Brane Backreaction: antidote to no-gos

Transcription

1 Brane Backreaction: antidote to no-gos Getting de Sitter (and flat) space unexpectedly w Leo van Nierop

2 Outline New tool: high codim back-reaction RS models on steroids

3 Outline New tool: high codim back-reaction RS models on steroids Honest-to-God higher-dim cosmology de Sitter no-go and inflationary models

4 Outline New tool: high codim back-reaction RS models on steroids Honest-to-God higher-dim cosmology de Sitter no-go and inflationary models Implications for naturalness issues Hierarchy problem Cosmological constant problem

5 New Tools arxiv: w Hoover & Tasinato arxiv: w Hoover, de Rham & Tasinato arxiv: w Bayntun and van Nierop

6 Extra dims and naturalness How can extra dimensions help? scalars in 4D need not be scalars in higher D eg f could arise as a component of A m or g mn

7 Extra dims and naturalness How can extra dimensions help? scalars in 4D need not be scalars in higher D lowering the gravity scale helps by lowering the UV scales to which one can be sensitive δm 2 = Λ4 M 4 2

8 Extra dims and naturalness Cod-1 backreaction scalars in 4D need (warping) not be scalars in Worked higher D out in makes low E detail for lowering the gravity physics scale local helps by lowering codimension-1 the UV scales to which in extra one can dims be sensitive How can extra dimensions help? back-reaction can be important for low energies (eg Randall-Sundrum models)

9 Higher-codimension back-reaction Why are higher codimensions harder? In d space dims massless fields vary as r 2-d and so tend to diverge at the source positions for d > 1

10 Higher-codimension back-reaction Why are higher codimensions harder? In d space dims massless fields vary as r 2-d and so tend to diverge at the source positions for d > 1 How is this dealt with? Source action dictates near-source boundary conditions

11 Higher-codimension back-reaction Why are higher codimensions harder? In d space dims massless fields vary as r d-2 and so tend to diverge at the source positions for d > 1 How is this dealt with? Source action dictates near-source boundary conditions E da Q

12 Higher-codimension back-reaction Why are higher codimensions harder? In d space dims massless fields vary as r d-2 and so tend to diverge at the source positions for d > 1 How is this dealt with? Source action dictates near-source boundary conditions r d 1 φ Peloso, Sorbo & Tasinato CB, Hoover, de Rham & Tasinato r S φ.and similarly for g mn etc

13 Codimension-2 back-reaction eg: for scalar-tensor theories CB, Hoover, de Rham & Tasinato given the action: if S = 1 2κ 2 g R + φ 2 + V(φ) + S b and ds 2 = e 2W g μν dx μ dx ν + dr 2 + e 2B dθ 2 with S b = g L b

14 Codimension-2 back-reaction Then the matching conditions are CB, Hoover, de Rham & Tasinato

15 Higher-codimension back-reaction Important complication Goldberger & Wise de Rham Divergences due to fields diverging at branes absorbed into renormalization of brane interactions

16 Higher-codimension back-reaction Important complication Divergences due to fields diverging at branes absorbed into renormalization of brane interactions A convenient regularization Codimension-1 cylinder with smooth interior

17 Higher-codimension back-reaction Baytun, CB & van Nierop Important complication Divergences due to fields diverging at branes absorbed into renormalization of brane interactions A convenient regularization Codimension-1 cylinder with smooth interior Matching rules are robust Can use any brane regularization at all to capture lowest multipole moment

18 Higher-codimension back-reaction Important complication Divergences due to fields diverging at branes absorbed into renormalization of brane interactions A convenient regularization Codimension-1 cylinder with smooth interior Matching rules are robust Baytun, CB & van Nierop Green, Shapere, Vafa & Yau Check: the result works well for D7s sourcing axio-dilaton and gravity in 10D Type IIB sugra Can use any brane regularization at all to capture lowest multipole moment

19 de Sitter solutions hep-th/ w Tolley, Hoover & Aghababaie arxiv: w Maharana, van Nierop, Nizami & Quevedo

20 Back-reaction vs de Sitter no-go Few honest-to-god extra dimensional inflationary cosmologies exist work in 4D effective theory work with moving branes in static backgrounds

21 Back-reaction vs de Sitter no-go Few honest-to-god extra dimensional inflationary cosmologies exist work in 4D effective theory Maldacena & Nunez; Wesley & Steinhardt work with moving branes in static backgrounds Lower-dimensional ds solutions in higher dimensions difficult to find Motivated no-go results for de Sitter solutions

22 Back-reaction vs de Sitter no-go de Sitter solutions and no-go results: Few honest-to-god extra dimensional cosmologies exist 1 if S = g R + L work in 4D 2κeffective 2 theory m + S source CB, Maharana, van Nierop, Nizami & Quevedo work with moving branes in static backgrounds and ds 2 = e 2W g μν dx μ dx ν + g mn dx m dx n What do extra dimensions do during inflation? Explicit then in absence example of from space-filling 6D supergravity fluxes with cod-2 sources 1 R = S 2κ 2 on shell + 2 e dw ds + g source μν dg μν

23 Back-reaction vs de Sitter no-go This term often vanishes de Sitter solutions and no-go results: Few honest-to-god extra dimensional cosmologies exist This 1 term has definite (AdS) sign if S = g R + L work in 4D 2κeffective 2 theory m + S source This term is often omitted work with moving branes in static backgrounds and ds 2 = e 2W g μν dx μ dx ν + dr 2 + e 2B dθ 2 What do extra dimensions do during inflation? Explicit then example from 6D supergravity with cod-2 sources 1 R = S 2κ 2 on shell + 2 e dw ds + g source μν dg μν

24 Back-reaction vs de Sitter no-go Source-brane matching conditions imply these de Sitter solutions and no-go results: terms exactly cancel one another for codim-2 Few honest-to-god extra dimensional cosmologies sources exist 1 if S = g R + L work in 4D 2κeffective 2 theory m + S source CB, Maharana, van Nierop, Nizami & Quevedo work with moving branes in static backgrounds and ds 2 = e 2W g μν dx μ dx ν + dr 2 + e 2B dθ 2 What do extra dimensions do during inflation? Explicit then example from 6D supergravity with cod-2 sources 1 R = S 2κ 2 on shell + 2 e dw ds + g source μν dg μν

25 Back-reaction vs de Sitter no-go Source-brane matching conditions imply these de Sitter solutions and no-go results: terms exactly cancel one another for codim-2 Few honest-to-god extra dimensional cosmologies sources exist 1 Aghababaie et al if S = g R + L work in 4D 2κeffective 2 theory m + S source This term is generically a total derivative for work with moving branes in static backgrounds and higher-dim ds 2 = esupergravities: 2W g μν dx μ dx ν S+ dr 2 + e 2B dθ 2 on shell = dω What do extra dimensions do during inflation? Explicit then example from 6D supergravity with cod-2 sources 1 R = S 2κ 2 on shell + 2 e dw ds + g source μν dg μν

26 Motivation This de can Source-brane Sitter have matching conditions imply these solutions de Sitter and sign no-go results: terms exactly cancel one another for codim-2 Few honest-to-god extra dimensional cosmologies sources exist For example, this 1 is a reason why if S = g R + L work in 4D 2κeffective 2 theory m + S the de Sitter compactifications of source 6D work sugra This with evade term is moving the generically no-go branes theorems a total derivative for in static backgrounds and higher-dim ds 2 = esupergravities: 2W g μν dx μ dx ν S+ dr 2 + e 2B dθ 2 on shell = dω What do extra dimensions do during inflation? Explicit then example from 6D supergravity with cod-2 sources 1 R = S 2κ 2 on shell + 2 e dw ds + g source μν dg μν Tolley et al

27 Motivation Because Source-brane de Sitter it is matching conditions imply these solutions a total derivative and no-go results: terms exactly cancel one another for codim-2 does Few not honest-to-god depend most extra of the dimensional cosmologies sources exist details of back-reacted 1 solutions. if S = g R + L work in 4D 2κeffective 2 theory m + S source This term is generically a total derivative for work with moving branes in static backgrounds and higher-dim ds 2 = esupergravities: 2W g μν dx μ dx ν S+ dr 2 + e 2B dθ 2 on shell = dω What do extra dimensions do during inflation? Explicit then example from 6D supergravity with cod-2 sources 1 R = S 2κ 2 on shell + 2 e dw ds + g source μν dg μν

28 Extra-dimensional Inflationary Models hep-th/ w Tolley & de Rham arxiv: w van Nierop

29 Higher-dimensional inflation What about time-dependent solutions? Must generalize brane matching conditions to case where on-brane geometry is not maximally symmetric Wish to solve higher-dimensional field equations exactly, including energetics of modulus stabilization

30 Higher-dimensional inflation What about time-dependent solutions? Must generalize brane matching conditions to case where on-brane geometry is not maximally symmetric Wish to solve higher-dimensional field equations exactly, including energetics of modulus stabilization For supersymmetric systems exact timedependent scaling solutions are known Can these be matched to sensible brane physics to see how brane properties control bulk fields?

31 Higher-dimensional inflation Nishino & Sezgin 6D Einstein-Maxwell-scalar system L = 1 2κ 2 R + φ 2 + e φ F mn F mn + Ce φ ds sign Brane-localized inflaton, c L b1 = T 1 + e φ χ 2 +V 1 e λχ + L b2 = T 2

32 Higher-dimensional inflation Tolley, CB, de Rham Exact time-dependent solution e φ = H 0 τ c+2 e φ(r) 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

33 Higher-dimensional inflation Tolley, CB, de Rham Exact time-dependent solution e φ = H 0 τ c+2 e φ(r) 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: S on shell = 2 φ

34 Higher-dimensional inflation Tolley, CB, de Rham Exact time-dependent solution e φ = H 0 τ c+2 e φ(r) 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

35 Higher-dimensional inflation Source-bulk matching: 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 CB & van Nierop Then c +2 = λχ 1 controls the slow roll and H 0 2 = λv 1 / χ λχ 1

36 Higher-dimensional inflation Extra dimensions grow slowly as the noncompact four dimensions inflate Inflation could help understand why extra dimensions have particular properties in our epoch

37 Higher-dimensional inflation Extra dimensions grow slowly as the noncompact four dimensions inflate Inflation could help understand why extra dimensions have particular properties in our epoch Evolving volume potentially allows the gravity scale to be high at horizon exit, but low at present (similar to Conlon, Kallosh, Linde & Quevedo)

38 Naturalness Issues arxiv: , arxiv: and arxiv: w van Nierop

39 An opportunity Goldberger & Wise Brane backreaction provides a mechanism for lifting flat directions described by bulk moduli Generalization of the Goldberger Wise mechanism for RS codimension-1 sources Can give exponentially large extra dimensions

40 An opportunity Brane backreaction provides a mechanism for lifting flat directions described by bulk moduli Generalization of the Goldberger Wise mechanism for RS codimension-1 sources Can give exponentially large extra dimensions Brane backreaction cleanly identifies how curvature depends on brane properties Luty & Ponton Backreaction competes with naïve tension terms

41 A puzzle For 6D flux-stabilized supergravity we have 1 2κ 2 R = S on shell = 1 2κ 2 2 φ δs b δφ and so R = 0 if no branes couple to 6D dilaton f Seems to imply geometry should be robustly flat, regardless of on-brane loops and perturbations This is superficially inconsistent with known perturbations of these geometries

42 A Simple Model 6D Einstein-Maxwell-scalar system

43 A Simple Model Carroll & Guica Aghababaie et al Simple solution

44 A Simple Model General Simple solution arguments An explicit realization Magnetic flux required to stabilize extra dimensions against gravitational collapse

45 A Simple Model General Simple solution arguments An explicit realization Labels flat direction (which exists due to shift symmetry or scale invariance)

46 A Simple Model General Simple solution arguments An explicit realization For later: notice radius is exponential in the flat direction f 0 in the SUSY case

47 A Simple Model Simple solution (including back-reaction)

48 A Simple Model Simple solution (non-susy case) Field equations Flux quantization

49 A Simple Model Simple solution (non-susy case)

50 A Simple Model Simple solution (non-susy case)

51 A Simple Model Simple solution (SUSY case) Field equations Flux quantization

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

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

54 Puzzle Resolution Resolution: subdominant effects in the brane action are important for flux quantization

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

56 Puzzle Resolution Non-SUSY result when F nonzero: Flat direction stabilized at: φ δt b QδΦ b = 0 b φ Vac energy there: ρ = δt b 2QδΦ b b φ

57 Puzzle Resolution SUSY result when F nonzero: Flat direction stabilized at: δt b 2QδΦ b φ δt b QδΦ b = 0 b φ

58 Puzzle Resolution SUSY result when F nonzero: Flat direction stabilized at: δt b Agrees 2QδΦwith b + 1 exact δt 2 φ b QδΦ b = 0 result for curvature b φ Vac energy there: ρ = δt b 2QδΦ b = 1 2 φ b δt b QδΦ b φ

59 Applications Three intriguing choices: Case 1: scale invariant (cf Weinberg s no-go theorem): if δt independent of φ and δφ = Ce φ then V(φ) = Ae 2φ

60 Applications 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

61 Applications 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

62 Applications 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

63 Conclusions

64 Conclusions Branes and brane back-reaction can have important implications for low-energy theory Little explored beyond codimension 1

65 Conclusions Branes and brane back-reaction can have important implications for low-energy theory Little explored beyond codimension 1 Vast unexplored territory Codim-2 back-reaction as big as brane effects Promising for naturalness issues (different parametric dependences in energy; unusual stability to quantum corrections; etc)

66 Conclusions Branes and brane back-reaction can have important implications for low-energy theory Little explored beyond codimension 1 Vast unexplored territory Codim-2 back-reaction asrge asect brane effects Potentially wide-ranging observational implications for Dark Energy cosmology, the LHC and elsewhere Promising for naturalness issues (different parametric dependences in energy; unusual stability to quantum corrections; etc)

67 Fin

68 Extra Slides

69 Calculation More general solutions

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

71 Calculation Sample solutions and so on

72 Calculation Brane-bulk boundary conditions:

73 Puzzle Resolution Non-SUSY result when F nonzero: Flat direction stabilized at: φ δt b QδΦ b = 0 b φ Vac energy there: ρ = δt b 2QδΦ b b φ

74 Puzzle Resolution SUSY result when F nonzero: Flat direction stabilized at: δt b 2QδΦ b φ δt b QδΦ b = 0 b φ

75 Puzzle Resolution SUSY result when F nonzero: Flat direction stabilized at: δt b 2QδΦ b φ δt b QδΦ b = 0 b φ Vac energy there: ρ = δt b 2QδΦ b = 1 2 φ b δt b QδΦ b φ