Crossing of Phantom Divide in F(R) Gravity Reference: Phys. Rev. D 79 , 083014 (2009) [arXiv:0810.4296 [hep-th]] Presenter : Kazuharu Bamba (National Tsing Hua University) Shin'ichi Nojiri (Nagoya University) Sergei D. Odintsov (ICREA and IEEC-CSIC) Collaborators : Chao-Qiang Geng (National Tsing Hua University) International Workshop on Dark Matter, Dark Energy and Matter-antimatter Asymmetry on November 21, 2009 at National Tsing Hua University
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Crossing of Phantom Divide in F(R) Gravity
Reference: Phys. Rev. D 79, 083014 (2009) [arXiv:0810.4296 [hep-th]]
International Workshop on Dark Matter, Dark Energy and Matter-antimatter Asymmetry
on November 21, 2009at National Tsing Hua University
I. Introduction
・ Current cosmic acceleration
・ F(R) gravity
・ Crossing of the phantom divide
II. Reconstruction of a F(R) gravity model with realizing the crossing of the phantom divide
III. Summary
< Contents >No. 2
We use the ordinary metric formalism, in which the connection is written by the differentiation of the metric.
*
I. Introduction No. 3
Recent observations of Supernova (SN) Ia confirmed that the current expansion of the universe is accelerating.
・
[Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999)][Riess et al. [Supernova Search Team Collaboration], Astron. . J. 116, 1009 (1998)]
There are two approaches to explain the current cosmic acceleration. [Copeland, Sami and Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006)]
・
< Gravitational field equation > Gö÷
Tö÷
: Einstein tensor: Energy-momentum tensor
: Planck mass
Gö÷ = ô2Tö÷
Gravity Matter
(1) General relativistic approach(2) Extension of gravitational theory
The scalar degree of freedom may acquire a large effective mass at terrestrial and Solar System scales, shielding it from experiments performed there.
‘‘Chameleon mechanism’’
(5) Existence of a matter-dominated stage and that of a late-time cosmic acceleration
m ñ RF 00(R)/F 0(R)
Combing local gravity constraints, it is shown that・
[Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)]
[Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)]
quantifies the deviation from the CDM model. Λ
has to be several orders of magnitude smaller than unity.
m
No. 13
(6) Stability of the de Sitter space
[Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)]
Rd Constant curvature in the de Sitter space
Fd = F(Rd)
F 0dF
00d
(F 0d)
2à 2FdF00d > 0 :
Linear stability of the inhomogeneous perturbations in the de Sitter space
・
Rd = 2Fd/F0d m < 1Cf.
(7) Free of curvature singularities
Existence of relativistic stars・
[Frolov, Phys. Rev. Lett. 101, 061103 (2008)]
[Kobayashi and Maeda, Phys. Rev. D 78, 064019 (2008)]
No. 14
[Kobayashi and Maeda, Phys. Rev. D 79, 024009 (2009)]
No. 15< Models of F(R) gravity >(a) Hu-Sawicki model [Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)]
(b) Starobinsky’s model [Starobinsky, JETP Lett. 86, 157 (2007)]
(c) Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)]
FHS(R) = R
FS(R) = R+
c2, c3 : Constants
Mö : Mass scale
p > 0 : Constant
õΛ0 1 +Λ2
0
R2ð ñàn
à 1
ô õ õ > 0, n > 0 : Constants
Λ0 Current cosmological constant
:
FAB(R) = 2R+
2a1 log cosh(aR)à tanh(b) sinh(aR)[ ]
a > 0, b : Constants
(d) Tsujikawa’s model
FT(R) = R à öRctanh Rc
Rð ñ[Tsujikawa, Phys. Rev. D 77, 023507 (2008)]]
Rc > 0 : Constants
No. 16
[Miranda, Joras, Waga and Quartin, Phys. Rev. Lett. 102, 221101 (2009)]
< Model with satisfying the condition (7) >
FMJWQ(R)
ë > 0, Rã > 0 : Constants
Cf. [de la Cruz-Dombriz, Dobado and Maroto, Phys. Rev. Lett. 103, 179001 (2009)]
Regarding the condition (7), under debate.・[Babichev and Langlois, arXiv:0904.1382 [gr-qc]]
[Upadhye and Hu, Phys. Rev. D 80, 064002 (2009)]
No. 17< Crossing of the phantom divide >Various observational data (SN, Cosmic microwave background radiation (CMB), BAO) imply that the effective EoS of dark energy may evolve from larger than -1 (non-phantom phase) to less than -1 (phantom phase). Namely, it crosses -1 (the crossing of the phantom divide).
・
[Alam, Sahni and Starobinsky, JCAP 0406, 008 (2004)][Nesseris and Perivolaropoulos, JCAP 0701, 018 (2007)]
wDE > à 1
Non-phantom phase(i)
wDE = à 1(ii)
Crossing of the phantom dividewDE < à 1(iii)
Phantom phase
wDE
wDE
à 1
0 t
tc
tc
: Time of the crossing of the phantom divide
[Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)]
< Models to account for the crossing of the phantom divide >
・ Two scalar field model, e.g., QuintomCanonical scalar field + phantom
[Feng, Wang and Zhang, Phys. Lett. B 607, 35 (2005)]
Gravitational field equations in the flat FRW background:
ú pwi
and are the sum of the energy density and pressure of matters with a constantEoS parameter , respectively, where i denotes some component of the matters.
þ = tmay be taken as because can be redefined properly.þ þ
aö gà(t) : Proper function
< Scale factor >
: Constant,
component(ö, ÷) = (0,0)
(ö, ÷) = (i, j) (i, j = 1, á á á,3)Trace part of components
・
・
No. 21
・
No. 22
úöi : Constant,
P(þ)We derive the solutions of and .Q(þ)
No. 23< II B. Explicit model >< Solution without matter >
,
í > 0, C > 0
t0 : Present time
pàæ
: Constants
: Arbitrary constants
gà(t)When , diverges.
þ = tþWe take as and only consider the period .・
: Big Rip singularity→∞・
No. 24
weff = úeff
peff
< Effective EoS >
,・
Hç > 0
Hç < 0
Hç = 0
Non-phantom phase
Crossing of the phantom divide
Phantom phase
(i)
(ii)
(iii) weff < à 1
weff = à 1
weff > à 1
< Hubble rate >
Cf.
1à F 0(R)/F 0(R0)( )Ωm
weffwDE '
ù weff
No. 25< Evolution of >
t < tc U(t) > 0
dtdU(t) < 0
decreases monotonously. U(t)
crosses -1. we
・
・
weff
:
ff
It evolves from positive to negative.
tc
tc
0 t tc
weff
à 1
0 t
U(t)
: Time of the crossing of the phantom divide
No. 26
If we have the solution , we can obtain t = t(R)
F(R) = P(t(R))R+Q(t(R)).
Scalar curvature:
< Forms of and >P Q
・
No. 27< Fig. 1 >
Behavior of as a function of for, , , ,
and [ ]. : Current curvatureR0
H0・ , : Present Hubble parameter
C = t0/ts( )2í+1 = 1/4
No. 28
t→ 0 (t/ts ü 1)(1)
F(R) ù A1Rà5í/2+1 P
j=æCjR
àìj/2
A1 = ,
(2) :
F(R) ù A2R7/2
:
A2 =
< Analytic form of >F(R)
[KB and Geng, Phys. Lett. B 679, 282 (2009)]
III. SummaryA scenario to explain the current accelerated expansion of the universe is to study a modified gravitational theory, such as F(R) gravity.
No. 29
We have reconstructed an explicit model of F(R) gravity with realizing the crossing of the phantom divide.
・
・
・
The Big Rip singularity appears.
Various observational data imply that the effective EoSof dark energy may evolve from larger than -1 (non-phantom phase) to less than -1 (phantom phase). Namely, it crosses -1 (the crossing of the phantom divide).
Around the Big Rip singularity, .F(R) ∝ R7/2
Backup Slides
H20
1aa = à 2
Ωm(1 + z)3 +ΩΛ
Ωm ñ3H2
0
ô2ú(t0)
ΩΛ ñ 3H20
Λ
From [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]
1 + z = aa0, z
z
mà
: Red shift
MDistance estimator
:
‘‘0’’ denotes quantities at the present time . t0
Flat cosmologyΛ
Ωm = 0.26
ΩΛ = 0.74
< SNLS data >
Ωm = 1.00ΩΛ = 0.00
Pure matter cosmologym
M
Apparent magnitude
Absolute magnitude
:
:
: Density parameter for Λ
: Density parameter for matter
No. BS1
From [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]
z
Flat cosmologyΛ
Δ(màM)
< Residuals for the best fit to a flat cosmology >Λ
Pure matter cosmology
No. BS2
F 0(R) = dF(R)/dR< Gravitational field equation >
0
: Covariant d'Alembertian
: Covariant derivative operator
< F(R) gravity >
S2ô2F(R)
: General Relativity
F(R) gravity
[Sotiriou and Faraoni, arXiv:0805.1726 [gr-qc]]
[Nojiri and Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)]
F(R) = R
[Capozziello and Francaviglia, Gen. Rel. Grav. 40, 357 (2008)]
[Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)]
[Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 170, 377 (2007)]
+
w(z) = w0 +w1 1+zz< Data fitting of >w(z)
1ûShaded region shows error.
No. BS9
[Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)]
For most observational probes (except the SNLS data), a low prior leads to an increased probability (mild trend) for the crossing of the phantom divide.
Ω0m (0.2 < Ω0m < 0.25)・
Ω0m : Current density parameter of matter
No. BS10
2û confidence level.
From [Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)]
SN gold data set+CMB+BAO SNLS data set+CMB+BAO
・ ・
< Data fitting of (2) >w(z) No. BS11
< Baryon acoustic oscillation (BAO) >
From [Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)]
Ωbh2 = 0.024
Ωmh2 = 0.12, 0.13, 0.14, 0.105
(From top to bottom)
Pure CDM model (No peak)
No. BS12
< Note on the reconstruction method >If we redefine and define , we obtain
This is equivalent to the original action.
Φ : Proper function
We have the choices in like a gauge symmetry and thus we can identify with time , i.e., , which can be interpreted as a gauge condition corresponding to the reparameterization of .
þþ t
・
・
No. BS13
< Reconstruction of an explicit model >Equation for without matter: P(þ)・
By redefining as P(þ)
The model
< Solutions >
,
No. BS14
(i)
> à 1
(ii)
(iii)
< à 1
Hç = 0
,
,
, weff = à 1
Non-phantom phase
Crossing of the phantom divide
Phantom phase
t→ 0
< Hubble rate > No. BS15
< Behavior of >U(t)
for ,t < tcU(t) > 0 í > 0
dtdU(t) < 0 for
・
・
decreases monotonously. It evolves from positive to negative. Consequently, crosses -1. U(t)
weff
because .
< Scalar curvature > No. BS16
< Analytic form of >F(R)
t→ 0(1) :
・ In the limit , t/ts ü 1
F(R) ù A1Rà5í/2+1 P
j=æCjR
àìj/2
A1 = ,
No. BS17
[KB and Geng, Phys. Lett. B 679, 282 (2009)]
・ For large R, namely, ,
(2) :
F(R) ù A2R7/2
A2 =
No. BS18
II C. Property of the singularity in the corresponding scalar field theory
Action in the Jordan frame
Equation of motion for :ø
ð
Action of F(R) gravity
We introduce two scalar fields: and .
There is a non-minimal coupling between and . ø R
F 0(ð) = dF(ð)/dð
ð ø
Equation of motion for :
・
・
No. BS19
Conformal transformation :
û
,
: Scalar field
・
Action in the Einstein frame There is no non-minimal coupling between a scalar field and . Rê
A hat denotes quantities in the Einstein frame.
We redefine as .ϕ・
Canonical scalar field theory
No. BS20
c1, n : Constants, M : Mass scale・
Around the Big Rip singularity, the reconstructed model of F(R) gravity behaves as .
・
dt dtê< Relation between and >
No. BS21
< Relation between and > t tê
If , the limit of corresponds to that of .
・
‘Finite-time’ Big Rip singularity in F(R) gravity(the Jordan frame)
‘Infinite-time’ singularity in the corresponding scalar field theory (the Einstein frame)
Conformal transformation
In the reconstructed model of F(R) gravity, around the Big Rip singularity, .
・
No. BS22
< Scale factor in the Einstein frame > aê têà á No. BS23
We have shown that the (finite-time) Big Rip singularity in the reconstructed model of F(R) gravity becomes the infinite-time singularity in the corresponding scalar field theory obtained through the conformal transformation.
・
No. BS24
< Further results >
It has been demonstrated that the scalar field theories describing the non-phantom phase (phantom one with the Big Rip singularity) can be represented as the theories of real (complex) F(R) gravity through the inverse (complex) conformal transformation.
・
We have examined that the quantum correction of massless conformally-invariant fields could be small when the crossing of the phantom divide occurs and the obtained solutions of the crossing of the phantom divide could be stable under the quantum correction, although it becomes important near the Big Rip singularity.
・
No. BS25
Appendix ARelation between scalar field theory and F(R) gravity
< Relation between scalar field theory and F(R) gravity >
à+ : Phantom case
Real conformal transformation :・
: Non-phantom case
Action in the Einstein-frame
Action in the Jordan-frame
can be expressed as by solving the equation of motion for :ÿÿ・
ÿ : Real scalar field
(1) Non-phantom (canonical) field[Capozziello, Nojiri and Odintsov, Phys. Lett. B 634, 93 (2006)]
Real F(R) gravity
Wà (ÿ) : Potential of ÿ
No. A2
No. A3(2) Phantom field[Briscese, Elizalde, Nojiri and Odintsov, Phys. Lett. B 646, 105 (2007)]
Complex conformal transformation :・
Action in the Jordan-frame
We can obtain the relation by solving the equation of motion for : ÿ
・
Complex F(R) gravity
Condition for to be real:R・
With the except of satisfying this relation, is complex. Wà (ÿ) R
Non-phantom (canonical) field theory
Phantom field theory
Real F(R) gravity
Complex F(R) gravity
Real conformaltransformation
Complex conformaltransformation
< Summary >
(except the special case)
No. A4
It has been demonstrated that the scalar field theories describing the non-phantom phase (phantom one with the Big Rip singularity) can be represented as the theories of real (complex) F(R) gravity through the inverse (complex) conformal transformation.
・
< Summary > No. A5
Appendix B
I. Stability under a quantum correction
II. Model of F(R) gravity with the transition from the de Sitter universe to the phantom phase
III. Summary
I. Stability under a quantum correctionQuantum effects produce the conformal anomaly: ・
: Square of 4d Weyl tensor
: Gauss-Bonnet invariant
In the flat FRW background, we find , .
: Real scalar
: Dirac spinor
: Vector fields: Gravitons
NN1/2
N1
can be arbitrary, e.g., we can choose
No. B2
NHD Higher derivative conformal scalars
:
・
Assuming andEnergy density and pressure from conformal anomaly, respectively
úA pAand :
the conservation law : ,
,
we find
Effective energy density and pressure from úF pF
・
・
We assume the Hubble rate at the phantom crossing is given by ・
No. B3
・
, : Dimensionless constant
・
f(R) plays the role of the effective cosmological constant
The quantum correction could be small when the crossing of the phantom divide occurs and the obtained solutions of the crossing of the phantom divide could be stable under the quantum correction. (The quantum correction becomes important near the Big Rip singularity.)
・
,
(Coming from the numerical constants: .)
No. B4
II. Model of F(R) gravity with the transition from the de Sitter universe to the phantom phase
Expanding the Kummer functions in and taking the first leading order in , we obtainz
No. B8
(2) Case with the cold dark matter We take into account the cold dark matter with its EoS . w = 0・We numerically solve the equation for . ・ P(þ)
Behavior of as a function of for, and .
úö
: Critical density
Current energy density of the cold dark matter
:
No. B9
< Behavior of and >
Behavior of and as a function of for , and .
t2sQ(Rà) Rà
No. B10P Q
We have reconstructed a model of F(R) gravity in which the transition from the de Sitter universe to the phantom phase can occur.
・
< Summary > No. B11
We have examined that the quantum correction of massless conformally-invariant fields could be small when the crossing of the phantom divide occurs and the obtained solutions of the crossing of the phantom divide could be stable under the quantum correction, although it becomes important near the Big Rip singularity.
・
Appendix CScalar field theory with realizing the constructed H(t)
< Scalar field theory with realizing the constructed >H(t)
ω(Φ) : Function of ,Φ W(Φ)Φ: Scalar field, : Potential of ΦIn the flat FRW background, the Einstein equations are given by
,
,
・
,
We define and in terms of a single function .ω(Φ) W(Φ) I(Φ)・
Solutions :
,
No. C2
Defining as , we obtain・ ÿ
ω(Φ) > 0
ω(Φ) < 0
à+
with realizing the phantom crossing in F(R) gravity H(t)・
No. C3
sign
sign
No. C4
Hç < 0 =⇒ ω(Φ) > 0
Hç = 0 =⇒ ω(Φ) = 0
Non-phantom phase
Crossing of the phantom divide
Phantom phase
(i)
(ii)
(iii)
: à
:
: Hç > 0 =⇒ ω(Φ) < 0 +
Cf. [Sanyal, arXiv:0710.3486 [astro-ph]]
It has been shown that for such a scalar field theory in the presence of the background cold dark matter, the crossing of the phantom divide can occur.
・
No. C5
sign
sign
・
< Summary > No. C6
We have studied the scalar field theory with realizing the constructed . H(t)