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METRIC PERTURBATIONS AND INFLATIONARY PHASE TRANSITIONS 0 D.CORMIER 0 Institute for Physics, University of Dortmund, 0 D-44221 Dortmund, Germany 2 E-mail: [email protected] n a R.HOLMAN J Physics Department, Carnegie Mellon University, 4 Pittsburgh PA 15213, U.S.A. 1 E-mail: [email protected] 1 We study the out of equilibrium dynamics of inflationary phase transitions and v compute the resulting spectrum of metric perturbations relevant to observation. 5 We show that simple single field models of inflation may produce an adiabatic 5 perturbation spectrum with a blue spectral tilt and that the precise spectrum 2 depends oninitialconditionsattheoutsetofinflation. 1 0 0 1 Field Evolution 0 / We work in a spatially flat Friedmann-Robertson-Walker universe with scale h factor a(t) and take the inflaton to be a real scalar field with Lagrangian p o- L= 1 µΦ Φ 3µ4 1µ2Φ2+ λΦ4 . (1) r 2∇ ∇µ −(cid:20) 2λ − 2 4! (cid:21) t s a We break up the field Φ into its expectation value, defined within the v: closed time path formalism,1 and fluctuations about that value: i X Φ(~x,t)=φ(t)+ψ(~x,t), φ(t) Φ(~x,t) . ≡h i r a ByimposingaHartreeresummation,2 wearriveatthefollowingequations of motion for the inflaton:3 a˙ λ λ φ¨+3 φ˙ µ2φ+ φ3+ ψ2 φ=0, (2) a − 6 2h i d2 a˙ d k2 λ λ 2 2 2 +3 + µ + ψ + φ f =0. (3) (cid:20)dt2 adt a2 − 2h i 2 (cid:21) k The fluctuation ψ2 is determined from the mode functions f : k h i d3k 2 2 ψ = f . (4) h i Z (2π)3| k| 1 For a(t0)=1, the initial conditions on the mode functions are 1 fk(t0)= , f˙k(t0)=( a˙(t0) iωk)fk(t0), (5) √2ω − − k with 2 2 2 λ 2 λ 2 R(t0) ω k µ + ψ + φ . k ≡ − 2h i 2 − 6 R(t0) is the initial Ricci scalar. For small k, we modify ωk either by means of a quench or by explicit deformation so that the frequecies are real.4,5 The gravitational dynamics are determined by the semi-classical Einstein equation.6 For a minimally coupled inflaton we have a˙2 8πG 1 1 1 = N φ˙2+ ψ˙2 + (~ψ)2 a2 3 (cid:20)2 2h i 2a2 D ∇ E 3µ4 1 λ 2 2 2 4 2 2 2 2 + µ φ + ψ + φ +3 ψ +6φ ψ , (6) 2λ − 2 h i 24 h i h i (cid:21) (cid:0) (cid:1) (cid:0) (cid:1) where G is Newton’s gravitationalconstant, and N d3k ψ˙2(t) f˙ 2 , (7) h i≡ Z (2π)3| k| 2 d3k ~ψ(t) k2 f 2 . (8) (cid:28)(cid:16)∇ (cid:17) (cid:29)≡ Z (2π)3 | k| Eachoftheseintegralsisregulatedusingacutoffwiththedivergencesabsorbed into a renormalizationof the parameters of the theory.4 A typical field evolution is depicted in Fig. 1. 2 Metric Perturbations 7 FollowingtheprocedureofMukhanov,FeldmanandBrandenberger, wearrive at the expression for the density contrast at mode re-entry:8 2 1 3µ4 1 2 2 2 δ (k) = µ φ + ψ | h | r75πMPl φ˙2+ ψ˙2 (cid:20) 2λ − 2 (cid:0) h i(cid:1) (cid:16) h i(cid:17) λ 1/2 4 2 2 2 2 + φ +6φ ψ +3 ψ 24 h i h i (cid:21) (cid:0) (cid:1) λµ2 4 2 2 4 2 2 2 2 µ φ + ψ φ +6φ ψ +3 ψ × (cid:20) h i − 3 h i h i (cid:0) (cid:1) (cid:0) (cid:1) λ2 1/2 6 4 2 2 2 2 2 3 + φ +15φ ψ +45φ ψ +15 ψ . (9) 36 h i h i h i (cid:21) (cid:0) (cid:1) 2 The computation of the tilt parameter n 1 is straightforward,given (9): s − d(ln δ (k)) n 1 | h | . (10) s− ≡ dln(k) (cid:12) (cid:12)k=aH (cid:12) As gravitational wave perturbations do no(cid:12)t directly interact with the in- flatonfield,theymayberelateddirectlytotheexpansionrate. Theamplitude 7 of gravitationalwaves is simply: 2 H δ (k) = , (11) | g | √3πMPl All expressions are to be evaluated when the given scale k first crosses the horizon, k =aH. An example perturbation spectrum is shown in Fig. 2, while Fig. 3 shows the dependence of the spectrum on the initial state for a number of possible evolutions. Both of these figures show distinct regions characterizedby a blue spectral tilt. Acknowledgments R.H. wassupportedin partby the DepartmentofEnergyContractDE-FG02- 91-ER40682. References 1. J. Schwinger, J. Math. Phys. 2, 407 (1961); L. V. Keldysh, Sov. Phys. JETP 20, 1018 (1965). 2. See, for example, A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York (1971). 3. D. Cormier, Non-Equilibrium Field Theory Dynamics in Inflationary Cosmology, hep-ph/9804449(1998). 4. D. Boyanovsky, D. Cormier, H.J. de Vega, R. Holman and P. Kumar Phys. Rev. D57, 2166 (1998). 5. D. Boyanovsky, D. Cormier, H.J. de Vega and R. Holman, Phys. Rev. D55, 3373 (1997). 6. N.D.BirrellandP.C.W.Davies,Quantum Fields in Curved Space, Cam- bridge Univ. Press, Cambridge, (1986). 7. V.F. Mukhanov, H.A. Feldman and R.H. Brandenberger, Phys. Rep. 215, 205 (1992). 8. D. Cormier and R. Holman, Spinodal Decomposition and Inflation: Dy- namics and Metric Perturbations, hep-ph/9912483(1999). 3 1 φ(t)/f0.5 0 21/2ψ<(t)>/f00000.....12345 0 2 µ 1.5 H(t)/ 1 0.5 0 0 50 100 150 200 µt Figure1: Themeanfieldφ(t)/f,thefluctuation<ψ2(t)>1/2/f,andtheHubbleparameter H(t)/µvs. twithφ(t0)=0.4H0/2π,φ˙(t0)=0,H0=2µ,λ/8π2=10−16,andf ≡µ 6/λ. p 10−4 10−5 δ||h 10−6 10−7 0.4 n−1s 0.20 4×1−00−.28 3×10−8 δ||g2×10−8 1×10−8 0 20 40 60 80 100 N Figure2: Thescalar amplitude δh,the scalartiltns−1,and thetensor amplitudeδg as a functionofthenumberofe-foldsN beforetheendofinflationthatthescalefirstcrossesthe horizon. ParametersareasinFig.1. 10−1 δ||6011110000−−−−5432 HHH000===µ24µµ 10−6 10−7 0.01 0.1 1 10 100 0.4 0.2 n−160 0 −0.2 −0.4 0.01 0.1 1 10 100 2πφ(t)/µ 0 Figure 3: The scalar amplitude δ60 and tilt n60 −1 of the scale crossing the horizon 60 e-folds before the inflation ends vs. 2πφ(t0)/µ with φ˙(t0)=0, λ/8π2 =10−16 and several valuesofH0. 4