Kouji Nakamura (NAOJ)

1. Construction of gauge-invariant variables for linear-order metric perturbation on general background spacetime References : K.N. Prog. Theor. Phys., 110(2003), 723. (arXiv:gr-qc/0303039). K.N. Prog. Theor. Phys., 113 (2005), 413. (arXiv:gr-qc/0410024). K.N. Phys. Rev. D 74 (2006), 101301R. (arXiv:gr-qc/0605107). K.N. Prog. Theor. Phys., 117 (2007), 17. (arXiv:gr-qc/0605108). K.N. Phys. Rev. D 80 (2009), 124021. (arXiv:0804.3840[gr-qc]). K.N. Prog. Theor. Phys. 121 (2009), 1321. (arXiv:0812.4865[gr-qc]). K.N. Adv. in Astron. 2010 (2010), 576273. (arXiv:1001.2621[gr-qc]). K.N. preprint (arXiv:1011.5272[gr-qc]). K.N. preprint (arXiv:1101.1147[gr-qc]). K.N. preprint. (arXiv:1103.3092[gr-qc]). Kouji Nakamura (NAOJ)

2. I. Introduction The higher order perturbation theory in general relativity has very wide physical motivation. • Cosmological perturbation theory • Expansion law of inhomogeneous universe (LCDM v.s. inhomogeneous cosmology) • 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. The first order approximation of our universe from a homogeneous isotropic one is revealed by the recent observations of the CMB. • To carry out this, it is necessary to discuss perturbation theories beyond linear order. • The second-order perturbation theory is one of such perturbation theories. (Bennett et al., (2003).) • It is suggested that the fluctuations are adiabatic and Gaussian at least in the first order approximation. • One of the next research is to clarify the accuracy of this result. • Non-Gaussianity, non-adiabaticity …and so on. • c.f. Non-Gaussianity is a topical subject also in Observations. • E. Komatsu, et al., APJ Supp. 180 (2009), 330; arXiv:1001.4538[astro-ph.CO]. • The first full-sky map of Planckwas • press-released on 5 July 2010!!! -----> 3

4. I. Introduction The higher order perturbation theory in general relativity has very wide physical motivation. • Cosmological perturbation theory • Expansion law of inhomogeneous universe (LCDM v.s. inhomogeneous cosmology) • 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.

5. It is worthwhile to formulate the higher-order gauge-invariant perturbation theory from general point of view. However, general relativistic perturbation theory requires very delicate treatments of “gauges”. • According to this motivation, we have been formulating the general relativistic second-order perturbation theory in a gauge-invariant manner. • General framework: • Framework of higher-order gauge-invariant perturbations: • K.N. PTP110 (2003), 723; ibid.113 (2005), 413. • Construction of gauge-invariant variables for the linear order metric perturbation: • K.N. arXiv:1011.5272[gr-qc]; 1101.1147[gr-qc]; 1103.3092[gr-qc]. • Application to cosmological perturbation theory : • Einstein equations : • K.N. PRD74 (2006), 101301R; PTP117 (2007), 17. • Equations of motion for matter fields: • K.N. PRD80 (2009), 124021. • Consistency of the 2nd order Einstein equations : • K.N. PTP121 (2009), 1321. • Summary of current status of this formulation: • K.N. Adv. in Astron. 2010 (2010), 576273. • Comparison with a different formulation: • A.J. Christopherson, K. Malik, D.R. Matravers, and K.N. arXiv:1101.3525 [astro-ph.CO]

6. metric on PS : metric perturbation : , metric on BGS : metric expansion : Our general framework of the second-order gauge invariant perturbation theory is based on a single assumption. linear order (assumption, decomposition hypothesis) : 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 . In cosmological perturbations, this is almost correct and we may choose as (longitudinal gauge, J. Bardeen (1980))

7. Problems in decomposition hypothesis • In cosmological perturbations, ..... • Background metric : • Zero-mode problem : • This decomposition is based on the existence of Green functions , , . • In our formulation, we ignored the modes (zero modes) which belong to the kernel of the operators , , . • How to include these zero modes into our consideration? • On general background spacetime, … --->Generality problem : • Is the decomposition hypothesis also correct in general background spacetime? : metric on maximally symmetric 3-space : curvature constant associated with

8. In this talk, ..... • We partly resolve this generality problem using ADM decomposition.

9. V. Summary • In this talk, ..... • We partly resolve this generality problem using ADM decomposition. • In our proof, we assume the existence of Green functions of two derivative operator in the simple case : ----> the zero-mode problem remains and it should be examined carefully. • Although our main explanation is for the case , a similar argument is applicable to the general case in which (The Green function of is necessary.) ---> the zero-mode problem is more delicate in general case. • We may say that the decomposition hypothesis • is almost correct for the linear metric perturbation • on general background spacetime.

10. II. “Gauge” in general relativity (R.K. Sachs (1964).) • There are two kinds of “gauge” in general relativity. • The concepts of these two “gauge” are closely related to the general covariance. • “General covariance” : There is no preferred coordinate system in nature. • The first kind “gauge” is a coordinate system on a single spacetime manifold. • The second kind “gauge”appears in the perturbation theory. • This is a point identification between the physical spacetime and the background spacetime. • To explain this second kind “gauge”, we have to remind what we are doing in perturbation theory.

11. Physical spacetime (PS) Background spacetime (BGS) III. The second kind gauge in GR. (Stewart and Walker, PRSL A341 (1974), 49.) “Gauge degree of freedom” in general relativistic perturbations 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 and this identification is called “gauge choice” in perturbation theory.

12. 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.) Through these understanding of gauges and the gauge-transformation rules, we developed second-order gauge-invariant perturbation theory.

13. III. Construction of gauge invariant variables in higher order perturbations metric on PS : metric perturbation : , metric on BGS : metric expansion : Our general framework of the second-order gauge invariant perturbation theory WAS based on a single assumption. linear order (decomposition hypothesis) : 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 .

14. Under the gauge transformation , the vector field is transformed as where is gauge invariant part and is gauge variant part. 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 :

15. : 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

16. Comments : gauge-variant parts v.s. gauge-invariant parts • As a corollary of these decomposition formulae, any order-by-order perturbative equation is automatically given in gauge-invariant form. (Gauge-variant parts are unphysical.) • The decomposition of the metric perturbation into gauge-invariant and gauge-variant parts is not unique. (This corresponds to the fact that there are infinitely many gauge • fixing procedure. Christopherson, et al., arXiv:1101.3525 [astro-ph.CO]) • Gauge-variant parts of metric perturbations also play an important role in the systematic construction of gauge-invariant variables for any perturbations. (In this sense, gauge-variant parts are also necessary.)

17. III. Construction of gauge invariant variables in higher-order perturbations metric on PS : metric perturbation : , metric on BGS : metric expansion : Our general framework of the second-order gauge invariant perturbation theory WAS based on a single assumption. linear order (decomposition hypothesis) : 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 . This decomposition hypothesis is an important premise of our general framework of higher-order gauge-invariant perturbation theory.

18. IV. Construction of gauge-invariant variables for linear metric perturbations on general background spacetime metric on PS : metric perturbation : , metric on BGS : (K.N. arXiv:1011.5272[gr-qc]) metric expansion : , . ADM decomposition of BGS : , . Gauge-transformation for the linear metric perturbation For simplicity, we first consider the case , (extrinsic curvature)

19. [cf. J. W. York Jr., JMP14 (1973), 456; AIHP21 (1974), 319.] Decomposition of the linear metric perturbation Here, and are the variables whose gauge transformation rules are given by We assume the existence of these variables at the starting point, but this existence is confirmed later soon. To accomplish this decomposition, we have to assume the existence of the Green functions of the derivative operators , : Ricci curvature on . This situation is equivalent to the case of cosmological perturbations. -----> zero-mode problem.

20. Gauge-variant parts of The above construction of gauge-variant parts implies that we may start from the decomposition of the components of : Gauge transformation rule : We have confirmed the existence of these variables, which was assumed at the starting point. Even if we start from this decomposition of , we reach to the same conclusion.

21. Gauge-invariant parts of Expression of the components of the original in terms of gauge-invariant and gauge-variant variables. • vector mode : • tensor mode : • scalar mode : Defining the gauge-invariant and gauge-variant part as The above components are summarized as the covariant form :

22. Inspecting these gauge-transformation rules, we construct gauge-variant, and gauge-invariant variables as in the previous simple case General case (1/4) (K.N. arXiv:1101.1147[gr-qc], K.N. arXiv:1103.3092[gr-qc]) • Gauge-transformation rules:

23. General case (2/4) Through the similar logic to the simple case , we can derive the following decomposition of hti and hij : We may start from this decomposition to derive the gauge- transformation rules for perturbative variables.

24. General case (3/4) • We can derive gauge-transformation rules for variables as To reach to these gauge-transformation rules, the Green function of the elliptic derivative operator is necessary.

25. General case (4/4) • Gauge-variant variables : • Gauge-invariant variables : • Definitions of gauge-invariant and gauge-variant parts : ---->

26. V. Summary • In this talk, ..... • We partly resolve this generality problem using ADM decomposition. • In our proof, we assume the existence of Green functions of two derivative operator in the simple case : ----> the zero-mode problem remains and it should be examined carefully. • Although our main explanation is for the case , a similar argument is applicable to the general case in which (The Green function of is necessary.) ---> the zero-mode problem is more delicate in general case. • We may say that the decomposition hypothesis • is almost correct for the linear metric perturbation • on general background spacetime.