Download pptx - Nonlinear superhorizon perturbations (gradient expansion) in Horava-Lifshitz gravity

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



Horava gravity



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



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))



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.)



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)



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.



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)



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



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)

?×



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



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



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)



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))



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



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



(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



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



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



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


i + 3A ji@jðà 2


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)



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


i + 3A ji@jðà 2


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



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



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



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



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



Thank you for your attention