aa r X i v : . [ g r- q c ] J a n November 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
PALATINI f ( R ) COSMOLOGY SEOKCHEON LEE ∗ Institute of Physics, Academia Sinica,Taipei, Nankang, 11529, Republic of [email protected]
Received (Day Month Year)Revised (Day Month Year)We investigate the modified gravity theories in terms of the effective dark energy models.We compare the cosmic expansion history and the linear growth in different models.We also study the evolution of linear cosmological perturbations in modified theories ofgravity assuming the Palatini formalism. We find the stability of the superhorizon metricevolution depends on models. We also study the matter density fluctuation in the generalgauge and show the differential equations in super and sub-horizon scales.
Keywords : Modified Gravity; Palatini Formalism.PACS Nos.: include PACS Nos.
1. Introduction
Type Ia supernova distance-redshift measurement shows that the present expansionof the universe is accelerating 1. To explain this phenomena we need to introducea homogeneous component of energy with a negative pressure, dubbed dark energy2, or a modification of gravity 3 , , , ∗ Institute of Physics, Academia Sinica, Taipei, Taiwan 11529, R.O.C.1 ovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Seokcheon Lee
In the next section we introduce the parametrization of modified gravities as aneffective dark energy. We review the Palatini f(R) gravity in section 3. In section 4,we investigate the linear perturbation of f(R) gravity. We also derive the stabilityequation of metric fluctuations in the high curvature limit and show the stability ina specific model. We show the evolutions of the metric fluctuation and the densityfluctuation in the superhorizon and the subhorizon scales in section 5. We reach ourconclusions in section 6.
2. Modified Gravity as An Effective Dark Energy
In general, the modified gravity can be expressed in the following form H − δH = 8 πG ρ m , (1)where δH represents the modification to the Friedmann equation of general relativ-ity and ρ m is the matter density. We can rewrite this equation as the effective darkenergy term H = 8 πG ρ m + ρ DE ! , (2)where we define the effective dark energy density as 8 πGρ DE = 3 δH . From thisequation (2) we can find the effective equation of state of dark energy ω DE = − − d ln δHd ln a . (3)We can also express the expansion history of the universe in very simple way forthe specific case. If δH is proportional to H (i.e. δH = αH ), then we can find theHubble parameter H ( z ) HH = 12 αH + r(cid:16) αH (cid:17) + 4Ω (0)m (1 + z ) ! . (4) Table 1. Comparison of δH and ω DE in BD, DGP, metric formalism f ( R ) theory, and Palatini formalism f ( R ) models. F ( R ) = ∂f ( R ) ∂R and F is the present value of F ( R ). δH ω DE BD ω BD φ φ − H ˙ φφ − ¨ φφ − H ˙ φφ + ω BD ˙ φ φ ω BD2 ˙ φ φ − H ˙ φφ DGP Hr − m Metric f ( R ) F “ ( F R − f ) − H ˙ F + 3 H ( F − F ) ” − F − H ˙ F − H ( F − F ) F R − f − H ˙ F +6 H ( F − F ) Palatini f ( R ) F “ ( F R − f ) + ¨ F + H ˙ F − F ˙ F F + 3 H ( F − F ) ” − F − H ˙ F − F ˙ F F − H ( F − F ) F R − f +3 ¨ F +3 H ˙ F − F ˙ F F +6 H ( F − F ) In table 1, we show the expressions of the modification term to the Friedmannequation ( δH ) and the effective equation of state of dark energy ( ω DE ) for theovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Palatini f ( R ) Cosmology (Paper’s Title) different modified gravity models. BD represents Brans-Dicke theory, DGP standsfor the brane model of Dvali, Gabadadze, and Porrati 4, metric f ( R ) means the f ( R ) theory in the metric formalism 5, and Palatini f ( R ) does that in the Palatiniformalism 6. In the Palatini formalism we can relate the functions f ( R ) and F ( R )to the matter energy density ρ m . Table 2. Comparisons of H ( z ) and evolution of the linear perturbation δ m with the effective gravitational con-stant G eff in each model. In metric and Palatini formalism the perturbation calculations are held for subhorizonscale (i.e. ka > H ). Where Q = − F ,R F k a and F ′ ( T ) = ∂F ( T ) ∂T . H ( z ) H δ m G eff BD r δHH + φ φ Ω (0)m (1 + z ) ¨ δ + 2 H ˙ δ − πG eff ρδ = 0 ω BD +42 ω BD +3 ! φ DGP r H + r“ r H ” + 4Ω (0)m (1 + z ) ! ¨ δ + 2 H ˙ δ − πG eff ρδ = 0 G “ r Hω DE ] ” Metric f ( R ) r δHH + Ω (0)m (1 + z ) ¨ δ + 2 H ˙ δ − πG eff ρδ = 0 − Q )2 − Q GF
Palatini f ( R ) r δHH + Ω (0)m (1 + z ) ¨ δ + H ˙ δ − πG eff ρδ = 0 − π F ′ F +3 F ′ ρ k a We show the expansion rate H ( z ) /H where H is the present value of theHubble parameter, the evolution equation for the energy density perturbation δ m ,and the effective gravitational constant G eff for different models in table 2. Theevolution equations of δ m for the f ( R ) models are suitable for the subhorizon scale.The scale dependence on G eff is the one of the major differences compared to thegeneral relativity.
3. Palatini f(R) Gravity
We consider a modification to the Einstein-Hilbert action assuming the Palatiniformalism, where the metric g µν and the torsionless connection ˆΓ αµν are independentquantities and the matter action depends only on metric S = Z d x √− g " κ f (cid:16) ˆ R ( g µν , ˆΓ αµν ) (cid:17) + L m ( g µν , ψ ) , (5)where ψ are matter fields. Then the Ricci tensor is defined solely by the connectionˆ R µν = ˆΓ αµν, α − ˆΓ αµα, ν + ˆΓ ααβ ˆΓ βµν − ˆΓ αµβ ˆΓ βαν , (6)whereas the scalar curvature is given by ˆ R = g µν ˆ R µν . Then we can find the Riccitensor and the scalar curvature by using the metric relationˆ R µν = R µν + 32 1 F ∇ µ F ∇ ν F − F ∇ µ ∇ ν F − g µν F (cid:3) F , (7)ˆ R = R − F (cid:3) F + 32 1 F ( ∂F ) . (8)ovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Seokcheon Lee
We can derive the field equation of f(R) gravity in the Palatini formalism from theabove action (5) F ( ˆ R ) ˆ R µν − g µν f ( ˆ R ) = 8 πGT µν , (9)where F ( ˆ R ) = ∂f ( ˆ R ) /∂ ˆ R and the matter energy momentum tensor is given as usualform T µν = − √− g δ ( √− g L m ) δg µν . From the above equations we can get the generalizedEinstein equation G µν = 8 πGT µν + (1 − F ) R µν −
32 1 F ∇ µ F ∇ ν F + ∇ µ ∇ ν F + 12 ( f − R ) g µν + 12 g µν (cid:3) F . (10)From the above equation we can derive the modified Friedmann equations3 H = 8 πG F ρ + f F − H ˙ FF −
34 ˙ F F , (11) − H = 8 πGF ( ρ + p ) + ¨ FF − H ˙ FF −
32 ˙ F F , (12)where dot denote derivatives with respect to t .
4. Linear Perturbation and Stability in Palatini f(R) Gravity
The line element in the conformal Newtonian gauage is given by ds = a ( τ ) " − (cid:16) τ, ~x ) (cid:17) dτ + (cid:16) − τ, ~x ) (cid:17) dx i dx i . (13)The main modifications for viable models with stable high curvature limits happenwell during the matter dominated epoch and we can take the components of theenergy momentum tensor as T = − ρ (1 + δ ) , T i = ρ∂ i q , T ij = 0 . (14)From the equation (10), we can find the perturbed Einstein equation F δG µν = κ δT µν − R µν δF − δ ( ∇ µ F ∇ ν F ) F + 32 ∇ µ F ∇ ν FF δF + δ ( ∇ µ ∇ ν F )+ (cid:3) FF δF + 34 δ ( ∂F ) F −
32 ( ∂F ) F δF − δ (cid:3) F ! δ µν , (15)where we use δf ( ˆ R ) = F ( ˆ R ) δ ˆ R . If we consider the ij -component of the perturbedequation, then we can find Φ − Ψ = δFF , (16)ovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee
Palatini f ( R ) Cosmology (Paper’s Title) where we assume the null anisotropic stress. If we use the above equation (16), thenwe can express the other components of the perturbed Einstein equation (15)3 H " Φ ′ + Ψ ′ + 12 F ′ F (Φ ′ + Ψ ′ ) + (cid:16) F ′′ F − F ′ F + 12 H ′ H F ′ F + 12 F ′ F + H ′ H + 1 (cid:17) Φ+ (cid:16) − F ′′ F + F ′ F − H ′ H F ′ F + 32 F ′ F − H ′ H + 1 (cid:17) Ψ + k a (Φ + Ψ) = − κ ρF δ , (17) H " Φ ′ + Ψ ′ + Φ + Ψ + 12 F ′ F (Φ + Ψ) = − κ ρF q , (18)3 H " Φ ′′ + Ψ ′′ + (cid:16) H ′ H (cid:17) Φ ′ + (cid:16) F ′ F + 4 + H ′ H (cid:17) Ψ ′ + (cid:16) − F ′′ F − (2 + H ′ H ) F ′ F + H ′ H + 3 (cid:17) Φ + (cid:16) F ′′ F + (6 + 3 H ′ H ) F ′ F + 3 H ′ H + 3 (cid:17) Ψ = 0 . (19)where primes denote derivatives with respect to ln a . To capture the metric evolu-tion, let us introduce two parameters as in the reference 8: θ the deviation from ζ conservation and ǫ the deviation from the superhorizon metric evolution ζ ′ = Φ ′ + Ψ − H ′ q = − H ′ H kaH ! Bθ , (20)Φ ′′ + Ψ ′ − H ′′ H ′ Φ ′ + H ′ H − H ′′ H ′ ! Ψ = − kaH ! Bǫ , (21)where we define the dimensionless quantity B = F ′ F HH ′ . (22)ovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Seokcheon Lee
From the above equations, we can find the expression for θ and ǫ , H ′ H kaH ! Bθ = 12 " B ′ B + 32 H ′ H B + H ′′ H ′ + 4 (Φ − Ψ) + 12 " − H ′ H B ′ + (cid:16) H ′ H − H ′′ H (cid:17) B + 12 H ′ H B Hq (23) kaH ! Bǫ = − " B ′ B + (cid:16) H ′′ H ′ + H ′ H + 9 (cid:17) B ′ B + H ′ H B ′ + H ′ H B + (cid:16) H ′′ H + 32 H ′ H +6 H ′ H (cid:17) B + H ′′ H + 3 H ′′ H ′ + 13 H ′′ H ′ + H ′ H + 9 + 2 (cid:16) H ′′ H ′ + H ′ H + 3 (cid:17) B (Φ − Ψ) + " H ′ H B ′ + 12 H ′ H B + (cid:16) H ′′ H + 32 H ′ H (cid:17) B Ψ − " H ′ H B + H ′′ H ′ + H ′ H + 3 κ ρF H q (24)From equation (23), we can recover the conservation of Newtonian gauge whenΦ = Ψ and F is a constant.In addition to these equations, we can find very useful equation from the struc-ture equation (9) δFF = − F ′ F δ = Φ − Ψ . (25)From this equation we can find the evolution equation of matter density fluctuation δ ′′ + B ′ B + H ′ H B + 2 H ′′ H ′ − H ′ H + 3 ! δ ′ + H ′ H B ′ − B ′ B − h H ′′ H ′ + 2 H ′ H + 4 i B ′ B + H ′ H B + h H ′′ H − H ′ H + 5 H ′ H i B + h − H ′′ H − H ′′ H ′ − H ′′ H ′ + H ′ H − H ′ H + 6 i − h H ′′ H ′ + H ′ H + 3 i B ! δ = − ′ . (26)Compared with previous works 9, we do not specify the gauge of matter density tosolve the matter density fluctuation.Unstable metric fluctuations can create order unity effects that invalidate thebackground expansion history. We can derive the evolution equation of the deviationparameter 10. If we differentiate the equation (24) and consider the evolution in theovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Palatini f ( R ) Cosmology (Paper’s Title) superhorizon scale, then we have ǫ ′′ + B ′ B + H ′ H B + H ′′ H ′ − H ′ H − ! ǫ ′ + − B ′ B + 2 H ′ H B ′ − h H ′′ H ′ + 4 H ′ H + 9 i B ′ B + 12 H ′ H B + h H ′′ H ′ − H ′ H + 32 H ′ H i B + Q ′ + h − H ′′ H ′ − H ′ H + 1 i Q + 4 H ′ H + 6 H ′ H + 7 − QB ! ǫ = 1 B F (Ψ , Φ , Hq ) , (27)where we use equations (19) and (21) and F (Ψ , Φ) is the source function for thedeviation ǫ and define Q as Q = H ′′ H ′ + H ′ H + 3 . (28)The above equation is different from that of the metric formalism 8. The stability of ǫ depends on the sign of the coefficient of the term proportional to ǫ . In the metricformalism ǫ is stable as long as B >
0. However, the stability is complicate andneed to be checked for each model in the Palatini formalism.
A particular example : f ( ˆ R ) = β ˆ R n We demonstrate the general consideration of the previous subsection with a specificchoice for the nonlinear Lagrangian, f ( ˆ R ) = β ˆ R n , where n = 0 , ,
3. The back-ground is simply described by a constant effective equation of state in this model.The Hubble parameter scales as H ∼ a − /n . Then it is easy to write it with itsderivatives in terms of ln aH ′ H = − n , H ′′ H = − n ! . (29)Here the scalar curvature is ˆ R = 3(3 − n ) H / (2 n ). From this fact, we can also findthe derivatives of F with respect to ln aF ′ F = F ′′ F ′ = 3(1 − n ) n , F ′′ F = F ′′′ F ′ = − n ) n ! . (30)If we use above equations (29) and (30) into (24), then we find that the deviationfrom the superhorizon metric evolution is null, ǫ = 0.
5. Evolutions of Metric and Matter Density5.1.
Superhorizon evolution
We consider the metric evolution in superhorizon sized, k/ ( aH ) ≪
1. In this case,the anisotropy relation of the equation (23) becomesΦ − Ψ ≃ (cid:16) B + A (cid:17) H ′ q , (31)ovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Seokcheon Lee where A is given by A = − B (cid:16) B H ′ H + 5 (cid:17)(cid:16) B ′ B + H ′ H B + H ′′ H ′ + 4 (cid:17) . (32)From the above equations we can find the superhorizon evolution equation of Φ and δ Φ ′′ + B ′ B + 2 H ′ H B + H ′′ H ′ − H ′ H + 4 − C ! Φ ′ + B ′ B + H ′ H B + H ′′ H ′ + 3 − C ! Φ ≃ , BB + A ! δ ′′ + B ′ B + H ′ H B + 2 B ′ A − BA ′ ( B + A ) − H ′ H BB + A + 2 H ′′ H ′ − H ′ H + 3 ! δ ′ + H ′ H B ′ − B ′ B − h H ′′ H ′ + 2 H ′ H + 4 i B ′ B + H ′ H B + h H ′′ H − H ′ H + 5 H ′ H i BB ′′ A − BA ′′ ( B + A ) − B ′ A − BA ′ )( B + A ) ( B ′ + A ′ )( B + A ) − H ′ H B ′ A − BA ′ ( B + A ) − H ′ H ! ′ BB + A + h − H ′′ H − H ′′ H ′ − H ′′ H ′ + H ′ H − H ′ H + 6 i − h H ′′ H ′ + H ′ H + 3 i B ! δ = 0 , (33)where C is defined as C = 1 B + A + 1 " B ′ B + 2 H ′ H B + 2 H ′′ H ′ − H ′ H + 3 . (34) Superhorizon evolution in a particular example
Now we can check the evolution equations in the previous subsection in a particularcase, f ( ˆ R ) ∼ ˆ R n . In this case, we can simplify the following quantities B = 2( n − , A = − n −
1) = − B , C = 32 n = − H ′ H . (35)From this, we can also simplify the evolution equations (33) and (33)Φ ′′ + 9 − n n Φ ′ = 0 , (36) δ ′ = 0 . (37)The Newtonian potential Φ = constant is a solution to the equation. Also the matterdensity fluctuation has the same for as general relativity, δ = constant. Subhorizon evolution
For subhorizon scales where k/aH ≫ k (Φ + Ψ) ≃ − κ a ρF δ . (38)ovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Palatini f ( R ) Cosmology (Paper’s Title) If we use equations (25) and (38), then we have3Ψ ≃ − κ ρF H a H k + F ′ F ! δ ≃ F ′ F δ ≃ − . (39)We can find the evolution equations of Φ and δ Φ ′′ + B ′ B + 52 H ′ H B + H ′′ H ′ + H ′ H + 6 ! Φ ′ − B ′ B + h H ′′ H ′ − i B ′ B − H ′ H B ′ − H ′ H B + h H ′′ H − H ′′ H ′ − H ′ H i B + h H ′′ H ′ − H ′′ H ′ + 10 H ′ H + 6 i + h H ′′ H ′ + 2 H ′ H + 6 i B ! Φ ≃ . (40) δ ′′ + B ′ B + 32 H ′ H B + 2 H ′′ H ′ − H ′ H + 3 ! δ ′ + H ′ H B ′ − B ′ B − h H ′′ H ′ + 2 H ′ H + 4 i B ′ B + H ′ H B + h H ′′ H − H ′ H + 5 H ′ H i B + h − H ′′ H − H ′′ H ′ − H ′′ H ′ + H ′ H − H ′ H + 6 i − h H ′′ H ′ + H ′ H + 3 i B ! δ = 0 . (41) Subhorizon evolution in a particular example
We can use the previous relation (35) into the evolution equations (40) and (41)Φ ′′ + 3(3 − n )2 n Φ ′ + 3(14 n + 19 n − n Φ = 0 (42) δ ′′ + 3(2 − n )2 n δ ′ = 0 (43)The subhorizon scale evolutions of Φ and δ show different behaviors from those ofgeneral relativity as expected 11.
6. Conclusions
We investigate the different modified gravity models as an effective dark energymodels. We can parameterize the effective equation of state of dark energy in termsof modified gravities. We show the evolution of energy density fluctuation comparedto that of general relativity.We have analyzed the stability of metric fluctuations by checking the cosmologi-cal evolution of linear perturbations in Palatini f(R) gravity. We have also consideredthe matter density fluctuation in the Newtonian gauge.We have shown that the stability of metric fluctuations in the high redshiftlimit of high curvature is not simply expressed. We need to check each model forthe stability. However, we have found that the deviation from the superhorizonovember 1, 2018 4:49 WSPC/INSTRUCTION FILE SeokcheonLee Seokcheon Lee metric evolution is null for a specific choice of the nonlinear Einstein-Hilbert action, f ( ˆ R ) ∼ ˆ R n and stability of this model is guaranteed.We have investigated the evolution equations of Newtonian potential and matterdensity contrast in super and sub-horizon scales. In the specific model, superhorizonevolutions of Newtonian potential and matter density fluctuation are same to thoseof general relativity. However, subhorizon evolutions show the different behaviorsfrom the general relativity case. This will give us the method to probe the possibilityof f ( R ) theory. Acknowledgments
We thank CosPA2007 organizing committee for their hospitality and for organizingsuch a nice meeting.
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