Origin of brane cosmological constant in warped geometry models - PowerPoint PPT Presentation

Origin of brane cosmological constant in warped geometry models Soumitra SenGupta IACS, Kolkata 26 July, 2013 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 1 / 51

The talk is based on my collaborations with Sayantani Lahiri and Saurya Das and Debaprasad Maity S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 2 / 51

Background Hierarchy Problem Vast disparity between the weak and Planck scale – Gauge hierarchy problem δ m 2 H ∼ Λ 2 where Λ is the cutoff scale say Planck scale To keep m H within Tev, one needs extreme fine tuning ∼ 10 − 32 UNNATURAL Challenge for standard model – Extra-dimensions ? ADD model and RS model S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 3 / 51

Warped model in 5-dimension Warped Geometry – Randall-Sundrum Model The Einstein action in 5 dimensional ADS 5 space � 1 d 5 x √− g 5 [ R − Λ] S = 16 G 5 Compactify the extra coordinate y = r φ on S 1 / Z 2 orbifold Identify φ to − φ i.e lower semi-circle to upper semi circle Place two 3-branes at the two orbifold fixed points φ = 0 , π r is the radius of S 1 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 4 / 51

Warped model in 5-dimension Planck Visible brane brane Compact coordinate y The Z 2 orbifolded coordinate y = r φ with 0 ≤ φ ≤ π and r is the radius of the S 1 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 5 / 51

Warped model in 5-dimension Action S = S Gravity + S vis + S hid � √ d 4 x r d φ − G [2 M 3 R − S Gravity = Λ ] ���� 5 − dim � d 4 x √− g vis [ L vis − V vis ] S vis = � d 4 x √− g hid [ L hid − V hid ] S hid = S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 6 / 51

Warped model in 5-dimension Metric ansatz: ds 2 = e − A ( φ ) η µν dx µ dx ν + r 2 d φ 2 Computing the warp factor A ( y ) Warp factor and the brane tensions are found by solving the 5 dimensional Einstein’s equation with orbifolded boundary conditions A = 2 kr φ V hid = − V vis = 24 M 3 k and − Λ k 2 = 24 M 3 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 7 / 51

Warped model in 5-dimension Warping ( m H ) 2 = e − 2 A | φ = π = e − 2 kr π ∼ (10 − 16 ) 2 m 0 ⇒ kr = 16 π ln(10) = 11 . 6279 ← RS value with k ∼ M P and r ∼ l P So hierarchy problem is resolved without introducing any new scale Gravity Gravity + SM r φ φ =0 = π φ Hidden brane Our universe (Visible brane) S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 8 / 51

Warped model in 5-dimension Remarks Warped geometry models have been construcetd in a string background – 1 ’Throat geometry’ with fluxes to stabilise the moduli RS model is a simple field theoretic description which captures the essential 2 idea of warped geometry and very useful in estimating various signatures of such models in particle phenomenology/cosmology Modulus can be stabilised by Goldberger-Wise mechanism 3 It is defined on a flat/static visible brane with zero cosmological constant 4 Can we generalize it to include non-flat branes? 5 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 9 / 51

Warped model in 5-dimension In the original RS scenario, it was proposed that the visible 3-brane being flat has zero cosmological constant. ds 2 = e − 2 kry η µν dx µ dx ν + r 2 dy 2 But such model was generalized to Ricci flat spaces : R µν = 0 and the warp factor turned out to be the same as obtained by RS See Chamblin, Hawking, Real : Phys.Rev.D, 61,065007 (2000) S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 10 / 51

Warped model in 5-dimension In our work we demonstrate that the condition of zero cosmological constant can be relaxed and a more general warp factor can be obtained S.Das, D.Maity, S.SenGupta S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 11 / 51

Warped model in 5-dimension Generalized Randall Sundrum braneworld with constant modulus Generalize the RS model to non-flat brane scenario with constant radion field The metric : ds 2 = e − 2 A ( y ) g µν dx µ dx ν + dy 2 The induced metric q µν ( x , y ) in the previous section is now taken as : e − 2 A ( y ) g µν The action is : � � √ d 4 x √− g i V i − G ( M 3 R − Λ 5 ) + d 5 x S = S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 12 / 51

Warped model in 5-dimension The bulk Einstein’s equations away from the 3-branes are as follows : = − Λ 5 (4) G µν − g µν e − 2 A � − 6 A ′ 2 + 3 A ′′ � 2 M 3 g µν e − 2 A and − 1 2 e 2 A (4) R + 6 A ′ 2 = − Λ 5 2 M 3 with the boundary conditions : ǫ i A ′ ( y ) = 12 M 3 V i ǫ pl = − ǫ vis = 1 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 13 / 51

Warped model in 5-dimension Rearranging terms we get, (4) G µν = − Ω g µν This is the effective four dimensional Einstein’s equation with Ω is the induced cosmological constant S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 14 / 51

Warped model in 5-dimension Negative Ω – ADS case Define the parameter ω 2 ≡ − Ω / 3 k 2 ≥ 0 The solution for the warp factor, � ln ω � e − A = ω cosh c 1 + ky The above solution is an exact solution for the warp factor in presence of Ω. The RS solution A = ky is recovered in the limit ω → 0. S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 15 / 51

Warped model in 5-dimension Positive Ω – de Sitter � � ln c 2 e − A = ω pl sinh − ˜ k | y | ω pl � k 2 with c 2 = 1 + pl = Ω pl / 3˜ where ω 2 1 + ω 2 pl Once again for ω → 0 we retrieve RS solution The brane tensions on both the branes are: � � � c 2 � c 2 2 + ω 2 2 + ω 2 V vis = − 12 M 3 ˜ , V pl = 12 M 3 ˜ pl vis k k c 2 2 − ω 2 c 2 2 − ω 2 vis pl Here the brane tension in one brane is always positive while the other is negative just as in RS case In this case, there are no bounds on ω 2 , i.e. the (positive) cosmological constant can be of arbitrary magnitude. S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 16 / 51

Warped model in 5-dimension � N -32 -32.5 B DS ADS -33.5 I II III -34.5 A 40 x 36.5 37.5 38.5 39.5 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 17 / 51

Warped model in 5-dimension From region I in FIG.1 and it is easy to observe that a small and positive value of the cosmological constant which corresponds to the observed value ∼ 10 − 124 in Planckian unit indicates a value for x i.e kr π very very close to the RS value 36 . 84 S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 18 / 51

Warped model in 5-dimension Here by generalizing the RS model with a non-vanishing cosmological constant on the visible brane we show that Issue of smallness of cosmological constant, smallness of the factor in gauge 1 hierarchy and brane tensions are intimately related in a generalized Randall-Sundrum (RS) type of warped geometry model . Exact solution for the warp factors are determined for both DS and ADS 2 cases. Region of positive cosmological constant on the visible 3-brane ( de-Sitter) 3 strictly implies negative brane tension However visible brane with negative cosmological constant ( anti de-Sitter) admits of both positive and negative brane tension. S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 19 / 51

Warped model in 5-dimension For both the cases the desired warping from Planck to Tev scale can be 1 achieved as a proper resolution of the gauge hierarchy problem. The magnitude of the negative induced cosmological constant on the 3-brane 2 has an upper bound ∼ 10 − 32 in Planck unit For a very tiny but negative value of the induced cosmological constant the 3 hierarchy problem can be resolved for two different values of the modulus, one of which corresponds to a positive tension Tev brane alongwith the positive tension Planck brane. In the other region namely Ω > 0 the Tev brane tension turns out to be 4 necessarily negative . The modulus value corresponding to the observed value of the cosmological constant lies very close to the RS value. S.SenGupta (IACS, Kolkata, India) () Origin of brane cosmological Constant in warped geometry models 20 / 51