aainstitutetext:  International Centre for Theoretical Sciences (ICTS-TIFR),
Tata Institute of Fundamental Research, Shivakote, Hesaraghatta, Bangalore
560089, India.

# Influence Phase of a dS Observer I : Scalar Exchange

R. Loganayagam a Omkar Shetye<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Inspired by real-time computations in AdS black holes, we propose a method to
obtain the influence phase of a cosmological observer by calculating the on-
shell action on a doubled spacetime geometry. The influence phase is the
effective action for an open system: for a dS static patch observer coupled to
a scalar field it incorporates the radiation reaction due to the bulk fields
and their dS Hawking radiation. For a general extended source in dS, we
describe how to account for finite size effects. In the long-time limit, we
get a Markovian open quantum system susceptible to cosmological fluctuations,
whereas the short-time limit reproduces the worldline theory of flat-space
radiation reaction. We also present a fully covariantised form for the cubic
corrections to the radiation reaction in even spacetime dimensions, including
Hubble contributions, and find an intriguing recursive structure across
dimensions.

##  1 Introduction

Over the last few decades, many independent lines of evidence have converged
on the fact that our universe has a positive cosmological
constantCarroll:2000fy ; Peebles:2002gy ; Frieman:2008sn ; Weinberg:2013agg .
This has presented a difficult conundrum for those who want to think about the
relation between gravity and quantum mechanicsWitten:2001kn ; Bousso_2004 .
Among the most fruitful ideas coming out of research in quantum gravity has
been holography, i.e. the statement that a gravitational theory is equivalent
to a quantum system living on its boundary. However, spacetimes with positive
cosmological constant do not have any time-like boundaries for a dual quantum
system to live in. Thus, it seems, that gravity in such spacetimes cannot have
a holographic dual theory (or at least we cannot have a dual which is a
conventional quantum dynamical system).

One attempt to overcome this obstacle is as followsAnninos:2011af ;
Nakayama:2011qh : imagine a lone observer probing such a spacetime. The
worldline of such an observer can then be thought of as a time-like boundary
where a possible holographic description might reside. This is the idea of
_solipsistic holography_ , which posits that a quantum system111Perhaps a
large $N$ matrix model as in BFSS dualityBanks:1996vh  (See Taylor:2001vb ;
Ydri:2017ncg  for a review). living on such a worldline encodes the quantum
theory of gravity that describes the universe. To claim that the information
about the entire universe can be gleaned from a single worldline within it
might seem speculative: but it is pertinent to remember that all existing
knowledge about our universe can be traced ultimately to measurements around
the earth. Thus, we might want to assess the viability of such a proposal by
examining it further.222Another alternative approach is to focus not on the
observables but rather on meta-observables like the global wave-function of
the universeStrominger:2001pn ; Maldacena:2002vr ; Marolf:2008hg ;
Anninos:2011ui ; Anninos:2017eib ; Hogervorst:2021uvp ; Chakraborty:2023yed ;
Loparco:2023rug . As has been emphasised in these references, this approach is
especially suited to model the physics of inflation, with us serving as meta-
observers to some extent. How the spacetime dynamics gets encoded in the
proposed dual is not yet entirely clear, though much progress has been
achieved over the past few yearsArkani-Hamed:2018kmz ; Sleight:2019hfp ;
Goodhew:2020hob ; Baumann:2022jpr .

Any object in a dynamical spacetime influences and is influenced by its
surroundings. In this sense, any gravitational observer should be thought of
as an open quantum system constantly interacting with the rest of the
universe. On the quantum mechanical side, the emergence of the open system is
due to integrating out observer’s internal degrees of freedom. This is the
cosmological analogue of the fluid-gravity correspondenceHubeny:2011hd . In
the AdS/CFT context, on the gravity side, fluid dynamics emerges by
integrating out the physics in radial direction, whereas on the gauge theory
side, it is a consequence of coarse-graining quarks and gluons. There is, by
now, non-trivial evidence supporting this statement, including precise
matching of anomalous effects on both sidesJensen:2013kka ; Jensen:2013rga .
In a similar vein, one might ask how we could go about checking the
cosmological version of this statement.

The central challenge in answering this question is twofold: first, to derive
an open quantum system on the worldline from the ambient dynamics. As we shall
see, a precise definition of this first step already involves some
work.333Systematic description of observers in the middle of a spacetime (as
opposed to asymptotic observers) is well-known to be a hard problem. Some of
the approaches to the AdS version of this question, starting from the CFT
side, can be found in Hamilton:2006fh ; Papadodimas:2012aq ; Maxfield:2017rkn
; Jafferis:2020ora . It would be interesting to extend these ideas to take
into account the open nature of the observer, as we do here.  More precisely,
what we need is a cosmological analogue of GKPW prescription in AdS/CFT that
will allow us to derive the open system for an observer. This note is aimed at
addressing this issue.

The second step would be to construct a dual unitary quantum system that,
after integrating out appropriate degrees of freedom, leads to the same open
theory as gravity. This might be a hard undertaking: after all, even in the
fluid-gravity correspondence, to derive the fluid dynamics from a strongly
coupled gauge theory is practically impossible. But, since we are dealing with
a quantum mechanical system here, there is reason for hope. One immediate goal
would be to check whether the putative open quantum system derived in the
first step shows the right structural features to admit a solipsistic
interpretation. We will postpone further thoughts on this issue to the
discussion section.

Let us return now to the issue of constructing the open system on the
gravitational side. Imagine a universe described by a dynamical spacetime
along with a variety of fields living on it. A local observer in such a theory
may be modelled as a source for these fields: a source that emits/radiates as
well as a source that absorbs/detects. Any autonomous motion of the observer
is then accompanied by an outgoing radiation and an associated radiation
reaction. This results in a dissipation of the observer’s energy, and we seek
an open quantum system should describe this physics. The open quantum theory
on the world line should also describe the influence of the incoming radiation
from the rest of the universe. As we shall elaborate on later, this incoming
radiation also includes the Hawking radiation from the Hubble horizon.444See
Chandrasekaran:2022cip  for an analysis of dS observer from the point of view
of von Neumann algebras. It would be interesting to link such an analysis to
the ideas discussed in this note, e.g., one may ask how the physics of
radiation reaction is encoded within von Neumann algebras. Another algebraic
statement of potential interest is the ‘time-like tube
theorem’borchers1961vollstandigkeit ; araki1963generalization ;
Strohmaier:2023opz ; Witten:2023qsv , but, again, it is unclear to us how such
formal statements relate to the description of dS observer as an open system.

Quite independently of such holographic quests, the worldline open quantum
theory under question shows up in a variety of concrete physical questions. As
an example, worldline EFT has emerged as a useful way to organise the post-
Newtonian expansion of a binary system radiating gravitational
wavesGoldberger:2004jt ; Goldberger:2007hy ; Porto:2016pyg ; Levi:2018nxp ;
Barack:2018yvs . The basic idea in such approaches is to systematically
integrate out the short-distance gravitational physics that binds the binaries
to get an effective theory that describes the inspiral process. Due to the
radiation of gravitational waves, ultimately such a binary is also an open
system of the type described above. These ideas can be generalised into a
cosmological setting where, for example, a worldline EFT which takes also the
expansion of the universe into account might be useful in studying the
dynamics of galactic formation, cooling and mergers.555See Porto:2013qua  for
the role played by worldline methods in the effective field theory(EFT) of
large scale structure(LSS). A motivation of this note is to describe an
approach that might help us systematically derive such an EFT.

We will conclude this preamble with a broad outline of what follows: in
section §2, we begin by describing the basic geometric set-up used in deriving
the open quantum mechanics associated with the cosmological observer. The
prescription we propose is inspired by the recent developments in real-time
AdS/CFTSkenderis:2008dg ; Skenderis:2008dh ; Glorioso:2018mmw ; deBoer:2018qqm
; Chakrabarty:2019aeu ; Jana:2020vyx ; Chakrabarty:2020ohe ; Ghosh:2020lel ;
He:2022jnc ; Loganayagam:2022zmq  that have led to systematic derivation of
open quantum systems by integrating out a thermal holographic CFT bath. The
essential idea here is a real-time version of Gibbons-Hawking
procedureGibbons:1976ue : one proposes an appropriate semi-classical geometry
only containing the relevant region (BH exterior for Gibbons-Hawking, dS
static patch in the current problem), and computes the path integral in a
saddle-point approximation by evaluating on-shell action. We will argue that
such a prescription leads to an answer which correctly encodes both the
radiation reaction and Hawking radiation from the Hubble horizon.

The problem of cosmological observer exhibits broad structural similarities to
the AdS case, which we exploit. But we also find significant differences: for
one, much of the standard holographic machinery (e.g. GKPW prescription,
counter-term procedure) available on the AdS side is simply absent. We outline
a regularisation procedure that gives finite answers in §2, relegating the
details to appendices.

In section §3 we use our prescription to derive the open effective
action/influence phase for an observer coupled to a class of generalised free
scalar fields. Next, we examine the flat space limit of our influence phase in
§4 demonstrating how the already known expressions of the flat space radiation
reaction are reproduced in this limit. We also compute the leading
cosmological corrections to the Abraham-Lorentz-Dirac radiation-reaction
force. In the penultimate section §5, we sketch how interactions can be
incorporated into our formalism. We conclude with a summary and a discussion
of further directions in §6.

To enable readability, we confine ourselves to describing the basic physical
ideas as well as the central results in the main sections. Much of the
relevant technical details are presented in appendices. The first appendix §A
is a review of multipole expansion in flat spacetime, along with a description
of the multipole expansion in terms of symmetric trace-free(STF) tensors. Much
of this is standard material just cast into a notation convenient for our
purposes. In the next appendix §B, we review the outgoing scalar solutions in
dS and describe a counterterm procedure to deal with point-like sources placed
at the centre of the static patch. In appendix §C, we show that the
counterterm procedure extends to the most general scalar configurations and
describe how to deal with extended sources. The discussion in these two
appendices culminates in an effective action describing arbitrary scalar
sources in the dS static patch. The next appendix §D specialises to point-like
observers in arbitrary motion in $dS_{d+1}$ with $d$ odd: we show that our
effective action evaluated for such sources re-assembles into a generally
covariant radiation-reaction force with Hubble corrections.

##  2 The cosmological influence phase $\boldsymbol{S_{\text{CIP}}}$

Our goal is to describe the experience of an observer in an expanding
spacetime. This, in turn, will help us in understanding the spacetime itself.
In particular, we want to ask how to construct the open quantum system that
describes the cosmological observer. In its full generality, this is a
difficult problem, but we can start with a simple model for the observer. We
can think of the observer as a single worldline undergoing absorption and
emission processes. So the observer is privy to 3 kinds of data:

  * •

Outgoing radiation: Emission data along with the outgoing propagator tells us
the field values at a later time in the spacetime.

  * •

Incoming radiation: The fields in the past can be reconstructed by using an
incoming propagator given the absorption data.

  * •

Fluctuations: The observer will also be sensitive to _cosmic noise_ , which
shows up in the absorption data.

EmissionAbsorptionincoming propagatornoiseoutgoingpropagator Figure 1: A
cosmological observer can access 3 kinds of data: radiation due to its own
emissions, incoming radiation from sources in the environment and noise.

This is reminiscent of the motion of a Brownian particle in Langevin theory. A
pollen grain in water is sensitive not only to coarse-grained currents in the
water (analogous to the incoming radiation) but also to fluctuations arising
from the motion of water molecules. Finally, the motion of the Brownian
particle can influence the dynamics of water as well (analogous to outgoing
radiation).

The dynamics of such open quantum systems can be derived by the path integral
prescription of Feynman and VernonFEYNMAN1963118  describing the density
matrix evolution. According to the authors of FEYNMAN1963118 , the effective
description of the open system can be derived starting from two non-
interacting copies of each of the system as well as the environment
(describing the combined density matrix). Integrating out two copies of the
environment then induces new interactions between the copies of the system,
resulting in a non-unitary evolution of the system state. These terms
constitute the _influence phase_ , which encodes completely the effect of the
environment on the system. Applying this insight to the question at hand, we
conclude that all cosmological effects on an observer(the system) are
succinctly summarised in a _cosmological influence phase_ $S_{\text{CIP}}$.

What does $S_{\text{CIP}}$ depend on? It should depend on how effective the
observer is at emitting/absorbing radiation of a given frequency $\omega$ and
a given multipole type $\mathbb{L}$. Say we have two sets of functions
$\mathcal{J}_{A}(\omega,\mathbb{L})$ and $\mathcal{J}_{D}(\omega,\mathbb{L})$
characterising the emission/absorption efficiency of the observer. From the
Feynman-Vernon viewpoint, $\mathcal{J}_{A}(\omega,\mathbb{L})$ and
$\mathcal{J}_{D}(\omega,\mathbb{L})$ have the following interpretation: to
begin with, we have two copies of the observer (left/right), each probing
their copy of the universe via their respective multipole moments
$\mathcal{J}_{L}(\omega,\mathbb{L})$ and $\mathcal{J}_{R}(\omega,\mathbb{L})$
respectively. The influence phase, which results from integrating out the
universe, then depends on the average

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

as well as the difference

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

of these two multipole moments. The fact that the average/difference sources
characterise its emissive/absorptive properties is a well-known feature of the
Feynman-Vernon formalismSchwinger:1960qe ; Keldysh:1964ud ; kamenev_2011 ;
Sieberer:2015svu : this fact can ultimately be traced to the past/future
boundary conditions on the two copies imposed within this formalism. To
conclude, the cosmology as seen by an observer with multipole moments
$\mathcal{J}_{A}(\omega,\mathbb{L})$ and $\mathcal{J}_{D}(\omega,\mathbb{L})$
is encoded in a single influence functional
$S_{\text{CIP}}\left[\mathcal{J}_{A}(\omega,\mathbb{L}),\mathcal{J}_{D}(\omega,\mathbb{L})\right]$.
In terms of the Schwinger-Keldysh path integral of quantum gravity, we can
write

$e^{iS_{\text{CIP}}}\equiv\int[d\varphi_{R}][d\varphi_{L}]\
e^{iS_{g}[\varphi_{R},\mathcal{J}_{R}]-iS_{g}[\varphi_{L},\mathcal{J}_{L}]}\
,$ (1)

where $\varphi_{L,R}$ denote the bra/ket copy of the bulk quantum fields in
cosmology (including the spacetime metric) and $S_{g}[\varphi,\mathcal{J}]$ is
the full gravitational action in the background of an observer with multipole
moments $\mathcal{J}$. The above path integral should then be interpreted in a
wilsonian sense: we want to integrate out the fast modes of quantum
gravitational theory, while freezing the slow degrees of freedom of the
observer, and obtain an effective action which describes the open dynamics of
such an observer.

The cosmological influence phase $S_{\text{CIP}}$ is a direct observable.
Given an expanding universe, assuming we have a sufficiently long-lived
observer with arbitrary multipole moments in some region, the force on an
observer due to radiation reaction as well as radiation reception can directly
be measured. This force serves to determine all terms in the ‘effective
action’ $S_{\text{CIP}}$ that encodes the influence of the ambient universe.
All the _real_ observables of astrophysics and cosmology, e.g. the sky maps at
different frequencies, can be incorporated this way into the absorptive part
of $S_{\text{CIP}}$.

From this viewpoint, all cosmological calculations should, in principle, be
recast in terms of $S_{\text{CIP}}$ to connect them with observations. This is
already implicit in the existing approaches to cosmology: for example, the
final step in CMB power spectrum computation is to expand it in spherical
harmonics centred around us. Phrasing observables in terms of $S_{\text{CIP}}$
makes explicit this observer-dependence (which is probably essential for
defining observables _within_ a quantum spacetime). Talking in terms of a
single functional $S_{\text{CIP}}$ may also be convenient for effective field
theory (EFT) based approaches to cosmology based on direct observables (e.g.
those based on classifying sources in the red-shift spaceSenatore:2014vja ;
Schmittfull:2020trd ; Philcox:2022frc ). More ambitiously, one may conceive of
a bootstrap program based on the cosmological influence phase that complements
existing proposals for cosmological bootstrapArkani-Hamed:2018kmz ;
Sleight:2019hfp ; Sleight:2021plv ; Baumann:2022jpr ; Hogervorst:2021uvp ;
Loparco:2023rug .

What are the general principles that constrain $S_{\text{CIP}}$? First of all,
when $\mathcal{J}_{D}(\omega,\mathbb{L})$ is set to zero, $S_{\text{CIP}}$
should vanish. This statement arises from the microscopic unitarity of the
environment: if the two copies of the observer in Feynman-Vernon formalism
introduce identical perturbations into the environment, their effect cancels
out of all correlatorskamenev_2011 . From the viewpoint of the observer, the
above condition is equivalent to the conservation of the observer density
matrix’s trace. Apart from this, there are also constraints on
$S_{\text{CIP}}$ coming from causality. For example, causality implies that
the coefficient of
$\mathcal{J}_{D}^{*}(\omega,\mathbb{L})\mathcal{J}_{A}(\omega,\mathbb{L})$ is
analytic in the upper half plane of complex $\omega$ : this coefficient is the
retarded correlator on the worldline of the observerSchwinger:1960qe ;
Keldysh:1964ud ; Caldeira:1982iu ; kamenev_2011 ; breuer2002theory ;
Sieberer:2015svu . A similar statement holds for the coefficients of any term
of the form
$\mathcal{J}_{D}^{*}(\omega,\mathbb{L})\prod_{k}\mathcal{J}_{A}(\omega_{k},\mathbb{L}_{k})$.

Evaluation of the influence phase requires us to know the real-time or
Schwinger-Keldysh(SK) propagators of the environment. It is unclear how to
perform such computations in generic cosmological spacetimes, especially if
gravity is also to be quantised. We will show that for an observer in dS, this
computation can be geometrised roughly akin to recent implementations of SK
path integrals in case of AdS black holesSkenderis:2008dg ; Skenderis:2008dh ;
Glorioso:2018mmw ; deBoer:2018qqm ; Chakrabarty:2019aeu ; Jana:2020vyx ;
Chakrabarty:2020ohe ; Ghosh:2020lel ; He:2022jnc ; Loganayagam:2022zmq . The
hope then is that one can later generalise it beyond dS to incorporate full
FRW cosmology.

Specifically, in the case of dS, we conjecture that the computation of
cosmological influence phase $S_{\text{CIP}}$ is dominated by a geometric
saddle point built out of two copies of the static patch stitched together at
their horizons. We will call this doubled geometry, the dS Schwinger-
Keldysh(dS-SK) spacetime. In the rest of this subsection, we will describe
this geometry in more detail before moving to the evidence for our conjecture
in the subsequent sections.

Let us begin by setting up the basic notation required: consider a
$(d+1)$-dimensional dS spacetime dSd+1 whose Penrose diagram is shown in
Fig.2. A horizontal slice (i.e., a constant time slice) in this diagram
denotes the prime-meridian on a spatial sphere $S^{d}$, with the two ends
denoting the poles. Each point in the horizontal slice corresponds to a sphere
$S^{d-1}$, which shrinks to a point near the poles. The first example we will
consider is a co-moving dS observer whom we place at the south pole. Our focus
will be on the static patch of such an observer, i.e., the patch between the
past and future cosmological horizons of the observer. We will later describe
a more general class of observers spread arbitrarily over this static patch,
modelled as a sequence of spherical shells around south pole (Fig.2).

We will find it convenient to work with _outgoing_ Eddington-Finkelstein
coordinates on the static patch. The metric in this coordinate system takes
the form

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

Here $H$ is the Hubble constant of dS spacetime, $r$ is the radial distance
from the observer, $u$ denotes the outgoing time labelling the outgoing waves
and $d\Omega_{d-1}^{2}$ is the line element on a unit $S^{d-1}$ sphere. The
south-pole observer sitting at $r=0$ sees a future horizon at $r=1/H$ where
the outgoing coordinates are well-behaved. In most of what follows, we will
set $H=1$ for convenience and restore it later when we examine the flat space
(i.e. $H\to 0$) limit.

$\mathscr{I}^{+}$$\mathscr{I}^{-}$NorthPoleSouthPole$\mathscr{I}^{+}$$\mathscr{I}^{-}$NorthPoleSouthPole$\displaystyle
r=r_{c}$ Figure 2: Penrose diagrams of dS with the static patch of the south
pole observer shown in green. Constant $u$ slices are shown in blue. Left : a
localised observer at the south pole whose worldline is thickened to a world-
tube(orange) of radius $r_{c}$. Right : an extended observer modelled as a
sequence of spherical shells of radius $r_{i}$ with $i=1,\ldots,N$.

We now turn to the model of the observer: Conceptually, the simplest model is
that of a point particle with specified multipole moments sitting at $r=0$.
However, such a model needs to be regulated with appropriate counter-terms to
allow the computation of radiation reaction effects. To this end, we will take
the observer to be a small sphere of radius $r_{c}$ and thicken its worldline
into a time-like ‘world-tube’. The point particle limit then corresponds to
taking $r_{c}\to 0$ limit _after_ the addition of counter-terms: both the
Green functions and required counter-terms can be determined exactly for a dS
observer coupled to generalised free scalar fields. The radius $r_{c}$ then
acts like a UV regulator for the problem.

Apart from the formal requirements of regularisation, we are also interested
in the actual problem of an extended observer of a finite size. In such a
case, there are no divergences. Nevertheless, finite counter-terms are needed
to renormalise the bare parameters into the physically measured properties of
the observer. As mentioned before, a simple model of the extended observer is
a sequence of spherical shells of radius $r_{i}$ with $i=1,\ldots,N$: their
complement in the static patch is then the rest of the universe to be
integrated out. We can define multipole moments for such extended observers
and still write down a cosmological influence phase as a function of those
multipole moments. The locality in radial direction gets obscured in such a
description: this is however natural in the solipsistic viewpoint where radial
locality is an approximate/emergent property of the dual quantum mechanics.

LRHorizon$\displaystyle\zeta=0$$\displaystyle\zeta=1$ Figure 3: The two
sheeted complex dS-SK geometry can be thought of as two static patches
smoothly connected at the future horizon. The radial contour along an outgoing
Eddington-Finkelstein slice (i.e., a constant $u$ slice) is shown in blue. The
radial contour has an outgoing R branch and an incoming L branch.
Re($\displaystyle r$)Im($\displaystyle
r$)$\displaystyle\zeta(0-i\epsilon)=0$$\displaystyle\zeta(0+i\epsilon)=1$$\displaystyle
r=H^{-1}$RL Figure 4: The branch cut structure of $\zeta(r)$ in the complex
$r$ plane at fixed $u$: branch-cut shown as a wiggly line. We also show the
_clockwise_ dS-SK radial contour running from $\zeta=1$ to $\zeta=0$ (the blue
curve in this figure and in Fig.3). The $\text{Im}\ r>0$ branch is the time-
ordered/right branch, whereas the $\text{Im}\ r<0$ branch is the anti-time-
ordered/left branch.

We now turn to our conjecture for the dS-SK geometry, i.e., the semi-classical
saddle point that dominates the quantum gravity path integral for
$S_{\text{CIP}}$. What we seek is a real-time analogue of the Gibbons-Hawking-
constructionGibbons:1977mu  as well as gr-SK construction in
AdSSkenderis:2008dg ; Skenderis:2008dh ; Glorioso:2018mmw ; deBoer:2018qqm ;
Chakrabarty:2019aeu ; Ghosh:2020lel , which would compute for us the
cosmological influence phase. Here is the geometry we propose: take two copies
of the static patch and stitch them together smoothly at the future horizon
(Fig.3). To parametrise this geometry, we complexify the radial coordinate and
think of dS-SK as a co-dimension one contour in the complex $r$ plane (Fig.4).
To make this precise, let us define a _mock tortoise coordinate_ $\zeta$ as
follows:

$\zeta(r)=\frac{1}{i\pi}\int\limits^{0-i\epsilon}_{r}\frac{dr^{\prime}}{1-r^{\prime
2}}=\frac{1}{2\pi i}\ln\left(\frac{1-r}{1+r}\right)$ (3)

This integral has logarithmic branch points at $r=\pm 1$ and we choose its
branch-cut to be over the interval $r\in[-1,1]$ on the real line. As shown in
Fig.4, our normalisation is such that, if we begin from $0+i\epsilon$ (i.e.,
just above the midpoint of the branch-cut) and then go clockwise around the
branch cut to $0-i\epsilon$ (i.e., just above the midpoint of the branch-cut),
we pick up a discontinuity in $\zeta$ equal to negative unity. The choice of
the overall constant in (3) is such that the real part is 1 on the $R$ static
patch (the $r+i\epsilon$ contour), and the real part falls to zero as we move
clockwise and turn to traverse the $L$ boundary(the $r-i\epsilon$ contour).666
The reader should note the use of clockwise contours in the complex $r$ plane
for dS, in contrast to the counter-clockwise contours used in the AdS black-
brane case. This fact means that we need to be careful to add appropriate
minus signs whenever we use the residue theorem, but this inconvenience seems
unavoidable given the standard time orientations of the Schwinger-Keldysh
contour.

We are now ready to state our prescription:

$\textbf{Cosmological influence phase}=\textbf{On-shell gravitational action
of the dS-SK geometry}\ .$ (4)

To be clear, on both sides of this equality, we treat observer(s) as
prescribed sources, viz., we take it off-shell by freezing its dynamics. Both
sides can then be thought of as functionals of the observer multipole moments
that emit/detect fields. In the dual quantum mechanics, these multipole
moments should be thought of as the ‘slow macroscopic degrees of freedom’
whose influence phase is computed by integrating out the ‘fast microscopic
degrees of freedom’. The solipsistic holography would then imply that we can
replace the LHS in the above equality with such an influence phase computed in
the dual quantum mechanics. The above statement can then be thought of as
giving a GKPW-like prescriptionGubser:1998bc ; Witten:1998qj  for solipsistic
holography. The primary aim of this note is to exhibit simple example systems
where we can show that the above prescription yields sensible answers.

Before we turn to examples, we would like to comment on an interesting
philosophical point: In this geometric picture, the cosmology reduces entirely
to the static patch accessible to the observer, bypassing questions about the
rest of the universe (or multi-verse as the case may be). We think of this
focus on actual observables as a desirable feature of our proposal, in
contrast to traditional descriptions of quantum gravity in dS spacetime
phrased in terms of global questions. In the AdS black-brane case,
gravitational Schwinger-Keldysh geometry (and its Gibbons-Hawking predecessor)
divorces the phenomenology of the exterior from speculations about singularity
and BH interior. In a similar vein, our geometric proposal aims at isolating
the physics of the static patch from speculations about super-horizon modes,
side-stepping the measure problem in cosmology. Our saddle point geometry can
be thought of as a way to implement the causal-diamond-based cosmological
measures ala BoussoBousso:2006ev ; Bousso:2006xc .

##  3 Computing $\boldsymbol{S_{\text{CIP}}}$ from on-shell effective action

In the following sections, we will evaluate the on-shell action on the dS-SK
geometry described above and show that we get meaningful semi-classical
results for the cosmological influence phase $S_{\text{CIP}}$. We will do this
in three parts: First, in this section, we will describe a class of systems
where observers act as sources for scalar fields. We will describe how the on-
shell action can be computed for these systems to yield $S_{\text{CIP}}$.
Next, in Sec§4, we will argue how $S_{\text{CIP}}$ does indeed capture the
physics of radiation reaction for a moving dS observer. Finally, in Sec§5, we
will describe how field interactions could be taken into account.

Let us begin by examining the mode decomposition in dS-SK geometry. Outgoing
modes of frequency $\omega$ in the static patch have the form:

$\mathfrak{f}(r,\omega,\ell)\ \mathscr{Y}_{\mathbb{L}}(\Omega)\ e^{-i\omega
u}.$ (5)

where $\mathfrak{f}(r,\omega,\ell)$ is an analytic function of $r$ in the
region $0<r\leq 1$: since we are working in outgoing Eddington-Finkelstein
coordinates, analyticity near $r=1$ is equivalent to the outgoing boundary
condition. Here, the field is decomposed into spherical harmonics
$\mathscr{Y}_{\mathbb{L}}(\Omega)$ on $\mathbb{S}^{d-1}$ with labels
$\mathbb{L}\equiv\left\\{\ell,\vec{m}\right\\}$. The spherical harmonics with
label $\ell$ are eigenfunctions of the sphere Laplacian with eigenvalue
$-\ell(\ell+d-2)$. Given the analyticity of $\mathfrak{f}(r,\omega,\ell)$, the
outgoing modes can be analytically continued to the complex $r$ plane without
any branch cuts. Consequently, on the dS-SK geometry, the outgoing modes
become modes which are _identical_ in the right/left branches of the static
patch.

The incoming modes are readily found by time reversing the outgoing mode.
Time-reversal isometry of dS-SK geometry is implemented by taking $u\to 2\pi
i\zeta-u$ and $\omega\to-\omega$. We then get an incoming mode of the form

$\mathfrak{f}(r,-\omega,\ell)\ \mathscr{Y}_{\mathbb{L}}(\Omega)\
e^{-2\pi\omega\zeta-i\omega u}=\mathfrak{f}^{\ast}(r,\omega,\ell)\
\mathscr{Y}_{\mathbb{L}}(\Omega)\ e^{-2\pi\omega\zeta-i\omega u}.$

To get the last equality, we have assumed $\mathfrak{f}(r,\omega,\ell)$ to be
a Fourier transform of a real function. The reader should note here the
presence of the non-analytic factor $e^{-2\pi\omega\zeta}$, thus resulting in
a branch cut for the incoming mode. The incoming mode hence picks up a factor
of $e^{2\pi\omega}$ if the argument crosses the branch cut from above
($\zeta=1$) to below $(\zeta=0)$, i.e., as we move from right to the left
static patch. As we will see below, this is indeed the appropriate Boltzmann
factor for the static patch, encoded automatically in the incoming modes.

Consider a free scalar field theory on dSd+1. Let
$G^{\text{Out}}_{\mathcal{N}}(r,\omega,\ell)$ denote the radial part of the
_outgoing_ boundary-to-bulk Green function, i.e., the outgoing field created
by a unit point source placed at the south pole. Here, and in what follows, we
use the subscript $\mathcal{N}$ to denote the exponent that characterises the
near origin behaviour of the scalar field. More precisely, we define
$G^{\text{Out}}_{\mathcal{N}}(r,\omega,\ell)$ as the solution of an
appropriate radial ODE that obeys the following boundary conditions: at the
worldline, we impose a Dirichlet condition

$\begin{split}\lim_{r\to
0}r^{\nu+\frac{\mathcal{N}-1}{2}}G^{\text{Out}}_{\mathcal{N}}(r,\omega,\ell)&=1\
,\end{split}$ (6)

where we have defined $\nu\equiv\ell+\frac{d}{2}-1$ and taken the behaviour of
the Green function to be $r^{-\nu-\frac{\mathcal{N}-1}{2}}$ near the source.
As an example, for a massless minimal scalar field, we have the fall-off
$r^{-(\ell+d-2)}$ corresponding to $\mathcal{N}=d-1$.

Apart from the above condition imposed at the origin, we impose
analyticity/outgoing boundary conditions at the dS horizon $(r=1)$. Note
however that this is _not_ the appropriate solution on the dS-SK geometry: its
boundaries are not the worldline $+$ dS horizon but rather the right/left
worldlines. It is then more natural to impose a _double_ Dirichlet boundary
condition. To this end, we begin with the most general linear combination of
outgoing/incoming modes for the radial part

$\begin{split}\varphi_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})=-G^{\text{Out}}_{\mathcal{N}}(r,\omega,\ell)\mathcal{J}_{\bar{F}}(\omega,\mathbb{L})+e^{2\pi\omega(1-\zeta)}G^{\text{Out}*}_{\mathcal{N}}(r,\omega,\ell)\mathcal{J}_{\bar{P}}(\omega,\mathbb{L})\
.\end{split}$ (7)

Here the subscripts $F$ and $P$ denote the sources that radiate to the future
and detectors that absorb from the past respectively. We use $\zeta$ to
indicate the radial argument of $\varphi_{{}_{\mathcal{N}}}$ to emphasise that
this general linear combination takes two different values in the two branches
of dS-SK geometry.

LR$\displaystyle\zeta=0$$\displaystyle\zeta=1$LR$\displaystyle\zeta=0$$\displaystyle\zeta=1$LR$\displaystyle\zeta=0$$\displaystyle\zeta=1$$\displaystyle
g_{R}$$\displaystyle g_{L}$$\displaystyle\mathbb{G}$ Figure 5: Propagators in
dS-SK geometry: the boundary to bulk propagators are denoted in red and the
bulk to bulk propagator is denoted in brown.

The coefficients $\mathcal{J}_{\bar{F}},\mathcal{J}_{\bar{P}}$ appearing above
can be linked to the left/right sources via the double Dirichlet condition,
i.e., at the left/right copy of the worldlines, we impose

$\begin{split}\mathcal{J}_{L}(\omega,\mathbb{L})\equiv\lim_{\zeta\to
0}r^{\nu+\frac{\mathcal{N}-1}{2}}\varphi_{{}_{\mathcal{N}}}&=-\mathcal{J}_{\bar{F}}(\omega,\mathbb{L})+e^{2\pi\omega}\mathcal{J}_{\bar{P}}(\omega,\mathbb{L})\
,\\\ \mathcal{J}_{R}(\omega,\mathbb{L})\equiv\lim_{\zeta\to
1}r^{\nu+\frac{\mathcal{N}-1}{2}}\varphi_{{}_{\mathcal{N}}}&=-\mathcal{J}_{\bar{F}}(\omega,\mathbb{L})+\mathcal{J}_{\bar{P}}(\omega,\mathbb{L})\
.\end{split}$ (8)

Using this, we can then rewrite Eq.(7) as
$\varphi_{{}_{\mathcal{N}}}(r,\omega,\mathbb{L})=g_{R}\mathcal{J}_{R}-g_{L}\mathcal{J}_{L}$,
where $g_{R,L}(\zeta,\omega,\mathbb{L})$ denote the right/left boundary-to-
bulk propagators on the dS-SK geometry (see Figure 5). Our use of the symbol
$\mathcal{J}$ here is a deliberate allusion to the observer’s multipole
moments. Inverting the above relations, we obtain

$\displaystyle\begin{split}\mathcal{J}_{\bar{F}}(\omega,\mathbb{L})&\equiv-\Bigl{\\{}(1+n_{\omega})\mathcal{J}_{R}(\omega,\mathbb{L})-n_{\omega}\mathcal{J}_{L}(\omega,\mathbb{L})\Bigr{\\}}=-\mathcal{J}_{A}(\omega,\mathbb{L})-\left(n_{\omega}+\frac{1}{2}\right)\mathcal{J}_{D}(\omega,\mathbb{L})\\\
\mathcal{J}_{\bar{P}}(\omega,\mathbb{L})&\equiv-
n_{\omega}\Bigl{\\{}\mathcal{J}_{R}(\omega,\mathbb{L})-\mathcal{J}_{L}(\omega,\mathbb{L})\Bigr{\\}}=-n_{\omega}\
\mathcal{J}_{D}(\omega,\mathbb{L})\ .\end{split}$ (9)

Here we have introduced the average/difference sources
$\mathcal{J}_{A}\equiv\frac{1}{2}\mathcal{J}_{R}+\frac{1}{2}\mathcal{J}_{L}$
and $\mathcal{J}_{D}\equiv\mathcal{J}_{R}-\mathcal{J}_{L}$. We note here the
natural appearance of the Bose-Einstein factor

$\displaystyle\begin{split}n_{\omega}\equiv\frac{1}{e^{2\pi\omega}-1}\
.\end{split}$ (10)

Such a factor arises naturally by solving the detailed-balance constraint
$1+n_{\omega}=e^{2\pi\omega}n_{\omega}$ which equates the probability of
spontaneous/stimulated emission by the source to the absorption probability.
The appearance of such a factor is an evidence that dS-SK contour naturally
incorporates the thermality of Hawking radiation emitted from the dS
horizonGibbons:1977mu .

Given the solution determined in terms of the multipole moments, we can
compute the on-shell action. A scalar system with a requisite exponent is
given by an action

$S=-\frac{1}{2}\int d^{d+1}x\sqrt{-g}\
r^{\mathcal{N}+1-d}\left\\{(\partial\Phi_{\mathcal{N}})^{2}+\frac{\Phi_{\mathcal{N}}^{2}}{4r^{2}}\left[(d+\mathcal{N}-3)(d-\mathcal{N}-1)-r^{2}\left(4\mu^{2}-(\mathcal{N}+1)^{2}\right)\right]\right\\}\
.$ (11)

We will refer to this as a _designer_ scalar system with a radially varying
dilaton, an appropriate centrifugal potential term, and a mass term. Our
motivation to consider this class of actions is that, at specific values of
$\mathcal{N}$ and $\mu$, the above action captures the physics of different
field theories. For example, a massive KG scalar of mass $m$ corresponds to
setting $\mathcal{N}=d-1$ and
$4m^{2}=(\mathcal{N}+1)^{2}-4\mu^{2}=d^{2}-4\mu^{2}$ in the above action.
Another example is the KG scalar field with a conformal mass: this corresponds
to setting $\mathcal{N}=d-1$ and $\mu=\frac{1}{2}$.

Further, such actions with different values of $\mathcal{N}$ and $\mu$ arise
naturally when considering scalar-vector-tensor spherical harmonic
decompositions of Maxwell as well as linearised Einstein equations about dS
backgroundMukohyama:2000ui ; Kodama:2003kk ; Ishibashi:2004wx . More
precisely, the radial ODEs in all these sectors coincide with the radial ODE
obtained from the above action for some value of $\mathcal{N}$ and $\mu$ (See
table 1).777We note that, for such massless fields, the exponent $\mathcal{N}$
and the parameter $\mu$ in all sectors are related by the condition
$4\mu^{2}=(\mathcal{N}+1)^{2}$, i.e., they do not have the last term given in
Eq.(11). For these reasons, we consider it worthwhile to study the influence
phase obtained by integrating out such designer scalars.

Table 1: $\mathcal{N},\mu$ values for different massless fields | KG Scalar | EM Vector | EM Scalar | Gravity Tensor | Gravity Vector | Gravity Scalar
---|---|---|---|---|---|---
$\mathcal{N}$ | $d-1$ | $d-3$ | $3-d$ | $d-1$ | $1-d$ | $3-d$
$\mu$ | $\frac{d}{2}$ | $\frac{d}{2}-1$ | $\frac{d}{2}-2$ | $\frac{d}{2}$ | $\frac{d}{2}-1$ | $\frac{d}{2}-2$

The radial ODE for designer scalar systems can be solved exactly in terms of
hypergeometric functions. The outgoing boundary to bulk Green function
satisfying the boundary condition in Eq.(6) is given byBunch:1978yq ; Lopez-
Ortega:2006aal ; Anninos:2011af

$\begin{split}G^{\text{Out}}_{\mathcal{N}}(r,\omega,\mathbb{L})&=r^{\nu-\frac{\mathcal{N}}{2}}(1+r)^{-i\omega}\\\
&\quad\times\frac{\Gamma\left(\frac{1+\nu-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu-i\omega}{2}\right)}{\Gamma(1-i\omega)\Gamma\left(1+\nu\right)}\
{}_{2}F_{1}\left[\frac{1+\nu-\mu-i\omega}{2},\frac{1+\nu+\mu-i\omega}{2};1-i\omega;1-r^{2}\right]\
,\end{split}$ (12)

where we have used $\nu\equiv\ell+\frac{d}{2}-1$. This solution is manifestly
analytic near $r=1$ and thus this is an outgoing solution. The solution on the
dS-SK geometry can then be written as

$\begin{split}\Phi_{\mathcal{N}}(\zeta,u,\Omega)=\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\varphi_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\mathscr{Y}_{\mathbb{L}}(\Omega)\
e^{-i\omega u}\end{split}$ (13)

with the radial part $\varphi_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})$
being given by Eq.(7). This expression can then be substituted back into the
designer scalar action given in Eq.(11). The resultant on-shell action itself
is formally divergent but can be rendered finite with counterterms localised
at the worldlines. We will refer the reader to appendices B and C for the
technical details of how this is done. The end result of this evaluation can
be cast into the form

$\begin{split}S_{\text{CIP}}&=-\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}K_{\text{Out}}(\omega,\ell)\
[\mathcal{J}_{R}-\mathcal{J}_{L}]^{*}\
[(1+n_{\omega})\mathcal{J}_{R}-n_{\omega}\mathcal{J}_{L}]\\\
&=\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\frac{K_{\text{Out}}(\omega,\ell)}{1+n_{\omega}}\
\mathcal{J}_{\bar{P}}^{*}\mathcal{J}_{\bar{F}}=-\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}K_{\text{Out}}(\omega,\ell)\
\mathcal{J}_{D}^{*}\
\Bigl{[}\mathcal{J}_{A}+\left(n_{\omega}+\frac{1}{2}\right)\mathcal{J}_{D}\Bigr{]}\
,\end{split}$ (14)

where we have written the answer in right/left, past/future as well as the
average/difference basis. Here, $K_{\text{Out}}$ is the boundary 2-point
function encoding the effects of radiation reaction (explicit expressions are
provided below). Notable features of this action are as follows:

  1. 1.

The absence of
$\mathcal{J}_{A}^{*}(\omega,\mathbb{L})\mathcal{J}_{A}(\omega,\mathbb{L})$
term is an expected consequence of the collapse rule which demands that the
influence phase go to zero when the sources on the two sides are the same.

  2. 2.

The $\mathcal{J}_{D}^{*}(\omega,\mathbb{L})\mathcal{J}_{A}(\omega,\mathbb{L})$
coefficient is imaginary, implying that this term is purely dissipative. We
will show in the next section that this term captures the radiation reaction
experienced by the observer.

  3. 3.

The noise term captured by the
$\mathcal{J}_{D}^{*}(\omega,\mathbb{L})\mathcal{J}_{D}(\omega,\mathbb{L})$
coefficient is proportional to the dissipative term with a factor. This factor
is correctly picked out by our geometry so as to satisfy the Kubo-Martin-
Schwinger (KMS) condition for dS.

Hence, we claim that this action correctly captures the effect of the
environment on the observer. In fact, this is exactly the form expected out of
two-point functions of thermal systemsChaudhuri:2018ymp  and matches with
analogous expressions in holographic open systemsChakrabarty:2019aeu ;
Jana:2020vyx .

Table 2: $\tau_{dS}$ for $\mu=\frac{d}{2}$ (Massless KG scalar, Gravity tensor sector) $\mu=\frac{d}{2}$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=5$
---|---|---|---|---|---
$d=3$ | $4$ | $1$ | $\frac{64}{225}$ | $\frac{4}{49}$ | $\frac{256}{11025}$
$d=4$ | $\frac{9\pi^{2}}{16}$ | $1$ | $\frac{25\pi^{2}}{1024}$ | $\frac{1}{16}$ | $\frac{441\pi^{2}}{262144}$
$d=5$ | $\frac{64}{9}$ | $1$ | $\frac{256}{1225}$ | $\frac{4}{81}$ | $\frac{16384}{1334025}$
$d=6$ | $\frac{225\pi^{2}}{256}$ | $1$ | $\frac{1225\pi^{2}}{65536}$ | $\frac{1}{25}$ | $\frac{3969\pi^{2}}{4194304}$
$d=7$ | $\frac{256}{25}$ | $1$ | $\frac{16384}{99225}$ | $\frac{4}{121}$ | $\frac{65536}{9018009}$
$d=8$ | $\frac{1225\pi^{2}}{1024}$ | $1$ | $\frac{3969\pi^{2}}{262144}$ | $\frac{1}{36}$ | $\frac{9801\pi^{2}}{16777216}$
$d=9$ | $\frac{16384}{1225}$ | $1$ | $\frac{65536}{480249}$ | $\frac{4}{169}$ | $\frac{1048576}{225450225}$
$d=10$ | $\frac{99225\pi^{2}}{65536}$ | $1$ | $\frac{53361\pi^{2}}{4194304}$ | $\frac{1}{49}$ | $\frac{1656369\pi^{2}}{4294967296}$
$d=11$ | $\frac{65536}{3969}$ | $1$ | $\frac{1048576}{9018009}$ | $\frac{4}{225}$ | $\frac{4194304}{1329696225}$
Table 3: $\tau_{dS}$ for $\mu=\frac{d}{2}-1$ (EM/Gravity vector sector) $\mu=\frac{d}{2}-1$ | $\ell=0$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=5$
---|---|---|---|---|---|---
$d=3$ | $1$ | $1$ | $\frac{4}{9}$ | $\frac{4}{25}$ | $\frac{64}{1225}$ | $\frac{64}{3969}$
$d=4$ | $\frac{\pi^{2}}{4}$ | $1$ | $\frac{9\pi^{2}}{256}$ | $\frac{1}{9}$ | $\frac{225\pi^{2}}{65536}$ | $\frac{1}{100}$
$d=5$ | $4$ | $1$ | $\frac{64}{225}$ | $\frac{4}{49}$ | $\frac{256}{11025}$ | $\frac{64}{9801}$
$d=6$ | $\frac{9\pi^{2}}{16}$ | $1$ | $\frac{25\pi^{2}}{1024}$ | $\frac{1}{16}$ | $\frac{441\pi^{2}}{262144}$ | $\frac{1}{225}$
$d=7$ | $\frac{64}{9}$ | $1$ | $\frac{256}{1225}$ | $\frac{4}{81}$ | $\frac{16384}{1334025}$ | $\frac{64}{20449}$
$d=8$ | $\frac{225\pi^{2}}{2566}$ | $1$ | $\frac{1225\pi^{2}}{65536}$ | $\frac{1}{25}$ | $\frac{3969\pi^{2}}{4194304}$ | $\frac{1}{441}$
$d=9$ | $\frac{256}{25}$ | $1$ | $\frac{16384}{99225}$ | $\frac{4}{121}$ | $\frac{65536}{9018009}$ | $\frac{64}{38025}$
$d=10$ | $\frac{1225\pi^{2}}{1024}$ | $1$ | $\frac{25\pi^{2}}{1024}$ | $\frac{1}{16}$ | $\frac{441\pi^{2}}{262144}$ | $\frac{1}{225}$
$d=11$ | $\frac{16384}{1225}$ | $1$ | $\frac{65536}{480249}$ | $\frac{4}{169}$ | $\frac{1048576}{225450225}$ | $\frac{64}{65025}$
Table 4: $\tau_{dS}$ for $\mu=\frac{d}{2}-2$ (EM/Gravity scalar sector) $\mu=\frac{d}{2}-2$ | $\ell=0$ | $\ell=1$ | $\ell=2$ | $\ell=3$ | $\ell=4$ | $\ell=5$
---|---|---|---|---|---|---
$d=3$ | $1$ | $1$ | $\frac{4}{9}$ | $\frac{4}{25}$ | $\frac{64}{1225}$ | $\frac{64}{3969}$
$d=4$ | $1$ | $\frac{\pi^{2}}{16}$ | $\frac{1}{4}$ | $\frac{9\pi^{2}}{1024}$ | $\frac{1}{36}$ | $\frac{225\pi^{2}}{262144}$
$d=5$ | $1$ | $\frac{4}{9}$ | $\frac{4}{25}$ | $\frac{64}{1225}$ | $\frac{64}{3969}$ | $\frac{256}{53361}$
$d=6$ | $1$ | $\frac{9\pi^{2}}{256}$ | $\frac{1}{9}$ | $\frac{225\pi^{2}}{65536}$ | $\frac{1}{100}$ | $\frac{1225\pi^{2}}{4194304}$
$d=7$ | $1$ | $\frac{64}{225}$ | $\frac{4}{49}$ | $\frac{256}{11025}$ | $\frac{64}{9801}$ | $\frac{16384}{9018009}$
$d=8$ | $1$ | $\frac{25\pi^{2}}{1024}$ | $\frac{1}{16}$ | $\frac{441\pi^{2}}{262144}$ | $\frac{1}{225}$ | $\frac{2025\pi^{2}}{16777216}$
$d=9$ | $1$ | $\frac{256}{1225}$ | $\frac{4}{81}$ | $\frac{16384}{1334025}$ | $\frac{64}{20449}$ | $\frac{65536}{81162081}$
$d=10$ | $1$ | $\frac{1225\pi^{2}}{65536}$ | $\frac{1}{25}$ | $\frac{3969\pi^{2}}{4194304}$ | $\frac{1}{441}$ | $\frac{245025\pi^{2}}{4294967296}$
$d=11$ | $1$ | $\frac{16384}{99225}$ | $\frac{4}{121}$ | $\frac{65536}{9018009}$ | $\frac{64}{38025}$ | $\frac{1048576}{2606204601}$

The exact expressions for $K_{\text{Out}}$ depend on whether the number of
spatial dimensions $d$ is odd or even. For $d$ odd, we have Lopez-
Ortega:2006aal ; Anninos:2011af

$\begin{split}K_{\text{Out}}|_{\text{Odd d}}&=-e^{i\nu\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\frac{\Gamma\left(\frac{1+\nu-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+\nu+\mu-i\omega}{2}\right)}{\Gamma\left(\frac{1-\nu+\mu-i\omega}{2}\right)\Gamma\left(\frac{1-\nu-\mu-i\omega}{2}\right)}\
,\end{split}$ (15)

and for $d$ even, we get

$\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}$ (16)

where $\psi^{(0)}(z)\equiv\frac{d}{dz}\ln\Gamma(z)$ is the di-gamma function
and the function $\Delta_{\mathcal{N}}$ is defined via

$\begin{split}\Delta_{\mathcal{N}}(n,\mu,\omega)&\equiv\frac{(-)^{n}}{\Gamma(n)^{2}}\frac{\Gamma\left(\frac{1+n-\mu-i\omega}{2}\right)\Gamma\left(\frac{1+n+\mu-i\omega}{2}\right)}{\Gamma\left(\frac{1-n+\mu-i\omega}{2}\right)\Gamma\left(\frac{1-n-\mu-i\omega}{2}\right)}=\frac{1}{\Gamma(n)^{2}}\prod\limits_{k=1}^{n}\left[\frac{\omega^{2}}{4}+\frac{1}{4}(\mu-n+2k-1)^{2}\right]\\\
&=\Delta_{\mathcal{N}}^{\ast}(n,\mu,\omega)\ .\end{split}$ (17)

The important fact to note about these expressions is that, for all values of
$\mu$ appearing in table 1 except $\mu=\frac{d}{2}$, we get a nice small
$\omega$ expansion. For $\mu=\frac{d}{2}$, we still get a small $\omega$
expansion for all $\ell>0$: only the $\ell=0$ term has a $1/\omega$ behaviour
at small $\omega$. The physical interpretation of these statements is this: in
all these cases except $\ell=0,\mu=\frac{d}{2}$, one obtains a Markovian open
system at small $\omega$, i.e., a cosmically old observer in dS does not
retain any memory of its past.888The mild breakdown of small $\omega$
expansion in $\mu=\frac{d}{2}$ gives a tail term in the radiation reaction.
This has been previously noted in Burko:2002ge . This tail term can be avoided
either by turning off the monopole moment or by giving the scalar a small
mass. This is an interesting observation, especially in even $d$ where the
corresponding flat spacetime problem has memory termsBirnholtz:2013nta . This
suggests that _the radiation reaction problem in an expanding spacetime is
perhaps better behaved than the one in flat spacetime._ In dual quantum
mechanics, this predicts that a clean separation of slow/fast degrees of
freedom should be possible, at least in the leading large $N$ approximation.

We will now argue that the fluctuations also admit a small $\omega$ expansion.
To this end, we use $1+n_{\omega}+n_{-\omega}=0$ to rewrite the cosmological
influence phase as

$\begin{split}S_{\text{CIP}}=-\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\Bigl{[}K_{\text{Out}}(\omega,\ell)\
\mathcal{J}_{D}^{*}\mathcal{J}_{A}+\frac{1}{2}\left(n_{\omega}+\frac{1}{2}\right)[K_{\text{Out}}(\omega,\ell)-K_{\text{Out}}(-\omega,\ell)]\
\mathcal{J}_{D}^{*}\mathcal{J}_{D}\Bigr{]}\ .\end{split}$ (18)

Since $\omega\ n_{\omega}$ has a regular small $\omega$ expansion, we conclude
from the above expression that $S_{\text{CIP}}$ has a regular small frequency
expansion provided $K_{\text{Out}}$ has such an expansion. Up to 1st order in
$\omega$, we have

$\begin{split}K_{\text{Out}}=K_{\text{Out}}|_{\omega=0}-i\ \omega\
\tau_{dS}+\ldots\end{split}$ (19)

where $\tau_{dS}$ can be interpreted as the cosmological decay time-scale for
slowly varying multipole moments in dS.999We tabulate $\tau_{dS}$ for various
cases of interest in tables 2, 3 and 4. Due to the dS version of fluctuation-
dissipation theorem, this is also proportional the variance of the Hubble
Hawking noise. This fact can be gleaned from the leading
$\mathcal{J}_{D}^{*}\mathcal{J}_{D}$ term in the cosmological influence phase:

$\begin{split}S_{\text{CIP}}\supset
i\sum_{\mathbb{L}}\frac{\tau_{dS}}{2\pi}\int\frac{d\omega}{2\pi}\mathcal{J}_{D}^{*}\mathcal{J}_{D}\
.\end{split}$ (20)

Using the Hubbard-Stratonovich transformation, we can think of this term
arising from integrating out a noise field with a time-domain action:

$\begin{split}\sum_{\mathbb{L}}\int
du\left[\frac{i}{2}\frac{\pi}{\tau_{dS}}\mathcal{N}^{2}(u)+\mathcal{J}_{D}(u)\mathcal{N}(u)\right]\
.\end{split}$ (21)

The first term here then shows that $\mathcal{N}(u)$ behaves like a Gaussian
noise field with variance $\frac{\tau_{dS}}{\pi}$.

We will conclude this section by describing how the above analysis can be
readily generalised to extended sources in dS, modelled as a sequence of
spherical shells. The main technical novelty is that we need the dS bulk-to-
bulk propagator to compute the radial part of the field. The expression in
Eq.(7) is then replaced by a radial contour integral

$\begin{split}\varphi_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})&=\oint
r_{0}^{\mathcal{N}}dr_{0}\
\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})\varrho_{{}_{\mathcal{N}}}(\zeta_{0},\omega,\mathbb{L})\
,\end{split}$ (22)

where $\varrho_{{}_{\mathcal{N}}}(\zeta_{0},\omega,\mathbb{L})$ is a scalar
source spread out over dS-SK geometry and
$\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})$ is the contour-ordered bulk-
to-bulk propagator. It is regular everywhere except at $\zeta=\zeta_{0}$ where
its radial derivative has a prescribed discontinuity. Further, we require
regularity at the center of the right/left static patches, viz.,

$\begin{split}\lim_{\zeta\to
0}r^{\nu+\frac{\mathcal{N}-1}{2}}\mathbb{G}=\lim_{\zeta\to
1}r^{\nu+\frac{\mathcal{N}-1}{2}}\mathbb{G}=0.\end{split}$ (23)

These conditions uniquely determine the bulk-to-bulk propagator as specific
combinations of the outgoing/incoming waves on either side of the source point
$\zeta_{0}$. An explicit expression in terms of the right/left boundary-to-
bulk propagators is (See appendix C)

$\begin{split}\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})&=\frac{1}{W_{LR}(\zeta_{0},\omega,\mathbb{L})}g_{R}(\zeta_{\succ},\omega,\mathbb{L})g_{L}(\zeta_{\prec},\omega,\mathbb{L})\\\
&\equiv\frac{1}{W_{LR}(\zeta_{0},\omega,\mathbb{L})}\begin{cases}g_{R}(\zeta,\omega,\mathbb{L})g_{L}(\zeta_{0},\omega,\mathbb{L})&\quad\text{if}\;\zeta\succ\zeta_{0}\\\
g_{L}(\zeta,\omega,\mathbb{L})g_{R}(\zeta_{0},\omega,\mathbb{L})&\quad\text{if}\;\zeta\prec\zeta_{0}\end{cases}\
.\end{split}$ (24)

Here the symbols $\succ$ and $\prec$ denote comparison using the radial
contour ordering of dS-SK contour. 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 .

Once we have the bulk-to-bulk Green function, the on-shell effective action in
terms of the extended sources can be computed to be

$\begin{split}S|_{\textbf{On-
shell}}&=\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}\oint
r^{\mathcal{N}}dr\ \oint r_{0}^{\mathcal{N}}dr_{0}\
[\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})]^{*}\mathbb{G}(\zeta|\zeta_{0},\omega,\mathbb{L})\varrho_{{}_{\mathcal{N}}}(\zeta_{0},\omega,\mathbb{L})\
.\end{split}$ (25)

This is the familiar statement that, in free theories, on-shell action reduces
to double integral over sources with an appropriate Green function serving as
the kernel. The above expression can then be evaluated for a sequence of shell
sources by performing the radial contour integrals.

We find that the end result of this computation can be written as

$\begin{split}S|_{\textbf{On-
shell}}=S_{\text{CIP}}^{\text{Pt}}+S_{\text{Int}}\ ,\end{split}$ (26)

where $S_{\text{CIP}}^{\text{Pt}}$ is the cosmological influence phase of
Eq.(14), computed for the point-like source. To get this form, we should
define the multipole moments of the extended source via

$\begin{split}\mathcal{J}_{R}(\omega,\mathbb{L})\equiv\int_{R}dr\
r^{\mathcal{N}}\Xi_{n}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
,\\\ \mathcal{J}_{L}(\omega,\mathbb{L})\equiv-\int_{L}dr\
r^{\mathcal{N}}\Xi_{n}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
.\end{split}$ (27)

Here the integrals are over the right/left open static patches and
$\Xi_{n}(r,\omega,\mathbb{L})$ is a smearing function given in Eq.(183). The
remaining terms in the on-shell action (denoted by $S_{\text{Int}}$) encode
the conservative self-interactions of the extended source:

$\begin{split}S_{\text{Int}}&=\frac{1}{2}\sum_{\mathbb{L}}\int\frac{d\omega}{2\pi}[\mathcal{J}_{R}^{\ast}\overline{\varphi}_{R,\text{Int}}-\mathcal{J}_{L}^{\ast}\overline{\varphi}_{L,\text{Int}}]\
.\end{split}$ (28)

Here $\overline{\varphi}_{R/L,\text{Int}}$ denote appropriately radially-
averaged mean fields in the right/left static patch which couple to the
multipole moments defined in Eq.(27). Their explicit form is

$\begin{split}\overline{\varphi}_{R,\text{Int}}(\omega,\mathbb{L})\equiv\int_{R}dr\
r^{\mathcal{N}}\Xi_{nn}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
,\\\ \overline{\varphi}_{L,\text{Int}}(\omega,\mathbb{L})\equiv-\int_{L}dr\
r^{\mathcal{N}}\Xi_{nn}(r,\omega,\mathbb{L})\
\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})\
.\end{split}$ (29)

Here $\Xi_{nn}(r,\omega,\mathbb{L})$ is a time-reversal invariant Green
function given in Eq.(184).

In the next section, we will describe how these results for extended sources
can be used to compute the radiation reaction force felt by a dS observer in
arbitrary motion. To that end, it is convenient to shift back to the standard
time domain: we remind the reader that, till now, we have been working in the
frequency domain dual to the outgoing EF time $u$. This is related to sources
data on standard time slices via

$\begin{split}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})=\int du\
e^{i\omega u}\widetilde{\varrho}_{{}_{\mathcal{N}}}(\zeta,t,\mathbb{L})=\int
dt\ e^{i\omega
t}\left(\frac{1-r}{1+r}\right)^{\frac{i\omega}{2}}\widetilde{\varrho}_{{}_{\mathcal{N}}}(\zeta,t,\mathbb{L})\
,\end{split}$ (30)

where we have used $u=t+\frac{1}{2}\ln\left(\frac{1-r}{1+r}\right)$. In other
words, the combination
$\left(\frac{1-r}{1+r}\right)^{-\frac{i\omega}{2}}\varrho_{{}_{\mathcal{N}}}(\zeta,\omega,\mathbb{L})$
appearing in the above definitions is just the Fourier transform with respect
to standard time. Thus, the radiative multipole moments defined in Eq.(27) can
equivalently well be thought of as being computed via standard time slices,
viz.,

$\begin{split}\mathcal{J}_{R}(\omega,\mathbb{L})\equiv\int dt\ e^{i\omega
t}\int_{R}dr\ r^{\mathcal{N}}\Xi_{n}(r,i\partial_{t},\mathbb{L})\
\widetilde{\varrho}_{{}_{\mathcal{N}}}(\zeta,t,\mathbb{L})\ ,\\\
\mathcal{J}_{L}(\omega,\mathbb{L})\equiv-\int dt\ e^{i\omega t}\int_{L}dr\
r^{\mathcal{N}}\Xi_{n}(r,i\partial_{t},\mathbb{L})\
\widetilde{\varrho}_{{}_{\mathcal{N}}}(\zeta,t,\mathbb{L})\ .\end{split}$ (31)

Similar statements hold for the radially-averaged mean fields
$\overline{\varphi}_{R/L,\text{Int}}$.

##  4 Radiation reaction and flat space limit

We will turn to the physics of dS radiation reaction(RR), as encoded in the
cosmological influence phase $S_{\text{CIP}}$. For simplicity, we will
consider an arbitrarily moving point-like source of a KG scalar field (i.e.,
the $\mathcal{N}=d-1$ case). In particular, this means that we will no longer
consider the cases of scalars coming from the harmonic decomposition of
EM/linearised gravity: the dS RR forces for such cases will be dealt with
elsewheredSSKvec . The reason for this restriction is as follows: the analysis
of RR force for EM/gravity requires extending the dS multipole expansion to
vector/tensor symmetric-trace-free tensors, as well as keeping track of
additional velocity dependences in the multipole moments, a task better done
elsewhere. Further, we will confine ourselves to $dS_{d+1}$ with odd values of
$d$, where the flat spacetime RR force is known to time-localBirnholtz:2013ffa
: in these cases, we can compute the RR force as a local expression in a low
curvature (or small $H$) expansion.101010We review the derivation of flat
spacetime RR force in appendix A of this work. To ensure clarity in the near-
flat limit, we will restore Hubble constant $H$ explicitly in what follows.

Radiative Multipole momentsTwo-point function Figure 6: RR computation has two
main ingredients: radiative multipole moments and a two-point function
describing how one multipole moment affects another. The black solid line
denotes the trajectory of the source.

Scalar self-force calculations in curved spacetime are a simple setting to
understand the intricacies of how RR is affected by curvature back-
scatteringGaltsov:1982hwm ; Ori:1995ed ; Ori:1997be ; Quinn:2000wa ;
EricPoisson_2004 . It is a simple version of the more observationally relevant
problem of gravitational RR force in BH backgrounds, especially in the context
of extreme mass ratio inspiralsBarack:2018yvs .

Before discussing the results, let us first understand the different
components of an action that describe RR. RR is a dissipative term that is
found in the
$\mathcal{J}^{*}_{D}(\omega,\mathbb{L})\mathcal{J}_{A}(\omega,\mathbb{L})$
term of the action. This term is a product of two radiative multipole moments
and a two-point function connecting them, as shown in Figure 6. In Equation
(14), this retarded two-point function is defined as $K_{\text{Out}}$. The
radiative multipole moment $\mathcal{J}$ is obtained by smearing the source
distribution with an appropriate function $\Xi_{n}$ (See Eq.(31)).

Now, let us consider the RR on a single particle. For this, we assume that the
particle’s trajectory is close to the south pole ($rH\ll 1$), the velocity is
small ($v\ll c$), and the wavelength of the radiation is much smaller than the
cosmological scale ($\omega\gg H$), but much larger than the observer’s length
scales ($r\omega\ll 1$). Together with the multipole expansion, these
approximations lead to the post-Newtonian expansion of the RR, as shown in
Figure 7. In the doubled geometry, this corresponds to two charges moving
along their trajectories near their respective south poles. The influence
phase on a point charge is then written in terms of the average and difference
of positions of the two charges, as well as their time derivatives.

←r→$\displaystyle\frac{2\pi}{\omega}$$\displaystyle\frac{1}{H}$$\displaystyle\vec{{\color[rgb]{0.25,0.46,0.02}\definecolor[named]{pgfstrokecolor}{rgb}{0.25,0.46,0.02}v}}$
Figure 7: The RR is given in a post-Newtonian expansion: the velocity is taken
to be small ($v\ll 1$) and the trajectory is centred about the south pole
($\omega r\ll 1$) while the near-flat expansion requires that the curvature
effects are small(i.e. $H\ll\omega$ and $rH\ll 1$).

To obtain the correct flat space answer, we need to check if both the two-
point function and the smearing function reduce to their flat space analogues
as $H\to 0$. This can be checked from the $H$ expansions of those quantities.
We can also obtain further curvature corrections and establish a near-flat
expansion. To illustrate how to obtain this expansion easily for any desired
order in the Hubble constant, we provide detailed calculations of these
expansions in Appendix D.1. For the reader’s convenience, we also give a
review of flat space RR in Appendix A.3.

Given the near expansion of the action, we can, in a controlled fashion,
calculate the curvature corrections to the RR force. This force is given by
the variation of the Lagrangian with respect to $x_{D}$. The leading term in
the PN expansion of the flat space RR is a scalar version of the Abraham-
Lorentz-Dirac force and stems from the dipole moment of the particle. This
term, in arbitrary $d$, with the first Hubble correction, we find to be:

$\displaystyle\begin{split}F^{i}_{\text{ALD}}&=\frac{(-1)^{\frac{d+1}{2}}}{|\mathbb{S}^{d-1}|d!!(d-2)!!}\Bigg{\\{}\partial^{d}_{t}x^{i}-H^{2}\frac{d}{6}\left(d^{2}-1\right)\partial^{d-2}_{t}x^{i}\Bigg{\\}}\
.\end{split}$ (32)

This expression gives an equation of motion that is third-order in derivative
for $dS_{4}$. Even higher time derivatives show up if we include higher-order
post-Newtonian corrections. This is a known effect in flat space calculations.
In the appendix D, we give a detailed calculation of 2nd order post-Newtonian
corrections along with Hubble corrections.

The overall sign of the leading term in the force agrees with the fact that
this force is dissipative rather than anti-dissipative. To understand this,
consider a 1d oscillator with an RR force of the form

$\frac{d^{2}x}{dt^{2}}+\omega_{0}^{2}x=\lambda(-1)^{\frac{d+1}{2}}\frac{d^{d}x}{dt^{d}}\
,$ (33)

where we will assume that $d$ is odd and $\lambda>0$. We would now like to
argue that the RR force is dissipative. To see this, we note that the above
equation is equivalent to a dispersion relation
$\omega^{2}=\omega_{0}^{2}-i\lambda\omega^{d}$, which can be solved
approximately to give $\omega\approx\omega_{0}-i\lambda\omega_{0}^{d}$. Since
the imaginary part of $\omega$ is negative, we conclude that the above force
is indeed dissipative.

As noted in Birnholtz:2013nta , the terms in the flat space PN expansion of
the RR force add up to give a Poincare covariant expression. This is a non-
trivial check for the accuracy of the result as both the structure of the
multipole PN expansion and the requirement that it sums up to a covariant
result leave little room for error. We similarly find that the curvature
corrections obtained along with the flat space results are also tightly
constrained: the contributions from our influence phase non-trivially sum up
to expressions covariant under dS metric.

We refer the reader to appendix D for a detailed enumeration of the terms in
the PN expansion. The final RR force then takes the form

$F^{\mu}_{\text{RR}}\equiv\frac{(-)^{\frac{d-1}{2}}}{|\mathbb{S}^{d-1}|(d-2)!!}f^{\mu}$

where $f^{\mu}$ has an expansion of the form

$\begin{split}f^{\mu}_{d}&={}^{0}f^{\mu}_{d}-\frac{H^{2}}{4\times 3!}c_{h}\
{}^{0}f^{\mu}_{d-2}+\frac{H^{4}}{8\times
6!}[5c_{h}^{2}-40(d+2)c_{h}+32(d+2)(d^{2}-1)]\ {}^{0}f^{\mu}_{d-4}+O(H^{6})\
.\end{split}$ (34)

Here $c_{h}\equiv 12\mu^{2}+d^{2}-4$ contains the information about the mass
of the scalar and the combinations ${}^{0}f_{d}^{\mu}$’s for odd values of $d$
are (we have listed the expressions up to $d=11$ in appendix D):

$\begin{split}{}^{0}f^{\mu}_{1}&\equiv-v_{\nu}\ ,\\\
{}^{0}f^{\mu}_{3}&\equiv\frac{P^{\mu\nu}}{3!!}\left\\{-a_{\nu}^{(1)}\right\\}-\frac{H^{2}}{3!!}\left\\{v_{\nu}\right\\}\
,\\\ {}^{0}f^{\mu}_{5}&\equiv\frac{P^{\mu\nu}}{5!!}\left\\{-a_{\nu}^{(3)}+5\
(a\cdot a)\ a_{\nu}^{(1)}+10\ (a\cdot a^{(1)})\
a_{\nu}\right\\}-H^{2}\frac{P^{\mu\nu}}{5!!}\left\\{a_{\nu}^{(1)}\right\\}+\frac{H^{4}}{5!!}\left\\{-v_{\nu}\right\\}\
,\\\ {}^{0}f^{\mu}_{7}&\equiv\frac{P^{\mu\nu}}{7!!}\left\\{-a_{\nu}^{(5)}+14\
(a\cdot a)\ a_{\nu}^{(3)}+70\ (a\cdot a^{(1)})\ a_{\nu}^{(2)}+84\ (a\cdot
a^{(2)})\ a_{\nu}^{(1)}+42\ (a\cdot a^{(3)})\ a_{\nu}\right.\\\
&\left.\qquad\qquad+\frac{224}{3}\ (a^{(1)}\cdot a^{(1)})\ a_{\nu}^{(1)}+105\
(a^{(1)}\cdot a^{(2)})\ a_{\nu}+O(a^{5})\right\\}\\\
&\quad-H^{2}\frac{P^{\mu\nu}}{7!!}\left\\{a_{\nu}^{(3)}+15\ (a\cdot a)\
a_{\nu}^{(1)}+37\ (a\cdot a^{(1)})\
a_{\nu}\right\\}+H^{4}\frac{P^{\mu\nu}}{7!!}\left\\{-a_{\nu}^{(1)}\right\\}-\frac{H^{6}}{7!!}\left\\{v_{\nu}\right\\}\
.\\\ \end{split}$ (35)

Here $v^{\mu}=\frac{dx^{\mu}}{d\tau}$ is the proper velocity of the particle
computed using dS metric, $a^{\mu}\equiv\frac{D^{2}x^{\mu}}{D\tau^{2}}$ is its
proper acceleration and $P^{\mu\nu}\equiv g^{\mu\nu}+v^{\mu}v^{\nu}$ is the
transverse projector to the worldline. We use
$a_{\mu}^{(k)}\equiv\frac{D^{k}a^{\mu}}{D\tau^{k}}$ to denote the proper-time
derivatives of the velocity. All the spacetime dot products are computed using
dS metric.

One remarkable feature of the above formula for radiation reaction is the
recursive nature of the Hubble corrections. One can see that the $O(H^{2k})$
correction to the force in $d$ dimensions is related to the RR force in $d-2k$
dimensions. It would be interesting to see whether there are specific quantum
mechanical models which can reproduce such a recursive structure.

One of the consequences of this recurrence is that the $H^{d-1}$ terms in
$dS_{d+1}$ resembles the RR effects in $d=1$ flat space. The flat space $d=1$
massless scalar RR was explored in Burko:2002gf . However, as noted there, it
is inconsistent to assume a constant coupling for a particle coupled to a
massless scalar in 2D flat space. Similar issues emerge at $O(H^{2})$ in $d=3$
dSBurko:2002ge  and in general in any $d$ at $O(H^{d-1})$ due to the
aforementioned recurrence relation. This is, in turn, related to the breakdown
of the small $\omega$ expansion of the $K_{\text{Out}}$ noted in footnote 8:
an issue that can be cured by turning on a small mass for the scalar.

We have checked that the flat limit of the RR force coincides with the
covariant expressions derived in Birnholtz:2013nta . However, there are sign
mismatches with expressions of Galtsov:2007zz 111111This sign mismatch was
noted by Birnholtz:2013nta  as well.. The expressions at order $H^{2}$ do not
match the general curved space force in Galtsov:2007zz  restricted to dS.
Since our methods differ significantly from Galtsov:2007zz , we are unable to
comment further on the specific source of disagreement.

##  5 Interactions

In this section, we will describe how the computation of on-shell action can
be extended beyond the free-field examples. In particular, we would like to
check that our prescription in Eq.(4) equating the cosmological influence
phase to on-shell action, still works after we include interactions. We will
check this in a simple example: $\varphi_{{}_{\mathcal{N}}}^{3}$ theory in
dS4. However, as will be clear below, our arguments can be easily adapted to
set up perturbative diagrammatics for arbitrary interactions.

For $\varphi_{{}_{\mathcal{N}}}^{3}$ theory in dS4, we should simply evaluate
the bulk on-shell action with the cubic interaction term. At leading order in
perturbation theory, the cubic contribution to $S_{\text{CIP}}$ is obtained by
substituting the free field solutions into interaction terms of the
action.121212The argument here is similar to the one given in appendix $C$ of
Jana:2020vyx  for gr-SK geometry. The interaction term should then be
integrated over the full dS-SK geometry: this is the dS version of a Witten
diagram vertex.

We saw in Sec§3 that the $S_{\text{CIP}}$ we derived satisfies constraints due
to SK collapse and KMS conditions. That this should be true for interacting
theories as well is not clear a priori, but we will now show that these
constraints are still satisfied at least at the level of contact diagrams.
This is most easily seen in terms of the $P-F$ basis multipole moments defined
in Eq.(9). In terms of these multipole moments, SK collapse and KMS conditions
are equivalent to showing that there are no terms in the action with only
$\mathcal{J}_{\bar{F}}$ or only $\mathcal{J}_{\bar{P}}$Chaudhuri:2018ymp .

To check these conditions, we use Eq.(7) and write the vertex contribution to
the on-shell action as

$\displaystyle\begin{split}-\frac{\lambda_{3}}{3!}\int
d^{3+1}x&\;\varphi_{{}_{\mathcal{N}}}^{3}=\sum_{\ell_{i},m_{i}}\text{Gaunt}(\ell_{i},m_{i})\int_{\omega_{1},\omega_{2},\omega_{3}}\delta(\omega_{1}+\omega_{2}+\omega_{3})\\\
&\times\left[\mathcal{I}_{FFF}(\omega_{i},\ell_{i},m_{i})+\mathcal{I}_{FFP}(\omega_{i},\ell_{i},m_{i})+\mathcal{I}_{FPP}(\omega_{i},\ell_{i},m_{i})+\mathcal{I}_{PPP}(\omega_{i},\ell_{i},m_{i})\right]\
.\end{split}$ (36)

Here the index $i$ runs over $\\{1,2,3\\}$ and $\text{Gaunt}(\ell_{i},m_{i})$
are the Gaunt coefficients coming from the integral of 3 spherical harmonics
over the sphere (see equation 34.3.22 of NIST:DLMF ). Time-translation
invariance implies that the three frequencies $\omega_{1}$, $\omega_{2}$ and
$\omega_{3}$ are constrained by an energy-conserving $\delta$ function. The
contributions to the cubic influence phase are given by radial contour
integrals, viz.,

$\displaystyle\begin{split}\mathcal{I}_{FFF}(\omega_{i},\ell_{i},m_{i})&\equiv\frac{\lambda_{3}}{3!}\oint_{\zeta}G^{\text{Out}}_{\mathcal{N}}(\zeta,\omega_{1},\ell_{1})\
G^{\text{Out}}_{\mathcal{N}}(\zeta,\omega_{2},\ell_{2})\
G^{\text{Out}}_{\mathcal{N}}(\zeta,\omega_{3},\ell_{3})\ \\\ &\hskip
85.35826pt\times\mathcal{J}_{\bar{F}}(\omega_{1},\mathbb{L}_{1})\mathcal{J}_{\bar{F}}(\omega_{2},\mathbb{L}_{2})\mathcal{J}_{\bar{F}}(\omega_{3},\mathbb{L}_{3})\
,\\\
\mathcal{I}_{FFP}(\omega_{i},\ell_{i},m_{i})&\equiv-\frac{\lambda_{3}}{2!}\oint_{\zeta}e^{2\pi\omega_{3}(1-\zeta)}\
G^{\text{Out}}_{\mathcal{N}}(\zeta,\omega_{1},\ell_{1})\
G^{\text{Out}*}_{\mathcal{N}}(\zeta,\omega_{2},\ell_{2})\
G^{\text{Out}*}_{\mathcal{N}}(\zeta,\omega_{3},\ell_{3})\ \\\ &\hskip
85.35826pt\times\mathcal{J}_{\bar{F}}(\omega_{1},\mathbb{L}_{1})\mathcal{J}_{\bar{P}}(\omega_{2},\mathbb{L}_{2})\mathcal{J}_{\bar{P}}(\omega_{3},\mathbb{L}_{3})\
,\\\
\mathcal{I}_{FPP}(\omega_{i},\ell_{i},m_{i})&\equiv\frac{\lambda_{3}}{2!}\oint_{\zeta}e^{2\pi(\omega_{2}+\omega_{3})(1-\zeta)}\
G^{\text{Out}}_{\mathcal{N}}(\zeta,\omega_{1},\ell_{1})\
G^{\text{Out}*}_{\mathcal{N}}(\zeta,\omega_{2},\ell_{2})\
G^{\text{Out}*}_{\mathcal{N}}(\zeta,\omega_{3},\ell_{3})\ \\\ &\hskip
85.35826pt\times\mathcal{J}_{\bar{F}}(\omega_{1},\mathbb{L}_{1})\mathcal{J}_{\bar{P}}(\omega_{2},\mathbb{L}_{2})\mathcal{J}_{\bar{P}}(\omega_{3},\mathbb{L}_{3})\
,\\\
\mathcal{I}_{PPP}(\omega_{i},\ell_{i},m_{i})&\equiv-\frac{\lambda_{3}}{3!}\oint_{\zeta}e^{2\pi(\omega_{1}+\omega_{2}+\omega_{3})(1-\zeta)}\
G^{\text{Out}*}_{\mathcal{N}}(\zeta,\omega_{1},\ell_{1})\
G^{\text{Out}*}_{\mathcal{N}}(\zeta,\omega_{2},\ell_{2})\
G^{\text{Out}*}_{\mathcal{N}}(\zeta,\omega_{3},\ell_{3})\ \\\ &\hskip
85.35826pt\times\mathcal{J}_{\bar{P}}(\omega_{1},\mathbb{L}_{1})\mathcal{J}_{\bar{P}}(\omega_{2},\mathbb{L}_{2})\mathcal{J}_{\bar{P}}(\omega_{3},\mathbb{L}_{3})\
.\\\\[8.61108pt] \end{split}$ (37)

We now note that, since $G^{\text{Out}}_{\mathcal{N}}$ is analytic, the
integrands in $\mathcal{I}_{FFF}$ and $\mathcal{I}_{PPP}$ are analytic (to see
the latter, we use energy conservation). This, in turn, implies that these
integrals evaluate to zero by Cauchy’s theorem. It is now evident that this
argument generalises to all contact diagrams of
$\varphi_{{}_{\mathcal{N}}}^{n}$ type, thus demonstrating our claim about SK
collapse and KMS conditions. A similar argument in the AdS blackhole case has
been checked also for exchange diagramsLoganayagam:2022zmq ; OpenEFT  and it
would be interesting to check whether a similar claim holds here. Further, it
would also be interesting to study the correction to the radiation reaction
due to such non-linear interactionsdSSKint .

##  6 Summary and Discussion

In this work, we have proposed a de Sitter-Schwinger Keldysh(dS-SK) geometry
formed by two copies of the static patch stitched together at their future
horizons. We then showed how the influence phase of a dS observer could be
obtained by evaluating the on-shell action on this geometry. Our proposal
yields results that pass a variety of checks: first, from a broad structural
point of view, it satisfies the constraints imposed on it from bulk unitarity
(SK collapse) and the dS version of Kubo-Martin-Schwinger (KMS) conditions.

Another check is the flat space limit, where we showed that, for point-like
sources, the dissipative part of the action correctly produces the flat space
radiation reaction. This also allows us to calculate Hubble corrections to the
radiation reaction in odd spatial dimensions, and show that they combine into
generally covariant expressions on the dS background, which serves another
non-trivial check on our computation. As a technical aside, we have also shown
how we can counter-term the influence phase for localised sources with
multipole moments by using a Dirac-Deitweiler-Whiting type decomposition of
the dS Green functions.

Our analysis can readily be extended to interactions, following techniques
invented in the AdS contextJana:2020vyx ; Jana:2021xyn ; Loganayagam:2022zmq ;
OpenEFT , as we sketched in the main text. This aspect will be explored in
detail elsewheredSSKint . It would also be interesting to explore whether the
familiar tools of conformal invariance, e.g., conformal block decomposition,
can shed more light on the structure of radiation reaction at a non-linear
level. On the face of it, the presence of the observer breaks the dS
isometries to just rotations/time-translations around the observer’s
worldline. But the re-emergence of the full dS isometry in the effective
action that we described above, suggests that conformal techniques could be
fruitfully exploited to understand the structure of Hubble corrections to the
radiation reaction. There is also the question of extending our analysis to
gauge theories and linearised gravity, which we shall pursue in a subsequent
workdSSKvec .

In this work, we have advocated a point of view that the real-world cosmology
is fruitfully framed in terms of a cosmological influence phase
$S_{\text{CIP}}$ for an observer’s worldline. It is interesting to ask whether
realistic FLRW cosmology from $\Lambda$-CDM and the CMB phenomenology can
indeed be rewritten in these terms. To this end, it would be interesting to
extend our analysis to time-varying cosmological spacetimes: perhaps, one
should begin by extending our framework to simpler time-dependent extensions
involving sudden/adiabatic approximations. More broadly, we can enquire of the
role played by radiation reaction in cosmology. Understanding the
gravitational radiation reaction at the galactic/extra-galactic scales might
be crucial to predict the stochastic gravitational wave backgroundYan:2023slm
; Yang:2023eqi ; Saeedzadeh:2023biq .

Much of what we say about dS radiation reaction can readily be adapted to the
AdS case, with a change of signs. This statement is expected to be true at
short times, where the cosmological constant can be treated perturbatively,
and its sign does not result in any qualitatively new features. Thus, at short
times, we expect generally covariant expressions for the radiation reaction
felt by an AdS observer, very similar to the ones we derive in this work.
However, we expect qualitative differences at long time-scales due to
reflection at AdS asymptotia, resulting in long-time tails in radiation
reaction. Further, we do not expect an analogue of dS Hawking radiation in
AdS. It might be worthwhile to make these intuitions more precise and
understand the dual CFT interpretation of these statements. This would be a
good test of the existing proposals describing bulk observers within
AdS/CFTPapadodimas:2012aq ; Maxfield:2017rkn ; Jafferis:2020ora .

We began this note by motivating our work in the context of solipsistic
holography. We see the results here as a first step towards constructing an
open system whose details can be compared against proposed dual quantum
mechanical models. Following the examples like BFSS matrix modelBanks:1996vh
131313See Taylor:2001vb ; Ydri:2017ncg  for a review and Susskind:2021dfc ;
Brahma:2022ikl  for matrix model proposals in dS. , it is natural to expect
some sort of a large $N$ matrix quantum mechanics to give rise to the same
influence phase as what we derive here. To check this, it would be good to
construct a formalism for computing the influence phase of slow macroscopic
observables in a large $N$ matrix model: our computations suggest that a clean
separation of slow/fast modes is possible at least when there is a dual
gravity description. These slow observables describing the dS observer should
not be entirely gauge-invariant but rather have the structure of partially
gauge-fixed probesFerrari:2013aba ; Ferrari:2013waa ; Ferrari:2016vcl .
Whether this is so is yet to be seen.

## Acknowledgements

We would like to thank Ofek Birnholtz, Tuneer Chakraborty, Chandramouli
Chowdhury, Victor Godet, Chandan Jana, Godwin Martin, Shiraz Minwalla,
Priyadarshi Paul, Suvrat Raju, Mukund Rangamani, Joseph Samuel, Ashoke Sen,
Shivam Sharma, Akhil Sivakumar, Sandip Trivedi and Spenta Wadia for valuable
discussions. RL would like to thank the organisers of All Lambdas Holography @
Prague 2021 online workshop for discussions related to this work. We
acknowledge support of the Department of Atomic Energy, Government of India,
under project no. RTI4001, and would also like to acknowledge our debt to the
people of India for their steady and generous support to research in the basic
sciences.

##  Appendix A STF tensors and multipole expansion

We will begin by reviewing the notion of symmetric trace-free tensors, which
are the appropriate tools to discuss multipole expansion. The $d=3$ version of
this story is discussed in a variety of places.141414See poisson2014gravity
for a textbook discussion. We will refer the reader to higuchi1986A ;
Rubin:1984tc ; rubin1984eigenvalues ; Frye:2012jj ; Birnholtz:2013ffa ;
Birnholtz:2013nta ; Bhattacharyya:2016nhn ; PhysRevD.107.104031  for a
discussion of STF tensors in general dimensions. The generalisation to
arbitrary dimensions is straightforward, if somewhat involved. In the course
of this work, we had to use a variety of identities involving STF tensors in
arbitrary dimensions scattered across these references. The goal of this
section is to review this theory for the reader’s benefit.

We will begin with a more traditional account of electrostatic multipole
expansion in $\mathbb{R}^{d}$ via orthonormal spherical harmonics on
$\mathbb{S}^{d-1}$. This is the generalisation of familiar multipole expansion
in $d=3$, and we will use it to set the stage for a more modern account of
multipole expansion using symmetric, trace-free (STF) tensors in the later
subsections. We conclude this appendix with a discussion of radiation reaction
in flat spacetime using these tools.

###  A.1 Orthonormal Spherical harmonics on $\boldsymbol{\mathbb{S}^{d-1}}$

Let us begin by considering the problem of electrostatics in $\mathbb{R}^{d}$.
Our goal in this subsection would be to describe the multipole expansion in
this case, given an orthonormal basis of spherical harmonics on
$\mathbb{S}^{d-1}$. Later in this subsection, we will give an explicit
construction of such an orthonormal basis, which can, in principle, be used in
explicit computations.

Given a charge distribution $\rho(\vec{r})$, the electric potential produced
by such a distribution is given in terms of the Newton-Coulomb integral

$\phi(\vec{r})=\int
d^{d}r_{0}\frac{\rho(\vec{r}_{0})}{(d-2)|\mathbb{S}^{d-1}||\vec{r}-\vec{r}_{0}|^{d-2}}\
.$ (38)

Here, we have denoted the volume of a unit sphere $\mathbb{S}^{d-1}$ via

$|\mathbb{S}^{d-1}|\equiv\frac{2\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2})}\ ,$
(39)

and have fixed our normalisations such that the Poisson equation takes the
form $\nabla^{2}\phi=-\rho$. While the above integral is indeed the right, a
more useful answer is obtained by performing a multipole expansion of the
Newton-Coulomb potential in terms of Legendre polynomials. In $d=3$, this is a
well-known statement from undergraduate physics courses, and we will now
describe a quick way to generalise this statement to arbitrary dimensions.

To this end, consider a simple problem where the answer due to multipole
expansion is straightforward: we imagine a spherical shell of radius $R$ in
$\mathbb{R}^{d}$ carrying a surface charge density
$\sigma_{\ell\vec{m}}(\hat{r})$ proportional to a spherical harmonic
$\mathscr{Y}_{\ell\vec{m}}(\hat{r})$, i.e., a spherical harmonic which under
the sphere laplacian has an eigenvalue $-\ell(\ell+d-2)$ and we use $\vec{m}$
to denote the additional labels required to furnish an orthonormal basis
within this eigenspace. The above eigenvalue follows from demanding that
$r^{\ell}\mathscr{Y}_{\ell\vec{m}}(\hat{r})$ be a harmonic function
annihilated by the Laplacian operator

$\begin{split}\nabla^{2}_{\mathbb{R}^{d}}\equiv\frac{1}{r^{d-1}}\frac{\partial}{\partial
r}r^{d-1}\frac{\partial}{\partial
r}+\frac{1}{r^{2}}\nabla^{2}_{\mathbb{S}^{d-1}}\ .\end{split}$ (40)

By symmetry, the potential due to such a problem should also be proportional
to the same spherical harmonic as the charge distribution. The potential
should be a harmonic function for $r\neq R$, regular at the origin, vanishing
at infinity, be continuous at $r=R$ but have a derivative discontinuity at the
shell equal to the charge density. These requirements uniquely determine the
solution to be

$\begin{split}\frac{R\sigma_{\ell\vec{m}}(\hat{r})}{(2\ell+d-2)}\Bigl{\\{}\Theta(r<R)\frac{r^{\ell}}{R^{\ell}}+\Theta(r>R)\frac{R^{\ell+d-2}}{r^{\ell+d-2}}\Bigr{\\}}\
.\end{split}$ (41)

This answer can be generalised to an arbitrary charge distribution, once it is
realised that any distribution can be built shell by shell and $\ell$ by
$\ell$. Using an orthonormal basis of spherical harmonics to do the projection
to every $\ell$, we can then write the potential for an arbitrary charge
distribution as

$\begin{split}\int d^{d}r_{0}\ \rho(\vec{r}_{0})\
\sum_{\ell\vec{m}}\frac{\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}^{\ast}_{\ell\vec{m}}(\hat{r}_{0})}{(2\ell+d-2)r_{0}^{d-2}}\Bigl{\\{}\Theta(r<r_{0})\frac{r^{\ell}}{r_{0}^{\ell}}+\Theta(r>R)\frac{r_{0}^{\ell+d-2}}{r^{\ell+d-2}}\Bigr{\\}}\
.\end{split}$ (42)

Comparing this against the Newton-Coulomb integral, we obtain the multipole
expansion formula in $\mathbb{R}^{d}$ :

$\begin{split}&\frac{1}{(d-2)|\mathbb{S}^{d-1}||\vec{r}-\vec{r}_{0}|^{d-2}}\
\\\
&\quad=\sum_{\ell\vec{m}}\frac{\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}^{\ast}_{\ell\vec{m}}(\hat{r}_{0})}{(2\ell+d-2)r_{0}^{d-2}}\Bigl{\\{}\Theta(r<r_{0})\frac{r^{\ell}}{r_{0}^{\ell}}+\Theta(r>R)\frac{r_{0}^{\ell+d-2}}{r^{\ell+d-2}}\Bigr{\\}}\
.\end{split}$ (43)

If we define the spherical multipole moments of the charge distribution
$\rho(\vec{r})$ by

$\begin{split}q_{{}_{\ell\vec{m}}}\equiv\frac{1}{2\ell+d-2}\int d^{d}r_{0}\
\rho(\vec{r}_{0})\ r_{0}^{\ell}\mathscr{Y}^{\ast}_{\ell\vec{m}}(\hat{r}_{0})\
,\end{split}$ (44)

we can write the potential far-outside the charge distribution as

$\begin{split}\sum_{\ell\vec{m}}\frac{1}{r^{\ell+d-2}}\
q_{{}_{\ell\vec{m}}}\mathscr{Y}_{\ell\vec{m}}(\hat{r})\ .\end{split}$ (45)

This is the basic content of multipole expansion in electrostatics. However,
to actually compute these multipole moments for a give charge distribution
$\rho(\vec{r})$, we will need the explicit form of the spherical harmonics
$\mathscr{Y}_{\ell\vec{m}}(\hat{r})$ on $\mathbb{S}^{d-1}$: we will now
proceed to address this in the rest the subsection.

The first step in constructing the spherical harmonics is to derive the most
symmetric among them: the Legendre polynomials. We will do this by recasting
the above expansion in terms of the Legendre polynomial. In the formula above,
the sum over orthonormal spherical harmonics of a given $\ell$ can be
performed through a higher dimensional generalisation of the addition theorem
for spherical harmonics, viz.,

$\begin{split}\sum_{\vec{m}}\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}^{\ast}_{\ell\vec{m}}(\hat{r}_{0})=\frac{N_{HH}(d,\ell)}{|\mathbb{S}^{d-1}|}P_{\ell}(d,\hat{r}\cdot\hat{r}_{0})\
.\end{split}$ (46)

Here $N_{HH}(d,\ell)$ is the number of orthonormal spherical harmonics of
degree $\ell$, with the notation here inspired by the fact that it is also the
number of linearly independent, homogeneous, harmonic polynomials (HHPs) of
degree $\ell$ in $\mathbb{R}^{d}$. We will elaborate on this and get an
explicit expression for $N_{HH}(d,\ell)$ below. For now, we move on to note
that $P_{\ell}(d,x)$ is the generalisation of the Legendre polynomial to
$\mathbb{R}^{d}$: it is the unique spherical harmonic invariant under
$SO(d-1)$ rotations which keep two poles of $\mathbb{S}^{d-1}$ fixed and is
normalised to unity at the north pole, i.e., $P_{\ell}(d,x=1)\equiv 1$.

With the above definitions, we can argue for the above addition theorem as
follows: first of all, the sum over orthonormal spherical harmonics of a given
$\ell$ should be a spherical harmonic which only depends on the relative
orientation of $\hat{r}$ and $\hat{r}_{0}$ and hence, the above sum should be
proportional to $P_{\ell}(d,\hat{r}\cdot\hat{r}_{0})$. The constant of
proportionality can then be fixed by setting $\hat{r}=\hat{r}_{0}$ and
integrating over the sphere $\mathbb{S}^{d-1}$ using orthonormality.

As a corollary of the above addition theorem, we note the following formula
for the inner product between Legendre harmonics of two different
orientations:

$\begin{split}\int_{\mathbb{S}^{d-1}}P_{\ell}(d,\hat{r}\cdot\hat{r}_{0})P_{\ell^{\prime}}(d,\hat{r}\cdot\hat{r}^{\prime}_{0})=\delta_{\ell\ell^{\prime}}\frac{|\mathbb{S}^{d-1}|}{N_{HH}(d,\ell)}P_{\ell}(d,\hat{r}_{0}\cdot\hat{r}^{\prime}_{0})\
.\end{split}$ (47)

This statement follows directly by the use of addition theorem followed by the
fact that $\mathscr{Y}_{\ell\vec{m}}(\hat{r})$ are assumed to be
orthonormal.For $\hat{r}_{0}=\hat{r}_{0}^{\prime}$, we get the Legendre
orthogonality relation

$\begin{split}\int_{0}^{\pi}d\vartheta\ \sin^{d-2}\vartheta\
P_{\ell}(d,\cos\vartheta)P_{\ell^{\prime}}(d,\cos\vartheta)=\delta_{\ell\ell^{\prime}}\frac{|\mathbb{S}^{d-1}|}{|\mathbb{S}^{d-2}|N_{HH}(d,\ell)}\
.\end{split}$ (48)

With the addition theorem, we can recast the multipole expansion in terms of
the Legendre polynomial as151515This expansion is often used to define
Gegenbauer polynomials $C^{\mu}_{\ell}(z)$, which differ from the generalised
Legendre polynomials introduced here merely by an overall normalisation. These
polynomials are also proportional to the associated Legendre functions. The
explicit relations are given by
$\begin{split}C^{\frac{d}{2}-1}_{\ell}(z)\equiv(d-2)\frac{N_{HH}(d,\ell)}{2\ell+d-2}P_{\ell}(d,z)\
,\quad
P_{\lambda}^{-\mu}(z)\equiv\frac{\left(\sqrt{1-z^{2}}\right)^{\mu}}{2^{\mu}\mu!}P_{\lambda-\mu}(2\mu+3;z)\
.\end{split}$ (49)

$\begin{split}&\frac{1}{(d-2)|\vec{r}-\vec{r}_{0}|^{d-2}}\ \\\
&\quad=\sum_{\ell}\frac{N_{HH}(d,\ell)P_{\ell}(d,\hat{r}\cdot\hat{r}_{0})}{(2\ell+d-2)r_{0}^{d-2}}\Bigl{\\{}\Theta(r<r_{0})\frac{r^{\ell}}{r_{0}^{\ell}}+\Theta(r>R)\frac{r_{0}^{\ell+d-2}}{r^{\ell+d-2}}\Bigr{\\}}\
.\end{split}$ (50)

As is well-known in $d=3$ case, this series expansion can be used to derive an
explicit expression for $P_{\ell}(d,x)$.

The steps involved are as follows: we take the case $r_{0}<r$, set
$t=\frac{r_{0}}{r}<1$ and $x=\hat{r}\cdot\hat{r}_{0}$ to write

$\begin{split}N_{HH}(d,\ell)P_{\ell}(d,x)&=(2\ell+d-2)\times\text{Coefficient
of $t^{\ell}$ in }\frac{1}{(d-2)(1-2xt+t^{2})^{\frac{d}{2}-1}}\ .\end{split}$
(51)

To extract the $t^{\ell}$ coefficient, we use

$\begin{split}\frac{(2\ell+d-2)}{(d-2)(1-2xt+t^{2})^{\frac{d}{2}-1}}&=\frac{\ell+\frac{d}{2}-1}{\Gamma(\frac{d}{2})}\int_{0}^{\infty}ds\
s^{\frac{d}{2}-2}\ e^{-s+2xst-st^{2}}\ ,\end{split}$ (52)

expand the exponentials involving $t$ and integrate to obtain

$\begin{split}N_{HH}(d,\ell)P_{\ell}(d,x)&=\frac{\ell+\frac{d}{2}-1}{\Gamma(\frac{d}{2})}\sum_{k}\int_{0}^{\infty}ds\
s^{\frac{d}{2}-2}\
e^{-s}\frac{(2xs)^{\ell-2k}}{(\ell-2k)!}\frac{(-s)^{k}}{k!}\\\
&=\frac{2^{\ell}\Gamma\left(\ell+\frac{d}{2}\right)}{\Gamma(\frac{d}{2})}\sum_{k}\frac{\Gamma\left(\ell+\frac{d}{2}-1-k\right)}{\Gamma(\ell+\frac{d}{2}-1)}\frac{(-)^{k}}{2^{2k}k!}\frac{x^{\ell-2k}}{(\ell-2k)!}\
.\end{split}$ (53)

Here, the sum over $k$ runs from $k=0$ and until the combination $\ell-2k$ is
non-negative. Defining the normalisation factor161616The interpretation of
this ubiquitous normalisation factor will become clearer when we describe STF
tensors in the next subsection. For now, we will note that
$\mathcal{N}_{d,\ell}$ is an inverse integer which has the following alternate
forms
$\begin{split}\mathcal{N}_{d,\ell}\equiv\frac{|\mathbb{S}^{d+2\ell-1}|}{|\mathbb{S}^{1}|^{\ell}|\mathbb{S}^{d-1}|}=\frac{(d-2)!!}{(d+2\ell-2)!!}\
.\end{split}$ (54)

$\begin{split}\mathcal{N}_{d,\ell}\equiv\frac{\Gamma(\frac{d}{2})}{2^{\ell}\Gamma\left(\ell+\frac{d}{2}\right)}\
,\quad\nu\equiv\frac{d}{2}+\ell-1\ ,\end{split}$ (55)

we finally obtain an explicit expression for the generalised Legendre
polynomial as

$\begin{split}\mathcal{N}_{d,\ell}N_{HH}(d,\ell)P_{\ell}(d,x)&=\sum_{k}\frac{\Gamma\left(\nu-k\right)}{2^{2k}k!\Gamma(\nu)}\frac{(-)^{k}x^{\ell-2k}}{(\ell-2k)!}\
.\end{split}$ (56)

Incidentally, the same expansion at $x=1$ also gives the number of orthonormal
spherical harmonics of degree $\ell$ as

$\begin{split}N_{HH}(d,\ell)&=\frac{2\ell+d-2}{d-2}\times\text{Coefficient of
$t^{\ell}$ in
}\frac{1}{(1-t)^{d-2}}=\frac{2\ell+d-2}{d-2}\binom{\ell+d-3}{\ell}\
.\end{split}$ (57)

This finishes our construction of the Legendre harmonic on $\mathbb{S}^{d-1}$.

Next, we will give a recursive construction of a complete orthonormal basis of
spherical harmonics on $\mathbb{S}^{d-1}$, just using the Legendre polynomials
constructed above. We begin with an explicit spherical coordinate system in
$\mathbb{R}^{d}$ given by

$\begin{split}x_{1}&=r\ \sin\vartheta_{d-2}\ \sin\vartheta_{d-3}\ \ldots\
\sin\vartheta_{2}\ \sin\vartheta_{1}\ \cos\varphi\ ,\\\ x_{2}&=r\
\sin\vartheta_{d-2}\ \sin\vartheta_{d-3}\ \ldots\ \sin\vartheta_{2}\
\sin\vartheta_{1}\ \sin\varphi\ ,\\\ x_{3}&=r\ \sin\vartheta_{d-2}\
\sin\vartheta_{d-3}\ \ldots\ \sin\vartheta_{2}\ \cos\vartheta_{1}\ ,\\\
x_{4}&=r\ \sin\vartheta_{d-2}\ \sin\vartheta_{d-3}\ \ldots\ \cos\vartheta_{2}\
,\\\ &\ldots\ ,\\\ x_{d-2}&=r\ \sin\vartheta_{d-2}\ \sin\vartheta_{d-3}\
\cos\vartheta_{d-4}\ ,\\\ x_{d-1}&=r\ \sin\vartheta_{d-2}\
\cos\vartheta_{d-3}\ ,\\\ x_{d}&=r\ \cos\vartheta_{d-2}\ .\end{split}$ (58)

Here the radius $r$ varies from $0$ to $\infty$ whereas the allowed values of
angles is $\vartheta_{i}\in[0,\pi]$ and $\varphi\in[0,2\pi)$. In these
coordinates, we can write the metric of $\mathbb{S}^{d-1}$ as

$\begin{split}d\Omega_{d-1}^{2}&=d\vartheta_{d-2}^{2}+\sin^{2}\vartheta_{d-2}d\Omega_{d-2}^{2}\\\
&=d\vartheta_{d-2}^{2}+\sin^{2}\vartheta_{d-2}d\vartheta_{d-3}^{2}+\sin^{2}\vartheta_{d-2}\sin^{2}\vartheta_{d-3}d\vartheta_{d-4}^{2}+\ldots\\\
&\quad+\prod_{k=j+1}^{d-2}\sin^{2}\vartheta_{k}\
d\vartheta_{j}^{2}+\ldots+\prod_{k=1}^{d-2}\sin^{2}\vartheta_{k}\
d\varphi^{2}.\end{split}$ (59)

The volume form

$\begin{split}\int_{\mathbb{S}^{d-1}}(\ldots)\equiv\int d\vartheta_{1}\wedge
d\vartheta_{2}\ldots d\vartheta_{d-2}\wedge d\varphi\
\prod_{k=1}^{d-2}\sin^{k}\vartheta_{k}\ (\ldots)\ .\end{split}$ (60)

We are interested in constructing an orthonormal basis of spherical harmonics
in these coordinates. As we described above, the simplest spherical harmonic
is the Legendre harmonic $P_{\ell}(d,\cos\vartheta_{d-2})$ which depends only
on $\vartheta_{d-2}$. It obeys the second-order ODE

$\begin{split}\left[\frac{1}{\sin^{d-2}\vartheta}\frac{d}{d\vartheta}\sin^{d-2}\vartheta\frac{d}{d\vartheta}+\ell(\ell+d-2)\right]P_{\ell}(d,\cos\vartheta)=0\
.\end{split}$ (61)

The function $P_{\ell}(d,\cos\vartheta)$ is the unique $\ell^{th}$ degree
polynomial in $\cos\vartheta$ that solves the above ODE and is normalised to
$P_{\ell}(d,\cos\vartheta=1)=1$. In general, spherical harmonics of degree
$\ell$ obey the eigenvalue equation
$\left[\nabla^{2}_{\mathbb{S}^{d-1}}+\ell(\ell+d-2)\right]\mathcal{S}_{\ell}(\Omega_{d-1})=0$,
or in more detail

$\begin{split}\left[\frac{1}{\sin^{d-2}\vartheta_{d-2}}\frac{\partial}{\partial\vartheta_{d-2}}\sin^{d-2}\vartheta_{d-2}\frac{\partial}{\partial\vartheta_{d-2}}+\frac{1}{\sin^{2}\vartheta_{d-2}}\nabla^{2}_{\mathbb{S}^{d-2}}+\ell(\ell+d-2)\right]\mathcal{S}_{\ell}(\Omega_{d-1})=0\
.\end{split}$ (62)

This equation can be solved via a separation of variables ansatz

$\begin{split}\mathcal{S}_{\ell}=(\sin\vartheta_{d-2})^{m}P_{\ell-m}(d+2m,\cos\vartheta_{d-2})\widehat{\mathcal{S}}_{m}(\Omega_{d-2})\
,\end{split}$ (63)

for a non-negative integer $0\leq m\leq\ell$. Substituting this ansatz into
the equation above yields the eigenvalue equation
$\left[\nabla^{2}_{\mathbb{S}^{d-2}}+m(m+d-3)\right]\widehat{\mathcal{S}}_{m}(\Omega_{d-2})=0$
in the lower dimensional sphere, i.e., the function
$\widehat{\mathcal{S}}_{m}(\Omega_{d-2})$ is actually a spherical harmonic of
degree $m$ on $\mathbb{S}^{d-2}$. This gives rise to

$\begin{split}\sum_{m=0}^{\ell}N_{HH}(d-1,m)=N_{HH}(d,\ell)\end{split}$ (64)

number of spherical harmonics of degree $\ell$ on $\mathbb{S}^{d-1}$ (to get
the above equality, we have used Eq.(57)). Recursing this construction, we get
a set of spherical harmonics of the form

$\begin{split}\mathcal{C}_{\ell\vec{m}}\ e^{\pm
im_{{}_{1}}\varphi}\left[\prod_{k=1}^{d-2}(\sin\vartheta_{k})^{m_{k}}P_{m_{k+1}-m_{k}}(k+2+2m_{k},\cos\vartheta_{k})\right]_{m_{d-1}=\ell}\
,\end{split}$ (65)

one for every non-decreasing sequence of non-negative integers

$\begin{split}0\leq m_{{}_{1}}\leq m_{2}\ldots\leq m_{d-2}\leq m_{d-1}=\ell\
.\end{split}$ (66)

Here $\mathcal{C}_{\ell\vec{m}}$ is a normalisation constant which we shall
determine below.

We will now argue that these spherical harmonics form an orthonormal set: any
two harmonics with distinct $e^{i\varphi}$ factors are evidently orthogonal.
Thus, we need to address only the case where $e^{i\varphi}$ factors are the
same. Without loss of generality, let us assume that the dependences on
$\vartheta_{k}$ for all $k<i$ are also the same between the two spherical
harmonics for some $i<d-1$, and they differ first on their $\vartheta_{i}$
dependence, i.e., we consider two spherical harmonics in the above set with
$m_{k}=m^{\prime}_{k}$ for all $k\leq i$, but have $m_{i+1}\neq
m_{i+1}^{\prime}$. The inner product between these two spherical harmonics
then has a factor

$\begin{split}\int_{0}^{\pi}d\vartheta_{i}(\sin\vartheta_{i})^{i+2m_{i}}P_{m_{i+1}-m_{i}}(i+2+2m_{i},\cos\vartheta_{i})P_{m^{\prime}_{i+1}-m_{i}}(i+2+2m_{i},\cos\vartheta_{i})\
,\end{split}$ (67)

which then vanishes using Legendre orthogonality (see Eq.(48)) on
$\mathbb{S}^{i+2m_{i}+1}$. The mutual orthogonality along with the counting in
Eq.(64) proves then that we have indeed constructed a complete set of
spherical harmonics of degree $\ell$ on $\mathbb{S}^{d-1}$.

We will conclude this discussion by normalising the spherical harmonics
constructed above. The norm computation reduces to a product integral like the
one above, which can then be evaluated using Eq.(48). Thus, the normalisation
of the spherical harmonic given in Eq.(65) is given by

$\begin{split}|\mathcal{C}_{\ell\vec{m}}|^{-2}&\equiv
2\pi\prod_{i=1}^{d-2}\int_{0}^{\pi}d\vartheta_{i}(\sin\vartheta_{i})^{i+2m_{i}}P^{2}_{m_{i+1}-m_{i}}(i+2+2m_{i},\cos\vartheta_{i})\\\
&=2\pi\prod_{i=1}^{d-2}\frac{|\mathbb{S}^{i+2m_{i}+1}|}{|\mathbb{S}^{i+2m_{i}}|N_{HH}(i+2m_{i}+2,m_{i+1}-m_{i})}\
,\end{split}$ (68)

With this, we have a concrete realisation of the orthonormal spherical
harmonics $\mathscr{Y}_{\ell\vec{m}}(\hat{r})$, using which multipole moments
could be computed for a given charge distribution.

###  A.2 STF tensors in $\boldsymbol{\mathbb{R}^{d}}$ and cartesian multipole
moments

Till now, we have described the multipole expansion in terms of an orthonormal
basis of spherical harmonics $\mathscr{Y}_{\ell\vec{m}}(\hat{r})$ and the
corresponding spherical multipole moments $q_{{}_{\ell\vec{m}}}$. We will now
describe an alternate formalism based on a more symmetric, but over-complete
basis of spherical harmonics made of Legendre polynomials about arbitrary
directions (we will call this basis an STF basis). A general spherical
harmonic in STF basis is naturally described by symmetric trace-free (STF)
tensors with constant cartesian components.

For definiteness, we consider spherical harmonics of the form

$\begin{split}\mathcal{N}_{d,\ell}N_{HH}(d,\ell)P_{\ell}(d,\hat{\kappa}\cdot\hat{r})&=\sum_{k}\frac{\Gamma\left(\nu-k\right)}{2^{2k}k!\Gamma(\nu)}\frac{(-)^{k}(\hat{\kappa}\cdot\hat{r})^{\ell-2k}}{(\ell-2k)!}\\\
&=\frac{1}{\ell!}\hat{\kappa}_{i_{1}}\hat{\kappa}_{i_{2}}\ldots\hat{\kappa}_{i_{\ell}}\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}\\\
&=\frac{1}{\ell!}\hat{\kappa}_{i_{1}}\hat{\kappa}_{i_{2}}\ldots\hat{\kappa}_{i_{\ell}}\hat{r}^{j_{1}}\hat{r}^{j_{2}}\ldots\hat{r}^{j_{\ell}}\Pi^{<i_{1}i_{2}\ldots
i_{\ell}>}_{<j_{1}j_{2}\ldots j_{\ell}>}\ ,\end{split}$ (69)

where $\hat{\kappa}$ is an arbitrary unit vector, and in the last line we have
written the spherical harmonic as a projected contraction of two tensors. The
angular bracket here denotes the symmetric trace-free (STF) projection and
$\Pi$ is the STF-projector. An explicit expression that follows from the above
definition is

$\begin{split}\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}&=\sum_{k}\frac{(-)^{k}\Gamma\left(\nu-k\right)}{2^{k}\Gamma(\nu)}\\\
&\times\Bigl{\\{}\hat{r}^{i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell-2k}}\delta^{i_{\ell+1-2k}i_{\ell+2-2k}}\ldots\delta^{i_{\ell-1}i_{\ell}}+\text{distinct
index permutations}\Bigr{\\}}\ .\end{split}$ (70)

Here the sum within the curly braces sums over all index permutations of the
set $\\{i_{1},\ldots,i_{\ell}\\}$ which give distinct answers. The number of
such distinct permutations can be counted as follows: there are
$\binom{\ell}{2k}$ ways of choosing the subset of indices that go into
Kronecker deltas, and $\frac{(2k)!}{2^{k}k!}=(2k-1)!!$ distinct ways of
pairing a given subset.171717The number of pairings can be counted as follows:
the $(2k)!$ ways to permute the subset of indices on Kronecker deltas.
Exchanging an index within a pair, as well as permuting the pair as a whole
does not change the final resultant pairings, i.e., there is a
$(\mathbb{Z}_{2})^{k}\times\mathbb{S}_{k}$ automorphism group which acts
freely and transitively on the equivalence class of permutations which result
in a given pairing. We hence obtain the number of distinct pairings by
dividing out the cardinality of the automorphism group. Thus, the total number
of distinct permutations is
$(2k-1)!!\binom{\ell}{2k}=\frac{\ell!}{2^{k}k!(\ell-2k)!}$. With this counting
of distinct permutations, it is then easy to check that contracting
$\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}$ with
$\frac{1}{\ell!}\hat{\kappa}_{i_{1}}\hat{\kappa}_{i_{2}}\ldots\hat{\kappa}_{i_{\ell}}$
does give
$\mathcal{N}_{d,\ell}\mathcal{N}_{HH}(d,\ell)P_{\ell}(d,\hat{\kappa}\cdot\hat{r})$.
The STF projector can also be given a closed-form expression as:

$\begin{split}\Pi^{<i_{1}i_{2}\ldots i_{\ell}>}_{<j_{1}j_{2}\ldots
j_{\ell}>}&=\sum_{k}\frac{(-)^{k}\Gamma\left(\nu-k\right)}{2^{k}k!(\ell-2k)!\Gamma(\nu)}\\\
&\times\delta^{(i_{1}}_{(j_{1}}\delta^{i_{2}}_{j_{2}}\ldots\delta^{i_{\ell-2k}}_{j_{\ell-2k}}\delta^{i_{\ell+1-2k}i_{\ell+2-2k}}\ldots\delta^{i_{\ell-1}i_{\ell})}\delta_{j_{\ell+1-2k}j_{\ell+2-2k}}\ldots\delta_{j_{\ell-1}j_{\ell})}\
,\end{split}$ (71)

where the $(i_{1}\ldots i_{\ell})$ denotes a symmetric projection. To
elucidate the arguments above, we will now write down the explicit expressions
of $\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}$ for $\ell\leq
5$. We have

$\begin{split}&\hat{r}^{<i_{1}>}\equiv\hat{r}^{<i_{1}>}\ ,\
\hat{r}^{<i_{1}}\hat{r}^{i_{2}>}\equiv\hat{r}^{i_{1}}\hat{r}^{i_{2}}-\frac{1}{d}\delta^{i_{1}i_{2}}\\\
&\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{3}>}\equiv\hat{r}^{i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{3}}-\frac{1}{d+2}\left(\hat{r}^{i_{1}}\delta^{i_{2}i_{3}}+\hat{r}^{i_{2}}\delta^{i_{1}i_{3}}+\hat{r}^{i_{3}}\delta^{i_{1}i_{2}}\right)\
,\\\ \end{split}$ (72)

for $\ell\leq 3$. For $\ell=4$, we have

$\begin{split}&\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{3}}\hat{r}^{i_{4}>}\equiv\hat{r}^{i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{3}}\hat{r}^{i_{4}}\\\
&\quad-\frac{1}{d+4}\left(\hat{r}^{i_{1}}\hat{r}^{i_{2}}\delta^{i_{3}i_{4}}+\hat{r}^{i_{1}}\hat{r}^{i_{3}}\delta^{i_{2}i_{4}}+\hat{r}^{i_{1}}\hat{r}^{i_{4}}\delta^{i_{2}i_{3}}+\hat{r}^{i_{2}}\hat{r}^{i_{3}}\delta^{i_{1}i_{4}}+\hat{r}^{i_{2}}\hat{r}^{i_{4}}\delta^{i_{1}i_{3}}+\hat{r}^{i_{3}}\hat{r}^{i_{4}}\delta^{i_{1}i_{2}}\right)\\\
&\quad+\frac{1}{(d+4)(d+2)}\left(\delta^{i_{1}i_{2}}\delta^{i_{3}i_{4}}+\delta^{i_{1}i_{3}}\delta^{i_{2}i_{4}}+\delta^{i_{1}i_{4}}\delta^{i_{2}i_{3}}\right)\
,\\\ \end{split}$ (73)

and for $\ell=5$, we get

$\begin{split}&\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{3}}\hat{r}^{i_{4}}\hat{r}^{i_{5}>}\equiv\hat{r}^{i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{3}}\hat{r}^{i_{4}}\hat{r}^{i_{5}}\\\
&\quad-\frac{1}{d+6}\Bigl{(}\hat{r}^{i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{3}}\delta^{i_{4}i_{5}}+\hat{r}^{i_{1}}\hat{r}^{i_{2}}\hat{r}^{i_{4}}\delta^{i_{3}i_{5}}+\hat{r}^{i_{1}}\hat{r}^{i_{3}}\hat{r}^{i_{4}}\delta^{i_{2}i_{5}}+\hat{r}^{i_{2}}\hat{r}^{i_{3}}\hat{r}^{i_{4}}\delta^{i_{1}i_{5}}\Bigr{.}\\\
&\quad\Bigl{.}+\hat{r}^{i_{5}}\hat{r}^{i_{1}}\hat{r}^{i_{2}}\delta^{i_{3}i_{4}}+\hat{r}^{i_{5}}\hat{r}^{i_{1}}\hat{r}^{i_{3}}\delta^{i_{2}i_{4}}+\hat{r}^{i_{5}}\hat{r}^{i_{1}}\hat{r}^{i_{4}}\delta^{i_{2}i_{3}}+\hat{r}^{i_{5}}\hat{r}^{i_{2}}\hat{r}^{i_{3}}\delta^{i_{1}i_{4}}+\hat{r}^{i_{5}}\hat{r}^{i_{3}}\hat{r}^{i_{4}}\delta^{i_{1}i_{2}}\Bigr{)}\\\
&\
+\frac{1}{(d+6)(d+4)}\Bigl{(}\hat{r}^{i_{1}}\delta^{i_{2}i_{3}}\delta^{i_{4}i_{5}}+\hat{r}^{i_{1}}\delta^{i_{2}i_{4}}\delta^{i_{3}i_{5}}+\hat{r}^{i_{1}}\delta^{i_{2}i_{5}}\delta^{i_{3}i_{4}}\Bigr{.}\\\
&\
+\Bigl{.}\hat{r}^{i_{2}}\delta^{i_{1}i_{3}}\delta^{i_{4}i_{5}}+\hat{r}^{i_{2}}\delta^{i_{1}i_{4}}\delta^{i_{3}i_{5}}+\hat{r}^{i_{2}}\delta^{i_{1}i_{5}}\delta^{i_{3}i_{4}}+\hat{r}^{i_{3}}\delta^{i_{1}i_{2}}\delta^{i_{4}i_{5}}+\hat{r}^{i_{3}}\delta^{i_{1}i_{4}}\delta^{i_{2}i_{5}}+\hat{r}^{i_{3}}\delta^{i_{1}i_{5}}\delta^{i_{2}i_{4}}\Bigr{.}\\\
&\
\Bigl{.}+\hat{r}^{i_{4}}\delta^{i_{1}i_{2}}\delta^{i_{3}i_{5}}+\hat{r}^{i_{4}}\delta^{i_{1}i_{3}}\delta^{i_{2}i_{5}}+\hat{r}^{i_{4}}\delta^{i_{1}i_{5}}\delta^{i_{2}i_{3}}+\hat{r}^{i_{5}}\delta^{i_{1}i_{2}}\delta^{i_{3}i_{4}}+\hat{r}^{i_{5}}\delta^{i_{1}i_{3}}\delta^{i_{2}i_{4}}+\hat{r}^{i_{5}}\delta^{i_{1}i_{4}}\delta^{i_{2}i_{3}}\Bigr{)}\
.\end{split}$ (74)

The reader can check that the expressions in RHS are completely symmetric
under permutations of indices, and vanish if we take a trace over any two
indices. Further, our counting of distinct permutations can also be checked
for every term written above.

A more succinct way to summarise the permutations/symmetrisations described
above is to work instead with the homogeneous harmonic polynomials (HHPs) in
cartesian coordinates

$\begin{split}x^{<i_{1}}x^{i_{2}}\ldots x^{i_{\ell}>}&\equiv r^{\ell}\
\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}=\left[\sum_{k=0}^{\lfloor\frac{\ell}{2}\rfloor}\
\frac{\Gamma\left(\nu-k\right)}{k!\
\Gamma\left(\nu\right)}\left(\frac{r}{2}\right)^{2k}(-\nabla^{2})^{k}\right]_{\nu=\frac{d}{2}+\ell-1}x^{i_{1}}x^{i_{2}}\ldots
x^{i_{\ell}}\ .\end{split}$ (75)

The relation to generalised Legendre polynomials then follows from

$\begin{split}\frac{1}{\ell!}\kappa_{i_{1}}\kappa_{i_{2}}\ldots\kappa_{i_{\ell}}x^{<i_{1}}x^{i_{2}}\ldots
x^{i_{\ell}>}&\equiv\left[\sum_{k=0}^{\lfloor\frac{\ell}{2}\rfloor}\
\frac{\Gamma\left(\nu-k\right)}{k!\
\Gamma\left(\nu\right)}\left(\frac{r}{2}\right)^{2k}(-\nabla^{2})^{k}\right]_{\nu=\frac{d}{2}+\ell-1}\frac{(\vec{\kappa}\cdot\vec{r})^{\ell}}{\ell!}\\\
&=\left[\sum_{k=0}^{\lfloor\frac{\ell}{2}\rfloor}\
\frac{\Gamma\left(\nu-k\right)}{k!\
\Gamma\left(\nu\right)}\left(-\frac{\kappa^{2}r^{2}}{4}\right)^{k}\frac{(\vec{\kappa}\cdot\vec{r})^{\ell-2k}}{(\ell-2k)!}\
\right]_{\nu=\frac{d}{2}+\ell-1}\\\
&=\mathcal{N}_{d,\ell}N_{HH}(d,\ell)(\kappa
r)^{\ell}P_{\ell}\left(d,\hat{\kappa}\cdot\hat{r}\right)\ ,\end{split}$ (76)

where, in the last step, we have used Eq.(56). The STF basis for multipole
expansion in flat spacetime is often introduced in terms of these cartesian
HHPs (See e.g.poisson2014gravity ). In dS spacetime (and more generally in
cosmology), the absence of global cartesian coordinates limits their scope.
The STF basis for spherical harmonics is, however, a useful tool for multipole
expansion in such spacetimes, since isotropy is still a true symmetry.

We will now describe how the STF basis relates to the description of spherical
harmonics given before. For any given $\ell$, we can form

$\begin{split}N_{H}(d,\ell)\equiv\binom{\ell+d-1}{\ell}\end{split}$ (77)

number of STF harmonics of the form
$\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}$. The above binomial
coefficient counts the number of ways $d$ directions can be filled into $\ell$
indices. The combinatorics here is identical to the bose-counting problem
familiar from elementary statistical mechanics, where one counts the ways in
which $d$ bosons could be filled into $\ell$ degenerate energy levels. All the
STF harmonics are not however linearly independent, they obey
$N_{H}(d,\ell-2)$ number of conditions of the form

$\begin{split}\delta_{i_{1}i_{2}}\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}=0\
.\end{split}$ (78)

They hence span a vector space of spherical harmonics of dimension

$\begin{split}N_{H}(d,\ell)-N_{H}(d,\ell-2)=N_{HH}(d,\ell)\ ,\end{split}$ (79)

where the equality folows by using the explicit forms in Eqs.(57) and (77).
This shows that the harmonics
$\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}$ indeed form an
overcomplete basis of spherical harmonics of degree $\ell$.

The completeness means the following: say we are given a spherical harmonic
$\mathscr{Y}_{\ell}(\hat{r})$ of degree $\ell$ on $\mathbb{S}^{d-1}$. We can
then define a symmetric trace-free (STF) tensor $\mathscr{Y}_{i_{1}i_{2}\ldots
i_{\ell}}$ of rank $\ell$ in $\mathbb{R}^{d}$ such that

$\begin{split}\mathscr{Y}_{\ell}(\hat{r})=\frac{1}{\ell!}\mathscr{Y}_{i_{1}i_{2}\ldots
i_{\ell}}\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}\
.\end{split}$ (80)

The orthonormal basis of spherical harmonics constructed in the previous
subsection then defines an orthonormal set of STF tensors

$\begin{split}\mathscr{Y}_{\ell\vec{m}}(\hat{r})=\frac{1}{\ell!}\mathscr{Y}^{(\ell\vec{m})}_{i_{1}i_{2}\ldots
i_{\ell}}\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}\
.\end{split}$ (81)

Further, the inner product on the space of STF tensors is induced from the
standard inner product on the space of functions on $\mathbb{S}^{d-1}$. To get
an explicit expression, consider the following integral

$\begin{split}\int_{\hat{r}\in\mathbb{S}^{d-1}}\frac{1}{\ell!}&\kappa_{i_{1}}\kappa_{i_{2}}\ldots\kappa_{i_{\ell}}\hat{r}^{<i_{1}}\hat{r}^{i_{2}}\ldots\hat{r}^{i_{\ell}>}\times\frac{1}{\ell!}\bar{\kappa}_{j_{1}}\bar{\kappa}_{j_{2}}\ldots\bar{\kappa}_{j_{\ell}}\hat{r}^{<j_{1}}\hat{r}^{j_{2}}\ldots\hat{r}^{j_{\ell}>}\\\
&=[\mathcal{N}_{d,\ell}N_{HH}(d,\ell)]^{2}(\kappa\bar{\kappa})^{\ell}\int_{\hat{r}\in\mathbb{S}^{d-1}}P_{\ell}(d,\hat{\kappa}\cdot\hat{r})P_{\ell}(d,\hat{\bar{\kappa}}\cdot\hat{r})\\\
&=\mathcal{N}^{2}_{d,\ell}N_{HH}(d,\ell)|\mathbb{S}^{d-1}|(\kappa\bar{\kappa})^{\ell}P_{\ell}(d,\hat{\kappa}\cdot\hat{\bar{\kappa}})\\\
&=\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|\
\frac{1}{\ell!}\kappa_{<i_{1}}\kappa_{i_{2}}\ldots\kappa_{i_{\ell}>}\bar{\kappa}^{<i_{1}}\bar{\kappa}^{i_{2}}\ldots\bar{\kappa}^{i_{\ell}>}\
.\end{split}$ (82)

For example, the STF tensors corresponding to the orthonormal spherical
harmonics have an inner product given by

$\begin{split}\frac{\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|}{\ell!}\mathscr{Y}_{(\ell\vec{m}^{\prime})}^{\ast<i_{1}i_{2}\ldots
i_{\ell}>}\mathscr{Y}^{(\ell\vec{m})}_{<i_{1}i_{2}\ldots
i_{\ell}>}=\delta^{\vec{m}}_{\vec{m}^{\prime}}\ .\end{split}$ (83)

We recognise $\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|$ here as the conversion
factor between the STF tensor inner product and the standard functional inner
product between the spherical harmonics. The same factor also appears in the
statement of spherical harmonic addition theorem, stated in terms of STF
tensors:

$\begin{split}\frac{\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|}{\ell!}\sum_{\vec{m}}\mathscr{Y}_{(\ell\vec{m})}^{\ast<i_{1}i_{2}\ldots
i_{\ell}>}\mathscr{Y}^{(\ell\vec{m})}_{<j_{1}j_{2}\ldots
j_{\ell}>}=\Pi^{<i_{1}i_{2}\ldots i_{\ell}>}_{<j_{1}j_{2}\ldots j_{\ell}>}\
.\end{split}$ (84)

This important relation can be proved in many ways: one way is to use Eq.(69)
to convert the standard addition theorem into STF tensors. Another ab initio
derivation is to first argue that LHS should be proportional to RHS for
symmetry reasons and then fix the normalisation by using the orthonormality
relation Eq.(83).

###  A.3 Green functions in Minkowski spacetime

We will begin by briefly reviewing the Green functions of the wave operator
(i.e., the massless scalar operator) in $\mathbb{R}^{d,1}$. This theory is
standard, although the notations and normalisations for Green functions in
$d\neq 2,3$ are non-standard. Thus, this subsection mainly serves to establish
our notation. We will state our results with an eye towards their
generalisation to dS Green functions.

We begin with the unique spherically symmetric eigenfunction of the Laplacian
in $\mathbb{R}^{d}$ with eigenvalue $-\omega^{2}$ :

$\begin{split}J_{0}(d,\omega
r)&\equiv{}_{0}F_{1}\left[\frac{d}{2},-\frac{\omega^{2}r^{2}}{4}\right]\
.\end{split}$ (85)

We can construct a whole tower of descendants from this eigenfunction by
taking an STF derivative

$\begin{split}\mathscr{Y}_{\ell}(-\vec{\nabla})J_{0}(d,\omega
r)&\equiv\omega^{\ell}\ J_{\ell}(d,\omega r)\
\mathscr{Y}_{\ell}(\vec{n})=\omega^{\nu-\frac{d}{2}+1}\ J_{\ell}(d,\omega r)\
\mathscr{Y}_{\ell}(\vec{n})\ ,\end{split}$ (86)

where we have defined (we remind the reader that
$\nu\equiv\ell+\frac{d}{2}-1$)

$\begin{split}J_{\ell}(d,\omega
r)&\equiv\Gamma\left(\frac{d}{2}\right)\left(\frac{\omega
r}{2}\right)^{1-\frac{d}{2}}\ J_{\nu}(\omega
r)\equiv\frac{\Gamma(d/2)}{\Gamma(1+\nu)}\left(\frac{\omega
r}{2}\right)^{\nu-\frac{d}{2}+1}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\ .\end{split}$ (87)

The notation is motivated by the fact that the functions that appear here
generalise the Bessel J functions in the $d=2$ version of the above problem.
We can also define the functions analogous to Neumann and Hankel functions. We
will define the _Neumann Green function_ via

$\begin{split}&N_{\ell}(d,\omega r)\equiv-\frac{1}{4}\frac{Y_{\nu}(\omega
r)}{(2\pi\omega r)^{\frac{d}{2}-1}}\\\
&=\frac{\Gamma(\nu)}{(4\pi)^{d/2}}\left(\frac{\omega
r}{2}\right)^{-\nu-\frac{d}{2}+1}\Biggl{\\{}{}_{0}F_{1}\left[1-\nu,-\frac{\omega^{2}r^{2}}{4}\right]-\frac{\pi\cot\nu\pi}{\Gamma(\nu)\Gamma(1+\nu)}\left(\frac{\omega
r}{2}\right)^{2\nu}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\Biggr{\\}}\
.\end{split}$ (88)

In the above definition, for half-integer $\nu$ (i.e., for $d$ odd), we can
set $\cot\nu\pi=0$, whereas for integer $\nu$, the divergence in the
$\cot\nu\pi$ cancels the divergence in the first term and this formula should
be interpreted as a limit. Green functions are normalised such that

$\begin{split}-(\vec{\nabla}^{2}+\omega^{2})[\omega^{\nu+\frac{d}{2}-1}N_{\ell}(d,\omega
r)\mathscr{Y}_{\ell}(\vec{n})]=\mathscr{Y}_{\ell}(-\vec{\nabla})\delta^{d}(\vec{r})\
.\end{split}$ (89)

Since the RHS here is a multipole source, the combination
$\omega^{\nu+\frac{d}{2}-1}N_{\ell}(d,\omega r)\mathscr{Y}_{\ell}(\vec{n})$
should then be interpreted as the amplitude of standing wave sourced by such a
multipole source. We term this a _standing wave_ since it is an even function
of frequency. In contrast, the outgoing/ingoing waves are denoted by
$\omega^{\nu+\frac{d}{2}-1}H_{\ell}^{\pm}(d,\omega r)$ respectively. We will
refer to them as _Hankel Green functions_. Given a spherical harmonic
$\mathscr{Y}_{\ell}(\vec{n})$ of degree $\ell$ on $\mathbb{S}^{d-1}$, both
these Green functions satisfy

$\begin{split}-(\vec{\nabla}^{2}+\omega^{2})[\omega^{\nu+\frac{d}{2}-1}H_{\ell}^{\pm}(d,\omega
r)\mathscr{Y}_{\ell}(\vec{n})]=\mathscr{Y}_{\ell}(-\vec{\nabla})\delta^{d}(\vec{r})\
.\end{split}$ (90)

The outgoing/ingoing conditions are imposed by taking $H_{\ell}^{\pm}(d,\omega
r)$ to be analytic in the upper/lower half plane of complex frequency
respectively. The notation here is again motivated by the fact that these
functions generalise the Hankel functions in $d=2$ (up to normalisations).
Their explicit forms are given by

$\begin{split}&H_{\ell}^{\pm}(d,\omega r)\equiv\frac{\pm
i}{4}\frac{H^{1,2}_{\nu}(\omega r)}{(2\pi\omega r)^{\frac{d}{2}-1}}\equiv
N_{\ell}(d,\omega r)\pm\frac{i\pi}{\Gamma(d/2)(4\pi)^{d/2}}J_{\ell}(d,\omega
r)\\\ &=\frac{\Gamma(\nu)}{(4\pi)^{d/2}}\left(\frac{\omega
r}{2}\right)^{-\nu-\frac{d}{2}+1}\Biggl{\\{}{}_{0}F_{1}\left[1-\nu,-\frac{\omega^{2}r^{2}}{4}\right]\pm(1\pm
i\cot\nu\pi)\frac{2\pi i}{\Gamma(\nu)^{2}}\frac{1}{2\nu}\left(\frac{\omega
r}{2}\right)^{2\nu}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\Biggr{\\}}\
.\end{split}$ (91)

As in the case of Neumann Green functions, for half-integer $\nu$ (i.e., for
$d$ odd), we can set $\cot\nu\pi=0$, whereas for integer $\nu$ the above
expression is indeterminate and should be interpreted as a limit.

As in the case of Bessel J functions, these Green functions could also be
obtained by STF-differentiating their corresponding primary eigenfunction at
$\ell=0$, viz.,

$\begin{split}\mathscr{Y}_{\ell}(-\vec{\nabla})N_{0}(d,\omega
r)&=\omega^{\ell}\ N_{\ell}(d,\omega r)\
\mathscr{Y}_{\ell}(\vec{n})=\omega^{\nu-\frac{d}{2}+1}\ N_{\ell}(d,\omega r)\
\mathscr{Y}_{\ell}(\vec{n})\ ,\\\
\mathscr{Y}_{\ell}(-\vec{\nabla})H^{\pm}_{0}(d,\omega r)&=\omega^{\ell}\
H^{\pm}_{\ell}(d,\omega r)\
\mathscr{Y}_{\ell}(\vec{n})=\omega^{\nu-\frac{d}{2}+1}\
H^{\pm}_{\ell}(d,\omega r)\ \mathscr{Y}_{\ell}(\vec{n})\ .\end{split}$ (92)

A related statement is the _multipole-expansion_ of these Green functions,
which, in our normalisations, takes the following form:

$\begin{split}&J_{0}(d,\omega|\vec{r}-\vec{r}_{0}|)=\sum_{\ell
m}|\mathbb{S}^{d-1}|\
\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}_{\ell\vec{m}}(\hat{r}_{0})^{\ast}J_{\ell}(d,\omega
r)J_{\ell}(d,\omega r_{0})\ ,\\\
&N_{0}(d,\omega|\vec{r}-\vec{r}_{0}|)=\sum_{\ell m}|\mathbb{S}^{d-1}|\
\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}_{\ell\vec{m}}(\hat{r}_{0})^{\ast}\\\
&\qquad\times\Bigl{\\{}\Theta(r<r_{0})J_{\ell}(d,\omega r)N_{\ell}(d,\omega
r_{0})+\Theta(r>r_{0})J_{\ell}(d,\omega r_{0})N_{\ell}(d,\omega r)\Bigr{\\}}\
,\\\
&H_{0}^{\pm}(d,\omega|\vec{r}-\vec{r}_{0}|)=\sum_{\ell\vec{m}}|\mathbb{S}^{d-1}|\
\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}_{\ell\vec{m}}(\hat{r}_{0})^{\ast}\\\
&\qquad\times\Bigl{\\{}\Theta(r<r_{0})J_{\ell}(d,\omega
r)H_{\ell}^{\pm}(d,\omega r_{0})+\Theta(r>r_{0})J_{\ell}(d,\omega
r_{0})H_{\ell}^{\pm}(d,\omega r)\Bigr{\\}}\ .\end{split}$ (93)

Here, the set of functions $\mathscr{Y}_{\ell\vec{m}}(\hat{r})$ for different
$\vec{m}$ denote an orthonormal basis of $\mathbb{S}^{d-1}$ spherical
harmonics of degree $\ell$. Further, in the equation above, the symbol

$\begin{split}|\mathbb{S}^{d-1}|\equiv\frac{2\pi^{\frac{d}{2}}}{\Gamma\left(\frac{d}{2}\right)}\
\end{split}$ (94)

denotes the volume of the unit sphere. The argument for the above expansion is
well-known within the theory of Green functions: we first expand the LHS in
terms of eigenfunctions and then fix the coefficients by demanding continuity
and a unit jump in the radial derivative. The jump can be readily evaluated
using the Wronskian formulae181818Our Wronskian convention is
$\mathcal{W}[f(z),g(z)]\equiv f\partial_{z}g-g\partial_{z}f$.

$\begin{split}\mathcal{W}[N_{\ell}(d,z),J_{\ell}(d,z)]=\mathcal{W}[H^{\pm}_{\ell}(d,z),J_{\ell}(d,z)]=\frac{1}{|\mathbb{S}^{d-1}|z^{d-1}}\
.\end{split}$ (95)

We will be interested here in the multipole expansion of the retarded/outgoing
Green function $\omega^{d-2}H_{0}^{+}(d,\omega|\vec{r}-\vec{r}_{0}|)$, which,
using the relations quoted earlier, we can rewrite entirely in terms of
${}_{0}F_{1}$ functions:

$\begin{split}&\omega^{d-2}H_{0}^{+}(d,\omega|\vec{r}-\vec{r}_{0}|)=\omega^{d-2}H_{0}^{+}(d,\omega|\vec{r}_{0}-\vec{r}|)\\\
&\
=\frac{i\pi}{2}\sum_{\ell\vec{m}}\frac{(rr_{0})^{\nu-\frac{d}{2}+1}}{\Gamma(1+\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}\
\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}_{\ell\vec{m}}(\hat{r}_{0})^{\ast}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r_{0}^{2}}{4}\right]\\\
&\quad+\sum_{\ell\vec{m}}\frac{1}{2\nu}\frac{r_{<}^{\nu-\frac{d}{2}+1}}{r_{>}^{\nu+\frac{d}{2}-1}}\
\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}_{\ell\vec{m}}(\hat{r}_{0})^{\ast}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r_{<}^{2}}{4}\right]\\\
&\qquad\times\Biggl{\\{}{}_{0}F_{1}\left[1-\nu,-\frac{\omega^{2}r_{>}^{2}}{4}\right]-\frac{\pi\cot\nu\pi}{\Gamma(\nu)\Gamma(1+\nu)}\left(\frac{\omega
r_{>}}{2}\right)^{2\nu}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r_{>}^{2}}{4}\right]\Biggr{\\}}\
.\end{split}$ (96)

Here we have used a commonly used notation in such expansions, viz.,

$\begin{split}r_{>}\equiv\text{Max}(r,r_{0})\ ,\quad
r_{<}\equiv\text{Min}(r,r_{0})\ .\end{split}$ (97)

Further, we have also separated out the real and the imaginary parts of the
radial functions.

Consider double integrals of the form

$\begin{split}\int d^{d}r\ \rho_{1}(\vec{r},\omega)\int d^{d}r_{0}\
\rho_{2}(\vec{r}_{0},\omega)\ f(d,\omega|\vec{r}-\vec{r}_{0}|)\end{split}$
(98)

where $f$ could be any one of the functions discussed above. Using the
multipole expansion, such a double integral can be decomposed into an infinite
sum of factorised integrals, one each for every spherical harmonic. For the
practical computation of radiation reaction, it is then convenient to convert
the spherical harmonic sum into an STF expression using Eq.(84).

Let us illustrate the above remarks by computing the flat spacetime scalar
radiation reaction, which, in a slightly different notation, is explained in
detail in referencesBirnholtz:2013ffa ; Birnholtz:2013nta . Say we have an
extended scalar source whose emissive part at frequency $\omega$ is the
average source $\rho_{A}(\omega,\vec{r})$, and whose absorptive part is the
difference source $\rho_{D}(\omega,\vec{r})$. Ignoring all fluctuation
effects, the flat spacetime influence phase for this source, after integrating
out the massless scalar field about vacuum, can be written down as

$\begin{split}S_{RR}^{\text{bare}}=\int\frac{d\omega}{2\pi}\int d^{d}r_{0}\int
d^{d}r\ [\rho_{D}(\vec{r}_{0},\omega)]^{\ast}\ \rho_{A}(\vec{r},\omega)\
\omega^{d-2}H_{0}^{+}(d,\omega|\vec{r}_{0}-\vec{r}|)\ .\end{split}$ (99)

Here $\omega^{d-2}H_{0}^{+}(d,\omega|\vec{r}_{0}-\vec{r}|)$ is the outgoing
Green function for the scalar field, and the superscript ‘bare’ indicates that
this expression is divergent and has to be counter-termed before it makes
sense. We do not give here a derivation of the above influence phase except
the heuristic that the above action describes causal propagation of a free
scalar about the Minkowski vacuum. The above influence phase is also natural
if one applies the original Feynman-Vernon argument in FEYNMAN1963118  for
harmonic oscillators to each Minkowski mode of the scalar field and sums the
result. A more proper derivation should involve a careful discussion of the
fall-offs near space-like, time-like, and null asymptotia. We do not attempt
such a discussion here because, as we shall see later, dS-SK geometry
naturally incorporates such boundary conditions. Our dS answer in an
appropriate limit will reduce to the above result.

We substitute the multipole expansion Eq.(96) into the influence phase
$S^{\text{bare}}_{RR}$. For simplicity, we will take the number of spatial
dimensions (i.e., $d$) to be odd, so that $\nu\equiv\ell+\frac{d}{2}-1$ is a
half-integer (and $\cot\nu\pi=0$). The above action then has two sets of
terms: the first set of terms, odd under time reversal, are

$\begin{split}&\sum_{\ell\vec{m}}\int\frac{d\omega}{2\pi}\frac{i\pi}{2}\left(\frac{\omega}{2}\right)^{2\nu}\frac{1}{\Gamma(1+\nu)^{2}}\\\
&\times\int
d^{d}r_{0}\left\\{\rho_{D}(\vec{r}_{0},\omega)r_{0}^{\nu-\frac{d}{2}+1}\mathscr{Y}_{\ell\vec{m}}(\hat{r}_{0})\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r_{0}^{2}}{4}\right]\right\\}^{\ast}\\\
&\qquad\times\int d^{d}r\
\left\\{\rho_{A}(\vec{r},\omega)r^{\nu-\frac{d}{2}+1}\mathscr{Y}_{\ell\vec{m}}(\hat{r})\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\right\\}\
.\end{split}$ (100)

The combinations appearing in the second and the third line are the Bessel-
smeared radiative multipole moments191919The reader should compare this
definition against electrostatic multipole moments defined in Eq.(44),
remembering $\nu\equiv\ell+\frac{d}{2}-1$.  of the sources, i.e.,

$\begin{split}\mathcal{J}_{A}(\omega,\ell,\vec{m})\equiv\frac{1}{2\nu}\int
d^{d}r\ \rho_{A}(\vec{r},\omega)\
r^{\nu-\frac{d}{2}+1}\mathscr{Y}_{\ell\vec{m}}(\hat{r})\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\ ,\\\
\mathcal{J}_{D}(\omega,\ell,\vec{m})\equiv\frac{1}{2\nu}\int d^{d}r\
\rho_{D}(\vec{r},\omega)\
r^{\nu-\frac{d}{2}+1}\mathscr{Y}_{\ell\vec{m}}(\hat{r})\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\ .\end{split}$ (101)

We can then write the time reversal odd terms in the form

$\begin{split}&\sum_{\ell\vec{m}}\int\frac{d\omega}{2\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}\mathcal{J}_{D}^{\ast}(\omega,\mathbb{L})\mathcal{J}_{A}(\omega,\mathbb{L})\
.\end{split}$ (102)

Given that ${}_{0}F_{1}$ functions are completely regular when their first
argument is positive (i.e., when $1+\nu=\ell+\frac{d}{2}>0$, we conclude that
these multipole moments are finite, even for point-like sources. Hence, each
term in Eq.(100) is finite. The reader should contrast this with the second
set of terms, even under time reversal:

$\begin{split}&\sum_{\ell\vec{m}}\int\frac{d\omega}{2\pi}\int d^{d}r\int
d^{d}r_{0}\ [\rho_{D}(\vec{r},\omega)]^{\ast}\ \rho_{A}(\vec{r},\omega)\\\
&\qquad\times\frac{1}{2\nu}\frac{r_{<}^{\nu-\frac{d}{2}+1}}{r_{>}^{\nu+\frac{d}{2}-1}}\
\mathscr{Y}_{\ell\vec{m}}(\hat{r})\mathscr{Y}_{\ell\vec{m}}(\hat{r}_{0})^{\ast}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r_{<}^{2}}{4}\right]{}_{0}F_{1}\left[1-\nu,-\frac{\omega^{2}r_{>}^{2}}{4}\right]\
.\end{split}$ (103)

which are divergent due to the Green functions ${}_{0}F_{1}(1-\nu,\ldots)$.
Fortunately, since these are all even under time reversal, one can counter-
term away these terms. In other words, these terms in the influence phase
serve to renormalise the non-dissipative terms already present in the action
of the source.

Let us return to the terms in Eq.(100): they are odd in $\omega$, and hence
_cannot_ be countertermed or absorbed into the non-dissipative action. We can
simplify these remaining terms by substituting the STF definition of spherical
harmonics (see Eq.(81)) and invoking the STF addition theorem in Eq.(84). We
then get the radiation-reaction influence phase as

$\begin{split}S_{RR}^{\text{Odd
$d$}}&=\sum_{\ell}\int\frac{d\omega}{2\pi}\frac{i\pi}{2\mathcal{N}_{d,\ell}|\mathbb{S}^{d-1}|}\left(\frac{\omega}{2}\right)^{2\nu}\frac{1}{\Gamma(1+\nu)^{2}}\frac{1}{\ell!}\Pi^{<i_{1}i_{2}\ldots
i_{\ell}>}_{<j_{1}j_{2}\ldots j_{\ell}>}\\\ &\times\int
d^{d}r_{0}\left\\{\rho_{D}(\vec{r}_{0},\omega)x_{0}^{j_{1}}x_{0}^{j_{2}}\ldots
x_{0}^{j_{\ell}}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r_{0}^{2}}{4}\right]\right\\}^{\ast}\\\
&\qquad\times\int d^{d}r\
\left\\{\rho_{A}(\vec{r},\omega)x_{i_{1}}x_{i_{2}}\ldots x_{i_{\ell}}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\right\\}\
.\end{split}$ (104)

Here we recognise the STF multipole moments of the sources’ absorptive and
emissive parts. We will find it convenient to define our STF multipole moments
as

$\begin{split}\mathcal{Q}_{A,STF}^{i_{1}\ldots
i_{\ell}}(\omega)\equiv\frac{1}{2\nu}\Pi^{<i_{1}i_{2}\ldots
i_{\ell}>}_{<j_{1}j_{2}\ldots j_{\ell}>}\int d^{d}r\
\rho_{A}(\vec{r},\omega)x^{j_{1}}x^{j_{2}}\ldots x^{j_{\ell}}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\ ,\\\
\mathcal{Q}_{D,STF}^{i_{1}\ldots
i_{\ell}}(\omega)\equiv\frac{1}{2\nu}\Pi^{<i_{1}i_{2}\ldots
i_{\ell}>}_{<j_{1}j_{2}\ldots j_{\ell}>}\int d^{d}r\
\rho_{D}(\vec{r},\omega)x^{j_{1}}x^{j_{2}}\ldots x^{j_{\ell}}\
{}_{0}F_{1}\left[1+\nu,-\frac{\omega^{2}r^{2}}{4}\right]\ .\end{split}$ (105)

In terms of these STF multipole moments, the action for radiation reaction
takes the form

$\begin{split}S_{RR}^{\text{Odd
$d$}}&=\sum_{\ell\vec{m}}\int\frac{d\omega}{2\pi}\frac{2\pi
i}{\Gamma(\nu)^{2}}\left(\frac{\omega}{2}\right)^{2\nu}\mathcal{J}_{D}^{\ast}(\omega,\mathbb{L})\mathcal{J}_{A}(\omega,\mathbb{L})\\\
&=\sum_{\ell}\int\frac{d\omega}{2\pi}\frac{2\pi