Cosmology from Brane Backreaction

Transcription

1 Cosmology from Brane Backreaction Higher codimension branes and their bulk interactions w Leo van Nierop

2 Outline Motivation Extra-dimensional cosmology Setup A 6D example Calculation Maximally symmetric geometries FRW geometries

3 Motivation Few honest-to-god extra-dimensional cosmologies exist work in 4D effective theory work with moving branes in static backgrounds

4 Motivation Few honest-to-god extra dimensional cosmologies exist work in 4D effective theory work with moving branes in static backgrounds What do extra dimensions do during inflation?

5 Motivation Few honest-to-god extra dimensional cosmologies exist work in 4D effective theory work with moving branes in static backgrounds What do extra dimensions do during inflation? Brane back-reaction can be important RS models include codimension-1 back-reaction

6 Setup 6D Einstein-Maxwell-scalar system

7 Setup Simple solution

8 Setup Simple solution (including back-reaction)

9 Setup Simple solution (non-susy case) Field equations Flux quantization

10 Setup Simple solution (non-susy case)

11 Setup Simple solution (non-susy case)

12 Setup Simple solution (SUSY case) Field equations Flux quantization

13 Setup In SUSY case, how does system respond to changes in brane tension? Flux quantization: Obstructs T to dt

14 Setup In SUSY case, how does system respond to changes in brane tension? Flux quantization: Obstructs T to dt On other hand, general argument:

15 Setup Resolution: subdominant effects in the brane action are important for flux quantization n g = F + 1 2π New function F has interpretation as branelocalized flux b Φ b e φ

16 Calculation More general solutions

17 Calculation Perturb brane properties To evade time-dependence add current Find general solution to linearized equations

18 Calculation Sample solutions and so on

19 Calculation Brane-bulk boundary conditions:

20 Calculation Non-SUSY result: φ δt b QδΦ b = 0 b φ ρ = δt b 2QδΦ b b φ

21 Calculation SUSY result: δt b 2QδΦ b φ δt b QδΦ b = 0 b φ

22 Calculation SUSY result: δt b 2QδΦ b φ δt b QδΦ b = 0 b φ ρ = δt b 2QδΦ b = 1 2 φ b δt b QδΦ b φ

23 Calculation Three intriguing choices: Case 1: scale invariant: if δt independent of φ and δφ = Ce φ then V(φ) = Ae 2φ

24 Calculation Three intriguing choices: Case 1: scale invariant: if δt independent of φ and δφ = Ce φ then V(φ) = Ae 2φ Case 2: exponentially large volume: δt b = A + B (φ + v) 2 with v ~ 50 then r = Le φ/2 L

25 Calculation Three intriguing choices: Case 3: parametrically small vacuum energy: δt b and δφ b both independent of φ then ρ = 0 and φ adjusts to satisfy flux quantization condition

26 Calculation Three intriguing choices: Case 3: parametrically small vacuum energy: δt b and δφ b both independent of φ then ρ = 0 and φ adjusts to satisfy flux quantization condition Brane action independent of φ stable against brane loops Bulk loops generate corrections of order e 2φ = 1 r 4

27 Higher-dimensional inflation What about cosmological solutions? Must generalize brane matching conditions to case where on-brane geometry is not maximally symmetric Must solve the higher-dimensional field equations exactly

28 Higher-dimensional inflation What about cosmological solutions? Must generalize brane matching conditions to case where on-brane geometry is not maximally symmetric Must solve the higher-dimensional field equations exactly For supersymmetric system exact timedependent scaling solutions are known Can these be matched to sensible brane physics to see how brane properties control bulk fields?

29 Higher-dimensional inflation 6D Einstein-Maxwell-scalar system Brane-localized inflaton, c L b1 = T 1 + e φ χ 2 +V 0 + V 1 e λχ + L b2 = T 2

30 Higher-dimensional inflation Tolley, CB, de Rham Exact time-dependent solution e φ = H 0 τ c+2 ds 2 = H 0 τ c g mn dx m dx n + τ 2 g ij dx i dx j FRW time in 4D Einstein frame dt = H 0 τ c+1 dτ

31 Higher-dimensional inflation Tolley, CB, de Rham Exact time-dependent solution e φ = H 0 τ c+2 ds 2 = H 0 τ c g mn dx m dx n + τ 2 g ij dx i dx j FRW time in 4D Einstein frame dt = H 0 τ c+1 dτ If c = -2 then a(t) = e H 0 t and r constant 4D de Sitter geometry: evades no-go results due to near-brane asymptotics

32 Higher-dimensional inflation Tolley, CB, de Rham Exact time-dependent solution e φ = H 0 τ c+2 ds 2 = H 0 τ c g mn dx m dx n + τ 2 g ij dx i dx j FRW time in 4D Einstein frame dt = H 0 τ c+1 dτ If c -2 then a(t) = (H 0 t) p and r(t) = (H 0 t) 1/2 with p = (c + 1)/(c + 2) accelerated expansion if p > 1 and so c < -2

33 Higher-dimensional inflation Connection to brane inflaton, and how does it end? Add inflaton c evolution to the equations L b = T + e φ χ 2 +V 0 + V 1 e λχ + χ = χ 0 + χ 1 ln H 0 τ Then c +2 = λχ 1 controls the slow roll and H 0 2 = λv 1 / χ λχ 1

34 Conclusions Relatively little is known about explicitly higher-dimensional cosmology Higher-dimensional inflation with evolving x-dims

35 Conclusions Relatively little is known about explicitly higher-dimensional cosmology Higher-dimensional inflation with evolving x-dims Branes and brane back-reaction can have important implications for low-energy theory Little explored beyond codimension 1 Different parametric dependences in energy: unusual stability to quantum corrections

36 Conclusions Relatively little is known about explicitly higher-dimensional cosmology Higher-dimensional inflation with evolving x-dims Branes and brane back-reaction can have important implications for low-energy theory Potentially wide-ranging observational implications for Dark Energy cosmology, the LHC and elsewhere Little explored beyond codimension 1 Different parametric dependences in energy: unusual stability to quantum corrections