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

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### 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

Horava gravity

Motivation: renormalizable theory of gravitation

Symmetry of this theory: foliation-presearvingdiffeomorphism

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"

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Quantum field theory is developed by the experiment.

General relativityis consistent with the observation of universe.

Combining them (quantum gravity), we have problems.

Non-renormalization

Scalar field (for simplicity)

In UV (b→0), for n>4, this becomes infinity.

Action of general relativity

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Motivation of Horava gravity (Horava 2009)

Idea of Horava

Change the relation between scalings time coordinate and spatial coordinate.

(Lifshitz scaling)

Able to realize it, introducing following action (scalar field example for simplicity)

If z≧3, all terms are renormalizable

(In UV, b→0, this goes to 0.)

In Horava-Lifshitz theory, this technicque is applied to gravity theory

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Foliation-preserving diffeomorphism

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

(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"

Foliation-preserving diffeomorphism

4 dim. spacetime

Surface (3 dim.)

Surface (3 dim.)

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"

metric

Basic variables

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.

In 3-dim space, can be expressed in terms of

Gravitational operators invariant under foliation-preserving diffeomorphism

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Kinetic terms

(GR limit: λ→１)

Potential terms

Three dimensional curvature

z=0 term

z=1 term

z=2 term

z=3 term

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"

Number of physical degree of freedom

9 local variables and 1 global variable

3 local constraint and 1 global constraint

3 local gauge and 1 global gauge

3 physical degree of freedom: 2 tensor gravitons and 1 scalar graviton

Whole-volume Integration of scalar graviton is constrained.

Scalar graviton

If it becomes ghost. So must be in range or .

In linear analysis, gravitational force change.

But it becomes strongly coupled in GR limit

(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"

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

×

Additional degree of freedom

(additional force)

Graviton in general relativity

?

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"

Difficult to solve non-linear equation

Need simplification or approximation

How?

Imposing symmetry of solution

Homogenity and isotropy

FLRW universe

Star and Black Hole

Static and spherical symmetry

Expansion w.r.t. other small variables than amplitude of perturbation

Gradient expansion

Concentrating only on superhorizon scale

Small scale:

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Motivationof our work

Linear analysis

2 tensor graviton

1 scalar graviton

Gravitational force become stronger??

Vainshtein effect

Is theory reduced to GR?

GR limit

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"

and

Our result

(Starobinsky(1985), Nambu and Taruya (1996))

Phys.Rev. D84 (2011) 064025

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

(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)

Gradient expansion

Perturbative approach

Small parameter

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Separate universe approach (δN)

0th order of gradient expansion

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.

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

Spatial point

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Considering the case where

higher order terms are

generic form.

ADM metric

Action

Projectability condition

Gauge fixing

(Gaussian normal)

Decomposition of spatial metric and extrinsic curvature

and are symmetric tensor

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

EOM: and definition of extrinsic curvature

There are no discontinuity in

the limit of

Constraint equation:

conservation law induced by 3-dimensional spatial diffeomorphism (Bianchi equation)

Spatial covariant derivative

compatible with

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

of

and are symmetric tensor

EOM of

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Order analysis

①

②

③

Suppose that

(no gravitational wave)

In most of analyses of GR this condition is imposed.

④

⑤

⑤

①

In sum

depends only on time

Defining as

④

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Constraint and EOMs

①

0th order equation

②

③

②

④

integrating

⑤

Friedmann eq.

Cosmological constant

Effective Dark matter

(Shinji Mukohyama 2009)

Integration constant

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"

Constraint and EOMs

①

②

③

④

nth order equation

⑤

Evolution equation

②

③

④

⑤

constraint

①

Bianchi equation

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

solutions

Integration constant

can be absorbed into 0th order counterparts

Constraint equation

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

nth order solutions

Integration constants can be absorbed into

nth order constraint is automatically satisfied

inductive method

No pathology in GR limit

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

definition

0th order

(constant in time)

1storder

Curvature perturbation is conserved up to first order in gradient expansion

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

In 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 ofVainshtein effect

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

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