Probing signatures of modified gravity models of dark energy

1. Probing signatures of modified gravity models of dark energy Shinji Tsujikawa (Tokyo University of Science)

2. (Part 1) (Part 2) (Part 3)

3. Dark energy About 70% of the energy density today consists of dark energy responsible for the cosmic acceleration. (Equation of state around )

4. Theoretical models of dark energy • Simplest model: Cosmological constant: If the cosmological constant originates from a vacuum energy, it is enormously larger than the energy scale of dark energy. • Other dynamical dark energy models: Modified matter models Quintessence, k-essence, chaplygin gas, tachyon,… (ii) Modified gravity models f(R) gravity, scalar-tensor theory, Braneworld, Galileon,… These dynamical dark energy models give rise to a time-varying w. Please see the review of Copeland, Sami and S.T. (2006).

5. Modified gravity models of dark energy (i) Cosmological scales (large scales) Modification from General Relativity (GR) can be allowed. This gives rise to a number of observational signatures such as (i) Peculiar dark energy equation of state (ii) Impact on large scale structure, weak lensing, and CMB. Beyond GR (ii) Solar system scales (small scales) GR+small corrections The models need to be close to GR from solar system experiments.

6. Concrete modified gravity models (i) f(R) gravity The Lagrangianfis a function of the Ricci scalar R: (ii) Scalar-tensor theory A branch of this theory is Brans-Dicke theory: (iii) Gauss-Bonnet gravity or (iv) DGP braneworld Self-accelerating solutions on the 3-brane in 5-dimensional Minkowski bulk. (v) Galileon gravity The field Lagrangian is restricted to satisfy the Galilean symmetry:

7. Recovery of GR behavior on small scales Two mechanisms are known. (i) Chameleon mechanism Khoury and Weltman, 2004 The effective mass of a scalar field degree of freedom is density-dependent. Effective potential: Massive (local region) Massless (cosmological region) The field does not propagate freely in the regions of high density.

8. Chameleon mechanism in f(R) dark energy models Viable f(R) dark energy models have been constructed to satisfy local gravity constraints in the regions of high density. (Starobinsky, 2007) Massive (in the regions of high density) Potential in the Einstein frame The field does not propagate freely. Massless (in the regions of low density)

9. Simplest modified gravity: Brans-Dicke theory (i) (original BD theory, 1961) Solar system constraints give Hardly distinguishable from GR. (ii) with the field mass: As long as the potential is massive in the regions of high density, local gravity constraints can be satisfied by the chameleon mechanism. • f(R) gravity ( ): Cappozzielo and S.T. n > 0.9 • : p > 0.7 S.T. et al.

10. (ii)Vainshtein mechanism Scalar-field self interaction such as allows the possibility to recover the GR behavior at high energy (without a field potential) This type of self interaction was considered in the context of `Galileon’ cosmology (Nicolis et al.) The field Lagrangian is restricted to satisfy the `Galilean’ symmetry: The field equation can be kept to second-order. The field can be nearly frozen in the regions of high density.

11. Observational signatures of modified gravity From the observations of supernovae only, it is not easy to distinguish modified gravity models from the LCDM model. • Other constraints on dark energy • Large-scale structure • Weak lensing • CMB • Baryon oscillations The evolution of matter density perturbations can allow us to distinguish modified gravity models from the LCDM. The modification of gravity leads to the modification of the growth rate of perturbations.

12. Matter perturbations in general dark energy models where This action includes most of dark energy models such as f(R) gravity, scalar-tensor theory, quintessence, k-essence,… For most of modified gravity theories the Lagrangian takes the form: We can define two masses that come from the modification of gravity and from the scalar field. Gravitational: Scalar field: For quintessence ( )

13. Matter perturbations under a quasi-static approximation On sub-horizon scales (k>>aH), the main contribution to the matter perturbation equation is the terms including We then obtain S.T., 2007 De Felice, Mukohyama, S.T., to appear. ____ where and Massive limits:

14. Brans-Dicke theory with Brans-Dicke parameter The effective gravitational coupling is where • The GR limit ( ) or massive limit ( ) During the early matter era • The massless limit ( ) Modified growth rate During the late matter era In f(R) gravity ( ),

15. Matter power spectra BD theory with the potential (Q=0.7, p=0.6) P ( Q is related with via ) LCDM Starobinsky’sf(R) model with n=2 k [h/Mpc]

16. Gravitational potentials Perturbed metric in the longitudinal gauge We introduce the effective gravitational potential Under the quasi-static approximation we have When it follows that In the massless regime in BD theory one has in f(R) gravity (matter era)

17. The effect of modified gravity on weak lensing Let us consider the shear power spectrum in BD with the potential: where (Q: coupling between field and matter in the Einstein frame) The shear spectrum compared to the LCDM model is where Larger Q LCDM (S.T. and Tatekawa, 2008)

18. Field self-interaction in generalized BD theories (without the field potential) The de Sitter solution exists for the choice The BD theory corresponds to n=2. • The viable parameter space (i) Required to avoid the negative gradient instability and for the existence of a matter era. (ii) Required to avoid ghosts. (iii) Required to realize the late-time de Sitter solution.

19. Background cosmological evolution The field is nearly frozen during radiation and matter eras. The GR behavior can be recovered by the field self interaction.

20. The field propagation speed Allowed region The dotted line shows the border between the sub-luminal and super-luminal regimes.

21. Distinguished observational signatures The effective gravitational potential can grow even if the matter perturbation decays during the accelerated epoch. Kobayashi, Tashiro, Suzuki, 2009 Anti-correlations in the cross-correlation of the Integrated Sachs-Wolfe Effect and large-scale structure LCDM This can provide a tight constraint on this model in future observations. Anti- correlation

22. Gauss-Bonnet gravity A. De Felice, D. Mota, S.T. (2009) where Considering the perturbations of a perfect fluid with an equation of state w, the speed of propagation is (normal one) Negative for This leads to the violent instability of perturbations of the fluid during radiation and matter eras.

23. Summary of modified gravity models of dark energy (i) f(R) gravity It is possible to construct viable models such as The modified growth of matter perturbation gives the bound (ii) Brans-Dicke theory One can design a field potential to satisfy cosmological and local gravity constraints (through the chameleon mechanism) (iii) Gauss-Bonnet gravity and Incompatible with observations and experiments (iv) Generalized Bran-Dicke theory with a field self interaction Anti-correlation of the ISW effect and LSS can distinguish this model. (v) DGP model Incompatible with observations, the ghost is present.

24. Conclusions and outlook • Modified gravity models of dark energy are distinguished • from other models in many aspects. • In particular the growth rate of matter perturbations gets larger • than that in the LCDM model. in the LCDM model In viable f(R) models the growth index today can be as small as For Brans-Dicke model with a potential, is even smaller than that in f(R) gravity. • The joint observational analysis based on the LSS, weak • lensing, ISW-LSS correlation data in future will be useful to • constrain modified gravity models.