Dynamical Domain Wall and Localization

Transcription

1 Dynamical Domain Wall and Localization Shin ichi Nojiri Department of Physics & Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya Univ. Typeset by FoilTEX 1

2 Based on Y. Toyozato, K. Bamba and S. Nojiri, Scalar Domain Wall as the Universe, Phys. Rev. D 87 (2013) 6, [arxiv: [hep-th]]. M. Higuchi and S. Nojiri, Reconstruction of Domain Wall Universe and Localization of Gravity, Gen. Rel. Grav. 46 (2014) 11, 1822 [arxiv: [hep-th]]. Y. Toyozato, M. Higuchi and S. Nojiri, Dynamical Domain Wall and Localization, arxiv: Construction of the dynamical domain wall universe, where the four dimensional FRW universe is realized on the domain wall in the five dimensional space-time. Localization of the fields (graviton, chiral spinor, vector,...) Typeset by FoilTEX 2

3 Introduction Our universe could be a brane or domain wall embedded in a higher dimensional space-time. K. Akama, An Early Proposal of Brane World, Lect. Notes Phys. 176 (1982) 267 [hep-th/ ]. V. A. Rubakov and M. E. Shaposhnikov, Do We Live Inside a Domain Wall?, Phys. Lett. B 125 (1983) 136. D-brane in string theories. J. Dai, R. G. Leigh and J. Polchinski, New Connections Between String Theories, Mod. Phys. Lett. A 4 (1989) J. Polchinski, Tasi lectures on D-branes, hep-th/ Typeset by FoilTEX 3

4 Brane world scenario. L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension, Phys. Rev. Lett. 83, 3370 (1999) [hepph/ ]. L. Randall, R. Sundrum, An Alternative to compactification, Phys. Rev. Lett. 83, (1999). [hep-th/ ]. Dynamics of brane? Brane: a limit where the thickness of domain wall vanishes Dynamics of domain wall Typeset by FoilTEX 4

5 Domain wall model with two scalar fields General FRW in the five dimensional space-time, the metric is given by ds 2 =dw 2 + L 2 e u(w,t) ds 2 FRW. { } ds 2 FRW = dt 2 + a (t) 2 dr 2 1 kr 2 + r2 dθ 2 + r 2 sin 2 θdϕ 2. Action with two scalar fields ϕ and χ: S ϕχ = d 5 x { R g 2κ A(ϕ, χ) µϕ µ ϕ B(ϕ, χ) µ ϕ µ χ 1 } 2 C(ϕ, χ) µχ µ χ V (ϕ, χ). Typeset by FoilTEX 5

6 Energy-momentum tensor T ϕχ µν =g µν { 1 2 A(ϕ, χ) ρϕ ρ ϕ B(ϕ, χ) ρ ϕ ρ χ 1 } 2 C(ϕ, χ) ρχ ρ χ V (ϕ, χ) + A(ϕ, χ) µ ϕ ν ϕ + B(ϕ, χ) ( µ ϕ ν χ + ν ϕ µ χ) + C(ϕ, χ) µ χ ν χ. Field equations read 0 = 1 2 A ϕ µ ϕ µ ϕ + A µ µ ϕ + A χ µ ϕ µ χ + ( B χ 1 ) 2 C ϕ µ χ µ χ + B µ µ χ V ϕ, 0 = ( 12 ) A χ + B ϕ µ ϕ µ ϕ + B µ µ ϕ C χ µ χ µ χ + C µ µ χ + C ϕ µ ϕ µ χ V χ. A ϕ = A(ϕ, χ)/ ϕ, etc. We now choose ϕ = t and χ = w. Typeset by FoilTEX 6

7 We can construct a model to realize the arbitrary metric in the form, ds 2 = dw 2 + L 2 e u(w,t) ds { 2 FRW, ds 2 FRW = dt2 + a (t) 2 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ }, 2 1 kr 2 by choosing (H (1/a) (da/dt)) A = 1 ( ) 2k ( u)2 κ 2 ü 2Ḣ + a2 2 + uh B = 3u 2κ 2 L 2 ( u + 2H), eu C = 1 κ 2 ( 3 2 u + V = 1 ( κ 2 3 ( u + 2 4, 2k L 2 e u a 2 1 ( 2e u ü + 2Ḣ + ( u)2 + 5 uh + 6H 2)), (u ) 2 ) + 3k L 2 e u a L 2 e u ( 3ü + 6Ḣ + 3 ( u) u + 18H 2)). Einstein equation is satisfied. Field equations are nothing but the Bianchi identities. Typeset by FoilTEX 7

8 If any eigenvalue of the matrix ( A a ghost. Example without ghost B B C ) is negative, there appears a(t) t h 0, e u(w,t) = W (w) T (t), T (t) = T 1 t 1 3h 0 + T 2 t 2h 0, A = 3 2κ 2 W (w) = e 1+ w2 w 2 0, ( T (t) T (t) + 2h ) 2 0 > 0, B = 0, C = 3 t 2κ 2 1 w 0 2 ( 1 + w2 w 2 0 )3 2 > 0. No ghost. General FRW universe can be realized by the Brans-Dicke type model. Typeset by FoilTEX 8

9 Localization of graviton Perturbation g µν g µν + h µν. Assume h µν = 0 if µ or ν 1, 2, 3. Einstein eq. equation for graviton in five dimensions: ( 0 = [ 2w u u 2 + L 2 e u ü + ȧ u a + u t + 2ä a + ȧ a t t 2 + )] a 2 h ij. By assuming h ij (w, x) = e u(w,t) ĥ ij (x), 0 = ( 2ȧ u a u t + 2ä a + ȧ a t t 2 + ) a 2 ĥ ij. If u goes to minus infinity sufficiently rapidly for large w, h ij (w, x) is normalized in the direction of w and therefore there occurs the localization of graviton. Typeset by FoilTEX 9

10 Graviton localized on the domain wall 0 = ( 2ȧ u a u t + 2ä a + ȧ a t t 2 + ) a 2 ĥ ij. In the four dimensional FRW space-time, 0 = ( 2ä a + ȧ a t t 2 + ) a 2 h ij. If u ( 2ȧ a t) ĥij = 0, two expressions coincide with each other. If u 0, there could appear some corrections proportional to u. Typeset by FoilTEX 10

11 Localization of chiral spinor There was an attempt to localize the fermion on the domain wall has been failed. D. P. George, M. Trodden and R. R. Volkas, Extra-dimensional cosmology with domain-wall branes, JHEP 0902 (2009) 035 [arxiv: [hep-ph]]. Dirac equation in five dimensions Γ M M Ψ + fχ (w) Ψ = 0. fχ: a function of the scalar field χ = w, general yukawa interaction. M : covariant derivative: M def = M ω ABMΓ AB, Γ AB def = 1 2 [ Γ A, Γ B]. Typeset by FoilTEX 11

12 Ψ = η (t, w) ψ (x) (ψ: four dimensional Dirac spinor) Γ M M Ψ + fχ (w) Ψ { =L 1 e u/2 Γˆ0 0 ψ + 1 } a (t) Γî i ψ η + { 5 η + u (t, w) η} Γˆ5 ψ + fχ (w) ηψ + L 1 e u/2 { 0 η Let C 1 (w) and D 1 (t) be arbitrary functions. ( 1 2 u + ȧ ) } η Γˆ0 ψ. a η (t, w) def = ζ (t) λ (w) g (t, w) g (t, w) def = exp [ C1 (w) + D 1 (t) u (t, w) ], Typeset by FoilTEX 12

13 By assuming Γˆ5 ψ = ±ψ, Γ M M Ψ + fχ (w) Ψ { =L 1 e u/2 Γˆ0 0 ψ + 1 } a (t) Γî i ψ η { ± ζg 5 λ (w) + C 1 (w) λ (w) ± fχ } (w) λ (w) ψ { ( 3 + L 1 e u/2 ȧ λg 0 ζ (t) + 2 a 1 ) 4 u (t, w) + Ḋ1 (t) e u(t,w) = T (t) W (w) and using the Dirac equation, λ (w) = C 3 exp [ C 1 (w) f ζ (t) = C 4 a 3/2 T 1/4 exp [ D 1 (t)]. ] dw χ (w) } ζ (t) Γˆ0 ψ. Typeset by FoilTEX 13

14 η (t, w) =C 3 C 4 a 3/2 T 1/4 exp =C 5 a 3/2 T 3/4 W 1 exp [ u (t, w) f [ f ] dw χ (w) ] dw χ (w), χ (w) = w η (t, w) = C 5 a 3/2 T 3/4 W 1 exp [ f ] 2 w2. Condition of localization I = + + dw e 3u/2 η 2 = C5a 2 3 dw W 1/2 exp [ fw 2] <. Typeset by FoilTEX 14

15 Fermion can be localized on the domain wall and the localized fermion can be chiral or anti-chiral. In D. P. George, M. Trodden and R. R. Volkas, Extra-dimensional cosmology with domain-wall branes, JHEP 0902 (2009) 035 [arxiv: [hep-ph]] It was assumed that the warp factor does not depend on the time but we have used the time-dependent warp factor. Typeset by FoilTEX 15

16 Localization of vector field Action of vector field S V = d 5 x g { 14 F MNF MN 12 m(χ)2 A M A M }, F MN = M A N N A M. Background ds 2 =dw 2 + L 2 W (w)t (t)ds 2 FRW, ds 2 FRW = dt 2 + a(t) 2 3 ( dx i ) 2. i=1 Typeset by FoilTEX 16

17 S V = d 5 x { 1 2 L2 W (w)t (t)a(t) 3 F L2 W (w)t (t)a(t)f 2 5i a(t)f 2 0i 1 4 a(t) 1 F 2 ij 1 2 m(χ)2 ( L 4 W (w) 2 T (t) 2 a(t) 3 A 2 5 L 2 W (w)t (t)a(t) 3 A L 2 W (w)t (t)a(t)a 2 i )}. Typeset by FoilTEX 17

18 Variation of A 5, A 0, and A i 0 =L 2 ( W (w) 0 T (t)a(t) 3 ( 5 A 0 0 A 5 ) ) L 2 W (w)t (t)a(t) ( 5 i A i i 2 ) A 5 m(χ) 2 L 4 W (w) 2 T (t) 2 a(t) 3 A 5, 0 = L 2 T (t)a(t) 3 5 (W (w) ( 5 A 0 0 A 5 )) + a(t) ( 0 i A i i 2 ) A 0 + m(χ) 2 L 2 W (w)t (t)a(t) 3 A 0, 0 =L 2 T (t)a(t) w (W (w) ( 5 A i i A 5 )) 0 (a ( 0 A i i A 0 )) a(t) 1 ( i j A j j 2 ) A i m(χ) 2 L 2 W (w)t (t)a(t)a i. Typeset by FoilTEX 18

19 Assuming A 5 = 0, A µ = X(w)C µ (x ν ), µ, ν = 0, 1, 2, 3, and choosing m (χ = w) 2 = (W (w)x (w)) W (w)x(w) 0 = 5 X(w) { 0 ( T (t)a(t) 3 C 0 ) T (t)a(t) i C i }, 0 = 0 i C i 2 i C 0, 0 = 0 (a(t) ( 0 C i i C 0 )) + a(t) 1 ( i j C j 2 j C i ). Last two eqs. : nothing but the field equations of the vector field in four dimensions. First eq. : a gauge condition, which is a generalization of the Landau gauge, µ A µ = 0., Typeset by FoilTEX 19

20 If X(w) 0 sufficiently when w, A µ : normalizable. Localization of vector field on the domain wall. In case non-abelian gauge theory? (In general, non-normalizable) (W (w)x (w)) = 0 m(χ) = 0. Typeset by FoilTEX 20

21 Summary We have constructed the dynamical domain wall, where arbitrary four dimensional FRW universe is realized on the domain wall in the five dimensional space-time. Graviton, Chiral Spinor, and Vector can localize on the brane. Typeset by FoilTEX 21

22 Graviton in FRW Universe in Four Dimensions Perturbation g µν g µν + h µν δr µν = 1 2 [ µ ρ h νρ + ν ρ h µρ 2 h µν µ ν ( g ρλ h ρλ ) 2R λ ρ ν µh λρ + R ρ µh µν + R ρ νh ρµ ], δr = h µν R µν + µ ν h µν 2 (g µν h µν ). Imposing the gauge condition µ h µν = g µν h µν = 0 Einstein equation R µν 1 2 g µνr = κ 2 T µν 1 2 [ 2 h µν 2R λ ρ ν µh λρ + R ρ µh µν + R ρ νh ρµ h µν R + g µν R ρλ h ρλ ] = κ 2 δt µν. Typeset by FoilTEX 22

23 By using the formulation in S. Nojiri and S. D. Odintsov, Unifying phantom inflation with latetime acceleration: Scalar phantom-non-phantom transition model and generalized holographic dark energy, Gen. Rel. Grav. 38 (2006) 1285 [hep-th/ ]. consider scalar field theory S ϕ = d 4 x gl ϕ, L ϕ = 1 2 ω(ϕ) µϕ µ ϕ V (ϕ), T µν = ω(ϕ) µ ϕ ν ϕ + g µν L ϕ, δt µν = h µν L ϕ g µνω(ϕ) ρ ϕ λ ϕh ρλ. Typeset by FoilTEX 23

24 Because we are now interested in the graviton, we may assume h µν = 0 except the components with µ, ν = 1, 2, 3. In the FRW universe (k = 0), assume ϕ = t, FRW equations 3 ȧ 2 κ 2 a 2 =ω 2 + V, 1 κ 2 ω = 1 κ 2 ( 2ä 0 = ( ) 2ä a + ȧ2 a 2 ) = ω 2 V, 4ȧ2 a + a 2 ( 2ä a + ȧ a t t 2 + a 2, V = 1 κ 2 (ä ) h ij. ) a + ȧ2 a 2. Typeset by FoilTEX 24

25 Explicit expressions of connections and curvatures in five dimensions Metric g AB = L 2 e u(w,t) L 2 e u(w,t) a(t) 2 1 kr 2 L 2 e u(w,t) a(t) 2 r 2 Connections Γ t tt = 1 2 u, Γw tt = 1 2 L2 e u u, Γ r rt = Γθ θt = Γϕ ϕt = ȧ Γ t ij = L 2 e u (ȧ a + 1 ) 2 u L 2 e u(w,t) a(t) 2 r 2 sin 2 θ. 1 a u, Γt tw = Γr rw = Γθ θw = Γϕ ϕw = 1 2 u, g ij, Γ r rr = kr 1 kr 2, Γw ij = 1 2 u g ij, Γ θ θr = Γϕ ϕr = 1 r, Γ r θθ = r(1 kr2 ), Γ ϕ ϕθ = cot θ, Γr ϕϕ = r(1 kr2 ) sin 2 θ, Γ θ ϕϕ = cos θ sin θ. Typeset by FoilTEX 25

26 Ricci curvatures R tt = R ij = [ 12 u u ( L 2 e u ü + ȧ u )] a + 2ä g tt, a [ 12 u u ( L 2 e u ü + 5ȧ u 4ȧ2 a + 2ä a + a 2 + u2 + 4 k )] a 2 R ww = 2u u 2 R tw = 3 2 u. g ij, Scalar curvature ( R = 4u 5u 2 + 3L 2 e u ü u2 + 3ȧ u 2ȧ2 a + 2ä a + a k ) a 2. Typeset by FoilTEX 26

27 Explicit form of spin connection Vierbein field e ˆB, g MN = def = eâm η Â ˆB e ˆB N eâm = Le u/ Le u/2 a Le u/2 a Le u/2 a eâm def = ηâ ˆB e ˆB M = Le u/ Le u/2 a Le u/2 a Le u/2 a 0, Typeset by FoilTEX 27

28 e M Â = def = eân g MN L 1 e u/ L 1 e u/2 a L 1 e u/2 a L 1 e u/2 a Ricci rotation coefficients and spin connections def Ω ABM = ( ) A eĉb B eĉa eĉm = Ω BAM, ω ABM def = 1 2 (Ω ABM Ω BMA Ω MAB ) = ω BAM. Typeset by FoilTEX 28

29 Non-vanishing components Ω 050 = Ω 500 = 1 2 u L 2 e u ( ) 1 Ω 0ij = Ω i0j = 2 ua2 + ȧa L 2 e u δ ij, Ω 5ij = Ω i5j = 1 2 u L 2 e u a 2 δ ij, ω 050 = ω 500 = 1 2 u L 2 e u, ( ) 1 ω 0ij = ω i0j = 2 ua2 + ȧa L 2 e u δ ij, ω 5ij = ω i5j = 1 2 u L 2 e u a 2 δ ij. Typeset by FoilTEX 29