contribution to the renormalized self-mass is,

$\displaystyle-iM^{2}_{\rm
ctm}(x;x^{\prime})=-i\alpha_{1}\mathcal{D}_{x}\mathcal{D}_{x^{\prime}}\Bigl{[}\frac{\delta^{D}(x\\!-\\!x^{\prime})}{a_{x}^{D}}\Bigr{]}$
(241) $\displaystyle\hskip
93.89418pt+i\alpha_{2}R\mathcal{D}_{x}\delta^{D}(x\\!-\\!x^{\prime})-i\alpha_{3}R\partial_{0}\Bigl{[}a^{D-2}\partial_{0}\delta^{D}(x\\!-\\!x^{\prime})\Bigr{]}.\qquad$

From relations (228), (236) and (239) we see that all divergences can be
expressed in terms of two constants,

$\displaystyle K$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\frac{\mu^{D-4}\Gamma(\frac{D}{2}\\!-\\!1)}{8(D\\!-\\!3)(D\\!-\\!4)\pi^{\frac{D}{2}}}=\frac{\mu^{D-4}}{8\pi^{2}}\\!\times\\!\frac{1}{D\\!-\\!4}+O\Bigl{(}(D\\!-\\!4)^{0}\Bigr{)}\;,\qquad$
(242) $\displaystyle\mathcal{K}$ $\displaystyle\\!\\!\\!\equiv\\!\\!\\!$
$\displaystyle\frac{\mu^{D-4}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{\Gamma(\frac{D}{2})}\\!\times\\!\tfrac{\pi}{2}{\rm
cot}(\tfrac{D\pi}{2})=\frac{\mu^{D-4}}{8\pi^{2}}\\!\times\\!\frac{1}{D\\!-\\!4}+O\Bigl{(}(D\\!-\\!4)^{0}\Bigr{)}\;.\qquad$
(243)

The counterterms required to renormalize the original, gauge-dependent self-
mass are $(\alpha_{1})_{0}=0$ and [13],

$R(\alpha_{2})_{0}=-\kappa^{2}H^{2}\\!\times\\!(D\\!-\\!2)K\quad,\quad
R(\alpha_{3})_{0}=-\kappa^{2}H^{2}\Bigl{[}2D\mathcal{K}-(D\\!-\\!2)(D\\!+\\!4)K\Bigr{]}\;.$
(244)

From expressions (226-227) we see that the $-iV_{1b,c}$ diagrams have
$(\alpha_{1})_{1}=0=(\alpha_{3})_{1}$ and,

$R(\alpha_{2})_{1}=\kappa^{2}H^{2}\\!\times\\!D(D\\!-\\!1)\mathcal{K}\;.$
(245)

The $-iV_{\overline{\overline{4}}}$ diagrams (220), (221) and (224) imply
$(\alpha_{3})_{4}=0$ and,

$(\alpha_{1})_{4}=-\tfrac{\kappa^{2}}{3}(\tfrac{13D-46}{D-2})K\qquad,\qquad
R(\alpha_{2})_{4}=\kappa^{2}H^{2}\Bigl{[}-2G_{1}(D)+(D\\!-\\!2)\Bigr{]}K\;,$
(246)

where $G_{1}(D)$ was defined in expression (222). Finally, the
$-iV_{\overline{5}}$ diagrams (155-156) imply $(\alpha_{3})_{5}=0$ and,

$(\alpha_{1})_{5}=-\kappa^{2}K\qquad,\qquad
R(\alpha_{2})_{5}=2\kappa^{2}H^{2}\mathcal{K}\;.$ (247)

Summing the contributions (244-247) gives,

$\displaystyle\alpha_{1}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{2\kappa^{2}}{8\pi^{2}}\\!\times\\!\frac{\mu^{D-4}}{D\\!-\\!4}+O(1)\;,\qquad$
(248) $\displaystyle R\alpha_{2}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{15\kappa^{2}H^{2}}{8\pi^{2}}\\!\times\\!\frac{\mu^{D-4}}{D\\!-\\!4}+O(1)\;,\qquad$
(249) $\displaystyle R\alpha_{3}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle+\frac{8\kappa^{2}H^{2}}{8\pi^{2}}\\!\times\\!\frac{\mu^{D-4}}{D\\!-\\!4}+O(1)\;.\qquad$
(250)

##  6 Conclusions

No one disputes that an epoch of primordial inflation would produce a vast
ensemble of cosmological scale gravitons [2]. At some level these quanta must
interact with themselves and with other particles to induce changes in
kinematics and in long range forces. A fascinating aspect of these changes is
that the continual production of infrared gravitons causes them to grow
without bound in time and sometimes also in space. During a very long period
of inflation, this growth can overwhelm the small coupling constants to
produce significant effects which might persist to the present day [50].
However, explicit computations [9, 10, 11, 12, 13, 14] have been criticized as
potentially unphysical owing to gauge dependence [23, 24, 25, 26].

Opinions differ on how to deal with gauge dependence. Some people favor
replacing gauge-dependent Green’s functions with expectation values of
generally coordinate invariant operators [51, 52, 53, 54, 55, 56, 57, 35, 58,
59, 60]. We have instead pursued the approach of combining bits and pieces of
gauge-dependent Green’s functions so as to achieve a gauge-independent result
based on the flat space S-matrix.

In our view the usual effective field equations are gauge-dependent because
they ignore quantum gravitational correlations with the source, which excites
the effective field, and with the observer which detects it. We restore these
correlations by first forming the position-space amplitudes that represent
quantum gravitational corrections to the $t$-channel scattering between two
massive fields by the exchange of the light field under study (in this case, a
massless, minimally coupled scalar). These amplitudes involve 2-point, 3-point
and 4-point correlators. Our second step is to simplify the 3-point and
4-point correlators to 2-point form using a series of relations for extracting
the $t$-channel contributions, first derived in flat space by Donoghue and
collaborators [43, 3, 44, 45]. Our final step is to insert exchange
propagators of the light field so that we can regard each of the 2-point
contributions as a correction to the 1PI 2-point function of the light field.
This has been explicitly carried out on flat space background, and verified to
be independent of the gauge, for 1-graviton loop corrections to a massless,
minimally coupled scalar [41] and for 1-graviton loop corrections to
electromagnetism [61]. The current work generalizes this procedure to de
Sitter background for 1-graviton loop corrections to a massless, minimally
coupled scalar.

Our de Sitter analysis has been enormously simplified by correspondence with
its flat space analog [41]. In particular, we used the very same Lagrangian
(3), and the very same five diagrams (Figures 4-8) which can potentially
enhance $t$-channel scattering between massive sources. The de Sitter
generalizations (35-36) and (39-40) of the Donoghue Identities were motivated
by minimally preserving the transformation properties of the flat space
results. One new feature of our analysis is the recognition in section 4 of
two classes of the five diagrams which can be consolidated to give two
residual amplitudes,
$-iV_{\overline{\overline{4}}}(x;x^{\prime};y;y^{\prime})$ — given in
expressions (100-103) — and $-iV_{\overline{5}}(x;x^{\prime};y;y^{\prime})$ —
given in expressions (104-107). This consolidation is exact, independent of
the Donoghue Identities, and valid for any gauge on any cosmological
background. Of course the final reduction of these diagrams to 2-point form in
sections 5.2 and 5.3 does require a specific gauge on de Sitter and the
Donoghue Identities. Our final primitive results (155-156), (220), (221),
(224) and (226-227) are expressed in terms of the four classes of 2-point
diagrams defined in Figure 2. Renormalization is carried out in section 5.4
using the same three counterterms (240) of the original, gauge-dependent
computation [13]. Our final results for the gauge-independent coefficients of
these counterterms are given in equations (248-250).

Of course the reason for purging the scalar effective field equation of gauge
dependence is to confirm or refute the reality of the large logarithmic
correction that was found (using the simplest gauge) to the scalar exchange
potential [13]. Recall the relation that was found in the gauge-dependent
calculation between large logarithms in the exchange potential and the
curvature-dependent field strength renormalization
$-\frac{1}{2}\alpha_{2}R\partial_{\mu}\phi\partial_{\nu}\phi
g^{\mu\nu}\sqrt{-g}$ [13]. Because the renormalized, gauge invariant self-mass
has the same functional form as its gauge-dependent ancestor, this relation
should continue to pertain. That means we can read off the gauge-independent
logarithm from our result (249) for the coefficient $\alpha_{2}R$. The fact
that it is nonzero contradicts the hypothesis that graviton-induced logarithms
are a gauge artifact.

We stress the importance of making a direct check of gauge dependence by re-
doing the computation in the de Sitter generalization [62] of the same
2-parameter family of gauges which was employed to check the flat space result
[41]. One reason is that there may be diagrams, such as external line
corrections, which do not enhance $t$-channel exchange in flat space but may
do so in de Sitter. Another reason is that we know of no way to derive the de
Sitter Donoghue Identities (35-36) and (39-40). Their flat space antecedents
could be (and were) derived [43, 3, 44, 45] by computing the scattering
amplitudes in momentum space and then extracting the leading contributions for
small $t$. What we did is to first express the same relations in flat position
space (which is exact) and then generalize them to de Sitter using minimal
coupling (which is a guess). We will not feel completely confident until the
gauge independence of the final result has been confirmed.

Acknowledgements

DG was supported by the European Union and the Czech Ministry of Education,
Youth and Sports (Project: MSCA Fellowship CZ FZU I — CZ.02.01.01/00/22
010/0002906). SPM was supported by Taiwan NSTC grants 111-2112-M-006-038 and
112-2112-M-006-017. TP was supported by the D-ITP consortium, a program of the
Netherlands Organization for Scientific Research (NWO) that is funded by the
Dutch Ministry of Education, Culture and Science (OCW). RPW was supported by
NSF grant PHY-2207514 and by the Institute for Fundamental Theory at the U. of
Florida.

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