Dark energy About 70% of the energy density today consists of dark energy responsible for the cosmic acceleration. (Equation of state around )
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: Quintessence, k-essence, chaplygin gas, tachyon,… (i)Modified matter models (ii) Modified gravity models f(R) gravity, scalar-tensor theory, Braneworld, Galileon,…
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. (ii) Solar system scales (small scales) The models need to be close to GR from solar system experiments. GR+small corrections Beyond GR
Concrete modified gravity models (i) f(R) gravity The Lagrangian f is a function of the Ricci scalar R: (ii) Scalar-tensor theory A branch of this theory is Brans-Dicke theory: (iv) DGP braneworld Self-accelerating solutions on the 3-brane in 5-dimensional Minkowski bulk. (iii) Gauss-Bonnet gravity (v) Galileon gravity or The field Lagrangian is restricted to satisfy the Galilean symmetry:
Recovery of GR behavior on small scales (i) Chameleon mechanism Two mechanisms are known. Khoury and Weltman, 2004 The effective mass of a scalar field degree of freedom is density-dependent. Massive (local region) Massless (cosmological region) The field does not propagate freely in the regions of high density. Effective potential:
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) Massless (in the regions of low density) Potential in the Einstein frame The field does not propagate freely.
Simplest modified gravity: Brans-Dicke theory (i) (original BD theory, 1961) Solar system constraints give (ii) 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. with the field mass:
(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.
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.
Matter perturbations in general dark energy models 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: where We can define two masses that come from the modification of gravity and from the scalar field. Gravitational: Scalar field: For quintessence ( )
On sub-horizon scales (k>>aH), the main contribution to the matter perturbation equation is the terms including Matter perturbations under a quasi-static approximation We then obtain S.T., 2007 De Felice, Mukohyama, S.T., to appear. whereand Massive limits:
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 ( ) During the late matter era In f(R) gravity ( ), Modified growth rate
Matter power spectra P k [h/Mpc] LCDM Starobinsky’s f(R) model with n=2 BD theory with the potential (Q=0.7, p=0.6) ( Q is related with via )
Gravitational potentials Perturbed metric in the longitudinal gauge We introduce the effective gravitational potential Under the quasi-static approximation we have Whenit follows that In the massless regime in BD theory one has (matter era) in f(R) gravity
The effect of modified gravity on weak lensing Let us consider the shear power spectrum in BD with the potential: where LCDM Larger Q The shear spectrum compared to the LCDM model is where (S.T. and Tatekawa, 2008) (Q: coupling between field and matter in the Einstein frame)
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.
Background cosmological evolution The field is nearly frozen during radiation and matter eras. The GR behavior can be recovered by the field self interaction.
The field propagation speed Allowed region The dotted line shows the border between the sub-luminal and super-luminal regimes.
Distinguished observational signatures The effective gravitational potential can grow even if the matter perturbation decays during the accelerated epoch. Kobayashi, Tashiro, Suzuki, 2009 This can provide a tight constraint on this model in future observations. Anti-correlations in the cross-correlation of the Integrated Sachs-Wolfe Effect and large-scale structure LCDM Anti- correlation
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 Negative for
(i) f(R) gravity Summary of modified gravity models of dark energy 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.
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.