1 / 27

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

talisa
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. Outline 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"

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

  4. Quantum 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"

  5. 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"

  6. 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"

  7. 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"

  8. Dynamical variables 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"

  9. Action 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"

  10. Linear analysis 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"

  11. 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 × 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"

  12. Non-linear analysis 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"

  13. Linear analysis 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"

  14. 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"

  15. 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) Gradient expansion Perturbative approach Small parameter Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

  16. 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"

  17. setup 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"

  18. Basic equations 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"

  19. Consistency check of and are symmetric tensor EOM of Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

  20. Constraint and EOMs 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"

  21. Equations in 0th order 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"

  22. Equations in each order 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"

  23. EOMs solutions Integration constant can be absorbed into 0th order counterparts Constraint equation Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

  24. nth order equation 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"

  25. Curvature perturbation 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"

  26. Summary 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"

  27. Thank you for your attention Keisuke Izumi "Nonlinear superhorizon perturbations in Horava-Lifshitz gravity"

More Related