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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. Outline. Horava gravity. Motivation: renormalizable theory of gravitation. - PowerPoint PPT Presentation

Nonlinear analysis in Horava gravity

Nonlinear superhorizon perturbations(gradient expansion) in Horava-Lifshitz gravityKeisuke Izumi (LeCosPA)Collaboration with Shinji Mukohyama(IPMU)Phys.Rev. D84 (2011) 064025OutlineHorava gravityGradient expansion and our resultKeisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 2Motivation: renormalizable theory of gravitationSymmetry of this theory: foliation-presearving diffeomorphism ActionLinear analysis and importance of non-linearityApproximationIntuitive understanding in 0th orderApplication to Horava theory and our result Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 3Horava gravityKeisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 4Scalar field (for simplicity)

In UV (b0), for n>4, this becomes infinity. Quantum gravityGeneral 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

Action of general relativity

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 5

If z3, all terms are renormalizable(In UV, b0, this goes to 0.) 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)In Horava-Lifshitz theory, this technicque is applied to gravity theory

5Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 6To obtain power-counting renormalizable theory Order of only spatial derivative must be higherWe must abandon 4-dim diffeomorphism invarianceHorava theory has foliation-preserving diffeomorphism invariance

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

(This might be minimum change.)Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 7Foliation-preserving diffeomorphismIn 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

Surface (3 dim.)

Surface (3 dim.)

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 8Gravitational operators invariant under foliation-preserving diffeomorphism

Basic variablesmetric

Lapse depends only on timeprojectability condition

Action must be constructed by operators invariant under foliation preserving diffeomorphism. In 3-dim space, can be expressed in terms of

Dynamical variablesIt is natural because time reparametrization is related to transformation of lapse function.Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 9ActionPotential termsz=3 term

By the Bianchi identity, other terms can be transformed into above expression z=2 termz=0 term

Kinetic terms

(GR limit: )Three dimensional curvaturez=1 term

Higher order potential term can be added if you wantIn my talk, we do not fix form of potential terms.

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 10Linear analysisNumber of physical degree of freedom9 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 gravitonWhole-volume Integration of scalar graviton is constrained. Scalar gravitonIf it becomes ghost. So must be in range or .

In 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)

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 11Vainshtein mechanismDVZ 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 gravitonsGraviton in general relativity1 scalar gravitonAdditional degree of freedom (additional force)?Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 12Non-linear analysisDifficult to solve non-linear equationNeed simplification or approximationHow?Imposing symmetry of solutionHomogenity and isotropy FLRW universeStatic and spherical symmetry Expansion w.r.t. other small variables than amplitude of perturbationGradient expansionConcentrating only on superhorizon scale Small scale:

Star and Black HoleKeisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 13Motivation of our work

Scalar graviton becomes strongly coupledVainshtein effectIs theory reduced to GR?Linear analysis2 tensor graviton1 scalar gravitonGravitational force become stronger??GR limitUsual metric perturbation breaks down. We must do full non-linear analysis, but it is difficult.Gradient expansionLinear analysisKeisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 14Gradient expansion and Our result(Starobinsky (1985), Nambu and Taruya (1996))Phys.Rev. D84 (2011) 064025Gradient 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

(small parameter)Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 15

Gradient expansionPerturbative approachSmall parameter

(Starobinsky (1985), Nambu and Taruya (1996))15Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 16Separate universe approach (N)0th order of gradient expansion

Ignoring spatial derivative termEOM is completely the same as that of homogeneous universe.If local shear can be neglected in this order, EOM is of FLRW. magnifying glassHorizon scalecharacteristic scaleLooks homogeneouscharacteristic scale is much larger thanhorizon scale, so dynamics in each region does not interact with each other.

amplitudeSpatial point

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 17setupADM metric

Action

Considering the case where higher order terms are generic form.Projectability condition

Gauge fixing

(Gaussian normal)Decomposition of spatial metric and extrinsic curvature

and are symmetric tensorKeisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 18

Basic equationsEOM: and definition of extrinsic curvatureconservation law induced by 3-dimensional spatial diffeomorphism (Bianchi equation)

Spatial covariant derivative compatible with

Constraint equation:

There are no discontinuity in the limit of

Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 19Consistency check

and are symmetric tensorEOM of

ofKeisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 20Order analysisSuppose that

(no gravitational wave)In most of analyses of GR this condition is imposed.

Constraint and EOMs

depends only on time

Defining as

In sumKeisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity" 21Equations in 0th order0th order equation

integrating

Integration constantCosmological constantEffective Dark matter (Shinji Mukohyama 2009)

Constraint and EOMs

Friedmann eq.Due to projectability condition, we dont 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" 22Equations in each order

Constra

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