Transcript Diapositive 1 - Tokyo Institute of Technology

Brane world
cosmological perturbations
and massive gravity
Cédric Deffayet
(IAP, GReCO)
Tokyo 2003
1/ DGP model (in 5D)
or « brane induced gravity »
and « massive gravity »
2/ Well posed brane World
cosmological perturbations
3/ van Dam-Veltman-Zakharov
discontinuity on FLRW spacetime
Use of DGP brane world
model as a tool to
investigate problems of
« massive gravity »
1. The DGP model.
Dvali-Gabadadze-Porrati ‘00
Consider a 3-brane in a 5D bulk with action
S = S(5) + S(4) + Smatter
with:
Bulk action:
Brane
action
S(5) =
3
M (5)
S(4) =
R4
p
d xdy g(5) R( 5)
2R 4
MP d x
R4 p
Smatter = d x
p
g(4) R(4)
gL matter
The bulk is taken to be Minkowskian (infinite and
flat). This can be generalized to more complicated bulks.
Then the Newtonian potential on the brane
behaves as
V(r ) /
V(r ) /
1
r
1
4D behavior at small distances
5D behavior at large distances
r2
The crossover distance between the two
regimes is given by
This enables to gets a 4D looking theory of gravity
out of one which is not, irrespectively of the
compactness (Kaluza-Klein) or “curvature” (RandallSundrum) of the bulk.
But tensorial structure of the
Graviton propagator ….
A small fluctuation h ö÷ over a flat background ñö÷
obeyes the equation of motion
3
M (5) @A @A h ö÷ +
2
M Pî (y)@ú@úh ö÷
… Is that of a
massive graviton !
ð
ñ
= î (y) Tö÷ à 13ñö÷T
+
2
y
M Pî (y)@ö@÷hy
Brane is located in
y=0
Leads to the van Dam-Veltman-Zakharov
discontinuity on Minkowski background!
NB: in terms of KK modes
V(r ) ø
R1
dm
eà mr
0 4+ m2r c2 r
vDVZ discontinuity
and
Pauli-Fierz action
Pauli-Fierz action: second order action
for a massive spin two h ö ÷
R4 p
2
ö÷
2
à
à
d x g(R 2Ë) + m (hö÷h
h)
second order in h
Only Ghost-free (quadratic) action for a
massive spin two Pauli, Fierz
Coupling the graviton with a conserved energy-momentum tensor
R4 p
ö÷
Sint = d x ghö÷T
h
ö÷
R ö÷ëì
= D
(x à x 0)Tëì (x 0)d4x 0
The amplitude between two conserved sources T and S
R 4 ö÷
is given by
A =
d xS (x)hö÷(x)
This amplitude is given by
Lichnerovitz
operator
Higuchi,Porrati,
Kogan-Mouslopoulos-Papazoglou
h
A / Sö ÷(É (L2) à 2Ë + m 2) à 1Töð÷
ñ
i
2
1
m
à S(à r 2 à 2Ë + m2)à 1 2 + 2(2Ëà 3m2) T
If Ë = 0 :
1
1
à
2
6
=
1
3
Discrete difference with the massless case (1/2). This stays
in the limit m2 ! 0: van Dam, Veltman, Zakharov discontinuity
2
6
If Ë= 0 and m ! 0 :
No discontinuity at classical level (Higuchi,Porrati,
Kogan-Mouslopoulos-Papazoglou), NB: in dS, no unitarity for small m
Can it be generalized to other curved background?
Here we want to use DGP model to investigate
this issue on FLRW background.
The rôle of m2 is played in DGP model by rc-1
So we are interested in the limit rc >> H-1
Homogeneous cosmology of DGP model
C.D. Phys. Lett. B
Energy momentum tensor
The standard techniques for deriving brane cosmology
of brane « real matter »
can be applied here as well
(4)
Gö÷
= Í ö÷ à Eö÷ with Í
Shiromizu, Maeda, Sasaki
ö÷
)
(4)
2
quadratic in Sö ÷ = T(M
ö÷ à Gö÷ =ô(4)
Quadratic equation for G(4)
ö÷ leading to the Friedmann
Equation for DGP brane world (Z2 symmetric):
Total energy momentum
Effective
tensor of the brane
energy
Asymptotes to
“brane effective matter”
momentum
standard
Friedmann
equation tensor
for smallof
Weyl
Hubble radius
induced
With ï = æ
1
“fluid”
gravity
C.D. Phys.Rev D.+ unpublished
The most general scalar perturbations of the 5D metric gAB
read (with background ds2= -n2 dt2 + a2 dxi dxj ij + dy2)
•7 scalar variables: A,B,R,E,By,Ay,Ayy
•3 gauge transformations
) 4 Gauge Invariant variables all related to a single master
variable .  obeys a master equation which solves all the
bulk Einstein equations (Mukohyama; Kodama,Ishibashi,Seto)
Note, that in the following, we will always consider quantities as Fourier
Transformed with respect to brane spatial coordianates i.e.  ! k2
Mukohyama, Kodama,Ishibashi,Seto;
Langlois, Maartens, Sasaki, Wands;
Langlois; Bridgman,Malik,Wands; Marteens
Scalar brane world cosmological perturbations
1.Bulk metric
Scalar brane world cosmological perturbations
2.Brane matter and induced metric
•Brane induced metric, choose a 4D longitudinal gauge:
•Brane localized sources
The scalar perturbations of which can be decomposed as
Density
perturbation
Momentum
perturbation
pressure
perturbation
Anisotropic
stress
The same decomposition holds for
•Real brane matter
•The Weyl fluid
•The effective energy momentum tensor of induced gravity
3.Relations between brane quantities
and master variable
In the remaining, we will assume  (M)=0
(vanishing anisotropic stress for real matter)
Then, all of the quantities of interest on the brane
( (M),  q(M),  P(M), , ,  (),  P(),  q(),  ())
are expressible in term of the master variable 
e.g.:
Note that those expression only contain 0th and 1st order y
derivatives of , all the projected Einstein equations are
identically satisfied (whatever )
The coefficients C are given by…
… some background dependent expressions
Where the only dependence
in rc comes in:
…
To summarize, when  (M) vanishes :
² On the brane, all quantities of interest are solely
expressible in term of 
The problem is entirely solved once one knows 
² In the bulk,  obeyes the master equation
) in addition to initial data, need for a boundary condition
on the brane. This is provided e.g. by an equation of state
(and not only by junction conditions!), i.e. a relation
between (M), P(M), q(M),…
e.g. ² for adiabatic perturbations:
² for a scalar field:
(B background expectation value of the scalar)
This lead to a « non local » boundary condition, i.e. one
containing derivatives along the brane
(Kodama,Ishibashi,Seto for RS model)
One has to solve the following differential problem
Associated with Brane Cosmological perturbations
Y
1/ The master equation
Reading in characteristic coordinates
O
2/ A « non local »
boundary condition on the brane
brane
Brane
cosmic
time
X
bulk
with r>1, and (0,r)=0
3/ Initial data to be provided in the bulk
The only non standard aspect of this problem is the
Nonl locality of the boundary condition
This can however be recast into a
NB: this system
has
more standard form
one ingoing
i  j 
•Define
u
=

characteristic (i,j)
for one
X Y
Set by the boundary
with 0condition
· i · r, 0 · j · r.
condition
boundary
) (r+1)(r+2)/2 unknowns u(i,j)
•The master equation is then recast into
Linear hyperbolic system of order (r+1)(r+2)/2
With characteristic curves X= const, Y=const
NB: the brane is always non characteristic
1. Non characteristic initial curve:
•If one specify for  Cauchy type of data on
the initial curve (i.e.  and n ) then,
because  obeys an order 2 PDE (the
master equation) then all the higher order
derivatives of  are known on the initial
curve
) all the ui,j are known there
•Note that this specification is in general
not compatible with the boundary condition
in O which is a constraint between the
functions ui,j in O
Y
X

n
Y
2.Characteristic initial curve
Xk 
O
X

•One can only specify on the initial curve the value
of , and in O (and only in O), the values of all the
derivatives kX. (k· r) Then the ui,j are known on
the initial curve.
• Here as well one should take care on the
compatibility with the boundary condition on the
brane.
In both cases 1. and 2., the hyperbolic problem
is well posed. I.e. There is a unique solution for the ui,j
(hence for ) and this solution depends continuously
on any parameters initial and boundary data may
depend on.
vDVZ (dis)continuity on FLRW space-times
C.D. in preparation
1. Brane world cosmological perturbations
and vDVZ discontinuity on Minkowski
background
2. Brane world cosmological perturbations
and vDVZ continuity on non FLRW spacetime with non vanishing scalar curvature
Setting H to zero, a(b) constant, and keeping rc to be
finite in the previous expressions, for cosmological
perturbation in DGP model, yield:
Remember that
onethat
has(b) differs in general from (b) while
Note
we only considered a perfect fluid source!
This remains true even if we send rc to infinity
The difference between (b) and (b) comes
(after setting H to zero) and is a sign of the
from the Weyl fluid anisotropic stress
vDVZ discontinuity!
Let us look at the gravitational potential generated
by a static source on a Minkowski background.
If we neglect the time derivatives in the previous
expressions, they reduce to
That we recast into
(defining 4  =  ,
and setting a(b) to
one)
While the master equation + the equation for
(M) can be recast in the unique equation
This was solved by Dvali, Gabadaze, Porrati, with the
result that at distances smaller that rc,  obeyes
I.e. at small distances 
obeyes the same equation
as the usual Newtonian
potential…
vDVZ discontinuity!
Standard Poisson
equation with standard
normalization
… But
Let us now turn to general FLRW background,
We are interested in the limit: H finite (but time
dependent), r c ! 1
(while keeping ô 2
(4)
fixed)
In this limit:
The background geometry stays perfectly regular,
and was shown to obey the standard 4D
Friedmann equations.
The master equation (which does not
depend on r c ) stays untouched.
The boundary condition for the master
variable has a well behaved limit, as
long as the scalar curvature of the
background FLRW space-time does not
vanish
) the differential problem associated with the cosmological
perturbations has a well defined limit, which, once initial
data are specified, unables to compute  everywhere.
Once  is known, all the 4D perturbations on the
brane are known, and they all have a well behaved
limit (as long as the FLRW background has a non
vanishing scalar curvature).
E.g. (b) is given by
But, one can verify explicitly that (b)0, (b)0,  (M)0,  q(M)0
and  P(M)0 obey the Standard 4D perturbed Einstein
equation ) the vDVZ discontinuity has disappeared!
(in particular (b)0 = (b)0)
Summary:
• Despite the non local aspect of the brane
boundary condition for scalar perturbation, the
differential problem associated with it can be
shown to be well posed
•vDVZ continuity on general FLRW background
for the case of matter fields with vanishing
anisotropic stress and adiabatic perturbations (or
scalar fields)
Interesting open question: interplay
between cosmological disappearance
of vDVZ (via  or FLRW) and brane
bending effect (à la Vainshtein)?