UFIFT-QG-21-01 , CCTP-2020-4

Graviton Self-Energy from Gravitons in Cosmology

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

Although matter contributions to the graviton self-energy
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ must be separately
conserved on $x^{\mu}$ and ${x^{\prime}}^{\mu}$, graviton contributions obey
the weaker constraint of the Ward identity, which involves a divergence on
both coordinates. On a general homogeneous and isotropic background this leads
to just four structure functions for matter contributions but nine structure
functions for graviton contributions. We propose a convenient parameterization
for these nine structure functions. We also apply the formalism to explicit
one loop computations of
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ on de Sitter
background, one of the contributions from a massless, minimally coupled scalar
and the other for the contribution from gravitons in the simplest gauge. We
also specialize the linearized, quantum-corrected Einstein equation to the
graviton mode function and to the gravitational response to a point mass.

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

This paper is dedicated to Stanley Deser on the occasion of his 90th birthday.

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

The graviton self-energy
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ is the 1PI (one-
particle-irreducible) 2-graviton function. It can be used to quantum-correct
the linearized Einstein equation,

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

where $h_{\mu\nu}(x)$ is the graviton field, $\mathcal{L}^{\mu\nu\rho\sigma}$
is the Lichnerowicz operator in the appropriate background geometry,
$T^{\mu\nu}_{\rm lin}$ is the linearized stress tensor density and
$\kappa^{2}\equiv 16\pi G$ is the loop counting parameter of quantum gravity.
Equation (1) can be used to study how quantum corrections change the
propagation of gravitational radiation and also how they affect gravitational
forces. Note that equation (1) does not quite represent a semi-classical
approach to gravity because the graviton self-energy receives contributions
from the 0-point fluctuations of gravity as well as matter.

Quantum corrections on flat space background make no change at all to the
kinematics of gravitons, and corrections to gravitational potentials only
become significant at the Planck length [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
12, 13, 14, 15, 16, 17]. However, the situation can be very different in
cosmology, especially during primordial inflation. Accelerated expansion rips
light scalars and gravitons from the vacuum, causing secular enhancements of
the graviton field strength [18], and changes in gravitational potentials that
grow nonperturbatively strong at large distances and at late times [19].

The purpose of this paper is to develop a technique for representing the
graviton self-energy on a general homogeneous, isotropic and spatially flat
background, with scale factor $a(\eta)$ and Hubble parameter $H(\eta)$,

$ds^{2}=a^{2}(\eta)\Bigl{[}-d\eta^{2}+d\vec{x}\\!\cdot\\!d\vec{x}\Bigr{]}\qquad\Longrightarrow\qquad
H(\eta)\equiv\frac{\partial_{0}a}{a^{2}}\;.$ (2)

Our representation consists of a sum of tensor differential operators acting
on four scalar structure functions. Similar representations have been already
given for matter contributions to the graviton self-energy on de Sitter
background [20, 21] but it is cumbersome to infer the structure functions from
the primitive result, and then use them in the effective field equation (1).
Another problem is that graviton contributions to the graviton self-energy
require five new structure functions.

To understand the difference between matter contributions and those from
gravity itself, first use general tensor analysis on the background (2) to
construct 21 basis tensors $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}]$ 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
$\overline{\partial}^{\mu}\equiv\partial^{\mu}+\delta^{\mu}_{~{}0}\partial_{0}$.
The graviton self-energy can be expressed as a sum of these operators acting
on scalar functions of $\eta$, $\eta^{\prime}$ and
$\|\vec{x}-\vec{x}^{\prime}\|$,

$-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\sum_{i=1}^{21}\Bigl{[}\mbox{}^{\mu\nu}\mathcal{D}_{i}^{\rho\sigma}\Bigr{]}\\!\times\\!T^{i}(x;x^{\prime})\;.$
(3)

The 21 basis tensors are listed in Table 1.

$i$ | $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}_{i}]$ | $i$ | $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}_{i}]$ | $i$ | $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}_{i}]$
---|---|---|---|---|---
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 (3). Note that the pairs
$(3,4)$, $(5,6)$, $(7,8)$, $(10,11)$, $(14,15)$, $(16,17)$ and $(19,20)$ are
related by reflection.

Now note that 7 of the scalar coefficient functions are related by reflection
invariance,

$-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=-i\Bigl{[}\mbox{}^{\rho\sigma}\Sigma^{\mu\nu}\Bigr{]}(x^{\prime};x)\;.$
(4)

The various relations are listed 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 (3) which are related by
reflection.

The 14 algebraically independent scalar coefficient functions
$T^{i}(x;x^{\prime})$ are related by differential equations whose number
depends upon whether the contributions to
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ come from matter or
from gravity itself. To understand these relations it is useful to define the
Ward Operator,

$\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\equiv\delta^{\mu}_{~{}(\alpha}\partial_{\beta)}+Ha\delta^{\mu}_{~{}0}\eta_{\alpha\beta}\;.$
(5)

Because matter interacts with gravity through its conserved stress tensor,
matter contributions to the graviton self-energy must be annihilated by the
Ward operator acting on either point,

$0=\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\times-i\Bigl{[}\mbox{}^{\alpha\beta}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=0=\sum_{i=1}^{10}\Bigl{[}\mbox{}^{\mu}\mathcal{D}^{\rho\sigma}\Bigr{]}\times
S^{i}(x;x^{\prime})\;.$ (6)

The 10 independent tensor factors $[\mbox{}^{\mu}\mathcal{D}^{\rho\sigma}]$
are listed in Table 3.

$i$ | $[\mbox{}^{\mu}\mathcal{D}_{i}^{\rho\sigma}]$ | $i$ | $[\mbox{}^{\mu}\mathcal{D}_{i}^{\rho\sigma}]$
---|---|---|---
$1$ | $\delta^{\mu}_{~{}0}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$ | $6$ | $2\overline{\eta}^{\mu(\rho}\delta^{\sigma)}_{~{}~{}0}$
$2$ | $2\delta^{\mu}_{~{}0}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}$ | $7$ | $2\overline{\partial}^{\mu}\overline{\partial}^{(\rho}\delta^{\sigma)}_{~{}~{}0}$
$3$ | $\delta^{\mu}_{~{}0}\overline{\eta}^{\rho\sigma}$ | $8$ | $2\overline{\eta}^{\mu(\rho}\overline{\partial}^{\sigma)}$
$4$ | $\delta^{\mu}_{~{}0}\overline{\partial}^{\rho}\overline{\partial}^{\sigma}$ | $9$ | $\overline{\partial}^{\mu}\overline{\eta}^{\rho\sigma}$
$5$ | $\overline{\partial}^{\mu}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$ | $10$ | $\overline{\partial}^{\mu}\overline{\partial}^{\rho}\overline{\partial}^{\sigma}$
Table 3: Scalar coefficient functions in expression (3) which are related by
reflection.

From (6) we see that matter contributions to the graviton self-energy are
characterized by $14-10=4$ independent structure functions. Gravity does not
interact with itself through a conserved vertex. Hence graviton contributions
to the graviton self-energy obey the weaker condition that they are
annihilated by acting the Ward operator on both points,

$\displaystyle
0=\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\\!\times\\!\mathcal{W}^{\rho}_{~{}\gamma\delta}(x^{\prime})\\!\times\\!-i\Bigl{[}\mbox{}^{\alpha\beta}\Sigma^{\gamma\delta}\Bigr{]}(x;x^{\prime})=\delta^{\mu}_{~{}0}\delta^{\rho}_{~{}0}\\!\times\\!R^{1}(x;x^{\prime})+\overline{\eta}^{\mu\rho}$
(7) $\displaystyle\times
R^{2}(x;x^{\prime})+\delta^{\mu}_{~{}0}\overline{\partial}^{\rho}\\!\times\\!R^{3}(x;x^{\prime})+\overline{\partial}^{\mu}\delta^{\rho}_{~{}0}\\!\times\\!R^{4}(x;x^{\prime})+\overline{\partial}^{\mu}\overline{\partial}^{\rho}\\!\times\\!R^{5}(x;x^{\prime})\;.\qquad$

Because expression (7) involves 5 independent tensors we see that graviton
contributions to $-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$
require $14-5=9$ structure functions. Our purpose is to propose a convenient
representation for these structure functions and to elucidate their role in
the effective field equation (1).

Section 2 derives insights from the vacuum polarization, on flat space and in
cosmology, and from the graviton self-energy on flat space. Our representation
is given in section 3. We also work out the equations for quantum corrections
to the graviton mode function, and for the two potentials that describe the
response to a point mass. Section 4 derives explicit results on de Sitter
background for a dimensionally regulated computation of the contribution from
a massless, minimally coupled (MMC) scalar [20], and for a $D=4$ computation
of the contribution from gravitons away from coincidence
($x^{\mu}\neq{x^{\prime}}^{\mu}$) [22]. Section 5 discusses how to extend the
$D=4$ computation to a fully renormalized result. Our conclusions comprise
section 6.

##  2 Other Bi-Tensor 1PI 2-Point Functions

The purpose of this section is to motivate our representation for the graviton
self-energy in cosmology by reviewing simpler bi-tensor 1PI 2-point functions
and simpler backgrounds. The section begins with the vacuum polarization on
flat space background. We then turn to the graviton self-energy on flat space
background. The section concludes with the vacuum polarization on a general
cosmological background (2).

###  2.1 Vacuum Polarization on Flat Space

The 1PI 2-photon function $i[\mbox{}^{\mu}\Pi^{\rho}](x;x^{\prime})$ has the
evocative name, “vacuum polarization”. A cumbersome and foolish way of
expressing it would be to give all $4^{2}=16$ of its tensor components as
functions of the two points $x^{\mu}$ and ${x^{\prime}}^{\mu}$. A much better
way is to consolidate the number of functions by using general tensor analysis
and reflection invariance. On flat space background this results in the form,

$i\Bigl{[}\mbox{}^{\mu}\Pi^{\rho}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\eta^{\mu\rho}\\!\times\\!A(\Delta
x^{2})+\partial^{\mu}\partial^{\rho}\\!\times\\!B(\Delta x^{2})\;,$ (8)

where the invariant interval in a Feynman propagator is,

$\Delta
x^{2}\equiv\Bigl{\|}\vec{x}\\!-\\!\vec{x}^{\prime}\Bigr{\|}^{2}-\Bigl{(}|\eta\\!-\\!\eta^{\prime}|\\!-\\!i\epsilon\Bigr{)}^{2}\;.$
(9)

Because photons couple to a conserved current the vacuum polarization is
transverse on each index,

$0=\partial_{\mu}\\!\times\\!i\Bigl{[}\mbox{}^{\mu}\Pi^{\rho}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\partial^{\rho}\Bigl{[}A(\Delta
x^{2})+\partial^{2}B(\Delta x^{2})\Bigr{]}\;.$ (10)

Conservation implies $A=-\partial^{2}B$, which allows us to express the flat
space vacuum polarization in terms of a single structure function,

$i\Bigl{[}\mbox{}^{\mu}\Pi^{\rho}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\Bigl{[}\partial^{\mu}\partial^{\rho}-\eta^{\mu\rho}\partial^{2}\Bigr{]}B(\Delta
x^{2})\equiv\Pi^{\mu\rho}B(\Delta x^{2})\;.$ (11)

It is more usual in the literature of quantum field theory to refer to the
structure function $B(\Delta x^{2})$ by the symbol $i\Pi(\Delta x^{2})$.

###  2.2 Graviton Self-Energy on Flat Space

The advantages of using structure functions are even greater for the graviton
self-energy. It would be fatuous to express this by giving all $4^{4}=256$
components. Just as with the vacuum polarization, it is more efficient to
exploit symmetries of the background, reflection invariance to express
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ in terms of five basis
tensors,

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\eta^{\mu\nu}\eta^{\rho\sigma}\\!\times\\!A(\Delta
x^{2})+\eta^{\mu(\rho}\eta^{\sigma)\nu}\\!\times\\!B(\Delta
x^{2})+\Bigl{[}\eta^{\mu\nu}\partial^{\rho}\partial^{\sigma}$ (12)
$\displaystyle+\partial^{\mu}\partial^{\nu}\eta^{\rho\sigma}\Bigr{]}\\!\times\\!C(\Delta
x^{2})+\partial^{(\mu}\eta^{\nu)(\rho}\partial^{\sigma)}\\!\times\\!D(\Delta
x^{2})+\partial^{\mu}\partial^{\nu}\partial^{\rho}\partial^{\sigma}\\!\times\\!E(\Delta
x^{2})\;.\qquad$

In this expression and henceforth parenthesized indices are symmetrized, for
example,
$\eta^{\mu(\rho}\eta^{\sigma)\nu}\equiv\frac{1}{2}[\eta^{\mu\rho}\eta^{\nu\sigma}+\eta^{\mu\sigma}\eta^{\nu\rho}]$.

####  2.2.1 From Matter

Because matter couples to gravitons through its conserved stress-energy
tensor, matter contributions to the graviton self-energy must be transverse on
each index,

$\displaystyle
0=\partial_{\nu}\\!\times\\!-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\partial^{\mu}\eta^{\rho\sigma}\Bigl{[}A\\!+\\!\partial^{2}C\Bigr{]}$
(13) $\displaystyle\hskip
99.58464pt+\eta^{\mu(\rho}\partial^{\sigma)}\Bigl{[}B\\!+\\!\frac{1}{2}\partial^{2}D\Bigr{]}+\partial^{\mu}\partial^{\rho}\partial^{\sigma}\Bigl{[}C\\!+\\!\frac{1}{2}D\\!+\\!\partial^{2}E\Bigr{]}\;.\qquad$

Conservation (13) allows us to express $A$, $C$ and $E$ in terms of $C$ and
$D$,

$\Bigl{(}{\rm Eqn.\ \ref{flatSigmamattercons}}\Bigr{)}\Longrightarrow
A=-\partial^{2}C\quad,\quad B=-\frac{1}{2}\partial^{2}D\quad,\quad
E=-\frac{1}{\partial^{2}}\Bigl{(}C\\!+\\!\frac{1}{2}D\Bigr{)}\;.$ (14)

Substituting (14) in (12) results in another familiar form,

$\Bigl{(}{\rm Eqn.\
\ref{flatSigmamattercons}}\Bigr{)}\Longrightarrow-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\Pi^{\mu\nu}\Pi^{\rho\sigma}\Bigl{(}-\frac{1}{\partial^{2}}C\Bigr{)}+\Pi^{\mu(\rho}\Pi^{\sigma)\nu}\Bigl{(}-\frac{1}{2\partial^{2}}D\Bigr{)}\;,$
(15)

where
$\Pi^{\alpha\beta}\equiv\partial^{\alpha}\partial^{\beta}-\eta^{\alpha\beta}\partial^{2}$
was introduced in (11).

####  2.2.2 From Gravitons

Because the couplings of gravitons to themselves are not conserved, the
divergence on a single index group does not vanish. Of course one can still
use general tensor analysis to parameterize it in terms of three scalar
functions,

$\partial_{\nu}\times-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\partial^{\mu}\eta^{\rho\sigma}F(\Delta
x^{2})+\eta^{\mu(\rho}\partial^{\sigma)}G(\Delta
x^{2})+\partial^{\mu}\partial^{\rho}\partial^{\sigma}\mathcal{H}(\Delta
x^{2})\;.$ (16)

The Ward identity requires gravitational contributions to the graviton self
energy to vanish when a divergence is taken on both index groups,

$0=\partial_{\nu}\partial_{\sigma}\\!\times\\!-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\eta^{\mu\rho}\Bigl{[}\frac{1}{2}\partial^{2}G\Bigr{]}+\partial^{\mu}\partial^{\rho}\Bigl{[}F\\!+\\!\frac{1}{2}G\\!+\\!\partial^{2}\mathcal{H}\Bigr{]}\;.$
(17)

Expression (17) implies,

$\Bigl{(}{\rm Eqn.\
\ref{flatSigmagravitycons1}}\Bigr{)}\qquad\Longrightarrow\qquad
F=-\partial^{2}\mathcal{H}\qquad,\qquad G=0\;.$ (18)

Of course the initial parameterization (12) of the graviton self-energy
pertains to both gravitational and matter contributions. Hence expression (13)
is still valid for the result of a single divergence. Comparing (13) with (16)
allows us to express the scalar coefficient functions $A$, $B$ and $E$ in
terms of $C$, $D$ and $\mathcal{H}$,

$\Bigl{(}{\rm Eqn.\ \ref{flatSigmagravitycons1}}\Bigr{)}\Longrightarrow
A=-\partial^{2}\Bigl{(}C\\!+\\!\mathcal{H}\Bigr{)}\;,\;B=-\frac{\partial^{2}D}{2}\;,\;E=\frac{1}{\partial^{2}}\Bigl{(}-C\\!-\\!\frac{D}{2}\\!+\\!\mathcal{H}\Bigr{)}\;.$
(19)

Substituting (19) into (12) gives,

$\displaystyle\Bigl{(}{\rm Eqn.\
\ref{flatSigmagravitycons1}}\Bigr{)}\qquad\Longrightarrow\qquad-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
flat}\Bigr{]}(x;x^{\prime})=\Pi^{\mu\nu}\Pi^{\rho\sigma}\Bigl{(}-\frac{1}{\partial^{2}}C\Bigr{)}$
(20) $\displaystyle\hskip
56.9055pt+\Pi^{\mu(\rho}\Pi^{\sigma)\nu}\Bigl{(}-\frac{1}{2\partial^{2}}D\Bigr{)}+\Bigl{[}\partial^{\mu}\partial^{\nu}\partial^{\rho}\partial^{\sigma}\\!-\\!\eta^{\mu\nu}\eta^{\rho\sigma}\partial^{4}\Bigr{]}\Bigl{(}\frac{1}{\partial^{2}}\mathcal{H}\Bigr{)}\;.\qquad$

####  2.2.3 An Explicit Example

In section 4 we will reconstruct the structure functions functions from an
explicit computation of
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ on de Sitter
background [22]. That result was derived using a de Sitter breaking gauge in
which the graviton propagator consists of three constant tensor factors,
constructed from $\eta^{\mu\nu}$ and $\delta^{\mu}_{0}$, which multiply scalar
propagators whose expansions in $D=4$ spacetime dimensions have at most two
terms [23, 24]. In 1979 Capper used the flat space limit of this same gauge,
with dimensional regularization, to compute the one loop contribution to the
graviton self-energy [25],

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma_{\rm
flat}^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\frac{-\kappa^{2}}{4(D^{2}\\!-\\!1)}\Biggl{\\{}T_{1}\partial^{\mu}\partial^{\nu}\partial^{\rho}\partial^{\sigma}+T_{2}\eta^{\mu\nu}\eta^{\rho\sigma}\partial^{4}+2T_{3}\eta^{\mu(\rho}\eta^{\sigma)\nu}\partial^{4}$
(21) $\displaystyle\hskip
28.45274ptT_{4}\Bigl{[}\eta^{\mu\nu}\partial^{2}\partial^{\rho}\partial^{\sigma}+\partial^{\mu}\partial^{\nu}\eta^{\rho\sigma}\partial^{2}\Bigr{]}+4T_{5}\partial^{(\mu}\eta^{\nu)(\rho}\partial^{\sigma)}\partial^{2}\Biggr{\\}}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}.\quad$

Here $i\Delta(x;x^{\prime})$ is the massless propagator in flat space,

$i\Delta(x;x^{\prime})=\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{4\pi^{\frac{D}{2}}}\frac{1}{{\Delta
x}^{D-2}}\;.$ (22)

Capper’s results for the coefficients $T_{i}(D)$ are [25],

$\displaystyle T_{1}(D)$ $\displaystyle=$
$\displaystyle\frac{9}{16}D^{4}-\frac{21}{16}D^{3}-\frac{9}{8}D^{2}\;,$ (23)
$\displaystyle T_{2}(D)$ $\displaystyle=$
$\displaystyle\frac{\frac{9}{16}D^{5}\\!-\\!\frac{39}{16}D^{4}\\!-\\!\frac{25}{8}D^{3}\\!+\\!\frac{123}{8}D^{2}\\!+\\!\frac{33}{4}D\\!-\\!8}{D\\!-\\!2}\;,\qquad$
(24) $\displaystyle T_{3}(D)$ $\displaystyle=$
$\displaystyle\frac{\frac{1}{4}D^{4}\\!+\\!\frac{17}{16}D^{3}\\!-\\!\frac{97}{16}D^{2}\\!-\\!\frac{17}{8}D\\!+\\!4}{D\\!-\\!2}=-T_{5}(D)\;,\qquad$
(25) $\displaystyle T_{4}(D)$ $\displaystyle=$
$\displaystyle\frac{-\frac{9}{16}D^{5}\\!+\\!\frac{43}{16}D^{4}\\!+\\!\frac{15}{8}D^{3}\\!-\\!\frac{119}{8}D^{2}\\!-\\!\frac{25}{4}D\\!+\\!8}{D\\!-\\!2}\;,\qquad$
(26)

Comparing expressions (12) and (21) allows us to identify two of the structure
functions,

$-\frac{1}{\partial^{2}}C=\frac{\kappa^{2}T_{4}}{4(D^{2}\\!-\\!1)}\Bigl{[}i\Delta^{2}(x;x^{\prime})\Bigr{]}^{2}\quad,\quad-\frac{1}{2\partial^{2}}D=\frac{\kappa^{2}4T_{5}}{4(D^{2}\\!-\\!1)}\Bigl{[}i\Delta^{2}(x;x^{\prime})\Bigr{]}^{2}\;.$
(27)

The final structure function derives from a comparison of (16) with the
divergence of (21),

$\frac{1}{\partial^{2}}H=-\frac{\kappa^{2}(T_{1}\\!+\\!T_{4}\\!+\\!2T_{5})}{4(D^{2}\\!-\\!1)}\Bigl{[}i\Delta^{2}(x;x^{\prime})\Bigr{]}^{2}\;.$
(28)

Substituting (27) and (28) in (20) gives,

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma_{\rm
flat}^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\frac{-\kappa^{2}}{4(D^{2}\\!-\\!1)}\Biggl{\\{}-T_{4}\Pi^{\mu\nu}\Pi^{\rho\sigma}-2T_{5}\Pi^{\mu(\rho}\Pi^{\sigma)\nu}$
(29) $\displaystyle\hskip
71.13188pt+(T_{1}\\!+\\!T_{4}\\!+\\!2T_{5})\Bigl{[}\partial^{\mu}\partial^{\nu}\partial^{\rho}\partial^{\sigma}\\!-\\!\eta^{\mu\nu}\eta^{\rho\sigma}\partial^{4}\Bigr{]}\Biggr{\\}}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}.\qquad$

Expression (29) is the dimensionally regulated, primitive contribution. To
renormalize we first isolate ultraviolet divergences using the expansion [26,
27],

$\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}=\frac{\Gamma(\frac{D}{2}\\!-\\!1)}{4\pi^{\frac{D}{2}}}\frac{\mu^{D-4}i\delta^{D}(x\\!-\\!x^{\prime})}{2(D\\!-\\!3)(D\\!-\\!4)}-\frac{\partial^{2}}{32\pi^{4}}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{]}+O(D\\!-\\!4)\;.$ (30)

At $D=4$ the coefficients $T_{4}(4)=-\frac{23}{2}$ and
$T_{5}(4)=-\frac{61}{4}$ are nonzero, but the final coefficient vanishes,

$T_{1}(D)\\!+\\!T_{4}(D)\\!+\\!2T_{5}(D)=-\frac{1}{4}(D\\!-\\!4)D(D\\!+\\!1)\;.$
(31)

This means that renormalization requires only the Ricci-squared and Weyl-
squared counterterms. Note also that (29) must be recovered in the flat space
limit of the graviton self-energy on de Sitter background [22]. The final,
unregulated result is,

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma_{\rm
ren}^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\frac{\kappa^{2}}{1920\pi^{4}}\Bigl{[}\frac{23}{2}\Pi^{\mu\nu}\Pi^{\rho\sigma}+\frac{61}{2}\Pi^{\mu(\rho}\Pi^{\sigma)\nu}\Bigr{]}\partial^{2}\Bigl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Bigr{]}$ (32) $\displaystyle\hskip
128.0374pt+\frac{\kappa^{2}}{96\pi^{2}}\Bigl{[}\partial^{\mu}\partial^{\nu}\partial^{\rho}\partial^{\sigma}\\!-\\!\eta^{\mu\nu}\eta^{\rho\sigma}\partial^{4}\Bigr{]}i\delta^{4}(x\\!-\\!x^{\prime})\;.\qquad$

Except for their numerical coefficients, the two nonlocal terms on the first
line of (32) could have come from matter contributions; the local term on the
last line is only possible from gravitational contributions to the graviton
self-energy because its divergence on a single index group is nonzero,
$\partial_{\nu}[\partial^{\mu}\partial^{\nu}\partial^{\rho}\partial^{\sigma}-\eta^{\mu\nu}\eta^{\rho\sigma}\partial^{4}]=\partial^{\mu}\partial^{2}\Pi^{\rho\sigma}$.
In cosmological backgrounds (2) we will see that distinctly gravitational
contributions are much more varied, that they can harbor divergences, and that
they can be nonlocal.

###  2.3 Vacuum Polarization in Cosmology

Now consider the photon self-energy (aka, the “vacuum polarization”) on a
general cosmological background (2). The symmetries of cosmology are
homogeneity and isotropy. This means that the initial reduction involves two
additional tensors from (8) and that the coefficient functions depend on
$\eta$, $\eta^{\prime}$ and $\|\vec{x}-\vec{x}^{\prime}\|$. When account is
also taken of reflection invariance we can write,111Rather than factors of the
spatial gradient $\partial^{i}$, the primitive expression contains one or two
factors of the spatial coordinate interval $\Delta x^{i}\equiv
x^{i}-{x^{\prime}}^{i}$ multiplied by functions of $\|\Delta\vec{x}\|^{2}$.
These are then written in terms of spatial gradients using the identities,
$\displaystyle\Delta x^{i}f(\|\vec{x}\|^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial^{i}I[f]\;,$ $\displaystyle\Delta x^{i}\Delta
x^{j}f(\|\vec{x}\|^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\partial^{i}\partial^{j}I^{2}[f]-\frac{1}{2}\delta^{ij}I[f]\;,$
where $I[f]$ represents the indefinite integral of $f(\|\vec{x}\|^{2})$ with
respect to $\|\vec{x}\|^{2}$.

$\displaystyle i\Bigl{[}\mbox{}^{\mu}\Pi^{\rho}_{\rm
cos}\Bigr{]}(x;x^{\prime})=\overline{\eta}^{\mu\rho}A(x;x^{\prime})+\delta^{\mu}_{~{}0}\delta^{\rho}_{~{}0}B(x;x^{\prime})$
(33) $\displaystyle\hskip
99.58464pt+\delta^{\mu}_{~{}0}\overline{\partial}^{\rho}C(x;x^{\prime})-\overline{\partial}^{\mu}\delta^{\rho}_{~{}0}C(x^{\prime};x)+\overline{\partial}^{\mu}\overline{\partial}^{\rho}D(x;x^{\prime})\;.\qquad$

The scalar coefficient functions $A$, $B$ and $D$ are all reflection
invariant,

$A(x;x^{\prime})=A(x^{\prime};x)\quad,\quad
B(x;x^{\prime})=B(x^{\prime};x)\quad,\quad D(x;x^{\prime})=D(x^{\prime};x)\;.$
(34)

We also remind the reader that the presence of a bar over a tensor indicates
suppression of its temporal components,

$\overline{\eta}^{\mu\rho}\equiv\eta^{\mu\rho}+\delta^{\mu}_{~{}0}\delta^{\rho}_{~{}0}\qquad,\qquad\overline{\partial}^{\mu}\equiv\partial^{\mu}+\delta^{\mu}_{~{}0}\partial_{0}\;.$
(35)

Because the vacuum polarization is a bi-vector density, its divergence on each
index group must vanish, on any background geometry. On cosmological
backgrounds (2) the divergence produces two independent tensors,

$\displaystyle
0=\partial_{\mu}\\!\times\\!i\Bigl{[}\mbox{}^{\mu}\Pi^{\rho}_{\rm
cos}\Bigr{]}(x;x^{\prime})=\overline{\partial}^{\rho}\Bigl{[}A(x;x^{\prime})+\partial_{0}C(x;x^{\prime})$
(36) $\displaystyle\hskip
113.81102pt+\nabla^{2}D(x;x^{\prime})\Bigr{]}+\delta^{\rho}_{~{}0}\Bigl{[}\partial_{0}B(x;x^{\prime})\\!-\\!\nabla^{2}C(x^{\prime};x)\Bigr{]}\;,\qquad$

where $\nabla^{2}\equiv\partial^{i}\partial^{i}$ is the flat space Laplacian.
Expression (36) allows us to solve for two of the coefficient functions,

$A(x;x^{\prime})=-\partial_{0}C(x;x^{\prime})-\nabla^{2}D(x;x^{\prime})\qquad,\qquad
B(x;x^{\prime})=\frac{\nabla^{2}}{\partial_{0}}C(x^{\prime};x)\;.$ (37)

The reflection invariance (34) of $A$ and $B$ also implies an important
relation for reflecting $C(x;x^{\prime})$,

$\partial_{0}C(x;x^{\prime})=\partial_{0}^{\prime}C(x^{\prime};x)\qquad\Longrightarrow\qquad
C(x^{\prime};x)=\frac{\partial_{0}}{\partial_{0}^{\prime}}C(x;x^{\prime})\;.$
(38)

Note that $\frac{1}{\partial_{0}^{\prime}}C(x;x^{\prime})$ is reflection
invariant.

Substituting (37) and (38) into (33) shows how the cosmological vacuum
polarization can be expressed using two structure functions,

$\displaystyle i\Bigl{[}\mbox{}^{\mu}\Pi^{\rho}_{\rm
cos}\Bigr{]}(x;x^{\prime})=\Bigl{[}-\overline{\eta}^{\mu\rho}\partial_{0}\partial_{0}^{\prime}\\!+\\!\delta^{\mu}_{~{}0}\partial_{0}^{\prime}\overline{\partial}^{\rho}\\!-\\!\overline{\partial}^{\mu}\delta^{\rho}_{~{}0}\partial_{0}\\!+\\!\delta^{\mu}_{~{}0}\delta^{\rho}_{~{}0}\nabla^{2}\Bigr{]}\frac{1}{\partial_{0}^{\prime}}C(x;x^{\prime})$
(40) $\displaystyle\hskip
199.16928pt+\Bigl{[}\overline{\partial}^{\mu}\overline{\partial}^{\rho}\\!-\\!\overline{\eta}^{\mu\rho}\nabla^{2}\Bigr{]}D(x;x^{\prime})\;,\qquad$
$\displaystyle\hskip
56.9055pt=\Bigl{[}\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu\sigma}\eta^{\nu\rho}\Bigr{]}\partial_{\nu}\partial_{\sigma}^{\prime}\Bigl{[}\frac{1}{\partial_{0}^{\prime}}C(x;x^{\prime})\Bigr{]}$
$\displaystyle\hskip
99.58464pt+\Bigl{[}\overline{\eta}^{\mu\nu}\overline{\eta}^{\rho\sigma}\\!-\\!\overline{\eta}^{\mu\sigma}\overline{\eta}^{\nu\rho}\Bigr{]}\partial_{\nu}\partial_{\sigma}^{\prime}\Bigl{[}D(x;x^{\prime})\\!-\\!\frac{1}{\partial_{0}^{\prime}}C(x;x^{\prime})\Bigr{]}\;.\qquad$

The representation (40) was first employed in studying charged MMC scalar
contributions to the vacuum polarization on de Sitter background [28, 29]. The
procedure for transforming to other representations has been given in detail
[30, 31].

##  3 Graviton Self-Energy in Cosmology

The purpose of this section is to present our formalism for representing the
graviton self-energy in cosmology. We give a unified derivation which applies
to the cases of matter contributions and to those from gravity itself. We then
specialize the effective field equations to the cases of the graviton mode
function and to the two potentials that represent the gravitational response
to a point mass.

###  3.1 Our Representation

As discussed in the Introduction, the symmetries of cosmology permit us to
represent the graviton self-energy as the sum (3) of the 21 tensor
differential operators $[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}_{i}]$ of
Table 1 acting on scalar coefficient functions $T^{i}(x;x^{\prime})$,

$-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\sum_{i=1}^{21}\Bigl{[}\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}\Bigr{]}\\!\times\\!T^{i}(x;x^{\prime})\;.$
(41)

Seven of the coefficient functions are related to others by reflection
invariance, as described in Table 2. Acting the Ward operator (5) on a single
index group produces a sum (6) of the 10 tensor differential operators of
Table 3 acting on scalar coefficient functions $S^{i}(x;x^{\prime})$,

$\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\\!\times\\!-i\Bigl{[}\mbox{}^{\alpha\beta}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\sum_{i=1}^{10}\Bigl{[}\mbox{}^{\mu}\mathcal{D}^{\rho\sigma}\Bigr{]}\\!\times\\!S^{i}(x;x^{\prime})\;.$
(42)

Relation (6) can be used to express each of the $S^{i}(x;x^{\prime})$ in terms
of the 14 algebraically distinct $T^{j}(x;x^{\prime})$. The expansions are
given in Table 4.

$S^{i}(x;x^{\prime})$ | Expansion in $T^{j}=T^{j}(x;x^{\prime})$ and $T^{jR}=T^{j}(x^{\prime};x)$
---|---
$S^{1}$ | $(D\\!-\\!1)aHT^{3}+(\partial_{0}\\!-\\!aH)T^{13}-\frac{1}{2}\nabla^{2}T^{14R}+aH\nabla^{2}T^{16R}$
$S^{2}$ | $(\frac{D-1}{2})aHT^{5}+\frac{1}{4}T^{9}-\frac{1}{2}aHT^{10R}$
|
$+\frac{1}{2}(\partial_{0}-aH)T^{14}+\frac{1}{4}\nabla^{2}T^{18}-\frac{1}{2}aH\nabla^{2}T^{19R}$
$S^{3}$ | $(D\\!-\\!1)aHT^{1}+aHT^{2}+(\partial_{0}\\!-\\!aH)T^{3R}-\frac{1}{2}\nabla^{2}T^{5R}+aH\nabla^{2}T^{7R}$
$S^{4}$ | $(D\\!-\\!1)aHT^{7}+\frac{1}{2}T^{10}+aHT^{12}$
|
$+(\partial_{0}\\!-\\!aH)T^{16}+\frac{1}{2}\nabla^{2}T^{19}+aH\nabla^{2}T^{21}$
$S^{5}$ | $T^{3}-\frac{1}{2}\partial_{0}T^{14R}+\nabla^{2}T^{16R}$
$S^{6}$ | $\frac{1}{4}\partial_{0}T^{9}-\frac{1}{4}\nabla^{2}T^{10R}$
$S^{7}$ | $\frac{1}{2}T^{5}-\frac{1}{4}T^{10R}+\frac{1}{4}\partial_{0}T^{18}-\frac{1}{2}\nabla^{2}T^{19R}$
$S^{8}$ | $\frac{1}{2}T^{2}+\frac{1}{4}\partial_{0}T^{10}+\frac{1}{4}\nabla^{2}T^{12}$
$S^{9}$ | $T^{1}-\frac{1}{2}\partial_{0}T^{5R}+\nabla^{2}T^{7R}$
$S^{10}$ | $T^{7}+\frac{1}{2}T^{12}+\frac{1}{2}\partial_{0}T^{19}+\nabla^{2}T^{21}$
Table 4: Expansion of the coefficients $S^{i}(x;x^{\prime})$ of equation (6)
in terms of the coefficients $T^{j}(x;x^{\prime})$ of the initial
representation (3).

Our strategy for representing the $T^{i}(x;x^{\prime})$ is motivated by the
flat space analog considered in section 2.2. It is the same for both matter
and gravity: we use the ten relations of Table 4 to solve for the coefficient
functions in terms of the functions $S^{i}$ and a “minimal” set of $T^{i}$’s
consisting of $T^{12}$, $T^{16}$, $T^{18}$ and $T^{19}$. Each of the ten
$S^{i}$ must vanish for matter contributions, whereas they can be nonzero for
contributions from gravity itself. However, because we must get zero from
acting the Ward operator on both index groups, and because this action results
in five distinct tensor operators (7), the ten $S^{i}$ are subject to five
relations given in Table 5.

$R^{i}(x;x^{\prime})$ | Expansion in $S^{j}=S^{j}(x;x^{\prime})$
---|---
$R^{1}$ | $(\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime})S^{1}-\nabla^{2}S^{2}+(D\\!-\\!1)a^{\prime}H^{\prime}S^{3}+a^{\prime}H^{\prime}\nabla^{2}S^{4}$
$R^{2}$ | $\partial_{0}^{\prime}S^{6}-\nabla^{2}S^{8}$
$R^{3}$ | $\partial_{0}^{\prime}S^{2}-S^{3}-\nabla^{2}S^{4}$
$R^{4}$ | $\\!(\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime})S^{5}\\!-\\!S^{6}\\!-\\!\nabla^{2}S^{7}\\!+\\!2a^{\prime}H^{\prime}S^{8}\\!+\\!(D\\!-\\!1)a^{\prime}H^{\prime}S^{9}\\!+\\!a^{\prime}H^{\prime}\nabla^{2}S^{10}\\!$
$R^{5}$ | $\partial_{0}^{\prime}S^{7}-S^{8}-S^{9}-\nabla^{2}S^{10}$
Table 5: Expansion of the coefficients $R^{i}(x;x^{\prime})=0$ of the Ward
identity (7) in terms of the coefficients $S^{j}(x;x^{\prime})$ of the action
(6) of the Ward operator on a single index group.

We can eliminate $S^{3}$ using $R^{3}(x;x^{\prime})=0$,

$S^{3}=\partial_{0}^{\prime}S^{2}-\nabla^{2}S^{4}\;.$ (43)

Combining this with $R^{1}(x;x^{\prime})=0$ gives $S^{1}$,222The inverse of
$\partial_{0}^{\prime}-a^{\prime}H^{\prime}$ can be expressed as a simple
integral with respect to $\eta^{\prime}$,
$\frac{1}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}=a^{\prime}\\!\times\\!\frac{1}{\partial_{0}^{\prime}}\\!\times\\!\frac{1}{a^{\prime}}\;.$
(44)

$S^{1}=\frac{1}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}\Bigl{\\{}-\Bigl{[}(D\\!-\\!1)a^{\prime}H^{\prime}\partial_{0}^{\prime}\\!-\\!\nabla^{2}\Bigr{]}S^{2}+(D\\!-\\!2)a^{\prime}H^{\prime}\nabla^{2}S^{4}\Bigr{\\}}\;.$
(45)

The relations $R^{2}(x;x^{\prime})=0$ and $R^{5}(x;x^{\prime})=0$ imply,

$S^{6}=\frac{\nabla^{2}}{\partial_{0}^{\prime}}S^{8}\qquad,\qquad
S^{9}=\partial_{0}^{\prime}S^{7}-S^{8}-\nabla^{2}S^{10}\;.$ (46)

and substituting (46) in $R^{4}(x;x^{\prime})=0$ gives,

$\displaystyle
S^{5}=\frac{1}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}\Biggl{\\{}-\Bigl{[}(D\\!-\\!1)a^{\prime}H^{\prime}\partial_{0}^{\prime}\\!-\\!\nabla^{2}\Bigr{]}S^{7}$
(47) $\displaystyle\hskip
99.58464pt+\Bigl{[}(D\\!-\\!3)a^{\prime}H^{\prime}\\!+\\!\frac{\nabla^{2}}{\partial_{0}^{\prime}}\Bigr{]}S^{8}+(D\\!-\\!2)a^{\prime}H^{\prime}\nabla^{2}S^{10}\Biggr{\\}}\;.\qquad$

So our structure functions consist of $T^{12}$, $T^{16}$, $T^{18}$ and
$T^{19}$, plus (for graviton contributions) $S^{2}$, $S^{4}$, $S^{7}$, $S^{8}$
and $S^{10}$.

Because the ten relations in Table 4 are coupled they are best solved in four
stages. First use the relations for $S^{5}$, $S^{8}$, $S^{9}$ and $S^{10}$ to
write,

$\displaystyle T^{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{0}T^{14R}-\nabla^{2}T^{16R}+S^{5}\;,\qquad$
(48) $\displaystyle T^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{0}T^{10}-\frac{1}{2}\nabla^{2}T^{12}+2S^{8}\;,\qquad$
(49) $\displaystyle T^{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{0}T^{5R}-\nabla^{2}T^{7R}+S^{9}\;,\qquad$
(50) $\displaystyle\nabla^{2}T^{21}$ $\displaystyle=$
$\displaystyle-T^{7}-\frac{1}{2}T^{12}-\frac{1}{2}\partial_{0}T^{19}+S^{10}\;.\qquad$
(51)

Because $T^{21}$ and $T^{12}$ are symmetric we can use relation 51) to solve
for the anti-symmetric part of $T^{7}\equiv T^{7S}+T^{7A}$,

$T^{7A}\equiv\frac{1}{2}(T^{7}-T^{7R})=-\frac{1}{4}(\partial_{0}T^{19}-\partial_{0}^{\prime}T^{19R})+S^{10A}\;.$
(52)

The next step is using the $S^{2}$ and $S^{7}$ relations to solve for $T^{5}$
and $T^{9}$,

$\displaystyle T^{5}$ $\displaystyle=$
$\displaystyle\frac{1}{2}T^{10R}-\frac{1}{2}\partial_{0}T^{18}+\nabla^{2}T^{19R}+2S^{7}\;,\qquad$
(53) $\displaystyle T^{9}$ $\displaystyle=$
$\displaystyle-(D\\!-\\!3)aHT^{10R}-2(\partial_{0}\\!-\\!aH)T^{14}+\Bigl{[}(D\\!-\\!1)aH\partial_{0}\\!-\\!\nabla^{2}\Bigr{]}T^{18}$
(54) $\displaystyle\hskip
85.35826pt-2(D\\!-\\!2)aH\nabla^{2}T^{19R}+4S^{2}-4(D\\!-\\!1)aHS^{7}\;.\qquad$

Relation (53) could be used in (50) to reduce $T^{1}$ but we postpone this. In
the 3rd stage the $S^{1}$ and $S^{4}$ relations give $T^{13}$ and $T^{10}$,

$\displaystyle T^{13}$ $\displaystyle=$
$\displaystyle\frac{1}{\partial_{0}\\!-\\!aH}\Biggl{\\{}-\frac{1}{2}\Bigl{[}(D\\!-\\!1)aH\partial_{0}-\nabla^{2}\Bigr{]}T^{14R}$
(55) $\displaystyle\hskip
85.35826pt+(D\\!-\\!2)aH\nabla^{2}T^{16R}+S^{1}-(D\\!-\\!1)aHS^{5}\Biggr{\\}}\;,\qquad$
$\displaystyle T^{10}$ $\displaystyle=$
$\displaystyle-2(D\\!-\\!2)aHT^{7}-aHT^{12}$ (56) $\displaystyle\hskip
56.9055pt-2(\partial_{0}\\!-\\!aH)T^{16}\\!+\\!(aH\partial_{0}\\!-\\!\nabla^{2})T^{19}\\!+\\!2S^{4}\\!-\\!2aHS^{10}.\qquad$

We now use relations (53) and (56) to update $T^{1}$, $T^{2}$, $T^{5}$ and
$T^{9}$,

$\displaystyle T^{1}$ $\displaystyle=$
$\displaystyle-\Bigl{[}\Bigl{(}\frac{D\\!-\\!2}{2}\Bigr{)}\partial_{0}aH\\!+\\!\nabla^{2}\Bigr{]}T^{7}-\frac{1}{4}\partial_{0}aHT^{12}-\frac{1}{2}\partial_{0}(\partial_{0}\\!-\\!aH)T^{16}$
(57) $\displaystyle\hskip
42.67912pt-\frac{1}{4}\partial_{0}\partial_{0}^{\prime}T^{18}+\frac{1}{4}(\partial_{0}aH\\!-\\!\nabla^{2})\partial_{0}T^{19}+\frac{1}{2}\nabla^{2}\partial_{0}^{\prime}T^{19R}$
$\displaystyle\hskip
42.67912pt+\frac{1}{2}\partial_{0}S^{4}+\partial_{0}^{\prime}S^{7}+\partial_{0}S^{7R}-S^{8}-\frac{1}{2}\partial_{0}aHS^{10}-\nabla^{2}S^{10R}\;,\qquad$
$\displaystyle T^{2}$ $\displaystyle=$
$\displaystyle(D\\!-\\!2)\partial_{0}aHT^{7}+\frac{1}{2}(\partial_{0}aH\\!-\\!\nabla^{2})T^{12}+\partial_{0}(\partial_{0}\\!-\\!aH)T^{16}$
(58) $\displaystyle\hskip
73.97733pt-\frac{1}{2}(\partial_{0}aH\\!-\\!\nabla^{2})\partial_{0}T^{19}-\partial_{0}S^{4}+2S^{8}+\partial_{0}aHS^{10}\;,\qquad$
$\displaystyle T^{5}$ $\displaystyle=$
$\displaystyle-(D\\!-\\!2)a^{\prime}H^{\prime}T^{7R}-\frac{1}{2}a^{\prime}H^{\prime}T^{12}-(\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime})T^{16R}-\frac{1}{2}\partial_{0}T^{18}$
(59) $\displaystyle\hskip
56.9055pt+\frac{1}{2}(a^{\prime}H^{\prime}\partial_{0}^{\prime}\\!+\\!\nabla^{2})T^{19R}+S^{4R}+2S^{7}-a^{\prime}H^{\prime}S^{10R}\;,\qquad$
$\displaystyle T^{9}$ $\displaystyle=$ $\displaystyle
2(D\\!-\\!3)(D\\!-\\!2)aHa^{\prime}H^{\prime}T^{7R}+(D\\!-\\!3)aHa^{\prime}H^{\prime}T^{12}-2(\partial_{0}\\!-\\!aH)T^{14}$
(60) $\displaystyle\hskip
28.45274pt+2(D\\!-\\!3)aH(\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime})T^{16R}\\!+\\!\Bigl{[}(D\\!-\\!1)aH\partial_{0}\\!-\\!\nabla^{2}\Bigr{]}T^{18}$
$\displaystyle\hskip
28.45274pt-\Bigl{[}(D\\!-\\!3)aHa^{\prime}H^{\prime}\partial_{0}^{\prime}\\!+\\!(D\\!-\\!1)aH\nabla^{2}\Bigr{]}T^{19R}+4S^{2}$
$\displaystyle\hskip
28.45274pt-2(D\\!-\\!3)aHS^{4R}\\!-\\!4(D\\!-\\!1)aHS^{7}\\!+\\!2(D\\!-\\!3)aHa^{\prime}H^{\prime}S^{10R}\;.\qquad$

The final stage begins by noting that the $S^{3}$ and $S^{5}$ relations can be
expressed in terms of two functions $A(x;x^{\prime})$ and $B(x;x^{\prime})$,

$\displaystyle S^{3}$ $\displaystyle\qquad\Longrightarrow\qquad$
$\displaystyle
0=-\Bigl{[}(D\\!-\\!3)aH\partial_{0}\\!+\\!\nabla^{2}\Bigr{]}A+\partial_{0}^{\prime}B\;,\qquad$
(61) $\displaystyle S^{6}$ $\displaystyle\qquad\Longrightarrow\qquad$
$\displaystyle
0=+\Bigl{[}(D\\!-\\!3)\partial_{0}aH\\!+\\!\nabla^{2}\Bigr{]}A^{R}-\partial_{0}B\;.\qquad$
(62)

The functions $A$ and $B$ are,

$\displaystyle A$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}(D\\!-\\!2)aHT^{7}+\frac{1}{4}aHT^{12}+\frac{1}{2}(\partial_{0}\\!-\\!aH)T^{16}-\frac{1}{4}(aH\partial_{0}\\!-\\!\nabla^{2})T^{19}$
(63) $\displaystyle\hskip
184.9429pt-\frac{1}{2}S^{4}-\frac{1}{\partial_{0}}S^{8R}+\frac{1}{2}aHS^{10}\;,\qquad$
$\displaystyle B$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}(\partial_{0}\\!-\\!aH)T^{14}-\frac{1}{4}\Bigl{[}(D\\!-\\!1)aH\partial_{0}\\!-\\!\nabla^{2}\Bigr{]}T^{18}+\frac{(D\\!-\\!2)}{2}aH\nabla^{2}T^{19R}$
(64) $\displaystyle\hskip
125.19194pt-S^{2}+(D\\!-\\!1)aHS^{7}-(D\\!-\\!3)aH\frac{1}{\partial_{0}^{\prime}}S^{8}\;.\qquad$

We first solve equation (61) for $T^{14}$,

$\displaystyle
T^{14}=\frac{1}{\partial_{0}\\!-\\!aH}\Biggl{\\{}\frac{1}{2}\Bigl{[}(D\\!-\\!1)aH\partial_{0}\\!-\\!\nabla^{2}\Bigr{]}T^{18}-(D\\!-\\!2)aH\nabla^{2}T^{19R}+2S^{2}$
(65) $\displaystyle\hskip
14.22636pt-2(D\\!-\\!1)aHS^{7}+2(D\\!-\\!3)aH\frac{1}{\partial_{0}^{\prime}}S^{8}+\Bigl{[}(D\\!-\\!3)aH\partial_{0}\\!+\\!\nabla^{2}\Bigr{]}\frac{1}{\partial_{0}^{\prime}}A\Biggr{\\}}.\qquad$

The final relation derives from combining (61) with (62),

$\partial_{0}\Bigl{(}{\rm
Eqn.}~{}\ref{S3final}\Bigr{)}-\partial_{0}^{\prime}\Bigl{(}{\rm
Eqn.}~{}\ref{S6final}\Bigr{)}=\Bigl{[}(D\\!-\\!3)\partial_{0}aH\\!+\\!\nabla^{2}\Bigr{]}\Bigl{[}-\partial_{0}A+\partial_{0}^{\prime}A^{R}\Bigr{]}=0\;.$
(66)

It follows that $\partial_{0}A=\partial_{0}^{\prime}A^{R}$, which allows us to
solve for the symmetric part of $T^{7}$. Combining this with the antisymmetric
part (52) gives,

$T^{7}=-\frac{\frac{1}{2}T^{12}}{D\\!-\\!2}-\frac{1}{2}\partial_{0}T^{19}+S^{10}-\frac{2}{D\\!-\\!2}\frac{1}{(\partial_{0}aH\\!-\\!\partial_{0}^{\prime}a^{\prime}H^{\prime})}\Bigl{[}\partial_{0}\Delta
A\\!-\\!\partial_{0}^{\prime}\Delta A^{R}\Bigr{]}\;.$ (67)

Here the residual part of $A$ is,

$\displaystyle\Delta
A\equiv\frac{1}{2}(\partial_{0}\\!-\\!aH)T^{16}-\frac{1}{4}\Bigl{[}(D\\!-\\!1)aH\partial_{0}\\!-\\!\nabla^{2}\Bigr{]}T^{19}$
(68) $\displaystyle\hskip
167.87108pt-\frac{1}{2}S^{4}-\frac{1}{\partial_{0}}S^{8R}+\frac{(D\\!-\\!1)}{2}aHS^{10}\;.\qquad$

We should comment on how to invert the differential operator
$\mathcal{D}=\partial_{0}aH-\partial_{0}^{\prime}a^{\prime}H^{\prime}$. This
is accomplished by first factoring out $aH\times a^{\prime}H^{\prime}$,

$\mathcal{D}=\frac{1}{aH\,a^{\prime}H^{\prime}}\times\frac{1}{aH\partial_{0}\\!-\\!a^{\prime}H^{\prime}\partial_{0}^{\prime}}\times
aH\,a^{\prime}H^{\prime}\;.$ (69)

Now change the time variable from $\eta$ to $u$ such that,

$du\equiv\frac{d\eta}{aH}\;.$ (70)

By employing “lightcone” variables,

$u_{\pm}\equiv\frac{1}{2}(u\pm
u^{\prime})\qquad\Longrightarrow\qquad\frac{\partial}{\partial
u}-\frac{\partial}{\partial u^{\prime}}=\frac{\partial}{\partial u_{-}}\;,$
(71)

we can express (69) as an integration with respect to $u_{-}$,

$\frac{1}{\partial_{0}aH\\!-\\!\partial_{0}^{\prime}a^{\prime}H^{\prime}}=\frac{1}{aH\,a^{\prime}H^{\prime}}\times\int\\!\\!du_{-}\times
aH\,a^{\prime}H^{\prime}\;.$ (72)

For the important special case of de Sitter we have $u=-\frac{1}{2}\eta^{2}$.

Our final expressions for the coefficient functions can be simplified by using
two symmetric auxiliary functions to absorb all the terms involving the
inverse of $\mathcal{D}$. We first define the (not necessarily symmetric)
function $\gamma(x;x^{\prime})$,

$\displaystyle\gamma\equiv\partial_{0}(\partial_{0}\\!-\\!aH)T^{16}-\frac{1}{2}\Bigl{[}(D\\!-\\!1)\partial_{0}aH\\!-\\!\nabla^{2}\Bigr{]}\partial_{0}T^{19}$
(73) $\displaystyle\hskip
170.71652pt-\partial_{0}S^{4}-2S^{8R}+(D\\!-\\!1)\partial_{0}aHS^{10}\;.\qquad$

The two symmetric functions are,

$\displaystyle\alpha$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\mathcal{D}}\Bigl{[}\gamma-\gamma^{R}\Bigr{]}\;,\qquad$
(74) $\displaystyle\beta$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\Bigl{(}\gamma\\!+\\!\gamma^{R}\Bigr{)}-\frac{1}{2}\Bigl{(}\partial_{0}aH\\!+\\!\partial_{0}^{\prime}a^{\prime}H^{\prime}\Bigr{)}\alpha\;.\qquad$
(75)

Note that the function $\beta(x;x^{\prime})$ can be written in two different
ways,

$\beta=\gamma-\partial_{0}aH\alpha=\gamma^{R}-\partial_{0}^{\prime}a^{\prime}H^{\prime}\alpha\;.$
(76)

Also note that we can eliminate $T^{16}$,

$\displaystyle
T^{16}=\frac{1}{\partial_{0}\\!-\\!aH}\frac{1}{\partial_{0}}\Biggl{\\{}\beta+\partial_{0}aH\alpha+\frac{1}{2}\Bigl{[}(D\\!-\\!1)\partial_{0}aH\\!-\\!\nabla^{2}\Bigr{]}\partial_{0}T^{19}$
(77) $\displaystyle\hskip
156.49014pt+\partial_{0}S^{4}+2S^{8R}-(D\\!-\\!1)\partial_{0}aHS^{10}\Biggr{\\}}.\qquad$

The notation can be further simplified by introducing symbols to stand for
three differential operators and an inverse operator that occur repeatedly,

$\displaystyle D_{0}\equiv\partial_{0}aH\qquad$ , $\displaystyle\qquad
D_{1}\equiv(D\\!-\\!1)\partial_{0}aH-\nabla^{2}\;,$ (78)
$\displaystyle\mathcal{I}\equiv\frac{1}{\partial_{0}\\!-\\!aH}\frac{1}{\partial_{0}}\qquad$
, $\displaystyle\qquad D_{3}\equiv(D\\!-\\!3)\partial_{0}aH+\nabla^{2}\;.$
(79)

Giving any of these operators a prime indicates that it is constructed from
the same quantities at ${x^{\prime}}^{\mu}$ instead of $x^{\mu}$, for example,
$D_{0}^{\prime}=\partial_{0}^{\prime}a^{\prime}H^{\prime}$. With these
definitions our final expressions for the algebraically independent
coefficient functions are,

$\displaystyle
T^{1}=\frac{\nabla^{2}T^{12}}{2(D\\!-\\!2)}-\frac{1}{4}\partial_{0}\partial_{0}^{\prime}T^{18}+\frac{1}{2}\nabla^{2}\Bigl{(}\partial_{0}T^{19}\\!+\\!\partial_{0}^{\prime}T^{19R}\Bigr{)}+\frac{\nabla^{2}\alpha}{D\\!-\\!2}-\frac{1}{2}\beta$
(80) $\displaystyle\hskip
82.51282pt+\Bigl{(}\partial_{0}^{\prime}S^{7}\\!+\\!\partial_{0}S^{7R}\Bigr{)}-\Bigl{(}S^{8}\\!+\\!S^{8R}\Bigr{)}-\nabla^{2}\Bigl{(}S^{10}\\!+\\!S^{10R}\Bigr{)}\;,\qquad$
$\displaystyle
T^{2}=-\frac{1}{2}\nabla^{2}T^{12}+\beta+2\Bigl{(}S^{8}\\!+\\!S^{8R}\Bigr{)}\;,$
(81) $\displaystyle
T^{3}=\mathcal{I}^{\prime}\Biggl{\\{}\frac{1}{4}D_{1}^{\prime}\partial_{0}\partial_{0}^{\prime}T^{18}-\frac{1}{2}(D\\!-\\!2)\nabla^{2}D_{0}^{\prime}\partial_{0}T^{19}-\frac{1}{2}\nabla^{2}D_{1}^{\prime}\partial_{0}^{\prime}T^{19R}-\nabla^{2}D_{0}^{\prime}\alpha$
(82)
$\displaystyle+\Bigl{(}\frac{1}{2}D_{1}^{\prime}\\!-\\!D_{0}^{\prime}\Bigr{)}\beta+\partial_{0}\partial_{0}^{\prime}S^{2R}-\nabla^{2}\partial_{0}^{\prime}S^{4R}-D_{1}^{\prime}\partial_{0}^{\prime}S^{7}-(D\\!-\\!1)D_{0}^{\prime}\partial_{0}S^{7R}$
$\displaystyle\hskip
11.38092pt+D_{3}^{\prime}S^{8}+(D\\!-\\!3)D_{0}^{\prime}S^{8R}+(D\\!-\\!2)\nabla^{2}D_{0}^{\prime}S^{10}+(D\\!-\\!1)\nabla^{2}D_{0}^{\prime}S^{10R}\Biggr{\\}},\qquad$
$\displaystyle
T^{5}=-\frac{1}{2}\partial_{0}T^{18}+\nabla^{2}T^{19R}-\frac{1}{\partial_{0}^{\prime}}\beta+2S^{7}-\frac{2}{\partial_{0}^{\prime}}S^{8}\;,$
(83) $\displaystyle
T^{7}=-\frac{T^{12}}{2(D\\!-\\!2)}-\frac{1}{2}\partial_{0}T^{19}-\frac{\alpha}{D\\!-\\!2}+S^{10}\;,$
(84) $\displaystyle
T^{9}=-\frac{2\nabla^{2}}{\partial_{0}\partial_{0}^{\prime}}\beta\;,$ (85)
$\displaystyle
T^{10}=-\frac{2}{\partial_{0}}\beta-\frac{4}{\partial_{0}}S^{8R}\;,$ (86)
$\displaystyle
T^{13}=\mathcal{I}\mathcal{I}^{\prime}\Biggl{\\{}-\frac{1}{4}D_{1}D_{1}^{\prime}\partial_{0}\partial_{0}^{\prime}T^{18}+\frac{1}{2}(D\\!-\\!2)\nabla^{2}\Bigl{[}D_{1}D_{0}^{\prime}\partial_{0}T^{19}\\!+\\!D_{0}D_{1}^{\prime}\partial_{0}^{\prime}T^{19R}\Bigr{]}$
(87)
$\displaystyle+(D\\!-\\!2)\nabla^{2}D_{0}D_{0}^{\prime}\alpha\\!-\\!\frac{1}{2}\Bigl{[}(D\\!-\\!3)D_{1}D_{3}^{\prime}\\!-\\!D_{3}\nabla^{2}\Bigr{]}\beta\\!-\\!D_{1}^{\prime}\partial_{0}\partial_{0}^{\prime}S^{2}\\!-\\!D_{1}\partial_{0}\partial_{0}^{\prime}S^{2R}$
$\displaystyle+(D\\!-\\!2)\nabla^{2}\Bigl{[}D_{0}^{\prime}\partial_{0}S^{4}\\!+\\!D_{0}\partial_{0}^{\prime}S^{4R}\Bigr{]}+(D\\!-\\!1)\Bigl{[}D_{0}D_{1}^{\prime}\partial_{0}^{\prime}S^{7}\\!+\\!D_{0}^{\prime}D_{1}\partial_{0}S^{7R}\Bigr{]}$
$\displaystyle-(D\\!-\\!3)\Bigl{[}D_{0}D_{1}^{\prime}S^{8}\\!+\\!D_{0}D_{1}S^{8R}\Bigr{]}\\!-\\!(D\\!-\\!2)(D\\!-\\!1)\nabla^{2}D_{0}D_{0}^{\prime}(S^{10}\\!+\\!S^{10R})\Biggr{\\}},\qquad$
$\displaystyle
T^{14}=\mathcal{I}\Biggl{\\{}\frac{1}{2}D_{1}\partial_{0}T^{18}-(D\\!-\\!2)\nabla^{2}D_{0}T^{19R}+\frac{D_{3}}{\partial_{0}^{\prime}}\beta$
(88) $\displaystyle\hskip
122.34692pt+2\partial_{0}S^{2}-2(D\\!-\\!1)D_{0}S^{7}+2(D\\!-\\!3)\frac{D_{0}}{\partial_{0}^{\prime}}S^{8}\Biggr{\\}},\qquad$
$\displaystyle
T^{21}=-\frac{1}{2}\Bigl{(}\frac{D\\!-\\!3}{D\\!-\\!2}\Bigr{)}\frac{1}{\nabla^{2}}T^{12}+\frac{1}{D\\!-\\!2}\frac{1}{\nabla^{2}}\alpha\;.$
(89)

The cumbersome nature of these expressions prompts several comments on the
issue of accuracy. First, the flat space limit agrees with the decomposition
of section 2.2. Second, our results for $T^{1}(x;x^{\prime})$,
$T^{2}(x;x^{\prime})$, $T^{9}(x;x^{\prime})$ and $T^{13}(x;x^{\prime})$ are
reflection symmetric as they should be. Finally, the contributions
proportional to $T^{12}$, $T^{18}$, $T^{19}$, $T^{19R}$, $\alpha$, and $\beta$
are each separately annihilated by the action of the Ward operator (5) on
either index group, while the contributions proportional to $S^{2}$, $S^{4}$,
$S^{7}$, $S^{8}$ and $S^{10}$ are annihilated when the Ward operator is acted
on both index groups.

###  3.2 Ricci and Weyl Operators

Our results (77) and (80-89) can be reorganized into a sum of products of
tensor differential operators acting on the fundamental structure functions.
Each of these tensor differential operators is separately annihilated by the
action of either one or two Weyl operators. All the operators descend from
$3+1$ decomposing the same products of the transverse projection operator that
we encountered in section 2.2,

$\Pi^{\mu\nu}\equiv\partial^{\mu}\partial^{\nu}-\partial^{2}\eta^{\mu\nu}=\Pi_{A}^{\mu\nu}+\Pi_{B}^{\mu\nu}\;,$
(90)

where the two projectors are,

$\displaystyle\Pi_{A}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\overline{\partial}^{\mu}\overline{\partial}^{\nu}-\nabla^{2}\overline{\eta}^{\mu\nu}\;,$
(91) $\displaystyle\Pi_{B}^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\nabla^{2}\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}-2\partial_{0}\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}+\partial_{0}^{2}\overline{\eta}^{\mu\nu}\;.$
(92)

What we term the Ricci operators are simple extensions of (91) and (92),

$\displaystyle\Pi_{A}^{\mu\nu}\longrightarrow\overline{\partial}^{\mu}\overline{\partial}^{\nu}-\nabla^{2}\overline{\eta}^{\mu\nu}+\frac{(D\\!-\\!2)\nabla^{2}}{\partial_{0}\\!-\\!aH}aH\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\equiv\mathcal{R}_{A}^{\mu\nu}(x)\;,$
(93)
$\displaystyle\Pi_{B}^{\mu\nu}\longrightarrow\frac{\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}}{\partial_{0}\\!-\\!aH}\Bigl{[}\nabla^{2}\\!-\\!(D\\!-\\!1)aH\partial_{0}\Bigr{]}\partial_{0}\\!-\\!2\partial_{0}\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\\!+\\!\partial_{0}^{2}\overline{\eta}^{\mu\nu}\equiv\mathcal{R}_{B}^{\mu\nu}(x)\;.$
(94)

It is straightforward to verify that acting the Ward operator annihilates each
Ricci operator,

$\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\\!\times\\!\mathcal{R}_{A}^{\alpha\beta}(x)=0=\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\\!\times\\!\mathcal{R}_{B}^{\alpha\beta}(x)\;.$
(95)

The related Weyl operators come from extending products of two projection
operators. The simplest is purely spatial,

$\Pi_{A}^{\mu(\rho}\\!\times\\!\Pi_{A}^{\sigma)\nu}\longrightarrow\Pi_{A}^{\mu(\rho}\Pi_{A}^{\sigma)\nu}-\frac{\Pi_{A}^{\mu\nu}\Pi_{A}^{\rho\sigma}}{D\\!-\\!2}\equiv\mathcal{C}_{AA}^{\mu\nu\rho\sigma}\;.$
(96)

Because $\mathcal{C}_{AA}^{\mu\nu\rho\sigma}$ is both transverse and
traceless, it is annihilated when the Ward operator acts on either index
group,

$\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\\!\times\\!\mathcal{C}_{AA}^{\alpha\beta\rho\sigma}=0=\mathcal{W}^{\rho}_{~{}\gamma\delta}(x^{\prime})\\!\times\\!\mathcal{C}_{AA}^{\mu\nu\gamma\delta}\;.$
(97)

The second Weyl operator comes from extending the product of two $B$-type
projectors,

$\displaystyle\Pi_{B}^{\mu(\rho}\\!\times\\!\Pi_{B}^{\sigma)\nu}\longrightarrow\mathcal{C}_{BB}^{\mu\nu\rho\sigma}(x;x^{\prime})\equiv\partial_{0}^{2}{\partial_{0}^{\prime}}^{2}\overline{\eta}^{\mu(\rho}\overline{\eta}^{\sigma)\nu}-2\partial_{0}{\partial_{0}^{\prime}}^{2}\delta^{(\mu}_{~{}~{}0}\overline{\eta}^{\nu)(\rho}\overline{\partial}^{\sigma)}$
(98)
$\displaystyle+2\partial_{0}^{2}\partial_{0}^{\prime}\overline{\partial}^{(\mu}\overline{\eta}^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}-2\partial_{0}\partial_{0}^{\prime}\nabla^{2}\delta^{(\mu}_{~{}~{}0}\overline{\eta}^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}-2\partial_{0}\partial_{0}^{\prime}\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}$
$\displaystyle-\frac{\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}}{\partial_{0}\\!-\\!aH}aH\partial_{0}^{2}\Bigl{[}{\partial_{0}^{\prime}}^{2}\overline{\eta}^{\rho\sigma}\\!\\!+\\!2\partial_{0}^{\prime}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}\Bigr{]}-\Bigl{[}\partial_{0}^{2}\overline{\eta}^{\mu\nu}\\!\\!-\\!2\partial_{0}\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\Bigr{]}\frac{\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}a^{\prime}H^{\prime}{\partial_{0}^{\prime}}^{2}$
$\displaystyle+\frac{\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}}{\partial_{0}\\!-\\!aH}\partial_{0}\Bigl{[}\partial_{0}^{\prime}\overline{\partial}^{\rho}\overline{\partial}^{\sigma}\\!\\!+\\!2\nabla^{2}\delta^{(\rho}_{~{}~{}0}\overline{\partial}^{\sigma)}\Bigr{]}\partial_{0}^{\prime}+\Bigl{[}\partial_{0}\overline{\partial}^{\mu}\overline{\partial}^{\nu}\\!\\!-\\!2\nabla^{2}\delta^{(\mu}_{~{}~{}0}\overline{\partial}^{\nu)}\Bigr{]}\partial_{0}\frac{\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}\partial_{0}^{\prime}$
$\displaystyle+\frac{\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}}{(\partial_{0}\\!-\\!aH)(\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime})}\\!\Bigl{[}(D\\!-\\!1)aH\partial_{0}a^{\prime}\\!H^{\prime}\partial_{0}^{\prime}\\!\\!-\\!(aH\partial_{0}\\!+\\!a^{\prime}H^{\prime}\partial_{0}^{\prime})\nabla^{2}\\!\\!+\\!\\!\nabla^{4}\\!\Bigr{]}\partial_{0}\partial_{0}^{\prime}.\quad$

One can also show that $\mathcal{C}_{BB}^{\mu\nu\rho\sigma}$ is annihilated by
the Ward operator acting on either index group,

$\mathcal{W}^{\mu}_{~{}\alpha\beta}(x)\\!\times\\!\mathcal{C}_{BB}^{\alpha\beta\rho\sigma}(x;x^{\prime})=0=\mathcal{W}^{\rho}_{~{}\gamma\delta}(x^{\prime})\\!\times\\!\mathcal{C}_{BB}^{\mu\nu\gamma\delta}(x;x^{\prime})\;.$
(99)

We can use the Ricci and Weyl operators to express matter contributions to the
graviton self-energy,

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
mat}\Bigr{]}(x;x^{\prime})=-\frac{1}{2}\mathcal{C}_{AA}^{\mu\nu\rho\sigma}\\!\times\\!\frac{1}{\nabla^{2}}T^{12}-\frac{1}{4}\mathcal{R}_{B}^{\mu\nu}(x)\\!\times\\!\mathcal{R}_{B}^{\rho\sigma}(x^{\prime})\\!\times\\!\frac{1}{\partial_{0}\partial_{0}^{\prime}}T^{18}$
(100)
$\displaystyle+\frac{1}{2}\Bigl{[}\mathcal{R}_{B}^{\mu\nu}(x)\\!\times\\!\mathcal{R}_{A}^{\rho\sigma}(x^{\prime})\\!\times\\!\frac{1}{\partial_{0}}T^{19}+\mathcal{R}_{A}^{\mu\nu}(x)\\!\times\\!\mathcal{R}_{B}^{\rho\sigma}(x^{\prime})\\!\times\\!\frac{1}{\partial_{0}^{\prime}}T^{19R}\Bigr{]}$
$\displaystyle+\frac{\mathcal{R}_{A}^{\mu\nu}(x)\\!\times\\!\mathcal{R}_{A}^{\rho\sigma}(x^{\prime})}{D\\!-\\!2}\frac{1}{\nabla^{2}}\alpha+\Bigl{[}\mathcal{C}_{BB}^{\mu\nu\rho\sigma}(x;x^{\prime})\\!-\\!\frac{1}{2}\mathcal{R}_{B}^{\mu\nu}(x)\\!\times\\!\mathcal{R}_{B}^{\rho\sigma}(x^{\prime})\Bigr{]}\frac{1}{\partial_{0}^{2}{\partial_{0}^{\prime}}^{2}}\beta\;.\qquad$

Expressing contributions from gravity itself requires the additional operators
formed from suppressing the temporal components of
$\mathcal{C}_{BB}^{\mu\nu\rho\sigma}(x;x^{\prime})$ on either $x^{\mu}$ of
${x^{\prime}}^{\mu}$,

$\mathcal{C}_{BB}^{\overline{\mu}\overline{\nu}\rho\sigma}(x;x^{\prime})\equiv\frac{\overline{\delta}^{\mu}_{~{}\alpha}\overline{\delta}^{\nu}_{~{}\beta}}{\partial_{0}^{2}}\mathcal{C}_{BB}^{\alpha\beta\rho\sigma}(x;x^{\prime})\;\;,\;\;\mathcal{C}_{BB}^{\mu\nu\overline{\rho}\overline{\sigma}}(x;x^{\prime})\equiv\frac{\overline{\delta}^{\rho}_{~{}\gamma}\overline{\delta}^{\sigma}_{~{}\delta}}{{\partial_{0}^{\prime}}^{2}}\mathcal{C}_{BB}^{\mu\nu\gamma\delta}(x;x^{\prime})\;,$
(101)

where
$\overline{\delta}^{\alpha}_{~{}\beta}\equiv\delta^{\alpha}_{~{}\beta}-\delta^{\alpha}_{~{}0}\delta^{0}_{~{}\beta}$
is the spatial unit matrix. The contributions from gravity itself include all
of the same terms in (100) with the addition of terms involving the structure
functions $S^{2}$, $S^{4}$, $S^{7}$, $S^{8}$ and $S^{10}$,

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
grav}\Bigr{]}(x;x^{\prime})=\Bigl{(}{\rm Eqn.}~{}\ref{mattersigma}\Bigr{)}$
(102)
$\displaystyle+\frac{\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}}{\partial_{0}\\!-\\!aH}\Bigl{[}\mathcal{R}_{B}^{\rho\sigma}(x^{\prime})\frac{1}{\partial_{0}^{\prime}}S^{2}\\!\\!+\\!\mathcal{R}_{A}^{\rho\sigma}(x^{\prime})S^{4}\Bigr{]}\\!+\\!\frac{\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}\Bigl{[}\mathcal{R}_{B}^{\mu\nu}(x)\frac{1}{\partial_{0}}S^{2R}\\!\\!+\\!\mathcal{R}_{A}^{\mu\nu}(x)S^{4R}\Bigr{]}$
$\displaystyle+\Bigl{[}\overline{\eta}^{\mu\nu}\\!-\\!\frac{(D\\!-\\!1)\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}}{\partial_{0}\\!-\\!aH}aH\Bigr{]}\Bigl{[}\mathcal{R}_{B}^{\rho\sigma}(x^{\prime})\frac{1}{\partial_{0}^{\prime}}S^{7}+\mathcal{R}_{A}^{\rho\sigma}(x^{\prime})S^{10}\Bigr{]}$
$\displaystyle\hskip
14.22636pt+\Bigl{[}\overline{\eta}^{\rho\sigma}\\!-\\!\frac{(D\\!-\\!1)\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}a^{\prime}H^{\prime}\Bigr{]}\Bigl{[}\mathcal{R}_{B}^{\mu\nu}(x)\frac{1}{\partial_{0}}S^{7R}+\mathcal{R}_{A}^{\mu\nu}(x)S^{10R}\Bigr{]}$
$\displaystyle\hskip
28.45274pt+\Biggl{\\{}\mathcal{C}_{BB}^{\overline{\mu}\overline{\nu}\rho\sigma}(x;x^{\prime})-\Bigl{[}\overline{\eta}^{\mu\nu}-\frac{(D\\!-\\!3)\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}}{\partial_{0}\\!-\\!aH}aH\Bigr{]}\mathcal{R}_{B}^{\rho\sigma}(x^{\prime})\Biggr{\\}}\frac{1}{{\partial_{0}^{\prime}}^{2}}S^{8}$
$\displaystyle\hskip
42.67912pt+\Biggl{\\{}\mathcal{C}_{BB}^{\mu\nu\overline{\rho}\overline{\sigma}}(x;x^{\prime})-\mathcal{R}_{B}^{\mu\nu}(x)\Bigl{[}\overline{\eta}^{\rho\sigma}-\frac{(D\\!-\\!3)\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}}{\partial_{0}^{\prime}\\!-\\!a^{\prime}H^{\prime}}a^{\prime}H^{\prime}\Bigr{]}\Biggr{\\}}\frac{1}{\partial_{0}^{2}}S^{8R}.\qquad$

###  3.3 Effective Field Equations

Of course the point of developing this representation for
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ is to facilitate
solving the effective field equation (1). We here adapt it to two important
special cases:

  1. 1.

Plane wave gravitons; and

  2. 2.

The response to a point mass.

In the first case the graviton field $h_{\mu\nu}(x)$ takes the form,

${\rm Gravitons}\Longrightarrow
h_{\mu\nu}=u(\eta,k)e^{i\vec{k}\cdot\vec{x}}\epsilon_{\mu\nu}(\vec{k})\quad,\quad\epsilon_{0\mu}=0=k_{i}\epsilon_{i\mu}=\epsilon_{ii}\;,$
(103)

and what we seek is an equation for the mode function $u(\eta,k)$. For the
second the graviton field takes the form,

${\rm Potentials}\Longrightarrow\kappa h_{00}=-2\Psi(\eta,r)\;,\;\kappa
h_{0i}=0\;,\;\kappa h_{ij}=-2\Phi(\eta,r)\delta_{ij}\;,$ (104)

and what we seek are equations for the two potentials $\Psi(\eta,r)$ and
$\Phi(\eta,r)$.

####  3.3.1 The Graviton Mode Function

On a general cosmological background the Lichnerowicz operator receives
contributions from whatever matter source supports the geometry. On de Sitter
background its action on a general graviton field takes the form [19],

$\mathcal{L}^{\mu\nu\rho\sigma}h_{\rho\sigma}=\partial_{\alpha}\Bigl{[}a^{2}\mathcal{L}^{\mu\nu\rho\sigma\alpha\beta}\partial_{\beta}h_{\rho\sigma}\Bigr{]}+\partial_{\alpha}\Bigl{[}Ha^{3}\eta^{\mu\nu}h^{\alpha
0}\Bigr{]}-Ha^{3}\eta^{0(\mu}\partial^{\nu)}h\;,$ (105)

where the tensor factor is,

$\mathcal{L}^{\mu\nu\rho\sigma\alpha\beta}=\frac{1}{2}\eta^{\alpha\beta}(\eta^{\mu(\rho}\eta^{\sigma)\nu}\\!-\\!\eta^{\mu\nu}\eta^{\rho\sigma})\\!+\\!\frac{1}{2}\eta^{\mu\nu}\eta^{\rho(\alpha}\eta^{\beta)\sigma}\\!+\\!\frac{1}{2}\eta^{\rho\sigma}\eta^{\mu(\alpha}\eta^{\beta)\nu}\\!-\\!\eta^{\alpha(\rho}\eta^{\sigma)(\mu}\eta^{\nu)\beta}.$
(106)

Acting the Lichnerowicz operator on a plane wave graviton (103) gives,

$\mathcal{L}^{\mu\nu\rho\sigma}u(\eta,k)e^{i\vec{k}\cdot\vec{x}}\epsilon_{\rho\sigma}=-\frac{1}{2}a^{2}\Bigl{[}\partial_{0}^{2}+2aH\partial_{0}+k^{2}\Bigr{]}u(\eta,k)\times
e^{i\vec{k}\cdot\vec{x}}\epsilon^{\mu\nu}\;.$ (107)

Gravitons have zero stress tensor. Because their polarization tensor
$\epsilon_{\mu\nu}$ is purely spatial, transverse and also traceless, the only
one of the coefficient functions $T^{i}(x;x^{\prime})$ that contributes is
$T^{2}(x;x^{\prime})$,333Of course $T^{2}(x;x^{\prime})$ may vanish for some
theories, however, it is generally nonzero because gravity couples to all
fields and the 0-point fluctuations of these fields can make nonzero
contributions to $-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ even
if the stress-energy of the background vanishes.

$\int\\!\\!d^{4}x^{\prime}\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})h_{\rho\sigma}(x^{\prime})=\epsilon^{\mu\nu}(\vec{k})e^{i\vec{k}\cdot\vec{x}}\int\\!\\!d^{4}x^{\prime}\,iT^{2}(x;x^{\prime})u(\eta^{\prime},k)e^{-i\vec{k}\cdot\Delta\vec{x}}\;.$
(108)

Hence the equation for corrections to the graviton mode function is,

$-\frac{1}{2}a^{2}\Bigl{[}\partial_{0}^{2}+2aH\partial_{0}+k^{2}\Bigr{]}u(\eta,k)=\int\\!\\!d^{4}x^{\prime}\,iT^{2}(x;x^{\prime})u(\eta^{\prime},k)e^{-i\vec{k}\cdot\Delta\vec{x}}\;.$
(109)

This is the same for matter and gravity contributions, however, what $T^{2}$
is in terms for the fundamental structure functions differs according to
relation (81). Note that it will be necessary to extract a number of
derivatives from the structure functions; primed derivatives being acted on
the mode function and unprimed derivatives pulled outside the integration.

####  3.3.2 Response to A Point Mass

Acting the Lichnerowicz operator on the potentials (104) produces,

$\displaystyle\mathcal{L}^{\mu\nu\rho\sigma}\Bigl{[}-2\delta^{0}_{\rho}\delta^{0}_{\sigma}\Psi-2\overline{\eta}_{\rho\sigma}\Phi\Bigr{]}=a^{2}\delta^{\mu}_{0}\delta^{\nu}_{0}\Bigl{[}6a^{2}H^{2}\Psi-2(\nabla^{2}\\!-\\!3aH\partial_{0})\Phi\Bigr{]}$
(110) $\displaystyle\hskip
14.22636pt+2a^{2}\delta^{(\mu}_{0}\overline{\partial}^{\nu)}\Bigl{[}2aH\Psi+2\partial_{0}\Phi\Bigr{]}+a^{2}\overline{\partial}^{\mu}\overline{\partial}^{\nu}\Bigl{[}\Psi-\Phi\Bigr{]}$
$\displaystyle\hskip
42.67912pt+a^{2}\overline{\eta}^{\mu\nu}\Bigl{[}-(\nabla^{2}\\!+\\!2aH\partial_{0}\\!+\\!6a^{2}H^{2})\Psi+(\nabla^{2}\\!-\\!4aH\partial_{0}\\!-\\!2\partial_{0}^{2})\Phi\Bigr{]}\;.\qquad$

The potentials $\Psi(\eta,r)$ and $\Phi(\eta,r)$ are the response to a static
point mass $M$ whose linearized stress tensor is,

$\frac{1}{2}\kappa T^{\mu\nu}_{\rm lin}(\eta,\vec{x})=-\frac{1}{2}\kappa
Ma(\eta)\delta^{3}(\vec{x})\delta^{\mu}_{0}\delta^{\nu}_{0}\;.$ (111)

The zeroth order response is,

$\Psi_{0}(\eta,r)=\Phi_{0}(\eta,r)=-\frac{GM}{a(\eta)r}\;.$ (112)

Loop corrections are sourced by the integral of the self-energy against lower
order response,

$\mathcal{L}^{\mu\nu\rho\sigma}h_{\rho\sigma}(x)=\int\\!\\!d^{4}x^{\prime}\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})h_{\rho\sigma}(x^{\prime})\equiv\mathcal{S}^{\mu\nu}(x)\;.$
(113)

Although we have worked out all components of $\mathcal{S}^{\mu\nu}(x)$, the
only necessary ones are $\mathcal{S}^{0i}$ — which is $\partial_{i}$ of
something — and the $\partial_{i}\partial_{j}$ part of $\mathcal{S}^{ij}$.
Comparison with (110) implies that the two potentials obey,

$\displaystyle
4a^{3}H\Psi+4a^{2}\partial_{0}\Phi=2\\!\int\\!\\!d^{4}x^{\prime}\Biggl{\\{}iT^{14}(x^{\prime};x)\Psi(x^{\prime})$
(114) $\displaystyle\hskip
85.35826pt+\Bigl{[}3iT^{5}(x^{\prime};x)\\!-\\!iT^{10}(x;x^{\prime})\\!-\\!iT^{19}(x;x^{\prime})\nabla^{2}\Bigr{]}\Phi(x^{\prime})\Biggr{\\}},\qquad$
$\displaystyle
a^{2}\Psi-a^{2}\Phi=-2\\!\int\\!\\!d^{4}x^{\prime}\Biggl{\\{}iT^{16}(x^{\prime};x)\Psi(x^{\prime})$
(115) $\displaystyle\hskip
156.49014pt+\Bigl{[}3iT^{7}(x^{\prime};x)\\!+\\!iT^{12}(x;x^{\prime})\Bigr{]}\Phi(x^{\prime})\Biggr{\\}}.\qquad$

The same comments apply to these results as for the mode equation (109): these
equations are valid for any contribution to the graviton self-energy, although
what those contributions are in terms of the fundamental structure functions
varies from matter to gravity according to relations (83), (84) and (86). And
one should also note that derivatives will be extracted from the structure
functions, with primed ones partially integrated onto the potentials and
unprimed ones taken outside the integration.

##  4 Explicit Examples on de Sitter

The previous section described our formalism for representing the graviton
self-energy in cosmology. The purpose of this section is to put this formalism
in context with two explicit one loop results obtained on de Sitter
background. As an example of matter contributions (100) we consider the
dimensionally regulated result from a loop of massless, minimally coupled
scalars [20]. The more complex relation for gravity itself (102) is
exemplified by an old $D=4$ computation [22] that was made in the simplest
gauge [23] before it was understood how to apply dimensional regularization.

###  4.1 Contributions from a MMC Scalar

Suppose that $S[\varphi,g]$ represents the sum of the scalar and gravitational
actions, and that $\Delta S[g]$ stands for the counter-action. The one scalar
loop contributions to the graviton self-energy can be expressed as the
expectation value of the sum of three variational derivatives of these
quantities,

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\Biggl{\langle}\Omega\Biggl{|}T^{*}\Biggl{[}\Bigl{[}\frac{i\delta
S[\varphi,g]}{\delta
h_{\mu\nu}(x)}\Bigr{]}_{\varphi\varphi}\Bigl{[}\frac{i\delta
S[\varphi,g]}{\delta h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{\varphi\varphi}$
(116) $\displaystyle\hskip
85.35826pt+\Bigl{[}\frac{i\delta^{2}S[\varphi,g]}{\delta h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{\varphi\varphi}+\Bigl{[}\frac{i\delta^{2}\Delta
S[g]}{\delta h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{1}\Biggr{]}\Biggr{|}\Omega\Biggr{\rangle},\qquad$

where the subscripts indicate how many of the weak fields are retained and the
$T^{*}$-ordering symbol means any derivatives are taken after time ordering
the operators. Figure 2 shows the associated Feynman diagrams.

Figure 1: One scalar loop contributions to the graviton self-energy, shown in
the same order, left to right, as the three contributions to (116). Scalar
lines are straight and graviton lines are wavy.

The 4-point diagram (the central one of Figure 1) can be exactly canceled by
the counterterm diagram (the right hand one of Figure 1). The 3-point diagram
(the left hand one of Figure 1) takes the form [20, 19],

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
3pt}\Bigr{]}(x;x^{\prime})=(aa^{\prime})^{D+2}\Biggl{\\{}D^{\mu}{D^{\prime}}^{(\rho}y\,{D^{\prime}}^{\sigma)}D^{\nu}y\,\alpha(y)$
(117) $\displaystyle\hskip
14.22636pt+D^{(\mu}y\,D^{\nu)}{D^{\prime}}^{(\rho}y\,{D^{\prime}}^{\sigma)}y\,\beta(y)+D^{\mu}y\,D^{\nu}y\,{D^{\prime}}^{\rho}y\,{D^{\prime}}^{\sigma}y\,\gamma(y)$
$\displaystyle\hskip
42.67912pt+H^{4}g^{\mu\nu}{g^{\prime}}^{\rho\sigma}\,\delta(y)+H^{2}\Bigl{[}g^{\mu\nu}{D^{\prime}}^{\rho}y\,{D^{\prime}}^{\sigma}y+D^{\mu}y\,D^{\nu}y\,{g^{\prime}}^{\rho\sigma}\Bigr{]}\epsilon(y)\Biggr{\\}}.\qquad$

Here $y\equiv aa^{\prime}H^{2}\Delta x^{2}$ and its covariant derivatives are,

$\displaystyle
D^{\mu}y=-\frac{H}{a}\Bigl{[}y\delta^{\mu}_{~{}0}\\!-\\!2a^{\prime}H\Delta
x^{\mu}\Bigr{]}\qquad,\qquad{D^{\prime}}^{\rho}y=-\frac{H}{a^{\prime}}\Bigl{[}y\delta^{\rho}_{~{}0}\\!+\\!2aH\Delta
x^{\rho}\Bigr{]}\;,$ (118) $\displaystyle
D^{\mu}{D^{\prime}}^{\rho}y=\frac{H^{2}}{aa^{\prime}}\Bigl{[}y\delta^{\mu}_{~{}0}\delta^{\rho}_{~{}0}\\!+\\!2a\delta^{\mu}_{~{}0}H\Delta
x^{\rho}\\!-\\!2H\Delta
x^{\mu}a^{\prime}\delta^{\rho}_{~{}0}\\!-\\!2\eta^{\mu\rho}\Bigr{]}\;.$ (119)

The various coefficients in expression (117) are given in terms of a single
function $A(y)$ whose first derivative is [26, 27],

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

The functions $\alpha(y)$, $\beta(y)$, $\gamma(y)$, $\delta(y)$ and
$\epsilon(y)$ are,444Note that the scalar functions $\alpha(y)$, $\beta(y)$
and $\gamma(y)$ are unrelated to the bi-scalar densities
$\alpha(x;x^{\prime})$, $\beta(x;x^{\prime})$ and $\gamma(x;x^{\prime})$
defined in equations (73-75).

$\alpha(y)=-\frac{\kappa^{2}}{2}{A^{\prime}}^{2}\;\;,\;\;\beta(y)=-\kappa^{2}A^{\prime}A^{\prime\prime}\;\;,\;\;\gamma(y)=-\frac{\kappa^{2}}{2}{A^{\prime\prime}}^{2}\;,$
(121)
$\delta(y)=-\frac{\kappa^{2}}{8}\Bigl{[}(4y\\!-\\!y^{2})^{2}{A^{\prime\prime}}^{2}\\!+\\!2(2\\!-\\!y)(4y\\!-\\!y^{2})A^{\prime}A^{\prime\prime}\\!+\\![4(D\\!-\\!4)\\!-\\!(4y\\!-\\!y^{2})]{A^{\prime}}^{2}\Bigr{]},$
(122)
$\epsilon(y)=\frac{\kappa^{2}}{4}\Bigl{[}(4y\\!-\\!y^{2}){A^{\prime\prime}}^{2}+2(2\\!-\\!y)A^{\prime}A^{\prime\prime}-{A^{\prime}}^{2}\Bigr{]}\;.$
(123)

We can express (117) in the basis of Table 1 by first $3+1$ decomposing the
tensors of expressions (118-119),

$\eta^{\mu\nu}=\overline{\eta}^{\mu\nu}-\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\qquad,\qquad\Delta
x^{\mu}=\overline{\Delta x}^{\mu}+\delta^{\mu}_{~{}0}\Delta\eta\;,$ (124)

where $\Delta\eta\equiv x^{0}-{x^{\prime}}^{0}$. Factors of $\overline{\Delta
x}^{\mu}$ are then expressed as derivatives using the rules,

$\displaystyle\overline{\Delta x}^{\alpha}f(\Delta
x^{2})=\frac{\overline{\partial}^{\mu}}{2}I[f]$ ,
$\displaystyle\overline{\Delta x}^{\alpha}\overline{\Delta x}^{\beta}f(\Delta
x^{2})=\frac{\overline{\partial}^{\alpha}\overline{\partial}^{\beta}}{4}I^{2}[f]\\!-\\!\frac{\overline{\eta}^{\alpha\beta}}{2}I[f]\;,\qquad$
(125) $\displaystyle\overline{\Delta x}^{\alpha}\overline{\Delta
x}^{\beta}\overline{\Delta x}^{\gamma}f(\Delta x^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{8}\overline{\partial}^{\alpha}\overline{\partial}^{\beta}\overline{\partial}^{\gamma}I^{3}[f]-\frac{3}{4}\overline{\eta}^{(\alpha\beta}\overline{\partial}^{\gamma)}I^{2}[f]\;,\qquad$
(126) $\displaystyle\overline{\Delta x}^{\alpha}\overline{\Delta
x}^{\beta}\overline{\Delta x}^{\gamma}\overline{\Delta x}^{\delta}f(\Delta
x^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{16}\overline{\partial}^{\alpha}\overline{\partial}^{\beta}\overline{\partial}^{\gamma}\overline{\partial}^{\delta}I^{4}[f]-\frac{3}{4}\overline{\eta}^{(\alpha\beta}\overline{\partial}^{\gamma}\overline{\partial}^{\delta)}I^{3}[f]$
(127) $\displaystyle\hskip
113.81102pt+\frac{3}{4}\overline{\eta}^{(\alpha\beta}\overline{\eta}^{\gamma\delta)}I^{2}[f]\;.\qquad$

Here the operator $I[f]$ stands for the indefinite integral of $f(\Delta
x^{2})$ with respect to $\Delta x^{2}$. Table 6 gives the coefficient
functions.

$i$ | $T^{i}(x;x^{\prime})$
---|---
$1$ | $\delta-2(a^{2}+{a^{\prime}}^{2})H^{2}I[\epsilon]+4a^{2}{a^{\prime}}^{2}H^{4}I^{2}[\gamma]$
$2$ | $4\alpha-4aa^{\prime}H^{2}I[\beta]+8a^{2}{a^{\prime}}^{2}I^{2}[\gamma]$
$3$ | $-\delta\\!+\\!(y\\!+\\!2aH\Delta\eta)^{2}\epsilon$
|
$-2{a^{\prime}}^{2}H^{2}I[\alpha\\!+\\!(2aH\Delta\eta\\!+\\!y)\beta\\!+\\!(4a^{2}H^{2}\Delta\eta^{2}\\!+\\!4aH\Delta\eta
y\\!+\\!y^{2})\gamma\\!-\\!\epsilon]$
$5$ | $-2a{a^{\prime}}^{2}H^{3}I^{2}[\beta\\!+\\!2(2aH\Delta\eta\\!+\\!y)\gamma]+2aHI[(2aH\Delta\eta\\!+\\!y)\epsilon]$
$7$ | $-2a^{2}{a^{\prime}}^{2}H^{4}I^{3}[\gamma]+a^{2}H^{2}I^{2}[\epsilon]$
$9$ | $-4[2\\!+\\!y\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2}]\alpha-2[y^{2}\\!-\\!2aa^{\prime}H^{2}\Delta\eta^{2}(2\\!-\\!y)]\beta$
|
$+2aa^{\prime}H^{2}I[2\alpha\\!+\\!(2\\!+\\!3y)\beta\\!+\\!4aa^{\prime}H^{2}\Delta\eta^{2}\beta\\!+\\!4y^{2}\gamma\\!-\\!8aa^{\prime}H^{2}\Delta\eta^{2}(2\\!-\\!y)\gamma]$
$10$ | $2aHI[-2\alpha\\!+\\!(2a^{\prime}H\Delta\eta\\!-\\!y)\beta]+4a^{2}a^{\prime}H^{3}I^{2}[\beta\\!-\\!2(2a^{\prime}H\Delta\eta\\!-\\!y)\gamma]$
$12$ | $2aa^{\prime}H^{2}I^{2}[\beta]-8a^{2}{a^{\prime}}^{2}H^{4}I^{3}[\gamma]$
|
$(2\\!+\\!y\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2})^{2}\alpha+(2\\!+\\!y\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2})$
$13$ | $\times[y^{2}\\!-\\!2aa^{\prime}H^{2}\Delta\eta^{2}(2\\!-\\!y)]\beta+(y^{2}\\!-\\!2aa^{\prime}H^{2}\Delta\eta^{2}(2\\!-\\!y)]^{2}\gamma$
|
$+\delta-2[y^{2}\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2}(2\\!+\\!y)+2a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4}]\epsilon$
|
$2aHI[(2\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2})\alpha\\!+\\!y\alpha]\\!-\\!4aa^{\prime}H^{2}\Delta\eta(1\\!+\\!aH\Delta\eta\\!+\\!aa^{\prime}H^{2}\Delta\eta^{2})I[\beta]$
$14$ | $+2aHI[(1\\!-\\!a^{\prime}H\Delta\eta\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2})y\beta\\!+\\!y^{2}\beta]+8a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{3}I[(2\\!-\\!y)\gamma]$
|
$-2aHI[2aa^{\prime}H^{2}\Delta\eta^{2}(2\\!-\\!y)y\gamma\\!+\\!(2a^{\prime}H\Delta\eta\\!-\\!y)y^{2}\gamma\\!+\\!(2aH\Delta\eta\\!+\\!y)\epsilon]$
$16$ | $H^{2}I^{2}[a^{2}\\{\alpha\\!-\\!(2\\!-\\!y)\beta\\!+\\!(2\\!-\\!y)^{2}\gamma\\!-\\!\epsilon\\}\\!+\\!aa^{\prime}\\{2\beta\\!-\\!4(2\\!-\\!y)\gamma\\}\\!+\\!4{a^{\prime}}^{2}\gamma]$
$18$ | $aa^{\prime}H^{2}I^{2}[-2\alpha+3(2-y)\beta-32\gamma+4(4y-y^{2})\gamma]$
| $+(a^{2}+{a^{\prime}}^{2})H^{2}I^{2}[-4\beta+8(2-y)\gamma]$
$19$ | $a^{2}a^{\prime}H^{3}I^{3}[-\beta+2(2-y)\gamma]-4a{a^{\prime}}^{2}I^{3}[\gamma]$
$21$ | $a^{2}{a^{\prime}}^{2}H^{4}I^{4}[\gamma]$
Table 6: Scalar contributions to the coefficient functions
$T^{i}(x;x^{\prime})$. The de Sitter length function is
$y=aa^{\prime}H^{2}\Delta x^{2}$, the various functions of it such as
$\alpha(y)$ are defined in expressions (121-123), and the operator “$I$”
indicates indefinite integration with respect to $\Delta x^{2}$.

It remains to comment on ultraviolet divergences and renormalization. Table 6
gives dimensionally regulated, primitive results. Comparing Table 6 with
expressions (120) and (121-123) reveals that the fundamental structure
functions have the following leading behaviors near coincidence:

$T^{12}\sim\frac{1}{\Delta x^{2D-2}}\;\;,\;\;T^{16}\sim\frac{1}{\Delta
x^{2D-2}}\;\;,\;\;T^{18}\sim\frac{1}{\Delta
x^{2D-2}}\;\;,\;\;T^{19}\sim\frac{\Delta\eta}{\Delta x^{2D-2}}\;.$ (128)

It must be recalled that the ultimate goal is to integrate
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ of ${x^{\prime}}$ in
the quantum-corrected, linearized Einstein equation (1). Hence an expression
such as $1/\Delta x^{2D-2}$ is quadratically divergent, while $1/\Delta
x^{2D-4}$ is logarithmically divergent. We localize these divergences by
extracting derivatives until the integrable power of $1/\Delta x^{2D-6}$ is
reached, then adding zero in the form of the massless scalar propagator
equation [26],

$\displaystyle\frac{1}{\Delta
x^{2D-2}}=\frac{\partial^{2}}{2(D\\!-\\!2)^{2}}\frac{1}{\Delta
x^{2D-4}}=\frac{\partial^{4}}{4(D\\!-\\!2)^{2}(D\\!-\\!3)(D\\!-\\!4)}\frac{1}{\Delta
x^{2D-6}}\;,$ (131)
$\displaystyle=\frac{\partial^{4}}{4(D\\!-\\!2)^{2}(D\\!-\\!3)(D\\!-\\!4)}\Biggr{[}\frac{1}{\Delta
x^{2D-6}}\\!-\\!\frac{\mu^{D-4}}{\Delta x^{D-2}}\Biggr{]}$
$\displaystyle\hskip
170.71652pt+\frac{\mu^{D-4}\pi^{\frac{D}{2}}\partial^{2}i\delta^{D}(x\\!-\\!x^{\prime})}{(D\\!-\\!2)^{2}(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)},\qquad$
$\displaystyle=\frac{\mu^{D-4}\pi^{\frac{D}{2}}\partial^{2}i\delta^{D}(x\\!-\\!x^{\prime})}{(D\\!-\\!2)^{2}(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{\partial^{4}}{32}\Biggl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Biggr{]}+O(D\\!-\\!4)\;.\qquad$

We can similarly write,

$\frac{\Delta\eta}{\Delta
x^{2D-2}}=\frac{\mu^{D-4}\pi^{\frac{D}{2}}\partial_{0}i\delta^{D}(x\\!-\\!x^{\prime})}{(D\\!-\\!2)(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{\partial_{0}\partial^{2}}{16}\Biggl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Biggr{]}+O(D\\!-\\!4)\;.$ (132)

Renormalization is accomplished by using local counterterms to cancel the
divergent delta functions in expressions (131-132). (This sometimes leaves
local residuals proportional to $\ln(a)$.) Because the derivatives on the
contributions act on functions of $x^{\mu}-{x^{\prime}}^{\mu}$, they can
either be maintained as unprimed derivatives and pulled outside the
${x^{\prime}}^{\mu}$ integration of the quantum-corrected Einstein equation
(1), or they can be reflected into primed derivatives
($\partial_{\mu}\rightarrow-\partial_{\mu}^{\prime}$) and then partially
integrated onto the graviton field $h_{\rho\sigma}(x^{\prime})$.

###  4.2 Contributions from Gravitons

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

$\displaystyle-i\Bigl{[}\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}\Bigr{]}(x;x^{\prime})=\Biggl{\langle}\Omega\Biggl{|}T^{*}\Biggl{[}\Bigl{[}\frac{i\delta
S[g]}{\delta h_{\mu\nu}(x)}\Bigr{]}_{hh}\Bigl{[}\frac{i\delta S[g]}{\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{hh}+\Bigl{[}\frac{i\delta S[g]}{\delta
h_{\mu\nu}(x)}\Bigr{]}_{\overline{\theta}\theta}$ (133) $\displaystyle\hskip
8.5359pt\times\Bigl{[}\frac{i\delta S[g]}{\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{\overline{\theta}\theta}+\Bigl{[}\frac{i\delta^{2}S[g]}{\delta
h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{hh}+\Bigl{[}\frac{i\delta^{2}\Delta
S[g]}{\delta h_{\mu\nu}(x)\delta
h_{\rho\sigma}(x^{\prime})}\Bigr{]}_{1}\Biggr{]}\Biggr{|}\Omega\Biggr{\rangle},\qquad$

where the subscripts indicate how many graviton fields are retained and the
$T^{*}$-ordering symbol means any derivatives are taken after time ordering
the operators. Figure 2 shows the associated Feynman diagrams.

Figure 2: Diagrams contributing to the one loop graviton self-energy, shown in
the same order, left to right, as the three contributions to (133). Graviton
lines are wavy and ghost lines are dashed.

The actual computation [22] was made in $D=4$ dimensions before it was
understood how to employ dimensional regularization, so it can only be used
away from coincidence. At the end of this section we discuss how it might be
extended to recover the full result. The computation was considerably more
difficult than deriving the scalar contribution of section 4.1. It differs
from the scalar result in three ways:

  * •

It breaks de Sitter invariance, both inessentially through the use of a de
Sitter breaking gauge [23, 24] and unavoidably through physical de Sitter
breaking for gravitons [32, 33, 34, 35, 36] whose kinematics are the same as
those of the massless, minimally coupled scalar [37, 38];

  * •

The coefficient functions $T^{i}(x;x^{\prime})$ involve not only the two scale
factors and powers of $\Delta x^{2}$, but also up to a single factor of
$\ln(H^{2}\Delta x^{2})$; and

  * •

The self-energy is not annihilated by the action of a single Ward operator so
the coefficients $S^{i}(x;x^{\prime})$ are nonzero.

The second point means that we can decompose the coefficient functions $T^{i}$
and $S^{i}$ into parts with and without a factor of $\ln(H^{2}\Delta x^{2})$,

$\displaystyle T^{i}(x;x^{\prime})$ $\displaystyle=$ $\displaystyle
T^{i}_{N}(x;x^{\prime})+T^{i}_{L}(x;x^{\prime})\\!\times\\!\ln(H^{2}\Delta
x^{2})\;,$ (134) $\displaystyle S^{i}(x;x^{\prime})$ $\displaystyle=$
$\displaystyle
S^{i}_{N}(x;x^{\prime})+S^{i}_{L}(x;x^{\prime})\\!\times\\!\ln(H^{2}\Delta
x^{2})\;.$ (135)

Tables 7 and 8 give our results for the algebraically independent
$T^{i}_{N}(x;x^{\prime})$ and $T^{i}_{L}(x;x^{\prime})$, respectively. Tables
9 and 10 do the $S^{i}_{N}(x;x^{\prime})$ and $S^{i}_{L}(x;x^{\prime})$.

$i$ | $T^{i}_{N}(x;x^{\prime})$
---|---
$1$ | $\frac{8(\frac{92}{5}-77aa^{\prime}H^{2}\Delta\eta^{2}-12a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4})}{\Delta x^{8}}-\frac{\frac{4}{3}aa^{\prime}(55+203aa^{\prime}H^{2}\Delta\eta^{2}+48a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4})H^{2}}{\Delta x^{6}}$
|
$-\frac{a^{2}{a^{\prime}}^{2}(19+22aa^{\prime}H^{2}\Delta\eta^{2})H^{4}}{\Delta
x^{4}}$
$2$ | $\frac{\frac{1952}{5}-416aa^{\prime}H^{2}\Delta\eta^{2}}{\Delta x^{8}}-\frac{\frac{16}{3}aa^{\prime}(8+7aa^{\prime}H^{2}\Delta\eta^{2}-6a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4})H^{2}}{\Delta x^{6}}$
|
$+\frac{4a^{2}{a^{\prime}}^{2}(14+3aa^{\prime}H^{2}\Delta\eta^{2})H^{4}}{\Delta
x^{4}}$
$3$ | $-\frac{\frac{64}{5}[23+(55a+90a^{\prime})H\Delta\eta]\Delta\eta^{2}}{\Delta x^{10}}-\frac{8[23+7(16a+13a^{\prime})H\Delta\eta-(a^{2}-36{a^{\prime}}^{2})H^{2}\Delta\eta^{2}]}{\Delta x^{8}}$
|
$-\frac{4[52a^{2}+117aa^{\prime}-\frac{163}{3}{a^{\prime}}^{2}-16(a^{3}-{a^{\prime}}^{3})H\Delta\eta]H^{2}}{\Delta
x^{6}}-\frac{aa^{\prime}(18a^{2}+147aa^{\prime}-20{a^{\prime}}^{2})H^{4}}{\Delta
x^{4}}$
$5$ | $\frac{\frac{16}{5}[23+(55a+90a^{\prime})H\Delta\eta]\Delta\eta}{\Delta x^{8}}+\frac{\frac{8}{3}[86a+a^{\prime}+(5a^{2}+12{a^{\prime}}^{2})H\Delta\eta]H}{\Delta x^{6}}-\frac{2a(6a^{2}-33aa^{\prime}+4{a^{\prime}}^{2})H^{3}}{\Delta x^{4}}$
$7$ | $-\frac{\frac{4}{15}[23+(55a+90a^{\prime})H\Delta\eta]}{\Delta x^{6}}+\frac{(\frac{1}{3}a^{2}-6aa^{\prime}+8{a^{\prime}}^{2})H^{2}}{\Delta x^{4}}+\frac{aa^{\prime}(2a^{2}-6aa^{\prime}+{a^{\prime}}^{2})H^{4}}{\Delta x^{2}}$
$9$ | $-\frac{\frac{128}{5}(61-55aa^{\prime}H^{2}\Delta\eta^{2})\Delta\eta^{2}}{\Delta x^{10}}-\frac{16(61+10aa^{\prime}H^{2}\Delta\eta^{2}-2a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4})}{\Delta x^{8}}$
| $-\frac{8aa^{\prime}(70-aa^{\prime}H^{2}\Delta\eta^{2})H^{2}}{\Delta
x^{6}}-\frac{12a^{2}{a^{\prime}}^{2}H^{4}}{\Delta x^{4}}$
$10$ | $\frac{\frac{16}{5}(61-40aa^{\prime}H^{2}\Delta\eta^{2})\Delta\eta}{\Delta x^{8}}+\frac{\frac{16}{3}[20a-16a^{\prime}+(9a-2a^{\prime})a^{\prime}H\Delta\eta]H}{\Delta x^{6}}+\frac{2a^{\prime}(a^{2}-19aa^{\prime}-2{a^{\prime}}^{2})H^{3}}{\Delta x^{4}}$
$12$ | $-\frac{\frac{8}{15}(61-20aa^{\prime}H^{2}\Delta\eta^{2})}{\Delta x^{6}}-\frac{\frac{2}{3}aa^{\prime}(16+5aa^{\prime}H^{2}\Delta\eta^{2})H^{2}}{\Delta x^{4}}+\frac{2a^{2}{a^{\prime}}^{2}(4-aa^{\prime}H^{2}\Delta\eta^{2})H^{4}}{\Delta x^{2}}$
$13$ | $\frac{5376\Delta\eta^{4}}{\Delta x^{12}}+\frac{64(84+21aa^{\prime}H^{2}\Delta\eta^{2})\Delta\eta^{2}}{\Delta x^{10}}+\frac{8(126+434aa^{\prime}H^{2}\Delta\eta^{2}+53a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4})}{\Delta x^{8}}$
|
$+\frac{4aa^{\prime}(409+36aa^{\prime}H^{2}\Delta\eta^{2}-24a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4})H^{2}}{\Delta
x^{6}}+\frac{a^{2}{a^{\prime}}^{2}(557+24aa^{\prime}H^{2}\Delta\eta^{2})H^{4}}{\Delta
x^{4}}$
$14$ | $-\frac{\frac{5376}{5}\Delta\eta^{3}}{\Delta x^{10}}-\frac{16[42+(19a-15a^{\prime})H\Delta\eta]\Delta\eta}{\Delta x^{8}}$
| $+\frac{8[-\frac{238}{3}a+56a^{\prime}+\frac{1}{3}(13a^{2}+48a^{\prime
2})H\Delta\eta-6aa^{\prime}(3a\\!-\\!4a^{\prime})H^{2}\Delta\eta^{2}]H}{\Delta
x^{6}}+\frac{2a(8a^{2}-139aa^{\prime}-6{a^{\prime}}^{2})H^{3}}{\Delta x^{4}}$
$16$ | $\frac{\frac{336}{5}\Delta\eta^{2}}{\Delta x^{8}}+\frac{4[\frac{13}{3}-(a+6a^{\prime})H\Delta\eta)]}{\Delta x^{6}}+\frac{(\frac{67}{3}a^{2}+14aa^{\prime}+6{a^{\prime}}^{2})H^{2}}{\Delta x^{4}}-\frac{aa^{\prime}(2a^{2}-10aa^{\prime}+3{a^{\prime}}^{2})H^{4}}{\Delta x^{2}}$
$18$ | $\frac{\frac{1344}{5}\Delta\eta^{2}}{\Delta x^{8}}+\frac{8(\frac{29}{3}+6aa^{\prime}H^{2}\Delta\eta^{2})}{\Delta x^{6}}+\frac{2aa^{\prime}(-46+5aa^{\prime}H^{2}\Delta\eta^{2})H^{2}}{\Delta x^{4}}$
$19$ | $-\frac{\frac{112}{5}\Delta\eta}{\Delta x^{6}}+\frac{(\frac{26}{3}a+8a^{\prime})H}{\Delta x^{4}}-\frac{a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta}{\Delta x^{2}}$
$21$ | $\frac{\frac{14}{5}}{\Delta x^{4}}$
Table 7: Contributions to $T^{i}(x;x^{\prime})$ that do not contain factors of $\ln(H^{2}\Delta x^{2})$. Each of the tabulated terms must be multiplied by $-\frac{\kappa^{2}}{64\pi^{4}}$. $i$ | $T^{i}_{L}(x;x^{\prime})$
---|---
$1$ | $\frac{16aa^{\prime}(a^{2}+{a^{\prime}}^{2})H^{4}\Delta\eta^{2}}{\Delta x^{6}}-\frac{4aa^{\prime}(a^{2}-4aa^{\prime}+{a^{\prime}}^{2})H^{4}}{\Delta x^{4}}+\frac{12a^{3}{a^{\prime}}^{3}H^{6}}{\Delta x^{2}}$
$2$ | $-\frac{32aa^{\prime}(a^{2}+{a^{\prime}}^{2})H^{4}\Delta\eta^{2}}{\Delta x^{6}}-\frac{16a^{2}{a^{\prime}}^{2}H^{4}}{\Delta x^{4}}-\frac{4a^{3}{a^{\prime}}^{3}H^{6}}{\Delta x^{2}}$
$3$ | $-\frac{16a^{3}{a^{\prime}}^{2}H^{5}\Delta\eta^{3}}{\Delta x^{6}}+\frac{4a^{2}{a^{\prime}}^{2}(3a-2a^{\prime})H^{5}\Delta\eta}{\Delta x^{4}}-\frac{16a^{3}{a^{\prime}}^{3}H^{6}}{\Delta x^{2}}$
$5$ | $\frac{8a^{2}a^{\prime}(a+a^{\prime})H^{4}\Delta\eta}{\Delta x^{4}}+\frac{12a^{3}{a^{\prime}}^{2}H^{5}}{\Delta x^{2}}$
$7$ | $-\frac{2aa^{\prime}(2a^{2}+aa^{\prime}+{a^{\prime}}^{2})H^{4}}{\Delta x^{2}}$
$9$ | $-\frac{1536aa^{\prime}H^{2}\Delta\eta^{4}}{\Delta x^{10}}-\frac{96(a^{2}+10aa^{\prime}+{a^{\prime}}^{2})H^{2}\Delta\eta^{2}}{\Delta x^{8}}-\frac{32(a^{2}+aa^{\prime}+{a^{\prime}}^{2})H^{2}}{\Delta x^{6}}+\frac{4a^{2}{a^{\prime}}^{2}H^{4}}{\Delta x^{4}}$
$10$ | $\frac{192aa^{\prime}H^{2}\Delta\eta^{3}}{\Delta x^{8}}+\frac{16(a^{2}+2aa^{\prime}+3{a^{\prime}}^{2})H^{2}\Delta\eta}{\Delta x^{6}}-\frac{4a^{\prime}(a^{2}-aa^{\prime}+2{a^{\prime}}^{2})H^{3}}{\Delta x^{4}}+\frac{4a^{2}{a^{\prime}}^{3}H^{5}}{\Delta x^{2}}$
$12$ | $-\frac{32aa^{\prime}H^{2}\Delta\eta^{2}}{\Delta x^{6}}-\frac{4(3a^{2}-4aa^{\prime}+3{a^{\prime}}^{2})H^{2}}{\Delta x^{4}}+\frac{2aa^{\prime}(2a^{2}+5aa^{\prime}+2{a^{\prime}}^{2})H^{4}}{\Delta x^{2}}$
$13$ | $-\frac{1536aa^{\prime}H^{2}\Delta\eta^{4}}{\Delta x^{10}}+\frac{96(a^{2}-14aa^{\prime}+{a^{\prime}}^{2})H^{2}\Delta\eta^{2}}{\Delta x^{8}}+\frac{32aa^{\prime}(-3+7aa^{\prime}H^{2}\Delta\eta^{2}+a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4})H^{2}}{\Delta x^{6}}$
| $-\frac{4aa^{\prime}(4a^{2}-19aa^{\prime}+4{a^{\prime}}^{2})H^{4}}{\Delta
x^{4}}+\frac{24a^{3}{a^{\prime}}^{3}H^{6}}{\Delta x^{2}}$
$14$ | $\frac{384aa^{\prime}H^{2}\Delta\eta^{3}}{\Delta x^{8}}-\frac{32(a^{2}-6aa^{\prime}-{a^{\prime}}^{2})H^{2}\Delta\eta}{\Delta x^{6}}-\frac{8a(2a^{2}+aa^{\prime}-{a^{\prime}}^{2})H^{3}}{\Delta x^{4}}-\frac{16a^{3}{a^{\prime}}^{2}H^{5}}{\Delta x^{2}}$
$16$ | $-\frac{32aa^{\prime}H^{2}\Delta\eta^{2}}{\Delta x^{6}}+\frac{4(a^{2}-3{a^{\prime}}^{2})H^{2}}{\Delta x^{4}}+\frac{2aa^{\prime}(2a^{2}-{a^{\prime}}^{2})H^{4}}{\Delta x^{2}}$
$18$ | $-\frac{96aa^{\prime}H^{2}\Delta\eta^{2}}{\Delta x^{6}}-\frac{12(a^{2}+{a^{\prime}}^{2})H^{2}}{\Delta x^{4}}-\frac{10a^{2}{a^{\prime}}^{2}H^{4}}{\Delta x^{2}}$
$19$ | $\frac{8aa^{\prime}H^{2}\Delta\eta}{\Delta x^{4}}+\frac{2aa^{\prime}(a-3a^{\prime})H^{3}}{\Delta x^{2}}$
$21$ | $0$
Table 8: Contributions to $T^{i}(x;x^{\prime})$ that contain factors of $\ln(H^{2}\Delta x^{2})$. Each tabulated term must be multiplied by $-\frac{\kappa^{2}}{64\pi^{4}}$. $i$ | $S^{i}_{N}(x;x^{\prime})$
---|---
| $\frac{2560a^{\prime}H\Delta\eta^{4}}{\Delta
x^{12}}+\frac{128[42a-6a^{\prime}+(9a^{2}+8{a^{\prime}}^{2})H\Delta\eta]H\Delta\eta^{2}}{\Delta
x^{10}}$
$1$ | $+\frac{16[303a-193a^{\prime}+(73a^{2}-33{a^{\prime}}^{2})H\Delta\eta+(12a^{3}+18{a^{\prime}}^{3})H^{2}\Delta\eta^{2}]H}{\Delta x^{8}}$
|
$+\frac{8(20a^{3}+41a^{2}a^{\prime}+50a{a^{\prime}}^{2}-14{a^{\prime}}^{3})H^{3}}{\Delta
x^{6}}+\frac{8a^{2}{a^{\prime}}^{2}(5a+4a^{\prime})H^{5}}{\Delta x^{4}}$
$2$ | $-\frac{256a^{\prime}H\Delta\eta^{3}}{\Delta x^{10}}-\frac{16[(42a-16a^{\prime})H\Delta\eta+(9a^{2}+6{a^{\prime}}^{2})H^{2}\Delta\eta^{2}]}{\Delta x^{8}}$
|
$-\frac{8H^{2}(17a^{2}+\frac{53}{3}aa^{\prime}-5{a^{\prime}}^{2}+4a^{3}H\Delta\eta)}{\Delta
x^{6}}-\frac{2a^{2}a^{\prime}H^{4}(5a^{\prime}+12a)}{\Delta x^{4}}$
$3$ | $\frac{-384(a^{2}-3aa^{\prime}-a^{\prime 2})H^{2}\Delta\eta^{3}}{\Delta x^{10}}-\frac{16[57a+7a^{\prime}-(5a^{2}-{a^{\prime}}^{2})H\Delta\eta+12{a^{\prime}}^{3}H^{2}\Delta\eta^{2}]H}{\Delta x^{8}}$
|
$+\frac{8(2a^{3}-\frac{65}{3}a{a^{\prime}}^{2}+9{a^{\prime}}^{3})H^{3}}{\Delta
x^{6}}-\frac{2a^{2}{a^{\prime}}^{2}(4a+11a^{\prime})H^{5}}{\Delta x^{4}}$
$4$ | $\frac{32a^{\prime}H\Delta\eta^{2}}{\Delta x^{8}}+\frac{8H[11a-5a^{\prime}+(4a^{2}+\frac{1}{3}{a^{\prime}}^{2})H\Delta\eta]}{\Delta x^{6}}$
| $+\frac{2H^{3}a[4a^{2}+2aa^{\prime}-\frac{11}{3}{a^{\prime}}^{2}]}{\Delta
x^{4}}+\frac{H^{5}a^{2}{a^{\prime}}^{2}(4a\\!+\\!a^{\prime})}{\Delta x^{2}}$
$5$ | $-\frac{256a^{\prime}H\Delta\eta^{3}}{\Delta x^{10}}+\frac{16[6a-16a^{\prime}+(3a^{2}-8{a^{\prime}}^{2})H\Delta\eta]H\Delta\eta}{\Delta x^{8}}$
|
$+\frac{8[7a^{2}-aa^{\prime}-\frac{11}{3}{a^{\prime}}^{2}-6{a^{\prime}}^{3}H\Delta\eta]H^{2}}{\Delta
x^{6}}-\frac{4a{a^{\prime}}^{2}(a-2a^{\prime})H^{4}}{\Delta x^{4}}$
$6$ | $-\frac{256(3a-2a^{\prime})H\Delta\eta^{2}}{\Delta x^{10}}-\frac{4[90a-50a^{\prime}+(27a^{2}+3{a^{\prime}}^{2})H\Delta\eta]H}{\Delta x^{8}}-\frac{6aa^{\prime}(9a-a^{\prime})H^{3}}{\Delta x^{6}}$
$7$ | $\frac{32a^{\prime}H\Delta\eta^{2}}{\Delta x^{8}}-\frac{8H(a^{3}-a^{2}a^{\prime}-\frac{10}{3}a{a^{\prime}}^{2}+2{a^{\prime}}^{3})}{aa^{\prime}\Delta x^{6}}+\frac{H^{3}aa^{\prime}(a-5a^{\prime})}{2\Delta x^{4}}$
$8$ | $\frac{32(3a-2a^{\prime})H\Delta\eta}{\Delta x^{8}}+\frac{(18a^{2}-20aa^{\prime}+\frac{38}{3}{a^{\prime}}^{2})H^{2}}{\Delta x^{6}}+\frac{a{a^{\prime}}^{2}(a^{\prime}-\frac{9}{2}a)H^{4}}{\Delta x^{4}}$
$9$ | $\frac{48(a^{2}-3aa^{\prime}-{a^{\prime}}^{2})H^{2}\Delta\eta^{2}}{\Delta x^{8}}+\frac{8(-2a^{2}+2aa^{\prime}-3{a^{\prime}}^{2}+4{a^{\prime}}^{3}H\Delta\eta)H^{2}}{\Delta x^{6}}+\frac{2a{a^{\prime}}^{2}(2a-3a^{\prime})H^{4}}{\Delta x^{4}}$
$10$ | $-\frac{16a^{\prime}H\Delta\eta}{3\Delta x^{6}}+\frac{2H^{2}a^{\prime}(a-\frac{1}{3}a^{\prime})}{\Delta x^{4}}+\frac{H^{4}a^{2}{a^{\prime}}^{2}}{2\Delta x^{2}}$
Table 9: Contributions to $S^{i}(x;x^{\prime})$ that do not contain a factor of $\ln(H^{2}\Delta x^{2})$. Each of the tabulated terms must be multiplied by $-\frac{\kappa^{2}}{64\pi^{4}}$. $i$ | $S^{i}_{L}(x;x^{\prime})$
---|---
$1$ | $\frac{384a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{3}}{\Delta x^{8}}+\frac{64aa^{\prime}(a^{2}+5aa^{\prime}-3{a^{\prime}}^{2})H^{4}\Delta\eta}{\Delta x^{6}}+\frac{64a^{3}{a^{\prime}}^{2}H^{5}}{\Delta x^{4}}$
$2$ | $-\frac{64a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{2}}{\Delta x^{6}}-\frac{16a^{3}a^{\prime}H^{4}}{\Delta x^{4}}$
$3$ | $\frac{192a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{3}}{\Delta x^{8}}+\frac{32aa^{\prime}(a^{2}-2aa^{\prime}+4{a^{\prime}}^{2})H^{4}\Delta\eta}{\Delta x^{6}}-\frac{24a^{3}{a^{\prime}}^{2}H^{5}}{\Delta x^{4}}$
$4$ | $\frac{8a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta}{\Delta x^{4}}+\frac{4a^{3}{a^{\prime}}^{2}H^{5}}{\Delta x^{2}}$
Table 10: Nonzero parts of $S^{i}(x;x^{\prime})$ which are proportional to
$\ln(H^{2}\Delta x^{2})$. (The cases of $i=5,6,7,8,9,10$ vanish.) Each of the
tabulated terms must be multiplied by $-\frac{\kappa^{2}}{64\pi^{4}}$.

We also need the auxiliary functions defined in expressions (73), (74) and
(75). The antisymmetric part of $\gamma(x;x^{\prime})$ is,

$\displaystyle\gamma-\gamma^{R}=(\eta\\!+\\!\eta^{\prime})\Delta\eta\Biggl{\\{}-\frac{5376aa^{\prime}H^{2}\Delta\eta^{2}}{5\Delta
x^{10}}-\frac{8aa^{\prime}H^{2}[359\\!+\\!124aa^{\prime}H^{2}\Delta\eta^{2}]}{5\Delta
x^{8}}$ (136)
$\displaystyle-\frac{4a^{2}{a^{\prime}}^{2}H^{4}[181\\!+\\!60aa^{\prime}H^{2}\Delta\eta^{2}]}{15\Delta
x^{6}}+\frac{a^{3}{a^{\prime}}^{3}H^{6}[24\\!+\\!11aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{4}}$
$\displaystyle-\frac{2a^{4}{a^{\prime}}^{4}H^{8}[5\\!-\\!2aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{2}}+\Biggl{[}\frac{96a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{2}}{\Delta
x^{8}}+\frac{8a^{2}{a^{\prime}}^{2}H^{4}[6\\!+\\!7aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{6}}$
$\displaystyle+\frac{2a^{3}{a^{\prime}}^{3}H^{6}[1\\!+\\!7aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{4}}-\frac{a^{4}{a^{\prime}}^{4}H^{8}[31\\!+\\!8aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{2}}\Biggr{]}\ln(H^{2}\Delta x^{2})\Biggl{\\}}.\qquad$

This implies that $\alpha(x;x^{\prime})$ and $\beta(x;x^{\prime})$ are,

$\displaystyle\alpha=\frac{\frac{672}{5}\Delta\eta^{2}}{\Delta
x^{8}}+\frac{\frac{764}{15}\\!+\\!\frac{16}{3}aa^{\prime}H^{2}\Delta\eta^{2}}{\Delta
x^{6}}-\frac{aa^{\prime}H^{2}[\frac{16}{3}\\!+\\!\frac{5}{3}aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{4}}$ (137)
$\displaystyle+\frac{a^{2}{a^{\prime}}^{2}H^{4}[4\\!-\\!aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{2}}+\Biggl{[}-\frac{16aa^{\prime}H^{2}\Delta\eta^{2}}{\Delta
x^{6}}-\frac{2aa^{\prime}H^{2}[2\\!+\\!3aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{4}}$ $\displaystyle\hskip
128.0374pt+\frac{a^{2}{a^{\prime}}^{2}H^{4}[9\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{2}}\Biggr{]}\ln(H^{2}\Delta x^{2}),\qquad$
$\displaystyle\beta=-\frac{[\frac{3904}{5}\\!-\\!704aa^{\prime}H^{2}\Delta\eta^{2}]\Delta\eta^{2}}{\Delta
x^{10}}-\frac{[\frac{488}{5}\\!+\\!272aa^{\prime}H^{2}\Delta\eta^{2}\\!-\\!80a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4}]}{\Delta
x^{8}}$ (138)
$\displaystyle-\frac{aa^{\prime}H^{2}[\frac{280}{3}\\!+\\!\frac{332}{3}aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{6}}\\!+\\!\frac{4a^{2}{a^{\prime}}^{2}H^{4}(6\\!-\\!aa^{\prime}H^{2}\Delta\eta^{2})}{\Delta
x^{4}}\\!+\\!\Biggl{[}-\frac{768aa^{\prime}H^{2}\Delta\eta^{4}}{\Delta
x^{10}}$
$\displaystyle-\frac{144aa^{\prime}H^{2}\Delta\eta^{2}[4\\!+\\!aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{8}}-\frac{8aa^{\prime}H^{2}[6\\!+\\!8aa^{\prime}H^{2}\Delta\eta^{2}\\!+\\!2a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta^{4}]}{\Delta
x^{6}}$ $\displaystyle\hskip
82.51282pt+\frac{2a^{2}{a^{\prime}}^{2}[1\\!+\\!2aa^{\prime}H^{2}\Delta\eta^{2}]}{\Delta
x^{4}}-\frac{4a^{3}{a^{\prime}}^{3}H^{6}}{\Delta
x^{2}}\Biggr{]}\ln(H^{2}\Delta x^{2}).\qquad$

Note that none of the structure functions differ in form from the primitive
contributions. That is not some miracle of the de Sitter background; one can
see that it must be true generally from equations (81) — which could be used
to infer $\beta(x;x^{\prime})$ — and equation (84) — which could be used to
infer $\alpha(x;x^{\prime})$. The absence of new functional forms is quite
unlike what happened in previous representations of the scalar result [20,
19], neither of which could even be applied to contributions from gravity or
to general cosmological backgrounds.

####  4.2.1 Recovering the Local Terms

The previous results determine the nine structure functions for all
${x^{\prime}}^{\mu}\neq x^{\mu}$. However, there are still potentially
important local contributions proportional to $i\delta^{4}(x-x^{\prime})$.
These terms dominate the fermion wave function [39, 40] and the photon field
strength [41, 42], and they make an important contribution to electromagnetic
forces [43], so it is worth explaining how they can be recovered. Of course we
could simply re-do the computation using dimensional regularization from the
beginning using the $D$-dependent propagators [24] and vertices [23], but we
have in mind a simpler approach based on understanding the three sources of
local contributions:555The reader is free to dismiss the comments of this
subsection as conjectural. However, they are based on the authors’ great
familiarity with the $D=4$ computation of the graviton loop contribution [22],
and the close simularity of that contribution to the general $D$ scalar loop
contribution [20] reported in section 4.1. In particular, the graviton 3-point
vertex takes the same $\kappa a^{D-2}h\partial h\partial h$ form as the
scalar-graviton vertex $\kappa a^{D-2}h\partial\varphi\partial\varphi$. The
most important part of the $D$-dimensional graviton propagator [23, 24] is
also just some constant tensors times the scalar propagator.

  * •

From renormalization;

  * •

From the 4-point diagram of Figure 2; and

  * •

From differentiated propagators in the two 3-point diagrams of Figure 2.

Renormalization is the simplest case to understand. From Tables 7 and 8 we see
that the most singular parts of the fundamental coefficient functions near
coincidence are,

$T^{12}\sim\frac{1}{\Delta x^{6}}\;\;,\;\;T^{16}\sim\frac{1}{\Delta
x^{6}}\;\;,\;\;T^{18}\sim\frac{1}{\Delta
x^{6}}\;\;,\;\;T^{19}\sim\frac{\Delta\eta}{\Delta x^{6}}\;.$ (139)

In general $D$ the ghost and graviton propagators involve functions of the
same form as (120) in the scalar propagator [24],

$i\Delta(x;x^{\prime})\sim\Bigl{(}\frac{1}{aa^{\prime}\Delta
x^{2}}\Bigr{)}^{\frac{D}{2}-1}+\Bigl{(}\frac{1}{aa^{\prime}\Delta
x^{2}}\Bigr{)}^{\frac{D}{2}-2}+\dots$ (140)

The generic vertex involves a factor of $a^{D-2}$ with two derivatives [23].
So comparison with the leading scalar divergences (128) means that $D=4$
results like $1/\Delta x^{6}$ correspond to $1/\Delta x^{2D-2}$ in the
dimensionally regulated theory, and all $D$-dependent powers of the scale
factor cancel between the two propagators and the two vertices. Hence we can
extend the $D=4$ results from Tables 7, 8, 9 and 10 by the replacements such
as,

$\displaystyle\frac{1}{\Delta x^{6}}$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\mu^{D-4}\pi^{\frac{D}{2}}\partial^{2}i\delta^{D}(x\\!-\\!x^{\prime})}{(D\\!-\\!2)^{2}(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{\partial^{4}}{32}\Biggl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Biggr{]}\;,\qquad$ (141)
$\displaystyle\frac{\Delta\eta}{\Delta x^{6}}$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\mu^{D-4}\pi^{\frac{D}{2}}\partial_{0}i\delta^{D}(x\\!-\\!x^{\prime})}{(D\\!-\\!2)(D\\!-\\!3)(D\\!-\\!4)\Gamma(\frac{D}{2}\\!-\\!1)}-\frac{\partial_{0}\partial^{2}}{16}\Biggl{[}\frac{\ln(\mu^{2}\Delta
x^{2})}{\Delta x^{2}}\Biggr{]}\;,\qquad$ (142)

times non-negative integer powers of $a$ and $a^{\prime}$. Because
counterterms are proportional to $a^{D-4}i\delta^{D}(x-x^{\prime})$ times non-
negative powers of $a$ and $a^{\prime}$, it is possible to predict the finite
factors of $\ln(a)\delta^{4}(x-x^{\prime})$ that remain after renormalization.
Note that we can also predict how to extract derivatives from the finite,
nonlocal parts of the structure functions.

All the 4-point diagrams are local, and they are simple enough to compute
directly. In dimensional regularization any $D$-dependent power of $\Delta
x^{2}$ vanishes at coincidence. The coincident propagator comes from the
integer sums in (120) and related propagator functions. These will never
contribute $D$-dependent powers of $a$, hence the factor of $a^{D-2}$ from the
4-point agrees with the $D$-dependent factor of $a$ from the counterterm, so
no finite factors of $\ln(a)$ can arise from this source. However, the
coincident propagator can produce an easily-predictable and ultraviolet finite
factor of $\ln(a)$.

Without regard to the tensor structure we can see that the generic 3-point
contribution takes the form,

$\Bigl{(}i\kappa a^{D-2}\partial^{2}\Bigr{)}\times
i\Delta(x;x^{\prime})i\Delta(x;x^{\prime})\times\Bigr{(}i\kappa{a^{\prime}}^{D-2}{\partial^{\prime}}^{2}\Bigr{)}\;.$
(143)

Acting two times derivatives on a propagator produces a delta function [26],

$\partial_{\mu}\partial_{\nu}^{\prime}i\Delta(x;x^{\prime})=\frac{\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}i\delta^{D}(x\\!-\\!x^{\prime})}{a^{D-2}}+{\rm
Nonlocal\ Terms}\;.$ (144)

The other propagator is taken to coincidence by the delta function, so the
same considerations apply to it as for the 4-point contributions considered
above. It turns out that acting the derivatives to produce the nonlocal terms
is the rate-limiting step of the computation, so it is considerably simpler to
access the local $\ln(a)$ term than to derive the dimensionally regulated
nonlocal contributions.

####  4.2.2 The Gauge Issue

Graviton propagators require gauge fixing. The calculation [22] reported in
section 4.2 was performed by adding a gauge fixing functional whose
$D$-dimensional extension is [23, 24],

$\mathcal{L}_{GF}=-\frac{a^{D-2}}{2}\eta^{\mu\nu}F_{\mu}F_{\nu}\;\;,\;\;F_{\mu}=\eta^{\rho\sigma}\Bigl{(}h_{\mu\rho,\sigma}-\frac{1}{2}h_{\rho\sigma,\mu}+(D\\!-\\!2)aHh_{\mu\rho}\delta^{0}_{~{}\sigma}\Bigr{)}\;.$
(145)

The special feature of this gauge is that it makes the propagator take the
form of a sum of three constant tensor factors times scalar propagators,

$i\Bigl{[}\mbox{}_{\mu\nu}\Delta_{\rho\sigma}\Bigr{]}(x;x^{\prime})=\sum_{I=A,B,C}\Bigl{[}\mbox{}_{\mu\nu}T^{I}_{\rho\sigma}\Bigr{]}\times
i\Delta_{I}(x;x^{\prime})\;.$ (146)

Here the constant tensor factors are,

$\displaystyle\Bigl{[}\mbox{}_{\mu\nu}T^{A}_{\rho\sigma}\Bigr{]}=2\overline{\eta}_{\mu(\rho}\overline{\eta}_{\sigma)\nu}-\frac{2}{D\\!-\\!3}\overline{\eta}_{\mu\nu}\overline{\eta}_{\rho\sigma}\quad,\quad\Bigl{[}\mbox{}_{\mu\nu}T^{B}_{\rho\sigma}\Bigr{]}=-4\delta^{0}_{~{}(\mu}\overline{\eta}_{\nu)(\rho}\delta^{0}_{~{}\sigma)}\;,\qquad$
(147)
$\displaystyle\Bigl{[}\mbox{}_{\mu\nu}T^{C}_{\rho\sigma}\Bigr{]}=\frac{2E_{\mu\nu}E_{\rho\sigma}}{(D\\!-\\!2)(D\\!-\\!3)}\quad,\quad
E_{\mu\nu}\equiv(D\\!-\\!3)\delta^{0}_{~{}\mu}\delta^{0}_{~{}\nu}+\overline{\eta}_{\mu\nu}\;.\qquad$
(148)

And the three scalar propagators are all related to the function $A(y)$ of
expression (120),

$\displaystyle i\Delta_{A}(x;x^{\prime})$ $\displaystyle=$ $\displaystyle
A(y)+k\ln(aa^{\prime})\qquad
k\equiv\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\;,$
(149) $\displaystyle i\Delta_{B}(x;x^{\prime})$ $\displaystyle=$
$\displaystyle
B(y)\equiv-\frac{[(4y\\!-\\!y^{2})A^{\prime}(y)\\!+\\!(2\\!-\\!y)k]}{2(D\\!-\\!2)}\;,$
(150) $\displaystyle i\Delta_{C}(x;x^{\prime})$ $\displaystyle=$
$\displaystyle C(y)\equiv\frac{1}{2}(2\\!-\\!y)B(y)+\frac{k}{D\\!-\\!3}\;.$
(151)

The flat space limit is obtained by taking the scale factor to unity and the
Hubble parameter to zero. In this limit our de Sitter gauge reduces to,

$\mathcal{L}_{GF}\longrightarrow-\frac{1}{2}\eta^{\mu\nu}F_{\mu}F_{\nu}\qquad,\qquad
F_{\mu}=\eta^{\rho\sigma}\Bigl{(}h_{\mu\rho,\sigma}-\frac{1}{2}h_{\rho\sigma,\mu}\Bigr{)}\;,$
(152)

and the corresponding propagator becomes,

$i\Bigl{[}\mbox{}_{\mu\nu}\Delta_{\rho\sigma}\Bigr{]}(x;x^{\prime})\longrightarrow\Bigl{(}2\eta_{\mu(\rho}\eta_{\sigma)\nu}-\frac{2}{D\\!-\\!2}\eta_{\mu\nu}\eta_{\rho\sigma}\Bigr{)}\times
i\Delta(x;x^{\prime})\;,$ (153)

where the massless scalar propagator of flat space was defined in expression
(22). This is precisely the gauge Capper employed to derive the results
reported in equations (21-26) [25]. It is straightforward to check that the
flat space limits of the de Sitter results reported in Tables 7 and 8 agree
with the specialization to $D=4$ of Capper’s results (21-26).

The advantages of our de Sitter gauge (145) are so great that it has been used
for nine [22, 44, 39, 45, 46, 41, 47, 48, 49] of the ten graviton loops which
have so far been computed de Sitter background. The exception was a year-long
tour de force made to check for gauge dependence in the vacuum polarization
[50] using a cumbersome, 1-parameter family of de Sitter invariant gauges
[51]. It would be quite challenging re-computing the graviton self-energy in
this family of gauges. We have instead devised a 2-parameter deformation of
the de Sitter breaking gauge (145) [52],

$\mathcal{L}^{\alpha\beta}_{GF}=-\frac{a^{D-2}}{2\alpha}\eta^{\mu\nu}F_{\mu}F_{\nu}\;\;,\;\;F_{\mu}=\eta^{\rho\sigma}\Bigl{(}h_{\mu\rho,\sigma}-\frac{\beta}{2}h_{\rho\sigma,\mu}+(D\\!-\\!2)aHh_{\mu\rho}\delta^{0}_{~{}\sigma}\Bigr{)}\;.$
(154)

Although we have not yet computed the graviton self-energy in this gauge,
Capper derived a result for its flat space limit [25]. The final, renormalized
result takes the same form as (32) but with the numerical coefficients
changed. In the general gauge the coefficient of
$\Pi^{\mu\nu}\Pi^{\rho\sigma}$ becomes [25],

$\frac{23}{2}\longrightarrow\frac{45}{2}\alpha^{2}\\!+\\!\frac{113}{2}\\!+\\!\frac{15}{2}\frac{(\alpha\\!-\\!3)^{2}}{(\beta\\!-\\!2)^{4}}\\!-\\!\frac{135}{2}\frac{(\alpha\\!-\\!3)}{(\beta\\!-\\!2)^{3}}\\!-\\!\frac{25}{2}\frac{(2\alpha\\!-\\!19)}{(\beta\\!-\\!2)^{2}}\\!+\\!\frac{5}{2}\frac{(11\alpha\\!+\\!59)}{(\beta\\!-\\!2)}\;.$
(155)

The coefficient of $\Pi^{\mu(\rho}\Pi^{\sigma)\nu}$ becomes [25],

$\frac{61}{2}\longrightarrow\frac{15}{2}\alpha^{2}\\!+\\!\frac{45}{4}\alpha\\!-\\!\frac{43}{4}\\!+\\!\frac{5}{2}\frac{(\alpha\\!-\\!3)^{2}}{(\beta\\!-\\!2)^{4}}\\!-\\!\frac{105}{4}\frac{(\alpha\\!-\\!3)}{(\beta\\!-\\!2)^{3}}\\!-\\!\frac{5}{4}\frac{(\alpha\\!-\\!51)}{(\beta\\!-\\!2)^{2}}\\!+\\!\frac{5}{4}\frac{(9\alpha\\!-\\!11)}{(\beta\\!-\\!2)}\;.$
(156)

Expressions (155) and (156) can be made arbitrarily positive by taking $\beta$
near 2. They do seem to be bounded below, but they can definitely change sign,
and there are two real solutions which cause them both two vanish,

$\displaystyle\alpha\simeq 0.551886\qquad$ , $\displaystyle\qquad\beta\simeq
1.42999\;,$ (157) $\displaystyle\alpha\simeq 2.24351\qquad$ ,
$\displaystyle\qquad\beta\simeq 1.78159\;.$ (158)

The gauge dependence we have exhibited in the flat space results (155-156)
must of course be present in the de Sitter result. However, it still is not
clear what happens to the parts of
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ which represent the
effects of inflationary gravitons. To understand this better, consider the
contributions to $T^{19}(x;x^{\prime})$ from Tables 7 and 8,

$\displaystyle
T^{19}(x;x^{\prime})=-\frac{\kappa^{2}}{64\pi^{4}}\Biggl{\\{}-\frac{\frac{112}{5}\Delta\eta}{\Delta
x^{6}}+\frac{(\frac{26}{3}a\\!+\\!8a^{\prime})H}{\Delta
x^{4}}-\frac{a^{2}{a^{\prime}}^{2}H^{4}\Delta\eta}{\Delta x^{2}}$ (159)
$\displaystyle\hskip
99.58464pt+\Biggl{[}\frac{8aa^{\prime}H^{2}\Delta\eta}{\Delta
x^{4}}+\frac{2aa^{\prime}(a\\!-\\!3a^{\prime})H^{3}}{\Delta
x^{2}}\Biggr{]}\ln(H^{2}\Delta x^{2})\Biggr{\\}}\;.\qquad$

The flat space result consists of just the first term. It is this term whose
coefficient can be driven to infinity, or made to vanish by the gauge
dependence of (155-156). This term has no effect on the graviton mode
function, and induces fractional corrections to the potentials of the form
$\kappa^{2}/r^{2}$. How the parameters $\alpha$ and $\beta$ of the general de
Sitter gauge (154) affect the other terms is not known. These other terms can
potentially change the graviton mode function, and they typically induce
fractional changes in the potentials of the form $GH^{2}$ times large temporal
and/or spatial logarithms.

In flat space Donoghue and collaborators have shown how to extract unique,
gauge independent results for the fractional $\kappa^{2}/r^{2}$ correction to
the potentials [14, 15]. Their technique [6, 8] is to first compute the one
loop scattering amplitude between two massive particles, then use inverse
scattering theory to infer the exchange potential. It was recently discovered
that this process can be short-circuited in order to directly purge the vacuum
polarization of gauge dependence [53]. The procedure is to assemble the same
diagrams whose sum would produce the scattering amplitude, however, one works
in position space and employs a series of identities which permit the higher
point diagrams to be viewed as corrections to the 1PI 2-point function. For
example, one of the many diagrams which contribute to the scattering of two
massive scalars consists of two graviton lines emerging from the vertex at
${x^{\prime}}^{\mu}$ and attaching to the other massive scalar at points
$x^{\mu}$ and $y^{\mu}$. This diagram does not have the 2-point topology to be
viewed as a contribution to the graviton self-energy, however, Donoghue and
collaborators have derived a series of reductions that capture the nonanlytic
parts of the full amplitude which are responsible for infrared phenomena [6,
8, 54, 55]. If $i\Delta_{m}(x;x^{\prime})$ denotes the massive scalar
propagator then the relevant Donoghue Identity for the 3-point contribution
just described is [53],

$i\Delta_{m}(x;y)i\Delta(x;x^{\prime})i\Delta(y,x^{\prime})\longrightarrow\frac{i\delta^{D}(x\\!-\\!y)}{2m^{2}}\Bigl{[}i\Delta(x;x^{\prime})\Bigr{]}^{2}\;.$
(160)

Applying (160) reduces the 3-point contribution to a 2-point form which can be
viewed as a correction to the graviton self-energy. When all such corrections
are combined, dependence upon $\alpha$ and $\beta$ drops out and one is left
with a unique and gauge independent result [53].

##  5 Epilogue

Quantum corrections from inflationary gravitons [39, 45, 46, 41, 47, 50, 48,
49] modify how other particles propagate [40, 56, 42, 57, 58, 59], and the
force laws they mediate [43]. At one loop order these results involve a single
graviton propagator, and it is principally the “tail” part of this propagator
that engenders the most interesting effects [60, 61]. Quantum gravity
corrections to gravity itself are even more interesting because they involve
two graviton propagators at one loop order. The potential for gravity to
mediate more interesting effects than matter can be seen from the factor of
$\ln(H^{2}\Delta x^{2})$ which multiplies all the contributions of Table 8,
and is absent from the analogous scalar contributions of Table 6.

The 1PI 2-graviton function
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ quantifies corrections
to linearized gravity from matter and from gravity itself. The point of this
paper has been to develop a representation for the tensor structure of this
object in terms of differential operators acting on structure functions.
Matter contributions must be annihilated when the Ward operator (5) acts on
either coordinate, but gravity contributions are only annihilated when the
Ward operator acts on both coordinates. On flat space background one requires
two structure functions for matter contributions and three for contributions
from gravity, as in expression (29). The absence of time translation
invariance and Lorentz invariance in cosmology means that four structure
functions are required for matter contributions whereas nine are needed for
contributions from gravity. Our representations are given in expressions (100)
for matter, and (102) for gravity.

Quantum field theory computations typically express
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ as a linear
combination of basis tensors which do not individually obey the relevant Ward
identity. For example, the flat space result (21) was originally reported [25]
using a basis of five tensors, which can then be organized into three
combinations (29) that obey the Ward identity. This procedure for passing from
raw results to structure functions is known as reconstruction. Our
reconstruction procedure for cosmology is based on first recasting the
primitive result as a sum (3) of the 21 tensor differential operators
$[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}]$ listed in Table 1, each acting on
a scalar coefficient function $T^{i}(x;x^{\prime})$. The
$[\mbox{}^{\mu\nu}\mathcal{D}^{\rho\sigma}]$ are constructed from
$\delta^{\mu}_{~{}0}$ and the spatial parts of the Minkowski metric
$\overline{\eta}^{\mu\nu}\equiv\eta^{\mu\nu}+\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}$
and the derivative operator
$\overline{\partial}^{\mu}\equiv\partial^{\mu}+\delta^{\mu}_{~{}0}\partial_{0}$.
A typical example is furnished by the scalar contribution (117) on de Sitter
background [20, 19]. We first $3+1$ decompose derivatives (118-119) of the de
Sitter length function, then express factors of the spatial coordinate
interval $\vec{x}-\vec{x}^{\prime}$ as gradients using relations (125-127).
Our fundamental structure functions for matter are $T^{12}(x;x^{\prime})$,
$T^{16}(x;x^{\prime})$, $T^{18}(x;x^{\prime})$ and $T^{19}(x;x^{\prime})$. For
gravity we express one action of the Ward operator in the form (6), involving
the ten basis tensors of Table 3 acting on coefficient functions $S^{i}(x;x)$.
The expansion of each $S^{i}$ in terms of the $T^{i}$ is given in Table 4.
Because acting the Ward operator a second time must produce zero, the ten
$S^{i}$ obeys the five relations given in Table 5. We take the five new
structure functions for gravity to be $S^{2}(x;x^{\prime})$,
$S^{4}(x;x^{\prime})$, $S^{7}(x;x^{\prime})$, $S^{8}(x;x^{\prime})$ and
$S^{10}(x;x^{\prime})$. We have found it convenient to group some of the
fundamental structure functions ($T^{16}$, $T^{19}$, $S^{4}$, $S^{8}$ and
$S^{10}$) into two symmetric auxiliary functions, $\alpha(x;x^{\prime})$ and
$\beta(x;x^{\prime})$, which are defined in expressions (73-75). Our final
representations for the self-energy in terms of the fundamental structure
functions are expressions (100) and (102).

The formalism we have derived for representing the graviton self-energy
improves on previous results [20, 21] in three ways:

  * •

It applies for contributions from gravitons in addition to contributions from
matter;

  * •

It is valid for any cosmological background (2), not just for de Sitter; and

  * •

Its structure functions involve the same functional forms as the primitive
result.

One can appreciate the final point from the explicit results for a loop of
massless, minimally coupled scalars [20, 19]. Primitive contributions to the
$T^{i}(x;x^{\prime})$ consist of sums of products of non-negative powers of
the two scale factors and the temporal separation $\Delta\eta$, times inverse
powers of the Poincaré interval $\Delta x^{2}\equiv(x-x^{\prime})^{2}$.
Because the fundamental structure functions of this new representation are
just $T^{12}$, $T^{16}$, $T^{18}$ and $T^{19}$, they of course have the same
form. One might worry about the auxiliary functions $\alpha$ and $\beta$, but
expressions (81) and (84) guarantee that they involve no new functional forms.
Compare that with what happens in the simplest of the previous
representations. The renormalized spin zero structure function roughly
equivalent to $T^{18}$ and a combination of $T^{16}$ and $T^{19}$ is [21],

$\displaystyle
F_{0R}=\frac{\kappa^{2}}{9(4\pi)^{4}}\Biggl{\\{}\frac{\partial^{2}}{2}\Biggl{[}\frac{\ln(H^{2}\Delta
x^{2})}{\Delta
x^{2}}\Biggr{]}+a^{2}{a^{\prime}}^{2}H^{4}\Biggl{[}-\frac{6}{y}\\!+\\!6$ (161)
$\displaystyle\hskip
128.0374pt\Bigl{(}-\frac{2}{y}\\!+\\!6\\!-\\!\frac{4}{4\\!-\\!y}\Bigr{)}\ln\Bigl{(}\frac{y}{4}\Bigr{)}+\frac{3}{2}(2\\!-\\!y)\Psi(y)\Biggr{\\}},\qquad$

where $y\equiv aa^{\prime}H^{2}\Delta x^{2}$ is the de Sitter length function
and we define

$\Psi(y)\equiv\frac{1}{2}\ln^{2}\Bigl{(}\frac{y}{4}\Bigr{)}-\ln\Bigl{(}1\\!-\\!\frac{y}{4}\Bigr{)}\ln\Bigl{(}\frac{y}{4}\Bigr{)}-{\rm
Li}_{2}\Bigl{(}\frac{y}{4}\Bigr{)}\;,$ (162)

where ${\rm Li}_{2}(x)\equiv-\int_{0}^{x}dt\ln(1-t)/t$ is the dilogarithm
function. The same exotic functional forms appear in the two tensor structure
functions, $F_{2R}(x;x^{\prime})$ and $G_{2R}(x;x^{\prime})$, which are
roughly equivalent to $T^{12}(x;x^{\prime})$ and a different combination of
$T^{16}$ and $T^{19}(x;x^{\prime})$. Deriving these structure functions from
the primitive result for
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ is a major undertaking
because it entails solving partial differential equations. Those equations
were barely tractable for the scalar contributions owing to the absence of de
Sitter breaking, but they become hopelessly complicated when de Sitter
invariance is lost with graviton contributions. Finally, it is of course
difficult using the exotic structure functions to solve the effective field
equations. The new formalism obviates all of these problems.

The point of devising this representation is to solve the effective field
equations. Section 3.3 specializes the effective field equations for a
graviton contribution to the cases of a spatial plane wave graviton (109) and
the two scalar potentials (114-115) which represent the gravitational response
to point mass. Both of these things have already been computed (using the old
formalism) for the contribution of a massless, minimally coupled scalar [20,
19]. Although there are no changes in the graviton mode function [20], the
response to a point mass acquires corrections which grow at late times and
large distances [19],

$\displaystyle\Psi(\eta,r)=-\frac{GM}{ar}\Biggl{\\{}1+\frac{G}{20\pi
a^{2}r^{2}}$ (163) $\displaystyle\hskip
56.9055pt+\frac{GH^{2}}{\pi}\Bigl{[}-\frac{1}{30}\ln(a)-\frac{3}{10}\ln(aHr)\Bigr{]}+O(G^{2})\Biggr{\\}},\qquad$
$\displaystyle\Phi(\eta,r)=-\frac{GM}{ar}\Biggl{\\{}1-\frac{G}{60\pi
a^{2}r^{2}}$ (164) $\displaystyle\hskip
56.9055pt+\frac{GH^{2}}{\pi}\Bigl{[}-\frac{1}{30}\ln(a)-\frac{3}{10}\ln(aHr)+\frac{2}{3}aHr\Bigr{]}+O(G^{2})\Biggr{\\}}.\qquad$

We are now in a position to study what gravity does to itself. Applying the
Hartree approximation indicates that inflationary gravitons enhance the
“electric” components of the Weyl field strength [18],

$C^{\rm
1~{}loop}_{0i0j}(\eta,k)\longrightarrow-\frac{8}{\pi}GH^{2}\ln(a)\times C^{\rm
tree}_{0i0j}(\eta,k)\;.$ (165)

It would be very interesting to extend the $D=4$ results of Tables 7, 8, 9 and
10 to recover fully renormalized results, and then employ them to solve
equations (109) and (114-115).

Acknowledgements

This work was partially supported by the European Union’s Seventh Framework
Programme (FP7-REGPOT-2012-2013-1) under grant agreement number 316165; by the
European Union’s Horizon 2020 Programme under grant agreement 669288-SM-GRAV-
ERC-2014-ADG; by NSF grants PHY-1506513 and 1806218; and by the UF’s Institute
for Fundamental Theory.

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