# Characterization of the Global Bias Problem in Aerial Federated Learning

Ruslan Zhagypar  Nour Kouzayha  Hesham ElSawy  Hayssam Dahrouj  and Tareq Y.
Al-Naffouri  R. Zhagypar, N. Kouzayha, and T. Y. Al-Naffouri are with King
Abdullah University of Science and Technology, Thuwal, Saudi Arabia (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>tareq.alnaffouri@kaust.edu.sa).H. ElSawy is with the School of Computing,
Queen’s University, ON, Canada (e-mail: hesham.elsawy@queensu.ca).H. Dahrouj
is with the Department of Electrical Engineering, University of Sharjah,
United Arab Emirates (e-mail: hayssam.dahrouj@gmail.com).

###### Abstract

Unmanned aerial vehicles (UAVs) mobility enables flexible and customized
federated learning (FL) at the network edge. However, the underlying
uncertainties in the aerial-terrestrial wireless channel may lead to a biased
FL model. In particular, the distribution of the global model and the
aggregation of the local updates within the FL learning rounds at the UAVs are
governed by the reliability of the wireless channel. This creates an
undesirable bias towards the training data of ground devices with better
channel conditions, and vice versa. This paper characterizes the global bias
problem of aerial FL in large-scale UAV networks. To this end, the paper
proposes a channel-aware distribution and aggregation scheme to enforce equal
contribution from all devices in the FL training as a means to resolve the
global bias problem. We demonstrate the convergence of the proposed method by
experimenting with the MNIST dataset and show its superiority compared to
existing methods. The obtained results enable system parameter tuning to
relieve the impact of the aerial channel deficiency on the FL convergence
rate.

###### Index Terms:

Federated learning, Unmanned aerial vehicle (UAV), stochastic geometry,
unreliable wireless channel, PCP, MCP, scheduling.

##  I Introduction

The development of the sixth generation (6G) of wireless networks enables the
joint use of unmanned aerial vehicles (UAVs) and federated learning (FL), so
as to harness the prospects of learning on the edge via aerial platforms. The
potential of UAVs coupled with the FL algorithm has been recognized in [1][2].
The FL algorithm facilitates the progress of the Internet of drones and
drones-as-a-service [3]. Moreover, the FL-based edge computing in a UAV
network enhances client quality-of-service and maintains privacy [4]. However,
the high communication cost between an FL aggregator and learners remains the
main obstacle to practical implementation. Thus, the work in [5] reduces this
cost by clients clustering and transfer learning at the price of smaller
accuracy. Authors of [6] advocate the role of UAVs in FL over terrestrial base
stations (BS) due to UAVs mobility, which allows for customized learning from
a specific group of devices. Not only do UAVs offer more extensive coverage,
but their on demand deployment also makes them a cost-effective solution [7].

The basic principle of the FL algorithm is to send the local parameters to an
aggregator and receive back the updated global model parameters. Since these
transmissions occur over wireless channels, the FL operation becomes
eventually governed by the conditions of the channel. Due to the unreliable
and stochastic nature of the wireless channel, some local updates may be lost
after transmission at the aggregator. Furthermore, the global model
broadcasting might not reach the scheduled devices on account of UAVs’ limited
energy resources to compensate for channel losses. These events create an
undesirable bias towards devices with more stable wireless connectivity, which
drifts the global FL model away from optimality. As a result, the model would
suffer from inefficient learning in terms of lower accuracy and slower
convergence. To this end, this paper considers a single-tier network
consisting of UAVs that act as aggregators and ground devices that execute
local FL learning. The paper then characterizes the global bias problem by
deriving the UAV’s download and clients’ upload success probabilities, and by
integrating such metrics in the aggregation step of the FL process.

In the existing literature, several studies attempt to account for the
wireless channel’s impact on the convergence of FL algorithms in terrestrial
networks [8][9]. To this end, stochastic geometry is utilized as the main
mathematical framework to provide analytical formulations of key performance
metrics that assist the FL algorithms. For instance, in [10], the authors
calculate the client update success probability and highlight its impact along
with resource constraints on the learning performance. Theoretical analysis
made in [11] and [12] show that the spatial convergence of FL depends on the
number of devices with high uplink (UL) success probability. Authors of [13]
apply the framework to explore the effect of dynamic signal-to-interference-
plus-noise ratio (SINR) threshold on learning.

In view of prior work, most studies assume a guaranteed downlink (DL)
transmission because of sufficient power resources of the terrestrial BSs.
However, the perfect global model reception is no longer true in the setting
of aerial wireless networks due to UAVs’ limited energy capacity. Thus,
despite the high potential of UAVs in flexible edge intelligence, no study
specifically solves the global bias problem of FL in the setting of UAV-
assisted wireless networks. To the best of the authors’ knowledge, this is the
first work that bridges this gap by optimizing a reliable FL algorithm for
aerial networks.

In this work, we propose a UAV-assisted FL algorithm in which UAVs provide an
intermediate model aggregation from the sky and communicate with ground
devices through unreliable wireless channels. Inspired by the results in [10],
we jointly characterize the UAV’s download and clients’ upload success
probabilities and integrate them in the aggregation step of the FL to counter
the mentioned bias problem. The main contributions of this work can,
therefore, be summarized as follows: 1) Developing an analytical framework for
modeling and analysis of aerial FL in large-scale UAV networks. The analytical
model is used to derive a tractable expression for the joint DL and UL success
probability as a function of the different system parameters. 2) Unbiasing the
FL algorithm by integrating the joint success probability in the aggregation
step. By experimenting on the MNIST dataset, we validate the convergence of
the proposed FL and show its superiority over the existing FL algorithms. 3)
Based on the proposed framework, we investigate the impact of different system
parameters including the height of UAVs and the effects of different
environments. The devised insights can be used to tune the system parameters
to achieve optimal learning performance.

Figure 1: Illustration of the system model and an example of MCP with $N=6$ in
a cluster of radius $R=100$ m.

##  II System Model

We consider a single-tier network consisting of UAVs that act as aggregators
and ground devices that execute local FL learning, as illustrated in Fig. 1.
As devices performing the FL with the same UAV are physically close to each
other, we utilize a Matern Cluster Process (MCP) to model the network. This
model also represents the case where a UAV flies up to a cluster of devices
and hovers over it to conduct an FL training on their datasets. The UAVs are
considered the parent points and their locations follow a homogeneous Poisson
Point Process (PPP) $\Phi$ with intensity $\lambda$. Around each UAV located
at $u\in\Phi$, a set $D_{u}$ of $N$ devices are independently and identically
distributed (iid) forming a cluster of radius $R$. An example of the MCP model
with a cluster radius of $R=100$ m and $N=6$ is illustrated in the foreground
of Fig. 1. Each UAV orchestrates a separate FL process by aggregating the
local updates from the devices within its cluster.

As the aerial links are affected with the blockages in the environment and
since the devices are considered at the ground level, we use the approximation
of the line-of-sight (LOS) probability proposed in [14]. Specifically, the
probability of having a LOS link between the UAV and a device $k$ located at a
horizontal distance $r_{k}$ is given by:

$P_{L}(r_{k})=\frac{1}{1+a\exp{(-b[\frac{180}{\pi}\arctan{(h/r_{k})}-a])}},$
(1)

where $a$ and $b$ are constants that depend on the environment and are given
in [15, Table I], and $h$ is the elevation height of the UAV. The non line-of-
sight (NLOS) probability is thus given as $P_{N}(r_{k})=1-P_{L}(r_{k})$. As a
result, the set of UAVs is divided into two subsets with intensities
$P_{L}(r_{k})\lambda$ and $P_{N}(r_{k})\lambda$. We assume different path loss
exponents and fading parameters for LOS and NLOS transmissions.

We assume that both the UAVs and devices can perform directional beamforming
to compensate for propagation path loss. Hence a sectored antenna model is
assumed at the UAVs and ground devices, where $M_{s}$ and $m_{s}$ are the main
lobe gain and side lobe gain, respectively, and $s\in\\{u,d\\}$ is the index
that denotes the UAV and the device, respectively. We assume that the UAV and
devices can steer their antennas to maximize the directionality gain. Thus,
the gain on the desired link achieves the maximum value of $G_{0}=M_{u}M_{d}$.
On the other side, the beam directions of the interfering links are uniformly
distributed over $[0,2\pi)$. Thus, we can formulate the directionality gain of
an interfering link as shown in Table I where the probabilities are functions
of the main lobe beamwidth $\theta_{s}$ for $s\in\\{u,d\\}$.

TABLE I: Four antenna patterns

$i$ | 1 | 2 | 3 | 4
---|---|---|---|---
$G_{i}$ | $M_{u}M_{d}$ | $M_{u}m_{d}$ | $m_{u}M_{d}$ | $m_{u}m_{d}$
$p_{i}$ | $(\frac{\theta_{u}}{2\pi})(\frac{\theta_{d}}{2\pi})$ | $(\frac{\theta_{u}}{2\pi})(1-\frac{\theta_{d}}{2\pi})$ | $(1-\frac{\theta_{u}}{2\pi})(\frac{\theta_{d}}{2\pi})$ | $(1-\frac{\theta_{u}}{2\pi})(1-\frac{\theta_{d}}{2\pi})$

We account for the impact of the distance-dependent path-loss and small-scale
fading modeled by the Nakagami-$m$ distribution. Universal frequency reuse is
used across different clusters and $M$ resource blocks (RBs) are available at
each UAV. Due to the scarcity of resources ($M\leq N$), we apply random
resource scheduling without replacement for devices in each cluster. The
scheduling also eliminates intra-cluster interference. We utilize the SINR as
a metric to characterize the UL channel used for transmitting local parameters
from devices to UAVs. Thus, the UL SINR is given by:

$\text{SINR}^{\text{UL}}_{z}=\frac{P_{d}G_{0}|h_{k}|_{z}^{2}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{I_{\text{UL}}+n_{0}^{2}},\vspace{-0.1cm}$
(2)

where $z\in\\{{L,N}\\}$ denotes the LOS and NLOS links, $\alpha$ is the path-
loss coefficient, $P_{d}$ is the transmission power of a device, $G_{0}$ is
the directionality gain of the desired link, $|h_{k}|^{2}_{z}$ is the power of
the normalized small-scale Nakagami-$m$ fading, $r_{k}$ is the distance
between projections of a selected UAV and transmitting device $k$, $n_{0}^{2}$
is the noise power, and $I_{\text{UL}}$ represents the interference from
inter-cluster devices.

The DL channel used for transmitting the global parameters from UAVs to
devices is also prone to errors due to interference, propagation path-loss,
and noise. The SINR is also utilized to characterize the DL channel, which is
given by

$\text{SINR}^{\text{DL}}_{z}=\frac{P_{u}G_{0}|h_{k}|_{z}^{2}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{I_{\text{DL}}+n_{0}^{2}},\vspace{-0.1cm}$
(3)

where $P_{u}$ is the transmission power of a UAV, and $I_{\text{DL}}$ is the
interference from other UAVs.

Fig. 1 demonstrates the projections of necessary distances for the
characterization of inter-cluster interference. The UAV of the typical
cluster, $u_{0}\in\Phi$, is positioned right above the origin at an altitude
$h$. Thus, $g=||d||,d\in D_{u\backslash u_{0}}$ represents the Euclidean
distance to a device in an interfering cluster. Further, let $q=||u||$, where
$u\in\Phi\backslash u_{0}$ denotes the distance to an interfering UAV. The
conditional distance $g|q$ distribution for MCP is given as [16]:

$\displaystyle\ \ \ \ f_{Q}^{\text{MAT}}(g|q)=\frac{2g}{\pi
R^{2}}\arccos{\left(\frac{g^{2}+q^{2}-R^{2}}{2gq}\right)}\textbf{U}(g-|R-q|)$
(4) $\displaystyle\times\textbf{U}(R+q-g)+\frac{2g}{\pi
R^{2}}\textbf{U}(R-q-g),$

with $\textbf{U}(\cdot)$ is the unit step function.

##  III FL Algorithm

In the FL scenario, geographically dispersed devices interact with a central
server, located at the UAV, to train a common learning model. For learning a
statistical model from the distributed data, the central server aims to solve
the following optimization problem:

$\min_{w}\ F(w)=\sum^{N}_{k=1}p_{k}F_{k}(w),$ (5)

where $w$ is the learning model parameter, $p_{k}=n_{k}/n$ represents the
weight of device $k$ with $n_{k}$ samples in its local dataset
$\mathscr{D}_{k}$, and $n$ is the total number of data samples. $F_{k}(w)$
denotes the local loss function at device $k$, given by:

$F_{k}(w)=\frac{1}{n_{k}}\sum_{x\in\mathscr{D}_{k}}f(w,x),$ (6)

where $f(w,x)$ is the point local loss function for data sample $x$. It should
be noted that the dataset $\mathscr{D}_{k}$ is non-iid across different
devices. Since the aggregator cannot directly solve (5), an iterative approach
should be applied. Hence, we propose the algorithm shown in Algorithm 1, which
solves the FL problem in (5) while considering the unreliable DL global model
distribution and UL local models aggregation in aerial setting. The
aggregation step is the key feature of Algorithm 1 when compared to existing
methods. It is worth noting that the work in [10] assumes perfect DL reception
and erroneous UL channel for terrestrial settings. In conventional FedAvg
algorithms [17], the global aggregation rule at the server does not consider
any channel unreliability and is performed as

$\vspace{-0.1cm}w_{t+1}=\sum^{N}_{k=1}p_{k}w^{k}_{t+1},$ (7)

where $w^{k}_{t}$ is the local model parameter of device $k$ at time $t$.
Since in the proposed method we allow for $E$ number of local stochastic
gradient descent (SGD) iterations, the value of $w^{k}_{t}$ takes the
following form:

$w^{k}_{t}=\begin{cases}w_{0}&\text{if }t=0\\\ w_{t}&\text{if
}t\in\\{E,2E,3E,...\\}\\\ v^{k}_{t}&\text{otherwise},\end{cases}$ (8)

where the second case denotes the local initialization stages, and $v^{k}_{t}$
is the locally updated model parameter. Note that the local learning in
Algorithm 1 is performed in batches of $\xi_{t}^{k}$ on device $k$ at time
$t$.

Algorithm 1 FL for UAV-assisted wireless networks

Server Executes:

initialize $w_{0}$

for each round $t=0,1,2,...$ do

$S_{t}$ $\leftarrow$ (random set of $M$ clients out of $N$)

Broadcast $w_{t}$ to the clients

for each client $k\in S_{t}$ in parallel do

$v^{k}_{t+1}\leftarrow$ LocalUpdate($k$, $w_{t}$)

Calculate the joint success probability $J_{k}$ using (10)

end for

$w_{t+1}=w_{t}+\sum_{k=1}^{N}\sum_{b=1}^{M}\frac{p_{k}}{q_{k}J_{k}}\mathbbm{1}(\text{SINR}_{k,b}^{\text{DL}}>\tau_{\text{DL}},\
\text{SINR}_{k,b}^{\text{UL}}>\tau_{\text{UL}})(v^{k}_{t+1}-w_{t})$
//aggregation step

end for

LocalUpdate($k$, $w_{t}$):

$\xi_{t}^{k}\leftarrow$ (split $\mathscr{D}_{k}$ into batches)

for each local epoch $i$ from 0 to $E-1$ do

$w^{k}_{t+i}=w^{k}_{t}-\eta_{t+i}\nabla F_{k}(w^{k}_{t+i},\xi_{t}^{k})$

end for

Transmit $w^{k}_{t+E}$ to the server

In this paper, we apply random scheduling in the following manner: the UAV
selects uniformly $M$ out of $N$ devices without replacement, thereby
guaranteeing that each device uses at most one RB. Hence, on average, the UAV
allocates a RB to device $k$ with probability $q_{k}$. Thus, $q_{k}$ is
defined as

$q_{k}=\mathbb{E}\left[\sum^{M}_{b=1}\mathbbm{1}(k\in
S_{t}(b))\right]=\frac{M}{N},$ (9)

where $S_{t}(b)$ represents a set of devices scheduled at time $t$ for a RB
$b$. The indicator function in Algorithm 1 captures the unreliability of the
wireless channel during UL/DL transmission. To counteract the impact of the
wireless channel, the weights of local updates are divided by the joint
success probability $J_{k}$ derived in Theorem 1 below. The proof of
convergence of the proposed algorithm is similar to the one shown in [10], and
is omitted in this paper due to space limitations.

######  Theorem 1.

In a UAV-assisted wireless network modeled as an MCP, the joint success
probability for device $k$ defined as the probability that SINR for both DL
and UL exceed predefined thresholds $\tau_{\text{DL}}$ and $\tau_{\text{UL}}$,
respectively, is given as:

$\displaystyle J_{k}$
$\displaystyle=\mathbb{E}_{z\in\\{{L,N}\\}}[\mathbb{P}[\text{SINR}_{z}^{\text{DL}}>\tau_{\text{DL}},\text{SINR}_{z}^{\text{UL}}>\tau_{\text{UL}}|z,k\in
S_{t}]]$ (10) $\displaystyle=P_{L}(r_{k}|k\in S_{t})\times
J_{k}^{L}+P_{N}(r_{k}|k\in S_{t})\times J_{k}^{N},$

where

$\displaystyle J_{k}^{z}$
$\displaystyle=\sum_{j=1}^{m_{z}}\binom{m_{z}}{j}(-1)^{j+1}$ (11)
$\displaystyle\exp\left({\frac{-j\eta_{z}n_{0}^{2}\tau_{\text{DL}}}{P_{u}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)\mathcal{L}_{\text{DL}}\left({\frac{-j\eta_{z}\tau_{\text{DL}}}{P_{u}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)$
$\displaystyle\times\sum_{j=1}^{m_{z}}\binom{m_{z}}{j}(-1)^{j+1}$
$\displaystyle\exp\left({\frac{-j\eta_{z}n_{0}^{2}\tau_{\text{UL}}}{P_{d}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)\mathcal{L}_{\text{UL}}\left({\frac{-j\eta_{z}\tau_{\text{UL}}}{P_{d}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right),$

where $z\in\\{{L,N}\\}$, $k\in S_{t}$ indicates that device $k$ is scheduled
for a RB at time $t$, $m_{z}$ is the Nakagami fading parameter for $z$ link
with $\eta_{z}=m_{z}(m_{z}!)^{-1/m_{z}}$, $\tau_{\text{DL}}$ and
$\tau_{\text{UL}}$ are SINR thresholds for DL and UL transmissions
respectively, and $\mathcal{L}_{\text{DL}}(\cdot)$ and
$\mathcal{L}_{\text{UL}}(\cdot)$ are the Laplace transforms of DL and UL
inteferences, respectively, and are provided in the next two lemmas.

###### Proof:

See Appendix A. ∎

The Laplace transforms of DL and inter-cluster UL interferences are formulated
in the following lemmas:

######  Lemma 1.

In a UAV-assisted wireless network modeled as an MCP, the Laplace transform of
the DL interference is given by:

$\displaystyle\mathcal{L}_{\text{DL}}(s)=\prod_{z\in\\{{L,N}\\}}\exp\left(-2\pi\lambda\int_{0}^{\infty}\left[1-\sum^{4}_{i=1}p_{i}\right.\right.$
$\displaystyle\left.\left.\left(1+\frac{sP_{u}G_{i}(g^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{m_{z}}\right)^{-m_{z}}\right]qP_{z}(q)\,dq\right),$

###### Proof:

See Appendix B. ∎

######  Lemma 2.

The Laplace transform of the inter-cluster UL interference in a UAV-assisted
network modeled as MCP is given by:

$\displaystyle\mathcal{L}_{\text{UL}}(s)=\prod_{z\in\\{{L,N}\\}}\exp$
$\displaystyle\left(-2\pi\lambda\left[\int_{0}^{R}\left[1-[O^{z}_{e1}(s,q)]^{\bar{N}}\right]q\,dq\right.\right.$
$\displaystyle\left.\left.+\int_{R}^{\infty}\left[1-[O_{e2}^{z}(s,q)]^{\bar{N}}\right]q\,dq\vphantom{\int_{0}^{R}\left[1-[O_{e1}(s,q)]^{\bar{N}}\right]q\,dq}\right]\right),$

where $\bar{N}$ represents the number of interfering devices in a neighboring
cluster, which is $\bar{N}=1$ due to the considered scheduling scheme and

$\displaystyle
O_{e1}^{z}(s,q)=\int_{|q-R|}^{R+q}\sum^{4}_{i=1}p_{i}\left(1+\frac{sP_{d}G_{i}(g^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{m_{z}}\right)^{-m_{z}}$
$\displaystyle\times\frac{2g}{\pi
R^{2}}\arccos{\left(\frac{g^{2}+q^{2}-R^{2}}{2gq}\right)}\,dq$
$\displaystyle+\int_{0}^{R-q}\sum^{4}_{i=1}p_{i}\left(1+\frac{sP_{d}G_{i}(g^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{m_{z}}\right)^{-m_{z}}\times\frac{2g}{R^{2}}\,dq,$

and

$\displaystyle
O_{e2}^{z}(s,q)=\int_{|q-R|}^{R+q}\sum^{4}_{i=1}p_{i}\left(1+\frac{sP_{d}G_{i}(g^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{m_{z}}\right)^{-m_{z}}$
$\displaystyle\times\frac{2g}{\pi
R^{2}}\arccos{\left(\frac{g^{2}+q^{2}-R^{2}}{2gq}\right)}\,dq.$

###### Proof:

See Appendix C. ∎

##  IV Numerical Results

TABLE II: System parameter values

$N,M=$ 100, 90 |  $R,h=$ 100, 120 m | $\lambda=2/(\pi 150^{2})~{}\text{UAV}/\text{m}^{2}$
---|---|---
$P_{d},P_{u}=$ 0.1, 0.25 W |  $\alpha_{L},\alpha_{N}$ = 2.1, 3.6 |  $n_{0}^{2}=4.14\times 10^{-6}$ W
$a,b=$ 9.61, 0.16 |  $m_{L},m_{N}$ = 3, 1 |  $\tau_{\text{DL}},\tau_{\text{UL}}=$ 15, 0 dB
$M_{u}=$ 10 dB |  $M_{d}=$ 5 dB |  $m_{u},m_{d}=$ -1, -3 dB

$20$$40$$60$$80$$100$$120$$140$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$UAV
height, (m) Coverage probability Joint (simulation)Joint (analytic)Uplink
(simulation)Uplink (analytic)

Figure 2: Coverage probability plot for parameter values in Table II.

In this section, we provide analytical results and Monte-Carlo simulations to
validate the accuracy of the joint success probability given in Theorem 1.
Furthermore, we present simulation results for the proposed FL algorithm in
UAV-assisted networks. Unless otherwise mentioned, the simulation parameters
are presented in Table II. We compare our FL aggregator (i.e., that accounts
for the joint DL/UL unreliability) with the traditional FedAvg algorithm and
with [10] that considers the UL unreliability only. Fig. 2 plots the coverage
probabilities for both joint DL and UL, and UL-only instances as a function of
the UAV height. It can be seen clearly that the results of the analytic
expression match the Monte-Carlo simulations, which proves the validity of the
conducted analysis. The figure also shows the prominent impact of the DL on
the transmission success probability when compared with the UL-only case.
Interestingly, the figure shows an optimal height for the UAV, which strikes a
tradeoff between improving LOS probability and aggravating propagation loss as
height increases.

$0$$50$$100$$150$$200$$250$$0.6$$0.8$$1$$1.2$$1.4$$1.6$$1.8$$2$$2.2$Communication
rounds Global loss JointUL-onlyFedAvg

Figure 4a: Global loss across communication rounds.

$0$$50$$100$$150$$200$$250$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$Communication
rounds Training accuracy JointUL-onlyFedAvg

Figure 4b: Training accuracy across communication rounds.

12345710$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$E$ Testing accuracy
JointUL-only

Figure 5: Testing accuracy for different local SGD iteration values.

$0$$50$$100$$150$$200$$250$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$Communication
rounds Training accuracy h = 25 mh = 50 mh = 120 m

Figure 6: Training accuracy for different UAV heights.

$0$$50$$100$$150$$200$$250$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$Communication
rounds Training accuracy SuburbanUrbanDense urbanHigh-rise urban

Figure 7a: Training accuracy for different environments at $h=25$ m.

$0$$50$$100$$150$$200$$250$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$Communication
rounds Training accuracy SuburbanUrbanDense urbanHigh-rise urban

Figure 7b: Training accuracy for different environments at $h=120$ m.

To evaluate the performance of the proposed FL algorithm, we rely on the
global loss and the training and testing accuracies of the global model. We
perform FL with the number of local SGD iterations of $E=2$ and batch size of
$\xi_{t}^{k}=64$ until the convergence of the global model. Data from the
MNIST dataset is distributed in a non-iid manner across all devices within a
cluster. The result in Fig. 5 shows the global loss versus the number of
communication rounds between a UAV and the associated devices. It can be
noticed that the proposed method drastically reduces the loss compared to the
method based on only erroneous UL channel and the conventional FedAvg. The
superiority of the proposed method is also reflected in accuracy plots in Fig.
5. The proposed method reaches the convergence much faster and provides around
25% improvement as compared to the UL-only scenario. The results evidence the
importance of accounting for the impact of wireless channels and considering
both DL and UL on FL.

The effect of changing the number of local iterations is studied and the
results are given in the bar plot in Fig. 5. One can observe that there is an
optimal value for local iterations which is $E=2$. At this value, both methods
using joint and UL-only success probabilities demonstrate the highest testing
accuracies. It is noted that the proposed joint method always performs better
than the method relying on only UL success probability.

The developed model allows to investigate the impact of different system
parameters on the learning performance. Fig. 7b illustrates that the highest
accuracy corresponds to the optimal UAV height of $h=50$ m, which is supported
by the observations from Fig. 2. The reason for such trend is that the higher
coverage probability value, the more devices participate in each round of FL.
In Fig. 7b and Fig. 7b, we plot the training accuracy values in four
environments for the UAV heights of $h=25$ m and $h=120$ m, respectively. We
can notice that the impact of the UAV height on the accuracy of the FL
algorithm is not the same for the considered environments. For instance, the
FL training accuracy in the suburban environment is the highest for $h=25$ m
and lowest for $h=120$ m. The impact of the height in the suburban environment
is always negative since the LOS probability is almost 1 even at low heights.
Thus, the increase of the height only affects the path-loss that leads to the
drop in FL accuracy. We observe an opposite trend for the high-rise urban
setting due to a low value of LOS probability at low and a high value at high
altitudes. For the high rise urban case, the LOS probability is low at low UAV
altitudes, which leads to low FL accuracy. When the UAV flies at higher
altitudes, the LOS probability starts to increase. This increase compensates
the deterioration caused by increasing path-loss. Thus, the FL accuracy
improves for higher UAV heights. Such behavior happens up to some extend after
which the path-loss factor dominates again when the LOS probability tends to
1, which results in an accuracy drop. These results signify the need for
optimizing the UAV position in addition to other design parameters to improve
the FL accuracy in aerial networks, subjects that are expected to be at the
forefront of future networks design and analysis.

##  V Conclusion

In this paper, we examine the performance of the FL algorithm in aerial UAV-
assisted networks. The unreliable and resource-constrained wireless channel
between aggregating UAVs and devices makes the learning inefficient due to
biasing the global model towards devices with better channel conditions. Using
tools from stochastic geometry, the joint upload and download success
probability for FL in UAV-assisted networks is computed and used to unbias the
FL aggregation rule. The proposed FL algorithm not only surpasses the existing
methods but also enables the adjustment of system parameters for optimum
learning performance.

## Appendix A

We derive the joint success probability based on the fact that probabilities
of SINR for DL and UL exceeding the thresholds become independent when we
condition on the serving distance. The serving distance remains the same for
both DL and UL transmissions since we consider a single-tier network where
devices always associate with UAVs in the centers of clusters. Thus, after
considering LOS and NLOS possibilities for the serving links, the overall
joint success probability is derived as

$\displaystyle J_{k}$
$\displaystyle=\mathbb{E}_{z\in\\{{L,N}\\}}[\mathbb{P}[\text{SINR}_{z}^{\text{DL}}>\tau_{\text{DL}},\text{SINR}_{z}^{\text{UL}}>\tau_{\text{UL}}|k\in
S_{t}]]$ (12) $\displaystyle=P_{L}(r_{k}|k\in S_{t})\times
J_{k}^{L}+P_{N}(r_{k}|k\in S_{t})\times J_{k}^{N},$

where

$\displaystyle J_{k}^{z}$
$\displaystyle=\mathbb{P}[\text{SINR}_{z}^{\text{DL}}>\tau_{\text{DL}}|k\in
S_{t}]\times\mathbb{P}[\text{SINR}_{z}^{\text{UL}}>\tau_{\text{UL}}|k\in
S_{t}]$ (13)
$\displaystyle=\mathbb{E}_{I_{\text{DL}}}\left[\mathbb{P}\left[|h_{k}|_{z}^{2}>{\frac{\tau_{\text{DL}}(n_{0}^{2}+I_{\text{DL}})}{P_{u}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right]\right]$
$\displaystyle\ \ \ \ \ \ \ \ \ \
\times\mathbb{E}_{I_{\text{UL}}}\left[\mathbb{P}\left[|h_{k}|_{z}^{2}>{\frac{\tau_{\text{UL}}(n_{0}^{2}+I_{\text{UL}})}{P_{d}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right]\right]$
$\displaystyle\stackrel{{\scriptstyle\text{(a)}}}{{=}}\left(1-\mathbb{E}_{I_{\text{DL}}}\left[\left(1-\exp\left({\frac{-\eta_{z}\tau_{\text{DL}}(n_{0}^{2}+I_{\text{DL}})}{P_{u}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)\right)^{m_{z}}\right]\right)$
$\displaystyle\times\left(1-\mathbb{E}_{I_{\text{UL}}}\left[\left(1-\exp\left({\frac{-\eta_{z}\tau_{\text{UL}}(n_{0}^{2}+I_{\text{UL}})}{P_{d}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)\right)^{m_{z}}\right]\right)$
$\displaystyle\stackrel{{\scriptstyle\text{(b)}}}{{=}}\sum_{j=1}^{m_{z}}\binom{m_{z}}{j}(-1)^{j+1}\exp\left({\frac{-j\eta_{z}n_{0}^{2}\tau_{\text{DL}}}{P_{u}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)$
$\displaystyle\mathbb{E}_{I_{\text{DL}}}\left[\exp{\left({\frac{-j\eta_{z}\tau_{\text{DL}}}{P_{u}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)}\right]$
$\displaystyle\times\sum_{j=1}^{m_{z}}\binom{m_{z}}{j}(-1)^{j+1}\exp\left({\frac{-j\eta_{z}n_{0}^{2}\tau_{\text{UL}}}{P_{d}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)$
$\displaystyle\mathbb{E}_{I_{\text{UL}}}\left[\exp{\left({\frac{-j\eta_{z}\tau_{\text{UL}}}{P_{d}G_{0}(r_{k}^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}}\right)}\right],$

where (a) is the result of using the cumulative distribution function (CDF) of
the normalized gamma fading coefficient, (b) is obtained by using the binomial
expansion. The final result is obtained by defining the expectations as the
Laplace transforms of DL and UL interference.

## Appendix B

The overall DL intereference consists of LOS and NLOS components:
$I_{\text{DL}}=I_{\text{DL}}^{L}+I_{\text{DL}}^{N}$. When evaluating the
exponential function for $I_{\text{DL}}$, the result becomes the product of
the two components. Thus, the Laplace transform of the DL interference can be
derived as

$\displaystyle\mathcal{L}_{\text{DL}}(s)$
$\displaystyle=\prod_{z\in\\{{L,N}\\}}\mathbb{E}\left[\exp{\left(-s\sum_{u\in\Phi^{z}\backslash
u_{0}}P_{u}G_{u}|h_{u}|_{z}^{2}(q^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}\right)}\right]$
$\displaystyle=\prod_{z\in\\{{L,N}\\}}\mathbb{E}_{\Phi^{z}}\left[\prod_{u\in\Phi^{z}\backslash
u_{0}}\mathbb{E}_{\begin{subarray}{c}G_{u},\\\
|h_{u}|_{z}^{2}\end{subarray}}\left[\exp\left(-sP_{u}G_{u}|h_{u}|_{z}^{2}\right.\right.\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.\left.\left.\times(q^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}\right)\right]\right]$
$\displaystyle\stackrel{{\scriptstyle\text{(a)}}}{{=}}\prod_{z\in\\{{L,N}\\}}\exp\left(-2\pi\lambda\int_{0}^{\infty}\left[1-\mathbb{E}_{\begin{subarray}{c}G_{u},\\\
|h_{u}|_{z}^{2}\end{subarray}}\left[\exp\left(-sP_{u}G_{u}|h_{u}|_{z}^{2}\right.\right.\right.\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.\left.\left.\left.\times(q^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}\right)\right]\right]qP_{z}(q)\,dq\right),$

where (a) is the result of applying the probability generating functional of
PPP for UAV locations. The final form is obtained from applying the moment
generating function of the gamma distribution characterizing the small-scale
fading gain and from averaging over the dimensionality gain using the
probability mass function (PMF) in Table I.

## Appendix C

Similar to DL interference, the UL intereference is comprised of two
compoenents: $I_{\text{UL}}=I_{\text{UL}}^{L}+I_{\text{UL}}^{N}$. Thus, the
Laplace transform of the UL interference can be derived as

$\displaystyle\mathcal{L}_{\text{UL}}(s)=\prod_{z\in\\{{L,N}\\}}\mathbb{E}\left[\exp\left(-s\sum_{u\in\Phi^{z}\backslash
u_{0}}\sum_{d\in D_{u}}P_{d}G_{d}|h_{d}|_{z}^{2}\right.\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \left.\left.\times(g^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle\text{(a)}}}{{=}}\prod_{z\in\\{{L,N}\\}}\mathbb{E}_{u}\left[\prod_{u\in\Phi^{z}\backslash
u_{0}}\mathbb{E}_{d}\left[\prod_{d\in
D_{u}}\sum^{4}_{i=1}p_{i}\left(1\right.\right.\right.$ $\displaystyle\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.\left.\left.+\frac{sP_{d}G_{i}(g^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{m_{z}}\right)^{-m_{z}}\middle|q\right]\right]$
$\displaystyle\stackrel{{\scriptstyle\text{(b)}}}{{=}}\prod_{z\in\\{{L,N}\\}}\exp{\left(-2\pi\lambda_{v}\int_{0}^{\infty}\left[1-[O_{e}^{z}(s,q)]^{\bar{N}}\right]qP_{z}(q)\,dq\right)}$

where

$\displaystyle O_{e}^{z}(s,q)$
$\displaystyle=\int_{0}^{\infty}\sum^{4}_{i=1}p_{i}\left(1+\frac{sP_{d}G_{i}(g^{2}+h^{2})^{-\frac{\alpha_{z}}{2}}}{m_{z}}\right)^{-m_{z}}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times
f_{Q}^{\text{MAT}}(g|q)\,dg\vphantom{\left(1+\frac{sP_{d}(g^{2}+h^{2})^{-\frac{\alpha}{2}}}{m}\right)^{-m}},$

where (a) is obtained by applying the moment generating function of the small-
scale fading gain. (b) is the result of the probability generating functional
of PPP for UAV locations. The final result follows from applying the piece-
wise expression of the conditional distance distribution
$f_{Q}^{\text{MAT}}(g|q)$ given in (4).

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# A Promising New Dark Matter Candidate, and Implications for the Search for
Extraterrestrial Intelligence (SETI)

A. Prillfool<EMAIL_ADDRESS>St. Cedd’s College, University of
Cambridge,  A. A. Stoffers  Kavli Institute of Cosmology, University of
Cambridge, Madingley Road, Cambridge CB3 0HA, UK  I. Juodžbalis  Kavli
Institute of Cosmology, University of Cambridge, Madingley Road, Cambridge CB3
0HA, UK  M. S. Bothwell  Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, UK  A. J. Wojcik  Electrical Engineering
Division, Department of Engineering, University of Cambridge, 9 JJ Thomson
Avenue, Cambridge CB3 0FA, UK

###### Abstract

We present a speculative exploration of the properties of a proposed new Dark
Matter (DM) candidate in a heretofore under-explored region of parameter
space. Our proposed ultra-cold candidatae has been a matter of speculation for
some time,and has recently been tentatively identified via direct-detection.
While unconventional, demonstrated existence of this DM candidate would have
wide-ranging implications for a range of fields, from particle cosmology to
exobiology and the Search for Extraterrestrial Life (SETI).

General – extraterrestrial intelligence: Cosmology – dark matter: General –
theology

##  1 INTRODUCTION: Dark Matter - A failed search

For nearly a century, the dark matter problem has been a fundamental issue in
astrophysics, bearing a striking resemblance to the historical challenge of
detecting unseen planets. Anomalies in large astrophysical systems can be
interpreted in two ways: as either challenges to the established laws of
gravitation and general relativity or as evidence of invisible, dark objects.
Initially, it took decades for the scientific community to recognize dark
matter as a genuine issue. Today, comprehensive data from the study of
microwave background radiation, supernova distance measurements, and large-
scale galaxy surveys have reinforced the standard model of cosmology. This
model suggests that the universe’s structure emerged from the gravitational
amplification of minor density fluctuations, facilitated by cold dark matter
(CDM). The presence of dark matter is crucial; without it, the density
contrasts observed in the universe today could not have evolved from the
modest amplitude of density fluctuations seen in the cosmic microwave
background.
Candidates for dark matter vary widely, spanning 90 orders of magnitude in
mass, from ultralight bosons to massive primordial black holes. Interest in
the latter has surged following the detection of gravitational waves from
black hole mergers by the LIGO and Virgo observatories Bird et al. (2016),
Clesse and García-Bellido (2017). For many years, Weakly Interacting Massive
Particles (WIMPs) were considered ideal candidates for dark matter. They fit
well within the weak-interaction mass scale (approximately 10 GeV to 1 TeV)
and could naturally emerge with the correct relic abundance in the early
universe Bertone et al. (2005). Moreover, WIMPs offered a potential solution
to the hierarchy problem De Gouvêa et al. (2014), a major focus of particle
physics for the last four decades. However, despite extensive searches, WIMPs
have not been detected.
Supersymmetry (SUSY) proposes a partner particle for every particle in the
standard model, with the partners of electroweak bosons being potential WIMP
candidates and thus suitable for dark matter. However, most of the parameter
space of natural simple SUSY models have been largely discounted, leading to
questions about the extent of fine-tuning we are willing to accept in pursuit
of discovering SUSY (Ross et al., 2017).

While primordial black holes have been considered as dark matter candidates,
stringent constraints from various observational probes, such as microlensing
and the cosmic microwave background, have largely ruled out their viability
over broad mass ranges, casting doubt on their prevalence in the universe
Sasaki et al. (2018). Similarly, while axions emerged as promising candidates
due to their potential to solve the strong CP problem, recent experimental
bounds and theoretical considerations have narrowed the allowed parameter
space, challenging their capacity to constitute the entirety of dark matter
Irastorza and Redondo (2018). As existing dark matter candidates face
increasing scrutiny, there is a growing impetus to explore alternative
theoretical frameworks and experimental avenues in the quest for novel dark
matter candidates that can better accommodate observational constraints. Next,
we will introduce an alternative dark matter candidate in Sec. 2 and explore
its potential implications in Sec. 3.

Figure 1: A range of proposed dark matter candidates, with their expected mass
scale. It can readily be seen that SOUL occupies a yet unexplored area of the
mass range.

##  2 An alternative DM candidate

Several authors have attempted to ascertain the mass of the human soul, most
famously MacDougall (1907), who compared pre- and post-mortem mass
measurements of the human body, reporting a value of $1.2\times 10^{28}$
MeV/$c^{2}$ (=21 grams; an estimate of the error on this value was not
reported by the original author). Though the specific methodology employed has
been called into question, it is undoubtedly true that throughout the 20th
Century a range of experimental and theoretical efforts have attempted to
improve this value, with the most recent estimates, utilizing self-consistent
treatment of neuron binding energies and brain masses placing the estimate at
a much higher value of $10^{30}$ GeV/$c^{2}$ (Kozłowski, 2018). A likely
reason for this wide range of estimated masses is the inherent difficulty of
performing direct measurements on this elusive substance as its coupling to
baryonic matter must be very weak, otherwise its existence would have been
well established long ago (given that every human being, presumably, possesses
a soul). This means that the soul has two of the fundamental properties of DM
– its might mass and its relative undetectability – and thus makes for a
promising candidate.

This is particularly true in light of the fact that the vast majority of
standard DM candidates are in tension with observations of Spontaneous Human
Combustion (SHC), as discussed by Hessman and Wheeler (2023) (notably, this
paper was published exactly one year ago today). Using SHC Hessman and Wheeler
(2023) find an estimated DM particle mass of $5.6\times 10^{27}$ GeV/$c^{2}$,
which falls comfortably within the mass range discussed above.

Of course, it is important to avoid anthropocentrism – it is exceedingly
unlikely that specifically human souls comprise a significant fraction of the
total DM content of the Universe. It is far more likely that other similarly-
ensouled beings represent the bulk of the missing matter. We therefore avoid
the anthropocentric (and indeed Christocentric) term ‘soul’. We instead
propose the existence of a Sentient/Organic Universal Life-force field (which
we will henceforth refer to as a ‘SOUL’), with the corresponding soulon
particle of $10^{25}<M_{S}<10^{30}$ GeV/$c^{2}$, which is compared to other
proposed DM candidates in Figure 1. As can be seen there, the SOUL occupies a
place in the parameter space not covered by other models, further cementing
its potential as a viable DM candidate.

##  3 IMPLICATIONS OF THE SOUL

Figure 2: Illustrative figure showcasing the proposed ratio of galaxy mass to
dark matter halo mass plotted against probable masses of the SOUL. The shaded
regions represent diverse interpretations of observational data and
theoretical predictions, offering insights into the potential relationship
between galaxies and dark matter. Each shading delineates distinct hypotheses
and models.

In order to examine the implications of our DM model we first perform a simple
order of magnitude estimation in order to quantify the number of SOULs in the
Milky Way galaxy.

Taking the DM halo mass of the Milky way to be of order $10^{69}$ GeV/$c^{2}$
(Watkins et al., 2019), we find that the number of SOULs in the galaxy can
range between $10^{39}\leq N_{S}\leq 10^{44}$, depending on the exact model
assumed. This greatly outnumbers the number of stars in the Milky Way, which
can be estimated to be on the order of $10^{11}$ (Licquia and Newman, 2015).
We estimate the density of SOULs throughout the Milky Way to be approximately
$10^{28}\leq N_{S}\leq 10^{33}$ per star system.

It is clear, therefore, the number of SOULs required to explain the DM halo of
our galaxy likely can not be supplied by intelligent civilizations similar to
ours. Assuming each civilization to be similar to our own in terms of the
number of conscious individuals – approximately $10^{11}$ humans have ever
lived – even the extreme situation of one intelligent civilization per star
system could explain only a tiny ($10^{-32}$ to $10^{-27}$) fraction of the
total Milky Way DM halo mass.

While some might consider this discrepancy a fatal strike against our theory,
we prefer to interpret this tension as potential evidence for new physics.
Throughout the rest of this paper we further consider the possible
interpretations and implications of our model, particularly where the
distribution and the true origin of SOULs is concerned. In the following
sections we examine several likely scenarios, colourfully illustrated in
Figure 2, many of which have been investigated since the dawn of human
history. Until now however, they could not be properly evaluated due to a
paucity of empirical evidence.

###  3.1 The Boltzmann interpretation

The concept of the Boltzmann brain is used to capture the vastness of possible
arrangements of matter allowed by the laws of thermodynamics. Given a
sufficiently long time, structures of any complexity are allowed to form,
including fully functional human brains. We further expand on this
philosophical tool, showing that such structures can occur much more
prominently. According to our model, human souls are formed of a large
quantity of bound soulons, which we call soulon clouds. It is possible
therefore, that such clouds may also form spontaneously in space due to
binding of large quantities of free soulons. We call such structures ANGELS
(Aggregated Nexus for Global Exchange of Life-force and Soulons). Since
soulons allow for transferring consciousness and participate in the thought
process, it is possible that the free ANGELS could act similarly to the
Boltzmann brains, experiencing dream-like thought and hallucinations. Such
ANGELS float through, and exchange particles with, the underlying field of
free soulons mentioned earlier, a possible origin of which is further explored
in the next section.

###  3.2 Cosmic Theological Background scenario

Another intriguing possibility is that the SOULs unaccounted for by the
presence of intelligent life may constitute a single separate intelligence.
This idea was first put forward by Giordano Bruno in the 16th century (Bruno,
1584a, b). These seminal works were among the first to introduce the idea of
extraterrestrial life and also brought to bear the idea that, what was defined
as ’God’ at the time, was uniformly distributed across all space rather than
confined to a particular location. The latter idea ended rather poorly for the
author as the mathematical and experimental machinery at the time was
inadequate to robustly establish it. However, it now presents an intriguing
framework within which our results can be interpreted.

To proceed, we shall consider the substance proposed by Bruno as a kind of
scalar field with soulons corresponding to its excitations. Such a field would
undoubtedly permeate the entirety of the observable Universe, constituting a
Cosmic Theological Background (CTB). The CTB could serve as the origin of the
free soulons discussed in the previous section and contribute the vast
majority of observed DM mass in the Universe. However, the precise treatment
of the dynamics and vacuum fluctuations of CTB is beyond the scope of this
work, particularly as there are further interpretations of the SOUL possible.

###  3.3 The panpsychist interpretation

An alternate, intriguing possibility is the so-called ‘panpsychist
interpretation’ – that consciousness, rather than being an emergent property
of matter arranged to a specific degree of complexity, instead represents a
truly fundamental component of the Universe. This idea dates back to
antiquaty, being first proposed by the pre-Socratic philosophers (see, e.g.,
Thales of Miletus et al., 600BC), and being carried forward in the modern era
by thinkers including Spinoza and Schopenhauer. Present-day philosophers
including Goff (2021) have argued that the “fundamental constituents of
reality – perhaps electrons and quarks – have incredibly simple forms of
experience”. Having forms of experience – that there is something which it is
like to ‘be’ a fundamental particle – would imply that these particles are
necessarily endowed with SOULs (though of course it seems likely that the SOUL
of a proton is less massive than the SOUL of a human.)

We return now to our previous estimate of the number of SOULs per star system,
$10^{28}\leq N_{S}\leq 10^{33}$. The number of atoms in our Solar System is of
the order $10^{57}$. Assuming the panpsychist interpretation to be correct,
this suggests that each atom is endowed with $10^{-29}\leq N_{S}\leq 10^{-24}$
of a SOUL. We note that, given that the number of atoms in a human body is
approximately $10^{28}$, this ‘atomic SOUL mass’ range is in good agreement
with the observation that humans are in possession of 1 SOUL each. This may be
counted as a verified prediction of the Panpsychist Dark Matter (PDM) model.

##  4 Conclusions and possibilities for further work

We have put forward a novel solution to the observational ‘missing matter’
problem. While this short preliminary work is only a first exploratory study,
it is clear that this avenue has the potential to revolutionise both the
cosmological understanding of dark matter and the Search for Extraterrestrial
Intelligence.

In addition to the above it should also be noted that the relatively high mass
of the SOUL implies that the birth and death of an ensouled being emits a
significant amount of gravitational waves. An exact treatment of the tensor
perturbation produced by such events results in a particularly elegant
description of the geometry of mortality, however, this description is too
large to fit in this paper and is therefore beyond the scope of this work.
Nevertheless, this implies that the future in the search for intelligent life
may lie with the next generation of gravitational wave instruments rather than
the more conventionally used optical/infrared telescopes.

## ACKNOWLEDGEMENTS

This work was supported by the cocktail menu at the Astronomer restaurant in
Cambridge, without which the productive discussions which led to this paper
would not have occurred.
Additionally, we extend our deepest gratitude to S. Gautama for generously
financing our research on the dark matter candidate SOUL. This support not
only made our work possible but also illuminated our path towards uncovering
the mysteries of the universe

## CONFLICT OF INTEREST

The authors would like to declare no conflicts of interest. However, in case
of the CTB or the PDM scenarios holding true, the possibility of the authors
being influenced by these underlying fields cannot be discounted.

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