deterministic events. Still, the rest of the paper views broadcasts and other
objects as unpredictable. This does not mean that the intelligences behind
them have no specific motivations. Random processes can be used to describe
human behavior by focusing on statistical trends rather than the unpredictable
specific outcomes resulting from complex motivations (e.g., Jusup et al.,
2022); the same seems plausible for ETIs.

The most basic properties of metasocieties and societies are where they are,
when they start, and how long they last. The first two of these likely involve
stochastic elements. Potential habitats are scattered randomly through the
galaxy, and the timing of ETI evolution likely involves some contingency,
although subsequent interstellar expansion might not.

Broadcasts can have predictable structure, though. Predictability can be a
byproduct of how a transmitter operates. A narrowbeam transmitter on a
rotating world sweeps past the Earth at regular intervals, for example (Gray &
Ellingsen, 2002). ETIs could even exploit periodicity to encode information or
make the artificial nature of their broadcasts obvious (as in Borra, 2012;
Harp et al., 2018).

However, although broadcasts may have predictable features, their properties
are not fully determined – otherwise, we would simply aim our telescopes where
and when we expect a broadcast instead of sifting through the haystack.
Broadcasts plausibly turn on and off at unpredictable times, the result of
unknowable social factors. Randomness in the host galaxy also induces
randomness in broadcasts. Each host star has random light travel delay times,
parallaxes, and Doppler shifts, and sits in a turbulent interstellar medium
with random dispersion and scintillation. A faint periodic broadcast might
only be observable during rare moments when scintillation magnifies it to a
detectable level (Cordes et al., 1997).

The real issue is whether the treatment of broadcasts as independent isolated
bursts of radiation is inadequate, and thus the haystack is defined according
to the wrong parameters. Introducing dependence is one way to address this: in
a periodic train, knowing two subsequent arrivals allows us to predict the
others. Often there is a natural parameterization in which we can treat the
entire complex of signals as a single broadcast, however, like defining a
periodic train by its period and phase. Although the specific distributions
used in this paper would not apply, the general ideas of the analysis still
do.

###  4.2 Independence

I also assume the properties of two sibling objects are independent of each
other, conditionalized on the shared parent’s parameter tuple. Likewise, the
number of children objects in a parent is conditionally independent of each
child object’s parameters. Thus, each broadcast is assumed to be independent
of every other, except in the sense they depend on a shared ancestor (society,
metasociety, galaxy, or the universe); societies are independent of each
other, aside from _their_ shared ancestors; and so on.

Independence is by no means obvious. On the broadcast level, although beacons
and noncommunicative broadcasts are plausibly completely independent,
intragalactic communications are part of a web of messages, providing a
natural mechanism for coordination. Broadcasts may be more common if there are
more potential receiving societies, and some may even be direct replies to
others. Metasocieties can converge to a common broadcast protocol, like
designated frequency channels. Dependence could extend to other object types –
if interstellar expansion is rampant, the existence of a society around one
star strongly suggests the existence of them around nearby stars, and so on.

The framework handles all such seeming dependence by moving the shared
properties up the tree. Instead of viewing sibling objects as mutually
dependent, we instead postulate that the “dependent quantities” are actually
properties of the ancestor itself. Thus, the common properties all follow from
the ancestral tuple, which governs the populations within it.
_Conditionalized_ on that tuple, the siblings are independent of each other.
The difference is that the population does not respond to its own
fluctuations.

Consider: what if each galaxy has exactly one society arising at a random time
since the Big Bang and the society’s lifespan is always one hundred years?
Without knowing when the society existed, a single broadcast from that society
could occur any time in cosmic history. Once we know when one broadcast
occurred, however, any others had to have been made to within one hundred
years because there was no broadcasting society at other times – a failure of
independence. However, that is only because the broadcast tells us when the
society appeared. If we already knew that, learning when one broadcast happens
does not necessarily give us any new information – the broadcast times can be
independent conditional on societal parameters.

The model can be altered to include different levels to allow for still
further degrees of dependence, but it nonetheless assumes that everything can
be grouped into discrete objects sorted into hierarchical levels. Perhaps ETI
distributions are more like scale-free fractals with a continuous range of
dependence. This would necessitate a different kind of framework.

Just because the properties of one object are conditionally independent of
another’s does not mean that the parameters of a single object are mutually
independent. Massive stars are younger on average; metasocieties encompassing
more worlds could live longer; “gregarious” societies could transmit
broadcasts at a higher rate and make them brighter on average; and so on. This
just means that the joint distribution is not separable into the product of
the PDFs of the variables; like when all the objects fall along a line or
surface in the haystack. It prevents us from modeling the aggregate emission
with compound Poisson statistics. The distribution can still be calculated,
however, and we still find the mean with Campbell’s formula (equation 4). If
the societal population is Poissonian, equation 10 allows us to find the
variance.

##  5 The astrophysical context of ETIs: The universe, galaxies, and stars

Table 6: Shared notation for objects

Quantity |  $\mathrm{J}$-Type | Universe | Galaxy | Metasociety | Society | Broadcast | Star
---|---|---|---|---|---|---|---
Random object index | $\mathrm{J}$ | $\mathrm{U}$ | $\mathrm{G}$ | $\mathrm{M}$ | $\mathrm{C}$ | $\mathrm{B}$ | $\star$
Realized object index | $\mathrm{j}$ | $\mathrm{u}$ | $\mathrm{g}$ | $\mathrm{m}$ | $\mathrm{c}$ | $\mathrm{b}$ |  $\lozenge$
Parameter tuple | $\boldsymbol{\mathnormal{w}}_{\mathrm{J}},\boldsymbol{\mathnormal{w}}_{\mathrm{j}}$ | $\boldsymbol{\mathnormal{w}}_{\mathrm{U}},\boldsymbol{\mathnormal{w}}_{\mathrm{u}}$ | $\boldsymbol{\mathnormal{w}}_{\mathrm{G}},\boldsymbol{\mathnormal{w}}_{\mathrm{g}}$ | $\boldsymbol{\mathnormal{w}}_{\mathrm{M}},\boldsymbol{\mathnormal{w}}_{\mathrm{m}}$ | $\boldsymbol{\mathnormal{w}}_{\mathrm{C}},\boldsymbol{\mathnormal{w}}_{\mathrm{c}}$ | $\boldsymbol{\mathnormal{w}}_{\mathrm{B}},\boldsymbol{\mathnormal{w}}_{\mathrm{b}}$ | $\boldsymbol{\mathnormal{w}}_{\star},\boldsymbol{\mathnormal{w}}_{\scalebox{0.75}{$\lozenge$}}$
Position | $\boldsymbol{\mathnormal{r}}_{\mathrm{J}},\boldsymbol{\mathnormal{r}}_{\mathrm{j}}$ |  $\cdots$ | $\boldsymbol{\mathnormal{r}}_{\mathrm{G}},\boldsymbol{\mathnormal{r}}_{\mathrm{g}}$ | $\boldsymbol{\mathnormal{r}}_{\mathrm{M}},\boldsymbol{\mathnormal{r}}_{\mathrm{m}}$ | $\boldsymbol{\mathnormal{r}}_{\mathrm{C}},\boldsymbol{\mathnormal{r}}_{\mathrm{c}}$ | $\boldsymbol{\mathnormal{r}}_{\mathrm{B}},\boldsymbol{\mathnormal{r}}_{\mathrm{b}}$ | $\boldsymbol{\mathnormal{r}}_{\star},\boldsymbol{\mathnormal{r}}_{\scalebox{0.75}{$\lozenge$}}$
Size (volume) | $\mathnormal{v}_{\mathrm{J}},\mathnormal{v}_{\mathrm{j}}$ | $\mathnormal{v}_{\mathrm{U}},\mathnormal{v}_{\mathrm{u}}$ | $\mathnormal{v}_{\mathrm{G}},\mathnormal{v}_{\mathrm{g}}$ | $\mathnormal{v}_{\mathrm{M}},\mathnormal{v}_{\mathrm{m}}$ |  $\cdots$ |  $\cdots$ |  $\cdots$
Starting time | $\mathnormal{\vartheta}_{\mathrm{J}},\mathnormal{\vartheta}_{\mathrm{j}}$ | $\mathnormal{\vartheta}_{\mathrm{U}},\mathnormal{\vartheta}_{\mathrm{u}}$ | $\mathnormal{\vartheta}_{\mathrm{G}},\mathnormal{\vartheta}_{\mathrm{g}}$ | $\mathnormal{\vartheta}_{\mathrm{M}},\mathnormal{\vartheta}_{\mathrm{m}}$ | $\mathnormal{\vartheta}_{\mathrm{C}},\mathnormal{\vartheta}_{\mathrm{c}}$ | $\mathnormal{\vartheta}_{\mathrm{B}},\mathnormal{\vartheta}_{\mathrm{b}}$ | $\vartheta_{\star},\vartheta_{\scalebox{0.75}{$\lozenge$}}$
Duration | $\mathnormal{\tau}_{\mathrm{J}},\mathnormal{\tau}_{\mathrm{j}}$ | $\mathnormal{\tau}_{\mathrm{U}},\mathnormal{\tau}_{\mathrm{u}}$ | $\mathnormal{\tau}_{\mathrm{G}},\mathnormal{\tau}_{\mathrm{g}}$ | $\mathnormal{\tau}_{\mathrm{M}},\mathnormal{\tau}_{\mathrm{m}}$ | $\mathnormal{\tau}_{\mathrm{C}},\mathnormal{\tau}_{\mathrm{c}}$ | $\mathnormal{\tau}_{\mathrm{B}},\mathnormal{\tau}_{\mathrm{b}}$ | $\tau_{\star},\tau_{\scalebox{0.75}{$\lozenge$}}$
Arbitrary singleton variable | $\mathnormal{\kappa}_{\mathrm{J}},\mathnormal{\kappa}_{\mathrm{j}}$ | $\mathnormal{\kappa}_{\mathrm{U}},\mathnormal{\kappa}_{\mathrm{u}}$ | $\mathnormal{\kappa}_{\mathrm{G}},\mathnormal{\kappa}_{\mathrm{g}}$ | $\mathnormal{\kappa}_{\mathrm{M}},\mathnormal{\kappa}_{\mathrm{m}}$ | $\mathnormal{\kappa}_{\mathrm{C}},\mathnormal{\kappa}_{\mathrm{c}}$ | $\mathnormal{\kappa}_{\mathrm{B}},\mathnormal{\kappa}_{\mathrm{b}}$ | $\mathnormal{\kappa}_{\star},\mathnormal{\kappa}_{\scalebox{0.75}{$\lozenge$}}$
Distribution (intensity) | $\mathnormal{\Psi}^{\mathrm{J}}$ | $\mathnormal{\Psi}^{\mathrm{U}}$ | $\mathnormal{\Psi}^{\mathrm{G}}$ | $\mathnormal{\Psi}^{\mathrm{M}}$ | $\mathnormal{\Psi}^{\mathrm{C}}$ | $\mathnormal{{\Psi}}^{\mathrm{B}}$ | $\mathnormal{\Psi}^{\star}$
Number of objects | $\mathnormal{N}^{\mathrm{J}}$ |  $\cdots$ | $\mathnormal{N}^{\mathrm{G}}$ | $\mathnormal{N}^{\mathrm{M}}$ | $\mathnormal{N}^{\mathrm{C}}$ | $\mathnormal{N}^{\mathrm{B}}$ | $\mathnormal{N}^{\star}$
Haystack (space state) | $\mathnormal{W}^{\mathrm{J}}$ | $\mathnormal{W}^{\mathrm{U}}$ | $\mathnormal{W}^{\mathrm{G}}$ | $\mathnormal{W}^{\mathrm{M}}$ | $\mathnormal{W}^{\mathrm{C}}$ | $\mathnormal{W}^{\mathrm{B}}$ | $\mathnormal{W}^{\star}$
Random sample (point process) | $\mathnormal{\Sigma}^{\mathrm{J}}$ | $\mathnormal{\Sigma}^{\mathrm{U}}$ | $\mathnormal{\Sigma}^{\mathrm{G}}$ | $\mathnormal{\Sigma}^{\mathrm{M}}$ | $\mathnormal{\Sigma}^{\mathrm{C}}$ | $\mathnormal{\Sigma}^{\mathrm{B}}$ | $\mathnormal{\Sigma}^{\star}$
Realized sample | $\mathnormal{S}^{\mathrm{J}}$ | $\mathnormal{S}^{\mathrm{U}}$ | $\mathnormal{S}^{\mathrm{G}}$ | $\mathnormal{S}^{\mathrm{M}}$ | $\mathnormal{S}^{\mathrm{C}}$ | $\mathnormal{S}^{\mathrm{B}}$ | $\mathnormal{S}^{\star}$
Temporal rate per star | $\mathnormal{\Gamma}^{\mathrm{J}}$ |  $\cdots$ |  $\cdots$ | $\mathnormal{\Gamma}^{\mathrm{M}}$ | $\mathnormal{\Gamma}^{\mathrm{C}}$ | $\mathnormal{\Gamma}^{\mathrm{B}}$ |  $\cdots$
Abundance per star | $\mathnormal{\Xi}^{\mathrm{J}}$ |  $\cdots$ |  $\cdots$ | $\mathnormal{\Xi}^{\mathrm{M}}$ | $\mathnormal{\Xi}^{\mathrm{C}}$ | $\mathnormal{\Xi}^{\mathrm{B}}$ |  $\cdots$
Total temporal (formation) rate | $\breve{\mathnormal{\Gamma}}^{\mathrm{J}}$ |  $\cdots$ |  $\cdots$ | $\breve{\mathnormal{\Gamma}}^{\mathrm{M}}$ | $\breve{\mathnormal{\Gamma}}^{\mathrm{C}}$ | $\breve{\mathnormal{\Gamma}}^{\mathrm{B}}$ | $\breve{\mathnormal{\Gamma}}^{\star}$
Arbitrary aggregate variable | $\mathnormal{K}^{\mathrm{J}}$ |  $\cdots$ | $\mathnormal{K}^{\mathrm{G}}$ | $\mathnormal{K}^{\mathrm{M}}$ | $\mathnormal{K}^{\mathrm{C}}$ | $\mathnormal{K}^{\mathrm{B}}$ | $\mathnormal{K}^{\star}$

Note. — The $\mathrm{J}$ type objects are generic, used for general formulae,
as are the $\mathrm{K}$ and $\mathrm{L}$ types.

Singleton variables (upper half) are here marked with the uppercase index for
a type of object (e.g., $\mathrm{B}$ for broadcasts), if the variable is
treated as a general parameter instead of describing a fixed object. A
lowercase index (defaults given for each type) in a singleton variable
indicates the variable refers to a realized object with fixed parameters.
Other indices may be substituted, however, if they are defined for realized
objects. Singleton variables may also specify a quantity window.

Aggregate variables (lower half) can also be marked with an object selection,
and a quantity window if different from object window. By default, the windows
are the ALL ($\infty$) window and the universe (${\mathrm{U}}$) as a host.

###  5.1 The universe

The universe is the root node of the tree (Figure 1), the “object” that
contains all the other objects as descendants. Even the universe has a
parameter tuple, $\boldsymbol{\mathnormal{w}}_{\mathrm{U}}$, and a haystack,
the space of all such tuples $\mathnormal{W}^{\mathrm{U}}$ (Table 6 lists
relevant notation.). Cosmological parameters like the Hubble constant can be
used to define this haystack, but in this work, I assume these are set to
fixed values. Instead, the model parameters describe the descendant
populations of ETIs and their broadcasts. For instance, suppose we thought
that in every transmitter population, the luminosity distribution follows a
power-law distribution, all with the same slope $\mathnormal{\gamma}$. We do
not know what the slope is, however. Then we can consider
$\mathnormal{\gamma}_{\mathrm{U}}$ as a cosmic-level parameter, one describing
all ETI populations in the Universe. In a given model universe ${\mathrm{u}}$,
it takes on a value $\mathnormal{\gamma}_{\mathrm{u}}$, which all the lower-
level populations inherit, and then we make calculations for each
$\mathnormal{\gamma}_{\mathrm{u}}$ variate to compare predictions.

As discussed in Section 3.1.2, no universe distribution is considered here,
only the consequences of different $\boldsymbol{\mathnormal{w}}_{\mathrm{U}}$
for observed populations.

###  5.2 Galaxies as domains for ETIs

ETIs can coordinate their properties on large scale through replication and
communication, but there must be a limit to how far these processes can
operate. We can imagine the Universe ends up broken up into a patchwork of
regions over which coordination succeeded, much like the magnetic domains in a
piece of iron (see Olson, 2015). In this series, a domain is an astrophysical
region so large and isolated that ETIs cannot spread outside of it, completely
confining any metasocieties. Depending on how difficult space travel is, a
domain might be as small as a planetary body or as large as the entire Hubble
volume. Every domain has sites where individual societies can reside. This
work treats galaxies as the natural scale for domains.262626IIf intergalactic
travel or communication is allowed, galactic metasocieties could fuse into an
intergalactic metasociety on the domain scale. If cohesive galactic
submetasocieties persist, then these would be the children of the top-level
intergalactic metasocieties, instead of being the parents. We could also
consider galaxies analogously to stars in Section 5.3.

Although they are home to entire populations of discrete stars and worlds,
galaxies themselves form a cosmic population with a distribution

$\displaystyle{\mathnormal{\Psi}_{\mathrm{U}}^{\mathrm{G}}(\boldsymbol{\mathnormal{w}}_{\mathrm{G}}|\boldsymbol{\mathnormal{w}}_{\mathrm{U}})}$
$\displaystyle\equiv\frac{d\left\langle{\mathnormal{N}_{\mathrm{U}}^{\mathrm{G}}}\right\rangle}{d\boldsymbol{\mathnormal{w}}_{\mathrm{G}}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{G}}|\boldsymbol{\mathnormal{w}}_{\mathrm{U}})}$
$\displaystyle=\frac{d^{3}\left\langle\mathnormal{N}_{\mathrm{U}}^{\mathrm{G}}\right\rangle}{d\mathnormal{N}_{\mathrm{G}}^{\star}d\boldsymbol{\mathnormal{r}}_{\mathrm{G}}d\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{G}}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{G}}|\boldsymbol{\mathnormal{w}}_{\mathrm{U}})}.$
(62)

The two parameters I single out are the number of stars in the entire galaxy,
$\mathnormal{N}_{\mathrm{G}}^{\star}$, and its position
$\boldsymbol{\mathnormal{r}}_{\mathrm{G}}$, with
$\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{G}}$ as a catch-all for any
remaining parameters. All other things being equal, galaxies with more stars
have more habitats in which intelligent life can evolve, increasing the mean
galaxy-wide rate at which metasocieties appear. It takes time for intelligence
to evolve, thus the cosmic epoch of the galaxy matters; distance also directly
affects observables. There are many other factors that could affect the
evolution and spread of ETIs: star formation history, metallicity
distribution, stellar density and velocity dispersion, nuclear activity, and
the nature of the interstellar medium all could be important (e.g., Gonzalez
et al. 2001; Lineweaver et al. 2004; Dayal et al. 2015; Di Stefano & Ray 2016;
Gowanlock 2016; Balbi & Tombesi 2017; Lingam et al. 2019; Lacki 2021).

It is also possible the development of new metasocieties is affected by their
predecessors, and thus there are techno-historical parameters for each galaxy
as well. The science-fiction trope wherein relics of a long-dead ETI yield
advanced technologies for practical interstellar travel (e.g., the “subway”
system in Sagan 1985) comes to mind – later ETIs would have an easier time
spreading than the first ones. Or, early ETIs might instead launch self-
replicating probes that actively destroy all new ETIs (see Brin, 1983, and
discussion therein). Even so, we might regard all these ETIs as being part of
a single metasociety, albeit one whose societies are highly clustered in time,
their appearances poorly described by a Poisson process (see Section 6.3).

Observational programs only draw a sample of the cosmic population. This
sample is fixed in targeted observations typical of most extragalactic radio
SETI efforts to date (Shostak et al., 1996; Gray & Mooley, 2017). Large-scale
cosmic surveys may be viewed as drawing random samples from the Universe
(Martínez & Saar, 2002). Random samples of background galaxies are also
present in targeted pointings at nearby objects (see Garrett & Siemion, 2023).
When the galaxies being studied are already known, we conditionalize on the
known galaxy sample; when treating galaxies statistically, there is a galactic
contribution to the sampling variance.

###  5.3 Stars: A tracer of ETIs?

Planetary systems around stars are commonly regarded as the origin sites of
ETIs. The properties of a sun can bear on whether a metasociety originates
there.272727HHypothetically, ETIs could originate elsewhere; rogue planets
detached from any star may even be habitable (e.g., Stevenson, 1999; Abbot &
Switzer, 2011; Badescu, 2011). On the whole, most baryonic bodies like planets
should trace the stellar mass on large scales, modulated by other quantities,
as discussed below. The star needs to live long enough for intelligence to
evolve, for one (Huang, 1959; Carter, 1983; Livio, 1999), and its luminosity
should be steady enough that a planet can remain habitable for geological
times (Hart, 1979; Kasting et al., 1993). There is a continuing debate on
whether planets around M dwarfs can develop life, against the backdrop of
planetary tidal locking and large stellar flares (Shields et al. 2016 and
references therein); the space weather around them might discourage radio
communications if a society evolves (Ćirković & Vukotić, 2020). Stellar
parameters could also affect whether a planetary system is settled: mass and
metallicity are correlated with number and types of attendant planets (e.g.,
Johnson et al., 2010; Mulders et al., 2015); exotic types of stars might draw
ETIs interested in astroengineering (Dyson, 1963; Learned et al., 2012;
Chennamangalam et al., 2015; Semiz & Oğur, 2015; Osmanov, 2016; Imara & Di
Stefano, 2018; Lacki, 2020; Lingam & Loeb, 2020); and so on. For a more
precise, fine-grained treatment, we consider stellar systems instead of stars,
with the distribution including quantities describing the architecture of the
system: how many stars, in what orbits; the number, sizes, and locations of
planets; their satellites; and other variables that could be relevant to
whether there are inhabitants (e.g., Ward & Brownlee, 2000).

Societies and their broadcasts may be concentrated in regions of high stellar
density, without necessarily being hosted by stars. On a galactic scale, we
may expect baryonic resources, stellar energy, and interstellar habitats like
interstellar objects to trace the stellar mass distribution. These could be
useful for ETIs in astroengineering projects. However, ETIs could migrate to
the cores or outskirts of galaxies, driven by the possibility of more
efficient collaboration (Kardashev, 1985; Smart, 2009), more optimal
thermodynamic environments (Ćirković & Bradbury, 2006), or large-scale
gradients in habitats. Some possible “attractor” objects trace young stars
instead of total stellar mass, like high-mass X-ray binaries (as in Vidal,
2011), maybe even concentrating ETIs into a few clumps while leaving the rest
of the galaxy empty. Even if they roughly follow stars, the dependence need
not be linear; they might be far more prevalent in regions of higher stellar
density because interstellar travel is easier, for example (Di Stefano & Ray,
2016; Gajjar et al., 2021; Lacki, 2021). In the absence of secure knowledge of
the relevant tradeoffs, I posit that the density of societies in inhabited
galaxies follows the stars on a large scale.

Although they do not fit neatly into the “tree” of objects, it is useful to
treat stars as another class of objects with its own point process. The stars
are points scattered in a “stellar haystack” $\mathnormal{W}^{\star}$,
described by parameter tuples $\boldsymbol{\mathnormal{w}}_{\star}$. Like
other types of objects, windows select stars based on position in their
haystack. Galaxies host stars, but stars might also be regarded as hosted by
metasocieties and societies if their planetary system is home to an
ETI.282828FFormally, we could use projections of the position and time of the
“host” (meta)society to pick out stars (Section 3.2.1). In any case, because
societies are localized in single planetary systems or interstellar space,
$\mathnormal{N}_{\mathrm{C}}^{\star}$ is zero, one, or perhaps up to a few in
multiple star systems, depending on whether we are considering stars _per se_
or stellar systems. Thus, we can consider the stellar sample
${\mathnormal{\Sigma}_{\mathsf{x},\mathrm{J}}^{\star}}$ drawn by a selection
$(\mathsf{x},{\mathrm{J}})$, consisting of
${\mathnormal{N}_{\mathsf{x},\mathrm{J}}^{\star}}$ stars.

The stellar properties considered here are current position
$\boldsymbol{\mathnormal{r}}_{\star}$, birth time $\vartheta_{\star}$, and
initial mass $\mathnormal{m}_{\star}$. Lifespan $\tau_{\star}$ is regarded as
a function of $\mathnormal{m}_{\star}$. The stars then have a distribution
function

${\mathnormal{\Psi}_{\mathrm{J}}^{\star}(\boldsymbol{\mathnormal{w}}_{\star}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}=\frac{d^{4}\left\langle{\mathnormal{N}_{\mathrm{J}}^{\star}}\right\rangle}{d\boldsymbol{\mathnormal{r}}_{\star}d\vartheta_{\star}d\mathnormal{m}_{\star}d\boldsymbol{\mathnormal{w}}^{\prime}_{\star}}{(\boldsymbol{\mathnormal{w}}_{\star}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})},$
(63)

a joint stellar density, initial mass function, and star formation history.
Any other parameters are absorbed into
$\boldsymbol{\mathnormal{w}}^{\prime}_{\star}$. This distribution can be used
as a term in metasocietal, societal, or broadcast distribution to introduce a
dependence on stellar properties.

##  6 The ETIs themselves: metasocieties and societies

Table 7: Other variable notation for objects  Notation | Explanation
---|---
Galaxies and stars
$\mathnormal{m}_{\star}$ |  MMass of individual star
$\mathnormal{M}_{\mathsf{t},\mathrm{G}}^{\star}$ |  TTotal stellar mass of galaxy at one epoch
Metasocieties and societies
$\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}$ |  Fraction of societies in a metasociety ${\mathrm{M}}$ that are communicative (broadcasting)
$\boldsymbol{\mathnormal{w}}_{\star}^{\mathrm{M}}$ |  Parameter tuple for the star from which the metasociety first emerged
$\breve{\mathnormal{\Gamma}}_{\star}^{\mathrm{M}}$ |  MMean rate of metasocietal origin events around star
$\breve{\mathnormal{\Gamma}}_{\star}^{\mathrm{C}}$ |  MMean rate of societal origin events around star
Broadcasts
$\mathnormal{\varpi}_{\mathrm{B}}$ |  Quantity describing polarization of broadcast; in general, it is a vector or matrix
$\mathnormal{\zeta}_{\mathsf{p}|\mathrm{B}}(p)$ |  Fraction of broadcast emission in polarization $p$
$\mathnormal{\zeta}_{\mathsf{x}|\mathrm{B}}$ |  Sum of $\mathnormal{\zeta}_{\mathsf{p}|\mathrm{B}}$ over all polarizations in $\mathsf{\Pi}_{\mathsf{x}}$
$\mathnormal{\omega}_{\mathrm{B}}$ |  Solid angle that broadcast is emitted into
$\mathnormal{\beta}_{\mathrm{B}}$ |  Total bandwidth of broadcast in the source frame, frequency span over the entire duration
$\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}$ |  Instantaneous bandwidth of broadcast in the source frame
$\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}$ |  Bandwidth covered by broadcast over the duration of ${\mathsf{x}}$
$\mathnormal{\upsilon}_{\mathrm{B}}$ |  Central frequency of broadcast in the source frame
$\mathnormal{\delta}_{\mathrm{B}}$ |  Drift rate of broadcast in the source frame
${\mathnormal{\Lambda}_{\mathrm{J}}^{\mathrm{B}}}$ |  MMean temporal rate of broadcasts in host ${{\mathrm{J}}}$ per unit frequency per star
${\mathnormal{Z}_{\mathrm{J}}^{\mathrm{B}}}$ |  MMean number of broadcasts in host ${{\mathrm{J}}}$ per unit frequency per star
${\breve{\mathnormal{\Lambda}}_{\mathrm{J}}^{\mathrm{B}}}$ |  MMean total temporal rate of broadcasts in host ${{\mathrm{J}}}$ per unit frequency
${\breve{\mathnormal{Z}}_{\mathrm{J}}^{\mathrm{B}}}$ |  MMean total number of broadcasts in host ${{\mathrm{J}}}$ per unit frequency
⊕ |  Superscript indicating observer-frame quantity

###  6.1 Metasocieties

Intelligences are organized into metasocieties and societies. An inhabited
galaxy contains one or more metasocieties at the present moment:
${\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}\geq 1$. A
metasociety is a collection of interacting ETI societies shaped by a shared
history. A society and all of its “offspring” always are part of the same
metasociety for our purposes here. The technosignatures of a metasociety have
a shared influence from a common origin. Metasocieties, unlike individual
societies, can be extended over galactic scales.

The nature and evolution of metasocieties is one of the most speculative areas
of SETI, with direct bearing on the Fermi Paradox and expected
technosignatures. A metasociety might consist of only one planet-confined
society, a loose galactic network of worlds, an expanding interstellar
settlement front, or a dense equilibrium state, among other possibilities
(section 6.3).

Given the diversity of hypotheses, huge uncertainties, and the possible
amplification of stochastic effects by initial exponential growth, there is no
one obviously correct single set of parameters to describe them. I
characterize them with these parameters:

  * •

All metasocieties are regarded as having a single origin, arising at a single
well-defined time $\mathnormal{\vartheta}_{\mathrm{M}}$, at a single location.
The origin site is indicated by a parameter tuple
$\boldsymbol{\mathnormal{w}}_{\star}^{\mathrm{M}}$ describing the first
inhabited host star (see section 5.3). The full distribution includes a rate
of metasociety appearance per star in an empty galaxy, even allowing us to
establish a functional dependence on stellar properties. However, the number
of current metasocieties does not necessarily scale with
$\left\langle\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}\right\rangle$ if
they are sufficiently common and cover the entire galaxy (section 6.3). Actual
metasocieties may form as a fusion of smaller metasocieties (Kardashev, 1985;
Forgan, 2017); the “origin” might be for the first or perhaps a dominant
component, but maybe the origins of the others affect the result as well.

  * •

All metasocieties cover a finite volume $\mathnormal{v}_{\mathrm{M}}$
containing a finite number of astrophysical objects centered at position
$\boldsymbol{\mathnormal{r}}_{\mathrm{M}}$. Not all regions with the same
spatial volume are equivalent – maybe there are more societies in a volume
that includes the core or disk of a galaxy than one that only includes the
distant halo. An alternative is the mean number of stars covered by a
metasociety, $\left\langle\mathnormal{N}_{\mathrm{M}}^{\star}\right\rangle$.
In the (perhaps simplistic) scenarios I discuss, however, the metasociety is
either restricted to one stellar system with $\mathnormal{v}_{\mathrm{M}}\to
0$ or pervades the entire galaxy. Conceivably, multiple metasocieties might
overlap, perhaps inhabiting different habitats (e.g., planetary surfaces
versus deep space versus compact objects).

  * •

The lifespan of the metasociety, $\mathnormal{\tau}_{\mathrm{M}}$.
Metasocieties may have effectively undefined lifespans, however, especially if
they cover entire galaxies, continuously regenerating through internal
replication.292929Of course, in the very long run, all metasocieties will
perish in our current understanding of cosmology, but there will be hydrogen-
burning stars to host them for $10$ trillion years (Laughlin et al., 1997),
much less more exotic remnants (Adams & Laughlin, 1997).

  * •

The mean fraction of societies within the metasociety that are communicative,
$\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}$.

  * •

Other parameters describing its internal distribution of communicative
societies, $\mathnormal{\boldsymbol{w}}^{\mathrm{C}}_{\mathrm{M}}$, and
metasociety-wide properties of broadcasts
$\mathnormal{\boldsymbol{w}}^{\mathrm{B}}_{\mathrm{M}}$. Most obviously, this
can include something that governs the number of communicative societies, like
the rate of appearance of new societies per star. A metasociety necessarily
has at least one society over its history, but the number of communicative
societies may be zero if most societies are unable or unwilling to make
broadcasts. For broadcasts, this could include a typical energy scale or
typical rate per society. These are all lumped under a residual tuple,
$\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}$.

Under the assumption that metasocieties are defined by these specific
properties, the adopted metasocietal distribution has the form:

${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}=\frac{d^{7}\left\langle{\mathnormal{N}_{\mathrm{G}}^{\mathrm{M}}}\right\rangle}{d\mathnormal{\vartheta}_{\mathrm{M}}d\mathnormal{\tau}_{\mathrm{M}}d\boldsymbol{\mathnormal{r}}_{\mathrm{M}}d\mathnormal{v}_{\mathrm{M}}d\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}d\boldsymbol{\mathnormal{w}}_{\star}^{\mathrm{M}}d\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}.$
(64)

Depending on scenario, metasocieties may interact in complicated ways, so the
metasocietal point process
${\mathnormal{\Sigma}_{\mathrm{G}}^{\mathrm{M}}(\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}$
can be far from Poissonian (see discussion in Section 6.3).

Although most of these parameters are regarded as single numbers, the
framework is sufficiently general that they can be replaced with subtuples,
expanding the haystack with more dimensions. Metasocietal size is treated here
as a fixed value, if only as a placeholder because the volume does not come
into play in this paper. While interstellar expansion can be quick on
cosmological timescales, it still takes time and we could catch it in progress
(Zackrisson et al., 2015). To account for that, we could replace “size” with
parameters describing the velocity of the expansion front (Jones, 1981),
stellar diffusion parameters (Carroll-Nellenback et al., 2019), initial low
filling factors due to percolation (Landis, 1998), and so on. The origin
star’s parameter tuple could be expanded to model the star’s trajectory
through the galaxy, including its initial velocity and perhaps artificial
propulsion (Badescu & Cathcart, 2006), if we regard its position as remaining
important. The communicative fraction
$\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}$ could be replaced with parameters
for a function describing communicativeness as a function of societal age
(Sagan, 1973). Many such complications can be postulated, and the distribution
altered to treat them.

###  6.2 Societies

In this series, a society is considered a localized, independent entity with
its infrastructure.303030TThough “society” may suggest distinct individuals
cooperating toward common goals, the concept applies even if these
(inter)planetary entities are solitary beings, group minds, ecospheres, or
even large collections of noninteracting intelligences. A communicating
society is one that emits broadcasts. In this view, a “society” does not just
include sentient beings, but the technology they produce. Thus, a society may
produce broadcasts long after its inhabitants have perished; as many have
noted, the longevity of technosignatures themselves determines the number of
detectable ETIs (e.g., Carrigan 2012; Balbi & Ćirković 2021; Wright et al.
2022). Conversely, a communicating society may effectively disappear when it
loses interest in broadcasting, even while the society itself thrives (Sagan
1973; Smart 2009).

Classically, SETI considers single planetary systems or worlds as unitary
bodies, naturally imposing a discreteness. The society level could be much
finer, though, including individual interstellar vehicles and inhabited
interstellar objects, or distinct organizations on a single world. The most
relevant factor here is the induced clumping of broadcasts; a vast number of
interstellar transmitting entities could reduce the Poisson fluctuations of
inhabited galaxies. However, there could plausibly be more “levels,” each
imposing its discreteness effects, requiring more complex models.313131What
are called “civilizations” or “societies” in SETI are planetary metasocieties;
all the diverse cultures, institutions, and peoples of a planet are
amalgamated into one entity. This single entity could have a much longer
lifespan than any society as sociologically defined. But a single world or
planetary system can be far more tightly integrated than an entire galaxy, so
there is motivation to treat them as one unit.

Each society starts at time $\mathnormal{\vartheta}_{\mathrm{C}}$ and survives
for a finite duration $\mathnormal{\tau}_{\mathrm{C}}$. It is located at
$\boldsymbol{\mathnormal{r}}_{\mathrm{C}}$; when considering very long
timescales, we can instead use parameters describing its trajectory. Thus I
adopt

${\mathnormal{\Psi}_{\mathrm{M}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}=\frac{d^{4}\left\langle{\mathnormal{N}_{\mathrm{M}}^{\mathrm{C}}}\right\rangle}{d\mathnormal{\vartheta}_{\mathrm{C}}d\mathnormal{\tau}_{\mathrm{C}}d\boldsymbol{\mathnormal{r}}_{\mathrm{C}}d\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{C}}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}$
(65)

as the societal distribution for a metasociety ${\mathrm{M}}$. Any other
parameters, including those describing a society’s broadcast population, are
collected into $\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{C}}$. As with
metasocieties, individual parameters listed here can be replaced with sets of
parameters (e.g., describing the society’s trajectory instead of just its
position). If a metasociety has more than one society, the appearance of those
societies is assumed to be Poissonian. Furthermore, all the properties of the
societies are assumed to be independent of one another, including their
broadcast distributions.

Long-lived interstellar metasocieties are plausibly in an equilibrium state,
motivating a single $\mathnormal{\Xi}_{\mathrm{M}}^{\mathrm{C}}$ for each one.
If ETIs tend to fill available habitats, then they can do so in much less than
1 Gyr. Sustainability arguments suggest that metasocieties necessarily have to
self-regulate to avoid resource depletion (Fogg, 1987; Haqq-Misra & Baum,
2009). The reality may be more complicated than a simple Poissonian
equilibrium process, however, with many hierarchical levels of organization
that this model neglects. Perhaps metasocieties are highly chaotic, with a
turbulent series of spikes, booms, plateaus, and crashes. Metasocieties that
consist of several submetasocieties knit together by messages in relics could
be extremely intermittent. Percolation hypotheses suggest that metasocieties
may be spatially inhomogeneous as well (Landis, 1998), albeit subject to
stellar mixing (Wright et al., 2014) and societal turnover (Wiley, 2011).
Without interstellar travel, even a multitude of independent transmitting
programs is “clumped” onto the rare broadcasting host worlds that presumably
dominate the Poissonian fluctuations. More detailed models are necessary to
treat effects like these.

###  6.3 Scenarios for metasocietal evolution

####  6.3.1 The “classical” scenario: metasocieties as societies

The default scenario in SETI theory is that societies are confined to one
solar system, surviving for a continuous span of time before disappearing
forever without replication (e.g., Oliver & Billingham, 1971). Thus, each
metasociety covers just one stellar system and contains just one society,
which may or may not be communicative. Each (meta)society is regarded as
appearing and behaving independently of the others. The adopted metasocietal
distribution function is

${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}=\updelta(\boldsymbol{\mathnormal{r}}_{\mathrm{M}}-\boldsymbol{\mathnormal{r}}_{\star}^{\mathrm{M}})\updelta(\mathnormal{v}_{\mathrm{M}}){\uppsi\left[\mathnormal{\tau}_{\mathrm{M}},\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}},\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}\right]_{\mathrm{G}}}\\\
\cdot~{}\breve{\mathnormal{\Gamma}}_{\star}^{\mathrm{M}}(\mathnormal{\vartheta}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\star}^{\mathrm{M}}){\mathnormal{\Psi}_{\mathrm{G}}^{\star}}(\boldsymbol{\mathnormal{w}}_{\star}^{\mathrm{M}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{G}}}).$
(66)

The metasociety thus has zero size ($\mathnormal{v}_{\mathrm{M}}=0$) and its
position is identical to that of the star it originated from. The number of
metasocieties is proportional to the stellar distribution, but it is modulated
by the number rate of metasocieties appearing around each star at a given time
after its birth. That term includes dependence on the stellar parameter tuple
– factors like the star’s position in a galaxy, mass, and metallicity might
all affect the prevalence of metasocieties. Note also that the delay-time
distribution does not generally integrate to $1$; it is vastly less if ETIs
are rare, but can be more than $1$ if technological societies evolve several
times in a given planetary system (Wright, 2018; Schmidt & Frank,
2019).323232A simple societal intensity distribution does not contain enough
information to model the clustering of metasocieties that happens when they
evolve multiple times on rare worlds (e.g., those with complex multicellular
life). An additional level for inhabited worlds can be added to the tree to
address that. Nor does equation 66 account for what happens if subsequent
metasocieties around the same star overlap; it is presumed that their
lifespans are much shorter than the gap time between them.

With only one society per metasociety, there is no coordination or dependence
between different societies. Whether or not a metasociety’s single society is
communicative is random:

${\mathnormal{N}_{\mathrm{M}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}\sim\mathrm{Bern}({\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}}),$
(67)

with a probability ${\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}}$ determined by
$\boldsymbol{\mathnormal{w}}_{\mathrm{M}}$, the “communicative” factor in the
Drake equation. The distribution of metasocieties (and societies) is entirely
set by galactic properties. For a realized galaxy they are treated as a
Poissonian process because of this independence (as in Glade et al., 2012):

$\displaystyle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}(\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}\sim\mathrm{Pois}(\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle)$
$\displaystyle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}\sim\mathrm{Pois}(\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle)\sim\mathrm{Pois}({\left\langle\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}\right\rangle_{\mathsf{x},\mathrm{G}}}\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle),$
(68)

where the marking and thinning of societies by their communicativeness
preserves the Poisson character. Communicative societies inherit the lifespan
and origin point of their parent metasociety, leading to a societal
distribution

${\mathnormal{\Psi}_{\mathrm{M}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}={\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}}\updelta(\mathnormal{\vartheta}_{\mathrm{C}}-{\mathnormal{\vartheta}_{\mathrm{M}}})\updelta(\mathnormal{\tau}_{\mathrm{C}}-{\mathnormal{\tau}_{\mathrm{M}}}){\updelta(\boldsymbol{\mathnormal{r}}_{\mathrm{C}}-{\boldsymbol{\mathnormal{r}}_{\mathrm{M}}})}{\uppsi\left[\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{C}}\right]_{\mathrm{M}}}.$
(69)

As long as
${\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}}$ is
much shorter than the galaxy’s evolutionary timescales,

$\displaystyle\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}}(t)\right\rangle\approx{\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}}}(t){\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}}{\left\langle\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}(t)\right\rangle}$
$\displaystyle\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{C}}}(t)\right\rangle\approx{\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}}}(t){\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}\left\langle\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}\right\rangle_{\mathrm{G}}\left\langle\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}(t)\right\rangle}.$
(70)

Thus,
${\mathnormal{\Xi}_{\mathrm{G}}^{\mathrm{M}}\approx\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}}\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}}$.
The rate and abundance themselves depend on factors like star formation rate
and the delay-time distribution (Section 6.4).

####  6.3.2 Galactic clubs as metasocieties

In the galactic club scenario, the first communicative society successfully
contacts the succeeding ones through either remote transmissions or
nonreplicating probes, establishing norms and protocols (Bracewell 1975). The
protocols become locked in, with each new communicative society conforming to
the precedent set by the previous ones. The implicit coordination results in a
single galaxy-spanning metasociety, even though the societies themselves
remain planetbound, aside possibly from some automated probes. The classical
scenario is a special kind of galactic club scenario; it simply interprets
“metasociety” differently.

Much of the metasocietal distribution function is a formality. A metasociety’s
size and position are simply those of the host galaxy. A galactic club is born
with the first society, its origin star is the site of that society, and it
endures for $\mathnormal{\tau}_{\infty}$, an arbitrary timescale longer than
the current age of the galaxy. The number of metasocieties in a given host
galaxy is a Bernoulli random variable, simply determined by the societies:
$\mathnormal{P}({\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}=1)=\mathnormal{P}({\mathnormal{N}_{\mathsf{h},\mathrm{G}}^{\mathrm{C}}(t)}\geq
1)$. The way the metasociety influences its children is through a single
metasociety-wide $\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}$, and the
distribution of any additional parameters in
$\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}$ describing shared societal
and broadcast properties.333333FFor example, perhaps each society broadcasts
only at a frequency assigned by the metasociety according to some
distribution. Or perhaps the “culture” of the galactic club can bias the
lifespans of its member societies.

The galactic club metasociety is therefore largely an emergent phenomenon,
considered here only as a mechanism to coordinate societal and broadcast
properties. Societies are the fundamental entities driving the galactic club.
Despite the seeming circular dependence between the metasociety and its hosted
societies, in practice, the dependence is one-way for any single parameter.
The births of societies are entirely independent of each other, so if we
restrict the societal haystack to just
$\boldsymbol{\mathnormal{r}}_{\mathrm{C}}$ and
$\mathnormal{\vartheta}_{\mathrm{C}}$, a realized host galaxy’s population is
a Poisson point process. Meanwhile, the metasociety’s
$\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}$ and
$\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}$ are chosen and then used
to mark each society with its other parameters. The societal haystack is then
expanded into the remaining dimensions, and thinned according to which ones
are communicative. Finally, the origin of the metasociety is identified with
the first (communicative) society.

When we consider a host galaxy with fixed properties and conditionalize on any
metasocietal parameters governing societal/broadcast distributions, the
societies are modeled with a Poisson point process:

${\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{G}}|\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}})}\sim\mathrm{Pois}(\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}|\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}}\right\rangle).$
(71)

The societal distribution includes a dependence on the stellar distribution,
because they originate around stars:

${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}},\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}})}={\uppsi\left[\mathnormal{\tau}_{\mathrm{C}},\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}\right]_{\mathrm{G}}}\\\
\cdot~{}\int_{\mathnormal{W}^{\star}}\updelta(\boldsymbol{\mathnormal{r}}_{\mathrm{C}}-{\boldsymbol{\mathnormal{r}}_{\star}})\breve{\mathnormal{\Gamma}}_{\star}^{\mathrm{C}}(\mathnormal{\vartheta}_{\mathrm{C}}|{\boldsymbol{\mathnormal{w}}_{\star},\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}}){\mathnormal{\Psi}_{\mathrm{G}}^{\star}}({\boldsymbol{\mathnormal{w}}_{\star}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}},\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}})d\boldsymbol{\mathnormal{w}}_{\star}.$
(72)

Actual galactic metasocieties may be more complicated, even in the absence of
interstellar travel. Forgan (2017) argues that instead multiple “galactic
cliques” would arise, although they may in turn contact each other and fuse
through a long period of inter-negotiation. There may in fact be a patchwork
of different technosignature footprints in a galaxy, on intermediate spatial
scales.343434This might also be true of expansive metasocieties.

####  6.3.3 Expansive metasocieties

Common interstellar travel allows for galaxy-spanning metasocieties. In an
expansive metasociety, the metasociety spreads throughout a galaxy in a
(cosmologically) minuscule time, treated as instantaneous here.353535IIf a
galaxy is seeded with life by direct panspermia (Crick & Orgel, 1973), in turn
evolving ETIs, it could be viewed as hosting an expansive metasociety that
takes billions of years to develop. The technosignature population of the
galaxy effectively undergoes a “phase transition” (see Ćirković & Vukotić,
2008). Only one metasociety is allowed at a time in a galaxy. Like the
galactic club, the size and position of the metasociety is that of the galaxy
itself. A metasociety may persist once established as internal migration
reestablishes societies in locations that have fallen, in which case we can
set
${\uppsi\left[\mathnormal{\tau}_{\mathrm{M}}\right]_{\mathrm{G}}}=\updelta(\mathnormal{\tau}_{\mathrm{M}}-\mathnormal{\tau}_{\infty})$.
Or, perhaps some internal process causes them to vanish in mere millions of
years or less (as in Haqq-Misra & Baum, 2009; Prantzos, 2020). Note this
lifespan is likely far greater than that of individual societies; a galaxy-
wide catastrophe is required for a galaxy-spanning metasociety to collapse.

Each metasociety arises from a single stellar system, but because only one
metasociety exists at a time, the realized rate can have a complicated
dependence. If a new society evolves independently in an already established
metasociety, there are many possible outcomes: it could be assimilated into it
without affecting its properties appreciably, it could “rejuvenate” a
metasociety and extend its lifespan, or perhaps it could subsume the extant
metasociety with its own. There could be different metasocieties coexisting,
or aggressive metasocieties could inhibit the evolution of intelligence on all
other planets in the galaxy, and so on. Each of these affects the metasocietal
distribution in different ways. The key point is that this interaction induces
a _dependent_ thinning on the metasocietal point process.

A fairly simple subscenario is one where metasocieties are exclusive and
inhibitory: once a metasociety appears, it persists without interference for
its entire lifespan, then vanishes. No other metasociety can arise during its
reign. This means
${\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(\mathnormal{\vartheta}_{\mathrm{M}},\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}$
is a Bernoulli variable, $1$ if the galaxy has a metasociety at
$t=\mathnormal{\vartheta}_{\mathrm{M}}$ and $0$ if it is empty. To model these
metasocieties, define an auxiliary distribution describing their appearance
rate in the absence of inhibition, if
${\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(\mathnormal{\vartheta}_{\mathrm{M}})}$
just happened to be $0$:

${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}=0}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}=\updelta(\boldsymbol{\mathnormal{r}}_{\mathrm{M}}-{\boldsymbol{\mathnormal{r}}_{\mathrm{G}}})\updelta(\mathnormal{v}_{\mathrm{M}}-{\mathnormal{v}_{\mathrm{G}}}){\uppsi\left[\mathnormal{\tau}_{\mathrm{M}},\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}},\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{M}}\right]_{\mathrm{G}}}\\\
\cdot~{}\breve{\mathnormal{\Gamma}}_{\star}^{\mathrm{M}=0}(\mathnormal{\vartheta}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\star}^{\mathrm{M}}){\mathnormal{\Psi}_{\mathrm{G}}^{\star}}(\boldsymbol{\mathnormal{w}}_{\star}^{\mathrm{M}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{G}}}).$
(73)

with
$\breve{\mathnormal{\Gamma}}_{\star}^{\mathrm{M}=0}=d{\left\langle\mathnormal{N}_{\star}^{\mathrm{M}}|\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(\mathnormal{\vartheta}_{\mathrm{M}})=0\right\rangle_{\mathrm{G}}}/d\mathnormal{\vartheta}_{\mathrm{M}}$.
In an unpopulated galaxy, metasocieties originate around stars much like they
do in the classical scenario. Without any interaction, their population is
Poisson given the galaxy’s properties. Hence, there is again a stellar
distribution function and a rate per star. This counterfactual distribution
can be marginalized to get an effective rate of appearance
${\breve{\mathnormal{\Gamma}}_{\mathrm{G}}^{\mathrm{M}=0}}\equiv
d\left\langle{\mathnormal{N}_{\mathrm{G}}^{\mathrm{M}}|\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(\mathnormal{\vartheta}_{\mathrm{M}})=0}\right\rangle/d\mathnormal{\vartheta}_{\mathrm{M}}$.
If extant metasocieties have lifespans unaffected by any independent “second
origins” in their domains, then the distribution function is

${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}=(1-\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(\mathnormal{\vartheta}_{\mathrm{M}})}\right\rangle){\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}=0}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}.$
(74)

It is found by integrating ${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}}}$
with origin time $\mathnormal{\vartheta}_{\mathrm{M}}$ restricted by duration
$\mathnormal{\tau}_{\mathrm{M}}$
($t-\mathnormal{\tau}_{\mathrm{M}}\leq\mathnormal{\vartheta}_{\mathrm{M}}\leq
t$; section 3.8). Differentiating by $\mathnormal{\vartheta}_{\mathrm{M}}$
turns this into a partial differential equation.

A few simple results are evident. When every expansive metasociety is
persistent, $\mathnormal{\tau}_{\mathrm{M}}\to\mathnormal{\tau}_{\infty}$. A
galaxy then has an expansive metasociety if and only if one has ever appeared
any time during its history. The only way it lacks one is if no society
develops during any interval of time. Because the first society in a realized
galaxy arises according to a Poisson process, the time until one appears has
an exponential distribution governed by
${\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}=0}\equiv\breve{\mathnormal{\Gamma}}_{\mathrm{G}}^{\mathrm{M}=0}/\left\langle\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}(\mathnormal{\vartheta}_{\mathrm{M}})\right\rangle}$.
The mean number of metasocieties is

$\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}}(t)\right\rangle=1-\exp\left(-\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}}(t)\right\rangle{\mathnormal{\Xi}_{\mathrm{G}}^{\mathrm{M}=0}}(t{|\boldsymbol{\mathnormal{w}}_{\mathrm{G}}})\right),$
(75)

defined in terms of the effective abundance per star

${\mathnormal{\Xi}_{\mathrm{G}}^{\mathrm{M}=0}}(t{|\boldsymbol{\mathnormal{w}}_{\mathrm{G}}})=\frac{1}{\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}}(t)\right\rangle}\int_{{\mathnormal{\vartheta}_{\mathrm{G}}}}^{t}{\breve{\mathnormal{\Gamma}}_{\mathrm{G}}^{\mathrm{M}=0}}(\mathnormal{\vartheta}_{\mathrm{M}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{G}}})d\mathnormal{\vartheta}_{\mathrm{M}},$
(76)

in turn using the effective rate found by marginalizing
${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}=0}}$.

But what if metasocieties are short-lived compared to the galaxy’s age? If
${\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}=0}}(\mathnormal{\vartheta}_{\mathrm{M}})$
is roughly constant given a $\boldsymbol{\mathnormal{w}}_{\mathrm{G}}$, then
the evolution should be described by

$\frac{d\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}}{(t)}\right\rangle}{dt}\approx{\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}=0}}\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}}\right\rangle(1-\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}\right\rangle)-\frac{\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}\right\rangle}{{\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}}},$
(77)

with the first term on the right side describing the modified appearance rate,
and the second term describing their disappearance. The natural equilibrium
number of metasocieties is

$\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}\right\rangle=[1+({\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}=0}}\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\star}(t)}\right\rangle{\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}})^{-1}]^{-1}.$
(78)

The convergence to $1$ in equation 78 is slower because all galaxies start
with ${\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}=0$. If
${\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}}$ is
long on cosmological scales, then there has not been time to reach the second
“gap” between metasocieties – each galaxy has either avoided having any or is
in the era of its first metasociety. In contrast, if
${\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}}$ is
short, then each galaxy has already had a random sequence of metasocieties and
gaps. Equation 78 thus describes the duty cycle of expansive metasocieties
once the transient initial condition has vanished. I adopt equation 75 unless
otherwise stated, but to order of magnitude the abundance of metasocieties
should be the same.

Expansive metasocieties differ from galactic clubs in that the number of
communicative societies saturates at a very high equilibrium level once they
become established – potentially in the billions or more. We have a
metasociety-level societal distribution of

${\mathnormal{\Psi}_{\mathrm{M}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{M}})=\mathnormal{\Gamma}_{\mathrm{M}}^{\mathrm{C}}(\mathnormal{\vartheta}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}\uppsi\left[\mathnormal{\tau}_{\mathrm{C}},\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{C}}\right]_{\mathrm{M}}\\\
{\frac{d\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{M}}^{\star}(\mathnormal{\vartheta}_{\mathrm{C}})}\right\rangle}{d\boldsymbol{\mathnormal{r}}_{\star}}(\boldsymbol{\mathnormal{r}}_{\mathrm{C}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{M}}}).}$
(79)

Furthermore, ${\mathnormal{\Gamma}_{\mathrm{M}}^{\mathrm{C}}}$ is not only
expected to be larger than but _independent_ of
${\mathnormal{\Gamma}_{\mathrm{G}}^{\mathrm{M}=0}}$. The number of
communicative societies in an expansive metasociety is treated separately.
They form a Poisson point process, dependent on the metasociety’s properties:

${\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}\sim\mathrm{Pois}(\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle)$
(80)

with
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle\approx{\mathnormal{\Gamma}_{\mathrm{M}}^{\mathrm{C}}\left\langle\mathnormal{\tau}_{\mathrm{C}}\right\rangle_{\mathrm{M}}\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\star}\right\rangle}$
when $\mathsf{T}_{\mathsf{x}}\ll\mathnormal{\tau}_{\mathrm{C}}$. This means
${\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{C}}(t)}$ has a mixture
distribution, with a
$(1-\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}{(t)}}\right\rangle)$
probability of being $0$ and a
$\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}\right\rangle$
probability of being
${\mathnormal{N}_{\mathsf{t},\mathrm{M}}^{\mathrm{C}}(t)}$, yielding:

$\mathnormal{P}({\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{C}}(t)}\geq
1)=\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}\right\rangle[1-\exp(-{\left\langle\mathnormal{N}_{\mathsf{t},\mathrm{M}}^{\mathrm{C}}(t)\right\rangle_{\mathrm{G}}})].$
(81)

The independence results in interesting effects: in particular, when
$\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{M}}(t)}\right\rangle\ll
1$, the probability of detecting a broadcast can vary quadratically with the
number of stars in a galaxy (Section 12).

The qualitative characteristics of galactic clubs and persistent expansive
metasocieties converge when
$\left\langle{\mathnormal{N}_{\mathrm{G}}^{\mathrm{M}}}\right\rangle$
approaches $1$. In both cases, metasociety-wide traits can be treated as
galactic traits and societies form a Poisson point process in a realized
galaxy. A galactic club can even be regarded as an expansive metasociety that
spreads through telecommunications instead of settlement, albeit with much
lower population densities in the absence of replication. A single metasociety
approximation, with ${\mathnormal{N}_{\mathrm{G}}^{\mathrm{M}}}=1$, serves to
work for both scenarios when ETIs are not very rare.

###  6.4 The Drake Equation in the formalism

The Drake equation applies to the classical scenario when the galaxy is in a
steady state (Drake, 1965; Glade et al., 2012). New stars are formed at a
constant rate, and as they age over billions of years, some fraction of those
host single ETI societies of limited lifespan. To implement it simplistically,
if an ETI evolves, it must appear exactly $\overline{\mathnormal{t}}$ after
the star’s birth: the delay-time distribution is

$\breve{\mathnormal{\Gamma}}_{\star}^{\mathrm{M}}=\mathnormal{f}_{\star}^{\mathrm{M}}\updelta((\mathnormal{\vartheta}_{\mathrm{M}}-\vartheta_{\star})-\overline{\mathnormal{t}}).$
(82)

Stars need to maintain a stable luminosity long enough for intelligence to
evolve on Earth-like planets, excluding massive stars. However, let us imagine
for simplicity that all stars with
$\mathnormal{m}_{\star}\leq\mathnormal{m}_{\star}^{H}$ are equally likely to
host an ETI. This probability $\mathnormal{f}_{\star}^{\mathrm{M}}$ includes
all terms in the Drake equation relating to the number of planets,
habitability, and the evolution of life and intelligence. Then we can apply
the above to the classical metasociety distribution (equation 73).

We want to calculate the number of communicative societies existing at any one
time in the galaxy. First, let us calculate the galactic-level societal
distribution, using equation 15. We convolve
$\mathnormal{\Psi}_{\mathrm{M}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{M}})$
with
${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{M}}}(\boldsymbol{\mathnormal{w}}_{\mathrm{M}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{G}}})$.
Directly plugging these into that equation yields a formidable integral over
all stellar and metasocietal parameters. Actually, most of the variables are
effectively nuisance parameters, and the expression is simplified by the many
delta functions and the separable integrals. In the end, we are left with

${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{C}}(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}={\left\langle\mathnormal{f}_{\star}^{\mathrm{M}}\right\rangle_{\mathrm{G}}}{\left\langle\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}\right\rangle_{\mathrm{G}}\uppsi\left[\boldsymbol{\mathnormal{r}}_{\star}^{\mathrm{M}}\right]_{\mathrm{G}}(\boldsymbol{\mathnormal{r}}_{\mathrm{C}})\uppsi\left[\mathnormal{\tau}_{\mathrm{M}}\right]_{\mathrm{G}}(\mathnormal{\tau}_{\mathrm{C}})}\\\
\cdot~{}{\uppsi\left[\boldsymbol{\mathnormal{w}}^{\prime}_{\mathrm{C}}\right]_{\mathrm{G}}}{\breve{\mathnormal{\Gamma}}^{\star}_{\mathrm{G}}(\mathnormal{\vartheta}_{\mathrm{M}}-\overline{\mathnormal{t}})}F[\mathnormal{m}_{\star}^{\mathrm{M}}](\mathnormal{m}_{\star}^{H}).$
(83)

In other words, the communicative societies trace the stars in space, each
existing for the lifespan of its metasociety, and they appear at a rate
proportional to the star formation rate $\overline{\mathnormal{t}}$ ago,
including only stars that last long enough for ETIs to evolve.

The expected number of currently active communicative societies
$\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{C}}}(t)\right\rangle$
is the integral of ${\mathnormal{\Psi}_{\mathrm{G}}^{\mathrm{C}}}$ with
$t-\mathnormal{\tau}_{\mathrm{C}}\leq\mathnormal{\vartheta}_{\mathrm{C}}\leq
t$ (section 3.8). Now let us suppose that the star formation rate of the
galaxy has been constant, that (meta)societies are always much younger than
the galaxy, and that
$F[\mathnormal{m}_{\star}^{\mathrm{M}}](\mathnormal{m}_{\star}^{H})\approx 1$
because most stars are low mass and likely to survive for the required
billions of years. Then we finally get,

$\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{G}}^{\mathrm{C}}}(t)\right\rangle\approx{\breve{\mathnormal{\Gamma}}^{\star}_{\mathrm{G}}}{\left\langle\mathnormal{f}_{\star}^{\mathrm{M}}\right\rangle_{\mathrm{G}}}{\left\langle\mathnormal{f}^{\mathrm{C}}_{\mathrm{M}}\right\rangle_{\mathrm{G}}\left\langle\mathnormal{\tau}_{\mathrm{M}}\right\rangle_{\mathrm{G}}}.$
(84)

This is the Drake equation rewritten in the formalism.

Including different delay-time distributions, galactic habitability,
dependence on stellar mass, and other effects is fairly straightforward. We
simply change the
$d{\left\langle\mathnormal{N}_{\star}^{\mathrm{M}}\right\rangle_{\mathrm{G}}}/d{}^{\star}_{\mathrm{M}}$
distribution to include dependencies on these parameters of the host star. A
spread in delay times is necessary when treating quiescent galaxies where the
star formation rate was quenched about $10$ billion years ago.

##  7 Broadcasts: General considerations

A broadcast is an artificial release of energy at a discrete site through some
specified time and frequency range (see Tables 6–8 for notation). Not every
technosignature is a broadcast: unpowered artifacts in the solar system
(Freitas & Valdes 1985; Rose & Wright 2004; Davies & Wagner 2013) and
anomalous atmospheric compositions resulting from industrial pollution
(Whitmire & Wright, 1980; Lin et al., 2014) are not broadcasts. On the other
hand, not every broadcast need be an attempt at communication – they include
directed energy transmission for power beaming (Inoue & Yokoo, 2011; Benford &
Benford, 2016), propulsion (Lingam & Loeb, 2017), or remote sensing (Scheffer,
2014). Broadly speaking, waste heat from megastructures (Dyson, 1960) and
exhaust radiation from vehicles (Harris, 1986) are broadcasts too.

The modulation of starlight by astronomical-scale megastructures (Arnold 2005;
Chennamangalam et al. 2015; Wright et al. 2016; Zackrisson et al. 2018; Lacki
2019; Suazo et al. 2022) is an interesting case – here, the technosignature is
the blocking of an energy release that normally would happen. These might be
viewed as negative energy broadcasts. Searches for galactic-scale obscuration
of starlight (Annis, 1999; Zackrisson et al., 2015) use a collective bound as
in Paper II, much as waste heat searches look for the positive energy release
from the reprocessing of this missing starlight.

###  7.1 Basic parameters to describe broadcasts

The basic considerations for whether a broadcast is detectable are where it
is, how bright it becomes, when it happens, and how it behaves. The first
question is answered by the broadcast position
$\boldsymbol{\mathnormal{r}}_{\mathrm{B}}$.

Fundamentally, the brightness is controlled by the total energy released
$\mathnormal{\epsilon}_{\mathrm{B}}$ and its distribution into the
transmitter’s sky over solid angle $\mathnormal{\omega}_{\mathrm{B}}$ (refer
to Table 8 for emission-related variables). These can be considered
separately, but I use the total effective isotropic energy
$\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}\equiv 4\pi
d\mathnormal{\epsilon}_{\mathrm{B}}/d\mathnormal{\omega}_{\mathrm{B}}$ of the
broadcast to describe the brightness, evaluated toward the direction of the
observer. In the absence of beaming,
$\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}=\mathnormal{\epsilon}_{\mathrm{B}}$.
When broadcasts are beamed, the
$\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}$ distribution has a large peak
near zero for off-axis broadcasts. Yet even on-axis beamed broadcasts may have
a wide range of $\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}$ because of
different beaming angles or simply different
$\mathnormal{\epsilon}_{\mathrm{B}}$. Previous SETI works have essentially
considered two basic classes of
$\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}$ distributions – the
monoenergetic -distribution with a characteristic
$\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}$ and power-law distributions
(e.g., Drake et al. 1973; Gulkis 1985; Dreher 2004). Both narrow and broad
distributions seem plausible at this point, with the former favored by
possible engineering constraints on deliberate transmissions and the latter
motivated by the possible diversity of ETI capabilities and goals. Each
broadcast’s emission may also have distinct degrees and states of
polarization. The polarization properties are encapsulated in
$\mathnormal{\varpi}_{\mathrm{B}}$, which should be understood as a Stokes
vector or similar representation.

Surveys are sensitive to particular kinds of time-frequency properties –
spectral lines or pulses, for example. Although the time-frequency behavior
may be endlessly complex, I model it with five basic parameters. Broadcasts
are limited in time, with a source-frame duration
$\mathnormal{\tau}_{\mathrm{B}}$, and in frequency, with a constant source-
frame instantaneous bandwidth $\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}$.
Broadcasts also start at times $\mathnormal{\vartheta}_{\mathrm{B}}$ and are
centered on source-frame frequencies $\mathnormal{\upsilon}_{\mathrm{B}}$.
Finally, a drift rate $\mathnormal{\delta}_{\mathrm{B}}$ describes the
skewness of the broadcast in the time-frequency space. The broadcast is
entirely contained in the time range
$[\mathnormal{\vartheta}_{\mathrm{B}},\mathnormal{\vartheta}_{\mathrm{B}}+\mathnormal{\tau}_{\mathrm{B}}]$
and frequency range
$[\mathnormal{\upsilon}_{\mathrm{B}}-\mathnormal{\beta}_{\mathrm{B}}/2,\mathnormal{\upsilon}_{\mathrm{B}}+\mathnormal{\beta}_{\mathrm{B}}/2]$,
where the total bandwidth $\mathnormal{\beta}_{\mathrm{B}}$ is derived from
$\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}$ and
$\mathnormal{\delta}_{\mathrm{B}}$
($\mathnormal{\beta}_{\mathrm{B}}=\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}+\mathnormal{\tau}_{\mathrm{B}}|\mathnormal{\delta}_{\mathrm{B}}|$
when they are constant).

These quantities define a haystack in which broadcasts are scattered. Most of
them match the quantities defining haystacks in Tarter (2007) and Wright et
al. (2018). A separate modulation parameter is missing here, not being
directly relevant for energy or photon detection, but bandwidth and drift rate
both can be considered kinds of modulation. The main differences are in the
treatment of time, reduced to a single quantity in previous works. Wright et
al. (2018) interpret this as a repetition period, while the duty cycle and the
longevity of the transmitter are ignored. Including periodicity would
necessitate adding at least one more dimension to the ones here (see
discussion in Section 4.1). Bursts repeating at random intervals (Kipping &
Gray, 2022), however, require no new dimensions – the mean rate is simply
considered a property of the hosting society.

###  7.2 Selection of broadcasts

When is a broadcast sampled? Primary considerations are whether a broadcast
“touches” the window in physical spacetime (Section 3.8) and frequency, with a
“cross section” set by a duration and bandwidth. A broadcast has an effective
bandwidth $\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}$ during window
${\mathsf{x}}$, a quantity accounting for a broadcast’s spectral evolution. In
some cases, the broadcast’s spectrum is unchanging (like the box model;
section 9), in which case we can simply use
$\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}={\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}}$.
For the purposes of selection, I adopt

$\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}=\operatorname{diam}\left\\{\nu\Big{|}{\frac{d^{2}\mathring{\mathnormal{\epsilon}}_{\mathsf{x}|\mathrm{B}}}{dtd\nu}}(t,\nu|\boldsymbol{\mathnormal{w}}_{\mathrm{B}})>0\right\\}$
(85)

where
$\operatorname{diam}\mathnormal{A}={\max[}|a_{1}-a_{2}|\,|a_{1},a_{2}\in\mathnormal{A}{]}$.363636$\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}$
and $\mathnormal{\beta}_{\mathsf{x}|\mathrm{B}}$ are distinct – the latter
accounts for the frequency, sky field, and so on as well in restricting the
bandwidth. A broadcast at a frequency never covered by ${\mathsf{x}}$ has
$\mathnormal{\beta}_{\mathsf{x}|\mathrm{B}}{=0}$ and a drifting signal that
just barely “touches” ${\mathsf{x}}$ has
$\mathnormal{\beta}_{\mathsf{x}|\mathrm{B}}\ll\mathsf{B}_{\mathsf{x}}$, but
neither situation affects $\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}$. In
all cases,
$\mathnormal{\beta}_{\mathrm{B}}\geq\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}\geq\mathnormal{\beta}_{\mathsf{x}|\mathrm{B}}$.

I regard the selection of a broadcast by window ${\mathsf{x}}$ as a binary
decision based on its time and frequency properties:

$\mathsf{P}_{\mathsf{x}}^{\mathrm{B}}(\boldsymbol{\mathnormal{w}}_{\mathrm{B}})=\mathbb{I}[-\mathnormal{\tau}_{\mathrm{B}}\leq\mathnormal{\vartheta}_{\mathrm{B}}-\mathsf{\Theta}_{\mathsf{x}}\leq\mathsf{T}_{\mathsf{x}}]\\\
\cdot~{}\mathbb{I}[|\mathnormal{\upsilon}_{\mathrm{B}}-\mathsf{\Upsilon}_{\mathsf{x}}|\leq(1/2)(\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}+\mathsf{B}_{\mathsf{x}})]$
(86)

The presumption is that the broadcast’s ancestors also are chosen by
${\mathsf{x}}$, as required (Section 3.2.3), because if the broadcast is
active at some time and place, so are all its hosts – so ${\mathsf{x}}$ must
not impose any filters on characteristics specific to society, metasociety, or
galaxy. Polarization characteristics are not considered, because of the likely
crosstalk between the broadcast’s polarization and the observed polarizations,
at least for linear polarization.

###  7.3 Energy and photons emitted by broadcasts

A broadcast is detected through its emission (Table 8). This can be measured
in various forms, including energy and photons. Each broadcast has an
(effective isotropic) energy output per polarization per unit time and
frequency

$\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}(t,\nu,p)\equiv\frac{d^{2}\mathring{\mathnormal{\epsilon}}_{p;\mathrm{B}}}{dtd\nu}(t,\nu,p)=\mathnormal{\zeta}_{\mathsf{p}|\mathrm{B}}(p)\mathring{\mathnormal{\ell}}_{\nu;\mathrm{B}}(t,\nu).$
(87)

I assume that the degree and state of polarization are independent of time and
frequency. Thus, I separate $\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}$
into the fraction of energy emitted in polarization $p$,
$\mathnormal{\zeta}_{\mathsf{p}|\mathrm{B}}(p)$, and the spectral luminosity,
$\mathring{\mathnormal{\ell}}_{\nu;\mathrm{B}}$.373737For example, a broadcast
with linear polarization fraction
$\mathnormal{\varpi}_{\mathrm{lin};\mathrm{B}}$ has
$\mathnormal{\zeta}_{\mathsf{p}|\mathrm{B}}(p_{\mathrm{lin}})=(1/2)(1-\mathnormal{\varpi}_{\mathrm{lin};\mathrm{B}})+\mathnormal{\varpi}_{\mathrm{lin};\mathrm{B}}\cos^{2}{\phi_{p_{\mathrm{lin}};\mathrm{B}}}$,
where ${\phi_{p_{\mathrm{lin}};\mathrm{B}}}$ is the relative angle between
broadcast’s polarization angle and $p_{\mathrm{lin}}$. The same basic ideas
can be applied to broadcasts in other messengers, like neutrinos and gravity
waves.

Of course, we do not collect all of the energy emitted by a broadcast, only
that which falls on our instruments within a time and frequency window in set
of observed polarizations. Out of a total effective isotropic emission
$\mathring{\mathnormal{\varrho}}_{\mathrm{B}}$, the broadcast releases only a
limited amount $\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}$
within these constraints, zero if the broadcast is not in the sample. The
amount of energy released within the time, frequency, and polarization
constraints for ${\mathsf{x}}$ is

$\mathring{\mathnormal{\epsilon}}_{\mathsf{x}|\mathrm{B}}=\int_{\mathsf{\Upsilon}_{\mathsf{x}}-\mathsf{B}_{\mathsf{x}}/2}^{\mathsf{\Upsilon}_{\mathsf{x}}+\mathsf{B}_{\mathsf{x}}/2}\int_{\mathsf{\Theta}_{\mathsf{x}}}^{\mathsf{\Theta}_{\mathsf{x}}+\mathsf{T}_{\mathsf{x}}}\sum_{p\in\mathsf{\Pi}_{\mathsf{x}}}\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}(t,\nu,p)dtd\nu$
(88)

When the polarization properties are independent of time and frequency, the
sum over polarizations is replaced by a single factor
$\mathnormal{\zeta}_{\mathsf{x}|\mathrm{B}}$, such that

${\mathnormal{\zeta}_{\mathsf{x}|\mathrm{B}}=\sum_{p\in\mathsf{\Pi}_{\mathsf{x}}}\mathnormal{\zeta}_{\mathsf{p}|\mathrm{B}}(p|\mathnormal{\varpi}_{\mathrm{B}}).}$
(89)

Note that the frequency and time variables are all in the _source
frame_.383838An object at redshift ${z}$ is observed at Earth to have
$\mathsf{\Upsilon}_{\mathsf{x}}=\mathsf{\Upsilon}_{\mathsf{x}}^{\oplus}(1+{z})$,
$\mathsf{B}_{\mathsf{x}}=\mathsf{B}_{\mathsf{x}}^{\oplus}(1+{z})$, and
$\mathsf{T}_{\mathsf{x}}=\mathsf{T}_{\mathsf{x}}^{\oplus}/(1+{z})$.

When dealing with photon detectors, the relevant quantity is the number of
photons emitted within the observational window:

$\mathring{\mathnormal{q}}_{\mathsf{x}|\mathrm{B}}\equiv\int_{\mathsf{\Upsilon}_{\mathsf{x}}-\mathsf{B}_{\mathsf{x}}/2}^{\mathsf{\Upsilon}_{\mathsf{x}}+\mathsf{B}_{\mathsf{x}}/2}\int_{\mathsf{\Theta}_{\mathsf{x}}}^{\mathsf{\Theta}_{\mathsf{x}}+\mathsf{T}_{\mathsf{x}}}\sum_{p\in\mathsf{\Pi}_{\mathsf{x}}}\frac{\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}(t,\nu,p)}{h\nu}dtd\nu,$
(90)

where $h$ is Planck’s constant.

Even though the time domain and frequency domain amplitudes are related by the
Fourier transform, $\mathring{\mathnormal{\ell}}_{\nu;\mathrm{B}}$ can have
extremely complex dependence on time and frequency.393939Formally,
$\mathnormal{\beta}_{\mathrm{B}}$ and $\mathnormal{\tau}_{\mathrm{B}}$ cannot
both be finite; applying a rectangular window to one domain convolves the
other by a $\operatorname{sinc}$ function that is nonzero over an infinite
range. Many observations integrate over
$\mathsf{T}_{\mathsf{o}}\mathsf{B}_{\mathsf{o}}\gg 1$, resulting in negligible
leakage. These include all practical observations in the optical or higher
energies, and “filterbank” data products in radio SETI which also employ
polyphase filterbank techniques to reduce sidebands (Price, 2021). This
motivates the later use of the box and chord models as simplifications.

Table 8: Notation for emission quantities

Quantity | Arbitrary | Energy | Power | Photons | Note
---|---|---|---|---|---
Emission from broadcast | $\mathnormal{\varrho}_{\mathrm{B}}$ | $\mathnormal{\epsilon}_{\mathrm{B}}$ | $\mathnormal{\ell}_{\mathrm{B}}$ | $\mathnormal{q}_{\mathrm{B}}$ | a, b
Effective isotropic emission from broadcast | $\mathring{\mathnormal{\varrho}}_{\mathrm{B}}$ | $\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}$ | $\mathring{\mathnormal{\ell}}_{\mathrm{B}}$ | $\mathring{\mathnormal{q}}_{\mathrm{B}}$ | b
…per unit frequency | $\mathring{\mathnormal{\varrho}}_{\nu;\mathrm{B}}$ | $\mathring{\mathnormal{\epsilon}}_{\nu;\mathrm{B}}$ | $\mathring{\mathnormal{\ell}}_{\nu;\mathrm{B}}$ | $\mathring{\mathnormal{q}}_{\nu;\mathrm{B}}$ | c
…per polarization $p$ | $\mathring{\mathnormal{\varrho}}_{p;\mathrm{B}}$ | $\mathring{\mathnormal{\epsilon}}_{p;\mathrm{B}}$ | $\mathring{\mathnormal{\ell}}_{p;\mathrm{B}}$ | $\mathring{\mathnormal{q}}_{p;\mathrm{B}}$ | c
…per unit frequency per polarization $p$ | $\mathring{\mathnormal{\varrho}}_{\nu,p;\mathrm{B}}$ | $\mathring{\mathnormal{\epsilon}}_{\nu,p;\mathrm{B}}$ | $\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}$ | $\mathring{\mathnormal{q}}_{\nu,p;\mathrm{B}}$ | c
…occurring during window ${\mathsf{x}}$ | $\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}$ | $\mathring{\mathnormal{\epsilon}}_{\mathsf{x}|\mathrm{B}}$ | $\mathring{\mathnormal{\ell}}_{\mathsf{x}|\mathrm{B}}$ | $\mathring{\mathnormal{q}}_{\mathsf{x}|\mathrm{B}}$ | c
Effective isotropic aggregate emission from broadcast population selected by $(\mathsf{x},{\mathrm{J}})$ | ${\mathring{\mathnormal{R}}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathring{\mathnormal{E}}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathring{\mathnormal{L}}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathring{\mathnormal{Q}}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | b
Effective isotropic background emission in window ${\mathsf{x}}$ | $\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathfrak{n}}$ | $\mathring{\mathnormal{\epsilon}}_{\mathsf{x}|\mathfrak{n}}$ | $\mathring{\mathnormal{\ell}}_{\mathsf{x}|\mathfrak{n}}$ | $\mathring{\mathnormal{q}}_{\mathsf{x}|\mathfrak{n}}$ | b
Effective isotropic total emission in window ${\mathsf{x}}$ from object ${{\mathrm{J}}}$ | ${\mathring{\mathnormal{R}}_{\mathsf{x},\mathrm{J}}}$ | ${\mathring{\mathnormal{E}}_{\mathsf{x},\mathrm{J}}}$ | ${\mathring{\mathnormal{L}}_{\mathsf{x},\mathrm{J}}}$ | ${\mathring{\mathnormal{Q}}_{\mathsf{x},\mathrm{J}}}$ | b
Emission distance | ${d}_{\mathnormal{\varrho};\mathrm{B}}$ | ${d}_{\mathnormal{\epsilon};\mathrm{B}}$ | ${d}_{\mathnormal{\ell};\mathrm{B}}$ | ${d}_{\mathnormal{q};\mathrm{B}}$ |
Dilution factor for broadcast | ${y}_{\mathnormal{\varrho};\mathrm{B}}$ | ${y}_{\mathnormal{\epsilon};\mathrm{B}}$ | ${y}_{\mathnormal{\ell};\mathrm{B}}$ | ${y}_{\mathnormal{q};\mathrm{B}}$ |
Transmittance factor within window ${\mathsf{x}}$ | ${\chi}_{\mathnormal{\varrho};\mathsf{x}|\mathrm{B}}$ | ${\chi}_{\mathnormal{\epsilon};\mathsf{x}|\mathrm{B}}$ | (${\chi}_{\mathnormal{\epsilon};\mathsf{x}|\mathrm{B}}$) | ${\chi}_{\mathnormal{q};\mathsf{x}|\mathrm{B}}$ |
Fluence (flux) from single broadcast in window ${\mathsf{x}}$ | $\mathfrak{u}_{\mathsf{x}|\mathrm{B}}$ | $\mathfrak{h}_{\mathsf{x}|\mathrm{B}}$ | $\mathfrak{f}_{\mathsf{x}|\mathrm{B}}$ | $\mathfrak{g}_{\mathsf{x}|\mathrm{B}}$ | d
Aggregate fluence (flux) from population of broadcasts selected by $(\mathsf{x},{\mathrm{J}})$ | ${\mathfrak{U}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathfrak{H}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathfrak{F}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathfrak{G}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | d
Background fluence (flux) in window ${\mathsf{x}}$ | $\mathfrak{u}_{\mathsf{x}|\mathfrak{n}}$ | $\mathfrak{h}_{\mathsf{x}|\mathfrak{n}}$ | $\mathfrak{f}_{\mathsf{x}|\mathfrak{n}}$ | $\mathfrak{g}_{\mathsf{x}|\mathfrak{n}}$ | d
Total fluence (flux) including background in window ${\mathsf{x}}$ from object ${{\mathrm{J}}}$ | ${\mathfrak{U}_{\mathsf{x},\mathrm{J}}}$ | ${\mathfrak{H}_{\mathsf{x},\mathrm{J}}}$ | ${\mathfrak{F}_{\mathsf{x},\mathrm{J}}}$ | ${\mathfrak{G}_{\mathsf{x},\mathrm{J}}}$ | d
Instrument-measured quantity for single broadcast in window ${\mathsf{x}}$ | $\mathfrak{m}_{\mathsf{x}|\mathrm{B}}$ | $\mathfrak{e}_{\mathsf{x}|\mathrm{B}}$ | $\mathfrak{l}_{\mathsf{x}|\mathrm{B}}$ | $\mathfrak{q}_{\mathsf{x}|\mathrm{B}}$ |
Aggregate measured quantity from population of broadcasts selected by $(\mathsf{x},{\mathrm{J}})$ | ${\mathfrak{M}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathfrak{E}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathfrak{L}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ | ${\mathfrak{Q}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ |
Background measured quantity in window ${\mathsf{x}}$ | $\mathfrak{m}_{\mathsf{x}|\mathfrak{n}}$ | $\mathfrak{e}_{\mathsf{x}|\mathfrak{n}}$ | $\mathfrak{l}_{\mathsf{x}|\mathfrak{n}}$ | $\mathfrak{q}_{\mathsf{x}|\mathfrak{n}}$ |
Total instrument-measured quantity in window ${\mathsf{x}}$ | $\mathfrak{M}_{\mathsf{x}}$ | $\mathfrak{E}_{\mathsf{x}}$ | $\mathfrak{L}_{\mathsf{x}}$ | $\mathfrak{Q}_{\mathsf{x}}$ |

aafootnotemark:

When no quantity window is given, the ALL window is assumed to apply; so
${\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}}$ is the total effective
isotropic energy released over the entire lifespan, frequency range, and all
polarizations from broadcast ${{\mathrm{B}}}$. bbfootnotetext: The variable
listed under power is the effective isotropic radiated power (EIRP).
ccfootnotetext: Any emission variable, aggregate or singleton, can be
substituted with these modifiers to the same effect. Thus,
${\mathring{\mathnormal{L}}_{p;\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ is the
aggregate single-polarization EIRP from the broadcasts in
${\mathnormal{\Sigma}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$;
$\mathfrak{f}_{\nu;\mathrm{B}}$ is the energy flux per unit frequency of a
single random broadcast ${\mathrm{B}}$, unfiltered by any window.
ddfootnotetext: The variable listed under power is energy flux; all others are
types of fluence.

###  7.4 Distances, fluences, and dilutions

Observable quantities depend not just on the intrinsic properties of the
broadcast, but where it is – its distance (${d}_{\mathrm{B}}$) and position in
the sky ($\boldsymbol{\theta}_{\mathrm{B}}$). The difference in redshift
between broadcasts in the same galaxy is insignificant, so I set them equal
(${z}_{\mathrm{B}}={z}_{\mathrm{G}}$). Variation in broadcast distances is
important for the Milky Way, which after all is another galaxy with some
population of broadcasts (even if it is empty aside from our own). For
extragalactic systems that are not Galactic satellites, however, the size of
the target galaxy is negligible compared to its distance, and we can
approximate the distance as that to the galaxy itself
(${d}_{\mathrm{B}}={d}_{\mathrm{G}}$).

Aside from the cuts imposed by the selections, only the emission incident on
our detector can be measured. The fluence
$\mathfrak{u}_{\mathsf{x}|\mathrm{B}}$ of a broadcast is the amount of
emission per unit area integrated over all combinations of time, frequency,
and polarization covered by window ${\mathsf{x}}$. Specific types of fluence
include the energy fluence $\mathfrak{h}_{\mathsf{x}|\mathrm{B}}$ where the
emission is quantified as energy and the photon fluence
$\mathfrak{g}_{\mathsf{x}|\mathrm{B}}$ which counts the expected number of
photons (Table 8).

The distance determines the flux and fluence of a broadcast with a given
amount of emission. The energy fluence can be found using the “energy
distance” ${d}_{\mathnormal{\epsilon}}=\sqrt{1+{z}}{d}_{M}$ which accounts for
redshift, correcting the transverse angular distance ${d}_{M}$ (see Hogg 1999
for how to calculate ${d}_{M}$). Likewise, the photon fluence uses a “photon
distance” ${d}_{\mathnormal{q}}$ equal to ${d}_{M}$.

The observed emission is also partly blocked by opacity from dust and gas on
the sightline, plus the Earth’s atmosphere, a range-limiting issue in optical
SETI (Howard et al., 2004). Only a fraction
${\chi}_{\mathnormal{\varrho};\mathsf{x}|\mathrm{B}}$ of
$\mathnormal{\varrho}$ emission remains after absorption and scattering, the
transmittance factor, which can vary with both position and frequency. The
frequency dependence means that windows at different frequencies have
different transmittances. At the usual radio frequencies observed in SETI,
however, ${\chi}_{\mathnormal{\varrho};\mathsf{x}|\mathrm{B}}$ is very near
$1$.

It is convenient to express the inverse-square law and redshift effects as a
“dilution factor,”
${y}_{\mathnormal{\varrho};\mathrm{B}}=1/(4\pi{{d}_{\mathnormal{\varrho};\mathrm{B}}}^{2})$.
For emission of type $\mathnormal{\varrho}$,

$\mathfrak{u}_{\mathsf{x}|\mathrm{B}}=\frac{\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}{{\chi}_{\mathnormal{\varrho};\mathsf{x}|\mathrm{B}}}}{4\pi{{d}_{\mathnormal{\varrho};\mathrm{B}}}^{2}}=\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}{{\chi}_{\mathnormal{\varrho};\mathsf{x}|\mathrm{B}}}{y}_{\mathnormal{\varrho};\mathrm{B}}.$
(91)

In principle, the $\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}$
and ${y}_{\mathnormal{\varrho};\mathrm{B}}$ are not independent – although the
intrinsic properties of the broadcast are modeled as independent, the cuts
depend on $\mathsf{T}_{\mathsf{x}}$ and $\mathsf{B}_{\mathsf{x}}$, which are
affected by redshift. Even the Doppler shifts of broadcasts should have a
slight effect on $\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}$.
Transmittance, when it is not near $1$, is likely to be much lower in regions
of high obscuration and thus is certainly not independent of
${y}_{\mathnormal{\varrho};\mathrm{B}}$. However, for practical purposes, with
the emission and dilution independent,

${\left\langle{\mathfrak{u}_{\mathsf{x}|\mathrm{B}}}^{n}\right\rangle_{\mathsf{x},\mathrm{J}}=\left\langle{\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}}\vphantom{\mathring{\mathnormal{\varrho}}_{\mathsf{x}|\mathrm{B}}}^{n}\right\rangle_{\mathsf{x},\mathrm{J}}\left\langle{{{\chi}_{\mathnormal{\varrho};\mathsf{x}|\mathrm{B}}}^{n}}{{y}_{\mathnormal{\varrho};\mathrm{B}}}^{n}\right\rangle_{\mathrm{J}}}$
(92)

for any arbitrary exponent $n$.

##  8 Populations of broadcasts

###  8.1 The broadcast distribution function

In this series, the broadcast distribution function takes the form

${\mathnormal{{\Psi}}_{\mathrm{J}}^{\mathrm{B}}(\boldsymbol{\mathnormal{w}}_{\mathrm{B}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})}=\frac{d^{8}\left\langle{\mathnormal{N}_{\mathrm{J}}^{\mathrm{B}}}\right\rangle}{d\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}d\mathnormal{\varpi}_{\mathrm{B}}d\mathnormal{\tau}_{\mathrm{B}}d\mathnormal{\vartheta}_{\mathrm{B}}d\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}d\mathnormal{\upsilon}_{\mathrm{B}}d\mathnormal{\delta}_{\mathrm{B}}{d\boldsymbol{\mathnormal{r}}_{\mathrm{B}}}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{B}}|\boldsymbol{\mathnormal{w}}_{\mathrm{J}})},$
(93)

using the variables identified in Section 7.1. The broadcast population is a
Cox point process, and the broadcast population of a realized society in
particular is a Poisson point process. The instantaneous rate of broadcasts
per frequency per star is especially useful:

${\mathnormal{\Lambda}_{\mathrm{J}}^{\mathrm{B}}}(t,\nu)\equiv{\frac{1}{\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{J}}^{\star}}(t)\right\rangle}\frac{d^{2}\left\langle{\mathnormal{N}_{\mathrm{J}}^{\mathrm{B}}}\right\rangle}{d\mathnormal{\vartheta}_{\mathrm{B}}d\mathnormal{\upsilon}_{\mathrm{B}}}(\mathnormal{\vartheta}_{\mathrm{B}}=t,\mathnormal{\upsilon}_{\mathrm{B}}=\nu)}{,}$
(94)

as is the frequency abundance per star,

${\mathnormal{Z}_{\mathrm{J}}^{\mathrm{B}}}(t,\nu)={\frac{1}{\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{J}}^{\star}}(t)\right\rangle}\frac{d\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{J}}^{\mathrm{B}}}(t)\right\rangle}{d\mathnormal{\upsilon}_{\mathrm{B}}}(\mathnormal{\upsilon}_{\mathrm{B}}=\nu).}$
(95)

The stellar rate ${\mathnormal{\Gamma}_{\mathrm{J}}^{\mathrm{B}}}$ and the
stellar abundance ${\mathnormal{\Xi}_{\mathrm{J}}^{\mathrm{B}}}$ are
additionally marginalized over $\mathnormal{\upsilon}_{\mathrm{B}}$, as
defined as in Section 3.2.4. It is more convenient to use the total abundances
and rates, the derivatives without dependence on
$\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{J}}^{\star}(\nu)}\right\rangle$,
when considering individual societies around single star systems:
${\breve{\mathnormal{\Lambda}}_{\mathrm{C}}^{\mathrm{B}}}\equiv
d^{2}\left\langle{\mathnormal{N}_{\mathrm{C}}^{\mathrm{B}}}\right\rangle/d\mathnormal{\vartheta}_{\mathrm{B}}d\mathnormal{\upsilon}_{\mathrm{B}}$,
${\breve{\mathnormal{Z}}_{\mathrm{C}}^{\mathrm{B}}}\equiv
d\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{C}}^{\mathrm{B}}(t)}\right\rangle/d\mathnormal{\upsilon}_{\mathrm{B}}$,
${\breve{\mathnormal{\Gamma}}_{\mathrm{C}}^{\mathrm{B}}}\equiv
d\left\langle{\mathnormal{N}_{\upnu,\mathrm{C}}^{\mathrm{B}}(\nu)}\right\rangle/d\mathnormal{\vartheta}_{\mathrm{B}}$,
${\breve{\mathnormal{\Xi}}_{\mathrm{C}}^{\mathrm{B}}}\equiv\left\langle{\mathnormal{N}_{\mathsf{t}\cap\upnu,\mathrm{C}}^{\mathrm{B}}(t,\nu)}\right\rangle$.

Specific examples of the broadcast distribution are given in Sections 9 and
10.

###  8.2 Interchangeability

When I compute properties of aggregate observables in this paper, I apply
results for compound Poisson random variables. However, to do this, the
variables being summed need to be identically distributed. Basically, each
society is assumed to have the “same” broadcast distribution as the others, an
“average” distribution. Likewise, each metasociety has the “same” societal and
broadcast distributions as any other in the galaxy. The _interchangeability_
assumption is that all distributions of an object type are the same, except
for translations in spacetime (because objects are located in different places
and start at different times). The broadcast distribution in every society
(regardless of metasociety) in galaxy ${\mathrm{G}}$ has the form:

$\mathnormal{{\Psi}}_{\mathrm{C}}^{\mathrm{B}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{B}}|\boldsymbol{\mathnormal{w}}_{\mathrm{C}})}=f_{{\mathrm{G}}}^{\mathrm{B}}(\mathring{\mathnormal{\epsilon}}_{\mathrm{B}},\mathnormal{\varpi}_{\mathrm{B}},\mathnormal{\tau}_{\mathrm{B}},\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}},\mathnormal{\upsilon}_{\mathrm{B}},\mathnormal{\delta}_{\mathrm{B}})\\\
\cdot\updelta(\boldsymbol{\mathnormal{r}}_{\mathrm{B}}-\boldsymbol{\mathnormal{r}}_{\mathrm{C}})\cdot~{}\mathbb{I}[0\leq\mathnormal{\vartheta}_{\mathrm{B}}-\mathnormal{\vartheta}_{\mathrm{C}}\leq\mathnormal{\tau}_{\mathrm{C}}],$
(96)

and every metasociety has a societal distribution

$\mathnormal{\Psi}_{\mathrm{M}}^{\mathrm{C}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{C}}|\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}=f_{{\mathrm{G}}}^{\mathrm{C}}(\mathnormal{\tau}_{\mathrm{C}})\cdot\mathbb{I}[0\leq\mathnormal{\vartheta}_{\mathrm{B}}-\mathnormal{\vartheta}_{\mathrm{C}}\leq\mathnormal{\tau}_{\mathrm{C}}],$
(97)

with fixed functions $f_{{\mathrm{G}}}^{\mathrm{B}}$ and
$f_{{\mathrm{G}}}^{\mathrm{C}}$.

Societies are selected at different points of their lifecycles, but for the
compound Poisson distribution to be applicable, the societal broadcast
distribution must be time-stationary. If, for example, societies lose interest
in communicating as they age, then young societies have a higher
${\mathnormal{\Lambda}_{\mathrm{C}}^{\mathrm{B}}}$ than older ones.
Interchangeability does not allow this. Thus there can be no further
dependence on
$\mathnormal{\vartheta}_{\mathrm{B}}-\mathnormal{\vartheta}_{\mathrm{C}}$ in
equation 96, as long as the broadcast is within bounds. Likewise, the societal
distribution is stationary. More generally, there cannot be any edge effects
wherein societies or metasocieties spend different amounts of time in the
selection window; thus all lifespans must either be much longer or much
shorter than the window’s duration.

Real societies and metasocieties may be incredibly diverse, each with its own
unique array of technosignatures. Societies and broadcasts are represented by
Cox processes (Section 2.5). Because these are generally nonergodic, a single
society or metasociety is a very poor representative of the ensemble
properties of the underlying distribution – we only see one realization of the
distribution, which itself is a random variable. If half of societies
broadcast in radio and the rest in gamma rays, and there is only a single
society that happens to be radio-broadcasting, we will not be able to “see”
the true diversity in broadcasts. The observed sample is thus subjected to
stochasticity when there are only a few societies. A large sample of societies
can sample the entire gamut of broadcasts. In that sense, all the broadcasts
can be pooled together into a galactic broadcast distribution, although the
effects of the “clumping” remain underestimated by the compound Poisson
distribution.

There are expressions for the mean and variances when dealing with diverse
populations, but they are very complicated because of the nesting selection-
relative averages and variances (equation 45). It is more insightful to work
directly with the distributions themselves when they are available (section
8.6).

Table 9 lists equations for some important quantities characterizing broadcast
populations, given the assumptions of independence and interchangeability.

###  8.3 The number of broadcasts

The number of broadcasts intercepted by ${\mathsf{x}}$ from a society
${\mathrm{C}}\in{\mathnormal{\Sigma}_{\mathsf{x},\mathrm{J}}^{\mathrm{C}}}$ is
$\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}(\boldsymbol{\mathnormal{w}}_{\mathrm{C}})$,
a Poisson random variable given $\boldsymbol{\mathnormal{w}}_{\mathrm{C}}$.
Its mean is given by the number of broadcasts that “touch” the ${\mathsf{x}}$
window:

$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle=\idotsint_{\mathsf{\Upsilon}_{\mathsf{x}}-(\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}+\mathsf{B}_{\mathsf{x}})/2}^{\mathsf{\Upsilon}_{\mathsf{x}}+(\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}+\mathsf{B}_{\mathsf{x}})/2}\int_{\mathsf{\Theta}_{\mathsf{x}}-\mathnormal{\tau}_{\mathrm{B}}}^{\mathsf{\Theta}_{\mathsf{x}}+\mathsf{T}_{\mathsf{x}}}\\\
{\mathnormal{{\Psi}}_{\mathrm{C}}^{\mathrm{B}}}(\boldsymbol{\mathnormal{w}}_{\mathrm{B}}{|\boldsymbol{\mathnormal{w}}_{\mathrm{C}}})d\mathnormal{\vartheta}_{\mathrm{B}}d\mathnormal{\upsilon}_{\mathrm{B}}d\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}d\mathnormal{\varpi}_{\mathrm{B}}d\mathnormal{\tau}_{\mathrm{B}}d\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}d\boldsymbol{\mathnormal{r}}_{\mathrm{B}}d\mathnormal{\delta}_{\mathrm{B}}.$
(98)

If societies are long-lived compared to $\mathnormal{\tau}_{\mathrm{B}}$, and
if ${\mathnormal{\Lambda}_{\mathrm{C}}^{\mathrm{B}}}$ varies slowly enough in
time and frequency compared to common values of
$\mathnormal{\tau}_{\mathrm{B}}$ and
$\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}$,

$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle\approx{\breve{\mathnormal{\Lambda}}_{\mathrm{C}}^{\mathrm{B}}}(\mathsf{\Theta}_{\mathsf{x}},\mathsf{\Upsilon}_{\mathsf{x}}){\cdot}(\mathsf{B}_{\mathsf{x}}+{\left\langle\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}\right\rangle_{\mathrm{C}}})(\mathsf{T}_{\mathsf{x}}+{\left\langle\mathnormal{\tau}_{\mathrm{B}}\right\rangle_{\mathrm{C}}}){.}$
(99)

Table 9: Equations for aggregate variables under standard assumptions

Variable | Galactic Club | Expansive Metasociety
---|---|---
Societal variables
$\mathnormal{P}({\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\geq 1)$ | $1-\exp\left(-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle\right)$ | $1-\exp\left(-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle\right)$
$\mathbb{V}\left[{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle$
$\left\langle{\mathnormal{K}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle{\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}}$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle{\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}}$
$\mathbb{V}\left[{\mathnormal{K}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle{\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}^{2}\right\rangle_{\mathrm{C}}}$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}}\right\rangle{\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}^{2}\right\rangle_{\mathrm{C}}}$
Metasocietal variables
$\mathnormal{P}({\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\geq 1)$ |  $\cdots$ | $1-\exp\left(-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle\right)$
$\mathbb{V}\left[{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right]$ |  $\cdots$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle$
$\left\langle{\mathnormal{K}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle$ |  $\cdots$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{C}}\right\rangle$
$\mathbb{V}\left[{\mathnormal{K}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right]$ |  $\cdots$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{C}}^{2}\right\rangle$
$\mathnormal{P}({\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\geq 1)$ |  $\cdots$ | $1-\exp\left[-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle\left(1-e^{-\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle}\right)\right]$
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\right\rangle$ |  $\cdots$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle$
$\mathbb{V}\left[{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\right]$ |  $\cdots$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\right\rangle\left[1+\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle\right]$
$\left\langle{\mathnormal{K}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\right\rangle$ |  $\cdots$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}$
$\mathbb{V}\left[{\mathnormal{K}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\right]$ |  $\cdots$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}}\right\rangle\left[\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}^{2}\right\rangle_{\mathrm{C}}+\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}^{2}\right]$
Galactic variables
$\mathnormal{P}({\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\geq 1)$ | $1-\exp\left(-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\right)$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle\left[1-\exp\left(-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\right)\right]$
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}\right\rangle$
$\mathbb{V}\left[{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\left[1+\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}\right\rangle\left(1-\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}\right\rangle\right)\right]$
$\left\langle{\mathnormal{K}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{C}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{C}}\right\rangle$
$\mathbb{V}\left[{\mathnormal{K}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{C}}^{2}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\left[\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{C}}^{2}\right\rangle+\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle(1-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle)\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{C}}\right\rangle^{2}\right]$
$\mathnormal{P}({\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\geq 1)$ | $1-\exp\left[-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\left(1-e^{-\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle}\right)\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle\left[1-\exp\left(-\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}\right\rangle\left(1-e^{-\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle}\right)\right)\right]$
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle$
$\mathbb{V}\left[{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\left[1+\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\left[1+\left(\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle+(1-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle)\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}\right\rangle\right)\right]$
$\left\langle{\mathnormal{K}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}$
$\mathbb{V}\left[{\mathnormal{K}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\left[\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}^{2}\right\rangle_{\mathrm{C}}+\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}^{2}\right]$ | $\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\left[\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}^{2}\right\rangle_{\mathrm{C}}+\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}^{2}\left(\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle+\left(1-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle\right)\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}\right\rangle\right)\right]$

Note. — These expressions are calculated under the assumptions of independence
and interchangeability. Note that
${\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}\right\rangle_{\mathrm{L}}}\equiv{\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{K}}^{\mathrm{J}}\right\rangle_{\mathsf{x},\mathrm{L}}}$
and
${\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathrm{K}}}\equiv{\left\langle\mathnormal{\kappa}_{\mathsf{x}|\mathrm{J}}\right\rangle_{\mathsf{x},\mathrm{K}}}$
by convention (section 3.4.2).

The number of broadcasts is a compound random sum, with means and variances
given in the table. As long as
$\mathnormal{\tau}_{\mathrm{B}}\ll\mathnormal{\tau}_{\mathrm{C}},\mathnormal{\tau}_{\mathrm{M}}$
and ${\mathnormal{\Lambda}_{\mathrm{G}}^{\mathrm{B}}}$ is slowly varying,

$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\approx{\mathnormal{\Lambda}_{\mathrm{G}}^{\mathrm{B}}}(\mathsf{\Theta}_{\mathsf{x}},\mathsf{\Upsilon}_{\mathsf{x}}){\cdot\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\star}\right\rangle}(\mathsf{B}_{\mathsf{x}}+{\left\langle\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}\right\rangle_{\mathrm{G}}})\\\
\cdot(\mathsf{T}_{\mathsf{x}}+{\left\langle\mathnormal{\tau}_{\mathrm{B}}\right\rangle_{\mathrm{G}}}).$
(100)

It is scaled by the mean rate that broadcasts occur per unit frequency per
star, which is

${\mathnormal{\Lambda}_{\mathrm{G}}^{\mathrm{B}}}=\begin{cases}{\mathnormal{\Xi}_{\mathrm{G}}^{\mathrm{C}}\left\langle\breve{\mathnormal{\Lambda}}_{\mathrm{C}}^{\mathrm{B}}\right\rangle_{\mathrm{G}}}&\text{(Galactic
club)}\\\
{\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}\right\rangle\left\langle\mathnormal{\Xi}_{\mathrm{M}}^{\mathrm{C}}\right\rangle_{\mathrm{G}}\left\langle\breve{\mathnormal{\Lambda}}_{\mathrm{C}}^{\mathrm{B}}\right\rangle_{\mathrm{G}}}&\text{(Expansive
interstellar)}\end{cases}$ (101)

This abundance is a key quantity in SETI. The “clumping” into societies and
metasocieties adds additional sampling variance, more so in the expansive
interstellar scenario where
${\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}$ is independent of
$\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}$. When fewer than one
broadcast is typically selected per society, the probability that one
broadcast is intercepted is

$P({\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\geq
1)\approx\begin{cases}1-\exp(-\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle)\\\
\hfill\text{(Galactic club)}\\\
\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle[1-\exp(-\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{B}}\right\rangle)]\\\
\hfill\text{(Expansive interstellar)}\end{cases}.$ (102)

In the opposite limit of many selected broadcasts per society, this
probability is simply the probability that one communicative society is
selected. The variance in the number of broadcasts per metasociety is then
mainly from the Poissonian clumping into societies, approaching
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{M}}^{\mathrm{C}}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle^{2}$
(full expressions are given in Table 9).

###  8.4 Aggregate emission and fluence

The total single-polarization spectral luminosity from the broadcasts in a
sample ${\mathnormal{\Sigma}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ is

${\mathring{\mathnormal{L}}_{\nu,p;\mathsf{x},\mathrm{J}}^{\mathrm{B}}}(t,\nu,p)={\sum_{\boldsymbol{\mathnormal{w}}_{\mathrm{B}}\in\mathnormal{\Sigma}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}}\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}(t,\nu,p)$
(103)

Of course, ${\mathnormal{\Sigma}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ itself
is random, even when ${\mathrm{J}}$ itself is specified. The usual conditions
for broadcast selection by the window ${\mathsf{x}}$ (equation 86) define the
bounds of integration to be used in equation 41. Under the interchangeability-
independence assumptions,

$\left\langle{\mathring{\mathnormal{L}}_{\nu,p;\mathsf{x},\mathrm{J}}^{\mathrm{B}}}(t,\nu,p)\right\rangle\approx{\mathnormal{\Lambda}_{\mathrm{J}}^{\mathrm{B}}}(t,\nu){\cdot\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{J}}^{\star}\right\rangle}{\left\langle\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}(t,\nu,p)\right\rangle_{\mathsf{x},\mathrm{J}}}\\\
\cdot~{}(\mathsf{B}_{\mathsf{x}}+{\left\langle\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}\right\rangle_{\mathrm{J}}})(\mathsf{T}_{\mathsf{x}}+{\left\langle\mathnormal{\tau}_{\mathrm{B}}\right\rangle_{\mathrm{J}}}).$
(104)

Strictly speaking, this is not necessarily a compound Poisson variable because
${\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}(\boldsymbol{\mathnormal{w}}_{\mathrm{G}})}$
is Bernoulli in the expansive metasociety scenario. It reaches the compound
Poisson limit when
$\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}\right\rangle=1$
and the galaxy as a whole can be identified with the metasociety (Table 9).

The aggregate fluence is a similar compound sum under these assumptions. A
fluence ${\mathfrak{U}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ of any type from
the sample is found by summing fluences $\mathfrak{u}_{\mathsf{x}|\mathrm{B}}$
of the same type from individual broadcasts in a sample. Examples include the
total energy fluence ${\mathfrak{H}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ and
total photon fluence ${\mathfrak{G}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ of
a sample, composed of the energy fluences
$\mathfrak{h}_{\mathsf{x}|\mathrm{B}}$ and photon fluences
$\mathfrak{g}_{\mathsf{x}|\mathrm{B}}$ of individual broadcasts. We have:

$\left\langle{\mathfrak{U}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}\right\rangle\approx{\mathnormal{\Lambda}_{\mathrm{J}}^{\mathrm{B}}(\mathsf{\Theta}_{\mathsf{x}},\mathsf{\Upsilon}_{\mathsf{x}})}{\cdot\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{J}}^{\star}\right\rangle}(\mathsf{B}_{\mathsf{x}}+{\left\langle\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}\right\rangle_{\mathrm{J}}})\\\
{\cdot(\mathsf{T}_{\mathsf{x}}+{\left\langle\mathnormal{\tau}_{\mathrm{B}}\right\rangle_{\mathrm{J}}})\left\langle\mathfrak{u}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{J}}}.$
(105)

The clumping of broadcasts into societies can have a big effect on the
variance of ${\mathfrak{U}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$ (Table 9).
When a window typically samples $\ll 1$ broadcast per society or the variance
in $\mathfrak{u}_{\mathsf{x}|\mathrm{B}}$ is sufficiently large, the variance
in an inhabited galaxy ${\mathrm{G}}$ with
${\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{M}}}=1$ approaches what we
would expect if ${\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}$ were
Poissonian. In the opposite limit,
$\mathbb{V}\left[{\mathfrak{U}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right]\approx\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}\right\rangle\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle\left\langle\mathfrak{u}_{\mathsf{x}|\mathrm{B}}\right\rangle_{\mathrm{C}}^{2}=\left\langle{\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{C}}}\right\rangle\left\langle\mathfrak{U}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle^{2}$.
The societies become a population of “standard candles,” and the fluctuations
are just shot noise in their number. This limit requires many broadcasts per
society just within a window, all becoming confused, which can be a tall order
(Paper II).

####  8.4.1 What do we expect the aggregate emission to look like?

Although we usually envision broadcasts to resemble isolated “spikes” in
$\mathring{\mathnormal{L}}_{\nu}$, on coarse enough time-frequency scales,
this roughness blurs out. The mean spectrum, as shown by equation 104, is
proportional to ${\mathnormal{\Lambda}_{\mathrm{G}}^{\mathrm{B}}}$ itself, as
long as it varies slowly enough and a mean
$\mathring{\mathnormal{\ell}}_{\nu,p;\mathrm{B}}$ exists. If there is no
“magic frequency” (unlike Drake & Sagan, 1973; Blair & Zadnik, 1993) or “magic
time” (unlike Pace & Walker, 1975; Makovetskii, 1977; Corbet, 2003; Nishino &
Seto, 2018) that ETIs in the target galaxy prefer to broadcast at,
${\mathnormal{\Lambda}_{\mathrm{G}}^{\mathrm{B}}}$ is most likely a smooth
function, perhaps a power law.

Supposing that the $\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}$
distribution has a mean, over large bandwidths and long durations the
aggregate emission converges to a constant luminosity source with a smooth
spectrum. In other words, it will resemble diffuse nonthermal emission
commonly seen in objects like active galactic nuclei. Like any population of
discrete sources (e.g., Tonry & Schneider, 1988), there are Poissonian
fluctuations in the emission, far greater than those from the immense number
of particles contributing to natural diffuse emission. With enough broadcasts
being added together, the fluctuations also become Gaussian by the central
limit theorem. This motivates use of the total emission of galaxies to set
constraints on broadcasts, as in Paper II.

If the $\mathring{\mathnormal{\epsilon}}_{\mathrm{B}}$ distribution has no
mean, or if broadcasts are so rare that confusion does not occur for any
feasible sample, the aggregate emission remains “spiky,” with strongly non-
Gaussian fluctuations that could be picked out by conventional search
strategies. Nonetheless, there still could be a background of fainter
broadcasts for which aggregate constraints still apply.

Even if there is a magic frequency, the broadcasts from an extended
metasociety may be smeared out in received frequency because of velocity
differences between the different transmitter sites. Then the aggregate
emission will appear much like natural line emission, although possibly at a
frequency corresponding to no natural transition. The luminosity of candidate
“lines” in the galaxy’s emission sets constraints on such broadcast
populations.

###  8.5 The diffuse approximation

Because societies are discrete, they clump broadcasts. However, including the
societal level in the tree greatly complicates the analysis, compounding the
Poissonian character of the number of broadcasts in collections of societies.
The _diffuse approximation_ ignores the discreteness of societies, instead
imagining the transmitters being spread diffusely across the galaxy. The
variance in ${\mathnormal{N}_{\mathsf{x},\mathrm{G}}^{\mathrm{B}}}$ is thus
underestimated. How large this correction is depends on the nature of the
discretization, but the diffuse approximation gives good results when there
are many societies, few of which emit a detectable broadcast
(${\left\langle\mathnormal{N}_{\mathsf{x},\mathrm{C}}^{\mathrm{B}}\right\rangle_{\mathrm{J}}}\ll
1$). This is appropriate when using very fine observations (e.g., narrow
channels for narrowband lines) except in the most extreme cases – far, far
into the confusion regime for heavily populated galaxies.

Practically speaking, the diffuse approximation treats the broadcasts as the
children of metasocieties (expansive metasociety scenario) or galaxies
(classical and Galactic club scenario). Thus, the diffuse approximation works
directly with the metasociety’s or galaxy’s broadcast distribution instead of
building those up from societies’ broadcast distributions. Just as the
societal distribution posits that societies trace stars (Section 5.3 & 6.3),
in the diffuse approximation, that dependence is shifted down to the broadcast
distribution. In the end, the broadcast distribution for the effective parent
${{\mathrm{J}}}$ in the diffuse approximation is found by substituting

$\breve{\mathnormal{\Lambda}}_{\mathrm{C}}^{\mathrm{B}}\updelta(\boldsymbol{\mathnormal{r}}_{\mathrm{B}}-\boldsymbol{\mathnormal{r}}_{\mathrm{C}})\to{\mathnormal{\Lambda}_{\mathrm{J}}^{\mathrm{B}}}\frac{d\left\langle{\mathnormal{N}_{\mathsf{t},\mathrm{J}}^{\star}}\right\rangle}{d\boldsymbol{\mathnormal{r}}_{\star}}(\boldsymbol{\mathnormal{r}}_{\mathrm{B}});$
(106)

the substitution can also apply to the broadcast abundance per frequency
variable.

###  8.6 Aggregate luminosities in diverse, noninterchangeable hosts: an
example

Diversity among host objects increases the sampling variance of aggregate
quantities, and the variance increases as it manifests in higher-level
ancestors. This section presents a simple example: how would the cosmic-
relative variance in an aggregate metasocietal luminosity
$\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}$ change if the
luminosities were shared among different types of hosts? To keep the example
focused, the number distributions
$\mathnormal{N}_{\mathrm{C}}^{\mathrm{B}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{C}})}\sim\mathrm{Pois}(\left\langle\mathnormal{N}_{\mathrm{C}}^{\mathrm{B}}\right\rangle)$
and
$\mathnormal{N}_{\mathrm{M}}^{\mathrm{C}}{(\boldsymbol{\mathnormal{w}}_{\mathrm{M}})}\sim\mathrm{Pois}(\left\langle\mathnormal{N}_{\mathrm{M}}^{\mathrm{C}}\right\rangle)$
are strictly the same among different societies and metasocieties. In case 1,
every metasociety and society is interchangeable, with

$\uppsi\left[\mathring{\mathnormal{\ell}}_{\mathrm{B}}\right]_{\mathrm{C}}(\ell)=\exp(-\ell/\overline{\mathnormal{\ell}})/\overline{\mathnormal{\ell}}.$
(107)

In case 2, all the broadcasts in an individual society have the same
luminosity, but this characteristic luminosity is exponentially distributed
between societies:

$\uppsi\left[\mathring{\mathnormal{\ell}}_{\mathrm{B}}\right]_{\mathrm{C}}(\ell)=\updelta(\ell-\overline{\mathnormal{\ell}}_{\mathrm{C}})~{}\text{and}~{}\uppsi\left[\overline{\mathnormal{\ell}}_{\mathrm{C}}\right]_{\mathrm{M}}(\ell)=\exp(-\ell/\overline{\mathnormal{\ell}})/\overline{\mathnormal{\ell}}.$
(108)

In case 3, each metasociety has decreed all broadcasts have the same
luminosity, but different metasocieties decide on different luminosities, with
an exponential distribution:

$\uppsi\left[\mathring{\mathnormal{\ell}}_{\mathrm{B}}\right]_{\mathrm{M}}(\ell)=\updelta(\ell-\overline{\mathnormal{\ell}}_{\mathrm{M}})~{}\text{and}~{}\uppsi\left[\overline{\mathnormal{\ell}}_{\mathrm{M}}\right]_{\mathrm{U}}(\ell)=\exp(-\ell/\overline{\mathnormal{\ell}})/\overline{\mathnormal{\ell}}.$
(109)

These can be compared with case 0, in which all broadcasts have the same
luminosity:
$\uppsi\left[\mathring{\mathnormal{\ell}}_{\mathrm{B}}\right]_{\mathrm{C}}(\ell)=\updelta(\ell-\overline{\mathnormal{\ell}})$.
In case 0, broadcasts, societies, and metasocieties are all interchangeable.

There are slightly different approaches to calculating the variance in each
case, but it involves building up from broadcast-level averages to the cosmic
level. All the broadcasts in a society are drawn from the same luminosity
distribution – a degenerate one in cases 0, 2, and 3, and an exponential one
in case 1. Thus, the aggregate luminosity of broadcasts in a realized society
is a compound Poisson variable. In cases 0, 1, and 3, the societies are
interchangeable, so we can then apply a compound Poisson distribution again.
However, in case 2, the societies have different luminosity distributions
depending on their $\overline{\mathnormal{\ell}}_{\mathrm{C}}$. To advance to
the metasocietal-level variables, we have to integrate over
$\uppsi\left[\overline{\mathnormal{\ell}}_{\mathrm{C}}\right]_{\mathrm{M}}$,
which is a sort of marginalized societal haystack (equation 10). Finally, in
cases 0–2, all metasocieties are interchangeable, and so
$\left\langle\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right\rangle=\left\langle\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right\rangle_{\mathrm{U}}$
and
$\mathbb{V}\left[\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right]=\mathbb{V}\left[\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right]_{\mathrm{U}}$.
However, in case 3, we need to find
$\mathbb{V}\left[\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right]_{\mathrm{U}}=\left\langle(\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}})^{2}\right\rangle_{\mathrm{U}}-\left\langle\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right\rangle_{\mathrm{U}}^{2}$
by averaging over
$\uppsi\left[\overline{\mathnormal{\ell}}_{\mathrm{C}}\right]_{\mathrm{U}}$
(equation 43), effectively a metasocietal haystack.

In all four cases, the mean aggregate luminosity is the same,
$\left\langle\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right\rangle_{\mathrm{U}}=\left\langle\mathnormal{N}_{\mathrm{M}}^{\mathrm{B}}\right\rangle\overline{\mathnormal{\ell}}$,
because the mean number of broadcasts and the mean broadcast luminosity are
the same. The variance, however, increases from one case to the next:

$\mathbb{V}\left[\mathring{\mathnormal{L}}_{\mathrm{M}}^{\mathrm{B}}\right]_{\mathrm{U}}=\begin{cases}2\overline{\mathnormal{\ell}}^{2}\left\langle\mathnormal{N}_{\mathrm{M}}^{\mathrm{B}}\right\rangle(\frac{1}{2}+\frac{1}{2}\left\langle\mathnormal{N}_{\mathrm{C}}^{\mathrm{B}}\right\rangle)&(\text{Case
0})\\\
2\overline{\mathnormal{\ell}}^{2}\left\langle\mathnormal{N}_{\mathrm{M}}^{\mathrm{B}}\right\rangle(1+\frac{1}{2}\left\langle\mathnormal{N}_{\mathrm{C}}^{\mathrm{B}}\right\rangle)&(\text{Case
1})\\\
2\overline{\mathnormal{\ell}}^{2}\left\langle\mathnormal{N}_{\mathrm{M}}^{\mathrm{B}}\right\rangle(1+\left\langle\mathnormal{N}_{\mathrm{C}}^{\mathrm{B}}\right\rangle)&(\text{Case
2})\\\
2\overline{\mathnormal{\ell}}^{2}\left\langle\mathnormal{N}_{\mathrm{M}}^{\mathrm{B}}\right\rangle(1+\left\langle\mathnormal{N}_{\mathrm{C}}^{\mathrm{B}}\right\rangle+\frac{1}{2}\left\langle\mathnormal{N}_{\mathrm{M}}^{\mathrm{B}}\right\rangle)&(\text{Case
3})\end{cases}$ (110)

In case 0, the variance is totally due to compound Poissonian fluctuations in
the number of broadcasts, supplemented by intrinsic variance in the luminosity
in case 1. All the broadcasts in a society share the same luminosity in cases
2 and 3, amplifying the fluctuations; in case 3, as the shared luminosity
$\overline{\mathnormal{\ell}}_{\mathrm{M}}$ varies, the entire aggregate
luminosity does as well.

This example only scratches the surface of the subject, but the methods in
sections 2 and 3 can be applied to these problems even when the compound
Poisson distribution does not apply.

##  9 The box model

In the box model, broadcasts are simple contiguous “boxes” in time-frequency
space. Each box covers the frequency range
$|\nu-\mathnormal{\upsilon}_{\mathrm{B}}|\leq\overline{\mathnormal{\beta}}/2$
and time range $0\leq
t-\mathnormal{\vartheta}_{\mathrm{B}}\leq\overline{\mathnormal{\tau}}$.
Because these boxes are not skewed, the drift rate
$\mathnormal{\delta}_{\mathrm{B}}$ is $0$. Furthermore, all the boxes are
assumed to have identical bandwidths $\overline{\mathnormal{\beta}}$ and
durations $\overline{\mathnormal{\tau}}$ (Figure 7). The distribution is
assumed to be stationary, in that $\mathnormal{\vartheta}_{\mathrm{B}}$ and
$\mathnormal{\upsilon}_{\mathrm{B}}$ have uniform distributions over any
observable range, with

${\mathnormal{{\Psi}}_{\mathrm{C}}^{\mathrm{B}}}={\breve{\mathnormal{\Lambda}}_{\mathrm{C}}^{\mathrm{B}}}\uppsi\left[\mathring{\mathnormal{\epsilon}}_{\mathrm{B}},\mathnormal{\varpi}_{\mathrm{B}}\right]_{\mathrm{C}}\updelta(\mathnormal{\tau}_{\mathrm{B}}-\overline{\mathnormal{\tau}})\updelta(\mathnormal{\beta}_{\mathsf{t}|\mathrm{B}}-\overline{\mathnormal{\beta}})\updelta(\mathnormal{\delta}_{\mathrm{B}}){\updelta(\boldsymbol{\mathnormal{r}}_{\mathrm{B}}-\boldsymbol{\mathnormal{r}}_{\mathrm{C}}).}$
(111)

In the box model,
$\mathnormal{\beta}_{\mathsf{x}{:}t|\mathrm{B}}=\overline{\mathnormal{\beta}}$
(equations 85 and 98).

The selection itself is also a contiguous box in time-frequency space,
spanning the frequency range
$|\nu-\mathsf{\Upsilon}_{\mathsf{x}}|\leq\mathsf{B}_{\mathsf{x}}/2$ and time
range $0\leq t-\mathsf{\Theta}_{\mathsf{x}}\leq\mathsf{T}_{\mathsf{x}}$.
Sample properties in the box model depend on the number and amount of overlap
between broadcast boxes and the sample box in time-frequency space (Figure 7).

Consider a sample
${\mathnormal{\Sigma}_{\mathsf{x},\mathrm{J}}^{\mathrm{B}}}$, consisting of
broadcasts within a field of
${\mathnormal{N}_{\mathsf{x},\mathrm{J}}^{\star}}$ stars with a time and
frequency response described by a box. The number of broadcasts expected from
a host ${{\mathrm{J}}}$ is proportional to the rate per star per unit
frequency, number of stars, and the temporal and frequency “cross sections”
resulting from the combined width of the window and the broadcasts: