aainstitutetext: Department of Physics, Faculty of Natural Sciences, NTNU,
Norwegian University of Science and Technology, Høgskoleringen 5, N-7491
Trondheim, Norwaybbinstitutetext: Faculty of Science and Technology,
University of Stavanger, 4036 Stavanger, Norway

# Quantum corrections to slow-roll inflation: scalar and tensor modes

Jens O. Andersen a,b,1 Magdalena Eriksson,111Corresponding author. b Anders
Tranberg<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Inflation is often described through the dynamics of a scalar field, slow-
rolling in a suitable potential. Ultimately, this inflaton must be identified
with the expectation value of a quantum field, evolving in a quantum effective
potential. The shape of this potential is determined by the underlying tree-
level potential, dressed by quantum corrections from the scalar field itself
and the metric perturbations. Following Herranen:2015aja , we compute the
effective scalar field equations and the corrected Friedmann equations to
quadratic order in both scalar field, scalar metric and tensor perturbations.
We identify the quantum corrections from different sources at leading order in
slow-roll, and estimate their magnitude in benchmark models of inflation. We
comment on the implications of non-minimal coupling to gravity in this
context.

##  1 Introduction

Cosmological observations show that the very early Universe underwent a stage
of accelerating expansion Aghanim:2018eyx . This is possible when the matter
in the Universe exhibits some specific thermodynamical properties, often
phrased in terms of the equation of state. An inflationary equation of state
emerges naturally, if during this inflationary epoch, the thermodynamics was
dominated by a scalar field degree of freedom (fundamental or composite),
evolving in an appropriate potential. Fairly generically, if the field is
initially displaced far from its equilibrium value, it will slow-roll back to
the potential minimum in such a way that inflation is achieved. In this well-
known and elegant formalism, the homogeneous scalar field is treated as a
single classical degree of freedom $\phi$, and the combined system of field
equation and Einstein (Friedmann) equations for the cosmological expansion may
be readily solved (see e.g. Lyth:2009zz ). In a homogeneous Friedmann-
Lemaître-Robertson-Walker (FLRW) background, field and metric perturbations
are introduced, and their spectra can be computed in a straightforward way.
Through a well-established procedure, these primordial quantum fluctuations
can be shown to seed the temperature fluctuations of the CMB, and hence the
formation of structure in the Universe. Direct comparison with observations
allows us to constrain the parameters and form of the scalar field potential
Akrami:2018odb .

Ultimately, the scalar field must be treated quantum mechanically, and then an
adjustment of the terminology is required. The object $\phi$ may be identified
as the time-dependent expectation value of the scalar field.222Or, according
to taste, as the mean field, order parameter, one-point function or
condensate. This degree of freedom now evolves in the quantum effective
potential, and the minimum of this potential corresponds to the thermodynamic
equilibrium state.

An important distinction is that this is not an effective potential in the
sense of a low-energy effective theory, where degrees of freedom above a
certain cutoff have been integrated out. Such effective potentials are often
invoked in inflationary model-building to motivate a wide range of functional
forms of classical potentials. The quantum effective potential is the free
energy of the system, once all quantum and thermal fluctuations have been
included at all scales (see for instance peskin ), and for cosmological
applications it includes gravitational corrections as well. From the quantum
effective action, equations of motion for both the scalar mean field and the
metric may be derived, and these then include all quantum corrections and
hence all the thermodynamical information for the system.333The quantum
effective action reduces to the quantum effective potential if the mean fields
are constant, in which case it is given by the classical potential plus the
quantum corrections to it.

In practice, computing this quantum effective potential in an expanding
cosmological background is possible, but less straightforward than in
Minkowski space (see e.g. Birrell ; toms ). As in Minkowski space, an
expansion in terms of Feynman diagrams is introduced and truncated. The issue
of renormalisation arises, and just as for Minkowski space, consistency limits
the set of viable tree-level potentials.

An alternative, but equivalent approach to the quantum dynamics of inflation
is to introduce quantum corrections at the level of the operator equations of
motion. Starting from the classical Friedmann and scalar field equations, we
may expand in powers of the perturbations around the FLRW solution, and by
taking quantum averages generate a Schwinger-Dyson-like set of evolution
equations for the correlators. These may then in principle be solved (see for
instance Markkanen:2012rh ; Herranen:2016xsy ).

As a consequence of gravity’s highly non-linear properties, including metric
perturbations in this scheme is technically challenging, and wholesale
resummations have not been performed. The most common avenue is to include the
effect of the homogeneous time-dependent metric in the computation of scalar
field correlators and quantum effective actions, leading to higher order
curvature contributions to the system. One recurring result from this
procedure is that the ”potential” appearing in the energy, the pressure and
the force of the Friedmann and scalar field equations receive different
corrections. This means that the ubiquitous slow-roll formalism must be
adjusted, and a number of standard identities correct to some order in a slow-
roll expansion, are in addition also correct only at tree-level (or some order
in perturbations) Herranen:2013raa ; Bilandzic:2007nb .

In Herranen:2015aja , scalar metric perturbations were introduced in a
computation of quantum-corrected Friedmann and scalar field equations, to
leading order in slow-roll and leading order in fluctuations. The constrained
quantisation of the field and metric degrees of freedom was given particular
emphasis, and the result was a manageable set of evolution equations. The
scalar fluctuations came out slow-roll suppressed, and the numerical effect of
including them turned out to be negligible, except near the end of inflation,
and then only for small-field inflation.

In the present work, we revisit this picture and include the tensor
perturbations, for which we compute the corresponding leading order
corrections to both scalar field evolution and Friedmann equations. We will
find that these new corrections enter differently in the evolution than the
scalar metric perturbations, and that they are larger than the scalar
corrections.

We also consider non-minimal coupling between the scalar field and the
curvature, and although we will not carry this complication along throughout
the calculation, we will be able to illustrate how a non-minimal coupling may
affect the results.

Quantum effects during and after inflation have received significant attention
in many different contexts and in many different guises. Prominent examples
are non-Gaussian effects in cosmological observables due to self-interactions
of metric perturbations (see e.g. Maldacena:2002vr ; Bartolo:2004if );
logarithmic infrared divergences of correlators in de Sitter and slow-roll
backgrounds and methods to handle them (see Bartolo:2007ti ; Seery:2010kh  for
a review), including the stochastic approximation Starobinsky:1994bd ;
Tsamis:2005hd ; and effective actions for the inflaton field itself (without
metric perturbations), in various resummation schemes, both in an adiabatic
expansion (around Minkowski space $H=0$) toms ; Tranberg:2008ae  and a slow-
roll expansion (around de Sitter space $\dot{H}=0$) Sloth:2006az ;
Seery:2007we ; Riotto:2008mv ; Herranen:2013raa , as well as in strict de
Sitter space (see e.g. Serreau:2011fu ; Gautier:2013aoa ; Gautier:2015pca ).

What we are concerned with here is yet another context; quantum corrections to
evolution equations of the inflaton and metric between horizon exit and the
end of inflation, taking into account both field and metric fluctuations.
These fluctuations influence the relation between the slow-roll parameters at
the end of inflation and at horizon crossing, which in turn enter in
observables in the sky.

The paper is organised as follows. In section 2 we set up our action, the
standard Friedmann and scalar equations at tree-level. We then introduce
metric fluctuations in Newtonian gauge and compute the field equations for
them. These are then quantised in sections 2.3 and 2.4. We proceed to derive
the corrected Friedmann and scalar field equation to second order in
perturbations in section 3 and discuss renormalisation. In section 4, we
briefly consider the magnitude of corrections and estimate the effect of
quantum corrections on the inflationary dynamics. We conclude in section 5.
Details of the calculations are listed in Appendices A–D for completeness.

##  2 Background and field equations in Newtonian gauge

We consider a single inflaton field $\phi$ non-minimally coupled to gravity
with action of the form (following the sign convention (+++) of Wheeler )

$S=\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\left[\tfrac{1}{2}M_{\text{Pl}}^{2}F({\phi})R-\tfrac{1}{2}{g}^{\mu\nu}{\phi}_{,\mu}{\phi}_{,\nu}-V({\phi})\right]\
,$ (1)

where $M_{\text{Pl}}^{2}\equiv(8\pi G)^{-1}$ is the reduced Planck mass and
where for the moment $F(\phi)$ and the tree-level potential $V(\phi)$ are kept
general.444 In some cases, it can be convenient by a conformal transformation
to perform the calculation in the Einstein frame and then transform the
results back again. In the Einstein frame the non-minimal coupling is absent
but the field potential and normalisation are different. We have chosen to
stay in the Jordan frame throughout.  The scalar field $\phi$ lives in a
spacetime defined by the metric $\hat{g}_{\mu\nu}$, and we will assume that
field and metric may be written as

$\displaystyle\phi(x)$ $\displaystyle=\phi(t)+\delta\phi(t,x^{i})\ ,$ (2)
$\displaystyle\hat{g}_{\mu\nu}(x)$ $\displaystyle=g_{\mu\nu}(t)+\delta
g_{\mu\nu}(t,x^{i})\ ,$

where $g_{\mu\nu}$ is the flat FLRW metric;
$g_{\mu\nu}=\textrm{diag}[-1,a^{2}(t)\delta_{ij}]$, with $a(t)$ being the
scale factor and where $t$ denotes cosmic time. Latin indices run from one to
three and Greek indices from zero to three.

The metric perturbations may be decomposed into scalar, vector, and tensor
modes. It is common to immediately discard the vector perturbations, as they
decay away in an expanding Universe. We will do so here as well. Choosing to
work in Newtonian gauge, we may write for the line element

$\mathop{}\\!\mathrm{d}s^{2}=-(1+2\Phi)\mathop{}\\!\mathrm{d}t^{2}+a^{2}[1+2(-\Psi\delta_{ij}+E_{ij})]\mathop{}\\!\mathrm{d}x^{i}\mathop{}\\!\mathrm{d}x^{j}\
,$ (3)

where $\Phi$ and $\Psi$ are scalar potentials and $E_{ij}$ is a traceless,
transverse matrix containing the tensor perturbations, i.e. $E_{ij}$ satisfies
$\partial^{i}E_{ij}=0$ and $E^{i}{}_{i}=0$.

The Einstein field equations are obtained by variation of the action in Eq.
(1) with respect to the metric, which yields

$G_{\mu\nu}\equiv
R_{\mu\nu}-\tfrac{1}{2}Rg_{\mu\nu}=\frac{1}{M_{\text{Pl}}^{2}}\frac{1}{F(\phi)}\tilde{T}_{\mu\nu}\equiv\frac{1}{M_{\text{Pl}}^{2}}{T}_{\mu\nu}\
,$ (4)

where

$\tilde{T}_{\mu\nu}\equiv\phi_{,\mu}\phi_{,\nu}-g_{\mu\nu}[\tfrac{1}{2}g^{\rho\kappa}\phi_{,\rho}\phi_{,\kappa}+V(\phi)]+F(\phi)_{;\mu,\nu}-g_{\mu\nu}\Box
F(\phi)\ .$ (5)

Here and in the following a comma subscript denotes a partial derivative, a
semi-colon denotes a covariant derivative and the ”box” operator is the
d’Alembertian in curved spacetime

$\Box=\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}\partial^{\mu}\ .$ (6)

The effective energy-momentum tensor ${T}_{\mu\nu}$ is covariantly conserved,
and we will assume that $F(\phi)\neq 0$. Variation of the action with respect
to $\phi$ yields the scalar field equation

$\Box{\phi}+\tfrac{1}{2}M_{\text{Pl}}^{2}F_{,{\phi}}R-V_{,{\phi}}=0\ .$ (7)

Next, we insert the perturbed fields of Eqs. (2) and (3) into Eqs. (4)–(7) and
extract equations to zeroth order in perturbations (the ”classical” Friedmann
and inflaton equations); to first order in perturbations (the mode equations
for the field and metric scalar and tensor perturbations in the background
FLRW metric); and to quadratic order in perturbations (the ”quantum-corrected”
Friedmann and inflaton equations).

The zeroth-order equations will provide a slow-roll background in which to
solve the first-order (linear) mode equations. These will in turn be quantised
and inserted into the quadratic-order equations, in order to explicitly
compute the quantum corrections to the cosmological evolution.

###  2.1 Zeroth order in perturbations: the classical equations

To zeroth order the Friedmann equation (4) and the scalar field equation (7)
reduce to the familiar expressions

$\displaystyle 3H^{2}$
$\displaystyle=\frac{1}{M_{\text{pl}}^{2}}\frac{1}{F}\left(\tfrac{1}{2}\dot{{\phi}}^{2}+V\right)-3H\frac{\dot{F}}{F}\
,$ (8) $\displaystyle 2\dot{H}+3H^{2}$
$\displaystyle=-\frac{1}{M_{\text{Pl}}^{2}}\frac{1}{F}\left(\tfrac{1}{2}\dot{{\phi}}^{2}-V\right)-2H\frac{\dot{F}}{F}-\frac{\ddot{F}}{F}\
,$ (9) $\displaystyle 0$
$\displaystyle=\ddot{\phi}+3H\dot{\phi}+V_{,\phi}-3M_{\text{Pl}}^{2}F_{,\phi}(\dot{H}+2H^{2})\
,$ (10)

where $H=\dot{a}/a$ and the right-hand side of Eqs. (8) and (9) represents
energy density and pressure, respectively. Here the dots are derivatives with
respect to cosmic time $t$ and we have inserted $R=6(\dot{H}+2H^{2})$, the
FLRW Ricci scalar. We note that in the limit of Einstein gravity, $F(\phi)=1$,
we recover the standard Friedmann equations. Also, since we assume that
$F\equiv F(\phi)$, we have $\dot{F}\propto\dot{\phi}$,
$\ddot{F}\propto\ddot{\phi},\dot{\phi}^{2}$. We also note that the same object
$V$ (or $V_{,\phi}$) appears in all three equations, which then allows for the
direct application of the slow-roll formalism.

It will therefore be convenient to introduce the dimensionless slow-roll
parameters

$\displaystyle\epsilon_{H}=-\frac{\dot{H}}{H^{2}}=1-\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}\
,\qquad\delta_{H}=-\frac{\ddot{\phi}}{H\dot{\phi}}=1-\frac{{\phi}^{\prime\prime}}{\mathcal{H}{\phi}^{\prime}}\
,\qquad\epsilon_{F}=\frac{1}{2}\frac{\dot{F}}{HF}=\frac{1}{2}\frac{F^{\prime}}{\mathcal{H}F}\
.$ (11)

The primes refer to derivatives with respect to conformal time $\eta$, where

$\displaystyle a(\eta)\mathop{}\\!\mathrm{d}\eta=\mathop{}\\!\mathrm{d}t\
,\qquad\eta=\int\frac{\mathop{}\\!\mathrm{d}t}{a}\
,\qquad\mathcal{H}=\frac{a^{\prime}}{a}\ .$ (12)

Inflation is equivalent to the slow-roll parameters being small, i.e. less
than unity, and evolving slowly in time.

One may apply the slow-roll formalism to solve Eqs. (8)–(10) to any order in
slow-roll parameters one chooses. However, as was emphasised in
Herranen:2013raa ; Bilandzic:2007nb , at the quantum level, corrections
manifest themselves differently in different relations, and the effective
potential $V$ is no longer the same. Hence, when including quantum
corrections, some of the elegance of the slow-roll formalism is lost. We will
see this explicitly below.

###  2.2 First order in perturbations: equations of motion for the
fluctuations

The perturbations of the scalar field and the metric Eq. (2) give rise to
linear perturbations in $F$, $G_{\mu\nu}$ and $\tilde{T}_{\mu\nu}$, which we
denote by $\delta F$, $\delta G_{\mu\nu}$ and $\delta\tilde{T}_{\mu\nu}$,
respectively. The Einstein equation can then be written as

$\displaystyle(F+\delta F)(G_{\mu\nu}+\delta
G_{\mu\nu})=\frac{1}{M_{\text{Pl}}^{2}}(\tilde{T}_{\mu\nu}+\delta\tilde{T}_{\mu\nu}),$
(13)

Using the zeroth-order equations, one immediately finds (now in mixed index
notation)

$\delta
G^{\mu}{}_{\nu}=\frac{1}{M_{\text{Pl}}^{2}}\frac{1}{F}\left(\delta\tilde{T}^{\mu}{}_{\nu}-\frac{\delta
F}{F}\tilde{T}^{\mu}{}_{\nu}\right)\ ,$ (14)

with the understanding, that since $F$ is a function of $\phi$ only, $\delta
F\equiv F_{,\phi}\delta\phi$. The 00- and $0i$-components of Eq. (14) are

$\displaystyle 3H(\dot{\Psi}+H\Phi)-\frac{1}{a^{2}}\nabla^{2}\Psi$
$\displaystyle=$
$\displaystyle\frac{1}{2M_{\text{Pl}}^{2}F}\left(-\dot{\phi}\dot{\delta\phi}+\dot{\phi}^{2}\Phi-
V_{,\phi}\delta\phi\right)-\frac{3\dot{F}}{2F}(\dot{\Psi}+2H\Phi)$
$\displaystyle+\frac{3H}{2}\frac{\dot{\delta
F}}{F}-\frac{1}{2a^{2}}\nabla^{2}\frac{\delta F}{F}+\frac{1}{2}\frac{\delta
F}{F}\left(\frac{\tfrac{1}{2}\dot{{\phi}}^{2}+V}{M_{\text{Pl}}^{2}F}-3H\frac{\dot{F}}{F}\right)\;,$
$\displaystyle\dot{\Psi}+H\Phi$ $\displaystyle=$
$\displaystyle\frac{1}{2M_{\text{Pl}}^{2}F}\dot{\phi}\delta\phi+\frac{1}{2}\frac{\dot{\delta
F}}{F}-\frac{H}{2}\frac{\delta F}{F}-\frac{1}{2}\frac{\dot{F}}{F}\Phi\ .$ (16)

We note that in the minimal coupling case $F(\phi)=1$, the relations simplify
substantially. The $ii$-component gives us

$\displaystyle\ddot{\Psi}+3H\dot{\Psi}+H\dot{\Phi}+(2\dot{H}+3H^{2})\Phi$
$\displaystyle=$
$\displaystyle\frac{1}{2M_{\text{Pl}}^{2}F}\Big{(}\dot{\phi}\dot{\delta\phi}-\dot{\phi}^{2}\Phi-
V_{,\phi}\delta\phi\Big{)}-\frac{\ddot{F}}{F}\Phi+H\frac{\dot{\delta F}}{F}$
(17)
$\displaystyle-\frac{1}{2}\frac{\dot{F}}{F}(4H\Phi+\dot{\Phi}+2\dot{\Psi})+\frac{1}{2}\frac{\ddot{\delta
F}}{F}-\frac{1}{2a^{2}}\nabla^{2}\left(\frac{\delta F}{F}\right)$
$\displaystyle-\frac{1}{2}\frac{\delta
F}{F}\left(\frac{\tfrac{1}{2}\dot{\phi}^{2}-V}{M^{2}_{\text{Pl}}F}+2H\frac{\dot{F}}{F}+\frac{\ddot{F}}{F}\right)\
,$

Again, the non-minimal coupling is responsible for substantial complexity. For
the off-diagonal components $i\neq j$ of the perturbed Einstein equation (14),
we may treat the longitudinal and transverse components separately, to find

$\displaystyle\frac{1}{a^{2}}\partial^{i}\partial_{j}(\Psi-\Phi)$
$\displaystyle=\partial^{i}\partial_{j}\left(\frac{1}{a^{2}}\frac{\delta
F}{F}\right)\ ,$ (18)
$\displaystyle\ddot{E}^{i}{}_{j}+3H\dot{E}^{i}{}_{j}-\frac{1}{a^{2}}\nabla^{2}E^{i}{}_{j}$
$\displaystyle=-\frac{\dot{F}}{F}\dot{E}^{i}{}_{j}\ .$ (19)

We make a few important observations. The right-hand side of Eq. (18) is an
anisotropic stress, and in the presence of a space-dependent non-minimal
coupling, it is nonzero. The equality $\Psi=\Phi$ is often used to simplify
the system of equations, but since $\delta F\propto\delta\phi$ is in general
space-dependent, this is no longer possible. In the absence of non-minimal
coupling ($F(\phi)=1$), this relation is recovered. For Einstein gravity, Eq.
(19) is a free wave equation with cosmological redshift. However, once $F$
becomes time-dependent, an additional damping/amplification is introduced,
with explicit dependence on $\dot{\phi}$. The new damping is proportional to
the slow-roll parameter $\epsilon_{F}$.

By combining Eqs. (2.2) and (17), we arrive at the scalar mode equation

$\displaystyle\ddot{\Psi}+6H\dot{\Psi}+$ $\displaystyle
H\dot{\Phi}+2(\dot{H}+3H^{2})\Phi-\frac{1}{a^{2}}\nabla^{2}\Psi=-\frac{V_{,\phi}\delta\phi}{M_{\text{Pl}}^{2}F}-\frac{\ddot{F}}{F}\Phi-\frac{\dot{F}}{2F}\left[5(2H\Phi+\dot{\Psi})+\dot{\Phi}\right]$
$\displaystyle+\frac{1}{2}\frac{\ddot{\delta
F}}{F}+\frac{5H}{2}\frac{\dot{\delta
F}}{F}-\frac{1}{a^{2}}\nabla^{2}\left(\frac{\delta
F}{F}\right)+\frac{1}{2}\frac{\delta
F}{F}\left(\frac{2V}{M_{\text{Pl}}^{2}F}-\frac{\ddot{F}}{F}-5H\frac{\dot{F}}{F}\right)\
,$ (20)

and inserting Eq. (18) into Eq. (16), we obtain

$\dot{\Psi}+\left(H+\frac{\dot{F}}{2F}\right)\Psi=\frac{1}{2}\left(\frac{1}{M_{\text{Pl}}^{2}F}\dot{\phi}\delta\phi+\frac{\dot{\delta
F}}{F}+H\frac{\delta F}{F}+\frac{\dot{F}\delta F}{F^{2}}\right)\ .$ (21)

At this stage, one procedure could be to solve Eq. (21) for $\delta\phi$ and
substitute the result into Eq. (20). An elegant field redefinition allows us
to absorb the anisotropic stress in Eq. (18) into new variables $\check{\Psi}$
and $\check{\Phi}$,

$\check{\Psi}\equiv-\Psi+\frac{\delta F}{2F}\
,\qquad\check{\Phi}\equiv\Phi+\frac{\delta F}{2F}\
,\qquad\check{\Psi}+\check{\Phi}=0\ ,$ (22)

so that Eq. (20) becomes homogeneous Hwang:1990re ; hwang0 . This is what we
need to carry out the quantisation of the modes. However, eventually, we will
be interested in the Friedmann and inflaton equations to quadratic order, and
these turn out not to be easily written in terms of these new variables (see
Appendix B). In effect, the procedure is ruined in the general case by the
appearance of $\dot{\delta\phi}$ in Eq. (21).

The equation of motion for the scalar field perturbations is found to be

$\ddot{\delta\phi}+3H\dot{\delta\phi}-\frac{1}{a^{2}}\nabla^{2}\delta\phi+V_{,\phi\phi}\delta\phi-\tfrac{1}{2}M_{\text{Pl}}^{2}RF_{,\phi\phi}\delta\phi=-2V_{,\phi}\Phi+\dot{\phi}(\dot{\Phi}+3\dot{\Psi})+M_{\text{Pl}}^{2}F_{,\phi}(R\Phi+\tfrac{1}{2}\delta
R)\ ,$ (23)

where the perturbation of the Ricci scalar is given by

$\delta
R=-12(\dot{H}+2H^{2})\Phi-6H(\dot{\Phi}+4\dot{\Psi})-6\ddot{\Psi}+\frac{2}{a^{2}}\nabla^{2}(2\Psi-\Phi)\
.$ (24)

The above equations capture the dynamics of the fluctuating fields, and we
will proceed by quantising them in the next two sections.

###  2.3 Quantisation of the tensor modes

The field equations for the inflaton fluctuations and the metric scalar
perturbations are too complicated to quantise in the case of a general $F$.
But the tensor fluctuation equation is relatively peaceful, so we will
consider that first, before we specialise to $F(\phi)=1$. We have from Eq.
(19) that in conformal time

$\displaystyle
E_{ij}^{\prime\prime}+2\mathcal{H}\left(1+\epsilon_{F}\right)E_{ij}^{\prime}-\nabla^{2}E_{ij}=0\
.$ (25)

Introducing $z_{g}=\sqrt{F}a$ and $v\delta_{ij}=z_{g}E_{ij}$ (for each
instance of $i,j$), this reduces in momentum space $v(q)$ to

$\displaystyle
v^{\prime\prime}+\left(q^{2}-\frac{z_{g}^{\prime\prime}}{z_{g}}\right)v=0\ .$
(26)

Using the slow-roll expression for conformal time

$\displaystyle\eta=-\frac{1}{aH}\frac{1}{1-\epsilon_{H}}=-\frac{1}{\mathcal{H}}\frac{1}{1-\epsilon_{H}}\
,$ (27)

we find

$\displaystyle\frac{z_{g}^{\prime\prime}}{z_{g}}=\frac{1}{\eta^{2}}\frac{(1+\epsilon_{F})(2-\epsilon_{H}+\epsilon_{F})}{(1-\epsilon_{H})^{2}}\equiv\frac{n}{\eta^{2}}\
.$ (28)

Treating $n$ as a constant, the solution to Eq. (26) is

$\displaystyle v(\eta,q)=\sqrt{|\eta|}\left[c_{1}({\bf
q})H_{\nu}^{(1)}(q|\eta|)+c_{2}({\bf q})H_{\nu}^{(2)}(q|\eta|)\right]\
,\qquad\nu=\sqrt{n+\frac{1}{4}}\ ,$ (29)

where $H_{\nu}^{(1)}$ and $H_{\nu}^{(2)}$ are Hankel functions of the first
and second kind, respectively. Choosing the Bunch-Davies vacuum, the
coefficients are set to $c_{1}({\bf q})=0$ and $c_{2}({\bf q})=1$ Birrell . We
still need the overall normalisation of the mode and so we write the quantised
tensor field as

$\displaystyle E_{ij}(\eta,{\bf
x})=\int\frac{\mathop{}\\!\mathrm{d}^{3}q}{(2\pi)^{3/2}}\sum_{\lambda=+,\times}\left[\hat{a}_{\lambda{\bf
q}}\tilde{h}_{\lambda{\bf q}}(\eta)e_{ij}^{(\lambda)}({\bf q})e^{i{\bf q\cdot
x}}+\textrm{h.c.}\right]\ ,$ (30)

where ${a}_{\lambda{\bf q}}$ and ${a}^{\dagger}_{\lambda{\bf q}}$ are
annihilation and creation operators and $\tilde{h}_{\lambda{\bf q}}$ are
scalar functions. The subscripts $+,\times$ refer to the two polarisation
states of the polarisation tensors $e^{(\lambda)}_{ij}$, that satisfy

$\displaystyle e^{(\lambda)}_{ij}({\bf q})e^{(\lambda^{\prime})}_{ij}({\bf
q})=2\delta_{\lambda\lambda^{\prime}}\ ,\qquad
e^{(\lambda)}_{ij}=e_{ji}^{(\lambda)}\ ,\qquad e^{(\lambda)i}{}_{i}=0\
,\qquad\partial^{i}e_{ij}^{(\lambda)}=0\ .$ (31)

The creation and annihilation operators satisfy the commutation relation

$\displaystyle[\hat{a}_{\lambda{\bf q}},\hat{a}^{\dagger}_{\lambda{\bf
q^{\prime}}}]=\delta_{\lambda\lambda^{\prime}}\delta^{3}({\bf q-q^{\prime}})\
,$ (32)

with all other commutators vanishing. Inserting the ansatz of Eq. (30) into
the action, one finds that the quadratic contributions $S_{\text{t}}$ from the
tensor may be written

$\displaystyle
S_{\text{t}}=M_{\text{Pl}}^{2}\int\mathop{}\\!\mathrm{d}\eta\mathop{}\\!\mathrm{d}^{3}xa^{2}F\sum_{\lambda}[\hat{h}^{{}^{\prime}2}-\hat{h}_{\lambda}{}^{,i}\hat{h}_{\lambda,i}]\
,$ (33)

where

$\displaystyle\hat{h}_{\lambda}(\eta,{\bf
x})=\int\frac{\mathop{}\\!\mathrm{d}^{3}q}{(2\pi)^{3/2}}\left[\hat{a}_{\lambda{\bf
q}}\tilde{h}_{\lambda{\bf q}}(\eta)e^{i{\bf q\cdot x}}+\textrm{h.c.}\right]\
.$ (34)

The canonical quantisation condition is

$\displaystyle[\hat{h}_{\lambda}(\eta,{\bf}),\hat{\pi}_{\lambda}(\eta,{\bf
x^{\prime}})]=i\delta^{3}({\bf x-x^{\prime}})\ ,$ (35)

and defining the conjugate momentum

$\displaystyle\hat{\pi}_{\lambda}(\eta,{\bf
x})=\frac{\partial\mathcal{L}}{\partial\hat{h}_{\lambda}^{{}^{\prime}}}=2a^{2}M_{\text{Pl}}^{2}F\hat{h}_{\lambda}^{{}^{\prime}}\
,$ (36)

together with Eq. (32), we may write down the Wronskian for the mode functions
themselves as

$\displaystyle\tilde{h}_{\lambda{\bf
q}}(\eta)\tilde{h}^{{}^{\prime}*}_{\lambda{\bf
q}}(\eta)-\tilde{h}^{*}_{\lambda{\bf
q}}(\eta)\tilde{h}^{{}^{\prime}}_{\lambda{\bf
q}}(\eta)=\frac{i}{2a^{2}M_{\text{Pl}}^{2}F}\ .$ (37)

This gives us the final normalisation

$\displaystyle\tilde{h}_{\lambda{\bf
q}}(\eta)=\sqrt{\frac{\pi|\eta|}{8a^{2}M_{\text{Pl}}^{2}F}}H_{\nu}^{(2)}(q|\eta|)\
,$ (38)

where $\nu$ is approximated as

$\nu\approx\tfrac{3}{2}+\epsilon_{F}+\epsilon_{H}+\mathcal{O}(\epsilon^{2})\
,$ (39)

to first order in slow-roll parameters. We see that the effect of non-minimal
coupling on the evolution of the tensor modes enters through $\epsilon_{F}$
and the overall normalisation.

###  2.4 Quantisation of the scalar modes

Since it is not possible to express all scalar-perturbation dependence in the
quadratic-order quantum corrections solely in terms of the new variable
$\hat{\Psi}$ of Eq. (22), we now consider the Einstein gravity limit where
$F(\phi)=1$. In this case, Eq. (20) reduces to

$\ddot{\Psi}+H(1+2\delta_{H})\dot{\Psi}+2H^{2}(\delta_{H}-\epsilon_{H})\Psi-\frac{1}{a^{2}}\nabla^{2}\Psi=0\
,$ (40)

using Eqs. (10) and (21) and the slow-roll parameters in Eq. (11). The scalar
field equations (2.2)–(18) in combination with the equation of motion for
$\delta\phi$ in Eq. (23) is an overdetermined system, and hence requires a
constrained quantisation procedure Dirac:1950pj . To this end we introduce the
canonical momenta

$\displaystyle\pi_{\Phi}$
$\displaystyle\equiv\frac{\partial\mathcal{L}}{\partial\Phi^{\prime}}=0\ ,$
(41) $\displaystyle\pi_{\Psi}$
$\displaystyle\equiv\frac{\partial\mathcal{L}}{\partial\Psi^{\prime}}=-6a^{2}M_{\text{Pl}}^{2}[\Psi^{\prime}+\mathcal{H}(\Phi+\Psi)]\
,$ (42) $\displaystyle\pi_{\delta\phi}$
$\displaystyle\equiv\frac{\partial\mathcal{L}}{\partial\delta\phi^{\prime}}=a^{2}[\delta\phi^{\prime}-\phi^{\prime}(\Phi+3\Psi)]\
,$ (43)

and encode the constraints of the field equations by introducing the conjugate
momenta

$\displaystyle\chi_{1}$ $\displaystyle\equiv\pi_{\Phi}\ ,$ (44)
$\displaystyle\chi_{2}$
$\displaystyle\equiv\nabla^{2}\Psi+3\mathcal{H}^{\prime}\Psi-\frac{1}{2M_{\text{Pl}}^{2}}\left(a^{2}V_{,\phi}\delta\phi+\frac{\phi^{\prime}}{a^{2}}\pi_{\delta\phi}-\frac{\mathcal{H}}{a^{2}}\pi_{\Psi}\right)\
,$ (45) $\displaystyle\chi_{3}$
$\displaystyle\equiv\mathcal{H}\Psi+\frac{1}{2M_{\text{Pl}}^{2}}\left(\phi^{\prime}\delta\phi+\frac{1}{3a^{2}}\pi_{\Psi}\right)\
,$ (46) $\displaystyle\chi_{4}$ $\displaystyle\equiv\Phi-\Psi\ ,$ (47)

where then

$\chi_{\alpha}=0\ ,\qquad\alpha=1,2,3,4.$ (48)

The constrained quantisation is realised by way of the Dirac bracket defined
as

$[A,B]_{\text{D}}\equiv[A,B]_{\text{P}}-[A,\chi_{m}]_{\text{P}}(C^{-1})_{mn}[\chi_{n},B]_{\text{P}}\
,\qquad m,n=2,3,$ (49)

where the Poisson bracket is defined as

$[A,B]_{\text{P}}\equiv\sum_{\varphi=\Psi,\delta\phi}\left(\frac{\partial
A}{\partial\varphi}\frac{\partial B}{\partial\Pi_{\varphi}}-\frac{\partial
B}{\partial\varphi}\frac{\partial A}{\partial\Pi_{\varphi}}\right)\ ,$ (50)

and $C_{mn}$ is a non-singular constraint matrix

$C_{mn}\equiv[\chi_{m},\chi_{n}]_{\text{P}}\ .$ (51)

The quantised variables will then have equal-time commutation relations given
by

$[A,B]=i[A,B]_{\text{D}}\ ,$ (52)

for which we obtain

$[\Psi(\mathbf{x}),\Psi(\mathbf{x^{\prime}})]=[\delta\phi(\mathbf{x}),\delta\phi(\mathbf{x^{\prime}})]=0\
,\qquad[\Psi(\mathbf{x}),\Psi^{\prime}(\mathbf{x^{\prime}})]=i\frac{\phi^{\prime
2}}{4M_{\text{Pl}}^{4}a^{2}\nabla^{2}}\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$ (53)

and where the resulting commutation relations for $\Psi,\,\delta\phi$ combined
with the conjugate momenta $\pi_{\Psi},\,\pi_{\delta\phi}$ are listed in
Appendix A.

The field $\Psi$ can be promoted to an operator and decomposed in terms of
mode functions $\tilde{f}_{\mathbf{k}}(\eta)$ as

$\hat{\Psi}(\eta,\mathbf{x})=\int\frac{\mathop{}\\!\mathrm{d}^{3}k}{(2\pi)^{3/2}}\left(\hat{b}_{\mathbf{k}}\tilde{f}_{\mathbf{k}}(\eta)e^{i\mathbf{k}\cdot\mathbf{x}}+\text{h.c.}\right)\
,$ (54)

where the creation and annihilation operators
$\hat{b}_{\mathbf{k}},\hat{b}_{\mathbf{k}}^{\dagger}$ fulfil the standard
commutation relations

$[\hat{b}_{\mathbf{k}},\hat{b}_{\mathbf{k^{\prime}}}]=[\hat{b}_{\mathbf{k}}^{\dagger},\hat{b}_{\mathbf{k^{\prime}}}^{\dagger}]=0\
,\qquad[\hat{b}_{\mathbf{k}},\hat{b}_{\mathbf{k^{\prime}}}^{\dagger}]=\delta^{3}(\mathbf{k}-\mathbf{k^{\prime}})\
,$ (55)

so that the equation of motion for the mode functions becomes

$\tilde{f}_{\mathbf{k}}^{\prime\prime}-\frac{2\delta_{H}}{\eta}\tilde{f}_{\mathbf{k}}^{\prime}+\left(\frac{2(\delta_{H}-\epsilon_{H})}{\eta^{2}}+k^{2}\right)\tilde{f}_{\mathbf{k}}=0\
.$ (56)

Here we have approximated the Hubble parameter $\mathcal{H}\simeq-1/\eta$ to
zeroth order in slow-roll. From the equal-time commutation relation in Eq.
(55), the Wronskian is calculated to

$\tilde{f}_{\mathbf{k}}(\eta)\tilde{f}_{\mathbf{k}}^{\prime*}(\eta)-\tilde{f}_{\mathbf{k}}^{*}(\eta)\tilde{f}_{\mathbf{k}}^{\prime}(\eta)=i\left(\frac{\phi^{\prime}}{2M_{\text{Pl}}^{2}a|\mathbf{k}|}\right)^{2}\
.$ (57)

The solution to the mode equation (56) is then given by

$\tilde{f}_{\mathbf{q}}(\eta)=\frac{\sqrt{\pi|\eta|}}{2}\frac{\phi^{\prime}}{2M_{\text{Pl}}^{2}a|\mathbf{k}|}H_{\upsilon}^{(2)}(k|\eta|)=\sqrt{\frac{\pi\epsilon_{H}}{8}}\frac{\mathcal{H}}{M_{\text{Pl}}^{2}a|\mathbf{k}|}(-\eta)^{1/2}H_{\upsilon}^{(2)}(-k\eta)\
,$ (58)

where the index $\upsilon$ is approximated as

$\upsilon=\tfrac{1}{2}\sqrt{1+8\epsilon_{H}-4\delta_{H}}\approx\tfrac{1}{2}+2\epsilon_{H}-\delta_{H}\
.$ (59)

##  3 Quantum-corrected equations of motion in the Einstein-gravity limit

Taking the vacuum expectation value of the perturbed equations of motion, we
calculate the quantum corrections in the Einstein gravity limit $F(\phi)=1$.
The second-order terms of the Friedmann and mean-field equations are listed in
Appendix B (with $F$ general), where the number of terms reduces significantly
after taking the vacuum expectation value. In the end, the equations may be
expressed in terms of just a few two-point correlators, which we will
therefore consider first.

###  3.1 Correlation functions and renormalisation

The quantum-corrected Friedmann and mean-field equations can be expressed in
terms of the two-point correlation functions
$\braket{\varphi^{2}},\,\braket{\varphi\nabla^{2}\varphi}$ and
$\braket{\varphi\nabla^{4}\varphi}$, with $\varphi=E_{ij},\Psi$, by using the
relations in Appendix C. Having obtained the mode solutions for the tensors in
Eq. (38) and scalars in Eq. (58), these correlators can now be calculated
explicitly. By use of the field decompositions in Eqs. (30) and (32), we
obtain for the tensor correlators:

$\displaystyle\begin{aligned}
\mathllap{C_{0}}&\equiv\braket{E_{ij}E_{ij}}=\frac{1}{2\pi^{2}}\frac{H^{2}}{M_{\text{Pl}}^{2}F}\bigg{[}\left(\frac{1}{2(\epsilon_{H}+\epsilon_{F})}+\log
2+\psi(\tfrac{3}{2})\right)\left(-1+\Lambda_{\text{IR}}^{-2(\epsilon_{H}+\epsilon_{F})}\right)\\\
&\hskip
128.0374pt+\tfrac{1}{2}\Lambda_{\text{UV}}^{2}+\log\Lambda_{\text{UV}}\bigg{]}+\mathcal{O}(\epsilon)\
,\end{aligned}$ (60) $\displaystyle\mathllap{C_{2}}$
$\displaystyle\equiv\braket{E_{ij}\nabla^{2}E_{ij}}=-\frac{1}{4\pi^{2}}\frac{a^{2}H^{4}}{M_{\text{Pl}}^{2}F}\left[\tfrac{1}{2}\Lambda_{\text{UV}}^{4}+\Lambda_{\text{UV}}^{2}-\tfrac{1}{2}\Lambda_{\text{IR}}^{4}-\Lambda_{\text{IR}}^{2}\right]\
,$ (61)

where $\psi(x)$ denotes the digamma function and $\Lambda_{\rm IR,UV}$ are
infrared and ultraviolet cutoffs respectively, see the details in Appendix D.
For the tensor modes, we have for illustration retained the non-minimal
coupling through $\epsilon_{F}$ and the overall normalisation $1/F$.

For the scalars we obtain

$\displaystyle
D_{0}\equiv\braket{\Psi^{2}}=\frac{\epsilon_{H}H^{2}}{8\pi^{2}M_{\text{Pl}}^{2}}\left[\left(\frac{1}{2(2\epsilon_{H}-\delta_{H})}+\log
2+\psi(\tfrac{1}{2})\right)\left(-1+\Lambda_{\text{IR}}^{-2(2\epsilon_{H}-\delta_{H})}\right)+\log\Lambda_{\text{UV}}\right]+\mathcal{O}(\epsilon)\
,$ (62) $\displaystyle
D_{2}\equiv\braket{\Psi\nabla^{2}\Psi}=-\frac{\epsilon_{H}a^{2}{H}^{4}}{16\pi^{2}M_{\text{Pl}}^{2}}\left(\Lambda_{\text{UV}}^{2}-\Lambda_{\text{IR}}^{2}\right)+\mathcal{O}(\epsilon)\
,$ (63) $\displaystyle
D_{4}\equiv\braket{\Psi\nabla^{4}\Psi}=\frac{\epsilon_{H}a^{4}{H}^{6}}{32\pi^{2}M_{\text{Pl}}^{2}}\left(\Lambda_{\text{UV}}^{4}-\Lambda_{\text{IR}}^{4}\right)+\mathcal{O}(\epsilon)\
.$ (64)

We notice that the scalar correlators each have an overall factor of
$\epsilon_{H}$, which is absent in the tensor correlators. This is a direct
consequence of the mode function normalisation, where the constrained
quantisation procedure of the scalars gives an additional factor of
$\phi^{\prime 2}$ in the commutation relations in Eq. (53). However, we will
see that once the correlators are reinstated in the equations of motion, they
appear at the same order in slow-roll.

The correlators have both IR and UV divergences. A careful treatment of the UV
involves applying dimensional regularisation, which removes all power law
divergences, and where logarithmic divergences are absorbed into counterterms
for higher-order invariant operators $R^{2},R^{\mu\nu}R_{\mu\nu}$, etc.
Markkanen:2013nwa . The computations in Appendix D employ a simpler cutoff
regularisation to identify where UV and IR divergences appear, after which we
assume that the UV can be dealt with, so that only the physical IR effects
remain. Based on this discussion, the tensor correlators will hereafter be
taken to be

$\displaystyle C_{0}$
$\displaystyle=-\frac{H^{2}}{4\pi^{2}M_{\text{Pl}}^{2}F(\epsilon_{H}+\epsilon_{F})}\left(1-\Lambda_{\text{IR}}^{-2(\epsilon_{H}+\epsilon_{F})}\right)\
,$ (65) $\displaystyle C_{2}$
$\displaystyle=\frac{a^{2}H^{4}}{4\pi^{2}M_{\text{Pl}}^{2}F}(\tfrac{1}{2}\Lambda_{\text{IR}}^{4}+\Lambda_{\text{IR}}^{2})\
,$

and the scalar correlators by

$\displaystyle D_{0}$
$\displaystyle=-\frac{\epsilon_{H}H^{2}}{16\pi^{2}M_{\text{Pl}}^{2}(2\epsilon_{H}-\delta_{H})}\left(1-\Lambda_{\text{IR}}^{-2(2\epsilon_{H}-\delta_{H})}\right)\
,$ (66) $\displaystyle D_{2}$
$\displaystyle=\frac{\epsilon_{H}a^{2}{H}^{4}}{16\pi^{2}M_{\text{Pl}}^{2}}\Lambda_{\text{IR}}^{2}\
,$ $\displaystyle D_{4}$
$\displaystyle=-\frac{\epsilon_{H}a^{4}{H}^{6}}{32\pi^{2}M_{\text{Pl}}^{2}}\Lambda_{\text{IR}}^{4}\
.$

Furthermore, we should only keep terms of leading order in slow-roll, since
the mode functions are only valid to that order. The correlators $C_{2},D_{2}$
and $D_{4}$ are proportional to the infrared cutoff, and as we imagine
$\Lambda_{\text{IR}}\ll 1$, they give negligible contributions (and are set to
zero). This leaves the IR-divergent $C_{0},D_{0}$, for which we may write

$\displaystyle 1-\Lambda_{\text{IR}}^{-2(\epsilon_{H}+\epsilon_{F})}$
$\displaystyle=2(\epsilon_{H}+\epsilon_{F})|\log\Lambda_{\text{IR}}|+\mathcal{O}(\epsilon^{2})\
,$ (67) $\displaystyle 1-\Lambda_{\text{IR}}^{-2(2\epsilon_{H}-\delta_{H})}$
$\displaystyle=2(2\epsilon_{H}-\delta_{H})|\log\Lambda_{\text{IR}}|+\mathcal{O}(\epsilon^{2})\
.$

This yields

$\displaystyle
C_{0}=-\frac{1}{2\pi^{2}}\frac{H^{2}}{M_{\text{Pl}}^{2}F}|\log\Lambda_{\text{IR}}|+\mathcal{O}(\epsilon)\
,$ (68) $\displaystyle
D_{0}=-\frac{\epsilon_{H}}{8\pi^{2}}\frac{H^{2}}{M_{\text{Pl}}^{2}}|\log\Lambda_{\text{IR}}|+\mathcal{O}(\epsilon^{2})\
.$ (69)

We will in the following count $C_{0}$ as $\mathcal{O}(1)$ and $D_{0}$ as
$\mathcal{O}(\epsilon)$. Below we will have a further analysis of the
magnitude of the IR-cutoff.

###  3.2 Friedmann equations

Perturbing the Einstein equations to quadratic order in fluctuations, the
quantum-corrected equation will take the form

$G^{\mu}{}_{\nu}=\frac{1}{M_{\text{Pl}}^{2}}(\tilde{T}^{\mu}{}_{\nu}+\braket{\delta_{2}\tilde{T}^{\mu}{}_{\nu}})-\braket{\delta_{2}G^{\mu}{}_{\nu}}\
,$ (70)

where the quadratic-order contributions $\delta_{2}\tilde{T}_{\mu\nu}$ and
$\delta_{2}G_{\mu\nu}$ are given in Appendix B. For the 00-component of Eq.
(70), we obtain

$\displaystyle 3H^{2}$
$\displaystyle=\frac{1}{M_{\text{Pl}}^{2}}\left[\tfrac{1}{2}\dot{\sigma}^{2}+V-\braket{\delta_{2}\tilde{T}^{0}{}_{0}}\right]+\braket{\delta_{2}G^{0}{}_{0}}$
(71)
$\displaystyle=\frac{1}{M_{\text{Pl}}^{2}}\left[\tfrac{1}{2}\dot{\sigma}^{2}+V+\tfrac{1}{2}\braket{\dot{\delta\phi}^{2}}+\frac{1}{2a^{2}}\braket{(\nabla\delta\phi)^{2}}-2\dot{\phi}\braket{\Phi\dot{\delta\phi}}+2\dot{\phi}^{2}\braket{\Phi^{2}}+\tfrac{1}{2}V_{,\phi\phi}\braket{\delta\phi^{2}}\right]$
$\displaystyle\quad+\tfrac{1}{2}\braket{\dot{E}^{ij}\dot{E}_{ij}}+4H\braket{E^{ij}\dot{E}_{ij}}-12H^{2}D_{0}-3\braket{\dot{\Psi}^{2}}-\frac{1}{a^{2}}\left[5D_{2}+\tfrac{3}{2}C_{2}\right]$
$\displaystyle=\frac{1}{M_{\text{Pl}}^{2}}\left(\tfrac{1}{2}\dot{\sigma}^{2}+V\right)+\frac{H^{2}}{\epsilon_{H}}\bigg{[}\tfrac{1}{4}\epsilon_{H}\frac{\partial_{t}^{2}C_{0}}{H^{2}}+\tfrac{19}{8}\epsilon_{H}\frac{\partial_{t}C_{0}}{H}-\tfrac{7}{4}\epsilon_{H}\frac{1}{a^{2}}\frac{C_{2}}{H^{2}}$
$\displaystyle\quad+(-\tfrac{3}{2}\epsilon_{H}+\tfrac{1}{2}\delta_{M})\frac{\partial_{t}^{2}D_{0}}{H^{2}}+(-6\delta_{H}+\tfrac{9}{2}\epsilon_{H}+\tfrac{3}{2}\delta_{M})\frac{\partial_{t}D_{0}}{H}+(-12\epsilon_{H}+\delta_{M})D_{0}$
$\displaystyle\quad-\frac{1}{a^{2}}\frac{\partial_{t}^{2}D_{2}}{H^{4}}-(2+\delta_{H})\frac{1}{a^{2}}\frac{\partial_{t}D_{2}}{H^{3}}-(1+2\delta_{H}+4\epsilon_{H}+\delta_{M})\frac{1}{a^{2}}\frac{D_{2}}{H^{2}}+\frac{1}{a^{4}}D_{4}\bigg{]}\
,$

where we used the constraint relations in Eqs. (16) and (18) to express $\Phi$
and $\delta\phi$ in terms of $\Psi$. Furthermore, we have introduced a new
slow-roll parameter

$\delta_{M}\equiv\frac{V_{,\phi\phi}}{H^{2}}\simeq 3(\delta_{H}+\epsilon_{H})\
,$ (72)

where the last relation follows from Eqs. (10) and (11), and is correct at
leading order in slow-roll. We have also used that

$\epsilon_{H}H^{2}=\frac{\dot{\phi}^{2}}{2M_{\text{Pl}}^{2}}\ ,$ (73)

which can be obtained by combining Eqs. (8) and (9) with Eq. (11). For the
spatial component of the Friedmann equations with $i=j$, we get

$\displaystyle 2\dot{H}+3H^{2}$
$\displaystyle=\frac{1}{M_{\text{Pl}}^{2}}\left[-\tfrac{1}{2}\dot{\phi}^{2}+V-\tfrac{1}{2}\braket{\dot{\delta\phi}^{2}}-\frac{1}{2a^{2}}\braket{(\nabla\delta\phi)^{2}}+2\dot{\phi}\braket{\Phi\dot{\delta\phi}}-2\dot{\phi}^{2}\braket{\Phi^{2}}+\tfrac{1}{2}V_{,\phi\phi}\braket{\delta\phi^{2}}\right]$
(74)
$\displaystyle\quad-\tfrac{1}{2}\braket{\dot{E}^{ij}\dot{E}_{ij}}-\frac{5}{2a^{2}}\braket{E^{ij}\nabla^{2}E_{ij}}-4(2\dot{H}+3H^{2})\braket{\Psi}^{2}-8H\braket{\Psi\dot{\Psi}}-\braket{\dot{\Psi}^{2}}+\frac{1}{a^{2}}\braket{\Psi\nabla^{2}\Psi}$
$\displaystyle=\frac{1}{M_{\text{Pl}}^{2}}\left(-\tfrac{1}{2}\dot{\phi}^{2}+V\right)+\frac{H^{2}}{\epsilon_{H}}\bigg{[}-\tfrac{1}{4}\epsilon_{H}\frac{\partial_{t}^{2}C_{0}}{H^{2}}-\tfrac{3}{8}\epsilon_{H}\frac{\partial_{t}C_{0}}{H}-\tfrac{9}{4}\epsilon_{H}\frac{1}{a^{2}}\frac{C_{2}}{H^{2}}$
$\displaystyle\quad+(-\tfrac{1}{2}\epsilon_{H}+\delta_{M})\frac{\partial_{t}^{2}D_{0}}{H^{2}}+(6\delta_{H}-\tfrac{21}{2}\epsilon_{H}+\tfrac{3}{2}\delta_{M})\frac{\partial_{t}D_{0}}{H}+(-12\epsilon_{H}+\delta_{M})D_{0}$
$\displaystyle\quad+\frac{1}{a^{2}}\frac{\partial_{t}^{2}D_{2}}{H^{4}}+(2+\delta_{H})\frac{1}{a^{2}}\frac{\partial_{t}D_{2}}{H^{3}}+(1+2\delta_{H}+4\epsilon_{H}-\delta_{M})\frac{1}{a^{2}}\frac{D_{2}}{H^{2}}-\frac{1}{a^{4}}\frac{D_{4}}{H^{4}}\bigg{]}\
.$

For illustration, we have included contributions to leading order in slow-roll
and quantum corrections, but also (parts of) the higher order contributions in
slow-roll. However, for consistency, we must again truncate the whole
expression at leading order. Remembering also that $D_{2}\simeq D_{4}\simeq
C_{2}\simeq 0$, we then find:

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{M_{\text{Pl}}^{2}}\left(\tfrac{1}{2}\dot{\phi}^{2}+V\right)+H^{2}\left(-12+\frac{\delta_{M}}{\epsilon_{H}}\right)D_{0}+\tfrac{19}{8}H\partial_{t}C_{0}\
,$ (75) $\displaystyle 2\dot{H}+3H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{M_{\text{Pl}}^{2}}\left(-\tfrac{1}{2}\dot{\phi}^{2}+V\right)+H^{2}\left(-12+\frac{\delta_{M}}{\epsilon_{H}}\right)D_{0}-\tfrac{3}{8}H\partial_{t}C_{0}\
.$ (76)

We recall that $C_{0}\sim\mathcal{O}(1)$, while
$D_{0}\sim\mathcal{O}(\epsilon)\sim\partial_{t}C_{0}$. Hence, we see that
scalar and tensor contributions to the effective potential appear at the same
order $\mathcal{O}(\epsilon)$, even though the correlators themselves are of
different order.

The corrections to the classical relations involve terms that could equally be
grouped with the left-hand side ($\propto H^{2}$) and the right-hand side
($\phi$-dependent). Interpreting the quantum corrections as corrections to the
potential, we see that the resulting effective potential is different in the
two Friedmann equations. At this level in slow-roll, this difference
originates from the tensor contributions only, while the scalar contributions
are the same.

###  3.3 Mean-field equation

The perturbed mean-field equation (7) is displayed in Eq. (113), which in the
Einstein gravity limit reduces to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}+\ddot{\phi}(6\Psi^{2}+E^{i}{}_{j}E^{j}{}_{i})+\dot{\phi}\big{[}3H(6\Psi^{2}+E^{i}{}_{j}E^{j}{}_{i})+12\Psi\dot{\Psi}+2\dot{E}^{i}{}_{j}E^{j}{}_{i}\big{]}$
(77)
$\displaystyle-4\ddot{\delta\phi}\Psi-4\dot{\delta\phi}[3H\Psi+\dot{\Psi}]+\frac{2}{a^{2}}\partial_{i}[E^{ij}\partial_{j}\delta\phi]-V_{,\phi}(2\Psi^{2}+E^{i}{}_{j}E^{j}{}_{i})$
$\displaystyle-2V_{,\phi\phi}\Psi\delta\phi+\tfrac{1}{2}V_{,\phi\phi\phi}\delta\phi^{2}\;.$

Taking the vacuum expectation value of this, we arrive at the quantum-
corrected equation

$\displaystyle 0$
$\displaystyle=\ddot{\phi}\left[1+\frac{1}{\epsilon_{H}}\left(\epsilon_{H}C_{0}+(14\epsilon_{H}-8\delta_{H})D_{0}-4(1+\delta_{H})\frac{\partial_{t}D_{0}}{H}-2\frac{\partial_{t}^{2}D_{0}}{H^{2}}+\frac{4}{a^{2}}\frac{D_{2}}{H^{2}}\right)\right]$
(78)
$\displaystyle\quad+3H\dot{\phi}\bigg{[}1+\frac{1}{3\epsilon_{H}}\left(3\epsilon_{H}C_{0}+\epsilon_{H}\partial_{t}C_{0}+(18\epsilon_{H}+2\delta_{M})D_{0}-(4\epsilon_{H}-\delta_{M})\frac{\partial_{t}D_{0}}{H}-\frac{4}{a^{2}}\frac{D_{2}}{H^{2}}-\frac{4}{a^{2}}\frac{\partial_{t}D_{2}}{H^{3}}\right)\bigg{]}$
$\displaystyle\quad+V_{,\phi}(1+6D_{0}-C_{0})+\frac{M_{\text{Pl}}^{2}}{\epsilon_{H}}V_{,\phi\phi\phi}\left[(1+2\delta_{H}-2\epsilon_{H})D_{0}+(\tfrac{3}{2}+\delta_{H})\frac{\partial_{t}D_{0}}{H}+\frac{1}{2}\frac{\partial_{t}^{2}D_{0}}{H^{2}}-\frac{1}{a^{2}}\frac{D_{2}}{H^{2}}\right]\
.$

This equation can be derived via two routes. Either using the field equation
(16) to re-express $\ddot{\delta\phi}$ in terms of $\Psi$, or by using Eq.
(23). Here we have chosen the latter way, as it is more straightforward,
although both methods must yield the same result at leading order in slow-roll
parameters. We may further truncate in slow-roll, and again set $D_{2}\simeq
D_{4}\simeq C_{2}\simeq 0$, to obtain

$\displaystyle 0$
$\displaystyle=\ddot{\phi}\left[1+C_{0}+\left(14-8\frac{\delta_{H}}{\epsilon_{H}}\right)D_{0}\right]+3H\dot{\phi}\bigg{[}1+C_{0}+\tfrac{1}{3}\partial_{t}C_{0}+\left(6\epsilon_{H}+\frac{2}{3}\frac{\delta_{M}}{\epsilon_{H}}\right)D_{0}\bigg{]}$
(79)
$\displaystyle\quad+V_{,\phi}(1-C_{0}+6D_{0})+M_{\text{Pl}}^{2}V_{,\phi\phi\phi}\frac{1+2\delta_{H}-2\epsilon_{H}}{\epsilon_{H}}D_{0}\
.$

Here we notice that the quantum corrections already appear at
$\mathcal{O}(1)$, and treating $\ddot{\phi}$ as higher order, we may further
truncate to get

$\displaystyle 0$
$\displaystyle=3H\dot{\phi}\left(1+C_{0}\right)+V_{,\phi}(1-C_{0})+M_{\text{Pl}}^{2}V_{,\phi\phi\phi}\frac{D_{0}}{\epsilon_{H}}\
.$ (80)

As for the Friedmann equation, corrections from scalars and tensor enter at
the same order, but the tensor contributions now enter as a multiplicative
correction to the potential and the damping (Hubble) rate.

Comparing to the classical field equation (where $C_{0}=D_{0}=0$), it is not
possible to identify the corrections as simply modifying the potential. On the
other hand, we note that for $C_{0}=0$ (ignoring tensors), when comparing to
Eq. (75), the term $V_{,\phi\phi\phi}D_{0}/\epsilon$ may be related to (the
derivative of) $\delta_{M}D_{0}/\epsilon$, provided $D_{0}/\epsilon$ is
assumed to not depend on $\phi$. Once $C_{0}\neq 0$ this correspondence is
lost, showing that tensor modes make a qualitative difference.

##  4 Magnitude of corrections: examples

In this section, we estimate the magnitude of the quantum corrections to the
mean-field and Friedmann equations for large-field monomial inflation and
quartic hilltop inflation. We wish to compute their values during inflation,
and we choose the time of horizon crossing, which we take to be $N=50-60$
$e$-foldings before the end of inflation.

First, we need a prescription to evaluate the IR-divergent correlators. As
noted in section 3.1, with a small IR cutoff $\Lambda_{\text{IR}}\rightarrow
0$, all correlators become negligible except $C_{0}$ and $D_{0}$, which
diverge logarithmically. The IR logarithm can be related to the number of
$e$-foldings during inflation. If we assume that the IR cutoff is set to
exclude superhorizon modes from the loop integrals, i.e. so that the comoving
momenta is cut off by the initial Hubble radius as $k\geq
a_{\text{in}}H_{\text{in}}$, then for $x\equiv-k\eta\geq\Lambda_{\text{IR}}$
the cutoff is

$\Lambda_{\text{IR}}=\frac{a_{\text{in}}H_{\text{in}}}{aH(1-\epsilon_{H})}\ .$
(81)

With approximately constant $\epsilon_{H}$, the scale factor and Hubble
parameter are solved by

$a=a_{\text{in}}e^{N}\ ,\qquad H=H_{\text{in}}e^{-\epsilon_{H}N}\ ,\qquad
N\equiv\int_{t_{\text{in}}}^{t}\mathop{}\\!\mathrm{d}t^{\prime}H(t^{\prime})\
,$ (82)

where $N$ is the number of $e$-foldings from horizon exit to the end of
inflation. The assumption $\epsilon_{H}\simeq\text{const.}$ signifies that we
are working at leading order in slow-roll. Inserting the solutions of Eq. (82)
into Eq. (81), the IR limit can be written

$\Lambda_{\text{IR}}=\frac{1}{1-\epsilon_{H}}e^{-(1-\epsilon_{H})N}\ ,$ (83)

so that for $\Lambda_{\text{IR}}\ll 1$ its logarithm can be approximated to

$|\log\Lambda_{\text{IR}}|\simeq(1-\epsilon_{H})N\ .$ (84)

The role of IR-divergences and ways to deal with them are discussed for
instance in Tsamis:2005hd ; Seery:2010kh .

Now, when comparing the magnitude of the tensor and scalar corrections, we
must consider the minimal-coupling limit in which the tensor correlator of Eq.
(65) is given by

$C_{0}=\frac{1}{2\pi^{2}}\frac{H^{2}}{M_{\text{Pl}}^{2}}N_{\text{t}}\ ,$ (85)

where we have introduced a number $N_{\text{t}}$ related to $N$ as

$N_{\text{t}}\equiv-\frac{1}{2\epsilon_{H}}\left(1-e^{2\epsilon_{H}|\log\Lambda_{\text{IR}}|}\right)\simeq-\frac{1}{2\epsilon_{H}}\left(1-e^{2\epsilon_{H}N}\right)\
.$ (86)

Similarly for the scalars, we write

$D_{0}=\frac{\epsilon_{H}}{8\pi^{2}}\frac{H^{2}}{M_{\text{Pl}}^{2}}N_{\text{s}}\
,$ (87)

introducing the quantity $N_{\text{s}}$ according to

$N_{\text{s}}\equiv-\frac{1}{2(2\epsilon_{H}-\delta_{H})}\left(1-e^{2(2\epsilon_{H}-\delta_{H})|\log\Lambda_{\text{IR}}|}\right)\simeq-\frac{1}{2(2\epsilon_{H}-\delta_{H})}\left(1-e^{2(2\epsilon_{H}-\delta_{H})N}\right)\
.$ (88)

We can relate the combination of slow-roll parameters to the measured spectral
index (see below for the definition of the ”potential” slow-roll parameters
$\epsilon_{V}$, $\delta_{V}$) according to

$-0.035=n_{\text{s}}-1\equiv\tfrac{2}{3}\delta_{M}-6\epsilon_{H}=2\delta_{H}-4\epsilon_{H}\simeq
2\delta_{V}-6\epsilon_{V}\ .$ (89)

Using this in Eq. (88), we find for the range $N=50-60$ that
$N_{\text{s}}\simeq 136-205$. It is less straightforward to infer
$\epsilon_{H}$ from $n_{\text{s}}$. If $\delta_{H}\simeq
2\epsilon_{H}\simeq(n_{\text{s}}-1)/2$, then $N_{\text{t}}=33-37$ or $80-108$
depending on the sign of $\epsilon_{H}$. There are also models where
$\epsilon_{H}\ll\delta_{H}$ during inflation, in which case we may expand
$N_{\text{t}}$ so that $N_{\text{t}}\simeq N$. We conclude that
$N_{\text{s,t}}\simeq 100$, within a factor of a few. This will be sufficient
for our estimates.

###  4.1 Monomial models

The simplicity of monomial potentials makes them a natural first choice to
apply our calculations to. Hence, we consider a general power-law inflaton
potential of the form

$V(\phi)=\frac{\lambda_{p}}{p!}\frac{\phi^{p}}{M_{\text{Pl}}^{p-4}}\ ,$ (90)

where $\lambda_{p}$ is a dimensionless scalar self-coupling. The corresponding
slow-roll parameters are

$\epsilon_{V}=\frac{1}{2}\left(\frac{pM_{\text{Pl}}}{\phi}\right)^{2}\
,\qquad\delta_{V}=p(p-1)\left(\frac{M_{\text{Pl}}}{\phi}\right)^{2}\ ,$ (91)

The requirement for slow-roll inflation is $\epsilon_{H}\simeq\epsilon_{V}<1$,
which determines the end of inflation to be $\phi/M_{\text{Pl}}=p/\sqrt{2}$.
We estimate the magnitude of the correlators at horizon exit for which

$\left(\frac{\phi_{*}}{M_{\text{p}}}\right)^{2}=2pN\left(1+\frac{p}{4N}\right)\
.$ (92)

The number of $e$-foldings between horizon crossing and the end of inflation
is given by the scale of inflation and the thermal history of the Universe
after inflation. As indicated, we will simply assume that $N$ is in the
interval $50-60$.

The spectral index is given by

$n_{\text{s}}-1=2\delta_{H}-4\epsilon_{H}\simeq
2\delta_{V}-6\epsilon_{V}=-\frac{p+2}{2N\left(1+\frac{p}{4N}\right)}.$ (93)

For $p=2$ and $N=57$ we get the observed value of $n_{\text{s}}\simeq 0.965$
Akrami:2018odb . This determines $\epsilon_{H}\simeq 0.00875$ and
$N_{\text{t}}\simeq 96$, $N_{\text{s}}\simeq 175$. The tensor correlator in
Eq. (85) is then of the order

$C_{0}\simeq 4.8\frac{H^{2}}{M_{\text{Pl}}^{2}}\;.$ (94)

Similarly for the scalars, we use the relation of Eq. (89) in Eq. (87), to
obtain

$D_{0}\simeq 2.2\frac{\epsilon_{H}H^{2}}{M_{\text{Pl}}^{2}}\ .$ (95)

For $p=4$, we need $N=85$ to get the correct spectral index. This is ruled out
by observations, but for illustration, this then gives $\epsilon_{H}\simeq
0.012$, $N_{\text{t}}\simeq 259$, $N_{\text{s}}\simeq 507$ and

$C_{0}\simeq 13.1\frac{H^{2}}{M_{\text{Pl}}^{2}}\ ,\qquad D_{0}\simeq
6.4\frac{\epsilon_{H}H^{2}}{M_{\text{Pl}}^{2}}\ .$ (96)

In the mean-field equation (80), we have

$\displaystyle 0$
$\displaystyle=3H\dot{\phi}\left(1+C_{0}\right)+V_{,\phi}(1-C_{0})+M_{\text{Pl}}^{2}V_{,\phi\phi\phi}\frac{D_{0}}{\epsilon_{H}}\
.$ (97)

With $p=2$ then $V_{,\phi\phi\phi}=0$, and so scalar fluctuations give no
corrections at leading order. The tensor modes do, although they are
numerically very small. For $p=4$ then $V_{,\phi\phi\phi}=\Lambda_{4}\phi$, so
both scalars and tensors contribute, and we may write

$\frac{V_{,\phi}C_{0}}{M_{\text{Pl}}^{2}V_{,\phi\phi\phi}\frac{D_{0}}{\epsilon_{H}}}=\frac{2}{3}\frac{\phi^{2}_{*}}{M_{\rm
Pl}^{2}}\frac{N_{\text{t}}}{N_{\text{s}}}\ ,$ (98)

which in this case is $\sim\mathcal{O}(10^{2})$. Hence, the tensor
contributions dominate.

But of course, the overall magnitude of the corrections is controlled by the
quantity $H^{2}/M^{2}_{\rm Pl}$, which is related to the overall amplitude of
the CMB (scalar) spectrum and constrained to be strictly smaller than
$(2.5\times 10^{-5})^{2}$ Akrami:2018odb . Since $N_{\text{s}}$ is fixed by
$n_{\rm s}-1$ and $N$, large corrections can only arise if $N_{\text{t}}$
would become large in a small-$\epsilon_{H}$ regime. In order for $C_{0}$ to
be comparable to unity, then $N_{\text{t}}\simeq N$ must be of order
$10^{10}$.

| $\phi^{2}$ | $\phi^{4}$ | $V_{0}(1-\lambda_{4}\phi^{4})|_{N=50}$ | $V_{0}(1-\lambda_{4}\phi^{4})|_{N=60}$
---|---|---|---|---
$C_{0}$ [${H^{2}}/{M_{\text{Pl}}^{2}}$] | 4.8 | 13.1 | 2.8 | 3.1
$D_{0}$ [${H^{2}}/{M_{\text{Pl}}^{2}}$] | $2.2\epsilon_{H}$ | $6.4\epsilon_{H}$ | $1.7\epsilon_{H}$ | $2.6\epsilon_{H}$
$\Delta\rho$ [${H^{4}}/{M_{\text{Pl}}^{2}}$] | $-0.26$ | $-0.86$ | $-0.12$ | $-0.15$
$\Delta p$ [${H^{4}}/{M_{\text{Pl}}^{2}}$] | $-0.025$ | $-0.016$ | $-0.10$ | $-0.14$
Table 1: Estimates of the tensor and scalar correlators ($C_{0}$ resp.
$D_{0}$) as well as the Friedmann corrections in Eq. (99) for two monomial
potentials and the quartic hilltop potential of Eq. (100). For the quartic
hilltop potential the parameter $\lambda_{4}$ is estimated given a selected
value of the number of $e$-foldings $N$.

The corrections to the Friedmann equations become (taking $N_{\text{t,s}}$ to
be constant)

$\displaystyle\Delta\rho\equiv
H^{2}\left[\left(-9\epsilon_{H}+3\delta_{H}\right)\frac{N_{\text{s}}}{8\pi^{2}}-\frac{19}{8}\epsilon_{H}\frac{N_{\text{t}}}{\pi^{2}}\right]\frac{H^{2}}{M_{\rm
Pl}^{2}}\ ,$ (99) $\displaystyle\Delta p\equiv
H^{2}\left[\left(-9\epsilon_{H}+3\delta_{H}\right)\frac{N_{\text{s}}}{8\pi^{2}}+\frac{3}{8}\epsilon_{H}\frac{N_{\text{t}}}{\pi^{2}}\right]\frac{H^{2}}{M_{\rm
Pl}^{2}}\ ,$

for Eq. (75) and (76) respectively. Estimates of these corrections are
summarised in Table 1 for the potentials considered in this section. For both
of the monomial potentials the corrections are of order
$\Delta\rho\sim\mathcal{O}(10^{-1}{H^{2}}/{M_{\text{Pl}}^{2}})$ and $\Delta
p\sim\mathcal{O}(10^{-2}{H^{2}}/{M_{\text{Pl}}^{2}})$, which are again
completely negligible.

###  4.2 Quartic hilltop

A general hilltop model Boubekeur:2005zm  of the inflaton potential can be
written

$V(\phi)=V_{0}\left[1-\lambda_{p}\left(\frac{\phi}{M_{\text{Pl}}}\right)^{p}+\dots\right]\
,\quad p\geq 2\ ,$ (100)

where $V_{0}$ is a constant energy density scale, $\lambda_{p}$ is some
parameter, $p$ is often an integer, and the dots indicate that some higher
power-law terms must kick in at larger values of $\phi$, for the potential to
be bounded from below. We will take $p=4$.

The quantum contributions for this model can be estimated in much the same way
as before. In fact, because $N_{\text{s}}$ is a function of the slow-roll
parameters in the combination $2\delta_{H}-4\epsilon_{H}=n_{\text{s}}-1$, if
we insist that the spectral index is the observed one, then $N_{\text{s}}$ is
independent of the inflation model.

Since the potential in Eq. (100) is a constant plus a monomial, the
expressions in Eqs. (98) and (99) will be left unchanged, and to find
$N_{\text{t}}$ we only need to compute $\epsilon_{H}\simeq\epsilon_{V}$ at
horizon crossing (since $\delta_{H}$ follows from $n_{\text{s}}-1$ and
$\epsilon_{H})$.

With $p=4$, the potential slow-roll parameters are

$\epsilon_{V}(\phi)=\frac{8\lambda_{4}^{2}\left(\frac{\phi}{M_{\rm
Pl}}\right)^{6}}{\left(1-\lambda_{4}\left(\frac{\phi}{M_{\rm
Pl}}\right)^{4}\right)^{2}}\
,\qquad\delta_{V}(\phi)=-\frac{12\lambda_{4}\left(\frac{\phi}{M_{\rm
Pl}}\right)^{2}}{1-\lambda_{4}\left(\frac{\phi}{M_{\rm Pl}}\right)^{4}}\ ,$
(101)

and we define the end of inflation by $\epsilon_{V}(\phi_{\text{e}})\equiv 1$.
From the definition of $N$ we then find

$\left(\frac{\phi_{*}}{M_{\rm
Pl}}\right)^{2}+\frac{1}{\lambda_{4}}\left(\frac{\phi_{*}}{M_{\rm
Pl}}\right)^{-2}=8N+\left(\frac{\phi_{\text{e}}}{M_{\rm
Pl}}\right)^{2}+\frac{1}{\lambda_{4}}\left(\frac{\phi_{\text{e}}}{M_{\rm
Pl}}\right)^{-2},$ (102)

from which we compute $\epsilon_{V}(\phi_{*})$ and $\delta_{V}(\phi_{*})$. For
$N=50$ we find the values $\lambda_{4}=4.3\times 10^{-6}$,
$(\phi_{*})^{2}=200M_{\rm Pl}^{2}$, $N_{\text{t}}=55$ and $N_{\text{s}}=136$.
With $N=60$ then we find $\lambda_{4}=3.2\times 10^{-5}$,
$(\phi_{*})^{2}=40M_{\rm Pl}^{2}$, $N_{\text{t}}=62$ and $N_{\text{s}}=205$.
The fraction of Eq. (98) then becomes about $50$ for $N=50$ and $8$ for
$N=60$. The Friedmann corrections of are of order $\Delta\rho,\Delta
p\sim\mathcal{O}(10^{-1}{H^{2}}/{M_{\text{Pl}}^{2}})$ as listed in Table 1.

##  5 Conclusions and outlook

We have computed the leading-order (in slow-roll and quantum fluctuations)
corrections to the inflaton equation of motion and the Friedmann equations
during inflation. We have included the fluctuations in the inflaton field, and
both scalar and tensor fluctuations in the metric. Starting out including a
non-minimal coupling to gravity $F(\phi)$, we solved the mode equation for the
tensor modes, and were able to derive expressions for the scalar mode
equations and the corrected Friedmann and field equations. We then proceeded
in the limit $F(\phi)=1$ to explicitly compute the leading-order corrections
to the evolution equations.

We found that both the tensor and scalar correlators enter at leading order
for a self-interacting inflaton, the scalars as an additive contribution, the
tensors as multiplicative contributions. The tensors contribute also for a
quadratic inflaton potential, where the scalar contribution decouples. While
the tensor contribution is larger by one or two orders of magnitude, the
overall size of the corrections is very small.

The correlators are logarithmically IR-divergent, and we introduced an ad hoc
IR-cutoff in order to estimate their magnitude. Infrared divergences in
Minkowski space are well-known from finite-temperature and finite-density
calculations. They are more severe in the former case due to phase-space
effects (dimensional reduction), but in both cases they are unphysical in the
sense that the real physical medium effects are screening them. This is
resolved by resummation of classes of certain diagrams to all orders in
perturbation theory, for example by using the resummation program of braaten
or the 2PI effective action formalism 2pi . A mass can also be generated non-
perturbatively in curved spacetime Serreau:2011fu ; Herranen:2013raa  that
screens infrared divergences.555The mass is non-perturbative in the sense that
it is not seen at any finite order in perturbation theory, but shows up after
summing classes of diagrams to all orders. For example, in de Sitter space,
the generated mass for a classical massless scalar field is of order $H$. In
Minkowski space, there are quantum fluctuations at all scales, however, in de
Sitter space, the scale is set by $H$ and this is the scale of the important
quantum fluctuations.

Since quantum corrections during inflation seems to anyway be suppressed by
$H^{2}/M_{\rm Pl}^{2}$, computing them with high accuracy may seem futile.
However, quantum and thermal corrections are known to be important for a
number of phenomena in cosmology, with examples including infrared effects in
de Sitter space Serreau:2011fu ; Gautier:2013aoa ; Gautier:2015pca ,
corrections to the effective potential of the Standard Model Herranen:2014cua
; Herranen:2015ima  and thermalisation Tranberg:2008ae .

Although significantly more challenging than field theory in a classical FLRW
background, exploring metric corrections to the field equations is part of
that story.

Much work remains to be done on resummation of the IR-divergences, including
metric fluctuations and away from de Sitter space, as well as the further
inclusion of non-minimal coupling to gravity and other more general theories
of gravity. Also, although in this work we have insisted that $\phi$ is the
inflaton, quantum corrections arise for spectator fields as well (curvaton,
other matter fields), due the background metric and its fluctuations. Although
the Friedmann equations remain dominated by the inflaton, each subdominant
spectator field acquires its own effective equation of motion. This opens up a
number of new model-building opportunities to explore.

##  Appendix A Scalar commutation relations

For completeness, we list a number of commutation relations in addition to
those given in Eq. (53). They are

$\displaystyle=i3a^{2}\left(\phi^{\prime}+\frac{2(3\mathcal{H}^{\prime}\phi^{\prime}+a^{2}\mathcal{H}V_{,\phi})}{\nabla^{2}}\right)\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$ (103)
$\displaystyle[\Psi(\mathbf{x}),\pi_{\delta\phi}(\mathbf{x^{\prime}})]$
$\displaystyle=-i\frac{3\mathcal{H}\phi^{\prime}+a^{2}V_{,\phi}}{2M_{\text{Pl}}^{2}\nabla^{2}}\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$
$\displaystyle[\delta\phi(\mathbf{x}),\pi_{\delta\phi}(\mathbf{x^{\prime}})]$
$\displaystyle=i\left(1+\frac{3\phi^{\prime
2}}{M_{\text{Pl}}^{2}\nabla^{2}}\right)\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$ $\displaystyle[\delta\phi(\mathbf{x}),\pi_{\Psi}(\mathbf{x^{\prime}})]$
$\displaystyle=i\frac{6\mathcal{H}\phi^{\prime}}{\nabla^{2}}\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$ $\displaystyle[\Psi(\mathbf{x}),\pi_{\Psi}(\mathbf{x^{\prime}})]$
$\displaystyle=-i\frac{3\phi^{\prime
2}}{2M_{\text{Pl}}^{2}\nabla^{2}}\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$ $\displaystyle[\Psi(\mathbf{x}),\delta\phi(\mathbf{x^{\prime}})]$
$\displaystyle=i\frac{\phi^{\prime}}{2M_{\text{Pl}}^{2}a^{2}\nabla^{2}}\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$
$\displaystyle[\delta\phi(\mathbf{x}),\delta\phi^{\prime}(\mathbf{x^{\prime}})]$
$\displaystyle=\frac{i}{a^{2}}\left(1+\frac{3\phi^{\prime
2}}{2M_{\text{Pl}}^{2}\nabla^{2}}\right)\delta^{3}(\mathbf{x}-\mathbf{x^{\prime}})\
,$

where it is implicit that they apply at equal times.

##  Appendix B Second-order equations with general non-minimal coupling

We present the Friedmann and mean-field equations with a general non-minimal
coupling to quadratic order in perturbations to illustrate the (rather non-
trivial) dependence on $F$. A central ingredient to these equations is the
perturbed inverse metric, which can be found from a Taylor expansion. To
quadratic order in perturbations, we find

$\displaystyle(g+\delta g)^{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}-\delta g^{\mu\nu}+\delta g^{\mu\lambda}\delta
g_{\lambda}^{\,\,\,\nu}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}1+2\Phi-4\Phi^{2}&&0\\\
0&&\frac{1}{a^{2}}\left[\delta^{ij}+2(\Psi\delta^{ij}-E^{ij})+4\Psi^{2}\delta^{ij}+4E^{ik}E_{\,\,\,k}^{j}-8\Psi
E^{ij}\right]\end{pmatrix}\;,$

where indices on the perturbed metric are raised as $\delta
g^{\mu\nu}=-g^{\alpha\mu}g^{\nu\beta}\delta g_{\mu\nu}$.

Similarly, using $\det g=\exp(\operatorname{tr}(\ln g))$ and Taylor expanding,
the perturbed metric determinant is given by

$\displaystyle\sqrt{-(g+\delta g)}$ $\displaystyle=$
$\displaystyle\sqrt{-g}\left(1+\tfrac{1}{2}\delta
g^{\mu}{}_{\mu}+\tfrac{1}{8}\delta g^{\mu}{}_{\mu}\delta
g^{\nu}{}_{\nu}-\tfrac{1}{4}\delta g^{\mu\nu}\delta g_{\nu\mu}\right)$ (105)
$\displaystyle=$ $\displaystyle
a^{3}\big{(}1+\Phi-3\Psi+E^{i}{}_{i}-\tfrac{1}{2}\Phi^{2}-3\Phi\Psi+\Phi
E^{i}{}_{j}$ $\displaystyle+\tfrac{3}{2}\Psi^{2}-\Psi
E^{i}{}_{i}+\tfrac{1}{2}E^{i}{}_{i}E^{j}{}_{j}-E^{ij}E_{ji}\big{)}\ ,$

where $\delta g^{\mu}{}_{\mu}=g^{\mu\nu}\delta g_{\nu\mu}$. Eqs. (B) and (105)
are then inserted into the expressions for the energy-momentum tensor in Eq.
(5) and mean-field equation (7), whose resulting expressions are truncated at
quadratic order.

With a general $F$, the perturbed Einstein field equations to quadratic order
read

$(F+\delta F+\delta_{2}F)(G_{\mu\nu}+\delta
G_{\mu\nu}+\delta_{2}G_{\mu\nu})=\frac{1}{M_{\text{Pl}}^{2}}(\tilde{T}_{\mu\nu}+\delta\tilde{T}_{\mu\nu}+\delta_{2}\tilde{T}_{\mu\nu})\
,$ (106)

from which the quantum-corrected equations take the form

$G_{\mu\nu}=\frac{1}{M_{\text{Pl}}^{2}}\frac{1}{F}\left(\tilde{T}_{\mu\nu}+\braket{\delta_{2}\tilde{T}_{\mu\nu}}\right)-\braket{\delta_{2}G_{\mu\nu}}-\left\langle\frac{\delta
F}{F}\delta
G_{\mu\nu}\right\rangle-\left\langle\frac{\delta_{2}F}{F}\right\rangle
G_{\mu\nu}\ .$ (107)

Specialising to the FLRW metric in Newtonian gauge, at linear order, the
perturbations to the Einstein tensor read

$\displaystyle{\delta G^{0}{}_{0}}$
$\displaystyle=6H^{2}\Phi+6H\dot{\Psi}-\frac{2}{a^{2}}\nabla^{2}\Psi\ ,$ (108)
$\displaystyle{\delta G^{i}{}_{j}}$
$\displaystyle=\left[6H^{2}\Phi+4\dot{H}\Phi+2H\dot{\Phi}+6H\dot{\Psi}+2\ddot{\Psi}+\frac{1}{a^{2}}\nabla^{2}(\Phi-\Psi)\right]\delta^{i}{}_{j}$
$\displaystyle\quad+\frac{1}{a^{2}}\partial_{j}\partial^{i}(\Psi-\Phi)+\ddot{E}^{i}{}_{j}+3H\dot{E}^{i}{}_{j}-\frac{1}{a^{2}}\nabla^{2}E^{i}{}_{j}\
,$

and the quadratic-order contributions are given by

$\displaystyle{\delta_{2}G^{0}{}_{0}}$
$\displaystyle=\tfrac{1}{2}\dot{E}_{ij}\dot{E}^{ij}+4HE^{ij}\dot{E}_{ij}-12H{}^{2}\Phi{}^{2}-12H\Phi\dot{\Psi}+12H\Psi\dot{\Psi}-3\dot{\Psi}^{2}$
(109)
$\displaystyle\quad+\frac{1}{a^{2}}\big{[}-8\Psi\nabla^{2}\Psi-3(\nabla\Psi)^{2}+2E^{ij}\partial_{j}\partial_{i}\Psi-2E^{ij}\nabla^{2}E{}_{ij}+\partial_{j}E{}_{ik}\partial^{k}E^{ij}-\tfrac{3}{2}\partial_{k}E{}_{ij}\partial^{k}E^{ij}\big{]}\
,$

respectively

$\displaystyle{\delta_{2}G^{i}{}_{j}}$
$\displaystyle=-2\dot{E}^{ik}\dot{E}_{jk}-2E^{ik}\ddot{E}{}_{jk}+\tfrac{3}{2}\dot{E}_{kl}\dot{E}^{kl}\delta^{i}{}_{j}+2E^{kl}\ddot{E}{}_{kl}\delta^{i}{}_{j}-6HE^{ik}\dot{E}_{jk}+6HE^{kl}\dot{E}_{kl}\delta^{i}{}_{j}$
(110)
$\displaystyle\quad-2\ddot{E}^{i}{}_{j}\Phi-6H\dot{E}^{i}{}_{j}\Phi-12\delta^{i}{}_{j}H{}^{2}\Phi{}^{2}-8\delta^{i}{}_{j}\dot{H}\Phi{}^{2}-\dot{E}^{i}{}_{j}\dot{\Phi}-8\delta^{i}{}_{j}H\Phi\dot{\Phi}+2\ddot{E}^{i}{}_{j}\Psi$
$\displaystyle\quad+6H\dot{E}^{i}{}_{j}\Psi+\dot{E}^{i}{}_{j}\dot{\Psi}+6HE^{i}{}_{j}\dot{\Psi}-12\delta^{i}{}_{j}H\Phi\dot{\Psi}-2\delta^{i}{}_{j}\dot{\Phi}\dot{\Psi}+12\delta^{i}{}_{j}H\Psi\dot{\Psi}+\delta^{i}{}_{j}\dot{\Psi}^{2}$
$\displaystyle\quad+2E^{i}{}_{j}\ddot{\Psi}-4\delta^{i}{}_{j}\Phi\ddot{\Psi}+4\delta^{i}{}_{j}\Psi\ddot{\Psi}+\frac{1}{a^{2}}\partial^{i}E^{kl}\partial_{j}E{}_{kl}+\frac{1}{a^{2}}\partial^{i}\Phi\partial_{j}\Phi-\frac{1}{a^{2}}\partial^{i}\Psi\partial_{j}\Phi-\frac{1}{a^{2}}\partial^{i}\Phi\partial_{j}\Psi$
$\displaystyle\quad+\frac{3}{a^{2}}\partial^{i}\Psi\partial_{j}\Psi+\frac{2}{a^{2}}E^{kl}\partial_{j}\partial^{i}E{}_{kl}+\frac{2}{a^{2}}\Phi\partial_{j}\partial^{i}\Phi-\frac{2}{a^{2}}\Psi\partial_{j}\partial^{i}\Phi+\frac{4}{a^{2}}\Psi\partial_{j}\partial^{i}\Psi+\frac{2}{a^{2}}E{}_{j}{}^{k}\partial_{k}\partial^{i}\Psi$
$\displaystyle\quad+\frac{2}{a^{2}}E^{ik}\partial_{k}\partial_{j}\Phi-\frac{4}{a^{2}}\Psi\nabla^{2}E^{i}{}_{j}-\frac{2}{a^{2}}\delta^{i}{}_{j}\Phi\nabla^{2}\Phi+\frac{2}{a^{2}}\delta^{i}{}_{j}\Psi\nabla^{2}\Phi-\frac{2}{a^{2}}E^{i}{}_{j}\nabla^{2}\Psi$
$\displaystyle\quad-\frac{4}{a^{2}}\delta^{i}{}_{j}\Psi\nabla^{2}\Psi+\frac{1}{a^{2}}\partial^{i}E{}_{jk}\partial^{k}\Phi+\frac{1}{a^{2}}\partial_{j}E^{i}{}_{k}\partial^{k}\Phi-\frac{1}{a^{2}}\partial_{k}E^{i}{}_{j}\partial^{k}\Phi-\frac{1}{a^{2}}\delta^{i}{}_{j}\partial_{k}\Phi\partial^{k}\Phi+\frac{1}{a^{2}}\partial^{i}E{}_{jk}\partial^{k}\Psi$
$\displaystyle\quad+\frac{1}{a^{2}}\partial_{j}E^{i}{}_{k}\partial^{k}\Psi-\frac{3}{a^{2}}\partial_{k}E^{i}{}_{j}\partial^{k}\Psi-\frac{2}{a^{2}}\delta^{i}{}_{j}\partial_{k}\Psi\partial^{k}\Psi-\frac{2}{a^{2}}E^{kl}\partial_{l}\partial^{i}E{}_{jk}-\frac{2}{a^{2}}E^{kl}\partial_{l}\partial_{j}E^{i}{}_{k}$
$\displaystyle\quad+\frac{2}{a^{2}}E^{kl}\partial_{l}\partial_{k}E^{i}{}_{j}-\frac{2}{a^{2}}E^{kl}\delta^{i}{}_{j}\partial_{l}\partial_{k}\Phi+\frac{2}{a^{2}}E^{ik}\nabla^{2}E{}_{jk}-\frac{2}{a^{2}}\partial_{k}E{}_{jl}\partial^{l}E^{ik}$
$\displaystyle\quad+\frac{2}{a^{2}}\partial_{l}E{}_{jk}\partial^{l}E^{ik}-\frac{2}{a^{2}}E^{kl}\delta^{i}{}_{j}\nabla^{2}E{}_{kl}+\frac{1}{a^{2}}\delta^{i}{}_{j}\partial_{l}E{}_{km}\partial^{m}E^{kl}-\frac{3}{2a^{2}}\delta^{i}{}_{j}\partial_{m}E{}_{kl}\partial^{m}E^{kl}\
.$

We keep in mind that since the tensor and scalar degrees of freedom propagate
separately, the two-point correlators that mix these, e.g. $\braket{\delta\phi
E_{ij}}$, vanish. Turning to the energy-momentum tensor, we determine the
00-component to

$\displaystyle\delta_{2}\tilde{T}^{0}{}_{0}$
$\displaystyle=-\tfrac{1}{2}\dot{\delta\phi}^{2}-\tfrac{1}{2a^{2}}(\nabla\delta\phi)^{2}+2\dot{\phi}\Phi\dot{\delta\phi}-2\dot{\phi}^{2}\Phi^{2}-\tfrac{1}{2}V_{,\phi\phi}\delta\phi^{2}$
(111)
$\displaystyle\quad+3H\dot{\delta_{2}F}-\frac{1}{a^{2}}\nabla^{2}\delta_{2}F-3\dot{\delta
F}(2H\Phi+\dot{\Psi})$
$\displaystyle\quad-\frac{1}{a^{2}}\left[\nabla(2\Phi-\Psi)\nabla\delta
F+2\Psi\nabla^{2}\delta F-2\partial_{i}(E^{ij}\partial_{j}\delta F)\right]$
$\displaystyle\quad+2\dot{F}\big{[}6H\Phi^{2}+3(\Phi-\Psi)\dot{\Psi}-\dot{E}^{i}{}_{j}E^{j}{}_{i}\big{]}\
,$

and for the $ij$-component, we get

$\displaystyle\delta_{2}\tilde{T}^{i}{}_{j}$
$\displaystyle=\frac{1}{a^{2}}\delta\phi^{,i}\delta\phi_{,j}+\big{[}\tfrac{1}{2}\dot{\delta\phi}^{2}-\tfrac{1}{2a^{2}}(\nabla\delta\phi)^{2}-2\dot{\phi}\Phi\dot{\delta\phi}+2\dot{\phi}^{2}\Phi^{2}-\tfrac{1}{2}V_{,\phi\phi}\delta\phi^{2}\big{]}\delta^{i}{}_{j}$
(112)
$\displaystyle\quad+\frac{1}{a^{2}}\partial^{i}\partial_{j}\delta_{2}F+\big{[}\ddot{\delta_{2}F}+2H\dot{\delta_{2}F}-\frac{1}{a^{2}}\nabla^{2}\delta_{2}F\big{]}\delta^{i}{}_{j}$
$\displaystyle\quad-2\ddot{\delta F}\Phi-\dot{\delta
F}[(4H\Phi+\dot{\Phi}+\dot{\Psi})\delta^{i}{}_{j}+2\dot{E}^{i}{}_{j}]$
$\displaystyle\quad+\frac{1}{a^{2}}[\partial^{i}(-\Psi\delta^{l}{}_{j}+E^{l}{}_{j})+\partial_{j}(-\Psi\delta^{li}+E^{li})-\partial^{l}(-\Psi\delta_{j}{}^{i}+E_{j}{}^{i})]\partial_{l}\delta
F$
$\displaystyle\quad+\frac{2}{a^{2}}(\Psi\delta^{ik}-E^{ik})\partial_{k}\partial_{j}\delta
F-\frac{1}{a^{2}}\left[\nabla(\Phi-\Psi)\nabla\delta F+2\Psi\nabla^{2}\delta
F-2\partial_{k}(E^{kl}\partial_{l}\delta F)\right]\delta^{i}{}_{j}$
$\displaystyle\quad+4\ddot{F}\Phi^{2}\delta^{i}{}_{j}+\dot{F}\big{(}8H\Phi^{2}+8\Phi\dot{\Psi}-2\Psi\dot{\Psi}+4\Phi\dot{\Phi}-2\dot{E}^{k}{}_{l}E^{l}{}_{k}\big{)}\delta^{i}{}_{j}$
$\displaystyle\quad+2\dot{F}\big{[}(\Phi-2\Psi)\dot{E}^{i}{}_{j}-2\dot{\Psi}E^{i}{}_{j}+2E^{ik}\dot{E}_{jk}\big{]}\
.$

Finally, for the mean-field equation, we obtain

$\displaystyle 0$ $\displaystyle=\ddot{\phi}+3H\dot{\phi}+V_{,\phi}$ (113)
$\displaystyle\quad+\ddot{\phi}\big{[}\tfrac{3}{2}(\Phi+\Psi)^{2}+E^{i}{}_{j}E^{j}{}_{i}\big{]}+\dot{\phi}\big{[}3H\big{(}\tfrac{3}{2}(\Phi+\Psi)^{2}+E^{i}{}_{j}E^{j}{}_{i}\big{)}+3(\Phi+\Psi)(\dot{\Phi}+\dot{\Psi})+2\dot{E}^{i}{}_{j}E^{j}{}_{i}\big{]}$
$\displaystyle\quad-\ddot{\delta\phi}(\Phi+3\Psi)-\dot{\delta\phi}[3H(\Phi+3\Psi)+\dot{\Phi}+3\dot{\Psi}]-\frac{1}{a^{2}}\big{(}\nabla[(\Phi-\Psi)\nabla\delta\phi]-2\partial_{i}[E^{ij}\partial_{j}\delta\phi]\big{)}$
$\displaystyle\quad+\big{(}-\tfrac{1}{2}\Phi^{2}-3\Phi\Psi+\tfrac{3}{2}\Psi^{2}-E^{i}{}_{j}E^{j}{}_{i}\big{)}V_{,\phi}+(\Phi-3\Psi)V_{,\phi\phi}\delta\phi+\tfrac{1}{2}V_{,\phi\phi\phi}\delta\phi^{2}$
$\displaystyle\quad-\tfrac{1}{2}\left[\big{(}-\tfrac{1}{2}\Phi^{2}-3\Phi\Psi+\tfrac{3}{2}\Psi^{2}-E^{i}{}_{j}E^{j}{}_{i}\big{)}F_{,\phi}+(\Phi-3\Psi)\delta
F_{,\phi}+\delta_{2}F_{,\phi}\right]R$
$\displaystyle\quad-\tfrac{1}{2}\left[(\Phi-3\Psi)F_{,\phi}+\delta
F_{,\phi}\right]\delta R-\tfrac{1}{2}F_{,\phi}\delta_{2}R\ ,$

where $\delta R$ is given in Eq. (24) and the second-order perturbation to the
Ricci scalar is given by

$\displaystyle\delta_{2}R$ $\displaystyle=$
$\displaystyle-3\dot{E}_{ij}\dot{E}^{ij}-4E_{ij}\ddot{E}^{ij}-16HE_{ij}\dot{E}^{ij}+\frac{1}{a^{2}}\left(4E_{ij}\nabla^{2}E^{ij}-2\partial_{j}E_{ik}\partial^{k}E^{ij}+3\partial_{k}E_{ij}\partial^{k}E^{ij}\right)$
(114)
$\displaystyle+24(\dot{H}+2H^{2})\Phi^{2}+24H\Phi\dot{\Phi}+48H(\Phi-\Psi)\dot{\Psi}+6\dot{\Phi}\dot{\Psi}+12(\Phi-\Psi)\ddot{\Psi}$
$\displaystyle+\frac{2}{a^{2}}\left[2(\Phi-\Psi)\nabla^{2}\Phi+8\Psi\nabla^{2}\Psi+(\nabla\Phi)^{2}+3(\nabla\Psi)^{2}+\partial_{i}\Psi\partial^{i}\Phi\right]\;.$

##  Appendix C Correlator relations

The various quadratic-order terms in the Friedmann equations can be related to
the two-point functions of the quantised fields via correlator relations.
These relations can be obtained by differentiation and by use of the fields’
equations of motion. The expectation values at quadratic order are defined to
be symmetric. For the tensors we have that

$\displaystyle\braket{\dot{E}_{ij}E^{ij}}$
$\displaystyle\equiv\tfrac{1}{2}\braket{\dot{E}_{ij}E^{ij}+{E}_{ij}\dot{E}^{ij}}=\tfrac{1}{2}\partial_{t}\braket{{E}_{ij}E^{ij}}\
,$ (115) $\displaystyle\braket{\dot{E}_{ij}\dot{E}^{ij}}$
$\displaystyle=\tfrac{1}{2}\partial_{t}^{2}\braket{{E}_{ij}E^{ij}}-\tfrac{1}{2}\braket{\ddot{E}_{ij}{E}^{ij}+{E}_{ij}\ddot{E}^{ij}}\
,$ (116) $\displaystyle\braket{\ddot{E}_{ij}E^{ij}}$
$\displaystyle=-H(\tfrac{3}{2}+\epsilon_{F})\partial_{t}\braket{{E}_{ij}E^{ij}}+\frac{1}{a^{2}}\braket{E_{ij}\nabla^{2}E^{ij}}\
,$ (117)

where the last relation is obtained from the tensors equation of motion (25).
The scalar degrees of freedom are constrained via the field equations (16) and
(18), that in the Einstein limit can be expressed in terms of the single field
$\Psi$. The correlator relations of $\Psi$ are then obtained as

$\displaystyle\braket{\dot{\Psi}\Psi}$
$\displaystyle=\tfrac{1}{2}\partial_{t}D_{0}\ ,$ (118)
$\displaystyle\braket{\ddot{\Psi}\Psi}$
$\displaystyle=-\tfrac{1}{2}H(1+2\delta_{H})\partial_{t}D_{0}-2H^{2}(\delta_{H}-\epsilon_{H})D_{0}+\frac{1}{a^{2}}D_{2}\
,$ $\displaystyle\braket{\dot{\Psi}^{2}}$
$\displaystyle=\tfrac{1}{2}\partial_{t}^{2}D_{0}-\braket{\ddot{\Psi}\Psi}\ ,$
$\displaystyle\braket{\ddot{\Psi}\dot{\Psi}}$
$\displaystyle=-H(1+2\delta_{H})\braket{\dot{\Psi}^{2}}-H^{2}(\delta_{H}-\epsilon_{H})\partial_{t}D_{0}+\frac{1}{2a^{2}}\partial_{t}D_{2}\
,$ $\displaystyle\braket{\dddot{\Psi}\Psi}$
$\displaystyle=-\tfrac{1}{2}\left[\partial_{t}[H(1+2\delta_{H})]+2H^{2}(\delta_{H}-\epsilon_{H})-H^{2}(1+2\delta_{H})^{2}\right]\partial_{t}D_{0}$
$\displaystyle\quad+\left[2H^{3}(1+2\delta_{H})(\delta_{H}-\epsilon_{H})-2\partial_{t}[H^{2}(\delta_{H}-\epsilon_{H})]\right]D_{0}+\frac{1}{2a^{2}}\partial_{t}D_{2}-H(3+2\delta_{H})\frac{1}{a^{2}}D_{2}\
,$ $\displaystyle\braket{\ddot{\Psi}^{2}}$
$\displaystyle=H^{2}(1+2\delta_{H})^{2}\braket{\dot{\Psi}^{2}}+2H^{3}(1+2\delta_{H})(\delta_{H}-\epsilon_{H})\partial_{t}D_{0}+4H^{4}(\delta_{H}-\epsilon_{H})^{2}D_{0}$
$\displaystyle\quad-H(1+2\delta_{H})\frac{1}{a^{2}}\partial_{t}D_{2}-4H^{2}(\delta_{H}-\epsilon_{H})\frac{1}{a^{2}}D_{2}+\frac{1}{a^{4}}D_{4}\
,$

using the equation of motion for the scalar metric perturbation in Eq. (40).
Again, all of the correlators should be understood to have equal time and
space arguments.

##  Appendix D Calculation of two-point correlators

From its mode decomposition in Eq. (30), the tensor correlator is given by

$C_{0}\equiv\braket{E_{ij}E^{ij}}=\int\frac{\mathop{}\\!\mathrm{d}^{3}q}{(2\pi)^{3/2}}\int\frac{\mathop{}\\!\mathrm{d}^{3}p}{(2\pi)^{3/2}}\sum_{\lambda}\sum_{\lambda^{\prime}}e_{ij}^{(\lambda)}e^{(\lambda^{\prime})*ij}\langle\hat{a}_{\lambda\mathbf{q}}\hat{a}^{\dagger}_{\lambda^{\prime}\mathbf{p}}\rangle\tilde{h}_{\lambda\mathbf{q}}\tilde{h}_{\lambda^{\prime}\mathbf{p}}\
,$ (119)

where $\tilde{h}_{\lambda\mathbf{q}}$ is given in Eq. (38). We have already
assumed that we are in the vacuum state, where the expectation value of the
number operators
$\hat{a}^{\dagger}_{\lambda\mathbf{q}}\hat{a}_{\lambda\mathbf{q}}$ vanish.
Then inserting

$[\hat{a}_{\lambda\mathbf{q}},\hat{a}^{\dagger}_{\lambda\mathbf{q}}]=\delta_{\lambda\lambda^{\prime}}\delta^{3}(\mathbf{q}-\mathbf{q^{\prime}}),$
(120)

we arrive at

$C_{0}\equiv\braket{E_{ij}E^{ij}}=\int\frac{\mathop{}\\!\mathrm{d}^{3}q}{(2\pi)^{3}}\sum_{\lambda}e_{ij}^{(\lambda)}e^{(\lambda)*ij}|\tilde{h}_{\lambda\mathbf{q}}|=\frac{4}{2\pi^{2}}\int\mathop{}\\!\mathrm{d}q\,q^{2}|\tilde{h}_{\lambda\mathbf{q}}|^{2}\;,$
(121)

In the last line, we have summed over $i,j$ and $\lambda$ (giving a factor of
4) and used that the mode functions $\tilde{h}_{\lambda\mathbf{q}}$ only
depend on the length of ${\bf q}$, and not on $\lambda$.

By introducing the new variable $x\equiv-q\eta$, as $\eta<0$ inside the
horizon, Eq. (119) can be written as

$C_{0}(\eta)=\frac{1}{4\pi}\frac{1}{M^{2}_{\rm
Pl}a^{2}F}(-\eta)^{-2}\int_{\Lambda_{\text{IR}}}^{\Lambda_{\text{UV}}}\mathop{}\\!\mathrm{d}xx^{2}|H_{\nu}^{(2)}(-x)|^{2}\
,$ (122)

where we have chosen a branch such that
$|H_{\nu}^{(1)}(x)|^{2}=|H_{\nu}^{(2)}(-x)|^{2}$ for $x>0$. We split the
remaining radial integration interval into three according to

$\int_{\Lambda_{\text{IR}}}^{\Lambda_{\text{UV}}}\mathop{}\\!\mathrm{d}x=\int_{\Lambda_{\text{IR}}}^{\kappa_{\text{IR}}}\mathop{}\\!\mathrm{d}x+\int_{\kappa_{\text{IR}}}^{\kappa_{\text{UV}}}\mathop{}\\!\mathrm{d}x+\int_{\kappa_{\text{UV}}}^{\Lambda_{\text{UV}}}\mathop{}\\!\mathrm{d}x\
,$ (123)

with $\Lambda_{\text{IR}}\ll\kappa_{\text{IR}}\ll
1\ll\kappa_{\text{UV}}\ll\Lambda_{\text{UV}}$ so that we can use different
approximations of the Hankel functions $H_{\nu}^{(1)}(x)$ appropriate for each
interval. In the low-momentum (IR) region, we use the asymptotic expansion

$H^{(1)}_{\nu,\text{IR}}(x)\equiv-\frac{i}{\pi}\left(\frac{2}{x}\right)^{\nu}\Gamma(\nu)+\dotso\
,$ (124)

and in the intermediate-momentum region, we may let $\epsilon_{H,F}\rightarrow
0$, so that the relevant Hankel function for the tensors reduces to

$H^{(1)}_{{\frac{3}{2}}}(x)=-\sqrt{\frac{2}{\pi x^{3}}}(i+x)e^{ix}\ .$ (125)

In the large-momentum (UV) region we use the large-$|x|$ expansion of the
Hankel function:

$H_{\nu,\text{UV}}^{(1)}(x)\equiv\frac{2}{\pi}\left[\frac{1}{x}-\frac{1-4\nu^{2}}{8x^{3}}\right]+\mathcal{O}(x^{-7/2})\;.$
(126)

Calculating the contributions from each momentum interval, the correlator in
Eq. (119) becomes

$\displaystyle C_{0}\equiv\braket{E_{ij}E^{ij}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\frac{H^{2}}{M_{\text{Pl}}^{2}F}\bigg{[}\left(\frac{1}{2(\epsilon_{F}+\epsilon_{H})}+\log
2+\psi(\tfrac{3}{2})\right)\left(-1+\Lambda_{\text{IR}}^{-2(\epsilon_{H}+\epsilon_{F})}\right)$
(127)
$\displaystyle+\tfrac{1}{2}\Lambda_{\text{UV}}^{2}+\log\Lambda_{\text{UV}}\bigg{]}+\mathcal{O}(\epsilon)\;,$

where we have switched back to cosmic time and $\psi(x)$ denotes the digamma
function. For the correlator involving two gradients, we have that

$\displaystyle C_{2}\equiv\braket{E_{ij}\nabla^{2}E^{ij}}$ $\displaystyle=$
$\displaystyle-4\int\frac{\mathop{}\\!\mathrm{d}^{3}q}{(2\pi)^{3}}q^{2}|\tilde{h}_{\lambda\mathbf{q}}|^{2}$
(128) $\displaystyle=$
$\displaystyle-\frac{1}{4\pi^{2}}\frac{a^{2}H^{4}}{M_{\text{Pl}}^{2}F}\left[\tfrac{1}{2}\Lambda_{\text{UV}}^{4}+\Lambda_{\text{UV}}^{2}-\tfrac{1}{2}\Lambda_{\text{IR}}^{4}-\Lambda_{\text{IR}}^{2}\right]\;,$

having performed the same procedure as for $C_{0}$. Note that keeping the non-
minimal slow-roll parameter $\epsilon_{F}$ is straightforward for the tensors
contributions.

For the scalars we work in the limit $F(\phi)=1$, so that the first correlator
is given by

$D_{0}(\eta)\equiv\braket{\Psi^{2}}=\int\frac{\mathop{}\\!\mathrm{d}^{3}k}{(2\pi)^{3}}|f_{\mathbf{k}}|^{2}=\frac{1}{16\pi}\frac{\epsilon_{H}\mathcal{H}^{2}}{M_{\text{Pl}}^{2}a^{2}}\int_{\Lambda_{\text{IR}}}^{\Lambda_{\text{UV}}}\mathop{}\\!\mathrm{d}x|H^{(2)}_{\upsilon}(-x)|^{2}\;,$
(129)

and with the mode function given by Eq. (58). We then perform the same
evaluation as for the tensors, with the difference being that the
approximation to the Hankel function in the intermediate-momentum region reads

$H^{(1)}_{{\frac{1}{2}}}(x)=-\sqrt{\frac{2}{\pi x}}e^{ix}\ ,$ (130)

when $\epsilon_{H},\delta_{H}\rightarrow 0$. For the various loop integrals we
then obtain

$\displaystyle D_{0}$
$\displaystyle=\frac{\epsilon_{H}H^{2}}{8\pi^{2}M_{\text{Pl}}^{2}}\left[\left(\frac{1}{2(2\epsilon_{H}-\delta_{H})}+\log
2+\psi(\tfrac{1}{2})\right)\left(-1+\Lambda_{\text{IR}}^{-2(2\epsilon_{H}-\delta_{H})}\right)+\log\Lambda_{\text{UV}}\right]+\mathcal{O}(\epsilon)\
,$ (131) $\displaystyle D_{2}$
$\displaystyle=-\frac{\epsilon_{H}a^{2}{H}^{4}}{16\pi^{2}M_{\text{Pl}}^{2}}\left(\Lambda_{\text{UV}}^{2}-\Lambda_{\text{IR}}^{2}\right)+\mathcal{O}(\epsilon)\
,$ (132) $\displaystyle D_{4}$
$\displaystyle=\frac{\epsilon_{H}a^{4}{H}^{6}}{32\pi^{2}M_{\text{Pl}}^{2}}\left(\Lambda_{\text{UV}}^{4}-\Lambda_{\text{IR}}^{4}\right)+\mathcal{O}(\epsilon)\
.$ (133)

The correlators in Eqs. (127), (128) and Eqs. (131)–(133) all contain powers
and logarithms of the UV and IR cutoffs. Had we chosen to carry out the
computation in dimensional regularisation, all powers would have immediately
disappeared, leaving us only with the logarithmic divergences both in the UV
and IR. The logarithmic UV divergences should be cancelled by counterterms, as
in e.g. Markkanen:2013nwa . The IR divergences, on the other hand, are
physical in the sense that they indicate that a higher order effect (such as
an infinite resummation) will generate a regulator (such as a gap)
Sloth:2006az ; Serreau:2011fu . With this in mind, we find it illuminating
through the use of a simple cutoff regularisation to identify where divergence
show up, but allow ourselves to assume that UV-divergences may be removed
through a proper choice of counterterms. Simply discarding the UV-divergent
terms amounts to a MS-like renormalisation condition.

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