Warped Brane-worlds in 6D Flux Compactification

Transcription

1 Warped Brane-worlds in D Flux Compactification Lefteris Papantonopoulos National Technical University of Athens

2 Plan of the Talk Brane-worlds Why -Dimensions? Codimension-1 branes Codimension- branes D Supergravity Conclusion

3 Brane-worlds 1. The size of the extra dimensions where can be much bigger than the Planck length. Our world is a 3-brane ( with a 3+1 dim world volume) extending in a 4+d dimensional manifold

4 3. The particles of the SM are localized to the brane 4. Gravity propagates everywhere

5 Old Idea: The 4+d dimensional Planck mass M= 4 dimensional Planck = M P = Gev New Idea: M can be much smaller than M P We want M to be of the order of a 1-10 Tev

6 This gives a new push to the hierarchy problem : Why the electroweak scale is so much ( 1 orders of magnitude) smaller than Planck mass? Because the Planck Mass is not fundamental. It is a derived Quantity!

7 The Fundamental mass M and M p + d 4+ d S = M d x G R + d 4 d = M V d x gr (4 + d ) (4) M = M + d V P d

8 4dimen gravity is weak because V is large! M p 19 = 10 Gev if M = 1 Tev, Then 15 d = 1 V d= 1 10 cm d = V (0.1 mm) d= d = 3 V (10 cm) d= 3 3

9 Why D= is interesting? M of the order of a Tev requires the size of the extra dimensions to be in the sub-mm range. We should see D effects at sub-mm Scales!

10 4. More Theoretical Reasons: Are there special features in 5D brane-worlds? is Is \ there any special reason Can we find a brane-world model describing gravity in D? D Brane-world Models RS-type RS-type Codimension-1 branes (4-brane in D) Codimension- branes (3-brane in D) D Supergravity brane models

11 Codimension-1 branes Consider the D metric: ds = n (, t y, zdt ) + a(, ty, zd ) Σ + b(, ty, zdy ) κ + d (, t y, z) dz If the brane is at z 0 and the energy-momentum tensor M = M ( B ) + M ( b ) N N N T T T where δ Mb () ( z z0) T N = diag( ρ, p, p, p, p%,0) d

12 We get a generalization of RS model in D Cosmology P. Kanti, R. Madden, K. Olive, hep-th/ B. Cuadros-Melgar, E.P, hep-th/05019 If a ( t ) = b ( t ) = R ( t ) We get the Friedmann-like equation: κ 4 κ 4 κ 4 () ρ () ρ () T R&& R& k + 3 = 3 p 3 R R R d

13 Remarks: Unconventional ρ term with a coefficient adjusted to D We should solve the D bulk equations to get the Friedmann equation Four-dimensional universe. A mechanism is needed to keep the 4 th dimension small

14 You get the same results if the 4-brane is moving in a static bulk described by the metric z ds h z dt d h z dz l 1 = ( ) + Σ ( ) κ + Where the metric of 4D space time is dr d S = + r d W + (1 - k r ) d y 1 - kr k and the function hz ( ) z = k+ - l M 3 z is written in the Schwarzschild coordinates

15 The two metrics are the same if we identify n ( z ) = h ( z ) z a ( z ) = b ( z ) = l - 1 ( ) = ( ) d z h z Then, using the junction conditions we get - h - R&& æ 3 ö = - k ç r - p h R çè 4 + & ø h + R& k - = R 4 From where we get the Friedmann-like equation in D r

16 What happens if a ( t) b( t) The equations of motion of the brane are 3 d d R d R z a ad z b bd && 1 + d - R& && k = ( 3( r p ) % p) 1 + d R& k = - 8 ( r + p - % p) 1 + d k R& = ( p - % p ) ( r ) Then we get the following relation between scale factors With B b ( t ) = a ( t ) B = 1-3 w + 3 w% 1 + w - w%

17 Codimension- branes General Features: J. Cline, J.Descheneau, M. Giovannini, Vinet, hep-th/ S. Carroll, M. Guica, hep-th/ The brane vacuum energy does not curve the brane world-volume, simply induces a deficit angle in the bulk around the brane. Hope to solve the Cosmological Constant Problem! - Brane energy-momentum tensor ~ induced metric We cannot recover ordinary 4d gravity on the brane!

18 Try to introduce non-trivial energy-momentum tensor on the brane: Singularities appear in the metric around the brane To cure them Either Introduce a cut-off (brane thickness) or S. Kanno, J. Soda, hep-th/ J. Vinet, J. Cline, hep-th/ I. Navarro, J. Santiago, hep-th/ Modify your gravitational dynamics P. Bostock, R. Gregory, I. Navarro, J. Santiago, hep-th/

19 Minimum modification: A. Papazoglou, E.P., hep-th/ M () 4 (4) δ () r S = d x GR rc d x gr + π L d() r LBulk d xlbulk + òd x + ò p L With rc = M / M the cross-over scale 4 Consider the metric: ds = g ( x, r) dx dx + dr + L ( x, r) d µ v θ µν

20 And expand around the brane: L x r x r O r (, ) = β ( ) + ( ) Then you get the Einstein equation 1 G = T + π (1 β ) g (4) ( br ) µ ν 4 µν µν rc M rc with β the deficit angle and Μ = Μ = r Μ Pl 4 c Λ = λ πμ (1 β ) 4 There is however a relation between the energy-momentum tensors of the bulk and brane ( B) r 1 ( br) µ Τ r = Τ 8 µ + πμ (1 β ) r c

21 We can add a GB term in the action M S = S+ α d x R R + R () 4 () () Then the Einstein equation on the brane is ( r 8 (1 ) ) c + π β α ( ) MN MNKΛ 1 π (1 β ) (4) ( br ) G = T + g µν 4 M µν r 8 (1 ) µν c + π β α with Pl 4 ( r 8 (1 ) ) c π β α Μ = Μ + Λ = λ π Μ (1 β ) There is also the relation R + α R R + R = T ( 4 ) Λ (4) (4) (4) (4) ( B) r MN MNK r M

22 We can see the effect of the constrained relations in cosmology Consider a time-dependent metric A. Papazoglou, E.P., hep-th/ r ds = N (, t r) dt + A (, t r) dx + dr + L ( x, r) dθ Then the Friedmann equations for the induced gravity case are 3 a& a = ρ M P l a&& a& ρ + = w a a M c P l And the constrained relation gives w c a& w& c + 3(1 + w c ) ρ = 0 ρ a There is no solution with w constant! c

23 Brane-worldsin D Supergravity + General Features: Consistent theories coming from higher dimensional Supergravity Exact solutions of 10D Supergravity theories - Because of supersymmetry and anomaly cancellation The D theory is highly constrained In spite of that it has a rich enough field content A. Salam, E. Sezgin, Phys. Lett.B147, 47 (1984) S. Randjbar-Daemi, A. Salam, E. Sezgin, J. Strathdee, Phys. Lett.B151, 351 (1985)

24 D Supergravity Romans Supergravity g B Bosonic part f A m n M N a M Salam-Sezgin Supergravity Supergravity-tensor multiplet g mn i y, MN, B, f, Chiral D matter M c i Fermionic part y c i M i A M, l, F, Y i a a

25 Introducing Branes G. Gibbons, R. Guven, C. Pope, hep-th/ C. Burgess, F. Quevedo, G. Tasinato, I. Zavala, hep-th/ Y. Aghababale et. al., hep-th/ Find solutions of warped brane-world compactifications of D Supergravity Solve D coupled Einstein-Maxwell-dilaton equations of both Romans and Salam-Sezgin Supergravities General Features of the solutions The warping is power law dependent The singularities sources by the branes are jumps for codimension-1 and conical for the codimension- branes Recover 4D gravity on the brane The cosmology of the Warped Flux Compactification Of D supergravity is still poorly understood

26 A D brane-world model Consider the action 4 M æ 1 MN ö I= ò d x - g ç R- L- F F MN çè ø The equations of motion are 1 æ 1 ö = ç L + - ø K RMN g 4 MN FMN FN F gmn M çè 8 M ( MN gf ) - = 0

27 Background solution S. Mukohyama, Y. Sendouda, H. Yoshiguchi, S. Kinishita hep-th/ dr ds r dx dx fd f m v = h mn + + b f b F r f = - 4 r L M f r r ( ) = b b r m r Assume there are two roots of f ( r ) 0 < r < r - +

28 Place two 3-branes at r + Identify and r - North Brane pe South Brane The deficit angle is related to the brane tensions by e = 4GT

29 What is the large distance behaviour of gravity in this brane-world model? Study how the linear perturbations of the brane energy-momentum tensor effect the bulk geometry and the background gauge field But For codimension- singularities we can not accommodate energy-momentum tensor different than pure tension We need to promote the brane to a thick defect Find a way to regularize the conical singularity

30 Brane-world D flux compactification with 4-branes The action is M. Peloso, L. Sorbo, G. Tasinato, hep-th/0030 S = S + S + S o i str æ 4 1 AB ö Soi, = ò d x - g ç M R- Loi, - F F AB çè 4 ø æ 5 u ö S ( )( M M) str =- d x - gç l s + Ms - eam s - ea ò Background Bulk Solution ç çè ø Minkowski times D Compact Space

31 Codimension-1 4-brane North spherical cap q Regularization of conical singularities

32 Brane content To compensate the discontinuities in the geometry and the gauge field we introduce energy-momentum tensor localized on the 4-brane. To take care the jump in the Maxwell equations there must be on the brane a coupling to the gauge field! The induced metric on the 4-brane is ds 5 = h dx dx + cos q df m v mn The energy-momentum tensor on the 4-brane is æ u ö mn = - ç l s + f s - f s - h mn f f ( )( ) S ea ea ç çè ø æ u ö f f = - ç l s + f s - f s - g f f f f ( )( ) S ea ea ç çè ø where s is a scalar field acting as a goldstone mode

33 Results Studying linear perturbations Einstein gravity is reproduced at large distances Correction terms appear at scales comparable to the size of compactification manifold However The regularization of the conical singularity is model dependent Cosmology: Can we get a cosmological evolution in these D supergravity models?

34 Warped Regularization of D Einstein-Maxwell System with Codimension-1 Branes Define r r r a - = and make a coordinate transformation = r z( r; a ) with + The solution becomes + 1 z( r; a ) = (1- a ) + (1 + a ) L j = r+ (1- a ) f M 4 [ r ] é dr ds r z r dx dx R f r d ê ëf( r; a ) m v = + ( ; ah ) mn + ê + b ( ; a) j with Fr j = - b RM a 1 a. 3 4 f f ( r; a ) = 4 R, R = a z ( r ; a ) M 4 r+ (1- a ) L A.Papazoglou V. Zamarias E. P. Work in progress ù ú û

35 We suppose f Î [ 0, p) which fixes the constant b in terms of of the brane tension. Expand the metric around r =± 1 ds r z v dx dx R é dr X r d ë m» + ( ± 1; ah ) mn + + b ± (1 m ) j X ê ± (1 mr) ù ú û with 8 3 X X The deficit angles are + - = = 5 + 3a - 8 a - - 0(1 a )(1 a 3 ) a - 5 a a (1 a )(1 a ) d = p(1 - b ) b = bx ± ± T = with M d 4 ± ± ± ± These singularities are supported by codimension- brane contributions with tensions related to the deficit angles as

36 The flux of the gauge field is quantized and this gives a quantization condition for the deficit angle. The gauge fields in the two patches are A A N j S j a æ1 ö = brm a. 1 3 ç a 3(1 - a) çèz ø a æ1 1 ö = brm a. 3 ç a 3(1 - a) çè z a ø Single valuedness of the gauge transformation at the overlapping region gives b Y = = M N e R Y - a - a 15 a (1- a ) 3 5 (1 ) (1 ) 3 When a 1 then Y 1 as expected, N = ± 1, ±,...

37 The metrics in the caps r1 r 1 are described by the solution and < < - 1< r< r é dr ds z r dx dx R f r d ê ëf ( r; a ) m v = ( ; a) hmn + ê i + bi ( ; a) j ù ú û F =- b R M rj i i a 1 a a z ( r; a) In order that there is no deficit angle in the caps we should demand b 1 1 X = + and b = 1 X -

38 Then we get the same quantization conditions for the gauge fields as before, while we get also a quantization condition for the Goldstone fields s a = n j n =± 1, ±,... a, a The induced metrics on the two 4-branes are ds5( ) = z ( r ; a) h dx dx + R b f ( r ; a) dj m v i i mn i i i Calculating the junction conditions we get the following quantization conditions N n1 = w( a ), n = n1 - N With the function w( a ) to be 8 æ5(1 - a ) ö 3 w( a) = - a 3 5 (1- a ) çè8(1 - a ) ø

39 As a 1 the wa ( ) 1 as expected. The above relation imposes a restriction to the values of the admissible warpings a since 3/4 < w( a ) < 5/4 The brane tensions are l M a 1 1- a 1ö æ1 1 ö , =± 8 z.. 4 a r= r - 1, è ç 1- a z 1- a z ø èç R0 R 1,ø 0(1 - a ) f æ While the total tension is T = 4 p R b f ( r ; a ) l 1, 0 0 1, 1,

40 Cosmology Move the 4-brane in the static bulk Cosmological evolution will be induced on the 4-brane with the junction conditions playing the role of the equation of motion of the 4-brane Warping of the space is necessary to induce cosmological evolution on the 4-brane However The brane-universe is 4-dimensional Stability issues

41 Conclusion For codimension- brane-worlds we need a kind of regularization in order to consider matter on the brane It is very difficult to get cosmological evolution on the brane for the codimension- branes D Supergravity models have interesting features and hopefully they can give some interesting cosmology