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INHOMOGENEITIES IN THE EARLY UNIVERSE

arXiv:gr-qc/9303036v1 30 Mar 1993January 1993DECOHERENCE FUNCTIONALANDINHOMOGENEITIES IN THE EARLY UNIVERSE1R. Laflamme ∗& 2A.

Matacz ∗∗1Theoretical Astrophysics, T-6, MSB288,Los Alamos National Laboratory,Los Alamos, NM87545,U.S.A.1 Department of Physics,University of Adelaide,PO Box 498 Adelaide,Australia 5001.Abstract. We investigate the quantum to classical transition of small inhomogeneousfluctuations in the early Universe using the decoherence functional of Gell-Mann and Har-tle.

We study two types of coarse graining; one due to coarse graining the value of thescalar field and the other due to summing over an environment. We compare the resultswith a previous study using an environment and the off-diagonal rule proposed by Zurek.We show that the two methods give different results.∗Email: laf@tdo-serv.lanl.gov∗∗Email: amatacz@physics.adelaide.edu.au1

1) Introduction.The gravitational instability picture for galaxy formation assumes that the early Uni-verse started with a very smooth background on which small density fluctuations weresuperimposed. It is these small fluctuations which are ultimately responsible for the struc-ture in the present Universe.

They have been amplified by the gravitational interactionsince the beginning of the matter dominated era and produced the galaxies we see.In the sixties and seventies no theories were able to predict the existence of theseperturbations, they were just postulated to be there. Zeldovich1 and Harrisson2 suggestedthat in order to fit the observation the initial spectrum of these perturbations must beroughly scale free.

In 1980 Guth3 proposed the Inflationary scenario to solve the horizon,flatness and monopole problems of the Big Bang. This scenario asserts that the Univsersewent through a phase of very rapid expansion in its very early stage.

The Universe wouldhave expanded by a factor of at least 1028 in a mere 10−32 seconds.It was soon realised that this very rapid expansion would have very interesting effecton fields especially the inhomogeneous part of the inflaton4,5. The state of small inhomo-geneities undergoes parametric amplification during the inflationary period as soon as as agiven mode crosses the Hubble radius.

This lead to a scale free spectrum. It was thereforesuggested that these inhomogeneities gave rise to the needed density fluctuations in theearly Universe.It was argued that the quantum expectation value of the square of the field < φ2 >can be interpreted as a statistical average of classical perturbations.

The argument usedby Guth & Pi2 was that (< φ2 >< π2φ >)1/2 >> ¯h and thus quantum mechanical affectsshould be negligible.Interestingly enough this consideration is not invariant under linear canonical trans-formations. To see this more clearly the Wigner function can be calculated.

In generalthe Wigner function is not positive and cannot represent a classical phase space densitydistribution but in the case of a gaussian state it is positive. So let’s assume it can give usan idea of the classical phase space distribution.

The Wigner function is defined asfw(φ, πφ) = 12πZd∆ei2ππφ∆ρ(φ −∆, φ + ∆)(1.1)2

where ρ is the state of the system. Figure 1 depicts the 1 −σ contour of the Wignerfunction for a mode k of a massless scalar field in the Bunch-Davies vacuum.

Initiallyit is an ellipse rotating with frequency k/2π whose amplitude is adiabatic. As soon asthe wavelength of the mode crosses the Hubble radius the ellipse stop rotating and getselongated in the momentum direction.

As seen from the picture the variance of φ and πφare such that (< φ2 >< π2φ >)1/2 >> ¯h but the surface of this ellipse remains ¯h. Using alinear canonical transformation so that ˜φ and ˜πφ are in the direction of the proper axis ofthe ellipse would give (< ˜φ2 >< ˜π2φ >)1/2 = ¯h.

It is therefore difficult to understand whythe quantum mechanical average can be substituted by a statistical one.Figure 1. 1 −σ contour of a Wigner function for a mode of a scalar field which hascrossed the Hubble radius.

The area remains ¯h even if (< φ2 >< π2φ >)1/2 >> ¯h.A lot of effort has recently been focused on understanding the transition betweenquantum and classical mechanics. It has been proposed that a measure of the classicalityof a system is obtained by investigating the off-diagonal terms of the density matrix6,7.In ref [8] such a criteria was used to investigate the classicality of the inhomogeneousquantum fluctuations in the inflationary period.

It was shown that these fluctuations were3

not classical if they were not interacting with an environment.A simple model of anenvironment represented by a single scalar field was constructed and it was shown that theoffdiagonal terms of the density matrix, in the configuration space basis, decreased rapidlyas soon as the mode left the Hubble radius. The main problem with this approach is theassumption that when the off-diagonal terms in configuration space vanish, the systembehaves classically.

The density matrix gives information about the field at a given instantin time but it does not indicate how a small cell in phase space evolves. It tells us only howthe sum of all these cells evolve.

Classical behaviour requires each small cell of phase spaceto evolve independently of the others, that is, for there to be no quantum interferencebetween different cells of phase space.In this paper we want to investigate a different approach to classicality, the one usingthe decoherence functional9,10. After introducing the method in section 2 we investigateits consequences for perturbations in the inflationary universe.

We study two types ofcoarse graining; one due to coarse graining of the value of the scalar field and the otherby summing over an environment as in ref [8]. We compare the coherence length of thedecoherence functional coarse grained from an environment with that of the density matrixand find striking differences.

We discuss and conclude in section 3.2) The Decoherence Functional.In recent years an approach to the quantum to classical transition has considered notthe state or eigenvalues of operators at a given time but rather focused on histories definedby a series of the value of fields at a different time9,10,11,12. The idea is that a necessarycondition for a system to be thought of as classical is that the probability sum rule fordifferent histories should be obeyed.

In other words interference should vanish. In such acase Griffith called them consistent histories.The tool to calculate the probability of an history is the decoherence functional.

Thefine grained decoherence functional for histories defined by the positions at all times canbe defined through the path integralD[h′, h] = δ(q′f −qf) exp i [S[q′(η)] −S[q(η)]] ρ(q′i, qi, ηi)(2.1)where S is the action for the given history. In order for two histories h′, h to be consistent4

the decoherence functional must be diagonal. Except for very special cases, fine grainedhistories will not decohere.

A possible way to get consistent histories is to coarse grainthem.A coarse graining of this decoherence functional can be obtained by looking at historieswith approximate position or momenta, or by summing over some field which is consideredan environment. It is the latter case which corresponds to the the decoherence studied byZeh7 and Zurek8.

A coarse graining can be defined asDc[h′, h] =Zh′ Dq′ZhDq δ(q′f −qf) exp i [S[q′(η)] −S[q(η)]] ρ(q′i, qi, ηi). (2.2)The path integral over q(η) is over all paths that start at qi at ηi, pass through the intervals∆1(η1), ∆2(η2)..., ∆n(ηn) at η1, η2...ηn and wind up at qf at time ηf.

Similarly for q′(η)which goes through primed interval but end at the same endpoint qf.We will evaluate (2.2) for a scalar field evolving in the early universe both for coarsegraining of the field or of an environment. A crucial question is how to model this envi-ronment.

Any realistic model will be very complicated and hard to analyze. However, thebasic physics should emerge from the simplest models.

Hence we use a model which can besolved exactly: the system is a real massless scalar field Φ1, (the inflaton), the environmentis taken to be a second massless real scalar field Φ2 interacting with Φ1 by their gradients.This will permit us to compare our results with the ones in ref.[8]. We consider the fieldsin the de Sitter phase of an expanding Universe with scale factor a(t) = exp(Ht), whereH is the Hubble constant.The action of system and environment isI =Zd4x√g 12(∂µΦ1)2 + (∂µΦ2)2 + 2c(∂µΦ1∂µΦ2)(2.3)where g is the determinant of the background metric with line element given byds2 = a2(−dη2 + dx2i ) .

(2.4)c is a constant measuring the strength of interaction between system and environment.We shall normalize the conformal time η such that η ranges between −∞and 0 anda = −(Hη)−1 with H−1 being the Hubble radius.5

We will study two types of coarse graining. The first one will consist in summingover the field Φ2 which mimicks the environment.

The second one will consist of coarsegraining the value of the field Φ1.Our Lagrangian is quadratic in the derivatives of the fields and can hence be diago-nalized using fields Φ+ and Φ−for which the interaction term disappears. The coherencesin the quantum state between Φ+ and Φ−are only given by the initial conditions.

Forexample, we could choose an initial state where these coherences vanish. In this case, apure state gives rise to a pure state reduced density matrix when summing over one ofthe fields.

Decoherence of one field cannot occur by summing over the other one. We,however, suppose that the inflaton and the environment do not form the diagonal basis.This assumption is reasonable since any inflaton field (whose reduced density matrix wewant) will interact with gravitational perturbations (part of the environment).We can expand the fields in harmonics in a box of fixed comoving volume (physicalvolume a3) and investigate a particular wavenumber k = (k2x + k2y + k2z)1/2.

As there is nocoupling between modes with different k, we can consider a single wavelength and dropthe index k for convenience. The Lagrangian reduces toL(q, ˙q, r, ˙r, η) = a2(η)2[ ˙q2 + ˙r2 −k2q2 −k2r2 + 2c( ˙q ˙r −k2qr)].

(2.5)For simplicity we will consider histories described by only two values of q’s, the value attime ηi and ηf. When a hamiltonian exists we ca n rewrite the decoherence functional inthe operator formalism asD(h1, h2) = Tr[U(ηf −ηi)P σq1r1ρ(ηi)P σq2r2U†(ηf −ηi)P σqfrf ].

(2.6)In order to compare with the results using the density matrix, we will look for historieswhere r is considered as an environment and at first q is fine grained. The two historiesthat we will consider are the define by starting at either q1 or q2 at ηi and ending up atqf.

The decoherence functional becomesD(h1, h2) = NZdr1dr2drfK∗(qf, rf, ηf; q2, r2, ηi)K(qf, rf, ηf; q1, r1, ηi)ρ(q1, r1; q2, r2, ηi)(2.7)6

and the propagator K isK(qf, rf, ηf; q1, r1, ηi) =ik32πH2xexp i2x"(q2f + r2f + 2cqfrf)yfH2η2f+ (q21 + r21 + 2cq1r1)yiH2ηi+ 2k3(qfq1 + rfr1 + cqfr1 + crfq1)H2(2.8)wherex = −k2ηfηi sin k∆+ k∆cos k∆−sin k∆(2.9a)yf = −k3ηfηi cos k∆−k2ηf sin k∆(2.9b)yi = −k3ηfηi cos k∆+ k2ηi sin k∆(2.9c)and ∆= ηf −ηi. If we assume that the initial to beψ(q, r, ηi) = a exp −b[q2 + r2 + 2αqr](2.10)where b is complex and α is real, we find thatD(h1, h2) = D exp i(1 −c2)2xyi(q21 −q22)H2η2i+ 2k3qf(q1 −q2)H2exp −(q21A + q22A∗+ q1q2B)(2.11)whereA =[b2(1 −α2) + bb∗(1 + c2 −2cα)]/(b + b∗)B = −2bb∗(c −α)2/(b + b∗)D =aa∗πb + b∗1/2k32πH2x.

(2.12)This decoherence functional predicts a certain coherence length Ldf, which is the maximumlength (in configuration space squared) between histories over which interference is notexponentially suppressed.However to get a better measure of the decoherence of thedecoherence functional, Ddf, we should divide Ldf by the probability width of the system7

Pdf. This is obtained by setting q1 = q2 in (2.11) and finding the length in configurationspace squared where the probability not exponentially suppressed.

We find thatDdf = PdfLdf= A + A∗−BA + A∗+ B = 1 +4bb∗(c −α)2(1 −α2)(b + b∗)2 . (2.13)When Ddf >> 1 we have significant histories decoherence.This measure for historiesdecoherence can be compared to the one used in ref[8], using the off–diagonal terms of thereduced density matrix which is given byρred(q1, q2, ηi) =Zψ(q1, r, ηi)ψ∗(q2, r, ηi)dr = aa∗πb + b∗1/2exp −(q21f + q22f ∗+ q1q2g)(2.14)whereg = −2α2bb∗/(b + b∗),f = (b2(1 −α2) + bb∗)/(b + b∗).

(2.15)In this case an analogous measure used wasDdm = PdmLdm= f + f ∗−gf + f ∗+ g = (b + b)2 −α2(b −b∗)2(1 −α2)(b + b∗)2. (2.16)This expression was analysed in ref[8] for the Bunch-Davies initial condition which corre-sponds toα = c,b =k22H2ηi(kηi + i).

(2.17)The limit of interest is long after Hubble crossing where |kηi| << 1 which implies thatb ≈k3/(2H2) −ik2/(2H2ηi). In this limit we see that Ddm >> 1, however Ddf = 1.Thus we have a situation where an arbitrarily large decoherence of the configuration spacedensity matrix corresponds to a maximally coherent decoherence functional.

The Bunch-Davies vacuum is the ground state. If we perturb the initial coupling away from c thenwe will have some histories decoherence as well as density matrix decoherence.

In thiscase the propagator will generate an imaginary contribution to α. This imaginary partwill modify (2.13) and (2.16) in a way that cancels the divergence which would otherwiseoccur for α ̸= c (α real) in the limit kηi →0.

In this case there may be closer relationship8

between the two decoherence measures. We can get an idea of the relative strengths of thetwo decoherence measures (for real α) by taking their ratios.

We findLdmLdf= A + A∗−Bf + f ∗−g = 1 +c(c −2α)bb∗(re b)2 + α2(im b)2 . (2.18)We can see that if c = 0 the two coherence lengths agree.

This can be easily seen form(2.7). The integral over rf will be proportional to δ(r1 −r2) and thus the decoherencefunctional is proportional to the initial density matrix.

It is rather surprising however thatturning on the interaction from c = 0 to c = 2α will increase the coherence of the decoher-ence functional relative to the density matrix. However for c > 2α the coherence lengthof the density matrix is larger than the coherence length of the decoherence functional.This shows that in this case there is no obvious correlation between the two decoherencemeasures and that the decoherence of the density matrix does not imply the decoherenceof the decoherence functional or vice-versa.Another possibility to obtain decoherence is to coarse grain our system in configurationspace, to which we turn now.

In this case the histories are not defined by precise valuesof q but by a range determined by σ (a variance) around a given value. The decoherencefunctional can then be obtained by integrating the fine grained one.

The Ps are projectorson a range 2σ of the fields. They are rather tedious to work with analytically.

It will beuseful to keep the analytical result simple so we use the gaussian pseudo-projectorsP σqiri =1π1/2σZ ∞−∞dzidri exp−(zi −qi)2σ2|zi, ri⟩⟨ri, zi|(2.19)They are not exactly projectors asP 2 ̸= P(2.20)but are a sufficiently good approximation for our purpose. Substituting (2.19) into (2.6)we getDc(h1, h2) =Zdz1dz2dz3 exph−(z1 −q1)2σ2−(z2 −q2)2σ2−(z3 −qf)2σ2iD(h1, h2) (2.21)9

where D(h1, h2) is given by (2.11). We choose the Bunch-Davies initial condition (2.17)for (2.11).

This is the most natural initial state to choose, it considerably simplifies thealgebra and it ensures by virtue of (2.13) that any decoherence obtained will not be dueto the environmental coarse grain. We can rewrite the result in terms of Q = q1 + q2 andδ = q1 −q2 and getDc(h1, h2) = N exp[a1Q2 + a2δ2 + a3q2f + a4Qδ + a5qfδ + a6Qqf](2.22)wherea1 = −12σ2 +14σ4M + M ∗+ 2VMM ∗−V 2a2 = −12σ2 +14σ4M + M ∗−2VMM ∗−V 2a3 = −V (M + M ∗−2V )σ2(MM ∗−V 2)a4 =M −M ∗2σ4(MM ∗−V 2)a5 =iV 1/2(M + M ∗−2V )σ3(MM ∗−V 2)a6 =iV 1/2(M −M ∗)σ3(MM ∗−V 2)(2.23)withM = 1σ2 + (1 −c2)b∗+ i(1 −c2)yi2xH2η2i+ (1 −c2)2k6σ24x2H4V =(1 −c2)2k6σ24x.

(2.24)Investigating (2.22-24) shows that coarse grained histories that are determined bytheir approximate positions at various times are not exactly consistent. Exact decoher-ence is rather difficult to obtain so we investigate approximate decoherence.

Histories areapproximatively consistent if|ReD(h1, h2)| < εMin[ReD(h1, h1), ReD(h2, h2)](2.25)10

This only means that the off-diagonal are much smaller than its corresponding diagonalpart and thus the classical sum rules applies approximatively. ε controls how good theapproximation is.

If we consider symmetrical histories (q1 = −q2, qf = 0) then it is easyto see from (2.22) that (2.25) translates mathematically asa1a2<< 1. (2.26)We also want to be able to interpret the quantum mechanical average of operators as astatistical one.

This implies that the coarse grain should be smaller than the fluctuations(∆q)2 of the field. For the Bunch Davies vacuum (2.17), in the long after Hubble crossinglimit, (∆q)2 →H2/k3 hence we requireσ2 << H2k3 .

(2.27)In general (2.23) will be very long expressions. However they simplify greatly in thelate time limit which implies that from (2.9a,2.9c) yi →−k3η2i and x →−k3∆3(3ηiηf +∆2).We further consider the limit ∆→0, ηi →0 and ηf →0.

We can take this limit whilekeeping an arbitrary constant proper time interval, δt since dηdt = −Hη. In this limit wefind thata1 →−12σ2 + 1σ4 3σ2 + (1 −c2)k3H2−1a2 →−12σ2a3 →−(1 −c2)σ22H2 + (1 −c2)k3σ23H2 + (1 −c2)k3σ2a4 →a5 →0a6 →2(1 −c2)σ4 3σ2 + (1 −c2)k3H2−1.

(2.28)For our model in the late time limit (2.26) becomesa1a2≈1/3. (2.29)11

Equation (2.29) tells us that there is weak decoherence but not a significant amount. To geta better feel for this number it is worth comparing it to the long before Hubble crossinglimit which gives a1/a2 = 1.Thus the after Hubble crossing limit does lead to somedecoherence but not a significant amount.Assuming decoherent histories (δ = 0, q = Q/2) we find that (2.22) becomes, using(2.28) and (2.27)Dc(h, h) ≈N exp −23σ2 (q −qf)2.

(2.30)Thus at late times the histories would be peaked about qf ≈q which is exactly the behaviorof the classical motion.3) Discussion and conclusion.We see from (2.16) that the decoherence in the density matrix is due purely to thephase of the wave-function. This dependence on the phase is interesting since the phasecan always be changed by a point transformation on the Lagrangian.

We can see this asfollows. A point transformation will transform the Lagrangian asL(⃗q(t), ˙⃗q(t)) →L(⃗q(t), ˙⃗q(t)) −ddtf(⃗q(t), t)(3.1)which in turn means that the action transforms asS[q(t)] →S[q(t)] −f(⃗qf, tf) + f(⃗qi, ti)(3.2)This point transformation doesn’t affect the classical equation of motion because they arederived from the stationary action condition δS = S[q(t)] −S[q(t) + δq(t)] = 0 whereδq(t) vanishes at the endpoints.

However from the general expression U(⃗qf, tf; ⃗qi, ti) =N Ppaths eiS for the quantum propagator we can see that under the transformation (3.2)the quantum propagator transforms asU(⃗qf, tf; ⃗qi, ti) →e−if(⃗qf ,tf )U(⃗qf, tf; ⃗qi, ti)eif(⃗qi,ti)(3.3)which in turn means that the wave function transforms asψ(⃗q, t) →e−if(⃗q,t)ψ(⃗q, t)(3.4)12

Physics is generally considered invariant under the point transformation (3.1) becauseexpectation values of functions of q and the physical momenta ˙q are invariant (it is im-portant to remember that the canonical momenta does change with (3.2)). However thereduced density matrix of a subsystem in is not invariant to these point transformations.A point transformation is exactly what is being done when surface terms are dropped ina lagrangian.

Using (3.3) and (3.2) we can see that the decoherence functional (2.7) isinvariant under point transformations. This is an important difference between the twoformalisms.In models with more general couplings we should expect decoherence of the reduceddensity matrix to depend not only on the phase but also on the real part of the exponent ofthe wave function.

In this case there might be a simpler relation between the decoherencefunctional and the evolution of the density matrix.It is also interesting to investigate the influence functional for this model. Naively wemight relate a diagonal decoherence functional with the existence of a noise kernel in theinfluence functional.

Consider (2.2) where q →(q, r), q′ →(q′, r′) and the r, r′ coordinatesare completely coarse-grained out. In this case (2.2) becomesD[h′, h] =Zh′ DqZhDq′Zdqidq′idqfdq′f δ(qf −q′f) exp i [Sf[q(η)] −Sf[q′(η)]] F[q(η), q′(η)](3.5)where F[q(η), q′(η)] the influence functional isF[q(η), q′(η)] =Zdridr′idrfdr′f δ(rf −r′f)ρ(qi, ri; q′i, r′i, ηi)×Z (rf ,r′f ,ηf)(ri,r′i,ηi)DrDr′ exp i [Sf[r(η)] + Si[q(η), r(η)] −Sf[r′(η)] −Si[q′(η), r′(η)]] .

(3.6)For our model (2.5) with the Bunch-Davies initial condition (2.17) we find that the influencefunctional isF[q(η), q′(η)] = expic22Z ηfηia2( ˙q′2 −k2q′2) −ic22Z ηfηia2( ˙q2 −k2q2)× exp"−(1 −c2)biq2i −(1 −c2)b∗i q′2i −c2bfb∗f(qf −q′f)2bf + b∗f#. (3.7)13

A striking feature of (3.7) is the absence of a noise kernel that is typically associated withdecoherence. This is due to the very special form of interaction we have chosen.

The resultis that in (3.5), there is no exponential suppression of widely separated histories and henceno histories decoherence. The influence functional will still not have a noise kernel even ifthe initial state does not have c = α.

This shows that the absence of noise kernel does notimply a coherent evolution.We have shown that the decoherence functional shows some decoherence for the in-teraction given in eq. (2.3) for a wide selection of initial states.

We have also shown thatthere is a surprising result for the case c = α as we have already mentioned. This casesurely needs further study in order to understand why a mixed state can lead to a maxi-mally coherent (factorizable) decoherence functional.

We also considered the possibility ofdecoherence after Hubble crossing though coarse graining the system field. We found thisled to weak decoherence after Hubble crossing but probably not enough for an effectivequantum to classical transition.14

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