Gravitational perturbations on branes

Transcription

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2 Content 1. Introduction and Motivation 2. Braneworld solutions in various gravities 2.1 General relativity 2.2 Scalar-tensor gravity 2.3 f(r) gravity 3. Gravitational perturbation on branes 4. The effective gravitational potential 5. Summary and further problems

3 1. Introduction Extra dimensions: Kaluza-Klein (KK) theory String/M theory

4 1. Introduction Infinite extra dimensions: Domain wall (DW) scenario [Akama, Rubakov, Shaposhnikov, 1983] ds 2 = η µν dx µ dx ν +dy 2 Our 4D world is a DW embedded in 5D flat space-time Generated by a scalar field: φ(y) = v 0 tanh(ky) Fermions can be localized on the DW by Yukawa coupling ηφ ΨΨ Newton s law can not be recovered on the DW U(r) 1 r 2

5 1. Introduction Thin braneworld scenario [Randall and Sundrum (RS), 1999] ds 2 = e 2k y η µν dx µ dx ν +dy 2 Our 4D world is a brane embedded in a 5D space-time Fermions can be localized on brane by mass term : ηmǫ(y) ΨΨ Newton s law can be recovered( on brane: ) m U(r) = G 1 m 2 N r 1+ 1 k 2 r 2 The energy density: ρ(y) σδ(y) Warp factor 2 A Energy density Ρ y Figure: Thin brane y

6 1. Introduction Thick braneworld scenario (Domain Wall) ds 2 = e 2A(y) ĝ µν (x) dx µ dx ν +dy 2 Infinite but warped extra dimension Braneworlds are generated by scalar fields, e.g. φ(y) = v 0 tanh(ky) Fermions can be localized on the brane by Yukawa coupling ηφ ΨΨ for odd φ(y) Warp factor 2 A Energy density Ρ y y

7 1. Motivation Motivation: Are thick braneworld systems stable especially in higher-derivative gravity theories? Can Newton s law be recovered on the brane? What s the correction to Newton s law? To answer these questions, we would like to Construct thick braneworld models in various gravity theories. Study gravitational perturbations of the brane systems. Calculate the effective gravitational potential.

8 Content 1. Introduction and Motivation 2. Braneworld solutions in various gravities 2.1 General relativity 2.2 Scalar-tensor gravity 2.3 f(r) gravity 3. Gravitational perturbation on branes 4. The effective gravitational potential 5. Summary and further problems

9 2. Braneworld solutions in various gravities The action describing a braneworld system is given by S = S Gravity +S Matter. (1) The metric of the background space-time is assumed as ds 2 = e 2A(y) ĝ µν (x) dx µ dx ν +dy 2 [ (2) = e 2A(z) ĝ µν (x) dx µ dx ν +dz 2]. (3) Warp factor 2 A Energy density Ρ y y

10 2.1a ds brane in general relativity For the 5D action S = d 5 dx g [ 1 2 R 1 ] 2 gmn M φ N φ V(φ) (4) with the potential V(φ) = V 0 (cos φ φ 0 ) 2(1 δ), (5) a ds domain wall was found in [JMP31(1990)2683],[PRD60(1999)065011],[Wang, PRD66(2002)024024]: ds 2 = e 2A(z)( dt 2 +e 2Ht dx i dx i +dz 2), (6) e 2A(z) = sech 2δ( Hz ), (0 < δ < 1,H > 0) (7) δ ( φ(z) = φ 0 arctan sinh Hz ). (8) δ

11 2.1b AdS brane in general relativity For the 5D action S = d 5 dx g [ 1 2 R 1 ] 2 gmn M φ N φ V(φ) (9) with the potential V(φ) = 3(1+3δ)H2 2δ ( cosh φ φ 0 ) 2(1 δ) (10) an AdS domain wall was found in [Wang, PRD66(2002)024024]: ds 2 = e 2A(z)[ e 2Hx 3 ( dt 2 +dx 2 1 +dx 2 2)+dx 2 3 +dz 2],(11) e 2A(z) = cos ( Hz/δ ) 2δ,(δ > 1, or δ < 0) (12) φ(z) = φ 0 arcsinh ( tan(hz/δ) ). (13)

12 2.1c Flat brane in general relativity For the 5D action S = d 5 x [ 1 g 2κ 2 R ( φ)2 1 ] 2 ( π)2 e 2 b/3 π V(φ,π) (14) with the potential V(φ) = 1 2 ( W ( ) W(φ) = vaφ 1 φ2, 3v 2 φ )2 4 b 6 W2, (15) a flat domain wall was given in [Fu, Liu and Guo, PRD84(2011)044036]: ds 2 = e 2A(z) (η µν dx µ dx ν +dz 2 ), (16) φ(z) = v tanh(az), π(z) = 3 A(z), (17) A(z) = v2 ( lncosh 2 (az)+ 1 ) 9 2 tanh2 (az), (18) Note that the physical length of the extra dimension is finite, although < z <.

13 2.1d Flat brane in general relativity For the φ 6 potential V(φ) = v 0 +g 1 φ 2 g 2 φ 4 +g 3 φ 6, we get the numerical solution of deformed flat domain walls [Fu, Liu and Guo, PRD85(2012)084023]: The warp factor e 2A(z) The scalar field φ(z) g = g 1 = 2.6 g 1 = m 10 e 2A(z) π(z) z The dilaton field π(z) g 1 = 2 g 1 = 2.6 g 1 = m φ(z) T 00 (z) g 1 = 2 g 1 = 2.6 g 1 = z The energy density T 00 (z) g 1 = 2 g 1 = 2.6 g 1 = m z m z 5 10

14 2.2 Scalar-tensor brane Consider the action in scalar-tensor gravity S 5 = d 5 x ( 1 g 2 F(φ)R 1 ) 2 ( φ)2 V(φ) (19) Solution 1: For F(φ) = 1 αφ 2 and V = aφ 2 +bφ 2 +c, the scalar-tensor brane solution is given by [C. Bogdanos, et al, Phys. Rev. D74 (2006) ] e A(y) = sech 6 α 1 (ky), (20) φ = φ 0 tanh(ky) = 3(1 6α) tanh(ky). α(1 2α) (21) (22)

15 2.2 Scalar-tensor brane Solution 2: For F(φ) = φ (1 φ2 0 3 )cosh( 3φ φ 0 ) and V(φ) = 1 2 φ2 0κ 2 cos 2 φ + 4κ2 φ 0 3 ( ) 3 φ 2 2φ 3φ 0 sin sinh 3 φ 0 φ 0 ) 3φ κ (2φ (6 2φ 2 0)cosh sin 2 φ. (23) φ 0 φ 0 the scalar-tensor brane solution is [Chen, Liu, et al, JHEP 1205 (2012) 108] e A(y) = sech(ky), (24) φ = φ 0 φ 0 arctan(sinh(ky)). (25) (26)

16 2.3 f(r)-brane Now we consider the 5D action in f(r) gravity: S = d 5 x ( 1 g 2κ 2 f(r) 1 ) 5 2 M φ M φ V(φ). (27) Solution 1: For f(r) = R +γr 2, (28) V(φ) = λ(φ 2 v 2 ) 2 +Λ 5, (29) the flat f(r)-brane solution is [Liu, Zhong, Zhao and Li, JHEP1106(2011)135]: ds 2 = e 2A(y) η µν dx µ dx ν +dy 2, (30) e 2A(y) = sech 2 (ky), (31) φ(y) = v tanh(ky). (32)

17 2.3 f(r)-brane Solution 2: For [ f(r) = R +α 2(R +6k 2 )cos(h(r)) ] + 3(R 8k 2 )(R +20k 2 ) sin(h(r)), (33) where H(R) = V(φ) = 3k2 [ ( 8 5cos 8 3 ( R 8k 2 + R+20k 2 2 7c ] 3, (34) φ ) φ 0 ), the solution is e 2A(y) = sech 2 (ky), (35) φ(y) = [ ( ky ) ] 6φ 0 arctan tanh. (36) 2 We have set κ 2 5 = 2.

18 2.3 f(r)-brane Solution 3: For the same f(r) (33) but the following potential: the solution is V(φ) = k2 [(3φ 2 +χ 2 9v 2 /2) 2 +8φ 2 χ 2] 12 4k2 [ v 2 3 φ φ 2 χ 2] 2, (37) 6 e 2A(y) = sech 2 (ky), (38) 3 φ(y) = v tanh(ky), (39) 2 3 χ(y) = vsech(ky). (40) 2

19 Content 1. Introduction and Motivation 2. Braneworld solutions in various gravities 2.1 General relativity 2.2 Scalar-tensor gravity 2.3 f(r) gravity 3. Gravitational perturbation on branes 4. The effective gravitational potential 5. Summary and further problems

20 3. Gravitational perturbation on branes Next, we consider the linear perturbations of the brane solutions. δg MN = e 2A(z) h MN, δφ i, (41) Make the following decompositions: h µz = µ F +G µ, (42) h µν = η µν P + µ ν Q +2 (µ C ν) + h µν, (43) where P, Q, F, and h 55 are scalar perturbations, C µ,g µ are transverse vector perturbations: µ C µ = 0 = µ G µ, (44) h µν is transverse and traceless tensor perturbation: ν h µν = 0 = h µ µ. (45)

21 3.1 Tensor perturbation in GR, ST, f(r) gravities The tensor perturbation of the background metric is ds 2 = e 2A(z) [(ĝ µν + h µν (x,z))dx µ dx ν +dz 2 ], (46) where h µν satisfies the transverse traceless (TT) condition [DeWolfe, Freedman, Gubser and Karch, PRD62(2000)046008]: h µ µ = 0 = ν hµν. (47) The equation for h µν is [PRD62(2000)046008] [Yang, Liu, et al, PRD 86(2012)127502] [Liu, Zhong, Zhao and Li, JHEP1106(2011)135], [Zhong, Liu and Yang, PLB699(2011)398] [ (4) + 2 z +3A z ] hµν = 0 ( (5) hµν = 0) for GR (R) [ (4) + 2 z +3A z +(lnf(φ)) z ] hµν = 0 for ST (F(φ)R) [ (4) + 2 z +3A z +3A (lnf R ) z ] hµν = 0 for f(r) (f(r))

22 3.1 Tensor perturbation in GR, ST, f(r) gravities By performing the following decomposition h µν (x,z) = e 3 2 B(z) e ikx ε µν (k) h(z), (k 2 = m 2 ) (48) where ε µν satisfies the TT condition: ε µ µ = k ν ε µν = 0, and A for GR B(z) = A+ 1 3 lnf(φ) for ST (49) A+ 1 3 lnf R for f(r) we obtain the equation for the KK modes h(z) [ 2 z +V g (z) ] h(z) = m 2 h(z), (50) where V g (z) = 3 2 A A 2 for GR 3 2 A A (lnf) (lnf) A (lnf) for ST 3 2 A A A f R fr 1 4 f 2 R f 2 R f R f R for f(r) (5

23 3.1 Tensor perturbation in GR, ST, f(r) gravities One can also factorize the Schrodinger-like equation (50) as the form K K h(z) = m 2 h(z): z A for GR K = z A (lnf) for ST (52) z A (lnf R) for f(r) so there is no gravitational massive mode with m 2 < 0, and the solutions in these theories are stable. The solution of gravitational zero mode (4D massless graviton) is e 3 2 A(z) for GR h 0 (z) e 3 2 A(z) (lnf) 1/2 for ST (53) e 3 2 A(z) (lnf R ) 1/2 for f(r)

24 3.2 Gravity localization on various branes 1. ds brane in GR [Wang, PRD66(2002)024024], [Liu, et al, JCAP02(2009)003] The potential is V g (z) = 3H2 [ ] 3δ (2+3δ)sech 2 (Hz/δ) (0 < δ < 1). (54) 4δ The gravitational zero mode h 0 (z) sech 3δ/2 (Hz/δ) and first exited KK mode h 1 (z) sech 3δ/2 (Hz/δ)sinh(Hz/δ) with mass m 2 1 = (3δ 1)H2 /δ 2 can be localized on the ds brane. V g,h z V g,h 0,h z < δ = 0.5 2/3, H = 3, 2/3 < δ = 0.9 < 1

25 3.2 Gravity localization on various branes 2. AdS brane in GR [Liu, Guo, Fu and Li, PRD84(2011)044033] The potential is [ ] V g (z) = 3H2 3δ +(2+3δ)sec 2 (Hz/δ) 4δ (δ > 1) (55) A series of massive gravity KK modes are localized on the AdS brane: h n(z) = cos(hz/δ) 1+3δ/2 2F 1 (1 n,1+n+3δ, 3(1+δ) 2, 1 sin[ H z] ) δ ;, 2 m n = (H/δ) n(n+3δ) (n 1,δ > 1). (56) V g z h n n 1,2, z m n m n

26 3.2 Gravity localization on various branes 3a. Flat brane in GR [Fu, Liu and Guo, PRD84(2011)044036] The potential is V g (z) = 1 36 a2 v 2[ v 2 Sech[az] 6 +3 ( 6+v 2) Sech[az] 4 4v 2] The gravitational zero mode can be localized on the brane z -0.5 V g -1 h z

27 3.2 Gravity localization on various branes 3b. Deformed flat brane in GR [Fu, Liu and Guo, PRD84(2011)044036] The gravitational zero mode can be localized on the brane The potential for gravity V g (z) The zero mode for gravity ξ 0 (z) g 1 = 2 g = g 1 = V g (z) ξ 0 (z) g 1 = g = g 1 = z z

28 3.2 Gravity localization on various branes 4. Scalar-tensor brane [Yang, Liu, et al, PRD 86(2012)127502] The potential is V g (z) = 15κ2( 14+37κ 2 z 2 +28κ 4 z 4 +4κ 6 z 6) 4(5+7κ 2 z 2 +2κ 4 z 4 ) 2. (57) The gravitational zero mode can be localized on the negative tension brane: κ 5+2κ h 0 (z) = 2 z 2 8 (1+κ 2 z 2 ) 5/4. (58) V g,h z

29 3.2 Gravity localization on various branes 5. f(r) brane [Liu, Zhong, Zhao and Li, JHEP1106(2011)135],[Zhong, Liu and Yang, PLB699(2011)398] The potential is V g (z) = 15k2( 14+37k 2 z 2 +28k 4 z 4 +4k 6 z 6) 4(5+7k 2 z 2 +2k 4 z 4 ) 2 (59) The gravitational zero mode can be localized on the brane: k 5+2k h 0 (z) = 2 z 2 8 (1+k 2 z 2 ) 5/4, (60) V g,h z

30 Content 1. Introduction and Motivation 2. Braneworld solutions in various gravities 2.1 General relativity 2.2 Scalar-tensor gravity 2.3 f(r) gravity 3. Gravitational perturbation on branes 4. The effective gravitational potential 5. Summary and further problems

31 4. The effective gravitational potential The effective gravitational potential between two point-like sources of mass M 1 and M 2 is from the contributions of the zero mode and the massive discrete and/or continuum KK modes, and can be expressed as [NPB ] U(r) = G N M 1 M 2 r + M 1M 2 M 3 ne mnr h n (0) 2. (61) r

32 4. The effective gravitational potential 1. ds brane in GR: For 0 < δ < 2 3, one bound state. The potential is U(r) = G N M 1 M 2 r = G N M 1 M 2 r + M 1M 2 M 3 e 3Hr 2 +η 1 M 3 dm e mr 3H/2 r h(0) 2 (62) M 1 M 2 r 2 (63) For 2 3 < δ < 1, two bound states. The potential is U(r) = G N M 1 M 2 r = G N M 1 M 2 r + M 1M 2 M 3 e 3Hr 2 +η 2 M 3 e m1r r M 1 M 2 r 2. h1 (0) 2 + M 1M 2 M 3 dm e mr 3H/2 r The correction to Newtonian potential at short distance h(0) 2 U(r) 1 r2. (64)

33 4. The effective gravitational potential 2. AdS brane in GR: There are infinite bound states. The potential is U(r) = G N M 1 M 2 r = G N M 1 M 2 r + M 1M 2 M 3 + M 1M 2 M 3 n=1 e mnr r h 2 (0) 2 (e 2 H δ r 1)r h n (0) 2 (65) (66) The correction to Newtonian potential at short distance U(r) 1 r2. (67)

34 4. The effective gravitational potential 3. Flat branes in GR and f(r) gravities: There is one bound state. The potential is U(r) = G N M 1 M 2 r + M 1M 2 M 3 0 dm e mr r h(0) 2. (68) The continuous KK spectrum contributes a correction to the Newtonian potential U(r) 1 r3. (69)

35 Content 1. Introduction and Motivation 2. Braneworld solutions in various gravities 2.1 General relativity 2.2 Scalar-tensor gravity 2.3 f(r) gravity 3. Gravitational perturbation on branes 4. The effective gravitational potential 5. Summary and further problems

36 5. Summary and further problems Summary: Brane world (domain wall) solutions in GR, ST, and f(r) gravity theories are obtained. Tensor perturbations are stable (m 2 n 0). Gravitational zero mode is localized on the branes for GR, ST, f(r). Newtonian potential can be recovered on the branes, and the correction to Newtonian potential is 1/r 2 or 1/r 3.

37 5. Summary and further problems Further problems: Stability of scalar perturbations in f(r) gravity. Complicated higher-derivative equations for scalars Shrödinger-like equations

38 5. Summary and further problems Further problems: Multi extra dimensions, g MN dx M dx N = e 2A(y) γ µν (z)dx µ dx ν +g ab (y)dy a dy b, (70) δg MN = g MN +h MN. (71) A complete analysis for tensor, vector, and scalar modes has been done in Einstein gravity in R.G. Cai and L.M. Cao, Generalized Formalism in Gauge-Invariant Gravitational Perturbations, Phys.Rev. D88 (2013) , which can be applied to black holes, cosmology, and brane worlds.

39 Thank you!