Tsujikawa, Phys. Rev. D 77, 023507 (2008)

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Conditions for the viability of f(R) gravity >(1) f 0 (R) > 0f 0 (R) ñ df(R)/dR(2)(3)f 00 (R) > 0f 00 (R) ñ d 2 f(R)/dR 2(4) 0 0 GM 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, 083504 (2007)][Amendola and Tsujikawa, Phys. Lett. B 660, 125 (2008)][Faraoni and Nadeau, Phys. Rev. D 75, 023501 (2007)](5) Constraints from the violation of the equivalence principle(Solar-system constraints) [Chiba, Phys. Lett. B 575, 1 (2003)][Chiba, Smith and Erickcek, Phys. Rev. D 75, 124014 (2007)]Scale-dependence ‘‘Chameleon mechanism’’M = M(R) :Cf. [Khoury and Weltman, Phys. Rev. D 69, 044026 (2004)]No. 12Positivity of the effective gravitational couplingStability condition:: Gravitational constantM :Mass of a new scalar degree of freedom (“scalaron”)in the weak-field regime.Existence of a matterdominatedstagef(R) → R à 2Λ for R ý R 0 .R 0 : Current curvature, Λ : Cosmological constant Stability of the latetimede Sitter pointCf. For general relativity,m =0.
Models of f(R) gravity (examples) >[Hu and Sawicki, Phys. Rev. D 76, 064004 (2007)](i) Hu-Sawicki modelCf.No. 13[Nojiri and Odintsov, Phys. Lett. B 657, 238(2007); Phys. Rev. D 77, 026007 (2008)]: Constant parameters(ii) Starobinsky’s model [Starobinsky, JETP Lett. 86, 157 (2007)]: Constant parameters(iii) Tsujikawa’s model [Tsujikawa, Phys. Rev. D 77, 023507 (2008)](iv) Exponential gravity model: Constant parameters[Cognola, Elizalde, Nojiri, Odintsov,Sebastiani and Zerbini, Phys. Rev. D 77,046009 (2008)][Linder, Phys. Rev. D 80, 123528 (2009)]: Constant parameters
Page 1: Equation of state for dark energyin
Page 4 and 5: (1) General relativistic approach
Page 6 and 7: m à:mMMDistanceestimator::Apparent
Page 9 and 10: Canonical scalar field >⎧S þ =
Page 11: ・In the flat FLRW background, gra
Page 15 and 16: Data fitting of w(z) >zw(z)=w 0 + w
Page 17 and 18: w DEw DE = à 1< Cosmological evolu
Page 19 and 20: II. Future crossing of the phantom
Page 21 and 22: Future evolutions of 1+w DE as func
Page 23 and 24: III. Equation of state for dark ene
Page 25 and 26: Combined f(T) theory >No. 25u(> 0):
Page 27 and 28: A. Non-local gravity< Action >g =de
Page 29 and 30: ・In the flat FLRW space-time, we
Page 31 and 32: ・The finite-time future singulari
Page 33 and 34: ・We estimate the present valueof
Page 35 and 36: Future evolutions of H as functions
Page 37 and 38: ・・In the future ( ), the crossi
Page 39 and 40: Conditions for the viability of f(R
Page 41 and 42: We solve Equations (1) and (2) by i
Page 43 and 44: Equation of state for (the componen
Page 45 and 46: No. 32We consider only non-relativi
Page 47 and 48: Gravitational field equations in th
Page 49: ・By using and ,No. 39・We take a
Page 52 and 53: Future crossing of the phantom divi
Page 54 and 55: B. Relations between the model para
Page 56 and 57: IV. Effective equation of state for
Page 58 and 59: (2) Extension of gravitational theo
Page 60 and 61: Conditions for the viability of f(R
Page 62 and 63:
(5) Existence of a matter-dominated
Page 64 and 65:
Conclusions of Sec. II >・We have
Page 66 and 67:
Conclusions of Sec. IV >No. 64・
Page 68 and 69:
・We assume the flat FLRW space-ti
Page 70 and 71:
No. 45p =0.001p =0.01p = à 0.1p =0
Page 72 and 73:
(b). Logarithmic f(T) theory >No. 4
Page 74 and 75:
The best-fit values >No. 42The mini
Page 76 and 77:
(4) Solar system constraintsf(R) gr
Page 78 and 79:
(4) Stability of the late-time de S
Page 80 and 81:
Data fitting of w(z) (3) >No. B-7Fr
Page 82 and 83:
Bekenstein-Hawking entropy on the a
Page 84 and 85:
Hu-Sawicki modelStarobinsky’s mod
Page 86 and 87:
(a). Exponential f(T) theory >No. 1
Page 88 and 89:
(c). Combined f(T) theory >No. 16
Page 90 and 91:
p : Constantú Mand P M : Energy de
Page 92 and 93:
IV. Effective equation of state for
Page 94 and 95:
・It is known that the finite-time
Page 96 and 97:
Other models >No. A-10・Appleby-Ba
Page 98 and 99:
w effNo. A-13< Cosmological evoluti
Page 100 and 101:
Cosmological evolutions of , and in
Page 102 and 103:
Initial conditions >Models of (i),
Page 104 and 105:
No. A-21・・ By examining the cos
Page 106 and 107:
No. A-23
Page 108 and 109:
Second law of thermodynamics >[KB a
Page 110 and 111:
Residuals for the best fit to a fla
Page 112 and 113:
5-year WMAP data on >・ For the fl
Page 114 and 115:
Appendix B
Page 116 and 117:
Recent work >No. B-3[Martinelli, Me
Page 118:
(5) Existence of a matter-dominated
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Tsujikawa, Phys. Rev. D 77, 023507 (2008)

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