nonlinear superhorizon perturbations (gradient expansion) in horava-lifshitz gravity
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Nonlinear superhorizon perturbations(gradient expansion)
in Horava-Lifshitz gravity
Keisuke Izumi (LeCosPA)Collaboration with Shinji Mukohyama(IPMU)
Phys.Rev. D84 (2011) 064025
泉 圭介
OutlineHorava gravity
Gradient expansion and our result
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 2
Motivation: renormalizable theory of gravitationSymmetry of this theory: foliation-presearving diffeomorphism ActionLinear analysis and importance of non-linearity
Approximation
Intuitive understanding in 0th order
Application to Horava theory and our result
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 3
Horava gravity
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 4
Scalar field (for simplicity)
S =Rdtdx3(à þ@2
tþ + þr 2þ + V(þ))
x ! bxt ! bt (E ! bà 1E)
þ ! bà 1þ dtdx3þn ! b4à ndtdx3þn
In UV (b→0), for n>4, this becomes infinity.
Quantum gravity
General relativity is consistent with the observation of universe.Quantum field theory is developed by the experiment.
Combining them (quantum gravity), we have problems.
Non-renormalization
R ø @È È ø gà 1@g
Action of general relativityRdtd3x à gp R
V(þ) û
1+ 3à n
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 5
S =Rdtdx3(à þ@2
tþ + þ(r 2)zþ + V(þ))
x ! bxt ! bzt (E ! bà zE)
þ ! bà 23àz
þ dtdx3þn ! bz+3à 2(3àz)n
dtdx3þn
If z 3, all terms are renormalizable≧(In UV, b→0, this goes to 0.)
Motivation of Horava gravity (Horava 2009)
Idea of Horava
t ! bzt; x ! bxChange the relation between scalings time coordinate and spatial coordinate.
(Lifshitz scaling)
Able to realize it, introducing following action (scalar field example for simplicity)
In Horava-Lifshitz theory, this technicque is applied to gravity theory
V(þ) ûZ + 3à 2
(3à z)n
S =Rdtdx3(à þ@2
tþ + þr 2þ + þ(r 2)zþ + V(þ))
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 6
To obtain power-counting renormalizable theory
Order of only spatial derivative must be higher
We must abandon 4-dim diffeomorphism invariance
Horava theory has foliation-preserving diffeomorphism invariance
xi ! xài(xj ; t) t ! tà(t)
In 4-dim manifold, time-constant surfaces are physically embedded. We can reparameterize time and each time constant surface has 3-dim diffeomorphism.
Foliation-preserving diffeomorphism
S = Rdtdx3(à þ@2tþ + þ(r 2)zþ + V(þ))
(This might be minimum change.)
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 7
Foliation-preserving diffeomorphism
In 4-dim manifold, time-constant surfaces are physically embedded. We can reparameterize time and each time constant surface has 3-dim diffeomorphism invariance.
4 dim. spacetime
ds2 = à N2dt2 + gij(dxi + N idt)(dxj + N jdt)
t = 0
t = 1
t0= 0
t0= 2t = 0 Surface (3 dim.)
dl2 = gijdxidxj
dl2 = g0ijdx0idx0j
t = 1 Surface (3 dim.)dl2 = gijdxidxj
dl2 = g00ijdx0idx0j
x ! x0(x; t = 1)
x ! x0(x; t = 0)
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 8
Gravitational operators invariant under foliation-preserving diffeomorphism
ds2 = à N2dt2 + gij(dxi + N idt)(dxj + N jdt)Basic variables
metric
N i = N i(t;xk) ; gij = gij(t;xk) ; N(t)
Lapse depends only on time projectability condition
Ndt; gp d3x; gij ; R ij ; K ij ; r i
Action must be constructed by operators invariant under foliation preserving diffeomorphism.
In 3-dim space, can be expressed in terms of R ij kl R ij
Dynamical variables
It is natural because time reparametrization is related to transformation of lapse function.
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 9
Action
Potential terms
z=3 termRdtdx3N gp ( R3 RR ijR j i R i
jRjkR
ki)
r iRr iR R ijr ir jR )
By the Bianchi identity, other terms can be transformed into above expression
z=2 term
z=0 term
Rdtdx3N gp ( R2 R ijR j i )
Rdtdx3N gp R
Kinetic termsRdtdx3N gp (K ijK j i à õK 2) (GR limit: λ→1 )
Three dimensional curvaturez=1 termRdtdx3N gp Ë
Higher order potential term can be added if you wantIn my talk, we do not fix form of potential terms.
K ij = 2N1 (gçij + r iN j + r jN i)
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 10
Linear analysisNumber of physical degree of freedom
9 local variables and 1 global variableN i(t;xk) ; gij(t;xk) N(t)3 local constraint and 1 global constraint
î Niî I g = 0 î N
î I g = 03 local gauge and 1 global gaugexi ! xài(xj ; t) t ! tà(t)
3 physical degree of freedom: 2 tensor gravitons and 1 scalar gravitonWhole-volume Integration of scalar graviton is constrained.
Scalar gravitonIf it becomes ghost. So must be in range or .3
1 < õ < 1In linear analysis, gravitational force change. But it becomes strongly coupled in GR limit Strong interaction might help recovery to GR like Vainshtein mechanism?
We need non-linear analysis
(Charmousis et al. 2009, Koyama et al. 2010)õ ! 1
õ 31 > õ õ > 1
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 11
Vainshtein mechanism
DVZ discontinuity (H.v.Dam, M.J.G Veltman ‘70 and V.I.Zakharov ‘70)
In most of modified gravity, extra propagating modes appear.
Massless limit is not reduced to general relativity in linear analysis.
Non-linear effect is important in some theories and theories are reduced to general relativity.
Vainshtain mechanism (Vainshtein 1972)
In case of Horava gravity2 tensor gravitons
Graviton in general relativity
1 scalar graviton
Additional degree of freedom (additional force)
?×
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 12
Non-linear analysisDifficult to solve non-linear equation
Need simplification or approximationHow?
Imposing symmetry of solution
Homogenity and isotropy FLRW universe
Static and spherical symmetry
Expansion w.r.t. other small variables than amplitude of perturbation
Gradient expansionConcentrating only on superhorizon scale
Small scale:
L
1=(LH)
Star and Black Hole
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 13
Motivation of our work
õ ! 1 : Scalar graviton becomes strongly coupled
Vainshtein effect
Is theory reduced to GR?
2 tensor graviton1 scalar graviton Gravitational force become stronger??
GR limit
Usual metric perturbation breaks down. We must do full non-linear analysis, but it is difficult.
Gradient expansion
Linear analysis
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 14
Gradient expansion and Our result
(Starobinsky (1985), Nambu and Taruya (1996))
Phys.Rev. D84 (2011) 064025
Gradient expansionMethod 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
1=LH ø @x=H ø ï ü 1 (small parameter)
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 15
î 0
î 1
î 2
ï0 ï1 ï2
Gradient expansion
Perturbative approach
Small parameter
î = î ú=ú
(Starobinsky (1985), Nambu and Taruya (1996))
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 16
Separate universe approach (δN)0th order of gradient expansion (ï ! 0)Ignoring spatial derivative term
EOM is completely the same as that of homogeneous universe.If local shear can be neglected in this order, EOM is of FLRW.
magnifying glass
Horizon scale
characteristic scale
Looks homogeneous
characteristic scale is much larger thanhorizon scale, so dynamics in each region does not interact with each other.
amplitude
Spatial point
1=LH ø @x=H ø ï ü 1
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 17
setupADM metric
ds2 = à N2dt2 + gij(dxi + N idt)(dxj + N jdt)Action
I g = RNdt gp dx3(K ijK ij à õK 2 à 2Ë + R + L z>1)
Considering the case where higher order terms are generic form.
Projectability condition
N = N(t)
Gauge fixing
N = 1; N i = 0 (Gaussian normal)
Decomposition of spatial metric and extrinsic curvature
gij = a2(t)e2ð(t;x)í ij(t;x)K i
j = 31K (t;x) + A i
j(t;x)detí = 1 A i
i = 0í ij í ikAk
jand are symmetric tensor
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 18
(3õ à 1)@tK = à 21(3õ à 1)K 2 à 2
3A ijA
ji à Z
@tð = à a@ta + 3
1K
@tA ij = à K A i
j + Zij à 3
1Zî ij
@tí ij = 2í ikAkj
Basic equationsEOM: and definition of extrinsic curvature
conservation law induced by 3-dimensional spatial diffeomorphism (Bianchi equation)
D iZij = 0 D i : Spatial covariant derivative
compatible with gij
Constraint equation:
î gijî I g = 0
î Niî I g = 0
@jA ji + 3A j
i@jðà 21A j
l í lk@ií j k à 31(3õ à 1)@iK = 0
There are no discontinuity in the limit of õ ! 1
Zij = î gij
î gp Lp
L p = à 2Ë + R + L Z>1
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 19
Consistency checkdetí = 1 A i
i = 0í ij í ikAk
jand are symmetric tensor
EOM of detí ; A ii; í ij à í j i and í ikAk
j à í j kAki
of
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 20
Order analysis
Suppose that @tí ij = O(ï) (no gravitational wave)
In most of analyses of GR this condition is imposed.
(3õ à 1)@tK = à 21(3õ à 1)K 2 à 2
3A ijA
ji à Z
@tð = à a@ta + 3
1K
@tA ij = à K A i
j + Zij à 3
1Zî ij
@tí ij = 2í ikAkj
Constraint and EOMs@jA j
i + 3A ji@jðà 2
1A jl í lk@ií j k à 3
1(3õ à 1)@iK = 0 ①
②
③
④
⑤
⑤
A ij = O(ï)
①
@iK = O(ï2)K (0) depends only on time
a(t)Defining as 3 a(t)@ta(t) = K (0)(ñ 3H(t))
④
@tð = O(ï)
ð = ð(0)(x) + ïð(1)(t;x) + áááí ij = f ij (x) + ïí (1)
ij (t;x) + áááK = 3H(t) + ïK (1)(t;x) + áááA ij = ïA (1)
ij (t;x) + ááá
In sum
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 21
Equations in 0th order
0th order equation
② (3õ à 1)à@tH + 2
3H2á = Ë
integrating
3H2 = 3õà 12Ë + a3
C
C : Integration constant
Cosmological constantEffective Dark matter (Shinji Mukohyama 2009)
(3õ à 1)@tK = à 21(3õ à 1)K 2 à 2
3A ijA
ji à Z
@tð= à a@ta + 3
1K
@tA ij = à K A i
j + Zij à 3
1Zî ij
@tí ij = 2í ikAkj
Constraint and EOMs@jA j
i + 3A ji@jðà 2
1A jl í lk@ií j k à 3
1(3õ à 1)@iK = 0 ①
②
③
④
⑤Friedmann eq.
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" 22
Equations in each order
(3õ à 1)@tK = à 21(3õ à 1)K 2 à 2
3A ijA
ji à Z
@tð = à a@ta + 3
1K
@tA ij = à K A i
j + Zij à 3
1Zî ij
@tí ij = 2í ikAkj
Constraint and EOMs@jA j
i + 3A ji@jðà 2
1A jl í lk@ií j k à 3
1(3õ à 1)@iK = 0 ①
②
③
④
⑤nth order equation
aà3@t(a3K ) = à 21P
p=1nà 1K (p)K (nà p)à 2(3õà 1)
3 Pp=1nà1A (p)i
jA(nàp)j
i à 3õà 1Z(n)
@tð(n) = 31K (n)
aà3@tàa3A (n)i
já
= à Pp=1nà1K (p)A (nàp)i
j + Z(n)ij à 3
1Z(n)î ij
@tí (n)ij = 2P
p=1nàp í (p)
ik A (nà1)kj
aà3@tàa3K (n)á
①
②
③
④ ⑤
Evolution equation
constraint@jA (n)j
i + 3Pp=1nà 1A (p)j
i@jð(nà p) à 21P
p=1n P
q=0nàpA (p)j
lí (q)lk@ií (nà pàq)j k à 3
1(3õ à 1)@iK (n) = 0Bianchi equation
@jZ(n)ji + 3P
p=1nà 1àZ(p)j
i à 31Z(p)î j
iá@jð(nà p) à 2
1Pp=1n P
q=0nàpZ(p)j
lí(q)lk@ií (nàpà q)
j k = 0
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 23
O(ï)EOMs
@tða3A (1)i
j
ñ= 0
@tða3K (1)
ñ= 0
@tð(1) = 31K (1)
@tí (1)ij = 2f ikA (1)k
j
solutions
K (1) = a(t)3C(1)(x) A (1)i
j = a(t)3C(1)i
j (x)
ð(1) = 3C(1)(x) R
tint
a(tà)3dtà + ð(1)
in (x)
í (1)ij = 2f ikC(1)k
jR
tint
a(tà)3tà + í (1)
in;ij(x)
C(1);C(1)ij ;ð
(1)in ; í (1)
in;ij : Integration constant
ð(1) and í (1)in;ij can be absorbed into 0th order counterparts
Constraint equation
@jC(1)ji + 3C(1)j
i@jð(0) à 21C(1)j
lf lk@if j k à 31(3õ à 1)@iC(1) = 0
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 24
O(ïn) (n õ 2)nth order equation
aà 3@t(a3K ) = à 21P
p=1nà 1K (p)K (nà p)à 2(3õà 1)
3 Pp=1nà 1A (p)i
jA(nà p)j
i à 3õà 1Z(n)
@tð(n) = 31K (n)
aà 3@tàa3A (n)i
já
= à Pp=1nà 1K (p)A (nà p)i
j + Z(n)ij à 3
1Z(n)î ij
@tí (n)ij = 2P
p=1nàp í (p)
ik A (nà1)kj
aà 3@tàa3K (n)á
K (n) = a(t)31 R
tint dtàa(tà)3
ðà 2
1Pp=1nà 1K (p)K (nàp)à 2(3õà1)
3 Pp=1nà1A (p)i
jA(nà p)j
i à 3õà 1Z(n)
ñ
A (n)ij = a(t)3
1 Rtint dtàa(tà)3
ðà P
p=1nà1K (p)A(nàp)i
j + Z(n)ij à 3
1Z(n)î ij
ñ
nth order solutions
ð(n) = 31R
tint dtàK (n)
í (n)ij = 2R
tint dtàP p=1
nàp í (p)ik A (nà1)k
jIntegration constants can be absorbed into
C(1);C(1)ij ;ð(0); f ij
nth order constraint is automatically satisfied inductive method
No pathology in GR limit õ ! 1
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 25
Curvature perturbationdefinition
R(t;x) ñ ð(tà;x) + lnð
a(t)a(tà)
ñ
ú(0)dm(tà) + î údm(tà;x) = ú(0)
dm(t)
ú(0)dm(t) ñ 3M 2
plH2 à 3õà12M2
pl Ë
î údm(t;x) ñ 2M2
plðR + 3
2(K 2 à 9H2) à A ijA
ji
ñ
0th order
R (0) = ð(0)(x) (constant in time)1st order
R (1) =ð(1) + Htà= C(1)ð R
a3dt à a3@tH
Hñ
@tR (1) =3a3(@tH)2C(1)H (@2
tH + 3H@tH) = 0
Curvature perturbation is conserved up to first order in gradient expansion
Keisuke Izumi "Nonlinear superhorizon perturbations in
Horava-Lifshitz gravity" 26
Summaryõ ! 1In GR limit
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" 27
Thank you for your attention