1. Introduction. [Arkani-Hamed, Dimopoulos, Dvali]

Transcription

1 2014 Ï Ò (í «) Ò Ò Ù Åǽ À

2 Content 1. Introduction and Motivation 2. f(r)-brane model and solutions 3. Localization of gravity on f(r)-brane 4. Gravity resonances on f(r)-brane 5. Corrections to Newton potential 6. Conclusion and open issues This talk is based on: [1]H Guo, YX Liu, et al, EPL 97(2012)60003 [ ]. [2] Y Zhong, YX Liu, and K Yang, PLB 699 (2011) 398 [ ]. [3] YX Liu, Y Zhong, et al, JHEP 1106 (2011) 135 [ ]. [4] H Liu, H Lu, and ZL Wang, JHEP 1202 (2012) 083 [ ]. [5] D. Bazeia et al, PLB 729 (2014) 127 [ ]. [6] ZG Xu, YX Liu, and Y Zhong, Gravity resonances on f(r) brane.

3 1. Introduction 1921,1926: Kaluza-Klein (KK) Theory Compact extra dimensions In order to unify 4D gravity and 4D electromagnetism 1983: Domain wall Scenario [Akama, Rubakov, Shaposhnikov] Infinite extra dimension Our 4D world is a brane embedded in 5D flat space-time Generated by a scalar field: φ(y) = v 0 tanh(ky) Fermions can be localized on DW by Yukawa coupling η ΨφΨ Newton potential cannot be recovered on DW: U(r) 1/r : Large Extra Dimensions (ADD Brane Scenario) [Arkani-Hamed, Dimopoulos, Dvali] Compact but very large extra dimensions To solve the gauge hierarchy problem Newton potential can be recovered on brane: U(r) 1/r when r > R ED

4 1. Introduction and motivation 1999: Warped Extra Dimension (RS Brane Scenario) [Randall and Sundrum] ds 2 = e 2k y η µν (x) dx µ dx ν +dy 2 Our 4D world is a brane embedded in a 5D space-time SM fields are assumed to be confined on brane, and gravity propagates in the whole space-time To solve the gauge hierarchy and cosmological problems Fermions can be localized on brane by mass term : ηmǫ(y) ΨΨ Newton potential: U(r) 1 ( ) r 1+ 1 k 2 r 2 Thin braneworld model: ρ(y) σδ(y) Warp factor 2 A Energy density Ρ y y

5 1. Introduction and motivation 1999 Now: Thin and Thick braneworlds (Domain Walls) [Bazeia, Cai, Cao, Csaki, DeWolfe, Freedman, Hollowood, Giovannini, Goldberger, Gremm, Gubser, Kodama, Lu, Rubakov, Schnabl, Wang, Wu,...] 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 q (ky) Fermions can be localized on the brane by Yukawa coupling ηφ ΨΨ Warp factor 2 A Energy density Ρ y y

6 1. Introduction and motivation A lot of braneworld models were considered in general relativity. However, GR suffers various troublesome theoretical problems: dark matter/energy, nonrenormalization, singularity... There are some modified theories of gravity, such as scalar-tensor, f(r) and other higher derivative gravity (critical gravity), Horava-Lifschitz gravity, massive gravity, bimetric gravity, f(t),... which are useful and important to understand the character of gravity. f(r) gravity is a simple higher derivative theory.

7 1. Introduction and motivation Three formalisms of gravity theories: Metric-affine formalism: Γ λ µν and g µν independent, S G = S G [g µν,γ λ µν ], S M = S M [g µν,γ λ µν,ψ]. (1) gkl δs M The hypermomentum: P MN 2. δγ P MN T MN does not represents the usual meaning of a energy-momentum-stress tensor, the hypermomentum also describes matter characteristics. Palatini formalism: Γ λ µν and g µν independent, S G = S G [g µν,γ λ µν], S M = S M [g µν,ψ]. (2) Metric formalism: Γ λ µν = { λ µν }, only gµν, S G = S G [g µν ], S M = S M [g µν,ψ]. (3)

8 1. Introduction Classification of f (R) theories of gravity Taken from [T.P. Sotiriou, CQG 23(2006)1253]. 4Œ f(r).þúå

9 1. Introduction and motivation Motivation: In braneworld model, (3+1)-dimensional massless graviton is the tensor zero mode of bulk gravity. Are gravitational perturbations stable in higher derivative gravities? Is the massless graviton localized on the brane? Can Newton potential be recovered on the brane? What s the correction to Newton potential? We will consider f(r) gravity as an example. For a more complex case in critical gravity, see Feng-Wei Chen s talk.

10 Content 1. Introduction and Motivation 2. f(r)-brane model and solutions 3. Localization of gravity on f(r)-brane 4. Gravity resonances on f(r)-brane 5. Corrections to Newton potential 6. Conclusion and open issues

11 2. f(r)-brane model and solutions The action describing a braneworld system is given by S = S Gravity +S Matter (4) The metric of the background space-time is assumed as ds 2 = e 2A(y) ĝ µν (x) dx µ dx ν +dy 2, (5) where e 2A(y) is called warp factor. Warp factor 2 A Energy density Ρ y y

12 2. f(r)-brane model and solutions For the 5D action in metric f(r) gravity S = d 5 x ( 1 g 2κ 2 f(r) 1 ) 5 2 M φ M φ V(φ), (6) the EoMs are given by φ +4A φ = V φ, (7) f +2f R ( 4A 2 +A ) 6f R A 2f R = κ 2 5(φ 2 +2V), (8) f 8f R ( A +A 2) +8f R A = κ 2 5(φ 2 2V). (9) For f(r) = R +αr 2 and V(φ) = λ(φ 2 v 2 ) 2 +Λ 5, a flat f(r)-brane was obtained [Liu and Zhong et al, JHEP 1106(2011)135]: e A(y) = sech(ky), (10) φ(y) = vtanh(ky). (11)

13 2. f(r)-brane model and solutions A family of brane solutions in metric f(r) gravity with f(r) = R +αr 2 were found in [Bazeia et al, PLB 729(2014)127]. The warp factor is assumed as e A(y) = sech B (ky). (B > 0) (12) The Ricci tensor at boundary of extra dimension R MN (y ± ) 4B 2 k 2 g MN Λ eff g MN. The derivative of the scalar field { φ 2 = Bk 2 sech 2 3 (ky) 2 4αk2[ 5B 2 +16B +8 (5B 2 +32B +12)sech 2 (ky) ] }. φ 2 0 implies 3 32(1+4B)k 2 α 1 α α 2 3 8(8+16B +5B 2 )k2. (13)

14 2. f(r)-brane model and solutions When α = α 1, the solution is φ(y) = v 1 [1 sech(ky)]sign(y), (14) V(φ) = c 2 ( φ v1 ) 2 [ ( φ v1 ) 2 c1 ] c 0. (15) when α = 0 φ(y) = [ ( ky 6B arctan tanh 2 ( ) 2 V(φ) = d 1 cos 2 3B φ )], (16) d 0. (17) when α = α 2, the result is just the one found in [Liu and Zhong et al, JHEP 1106(2011)135] but with arbitrary B in e A(y) = sech B (ky) φ(y) = v 2 tanh(ky), (18) V(φ) = λ 1 (φ 2 v 2 2 )2 λ 0. (19)

15 Content 1. Introduction and Motivation 2. f(r)-brane model and solutions 3. Localization of gravity on f(r)-brane 4. Gravity resonances on f(r)-brane 5. Corrections to Newton potential 6. Conclusion and open issues

16 3. Localization of gravity on f(r)-brane Next, we consider the linear perturbations of the brane solutions. 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 ], (20) where h µν satisfies the transverse traceless (TT) condition [DeWolfe, Freedman, Gubser and Karch, PRD62(2000)046008]: h µ µ = 0 = ν h µν. (21) The equation for h µν is [PRD62(2000)046008] [Yang, Liu, et al, PRD 86(2012)127502] [Liu, Zhong, et al, JHEP1106(2011)135], [Zhong, Liu and Yang, PLB699(2011)398] { ( (4) + z 2 +3A z ) hµν ( = 0 for GR (4) + z 2 +3A z + zf R f R z ) hµν = 0 for metric f(r) (22) where f R df/dr.

17 3. Localization of gravity on f(r)-brane By performing the following decomposition h µν (x,z) = e ikx ε µν (k)e 3 2 A(z) f 1 2 R (z)ψ m(z), (k µ µ = m2 ) (23) where ε µν satisfies the TT condition: ε µ µ = k ν ε µν = 0, we obtain the equation for the gravity KK modes Ψ m (z) [ 2 z +V g (z) ] Ψ m (z) = m 2 Ψ m (z), (24) where V g (z) = { 3 2 A A 2 for GR 3 2 A A A f R fr 1 4 f 2 R f 2 R f R f R for f(r) (25)

18 3. Localization of gravity on f(r)-brane One can also factorize the Schrodinger-like equation (24) as the form m 2 Ψ m (z) = P PΨ m (z): { ( z + 3 m 2 2 Ψ m (z) = ( A )( z A ) Ψ m (z) for GR z A + 1 f R 2 f R )( z A + 1 f R 2 f R )Ψ m (z) for f(r) so there is no tachyonic gravitational mode with m 2 < 0, and the solutions in these theories are stable.

19 3. Localization of gravity on f(r)-brane For critical gravity, S = 1 2κ 2 d 5 x g the fluctuation equations are (R 3Λ 0 +αr 2 +βr MN R MN +γl GB ) +S Matter G µν (L) Λ 0e 2A h µν +αe µν (1)(L) +βe µν (2)(L) 1 2 γh(l) µν = κ 2 T µν, (L) (26) G µν (L) = 1 [ (4) h µν +e 2A h µν 2 +4A e 2A h ] µν +e2a (4A 2 +A ) h µν, E (1)(L) µν = 4(2A +5A 2 ) (4) hµν +4(2A +5A 2 )e 2A h µν +8e 2A (A +9A A +10A 3 ) h µν 8e 2A (2A +16A A +12A 2 +37A 2 A +5A 4 ) h µν, E µν (2)(L) = 1 2 e 2A (4) (4) h µν (4) h µν 2A (4) h µν 4e2A A 2 h µν +2(A +3A 2 ) (4) h µν 1 e2a h µν 2 4e2A A h µν +e 2A (40A 3 +16A A +2A ) h µν e 2A (5A +40A A +30A 2 72A 4 +20A A 2 ) h µν. It is unclear whether the tensor perturbation is stable.

20 3. Localization of gravity on f(r)-brane Localization of gravity zero mode The potential is W(z(y)) = + ( 1 4 k2 sech 2B (ky) 15B 2 (2 + 3B)(4 + 5B)sech 2 (ky) 128B(2 + 5B)(1 + 16Bk 2 α)k 2 α (1 + 8B(4 + 5B)k 2 α + (1 40B 2 k 2 α)cosh(2ky)) 2 16(1 + 2B)(1 + 16Bk 2 α) 1 + 8B(4 + 5B)k 2 α + (1 40B 2 k 2 α)cosh(2ky) ). (27) The gravity zero mode (m=0) can be localized on the f(r)-brane: Ψ 0 (z) e 3 A(z) + 2 f R (z), Ψ 0(z) 2 dz <. Such a mode is identified as our 4D graviton. Its localization is the physical reason why gravity still behaves as four dimensional at the brane.

21 3. Localization of gravity on f(r)-brane Special f(r) brane (B = 1 and α = α 2 ) [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 (28) The gravity zero mode can be localized on the brane: k 5+2k Ψ 0 (z) = 2 z 2 8(1+k 2 z 2 ) 5/4. (29) V g z z z z

22 3. Localization of gravity on f(r)-brane The effective potential for general f(r) brane (f(r) = R +αr 2 ) W z W z W z (a) α 1 α < α s, (b) α 1 α < α 2 (c) B > 2,α 2 < α α 2 (c) When B > 2 and α 2 < α α 2, W has a singularity. α (1 + 4B)k 2,α 1 = 3 + 9B 8k 2 ( B + 49B 2 ),α 2 = 1 40B 2 k 2, α 3 2 8(8 + 16B + 5B 2 )k 2, α 1 < α 1 < 0 < α 2 < α 2.

23 Content 1. Introduction and Motivation 2. f(r)-brane model and solutions 3. Localization of gravity on f(r)-brane 4. Gravity resonances on f(r)-brane 5. Corrections to Newton potential 6. Conclusion and open issues

24 4. Gravity resonances on f(r)-brane The contribution of a massive KK mode to the effective gravitational potential U(r) is U(r) = M 1M 2 M 3 e mr r Ψ m (0) 2. (30) To get the numerical solutions of gravity KK modes, we impose the following conditions: Ψ even m (0) = c 0, z Ψ even m (0) = 0. We can normalize the KK modes with Ψ m (z ) 0.5cos(mz), (31) and so Ψ m (z = 0) can be fixed by the EOM. Then the peaks in the curve Ψ m (0) m indicate the existence of resonant KK modes, which would give non-trivial contribution to the Newton potential.

25 4. Gravity resonances on f(r)-brane Potential W(z) and KK modes Ψ m (0) at the brane with B = 2, α = α W 3 z m m m z m z z z

26 4. Gravity resonances on f(r)-brane General f(r) brane (B > 2, α 2 < α < α 2) W z m m There are a series of resonant KK modes m z z m z z -2-4 m z z -2-4 m z z The appearance of resonances would have nontrivial contribution to Newton potential.

27 Content 1. Introduction and Motivation 2. f(r)-brane model and solutions 3. Localization of gravity on f(r)-brane 4. Gravity resonances on f(r)-brane 5. Corrections to Newton potential 6. Conclusion and open issues

28 5. Corrections to Newton potential The effective gravitational potential U(r) between two point-like sources of mass M 1 and M 2 on the brane separated by a distance r is from the contributions of the zero mode and the massive continuum KK modes, and can be expressed as [NPB581(2000)309] U(r) = G N M 1 M 2 r = G N M 1 M 2 r M 1M 2 M ( k Randall-Sundrum braneworld: U(r) = G N M 1 M 2 r 0 Ψ m (0) m/k, for m < k, 0 dm e mr Ψ m (0) 2 r ) dm e mr Ψ m (0) 2.(32) ( 1+ 1 ) k 2 r 2, U(r) 1 r3. (33)

29 5. Corrections to Newton potential ds brane in GR [A Wang, PRD 66(2002)02402]; [Guo and Liu, EPL 97(2012)60003]: 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 Ψ m (0) 2 (34) M 1 M 2 r 2 (35) 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. Ψ 1 (0) 2 M 1M 2 M 3 dm e mr 3H/2 r The correction to Newtonian potential at short distance U(r) 1 r2. (36) Ψ m (0)

30 5. Corrections to Newton potential AdS brane in GR [A Wang, PRD 66(2002)02402]; [Guo and Liu, EPL 97(2012)60003]: 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 Ψ 2 (0) 2 (e 2 H δ r 1)r Ψ n (0) 2 (37) (38) The correction to Newtonian potential at short distance U(r) 1 r2. (39)

31 5. Corrections to Newton potential f(r)-brane with no resonances 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 Ψ m (0) 2. (40) The continuous KK spectrum contributes a correction to the Newtonian potential { 1/r 3 long distance U(r) 1/r 2 short distance (41) f(r)-brane with resonances Under investigation {??? long distance U(r) 1/r 2 short distance (42)

32 Content 1. Introduction and Motivation 2. f(r)-brane model and solutions 3. Localization of gravity on f(r)-brane 4. Gravity resonances on f(r)-brane 5. Corrections to Newton potential 6. Conclusion and open issues

33 6. Conclusion and open issues Summary: Brane world (domain wall) solutions in metric f(r) gravity are obtained. Tensor perturbation is stable in metric f(r) gravity. Gravitational zero mode is localized on the f(r)-brane. There are gravitational resonant modes, they are quasi-localized on the f(r)-brane. Newtonian potential can be recovered on the branes, and the correction to Newtonian potential is 1/r 2 or 1/r 3.

34 6. Conclusion and open issues Open issues: ds/ads brane world solutions in metric f(r) gravity. The correction of resonant KK modes to Newton potential. Scalar perturbations in f(r) gravity.

35 Thank you for your listening!