# The sensitivity of past and near-future lunar radio experiments to ultra-
high-energy cosmic rays and neutrinos

J.D. Bray<EMAIL_ADDRESS>School of Chemistry & Physics, Univ.
of Adelaide, SA 5005, Australia  CSIRO Astronomy & Space Science, Marsfield,
NSW 2122, Australia  JBCA, School of Physics & Astronomy, Univ. of Manchester,
Manchester M13 9PL, UK

###### Abstract

Various experiments have been conducted to search for the radio emission from
ultra-high-energy particles interacting in the lunar regolith. Although they
have not yielded any detections, they have been successful in establishing
upper limits on the flux of these particles. I present a review of these
experiments in which I re-evaluate their sensitivity to radio pulses,
accounting for effects which were neglected in the original reports, and
compare them with prospective near-future experiments. In several cases, I
find that past experiments were substantially less sensitive than previously
believed. I apply existing analytic models to determine the resulting limits
on the fluxes of ultra-high-energy neutrinos and cosmic rays. In the latter
case, I amend the model to accurately reflect the fraction of the primary
particle energy which manifests in the resulting particle cascade, resulting
in a substantial improvement in the estimated sensitivity to cosmic rays.
Although these models are in need of further refinement, in particular to
incorporate the effects of small-scale lunar surface roughness, their
application here indicates that a proposed experiment with the LOFAR telescope
would test predictions of the neutrino flux from exotic-physics models, and an
experiment with a phased-array feed on a large single-dish telescope such as
the Parkes radio telescope would allow the first detection of cosmic rays with
this technique, with an expected rate of one detection per 140 hours.

###### keywords:

ultra-high-energy neutrinos , ultra-high-energy cosmic rays , radio detection
, Moon

††journal: Astroparticle Physics

##  1 Introduction

Observations of ultra-high-energy (UHE; $>10^{18}$ eV) cosmic rays (CRs), and
attempts to detect their expected counterpart neutrinos, are hampered by their
extremely low flux. The detection of a significant number of UHE particles
requires the use of extremely large detectors, or the remote monitoring of a
large volume of a naturally-occurring detection medium. One approach,
suggested by Dagkesamanskii and Zheleznykh [1], is to make use of the lunar
regolith as the detection medium by observing the Moon with ground-based radio
telescopes, searching for the Askaryan radio pulse produced when the
interaction of a UHE particle initiates a particle cascade [2]. The high time
resolution required to detect this coherent nanosecond-scale pulse puts these
efforts in a quite different regime to conventional radio astronomy.

Since the first application of this lunar radio technique with the Parkes
radio telescope [3], many similar experiments have been conducted, none of
which has positively detected a UHE particle. Consequently, these experiments
have placed limits on the fluxes of UHECRs and neutrinos. To determine these
limits, each experiment has developed an independent calculation of its
sensitivity to radio pulses and, in most cases, an independent model for
calculating the resulting aperture for the detection of UHE particles. This
situation calls for further work in two areas, both of which are addressed
here: the recalculation of the radio sensitivity of past experiments in a
common framework, incorporating all known experimental effects, and the
calculation of the resulting apertures for both UHECRs and neutrinos using a
common analytic model.

An additional benefit of this work is to provide a comprehensive description
of the relevant experimental considerations, with past experiments as case
studies, to support future work in this field. To that end, I also present
here a similar analysis of the radio sensitivity and particle aperture for
several possible future lunar radio experiments. The most sensitive telescope
available for the application of this technique for the forseeable future will
be the Square Kilometre Array (SKA), prospects for which have been discussed
elsewhere [4], but phase 1 of this instrument is not scheduled for completion
until 2023; in this work, I instead evaluate three proposed experiments that
could be carried out in the near future ($<5$ yr) with existing radio
telescopes. Most other experiments that could be conducted with existing radio
telescopes will resemble one of these.

This work is organised as follows. In Sec. 2 I address the calculation of the
sensitivity of radio telescopes to coherent pulses, obtaining a similar result
to Eq. 2 of Gorham et al. [5], but incorporating a wider range of experimental
effects. This provides the theoretical basis for the re-evaluation in Sec. 3
of past lunar radio experiments, in which I calculate a common set of
parameters to represent their sensitivity to a lunar-origin radio pulse.
Alongside these, I calculate the same parameters for proposed near-future
experiments.

In Sec. 4 I discuss the calculation of the sensitivity of lunar radio
experiments to UHE particles. For each of the experiments evaluated in Sec. 3,
I calculate the sensitivity to neutrinos based on the analytic model of Gayley
et al. [6], and the sensitivity to UHECRs based on the analytic model of Jeong
et al. [7]. Finally, in Sec. 5, I briefly discuss the implications for future
work in this field.

##  2 Sensitivity to coherent radio pulses

The sensitivity of a radio telescope is characterised by the system equivalent
flux density (SEFD), conventionally measured in janskys (1 Jy = $10^{-26}$ W
m-2 Hz-1), which is given by

$\langle F\rangle=2\,\frac{k\,T_{\rm sys}}{A_{\rm eff}}$ (1)

where $k$ is Boltzmann’s constant, $T_{\rm sys}$ the system temperature and
$A_{\rm eff}$ the effective aperture (i.e. the total collecting area of the
telescope multiplied by the aperture efficiency). In the context of a lunar
radio experiment, the system temperature is typically dominated by thermal
radiation from the Moon — or, at lower frequencies, by Galactic background
emission — with a smaller contribution from internal noise in the radio
receiver. However, the strength of a coherent pulse, such as the Askaryan
pulse from a particle cascade, is expressed in terms of a spectral electric
field strength, in e.g. V/m/Hz. To describe the sensitivity of a radio
telescope to a coherent pulse, we must relate this quantity to the parameters
in Eq. 1.

The factor of two in Eq. 1 occurs because the flux contains contributions from
two polarisations, whether these are considered as orthogonal linear
polarisations or as opposite circular polarisations (left and right circular
polarisations; LCP and RCP). The bolometric flux density in a single
polarisation is given by the time-averaged Poynting vector

$\langle S\rangle=\frac{E_{\rm rms}^{2}}{Z_{0}}$ (2)

where $E_{\rm rms}$ is the root mean square (RMS) electric field strength in
that polarisation, and $Z_{0}$ is the impedance of free space. If the received
radiation has a flat spectrum over a bandwidth $\Delta\nu$, the total spectral
flux density is found by averaging the combined bolometric flux density in
both polarisations over the band, giving us

$\displaystyle\langle F\rangle$ $\displaystyle=2\,\frac{\langle
S\rangle}{\Delta\nu}$ (3) $\displaystyle=2\,\frac{E_{\rm
rms}^{2}}{Z_{0}\,\Delta\nu}$ from Eq. 2 (4)

which is the SEFD again. Combining Eqs. 1 and 4 shows that

$E_{\rm rms}=\left(\frac{k\,T_{\rm sys}\,Z_{0}\,\Delta\nu}{A_{\rm
eff}}\right)^{1/2}.$ (5)

It is also useful to define

$\displaystyle\mathcal{E}_{\rm rms}$ $\displaystyle=\frac{E_{\rm
rms}}{\Delta\nu}$ (6) $\displaystyle=\left(\frac{k\,T_{\rm sys}\,Z_{0}}{A_{\rm
eff}\,\Delta\nu}\right)^{1/2}$ from Eq. 5, (7)

the equivalent RMS spectral electric field for this bandwidth, although for
incoherent noise it should be borne in mind that, unlike the flux density, the
spectral electric field varies with the bandwidth. This is in contrast to the
behaviour of coherent pulses, for which the spectral electric field is
bandwidth-independent, and the flux density scales with the bandwidth.

The sensitivity of an experiment to detect a coherent radio pulse can be
expressed as $\mathcal{E}_{\rm min}$, a threshold spectral electric field
strength above which a pulse would be detected. This is typically measured
with respect to $\mathcal{E}_{\rm rms}$, in terms of a significance threshold
$n_{\sigma}$. Note that the addition of thermal noise will increase or
decrease the amplitude of a pulse, so that $\mathcal{E}_{\rm min}$ is actually
the level at which the detection probability is 50% rather than an absolute
threshold, but this distinction becomes less important when $n_{\sigma}$ is
large. $\mathcal{E}_{\rm min}$ further depends on the position of the pulse
origin within the telescope beam, as

$\mathcal{E}_{\rm
min}(\theta)=f_{C}\,\frac{n_{\sigma}}{\alpha}\sqrt{\frac{\eta}{\mathcal{B}(\theta)}}\,\mathcal{E}_{\rm
rms}$ (8)

where $\mathcal{B}(\theta)$ is the beam power at an angle $\theta$ from its
axis, normalised to $\mathcal{B}(0)=1$ and assumed here to be radially
symmetric (e.g. an Airy disk). This same equation is used to calculate
$\mathcal{E}_{\rm max}$ as described in Sec. 3. The factor $\eta$ is the ratio
between the total pulse power and the power in the chosen polarisation
channel, typically found as

$\eta=\begin{cases}2&\mbox{for circular polarisation}\\\
1/\cos^{2}\phi&\mbox{for linear polarisation}\end{cases}$ (9)

with $\phi$ the angle between the receiver and a linearly polarised pulse such
as that expected from the Askaryan effect. The term $\alpha$ is the proportion
of the original pulse amplitude recovered after inefficiencies in pulse
reconstruction, as described in Sec. 2.1. The remaining factor, $f_{C}$,
accounts for the improvement in sensitivity from combining $C$ independent
channels with a threshold of $n_{\sigma}$ in each, as described in Sec. 2.2.

The behaviour of coherent pulses as described above is quite different to that
of conventional radio astronomy signals. As a consequence of Eq. 7,
sensitivity to coherent pulses scales as $\sqrt{A_{\rm eff}\Delta\nu}$ in
electric field and hence as $A_{\rm eff}\Delta\nu$ in power, whereas
sensitivity to incoherent signals scales as $A_{\rm eff}\sqrt{\Delta\nu}$ in
power. Fundamentally, this is because the signal of a coherent pulse combines
coherently both across the collecting area of the telescope and across its
frequency range, while most radio astronomy signals combine coherently across
the collecting area and incoherently across frequency. Because of this
difference it is not entirely appropriate to represent a detection threshold
in terms of an equivalent flux density, as the flux density of a coherent
pulse depends on its bandwidth, which defeats the purpose of using a spectral
(rather than bolometric) measure such as flux density in the first place.
However, this quantity is occasionally reported in the literature, so I
calculate it in several cases for comparative purposes; ensuring, to the best
of my ability, that both values are calculated for the same bandwidth, so that
the comparison is valid. For a polarised pulse at the detection threshold,
with spectral electric field $\mathcal{E}_{\rm min}$ and total electric field
$E_{\rm min}=\mathcal{E}_{\rm min}\Delta\nu$, the equivalent flux can be found
similarly to Eq. 4 — omitting the factor of 2, as the pulse appears in only a
single polarisation — as

$F_{\rm min}=\frac{\mathcal{E}_{\rm min}^{2}\,\Delta\nu}{Z_{0}}.$ (10)

###  2.1 Amplitude recovery efficiency

The spectral electric field $\mathcal{E}$ of a pulse is, in general, a complex
quantity. For a coherent pulse, its phase is constant across all frequencies.
If this phase is zero, then the time-domain function $E(t)$ has its power
concentrated at a single point in time with peak amplitude
$|\mathcal{E}|\Delta\nu$, as implicitly assumed in the above discussion.
However, an Askaryan pulse has a phase close to the worst-case value of
$\pi/2$ [8], for which it takes on a bipolar profile with the power split
between the poles, causing the peak amplitude to be reduced by a factor
$\sim\sqrt{2}$. If this pulse is recorded directly without correcting the
phase, this gives $\alpha\sim 0.71$. If the signal undergoes frequency
downconversion, the phase is randomised, giving $\alpha$ somewhere between
this value and unity [9].

A pulse originating from the Moon is smeared out in time, also reducing its
peak amplitude, by dispersion as it passes through the Earth’s ionosphere. The
frequency-dependent delay is

$\Delta t=1.34\times 10^{9}\left(\frac{\rm STEC}{\rm
TECU}\right)\left(\frac{\nu}{\rm Hz}\right)^{-2}{\rm s}$ (11)

where STEC is the electron column density or slant total electron content
measured in total electron content units (${\rm 1~{}TECU}=10^{16}$ electrons
m-2). Typical values are in the range 5–100 TECU, depending on the time of
day, season, solar magnetic activity cycle, and slant angle through the
ionosphere.

When a signal is converted to digital samples with a finite sampling rate, the
peak amplitude is further reduced, because the sampling times do not
necessarily correspond to the peak in the original analog signal [10]. This
effect can be mitigated by oversampling the analog signal, or by interpolating
the digital data [11]. For a coherent ${\rm sinc}$-function pulse with no
oversampling or interpolation, the worst case corresponds to sampling times
equally spaced either side of the peak, giving a value for $\alpha$ of ${\rm
sinc}(0.5)=0.64$.

The interaction between these effects is complex, and not susceptible to a
simple analytic treatment. I have instead developed a simulation to find a
representative value of $\alpha$ for a given experiment, described in App. A.

###  2.2 Combining channels

Some coherent pulse detection experiments combine the signals from multiple
channels, which may be different polarisations, frequency bands, antennas, or
any combination of these. In this context, I take $\Delta\nu$ to be the
bandwidth of a single channel, and Eq. 8 with $f_{C}=1$ gives the threshold
for a single channel on its own. The sensitivity of the combined signal
depends critically on whether there is phase coherence between the channels,
and whether they are combined coherently (i.e. direct summation of voltages)
or incoherently (summing the squared voltages, or power). The scaling of the
sensitivity for $C$ independent identical channels is as described below.

Coherent channels, coherent combination

In this case, the pulses in each channel combine coherently, and the
combination acts as a single channel with bandwidth $C\,\Delta\nu$. The
threshold in voltage thus scales as $f_{C}=C^{-1/2}$.

Coherent channels, incoherent combination

Squaring the voltages in this case converts them to the power domain, in which
the sensitivity scales as $C^{1/2}$. The sensitivity in the voltage domain
scales as the square root of this, or $C^{1/4}$, and hence $f_{C}=C^{-1/4}$.

Incoherent channels, coherent combination

Since there is no phase coherence between the pulses in different channels,
they sum incoherently, in the same way as the noise. The signal-to-noise ratio
therefore does not scale with the number of channels, so $f_{C}=1$.

Incoherent channels, incoherent combination

Squaring the voltages converts them to the power domain, in which the
sensitivity scales as $C^{1/2}$, regardless of the original phases. The
sensitivity in the voltage domain therefore scales as $C^{1/4}$, and hence
$f_{C}=C^{-1/4}$.

Conventional radio astronomy operates in the first regime for the combination
of multiple antennas, as the signal is coherent across the collecting area;
and in the last regime for the combination of multiple frequency channels, as
most astronomical radio signals are not coherent across a range of
frequencies.

Care must be taken in defining the significance threshold $n_{\sigma}$ when
the signal is in the power domain. For a voltage-domain signal $s$, which has
a Gaussian distribution, the significance is defined simply in terms of the
peak and RMS signal values as $n_{\sigma}=s_{\rm peak}/s_{\rm rms}$. If this
signal is squared to produce the power-domain signal $S$, it has a $\chi^{2}$
distribution with one degree of freedom, and the significance is instead found
as $n_{\sigma}=(S_{\rm peak}/{}\mkern 3.0mu\overline{\mkern-3.0muS})^{1/2}$ in
terms of the mean value ${}\mkern 3.0mu\overline{\mkern-3.0muS}$, since
$S_{\rm peak}=s_{\rm peak}^{2}$ and ${}\mkern
3.0mu\overline{\mkern-3.0muS}=s_{\rm rms}^{2}$. The ratio $S_{\rm
peak}/{}\mkern 3.0mu\overline{\mkern-3.0muS}$ is the same as the ratio between
the equivalent flux density of the pulse (from Eq. 10) and the mean background
flux in a single polarisation (i.e. half the SEFD). When $C$ identical
independent power-domain channels are summed, the resulting signal has a
$\chi^{2}$ distribution with $C$ degrees of freedom, but the scaling factor
$f_{C}$ corrects for this, with $n_{\sigma}$ remaining the significance in a
single channel.

Some experiments operate with multiple channels, but do not combine them
either coherently or incoherently as described above. Instead, they combine
them in coincidence, requiring a pulse to be simultaneously detected in all
channels simultaneously. This increases the effective detection threshold:
taking $f_{C}=1$ gives the threshold $\mathcal{E}_{\rm min}$ at which the
detection probability is 50%, due to Gaussian thermal noise increasing or
decreasing the pulse amplitude, but the probability of simultaneous detection
in $C$ channels is only $2^{-C}$. To scale $\mathcal{E}_{\rm min}$ so that the
detection probability remains 50%, for $C$ identical independent channels, we
require $f_{C}$ such that

$\prod_{i=1}^{C}\left(\int_{n_{\sigma}(1-f_{C})}^{\infty}\\!\frac{ds_{i}}{\sqrt{2\pi}}e^{-s_{i}^{2}/2}\right)=0.5$
(12)

where the integral is over the Gaussian-distributed voltage-domain signal
$s_{i}$ in each channel. Solving for $f_{C}$ gives us

$f_{C}=1-\frac{\sqrt{2}}{n_{\sigma}}\,{\rm
erf}^{-1}{\left(1-2^{(C-1)/C}\right)}$ (13)

where ${\rm erf}^{-1}$ is the inverse of the standard error function. The
value of $f_{C}$ approaches unity for large $n_{\sigma}$, for which the
effects of thermal noise become insignificant, and for small $C$.

##  3 Past and near-future lunar radio experiments

Lunar radio experiments have been carried out with a diverse range of
telescopes, with a variety of different receivers and trigger schemes to
balance their sensitivity with their ability to exclude radio-frequency
interference (RFI). Here I attempt to represent them with a unified set of
parameters, so their sensitivity to UHE particles can be calculated with the
analytic models used in Sec. 4. Although this representation is inevitably
only an approximation to the inputs to numerical simulations (e.g. [12]), it
lends itself more easily to use in future models. This work is similar in
concept to previous work by Jaeger et al. [13], but contains a more detailed
analysis of previous experiments, including all the effects described in Sec.
2. I determine the following parameters.

Observing frequency: $\nu$

I take this to be the central frequency of the triggering band. Generally
speaking, a lower frequency results in a larger effective aperture for UHE
particles, while a higher frequency reduces the threshold detectable particle
energy. As the analytic models used in this work all assume a small fractional
bandwidth, I also report the width $\Delta\nu$ of the triggering band as an
indication of the accuracy of this assumption. However, this does not include
the secondary 1.4 GHz band of the Kalyazin experiment (see Sec. 3.3).

Minimum spectral electric field: $\mathcal{E}_{\rm min}$

This is the spectral electric field strength of a coherent pulse for which the
detection probability is 50%, as described in Sec. 2; its interpretation as an
absolute threshold will slightly underestimate the sensitivity for weaker
pulses and overestimate it for stronger ones. An Askaryan pulse from a lunar
UHE particle interaction is expected to have linear polarisation oriented
radially to the Moon, and to originate from the lunar limb [12]. For telescope
beams pointed at the limb of the Moon I use the minimum value
$\mathcal{E}_{\rm min}=\mathcal{E}_{\rm min}(0)$ at the centre of the beam;
otherwise, I take $\mathcal{E}_{\rm min}(\theta_{\rm L})$ at the closest point
on the limb. I represent the pulse reconstruction efficiency with the mean
value ${}\mkern 0.5mu\overline{\mkern-0.5mu\alpha}$ for a flat-spectrum pulse,
calculated with the simulation described in App. A.

Limb coverage: $\zeta$

A single telescope beam typically covers only part of the Moon, which reduces
the probability of detecting a UHE particle. As the probability of detection
is dominated by radio pulses originating from the outermost fraction of the
lunar radius, at least at higher frequencies [14], I take the effective
coverage to be the fraction of the circumference of the lunar limb within the
beam, multiplied by the number of beams $n_{\rm beams}$ when there are
multiple similar beams pointed at different parts of the limb. For this
purpose, I consider a point on the limb to be within the beam if the effective
threshold $\mathcal{E}_{\rm min}(\theta)$ in that direction is no more than
$\sqrt{2}$ times the minimum threshold $\mathcal{E}_{\rm min}$ as defined
above. For a beam pointed at the limb, this corresponds to the commonly-used
full width at half maximum (FWHM) beam size. The analytic models used in this
work assume full sensitivity within this beam and zero outside of it, which
will slightly overestimate the sensitivity to weaker pulses near the detection
threshold, which cannot be detected throughout the beam, and underestimate the
sensitivity to stronger pulses, which can be detected even when they are
slightly outside of it. Where available, I have used the dates of observations
to determine the median apparent size of the Moon when calculating the limb
coverage, although this has only a minor effect on the result: the apparent
size of the Moon varies across the range 29–34′, but most experiments provide
a fairly even sampling of this range, so their median values are within 1′ of
one another.

Effective observing time: $t_{\rm obs}$

This is the effective time spent observing the Moon after allowing for
inefficiency in the trigger algorithm, instrumental downtime while data is
being stored, and the false positive rates of anti-RFI cuts.

Some experiments have used an anticoincidence filter in which they exclude any
event which is detected in multiple receivers pointed at different parts of
the sky, as these are typically caused by local RFI detected through the
antenna sidelobes. These filters are critical for excluding pulsed RFI which
might otherwise be misidentified as a lunar-origin pulse, but they also have
the potential to misidentify a sufficiently intense lunar-origin pulse as RFI,
which may substantially decrease the sensitivity of an experiment to UHE
particles [15]. To reflect this, for these experiments I calculate another
quantity.

Maximum spectral electric field: $\mathcal{E}_{\rm max}$

This is the spectral electric field strength of a coherent pulse which, if
detected in one beam, would have a 50% chance of also being detected through a
sidelobe of another beam and hence being misidentified as RFI. It is otherwise
defined similarly to $\mathcal{E}_{\rm min}$, and calculated with Eq. 8 with
$n_{\sigma}$ as the significance level for exclusion and $\mathcal{B}(\theta)$
as the sidelobe power of one beam at the centre of another. A lunar-origin
pulse is considered to be detected and identified as such only if its spectral
electric field strength is between $\mathcal{E}_{\rm min}$ and
$\mathcal{E}_{\rm max}$.

I derive these values for past experiments in Secs. 3.1–3.8, calculating them
separately for each pointing if the experiment used multiple pointing
strategies. I also consider possible near-future experiments in Secs.
3.9–3.11. The results are presented in Table 1, and are used in the rest of
this work.

Table 1: Observation parameters for past and near-future lunar radio experiments. Experiment | Pointing | $\nu$ | $\Delta\nu$ | $\mathcal{E}_{\rm min}$ | $\mathcal{E}_{\rm max}$ | $\zeta$ | $t_{\rm obs}$
---|---|---|---|---|---|---|---
$(\times n_{\rm beams})$ | (MHz) | (MHz) | ($\mu$V/m/MHz) | (%) | (hr)
GLUE | limb | 2200 | 150 | 0.0221 | 0.3695 | 11 | 73.5
half-limb | 2200 | 150 | 0.0500 | 0.2527 | 20 | 39.9
centre | 2200 | 150 | 0.4737 | 0.2527 | 100 | 10.3
Kalyazin | limb | 2250 | 120 | 0.0235 | — | 7 | 31.3
LUNASKA | limb | 1500 | 600 | 0.0153 | — | 36 | 13.6
ATCA | centre | 1500 | 600 | 0.0207 | — | 100 | 12.6
NuMoon | limb ($\times 2$) | 141 | 55 | 0.1453 | — | 14 | 46.7
RESUN | limb ($\times 3$) | 1425 | 100 | 0.0549 | — | 100 | 200.0
LUNASKA | limb ($\times 2$) | 1350 | 300 | 0.0053 | 0.0241 | 16 | 127.2
Parkes | half-limb | 1350 | 300 | 0.0142 | 0.0489 | 15 | 99.4
Future experiments
LOFAR | face ($\times 50$) | 166 | 48 | 0.0313 | 0.0768 | 100 | 183.3
Parkes PAF | limb ($\times 12$) | 1250 | 1100 | 0.0043 | 0.0303 | 100 | 170.0
AuScope | centre | 2300 | 200 | 0.0830 | — | 100 | 2900.0

###  3.1 Parkes

The first lunar radio experiment was conducted with the 64 m Parkes radio
telescope in January 1995 [3, 16]. They observed for 10 hours with a receiver
that Nyquist-sampled the frequency range 1175–1675 MHz in dual circular
polarisations. The storage of this data was triggered when a threshold was
exceeded by the power in both of two subbands, each of width 100 MHz in a
single polarisation, centred on 1325 MHz and 1525 MHz, at a delay offset
corresponding to that expected from ionospheric dispersion. This last
criterion was effective in discriminating against terrestrial RFI. However,
they calculated the relative dispersive delay across a band $\Delta\nu$ as

$\Delta t=0.012\left(\frac{\Delta\nu}{\rm Hz}\right)\left(\frac{\rm STEC}{{\rm
electrons}~{}{\rm cm}^{-2}}\right)\left(\frac{\nu}{\rm Hz}\right)^{-3}{\rm s}$
(14)

whereas, to be equivalent (for small $\Delta\nu$) to Eq. 11, the leading
constant should be 0.00268 [17]. Consequently, the 10 ns dedispersive delay
they introduced between the two subbands exceeded the required value by a
factor of $\sim 4$. Since the delay error is comparable to the 10 ns length of
a band-limited pulse in a 100 MHz subband, a lunar-origin Askaryan pulse would
have no significant overlap between the two subbands, and would not meet the
trigger criteria. Even if such a pulse were recorded, it would be excluded by
later tests on the stored full-band data, which required that a pulse display
an increased amplitude when ‘correctly’ dedispersed. This experiment was
therefore not appreciably sensitive to UHE particles.

The telescope beam for this experiment was directed at the centre of the Moon,
reflecting the contemporary expectation that this was the most likely point at
which to detect the Askaryan pulse from an interacting UHE neutrino [1].
Because of this, the beam had only minimal sensitivity at the lunar limb,
where detectable Askaryan pulses are now known to be most likely to originate,
which limits its sensitivity to UHE particles [18], even if the dedispersion
problem described above is ignored. This experiment did, however, serve an
important role in triggering further work in this field.

###  3.2 GLUE

The Goldstone Lunar Ultra-high-energy Neutrino Experiment (GLUE) made use of
the 34 m DSS13 and 70 m DSS14 antennas at the Goldstone Deep Space
Communications Complex in a series of observations over 2000–2003, with a
total of 124 hours of effective observing time [19, 5, 20]. They observed
around 2.2 GHz on both antennas, forming two non-overlapping 75 MHz RCP
channels on DSS13, and a 40 MHz LCP channel and a 150 MHz RCP channel (later
two 75 MHz RCP channels) on DSS14. Each channel was triggered by a peak in the
signal power as measured by a square-law detector. A global trigger, causing
an event to be stored, required a coincidence between all four (or five)
channels within a 300 $\mu$s time window. Subsequent cuts eliminated RFI by
tightening the coincidence timing criteria, aided considerably by the 22 km
baseline between the two antennas, as well as by excluding extended pulses,
pulses clustered in time, and pulses detected by an off-axis 1.8 GHz receiver
on DSS14. A range of beam pointings were used, ranging from the centre to the
limb of the Moon, reflecting the realisation that Askaryan pulses were most
likely to be observed from the limb.

Williams [20] excluded thermal noise by applying significance cuts at
$n_{\sigma}=4$ (DSS13 RCP), $n_{\sigma}=6$ (DSS14 RCP) and $n_{\sigma}=3$
(DSS14 LCP), with these thresholds chosen by scaling based on bandwidth (but
not on collecting area) to equalise their sensitivity, and considered these,
rather than the trigger thresholds, to define the sensitivity of the
experiment. The trigger thresholds are not straightforward to determine, as
they depend on the characteristics of the signal output of the square-law
detectors, but I assume that the $\sim 10$ ns integration time of the square-
law detectors effectively removes any dependence on the phase of the original
signal while not further smearing out any peaks, and take the output to be the
square of the signal envelope. This analog output was searched for peaks by
SR400 discriminators which act on a continuous signal [21], and so are not
subject to the amplitude loss from a finite sampling rate described in Sec.
2.1. Given these assumptions, the 30 kHz single-channel trigger rates for
DSS13 RCP and DSS14 RCP imply thresholds equivalent to $n_{\sigma}=4.2$ and
$4.4$ respectively in the original unsquared voltages, and the 45 kHz trigger
rate for DSS14 LCP implies $n_{\sigma}=4.0$ (from Ref. [9], Eq. 46). I
therefore find that the trigger thresholds are higher than the cut thresholds,
and thus limit the sensitivity, for the DSS13 RCP and DSS14 LCP channels. Note
that my assumptions, and the insignificance of dispersion at this experiment’s
high observing frequency, imply $\alpha=1$. If my assumptions are invalid then
the true trigger thresholds will be lower than found here, but the amplitude
reconstruction efficiency $\alpha$ will be decreased, leading to a net
increase in the effective threshold and a decrease in the sensitivity of this
experiment.

Due to the range of different channels used in the coincidence trigger
requirement, the scaling relation in Sec. 2.2 is not directly applicable:
instead, the threshold is determined by the least sensitive channel or
channels. Most of the observing time for this experiment was spent with both
antennas pointed on the limb of the Moon, in which configuration the least
sensitive channels are those of DSS13 RCP: given the reported values of 105 K
for the system temperature and 75% for the aperture efficiency, I find them by
Eq. 7 to have $\mathcal{E}_{\rm rms}=0.0033$ $\mu$V/m/MHz. Under the
assumption that any event which exceeds the trigger threshold on both DSS13
RCP channels will almost certainly also trigger the more sensitive channels,
Eq. 13 can then be applied to find that the coincidence requirement between
the two DSS13 RCP channels gives $f_{C}=1.13$.

From Eq. 8, taking the above values and $\eta=2$ for circular polarisation, I
find $\mathcal{E}_{\rm min}=0.022$ $\mu$V/m/MHz at the centre of the beam.
Note that this is higher (less sensitive) than the value $0.00914$
$\mu$V/m/MHz found by Williams [20], which was based on the cut threshold
(rather than the trigger threshold) and the more sensitive 150 MHz DSS14 RCP
channel. Fig. 1 shows the relationship between the cut and trigger thresholds,
calculating $\mathcal{E}_{\rm min}(\theta)$ for all channels through the same
procedure as above and assuming an Airy disk beam shape. Although the DSS14
LCP channel is more sensitive than DSS13 RCP, its beam is narrower, so it
limits the effective beam width to 11′, giving a limb coverage of 11%.

Figure 1: Threshold electric field strength $\mathcal{E}_{\rm min}(\theta)$
over angle $\theta$ from the beam axis for different channels of the GLUE
experiment, for a limb pointing. Solid lines show the trigger thresholds I
calculate for each channel, with the dashed line showing the threshold for a
coincidence on both DSS13 RCP channels, while dotted lines show thresholds
based on the cuts of Williams [20]. The cut threshold calculated by Williams
for DSS14 RCP at the centre of the beam (starred) corresponds closely to my
curve. The sensitivity is determined by the highest threshold, which is a
trigger threshold (rather than a cut threshold) across the entire beam. I take
$\mathcal{E}_{\rm min}$ at the centre of the beam to be given by the two-
channel coincidence requirement for DSS13 RCP, as described in the text, and
the beam width to be that at which the trigger threshold for the DSS14 LCP
channel reaches $\sqrt{2}$ times this value, as shown.

The GLUE experiment spent a shorter period of time (see Table 1) pointing
either directly at the lunar centre, or in a half-limb position offset 0.125∘
from this. In these cases, the DSS14 antenna was deliberately defocused, which
reduced its aperture efficiency but improved its sensitivity on the limb of
the Moon. The degree of defocusing was chosen to match the DSS13 beam size, so
under these circumstances I treat DSS14 as a 34 m antenna, and find the
sensitivity to be limited by the 40 MHz DSS14 LCP channel. As there is only
one such channel, $f_{C}=1$. Given the reported system temperatures of 170 K
(half-limb) and 185 K (centre), I find $\mathcal{E}_{\rm rms}$ in this channel
to be 0.0057 and 0.0059 $\mu$V/m/MHz respectively.

The sensitivity in these cases, however, is dramatically affected by the large
angle between the beam centre and the lunar limb. Assuming an Airy disk beam
shape and an apparent lunar size of 31′, the beam power at the closest point
on the lunar limb is 40.7% for a half-limb pointing, and only 0.5% for a
centre pointing. Including these factors as $\mathcal{B}(\theta_{L})$ in Eq.
8, I obtain values for $\mathcal{E}_{\rm min}$ of 0.050 and 0.474 $\mu$V/m/MHz
respectively, greatly increasing the threshold relative to that for a limb
pointing. The advantage of these configurations is that the limb coverage is
increased: 20% for a half-limb pointing, and 100% for a centre pointing since
the beam is equally sensitive to the entire limb.

The off-axis 1.8 GHz receiver on DSS14 used to identify RFI was operated
throughout the experiment and, for most of the data, a cut was applied to
exclude events in which this receiver detected a significant increase in noise
power. Since a lunar-origin pulse could be detected through a sidelobe of its
beam, this cut places an upper limit on the intensity of a pulse that could be
identified by this experiment. The cut was applied to the power averaged over
1 $\mu$s, which is $80\times$ the Nyquist sampling interval for the 40 MHz
bandwidth of the receiver; hence, a band-limited pulse would need an amplitude
of $\sqrt{80}\sigma$ to increase the averaged power by a factor of two, which
was the threshold for the cut. I assume a system temperature for the receiver
of only 30 K, as it was offset from the main beam by 0.5∘ and hence not
directed at the Moon. Due to this offset, it was only minimally sensitive to a
lunar-origin pulse: the beam power $\mathcal{B}(\theta)$ of a 1.8 GHz Airy
disk at 0.5∘ is only 0.16% for DSS14, or 1.43% when defocused. Combining these
parameters with Eq. 8, the threshold $\mathcal{E}_{\rm max}$ for exclusion of
a pulse by this effect is 0.370 $\mu$V/m/MHz, or 0.253 $\mu$V/m/MHz when DSS14
was defocused. Since this latter value is below the detection threshold
$\mathcal{E}_{\rm min}$ for the centre-pointing configuration, I conclude that
this configuration was not sensitive to UHE particles, as any pulse from the
limb of the Moon which was detected in the primary DSS14 beam would also be
detected in the off-axis receiver and thus be excluded as RFI.

There are substantial uncertainties associated with this analysis of the
effects of the anti-RFI cut with the off-axis receiver. The exclusion
threshold is highly sensitive to the assumed system temperature and beam
shape, and realistically it will vary with the power of the off-axis beam at
different points on the limb, rather than taking a single value (for the
centre of the on-axis beam) as assumed here. There is a less serious
approximation involved in conflating the 2.2 GHz primary observing frequency
with the 1.8 GHz frequency of the off-axis receiver, effectively assuming that
an Askaryan pulse will have a flat spectrum across this frequency range.
Finally, this anti-RFI cut was not applied to all of the data, so some
fraction of the observing time will be free of this effect. However, this is
the best representation of this effect that can be achieved with the chosen
set of parameters, and I expect it to be at least approximately correct. Note
that the complete exclusion of the centre-pointing configuration makes little
difference to the total sensitivity of the GLUE experiment, as only a small
fraction of the observing time was spent in this configuration, and previous
work which neglected the anti-RFI cut [12] has already shown that this
configuration had only minimal sensitivity to UHE neutrinos.

###  3.3 Kalyazin

Beresnyak et al. [22] conducted a series of lunar radio observations with the
64 m Kalyazin radio telescope, with an effective duration of 31 hours, using
120 MHz of bandwidth (RCP only) at 2.25 GHz. Pulses in this band triggered the
storage of buffered data both for this channel and for a 50 MHz band with dual
circular polarisations at 1.4 GHz. RFI was excluded by requiring a
corresponding pulse to be visible in both polarisations at 1.4 GHz at a delay
corresponding to the expected ionospheric dispersion, along with further cuts
on the pulse shape and the clustering of their times of arrival. Of 15,000
events exceeding the 2.25 GHz trigger threshold of 13.5 kJy, none met these
criteria.

Interpreting this trigger threshold as an equivalent total flux density in
both polarisations, it is equivalent by Eq. 10 to a threshold of 0.0206
$\mu$V/m/MHz in a radially-aligned linear polarisation. (If it is instead
interpreted as the flux density in the RCP channel alone, the electric field
threshold will be increased by a factor of $\sqrt{2}$.) This value for
$\mathcal{E}_{\rm min}$ neglects several of the scaling factors in Eq. 8,
which I will now apply. For a single channel in a beam directed at the limb,
$f_{C}=\mathcal{B}(\theta)=1$, so only $\alpha$ needs to be calculated to
compensate for inefficiency in reconstruction of the peak pulse amplitude.

Dispersion is negligible at 2.25 GHz over the relatively narrow band of this
experiment. The trigger system is described as having a time resolution of 2
ns, which I take to be the sampling interval, giving a sampling rate of 500
Msample/s, compared with a Nyquist rate of 240 Msample/s. This oversampling
substantially mitigates the signal loss from a finite sampling rate. (Note
that this sampling rate is lower than the maximum 2.5 Gsample/s rate of the
TDS 3034 digital oscillograph used in this experiment [23]; possibly it was
set to less than the maximum value, or the trigger algorithm only processed
every fifth sample. In any case, the improvement in sensitivity from further
oversampling is minimal.) Due to the frequency downconversion, the final phase
of the pulse is essentially random, as described in Sec. 2.1. I simulate these
effects as described in App. A, assuming the downconverted signal to be at
baseband (0–120 MHz), and find a mean signal loss of 13% (i.e. $\alpha=0.87$),
almost entirely from this last effect. Applying this correction, I find an
effective threshold of $\mathcal{E}_{\rm min}=0.0235$ $\mu$V/m/MHz, equivalent
to $F_{\rm min}=17.6$ kJy.

For a pulse to be detected by this experiment it must also have sufficient
amplitude to be visible in the 1.4 GHz band, to distinguish it from RFI.
Assuming a system temperature of 120 K and an aperture efficiency of 60%, both
polarisations at this frequency have a noise level of $\mathcal{E}_{\rm
rms}=0.0025$ $\mu$V/m/MHz. Given $\eta=2$ for circular polarisation and
$\alpha=0.90$ for this band calculated as above, a pulse with an amplitude
matching the threshold $\mathcal{E}_{\rm min}$ at 2.25 GHz would be visible at
1.4 GHz with a significance of $n_{\sigma}=5.9$ in each polarisation. This
exceeds the $\sim 4\sigma$ maximum level expected from thermal noise for the
15,000 stored events, making it sufficient to confirm the detection of a
pulse. The coincidence requirement is thus not the limiting factor on the
sensitivity of this experiment, which is instead determined entirely by the
trigger threshold at 2.25 GHz. Note, however, that I have assumed a flat pulse
spectrum between 1.4 GHz and 2.25 GHz: a pulse could still fail the
coincidence requirement if its spectrum peaked toward the latter frequency. I
have also neglected the scaling factor $f_{C}$ for the coincidence requirement
between the 2.25 GHz band and both 1.4 GHz channels, and my assumptions for
the system temperature and aperture efficiency may be inaccurate, but these
effects are unlikely to reduce the significance of a pulse so much that its
detection cannot be confirmed.

This experiment observed a point offset from the lunar centre by 14′,
effectively on the limb. The resulting limb coverage for the 2.25 GHz beam,
with an FWHM of 7′, is 7%. The 1.4 GHz beam is larger than this, and is thus
able to confirm a detection anywhere within the 2.25 GHz beam, so it does not
further constrain the limb coverage. Dagkesamanskii et al. [24] report further
observations with a new recording system and a lower trigger threshold, but do
not provide enough detail to evaluate the sensitivity of these observations,
so they are not included here.

###  3.4 LUNASKA ATCA

The Lunar Ultra-high-energy Neutrino Astrophysics with the Square Kilometre
Array (LUNASKA) project conducted lunar radio observations with three of the
22 m antennas of the Australia Telescope Compact Array (ATCA), requiring a
three-way coincidence for a successful detection, in February and May 2008
[25, 10]. The pointing of the telescope in the two observation runs was at the
centre and the limb of the Moon respectively, with a total effective duration
of 26 hours. The radio frequency range was 1.2–1.8 GHz, with an analog
dedispersion filter to compensate for ionospheric dispersion over this wide
band, and sampling at 2.048 Gsample/s which aliased the signal from the
1.024–2.048 GHz range to 0–1.024 GHz.

They report a median threshold over their observations of 0.0153 $\mu$V/m/MHz,
not significantly different between the two observing runs, possibly because
the reduced thermal emission from the Moon in the limb pointing of May 2008
was counteracted by the introduction of an anti-RFI filter that removed part
of the band. Their figure already includes most of the effects considered
here: it is averaged over a range of linear polarisation alignments, scaled
for a 50% detection probability given the requirement of a three-way
coincidence, and increased to compensate for the signal loss from the finite
sampling rate, and from the mismatch between the fixed dedispersion
characteristic of their filter and the varying ionospheric STEC. These last
two effects are treated with greater sophistication than in this work, because
they simulate them for pulses with a range of spectra, rather than only for a
flat spectrum. They implicitly assume the pulse to have a base phase of zero,
whereas the inherent phase of an Askaryan pulse is close to the worst-case
value of $\pi/2$ [11], which will be preserved when the signal is
downconverted by aliasing rather than by mixing with a local oscillator
signal, but the original phase will most likely be near-completely randomised
by the remnant dispersion, which is included in their calculation.

I therefore adopt their threshold of 0.0153 $\mu$V/m/MHz without modification
as $\mathcal{E}_{\rm min}(0)$, the threshold at the centre of the beam. For
the limb pointing, I take this value directly as $\mathcal{E}_{\rm min}$, and
use the apparent lunar size of 30′ and an FWHM beam size of 32′ when averaged
over the band from the empirical model of Wieringa and Kesteven [26], which
should provide a more precise result than an Airy disk in this case, to find
the limb coverage to be 36%. For the centre pointing, the same model gives a
beam power at the limb of $\mathcal{B}(\theta_{L})=55.1$% and hence a
threshold of $\mathcal{E}_{\rm min}(\theta_{L})=0.0207$ $\mu$V/m/MHz, with
equal sensitivity around the entire limb.

###  3.5 NuMoon

The NuMoon project [27] conducted a series of lunar radio observations from
June 2007 to November 2008 with the Westerbork Synthesis Radio Telescope
(WSRT), using the PuMa-II backend [28] to combine the signals from eleven of
its fourteen 25 m antennas to form two tied-array beams pointing at opposite
sides of the Moon, in four overlapping 20 MHz bands covering the effective
frequency range 113–168 MHz. They recorded baseband data continuously during
their observations, and retroactively applied dedispersion and a series of
cuts to remove RFI based on pulse width, regular timing, and coincidence
between the two beams. The effective observing time was 46.7 hours, spread out
over 14 observing runs.

They represented their sensitivity in terms of a parameter $S$ which is a
measure of the power in a single beam summed across all four bands, both
polarisations, and five samples (125 ns) in time, such that $S=8$ corresponds
to the mean power or SEFD. The summation over time compensates for uncertainty
in the STEC during the observations, which leads to some remnant dispersion or
excess dedispersion extending a pulse. The events remaining after cuts show a
large excess over the distribution expected from thermal noise, the most
significant event having $S=76$ compared to an expected maximum of $S\sim 30$,
with hundreds of other events falling between these two values. Due to the
large number of these events, they are unlikely to originate from UHE
particles interacting in the Moon, but they are not positively identified as
RFI, and so they limit the sensitivity of this experiment: the detection
threshold must be raised to exclude them.

Due to the low observing frequency of this experiment, dispersion is a large
effect, and even small errors in the STEC used for dedispersion can lead to
pulses being extended in time beyond a five-sample window, preventing the
parameter $S$ from recording their entire power. Buitink et al. [27] simulated
this effect and found that a pulse with an original power equivalent to $S>90$
would have a $>50$% probability of being detected with power in excess of the
most significant event actually recorded in the experiment. This value of $S$
defines the significance threshold, equivalent in the voltage domain to
$n_{\sigma}=\sqrt{90/8}=3.4$. The detection efficiency declines again for
stronger pulses, as they may have sufficient power dispersed over a sufficient
interval to be excluded by the cut on pulse width, but the threshold width for
this cut was chosen to minimise this effect, and I neglect it here.

Since the tied-array beams were formed coherently, I treat all antennas, for a
single polarisation and 20 MHz band, as a single channel. For eleven antennas
each with a diameter of 25 m, and with an aperture efficiency of 33% for the
Low Frequency Front End (LFFE) receivers used in this experiment [29], the
total effective area is 1782 m2. Buitink et al. [27] give a range for the
system temperature of 400–700 K, with the range being due to the varying
contribution from Galactic background noise; I take the central value of 550
K. Given these parameters, I calculate from Eq. 7 the value of
$\mathcal{E}_{\rm rms}$ for a single 20 MHz band in a single polarisation as
0.020 $\mu$V/m/MHz.

All $C=8$ channels (two polarisations and four frequency bands) for a single
beam were separately downconverted to baseband signals, introducing arbitrary
phase factors which were not calibrated, so there is no phase coherence
between them. This is irrelevant, however, because they were combined in the
power domain, which puts this experiment in the fourth regime described in
Sec. 2.2, so that the sensitivity scales as $f_{C}=C^{-1/4}$ regardless of
phase coherence. I modify this slightly because the bands were overlapping and
thus not completely independent, and instead take $f_{C}$ based on the ratio
between a single 20 MHz band and the 55 MHz total bandwidth, with an
additional factor of 2 for the combination of polarisations, as $(2\times
55/20)$${}^{-1/4}=0.65$. This is slightly optimistic, as the combination of
the bands applies a suboptimal uneven weighting between overlapping and non-
overlapping frequency ranges, but this discrepancy should be minor.

The threshold in $S$ already incorporates the effects of dispersion, and the
averaging of power over five consecutive samples will minimise the loss of
pulse amplitude through finite sampling and randomisation of the pulse phase,
so I do not calculate $\alpha$ as in Sec. 2.1. The amplitude of a pulse will,
however, be decreased when it is averaged in time, and I take
$\alpha=1/\sqrt{5}$ to reflect this. The summing of power between
polarisations ensures that $\eta=2$ regardless of the alignment between the
linear polarisations of the receivers and of the pulse, the latter of which is
in this case strongly frequency-dependent due to Faraday rotation. Given these
parameters, and with $n_{\sigma}$ as calculated earlier, I calculate from Eq.
8 the threshold electric field for this experiment to be 0.136 $\mu$V/m/MHz,
equivalent by Eq. 10 to a flux density over the 55 MHz bandwidth of 272 kJy.
The originally-reported value was 240 kJy, but this was for a detection
efficiency of 87.5% (rather than 50%) and assumed perfect aperture efficiency,
which will respectively increase and decrease the threshold.

The limb coverage is dependent on the shape of the tied-array beams, which is
the Fourier transform of the instantaneous u-v coverage of the telescope. The
WSRT is a linear array, which results in an elongated beam oriented
perpendicular to the array axis. The tied-array beam is further tapered by the
primary beam of a single antenna, but this is extremely wide (FWHM of 5∘) and
so does not significantly affect the tied-array beam power around the Moon.
The scale of the beam pattern is determined by the angle between the Moon and
the east-west array axis, which determines the projected array length; I take
this angle to be 65∘, which is its median value during the scheduled time
listed for this experiment in the WSRT schedule
archive111http://www.astron.nl/wsrt-schedule. The eleven WSRT antennas used in
this experiment consisted of nine of the ten fixed antennas with regular 144 m
spacing (RT0–RT4 and RT6–RT9), and two of the four moveable antennas (RTA and
RTB), which are respectively 36 m and 90 m distant from the last fixed antenna
when the array is in the “Maxi-Short” configuration used in this experiment. I
calculate the beam shape based on the u-v coverage of these antennas,
neglecting the minor effect of any phase errors between antennas in forming
the tied-array beams, with the results shown in Fig. 2: each beam has an FWHM
size of 4.2′ in the direction parallel to the array, and is highly elongated
in the transverse direction.

Figure 2: WSRT beams as used in the NuMoon experiment, averaged across the
four bands, for the Moon at the median angle of 65∘ from the WSRT array axis.
Solid lines show the two tied-array beams, pointed at opposite sides of the
Moon; the strong sidelobes at 50–60′ are due to the regular spacing of the
majority of the WSRT antennas, with the sidelobe width due to the large
fractional bandwidth. The upper dashed line shows the primary beam of a single
WSRT antenna, assumed to be an Airy disk. The lower dashed line shows the mean
sidelobe level corresponding to 1/11 of the primary beam power, expected for
random incoherent combination of the signals from eleven antennas. Starred
points show the power of each beam at the centre of the other (the cross-beam
power), which is 27.5%. The overlapping positions of the FWHM beams with
respect to the Moon are shown above the plot; in the transverse direction
(vertical in this figure) they will extend out to the 5∘ scale of the primary
beam.

From the original pointing data for this experiment [30], I find that the
separation between the beams was scaled during each observation to match the
changing resolution of the array. The 2.8′ separation between the centres of
the beams shown in Fig. 2 is for the resolution when the Moon is at 65∘ to the
array axis, as assumed for the calculation of the beam pattern. Since this is
less than the FWHM beam size, the FWHM beams overlap as shown; and since the
scaling of the beam separation matches that of the beam pattern, the
proportional overlap will be constant throughout the observations. Counting
the overlap region only once, the fraction of the limb covered by the two
beams is 14%. Given the low observing frequency of this experiment, at which
the Askaryan pulse from a particle cascade is very broadly beamed and hence
may be detected away from the limb of the Moon, it is arguable that the metric
should instead be the fraction of the nearside lunar surface area within the
FWHM beams, which is 21% in this pointing configuration. By either of these
metrics, the coverage is substantially lower than the figure of 67% given in
the original report.

The original report of this experiment also neglected the possibility of a
lunar-origin pulse being simultaneously detected in both beams, leading to it
being excluded by the anticoincidence cut. A pulse was considered to be
detected, and hence eligible for the anticoincidence cut, if it exceeded a
threshold of $S=20$ or $n_{\sigma}=\sqrt{20/8}=1.58$ in the combined power in
both polarisations, simultaneously in all four bands. The scaling factor
$f_{C}$ must therefore be calculated as the product of factors corresponding
to both methods of combining channels described in Sec. 2.2: one for the
incoherent combination of the two polarisation channels, and one for the
required coincidence between the four bands. The first of these is $2^{-1/4}$
for the two polarisations, as in the earlier calculation of $\mathcal{E}_{\rm
min}$ for this experiment. For the second factor Eq. 13 cannot be used
directly, as the channels being combined in coincidence do not have a Gaussian
distribution: they have a $\chi^{2}$ distribution with ten degrees of freedom
(for the incoherent sum of two polarisations and five consecutive samples in
time), and are in the power domain. Instead, I approximate this distribution
with a Gaussian distribution with equal variance, and apply Eq. 13 with $C=4$
bands and a significance of $\sqrt{2\times 10}\,n_{\sigma}^{2}$ (with the
factor of 2 for the variance of a $\chi^{2}$ distribution, the factor of 10
for the number of degrees of freedom, and the square of $n_{\sigma}$ to
convert to the power domain), taking the square root of the result to return
it to the voltage domain. This gives a value of $1.04$ for the factor of
$f_{C}$ describing the four-band coincidence requirement, which I multiply by
the factor of $2^{-1/4}$ for the combination of the two polarisation channels
to find a combined value of $f_{C}=0.88$. Finally, a lunar-origin pulse
detected at the centre of one beam will be detected in the other beam with its
intensity scaled by the power $\mathcal{B}(\theta)$ of the second beam at this
point, which is shown in Fig. 2 to be 27.5%.

Applying these values for $n_{\sigma}$, $f_{C}$ and $\mathcal{B}(\theta)$ in
Eq. 8, with $\eta=2$ and $\alpha=1/\sqrt{5}$ as in the calculation of
$\mathcal{E}_{\rm min}$, I find the maximum detectable pulse strength to be
$\mathcal{E}_{\rm max}=0.165$ $\mu$V/m/MHz. As this exceeds $\mathcal{E}_{\rm
min}$ by a factor of only 1.2, a lunar-origin pulse must have a strength
within a quite narrow range for it to be detected without being excluded as
RFI, which severely limits the sensitivity of this experiment. As for the GLUE
centre-Moon pointing discussed in Sec. 3.2, I note that the exclusion
threshold will vary across the beam, so it may be less restrictive at some
points. The contribution from thermal noise may also assist in some cases by
chance, elevating the power of a lunar-origin pulse in one beam by a greater
degree than for the other beam, though this effect is limited by the fact that
both tied-array beams are derived from the same set of receivers, so their
noise will be strongly correlated. However, these are minor effects which only
provide a benefit under limited circumstances, and are detrimental at other
times; the parameter values derived above are the best representation of the
average sensitivity of this experiment that can be achieved within the
framework used here.

####  3.5.1 Without anticoincidence cut

Since the anticoincidence cut so strongly limits the sensitivity of the NuMoon
experiment, it is worth considering the sensitivity of this experiment if this
cut had not been applied. With the anticoincidence cut omitted, the most
significant event remaining has an amplitude of $S=86$ (rather than $S=76$).
Assuming linear behaviour in the signal path, this implies that the threshold
for a 50% detection rate in excess of this amplitude is at $S=102$ (rather
than $S=90$) which leads, through the same procedure as describe above, to an
electric field threshold of $\mathcal{E}_{\rm min}=0.145$ $\mu$V/m/MHz.

All other parameters are identical in this case, except for $\mathcal{E}_{\rm
max}$, which is not defined. This set of parameters leads to a minor ($<10$%)
increase in the minimum detectable UHE particle energy, but overall a
substantial increase in the effective sensitivity to UHE particles, if the
experiment is interpreted without the anticoincidence cut. I therefore use
these modified parameters to represent the NuMoon experiment in Table 1 and
Sec. 4.

###  3.6 RESUN

The Radio EVLA Search for Ultra-high-energy Neutrinos (RESUN) project
conducted lunar radio observations with the Expanded Very Large Array (EVLA)
for a total of 200 hours between September and November 2009 [13]. At the
time, this telescope consisted of a mix of antennas of the EVLA and of its
predecessor, the Very Large Array (VLA), but the receiver systems of the
unupgraded antennas were unable to maintain a linear response up to the large
amplitudes required to detect an Askaryan pulse, so this experiment was
conducted only with the upgraded EVLA antennas. They used three subarrays of
four 25 m antennas each, with each subarray pointing at a different point on
the lunar limb; given the FWHM beam size of $\sim 30$′, this achieves coverage
of the entire limb. For each antenna there were two 50 MHz bands centred on
1385 MHz and 1465 MHz, in dual circular polarisations, with all four channels
converted to baseband and coherently summed; the experiment aimed to detect a
coincident pulse with appropriate timing on all four antennas of a single
subarray. No such pulses were detected with a significance exceeding
$n_{\sigma}=4.1$, consistent with the expectation from thermal noise.

The coherent sum between two circular polarisations effectively constructs a
single linear polarisation, with its orientation determined by the relative
phase of the two input channels. Since this phase was not calibrated in this
experiment, the resulting orientation is arbitrary. A pulse with a particular
linear polarisation (e.g. radial to the Moon, as expected for an Askaryan
pulse) will be detected in both circular polarisations with effectively random
phases, and so it will not sum coherently when these two channels are
combined. Since the two frequency bands also have arbitrary phase offsets,
introduced when they are separately downconverted to baseband, the combination
of all four channels (two polarisations in each of two bands) on each antenna
is in the third regime described in Sec. 2.2, and there is no advantage in
sensitivity over a single channel; i.e. $f_{C}=1$, and the value for
$n_{\sigma}$ given above is the significance both in the combined signal and
in a single channel. If the signals in each channel had been squared before
they were summed then the experiment would have been in the fourth regime,
improving the sensitivity (in the voltage domain) by a factor of $\sqrt{2}$.

Adopting the assumptions from Jaeger et al. [13] of $T_{\rm sys}=120$ K and
$A_{\rm eff}=343$ m2 for a single antenna (implying an aperture efficiency of
70%), the noise level in a single 50 MHz channel is $\mathcal{E}_{\rm
rms}=0.0060$ $\mu$V/m/MHz, from Eq. 7. The combined baseband signal, which is
Nyquist-sampled at 100 Msample/s, is subject to inefficiency in amplitude
reconstruction from the finite sampling rate and ambiguity of the pulse phase
as described in Sec. 2.1, for which I find $\alpha=0.79$ with the simulation
from App. A, with dispersion having a negligible effect over this bandwidth.
The four-antenna coincidence requirement at an $n_{\sigma}=4.1$ level
increases the threshold by a factor $f_{C}=1.24$ by Eq. 13. With $\eta=2$ for
circular polarisation, applying these factors in Eq. 8 gives a detection
threshold of 0.055 $\mu$V/m/MHz. This is substantially higher than the
originally-reported value of 0.017 $\mu$V/m/MHz, which was based on the
assumption that the signal would combine coherently between all four channels.
Note, however, that the original publication incorporated the effects of the
coincidence requirement when determining the resulting limit on the UHE
neutrino flux rather than incorporating it into the reported electric field
threshold, which explains part of the difference.

###  3.7 LaLuna

The LaLuna project (Lovell attempts Lunar neutrino acquisition) conducted
preliminary observations with the 76 m Lovell telescope in November 2009 and
May 2010, with an effective time of 1 hour spent observing the lunar limb
[31]. They observed at 1418 MHz with 32 MHz of bandwidth, recording pulses
that occurred in either circular polarisation, and discriminated against
circularly-polarised RFI by requiring that a pulse should appear in both
polarisations simultaneously. However, they detected 6 pulses meeting this
criterion, with no further means to determine whether they were of lunar
origin and no reported upper limit on their amplitude, so no limit can be set
from this experiment on the flux of UHE particles. Spencer et al. [31] have
proposed improving on this by searching for coincident pulses with additional
widely-spaced telescopes, usually used for Very Long Baseline Interferometry
(VLBI), similar to the prospective experiment described in Sec. 3.11.

###  3.8 LUNASKA Parkes

In a continuation of the LUNASKA project, further lunar radio observations
were conducted with the 64 m Parkes radio telescope in April–September 2010
[11, 15], using the frequency range 1.2–1.5 GHz with the Parkes 21 cm
multibeam receiver [32] for an effective observing time of 127 hours.
Interpolation and dedispersion were performed in real time with the Bedlam
backend [9], based on real-time measurements of ionospheric conditions.
Multiple beams were pointed at different points on the limb of the Moon, with
a real-time anticoincidence filter to exclude RFI. Further cuts refined the
anticoincidence criteria, as well as excluding pulses with excessive width or
clustering in their times of arrival. After these cuts, and compensating for
the effects described in Sec. 2.1, there were no events with a significance in
excess of $n_{\sigma}=8.6$, which is consistent with the expected thermal
noise.

The pointing strategy of this experiment placed two beams slightly off the
limb of the Moon to reduce their system temperature by minimising the lunar
thermal radiation they received, as shown in Fig. 3. For each of these beams,
one of their orthogonal linear polarisations was oriented radially to the
Moon, to match the expected polarisation of an Askaryan pulse. For 99 hours of
the observations an additional beam was placed in a half-limb position,
sacrificing sensitivity for slightly improved limb coverage. There were always
four beams in total: the remaining one or two were pointed off-Moon to reduce
their system temperature and make them more sensitive to RFI, to improve the
effectiveness of the anticoincidence filter.

Figure 3: Typical pointing configuration for the LUNASKA Parkes experiment.
Crosses in each beam indicate the orientation of the linear polarisations. I
assume events in the half-limb beam to be most likely to be multiply detected
with one of the adjacent limb beams, and events in the limb beams to be most
likely to be multiply detected with the highly sensitive off-Moon beam.

Due to the real-time processing, the trigger threshold was sufficiently low
that any events exceeding $n_{\sigma}=8.6$ would have been recorded, so it is
this significance that determines the sensitivity of the experiment. The
reported electric field thresholds based on this significance already include
all of the effects considered here, and the limb coverage is determined with
the same approach, so I adopt these values unchanged in Table 1. Note that the
calculation in this case involves scaling the sensitivity by the beam power
$\mathcal{B}(\theta_{L})$ at the closest point on the limb, and the values
$\eta=1$ (limb beams) and $\eta=2$ (half-limb beam) have been adopted because
of their respective polarisation alignments.

The strictest anticoincidence cut was applied at a level of $n_{\sigma}=4.5$,
which imposes a limit on the strongest event which could be detected without
appearing in multiple beams and being excluded as RFI. I consider this limit
for each beam to be determined by the most sensitive adjacent beam (see Fig.
3), as these will have the most strongly overlapping sidelobes. For the limb
beams, this means the limit is determined by the off-Moon beam, for which
$\mathcal{E}_{\rm rms}=0.00038$ $\mu$V/m/MHz based on Eq. 7 and the system
temperature in this beam. With a sidelobe power of 0.5% [15], using Eq. 8,
this gives a value for $\mathcal{E}_{\rm max}$ of 0.0241 $\mu$V/m/MHz. For the
half-limb beam, the limit is determined by the adjacent limb beams, for which
$\mathcal{E}_{\rm rms}=0.00054$ $\mu$V/m/MHz and hence $\mathcal{E}_{\rm
max}=0.0489$ $\mu$V/m/MHz, where I have again used $\eta=2$ to represent the
misalignment between the receiver polarisation and the radius of the Moon in
the half-limb beam.

###  3.9 LOFAR

Singh et al. [33] have proposed lunar radio observations with the Low
Frequency Array (LOFAR), a recently-constructed radio telescope which consists
of a network of phased arrays, with all beamforming accomplished
electronically rather than with movable antennas. Under their scheme, each of
the 24 stations in the core of LOFAR would form a beam covering the entire
Moon, and these signals would be combined to form 50 higher-resolution tied-
array beams covering the face of the Moon. RFI would be excluded in real time
by anticoincidence criteria applied between the tied-array beams. The trigger
algorithm would be based on a subset of the frequency channels of the high-
band antennas (HBAs), and would trigger the storage of buffered data from the
rest of the HBA band and from non-core stations of the telescope, allowing
greater sensitivity for confirmation of events. They consider triggering
algorithms based on different subsets of the HBA frequency range; I take their
‘HiB’ case, for which the effects of dispersion are minimised, and hence they
find the highest detection efficiency. This case corresponds roughly to the
highest-frequency 244 channels within the usable HBA band, each of width 195
kHz, and it is this 142–190 MHz frequency range that is shown in Table 1.
Their sensitivity calculation, however, is based on the entire HBA band of
110–190 MHz, and it is this bandwidth that I use as $\Delta\nu$ for the
calculation below.

The effective aperture for a single LOFAR HBA is

$\displaystyle A_{\rm eff}$
$\displaystyle=\min(\frac{\lambda^{2}}{3},1.5625{\rm~{}m}^{2})$ per antenna
(15)

or 1.09 m2 at the centre of the HiB band. The core region of LOFAR contains 24
stations, each with 2 HBA fields of 24 tiles each, with each tile consisting
of 16 antennas, so its total effective aperture will be 20,025 m2. However,
unlike the steerable dish antennas used in the other experiments considered
here, the phased arrays of LOFAR maintain a fixed orientation on the ground,
and will have a reduced projected area for a source away from zenith. From the
LOFAR site, the Moon reaches a maximum elevation of 56∘, at which the
projected area is reduced to 16,600 m2. I use this value for the effective
aperture, assuming that observations can be scheduled close to transit at the
optimum point in the Moon’s orbit.

The system temperature contains contributions from instrumental noise and
Galactic synchrotron emission:

$T_{\rm sys}=T_{\rm inst}+T_{\rm sky,0}\left(\frac{\lambda}{\rm
1~{}m}\right)^{2.55}$ (16)

where $T_{\rm inst}=200$ K and $T_{\rm sky,0}=60$ K, so the Galactic
background sets a sky temperature of 270 K at the centre of the HiB band, for
a total system temperature of $T_{\rm sys}=470$ K. In this application, the
sky temperature will be influenced by the Moon, which will occult some
fraction of the Galactic background and replace it with its own thermal
emission, but the Moon will occupy only a small fraction of the beam, and its
temperature of 230 K [34] is similar to that of the Galactic background, so
this makes little difference. With the effective aperture and system
temperature derived above, $\mathcal{E}_{\rm rms}$ can be found by Eq. 7 to be
0.0018 $\mu$V/m/MHz.

The proposed trigger algorithm averages the signal power over a number of
consecutive samples with the threshold chosen so that the background trigger
rate from thermal noise is one per minute, to minimise the effect of the 5 s
of dead time while storing the data after each trigger. Singh et al. [33] find
the optimum window length to be 15 samples, finding for this case a detection
efficiency of 50% at a pulse amplitude of $n_{\sigma}=11.0$, assuming perfect
dedispersion. When there is an uncertainty in the STEC of $\pm 1$ TECU,
causing the dispersion to be imperfect, they find their parameter $S_{80}$
(equivalent to $n_{\sigma}$, but for 80% detection efficiency) to be increased
by 14%, so I scale $n_{\sigma}$ by the same ratio, to $12.6$. Achieving this
precision in the STEC measurement will require an improvement over that
achieved in the LUNASKA Parkes experiment, which found typical uncertainties
of $\pm 2$ TECU in retrospective TEC maps based on Global Positioning System
(GPS) data, or $\pm 4$ TECU in real-time ionosonde data [11]. This improvement
may be achieved by interpolating directly between real-time line-of-sight GPS
measurements, which are accurate to better than 0.1 TECU [35], or by measuring
the Faraday rotation of polarised lunar radio emission passing through the
ionosphere [36]. Alternatively, if sufficient processing power is available,
multiple copies of the signal could be dedispersed for different STECs and
searched independently for pulses as suggested by Romero-Wolf et al. [37], at
the cost of an increased trigger threshold required to maintain the same
trigger rate from thermal noise.

The simulations of Singh et al. [33] are more comprehensive than those in this
work for their signal-processing strategy, so I assume all the effects
described in Sec. 2.1 to be incorporated into the significance threshold given
above, and apply no further corrections for the amplitude recovery efficiency
(i.e. $\alpha=1$). Singh et al. describe a triggering algorithm which operates
individually on each polarisation channel, so I take $f_{C}=1$. Since the
pulse power at this frequency is split between linear polarisations by Faraday
rotation, I take $\eta=2$. Combining these with Eq. 8, the trigger threshold
is $\mathcal{E}_{\rm min}=0.031$ $\mu$V/m/MHz. Since a trigger causes the
storage of buffered data for the entire telescope, which improves over the
data available for the trigger by a factor of $\sim 2$ in both collecting area
and bandwidth, there is ample sensitivity to confirm the detection of an
Askaryan pulse in retrospective analysis, so this trigger threshold defines
the sensitivity of the experiment. This threshold spectral electric field is
equivalent, for the full HBA band, to a flux density threshold of 12 kJy,
compared to the value of 26 kJy determined by Singh et al., although their
reported threshold is for a detection efficiency of 80% and averaged over the
FWHM beam, both of which will increase its value.

The anticoincidence criteria applied between the tied-array beams places an
upper limit on the power of a pulse which can be detected without appearing in
multiple beams, and hence being excluded as RFI. This can be mitigated by
applying anticoincidence criteria only between widely-separated beams, to
reduce the overlap between their beam patterns. Singh et al. [33] find the
beam patterns to be complex, with different variation in azimuth and zenith
angles, so I instead represent them with the theoretical mean sidelobe power
level corresponding to the incoherent combination of the signals from 24
stations, which is $\mathcal{B}(\theta)=1/24=4.2$%. Since the stations of the
LOFAR core have a much less regular distribution than the antennas of the
WSRT, this is likely to be a better approximation than it is for the WSRT
tied-array beams in Fig. 2. I assume that RFI can be effectively excluded by
setting the anticoincidence significance threshold at half the trigger
threshold, consistent with the results from the LUNASKA Parkes experiment
[11], so I take $n_{\sigma}=6.3$ for the exclusion threshold. The experience
of Buitink et al. [27] suggests that this is insufficient to deal with the
increased RFI at low observing frequencies, but the use of a two-dimensional
array in this case rather than the one-dimensional WSRT may counteract this,
as it avoids the strong sidelobes a one-dimensional array has on the RFI-rich
horizon. Combining these values with Eq. 8 gives us $\mathcal{E}_{\rm
max}=0.077$ $\mu$V/m/MHz.

Assuming a duration of 200 hours, comparable with previous lunar radio
experiments, the 5 s per minute of dead time after each trigger results in an
effective observing time of 183 hours. Since LOFAR is electronically steered,
and additional beams can be formed with sufficient signal-processing hardware,
with future upgrades it may be possible to achieve much greater observing
times by observing commensally with other projects: the Moon is above 30∘ in
elevation from the LOFAR site for 1,490 hours per year, or 1,360 hours after
allowing for dead time.

###  3.10 Parkes PAF

Bray et al. [38] have proposed continued observations with the Parkes radio
telescope using one of the phased array feed (PAF) receivers developed for the
Australian Square Kilometre Array Pathfinder (ASKAP). These receivers [39]
combine the signals from elements in the focal plane to form multiple beams
within the field of view of the antenna. This would allow a lunar radio
experiment to improve over the previous LUNASKA Parkes experiment with the 21
cm multibeam receiver (Sec. 3.8) by forming beams around the entire limb of
the Moon, rather than the limited coverage shown in Fig. 3. The frequency
range of these receivers is 0.7–1.8 GHz, not all of which will be processed
for the 36 antennas of ASKAP, but the use of a single receiver on the 64 m
Parkes antenna could justify processing the entire band. The major
disadvantage of these receivers is their high system temperature ($\sim 50$
K), but this is less significant for a lunar radio experiment because the
total system temperature is dominated by lunar thermal emission. Apart from
the new receiver, this experiment would function similarly to the LUNASKA
Parkes experiment, with real-time dedispersion and anticoincidence filtering
between the beams to exclude RFI. I assume a duration of 200 hours as for
LOFAR in Sec. 3.9, but with a duty cycle of only 85%, consistent with the loss
of effective observing time from data storage and false positive rates of
anti-RFI cuts in the LUNASKA Parkes experiment.

The positioning of the beams relative to the limb is a trade-off between beam
power on the limb and lunar thermal noise. Assuming the beams to be positioned
slightly away from the Moon, as for the limb beams in Fig. 3, approximately 12
beams are required to achieve complete limb coverage. As the base system
temperature for the ASKAP PAFs is $\sim 25$ K higher than that of the receiver
used for the LUNASKA Parkes experiment, I take the total system temperature to
be increased by this amount relative to the limb beams in that experiment,
which gives $T_{\rm sys}=80$ K. The effective aperture of the 64 m Parkes
antenna with a PAF, given the stated 80% aperture efficiency of these
receivers, is 2,574 m2. By Eq. 7 the noise level can then be found to be
$\mathcal{E}_{\rm rms}=0.00038$ $\mu$V/m/MHz.

The pointing assumed above, 4′ from the lunar limb, implies a beam power of
77.7% at the closest point on the limb, assuming an Airy disk and averaging
across the band. I assume the native orthogonal linear polarisations of the
receiver to be coherently summed with an appropriate phase offset to form
channels with linear polarisations aligned radially to the Moon for each beam,
implying $\eta=1$. This neglects the effects of Faraday rotation, which is not
very significant for this frequency range: under typical conditions (STEC of
20 TECU; projected geomagnetic field of 50 $\mu$T along the line of sight) the
polarisation of a lunar-origin pulse will be subjected to a differential
rotation of 23∘ between the minimum and maximum frequencies, corresponding to
a $\sim 1$% loss of signal power for a receiver oriented to match the
polarisation at the centre of the band.

Assuming an STEC uncertainty of 1 TECU, as for LOFAR in Sec. 3.9, and also
assuming effectively-complete interpolation and formation of the signal
envelope, the signal recovery efficiency determined by the simulation in App.
A is $\alpha=0.89$. Taking a significance threshold of $n_{\sigma}=8.8$, which
is the expected maximum level of the thermal noise in 12 channels over the
assumed observing time (from Ref. [9], Eq. 46), Eq. 8 then gives
$\mathcal{E}_{\rm min}=0.0043$ $\mu$V/m/MHz. As for the LUNASKA Parkes
experiment, partial optimisation of the signal in real time should allow the
trigger threshold to be set low enough that any events exceeding this
threshold are stored, so that the sensitivity of the experiment is determined
by this value for $\mathcal{E}_{\rm min}$ determined for a fully-optimised
signal.

As for other experiments using an anticoincidence filter to exclude RFI, the
possibility of a lunar-origin pulse being detected in multiple beams places an
upper limit on the detectable pulse strength. I take the sidelobe beam power
to be 0.5%, the same as for the Parkes 21 cm multibeam receiver. As for LOFAR
in Sec. 3.9, I assume an anticoincidence significance threshold of half the
trigger threshold, or $n_{\sigma}=4.4$, consistent with the successful
exclusion of RFI in the LUNASKA Parkes experiment. Combining these values with
Eq. 8, I find $\mathcal{E}_{\rm max}=0.030$ $\mu$V/m/MHz.

###  3.11 AuScope

The AuScope VLBI array [40] is a recently-completed array of three 12 m
antennas with baselines ranging from 2,360 km to 3,432 km. Its primary purpose
is geodesy, observing fixed radio sources in order to improve the precision of
the terrestrial and celestial reference frames, but it may also be used for
observational radio astronomy. It is less heavily subscribed than the other
telescopes considered for lunar radio experiments, so longer observation times
are possible: during each year the Moon is visible from all three antennas for
2,900 hours, which I take as the observing time.

Each antenna is equipped with a combined S- and X-band receiver with dual
circular polarisations. Of these bands, only the S band is useful in this
application, with a frequency range of 2.2–2.4 GHz. The beam at this frequency
is larger than the Moon, with an FWHM Airy disk size of $\sim 38$′, indicating
that the optimum observing strategy is to point at the centre of the Moon in
order to achieve equal sensitivity around the entire lunar limb. Using the
lunar thermal emission model of Moffat [41] as applied in Ref. [11], the Moon
contributes 69 K to the system temperature in this pointing configuration, for
a total system temperature of 154 K when combined with the 85 K base level of
the receivers. With the reported aperture efficiency of 60%, the effective
aperture for each antenna is 69 m2, so from Eq. 7 I find the noise level in a
single polarisation channel to be $\mathcal{E}_{\rm rms}=0.0076$ $\mu$V/m/MHz.

The simplest way to perform this experiment is to search for coincident pulses
on all six channels (two polarisations on each of three antennas). Each
antenna would be monitored for a linearly-polarised Askaryan pulse appearing
simultaneously in both circular polarisation channels, which would trigger the
storage of voltage data for the event. This would eliminate the majority of
the RFI, as in the GLUE and LaLuna experiments, so the resulting trigger rate
should be dominated by thermal noise. These stored events would then be
compared retrospectively, to find any coincident events on all three antennas
with relative times of arrival indicating that they originated from the Moon.
As RFI sources are unlikely to be simultaneously visible to such widely-
separated antennas, this criterion should provide effectively-complete
rejection of RFI.

To find the effective significance threshold, I consider the trigger rates
$R_{1}$ in a single polarisation channel, $R_{2}$ for the rate of coincidences
between both polarisations channels on a single antenna, and $R_{6}$ for six-
fold coincidences between both polarisations on all three antennas with
reconstructed pulse origins on the Moon. The last two of these are related by

$R_{6}=R_{2}^{3}\,W^{2}$ (17)

where $W$ is the time window corresponding to the range of arrival directions
across the face of the Moon, typically $\sim 30$–100 $\mu$s over these
baselines. Setting $R_{6}$ equivalent to a single detection in the observing
time of the experiment, to obtain the expected level of the thermal noise, I
find $R_{2}$ to be 0.1–0.3 Hz. This is the required trigger rate on each
antenna for the sensitivity to be limited by thermal noise rather than by the
trigger threshold, and is sufficiently low that the minimal data required on
each trigger can be recorded without incurring significant dead time. The
relation to the trigger rate $R_{1}$ in a single polarisation channel is

$R_{2}=R_{1}^{2}\,\frac{1}{\Delta\nu},$ (18)

assuming that the delay between the two polarisation channels can be
calibrated to a precision comparable to the scale of the inverse of the
bandwidth $\Delta\nu$, resulting in typical $R_{1}$ values in the range 5–8
kHz. If the inter-polarisation delay can be calibrated to a small fraction of
the inverse bandwidth, then the two channels could be summed incoherently (in
the the fourth regime described in Sec. 2.2) rather than being operated in
coincidence, allowing an improvement in sensitivity by a factor $2^{1/4}$, but
I do not assume this here.

This trigger rate $R_{1}$ makes it possible to find the trigger threshold in a
single polarisation channel for which a single global coincidence is expected
from thermal noise, equivalent to the limiting significance threshold
$n_{\sigma}$ of the experiment. I assume effectively-complete interpolation
and formation of the signal envelope, implying $\alpha=1$, given that
dispersion is negligible at this observing frequency. The trigger threshold
for the signal envelope can then be found (from Ref. [9], Eq. 46) as
$n_{\sigma}=4.8$, with no significant variation across the range of values
found for $R_{1}$. Given a beam power of $\mathcal{B}(\theta_{L})=62$% on the
limb for an Airy disk centred on the Moon, $\eta=2$ for circular polarisation,
and a scaling factor $f_{C}=1.26$ for the required six-channel coincidence
from Eq. 13, I find $\mathcal{E}_{\rm min}$ from Eq. 8 to be 0.0083
$\mu$V/m/MHz.

The feature that most clearly distinguishes this potential experiment from the
others described here is the length of the baselines between the antennas.
Apart from improving the efficacy of RFI rejection, this also allows the
position on the Moon of the particle cascade responsible for a detected pulse
to be determined with high precision, which is a vital piece of information
for determining the direction of origin of the primary UHE particle. The
disadvantage of the long baselines is the statistical penalty imposed by the
increased search space for a coincident pulse, which leads to a threshold
significance (as calculated above) higher than that for the otherwise similar
RESUN experiment. An additional concern is that the narrowly-directed Askaryan
pulse may not be visible to all of the antennas, which are separated by up to
0.5∘ as seen from the Moon. However, the angular scale $\Delta\theta$ of the
Askaryan radiation pattern at this observing frequency is 2.4∘ (see Eq. 8 of
Ref. [42]), larger than the separation between antennas, so this does not pose
a significant problem.

##  4 Sensitivity to ultra-high-energy particles

The first detailed estimation of the particle aperture of a lunar radio
experiment comes from the Monte Carlo simulations of Gorham et al. [19], which
were followed by further simulations by Beresnyak [43], Scholten et al. [44],
Panda et al. [45] and James and Protheroe [12], and an analytic approach by
Gayley et al. [6]. Comparing these models is difficult, because the code for
each simulation is generally not published, and reimplementing them from their
published descriptions is laborious, but it is possible to compare their
published results when several models have been applied to the same
experiment. The most detailed simulations to date, those of James and
Protheroe, find results that are more pessimistic (lower aperture) than those
reported for the GLUE experiment [5] (simulations from Ref. [19]) by around an
order of magnitude, more pessimistic than those reported for the NuMoon
experiment [27] (simulations from Ref. [44]) by a similar factor [46], and
approximately consistent [25] with those reported for the Kalyazin experiment
[22] (simulations from Ref. [43]). Gayley et al. also calculate the aperture
for the GLUE experiment with their analytic model, finding results consistent
with those of James and Protheroe.

Perfect agreement between these models is not expected, as they make different
physical assumptions regarding the spectrum and angular distribution of
Askaryan radiation, the physical properties of the lunar regolith, etc.
However, even with these assumptions matched as closely as possible between
different simulations, there remain in some cases discrepancies in the results
(see App. A of Ref. [25]), which may be due to errors in their implementation
in software. The analytic model of Gayley et al. avoids this problem because
its published version includes the complete derivation of its final result,
allowing it to be rigorously checked by other researchers. However, it makes
several approximations in order to obtain a result in closed form, such as
assuming constant elasticity for neutrino-nucleon interactions, and a constant
transmission coefficient for radiation passing through the regolith-vacuum
boundary, which may affect its accuracy.

The use of lunar radio observations was originally suggested by Dagkesamanskii
and Zheleznykh [1] primarily for the detection of neutrinos, and most of the
above models were originally developed with this purpose in mind, neglecting
the possibility of detecting UHECRs. The simulations of Scholten et al. and
James and Protheroe have been applied to calculating the aperture for the
detection of UHECRs, and the analytic model of Gayley et al. has been adapted
to this purpose by Jeong et al. [7]. However, none of these models have been
compared in this context.

In this section, I calculate the sensitivity of the lunar radio experiments
listed in Sec. 3 to both neutrinos (Sec. 4.1) and UHECRs (Sec. 4.2), based on
the analytic models of Gayley et al. and Jeong et al. respectively, with some
modifications as described in the corresponding sections. The implementation
of these models is described in detail in App. B, and the parameters used
listed in Table 1. For the case of neutrinos, I compare the results with those
from the simulations of James and Protheroe in greater detail than previous
work, in Sec. 4.1.1.

The models used here do not include any correction for the effects of small-
scale lunar surface roughness, which may cause a large (more than an order of
magnitude) increase in aperture at high particle energies, at least at high
frequencies [10]. Accordingly, the results in this section may be taken as a
comparison of lunar radio experiments, but should not be taken as a precise
measure of their absolute sensitivity. Further development of aperture models
— either these analytic models, or simulations — is strongly motivated.

For experiments with only a minimum threshold electric field $\mathcal{E}_{\rm
min}$, the models described in App. B can be applied directly, finding the
aperture due to the detection of events with electric field
$\mathcal{E}>\mathcal{E}_{\rm min}$. For experiments which also have a maximum
threshold electric field $\mathcal{E}_{\rm max}$, I find the aperture as

$A(E)=A(E;\mathcal{E}_{\rm min})-A(E;\mathcal{E}_{\rm max}),$ (19)

which excludes events which would be detected with electric field
$\mathcal{E}>\mathcal{E}_{\rm max}$. When $\mathcal{E}_{\rm
min}>\mathcal{E}_{\rm max}$, as for the centre-pointing configuration of the
GLUE experiment, the aperture is zero.

The aperture $A_{\rm P}(E)$ can be found separately for each pointing
configuration P used in an experiment. The total exposure for an experiment is
found by summing the exposure for each pointing, as

$X\\!(E)=\sum_{\rm P}A_{\rm P}(E)\,t_{\rm obs,P}.$ (20)

The 90%-confidence model-independent limit set by the experiment to a diffuse
isotropic particle flux, assuming zero detected events, is then

$\frac{dF_{\rm iso}}{dE}<\frac{2.3}{E\,X\\!(E)}$ (21)

where the factor of 2.3 is the mean of a Poisson distribution for which there
is a 10% probability of zero detections.

###  4.1 Neutrinos

I find the sensitivity of lunar radio experiments to neutrinos using the model
of Gayley et al. [6], with one modification for consistency with the
simulations of James and Protheroe [12]. The two models are otherwise
consistent in their assumptions, but they differ in the way they treat the
composition of the Moon. James and Protheroe assume a surface regolith layer
of depth 10 m underlaid by a sub-regolith layer of effectively infinite depth,
both of which are characterised by their density $\rho$, their refractive
index $n_{r}$, and their electric field attenuation length for radio waves
$L_{\gamma}$, defined in terms of $\lambda$, the radio wavelength in vacuum.
Values for these parameters are given in Table 2. Gayley et al. make the
simplifying assumption that all detectable particle cascades occur in the
regolith, for which they take the same values as James and Protheroe for
$\rho$ and $n_{r}$, but for $L_{\gamma}$ they give an expression equivalent to
29$\lambda$, matching the value used byJames and Protheroe for the sub-
regolith layer. I modify the model of Gayley et al. by instead taking
$L_{\gamma}=60\lambda$, matching the value that James and Protheroe use for
the surface regolith layer.

Table 2: Regolith parameters in different neutrino aperture models.

Model | layer | $\rho$ | $n_{r}$ | $L_{\gamma}$
---|---|---|---|---
(g cm-3) |  |
James and Protheroe [12] | regolith | 1.8 | 1.73 | 60$\lambda$
sub-regolith ${}^{\textrm{a}}$ | 3.0 | 2.50 | 29$\lambda$
Gayley et al. [6] | regolith | 1.8 | 1.73 | 29$\lambda$
this work | regolith | 1.8 | 1.73 | 60$\lambda$

  * 1.

${}^{\textrm{a}}$ Below depth of 10 m.

This value for $L_{\gamma}$ corresponds to a loss tangent of $1/60\pi
n_{r}=0.003$. The loss tangent of the regolith is determined primarily by the
(depth-dependent) density and the abundances of FeO and TiO2, with this value
equivalent to a combined abundance of $\sim 10$% at the surface (see Fig. 6 of
Ref. [47]), which is a reasonable approximation for the varied abundance over
the surface of the Moon [48]. At a depth of 10 m or more the loss tangent is
roughly doubled, corresponding to the halved value of $L_{\gamma}$ that James
and Protheroe use for the sub-regolith layer.

By matching the parameters used by James and Protheroe for the surface
regolith layer, I should find an equal contribution to the effective aperture
from neutrinos interacting in this volume, but I should find a different
contribution from the volume represented by the sub-regolith layer. Compared
to their work, the value used here for the attenuation length of the sub-
regolith layer is 2.1 times larger, leading to a corresponding increase in the
detector volume, while the value for the density of this layer is 1.7 times
smaller, leading to a corresponding decrease in the neutrino interaction rate;
combined, these should lead to the neutrino aperture of the sub-regolith layer
being overestimated here by a factor of 1.2. The analytic model used here also
neglects the transmission losses at the regolith/sub-regolith interface
modelled by James and Protheroe, which will cause it to further overestimate
the aperture contribution from the sub-regolith layer. These inaccuracies will
be most significant for low radio frequencies and high neutrino energies, for
which the sub-regolith contributes the largest fraction of the total aperture.

####  4.1.1 Comparison of analytic and simulation results

Figure 4: Comparison of neutrino apertures from the analytic model used in
this work (thin lines) and previously-reported apertures (thick lines) from
the simulations of James and Protheroe [12], for the LUNASKA ATCA experiment
[10] (left) and the LUNASKA Parkes experiment [15] (right), for a range of
pointings (solid, dashed, dash-dotted). The ratio between apertures from
analytic and simulation results (lower plots) shows that, compared to
simulations, the analytic model tends to underestimate the aperture at low and
high neutrino energies, but is approximately accurate at intermediate
energies.

The originally-reported apertures for the LUNASKA ATCA and LUNASKA Parkes
experiments are based on the simulations of James and Protheroe, so the level
of agreement between these and the apertures calculated in this work may be
taken as a measure of the accuracy of the simplifying assumptions used in the
model of Gayley et al., and the further assumptions made in my implementation
thereof. For the LUNASKA ATCA experiment, this includes the assumption of a
flat bandpass made in this work, as a piecewise linear approximation to the
bandpass was used in calculating the originally-reported limit; for the
LUNASKA Parkes experiment, with a narrower band, a flat bandpass is assumed in
both the original report and this work.

A comparison of the apertures from the original reports and in this work is
shown in Fig. 4. For both experiments, the apertures derived in this work
indicate a higher neutrino energy threshold than those from the original
reports, agree approximately at slightly higher energies, and (in most cases)
indicate a lower aperture than the original reports at higher energies. The
form of this deviation matches that found in a previous comparison [6] for the
GLUE experiment, though an absolute comparison is difficult, as no explanation
is given by Gayley et al. for their choice of the limb coverage parameter
$\zeta$.

The simplest explanation for the first discrepancy — the increased energy
threshold in the analytic model — is that it is due to the variable
inelasticity of neutrino-nucleon interactions (e.g. [49]): the interactions of
lower-energy neutrinos may be detectable only when a large fraction of their
energy is manifested in the resulting hadronic particle cascade, rather than
the flat rate of 20% assumed in this work, resulting in a lower detectable
neutrino energy threshold for models (such as those of James and Protheroe)
which include this effect.

Alternatively, the first discrepancy may also be due to the charged leptons
(electrons, muons and taus) produced by neutrino-nucleon charged-current
interactions, which are also neglected in this work. These particles typically
carry $\sim 80$% of the energy of the primary neutrino, and are thus capable
of initiating a particle cascade which is detectable even when the primary
hadronic cascade (with the remaining $\sim 20$% of the energy) is below the
detection threshold; however, muons and taus do not generally initiate a
single cascade containing the majority of their energy, and the
electromagnetic cascade initiated by a UHE electron is elongated by the LPM
effect [50, 51] causing the resulting Askaryan radiation to be directed in a
very narrow cone, and hence are unlikely to be detected. Consequently, these
secondary leptons make only a minor ($\sim 10$%) contribution [12] to the
neutrino aperture in the energy range in which the primary hadronic cascade is
detectable, but the possibility of detecting the electromagnetic cascade from
a charged-current interaction of an electron neutrino provides some minimal
sensitivity down to a lower threshold neutrino energy than would otherwise be
the case, matching the observed discrepancy in the threshold. This is also
consistent with James and Protheroe, who find the fractional contribution to
the neutrino aperture of these primary electromagnetic cascades to be larger
for lower neutrino energies. However, this contribution was omitted from the
simulations for the LUNASKA Parkes experiment, so it can only assist in
explaining the discrepancy seen for the LUNASKA ATCA experiment.

The second discrepancy — the decreased neutrino aperture at high energies in
the analytic model — is in the wrong direction and probably much too large to
be explained by the different treatment of the sub-regolith layer. One
possible explanation is that it is a consequence of the small-angle
approximations made by Gayley et al. [6], under the assumption that a particle
cascade is only detectable from a point very close to the Cherenkov angle,
which becomes less accurate at higher energies. Part of the discrepancy may
also be caused by the way the aperture calculation in Eq. 19 incorporates the
maximum threshold $\mathcal{E}_{\rm max}$, which is a more significant
constraint at higher energies; this is supported by the lesser discrepancy
found for the LUNASKA ATCA experiment, which did not apply an anticoincidence
filter and therefore had no maximum threshold. Finally, the discrepancy may be
largely due to the assumption of a fixed limb coverage parameter $\zeta$: at
high energies, particle cascades may be visible outside the fraction of the
lunar limb covered by the primary telescope beam, through the beam sidelobes,
which is neglected in the analytic model. This explanation is supported by the
absence of this discrepancy for the Moon-centre pointing of the LUNASKA ATCA
experiment, for which I take $\zeta=100$%. Future refinement of the analytic
model might benefit from incorporating an energy-dependent limb coverage
parameter $\zeta(E)$ to correct for this effect. Note that all of the
prospective future experiments considered in Secs. 3.9–3.11 have 100% limb
coverage, so this effect should not apply to them.

Most importantly, the analytic model of Gayley et al. [6] as implemented in
this work produces apertures which are consistent with the simulations of
James and Protheroe [12] at intermediate energies, around the region of
maximum sensitivity to an $E_{\nu}^{-2}$ neutrino spectrum. The apertures in
this region are consistent within a factor of two, which may be taken as the
uncertainty associated with the implementation of this model of the neutrino
aperture. This is smaller than the uncertainties associated with the neutrino-
nucleon cross-section [49], or with small-scale lunar surface roughness [10].

####  4.1.2 Comparison of different experiments

The neutrino apertures that I calculate for the experiments in Sec. 3 are
shown in Fig. 5. They show trends that are familiar from previous work, but
worth revisiting. The aperture for each experiment increases rapidly above
some threshold neutrino energy for which the Askaryan radio pulse is strong
enough to detect, and continues to increase, more slowly, at higher energies,
both due to the increased radio pulse strength which allows a cascade to be
detected deeper in the regolith, and because the neutrino-nucleon cross-
section increases with energy, making down-going neutrinos more likely to
interact in the regolith. By comparison with Table 1, we see that minimum
detectable neutrino energy is determined by $\mathcal{E}_{\rm min}$, and the
aperture for higher-energy neutrinos is determined by the limb coverage
$\zeta$. Lower-frequency experiments (NuMoon and LOFAR) have a larger
aperture, as they can detect cascades over a wider range of angles or at
greater depths beneath the lunar surface, although this latter effect may be
overestimated here due to the optimistic assumptions regarding the sub-
regolith layer. The parameter $\mathcal{E}_{\rm max}$ has little effect on the
aperture, implying that the detectable cascades are dominated by those
producing radio pulses with amplitudes only slightly exceeding
$\mathcal{E}_{\rm min}$.

Figure 5: Neutrino apertures for the experiments listed in Sec. 3, calculated
with the analytic model used in this work. For experiments which used multiple
pointing configurations, on the limb, half-limb or centre of the Moon, the
aperture for each pointing is shown individually.

For past experiments, the corresponding limits on the diffuse neutrino flux
are shown in Fig. 6, compared to the limits originally reported for each
experiment. For future experiments, limits are shown in Fig. 7, along with
predicted neutrino fluxes from the decay of superheavy particles from kinks in
cosmic strings in the model of Lunardini and Sabancilar [52]. These are the
most optimistic predictions not yet excluded by other (non-lunar) neutrino
detection experiments; this is the class of models which are most suited to
being tested by lunar radio experiments. For the most optimistic of the fluxes
shown in this figure, the LOFAR experiment would expect to detect 5.1
neutrinos in a nominal 200 hours of observing time, or exclude it with a
confidence of 99% if no neutrinos were detected.

Figure 6: Limits on the diffuse neutrino flux set by the past experiments
listed in Sec. 3. Solid lines show the limits derived in this work based on
the parameters in Table 1, while dotted lines show previously-reported limits
for the Parkes [18], GLUE [5], Kalyazin [22], LUNASKA ATCA [10], NuMoon [27],
RESUN [13] and LUNASKA Parkes [15] experiments. In the case of the Kalyazin
experiment, this is a model-dependent limit for an $E_{\nu}^{-2}$ neutrino
spectrum, and has been rescaled from 95% to 90% confidence. Figure 7: Limits
on the diffuse neutrino flux that may be set by the near-future experiments
listed in Sec. 3, for the nominal observing times given in the text. Dashed
lines show the potential limits derived in this work based on the parameters
in Table 1, while solid lines (unlabelled) show the limits set by past
experiments from Fig. 6. Dash-dotted lines show models of the potential
neutrino flux from kinks in cosmic strings [52].

The limits found in this work for past experiments, shown in Fig. 6, are
generally less constraining than those originally reported for each
experiment; in some cases, dramatically so. This may result from differences
between the original analysis and the re-analysis in this work either in the
calculation of the sensitivity of the experiment to coherent radio pulses, or
in the model used to translate this radio sensitivity to a neutrino aperture.
To discriminate between these possibilities, Fig. 8 also shows, for selected
experiments, neutrino limits calculated with the aperture model used in this
work, but with the radio sensitivity from the original reports. For the GLUE
experiment, the limits I calculate for this plot are for the limb pointing
only, as this is the only configuration for which Williams [20] reports the
radio detection threshold ($\mathcal{E}_{\rm min}=0.00914$ $\mu$V/m/MHz) — but
this configuration was used for a majority (59%) of the total observing time
for this experiment, and had a lower radio detection threshold than other
pointings, so the limit set by this pointing alone is close to that for the
entire experiment. For NuMoon, the reported flux density threshold $F_{\rm
min}=240$ kJy was converted to a minimum spectral electric field
$\mathcal{E}_{\rm min}=0.128$ $\mu$V/m/MHz with Eq. 10, using the 55 MHz
bandwidth of the experiment, and the limb coverage of $\zeta=0.67$ was taken
from the original report [27]. For RESUN, the originally-reported radio
detection threshold is $\mathcal{E}_{\rm min}=0.017$ $\mu$V/m/MHz [13]. All
other parameters for the radio sensitivity of these experiments are as given
in Table 1.

Figure 8: Limits on the diffuse neutrino flux set by selected past
experiments, showing versions of each limit calculated with different models,
to illustrate the effects of the choice of model at each stage of the
calculation. As in Fig. 6, solid lines show limits derived in this work, and
dotted lines show limits from the original reports [5, 27, 13]. Dashed lines
show limits calculated with the neutrino aperture model used in this work, but
based on the radio pulse detection thresholds from the original reports, as
described in the text. The upper solid line for the NuMoon experiment shows
the limit after allowing for the effect of the anticoincidence filter between
the two on-Moon beams described in Sec. 3.5 (i.e. without the modified
analysis in Sec. 3.5.1).

For the GLUE experiment, the neutrino limit calculated in this work with the
originally-reported radio sensitivity is more similar to the limit calculated
with the revised radio sensitivity from Sec. 3.2 than to the limit from the
original report. This indicates that the bulk of the discrepancy is due to the
relative optimism of the simulations of Gorham et al. [19], as previously
found by James and Protheroe [12] and Gayley et al. [6]. The limit calculated
here with the radio detection threshold and lunar coverage from the original
report of the NuMoon experiment is a factor $\sim 6$ less constraining than
that reported by Buitink et al. [27], roughly matching a factor $\sim 10$
found by James et al. [46] in a similar test with their own aperture model.
The limit is relaxed by a further factor $\sim 5$ when using the revised radio
sensitivity derived in Sec. 3.5, in proportion with the decrease in the
estimated lunar coverage, and by a final factor $\sim 5$, or more at higher
energies, if the radio sensitivity is calculated with the parameter
$\mathcal{E}_{\rm max}$ based on the anticoincidence cut applied in this
experiment (i.e. neglecting the modified analysis in Sec. 3.5.1). For the
RESUN experiment, the limit from the original report and the limit calculated
here based on the same radio detection threshold use almost the same aperture
model, but the differences (in the treatment of the regolith, and of thermal
noise) cause the latter to be slightly (factor $\sim 1.5$) more constraining.
The reduced sensitivity to neutrinos shown for this experiment in Fig. 6 is
therefore entirely due to the revised radio sensitivity calculated in Sec.
3.6.

###  4.2 Cosmic rays

I estimate the sensitivity of lunar radio experiments to CRs using the model
of Jeong et al. [7], with one simple but highly significant modification.
Jeong et al. based their model for the CR aperture on the model of Gayley et
al. [6] for the neutrino aperture, which correctly took the energy of a
neutrino-initiated hadronic particle cascade to be $\sim 20$% of the original
neutrino energy, as described in Sec. 4.1.1. For CRs, however, 100% of the CR
energy goes into a hadronic particle cascade. The result of this correction is
to increase the expected radio pulse amplitude, and thus to decrease the
detection threshold in the CR energy, by a factor of five. Note that other
models [44, 12] already assume 100% of the CR energy to go into a hadronic
particle cascade, so no modification is implied to results based on these
models.

The CR apertures that I calculate for the experiments in Sec. 3 are shown in
Fig. 9, and display several differences from the neutrino apertures in Fig. 5.
Because all CRs interact very close to the lunar surface, and at sufficiently
high energies they are almost all detectable, the CR aperture increases only
slowly at high energies. For experiments with a maximum threshold
$\mathcal{E}_{\rm max}$, the aperture decreases at high energies, implying
that the Askaryan radio pulses from these events are dominated by strong
pulses which may be rejected by anticoincidence criteria. As in Fig. 5, the
low-frequency experiment with LOFAR has a larger maximum aperture than other
experiments, though in this case this is purely because a cascade may be
detected from a broader range of angles.

Figure 9: CR apertures for the experiments listed in Sec. 3, calculated with
the analytic model used in this work. As in Fig. 5, apertures for each
pointing configuration are shown individually. Note the characteristic
decrease in the aperture at high energies for experiments which apply
anticoincidence rejection, and hence have a defined maximum radio threshold
$\mathcal{E}_{\rm max}$ (see Table 1).

The corresponding limits on the diffuse CR flux are shown in Fig. 10, compared
to the only such limit that has been previously published, for the NuMoon
experiment [53]. As in Sec. 4.1.2, the limit found for this experiment in this
work is significantly less constraining.

Figure 10: Limits on the diffuse CR flux set by the experiments listed in Sec.
3. Solid lines show the limits derived in this work based on the parameters in
Table 1, while a dotted line shows the previously-reported limit for the
NuMoon experiment [53], the only one of these experiments for which such a
limit has been published. Dashed lines show the limits that may be set by
near-future experiments, for the nominal observing times given in the text.
The measured flux shown is from observations by the Pierre Auger Observatory
[54], with a 22% systematic uncertainty $\sigma_{\rm sys}$ in the energy
scale, and the corresponding limit at higher energies (dotted) is based on its
contemporary exposure of 12,790 km2 sr yr (now 66,000 km2 sr yr [55]), with
the same definition as the other limits.

Of the past lunar radio experiments shown here, the LUNASKA Parkes experiment
came closest to being able to detect the known CR spectrum, with 0.09 events
expected to be detected based on a parameterisation of the spectrum [54], or a
range of 0.04–0.19 events corresponding to the 22% systematic uncertainty in
the energy scale; it is therefore unsurprising that this experiment did not
detect any events. The prospective Parkes PAF experiment shown here would
expect to detect 1.4 events (uncertainty range from energy scale of 0.7–2.8
events) in a nominal 200 hours of observing time. These numbers will, however,
depend strongly on the effects of small-scale lunar surface roughness, which
are neglected here but will dominate the uncertainty.

##  5 Discussion

This work indicates that past lunar radio experiments are in some cases less
sensitive than initially believed, both in their sensitivity to radio pulses
and in their consequent sensitivity to the UHE particle flux. This underscores
the need for these experiments to be conducted with a proper appreciation of
the specialised requirements for the detection of coherent radio pulses, and
for all experimental details to be fully reported so that they can be re-
evaluated by other researchers; it remains to be seen whether other effects
will be discovered that further affect the sensitivity of the experiments
considered here. Ideally, it is also desirable for multiple experiments to be
conducted with different techniques, to minimise the possibility that a single
oversight will lead to the acceptance of an incorrect result.

Previous comparisons between low- and high-frequency lunar radio experiments
have generally found the larger particle apertures of the former to be a
decisive advantage [44, 12, 6]. However, these comparisons have generally
assumed frequency-independent radio sensitivity. The comparison in Table 1
indicates that low-frequency experiments, due to a combination of high system
temperatures and increased ionospheric dispersion, typically have an increased
radio pulse threshold. This is likely to remain the case for the near-future
experiments considered here, until the advent of the SKA, for which the
extremely large collecting area of its low-frequency component results in
sensitivity similar to that of the high-frequency component [4].

The application of existing analytic aperture models indicates that an
experiment with 200 hours of observing time on the Parkes radio telescope,
using a phased-array feed, would detect an average of 1.4 UHECRs, and an equal
observing time with LOFAR could exclude UHE neutrino spectra predicted by
exotic-physics models (e.g. [52]) with up to 99% confidence for the most
optimistic predictions. (The correction applied in Sec. 4.2 to the model of
Jeong et al. [7] reinforces their conclusion that, in the absence of neutrinos
from such models, lunar radio experiments will detect UHECRs well before they
detect the more confidently expected cosmogenic neutrino flux.) Note that
these observing times are nominal values, representing a comparable effort to
previous experiments. The likely prospect of the first UHECR detection with
this technique, in particular, could justify a longer experiment; ignoring the
uncertainties in the detection rate of one UHECR per 140 hours, 1,000 hours of
observations with a phased-array feed on the Parkes radio telescope would
detect an average of 7 UHECRs, with a 99.9% probability of at least one
detection.

Future theoretical work in this field should seek to refine these predictions
through further development of CR and neutrino aperture models, either by
improving the analytic models used here or through new simulations, in
particular to properly represent the effects of small-scale lunar surface
roughness. The parameters derived in this work to describe lunar radio
experiments allow the easy application of future models to recalculate the
sensitivity to UHE particles of past experiments, or to predict the
sensitivity of new ones.

## Acknowledgements

I would like to thank S. Buitink, B. Stappers and R. Smits for further
information on the NuMoon observations with the WSRT, as well as T. R. Jaeger
and R. L. Mutel for further information on the RESUN observations with the
EVLA. I would also like to thank J. E. J. Lovell and J. M. Dickey for
information about the AuScope VLBI array, and R. D. Ekers for general
discussions regarding lunar radio experiments. Finally, I would like to thank
an anonymous referee for several helpful comments. This research was supported
by the Australian Research Council’s Discovery Project funding scheme (project
number DP0881006) and by the European Research Council’s Starting Grant
funding scheme (project number 307215, LODESTONE).

##  A Simulation of amplitude recovery efficiency

This appendix describes a procedure for determining a representative value for
$\alpha$, the scaling factor in Eq. 8 that accounts for inefficiencies in
reconstruction of the amplitude of a coherent pulse. It simulates the phase,
dispersion and sampling of a pulse, which are the three properties discussed
in Sec. 2.1.

The time-domain profile of the pulse is represented here as $s(t)$, and its
Fourier transform and frequency-domain equivalent as $S(\nu)$. Properly,
$S(\nu)$ is a Hermitian quantity defined for both positive and negative
frequencies, but this procedure describes only the operations on the former,
omitting the conjugate operations on the latter. The frequency $\nu$
represents the baseband or intermediate frequency $\nu_{\textsc{if}}$ at which
the pulse is processed, with the original radio frequency referred to
explicitly as $\nu_{\textsc{rf}}$. These are related by
$\nu_{\textsc{if}}=|\nu_{\textsc{rf}}-\nu_{\textsc{lo}}|$, where
$\nu_{\textsc{lo}}$ is the frequency of the local oscillator used for
frequency downconversion.

The procedure consists of the steps outlined below.

  1. 1.

Define a flat pulse spectrum

$S(\nu)=\begin{cases}1&\mbox{for $\nu_{\rm min}<\nu_{\textsc{rf}}<\nu_{\rm
max}$}\\\ 0&\mbox{otherwise}\end{cases}$ (22)

between minimum and maximum radio frequencies $\nu_{\rm min}$ and $\nu_{\rm
max}$.

  2. 2.

Perform an inverse Fourier transform to convert $S(\nu)$ to the time domain,
and find its maximum value

$s_{\rm norm}={\rm max}(\mathscr{F}^{-1}[S(\nu)])$ (23)

which will be used for normalisation.

  3. 3.

Discarding the time-domain function calculated in the previous step, perform
the transform

$S(\nu)\rightarrow i\,S(\nu)$ (24)

to represent the inherent phase of an Askaryan pulse.

  4. 4.

Disperse the pulse by applying dispersion based on the radio frequency

$S(\nu)\rightarrow e^{i\phi_{d}(\nu)}S(\nu)$ (25)

where the dispersive phase is

$\displaystyle\phi_{d}(\nu)$
$\displaystyle=-2\pi\int_{\infty}^{\nu_{\textsc{rf}}}\\!d\nu_{\textsc{rf}}\,\Delta
t$ (26) or, per Eq. 11, $\displaystyle\phi_{d}(\nu)$ $\displaystyle=2\pi\times
1.34\times 10^{9}\left(\frac{\rm STEC}{\rm
TECU}\right)\biggl{(}\frac{\nu_{\textsc{rf}}}{\rm Hz}\biggr{)}^{-1}$ (27)

determined by the STEC or electron column density in the ionosphere.

  5. 5.

Apply a small frequency-independent phase $\phi_{r}$

$S(\nu)\rightarrow e^{i\phi_{r}}S(\nu)$ (28)

to represent the random phase introduced by frequency downconversion.

  6. 6.

Find the time-domain representation of the signal as the inverse Fourier
transform

$s(t)=\mathscr{F}^{-1}[S(\nu)].$ (29)

  7. 7.

Replace the signal with its envelope

$s(t)\rightarrow\left(s(t)^{2}+\mathscr{H}[s(t)]^{2}\right)^{1/2}$ (30)

which is the norm of the original signal and its Hilbert transform.

  8. 8.

Choose sampling times

$\displaystyle t_{s}$ $\displaystyle=t_{0}+n\,\Delta t_{s}$ $\displaystyle{\rm
for~{}}n\in\mathbb{Z}$ (31)

where $\Delta t_{s}$ is the sampling interval and $t_{0}$ is a small arbitrary
offset to represent the unknown time of arrival of the pulse.

  9. 9.

Find the maximum sampled amplitude

$s_{\rm max}={\rm max}(|s(t_{s})|).$ (32)

  10. 10.

Loop through steps (8)–(9), taking values of $t_{0}$ uniformly distributed
between $0$ and $\Delta t_{s}$.

  11. 11.

Loop through steps (28)–(10), taking values of $\phi_{r}$ uniformly
distributed between $0$ and $\pi$.

  12. 12.

Find the mean of the peak amplitudes found in step (9), and normalise it to
give

${}\mkern 0.5mu\overline{\mkern-0.5mu\alpha}=\frac{{}\mkern
1.5mu\overline{\mkern-1.5mus}_{\rm max}}{s_{\rm norm}}$ (33)

which can be used as a representative value for $\alpha$.

This is the complete procedure incorporating all the effects described in Sec.
2.1, not all of which will be relevant for a single experiment. For example,
if an experiment directly Nyquist-sampled the radio-frequency signal (i.e.
$\nu_{\textsc{lo}}=\nu_{\textsc{rf}}$) steps (28) and (11) would be omitted,
if it triggered directly on the voltage rather than forming the signal
envelope step (30) would be omitted, and if it operated at a high radio
frequency it would be reasonable to omit the dispersion applied in step (4).
For experiments (as in Secs. 3.5 and 3.9) which average the power over a
series of consecutive samples, the approach used here is insufficient, and
simulations such as those of Buitink et al. [27] or Singh et al. [33] are
required.

##  B Analytic calculation of particle aperture

This appendix describes the implementation of the models of Gayley et al. [6]
and Jeong et al. [7] for the analytic calculation of the apertures of lunar
radio experiments to ultra-high-energy neutrinos and cosmic rays respectively.
Although the derivation of these models is described in detail in the original
articles, the straightforward guide to their implementation presented here may
also be useful to other researchers. I have restricted myself here to only
occasional comments on the physical meaning of the variables derived as
intermediate results, and still fewer regarding the approximations involved in
obtaining the final closed-form results.

I have made two significant changes to the original models. In the model of
Gayley et al., I have increased the assumed electric field dissipation length
in the lunar regolith by a factor of $\sim 2$, as discussed in Sec. 4.1. In
the model of Jeong et al., I have assumed that 100% (rather than 20%) of the
energy of an interacting cosmic ray goes into the resulting hadronic particle
cascade, as discussed in Sec. 4.2. Apart from this, I have made only minor
changes for the sake of consistency of notation.

The physical constants required for the analytic aperture calculation are
defined in Table 3. The other required parameters are those calculated in Sec.
3. The observing frequency $\nu$ and the threshold electric field
$\mathcal{E}_{\rm min}$ are used as in the original models. Gayley et al.
scale their results by the limb coverage $\zeta$; here this dependence has
been explicitly inserted into the aperture calculation. For the role of the
remaining parameters $\mathcal{E}_{\rm max}$ and $t_{\rm obs}$ in calculating
the sensitivity of a lunar radio experiment, see Sec. 4; $\mathcal{E}_{\rm
max}$ is substituted for $\mathcal{E}_{\rm min}$ here when calculating the
aperture $A(E;\mathcal{E}_{\rm max})$.

Table 3: Constants used in analytic aperture calculation.

Symbol | Value | Meaning
---|---|---
$d$ |  $3.8\times 10^{8}$ m | distance to Moon
$R$ |  $1.738\times 10^{6}$ m | radius of Moon
$t_{\parallel}$ | 0.6 | transmission coefficient ${}^{\textrm{a}}$
$n_{r}$ | 1.73 | refractive index of regolith ${}^{\textrm{b}}$
$c$ |  $3\times 10^{8}$ m/s | speed of light

  * 1.

${}^{\textrm{a}}$ Averaged over variation with the angle of incidence, as
shown in Fig. 2 of Gayley et al. [6].

  * 2.

${}^{\textrm{b}}$ Within the range measured by Olhoeft and Strangway [47], and
consistent with James and Protheroe [12].

Different parts of the aperture calculation depend on results from widely-
separated areas of physics:

  * 1.

steps (35) and (36) are based on the particle cascade simulations of Alvarez-
Muñiz et al. [42];

  * 2.

step (39) is based on the model of the lunar surface developed by Shepard et
al. [56] from radar scattering measurements;

  * 3.

step (40) (Sec. B.1 only) is based on the radio attenuation measurements of
Olhoeft and Strangway [47] as discussed in Sec. 4.1; and

  * 4.

steps (41) and (42) (Sec. B.1 only) use a parameterisation of the neutrino-
nucleon cross-section based on Gandhi et al. [57].

To substitute an alternative model for any of these aspects of the aperture
calculation, these are the corresponding steps that must be modified.

###  B.1 Neutrinos

The aperture of a lunar radio experiment to neutrinos with energy $E_{\nu}$ is
determined as follows.

  1. 1.

Find the shower energy

$E_{\textsc{s}}=0.2\,E_{\nu}$ (34)

based on the assumption that 20% of the energy of the primary neutrino goes
into the resulting hadronic particle cascade.

  2. 2.

Find the peak electric field, from Eq. 18 of Gayley et al.,

$\begin{split}\mathcal{E}_{0}=0.0845\,\frac{\rm V}{\rm
m\,MHz}\left(\frac{d}{\rm
m}\right)^{\\!\\!-1}\\!\left(\frac{E_{\textsc{s}}}{10^{18}{\rm~{}eV}}\right)\\\
\times\biggl{(}\frac{\nu}{\rm GHz}\biggr{)}\\!\left(1+\biggl{(}\frac{\nu}{\rm
2.32~{}GHz}\biggr{)}^{\\!1.23}\right)^{\\!\\!-1}\end{split}$ (35)

which would be observed by a detector precisely on the Cherenkov cone of the
cascade.

  3. 3.

Characterise the width of the Cherenkov cone with the angle, from Eq. 19 of
Gayley et al.,

$\Delta_{0}=0.05\left(\frac{\nu}{\rm
GHz}\right)^{\\!\\!-1}\left(1+0.075\log_{10}\\!\left(\frac{E_{\textsc{s}}}{10^{19}{\rm~{}eV}}\right)\right)^{\\!\\!-1}$
(36)

which is its $1/e$ half-width.

  4. 4.

Find the dimensionless parameter, from Eq. 32 of Gayley et al.,

$f_{0}=\sqrt{\ln\\!\left(\frac{\mathcal{E}_{0}\,t_{\parallel}}{\mathcal{E}_{\rm
min}}\right)}$ (37)

which describes how far the detector can be from the Cherenkov cone while
observing an electric field in excess of the threshold $\mathcal{E}_{\rm
min}$.

  5. 5.

The maximum possible aperture to an isotropic flux of neutrinos, if the Moon
were a perfect detector, from Eq. 9 of Gayley et al., is

$A_{0}=4\pi^{2}R^{2}$ (38)

in dimensions of area multiplied by solid angle.

  6. 6.

Characterise the roughness of the lunar surface at the relevant wavelength
scale with the angle, from Eq. 3 of Gayley et al.,

$\sigma_{0}=\sqrt{2}\tan^{-1}\\!\left(0.14\,\biggl{(}\frac{\nu}{\rm
GHz}\biggr{)}^{\\!\\!0.22}\right)$ (39)

which is the $1/e$ half-width of the assumed Gaussian distribution of
unidirectional surface slopes.

  7. 7.

Find the electric field dissipation length (twice the power dissipation length
or photon mean free path) as

$L_{\gamma}=60\lambda$ (40)

where $\lambda=c/\nu$ is the vacuum radio wavelength. As discussed in Sec.
4.1, this is different to the expression given by Eq. 25 of Gayley et al.

  8. 8.

Take the neutrino attenuation length, from Eq. 26 of Gayley et al., as

$L_{\nu}=122{\rm~{}km}\left(\frac{E_{\nu}}{10^{20}{\rm~{}eV}}\right)^{\\!\\!-1/3}$
(41)

in the lunar regolith.

  9. 9.

For up-going neutrinos, which pass through the Moon before interacting in the
regolith, calculate from Eq. 37 of Gayley et al.

$\alpha_{0}=0.03\left(\frac{E}{10^{20}{\rm~{}eV}}\right)^{\\!\\!-1/3},$ (42)

the maximum upward angle with respect to the large-scale surface for which a
neutrino can typically penetrate the lunar secant without being attenuated.
This expression incorporates the contribution from higher-energy neutrinos
which lose energy in neutral-current interactions, making it sensitive to the
neutrino spectrum which is assumed to be $\propto E_{\nu}^{-2}$; but, as
discussed by Gayley et al., the dependency is only weak.

  10. 10.

Find the angular acceptance parameters describing contributions to the
neutrino aperture, defined in Eqs. 55–57 of Gayley et al.:

$\displaystyle\Psi_{\rm ds}$ $\displaystyle=f_{0}\Delta_{0}$ (43) for down-
going neutrinos that would be detected on a smooth Moon, due to the width of