PROBING SIGNATURES OF MODIFIED GRAVITY MODELS OF DARK ENERGY

Shinji Tsujikawa (Tokyo University of Science)

(Part 1)

(Part 2)(Part 3)

Dark energy About 70% of the energy density today consists ofdark 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,…

These dynamical dark energy models give rise to a time-varying w.Please see the review of Copeland, Sami and S.T. (2006).

(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 GRfrom solar system experiments.

GR+small corrections

Beyond GR

Concrete modified gravity models

(i) f(R) gravityThe 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 freelyin 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 giveHardly distinguishablefrom GR.

(ii)

As long as the potential is massive in the regions of high density, local gravity constraints can be satisfiedby 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 behaviorat 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., 2007De Felice, Mukohyama, S.T., to appear.

where andMassive limits:

Brans-Dicke theory withBrans-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]

LCDMStarobinsky’s f(R) modelwith n=2

BD theory with the potential

(Q=0.7, p=0.6)

( Q is related with via

)

Gravitational potentialsPerturbed 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

(matter era)in f(R) gravity

The effect of modified gravity on weak lensingLet us consider the shear power spectrum in BD with the potential:

where

LCDMLarger Q

The shear spectrum comparedto 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 frozenduring 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 borderbetween 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-scalestructure

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

(normal one)

Negative for

This leads to the violent instability of perturbations of the fluid during radiation and matter eras.

(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.