Crossing of Phantom Divide in F(R) Gravity

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 at National Tsing Hua University on November 21, 2009 Presenter : Kazuharu Bamba (National Tsing Hua University) Collaborators : Chao-Qiang Geng (National Tsing Hua University) Shin'ichi Nojiri (Nagoya University) Sergei D. Odintsov (ICREA and IEEC-CSIC)

< Contents > No. 2 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 * We use the ordinary metric formalism, in which the connection is written by the differentiation of the metric.

I. Introduction Recent observations of Supernova (SN) Ia confirmed that the current expansion of the universe is accelerating. No. 3 [Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, 565 (1999)] [Riess et al. [Supernova Search Team Collaboration], Astron.. J. 116, 1009 (1998)] [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)] 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 > G ö G ö = ô 2 T ö Gravity Matter (1) General relativistic approach T ö (2) Extension of gravitational theory : Einstein tensor : Energy-momentum tensor : Planck mass Dark Energy

(1) General relativistic approach Cosmological constant Scalar field : [Chiba, Sugiyama and Nakamura, Mon. Not. Roy. Astron. Soc. 289, L5 (1997)] [Caldwell, Dave and Steinhardt, Phys. Rev. Lett. 80, 1582 (1998)] K-essence [Chiba, Okabe and Yamaguchi, Phys. Rev. D 62, 023511 (2000)] [Armendariz-Picon, Mukhanov and Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)] Tachyon X matter, Quintessence Cf. Pioneering work: [Fujii, Phys. Rev. D 26, 2580 (1982)] Phantom Non canonical kinetic term String theories [Padmanabhan, Phys. Rev. D 66, 021301 (2002)] Chaplygin gas Wrong sign kinetic term [Caldwell, Phys. Lett. B 545, 23 (2002)] p = à A/ú [Kamenshchik, Moschella and Pasquier, Phys. Lett. B 511, 265 (2001)] No. 4 Canonical field A>0 : Constant ú : Energy density : Pressure p

(2) Extension of gravitational theory F(R) gravity F(R) Scalar-tensor theories DGP braneworld scenario Cf. Application to inflation: [Starobinsky, Phys. Lett. B 91, 99 (1980)] : Arbitrary function of the Ricci scalar [Capozziello, Cardone, Carloni and Troisi, Int. J. Mod. Phys. D 12, 1969 (2003)] [Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, 043528 (2004)] [Nojiri and Odintsov, Phys. Rev. D 68, 123512 (2003)] f 1 (þ)r [Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)] [Deffayet, Dvali and Gabadadze, Phys. Rev. D 65, 044023 (2002)] R No. 5 f i (þ) : Arbitrary function (i =1, 2) of a scalar field þ [Boisseau, Esposito-Farese, Polarski and Starobinsky, Phys. Rev. Lett. 85, 2236 (2000)] [Gannouji, Polarski, Ranquet and Starobinsky, JCAP 0609, 016 (2006)] Ghost condensates [Arkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405, 074 (2004)] Higher-order curvature term f 2 (þ)g Gauss-Bonnet term with a coupling to a scalar field: G ñ R 2 à [Nojiri, Odintsov and Sasaki, Phys. Rev. D 71, 123509 (2005)] : Ricci curvature tensor : Riemann tensor

< Flat Friedmann-Robertson-Walker (FRW) space-time > < Equation for a(t) with a perfect fluid > ä = à ô 2 ( ú +3p) a 6 If ú +3p<0 ä > 0 < Equation of state (EoS) > w ñ ú p Continuity equation úç+3h(1 + w)ú =0 : Hubble parameter w a (t) : Scale factor T ö =diag(ú, p, p, p) ú : Energy density, Condition for accelerated expansion p : Pressure : Accelerated expansion Cf. Cosmological constant a t 3(1+w) 2 ú a à3(1+w) : w<à 3 1 No. 6 w = à 1

< 5-year WMAP data on the current value of > From [Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009), arxiv:0803.0547 [astro-ph]] ô Ω Λ ñ 2 ú (0) 3H 2 0 ú (0) c ú (0) Λ w Λ = ú (0) : Critical density : ú (0) Λ Current energy density of dark energy c No. 7 HST : Hubble Space Telescope Key Project Baryon acoustic oscillation (BAO) w For the flat universe, constant : : à 0.14 < 1+w<0.12 (95% CL) Special pattern in the large-scale correlation function of Sloan Digital Sky Survey (SDSS) luminous red galaxies (From WMAP+BAO+SN) Cf. (68% CL)

From [Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009), arxiv:0803.0547 [astro-ph]] (68.3% CL) (95.4% CL) No. 8 For the flat universe, a variable EoS : à 0.33 < 1+w 0 < 0.21 (95% CL) w(z) w 0 = w(a =1) z>z trans approaches to -1. :

< Canonical scalar field > S þ = d 4 x à g â à 1 g ö ã 2 ö þ þ à V(þ) No. 9 g =det(g ö ) ú = 2 1 þ ç 2 + V(þ), þ ç 2 ü V(þ) V(þ) þ : Scalar field : Potential of þ For a homogeneous scalar field þ = þ(t) : p w þ = þ þ = ç 2 à2v(þ) úþ þ ç 2 +2V(þ) p = 2 1 þ ç 2 à V(þ) w þ ùà1 If,. Accelerated expansion can be realized.

< F(R) gravity > F(R) S 2ô 2 F(R)=R : General Relativity F(R) gravity [Nojiri and Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)] [Capozziello and Francaviglia, Gen. Rel. Grav. 40, 357 (2008)] [Sotiriou and Faraoni, arxiv:0805.1726 [gr-qc]] Example : F(R) R n (n6=1) à2n 2 +3nà1 a t q, q = nà2 6n w eff = à 2 à7nà1 6n 2 à9n+3 q > 1 No. 10 g =det(g ö ) [Capozziello, Carloni and Troisi, Recent Res. Dev. Astron. Astrophys. 1, 625 (2003)] If, accelerated expansion can be realized. (For n =3/2 or n = à 1, q =2 and w eff = à 2/3.)

< Conditions for the viability of F(R) gravity > (1) F 0 (R) > 0 (2) (3) Positivity of the effective gravitational coupling G eff = G 0 /F 0 (R) > 0 (The graviton is not a ghost.) F 00 (R) > 0 G 0 : Gravitational constant [Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)] Stability condition: M 2 ù 1/(3F 00 (R)) > 0 M : Mass of a new scalar degree of freedom (called the scalaron ) in the weak-field regime. (The scalaron is not a tachyon.) F(R) R à 2Λ for R ý R 0 R 0 No. 11 F 0 (R)=dF(R)/dR : Current curvature Λ : Cosmological constant Realization of the Λ CDM-like behavior in the large curvature regime Λ Standard cosmology [ + Cold dark matter (CDM)]

(4) Solar system constraints F(R) gravity [Bertotti, Iess and Tortora, Nature 425, 374 (2003).] Equivalent [Chiba, Phys. Lett. B 575, 1 (2003)] Brans-Dicke theory with ω BD =0 ω BD : Brans-Dicke parameter Observational constraint: ω BD > 40000 No. 12 [Erickcek, Smith and Kamionkowski, Phys. Rev. D 74, 121501 (2006)] [Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)] However, if the mass of the scalar degree of freedom M is large, namely, the scalar becomes short-ranged, it has no effect at Solar System scales. M = M(R) Scale-dependence: Chameleon mechanism Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)] The scalar degree of freedom may acquire a large effective mass at terrestrial and Solar System scales, shielding it from experiments performed there.

(5) Existence of a matter-dominated stage and that of a late-time cosmic acceleration Combing local gravity constraints, it is shown that m ñ RF 00 (R)/F 0 (R) has to be several orders of magnitude smaller than unity. m quantifies the deviation from the Λ CDM model. [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)] [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)] (6) Stability of the de Sitter space F d = F(R d ) (Fd 0)2 à 2F d F 00 d R d Fd 0F > 0 d 00 : No. 13 Constant curvature in the de Sitter space Linear stability of the inhomogeneous perturbations in the de Sitter space Cf. R d =2F d /F 0 d m<1 [Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)]

(7) Free of curvature singularities Existence of relativistic stars [Frolov, Phys. Rev. Lett. 101, 061103 (2008)] No. 14 [Kobayashi and Maeda, Phys. Rev. D 78, 064019 (2008)] [Kobayashi and Maeda, Phys. Rev. D 79, 024009 (2009)]

< Models of F(R) gravity > (a) Hu-Sawicki model [Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)] F HS (R)=R c 2, c 3 : Constants M ö : Mass scale p>0 : Constant (b) Starobinsky s model [Starobinsky, JETP Lett. 86, 157 (2007)] ôð ñ àn R F S (R)=R + õλ 0 1+ 2 Λ 2 à 1 Λ 0 0 : No. 15 õ õ>0, n> 0 : Constants Current cosmological constant (c) Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)] F AB (R)= R 2 + 1 2a log [ cosh(ar) à tanh(b) sinh(ar) ] a>0, b : Constants (d) Tsujikawa s model [Tsujikawa, Phys. Rev. D 77, 023507 (2008)]] F T (R)=R à ör c tanh ð R Rc ñ R c > 0 : Constants

Regarding the condition (7), under debate. < Model with satisfying the condition (7) > F MJWQ (R) ë>0, R ã > 0 : Constants No. 16 [Babichev and Langlois, arxiv:0904.1382 [gr-qc]] [Upadhye and Hu, Phys. Rev. D 80, 064002 (2009)] [Miranda, Joras, Waga and Quartin, Phys. Rev. Lett. 102, 221101 (2009)] Cf. [de la Cruz-Dombriz, Dobado and Maroto, Phys. Rev. Lett. 103, 179001 (2009)]

< Crossing of the phantom divide > No. 17 Various observational data (SN, Cosmic microwave background radiation (CMB), BAO) imply that the effective EoS of dark energy w DE 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). w DE à 1 t c 0 t t c : Time of the crossing of the phantom divide [Alam, Sahni and Starobinsky, JCAP 0406, 008 (2004)] [Nesseris and Perivolaropoulos, JCAP 0701, 018 (2007)] [Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)] (i) w DE > à 1 Non-phantom phase (ii) w DE = à 1 Crossing of the phantom divide (iii) w DE < à 1 Phantom phase

< Models to account for the crossing of the phantom divide > No. 18 Two scalar field model, e.g., Quintom Canonical 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)]

We reconstruct an explicit model of F(R) gravity with realizing the crossing of the phantom divide. No. 19 < Preceding work > F(R)=(R 1/c à Λ) c Example: c, Λ : Constants c =1.8 < Recent work > [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)] Phantom phase Non-phantom phase [Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, 123516 (2009)] Cf. [Nozari and Azizi, Phys. Lett. B 680, 205 (2009)] It has been shown that in the Hu-Sawicki model, the transition from the phantom phase to the non-phamtom one can also occur.

II. Reconstruction of a F(R) gravity model with realizing the crossing of the phantom divide < II A. Reconstruction method > < Action > Equation of motion for þ : No. 20 [Nojiri and Odintsov, Phys. Rev. D 74, 086005 (2006)] By introducing proper functions P(þ) and Q(þ), we find : Matter Lagrangian þ : Scalar field < Gravitational field equation > : Covariant d'alembertian : Covariant derivative operator : Energy-momentum tensor of matter

Gravitational field equations in the flat FRW background: (ö, )=(0, 0)component No. 21 Trace part of (ö, )=(i, j) (i, j =1, ááá, 3) components ú p and are the sum of the energy density and pressure of matters with a constant EoS parameter w i, respectively, where i denotes some component of the matters. þ may be taken as þ = t because þ can be redefined properly. < Scale factor > aö : Constant, gà(t) : Proper function

No. 22 úö i : Constant, We derive the solutions of P(þ) and Q(þ).

< II B. Explicit model > < Solution without matter > No. 23, í>0, C > 0 : Constants t 0 : Present time pà æ : Arbitrary constants When, gà(t) diverges. þ þ = t : Big Rip singularity We take as and only consider the period.

< Effective EoS > w eff = p eff úeff, Cf. w DE ' No. 24 w eff 1à( F 0 (R)/F 0 (R 0 ))Ω m ù w eff (i) H ç < 0 w eff > à 1 Non-phantom phase (ii) H ç =0 w eff = à 1 (iii) H ç > 0 w eff < à 1 < Hubble rate > Crossing of the phantom divide Phantom phase

< Evolution of w eff > No. 25 U(t) t<t c : U(t) > 0 du(t) < 0 dt t c : Time of the crossing of the phantom divide t c U(t) decreases monotonously. It evolves from positive to negative. w eff 0 t à 1 0 t t c w e ff crosses -1.

< Forms of P and Q > No. 26 Scalar curvature: If we have the solution, we can obtain t = t(r) F(R)=P(t(R))R + Q(t(R)).

< Fig. 1 > No. 27 Behavior of as a function of for,,,, and [ C = ( t 0 /t s ) 2í+1 =1/4 ]. R 0 : Current curvature, H 0 : Present Hubble parameter

< Analytic form of F(R) > No. 28 (1) t 0 (t/t s ü 1) F(R) ù A 1 R à5í/2+1 : P j=æ C j R àì j/2 A 1 =, (2) : [KB and Geng, Phys. Lett. B 679, 282 (2009)] F(R) ù A 2 R 7/2 A 2 =

III. Summary No. 29 A scenario to explain the current accelerated expansion of the universe is to study a modified gravitational theory, such as F(R) gravity. Various observational data imply that the effective EoS of dark energy may evolve from larger than -1 (nonphantom phase) to less than -1 (phantom phase). Namely, it crosses -1 (the crossing of the phantom divide). We have reconstructed an explicit model of F(R) gravity with realizing the crossing of the phantom divide. The Big Rip singularity appears. Around the Big Rip singularity,. F(R) R 7/2

Backup Slides

m à : m M M Distance estimator : : Apparent magnitude Absolute magnitude < SNLS data > Pure matter cosmology Ω m =1.00 Ω Λ =0.00 z No. BS1 Flat Λ cosmology Ω m =0.26 Ω Λ =0.74 1 H 2 0 ä Ω = a à m 2 (1 + z) 3 + Ω Λ ô Ω m ñ 2 ú(t 0 ) 3H 2 0 Λ Ω Λ ñ 3H 2 0 : Density parameter for matter : Density parameter for Λ From [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)] 1+z = a a 0, z : Red shift 0 denotes quantities at the present time. t 0

< Residuals for the best fit to a flat Λ cosmology > Δ(m à M) No. BS2 Flat Λ cosmology z Pure matter cosmology From [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]

< F(R) gravity > S F(R)=R F(R) 2ô 2 : General Relativity F(R) gravity [Nojiri and Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)] [Capozziello and Francaviglia, Gen. Rel. Grav. 40, 357 (2008)] [Sotiriou and Faraoni, arxiv:0805.1726 [gr-qc]] No. BS3 < Gravitational field equation > F 0 (R)=dF(R)/dR 0 : Covariant d'alembertian : Covariant derivative operator

In the flat FRW background, gravitational field equations read, ú eff, p eff No. BS4 : Effective energy density and pressure from the term F(R) à R Example : F(R) R n (n6=1) à2n 2 +3nà1 a t q, q = nà2 w eff = à 6n 2 à9n+3 q>1 6n 2 à7nà1 [Capozziello, Carloni and Troisi, Recent Res. Dev. Astron. Astrophys. 1, 625 (2003)] If, accelerated expansion can be realized. (For n =3/2 or n = à 1, q =2 and w eff = à 2/3.)

We reconstruct an explicit model of F(R) gravity with realizing the crossing of the phantom divide. No. BS5 < Preceding work > From [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)] F(R)=(R 1/c à Λ) c Example: c, Λ : Constants c =1.8 Phantom phase Non-phantom phase

< 5-year WMAP data on > For the flat universe, constant : à 0.14 < 1+w<0.12 For a variable EoS : w [Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 180, 330 (2009), arxiv:0803.0547 [astro-ph]] w (95% CL) à 0.33 < 1+w 0 < 0.21 (95% CL) No. BS6 (From WMAP+BAO+SN) Baryon acoustic oscillation (BAO) : Special pattern in the large-scale correlation function of Sloan Digital Sky Survey (SDSS) luminous red galaxies w 0 = w(a =1) : Dark energy density tends to a constant value Cf. Dark Energy : Dark Matter : Baryon : (68% CL) ô Ω i ñ 2 ú (0) 3H 2 0 i = Λ,c,b ú (0) c ú (0) i i = ú (0) : Critical density c

< F(R) gravity > S F(R) 2ô 2 F(R) gravity No. BS7 Cf. F(R)=R [Nojiri and Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007)] [Capozziello and Francaviglia, Gen. Rel. Grav. 40, 357 (2008)] [Sotiriou and Faraoni, arxiv:0805.1726 [gr-qc]] : General Relativity ö F(R)=Rà 2(n+1) R n a t q (2n+1)(n+1), q = n+2 2(n+2) w eff = à 1+ 3(2n+1)(n+1) n =1 q =2 w eff = à 2/3 For, and. [Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, 043528 (2004)] ö : Mass scale, n : Constant Second term become R important as decreases. Accelerated expansion

(5) Existence of a matter-dominated stage and that of a late-time cosmic acceleration Analysis of m(r) curve on the (r, m) plane m ñ RF 00 (R)/F 0 (R), r ñàrf 0 (R)/F(R) Presence of a matter-dominated stage and at m(r) ù +0 dm > à 1 dr Presence of a late-time acceleration (i) (ii) 3 m(r) =à r à 1, à1 <mô 1 2 0 <mô 1 r = à 2 at r ùà1 and Combing local gravity constraints, we obtain dm dr < à 1 m(r) =C(à r à 1) p with p>1 as r à 1. C>0, p : Constants No. BS8 [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)] [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)]

z w(z)=w 0 + w 1 1+z < Data fitting of w(z) > No. BS9 From [Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)] SN gold data set [Riess et al. [Supernova Search Team Collaboration], Astrophys. J. 607, 665 (2004)] SNLS data set [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)] Shaded region shows 1û error. + Cosmic microwave background radiation (CMB) data [Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 170, 377 (2007)] SDSS baryon acoustic peak (BAO) data [Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)]

For most observational probes (except the SNLS data), a low Ω 0m prior (0.2 < Ω 0m < 0.25) leads to an increased probability (mild trend) for the crossing of the phantom divide. No. BS10 Ω 0m : Current density parameter of matter [Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)]

< Data fitting of w(z) (2) > No. BS11 From [Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)] SN gold data set+cmb+bao SNLS data set+cmb+bao 2û confidence level.

< Baryon acoustic oscillation (BAO) > No. BS12 Pure CDM model (No peak) Ω m h 2 =0.12, 0.13, 0.14, 0.105 (From top to bottom) Ω b h 2 =0.024 From [Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)]

< Note on the reconstruction method > If we redefine and define No. BS13, we obtain Φ : Proper function This is equivalent to the original action. þ We have the choices in like a gauge symmetry and thus we can identify þ with time t, i.e.,, which can be interpreted as a gauge condition corresponding to the reparameterization of.

< Reconstruction of an explicit model > Equation for P(þ) without matter: No. BS14 By redefining P(þ) as The model < Solutions >,

< Hubble rate > No. BS15 (i) t 0 Non-phantom phase, > à 1 (ii) H ç =0, w eff = à 1 Crossing of the phantom divide (iii) Phantom phase, < à 1

< Scalar curvature > No. BS16 < Behavior of U(t) > for t<t c, U(t) > 0 because í > 0. du(t) dt < 0 for U(t) decreases monotonously. It evolves from positive to negative. Consequently, crosses -1. w eff

< Analytic form of F(R) > (1) t 0 : No. BS17 In the limit t/t s ü 1, F(R) ù A 1 R à5í/2+1 P j=æ [KB and Geng, Phys. Lett. B 679, 282 (2009)] C j R àì j/2 A 1 =,

(2) : No. BS18 For large R, namely,, F(R) ù A 2 R 7/2 A 2 =

II C. Property of the singularity in the corresponding scalar field theory Action of F(R) gravity No. BS19 We introduce two scalar fields: and. ð ø Action in the Jordan frame There is a non-minimal coupling between ø and. R Equation of motion for ø : ð Equation of motion for : F 0 (ð)=df(ð)/dð

No. BS20 Conformal transformation : Action in the Einstein frame, û : Scalar field There is no non-minimal coupling between a scalar field and R ê. We redefine ϕ as. A hat denotes quantities in the Einstein frame. Canonical scalar field theory

c 1,n: Constants, M : Mass scale No. BS21 Around the Big Rip singularity, the reconstructed model of F(R) gravity behaves as. dt dtê < Relation between and >

< Relation between t and tê > No. BS22 If, the limit of corresponds to that of. In the reconstructed model of F(R) gravity, around the Big Rip singularity,. Finite-time Big Rip singularity in F(R) gravity (the Jordan frame) Conformal transformation Infinite-time singularity in the corresponding scalar field theory (the Einstein frame)

< Scale factor aê àá tê in the Einstein frame > No. BS23

No. BS24 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.

< Further results > No. BS25 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.

Appendix A Relation between scalar field theory and F(R) gravity

< Relation between scalar field theory and F(R) gravity > Action in the Einstein-frame (1) Non-phantom (canonical) field Real conformal transformation : ÿ : Real scalar field à No. A2 : Non-phantom case + : Phantom case W à (ÿ) : Potential of ÿ [Capozziello, Nojiri and Odintsov, Phys. Lett. B 634, 93 (2006)] Action in the Jordan-frame Real F(R) gravity ÿ can be expressed as by solving the equation of motion for ÿ :

(2) Phantom field No. A3 [Briscese, Elizalde, Nojiri and Odintsov, Phys. Lett. B 646, 105 (2007)] Complex conformal transformation : Action in the Jordan-frame Complex F(R) gravity We can obtain the relation by solving the equation of motion for ÿ:

Condition for R to be real: No. A4 With the except of W à (ÿ) satisfying this relation, R is complex. < Summary > Non-phantom (canonical) field theory Phantom field theory Real conformal transformation Complex conformal transformation Real F(R) gravity Complex F(R) gravity (except the special case)

< Summary > 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. 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 correction Quantum effects produce the conformal anomaly: No. B2 : Square of 4d Weyl tensor : Gauss-Bonnet invariant In the flat FRW background, we find,. can be arbitrary, e.g., we can choose N HD : N N 1/2 N 1 : Real scalar : Dirac spinor : Vector fields : Gravitons Higher derivative conformal scalars

Assuming and the conservation law :, we find ú A p A and : Energy density and pressure from conformal anomaly, respectively No. B3 Effective energy density ú F and pressure p F from, We assume the Hubble rate at the phantom crossing is given by

, : Dimensionless constant (Coming from the numerical constants:.) No. B4 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.)

II. Model of F(R) gravity with the transition from the de Sitter universe to the phantom phase Hubble rate : (1) : : Constant g 0 > 0, g 1 > 0, t s > 0 : Constants Asymptotically de Sitter space No. B5 (2) : : Big Rip singularity Phantom phase

(1) Case with neglecting matter No. B6 < Solutions >, C æ : Dimensionless constants F K : Kummer functions (confluent hypergeometric function)

No. B7 We have taken and therefore. (1) (2) : : : Constant de Sitter phase In this limit, because R diverges.

Expanding the Kummer functions in and taking the first leading order in, we obtain z 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). úö : No. B9 Current energy density of the cold dark matter : Critical density Behavior of as a function of, and. for

< Behavior of P and Q > No. B10 Behavior of and t 2 sq(r à ) as a function of for, and. R à

< Summary > 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. B11 We have reconstructed a model of F(R) gravity in which the transition from the de Sitter universe to the phantom phase can occur.

Appendix C Scalar field theory with realizing the constructed H(t)

< Scalar field theory with realizing the constructed H(t) > No. C2 Φ : Scalar field, ω(φ) : Function of, W(Φ) : Potential of Φ In the flat FRW background, the Einstein equations are given by Φ,,, We define ω(φ) and W(Φ) in terms of a single function I(Φ)., Solutions :

Defining ÿ as, we obtain No. C3 ω(φ) > 0 ω(φ) < 0 à + sign sign H(t) with realizing the phantom crossing in F(R) gravity

No. C4

(i) H ç < 0 = ω(φ) > 0 : à Non-phantom phase No. C5 sign (ii) : H ç =0 = ω(φ)=0 Crossing of the phantom divide (iii) : H ç > 0 = ω(φ) < 0 + Phantom phase sign 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.

< Summary > We have studied the scalar field theory with realizing the constructed H(t). No. C6