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arXiv:1011

.1630v2

[hep-th]

19Dec2010

UTTG-04-10

Ultraviolet Divergences in Cosmological Correlations

Steven Weinberg*

Theory Group, Department of Physics, University of Texas

Austin, TX, 78712

Abstract

A method is developed for dealing with ultraviolet divergences in calcula-tions of cosmological correlations, which does not depend on dimensionalregularization. An extended version of the WKB approximation is used toanalyze the divergences in these calculations, and these divergences are con-trolled by the introduction of PauliVillars regulator fields. This approachis illustrated in the theory of a scalar field with arbitrary self-interactionsin a fixed flat-space RobertsonWalker metric with arbitrary scale factora(t). Explicit formulas are given for the counterterms needed to cancel all

dependence on the regulator properties, and an explicit prescription is givenfor calculating finite regulator-independent correlation functions. The pos-sibility of infrared divergences in this theory is briefly considered.

*Electronic address: weinberg@physics.utexas.edu

1

http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2http://arxiv.org/abs/1011.1630v2

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I. INTRODUCTION

Much effort has been expended in recent years in the calculation of quan-tum effects on cosmological correlations produced during inflation. Thesecalculations are complicated by the occurrence of ultraviolet divergences,which have typically been treated by the method of dimensional regular-ization. Unfortunately, this method has several drawbacks. It is difficultor impossible to employ dimensional regularization unless the analytic formof the integrand as a function of wave number is explicitly known, so cal-culations have generally relied on an assumption of slow roll inflation, oreven strictly exponential inflation. Also, even where an analytic form of theintegrand is known, dimensional regularization can be tricky. Senatore andZaldarriaga[1] have shown that there are terms in correlation functions thatwere omitted in work by other authors[2],[3].

This article will describe a method of dealing with ultraviolet diver-gences in cosmological correlations, without dimensional regularization. Forthe purposes of regularization of infinities, we employ a generally covariantversion of PauliVillars regularization[4]. In order to calculate the coun-terterms that are needed to cancel infinities when the regulator masses goto infinity, we introduce an extended version of the WKB approximation(keeping not only terms of leading order in wavelength), which works welleven when the wave number dependence of the integrand is not explicitlyknown, and can therefore be applied for an arbitrary history of expansionduring inflation.

This method is described here in a classic model, the fluctuations of areal scalar field in a fixed general RobertsonWalker metric. This is simpleenough to illustrate the use of the method without the general idea beinglost in the complications of quantum gravity, and yet sufficiently generalso that we can see how to deal with an arbitrary expansion history. Aswe shall see, these methods yield a prescription for calculating correlationfunctions that are not only free of ultraviolet divergences, but independentof the properties of the regulator fields.

II. THE MODEL

We consider the theory of a single real scalar field (x) in a fixed metricg(x), with Lagrangian density

L = Detg 12

g V()

, (1)

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where V() is a general potential. The modifications in this Lagrangian

needed to introduce counterterms and regulator fields will be discussed inSections III and IV, respectively.

This theory will be studied in the case of a general flat-space RobertsonWalker metric:

g00 = 1, g0i = 0, gij =a2(t) ij , (2)with a(t) a fixed function (unrelated to V()), which is arbitrary exceptthat we assume that a(t) increases monotonically from a value that vanishesfor t . The field equation is then

+ 3H a22 + V() = 0, (3)where as usual H a/a is the expansion rate. We define a fluctuationby writing

(x, t) = (t) + (x, t), (4)

where (t) is a position-independent c-number solution of the field equation:

+ 3H + V() = 0 . (5)

Our calculations will be done using an interaction picture, in which thetime-dependence of is governed by the part of the Hamiltonian quadraticin , so that satisfies a linear differential equation

+ 3H a2

2

+ V

() = 0. (6)

The commutation relations of are

[(x, t), (y, t)] =ia3(t)3(x y), (7)[(x, t), (y, t)] = [(x, t), (y, t)] = 0. (8)

The fluctuation can therefore be expressed as

(x, t) =

d3q

eiqx(q)uq(t) + eiqx(q)uq(t)

, (9)

where (q) is an operator satisfying the familiar commutation relations

[(q), (q)] =3(q q), [(q), (q)] = 0, (10)and uq(t) satisfies the differential equation

uq+ 3Huq+ a2q2uq+ V

()uq = 0 (11)

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and the initial condition, that for t ,

uq(t) 1(2)3/2a(t)

2q

exp

iq

Tt

dt

a(t)

, (12)

whereT is an arbitrary fixed time. (The commutation relations (10) andthe initial condition (12) ensure that the commutation relations (7) and(8) are satisfied for t . The three commutators in these commutationrelations satisfy coupled first-order differential equations in time, which withthis initial condition imply that the commutation relations are satisfied forall times.)

According to the inin formalism[5], the vacuum expectation value ofa product

OH(t) of Heisenberg picture fields and their derivatives, all at

time t, is given by1

OH(t)VAC=

Texp

i

t

HI(t)dt

OI(t)Texpi t

HI(t)dt

0(13)

where 0 denotes the expectation value in a bare vacuum state annihi-lated by (q); T and Tdenote time-ordered and anti-time-ordered prod-ucts;OI(t) is the operatorO(t) expressed in terms of interaction picturefluctuations; and HI is the interaction Hamiltonian, the sum of terms inthe Hamiltonian of third and higher order in the fluctuations, expressed interms of the interaction-picture fluctuation :

HI a3

d3x

16

V()3 + 124

V()4 + . . .

(14)

We will evaluate Eq. (13) as an expansion in the number of loops. Ifwe like, we can introduce a loop-counting parameter g by writing V() =g2F(g), with F(z) a g-independent function ofz, so that the number offactors ofg in a diagram with L loops and Eexternal scalar lines is

# = 2L 2 + E . (15)

Thus an expansion in the number of loops is the same as a series in powersofg2.

1It will be implicitly understood that the contours of integration over time are distortedat very early times to provide exponential convergence factors, as described in ref. [3].

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III. ONE-LOOP COUNTERTERMS

Infinities are encountered when calculating loop contributions to (13) inthis model. As in flat space, they can be canceled by introducing suitablecounterterms into the Lagrangian. (When regulator fields are introduced,the counterterms instead cancel dependence on the regulator properties.)But the Lagrangian cannot know what metric will be adopted, or the clas-sical field around which the field is to be expanded, so neither can thecounterterms. Thus we must return to the generally covariant form (1) ofthe Lagrangian in analyzing the possible counterterms that may be neededand employed.

The general one-loop one-particle-irreducible diagram consists of a loopinto which are inserted a number of vertices, to each of which is attached anynumber of external lines. An insertion with Nexternal lines is given by the(N+ 2)th derivative ofV() with respect to at = , so the countertermin the Lagrangian can only be a function of V(), and of g and itsderivatives. Furthermore, the operators appearing in a counterterm neededto cancel infinities can only be of dimensionality (in powers of energy) fouror less. ButV() has dimensionality two, so the only generally covariantcounterterm satisfying these conditions is of the form2

L1 loop =Detg A V() + B[V()]2 + C R V() , (16)

where R is the usual scalar curvature, and A, B, and Care constants that

depend on the cutoff (that is, on the regulator masses), but not on thepotential. Dimensional analysis tells us that in the absence of regulator fieldsA is quadratically divergent, while B and Care logarithmically divergent.

If we now specialize to the RobertsonWalker metric (2), and write thescalar field as in (4), this counterterm becomes (aside from a c-number term)

L1 loop = a3

A

V()+1

2V()2 + . . .

+B

2V()V()+ [V2() + V()V()]2 + . . .

(6H+ 12H2)CV() +1

2

V()2 + . . . . (17)2This argument does not rule out an additional term proportional toDetg V ()g, but one-loop diagrams do not generate ultraviolet divergent

terms with spacetime derivatives acting on external line wave functions.

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These terms are of one-loop order, and hence to that order are to be used

only in the tree approximation, with a new term in the interaction Hamil-tonian given by

HI=

d3x L1 loop . (18)

The terms shown explicitly in Eq. (17) are the only counterterms in Eq. (18)that contribute in one-loop order to the one-point and two-point functions.

IV. REGULATORS

The counterterm (17) is certainly not the most general counterterm thatwould be consistent with the symmetries of the RobertsonWalker metric.For instance, if we didnt know anything about general covariance, we wouldhave no reason to expect that H and H2 should occur in the linear com-bination R =6H 12H2. In order to be sure that the divergences weencounter will be of a form that can be canceled by the counterterm (17),although we do our calculations for the RobertsonWalker metric (2), weshall adopt a regulator scheme derived from a generally covariant theory.

The usual approach to this problem is to use dimensional regularization,which we wish to avoid for reasons given in Section I. There are other meth-ods of regularization that have been extensively applied to the evaluation ofexpectation values of operators like the energy-momentum tensor in curvedspacetimes[6] but not as far as I know to the calculation of cosmologicalcorrelations.

One such method is covariant point-splitting[7]. This method is wellsuited to the calculation of expectation values of bilinear operators, wherethe ultraviolet divergence arises from the confluence of the arguments of thetwo operators. Because it is a covariant method, it can be implemented by arenormalization of the bilinear operator that respects its transformation andconvergence properties. It seems difficult to apply covariant point-splittingto the calculation of cosmological correlations, where one integrates over theseparation of the spacetime arguments of the interaction Hamiltonian.

There is another widely used method known as adiabatic regulariza-tion[8]. In this method, one subtracts from the integrand its asymptotic formfor large wave numbers, as determined by an extended version of the WKB

method. Experience has shown that though not covariant, this methodyields the same results for expectation values of bilinear operators as covari-ant point-splitting[9]. But adiabatic regularization affects the contributionof small as well as large internal wave numbers, so it seems unlikely thatit can be applied to the calculation of cosmological correlations, where for

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some diagrams the contribution of small virtual wave numbers to correla-

tion functions depends in a complicated way on external wave numbers, sothat adiabatic regularization cannot be implemented by the introduction ofgenerally covariant counterterms in the Lagrangian.

We will instead here employ a generally covariant version of PauliVillarsregularization[4], which like covariant point splitting and adiabatic regular-ization has previously been applied to the calculation of expectation values.For the theory studied here, the Lagrangian (1) is modified to read

L =Detg

12

g 12

n

Zn

gnn+ M2n

2n

V +

n

n, (19)wheren are regulator fields, and Zn andMn are real non-zero parameters.In order to eliminate ultraviolet divergences up to some even order D, wemust take the Zn and regulator masses Mn to satisfyn

Z1n = 1,n

Z1n M2n = 0,n

Z1n M4n = 0 , . . . ,n

Z1n MDn = 0.

(20)For instance, if there were only logarithmic divergences then D = 0, and wewould only need one regulator field, with Z1= 1. In one-loop calculationsthe maximum degree of divergence is quadratic, i.e. D = 2, and to satisfy

the conditions (20) we need at least two regulator fields. In our calculationswe will not need to make a specific choice of the number of regulator fields,but only assume that there are enough to satisfy Eq. (20).

The coefficients A, B, and C in the one-loop counterterm (17) will begiven values depending on the Zn and Mn, such that all expectation values(13) approach finite limits independent of the Znand Mn, as theMnbecomeinfinite. As we will see, this condition not only fixes the terms in A,B , andC that are proportional to logarithms of regulator masses and the term inA that is proportional to squares of regulator masses, but also the termsin A, B, and Cthat depend on regulator masses only through their ratios,and hence that remain fixed as the regulator mass scale goes to infinity.

The only terms in A, B, and Cthat will not be fixed by this condition arefinite terms independent of regulator properties, which of course representthe freedom we have to change the parameters in the potential or to add anon-minimal coupling of the scalar field to the curvature.

The regulator fields n like the physical field are written as classical

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fields plus fluctuations

n(x, t) =n(t) + n(x, t). (21)

The classical fields satisfy the coupled field equations

+ 3H + V

+n

n

= 0 (22)

n+ 3Hn+ Z1n V

+n

n

+ M2nn= 0. (23)

We assume throughout that the regulator masses Mn are all much larger

than H(t) and V

(t)

1/2over the whole range from t to the

time t = t at which the correlations are measured. In consequence, theclassical field equations (22) and (23) have a solution in which all the nare less than by factors of order H2/M2n and|V()|/M2n, and so may beneglected. We adopt this solution for the classical fields. In particular, thefield then satisfies the original classical field equation (5).

In dealing with internal lines, it is convenient to lump together the phys-ical field fluctuation and the fluctuations n in the regulator fields, byintroducing an index N (and likewise M, etc.) such thatN is the phys-ical field fluctuation for N = 0 and is a regulator field fluctuation forN = n 1, both in the interaction picture. The general field fluctuationssatisfy the coupled field equations

N+ 3HN a22N+ M2NN+ Z1N V()M

M = 0, (24)

where Z0= 1 and M0= 0. The commutation relations of the are

[N(x, t), M(y, t)] =ia3(t)3(x y)Z1N NM , (25)

[N(x, t), M(y, t)] = [N(x, t), M(y, t)] = 0. (26)

The general fluctuation can therefore be expressed as

N(x, t) =M d

3q eiqxM(q)u

MNq(t) + e

iqxM(q)uMNq(t) , (27)

where N(q) satisfy the commutation relations

[N(q), M(q

)] =3(q q)Z1N NM , [N(q), M(q)] = 0, (28)

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and the uMNq (t) are solutions of Eq. (24):

uMNq+ 3HuMNq+ a

2q2uMNq+ M2Nu

MNq + Z

1N V

()L

uMLq = 0 (29)

distinguished by the initial condition, that for t ,

uMNq(t) 1

(2)3/2a3/2(t)

2Nq (t)MN exp

i tT

Nq(t) dt

, (30)

where

Nq(t)

q2

a2(t)+ M2N

1/2. (31)

The N(q) are all taken to annihilate the vacuum. The two-point functionsappearing in propagators are then given by

N(x1, t1)M(x2, t2)0=K

d3q eiq(x1x2)Z1K u

KNq (t1) u

KMq(t2).

(32)In calculating one-loop graphs, we must integrate over one or more times

ti associated with vertices, and over a single co-moving wave number q.There are two ranges ofq |q| where the integrand is greatly simplified.

In the first range, q/a(t) (and hence all q/a(ti)) is much greater than

H(t) and V(t)1/2 for all t t, as well as much greater than thephysical wave numbers associated with external lines, though q/a(t) is notnecessarily greater than the regulator masses. In this range, we can reli-ably evaluate the integrand in an extended version of the WKB approxi-mation, described in an Appendix. Any term that would be convergent inthe absence of cancelations among the physical and regulator fields makes anegligible contribution to the integral over this range.

In the second range, q/a(t) is much less than the regulator masses,

though it is not necessarily less thanH(t) orV(t)1/2 or the physical

wave numbers associated with external lines. In this range, it is safe to ig-nore the regulator fields. (We do not have to worry about the contribution

of timest

so much earlier than t thatq/a(t

) is of the order of the regulatormasses, because this contribution is exponentially suppressed by the rapidoscillation of the integrand at these early times.)

It is crucially important to our method of calculation that, because we as-

sume that the regulator masses are much larger than H(t) andV(t)1/2

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and the physical wave numbers associated with external lines, these ranges

of wave number overlap. We can therefore separate the range of integra-tion of co-moving wave number by introducing a quantity Q in the overlapregion, so that Q/a(t) is much less than all regulator masses, and much

greater than H(t) andV(t)1/2 and the physical wave numbers asso-

ciated with external lines. We can evaluate the integral over q Qignoringthe regulators, and overq Qby using the WKB approximation. No errorsare introduced by this procedure in the final result, because we are takingthe regulator masses to be arbitrarily large compared with Q/a(t), which

is taken to be arbitrarily large compared with H(t) orV(t)1/2 for

t tor the physical wave numbers associated with external lines, so termsproportional to quantities like Q/M

na(t) orH a(t)/Qare entirely negligible.

It should be emphasized thatQis neither an infrared nor an ultravioletcutoff, but simply a more-or-less arbitrary point at which we choose to splitthe range of integration. As long as Q is chosen in the overlap of the tworegions defined in the previous paragraphs, the sum of the integrals overq Q and q Q will automatically be independent ofQ.

V. THE TWO-POINT FUNCTION

To demonstrate the use of the methods described in the previous section,and to evaluate the coefficients A, B, and C in the counterterm (16), wewill now calculate the one-loop corrections to the vacuum expectation value

of the productH(y, t) H(z, t) of Heisenberg picture fields. This can bewritten in terms of a Greens function Gp(t), as

H(y, t) H(z, t)VAC=

d3p exp

ip (y z)

Gp(t). (33)

Leaving aside vacuum fluctuations and counterterms, there are three one-loop diagrams, shown in Figure 1. In this section we will consider onlythe one-particle-irreducible diagrams, I and II; these will suffice to allow usin Section VI to fix the coefficients A, B, and C in the counterterm (16).Diagram III will be dealt with in Section VII.

Diagram I

By the usual rules of the inin formalism, after integrating over spatialcoordinates, the contribution of diagram I to the two-point function is

GIp(t) = 2(2)6Re t

dt1 a3(t1) V

(t1) t

dt2 a

3(t2) V

(t2)

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I

II

III

Figure 1: Diagrams for the two-point function.

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KLMNMNZ1K Z

1L d3q

(t1 t2)u2p(t)up(t1)up(t2)uKMq(t1)uKMq(t2)uLNq (t1)uLNq (t2)

12|up(t)|2up(t1)up(t2)uKMq(t1)uKMq(t2)uLNq (t1)uLNq (t2)

, (34)

where q |q| and q |q p|. The first term in the square brackets arisesfrom diagrams in which the vertices come either both from the time-orderedproduct or both from the anti-time-ordered product in Eq. (13), while thesecond term arises from diagrams in which one vertex comes from the time-ordered product and the other from the anti-time-ordered product.

As described at the end of the previous section, to calculate GIp(t) wedivide the region of integration over q |q| into the ranges q < Q andq > Q, where Q is chosen so that Q/a(t) is much less than all regulatormasses but much greater than p/a(t) and H(t) and|V((t))|1/2 for allt t.

For q < Q, we can ignore the regulators, and set K, L, M, N, M, N

all equal to zero, with u00q just equal to the wave function uq in the absenceof regulators. This contribution takes the form

GI,Qp (t) = 2Re t

dt1V

(t1) t1

dt2 V

(t2) u2p(t)u

p(t1)u

p(t2)

KL

Z1K Z1L

q>Q

d3q

4

Kq (t1)Kq (t2)Lq(t1)Lq(t2)

expi t1t2

[Kq (t) + Lq(t

)] dt

. (36)

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Note that we have dropped the distinction between q and q, because p is

negligible compared with qfor q > Q. We have also dropped the contributionof the second term in Eq. (34), because this term converges for each KandL, and so makes a negligible contribution to the integral over values q > Q.

The contribution of values of t2 at any fixed time less than t1 is alsonegligible, because of the rapid oscillation of the final factor. But thereis an important contribution from values of t2 that are so close to t1 that(t1 t2)Q/a(t1) is not large. This contribution can be evaluated by settingt2 = t1 everywhere except in the range of integration in the exponential, sothat

GI,>Qp (t) = 2Re t

dt1V

(t1) t1

dt2V

(t1)

u2p(t)u2p (t1)

KL

Z1K Z1L

q>Q

d3q4Kq (t1)Lq(t1)

exp(i(t1 t2)[Kq (t1) + Lq(t1)])= t

dt1V

(t1)2

Im

u2p(t)u2p (t1)

KL

Z1K Z1L

q>Q

d3q

2Kq (t1)Lq(t1)[Kq (t1) + Lq(t1)] .(37)

The integral over q converges because

KZ1K = 0. This integral receives

contributions from terms where K and L are both regulator fields mand n, or are a regulator field n and a physical field 0 = , or are two

physical fields. Adding these contributions gives

GI,>Qp (t) =

t

dt1a3(t1) V

(t1)2

Im

u2p(t)u2p (t1)

mn

Z1m Z1n

M2nln Mn M2mln MmM2n M2m

+ 2n

Z1n ln Mn+ ln

Q

a(t1)

. (38)

Note that, because

n Z1n =1, this is independent of the units used to

measure Q and the regulator masses, as long as the same units are used inall logarithms.

Diagram II

By the usual rules of the inin formalism, after integrating over spatialcoordinates, the contribution of diagram II to the two-point function (33) isgiven by

GIIp (t) = (2)3 t

dt1 a3(t1) V

(t1)

Im

u2p(t)u2p (t1)

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KNNZ1K d3q uKNq (t1)uKNq(t1). (39)

We again divide the range of integration over q |q| into the ranges q < Qandq Q, whereQ is chosen so that Q/a(t) is much less than all regulatormasses but much greater than p/a(t) and H(t) and|V((t))|1/2 for allt t1.

Forq < Qwe can ignore the regulators, and set K,N, andN all equal tozero, withu00q just equal to the wave functionuq in the absence of regulators.This contribution takes the form

GII,Qp (t) of zeroth order in V(), we note

that in the absence of the potential, uKN(t1) is proportional to KN:

uKNq(t1) =NKuNq(t1), (41)

whereuNq + 3HuNq+ (q

2/a2)uNq+ M2NuNq = 0. (42)

This contribution is

GII,>Q,0p (t) = (2)3 t

dt1a3(t1) V

(t1)

Im

u2p(t)u2p (t1)

N

Z1N

d3q|uNq (t1)|2 . (43)

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The integrand is given by an asymptotic expansion derived in the Appendix.

For both q2

/a2

(t1) and M2Nmuch greater than both H

2

(t1) and H(t1), wehave

|uNq |2 12Nqa3(2)3

1 +

H+ 2H2

22Nq+

(H+ 3H2)M2N44Nq

5H2M4N

86Nq

,

(44)

where, as before,2Nq(t1) =

q/a(t1)2

+ M2N. The integral over q converges

because

NZ1N =

NZ1N M

2N = 0. The sum over N receives contri-

butions from terms where N is a regulator field n or the physical field0= . Adding these contributions gives

GII,>Q,0p

(t) = t

dt1

a3(t1

) V(t1

) Imu2p

(t)u2p

(t1

)

n

Z1n M2nln Mn+

H(t1) + 2H2(t1)5

6n

Z1n ln Mn

Q2

a2(t1)

H(t1) + 2H2(t1)

ln

Q

a(t1)

. (45)

The regulator-dependent term arising from diagram II that are of firstorder in V() can be calculated by applying the rules of the in-in for-malism a diagram like that of diagram II, but with a V insertion in theloop. This gives

GII,>Q,1p (t) = (2)6 t

dt1a3(t1) V

(t1) t

dt2a3(t2) V

(t2)

KLMNMN

Z1K Z1L

q>Q

d3qRe

u2p(t)u

p(t1)u

p(t1)

(t1 t2)uKMq(t1)uKMq(t2)uLNq(t1)uLNq(t2) + 1 2

. (46)

(This contribution is produced only by terms in which both interactionscome from the time-ordered product in Eq. (13), or both from the anti-time-ordered product. As in the case of diagram I, the other terms make anegligible contribution to the part of the integral with q > Q.) The individ-

ual terms in Eq. (46) are only logarithmically divergent, so we can evaluatethis using the leading term (30) in the WKB approximation. Following thesame limiting procedure as for diagram I, we find

GII,>Q,1p (t) =

t

dt1a3(t1) V

(t1)

V

(t1)

Im

u2p(t)u2p (t1)

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mn Z1m Z

1n

M2nln Mn M2mln MmM2n M2m

+ 2n

Z1n ln Mn+ ln Qa(t1) . (47)

Total 1PI Amplitude

The complete contribution of the two one-particle irreducible diagramsis given by the sum of the terms (35), (38), (40), (45), and (47):

G1PIp (t) = 2(2)6 t

dt1a3(t1)V

(t1) t1

dt2a

3(t2)V

(t2)

Re

u2p(t)up(t1)u

p(t2)

q

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where

G = a3 AV() + 2B[V2() + V()V()] 6C(H+ 2H2)V()(50)

According to the rules of the in-in formalism, this makes a contributionto the two-point function (33) given by

G1PIp (t) = 2(2)3 t

dt1G(t1)Im{u2p(t)u2p (t1)} . (51)

Comparing Eqs. (50) and (51) with (48), we see that in order to cancelthe dependence of the one-particle irreducible two-point function on theregulator properties, we need

A = 1162

n

Z1n M2nln Mn+

2A

(52)

B = 1

322

nm

Z1n Z1m

M2nln(Mn/B) M2mln(Mm/B)

M2n M2m

+2n

Z1n ln(Mn/B)

(53)

C = 1962

5

6n

Z1n ln

MnC

. (54)

(The first term in Eq. (52) does not depend on the units used for regulator

masses in the logarithm, because

n Z1n M

2n = 0.) HereA, B, and C

are unknown mass parameters. The presence of these parameters shouldnot be seen as a drawback of this method; they reflect the real freedom wehave to add finite regulator-independent terms to the original Lagrangianproportional toV() or V 2() or R V().

Adding Eqs. (48) and (51) gives our final answer for the one-particle-irreducible part of the two-point function

G1PIp (t) + G1PIp (t) =

2(2)6 t

dt1a3(t1)V

(t1) t1

dt2a

3(t2)V

(t2)

Reu

2p(t)u

p(t1)u

p(t2)

q

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|up(t)

|2Reup(t1)up(t2) d3q uq(t1)uq (t1)uq(t2)uq (t2)

+

(2)3 t

dt1a3(t1)V

(t1)

Im

u2p(t)up(t1)

q

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where

I(t2) 12

a3(t2)V

(t2)

d3q

KNN

uKNq(t2)uKNq(t2) + F(t2). (59)

In the first term in the square brackets in Eq. (58), both vertices comefrom the time-ordered product in Eq. (13), while in the second term, ver-tex 1 comes from the time-ordered product and vertex 2 from the anti-time-ordered product; in the complex conjugate time-ordered and anti-time-ordered products are interchanged.

There is no problem here with ultraviolet divergences coming from theintegral overq. Following the same procedure as in our treatment of diagramII in the preceeding two sections, we have

I(t2) = 12

a3(t2)V

(t2)

q

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Despite the ambiguity in u0(t) and the inapplicability of the BunchDavies

condition for q = 0, the function G(t1, t2) is perfectly well-defined. It is thesolution of the second-order differential equation

d2

dt21+ 3H(t1)

d

dt1+ V

(t1)

G(t1, t2) = 0, (63)

subject to initial conditions dictated by the commutation relations (7) and(8):

G(t2, t2) = 0, (64) d

dt1G(t1, t2)

t1=t2

=a3(t2). (65)

The only property of the vacuum state used here is that it has zero momen-tum and unit norm. The general solution is

G(t1, t2) =u(t1) u(t2)

t1t2

dt

a3(t) u2(t) , (66)

where u(t) is any solution of the q= 0 wave equation

u + 3Hu + V()u= 0, (67)

that does not vanish between t1 and t2. (For instance, for a general poten-tial and a de Sitter metric, we can take u = , which does not vanish intypical inflationary models.) Putting this together, we have the one-particle-reducible contribution to the two-point function (33):

G1PRp = 2(2)3 t

dt1a3(t1) V

(t1)

Im{u2p(t)u2k (t1)}

t1

dt2 G(t1, t2)I(t2). (68)

VIII. THE ONE-POINT FUNCTION

In Section II we defined as the departure of the field from itsclassicalvalue , not from its meanvalue, so we must expect to have anon-vanishing expectation value. As we will see, this is closely related toquantities calculated in the previous section.

According to the general diagrammatic rules, the vacuum expectationvalue of the Heisenberg picture scalar field fluctuation in one-loop order is

H(y, t)one loopVAC = i

d3x1

t

dt1(y, t) (x1 , t1)0I(t1) + c.c. ,(69)

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withIgiven by Eq. (60) representing the insertion of a loop or a countertermat the end of the single incoming line. In the term shown in Eq. (69) thesingle vertex comes from the time-ordered product in Eq. (13); in its complexconjugate, the vertex comes from the anti-time-ordered product. The twoterms together involve the commutator of the field perturbations, so theone-point function may be written in terms of the function G defined byEq. (62):

H(y, t)one loopVAC = t

dt1G(t, t1)I(t1). (70)

We see now that the contribution (68) of the one-particle-reducible diagramsto the two-point function may be simply expressed in terms of the meanfluctuation:

G1PRp (t) = t

dt1 a

3(t1) V

(t1)

H(0, t1)one loopVAC Im{u2k(t)u2k (t1)} . (71)This is the same as would be given by adding an interaction obtained byshifting by its expectation value:

HI(t) =1

2a3(t) V

(t)

H(0, t)one loopVAC

d3x 2(x, t). (72)

IX. INFRARED DIVERGENCES?

Although the model treated in this paper is intended to provide an il-lustration of a method of dealing with ultraviolet divergences, it may be ofsome interest to look into the possible presence of infrared divergences inthis model. For any fixed co-moving wave number q, the evolution of thewave function uq(t) defined by Eqs. (11) and (12) becomes q-independentonceq/a(t) drops belowH(t), so the behavior of the wave function for fixed

tand q 0 is determined by the behavior ofV

(t)

andH(t) fort 0.We can distinguish two cases in which this problem is greatly simplified.

Expansion-dominated:

If

V

(t)

H2(t) fort 0, then as long as this inequality is satisfied,

we can drop the potential term in Eq. (11), which then becomes the sameas the differential equation for tensor fluctuations. It is well known[10]in this case that if H(t) H2(t) as t 0, then the wave functionuq(t1) at a fixed time t1 goes as q

3/2 for q/a(t1) H(t1). This q-dependence is unaffected even ifH2(t) drops below

V(t)at some time21

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after q/a drops below H, since the evolution of the wave function at such

times is q-independent. So (taking

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Beyond the simple model discussed here, of a scalar field in a fixed met-

ric, there is the more realistic problem of scalar and tensor fluctuations ina theory of coupled scalar and gravitational fields. This is more compli-cated, because even in one-loop order there are quartic as well as quadraticand logarithmic ultraviolet divergences. That alone should not prevent themethod described here from being applicable to realistic theories, at leastfor one-loop graphs, since divergences of any order can be eliminated byincluding enough regulator fields.

A more serious problem is the difficulty of introducing regulator fields forthe graviton propagator. (This problem is of course avoided in theories withlarge numbers of matter fields, where matter loops dominate over gravitonloops.) If the only vertices that involve gravitons have a single graviton line

attached to matter lines, then we can introduce regulators for the gravi-ton propagator by coupling heavy tensor fields with suitable Z-factors tothe energy-momentum tensor. But it is not clear how to deal with graphscontaining vertices to which are attached two or more graviton lines.

This raises the question whether PauliVillars regularization is reallynecessary. The final results (55) and (68) for the one-particle irreducibleand reducible parts of the two-point function could almost have been guessedwithout introducing regulator fields. It would only be necessary to intro-duce an ultraviolet cut-off at a sufficiently large co-moving wave numberQ, calculate the Q-dependence of the resulting two-point function by usingthe WKB methods described in this paper, and then introduce a countert-erm of form (16), with A, B, and Cchosen as functions ofQ to cancel the

Q-dependence found in this way. (This is not the adiabatic regularizationprocedure mentioned in Section IV, even though both procedures use WKBmethods, because with a cut-off at Q only the part of the integrand forinternal wave numbers larger than Q is affected.) Of course, this procedureleaves finite terms in A,B , andCundetermined, but they are undeterminedanyway, since they represent the real possibility of changing the original La-grangian by adding corrections to the potential and adding a coupling ofthe scalar field to the spacetime curvature. The cut-off introduced in thisway would not respect general covariance, but apparently one would get thecorrect results (55) and (68) anyway.

There is something mysterious about this. The actual calculations in

this paper were done for a fixed RobertsonWalker metric, Eq. (2). Theywould have been done in the same way by someone who had never heardof general covariance. Yet the infinities turned out to depend on H andHonly in the combination H+ 2H2, proportional to the scalar spacetimecurvature. We can understand this for a generally covariant regularization

23

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procedure, like PauliVillars regularization, because in that case general

covariance is broken only by the background, which presumably does notaffect ultraviolet divergences. But how do these calculations know that theyare supposed to give infinities that can be canceled by counterterms thatare generally covariant, when we use a non-covariant cutoff on the internalwave number instead of introducing regulator fields?

ACKNOWLEDGMENTS

I am grateful for discussions with Joel Meyers, Emil Mottola, and RichardWoodard. This material is based upon work supported by the National Sci-ence Foundation under Grant Numbers PHY-0969020 and PHY-0455649and with support from The Robert A. Welch Foundation, Grant No. F-

0014.APPENDIX: THE EXTENDED WKB APPROXIMATION

We wish to find an asymptotic expression for the solution uq(t) of the dif-ferential equation

uq(t) + 3H(t)uq(t) +

q2/a2(t)

uq(t) + M2uq(t) = 0 (A.1)

subject to the initial condition, that for t 0,

uq(t) 1(2)3/2a(t)

2q

exp

iq

Tt

dt/a(t)

. (A.2)

(The effects of the potential are treated separately in Section V.) We areinterested in the behavior of uq(t) at a fixed time t, when q/a(t) is muchlarger than H(t), but not necessarily greater than M.

As an ansatz, we take

uq(t) 1(2)3/2a3/2(t)

2(t)exp

i

Tt

(t)dt

1 +f(t)

(t)+

g(t)

2(t)+ O(3)

(A.3)

with f,g, etc. of zeroth order in qand M, and

(t)

q2/a2(t) + M2 . (A.4)

This clearly satisfies the initial condition (A.2). The differential equation(A.1) is satisfied by (A.3) to order 3/2 and 1/2, while the terms in (A.1)of order 1/2 (countingMas being the same order as ) give

d

dt

f

=

i

2

H+ 2H2 +

3H2M2

22 5M

4H2

44 +

HM2

22

. (A.5)

24

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The terms in (A.1) of order 3/2 are more complicated, but fortunately

we only need these terms in|uq|2

, and for this purpose we can avoid havingto work out these terms by using the time-dependence of the Wronskian:

uquq uquq 1

a3 . (A.6)

Using (A.3) gives

2(2)3a3

uquq uquq

= 2i 4iRe f

+2i

d

dt

Imf

2i |f|

2

2 4iRe g

2 + O(3). (A.7)

Now, Eq. (A.5) shows that d/dt(f /) is imaginary, so since f(t)/(t)vanishes for t 0, f(t)/(t) and hence f(t) is imaginary for all t. Thefirst term on the right-hand side of Eq. (A.7) is constant, and the secondterm vanishes, so the constancy of this quantity requires the vanishing ofthe terms of order 2:

|f|2 + 2Re g= ddt

Imf

(A.8)

But this is just what we need, for Eq. (A.3) (with f imaginary) gives

|uq(t)|2 12(t)(2)3a3(t)

1 +

|f(t)|2 + 2Re g(t)2(t)

. (A.9)

Together with Eqs. (A.5) and (A.8), this gives the result used in evaluatingdiagram II in Section V.

|uq|2 12a3(2)3

1 +

H+ 2H2

22 +

( H+ 3H2)M2

44 5H

2M4

86

. (A.10)

-

1. L. Senatore and M. Zaldarriaga, [0912.2734].

2. S. Weinberg, Phys. Rev. D 74, 023508 (2006) [hep-th/0605244]; K.Chaicherdsakul, Phys. Rev. D 75, 063522 (2007) [hep-th/0611352];

D. Seery, JCAP 0711, 025 (2007) [0707.3377]; E. Dimastrogiovanniand N. Bartolo, JCAP 0811, 016 (2008) [0807.2709]; P. Adshead, R.Easther, and E. A. Lim, Phys. Rev. D 79, 063504 (2009) [0809.4008];X. Gao and F. Xu, JCAP 0907, 042 (2009) [0905.0405]; D. Campo,[0908.3642].

25

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3. P. Adshead, R. Easther, and E. A. Lim, [0904.4207].

4. W. Pauli and F. Villars, Rev. Mod. Phys. 75, 434 (1949). Pauli-Villars regularization has been applied to the problem of calculatingthe expectation value of the energy-momentum tensor in a curvedspacetime, by C. Bernard and A. Duncan, Ann. Phys. 107, 201 (1977)and A. Vilenkin, Nuovo Cimento 44 A, 441 (1977), but not as far asI know in the more complicated problem of calculating cosmologicalcorrelations. The present work was done as a result of preparing acourse on quantum field theory given in Spring 2010.

5. J. Schwinger, Proc. Nat. Acad. Sci. US 46, 1401 (1960); J. Math.Phys. 2, 407 (1961); K. T. Mahanthappa, Phys. Rev. 126, 329

(1962); P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4,1, 12 (1963); L. V. Keldysh, Soviet Physics JETP 20, 1018 (1965);D. Boyanovsky and H. J. de Vega, Ann. Phys. 307, 335 (2003); B.DeWitt, The Global Approach to Quantum Field Theory(ClarendonPress, Oxford, 2003): Sec. 31. For a review, with applications tocosmological correlations, see S. Weinberg, Phys. Rev. D72, 043514(2005)[hep-th/0506236].

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7. S. M. Christensen, Phys. Rev. D 14, 2490 (1976).

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9. P. R. Anderson and L. Parker, Phys. Rev. D 36, 2963 (1987); S. Habib,C. MOlin-Paris and E. Mottola, Phys. Rev. D 61, 024010 (1999); P.Anderson, W. Eaker, S.Habib, C. Molina-Paris, and E.Mottola, Phys.Rev. D 62, 124019 (2000)

10. For a textbook treatment, see S. Weinberg, Cosmology (Oxford Uni-versity Press, 2008), Sec. 10.3.

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http://arxiv.org/abs/hep-th/0506236http://arxiv.org/abs/hep-th/0506236