２次ゲージ不変摂動論定式化の進行状況

1. ２次ゲージ不変摂動論定式化の進行状況 Kouji Nakamura (Grad. Univ. Adv. Stud. (NAOJ)) References : K.N. Prog. Theor. Phys., vol.110 (2003), 723. (gr-qc/0303039). K.N. Prog. Theor. Phys., vol.113 (2005), 413. (gr-qc/0410024). K.N. Phys. Rev. D 74 (2006), 101301R. (gr-qc/0605107). K.N. Prog. Theor. Phys., vol.117 (2007), 1 7. (gr-qc/0605108). + a

2. I. Introduction The second order perturbation theory in general relativity has very wide physical motivation. • Cosmological perturbation theory • Expansion law of inhomogeneous universe (backreaction effect, averaging problem) • Non-Gaussianity in CMB (beyond WMAP) • Black hole perturbations • Radiation reaction effects due to the gravitational wave emission. • Close limit approximation of black hole - black hole collision (Gleiser, et.al (1996)) • Perturbation of a star (Neutron star) • Rotation – pulsation coupling (Kojima 1997) There are many physical situations to which higher order perturbation theory should be applied.

3. It is worthwhile to formulate the higher order gauge invariant perturbation theory from general point of view. However, general relativistic perturbation theory requires more delicate treatments of “gauges”. • In this poster presentation, we show … • General framework of the second-order gauge-invariant perturbation theory. (K.N. PTP, 110 (2003), 723; ibid, 113 (2005), 413.) • Applications • Second-order cosmological perturbations (K.N. PRD, 74 (2006), 101301R; PTP,117 (2007), 17.) • Extension to include the first-order vector and tensor modes. (+a)

4. Physical spacetime (PS) Background spacetime (BGS) II. The second kind gauge in GR. (Stewart and Walker, PRSL A341 (1974), 49.) “Gauge degree of freedom” in general relativistic perturbations (i.e., “gauge” of the second kind) arises due to general covariance. In any perturbation theories, we always treat two spacetimes : • Physical Spacetime (PS); • Background Spacetime (BGS). In perturbation theories, we always write equations like Through this equation, we always identify the points on these two spacetimes. This identification is called “gauge choice” in perturbation theory.

5. Physical spacetime (PS) , , Background spacetime (BGS) , The gauge choice is not uniqueby virtue of general covariance. General covariance: • “There is no preferred coordinates in nature” (intuitively). Gauge transformation : • The change of the point identification map. • Different gauge choice : • Representation of physical variable : • Gauge transformation :

6. Gauge transformation rules of each order • Expansion of gauge choices : We assume that each gauge choice is an exponential map. -------> • Expansion of the variable : • Order by order gauge transformation rules : (Sonego and Bruni, CMP, 193 (1998), 209.) Based on this understanding of gauges, we develop second-order gauge-invariant perturbation thoery.

7. metric on BGS : linear order (assumption) : Suppose that the linear order perturbation is decomposed as so that the variable and are the gauge invariant and the gauge variant parts of , respectively. These variables are transformed as under the gauge transformation . III. Gauge invariant variables ( e : parameters for perturbations) metric on PS : metric perturbation : , Inspecting the above gauge transformation rules, we can define the gauge invariant variables for the metric perturbations and the other matter fields. metric expansion :

8. Background metric : metric on maximally symmetric 3-space metric perturbation decomposition of linear perturbation Uniqueness of this decomposition ---> Existence of Green functions , , : curvature constant associated with the metric Cosmological perturbations

9. where . gauge invariant variables : where Gauge variant and invariant variables of linear order metric perturbation. gauge variant variables : (J. Bardeen (1980))

10. Under the gauge transformation the vector field is transformed as is gauge variant part. where is gauge invariant part and Second order : Once we accept the above assumption for the linear order metric perturbation , we can always decompose the second order metric perturbations as follows : Components of gauge invariant variable in cosmological perturbations :

11. : gauge invariant part : gauge variant part Perturbations of an arbitrary matter field Q : Using gauge variant part of the metric perturbation of each order, gauge invariant variables for an arbitrary fields Q other than metric are defined by First order perturbation of Q : Second order perturbation of Q : These implies that each order perturbation of an arbitrary field is always decomposed as

12. , , , Energy momentum tensor (perfect fluid) • First order gauge invariant variables • Second order gauge invariant variables

13. , , : gauge invariant part : gauge variant part Perturbations of Einstein tensor and Energy momentum tensor • First order : • Second order :

14. linear order : second order : IV. Gauge Invariant Einstein equations We impose the Einstein equation of each order, Then, the Einstein equation of each order is necessarily given in terms of gauge invariant variables : , . We do not have to care about gauge degree of freedom at least in the level where we concentrate only on Einstein equations.

15. First order Einstein equations , • Components of perturbation of the fluid four-velocity : • energy density perturbation : , , • velocity perturbation (scalar) : , • Pressure perturbation : , • traceless part of the spatial component of Einstein equation , • Vector mode (evolution and constraint equations) . , , • tensor mode (evolution equations)

16. Second order Einstein equations (scalar) • components of perturbation of the fluid four-velocity , , • energy density perturbation . . • pressure perturbation . • velocity perturbation . • traceless part of the spatial component of Einstein equation .

17. Mode coupling : : scalar-tensor : scalar-scalar : scalar-vector : tensor-tensor : vector-vector : vector-tensor Second order Einstein equations (source terms1)

18. Mode coupling : : scalar-tensor : scalar-scalar : scalar-vector : tensor-tensor : vector-vector : vector-tensor Second order Einstein equations (source terms 2)

19. Second order Einstein equations (source terms 3)

20. Second order Einstein equations (scalar master eq.) • Eq. of state (second order) • Master equation These equations will be useful to discuss the non-linear effects in CMB.

21. Second order Einstein equations (vector, tensor) • vector mode (rotational part of the velocity perturbation) • vector mode (evolution equation) • tensor mode (evolution equation) These equations imply that the first order scalar perturbation may generate the second order vector and tensor modes.

22. V. Summary Based on the general framework of the general relativistic second order perturbations in [K.N., PTP 110 (2003), 723; ibid, 113 (2005), 413.], we have derived the Einstein equations for the second order gauge invariant variables in cosmological perturbations. [cf. K.N., PRD74(2006), 101301(R); PTP 117 (2007), 17.] In this derivation, we did not ignore any variables of metric perturbations and showed that, in principle, all types of mode couplings arise in the second order. This implies that we can extend the above general framework of the second order perturbation theory to more realistic situations of cosmology. • We are going to extend our formulation shown here to more realistic situation in cosmology. • We can also expect that the above framework has very many applications.

23. List of application candidates (1) • Second-order cosmological perturbation theory (in progress) • Ignoring the first order vector- and tensor-modes • Single perfect fluid system. (OK) • Single scalar field system. (OK) • Extension of our formulation to include the first order vector- and tensor-modes. • Single perfect fluid system (almost OK, (JPS ’07A)) --> (OK) • Single scalar field system (in progress) • Extensions to imperfect fluid system and to the multi-fields system (in progress) • Extensions to the Einstein-Boltzmann system • Nonlinear effects in CMB physics. • Vector and tensor modes generated by the density perturbation (which should exist.)

24. There are many applications to which our formulation should be applied. I want to clarify these problems step by step. List of application candidates (2) • Radiation reaction effect of the gravitational wave emission. • Schwarzschild case ... Problems in the treatments of l=0,1 modes. • Kerr case ... ??? • The correspondence between observables in experiments (observation) and gauge invariant variables defined here. • Ex. The relation between gauge invariant variables and phase difference in the laser interferometer for GW detection. • Post-Minkowski expansion alternative to post Newtonian expansion (post-Minkowski description of a binary system). • Second-order perturbation of the Einstein tensor is already given!!! • But we have to specify the energy momentum tensor of a binary system. • … etc.