CCTP-2024-8

UFIFT-QG-24-05

The Third Structure Function

S. P. Miao1⋆, N. C. Tsamis2†, R. P. Woodard3‡ and B. Yesilyurt3∗

1 Department of Physics, National Cheng Kung University,
No. 1 University Road, Tainan City 70101, TAIWAN

2 Institute of Theoretical Physics & Computational Physics,
Department of Physics, University of Crete,
GR-700 03 Heraklion, HELLAS

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

ABSTRACT

We re-consider the graviton self-energy induced by a loop of massless,
minimally coupled scalars on de Sitter background. On flat space background it
can be represented as a sum of two tensor differential operators acting on
scalar structure functions. On a general background these tensor differential
operators can be constructed from the linearized Ricci scalar and the
linearized Weyl tensor. However, in cosmology one requires a third
contribution which we derive here.

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

⋆ e-mail<EMAIL_ADDRESS>
† e-mail<EMAIL_ADDRESS>
‡ e-mail<EMAIL_ADDRESS>
∗ e-mail<EMAIL_ADDRESS>
##  1 Prologue

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

$\mathcal{D}^{\mu\nu\rho\sigma}\kappa h_{\rho\sigma}(x)=8\pi
GT^{\mu\nu}(x)+\int\\!\\!d^{4}x^{\prime}\,[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})\,\kappa
h_{\rho\sigma}(x^{\prime})\;\;,$ (1)

where $D^{\mu\nu\rho\sigma}$ is the Lichnerowicz operator, $\kappa^{2}\equiv
16\pi G$ is the loop-counting parameter of quantum gravity and $T^{\mu\nu}$ is
$-2/\kappa$ times the variation of the matter action with respect to the
graviton field $h_{\mu\nu}$. This equation describes quantum corrections to
free gravitational radiation, and to the gravitational response to specific
matter sources. Although flat space background shows no change to
gravitational radiation, and only small corrections to gravitational
potentials [1, 2, 3], there can be dramatic effects on backgrounds such as de
Sitter which generate copious particle production [4, 5, 6, 7, 8].

The graviton self-energy is a 2nd rank, bi-tensor density on whatever
background is under study, and the struggle to find a simple and useful way of
representing its tensor structure has been long and confusing [9, 10, 11, 12].
An additional complication is the distinction between matter loop
contributions, which obey a conservation relation at either point, and
graviton loop corrections, which only obey the weaker relation of conservation
at both points. We shall here consider the simpler case of matter loop
corrections.

On flat, $D$-dimensional spacetime the tensor structure of matter
contributions can be represented by spin zero and spin two combinations of the
transverse projection operator,
$\Pi^{\mu\nu}\equiv\partial^{\mu}\partial^{\nu}-\eta^{\mu\nu}\partial^{2}$
[3],

$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
flat}](x;x^{\prime})=\Pi^{\mu\nu}\Pi^{\rho\sigma}F_{0}(x-x^{\prime})+\Bigl{[}\Pi^{\mu(\rho}\Pi^{\sigma)\nu}-\tfrac{1}{D-1}\Pi^{\mu\nu}\Pi^{\rho\sigma}\Bigr{]}F_{2}(x-x^{\prime})\;,$
(2)

where parenthesized indices are symmetrized. For a massless, minimally coupled
scalar in $D=4$ the renormalized, Schwinger-Keldysh structure functions are
[13, 14, 15],

$\displaystyle iF_{0}(\Delta x)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\kappa^{2}}{2^{10}\\!\cdot\\!3^{2}\\!\cdot\\!\pi^{3}}\\!\times\\!\partial^{4}\Bigl{\\{}\theta(\Delta
t\\!-\\!\Delta r)\Bigl{(}\ln[\mu^{2}(\Delta t^{2}\\!-\\!\Delta
r^{2})]\\!-\\!1\Bigr{)}\Bigr{\\}},\qquad$ (3) $\displaystyle iF_{2}(\Delta x)$
$\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{\kappa^{2}}{2^{11}\\!\cdot\\!3\\!\cdot\\!5\\!\cdot\\!\pi^{3}}\\!\times\\!\partial^{4}\Bigl{\\{}\theta(\Delta
t\\!-\\!\Delta r)\Bigl{(}\ln[\mu^{2}(\Delta t^{2}\\!-\\!\Delta
r^{2})]\\!-\\!1\Bigr{)}\Bigr{\\}},\qquad$ (4)

where $\Delta t\equiv t-t^{\prime}$, $\Delta
r\equiv\|\vec{x}-\vec{x}^{\prime}\|$ and $\mu$ is the renormalization scale.

We seek a similar representation on de Sitter background in conformal
coordinates,

$ds^{2}=a^{2}\Bigl{[}-d\eta^{2}+d\vec{x}\\!\cdot\\!d\vec{x}\Bigr{]}\qquad,\qquad
a(\eta)\equiv-\frac{1}{H\eta}\;,$ (5)

where the Hubble constant $H$ is related to the cosmological constant
$\Lambda\equiv(D-1)H^{2}$. We will continue to employ the flat space
convention that indices are raised and lowered using the Minkowski metric
($\partial^{\mu}\equiv\eta^{\mu\nu}\partial_{\nu}$). The appropriate de Sitter
generalization of transversality is,

$W^{\mu}_{~{}\alpha\beta}\\!\times\\!-i[\mbox{}^{\alpha\beta}\Sigma^{\rho\sigma}](x;x^{\prime})=0={W^{\prime}}^{\rho}_{~{}\gamma\delta}\\!\times\\!-i[\mbox{}^{\mu\nu}\Sigma^{\gamma\delta}](x;x^{\prime})\;,$
(6)

where the Ward operator is [11],

$W^{\mu}_{~{}\alpha\beta}\equiv\delta^{\mu}_{~{}(\alpha}\partial_{\beta)}+aH\delta^{\mu}_{~{}0}\eta_{\alpha\beta}\;.$
(7)

The spin zero tensor in relation (2) can be generalized to de Sitter using a
tensor differential operator $\mathcal{F}^{\mu\nu}$,

$\Pi^{\mu\nu}\Pi^{\rho\sigma}\longrightarrow\mathcal{F}^{\mu\nu}\\!\times\\!{\mathcal{F}^{\prime}}^{\rho\sigma}\;.$
(8)

One defines this by first expanding the Ricci tensor to first order in the
graviton field $g_{\mu\nu}\equiv a^{2}[\eta_{\mu\nu}+\kappa h_{\mu\nu}]$,

$R-D\Lambda\equiv\tfrac{1}{a^{2}}\\!\times\\!\overline{\mathcal{F}}^{\mu\nu}\\!\times\\!\kappa
h_{\mu\nu}+O(h^{2})\;.$ (9)

The operator $\mathcal{F}^{\mu\nu}$ is obtained by partially integrating
$\overline{\mathcal{F}}^{\mu\nu}$,

$\int\\!\\!d^{D}x\,f_{1}(x)\\!\times\\!\overline{\mathcal{F}}^{\mu\nu}f_{2}(x)\equiv\int\\!\\!d^{D}x\,\mathcal{F}^{\mu\nu}f_{1}(x)\\!\times\\!f_{2}(x)\;.$
(10)

The final result is an obvious de Sitter generalization of $\Pi^{\mu\nu}$,

$\displaystyle\mathcal{F}^{\mu\nu}=\partial^{\mu}\partial^{\nu}-\eta^{\mu\nu}\partial^{2}$
(11) $\displaystyle\hskip
42.67912pt+(D\\!-\\!1)aH\Bigl{[}2\delta^{(\mu}_{~{}~{}0}\partial^{\nu)}+(D\\!-\\!2)aH\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}-\eta^{\mu\nu}\partial_{0}-\eta^{\mu\nu}aH\Bigr{]}.\qquad$

It is simple to check that
$\mathcal{W}^{\mu}_{~{}\alpha\beta}\times\mathcal{F}^{\alpha\beta}=0$.

The spin two tensor in relation (2) can be generalized to de Sitter using a
tensor differential operator
$\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}$,

$\Pi^{\mu(\rho}\Pi^{\sigma)\nu}-\tfrac{1}{D-1}\Pi^{\mu\nu}\Pi^{\rho\sigma}\longrightarrow\Bigl{(}\frac{D\\!-\\!2}{D\\!-\\!3}\Bigr{)}\\!\times\\!\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}\\!\times\\!{\mathcal{C}^{\prime}}^{\alpha\beta\gamma\delta\rho\sigma}\;.$
(12)

One defines $\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}$ by
expanding the Weyl tensor to first order in the graviton field,

$\eta_{\alpha\epsilon}C^{\epsilon}_{~{}\beta\gamma\delta}\equiv\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}\\!\times\\!\kappa
h_{\mu\nu}+O(h^{2})\;.$ (13)

It is not necessary to partially integrate because each term in
$\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}$ contains two
derivatives, with no scale factors. An explicit form can be found in published
works [10, 11]. Like $\mathcal{F}^{\mu\nu}$, it is annihilated by the Ward
operator,

$\mathcal{W}^{\lambda}_{~{}\mu\nu}\\!\times\\!\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}=0\;.$
(14)

It turns out that representing the graviton self-energy on cosmological
backgrounds such as (5) requires a third structure function [11], and finding
a simple representation for this has proven to be difficult. Because relation
(14) pertains for any choices of the first four indices of
$\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}$, conservation
(6) would apply were we to contract any constant tensor into the product,

$\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}\\!\times\\!{\mathcal{C}^{\prime}}_{\kappa\lambda\theta\phi}^{~{}~{}~{}~{}~{}\rho\sigma}\;.$
(15)

The particular choice giving rise to (12) is the product of four Minkowski
metrics,

$\eta^{\alpha\kappa}\\!\times\\!\eta^{\beta\lambda}\\!\times\\!\eta^{\gamma\theta}\\!\times\\!\eta^{\delta\phi}\;.$
(16)

So a natural thought is to define a distinct projector by replacing (16) with
another constant tensor, such as the product of four spatial metrics,

$\overline{\eta}^{\mu\nu}\equiv\eta^{\mu\nu}+\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\;.$
(17)

However, when this is done the structure functions become quite complicated
[11] and difficult to use [5]. Further, assuming this form for the third
projector seems ill-considered because, like the de Sitter generalizations (8)
and (12), it is a 4th order tensor differential operator. Because the need for
the third projector is tied to the nonzero Hubble parameter of cosmological
backgrounds, it would make more sense to seek a 2nd order tensor differential
operator, with two factors of $H$ in place of the two missing derivatives.

The point of this work is to identify a 3rd contribution to the graviton self-
energy as $H^{2}$ times some second order differential operator, acting on
simple structure functions. In Section 2 we begin with a recent computation of
the massless, minimally coupled scalar loop contribution to
$-i[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ and extract the terms
coming from the spin 0 and spin 2 projectors (8) and (12). In Section 3 we
promote the Lichnerowicz operator into a bi-tensor. In Section 4 it is made
the basis for expressing the 3rd contribution. Our conclusions comprise
Section 5.

##  2 The First Two Structure Functions

The purpose of this section is to express the previously computed result for
the (massless, minimally coupled scalar contribution to the) graviton self-
energy [8] as a sum of the contributions from the first two terms plus a
separately conserved residual. It is this residual which will eventually be
expressed as the 3rd structure function. We begin by introducing some useful
notation. Then the previous result for
$[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$ is given. The section
closes by extracting the parts associated with the (8) and (12) projectors to
leave the residual we seek to study.

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

The massless, minimally coupled scalar contribution to the graviton self-
energy on de Sitter background was first computed more than a decade ago [10].
However, it was not then realized that a finite renormalization of the
cosmological constant is required to prevent the appearance of a delta
function obstruction to conservation [16]. This mistake was compounded by
choosing to express the structure function in terms of projectors which are
annihilated by the Ward operator [11]. The two errors might have canceled out
but it turns out they do not [8]. In order to check, and to facilitate a
subsequent resummation [17], the result was recomputed in the safest way
possible, by expressing it as a sum of the 21 (non-conserved) tensor
differential operators given in Table 1, whose form is dictated by the
homogeneity and isotropy of cosmology.

$i$ | $T^{i}(a,a^{\prime},\partial)$ | $T_{A}^{i}(a,a^{\prime},\partial)$
---|---|---
1 | $-aa^{\prime}\ln(aa^{\prime})[\partial^{2}+2aa^{\prime}H^{2}]$ | $-\frac{1}{4}a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}\partial_{0}^{2}$
| $-3aa^{\prime}\partial^{2}-aa^{\prime}\partial_{0}^{2}-6a^{2}{a^{\prime}}^{2}H^{2}$ |
2 | $aa^{\prime}\ln(aa^{\prime})[\partial^{2}+2aa^{\prime}H^{2}]$ | $0$
| $+aa^{\prime}\partial^{2}+3a^{2}{a^{\prime}}^{2}H^{2}$ |
3 | $2a^{3}a^{\prime}H^{2}[\ln(aa^{\prime})+\frac{9}{2}]$ | $0$
4 | $2a{a^{\prime}}^{3}H^{2}[\ln(aa^{\prime})+\frac{9}{2}]$ | $0$
5 | $-2a^{2}a^{\prime}H[\ln(aa^{\prime})+4]$ | $\frac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}$
6 | $2a{a^{\prime}}^{2}H[\ln(aa^{\prime})+4]$ | $\frac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}$
7 | $aa^{\prime}[\ln(aa^{\prime})+3]$ | $-\frac{1}{2}a^{2}a^{\prime}H\partial_{0}\partial^{2}+\frac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}$
8 | $aa^{\prime}[\ln(aa^{\prime})+3]$ | $\frac{1}{2}a{a^{\prime}}^{2}H\partial_{0}\partial^{2}+\frac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}$
9 | $2a^{2}{a^{\prime}}^{2}H^{2}[\ln(aa^{\prime})+1]$ | $0$
10 | $-2a{a^{\prime}}^{2}H[\ln(aa^{\prime})+2]$ | $0$
11 | $2a^{2}a^{\prime}H[\ln(aa^{\prime})+2]$ | $0$
12 | $-2aa^{\prime}[\ln(aa^{\prime})+1]$ | $-a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}$
18 | $0$ | $-3a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}$
19 | $0$ | $a^{2}a^{\prime}H\partial^{2}$
20 | $0$ | $-a{a^{\prime}}^{2}H\partial^{2}$
21 | $0$ | $\frac{1}{2}aa^{\prime}\partial^{2}-a^{2}{a^{\prime}}^{2}H^{2}$
Table 2: Coefficients $T^{i}(a,a^{\prime},\partial)$ and
$T^{i}_{A}(a,a^{\prime},\partial)$ which appear in expression (19).

Finite remainders from the $a^{D-4}R^{2}$ and $a^{D-4}C^{2}$ counterterms give
rise to factors of $\ln(aa^{\prime})\delta^{4}(\Delta x)$ acted on by the de
Sitter spin 0 and spin 2 projectors (8) and (12). The remaining terms involve
$\delta^{4}(\Delta x)$ and two other functions,

$f_{A}(\Delta x)\equiv\theta(\Delta\eta\\!-\\!\Delta r)\quad,\quad
f_{B}(\Delta x)\equiv\partial^{4}\Bigl{\\{}\theta(\Delta\eta\\!-\\!\Delta
r)\Bigl{(}\ln[\mu^{2}(\Delta\eta^{2}\\!-\\!\Delta
r^{2})]\\!-\\!1\Bigr{\\}}\;,$ (18)

where $\Delta\eta\equiv\eta-\eta^{\prime}$ and $\Delta
r\equiv\|\vec{x}-\vec{x}^{\prime}\|$. Coefficient functions of the two scale
factors and the derivative operator (given in Tables 2 and 3) times the basis
tensors of Table 1 act on these three functions [8],

$\displaystyle[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})=\frac{-\kappa^{2}}{2^{7}\\!\cdot\\!3\\!\cdot\\!\pi^{2}}\Bigl{[}\tfrac{1}{3}\mathcal{F}^{\mu\nu}\\!\\!\times\\!{\mathcal{F}^{\prime}}^{\rho\sigma}\\!\\!+\\!\tfrac{1}{5}\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}\\!\\!\times\\!{\mathcal{C}^{\prime}}^{\alpha\beta\gamma\delta\rho\sigma}\Bigr{]}\Bigl{[}\ln(aa^{\prime})\delta^{4}(\Delta
x)\Bigr{]}$ (19)
$\displaystyle+\frac{\kappa^{2}H^{2}}{2^{7}\\!\cdot\\!3\\!\cdot\\!\pi^{3}}\sum_{i=1}^{21}\Bigl{\\{}-2\pi
T^{i}(a,a^{\prime},\partial)[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]\delta^{4}(\Delta
x)+T^{i}_{A}(a,a^{\prime}\\!,\partial)[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]f_{A}(\Delta
x)$ $\displaystyle\hskip
196.32384pt+T^{i}_{B}(a,a^{\prime}\\!,\partial)[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]f_{B}(\Delta
x)\Bigr{\\}}.\qquad$

$i$ | $T_{B}^{i}(a,a^{\prime},\partial)$ | $i$ | $T_{B}^{i}(a,a^{\prime},\partial)$
---|---|---|---
1 | $-\frac{3\partial^{4}}{80H^{2}}+\frac{aa^{\prime}\partial^{2}}{8}+\frac{aa^{\prime}\partial_{0}^{2}}{8}$ | 12 | $\frac{\partial^{2}}{40H^{2}}+\frac{aa^{\prime}}{2}$
2 | $-\frac{\partial^{4}}{80H^{2}}-\frac{aa^{\prime}\partial^{2}}{4}-\frac{a^{2}{a^{\prime}}^{2}H^{2}}{2}$ | 13 | $-\frac{3a^{2}{a^{\prime}}^{2}H^{2}}{2}$
3 | $\frac{{a^{\prime}}^{2}\partial^{2}}{4}+\frac{3a{a^{\prime}}^{2}H\partial_{0}}{4}+\frac{a^{2}{a^{\prime}}^{2}H^{2}}{2}$ | 14 | $\frac{3a^{2}a^{\prime}H}{2}$
4 | $\frac{a^{2}\partial^{2}}{4}-\frac{3a^{2}a^{\prime}H\partial_{0}}{4}+\frac{a^{2}{a^{\prime}}^{2}H^{2}}{2}$ | 15 | $-\frac{3a{a^{\prime}}^{2}H}{2}$
5 | $-\frac{a^{\prime}\partial^{2}}{4H}-\frac{3aa^{\prime}\partial_{0}}{4}+\frac{a{a^{\prime}}^{2}H}{4}$ | 16 | $-\frac{a^{2}}{4}$
6 | $\frac{a\partial^{2}}{4H}-\frac{3aa^{\prime}\partial_{0}}{4}-\frac{a^{2}a^{\prime}H}{4}$ | 17 | $-\frac{{a^{\prime}}^{2}}{4}$
7 | $\frac{3\partial^{2}}{80H^{2}}+\frac{a\partial_{0}}{8H}-\frac{a^{2}}{8}$ | 18 | $\frac{3aa^{\prime}}{2}$
8 | $\frac{3\partial^{2}}{80H^{2}}-\frac{a^{\prime}\partial_{0}}{8H}-\frac{{a^{\prime}}^{2}}{8}$ | 19 | $-\frac{a}{4H}$
9 | $-\frac{3a^{2}{a^{\prime}}^{2}H^{2}}{2}$ | 20 | $\frac{a^{\prime}}{4H}$
10 | $\frac{a^{2}a^{\prime}H}{2}$ | 21 | $-\frac{1}{20H^{2}}$
11 | $-\frac{a{a^{\prime}}^{2}H}{2}$ |  |
Table 3: Coefficient functions $T^{i}_{B}(a,a^{\prime},\partial)$ which appear
in expression (19).

Note that we do not need to distinguish
$\partial^{\prime}_{\mu}\rightarrow-\partial_{\mu}$ when acting on functions
of $\Delta x^{\mu}=x^{\mu}-{x^{\prime}}^{\mu}$. Note also that the three
functions obey,

$\partial^{4}f_{A}=8\pi\delta^{4}(\Delta
x)\;\;,\;\;[\Delta\eta\partial^{2}+2\partial_{0}]f_{A}=0\;\;,\;\;\Delta\eta
f_{B}=-2\partial_{0}\partial^{2}f_{A}\;.$ (20)

$i$ | $\Delta T^{i}(a,a^{\prime},\partial)$ | $\Delta T^{i}_{A}(a,a^{\prime},\partial)$ | $\Delta T^{i}_{B}(a,a^{\prime},\partial)$
---|---|---|---
1 | $\\!\\!\\!-aa^{\prime}\ln(aa^{\prime})[\partial^{2}\\!+\\!2aa^{\prime}H^{2}]\\!\\!\\!$ | $0$ | $\\!\\!\\!\tfrac{1}{4}aa^{\prime}\partial^{2}\\!+\\!\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$
| $\\!\\!\\!-3aa^{\prime}\partial^{2}\\!-\\!7a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$ |  |
2 | $\\!\\!\\!aa^{\prime}\ln(aa^{\prime})[\partial^{2}\\!+\\!2aa^{\prime}H^{2}]\\!\\!\\!$ | $0$ | $\\!\\!\\!-\tfrac{1}{4}aa^{\prime}\partial^{2}\\!-\\!\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$
| $\\!\\!\\!+aa^{\prime}\partial^{2}\\!+\\!3a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$ |  |
3 | $\\!\\!\\!a^{3}a^{\prime}H^{2}[2\ln(aa^{\prime})\\!+\\!9]\\!\\!\\!$ | $0$ | $\\!\\!\\!-\tfrac{1}{4}a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$
4 | $\\!\\!\\!a{a^{\prime}}^{3}H^{2}[2\ln(aa^{\prime})\\!+\\!9]\\!\\!\\!$ | $0$ | $\\!\\!\\!-\tfrac{1}{4}a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$
5 | $\\!\\!\\!-a^{2}a^{\prime}H[2\ln(aa^{\prime})\\!+\\!8]\\!\\!\\!$ | $\\!\\!\\!\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}\\!\\!\\!$ | $\\!\\!\\!\tfrac{1}{4}a{a^{\prime}}^{2}H\\!+\\!\tfrac{1}{4}a^{2}a^{\prime}H\\!\\!\\!$
6 | $\\!\\!\\!a{a^{\prime}}^{2}H[2\ln(aa^{\prime})\\!+\\!8]\\!\\!\\!$ | $\\!\\!\\!\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}\\!\\!\\!$ | $\\!\\!\\!-\tfrac{1}{4}a{a^{\prime}}^{2}H\\!-\\!\tfrac{1}{4}a^{2}a^{\prime}H\\!\\!\\!$
7 | $\\!\\!\\!aa^{\prime}[\ln(aa^{\prime})\\!+\\!3]\\!\\!\\!$ | $\\!\\!\\!\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}\\!\\!\\!$ | $\\!\\!\\!-\tfrac{1}{4}aa^{\prime}$
8 | $\\!\\!\\!aa^{\prime}[\ln(aa^{\prime})\\!+\\!3]\\!\\!\\!$ | $\\!\\!\\!\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}\\!\\!\\!$ | $\\!\\!\\!-\tfrac{1}{4}aa^{\prime}\\!\\!\\!$
9 | $\\!\\!\\!a^{2}{a^{\prime}}^{2}H^{2}[2\ln(aa^{\prime})\\!+\\!2]\\!\\!\\!$ | $0$ | $\\!\\!\\!-\tfrac{3}{2}a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$
10 | $\\!\\!\\!-a{a^{\prime}}^{2}H[2\ln(aa^{\prime})\\!+\\!4]\\!\\!\\!$ | $0$ | $\\!\\!\\!\tfrac{1}{2}a^{2}a^{\prime}H\\!\\!\\!$
11 | $\\!\\!\\!a^{2}a^{\prime}H[2\ln(aa^{\prime})\\!+\\!4]\\!\\!\\!$ | $0$ | $\\!\\!\\!-\tfrac{1}{2}a{a^{\prime}}^{2}H\\!\\!\\!$
12 | $\\!\\!\\!-aa^{\prime}[2\ln(aa^{\prime})\\!+\\!2]\\!\\!\\!$ | $\\!\\!\\!-a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}\\!\\!\\!$ | $\\!\\!\\!\tfrac{1}{2}aa^{\prime}\\!\\!\\!$
18 | $0$ | $\\!\\!\\!-3a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}\\!\\!\\!$ | $0$
19 | $0$ | $\\!\\!\\!a^{2}a^{\prime}H\partial^{2}\\!\\!\\!$ | $0$
20 | $0$ | $\\!\\!\\!-a{a^{\prime}}^{2}H\partial^{2}\\!\\!\\!$ | $0$
21 | $0$ | $\\!\\!\\!\tfrac{1}{2}aa^{\prime}\partial^{2}\\!-\\!a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$ | $0$
Table 4: Tabulation of the residual coefficient functions which appear in
expression (21).

Expression (19) for the graviton self-energy was employed because it is safe,
however, it can easily be simplified. The key is recognizing the function
$f_{B}(\Delta x)$ in the flat space structure functions (3) and (4), if we
identify the flat space time $t$ with conformal time $\eta$. Because the de
Sitter result (19) must degenerate to its flat space ancestor (2) for $H=0$,
we can extract terms proportional to the projectors (8) and (12), acting on
$f_{B}(\Delta x)$,

$\displaystyle[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})=\frac{-\kappa^{2}}{2^{10}\\!\cdot\\!3\\!\cdot\\!\pi^{3}}\Bigl{[}\tfrac{1}{3}\mathcal{F}^{\mu\nu}\\!\\!\times\\!{\mathcal{F}^{\prime}}^{\rho\sigma}\\!\\!+\\!\tfrac{1}{5}\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}\\!\\!\times\\!{\mathcal{C}^{\prime}}^{\alpha\beta\gamma\delta\rho\sigma}\Bigr{]}\Bigl{[}8\pi\ln(aa^{\prime})$
(21) $\displaystyle\times\delta^{4}(\Delta x)\\!+\\!f_{B}(\Delta
x)\Bigr{]}+\frac{\kappa^{2}H^{2}}{2^{7}\\!\cdot\\!3\\!\cdot\\!\pi^{3}}\sum_{i=1}^{21}\Bigl{\\{}-2\pi\Delta
T^{i}(a,a^{\prime},\partial)[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]\delta^{4}(\Delta
x)$ $\displaystyle\hskip 28.45274pt+\Delta
T^{i}_{A}(a,a^{\prime}\\!,\partial)[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]f_{A}(\Delta
x)+\Delta
T^{i}_{B}(a,a^{\prime}\\!,\partial)[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]f_{B}(\Delta
x)\Bigr{\\}}.\qquad$

Table 4 gives the residual coefficient functions. Note that the same
combination of $8\pi\ln(aa^{\prime})\delta^{4}(\Delta x)+f_{B}(\Delta x)$ in
both structure functions arises because of the incomplete cancellation between
primitive divergences (which contain no $D$-dependent factors of $a$) and
countermterms (which inherit an $a^{D}$ from the measure factor) [8],

$\frac{(\frac{H}{2})^{D-4}}{D\\!-\\!4}-\frac{(\mu
a)^{D-4}}{D\\!-\\!4}=-\ln\Bigl{(}\frac{2\mu a}{H}\Bigr{)}+O(D\\!-\\!4)\;.$
(22)

##  3 The Lichnerowicz Operator as a Bi-Tensor

The purpose of this section is to extract from the summed terms of expression
(21) a 3rd contribution proportional to the combination
$8\pi\ln(aa^{\prime})\delta^{4}(\Delta x)+f_{B}(\Delta x)$. This is done by
promoting the Lichnerowicz operator of equation (1) into a bi-tensor
$[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]$. The simplest choice permits us
to represent the logarithm parts of the $\Delta T^{i}$ sum in (21) as
proportional to $[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\times
8\pi\delta^{4}(\Delta x)$. We then add factors proportional to $\Delta\eta$ to
the $\Delta T^{i}_{B}(a,a^{\prime},\partial)$ coefficient functions of Table
4, and use the final relation of (20), to express the $\Delta T^{i}_{B}$ sum
in (21) as proportional to $[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\times
f_{B}(\Delta)$, plus a sum of terms proportional to $f_{A}(\Delta x)$. The
section ends by noting that
$W^{\mu}_{~{}\alpha\beta}\times[\mbox{}^{\alpha\beta}\mathcal{L}^{\rho\sigma}]\neq
0$, so that the new contribution is not conserved without the $\delta T^{i}$
and $\delta T^{i}_{A}$ residues.

On de Sitter background the Lichnerowicz operator is,

$\displaystyle\mathcal{D}^{\mu\nu\rho\sigma}h_{\rho\sigma}=\tfrac{1}{2}a^{2}\Bigl{[}\partial^{2}h^{\mu\nu}-\eta^{\mu\nu}\partial^{2}h+\eta^{\mu\nu}\partial^{\rho}\partial^{\sigma}h_{\rho\sigma}+\partial^{\mu}\partial^{\nu}h-2\partial^{\rho}\partial^{(\mu}h^{\nu)}_{~{}~{}\rho}\Bigr{]}$
(23) $\displaystyle\hskip
14.22636pt+a^{3}H\Bigl{[}\eta^{\mu\nu}\partial_{0}h-\partial_{0}h^{\mu\nu}-2\eta^{\mu\nu}\partial^{\rho}h_{\rho
0}+2\partial^{(\mu}h^{\nu)}_{~{}~{}0}\Bigr{]}+3a^{4}H^{2}\eta^{\mu\nu}h_{00}\;.\qquad$

We seek to define a bi-tensor operator
$[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]$ such that,

$\int\\!\\!d^{4}x^{\prime}\,\Bigl{\\{}[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\\!\times\\!\delta^{4}(\Delta
x)\Bigr{\\}}\\!\times\\!h_{\rho\sigma}(x^{\prime})=\mathcal{D}^{\mu\nu\sigma\rho}h_{\rho\sigma}(x)\;.$
(24)

A number of choices are possible but the simplest is,

$[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\equiv\Bigl{\\{}[\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu(\rho}\eta^{\sigma)\nu}]\partial\\!\cdot\\!\partial^{\prime}+2{\partial^{\prime}}^{(\mu}\eta^{\nu)(\rho}\partial^{\sigma)}+{\partial^{\prime}}^{\mu}{\partial^{\prime}}^{\nu}\eta^{\rho\sigma}+\eta^{\mu\nu}\partial^{\rho}\partial^{\sigma}\Bigr{\\}}\tfrac{aa^{\prime}}{2}\;.$
(25)

$i$ | $\delta T^{i}(a,a^{\prime},\partial)$ | $\delta T^{i}_{A}(a,a^{\prime},\partial)$ | $[\mbox{}^{\mu\nu}D_{i}^{\rho\sigma}]$
---|---|---|---
1 | $-3aa^{\prime}\partial^{2}\\!-\\!4a^{2}{a^{\prime}}^{2}H^{2}$ | $-\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}^{2}\partial^{2}$ | $\eta^{\mu\nu}\eta^{\rho\sigma}$
2 | $aa^{\prime}\partial^{2}$ | $\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}^{2}\partial^{2}$ | $\eta^{\mu(\rho}\eta^{\sigma)\nu}$
3 | $6a^{3}a^{\prime}H^{2}$ | $-a^{3}{a^{\prime}}^{2}H^{3}\partial_{0}\partial^{2}$ | $\eta^{\mu\nu}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$
4 | $6a{a^{\prime}}^{3}H^{2}$ | $a^{2}{a^{\prime}}^{3}H^{3}\partial_{0}\partial^{2}$ | $\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\eta^{\rho\sigma}$
5 | $-6a^{2}a^{\prime}H$ | $a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}$ | $\eta^{\mu\nu}\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}$
6 | $6a{a^{\prime}}^{2}H$ | $a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}$ | $\delta^{(\mu}_{~{}~{}0}\partial^{\nu)}\eta^{\rho\sigma}$
7 | $3aa^{\prime}$ | $\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}$ | $\eta^{\mu\nu}\partial^{\rho}\partial^{\sigma}$
8 | $3aa^{\prime}$ | $\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}\\!\\!\\!$ | $\partial^{\mu}\partial^{\nu}\eta^{\rho\sigma}$
9 | $-2a^{2}{a^{\prime}}^{2}H^{2}$ | $0$ | $\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}$
10 | $-2a{a^{\prime}}^{2}H$ | $-a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}$ | $\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\partial^{\sigma)}$
11 | $2a^{2}a^{\prime}H$ | $-a^{2}{a^{\prime}}^{2}H^{2}\partial_{0}\partial^{2}$ | $\partial^{(\mu}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}$
12 | $-2aa^{\prime}$ | $-a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}$ | $\partial^{(\mu}\eta^{\nu)(\rho}\partial^{\sigma)}$
18 | $0$ | $\\!\\!\\!-3a^{2}{a^{\prime}}^{2}H^{2}\partial^{2}$ | $\delta^{(\mu}_{~{}~{}0}\partial^{\nu)}\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}$
19 | $0$ | $\\!\\!\\!a^{2}a^{\prime}H\partial^{2}\\!\\!\\!$ | $\delta^{(\mu}_{~{}~{}0}\partial^{\nu)}\partial^{\rho}\partial^{\sigma}$
20 | $0$ | $\\!\\!\\!-a{a^{\prime}}^{2}H\partial^{2}\\!\\!\\!$ | $\partial^{\mu}\partial^{\nu}\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}$
21 | $0$ | $\\!\\!\\!\tfrac{1}{2}aa^{\prime}\partial^{2}\\!-\\!a^{2}{a^{\prime}}^{2}H^{2}\\!\\!\\!$ | $\partial^{\mu}\partial^{\nu}\partial^{\rho}\partial^{\sigma}$
Table 5: Tabulation of the residues $\delta T^{i}(a,a^{\prime},\partial)$ and
$\delta T^{i}_{A}(a,a^{\prime},\partial)$ which appear in expression (26).

$\displaystyle[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})=\frac{-\kappa^{2}}{2^{10}\\!\cdot\\!3\\!\cdot\\!\pi^{3}}\Bigl{[}\tfrac{1}{3}\mathcal{F}^{\mu\nu}\\!\\!\times\\!{\mathcal{F}^{\prime}}^{\rho\sigma}\\!\\!+\\!\tfrac{1}{5}\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}\\!\\!\times\\!{\mathcal{C}^{\prime}}^{\alpha\beta\gamma\delta\rho\sigma}\\!\\!+4H^{2}[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\Bigr{]}$
(26) $\displaystyle\hskip
28.45274pt\times\Bigl{[}8\pi\ln(aa^{\prime})\delta^{4}(\Delta
x)\\!+\\!f_{B}(\Delta
x)\Bigr{]}-\frac{\kappa^{2}H^{2}}{2^{7}\\!\cdot\\!3\\!\cdot\\!\pi^{3}}\sum_{i=1}^{21}\Bigl{\\{}2\pi\delta
T^{i}(a,a^{\prime},\partial)$ $\displaystyle\hskip
99.58464pt\times[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]\delta^{4}(\Delta
x)-\delta
T^{i}_{A}(a,a^{\prime},\partial)[\mbox{}^{\mu\nu}D^{\rho\sigma}_{i}]f_{A}(\Delta
x)\Bigr{\\}}.\qquad$

Unlike the Ricci and Weyl contributions to expression (26), the terms
proportional to the Lichnerowicz operator are not independently conserved.
This is evident from the action of the Ward operator,

$\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\times\\![\mbox{}^{\alpha\beta}\mathcal{L}^{\rho\sigma}]=\delta^{\mu}_{~{}0}\Bigl{\\{}\tfrac{1}{2}\eta^{\rho\sigma}\partial\\!\cdot\\!\partial^{\prime}+\tfrac{1}{2}\partial^{\rho}\partial^{\sigma}$
(27) $\displaystyle\hskip
85.35826pt+\tfrac{1}{2}\eta^{\rho\sigma}\partial^{\prime}_{0}aH\\!+\\!\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}aH\\!+\\!\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}\,a^{2}H^{2}\Bigr{\\}}a^{2}{a^{\prime}}^{2}H^{2}\Delta\eta\;.\qquad$

Neither the $\delta^{4}(\Delta x)$ term nor the $f_{B}(\Delta x)$ contribution
is independently conserved, nor is their sum. Conservation only follows when
one includes the sums over the “residual” terms involving $\delta
T^{i}(a,a^{\prime},\partial)$ and $\delta T^{i}_{A}(a,a^{\prime},\partial)$.

##  4 Representing the Residue

The purpose of this section is to recognize the remaining sum in expression
(26) as tensor differential operators acting on $\delta^{4}(\Delta x)$ and
$f_{A}(\Delta x)$. We begin by summing the $\delta T^{i}$ terms to a
relatively compact form, then we do the same for the sum of the $T^{i}_{A}$
terms. This prompts the definition of two new tensor differential operators,
$[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]$ and
$[\mbox{}^{\mu\nu}\mathcal{J}^{\rho\sigma}]$, in terms of which the graviton
self-energy is given its final form. The section closes with a consideration
of conservation.

The $\delta T^{i}(a,a^{\prime},\partial)$ terms can be represented as,

$\displaystyle-2\pi\sum_{i=1}^{21}\delta
T^{i}(a,a^{\prime},\partial)[\mbox{}^{\mu\nu}D_{i}^{\rho\sigma}]\delta^{4}(\Delta
x)=-2\pi\Bigl{\\{}[3\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu(\rho}\eta^{\sigma)\nu}]\partial\\!\cdot\\!\partial^{\prime}$
(29) $\displaystyle\hskip
85.35826pt+3[{\partial^{\prime}}^{\mu}{\partial^{\prime}}^{\nu}\eta^{\rho\sigma}\\!+\\!\eta^{\mu\nu}\partial^{\rho}\partial^{\sigma}]+2{\partial^{\prime}}^{(\mu}\eta^{\nu)(\rho}\partial^{\sigma)}\Bigr{\\}}\Bigl{[}aa^{\prime}\delta^{4}(\Delta
x)\Bigr{]}$ $\displaystyle\hskip 56.9055pt-4\pi
a^{2}{a^{\prime}}^{2}H^{2}\Bigl{\\{}\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu(\rho}\eta^{\sigma)\nu}\\!-\\!2\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}\Bigr{\\}}\delta^{4}(\Delta
x)\;,\qquad$
$\displaystyle=-4\pi[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\delta^{4}(\Delta
x)-4\pi\Bigl{\\{}\eta^{\mu\nu}\eta^{\rho\sigma}\partial\\!\cdot\\!\partial^{\prime}\\!+\\!{\partial^{\prime}}^{\mu}{\partial^{\prime}}^{\nu}\eta^{\rho\sigma}\\!+\\!\eta^{\mu\nu}\partial^{\rho}\partial^{\sigma}\Bigr{\\}}\Bigl{[}aa^{\prime}\delta^{4}(\Delta
x)\Bigr{]}$ $\displaystyle\hskip
56.9055pt-\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\Bigl{\\{}\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu(\rho}\eta^{\sigma)\nu}\\!-\\!2\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}\Bigr{\\}}\partial^{4}f_{A}(\Delta
x)\;.\qquad$

When the three terms on the final line of (29) are added to the sum over
$\delta T^{i}_{A}$ we express the result as a new tensor differential
operator,

$\displaystyle-\tfrac{1}{2}[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]f_{A}(\Delta
x)=\sum_{i=1}^{21}\delta
T^{i}(a,a^{\prime},\partial)[\mbox{}^{\mu\nu}D_{i}^{\rho\sigma}]f_{A}(\Delta
x)$ (30) $\displaystyle\hskip
56.9055pt-\tfrac{1}{2}a^{2}{a^{\prime}}^{2}H^{2}\Bigl{\\{}\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu(\rho}\eta^{\sigma)\nu}\\!-\\!2\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}\Bigr{\\}}\partial^{4}f_{A}(\Delta
x)\;.\qquad$

This operator is,

$\displaystyle[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]\equiv
a^{2}{a^{\prime}}^{2}H^{2}\Bigl{\\{}\Pi^{\mu\nu}\Pi^{\rho\sigma}\\!\\!-\\!\Pi^{\mu(\rho}\Pi^{\sigma)\nu}\\!\\!+\\![\eta^{\mu\nu}\partial_{0}\\!-\\!2\delta^{(\mu}_{~{}~{}0}\partial^{\nu)}][\eta^{\rho\sigma}\partial_{0}\\!-\\!2\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}]\partial^{2}-$
(31)
$\displaystyle\Bigl{[}\eta^{\mu(\rho}\eta^{\sigma)\nu}\partial_{0}^{2}\\!\\!-\\!2\delta^{(\mu}_{~{}0}\eta^{\nu)(\rho}\partial^{\sigma)}\partial_{0}\\!\\!-\\!2\partial^{(\mu}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}0}\partial_{0}\\!\\!+\\!2\delta^{(\mu}_{~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}0}\partial^{2}\\!\\!+\\!2\delta^{(\mu}_{~{}0}\partial^{\nu)}\delta^{(\rho}_{~{}0}\partial^{\sigma)}\Bigr{]}\partial^{2}$
$\displaystyle\hskip
42.67912pt-\partial^{\mu}\partial^{\nu}{\partial^{\prime}}^{\rho}{\partial^{\prime}}^{\sigma}\\!\Bigl{[}\frac{\partial^{2}}{aa^{\prime}H^{2}}\\!-\\!2\Bigr{]}\\!+\\!2H[a\eta^{\mu\nu}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}\\!-\\!a^{\prime}\delta^{\mu}_{~{}0}\delta^{\nu}_{~{}0}\eta^{\rho\sigma}]\partial_{0}\partial^{2}\\!\Bigr{\\}}.\qquad$

We can give the graviton self-energy a relatively compact form by defining a
final tensor differential operator,

$[\mbox{}^{\mu\nu}\mathcal{J}^{\rho\sigma}]\equiv\Bigl{\\{}\eta^{\mu\nu}\eta^{\rho\sigma}\partial\\!\cdot\\!\partial^{\prime}\\!+\\!{\partial^{\prime}}^{\mu}{\partial^{\prime}}^{\nu}\eta^{\rho\sigma}\\!+\\!\eta^{\mu\nu}\partial^{\rho}\partial^{\sigma}\Bigr{\\}}aa^{\prime}\;.$
(32)

The resulting form for $[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}](x;x^{\prime})$
is,

$\displaystyle[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}]=\frac{-\kappa^{2}}{2^{10}\\!\cdot\\!3\\!\cdot\\!\pi^{3}}\Bigl{[}\tfrac{1}{3}\mathcal{F}^{\mu\nu}{\mathcal{F}^{\prime}}^{\rho\sigma}\\!\\!\\!+\\!\tfrac{1}{5}\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}~{}\mu\nu}{\mathcal{C}^{\prime}}^{\alpha\beta\gamma\delta\rho\sigma}\Bigr{]}\\!\\!\Bigl{[}8\pi\ln(aa^{\prime})\delta^{4}(\Delta
x)\\!+\\!\\!f_{B}(\Delta x)\Bigr{]}$ (33) $\displaystyle\hskip
56.9055pt-\frac{\kappa^{2}H^{2}}{2^{8}\\!\cdot\\!3\\!\cdot\\!\pi^{3}}\Bigl{\\{}[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\Bigl{[}8\pi[\ln(aa^{\prime})\\!+\\!1]\delta^{4}(\Delta
x)+f_{B}(\Delta x)\Bigr{]}$ $\displaystyle\hskip
142.26378pt+[\mbox{}^{\mu\nu}\mathcal{J}^{\rho\sigma}]8\pi\delta^{4}(\Delta
x)+[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]f_{A}(\Delta x)\Bigr{\\}}.\qquad$

Although the combination of all the “new” terms on the last two lines of
expression (33) are conserved, the only separately conserved constituent is
$[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]\delta^{4}(\Delta x)$. Its
conservation follows from relation (27). The action of the Ward operator on
the $f_{B}$ term is,

$\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\times\\![\mbox{}^{\alpha\beta}\mathcal{L}^{\rho\sigma}]f_{B}(\Delta
x)=8\pi
a^{2}{a^{\prime}}^{2}H^{2}\delta^{\mu}_{~{}0}\eta^{\rho\sigma}\partial_{0}\delta^{4}(\Delta
x)-a^{2}{a^{\prime}}^{2}H^{2}\delta^{\mu}_{~{}0}\Bigl{\\{}\partial^{\rho}\partial^{\sigma}$
(34)
$\displaystyle-2aH\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}\\!\\!+\\!2a^{2}H^{2}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}\\!+\\!\eta^{\rho\sigma}\\!\Bigl{[}aH\partial_{0}\\!-\\!2a^{\prime}H\partial_{0}\\!-\\!2aa^{\prime}H\Bigr{]}\\!\Bigr{\\}}\partial_{0}\partial^{2}\\!f_{A}(\Delta
x)\,.\qquad$

This is very largely canceled by the
$[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]f_{A}(\Delta x)$ term,

$\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\\!\times\\!\\![\mbox{}^{\alpha\beta}\mathcal{K}^{\rho\sigma}]f_{A}(\Delta
x)=-8\pi
a^{2}{a^{\prime}}^{2}\\!\Bigl{\\{}\\!H^{2}\delta^{\mu}_{~{}0}\eta^{\rho\sigma}\partial_{0}\\!\\!+\\!(\partial^{\mu}\\!+\\!aH\delta^{\mu}_{~{}0}){\partial^{\prime}}^{\rho}\\!{\partial^{\prime}}^{\sigma}\\!\\!\frac{1}{aa^{\prime}}\\!\\!\Bigr{\\}}\delta^{4}(\Delta
x)$ (35) $\displaystyle\hskip
56.9055pt+a^{2}{a^{\prime}}^{2}H^{2}\delta^{\mu}_{~{}0}\Bigl{\\{}\partial^{\rho}\partial^{\sigma}\\!-\\!2aH\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}\\!+\\!2a^{2}H^{2}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}$
$\displaystyle\hskip
113.81102pt+\eta^{\rho\sigma}\Bigl{[}aH\partial_{0}\\!-\\!2a^{\prime}H\partial_{0}\\!-\\!2aa^{\prime}H^{2}\Bigr{]}\Bigr{\\}}\partial_{0}\partial^{2}f_{A}(\Delta
x).\qquad$

The remaining contributions are proportional to $\delta^{4}(\Delta x)$,

$\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\times\\![\mbox{}^{\alpha\beta}\mathcal{J}^{\rho\sigma}]\Bigl{[}8\pi\delta^{4}(\Delta
x)\Bigr{]}=8\pi(\partial^{\mu}\\!+\\!4aH\delta^{\mu}_{~{}0})\partial^{\rho}\partial^{\sigma}\Bigl{[}aa^{\prime}\delta^{4}(\Delta
x)\Bigr{]}$ (36) $\displaystyle\hskip 28.45274pt+8\pi
H\delta^{\mu}_{~{}0}\eta^{\rho\sigma}\partial\\!\cdot\\!\partial^{\prime}\Bigl{[}a{a^{\prime}}^{2}\delta^{4}(\Delta
x)\Bigr{]}+8\pi
H^{2}\delta^{\mu}_{~{}0}\eta^{\rho\sigma}\partial_{0}^{\prime}\Bigl{[}a^{2}{a^{\prime}}^{2}\delta^{4}(\Delta
x)\Bigr{]}\;,\qquad$ $\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\times\\![\mbox{}^{\alpha\beta}\mathcal{L}^{\rho\sigma}]\Bigl{[}8\pi\ln(aa^{\prime})\delta^{4}(\Delta
x)\Bigr{]}=-8\pi
H\delta^{\mu}_{~{}0}(\eta^{\rho\sigma}\partial\\!\cdot\\!\partial^{\prime}\\!+\\!\partial^{\rho}\partial^{\sigma})[a^{3}\delta^{4}(\Delta
x)]$ (37) $\displaystyle\hskip 11.38092pt-8\pi
H^{2}\delta^{\mu}_{~{}0}\Bigl{[}\eta^{\rho\sigma}\partial^{\prime}_{0}\\!+\\!2\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}\Bigr{]}[a^{4}\delta^{4}(\Delta
x)]\\!-\\!16\pi
H^{3}\delta^{\mu}_{~{}0}\delta^{\rho}_{~{}0}\delta^{\sigma}_{~{}0}a^{5}\delta^{4}(\Delta
x)\;.\qquad$

##  5 Epilogue

On flat space background, matter contributions to the graviton self-energy
take the form (2) of two 4th order tensor differential operators acting on
scalar structure functions. One of the 4th order operators has spin zero and
the other has spin two, and each of them is transverse, so that each
contribution is conserved, no matter what the form of the scalar structure
functions. On de Sitter background (5) these operators generalize to (8) and
(12), respectively. They are based on tensor differential operators (10) and
(13), which are constructed from the linearized Ricci and Weyl tensors,
respectively. These operators are each annihilated by the Ward operator (7),
so their contributions are again conserved, no matter what structure function
they act upon. These contributions to the graviton self-energy are given in
expression (21), whose scalar structure functions agree with those of flat
space (3) and (4), except for factors of
$8\pi\ln(aa^{\prime})\delta^{4}(\Delta x)$, which derive from renormalization
(22).

Our concern in this paper is with the new contributions which arise from the
cosmological background. They appear in expression (21) as sums of functions
of the two scale factors and the derivative operator, given in Table 4,
multiplying the 21 tensor differential operators given in Table 1. Our goal
has been to represent them as a 2nd order tensor differential operator acting
on $\delta^{4}(\Delta x)$ and the functions $f_{A}(\Delta x)$ and
$f_{B}(\Delta x)$ defined in expression (18). The alternative is to represent
the new terms using a 4th order tensor differential operator constructed from
contracting two factors of (13) with constant tensors such as the spatial
Minkowski metric (17). The result is very complicated structure functions
[11], which are difficult to use in solving the effective field equations [5].

Our final form is expression (33), which is based on the Lichnerowicz bi-
tensor $[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]$, given in (25). However,
two additional tensor differential operators are required:
$[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]$, given in (31), and
$[\mbox{}^{\mu\nu}\mathcal{J}^{\rho\sigma}]$, given in (32). Each of these
tensor differential operators acts on a different combination of
$\delta^{4}(\Delta x)$, $f_{A}(\Delta x)$ and $f_{B}(\Delta x)$, and relations
(34-37) show that they must be combined so as to produce a conserved result.

Consideration of Table 4 suggests that we consider moving all the
$\partial_{0}\partial^{2}$ contributions in the $\Delta T^{i}_{A}$ column to
the $\Delta T^{i}_{B}$ column using relation (20). One would then define a new
Lichnerowicz bi-tensor to preserve the form
$[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}][8\pi\ln(aa^{\prime})\delta^{4}(\Delta
x)+f_{B}(\Delta x)]$. There would be corresponding changes in the residual
$\delta^{4}(\Delta x)$ and $f_{A}(\Delta x)$ contributions. We explore this
possibility in the Appendix. The choice seems to be between having a simple
Lichnerowicz bi-tensor, with a complicated
$[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]$, or the converse.

This form (33), with different multiples of the three contributions, must
pertain for any matter contribution to the graviton self-energy on de Sitter
background. For example, the conformal invariance of $D=4$ electrodynamics
restricts the contribution from a single loop of photons to just the Weyl term
[4],

$[\mbox{}^{\mu\nu}\Sigma^{\rho\sigma}_{\rm
EM}](x;x^{\prime})=-\frac{\kappa^{2}}{2^{8}\\!\cdot\\!5\\!\cdot\\!\pi^{3}}\,\mathcal{C}_{\alpha\beta\gamma\delta}^{~{}~{}~{}~{}\mu\nu}\\!\times\\!{\mathcal{C}^{\prime}}^{\alpha\beta\gamma\delta\rho\sigma}\Bigl{[}8\pi\ln(aa^{\prime})\delta^{4}(\Delta
x)\\!+\\!f_{B}(\Delta x)\Bigr{]}.$ (38)

It would be interesting to work out the contributions from other matter
theories, such as fermions and non-minimally coupled scalars. Also, it should
be noted that the 3rd structure function is bound to occur in the graviton
self-energy from a loop of gravitons, although there will be additional
contributions as well.

##  6 Appendix

The purpose of this appendix is to explore the consequences of using the
identity $\partial_{0}\partial^{2}f_{A}(\Delta x)=-\tfrac{1}{2}\Delta\eta
f_{B}(\Delta x)$ to simplify the $\Delta T^{i}_{A}(a,a^{\prime},\partial)$
column of Table 4. If we want the modified $\Delta
T^{i}_{B}(a,a^{\prime},\partial)$ terms to still sum to “the Lichnerowicz
operator” acting on $f_{B}(\Delta x)$ this obviously entails a redefinition of
$[\mbox{}^{\mu\nu}\mathcal{L}^{\rho\sigma}]$,

$\displaystyle\left[\mbox{}^{\mu\nu}\\!\mathcal{L}^{\rho\sigma}_{\rm
new}\right]=\frac{aa^{\prime}}{2}\Bigl{\\{}\left(\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu(\rho}\eta^{\sigma)\nu}\right)\left[\partial\\!\cdot\\!\partial^{\prime}\\!-\\!2aa^{\prime}H^{2}\right]\\!+\\!\eta^{\mu\nu}[\partial^{\rho}\partial^{\sigma}\\!-\\!2a^{\prime}H\delta^{(\rho}_{~{}0}\partial^{\sigma)}]$
(39) $\displaystyle\hskip
28.45274pt+\eta^{\rho\sigma}[\partial^{\prime\mu}\partial^{\prime\nu}\\!-\\!2aH\delta^{(\mu}_{~{}~{}0}\partial^{\prime\nu)}]+2\Bigl{[}\partial^{\prime(\mu}\eta^{\nu)(\rho}\partial^{\sigma)}\\!-\\!aH\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\partial^{\sigma)}$
$\displaystyle\hskip
136.5733pt-a^{\prime}H\partial^{\prime(\mu}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}\\!+\\!3aa^{\prime}H^{2}\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}\Bigr{]}\Bigr{\\}}.\qquad$

There is no change in $[\mbox{}^{\mu\nu}\mathcal{J}^{\rho\sigma}]$ from (32),
but $[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}]$ is simplified to,

$\displaystyle[\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}_{\rm
new}]=a^{2}{a^{\prime}}^{2}H^{2}\Bigl{\\{}\Pi^{\mu\nu}\Pi^{\rho\sigma}\\!-\\!\Pi^{\mu(\rho}\Pi^{\sigma)\nu}\\!-\\!\partial^{\mu}\partial^{\nu}{\partial^{\prime}}^{\rho}{\partial^{\prime}}^{\sigma}\Bigl{(}\tfrac{\partial^{2}}{aa^{\prime}H^{2}}\\!-\\!2\Bigr{)}$
(40) $\displaystyle\hskip
54.06006pt-2\delta^{(\mu}_{~{}~{}0}\partial^{\nu)}\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}\partial^{2}\\!-\\![\eta^{\mu\nu}\eta^{\rho\sigma}\\!-\\!\eta^{\mu(\rho}\eta^{\sigma)\nu}\\!+\\!2\delta^{(\mu}_{~{}~{}0}\eta^{\nu)(\rho}\delta^{\sigma)}_{~{}~{}0}]\partial^{4}\Bigr{\\}}.\qquad$

The full result (33) then preserves its form.

We conclude this appendix by showing how conservation works with the new
operators. Because $[\mbox{}^{\mu\nu}\mathcal{J}^{\rho\sigma}]$ is unaffected
there is of course no change in relation (36). The analog of (34) is ,

$\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\times\\![\mbox{}^{\alpha\beta}\mathcal{L}^{\rho\sigma}_{\rm
new}]f_{B}(\Delta x)=8\pi
a^{2}{a^{\prime}}^{2}H^{2}\eta^{\mu(\rho}\delta^{\sigma)}_{~{}~{}0}\partial_{0}\delta^{4}(\Delta
x)$ (41) $\displaystyle\hskip
28.45274pt-2a^{2}{a^{\prime}}^{2}H^{2}\Bigl{\\{}\tfrac{1}{2}\partial^{\mu}\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}\\!+\\!aH\partial^{\mu}\eta^{\rho\sigma}\\!-\\!aH\eta^{\mu(\rho}\partial^{\sigma)}\Bigr{\\}}\partial_{0}\partial^{2}f_{A}(\Delta
x)\;.\qquad$

The $f_{A}(\Delta x)$ terms of this are canceled by the analog of (35),

$\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\times\\![\mbox{}^{\mu\nu}\mathcal{K}^{\rho\sigma}_{\rm
new}]f_{A}(\Delta
x)=a^{2}{a^{\prime}}^{2}H^{2}\Bigl{\\{}\partial^{\mu}\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}\\!+\\!2aH\partial^{\mu}\eta^{\rho\sigma}\\!-\\!2aH\eta^{\mu(\rho}\partial^{\sigma)}\Bigr{\\}}$
(42) $\displaystyle\times\partial_{0}\partial^{2}f_{A}(\Delta x)-8\pi
aa^{\prime}\Bigl{\\{}(\partial^{\mu}\\!+\\!2aH\delta^{\mu}_{~{}0})\partial^{\rho}\partial^{\sigma}\\!-\\!a^{\prime}H(\partial^{\mu}\\!+\\!2aH\delta^{\mu}_{~{}0})\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}$
$\displaystyle+aa^{\prime}H^{2}(\partial^{\mu}\\!\\!+\\!aH\delta^{\mu}_{~{}0})\eta^{\rho\sigma}\\!\\!\\!-\\!aa^{\prime}H^{2}\eta^{\mu(\rho}\partial^{\sigma)}\\!\\!+\\!aa^{\prime}H^{2}(\partial_{0}\\!+\\!2aH)\eta^{\mu(\rho}\delta^{\rho)}_{~{}~{}0}\\!\Bigr{\\}}\delta^{4}(\Delta
x).\qquad$

The $\delta^{4}(\Delta x)$ terms are canceled by (36) and the analog of (37),

$\displaystyle
W^{\mu}_{~{}\alpha\beta}\\!\\!\times\\!\\![\mbox{}^{\alpha\beta}\\!\mathcal{L}^{\rho\sigma}_{\rm
new}]\\!\Bigl{[}8\pi\ln(aa^{\prime})\delta^{4}(\Delta
x)\\!\Bigr{]}\\!\\!=\\!8\pi
aa^{\prime}H\Bigl{\\{}\\!aa^{\prime}H\partial^{\mu}\eta^{\rho\sigma}\\!\\!\\!-\\!aa^{\prime}H\eta^{\mu(\rho}\partial^{\sigma)}\\!\\!-\\!a\delta^{\mu}_{0}\partial^{\rho}\partial^{\sigma}$
(43) $\displaystyle\hskip
14.22636pt+2a^{2}a^{\prime}H^{2}\eta^{\mu(\rho}\delta^{\sigma)}_{~{}0}\\!\\!+\\!2aa^{\prime}H\delta^{\mu}_{~{}0}\delta^{(\rho}_{~{}~{}0}\partial^{\sigma)}\\!\\!+\\!\delta^{\mu}_{~{}0}\eta^{\rho\sigma}a^{\prime}(\partial^{2}\\!\\!+\\!3aa^{\prime}H^{2})\\!\Bigr{\\}}\delta^{4}(\Delta
x).\qquad$

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