Lecture on Modified Gravity Theories

Transcription

1 Lecture on Modified Gravity Theories Main references: [1] K. Bamba, S. Nojiri, S. D. Odintsov and M. Sasaki, Gen. Relativ. Gravit. 44, 1321 (2012) [arxiv: [hep-th]]. [2] K. Bamba, R. Myrzakulov, S. Nojiri and S. D. Odintsov, Phys. Rev. D 85, (2012) [arxiv: [gr-qc]]. 5th Joint NCTS/FGCPA-LeCosPA Meeting on Dark Energy, Room 812, Astro-Math Building, National Taiwan University, 20th October, 2012 Presenter : Kazuharu Bamba (KMI, Nagoya University) Collaborators : Ratbay Myrzakulov (EICTP, Eurasian National University) Shin'ichi Nojiri (KMI and Dep. of Physics, Nagoya University), Sergei D. Odintsov (ICREA and IEEC-CSIC), Misao Sasaki (YITP, Kyoto University and KIAS)

2 < Contents > No. 2 I. Introduction Current cosmic acceleration Scalar field theory F(R) gravity II. Finite-time future singularities in non-local gravity III. Finite-time future singularities and cosmology in f(t) gravity IV. Summary

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)] [Caldwell and Kamionkowski, Ann. Rev. Nucl. Part. Sci. 59, 397 (2009)] [Amendola and Tsujikawa, Dark Energy (Cambridge University press, 2010)] [Li, Li, Wang and Wang, Commun. Theor. Phys. 56, 525 (2011)] [KB, Capozziello, Nojiri and Odintsov, Astrophys. Space Sci. 342, 155 (2012)] < Gravitational field equation > G ö G ö = ô 2 T ö T ö Gravity Matter (1) General relativistic approach (2) Extension of gravitational theory 2011 Nobel Prize in Physics : Einstein tensor : Energy-momentum tensor, Dark Energy : Planck mass

4 (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 [Caldwell, Phys. Lett. B 545, 23 (2002)] Non canonical kinetic term String theories The mass squared is negative. [Padmanabhan, Phys. Rev. D 66, (2002)] (Generalized) Chaplygin gas Wrong sign kinetic term A>0, [Kamenshchik, Moschella and Pasquier, Phys. Lett. B 511, 265 (2001)] [Bento, Bertolami and Sen, Phys. Rev. D 66, (2002)] * Equation of state (EoS) P = à A/ú u No. 4 Canonical field u : Constants ú : Energy density : Pressure P (u =1)

5 (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 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 : 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

6 DGP braneworld scenario [Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)] [Deffayet, Dvali and Gabadadze, Phys. Rev. D 65, (2002)] Non-local gravity Quantum effects Galileon gravity [Nicolis, Rattazzi and Trincherini, Phys. Rev. D 79, (2009)] Longitudinal graviton (a branebending mode þ ) [Nojiri and Odintsov, Phys. Lett. B 659, 821 (2008)] f(t) gravity The equations of motion are invariant under the Galilean shift: No. 6 [Deser and Woodard, Phys. Rev. Lett. 99, (2007)] : Extended teleparallel Lagrangian described by the torsion scalar [Bengochea and Ferraro, Phys. Rev. D 79, (2009)] T. [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 the Levi-Civita connection. [Hayashi and Shirafuji, Phys. Rev. D 19, 3524 (1979) [Addendum-ibid. D 24, 3312 (1982)]] 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 Massive gravity [de Rham and Gabadadze, Phys. Rev. D 82, (2010)] [de Rham and Gabadadze and Tolley, Phys. Rev. Lett. 106, (2011)] Review: [Hinterbichler, Rev. Mod. Phys. 84, 671 (2012)]

7 < Flat Friedmann-Lema tre-robertson-walker (FLRW) space-time > a (t) No. 7 : 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

8 m à : m M 1 H 2 0 M Distance estimator : : Apparent magnitude Absolute magnitude < Fig. 1 > 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 < SNLS data > : Density parameter for matter : Density parameter for Λ Pure matter cosmology Ω m =1.00 Ω Λ =0.00 z From [Astier et al. [The SNLS Collaboration], Astron. Astrophys. 447, 31 (2006)]. 1+z = a a 0, z No. 8 Flat Λ cosmology Ω m =0.26 Ω Λ =0.74 a 0 =1 : Red shift 0 denotes quantities at the present time. t 0

9 < Baryon acoustic oscillation (BAO) > Pure CDM model (No peak) No. 9 Special pattern in the large-scale correlation function of Sloan Digital Sky Survey (SDSS) luminous red galaxies Ω m h 2 =0.12, 0.13, 0.14, (From top to bottom) Ω b h 2 =0.024 < Fig. 2 > From [Eisenstein et al. [SDSS Collaboration], Astrophys. J. 633, 560 (2005)].

10 < 7-year WMAP data on the current value of > w No. 10 K =0 From [E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arxiv: [astro-ph.co]]]. Hubble constant ( ) measurement H 0 < Fig. 3 > For the flat universe, constant w : Cf. Ω Λ = (68% CL) K Ω K ñ (a0 H 0 ) 2 (68% CL) (95% CL) : Time delay distance K =0 : Flat universe (From : Density parameter for the curvature.)

11 (68% CL) (95% CL) Time-dependent w No. 11 From [E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011) [arxiv: [astro-ph.co]]]. < Fig. 4 > For the flat universe, a variable EoS : w 0 : Current value of w (From 1 a = 1+z z : Redshift.),

12 < Canonical scalar field > S þ = d 4 x à g â à 1 g ö ã 2 ö þ þ à V(þ) No. 12 g =det(g ö ) V(þ) þ : Scalar field : Potential of þ For a homogeneous scalar field þ = þ(t) : ú þ = 2 1 þ ç 2 + V(þ), P þ = 2 1 þ ç 2 à V(þ) P w þ = þ þ = ç 2 à2v(þ) úþ þ ç 2 +2V(þ) þ ç 2 ü V(þ) w þ ùà1 If,. Accelerated expansion can be realized.

13 < F(R) gravity > F(R) S 2ô 2 < Gravitational field equation > F(R) gravity F(R)=R General Relativity No. 13 [Nojiri and Odintsov, Phys. Rept. 505, 59 (2011) [arxiv: [gr-qc]]; Int. J. Geom. Meth. Mod. Phys. 4, 115 (2007) [arxiv:hep-th/ ]] [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)] [Capozziello and Faraoni, Beyond Einstein Gravity (Springer, 2010)] [Capozziello and De Laurentis, Phys. Rept. 509, 167 (2011)] [Clifton, Ferreira, Padilla and Skordis, Phys. Rept. 513, 1 (2012)] : 0 F 0 (R)=dF(R)/dR : Covariant d'alembertian : Covariant derivative operator

14 In the flat FLRW background, gravitational field equations read ú eff, P eff, P eff : Effective energy density and No. 14 pressure from the term F(R) à R P eff P eff Example : F(R) R n (n6=1) à2n 2 +3nà1 a t q, q = nà2 6n 2 à7nà1 w eff = à 6n 2 à9n+3 [Capozziello, Carloni and Troisi, Recent Res. Dev. Astron. Astrophys. 1, 625 (2003)] q > 1 If, accelerated expansion can be realized. (For n =3/2 or n = à 1, q =2 and w eff = à 2/3.)

15 < 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 <mñrf 00 (R)/F 0 (R) < 1 Positivity of the effective gravitational coupling G eff = G/F 0 (R) > 0 G M 2 ù 1/(3F 00 (R)) > 0 Stability condition: [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) : : 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. No. 15 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)]

16 < Models of F(R) gravity (examples) > [Hu and Sawicki, Phys. Rev. D 76, (2007)] (i) Hu-Sawicki model F HS (ii) Starobinsky s model [Starobinsky, JETP Lett. 86, 157 (2007)] Cf. : Constant parameters No. 16 [Nojiri and Odintsov, Phys. Lett. B 657, 238 (2007); Phys. Rev. D 77, (2008)] F S : Constant parameters (iii) Hyperbolic model [Tsujikawa, Phys. Rev. D 77, (2008)] F H (iv) Exponential gravity model F E R H R H e R H : Constant parameters [Cognola, Elizalde, Nojiri, Odintsov, Sebastiani and Zerbini, Phys. Rev. D 77, (2008)] [Linder, Phys. Rev. D 80, (2009)] : Constant parameters

17 (v) 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 No. 17 (vi) 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)]

18 < Fig. 5 > Ω DE Ω m Ω r No. 18 < Cosmological evolutions of, and in the exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. F E (R)= e ì =1.8

19 w DE < Cosmological evolution of in the exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. No. 19 w DE (z =0)=à0.93 F E (R)= (< à 1/3) e w DE = à 1 Crossing in the past Crossing of the phantom divide < Fig. 6 >

20 II. Finite-time future singularities in non-local gravity No. 20 Non-local gravity produced by quantum effects [Deser and Woodard, Phys. Rev. Lett. 99, (2007)] [Nojiri and Odintsov, Phys. Lett. B 659, 821 (2008)] This theory can explain the current accelerated expansion of the universe. It is known that so-called matter instability occurs in F(R) gravity. [Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)] This implies that the curvature inside matter sphere becomes very large and hence the curvature singularity could appear. [Arbuzova and Dolgov, Phys. Lett. B 700, 289 (2011)] Cf. [KB, Nojiri and Odintsov, Phys. Lett. B 698, 451 (2011)] It is important to examine whether there exists the curvature singularity, i.e., the finite-time future singularities in non-local gravity.

21 A. Non-local gravity < Action > g =det(g ö ) f : Some function : Metric tensor No. 21 Λ : Cosmological constant Non-local gravity By introducing two scalar fields and, we find By the variation of the action in the first expression over ø, we obtain (or ) Substituting this equation into the action in the first expression, one re-obtains the starting action. : Covariant derivative operator : Covariant d'alembertian : Matter Lagrangian Q : Matter fields

22 < Gravitational field equation > No. 22 The variation of the action with respect to ñ gives 0 : Energy-momentum tensor of matter (prime) : Derivative with respect to < Flat Friedmann-Lema tre-robertson-walker (FLRW) metric > a(t) : Scale factor ñ We consider the case in which the scalar fields depend on time. ñ ø and only

23 Gravitational field equations in the FLRW background: No. 23 : Hubble parameter and : Energy density and pressure of matter, respectively. For a perfect fluid of matter: < Equations of motion for ñ and ø >

24 B. Finite-time future singularities No. 24 In the flat FLRW space-time, we analyze an asymptotic solution of the gravitational field equations in the limit of the time t s when the finite-time future singularities appear. We consider the case in which the Hubble parameter is expressed as When, Scale factor h s : Positive constant q : Non-zero constant larger than -1 * We only consider the period. a s : Constant

25 By using and, No. 25 We take a form of f(ñ) as. ñ c : Integration constant, : Non-zero constants By using and, ø c : Integration constant There are three cases.,,,

26 We examine the behavior of each term of the gravitational field equations in the limit, in particular that of the leading terms, and study the condition that an asymptotic solution can be obtained. the leading term For case, ø c =1 vanishes in both gravitational field For case, equations. No. 26 Thus, the expression of the Hubble parameter can be a leading-order solution in terms of for the gravitational field equations in the flat FLRW space-time. This implies that there can exist the finite-time future singularities in non-local gravity.

27 C. Relations between the model parameters and the property of the finite-time future singularities f s and û characterize the theory of non-local gravity. h s, t q s and specify the property of the finite-time future singularity. ñ c ø c and determine a leading-order solution in terms of for the gravitational field equations in the flat FLRW space-time. When, * ú eff P eff for, for and, for, H ú eff ú s for, asymptotically becomes finite and also asymptotically approaches a finite constant value. for,,, No. 27 and correspond to the total energy density and pressure of the universe, respectively.

28 It is known that the finite-time future singularities can be classified in the following manner: Type I ( Big Rip ): Type II ( sudden ): Type III: Type IV: * * * In the limit, [Nojiri, Odintsov and Tsujikawa, Phys. Rev. D 71, (2005)],, The case in which ú eff and P eff becomes finite values at is also included.,,,,,, H Higher derivatives of diverge. ú eff Peff The case in which and/or asymptotically approach finite values is also included. No. 28

29 The finite-time future singularities described by the expression of H in nonlocal gravity have the following properties: For, Type I ( Big Rip ) For, Type II ( sudden ) For, Type III No. 29 < Table 1 > f(ñ) H Range and conditions for the value of parameters of,, and and ø c in order that the finite-time future singularities can exist. ñ c

30 III. Finite-time future singularities in f(t) gravity No. 30 < Formulations in teleparallelism > : Torsion tensor : Contorsion tensor : Torsion scalar e A (x ö ) : Orthonormal tetrad components * * ñ AB : Monkowski metric An index A runs over 0, 1, 2, 3 for the tangent space at each point of the manifold. x ö ö and are coordinate indices on the manifold and also run over 0, 1, 2, 3, and e A (x ö ) forms the tangent vector of the manifold. Instead of the Ricci scalar for the Lagrangian density in general relativity, the teleparallel Lagrangian density is described by the torsion scalar. T R

31 < Modified teleparallel action for f(t) theory > Action No. 31 S : Matter Lagrangian : Energy-momentum tensor of matter Gravitational field equation [Bengochea and Ferraro, Phys. Rev. D 79, (2009)] * A prime denotes a derivative with respect to T. The gravitational field equation in f(t) gravity is 2nd order, although it is 4th order in F(R) gravity. We assume the flat FLRW space-time with the metric,,

32 < Cosmological evolutions of Ω DE, Ω m and Ω r > No. 32 Ω DE f(t)=t + Ω r u(> 0) : Positive constant ô 2, < Fig. 7 > Ω m The model contains only one parameter u if one has the value of Ω (0). From [KB, Geng, Lee and Luo, JCAP 1101, 021 (2011)]. u =1 m

33 < Cosmological evolutions of w DE > No. 33 f(t)=t + u =0.5 (dash-dotted line) u =0.8 (dashed line) u(> 0) : Positive constant ô 2, u =1 (solid line) From [KB, Geng, Lee and Luo, JCAP 1101, 021 (2011)]. w DE = à 1 Crossing of the phantom divide < Fig. 8 >

34 < Finite-time future singularities > Gravitational field equations in the flat FLRW background No. 34, Continuity equation Effective EoS,,,

35 Hubble parameter No. 35 Scale factor

36 < Table 2 > Conditions to produce the finite-time future singularities in the limit of t t s. No. 36

37 No. 37, Gravitational field equations : Consistency condition (Friedmann equation) Power-law model,

38 < Removing the finite-time future singularities > No. 38 Power-law correction term

39 < Table 3 > Necessary conditions for the appearance of the finitetime future singularities on a power-law f(t) model and those for the removal of the finite-time future singularities on a power-law correction term f c (T)=BT ì. No. 39

40 < f(t) gravity models realizing cosmologies > (a) (Power-law) Inflation No. 40, or Power-law inflation is realized. (b) Λ CDM model, Λ > 0,, Exponential expansion is realized.

41 : Deceleration parameter : Jerk parameter : Snark parameter No. 41 [Chiba and Nakamura, Prog. Theor. Phys. 100, 1077 (1998)] [Sahni, Saini, Starobinsky and Alam, JETP Lett. 77, 201 (2003) [Pisma Zh. Eksp. Teor. Fiz. 77, 249 (2003)]] Λ CDM model [Komatsu et al. [WMAP Case that the flat universe and w DE is constant: = These four parameter can be used to test models. Collaboration], Astrophys. J. Suppl. 192, 18 (2011)], w DE Case that the flat universe and is time-dependent with the linear form:

42 (c) Little Rip cosmology A scenario to avoid the Big Rip singularity. ú DE increases in time with w DE < à 1 and approaches w DE = à 1.,, w DE or No. 42 asymptotically [Frampton, Ludwick and Scherrer, Phys. Rev. D 84, (2011)] [Frampton, Ludwick, Nojiri, Odintsov and Scherrer, Phys. Lett. B 708, 204 (2012)], [Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011)] : Current value of H, [Freedman et al. [HST Collaboration], Astrophys. J. 553, 47 (2001)]

43 No. 43 Λ CDM model = Current values of the four parameters,, If we take ÿ ü 1 enough for the deviation of the values of the four parameters (w DE,q dec,j,s) from those for the Λ CDM model (à 1, à 1, 1, 0) to be very small, this Little Rip model can be compatible with the Λ CDM model.

44 (d) Pseudo Rip cosmology A phantom scenario with the universe approaching de Sitter phase. H(t) approaches to a finite value in the limit t. This behavior is different from Little Rip cosmology. No. 44 [Frampton, Ludwick and Scherrer, Phys. Rev. D 85, (2012)] [Astashenok,, Nojiri, Odintsov and Yurov, Phys. Lett. B 709, 396 (2012)],,,, [Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011)] [Freedman et al. [HST Collaboration], Astrophys. J. 553, 47 (2001)]

45 , No. 45 Λ CDM model = Current values of the four parameters,,, We can take an appropriate value of î in order for the deviation of the values of the four parameters (w DE,q dec,j,s) from those for the Λ CDM model (à 1, à 1, 1, 0) to be very small, so that this Pseudo-Rip model can be consistent with the Λ CDM model.

46 < Table 4 > Forms of H and f(t) with realizing (a) inflation in the early universe, (b) the Λ CDM model, (c) Little Rip cosmology and (d) Pseudo-Rip cosmology. No. 46

47 Inertial force on a particle with mass m in the expanding universe No. 47 [Frampton, Ludwick and Scherrer, Phys. Rev. D 84, (2011)] [Frampton, Ludwick, Nojiri, Odintsov and Scherrer, Phys. Lett. B 708, 204 (2012)] We provide that two particles are bound by a constant force F b. When F inert (> 0) and F inert >F b, the two particles become unbound and hence the bound structure is dissociated. For a Big Rip singularity For Little Rip cosmology

48 For Pseudo Rip cosmology No. 48 < Earth-Sun (ES) system > F inert asymptotically approaches to a finite value. Condition for the disintegration of the ES system,, If this condition is met, the disintegration of the ES system can occur much before arriving at de Sitter universe, so that the Pseudo-Rip scenario can be realized. * The constraint from the current value of w DE is much weaker as

49 IV. Summary We have discussed modified gravitational theories to explain the current accelerated expansion of the universe, so-called dark energy problem. We have explicitly shown that three types of the finitetime future singularities (Type I, II and III) can occur in non-local gravity and examined their properties. We have illustrated that there appear finite-time future singularities (Type I and IV) in f(t) gravity and reconstructed an f(t) gravity model with realizing the finite-time future singularities. We have verified that a power-law type correction term ( ì>1 ) such as a T 2 term can remove the finite-time future singularities in f(t) gravity. This is the same feature as in F(R) gravity. T ì No. 49

50 We have derived the expressions of f(t) gravity models in which (a) Inflation, (b) Λ CDM model, (c) Little Rip cosmology, and (d) Pseudo Rip Cosmology can be realized. No. 50

51 Backup Slides

52 Gravitational field equations in the flat FLRW background: No. 32 * A prime denotes a derivative with respect to. * We consider only non-relativistic matter (cold dark matter and baryon) with and. Continuity equation:

53 < Combined f(t) model > From [KB, Geng, Lee and Luo, JCAP 1101, 021 (2011)]. u(> 0) No. 33 : Positive constant u =0.5 (dash-dotted line) u =0.8 (dashed line) Logarithmic term Exponential term u =1 (solid line) w DE = à 1 Crossing of the phantom divide The model contains only one parameter u if one has the value of. Ω (0) m

54 R 2 Cosmological consequences of adding an term We explore whether the addition of an R 2 term removes the finite-time future singularities in non-local gravity. < Action > No. 24 Gravitational field equations in the flat FLRW background: In the limit, The leading terms do not vanish. The additional R 2 term can remove the finite-time future singularity.

55 < 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)]

56 IV. Effective equation of state for the universe and phantom-divide crossing A. Cosmological evolution of the effective equation of state for the universe The effective equation of state for the universe : No. 20 H ç < 0 H ç =0 : The non-phantom (quintessence) phase w eff > à 1 w eff = à 1 Phantom crossing, H ç > 0 : The phantom phase w eff < à 1

57 We examine the asymptotic behavior of w eff in the limit by taking the leading term in terms of. No. 21 q > 1 For [Type I ( Big Rip ) singularity], we ff evolves from the non-phantom phase or the phantom one and asymptotically approaches. w eff = à 1 0 < q < 1 For [Type III singularity], w eff evolves from the non-phantom phase to the phantom one with realizing a crossing of the phantom divide or evolves in the phantom phase. The final stage is the eternal phantom phase. à 1 < q < 0 For [Type II ( sudden ) singularity], at the final stage. w eff > 0

58 We estimate the present value of. w eff For case, * No. 23 We regard at the present time because the energy density of dark energy is dominant over that of nonrelativistic matter at the present time. : The present time has the dimension of : Current value of H, [Freedman et al. [HST Collaboration], Astrophys. J. 553, 47 (2001)] 0 < q < 1 For,. For à 1 < q < 0, w eff > 0. w eff In our models, can have the present observed value of w DE.

59 < Crossing of the phantom divide > No. 20 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

60 < Data fitting of w(z) > z w(z)=w 0 + w 1 1+z No. 21 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)]

61 Continuity equation: No. 44 We define a dimensionless variable : : Evolution equation of the universe < (a). Exponential f(t) theory > p : Constant p =0 The case in which corresponds to the CDM model. This theory contains only one parameter p if the value of Ω (0) m is given. Λ

62 < (c). Combined f(t) theory > No. 48 u(> 0) : Positive constant ( ) u =0.5 (dash-dotted line) u =0.8 (dashed line) Logarithmic term Exponential term u =1 (solid line) w DE = à 1 Crossing of the phantom divide The model contains only one parameter u if one has the value of. Ω (0) m

63 < 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. 14 Standard cosmology [ Λ + Cold dark matter (CDM)]

64 (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. 15

65 (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. 16 For general relativity, m =0. [Faraoni and Nadeau, Phys. Rev. D 75, (2007)]

66 (4) Stability of the late-time de Sitter point 0 <mñ Rf 00 (R)/f 0 (R) < 1 [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, (2007)] For general relativity, [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)] m =0. [Faraoni and Nadeau, Phys. Rev. D 75, (2007)] m quantifies the deviation from the Λ CDM model. (5) Constraints from the violation of the equivalence principle M = M(R) Chameleon mechanism Scale-dependence No. 15 Cf. [Khoury and Weltman, Phys. Rev. D 69, (2004)] If the mass of the scalar degree of freedom is large, namely, the scalar becomes short-ranged, it has no effect at Solar System scales. The scalar degree of freedom may acquire a large effective mass at terrestrial and Solar System scales, shielding it from experiments performed there. (6) Solar-system constraints [Chiba, Phys. Lett. B 575, 1 (2003)] M [Chiba, Smith and Erickcek, Phys. Rev. D 75, (2007)]

67 Modified Friedmann equations in the flat FLRW background: No. 42 A prime denotes a derivative with respect to. We consider only non-relativistic matter (cold dark matter and baryon) with and. Continuity equation: We define a dimensionless variable : : Evolution equation of the universe

68 <(a). Exponential f(t) theory > No. 45 p =0.001 p =0.01 p = à 0.1 p =0.1 p = à 0.01 p>0 p<0 p = à 0.001

69 < Equation of state for (the component corresponding to) dark energy > No. 29 P w DE ñ DE úde < Continuity equation for dark energy >

70 < Future evolutions of 1+w DE as functions of z > No. 32 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)

71 < Future evolutions of H as functions of z > No. 34 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

72 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. 38 < 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

73 < The best-fit values > No. 50 The minimum ( ) of the combined f(t) theory is slightly smaller than that of the CDM model. Contours of, and confidence levels in the plane from SNe Ia, BAO and CMB data. The plus sign depicts the best-fit point. Λ ÿ 2 ÿ 2 min The combined f(t) theory can fit the observational data well.

74 < Data fitting of w(z) No. 22 (2) > From [Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)]. SN gold data set+cmb+bao SNLS data set+cmb+bao 2û confidence level.

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.

76 IV. Effective equation of state for the universe and the finite-time future singularities in non-local gravity Non-local gravity produced by quantum effects [Deser and Woodard, Phys. Rev. Lett. 99, (2007)] There was a proposal on the solution of the cosmological constant problem by non-local modification of gravity. [Arkani-Hamed, Dimopoulos, Dvali and Gabadadze, arxiv:hep-th/ ] Recently, an explicit mechanism to screen a cosmological constant in non-local gravity has been discussed. [Nojiri, Odintsov, Sasaki and Zhang, Phys. Lett. B 696, 278 (2011)] Recent related reference: [Zhang and Sasaki, arxiv: [gr-qc]] It is known that so-called matter instability occurs in F(R) gravity. [Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)] This implies that the curvature inside matter sphere becomes very large and hence the curvature singularity could appear. It is important to examine whether there exists the curvature singularity, i.e., the finite-time future singularities in non-local gravity. No. 51 [Arbuzova and Dolgov, Phys. Lett. B 700, 289 (2011)]

77 Appendix

78 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)=

79 < 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

80 In the flat FLRW background, gravitational field equations read ú eff, p eff, : Effective energy density and No. 13 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.)