Kaluza-Klein Braneworld Cosmology S Kanno, D Langlois, MS & J Soda, PTP118 (2007) 701 [arXiv: ] Misao Sasaki YITP, Kyoto University

1 Introduction String theory can be consistently formulated only in 10-dim spacetime. In other words, string theory predicts extra dimensions. We have to compactify the extra dimensions to reconcile string theory with the realistic world. We perhaps need String Theory in order to answer this question. Kaluza-Klein compactification Braneworld 2 possibilities An ultimate question in cosmology is How the Universe was born On the other hand, the conventional cosmology is formulated in 4- dimensions.

2 Kaluza-Klein (KK) Compactification Extra dimensions are too small for us to detect How small? ＜ cm from collider experiments. The size of extra dimensions 2-dimensional stick1-dimensional stick

3 Braneworld Akama (1982), Rubakov & Shaposhnikov (1983) Horava & Witten (1996), Arkani-Hamed et al (1998) Randall & Sundrum (1999) e, p, photon, etc graviton Our universe is a 4D (mem)brane embedded in 10D Matter is confined on the brane Gravitons propagate freely in extra dimensions How small? ≲ 0.1mm from gravity experiments. The size of extra dimensions

4 Randall-Sundrum (RS) Model Z 2 symmetry Brane tension RSI Model RSII Model Self-gravity of Codimension-1 brane can be consistently taken into account. However, we need higher codimension branes to reconcile string theory with the braneworld. Randall & Sundrum (1999) exponential warping ～ 0.1mm Cosmological Constant AdS curvature radius

5 Towards Higher Dimensional Braneworld Higher Codimension Braneworld “KK Braneworld”KK compactification on the brane : ・ Codimension-2 branes: conical singularity (cf. cosmic strings in 4D) difficult to put matter on the brane ・ Codimension-3 or more branes: brane=singularity “black brane” (cf. black holes in 4D) We need regularized braneworld No successful cosmological model so far A possible resolution example in 10 dimensions: 9-dim braneworld = 4-dim + 5-dim internal space

6 Our aim Standard Friedmann eq. RS model (codimension 1 braneworld) How about in KK braneworld ? 5dim. curvature radius Constant of integration 5dim. gravitational coupling constant

7 4D Effective Theory E   SMS Formalism Gauss equation metric Junction Condition Matter on the brane Weyl tensor Shiromizu, Maeda and Sasaki (2000) This rate of change = Kμν 4dim. spacetime The SMS formalism is to write down the effective Einstein equation on the brane without solving the bulk. ∙∙∙ extrinsic curvature

8 RS Braneworld Cosmology 4D Effective Theory Dark Radiation Unknown! High Energy Effect Bulk Effect Homogeneous & Isotropic Cosmology High Energy Effect Integration Constant

9 Kaluza-Klein Braneworld Action Metric Bulk action Brane tension Brane actionBulk matter d+1-dim. Cosmological Constant d-dim. spacetime

10 Effective equation Einstein eq. Weyl tensor EM tensor on the brane Bulk EM tensor (0~d-1)

11 Effective equation Junction Condition Einstein eq. Projected Weyl tensor

12 Effective equation d-dimensional SMS equation High Energy Effects Information of bulk Bulk matter EM tensor on the brane Brane tension

13 KK Braneworld Cosmology (4+n)-dim metric n-dim internal coordinates 3 ordinary spatial coordinates We ignore the bulk matter for simplicity ex.) n=1 (cf. d=4+n)

14 Equations and Independent Variables Because of the symmetry, EM tensor takes the form Energy density Pressure Internal Pressure Variables: Einstein equation (8) MetricEM tensor Projected Weyl tensor We need one more condition. We assume an anisotropic fluid Equation of state (2) Traceless condition for E μν Bianchi identity (3) (1) (2) Weyl energy density Weyl Pressure Weyl Internal Pressure constant

15 Staticity assumption We put The system of eoms becomes closed. SMS eq. What we want is the effective Friedmann equation ・・・ Conservation law We need to know assumed to be stabilized by some mechanism.

16 Solving the Bianchi identity Bianchi identity or need to be considered separately.

17 Effective Friedmann Equation: the case The effective Friedmann equation Newton constant depends on the equation of state. KK cosmology KK Braneworld cosmology Forwe have to assume & 5D RS model case (n=0) transient instability around the matter-radiation equality?

18 Radiation-dominated case Logarithmic dependence The effective Newton constant could become negative The dark radiation component should be chosen appropriately in order to realize a sensible cosmology on the brane. The effective Friedmann equation Integration constant The Weyl tensor in the bulk geometry is constrained.

19 The case: The effective Friedmann equation Logarithmic dependence Integration constant This can have some impact at high energy.

20 Why KK braneworld is different? RS model Due to the homogeneity and isotropy of the brane, + symmetry Anisotropy of the brane affects the bulk geometry comes into the effective eq. on the brane even at low energies KK braneworld Bianchi identity Staticity assumption constrains the bulk geometry : Dark radiation : Non-trivial in 4-dim

21 Conclusion We have obtained the effective Friedman equation for the Kaluza-Klein braneworld We found that Friedman equation depends on the matter on the brane. Namely, the Newton gravitational “constant” varies as the equation of state changes. In particular, at the radiation dominant stage, the Newton constant has a logarithmic time dependence. When matter dominates this time dependence disappears. This can be regarded as a new stabilization mechanism of the Newton constant. We have found a possible transient instability which might lead to interesting cosmological consequences. We need to check Newton force.