Transcript Nonlinear analysis in Horava gravity

Nonlinear superhorizon perturbations
(gradient expansion)
in Horava-Lifshitz gravity
泉 圭介
Keisuke Izumi (LeCosPA)
Collaboration with Shinji Mukohyama(IPMU)
Phys.Rev. D84 (2011) 064025
Outline
Horava gravity
Motivation: renormalizable theory of gravitation
Symmetry of this theory: foliation-presearving diffeomorphism
Action
Linear analysis and importance of non-linearity
Gradient expansion and our result
Approximation
Intuitive understanding in 0th order
Application to Horava theory and our result
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Horava gravity
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
3
Quantum gravity
Quantum field theory is developed by the experiment.
General relativity is consistent with the observation of universe.
Combining them (quantum gravity), we have problems.
Non-renormalization
Scalar field (for simplicity)
R
2
2
r
S = dtdx 3(à þ@
þ
+
þ
þ + V(þ))
t
x ! bx
t ! bt (E ! bà 1E)
þ ! bà 1þ
1+ 3à n
V(þ) û
dtdx3þn ! b4à ndtdx3þn
In UV (b→0), for n>4, this becomes infinity.
Action of general relativity
R 3 p
dtd x à gR
Rø @
È
È ø gà 1@
g
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Motivation of Horava gravity
(Horava 2009)
Idea of Horava
Change the relation between scalings time coordinate and spatial coordinate.
t ! bzt; x ! bx
(Lifshitz scaling)
Able to realize it, introducing following action (scalar field example for simplicity)
R
2
2 z
r
S = dtdx 3(à þ@
þ
+
þ(
) þ + V(þ))
t
t ! bzt (E ! bà zE)
x ! bx
þ!
à
b
3à z
2
þ
Z + 3à
(3à z)
2 n
V(þ) û
(3à z)n
z+ 3à 2
3 n
dtdx þ ! b
dtdx 3þn
If z≧3, all terms are renormalizable
(In UV, b→0, this goes to 0.)
R
2
2
2 z
r
r
S = dtdx 3(à þ@
þ
+
þ
þ
+
þ(
) þ + V(þ))
t
In Horava-Lifshitz theory, this technicque is applied to gravity theory
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Foliation-preserving diffeomorphism
To obtain power-counting renormalizable theory
Order of only spatial derivative must be higher
R
2
2 z
r
S = dtdx 3(à þ@
þ
+
þ(
) þ + V(þ))
t
We must abandon 4-dim diffeomorphism invariance
Horava theory has foliation-preserving diffeomorphism invariance
xi ! à
x i (x j ; t)
t! à
t (t)
(This might be minimum change.)
In 4-dim manifold, time-constant surfaces are physically embedded.
We can reparameterize time and
each time constant surface has 3-dim diffeomorphism.
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Foliation-preserving diffeomorphism
4 dim. spacetime
t = 1 Surface (3 dim.)
dl 2 = gi j dx i dx j
x ! x 0(x; t = 1)
00
dl 2 = gij dx 0i dx 0j
t 0== 12
t = 0 Surface (3 dim.)
tt0=
=0
0
dl 2 = gi j dx i dx j
x!
ds2 = à N 2dt 2 + gi j (dx i + N i dt)(dx j + N j dt)
x 0(x; t = 0)
0
dl 2 = gij dx 0i dx 0j
In 4-dim manifold, time-constant surfaces are physically embedded.
We can reparameterize time and
each time constant surface has 3-dim diffeomorphism invariance.
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Dynamical variables
metric
ds2 = à N 2dt 2 + gi j (dx i + N i dt )(dx j + N j dt )
Basic variables
N i = N i (t ; x k) ; gi j = gi j (t ; x k) ; N (t )
Lapse depends only on time
projectability condition
It is natural because time reparametrization is related to
transformation of lapse function.
Action must be constructed by operators invariant under
foliation preserving diffeomorphism.
Gravitational operators invariant under foliation-preserving diffeomorphism
p 3
Ndt;
gd x; gij ; Rij ; K ij ; r i
In 3-dim space, Ri j kl can be expressed in terms of R i j
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Action
K ij =
1
çi j
2N(g
+ r iNj + r j Ni)
Kinetic terms
R
p
3
(GR limit: λ→１)
dtdx N g(K ij K j i à õK 2)
Potential terms
Three dimensional curvature
z=0 term
z=1 term
R
p
R
3
p
3
dtdx
N
gR
dtdx N gË
z=2 term
i
ij
R
r
r
r ir j R )
p
R
R
R
i
3
2
ji
dtdx N g( R
Rij R
)
z=3 term
R
p
i
j
k
dtdx 3N g( R3
RRij Rj i
R j R kR i )
By the Bianchi identity, other terms can be transformed into above expression
Higher order potential term can be added if you want
In my talk, we do not fix form of potential terms.
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
9
Linear analysis
Number of physical degree of freedom
9 local variables N i (t; x k) ; gij (t; x k) and 1 global variable N(t)
3 local constraint î I g
3 local gauge x i !
î Ni
=0
and 1 global constraint
îIg
îN =
0
x i (x j ; t) and 1 global gauge t ! tà(t)
à
3 physical degree of freedom: 2 tensor gravitons and 1 scalar graviton
Whole-volume Integration of scalar graviton is constrained.
Scalar graviton
If
1
3
< õ < 1 it becomes ghost. So õ must be in range
1
3
> õ or õ > 1 .
In linear analysis, gravitational force change.
But it becomes strongly coupled in GR limit õ ! 1 (Charmousis et al. 2009, Koyama et al. 2010)
Strong interaction might help recovery to GR like Vainshtein mechanism?
We need non-linear analysis
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Vainshtein mechanism
In most of modified gravity, extra propagating modes appear.
Massless limit is not reduced to general relativity in linear analysis.
DVZ discontinuity (H.v.Dam, M.J.G Veltman ‘70 and V.I.Zakharov ‘70)
In case of Horava gravity
1 scalar graviton
2 tensor gravitons
Graviton in general relativity
×
Additional degree of freedom
(additional force)
Non-linear effect is important in some theories
and theories are reduced to general relativity.
?
Vainshtain mechanism (Vainshtein 1972)
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Non-linear analysis
Difficult to solve non-linear equation
Need simplification or approximation
How?
Imposing symmetry of solution
Homogenity and isotropy
FLRW universe
Static and spherical symmetry
Star and Black Hole
Expansion w.r.t. other small variables than amplitude of perturbation
Gradient expansion
Concentrating only on superhorizon scale L
Small scale: 1=(LH)
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
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Motivation of our work
Linear analysis
2 tensor graviton
1 scalar graviton
Gravitational force become stronger??
Vainshtein effect
Is theory reduced to GR?
GR limit
õ ! 1 : Scalar graviton becomes strongly coupled
Usual metric perturbation breaks down.
We must do full non-linear analysis, but it is difficult.
Gradient expansion
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
13
Gradient expansion
and
Our result
(Starobinsky (1985), Nambu and Taruya (1996))
Phys.Rev. D84 (2011) 064025
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
14
Gradient expansion
(Starobinsky (1985), Nambu and Taruya (1996))
Method to analyze the full non-linear dynamics at large scale
Suppose that characteristic scale L of deviation is
much larger than Hubble horizon scale 1/H
(small parameter)
1=LH ø @
x=H ø ï ü 1
Gradient expansion
ï0
Perturbative approach
Small parameter
î = î ú=ú
ï1
ï2
î0
î1
î2
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
15
Separate universe approach (δN)
0th order of gradient expansion (ï ! 0)
Ignoring spatial derivative term
1=LH ø @
x=H ø ï ü 1
EOM is completely the same as that of homogeneous universe.
If local shear can be neglected in this order, EOM is of FLRW.
Looks homogeneous
magnifying glass
characteristic scale is much larger than
horizon scale, so dynamics in each region
does not interact with each other.
amplitude
characteristic scale
Horizon scale
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
Spatial point
16
setup
ADM metric
ds2 = à N 2dt 2 + gi j (dx i + N i dt)(dx j + N j dt)
Considering the case where
higher order terms are
generic form.
Action
R
p
I g = Ndt gdx 3(K ij K ij à õK 2 à 2Ë + R + L z> 1)
Projectability condition
N = N(t)
Gauge fixing
N = 1; Ni = 0 (Gaussian normal)
Decomposition of spatial metric and extrinsic curvature
gij = a2(t)e2ð(t;x) í ij (t; x)
i
1
i
K j = 3K(t; x) + Aj (t; x)
detí = 1
í ij
i
Ai = 0
and í ikA kj are symmetric tensor
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
17
Basic equations
EOM: î I g
î gij
= 0 and definition of extrinsic curvature
(3õ à 1)@
tK = à
1
2(3õ
2
à 1)K à
3 i j
2Aj Ai
àZ
i
i
i
1
i
@
tA j = à KA j + Zj à 3Zî j
@
tð = à
=
î
p
gL p
î gij
L p = à 2Ë + R + L Z> 1
@
ta
1
+
a
3K
There are no discontinuity in
the limit of õ ! 1
k
@
tí ij = 2í ikA j
Constraint equation: î I g
î Ni
j
i
Zj
j
1
j
=0
1
lk
@
j Ai + 3Ai @
j ð à 2Al í @
i í j k à 3(3õ à 1)@
iK = 0
conservation law induced by 3-dimensional spatial diffeomorphism (Bianchi equation)
i
D i : Spatial covariant derivative
compatible with gi j
D i Zj = 0
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
18
Consistency check
of
EOM of
detí = 1
í ij
i
Ai = 0
and í ikA kj are symmetric tensor
i
detí ; Ai ; í ij à í j i
k
k
and í ikAj à í j kAi
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
19
Order analysis
Constraint and EOMs
j
j
1 j lk
1
①
@
j Ai + 3Ai @
j ð à 2Al í @
i í j k à 3(3õ à 1)@
iK = 0
1
3 i j
2
(3õ à 1)@
tK = à 2(3õ à 1)K à 2Aj Ai à Z ②
Suppose that @
(no gravitational wave)
tí ij = O(ï)
⑤
i
Aj
i
i
i
1
i
@
tA j = à KA j + Zj à 3Zî j ③
In most of analyses of GR this condition is imposed.
@
ta
1
a + 3K
④
k
⑤
@
tí ij = 2í ikA j
= O(ï)
①
In sum
ð = ð(0) (x) + ïð(1) (t; x) + ááá
2
@
i K = O(ï )
K (0) depends only on time
Defining a(t) as 3@ta( t) = K ( 0) ( ñ 3H (t ))
a( t)
④
@
tð = à
( 1)
í i j = f i j (x ) + ïí i j (t; x ) + ááá
K = 3H(t) + ïK (1) (t; x) + ááá
( 1)
A i j = ïA i j (t; x) + ááá
@
tð = O(ï)
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
20
Equations in 0th order
Constraint and EOMs
j
j
1 j lk
1
①
@
j Ai + 3Ai @
j ð à 2Al í @
i í j k à 3(3õ à 1)@
iK = 0
0th order equation
②
1
3 i j
2
(3õ à 1)@
tK = à 2(3õ à 1)K à 2Aj Ai à Z ②
à
á
3 2
(3õ à 1) @
H
+
H
= Ë
t
2
i
i
i
1
i
@
tA j = à KA j + Zj à 3Zî j ③
@
tð = à
integrating
3H 2 =
2Ë
3õà 1
+
C
a3
Cosmological constant
C : Integration constant
Friedmann eq.
@
ta
1
a + 3K
④
k
⑤
@
tí ij = 2í ikA j
Effective Dark matter
(Shinji Mukohyama 2009)
Due to projectability condition, we don’t have (00) component of Einstein eq..
However, we have Bianchi identity.
(In 0th order, correction terms such as R^2 can be negligible.)
Integrating Bianchi identity, we can obtain Friedmann eq. with dark matter as
Integration constant. (Shinji Mukohyama 2009)
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
21
Equations in each order
Constraint and EOMs
j
j
1 j lk
1
①
@
j Ai + 3Ai @
j ð à 2Al í @
i í j k à 3(3õ à 1)@
iK = 0
1
3 i j
2
(3õ à 1)@
tK = à 2(3õ à 1)K à 2Aj Ai à Z ②
i
i
i
1
i
@
tA j = à KA j + Zj à 3Zî j ③
nth
@
tð = à
order equation
Evolution equation
@
ta
1
a + 3K
④
⑤
@
tí ij = 2í ikA j
à 3 3(n) á
P nà 1 (p)i (nà p)j
1 P nà 1 (p) (nà p)
3
àa
3 à 3@
à
à
à
(a
K
)
=
K
K
A jA
t
a @
t a K
2
2(3õà 1)
p= 1
p= 1
i
à 3 ( n) i á
P nà 1 ( p) ( nà p) i
( n) i
1 ( n) i
à
à
aà 3@
a
A
=
K
A
+
Z
îj
t
3Z
j
p= 1
j
j
P n à p ( p) ( n à 1) k
(n)
1 (n)
(n)
⑤
@
í
=
2
í A
t
@
ð
=
K
t
ij
p= 1 i k
j
3
②
③
④
k
Z(n)
3õà 1
constraint
①
@
jA
( n) j
i
P
+ 3
nà 1 ( p) j
( nà p)
A
@
ð
j
p= 1
i
à
P nà p ( p) j ( q) l k (nà pà q)
1P n
A lí
@
ií j k
2
p= 1
q= 0
(n)
à 13(3õ à 1)@
= 0
iK
Bianchi equation
@
jZ
( n) j
i
P
+3
à ( p) j
á (nà p) 1P
1
j
à 2
Z i à 3Z ( p) î i @
jð
nà 1
p= 1
P nà p ( p) j ( q)l k ( nà pà q)
n
Z lí
@
ií j k
p= 1
q= 0
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
= 0
22
O(ï)
EOMs
solutions
ð
ñ
3 ( 1)
@t a K
= 0
ð
ñ
3 ( 1) i
@
= 0
t a A
j
K
1
( 1)
C (1) ( x)
a( t) 3
(1)
(1)k
j
( 1) k
j
A
C(1) (x) Rt dtà
+
3
tin a(tà) 3
í ij = 2f ikC
@
t í i j = 2f i k A
ð( 1)
=
ð(1) =
(1)
@
= 3K (1)
tð
( 1)
( 1)
( 1) i
C( 1) ; C
j
Rt
C
(1)i
j
=
(1)i
j
(x)
a(t) 3
(1)
ðin (x)
à
t
+
tin a(tà) 3
(1)
í in;ij (x)
( 1)
( 1)
; ði n ; í i n;i j : Integration constant
can be absorbed into 0th order counterparts
and í i n;i j
Constraint equation
( 1) j
@
jC
( 1) j
i
+ 3C
1 ( 1) j l k
1
( 0)
( 1)
à
à
à
@
ð
C
f
@
f
(3õ
1)@
C
= 0
j
i
j
k
i
2
3
i
l
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
23
n
O(ï )
(n õ 2)
nth order equation
à 3 3(n) á
àa
3 à 3@
a @
K K) =
t at (a
à
1 P nà 1 (p) (nà p)
3
à
K
K
2
2(3õà 1)
p= 1
à 3 ( n) i á
P
à
a @
=
t a A
j
nà 1 ( p) ( nà p) i
K A
p= 1
j
à3
(n)
@
=
tð
(n)
@
tí i j =
K
=
( n) i
ð
1 Rt
à à3
a(t) 3 t in dt a(t )
à
Z(n)
3õà 1
à 13Z ( n) î ij
à
ð
1 P nà 1 ( p) ( nà p)
3
à 2( 3õà
K K
2
1)
p= 1
P nà 1 ( p) (nà p) i
1 Rt
A j = a(t) 3 t in dtàa(tà) 3 à
K A
p= 1
j
R
1 t
ð( n ) = 3 t in dtàK ( n )
Rt P nà p (p) (nà 1)k
(n)
í ij = 2 tin dtà p= 1 í ik A
j
( n) i
j
nà 1 (p)i
(nà p)j
A
A
p= 1
j
i
1 (n)
3K
P n à p ( p) ( n à 1) k
2 p= 1 í i k A
j
nth order solutions
(n)
+Z
P
+Z
P
( n)i
j
à
nà 1 (p) i
( nà p)j
A jA
p= 1
i
1 ( n) i
îj
3Z
ñ
Z (n)
3õà 1
à
ñ
Integration constants can be absorbed into
nth order constraint is automatically satisfied
( 1) i
C( 1) ; C
( 0)
;
ð
; f ij
j
inductive method
No pathology in GR limit õ ! 1
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
24
Curvature perturbation
definition
ð
R (t ; x ) ñ ð(tà; x ) + ln
a( à
t)
a( t)
ñ
( 0)
( 0)
údm(tà) + î údm(tà; x) = údm(t)
(0)
údm(t)
ñ
2
3M pl H 2
2
M ð
î údm(t ; x ) ñ
pl
2
2M
à
R+
2
pl
3õà 1Ë
2
2
(K
3
2
à 9H ) à
i j
Aj Ai
ñ
0th order
R ( 0) = ð( 0) (x )
(constant in time)
1st order
R ( 1) = ð( 1) + H tà = C( 1)
ðR
dt
H
à
3
3
a
a @tH
ñ
(1)
C H
2
(1)
@
= 3a3(@tH ) 2(@
H + 3H@
tR
t H) = 0
t
Curvature perturbation is conserved up to first order in gradient expansion
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
25
Summary
In GR limit õ ! 1
Scalar graviton becomes strongly coupled.
We need fully non-linear analysis.
gradient expansion: fully non-linear analysis of
superhorizon cosmological perturbation
We can not see any pathological behavior in GR limit and
theory is reduced to GR+DM.
Analogue of
Vainshtein effect
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
26
Thank you for your attention
Keisuke Izumi
"Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity"
27