Future crossing of the phantom divide in f(r) gravity

Transcription

1 Future crossing of the phantom divide in f(r) gravity Reference: K. Bamba, C. Q. Geng and C. C. Lee, JCAP 1011, 001 (2010) [arxiv: [astro-ph.co]]. Two-day workshop in the YITP molecule-type workshop Cosmological Perturbation and Cosmic Microwave Background, Panasonic Auditorium at Yukawa Institute for Theoretical Physics on 22nd March, 2011 Presenter : Kazuharu Bamba (KMI, Nagoya University) Collaborators : Chao-Qiang Geng (National Tsing Hua University) Chung-Chi Lee (National Tsing Hua University)

2 < Contents > No. 2 I. Introduction Current cosmic acceleration f(r) gravity Crossing of the phantom divide II. Cosmological evolution in f(r) gravity III. Future crossing of the phantom divide IV. Summary * We use the ordinary metric formalism, in which the connection is written by the differentiation of the metric.

3 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)] [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)] [Tsujikawa, arxiv: [astro-ph.co]] < 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

4 (1) General relativistic approach Cosmological constant Scalar fields: X matter, Quintessence, Phantom, K-essence, Tachyon. Fluid: Chaplygin gas f(r) : (2) Extension of gravitational theory f(r) gravity Scalar-tensor theories No. 4 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, (2004)] [Nojiri and Odintsov, Phys. Rev. D 68, (2003)] Ghost condensates Higher-order curvature term DGP braneworld scenario f(t) gravity Galileon gravity R [Arkani-Hamed, Cheng, Luty and Mukohyama, JHEP 0405, 074 (2004)] : Gauss-Bonnet term f(g) gravity [Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)] [Bengochea and Ferraro, Phys. Rev. D 79, (2009)] [Linder, Phys. Rev. D 81, (2010) [Erratum-ibid. D 82, (2010)]] [Nicolis, Rattazzi and Trincherini, T : torsion scalar Phys. Rev. D 79, (2009)] G

5 < Flat Friedmann-Lema tre-robertson-walker (FLRW) space-time > a (t) No. 5 : Scale factor < Equation for a(t) with a perfect fluid > ä = à ô 2 ( 1+3w)ú a 6 w ñ ú P : Equation of state (EoS) T ö =diag(ú, P, P, P) ú : Energy density P : Pressure ä > 0 : Accelerated expansion w<à 3 1 Cf. Cosmological constant : Condition for accelerated expansion w = à 1

6 < 7-year WMAP data on the current value of > K =0 For the flat universe, constant w : Cf. Ω Λ = (68% CL) (68% CL) (95% CL) (From w K Ω K ñ (a0 H 0 ) 2 No. 6 From [E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arxiv: [astro-ph.co]]]. Hubble constant ( H 0 ) measurement Baryon acoustic oscillation (BAO) : Special pattern in the large-scale correlation function of Sloan Digital Sky Survey (SDSS) luminous red galaxies : Time delay distance : Density parameter for the curvature K =0 : Flat universe.)

7 (68% CL) (95% CL) Time-dependent w No. 7 From [E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arxiv: [astro-ph.co]]]. For the flat universe, a variable EoS : w 0 : Current value of w (From 1 a = 1+z z : Redshift.),

8 < f(r) gravity > No. 8 S f(r) 2ô 2 f(r) gravity Cf. f(r)=r : General Relativity [Nojiri and Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007) [arxiv:hep-th/ ]; arxiv: [gr-qc]] [Capozziello and Francaviglia, Gen. Rel. Grav. 40, 357 (2008)] [Sotiriou and Faraoni, Rev. Mod. Phys. 82, 451 (2010)] [De Felice and Tsujikawa, Living Rev. Rel. 13, 3 (2010)]

9 < Conditions for the viability of f(r) gravity > (1) f 0 (R) > 0 f 0 (R) ñ df(r)/dr (2) (3) f 00 (R) > 0 f 00 (R) ñ d 2 f(r)/dr 2 (4) 0 <mñ Rf 00 (R)/f 0 (R) < 1 G eff = G/f 0 (R) > 0 G M 2 ù 1/(3f 00 (R)) > 0 [Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)] [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, (2007)] [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)] [Faraoni and Nadeau, Phys. Rev. D 75, (2007)] (5) Constraints from the violation of the equivalence principle Scale-dependence Chameleon mechanism M = M(R) : No. 9 Positivity of the effective gravitational coupling Stability condition: : Gravitational constant M : Mass of a new scalar degree of freedom ( scalaron ) in the weak-field regime. Existence of a matterdominated stage f(r) R à 2Λ for R ý R 0. R 0 : Current curvature, Λ : Cosmological constant Stability of the latetime de Sitter point Cf. For general relativity, m =0. Cf. [Khoury and Weltman, Phys. Rev. D 69, (2004)] (6) Solar-system constraints [Chiba, Phys. Lett. B 575, 1 (2003)] [Chiba, Smith and Erickcek, Phys. Rev. D 75, (2007)]

10 < Models of f(r) gravity (examples) > [Hu and Sawicki, Phys. Rev. D 76, (2007)] (i) Hu-Sawicki model Cf. No. 10 [Nojiri and Odintsov, Phys. Lett. B 657, 238 (2007); Phys. Rev. D 77, (2008)] : Constant parameters (ii) Starobinsky s model [Starobinsky, JETP Lett. 86, 157 (2007)] : Constant parameters (iii) Tsujikawa s model [Tsujikawa, Phys. Rev. D 77, (2008)] (iv) Exponential gravity model : Constant parameters [Cognola, Elizalde, Nojiri, Odintsov, Sebastiani and Zerbini, Phys. Rev. D 77, (2008)] [Linder, Phys. Rev. D 80, (2009)] : Constant parameters

11 < Other model > No. 11 Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)] f AB (R)= R b1 log [ cosh(b 1 R) à tanh(b 2 ) sinh(b 1 R) ] b 1 (> 0), b 2 : Constant parameters

12 < Crossing of the phantom divide > No. 12 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 z c 0 z z c Redshift: z ñ 1 a à 1 : Red shift at 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)] (a) w DE > à 1 Non-phantom phase (b) w DE = à 1 Crossing of the phantom divide (c) w DE < à 1 Phantom phase

13 It is known that in several viable f(r) gravity models, the crossing of the phantom divide can occur in the past. (i) Hu-Sawicki model [Hu and Sawicki, Phys. Rev. D 76, (2007)] [Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, (2009)] Cf. [Nozari and Azizi, arxiv: [gr-qc]] (ii) Starobinsky s model (iv) Exponential gravity model [Motohashi, Starobinsky and Yokoyama, Prog. Theor. Phys. 123, 887 (2010); Prog. Theor. Phys. 124, 541 (2010)] [Linder, Phys. Rev. D 80, (2009)] [KB, Geng and Lee, JCAP 1008, 021 (2010) arxiv: [astro-ph.co]] Appleby-Battye model No. 13 [Appleby, Battye and Starobinsky, JCAP 1006, 005 (2010)]

14 w DE < Cosmological evolution of in the exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. No. 14 w DE (z =0)=à 0.93 (< à 1/3) f E (R)= w DE = à 1 Crossing in the past Crossing of the phantom divide

15 We explicitly demonstrate that the future crossings of the phantom divide line w DE = à 1 are the generic feature in the existing viable f(r) gravity models. No. 15 Recent related study on the future crossings of the phantom divide: [Motohashi, Starobinsky and Yokoyama, arxiv: [astro-ph.co]]

16 II. Cosmological evolution in f(r) gravity < Action > g =det(g ö ) f : : Metric tensor Arbitrary function of R No. 16 : Action of matter < Gravitational field equation > : Matter fields : Energy-momentum tensor of all perfect fluids of matter R ö : Ricci tensor : Covariant derivative operator : Covariant d'alembertian

17 < Flat Friedmann-Lema tre-robertson-walker (FLRW) space-time > a(t) : Scale factor No. 17 Gravitational field equations in the FLRW background: : Hubble parameter ú M and P M : Energy density and pressure of all perfect fluids of matter, respectively, < Analysis method > [Hu and Sawicki, Phys. Rev. D 76, (2007)] (1) Ricci scalar: (2) 0 (prime): Derivative with respect to R

18 We solve Equations (1) and (2) by introducing the following variables: No. 18 ú DE ú m ú r (0) denotes the present values. : Energy density of dark energy : Energy density of non-relativistic matter (cold dark matter and baryon) : Energy density of radiation (3) (4)

19 Combining Equations (3) and (4), we obtain No. 19 : Equation for y H

20 < Equation of state for (the component corresponding to) dark energy > No. 20 P w DE ñ DE úde < Continuity equation for dark energy >

21 < Bekenstein-Hawking entropy on the apparent horizon in the flat FLRW background > S = 4G A : Bekenstein-Hawking entropy No. 21 rà=1/h : : Area of the apparent horizon Radius of the apparent horizon in the flat FLRW space-time It has been shown that it is possible to obtain a picture of equilibrium thermodynamics on the apparent horizon in the FLRW background for f(r) gravity due to a suitable redefinition of an energy momentum tensor of the dark component that respects a local energy conservation. [KB, Geng and Tsujikawa, Phys. Lett. B 688, 101 (2010)] In this picture, the horizon à á entropy is simply expressed as S = ù/ GH 2.

22 III. Future crossing of the phantom divide (i) Hu-Sawicki model p =1 c 1 =1, c 2 =1 (ii) Starobinsky s model [Hu and Sawicki, Phys. Rev. D 76, (2007)] [Starobinsky, JETP Lett. 86, 157 (2007)] n =2 õ =1.5 No. 22 (iii) Tsujikawa s model [Tsujikawa, Phys. Rev. D 77, (2008)] ö =1 (iv) Exponential gravity model [Cognola, Elizalde, Nojiri, Odintsov, Sebastiani and Zerbini, Phys. Rev. D 77, (2008)] [Linder, Phys. Rev. D 80, (2009)] ì =1.8

23 < Future evolutions of 1+w DE as functions of z > No. 23 Exponential gravity model 1+w DE Crossings in the future 0 1+w DE =0 Crossing of the phantom divide Redshift: z ñ a 1 à 1 ( z<0 : Future)

24 Hu-Sawicki model Starobinsky s model No w DE Redshift: z ñ a 1 à 1 ( z<0: Future) Tsujikawa s model Exponential gravity model Crossings in the future 0 1+w DE =0 Crossing of the phantom divide

25 < Future evolutions of H as functions of z > No. 25 Exponential gravity model Oscillatory behavior : ñ H(z=à1) H0 f denotes the value at the final stage z = à 1. : Present value of the Hubble parameter

26 Hu-Sawicki model Starobinsky s model No. 26 ñ H(z=à1) H0 Tsujikawa s model Exponential gravity model Oscillatory behavior : Present value of the Hubble parameter

27 < Future evolutions of S as functions of z > No. 27 Exponential gravity model ñ S(z=à1) S0 Oscillating behavior :Present value of the horizon entropy

28 Hu-Sawicki model Starobinsky s model No. 28 ñ S(z=à1) S0 Tsujikawa s model Exponential gravity model Oscillating behavior : Present value of the horizon entropy

29 In the future ( ), the crossings of the phantom divide are the generic feature for all the existing viable f(r) models. As z decreases ( ), dark energy becomes much more dominant over non-relativistic matter ( ). No. 29 < Effective equation of state for the universe > : Total energy density of the universe : Total pressure of the universe P DE P m P r : Pressure of dark energy : Pressure of non-relativistic matter (cold dark matter and baryon) : Pressure of radiation

30 H w eff = à 1 à 3 2H ç 2 (a) (b) H ç < 0 w eff > à 1 H ç =0 w eff = à 1 w eff < à 1 Non-phantom phase Crossing of the phantom divide (c) H ç > 0 Phantom phase The physical reason why the crossing of the phantom divide appears in the farther future ( ) is that the sign of H ç changes from negative to positive due to the dominance of dark energy over non-relativistic matter. As in the farther future, w DE oscillates around the phantom divide line w DE = à 1 because the sign of H ç changes and consequently multiple crossings can be realized. No. 30

31 S H à2 H Since, the oscillating behavior of comes from that of. Cf. However, it should be emphasized that although S decreases in some regions, the second law of thermodynamics in f(r) gravity can be always satisfied. S This is because is the cosmological horizon entropy and it is not the total entropy of the universe including the entropy of generic matter. It has been shown that the second law of thermodynamics can be verified in both phantom and non-phantom phases for the same temperature of the universe outside and inside the apparent horizon. S [KB and Geng, JCAP 1006, 014 (2010)] No. 31

32 IV. Summary We have explicitly shown that the future crossings of the phantom divide are the generic feature in the existing viable f(r) gravity models. We have also illustrated that the cosmological horizon entropy oscillates with time due to the oscillatory behavior of the Hubble parameter. The new cosmological ingredient obtained in this study is that in the future the sign of H ç changes from negative to positive due to the dominance of dark energy over non-relativistic matter. This is a common physical phenomena to the existing viable f(r) models and thus it is one of the peculiar properties of f(r) gravity models characterizing the deviation from the Λ CDM model. No. 32

33 Backup Slides A

34 < f(r) gravity > No. BS-A1 S f(r) 2ô 2 f(r) gravity f(r)=r : General Relativity [Nojiri and Odintsov, Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007) [arxiv:hep-th/ ]; arxiv: [gr-qc]] [Capozziello and Francaviglia, Gen. Rel. Grav. 40, 357 (2008)] [Sotiriou and Faraoni, Rev. Mod. Phys. 82, 451 (2010)] [De Felice and Tsujikawa, Living Rev. Rel. 13, 3 (2010)] < Gravitational field equation > f 0 (R)=df(R)/dR : Covariant d'alembertian : Covariant derivative operator

35 In the flat FLRW background, gravitational field equations read, ú eff, P eff No. BS-A2 : Effective energy density and pressure from the term f(r) à R Example: ö f(r)=rà 2(n+1) R n [Carroll, Duvvuri, Trodden and Turner, Phys. Rev. D 70, (2004)] ö : Mass scale, n : Constant Second term become a t q (2n+1)(n+1), q = important as R decreases. n+2 If q>1, accelerated 2(n+2) w eff = à 1+ 3(2n+1)(n+1) expansion can be realized. (For n =1, q =2 and w eff = à 2/3.)

36 z w(z)=w 0 + w 1 1+z < Data fitting of w(z) (1) > No. BS-A3 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)]

37 < Data fitting of w(z) (2) > No. BS-A4 From [Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)]. SN gold data set+cmb+bao SNLS data set+cmb+bao 2û confidence level.

38 Appendix A

39 (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, (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, (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. A-1 Canonical field A>0 : Constant ú : Energy density : Pressure p

40 (2) Extension of gravitational theory f(r) gravity f(r) Scalar-tensor theories f(g) gravity 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, (2004)] [Nojiri and Odintsov, Phys. Rev. D 68, (2003)] f 1 (þ)r R 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 No. A-2 : Ricci curvature [Nojiri, Odintsov and Sasaki, Phys. Rev. D 71, (2005)] tensor R : Riemann + f(g) 2ô 2 ô 2 ñ 8ùG tensor [Nojiri and Odintsov, Phys. Lett. B 631, 1 (2005)] G : Gravitational constant

41 DGP braneworld scenario [Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)] [Deffayet, Dvali and Gabadadze, Phys. Rev. D 65, (2002)] f(t) gravity : Extended teleparallel Lagrangian density described by the torsion scalar T. [Bengochea and Ferraro, Phys. Rev. D 79, (2009)] [Linder, Phys. Rev. D 81, (2010) [Erratum-ibid. D 82, (2010)]] Teleparallelism : One could use the Weitzenböck connection, which has no curvature but torsion, rather than the curvature defined by Galileon gravity the Levi-Civita connection. No. A-3 [Hayashi and Shirafuji, Phys. Rev. D 19, 3524 (1979) [Addendum-ibid. D 24, 3312 (1982)]] [Nicolis, Rattazzi and Trincherini, Phys. Rev. D 79, (2009)] Recent review: [Tsujikawa, Lect. Notes Phys. 800, 99 (2010)] Longitudinal graviton (i.e. a branebending mode þ ) The equations of motion are invariant under the Galilean shift: One can keep the equations of motion up to the second-order. This property is welcome to avoid the appearance of an extra degree of freedom associated with ghosts. : Covariant d'alembertian

42 < Flat Friedmann-Lema tre-robertson-walker (FLRW) space-time > a (t) < 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 : 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. A-4 w = à 1

43 < Canonical scalar field > S þ = d 4 x à g â à 1 g ö ã 2 ö þ þ à V(þ) No. A-5 g =det(g ö ) ú = 2 1 þ ç 2 + 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) þ ç 2 ü V(þ) w þ ùà1 If,. Accelerated expansion can be realized.

44 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. A-6 Flat Λ cosmology Ω m =0.26 Ω Λ = 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 a 0 =1 : Redshift 0 denotes quantities at the present time. t 0

45 < Conditions for the viability of f(r) gravity > (1) f 0 (R) > 0 f 00 (R) > 0 (2) [Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)] (3) Positivity of the effective gravitational coupling G eff = G 0 /f 0 (R) > 0 G 0 : Gravitational constant (The graviton is not a ghost.) 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 Λ : Current curvature Λ : Cosmological constant Realization of the CDM-like behavior in the large curvature regime No. A-7 Standard cosmology [ Λ + Cold dark matter (CDM)]

46 (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 [Erickcek, Smith and Kamionkowski, Phys. Rev. D 74, (2006)] [Chiba, Smith and Erickcek, Phys. Rev. D 75, (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) Chameleon mechanism Scale-dependence: ω BD : Brans-Dicke parameter Observational constraint: ω BD > Cf. [Khoury and Weltman, Phys. Rev. D 69, (2004)] The scalar degree of freedom may acquire a large effective mass at terrestrial and Solar System scales, shielding it from experiments performed there. No. A-8

47 (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, (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 fd 00 R d fd 0f > 0 d 00 : 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 No. A-9 [Faraoni and Nadeau, Phys. Rev. D 75, (2007)]

48 < Other models > No. A-10 Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)] f AB (R)= R b1 log [ cosh(b 1 R) à tanh(b 2 ) sinh(b 1 R) ] b 1 (> 0), b 2 : Constant parameters Cf. Power-law model f(r)=r à ör v ö(> 0) : Constant parameter [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, (2007)] [Li and Barrow, Phys. Rev. D 75, (2007)] 0 <v<10 à10 : Constant parameter (close to 0) [Capozziello and Tsujikawa, Phys. Rev. D 77, (2008)]

49 S = 4G A : Bekenstein-Hawking horizon entropy in the Einstein gravity No. A-11 : Wald entropy in modified gravity theories : Area of the apparent horizon Wald introduced a horizon entropy associated with a Noether charge in the context of modified gravity theories. This is equivalent to S ê = A/(4G eff ). S ê The Wald entropy is a local quantity defined in terms of quantities on the bifurcate Killing horizon. More specifically, it depends on the variation of the Lagrangian density of gravitational theories with respect to the Riemann tensor. G eff = G/F S ê : Effective gravitational coupling

50 Ω DE Ω m Ω r No. A-12 < Cosmological evolutions of, and in the exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. f E (R)= ì =1.8

51 w eff No. A-13 < Cosmological evolution of in the exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. f E (R)=

52 < Remarks > (a) The qualitative results do not strongly depend on the values of the parameters in each model. (b) The evolutions of the Wald entropy are similar to in the models of (i) (iv). S = 4G A : Bekenstein-Hawking horizon entropy in the Einstein gravity S ê No. A-14 Cf. [KB, Geng and Tsujikawa, Phys. Lett. B 688, 101 (2010)] S [KB, Geng and Lee, JCAP 1008, 021 (2010)] S ê = F(R)A 4G : Wald entropy in modified gravity theories including f(r) gravity (c) The numerical results in the Appleby-Battye model are similar to those in the Starobinsky model of (ii).

53 < Cosmological evolutions of, and in the exponential gravity model > S ö S êö H ö No. A-15 From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. f E (R)=

54 < Numerical results > Models of (i), (ii), (iii) and (iv) w DE (z =0)= = (-0.76, -0.82), (-0.83, -0.98), (-0.79, -0.80) and (-0.74, -0.80) z cross z p : : Value of z at the first future crossing of the phantom divide z Value of at the first sign change of from negative to positive =,,, and H ç No. A-16

55 < Second law of thermodynamics in equilibrium description > We consider the same temperature of the universe outside and inside the apparent horizon. < Gibbs equation > No. A-17 [KB and Geng, JCAP 1006, 014 (2010)] < Second law of thermodynamics > < Condition > As long as R>0, the second law of thermodynamics can be met in both non-phantom ( H ç < 0, w eff > à 1 ) and phantom ( H ç > 0, w eff < à 1 ) phases. Cf. [Gong, Wang and Wang, JCAP 0701, 024 (2007)] [Jamil, Saridakis and Setare, Phys. Rev. D 81, (2010)]

56 < Second law of thermodynamics > [KB and Geng, JCAP 1006, 014 (2010)] No. A-18 We assume the same temperature between the outside and inside of the apparent horizon. < Gibbs equation > : Entropy of total energy inside the horizon < Second law of thermodynamics > < Condition > [Wu, Wang, Yang and Zhang, Class. Quant. Grav. 25, (2008)] F > 0 because G eff = G/F > 0. The second law of thermodynamics in f(r) gravity can be satisfied in phantom (, ) as well as non-phantom ( H ç H ç > 0 w eff < à 1 < 0, w eff > à 1 ) phases. : Effective equation of state (EoS)

57 < Initial conditions > Models of (i), (ii), (iii) and (iv) We have taken the initial conditions at z = z 0, so that with, to ensure that they can be all close enough to the Λ CDM model with. In order to save the calculation time, the different values of z 0 mainly reflect the forms of the models, i.e., the power-law types of (i) and (ii) and the exponential ones of (iii) and (iv). At z = z 0, w DE = à 1. No. A-19

58 No. A-20 Models of (i), (ii), (iii) and (iv) =,, and [E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arxiv: [astro-ph.co]]] In the high z regime ( ),, in which f(r) gravity has to be very close to the ΛCDM model.

59 No. A-21 By examining the cosmological evolutions of and w DE as functions of the redshift z for the models, we have found that is close to its initial value of. y H This is because in the higher z regime, the universe is in the phantom phase ( w DE < à 1) and therefore, ú DE and y H increase (since ), whereas in the lower z regime, the universe is in the non-phantom (quintessence) phase ( w DE > à 1 ) and hence they decrease. Consequently, the above two effects cancel out.

60 Our results are not sensitive to the initial values of and. No. A-22 z 0 The initial condition of is due to that the f(r) gravity models at z = z 0 should be very close to the ΛCDM model, in which.

61 No. A-23

62 Backup Slides B

63 < Residuals for the best fit to a flat Λ cosmology > Δ(m à M) No. BS-B1 Flat Λ cosmology z Pure matter cosmology From [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]

64 < Baryon acoustic oscillation (BAO) > No. BS-B2 Pure CDM model (No peak) Ω m h 2 =0.12, 0.13, 0.14, (From top to bottom) Ω b h 2 =0.024 From [Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)]

65 < 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: [astro-ph]] w (95% CL) à 0.33 < 1+w 0 < 0.21 (95% CL) No. BS-B3 (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

66 No. BS-B4 In the flat FLRW background, gravitational field equations read, ú eff, p eff : 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.)

67 < Crossing of the phantom divide > 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)] (a) w DE > à 1 Non-phantom phase (b) w DE = à 1 Crossing of the phantom divide (c) w DE < à 1 Phantom phase No. BS-B5

68 Appendix B

69 (7) Free of curvature singularities Existence of relativistic stars [Frolov, Phys. Rev. Lett. 101, (2008)] No. B-1 [Kobayashi and Maeda, Phys. Rev. D 78, (2008)] [Kobayashi and Maeda, Phys. Rev. D 79, (2009)]

70 < Model with satisfying the condition (7) > No. B-2 F MJWQ (R) ë>0, R ã > 0 : Constants [Miranda, Joras, Waga and Quartin, Phys. Rev. Lett. 102, (2009)] Cf. [de la Cruz-Dombriz, Dobado and Maroto, Phys. Rev. Lett. 103, (2009)]

71 < Recent work > No. B-3 [Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, (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. From [Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, (2009)] n =1 F 0 (R 0 )=à 0.01 F 0 (R 0 )=à 0.03 F 0 (R 0 )=à 0.1 Ω m =0.24

72 We reconstruct an explicit model of F(R) gravity with realizing the crossing of the phantom divide. No. B-4 < 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

73 (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 <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. B-5 [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, (2007)] [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)]

74 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. Ω 0m : Current density parameter of matter No. B-6 [Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)]

75 < Data fitting of w(z) (3) > No. B-7 From [Zhao, Xia, Feng and Zhang, Int. J. Mod. Phys. D 16, 1229 (2007) [arxiv:astro-ph/ ]] 68% Best-fit confidence level 95% confidence level 157 gold SN Ia data set+wmap 3-year data+sdss with/without dark energy perturbations.