Kaluza-Klein Braneworld Cosmology

1. Kaluza-Klein Braneworld Cosmology Misao Sasaki YITP, Kyoto University S Kanno, D Langlois, MS & J Soda, PTP118 (2007) 701 [arXiv:0707.4510]

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

3. The size of extra dimensions ＜ 10-17 cm from collider experiments. Kaluza-Klein (KK) Compactification Extra dimensions are too small for us to detect How small? 2-dimensional stick 1-dimensional stick

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

5. ～ 0.1mm exponential warping Randall-Sundrum (RS) Model RSII Model RSI Model Randall & Sundrum (1999) Brane tension Cosmological Constant AdS Minkowski Minkowski AdS curvature radius Z2 symmetry Z2 symmetry 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.

6. KK compactification on the brane : “KK Braneworld” Towards Higher Dimensional Braneworld Higher Codimension Braneworld conical singularity (cf. cosmic strings in 4D) difficult to put matter on the brane ・Codimension-2 branes: ・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

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

8. 4D Effective Theory - E mn SMS Formalism Shiromizu, Maeda and Sasaki (2000) The SMS formalism is to write down the effective Einstein equation on the brane without solving the bulk. metric ∙∙∙ extrinsic curvature 4dim. spacetime Gauss equation Junction Condition Weyl tensor Matter on the brane E This rate of change = Kμν μν Weyl tensor

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

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

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

12. Projected Weyl tensor Effective equation Einstein eq. Junction Condition

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

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

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

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

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

18. 5D RS model case (n=0) For we have to assume & transient instability around the matter-radiation equality? Effective Friedmann Equation: the case The effective Friedmann equation Newton constant depends on the equation of state. KK Braneworld cosmology KK cosmology

19. The Weyl tensor in the bulk geometry is constrained. Radiation-dominated case The effective Friedmann equation Integration constant 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.

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

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

22. 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.