We revisit an extension of the well-known formalism for gauge-invariant
scalar metric fluctuations, to study the spectrums for both, the inflaton and
gauge invariant (scalar) metric fluctuations in the framework of a single field
inflationary model where the quasi-exponential expansion is driven by an
inflation which is minimally coupled to gravity. The proposal here examined is
valid also for fluctuations with large amplitude, but for cosmological scales,
where vector and tensor perturbations can be neglected and the fluid is
irrotacional.Comment: Version accepted in EPJC with new title. 11 pages, no figure

A. Albrecht

A. Berera

A. Guth

A.D. Linde

D.S. Salopek

E.W. Kolb

F. Finelli

G. Dvali

G. D’Amico

J.E. Madriz Aguilar

J.M. Bardeen

J.M. Stewart

K. Nakamura

M. Anabitarte

M. Bellini

M. Mijic

M.D. Pollock

Mariano Anabitarte

Mauricio Bellini

S. Habib

S. Mollerach

V. Acquaviva

V.F. Mukhanov

W.L. Power

English

arXiv

We revisit an extension of the well-known formalism for gauge-invariant scalar metric fluctuations to study the spectra for both the inflaton and gauge-invariant (scalar) metric fluctuations in the framework of a single-field inflationary model, in which the quasi-exponential expansion is driven by an inflaton which is minimally coupled to gravity. The proposal here examined is valid also for fluctuations with large amplitudes, but for cosmological scales, where vector and tensor perturbations can be neglected and the fluid is irrotacional. © 2009 Springer-Verlag / Società Italiana di Fisica. PB  - Springer-Verla

Anabitarte, M.

Bellini, M.

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

arXiv:0809.5280v3  [gr-qc]  6 Jan 2009A confirmation of agreement of different approaches for scalargauge-invariant metric perturbations during inflation1,2 Mariano Anabitarte∗ and 1,2 Mauricio Bellini †1 Departamento de F́ısica, Facultad de Ciencias Exactas y Naturales,Universidad Nacional de Mar del Plata,Funes 3350, (7600) Mar del Plata, Argentina.2 Consejo Nacional de Investigaciones Cient́ıficas y Técnicas (CONICET).AbstractWe revisit an extension of the well-known formalism for gauge-invariant scalar metric fluctua-tions, to study the spectrums for both, the inflaton and gauge invariant (scalar) metric fluctuationsin the framework of a single field inflationary model where the quasi-exponential expansion is drivenby an inflation which is minimally coupled to gravity. The proposal here examined is valid also forfluctuations with large amplitude, but for cosmological scales, where vector and tensor perturba-tions can be neglected and the fluid is irrotacional.∗ E-mail address: anabitar@mdp.edu.ar† E-mail address: mbellini@mdp.edu.ar1I. INTRODUCTIONThe big bang theory has provided a remarkably successful description of the evolutionof the universe and now rests upon solid observational foundations. Over the last decadescosmologists have also drawn up architectural sketches of a model which may account forthe origin and evolution of structure within this framework: primordial perturbations froman early epoch of inflation, which subsequently grow by gravitational instability to formgalaxies and larger structures. The standard picture of inflation introduced in 1981[1] re-lied on a scalar field, called the inflaton, which during inflation was assumed to have nointeraction with any other field. The inflationary scenario postulates that the universe un-derwent a phase of very rapid, accelerated expansion in its distant past. Observations haveprovided strong support for the paradigm. However, despite this success, the mechanismwhich drove the inflationary expansion has yet to be indentified. A multitude of inflationarymodels involving a broad range of energy scales have been discussed in the literature, in-cluding chaotic inflation[2], warm inflation[3], stochastic inflation[4], fresh inflation[5], braneinflation[6], STM inflation[7], and many others.The scalar fluctuations of the metric are associated with density perturbations. Theseare the spin-zero projections of metric perturbations and were induced by the vacuum fluc-tuations of the inflaton field during inflation. Furthermore, they played a crucial role in thegeneration of primordial inhomogeneities which gave rise to the large scale structure of thepresent day universe as well as the observed anisotropies of the cosmic microwave backgroundof radiation. Gauge invariance guarantees that equation for the fluctuations of the geometrydo not change when moving from one coordinate system to other. This allows us to formu-late the problem of the evolution for the amplitude of scalar metric perturbations around theFriedmann-Robertson-Walker (FRW) universe in a coordinate-independent manner at everymoment in time. The issue of gauge invariance becomes critical when attempt to analyzehow the scalar metric perturbations produced in the early universe influence a globally flat,isotropic and homogeneous universe on super Hubble scales. Space-time fluctuations can alsolead to decoherence of matter waves[10]. In the infrared (IR) sector these fluctuations canbe represented by a coarse-grained field, which describes a stochastic dynamics[11]. Withrespect to perturbative approaches, second order gauge invariant perturbations quantitieshave been calculated in [12] and third order perturbations are supposed to be negligible[13]2during inflation, but not for large fluctuations.In this work we examine a perturbed FRW metric using the proposal developed in[14],which is an extension of the well known linearized line element ds2 = (1+2ψ)dt2−a(t)2 (1−2ψ) d~r2, for a longitudinal gauge. Our results show a coincidence of different approaches forsmall fluctuations; the formalism developed in [14] and the standard method.II. FORMALISMWe consider a scalar field ϕ which is minimally coupled to gravity. The action of thesystem isI =∫d4x√−g[ R16πG+12gµνϕ,µϕ,ν − V (ϕ)], (1)where g = −e−4ψa6(t) is the determinant of the covariant metric tensor with componentsgµν (µ, ν run from 0 to 3) and V (ϕ) is the potencial related to the inflaton field. To studythe gauge invariant scalar metric fluctuations ψ (xα), we propose the following perturbedmetricdS2 = e2ψ dt2 − a2(t)e−2ψ d~r2, (2)where d~r2 = dx2 + dy2 + dz2 and ψ(t, ~r) is the scalar metric fluctuation. The metric (2)is the perturbed version of the background FRW one, which is spatially flat, isotropic,homogeneous and has a scalar curvature R̄ = 6[äa+(ȧa)2]. This metric describe non-perturbative gravitational fluctuations on cosmological scales, on which vector and tensorperturbations of the metric can be neglected and the fluid can be considered as irrotacional.Furthermore, using the continuity equation on large scales∂ρ∂τ= −3H (ρ+ P ) , (3)where H = ddτ[ln(a(t) e−ψ)]and dτ = eψdt. One can show that exists a conserved quantityin time at any order in perturbation theoryf = ln(ae−ψ)+13∫ ρ dρ′(P ′ + ρ′). (4)Considering the invariant ω, which characterizes the equation of state P = ω ρ. The pertur-bation δf = −ψ + 13(1+ω)ln (ρ/ρ̄), is a gauge-invariant quantity representing the non-linearextension of the curvature perturbation for adiabatic fluids on uniform energy density hy-persurfaces on superhorizon scales[14]. Here, (P̄ , ρ̄) denote the background pressure and3energy density and (P, ρ) denote pressure and energy density on the perturbed metric (2).The scalar curvature of this metric isR = 6e−2ψ[[äa+(ȧa)2]− ψ̈e4ψ − 5ψ̇(ȧa)+e4ψ3a2[∇2ψ −(~∇ψ)2]+ 3ψ̇2]. (5)Now we shall use the Lagrangian formalism to describe the dynamics of ϕ and ψ. Theequation of motion for ϕ isϕ̈+[3(ȧa)− 4ψ̇]ϕ̇− 1a2e4ψ∇2ϕ+ e2ψV ′(ϕ) = 0, (6)where V ′(ϕ) ≡ dVdϕ. The equation (6) is an operatorial one because ϕ is considered as aquantum scalar field. The equation of motion for ψ is[∂R∂ψ− ∂∂xµ∂R∂ψ,µ]−2R− 1√−g∂√−g∂t∂R∂ψ̇= 32πG[e−2ψϕ̇2 − e2ψ 1a2(~∇ϕ)2− V (ϕ)], (7)such that both, ϕ and ψ comply with the commutation relations[ϕ(t, ~r),Πϕ(t,~r)] = i δ(3) (~r −~r) , (8)[ψ(t, ~r),Πψ(t,~r)] = i δ(3) (~r −~r) , (9)Πϕ and Πψ being respectively the conjugate momentums for ϕ and ψ:Πϕ =∂L∂ϕ̇, Πψ =∂L∂ψ̇,such that L =√−g[R16πG+ 12gµνϕ,µϕ,ν − V (ϕ)]is the density Lagrangian of the system.To complete the description of the dynamics, we need to write the Einstein equations Gαβ =−8πGTαβ . Taking into account cartesian coordinates, the diagonal Einstein equations are− 2a2e4ψ∇2ψ + 6Hψ̇ − 3ψ̇2 + e4ψa2(~∇ψ)2+ 3H2= −8πG[ϕ̇22+ e4ψ12a2(~∇ϕ)2+ V (ϕ)e2ψ], (10)−8πG[3e−4ψ2ϕ̇2 − 12a2(~∇ϕ)2− 3e−2ψV]= − 2a2(~∇ψ)2− 24Hψ̇e−4ψ+15ψ̇2e−4ψ + 6e−4ψ(äa)− 6ψ̈e−4ψ − 3H2e−4ψ, (11)where Gαβ = Rαβ − 12Rgαβ is the Einstein tensor and T αβ = ϕ,αϕ,β − gαβ[12ϕ,ρϕ,ρ − V (ϕ)]is the Energy - Momentum tensor for a scalar field. On the other hand, the non-diagonalEinstein equations have the form1a∂∂xi[∂∂t(aψ)]− ∂ψ∂t∂ψ∂xi= 4π G∂ϕ∂t∂ϕ∂xi, (12)4so that, using the eq. (12) in eqs. (10) and (11), we obtain the exact equation of motion forψψ̈ + 7Hψ̇ − e4ψa2∇2ψ − 4ψ̇2 − 5e4ψ3a2(~∇ψ)2+ 8πGV (ϕ)e2ψ = −8πG3a2(~∇ϕ)2, (13)which describes the dynamics of ψ with arbitrary amplitude. However, it is very difficultto solve these equations in an exact manner. Notice that we have used the Einsteinequations, and not the Lagrange one, to obtain the dynamics of ψ. One could make theinverse procedure, because both manners to work are equivalent. However, in this case thecalculations with the Einstein equations are more simple.III. LINEAR APPROXIMATIONIn the weak field limit, it is sufficient to make a linear approximation on the scalar metricperturbations, so that one can write e±2ψ(xα) ≃ 1± 2ψ(xα) in the exact equations of motion(6) and (13), for ψ and ϕ. In this limit the metric (2) preserves gauge invariance and thelinearized line elementdS2 = (1 + 2ψ) dt2 − a(t)2 (1− 2ψ)d~r2,takes the form of a longitudinal gauge so that coordinate transformations induce difeomor-fism transformations[8].The equation of motion for the inflaton in its exact form is given by (6). However, in theweak field limit we can make the semi-classical approximation ϕ(xα) = 〈E|ϕ|E〉 + φ(xα).Here, 〈E|ϕ|E〉 = φc(t) is the expectation value of ϕ evaluated on the quantum state |E〉.Furthermore, in this limit the quantum fluctuations of the inflaton field are considered to bevery small and 〈E|φ|E〉 =〈E|φ̇|E〉= 0. With this representation we obtain the followingdynamical equations for the fluctuations of the inflaton field φ and the classical field φc:φ̈+ 3(ȧa)φ̇− 1a2∇2φ+ V ′′(φc)φ = 4ψ̇φ̇c + 2ψV ′(φc), (14)φ̈c + 3(ȧa)φ̇c + V′(φc) = 0. (15)The background Friedmann equations are3H2c = 8πG[φ̇2c2+ V (φc)], (16)5where V (φc) ≡ V (ϕ)|φc is the scalar potential evaluated on the classical background fieldφc(t) and the Hubble parameter with back-reaction effects included isH =ȧa≃ Hc(t) +4πG3Hc〈E| φ̇22+(~∇φ)22a2+∞∑n=1V (n)(φc)n!φn(xα) |E〉 . (17)When the metric fluctuations are small it is sufficient to make Hc ≃ ȧa , because the lastterm in the right hand of the expression (17) is negligible with respect to the first one.This approximation is valid only on large scales, which are super Hubble scales during theinflationary epoch. Furthermore, the primer denotes the derivative with respect to ϕ, suchthat V ′(φc) ≡ dV (ϕ)dϕ∣∣∣φc.Furthermore, the Einstein equations (12) now hold1a∂2∂xi∂t[aψ] = 4πG∂∂xi[φ̇c φ], (18)from which (once we have taken into account that 〈E|φ|E〉 = 0) we obtain that the evolutionfor the expectation value of ψ goes as〈E|ψ|E〉 ∼ a−1, (19)which decreases with the inverse of the scale factor of the universe. Finally, the linearizeddynamics of ψ can be obtained from the Einstein equations (10) and (11): δGµν = −8πG δT µνψ̈ +[H − 2 φ̈cφ̇c]ψ̇ − 1a2∇2ψ + 2[Ḣ − φ̈cφ̇cH]ψ = 0, (20)which, as one expects, is the same as the equation obtained in[9]. Note that the equation(20) is the equation (13) once linearized, with the constriction (18).IV. AN EXAMPLEIn this section we shall illustrate the formalism in the linearized approximation, when theexpansion is governed by a power-law expansion a = β tp. In this case the Hubble parameteris given by H = p/t and the classical field φc(t) isφc(t) = φ0[1− ln(H0tp)], (21)where the power p isp = 4πGφ20,6φ0 being the value of φc(t0) when inflation begins (i.e., at t = t0). The equation of state isgiven by ω = P̄ρ̄= −(p−2/3)p, ω being an invariant. Furthermore, the classical potential V (φc)isV (φc) =3H208πG(3p− 13p)e2φc/φ0 . (22)The solution for the ψ-modes, once normalized, areξk(t) =√π2√t(p− 1) H(2)µ[k t1−p(p− 1)β]×(tt0)−(p+2)/2, (23)where H(2)µ [x(t)] is the second kind Hankel function with µ = p+12(p−1) . Using the smallargument Hankel functions limit, we obtain that these modes have the following expressionon super Hubble scales:ξk(t)|k≫1/(aH) ≃ i√π4(p− 1) Γ[(p+ 1)2(p− 1)] [2(p− 1)βπ](p+1)2(p−1)k−(p+1)2(p−1) , (24)which is independent of time. The equation of motion for the modes of the inflaton field oncosmological scales can be approximated to¨̃ξk(t) +3pt˙̃ξk(t) +[k2β2 t2p+4 (6πGφ20 − 1)t2]ξ̃k(t)= i[(3p− 2)φ0t2]√πp− 1Γ(p+ 12(p− 1))[2β(p− 1)π](p+1)2(p−1)k−(p+1)2(p−1) , (25)which, on super Hubble scales, has the solutionξ̃k(t) ≃ t12(1−3p)×{A Jν2 [x(k, t)] +B Yν2 [x(k, t)]− iγβ√πp− 1Γ(p+ 12(p− 1))[2β(p− 1)π]p+12(p−1)k1−3p2(p−1)×[Jν2 [x(k, t)]∫dt t52(p−1) Yν2 [x(k, t)][Jν2 [x(k, t)] Yν1 [x(k, t)]−Yν2 [x(k, t)] Jν1 [x(k, t)]]−[Yν2 [x(k, t)]∫dt t52(p−1) Jν2 [x(k, t)][Jν2 [x(k, t)] Yν1 [x(k, t)]Yν2 [x(k, t)] Jν1 [x(k, t)]]]]}, (26)7where γ = 3(p − 2)φ0, 3p > m2 = 4(6πGφ20 − 1), ν2 =√9p2−6p+1−4m22(p−1), ν1 = 1 − ν2 andx(k, t) = kt(1−p)β(p−1). For x(t) ≪ 1 this solution can be written asξ̃k(t) ≃ −BπΓ (ν2)(k2β(p− 1))−ν2t−(1−p)ν2+12(1−3p)+ iβ(3p− 2)φ0Γ(p+ 12(p− 1)) [2β(p− 1)π]p+12(p−1)×[Γ (ν2 + 1) t−(ν1+1)(1−p)2Γ (ν1 + 1) [(ν2 − ν1)(p− 1)](k2β(p− 1))ν2−ν1×∞∑n=01[5p−3+2ν1(p−1)4(ν2−ν1)(1−p)+ 1][−Γ(ν2 + 1)Γ(ν1)Γ(ν2)Γ(ν1 + 1)x(k, t)]2n(ν2−ν1)+Γ(ν2) t−(2ν2+1)(1−p)2Γ(ν2 + 1) (ν2 − ν1) (p− 1)(k2β(p− 1))−2ν2×∞∑n=01[5p−3+2ν2(p−1)4(ν2−ν1)(p−1)+ 1][−Γ(ν1 + 1)Γ(ν2)Γ(ν1)Γ(ν2 + 1)x(k, t)]2n(ν1−ν2) , (27)where −1/2 < ν1 < 1/2 and 1/2 < ν2 < 3/2.Now we are interested in obtaining the spectrum of the ψ and ϕ squared-fluctuations.Their spectrums Pϕ(k, t) and Pψ(k, t) on cosmological scales, are given respectively by theexpressionsPϕ(k, t)|IR =k32π2(ξ̃kξ̃∗k)≃ k32π2[A21(t)k−2ν2 +[B1(t)k2(ν2−ν1)∞∑n=0[α1 x(k, t)]2n(ν2−ν1)(v1 + 1)+ B2(t)k2(ν1−ν2)∞∑n=0[α2 x(k, t)]2n(ν1−ν2)(v2 + 1)]2 , (28)Pψ(k, t)|IR =k32π2(ξkξ∗k) ≃14π(p− 1) Γ(p+ 12(p− 1))2 [2(p− 1)βπ]p+1p−1k3−p+1p−1 , (29)8whereB1(t) =β(3p− 2)φ0Γ(ν2 + 1)Γ(p+12(p−1))[2β(p− 1)]p+12(p−1)+ν2−ν12πp+12(p−1)Γ(ν1 + 1) [(ν2 − ν1)(p− 1)]t−(ν1+1)(1−p), (30)B2(t) =β(3p− 2)φ0Γ(ν2)Γ(p+12(p−1))[2β(p− 1)]p+12(p−1)+2ν22πp+12(p−1)Γ(ν2 + 1) [(ν2 − ν1)(p− 1)]t−(2ν2+1)(1−p), (31)A = −BΓ(ν2) tν2(p−1)+(1−3p)/2π [2β(p− 1)] , (32)α1 = −Γ(ν2 + 1) Γ(ν1)Γ(ν2) Γ(ν1 + 1), (33)α2 = −Γ(ν1 + 1) Γ(ν2)Γ(ν2) Γ(ν2 + 1), (34)v1 =5p− 3 + 2ν1(p− 1)4(ν2 − ν1)(1− p), (35)v2 =5p− 3 + 2ν2(p− 1)4(ν2 − ν1)(p− 1). (36)Notice that for p → ∞ Pψ(k, t)|IR goes as k2. An interesting result for Pϕ(k, t) is that itdepends on 3p < m2 = 4(6πGφ20 − 1) < p(2p− 1), and hence it is required that p > 2. Onthe other hand, for sufficiently large t the first term in (28) is dominant, so thatPϕ(k, t)|IR|t→∞ ∼ k3−2ν2 ,which approaches to a scale invariant spectrum for p→ ∞.V. FINAL REMMARKSIn this work we have studied an example of the formalism developed in [14], which is anextension of the well-known formalism for gauge-invariant scalar metric fluctuations duringinflation. The formalism here examined is valid also for fluctuations with large amplitude,but the equations are very difficult to be solved due to the non-linearity of the Einsteinand Lagrange equations. In the proposal here studied vector and tensor perturbations ofthe metric are neglected and the fluid is considered as irrotacional. Of course, the analysisis only valid in a cosmological context on super Hubble scales when the universe expandsadiabatically. We have confirmated that, for small fluctuations the linear approximationsgive us the same dynamics that for the standard method. In this work we have illustratedone example where the universe grows with a scale factor a(t) ∼ tp, (with p≫ 1). We found9that, for very large p, at the end of inflation the spectrum Pϕ(k, t)|IR|t→∞ becomes scaleinvariant on cosmological scales, but Pψ(k, t)|IR|t→∞ goes as k2. However, at the beginningof inflation it is not true, because the spectrum of the ϕ-fluctuations is altered by the modesof metric fluctuations ξ̃k(t). Furthermore 〈E|ψ|E〉 ∼ 1/a, so that we conclude that at theend of inflation (and later), next order effects due to metric fluctuations on cosmologicalscales should be negligible.AcknowledgementsThe authors acknowledge UNMdP and CONICET (Argentina) for financial support.[1] A. Guth, Phys. Rev. D23, 347 (1981);A. D. 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A confirmation of agreement of different approaches for scalar gauge-invariant metric perturbations during inflation

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A confirmation of agreement of different approaches for scalar
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A confirmation of agreement of different approaches for scalar gauge-invariant metric perturbations during inflation

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