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f(R) Gravity and its relation to the interaction between DE and DM

Bin WangShanghai Jiao Tong University

SNe Ia

The current universe isaccelerating!

LSS

CMB

Dark Energy

Simplest model of dark energy

Cosmological constant:

This corresponds to the energy scale

If this originates from vacuum energy in particle physics,

Huge difference compared to the present value!

(Equation of state: )

Cosmological constant problem

　　

There are two approaches to dark energy.

(i) Changing gravity (ii) Changing matter

f(R) gravity models,Scalar-tensor models,Braneworld models,Inhomogeneities, …..

Quintessence,K-essence,Tachyon,Chaplygin gas,…..

Are there some other models of dark energy?

(Einstein equations)

‘Changing matter’ models

To get the present acceleration most of these models are based upon scalar fields with a very light mass:

Quintessence, K-essence, Tachyon, phantom field, …

Flat

In super-symmetric theories the severe fine-tuning of the field potential is required.

(Kolda and Lyth, 1999)

The coupling of the field to ordinary matter should lead to observable long-range forces.

(Carroll, 1998)

‘Changing gravity’ models f(R) gravity, scalar-tensor gravity, braneworld models,..

Dark energy may originate from some geometricmodification from Einstein gravity.

The simplest model: f(R) gravity

model:

f(R) modified gravity models can be used for dark energy ?

R: Ricci scalar

Field Equations The field equation can be derived by varying the action with respect to

satisfies

Trace

The field equation can be written in the form

Field EquationsWe consider the spatially flat FLRW spacetime

the Ricci scalar R is given by

The energy-momentum tensor of matter is given by

The field equations in the flat FLRW background give

where the perfect fluid satisfies the continuity equation

f(R) gravity

GR Lagrangian: (R is a Ricci scalar)

Extensions to arbitrary function f (R)

f(R) gravity

The first inflation model (Starobinsky 1980) Starobinsky

Inflation is realized by the R term.2

Favored from CMB observations

Spectral index:

Tensor to scalar ratio:

N: e-foldings

　

f(R) dark energy: Example

Capozziello, Carloni and Troisi (2003)Carroll, Duvvuri, Trodden and Turner (2003)

It is possible to have a late-time acceleration as the second term becomes important as R decreases.

In the small R region we have

Late-time acceleration is realized.

(n>0)

Problems: Matter instability, perturbation instability, absence of matter dominated era, local gravity constraints…

Stability of dynamical systems

consider the following coupled differential equations for two variables x(t) and y(t):

Fixed or critical points (xc, yc) if

A critical point (xc, yc) is called an attractor when it satisfiesthe condition

Stability around the fixed points

consider small perturbations δx and δy around the critical point (xc, yc),

leads to

The general solution for the evolution of linear perturbations

E. Copeland, M. Sami, S.Tsujikawa, IJMPD (2006)

stability around the fixed points

Autonomous equations

We introduce the following variables:

Then we obtain and

where

The above equations are closed.

See review: S.Tsujikawa et al (2006,2010)

and

,

model:

The parameter

characterises a deviation from the model.

The constant m model corresponds to

(a)

(b)

(c) The model of Capozziello et al and Carroll et al:

This negative m case is excluded as we will see below.

The cosmological dynamics is well understood by the geometrical approach in the (r, m) plane.

(i) Matter point: P M

From the stability analysis around the fixed point, the existence of the saddle matter epoch requires

at

(ii) De-sitter point P A

For the stability of the de-Sitter point, we require

m1

Viable trajectories

Constant m model:

(another accelerated point)

Lists of cosmologically non-viable models(n>0)

…. many !

Lists of cosmologically viable models(0<n<1) Li and Barrow (2007)

Amendola and Tsujikawa. (2007)

Hu and Sawicki (2007)

Starobinsky (2007)

More than 200 papers were written about f(R) dark energy!

Conformal transformation

Under the conformal transformation

The Ricci scalars in the two frames have the following relation

whereThe action

is transformed as

for the choice introduce a new scalar field φ defined by

the action in the Einstein frame (The scalar is directly coupled to matter)

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the Lagrangian density of the field φ is given by

the energy-momentum tensor

The conformal factor is field-dependent.

Using

matterThe energy-momentum tensor of perfect fluids in the Einstein frame is given by

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consider the flat FLRW spacetime

The field equation can be expressed as

the scalar field and matter interacts with each other

• The f(R) action is transformed to

Matter fluid satisfies:

Coupled quintessence

where

Dark matter is coupled to the field

Is the model (n>0) cosmologically viable?

No!This model does not have a standard matter eraprior to the late-time acceleration.

(Einstein frame)

The model

The potential in Einstein frame is

The standard matter era is replaced by ‘phi matter dominated era’

For large field region,

Coupled quintessence withan exponential potential

:

(n>0)

Jordan frame:

Incompatible with observations

L. Amendola, D. Polarski, S.Tsujikawa, PRL (2007)

.

Inertia of EnergyMeshchersky’s equation

Mv dmvt v

Momentum Inertial drag Momentum transfer

energyRocket

He, Wang, Abdalla PRD(2010)

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Physical meaning of The conformal transformation

The equation of motion under such a transformation

where we have used D.20 in Wald’s book

for perfect fluid and drop pressure

where

it reduces to

comparing with the equation of motion of particles with varying mass

We have introduce a scalar field Γ which satisfies

This Γ can be rewritten asmass dilation rate due to the conformal transformation.

J.H.He, B.Wang, E.Abdalla, PRD(11)

For the FRW background with a scale factor a, we have

Pressure-less Matter:

Radiation:

To avoid: matter instability, instability in perturbation, absence of MD era, inability to satisfy local gravity constraints

What are general conditions for the cosmological viabilityof f(R) dark energy models?

S.Tsujikawa et al (07); W.Hu et al (07)

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(0<n<1) Li and Barrow (2007)

Amendola and Tsujikawa. (2007)

Hu and Sawicki (2007)

Starobinsky (2007)

Lists of cosmologically viable models constructed

To avoid: matter instability, instability in perturbation, absence of MD era, inability to satisfy local gravity constraints

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Construct the f(R) model in the Jordan frameFRW metric

take an expansion history in the Jordan frame that matches a DE model with equation of state w

For w=-1:

C and D are coefficients which will be determined by boundary conditions

J.H.He, B.Wang (2012)

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f(R) model should be “chameleon” type go back to the standard Einstein gravity in the

high curvature regionneed to set C = 0.

The solution turnsD above is a free parameter which characterizes the different f(R) models which have the same expansion history as that of the LCDM model.

is the complete Euler Gamma functionanalytic f(R) form

and D are two free dimensionless parameters,

avoid the short-timescale instabilityat high curvature, D<0 is requiredis satisfied

J.H.He, B.Wang (2012)

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Construct the f(R) model from the Einstein frameConformal transformationMotion of particle with varying mass

freedom in choosing

Solving dynamics in the Einstein frame

J.H.He, B.Wang,E.Abdalla, PRD(11)

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conformal dynamics in the Jordan frame

J.H.He, B.Wang,E.Abdalla, PRD(11)

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Perturbation theoryThe Jordan frame

The perturbed line element in Fourier space

The perturbed form of the modified Einstein equations

Inserting the line element, we can get the perturbation form of the modified Einstein equation

J.H.He, B.Wang (2012)

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perturbed form of the modified Einstein equations

Under the infinitesimal transformation, we can show that the perturbed quantities in the line element

Inserting into the perturbation equation, we find that under the infinitesimal transformation, the perturbation equations are covariant. They go back to the standard form when F → 1,δF → 0.

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The Newtonian gauge is defined by setting B=E=0 and these conditions completely fix the gauge

The perturbations in this gauge can be shown as gauge invariant.the Synchronous gauge is not completely fixed because the gauge

condition ψ = B = 0 only confines the gauge up to two arbitrary constants C1,C2

Usually, C2 is fixed by specifying the initial condition for the curvature perturbation in the early time of the universe and C1 can be fixed by setting the peculiar velocity of DM to be zero, v_m = 0. After fixing C1,C2, the Synchronous gauge can be completely fixed.

perturbations in different gauges are related

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in the Newtonian gaugeuse the Bardeen potentials Φ= φ, Ψ= ψ to represent the space time

perturbations. Consider the DM dominated period in the f(R) cosmology, we set P_m = 0 and δP_m = 0The perturbed Einstein equations

Where

From the equations of motion

matter perturbation evolves

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Perturbation theoryThe Einstein

frameIn the background

in perturbed spacetime

the symbols with “tilde” indicate the quantities in Einstein frameUnder the infinitesimal coordinate transformation, the perturbed

quantities ˜ψ, ˜φ behave as

in a similar way as in the Jordan frame

In the Newtonian gauge, the gauge conditions B=E=0

in the Synchronous gauge, the gauge conditions in the Einstein frame reads

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The perturbed equationIn the Einstein frame

J.H.He, B.Wang (2012)

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SUBHORIZION APPROXIMATION

When the modified gravity doesn’t show up

When the modified gravity becomes important

J.H.He, B.Wang (2012)

Modified Gravity

G

mm

Jordan frame Einstein frame

if we compare it with the Einstein gravity.

This is effectively equivalent to rescale the gravitational mass

the inertial mass in the Jordan frame is conserved so that the equation of motion for a free particle in the Jordan frame is described by change in the gravitational mass

changes the gravitational field

change the inertial frame.

inertial“frame-dragging”

inertial frameunchanged, inertial mass rescaled

Gravity Probe B

Mach principle

inertial“mass-dragging”

Understand the mass dilation

Conclusions 1. We reviewed the relation between the f(R) gravity and the interaction between DE and DM

2. We discussed viability condition for the f(R) model

3. We discussed the perturbation theory for the f(R) model

4. We further discussed the physical connection between the Jordan frame and the Einstein frame and the physical meaning of the mass dilation