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, 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

Dark Energy

[Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]

(1) General relativistic approach・ Cosmological constant

K-essence

Tachyon

[Caldwell, Dave and Steinhardt, Phys. Rev. Lett. 80, 1582 (1998)]

[Chiba, Okabe and Yamaguchi, Phys. Rev. D 62, 023511 (2000)] [Armendariz-Picon, Mukhanov and Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)]

[Padmanabhan, Phys. Rev. D 66, 021301 (2002)]

X matter, Quintessence

Non canonical kinetic term

String theories

・ Scalar field :

No. 4

・ Chaplygin gas[Kamenshchik, Moschella and Pasquier, Phys. Lett. B 511, 265 (2001)]

ú : Energy density: Pressurepp = àA/ú

A > 0 : Constant

Phantom[Caldwell, Phys. Lett. B 545, 23 (2002)]

Cf. Pioneering work: [Fujii, Phys. Rev. D 26, 2580 (1982)]

[Chiba, Sugiyama and Nakamura, Mon. Not. Roy. Astron. Soc. 289, L5 (1997)]

Wrong sign kinetic term

Canonical field

(2) Extension of gravitational theory・ F(R) gravity

: Arbitrary function of the Ricci scalarF(R) R

・ Scalar-tensor theories

・ DGP braneworld scenario

No. 5

[Capozziello, Cardone, Carloni and Troisi, Int. J. Mod. Phys. D 12, 1969 (2003)]

f1(þ)Rþ

: [Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, 043528 (2004)]

[Nojiri and Odintsov, Phys. Rev. D 68, 123512 (2003)]

[Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)]

・ Higher-order curvature term

[Nojiri, Odintsov and Sasaki, Phys. Rev. D 71, 123509 (2005)]

[Gannouji, Polarski, Ranquet and Starobinsky, JCAP 0609, 016 (2006)]

fi(þ) Arbitrary function of a scalar field

Gauss-Bonnet term with a coupling to a scalar field: Ricci curvature tensor

Riemann tensor

G ñ R2à

[Starobinsky, Phys. Lett. B 91, 99 (1980)]Cf. Application to inflation:

[Boisseau, Esposito-Farese, Polarski and Starobinsky, Phys. Rev. Lett. 85, 2236 (2000)]

[Deffayet, Dvali and Gabadadze, Phys. Rev. D 65, 044023 (2002)]

・ Ghost condensates [Arkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405, 074 (2004)]

(i = 1,2)

:

:

f2(þ)G

< Flat Friedmann-Robertson-Walker (FRW) space-time >)(ta : Scale factor

aa = à 6

ô2

ú+ 3p( )

Tö÷ = diag(ú, p, p, p)

ú : Energy density, : Pressurep

ú+ 3p < 0 a > 0

w ñ úp w < à 3

1

: Accelerated expansion

< Equation of state (EoS) >

Condition for accelerated expansion

No. 6

: Hubble parameter

úç + 3H(1 +w)ú = 0 a ∝ t3(1+w)2

ú ∝ aà3(1+w)

Continuity equation・

w

If・

< Equation for with a perfect fluid >)(ta

:

w = à 1Cf. Cosmological constant

< 5-year WMAP data on the current value of >

à 0.14 < 1 +w < 0.12 (95%CL)

・For the flat universe, constant :w

From [Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009), arXiv:0803.0547 [astro-ph]]

(From WMAP+BAO+SN)

w

ΩΛ ñ 3H20

ô2ú(0)Λ =

ú(0)c

ú(0)Λ

ú(0)c

ú(0)Λ Current energy density

of dark energy:

Hubble Space Telescope Key Project

: HST

: Critical density

(68%CL)Cf.

Baryon acoustic oscillation (BAO)

Special pattern in the large-scale correlation function of Sloan Digital Sky Survey (SDSS) luminous red galaxies

:

No. 7

à 0.33 < 1 +w0 < 0.21(95%CL)

・ For the flat universe, a variable EoS :

From [Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009), arXiv:0803.0547 [astro-ph]]

w0 = w(a = 1)

approaches to -1.

z > ztransw(z)

:

(68.3%CL)

(95.4%CL)

No. 8

< Canonical scalar field >

à 21gö÷∂öþ∂÷þà V(þ)

â ãSþ =

⎧⎭d4x à g√

g = det(gö÷)

ú =21þç2 + V(þ), p =

21þç2à V(þ)

: Potential of þV(þ)

þç2 ü V(þ) wþ ù à 1If , .

þ : Scalar field

Accelerated expansion can be realized.

・ For a homogeneous scalar field :þ = þ(t)

wþ = úþ

pþ =þç 2+2V(þ)

þç 2à2V(þ)

No. 9

< 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)]

・Example : F(R) ∝ Rn (n6=1)

a ∝ tq, q = nà2à2n2+3nà1

q > 1If , accelerated expansion can be realized.

weff = à 6n2à9n+36n2à7nà1

(For or , and .)n = 3/2 q = 2 weff = à 2/3

[Capozziello, Carloni and Troisi, Recent Res. Dev. Astron. Astrophys. 1, 625 (2003)]

n = à 1

g = det(gö÷)No. 10

No. 11< Conditions for the viability of F(R) gravity >(1)F 0(R) > 0

F 00(R) > 0(2)

(3) F(R)→Rà 2Λ Rý R0 R0for : Current curvature

[Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)]

Λ : Cosmological constant

Positivity of the effective gravitational coupling

Geff = G0/F0(R) > 0 G0 : Gravitational constant

・

Stability condition: M2 ù 1/(3F 00(R)) > 0

M Mass of a new scalar degree of freedom (called the “scalaron”) in the weak-field regime.

・

:

(The graviton is not a ghost.)

(The scalaron is not a tachyon.)

・Realization of the CDM-like behavior in the large curvature regime

Λ

Standard cosmology [ + Cold dark matter (CDM)]Λ

F 0(R) = dF(R)/dR

No. 12(4) Solar system constraints

EquivalentF(R) gravity Brans-Dicke theory

However, if the mass of the scalar degree of freedom is large, namely, the scalar becomes short-ranged, it hasno effect at Solar System scales.

[Chiba, Phys. Lett. B 575, 1 (2003)]

[Erickcek, Smith and Kamionkowski, Phys. Rev. D 74, 121501 (2006)]

[Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]

Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)]M = M(R)

・

・ Scale-dependence:

M

ωBD = 0with

Observational constraint: |ωBD| > 40000[Bertotti, Iess and Tortora, Nature 425, 374 (2003).]

ωBD : Brans-Dicke parameter

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)]

・ Scalar-tensor theories [Perivolaropoulos, JCAP 0510, 001 (2005)]

・ Single scalar field model with nonlinear kinetic terms[Vikman, Phys. Rev. D 71, 023515 (2005)]

・ String-inspired models[McInnes, Nucl. Phys. B 718, 55 (2005)]

[Cai, Zhang and Wang, Commun. Theor. Phys. 44, 948 (2005)]

No. 18

[Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)]< Preceding work >

F(R) = (R1/càΛ)c

c, Λ : ConstantsPhantom phase

Non-phantom phase

We reconstruct an explicit model of F(R) gravity with realizing the crossing of the phantom divide.

No. 19

c = 1.8Example: ・

< Recent work >

It has been shown that in the Hu-Sawicki model, the transition from the phantom phase to the non-phamtom one can also occur.

・

[Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, 123516 (2009)]

Cf. [Nozari and Azizi, Phys. Lett. B 680, 205 (2009)]

< II A. Reconstruction method >

II. Reconstruction of a F(R) gravity model with realizing the crossing of the phantom divide

No. 20

< Gravitational field equation >

Matter Lagrangian

< Action >

[Nojiri and Odintsov, Phys. Rev. D 74, 086005 (2006)]

P(þ)By introducing proper functions and , we find

þ : Scalar field

Equation of motion for :þ

Energy-momentum tensor of matter

:

Q(þ)

:

・

: Covariant d'Alembertian: Covariant derivative operator

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)]

No. BS3

・

,： Effective energy

density and pressure from the term F(R)àR

úeff , peff

・ Example : F(R) ∝ Rn (n6=1)

a ∝ tq, q = nà2à2n2+3nà1

q > 1If , accelerated expansion can be realized.

weff = à 6n2à9n+36n2à7nà1

(For or , and .)n = 3/2 q = 2 weff = à 2/3

[Capozziello, Carloni and Troisi, Recent Res. Dev. Astron. Astrophys. 1, 625 (2003)]

n = à 1

In the flat FRW background, gravitational field equations read No. BS4

From [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)]

< Preceding work >

F(R) = (R1/càΛ)c

c, Λ : Constants

Phantom phase

Non-phantom phase

We reconstruct an explicit model of F(R) gravity with realizing the crossing of the phantom divide.

c = 1.8Example: ・

No. BS5

< 5-year WMAP data on >

à 0.14 < 1 +w < 0.12 (95%CL)

・ For the flat universe, constant :w

à 0.33 < 1 +w0 < 0.21 (95%CL)

・ For a variable EoS :

w0 = w(a = 1)

Dark energy density tends to a constant value

:

[Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009), arXiv:0803.0547 [astro-ph]]

(From WMAP+BAO+SN)

No. BS6

Baryon acoustic oscillation (BAO) : Special pattern in the large-scale correlation function of Sloan Digital Sky Survey (SDSS) luminous red galaxies

Cf.

w

(68%CL)

Dark Energy : Dark Matter : Baryon :

Ωi ñ 3H20

ô2ú(0)i =

ú(0)c

ú(0)i

i = Λ, c, bú(0)c : Critical density

< F(R) gravity >

S 2ô2F(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)]

・ F(R) = Rà Rnö2(n+1)

a ∝ tq, q = n+2(2n+1)(n+1)

n = 1For , and .

weff = à 1 +3(2n+1)(n+1)

2(n+2)

Accelerated expansionq = 2 weff = à 2/3

[Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, 043528 (2004)]

Cf.

: Mass scale, ö n Second term become important as decreases.R

: Constant

No. BS7

(5) Existence of a matter-dominated stage and that of a late-time cosmic acceleration

Analysis of curve on the planem(r) (r,m)

m ñ RF 00(R)/F 0(R), r ñ àRF 0(R)/F(R)

m(r) ù + 0 drdm > à 1 r ù à 1

m(r) = à r à 1, 23

√à1 < m ô 1 dr

dm < à 1

0 < m ô 1 r = à 2

Presence of a matter-dominated stage・

Presence of a late-time acceleration・

and at

and

at(ii)

Combing local gravity constraints, we obtain

m(r) = C(à r à 1)p p > 1 as r→ à 1.

C > 0, p

with ・

(i)

: Constants

[Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)][Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)]

No. BS8

From [Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)]

[Riess et al. [Supernova Search Team Collaboration], Astrophys. J. 607, 665 (2004)]

[Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]

Cosmic microwave background radiation (CMB) data

SDSS baryon acoustic peak (BAO) data

SN gold data set SNLS data set

[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

: Constants

g0 > 0, g1 > 0, ts > 0

(1)

Hubble rate : ・

(2)

:

:

Asymptotically de Sitter space

Phantom phase

: Constant

: Big Rip singularity

・

No. B5

(1) Case with neglecting matter

・

< Solutions >

: Kummer functions (confluent hypergeometric function)FK

Cæ : Dimensionless constants

,

No. B6

We have taken and therefore .

(1)

(2)

: Constant de Sitter phase:

:

・

In this limit, because diverges. R・

No. B7

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)