1. Effective field theory approach to modified gravity with applications to inflation and dark energy Shinji Tsujikawa Hot Topics in General Relativity And Gravitation, Quy Nhon, 2015 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 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? Construction of renormalizable theory of gravity --short-distance modification of gravity

3. Horndeski theories Horndeski (1974) Deffayet et al (2011) Charmousis et al (2011) Kobayashi et al (2011) 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. Most general scalar-tensor theories with second-order equations

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. Constant time hypersurface We choose the unitary gauge on the flat FLRW cosmological background Then, the scalar perturbation can be absorbed into the gravitational sector: ADM metric We have several geometric tensors: Extrinsic curvature: 3-dimensional Ricci tensor:

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

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

8. Equations of motion for tensor perturbations 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-order tensor power spectrum is of 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: Conformal transformation Disformal transformation The GLPV action in the transformed frame reads with the relations among coefficients where Gleyzes et al, JCAP (2014) Bekenstein (1993) Same form as that in the original frame

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

13. Transformation to the Einstein frame 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) ST, JCAP (2014) Creminelli et al, PRL (2014), 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): __________________ Background _____________________ Perturbations __ Matter sector Three functionsFunctions 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 ___________ BackgroundPerturbations The perturbed line element in the longitudinal gauge is The four velocity of non-relativistic matter is

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

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

18. Effective gravitational coupling in Horndeski theories In the massless limit, the effective gravitational coupling in Horndeski theories reads ___________ Tensor contribution Scalar contribution 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. ST, 1505.02459 (2015) This correspond to the intrinsic modification of the gravitational part.

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

20. Summary and outlook