CCosmology in massive conformal gravity
F. F. Faria ∗ Centro de Ciˆencias da Natureza,Universidade Estadual do Piau´ı,64002-150 Teresina, PI, Brazil
Abstract
In this paper we find the cosmological solutions of the massiveconformal gravity field equations in the presence of matter fields. Inparticular, we show that the solution of negative curvature is in goodagreement with our universe.
PACS numbers: 04.20.-q, 04.20.Cv, 04.50.Kd* [email protected] a r X i v : . [ g r- q c ] M a r Introduction
The massive conformal gravity (MCG) is a conformally invariant theory ofgravity in which the gravitational action is the sum of the Weyl action withthe Einstein-Hilbert action conformally coupled to a scalar field. The MCGgravitational potential, which is composed by an attractive Newtonian po-tential and a repulsive Yukawa potential, describes well the rotation curves ofgalaxies without dark matter [1]. Furthermore, MCG is a consistent quantumtheory of gravity [2, 3, 4] and is free of the vDVZ discontinuity [5].Despite the promising results of MCG obtained so far, it is very impor-tant to test the theory with cosmological observations. The most acceptedcosmological model to explain the current accelerated expansion of the uni-verse is the ΛCDM model. This model is consistent with most of currentcosmological observations, but suffers from the cosmological constant prob-lem [6, 7]. Here we want to see if MCG explains the current acceleratinguniverse without this important problem.This paper is organized as follows. In Sec. 2 we describe the MCGcosmological field equations. In Sec. 3 we derive the energy-momentumtensor of a perfect fluid in MCG. In Sec. 4 we find the MCG cosmologicalsolutions. In Sec. 5 we compare the cosmological solutions of MCG withcosmological observations. Finally, in Sec. 6 we present our conclusions.
The total MCG action is given by [5] S = 1 k c (cid:90) d x √− g (cid:20) ϕ R + 6 ∂ µ ϕ∂ µ ϕ − αC αβµν C αβµν (cid:21) + 1 c (cid:90) d x L m , (1)where k = 32 πG/ c , α is a constant with dimension of length , ϕ is a scalarfield called dilaton, C αµβν = R αµβν + 12 (cid:0) δ αν R µβ − δ αβ R µν + g µβ R αν − g µν R αβ (cid:1) + 16 ( δ αβ g µν − δ αν g µβ ) R (2)is the Weyl tensor, R αµβν is the Riemann tensor, R µν = R αµαν is the Riccitensor, R = g µν R µν is the scalar curvature, and L m = L m ( g µν , Ψ) is theLagrangian density of the matter field Ψ. It is worth noting that the action11) is invariant under the conformal transformations˜ g µν = e θ ( x ) g µν , ˜ ϕ = e − θ ( x ) ϕ, ˜ L m = L m , (3)where θ ( x ) is an arbitrary function of the spacetime coordinates.The variation of (1) with respect to g µν and ϕ gives the MCG field equa-tions ϕ G µν + 6 ∂ µ ϕ∂ ν ϕ − g µν ∂ ρ ϕ∂ ρ ϕ + g µν ∇ ρ ∇ ρ ϕ − ∇ µ ∇ ν ϕ − αW µν = 12 k T µν , (4) (cid:18) ∇ µ ∇ µ − R (cid:19) ϕ = 0 , (5)where W µν = ∇ α ∇ β C µανβ − R αβ C µανβ (6)is the Bach tensor, G µν = R µν − g µν R (7)is the Einstein tensor, ∇ ρ ∇ ρ ϕ = 1 √− g ∂ ρ (cid:0) √− g∂ ρ ϕ (cid:1) (8)is the generally covariant d’Alembertian for a scalar field, and T µν = 2 √− g δ L m δg µν (9)is the matter energy-momentum tensor.Using the conformal invariance of the theory, we can impose the unitarygauge ϕ = ϕ = constant. In this case, the field equations (4) and (5) become ϕ G µν − αW µν = 12 k T µν , (10) R = 0 . (11)In addition, in the unitary gauge, the MCG line element ds = ( ϕ g µν ) dx µ dx ν reduces to ds = ( ϕ g µν ) dx µ dx ν . (12)The full cosmological content of MCG can be obtained from (10)-(12) withoutloss of generality. We will address this issue in section 4.2 Dynamical perfect fluid
We have seen in the previous section that the matter Lagrangian density hasto be conformally invariant in MCG. Most of the general coordinate invariantLagrangian densities of massless matter fields are conformally invariant too.In these cases we do not need to modify such Lagrangian densities. However,the same is not true for massive matter fields.Let us consider the real valued general coordinate invariant Lagrangiandensity for a fermion field ψ coupled to the metric g µν , which is given by L m = √− g (cid:20) i (cid:0) ψγ µ D µ ψ − D µ ψγ µ ψ (cid:1) − mψψ (cid:21) , (13)where m is the mass of the fermion field, ψ = ψ † γ is the adjoint fermionfield, D µ = ∂ µ + [ γ ν , ∂ µ γ ν ] / − [ γ ν , γ λ ]Γ λµν / λµν is the Levi-Civita connec-tion), and γ µ are the general relativistic Dirac matrices, which satisfy theanticommutation relation { γ µ , γ ν } = 2 g µν .With the help of the conformal transformations˜ g µν = e θ ( x ) g µν , ˜ ψ = e − θ ( x ) / ψ, ˜ γ µ = e − θ ( x ) γ µ , (14)its possible to verify that the massless term of the Lagrangian density (13) isconformally invariant whereas the massive term is not. However, consideringthe coupling of the fermion field ψ with both the metric g µν and a scalarHiggs field S , we can generalize (13) to the conformally invariant Lagrangiandensity [8] L m = −√− g (cid:34) S R + 12 ∂ µ S∂ µ S + 14! λS + i (cid:0) ψγ µ D µ ψ − D µ ψγ µ ψ (cid:1) + µSψψ (cid:35) , (15)where λ and µ are dimensionless coupling constants. Note that we have touse the additional conformal tansformation ˜ S = e − θ ( x ) S to show that (15) isconformally invariant.The variation of (15) with respect to S , ψ and ψ gives the field equations (cid:18) ∇ µ ∇ µ − R (cid:19) S + 16 λS + µψψ = 0 , (16) iγ µ D µ ψ + µSψ = 0 , (17)3 D µ ψγ µ − µSψ = 0 . (18)Substituting (15) into (9), and using (17) and (18), we obtain the energy-momentum tensor T µν = 16 g µν ∇ ρ S ∇ ρ S − ∇ µ S ∇ ν S + 13 S ∇ µ ∇ ν S − g µν S ∇ ρ ∇ ρ S + 16 S G µν + 14! g µν λS + T fµν , (19)where T fµν = i (cid:0) ψγ µ D ν ψ − D ν ψγ µ ψ + ψγ ν D µ ψ − D µ ψγ ν ψ (cid:1) (20)is the fermion energy-momentum tensor.By imposing the unitary gauge S = S , where S is a spontaneouslybroken constant expectation value for the Higgs field, and taking an inco-herent average of T fµν over all the modes propagating in a Robertson-Walkerbackground, we find that (19) reduces to the energy-momentum tensor of adynamical perfect fluid T µν = 16 S G µν + 14! g µν Λ + (cid:16) ρ + pc (cid:17) u µ u ν + g µν p, (21)where Λ = λS is an effective cosmological constant, ρ is the fluid density, p is the fluid pressure and u µ is the fluid four-velocity, which is normalized to u µ u µ = − c .Taking the trace of (21) and substituting into the trace of (10), we find − ϕ R = 12 k (cid:18) − S R + 16 Λ − c ρ + 3 p (cid:19) . (22)The additional use of (11) then gives the relationΛ = 6( c ρ − p ) . (23)Substituting this relation back into (21), we obtain T µν = 16 S G µν + T Tµν , (24)where T Tµν = (cid:16) ρ + pc (cid:17) u µ u ν + 14 g µν (cid:0) c ρ + p (cid:1) (25)is the traceless part of the kinematic perfect fluid energy-momentum tensor.Note that according to (24) the cosmological constant does not contributeto the dynamics of the MCG universe. This makes the theory free from thecosmological constant problem. 4 Cosmological solutions
By substituting (24) into (10), we find (cid:0) ϕ − ω (cid:1) G µν − α W µν = 16 πG c T Tµν , (26)where ω = 8 πG c S . (27)Then, using (11), (25), the FriedmannLemaˆıtreRobertsonWalker (FLRW)line element ds = − c dt + a ( t ) (cid:18) dr − Kr + r dθ + r sin θdφ (cid:19) , (28)and the fluid four-velocity u µ = ( ϕ c, , , aa = − πG eff c (cid:0) c ρ + p (cid:1) , (29) (cid:18) ˙ aa (cid:19) + Kc a = 4 πG eff c (cid:0) c ρ + p (cid:1) , (30)where the dot denotes d/dt , a = a ( t ) is the scale factor, K = -1, 0 or 1 is thespatial curvature, and G eff = Gϕ − ω (31)is an effective gravitational constant.The combination of (29) and (30) gives the energy continuity equation( ∇ µ T µν = 0) for the dynamical perfect fluid ddt (cid:2)(cid:0) c ρ + p (cid:1) a (cid:3) = 0 . (32)It follows from this equation that c ρ ( t ) + p ( t ) = (cid:0) c ρ + p (cid:1) (cid:16) a a (cid:17) , (33)where, from now on, the subscript 0 denotes values at the present time t .We can see from (33) that non-relativistic matter ( p = 0) and radiation(3 p = c ρ ) act the same way in the MCG universe.5e can write (30) in the usual formΩ + Ω K = 1 , (34)where Ω = 4 πG eff c H (cid:0) c ρ + p (cid:1) , Ω K = − Kc a H , (35)are dimensionless density parameters, and H = ˙ aa (36)is the Hubble constant. By using (33) in (30), we find (cid:18) ˙ aa H (cid:19) = Ω (cid:16) a a (cid:17) + Ω K, , (37)where Ω = 4 πG eff c H (cid:0) c ρ + p (cid:1) , Ω K, = − Kc a H . (38)The combination of (34) and (37) gives dt = dxH (1 − Ω + Ω x − ) / , (39)where x = a/a . It follows from (39) that the time at which light emittedfrom a cosmological source reaches the earth with redshift z is given by t = 1 H (cid:90) / (1+ z )0 dx (1 − Ω + Ω x − ) / , (40)where we considered that the zero of time corresponds to an infinite redshift,and 1 + z = a a . (41)By considering the redshift equal to zero in (40), we find the present ageof the MCG universe t = (cid:18) √ Ω − − (cid:19) H . (42)It follows from this equation that0 < t < H (43)6or a closed universe (Ω > t = 12 H (44)for a flat universe (Ω = 1), and12 H < t < H (45)for an open universe (Ω < H and t . Assuming that the MCG universe is open ( K = − a = a H Ω a + c , (46)whose solution is given by a ( t ) = a H √ Ω c (cid:34)(cid:18) c ta H √ Ω (cid:19) − (cid:35) / . (47)Substituting (47) into the deceleration parameter q = − ¨ aa ˙ a , (48)we find q ( t ) = a H Ω ( a H √ Ω + c t ) . (49)According to (47) and (49) the MCG universe begins with a big bang at t = 0 and continues to expand accelerated forever, becoming flat as t → ∞ .On scales above the electroweak scale, the Higgs field grows until ω acquires avalue close to ϕ near the Planck scale, which gives G eff → ∞ in the vicinityof the big bang. This behavior plays an essential role in the evolution of theearly MCG universe, including the resolution of the big bang singularity.7 Cosmological experimental tests
In order to compare the cosmological solution of MCG with cosmologicalobservations, we must find the luminosity distance d L ( z ) = a r ( z )(1 + z ) , (50)where r ( z ) is the radial distance of a cosmological light source that is observednow on earth with redshift z .In an open universe such as the MCG universe, the radial distance isdetermined by the equation of the radial worldline of a light ray (cid:90) r ( z )0 dr √ r = (cid:90) t t ( z ) c dta ( t ) . (51)By using (39) in (51), integrating both sides, substituting the result into (50),and making some algebra, we find H d L c = (cid:0) √ − Ω (cid:1) (1 + z ) − (cid:0) √ − Ω + √ z + Ω z (cid:1) (cid:0) √ − Ω + √ z + Ω z (cid:1) (cid:0) √ − Ω (cid:1) (cid:0) √ − Ω (cid:1) . (52)In order to estimate the values of H and Ω , we use the Union2.1 compi-lation with 580 SNIa data [9, 10, 11]. The procedure consists in compare, foreach SNIa at redshift z i , the observed distance modulus µ obs ( z i ) ( ≡ m obs − M ,where M is the absolute magnitude) with the theoretical distance modulus µ th ( z ) defined as µ th ( z ) = 5 log d L ( z ) + 25 , (53)where d L is measured in Mpc. The best-fit values of H and Ω are deter-mined by an iterative minimization of the function χ (Ω , H ) = (cid:88) i =1 [ µ obs ( z i ) − µ th ( z i )] σ i , (54)where σ i is the uncertainty on µ obs ( z i ). Although the χ procedure has a degeneracy between the Hubble constant and theabsolute magnitude, it is useful to estimate the values of the cosmological parameters witha good confidence level. In order to find more accurate estimates, we must use model-independent procedures such as the Bayesian approach. However, due to the greatercomplexity of these procedures, we will leave them for future works. H = 67 . ± .
29 km s − Mpc − , (55)Ω = 10 − , (56)with χ / dof = 0 .
99, where dof is the logogram of degree of freedom.Figure 1: Hubble diagram for the Union2.1 compilation with 580 SNIa data.The red line represents the best-fitted MCG model.Noting that the MCG gravitational potential describes well the rotationcurves of galaxies without needing to appeal to dark matter, we can con-sider that the current MCG universe is dominated by baryonic matter withestimated mass density ρ ≈ . × − g cm − and pressure p ≈
0. Us-ing these values, (55), and (56) in (38), we find the effective gravitationalconstant G eff ≈ . × − m kg − s − , (57)9nd the current scale factor a ≈ . × . (58)We can see from (57) that the effective gravitational constant decreases thecontribution of matter to the MCG cosmology.The substitution of (55) and (56) into (42) gives the current age of theMCG universe t = 14 . ± .
06 Gyr , (59)which is consistent with the 14 Gyr estimated from old globular clusters [12].Further analysis is needed to see if the MCG universe accommodates the ageof the old quasar APM 08279+5255 . We have shown in this paper that the negative curvature cosmological so-lution of MCG is compatible with the current accelerating universe withoutneeding to appeal to dark matter or dark energy. Additionally, the MCGcosmology is free of the cosmological constant and the big bang singularityproblems. The early MCG cosmology, in particular CMB production and nu-cleosynthesis, and the compatibility of the current age of the MCG universewith the age of old high redshift objects will be investigated in the future.
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