Upload
ophelia-horton
View
215
Download
0
Tags:
Embed Size (px)
Citation preview
1
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)
19
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
20
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)
24
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)
26
(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
27
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)
28
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)
29
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)
30
conformal dynamics in the Jordan frame
J.H.He, B.Wang,E.Abdalla, PRD(11)
31
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)
32
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.
33
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
34
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
35
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
36
The perturbed equationIn the Einstein frame
J.H.He, B.Wang (2012)
37
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