e quite different, since the universe may I. enter FORMULAE in a phantom phase [19], or may produce Cosmological oscillations along the cosmological evolution [27]. We will explore in the evolution, viable and the crossing of the phantom barrier in viable f(R) gravity in next sections. 1 F(R) 1 Hence, under these circumstances, the cF 1 (R/R (R) term HS ) n is able to reproduce both accelerating f(R) =R − R HS epochs of the universe evolution. Here c 2 (R/R we will HS ) n focus +1 , on f(R) the =R study + Rn (aR n − b) I. FORMULAE of1+cR two n . (1) I. FORMULAE models of this kind of viable gravity, proposed by Hu and Sawicki in Ref. [15], and Nojiri 1and Odintsov in Ref. [16], whose actions are given by, S EH = d 4 x √ −g R +2κ 2 L m → S = d 4 x √ −g 1 f(R)+2κ 2 I. FORMULAE c 1 (R/R HS ) n L m . (2) Hu-Sawicki model f(R) =R − R HS c 1 c HS ) n F HS (R) =−R HS c 2 (R/R HS ) n +1 , F NO(R) = Rn (aR n 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n − b) c 1+cR − b) n . (1) 1 (R/R f(R) =R − R I. FORMULAE HS ) n HS c 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n − b) 1+cR n . (2.10) T µν (m) = 2 δ ( √ 1+cR n . (1) 1 −gL m ) √ Here, {c 1 ,c 2 ,n} are free parameters where and R HS κ 2 ρ −g 0 m according δg µν (3) S EH = d 4 x √ −g c R +2κ 2 L m → S = d 4 x √ to −gRef. 1 (R/R HS ) n f(R)+2κ [15], while 2 f(R) =R − R L m {a, . b, c, n} (2) I. FORMULAE are free parameters S EH for = the d 4 second x √ −g HS model R +2κ 2 in(2.10). L m → The S first = model d 4 x √ −g in(2.10) f(R)+2κ has 2 c c 1 (R/R HS ) f(R) =R − R HS been L m . studied (2) c 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n − b) 1+cR n . (1) − b) 1+cR n . (1) in Ref. [7], where it was proven andthat then =1, universe c 1 evolution =2, c 2 = goes 1 through two different De (4) Sitter points, being one of them stableT and (m) the other one unstable, which can be identified to µν = −√ 2 δ ( √ −gL m ) the current accelerated f(R) era=R and−to the inflationary c 1 (R/R HS −g ) n epoch, respectively. Similarly, the second Initial conditions: HS model in(2.10) was studied in Ref. [18], c 2 also (R/Rwith HS ) n the +1 , δg µν (3) T f(R) =R + Rn (aR n − b) µν (m) = − 2 δ ( √ −gL m ) S EH = d 4 x √ −g R +2κ 2 L m √→ −g S = δg µν d 4 x √ −g f(R)+2κ 2 L m presence of an extra1+cR field, n . . (2) (3) (1) and was shown n =1, c 1 =2, c 2 = 1 (4) 1. Assuming "CDM model at z=0, S EH = d 4 x √ −g n =1, R +2κ 2 c 1 =2, c 2 = 1 L m → S = d 4 x √ −g f(R)+2κ 2 (4) T µν (m) = − 2 δ ( √ −gL m ) √ L m . (2) −g H(z = 0) = H 0 = 100 hkms −1 δg Mpc µν (3) −1 ,h=0.71 ± 0.03 (5) H(z = 0) = H 0 = 100 –4– hkms −1 Mpc −1 T µν (m) = −√ 2 δ ( √ ,h=0.71 ± 0.03 (5) n =1, c −gL m ) −g δg µν (3) H (z) = κ2 ρ 1 =2, c 2 = 1 m 2 H 0 (1 + z)h(z) , → H (0) = κ2 ρ 0 m = 3 (4) 2H 0 2 Ω0 m . (6) H(z = 0) = H 0 = 100 hkms −1 Mpc −1 ,h=0.71 ± 0.03 (5) n =1, c 1 =2, c 2 = 1 (4) 2. Assuming a phantom expansion at z=0, ρ m H (z) = κ2 H(z = 2 0) H = H 0 = 100 hkms −1 Mpc −1 0 (1 + z)h(z) , → H (0) = κ2 ρ 0 m = 3 ,h=0.71 2H 0 ± 20.03 Ω0 m . (6) (5) H (0) 0 (7)
Cosmological evolution in viable f(R) Hu-Sawicki and Nojiri-Odintsov models 1. "CDM initial conditions, Cosmological evolution in viable modified gravity 8 EoS parameter w(z) 0.2 w HS z 0.996 w HS z w NO z 0.998 w NO z 0.4 1.000 0.6 1.002 1.004 0.8 1.006 1.0 1.0 0.5 0.0 0.5 1.0 z (a) The EoS parameter will cross the phantom barrier in the future. 1.008 1.00 0.95 0.90 0.85 0.80 z (b) K. Bamba, C. Q. Geng and C. C. Lee, JCAP 1011, 001 (2010) DSG, Class.Quant.Grav. 30 095008 (2013)
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