arX

iv:1

606.

0750

0v1

[gr

-qc]

23

Jun

2016

Prepared for submission to JHEP

Evolution of Vacuum Bubbles Embeded in

Inhomogeneous Spacetimes

Florencia Anabella Teppa Pannia,a,1,2 Santiago Esteban Perez Bergliaffa,b

aGrupo de Astrofísica, Relatividad y Cosmología, Facultad de Ciencias Astronómicas y Geofísicas,

Universidad Nacional de La Plata, Paseo del Bosque s/n B1900FWA, La Plata, Argentina.bDepartamento de Física Teórica, Instituto de Física, Universidade do Estado de Rio de Janeiro,

CEP 20550-013, Rio de Janeiro, Brazil.

E-mail: fteppa@fcaglp.unlp.edu.ar, sepbergliaffa@gmail.com

Abstract: As a first step in the analysis of the influence of inhomogeneities in the evolu-

tion of an inflating region, we study the propagation of bubbles of new vacuum in a radially

inhomogeneous background filled with dust or radiation, and including a cosmological con-

stant. For comparison, we also analyse the cases with homogeneous dust and radiation

backgrounds. We show that the evolution of the bubble in the radiation environments is

always slower than in the dust cases, both for homogeneous and inhomogeneous ambients,

and leads to appreciable differences in the evolution of the proper radius of the bubble.

1Fellow of CONICET.2Corresponding author.

Contents

1 Introduction 1

2 The thin-shell formalism 2

2.1 Numerical evolution 6

3 Evolution in homogeneous backgrounds: dust vs. radiation 8

4 Evolution in inhomogeneous backgrounds 10

4.1 Evolution in an inhomogeneous dust background 10

4.2 Evolution in an inhomogeneous radiation background 12

5 General Discussion 13

A Lemaître’s geometry 15

1 Introduction

An inflationary phase can solve some of the issues present in the standard cosmological

model (such as the horizon and flatness problems). However it is not clear to what extent

this assertion depends on assuming that the pre-inflationary universe is described by a

homogeneous and isotropic space-time. In particular, the onset of inflation in the presence

of inhomogeneities has been discussed in [1–7], among others, where it was claimed that

highly homogeneous and isotropic initial conditions on a patch several times larger than the

horizon are necessary for inflation to start. Other authors [8–10] asseverate that inflation

is viable with inhomogeneous initial conditions (see [11] for a review).

Once inflation has started in a certain region, it remains to see whether the ambient

inhomogeneities affect its development, and in particular if it will smooth out large initial

inhomogeneities. These inquiries are relevant not only in the context of traditional early-

universe models of inflation, but also in the more speculative scenario of eternal inflation

and the string landscape, in which regions filled with new vacuum nucleate into an ambient

region, leading to inflationary patches in different environments. Since the nucleation does

not necessarily occur in vacuum-dominated regions, it is interesting to study the time

development of such patches in less symmetric ambients with different matter contents. In

this regard, the evolution of a vacuum bubble embedded in an inhomogeneous background

sourced by a pressureless fluid was analysed in [12–14], the background being described

there by the spherically-symmetric Lemaître-Tolman-Bondi (LTB) solution of the Einstein’s

equations (see for instance [15]). In this work we present a generalisation of the above-

mentioned analysis, in which there is an inhomogeneous distribution of radiation with

– 1 –

spherical symmetry in the ambient space-time, described by Lemaître’s solution [16]. The

internal and external regions are matched using the thin-shell formalism, based on the Israel

junction conditions [17]. The evolution of the bubble is computed numerically, together

with the evolution of the external metric. We also present here the evolution of a vacuum

bubble in the homogeneous and isotropic case, and show that such evolution in a radiation

background is markedly different from that of dust, in the sense that the growth of the

vacuum region is slower when radiation fills the space-time outside the bubble. We show

that this difference in the growth of the bubble is present both in external coordinates

and in the proper radius of the bubble, and subsists in the inhomogeneous cases (namely,

external space-times described by LTB and Lemaître’s solution).

The paper is organised as follows. In Section 2 the thin-shell formalism is described,

including a detailed characterisation of Lemaître’s solution. In order to analyse the different

effects of the matter content on the growth of the bubble, we start by comparing in Section 3

the evolution of vacuum regions in dust and radiation homogeneous backgrounds. The

inhomogeneous cases are studied in Section 4. General results and remarks conclusions are

discussed in Section 5.

2 The thin-shell formalism

We shall assume that the region separating the inflating patch and the environment can be

described by a thin shell, as opposed to a gradual profile. In order to solve Einstein’s equa-

tions in this case, we need to enforce the validity of Israel’s junction conditions [17], which

link the discontinuity in the extrinsic curvature across the shell to its energy-momentum

content. The thin-shell approximation together with the junction conditions have been

widely used to describe the evolution of cosmic bubbles in the context of inflationary mod-

els [12, 13, 18].

The two space-time regions, denoted here by M− and M+, are separated by a hy-

persurface Σ [19], modelled by a thin shell with negligible radius, whose matter content is

described by an energy-momentum tensor along with an equation of state. The metric asso-

ciated to the gravitational field produced by the thin shell is continuous, but its derivatives

are discontinuous through the shell. This implies that the affine connection is represented

by functions with step-like discontinuities, while the Riemann tensor becomes a Dirac delta

[20]. We restrict our analysis to the spherically symmetric case and a time-like shell (that

is, orthogonal to surfaces of constant time).

With the aim of studying the evolution of vacuum bubbles in the different ambient

space-times, we shall analyse cases in which the inner vacuum region is described by the de

Sitter metric, while the matter content of the outer region (dust or radiation) may have an

homogeneous or inhomogeneous distribution. The most general inhomogeneous spherically-

symmetric solution of Einstein’s equations with non-zero matter content is described by

Lemaître’s geometry, along with an equation of state for the outer fluid of the form p(t, r) =

λǫ(t, r). The homogeneous FLRW solution and the inhomogeneous dust solution can be

recovered, respectively, in the limits ǫ(t, r) = ǫ(t) and λ = 0. Following the notation used

– 2 –

in [12], the problem is then characterised as follows:1

• The inner vacuum region, M−, with non-zero cosmological constant Λ−, is described

by an isotropic and homogeneous metric. Using the coordinates (T, z, θ, φ), the line

element is given by

ds2|M− = dT 2 − b2(T )

(

dz2

1 + z2+ dΩ2

)

. (2.1)

The evolution of the scale factor b(T ) is determined by the equation(

db

dT

)

=

(

Λ−

3

)

b2(T ) + 1 . (2.2)

• The spherically-symmetric shell, Σ, is characterised by the line element

ds2|Σ = dτ2 − ρ(τ)2dΩ2 , (2.3)

where (τ, θ, φ) are coordinates on the bubble. We assume that the matter content on

the shell can be described by a perfect fluid with energy-momentum tensor given by

TΣµν = [σ(τ) + Π(τ)]uΣµu

Σν −Π(τ)gΣµν , (2.4)

where σ and Π denote, respectively, the density and the pressure of the fluid.

• The outer region, M+, is modelled by the Lemaître’s solution, with line element in

external coordinates (t, r, θ, φ) given by

ds2|M+ = eA(t,r)dt2 − eB(t,r)dr2 −R2(t, r)dΩ2 , (2.5)

and the following energy-momentum tensor:

Tµν = [ǫ(t, r) + p(t, r)]u+µ u+ν − p(t, r)g+µν − Λ+g

+µν . (2.6)

Einstein’s field equations for the outer metric are

R2(t, r)R′(t, r)ρ(t, r) = 2M ′(t, r) , (2.7)

R2(t, r)R(t, r)p(t, r) = −2M (t, r) , (2.8)

where M(t, r) is defined by

2M(t, r) = R(t, r)+e−A(t,r)R2(t, r)R(t, r)−e−B(t,r)R′2(t, r)R(t, r)−ΛR3(t, r)

3. (2.9)

The conservation of Tµν yields the following relations

A′(t, r) = − 2p′(t, r)

ρ(t, r) + p(t, r), (2.10)

eB(t,r) =R′2(t, r)

1 + 2E(r)exp

(

∫ t

t0

2R(t, r)p′(t, r)

[ǫ(t, r) + p(t, r)]R′(t, r)dt

)

, (2.11)

where E(r) is an arbitrary function related to the local curvature [21].

1Hereafter, the subscripts “-” and “+” indicate, respectively, inner and outer quantities.

– 3 –

It is important to note that although the metrics are expressed in different coordinate

systems, the angular coordinates coincide due to the spherical symmetry of the problem.

The continuity condition for the metric through the thin shell imposes the following

restrictions:

b(T, ζ) = ρ(τ) = R(t, r) , (2.12)

dT 2 −(

b2(T )

1 + ζ2

)

dζ2 = dτ2 = eA(t,r)dt2 − eB(t,r)dr2 , (2.13)

where all functions are evaluated on Σ. From eqs. (2.12) and (2.13) we can write (T, ζ)

and (t, r) as (T (τ), ζ(τ)) and (t(τ), r(τ)). However, since Lemaître’s solution is known only

numerically, it is convenient to describe the evolution of the shell in terms of the outer

coordinates (t, r, θ, φ). Hence, the evolution of Σ will be parametrised by t, instead of

τ . Following [12], we will denote x(t) the outer radial coordinate of the bubble, that is,

x(t) ≡ r|Σ.

The restrictions (2.12) and (2.13) lead to the Israel’s junction conditions, given by

[17–19, 22]:

− σ

2=[

Kθθ

]

, (2.14)

Π = [Kττ ] +

[

Kθθ

]

, (2.15)

dσ

dτ+

2

ρ

dρ

dτ(σ +Π) = −[T n

τ ] , (2.16)

where all quantities are functions of the proper time of the shell, τ , and [A] ≡ A+ − A−.

The tensor Kab is the extrinsic curvature of the shell, defined as Kab ≡ nα;βeαae

βb, and

T nτ ≡ e

ατT

βα nβ is the projection of the energy-momentum tensor of the inner/outer region

in the direction normal to the shell surface.

Equations (2.14)-(2.16) completely determine the evolution of the radius ρ, the density

σ and the pressure Π of the shell, and are coupled to Einstein’s equations which govern the

evolution of the inner and outer geometries. In particular, eq. (2.15) can be substituted

by an equation of state for the matter content on the shell, which is assumed of the form

Π = wσ. To study the evolution of the shell we need to write eqs. (2.14) and (2.16) in terms

of t and r, and for this the components of the extrinsic curvature must be calculated. The

projectors over the hypersurface Σ expressed in the external coordinates are

eατ =

(

dt

dτ,dx

dτ, 0, 0

)

, (2.17)

eαθ = (0, 0, 1, 0) , (2.18)

eαφ = (0, 0, 0, 1) . (2.19)

The velocity of the bubble is uα = eατ and the normal vector oriented to the outer region

is defined by the conditions uαnα = 0 and nαnα = −1.

For Lemaitre’s metric we have

nα = γ+eA(t,x)/2eB(t,x)/2

(

−(

dx

dτ

)

,

(

dt

dτ

)

, 0, 0

)

, (2.20)

– 4 –

where the parameter γ+ determines if the bubble is expanding (γ+ = 1) or contracting

(γ+ = −1). The projectors and the normal vector in terms of the inner coordinates can

be analogously defined. The angular components of the extrinsic curvature tensor then are

given by

(Kθθ)

− = γ−1

ρ√

1 + ζ2

(

ζbdz

dτ

db

dT+ (1 + z2)

dT

dτ

)

, (2.21)

(Kθθ)

+ = γ+1

ρe−A/2e−B/2

(

eAR′ dt

dτ+ eBR

dx

dτ

)

, (2.22)

where γ− = 1 (γ− = −1) if the radius expressed in internal coordinates is increasing

(decreasing). Following [12], equation (2.14) can be written as:

γ+

√

(

dρ

dτ

)2

−∆+ − γ−

√

(

dρ

dτ

)2

−∆− = −σρ

2. (2.23)

with

∆+ =R2

eA− R′2

eB= −1 +

(

2M

R3+

Λ+

3

)

ρ2 , (2.24)

∆− = −(1 + ζ2) + ζ2(

db

dT

)2

= −1 +Λ−

3ρ2 . (2.25)

Using expressions (2.24) and (2.25), eq. (2.23) can be then rewritten as

(

dρ

dτ

)2

= ρ2V 2 − 1 , (2.26)

where

V 2 ≡ Λ− +

[

σ

4+

1

σ

(

Λ+ − Λ−

3+

2M

R3

)]2

. (2.27)

Taking into account that over the bubble we have ρ(τ) = R(t(τ), x(τ)), and replacing

(dρ/dτ) with eq. (2.26), we get for eq. (2.14)

dx

dt=

−RR′ ±√

(R2V 2 − 1)[R′2eA − R2eB + eAeB(R2V 2 − 1)]

R′2 + eB(R2V 2 − 1). (2.28)

It only rests to rewrite eq. (2.16) in terms of the coordinates (t, x(t)). The projection of

the energy-momentum tensor normal to Σ is

(T nτ )

+ = −γ

(

dt

dτ

)(

dx

dτ

)

eA(t,x)/2eB(t,x)/2[ǫ(t, x) + p(t, x)]√

eA(t,x) − eB(t,x)(

dxdt

)2. (2.29)

On the other hand, since the bubble encloses a vacuum region, we have (T nτ )

− = 0. Hence

eq. (2.16) takes the form

dσ

dt= −2(1 + w)σ

R

R+ γ+(ǫ+ p)

(

dx

dt

)

eA/2eB/2

√

eA − eB(

dxdt

)2. (2.30)

– 5 –

Once the outer geometry is known, the coupled system given by eqs.(2.28) and (2.30) gives

the evolution for the shell in terms of the external coordinates (t, x(t)).

An additional restriction on the functions follows from requiring that the sign of the

argument of the square root in eq. (2.28) be positive [12]. This sets a lower limit for the

radial coordinate of the bubble, since x must satisfy the following constraint:

1

V 2(t, x(t))< R2(t, x(t)) . (2.31)

We will also assume that the matter on Σ satisfies the weak energy condition during all the

evolution, that is, σ > 0. This condition is equivalent to impose the following restrictions

δ+ − δ− >ρ2σ2

4, if γ = +1 , (2.32)

δ+ − δ− <ρ2σ2

4, if γ = −1 , (2.33)

with δ ≡ R2

eA− R′2

eB.

2.1 Numerical evolution

We have developed a numerical code to compute the evolution of the bubble, given by the

solution of equations (2.28) and (2.30). These equations are coupled to those determining

the evolution of the external geometry, which can be written as follows [23]

R = eA/2

[

2M

R+

Λ

3R2 − 1 +R′2e−B

]1/2

, (2.34)

M = −p

2R2R , (2.35)

ǫ = −p′R

R′− [ǫ+ p]

[

R′

R′+ 2

R

R

]

, (2.36)

B = 2

[

R′

R′+

Rp′

[ǫ+ p]R′

]

, (2.37)

with

M ′ =ǫ

2RR′ , (2.38)

A = −2

∫ r

0

p′

ǫ+ pdr , (2.39)

where the symbols ˙ and ′ indicate, respectively, derivatives with respect to t and r. The

pressure p is determined, at each instant, from the corresponding equation of state for the

outer matter content. The integration of the above system of partial differential equations

was implemented using the method of lines with a fourth order differentiation scheme [24].

The initial conditions to solve the problem are specified by giving the initial position

and density of the bubble (x0 and σ0, respectively), as well as the initial profiles for the

– 6 –

metric functions R(t0, r) and ǫ(t0, r), and the curvature E(r). We choose for our problem

the following initial profiles:

R(t0, r) = a0r , (2.40)

E(r) = −1

2

( r

k

)2, (2.41)

ǫ(t0, r) = ǫ0

[

1− δǫexp

(

−(r − r0)2

w20

)]

, (2.42)

with a0, k and ǫ0 arbitrary constants. The quantities δǫ, r0 and w0 characterise the inhomo-

geneous initial distribution of the background matter.2 In the case of homogeneous dust or

radiation we need to set δǫ = 0. Note that these three functions are sufficient to completely

determine the evolution of the outer geometry. The functions M(t0, r) and A(t0, r) are then

computed from eqs. (2.38) and (2.39), respectively, and eB(t0,r) = R′2(t0, r)/(1 + 2E(r)).

The parameter Λ− represents the vacuum energy of the region inside the bubble, and is

intrinsically related to the energy scale imposed by the inflationary models for the nucleation

process [25]. We choose Λ− ≃ 5× 10−5 to characterise the inner region, which corresponds

to an energy of order 1014 GeV in Planck units.

There are not a priori restrictions on the parameters Λ+ and ǫ0. We shall work with

ǫ0 = 10Λ− and Λ+ < ǫ0. Since ǫ0 represents the asymptotic value of the background density

away from the inhomogeneous region, the above choice ensures that at t = t0 the dynamics

of the external region is dominated by the term (2M/R) in eq. (2.34). Consequently, the

potential effects on the dynamics due to the background dust or radiation distributions

become more pronounced. In the opposite case, the Λ-dominated expansion would rapidly

dilute the background density, and thus becoming a de Sitter-de Sitter scenario. The

parameter Λ+ is allowed to take four representative values: Λ+ = 0,Λ−/2,Λ−, 2Λ−. Each

of these leads to a different dynamical behaviour, which will be analysed in the following

sections.

Finally, the initial conditions for the thin-shell are x0 = 15, and σ0 = 1 × 10−3. The

election of x0 is such that the nucleation of the bubble takes place at a point where the radial

derivative of ρ is non-negligible. In the most general case, in which the bubble expands in

an inhomogeneous background with non-zero pressure (described by Lemaître’s solution),

this choice implies a non-zero initial pressure gradient, whose influence on the background

evolution is briefly analysed in appendix A. The initial value for σ0 is chosen to satisfy the

constraint given by eq. (2.32). We consider the values w = 0, 1/3,−1 for the equation of

state parameter for the matter on the bubble.

We show in the next section the results for the numerical evolution of the vacuum bubble

embedded in different backgrounds. We will start with the simplest case of homogeneous

outer distribution, in order to study the effects of the radiation pressure over the bubble

evolution.

2It is also possible to introduce the inhomogeneous profile through the curvature function E(r), as

discussed in [12, 21].

– 7 –

3 Evolution in homogeneous backgrounds: dust vs. radiation

We shall study in this section the effects of two different homogeneous backgrounds on the

dynamics of the bubble. The first one corresponds to a content of pressureless matter, and

the second one to radiation. In both cases, the outer region is characterised by the isotropic

and homogeneous FLRW metric, with line element given by

ds2 = dt2 − a2(t)

(

1

1− kr2dr2 − r2dΩ2

)

. (3.1)

This metric can be recovered from the expression (2.5) when E(r) = −12kr

2, R(t, r) = a(t)r

and ǫ(t, r) = ǫ(t) (in this case we have that p′ = 0). Hence A(t, r) = 0 and B(t, r) =

R′2(t, r)/(1 + 2E(r)) = a2(t)/(1 − kr2). Equations (2.34)-(2.37), which determine the

evolution of the geometry, are simplified in the FLRW case to the following:

a2 =2M

ar3+

a2Λ+

3− k , (3.2)

M = −ar3p

2, (3.3)

ǫ = −3(ǫ+ p)a

a, (3.4)

along with the equations of state p = 0 (dust) and p = ǫ/3 (radiation). In the case of a

dust background, we also have M(t, r) ≡ 0.

Equations (2.28) and (2.30), which respectively determine the evolution of the radius

and density of the bubble, become

dx

dt=

−(1− kx2)a±√

(x2a2V 2 − 1)(1 − kx2)(a2V 2 − a2 − k)

x(a2V 2 − k), (3.5)

dσ

dt= −2(σ + pσ)

a

a+ γ+

(

dx

dt

)

a(ǫ+ p)√1− kx2

√

1− a2

1− kx2

(

dx

dt

)2

, (3.6)

with

V 2 = Λ− +

[

σ

4+

1

σ

(

Λ+ − Λ−

3+

2M

a3x3

)]2

. (3.7)

In Figure 1 we compare the evolution of these quantities for the dust and radiation

backgrounds. In both cases, the values of the parameter w in the equation of state of the

bubble are w = −1, 0, 1/3, indicated with different lines in each plot. The parameters which

characterise the geometry, as well as the initial conditions for the numerical evolution are

those detailed in section 2.1.

The curves showed in Figure 1 make evident that the value of the parameter Λ+ is

crucial for the evolution of the bubble. In those cases for which Λ+ < Λ−, the radius

expressed in external coordinates initially grows (due to the nonzero initial velocity, given

by eq. (3.5) evaluated at t = t0) but then decreases until it reaches the lower limit imposed

by the constraint (2.31), that is, x > 1/(aV ). On the other hand, if Λ− < Λ+, the bubble

grows indefinitely.

– 8 –

14

15

16

17

18

19

0 20 40 60 80 100

x(t)

t

Λ+= 0

dust (FLRW)

rad (FLRW)

w=1/3 w=0

w=-1

15

16

17

18

19

20

0 20 40 60 80 100

x(t)

t

Λ+=Λ−/2

dust (FLRW)

rad (FLRW)

w=1/3 w=0

w=-1

14

15

16

17

18

19

20

21

22

0 20 40 60 80 100

x(t)

t

Λ+=Λ−

dust (FLRW)

rad (FLRW)

w=1/3 w=0

w=-1

16

18

20

22

24

26

28

0 20 40 60 80 100

x(t)

t

Λ+=2Λ−

dust (FLRW)

rad (FLRW)

w=1/3 w=0

w=-1

Figure 1. Evolution of the radial (outer) coordinate of the bubble for different values of theparameter Λ+. In those cases for which Λ+ < Λ

−, the radius expressed in external coordinates

initially grows but then decreases. If Λ−< Λ+, the bubble grows indefinitely.

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 20 40 60 80 100

σ(τ)

t

Λ+= 0

dust (FLRW)

rad (FLRW)

w=1/3 w=0

w=-1

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 20 40 60 80 100

σ(τ)

t

Λ+= 2Λ−

dust (FLRW)

rad (FLRW)

w=1/3 w=0

w=-1

Figure 2. Evolution of the bubble density for the values Λ+ = 0 and Λ+ = 2Λ−

. For all cases, theevolution in the radiation background reaches lower density respect to the corresponding dust cases.There are not qualitative differences between the different values of the parameter of equation ofstate w. The evolution with Λ+ = Λ

−and Λ+ = Λ

−/2 shows similar behaviours.

Although the curves presented in Figure 1 for the radiation background case have a

similar qualitative behaviour to those corresponding to a dust background, the evolution in

the first case is slower. In other words, for all the examples considered, with the same initial

conditions, the radiation background slows down the evolution of the bubble. Figure 1 also

shows that the evolution of the bubble is strongly dependent on its matter content. In all

– 9 –

the examples we have studied, the value w = 0 yields a slower evolution than the case with

w = 1/3. This behaviour can be also interpreted as a consequence of the pressure generated

by the radiation content of the bubble. All the cases with parameter of equation of state

w = −1 rapidly reach the constraint imposed by equation (2.31), independently of the type

of fluid in the background.

A noticeable difference exists between the cases with Λ+ < Λ− and those correspond

to Λ− < Λ+. While the former present a maximum is the evolution curve of the radial

coordinate, the later cases have increasing x(t). We will return to this point below.

Notice also that for the case Λ+ > Λ−, for times large enough such that the matter

density of the background is diluted, the evolution tends asymptotically to that of de Sitter,

which can be obtained in a closed form [12, 26]. This constitutes a test for our numerical

code.

We show in Figure 2 the evolution of the bubble density for the limit values Λ+ = 0 and

Λ+ = 2Λ−. In both cases, the evolution of the bubble in the radiation background presents

lower values than the corresponding case with pressureless outer content of mater, in agree-

ment with the above-mentioned difference found in the radial coordinate x(t). However,

for each value of Λ+, there are no qualitative differences between the chosen values for w.

The evolution with Λ+ = Λ− and Λ+ = Λ−/2 (omitted here) presents similar features.

4 Evolution in inhomogeneous backgrounds

We will present in this section the evolution of a vacuum bubble embedded in dust or

radiation backgrounds described, respectively, by the LTB and Lemaître solutions, with

the aim of analysing the effects of inhomogeneous distribution of the background matter

content.

The evolution of a bubble in an inhomogeneous pressureless background, described by

the spherically- symmetric LTB solution, has been previously studied in [12–14]. We start

by reviewing this case, but using different initial conditions and inhomogeneous profiles.

Afterwards we will study the evolution of vacuum bubbles in inhomogeneous radiation

backgrounds, described by Lemaître’s solution. This problem has not been previously

studied and represents the most important contribution of this work.

4.1 Evolution in an inhomogeneous dust background

The evolution of a bubble immersed in a inhomogeneous dust background is described by

an outer region characterised by the LTB metric. The line element is

ds2 = dt2 − R′2(t, r)

1 + 2E(r)dr2 −R2(t, r)dΩ2 , (4.1)

which is a special case of eq. (2.5) for p(t, r) ≡ 0. The equations that determine the

evolution of the outer geometry become

R2 =2M

R+

Λ

3R2 + 2E(r) , (4.2)

ǫ = −ǫ

(

R′

R′+ 2

R

R

)

, (4.3)

– 10 –

14

15

16

17

18

19

20

21

0 10 20 30 40 50

x(t)

t

Λ+=Λ−/2

dust (FLRW)

dust (LTB)

w=1/3 w=0

w=-1

0.0x100

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

5 10 15 20 25 30 35 40 45

ε

r

Λ+ = Λ−/2

tt

16

18

20

22

24

0 10 20 30 40 50

x(t)

t

Λ+= 2Λ−

dust (FLRW)

dust (LTB)

w=1/3 w=0

w=-1

0.0x100

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

5 10 15 20 25 30 35 40 45

ε

r

Λ+ = 2Λ−

tttt

Figure 3. Evolution of the external radial coordinate of the bubble immerse in an inhomoge-neous background, described by the LTB solution. The evolution in the homogeneous background,included for comparison, corresponds to constant initial density equal to the asymptotic value ofthe inhomogeneous profile. The density radial profiles correspond to the dust inhomogeneous back-ground, and the crosses indicate the value of the radial coordinate of the bubble at the correspondingtime.

where M is a function of r only. The radius coordinate and the density of the bubble evolve

following

dx

dt=

−(1 + 2E)R ±√

(R2V 2 − 1)(1 + 2E)(2E − R2 +R2V 2)

R′(2E +R2V 2), (4.4)

dσ

dt= −2(σ + pσ)R

R+

(

dx

dt

)

(ǫ+ p)R′/√1 + 2E

√

1− R′2

(1+2E)

(

dxdt

)2, (4.5)

with

V 2 = Λ− +

[

σ

4+

1

σ

(

Λ+ − Λ−

3+

2M

R3

)]2

. (4.6)

In Figure 3 we show the evolution of the external radial coordinate of the bubble in an

inhomogeneous dust background, described by the LTB solution, for different values of the

parameter Λ+, and equations of state for the bubble with parameter wσ = −1, 0, 1/3. The

radial profiles shown in the right plots correspond to the density evolution, where the cross

represents the corresponding radial coordinate of the bubble in each instant. In order to

analyse the consequences of the evolution of the bubble in an inhomogeneous background,

– 11 –

we compare the curves with those obtained for the case of homogeneous dust background

with the same initial conditions. In the homogeneous cases, the initial constant density of

the external region corresponds to the outer asymptotic value of the inhomogeneous profile.

The curves show that the growth of the outer radial coordinate x(t) is qualitatively the same

in both cases. However, the inhomogeneous cases show a slower evolution. This behaviour

can be understood by the fact that the evolution in a inhomogeneous background starts in

a region with lower density when compared with the homogeneous case. Notice that the

case with Λ+ = 2Λ− is qualitative different from the others, and the previous analysis does

not apply. In this case, the difference (Λ+ − Λ−) dominates the evolution of the auxiliary

quantity V (t), and lower initial densities contribute to faster evolutions.

It follows from the plots that the evolution in inhomogeneous backgrounds can be de-

scribed by the equivalent homogeneous problem with equal initial conditions. This feature

of the evolution can be also understood by analysing the evolution of the radial profile for

the density of the background (right panels). The evolution of the background dilutes the

inhomogeneous external region and the radial coordinate will eventually follow an homoge-

neous evolution.

The plots in Figure 3 also show that the differences due to the values of the parameter

w (which determines the matter content on the bubble), are similar to those described in the

homogeneous cases presented in the previous section. Finally, it is important to highlight

that due to the inhomogeneous profiles, the evolution is quantitative different.

4.2 Evolution in an inhomogeneous radiation background

We have studied in the previous sections the possible effects on the evolution of vacuum

bubbles due to the pressure of an homogeneous radiation background (section 3), as well as

the consequences of the nucleation in an underdense region of the inhomogeneous dust radial

distribution (section 4.1). Motivated by the results found in those analysis, we will explore

in this section the problem which combines both effects, that is, the evolution of a vacuum

bubble in a radiation inhomogeneous background. The external geometry is described by

Lemaître’s solution, and the radius and density of the bubble obeys the evolution equations

(2.28) and (2.30).

Figures 4 and 5 show the growth of the radius of the bubble in external coordinates.

Each case is compared with the analogous cases of homogeneous radiation background

and inhomogeneous dust background, respectively. In the first case, the evolution of the

radial density profiles of the external region is presented in the corresponding right panels.

Contrary to the LTB solution, the evolution of the Lemaître geometry does not confine

the inhomogeneity to a fixed radial interval. Due to the effect of a non-zero pressure

gradient, the inhomogeneous profile is distorted to lower or higher values of the r coordinate,

according to the sign of p′ [27].

In Figure 4 the evolution of the bubble in homogeneous and inhomogeneous radiation

background are compared. We found similar features to those previously analysed for the

cases with homogeneous and inhomogeneous dust backgrounds (Figure 3). The evolution of

the bubble in the Lemaître background presents qualitatively the same behaviour of the case

of the homogeneous radiation background (note that the quantity V 2 does not depend on

– 12 –

14

15

16

17

18

19

0 5 10 15 20

x(t)

t

Λ+= Λ−/2

rad (Lem)

rad (FLRW)

w=1/3 w=0

w=-1

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

5 10 15 20 25 30 35 40 45

ε

r

Λ+ = Λ−/2

tt

15

16

17

18

19

20

0 5 10 15 20

x(t)

t

Λ+= 2 Λ−

rad (Lem)

rad (FLRW)

w=1/3 w=0

w=-1

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

5 10 15 20 25 30 35 40 45

ε

r

Λ+ = 2Λ−

tttt

Figure 4. Evolution of the radius of the bubble in the case of inhomogeneous radiation backgroundsdescribed by the Lemaître solution. The FLRW curves represent the evolution in an homogeneousradiation background with initial constant density equal to the outer asymptotic initial density ofthe ihnomogeneous cases. The inhomogeneous radiation radial profiles are described by Lemaître’ssolution. The crosses represent the radial coordinate of the bubble in the corresponding time.

the fluid pressure). Some qualitative differences can be seen for those cases with Λ− ≤ Λ+,

probably due to the motion of the bubble through the inhomogeneity (that can be tracked

with the crosses in the corresponding right panels). However, the evolution is not modified

qualitatively. The observed differences due to the matter content of the bubble are similar

to those found in the previous cases. Again, the bubbles with equation of state described

by w = −1 quickly reach the lower limit for the radial coordinate imposed by the constraint

(2.31).

The comparison between the evolution in both inhomogeneous cases (pure radiation and

pure dust) is shown in Figure 5. There exist quantitative (but not qualitative) differences

between both cases, analogous to those found in the comparison of the homogeneous cases

(Figure 1). For all cases, the evolution in radiation backgrounds is slower than in the dust

backgrounds, due to the pressure of the external fluid.

5 General Discussion

We presented a study of the evolution of vacuum bubbles in backgrounds with inhomo-

geneous dust or radiation contents, and compared it with the homogeneous cases. This

– 13 –

14

14.5

15

15.5

16

16.5

17

17.5

18

0 5 10 15 20

x(t)

t

Λ+= 0

dust (LTB)

rad (Lem)

w=1/3 w=0

w=-1

14

14.5

15

15.5

16

16.5

17

17.5

18

0 5 10 15 20

x(t)

t

Λ+= Λ−/2

dust (LTB)

rad (Lem)

w=1/3 w=0

w=-1

14.5

15

15.5

16

16.5

17

17.5

18

0 5 10 15 20

x(t)

t

Λ+= Λ−

dust (LTB)

rad (Lem)

w=1/3 w=0

w=-1

15

16

17

18

19

20

0 5 10 15 20

x(t)

t

Λ+= 2 Λ−

dust (LTB)

rad (Lem)

w=1/3 w=0

w=-1

Figure 5. Comparison between the evolution in inhomogeneous radiation backgrounds and in-homogeneous dust background. The geometry of the external region is described, respectively byLemaître’s and the LTB solution.

analysis is important in the context of inflationary models, to test whether vacuum regions

can evolve in the presence of inhomogeneities and, if so, how these affect the evolution.

We have developed a numerical code to compute the evolution of vacuum bubbles using

the thin-shell formalism. The problem involves the integration of a system of partial differ-

ential equations, to determine the evolution of the radial coordinate and the surface density

of the bubble together with the evolution of the geometry of the background. This geom-

etry is described by the FLRW metric in homogeneous cases, and the LTB and Lemaître

metrics for inhomogeneous dust and radiation cases, respectively. Our code reproduces

the results for dust backgrounds previously obtained by other authors [12, 13], and gener-

alises the problem to those cases with radiation backgrounds (for both homogeneous and

inhomogeneous distributions).

We computed the evolution for different values of the parameters Λ+ and w, which

characterise the external geometry and the matter content of the bubble, respectively. We

have obtained several results. The comparison between cases with homogeneous back-

grounds, described by the FLRW metric, shows that the radiation content in the external

region slows down the evolution of the bubble in the external coordinates (Figure 1). These

differences are quantitative, that is, the general behaviour of the evolution do not change.

The analysis of the inhomogeneous cases shows that the evolution is also delayed if the

bubble nucleates in a subdensity region, in both dust and radiation ambients. However, we

– 14 –

found that the inhomogeneous profile of the background only affects the evolution through

the initial conditions: once the evolution starts, the radius grows like in an homogeneous

background with lower initial density.

Regarding the matter content of the bubble, its time dependence markedly changes

with the parameter w. For all cases, we have found qualitatively similar evolutions for the

values w = 0 and w = 1/3. However, a radiation bubble expands faster than a dust bubble,

due to the radiation pressure of the bubble. The evolution with parameter w = −1 quickly

reaches the upper limit for the radial coordinate imposed by the equation (2.31). The

evolution of the density of the bubble do not change appreciably with the matter content

of the bubble.

Other differences can be noted for different values of the parameter Λ+. While all cases

with Λ− < Λ+ the evolution of the bubble is practically monotonous, for the cases with

Λ+ < Λ− the radial coordinate x(t) initially grows due to the velocity imprinted by the

initial conditions, but it decreases to lower values afterwards. This behaviour, however,

must not be interpreted as a collapse scenario, since x(t) represents the radial coordinate

of the bubble non-comoving with the background, and hence indicates the growth of the

bubble with respect to the expanding background.

The consequences of the above-mentioned differences can be also analysed considering

the proper radius of the bubble. In figure 6 we show the evolution of the function ρ(τ),

which represents the proper radius of the bubble expressed in its proper coordinates. The

evolution of the bubble is noticeably affected by the background. In those cases in which

the bubble is in a radiation ambient, the growth of ρ is slower than in the corresponding

dust case. Furthermore, the radial distribution of the radiation as well as the value of the

external cosmological constant also modify the evolution.

There are several possibilities for extensions of our work. Among them we shall mention

two. First, it would be interesting to develop a more detailed study of the dependence of the

evolution of the bubble with the initial profiles, to asses the issue of genericity of inflation

in inhomogeneous backgrounds. Second, the evolution in different backgrounds may leave

signatures in the inflating region. Since the bubble plays the role of a moving boundary

of this region and, as we have shown, the presence of inhomogeneities outside the bubble

modifies its motion, quantum fields inside the bubble will be indirectly influenced by the

external inhomogeneities. We hope to return to these issues in future publications.

A Lemaître’s geometry

The features present in the evolution of Lemaître’s solution can be qualitatively understood

following the discussion in ref. [27]. Let’s consider the evolution eq. (2.9), which can be

rewritten as

e−AR2 =2M

R+

1

3ΛR2 − 1 + (1 + 2E)exp

(

−2

∫

dtp′

(ρ+ p)

R

R′

)

. (A.1)

The l.h.s. is associated to the expansion rate of the external space-time. In regions in

which the initial profiles are such that the pressure gradient is large, the exponential will

– 15 –

15

16

17

18

19

20

21

22

23

24

5 10 15 20

ρ(τ)

τ

Λ+=Λ−/2, ω=1/3

dust (FLRW)rad (FLRW)dust (LTB)rad (Lem)

14

16

18

20

22

24

26

28

5 10 15 20

ρ(τ)

τ

Λ+=2Λ−, ω=1/3

dust (FLRW)rad (FLRW)dust (LTB)rad (Lem)

Figure 6. Evolution of the proper radius of the bubble expressed in terms of the proper time fordifferent outer space-times. In those cases in which the bubble is in a radiation ambient, the growthof ρ is slower than in the corresponding dust case.

decrease and the expansion rate of the shells with r = constant will be reduced. Relative

to these, shells with larger values of r will expand faster, leading to a drop in the gradient

of p, and eventually to a change of sign in p′. Negative values of p′ cause the increment of

the expansion rate, hence leading to acoustic oscillations, which were previously analysed

in ref. [27], and can be seen in figure 7. If the oscillations grow enough to change the sign

of R, then a collapse of the geometry could take place at different radial coordinates.

Unlike the LTB solution, the inhomogeneous regions are not confined to the initial

radial coordinate. This behaviour of the evolution of the geometry is a direct consequence

of the not-zero pressure gradient which characterises Lemaître’s solution (note that in the

particular case with p′ = 0, the metric functions reduce to the form A(t, r) = 0 and

eB(t,r) = R′(t,r)(1+2E(r)) , that is, the LTB limit is recovered).

Acknowledgements

FATP acknowledges support from CONICET and CLAF/ICTP. SEPB would like to ac-

knowledge support from FAPERJ and UERJ.

References

[1] T. Piran, Numerical relativity and cosmology, in Numerical relativity and cosmology (W. G.Unruh and G. W. Semenoff, eds.), vol. 219, pp. 261–282, NATO Advanced Science Institutes(ASI) Series C, 1988.

[2] D. S. Goldwirth and T. Piran, Spherical inhomogeneous cosmologies and inflation:

Numerical methods, Physical Review D 40 (Nov., 1989) 3263–3279.

[3] D. S. Goldwirth and T. Piran, Extending the realm of numerical relativity. A numerical study

of inhomogeneous inflation., Annales de Physique Colloque Supplement 14 (Dec., 1989)149–156.

[4] D. S. Goldwirth and T. Piran, Inhomogeneity and the onset of inflation,Physical Review Letters 64 (June, 1990) 2852–2855.

– 16 –

0

5e-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

5 10 15 20 25 30 35 40 45

ρ

r

t1t2t3t4t5t6t7t8t9

t10t11

-1e-05

-5e-06

0

5e-06

1e-05

1.5e-05

2e-05

2.5e-05

3e-05

5 10 15 20 25 30 35 40 45

dp/d

r

r

t1t2t3t4t5t6t7t8t9

t10t11

0

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25 30 35 40 45

dR/d

t

r

t1t2t3t4t5t6t7t8t9

t10t11

Figure 7. Evolution of the functions that characterise Lemaître’s geometry, namely the density, thepressure gradient and the expansion rate, from top to bottom, respectively. The labels ti indicatessuccessive time steps of the evolution. A pronounced pressure gradient causes the propagation ofthe inhomogeneous region away from the initial radial coordinate. If R eventually changes its sign,the geometry will represent a collapsing scenario.

[5] D. S. Goldwirth and T. Piran, Initial conditions for inflation,Physical Report 214 (May, 1992) 223–292.

[6] N. Deruelle and D. S. Goldwirth, Conditions for inflation in an initially inhomogeneous

universe, Physical Review D 51 (Feb., 1995) 1563–1568, [gr-qc/9409056].

[7] R. S. Perez and N. Pinto-Neto, Spherically symmetric inflation,Gravitation and Cosmology 17 (Apr., 2011) 136–140, [1205.3790].

[8] H. Kurki-Suonio, P. Laguna and R. A. Matzner, Inhomogeneous inflation: Numerical

evolution, Physical Review D 48 (Oct., 1993) 3611–3624, [astro-ph/9306009].

[9] A. Berera and C. Gordon, Inflationary initial conditions consistent with causality,Physical Review D 63 (Mar., 2001) 063505, [hep-ph/0010280].

[10] W. E. East, M. Kleban, A. Linde and L. Senatore, Beginning inflation in an inhomogeneous

universe, ArXiv e-prints (Nov., 2015) , [1511.05143].

[11] R. Brandenberger, Initial Conditions for Inflation - A Short Review, ArXiv e-prints (Jan.,2016) , [1601.01918].

[12] W. Fischler, S. Paban, M. Žanić and C. Krishnan, Vacuum bubble in an inhomogeneous

cosmology, Journal of High Energy Physics 5 (May, 2008) 41, [0711.3417].

[13] D. Simon, J. Adamek, A. Rakić and J. C. Niemeyer, Tunneling and propagation of vacuum

– 17 –

bubbles on dynamical backgrounds, JCAP 11 (Nov., 2009) 8, [0908.2757].

[14] A. Rakić, D. Simon, J. Adamek and J. Niemeyer, On the fate of vacuum bubbles on matter

backgrounds, Annalen der Physik 522 (Mar., 2010) 336–339, [0911.3910].

[15] K. Bolejko, A. Krasiński, C. Hellaby and M.-N. Célérier, Structures in the Universe by Exact

Methods: Formation, Evolution, Interactions. Oct., 2009.

[16] A. G. Lemaître, The expanding universe, General Relativity and Gravitation 29 (1997)641–680.

[17] W. Israel, Singular hypersurfaces and thin shells in general relativity,Il Nuovo Cimento B Series 10 44 (1966) 1–14.

[18] V. A. Berezin, V. A. Kuzmin and I. I. Tkachev, Dynamics of bubbles in general relativity,Physical Review D 36 (Nov., 1987) 2919–2944.

[19] Ø. Grøn and S. Hervik, Einstein’s General Theory of Relativity with Modern Applications in

Cosmology. Springer Science+Business Media, 2007.

[20] M. Visser, Lorentzian wormholes. From Einstein to Hawking. 1995.

[21] K. Bolejko, Radiation in the process of the formation of voids,Monthly Notices of the Royal Astron. Soc. 370 (Aug., 2006) 924–932, [astro-ph/0503356].

[22] N. Sakai and K.-I. Maeda, Junction conditions of Friedmann-Robertson-Walker space-times,Physical Review D 50 (Oct., 1994) 5425–5428, [gr-qc/9311024].

[23] A. H. A. Alfedeel and C. Hellaby, The Lemaître model and the generalisation of the cosmic

mass, General Relativity and Gravitation 42 (Aug., 2010) 1935–1952, [0906.2343].

[24] W. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations.Academic Press, 1991.

[25] A. H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems,Physical Review D 23 (Jan., 1981) 347–356.

[26] K.-W. Ng and S.-Y. Wang, Collapse of vacuum bubbles in a vacuum,Physical Review D 83 (Feb., 2011) 043512, [1006.3441].

[27] K. Bolejko and P. D. Lasky, Pressure gradients, shell-crossing singularities and acoustic

oscillations - application to inhomogeneous cosmological models,Monthly Notices of the Royal Astron. Soc. 391 (Nov., 2008) L59–L63, [0809.0334].

– 18 –