On the Fate of Bubble Universes on Non-Vacuum Backgrounds D ...

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On the Fate of Bubble Universes on Non-Vacuum Backgrounds D ...

INTERNATIONAL WORKSHOP ON COSMIC STRUCTURE AND EVOLUTION

SEPTEMBER 2009, BIELEFELD

ON THE FATE OF BUBBLE UNIVERSES ON

NON-VACUUM BACKGROUNDS

D. SIMON, J. ADAMEK, A. RAKIĆ & J. NIEMEYER, ARXIV:0908.2757 [GR-QC]

ALEKSANDAR RAKIĆ (ITPA WÜRZBURG)


MOTIVATION

V (φ)

non-vacuum

state

φ1

CONTEXT: TUNNELING ON THE LANDSCAPE

◮ Classically: field trapped in some vacuum, cannot move through ong>theong> landscape

◮ Quantum mechanically: vacua metastable - e.g. tunneling by CdL process

◮ CdL process: nucleation ong>ofong> bubbles ong>ofong> new vacuum that expand into ong>theong> old

vacuum - first order PT (a form ong>ofong> old inflation)

◮ Here: effects on a dS bubble when nucleated from a non-vacuum precursor state

◮ But why should one do that?

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

φ1

CONTEXT: TUNNELING ON THE LANDSCAPE

◮ Classically: field trapped in some vacuum, cannot move through ong>theong> landscape

◮ Quantum mechanically: vacua metastable - e.g. tunneling by CdL process

◮ CdL process: nucleation ong>ofong> bubbles ong>ofong> new vacuum that expand into ong>theong> old

vacuum - first order PT (a form ong>ofong> old inflation)

◮ Here: effects on a dS bubble when nucleated from a non-vacuum precursor state

◮ But why should one do that?

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

φ1

CONTEXT: TUNNELING ON THE LANDSCAPE

◮ Classically: field trapped in some vacuum, cannot move through ong>theong> landscape

◮ Quantum mechanically: vacua metastable - e.g. tunneling by CdL process

◮ CdL process: nucleation ong>ofong> bubbles ong>ofong> new vacuum that expand into ong>theong> old

vacuum - first order PT (a form ong>ofong> old inflation)

◮ Here: effects on a dS bubble when nucleated from a non-vacuum precursor state

◮ But why should one do that?

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

φ1

CONTEXT: TUNNELING ON THE LANDSCAPE

◮ Classically: field trapped in some vacuum, cannot move through ong>theong> landscape

◮ Quantum mechanically: vacua metastable - e.g. tunneling by CdL process

◮ CdL process: nucleation ong>ofong> bubbles ong>ofong> new vacuum that expand into ong>theong> old

vacuum - first order PT (a form ong>ofong> old inflation)

◮ Here: effects on a dS bubble when nucleated from a non-vacuum precursor state

◮ But why should one do that?

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

φ1

CONTEXT: TUNNELING ON THE LANDSCAPE

◮ Classically: field trapped in some vacuum, cannot move through ong>theong> landscape

◮ Quantum mechanically: vacua metastable - e.g. tunneling by CdL process

◮ CdL process: nucleation ong>ofong> bubbles ong>ofong> new vacuum that expand into ong>theong> old

vacuum - first order PT (a form ong>ofong> old inflation)

◮ Here: effects on a dS bubble when nucleated from a non-vacuum precursor state

◮ But why should one do that?

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

CONTEXT: TUNNELING ON THE LANDSCAPE

φ1

◮ Standard view: tunneling largely supressed + cosmic no-hair conjencture

◮ But ong>theong>re are interesting cases, e.g. when tunneling is very rapid, where ong>theong>

background is in non-vacuum

◮ Relevant for: chain inflation, DBI or resonance tunneling, ...

Freese & Spolyar, hep-ph/0412145, Watson, Perry, Kane & Adams, hep-th/0610054

Feldstein & Tweedie, hep-ph/0611286, Sarangi, Shiu & Shlaer, 0708.4375 [hep-th], ...

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

CONTEXT: TUNNELING ON THE LANDSCAPE

φ1

◮ Standard view: tunneling largely supressed + cosmic no-hair conjencture

◮ But ong>theong>re are interesting cases, e.g. when tunneling is very rapid, where ong>theong>

background is in non-vacuum

◮ Relevant for: chain inflation, DBI or resonance tunneling, ...

Freese & Spolyar, hep-ph/0412145, Watson, Perry, Kane & Adams, hep-th/0610054

Feldstein & Tweedie, hep-ph/0611286, Sarangi, Shiu & Shlaer, 0708.4375 [hep-th], ...

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

CONTEXT: TUNNELING ON THE LANDSCAPE

φ1

◮ Standard view: tunneling largely supressed + cosmic no-hair conjencture

◮ But ong>theong>re are interesting cases, e.g. when tunneling is very rapid, where ong>theong>

background is in non-vacuum

◮ Relevant for: chain inflation, DBI or resonance tunneling, ...

Freese & Spolyar, hep-ph/0412145, Watson, Perry, Kane & Adams, hep-th/0610054

Feldstein & Tweedie, hep-ph/0611286, Sarangi, Shiu & Shlaer, 0708.4375 [hep-th], ...

dS

φ2


MOTIVATION

V (φ)

non-vacuum

state

BUBBLES ON DYNAMICAL BACKGROUNDS

φ1

◮ Oong>theong>r context: Issue ong>ofong> initial conditions vs. onset ong>ofong> inflation

Goldwirth & Piran, Phys. Rept. 214 (1992) 223, Calzetta & Sakellariadou, Phys. Rev. D 45

(1992) 2802, Kurki-Suonio, Laguna & Matzner, astro-ph/9306009,

Deruelle & Goldwirth, gr-qc/9409056, Iguchi & Ishihara, gr-qc/9611047, ...

dS

φ2


MOTIVATION

TECHNIQUES

V (φ)

non-vacuum

state

φ1

◮ Interior ong>ofong> bubble is assumed to be dS - a toy ong>ofong> ong>theong> inflation ong>ofong> our universe

◮ Effect on nucleation rates - backgrounds: FRW power law inflation, radiation

dominated universes

Methods: semiclassical approach, extension ong>ofong> complex time path techniques

◮ Effect on bubble evolution on non-standard background: inhomogeneous LTB

spacetimes, FRW spacetime just undergoing a phase transition

Method: glueing spacetimes togeong>theong>r via Israel junction, semi-analytical evolution

dS

φ2


BUBBLE PROPAGATION ON DYNAMICAL BACKGROUND - METHODOLOGY

INTERIOR: DE SITTER (FLAT SLICING)

ds2 = −dt2

+ exp 2 dr Λ/3 t 2 + r 2dΩ2 , energy density Λ/(8π)

BUBBLE WALL (Σ)

EXTERIOR: LTB (, FRW)

ds 2 = −dτ 2 + R 2 dΩ 2 , stress-energy S ij = −σh ij

ds 2 = −dt 2 + (r∂r a(t,r) + a(t,r)) 2

dr

1 + 2E(r)

2 + a 2 (t,r)r 2 dΩ 2 , source Tµν = ρuµuν − Λgµν

2 ∂t a

a − 2E

a2r 2 = 2M

a3r 3 + Λ 2∂r M

3 (Eq. ong>ofong> motion) 8πρ =

a2r 2 (dust density)

(r∂r a+a)

3 free functions ong>ofong> ong>theong> model: E(r), M(r) and bang time

1 remaining gauge freedom to rescale r, choose r such that M(r) = 4π 3 Ar 3

This leaves effectively 2 free functions ong>ofong> ong>theong> model: curvature and bang time


BUBBLE PROPAGATION ON DYNAMICAL BACKGROUND - METHODOLOGY

ISRAEL JUNCTION CONDITIONS

1st condition: induced metrics hij = gµνe µ

i eν j must match on Σ:

+

hij ≡ h ij | Σ − h −

ij | Σ = 0

2nd condition: whenever ong>theong>re is a discontinuity in extrinsic curvature ong>ofong> Σ as

seen from M±, a surface layer ong>ofong> stress-energy Sij will be present:

8πSij =

Kij − hij [K ]


BUBBLE PROPAGATION ON DYNAMICAL BACKGROUND - METHODOLOGY

EVOLUTION EQS. IN EXTERIOR COORDINATES

Junction conditions give:



−(1 + 2E)¯r∂t a + (1 + 2E)(1 + 2V ) (¯r∂t a)

∂t¯r =

2

− 2E + 2V

(¯r∂¯r a + a)(2E − 2V )

and

(¯r∂¯r a + a)∂t

∂t σ = ρ

¯r


1 + 2E − (¯r∂¯r a + a) 2 (∂t¯r)2 with potential V and a geometrical constraint


Λ−

2V = −

3 +


A

3a3σ + Λ+


2

− Λ−

+ 2πσ R

24πσ 2

8πA

, 4πσ <

3a3 + Λ+


− Λ− 8π


3 3 ε

◮ Background dynamics (LTB, FRW) can only be solved for numerically – all eqs.

solved in ext. coordinates using JC; after solution is obtained one can again use

JC and express ong>theong> evolution in interior coordinates

◮ Limitation: σ becomes time dep.: it collects matter from ong>theong> background and when

ong>theong> bubble shrinks it reprovides exactly ong>theong> amount ong>ofong> matter determined by ong>theong>

background


BUBBLE PROPAGATION ON DYNAMICAL BACKGROUND - METHODOLOGY

EVOLUTION EQS. IN EXTERIOR COORDINATES

Junction conditions give:



−(1 + 2E)¯r∂t a + (1 + 2E)(1 + 2V ) (¯r∂t a)

∂t¯r =

2

− 2E + 2V

(¯r∂¯r a + a)(2E − 2V )

and

(¯r∂¯r a + a)∂t

∂t σ = ρ

¯r


1 + 2E − (¯r∂¯r a + a) 2 (∂t¯r)2 with potential V and a geometrical constraint


Λ−

2V = −

3 +


A

3a3σ + Λ+


2

− Λ−

+ 2πσ R

24πσ 2

8πA

, 4πσ <

3a3 + Λ+


− Λ− 8π


3 3 ε

◮ Background dynamics (LTB, FRW) can only be solved for numerically – all eqs.

solved in ext. coordinates using JC; after solution is obtained one can again use

JC and express ong>theong> evolution in interior coordinates

◮ Limitation: σ becomes time dep.: it collects matter from ong>theong> background and when

ong>theong> bubble shrinks it reprovides exactly ong>theong> amount ong>ofong> matter determined by ong>theong>

background


ρ/ǫvac

DE SITTER/LTB - HOMOGENEOUS LIMIT (I)

1

0.5

matter dominated universes possible

all universes vacuum dominated

contracting

expanding

forbidden

0

0 0.5 1 1.5 2 2.5

6πσ2 /ǫvac

HOMOGENEOUS LIMIT WITH DUST MATTER

◮ Already this setup has significant effects on ong>theong>

bubble evolution

ong>Bubbleong> nucleated in pure vacuum always expands

◮ Here: presence ong>ofong> matter makes it hard for ong>theong>

bubble to expand

◮ Force budget for bubble wall: surface tension

versus pressure support

∂ 2

t ¯r

∂t¯r=0 = 1

Λ+−Λ−


a 24πσ − 2πσ − 3σ

EARLY BEHAVIOUR OF BUBBLE NUCLEATED AT REST IN COMOVING FRAME

◮ Exterior has to be vacuum dominated in order to allow ong>theong> bubble to

reach ong>theong> inhomogeneity in ong>theong> first place

◮ This limits ong>theong> possibilities to study ong>theong> propagation ong>ofong> bubbles into an

inhomogeneous matter environment


DE SITTER/LTB - HOMOGENEOUS LIMIT (II)

H0¯r

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0.0 0.5 1.0 1.5

2.0 2.5 3.0

Λ+/3 t

HOMOGENEOUS LIMIT: NUMERICAL TRAJECTORIES

Λ−/Λ+

1.0

0.8

0.6

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0

4πσ0/H0

ong>Fateong> ong>ofong> bubble depends on nucleation size: small bubbles will shrink to zero, proper kinetic

energy becomes imaginary, ˙ R 2 |t=t 0 = H 2 0 R2 0 , where H0 ≡ ∂ t a

a (t0,¯r0) and R0 = a0(¯r0)¯r0

◮ The larger ong>theong> bubble ong>theong> more likely it sustains kinetic energy until ong>theong> background is

dominated by Λ+ and it converges to a finite coord. radius

◮ Right: ong>Fateong> ong>ofong> bubbles in parameter space (σ0,Λ−)


DE SITTER/LTB - INHOMOGENEOUS INITIAL DUST DENSITY


Λ+/3 ¯r

1.5

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Λ+/3 t

ρ(t, ¯r)/A

5

4

3

2

1

0

0.00 0.05 0.10 0.15

0.20 0.25 0.30

Λ+/3 t

◮ There are two ways to introduce inhomogeneity: 2E(r) = −k(r)r 2 or via a0(r) ≡ a(t0,r)

For given ρ0(r) ong>theong> initial scale factor is given by a 3 0

(r) = 3A

r 3

r 2

ρ 0 (r) dr

◮ Because ong>ofong> ong>theong> rapid dilution take an increasing prong>ofong>ile: ρ 0 (r) = Ar 3 /r 3 A with r A = Λ + /3 −1/2


DE SITTER/LTB - INHOMOGENEOUS INITIAL DUST DENSITY


Λ+/3 ¯r

1.5

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Λ+/3 t

ρ(t, ¯r)/A

5

4

3

2

1

0

0.00 0.05 0.10 0.15

0.20 0.25 0.30

Λ+/3 t

◮ There are two ways to introduce inhomogeneity: 2E(r) = −k(r)r 2 or via a0(r) ≡ a(t0,r)

For given ρ0(r) ong>theong> initial scale factor is given by a 3 0

(r) = 3A

r 3

r 2

ρ 0 (r) dr

◮ Because ong>ofong> ong>theong> rapid dilution take an increasing prong>ofong>ile: ρ 0 (r) = Ar 3 /r 3 A with r A = Λ + /3 −1/2


DE SITTER/LTB - INHOMOGENEOUS INITIAL DUST DENSITY


Λ+/3 ¯r

1.5

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Λ+/3 t

ρ(t, ¯r)/A

5

4

3

2

1

0

0.00 0.05 0.10 0.15

0.20 0.25 0.30

Λ+/3 t

◮ Recall: nucleation radius is determined by ong>theong> JC and ong>theong> parameters A,Λ+,Λ−,σ0, so:

1

¯r 2 = k(¯r0) + a

0

2 0 (¯r0)

2 ε0

− 2πσ0

3σ0


DE SITTER/LTB - INHOMOGENEOUS INITIAL DUST DENSITY


Λ+/3 ¯r

1.5

1.0

0.5

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Λ+/3 t

ρ(t, ¯r)/A

5

4

3

2

1

0

0.00 0.05 0.10 0.15

0.20 0.25 0.30

Λ+/3 t

◮ Smaller bubbles nucleate in a region where vacuum dominates over dust and can

ong>theong>refore expand

◮ Larger bubbles nucleate in a dust dominated background - as expected from ong>theong>

homogeneous case, ong>theong>y fail to expand

◮ Right: allthough ong>theong> dust prong>ofong>ile radially increases, ong>theong> background expansion

effectively leaves a decreasing prong>ofong>ile for ong>theong> expanding bubble


DE SITTER/LTB - INHOMOGENEOUS DUST + CURVATURE

√ k

Rcr


Λ+/3 ¯r

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.0 0.1 0.2 0.3 0.4 0.5

0.4

0.3

0.2

0.1

Λ+/3r

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Λ+/3 t


Λ−/3 ¯r

σ/σ0

1.4

1.2

1.0

0.8

0.6

0.4

0.0 0.5 1.0 1.5 2.0

1.005

1.004

1.003

1.002

1.001

Λ−/3 t

1.000

0.0 0.5 1.0 1.5 2.0

Λ−/3 t

◮ Curvature (or dust) inhomogeneity alone: raong>theong>r hopeless to get ong>theong> buble ong>theong>re,

appears to be in contrast to Fischler et al., 0711.3417 [hep-th]

◮ In order to enforce transition through inhomogeneity: use dust prong>ofong>ile from

analysis before, take smaller bubbles, put curvature right in front ong>ofong> bubble


DE SITTER/LTB - INHOMOGENEOUS DUST + CURVATURE

√ k

Rcr


Λ+/3 ¯r

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.0 0.1 0.2 0.3 0.4 0.5

0.4

0.3

0.2

0.1

Λ+/3r

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Λ+/3 t


Λ−/3 ¯r

σ/σ0

1.4

1.2

1.0

0.8

0.6

0.4

0.0 0.5 1.0 1.5 2.0

1.005

1.004

1.003

1.002

1.001

Λ−/3 t

1.000

0.0 0.5 1.0 1.5 2.0

Λ−/3 t

ong>Bubbleong> trajectories are affected notably (red, green) in exterior coordinates

◮ However: in interior coordinates ong>theong> effect vanishes

◮ A substantial effect on ong>theong> surface tension remains


DE SITTER/FRW WITH PHASE TRANSITION

H0¯r


Λ−/3 ¯r

2.0

1.5

1.0

0.5

0.0

0 2 4 6 8 10

0.8

0.6

0.4

0.2

H0t

w = 1/3 → w = −1

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Λ−/3 t

w = 1/3 → w = −1

H0¯r


Λ−/3 ¯r

1.5

1.0

0.5

0.0

0 2 4 6 8 10

0.8

0.6

0.4

0.2

H0t

w = −1 → w = 1/3

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Λ−/3 t

w = −1 → w = 1/3

◮ Here: background with vacuum energy and a perfect fluid undergoing a PT

◮ Significant effects on trajectories both in exterior and interior coordinates


TUNNELING AND PROPAGATION OF VACUUM BUBBLES ON DYNAMICAL

BACKGROUND, ARXIV:0908.2757 [GR-QC]

SUMMARY & CONCLUSIONS

◮ How much information about ong>theong> ‘initial state‘ survives ong>theong> transition?

◮ Used toy models (LTB, FRW) and junction machinery to approach this question

◮ Already a background with homogeneous matter has significant influence on bubble evolution

◮ In ong>theong> junction approach: effect ong>ofong> inhomogeneities (curvature + dust) can be seen from ong>theong>

outside - inside (physical) observervers see no difference

◮ Surface tension is clearly perturbed also as seen from ong>theong> inside - consequence ong>ofong> method

ong>Bubbleong>s trajectories on FRW background with PT show significant effects (also inside frame),

also clear effect on surface tension

◮ Could such effects be potentially observable?

◮ Similar to primordial bubble collisions ong>theong>re exists a characteristic disturbance ong>ofong> ong>theong> bubble

trajectory, see e.g. Aguirre, Johnson & Tysanner, 0811.0866 [hep-th]

◮ Such perturbations can break ong>theong> isotropy and homogeneity ong>ofong> ong>theong> bubble (‘cosmological axis

ong>ofong> evil‘), unique effects in ong>theong> CMB possible, c.f. Chang, Kleban & Levi, 0810.5128 [hep-th]

◮ Shortcoming: junction method, as a toy model we have fixed interior and exterior ab initio -

no proper modelling ong>ofong> energy transport through ong>theong> domain wall


POWER LAW INFLATING BACKGROUND: NUCLEATION RATE

ImS / ImSdS

1.0

0.9

0.8

0.7

0.6

proper time average

conformal time average

0.5

0.0 0.1 0.2


∂tH/H 2

POWER LAW INFLATION - NUMERICAL STUDY

ε

σ

1+α

η1

(∂¯r η)

η

2 − 1 = 2 ∂¯r η 1 + α

− 3

¯r η −

∂ 2 ¯r η

(∂¯r η) 2 − 1


POWER LAW INFLATING BACKGROUND: NUCLEATION RATE

ImS / ImSdS

1.0

0.9

0.8

0.7

0.6

proper time average

conformal time average

0.5

0.0 0.1 0.2


∂tH/H 2

POWER LAW INFLATION - NUMERICAL STUDY

◮ 3 different time scales in ong>theong> problem

◮ Inverse expansion rate H −1 (η): scale on which background changes significantly

◮ Change ong>ofong> expansion rate itself: |∂t H/H| −1

◮ These two are to be compared to bubble crossing time: 3σ/ε


POWER LAW INFLATING BACKGROUND: NUCLEATION RATE

ImS / ImSdS

1.0

0.9

0.8

0.7

0.6

proper time average

conformal time average

0.5

0.0 0.1 0.2


∂tH/H 2

POWER LAW INFLATION - NUMERICAL STUDY

◮ If bubble crossing time smallest scale: tunneling rate is well approximated by ong>theong>

Minkowskian calculation ImSMink = π2

12 ε¯r 4 0 = 27π2 σ4 4ε3 ◮ If bubble crossing time is not ≪ than Hubble time at nucleation H −1 (η0): two possibilities

◮ a) |∂t H/H| −1 still ≫ bubble crossing time: quasistatic approximation valid H = H (η0)

◮ b) Else: bubble crossing time is not small w.r.t. any oong>theong>r timescale: ong>theong>n tunneling process

’feels’ ong>theong> changing expansion rate - decay rate is modified significantly (enhancement)

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