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arXiv:hep-th/0506236v12

8Jun2005

UTTG-01-05

Quantum Contributions to Cosmological Correlations

Steven Weinberg

Theory Group, Department of Physics, University of Texas

Austin, TX, 78712

Abstract

The in-in formalism is reviewed and extended, and applied to the calcula-tion of higher-order Gaussian and non-Gaussian correlations in cosmology.Previous calculations of these correlations amounted to the evaluation oftree graphs in the in-in formalism; here we also consider loop graphs. Itturns out that for some though not all theories, the contributions of loopgraphs as well as tree graphs depend only on the behavior of the inflatonpotential near the time of horizon exit. A sample one-loop calculation ispresented.

Electronic address: [email protected]

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I. INTRODUCTION

The departures from cosmological homogeneity and isotropy observedin the cosmic microwave background and large scale structure are small,so it is natural that they should be dominated by a Gaussian probabilitydistribution, with bilinear averages given by the terms in the Lagrangianthat are quadratic in perturbations. Nevertheless, there is growing interestin the possibility of observing non-Gaussian terms in various correlationfunctions,1 such as an expectation value of a product of three temperaturefluctuations. It is also important to understand the higher-order correctionsto bilinear correlation functions, which appear in Gaussian correlations.

Until now, higher-order cosmological correlations have been calculated

by solving the classical field equations beyond the linear approximation.As will be shown in the Appendix, this is equivalent to calculating sumsof tree graphs, though in a formalism different from the familiar Feynmangraph formalism. For instance, Maldacena2 has calculated the non-Gaussianaverage of a product of three scalar and/or gravitational fields to first orderin their interactions, which amounts to calculating a tree graph consistingof a single vertex with 3 attached gravitational and/or scalar field lines.

This paper will discuss how calculations of cosmological correlations canbe carried to arbitrary orders of perturbation theory, including the quantumeffects represented by loop graphs. So far, loop corrections to correlationfunctions appear to be much too small ever to be observed. The present

work is motivated by the opinion that we ought to understand what ourtheories entail, even where in practice its predictions cannot be verifiedexperimentally, just as field theorists in the 1940s and 1950s took pains tounderstand quantum electrodynamics to all orders of perturbation theory,even though it was only possible to verify results in the first few orders.

There is a particular question that will concern us. In the familiar calcu-lations of lowest-order Gaussian correlations, and also in Maldacenas tree-graph calculation of non-Gaussian correlations, the results depended onlyon the behavior of the unperturbed inflaton field near the time of horizonexit. Is the same true for loop graphs? If so, it will be possible to calcu-lated the loop contributions with some confidence, but we can learn littlenew from such calculations. On the other hand, if the contribution of loopgraphs depends on the whole history of the unperturbed inflaton field, thencalculations become much more difficult, but potentially more revealing. Inthis case, it might even be that the loop contributions are much larger thanotherwise expected.

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see that the time derivatives of the right-hand sides are equal up to order

N. Eqs. (1) and (2) also give the same results for t to all orders, sothey give the same results for arbitrary t to order N.

III. THEORIES OF INFLATION

To make our discussion concrete, in this section we will take up a partic-ular class of theories of inflation. The reader who prefers to avoid details ofspecific theories can skip this section, and go on immediately to the generalanalysis of late-time behavior in the following section.

In this section we will consider theories of inflation with two kinds of

matter fields : a real scalar field (x, t) with a non-zero homogeneous ex-pectation value (t) that rolls down a potential V(), and any numberof real massless scalar fields n(x, t), which have only minimal gravitationalinteractions, and are prevented by unbroken symmetries from acquiring vac-uum expectation values. The real field serves as an inflaton whose energydensity drives inflation, while the n are a stand-in for the large number ofspecies of matter fields that will dominate the effects of loop graphs on thecorrelations of the inflaton field.

We follow Maldacena,2 adopting a gauge in which there are no fluctua-tions in the inflaton field, so that (x, t) = (t), and in which the spatialpart of the metric takes the form

gij =a2e2[exp ]ij , ii= 0, iij = 0 . (3)

where a(t) is the RobertsonWalker scale factor, ij(x, t) is a gravitationalwave amplitude, and (x, t) is a scalar whose characteristic feature is that

Standard counting arguments show that in these theories the number of factors of 8Gin any graph equals the number of loops of any kind, plus a fixed number that dependsonly on which correlation function is being calculated. Matter loops are numericallymore important than loops containing graviton or inflaton lines, because they carry anadditional factor equal to the number of types of matter fields.I am adopting Maldacenas notation, but the quantity he calls is more usually calledR. To first order in fields, the quantity usually called is defined as H/ , whilethe quantity usually called R is defined as + Hu. (Here the contribution of scalarmodes to gij is written in general gauges as 2a

2(ij+ 2/xixj), while and

are the perturbation to the total energy density and its unperturbed value, while u isthe perturbed velocity potential, which for a single inflaton field is u= / .) In thegauge used by Maldacena and in the present paper u= = 0, so since is defined hereas to first order in fields, it corresponds to the quantity usually called R. Outside thehorizon R and are the same.

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it is conserved outside the horizon,5 that is, for physical wave numbers that

are small compared with the expansion rate. The same is true ofij .The other components of the metric are given in the ArnowittDeser

Misner (ADM) formalism6 by

g00= N2 + gijNiNj , gi0= gijNj , (4)where N and Ni are auxiliary fields, whose time-derivatives do not appearin the action. The Lagrangian density in this gauge (with 8G 1) is

L = a3

2e3

N R(3) 2N V() + N1

Ej iE

ij (Eii)2

+ N1

2

+ N1

n

n N

i

in2 Na2e2[exp()]ij

ninjn

,

(5)

where

Eij 12

gij iNj jNi

, (6)

and bars denote unperturbed quantities. All spatial indices i, j, etc. arelowered and raised with the matrix gij and its reciprocal;i is the three-dimensional covariant derivative calculated with this three-metric; and R(3)

is the curvature scalar calculated with this three-metric:

R(3) =a2e2eijR(3)ij

.

The auxiliary fields N and Ni are to be found by requiring that the actionis stationary in these variables. This gives the constraint equations:

i

N1

Eij ijEkk

= N1n

jn

n Niin

, (7)

N2

R(3) 2V a2e2[exp()]ijn

injn

= EijE

ji

Eii

2

+ 2

+

n

n Niin2

(8)

For instance, to first order in fields (including field derivatives) the auxiliaryfields are the same as in the case of no additional matter fields2

N= 1 + /H , N i = 1a2H

i+ i2 , (9)

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where

HH2

= 2

2H2 , H a

a (10)

The fields in the interaction picture satisfy free-field equations. For wehave the Mukhanov equation:7

2

t2 +

d ln(a3)

dt

t a22= 0, (11)

The field equation for gravitational waves is

2ijt2

+ 3Hij

t a22ij = 0, (12)

and for the matter fields

2nt2

+ 3Hnt

a22n = 0. (13)

The fields in the interaction picture are then

(x, t) =

d3q

eiqx(q)q(t) + e

iqx(q)q (t)

, (14)

ij(x, t) =

d3q

eiqxeij(q, )(q, )q(t) + e

iqxeij(q, )(q, )q (t)

,

(15)n(x, t) =

d3q

eiqx(q, n)q(t) + e

iqx(q, n)q (t)

, (16)

where = 2 is a helicity index and eij(q, ) is a polarization tensor, while(q), (q, ), and (q, n) are conventionally normalized annihilation oper-ators, satisfying the usual commutation relations

(q) , (q)

= 3

q q

,

(q) , (q)

= 0. (17)

(q, ) , (q, )

=

3

q q

,

(q, ) , (q, )

= 0, (18)

and(q, n) , (q, n)

= nn

3

q q

,

(q, n) , (q, n)

= 0, (19)

Also, q(t), q(t), and q(t) are suitably normalized positive-frequency so-lutions of Eqs. (11)(13), with2 replaced withq2. They satisfy initial

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Dangerous interactions, which grow at late times no faster thana22,

and contain only fields, not time derivatives of fields.

These conditions are evidently met by the interaction (27), irrespective ofthe value of, and, as we shall see in Section VI, they are satisfied by allother interactions in the theories of Section III, but not in all theories.

Before proceeding to the proof, it should be noted that just as in Eq. (27),the space dimensionality 2enters in the interaction only in a factor

Detga2, so the question of whether or not a given theory satisfies the conditionsof this theorem does not depend on the value of 2. Thus this theorem hasthe corollary:

CorollaryThe integrals over the time coordinates of interactions convergeexponentially for t , essentially as dt/an(t) with n > 0, provided

that in 3 space dimensions all interactions are of one or the other of twotypes:

Safe interactions, that contain a number of factors ofa(t) (including2 factors of a for each time derivative and the 3 factors of a fromDetg) strictly less than +1, and

Dangerous interactions, which grow at late times no faster thana,andcontain only fields, not time derivatives of fields.

Here is the proof. As already mentioned, the reason that dangerous

interactions are not necessarily fatal has to do with how they enter intocommutators in Eq. (2). Because of the time-ordering in Eq. (2), any failureof convergence of the time integrals for t + in Nth-order perturba-tion theory must come from a region of the multi-time region of integra-tion in which, for some r, the time arguments tr, tr+1, . . . tN, all go toinfinity together. We will therefore have to count the number of factors ofa(tr), a(tr+1), . . . a(tN), treating them all as being of the same order of mag-nitude. (This does not take proper account of factors of log a, but as long asthe integral over tr, tr+1, . . . tN involves a negative total number of factorsofa, it converges exponentially fast no matter how many factors of log aarisefrom subintegrations.) Now, at least one of the fields or field time derivatives

in each term inH(ts) withr s Nmust appear in a commutator with oneof the fields in some other HI(ts) with s < s

N. So we need to considerthe commutators of fields at times which may be unequal, but are both late.In the sense described above, treating all a(tr), a(tr+1), . . . a(tN) as being

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of the same order of magnitude, ifa(t) increases more-or-less exponentially,

then the commutator of two fields or a field and a field time-derivative goesas a2, while the commutator of two field time-derivatives goes as a22.

For instance, the unequal-time commutators of the interaction-picturefields (14)(16) are

(x, t), (x, t)

=

d2p eip(xx

)

p(t)p(t

) p(t)p (t)

, (28)

ij(x, t), kl(x

, t)

=

d2p eip(xx

) ijkl(p)

p(t)p (t

) p(t)p(t)

,

(29)

n(x, t), m(x

, t)

= nm

d2p eip(xx

)

p(t)

p(t

) p(t)p(t),

(30)where ijkl(p)

eij(p, )ekl(p, ). The two asymptotic expansions given

in Eqs.(21(23) for each of the fields are both real aside from over-all factors,so neither by itself contributes to the commutators. On the other hand, theconstantsCp

op , Dp

op , and Ep

op are in general complex. (For instance,

in a strictly exponential expansion, inflation, the phase ofCpop is given by

a factorei .) The asymptotic expansions of the commutators at latetimes are therefore

(x1, t1), (x2, t2)

2i

d2pIm

Cpop

eip(x1x2)

t2t1

dt

a2(t)(t)

+p2 t2t1

dt

a2(t)(t) t a

22

(t

) (t

) dt t

dt

a2(t) (t)

+p2t1

dt1a2(t1) (t

1)

t2

dt2a2(t2) (t

2)

t2t1

a22(t)(t) dt + . . .

(ij(x1, t1), kl(x2, t2)

2i

d2pijkl(p) Im

Dp

op

eip(x1x2)

t2t1

dt

a2(t)

+p2 t2t1

dt

a2(t)

t

a22(t) dtt

dt

a2(t)

+p2

t1

dt1

a2

(t1)

t2

dt2

a2

(t2)

t2

t1

a22(t) dt + . . . ,(

n(x1, t1), m(x2, t2)

2i nm

d2pIm

Epop

eip(x1x2)

t2t1

dt

a2(t)

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+p2 t2t1

dt

a2(t) t a

22

(t

) dt t

dt

a2(t)

+p2t1

dt1a2(t1)

t2

dt2a2(t2)

t2t1

a22(t) dt + . . .

.

(

We see that the commutator of two fields vanishes essentially as a2 forlate times, and the same is true for the commutator of a field and its timederivative, but the commutators of two time derivatives arise only from thethird terms in the expansions (31)(33), and therefore go as a22. Thatis,

(x1, t1), (x2, t2)

2i d2pImCpop eip(x1x2)

p2

a2(t1) (t1)a2(t2) (t2)

t2t1

a22(t)(t) dt + . . .

,

and likewise for ij and n.Lets now add up the total number of factors ofa(tr), a(tr+1), . . . anda(tN)

in the integrand of Eq. (2), for some selection of terms in the interactionsH(ts) withr s N. Suppose that the selected term in H(ts) contains anexplicit factora(ts)

As, andBs factors of field time derivatives. Suppose alsothat in the inner N r+ 1 commutators in Eq. (2) there appear C com-mutators of fields with each other, C commutators of fields with field timederivatives, and C commutators of field time derivatives with each other.The number of field time derivatives that are not in these commutators issBsC2C, and these contribute a total 2

sBs+ 2C

+ 4C factorsofa. (All sums over s here run from r to N.) In addition, there are

sAs

factors ofa that appear explicitly in the interactions, and as we have seen,the commutators contribute2C 2C (2+ 2)C factors ofa. Hencethe total number of factors ofa(tr), a(tr+1), . . . anda(tN) in the integrandof Eq. (2) is

# =s

(As2Bs)2C (22)(C + C) =s

(As2Bs2+ 2)2C ,(34)

in which we have used the fact that the total number C+ C + C of com-mutators of the interactions H(tr), H(tr+1), . . . andH(tN) with each otherand with the field productQequals the number of these interactions. Underthe assumptions of this theorem, all interactions have As 2Bs 2 2.

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If any of them are safe in the sense that As

2Bs < 2

2, then # < 0,

and the integral over time converges exponentially fast. On the other hand,if all of them have As 2Bs = 2 2, then under the assumptions of thistheorem they all involve only fields, not field time derivatives, so the sameis true of the commutators of these interactions. In this case C > 0 and# = 2C

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mation of extending the time integrals to +

is not valid, and that these

integrals can be taken only to some time t late in inflation. The decreaseof the integrand at wave numbers p much greater than1/(t) would thenprovide the ultraviolet cut off that is still needed, but the correlation func-tions would exhibit the sort of time dependence that has been found in othercontexts by Woodard and his collaborators,3 and we would not be able todraw conclusions about correlations actually measured at times much closerto the present. The possible presence of such ultraviolet divergences thatare not removed by renormalization and field redefinition is an importantissue, which merits further study. But even if such ultraviolet divergencesare present, it would still be possible to calculate the non-polynomial partof the integrals over internal momenta which is not ultraviolet divergent (at

least in one loop order) even when the time t is taken to infinity. Such acalculation will be presented in Section VII.

V. AN EXAMPLE: EXPONENTIAL EXPANSION

To clarify the issues discussed at the end of the previous section, wewill examine a simple unphysical model, along with a revealing class ofgeneralizations.

First, consider a single real scalar field(x, t) in a fixed de Sitter metric.

In order to implement dimensional regularization, we work in 2 space di-mensions, letting

3/2 at the end of our calculation. The Lagrangian

density is taken as

L = 12

Detg g(1+2) = (1+2)

a2

22 a

22

2 ()2

,

(35)

Many theories are afflicted with infrared divergences, even when t is held fixed. Theinfrared divergences are attributed to the imposition of the unrealistic initial condition,that at early times all of infinite space is occupied by a BunchDavies vacuum. Theinfrared divergence can be eliminated either by taking space to be finite 8 or by changingthe vacuum.9 In any case, it is the appearance of uncancelled ultraviolet rather thaninfrared divergences when we integrate over internal wave numbers after taking the limitt that shows the impropriety of this interchange of limit and integral, because factors

of 1/a in the integrand are typically accompanied with factors of internal wave numbers,so that the 1/a factors do not suppress the integrand for large values ofa if the integralreceives contributions from arbitrarily large values of the internal wave number.

This model, and much of the analysis, was suggested to me by R. Woodard, privatecommunication.

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where a

eHt with H constant. (This of course can be rewritten as a free

field theory, but it is instructive nonetheless, and will be generalized laterin this section to interacting theories.) We follow the usual procedure ofdefining a canonical conjugate field =L/, constructing the Hamilto-nian densityH = L with expressed in terms of, dividingH intoa quadratic partH0 and interaction partHI, and then replacing inHIwith the interaction-picture I given by = [H0/]=I. This gives aninteraction

HI=

2

d2x

a

2

2

2

1 + 2, 2

+ a22()22

. (36)

(An anticommutator is needed in the first term to satisfy the requirementthat HIbe Hermitian.) This interaction satifies the conditions of the theo-rem proved in the previous section for any value of the space dimensionality2: the first term in the square brackets contains 2 4 factors ofa (count-ing a factor a2 for each time derivative, so it is safe, while the second termcontains 2 2 factors ofa, and is therefore dangerous, but it only involvesfields (including space derivatives), not their time derivatives, so thoughdangerous it still satisfies the conditions of our theorem.

To first order in , the expectation value(x, t) (x , t) is given by aone-loop diagram, in which a scalar field line is emitted and absorbed at thesame vertex, with the two external lines also attached to this vertex. Thisexpectation value receives contributions of three kinds:

i Terms in which no time derivatives act on the internal lines. This contribu-tion is the same as would be obtained by adding effective interactionsproportional to a22()2, a222, or a22, all of which satisfythe conditions of the theorem of the previous section. Thus it can-not affect the conclusion that the integral over the time argument ofHI(t1) converges exponentially at t1 = +, so that(x, t) (x, t)approaches a finite limit for t .

ii Terms in which time derivatives act on both ends of the internal line. Thisproduces an effective interaction proportional toa22, which violatesthe conditions of our theorem, but it can be removed by adding anR2Detgcounterterm in the Lagrangian. (This cancellation is notautomatic, because the condition of minimal coupling is not enforcedby any symmetry.)

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The ultraviolet divergent integral over p is the price we pay for the naugh-

tiness of taking the limit t before we integrate over p.In this model it is clear how to remedy the difficulty of calculating corre-

lation functions at late times. As already mentioned, the original Lagrangiandensity (35) actually describes a free field theory. This is made manifest bydefining a new scalar field

1 + 2 d , (44)

for which the Lagrangian density takes the form

L = 12

Detg g . (45)There is no problem in taking the late-time limit of the correlation function

d2eiq(xx)(x, t) (x, t) it is just 22H21()2/324q2. From

this point of view, the growth of the correlation function (42) at late times isa result of our perversity in calculating the correlation function of insteadof .

Can we find fields whose correlation functions have a constant limit atlate times in theories that satisfy the conditions of our theorem but are notequivalent to free field theories? The general answer is not known, but hereis a class of interacting field theories for which such renormalized fieldscan be found. This time we consider an arbitrary number of real scalar fieldsn(x, t) in a fixed de Sitter metric. The Lagrangian density is taken to have

the form of a non-linear -model:L = 1

2

nm

Detg gnm+ Knm() n m , (46)where Knm() is an arbitrary real symmetric matrix function of the n; is a coupling constant; and again aeHt with H constant. The Hamilto-nian derived from this Lagrangian density does satisfy the conditions of thetheorem of Section IV, whatever the function Knm().

To first order in , the same problem discussed earlier in this sectionarises from graphs in which an internal line of the field n is emitted andabsorbed from the same vertex, with a time derivative acting on just oneend of this line. Depending on what correlation function is being calculated,

the contribution of such graphs is proportional to various contractions ofpartial derivatives of the function

Am() n

Knm()

n. (47)

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Suppose we make a redefinition of the fields of first order in :

n n n(). (48)

This changes the matrix K to

Knm() =Knm() +n()

m+

m()

n, (49)

and so

Am() n

Knm()

n=Am() +

n

2n()

nm+n

m()

nn. (50)

Thus the fields n are renormalized, in the sense that to first order in correlation functions have finite limits at late times, provided that

n

2n()

nm+n

m()

nn= Am(). (51)

This can be solved by first solving the Poisson equation

n

2B()

nn= 1

2

n

An()

n(52)

and then solving a second Poisson equation

n

2m()

nn= Am() B()

m. (53)

Thus for at least to first order in this class of theories, it is always possibleto find a suitable set of renormalized fields.

Because we can take the limit t only for the correlation functionsof suitably defined fields (such as n in our example), the question naturallyarises, whether these are the fields whose correlation functions we want tocalculate. The answer is conditioned by the fact that astronomical observa-tions of the cosmic microwave background or large scale structure are madefollowing a long era that has intervened since the end of inflation, duringwhich things happened about which we know almost nothing, such as re-heating, baryon and lepton synthesis, and dark matter decoupling. The onlything that allows us to use observations to learn about inflation is that somequantities were conserved during this era, while fluctuation wave lengths

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were outside the horizon. These are the only quantities whose correlation

functions at the end of inflation can be interpreted in terms of current ob-servations. In the classical limit, the quantities that are conserved outsidethe horizon areandij, but we dont know whether this will be true whenquantum effects are taken into account. Still, we can expect that quantitiesare conserved only when there is some symmetry principle that makes themconserved, and whatever symmetry principle keeps some quantity conservedfrom the end of inflation to the time of horizon re-entry is likely also to keepit conserved from the time of horizon exit to the end of inflation. So we mayguess that the quantities whose correlation functions we will need to knoware just those whose correlation functions approach constant limits at theend of inflation.

VI. DANGEROUS INTERACTIONS IN INFLATIONARY

THEORIES

We now return to the semi-realistic theories described in Section III. Wewill show in this section that all interactions are of the type called for in thetheorem of Section IV; that is, they are all safe interactions that (in threespace dimensions) do not grow exponentially at late times (and in fact aresuppressed at late times at least by a factora1), or dangerous interactionscontaining only fields and not their time derivatives, which grow no fasterthat a at late times. Fortunately, as noticed by Maldacena2 in a differentcontext, for this purpose it is not necessary to solve the constraint equations(7) and (8), which are quite complicated especially when the n fields areincluded. Inspection of these equations shows that when we count , ij ,and n as of order a

2, the auxiliary fields N 1 and Ni are both also ofordera2. This is apparent in the first-order solution (9) of the constraintequations, but it holds to all orders in the fields. To calculate the quantityEj iE

ij (Eii)2 in Eq. (5), we note that

Eij =H ij+

ij+

1

2

e

teij1

2

iNj+ jNi

. (54)

The first term Hij is of order zero in a, while all other terms are of ordera2, so

Ej iEij (Eii)2 = 6H2 12H 4HkNk + O(a4) (55)

In counting powers ofa, note that the three-dimensional affine connection and Riccitensor are independent ofa, so the curvature scalar R(3) goes as a2. For instance, forij = 0, we have R

(3) = a2e2(42+ 2()2).

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(In deriving this result, we note that et

eii =

ii = 0.) The terms

in (5) of first order in N 1 all cancel as a consequence of the constraintequation (8), while terms of second order in N 1 in Eq. (5) (and in partic-ular ina3e3

2/2N and 3H3a3e3/2N) are suppressed by at least a factor

a3(a2)2, and are therefore safe. Therefore we can isolate all terms that arepotentially dangerous by setting N= 1, and find

L = a3

2e3

R(3) 2V() 6H2 12H 4HkNk

+ 2 a2e2[exp ()]ij

n

injn

+ O(a1),

(56)

We note thate3kNk =k(e3Nk), so this term vanishes when integratedover three-space, and therefore makes no contribution to the action. Theterm proportional to can be written

6a3e3H= t

2a3 H e3

+ a3e3

6H2 + 2H

.

The first term vanishes when integrated over time, so it gives no contributionto the action. To evaluate the remaining terms we use the unperturbedinflaton field equation, which (with 8G 1) gives H = 2/2, and theFriedmann equation, which gives 6H

2

= 2V+

2

. We then find a cancellation

V 3H2 +12

2

+ 6H2 + 2H= 0 .

Aside from terms that make no contribution to the action, the Lagrangiandensity is then

L = a3

2e3

R(3) a2e2[exp()]ij

n

injn

+ O(a1). (57)

We see that, at least in this class of theories, the dangerous terms that arenot suppressed by a factor a1 grow at most likea at late times, and involve

only fields, not their time derivatives, as assumed in the theorem of SectionIII.

It remains to be seen if in these theories, after integrating over times andtaking the limit t , the remaining integrals over internal wave numbers

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are made convergent by the same counterterms that eliminate ultraviolet di-

vergences in flat spacetime, and if not, whether they can be made convergentby suitable redefinitions of the fields and ij appearing in the correlationfunctions. This is left as a problem for further work.

Not all theories satisfy the conditions of the theorem of Section IV. Forinstance, a non-derivative interaction

DetgF() of the fields wouldhave +3 factors ofa, and hence would violate the condition that the totalnumber of factors of a (counting each time derivative as -2 factors) mustbe no greater than +1. The fields must be the Goldstone bosons of somebroken global symmetry in order to satisfy the conditions of our theorem ina natural way.

VII. A SAMPLE CALCULATION

As an application of the formalism described in this paper, we will nowcalculate the one-loop contribution to the correlation function of two fields,which is measured in the spectrum of anisotropies of the cosmic microwavebackground. As already mentioned, in the class of theories described inSection III, this two-point function is dominated by a matter loop, becausethere are many types of matter field and only one gravitational field. Wefirst consider the contribution of second order in the interaction (27). Itsaves a great deal of work if we use the interaction-picture field equations(11) and (13) to put this interaction in the form

H (t) =

d3x L (x, t) =A(t) + B(t) (58)

where

A = 2Ha5n

d3x 2n2 (59)

B =n

d3x

a

H a32

1

2(n)2 +1

2a22n

. (60)

In general, for an interaction Hamiltonian of the form (58), Eq. (2) can beput in the form

Q(t) =N=0

iN t

dtN

tN

dtN1 t2

dt1

HI(t1),

HI(t2),

HI(tN), QI(t)

, (61)

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where

HI(t) =eiB(t)

A(t)+B(t)+ieiB(t)

d

dteiB(t)

eiB(t) =A(t)+i[B(t), A(t)]+

i

2[B(t), B(t)]+. . .

(62)

QI(t) =eiB(t)QI(t)eiB(t) =QI(t)+i[B(t), QI(t)]12

[B(t), [B(t), QI(t)]]+. . . .

(63)To second order in an interaction of the form (58), the expectation value isthen

Q(t)2 = t

dt2

t2

dt1

A(t1),

A(t2), QI(t)

t

dt1

B(t1), A(t1) + B(t1)/2

, QI(t)

B(t),

B(t), QI(t)

, (64)

The Fourier transform of the second-order term in the expectation value ofa product of two s is then

d3x eiq(xx)

vac, in(x, t) (x, t)vac, in

2

= 32(2)9

q4 Re

t

a5(t2) (t2) H(t2) dt2

t2

a5(t1) (t1) H(t1) dt1

q(t1) q (t)

q(t2) q (t) q(t)q (t2)

N

d3p

d3p 3(p + p + q)

p(t1) p(t2) p(t1) p(t2)

+(2)3

4q4N

d3p

d3p 3(p + p + q)

(p p)2 |p(t)|2 |p(t)|2+ . . . (65)

whereN is the number of fields. We have shown here explicitly thecontribution of the first and third lines on the right-hand side of Eq. (64).The dots represent one-loop contributions of the second line, in which [B, A+B/2] plays the role of aseagull interaction, as well as one-loop terms

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of first order in the terms in Eq. (5), in both of which the integral

over internal wave number is q-independent, plus counterterms arising infirst order from interactions that cancel ultraviolet divergences in flat space,including

Detg RRandDetg R2 terms in the Lagrangian density

that are not included in Eq. (5).Though it has not been made explicit in this section, we use dimensional

regularization to remove infinities in the integrals over p and p at interme-diate stages in the calculation, and we now assume that the singularity asthe number of space dimensions approaches three is cancelled by the termsin Eq. (65) represented by dots, leaving it to future work to show that thisis the case. Then these integrals are dominated bypp q. As we haveseen, the integrals over time are then dominated by the time tq of horizon

exit, whenq/a(tq) H(tq). For simplicity, we will assume (for the first timein this paper) that the unperturbed inflaton field (t) is rolling very slowlydown the potential at time tq, so that the expansion near this time can beapproximated as strictly exponential, a(t)eHt . Then the wave functionsare

q(t) oqeiq

1 + iq

,

q(t) oq eiq

1 + iq

,

where is the conformal time

t

dt

a(t)

,

and the wave functions outside the horizon have modulus

|oq |2 = H2(tq)

2(2)3 q3 , |oq |2 =

H2(tq)

2(2)3 (tq) q3

Using these wave functions in Eq. (65) gives d3x eiq(xx

)

vac, in(x, t) (x, t)vac, in

2

=(8GH2(tq))

2N(2)3 d

3p d3p 3(p + p + q)

p p

q7 (p+p + q)+

(p p)216 q4p3p3

+ . . . (66)

with the dots having the same meaning as in Eq. (65).

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Simple dimensional analysis tells us that when the integral over internal

wave numbers of the first term in square brackets is made finite by dimen-sional regularization, it is converted to

d3p

d3p 3(p + p + q)

p p

p +p + q q4+F(), (67)

where is a measure of the difference between the space dimensionalityand three. The ultraviolet divergences in this integrals for = 0 gives thefunction F() a singularities as 0:

F() F0

+ F1, (68)

so that in the limit = 0 d3p

d3p 3(p + p + q)

p p

p+p + q =q4

F0ln q+ L

, (69)

where L is a divergent constant. We can easily calculate the coefficientF0of the logarithm. For this purpose, we note that, in general,

d3p

d3p 3(p + p + q)f(p, p, q) =

2

q

0

p dp

p+q|pq|

p dp f(p, p, q)

(70)To eliminate the divergence in the integral over p and p, we multiply by q

and differentiate six times with respect to q. A tedious but straightforwardcalculation gives

d6

dq6

q

d3p

d3p 3(p + p + q)

p p

p +p + q

= 8

q

Comparing this with the result of applying the same operation to Eq. (69)then gives F0= /15.

In contrast, the integral of the second term in square brackets in Eq. (66)is a sum of powers ofqwith divergent coefficients, but with no logarithmicsingularity in q. (This term would be eliminated if we calculated the ex-pectation value of a product of fields

exp(

iB)exp(iB) instead of.)

The terms represented by dots in Eq. (65) make contributions that are alsojust a sum of powers ofqwith divergent coefficients. We are assuming thatall ultraviolet divergences cancel, but we cannot find resulting finite powerterms without knowing the renormalized coefficients of the

DetgRR

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and

DetgR2 terms in the Lagrangian density. So we are left with the

result (now restoring a suitable power of 8G) that d3x eiq(xx

)

vac, in(x, t) (x, t)vac, in

2

=

8GH2(tq)2N

15(2)3q3

ln q+ C

(71)

withCan unknown constant. This may be compared with the classical (andclassic) result, that in slow roll inflation this correlation function takes theform

d3x ei

q(xx

)

vac, in(x, t) (x, t)vac, in0

= 8GH2(tq)

4(2)3|(tq)|q3 (72)

The one-loop correction (71) is smaller by a factor of order 8GH2N |(tq)|,so even ifN is 102 or 103 this correction is likely to remain unobservable.Still, it is interesting that even in the extreme slow roll limit, where H(tq)and (tq) are nearly constant, the factor ln qgives it a different dependenceon the wave number q.

ACKNOWLEDGMENTS

For helpful conversations I am grateful to K. Chaicherdsakul, S. Deser,W. Fischler, E. Komatsu, J. Maldacena, A. Vilenkin, and R. Woodard. Thismaterial is based upon work supported by the National Science Foundationunder Grants Nos. PHY-0071512 and PHY-0455649 and with support fromThe Robert A. Welch Foundation, Grant No. F-0014, and also grant supportfrom the US Navy, Office of Naval Research, Grant Nos. N00014-03-1-0639and N00014-04-1-0336, Quantum Optics Initiative.

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APPENDIX: THE IN-IN FORMALISM

1. Time Dependence

First, it is necessary to be precise about the origin of the time-dependenceof the fluctuation Hamiltonian in applications such as those encountered incosmology. Consider a general Hamiltonian system, with canonical variablesa(x, t) and conjugates a(x, t) satisfying the commutation relations

a(x, t), b(y, t)

= iab3(xy),

a(x, t), b(y, t)

=

a(x, t), b(y, t)

= 0,

(A.1)and the equations of motion

a(x, t) =iH[(t), (t)], a(x, t), a(x, t) =iH[(t), (t)], a(x, t).(A.2)

Here a is a compound index labeling particular fields and their spin com-ponents. The HamiltonianHis a functional of the a(x, t) and a(x, t) atfixed time t, which according to Eq. (A.2) is of course independent of thetime at which these variables are evaluated.

We assume the existence of a time-dependent c-number solution a(x, t),a(x, t), satisfying the classical equations of motion:

a(x, t) = H[(t),(t)]

a(x, t) , a(x, t) = H((t),(t)]

a(x, t) , (A.3)

and we expand around this solution, writing

a(x, t) = a(x, t) + a(x, t), a(x, t) = a(x, t) + a(x, t). (A.4)(In cosmology, a would describe the RobertsonWalker metric and theexpectation values of various scalar fields.) Of course, since c-numbers com-mute with everything, the fluctuations satisfy the same commutation rules(A.1) as the total variables:

a(x, t), b(y, t)

= iab3(xy),

a(x, t), b(x, t)

=

a(x, t), b(x, t)

= 0,

(A.5)When the Hamiltonian is expanded in powers of the perturbationsa(x, t)anda(x, t) at some definite time t, we encounter terms of zeroth and firstorder in the perturbations, as well as time-dependent terms of second and

higher order:

H[(t), (t)] = H[(t), (t)] +a

H[(t),(t)]

a(x, t) a(x, t] +

a

H[(t), (t)]

a(x, t) a(x, t)

+ H[(t), (t); t] , (A.6)

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where H[(t), (t); t] is the sum of all terms inH[(t)+(t),(t)+(t)]

of second and higher order in the (x, t) and/or(x, t).Now, althoughHgenerates the time-dependence ofa(x, t) anda(x, t),

it is Hrather than Hthat generates the time dependence ofa(x, t) anda(x, t). That is, Eq. (A.2) gives

a(x, t)+a(x, t) =i

H[(t), (t)], a(x, t)

, a(x, t)+a(x, t) =i

H[(t), (t)], a(x, t)

while Eqs. (A.5) and (A.3) give

i

b

d3y

H[(t),(t)]

b(y, t) b(y, t) +

b

d3y

H[(t),(t)]

b(y, t) b(y, t), a(x, t)

= a(x, t)

i

b

d3y

H[(t),(t)]

b(y, t) b(y, t) +

b

d3y

H[(t),(t)]

b(y, t) b(y, t), a(x, t)

= a(x, t).

Subtracting, we find

a(x, t) =i

H[(t), (t); t], a(x, t)

, a(x, t) =i

H[(t), (t); t], a(x, t)

.

(A.7)This then is our prescription for constructing the time-dependent Hamilto-nian H that governs the time-dependence of the fluctuations: expand theoriginal Hamiltonian H in powers of fluctuations and , and throwaway the terms of zeroth and first order in these fluctuations. It is this

construction that gives Han explicit dependence on time.

2. Operator Formalism for Expectation Values

We consider a general Hamiltonian system, of the sort described in theprevious subsection. It follows from Eq. (A.7) that the fluctuations at timet can be expressed in terms of the same operators at some very early timet0 through a unitary transformation

a(t) =U1(t, t0)a(t0) U(t, t0), a(t) =U

1(t, t0)a(t0) U(t, t0),(A.8)

where U(t, t0) is defined by the differential equation

ddt

U(t, t0) = i H[(t), (t); t] U(t, t0 ) (A.9)

and the initial conditionU(t0, t0) = 1. (A.10)

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In the application that concerns us in cosmology, we can take t0 =

, by

which we mean any time early enough so that the wavelengths of interestare deep inside the horizon.

To calculateU(t, t0), we now further decompose Hinto a kinematic termH0 that is quadratic in the fluctuations, and an interaction term HI:

H[(t), (t); t] =H0[(t), (t); t] + HI[(t), (t); t], (A.11)

and we seek to calculate Uas a power series in HI. To this end, we intro-duce an interaction picture: we define fluctuation operators Ia(t) andIa(t) whose time dependence is generated by the quadratic part of theHamiltonian:

Ia(t) =i

H0[I(t), I(t); t], Ia(t)

, Ia(t) =i

H0[I(t), I(t); t], Ia(t)

,(A.12)

and the initial conditions

Ia(t0) =a(t0), Ia(t0) =a(t0). (A.13)

Because H0 is quadratic, the interaction picture operators are free fields,satisfying linear wave equations.

It follows from Eq. (A.12) that in evaluating H0[I, I; t] we can take

the time argument ofI and I to have any value, and in particular wecan take it as t0, so that

H0[I

(t), I

(t); t] =H0[(t0), (t0); t], (A.14)

but the intrinsic time-dependence of H0 still remains. The solution ofEq. (A.12) can again be written as a unitary transformation:

Ia(t) =U10 (t, t0)a(t0)U0(t, t0),

Ia(t) =U

10 (t, t0)a(t0)U0(t, t0),

(A.15)with U0 defined by the differential equation

d

dtU0(t, t0) = i H0[(t0), (t0); t] U0(t, t0) (A.16)

and the initial condition

U0(t0, t0) = 1. (A.17)Then from Eqs. (A.9) and (A.16) we have

d

dt

U10 (t, t0)U(t, t0)

= iU10 (t, t0)HI[(t0), (t0); t]U(t, t0).

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distribution, except that the order of operators in bilinear averages has to

be the same as the order in which they appear in Eq. (A.22).) Expand-ing Eq. (A.22) as a sum of products of bilinear products leads to a set ofdiagrammatic rules, but one that is rather complicated.

In calculating the term inQ ofNth order in the interaction, we drawall diagrams with Nvertices. Just as for ordinary Feynman diagrams, eachvertex is labeled with a space and time coordinate, and has lines attachedcorresponding to the fields in the interaction. There are also external lines,one for each field operator in the product Q, labeled with the differentspace coordinates and the common time t in the arguments of these fields.All external lines are connected to vertices or other external lines, and allremaining lines attached to vertices are attached to other vertices. But there

are significant differences between the rules following from Eq. (A.22) andthe usual Feynman rules:

We have to distinguish b etween right and left vertices, arisingrespectively from the time-ordered product and the anti-time-orderedproduct. A diagram with N vertices contributes a sum over all 2N

ways of choosing each vertex to be a left vertex or a right vertex.Each right or left vertex contributes a factor ior +i, respectively, aswell as whatever coupling parameters appear in the interaction.

A line connecting two right vertices or a right vertex and an externalline, in which it is associated with field operatorsA(x, t) andB(y, t),

contributes a conventional Feynman propagatorT{A(x, t

)B(y, t

}.(It will be understood here and below, that in calculating propagatorsall fields A, B, etc. are taken in the interaction picture, and canbe Is and/or Is.) As a special case, if B is associated with anexternal line then t =t, and since t t, this isB(y, t)A(x, t).

A line connecting two left vertices, associated with field operatorsA(x, t) and B(y, t), contributes a propagatorT{A(x, t)B(y, t}.As a special case, ifB is associated with an external line then t =t,and this isA(x, t)B(y, t).

A line connecting a left vertex, in which it is associated with a fieldoperator A(x, t

), to a right vertex, in which it is associated with afield operator B(y, t), contributes a propagatorA(x, t)B(y, t).

We must integrate over all over the times t, t, . . ., associated with thevertices from t0 to t, as well as over all space coordinates associated

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with the vertices.

We must say a word about the disconnected parts of diagrams. A vac-uum fluctuation subdiagram is one in which each vertex is connected only toother vertices, not to external lines. Just as in ordinary quantum field the-ories, the sum of all vacuum fluctuation diagrams contributes a numericalfactor multiplying the contribution of diagrams in which vacuum fluctua-tions are excluded. But unlike the case of ordinary quantum field theory,this numerical factor is not a phase factor, but is simply

Texp

i

tt0

HI(t) dt

Texp

i

tt0

HI(t) dt

= 1. (A.23)

Hence in the in-in formalism all vacuum fluctuation diagrams automati-cally cancel. Even so, a diagram may contain disconnected parts which donot cancel, such as external lines passing through the diagram without in-teracting. Ignoring all disconnected parts gives what in the theory of noiseis known as the cumulants of expectation values,10 from which the full ex-pectation values can easily be calculated as a sum of products of cumulants.

4. Path Integral Derivation of the Diagrammatic Rules.It is often preferable use path integration instead of the operator for-

malism, in order to derive the Feynman rules directly from the Lagrangianrather than from the Hamltonian, or to make available a larger range ofgauge choices, or to go beyond perturbation theory. Going back to Eq. (1),

and following the same reasoning11 that leads from the operator formalismto the path-integral formalism in the calculation of S-matrix elements, wesee that the vacuum expectation value of any product Q(t) ofs and sat the same time t(now taking t0 = ) is

Q(t) =

x,t,a

dLa(x, t)

x,t,a

dLa(x, t)

2

x,t,a

dRa(x, t)

x,t,a

dRa(x, t)

2

exp

i t

dta

d3x La(x, t

)La(x, t) H[L(t), L(t); t]

expi

t

dt a d3x Ra(x, t)Ra(x, t)

HR(x, t), R(x, t); t x,a

La(x, t) Ra(x, t)

La(x, t) Ra(x, t)

Q

L(t), L(t)

0

L()

0

R()

. (A.24

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Here the functional 0[] is the wave function of the vacuum,12

0[()] exp1

2

a,b

d3x

d3y Eab(x, y)a(x, ) b(y, )

= exp

2

t

dt et a,b

d3x

d3y Eab(x, y) a(t) b(t)

,(A.25)

whereEab is a positive-definite kernel. For instance, for a real scalar field ofmass m,

E(x, y) 1(2)3

d3p eip(xy)

p2 + m2 . (A.26)

As is well known, if the Hamiltonian is quadratic in the canonical con-jugatesa with a field-independent coefficient in the term of second order,then we can integrate over the a by simply setting a =H/a, and

the quantitya

a(t)a(t

) H

(t), (t);t

in Eq. (A.24) then be-

comes the original Lagrangian. We will not pursue this here, but will rathertake up a puzzle that at first sight seems to throw doubt on the equivalenceof the path integral formula (A.24), when we do not integrate out the s,with the operator formalism.

The puzzle is that, although the propagators for lines connecting leftvertices to each other or right vertices to each other or left or right verticesto external lines are Greens functions of the sort that familiarly emerge

from path integrals, what are we to make of the propagators arising fromEq. (A.22) for lines connecting left vertices with right vertices? These are notGreens functions; that is, they are solutions ofhomogeneouswave equations,not of inhomogeneous wave equations with a delta function source. Aswe shall see, the source of these propagators lies in the delta functions inEq. (A.24). It is these delta functions that tie together the integrals overtheL variables and over the R variables, so that the expression (A.18) doesnot factor into a product of these integrals.

In analyzing the consequences of Eq. (A.24), it is convenient to condenseour notation yet further, and let a variable n(t) stand for all the a(x, t)and a(x, t), so that n runs over positions in space and whatever discrete

indices are used to distinguish different fields, plus a two-valued index thatdistinguishes from. With this understanding, Eq. (A.24) reads

Q(t) =t,n

dLn(t)

2

t,n

dRn(t)

2

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expi t

dt

L

L(t

), L(t

); t

exp

i t dt

L

R(t

), R(t

); t

n

Ln(t) Rn(t)

Q

L(t)

0

L()

0

R()

, (A.27)

where

L[(t), (t); t] a

d3x a(x, t

) a(x, t) H

(t), (t); t

.

(A.28)To expand in powers of the interaction, we split L into a term L0 that isquadratic in the fluctuations, plus an interaction termHI:

L= L0 HI , (A.29)where

L0[(t), (t); t ] =

a

d3xa(x, t

)a(x, t) H0

(t), (t); t

.

(A.30)As in calculations of the S-matrix, we will include the argument of theexponential in the vacuum wave functions along with the quadratic part ofthe Lagrangian, writing

t

dtL0[R(t

), R(t); t]

+i

2

ab

d3x

d3y Eab(x, y) Ra(x, t) Rb(y, t)

12

nn

t,t

DRnt,mtRn(t) Rn(t), (A.31) t

dt

L0[L(t), L(t

);t]

i2

ab

d3x

d3y Eab(x, y) La(x, t) Lb(y, t)

1

2nn

t,t

DLnt,ntLn(t) Ln(t) (A.32)

The vacuum wave function is the same for Land R, but it is combined herewith an exponential exp(i L0) for the Lnand an exponential exp(+i L0)

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with the time-ordering dictated by the +i in Eq. (A.31). The second

Eq. (A.36) is the complex conjugate of the first wave equation, whose solu-tion is the complex conjugate of Eq. (A.39):

iLLnt,nt = T{n(t) n(t)} . (A.40)

Eqs. (A.39) and (A.40) thus give the same propagators for lines connectingright vertices with each other or with external lines, and for lines connectingleft vertices with each other or with external lines, as we we encounteredin the operator formalism. Equations (A.37) tell us that RL and (RL)T

satisfy the homogeneous versions of the wave equations satisfied by RR andLL, but to find RL we also need an initial condition. This is provided by

the first of Eqs. (A.38), which in more detail readsiRLnm(t, t2) =i

LLnm(t, t2) = T{n(t)m(t2)} = m(t2)n(t) , (A.41)

in which we have used the fact that t > t2. This, together with the first ofEqs. (A.37), tells us that

iRLnm(t1, t2) = m(t2)n(t1) , (A.42)

which is the same propagator for internal lines connecting right vertices withleft vertices that we found in the operator formalism.

5. Tree Graphs and Classical Solutions.

We will now verify the remark made in Section I, that the usual approachto the calculation of non-Gaussian correlations, of solving the classical fieldequations beyond the linear approximation, simply corresponds to the cal-culation of tree diagrams in the in-in formalism. This is a well-knownresult13 in the usual applications of quantum field theory, but some modifi-cations in the usual argument are needed in the in-in formalism, in whichthe vacuum persistence functional is always unity whether or not we add acurrent term to the Lagrangian.

We begin by introducing a generating functional W[j, t, g] for correlationfunctions of fields at a fixed time t:

eW[J,t,g]/g vac, ine 1g a d3x a(x,t)Ja(x)vac, ing

, (A.43)

where Ja is an arbitrary current, and g a real parameter, with the sub-script g indicating that the expectation value is to be calculated using a

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Lagrangian density multiplied with a factor 1/g. (This is different from the

usual definition of the effective action, because here we are not introducingthe current into the Lagrangian.) The quantity of physical interest is ofcourseW[J,t, 1], from which expectation values of all products of fields canbe found by expanding in powers of the current.

Using Eq. (A.27), we can calculate Was the path integral

eW[J,t,g]/g =

L

L

R

R

expi

t

dt 1

gL[L, L; t

]

exp +i

t

dt 1

g

L[R, R; t]

[L(t) R(t)]

[L(t) R(t)] e 1g

a

d3x a(x,t)Ja(x)

vac[L()] vac[R()] (A.44)

The usual power-counting arguments13 show that the L loop contributionto W[J,t,g] has a g-dependence given by a factor gL. For g 0, W isthus given by the sum of all treegraphs. The integrals over L, L, L,L are dominated in the limit g 0 by fields where L is stationary, i.e.,where

L

= R

=classical

L =R= classical

with classical and classical the solutions of the classical field equationswith the initial conditions that the fields go to free fields such as (14)(16)satisfying the initial conditions (20) at t . Since theL and R fieldstake the same values at this stationary point, the action integrals cancel,and we conclude that

W[J,t, 1]

zero loops=a

d3x classicala (x, t) Ja(x). (A.45)

Expanding in powers of the current, this shows that in the tree approxima-tion the expectation value of any product of fields is to be calculated bytaking the product of the fields obtained by solving the non-linear classicalfield equations with suitable free-field initial conditions, as was to be proved.

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REFERENCES

1. For a review, see N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto,astro-ph/0406398.

2. J. Maldacena, JHEP0305, 013 (2003) (astro-ph/0210603). For otherwork on this problem, see A. Gangui, F. Lucchin, S. Matarrese,andS. Mollerach, Astrophys. J. 430, 447 (1994) (astro-ph/9312033); P.Creminelli, astro-ph/0306122; P. Creminelli and M. Zaldarriaga, astro-ph/0407059; G. I. Rigopoulos, E.P.S. Shellard, and B.J.W. van Tent,astro-ph/0410486; and ref. 3.

3. J. Schwinger, Proc. Nat. Acad. Sci. US 46, 1401 (1961). Also see

L. V. Keldysh, Soviet Physics JETP 20, 1018 (1965); B. DeWitt, TheGlobal Approach to Quantum Field Theory(Clarendon Press, Oxford,2003): Sec. 31. This formalism has been applied to cosmology by E.Calzetta and B. L. Hu, Phys. Rev. D 35, 495 (1987); M. Morikawa,Prog. Theor. Phys. 93, 685 (1995); N. C. Tsamis and R. Woodard,Ann. Phys. 238, 1 (1995); 253, 1 (1997); N. C. Tsamis and R.Woodard, Phys. Lett. B426, 21 (1998); V. K. Onemli and R. P.Woodard, Class. Quant. Grav. 19, 407 (2002); T. Prokopec, O.Tornkvist, and R. P. Woodard, Ann. Phys. 303, 251 (2003); T.Prokopec and R. P. Woodard, JHEP 0310, 059 (2003); T. Brunier,V.K. Onemli, and R. P. Woodard, Class. Quant. Grav. 22, 59 (2005),

but not (as far as I know) to the problem of calculating cosmologicalcorrelation functions.

4. F. Bernardeau, T. Brunier, and J-P. Uzam, Phys. Rev. D 69, 063520(2004).

5. The constancy of this quantity outside the horizon has been usedin various special cases by J. M. Bardeen, Phys. Rev. D22, 1882(1980); D. H. Lyth, Phys. Rev. D31, 1792 (1985). For reviews, seeJ. Bardeen, in Cosmology and Particle Physics, eds. Li-zhi Fang andA. Zee (Gordon & Breach, New York, 1988); A. R. Liddle and D. H.Lyth, Cosmological Inflation and Large Scale Structure (Cambridge

University Press, Cambridge, UK, 2000).

6. R. S. Arnowitt, S. Deser, and C. W. Misner, in Gravitation: An Intro-duction to Current Research, ed. L. Witten (Wiley, New York, 1962):227. This classic article is now available as gr-qc/0405109.

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7. V. S. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Physics

Reports 215, 203 (1992); E. D. Stewart and D. H. Lyth, Phys. Lett.B 302, 171 (1993).

8. N. C. Tsamis and R. Woodard, ref. 3.

9. A. Vilenkin and L. H. Ford, Phys. Rev. D. 26, 1231 (1982); A.Vilenkin, Nucl. Phys. B226, 527 (1983).

10. R. Kubo, J. Math. Phys. 4, 174 (1963).

11. See, e.g., S. Weinberg, The Quantum Theory of Fields Volume I(Cambridge, 1995): Sec. 9.1.

12. ibid., Sec. 9.2.

13. S. Coleman, in Aspects of Symmetry (Cambridge University Press,Cambridge, 1985): pp 139142.

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