No. 15 w ( z ) z < Data fitting of > w ( z ) = w 0 + w 1 1+ z From [Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)]. SN gold data set SNLS data set Shaded region [Riess et al . [Supernova Search [Astier et al . [The SNLS 1 û shows error. Team Collaboration], Collaboration], Astron. Astrophys. J. 607, 665 (2004)] Astrophys. 447, 31 (2006)] 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)]
No. 16 ・ 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, 064004 (2007)] [Martinelli, Melchiorri and Amendola, Phys. Rev. D 79, 123516 (2009)] Cf. [Nozari and Azizi, Phys. Lett. B 680, 205 (2009)] (ii) Starobinsky’s model [Motohashi, Starobinsky and Yokoyama, Prog. Theor. Phys. 123, 887 (2010); Prog. Theor. Phys. 124, 541 (2010)] (iv) Exponential gravity model [Linder, Phys. Rev. D 80, 123528 (2009)] [KB, Geng and Lee, JCAP 1008, 021 (2010) arXiv:1005.4574 [astro-ph.CO]] Appleby-Battye model Cf. [Appleby, Battye and Starobinsky, JCAP 1006, 005 (2010)]
w DE No. 17 < Cosmological evolution of in the exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. f E ( R ) = w DE ( z = 0) = à 0 . 93 ( < à 1 / 3) w DE = à 1 Crossing of the phantom divide Crossing in the past
No. 18 We explicitly demonstrate that the future crossings w DE = à 1 of the phantom divide line are the generic feature in the existing viable f ( R ) gravity models. ・ Recent related study on the future crossings of the phantom divide: [Motohashi, Starobinsky and Yokoyama, JCAP 1106, 006 (2011)]
II. Future crossing of the phantom divide in f ( R ) gravity No. 19 g = det( g ö÷ ) : Metric tensor < Action > f ： Arbitrary function R of : Action of matter : Matter fields < Gravitational field equation > ： Energy-momentum tensor of all R ö÷ : Ricci tensor perfect fluids of matter : Covariant derivative operator : Covariant d'Alembertian
No. 20 Gravitational field equations in the FLRW background: : Hubble parameter ú M P M and : Energy density and pressure of all perfect fluids of matter, respectively. P DE w DE ñ ú DE ・ Analysis method: [Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)]
1 + w DE z No. 21 < Future evolutions of as functions of > 1 + w DE Exponential gravity model Crossings in the future 1 + w DE = 0 0 Crossing of the phantom divide Redshift: 1 à 1 z ñ a z < 0 ( : Future)
No. 22 Hu-Sawicki model Starobinsky’s model 1 + w DE Redshift: 1 à 1 z ñ a z < 0 ( : Future) Tsujikawa’s model Exponential gravity model Crossings in the future 0 1 + w DE = 0 Crossing of the phantom divide
III. Equation of state for dark energy in f ( T ) theory No. 23 e A ( x ö ) : Orthonormal tetrad components A An index runs over 0, 1, 2, 3 for x ö the tangent space at each point of : Torsion tensor the manifold. ö ÷ and are coordinate indices on the manifold and also run over 0, 1, 2, 3, : Contorsion tensor e A ( x ö ) and forms the tangent vector : Torsion scalar of the manifold. R Instead of the Ricci scalar for the Lagrangian density in general relativity, the teleparallel Lagrangian density T is described by the torsion scalar . < Modified teleparallel action for f ( T ) theory > F ( T ) ñ T + f ( T )
No. 24 [Bengochea and Ferraro, Phys. : Gravitational field equation Rev. D 79, 124019 (2009)] T A prime denotes a derivative with respect to . * ・ We assume the flat FLRW space-time with the metric. Modified Friedmann equations in the flat FLRW background: , A prime denotes a derivative with respect to . * ・ We consider only non-relativistic matter (cold dark matter and baryon).
No. 25 < Combined f ( T ) theory > u ( > 0) : Positive constant Logarithmic Exponential u = 0 . 5 term term (dash-dotted line) u = 0 . 8 (dashed line) ・ The model contains only w DE = à 1 one parameter u if one has Crossing of u = 1 (solid line) the value the phantom Ω (0) of . m divide
No. 26 IV. Effective equation of state for the universe and the finite-time future singularities in non-local gravity produced by quantum effects Non-local gravity [Deser and Woodard, Phys. Rev. Lett. 99, 111301 (2007)] ・ 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)] It is important to examine whether there exists the curvature singularity, i.e., “ the finite-time future singularities ” in non-local gravity .
No. 27 A. Non-local gravity g = det( g ö÷ ) : Metric tensor f : Some function < Action > Λ : Cosmological constant Non-local gravity By introducing two scalar fields and , we find : Covariant derivative operator ・ By the variation of the action in the ø first expression over , we obtain : Covariant d'Alembertian (or ) : Matter Lagrangian Substituting this equation into the action in the Q ： Matter fields first expression, one re-obtains the starting action.
< Gravitational field equation > No. 28 : Energy-momentum tensor of matter ñ ・ The variation of the action with respect to gives ñ 0 : Derivative with respect to (prime) * ・ We assume the flat FLRW space-time with the metric and consider ñ ø the case in which the scalar fields and only depend on time. Gravitational field equations in the FLRW background: : Energy density and and ñ ø < Equations of motion for and > pressure of matter. ,
In the flat FLRW space-time, we analyze an asymptotic No. 29 solution of the gravitational field equations in the limit of t s the time when the finite-time future singularities appear. ・ We consider the case in which the Hubble parameter is expressed as h s : Positive constant, q : Non-zero constant larger than -1 ・ → ∞ When , . Scale factor : a s : Constant ñ c , ・ We have . ø c : Integration constants , f ( ñ ) ・ We take a form of as . : Non-zero constants
・ It is known that the finite-time future singularities can be classified in the following manner: [Nojiri, Odintsov and Tsujikawa, No. 30 Phys. Rev. D 71, 063004 (2005)] In the limit , , , Type I (“Big Rip”): ú eff P eff The case in which and becomes * finite values at is also included. , , Type II (“sudden”): , , Type III: , , Type IV: H Higher derivatives of diverge. * ú eff | P eff | The case in which and/or * asymptotically approach finite values is also included.
・ No. 31 The finite-time future singularities H described by the expression of in non- local gravity have the following properties: For , Type I (“Big Rip”) For , Type II (“sudden”) For , Type III ñ c f ( ñ ) H Range and conditions for the value of parameters of , , and * ø c and in order that the finite-time future singularities can exist.
No. 32 w eff We examine the asymptotic behavior of in the limit by taking the leading term in terms of . ・ w e q > 1 For [Type I (“Big Rip”) singularity], evolves from the non-phantom phase or the phantom one ff w eff = à 1 and asymptotically approaches . ・ 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], w eff > 0 at the final stage.
No. 33 We estimate the present value We regard at the present * w eff of . time because the energy density of dark energy is dominant over that of non- ・ For case , relativistic 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 , w eff In our models, can have the present à 1 < q < 0 w eff > 0 ・ w DE For , . observed value of .
V. Summary No. 34 ・ We have discussed modified gravitational theories in order to explain the current accelerated expansion of the universe, so-called dark energy problem. ・ We have investigated the equation of state for dark energy w DE in f ( R ) gravity as well as f ( T ) theory. The future crossings of the phantom divide line w DE = à 1 are the generic feature in the existing viable f ( R ) gravity models. The crossing of the phantom divide line can be realized in the combined f ( T ) theory. ・ We have studied the effective equation of state for the universe when the finite-time future singularities occur in non-local gravity.
H z < Future evolutions of as functions of > No. 23 Exponential gravity model Oscillatory behavior H ( z = à 1) ñ H 0 : ‘f’ denotes the value at the final stage z = à 1 . : Present value of the Hubble parameter
No. 24 Hu-Sawicki model Starobinsky’s model H ( z = à 1) ñ H 0 Tsujikawa’s model Exponential gravity model Oscillatory behavior : Present value of the Hubble parameter
No. 25 ・ In the future ( ), the crossings of the phantom divide are the generic feature for all the existing viable f ( R ) models. z ・ As decreases ( ), dark energy becomes much more dominant over non-relativistic matter ( ). < Effective equation of state for the universe > : Total energy density of the universe : Total pressure of the universe P DE : Pressure of dark energy P m Pressure of non-relativistic matter : (cold dark matter and baryon) P r : Pressure of radiation
ç No. 26 2 H w eff = à 1 à 3 H 2 ç < 0 w eff > à 1 (a) H Non-phantom phase Crossing of the ç = 0 w eff = à 1 H (b) phantom divide ç > 0 w eff < à 1 (c) Phantom phase H ・ The physical reason why the crossing of the phantom divide appears in the farther future ( ) ç H is that the sign of changes from negative to positive due to the dominance of dark energy over non-relativistic matter. w DE ・ As in the farther future, oscillates w DE = à 1 around the phantom divide line because ç H the sign of changes and consequently multiple crossings can be realized.
< Conditions for the viability of f ( R ) gravity > No. 12 (1) f 0 ( R ) > 0 Positivity of the effective gravitational coupling G eff = G/f 0 ( R ) > 0 f 0 ( R ) ñ df ( R ) /dR G : Gravitational constant M 2 ù 1 / (3 f 00 ( R )) > 0 f 00 ( R ) > 0 (2) Stability condition: f 00 ( R ) ñ d 2 f ( R ) /dR 2 [Dolgov and Kawasaki, Phys. Lett. B 573, 1 (2003)] Mass of a new scalar degree of freedom (“scalaron”) M : in the weak-field regime. Existence of a matter- f ( R ) → R à 2 Λ R ý R 0 (3) for . dominated stage R 0 : Current curvature, Λ : Cosmological constant Stability of the late- 0 < m ñ Rf 00 ( R ) /f 0 ( R ) < 1 (4) time de Sitter point [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)] Cf. For general relativity, [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)] m = 0 . [Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)] (5) Constraints from the violation of the equivalence principle M = M ( R ) Scale-dependence ： ‘‘Chameleon mechanism’’ Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)] (6) Solar-system constraints [Chiba, Phys. Lett. B 575, 1 (2003)] [Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]
< Flat Friedmann-Lema tre-Robertson-Walker (FLRW) No. 20 space-time > a ( t ) : Scale factor Gravitational field equations in the FLRW background: : Hubble parameter ú M P M Energy density and pressure of all perfect and : fluids of matter, respectively. < Analysis method > [Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)] (1) 0 ・ Ricci scalar: (prime): Derivative with R respect to (2)
No. 21 We solve Equations (1) and (2) by introducing the following variables: ・ ‘(0)’denotes the present values. ú DE : Energy density of dark energy ú m : Energy density of non-relativistic matter (cold dark matter and baryon) ú r : Energy density of radiation (3) (4)
No. 22 y H : Equation for Combining Equations (3) and (4), we obtain
No. 23 < Equation of state for (the component corresponding to) dark energy > P DE w DE ñ ú DE < Continuity equation for dark energy >
No. 31 A prime denotes a derivative * : Gravitational field equation T with respect to . [Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)] ・ We assume the flat FLRW space-time with the metric, , Modified Friedmann equations in the flat FLRW background: A prime denotes a derivative * with respect to .
No. 32 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
No. 36 < Gravitational field equation > : Energy-momentum tensor of matter ñ ・ The variation of the action with respect to gives 0 ñ : Derivative with respect to (prime) < Flat Friedmann-Lema tre-Robertson-Walker (FLRW) metric > a ( t ) : Scale factor ñ ø ・ We consider the case in which the scalar fields and only depend on time.
No. 37 Gravitational field equations in the FLRW background: : Hubble parameter : Energy density and pressure of matter, respectively. and For a perfect fluid of matter: ñ ø < Equations of motion for and >
No. 38 A. Finite-time future singularities In the flat FLRW space-time, we analyze an asymptotic solution t s of the gravitational field equations in the limit of the time when the finite-time future singularities appear. ・ We consider the case in which the Hubble parameter is expressed as h s : Positive constant q : Non-zero constant larger than -1 We only consider the period . ・ → ∞ When , Scale factor a s : Constant
・ By using and , No. 39 ñ c : Integration constant , f ( ñ ) ・ We take a form of as . : Non-zero constants ・ By using and , ø c : Integration constant There are three cases. , , ,
< Other model > No. 18 ・ Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)] R + 2 b 1 1 log cosh( b 1 R ) à tanh( b 2 )sinh( b 1 R ) f AB ( R ) = 2 [ ] b 1 ( > 0) , b 2 : Constant parameters
Future crossing of the phantom divide No. 30 (i) Hu-Sawicki model [Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)] p = 1 c 2 = 1 c 1 = 1 , (ii) Starobinsky’s model [Starobinsky, JETP Lett. 86, 157 (2007)] n = 2 õ = 1 . 5 (iii) Tsujikawa’s model [Tsujikawa, Phys. Rev. D 77, 023507 (2008)] ö = 1 [Cognola, Elizalde, Nojiri, Odintsov, (iv) Exponential gravity model Sebastiani and Zerbini, Phys. Rev. D 77, 046009 (2008)] [Linder, Phys. Rev. D 80, 123528 (2009)] ì = 1 . 8
No. 16 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. 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.
B. Relations between the model parameters and the property No. 17 of the finite-time future singularities f s û ・ and characterize the theory of non-local gravity. q h s t 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 , for , for and , , for , ú eff ú s H asymptotically becomes finite and also for , asymptotically approaches a finite constant value . , → ∞ for ,
B. Estimation of the current value of the effective equation of No. 22 state parameter for non-local gravity [Komatsu et al . [WMAP Collaboration], Astrophys. J. Suppl. 192, 18 (2011)] ・ The limit on a constant equation of by combining the data of Seven- state for dark energy in a flat Year Wilkinson Microwave universe has been estimated as Anisotropy Probe (WMAP) Observations with the latest distance measurements from the baryon acoustic oscillations (BAO) in the distribution of ・ For a time-dependent equation of state galaxies and the Hubble constant measurement. for dark energy, by using a linear form , w DE : Current value of w DE constraints on and : Derivative of have been found as from the combination of the , WMAP data with the BAO data, the Hubble constant measurement and the high-redshift SNe Ia data.
IV. Effective equation of state for the universe and No. 20 phantom-divide crossing A. Cosmological evolution of the effective equation of state for the universe ・ The effective equation of : state for the universe , ç < 0 H : The non-phantom (quintessence) phase w eff > à 1 ç = 0 w eff = à 1 Phantom crossing H ç > 0 : The phantom phase H w eff < à 1
No. 4 (1) General relativistic approach ・ Cosmological constant Canonical field ・ Scalar field : X matter , Quintessence [Chiba, Sugiyama and Nakamura, Mon. Not. Roy. Astron. Soc. 289, L5 (1997)] [Caldwell, Dave and Steinhardt, Phys. Rev. Lett. 80, 1582 (1998)] Cf. Pioneering work: [Fujii, Phys. Rev. D 26, 2580 (1982)] Phantom Wrong sign kinetic term [Caldwell, Phys. Lett. B 545, 23 (2002)] K-essence Non canonical kinetic term [Chiba, Okabe and Yamaguchi, Phys. Rev. D 62, 023511 (2000)] [Armendariz-Picon, Mukhanov and Steinhardt, Phys. Rev. Lett. 85, 4438 (2000)] Tachyon String theories A > 0 : Constant [Padmanabhan, Phys. Rev. D 66, 021301 (2002)] ú : Energy density ・ Chaplygin gas p = à A/ú p : Pressure [Kamenshchik, Moschella and Pasquier, Phys. Lett. B 511, 265 (2001)]
No. 5 (2) Extension of gravitational theory Cf. Application to inflation: ・ f ( R ) gravity [Starobinsky, Phys. Lett. B 91, 99 (1980)] f ( R ) R : 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)] f i ( þ ) [Nojiri and Odintsov, Phys. Rev. D 68, 123512 (2003)] : Arbitrary function þ of a scalar field ( i = 1 , 2) ・ Scalar-tensor theories f 1 ( þ ) R [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 tensor [Nojiri, Odintsov and Sasaki, Phys. Rev. D 71, 123509 (2005)] R + f ( G ) : Riemann ô 2 ñ 8 ùG ・ f ( G ) gravity 2 ô 2 tensor G : Gravitational constant [Nojiri and Odintsov, Phys. Lett. B 631, 1 (2005)]
・ DGP braneworld scenario No. 6 [Dvali, Gabadadze and Porrati, Phys. Lett B 485, 208 (2000)] [Deffayet, Dvali and Gabadadze, Phys. Rev. D 65, 044023 (2002)] ・ f(T) gravity Extended teleparallel Lagrangian density described by the : T torsion scalar . [Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)] [Linder, Phys. Rev. D 81, 127301 (2010) [Erratum-ibid. D 82, 109902 (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)]] ・ Galileon gravity [Nicolis, Rattazzi and Trincherini, Phys. Rev. D 79 , 064036 (2009)] Review: [Tsujikawa, Lect. Notes Phys. 800, 99 (2010)] þ Longitudinal graviton (i.e. a branebending mode ) ・ 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 : Covariant d'Alembertian freedom associated with ghosts. ・ Non-local gravity [Deser and Woodard, Phys. Quantum effects Rev. Lett. 99, 111301 (2007)]
< Conditions for the viability of f ( R ) gravity > No. 14 (1) f 0 ( R ) > 0 ・ Positivity of the effective gravitational coupling G eff = G 0 /f 0 ( R ) > 0 G 0 : Gravitational constant (The graviton is not a ghost.) f 00 ( R ) > 0 (2) [Dolgov and Kawasaki, Phys. Lett. B 573 , 1 (2003)] M 2 ù 1 / (3 f 00 ( R )) > 0 ・ Stability condition: 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 2 Λ (3) R ý R 0 R 0 for : Current curvature Λ : Cosmological constant Λ ・ Realization of the CDM-like behavior in the large curvature regime Λ Standard cosmology [ + Cold dark matter (CDM)]
(4) Solar system constraints No. 15 Brans-Dicke theory f ( R ) gravity ω BD = 0 with Equivalent ω BD : Brans-Dicke parameter [Bertotti, Iess and Tortora, Observational constraint: | ω BD | > 40000 Nature 425, 374 (2003).] [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)] ・ M However, 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. ・ M = M ( R ) ‘‘Chameleon mechanism’’ Scale-dependence: 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.
No. 16 (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 = 0 . ・ For general relativity, m Λ quantifies the deviation from the 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 ) d ) 2 à 2 f d f 00 ( f 0 d > 0 R d : Constant curvature f 0 d f 00 in the de Sitter space d ・ Linear stability of the inhomogeneous perturbations in the de Sitter space [Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)] R d = 2 f d /f 0 m < 1 Cf. d
Ω DE Ω m Ω r No. 19 < Cosmological evolutions of , and in the exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. f E ( R ) = ì = 1 . 8
< Conclusions of Sec. II > No. 41 ・ 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 ç H study is that in the future the sign of 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 Λ deviation from the CDM model.
< Conclusions of Sec. III > No. 51 ・ We have investigated the cosmological evolution in the exponential f ( T ) theory. The phase of the universe depends on the sign of the p p < 0( > 0) parameter , i.e., for the universe is always in the non-phantom (phantom) phase without the crossing of the phantom divide. ・ We have presented the logarithmic type f ( T ) model. It does not allow the crossing of the phantom divide. ・ To realize the crossing of the phantom divide, we have constructed an f ( T ) theory by combining the logarithmic and exponential terms. w DE > à 1 The crossing in the combined f ( T ) theory is from w DE < à 1 to , which is opposite to the typical manner in f ( R ) gravity models. This combined theory is consistent with the recent observational data of SNe Ia, BAO and CMB.
No. 64 < Conclusions of Sec. IV > ・ We have explicitly shown that three types of the finite- time future singularities (Type I, II and III) can occur in non-local gravity and examined their properties. ・ We have investigated the behavior of the effective equation of state for the universe when the finite- time future singularities occur.
No. 44 Continuity equation: ・ We define a dimensionless : variable : Evolution equation of the universe < (a). Exponential f ( T ) theory > p : Constant ・ p = 0 Λ The case in which corresponds to the CDM model. p Ω (0) ・ This theory contains only one parameter if the value of m is given.
A prime denotes a derivative T with respect to . : Gravitational field equation [Bengochea and Ferraro, Phys. Rev. D 79, 124019 (2009)] ・ No. 38 We assume the flat FLRW space-time with the metric, , Modified Friedmann equations in the flat FLRW background: A prime denotes a derivative with respect to . We consider only non-relativistic matter (cold dark matter and baryon) with and .
No. 39 Continuity equation: ・ We define a dimensionless : variable : Evolution equation of the universe
No. 45 <(a). Exponential f ( T ) theory > p = 0 . 001 p = 0 . 01 p = à 0 . 1 p = 0 . 1 p = à 0 . 01 p = à 0 . 001 p < 0 p > 0 ・ ・
No. 46 w DE = à 1 w DE does not cross the phantom divide line in the exponential f ( T ) theory. ・ p > 0 For , the universe always stays in the non-phantom w DE > à 1 p < 0 (quintessence) phase ( ), whereas for w DE < à 1 it in the phantom phase ( ). | | p ・ The larger is, the larger the deviation of the Λ exponential f ( T ) theory from the CDM model is. ・ z = 0 We have taken the initial conditions at as .
< (b). Logarithmic f ( T ) theory > No. 47 q ( > 0) : Positive constant ・ This theory contains only q one parameter if the value Ω (0) of is obtained. m w DE does not cross the phantom divide w DE = à 1 line .
Ω DE Ω m Ω r No. 41 < Cosmological evolutions of , and > Ω DE Radiation-dominated stage [ ] Ω r Matter-dominated stage Ω m Dark energy becomes dominant over matter z < 0 . 36 ( ). u = 1
No. 42 < The best-fit values > ÿ 2 ÿ 2 The minimum ( ) min of the combined f ( T ) theory is slightly smaller than that Λ of the CDM model. The combined f ( T ) theory can fit the observational data well. Contours of , and confidence levels in the plane from SNe Ia, BAO and CMB data. The plus sign depicts the best-fit point.
Backup Slides A
(4) Solar system constraints No. 15 Brans-Dicke theory f ( R ) gravity ω BD = 0 with Equivalent ω BD : Brans-Dicke parameter [Bertotti, Iess and Tortora, Observational constraint: | ω BD | > 40000 Nature 425, 374 (2003).] [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)] ・ M However, 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. ・ M = M ( R ) ‘‘Chameleon mechanism’’ Scale-dependence: 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.
No. 16 (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 Λ quantifies the deviation from the 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 ) d ) 2 à 2 f d f 00 ( f 0 d > 0 R d : Constant curvature f 0 d f 00 in the de Sitter space d ・ Linear stability of the inhomogeneous perturbations in the de Sitter space [Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)] R d = 2 f d /f 0 m < 1 Cf. d
No. 15 (4) Stability of the late-time de Sitter point 0 < m ñ Rf 00 ( R ) /f 0 ( R ) < 1 [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)] ・ [Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)] For general relativity, m = 0 . [Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)] m Λ quantifies the deviation from the CDM model. (5) Constraints from the violation of the equivalence M = M ( R ) principle ‘‘Chameleon mechanism’’ Scale-dependence Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)] M ・ 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)] [Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]
w ( z ) No. 22 < Data fitting of (2) > From [Alam, Sahni and Starobinsky, JCAP 0702, 011 (2007)]. SN gold data set+CMB+BAO SNLS data set+CMB+BAO ・ 2 û confidence level. ・
w ( z ) No. B-7 < Data fitting of (3) > From [Zhao, Xia, Feng and Zhang, Int. J. Mod. Phys. D 16, 1229 (2007) [arXiv:astro-ph/0603621]] Best-fit 68% confidence level 95% confidence level 157 “gold” SN Ia data set+WMAP 3-year data+SDSS with/without dark energy perturbations.
No. B-6 ・ For most observational probes (except the SNLS Ω 0m (0 . 2 < Ω 0m < 0 . 25) data), a low prior leads to an increased probability (mild trend) for the crossing of the phantom divide. Ω 0m : Current density parameter of matter [Nesseris and L. Perivolaropoulos, JCAP 0701, 018 (2007)]
No. 30 < Bekenstein-Hawking entropy on the apparent horizon in the flat FLRW background > A S = 4 G : Bekenstein-Hawking entropy : Area of the apparent horizon à = 1 /H r ： 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 entropy is simply à á S = ù/ GH 2 expressed as .
S z < Future evolutions of as functions of > No. 36 Exponential gravity model S ( z = à 1) ñ S 0 ： Present value of the horizon entropy Oscillating behavior
No. 37 Starobinsky’s model Hu-Sawicki model S ( z = à 1) ñ S 0 Tsujikawa’s model Exponential gravity model ： Present value of the horizon entropy Oscillating behavior
No. 40 S ∝ H à 2 S ・ Since , the oscillating behavior of comes H from that of . 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. Cf. 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. [KB and Geng, JCAP 1006, 014 (2010)]
No. 16 < (a). Exponential f ( T ) theory >
No. 16 < (b). Logarithmic f ( T ) theory >
No. 16 < (c). Combined f(T ) theory >
No. 16 < (c). Combined f(T ) theory >
p : Constant ú M P M and : Energy density and pressure of all perfect fluids of generic matter, respectively. ・ Initial conditions:
No. 48 < Combined f(T ) theory > u : Constant ( ) Logarithmic Exponential u = 0 . 5 term term (dash-dotted line) u = 0 . 8 (dashed line) ・ The model contains only w DE = à 1 one parameter u if one has Crossing of u = 1 (solid line) the value the phantom Ω (0) of . m divide
IV. Effective equation of state for the universe and the No. 51 finite-time future singularities in non-local gravity produced by quantum effects Non-local gravity [Deser and Woodard, Phys. Rev. Lett. 99, 111301 (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/0209227] 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:1108.2112 [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. [Arbuzova and Dolgov, Phys. Lett. B 700, 289 (2011)] It is important to examine whether there exists the curvature singularity, i.e., “ the finite-time future singularities ” in non-local gravity .
No. 59 C. Relations between the model parameters and the property of the finite-time future singularities f s û ・ and characterize the theory of non-local gravity. q h s t 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 , for , , for and , , for , ú eff ú s H asymptotically becomes finite and also for , asymptotically approaches a finite constant value . , → ∞ for ,
No. 60 ・ It is known that the finite-time future singularities can be classified in the following manner: In the limit , , , Type I (“Big Rip”): ú eff P eff The case in which and becomes * finite values at is also included. , , Type II (“sudden”): , , Type III: , , Type IV: H Higher derivatives of diverge. * ú eff | P eff | The case in which and/or [Nojiri, Odintsov and * Tsujikawa, Phys. Rev. asymptotically approach finite values is D 71, 063004 (2005)] also included.
Appendix A
< Other models > No. A-10 ・ Appleby-Battye model [Appleby and Battye, Phys. Lett. B 654, 7 (2007)] R + 2 b 1 1 log cosh( b 1 R ) à tanh( b 2 )sinh( b 1 R ) f AB ( R ) = 2 [ ] b 1 ( > 0) , b 2 : Constant parameters Cf. Power-law model [Amendola, Gannouji, Polarski and Tsujikawa, Phys. Rev. D 75, 083504 (2007)] f ( R ) = R à öR v [Li and Barrow, Phys. Rev. D 75, 084010 (2007)] ö ( > 0) : Constant parameter 0 < v < 10 à 10 : Constant parameter (close to 0) [Capozziello and Tsujikawa, Phys. Rev. D 77, 107501 (2008)]
No. A-11 : Bekenstein-Hawking horizon entropy A S = 4 G in the Einstein gravity : Wald entropy in modified gravity theories : Area of the apparent horizon ê ・ S Wald introduced a horizon entropy associated with a Noether charge in the context of modified gravity theories. ê S ・ 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. ê = A/ (4 G eff ) S This is equivalent to . G eff = G/F : Effective gravitational coupling
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 ) =
No. A-14 < Remarks > (a) The qualitative results do not strongly depend on the values of the parameters in each model. ê S S (b) The evolutions of the Wald entropy are similar to in the models of (i) ‐ (iv). Cf. [KB, Geng and Tsujikawa, Phys. Lett. B 688, 101 (2010)] [KB, Geng and Lee, JCAP 1008, 021 (2010)] : A Bekenstein-Hawking horizon entropy S = 4 G in the Einstein gravity : Wald entropy in modified gravity ê = F ( R ) A S 4 G theories including f ( R ) gravity (c) The numerical results in the Appleby-Battye model are similar to those in the Starobinsky model of (ii).
ö ö ê No. A-15 ö H S < Cosmological evolutions of , and in the S exponential gravity model > From [KB, Geng and Lee, JCAP 1008, 021 (2010)]. f E ( R ) =
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