p robing signatures of modified gravity models of dark energy shinji tsujikawa (tokyo university of...
TRANSCRIPT
PROBING SIGNATURES OF MODIFIED GRAVITY MODELS OF DARK ENERGY
Shinji Tsujikawa (Tokyo University of Science)
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.