Catalysing Vacuum Decay

Transcription

1 Catalysing Vacuum Decay Ruth Gregory Centre for Particle Theory + Ian Moss and Ben Withers JHEP

2 The Question The Coleman de Luccia instanton started a trend of understanding more complex and physically realistic tunnelling scenarios, including gravity and nonlinear field theory. CDL still the gold standard in computing probability of false vacuum decay, but? How dependent is amplitude on homogeneity?

3 Coleman Coleman originally showed that in limit of small energy difference transition modelled by a thin wall bubble. Amplitude determined by action of Euclidean tunneling solution: The Bounce ϕ F ϕ T ε Stationarity wrt R: ϕ T ϕ F Tunneling amplitude: Coleman, PRD (1977)

4 Coleman-de Luccia Includes gravity: o The instanton is a solution of the Euclidean Einstein equations with a bubble of flat space separated from ds space by a thin wall. o The wall radius is determined by the Israel junction conditions o The action of the bounce is the difference of the action of this wall configuration and a pure de Sitter geometry. Coleman and de Luccia, PRD (1980)

5 CDL Action The instanton looks like a truncated sphere: Israel conditions give truncation radius: hence bounce action:

6 Geometrical Picture ds de Sitter space is represented by a hyperboloid (sphere) in 5D Lorentzian (Euclidean) spacetime. The instanton is often represented by joining the virtual Euclidean geometry to the real Lorentzian geometry across a surface of time symmetry.

7 A more general look In general, the wall separates two different regions of spacetime, which are solutions to Einstein equations: The regions in general have different cosmological constants, and possibly a black hole mass. Bowcock, Charmousis, RG: CQG (2000)

8 Wall trajectory: Wall trajectories Israel junction conditions determine the equation of motion: Inputting the form of the trajectory gives a Friedmann like equation for R: Lorentz Euclid

9 CDL Wall Coleman-de Luccia has: Static patch Hence and Periodicity not the same as static patch HENCE CONICAL DEFICIT IN BOUNCE

10 Conical Deficits Familiar in Euclidean sections, SdS has a deficit/excess on at least one horizon:

11 Conical Actions The conical deficit is produced by a delta function in the Ricci tensor (caveat no transverse energy momentum, metric a product space) so can compute the action: Smooth out A: (Geroch-Traschen not!)

12 SdS Action Applying this to the SdS black hole now gives an interesting result. For a general periodicity: i.e. the result is independent of β

13 Back to walls Adding in a wall adds in a contribution to the action: So can compute the action of a bubble of Minkowski space inside SdS with a wall boundary: For CDL, M=0, and recover usual result.

14 Adding a black hole Can take more general instantons with black holes: e.g. a Minkowski bubble inside SdS Find numerically (except for unstable static solution R * ) See also Hiscock, PRD (1987)

15 Static solution This is the one example where we can have no conical deficit (hence another cross-check). Both methods give the bounce action:

16 Expanding Bubbles The general solution has an expanding bubble, and have to construct the wall numerically, determine its periodicity, then perform the action integral.

17 The bounce action drops as we add in an inhomogeneity: the larger the black hole, the lower the action. For light tension walls the effect is most pronounced. The periodicity of the bubble is (approx) a fixed ratio of the cosmological to black hole periodicity depending on σl The general instanton can have a remnant black hole, as well as a seed.

18 Have to calculate the generic action numerically here shown for σl =0.2. For a given seed mass M +, there is a preferred instanton with lowest action the flat interior for low mass, and static bounce for higher mass.

19 The picture is much the same for differing σl.

20 Can sum up by plotting remnant mass as a function of σl.

21 Tunneling v Evaporation: For Hawking evaporation: And tunneling: Ratio: i.e. less than unity for black holes above the Planck mass. Black holes in pure ds will evaporate before they seed decay, but if not quite ds (slow roll) accretion can dominate.

22 σ =0.1 CDL action: Trying on for size:? σ =2πGσ energy of wall in Planck units = 3 Λ = 3 8πGE (inverse) scale of false vacuum B CDL = π2 16 σ 4 4 G (1 + 4 σ 2 2 ) G E e.g GeV 2 G gives B CDL 10 6 For no remaining black hole, M + <M *, where M * is the static instanton mass: GM = 8 27 σγ R 4 B = πr2 h G 104 Black hole quite small nucleated bubble quite large.

23 Summary Have shown how to compute the action of a singular instanton, verified for known or special cases. Tunneling amplitude significantly enhanced in the presence of a black hole bubble forms around black hole and can remove it altovether. Evaporation beats tunneling in pure ds, however if in a slow roll background, accretion beats evaporation. However if no event horizon, amplitude enhanced even more.

24 Our universe?