Dilaton gravity at the brane with general matter-dilaton ...€¦ · Dilaton gravity at the brane with general matter-dilaton coupling. introductionat the brane: effective Einstein-like

introduction at the brane: effective Einstein-like equation inhomogeneous perfect fluid on the brane in AdS5? conclusions

Dilaton gravity at the brane

with general matter-dilaton coupling

Dominika Konikowska

University of Würzburg,

Institute for Theoretical Physics and Astrophysics

Bielefeld, 6. “Kosmologietag” – May 5th, 2011

Dominika Konikowska University of Würzburg, Institute for Theoretical Physics and Astrophysics

Dilaton gravity at the brane with general matter-dilaton coupling


Outline

introduction

at the brane: effective Einstein-like equation

inhomogeneous perfect fluid on the brane in AdS5?

conclusions




beyond General Relativity

I ongoing search for a unified description of

gravity

gauge interactions of the Standard Model

→ string theories as the most promising proposal

I low-energy effective action in string theories

dilaton (φ): a scalar field accompanying gravity

at the leading order (when restricted to gravity and the dilaton)

→ Einstein gravity coupled to the dilaton

I additional spatial dimensions

required by the string theories’ formulation

cannot be ‘big and standard’, have to be compactified or warped

→ dilaton gravity in a 5d brane scenario






gravity
















gravity














scalar-tensor theories of gravity & conformal frames

I dilaton gravity: a scalar-tensor theory of gravity

→ can be formulated in various conformally-related frames gravitational Lagrangians differ e.g. in the coefficient of the Ricci scalar

→ (generically) scalar field dependent coefficients

Einstein frame: L = 12κ R + · · · (coefficient: a constant)

Jordan frame: e.g. L = 116πφR + · · ·

(coefficient: a polynomial function of the scalar field)

string frame: e.g. L = e−φ α12 R + · · ·

(coefficient: an exponential function of the dilaton)

related (gµν & gµν ) by a conformal (Weyl) transformation: gµν = Ω(x)2 gµν




non-minimal matter-dilaton coupling

I if a matter term Lm is included into the Lagrangian in one frame

conformal transformation to another frame will change its coefficient

→ if constant in one frame, it will become dilaton dependent in the other

I which conformal frame is the natural physical frame?

no clear consensus

→ in which frame the matter-dilaton coupling should be minimal?

I thus: a general non-minimal coupling f (φ)Lm

of the dilaton to the brane matter Lagrangian

choice: working in the Einstein frame




Outline

introduction



conclusions




dilaton gravity at the brane with general matter-dilaton coupling

I 5d Lagrangian density:

L = α12

[R− 2

3∇σ∂σφ− 1

3 (∂φ)2]− V (φ) + f (φ)LB δB

f (φ)LB δB : brane localized term

→ position of the co-dimension 1 brane: Dirac delta type distribution δB

‘cosmological constant’-type term (λ) on the brane:

f (φ)LB = f (φ)Lm + λ(φ) (Lm : matter content of the universe)

I induced (projected) brane metric: hµν = gµν − nµnν (covariant approach)

nµ: vector field orthonormal to the brane at its position

gµν : Rµνρσ & ∇µ vs hµν : Rµνρσ & Dµ

I assume a Z2 symmetry for the bulk (with its fixed point at the brane position)

usually imposed ‘automatically’

crucial for the existence of the effective brane equations




dilaton gravity at the brane with general matter-dilaton coupling

I 5d Lagrangian density:

L = α12

[R− 2

3∇σ∂σφ− 1

3 (∂φ)2]− V (φ) + f (φ)LB δB

f (φ)LB δB : brane localized term

→ position of the co-dimension 1 brane: Dirac delta type distribution δB

‘cosmological constant’-type term (λ) on the brane:

f (φ)LB = f (φ)Lm + λ(φ) (Lm : matter content of the universe)

I induced (projected) brane metric: hµν = gµν − nµnν (covariant approach)

nµ: vector field orthonormal to the brane at its position

gµν : Rµνρσ & ∇µ vs hµν : Rµνρσ & Dµ

I assume a Z2 symmetry for the bulk (with its fixed point at the brane position)

usually imposed ‘automatically’

crucial for the existence of the effective brane equations





I consequently, the effective Einstein-like equation at the brane reads

Rµν − 12 hµνR = 8πG τµν − hµνΛ(φ) + f 2(φ)

4α21πµν − Eµν

+ 29 (∂µφ)(∂νφ)− 5

36 hµν (∂φ)2

G = −148πα2

1f (φ)λ (effective brane Newton’s constant)

τµν = hµν Lm−2 δLmδhµν , τφ = f ′(φ)

f (φ)Lm + δLm

δφ(brane localized sources)

Λ = 12α1

V − 14α2

1

[3f 2

4 τ2φ −

13λ

2 + 34λ′2 + 3f

2 λ′τφ

](eff. brane cosmol. const.)

πµν ≡ − τµρτρν + 13 τ τµν + 1

2 hµντσρ τρσ − 1

6 hµν τ2

bulk’s influence on the brane gravity: Eµν = nαhβµnγhδν Cαβγδ(bulk Weyl tensor projected on the brane)

I consistency condition (on the brane sources): Dλ(f (φ)τλµ

)= f (φ) τφ(∂µφ)

(brane: ‘generalized’ covariant conservation of the energy-momentum tensor)




Outline

introduction



conclusions




on the brane: spatial derivative of the energy density

I effective Einstein-like equation at the brane:



+ 29 (∂µφ)(∂νφ)− 5

36 hµν (∂φ)2

I assumptions:

bulk: anti de Sitter type spacetime: AdS5 → Eµν = 0

brane: perfect fluid→ τµν = ρm tµtν + pm γµν (ρm : (dark) matter & radiation)

I calculus ingredients

4d Bianchi identity: Dν(Rµν − 1

2 hµνR)

= 0

consistency condition: Dλ(f (φ)τλµ

)= f (φ) τφ(∂µφ)

I consequently, the spatial derivative of the energy density reads

ρm,i = −(

f ′f ρm − λ′

f

)φ,i +

α21

3f 2(ρm+pm)

[Dν∂iφ− φ−1φ,i Dν∂tφ

](∂νφ)




on the brane: spatial derivative of the energy density

I effective Einstein-like equation at the brane:



+ 29 (∂µφ)(∂νφ)− 5

36 hµν (∂φ)2

I assumptions:

bulk: anti de Sitter type spacetime: AdS5 → Eµν = 0

brane: perfect fluid→ τµν = ρm tµtν + pm γµν (ρm : (dark) matter & radiation)

I calculus ingredients

4d Bianchi identity: Dν(Rµν − 1

2 hµνR)

= 0

consistency condition: Dλ(f (φ)τλµ

)= f (φ) τφ(∂µφ)

I consequently, the spatial derivative of the energy density reads

ρm,i = −(

f ′f ρm − λ′

f

)φ,i +

α21

3f 2(ρm+pm)

[Dν∂iφ− φ−1φ,i Dν∂tφ

](∂νφ)




on the brane: late universe

I late universe approximationsI G ≈ const

experimental limits: G/G < 10−12 yr−1 (solar system, pulsar timing, CMB, BBN)

spatial variation of the fundamental constants

expected to be smaller than their time variation

also: typically φ,i . c−1s φ (where the speed of sound c2

s = wφ)

(any initial inhomogeneities of the dilaton washed out by inflation)

thus G,i . c−1s G, as G = G(φ)

→ ∂µG negligible

∂µG = 0 λ′(φ) = − f ′(φ)f (φ) λ(φ)

I φ ≈ const (induces variation of fundamental constants)

terms O(

(∂φ))

still treated as non-negligible

terms O(

(∂φ)D∂φ)

can be dropped, as φ φ2 expected(if φ /φ2: currently observed φ ≈ const would be another coincidence problem)




late universe: spatial derivative of the energy densityI hence for the late universe we obtain

ρm,i = − f ′f

(ρm + λ

f

)φ,i

spatial inhomogeneities in the matter energy density

are highly constrained

for the ‘popular’ assumptions of AdS5 bulk and perfect fluid on the brane

inhomogeneous perfect fluid (ρm,i 6= 0) on the brane is allowed

only if the matter on the brane non-minimally coupled to the dilaton (f ′ 6= 0)

I inhomogeneities in the matter energy density suppressed by small φ,i : φ0 . 2.5 H0 ≈ 1.8 · 10−10 yr−1

(model-independent) bound set by current observational data

analysis: deceleration parameter q0 < 0 for [ρ2m] [ρm]

with Ωtot = Ωm + ΩΛ + Ωφ = 1, Ωm0 > 0.2, H0 ≈ 70 kms·Mpc

φ,i . c−1s φ

→ φ0,i . 1.4 · 10−10 ly−1




late universe: spatial derivative of the energy densityI hence for the late universe we obtain

ρm,i = − f ′f

(ρm + λ

f

)φ,i

spatial inhomogeneities in the matter energy density

are highly constrained

for the ‘popular’ assumptions of AdS5 bulk and perfect fluid on the brane

inhomogeneous perfect fluid (ρm,i 6= 0) on the brane is allowed

only if the matter on the brane non-minimally coupled to the dilaton (f ′ 6= 0)

I inhomogeneities in the matter energy density suppressed by small φ,i : φ0 . 2.5 H0 ≈ 1.8 · 10−10 yr−1

(model-independent) bound set by current observational data

analysis: deceleration parameter q0 < 0 for [ρ2m] [ρm]

with Ωtot = Ωm + ΩΛ + Ωφ = 1, Ωm0 > 0.2, H0 ≈ 70 kms·Mpc

φ,i . c−1s φ

→ φ0,i . 1.4 · 10−10 ly−1




example: maximal observationally allowed values

I ingredients (late universe):

φ0,i ≈ 1.4 · 10−10 ly−1 (max.)

observations:∑

i Ωi = 1 up to 2%

dilaton gravity at the brane - modified Friedmann equation:∑i Ωi = 1 + Ωm

f〈ρm〉2λ (AdS5, perfect fluid; Ωm = 〈ρm〉/ρc )

thus for Ωmf〈ρm〉

2λ ≈ 2% (max.) → f 〈ρm〉/λ ≈ 0.14 (for Ωm = 0.28)

I thus spatial inhomogeneities in the energy density:

ρm,i ≈ − f ′f

(0.14 ρm + 〈ρm〉

)10−9 ly−1

→ is it possible to adjust f ′/f to fit our universe?. . .




Outline

introduction



conclusions




conclusions & outlook

dilaton gravity addressed in a 5d brane scenario

brane: non-minimal dilaton–matter coupling f (φ)

assumptions (popular in the literature):

bulk: anti de Sitter type spacetime

brane: perfect fluid (matter content of the universe)

energy density inhomogeneities constrained

non-minimal dilaton–matter coupling essential

can f ′/f be appropriately adjusted, so that

our universe’s structures can be described?

or: no pure AdS5 bulk if on the brane minimal matter-dilaton coupling?




backup & details




limits on the present value of φ: q analysis

I wφ = 53

wφ not necessarily like in the standard cosmology where w = [−1, 1]: an

effect of 5d→4d: effective 4d lagrangian ‘non-standard’;

e.o.s. depends on the effective action in 4d

different eq. of state than the standard +1 (free field) & that’s how it’s

supposed to be (i.e. wφ = 53 )

relative coeff. of (∂µφ)(∂νφ) & hµν(∂φ)2 non-standard: eff. brane

Einstein-like eq., where the coeff. of hµν(∂φ)2 modified during the derivation

of eff. brane eqs.

I limits on the present value of φ,i from φ

starting point??: φ & x · cs · φ,i , where x might depend on d , e.g. O(π±1)

then: φ,i . c−1s φ

this relation can be written in a precise form (with ‘=’) only for a specific

model




final approach

I on the other hand we haveδρm

〈ρm〉=

1〈ρm〉

∂ρm

∂xiδxi =

∂iρm

〈ρm〉δxi &

∂iρm

〈ρm〉≈ − f ′

f

(0.14

ρm

〈ρm〉+ 1)

10−9 ly−1

ρm,i〈ρm〉

≈ − f ′f

( ρm〈ρm〉

+ λf〈ρm〉

)φ,i

I time/duration of the experiment→ a horizon, within which we do notexpect bigger fluctuations (than the established limits)

for q < 0, Ωi = . . ., [ρ2] [ρ]: reasonably long cosmological past (till now)




why not to be perfectly happy with General Relativity?

I successful classical description of gravitational interactions

I passed numerous consistency and experimental tests

I however, not without shortcomingsI resistance to quantizationI no unification of gauge interactions and gravity

no connection whatsoever to the Standard Model

I an extended theory of gravity is searched for

→ with General Relativity as the main part of its structure




General Relativity: making the long story short

I currently observed 4 space-time dimensions

I vacuum Einstein equation of motion

Rµν − 12 R gµν = 0

I Einstein tensor: most general linear combination of rank 2 tensorsfulfilling conditions for a physically viable theory of gravity

I symmetric in its indicesI dependent on the metric (gµν ) and its first two derivatives only (Rµν

ρσ)I linear in the second derivatives of the metricI divergence-free (i.e. conserved)

I Einstein-Hilbert action ∫d4x√−g

12κR

→ linear in the Riemann tensorI κ = M−2

P




and what if ‘some’ extra dimensions existed . . .

I let’s relax a bit one of the conditions for the equations of motionI quasi-linearity in the second derivatives of the metric

more possibilities to satisfy the conditions, but: extra dimensions needed

I hence: higher-dimensional space-times

additional higher curvature terms can be considered

I introducing higher powers of the Riemann tensor into the gravity action

Einstein theory of gravity generalized

I for a given order in the Riemann tensor→ contribution to the action unique (up to overall normalization)

I quadratic contribution: Gauss-Bonnet term

LGB = R2 − 4RµνRµν +RµνρσRµνρσ

I generalized to higher orders by Lovelock




string theory behind the scenes?

I effective action obtained from string theories

higher derivative corrections to the gravity interactions

I first correction exactly of the form of the Gauss-Bonnet term

± local field redefinitions

I dilaton (φ): scalar field of the gravitational sector

I α′ expansion in the string theories (effective action)I α′ = M−2

s

higher order corrections also for the dilaton interactions




where to?

I brane-world ideology: Standard Model localized on a brane

→ embedded in higher(5-)dimensional space-time

how will the induced gravity look like at the brane?

I effective equations of motion: 4-dimensional

simply restricting full 5-dimensional equations? NO!

I certain quantities contributing to 5d equations equations of motion . . .

can be singular or discontinuous on the brane

non-trivial derivation of the effective equations at the brane

I carried out in the COVARIANT APPROACH

I no specific metric tensor chosenI no coordinate system imposed




effective equations of motion at the brane . . . what are they?

I brane fields:

→ the induced metric tensor hµν and the dilaton field φ

(both evaluated on the brane)

I brane localized sources: τµν and τφ

I effective brane equations of motion

should relate the dynamics of the brane fields to the brane sources

optimally, should be independent of the bulk solution




effective gravitational equations at the brane . . .

I important for the covariant approach:

the higher-dimensional analogous of the 3+1 Einstein eq. decomposition

I tensor equation of motion: identifying partsI tangential to the brane

→ projecting the tensor equation of motion on the brane hypersurfaceI perpendicular to the brane

→ projecting on the vector field nµ (twice)I mixed

I within the tensor and the scalar equations of motion

→ ‘removing’ the bulk associated quantities




projecting: tangential & perpendicular to the brane

I decomposition of relevant tensors under T & T antisymmetrization:

R∗∗∗∗ → R∗∗∗∗ − 2K∗∗ K∗∗ − 4(nn)∗∗

–LnK∗∗ − (KK )∗∗− 8(nD)∗∗K

∗∗

(∇∇)∗∗φ →[(DD)∗∗φ+ K∗∗ –Lnφ

]+ (nn)∗∗

–Ln

2φ− ae∇eφ

+2[(nD)∗∗–Lnφ− (nKD)∗∗φ

](∂φ)2 = (Dφ)2 + (–Lnφ)2

I gµν : Rµνρσ & ∇µ vs hµν : Rµνρσ & DµI Kµν : extrinsic curvature of the brane (hypersurface orthogonal to nµ)I –Ln: Lie derivative along nµ

I ae∇eφ = nf (∇f ne)(∇eφ) (‘non-typical’ structure; not in final results)

I shorthand again: (nn)∗∗ ≡ n∗n∗, (DD)∗∗ ≡ D∗D∗, (KK )∗∗ ≡ K∗x K x∗ ,

(nD)∗∗ ≡ 12 (n∗D∗ + n∗D∗), (nKD)∗∗ ≡ 1

2 (n∗K∗x Dx + n∗K x∗Dx )




effective gravitational equations at the brane . . . identifying problems

I the good

→ hµν , Rµν , (DD)µνφ, (Dφ)2, V (φ)

no work needed here, rejoice!

I the kind of bad

→ Kµν , –Lnφ

can be discontinuous when ‘crossing’ the brane

I and the slightly ugly

→ –LnKµν , –Ln2φ

can be singular on the brane

can have a finite contribution as well

all this information has to be properly taken into account

& the quest beginsDominika Konikowska University of Würzburg, Institute for Theoretical Physics and Astrophysics



effective gravitational equations at the brane . . . addressing problems

I terms discontinuous on the brane (leading to singularities)

→ Kµν , –Lnφ

junction (boundary) conditions for a given point xµ0 on the braneI integrating the d-dimensional equations of motion ‘across-the-brane’

i.e. in the direction perpendicular to the braneI and shrinking the interval: ‘infinitesimal across-the-brane integration’

I only some terms in the equations of motions 9 zeroI explicit brane contributions τµν & τφ proportional to δB

I terms containing second Lie derivatives: –Ln2φ and –LnKµν

jumps in the values of –Lnφ and Kµν determined→in terms of the brane sources τµν and τφ

I if Z2 symmetry is assumed: brane limits of –Lnφ and Kµν(brane located at the Z2 symmetry fixed point)




effective gravitational equations at the brane . . .

addressing the discontinuities’ problems

I terms singular on the brane→ –LnKµν , –Ln

2φ

I purely singular contributions already addressed by the junction conditionsI the smooth part has to be determined as well

(yields a finite contribution to the effective equations)

“brane limit of bulk equations system”




addressing the discontinuities’ problem: bulk equations system

I Eµν ≡ Cabcd haµnbhc

νnd , where Cabcd : bulk Weyl tensor

Eµν enters T ‖‖µν = 0

(so promising as effective gravitational equations on the brane)

→ to never leave it

I treating T ‖‖µν = 0 as effective gravitational equations on the brane . . .I single bulk associated variable Eµν describes the permanent influence of bulk theory on brane-world gravity

I not a closed system

bulk solutions essential to fully describe the gravity induced on the brane




effective gravitational equations at the brane: Nmax = 1, d = 5, Z2

I 4-dimensional brane

embedded in a 5-dimensional space-time with Z2 symmetry

how shall we do it?

I take the projected bulk tensor equation of motion

I enter the result of junction conditions analysis

[Kµν

]+

&[–Lnφ

]+

I enter the solution of the brane limit of bulk equations system

(h–LnK )− (KK )

&

–Ln

2φ− ae∇eφ

&

–LnKµν − (KK )µν




as for the technical details and tools...

I will gladly provide them whenever needed




effective brane equation

new terms appear as comparing to the standard Einstein equation

I terms depending on the dilaton field φ

typical of the gravity theories involving scalar fields

I quadratic contributions from the brane sources τµν and τφ

typical of the brane-world gravity theories

I last but not least - the bulk Weyl tensor projected on the brane:

Eµν = nαhβµnγhδν Cαβγδ

explicit influence of the bulk gravity solution on the brane dynamics

encoded in a single quantity

(no direct dependence on the bulk scalar solution)

Eµν : influence of the original higher-dimensional theory

on the effective 4-dimensional phenomenology




limits on the present value of φ: q analysis

I deceleration parameter: q = − aa H−2

I Friedmann eqs. (AdS5 bulk & perfect fluid matter on the brane) read

a2

a2+

ka2

=8πG

3

∑i=m,Λ,φ

ρi −4πG

3f (φ)

λρ2

m

aa

= −4πG

3

∑i=m,Λ,φ

ρi (1 + 3 wi ) +8πG

3f (φ)

λρ2

m

I hence for flat universe (k = 0) the deceleration parameter reads

q =

4πG3∑

i=m,Λ,φ ρi (1 + 3 wi )− 8πG3

f (φ)

λρ2

m

8πG3∑

i=m,Λ,φ ρi − 4πG3

f (φ)

λρ2

m




modified Friedmann equations: comments and complaints?

I part with the scalar factor only: exactly the same

I however, there are obviously quite relevant differencesI terms associated with dilaton appear, as well as mixing terms

due to the introduction of the additional field interacting with gravitonI terms quadratic in the brane localized terms

a ‘feature’ of higher-dimensional theoriesI Eµν : influence of the original higher-dimensional theory

on the effective 4-dimensional phenomenology




effective gravitational equations at the brane: arbitrary Nmax

I derivation procedure for the effective gravitational equations at the brane

→ established

works fine, we’ve just seen the example of N = 1 & d = 5

I and what about higher orders?

yes, more work→nevertheless, it is just the same procedureI take the projected bulk tensor equation of motionI enter the solution of the brane limit of bulk equations system

(h–LnK )− (KK )

&

–Ln2φ− ae∇eφ

&

–LnKµν − (KK )µν

I enter the result of junction conditions analysis

[Kµν

]+

&[–Lnφ

]+

I phenomenological applications?

ideas most welcome!




effective gravitational equations at the brane: arbitrary Nmax

I a physically viable higher order dilaton gravity

→ constructed

I derivation procedure for the effective gravitational equations at the brane

→ established

I for any order corrections (as shown for Nmax = 1)I no additional assumptions

a consistency condition, usually non-dynamicalI bulk Z2 symmetry

a scalar effective brane equation of motionEinstein-like brane equation: explicit dependence on the bulk gravity solution

I Friedmann equations modified / generalized




conclusions

I starting point: higher order dilaton gravity

→ natural to consider in higher-dimensional space-times

physically viable equations of motion: constructed

appropriate Lagrangian density: presented

I effective gravitational equations on the brane (co-dimension 1)

→ derivation procedure for arbitrary order N: established

details and results presented explicitly for N = 1 & d = 5

→ together with a cosmological example

modified Friedmann equations on 4d brane



Documents

Dilaton gravity at the brane with general matter-dilaton ...€¦ · Dilaton gravity at the brane with general matter-dilaton coupling. introductionat the brane: effective Einstein-like