Good things from Brane Backreaction

Good things from Brane Backreaction Codimension-2 Backreaction as a counterexample to almost everything w Leo van Nierop

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

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

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

New Tools arxiv:0705.3212 w Hoover & Tasinato arxiv:0812.3820 w Hoover, de Rham & Tasinato arxiv:0912.3039 w Bayntun and van Nierop

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

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

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

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

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

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

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

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

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

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

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

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

de Sitter solutions hep-th/0512218 w Tolley, Hoover & Aghababaie arxiv:1109.0532 w Maharana, van Nierop, Nizami & Quevedo

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

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

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

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 usually 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

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

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

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

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

Extra-dimensional Inflationary Models hep-th/0608083 w Tolley & de Rham arxiv:1108.2553 w van Nierop or not.

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?

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

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

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

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

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 1 3 + 2λχ 1

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

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)

Naturalness Issues arxiv:1012.2638, arxiv:1101.0152 and arxiv:1108.0345 w van Nierop

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

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

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

A Simple Model 6D Einstein-Maxwell-scalar system

A Simple Model Simple solution

A Simple Model Carroll & Guica Aghababaie et al General Simple solution arguments An explicit realization Magnetic flux required to stabilize extra dimensions against gravitational collapse

A Simple Model Carroll & Guica Aghababaie et al General Simple solution arguments An explicit realization Labels flat direction (which exists due to shift symmetry or scale invariance)

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

A Simple Model Simple solution (including back-reaction)

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

A Simple Model Simple solution (non-susy case)

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

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

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:

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

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

Puzzle Resolution Linearize with F using more general solutions: cut to the chase?

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

Puzzle Resolution Sample solutions and so on

Puzzle Resolution Brane-bulk boundary conditions:

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

Puzzle Resolution SUSY result when F nonzero: Flat direction stabilized at: δt b 2QδΦ b + 1 2 φ δt b QδΦ b = 0 b φ

Puzzle Resolution SUSY result when F nonzero: Flat direction stabilized at: δt b 2QδΦ b + 1 2 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

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

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

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

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

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 so stable against brane loops

Conclusions

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

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)

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)

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