Phantom crossing in viable f(R) theories
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International Journal of Modern Physics Dc (cid:13)
World Scientific Publishing Company
Phantom crossing in viable f ( R ) theories Kazuharu Bamba a ∗ , Chao-Qiang Geng a,b and Chung-Chi Lee a † a Department of Physics, National Tsing Hua University, Hsinchu, Taiwan, R.O.C. b National Center for Theoretical Sciences, Hsinchu, Taiwan, R.O.C.E-Mail addresses: [email protected], [email protected], [email protected]
We review the equation of state for dark energy in modified gravity theories. Inparticular, we summarize the generic feature of the phantom divide crossing in the pastand future in viable f ( R ) gravity models. Keywords : Modified theories of gravity; Equation of state; Dark energy; Cosmology.
To understand the late time acceleration universe 1, one of the interestingpossibilities is to consider a modified gravitational theory, such as f ( R ) grav-ity 2. To build up a viable f ( R ) gravity model, one needs to satisfy the fol-lowing conditions: (a) positivity of the effective gravitational coupling, (b) sta-bility of cosmological perturbations 3, (c) asymptotic behavior to the standardΛ-Cold-Dark-Matter (ΛCDM) model in the large curvature regime, (d) stabil-ity of the late-time de Sitter point 4, (e) constraints from the equivalence prin-ciple, and (f) solar-system constraints 5. Several viable models have been con-structed in the literature, such as the following popular ones:6 , , , , , , , f ( R ) Constant parameters(i) Hu-Sawicki R − c R HS ( R/R HS ) p c ( R/R HS ) p +1 c , c , p ( > R HS ( > R + λR S (cid:20)(cid:16) R R (cid:17) − n − (cid:21) λ ( > n ( > R S (iii) Tsujikawa R − µR T tanh (cid:16) RR T (cid:17) µ ( > R T ( > R − βR E (cid:0) − e − R/R E (cid:1) β , R E Recently, the cosmological observational data 14 seems to indicate the crossing ofthe phantom divide w DE = − ∗ Present address:Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, NagoyaUniversity, Nagoya 464-8602, Japan. † Talk presented by C.C. Lee at the 2nd International Workshop on Dark Matter, Dark Energyand Matter-Antimatter Asymmetry, Hsinchu, Taiwan, 5-6 Nov 2010.1 ctober 10, 2018 11:46 WSPC/INSTRUCTION FILE CCLee Bamba, Geng and Lee noticeable one is to use a phantom field with a negative kinetic energy term 15.Clearly, it surfers a serious problem as it is not stable at the quantum level. On theother hand, the crossing of the phantom divide can also be realized in the aboveviable f ( R ) models 6 , , , ,
16 without violating any stability conditions. This isprobably the most peculiar character of the modified gravitational models. Other f ( R ) gravity models with realizing a crossing 17 as well as multiple crossings 18 ofthe phantom boundary have also been examined.In this talk, we would like to review equation of state in f ( R ) gravity. In par-ticular, we show that the viable f ( R ) models generally exhibit the crossings of thephantom divide in the past as well as future 12 , f ( R ) gravity with matter is given by I = Z d x √− g f ( R )2 κ + I matter ( g µν , Υ matter ) , (1)where g is the determinant of the metric tensor g µν , I matter is the action of matterwhich is assumed to be minimally coupled to gravity, i.e., the action I is writtenin the Jordan frame, and Υ matter denotes matter fields. Here, we use the standardmetric formalism. By taking the variation of the action in Eq. (1) with respect to g µν , one obtains 2 F G µν = κ T (matter) µν − g µν ( F R − f ) + ∇ µ ∇ ν F − g µν (cid:3) F , (2)where G µν = R µν − (1 / g µν R is the Einstein tensor, F ( R ) ≡ df ( R ) /dR , ∇ µ is thecovariant derivative operator associated with g µν , (cid:3) ≡ g µν ∇ µ ∇ ν is the covariantd’Alembertian for a scalar field, and T (matter) µν is the contribution to the energy-momentum tensor from all perfect fluids of matter.From Eq. (2), we obtain the following gravitational field equations:3 F H = κ ρ M + 12 ( F R − f ) − H ˙ F , − F ˙ H = κ ( ρ M + P M ) + ¨ F − H ˙ F , (3)where H = ˙ a/a is the Hubble parameter, the dot denotes the time derivative of ∂/∂t , and ρ M and P M are the energy density and pressure of all perfect fluids ofmatter, respectively.The equation of state for dark energy is given by w DE ≡ P DE /ρ DE , (4)where ρ DE = 1 κ (cid:20)
12 (
F R − f ) − H ˙ F + 3 (1 − F ) H (cid:21) ,P DE = 1 κ (cid:20) −
12 (
F R − f ) + ¨ F + 2 H ˙ F − (1 − F ) (cid:16) H + 3 H (cid:17)(cid:21) . (5)ctober 10, 2018 11:46 WSPC/INSTRUCTION FILE CCLee Phantom crossing in viable f ( R ) theories In Figs. 1, 2 and 3, we depict the evolution of w DE , future evolutions of 1 + w DE ,and ˜ H ≡ ¯ H − ¯ H f with ¯ H ≡ H/H and ¯ H f ≡ H ( z = − /H , as functions of theredshift z ≡ /a − p = 1, c = 2 and c = 1, (ii)Starobinsky model for n = 2 and λ = 1 .
5, (iii) Tsujikawa model for µ = 1 and(iv) the exponential gravity model for β = 1 .
8, respectively, and the subscript ‘f’denotes the value at the final stage z = −
1. Note that the present time is z = 0and the future is − ≤ z <
0. The parameters used for each model in Figs. 1–3are the viable ones 19 ,
20. Several remarks are as follows: (a) the qualitative resultsdo not strongly depend on the values of the parameters in each model; (b) we havestudied the Appleby-Battye model 21, which is also a viable f ( R ) model, and wehave found that the numerical results are similar to those in the Starobinsky modelof (ii) as expected.We note that the present values of w DE ( z = 0) are -0.92, -0.97, -0.92 and -0.93 Fig. 1. Evolutions of the equation of state w DE as functions of the redshift z in (i) Hu-Sawickimodel for p = 1, c = 2 and c = 1, (ii) Starobinsky model for n = 2 and λ = 1 .
5, (iii) Tsujikawamodel for µ = 1 and (iv) the exponential gravity model for β = 1 .
8, respectively, where the thinsolid lines show w DE = − ctober 10, 2018 11:46 WSPC/INSTRUCTION FILE CCLee Bamba, Geng and Lee
Fig. 2. Future evolutions of 1 + w DE as functions of the redshift z . Legend is the same as Fig. 1. for the models of (i)–(iv), respectively. These values satisfy the present observa-tional constraints 22. Moreover, a dimensionless quantity H / (cid:16) κ ρ (0)m / (cid:17) can bedetermined through the numerical calculations, where ρ (0)m is the energy densityof non-relativistic matter at the present time. If we use the observational data onthe current density parameter of non-relativistic matter Ω (0)m ≡ ρ (0)m /ρ (0)crit = 0 . ρ (0)crit = 3 H /κ
22, we find that the present value of the Hubble parameter H = H ( z = 0) is H = 71km / s / Mpc 22 for all the models of (i)–(iv). Further-more, ¯ H f = 0.80 , 0.85 , 0.78 and 0.81 for the models of (i)–(iv), respectively, where¯ H f ≡ H ( z = − /H .It is clear from Figs. 1–3 that in the future ( − ≤ z . − . f ( R )models. By writing the first future crossing of the phantom divide and the firstsign change of ˙ H from negative to positive as z = z cross and z = z p , respectively,we find that ( z cross , z p ) α = ( − . , − . i , ( − . , − . ii , ( − . , − . iii and( − . , − . iv , where the subscript α represents the α th viable model. The valuesctober 10, 2018 11:46 WSPC/INSTRUCTION FILE CCLee Phantom crossing in viable f ( R ) theories H ~ H ~ H ~ H ~ Fig. 3. Future evolutions of ˜ H ≡ ¯ H − ¯ H f with ¯ H ≡ H/H and ¯ H f ≡ H ( z = − /H as functionsof the redshift z . Legend is the same as Fig. 1. of the ratio Ξ ≡ Ω m / Ω DE at z = z cross and z = z p are (Ξ( z = z cross ) , Ξ( z = z p )) α =(5 . × − , . × − ) i , (1 . × − , . × − ) ii , (4 . × − , . × − ) iii and(6 . × − , . × − ) iv , where Ω DE ≡ ρ DE /ρ (0)crit and Ω m ≡ ρ m /ρ (0)crit are the den-sity parameters of dark energy and non-relativistic matter (cold dark matter andbaryon), respectively. As z decreases ( − ≤ z . − . m / Ω DE . − ). As a result, onehas w DE ≈ w eff ≡ − − H/ (cid:0) H (cid:1) = P tot /ρ tot , where w eff is the effective equationof state for the universe, and ρ tot ≡ ρ DE + ρ m + ρ r and P tot ≡ P DE + P r are the totalenergy density and pressure of the universe, respectively. Here, ρ m(r) and P r are theenergy density of non-relativistic matter (radiation) and the pressure of radiation,respectively. The physical reason why the crossing of the phantom divide appearsin the farther future ( − ≤ z . − .
90) is that the sign of ˙ H changes from negativeto positive due to the dominance of dark energy over non-relativistic matter. As w DE ≈ w eff in the farther future, w DE oscillates around the phantom divide line w DE = − H changes and consequently multiple crossings canctober 10, 2018 11:46 WSPC/INSTRUCTION FILE CCLee Bamba, Geng and Lee be realized.Finally, we mention that in our numerical calculations, we have taken the initialconditions of z = 8 .
0, 8 .
0, 3 . . z = z , respec-tively, so that RF ′ ( z = z ) ∼ − with F ′ = dF/dR , to ensure that they can beall close enough to the ΛCDM model with RF ′ = 0.In this talk, we have explored the past and future evolutions of w DE in the viable f ( R ) gravity models and explicitly shown that the crossings of the phantom divideare the generic feature in these models. We have demonstrated that in the futurethe sign of ˙ H changes from negative to positive due to the dominance of dark energyover non-relativistic matter. This is a common physical phenomena to the existingviable f ( R ) models and thus it is one of the peculiar properties of f ( R ) gravitymodels characterizing the deviation from the ΛCDM model. Acknowledgments
The work is supported in part by the National Science Council of R.O.C. underGrant
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