Quantum radiative corrections to slow-roll inflation
NIKHEF-2007-09, SPIN-07-11, ITP-UU-07-18
Quantum radiative corrections to slow-roll inflation
Ante Bilandžić1, 2, ∗ and Tomislav Prokopec1, †
1Institute for Theoretical Physics (ITP) & Spinoza Institute, Utrecht University,
Leuvenlaan 4, Postbus 80.195, 3508 TD Utrecht, The Netherlands
2NIKHEF, Kruislaan 409, 1098 SJ Amsterdam, The Netherlands
We consider the nonminimally coupled λϕ4 scalar field theory in de Sitter space
and construct the renormalization group improved renormalized effective theory at
the one-loop level. Based on the corresponding quantum Friedmann equation and
the scalar field equation of motion, we calculate the quantum radiative corrections
to the scalar spectral index ns, gravitational wave spectral index ng and the ratio
r of tensor to scalar perturbations. When compared with the standard (tree-level)
values, we find that the quantum contributions are suppressed by λN2 where N
denotes the number of e-foldings. Hence there is an N2 enhancement with respect
to the näıve expectation, which is due to the infrared enhancement of scalar vacuum
fluctuations characterising de Sitter space. Since observations constrain λ to be very
small λ ∼ 10−12 and N ∼ 50−60, the quantum corrections in this inflationary model
are unobservably small.
PACS numbers: 98.80.-k, 98.80.Cq
1. INTRODUCTION
Given the fact that we live in the era of precision cosmology, it is important to establish a
framework within which the quantum radiative corrections for observables induced by the vacuum
fluctuations of matter fields can be calculated. Such radiative corrections may be important in
some inflationary models. In this paper we consider the nonminimal λϕ4 inflationary model which
includes a nonvanishing coupling ξ to the curvature scalar. We consider this model for simplicity;
once the framework for calculating the quantum (radiative) corrections is established, it can be
∗A.Bilandzic@students.uu.nl, anteb@nikhef.nl
†T.Prokopec@phys.uu.nl
http://arxiv.org/abs/0704.1905v2
mailto:A.Bilandzic@students.uu.nl, anteb@nikhef.nl
mailto:T.Prokopec@phys.uu.nl
quite easily applied to other inflationary models [1].
The minimally coupled λϕ4 inflationary model (with ξ = 0) is already more than two standard
deviations disfavored by cosmological observations [2]. However, for a certain choice of the coupling
ξ to the background, the resulting nonminimally coupled scalar field model can still match the
experimental data [3, 4, 5, 6], basically because the model then produces the spectral index of the
massive chaotic inflaton model. In this work we show that this is indeed the case, but only for a
rather limited values of ξ, namely for ξ which satisfies
≪ |ξ| ≪ 1
, (ξ < 0) , (1)
where Ñ ≃ N + 1 and N is the number of e-foldings.
If the condition |dH/dt| ≪ H2 is not fulfilled, then our framework is not applicable, because
we have constructed the de Sitter invariant scalar field propagator with the assumption that the
Ricci scalar R = 6(2H2 + dH/dt) ≃ 12H2, where H = H(t) denotes the Hubble parameter. This
condition is fulfilled in most of inflationary models and thus does not present a significant constraint
to our model.
During inflation the amplitude of field correlators at the classical level is suppressed by powers
of the Hubble parameter, but the quantum corrections to field correlators can depend on the whole
history of inflation, leaving hence the possibility that in a cumulative manner quantum corrections
can become important and even detectable by future experiments [7, 8, 9, 10, 11]. Such cumulative
effects are claimed to be present in the analysis done recently in Refs. [12, 13]. In our analysis no
such cumulative effects are present.
The quantum radiative corrections to slow-roll parameters in inflation have been firstly calculated
in Refs. [7, 8, 9]. The authors begin by considering single field inflationary models and subsequently
generalize their analysis to include the inflaton coupling to a light scalar and light fermionic field.
While Refs. [7, 8] consider quantum corrections to the equation of motion in momentum space, in
this work we make use of the effective action techniques. Our results are in a qualitative disagree-
ment with those of Refs. [7, 8, 9]. One important difference is in that in their analysis the authors
of [7, 8, 9] neglect the inflaton coupling to the background curvature (Ricci scalar), which within our
framework yields the dominant contribution to the quantum radiative corrections during slow-roll
inflation. A second important difference is that we made our analysis by using the de Sitter invari-
ant propagator, while the proper analysis should be conducted by making use of a scalar propagator
suitable for quasi-de Sitter spaces. Our method is based on the effective action approach. We arrive
at our one-loop effective action by making use of the position space propagator at coincidence.
Within this method we are able to use the well established machinery of dimensional regularization,
renormalization and renormalization group improvement of our resulting effective field theory. The
authors of Ref. [9] advocate the use of the dynamical renormalisation group method (DRG) [14].
In that novel method the secular terms, which induce a logarithmic growth (with conformal time)
of the mode functions, are resummed to yield the renormalisation group improved mode functions.
These improved mode functions exhibit regulated late time infrared divergences, rendering the mode
functions infrared finite.
More specifically, within our framework we obtain a quantum infrared enhancement to slow-roll
parameters which is, when compared to the classical values, proportional to the number of e-foldings
squared N2. This enhancement is due to the scalar field mass generated by the coupling to the
background curvature scalar. The authors of [7, 8, 9] obtain a quadratic enhancement but for the
λϕ4 model without including the inflaton coupling to the Ricci scalar. On the other hand, when
compared to the classical value, the quantum corrections generated by the λϕ4 interaction term are
enhanced in our framework only linearly by N .
It is by now a well established fact that quantum effects can have quite a dramatic impact
during inflation. An example is the breakdown of conformal invariance for the photons of scalar
electrodynamics, which has as a consequence a photon mass generation during inflation and a
generation of cosmological scale magnetic fields [15]. Similarly the quantum radiative effects break
conformal invariance of the fermions of the Yukawa theory in de Sitter space [16]. As a result
fermions acquire a mass during inflation [16, 17], having as a consequence a production of fermions
during inflation and possibly inflationary baryogenesis. Finally, the canonical coupling of gravitons
to fermions enhances the production of fermions during inflation [18].
The main result of our work is the quantum correction to the scalar spectral index (107–108).
The leading order contribution reads
(ns − 1)Q =
λÑ(ξ − 1/6)
(1− κ)2(1− 2
+O(λ ln(Ñ)) , (κ = 8Ñξ, Ñ ≈ N + 1− ξ/2) , (2)
which is to be compared with the classical contribution,
(ns − 1)C = −
. (3)
The leading contribution (2) originates solely from the resummation of the mass insertions generated
by the coupling to the background curvature. Since λ ∼ 10−12 the quantum contribution (2) is
indeed too small to be observable. Note that the condition (98) implies −Ñ/3 ≪ κ < 1, such that
κ can be large and negative. If this is the case the classical spectral index (3) becomes consistent
with the current CMB data [2]. We futhermore calculate the quantum corrections to the spectrum
of curvature perturbation, to the tensor spectral index, and to the ratio of the tensor-to-scalar
spectrum.
The present work is organized as follows. In Section 2 we first recall the basics of de Sitter space
and then sketch the derivation of the de Sitter invariant Chernikov-Tagirov propagator. In Section
3 we use this propagator and the techniques of dimensional regularization and renormalization
to derive the one-loop improved effective potential for our theory. This procedure requires one
counterterm for the quartic self-coupling constant λ and one for the coupling to the background ξ,
which are defined at an arbitrary scale ϕ0. In Section 4 we use the standard renormalization group
(RG) techniques to improve our effective potential. Having obtained the RG improved effective
potential, in Section 5 we calculate the corresponding quantum scalar field stress-energy tensor
in the slow-roll approximation. By making use of the quantum Friedmann equation and of the
scalar field equation of motion, we then develop our framework within which the quantum radiative
corrections from the vacuum matter fluctuations to slow-roll parameters can be calculated. In
particular, we organize the quantum radiative corrections to slow-roll parameters ǫ and η into two
distinct parts. The first part arises from the one-loop resummation of the mass insertions generated
by the quartic self-coupling in the presence of a scalar (inflaton) condensate, while the second
part arises from the resummation of the scalar mass insertions generated by the coupling to the
background. Both of these quantum corrections are suppressed by the coupling constant λ but
they are enhanced by the number of e-foldings squared. Based on these results we then calculate
the quantum radiative corrections to the observables: the spectrum of curvature perturbation and
its spectral index, the tensor spectral index and the ratio of tensor-to-scalar spectra. Finally, in
Section 6 we summarize our results and discuss their physical implications.
2. PROPAGATOR IN DE SITTER SPACE
2.1. de Sitter space
A four dimensional de Sitter space is perhaps best viewed as a 4-dimensional hyperboloid em-
bedded into the 5-dimensional Minkowski space-time with the line element,
ds25 = −dX20 + dX21 + dX22 + dX23 + dX24 . (4)
FIG. 1: The embedding of de Sitter space into a five dimensional flat space-time. The vertical line
corresponds to the time coordinate, X0 = T , and the radial coordinate R =
X21 +X
4 . At
each point (T,R) there is a unit 3-sphere S3, which is for the sake of clarity represented by a circle S1
erected at each point (T,R). The Hubble radius RH = 1/H is the coordinate distance R of the hyperboloid
from the origin at T = 0.
The embedded hyperboloid of de Sitter space is shown in Figure 1, and it is determined by
−X20 +X21 +X22 +X23 +X24 =
, (5)
where H denotes the Hubble parameter. The symmetry group of de Sitter space, SO(1, 4), is
manifest by this embedding. One defines the de Sitter invariant distance functions as,
Z(X ;X ′) = H2
A,B=0
ηABXAX
B = 1−
Y (X ;X ′) , ηAB = diag(−1, 1, 1, 1, 1) . (6)
We shall use the following flat 4-dimensional coordinates (which cover 1/2 of the de Sitter manifold),
sinh(Ht) +
eHt‖~x‖2 , (−∞ < t <∞) ,
Xi = e
Htxi , (−∞ < xi <∞, i = 1, 2, 3) ,
cosh(Ht)− H
eHt‖~x‖2 , (7)
in which the metric tensor reduces to the form
ds2 = −dt2 + a2d~x 2 , (8)
with the scale factor a = eHt. When written in terms of conformal time η, defined as adη = dt, the
metric tensor acquires the conformal form,
gµν = a
2(η)ηµν , a = −
(η < 0) , ηµν = diag(−1, 1, 1, 1) . (9)
The invariant distance functions Z(X ;X ′) ≡ z(x; x′) and Y (X ;X ′) ≡ y(x; x′) reduce in these
coordinates to the simple form,
z(x; x′) = 1− 1
y(x; x′) , y(x; x′) = aa′H2∆x2 , (10)
with a = a(η) = −1/(Hη), a′ = a(η′) = −1/(Hη′), and
∆x2(x; x′) = −(|η − η′| − iε)2 + ‖~x− ~x ′‖2 , (11)
where (for a later use) we introduced the infinitesimal parameter ε > 0, which defines how the poles
of the propagator (discussed in the next section) contribute. In these coordinates the curvature of
spatial sections vanishes, and thus they are also known as flat (Euclidean) coordinates, in which de
Sitter space appears as uniformly expanding.
By solving the relevant geodesic equations for x0 and xi (i = 1, 2, 3) one can show that the de
Sitter invariant distance function y = y(x; x′ ) is related to the geodesic distance ℓ = ℓ(x; x′ ) by the
following simple relation,
y(x; x′ ) = 4 sin2
Hℓ(x; x′ )
. (12)
2.2. Scalar propagator in de Sitter space
The dynamics of the scalar field are specified by the following tree-level action,
−gLϕ , (13)
with the Lagrangean,
−gLϕ =
gµν(∂µϕ)(∂νϕ)− Vb(ϕ)
, (14)
where
Vb(ϕ) =
ξbRϕ2 +
ϕ4 . (15)
In the above expression λb and ξb are the bare values of the quartic self-coupling and the nonminimal
coupling to the Ricci curvature scalar R, respectively, and g = det(gµν). For simplicity we set the
bare scalar mass mb = 0. The theory (13–15) is a simplified version of the Yukawa theory in de
Sitter background studied in Refs. [20? ]. An early related work can be found in Ref. [19].
The scalar propagator in a curved background space can be defined as the expectation value,
i∆(x; x′) = 〈x|
−g(�−m2ϕ − ξbRD)
|x′〉 ,
m2ϕ(ϕ) ≡
, (16)
where |x〉 is the eigenstate of the position operator x̂ (i.e. x̂|x〉 = x|x〉), � = (−g)−1/2∂µ(−g)1/2gµν∂ν
denotes the d’Alambertian and g = det[gµν ]. This Feynman propagator solves the following equation
in de Sitter space in general D space-time dimensions (needed for dimensional regularization and
renormalization),
−g(�−m2ϕ − ξbRD)i∆(x; x′) = iδD(x− x′) , (17)
where δD(x− x′) is the D-dimensional Dirac δ-distribution, RD = D(D − 1)H2 is the Ricci scalar
in a D dimensional de Sitter space, and H is the Hubble parameter.
The de Sitter invariant form of (17) is [22, 23, 24, 25]
(1− z2) d
−Dz d
m2ϕ + ξbRD
iG(y) =
iδD(x− x′)
, (18)
where the invariant propagator is defined as, iG(y) = i∆(x; x′). Here we made use of Eqs. (10–11)
and of
∂µ ≡ (∂µz)
δ 0µ y + 2a
′H∆xµ
. (19)
The properly normalized de Sitter invariant solution of Eq. (18), which near the light-cone and
in the massless limit reduces to the Hadamard form,
iG(y)
y→0−→ H
(2π)D/2
y2−D/2, y0
, (20)
is unique,
iG(y) =
) 2F1
(D − 1
+ νD,
D − 1
− νD;
, (21)
where
((D − 1)2
m2ϕ + ξbRD
. (22)
This is the Chernikov-Tagirov propagator for de Sitter space [22, 23] generalized to D space-time
dimensions. The pole prescription defined by the iε-prescription in Eq. (11) implies that the prop-
agator (21) corresponds to the time-ordered (Feynman) propagator. For a discussion of other
propagators relevant for expanding space-times in the Schwinger-Keldysh in-in formalism we refer
to [10, 25].
3. EFFECTIVE POTENTIAL
The one-loop effective action for a real scalar field reads,
Γϕ[gµν , ϕ] ≡
−gLϕ = SHE[gµν ] + Sϕ[gµν , ϕ] +
Tr ln
−g(�−m2ϕ − ξbRD)
, (23)
FIG. 2: The one-loop diagram (vacuum bubble) contributing to the scalar effective theory (23) in a curved
background.
where Tr refers to the space-time integration
dDx, Sϕ is the tree-level scalar field action (13) and
SHE denotes the Hilbert-Einstein action,
SHE = −
16πGN
−gRD . (24)
The last term in Eq. (23) represents the one-loop contribution to the effective action δ1Lϕ, whose
graphical representation is shown in Figure 2. We shall now evaluate the general expression (23) in
de Sitter space. It is convenient to differentiate the one-loop contribution δ1Lϕ with respect to the
scalar mass,
∂δ1Lϕ
−g(�−m2ϕ − ξbRD)
i∆(x; x) . (25)
Now making use of Eq. (21) one obtains,
∂δ1Lϕ
iG(y)|y→0 = −
(4π)D/2
) . (26)
Separating the divergent and finite contributions in (26) yields
∂δ1Lϕ
(4π)D/2
)(m2ϕ
− (D−2) + ξbD(D−1)
− 2(1− 6ξb)
, (27)
where we made use of,
D − 4
− (1− γE) +O(D − 4) , (28)
γE ≃ 0.577 is the Euler constant, RD=4 ≡ R = 12H2, and
+ 2(1−6ξb)
, (29)
where m2ϕ = λϕ
2/2 is defined in Eq. (16). When integrated, Eq. (27) gives the following contribution
to the effective Lagrangean,
δ1Lϕ = −
2(4π)D/2
HD−4m4ϕ −
(D−2)− ξbD(D−1)
HD−2m2ϕ
, w =
where the integral is an indefinite integral.
In order to renormalize our Lagrangean Lϕ we will add to it the counterterms λ0 and ξ0 and
apply the renormalization conditions which will determine the finite parts of those counterterms,
Lϕ, ren = −
gµν(∂µϕ)(∂νϕ)− Vren(ϕ) , (31)
where now
Vren(ϕ) =
ξbRϕ2 +
ξ0Rϕ2 − δ1Lϕ, (32)
and δ1Lϕ is given by Eq. (30).
We renormalize our Lagrangean at an arbitrary scale ϕ0,
= λb = λb + λ(ϕ0, H
2)− δ
4(δ1L)
δ(H2)δϕ2
= D(D−1)ξb = D(D −1)ξb +D(D −1)ξ(ϕ0, H2)−
δ3(δ1L)
δ(H2)δϕ2
, (33)
which yields
λ(ϕ0, H
2) = − 3
2(4π)D/2
Γ(1− D
)µD−4λ2b
0 + 24H
2(ξb − 1/6)
0 + 24H
2(ξb − 1/6)
ξ(ϕ0, H
2(4π)D/2
Γ(1− D
D − 2
D(D − 1)
0 + 24H
2(ξb − 1/6)
. (34)
From (33) it follows that the counterterms λ0 and ξ0 are given by
λ0 ≡ λ(ϕ0, 0) = −
2(4π)D/2
µD−4λ2b −
ξ0 ≡ ξ(ϕ0, 0) =
2(4π)D/2
)[ D−2
D(D − 1)
. (35)
Given thatH and ϕ are dynamical fields, the counterterm parameters λ0 and ξ0 must be independent
of H and ϕ, which is indeed satisfied by (35). Now making use of Eqs. (30), (32) and (35) we can
calculate the renormalized Lagrangean (31). The result is,
Lϕ, ren = −
gµν∂µϕ∂νϕ−
( 2H2
( 2H2
. (36)
This is the fully renormalized effective Lagrangean. We now consider the two asymptotic forms of
(36), first the ultraviolet (UV) limit.
The following asymptotic expansion of the di-gamma function, ψ(z) = (d/dz)[ln(Γ(z))], is then
useful (cf. Eq. (8.344) in [26]),
ψ(z) = ln(z)− 1
120z4
+O(z−6) , (37)
such that
= ln(w)−
+O(1/w3) , ν2 =
− w . (38)
Upon evaluating the integral in (27) one obtains
ln(w)−
ln(w) +O(1/w)
. (39)
Taking account of this, we can recast Eq. (36) to the form,
Lϕ = −
gµν∂µϕ∂νϕ−
2 + 24H2
ξb − 16
2 + 24H2
ξb − 16
+O(R3). (40)
This is the UV form of (36). In the limit H2 → 0 to flat Minkowski spacetime, the first line in (40)
reproduces the classical Coleman-Weinberg result [27].
To complete the analysis of the effective Lagrangean (36) we still need to consider the small field
limit of the integral in (36). Let us first consider the expansion which is applicable around the poles
of the di-gamma function, which are located at
+ n , n = 0, 1, 2, 3, . . . . (41)
This implies that the poles are located at
wn = −n(n + 1) (42)
and in the vicinity of the poles we can write
w = wn + δw . (43)
With the above definitions, a conformally coupled scalar field with ξ = 1/6 corresponds to n = 0
and a minimally coupled scalar field with ξ = 0 corresponds to n = 1. More generally we have,
(1− n)(2 + n)
, (n ≥ 0) , (44)
such that for n > 1 all ξn < 0. In particular, ξ2 = −1/3, ξ3 = −5/6, etc. Now we can expand ν as
ν = νn −
(δw)2
and by making use of
+ ℓ− ν
we can finally write
+ bn + cnδw +O((δw)2) . (47)
In the above expressions
an = n(n+ 1)(2n+ 1) ,
bn = −2n(n + 1)ψ(n+ 1)− (2n+ 1)−
n(n+ 1)
2n+ 1
2n+ 1
+ 2ψ(n+ 1) +
2n(n+ 1)
2n+ 1
−ψ′(1) + ψ′(n+ 1)−
2(2n+ 1)2
, (48)
and we made use of
= ψ(n + 1)− ψ(1) ,
= −ψ′(n + 1) + ψ′(1) , (49)
where ψ(1) = −γE = 0.577215... is the Euler constant, ψ′(1) = π2/6 and ψ′(z + 1) = ψ ′(z)− 1/z2.
We can now write the infrared limit of the renormalized Lagrangean (36). From Eqs. (47–48) it
follows that the integral in the last line of Eq. (36) has the infrared limit,
an ln(δwn) + bn(δwn) +
cn(δwn)
(δwn)
, (50)
where
δwn =
+ 12δξn , δξn = ξb +
(n− 1)(n+ 2) ≪ 1 (51)
and an, bn and cn are defined in Eq. (48).
4. RENORMALIZATION GROUP ANALYSIS
In deriving expression (36) we have introduced an arbitrary renormalization scale ϕ0 by definin-
ing the renormalization conditions (33). From this definition an arbitrary scale ϕ0 enters into the
expressions for the counterterms (35) and hence into the renormalized Lagrangean (36). How-
ever, as it was stressed in the classic paper of Coleman and Weinberg in 1973, the change of the
renormalization scale can only change the definitions of coupling constants, not the physics [27].
By applying the same reasoning in our case, we arrive at the following conclusion: a small change
in ϕ0 in the expression for the physical quantity of interest can always be compensated for by an
appropriate small change in λ and ξ. The convenient way of expressing this statement is
Veff(ϕ0, λ, ξ, ϕ) = 0, (52)
which is the standard Callan-Symanzik (CS) equation written for the theory at hand. The renor-
malization group functions βλ and βξ are given by the following relations,
βλ = −ϕ0
βξ = −ϕ0
. (53)
Within the one-loop approximation the renormalization group functions βλ and βξ are uniquely
determined as the coefficients of the divergent logarithmic terms appearing in the counterterms λ0
and ξ0 (35). It follows (writing λb and ξb from now on as λ and ξ, respectively):
. (54)
These expressions for βλ and βξ we will use to determine the running of λ and ξ with the scale ϕ0.
We shall now solve the Callan-Symanzik equation (52). From the theory of partial differential
equations we can make use of the method of characteristics [29, 30, 31, 32]. Applying this method
to (52) we can write down the solution to the Callan-Symanzik equation (52) as
Veff(ϕ0, λ, ξ, ϕ) = Veff(ϕ0(t), λ(t), ξ(t), ϕ(t)), (55)
where ϕ0(t), λ(t), ξ(t), ϕ(t) are the running parameters. The t-dependence of the running parameters
is given (to the order we are working in) by the following differential equations:
dϕ0(t)
= ϕ0(t),
dϕ(t)
dλ(t)
= βλ(λ(t)),
dξ(t)
= βξ(ξ(t), λ(t)). (56)
The boundary conditions (at t = 0) are,
ϕ0(0) = ϕ0, ϕ(0) = ϕ,
λ(0) = λ, ξ(0) = ξ. (57)
The solutions of the first two differential equations in (56) are trivial and read,
ϕ20(t) = ϕ
2t, ϕ(t) = ϕ . (58)
When combined with the previous results (54), the last two differential equations in (56) are solved
λ(t) =
1− 3λ
, (59)
ξ(t) =
+ (ξ −
, (60)
where here λ = λ(0) and ξ = ξ(0). These solutions imply that our model exhibits an infrared fixed
point, where the coupling constants are λFP = 0 and ξFP = 1/6. In this limit the theory possesses
an enhanced symmetry (conformal symmetry) and it can be reduced to a pure metric theory (see
Ref. [33] for a nonperturbative proof of this statement).
We stress that the parameter t in the above relations is completely arbitrary. The basic idea of
the renormalization group (RG) improvement of an effective potential is that we can choose t in
such a way that the perturbation series for the effective potential converges more rapidly. Indeed
by suitably choosing t one can extend the range of validity of the effective theory to a larger range
of the dynamical quantities H and ϕ by replacing the perturbative expression on the left-hand side
of (55) by its right-hand side. This is intimately related to the fact that the unimproved expression
for the effective potential is actually valid only for ϕ’s sufficiently close to ϕ0. Since the change of
the arbitrary scale ϕ0 corresponds just to a reparametrization of the coupling constants within our
theory, the unimproved effective potential is valid only near ϕ ∼ ϕ0 and H ∼ 0 and thus not a very
useful quantity. If we, on the other hand, require that the effective theory does not depend on ϕ0,
then the improved effective potential (which solves the Callan-Symanzik equation) remains valid
whenever the coupling constants are small.
Since the perturbation series for the effective potential is characterized by the occurrence of
powers of logarithmic terms, we choose1
t = ln
. (61)
The improved expression for the renormalized effective potential now becomes
VRG(ϕ) =
+6H2ϕ2
where λ and ξ are now t-dependent according to (59) and (60). We remark that even after the
improvement logarithmic terms still appear in the expression for the effective potential. As it can
be seen from Eq. (40), these logarithmic terms vanish in the limit when H → 0. Had we introduced
the renormalization scale H0 for the Hubble parameter and then solved the CS equation for this
case, we could in principle get rid off all logarithmic terms. However, as we shall see later, for the
calculation of the quantum radiative corrections to slow-roll parameters the expression (62) suffices
because our final results do not depend on H . With this in mind we use Eq. (62) in the next section.
For an alternative approach to the renormalisation group improved scalar effective theories in de
Sitter space see Refs. [34, 35].
1 The choice (61) is not unique. However, this is the unique choice for which ϕ0∂ϕ0 = −∂t when H2 → 0 in the
CS equation (52). For any other choice of t there is an additional prefactor in front of ∂t, and after dividing the
CS equation with that prefactor one can solve it as if the β functions are modified, which in turn will modify the
functions λ(t) and ξ(t). This in principle leads to a different effective RG improved theory, which however differs
from the one we use here only at higher orders in the coupling constants.
5. SLOW-ROLL PARAMETERS
In this section we calculate the quantum2 one-loop corrections to the slow-roll parameters ǫ and
η arising from the scalar matter vacuum fluctuations in inflation. Within the slow-roll approxi-
mation we can drop the kinetic term in the action because it is formally second order in slow-roll
parameters [28]. That implies that – within the slow-roll approximation – the leading contribution
to the stress-energy tensor is given by3
Tµν = −gµν
VRG(ϕ)−
δVRG(ϕ)
δ lnH
, (64)
where VRG(ϕ) is the improved renormalized effective potential (62). A straightforward calculation
yields,
Tµν = −gµν
A+ 3H2ϕ2B
, (65)
where we have introduced
A ≡ λ+ 3λ
X − 25
B ≡ ξ +
λ(ξ − 1
X − 4 + 1
36(ξ − 1
, (66)
X ≡ ln
. (67)
The above result for the stress-energy tensor we insert into the Einstein equation
Rµν −
gµνR = 8πGNTµν (68)
to obtain the following quantum Friedmann equation
3M2Pl
A + 3H2ϕ2B
, (69)
2 In this work when we refer to the ‘classical’ value of a parameter we mean its tree-level value. When we refer to
the ‘quantum correction’ we mean the one-loop contribution to the corresponding parameter.
3 In Eq. (64) we have neglected the tree-level contributions to the stress-energy tensor, which are proportional
to ξ(d/dt)2(ϕ2) and ξH(d/dt)(ϕ2). These terms can be neglected based on the observation that the condition
Ḣ ≪ H2 together with the slow-roll approximation imply
3ξHϕϕ̇ ≪ 1
V (ϕ) . (63)
For a derivation of this condition we refer to Ref. [36].
where M2Pl = 1/(8πGN). If we take from A and B, defined in (66), the leading (classical) contribu-
tions, then we can from (69) extract the classical Friedmann equation in the form,
H2C =
72M2Pl
1− ξ ϕ
, (70)
which in the limit ξ → 0 reduces to the well-known result. We shall use equation (70) to calculate
the number of e-foldings N in the next section.
In order to determine the slow-roll parameters we still need an expression for ϕ̇. From Eq. (62)
and the slow-roll form4 of the scalar field equation,
3Hϕ̇+
= 0 , (71)
we obtain
ϕ̇ = −W
, (72)
where
W ≡ ϕ
C + 12H2ϕD + 72
E , (73)
C ≡ λ+
(4π)4
D ≡ ξ +
λ(ξ − 1
X − 3 +
36(ξ − 1
λ2(ξ − 1
(4π)4
48(ξ − 1
λ(ξ − 1
(4π)4
. (74)
Upon inserting the leading contributions from Eq. (74) into Eq. (72), the classical expression for ϕ̇
follows immediately,
ϕ̇C = −
+ 12ξH2Cϕ
. (75)
It is important to note that with the above definitions
C = A+
βλ +O(λ3) ,
D = B +
βξ +O(λ2) . (76)
We keep the parameter E in (73) for completeness, although it yields only higher order contributions
comparing to other parameters defined by (66) and (74).
4 Within the slow-roll approximation we can drop the ϕ̈ term in Eq. (71) because that term is second order in
slow-roll parameters.
We now proceed by making use of the standard result for the spectrum of primordial curvature
perturbation [28]
PR(k) =
. (77)
In deriving this expression the canonical quantization of the inflaton field has been performed in
the standard way, by studying the evolution of small perturbations around the inflaton condensate.
Since in our approach the quantum corrections are calculated at the level of the effective potential,
which changes the on-shell structure of the theory but does not change the structure of Eq. (77), we
conjecture that Eq. (77) can be used without any further generalization for the calculation of the
one-loop quantum corrections to the spectrum of curvature perturbation and the implied slow-roll
parameters that arise from the matter vacuum fluctuations within the framework proposed in this
work. A proof of this conjecture is nevertheless desirable. The right-hand side of (77) is evaluated
at the horizon exit, at which k = aH , because during slow-roll inflation the Hubble parameter does
not change significantly over a few Hubble times [28].
The scalar spectral index ns is defined as
ns − 1 =
d lnPR
d ln k
, (78)
which after some algebra, by making use of (77) and (72), yields
ns − 1 = −
. (79)
By analogy with the standard result
ns − 1 = −6ǫ+ 2η , (80)
which is valid at the classical level for various inflationary models, we define
ǫ ≡ W
η ≡ 1
, (81)
such that equation (80) still holds for the quantum case. On the other hand, for the gravitational
wave spectrum we use the result
. (82)
The gravitational wave spectral index ng is defined as
d lnPg
d ln k
, (83)
from which it follows that within our framework,
ng = −
. (84)
After taking into account the definition of ǫ from (81) we obtain that the standard result,
ng = −2ǫ , (85)
remains valid for the quantum case as well. It is also convenient to introduce the ratio r between
the gravitational wave spectrum and the spectrum of primordial curvature perturbation,
r ≡ Pg
, (86)
which here turns into
9M2Pl
. (87)
With the above definitions the standard relation, r = 16ǫ is not any more satisfied at the quantum
level. However, we still expect to reproduce it at the classical limit (but see the discussion below).
The final result for ǫ and η, which includes both the classical and quantum contributions, we
write in the form
ǫ = ǫC + ǫQ ,
η = ηC + ηQ, (88)
and we separate the quantum contributions into the following two characteristic parts,
Qǫλ + βξ Qǫξ ,
Qηλ + βξ Qηξ . (89)
Although the two contributions are formally of the same order of magnitude, they have a different
origin. The former contribution in Eq. (89) arises as a result of the resummation of the mass
insertions m2ϕ = λϕ
2/2 generated by the quartic self-interaction in the presence of an inflaton
condensate. The latter contribution is a consequence of the resummation induced by the effective
mass parameter 12ξH2 generated by the inflaton field coupled to the background curvature. Now
we shall present our results, first the classical part.
5.1. Classical contributions ǫC and ηC
After a straightforward calculation, we arrive at
, (90)
where z and κ are defined by
z ≡ ϕ
κ ≡ ξz = ξ
. (91)
It is clear that in the limit when ξ → 0, i.e. when κ → 0, we recover the standard expressions for
the slow-roll parameters in the λϕ4 inflationary model; namely ǫ = 8M2Pl/ϕ
2 and η = 12M2Pl/ϕ
This is not surprising since in this limit our theory reduces precisely to that inflationary model.
We now introduce the number of e-foldings
N = −
Hdt , (92)
which somewhat surprisingly, when calculated classically (H → HC), gives the same result as the
λϕ4 inflationary model
ϕ2 − ϕ2end
. (93)
However, a mild ξ-dependence does enter the expression for N through the value of the inflaton
field at the end of inflation, ϕend, which is determined from the condition ǫC(ϕend) = 1. From (90)
it follows5
ϕ2end ≃ 4M2Pl(2− ξ), (94)
and finally
= 8Ñ , Ñ ≃ N + 1−
ξ . (95)
We shall use the above notation when writing the quantum contributions to slow-roll parameters.
5 The result (94) is valid to the leading order in ξ.
5.2. Quantum contributions ǫQ and ηQ
In calculating the quantum contributions to slow-roll parameters we must take into account the
observational constraint required by the near scale invariance of the spectrum
λϕ2 + 12ξH2
≪ H2 . (96)
In view of equations (41)-(48) the observational constraint (96) implies that, in order to study
the quantum radiative corrections to slow-roll parameters, we need the infrared limit of the RG
improved theory (62). This is the opposite limit from the ultraviolet limit in which our effective
theory reduces to the Coleman-Weinberg form (40) [27]. That means that in order to study the
quantum radiative corrections to slow-roll parameters, one needs to focus on the infrared radiative
corrections which are specific for (quasi-)de Sitter spaces, and completely absent in Minkowski space
(which is related by a conformal rescaling to our n = 0 case), and hence has a very different infrared
structure. In particular, the most singular term an/δw in (48) is absent in the conformal n = 0 case
(a0 = 0). In summary, that means that the infrared sector plays a crucial role in determining the
quantum corrections to slow-roll parameters.
After taking into account the constraint (96) and the expression (70) for the classical Friedmann
equation, we arrive at the condition
− 24ξ
≪ 1 , (97)
Together with the condition 8Ñξ < 1, this equation then gives,
≪ ξ <
. (98)
Recall that typically N (and hence also Ñ) is between 50 and 60 such that the term 9/(2Ñ) ∼
10−1 ≪ 1 in Eq. (98) can be to a good approximation neglected.
We proceed by writing approximately the relation (67) as (δw ≪ 1)
X = ln
− 2γE +
+O(δw) , (99)
from which it follows
= − 1
ζ , (100)
where we have introduced
ζ ≡ −
. (101)
Now the calculation of the slow-roll parameters ǫ and η is straightforward; here we present only our
final results. Some intermediate steps and results can be found in Appendix A.
For the quantum contribution (89) to the slow-roll parameter ǫ we obtain
Qǫλ =
1− 29
κ+ 17
(1− κ)2(1− 2
6− 5κ+ κ2
Qǫξ = −
(1− κ)2(1− 2
(1− κ)2
2− σ − 2κ+ 1
(1− κ)2
, (102)
and for the η slow-roll parameter defined in (89)
Qηλ =
1− 29
κ+ 17
(1− κ)2(1− 2
11− 10κ+ 3κ2
= 2Qǫλ −
1 + κ
Qηξ = −
(1− κ)2(1− 2
(1− κ)2
18− 8σ − 20κ+ 6κ2
(1− κ)2
= 2Qǫξ + 2 . (103)
In Eqs. (102) and (103) we abbreviated
z ≡ ϕ
= 8Ñ ,
κ ≡ zξ ,
+ γE + ln 6 . (104)
Since typically the number of e-foldings required during inflationary epoch ranges between 50 and
60, it is evident from (104) that z is of the order 5×102, which justifies the ordering of the quantum
corrections in powers of z. The leading contributions are the terms linear in z, and they are present
only in Qǫξ and Qηξ in (102) and (103). Both Qǫξ and Qηξ contain also the next-to-leading terms
of the order ln(z). The (subleading) terms of the order z0 are in fact the leading contributions to
Qǫλ and Qηλ in (102) and (103).
5.3. Tensor and scalar spectral indices ng and ns
We can now easily calculate the tensor and scalar spectral indices from the results for the slow-roll
parameters ǫ and η (90), (102–103).
Note first that the tensor spectral index ng (85) can be trivially obtained by summing the
classical (90) and quantum (102) contributions for ǫ, since from our definition (81) it follows that
Eq. (85) is valid also at the quantum level.
Next we consider the scalar spectral index ns. By making use of Eqs. (90) and (102–103) and
separating again the classical and quantum contributions as,
ns − 1 = (ns − 1)C + (ns − 1)Q , (105)
where
(ns − 1)Q =
Q(ns−1)λ + βξ Q(ns−1)ξ , (106)
we arrive at the classical scalar spectral index,
(ns − 1)C = −
, z = 8Ñ , κ = 8Ñξ . (107)
The quantum contributions are given by,
Q(ns−1)λ = −
1− 29
κ+ 17
(1− κ)2(1− 2
14− 10κ
, (108)
Q(ns−1)ξ =
(1− κ)2(1− 2
(1− κ)2
− 12− 8σ − 8κ
(1− κ)2
Notice that the quantum contributions (βλ/λ)Q(ns−1)λ and βξQ(ns−1)ξ are both much smaller than
the classical contribution (ns−1)C due to the fact that both βλ/λ and βξ are of the order of λ, which
is constrained by experimental data to be of the order of λ ∼ 10−12 (we provide a more precise
constraint for λ below). The quantum contributions can become significant only in an inflationary
model in which the relevant coupling constant can be as large as the order of 10−3. It is important to
notice that Q(ns−1)λ provides an infrared enhanced quantum correction proportional to the number
of e-foldings N , while, on the other hand, Q(ns−1)ξ contains a correction which is enhanced by N
when compared to the näıve expectation O(1/z) ∼ 1/N).
This shows that in the de Sitter invariant limit the quantum corrections to slow-roll parameters
accumulate only from the time the mode becomes super-Hubble until the end of inflation. For a
mode which exits horizon N e-foldings before the end of inflation, the whole history of inflation
before the Hubble exit (i.e. when the mode was sub-Hubble) is completely irrelevant and does
not contribute in a cumulative manner to the quantum corrections. This disagrees with the result
found in Refs. [12, 13], where it is claimed that the quantum loop corrections induce corrections to
the slow-roll parameters and spectral indices which depend on the total duration of inflation and
are thus enhanced when inflation lasts for a large number of e-foldings. This N2 enhancement is
the main result of our work, and it resolves the Weinberg’s dilemma [10, 11]: how big can be the
correction induced by the quantum fluctuations of light or massless scalar fields during inflation,
given the fact that the (equal time and space) scalar field correlator for a massless minimally coupled
scalar grows linearly with time during (de Sitter) inflation,
〈0|ϕ2(x) |0〉 = infinite +
ln a , (109)
〈0|ϕ2(x) |0〉
N , (110)
where here N denotes (minus) the number of e-foldings.
By comparing Eqs. (85) (102) and (108) we observe that the following curious relation holds,
(ng)Q = (ns − 1)Q +
2(1 + κ)
− 4βξ , (111)
such that the leading quantum contributions O(λÑ) and O(λ ln(Ñ)) to the tensor and scalar
spectral indices are equal. This approximate equality can be traced back to the fact that
ǫQ ≃ ηQ/2, from where it follows that [(W/H4)(dH2/dϕ)]Q ≃ [(1/H2)(dW/dϕ)]Q, or equiva-
lently [d(W/H2)/dϕ]Q ≃ 0. The expression W/H2 is proportional to the square-root of the ratio
r = Pg/PR, which according to Eq. (121) does not receive any quantum corrections that are am-
plified by N . That means that, because the tensor and scalar spectra are identically affected by
the quantum corrections enhanced by N , these leading corrections cancel in the ratio r. We do not
have a deeper insight to why that is the case. Note that the approximate equality (111) does not
hold for the corresponding classical parts (90) and (107). If inflationary models with large quantum
corrections are found, Eq. (111) could be used to resolve the quantum from classical contributions
to the tensor and scalar spectral indices.
5.4. The spectrum of curvature perturbation PR and the tensor-to-scalar ratio r
Next we consider the spectrum of curvature perturbation (77). As usual we decompose the
spectrum into the classical and quantum parts as,
PR = (PR)C + (PR)Q , (112)
where
(PR)Q ≡
QPRλ + βξQPRξ . (113)
Working within our framework we obtain,
(PR)C =
, (114)
(PR)C
1− 25
(1− κ)(1− 2
(1− κ)− σ −
(PR)C
(1− κ)(1− 2
, (115)
where Ñ = N +1− 1
ξ, z = 8Ñ and κ = 8ξÑ . This implies that, just like the spectral indices, when
compared to the classical contribution, the quantum contribution to the spectrum is suppressed as
λÑ2, and thus unobservably small for the model in consideration.
The spectrum of curvature perturbation is an observable quantity and the three year WMAP
data provide a strong constrain for it [2],
PR ≈ 29.5× 10−10A , A = 0.801+0.043−0.054 . (116)
In the case of a weak coupling to the background, i.e. when 8|ξ|Ñ ≪ 1 and Ñ ≈ N +1, Eqs. (114)
and (116) imply
λ50 ≈ 1.58× 10−12 , λ60 ≈ 9.25× 10−13 , (8|ξ|Ñ ≪ 1) , (117)
where the subscripts on λ denote the number of e-foldings N . We stress that these values for λ are
strictly speaking valid only for this particular model. Indeed, when we choose the coupling to the
background in the range,
≫ −ξ >
, (118)
then from Eq. (114) we see that the value of λ can be up to one order of magnitude larger,
λ50 ≃ 2.69× 10−11 × (−24ξ) , λ60 ≃ 1.88× 10−11 × (−24ξ) , (−24ξ ≪ 1) . (119)
These larger values of λ are still too small however to render the quantum effects observable.
In other inflationary models the relation for the spectrum of curvature perturbation (114) can
in general be different, hence allowing for models in which couplings are larger. Furthermore, the
quantum effects in some other models may be much stronger – of particular interest are hybrid
inflationary models [1].
Let us now consider the ratio r of the gravitational wave and curvature spectrum. From Eq. (87)
we obtain
r = rC + rQ
(120)
(1− κ) + 128βξ , (121)
where again z ≡ 8Ñ and κ ≡ zξ. We see that rC = 128/z, and after observing from (90) that
ǫC = 8/z in the limit when ξ → 0 (i.e. when κ → 0), we obtain the standard result r = 16ǫ.
This relation is violated both by the classical corrections from κ 6= 0 (ξ 6= 0) and by the quantum
contributions. In particular, when κ = 8Ñξ ≪ −2, one gets r ≃ 32ǫ. When compared with the
classical contribution (120), the quantum contribution to r (121) is as usual suppressed by λ, but
– in contrast to the slow-roll parameters ǫQ and ηQ – it is not enhanced by powers of N .
It is well known that the minimally coupled λϕ4 inflationary model is disfavored by observa-
tions [2] by about 2 standard deviations. This is not however in general the case with the non-
minimally coupled λϕ4 inflationary model. Indeed, from Eqs. (98) and (107) we infer that in the
range
≫ −ξ > 1
. (122)
the deviation of the classical spectral index of scalar curvature perturbation (107) from scale invari-
ance is reduced approximately by a factor of 2/3,
(ns − 1)C ≈ −
, (−ξ ≫ (8Ñ)−1) , (123)
while the value of rC (120) remains unchanged. This then implies that, similarly as in the minimally
coupled massive inflationary model, the λϕ4 model with ξ in the range (122) falls roughly at the
1σ contour of Figure 14 in Ref. [2], rendering these nonminimally coupled λϕ4 inflationary models
consistent with the three year WMAP data [2, 4, 5, 6]. We emphasize that, because ξ = ξ(ϕ0) runs
logaritmically towards its infrared fixed point ξFP = 1/6, it is natural to assume that ξ deviates from
zero. Indeed, even if we choose ξ = 0 (which corresponds to the coupling at some scale ϕ = ϕ0),
the running of ξ will induce the dominant quantum contributions to slow-roll parameters. Thus
for consistency it is necessary to consider the effects of nonminimal coupling, and choosing ξ in the
range (122) is a priory as natural as any other choice (different, of course, from ξ = 1/6).
In conclusion, we have found out that, even though the quantum effects to slow-roll parameters
are enhanced by the number of e-foldings squared, they are suppressed by the small coupling
constant λ. Due to the smallness of λ however, the quantum effects have a negligible impact on the
plot presented for example in Figure 14 of [2].
6. DISCUSSION
In this work we develop a quantum field theoretic framework within which the quantum correc-
tions to slow-roll parameters and observables from inflationary models can be calculated. Whe main
purpose of this paper is methodical, and we postpone a detailed study of the quantum corrections
to inflationary observables in various inflationary models to a future work [1].
We illustrate how our framework works by performing the relevant calculations in a concrete
inflationary model chosen for its simplicity. More specifically, we consider a λϕ4 inflationary
model (13–15) with a nonminimal coupling to the background curvature. Our formalism can be
quite straightforwardly generalized to other inflationary models. Within our λϕ4 model we cal-
culate the quantum corrections to the inflationary slow-roll parameters ǫ and η (89), (102–103),
based on which we derive the scalar spetral index ns (105–108), the tensor spectral index ng (85),
(103) and the spectrum of curvature perturbation PR (112–115). These corrections arise from the
one-loop scalar vacuum fluctuations during de Sitter inflation. We find that the dominant quantum
effects for the spectral indices (2), (85), (103) are suppressed as λN2 when compared to the classical
(tree-level) results (3), (90). The dominant quantum contribution arises from the inflaton coupling
to the background curvature.
Our theoretic framework can be improved in several aspects. For example, one could generalize
our calculation of the renormalization group improved effective action (62) to quasi-de Sitter spaces,
which are important since these spaces comprise a large fraction of inflationary models. Next one
should generalize our analysis to incorporate other matter degrees of freedom which would allow us
to incorporate a broad spectrum of inflationary models. Even more importantly one should study
the role of the interactions that couple matter and gravitational degrees of freedom.
Appendix A
Here we provide a more detailed derivation for the quantum contributions to slow-roll parameters
given by the relations (102) and (103). We begin by abbreviating the equalities in Eq. (66) as
A = λ + βλXA ,
B = ξ + βξXB , (124)
where now
X − 4 +
ξ − 1
. (125)
It is important to note that, in order to strictly follow our notation in Eq. (89), the last term in
the definition of XB actually contributes to both Qǫλ and Qηλ. The reason is that, after taking into
account the relations (54) for βλ and βξ, it follows immediately that,
ξ − 1
. (126)
The origin of this mixing is the peculiar 1/36 term in (62), which is suppressed by λ, but not by
ξ − 1
It is convenient to introduce the parameter α as follows,
, (127)
which, from the definitions in (91), can also be expressed as
. (128)
Assuming that ξϕ2/M2Pl < 1 is far enough from 1, from the quantum Friedmann equation (69) we
obtain the following expression,
1− βξαXB −
. (129)
Upon differentiating the quantum Friedmann equation (69) with respect to ϕ we get,
λ+ βλζA + 36
(ξ + βξζB)
1 + βλ
+ βξα
) . (130)
In deriving this expression we have used,
βξ , (131)
where dX/dϕ and ζ are given by Eqs. (100) and (101), respectively. In writing Eq. (130) we have
also introduced
ζA ≡ XA +
(1 + ζ) ,
ζB ≡ XB +
(1 + ζ) . (132)
In order to evaluate ǫ from (81) we still need the expression for W . With the above definitions,
from (73) it follows
1 + αξ +
+XA(1 + αξ)
+ βξα
+XB(1 + αξ)
. (133)
What remains to be done is to expand the denominator in (130) to the linear order in βλ and βξ and
then insert the resulting expression together with (129) and (133) into the definition of ǫ given in
(81). In this manner both the classical (90) and the quantum contributions (102) can be obtained.
To calculate η, we must determine dW/dϕ. After some algebra we arrive at
ξ − βλ
(XB +
λ+ βλ
+12H2
ξ + βξ
XB + ζ +
, (134)
where dH2/dϕ is given in (130). After expanding the denominator in (130) and after inserting (129)
and (134) into the definition for η given by (81), both the classical (90) and quantum contribution
(103) are obtained.
At the end, we summarize
− σ −
lnα− 3
ξ − 1
− σ ,
δw ≃ 36
+ 12ξ ,
ζ ≃ −
(3 + αξ)2
, (135)
where α is determined by (128), while σ is given by (104).
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	Introduction
	Propagator in de Sitter space
	de Sitter space
	Scalar propagator in de Sitter space
	Effective potential
	Renormalization group analysis
	Slow-roll parameters
	Classical contributions C and C
	Quantum contributions Q and Q
	Tensor and scalar spectral indices ng and ns
	The spectrum of curvature perturbation PR and the tensor-to-scalar ratio r
	Discussion
	Appendix A
	References
