Local Conservation Law and Dark Radiation in Brane Models Masato Minamitsuji Osaka University (D1) In collaboration with Misao Sasaki (YITP) October 30th _ 31st, 2003 @Tokyo Institute of Technology

§1.Introduction (a) For the case of static bulk Static Bulk; Effective cosmology on the brane ; (b) For the case of dynamical bulk In this case, we need to generalize the to the locally conserved quantity as introduced. We investigate the bulk geometry and brane cosmology with the energy inflow from the brane, in terms of the local conservation law.

§2. Locally Conservation Law in the dynamical Bulk Bulk = space-time with the maximally symmetric 3-space, 5D Einstein system with a negative cosmological constant as The flux vector field ; The locally conservation law ; 33

The locally conserved quantity; the matter part + the cosmological constant part We will focus on the matter part. The matter part is related to the local energy-momentum tensor by using the Einstein equation as, Weyl tensor (has only one non-trivial component) the locally conserved quantity part + the local matter part

§2.2 Brane cosmology in the dynamical bulk Brane = a singular boundary of the dynamical space-time. The brane moves along the trajectory parameterized by The junction condition + - symmetry The effective Friedmann equation on the brane is given by ◎The projected Weyl tensor Local conservation quantity part (“dark radiation”) + Remaining bulk energy-momentum part uniquely decomposed.

§3. Collapse of the Bulk - Null dust model In this section, we analyze the backreaction problem of the bulk geometry. For the first step, we approximate the bulk matter as null dust . §3-1. Set up Null dust: To satisfy the energy conservation law, and are determined as 　The energy variation rate along the null directions is given as where and are determined by the Einstein equations.

§3-2. Apparent horizons The and are related to the out- and in-going expansion of the congruence of the null geodesics, respectively. out-going expansion rate in-going expansion rate (i). Out-going apparent horizon (ii). In-going apparent horizon Einstein equations

The space-time is classified into 3 regions [I]. “ Normal region ”. [II]. “ Black Hole region ” . [III]. “ White Hole region ” . We discuss the condition of the formation of apparent horizons for the following two cases about the condition that the energy flux is turned on. (a) (b) For the situation (a) and (b), we can obtain the results about the possibility of formation of apparent horizons.

[I]. (a). (b). [II]. (a). (b). [III]. (a). 　(b).

§3-3. The case for the single null dust For the case that single in-falling null dust exists, the dynamical collapse of the bulk space-time is analyzed. The in-falling null dust Generated by the decay or collisions of matter on the brane 　 Transmitted along the constant lines . The bulk geometry exact solution Locally conserved quantity; In this case, we can analyze the bulk geometry and the effective cosmology on the brane. BH ?

Schematic View of Single Null Dust Collapse ○Location of apparent horizon ; ○Trajectory of apparent horizon ; The apparent horizon is space-like and becomes null for For example, considering null dust collapse as follows, (a) AdS : (b) AdS-Vaidya : (c) AdS-Sch. :

From the point of view of the bulk The proper time on the brane the advanced time in the bulk Now, we require that the brane trajectory is time-like and (i). (ii). (iii). (iv). The existence possibility of the brane in the region [I] ~ [III]; [I] The expanding and contracting can exist. [II] The contracting brane can exist. Limitation on [III] No time-like brane can exist.

From the point of view of the brane The dynamics on the brane is determined by the equations If the intensity is given or related to the quantities on brane, brane trajectory energy density on the brane locally conserved quantity self-consistently determined.

§3-4. The case for the double null dust It is worth mentioned that if the both of the null dust exists. In-falling flux exponentially enhanced by the out-going flux Out-going flux exponentially suppressed by the in-going flux. cross flow region ; Enhancement factor of in-falling flux Suppression of the outgoing null flux

for the AdS cosmological constant, Perturbative analysis for the AdS cosmological constant, For the closed chart ( ) case , For the constant intensity out-flow, we evaluate the enhancement factor near the future Cauchy horizon ( ) as ; No problem in the perturbative level.

§4. Summary and Future Works We would like to investigate the bulk geometry and brane cosmology in terms of the local conservation law. (a). For the bulk with maximally symmetric 3-space, we can exactly decompose the locally conserved quantity and the remaining bulk energy momentum. (b). For the case of the null dust collapse, we confirm the formation of apparent horizons and BH. (c).　For single null dust, the exact solution of bulk geometry exists and the dynamics on the brane is determined self-consistently . (d). For double null dust, the in-falling flux is enhanced by out-one. No problem for the perturbative level. The enhancement at dynamical level have to be analyzed. Future Works ; (a). Analysis of bulk collapse for the ingoing scalar wave and the dynamics on the brane . (b). Quantum effects with respect to the bulk BHs .