Dilaton gravity at the brane with general matter-dilaton coupling

Dilaton gravity at the brane with general matter-dilaton coupling University of Würzburg, Institute for Theoretical Physics and Astrophysics Bielefeld, 6. Kosmologietag May 5th, 2011

Outline introduction at the brane: effective Einstein-like equation inhomogeneous perfect fluid on the brane in AdS 5? conclusions

beyond General Relativity ongoing search for a unified description of gravity gauge interactions of the Standard Model string theories as the most promising proposal 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 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

scalar-tensor theories of gravity & conformal frames 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 = 1 2κ R + (coefficient: a constant) Jordan frame: e.g. L = 1 16π φ R + (coefficient: a polynomial function of the scalar field) string frame: e.g. L = e φ α 1 2 R + (coefficient: an exponential function of the dilaton) related (g µν & g µν) by a conformal (Weyl) transformation: g µν = Ω(x) 2 g µν

non-minimal matter-dilaton coupling if a matter term L m 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 which conformal frame is the natural physical frame? no clear consensus in which frame the matter-dilaton coupling should be minimal? thus: a general non-minimal coupling f (φ) L m of the dilaton to the brane matter Lagrangian choice: working in the Einstein frame

Outline introduction at the brane: effective Einstein-like equation inhomogeneous perfect fluid on the brane in AdS 5? conclusions

dilaton gravity at the brane with general matter-dilaton coupling 5d Lagrangian density: L = α 1 2 [R 2 3 σ σφ 1 3 ( φ)2] V (φ) + f (φ) L B δ B f (φ)l B δ B : brane localized term position of the co-dimension 1 brane: Dirac delta type distribution δ B cosmological constant -type term (λ) on the brane: f (φ)l B = f (φ)l m + λ(φ) (L m: matter content of the universe) induced (projected) brane metric: h µν = g µν n µn ν (covariant approach) n µ : vector field orthonormal to the brane at its position g µν: R ρσ µν & µ vs h µν: R ρσ µν & D µ assume a Z 2 symmetry for the bulk (with its fixed point at the brane position) usually imposed automatically crucial for the existence of the effective brane equations

at the brane: effective Einstein-like equation consequently, the effective Einstein-like equation at the brane reads R µν 1 2 hµνr = 8πG τµν hµνλ(φ) + f 2 (φ) π 4α 2 µν E µν 1 + 2 9 ( µφ)( νφ) 5 hµν ( φ)2 36 G = 1 f (φ)λ 48πα 2 1 τ µν = h µν L m 2 δlm δh µν, τ φ = f (φ) δlm Lm + f (φ) δφ [ ] 3f 2 4 τ φ 2 1 3 λ2 + 3 4 λ 2 + 3f 2 λ τ φ Λ = 1 V 1 2α 1 4α 2 1 π µν τ µρτν ρ + 1 3 τ τµν + 1 2 hµντσ ρ τσ ρ 1 6 hµν τ 2 (effective brane Newton s constant) bulk s influence on the brane gravity: E µν = n α h β µn γ h δ ν C αβγδ (brane localized sources) (eff. brane cosmol. const.) (bulk Weyl tensor projected on the brane) consistency condition (on the brane sources): D λ ( f (φ)τ λ µ ) = f (φ) τφ ( µφ) (brane: generalized covariant conservation of the energy-momentum tensor)

Outline introduction at the brane: effective Einstein-like equation inhomogeneous perfect fluid on the brane in AdS 5? conclusions

on the brane: spatial derivative of the energy density effective Einstein-like equation at the brane: assumptions: R µν 1 2 hµνr = 8πG τµν hµνλ(φ) + f 2 (φ) π 4α 2 µν E µν 1 + 2 9 ( µφ)( νφ) 5 hµν ( φ)2 36 bulk: anti de Sitter type spacetime: AdS 5 E µν = 0 brane: perfect fluid τ µν = ρ m t µt ν + p m γ µν calculus ingredients 4d Bianchi identity: D ν( R µν 1 2 hµνr) = 0 consistency condition: D λ ( f (φ)τ λ µ ) = f (φ) τφ ( µφ) (ρm: (dark) matter & radiation) consequently, the spatial derivative of the energy density reads ( ) [ ρ m, i = f ρ f m λ φ, f i + D ν i φ φ ] 1 φ, i D ν t φ ( νφ) α 2 1 3f 2 (ρ m+p m)

on the brane: late universe late universe approximations G const experimental limits: Ġ/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 1 s φ (where the speed of sound cs 2 = w φ) (any initial inhomogeneities of the dilaton washed out by inflation) thus G, i c 1 s Ġ, as G = G(φ) µg negligible µg = 0 λ (φ) = f (φ) f (φ) λ(φ) 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 φ 2 expected (if φ / φ 2 : currently observed φ const would be another coincidence problem)

late universe: spatial derivative of the energy density hence for the late universe we obtain ρ m, i = f ( ρm + λ ) φ,i f f spatial inhomogeneities in the matter energy density are highly constrained for the popular assumptions of AdS 5 bulk and perfect fluid on the brane inhomogeneous perfect fluid (ρ m, i 0) on the brane is allowed only if the matter on the brane non-minimally coupled to the dilaton (f 0) inhomogeneities in the matter energy density suppressed by small φ, i : φ 0 2.5 H 0 1.8 10 10 yr 1 (model-independent) bound set by current observational data analysis: deceleration parameter q 0 < 0 for [ρ 2 m ] [ρm] with Ω tot = Ω m + Ω Λ + Ω φ = 1, Ω m0 > 0.2, H 0 70 km s Mpc φ, i cs 1 φ φ 0, i 1.4 10 10 ly 1

example: maximal observationally allowed values ingredients (late universe): φ 0, i 1.4 10 10 ly 1 (max.) observations: i Ω i = 1 up to 2% dilaton gravity at the brane - modified Friedmann equation: i Ω f ρ i = 1 + Ω m m (AdS 2λ 5, perfect fluid; Ω m = ρ m /ρ c ) f ρ thus for Ω m m 2% (max.) f ρm /λ 0.14 (for Ωm = 0.28) 2λ thus spatial inhomogeneities in the energy density: ρ m, i f ( f 0.14 ρm + ρ ) m 10 9 ly 1 is it possible to adjust f /f to fit our universe?...

Outline introduction at the brane: effective Einstein-like equation inhomogeneous perfect fluid on the brane in AdS 5? 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 AdS 5 bulk if on the brane minimal matter-dilaton coupling?

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limits on the present value of φ: q analysis w φ = 5 3 w φ 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. w φ = 5 3 ) relative coeff. of ( µφ)( νφ) & h µν( φ) 2 non-standard: eff. brane Einstein-like eq., where the coeff. of h µν( φ) 2 modified during the derivation of eff. brane eqs. 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 cs 1 φ this relation can be written in a precise form (with = ) only for a specific model

final approach on the other hand we have δρ m ρ = 1 ρ m δx i = i ρ m m ρ m x i ρ δx i & m ( ) i ρ m ρ f 0.14 ρm m f ρ + 1 10 9 ly 1 m ρm, i ρ f ( ρm m f ρ + λ ) φ,i m f ρ m time/duration of the experiment a horizon, within which we do not expect bigger fluctuations (than the established limits) for q < 0, Ω i =..., [ρ 2 ] [ρ]: reasonably long cosmological past (till now)

why not to be perfectly happy with General Relativity? successful classical description of gravitational interactions passed numerous consistency and experimental tests however, not without shortcomings resistance to quantization no unification of gauge interactions and gravity no connection whatsoever to the Standard Model 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 currently observed 4 space-time dimensions vacuum Einstein equation of motion R µν 1 2 R gµν = 0 Einstein tensor: most general linear combination of rank 2 tensors fulfilling conditions for a physically viable theory of gravity symmetric in its indices dependent on the metric (g µν) and its first two derivatives only (R µν ρσ ) linear in the second derivatives of the metric divergence-free (i.e. conserved) Einstein-Hilbert action d 4 x g 1 2κ R linear in the Riemann tensor κ = M 2 P

and what if some extra dimensions existed... let s relax a bit one of the conditions for the equations of motion quasi-linearity in the second derivatives of the metric more possibilities to satisfy the conditions, but: extra dimensions needed hence: higher-dimensional space-times additional higher curvature terms can be considered introducing higher powers of the Riemann tensor into the gravity action Einstein theory of gravity generalized for a given order in the Riemann tensor contribution to the action unique (up to overall normalization) quadratic contribution: Gauss-Bonnet term L GB = R 2 4 R µνr µν + R µνρσr µνρσ generalized to higher orders by Lovelock

string theory behind the scenes? effective action obtained from string theories higher derivative corrections to the gravity interactions first correction exactly of the form of the Gauss-Bonnet term ± local field redefinitions dilaton (φ): scalar field of the gravitational sector α expansion in the string theories (effective action) α = M 2 s higher order corrections also for the dilaton interactions

where to? 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? effective equations of motion: 4-dimensional simply restricting full 5-dimensional equations? NO! 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 carried out in the COVARIANT APPROACH no specific metric tensor chosen no coordinate system imposed

effective equations of motion at the brane... what are they? brane fields: the induced metric tensor h µν and the dilaton field φ (both evaluated on the brane) brane localized sources: τ µν and τ φ 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... important for the covariant approach: the higher-dimensional analogous of the 3+1 Einstein eq. decomposition tensor equation of motion: identifying parts tangential to the brane projecting the tensor equation of motion on the brane hypersurface perpendicular to the brane projecting on the vector field n µ (twice) mixed within the tensor and the scalar equations of motion removing the bulk associated quantities

projecting: tangential & perpendicular to the brane decomposition of relevant tensors under T & T antisymmetrization: R R 2K K 4(nn) { LnK (KK ) } 8(nD) K { 2 Ln φ a e } eφ ( ) φ [ (DD) φ + K Lnφ] + (nn) +2 [ (nd) L nφ (nkd) φ ] ( φ) 2 = (Dφ) 2 + ( L nφ) 2 g µν: R µν ρσ & µ vs h µν: R µν ρσ & D µ K µν: extrinsic curvature of the brane (hypersurface orthogonal to n µ ) L n: Lie derivative along n µ a e eφ = n f ( f n e )( eφ) ( non-typical structure; not in final results) shorthand again: (nn) n n, (DD) D D, (KK ) K x K x, (nd) 1 2 (n D + n D ), (nkd) 1 2 (n K x D x + n K x D x )

effective gravitational equations at the brane... identifying problems the good h µν, R µν, (DD) µνφ, (Dφ) 2, V (φ) no work needed here, rejoice! the kind of bad K µν, L nφ can be discontinuous when crossing the brane and the slightly ugly L nk µν, L 2 n φ 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 begins

effective gravitational equations at the brane... addressing problems terms discontinuous on the brane (leading to singularities) K µν, L nφ junction (boundary) conditions for a given point x µ 0 on the brane integrating the d-dimensional equations of motion across-the-brane i.e. in the direction perpendicular to the brane and shrinking the interval: infinitesimal across-the-brane integration only some terms in the equations of motions zero explicit brane contributions τ µν & τ φ proportional to δ B terms containing second Lie derivatives: L n 2 φ and L nk µν jumps in the values of L nφ and K µν determined in terms of the brane sources τ µν and τ φ if Z 2 symmetry is assumed: brane limits of L nφ and K µν (brane located at the Z 2 symmetry fixed point)

effective gravitational equations at the brane... addressing the discontinuities problems terms singular on the brane L nk µν, L n 2 φ purely singular contributions already addressed by the junction conditions 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 E µν C abcd h a µn b h c νn d, where C abcd : bulk Weyl tensor E µν enters T µν = 0 to never leave it (so promising as effective gravitational equations on the brane) treating T µν = 0 as effective gravitational equations on the brane... single bulk associated variable E µν describes the permanent influence of bulk theory on brane-world gravity not a closed system bulk solutions essential to fully describe the gravity induced on the brane

effective gravitational equations at the brane: N max = 1, d = 5, Z 2 4-dimensional brane embedded in a 5-dimensional space-time with Z 2 symmetry how shall we do it? take the projected bulk tensor equation of motion enter the result of junction conditions analysis [ K µν ] + & [ L nφ ] + enter the solution of the brane limit of bulk equations system { (h L nk ) (KK ) } & { L n 2 φ a e eφ } & { L nk µν (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 terms depending on the dilaton field φ typical of the gravity theories involving scalar fields quadratic contributions from the brane sources τ µν and τ φ typical of the brane-world gravity theories last but not least - the bulk Weyl tensor projected on the brane: E µν = n α hµn β γ hν δ C αβγδ explicit influence of the bulk gravity solution on the brane dynamics encoded in a single quantity (no direct dependence on the bulk scalar solution) E µν: influence of the original higher-dimensional theory on the effective 4-dimensional phenomenology

limits on the present value of φ: q analysis deceleration parameter: q = ä a H 2 Friedmann eqs. (AdS 5 bulk & perfect fluid matter on the brane) read ȧ 2 a 2 + k a 2 = 8πG 3 ä a = 4πG 3 i=m,λ,φ i=m,λ,φ ρ i 4πG 3 f (φ) λ ρ2 m ρ i (1 + 3 w i ) + 8πG 3 f (φ) λ ρ2 m hence for flat universe (k = 0) the deceleration parameter reads q = 4πG 3 i=m,λ,φ ρ i (1 + 3 w i ) 8πG 3 8πG 3 i=m,λ,φ ρ i 4πG 3 f (φ) λ ρ2 m f (φ) λ ρ2 m

modified Friedmann equations: comments and complaints? part with the scalar factor only: exactly the same however, there are obviously quite relevant differences terms associated with dilaton appear, as well as mixing terms due to the introduction of the additional field interacting with graviton terms quadratic in the brane localized terms a feature of higher-dimensional theories E µν: influence of the original higher-dimensional theory on the effective 4-dimensional phenomenology

effective gravitational equations at the brane: arbitrary N max derivation procedure for the effective gravitational equations at the brane established works fine, we ve just seen the example of N = 1 & d = 5 and what about higher orders? yes, more work nevertheless, it is just the same procedure take the projected bulk tensor equation of motion enter the solution of the brane limit of bulk equations system { (h L nk ) (KK ) } & { L n 2 φ a e eφ } & { L nk µν (KK ) µν } enter the result of junction conditions analysis [ K µν ] + & [ L nφ ] + phenomenological applications? ideas most welcome!

effective gravitational equations at the brane: arbitrary N max a physically viable higher order dilaton gravity constructed derivation procedure for the effective gravitational equations at the brane established for any order corrections (as shown for N max = 1) no additional assumptions a consistency condition, usually non-dynamical bulk Z 2 symmetry a scalar effective brane equation of motion Einstein-like brane equation: explicit dependence on the bulk gravity solution Friedmann equations modified / generalized

conclusions starting point: higher order dilaton gravity natural to consider in higher-dimensional space-times physically viable equations of motion: constructed appropriate Lagrangian density: presented 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