UFIFT-QG-21-02 , CCTP-2021-02

How Inflationary Gravitons Affect Gravitational Radiation

L. Tan1⋆, N. C. Tsamis2† and R. P. Woodard1‡

1 Department of Physics, University of Florida,
Gainesville, FL 32611, UNITED STATES

2 Institute of Theoretical Physics & Computational Physics,
Department of Physics, University of Crete,
GR-710 03 Heraklion, HELLAS

ABSTRACT

We include the single graviton loop contribution to the linearized Einstein
equation. Explicit results are obtained for one loop corrections to the
propagation of gravitational radiation. Although suppressed by a minuscule
loop-counting parameter, these corrections are enhanced by the square of the
number of inflationary e-foldings. One consequence is that perturbation theory
breaks down for a very long epoch of primordial inflation. Another consequence
is that the one loop correction to the tensor power spectrum might be
observable, in the far future, after the full development of 21cm cosmology.

PACS numbers: 04.50.Kd, 95.35.+d, 98.62.-g

⋆ e-mail<EMAIL_ADDRESS>
† e-mail<EMAIL_ADDRESS>
‡ e-mail<EMAIL_ADDRESS>
##  1 Introduction

Primordial inflation produces a vast ensemble of infrared gravitons [1, 2]. If
the tensor power spectrum for modes of co-moving wave number $k$ is
$\Delta^{2}_{h}(k)$, then the occupation number for each polarization of wave
vector $\vec{k}$ is,

$N(\eta,k)=\frac{\pi\Delta^{2}_{h}(k)}{64Gk^{2}}\times a^{2}(\eta)\;,$ (1)

where $a(\eta)$ is the scale factor at conformal time $\eta$ and $G$ is
Newton’s constant. The aim of this paper is to study how these gravitons
change the kinematics of other gravitons.

The background geometry of cosmology can be well described using spatially
flat, conformal coordinates,

$ds^{2}=a^{2}(\eta)\Bigl{[}-d\eta^{2}+d\vec{x}\\!\cdot\\!d\vec{x}\Bigr{]}\qquad\Longrightarrow\qquad
H\equiv\frac{a^{\prime}}{a^{2}}\quad,\quad\epsilon\equiv-\frac{H^{\prime}}{aH^{2}}\;,$
(2)

where $H(\eta)$ is the Hubble parameter and $\epsilon(\eta)$ is the first slow
roll parameter. The special case of de Sitter ($\epsilon=0$, constant $H$ and
$a(\eta)=-1/H\eta$) is often considered as a reasonable paradigm for
inflation, and is attractive both because we possess analytic expressions for
the graviton propagator [3, 4] and because there is no mixing with whatever
matter fields support inflation [5, 6]. On this background the quantum-
corrected, linearized Einstein equation takes the form,

$\mathcal{D}^{\mu\nu\rho\sigma}h_{\rho\sigma}(x)-\int\\!\\!d^{4}x^{\prime}\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})h_{\rho\sigma}(x^{\prime})=\frac{1}{2}\kappa
T^{\mu\nu}_{\rm lin}(x)\;,$ (3)

where $\mathcal{D}^{\mu\nu\rho\sigma}$ is the gauge-fixed kinetic operator,
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ is the 1PI (one
particle irreducible) 2-graviton function, $T^{\mu\nu}_{\rm lin}(x)$ is the
linearized stress-energy tensor, $\kappa^{2}\equiv 16\pi G$ is the loop-
counting parameter of quantum gravity and
$h_{\mu\nu}\equiv(g_{\mu\nu}-a^{2}\eta_{\mu\nu})/\kappa$ is the graviton
field. In this paper we will show how to extract a fully renormalized result
for the one loop graviton self-energy from an old, unregulated computation
[7]. We then use this result to derive one loop corrections to the mode
functions of dynamical gravitons.

In section 2 we derive a renormalized result for the part of the graviton
self-energy which affects dynamical gravitons. Section 3 solves equation (3)
for one loop corrections to transverse-traceless, spatial plane waves. Our
conclusions comprise section 4. There we discuss the exciting possibility that
the one loop correction we find might be observable.

##  2 Quantum Linearized Einstein Equation

The purpose of this section is to give an explicit expression for the
linearized Einstein equation (3). We begin by defining the gauge-fixed kinetic
operator $\mathcal{D}^{\mu\nu\rho\sigma}$, explaining how we represent the
graviton self-energy, and working out $3+1$ decompositions of both. We then
describe the process through which an old, unregulated computation of the
graviton self-energy [7] can be used to recover most of the fully
renormalized, Schwinger-Keldysh result. The section closes with a direct
computation of the remaining contribution.

###  2.1 $3+1$ Decomposition

In $D=3+1$ dimensions the gauge-fixed kinetic operator is [3, 4],

$\mathcal{D}^{\mu\nu\rho\sigma}=\frac{1}{2}\eta^{\mu(\rho}\eta^{\sigma)\nu}D_{A}-\frac{1}{4}\eta^{\mu\nu}\eta^{\rho\sigma}D_{A}+2a^{4}H^{2}\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}\;,$
(4)

where the kinetic operator of a massless, minimally coupled scalar is,

$D_{A}=-a^{2}\Bigl{[}\partial_{0}^{2}+2aH\partial_{0}-\nabla^{2}\Bigr{]}\;.$
(5)

The $3+1$ components of $\mathcal{D}^{\mu\nu\rho\sigma}h_{\rho\sigma}$ are,

$\displaystyle\mathcal{D}^{00\rho\sigma}h_{\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{4}D_{A}(h_{00}+h_{kk})-2a^{4}H^{2}h_{00}\;,\qquad$ (6)
$\displaystyle\mathcal{D}^{0i\rho\sigma}h_{\rho\sigma}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}D_{B}h_{0i}\;,$ (7)
$\displaystyle\mathcal{D}^{ij\rho\sigma}h_{\rho\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}D_{A}\Bigl{[}h_{ij}+\frac{1}{2}\delta_{ij}(h_{00}-h_{kk})\Bigr{]}\;,\qquad$
(8)

where $D_{B}$ is the kinetic operator of a conformally coupled scalar,

$D_{B}=-a^{2}\Bigl{[}\partial_{0}^{2}+2aH\partial_{0}-\nabla^{2}+2a^{2}H^{2}\Bigr{]}\;.$
(9)

On a cosmological background (2) the graviton self-energy can be expressed as
a sum of 21 tensor differential operators
$[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}]$ acting on scalar functions of
$\eta$, $\eta^{\prime}$ and $\|\vec{x}-\vec{x}^{\prime}\|$ [8],

$-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\sum_{i=1}^{21}\Bigl{[}\mbox{}^{\mu\nu}\mathcal{D}_{i}^{\rho\sigma}\Bigr{]}\\!\times\\!T^{i}(x;x^{\prime})\;.$
(10)

By general tensor analysis the 21 basis tensors are constructed from
$\delta^{\mu}_{~{}0}$, the spatial part of the Minkowski metric
$\overline{\eta}^{\mu\nu}\equiv\eta^{\mu\nu}+\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}$
and the spatial derivative operator
$\overline{\partial}^{\mu}\equiv\partial^{\mu}+\delta^{\mu}_{~{}0}\partial_{0}$.
Table 1 lists the $[\mbox{}^{\mu\nu}\mathcal{D}_{i}^{\rho\sigma}]$.

$i$ | $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}_{i}]$ | $i$ | $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}_{i}]$ | $i$ | $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}_{i}]$
---|---|---|---|---|---
1 | $\overline{\eta}^{\mu\nu}\overline{\eta}^{\rho\sigma}$ | 8 | $\overline{\partial}^{\mu}\overline{\partial}^{\nu}\overline{\eta}^{\rho\sigma}$ | 15 | $\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$
2 | $\overline{\eta}^{\mu(\rho}\overline{\eta}^{\sigma)\nu}$ | 9 | $\delta^{(\mu}_{~{}~{}0}\overline{\eta}^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}$ | 16 | $\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\overline{\partial}^{\rho}\overline{\partial}^{\sigma}$
3 | $\overline{\eta}^{\mu\nu}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$ | 10 | $\delta^{(\mu}_{~{}~{}0}\overline{\eta}^{\nu)(\rho}\overline{\partial}^{\sigma)}$ | 17 | $\overline{\partial}^{\mu}\overline{\partial}^{\nu}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$
4 | $\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\overline{\eta}^{\rho\sigma}$ | 11 | $\overline{\partial}^{(\mu}\overline{\eta}^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}$ | 18 | $\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}$
5 | $\overline{\eta}^{\mu\nu}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}$ | 12 | $\overline{\partial}^{(\mu}\overline{\eta}^{\nu)(\rho}\overline{\partial}^{\sigma)}$ | 19 | $\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\overline{\partial}^{\rho}\overline{\partial}^{\sigma}$
6 | $\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\overline{\eta}^{\rho\sigma}$ | 13 | $\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$ | 20 | $\overline{\partial}^{\mu}\overline{\partial}^{\nu}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}$
7 | $\overline{\eta}^{\mu\nu}\overline{\partial}^{\rho}\overline{\partial}^{\sigma}$ | 14 | $\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}$ | 21 | $\overline{\partial}^{\mu}\overline{\partial}^{\nu}\overline{\partial}^{\rho}\overline{\partial}^{\sigma}$
Table 1: The 21 basis tensors used in expression (10). The pairs $(3,4)$,
$(5,6)$, $(7,8)$, $(10,11)$, $(14,15)$, $(16,17)$ and $(19,20)$ are related by
reflection.

Reflection invariance —
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})=-i[\mbox{}^{\rho\sigma}\Sigma^{\mu\nu}](x^{\prime};x)$
— relates the 7 pairs of $T^{i}(x;x^{\prime})$ given in Table 2.

$i$ | Relation | $i$ | Relation
---|---|---|---
$3,4$ | $T^{4}(x;x^{\prime})=+T^{3}(x^{\prime};x)$ | $14,15$ | $T^{15}(x;x^{\prime})=-T^{14}(x^{\prime};x)$
$5,6$ | $T^{6}(x;x^{\prime})=-T^{5}(x^{\prime};x)$ | $16,17$ | $T^{17}(x;x^{\prime})=+T^{16}(x^{\prime};x)$
$7,8$ | $T^{8}(x;x^{\prime})=+T^{7}(x^{\prime};x)$ | $19,20$ | $T^{20}(x;x^{\prime})=-T^{19}(x^{\prime};x)$
$10,11$ | $T^{11}(x;x^{\prime})=-T^{10}(x^{\prime};x)$ |  |
Table 2: Scalar coefficient functions in expression (10) which are related by
reflection.

The factor
$[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})h_{\rho\sigma}(x^{\prime})$
in expression (3) has the $3+1$ decomposition,

$\displaystyle\Bigl{[}\mbox{}^{00}\Sigma^{\rho\sigma}\Bigr{]}h_{\rho\sigma}\longrightarrow
iT^{4}h_{kk}+iT^{13}h_{00}+iT^{14}h_{0k,k}+iT^{16}h_{k\ell,k\ell}\;,$ (11)
$\displaystyle\Bigl{[}\mbox{}^{0i}\Sigma^{\rho\sigma}\Bigr{]}h_{\rho\sigma}\longrightarrow\frac{i}{2}\partial_{i}\Bigl{[}T^{6}h_{kk}+T^{15}h_{00}+T^{18}h_{0k,k}+T^{19}h_{k\ell,k\ell}\Bigr{]}$
(12) $\displaystyle\hskip
227.62204pt+\frac{i}{2}T^{9}h_{0i}+\frac{i}{2}T^{10}h_{ik,k}\;,\qquad$
$\displaystyle\Bigl{[}\mbox{}^{ij}\Sigma^{\rho\sigma}\Bigr{]}h_{\rho\sigma}\longrightarrow
i\delta_{ij}\Bigl{[}T^{1}h_{kk}+T^{3}h_{00}+T^{5}h_{0k,k}+T^{7}h_{k\ell,k\ell}\Bigr{]}+iT^{2}h_{ij}$
(13)
$\displaystyle+i\partial_{(i}\Bigl{[}T^{11}h_{j)0}\\!+\\!T^{12}h_{j)k,k}\Bigr{]}+i\partial_{i}\partial_{j}\Bigl{[}T^{8}h_{kk}\\!+\\!T^{17}h_{00}\\!+\\!T^{20}h_{0k,k}\\!+\\!T^{21}h_{k\ell,k\ell}\Bigr{]}.\qquad$

In these relations we have sometimes exploited spatial transition invariance
to partially integrate spatial derivatives from the coefficient functions
$T^{i}(x;x^{\prime})$ onto the graviton field.

###  2.2 The Quantum Correction

Let $S[g]$ represent the classical action of gravity, while
$S_{g}[h,\overline{\theta},\theta]$ stands for the ghost and gauge fixing
action, and $\Delta S[g]$ is the counter-action. The one loop graviton self-
energy can be expressed as the expectation value of the sum of four
variational derivatives of these quantities,

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\Biggl{\langle}\Omega\Biggl{|}T^{*}\Biggl{[}\Bigl{[}\frac{i\delta
S[g]}{\delta h_{\mu\nu}(x)}\Bigr{]}_{hh}\Bigl{[}\frac{i\delta S[g]}{\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{hh}+\Bigl{[}\frac{i\delta S[g]}{\delta
h_{\mu\nu}(x)}\Bigr{]}_{\overline{\theta}\theta}$ (14) $\displaystyle\hskip
8.5359pt\times\Bigl{[}\frac{i\delta S[g]}{\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{\overline{\theta}\theta}+\Bigl{[}\frac{i\delta^{2}S[g]}{\delta
h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{hh}+\Bigl{[}\frac{i\delta^{2}\Delta
S[g]}{\delta h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{1}\Biggr{]}\Biggr{|}\Omega\Biggr{\rangle}.\qquad$

The various subscripts indicate how many graviton fields contribute and the
$T^{*}$-ordering symbol signifies that any derivatives are taken after time
ordering. Figure 1 gives the relevant Feynman diagrams.

Figure 1: Diagrams contributing to the one loop graviton self-energy, shown in
the same order, left to right, as the four contributions to (14). Graviton
lines are wavy and ghost lines are dashed.

####  2.2.1 The $D=4$ Result

To understand the unregulated result [7] it helps to consider how a
computation of $-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ would
look in dimensional regularization. The 3-graviton and 4-graviton vertices
take the general form [3, 9],111 Ghost-antighost-graviton vertices are similar
to (15).

$\displaystyle\kappa a^{D-2}h\partial h\partial h\qquad$ ,
$\displaystyle\qquad\kappa Ha^{D-1}hh\partial h\;,$ (15)
$\displaystyle\kappa^{2}a^{D-2}hh\partial h\partial h\qquad$ ,
$\displaystyle\qquad\kappa^{2}Ha^{D-1}hhh\partial h\;.\qquad$ (16)

Without worrying about indices (which come in a dizzying variety of different
possibilities), contributions to the first two diagrams of Figure 1 have the
nonlocal structure,

$\kappa
a^{D-2}\times\partial\partial^{\prime}i\Delta(x;x^{\prime})\times\partial\partial^{\prime}i\Delta(x;x^{\prime})\times\kappa{a^{\prime}}^{D-2}\;,$
(17)

where $i\Delta(x;x^{\prime})$ is a ghost or graviton propagator, and one
derivative at each vertex could be exchanged for $H$ times the appropriate
scale factor. One should also bear in mind that if the external leg graviton
field happens to be differentiated then minus its derivative is acted on the
entire diagram. In contrast, the third diagram of Figure 1 is local,

$\kappa^{2}a^{D-2}\times\partial\partial^{\prime}i\Delta(x;x^{\prime})\times
i\delta^{D}(x\\!-\\!x^{\prime})\;,$ (18)

with the same stipulation about derivatives. The final diagram of Figure 1 is
also local,

$\frac{\kappa^{2}a^{D-4}}{D\\!-\\!4}\times\partial^{2}{\partial^{\prime}}^{2}\times
i\delta^{D}(x\\!-\\!x^{\prime})\;,$ (19)

where any number of the four derivatives could be exchanged for $H$ times a
scale factor.

The graviton and ghost propagators were defined by adding to the Lagrangian a
functional whose $D$-dimensional extension is [3, 4],

$\mathcal{L}_{GF}=-\frac{a^{D-2}}{2}\eta^{\mu\nu}F_{\mu}F_{\nu}\;\;,\;\;F_{\mu}=\eta^{\rho\sigma}\Bigl{(}h_{\mu\rho,\sigma}-\frac{1}{2}h_{\rho\sigma,\mu}+(D\\!-\\!2)aHh_{\mu\rho}\delta^{0}_{~{}\sigma}\Bigr{)}\;.$
(20)

The wonderful property of this gauge is that the gauge and graviton
propagators are sums of constant tensor factors times scalar propagators,

$\displaystyle i\Bigl{[}\mbox{}_{\mu}\Delta_{\rho}\Bigr{]}(x;x^{\prime})$
$\displaystyle=$ $\displaystyle\overline{\eta}_{\mu\rho}\times
i\Delta_{A}(x;x^{\prime})-\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}\times
i\Delta_{B}(x;x^{\prime})\;,$ (21) $\displaystyle
i\Bigl{[}\mbox{}_{\mu\nu}\Delta_{\rho\sigma}\Bigr{]}(x;x^{\prime})$
$\displaystyle=$
$\displaystyle\sum_{I=A,B,C}\Bigl{[}\mbox{}_{\mu\nu}T^{I}_{\rho\sigma}\Bigr{]}\times
i\Delta_{I}(x;x^{\prime})\;.$ (22)

The constant tensor factors $[\mbox{}_{\mu\nu}T^{I}_{\rho\sigma}]$ are,

$\displaystyle\Bigl{[}\mbox{}_{\mu\nu}T^{A}_{\rho\sigma}\Bigr{]}=2\overline{\eta}_{\mu(\rho}\overline{\eta}_{\sigma)\nu}-\frac{2}{D\\!-\\!3}\overline{\eta}_{\mu\nu}\overline{\eta}_{\rho\sigma}\quad,\quad\Bigl{[}\mbox{}_{\mu\nu}T^{B}_{\rho\sigma}\Bigr{]}=-4\delta^{0}_{~{}(\mu}\overline{\eta}_{\nu)(\rho}\delta^{0}_{~{}\sigma)}\;,\qquad$
(23)
$\displaystyle\Bigl{[}\mbox{}_{\mu\nu}T^{C}_{\rho\sigma}\Bigr{]}=\frac{2E_{\mu\nu}E_{\rho\sigma}}{(D\\!-\\!2)(D\\!-\\!3)}\quad,\quad
E_{\mu\nu}\equiv(D\\!-\\!3)\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}+\overline{\eta}_{\mu\nu}\;.\qquad$
(24)

The three scalar propagators are given in terms of the de Sitter length
function $y(x;x^{\prime})\equiv aa^{\prime}H^{2}\Delta x^{2}$ and a function
$A(y)$,

$\displaystyle i\Delta_{A}(x;x^{\prime})$ $\displaystyle=$ $\displaystyle
A(y)+k\ln(aa^{\prime})\qquad
k\equiv\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\;,$
(25) $\displaystyle i\Delta_{B}(x;x^{\prime})$ $\displaystyle=$ $\displaystyle
B(y)\equiv-\frac{[(4y\\!-\\!y^{2})A^{\prime}(y)\\!+\\!(2\\!-\\!y)k]}{2(D\\!-\\!2)}\;,$
(26) $\displaystyle i\Delta_{C}(x;x^{\prime})$ $\displaystyle=$ $\displaystyle
C(y)\equiv\frac{1}{2}(2\\!-\\!y)B(y)+\frac{k}{D\\!-\\!3}\;.$ (27)

The first derivative of $A(y)$ is [10, 11],

$\displaystyle
A^{\prime}(y)=-\frac{H^{D-2}}{4(4\pi)^{\frac{D}{2}}}\Biggl{\\{}\Gamma\Bigl{(}\frac{D}{2}\Bigr{)}\Bigl{(}\frac{4}{y}\Bigr{)}^{\frac{D}{2}}+\Gamma\Bigl{(}\frac{D}{2}\\!+\\!1\Bigr{)}\Bigl{(}\frac{4}{y}\Bigr{)}^{\frac{D}{2}-1}$
(28) $\displaystyle\hskip
56.9055pt+\sum_{n=0}^{\infty}\Biggl{[}\frac{\Gamma(n\\!+\\!\frac{D}{2}\\!+\\!2)}{\Gamma(n\\!+\\!3)}\Bigl{(}\frac{y}{4}\Bigr{)}^{n-\frac{D}{2}+2}-\frac{\Gamma(n\\!+\\!D)}{\Gamma(n\\!+\\!\frac{D}{2}\\!+\\!1)}\Bigl{(}\frac{y}{4}\Bigr{)}^{n}\Biggr{]}\Biggr{\\}}.\qquad$

Because the $y^{n}$ and $y^{n-\frac{D}{2}-2}$ terms cancel in $D=4$ dimensions
they can only contribute when multiplied by an ultraviolet divergence.

Divergences only occur from taking a propagator to coincidence, that is, by
setting ${x^{\prime}}^{\mu}=x^{\mu}$. It follows that we can take the
unregulated ($D=4$) limit in the first two diagrams of Figure 1 so long as
${x^{\prime}}^{\mu}\neq x^{\mu}$. Of course that same stipulation makes the
last two diagrams of Figure 1 vanish. Taking $D=4$ also results in a huge
simplification of the ghost and graviton propagators,

$\displaystyle
i\Bigl{[}\mbox{}_{\mu}\Delta^{D=4}_{\rho}\Bigr{]}(x;x^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\Biggl{\\{}\frac{\eta_{\mu\rho}}{aa^{\prime}\Delta
x^{2}}-\frac{1}{2}H^{2}\ln(H^{2}\Delta
x^{2})\overline{\eta}_{\mu\rho}\Biggr{\\}},\qquad$ (29) $\displaystyle
i\Bigl{[}\mbox{}_{\mu\nu}\Delta^{D=4}_{\rho\sigma}\Bigr{]}(x;x^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\Biggl{\\{}\frac{(2\eta_{\mu(\rho}\eta_{\sigma)\nu}\\!-\\!\eta_{\mu\nu}\eta_{\rho\sigma})}{aa^{\prime}\Delta
x^{2}}$ (30) $\displaystyle\hskip 56.9055pt-H^{2}\ln(H^{2}\Delta
x^{2})\Bigl{(}\overline{\eta}_{\mu(\rho}\overline{\eta}_{\sigma)\nu}\\!-\\!\overline{\eta}_{\mu\nu}\overline{\eta}_{\rho\sigma}\Bigr{)}\Biggr{\\}}.\qquad$

$i$ | Coefficients $T^{i}_{L}(x;x^{\prime})$ in expression (31)
---|---
$1$ | $8a^{2}{a^{\prime}}^{2}H^{4}\times[\frac{4\Delta\eta^{2}}{\Delta x^{6}}+\frac{1}{\Delta x^{4}}]+4a^{3}{a^{\prime}}^{3}H^{6}\times[\frac{4\Delta\eta^{4}}{\Delta x^{6}}-\frac{\Delta\eta^{2}}{\Delta x^{4}}+\frac{3}{\Delta x^{2}}]$
$2$ | $-16a^{2}{a^{\prime}}^{2}H^{4}\times[\frac{4\Delta\eta^{2}}{\Delta x^{6}}+\frac{1}{\Delta x^{4}}]-4a^{3}{a^{\prime}}^{3}H^{6}\times[\frac{8\Delta\eta^{4}}{\Delta x^{6}}+\frac{1}{\Delta x^{2}}]$
$3$ | $8a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\![\frac{\Delta\eta^{2}}{\Delta x^{4}}\\!-\\!\frac{2}{\Delta x^{2}}]\\!-\\!4a^{3}{a^{\prime}}^{2}H^{5}\\!\times\\![\frac{4\Delta\eta^{3}}{\Delta x^{6}}\\!-\\!\frac{\Delta\eta}{\Delta x^{4}}]$
$5$ | $16a^{2}{a^{\prime}}^{2}H^{4}\times\frac{\Delta\eta}{\Delta x^{4}}+4a^{3}{a^{\prime}}^{2}H^{5}\times[\frac{2\Delta\eta^{2}}{\Delta x^{4}}+\frac{3}{\Delta x^{2}}]$
$7$ | $-8a^{2}{a^{\prime}}^{2}H^{4}\times\frac{1}{\Delta x^{2}}-2a^{3}{a^{\prime}}^{3}H^{6}\times\frac{\Delta\eta^{2}}{\Delta x^{2}}-2a^{3}{a^{\prime}}^{2}H^{5}\times\frac{\Delta\eta}{\Delta x^{2}}$
$9$ | $-96aa^{\prime}H^{2}\times[\frac{16\Delta\eta^{4}}{\Delta x^{10}}+\frac{12\Delta\eta^{2}}{\Delta x^{8}}+\frac{1}{\Delta x^{6}}]-4a^{2}{a^{\prime}}^{2}H^{4}\times[\frac{24\Delta\eta^{4}}{\Delta x^{8}}+\frac{8\Delta\eta^{2}}{\Delta x^{6}}-\frac{1}{\Delta x^{4}}]$
$10$ | $\\!\\!96aa^{\prime}H^{2}\\!\times\\![\frac{2\Delta\eta^{3}}{\Delta x^{8}}\\!+\\!\frac{\Delta\eta}{\Delta x^{6}}]\\!+\\!12a^{2}{a^{\prime}}^{2}H^{4}\\!\times\\![\frac{4\Delta\eta^{3}}{\Delta x^{6}}\\!+\\!\frac{\Delta\eta}{\Delta x^{4}}]\\!+\\!a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\![\frac{8\Delta\eta^{3}}{\Delta x^{4}}\\!-\\!\frac{4\Delta\eta}{\Delta x^{2}}]\\!\\!$
| $-8a^{2}a^{\prime}H^{3}\\!\times\\![\frac{4\Delta\eta^{2}}{\Delta
x^{6}}\\!+\\!\frac{1}{\Delta x^{4}}]\\!-\\!4a^{3}a^{\prime
2}H^{5}\\!\times\\![\frac{2\Delta\eta^{2}}{\Delta x^{4}}\\!-\\!\frac{1}{\Delta
x^{2}}]$
$12$ | $-8aa^{\prime}H^{2}\\!\times\\![\frac{4\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{1}{\Delta x^{4}}]\\!-\\!2a^{2}{a^{\prime}}^{2}H^{4}\\!\times\\![\frac{6\Delta\eta^{2}}{\Delta x^{4}}\\!-\\!\frac{9}{\Delta x^{2}}]\\!+\\!4a^{3}{a^{\prime}}^{3}H^{6}\times\frac{\Delta\eta^{2}}{\Delta x^{2}}$
$13$ | $-96aa^{\prime}H^{2}\times[\frac{16\Delta\eta^{4}}{\Delta x^{10}}\\!+\\!\frac{12\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{1}{\Delta x^{6}}]+4a^{2}{a^{\prime}}^{2}H^{4}\times[\frac{24\Delta\eta^{4}}{\Delta x^{8}}\\!+\\!\frac{56\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{11}{\Delta x^{4}}]$
| $+8a^{3}{a^{\prime}}^{3}H^{6}\times[\frac{4\Delta\eta^{4}}{\Delta
x^{6}}-\frac{2\Delta\eta^{2}}{\Delta x^{4}}+\frac{3}{\Delta x^{2}}]$
$14$ | $192aa^{\prime}H^{2}\times[\frac{2\Delta\eta^{3}}{\Delta x^{8}}+\frac{\Delta\eta}{\Delta x^{6}}]+8a^{2}{a^{\prime}}^{2}H^{4}\times[\frac{4\Delta\eta^{3}}{\Delta x^{6}}-\frac{3\Delta\eta}{\Delta x^{4}}]$
| $-16a^{2}a^{\prime}H^{3}\times[\frac{4\Delta\eta^{2}}{\Delta
x^{6}}+\frac{1}{\Delta
x^{4}}]-16a^{3}{a^{\prime}}^{2}H^{5}\times[\frac{\Delta\eta^{2}}{\Delta
x^{4}}+\frac{1}{\Delta x^{2}}]$
$16$ | $-8aa^{\prime}H^{2}\\!\times\\![\frac{4\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{1}{\Delta x^{4}}]\\!+\\!2a^{2}{a^{\prime}}^{2}H^{4}\\!\times\\![-\frac{6\Delta\eta^{2}}{\Delta x^{4}}\\!+\\!\frac{1}{\Delta x^{2}}]\\!-\\!2a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\!\frac{\Delta\eta^{2}}{\Delta x^{2}}$
| $+16a^{2}a^{\prime}H^{3}\times\frac{\Delta\eta}{\Delta
x^{4}}+6a^{3}{a^{\prime}}^{2}H^{5}\times\frac{\Delta\eta}{\Delta x^{2}}$
$18$ | $-24aa^{\prime}H^{2}\times[\frac{4\Delta\eta^{2}}{\Delta x^{6}}+\frac{1}{\Delta x^{4}}]-2a^{2}{a^{\prime}}^{2}H^{4}\times[\frac{6\Delta\eta^{2}}{\Delta x^{4}}+\frac{5}{\Delta x^{2}}]$
$19$ | $8aa^{\prime}H^{2}\times\frac{\Delta\eta}{\Delta x^{4}}+6a^{2}{a^{\prime}}^{2}H^{4}\times\frac{\Delta\eta}{\Delta x^{2}}-4a^{2}a^{\prime}H^{3}\times\frac{1}{\Delta x^{2}}$
Table 3: Each tabulated term must be multiplied by
$-\frac{\kappa^{2}}{64\pi^{4}}$.

Because one of the propagators in the nonlocal diagrams (17) might be
undifferentiated, the functions $T^{i}(x;x^{\prime})$ of equation (10) take
the form,

$T^{i}(x;x^{\prime})\equiv
T^{i}_{N}(x;x^{\prime})+T^{i}_{L}(x;x^{\prime})\\!\times\\!\ln(H^{2}\Delta
x^{2})\;.$ (31)

where $T^{i}_{L}(x;x^{\prime})$ (given in Table 3) and
$T^{i}_{N}(x;x^{\prime})$ (given in Table 4) are functions of $a$,
$a^{\prime}$, $\Delta\eta\equiv\eta-\eta^{\prime}$ and inverse powers of the
Poincaré interval $\Delta
x^{2}\equiv\|\vec{x}-\vec{x}^{\prime}\|^{2}-(|\eta-\eta^{\prime}|-i\varepsilon)^{2}$.

$i$ | Coefficient $T^{i}_{N}(x;x^{\prime})$ in expression (31)
---|---
$1$ | $\\!\\!\frac{\frac{736}{5}}{\Delta x^{8}}\\!-\\!aa^{\prime}\\!H^{2}[\frac{616\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{\frac{220}{3}}{\Delta x^{6}}]\\!-\\!a^{2}{a^{\prime}}^{2}\\!H^{4}[\frac{96\Delta\eta^{4}}{\Delta x^{8}}\\!+\\!\frac{\frac{812}{3}\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{19}{\Delta x^{4}}]\\!-\\!a^{3}{a^{\prime}}^{3}\\!H^{6}[\frac{64\Delta\eta^{4}}{\Delta x^{6}}\\!+\\!\frac{22\Delta\eta^{2}}{\Delta x^{4}}]\\!\\!$
$2$ | $\\!\\!\frac{\frac{1952}{5}}{\Delta x^{8}}\\!-\\!aa^{\prime}H^{2}[\frac{416\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{\frac{128}{3}}{\Delta x^{6}}]\\!-\\!a^{2}{a^{\prime}}^{2}H^{4}[\frac{\frac{112}{3}\Delta\eta^{2}}{\Delta x^{6}}\\!-\\!\frac{56}{\Delta x^{4}}]\\!+\\!a^{3}{a^{\prime}}^{3}H^{6}[\frac{32\Delta\eta^{4}}{\Delta x^{6}}\\!+\\!\frac{12\Delta\eta^{2}}{\Delta x^{4}}]\\!\\!$
| $-184[\frac{\frac{8}{5}\Delta\eta^{2}}{\Delta x^{10}}\\!+\\!\frac{1}{\Delta
x^{8}}]\\!+\\!32aa^{\prime}H^{2}[\frac{36\Delta\eta^{4}}{\Delta
x^{10}}\\!+\\!\frac{14\Delta\eta^{2}}{\Delta
x^{8}}\\!-\\!\frac{\frac{43}{3}}{\Delta x^{6}}]$
$3$ | $-a^{2}{a^{\prime}}^{2}H^{4}[\frac{288\Delta\eta^{4}}{\Delta x^{8}}\\!-\\!\frac{\frac{1228}{3}\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{145}{\Delta x^{4}}]\\!+\\!4a^{3}{a^{\prime}}^{3}H^{6}[\frac{16\Delta\eta^{4}}{\Delta x^{6}}\\!+\\!\frac{5\Delta\eta^{2}}{\Delta x^{4}}]$
| $-8aH[\frac{232\Delta\eta^{3}}{\Delta
x^{10}}\\!+\\!\frac{203\Delta\eta}{\Delta
x^{8}}]\\!+\\!4a^{2}a^{\prime}H^{3}[\frac{74\Delta\eta^{3}}{\Delta
x^{8}}\\!-\\!\frac{\frac{319}{3}\Delta\eta}{\Delta
x^{6}}]\\!-\\!a^{3}{a^{\prime}}^{2}H^{5}\\!\times\\!\frac{38\Delta\eta}{\Delta
x^{4}}$
$5$ | $\frac{\frac{368}{5}\Delta\eta}{\Delta x^{8}}\\!-\\!32aa^{\prime}H^{2}[\frac{9\Delta\eta^{3}}{\Delta x^{8}}\\!-\\!\frac{\frac{4}{3}\Delta\eta}{\Delta x^{6}}]\\!+\\!4a^{2}{a^{\prime}}^{2}H^{4}[\frac{8\Delta\eta^{3}}{\Delta x^{6}}\\!-\\!\frac{\Delta\eta}{\Delta x^{4}}]$
| $+232aH[\frac{2\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{1}{\Delta
x^{6}}]\\!-\\!2a^{2}a^{\prime}H^{3}[\frac{\frac{28}{3}\Delta\eta^{2}}{\Delta
x^{6}}\\!-\\!\frac{23}{\Delta
x^{4}}]\\!-\\!a^{3}{a^{\prime}}^{2}H^{5}\\!\times\\!\frac{12\Delta\eta^{2}}{\Delta
x^{4}}$
$7$ | $-\frac{\frac{92}{15}}{\Delta x^{6}}\\!+\\!aa^{\prime}H^{2}[\frac{24\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{\frac{7}{3}}{\Delta x^{4}}]\\!+\\!a^{2}{a^{\prime}}^{2}H^{4}[\frac{8\Delta\eta^{2}}{\Delta x^{4}}\\!-\\!\frac{3}{\Delta x^{2}}]\\!+\\!a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\!\frac{\Delta\eta^{2}}{\Delta x^{2}}$
| $-aH\\!\times\\!\frac{\frac{116}{3}\Delta\eta}{\Delta
x^{6}}-a^{2}a^{\prime}H^{3}\\!\times\\!\frac{\frac{23}{3}\Delta\eta}{\Delta
x^{4}}+a^{3}{a^{\prime}}^{2}H^{5}\\!\times\\!\frac{\Delta\eta}{\Delta x^{2}}$
$9$ | $\\!\\!-16[\frac{\frac{488}{5}\Delta\eta^{2}}{\Delta x^{10}}\\!+\\!\frac{61}{\Delta x^{8}}]\\!+\\!16aa^{\prime}H^{2}[\frac{88\Delta\eta^{4}}{\Delta x^{10}}\\!-\\!\frac{10\Delta\eta^{2}}{\Delta x^{8}}\\!-\\!\frac{35}{\Delta x^{6}}]\\!+\\!4a^{2}{a^{\prime}}^{2}H^{4}[\frac{8\Delta\eta^{4}}{\Delta x^{8}}\\!+\\!\frac{2\Delta\eta^{2}}{\Delta x^{6}}\\!-\\!\frac{3}{\Delta x^{4}}]\\!\\!$
$10$ | $\frac{\frac{976}{5}\Delta\eta}{\Delta x^{8}}\\!-\\!16aa^{\prime}H^{2}[\frac{8\Delta\eta^{3}}{\Delta x^{8}}\\!-\\!\frac{\frac{23}{3}\Delta\eta}{\Delta x^{6}}]\\!-\\!2a^{2}{a^{\prime}}^{2}H^{4}[\frac{\frac{16}{3}\Delta\eta^{3}}{\Delta x^{6}}\\!-\\!\frac{23\Delta\eta}{\Delta x^{4}}]$
| $+a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\!\frac{4\Delta\eta^{3}}{\Delta
x^{4}}\\!+\\!aH\\!\times\\!\frac{\frac{64}{3}}{\Delta
x^{6}}\\!+\\!8a^{2}a^{\prime}H^{3}[\frac{\frac{4}{3}\Delta\eta^{2}}{\Delta
x^{6}}\\!-\\!\frac{5}{\Delta
x^{4}}]\\!-\\!a^{3}{a^{\prime}}^{2}H^{5}\\!\times\\!\frac{4\Delta\eta^{2}}{\Delta
x^{4}}$
$12$ | $-\frac{\frac{488}{15}}{\Delta x^{6}}\\!+\\!\frac{32}{3}aa^{\prime}H^{2}[\frac{\Delta\eta^{2}}{\Delta x^{6}}\\!-\\!\frac{1}{\Delta x^{4}}]\\!-\\!a^{2}{a^{\prime}}^{2}H^{4}[\frac{\frac{10}{3}\Delta\eta^{2}}{\Delta x^{4}}\\!-\\!\frac{8}{\Delta x^{2}}]\\!-\\!a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\!\frac{2\Delta\eta^{2}}{\Delta x^{2}}$
$13$ | $16[\frac{336\Delta\eta^{4}}{\Delta x^{12}}\\!+\\!\frac{336\Delta\eta^{2}}{\Delta x^{10}}\\!+\\!\frac{63}{\Delta x^{8}}]+4aa^{\prime}H^{2}[\frac{336\Delta\eta^{4}}{\Delta x^{10}}\\!+\\!\frac{868\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{409}{\Delta x^{6}}]$
| $+a^{2}{a^{\prime}}^{2}H^{4}[\frac{424\Delta\eta^{4}}{\Delta
x^{8}}\\!+\\!\frac{144\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{557}{\Delta
x^{4}}]\\!-\\!24a^{3}{a^{\prime}}^{3}H^{6}[\frac{4\Delta\eta^{4}}{\Delta
x^{6}}\\!-\\!\frac{\Delta\eta^{2}}{\Delta x^{4}}]$
$14$ | $-672[\frac{\frac{8}{5}\Delta\eta^{3}}{\Delta x^{10}}\\!+\\!\frac{\Delta\eta}{\Delta x^{8}}]\\!-\\!8aa^{\prime}H^{2}[\frac{30\Delta\eta^{3}}{\Delta x^{8}}\\!+\\!\frac{\frac{107}{3}\Delta\eta}{\Delta x^{6}}]\\!-\\!4a^{2}{a^{\prime}}^{2}H^{4}[\frac{16\Delta\eta^{3}}{\Delta x^{6}}\\!-\\!\frac{7\Delta\eta}{\Delta x^{4}}]$
| $-16aH[\frac{4\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{\frac{35}{3}}{\Delta
x^{6}}]\\!-\\!2a^{2}a^{\prime}H^{3}[\frac{\frac{68}{3}\Delta\eta^{2}}{\Delta
x^{6}}\\!+\\!\frac{137}{\Delta
x^{4}}]\\!+\\!a^{3}{a^{\prime}}^{2}H^{5}\\!\times\\!\frac{16\Delta\eta^{2}}{\Delta
x^{4}}$
$16$ | $4[\frac{\frac{84}{5}\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{\frac{13}{3}}{\Delta x^{6}}]\\!+\\!aa^{\prime}H^{2}[\frac{24\Delta\eta^{2}}{\Delta x^{6}}\\!+\\!\frac{\frac{127}{3}}{\Delta x^{4}}]\\!+\\!a^{2}{a^{\prime}}^{2}H^{4}[\frac{6\Delta\eta^{2}}{\Delta x^{4}}\\!+\\!\frac{5}{\Delta x^{2}}]$
| $-a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\!\frac{3\Delta\eta^{2}}{\Delta
x^{2}}\\!-\\!aH\\!\times\\!\frac{28\Delta\eta}{\Delta
x^{6}}\\!+\\!a^{2}a^{\prime}H^{3}\\!\times\\!\frac{\frac{49}{3}\Delta\eta}{\Delta
x^{4}}\\!+\\!a^{3}{a^{\prime}}^{2}H^{5}\\!\times\\!\frac{\Delta\eta}{\Delta
x^{2}}$
$18$ | $8[\frac{\frac{168}{5}\Delta\eta^{2}}{\Delta x^{8}}\\!+\\!\frac{\frac{29}{3}}{\Delta x^{6}}]\\!+\\!4aa^{\prime}H^{2}[\frac{12\Delta\eta^{2}}{\Delta x^{6}}\\!-\\!\frac{23}{\Delta x^{4}}]\\!+\\!a^{2}{a^{\prime}}^{2}H^{4}\\!\times\\!\frac{10\Delta\eta^{2}}{\Delta x^{4}}$
$19$ | $-\frac{\frac{112}{5}\Delta\eta}{\Delta x^{6}}-8aa^{\prime}H^{2}\\!\times\\!\frac{\Delta\eta}{\Delta x^{4}}-a^{2}{a^{\prime}}^{2}H^{4}\\!\times\\!\frac{\Delta\eta}{\Delta x^{2}}+aH\\!\times\\!\frac{\frac{50}{3}}{\Delta x^{4}}$
$21$ | $\frac{\frac{14}{5}}{\Delta x^{4}}$
Table 4: Each of the tabulated terms must be multiplied by
$-\frac{\kappa^{2}}{64\pi^{4}}$.

####  2.2.2 Recovering the Renormalized Result

We have developed a 4-step procedure for converting the unregulated
expressions of Tables 3 and 4 to fully renormalized results:

  1. 1.

Express $T^{i}_{L}(x;x^{\prime})$ as derivatives acting on integrable
functions;

  2. 2.

Pass the multiplicative factor of $\ln(H^{2}\Delta x^{2})$ through the
derivatives;

  3. 3.

Express the remainder $\Delta T^{i}_{L}(x;x^{\prime})$ from step 2, plus
$T^{i}_{N}(x;x^{\prime})$, as derivatives acting on integrable functions and
$1/\Delta x^{4}$; and

  4. 4.

Renormalize the factors of $1/\Delta x^{4}$ from step 3.

We will provide details and explain the rationale for each step below. As an
example, we also implement the steps on the crucial coefficient function
$T^{i}(x;x^{\prime})$ which controls corrections to the graviton mode
function,

$\displaystyle T^{2}_{L}(x;x^{\prime})=-\frac{\kappa^{2}\ln(H^{2}\Delta
x^{2})}{64\pi^{4}}\Biggl{\\{}a^{2}{a^{\prime}}^{2}H^{4}\Bigl{[}-\frac{64\Delta\eta^{2}}{\Delta
x^{6}}-\frac{16}{\Delta x^{4}}\Bigr{]}$ (32) $\displaystyle\hskip
184.9429pt+a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}-\frac{32\Delta\eta^{4}}{\Delta
x^{6}}-\frac{4}{\Delta x^{2}}\Bigr{]}\Biggr{\\}},\qquad$ $\displaystyle
T^{2}_{N}(x;x^{\prime})=-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}\frac{\frac{1952}{3}}{\Delta
x^{8}}+aa^{\prime}H\Bigl{[}-\frac{416\Delta\eta^{2}}{\Delta
x^{8}}-\frac{\frac{128}{3}}{\Delta x^{6}}\Bigr{]}$ (33) $\displaystyle\hskip
34.14322pt+a^{2}{a^{\prime}}^{2}H^{4}\Bigl{[}-\frac{\frac{112}{3}\Delta\eta^{2}}{\Delta
x^{6}}+\frac{56}{\Delta
x^{4}}\Bigr{]}+a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}\frac{32\Delta\eta^{4}}{\Delta
x^{6}}+\frac{12\Delta\eta^{2}}{\Delta x^{4}}\Bigr{]}\Biggr{\\}}.\qquad$

The rationale for Step 1 derives from the genesis of the logarithm
contribution to the coefficient functions (31). The only way a factor of
$\ln(H^{2}\Delta x^{2})$ can survive from one of the ghost or graviton
propagators (29-30) in a nonlocal contribution like (17), is for all the
derivatives to act on the other propagator. It is this differentiated
propagator, times the scale factors from the vertices, that contribute to
$T^{i}_{L}(x;x^{\prime})$. Hence we must be able to express each
$T^{i}_{L}(x;x^{\prime})$ as a sum of terms involving scale factors times a
number of derivatives acting on one of three integrable expressions,

$\frac{1}{\Delta x^{2}}\qquad,\qquad\frac{\Delta\eta}{\Delta
x^{2}}\qquad,\qquad\frac{\Delta\eta^{2}}{\Delta x^{2}}\;.$ (34)

Implementing Step 1 on expression (32) for $T^{i}_{L}(x;x^{\prime})$ gives,

$\displaystyle T^{2}_{L}(x;x^{\prime})=-\frac{\kappa^{2}\ln(H^{2}\Delta
x^{2})}{64\pi^{4}}\Biggl{\\{}a^{2}{a^{\prime}}^{2}H^{4}\\!\times\\!-8\partial_{0}^{2}\Bigl{(}\frac{1}{\Delta
x^{2}}\Bigr{)}$ (35) $\displaystyle\hskip
99.58464pt+a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}-4\partial_{0}^{2}\Bigl{(}\frac{\Delta\eta^{2}}{\Delta
x^{2}}\Bigr{)}\\!+\\!20\partial_{0}\Bigl{(}\frac{\Delta\eta}{\Delta
x^{2}}\Bigr{)}\\!-\\!\frac{16}{\Delta x^{2}}\Bigr{]}\Biggr{\\}}.\qquad$

In Step 2 we pass the overall multiplicative factor of $\ln(H^{2}\Delta
x^{2})$ through the derivatives to multiply the three integrable functions
(34). Implementing this on expression (35) for $T^{i}_{L}(x;x^{\prime})$
gives,

$\displaystyle
T^{2}_{L}(x;x^{\prime})=-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}a^{2}{a^{\prime}}^{2}H^{4}\\!\times\\!-8\partial_{0}^{2}\Bigl{(}\frac{\ln(H^{2}\Delta
x^{2})}{\Delta
x^{2}}\Bigr{)}+a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}-4\partial_{0}^{2}$ (36)
$\displaystyle\hskip
14.22636pt\times\Bigl{(}\frac{\Delta\eta^{2}\ln(H^{2}\Delta x^{2})}{\Delta
x^{2}}\Bigr{)}+20\partial_{0}\Bigl{(}\frac{\Delta\eta\ln(H^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{)}\\!-\\!\frac{16\ln(H^{2}\Delta x^{2})}{\Delta
x^{2}}\Bigr{]}\Biggr{\\}}\qquad$ $\displaystyle\hskip
42.67912pt-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}a^{2}{a^{\prime}}^{2}H^{4}\Bigl{[}-\frac{96\Delta\eta^{2}}{\Delta
x^{6}}-\frac{1}{\Delta
x^{4}}\Bigr{]}+a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\!-\frac{48\Delta\eta^{4}}{\Delta
x^{6}}\Biggr{\\}}.\qquad$

We can at this stage identify six integrable functions, with a factor of $2\pi
i$ extracted for future convenience,

$\displaystyle 2\pi iA_{1}\equiv\frac{\ln(H^{2}\Delta x^{2})}{\Delta
x^{2}}\qquad$ , $\displaystyle\qquad 2\pi iA_{2}\equiv\frac{1}{\Delta
x^{2}}\;,\qquad$ (37) $\displaystyle 2\pi
iB_{1}\equiv\frac{\Delta\eta\ln(H^{2}\Delta x^{2})}{\Delta x^{2}}\qquad$ ,
$\displaystyle\qquad 2\pi iB_{2}\equiv\frac{\Delta\eta}{\Delta
x^{2}}\;,\qquad$ (38) $\displaystyle 2\pi
iC_{1}\equiv\frac{\Delta\eta^{2}\ln(H^{2}\Delta x^{2})}{\Delta x^{2}}\qquad$ ,
$\displaystyle\qquad 2\pi iC_{2}\equiv\frac{\Delta\eta^{2}}{\Delta
x^{2}}\;.\qquad$ (39)

Implementing Step 2 allows us to express $T^{i}_{L}(x;x^{\prime})$ as a sum of
logarithm terms involving $A_{1}$, $B_{1}$ and $C_{1}$, which we term “Group
1”, plus a “remainder” $\Delta T^{i}_{L}(x;x^{\prime})$ which is devoid of
logarithms. For example, expression (36) for $T^{2}_{L}(x;x^{\prime})$ can be
written as,

$T^{2}_{L}=-\frac{i\kappa^{2}}{32\pi^{3}}\Biggl{\\{}-8a^{2}{a^{\prime}}^{2}H^{4}\partial_{0}^{2}A_{1}\\!-\\!4a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}\partial_{0}^{2}C_{1}\\!-\\!5\partial_{0}B_{1}\\!-\\!4A_{1}\Bigr{]}\Biggr{\\}}\\!+\\!\Delta
T^{2}_{L}\;,$ (40)

where the remainder term is,

$\Delta
T^{2}_{L}(x;x^{\prime})=-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}a^{2}{a^{\prime}}^{2}H^{4}\Bigl{[}-\frac{96\Delta\eta^{2}}{\Delta
x^{6}}-\frac{1}{\Delta
x^{4}}\Bigr{]}+a^{3}{a^{\prime}}^{3}H^{6}\\!\times\\!-\frac{48\Delta\eta^{4}}{\Delta
x^{6}}\Biggr{\\}}.$ (41)

Step 3 begins by combining $T^{i}_{N}(x;x^{\prime})$ with the remainder
$\Delta T^{i}_{L}(x;x^{\prime})$. For the case of $T^{2}(x;x^{\prime})$ this
consists of expressions (33) and (41),

$\displaystyle T^{2}_{N}(x;x^{\prime})+\Delta
T^{2}_{L}(x;x^{\prime})=-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}\frac{\frac{1952}{5}}{\Delta
x^{8}}+aa^{\prime}H\Bigl{[}-\frac{416\Delta\eta^{2}}{\Delta
x^{8}}-\frac{\frac{128}{3}}{\Delta x^{6}}\Bigr{]}$ (42) $\displaystyle\hskip
25.6073pt+a^{2}{a^{\prime}}^{2}H^{4}\Bigl{[}-\frac{\frac{400}{3}\Delta\eta^{2}}{\Delta
x^{6}}+\frac{40}{\Delta
x^{4}}\Bigr{]}+a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}-\frac{16\Delta\eta^{4}}{\Delta
x^{6}}+\frac{12\Delta\eta^{2}}{\Delta x^{4}}\Bigr{]}\Biggr{\\}}.\qquad$

The resulting expressions can harbor ultraviolet divergences. One can see this
by once again extracting derivatives,

$\displaystyle T^{2}_{N}(x;x^{\prime})+\Delta T^{2}_{L}(x;x^{\prime})$ (43)
$\displaystyle=-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}\partial^{4}\Bigl{(}\frac{\frac{61}{30}}{\Delta
x^{4}}\Bigr{)}+aa^{\prime}H\Bigl{[}-\partial_{0}^{2}\Bigl{(}\frac{\frac{52}{3}}{\Delta
x^{4}}\Bigr{)}+\partial^{2}\Bigl{(}\frac{\frac{10}{3}}{\Delta
x^{4}}\Bigr{)}\Bigr{]}+a^{2}{a^{\prime}}^{2}H^{4}$
$\displaystyle\times\Bigl{[}-\partial_{0}^{2}\Bigl{(}\frac{\frac{50}{3}}{\Delta
x^{2}}\Bigr{)}\\!+\\!\frac{\frac{220}{3}}{\Delta
x^{4}}\Bigr{]}\\!+\\!a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}-2\partial_{0}^{2}\Bigl{(}\frac{\Delta\eta^{2}}{\Delta
x^{2}}\Bigr{)}\\!+\\!16\partial_{0}\Bigl{(}\frac{\Delta\eta}{\Delta
x^{2}}\Bigr{)}\\!-\\!\frac{12}{\Delta x^{2}}\Bigr{]}\Biggr{\\}}.\qquad$

In addition to the three integrable functions (34) one also encounters
logarithmically divergent factors of $1/\Delta x^{4}$. We consign the
ultraviolet finite factors of $A_{2}$, $B_{2}$ and $C_{2}$ to “Group 2” and
reserve the factors of $1/\Delta x^{4}$ for further analysis,

$\displaystyle T^{2}_{N}\\!+\\!\Delta
T^{2}_{L}=-\frac{i\kappa^{2}}{32\pi^{3}}\Biggl{\\{}-\frac{50}{3}a^{2}{a^{\prime}}^{2}H^{4}\partial_{0}^{2}A_{2}\\!-\\!2a^{3}{a^{\prime}}^{3}H^{6}\Bigl{[}\partial_{0}^{2}C_{2}\\!-\\!8\partial_{0}B_{2}\\!+\\!6A_{2}\Bigr{]}\Biggr{\\}}$
(44)
$\displaystyle-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}\partial^{4}\Bigl{(}\frac{\frac{61}{30}}{\Delta
x^{4}}\Bigr{)}\\!-\\!aa^{\prime}H^{2}\Bigl{[}\frac{52}{3}\partial_{0}^{2}\\!-\\!\frac{10}{3}\partial^{2}\Bigr{]}\frac{1}{\Delta
x^{4}}\\!+\\!\frac{220}{3}a^{2}{a^{\prime}}^{2}H^{4}\Bigl{(}\frac{1}{\Delta
x^{4}}\Bigr{)}\Biggr{\\}}.\qquad$

Step 4 is devoted to reducing and renormalizing the factor of $1/\Delta
x^{4}$. In the dimensionally regulated computation it would derive from
$1/\Delta x^{2D-4}$, from which we extract a factor of $\partial^{2}$,

$\frac{1}{\Delta x^{4}}\longrightarrow\frac{1}{\Delta
x^{2D-4}}=\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{[}\frac{1}{\Delta
x^{2D-6}}\Bigr{]}\;.$ (45)

The factor of $1/\Delta x^{2D-6}$ this produces is integrable in $D=4$
dimensions, so we could take the unregulated limit except for the explicit
factor of $1/(D-4)$. We can add zero in the form of the propagator equation
for a massless scalar in flat space [10, 11],

$\displaystyle\frac{1}{\Delta
x^{4}}\longrightarrow\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{[}\frac{1}{\Delta
x^{2D-6}}\Bigr{]}$ (46)
$\displaystyle=\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{[}\frac{1}{\Delta
x^{2D-6}}-\frac{\mu^{D-4}}{\Delta
x^{D-2}}\Bigr{]}+\frac{\mu^{D-4}4\pi^{\frac{D}{2}}i\delta^{D}(x\\!-\\!x^{\prime})}{2(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)}\;.\qquad$

We can take the unregulated limit of the nonlocal part of (46),

$\frac{\partial^{2}}{2(D\\!-\\!3)(D\\!-\\!4)}\Bigl{[}\frac{1}{\Delta
x^{2D-6}}-\frac{\mu^{D-4}}{\Delta
x^{D-2}}\Bigr{]}\longrightarrow-\frac{\partial^{2}}{4}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{]}\equiv-\frac{\partial^{2}}{4}\Bigl{[}2\pi
iA_{3}\Bigr{]}\;.$ (47)

We call these ultraviolet finite terms “Group 3A”.

The local divergence in the final part of expression (46) requires
explanation. First, recall from the generic expression (17) that the nonlocal
diagrams of Figure 1 inherit a factor of $(aa^{\prime})^{D-2}$ from the two
vertices. Now, note from expressions (25-28) that the first terms in the
expansions of the dimensionally regulated ghost and graviton propagagors
(21-22) each have the form,

$\frac{H^{D-2}\Gamma(\frac{D}{2}\\!-\\!1)}{(4\pi)^{\frac{D}{2}}}\Bigl{(}\frac{4}{y}\Bigr{)}^{\frac{D}{2}-1}=\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{4\pi^{\frac{D}{2}}}\Bigl{(}\frac{1}{aa^{\prime}\Delta
x^{2}}\Bigr{)}^{\frac{D}{2}-1}\;.$ (48)

This means that the scale factors have no dependence on $D$, even in the fully
regulated expression. On the other hand, the counterterms — the final diagram
in Figure 1 — each inherit a factor of $a^{D}$ from the universal $\sqrt{-g}$.
Hence there is always a slight mismatch between the primitive divergences of
the nonlocal diagrams and the counterterms which cancel them. This mismatch
results in a local $\ln(a)$ in the renormalized, unregulated limit,

$\displaystyle\frac{\mu^{D-4}4\pi^{\frac{D}{2}}i\delta^{D}(x\\!-\\!x^{\prime})}{2(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{a^{D-4}\mu^{D-4}4\pi^{\frac{D}{2}}i\delta^{D}(x\\!-\\!x^{\prime})}{2(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)}$
(49) $\displaystyle\hskip
199.16928pt\longrightarrow-2\pi^{2}i\\!\times\\!\ln(a)\delta^{4}(x\\!-\\!x^{\prime})\;.\qquad$

These local terms are termed “Group 3B”.

####  2.2.3 The Schwinger-Keldysh Result

The factors of $\delta^{4}(x-x^{\prime})$ in Group 3B are already real and
causal, however, the various nonlocal factors (47) and (37-39) are not. This
is because Feynman diagrams represent in-out matrix elements rather than true
expectation values. The Schwinger-Keldysh formalism [12, 13, 14, 15, 16] is a
diagrammtic technique for computing true expectation values that is almost as
simple to use as Feynman diagrams. Expectation values of the graviton field
obey effective field equations which are nonlocal but real and causal [17, 18,
19].

It is simple to convert an in-out 1PI $N$-point function such as the graviton
self-energy to the Schwinger-Keldysh formalism. Without giving the derivation,
the rules are [20]:

  * •

Each spacetime point is assigned a polarity $\pm$.

  * •

One consequence is that every Feynman propagator $i\Delta(x;x^{\prime})$ gives
rise to four Schwinger-Keldysh propagators $i\Delta_{\pm\pm}(x;x^{\prime})$.
The $++$ propagator is the same as the Feynman propagator and the $--$
propagator is its complex conjugate. The $-+$ propagator is the free
expectation of the field at $x^{\mu}$ times the field at ${x^{\prime}}^{\mu}$,
and the $+-$ propagator is the free expectation value of the reverse-ordered
product.

  * •

Another consequence is that each vertex has a $\pm$ polarity. The $+$ vertices
are the same as those of the in-out formalism while the $-$ vertices are
complex cojugates.

  * •

A final consequence is that every in-out 1PI $N$-point function gives rise to
$2^{N}$ $N$-point functions in the Schwinger-Keldysh formalism.

  * •

The factor of $[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ in the
linearized quantum Einstein equation (3) is replaced by the sum of
$[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{++}](x;x^{\prime})$, which is the same
as the in-out result, and
$[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{+-}](x;x^{\prime})$.

  * •

On our simple background (2), one can infer the result for
$[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{+-}](x;x^{\prime})$ from that for
$[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ by dropping all the
local contributions of Group 3B, multiplying the nonlocal terms by $-1$, and
converting the coordinate interval $\Delta x^{2}$ from

$\Delta
x^{2}_{++}(x;x^{\prime})\equiv\Bigl{\|}\vec{x}-\vec{x}^{\prime}\Bigr{\|}^{2}-\Bigl{(}|\eta\\!-\\!\eta^{\prime}|-i\varepsilon\Bigr{)}^{2}\;,$
(50)

to

$\Delta
x^{2}_{+-}(x;x^{\prime})\equiv\Bigl{\|}\vec{x}-\vec{x}^{\prime}\Bigr{\|}^{2}-\Bigl{(}\eta\\!-\\!\eta^{\prime}+i\varepsilon\Bigr{)}^{2}\;.$
(51)

Implementing these rules is straightforward. First, recall that the only
dependence on the coordinate interval $\Delta x^{2}$ in the nonlocal results
of Groups 1, 2, and 3A comes through the integrable functions $A_{1-3}$,
$B_{12}$ and $C_{1-2}$, which were defined in expressions (37-39) and (47). We
can eliminate the factors of $1/\Delta x^{2}$ by extracting derivatives. For
example, the $++$ and $+-$ versions of $2\pi i\times A_{1}$ are,

$2\pi i\\!\times\\!A_{1}=\frac{\ln(H^{2}\Delta x^{2}_{+\pm})}{\Delta
x^{2}_{+\pm}}=\frac{\partial^{2}}{8}\Bigl{[}\ln^{2}(H^{2}\Delta
x^{2}_{+\pm})\\!-\\!2\ln(H^{2}\Delta x^{2}_{+\pm})\Bigr{]}\;.$ (52)

Because the scale factors and the derivatives are identical in the $++$ and
$+-$ contributions, we need just consider differences of logarithms,

$\displaystyle\ln(H^{2}\Delta x^{2}_{++})-\ln(H^{2}\Delta x^{2}_{+-})$
$\displaystyle=$ $\displaystyle 2\pi
i\\!\times\\!\theta(\Delta\eta\\!-\\!r)\;,$ (53)
$\displaystyle\ln^{2}(H^{2}\Delta x^{2}_{++})-\ln^{2}(H^{2}\Delta x^{2}_{+-})$
$\displaystyle=$ $\displaystyle 4\pi
i\\!\times\\!\theta(\Delta\eta\\!-\\!r)\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\;,\qquad$
(54)

where $r\equiv\|\vec{x}-\vec{x}^{\prime}\|$. The final reductions are,

$\displaystyle A_{1}$ $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle+\frac{\partial^{2}}{4}\Biggl{\\{}\theta(\Delta\eta\\!-\\!r)\Bigl{[}\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\\!-\\!1\Bigr{]}\Biggr{\\}}\;,\qquad$
(55) $\displaystyle B_{1}$ $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle-\frac{\partial_{0}}{2}\Biggl{\\{}\theta(\Delta\eta\\!-\\!r)\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\Biggr{\\}}\;,\qquad$
(56) $\displaystyle C_{1}$ $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle+\frac{\partial_{0}^{2}}{4}\Biggl{\\{}\theta(\Delta\eta\\!-\\!r)(r^{2}\\!-\\!\Delta\eta^{2})\Bigl{[}\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\\!-\\!1\Bigr{]}\Biggr{\\}}$
(57) $\displaystyle\hskip
156.49014pt+\frac{1}{2}\theta(\Delta\eta\\!-\\!r)\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\;,\qquad$
$\displaystyle A_{2}$ $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle+\frac{\partial^{2}}{4}\Biggl{\\{}\theta(\Delta\eta\\!-\\!r)\Biggr{\\}}\;,\qquad$
(58) $\displaystyle B_{2}$ $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle-\frac{\partial_{0}}{2}\Biggl{\\{}\theta(\Delta\eta\\!-\\!r)\Biggr{\\}}\;,\qquad$
(59) $\displaystyle C_{2}$ $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle+\frac{\partial_{0}^{2}}{4}\Biggl{\\{}\theta(\Delta\eta\\!-\\!r)(r^{2}\\!-\\!\Delta\eta^{2})\Biggr{\\}}+\frac{1}{2}\theta(\Delta\eta\\!-\\!r)\;,\qquad$
(60) $\displaystyle A_{3}$ $\displaystyle\\!\\!\\!\longrightarrow\\!\\!\\!$
$\displaystyle+\frac{\partial^{2}}{4}\Biggl{\\{}\theta(\Delta\eta\\!-\\!r)\Bigl{[}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\\!-\\!1\Bigr{]}\Biggr{\\}}\;.\qquad$
(61)

###  2.3 The 4-Point Contribution

The previous discussion concerned the two nonlocal diagrams of Figure 1, and
the local counterterms needed to renormalize them. There are also finite local
contributions from the 3rd diagram. It derives from the 42 4-graviton
interactions given in equation (4.1) of [9]. One connects two of the graviton
fields to the external legs and then replaces the remaining two fields by
graviton propagator. The procedure is tedious and we shall content ourselves
with simply sketching it and giving the final result.

As an example we reduce the first of the 42 interactions,

$\mathcal{S}_{1}\equiv\frac{\kappa^{2}}{32}\\!\int\\!\\!d^{D}x\,a^{D-2}h^{2}h_{,\theta}h^{,\theta}\;,$
(63)

where a comma denotes differentiation and the trace of the graviton field is
$h\equiv h^{\alpha}_{~{}\alpha}\equiv\eta^{\alpha\beta}h_{\alpha\beta}$. We
first take the variational derivatives of the action integral with respect to
$h_{\mu\nu}(x)$ and $h_{\rho\sigma}(x^{\prime})$ as in expression (14),

$\displaystyle\frac{i\delta^{2}\mathcal{S}_{1}}{\delta h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}=\frac{\kappa^{2}}{32}\eta^{\mu\nu}\eta^{\rho\sigma}\Biggl{\\{}-\partial_{\theta}\Bigl{[}2a^{D-2}h^{\alpha}_{~{}\alpha}(x)h^{\beta}_{~{}\beta}(x)\partial^{\theta}i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}$
(64)
$\displaystyle+4a^{D-2}h^{\alpha}_{~{}\alpha}(x)h^{\beta}_{~{}\beta,\theta}(x)\partial^{\theta}i\delta^{D}(x\\!-\\!x^{\prime})\\!-\\!4\partial^{\theta}\Bigl{[}a^{D-2}h^{\alpha}_{~{}\alpha}(x)h^{\beta}_{~{}\beta,\theta}(x)i\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}$
$\displaystyle\hskip
156.49014pt+2a^{D-2}h^{\alpha}_{~{}\alpha,\theta}(x)h^{\beta}_{~{}\beta,\theta}(x)\Big{]}i\delta^{D}(x\\!-\\!x^{\prime})\Biggr{\\}}.\qquad$

Now take the expectation value of the $T^{*}$-ordered product, which amounts
to replacing the remaining two graviton fields of each term by the appropriate
coincident (and sometimes differentiated) propagator,

$\displaystyle\Bigl{\langle}\Omega\Bigl{|}T^{*}\Bigl{[}\frac{i\delta^{2}\mathcal{S}_{1}}{\delta
h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}\Bigr{|}\Omega\Bigr{\rangle}=\frac{\kappa^{2}}{32}\eta^{\mu\nu}\eta^{\rho\sigma}\Biggl{\\{}-\partial_{\theta}\Biggl{[}2a^{D-2}\\!\times\\!i\\!\Bigl{[}\mbox{}^{\alpha}_{~{}\alpha}\Delta^{\beta}_{~{}\beta}\Bigr{]}\\!(x;x)$
(65)
$\displaystyle\times\partial^{\theta}i\delta^{D}(x\\!-\\!x^{\prime})\Biggr{]}\\!+\\!4a^{D-2}\\!\times\\!\partial^{\prime}_{\theta}i\\!\Bigl{[}\mbox{}^{\alpha}_{~{}\alpha}\Delta^{\beta}_{~{}\beta}\Bigr{]}\\!(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}\\!\\!\\!\\!\\!\\!\\!\\!\times\partial^{\theta}i\delta^{D}\\!(x\\!-\\!x^{\prime})\\!-\\!4\partial^{\theta}\Biggl{[}a^{D-2}$
$\displaystyle\times
i\delta^{D}\\!(x\\!-\\!x^{\prime})\\!\times\\!\partial^{\prime}_{\theta}i\\!\Bigl{[}\mbox{}^{\alpha}_{~{}\alpha}\Delta^{\beta}_{~{}\beta}\Bigr{]}\\!(x;x^{\prime})\\!\Biggr{]}\\!\\!+\\!2a^{D-2}i\delta^{D}\\!(x\\!-\\!x^{\prime})\\!\times\\!\partial_{\theta}\partial^{\prime\theta}i\\!\Bigl{[}\mbox{}^{\alpha}_{~{}\alpha}\Delta^{\beta}_{~{}\beta}\Bigr{]}\\!(x;x^{\prime})\\!\Biggr{\\}}.\qquad$

Finally, we express the tensor structure using the 21 basis tensors of Group
1,

$\displaystyle\eta^{\mu\nu}\eta^{\rho\sigma}$ $\displaystyle=$
$\displaystyle\Bigl{(}\overline{\eta}^{\mu\nu}\\!-\\!\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\Bigr{)}\Bigl{(}\overline{\eta}^{\rho\sigma}\\!-\\!\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}\Bigr{)}\;,$
(66) $\displaystyle=$
$\displaystyle\Bigl{[}\mbox{}^{\mu\nu}\mathcal{D}_{1}^{\rho\sigma}\Bigr{]}\\!-\\!\Bigl{[}\mbox{}^{\mu\nu}\mathcal{D}_{3}^{\rho\sigma}\Bigr{]}\\!-\\!\Bigl{[}\mbox{}^{\mu\nu}\mathcal{D}_{4}^{\rho\sigma}\Bigr{]}\\!+\\!\Bigl{[}\mbox{}^{\mu\nu}\mathcal{D}_{13}^{\rho\sigma}\Bigr{]}\;.\qquad$
(67)

The coincidence limits of the three propagators which appear in the graviton
propagator (22) are,

$\displaystyle i\Delta_{A}(x;x)=k\Bigl{[}-\pi{\rm cot}\Bigl{(}\frac{\pi
D}{2}\Bigr{)}\\!+\\!2\ln(a)\Bigr{]}\quad$ , $\displaystyle\quad
i\Delta_{B}(x;x)=-\frac{k}{D\\!-\\!2}\;,\qquad$ (68) $\displaystyle
i\Delta_{C}(x;x)=\frac{k}{(D\\!-\\!2)(D\\!-\\!3)}\quad$ , $\displaystyle\quad
k\equiv\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\;.\qquad$
(69)

Note that only the undifferentiated $A$-type propagator is ultraviolet
divergent in dimensional regularization. The undifferentiated $A$-type
propagator is also the only way to get a factor of $\ln(a)$. First derivatives
of coincident propagators are all finite,

$\partial_{\alpha}i\Delta_{A}(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}=aHk\delta^{0}_{~{}\alpha}\;\;,\;\;\partial_{\alpha}i\Delta_{B}(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}=0=\partial_{\alpha}i\Delta_{C}(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}\;.$
(70)

Mixed second derivatives are also finite,

$\displaystyle\partial_{\alpha}\partial^{\prime}_{\beta}i\Delta_{A}(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}$
$\displaystyle=$
$\displaystyle-\Bigl{(}\frac{D\\!-\\!1}{D}\Bigr{)}kH^{2}g_{\alpha\beta}\;,$
(71)
$\displaystyle\partial_{\alpha}\partial^{\prime}_{\beta}i\Delta_{B}(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}$
$\displaystyle=$ $\displaystyle\frac{1}{D}kH^{2}g_{\alpha\beta}\;,$ (72)
$\displaystyle\partial_{\alpha}\partial^{\prime}_{\beta}i\Delta_{C}(x;x^{\prime})\Bigl{|}_{x^{\prime}=x}$
$\displaystyle=$
$\displaystyle-\frac{2}{D(D\\!-\\!2)}kH^{2}g_{\alpha\beta}\;.$ (73)

Note that all primitive contributions have factors of $a^{D-2}$, $a^{D-1}$ or
$a^{D}$. The counterterms which absorb ultraviolet divergences possess the
very same dependence on $a$ so renormalization engenders no finite factors of
$\ln(a)$ the way it did for the nonlocal diagrams of expression (49). It does
produce factors of $\ln(H/\mu)$ but we report only the $\ln(a)$ contributions
to $iT^{2}(x;x^{\prime})$,

$\frac{\kappa^{2}\ln(a)}{32\pi^{2}}\\!\times\\!8a^{2}H^{2}(\partial_{0}\\!+\\!2aH)\partial_{0}\delta^{4}(x\\!-\\!x^{\prime})\;.$
(74)

We call this class of contributions “Group 4”.

##  3 The Effect on Gravitational Radiation

The purpose of this section is to solve (3) for $T^{\mu\nu}_{~{}\rm lin}(x)=0$
to derive one loop corrections to the graviton mode function. We begin by
deriving an equation for the graviton mode function, then consider the one
loop response to various sources. To simplify the computation we specialize
the various source terms to late times and small wave numbers. Finally, we
compute the source integrals and read off the late time result for the one
loop correction.

Dynamical gravitons correspond to zero stress tensor and the wave form,

$h_{\mu\nu}(x;\vec{k})=\epsilon_{\mu\nu}(\vec{k})u(\eta,k)e^{i\vec{k}\cdot\vec{x}}\qquad,\qquad
0=\epsilon_{\mu 0}=\epsilon_{ij}k_{j}=\epsilon_{ii}\;.$ (75)

The graviton polarization tensors $\epsilon_{\mu\nu}(\vec{k})$ is identical to
the usual ones from flat space. From their properties, and the $3+1$
decomposition (6-8) of the gauge-fixed kinetic operator, we find,

$\mathcal{D}^{\mu\nu\rho\sigma}h_{\rho\sigma}(x;\vec{k})=\epsilon^{\mu\nu}(\vec{k})e^{i\vec{k}\cdot\vec{x}}\\!\times\\!-\frac{1}{2}a^{2}\Bigl{[}\partial_{0}^{2}\\!+\\!2Ha\partial_{0}\\!+\\!k^{2}\Bigr{]}u(\eta,k)\;.$
(76)

The graviton wave form (75) implies $0=h_{00}=h_{0i}=h_{ij,j}$. Using these
relations in the $3+1$ decomposition (11-13) of the quantum source term
implies,

$\displaystyle\int\\!\\!d^{4}x^{\prime}\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})h_{\rho\sigma}(x^{\prime};\vec{k})$
(77) $\displaystyle\hskip
85.35826pt=\epsilon^{\mu\nu}(\vec{k})e^{i\vec{k}\cdot\vec{x}}\\!\times\\!\\!\int\\!\\!d^{4}x^{\prime}\,iT^{2}_{SK}(x;x^{\prime})u(\eta^{\prime},k)e^{-i\vec{k}\cdot(\vec{x}-\vec{x}^{\prime})}\;.\qquad$

Equating the left hand side (76) with the right hand side (77) gives the
graviton mode equation,

$-\frac{1}{2}a^{2}\Bigl{[}\partial_{0}^{2}\\!+\\!2Ha\partial_{0}\\!+\\!k^{2}\Bigr{]}u(\eta,k)=\int\\!\\!d^{4}x^{\prime}\,iT^{2}_{SK}(x;x^{\prime})u(\eta^{\prime},k)e^{-i\vec{k}\cdot\Delta\vec{x}}\;.$
(78)

The graviton mode equation (78) is valid to all orders, but we only possess
one loop results for the coefficient function $T^{2}_{SK}(x;x^{\prime})$.
Hence we must solve the equation perturbatively. The 0th order mode function,
and its asymptotic late time expansion, are,

$u_{0}(\eta,k)=\frac{H}{\sqrt{2k^{3}}}\Bigl{[}1-\frac{ik}{Ha}\Bigr{]}\exp\Bigl{[}\frac{ik}{Ha}\Bigr{]}\longrightarrow\frac{H}{\sqrt{2k^{3}}}\Bigl{[}1+\frac{k^{2}}{2H^{2}a^{2}}+\dots\Bigr{]}.$
(79)

The one loop correction we seek obeys the equation,

$-\frac{1}{2}a^{2}\Bigl{[}\partial_{0}^{2}\\!+\\!2Ha\partial_{0}\\!+\\!k^{2}\Bigr{]}u_{1}(\eta,k)=\int\\!\\!d^{4}x^{\prime}\,iT^{2}_{SK}(x;x^{\prime})u_{0}(\eta^{\prime},k)e^{-i\vec{k}\cdot\Delta\vec{x}}\equiv
S(\eta)\;.$ (80)

Finally, let us consider the relation between possible late time behaviors for
the quantum source $S(\eta)$ and the corresponding asymptotic late time form
of the one loop mode function,

$\displaystyle S=a^{4}\ln(a)\Longrightarrow
u_{1}\longrightarrow-\frac{\ln^{2}(a)}{3H^{2}}$ , $\displaystyle
S=a^{4}\Longrightarrow u_{1}\longrightarrow-\frac{2\ln(a)}{3H^{2}}\;,\qquad$
(81) $\displaystyle S=a^{3}\ln(a)\Longrightarrow
u_{1}\longrightarrow+\frac{\ln(a)}{H^{2}a}$ , $\displaystyle
S=a^{3}\Longrightarrow u_{1}\longrightarrow+\frac{1}{H^{2}a}\;,\qquad$ (82)
$\displaystyle S=a^{2}\ln(a)\Longrightarrow
u_{1}\longrightarrow+\frac{\ln(a)}{H^{2}a^{2}}$ , $\displaystyle
S=a^{2}\Longrightarrow u_{1}\longrightarrow+\frac{1}{H^{2}a^{2}}\;.$ (83)

Let us start with the local contributions of Groups 3B and 4,

$\displaystyle S_{3B}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\kappa^{2}\ln(a)}{32\pi^{2}}\Biggl{\\{}-\frac{61}{30}\partial^{4}u_{0}+aH^{2}\Bigl{[}\frac{52}{3}\partial_{0}^{2}\\!-\\!\frac{10}{3}\partial^{2}\Bigr{]}(au_{0})-\frac{220}{3}a^{4}H^{4}u_{0}\Biggr{\\}},\qquad$
(84) $\displaystyle S_{4}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{\kappa^{2}\ln(a)}{32\pi^{2}}\Biggl{\\{}8a^{2}H^{2}\partial_{0}^{2}u_{0}\\!+\\!16a^{3}H^{3}\partial_{0}u_{0}\Biggr{\\}},$
(85)

where $\partial^{2}\rightarrow-(\partial_{0}^{2}+k^{2})$. Expression (85) was
studied previously when making the Hartree approximation [21], but it is
simple enough to evaluate both local contributions (84-85) using relations
such as,

$\displaystyle\partial_{0}u_{0}(\eta,k)=\frac{H}{\sqrt{2k^{3}}}\\!\times\\!k^{2}\eta
e^{-ik\eta}$ ,
$\displaystyle\partial_{0}^{2}u_{0}(\eta,k)=\frac{H}{\sqrt{2k^{3}}}\Bigl{(}k^{2}\\!-\\!ik^{3}\eta\Bigr{)}e^{-ik\eta}\;,\qquad$
(86) $\displaystyle(\partial_{0}^{2}\\!+\\!k^{2})^{2}u_{0}(\eta,k)=0$ ,
$\displaystyle(\partial_{0}^{2}\\!+\\!k^{2})\Bigl{(}au_{0}(\eta,k)\Bigr{)}=2a^{3}H^{2}u_{0}(\eta,k)\;.\qquad$
(87)

Hence (84-85) are,

$\displaystyle S_{3B}(\eta)$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}H^{4}}{32\pi^{2}}\\!\times\\!a^{4}\ln(a)\\!\times\\!\Bigl{[}-32\\!-\\!\frac{52}{3}\frac{k^{2}}{a^{2}H^{2}}\Bigr{]}u_{0}(\eta,k)\;,\qquad$
(88) $\displaystyle S_{4}(\eta)$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}H^{4}}{32\pi^{2}}\\!\times\\!a^{4}\ln(a)\\!\times\\!\Bigl{[}0\\!-\\!\frac{8k^{2}}{a^{2}H^{2}}\Bigr{]}u_{0}(\eta,k)\;.\qquad$
(89)

However, to recover the leading late time behavior we can set
$\partial_{0}^{2}+k^{2}$ to just $\partial_{0}^{2}$, and $u_{0}(\eta,k)$ to
just $u_{0}(0,k)$,

$\displaystyle S_{3B}(\eta)$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\kappa^{2}H^{4}}{4\pi^{2}}\\!\times\\!u_{0}(0,k)\\!\times\\!-4a^{4}\ln(a)\;,\qquad$
(90) $\displaystyle S_{4}(\eta)$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\kappa^{2}H^{4}}{4\pi^{2}}\\!\times\\!u_{0}(0,k)\\!\times\\!0\;.\qquad$
(91)

Old results [22, 23] could be exploited to evaluate the nonlocal contributions
from Groups 1, 2 and 3A exactly, the same way we did with the local
contribution (88). However, if we are only interested in the leading behavior
at late times, as in expression (90), we can replace
$u_{0}(\eta^{\prime},k)e^{-i\vec{k}\cdot\Delta\vec{x}}$ with $u_{0}(0,k)$,
which means that $\partial^{2}$ degenerates to just $-\partial_{0}^{2}$. We
also change $\partial_{0}$ to $-\partial_{0}^{\prime}$ and partially integrate
to act on any primed scale factors, or give zero if there are no primed scale
factors. The source integrals for Groups 1, 2 and 3A are,

$\displaystyle
S_{1}\longrightarrow\frac{\kappa^{2}}{32\pi^{3}}\\!\times\\!u_{0}(0,k)\\!\times\\!\\!\int\\!\\!d^{4}x^{\prime}\,\theta(\Delta\eta\\!-\\!r)\Biggl{\\{}-144a^{3}{a^{\prime}}^{5}H^{8}\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]$
(92) $\displaystyle\hskip
28.45274pt+\Bigl{[}240a^{2}{a^{\prime}}^{6}H^{8}\\!+\\!48a^{3}{a^{\prime}}^{5}H^{8}\\!-\\!360a^{3}{a^{\prime}}^{7}H^{10}(r^{2}\\!-\\!\Delta\eta^{2})\Bigr{]}$
$\displaystyle\hskip
207.7052pt\times\Bigl{[}\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\\!-\\!1\Bigr{]}\Biggr{\\}},\qquad$
$\displaystyle
S_{2}\longrightarrow\frac{\kappa^{2}}{32\pi^{3}}\\!\times\\!u_{0}(0,k)\\!\times\\!\\!\int\\!\\!d^{4}x^{\prime}\,\theta(\Delta\eta\\!-\\!r)\Biggl{\\{}500a^{2}{a^{\prime}}^{6}H^{8}\\!-\\!72a^{3}{a^{\prime}}^{5}H^{8}$
(93) $\displaystyle\hskip
207.7052pt-180a^{3}{a^{\prime}}^{7}H^{10}(r^{2}\\!-\\!\Delta\eta^{2})\Biggr{\\}},\qquad$
$\displaystyle
S_{3A}\longrightarrow\frac{\kappa^{2}}{32\pi^{3}}\\!\times\\!u_{0}(0,k)\\!\times\\!\\!\int\\!\\!d^{4}x^{\prime}\,\theta(\Delta\eta\\!-\\!r)\Biggl{\\{}\Bigl{[}930a{a^{\prime}}^{7}H^{8}\\!-\\!550a^{2}{a^{\prime}}^{6}H^{8}\Bigr{]}$
(94) $\displaystyle\hskip
213.39566pt\times\Bigl{[}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!r^{2})]\\!-\\!1\Bigr{]}\Biggr{\\}}.\qquad$

The next step is to perform the spatial integrations,

$\displaystyle\int\\!\\!d^{3}x^{\prime}\,\theta(\Delta\eta\\!-\\!r)=\frac{4\pi}{3}\Delta\eta^{3}\theta(\Delta\eta)\;,$
(95)
$\displaystyle\int\\!\\!d^{3}x^{\prime}\,\theta(\Delta\eta\\!-\\!r)\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]=\frac{8\pi}{3}\Bigl{[}\ln(2H\Delta\eta)\\!-\\!\frac{4}{3}\Bigr{]}\Delta\eta^{3}\theta(\Delta\eta)\;,$
(96)
$\displaystyle\int\\!\\!d^{3}x^{\prime}\,\theta(\Delta\eta\\!-\\!r)(r^{2}\\!-\\!\Delta\eta^{2})=-\frac{8\pi}{15}\Delta\eta^{5}\theta(\Delta\eta)\;,$
(97)
$\displaystyle\int\\!\\!d^{3}x^{\prime}\,\theta(\Delta\eta\\!-\\!r)(r^{2}\\!-\\!\Delta\eta^{2})\ln[H^{2}(\Delta\eta^{2}\\!-\\!r^{2})]$
(98) $\displaystyle\hskip
153.6447pt=-\frac{8\pi}{15}\Bigl{[}2\ln(2H\Delta\eta)\\!-\\!\frac{31}{15}\Bigr{]}\Delta\eta^{5}\theta(\Delta\eta)\;.\qquad$

We also change variables from $\eta^{\prime}$ to
$a^{\prime}=-1/H\eta^{\prime}$,

$\int_{\eta_{i}}^{\eta}\\!\\!\\!\\!d\eta^{\prime}=\int_{1}^{a}\\!\\!\frac{da^{\prime}}{H{a^{\prime}}^{2}}\;,$
(99)

where the initial time $\eta_{i}=-1/H$ corresponds to unit scale factor. Note
also that $a^{\prime}H\Delta\eta=1-a^{\prime}/a$. These reductions allow us to
express (92-94) as,

$\displaystyle
S_{1}\longrightarrow\frac{\kappa^{2}H^{4}}{4\pi^{2}}u_{0}(0,k)\\!\\!\\!\int_{1}^{a}\\!\\!\\!\\!da^{\prime}\Biggl{\\{}\\!-48a^{3}\Bigl{(}\\!1\\!-\\!\frac{a^{\prime}}{a}\Bigr{)}^{\\!3}\\!\Bigl{[}\ln(2H\Delta\eta)\\!-\\!\frac{4}{3}\Bigr{]}\\!+\\!\Bigl{[}80a^{2}a^{\prime}\\!+\\!16a^{3}\Bigr{]}$
(100) $\displaystyle\hskip
14.22636pt\times\Bigl{(}1\\!-\\!\frac{a^{\prime}}{a}\Bigr{)}^{3}\Bigl{[}\ln(2H\Delta\eta)\\!-\\!\frac{11}{6}\Bigr{]}+48a^{3}\Bigl{(}1\\!-\\!\frac{a^{\prime}}{a}\Bigr{)}^{5}\Bigl{[}\ln(2H\Delta\eta)\\!-\\!\frac{23}{15}\Bigr{]}\Biggr{\\}},\qquad$
$\displaystyle
S_{2}\longrightarrow\frac{\kappa^{2}H^{4}}{4\pi^{2}}u_{0}(0,k)\\!\\!\\!\int_{1}^{a}\\!\\!\\!\\!da^{\prime}\Biggl{\\{}\Bigl{[}\frac{250}{3}a^{2}a^{\prime}\\!-\\!12a^{3}\Bigr{]}\Bigl{(}\\!1\\!-\\!\frac{a^{\prime}}{a}\Bigr{)}^{3}\\!\\!+\\!12a^{3}\Bigl{(}1\\!-\\!\frac{a^{\prime}}{a}\Bigr{)}^{5}\Biggr{\\}},\qquad$
(101) $\displaystyle
S_{3A}\longrightarrow\frac{\kappa^{2}H^{4}}{4\pi^{2}}u_{0}(0,k)\\!\\!\\!\int_{1}^{a}\\!\\!\\!\\!da^{\prime}\Bigl{[}310a{a^{\prime}}^{2}\\!-\\!\frac{550}{3}a^{2}a^{\prime}\Bigr{]}\Bigl{(}\\!1\\!-\\!\frac{a^{\prime}}{a}\Bigr{)}^{3}\\!\Bigl{[}\ln(2\mu\Delta\eta)\\!-\\!\frac{11}{6}\Bigr{]}.\qquad$
(102)

It remains to perform the temporal integrations. These can be done exactly but
we give only the coefficients of the leading $a^{4}\ln(a)$ dependence,

$\displaystyle S_{1}$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\kappa^{2}H^{4}}{4\pi^{2}}\\!\times\\!u_{0}(0,k)\\!\times\\!-4a^{4}\ln(a)\;,\qquad$
(103) $\displaystyle S_{2}$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\kappa^{2}H^{4}}{4\pi^{2}}\\!\times\\!u_{0}(0,k)\\!\times\\!0\;,\qquad$
(104) $\displaystyle S_{3A}$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\kappa^{2}H^{4}}{4\pi^{2}}\\!\times\\!u_{0}(0,k)\\!\times\\!4a^{4}\ln(a)\;.\qquad$
(105)

Combining these with the local results (90-91) we see that the late time
asymptotic form of the total source is,222It is interesting to note that the
terms of Groups 3A and 3B, which ultimately descended from factors of
$1/\Delta x^{4}$, cancel one another at leading order. It is also interesting
that the nonzero contribution from Group 1 derives entirely from $A_{1}$
because the $B_{1}$ and $C_{1}$ contributions also cancel.

$S(\eta)\longrightarrow\frac{\kappa^{2}H^{4}}{4\pi^{2}}\\!\times\\!u_{0}(0,k)\\!\times\\!-4a^{4}\ln(a)\;.$
(106)

In view of expression (81) it follows that the asymptotic form of the one loop
correction to the mode function is,

$u_{1}(\eta,k)\longrightarrow\frac{\kappa^{2}H^{2}}{4\pi^{2}}\\!\times\\!u_{0}(0,k)\\!\times\\!\frac{4}{3}\ln^{2}(a)\;.$
(107)

##  4 Epilogue

In section 2 of this paper we give a 4-step procedure for converting an old,
unregulated computation [7] of the single graviton loop contribution to the
self-energy $-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ to the
fully renormalized result, with only one additional computation needed for the
4-point contribution. The technique laid out in Section 2 has been used to
obtain results for all 21 of the coefficient functions $T^{i}(x;x^{\prime})$
in the representation (10), and these will be reported in a subsequent work
[24]. Here we give only the coefficient $T^{2}(x;x^{\prime})$ which enters the
equation (78) for the mode function $u(\eta,k)$ of plane wave gravitons (75).
In section 3 we applied this result to derive the asymptotic late time form
(107) for one loop corrections to $u(\eta,k)$.

There are two striking features about our result (107). First, it grows with
time. We saw in section 3 that this growth derives entirely from the “tail”
part of the graviton propagator, which is associated with the continual
production of gravitons with Hubble-scale momenta. The physical interpretation
is that the spin-spin coupling allows our test graviton to remain in
interaction with these gravitons, long after its own kinetic energy has red-
shifted to insignificance. The interaction produces a growth in the amplitude
for the simple reason that scattering between very long wavelength and shorter
wavelength particles tends to transfer momentum from the latter to the former.
Even though the loop counting parameter $\kappa^{2}H^{2}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}10^{-10}$ is
minuscule for actual inflation, this growth must eventually cause perturbation
theory to break down if inflation persists long enough.

The second striking feature of our result is that it implies a potentially
observable one loop correction to the tensor power spectrum,

$\Delta_{h}^{2}(k)\simeq\frac{16\hbar GH^{2}(t_{k})}{\pi
c^{3}}\Biggl{\\{}1+\frac{16\hbar GH^{2}(t_{k})}{3\pi
c^{3}}\\!\times\\!N_{k}^{2}+O(G^{2})\Biggr{\\}}\;,$ (108)

where $H(t_{k})$ is the value of the Hubble parameter at horizon crossing
(defined by $a(t_{k})H(t_{k})=k$) and $N_{k}$ is the number of e-foldings from
horizon crossing to the end of inflation.333Note that our result obeys the
bound on loop corrections to the primordial power spectrum derived by the late
Steven Weinberg [25, 26]. No one knows how small the tree order power spectrum
is, but if it is near the current upper bound of $10^{-10}$ [27] then the
large factor of $N_{k}^{2}$ in (108) might allow the one loop correction to be
resolved in the far future, after the full development of 21cm cosmology [28,
29, 30].

The great uncertainty in all this is the gauge issue. The graviton propagator
requires gauge fixing, and how one accomplishes that can affect loop
corrections. This is well known in flat space. For example, the numerical
coefficient $\frac{61}{30}$ in the first term of equation (84) derives from
flat space. When the graviton self-energy is re-computed in the most general
Poincaré invariant gauge ($\mathcal{L}=-\frac{1}{2\alpha}F_{\mu}F^{\mu}$, with
$F_{\mu}=\partial^{\nu}h_{\mu\nu}-\frac{\beta}{2}\partial_{\mu}h$), this
numerical coefficient becomes [31],

$\frac{61}{30}\longrightarrow\frac{1}{2}\alpha^{2}+\frac{3}{4}\alpha-\frac{43}{60}+\frac{1}{6}\frac{(\alpha\\!-\\!3)^{2}}{(\beta\\!-\\!2)^{4}}-\frac{7}{4}\frac{(\alpha\\!-\\!3)}{(\beta\\!-\\!2)^{3}}-\frac{1}{12}\frac{(\alpha\\!-\\!51)}{(\beta\\!-\\!2)^{2}}+\frac{1}{12}\frac{(9\alpha\\!-\\!11)}{(\beta\\!-\\!2)}\;.$
(109)

This particular term makes zero correction to $u(\eta,k)$, and the gauge
dependence of the distinctly de Sitter terms which produce (107) is not known.
Computations on de Sitter background are so difficult that all but one [32] of
the graviton loop results [9, 7, 33, 34, 35, 36, 37, 38, 39, 40] so far
obtained have been made using the simplest gauge [3, 4]. The single exception
[32] was a re-computation of the vacuum polarization [37] in a 1-parameter
family of de Sitter invariant gauges [41]. When that result was used to
compute quantum gravitational corrections the photon field strength [42], the
same asymptotic time dependence was found as for the simple, de Sitter
breaking gauge [43], but with a different numerical coefficient, and no
dependence at all on the gauge parameter. One must therefore assume a slight
gauge dependence in cosmological solutions to the naive effective field
equations.

John Donoghue has demonstrated how general relativity can be used as a low
energy effective field theory on flat space background to derive quantum
gravitational corrections to the long-range potentials carried by massless
particles such as photons and gravitons [44, 45]. The method is first to
compute the scattering amplitude between two massive particles that exchange
the massless field, then infer the exchange potential using inverse scattering
theory. That is how gauge independent, graviton loop corrections were derived
for the Newtonian [46, 47] and Coulomb [48] potentials.

It has recently been shown that Donoghue’s S-matrix technique can be applied
directly to produce gauge independent effective field equations, without
computing cosmologically unobservable scattering amplitudes [49]. The new
procedure employs position space versions of crucial identities Donoghue and
collaborators derived to isolate the nonlocal and nonanalytic parts of
scattering amplitudes that affect long-range potentials [45, 50]. These
“Donoghue Identities” reduce internal massive propagators to delta functions,
which causes higher-point contributions to 2-particle scattering to assume a
form that can be regarded as corrections to the 1PI 2-point function of the
massless exchange particle. The new technique was implemented at one loop
order for quantum gravitational corrections to a massless scalar on flat space
background to derive a result which is indpendent of the gauge parameters
$\alpha$ and $\beta$ [49]. The same technique has just been applied to quantum
gravitational corrections to electrodynamics on flat space background [51],
and strenuous efforts are underway to generalize it to de Sitter background
[52]. The two flat space exercises demonstrate that gauge independent results
typically show the same spacetime dependence as in a fixed gauge, but with
different numerical coefficients. We expect that the same thing will be found
on de Sitter background.

We should also comment on the breakdown of perturbation theory which must
occur over the course of a very long period of inflation during which $\ln(a)$
becomes larger than $1/\kappa H$. This would be relevant to $\Lambda$-driven
inflation in which inflation is driven by a positive cosmological constant,
without any scalar inflaton, and the self-gravitation of inflationary
gravitons gradually builds up to stop inflation [53, 54]. Of course one cannot
trust our one loop result result (107) when $\kappa^{2}H^{2}\ln^{2}(a)\sim 1$
because unknown higher loop corrections could be equally or even more
important. What is needed is a way to sum up the leading logarithms at each
order. Starobinsky has developed a stochastic formalism that accomplishes this
for the similar large logarithms produced in scalar potential models [55, 56].
This technique captures large logarithms that arise from the “tail” part of
the propagator [57], but it fails for large logarithms which have an
ultraviolet origin, as can sometimes happen with the derivative interactions
of quantum gravity [58]. The renormalization group was designed to sum up
large logarithms from the ultraviolet, but it fails to recover large
logarithms from the “tail” part of the propagator [59]. Our one loop result
(107) is a purely “tail” effect, but it is not known what happens at higher
loops, and one loop quantum gravitational corrections to electrodynamics are
known to derive from both sources [60]. It would be interesting to try summing
all large quantum gravitational logarithms by combining a variant of the
renormlization group with some version of Starobinsky’s technique.

Acknowledgements

This work was partially supported by the European Union’s Horizon 2020
Programme under grant agreement 669288-SM-GRAV-ERC-2014-ADG; by NSF grant
1912484; and by the UF’s Institute for Fundamental Theory.

## References

  * [1] A. A. Starobinsky, JETP Lett. 30, 682-685 (1979)
  * [2] A. A. Starobinsky, Sov. Astron. Lett. 11, 133 (1985)
  * [3] N. C. Tsamis and R. P. Woodard, Commun. Math. Phys. 162, 217-248 (1994) doi:10.1007/BF02102015
  * [4] R. P. Woodard, [arXiv:gr-qc/0408002 [gr-qc]].
  * [5] J. Iliopoulos, T. N. Tomaras, N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 534, 419-446 (1998) doi:10.1016/S0550-3213(98)00528-8 [arXiv:gr-qc/9801028 [gr-qc]].
  * [6] L. R. Abramo and R. P. Woodard, Phys. Rev. D 65, 063515 (2002) doi:10.1103/PhysRevD.65.063515 [arXiv:astro-ph/0109272 [astro-ph]].
  * [7] N. C. Tsamis and R. P. Woodard, Phys. Rev. D 54, 2621-2639 (1996) doi:10.1103/PhysRevD.54.2621 [arXiv:hep-ph/9602317 [hep-ph]].
  * [8] L. Tan, N. C. Tsamis and R. P. Woodard, [arXiv:2103.08547 [gr-qc]].
  * [9] N. C. Tsamis and R. P. Woodard, Annals Phys. 253, 1-54 (1997) doi:10.1006/aphy.1997.5613 [arXiv:hep-ph/9602316 [hep-ph]].
  * [10] V. K. Onemli and R. P. Woodard, Class. Quant. Grav. 19, 4607 (2002) doi:10.1088/0264-9381/19/17/311 [arXiv:gr-qc/0204065 [gr-qc]].
  * [11] V. K. Onemli and R. P. Woodard, Phys. Rev. D 70, 107301 (2004) doi:10.1103/PhysRevD.70.107301 [arXiv:gr-qc/0406098 [gr-qc]].
  * [12] J. S. Schwinger, J. Math. Phys. 2, 407-432 (1961) doi:10.1063/1.1703727
  * [13] K. T. Mahanthappa, Phys. Rev. 126, 329-340 (1962) doi:10.1103/PhysRev.126.329
  * [14] P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4, 1-11 (1963) doi:10.1063/1.1703883
  * [15] P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4, 12-16 (1963) doi:10.1063/1.1703879
  * [16] L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515-1527 (1964)
  * [17] K. c. Chou, Z. b. Su, B. l. Hao and L. Yu, Phys. Rept. 118, 1-131 (1985) doi:10.1016/0370-1573(85)90136-X
  * [18] R. D. Jordan, Phys. Rev. D 33, 444-454 (1986) doi:10.1103/PhysRevD.33.444
  * [19] E. Calzetta and B. L. Hu, Phys. Rev. D 35, 495 (1987) doi:10.1103/PhysRevD.35.495
  * [20] L. H. Ford and R. P. Woodard, Class. Quant. Grav. 22, 1637-1647 (2005) doi:10.1088/0264-9381/22/9/011 [arXiv:gr-qc/0411003 [gr-qc]].
  * [21] P. J. Mora, N. C. Tsamis and R. P. Woodard, JCAP 10, 018 (2013) doi:10.1088/1475-7516/2013/10/018 [arXiv:1307.1422 [gr-qc]].
  * [22] T. Prokopec, O. Tornkvist and R. P. Woodard, Annals Phys. 303, 251-274 (2003) doi:10.1016/S0003-4916(03)00004-6 [arXiv:gr-qc/0205130 [gr-qc]].
  * [23] L. D. Duffy and R. P. Woodard, Phys. Rev. D 72, 024023 (2005) doi:10.1103/PhysRevD.72.024023 [arXiv:hep-ph/0505156 [hep-ph]].
  * [24] L. Tan, N. C. Tsamis and R. P. Woodard, [arXiv:2206.11467 [gr-qc]].
  * [25] S. Weinberg, Phys. Rev. D 72, 043514 (2005) doi:10.1103/PhysRevD.72.043514 [arXiv:hep-th/0506236 [hep-th]].
  * [26] S. Weinberg, Phys. Rev. D 74, 023508 (2006) doi:10.1103/PhysRevD.74.023508 [arXiv:hep-th/0605244 [hep-th]].
  * [27] N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * [28] A. Loeb and M. Zaldarriaga, Phys. Rev. Lett. 92, 211301 (2004) doi:10.1103/PhysRevLett.92.211301 [arXiv:astro-ph/0312134 [astro-ph]].
  * [29] S. Furlanetto, S. P. Oh and F. Briggs, Phys. Rept. 433, 181-301 (2006) doi:10.1016/j.physrep.2006.08.002 [arXiv:astro-ph/0608032 [astro-ph]].
  * [30] K. W. Masui and U. L. Pen, Phys. Rev. Lett. 105, 161302 (2010) doi:10.1103/PhysRevLett.105.161302 [arXiv:1006.4181 [astro-ph.CO]].
  * [31] D. M. Capper, J. Phys. A 13, 199 (1980) doi:10.1088/0305-4470/13/1/022
  * [32] D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, Class. Quant. Grav. 32, no.19, 195014 (2015) doi:10.1088/0264-9381/32/19/195014 [arXiv:1504.00894 [gr-qc]].
  * [33] N. C. Tsamis and R. P. Woodard, Annals Phys. 321, 875-893 (2006) doi:10.1016/j.aop.2005.08.004 [arXiv:gr-qc/0506056 [gr-qc]].
  * [34] S. P. Miao and R. P. Woodard, Class. Quant. Grav. 23, 1721-1762 (2006) doi:10.1088/0264-9381/23/5/016 [arXiv:gr-qc/0511140 [gr-qc]].
  * [35] E. O. Kahya and R. P. Woodard, Phys. Rev. D 76, 124005 (2007) doi:10.1103/PhysRevD.76.124005 [arXiv:0709.0536 [gr-qc]].
  * [36] S. P. Miao, Phys. Rev. D 86, 104051 (2012) doi:10.1103/PhysRevD.86.104051 [arXiv:1207.5241 [gr-qc]].
  * [37] K. E. Leonard and R. P. Woodard, Class. Quant. Grav. 31, 015010 (2014) doi:10.1088/0264-9381/31/1/015010 [arXiv:1304.7265 [gr-qc]].
  * [38] S. Boran, E. O. Kahya and S. Park, Phys. Rev. D 90, no.12, 124054 (2014) doi:10.1103/PhysRevD.90.124054 [arXiv:1409.7753 [gr-qc]].
  * [39] S. Boran, E. O. Kahya and S. Park, Phys. Rev. D 96, no.2, 025001 (2017) doi:10.1103/PhysRevD.96.025001 [arXiv:1704.05880 [gr-qc]].
  * [40] D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, Phys. Rev. D 101, no.10, 106016 (2020) doi:10.1103/PhysRevD.101.106016 [arXiv:2003.02549 [gr-qc]].
  * [41] P. J. Mora, N. C. Tsamis and R. P. Woodard, J. Math. Phys. 53, 122502 (2012) doi:10.1063/1.4764882 [arXiv:1205.4468 [gr-qc]].
  * [42] D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, Class. Quant. Grav. 34, no.8, 085002 (2017) doi:10.1088/1361-6382/aa61da [arXiv:1609.00386 [gr-qc]].
  * [43] C. L. Wang and R. P. Woodard, Phys. Rev. D 91, no.12, 124054 (2015) doi:10.1103/PhysRevD.91.124054 [arXiv:1408.1448 [gr-qc]].
  * [44] J. F. Donoghue, Phys. Rev. Lett. 72, 2996-2999 (1994) doi:10.1103/PhysRevLett.72.2996 [arXiv:gr-qc/9310024 [gr-qc]].
  * [45] J. F. Donoghue, Phys. Rev. D 50, 3874-3888 (1994) doi:10.1103/PhysRevD.50.3874 [arXiv:gr-qc/9405057 [gr-qc]].
  * [46] N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, Phys. Rev. D 68, 084005 (2003) [erratum: Phys. Rev. D 71, 069904 (2005)] doi:10.1103/PhysRevD.68.084005 [arXiv:hep-th/0211071 [hep-th]].
  * [47] N. E. J. Bjerrum-Bohr, J. F. Donoghue and B. R. Holstein, Phys. Rev. D 67, 084033 (2003) [erratum: Phys. Rev. D 71, 069903 (2005)] doi:10.1103/PhysRevD.71.069903 [arXiv:hep-th/0211072 [hep-th]].
  * [48] N. E. J. Bjerrum-Bohr, Phys. Rev. D 66, 084023 (2002) doi:10.1103/PhysRevD.66.084023 [arXiv:hep-th/0206236 [hep-th]].
  * [49] S. P. Miao, T. Prokopec and R. P. Woodard, Phys. Rev. D 96, no.10, 104029 (2017) doi:10.1103/PhysRevD.96.104029 [arXiv:1708.06239 [gr-qc]].
  * [50] J. F. Donoghue and T. Torma, Phys. Rev. D 54, 4963-4972 (1996) doi:10.1103/PhysRevD.54.4963 [arXiv:hep-th/9602121 [hep-th]].
  * [51] S. Katuwal and R. P. Woodard, [arXiv:2107.13341 [gr-qc]].
  * [52] D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, JHEP 10, 096 (2019) doi:10.1007/JHEP10(2019)096 [arXiv:1908.06064 [gr-qc]].
  * [53] N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 474, 235-248 (1996) doi:10.1016/0550-3213(96)00246-5 [arXiv:hep-ph/9602315 [hep-ph]].
  * [54] N. C. Tsamis and R. P. Woodard, Int. J. Mod. Phys. D 20, 2847-2851 (2011) doi:10.1142/S0218271811020652 [arXiv:1103.5134 [gr-qc]].
  * [55] A. A. Starobinsky, Lect. Notes Phys. 246, 107-126 (1986) doi:10.1007/3-540-16452-9_6
  * [56] A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50, 6357-6368 (1994) doi:10.1103/PhysRevD.50.6357 [arXiv:astro-ph/9407016 [astro-ph]].
  * [57] N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 724, 295-328 (2005) doi:10.1016/j.nuclphysb.2005.06.031 [arXiv:gr-qc/0505115 [gr-qc]].
  * [58] S. P. Miao and R. P. Woodard, Class. Quant. Grav. 25, 145009 (2008) doi:10.1088/0264-9381/25/14/145009 [arXiv:0803.2377 [gr-qc]].
  * [59] R. P. Woodard, Phys. Rev. Lett. 101, 081301 (2008) doi:10.1103/PhysRevLett.101.081301 [arXiv:0805.3089 [gr-qc]].
  * [60] S. P. Miao, T. Prokopec and R. P. Woodard, Phys. Rev. D 98, no.2, 025022 (2018) doi:10.1103/PhysRevD.98.025022 [arXiv:1806.00742 [gr-qc]].