ φ 0 ₋dU dξ dξ = constant .

In the non-gravitational case, we could shift the potential to make sure that this constant was zero and we would find an expression for dφ_dρ that we would then use to compute B. In this case, however, we already acknowledged that shifting the potential would imply adding a cosmological constant, therefore this step is not possible. We will have to compute B considering two different cases: that we either live in a true vacuum, or in a false one.

4.1.3 Computation of B

As we did before, we are going to compute B splitting the integral in three parts. We can denote the radial coordinate of the surface of the bubble with ¯ξ, then the radius of curvature of the surface of the bubble is ¯ρ ≡ ρ( ¯ξ). The three regions are then:

1. inside the wall: 0 < ξ < ¯ξ ; 2. on the wall: ξ ' ¯ξ ;

3. outside the wall: ¯ξ < ξ < ∞ .

We are now going to compute the coefficient B, keeping in mind that B = SE(φ) − SE(φ+)

and that SE is SE(φ) = 4π2 Z ρ3U (φ) −3ρ κ dξ .

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• We live in a true vacuum state.

This is the case when the decay happens from a state with a positive energy density (U (φ+) =

) to one of zero energy density (U (φ−) = 0). The potential is given by

U (φ) = U0(φ) +

2a(φ + φ+) . 1. INSIDE THE WALL:

φ = φ− and 0 < ξ < ¯ξ, then B is BIN = SE(φ−) − SE(φ+) = 4π2 Z ξ¯ 0 ρ3U (φ−) | {z } =0 −3ρ κ dξ − 4π2 Z ξ¯ 0 ρ3U (φ+) | {z } = −3ρ κ dξ .

We are going to use the expression for ρ0 to perform a change of variable (ρ0)2 = 1 + κρ 2 3 1 2(φ± 0 )2− U (φ±) = 1 − κρ 2 3 U (φ±) , (4.15) - φ = φ− → 1 − κρ2 3 U (φ−) = 1 → dξ = dρ , - φ = φ+ → 1 − κρ2 3 U (φ+) = 1 − κ 3ρ 2 _→ _{dξ =} 1 − κ 3ρ 2 −1₂ dρ . BIN = 4π2 Z ξ¯ 0 − 3ρ κ dξ − 4π2 Z ξ¯ 0 ρ3 −3ρ κ dξ = −12π 2 κ Z ρ¯ 0 ρdρ − 4π2 Z ρ¯ 0 ρ3 −3ρ κ 1 − κ 3ρ 2 −1₂ dρ = −6π 2 κ ρ¯ 2₋ 12π2 κ2 1 − κ 3ρ 2 3₂ − 1 . 2. ON THE WALL:

ξ ' ¯ξ and ρ = ¯ρ. For this case we are going to use the thin wall approximation so that the potential becomes U (φ) = U0(φ) 2a(φ + φ+) ' U0(φ) . Then B is given by BON = 4π2 Z ¯ ρ3U (φ) − 3 ¯ρ κ dξ − 4π2 Z ¯ ρ3U (φ+) − 3 ¯ρ κ dξ = 4π2ρ¯3 Z [U (φ) − U (φ+)]dξ = 2π2ρ¯3S1, (4.16) where we defined S1 ≡ 2 Z [U (φ) − U (φ+)]dξ = Z p 2[U (φ) − U (φ+)]dφ .

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3. OUTSIDE THE WALL:

φ = φ+ and ¯ξ < ξ < ∞, then B is

BOU T = SE(φ+) − SE(φ+) = 0 .

The final result for B is given by the sum

Btv = BIN+ BON + BOU T = − 6π2 κ ρ¯ 2₋ 12π2 κ2 1 −κ 3ρ 2 3₂ − 1 + 2π2ρ¯3S1. (4.17)

We can determine ¯ρ by demanding dBtv

d ¯ρ = 0 dBtv dρ = 0 → 1 2ρκS¯ 1− 1 + r 1 −κ 3ρ¯ 2 _{= 0 ,} ¯ ρ = 12S1 3κS1+ 4 . (4.18)

Substituting it back in Btv we rewrite B and ρ in terms of the quantities found in chapter 3

Btv =

Bng

[1 + (R/2Λ)2_]2 , ρ =¯

R

1 + (R/2Λ)2 , (4.19)

where we indicated with Bng the result found in non gravitational case (3.18) and R was

found at the end of section 3.1 and Λ ≡ p3/(κ).

• We live in a false vacuum state.

In this case the decay happens from a space of zero energy density (U (φ+) = 0) to one of

negative energy density (U (φ−) = −). In particular the potential is given by

U (φ) = U0(φ) +

2a(φ − φ+) . 1. INSIDE THE WALL: φ = φ− and 0 < ξ < ¯ξ, then B is

BIN = SE(φ−) − SE(φ+) = 4π2 Z ξ¯ 0 ρ3U (φ−) | {z } =− −3ρ κ dξ − 4π2 Z ξ¯ 0 ρ3U (φ+) | {z } =0 −3ρ κ dξ .

We can change the integration variable using (4.15)

- φ = φ− → dξ = 1 + κ 3ρ 2 −1₂ dρ ,

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- φ = φ+ → dξ = dρ . BIN = 4π2 Z ξ¯ 0 − ρ3₋3ρ κ dξ − 4π2 Z ξ¯ 0 − 3ρ κ dξ = −4π2 Z ρ¯ 0 ρ3 + 3ρ κ 1 + κ 3ρ 2 −1₂ dρ + 4π2 Z ρ¯ 0 3ρ κdρ = 6π 2 κ ρ¯ 2₋ 12π 2 κ2 1 + κ 3ρ¯ 2 3₂ − 1 . 2. ON THE WALL:

Again ξ ' ¯ξ and ρ = ¯ρ, so the result is the same as before BON = 2π2ρ¯3S1.

3.OUTSIDE THE WALL:

φ = φ+ and ¯ξ < ξ < ∞, then B is

BOU T = SE(φ+) − SE(φ+) = 0 .

Finally, the coefficient B is given by

Bf v = BIN + BON + BOU T = 6π2 κ ρ¯ 2₋12π2 κ2 1 + κ 3ρ¯ 2 3₂ − 1 + 2π2ρ¯3S1. (4.20)

We can find ¯ρ as we did before and substitute it back in Bf v to find

Bf v =

Bng

[1 − (R/2Λ)2_]2, ρ =¯

R

1 − (R/2Λ)2. (4.21)

4.2 Is the thin wall approximation legit?

In the non-gravitational case, the thin wall approximation ( 1) was justified by the fact that we wanted the radius of the bubble R to be large enough to neglect the viscous damping term. We can compare the equations of motions for the two cases:

No Gravity: d 2_φ dρ2 + 3 ρ dφ dρ − dU dφ = 0 , With Gravity: d2φ dξ2 + 3 ρ dρ dξ dφ dξ − dU dφ = 0 . In both cases we used the thin wall approximation to compute B on the surface of the bubble, that is when ρ ' R. In the non gravitational case, we found the expression for R and we

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checked that, for 1, indeed we had R 1, which made the approximation legit. In the gravitational case instead, the term that we want to neglect is proportional to ρ_ρ0. We can find an expression for this term using equation (4.15)

(ρ0)2 = 1 + κρ 2 3 1 2(φ 0 )2− U (φ) → ρ 0 ρ 2 = 1 ρ2 + κ 3 1 2(φ 0 )2− U (φ) . (4.22)

If we want to make sure that ρ_ρ0 is small, just ensuring that 1_ρ 1 is not enough this time, we must also require that

κ 3 1 2(φ 0 )2− U (φ) 1 . Once again we need to consider the two different cases: • we live in a true vacuum state:

1. inside the wall: κ 3 1 2(φ− 0 )2− U (φ−) = 0 ; 2. on the wall: κ 3 1 2(φ 0 )2− U (φ) = const ;

3. outside the wall: κ 3 1 2(φ+ 0 )2− U (φ+) = −κ 3 . • we live in a false vacuum state:

1. inside the wall: κ 3 1 2(φ− 0 )2− U (φ−) = κ 3 ; 2. on the wall: κ 3 1 2(φ 0 )2− U (φ) = const ;

3. outside the wall: κ 3 1 2(φ+ 0 )2− U (φ+) = 0 .

It appears that, in both cases, we can overestimate the term in the following way: κ 3 1 2(φ 0₎2_{− U (φ)} ≤ κ 3 = 8πG 3 = 1 Λ2 ,

which allows us to rewrite (4.15) as ρ0 ρ 2 = 1 ρ2 + κ 3 1 2(φ 0 )2− U (φ) ≤ 1 ρ2 + 1 Λ2 .

Thus, in the gravitational case, the thin wall approximation is justified only if both ¯ρ and Λ are large, compared to the characteristic range of variation of φ. Note that, however, this puts no restriction on the ratio R_Λ, which is the ratio that indicates the importance of gravity.

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4.3 The importance of Gravity

We can discuss the values of Btv and Bf v for various values of the ratio R_Λ.

• We live in a true vacuum: Btv =

Bng

[1 + (R/2Λ)2_]2 , ρ =¯

R

1 + (R/2Λ)2 ,

- for R_Λ → 0 we find that ¯ρ ' R, which is the case without gravity; - in presence of gravity the radius of the bubble is smaller;

- also, gravity makes the materialization of the bubble more likely since it makes B smaller.

• We live in a false vacuum: Bf v =

Bng

[1 − (R/2Λ)2_]2, ρ =¯

R 1 − (R/2Λ)2,

- in this case, the presence of gravity increases B, making the materialization of the bubble less likely;

- at the moment when the bubble materializes, its radius will be bigger.

Note that for R = 2Λ, gravity stops the vacuum decay, as B → ∞. This corresponds to a value of R = 2Λ = 2 r 3 8πG = 3S1 → = 3 4κS1 2_. _(4.23)

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5 Thermal Derivation of B

In section 4, we derived the coefficient B by extending the calculations for the flat space case and assuming that, even with gravity, the bounce would still be O(4) invariant. In this section, however, we are going once again to derive B , by treating the problem as a thermal one [4]. In particular, we will work on the static patch of de Sitter spacetime and we will see how the Euclidean spherical symmetry of the bounce, this time, will be a result and not an assumption.

5.1 Flat spacetime

We will start by working out the calculations for a single particle in quantum mechanics and we will then extend the results to quantum field theory.

Figure 10: Potential V (x). The thin wall approximation can be expressed as Etop− Ef v

Ef v.

Consider the potential depicted in figure (10): in section 3 we shifted the potential so that the energy of the false vacuum state was zero. This time, however, we will not do that, but we will keep Ef v explicit in our calculations. We are going to consider a particle in a state

with energy kBT ≥ Ef v and we will try to find its decay rate, which is, in general, given by

the Boltzmann average of the probability current over a set of quantum states [14] Γ = 1

Z0

Z ∞

Ef v

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where we defined β ≡ (kBT )−1, T is the temperature of the system and the partition function Z0 is given by Z0 = Z ∞ Ef v e−β(E−Ef v)_{dE .}

Since we are interested in re-deriving only the argument of the exponential, to see that it is indeed B, we are going to omit the constants in front of the integral and redefine

Γ ≡ Z ∞

Ef v

ρ(E)Γ(E)e−β(E−Ef v)_{dE .}

We are going distinguish three cases:

1. kBT ' Ef v Etop: in this case the particle is in a metastable false vacuum and we

can use the WKB formula for tunnelling ρ(E)Γ(E) = 1 2π~exp − 2 ~ Z x2 x1 p 2(V (x) − E)dx ;

2. kBT ' Etop: in this case the WKB formula is no longer valid, but we can use the

transmission coefficient for a parabolic barrier ρ(E)Γ(E) = 1 2π~ 1 + exp −2π ~ω(E − Etop) −1 ,

where we approximated the top of the barrier with an upside down parabola, V (x) ' Etop−

1 2mω

2_x2_;

3. kBT > Etop: this is the trivial case where we are above the barrier and the probability

current is simply given by

ρ(E)Γ(E) = 1 2π~.

Let’s start with the tunnelling case. We can use the WKB formula to compute the probability current, and find that

ρ(E)Γ(E) = 1 2π~exp − 2 ~ Z x2 x1 p 2(V (x) − E)dx ≡ 1 2π~e −2B(E) ~ ,

where we identified B as we first defined it in (2.4), B(E) =

Z x2

x1

p

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Note that so far we always worked in natural units, thus from now on we will consider ~ = 1. We can compute Γ for the tunnelling case,

Γtunn =

Z ∞

Ef v

e−β(E−Ef v)_e−B(E)_{dE = e}−β(E∗−Ef v)_e−B(E∗)_. _(5.1)

At low temperatures, the integral is dominated by the stationary point E∗, that is the value

of the energy that maximizes the function in the integral. Therefore, E∗ must be the value

that minimizes the argument of the exponential

−β(E − Ef v) − B(E) ≡ f (E, x, x1, x2) .

To minimize the argument of the exponential we need to minimize a function f (E, x, x1, x2)

that depends on the path x, the energy E and the turning points x1 and x2. Therefore, we

need to consider the variations with respect to each of these quantities: • ∂f

∂x = 0 : this is true if x is a path that satisfies the equations of motion; • ∂f

∂x1

= ∂f ∂x2

= 0 : this is true because V (x1) = V (x2) = E, since x1 and x2 are the

turning points; • ∂f

∂E = 0 : this is true in the integral B(E) satisfies d dE Z x2 x1 p 2(V (x) − E)dx = −β 2 . In particular, we can work out this last condition to find

Z x2 x1 dx p2(V (x) − E) = Z x2 x1 dx √ ˙x = Z τ2 τ1 dτ = (τ2− τ1) ≡ ∆τ = β 2.

This result tells us that the tunnelling through the barrier takes an Euclidean time interval ∆τ = β/2. The continuation of the solution, past the end points, gives an Euclidean time-reversed solution back to the starting point: the bounce. Moreover, since we are in Euclidean time, this result implies that the solution is periodic in time with period T = β. Therefore, if we want to compute the tunnelling probability, for the metastable false vacuum, we need to:

1. find a solution to the Euclidean equations of motion with period β and x(0) = x1,

x(β/2) = x2 ;

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The final result is given by

Γtunn = e−β(E∗−Ef v)e−B(E∗) ∝ e−B(E∗), (5.2)

which is the same result that we found at the end of section 2.

Now let’s consider the second case, in which the particle is in the state with kBT ' Etop:

instead of tunnelling, we can compute the probability that the particle gets thermally excited all the way up and beyond the barrier. As we already mentioned, the WKB approximation is no longer valid in this case, but we can use the transmission coefficient for a parabolic barrier ρ(E)Γ(E) = 1 2π~ 1 + exp − 2π ~ω(E − Etop) −1 .

Notice that this expression agrees with both the WKB formula below the barrier, and the formula of the third case above the barrier. If we compute Γtherm we find:

Γtherm = Z ∞ Ef v ρ(E)Γ(E)e−β(E−Ef v)_{dE =} = 1 2π~ Z ∞ Ef v 1 + exp − 2π ~ω(E − Etop) −1

e−β(E−Ef v)_{dE ∝ e}−β(Etop−Ef v)_,

(5.3)

where, in the last step, we are neglecting the constants in front of the exponential. We can see that the rate for the particle that gets excited all the way up to the barrier, is proportional to the Boltzmann factor.

This was the case for a particle in a 1D quantum mechanical case, we can now extend it to Quantum Field Theory. Since the temperature is larger than zero, we have different possible paths over the barrier that satisfy the equations of motion and the one that dominates is the one of minimum action. Note that this might be a local minimum, as there may be other paths among the barrier that minimize the action but lead to different final states. The value E∗ is then the value of the energy on the saddle point of this path Esaddle, and we can

write an expression similar to (5.2):

Γtunn = e−β(Esaddle−Ef v)e−B(Esaddle)/~∝ e−B(Esaddle)/~. (5.4)

The three different cases that we described at the beginning of this section can be illustrated with the following scheme:

(a) in the tunnelling case, that is in the small temperature limit, the solution is the same as the bounce found by Coleman in [1];

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(b) this is the thermally assisted tunneling case and it looks like a deformed version of the zero temperature bounce: we can see that the continuation of the solution beyond β/2, the critical bubble configuration, is equivalent to the τ -reversed solution;

(c) this represents the third case, in which the state is thermally excited over the barrier and it’s τ -independent: every slice taken at constant τ corresponds to the critical bubble configuration.

Figure 11: The three types of bounces for flat spacetime. The horizontal direction repre-sents the three spatial components, while the vertical one reprerepre-sents Euclidean time. The highlighted areas represent the regions where the field is in the true vacuum. The red line represents the τ -constant slice that corresponds to the critical configuration of the bounce.

5.2 Curved spacetime

In section 4 we assumed that gravity would not change the O(4) Euclidean symmetry of the bounce and we started our calculation by writing the most generic O(4) invariant Euclidean metric. In this section, however, we will not make such an assumption but we will freeze the dynamic degrees of freedom of the metric and we will treat the geometry as a fixed de Sitter space, in particular we will work in the static patch. This choice is not casual, as false vacuum decay has always been associated with inflation4 _{and since we know that the early}

universe is well approximated by a de Sitter spacetime.

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5.2.1 Static patch

The FRW metric for a de Sitter spacetime is given by

ds2 = −dt2+ a2(t)dΣ2with a(t) = H−1cosh (Ht) ,

where the surface element is indicated by dΣ2 and a(t) is the scale factor. A free-falling observer is in causal contact with only a portion of de Sitter space, called the static patch. The time independent metric that describes the static path of de Sitter space is5

ds2 = −(1 − H2r2)dt2+ 1

(1 − H2_r2₎dr

2_{+ r}2_(dθ2_{+ sin}2_θdφ2_{) ,} _(5.5)

where H is the horizon

H = s 8πV (φf v) 3M2 P l .

We will write the 3D spatial part of the metric as hij ≡ gij and we will define the gtt

component as

−gtt = 1 − H2r2 ≡ A(r) .

Moreover, we will treat the scalar field on the static patch as a thermal state with temperature T = TdS = _2πH.

Let’s start then with the gravitational action S = Z _√ −g 1 2g µν_∂ µφ∂νφ − V (φ) − R 16πG d4x . In the static patch the Ricci scalar is R = 0 and the action becomes

S = Z _√ −g 1 2g µν_∂ µφ∂νφ − V (φ) d4x = Z dt Z d3x√h 1 2pA(r)φ˙ 2₊ 1 2 p A(r)hij∂iφ∂jφ − p A(r)V (φ) ≡ Z Ldt ,

where we used the fact that √−g =pA(r) · det(h) ≡pA(r)√h. This is the lagrangian of a field theory on a 3D curved space where the interaction between the fields depend on the position through the term pA(r). We can write the energy as

E = Z d3x √ h 1 2pA(r)φ˙ 2 + 1 2 p A(r)hij∂iφ∂jφ + p A(r)V (φ) . (5.6)

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As we said earlier, the field is in a thermal state with temperature TdS: we must look at

solutions to the equations of motion that are periodic in time with period T = (TdS)−1 = 2π_H.

The Euclidean action is given by SE = Z πΛ −πΛ dτ Z d3x√h 1 2pA(r) dφ dτ 2 +1 2 p A(r)hij∂iφ∂jφ + p A(r)V (φ) . (5.7) Note that, this time we choose to perform the integral from −β/2 to β/2, instead than from 0 to β. Therefore, the hypersurfaces at τ = −πH and at τ = πH give the configuration before tunnelling, while at τ = 0 the field is in the true vacuum state.

If we rewrite the metric (5.5) in Euclidean time we get ˜

gµνdxµdxν = (1 − H2r2)dτ2+

1

(1 − H2_r2₎dr

2_{+ r}2_(dθ2_{+ sin}2_θdφ2_{) ,} _(5.8)

with 0 < r < Λ. This is the metric for a 4-sphere and can be seen as embedded in a 5-dimensional Euclidean space, if we make the following identifications:

                       y0 _{= r sin θ cos φ} y1 _{= r sin θ cos φ} y2 _{= r cos θ} y3 ₌√_H−2_{− r}2_{cos (Hτ )} y4 =√H−2_{− r}2_{sin (Hτ )} .

We can rewrite the Euclidean action as SE = Z d4xpg˜ 1 2˜g µν_∂ µφ∂νφ + V (φ) . (5.9)

From this equation, the tunnelling rate can be calculated in the same way as we did before: we can find a solution to the Euclidean equations of motion which is periodic in τ and that minimizes SE and we can then use to compute the coefficient B, leading to the same

tunnelling probability of Coleman and De Luccia, Γtunn ∝ e−B.

In this way we derived a result that has a Euclidean O(4) symmetry without any initial assumption, as a confirmation of the intuition of Coleman and De Luccia. Moreover, by working in the static patch, the tunnelling will not be from the false vacuum state, but from a thermally excited state, which explains why the field never achieves its false vacuum value on the bounce.

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6 Covariant Boundary Conditions

In section 4 we wrote the equations of motion for the gravitational Euclidean Lagrangian without showing how we actually derived them. In particular, we used different components of the Einstein equations

Gµν = 8πGTµν. (6.1)

This is a covariant set of equations and the way in which they are derived, in most General Relativity books, is through variational principle of the Einstein-Hilbert action6 _[15]

SEH = 1 16πG Z M √ −gRd4x . Imposing δSEH = 0 leads to (16πG)δSEH = Z M −1 2 √ −ggλσδgλσgµνRµν+ √ −gRµνδgµν d4x + Z M √ −g∇σgµν∇σ(δgµν) − ∇λ(δgσλ) d4x .

If we rearrange the first integral we find the Einstein tensor, while for the second integral we can use Stokes’ theorem and write it as

Z M √ −g∇σgµν∇σ(δgµν) − ∇λ(δgσλ) d4x = Z ∂M √ γgµν∇σ(δgµν) − ∇λ(δgσλ) nσd3x , (6.2) where γ is the induced metric on the boundary ∂M of the volume M. We can see that, if we want to obtain the equations of motion from the variational principle δSEH = 0, we must

find a way to get rid of the boundary term. What we implicitly did in section 4, was to set the boundary contributions to vanish at infinity. Because of these boundary conditions, this way of deriving the Einstein equations is not covariant and in this section we would like to focus on finding a way that actually is. Let’s start by explaining what do we mean when we talk about covariant boundary conditions.

Consider the volume M to be a region with homogeneous vacuum energy density ρ and described by the coordinates t, r and the solid angle of the two-sphere Ω2. Then the boundary

∂M will be given by the surfaces of constant coordinates that limit M: tmin ≤ t ≤ tmax, rmin ≤ r ≤ rmax.

6_{Just by varying S}

EH we find the Einstein tensor Gµν. To retrieve the complete equations (6.1) we must

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Figure 12: Volume M with its boundary.

On ∂M we will impose a set of boundary conditions that we will indicate with B(t, r). General covariance requires every physical law to be independent of the coordinates we choose. If we perform a coordinate transformation (t, r) → (˜t, ˜r) and the volume becomes M → ˜M, then the physical laws derived in M with boundary conditions B(t, r) on ∂M must be the same as the laws derived in ˜M with boundary conditions ˜B(˜t, ˜r) on ∂ ˜M. This means that we will consider covariant those boundary conditions that transform as a scalar under a change of coordinates

B(t, r) → ˜B = B(˜t, ˜r) . (6.3) Therefore we must be particularly careful with the boundary conditions that we impose on coordinate-dependent quantities.

6.1 Inclusion of the Gibbons-Hawking-York term

If the only reason why we cannot derive Einstein equations in a covariant manner from varying the Einstein-Hilbert action is the presence of a boundary term, the first solution that might come to one’s mind is to get rid of it. A way to do so was derived first by York [16] in 1972 and later by Gibbons and Hawking [17], with the intent to evaluate the gravitational action, and from that the partition function, for metrics that contain singularities.

If we look at the usual Einstein-Hilbert action, we notice that the Ricci scalar R contains terms that are linear in the second derivatives of the metric. Since the path integral approach requires that the action depends only on first order derivatives of the metric, we must include a boundary term S = 1 16πG Z M √ −gRd4x + Z ∂M √ γBd3x .

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The term B is chosen such that it will vanish if the take the variations of the action with respect to the metric, for metrics that satisfy the Einstein equations. Therefore

B = − 1

8πGK + C ,

where K is the trace of the extrinsic curvature of ∂M and C is a term independent of the metric, that will be absorbed into the normalization of the measure on the space of all metrics once we will perform the path integral. Thus, the term that we need is

SGHY = − 1 8πG Z ∂M √ γKd3x , (6.4)

which is called Gibbons-Hawking-York term. Its variation is given by δSGHY = − 1 8πG Z ∂M δ [√γK] d3x = − 1 8πG Z ∂M [δ√γK +√γδK] = − 1 16πG Z ∂M √ γ[Kγµν− Kµν_{] δh} µν+gµν∇σ(δgµν) − ∇λ(δgσλ) nσ d3x . (6.5) If we include this term in the variation of the whole action, we can notice that the second term inside this integral cancels the boundary term in SEH, leaving us with

δ(SEH + SGHY) = 0 → 1 16πG Z ∂M √ γ [Kγµν− Kµν_{] δγ} µνd3x = 0 , (6.6)

which leads to the equations of motion if we impose Dirichlet boundary conditions on the induced metric: δγµν = 0. However, the induced metric depends on the coordinates choice,

thus imposing this condition is equivalent to picking a set of coordinates. Since we had in mind to derive the Einstein equations in a covariant way, we conclude that the variational principle it’s not the solution.

6.2 ADM formalism

An alternative approach to the Euclidean path integral one, is given by the ADM decomposi-tion7_{. So far we assumed spherical symmetry for the initial and final states of the bubble and}

for the path of least action between them. Let’s then start with the most general spherically symmetric metric [5]

ds2 = −N_t2(t, r)dt2+ L2(t, r) [dr + Nr(t, r)dt] 2

+ R2(t, r)dΩ2₂, (6.7)

7_{for more details about the ADM decomposition and the derivation the metric the reader can check the}

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where Nt and Nr are, respectively, the shift and the lapse and they are non-dynamical

functions that set the gauge. The function L(t, r) is the ratio of proper to coordinate length and, together with R(t, r), they are dynamical functions, which we would like to be covariant under coordinate transformation:

• R(t, r) is multiplied by a coordinates independent factor, thus it is a covariant quantity and the boundary conditions imposed on it will be covariant as well: R(t, r) → ˜R = R(˜t, ˜r);

• L(t, r), on the other hand, is multiplied by a term that transforms under coordinate changes in a not covariant way. Any boundary condition on L will not be covariant. In particular, L transforms in the following way:

L(t, r) → ˜L = s L2_(˜_{t, ˜}_r) ∂r ∂ ˜r + Nr(˜t, ˜r) ∂t ∂ ˜r 2 − Nt(˜t, ˜r) ∂t ∂ ˜r 2 .

We can check that, even using the metric (6.7), the Einstein-Hilbert action would still lead to a ill posed variational principle if we impose Dirichlet boundary conditions

δL|δM = δR|δM = δNt|δM= δNr|δM = 0 . (6.8)

Let’s start with the action SEH = 1 16πG Z M p −g(ADM )_Rd4_{x =} Z tmax tmin Z rmax rmin LEH(X, ˙X, X0, ¨X, ˙X0, X00)dtdr ,

where we defined LEH to be the lagrangian density integrated over the angular coordinates

dΩ2

2. We indicated with ˙X = dX

dt, X 0 ₌ dX

dr and X ∈ {L, R, Nt, Nr}. The determinant of the

metric is given by g(ADM ) = −N2 t + L2Nr2 L2Nr 0 0 L2_N r L2 0 0 0 0 R2 ₀ 0 0 0 R2sin2θ = −R4sin2θL2N_t2,

which, once integrated over the angular coordinates, gives Z p −g(ADM )_dΩ2 2 = 4πR 2_LN t. We can vary LEH δLEH = δLEH δX δX + δLEH δ ˙X δ ˙X + δLEH δX0 δX 0 +δLEH δ ¨X δ ¨X + δLEH δX00 δX 00 + δLEH δ ˙X0 δ ˙X 0

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and we can use integration by part to rearrange this expression and find δLEH = ∂r δLEH δX0 − ∂r δLEH δX00 − ∂t δLEH δ ˙X0 δX + δLEH δ ˙X0 δ ˙X + δLEH δX00 δX 0 + ∂t δLEH δ ˙X − ∂t δLEH δ ¨X − ∂r δLEH δ ˙X0 δX + δLEH δ ˙X0 δX 0 +δLEH δ ¨X δ ˙X + δLEH δX − ∂t δLEH δ ˙X − ∂r δLEH δX0 + ∂_t2 δLEH δ ¨X + ∂_r2 δLEH δX00 + ∂t∂r δLEH δ ˙X0 δX . Keeping in mind that X ∈ {L, R, Nt, Nr} we will introduce a sum over the index i and write

the variation of the action as δSEH = Z tmax tmin X i δLEH δX0 − ∂r δLEH δX00 − ∂t δLEH δ ˙X0 δX +δLEH δ ˙X0 δ ˙X + δLEH δX00 δX 0 rmax rmin dt + Z rmax rmin X i δLEH δ ˙X − ∂t δLEH δ ¨X − ∂r δLEH δ ˙X0 δX + δLEH δ ˙X0 δX 0 ₊δLEH δ ¨X δ ˙X tmax tmin dr+ Z M X i δL_EH δX − ∂t δL_EH δ ˙X − ∂r δL_EH δX0 + ∂_t2 δLEH δ ¨X + ∂_r2 δLEH δX00 + ∂t∂r δL_EH δ ˙X0 δXdrdt .

The first two integrals are the boundary terms, while the third integral contains the equations of motion. Note that Dirichlet boundary conditions δX = 0, like the ones of (6.8), are not enough to get rid of the boundary terms, since they depend on the variation of the derivatives: δ ˙X and δX0. This is another confirmation that the Einstein-Hilbert action doesn’t give a well posed variational principle with just Dirichlet boundary conditions.

We can derive and vary the ADM action to see if that would work. We start by adding to LEH the total derivatives of two terms: Ft and Fr. This is allowed because adding

total derivatives to the lagrangian only affects the boundary terms, and does not change the theory. LADM = LEH + dFr dr + dFt dt = 1 2G ( 2 ˙ R − NrR0 Nt [(NrLR)0− ∂t(LR)] + L Nt ( ˙R − NrR0)2− 2 1 −R 0 L (NtR)0+ Nt L [L 2_{− (R}0 )2] ) . (6.9)

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We can evaluate the conjugate momenta to the ADM variables: πX ≡ ∂ ˙ X p −g(ADM )_L ADM , πNt = 0 , πNr = 0 , πR= (NrLR)0− ∂t(LR) GNt , πL= (N R0− ˙R)R GNt . (6.10)

These allow us to write the ADM action in canonical form [6] SADM = Z M LADMd4x + SM = Z πLL + π˙ RR − N˙ tHt− NrHr dtdr − Z LADM BT dt , (6.11)

where we included in LADM

BT the terms that eliminate the second derivatives from the

la-grangian and that make sure that there will be no variations of the first derivatives. We also indicated the matter action with SM

SM = − Z M √ −gρd4x = − Z LNtR2sin θρdtdtdΩ22 = −4π Z tmax tmin Z rmax rmin R2LNtρdtdr . (6.12) The terms Hr and Ht are

Hr =R0πR− Lπ0L, Ht= GLπ_L2 2R2 − GπRπL R + 1 2G 2RR0 L 0 − (R 0₎2 L − L + 4πLR2ρ . (6.13)

As we did the variation of the Einstein-Hilbert action before, we can now vary the ADM action to find δSADM = Z M X i (e.o.m.)δXidrdt − Z rmax rmin [πRδR + πLδL] tmax tmin dr+ − Z tmax tmin R(L − R0₎ GL δNt+ LπLδNr+ NrπLδL + L(GNrπR+ Nt) − (NtR)0 GL δR rmax rmin dt . (6.14) Notice that under the boundary conditions (6.8) all the boundary terms vanish, ensuring that the ADM action gives a well posed variational principle. However, as we already mentioned, L is a quantity that depends on the choice of coordinates that we make: the boundary condition δL = 0 is not covariant. The ADM formalism alone cannot ensure that we derive Einstein equations in a covariant manner.

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6.3 Covariant boundary conditions for the ADM metric

We assessed that the ADM formalism gives us a well posed variational principle, but the boundary conditions are not covariant. A way to obtain covariant boundary conditions was suggested recently by Thomas Bachlechner [7]. We demand that there is a diffeomorphism that renders the ADM metric at the boundary to be the Schwarzschild-de Sitter metric

ds2|∂M= A(M, R)dT2+ A−1(M, R)dR2+ R2dΩ22|∂M, (6.15)

where T is the Killing time, M is the black hole mass and the function A(M, R) is given by A(M, R) = 1 − 2GM

R − 8πGρ

3 R

2_. _(6.16)

The Dirichlet boundary conditions in this case are given by

δM|∂M= δR|∂M= 0 . (6.17)

We can easily find the relations between these new variables and the ADM ones by comparing (6.7) with (6.15) and writing T and R as functions of (t, r)

dT = ∂T ∂tdt + ∂T ∂rdr , dR = ∂R ∂tdt + ∂R ∂rdr , ds2 = − A( ˙Tdt + T0dr)2+ A−1( ˙Rdt + R0dr)2+ R2dΩ2₂

=(−A ˙T2+ A−1R˙2)dt2+ (−AT02+ A−1R02)dr2+ 2(−A ˙TT0+ A−1RR˙ 0)dtdr + R2dΩ2₂ =(−N_t2+ L2N_r2)dt2+ L2dr2+ 2L2Nrdrdt + R2dΩ22, to find L = q A−1_(R0 )2_{− A(T}0₎2_, Nr= A−1RR˙ 0− A ˙TT0 A−1_(R0 )2_{− A(T}0₎2 , Nt = ˙ TR0− T0R˙ p A−1_(R0 )2_{− A(T}0₎2 . (6.18)

We can substitute these quantities in LADM (6.9) to find the new action

SG = Z M LADM + dFt dt + dFr dr dtdr = Z M 1 2G 2 ˙ R − NrR0 Nt [(NrLR)0− ∂t(LR)] + L Nt ( ˙R − NrR0)2 − 2(1 − R 0 L)(NtR) 0 +Nt L(L 2_{− R}02 ) − 8πGLNtρR2 dtdr = Z M M0[(N2_r− N2 t)R 0_{− N} rR] + ˙˙ M[ ˙R − NrR0] A(M, R)Nt dtdt = Z M πMM + π˙ RR − N˙ tHtG− NrHGr dtdr , (6.19)

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where we redefined the shift and the lapse as Nt ≡ −N_Lt and Nr ≡ Nr to write the action in

a canonical form. The Hamiltonian densities are given by HG

t ≡ A(M, R)πMπR+ A(M, R)−1M0R0, HGr ≡ πRR0+ πMM0. (6.20)

The variation of this actions is given by δSG = Z M X i ( e.o.m. )δXidrdt + Z rmax rmin [πRδR + πMδM] tmax tmin dr+ − Z tmax tmin NtM0+ NrAπR A(M, R) δR + NtR0+ NrAπM A(M, R) δM rmax rmin dt .

Thus, the covariant boundary conditions of (6.17) are enough to grant a well posed variational principle. Note that it has not been necessary to fix the lapse and the gauge to retrieve the equations of motion.

6.4 Importance of the boundary conditions

We are now going to consider the example of de Sitter space to illustrate the importance of boundary conditions when computing the action. Let’s start by identifying the metric (6.7) with the one of an FRW spacetime, if we make the following choice of gauge

Nt= 1 , Nr = 0 , R(t, r) = a(t) sin(r) , L(t, r) = a(t) ,

ds2_dS = −N_t2(t, r)dt2+ L2(t, r) [dr + Nr(t, r)dt]2+ R2(t, r)dΩ22 =

= −dt2+ a2(t)dr2 + a2(t) sin2(r)dΩ2₂,

(6.21) where a(t) is the scale factor, the momenta of (6.10) become

πL= −

1

Ga(t) ˙a(t) sin

2

(r) , πR= −

2

Ga(t) ˙a(t) sin(r) , (6.22) while the Hamiltonian densities of (6.13) are

Hr = R0πR− Lπ0L= R0 (NrLR)0 − ∂t(LR) GNt − L NrR 0 _{− ˙}_R GNt R !0 = = 1 Ga(t) cos(r)−∂t(a 2_{(t) sin(r)) −} 1 G− ˙a(t)a(t) sin 2_(r)0 = = 1 Ga

2_{(t) ˙a(t)2 sin(r) cos(r) −} 1

Ga

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Ht= GLπ2 L 2R2 − GπRπL R + 1 2G 2RR0 L 0 − (R 0₎2 L − L + 4πLR2ρ = = 4πρa3(t) sin2(r) + 1 2Ga(t) ˙a 2_{(t) sin}2_{(r) −} 1 Ga(t) ˙a 2_{(t) sin}2_(r) + 1 2G2a(t) cos

2_{(r) − 2a(t) sin}2_{(r) − a(t) cos}2_{(r) − a(t)} = − 1

2Ga(t) sin

2_(r)_˙a2_{(r) − 3 + 4πρa}3_{(t) sin}2_{(r) .}

At this point we can write the action (6.11) as SADM =

1 2G

Z

−3 ˙a2_{(t)a(t) sin}2_{(r) + a(t) cos}2_{(r) + a(t) − 2a(t) cos(r) − 8πGρa}3_{(t) sin}2_{(r) drdt ,}

(6.23) where we have already integrated over the solid angle dΩ2_{. We can vary the action with}

respect to a(t) to find the equations of motion for the scale parameter

˙a2(t)a(t) = a(t)H2a2(t) − 1 , 2a3(t)¨a(t) = a2(t)3H2a2(t) − ˙a2(t) − 1 , (6.24) where we defined the Hubble constant H ≡

q

8πGρ

3 . Solving the equations of motion, we

find an expression for a(t) for de Sitter spacetime

a(t) = H−1cosh(Ht) , which can be plugged in (6.23) to find

S_ADMdS =

Z _a(t) G 2 sin

2

(r/2)[cos(r) + 2] − 3H2a2(t) sin2(r) drdt . (6.25) At this point we can perform the integral, keeping in mind that different boundary conditions will yield to different result.

For example, evaluating the integral in Euclidean time t = iτ and with the following bound-ary conditions, −iπ 2 ≤ t ≤ iπ 2 , 0 ≤ r ≤ π , gives S_ADMdS = Z a(t) G 2 sin 2_{(r/2)[cos(r) + 2] − 3H}2_a2_{(t) sin}2_{(r) drdt} =2i Z H−1 0 Z π 0 a G2 sin 2_{(r/2)[cos(r) + 2] − 3H}2_a2_sin2_{(r) dr}da ˙a = π GH2 ,

where we made a change of integration variables from dt to da to perform the integral. This result is the well known entropy of de Sitter space. Choosing a different set of boundary

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conditions would have led to a different result, meaning that physical observables depend on the location of the boundary ∂M. We can employ this method to compute other observables, for example, tunnelling probabilities and decay rates, like in the problem of false vacuum decay. In particular we can compute the ratio between the probability per unit volume per unit time Γ of transitions between two different de Sitter spacetimes A and B with Hubble parameters HA and HB respectively

Γ|A→B

Γ|B→A

= eSdSADM(HB)−SADMdS (HA). (6.26)

This quantity is dependent on the choice of boundary ∂M that we make. In fact, if we use the covariant boundary conditions we defined in (6.3), we will find a different result. The gauge we fixed at the beginning of the section remains the same

Nt= − Nt L = 1 a(t), Nr= Nr = 0 , M = a(t) 2G 1 − H 2

a2+ ˙a2 sin(r) , R = R = a(t) sin(r) . However, with such a gauge choice, the integrand of the gravitational action (6.19) computed for the solution to the equations of motion of the scale factor, vanishes. This implies that equation (6.26) becomes

Γ|A→B = Γ|B→A, (6.27)

which means that, if we choose covariant boundary conditions, we find equal probability of tunnelling from a false vacuum state to a true vacuum one, and viceversa [7]. This statement clashes noticeably with the results found in section 4, yet we should not worry about it too much. The fact that the gravitational action vanishes for de Sitter space means that it is independent of the location of the boundary, which is in contrast with what we just said about observables. Physical observables depend on the boundary conditions we choose, therefore, if we change boundary conditions, the results of our calculations will transform accordingly. When picking covariant boundary conditions, however, the lagrangian density vanishes and we lose information about the problem. We can conclude that, this formulation of the action with covariant boundary conditions, even though it is an interesting exercise, it is not a useful tool when applied to the problem of false vacuum decay.

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7 Evolution of false vacuum bubbles

In this section we will see how to interpret the problem of false vacuum decay in a broader cosmological context. In particular, we will focus on some inflationary models which make use of the formalism we introduced so far. The theory of inflation has been elaborated as a possible solution to some problems of cosmology, like the flatness problem, or the horizon problem. The theory of false vacuum decay in quantum field theory, instead, had been researched with the intent of giving a plausible explanation to the ending of the inflationary scenario. On the other hand, the question still remained about the origin of inflation, especially about its initial conditions at the big bang.

7.1 Tunnelling from nothing

In 1982, Alexander Vilenkin proposed a cosmological model in which the universe is created by quantum tunnelling from nothing to a de Sitter space [9]. Vilenkin, with his idea of tunnelling from nothing, found an answer that does not require any initial or boundary condition.

Let’s suppose that the universe started in a symmetric vacuum state with energy density ρV

and that it could be described by the closed FRW metric ds2 = dt2− a2_(t) dr2 1 − r2 + r 2_dΩ2 , (7.1)

in which the scale factor a(t) can be found from the evolution equation ˙a2(t) + 1 = 8

3πGρVa

2

(t) . Solving this equation we find a de Sitter space

a(t) = H−1cosh(Ht) , (7.2) where H ≡p(8πGρV)/3. This expression describes a universe that:

- is contracting for t < 0; - is expanding for t > 0;

Aspects of False Vacuum Decay