i}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}\frac{1}{\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|}\frac{1}{\ell!}\mathcal{Q}_{D,STF}^{\ast<i_{1}i_{2}\ldots
i_{\ell}>}\mathcal{Q}^{A,STF}_{<i_{1}i_{2}\ldots i_{\ell}>}\ .\end{split}$
(106)

In the first line, we have quoted the answer in terms of the spherical
multipole moments for comparison. The multipole action above could also be
derived entirely by using Cartesian STF harmonics from the very beginning (See
Ross:2012fc  for a detailed derivation). Given the absence of Cartesian
coordinates valid everywhere on the static patch, we will employ a judicious
mix of spherical harmonic and STF harmonic expansions to compute the influence
phase. The flat spacetime derivation we have given here closely mimics the
strategy we will eventually use for dS.

Let us conclude this flat spacetime discussion by commenting on the case where
$d$ is even and $\nu\in\mathbb{Z}$. We will tackle this case by a dimensional
regularisation via analytic continuation in $\nu$. From our discussion of
multipole expansion, it is clear that the time reversal even terms in Eq.(103)
are the same for any $\nu$ and can be counter-termed away similarly.

The terms in Eq.(100), on the other hand, get multiplied by a factor of
$(1+i\cot\pi\nu)$ for a general $\nu$: this can be seen, e.g., in Eq.(96). The
$\cot\pi\nu$ factor leads to novel divergences as $\nu$ approaches an integer,
necessitating further counter-terms.

To compute the counter-terms as $\nu\to n\in\mathbb{Z}$, we need the following
expansion:

$(1+i\cot\pi\nu)\frac{2\pi
i}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2H}\right)^{2\nu}=\frac{1}{\Gamma(n)^{2}}\left(\frac{\omega}{2H}\right)^{2n}\left\\{\frac{2}{\nu-n}-4\psi^{(0)}(n)+\ln\left(\frac{\omega}{2H}\right)^{4}+O(\nu-n)\right\\}\
.$ (107)

Here $H$ is the characteristic scale for dimensional regularisation and
$\psi^{(0)}(x)\equiv\frac{d}{dx}\ln\Gamma(x)$ is the di-gamma function. Using
a version of modified minimal subtraction, we counter-term away the first two
terms inside the bracket of RHS. Thus, the influence phase due to radiation
reaction for even spatial dimensions is

$\begin{split}S_{RR}^{\text{Even
$d$}}&=\sum_{\ell}\int\frac{d\omega}{2\pi}\frac{1}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}\ln\left(\frac{\omega^{4}}{H^{4}}\right)\frac{1}{\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|}\frac{1}{\ell!}\mathcal{Q}_{D,STF}^{\ast<i_{1}i_{2}\ldots
i_{\ell}>}\mathcal{Q}^{A,STF}_{<i_{1}i_{2}\ldots i_{\ell}>}\ ,\end{split}$
(108)

where we have reset $n$ again everywhere to the variable $\nu$. What we have
here is a classical renormalisation group running of the multipole couplings
present in the world line action, i.e., an RGE induced by the classical
radiation reaction. Such classical RGE is, in fact, common in many radiation
reaction problems (See e.g. discussions in Goldberger:2004jt ;
Goldberger:2007hy ; Foffa:2011np ; Birnholtz:2013nta ). We will see later how
this non-local influence phase gets further modified in dS spacetime.

##  Appendix B Designer scalar in dS : Green functions, regularisation and
renormalistion

In this appendix, we aim to describe the scalar Green functions in dS
spacetime in some amount of detail. Our focus will be on a point-like observer
sitting on the south pole, and our Green functions are all hence ‘boundary-to-
bulk’ with the boundary being the world line at the south pole. The point-like
nature necessitates a careful discussion of regularisation, counter-terms
etc.: our discussion will closely parallel the flat spacetime discussion in
the previous appendix as well as the dS discussion in Lopez-Ortega:2006aal ;
Anninos:2011af . We will also confine ourselves to a single copy of the static
patch in this appendix, relegating the applications to dS-SK to the next
appendix.

We will work with outgoing Eddington Finkelstein(EF)
coordinatesSpradlin:2001pw  describing the static patch of dS spacetime dSd+1.
This spacetime is a solution of the Einstein equations with a positive
cosmological constant

$\Lambda=\frac{1}{2}d(d-1)\ .$ (109)

We have chosen units where the Hubble constant is unity. The spacetime metric
is

$ds^{2}=-2\ du\ dr-(1-r^{2})\ du^{2}+r^{2}d\Omega_{d-1}^{2}\ .$ (110)

Here $d\Omega_{d-1}^{2}$ denotes the metric on a unit $\mathbb{S}^{d-1}$. The
outgoing Eddington Finkelstein time $u$ is related to the more commonly used
time $t$ via $u=t-r_{\ast}$ where $r_{\ast}$ is the tortoise coordinate
defined via

$r_{\ast}\equiv-i\pi\zeta\equiv\int_{0}^{r}\frac{d\rho}{1-\rho^{2}}=\frac{1}{2}\ln\left(\frac{1+r}{1-r}\right)\
.$ (111)

The radial coordinate $r$ is centred around a static observer sitting at
$r=0$. We will mostly work with the frequency domain where the time dependence
of fields202020We note a slight inconsistency in our definitions when compared
to definitions in the appendix A. In appendix A, we fourier-transformed with
respect to standard time slices, whereas here in $dS$ we are fourier-
transforming with respect to outgoing EF time $u$. Since in flat spacetime
$u=t-r$, this means that all the flat space radial functions in appendix A
should be multiplied with a pre-factor of $e^{-i\omega r}$ before they can be
compared against the dS results described here. is taken to be $\sim
e^{-i\omega u}$. Further, we will decompose everything into appropriate
spherical harmonics on $\mathbb{S}^{d-1}$. The spherical harmonics are
labelled by the eigenvalue of the sphere Laplacian
$\nabla^{2}_{\mathbb{S}^{d-1}}$ which is $-\ell(\ell+d-2)$.

As described in the main text, we will consider a class of _designer_ scalar
systems in dS with an action

$S=-\frac{1}{2}\int d^{d+1}x\sqrt{-g}\
r^{\mathcal{N}+1-d}\left\\{\partial^{\mu}\Phi_{\mathcal{N}}\,\partial_{\mu}\Phi_{\mathcal{N}}+\frac{\Phi_{\mathcal{N}}^{2}}{4r^{2}}\left[(d+\mathcal{N}-3)(d-\mathcal{N}-1)-r^{2}\left(4\mu^{2}-(\mathcal{N}+1)^{2}\right)\right]\right\\}$
(112)

After we strip out the harmonic dependence in time/angles, the above action
results in a radial ODE of the form

$\begin{split}&\frac{1}{r^{\mathcal{N}}}D_{+}[r^{\mathcal{N}}D_{+}\varphi_{{}_{\mathcal{N}}}]+\omega^{2}\varphi_{{}_{\mathcal{N}}}\\\
&\qquad+\frac{1-r^{2}}{4r^{2}}\Bigl{\\{}(\mathcal{N}-1)^{2}-(d+2\ell-2)^{2}+[4\mu^{2}-(\mathcal{N}+1)^{2}]r^{2}\Bigr{\\}}\varphi_{{}_{\mathcal{N}}}=0\
.\end{split}$ (113)

Here $\varphi_{{}_{\mathcal{N}}}(r,\omega,\mathbb{L})$ is the radial part of
the field, the derivative operators $D_{\pm}\equiv(1-r^{2})\partial_{r}\pm
i\omega$, and the equation depends on the parameters
$\\{\mu,\mathcal{N},\ell\\}$ whose physical interpretation will be clear
momentarily.

The combination $(\mathcal{N}+1)^{2}-4\mu^{2}$ can be interpreted as a mass
term $4m^{2}$ for the scalar in Hubble units. The exponent $\mathcal{N}$
describes the auxiliary radial varying dilaton mentioned at the beginning of
this appendix. The index $\ell$ is associated with the eigenvalue of the
sphere laplacian. The expressions involved simplify considerably if we use,
instead of $\ell$, the following parameter:

$\displaystyle\begin{split}\nu&\equiv\frac{d}{2}+\ell-1\ .\end{split}$ (114)

For example, in terms of $\nu$, the eigenvalue of the sphere laplacian becomes
$(\frac{d}{2}-1)^{2}-\nu^{2}$. Since we will be concerned with the cases where
$d>2$ and $\ell\geq 0$, $\nu$ is a positive number. We can then rewrite the
above ODE as

$\begin{split}&\frac{1}{r^{\mathcal{N}}}D_{+}[r^{\mathcal{N}}D_{+}\varphi_{{}_{\mathcal{N}}}]+\omega^{2}\varphi_{{}_{\mathcal{N}}}\\\
&\qquad+\frac{1-r^{2}}{4r^{2}}\Bigl{\\{}(\mathcal{N}-1)^{2}-4\nu^{2}+[4\mu^{2}-(\mathcal{N}+1)^{2}]r^{2}\Bigr{\\}}\varphi_{{}_{\mathcal{N}}}=0\
.\end{split}$ (115)

It is instructive to rewrite the above ODE in terms of a new field $\psi\equiv
r^{\frac{\mathcal{N}}{2}}\varphi_{{}_{\mathcal{N}}}$ as

$\begin{split}(D_{+}^{2}+\omega^{2})\psi+\frac{1-r^{2}}{4r^{2}}\Bigl{\\{}1-4\nu^{2}+[4\mu^{2}-1]r^{2}\Bigr{\\}}\psi=0\
.\end{split}$ (116)

The absence of $\mathcal{N}$ in this ODE shows that $\mathcal{N}$ merely
controls the overall pre-factor. We also note a symmetry under
$\nu\mapsto-\nu$ and $\mu\mapsto-\mu$: either of these sign changes should map
one solution to the other.

###  B.1 Outgoing Green function

The above second-order radial ODE can be exactly solved in terms of
hypergeometric functions. The worldline to bulk outgoing Green function is
given byBunch:1978yq ; Lopez-Ortega:2006aal ; Anninos:2011af

$\begin{split}G^{\text{Out}}_{\mathcal{N}}(r,\omega,\mathbb{L})&=r^{\nu-\frac{\mathcal{N}}{2}}(1+r)^{-i\omega}\\\
&\quad\times\frac{\Gamma\left(\frac{1+\nu-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu-i\omega}{2}\right)}{\Gamma(1-i\omega)\Gamma\left(1+\nu\right)}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1-i\omega;1-r^{2}\right]\
.\end{split}$ (117)

Here we have fixed the overall normalisation by an appropriate boundary
condition to be described below. We will devote this subsection to a detailed
study of the above Green function.

We remind the reader that the hypergeometric function always has a nice series
expansion around the point where its last argument vanishes. It then follows
that the above solution is manifestly regular at the future horizon $r=1$
without any branch cuts or poles. An alternate form for the same function that
emphasises the small $r$ behaviour near the observer’s worldline is

$\begin{split}G^{\text{Out}}_{\mathcal{N}}&=r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}\\\
&\times\Bigl{\\{}{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\Bigr{.}\\\
&\Bigl{.}\quad\qquad-(1+i\cot\nu\pi)\widehat{K}_{\text{Out}}\frac{r^{2\nu}}{2\nu}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\Bigr{\\}}\
.\end{split}$ (118)

Here $\widehat{K}_{\text{Out}}$ is the worldline retarded Green function given
by the expressionLopez-Ortega:2006aal ; Anninos:2011af .

$\begin{split}\widehat{K}_{\text{Out}}(\omega,\ell)&\equiv
2\frac{\Gamma\left(\frac{1+\nu-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu-i\omega}{2}\right)\Gamma\left(1-\nu\right)}{\Gamma\left(\frac{1-\nu+\mu-i\omega}{2}\right)\Gamma\left(\frac{1-\nu-\mu-i\omega}{2}\right)\Gamma\left(\nu\right)(1+i\cot\nu\pi)}\\\
&=-e^{i\nu\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\frac{\Gamma\left(\frac{1+\nu-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu-i\omega}{2}\right)}{\Gamma\left(\frac{1-\nu+\mu-i\omega}{2}\right)\Gamma\left(\frac{1-\nu-\mu-i\omega}{2}\right)}\
.\end{split}$ (119)

The reason for choosing the normalisation of $\widehat{K}_{\text{Out}}$ this
way will become clear eventually. The above equation is the dS analogue of the
Hankel function decomposition into Neumann and Bessel functions. As in
Eq.(91), when $d$ is even and $\nu$ is an integer, the above expression should
be understood as a limit, with the $\cot\nu\pi$ divergence exactly cancelling
the divergence in the first term of Eq.(118).

The hypergeometric identity used for the above decomposition is

$\begin{split}&\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)\Gamma(a+b-c)}z^{a+b-c}{}_{2}F_{1}(a,b;c;1-z)\\\
&={}_{2}F_{1}(c-a,c-b;1+c-a-b;z)\\\
&\quad+z^{a+b-c}\frac{\Gamma(a)\Gamma(b)}{\Gamma(c)\Gamma(c-a-b)}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(c-a)\Gamma(c-b)}{}_{2}F_{1}(a,b;a+b-c+1;z)\
,\end{split}$ (120)

where we have taken

$\begin{split}a=\frac{1+\nu-\mu-i\omega}{2}\ ,\ b=\frac{1+\nu+\mu-i\omega}{2}\
,\ c=1-i\omega\ ,\ z=r^{2}\ .\end{split}$ (121)

In all these identities, we take the branch cuts of hypergeometric functions
as well as $(1+r)^{-i\omega}$ to be outside the open unit disk in the complex
$r$ plane. Thus, all these functions are analytic within the open static patch
and in turn, on the dS-SK contour.

With this new form for the outgoing Green function, it is straightforward to
obtain a near-origin expansion to all orders. The explicit expressions are
given by

$\begin{split}&{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\\\
&=\sum_{k=0}^{\infty}\frac{r^{2k}}{(2k)!}\frac{(\nu-\mu-i\omega-1+2k)!!}{(\nu-\mu-i\omega+1)!!}\frac{(\nu+\mu-i\omega-1+2k)!!}{(\nu+\mu-i\omega-1)!!}\frac{(2\nu)!!(2k-1)!!}{(2\nu+2k)!!}\
,\end{split}$ (122)

as well as

$\begin{split}&{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\\\
&=\sum_{k=0}^{\infty}\frac{(-r^{2})^{k}}{(2k)!}\frac{(\nu-\mu+i\omega-1)!!}{(\nu-\mu+i\omega-1-2k)!!}\frac{(\nu+\mu+i\omega-1)!!}{(\nu+\mu+i\omega-1-2k)!!}\frac{(2\nu-2-2k)!!(2k-1)!!}{(2\nu-2)!!}\
.\end{split}$ (123)

The second expansion can be interpreted literally only for $d$ odd (i.e., when
$\nu\in\mathbb{Z}+\frac{1}{2}$). For $d$ even, the above expansion (and most
of the discussion below) should be understood in a dimensionally regularised
sense.

The near-origin form of the outgoing Green function shows the normalisation

$\begin{split}\lim_{r\to
0}r^{\nu+\frac{\mathcal{N}-1}{2}}G^{\text{Out}}_{\mathcal{N}}&=1\
.\end{split}$ (124)

This condition can be thought of as the dS analogue of the condition on the
AdS boundary-to-bulk Green function. As in that case, the above condition
along with outgoing property/analyticity at the future horizon uniquely
determines $G^{\text{Out}}_{\mathcal{N}}$. Extending this analogy to AdS, we
can roughly read off the retarded worldline Green function
$\widehat{K}_{\text{Out}}$ by looking at the ratio of coefficients of the sub-
dominant solution to the dominant solution in the outgoing solution
$G^{\text{Out}}_{\mathcal{N}}$. This is essentially the Son-Starinets
prescriptionSon:2002sd  of AdS/CFT adapted to the present dS context. Such
analogies have been noted before in Anninos:2011af : our aim here is to give a
more systematic derivation of these statements, taking into account the
subtleties associated with divergences, regularisation, finite size effects,
etc.

To this end, let us begin with a physical interpretation of the outgoing Green
function $G^{\text{Out}}_{\mathcal{N}}$. If we are given that
$\varphi_{{}_{\mathcal{N}}}$ behaves at small $r$ near the worldline as

$\begin{split}\varphi_{{}_{\mathcal{N}}}(r,\omega,\mathbb{L})=\frac{\mathcal{J}(\omega,\mathbb{L})}{r^{\nu+\frac{1}{2}(\mathcal{N}-1)}}+\ldots\
,\end{split}$ (125)

where $\mathbb{S}_{\mathbb{L}}$ is a spherical harmonic on $\mathbb{S}^{d-2}$,
we then have a unique outgoing solution

$\varphi_{{}_{\mathcal{N}}}(r,\omega,\mathbb{L})=G^{\text{Out}}_{\mathcal{N}}(r,\omega,\mathbb{L})\mathcal{J}(\omega,\mathbb{L})$

describing the field that is radiated out of the worldline. This is the dS
analogue of the outgoing Hankel Green function in flat space.212121More
precisely, in outgoing EF coordinates the corresponding Green functions in
flat space are the outgoing Hankel Green functions given in Eq.(91) multiplied
with a prefactor of $e^{-i\omega r}$. See footnote 20 for an explanation for
this pre-factor.

The alternate form we have written down above in Eq.(118) is then the dS
version of the familiar statement222222We review this statement, for the
benefit of the reader, around Eq.(91). that the outgoing Hankel Green function
can be written as the sum of a Neumann Green function (which diverges near the
origin) and a Bessel J function (which is regular at the origin). Such a
decomposition of the outgoing Green function into a singular Green function
and a regular solution is a first step in Dirac’s approach to the self-
forceDirac:1938nz  (the curved space version is sometimes also termed as the
Detweiler-Whiting decompositionDetweiler:2002mi ). We will later show in
appendixC.5 that our answer matches in dS4 with the regular part quoted in
Burko:2002ge ; Poisson:2011nh  using the rules of Deitweiler-Whiting
decomposition.

###  B.2 Renormalised conjugate field and $\boldsymbol{K_{\text{Out}}}$

We will now turn to the question of deriving the worldline Green function
$K_{\text{Out}}$ from the outgoing Green function
$G^{\text{Out}}_{\mathcal{N}}$. As we will describe in detail below, the
physics here is that of radiation reaction and the main subtlety is how to
deal with divergences. Our main strategy here will be to define a renormalised
conjugate field which reduces to $K_{\text{Out}}$ near the source worldline.
The idea here is philosophically similar to other radiation reaction
computations in the literatureDetweiler:2002mi ; Goldberger:2004jt ;
Goldberger:2007hy ; Porto:2016pyg ; Levi:2018nxp ; Barack:2018yvs  as well as
the counter-term subtraction in AdS/CFTBalasubramanian:1999re . The
implementation is however sufficiently different that we provide a detailed
analysis below.

The radial ODE Eq.(115) can be derived by extremising the action

$\begin{split}S&=-\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint\frac{r^{\mathcal{N}}dr}{1-r^{2}}\Bigl{[}(D_{+}\varphi_{{}_{\mathcal{N}}})^{*}D_{+}\varphi_{{}_{\mathcal{N}}}-\omega^{2}\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}\Bigr{.}\\\
&\Bigl{.}\qquad-\frac{1-r^{2}}{4r^{2}}\Bigl{\\{}(\mathcal{N}-1)^{2}-4\nu^{2}+[4\mu^{2}-(\mathcal{N}+1)^{2}]r^{2}\Bigr{\\}}\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}\Bigr{]}+S_{ct}\
.\end{split}$ (126)

Here $S_{ct}$ denotes the counter-term action to be determined later. The
integration over $r$ ranges over the regulated dS-SK contour (clockwise from
the right static patch to the left static patch) and, in addition, we have
indicated an integration over all frequencies and a sum over spherical
harmonics. The reality condition in the Fourier domain takes the form

$\begin{split}\varphi_{{}_{\mathcal{N}}}^{*}(r,w,\ell,\vec{m})=\varphi_{{}_{\mathcal{N}}}(r,-\omega,\ell,-\vec{m})\
.\end{split}$ (127)

Here $\vec{m}$ denotes the additional labels appearing in the spherical
harmonic decomposition.

The canonical conjugate field for radial evolution is obtained by varying the
above action with respect to $\partial_{r}\varphi_{{}_{\mathcal{N}}}^{*}$
which yields $-r^{\mathcal{N}}D_{+}\varphi_{{}_{\mathcal{N}}}$ after we take
into account the fact that $\varphi_{{}_{\mathcal{N}}}^{*}$ and
$\varphi_{{}_{\mathcal{N}}}$ are related by the reality condition quoted
above. The minus sign in the canonical conjugate is because we are looking at
evolution along a space-like direction.

Taking into account the powers of $r$ multiplying the multipole moment
$\mathcal{J}$ in Eq.(125), the canonical conjugate of $\mathcal{J}$ should be
defined with the opposite power, viz., we should consider instead

$\begin{split}-r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\left[r^{\mathcal{N}}D_{+}\varphi_{{}_{\mathcal{N}}}\right]\
.\end{split}$ (128)

The canonical conjugate field of the radial evolution at the two regulated
boundaries is given by evaluating the above expressions at $r=r_{c}\pm
i\varepsilon$. Naively the $r_{c}\to 0$ limit should then yield the required
canonical conjugate that couples to the right/left point multipole source.
This limit however does not work: on a generic solution, the $r_{c}\to 0$
limit is beset with divergences. Appropriate counter-terms need to be added to
the above bare expression before a sensible $r_{c}\to 0$ limit can be taken.
The counter-terms arise from adding in a worldline counter-term action

$\begin{split}S_{ct}&=-\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}(r,\omega,\mathbb{L})\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}|_{\text{Bnd}}\
.\end{split}$ (129)

Here $|_{\text{Bnd}}$ refers to the fact that we add such a contribution at
every boundary. Being a boundary contribution, this addition does not change
the equations of motion for the scalar field. If the original variational
principle was defined with a Dirichlet boundary condition
$\delta\varphi_{{}_{\mathcal{N}}}|_{\text{Bnd}}=0$, the counterterm above does
not change that boundary condition. The reader might expect, from the
discussion at the end of appendix §A on flat spacetime, that additional
counter-terms will be required in even $d$ to deal with $\cot\pi\nu$
divergences. As we did there, we will first deal with singularities at the
sources before solving the divergences peculiar to even $d$.

In the above expression, we should take
$\mathscr{C}_{\mathcal{N}}(r,\omega,\mathbb{L})$ to be a real and even
function of $\omega$ to get a real counter-term action. Addition of this
worldline action modifies the canonical conjugate evaluated at the radial
boundaries to

$\begin{split}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\pi_{\mathcal{N}}\equiv-r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\left[r^{\mathcal{N}}D_{+}+r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}\right]\varphi_{{}_{\mathcal{N}}}\
.\end{split}$ (130)

The $\mathscr{C}_{\mathcal{N}}$ should then be chosen such that this object
evaluated at $r=r_{c}\pm i\varepsilon$ has a well-defined $r_{c}\to 0$ limit.

We will now determine $\mathscr{C}_{\mathcal{N}}$ by studying the outgoing
Green function (the counter-terms determined using a generic enough solution
should work for every other solution). As we shall see, the boundary value of
the _renormalised_ conjugate field in this case is the boundary Green function
$K_{\text{Out}}$. Before going into the details of the computation, it might
be useful to situate it in a familiar physical context.

In the case of electromagnetism, the worldline Green function $K_{\text{Out}}$
for a charged particle encodes the radiation reaction or self-force due to the
particle’s EM fields acting on itself. While this statement is broadly true,
it is clear that this idea has to be interpreted with some care. If we take
the bare electric field produced by the point charge and try to compute the
self-force on it naively, the calculation will be dominated by the Coulomb
divergence at the origin yielding an infinite answer.

A little bit of thought however reveals that these divergences merely serve to
relate the bare properties (e.g., mass) of the fictitious charge-free particle
to the properties of the actual physical particle. What we should do instead
is to compute the renormalised electric field felt by the particle after
adding counter-terms which shift the mass to the experimentally measured
value. This renormalised field associated with the radiation is determined
from the near field by imposing the outgoing boundary condition and can then
be used to compute the self-force of the particle.

With this physical example in mind, we can interpret the first term in
Eq.(118) as analogous to the Coulomb field in the near region whose divergent
contributions need to be removed by using counter-terms. It is only after this
is done that we can extract $\widehat{K}_{\text{Out}}$ as the renormalised
worldline Green function.

We will now demand that the renormalised conjugate field computed over the
first term in Eq.(118) vanish. This fixes the counter-term function
$\mathscr{C}_{\mathcal{N}}$ to be

$\begin{split}\frac{\mathscr{C}_{\mathcal{N}}}{1-r^{2}}&\equiv-r\frac{d}{dr}\ln\left\\{r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1-r^{2})^{-\frac{i\omega}{2}}{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\right\\}\
.\end{split}$ (131)

Here we take the branch cut of $(1-r^{2})^{-\frac{i\omega}{2}}$ to be away
from the open unit disc $|r|<1$ in the complex $r$ plane and, with this
choice, $\mathscr{C}_{\mathcal{N}}$ is analytic everywhere inside each copy of
the static patch, and has no discontinuity across the dS-SK branch-cut. While
it is not obvious from the expression above, we can invoke the Euler
transformation formula for the hyper-geometric function which states that

$\begin{split}{}_{2}F_{1}&\left[\frac{1\pm\nu+\mu+i\omega}{2},\frac{1\pm\nu-\mu+i\omega}{2};1\pm\nu;r^{2}\right]\\\
&=(1-r^{2})^{-i\omega}{}_{2}F_{1}\left[\frac{1\pm\nu+\mu-i\omega}{2},\frac{1\pm\nu-\mu-i\omega}{2};1\pm\nu;r^{2}\right]\
,\end{split}$ (132)

to conclude that $\mathscr{C}_{\mathcal{N}}$ is a real and even function of
$\omega$. Here we have taken the function to be analytic in static patch again
and hence $\mathscr{C}_{\mathcal{N}}$ has a well-behaved small $r$ expansion.
The first few terms in this expansion are given by

$\begin{split}\mathscr{C}_{\mathcal{N}}&=(1-r^{2})\left(\nu+\frac{1}{2}(\mathcal{N}-1)\right)+r^{2}\frac{(\nu-\mu-1)(\nu+\mu-1)-\omega^{2}}{2\nu-2}\\\
&+r^{4}\frac{[(\nu-\mu-1)^{2}+\omega^{2}][(\nu+\mu-1)^{2}+\omega^{2}]}{(2\nu-2)^{2}(2\nu-4)}\\\
&+r^{6}\frac{[(\nu-\mu-1)^{2}+\omega^{2}][(\nu+\mu-1)^{2}+\omega^{2}]}{(2\nu-2)^{3}(2\nu-4)(2\nu-6)}\\\
&\quad\times[(2\nu-2)(2\nu-4)-2(\nu-\mu-1)(\nu+\mu-1)+2\omega^{2}]\
+\ldots.\end{split}$ (133)

Note that all terms in the above expansion are indeed real and even functions
of $\omega$ as claimed. Note that all the $r$ and $\omega$ factors appear in
the numerator implying that this counter-term is local in time/radial
direction.

Now that we have the expression for the counter-term, it is straightforward to
compute the renormalised conjugate field evaluated over the outgoing Green
function. We obtain the following answer

$\begin{split}\pi_{\mathcal{N}}^{\text{Out}}&\equiv-\left[r^{\mathcal{N}}D_{+}+r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}\right]G^{\text{Out}}_{\mathcal{N}}\\\
&=(1+i\cot\pi\nu)\widehat{K}_{\text{Out}}\mathscr{Z}_{\mathcal{N}}(r,\omega)r^{\nu+\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\
,\end{split}$ (134)

where $\mathscr{Z}_{\mathcal{N}}(r,\omega)$ is a function given by the
expression

$\begin{split}\frac{\mathscr{Z}_{\mathcal{N}}}{1-r^{2}}&\equiv
1-\frac{r}{2\nu}\frac{d}{dr}\ln\left\\{\frac{{}_{2}F_{1}\left[\frac{1-\nu-\mu-i\omega}{2},\frac{1-\nu+\mu-i\omega}{2};1-\nu;r^{2}\right]}{{}_{2}F_{1}\left[\frac{1+\nu+\mu-i\omega}{2},\frac{1+\nu-\mu-i\omega}{2};1+\nu;r^{2}\right]}\right\\}\
.\end{split}$ (135)

This is also a real and even function of $\omega$ with a well-behaved series
expansion near the origin. We thus see that the renormalised conjugate field
of the outgoing wave is essentially its regular part, obtained after dropping
its singular part and then renormalised by a factor of
$\mathscr{Z}_{\mathcal{N}}$. Taking the $r\to 0$ limit yields

$\begin{split}\lim_{r\to
0}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\pi_{\mathcal{N}}^{\text{Out}}\equiv-\lim_{r\to
0}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\left[r^{\mathcal{N}}D_{+}+r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}\right]G^{\text{Out}}_{\mathcal{N}}=(1+i\cot\pi\nu)\widehat{K}_{\text{Out}}\
.\end{split}$ (136)

This then justifies our original definition for $\widehat{K}_{\text{Out}}$.

If $d$ is odd and $\nu\equiv\frac{d}{2}+\ell-1\in\mathbb{Z}+\frac{1}{2}$, we
can set $\nu$ to its actual value everywhere (i.e., remove dim-reg.) in our
result: the value of the renormalised conjugate field at the world line (which
we shall henceforth refer to by the symbol $K_{\text{Out}}$) is then finite.
We can then write

$\begin{split}K_{\text{Out}}|_{\text{Odd
d}}&=(1+i\cot\pi\nu)\widehat{K}_{\text{Out}}|_{\text{Odd
d}}=-e^{i\nu\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\frac{\Gamma\left(\frac{1+\nu-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu-i\omega}{2}\right)}{\Gamma\left(\frac{1-\nu+\mu-i\omega}{2}\right)\Gamma\left(\frac{1-\nu-\mu-i\omega}{2}\right)}\
.\end{split}$ (137)

For the massless case in odd $d$, we have $\mu,\nu\in\mathbb{Z}+\frac{1}{2}$
for all values of interest given in table 1. If we further assume that
$\mu\neq 1+\nu\equiv\frac{d}{2}+\ell$, the above expression is, in fact, an
odd polynomial of $i\omega$ with degree $2\nu$ (see table 5 for an
illustration). An interesting example is that of a conformally coupled scalar
in odd $d$, where we have a closed-form expression

$K_{\text{Out}}\Big{|}_{\mu=\frac{1}{2}}=\frac{(-1)^{\nu-\frac{1}{2}}}{(2\nu-2)!!^{2}}\prod\limits_{k=1}^{2\nu}\left[\nu+\frac{1}{2}-k-i\omega\right]$
(138)

In all such cases, for every multipole moment, the Hubble corrections for the
radiation correction terminate. Hence, we get a completely Markovian influence
phase with no memory/tail terms. Further, as we shall explain in detail in the
appendix D, for an arbitrarily moving point-like source, all the multipole
contributions add up nicely into a local generally covariant expression for
the radiation reaction force.

Table 5: $\frac{K_{\text{Out}}}{-i\omega}$ for $\mu\in\left\\{\frac{d}{2}-1,\frac{d}{2}-2\right\\}$ (gauge/gravity scalar/vector sectors)  $\mu=\frac{d}{2}-1$ | $\ell=0$ | $\ell=1$ | $\ell=2$
---|---|---|---
$d=3$ | $1$ | $\omega^{2}+1$ | $\frac{\omega^{4}}{9}+\frac{5\omega^{2}}{9}+\frac{4}{9}$
$d=5$ | $\omega^{2}+4$ | $\frac{\omega^{4}}{9}+\frac{10\omega^{2}}{9}+1$ | $\frac{\omega^{6}}{225}+\frac{7\omega^{4}}{75}+\frac{28\omega^{2}}{75}+\frac{64}{225}$
$d=7$ | $\frac{\omega^{4}}{9}+\frac{20\omega^{2}}{9}+\frac{64}{9}$ | $\frac{\omega^{6}}{225}+\frac{7\omega^{4}}{45}+\frac{259\omega^{2}}{225}+1$ | $\frac{\omega^{8}}{11025}+\frac{19\omega^{6}}{3675}+\frac{8\omega^{4}}{105}+\frac{3088\omega^{2}}{11025}+\frac{256}{1225}$
$\mu=\frac{d}{2}-2$ | $\ell=0$ | $\ell=1$ | $\ell=2$
$d=3$ | $1$ | $\omega^{2}+1$ | $\frac{\omega^{4}}{9}+\frac{5\omega^{2}}{9}+\frac{4}{9}$
$d=5$ | $\omega^{2}+1$ | $\frac{\omega^{4}}{9}+\frac{5\omega^{2}}{9}+\frac{4}{9}$ | $\frac{\omega^{6}}{225}+\frac{14\omega^{4}}{225}+\frac{49\omega^{2}}{225}+\frac{4}{25}$
$d=7$ | $\frac{\omega^{4}}{9}+\frac{10\omega^{2}}{9}+1$ | $\frac{\omega^{6}}{225}+\frac{7\omega^{4}}{75}+\frac{28\omega^{2}}{75}+\frac{64}{225}$ | $\frac{\omega^{8}}{11025}+\frac{13\omega^{6}}{3675}+\frac{19\omega^{4}}{525}+\frac{1261\omega^{2}}{11025}+\frac{4}{49}$

For the minimally coupled massless scalar ($\mu=\frac{d}{2}$), we still obtain
a polynomial $K_{\text{Out}}$ for all multipoles except the monopole
($\ell=0$) contribution. The monopole has an extra $1/\omega$ correction in
addition to the polynomial terms odd in $\omega$ (See tables 6 and 7). An
explicit expression for $\ell=0$ contribution is given by

$K_{\text{Out}}|_{\mu=1+\nu=\frac{d}{2}}=\frac{(d-2)^{2}}{i\omega}\cosh\frac{\pi\omega}{2}\
\frac{\Gamma\left(\frac{d-i\omega}{2}\right)\Gamma\left(\frac{d+i\omega}{2}\right)}{\Gamma\left(\frac{d}{2}\right)^{2}}$
(139)

The inverse omega that appears in the front of this expression suggests that
the correct variable for a low frequency expansion in this case is the time
integral of the scalar source rather than the source itself. Such a mild non-
Markovianity for minimally coupled scalars in dS has been noted
beforeBurko:2002ge ; Burko:2002gf , and we will review its physical
interpretation in appendix D.

Table 6: $i\omega K_{\text{Out}}$ $\mu=\frac{d}{2}$ | $\ell=0$
---|---
$d=3$ | $\omega^{2}+1$
$d=5$ | $\omega^{4}+10\omega^{2}+9$
$d=7$ | $\frac{\omega^{6}}{9}+\frac{35\omega^{4}}{9}+\frac{259\omega^{2}}{9}+25$
Table 7: $\frac{K_{\text{Out}}}{-i\omega}$ $\mu=\frac{d}{2}$ | $\ell=1$ | $\ell=2$
---|---|---
$d=3$ | $\omega^{2}+4$ | $\frac{\omega^{4}}{9}+\frac{10\omega^{2}}{9}+1$
$d=5$ | $\frac{\omega^{4}}{9}+\frac{20\omega^{2}}{9}+\frac{64}{9}$ | $\frac{\omega^{6}}{225}+\frac{7\omega^{4}}{45}+\frac{259\omega^{2}}{225}+1$
$d=7$ | $\frac{\omega^{6}}{225}+\frac{56\omega^{4}}{225}+\frac{784\omega^{2}}{225}+\frac{2561}{25}$ | $\frac{\omega^{8}}{11025}+\frac{4\omega^{6}}{525}+\frac{94\omega^{4}}{525}+\frac{12916\omega^{2}}{11025}+1$

For generic values of $(\mu,\nu)$, a small $\omega$ expansion of
$K_{\text{Out}}$ is easy to obtain by expanding out the gamma functions in
terms of polygamma functions. We get

$\begin{split}&K_{\text{Out}}|_{\text{Odd $d$}}=-e^{i\nu\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\frac{\Gamma\left(\frac{1+\nu-\mu}{2}\right)\Gamma\left(\frac{1+\nu+\mu}{2}\right)}{\Gamma\left(\frac{1-\nu+\mu}{2}\right)\Gamma\left(\frac{1-\nu-\mu}{2}\right)}\\\
&\times\exp\left\\{\sum\limits_{k=0}^{\infty}\frac{\left(\frac{-i\omega}{2}\right)^{k+1}}{(k+1)!}\left[\psi^{(k)}\left(\frac{1+\nu-\mu}{2}\right)+\psi^{(k)}\left(\frac{1+\nu+\mu}{2}\right)-\psi^{(k)}\left(\frac{1-\nu-\mu}{2}\right)-\psi^{(k)}\left(\frac{1-\nu+\mu}{2}\right)\right]\right\\}\
,\end{split}$ (140)

where $\psi^{(k)}(z)\equiv\frac{d^{k+1}}{dz^{k+1}}\ln\Gamma(z)$ is the
polygamma function. When both $\nu+\mu$ or $\nu-\mu$ are non-negative
integers, the terms in the above expressions become indeterminate and should
instead be interpreted as a limit. In such cases, explicit computations show
that the above exponential terminates yielding an odd polynomial in $\omega$
when $\nu$ is half-integer.

We will now comment on the even $d$/integer $\nu$ case. The $\cot\pi\nu$
diverges in this limit, and we need the analogue of Eq.(107) to figure out the
counter-terms needed to remove this divergence. The analogous expansion is
given by

$\begin{split}(1+i\cot(\pi\nu))\widehat{K}_{\text{Out}}&=\frac{(-)^{n}}{\Gamma(n)^{2}}\frac{\Gamma\left(\frac{1+n-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+n+\mu-i\omega}{2}\right)}{\Gamma\left(\frac{1-n+\mu-i\omega}{2}\right)\Gamma\left(\frac{1-n-\mu-i\omega}{2}\right)}\left[\frac{2}{\nu-n}\right.\\\
&\left.+\psi^{(0)}\left(\frac{1+n-\mu-i\omega}{2}\right)+\psi^{(0)}\left(\frac{1+n+\mu-i\omega}{2}\right)\right.\\\
&\left.+\psi^{(0)}\left(\frac{1-n-\mu-i\omega}{2}\right)+\psi^{(0)}\left(\frac{1-n+\mu-i\omega}{2}\right)-4\psi^{(0)}(n)+O(\nu-n)\right]\
.\end{split}$ (141)

As in the flat spacetime, we can counter-term away the first two terms, and
change the $n$ back to $\nu$. This yields the renormalised worldline Green
function asLopez-Ortega:2006aal ; Anninos:2011af

$\begin{split}K_{\text{Out}}|_{\text{Even
$d$}}&=\Delta_{\mathcal{N}}(\nu,\mu,\omega)\left[\psi^{(0)}\left(\frac{1+\nu-\mu-i\omega}{2}\right)+\psi^{(0)}\left(\frac{1+\nu+\mu-i\omega}{2}\right)\right.\\\
&\left.+\psi^{(0)}\left(\frac{1-\nu-\mu-i\omega}{2}\right)+\psi^{(0)}\left(\frac{1-\nu+\mu-i\omega}{2}\right)-4\psi^{(0)}(\nu)\right]\
,\end{split}$ (142)

where the function $\Delta_{\mathcal{N}}$ is defined below in Eq.(144). To get
this answer, we add to counterterm in Eq.(129), further terms of the form

$\begin{split}S_{ct,\text{Even}}&=\sum_{\mathbb{L}}\frac{1}{\nu-n}\int\frac{d\omega}{2\pi}r^{\mathcal{N}-1+2n}\Delta_{\mathcal{N}}(n,\mu,\omega)\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}|_{r_{c}}\
,\end{split}$ (143)

where $n=\ell+\frac{d}{2}-1$ and we have defined

$\begin{split}\Delta_{\mathcal{N}}(n,\mu,\omega)&\equiv\frac{(-)^{n}}{\Gamma(n)^{2}}\frac{\Gamma\left(\frac{1+n-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+n+\mu-i\omega}{2}\right)}{\Gamma\left(\frac{1-n+\mu-i\omega}{2}\right)\Gamma\left(\frac{1-n-\mu-i\omega}{2}\right)}=\frac{1}{\Gamma(n)^{2}}\prod\limits_{k=1}^{n}\left[\frac{\omega^{2}}{4}+\frac{1}{4}(\mu-n+2k-1)^{2}\right]\\\
&=\Delta_{\mathcal{N}}^{\ast}(n,\mu,\omega)\ .\end{split}$ (144)

Note that the explicit product form we give above is valid for
$n\in\mathbb{Z}_{+}$. This form shows that $\Delta_{\mathcal{N}}$ is a real
and even function of $\omega$, which is an essential condition for such a
counterterm to be admissible. With this counterterm, Eq.(142) is the dS
generalisation of the radiation reaction influence phase in flat spacetime
described by Eq.(108). The simple logarithmic running in flat spacetime is now
replaced by a more complicated RGE with the Hubble constant playing the role
of the IR cutoff. A low frequency expansion $K_{\text{Out}}$ can be obtained
by using the polygamma series expansion

$\displaystyle\begin{split}\psi^{(0)}&\left(\frac{1+\nu-\mu-i\omega}{2}\right)+\psi^{(0)}\left(\frac{1+\nu+\mu-i\omega}{2}\right)+\psi^{(0)}\left(\frac{1-\nu-\mu-i\omega}{2}\right)+\psi^{(0)}\left(\frac{1-\nu+\mu-i\omega}{2}\right)\\\
&=\sum\limits_{k=0}^{\infty}\frac{\left(\frac{-i\omega}{2}\right)^{k+1}}{(k+1)!}\left[\psi^{(k)}\left(\frac{1+\nu-\mu}{2}\right)+\psi^{(k)}\left(\frac{1+\nu-\mu}{2}\right)\right.\\\
&\hskip
113.81102pt\left.+\psi^{(k)}\left(\frac{1-\nu-\mu}{2}\right)+\psi^{(k)}\left(\frac{1-\nu+\mu}{2}\right)\right]\
,\end{split}$ (145)

which is well-defined except when any one of the poly-gamma arguments is a
negative integer. These results agree with the dS expressions derived in
Anninos:2011af .

We will conclude this section with a comment on the flat spacetime limit of
the expressions derived in this appendix. Intuitively, we expect that the
high-frequency modes in with $\omega\gg 1$ would be insensitive to the
cosmological constant, and would behave like Minkowski modes. This intuition
can indeed be made precise by examining the high frequency expansion of
$K_{\text{Out}}$. Using Stirling approximation for the Gamma functions, we can
indeed check the following valid for $\omega\gg 1$:

$\displaystyle\begin{split}K_{\text{Out}}\approx\left\\{\begin{array}[]{cc}\frac{2\pi
i}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}&\text{for $d$ odd
,}\\\
\frac{1}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}\ln\left(\frac{\omega^{4}}{H^{4}}\right)&\text{for
$d$ even .}\\\ \end{array}\right.\end{split}$ (146)

Comparing these limits with Eqs.(106) and (108), we conclude $K_{\text{Out}}$
is indeed the dS generalisation of the radiation reaction kernel.

##  Appendix C SK Green functions and the cosmological influence phase

We now turn to the problem of constructing the solution on the dS-SK spacetime
contour. The construction here closely parallels corresponding derivation in
AdSSkenderis:2008dg ; Skenderis:2008dh ; Glorioso:2018mmw ;
Chakrabarty:2019aeu ; Jana:2020vyx  and we include a concise summary here
mainly for completeness. The reader is encouraged to see these references for
a more extensive discussion and interpretation of the expressions quoted
below.

Our discussion in this appendix is structured as follows: we begin by
extending our discussion of counter-terms etc. to the _incoming_ Green
functions. Physically, such Green functions are relevant while describing the
effect of a distant source in the past of the observer. As will be derived
below, even if there are no sources present, an observer in dS spacetime sees
cosmic background radiation at the dS temperature. We will need the incoming
Green function to describe these waves.

###  C.1 Time reversal, incoming waves and their branch-cut

We would now like to argue that the renormalised conjugate field continues to
be finite for the Green function describing incoming waves. The incoming Green
function can be computed from the answers we already have by using the time
reversal isometry of the dS spacetime. The only non-trivial step involved is
to realise how the time reversal isometry acts on EF coordinates.

The action of time reversal is achieved by the diffeomorphism

$\begin{split}u\mapsto 2\pi i\zeta-u\ ,\ \omega\mapsto-\omega\ ,\end{split}$
(147)

where $\zeta$ is the mock tortoise coordinate introduced in Eq.(3). One can
check that this diffeomorphism preserves the metric in Eq.(110) and is hence
an isometry. The map $\omega\mapsto-\omega$ is necessary maintain the $\sim
e^{-i\omega u}$ factor in Fourier domain. The time reversal is hence achieved
by reversing $\omega$ and then multiplying all Fourier domain functions by a
factor $e^{-2\pi\omega\zeta}$.

Using the time reversal isometry the bulk to worldline Green function with
incoming boundary condition takes the form

$G^{\text{In}}_{\mathcal{N}}(r,\omega,\mathbb{L})=e^{-2\pi\omega\zeta}G^{\text{Out}*}_{\mathcal{N}}(r,\omega,\mathbb{L})$
(148)

Unlike $G^{\text{Out}}_{\mathcal{N}}$, the Green function
$G^{\text{In}}_{\mathcal{N}}$ has a branch-cut on the dS-SK contour, taking
different values in the left vs. right static patches. The near origin
expansion of $G^{\text{In}}_{\mathcal{N}}$ can be obtained by using the Euler
transformation in Eq.(132):

$\begin{split}G^{\text{In}}_{\mathcal{N}}(r,\omega,\mathbb{L})&\equiv
e^{-2\pi\omega\zeta}G^{\text{Out}*}_{\mathcal{N}}=e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}\times
r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}\\\
&\times\Bigl{\\{}{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\Bigr{.}\\\
&\Bigl{.}\quad\qquad-(1-i\cot\pi\nu)\widehat{K}_{\text{In}}\frac{r^{2\nu}}{2\nu}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\Bigr{\\}}\
.\end{split}$ (149)

The branch cuts of the explicit $(1\pm r)^{-i\omega}$ are chosen to lie
outside the open unit disc in the complex $r$ plane and a careful evaluation
of the pre-factor above yields

$\begin{split}e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}=\begin{cases}1\qquad&\text{L
contour}\\\ e^{-2\pi\omega}\qquad&\text{R contour .}\end{cases}\end{split}$
(150)

This shows explicitly the branch-cut and jump in the incoming Green function.
In the above equation, the symbol $\widehat{K}_{\text{In}}$ denotes the
worldline advanced Green function given by the expression

$\begin{split}\widehat{K}_{\text{In}}(\omega,\ell)&\equiv[\widehat{K}_{\text{Out}}(\omega,\ell)]^{\ast}=-e^{-2\pi
i\nu}\widehat{K}^{\text{Out}}(-\omega,\ell)\\\ &=e^{-i\nu\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\frac{\Gamma\left(\frac{1+\nu-\mu+i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu+i\omega}{2}\right)}{\Gamma\left(\frac{1-\nu+\mu+i\omega}{2}\right)\Gamma\left(\frac{1-\nu-\mu+i\omega}{2}\right)}\
.\end{split}$ (151)

The comments made in the context of $\widehat{K}_{\text{Out}}$ below Eq.(119)
apply also in this case. The decomposition in Eq.(149)

Given the above definition of $G^{\text{In}}_{\mathcal{N}}$, it is now
straightforward to compute the renormalised conjugate field. Since the
incoming mode has a branch cut, it behaves differently at the two boundaries.
Adding in the counterterm in Eq.(131), we get the renormalised conjugate field
as

$\begin{split}\pi_{\mathcal{N}}^{\text{In}}&\equiv-\left[r^{\mathcal{N}}D_{+}+r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}\right]G^{\text{In}}_{\mathcal{N}}\\\
&=-e^{-2\pi\omega\zeta}\left[r^{\mathcal{N}}D_{-}+r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}\right]G^{\text{Out}*}_{\mathcal{N}}\\\
&=e^{-2\pi\omega\zeta}\pi^{\text{Out}*}_{\mathcal{N}}.\end{split}$ (152)

Here we have used $D_{\pm}\equiv(1-r^{2})\partial_{r}\pm i\omega$ as well as
the property that
$D_{+}[e^{-2\pi\omega\zeta}\\#]=e^{-2\pi\omega\zeta}D_{-}[\\#]$. Using
Eq.(134), we obtain

$\begin{split}\pi_{\mathcal{N}}^{\text{In}}&=(1-i\cot\pi\nu)\widehat{K}_{\text{In}}e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}\mathscr{Z}_{\mathcal{N}}(r,\omega)\\\
&\quad\times
r^{\nu+\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\
.\end{split}$ (153)

As in the case of outgoing waves, we see again that the renormalised conjugate
field is the regular part of the incoming waves renormalised with the same
factor $\mathscr{Z}_{\mathcal{N}}(r,\omega)$. We can then take $r\to 0$ limit
above and below the branch cut to get

$\begin{split}\lim_{r\to
0}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\pi_{\mathcal{N}}^{\text{In}}=\begin{cases}(1-i\cot\pi\nu)\widehat{K}_{\text{In}}\qquad&\text{L
boundary}\ ,\\\
e^{-2\pi\omega}(1-i\cot\pi\nu)\widehat{K}_{\text{In}}\qquad&\text{R boundary}\
.\end{cases}\end{split}$ (154)

This shows that the counter-term we derived also works for the incoming waves.
When $d$ is odd and $\cot\pi\nu=0$, we can remove the dimensional
regularisation without any further counterterms. The analogue of Eq.(137) for
the incoming waves is

$\begin{split}K_{\text{In}}|_{\text{Odd
d}}&=(1-i\cot\pi\nu)\widehat{K}_{\text{In}}|_{\text{Odd
d}}=(K_{\text{Out}})^{\ast}|_{\text{Odd d}}=e^{-i\nu\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\frac{\Gamma\left(\frac{1+\nu-\mu+i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu+i\omega}{2}\right)}{\Gamma\left(\frac{1-\nu+\mu+i\omega}{2}\right)\Gamma\left(\frac{1-\nu-\mu+i\omega}{2}\right)}\
.\end{split}$ (155)

All our statements about $K_{\text{Out}}$ in odd $d$ apply mutatis mutandis to
$K_{\text{In}}$.

When $d$ is even and $\nu$ approaches an integer, there are additional
divergences due to $\cot\pi\nu$. We already encountered such divergences and
countertermed them away for outgoing waves. We have to check now that the
counterterms in Eq.(143) added to cancel such divergences out of outgoing
waves, work also for the incoming waves. To see this, we examine the expansion

$\begin{split}(1-i\cot\pi\nu)\widehat{K}_{\text{In}}(\nu)&=\Delta_{\mathcal{N}}(n,\mu,\omega)\left[\frac{2}{\nu-n}\right.\\\
&\left.+\psi^{(0)}\left(\frac{1+n-\mu+i\omega}{2}\right)+\psi^{(0)}\left(\frac{1+n+\mu+i\omega}{2}\right)\right.\\\
&\left.+\psi^{(0)}\left(\frac{1-n-\mu+i\omega}{2}\right)+\psi^{(0)}\left(\frac{1-n+\mu+i\omega}{2}\right)-4\psi^{(0)}(n)+O(\nu-n)\right]\
,\end{split}$ (156)

where $\Delta_{\mathcal{N}}(n,\mu,\omega)$ is given by Eq.(144). Here, we have
used crucially the fact that $\Delta_{\mathcal{N}}$ is a real, even function
of $\omega$.

From the above expression, we can see that the incoming conjugate field in
Eq.(153) is also rendered finite by the same counterterms as before.
Crucially, the monodromy factors of $e^{-2\pi\omega\zeta}$ work out correctly
to cancel the divergences near both the left/right world lines. We get the
final renormalised advanced worldline Green function as

$\begin{split}K_{\text{In}}|_{\text{Even
$d$}}&=\Delta_{\mathcal{N}}(\nu,\mu,\omega)\left[\psi^{(0)}\left(\frac{1+\nu-\mu+i\omega}{2}\right)+\psi^{(0)}\left(\frac{1+\nu+\mu+i\omega}{2}\right)\right.\\\
&\left.+\psi^{(0)}\left(\frac{1-\nu-\mu+i\omega}{2}\right)+\psi^{(0)}\left(\frac{1-\nu+\mu+i\omega}{2}\right)-4\psi^{(0)}(\nu)\right]\
,\end{split}$ (157)

To conclude, we have demonstrated a set of counterterms which result in finite
answers for conjugate fields evaluated over both outgoing as well as incoming
waves. The final renormalised conjugate field is given by

$\begin{split}\lim_{r\to
0}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\pi_{\mathcal{N}}^{\text{In}}=\begin{cases}K_{\text{In}}\qquad&\text{L
boundary}\ ,\\\ e^{-2\pi\omega}K_{\text{In}}\qquad&\text{R boundary}\
.\end{cases}\end{split}$ (158)

Since the most general solution on the dS-SK geometry is a linear combination
of outgoing/incoming waves, it follows that our counterterm prescription will
yield a finite answer for the cosmological influence phase.

###  C.2 Point-like sources and Green functions on dS-SK contour

In this subsection, we solve for the unique combination of outgoing and
incoming waves corresponding to a point source placed at the centre(s) of
left/right static patches in dS-SK geometry. As we will describe subsequently,
with some more effort, arbitrary extended sources on the dS-SK background can
also be dealt with.

We describe the point source problem first to introduce, within a simpler
setting, the ingredients needed for the extended sources. As we shall see, in
analogy with AdS, we can think of the problem of point sources placed at the
centre of the static patch as one involving boundary-to-bulk Green functions.
In contrast, the problem of extended sources is that of bulk-to-bulk Green
functions, and it is hence fairly more involved.

As described in the main text, the solution for the bulk field produced by
point-like sources is given by Eq.(7). Using Eq.(9), we then have

$\begin{split}\varphi_{{}_{\mathcal{N}}}=g_{R}\mathcal{J}_{R}-g_{L}\mathcal{J}_{L}\
,\end{split}$ (159)

where we have defined

$\begin{split}g_{L}&\equiv
n_{\omega}\Bigl{(}G^{\text{Out}}_{\mathcal{N}}-e^{2\pi\omega(1-\zeta)}G^{\text{Out}*}_{\mathcal{N}}\Bigr{)}\
,\\\
g_{R}&\equiv(1+n_{\omega})\Bigl{(}G^{\text{Out}}_{\mathcal{N}}-e^{-2\pi\omega\zeta}G^{\text{Out}*}_{\mathcal{N}}\Bigr{)}\
.\end{split}$ (160)

These are the dS analogues of the left/right _boundary-to-bulk_ Green
functions which tell us how left and right sources affect the solution on the
dS-SK geometry. They obey the Kubo-Martin-Schwinger (KMS) relation
$g_{R}(\zeta)=e^{2\pi\omega}g_{L}(1+\zeta)$ as well as the following boundary
conditions on the dS-SK contour:

$\begin{split}\lim_{\zeta\to 0}r^{\nu+\frac{\mathcal{N}-1}{2}}g_{L}=-1\
,&\quad\lim_{\zeta\to 0}r^{\nu+\frac{\mathcal{N}-1}{2}}g_{R}=0\ ,\\\
\lim_{\zeta\to 1}r^{\nu+\frac{\mathcal{N}-1}{2}}g_{L}=0\ ,&\quad\lim_{\zeta\to
1}r^{\nu+\frac{\mathcal{N}-1}{2}}g_{R}=1.\end{split}$ (161)

This result can be derived directly from the boundary condition in Eq.(124).
The above conditions imply that the Green function $g_{L,R}$ are two different
smooth interpolations between the homogeneous solution regular at the origin
on one side and a Green function with a source singularity on the other side.
Thus, $g_{R}$ is regular near the left boundary whereas $g_{L}$ is regular
near the right boundary.

The Green functions $g_{L,R}$ can be written down explicitly. Substituting
Eqs.(118) and (149) into Eq.(160), we get the following expressions:

$\begin{split}g_{L}&=n_{\omega}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}\\\
&\times\Bigl{\\{}\left[1-e^{2\pi\omega(1-\zeta)}\left(\frac{1-r}{1+r}\right)^{-i\omega}\right]\times{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\Bigr{.}\\\
&\quad-i\cot\pi\nu\left[\widehat{K}_{\text{Out}}+e^{2\pi\omega(1-\zeta)}\left(\frac{1-r}{1+r}\right)^{-i\omega}\widehat{K}_{\text{In}}\right]\times\frac{r^{2\nu}}{2\nu}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\\\
&\Bigl{.}\quad-\left[\widehat{K}_{\text{Out}}-e^{2\pi\omega(1-\zeta)}\left(\frac{1-r}{1+r}\right)^{-i\omega}\widehat{K}_{\text{In}}\right]\times\frac{r^{2\nu}}{2\nu}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\Bigr{\\}}\
,\\\ \end{split}$ (162)

and

$\begin{split}g_{R}&=(1+n_{\omega})r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}\\\
&\times\Bigl{\\{}\left[1-e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}\right]\times{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\Bigr{.}\\\
&\quad-i\cot\pi\nu\left[\widehat{K}_{\text{Out}}+e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}\widehat{K}_{\text{In}}\right]\times\frac{r^{2\nu}}{2\nu}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\\\
&\Bigl{.}\quad-\left[\widehat{K}_{\text{Out}}-e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}\widehat{K}_{\text{In}}\right]\times\frac{r^{2\nu}}{2\nu}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\Bigr{\\}}\
.\end{split}$ (163)

These equations describe the Dirac-Deitweiler-WhitingDirac:1938nz ;
Detweiler:2002mi  type decomposition of the left/right Green functions into a
singular solution which does not contribute to the radiation reaction, and a
regular solution (the terms in the last line of each equation) which
contribute to the finite influence phase.

Having said that, the reader should note that the expressions above are fairly
complicated, with an elaborate branch cut structure that cannot be easily
guessed a priori without the dS-SK prescription. These formulae are more
complicated by the fact that we are forced to work with dimensional
regularisation for even $d$. We will simplify the expressions for these dS
boundary-to-bulk propagators in the next subsection when we describe extended
sources. For present purposes, it is, however, sufficient to note the
following: despite the complexity of expressions, given that we have a
counterterm procedure that works both for outgoing and incoming waves, we are
guaranteed a finite renormalised conjugate field.

To see this explicitly, we construct the corresponding renormalised conjugate
field

$\begin{split}\pi_{\mathcal{N}}(\zeta,\omega,\ell)=-\pi^{\text{Out}}_{\mathcal{N}}(r,\omega,\ell)\mathcal{J}_{\bar{F}}+e^{2\pi\omega(1-\zeta)}\pi^{\text{Out}*}_{\mathcal{N}}(r,\omega,\ell)\mathcal{J}_{\bar{P}}=\pi_{R}(\zeta,\omega,\ell)\mathcal{J}_{R}-\pi_{L}(\zeta,\omega,\ell)\mathcal{J}_{L}\
,\end{split}$ (164)

with the left/right boundary-to-bulk Green functions for the conjugate field
defined by

$\displaystyle\begin{split}\pi_{L}(\zeta,\omega,\ell)&\equiv-\left[r^{\mathcal{N}}D_{+}+r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}\right]g_{L}(\zeta,\omega,\ell)=n_{\omega}\Bigl{(}\pi^{\text{Out}}_{\mathcal{N}}(r,\omega,\ell)-e^{2\pi\omega(1-\zeta)}\pi^{\text{Out}*}_{\mathcal{N}}(r,\omega,\ell)\Bigr{)}\
,\\\
\pi_{R}(\zeta,\omega,\ell)&\equiv-\left[r^{\mathcal{N}}D_{+}+r^{\mathcal{N}-1}\mathscr{C}_{\mathcal{N}}\right]g_{R}(\zeta,\omega,\ell)=(1+n_{\omega})\Bigl{(}\pi^{\text{Out}}_{\mathcal{N}}(r,\omega,\ell)-e^{-2\pi\omega\zeta}\pi^{\text{Out}*}_{\mathcal{N}}(r,\omega,\ell)\Bigr{)}\
.\end{split}$ (165)

The equality here follows from a logic similar to that used in Eq.(152). The
explicit forms of $\pi^{\text{Out}}_{\mathcal{N}}$ and
$e^{-2\pi\omega\zeta}\pi^{\text{Out}*}_{\mathcal{N}}$ are given in Eqs.(134)
and (153) respectively. Substituting them in, we get

$\begin{split}\pi_{L}&=n_{\omega}\left[(1+i\cot{\pi\nu})\widehat{K}_{\text{Out}}-e^{2\pi\omega(1-\zeta)}\left(\frac{1-r}{1+r}\right)^{-i\omega}(1-i\cot{\pi\nu})\widehat{K}_{\text{In}}\right]\mathscr{Z}_{\mathcal{N}}(\omega,r)\\\
&\quad\times
r^{\nu+\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\
,\\\
\pi_{R}&=(1+n_{\omega})\left[(1+i\cot{\pi\nu})\widehat{K}_{\text{Out}}-e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}(1-i\cot{\pi\nu})\widehat{K}_{\text{In}}\right]\mathscr{Z}_{\mathcal{N}}(\omega,r)\\\
&\quad\times
r^{\nu+\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\
.\end{split}$ (166)

This picks out the regular part of the solution on dS-SK contour renormalised
by $\mathscr{Z}_{\mathcal{N}}(\omega,r)$, as expected.

For $\nu\in\mathbb{Z}$, we should subtract the $\cot\pi\nu$ divergences using
further counterterms in Eq.(143): once this is done, we can relax the
dimensional regularisation and effectively replace

$(1+i\cot\pi\nu)\widehat{K}_{\text{Out}}\to K_{\text{Out}}\
,\quad(1-i\cot\pi\nu)\widehat{K}_{\text{In}}\to K_{\text{In}}\ .$

After this is done, we can take $r\to 0$ limit on both sides of the dS-SK
contour to get

$\begin{split}\lim_{r\to
0}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\pi_{\mathcal{N}}=\begin{cases}K_{LR}\mathcal{J}_{R}-K_{LL}\mathcal{J}_{L}\qquad&\text{L
boundary}\ ,\\\ K_{RR}\mathcal{J}_{R}-K_{RL}\mathcal{J}_{L}\qquad&\text{R
boundary}\ ,\end{cases}\end{split}$ (167)

where we have defined the Schwinger-Keldysh worldline Green functions defined
via

$\begin{split}K_{LL}\equiv
n_{\omega}K_{\text{Out}}-(1+n_{\omega})K_{\text{In}}\ ,&\quad
K_{LR}\equiv(1+n_{\omega})\Bigl{(}K_{\text{Out}}-K_{\text{In}}\Bigr{)}\ ,\\\
K_{RL}\equiv n_{\omega}\Bigl{(}K_{\text{Out}}-K_{\text{In}}\Bigr{)}\ ,&\quad
K_{RR}\equiv(1+n_{\omega})K_{\text{Out}}-n_{\omega}K_{\text{In}}\
.\end{split}$ (168)

These are exactly the expressions for the Schwinger-Keldysh two-point
functions of a bosonic system coupled to a thermal bathSchwinger:1960qe ;
FEYNMAN1963118 ; Caldeira:1982iu .

Now that we have the near origin values of both the generalised free scalar
system as well as its renormalised conjugate field, we are ready to compute
the influence phase of the observer in the saddle point approximation by
evaluating the on-shell action. We want to compute the action given in
Eq.(126) along with the counter-term in Eqs.(129) and (143) over the dS-SK
solution we found in Eq.(159). We begin with the full action

$\begin{split}S&=-\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint\frac{r^{\mathcal{N}}dr}{1-r^{2}}\Bigl{[}(D_{+}\varphi_{{}_{\mathcal{N}}})^{*}D_{+}\varphi_{{}_{\mathcal{N}}}-\omega^{2}(1-r^{2})\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}\Bigr{.}\\\
&\Bigl{.}\qquad-\frac{1}{4r^{2}}\Bigl{\\{}(\mathcal{N}-1)^{2}-4\nu^{2}+[4\mu^{2}-(\mathcal{N}+1)^{2}]r^{2}\Bigr{\\}}\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}\Bigr{]}+S_{ct}\
,\end{split}$ (169)

integrate by parts over the bulk terms and then use the equation of motion in
Eq.(115). This results in an on-shell action written purely in terms of
boundary terms:

$\begin{split}S_{\text{On-
Shell}}&=\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\varphi_{{}_{\mathcal{N}}}^{*}\pi_{\mathcal{N}}|_{\text{Bnd}}=-\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\Bigl{\\{}\mathcal{J}_{R}^{*}[K_{RR}\mathcal{J}_{R}-K_{RL}\mathcal{J}_{L}]-\mathcal{J}_{L}^{*}[K_{LR}\mathcal{J}_{R}-K_{LL}\mathcal{J}_{L}]\Bigr{\\}}\
,\end{split}$ (170)

where $\pi_{\mathcal{N}}$ is the renormalised conjugate field defined in
Eq.(130). Here we have used the fact that the integrand in the first step can
be written as a product

$\begin{split}[r^{\nu+\frac{1}{2}(\mathcal{N}-1)}\varphi_{{}_{\mathcal{N}}}]^{*}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}\pi_{\mathcal{N}}\
,\end{split}$ (171)

and each factor in this product has a finite limit as we remove the regulator
at the boundary (i.e. take $r_{c}\to 0$ limit). The dS-SK contour integral
$\oint$ runs clockwise from the right static patch to the left static patch,
thus resulting in the sign of the final expression above.

We can further simplify the above expression using the reality properties of
the multipole sources as well as $1+n_{\omega}+n_{-\omega}=0$. The
cosmological influence phase of the point-like dS observer can then be written
in the form given in Eq.(14).

###  C.3 Extended sources on dS-SK contour I : bulk-to-bulk propagator

In this section, we will describe the problem of a finite size observer within
dS spacetime. One motivation for such an exercise is to give a more physical
version of the regularisation, counter-terms and renormalisation described in
the previous sub-sections. We will see that indeed a finite size observer has
a renormalised cosmological influence phase, which, as its size is reduced,
approaches the result for a point-like observer. Apart from this formal
motivation, we are also interested in checking whether the conjectured dS-SK
saddle point correctly reproduces the finite size physics in dS. As we shall
see, this is also a way to naturally generalise our construction to a non-co-
moving observer with a peculiar velocity as well as to describe observers made
of multiple worldlines (or equivalently the case of a string or a membrane in
dS).

The main physics in all the above cases is that of relative time-delays: for
an extended source, its effective radiative multipole moments have to be
computed by adding up source strengths at various points with different time-
delays. This is necessary because the emitted wave takes a finite amount of
time to cross an extended source, and this wave-crossing time has to be
accounted for when adding up emissions from two farther ends of the source.
For spherical sources in flat space, this translates to modulating the source
with an appropriate Bessel J function in frequency domain. We will see below
that an analogous statement in dS emerges naturally out of dS-SK saddle-point
geometry.

Let us begin by describing our setup. Consider an extended source of the
generalised/designer scalar field in dS spacetime. This means modifying the
radial ODE in Eq.(115) by a source term of the form

$\begin{split}&\frac{1}{r^{\mathcal{N}}}D_{+}[r^{\mathcal{N}}D_{+}\varphi_{{}_{\mathcal{N}}}]+\omega^{2}\varphi_{{}_{\mathcal{N}}}\\\
&\qquad+\frac{1-r^{2}}{4r^{2}}\Bigl{\\{}(\mathcal{N}-1)^{2}-4\nu^{2}+[4\mu^{2}-(\mathcal{N}+1)^{2}]r^{2}\Bigr{\\}}\varphi_{{}_{\mathcal{N}}}+(1-r^{2})\varrho_{\mathcal{N}}(\zeta,\omega,\mathbb{L})=0\
.\end{split}$ (172)

In the context of dS-SK contour, we will let $\varrho_{\mathcal{N}}$ be a
general function over the saddle-point geometry, allowing it to even take
completely different values in the two copies of the static patch (i.e., as a
function of complex $r$, it is allowed to have a branch-cut along the static
patch). The $(\omega,\mathbb{L})$ arguments of $\varrho_{\mathcal{N}}$ imply
that we also allow the most general time/angle dependence.

The solution for the above ODE can then be written in terms of an appropriate
dS-SK contour-ordered, bulk-to-bulk Green function:

$\begin{split}\varphi_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})&=\oint
r_{0}^{\mathcal{N}}dr_{0}\
\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})\varrho_{{}_{\mathcal{N}}}(\zeta_{0},\omega,\mathbb{L})\
.\end{split}$ (173)

Here $\oint$ refers to the integral over clockwise dS-SK contour and
$\mathbb{G}$ is the radial Green function satisfying the appropriate boundary
conditions (which we will detail below).

According to our proposal in this note, the influence phase of the extended
source can be computed by solving the above ODE everywhere on dS-SK and then
substituting the solution into the action corresponding to the above ODE,
viz., by evaluating

$\begin{split}S&=-\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint\frac{r^{\mathcal{N}}dr}{1-r^{2}}\Bigl{[}(D_{+}\varphi_{{}_{\mathcal{N}}})^{*}D_{+}\varphi_{{}_{\mathcal{N}}}-\omega^{2}\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}\Bigr{.}\\\
&\Bigl{.}\qquad-\frac{1-r^{2}}{4r^{2}}\Bigl{\\{}(\mathcal{N}-1)^{2}-4\nu^{2}+[4\mu^{2}-(\mathcal{N}+1)^{2}]r^{2}\Bigr{\\}}\varphi_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}\Bigr{]}\\\
&\qquad+\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint r^{\mathcal{N}}dr\
\varphi_{{}_{\mathcal{N}}}^{*}\varrho_{{}_{\mathcal{N}}}+S_{ct}[\varrho_{{}_{\mathcal{N}}}]\
\end{split}$ (174)

on the Green function solution above. The last line in the action above gives
the source term and the counter-term parts of the action.232323The reader
should note that the counterterms used here for extended sources need not
(and, indeed will not) match with the counterterms used for point sources in
the previous subsections.  For a truly extended source, counter-terms are not
necessary for finiteness and their job is to provide the finite
renormalisation of the conservative part of the action.

Using the radial ODE above, on-shell action can be reduced to the following
simple form

$\begin{split}S|_{\textbf{On-
shell}}&=\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint
r^{\mathcal{N}}dr\
\varrho_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}|_{\textbf{On-
shell}}+S_{ct}[\varrho_{{}_{\mathcal{N}}}]\\\
&=\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint r^{\mathcal{N}}dr\
\oint r_{0}^{\mathcal{N}}dr_{0}\
[\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})]^{*}\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})\varrho_{{}_{\mathcal{N}}}(\zeta_{0},\omega,\mathbb{L})+S_{ct}[\varrho_{{}_{\mathcal{N}}}]\
.\end{split}$ (175)

Thus, once we solve for the bulk-to-bulk Green function $\mathbb{G}$, we can
substitute it into the above expression to obtain the dS-SK saddle point
answer for cosmological influence phase $S_{\text{CIP}}$. While it is not
immediately evident, we will demonstrate in the next subsection that _the
dissipative part of the influence phase for the extended sources computed from
the expression above, when written in terms of appropriate multipole moments,
takes a form identical to that for a point source derived before._ In addition
to this radiation reaction, for extended sources, we also expect conservative
interactions between its different internal parts.

Let us now derive an explicit expression for the bulk-to-bulk Green function
$\mathbb{G}$. The construction here is analogous to the one in vacuum
AdSWitten:1998qj , as well as the contour-ordered bulk-to-bulk Green function
in the SK contour corresponding to planar AdS black holesLoganayagam:2022zmq ;
OpenEFT . We will demand that this Green function be regular at the edges of
dS-SK contour, viz., we require that

$\begin{split}\lim_{\zeta\to
0}r^{\nu+\frac{\mathcal{N}-1}{2}}\mathbb{G}=\lim_{\zeta\to
1}r^{\nu+\frac{\mathcal{N}-1}{2}}\mathbb{G}=0.\end{split}$ (176)

Further, to be a Green function, it should obey the ODE

$\begin{split}&\frac{1}{r^{\mathcal{N}}}D_{+}[r^{\mathcal{N}}D_{+}\mathbb{G}]+\omega^{2}\mathbb{G}\\\
&\qquad+\frac{1-r^{2}}{4r^{2}}\Bigl{\\{}(\mathcal{N}-1)^{2}-4\nu^{2}+[4\mu^{2}-(\mathcal{N}+1)^{2}]r^{2}\Bigr{\\}}\mathbb{G}+\frac{1}{r^{\mathcal{N}}}(1-r^{2})\delta_{c}(r-r_{0})=0\
.\end{split}$ (177)

Here $\delta_{c}(r-r_{0})$ is the contour-ordered delta function on the dS-SK
contour. The above ODE implies that $\mathbb{G}$ is a solution of the
homogeneous radial ODE for $\zeta\neq\zeta_{0}$ with a unit discontinuity in
the conjugate field at $\zeta=\zeta_{0}$. We have already solved the
homogeneous radial ODE for point sources to construct the left and right
boundary-to-bulk Green functions in Eq.(160). These are solutions
characterised by the boundary conditions specified in Eq.(161).

Looking at Eq.(161), we conclude that, we should take $\mathbb{G}\propto
g_{R}$ near the left boundary and $\mathbb{G}\propto g_{L}$ near the right
boundary since these are the solutions that satisfy the necessary regularity
conditions in Eq.(176). Demanding continuity, we surmise that

$\begin{split}\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})&=\frac{1}{W_{LR}(\zeta_{0},\omega,\mathbb{L})}g_{R}(\zeta_{\succ},\omega,\mathbb{L})g_{L}(\zeta_{\prec},\omega,\mathbb{L})\\\
&\equiv\frac{1}{W_{LR}(\zeta_{0},\omega,\mathbb{L})}\begin{cases}g_{R}(\zeta,\omega,\mathbb{L})g_{L}(\zeta_{0},\omega,\mathbb{L})&\quad\text{if}\;\zeta\succ\zeta_{0}\\\
g_{L}(\zeta,\omega,\mathbb{L})g_{R}(\zeta_{0},\omega,\mathbb{L})&\quad\text{if}\;\zeta\prec\zeta_{0}\end{cases}\
.\end{split}$ (178)

Here the symbols $\succ$ and $\prec$ denote comparison using the radial
contour ordering of dS-SK contour. The unit discontinuity condition on the
conjugate field fixes the function $W_{RL}$ to be the Wronskian between right
and left boundary-to-bulk Green functions, viz.,

$\begin{split}W_{RL}(\zeta,\omega,\mathbb{L})&\equiv
g_{L}\pi_{R}-g_{R}\pi_{L}=(1+n_{\omega})e^{-2\pi\omega\zeta}\Bigl{(}G^{\text{Out}}_{\mathcal{N}}\pi^{\text{Out}*}_{\mathcal{N}}-G^{\text{Out}*}_{\mathcal{N}}\pi^{\text{Out}}_{\mathcal{N}}\Bigr{)}\\\
&=(1+n_{\omega})e^{-2\pi\omega\zeta}\left[(1-i\cot\pi\nu)\widehat{K}_{\text{In}}-(1+i\cot\pi\nu)\widehat{K}_{\text{Out}}\right]\
.\end{split}$ (179)

Here, the equality in the first line follows from Eqs.(160) and (165). The
last equality follows by substituting the expressions for
$G^{\text{Out}}_{\mathcal{N}}$ and $\pi^{\text{Out}}_{\mathcal{N}}$ from
Eqs.(118) and (134), and then invoking the following hypergeometric Wronskian
identity

$\begin{split}\mathscr{Z}_{\mathcal{N}}(r,\omega)&r^{\nu+\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\\\
&=\left(\frac{1-r}{1+r}\right)^{i\omega}\Biggl{\\{}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1+r)^{-i\omega}{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\Biggr{\\}}^{-1}\
.\end{split}$ (180)

This identity expresses a combination of the derivatives of hypergeometric
functions in terms of the hypergeometric functions, and such an identity can
be derived from a Wronskian like argument associated with the corresponding
radial ODE.

The reader should note an important subtlety in the statement above: the
Wronskian here is _not_ a constant function along radial direction, but rather
varies as we traverse the dS-SK contour. A similar subtlety was already noted
in the AdS context by Loganayagam:2022zmq . As we shall eventually see, the
extra $e^{-2\pi\omega\zeta}$ factor is here for a good physical reason: it
ensures that multipole moments that enter into cosmological influence phase
are computed using source distributions in standard time-slices, instead of
source distributions along Eddington-Finkelstein null time-slices.

To proceed further, we should now substitute the explicit forms of dS-SK
boundary-to-bulk propagators given in Eqs.(162) and (163) into the expression
for the bulk-to-bulk propagator in Eq.(178), and then perform the dS-SK
contour integral in Eq.(175). To this end, we first regroup the expressions
for $g_{L}$ and $g_{R}$ into somewhat more tractable expressions with clear
branch cut structures. For what follows, we will find it convenient to
separate out the solutions into a singular (non-normalisable) part $\Xi_{nn}$
vs a regular (normalisable) part $\Xi_{n}$, _using the renormalised world line
Green functions instead of the bare ones from the start_. The adjectives
singular/regular refer here to their behaviour near the worldline (i.e., near
$r=0$). To this end, let us begin by defining two functions $\Xi_{nn},\Xi_{n}$
implicitly via

$\begin{split}\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}G^{\text{Out}}_{\mathcal{N}}(r,\omega,\mathbb{L})&\equiv\Xi_{nn}(r,\omega,\mathbb{L})-K_{\text{Out}}\
\Xi_{n}(r,\omega,\mathbb{L})\ ,\\\
\left(\frac{1-r}{1+r}\right)^{\frac{i\omega}{2}}G^{\text{Out}*}_{\mathcal{N}}(r,\omega,\mathbb{L})&\equiv\Xi_{nn}(r,\omega,\mathbb{L})-K_{\text{In}}\
\Xi_{n}(r,\omega,\mathbb{L})\ ,\end{split}$ (181)

where $K_{\text{Out}}$ and $K_{\text{In}}$ are the final renormalised world
line Green functions. The above equality should be thought of as defining the
functions $\Xi_{n}(r,\omega,\mathbb{L})$ and $\Xi_{nn}(r,\omega,\mathbb{L})$
as analytic functions on the open static patch $0<r<1$, viz., in the equations
above, we align all the potential branch cuts away from the unit disc in
complex radius plane. The above equations can be inverted to give a direct
definition of these functions

$\begin{split}(K_{\text{In}}-K_{\text{Out}})\
\Xi_{n}&\equiv\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}G^{\text{Out}}_{\mathcal{N}}-\left(\frac{1-r}{1+r}\right)^{\frac{i\omega}{2}}G^{\text{Out}*}_{\mathcal{N}}\
,\\\ (K_{\text{In}}-K_{\text{Out}})\
\Xi_{nn}&\equiv\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}K_{\text{In}}G^{\text{Out}}_{\mathcal{N}}-\left(\frac{1-r}{1+r}\right)^{\frac{i\omega}{2}}K_{\text{Out}}G^{\text{Out}*}_{\mathcal{N}}\
.\end{split}$ (182)

Since $K_{\text{In}}(\omega,\ell)=K_{\text{Out}}(-\omega,\ell)$ and
$G^{\text{Out}*}_{\mathcal{N}}(\omega,\ell)=G^{\text{Out}}_{\mathcal{N}}(-\omega,\ell)$,
the above expressions imply that both $\Xi_{n}$ and $\Xi_{nn}$ are even
functions of $\omega$. Explicit expressions can be written down for these two
functions using Eq.(118). We have

$\begin{split}\Xi_{n}&\equiv\frac{1}{2\nu}r^{\nu-\frac{1}{2}(\mathcal{N}-1)}(1-r^{2})^{-\frac{i\omega}{2}}\frac{(1+i\cot\nu\pi)\widehat{K}_{\text{Out}}-(1-i\cot\nu\pi)\widehat{K}_{\text{In}}}{K_{\text{Out}}-K_{\text{In}}}\\\
&\qquad\times{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\
,\\\ \end{split}$ (183)

for the normalisable/regular mode and

$\begin{split}\Xi_{nn}&\equiv
r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1-r^{2})^{-\frac{i\omega}{2}}\Bigl{\\{}{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\Bigr{.}\\\
&-\
\frac{K_{\text{In}}(1+i\cot\nu\pi)\widehat{K}_{\text{Out}}-K_{\text{Out}}(1-i\cot\nu\pi)\widehat{K}_{\text{In}}}{K_{\text{In}}-K_{\text{Out}}}\\\
\
&\Bigl{.}\qquad\times\frac{r^{2\nu}}{2\nu}{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\Bigr{\\}}\
\end{split}$ (184)

for the non-normalisable/singular mode. One advantage of working with such
renormalised functions is that we can safely remove the dimensional
regularisation in the above expressions resulting in a finite limit. When $d$
is odd and $\nu\equiv\ell+\frac{d}{2}-1\in\mathbb{Z}+\frac{1}{2}$, we can
simply set $\cot\nu\pi=0$ and take $\widehat{K}_{\text{Out}}\to
K_{\text{Out}}$ and $\widehat{K}_{\text{In}}\to K_{\text{In}}$. All the $K$’s
then drop out of the above expression, and $\Xi_{nn}$ and $\Xi_{n}$ become
proportional to single hypergeometric functions.

When $d$ is even and $\nu\to n\in\mathbb{Z}$, we can use Eqs.(141) and (156)
to write

$\begin{split}(1+i\cot\nu\pi)\widehat{K}_{\text{Out}}&=\frac{2}{\nu-n}\Delta_{\mathcal{N}}(n,\mu,\omega)+K_{\text{Out}}+O(\nu-n)\
,\\\
(1-i\cot\nu\pi)K_{\text{In}}&=\frac{2}{\nu-n}\Delta_{\mathcal{N}}(n,\mu,\omega)+K_{\text{In}}+O(\nu-n)\
.\end{split}$ (185)

Using these expansions, the $K$s cancel out again and we are left with the
following limits:

$\begin{split}\Xi_{n}|_{\text{Even d}}&\equiv\lim_{\nu\to
n}\frac{1}{2\nu}r^{\nu-\frac{1}{2}(\mathcal{N}-1)}(1-r^{2})^{-\frac{i\omega}{2}}{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\
,\\\ \Xi_{nn}|_{\text{Even d}}&\equiv\lim_{\nu\to
n}r^{-\nu-\frac{1}{2}(\mathcal{N}-1)}(1-r^{2})^{-\frac{i\omega}{2}}\Biggl{\\{}{}_{2}F_{1}\left[\frac{1-\nu+\mu-i\omega}{2},\frac{1-\nu-\mu-i\omega}{2};1-\nu;r^{2}\right]\Biggr{.}\\\
&\Biggl{.}\qquad-\
\frac{r^{2\nu}}{\nu(\nu-n)}\Delta_{\mathcal{N}}(n,\mu,\omega)\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1+\nu;r^{2}\right]\Biggr{\\}}\
.\end{split}$ (186)

One can explicitly check that these limits exist and result in finite
expressions for both regular/singular modes when $d$ is even. To summarise,
Eq.(181) decomposes the outgoing/incoming Green functions into renormalised
pieces in any $d$.

We will now rewrite the full bulk-to-bulk propagator in Eq.(178) in terms of
these renormalised modes. We begin by rewriting the boundary-to-bulk
propagators: using Eq.(160), we obtain

$\begin{split}g_{L}&=n_{\omega}\left(\frac{1-r}{1+r}\right)^{\frac{i\omega}{2}}\Biggl{\\{}\left[1-e^{2\pi\omega(1-\zeta)}\left(\frac{1-r}{1+r}\right)^{-i\omega}\right]\Xi_{nn}\Biggr{.}\\\
&\qquad\Biggl{.}\qquad-\left[K_{\text{Out}}-K_{\text{In}}e^{2\pi\omega(1-\zeta)}\left(\frac{1-r}{1+r}\right)^{-i\omega}\right]\Xi_{n}\Biggr{\\}}\
,\\\
g_{R}&=(1+n_{\omega})\left(\frac{1-r}{1+r}\right)^{\frac{i\omega}{2}}\Biggl{\\{}\left[1-e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}\right]\Xi_{nn}\Biggr{.}\\\
&\qquad\Biggl{.}\qquad-\left[K_{\text{Out}}-K_{\text{In}}e^{-2\pi\omega\zeta}\left(\frac{1-r}{1+r}\right)^{-i\omega}\right]\Xi_{n}\Biggr{\\}}\
.\end{split}$ (187)

Substituting them back into Eq.(178), we get an explicit expression for the
bulk-to-bulk propagator of the form

$\begin{split}\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})&=\frac{1}{W_{LR}(\zeta_{0},\omega,\mathbb{L})}g_{R}(\zeta_{\succ},\omega,\mathbb{L})g_{L}(\zeta_{\prec},\omega,\mathbb{L})\\\
&=\frac{n_{\omega}e^{2\pi\omega\zeta_{0}}}{K_{\text{In}}-K_{\text{Out}}}\left(\frac{1-r}{1+r}\right)^{\frac{i\omega}{2}}\left(\frac{1-r_{0}}{1+r_{0}}\right)^{\frac{i\omega}{2}}\\\
&\times\Biggl{\\{}\left[1-e^{-2\pi\omega\zeta_{\succ}}\left(\frac{1-r_{\succ}}{1+r_{\succ}}\right)^{-i\omega}\right]\Xi_{nn}(r_{\succ})\Biggr{.}\\\
&\qquad\Biggl{.}\qquad-\left[K_{\text{Out}}-K_{\text{In}}e^{-2\pi\omega\zeta_{\succ}}\left(\frac{1-r_{>}}{1+r_{\succ}}\right)^{-i\omega}\right]\Xi_{n}(r_{\succ})\Biggr{\\}}\\\
&\times\Biggl{\\{}\left[1-e^{2\pi\omega(1-\zeta_{\prec})}\left(\frac{1-r_{\prec}}{1+r_{\prec}}\right)^{-i\omega}\right]\Xi_{nn}(r_{\prec})\Biggr{.}\\\
&\qquad\Biggl{.}\qquad-\left[K_{\text{Out}}-K_{\text{In}}e^{2\pi\omega(1-\zeta_{\prec})}\left(\frac{1-r_{\prec}}{1+r_{\prec}}\right)^{-i\omega}\right]\Xi_{n}(r_{\prec})\Biggr{\\}}\
.\end{split}$ (188)

Here, since all quantities are already renormalised, we have removed the
dimensional regularisation242424For odd $d$, we set $\cot\nu\pi=0$ and remove
the hats on $K$s. For even $d$, we use Eq.(185). in the Wronskian given in
Eq.(179). To conclude, given an arbitrary extended source on the dS-SK
geometry, the above bulk-to-bulk propagator, we can get the bulk field by
substituting the above bulk-to-bulk Green function into Eq.(173). Further, we
can also compute the on-shell action Eq.(175), which, according to our
prescription, should yield the influence phase of that extended source.

###  C.4 Extended sources on dS-SK contour II: Radiative multipoles

In this subsection, we would like to evaluate both the field and the influence
of an extended source. We will find it convenient to discretise the source
into a set of spherical shells around the centre of the right/left static
patches. Let $\zeta=1+\zeta_{i}$ characterise the radial position of the
$i^{th}$ spherical shell in the right patch, the same radial position in the
left patch is then characterised by $\zeta=\zeta_{i}$. We will let the $i$
vary over $1$ to $N_{s}$, where $N_{s}$ is the number of shells in each copy
of the static patch. We will take the strength of the scalar source on these
spherical shells to be

$\begin{split}r^{\mathcal{N}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})=\sum_{i}\sigma^{R}_{i}(\omega,\mathbb{L})\
\delta_{c}(\zeta|1+\zeta_{i})-\sum_{i}\sigma^{L}_{i}(\omega,\mathbb{L})\
\delta_{c}(\zeta|\zeta_{i})\ .\end{split}$ (189)

Here, as before, we work in frequency domain/orthonormal spherical harmonic
basis, and allow arbitrary time/angle dependence. Any arbitrary source
distribution confined within the open static patch can be approximated to any
desired accuracy as being built from such spherical shell sources. As we shall
see, such a discrete model regularises the divergences associated with the
self-interactions.

$\displaystyle\zeta=0$$\displaystyle\zeta=1$$1+\zeta_{1}$$\zeta_{1}$$1+\zeta_{3}$$1+\zeta_{2}$$\zeta_{2}$$\zeta_{3}$Re($\displaystyle
r$)Im($\displaystyle r$)$\displaystyle r=H^{-1}$$\displaystyle
R$$\displaystyle L$ Figure 8: Spherical shell sources centred around the
right/left static patches shown in the complex r plane. Their positions on the
$L$ contour are related to their position on the $R$ contour by the branch cut
discontinuity in $\zeta$.

We will begin by writing down the bulk field due to the spherical shell
sources described above. We have, using Eq.(173), a superposition of fields
produced by each shell source, i.e.,

$\begin{split}\varphi_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})&=\oint
r_{0}^{\mathcal{N}}dr_{0}\
\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})\varrho_{{}_{\mathcal{N}}}(\zeta_{0},\omega,\mathbb{L})\\\
&=\sum_{i}\frac{1}{W_{LR}(\zeta_{i},\omega,\mathbb{L})}\begin{cases}e^{2\pi\omega}g_{L}(\zeta,\omega,\mathbb{L})\
\Bigl{[}g_{R}(1+\zeta_{i},\omega,\mathbb{L})\
\sigma^{R}_{i}-g_{L}(1+\zeta_{i},\omega,\mathbb{L})\
\sigma^{L}_{i}\Bigr{]}&\quad\text{if}\;\zeta\prec 1+\zeta_{i}\ ,\\\ &\\\ \quad
g_{R}(\zeta_{i},\omega,\mathbb{L})\ \Bigl{[}g_{R}(\zeta,\omega,\mathbb{L})\
\sigma^{R}_{i}-g_{L}(\zeta,\omega,\mathbb{L})\
\sigma^{L}_{i}\Bigr{]}&\quad\text{if}\;1+\zeta_{i}\prec\zeta\prec\zeta_{i}\
,\\\ &\\\ \quad g_{R}(\zeta,\omega,\mathbb{L})\
\Bigl{[}g_{R}(\zeta_{i},\omega,\mathbb{L})\
\sigma^{R}_{i}-g_{L}(\zeta_{i},\omega,\mathbb{L})\
\sigma^{L}_{i}\Bigr{]}&\quad\text{if}\;\zeta\succ\zeta_{i}\
.\end{cases}\end{split}$ (190)

We remind the reader that $\prec$ and $\succ$ are comparisons using the radial
contour-ordering of the dS-Sk contour. We also remind the reader that $\zeta$
changes from $1$ to $0$, as we traverse the clockwise dS-SK contour, starting
from the right static patch (See Fig.8). The reader should note that the above
superposition of fields is continuous everywhere, but its derivative (and
hence the conjugate field) is discontinuous at each spherical shell, with the
discontinuity being determined by the strength of the scalar source at that
shell. This is expected since the bulk-to-bulk Green function was constructed
in the last subsection with precisely these boundary conditions in mind.

Given the above field, computing the on-shell action is straightforward. We
use Eq.(175) to write

$\begin{split}S|_{\textbf{On-
shell}}&=\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint
r^{\mathcal{N}}dr\
\varrho_{{}_{\mathcal{N}}}^{*}\varphi_{{}_{\mathcal{N}}}|_{\textbf{On-
shell}}\\\
&=\frac{1}{2}\sum_{ij\mathbb{L}}\int\frac{d\omega}{2\pi}\frac{g_{R}(\zeta_{i},\omega,\mathbb{L})}{W_{LR}(\zeta_{i},\omega,\mathbb{L})}\Bigl{\\{}\sigma^{R\ast}_{j}\
\Bigl{[}g_{R}(1+\zeta_{j},\omega,\mathbb{L})\sigma^{R}_{i}-g_{L}(1+\zeta_{j},\omega,\mathbb{L})\sigma^{L}_{i}\Bigr{]}\Bigr{.}\\\
&\qquad\Bigl{.}\qquad-\sigma^{L\ast}_{j}\
\Bigl{[}g_{R}(\zeta_{j},\omega,\mathbb{L})\sigma^{R}_{i}-g_{L}(\zeta_{j},\omega,\mathbb{L})\sigma^{L}_{i}\Bigr{]}\Bigr{\\}}\
.\end{split}$ (191)

Even though we are working with distributional sources/fields, given the
continuity of $\varphi_{{}_{\mathcal{N}}}$, the computation above is
unambiguous. Next, we substitute explicit forms of the boundary-to-bulk Green
functions as well as the Wronskian in terms of renormalised quantities. We
have, using Eqs.(187) and (150), the following set of equalities:

$\begin{split}W_{LR}(\zeta_{i},\omega,\mathbb{L})&=-(1+n_{\omega})\left(\frac{1-r_{i}}{1+r_{i}}\right)^{i\omega}[K_{\text{Out}}-K_{\text{In}}]\
,\\\
\frac{g_{R}(\zeta_{i},\omega,\mathbb{L})}{W_{LR}(\zeta_{i},\omega,\mathbb{L})}&=\left(\frac{1-r_{i}}{1+r_{i}}\right)^{-\frac{i\omega}{2}}\Xi_{n}(r_{i},\omega,\mathbb{L})\
,\\\
g_{L}(\zeta_{i},\omega,\mathbb{L})&=-\left(\frac{1-r_{i}}{1+r_{i}}\right)^{\frac{i\omega}{2}}\Biggl{\\{}\Xi_{nn}(r_{i},\omega,\mathbb{L})+\left[n_{\omega}K_{\text{Out}}-(1+n_{\omega})K_{\text{In}}\right]\Xi_{n}(r_{i},\omega,\mathbb{L})\Biggr{\\}}\
,\\\
g_{L}(1+\zeta_{i},\omega,\mathbb{L})&=-n_{\omega}\left(\frac{1-r_{i}}{1+r_{i}}\right)^{\frac{i\omega}{2}}\left[K_{\text{Out}}-K_{\text{In}}\right]\Xi_{n}(r_{i},\omega,\mathbb{L})\
,\\\
g_{R}(\zeta_{i},\omega,\mathbb{L})&=-(1+n_{\omega})\left(\frac{1-r_{i}}{1+r_{i}}\right)^{\frac{i\omega}{2}}\left[K_{\text{Out}}-K_{\text{In}}\right]\Xi_{n}(r_{i},\omega,\mathbb{L})\
,\\\
g_{R}(1+\zeta_{i},\omega,\mathbb{L})&=\left(\frac{1-r_{i}}{1+r_{i}}\right)^{\frac{i\omega}{2}}\Biggl{\\{}\Xi_{nn}(r_{i},\omega,\mathbb{L})-\left[(1+n_{\omega})K_{\text{Out}}-n_{\omega}K_{\text{In}}\right]\Xi_{n}(r_{i},\omega,\mathbb{L})\Biggr{\\}}\
.\end{split}$ (192)

Substituting these expressions back into the on-shell action yields the
following double sum:

$\begin{split}S|_{\textbf{On-
shell}}&=\frac{1}{2}\sum_{ij\mathbb{L}}\int\frac{d\omega}{2\pi}\left(\frac{1-r_{i}}{1+r_{i}}\right)^{-\frac{i\omega}{2}}\left(\frac{1-r_{j}}{1+r_{j}}\right)^{\frac{i\omega}{2}}\\\
&\qquad\times\Bigl{\\{}\Xi_{n}(r_{i},\omega,\mathbb{L})\
\Xi_{nn}(r_{j},\omega,\mathbb{L})\
[\sigma^{R\ast}_{j}\sigma^{R}_{i}-\sigma^{L\ast}_{j}\sigma^{L}_{i}]\Bigr{.}\\\
&\quad\qquad-\Xi_{n}(r_{i},\omega,\mathbb{L})\
\Xi_{n}(r_{j},\omega,\mathbb{L})\
K_{\text{Out}}(\sigma^{R}_{j}-\sigma^{L}_{j})^{\ast}[(1+n_{\omega})\sigma^{R}_{i}-n_{\omega}\sigma^{L}_{i}]\\\
&\quad\qquad\Bigl{.}-\Xi_{n}(r_{i},\omega,\mathbb{L})\
\Xi_{n}(r_{j},\omega,\mathbb{L})\
K_{\text{In}}(\sigma^{R}_{i}-\sigma^{L}_{i})[(1+n_{-\omega})\sigma^{R\ast}_{j}-n_{-\omega}\sigma^{L\ast}_{j}]\Bigr{]}\
.\end{split}$ (193)

Let us begin by interpreting the terms in the above double sum. We first note
that the last two lines in the above expression are related by the relabelling
$\omega\to-\omega$ and are hence equal. The physical meaning of the last two
lines is clarified by defining the _radiative multipole moments_ :

$\begin{split}\mathcal{J}_{R}(\omega,\mathbb{L})\equiv\sum_{i}\left(\frac{1-r_{i}}{1+r_{i}}\right)^{-\frac{i\omega}{2}}\Xi_{n}(r_{i},\omega,\mathbb{L})\
\sigma_{i}^{R}\equiv\int_{R}dr\ r^{\mathcal{N}}\Xi_{n}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
,\\\
\mathcal{J}_{L}(\omega,\mathbb{L})\equiv\sum_{i}\left(\frac{1-r_{i}}{1+r_{i}}\right)^{-\frac{i\omega}{2}}\Xi_{n}(r_{i},\omega,\mathbb{L})\
\sigma_{i}^{L}\equiv-\int_{L}dr\ r^{\mathcal{N}}\Xi_{n}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
.\end{split}$ (194)

The integrals here are performed over right/left _half_ of the dS-SK contour
respectively. We will also find it convenient to define the average/difference
multipole moments via

$\mathcal{J}_{A}(\omega,\mathbb{L})\equiv\frac{1}{2}[\mathcal{J}_{R}(\omega,\mathbb{L})+\mathcal{J}_{L}(\omega,\mathbb{L})]\
,$

and

$\mathcal{J}_{D}(\omega,\mathbb{L})\equiv\mathcal{J}_{R}(\omega,\mathbb{L})-\mathcal{J}_{L}(\omega,\mathbb{L})$

Here we deliberately use the same notation as we did for multipole moments in
flat spacetime (see Eq.(101)) and for point-like dS sources (See Eq.(8)). One
reason for this is as follows: the last two lines of Eq.(193) can be recast in
terms of the above definitions, into the cosmological influence phase of a
point-source

$\begin{split}S_{\text{CIP}}^{\text{Pt}}&\equiv-\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}K_{\text{Out}}(\mathcal{J}_{R}-\mathcal{J}_{L})^{\ast}[(1+n_{\omega})\mathcal{J}_{R}-n_{\omega}\mathcal{J}_{L}]=-\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}K_{\text{Out}}\
\mathcal{J}_{D}^{*}\
\Bigl{[}\mathcal{J}_{A}+\left(n_{\omega}+\frac{1}{2}\right)\mathcal{J}_{D}\Bigr{]}\
.\end{split}$ (195)

We recognise here the exact influence phase derived for a point-like dS
observer in Eq.(14), using a detailed counterterm procedure. More evidence for
this identification will be presented in appendix D, where we describe how
these multipole moments correctly reproduce the flat space answers with Hubble
corrections.

For now, we turn our attention to the remaining terms, viz., the first double
sum in Eq.(193). The presence of the singular Green solution $\Xi_{nn}$, as
well as the right/left factorised form of this sum, indicates that these terms
incorporate non-dissipative/conservative _self-energy_ corrections of the
extended source. The final on-shell action can then be written as
$S|_{\textbf{On-shell}}=S_{\text{CIP}}^{\text{Pt}}+S_{\text{Int}}$, where
$S_{\text{Int}}$ denotes the internal potential energy of the spherical
shells:

$\begin{split}S_{\text{Int}}&\equiv\frac{1}{2}\sum_{i\neq
j\mathbb{L}}\int\frac{d\omega}{2\pi}\left(\frac{1-r_{i}}{1+r_{i}}\right)^{-\frac{i\omega}{2}}\left(\frac{1-r_{j}}{1+r_{j}}\right)^{\frac{i\omega}{2}}\Xi_{n}(r_{i},\omega,\mathbb{L})\
\Xi_{nn}(r_{j},\omega,\mathbb{L})\
[\sigma^{R\ast}_{j}\sigma^{R}_{i}-\sigma^{L\ast}_{j}\sigma^{L}_{i}]\
.\end{split}$ (196)

Another instructive way to rewrite this potential energy contribution is to
define radially averaged mean fields on the right/left static patch via

$\begin{split}\overline{\varphi}_{R,\text{Int}}(\omega,\mathbb{L})\equiv\sum_{i}\left(\frac{1-r_{i}}{1+r_{i}}\right)^{-\frac{i\omega}{2}}\Xi_{nn}(r_{i},\omega,\mathbb{L})\
\sigma_{i}^{R}\equiv\int_{R}dr\ r^{\mathcal{N}}\Xi_{nn}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
,\\\
\overline{\varphi}_{L,\text{Int}}(\omega,\mathbb{L})\equiv\sum_{i}\left(\frac{1-r_{i}}{1+r_{i}}\right)^{-\frac{i\omega}{2}}\Xi_{nn}(r_{i},\omega,\mathbb{L})\
\sigma_{i}^{L}\equiv-\int_{L}dr\ r^{\mathcal{N}}\Xi_{nn}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
.\end{split}$ (197)

We can then rewrite the potential energy as that of multipole moments placed
in such an average field, viz.,

$\begin{split}S_{\text{Int}}&=\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}[\mathcal{J}_{R}^{\ast}\overline{\varphi}_{R,\text{Int}}-\mathcal{J}_{L}^{\ast}\overline{\varphi}_{L,\text{Int}}]\
.\end{split}$ (198)

###  C.5 Relation to Detweiler-Whiting decomposition

We will begin by writing down the non-normalisable solution $\Xi_{nn}$ and the
normalisable solution $\Xi_{n}$ in the Schwarzschild time valid for odd $d$ (
restoring all factors of $H$):

$\begin{split}\Xi_{nn}&\equiv
r^{-\nu-\frac{d}{2}+\frac{1}{2}(d-1-\mathcal{N})}(1-H^{2}r^{2})^{-\frac{i\omega}{2H}}\\\
&\quad\times{}_{2}F_{1}\left[\frac{1}{2}\left(1+\mu-\nu-\frac{i\omega}{H}\right),\frac{1}{2}\left(1-\mu-\nu-\frac{i\omega}{H}\right),1-\nu,H^{2}r^{2}\right]\
,\\\
\Xi_{n}&\equiv\frac{1}{2\nu}r^{\nu-\frac{d}{2}+1+\frac{1}{2}(d-1-\mathcal{N})}(1-H^{2}r^{2})^{-\frac{i\omega}{2H}}\\\
&\quad\times{}_{2}F_{1}\left[\frac{1}{2}\left(1+\mu+\nu-\frac{i\omega}{H}\right),\frac{1}{2}\left(1-\mu+\nu-\frac{i\omega}{H}\right),1+\nu,H^{2}r^{2}\right]\
,\\\ K_{\text{Out}}&\equiv
2\frac{\Gamma\left(\frac{1+\nu-\mu-\frac{i\omega}{H}}{2}\right)\Gamma\left(\frac{1+\nu+\mu-\frac{i\omega}{H}}{2}\right)\Gamma\left(1-\nu\right)}{\Gamma\left(\frac{1-\nu+\mu-\frac{i\omega}{H}}{2}\right)\Gamma\left(\frac{1-\nu-\mu-\frac{i\omega}{H}}{2}\right)\Gamma\left(\nu\right)}\
.\end{split}$ (199)

We have also quoted above the retarded two-point function on the world line.
The outgoing Green function can then be decomposed into $K_{\text{Out}}\
\Xi_{n}$ and $\Xi_{nn}$: we will now argue that this should be thought of as
the regular/singular Green functions ala Deitweiler-
Whiting(DW)Detweiler:2002mi  corresponding to dS spacetime.

The relation to DW decomposition is not prima facie clear, since DW formulated
their rules for general curved spacetimes in time domain, whereas the above
expressions are quoted in frequency domain. So, to substantiate our assertion,
we need to Fourier transform the complicated expressions above into time
domain, and then show that the DW axioms are satisfied. Rather than do that
exercise in general, we will content ourselves with showing how this works in
the particular example of a massless scalar field in $dS_{4}$, whose DW
decomposition is described in Burko:2002ge ; Poisson:2011nh .

The regular term for DW decomposition in this case was calculated by the
authors of Burko:2002ge  in FLRW-like coordinates as

$\begin{split}G_{R}&=\frac{\eta\eta^{\prime}}{2|\mathbf{x}-\mathbf{x}^{\prime}|}\left[\delta(\eta-\eta^{\prime}-|\mathbf{x}-\mathbf{x}^{\prime}|)-\delta(\eta-\eta^{\prime}+|\mathbf{x}-\mathbf{x}^{\prime}|)\right]\\\
&+\frac{1}{2}\left[\theta(\eta-\eta^{\prime}-|\mathbf{x}-\mathbf{x}^{\prime}|)+\theta(\eta-\eta^{\prime}+|\mathbf{x}-\mathbf{x}^{\prime}|)\right]\
,\end{split}$ (200)

To check this expression against $K_{\text{Out}}\Xi_{n}$, we will convert it
into static coordinates and then Fourier transform the result to frequency
domain.

The coordinate transformation between static and FLRW coordinates is given by

$\begin{split}\eta&=-\frac{e^{-Ht}}{\sqrt{1-r^{2}H^{2}}}\
,\quad\rho=\frac{re^{-Ht}}{\sqrt{1-r^{2}H^{2}}}\ .\end{split}$ (201)

We will assume the source to be at the origin $\rho^{\prime}=0$, so that only
the $\ell=0$ term survives by spherical symmetry. With this choice, $G_{R}$
becomes

$\begin{split}G_{R}&=\frac{e^{H(t-t^{\prime})}}{2r}\\\
&\times\left[\sqrt{\frac{1-Hr}{1+Hr}}\delta\left(t^{\prime}-t-\frac{1}{H}\ln\left(\sqrt{\frac{1-Hr}{1+Hr}}\right)\right)-\sqrt{\frac{1+Hr}{1-Hr}}\delta\left(t^{\prime}-t-\frac{1}{H}\ln\left(\sqrt{\frac{1+Hr}{1-Hr}}\right)\right)\right]\\\
&+\frac{1}{2}\left[\theta\left(t^{\prime}-t-\frac{1}{H}\ln\left(\sqrt{\frac{1-Hr}{1+Hr}}\right)\right)+\theta\left(t^{\prime}-t-\frac{1}{H}\ln\left(\sqrt{\frac{1+Hr}{1-Hr}}\right)\right)\right]\
.\end{split}$ (202)

This expression can be readily Fourier transformed with respect to
$t-t^{\prime}$ yielding

$\widetilde{G}_{R}=\frac{1}{2r}\left[\left(\frac{1-Hr}{1+Hr}\right)^{-\frac{i\omega}{2H}}-\left(\frac{1+Hr}{1-Hr}\right)^{-\frac{i\omega}{2H}}\right]-\frac{H^{2}}{2i\omega}\left[\left(\frac{1-Hr}{1+Hr}\right)^{-\frac{i\omega}{2H}}+\left(\frac{1+Hr}{1-Hr}\right)^{-\frac{i\omega}{2H}}\right]\
.$ (203)

Regularity near the origin is manifest in frequency domain. Further, the above
expression is also an odd function of the frequency $\omega$, signalling that
these terms encode the dissipation due to radiation reaction.

Similarly, we can consider the singular Green’s function quoted in
Burko:2002ge :

$\displaystyle\begin{split}G_{S}=\frac{\eta\eta^{\prime}}{2|\mathbf{x}-\mathbf{x}^{\prime}|}\left[\delta(\eta-\eta^{\prime}-|\mathbf{x}-\mathbf{x}^{\prime}|)+\delta(\eta-\eta^{\prime}+|\mathbf{x}-\mathbf{x}^{\prime}|)\right]\\\
+\frac{1}{2}\left[\theta(\eta-\eta^{\prime}-|\mathbf{x}-\mathbf{x}^{\prime}|)-\theta(\eta-\eta^{\prime}+|\mathbf{x}-\mathbf{x}^{\prime}|)\right]\end{split}$
(204)

whose Fourier transform at $\rho^{\prime}=0$ is

$\tilde{G}_{S}=\frac{1}{2r}\left[\left(\frac{1-Hr}{1+Hr}\right)^{-\frac{i\omega}{2H}}+\left(\frac{1+Hr}{1-Hr}\right)^{-\frac{i\omega}{2H}}\right]-\frac{H^{2}}{2i\omega}\left[\left(\frac{1-Hr}{1+Hr}\right)^{-\frac{i\omega}{2H}}-\left(\frac{1+Hr}{1-Hr}\right)^{-\frac{i\omega}{2H}}\right]\
.$ (205)

This expression has a $\sim\frac{1}{r}$ behaviour near the origin and is an
even function of $\omega$. The expressions in Eq.(203) and Eq.(203) can then
be matched against $K_{\text{Out}}\ \Xi_{n}$ and $\Xi_{nn}$ respectively. This
is done by taking Eq.(199), setting
$\mathcal{N}=d-1,\mu=\frac{d}{2},\nu=\ell+\frac{d}{2}-1$, and then taking the
limit $d=3$ and $\ell=0$.

##  Appendix D Radiation reaction due to light scalar fields

In this section, we will evaluate the radiation reaction force on a dS point
particle coupled to a scalar field. We will do this in a small curvature
approximation, i.e., we begin with the leading order result in flat
spacetimeBirnholtz:2013ffa ; Birnholtz:2013nta  and then systematically
correct it for curvature effects. In dS, Hubble constant $H$ parametrises the
deviation from flat spacetime, so the small curvature expansion is an
expansion in $H$. We will also work within a non-relativistic expansion and a
multipole expansion, and then eventually covariantise the final answer for the
radiation reaction(RR).

To this end, consider a point-like source moving along a time-like worldline
$x(\tau)$ in dS, where $\tau$ denotes the proper time of the source. We will
assume that the particle trajectory is close to the south pole ($rH\ll 1$) and
the radiation wavelength is taken to be much larger than the length scale of
the particle trajectory ($\omega r\ll 1$), but much smaller than the curvature
length scale ($\omega\gg H$). Further, we work in a non-relativistic limit
($v\ll 1$). Thus, we consider the following hierarchy of scales (See Fig.7):

$\displaystyle\begin{split}H\ll\omega\ll 1/r\ .\end{split}$ (206)

In analogy with flat spacetime, we will refer to this expansion as the post-
Newtonian(PN) expansion in dS.

The source density for a moving source in dS is given by

$\displaystyle\begin{split}\widetilde{\rho}(x^{\prime})=\int\delta^{d+1}(x-x^{\prime})d\tau=&\sqrt{1-H^{2}r^{2}-\frac{\dot{r}^{2}}{1-H^{2}r^{2}}+\dot{r}^{2}-v^{2}}\
\delta^{d}(\vec{x}-\vec{x}^{\prime})\ ,\end{split}$ (207)

where we have defined $v=\sqrt{\sum_{i=1}^{d}\dot{x}_{i}^{2}}$. Here the dots
denote the derivative with respect to the standard time $t$. The time-
dilation/length-contraction factor for the particle worldline can then be
expanded as follows:

$\begin{split}&\sqrt{1-H^{2}r^{2}-\frac{\dot{r}^{2}}{1-H^{2}r^{2}}+\dot{r}^{2}-v^{2}}\\\
&=-\sum_{n,s,k}\binom{n}{k}\frac{(2s+2n+2k-5)!!(2s+2k-5)!!}{2^{n+s-1}(2s+4k-5)!!\
n!(s-1)!}(Hr)^{2n}\dot{r}^{2k}v^{2s-2}\ .\end{split}$ (208)

Here the sum is over all non-negative integers, and the binomial coefficient
vanishes for all values of $k$ outside the range $0\leq k\leq n$. This
expansion describes the red-shift of the particle within dS spacetime in a
slow-motion approximation, assuming that both the cosmological redshift $Hr$
as well as the Doppler red-shifts due to peculiar motions (proportional to
$\dot{r}$ and $v$) are small. Our strategy below will be to use the above
expansion to compute the symmetric trace-free(STF) multipole moments of the
source, which can then be fed into the cosmological influence phase to compute
the RR force.

We caution the reader that the source form given in Eq.(207) is specific to
the KG scalar case with $\mathcal{N}=d-1$. This is _not_ the correct form of
the source for the scalar/vector/tensor sectors of EM field and gravity. For
such cases, the explicit form of sources involves extra velocity/time-dilation
factors, e.g., EM vector sector source is the electric current carried by the
particle which has additional velocity dependence not captured by Eq.(207).
Another related comment on EM/gravity sources is that the RR force coming from
just one of the sectors is not expected to be covariantBirnholtz:2013nta : one
should add in the contributions from all sectors to derive a covariant force
expression. To do this for EM/linearised gravity, we need a theory of
vector/tensor STF expansions (i.e., a formalism analogous to the one described
in appendix A). We will derive such a formalism elsewheredSSKvec : in this
note, we will limit our RR force analysis to KG scalars.

In the dS-SK geometry the above source will be doubled to a
$\widetilde{\rho}_{L}$ and a $\widetilde{\rho}_{R}$ coming from left/right
trajectories $x_{L}(\tau_{L})$ and $x_{R}(\tau_{R})$. The degrees of freedom
of our open system are thus two copies of the position of the particle and its
time derivatives:
$\\{x_{L},x_{R},\dot{x}_{L},\dot{x}_{R},\ddot{x}_{L},\ddot{x}_{R},\dots\\}$.
The scalar ALD force and its post-Newtonian corrections only require
expressions linear in $x_{D},\dot{x}_{D},\ddot{x}_{D},\dots$ which are the
difference in the positions and their derivatives. We will also keep terms up
to cubic powers of $x_{D}$. In this approximation, the average and difference
functions of the sources can be written in a simple way. Consider for
illustration, the average and difference functions of just the position:

$\displaystyle\frac{1}{2}\left[\mathfrak{f}\left(x_{A}+\frac{x_{D}}{2}\right)+\mathfrak{f}\left(x_{A}-\frac{x_{D}}{2}\right)\right]=\mathfrak{f}(x_{A})+\frac{x_{D}^{2}}{4}\frac{\partial^{2}\mathfrak{f}}{\partial
x_{A}^{2}}+O(x_{D}^{4})$ (209)
$\displaystyle\mathfrak{f}\left(x_{A}+\frac{x_{D}}{2}\right)-\mathfrak{f}\left(x_{A}-\frac{x_{D}}{2}\right)=x_{D}\frac{\partial\,\mathfrak{f}}{\partial
x_{A}}+\frac{x_{D}^{3}}{24}\frac{\partial^{3}\,\mathfrak{f}}{\partial
x_{A}^{3}}+O(x_{D}^{4})$ (210)

In general, the sources will be functions not only of the positions but also
their time derivatives: in such cases, the above formula should be interpreted
as a multi-variable Madhava-Taylor expansion.

We will now substitute the particle source Eq.(207) into the multipole moments
defined in Eq.(194) and obtain the Lagrangian for RR force in PN expansion. We
begin by expressing the influence phase in terms of STF moments of the
particle density: we proceed similarly to how we rewrote the RR influence
phase in flat spacetime (Eq.(102)) in terms of STF multipole moments
(Eq.(106)). Using the STF addition theorem in Eq.(84), we can rewrite the
dissipative part of Eq.(195) in time-domain as:

$\begin{split}S_{RR}^{\text{Odd
$d$}}&=\sum_{\ell}\int\frac{d\omega}{2\pi}\frac{K_{\text{Out}}}{\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|}\frac{1}{\ell!}\mathcal{Q}_{D,STF}^{\ast<i_{1}i_{2}\ldots
i_{\ell}>}\mathcal{Q}^{A,STF}_{<i_{1}i_{2}\ldots i_{\ell}>}\ .\end{split}$
(211)

where we have defined the time-domain STF multipole moments in dS as

$\begin{split}\mathcal{Q}_{A,STF}^{i_{1}\ldots
i_{\ell}}(t)\equiv\Pi^{<i_{1}i_{2}\ldots i_{\ell}>}_{<j_{1}j_{2}\ldots
j_{\ell}>}\int r^{d-1}dr\
\hat{r}^{j_{1}}\hat{r}^{j_{2}}\ldots\hat{r}^{j_{\ell}}\
\Xi_{n}(i\partial_{t},r)\widetilde{\rho}_{A}(t,\vec{r})\ ,\\\
\mathcal{Q}_{D,STF}^{i_{1}\ldots i_{\ell}}(t)\equiv\Pi^{<i_{1}i_{2}\ldots
i_{\ell}>}_{<j_{1}j_{2}\ldots j_{\ell}>}\int r^{d-1}dr\
\hat{r}^{j_{1}}\hat{r}^{j_{2}}\ldots\hat{r}^{j_{\ell}}\
\Xi_{n}(i\partial_{t},r)\widetilde{\rho}_{D}(t,\vec{r})\ .\end{split}$ (212)

We will now use the PN expansion for $\widetilde{\rho}_{D}$ in Eq.(208), the
expansion for $K_{\text{Out}}$ from Eq.(239) and the expansion for $\Xi_{n}$
from Eq.(222) respectively. Keeping all terms in the action up to quartic
order in amplitudes (i.e. in position $x$) and quartic order in Hubble
constant $H$, we get an effective Lagrangian of the form

$\begin{split}|\mathbb{S}^{d-1}|(d-2)!!\times(-1)^{\frac{d-1}{2}}L=&-[x_{i}]_{D}\mathbb{D}_{1}[x^{i}]_{A}+\frac{1}{2}\left[x_{i}x_{j}-\frac{x^{2}}{d}\delta_{ij}\right]_{D}\mathbb{D}_{2}\left[x^{i}x^{j}-\frac{x^{2}}{d}\delta_{ij}\right]_{A}\\\
&-\left\\{\frac{1}{2}(x_{i})_{D}\mathbb{D}_{1}^{X}[x^{i}x^{2}]_{A}+\frac{1}{2}(x^{2}x_{i})_{D}\mathbb{D}_{1}^{X}[x^{i}]_{A}\right\\}\\\
&+\left\\{\frac{1}{2}(x_{i})_{D}\mathbb{D}_{1}^{V}[x^{i}v^{2}]_{A}+\frac{1}{2}(v^{2}x_{i})_{D}\mathbb{D}_{1}^{V}[x^{i}]_{A}\
\right\\}\\\
&+\frac{1}{2d}[x^{2}]_{D}\mathbb{D}_{0}^{XX}[x^{2}]_{A}+\frac{1}{4}[v^{2}]_{D}\mathbb{D}_{0}^{VV}[v^{2}]_{A}\\\
&-\frac{1}{4}[x^{2}]_{D}\mathbb{D}_{0}^{XV}[v^{2}]_{A}-\frac{1}{4}[v^{2}]_{D}\mathbb{D}_{0}^{XV}[x^{2}]_{A}\
.\end{split}$ (213)

Here, we have the seven differential operators, each built out of a finite
number of time-derivatives with constant coefficients, and labelled by
$\\{\mathbb{D}_{1},\mathbb{D}_{2},\mathbb{D}_{1}^{X},\mathbb{D}_{1}^{V},\mathbb{D}_{0}^{XX},\mathbb{D}_{0}^{VV},\mathbb{D}_{0}^{XV}\\}$.
We use the subscripts of these operators to denote the multipole number,
whereas the superscripts are used to distinguish between different structures
occurring at the same multipole number. The explicit form of these operators
is tabulated in table 8. We note that terms beyond quadrupole moment do not
contribute to the quartic influence phase.

We can expand out the average and the difference multipole moments in terms of
the average/difference in the particle position by using the following
identities:

$\begin{split}[Z]_{d}[Y^{3}]_{a}&=Z_{d}Y_{a}^{3}+\frac{1}{4}Z_{d}\
(3Y_{d}^{2}Y_{a})\\\
[Z^{2}]_{d}[Y^{2}]_{a}&=(2Z_{d}Z_{a})Y_{a}^{2}+\frac{1}{4}(2Z_{d}Z_{a})Y_{d}^{2}\\\
[Z^{3}]_{d}[Y]_{a}&=(3Z_{a}^{2}Z_{d})Y_{a}+\frac{1}{4}Z_{d}^{3}Y_{a}\
.\end{split}$ (214)

After integration by parts, the above Lagrangian can be cast into the form:

$L=\frac{(-1)^{\frac{d-1}{2}}}{|\mathbb{S}^{d-1}|(d-2)!!}\left[F_{i}(x_{A})x^{i}_{D}+\frac{1}{4}N_{i}(x_{D})x^{i}_{A}\right]$
(215)

where $F^{i}$ are the Euler-Lagrange derivatives of the terms linear in
$x_{D}$ with respect to $x^{i}_{D}$. Similarly, $N^{i}$ are the Euler-Lagrange
derivatives of the terms linear in $x_{A}$ with respect to $x^{i}_{A}$.

The terms in the Lagrangian which are cubic in $x_{D}$ give rise to noise
terms $N^{i}$. These terms are different from pure noise terms, i.e. those
quartic in $x_{D}$. The $N^{i}$ contribute to the radiation reaction with a
$x^{i}_{A}$ dependent term. It should be noted that this noise is not thermal
in origin. Rather, the origin of this noise can be understood as follows:
Despite the scalar field coupling linearly to the multipole moments, the
moments themselves are non-linear functions of the positions. Hence, the open
system described in terms of position has extra noise terms.

Both the $F^{i}$ and the $N^{i}$ can be written in terms of the operators
given in table 8 as:

$\begin{split}F^{i}=&-\mathbb{D}_{1}[x^{i}]+x_{j}\mathbb{D}_{2}[x^{i}x^{j}]-\frac{x^{i}}{d}\mathbb{D}_{2}[x^{2}]\\\
&-\left\\{\frac{1}{2}\mathbb{D}_{1}^{X}[x^{i}x^{2}]+\frac{1}{2}x^{2}\mathbb{D}_{1}^{X}[x^{i}]+x^{i}x^{j}\mathbb{D}_{1}^{X}[x_{j}]\right\\}+\left\\{\frac{1}{2}\mathbb{D}_{1}^{V}[x^{i}v^{2}]+\frac{1}{2}v^{2}\mathbb{D}_{1}^{V}[x^{i}]-\partial_{t}\left(v^{i}x^{j}\mathbb{D}_{1}^{V}[x_{j}]\
\right)\right\\}\\\
&+\frac{x^{i}}{d}\mathbb{D}_{0}^{XX}[x^{2}]-\frac{1}{2}\partial_{t}\left(v^{i}\mathbb{D}_{0}^{VV}[v^{2}]\
\right)+\left\\{\frac{1}{2}\partial_{t}\left(v^{i}\mathbb{D}_{0}^{XV}[x^{2}]\
\right)-\frac{1}{2}x^{i}\mathbb{D}_{0}^{XV}[v^{2}]\right\\}\ ,\end{split}$
(216)

as well as

$\begin{split}N^{i}(x)=&x_{j}\mathbb{D}_{2}[x^{i}x^{j}]-\frac{x^{i}}{d}\mathbb{D}_{2}[x^{2}]\\\
&+\left\\{\frac{1}{2}\mathbb{D}_{1}^{X}[x^{i}x^{2}]+\frac{1}{2}x^{2}\mathbb{D}_{1}^{X}[x^{i}]+x^{i}x^{j}\mathbb{D}_{1}^{X}[x_{j}]\right\\}-\left\\{\frac{1}{2}\mathbb{D}_{1}^{V}[x^{i}v^{2}]+\frac{1}{2}v^{2}\mathbb{D}_{1}^{V}[x^{i}]-\partial_{t}\left(v^{i}x^{j}\mathbb{D}_{1}^{V}[x_{j}]\
\right)\right\\}\\\
&+\frac{x^{i}}{2d}\mathbb{D}_{0}^{XX}[x^{2}]-\frac{1}{2}\partial_{t}\left(v^{i}\mathbb{D}_{0}^{VV}[v^{2}]\
\right)+\left\\{\frac{1}{2}\partial_{t}\left(v^{i}\mathbb{D}_{0}^{XV}[x^{2}]\
\right)-\frac{1}{2}x^{i}\mathbb{D}_{0}^{XV}[v^{2}]\right\\}\end{split}$ (217)

The RR force $F^{i}$ then covariantises to the expression given in Eq.(34).
Beyond the expressions quoted in the main text, we have also covariantised RR
force expressions in $d=9,11$. they are given by

$\begin{split}{}^{0}f^{\mu}_{9}&\equiv\frac{P^{\mu\nu}}{9!!}\left\\{-a_{\nu}^{(7)}+30\
(a\cdot a)\ a_{\nu}^{(5)}+210\ (a\cdot a^{(1)})\ a_{\nu}^{(4)}+378\ (a\cdot
a^{(2)})\ a_{\nu}^{(3)}\right.\\\ &\left.\qquad\qquad\quad+420\ (a\cdot
a^{(3)})\ a_{\nu}^{(2)}+300\ (a\cdot a^{(4)})\ a_{\nu}^{(1)}+108\ (a\cdot
a^{(5)})\ a_{\nu}\right.\\\ &\left.\qquad\qquad+336\ (a^{(1)}\cdot a^{(1)})\
a_{\nu}^{(3)}+1050\ (a^{(1)}\cdot a^{(2)})\ a_{\nu}^{(2)}+960\ (a^{(1)}\cdot
a^{(3)})\ a_{\nu}^{(1)}+420\ (a^{(1)}\cdot a^{(4)})\ a_{\nu}\right.\\\
&\left.\qquad\qquad+675\ (a^{(2)}\cdot a^{(2)})\ a_{\nu}^{(1)}+756\
(a^{(2)}\cdot a^{(3)})\ a_{\nu}+O(a^{5})\right\\}\\\
&-H^{2}\frac{P^{\mu\nu}}{9!!}\left\\{a_{\nu}^{(5)}+97\ (a\cdot a)\
a_{\nu}^{(3)}+433\ (a\cdot a^{(1)})\ a_{\nu}^{(2)}+408\ (a\cdot a^{(2)})\
a_{\nu}^{(1)}+199\ (a\cdot a^{(3)})\ a_{\nu}\right.\\\
&\left.\qquad\qquad+339\ (a^{(1)}\cdot a^{(1)})\ a_{\nu}^{(1)}+448\
(a^{(1)}\cdot a^{(2)})\ a_{\nu}+O(a^{5})\right\\}\\\
&+H^{4}\frac{P^{\mu\nu}}{9!!}\left\\{-a_{\nu}^{(3)}+157\ (a\cdot a)\
a_{\nu}^{(1)}+296\ (a\cdot a^{(1)})\ a_{\nu}\right\\}+O(H^{6})\ ,\end{split}$
(218)

as well as

$\begin{split}{}^{0}f^{\mu}_{11}&\equiv\frac{P^{\mu\nu}}{11!!}\left\\{-a_{\nu}^{(9)}+55\
(a\cdot a)\ a_{\nu}^{(7)}+495\ (a\cdot a^{(1)})\ a_{\nu}^{(6)}+1188\ (a\cdot
a^{(2)})\ a_{\nu}^{(5)}\right.\\\ &\left.\qquad\qquad\quad+1848\ (a\cdot
a^{(3)})\ a_{\nu}^{(4)}+1980\ (a\cdot a^{(4)})\ a_{\nu}^{(3)}\right.\\\
&\left.\qquad\qquad\quad+1485\ (a\cdot a^{(5)})\ a_{\nu}^{(2)}+770\ (a\cdot
a^{(6)})\ a_{\nu}^{(1)}+220\ (a\cdot a^{(7)})\ a_{\nu}\right.\\\
&\left.\qquad\qquad+1056\ (a^{(1)}\cdot a^{(1)})\ a_{\nu}^{(5)}+4620\
(a^{(1)}\cdot a^{(2)})\ a_{\nu}^{(4)}\right.\\\ &\left.\qquad\qquad\quad+6336\
(a^{(1)}\cdot a^{(3)})\ a_{\nu}^{(3)}+5775\ (a^{(1)}\cdot a^{(4)})\
a_{\nu}^{(2)}+3520\ (a^{(1)}\cdot a^{(5)})\ a_{\nu}^{(1)}+1155\ (a^{(1)}\cdot
a^{(6)})\ a_{\nu}\right.\\\ &\left.\qquad\qquad+4455\ (a^{(2)}\cdot a^{(2)})\
a_{\nu}^{(3)}+10395\ (a^{(2)}\cdot a^{(3)})\ a_{\nu}^{(2)}+7700\ (a^{(2)}\cdot
a^{(4)})\ a_{\nu}^{(1)}+2970\ (a^{(2)}\cdot a^{(5)})\ a_{\nu}\right.\\\
&\left.\qquad\qquad+4928\ (a^{(3)}\cdot a^{(3)})\ a_{\nu}^{(1)}+4620\
(a^{(3)}\cdot a^{(4)})\ a_{\nu}\right\\}\\\
&-H^{2}\frac{P^{\mu\nu}}{11!}\left\\{a_{\nu}^{(7)}+342\ (a\cdot a)\
a_{\nu}^{(5)}+2294\ (a\cdot a^{(1)})\ a_{\nu}^{(4)}+3826\ (a\cdot a^{(2)})\
a_{\nu}^{(3)}\right.\\\ &\left.\qquad\qquad\quad+3737\ (a\cdot a^{(3)})\
a_{\nu}^{(2)}+2066\ (a\cdot a^{(4)})\ a_{\nu}^{(1)}+622\ (a\cdot a^{(5)})\
a_{\nu}\right.\\\ &\left.\qquad\qquad+3231\ (a^{(1)}\cdot a^{(1)})\
a_{\nu}^{(3)}+8490\ (a^{(1)}\cdot a^{(2)})\ a_{\nu}^{(2)}+5663\ (a^{(1)}\cdot
a^{(3)})\ a_{\nu}^{(1)}+1974\ (a^{(1)}\cdot a^{(4)})\ a_{\nu}\right.\\\
&\left.\qquad\qquad+3785\ (a^{(2)}\cdot a^{(2)})\ a_{\nu}^{(1)}+3210\
(a^{(2)}\cdot a^{(3)})\ a_{\nu}+O(a^{5})\right\\}\ ,\\\
&-H^{4}\frac{P^{\mu\nu}}{11!!}\left\\{a_{\nu}^{(5)}+1340\ (a\cdot a)\
a_{\nu}^{(3)}+6108\ (a\cdot a^{(1)})\ a_{\nu}^{(2)}+6148\ (a\cdot a^{(2)})\
a_{\nu}^{(1)}+2599\ (a\cdot a^{(3)})\ a_{\nu}\right.\\\
&\left.\qquad\qquad+5182\ (a^{(1)}\cdot a^{(1)})\ a_{\nu}^{(1)}+5876\
(a^{(1)}\cdot a^{(2)})\ a_{\nu}+O(a^{5})\right\\}\ .\end{split}$ (219)

We have not yet succeeded in finding similar covariant expressions for the
$N^{i}$.

###  D.1 Near flat expansions for odd $d$

In this subsection, we will describe how normalisable modes of the generalised
scalar equation in dS can be thought of as perturbations of the corresponding
Bessel J modes in the flat spacetime. In the context of radiation reaction
problems, these modes are essential in defining the radiative multipole
moments: their role is to appropriately smear the sources to take into account
time-delay effects. Once such an expansion is obtained, it is easy to find the
flat space expansion of the non-normalisable mode for even dimensional dS just
by analytical continuation.

The solution of the generalised scalar wave equation, regular at $r=0$, is
given by

$\begin{split}\Xi_{n}&\equiv\frac{1}{2\nu}r^{\nu-\frac{d}{2}+1+\frac{1}{2}(d-1-\mathcal{N})}(1-H^{2}r^{2})^{-\frac{i\omega}{2H}}\\\
&\quad\times{}_{2}F_{1}\left[\frac{1}{2}\left(1+\mu+\nu-\frac{i\omega}{H}\right),\frac{1}{2}\left(1-\mu+\nu-\frac{i\omega}{H}\right),1+\nu,H^{2}r^{2}\right]\
.\end{split}$ (220)

Here we have made all $H$ factors explicit so that the $H\to 0$ limit can
readily be taken. In such a limit, the above expression reduces to a Bessel
$J$ function. More explicitly, we will find it convenient to define a sequence
of scaled Bessel $J$ functions of the form

$\begin{split}\mathfrak{B}_{k}&\equiv\frac{r^{\nu-\frac{d}{2}+1+\frac{1}{2}(d-1-\mathcal{N})+2k}}{2\nu(\nu+1)\ldots(\nu+k)}\
{}_{0}F_{1}\left[1+k+\nu,-\frac{\omega^{2}r^{2}}{4}\right]=\frac{\Gamma(\nu)\
r^{\frac{1}{2}(1-\mathcal{N})+k}}{2(\omega/2)^{k+\nu}}J_{k+\nu}(\omega r)\
\end{split}$ (221)

in flat spacetime. In terms of these functions, the dS wavefunction $\Xi_{n}$
has a small $H$ expansion

$\begin{split}\Xi_{n}=\sum_{k=0}^{\infty}\mathfrak{p}_{k}(\nu,H^{2},\omega^{2})\mathfrak{B}_{k}\
,\end{split}$ (222)

with $\mathfrak{p}_{k}(H^{2},\omega^{2})$ being a homogeneous polynomial of
degree $k$ in the variables $H^{2}$ and $\omega^{2}$. An explicit expression
is given by

$\begin{split}\mathfrak{p}_{k}&\equiv\frac{H^{2k}}{k!}\sum_{m=0}^{k}(-)^{m}\binom{k}{m}\sum_{n=0}^{m}(-)^{n}\binom{m}{n}\sigma^{2k-2m}\frac{\Gamma(\alpha+m)\Gamma(1+\nu+m)}{\Gamma(\alpha+m-n)\Gamma(1+\nu+m-n)}\
\\\
&\quad\qquad\qquad\times\frac{\Gamma(\alpha+i\sigma+m-n)\Gamma(\alpha-i\sigma+m-n)}{\Gamma(\alpha+i\sigma)\Gamma(\alpha-i\sigma)}\
,\end{split}$ (223)

where we have defined the variables

$\begin{split}\alpha\equiv\frac{1}{2}(1+\nu-\mu)\
,\quad\sigma=\frac{\omega}{2H}\ .\end{split}$ (224)

A useful fact about these polynomials is the leading $H$ scaling at small $H$
of these polynomials, given by

$\begin{split}\mathfrak{p}_{3n-2}\ ,\ \mathfrak{p}_{3n-1}\ ,\
\mathfrak{p}_{3n}\ \propto H^{2n}\ .\end{split}$ (225)

Thus, to obtain an answer accurate up to $H^{2n}$, we only need the
polynomials till $\mathfrak{p}_{3n}$.

We will now outline a derivation for the above expansion as follows: first, we
use the Euler transformation on the hypergeometric functions to write

$\begin{split}\Xi_{n}&=\frac{1}{2\nu}r^{\nu-\frac{d}{2}+\frac{1}{2}(d-1-\mathcal{N})}(1-H^{2}r^{2})^{-\alpha}\
{}_{2}F_{1}\left[\alpha+i\sigma,\alpha-i\sigma,1+\nu,-\frac{H^{2}r^{2}}{1-H^{2}r^{2}}\right]\
,\end{split}$ (226)

where the variables $\alpha$ and $\sigma$ are as defined above. In the next
step, we employ the Mellin-Barnes representation of the hypergeometric
function, viz.NIST:DLMF ,

$\begin{split}{}_{2}F_{1}\left[a,b,c,x\right]=\int_{-i\infty}^{i\infty}\frac{dz}{2\pi
i}(-x)^{z}\Gamma(-z)\frac{\Gamma(a+z)\Gamma(b+z)\Gamma(c)}{\Gamma(a)\Gamma(b)\Gamma(c+z)}\
.\end{split}$ (227)

and expand the resultant integrand using

$\begin{split}(1-H^{2}r^{2})^{-\alpha-z}\left(H^{2}r^{2}\right)^{z}=\sum_{k=0}^{\infty}\frac{(H^{2}r^{2})^{k+z}}{k!}\frac{\Gamma\left[k+z+\frac{1}{2}(1+\nu-\mu)\right]}{\Gamma\left[\alpha+z+\frac{1}{2}(1+\nu-\mu)\right]}\
.\end{split}$ (228)

Shifting the Mellin-Barnes integration variable, we get the following Mellin-
integral representation for $\Xi_{n}$:

$\begin{split}\Xi_{n}&=\frac{1}{2\nu}r^{\nu-\frac{d}{2}+\frac{1}{2}(d-1-\mathcal{N})}\int_{-i\infty}^{i\infty}\frac{dz}{2\pi
i}\left(\frac{\omega
r}{2}\right)^{2z}\Gamma(-z)\frac{\Gamma(c)}{\Gamma(c+z)}\widetilde{\Xi}(z)\
,\end{split}$ (229)

where the Mellin-transform

$\begin{split}\widetilde{\Xi}(z)&\equiv\left(\frac{2H}{\omega}\right)^{2z}\sum_{k=0}^{\infty}(-)^{k}\binom{z}{k}\frac{\Gamma(a+z-k)\Gamma(b+z-k)\Gamma(c+z)}{\Gamma(a)\Gamma(b)\Gamma(c+z-k)}\frac{\Gamma(z+\alpha)}{\Gamma(z-k+\alpha)}\\\
&=\left(\frac{2H}{\omega}\right)^{2z}\frac{\Gamma(a+z)\Gamma(b+z)}{\Gamma(a)\Gamma(b)}{}_{3}F_{2}\left[\begin{array}[]{ccc}1-c-z,&-z,&1-\alpha-z\\\
1-a-z&,&1-b-z\end{array};1\right]\ .\end{split}$ (230)

Here, $\alpha\equiv\frac{1}{2}(1+\nu-\mu)$ and $a,b,c$ denote the parameters
of the hyper-geometric function appearing in Eq.(226). The Mellin-transform
$\widetilde{\Xi}(z)$ evaluated at integer $z$ is, in fact, a polynomial of
degree $z$ in the variable $(H/\omega)^{2}$: this can be gleaned from the fact
that the series above truncates in this case with polynomial coefficients.

To determine the polynomials $\mathfrak{p}_{n}$, we should compare the
polynomials $\widetilde{\Xi}(n)$ against the Mellin-transform of
$\sum_{k}\mathfrak{p}_{k}\mathfrak{B}_{k}$. This can be done using the Mellin-
Barnes representation of ${}_{0}F_{1}$, viz.NIST:DLMF ,

$\begin{split}{}_{0}F_{1}\left[c,x\right]=\int_{-i\infty}^{i\infty}\frac{dz}{2\pi
i}(-x)^{z}\Gamma(-z)\frac{\Gamma(c)}{\Gamma(c+z)}\ .\end{split}$ (231)

This, in turn, yields an expression of the form

$\begin{split}\widetilde{\Xi}(z)=\sum_{k=0}^{\infty}\left(\frac{2}{\omega}\right)^{2k}\mathfrak{p}_{k}(\nu,H^{2},\omega^{2})\
\frac{\Gamma(k-z)}{\Gamma(-z)}\ .\end{split}$ (232)

This series also truncates for integer $z$ and the above relation can then be
inverted to give

$\begin{split}\mathfrak{p}_{k}&=\frac{1}{k!}\left(\frac{\omega}{2}\right)^{2k}\sum_{m=0}^{k}(-)^{m}\binom{k}{m}\
\widetilde{\Xi}(m)\ .\end{split}$ (233)

The explicit expression quoted before follows from this equation. The first
few polynomials are given by

$\begin{split}\mathfrak{p}_{0}&=1\ ,\\\
\mathfrak{p}_{1}&=\frac{H^{2}}{2^{2}}(1+\nu+\mu)(1+\nu-\mu)\ ,\\\
\mathfrak{p}_{2}&=\frac{H^{2}}{2^{3}}\left\\{-\omega^{2}(2\nu+3)+\frac{H^{2}}{2^{2}}(1+\nu+\mu)(1+\nu-\mu)(3+\nu+\mu)(3+\nu-\mu)\right\\}\
,\\\ \mathfrak{p}_{3}&=\frac{H^{2}}{2^{2}\times
3!}\Biggl{\\{}\omega^{4}+\frac{H^{2}\omega^{2}}{2^{2}}[3\mu^{2}(2\nu+5)-(103+132\nu)-3(17\nu^{2}+2\nu^{3})]+H^{4}(\ldots)\Biggr{\\}}\
,\end{split}$ (234)

The polynomials $\mathfrak{p}_{4}$ and higher are proportional to $H^{4}$ and,
hence the above expressions are sufficient to obtain an answer accurate up to
order $H^{2}$ terms. To get terms accurate up to order $H^{4}$, we also need
the leading terms of the next three polynomials:

$\begin{split}\mathfrak{p}_{4}&=\frac{H^{4}}{2^{5}\times
4!}\Biggl{\\{}\omega^{4}[-4\mu^{2}+8\nu(2\nu+13)+157]+H^{2}(\ldots)\Biggr{\\}}\
,\\\ \mathfrak{p}_{5}&=\frac{H^{4}}{2^{2}\times
5!}\Biggl{\\{}\omega^{6}(5\nu+18)+H^{2}(\ldots)\Biggr{\\}}\ ,\\\
\mathfrak{p}_{6}&=\frac{H^{4}}{2^{7}\times
3^{2}}\Biggl{\\{}\omega^{8}+H^{2}(\ldots)\Biggr{\\}}\ .\end{split}$ (235)

The polynomials $\mathfrak{p}_{7}$ and higher are proportional to $H^{6}$, and
hence can be ignored at this order.

For odd values of $d$, the function $\Xi_{nn}$ is related to $\Xi_{n}$ simply
by the transformation: $\nu\to-\nu$. This allows us to also obtain the flat
space expansion for $\Xi_{nn}$ in odd $d$:

$\begin{split}\Xi_{nn}|_{\text{Odd
$d$}}=\sum_{k=0}^{\infty}\mathfrak{p}_{k}(-\nu,H^{2},\omega^{2})\mathfrak{G}_{k}\
,\end{split}$ (236)

where the functions $\mathfrak{G}_{k}$ are related to the $\mathfrak{B}_{k}$
by $\nu\to-\nu$:

$\begin{split}\mathfrak{G}_{k}&\equiv\frac{r^{-\nu-\frac{d}{2}+\frac{1}{2}(d-1-\mathcal{N})+2k}}{(-\nu+1)\ldots(-\nu+k)}\
{}_{0}F_{1}\left[1+k-\nu,-\frac{\omega^{2}r^{2}}{4}\right]=-2\nu\frac{\Gamma(-\nu)\
r^{\frac{1}{2}(1-\mathcal{N})+k}}{2(\omega/2)^{k-\nu}}J_{k-\nu}(\omega r)\
.\end{split}$ (237)

We will conclude by giving the near-flat/high-frequency expansion of
$K_{\text{Out}}$ in odd $d$. This can be achieved using Stirling
approximation, i.e.,

$\Gamma\left(z\right)\sim\exp\left\\{\left(z-\tfrac{1}{2}\right)\ln
z-z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}\right\\}\
,$ (238)

an approximation valid as long $z\to\infty$ away from the negative real axis.
We then obtain the following expansion for $K_{\text{Out}}$ in odd $d$:

$\begin{split}K_{\text{Out}}|_{\text{Odd $d$}}&=\frac{2\pi
i}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}\Bigg{[}1+(\nu^{2}+3\mu^{2}-1)\frac{\nu}{3!!}\frac{H^{2}}{\omega^{2}}\Bigg{.}\\\
&\quad+\frac{5\nu^{4}-4\nu^{3}+(30\mu^{2}-14)\nu^{2}-(60\mu^{2}-16)\nu+(45\mu^{2}-90\mu^{2}+21)}{2\times
3}\frac{\nu(\nu-1)}{5!!}\frac{H^{4}}{\omega^{4}}\\\
&\qquad+O\left(\frac{H^{6}}{\omega^{6}}\right)\Bigg{]}\ .\end{split}$ (239)

This expression describes how the radiation reaction kernel gets corrected due
to the non-zero cosmological constant.

Symbol | ${}^{0}f^{\mu}_{d}$ | ${}^{0}f^{\mu}_{d-2}$ | ${}^{0}f^{\mu}_{d-4}$
---|---|---|---
$\mathbb{D}_{1}$ | $\frac{\partial_{t}^{d}}{d!!}$ | $\frac{\partial_{t}^{d-2}}{(d-2)!!}$ | $\frac{\partial_{t}^{d-4}}{(d-4)!!}$
|  |  |
$\mathbb{D}_{2}$ | $\frac{\partial_{t}^{d+2}}{(d+2)!!}-\frac{H^{2}}{3!}(d+1)\frac{\partial_{t}^{d}}{d!!}+\frac{H^{4}}{5!}\frac{7}{3}(d^{2}-1)\frac{\partial_{t}^{d-2}}{(d-2)!!}$ | $\frac{\partial_{t}^{d}}{d!!}-\frac{H^{2}}{3!}(d-1)\frac{\partial_{t}^{d-2}}{(d-2)!!}+\frac{H^{4}}{5!}\frac{7}{3}(d-1)(d-3)\frac{\partial_{t}^{d-4}}{(d-4)!!}$ | $\frac{\partial_{t}^{d-2}}{(d-2)!!}$
|  |  |
$\mathbb{D}_{1}^{X}$ | $\frac{\partial_{t}^{d+2}}{(d+2)!!}+\frac{H^{2}}{3}(d-2)\frac{\partial_{t}^{d}}{d!!}-\frac{H^{4}}{45}(d^{2}-1)\frac{\partial_{t}^{d-2}}{(d-2)!!}$ | $\frac{\partial_{t}^{d}}{d!!}+\frac{H^{2}}{3}(d-4)\frac{\partial_{t}^{d-2}}{(d-2)!!}-\frac{H^{4}}{45}(d-1)(d-3)\frac{\partial_{t}^{d-4}}{(d-4)!!}$ | $\frac{\partial_{t}^{d-2}}{(d-2)!!}$
|  |  |
$\mathbb{D}_{1}^{V}$ | $\frac{\partial_{t}^{d}}{d!!}$ | $\frac{\partial_{t}^{d-2}}{(d-2)!!}$ | $\frac{\partial_{t}^{d-4}}{(d-4)!!}$
|  |  |
$\mathbb{D}_{0}^{XX}$ | $\frac{d+2}{2}\frac{\partial_{t}^{d+2}}{(d+2)!!}+\frac{H^{2}}{4!}2(5d^{2}-15d-2)\frac{\partial_{t}^{d}}{d!!}$ | $\frac{d+2}{2}\frac{\partial_{t}^{d}}{d!!}+\frac{H^{2}}{4!}2(5d^{2}-25d+2)\frac{\partial_{t}^{d-2}}{(d-2)!!}$ | $\frac{d+2}{2}\frac{\partial_{t}^{d-2}}{(d-2)!!}$
| $+\frac{H^{4}}{6!}(67d^{3}-526d^{2}+833d-14)$ | $+\frac{H^{4}}{6!}(67d^{3}-794d^{2}+2125d+42)$ |
|  |  |
$\mathbb{D}_{0}^{VV}$ | $\frac{\partial_{t}^{d-2}}{(d-2)!!}+\frac{H^{2}}{3!}(d-1)\frac{\partial_{t}^{d-4}}{(d-4)!!}+\frac{H^{4}}{5!}(d-1)(d-3)\frac{\partial_{t}^{d-6}}{(d-6)!!}$ | $\frac{\partial_{t}^{d-4}}{(d-4)!!}+\frac{H^{2}}{3!}(d-3)\frac{\partial_{t}^{d-6}}{(d-6)!!}+\frac{H^{4}}{5!}(d-3)(d-5)\frac{\partial_{t}^{d-6}}{(d-6)!!}$ | $\frac{\partial_{t}^{d-6}}{(d-6)!!}$
|  |  |
$\mathbb{D}_{0}^{XV}$ | $\frac{\partial_{t}^{d}}{d!!}+\frac{H^{2}}{2!}(d-3)\frac{\partial_{t}^{d-2}}{(d-2)!!}+\frac{H^{4}}{4!}(d-1)(d-7)\frac{\partial_{t}^{d-4}}{(d-4)!!}$ | $\frac{\partial_{t}^{d-2}}{(d-2)!!}+\frac{H^{2}}{2!}(d-5)\frac{\partial_{t}^{d-4}}{(d-4)!!}+\frac{H^{4}}{4!}(d-3)(d-9)\frac{\partial_{t}^{d-6}}{(d-6)!!}$ | $\frac{\partial_{t}^{d-4}}{(d-4)!!}$
|  |  |
Table 8: The differential operators that appear in dS radiation reaction (for
$d$ odd). We have divided up the sum into three columns where each column
combines into an expression covariant under dS isometries. The entries in the
second column must be multiplied by a relative factor of
$-\frac{H^{2}}{4\times 3!}c_{h}$ with $c_{h}\equiv 12\mu^{2}+d^{2}-4$ and then
added to the first column. Similarly, the third column should be multiplied by
a relative factor of $\frac{H^{4}}{8\times
6!}[5c_{h}^{2}-40(d+2)c_{h}+32(d+2)(d^{2}-1)]$ and then added to the first two
contributions. The sum of these contributions should be further multiplied by
a factor of $\frac{(-1)^{\frac{d-1}{2}}}{|S^{d-1}|(d-2)!!}$.

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