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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 of
tree graphs in the in-in formalism; here we also consider loop graphs. It
turns out that for some though not all theories, the contributions of loop
graphs as well as tree graphs depend only on the behavior of the inaton
potential near the time of horizon exit. A sample one-loop calculation is
presented.

Electronic address: weinberg@physics.utexas.edu

I. INTRODUCTION
The departures from cosmological homogeneity and isotropy observed
in the cosmic microwave background and large scale structure are small,
so it is natural that they should be dominated by a Gaussian probability
distribution, with bilinear averages given by the terms in the Lagrangian
that are quadratic in perturbations. Nevertheless, there is growing interest
in the possibility of observing non-Gaussian terms in various correlation
functions,
1
such as an expectation value of a product of three temperature
uctuations. It is also important to understand the higher-order corrections
to bilinear correlation functions, which appear in Gaussian correlations.
Until now, higher-order cosmological correlations have been calculated
by solving the classical eld equations beyond the linear approximation.
As will be shown in the Appendix, this is equivalent to calculating sums
of tree graphs, though in a formalism dierent from the familiar Feynman
graph formalism. For instance, Maldacena
2
has calculated the non-Gaussian
average of a product of three scalar and/or gravitational elds to rst order
in their interactions, which amounts to calculating a tree graph consisting
of a single vertex with 3 attached gravitational and/or scalar eld lines.
This paper will discuss how calculations of cosmological correlations can
be carried to arbitrary orders of perturbation theory, including the quantum
eects represented by loop graphs. So far, loop corrections to correlation
functions appear to be much too small ever to be observed. The present
work is motivated by the opinion that we ought to understand what our
theories entail, even where in practice its predictions cannot be veried
experimentally, just as eld theorists in the 1940s and 1950s took pains to
understand quantum electrodynamics to all orders of perturbation theory,
even though it was only possible to verify results in the rst 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 only
on the behavior of the unperturbed inaton eld near the time of horizon
exit. Is the same true for loop graphs? If so, it will be possible to calcu-
lated the loop contributions with some condence, but we can learn little
new from such calculations. On the other hand, if the contribution of loop
graphs depends on the whole history of the unperturbed inaton eld, then
calculations become much more dicult, but potentially more revealing. In
this case, it might even be that the loop contributions are much larger than
otherwise expected.
1
The appropriate formalism for dealing with this sort of problem is the
in-in formalism originally due to Schwinger.
3
Schwingers presentation is
somewhat opaque, so this formalism is outlined (and extended) in an Ap-
pendix. In section II we summarize those aspects of this formalism that
are needed for our present purposes. Section III introduces a class of the-
ories to serve as a basis of discussion, with a single inaton eld, plus any
number of additional massless scalar elds with only gravitational interac-
tions and vanishing expectation values. In Section IV we prove a general
theorem about the late time behavior of cosmological correlations at xed
internal as well as external wave numbers. Section V introduces a class
of unrealistic theories to illustrate the problems raised by the integration
over internal wave numbers, and how these problems may be circumvented.
In Section VI we return to the theories introduced in Section III, and we
show that the conditions of the theorem proved in Section IV are satised
for these theories. This means that, to all orders of perturbation theory, if
ultraviolet divergences cancel in the integrals over internal wave numbers,
then cosmological correlations do indeed depend only on the behavior of the
unperturbed ination eld near the time of horizon exit in the cases studied.
We can also nd other theories in which this result does not apply, as for
instance by giving the additional scalar elds a self-interaction. Section VII
presents a sample one-loop calculation of a cosmological correlation.
II. THE IN-IN FORMALISM IN COSMOLOGY
The problem of calculating cosmological correlation functions diers
from the more familiar problems encountered in quantum eld theory in
at least three respects:
We are not interested here in the calculation of S-matrix elements, but
rather in evaluating expectation values of products of elds at a xed
time.
Conditions are not imposed on the elds at both very early and very
late times, as in the calculation of S-matrix elements, but only at
very early times, when the wavelength is deep inside the horizon and
according to the Equivalence Principle the interaction picture elds
should have the same form (when expressed in terms of metric rather
than co-moving coordinates) as in Minkowski spacetime.
Although the Hamiltonian H that generates the time dependence of
the various quantum elds is constant in time, the time-dependence
2
of the uctuations in these elds are governed by a uctuation Hamil-
tonian

H with an explicit time dependence, which as shown in the
Appendix is constructed by expanding H around the unperturbed so-
lution of the eld equation, and discarding the terms of rst order in
the perturbations to the elds and their canonical conjugates.
Given a uctuation Hamiltonian

H, we want to use it to calculate expec-
tation values of some product Q(t) of eld operators, all at the same time t
but generally with dierent space arguments. As discussed in the Appendix,
the prescription of the in-in formalism is that
Q(t) =
__

T exp
_
i
_
t

H
I
(t) dt
__
Q
I
(t)
_
T exp
_
i
_
t

H
I
(t) dt
___
,
(1)
Here T denotes a time-ordered product;

T is an anti-time-ordered product;
Q
I
is the product Q in the interaction picture (with time-dependence gen-
erated by the part of

H that is quadratic in uctuations); and H
I
is the
interaction part of

H in the interaction picture. (This result is dierent
from that originally given by Maldacena
2
and other authors
4
, who left out
the time-ordering and anti-time-ordering, perhaps through a typographical
error. However, this makes no dierence to rst order in the interaction,
which is the approximation used by these authors in their calculations.) We
are here taking the time t
0
at which the uctuations are supposed to behave
like free elds as t
0
= , which is appropriate for cosmology because at
very early times the uctuation wavelengths are deep inside the horizon.
Eq. (1) leads to a fairly complicated diagrammatic formalism, described
in the Appendix. Unfortunately this formalism obscures crucial cancella-
tions that occur between dierent diagrams. For our present purposes, it is
more convenient to use a formula equivalent to Eq. (1):
Q(t) =

N=0
i
N
_
t

dt
N
_
t
N

dt
N1

_
t
2

dt
1

__
H
I
(t
1
),
_
H
I
(t
2
),
_
H
I
(t
N
), Q
I
(t)
_

___
, (2)
(with the N = 0 term understood to be just Q
I
(t)). This can easily be
derived from Eq. (1) by mathematical induction. Obviously Eqs. (1) and
(2) give the same results to zeroth and rst order in H
I
. If we assume that
the right-hand sides of Eqs. (1) and (2) are equal for arbitrary operators Q
up to order N 1 in H
I
, then by dierentiating these equations we easily
3
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, so
they 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 ination. The reader who prefers to avoid details of
specic theories can skip this section, and go on immediately to the general
analysis of late-time behavior in the following section.
In this section we will consider theories of ination with two kinds of
matter elds : a real scalar eld (x, t) with a non-zero homogeneous ex-
pectation value (t) that rolls down a potential V (), and any number
of real massless scalar elds
n
(x, t), which have only minimal gravitational
interactions, and are prevented by unbroken symmetries from acquiring vac-
uum expectation values. The real eld serves as an inaton whose energy
density drives ination, while the
n
are a stand-in for the large number of
species of matter elds that will dominate the eects of loop graphs on the
correlations of the inaton eld.

We follow Maldacena,
2
adopting a gauge in which there are no uctua-
tions in the inaton eld, so that (x, t) = (t), and in which the spatial
part of the metric takes the form

g
ij
= a
2
e
2
[exp ]
ij
,
ii
= 0,
i

ij
= 0 . (3)
where a(t) is the RobertsonWalker scale factor,
ij
(x, t) is a gravitational
wave 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 8G
in any graph equals the number of loops of any kind, plus a xed number that depends
only on which correlation function is being calculated. Matter loops are numerically
more important than loops containing graviton or inaton lines, because they carry an
additional factor equal to the number of types of matter elds.

I am adopting Maldacenas notation, but the quantity he calls is more usually called
R. To rst order in elds, the quantity usually called is dened as H/

, while
the quantity usually called R is dened as + Hu. (Here the contribution of scalar
modes to gij is written in general gauges as 2a
2
(ij +
2

/x
i
x
j
), while and
are the perturbation to the total energy density and its unperturbed value, while u is
the perturbed velocity potential, which for a single inaton eld is u = /

.) In the
gauge used by Maldacena and in the present paper u =

= 0, so since is dened here

as to rst order in elds, it corresponds to the quantity usually called R. Outside the
horizon R and are the same.
4
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 of
ij
.
The other components of the metric are given in the ArnowittDeser
Misner (ADM) formalism
6
by
g
00
= N
2
+g
ij
N
i
N
j
, g
i0
= g
ij
N
j
, (4)
where N and N
i
are auxiliary elds, whose time-derivatives do not appear
in the action. The Lagrangian density in this gauge (with 8G 1) is
L =
a
3
2
e
3
_
NR
(3)
2NV ( ) +N
1
_
E
j
i
E
i
j
(E
i
i
)
2
_
+N
1

2
+N
1

n
_

n
N
i

n
_
2
Na
2
e
2
[exp ()]
ij

n
_
,
(5)
where
E
ij

1
2
_
g
ij

i
N
j

j
N
i
_
, (6)
and bars denote unperturbed quantities. All spatial indices i, j, etc. are
lowered and raised with the matrix g
ij
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)
= a
2
e
2
_
e

_
ij
R
(3)
ij
.
The auxiliary elds N and N
i
are to be found by requiring that the action
is stationary in these variables. This gives the constraint equations:

i
_
N
1
_
E
i
j

i
j
E
k
k
__
= N
1

n
_

n
N
i

n
_
, (7)
N
2
_
R
(3)
2V a
2
e
2
[exp()]
ij

n
_
= E
i
j
E
j
i

_
E
i
i
_
2
+


2
+

n
_

n
N
i

n
_
2
(8)
For instance, to rst order in elds (including eld derivatives) the auxiliary
elds are the same as in the case of no additional matter elds
2
N = 1 +

/H , N
i
=
1
a
2
H

i
+
i

2

, (9)
5
where

H
H
2
=

2
2H
2
, H
a
a
(10)
The elds in the interaction picture satisfy free-eld equations. For we
have the Mukhanov equation:
7

t
2
+
_
d ln(a
3
)
dt
_

t
a
2

2
= 0 , (11)
The eld equation for gravitational waves is

ij
t
2
+ 3H

ij
t
a
2

ij
= 0 , (12)
and for the matter elds

n
t
2
+ 3H

n
t
a
2

n
= 0 . (13)
The elds in the interaction picture are then
(x, t) =
_
d
3
q
_
e
iqx
(q)
q
(t) +e
iqx

(q)

q
(t)
_
, (14)

ij
(x, t) =
_
d
3
q

_
e
iqx
e
ij
( q, )(q, )
q
(t) +e
iqx
e

ij
( q, )

(q, )

q
(t)
_
,
(15)

n
(x, t) =
_
d
3
q
_
e
iqx
(q, n)
q
(t) +e
iqx

(q, n)

q
(t)
_
, (16)
where = 2 is a helicity index and e
ij
( 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), with
2
replaced with q
2
. They satisfy initial
6
conditions, designed to make

/H,
ij
/

16G, and
n
behave like con-
ventionally normalized free elds at t :

(t)
q
(t)
H(t)


q
(t)

16G

q
(t)

1
(2)
3/2

2q a(t)
exp
_
iq
_

t
dt

a(t

)
_
. (20)
IV. LATE TIME BEHAVIOR
The question to be addressed in this section is whether the time integrals
in Eqs. (1) and (2) are dominated by times near horizon exit for general
graphs. This question is more complicated for loop graphs than for tree
graphs, such as that considered by Maldacena, because for loops there are
two dierent kinds of wave number: the xed wave numbers q associated
with external lines, and the internal wave numbers p circulating in loops,
over which we must integrate. It is only if the integrals over internal wave
numbers p are dominated by values of order p q that we can speak of
a denite time of horizon exit, when q/a p/a H. In this section we
will integrate rst over the time arguments in Eq. (2), holding the internal
wave numbers at xed values, and return at the end of this section to the
problems raised by the necessity of then integrating over the p s.
There is never any problem with the convergence of the time integrals
at very early times; all uctuations oscillate very rapidly for q/a H and
p/a H, suppressing the contribution of early times to the time integrals in
Eq. (2). To see what happens for late times, when q/a H and p/a H,
we need to count the powers of a in the contribution of late times in general
loop as well as tree graphs.
For this purpose, we need to consider the behavior of the coecient
functions appearing in the Fourier decompositions (14)(16) of the elds in
the interaction picture. In order to implement dimensional regularization,
we will consider these coecient functions in 2 space dimensions, returning
later to the limit 2 3. The coecient functions then obey dierential
equations obtained by replacing the space dimensionality 3 in Eqs. (11)(13)

In Newtonian gauge the quantity (x, t)

(t)/H(t) approaches the inaton eld uc-

tuation (t) for t .
7
with 2, as well as replacing the Laplacian with q
2
:
d
2

q
(t)
dt
2
+
_
_
d ln
_
a
2
(t)(t)
_
dt
_
_
d
q
(t)
dt
+
q
2
a
2
(t)

q
(t) = 0 , (21)
d
2

q
(t)
dt
2
+ 2H(t)
d
q
(t)
dt
+
q
2
a
2
(t)

q
(t) = 0 , (22)
d
2

q
(t)
dt
2
+ 2H(t)
d
q
(t)
dt
+
q
2
a
2

q
(t) = 0 . (23)
At late times, when q/a H, the solutions can be written as asymptotic
expansions in inverse powers of a:

q
(t)
o
q
_
1 +
_

t
q
2
dt

a
2
(t

)(t

)
_
t

a
22
(t

) (t

) dt

+. . .
_
+C
q
_
_

t
dt

a
2
(t

)(t

)
+q
2
_

t
dt

a
2
(t

)(t

)
_
t

a
22
(t

) (t

) dt

_

t

dt

a
2
(t

)(t

)
+. . .
_
(24)

q
(t)
o
q
_
1 +
_

t
q
2
dt

a
2
(t

)
_
t

a
22
(t

) dt

+. . .
_
+D
q
_
_

t
dt

a
2
(t

)
+q
2
_

t
dt

a
2
(t

)
_
t

a(t

) dt

_

t

dt

a
2
(t

)
+. . .
_
(25)

q
(t)
o
q
_
1 +
_

t
q
2
dt

a
2
(t

)
_
t

a
22
(t

) dt

+. . .
_
+E
q
_
_

t
dt

a
2
(t

)
+q
2
_

t
dt

a
2
(t

)
_
t

a
22
(t

) dt

_

t

dt

a
2
(t

)
+. . .
_
(26)
where
o
q
,
0
q
, and
o
q
are the limiting values of
q
(t),
q
(t), and
q
(t) (the
o superscript stands for outside the horizon) and C
q
, D
q
, and E
q
are
additional constants. In any kind of ination with sucient expansion, the
Robertson-Walker scale factor a will grow much faster than H or can

By t = in the limits of these integrals and elsewhere in this paper, we mean a time
still during ination, but suciently late so that a(t) is many e-foldings larger than its
value when q/a falls below H.
8
change, and Eqs. (24)(26) thus show that (at least for 2 2) the time
derivatives of
q
,
q
, and
q
all vanish for q/a H like 1/a
2
.
If an interaction involves enough factors of

,
ij
, and/or
n
so that
these 1/a
2
factors and any 1/a
2
factors from the contraction of space indices
more than compensate for the a
2
factor in the interaction from the square
root of the metric determinant, then the integral over the associated time
coordinate will converge exponentially fast at late times as well as at early
times, and therefore may be expected to be dominated by the era in which
the wavelength leaves the horizon. For instance, the extension of Eq. (5) to
2 space dimensions gives the interaction between a eld and a pair of
elds
L

=
a
22
2

a
22
2H

n
+a
22

i
_

H
a
2

n

n

a
2
2H

n

2
n
+
3a
2
2

n

2
n
. (27)
(The interaction Hamiltonian given by canonical quantization is just

_
d
2
x L

, but this simple relation does not always apply.) Counting a

factor a
2
for each

or
n
, the terms in this interaction go as a
22
, a
24
,
a
24
, a
26
, and a
24
, respectively. All these terms are safe for 2 < 4,
except for the rst, which for 2 > 2 grows exponentially at late times.
Because of the commutators in Eq. (2), the condition for a safe interac-
tion is actually less stringent than that it should decay exponentially with
time, and even a growing term that only involves elds rather than their time
derivatives, like the rst term in Eq. (27), may not destroy the convergence
of the time integrals. We will now prove the following:
Theorem The integrals over the time coordinates of interactions converge
exponentially for t , essentially as
_

dt/a
n
(t) with n > 0, provided
that in 2 space dimensions, all interactions are of one or the other of two
types:
Safe interactions, that contain a number of factors of a(t) (including
2 factors of a for each time derivative and the 2 factors of a from

Detg) strictly less than 2 2, and

9
Dangerous interactions, which grow at late times no faster than a
22
,
and contain only elds, not time derivatives of elds.
These conditions are evidently met by the interaction (27), irrespective of
the value of , and, as we shall see in Section VI, they are satised by all
other 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 2 enters in the interaction only in a factor

Detg
a
2
, so the question of whether or not a given theory satises the conditions
of this theorem does not depend on the value of 2. Thus this theorem has
the corollary:
Corollary The integrals over the time coordinates of interactions converge
exponentially for t , essentially as
_

dt/a
n
(t) with n > 0, provided
that in 3 space dimensions all interactions are of one or the other of two
types:
Safe interactions, that contain a number of factors of a(t) (including
2 factors of a for each time derivative and the 3 factors of a from

Detg) strictly less than +1, and

Dangerous interactions, which grow at late times no faster than a, and
contain only elds, not time derivatives of elds.
Here is the proof. As already mentioned, the reason that dangerous
interactions are not necessarily fatal has to do with how they enter into
commutators in Eq. (2). Because of the time-ordering in Eq. (2), any failure
of 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 t
r
, t
r+1
, . . . t
N
, all go to
innity together. We will therefore have to count the number of factors of
a(t
r
), a(t
r+1
), . . . a(t
N
), 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 as
the integral over t
r
, t
r+1
, . . . t
N
involves a negative total number of factors
of a, it converges exponentially fast no matter how many factors of log a arise
from subintegrations.) Now, at least one of the elds or eld time derivatives
in each term in H(t
s
) with r s N must appear in a commutator with one
of the elds in some other H
I
(t
s
) with s < s

N. So we need to consider
the commutators of elds at times which may be unequal, but are both late.
In the sense described above, treating all a(t
r
), a(t
r+1
), . . . a(t
N
) as being
10
of the same order of magnitude, if a(t) increases more-or-less exponentially,
then the commutator of two elds or a eld and a eld time-derivative goes
as a
2
, while the commutator of two eld time-derivatives goes as a
22
.
For instance, the unequal-time commutators of the interaction-picture
elds (14)(16) are
_
(x, t), (x

, t

)
_
=
_
d
2
p e
ip(xx

)
_

p
(t)

p
(t

)
p
(t

p
(t)
_
, (28)
_

ij
(x, t),
kl
(x

, t

)
_
=
_
d
2
p e
ip(xx

ijkl
( p)
_

p
(t)

p
(t

)
p
(t

p
(t)
_
,
(29)
_

n
(x, t),
m
(x

, t

)
_
=
nm
_
d
2
p e
ip(xx

)
_

p
(t)

p
(t

)
p
(t

p
(t)
_
,
(30)
where
ijkl
( p)

e
ij
( p, )e
kl
( p, ). The two asymptotic expansions given
in Eqs.(21(23) for each of the elds are both real aside from over-all factors,
so neither by itself contributes to the commutators. On the other hand, the
constants C
p

o
p
, D
p

o
p
, and E
p

o
p
are in general complex. (For instance,
in a strictly exponential expansion, ination, the phase of C
p

o
p
is given by
a factor e
i
.) The asymptotic expansions of the commutators at late
times are therefore
_
(x
1
, t
1
), (x
2
, t
2
)
_
2i
_
d
2
p Im
_
C
p

o
p
_
e
ip(x
1
x
2
)
_
_
t
2
t
1
dt

a
2
(t

)(t

)
+p
2
_
t
2
t
1
dt

a
2
(t

)(t

)
_
t

a
22
(t

) (t

) dt

_

t

dt

a
2
(t

) (t

)
+p
2
_

t
1
dt

1
a
2
(t

1
) (t

1
)
_

t
2
dt

2
a
2
(t

2
) (t

2
)
_
t

2
t

1
a
22
(t

)(t

) dt

+. . .
_
,
(31)
_

ij
(x
1
, t
1
),
kl
(x
2
, t
2
)
_
2i
_
d
2
p
ijkl
( p) Im
_
D
p

o
p
_
e
ip(x
1
x
2
)
_
_
t
2
t
1
dt

a
2
(t

)
+p
2
_
t
2
t
1
dt

a
2
(t

)
_
t

a
22
(t

) dt

_

t

dt

a
2
(t

)
+p
2
_

t
1
dt

1
a
2
(t

1
)
_

t
2
dt

2
a
2
(t

2
)
_
t

2
t

1
a
22
(t

) dt

+. . .
_
,
(32)
_

n
(x
1
, t
1
),
m
(x
2
, t
2
)
_
2i
nm
_
d
2
p Im
_
E
p

o
p
_
e
ip(x
1
x
2
)
_
_
t
2
t
1
dt

a
2
(t

)
11
+p
2
_
t
2
t
1
dt

a
2
(t

)
_
t

a
22
(t

) dt

_

t

dt

a
2
(t

)
+p
2
_

t
1
dt

1
a
2
(t

1
)
_

t
2
dt

2
a
2
(t

2
)
_
t

2
t

1
a
22
(t

) dt

+. . .
_
.
(33)
We see that the commutator of two elds vanishes essentially as a
2
for
late times, and the same is true for the commutator of a eld and its time
derivative, but the commutators of two time derivatives arise only from the
third terms in the expansions (31)(33), and therefore go as a
22
. That
is,
_

(x
1
, t
1
),

(x
2
, t
2
)
_
2i
_
d
2
p Im
_
C
p

o
p
_
e
ip(x
1
x
2
)

_
p
2
a
2
(t
1
) (t
1
)a
2
(t
2
) (t
2
)
_
t
2
t
1
a
22
(t

)(t

) dt

+. . .
_
,
and likewise for
ij
and
n
.
Lets now add up the total number of factors of a(t
r
), a(t
r+1
), . . . and a(t
N
)
in the integrand of Eq. (2), for some selection of terms in the interactions
H(t
s
) with r s N. Suppose that the selected term in H(t
s
) contains an
explicit factor a(t
s
)
As
, and B
s
factors of eld time derivatives. Suppose also
that in the inner N r + 1 commutators in Eq. (2) there appear C com-
mutators of elds with each other, C

commutators of elds with eld time

derivatives, and C

commutators of eld time derivatives with each other.

The number of eld time derivatives that are not in these commutators is

s
B
s
C

2C

, and these contribute a total 2

s
B
s
+2C

+4C

factors
of a. (All sums over s here run from r to N.) In addition, there are

s
A
s
factors of a that appear explicitly in the interactions, and as we have seen,
the commutators contribute 2C 2C

(2 +2)C

factors of a. Hence
the total number of factors of a(t
r
), a(t
r+1
), . . . and a(t
N
) in the integrand
of Eq. (2) is
# =

s
(A
s
2B
s
)2C(2 2)(C

+C

) =

s
(A
s
2B
s
2 +2)2C ,
(34)
in which we have used the fact that the total number C +C

+C

of com-
mutators of the interactions H(t
r
), H(t
r+1
), . . . and H(t
N
) with each other
and with the eld product Q equals the number of these interactions. Under
the assumptions of this theorem, all interactions have A
s
2B
s
2 2.
12
If any of them are safe in the sense that A
s
2B
s
< 2 2, then # < 0,
and the integral over time converges exponentially fast. On the other hand,
if all of them have A
s
2B
s
= 2 2, then under the assumptions of this
theorem they all involve only elds, not eld time derivatives, so the same
is true of the commutators of these interactions. In this case C > 0 and
# = 2C < 0, so again the integral over time converges exponentially fast.
In counting powers of a, we have held the wave numbers p associated
with internal lines xed, like the external wave numbers, because we are
integrating over time coordinates before we integrate over the internal wave
numbers. The integrals over time receive little contribution from values of
the conformal time
_

t
dt/a satisfying p 1 and q 1, be-
cause of the rapid oscillation of the integrand, and for theories satisfying the
conditions of our theorem they also receive little contribution from values of
with p 1 and q 1, because of the damping provided by negative
powers of a. (Note that when a(t) increases more or less exponentially, is
of the order of 1/aH.) Thus for these theories, we expect the integrals to
be dominated by times for which 1/ is in the range from the qs to the pss.
The question then is whether the integrals over the internal wave numbers
p are dominated by values of the order of the external wave numbers q? If
they are, then the results depend only on the history of ination around the
time of horizon exit, q 1, or in other words, q/a H.
Any integral over the internal wave numbers will in general take the form
of a polynomial in the external wave numbers, with coecients that may be
divergent, plus a nite term given by a convergent integral dominated by
internal wave numbers of the same order of magnitude as the xed external
wave numbers. An example of this decomposition is given in Section VII. In
particular, the integral over the wave number associated with an internal line
that begins and ends at the same vertex does not involve the external wave
numbers, so its contribution is purely a polynomial in the wave numbers of
the other lines attached to the same vertex.
Just as in dealing with ultraviolet divergences in at space quantum
eld theory, renormalization removes some of these ultraviolet divergent
polynomial terms, and others are removed by appropriate redenitions of
the eld operators. (Some examples are given in the next section.) Where
redenition of the eld operators is necessary, it is only products of the
redened renormalized eld operators whose expectation values may be
expected to give results that converge at late times. If, after all such renor-
malizations and redenitions, there remained ultraviolet divergences in the
integrals over internal wave numbers, we could conclude that the approxi-
13
mation of extending the time integrals to + is not valid, and that these
integrals can be taken only to some time t late in ination. The decrease
of the integrand at wave numbers p much greater than 1/(t) would then
provide the ultraviolet cut o that is still needed, but the correlation func-
tions would exhibit the sort of time dependence that has been found in other
contexts by Woodard and his collaborators,
3
and we would not be able to
draw conclusions about correlations actually measured at times much closer
to the present. The possible presence of such ultraviolet divergences that
are not removed by renormalization and eld redenition is an important
issue, which merits further study.

But even if such ultraviolet divergences

are present, it would still be possible to calculate the non-polynomial part
of the integrals over internal momenta which is not ultraviolet divergent (at
least in one loop order) even when the time t is taken to innity. Such a
calculation will be presented in Section VII.
V. AN EXAMPLE: EXPONENTIAL EXPANSION
To clarify the issues discussed at the end of the previous section, we
will examine a simple unphysical model, along with a revealing class of
generalizations.
First, consider a single real scalar eld (x, t) in a xed 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 =
1
2
_
Detg g

(1+
2
)

= (1+
2
)
_
a
2
2

2

a
22
2
()
2
_
,
(35)

Many theories are aicted with infrared divergences, even when t is held xed. The
infrared divergences are attributed to the imposition of the unrealistic initial condition,
that at early times all of innite space is occupied by a BunchDavies vacuum. The
infrared divergence can be eliminated either by taking space to be nite
8
or by changing
the vacuum.
9
In any case, it is the appearance of uncancelled ultraviolet rather than
infrared divergences when we integrate over internal wave numbers after taking the limit
t 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 of a if the integral
receives contributions from arbitrarily large values of the internal wave number.

This model, and much of the analysis, was suggested to me by R. Woodard, private
communication.
14
where a e
Ht
with H constant. (This of course can be rewritten as a free
eld theory, but it is instructive nonetheless, and will be generalized later
in this section to interacting theories.) We follow the usual procedure of
dening a canonical conjugate eld = L/ , constructing the Hamilto-
nian density H = L with expressed in terms of , dividing H into
a quadratic part H
0
and interaction part H
I
, and then replacing in H
I
with the interaction-picture
I
given by = [H
0
/]
=
I
. This gives an
interaction
H
I
=

2
_
d
2
x
_

a
2
2
_

2
1 +
2
,
2
_
+a
22
()
2

2
_
. (36)
(An anticommutator is needed in the rst term to satisfy the requirement
that H
I
be Hermitian.) This interaction saties the conditions of the theo-
rem proved in the previous section for any value of the space dimensionality
2: the rst term in the square brackets contains 2 4 factors of a (count-
ing a factor a
2
for each time derivative, so it is safe, while the second term
contains 2 2 factors of a, and is therefore dangerous, but it only involves
elds (including space derivatives), not their time derivatives, so though
dangerous it still satises the conditions of our theorem.
To rst order in , the expectation value (x, t) (x

, t) is given by a
one-loop diagram, in which a scalar eld line is emitted and absorbed at the
same vertex, with the two external lines also attached to this vertex. This
expectation 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 eective interactions
proportional to a
22
()
2
, a
22

2
, or a
2

2
, all of which satisfy
the conditions of the theorem of the previous section. Thus it can-
not aect the conclusion that the integral over the time argument of
H
I
(t
1
) converges exponentially at t
1
= +, so that (x, t) (x

, t)
approaches a nite limit for t .
ii Terms in which time derivatives act on both ends of the internal line. This
produces an eective interaction proportional to a
2

2
, which violates
the conditions of our theorem, but it can be removed by adding an
R
2

Detg counterterm in the Lagrangian. (This cancellation is not

automatic, because the condition of minimal coupling is not enforced
by any symmetry.)
15
iii Terms in which a time derivative acts on just one end of the internal
line. This produces an eective interaction proportional to a
2
,
which violates the conditions of our theorem, and cannot be removed
by adding a generally covariant counterterm to the Lagrangian.
To see in detail what trouble is caused by the third type of contribution,
note that the interaction picture scalar eld is given by a Fourier decompo-
sition like Eq. (16), with coecient functions

q
(t) =
e
i(2+1)/4
H
1/2
4

2q

H
(1)

(q) (q)

, (37)
where is the conformal time

_

t
dt

a(t

)
=
1
a(t)H
. (38)
The contribution of the third kind to the expectation value then has the
Fourier transform
_
d
2
x e
iq(xx

)
(x, t) (x

, t)
iii
=
_
H
21
32
2
_
3 _
2
q
_
4
4
_

0
dp
p
_
t

dt
1
a
2
(t
1
)
_
d
dt
1

(p
1
)

H
(1)

(p
1
)

2
_
Im
d
dt
1
_
_
(q
1
)

H
(1)

(q
1
)
_
2
_
(q)

H
(1)

(q)
_
2
_
(39)
Lets see what happens if we evaluate this by integrating rst over p and
then over t
1
from to late times, or vice versa.
To integrate rst over p, we can change the variable of integration from
p to z p
1
, in which case the rst derivative with respect to t
1
can be
replaced with d/dt
1
= (z/a
1

1
)(d/dz) = Hz(d/dz), while dp/p = dz/z.
Dimensional regularization (with 2 < 1) makes the function

H
(1)

(z)

vanish at z , while for > 0 it takes the value 2

()/ for z 0, so
_

0
dz
d
dz

H
(1)

(z)

2
=
_
2

()

_
2
,

Here and below we will not be careful to extend factors like 4 to 2 space dimensions.
This only aects constant terms that accompany any (2 3)
1
poles.
16
and therefore
_
d
2
x e
iq(xx

)
(x, t) (x

, t)
iii
= 4H
_
2

()

_
2
_
H
21
32
2
_
3 _
2
q
_
4

_
t

dt
1
a
2
(t
1
)Im
d
dt
1
_
_
(q
1
)

H
(1)

(q
1
)
_
2
_
(q)

H
(1)

(q)
_
2
_
(40)
For t
1
+ and t + (that is, 0 and
1
0), the integrand of
the integral over t
1
on the second line has the constant limit
a
2
(t
1
)Im
d
dt
1
_
_
(q
1
)

H
(1)

(q
1
)
_
2
_
(q)

H
(1)

(q)
_
2
_

4()
2
q
2

3
H
21
.
(41)
Thus for t , the correlation function (39) does not approach a constant,
but instead goes as
_
d
2
x e
iq(xx

)
(x, t) (x

, t)
iii

H
41
()
4
t
2(2)
104
q
2
. (42)
There is no pole here that prevents continuation to space dimensionality
2 = 3. From this point of view, integrating rst over p, the failure of the
correlation function to approach a nite limit at late times is due to the fact
already noted, that the integral over p produces an eective interaction that
does not satisfy the conditions of our theorem.
But suppose we rst integrate over t
1
from to +. Now there is no
problem with convergence at late times, because the original interaction does
satisfy the conditions of our theorem, but instead we now have a problem
with the convergence of the integral over p. It will be helpful to divide the
integral over p into an integral from 0 to q, where 1, and an integral
from q to innity. The rst integral obviously has no ultraviolet divergence,
and the vanishing of the rst time derivative in Eq. (39) for p 0 prevents
any infrared divergence. In the second integral p and 1/ are the only
magnitudes in the problem with which q can be compared, so for t +
and hence 0 we can evaluate the correlation function by letting q 0
and keeping only the leading term in q. Here again we can use the limiting
formula (41), now for q 0 instead of 0 and
1
0. The integral over
t
1
is then trivial, and we nd that for q 1/ the correlation function is
_
d
2
x e
iq(xx

)
(x, t) (x

, t)
iii

H
42
()
4
2(2)
102
q
2
_

q
dp
p
+ nite . (43)
17
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 diculty of calculating corre-
lation functions at late times. As already mentioned, the original Lagrangian
density (35) actually describes a free eld theory. This is made manifest by
dening a new scalar eld

_
_
1 +
2
d , (44)
for which the Lagrangian density takes the form
L =
1
2
_
Detg g

. (45)
There is no problem in taking the late-time limit of the correlation function
_
d
2
e
iq(xx

)
(x, t) (x

, t) it is just 2
2
H
21
()
2
/32
4
q
2
. From
this point of view, the growth of the correlation function (42) at late times is
a result of our perversity in calculating the correlation function of instead
of .
Can we nd elds whose correlation functions have a constant limit at
late times in theories that satisfy the conditions of our theorem but are not
equivalent to free eld theories? The general answer is not known, but here
is a class of interacting eld theories for which such renormalized elds
can be found. This time we consider an arbitrary number of real scalar elds

n
(x, t) in a xed de Sitter metric. The Lagrangian density is taken to have
the form of a non-linear -model:
L =
1
2

nm
_
Detg g

nm
+K
nm
()
_

m
, (46)
where K
nm
() is an arbitrary real symmetric matrix function of the
n
;
is a coupling constant; and again a e
Ht
with H constant. The Hamilto-
nian derived from this Lagrangian density does satisfy the conditions of the
theorem of Section IV, whatever the function K
nm
().
To rst order in , the same problem discussed earlier in this section
arises from graphs in which an internal line of the eld
n
is emitted and
absorbed from the same vertex, with a time derivative acting on just one
end of this line. Depending on what correlation function is being calculated,
the contribution of such graphs is proportional to various contractions of
partial derivatives of the function
A
m
()

n
K
nm
()

n
. (47)
18
Suppose we make a redenition of the elds of rst order in :

n

n

n
() . (48)
This changes the matrix K to

K
nm
() = K
nm
() +

n
()

m
+

m
()

n
, (49)
and so

A
m
()

K
nm
()

n
= A
m
() +

n
()

m
+

m
()

n
. (50)
Thus the elds
n
are renormalized, in the sense that to rst order in
correlation functions have nite limits at late times, provided that

n
()

m
+

m
()

n
= A
m
() . (51)
This can be solved by rst solving the Poisson equation

2
B()

n
=
1
2

n
A
n
()

n
(52)
and then solving a second Poisson equation

m
()

n
= A
m
()
B()

m
. (53)
Thus for at least to rst order in this class of theories, it is always possible
to nd a suitable set of renormalized elds.
Because we can take the limit t only for the correlation functions
of suitably dened elds (such as
n
in our example), the question naturally
arises, whether these are the elds whose correlation functions we want to
calculate. The answer is conditioned by the fact that astronomical observa-
tions of the cosmic microwave background or large scale structure are made
following a long era that has intervened since the end of ination, during
which things happened about which we know almost nothing, such as re-
heating, baryon and lepton synthesis, and dark matter decoupling. The only
thing that allows us to use observations to learn about ination is that some
quantities were conserved during this era, while uctuation wave lengths
19
were outside the horizon. These are the only quantities whose correlation
functions at the end of ination can be interpreted in terms of current ob-
servations. In the classical limit, the quantities that are conserved outside
the horizon are and
ij
, but we dont know whether this will be true when
quantum eects are taken into account. Still, we can expect that quantities
are conserved only when there is some symmetry principle that makes them
conserved, and whatever symmetry principle keeps some quantity conserved
from the end of ination to the time of horizon re-entry is likely also to keep
it conserved from the time of horizon exit to the end of ination. So we may
guess that the quantities whose correlation functions we will need to know
are just those whose correlation functions approach constant limits at the
end of ination.
VI. DANGEROUS INTERACTIONS IN INFLATIONARY
THEORIES
We now return to the semi-realistic theories described in Section III. We
will show in this section that all interactions are of the type called for in the
theorem of Section IV; that is, they are all safe interactions that (in three
space dimensions) do not grow exponentially at late times (and in fact are
suppressed at late times at least by a factor a
1
), or dangerous interactions
containing only elds and not their time derivatives, which grow no faster
that a at late times. Fortunately, as noticed by Maldacena
2
in a dierent
context, for this purpose it is not necessary to solve the constraint equations
(7) and (8), which are quite complicated especially when the
n
elds are
included. Inspection of these equations shows that when we count

,
ij
,
and
n
as of order a
2
, the auxiliary elds N 1 and N
i
are both also of
order a
2
.

This is apparent in the rst-order solution (9) of the constraint

equations, but it holds to all orders in the elds. To calculate the quantity
E
j
i
E
i
j
(E
i
i
)
2
in Eq. (5), we note that
E
i
j
= H
i
j
+

i
j
+
1
2
_
e

t
e

_
i
j

1
2
_

i
N
j
+
j
N
i
_
. (54)
The rst term H
i
j
is of order zero in a, while all other terms are of order
a
2
, so
E
j
i
E
i
j
(E
i
i
)
2
= 6H
2
12H

4H
k
N
k
+O(a
4
) (55)

In counting powers of a, note that the three-dimensional ane connection and Ricci
tensor are independent of a, so the curvature scalar R
(3)
goes as a
2
. For instance, for
ij = 0, we have R
(3)
= a
2
e
2
(4
2
+ 2()
2
).
20
(In deriving this result, we note that
_
e

t
e

_
i
i
=
ii
= 0.) The terms
in (5) of rst order in N 1 all cancel as a consequence of the constraint
equation (8), while terms of second order in N 1 in Eq. (5) (and in partic-
ular in a
3
e
3

2
/2N and 3H
3
a
3
e
3
/2N) are suppressed by at least a factor
a
3
(a
2
)
2
, and are therefore safe. Therefore we can isolate all terms that are
potentially dangerous by setting N = 1, and nd
L =
a
3
2
e
3
_
R
(3)
2V ( ) 6H
2
12H

4H
k
N
k
+


2
a
2
e
2
[exp ()]
ij

n
_
+O(a
1
) ,
(56)
We note that e
3

k
N
k
=
k
(e
3
N
k
), so this term vanishes when integrated
over three-space, and therefore makes no contribution to the action. The
term proportional to

can be written
6a
3
e
3
H

=

t
_
2a
3
H e
3
_
+a
3
e
3
_
6H
2
+ 2

H
_
.
The rst term vanishes when integrated over time, so it gives no contribution
to the action. To evaluate the remaining terms we use the unperturbed
inaton eld equation, which (with 8G 1) gives

H =

2
/2, and the
Friedmann equation, which gives 6H
2
= 2V +

2
. We then nd a cancellation
V 3H
2
+
1
2

2
+ 6H
2
+ 2

H = 0 .
Aside from terms that make no contribution to the action, the Lagrangian
density is then
L =
a
3
2
e
3
_
R
(3)
a
2
e
2
[exp ()]
ij

n
_
+O(a
1
) . (57)
We see that, at least in this class of theories, the dangerous terms that are
not suppressed by a factor a
1
grow at most like a at late times, and involve
only elds, not their time derivatives, as assumed in the theorem of Section
III.
It remains to be seen if in these theories, after integrating over times and
taking the limit t , the remaining integrals over internal wave numbers
21
are made convergent by the same counterterms that eliminate ultraviolet di-
vergences in at spacetime, and if not, whether they can be made convergent
by suitable redenitions of the elds and
ij
appearing in the correlation
functions. This is left as a problem for further work.
Not all theories satisfy the conditions of the theorem of Section IV. For
instance, a non-derivative interaction

DetgF() of the elds would

have +3 factors of a, and hence would violate the condition that the total
number of factors of a (counting each time derivative as -2 factors) must
be no greater than +1. The elds must be the Goldstone bosons of some
broken global symmetry in order to satisfy the conditions of our theorem in
a natural way.
VII. A SAMPLE CALCULATION
As an application of the formalism described in this paper, we will now
calculate the one-loop contribution to the correlation function of two elds,
which is measured in the spectrum of anisotropies of the cosmic microwave
background. As already mentioned, in the class of theories described in
Section III, this two-point function is dominated by a matter loop, because
there are many types of matter eld and only one gravitational eld. We
rst consider the contribution of second order in the interaction (27). It
saves a great deal of work if we use the interaction-picture eld equations
(11) and (13) to put this interaction in the form
H

(t) =
_
d
3
x L

(x, t) = A(t) +

B(t) (58)
where
A = 2Ha
5

n
_
d
3
x
2
n

2

(59)
B =

n
_
d
3
x
_
a
H
a
3

_ _
1
2
(
n
)
2
+
1
2
a
2

2
n
_
. (60)
In general, for an interaction Hamiltonian of the form (58), Eq. (2) can be
put in the form
Q(t) =

N=0
i
N
_
t

dt
N
_
t
N

dt
N1

_
t
2

dt
1

__

H
I
(t
1
),
_

H
I
(t
2
),
_

H
I
(t
N
),

Q
I
(t)
_

___
, (61)
22
where

H
I
(t) = e
iB(t)
_
A(t)+

B(t)+ie
iB(t)
_
d
dt
e
iB(t)
_
_
e
iB(t)
= A(t)+i[B(t), A(t)]+
i
2
[B(t),

B(t)]+. . .
(62)

Q
I
(t) = e
iB(t)
Q
I
(t)e
iB(t)
= Q
I
(t)+i[B(t), Q
I
(t)]
1
2
[B(t), [B(t), Q
I
(t)]]+. . . .
(63)
To second order in an interaction of the form (58), the expectation value is
then
Q(t)
2
=
_
t

dt
2
_
t
2

dt
1
__
A(t
1
),
_
A(t
2
), Q
I
(t)
___

_
t

dt
1
___
B(t
1
), A(t
1
) +

B(t
1
)/2
_
, Q
I
(t)
__

__
B(t),
_
B(t), Q
I
(t)
___
, (64)
The Fourier transform of the second-order term in the expectation value of
a product of two s is then
_
d
3
xe
iq(xx

)
_
vac, in

(x, t) (x

, t)

vac, in
_
2
=
32(2)
9
q
4
Re
_
t

a
5
(t
2
) (t
2
) H(t
2
) dt
2

_
t
2

a
5
(t
1
) (t
1
) H(t
1
) dt
1

q
(t
1
)

q
(t)
_

q
(t
2
)

q
(t)
q
(t)

q
(t
2
)
_
N
_
d
3
p
_
d
3
p

3
(p +p

+q)

p
(t
1
)

p
(t
2
)
p
(t
1
)

p
(t
2
)
+
(2)
3
4q
4
N
_
d
3
p
_
d
3
p

3
(p +p

+q)
(p p

)
2
|
p
(t)|
2
|
p
(t)|
2
+ . . . (65)
where N is the number of elds. We have shown here explicitly the
contribution of the rst 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 a seagull interaction, as well as one-loop terms

23
of rst order in the terms in Eq. (5), in both of which the integral
over internal wave number is q-independent, plus counterterms arising in
rst order from interactions that cancel ultraviolet divergences in at space,
including

Detg R

and

Detg R
2
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 innities in the integrals over p and p

at interme-
diate stages in the calculation, and we now assume that the singularity as
the number of space dimensions approaches three is cancelled by the terms
in Eq. (65) represented by dots, leaving it to future work to show that this
is the case. Then these integrals are dominated by p p

q. As we have
seen, the integrals over time are then dominated by the time t
q
of horizon
exit, when q/a(t
q
) H(t
q
). For simplicity, we will assume (for the rst time
in this paper) that the unperturbed inaton eld (t) is rolling very slowly
down the potential at time t
q
, so that the expansion near this time can be
approximated as strictly exponential, a(t) e
Ht
. Then the wave functions
are

q
(t)
o
q
e
iq
_
1 +iq
_
,

q
(t)
o
q
e
iq
_
1 +iq
_
,
where is the conformal time

_

t
dt
a(t)
,
and the wave functions outside the horizon have modulus
|
o
q
|
2
=
H
2
(t
q
)
2(2)
3
q
3
, |
o
q
|
2
=
H
2
(t
q
)
2(2)
3
(t
q
) q
3
Using these wave functions in Eq. (65) gives
_
d
3
xe
iq(xx

)
_
vac, in

(x, t) (x

, t)

vac, in
_
2
=
(8GH
2
(t
q
))
2
N
(2)
3
_
d
3
p
_
d
3
p

3
(p +p

+q)

_
p p

q
7
(p +p

+q)
+
(p p

)
2
16 q
4
p
3
p
3
_
+. . . (66)
with the dots having the same meaning as in Eq. (65).
24
Simple dimensional analysis tells us that when the integral over internal
wave numbers of the rst term in square brackets is made nite by dimen-
sional regularization, it is converted to
_
d
3
p
_
d
3
p

3
(p +p

+q)
p p

p +p

+q
q
4+
F() , (67)
where is a measure of the dierence between the space dimensionality
and three. The ultraviolet divergences in this integrals for = 0 gives the
function F() a singularities as 0:
F()
F
0

+F
1
, (68)
so that in the limit = 0
_
d
3
p
_
d
3
p

3
(p +p

+q)
p p

p +p

+q
= q
4
_
F
0
ln q +L
_
, (69)
where L is a divergent constant. We can easily calculate the coecient F
0
of the logarithm. For this purpose, we note that, in general,
_
d
3
p
_
d
3
p

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 dierentiate six times with respect to q. A tedious but straightforward
calculation gives
d
6
dq
6
_
q
_
d
3
p
_
d
3
p

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 F
0
= /15.
In contrast, the integral of the second term in square brackets in Eq. (66)
is a sum of powers of q with divergent coecients, but with no logarithmic
singularity in q. (This term would be eliminated if we calculated the ex-
pectation value of a product of elds

exp(iB) exp(iB) instead of .)
The terms represented by dots in Eq. (65) make contributions that are also
just a sum of powers of q with divergent coecients. We are assuming that
all ultraviolet divergences cancel, but we cannot nd resulting nite power
terms without knowing the renormalized coecients of the

DetgR

25
and

DetgR
2
terms in the Lagrangian density. So we are left with the
result (now restoring a suitable power of 8G) that
_
d
3
xe
iq(xx

)
_
vac, in

(x, t) (x

, t)

vac, in
_
2
=

_
8GH
2
(t
q
)
_
2
N
15(2)
3
q
3
_
ln q +C
_
(71)
with C an unknown constant. This may be compared with the classical (and
classic) result, that in slow roll ination this correlation function takes the
form
_
d
3
xe
iq(xx

)
_
vac, in

(x, t) (x

, t)

vac, in
_
0
=
8GH
2
(t
q
)
4(2)
3
|(t
q
)|q
3
(72)
The one-loop correction (71) is smaller by a factor of order 8GH
2
N|(t
q
)|,
so even if N is 10
2
or 10
3
this correction is likely to remain unobservable.
Still, it is interesting that even in the extreme slow roll limit, where H(t
q
)
and (t
q
) are nearly constant, the factor ln q gives it a dierent dependence
on 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. This
material is based upon work supported by the National Science Foundation
under Grants Nos. PHY-0071512 and PHY-0455649 and with support from
The Robert A. Welch Foundation, Grant No. F-0014, and also grant support
from the US Navy, Oce of Naval Research, Grant Nos. N00014-03-1-0639
and N00014-04-1-0336, Quantum Optics Initiative.
26
APPENDIX: THE IN-IN FORMALISM
1. Time Dependence
First, it is necessary to be precise about the origin of the time-dependence
of the uctuation Hamiltonian in applications such as those encountered in
cosmology. Consider a general Hamiltonian system, with canonical variables

a
(x, t) and conjugates
a
(x, t) satisfying the commutation relations
_

a
(x, t),
b
(y, t)
_
= i
ab

3
(xy) ,
_

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

a
(x, t),
b
(y, t)
_
= 0 ,
(A.1)
and the equations of motion

a
(x, t) = i
_
H[(t), (t)],
a
(x, t)
_
,
a
(x, t) = i
_
H[(t), (t)],
a
(x, t)
_
.
(A.2)
Here a is a compound index labeling particular elds and their spin com-
ponents. The Hamiltonian H is a functional of the
a
(x, t) and
a
(x, t) at
xed time t, which according to Eq. (A.2) is of course independent of the
time 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 the
expectation values of various scalar elds.) Of course, since c-numbers com-
mute with everything, the uctuations satisfy the same commutation rules
(A.1) as the total variables:
_

a
(x, t),
b
(y, t)
_
= i
ab

3
(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 perturbations
a
(x, t)
and
a
(x, t) at some denite time t, we encounter terms of zeroth and rst
order 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)
27
where

H[(t), (t); t] is the sum of all terms in H[

(t)+(t), (t)+(t)]
of second and higher order in the (x, t) and/or (x, t).
Now, although H generates the time-dependence of
a
(x, t) and
a
(x, t),
it is

H rather than H that generates the time dependence of
a
(x, t) and

a
(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
_
d
3
y
H[

(t), (t)]

b
(y, t)

b
(y, t) +

b
_
d
3
y
H[

(t), (t)]

b
(y, t)

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

a
(x, t)
i
_

b
_
d
3
y
H[

(t), (t)]

b
(y, t)

b
(y, t) +

b
_
d
3
y
H[

(t), (t)]

b
(y, t)

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


a
(x, t) .
Subtracting, we nd

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 uctuations: expand the
original Hamiltonian H in powers of uctuations and , and throw
away the terms of zeroth and rst order in these uctuations. It is this
construction that gives

H an explicit dependence on time.
2. Operator Formalism for Expectation Values
We consider a general Hamiltonian system, of the sort described in the
previous subsection. It follows from Eq. (A.7) that the uctuations at time
t can be expressed in terms of the same operators at some very early time
t
0
through a unitary transformation

a
(t) = U
1
(t, t
0
)
a
(t
0
) U(t, t
0
) ,
a
(t) = U
1
(t, t
0
)
a
(t
0
) U(t, t
0
) ,
(A.8)
where U(t, t
0
) is dened by the dierential equation
d
dt
U(t, t
0
) = i

H[(t), (t); t] U(t, t
0
) (A.9)
and the initial condition
U(t
0
, t
0
) = 1 . (A.10)
28
In the application that concerns us in cosmology, we can take t
0
= , by
which we mean any time early enough so that the wavelengths of interest
are deep inside the horizon.
To calculate U(t, t
0
), we now further decompose

H into a kinematic term
H
0
that is quadratic in the uctuations, and an interaction term H
I
:

H[(t), (t); t] = H
0
[(t), (t); t] +H
I
[(t), (t); t] , (A.11)
and we seek to calculate U as a power series in H
I
. To this end, we intro-
duce an interaction picture: we dene uctuation operators
I
a
(t) and

I
a
(t) whose time dependence is generated by the quadratic part of the
Hamiltonian:

I
a
(t) = i
_
H
0
[
I
(t),
I
(t); t],
I
a
(t)
_
,
I
a
(t) = i
_
H
0
[
I
(t),
I
(t); t],
I
a
(t)
_
,
(A.12)
and the initial conditions

I
a
(t
0
) =
a
(t
0
) ,
I
a
(t
0
) =
a
(t
0
) . (A.13)
Because H
0
is quadratic, the interaction picture operators are free elds,
satisfying linear wave equations.
It follows from Eq. (A.12) that in evaluating H
0
[
I
,
I
; t] we can take
the time argument of
I
and
I
to have any value, and in particular we
can take it as t
0
, so that
H
0
[
I
(t),
I
(t); t] = H
0
[(t
0
), (t
0
); t] , (A.14)
but the intrinsic time-dependence of H
0
still remains. The solution of
Eq. (A.12) can again be written as a unitary transformation:

I
a
(t) = U
1
0
(t, t
0
)
a
(t
0
)U
0
(t, t
0
) ,
I
a
(t) = U
1
0
(t, t
0
)
a
(t
0
)U
0
(t, t
0
) ,
(A.15)
with U
0
dened by the dierential equation
d
dt
U
0
(t, t
0
) = i H
0
[(t
0
), (t
0
); t] U
0
(t, t
0
) (A.16)
and the initial condition
U
0
(t
0
, t
0
) = 1 . (A.17)
Then from Eqs. (A.9) and (A.16) we have
d
dt
_
U
1
0
(t, t
0
)U(t, t
0
)
_
= iU
1
0
(t, t
0
)H
I
[(t
0
), (t
0
); t]U(t, t
0
) .
29
Using Eq. (A.15), this gives
U(t, t
0
) = U
0
(t, t
0
)F(t, t
0
) , (A.18)
where
d
dt
F(t, t
0
) = iH
I
(t)F(t, t
0
) , F(t
0
, t
0
) = 1 . (A.19)
and H
I
(t) is the interaction Hamiltonian in the interaction picture:
H
I
(t) U
0
(t, t
0
)H
I
[(t
0
), (t
0
); t]U
1
0
(t, t
0
) = H
I
[
I
(t),
I
(t); t]
(A.20)
The solution of equations like (A.19) is well known
F(t, t
0
) = T exp
_
i
_
t
t
0
H
I
(t) dt
_
(A.21)
where T indicates that the products of H
I
s in the power series expansion of
the exponential are to be time-ordered; that is, they are to be written from
left to right in the decreasing order of time arguments. The solution for the
uctuations in terms of the free elds of the interaction picture is then given
by Eqs. (A.8) and (A.15) as
Q(t) = F
1
(t, t
0
) Q
I
(t)F(t, t
0
)
=
_

T exp
_
i
_
t
t
0
H
I
(t) dt
__
Q
I
(t)
_
T exp
_
i
_
t
t
0
H
I
(t) dt
__
, (A.22)
where Q(t) is any (x, t) or (x, t) or any product of the s and/or s,
all at the same time t but in general with dierent space coordinates, and
Q
I
(t) is the same product of
I
(x, t) and/or
I
(x, t). Also,

T denotes
anti-time-ordering: products of H
I
s in the power series expansion of the
exponential are to be written from left to right in the increasing order of
time arguments.
3. Diagrammatic Formalism for Expectation Values
We want to use Eq. (A.22) to calculate the expectation value Q(t)of
the product Q(t) in a BunchDavies vacuum, annihilated by the positive-
frequency part of the interaction picture uctuations
I
and
I
. We can
use the familiar Wick theorem to express the vacuum expectation value of
the right-hand side of Eq. (A.22) as a sum over pairings of the
I
and
I
with each other. (This of course is the same as supposing the interaction-
picture elds in H
I
(t) and Q
I
(t) to be governed by a Gaussian probability
30
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 of
diagrammatic rules, but one that is rather complicated.
In calculating the term in Q of Nth order in the interaction, we draw
all diagrams with N vertices. Just as for ordinary Feynman diagrams, each
vertex is labeled with a space and time coordinate, and has lines attached
corresponding to the elds in the interaction. There are also external lines,
one for each eld operator in the product Q, labeled with the dierent
space coordinates and the common time t in the arguments of these elds.
All external lines are connected to vertices or other external lines, and all
remaining lines attached to vertices are attached to other vertices. But there
are signicant dierences between the rules following from Eq. (A.22) and
the usual Feynman rules:
We have to distinguish between right and left vertices, arising
respectively from the time-ordered product and the anti-time-ordered
product. A diagram with N vertices contributes a sum over all 2
N
ways of choosing each vertex to be a left vertex or a right vertex.
Each right or left vertex contributes a factor i or +i, respectively, as
well as whatever coupling parameters appear in the interaction.
A line connecting two right vertices or a right vertex and an external
line, in which it is associated with eld operators A(x, t

) and B(y, t

),
contributes a conventional Feynman propagator T{A(x, t

)B(y, t

}.
(It will be understood here and below, that in calculating propagators
all elds A, B, etc. are taken in the interaction picture, and can
be
I
s and/or
I
s.) As a special case, if B is associated with an
external line then t

= t, and since t

t, this is B(y, t)A(x, t

).
A line connecting two left vertices, associated with eld operators
A(x, t

) and B(y, t

), contributes a propagator

T{A(x, t

)B(y, t

}.
As a special case, if B is associated with an external line then t

= t,
and this is A(x, t

)B(y, t).
A line connecting a left vertex, in which it is associated with a eld
operator A(x, t

), to a right vertex, in which it is associated with a

eld operator B(y, t

), contributes a propagator A(x, t

)B(y, t

).
We must integrate over all over the times t

, t

, . . ., associated with the

vertices from t
0
to t, as well as over all space coordinates associated
31
with the vertices.
We must say a word about the disconnected parts of diagrams. A vac-
uum uctuation subdiagram is one in which each vertex is connected only to
other vertices, not to external lines. Just as in ordinary quantum eld the-
ories, the sum of all vacuum uctuation diagrams contributes a numerical
factor multiplying the contribution of diagrams in which vacuum uctua-
tions are excluded. But unlike the case of ordinary quantum eld theory,
this numerical factor is not a phase factor, but is simply
__

T exp
_
i
_
t
t
0
H
I
(t) dt
__ _
T exp
_
i
_
t
t
0
H
I
(t) dt
___
= 1 . (A.23)
Hence in the in-in formalism all vacuum uctuation diagrams automati-
cally cancel. Even so, a diagram may contain disconnected parts which do
not cancel, such as external lines passing through the diagram without in-
teracting. Ignoring all disconnected parts gives what in the theory of noise
is 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 Lagrangian
rather than from the Hamltonian, or to make available a larger range of
gauge choices, or to go beyond perturbation theory. Going back to Eq. (1),
and following the same reasoning
11
that leads from the operator formalism
to the path-integral formalism in the calculation of S-matrix elements, we
see that the vacuum expectation value of any product Q(t) of s and s
at the same time t (now taking t
0
= ) is
Q(t) =
_

x,t

,a
d
La
(x, t

x,t

,a
d
La
(x, t

)
2

x,t

,a
d
Ra
(x, t

x,t

,a
d
Ra
(x, t

)
2
exp
_
i
_
t

dt

a
_
d
3
x

La
(x, t

)
La
(x, t

)

H[
L
(t

),
L
(t

); t

]
__
exp
_
i
_
t

dt

a
_
d
3
x

Ra
(x, t

)
Ra
(x, t

)

H
_

R
(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)
32
Here the functional
0
[] is the wave function of the vacuum,
12

0
[()] exp
_
_

1
2

a,b
_
d
3
x
_
d
3
y E
ab
(x, y)
a
(x, )
b
(y, )
_
_
= exp
_
_

2
_
t

dt

e
t

a,b
_
d
3
x
_
d
3
y E
ab
(x, y)
a
(t

)
b
(t

)
_
_
, (A.25)
where E
ab
is a positive-denite kernel. For instance, for a real scalar eld of
mass m,
E(x, y)
1
(2)
3
_
d
3
p e
ip(xy)
_
p
2
+m
2
. (A.26)
As is well known, if the Hamiltonian is quadratic in the canonical con-
jugates
a
with a eld-independent coecient in the term of second order,
then we can integrate over the
a
by simply setting

a
=

H/
a
, and
the quantity

a

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 rather
take up a puzzle that at rst sight seems to throw doubt on the equivalence
of 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 left
vertices to each other or right vertices to each other or left or right vertices
to external lines are Greens functions of the sort that familiarly emerge
from path integrals, what are we to make of the propagators arising from
Eq. (A.22) for lines connecting left vertices with right vertices? These are not
Greens functions; that is, they are solutions of homogeneous wave equations,
not of inhomogeneous wave equations with a delta function source. As
we shall see, the source of these propagators lies in the delta functions in
Eq. (A.24). It is these delta functions that tie together the integrals over
the L variables and over the R variables, so that the expression (A.18) does
not factor into a product of these integrals.
In analyzing the consequences of Eq. (A.24), it is convenient to condense
our 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 dierent elds, plus a two-valued index that
distinguishes from . With this understanding, Eq. (A.24) reads
Q(t) =
_

t

,n
d
Ln
(t

,n
d
Rn
(t

2
33
exp
_
i
_
t

dt

L
_

L
(t

),

L
(t

); t

_
_
exp
_
i
_
t

dt

L
_

R
(t

),

R
(t

); t

_
_

Ln
(t)
Rn
(t)
_
_
Q
_

L
(t)
_

0
_

L
()
_

0
_

R
()
_
, (A.27)
where

L[(t

),

(t

); t

a
_
d
3
x
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

L
0
that is
quadratic in the uctuations, plus an interaction term

H
I
:

L =

L
0

H
I
, (A.29)
where

L
0
[(t

),

(t

); t

] =

a
_
d
3
x

a
(x, t

)
a
(x, t

)

H
0
_
(t

), (t

); t

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

dt

L
0
[
R
(t

),

R
(t

); t

]
+
i
2

ab
_
d
3
x
_
d
3
y E
ab
(x, y)
Ra
(x, t

)
Rb
(y, t

)
_

1
2

nn

,t

D
R
nt

,mt

Rn
(t

)
Rn
(t

) , (A.31)
_
t

dt

L
0
[
L
(t

),

L
(t

); t

i
2

ab
_
d
3
x
_
d
3
y E
ab
(x, y)
La
(x, t

)
Lb
(y, t

)
_

1
2

nn

,t

D
L
nt

,n

t

Ln
(t

)
Ln
(t

) (A.32)
The vacuum wave function is the same for
L
and
R
, but it is combined here
with an exponential exp(i
_

L
0
) for the
Ln
and an exponential exp(+i
_

L
0
)
34
for the
Rn
, which accounts for the dierent signs of the i terms in Eqs. (A.31)
and (A.32). (The factor e
t

in Eq. (A.25) is eectively equal to one for any

nite t

, and has therefore been dropped.) We also express the product of

delta functions in Eq. (A.27) as a Gaussian:

Ln
(t)
Rn
(t)
_
exp
_

n
_

Ln
(t)
Rn
(t)
_
2
_
= exp
_

nn

C
nt

,n

Ln
(t

)
Rn
(t

)
__

Ln
(t

)
Rn
(t

)
_
_
, (A.33)
where
C
nt

,n

t

1

nn
(t

t) (t

t) , (A.34)
and

is another positive innitesimal.

Following the usual rules for integrating a Gaussian times a polynomial,
the integral is given by a sum over diagrams as described above, but with
a line that connects right vertices with each other (or with external lines)
contributing a factor i
RR
nt

,n

t
, a line that connects left vertices with each
other (or with external lines) contributing a factor i
LL
nt

,n

t
, and a line
that connects a right vertex where it is associated with
n
(t

) with a left
vertex associated with
n
(t

) contributing a factor i
RL
nt

,n

t
, with the s
determined by the condition
_
iD
R
C C
C iD
L
C
_ _
i
RR
i
RL
i(
RL
)
T
i
LL
_
=
_
1 0
0 1
_
. (A.35)
This must hold whatever tiny value we give to

, and so
D
R

RR
= 1 , D
L

LL
= 1 , (A.36)
D
R

RL
= 0 , D
L
_

RL
_
T
= 0 , (A.37)
C
LL
= C
RL
, C
RR
= C(
RL
)
T
. (A.38)
The rst Eq. (A.36) is the usual inhomogeneous wave equation for the
propagator, whose solution as well known is
i
RR
nt

,n

t
= T{
n
(t

)
n
(t

)} , (A.39)
35
with the time-ordering dictated by the +i in Eq. (A.31). The second
Eq. (A.36) is the complex conjugate of the rst wave equation, whose solu-
tion is the complex conjugate of Eq. (A.39):
i
LL
nt

,n

t
=

T{
n
(t

)
n
(t

)} . (A.40)
Eqs. (A.39) and (A.40) thus give the same propagators for lines connecting
right vertices with each other or with external lines, and for lines connecting
left vertices with each other or with external lines, as we we encountered
in the operator formalism. Equations (A.37) tell us that
RL
and (
RL
)
T
satisfy the homogeneous versions of the wave equations satised by
RR
and

LL
, but to nd
RL
we also need an initial condition. This is provided by
the rst of Eqs. (A.38), which in more detail reads
i
RL
nm
(t, t
2
) = i
LL
nm
(t, t
2
) =

T{
n
(t)
m
(t
2
)} =
m
(t
2
)
n
(t) , (A.41)
in which we have used the fact that t > t
2
. This, together with the rst of
Eqs. (A.37), tells us that
i
RL
nm
(t
1
, t
2
) =
m
(t
2
)
n
(t
1
) , (A.42)
which is the same propagator for internal lines connecting right vertices with
left 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 approach
to the calculation of non-Gaussian correlations, of solving the classical eld
equations beyond the linear approximation, simply corresponds to the cal-
culation of tree diagrams in the in-in formalism. This is a well-known
result
13
in the usual applications of quantum eld theory, but some modi-
cations in the usual argument are needed in the in-in formalism, in which
the vacuum persistence functional is always unity whether or not we add a
current term to the Lagrangian.
We begin by introducing a generating functional W[j, t, g] for correlation
functions of elds at a xed time t:
e
W[J,t,g]/g

_
vac, in

e
1
g

a
_
d
3
x a(x,t)Ja(x)

vac, in
_
g
, (A.43)
where J
a
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
36
Lagrangian density multiplied with a factor 1/g. (This is dierent from the
usual denition of the eective action, because here we are not introducing
the current into the Lagrangian.) The quantity of physical interest is of
course W[J, t, 1], from which expectation values of all products of elds can
be found by expanding in powers of the current.
Using Eq. (A.27), we can calculate W as the path integral
e
W[J,t,g]/g
=
_

L
_

L
_

R
_

R
exp
_
i
_
t

dt

1
g

L[
L
,
L
; t

]
_
exp
_
+i
_
t

dt

1
g

L[
R
,
R
; t

]
_

[
L
(t)
R
(t)]

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

a
_
d
3
xa(x,t)Ja(x)


vac
[
L
()]
vac
[
R
()] (A.44)
The usual power-counting arguments
13
show that the L loop contribution
to W[J, t, g] has a g-dependence given by a factor g
L
. For g 0, W is
thus given by the sum of all tree graphs. The integrals over
L
,
L
,
L
,

L
are dominated in the limit g 0 by elds where

L is stationary, i.e.,
where

L
=
R
=
classical

L
=
R
=
classical
with
classical
and
classical
the solutions of the classical eld equations
with the initial conditions that the elds go to free elds such as (14)(16)
satisfying the initial conditions (20) at t . Since the L and R elds
take the same values at this stationary point, the action integrals cancel,
and we conclude that
_
W[J, t, 1]
_
zero loops
=

a
_
d
3
x
classical
a
(x, t) J
a
(x) . (A.45)
Expanding in powers of the current, this shows that in the tree approxima-
tion the expectation value of any product of elds is to be calculated by
taking the product of the elds obtained by solving the non-linear classical
eld equations with suitable free-eld initial conditions, as was to be proved.
37
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astro-ph/0406398.
2. J. Maldacena, JHEP 0305, 013 (2003) (astro-ph/0210603). For other
work on this problem, see A. Gangui, F. Lucchin, S. Matarrese,and
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5. The constancy of this quantity outside the horizon has been used
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227. This classic article is now available as gr-qc/0405109.
38
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(Cambridge, 1995): Sec. 9.1.
12. ibid., Sec. 9.2.
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Cambridge, 1985): pp 139142.
39