f(R) Gravity and its
relation to the
interaction between DE
and DM
Bin Wang
Shanghai Jiao Tong University
1
Dark Energy
SNe Ia
CMB
LSS
The current
universe is
accelerating!
Simplest model of dark energy
Cosmological constant:
(Equation of state:
)
This corresponds to the energy scale
If this originates from vacuum energy in particle physics,
Huge difference compared to the present value!
Cosmological constant problem
Are there some other models of dark energy?
There are two approaches to dark energy.
(Einstein equations)
(i) Changing gravity
f(R) gravity models,
Scalar-tensor models,
Braneworld models,
Inhomogeneities,
…..
(ii) Changing matter
Quintessence,
K-essence,
Tachyon,
Chaplygin gas,
…..
‘Changing matter’ models
Quintessence, K-essence,
Tachyon, phantom field, …
To get the present acceleration
most of these models are based upon
scalar fields with a very light mass:
The coupling of the field to ordinary
matter should lead to observable
long-range forces.
(Carroll, 1998)
In super-symmetric theories the severe fine-tuning
of the field potential is required.
(Kolda and Lyth, 1999)
Flat
‘Changing gravity’ models
f(R) gravity, scalar-tensor gravity,
braneworld models,..
Dark energy may originate from some geometric
modification from Einstein gravity.
The simplest model:
f(R) gravity
R: Ricci scalar
model:
f(R) modified gravity models can be used for dark energy ?
Field Equations
The field equation can be derived by varying
the action with respect to
Trace
The field equation can be written in
the form
satisfies
We consider the spatially flat FLRW spacetime
Field Equations
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
2
Inflation is realized by the R term.
Favored from CMB observations
Spectral index:
Tensor to scalar ratio:
N: e-foldings
f(R) dark energy: Example
(n>0)
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.
Problems: Matter instability, perturbation instability, absence of
matter dominated era, local gravity constraints…
E. Copeland, M. Sami,
S.Tsujikawa, IJMPD (2006)
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 satisfies
the 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
stability around the fixed points
Autonomous equations
See review:
S.Tsujikawa et al (2006,2010)
We introduce the following variables:
Then we obtain
where
and
,
and
The above equations are closed.
The parameter
characterises a deviation from the
(a)
model.
model:
(b) The constant m model corresponds to
(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
m1
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
Viable trajectories
(another accelerated point)
Constant m model:
More than 200 papers were written about f(R) dark energy!
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)
Conformal transformation
Under the conformal transformation
The Ricci scalars in the two frames have the following relation
where
The 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)
the Lagrangian density
of the field φ is given by
the energy-momentum
tensor
The conformal factor is field-dependent.
Using
matter
The energy-momentum tensor of perfect fluids in the Einstein frame is given by
19
consider the flat FLRW spacetime
The field equation can be expressed as
the scalar field and matter
interacts with each other
20
Is the model
(n>0) cosmologically viable?
No!
This model does not have a standard matter era
prior to the late-time acceleration.
• The f(R) action is transformed to
(Einstein
frame)
where
Matter fluid satisfies:
Dark matter is coupled to the field
Coupled
quintessence
(n>0)
The model
The potential in Einstein frame is
.
For large field region,
Coupled quintessence with
an exponential potential
The standard matter era is replaced by ‘phi matter dominated era’
:
Jordan frame:
Incompatible with observations
L. Amendola, D. Polarski, S.Tsujikawa, PRL (2007)
Inertia ofMeshchersky’s
Energy equation
Rocket
v
energy
vt
dm
v
M
Momentum
Inertial drag
Momentum
transfer
He, Wang, Abdalla
PRD(2010)
Physical meaning of
J.H.He, B.Wang, E.Abdalla, PRD(11)
The conformal transformation
The equation of motion under such a transformation
for perfect fluid and drop pressure
where we have used D.20
in Wald’s book
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 as
mass dilation rate
due to the conformal
24
transformation.
What are general conditions for the cosmological viability
of f(R) dark energy models?
S.Tsujikawa et al (07); W.Hu et al (07)
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
To avoid: matter instability, instability in perturbation, absence of
MD era, inability to satisfy local gravity constraints
Lists of cosmologically viable models constructed
(0<n<1)
Li and Barrow (2007)
Amendola and Tsujikawa. (2007)
Hu and Sawicki (2007)
Starobinsky (2007)
26
J.H.He, B.Wang (2012)
Construct the f(R) model in the Jordan frame
FRW 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
27
J.H.He, B.Wang (2012)
f(R) model should be “chameleon” type
go back to the standard Einstein gravity in the high
curvature region need to set C = 0.
The solution turns
D 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 function
analytic f(R) form
and D are two free dimensionless
parameters,
avoid the short-timescale instability
at high curvature, D<0 is required
is satisfied
28
J.H.He, B.Wang,E.Abdalla, PRD(11)
Construct the f(R) model from the Einstein frame
Conformal transformation
Motion of particle with varying mass
freedom in choosing
Solving dynamics in
the Einstein frame
29
conformal dynamics in the Jordan frame
30
J.H.He, B.Wang,E.Abdalla, PRD(11)
Perturbation theory
The 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
31
J.H.He, B.Wang (2012)
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
32
when F → 1,δF → 0.
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
33
in the Newtonian gauge
use 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 = 0
The perturbed Einstein equations
Where
From the equations of motion
matter perturbation evolves
34
Perturbation theory
The Einstein frame
In the background
in perturbed spacetime
the symbols with “tilde” indicate the quantities in Einstein frame
Under 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
35
The perturbed equation
In the Einstein frame
J.H.He, B.Wang (2012)
36
J.H.He, B.Wang (2012)
SUBHORIZION APPROXIMATION
When the modified gravity doesn’t show up
When the modified gravity becomes important
37
Modified
Gravity
Jordan frame
Mach principle
Einstein frame
m  m
G
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
Understand the
mass dilation
inertial
“massdragging”
change the
inertial frame.
inertial frame
unchanged,
inertial mass
rescaled
inertial
“framedragging”
Gravity
Probe B
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