gr-qc/9801028 CRETE-97-17 ENS-97/67 UFIFT-HEP-97-27 PERTURBATIVE QUANTUM GRAVITY AND NEWTON’S LAW ON A FLAT ROBERTSON-WALKER BACKGROUND 8 9 9 1 J. Iliopoulos n a J Laboratoire de Physique Th´eorique, Ecole Normale Sup´erieure 9 24, rue Lhomond, F-75231 Paris Cedex 05, FRANCE [email protected] 1 v 8 2 T. N. Tomaras , N. C. Tsamis 0 1 0 Department of Physics, University of Crete 8 and Research Center of Crete 9 P. O. Box 2208, GR-71003 Heraklion, GREECE / c [email protected] , [email protected] q - r g : R. P. Woodard v i X Department of Physics, University of Florida r Gainesville, FL 32611, USA a [email protected]fl.edu ABSTRACT We derive the Feynman rules for the graviton in the presence of a flat Robertson-Walker background and give explicit expressions for the propagator in the physically interesting cases of inflation, radiation domination, and matter domination. The aforementioned background is generated by a scalar field source which should be taken to be dynamical. As an elementary application,we compute the correctionsto the Newtoniangravitationalforceinthe presentmatter dominated era and conclude – as expected – that they are negligible except for the largest scales. 1. Introduction Of the non-trivial spacetime backgrounds, those of particular interest in cosmology are spatially homogeneous and isotropic. This is a direct consequence of the cosmological principle which states that there is no preferred position in the universe. It can be shown that the most general background satisfying the above requirements is the Robertson- Walker one [1]: 2 2 2 2 1 2 2 2 ds = dt +a (t) (1 kr ) dr +r dΩ . (1.1) − 2 − − n o b It is characterized by a function a(t) and a constant k which can be chosen to take on the values +1, 0, or 1. The function a(t) – known as the scale factor – is a measure of the − “radius” of the universe, while the constant k determines the spatial curvature. By far the most natural explanation of the observed homogeneity and isotropy in the present universe is the assumption of an inflationary era in the evolution of the universe. Duringthatera, thephysical distancebetweenobservers atrestonfixedspatialcoordinates increases superluminally. Accordingly, the observed universe was once so small that causal processes could have established an initial thermal equilibrium across it. This accounts for the fact that the cosmic microwave background radiation from different regions of the sky is seen to be in thermal equilibrium to within one part in 105 [2]. Since any decent amount of inflation * redshifts the spatial curvature to insignificance, we can restrict our study to the k = 0 geometries: 2 2 2 ds = dt +a (t) d~x d~x . (1.2) − · b Such geometries do not represent solutions of the Einstein equations in vacuum except in the case of flat space where the scale factor is equal to one. In all other cases a non-trivial stress tensor must be present. Homogeneity and isotropy constrain the stress tensor to have only diagonal elements consisting of the density ρ(t) and pressure p(t): * Bydecentwemeanatleastthe60e-foldingsofinflationneededtoexplaintheobserveddegreeofisotropy. 2 Tµν = [ ρ(t) p(t) p(t) p(t) ]. This form can arise either from a non-zero temperature matter theory or a non-trivial scalar field in zero temperature quantum field theory. To preserve the isotropy of space, the matter fields must be scalar since the Lorentz transformation properties of all other fields – like fermions or gauge bosons – exhibit preferred directions. The determination of the scalar field action which supports an arbitrary spatially flat Robertson-Walker background is given in Section 2. It is best to set-up the theory on an arbitrary spatial manifold of finite extent – for instance, T3 – in which case ∆xi R. We shall find it very convenient to utilize the ×ℜ ≤ conformal set of coordinates: 2 2 2 2 2 ds = dt +a (t) d~x d~x = Ω (η) dη +d~x d~x . (1.3a) − · − · (cid:16) (cid:17) b The relation between the two coordinate systems is given by: dt = Ω(η) dη ; a(t) = Ω(η) , (1.3b) so that a generic power law in co-moving coordinates: t s a(t) = , (1.4a) t (cid:16) 0(cid:17) takes the following form in conformal coordinates: s η 1 s Ω(η) = − , (1.4b) η (cid:16) 0(cid:17) with η = t (1 s) 1. The physically most distinguishable cases consist of the inflating 0 0 − − universe for s = + , the radiation dominated universe for s = 1 , the matter dominated ∞ 2 universe for s = 2 , and the flat universe for s = 0. We shall also assume throughout that 3 ∆x R and ∆η R since then the integral approximation to mode sums which appear ≪ ≪ because we work on T3 is excellent. * * In conformal coordinates light reaches the distance ∆x=R in time ∆η =R. 3 In order to study gravitational effects not associated with the smallest of scales in the class of backgrounds (1.2), we must develop the appropriate perturbative tools – this is accomplished in Section 3. As an elementary application of the perturbative results, we calculateinSection4theresponseofthegravitationalfieldduetoapointsourceinamatter dominated universe. The resulting corrections to the Newtonian long-range gravitational force are found to be negligible as expected. Our conclusions comprise Section 5. 2. Determination of the source background The dynamical system under consideration has a background action which consists of two parts: = + . (2.1) g m S S S b b b The gravitational action is obtained from the Einstein Lagrangian: g S b 1 = R √ g , (2.2a) g L 16πG − b while the matter action is based on a generic scalar field Lagrangian: m S b Lm = −21 ∂µϕ ∂νϕ gµν√−g − V(ϕ) √−g . (2.2b) b We obtain the following equations of motion for the gravitational field: µν µν G = (8πG) T , (2.3a) and for the scalar field: ∂V(ϕ) µν ∂µ √ g g ∂νϕ = √ g . (2.3b) − ∂ϕ − (cid:16) (cid:17) The Einstein tensor Gµν is determined by the geometry (1.2): * a˙ 2 a¨ a˙2 00 ij ij G = 3 ; G = g 2 + , (2.4) (cid:18)a(cid:19) − (cid:18) a a2 (cid:19) * Our metric has spacelike signature and Rµ = Γµ +... Greek indices run from 0 to 3, while Latin νρσ σν,ρ indices run from 1 to 3. Dots as superscripts denote differentiation with respect to the co-moving time t. We are also using units where c=h¯ =1. 4 and the covariantly conserved stress tensor Tµν by the matter sector of the theory: 2 δ Tµν ≡ √ g δgSµmν = (cid:16) gµα gνβ − 12gµν gαβ (cid:17) ∂αϕ ∂βϕ − gµν V(ϕ) , (2.5a) − µν T = 0 . (2.5b) ;ν The resulting density ρ(t) and pressure p(t) take the form: ˙ 2 T00 ρ = 1φ +V(φ) , (2.6a) ≡ 2 Tij p gij = gij b1φ˙ 2 bV(φ) . (2.6b) ≡ 2 − (cid:16) (cid:17) b b In order not to disturb the spatial homogeneity and isotropyof our background geometries, we have taken the scalar field to be a function of time only: ϕ = φ(t). b Using (2.4) and (2.6), the gravitational equations (2.3a) become: ˙ 2 1 a¨ a˙2 φ = + , (2.7a) 4πG (cid:18) −a a2 (cid:19) b 1 a¨ a˙2 V(φ) = +2 . (2.7b) 8πG (cid:18) a a2 (cid:19) b ˙ By satisfying equations (2.7), we determine the source parameters φ(t) and V(φ) – or, equivalently, ρ(t) and p(t) – which support the flat Robertson-Walkebr backgrounbd (1.2). In conformal coordinates the above equations take the form: * 2 ˙ 2 1 Ω′′ 2Ω′ φ = + , (2.8a) 4πG (cid:18) −Ω3 Ω4 (cid:19) b 2 1 Ω Ω ′′ ′ V(φ) = + , (2.8b) 8πG (cid:18) Ω3 Ω4 (cid:19) b where the primes over Ω indicate differentiation with respect to the conformal time η. It remains to be shown that the above choice for the source parameters is consistent with the scalar equation of motion (2.3b): ¨ 3a˙ ˙ ∂V(φ) φ+ φ+ = 0 . (2.9) a ∂φ b b b b ˙ 2 * Notice that in the case of de Sitter spacetime, s + , equations (2.7-8) have the proper limit: φ 0 and V(φ) (8πG)Λ−1, where Λ=3η−2. → ∞ → → 0 b b 5 This is a direct consequence of the conservation equation (2.5b): 3a˙ ρ˙ = (ρ+p) . (2.10) − a By substituting (2.6) in (2.10) we obtain: d ˙ 2 3a˙ ˙ 2 1φ +V(φ) = φ , (2.11) dt 2 − a (cid:16) (cid:17) b b b which is identical to (2.9). 3. The Feynman rules 3.1 The Lagrangian. The quantum fields are the graviton hµν(x) and the scalar φ(x): gµν gµν +κhµν , (3.1a) ≡ b ϕ φ+φ . (3.1b) ≡ b It turns out that – just like the case of de Sitter spacetime [3] – it is most convenient to organize perturbation theory in terms of the “pseudo-graviton” field, ψµν(x), obtained by conformally re-scaling the metric: 2 2 gµν Ω gµν Ω ηµν +κψµν . (3.2) ≡ ≡ (cid:16) (cid:17) e As usual, pseudo-graviton indices are raised and lowered with the Lorentz metric and κ2 16πG. The total Lagrangian is: ≡ 1 L = κ2 R √−g − 12 ∂µϕ ∂νϕ gµν√−g − V(ϕ) √−g . (3.3) By substituting (3.1b) in (3.3) we get: L = κ12 R √−g − 12φ′2 g00√−g − ∂µφ ∂νφ gµν√−g − 12∂µφ ∂νφ gµν√−g b b +∞ 1 ∂nV n (φ) φ √ g , (3.4) − n! ∂ϕn − nX=0 b 6 and we can proceed to organize according to the number of fields φ present. L The pure metric part of : L 1 ′ = R √ g 1φ 2 g00√ g V(φ) √ g , (3.5) L(0) κ2 − − 2 − − − b b in terms of the re-scaled metric is: 1 2 00 2 = Ω R+2 1+g ΩΩ +Ω g , (3.6) L(0) κ2 (cid:26) − ′′ ′ (cid:27) − (cid:16) (cid:17)(cid:16) (cid:17) p e e e whereweused (2.8)andignoredsurfaceterms. Afterperformingmanypartialintegrations, (3.6) can be cast in a form identical to that of de Sitter spacetime [3]: L(0) = −g gαβ gρσ gµν 21ψαρ,µ ψνσ,β − 21ψαβ,ρ ψσµ,ν + 14ψαβ,ρ ψµν,σ − 14ψαρ,µ ψβσ,ν Ω2 p h i ee e e −12 −g gρσ gµν ψρσ,µ ψνα (Ω2),α . (3.7) p ee e All the self-interactions of the graviton in the presence of an arbitrary scale factor Ω(η) can be obtained by expanding expression (3.7). The quadratic part of is: (0) L L((20)) = 21ψαρ,µ ψµρ,α − 21ψ,α ψαρ,ρ + 14ψ,α ψ,α − 41ψαρ,µ ψαρ,µ Ω2 h i ,µ +ψ ψ Ω Ω , (3.8) µ0 ′ µ where ψ ψ . µ ≡ The part of linear in φ is simpler: L L(1) = −φ′ ∂νφ g0ν√−g − ∂∂Vϕ(φ) φ √−g , (3.9a) 1b b µ0 = ξ ∂µφ g +ξ′ φ g . (3.9b) κ − − h ip e e In the last step we used the scalar background equation of motion (2.9) to derive the identity: ∂V 1 (φ) = ξ , (3.10a) ′ ∂ϕ −κΩ4 b 7 where we define ξ(η) as: ′ 2 ξ κΩ φ . (3.10b) ≡ b The following quadratic part emerges: L((21)) = ξ ∂µφ ψµ0 + 21 (ξφ)′ ψ . (3.11) We shall also need the part of which is quadratic in φ: L ∂2V L(2) = −21∂µφ ∂νφ gµν√−g − 12 ∂ϕ2(φ) φ2 √−g , (3.12a) = −21Ω2 ∂µφ ∂νφ gµν −g + 21Ω6bξ−1 ( Ω−4ξ′ )′ φ2 −g , (3.12b) p p e e e and the associated piece which is quadratic in the quantum fields: L((22)) = −12Ω2 ∂µφ ∂µφ + 12Ω6 ξ−1 ( Ω−4ξ′ )′ φ2 . (3.13) Inordertoproperly quantizethetheoryandcalculatethevariouspropagators, wemust fix the gauge. We shall accomplish this by adding the following term to the Lagrangian: LGF = −21ηµν Fµ Fν , (3.14a) where: Fµ = Ω ψµν,ν − 12Ω ψ,µ −2Ω′ ψµ0 +Ω−1 ηµ0 ξ φ . (3.14b) The resulting gauge-fixed quadratic Lagrangian is: (2) (2) (2) (2) = + + + LGF L(0) L(1) L(2) LGF Ω Ω =− 81ψ Ω(cid:20) ∂2 + Ω′′ (cid:21)Ω ψ + 14ψµν Ω(cid:20) ∂2 + Ω′′ (cid:21)Ω ψµν 2 Ω Ω Ω µ0 ′′ ′ 2 ′ +ψ Ω + Ω ψ +ψ Ω Ω ξ +2 ξ Ω φ (cid:20) − Ω Ω2 (cid:21) µ0 00 (cid:20) − − ′ Ω3 (cid:21) Ω + 1φ Ω ∂2 + ′′ +Ω 4 ξ2 +Ω4 ξ 1 ( Ω 4ξ ) Ω φ . (3.15) 2 (cid:20) Ω − − − ′ ′ (cid:21) 8 ρσ In terms of a kinetic operator D we get: µν Ω L(G2F) ≡+ 12ψµν Dµνρσ ψρσ +ψ00 Ω(cid:20) −Ω−2 ξ′ +2Ω3′ ξ (cid:21)Ω φ Ω + 1φ Ω ∂2 + ′′ +Ω 4 ξ2 +Ω4 ξ 1 ( Ω 4ξ ) Ω φ . (3.16) 2 (cid:20) Ω − − − ′ ′ (cid:21) where we have: Dµνρσ = 12δµ(ρ δνσ) − 14ηµν ηρσ − 21δµ0 δν0 δ0ρ δ0σ DA h i 0 (ρ σ) 0 0 ρ σ +δ δ δ D +δ δ δ δ D . (3.17) (µ ν) 0 B µ ν 0 0 B Parenthesized indices are symmetrized and a bar above a Lorentz metric or a Kronecker delta symbol means that the zero (i.e., η) component is projected out: 0 0 ν ν 0 ν ηµν ηµν +δµ δν ; δµ δµ δµ δ0 . (3.18) ≡ ≡ − The quadratic operators defined in (3.17) are given by: Ω 2 ′′ D Ω ∂ + Ω , (3.19a) A ≡ (cid:20) Ω (cid:21) 2 Ω Ω 2 ′′ ′ D Ω ∂ +2 Ω . (3.19b) B ≡ (cid:20) − Ω Ω2 (cid:21) Notice that D is the kinetic operator for a massless, minimally coupled scalar in the A presence of the gravitational background (1.3). 3.2 The Diagonal Variables. ThequadraticLagrangian(3.16)shouldbebroughtintodiagonalform. Thereismixing in the first term of (3.16) between ψ and ψ : ii 00 21ψµν Dµνρσ ψρσ = 21 21ψij DA ψij − 41(ψii −ψ00) DA (ψjj −ψ00)− 12ψ00 DA ψ00 n ψ D ψ +ψ D ψ , (3.20) 0i B 0i 00 B 00 − o which is removed by transforming to the field variable ζµν: 21ψµν Dµνρσ ψρσ = 12 12ζij DA ζij − 41ζii DA ζjj −ζ0i DB ζ0i +ζ00 DB ζ00 n o = 21 ζij DA ζrs 21δi(r δs)j − 14δij δrs +ζ0i DB ζ0r −δir n h i h i +ζ D ζ , (3.21) 00 B 00 o 9 where we defined: ψ ζ +δ ζ ψ ζ ψ ζ . (3.22) ij ij ij 00 0i 0i 00 00 ≡ ≡ ≡ There is also mixing between ζ and φ. The part of the quadratic Lagrangian involved 00 is: Ω = 1ζ D ζ +ζ Ω Ω 2 ξ +2 ′ ξ Ω φ Lmixing 2 00 B 00 00 (cid:20) − − ′ Ω3 (cid:21) Ω + 1φ Ω ∂2 + ′′ +Ω 4 ξ2 +Ω4 ξ 1 ( Ω 4ξ ) Ω φ . (3.23) 2 (cid:20) Ω − − − ′ ′ (cid:21) Wecanalwaysdiagonalizethissystembut, formost scalefactorsΩ(η), theresultingkinetic operators will be non-local. In this case the mode functions obey fourth order differential equations. However, for a class of scale factors which includes the generic power law (1.4), the system can be diagonalized without sacrificing locality or the second order character of the mode equations.* In this case, takes the form: mixing L 1 1 s = 1ζ Ω ∂2 + s Ω ζ +ζ Ω 2√s Ω φ Lmixing 2 00 (1 s)2 η2 00 00 1 s η2 h − i − 1 +1φ Ω ∂2 + 2s2 3s+2 Ω φ . (3.24) 2 (1−s)2 η2 h − i Let us call the diagonal variables χ and υ: ζ = cosθ χ sinθ υ , (3.25a) 00 − φ = sinθ χ+cosθ υ , (3.25b) and demand that the part of s involving χυ vanish: Lmixing 1 s 2√s = χ Ω sin(2θ)+ cos(2θ) Ω υ = 0 . (3.26) Lχυ η2 1 s n − o Of the two available solutions, it is sufficient to consider one of them: cos2θ = 1 ; sin2θ = s ; sinθ cosθ = √s , (3.27) 1+s 1+s −1+s * The condition that Ω(η) must obey is: −Ω ddη(cid:16)Ω−2 Ω′(cid:17) = cc12 tan2(cid:16)√c1c2 η+qcc21 θ(cid:17), where c1,c2 are constants and θ is an angle. 10