1. Kouji Nakamura (Grad. Univ. Adv. Stud. (NAOJ))
Gauge-invariant Formulation of the Second-order Cosmological Perturbations:
--- Single scalar field case ---
Kouji Nakamura
(Grad. Univ. Adv. Stud. (NAOJ))
References :
K.N. Prog. Theor. Phys., 110 (2003), (gr-qc/ ).
K.N. Prog. Theor. Phys., 113 (2005), (gr-qc/ ).
K.N. Phys. Rev. D 74 (2006), R (gr-qc/ ).
K.N. Prog. Theor. Phys., 117 (2007), (gr-qc/ ).
K.N. Proc. of LT7 (2008) (arXiv : [gr-qc]).
K.N. arXiv : [gr-qc]　　 rejected by PTP.
K.N. arXiv : [gr-qc]  accepted by PTP, last week.

2. I. Introduction
The second 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 by WMAP.
(Bennett et al., ApJ. Suppl. 148, 1 (2003).)
It is suggested that the fluctuations are adiabatic and Gaussian at least in the first order approximation.
One of the next theoretical research is to clarify the accuracy of this result.
Non-Gaussianity, non-adiabaticity …and so on.
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.
c.f. The detection of non-Gaussianity is a topical subject also in Observation.
E. Komatsu, et. al., arXiv: [astro-ph]

4. I. Introduction
The second 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
However, general relativistic perturbation theory
requires very delicate treatments of “gauges”.
It is worthwhile to formulate the
higher order gauge invariant perturbation
theory from general point of view.
According to this motivation, we have been formulating the general relativistic second-order perturbation theory in a gauge-invariant manner.
General framework:
K.N. PTP, 110 (2003), 723; ibid, 113 (2005), 413.
Application to cosmological perturbation theory :
Einstein equations (the first-order : scalar mode only).
K.N. PRD, 74 (2006), R; PTP,117 (2007), 17.
Equations of motion for matter fields :
K.N. arXiv: [gr-qc].
Consistency of the 2nd order Einstein equations including all modes (perfect fluid, scalar field).
K.N. arXiv: [gr-qc].

6. In this talk, ...
We will show the essence of our gauge-invariant formulation of the second-order cosmological perturbations through the single scalar field case (as a simple example of matter field).
Table of Contents
II. Taylor expansion of tensors on manifolds
III. “Gauge” in general relativity
IV. Gauge degree of freedom in perturbations
V. Gauge invariant variables
VI. Perturbative Einstein and Klein-Gordon Eqs.
VII. Summary

7. II. Taylor expansion of tensors on a manifold
The Taylor expansion of tensors is an approximated form of tensors at 　 (in　 　　) in terms of the variables at 　 (in 　 　　).
One parameter family of diffeomorphisms :
　　　　　　　　　　　　　,　　　　　　　　　　　　　
Taylor expansion of a function
Taylor expansion of a function is regarded as that of the diffeomorphism , and general arguments lead

8. The general form of the second order Taylor expansion is
and are the generators of the diffeomorphism and these represent the direction along which the Taylor expansion is carried out.
This equation is also regarded as the definitions of these generators.
This equation also gives a representation of the pull-back of general class of diffeomorphisms up to second order.
is extended to diffeomorphisms acting on all tensor fields.
This expression of the Taylor expansion is also extended to those of all tensor fields and this expression of the Taylor expansion is quite general.
(Sonego and Bruni, CMP, 193 (1998), 209.)
Through this general representation of diffeomorphisms, we develop general relativistic second order perturbation theory in gauge invariant manner.

9. When the function is a coordinate function ,
Example :
When the function is a coordinate function ,
Coordinate system

13. Correction term
by the exponential map
(Lie transport)

15. The general form of the second order Taylor expansion is
and are the generators of the diffeomorphism and these represent the direction along which the Taylor expansion is carried out.
This equation is also regarded as the definitions of these generators.
This equation also gives a representation of the pull-back of general class of diffeomorphisms up to second order.
is extended to diffeomorphisms acting on all tensor fields.
This representation of the Taylor expansion is also extended to those of all tensor fields and this expression of the Taylor expansion is quite general.
(Sonego and Bruni, CMP, 193 (1998), 209.)
Through this general representation of diffeomorphisms, we develop general relativistic second order perturbation theory in gauge invariant manner.

16. III. “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.

17. The first kind “gauge” = a coordinate system on a manifold.
The definition of a manifold includes following property :
: coordinate system of
a manifold M.
= a “gauge” choice.
: coordinate transformation
=“gauge” transformation
“Gauge” issue of the first
kind is “which coordinate
system is convenient?”.
The solution to this problem
depends on the problem which
we are concerning.
(or covariant theory)

18. III. “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.

19. IV. Gauge degree of freedom in perturbations
(Stewart and Walker, PRSL A341 (1974), 49.)
Physical spacetime (PS)
Background spacetime (BGS)
“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 different spacetimes.
This identification is called “gauge choice” in perturbation
theory.

20. The gauge choice is not unique by virtue of general covariance.
Physical spacetime (PS)
Background spacetime (BGS)
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 : , , ,

21. Gauge transformation rules of each order
(Bruni et. al.)
Expansion of gauge choices :
We assume that each gauge choice is an exponential map. ⇒
Expansion of the variable :
Order by order gauge transformation rules

22. V. Gauge invariant variables
Inspecting the above gauge transformation rules, we can define
the gauge invariant variables for the metric perturbations and the
other matter fields.
( e : parameters for perturbations)
metric perturbation :
metric on PS : ,
metric on BGS :
metric expansion :
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

23. Cosmological perturbation case
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

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

25. V. Gauge invariant variables
Inspecting the above gauge transformation rules, we can define
the gauge invariant variables for the metric perturbations and the
other matter fields.
( e : parameters for perturbations)
metric perturbation :
metric on PS : ,
metric on BGS :
metric expansion :
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

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

27. Decomposition of the second-order perturbation
Gauge transformation rule :
We define the new variable of second order :
by using linear order metric perturbation and
Gauge transformation rule for : ,
This rule is same as that of the linear metric perturbation.
Hence, we can decompose so that ,
Components of gauge invariant variable :

28. 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
: gauge invariant part
: gauge variant part

29. VI. Perturbative Einstein and Klein-Gordon Eqs.
Using the above definition of gauge invariant variables,
we derive the perturbative form of the curvatures in terms of
gauge invariant variables.
Applying the standard derivation of curvatures in the text
book (Wald (1984)), the relation between the Riemann curvatures
on BGS and PS is given by
where
Each order expression of the perturbative curvature is derived
by the direct expansion of this relation.

30. Energy momentum tensor (scalar field)
Perturbative expansion of the scalar field
First order gauge invariant variables
Second order gauge invariant variables

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

32. Gauge invariant parts of Einstein tensor are defined by
where .

33. Klein-Gordon equation ( )
Perturbations of the Klein-Gordon equation
Klein-Gordon equation ( )
Perturbative expansion of the Klein-Gordon equation :
We can show that each order perturbation of this Klein-Gordon equation is decomposed into the gauge-invariant and gauge-variant parts as
: gauge invariant part
: gauge variant part

34.  This formulation is self-consistent !
We have also shown that the perturbative expansion of the following variables are given in the similar form. (K.N., gr-qc/ )
Riemann　curvature
divergence of an arbitrary tensor of second rank (eq. of motion) ,
Scalar　curvature
Ricci curvature ,
Weyl curvature
The explicit formulae can be seen in the above reference. ,
Bianchi identity
We can check this order by order.
 This formulation is self-consistent !

35.  These decomposition formulae are universal!
Further, we also showed that the perturbative expansion of the following energy momentum tensor and equations of motion (the divergence of the energy momentum tensor) are explicitly decomposed into gauge-invariant and gauge-variant parts in the similar manner. (K.N., arXiv: [gr-qc])
Perfect fluid
Energy momentum tensor.
Energy continuity equation.
Euler equation.
Imperfect fluid
Energy momentum tensor.
Energy continuity equation.
Navier-Stokes equation.
Scalar field
Energy momentum tensor.
Klein-Gordon equation. ,
 These decomposition formulae are universal!

36. We impose the Einstein equation of each order,
Then, the Einstein equation of each order is automatically given
in terms of gauge invariant variables :
linear order : ,
second order : .
Further, each order perturbation of the Klein-Gordon equation is also automatically given in the gauge invariant form :
Linear order :
Background :
Second order :
We do not have to care about gauge degree of freedom at least
in the level where we concentrate only on the equations.

37. Background Einstein equations and Klein-Gordon equation
Hamiltonian constraint :
Evolution equation :
Klein-Gordon equation :
It is well-known that these three equations are not independent.

38. First order Einstein equations and Klein-Gordon equation
Master equation of the linear order Einstein equations :
Momentum constraint (scalar mode, vector mode) :
Potential perturbation :
traceless part of the spatial component of Einstein equation ,
tensor mode (evolution equations) ,
Klein-Gordon equation (this is not independent eq.):

39. Second order Einstein equations (scalar)
Master equation of the second-order Einstein equations :
Momentum constraint (scalar mode) :
potential perturbation :
traceless part of the spatial component of Einstein equation : .

40. Second order Einstein equations (source terms 1)
: scalar-scalar
: scalar-tensor
: tensor-tensor
Mode coupling :

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

42. Second order Einstein equations (vector- and tensor-mode)
Momentum constraint (vector mode) :
Evolution equation for vector mode :
tensor mode (evolution equation)
These equations imply that the first-order perturbations may generate the vector- and tensor-modes of the second order.

43. Second order Klein-Gordon equation
The second-order perturbation of the Klein-Gordon equation in the cosmological situation is given by
where
This second-order Klein-Gordon equation is not independent of the second-order Einstein equations.
We use this fact to check the consistency of all equations.

44. Consistency of the Einstein equations of the second order
Since momentum constraint for the vector mode is an initial value constraint, it should be consistent with the evolution equation of the vector mode. This consistency of equations leads an identity :
Further, through this identity, we can confirm that the all Einstein equations for the second-order scalar mode are consistent with each other.
The consistency between the Klein-Gordon equation and the Einstein equation of the second order leads an identity :
Actually, we have confirmed that these identities are guaranteed by the background Einstein equations and the first-order perturbations of the Einstein equation.
 Our derived set of equations for the second-order
perturbations are self-consistent.

45. VII. Summary
Based on the general framework of the general relativistic 2nd-order
perturbations in [K.N., PTP 110 (2003), 723; ibid, 113 (2005), 413.],
we have derived the all components of the 2nd-order Einstein
equations and equations of motion for matter fields in
gauge-invariant manner.
[ K.N., arXiv: [gr-qc]; arXiv: [gr-qc]. ]
Further, we have confirmed that the set of all components
of the second-order perturbation of the Einstein equations and
the equations of motion for matter field are self-consistent.
[K.N. arXiv: [gr-qc]]
Therefore, we may say that we are ready to discuss the physical
behaviors of the second-order cosmological perturbations.
The above general framework of 2nd-order perturbations does work in the case of cosmological perturbations, and it will have many applications.

46. List of Application (1)
Second-order cosmological perturbations (in progress)
Second-order perturbation of the Einstein equation :
In the case of the cosmological
perturbations, these terms are
almost completely derived.
The next task is to clarify the nature of the second-order perturbations of this energy momentum tensor.
Incomplete parts are in homogeneous modes!!!
At this moment, homogeneous modes are not included in our formulation, which should be included for completion.
Classical behaviors of the second-order perturbations.
This is a preliminary step to clarify the quantum behaviors of perturbations in inflationary universes.
Comparison with the long-wavelength approximations.
Multi-fluid or multi-field system
Einstein Boltzmann system (treatments of photon and neutrino) [N. Bartolo, et. al., ( ); C. Pitrou, et. al., (2008); L. Senatore, et. al. (2008).]
---> Non-linear effects in CMB physics.

47. List of Applications (2)
The correspondence between observables in experiments (observation) and gauge invariant variables defined here.
Example 1: The relation between the gauge invariant variables and
temperature perturbation of the second order.
Example 2: The relation between gauge invariant variables and
phase difference in the laser interferometer for GW
detection.
Post-Minkowski expansion for a binary system (alternative to post Newtonian expansion).
Second-order perturbation of the Einstein tensor is already given !!!
But we have to specify the energy momentum tensor of a binary system. (Some regularizations are necessary).
Applications to the black hole perturbation theories ...
etc ...
There are many applications to which our formulation should be applied.
I want to clarify these problems step by step.