Shinji Tsujikawa

1. HotTopicsinGeneralRelativity AndGravitation,QuyNhon, 2015 Effective field theory approach to modified gravity with applications to inflation and dark energy Shinji Tsujikawa Tokyo University of Science Collaboration with A. De Felice (Kyoto) L.Gergely(Sedged) R. Kase (TUS) K. Koyama (Portsmouth)

2. Motivation of going beyond General Relativity • Construction of renormalizable theory of gravity --short-distance modification of gravity Origin of inflation --it comes from some geometric effect or from a scalar field beyond the standard model of particle physics? Origin of dark energy (and dark matter) --the present cosmic acceleration may come from a large-distance modification of gravity?

3. Horndeski theories Horndeski (1974) Deffayet et al (2011) Charmousis et al (2011) Kobayashi et al (2011) Most general scalar-tensor theories with second-order equations The Lagrangian of Horndeski theories is constructed to keep the equations of motion up to second order, such that the theories are free from the Ostrogradski instability.

4. ADM decompositon of space-time We can start from a general action involving all the possible geometric scalar quantities appearing in the ADM formalism. The ADM formalism is based on the 3+1 decomposition of space-time. ADM metric We choose the unitary gauge on the flat FLRW cosmological background Then, the scalar perturbation can be absorbed into the gravitational sector: Constant time hypersurface We have several geometric tensors: Extrinsic curvature: 3-dimensional Ricci tensor:

5. Horndeski Lagrangian in the ADM Language Gleyzes et al (2013) Several scalar quantities can be constructed: Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories do NOT impose these conditions. The action in Horndeski and GLPV theories has the dependence

6. Inclusion of higher spatial derivatives In Horava-Lifshitz gravity, there are spatial derivatives like to realize an anisotropic scaling between time and spatial derivatives. In the healthy extension of Horava-Lifshitz grvaity, there are scalar quantities coming from the acceleration vector like Then, the general action implementing GLPV theories and Horava-Lifshitz gravity is

7. De Felice and ST (2014) Tensor perturbations intheEFTapproach where ____ ______________ Present in GLPV theories Higher spatial derivatives appearing in Horava gravity In the absence of higher spatial derivative, the no-ghost and no-instability conditions are

8. Equations of motion for tensor perturbations where where We can use this result to derive the tensor power spectrum generated during inflation.

9. Inflationary tensor power spectrum Then, the spatial derivatives can be treated as corrections, in which case the tensor power spectrum under the slow-roll approximation reads _______ ___________________ __________ Leading-order spectrum Slow-roll corrections Corrections from are much smaller than 1.

10. Einstein frame Is there a convenient frame in which the leading-ordertensor power spectrum isof the simpler form? We define the Einstein frame as the one in which the second-order action for tensor perturbation is of the same form as in GR, i.e., In the Einstein frame, the leading-order inflationary tensor spectrum should be of the form In GLPV theories it is possible to transform to the Einstein frame under the so-called disformal transformation.

11. Disformal transformations The structure of the GLPV action is preserved under the disformal transformation: ______ ___________ Bekenstein (1993) Conformal transformation Disformal transformation The GLPV action in the transformed frame reads Same form as that in the original frame with the relations among coefficients Gleyzes et al, JCAP (2014) where

12. Disformal invariance of cosmological perturbations Consider the perturbed metric ST (2014), See also Minamitsuji (2014) for the case where _ __ Curvature perturbations Tensor perturbations where

13. Transformation to the Einstein frame Creminelli et al, PRL (2014), ST, JCAP (2014) In GLPV theories, the next-to-leading order tensor power spectrum in the transformed frame is given by We can transform to the Einstein frame for the choice Then the tensor power spectrum reads Same as the GR tensor spectrum (Stewart and Lyth, 1993) where

14. Application to dark energy The EFT formalism was also applied to dark energy (Vernizzi’s talk). Usually, the quadratic-order EFT action is written of the form (Creminelli et al): __ __________________ _____________________ Matter sector Background Perturbations Three functions Functions If we specify the theories (e.g. Horndeski), there are explicit relations between the above EFT functions and the free parameters of theories. See Gleyzes et al (2013), ST(2014) The EFT formalism is also implemented in the CAMB code (Silvestri et al).

15. Cosmological perturbations in the presence of matter The scalar degree of freedom can give rise to • the late-time cosmic acceleration at the background level • interactions with the matter sector (CDM, baryons) We take into account non-relativistic matter with the energy density ____ _______ Background Perturbations The four velocity of non-relativistic matter is The perturbed line element in the longitudinal gauge is

16. Effective gravitational coupling with matter The growth rate of matter perturbations is constrained from peculiar velocities of galaxies in red-shift space distortion measurements. The gauge-invariant density contrast obeys where ___

17. Recent observations favor weak gravity on cosmological scales. Planck LCDM fit GR RSD fit

18. ST, 1505.02459 (2015) Effectivegravitational couplinginHorndeskitheories In the massless limit, the effective gravitational coupling in Horndeski theories reads ____ _______ Tensor contribution Scalar contribution This correspond to the intrinsic modification of the gravitational part. Always positive under the no-ghost and no-instability conditions: The necessary condition to realize weaker gravity than that in GR is The scalar-matter interaction always enhances the effective gravitational coupling, so the realization of weak gravity is quite limited in Horndeski theories.

19. A model realizing weak gravitybeyondtheHorndeskidomain ST (2015) where Black points are RSD data. In the scaling matter era, ____ Negative for It would be of interest to see the feature of weak gravity persists in future observations.

20. Summary and outlook