# Dark photon limits: a handbook

Andrea Caputo<EMAIL_ADDRESS>School of Physics and Astronomy, Tel-Aviv
University, Tel-Aviv 69978, Israel  Department of Particle Physics and
Astrophysics,Weizmann Institute of Science, Rehovot 7610001, Israel  Alexander
J. Millar<EMAIL_ADDRESS>The Oskar Klein Centre, Department of
Physics, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden  Nordita,
KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken
23, 10691 Stockholm, Sweden  Ciaran A. J. O’Hare<EMAIL_ADDRESS>ARC Centre of Excellence for Dark Matter Particle Physics, Sydney, NSW,
Australia  School of Physics, Physics Road, The University of Sydney, NSW 2006
Camperdown, Australia  Edoardo Vitagliano<EMAIL_ADDRESS>Department
of Physics and Astronomy, University of California, Los Angeles, California,
90095-1547, USA

###### Abstract

The dark photon is a massive hypothetical particle that interacts with the
Standard Model by kinetically mixing with the visible photon. For small values
of the mixing parameter, dark photons can evade cosmological bounds to be a
viable dark matter candidate. Due to the similarities with the electromagnetic
signals generated by axions, several bounds on dark photon signals are simply
reinterpretations of historical bounds set by axion haloscopes. However, the
dark photon has a property that the axion does not: an intrinsic polarisation.
Due to the rotation of the Earth, accurately accounting for this polarisation
is nontrivial, highly experiment-dependent, and depends upon assumptions about
the dark photon’s production mechanism. We show that if one does account for
the DP polarisation, and the rotation of the Earth, an experiment’s discovery
reach can be enhanced by over an order of magnitude. We detail the strategies
that would need to be taken to properly optimise a dark photon search. These
include judiciously choosing the location and orientation of the experiment,
as well as strategically timing any repeated measurements. Experiments located
at $\pm$35∘ or $\pm$55∘ latitude, making three observations at different times
of the sidereal day, can achieve a sensitivity that is fully optimised and
insensitive to the dark photon’s polarisation state, and hence its production
mechanism. We also point out that several well-known searches for axions
employ techniques for testing signals that preclude their ability to set
exclusion limits on dark photons, and hence should not be reinterpreted as
such.

††preprint: NORDITA-2021-036

##  I Introduction

The hypothesis that galactic dark matter (DM) halos are comprised of a cold
population of bosons is accumulating substantial interest in both theoretical
and experimental circles Essig:2013lka ; Battaglieri:2017aum ; Agrawal:2021dbo
. Examples of these DM candidates include the pseudoscalars— like the widely
popular QCD axion DiLuzio:2020wdo , or its generalisation, the axion-like
particle Masso:1995tw ; Masso:2002ip ; Ringwald:2012hr ; Ringwald:2012cu ;
Arvanitaki:2009fg ; Cicoli:2012sz ; Jaeckel:2010ni —as well as light scalars
Taylor:1988nw ; Hu:2000ke ; Magana:2012ph ; Lesgourgues:2002hk ; Hui:2016ltb ;
Damour:2010rp ; Damour:2010rm ; Piazza:2010ye , and vector particles
Holdom:1985ag ; Okun:1982xi ; Fayet:1980rr ; Georgi:1983sy . From an
experimental standpoint, one of the primary appeals of these ideas is that
many of them possess a coupling to electromagnetism that, while usually
suppressed by a high energy scale, is generically nonzero. This permits a
diverse array of laboratory experiments to directly detect them as galactic
DM. Much of the experimental activity has been driven towards the detection of
the axion. However many of the axion’s experimental signatures are shared by
another DM candidate—the dark photon (DP)111aka hidden photon, or paraphoton.
Jaeckel:2010ni ; Jaeckel:2013ija ; Fabbrichesi:2020wbt —which can be searched
for with very similar techniques Jaeckel:2007ch ; Horns:2012jf ;
Suzuki:2015sza ; Jaeckel:2015kea ; Knirck:2018ojz ; Brun:2019kak ;
Nguyen:2019xuh ; Andrianavalomahefa:2020ucg ; Tomita:2020usq ; Godfrey:2021tvs
; Baryakhtar:2018doz ; Chaudhuri:2018rqn ; Gelmini:2020kcu ; Arias:2012az .

The DP is the gauge boson of a new dark U(1) added to the Standard Model (SM)
gauge group, under which the SM fields are uncharged. This makes the DP nearly
unobservable, save for a small kinetic mixing with the visible photon that is
left in the theory at low energies Holdom:1985ag . The kinetic mixing leads to
photon-DP oscillations, reminiscent of neutrino oscillations, or the axion-
photon mixing relied upon by DM axion detectors known as haloscopes (see e.g.
Ref. Irastorza:2018dyq  for a review of experimental techniques). The primary
practical difference between the two is that axion-photon conversion requires
an applied magnetic field, whereas DP-photon mixing is an inherent feature of
the model—making the latter in some scenarios less demanding to search for.

While DPs lack as compelling a theoretical backing as the QCD axion
Peccei:1977hh ; Peccei:1977ur ; Weinberg:1977ma ; Wilczek:1977pj ; Kim:2008hd
, they can, at the very least, function as a viable cold DM candidates
Nelson:2011sf ; Arias:2012az , and several production mechanisms have been
proposed to generate a sufficient abundance of them in the early Universe
AlonsoAlvarez:2019cgw ; Graham:2015rva ; Kolb:2020fwh ; Ema:2019yrd ;
Ahmed:2020fhc ; Nakai:2020cfw ; Agrawal:2018vin ; Co:2018lka ; Bastero-
Gil:2018uel ; Co:2021rhi ; Long:2019lwl . So as a minimal extension of the SM,
the DP is therefore just as worthwhile a target for experimental investigation
as, say, an axion-like particle.222Indeed, DPs can also play the role of a
$Z^{\prime}$-like mediator to a dark sector, hence there is substantial
interest in searching for DPs in accelerators Hewett:2012ns ; Beacham:2019nyx
; Alemany:2019vsk ; Graham:2021 ; Lanfranchi:2020crw  and via searches for the
millicharged particles they may couple to Prinz:1998ua ; Ball:2016zrp ;
Magill:2018tbb ; Kelly:2018brz ; Harnik:2019zee ; Ball:2020dnx . See Ref.
Fabbrichesi:2020wbt  for a recent review of DPs that extends to their role in
contexts well outside of the scope of our work.

While many haloscopes designed to search for axions are also sensitive to DPs,
there are few, if any, dedicated exclusion analyses made by their respective
collaborations. The non-observation of axions of a particular mass is simply
reinterpreted as a constraint on DPs of the same mass, without fully taking
into account the subtle differences between the ways the two particles couple
to experiments. Most notably, the DP possesses a polarisation, and therefore
the direction of the $E$-field it generates will have nothing to do with the
experiment. In contrast, the $E$-field generated by axions has a preferred
direction provided by the applied magnetic field. Most experiments designed to
detect DM-induced $E$ or $B$-fields are sensitive to the directions of those
fields. The parallels and differences between the behaviours of axions and DPs
in haloscopes were first pointed out in 2012 by Ref. Arias:2012az .333The
haloscope limits from Ref. Arias:2012az  were updated very recently in Ref.
Ghosh:2021ard . Since then, some subtly erroneous results of the recasting
exercise detailed in that work have propagated through the literature.

However, even dedicated DP searches must reckon with the DP polarisation.
There are several reasons why this is fraught with difficulty. Firstly, the DP
polarisation distribution around the Earth will depends upon assumptions about
its production mechanism. Secondly, experiments operate in a rest frame that
rotates with respect to the DP field, meaning the preferred direction of
detection will vary over the day if one assumes any degree of coherence to the
underlying polarisation distribution. Since many experiments, especially those
designed to detect axions, are only sensitive to $E$-fields aligning with one
axis, this will make any DP interpretation of those experimental analyses
strongly orientation- and time-dependent. Approaches taken to address these
issues in past analyses have been overly simplistic at best, and plainly
incorrect at worst.

A large number of new DP searches, and an even larger number of proposals,
have appeared since 2012. It is therefore high time that the steps be laid out
to perform a more careful treatment of DP polarisation. We will show that with
some simple changes to the data-taking procedure (without changing the total
measurement time, and therefore cost, of the experiment), an experiment can
obtain vastly improved limits on DPs in the absence of detection, and would be
in a much more robust position to study the signal if in the presence of one.

In what follows, we will derive a set of mathematical formulae to compute
relevant quantities needed to account for the Earth’s rotation with respect to
the DP. These results take the form of conversion factors that quantify how
much an exclusion limit is impacted by the range of possible angles between
the DP polarisation and the axis or plane that the experiment is sensitive to.
We will see that the dependencies of these conversion factors entail certain
experimental configurations being less impacted by this uncertainty than
others. The fully optimised scenario is one in which the experimental
location, running time, and orientation all conspire to give a signal that is
effectively insensitive to the DP polarisation. For the location, we will see
that latitudes of $\lambda_{\rm lat}=\pm 35^{\circ}$ or $\pm 55^{\circ}$ are
optimal for experiments that are sensitive along the North-South or the
Zenith-Nadir axes respectively. The optimal measurement time is always one
sidereal day, since in this case the experiment samples all possible DP
polarisation angles available to it. However, we will explain that a
continuous measurement lasting the entire day is not necessary, and a close-
to-fully optimised discovery reach can be achieved if the experiment splits a
short $\mathcal{O}({\rm min})$-long measurement into three measurements spaced
evenly over the day. As mentioned above, we aim for all of our recommendations
to be incorporated relatively unintrusively via simple scheduling changes. So
we emphasise that we do not require these measurements to be conducted on the
same date—three measurements at different times of the sidereal day, spaced
months apart, would achieve the same result.

To begin, in Sec. II, we set the stage by giving a brief overview of existing
constraints on dark photons, all of which are displayed in Fig. 1. Then in
Sec. III we discuss ways in which DPs can be produced in the early Universe
with sufficient abundance to explain the DM, and in the process what these
production mechanisms imply about the present day DP polarisation distribution
around the Earth. Then in Sec. IV we walk through the electrodynamics
associated with detecting DPs, and in Sec. V we discuss various experimental
techniques. In Sec. VI we discuss the direction and time dependence inherent
in DP detection, and how this dictates the optimum scanning strategy. Finally,
in Sec. VII we determine how an experiment could use the daily modulation
signal to _measure_ the DP polarisation following a positive detection. We
summarise and conclude in Sec. VIII. All the figures created for this paper
can be reproduced using the code available at
https://github.com/cajohare/DarkPhotonCookbook with the exception of the DP
constraint plots and their associated data which are available at
https://cajohare.github.io/AxionLimits.

##  II Existing constraints on the dark photon

Figure 1: Current constraints on the DP’s mass, $m_{X}$, and kinetic mixing
parameter with the SM photon, $\chi$. The general colour-scheme is:
cosmological bounds in blue, experimental bounds in red, and astrophysical
bounds in green. The thick white line that divides the parameter space in two
is the upper limit for which DPs are a viable candidate for 100% of the DM.
The focus of this work are the experimental bounds that reach below this line.
Descriptions of each bound are given in Sec. II.

In Fig. 1 we show the current landscape of bounds on DPs lighter than 0.1
MeV.444Accelerator bounds on heavier DPs have been purposefully ignored
because the focus here is on DPs which can constitute DM in and of themselves.
We have chosen to show any bound set by a physical laboratory experiment in
red, those set using astrophysical data in green, and those set using
cosmological data in blue. We now briefly run through the sources of each
bound.

Many model-independent bounds on the existence of the DP in Nature have been
obtained through tests of the Coulomb $1/r^{2}$ force law, or, equivalently,
via bounds on the photon mass Goldhaber:2008xy . The ones we have shown here
are from Cavendish-like experiments Williams:1971ms ; Bartlett:1988yy ;
Tu:2005ge ; Kroff:2020zhp , Plimpton & Lawton’s experiment Plimpton:1936ont ;
Kroff:2020zhp , atomic spectroscopy Jaeckel:2010xx , atomic force microscopy
(AFM) Kroff:2020zhp , and, at the lightest masses displayed here, from the
static magnetic fields of the Earth Goldhaber:1971mr  and Jupiter Davis:1975mn
. Similarly, there are purely laboratory bounds on DPs set using light-
shining-through-walls (LSW) experiments, e.g. those run at ALPs Ehret:2010mh ,
SPring-8 Inada:2013tx  and UWA Povey:2010hs ; Parker:2013fxa , as well as the
microwave LSW experiments performed by ADMX Wagner:2010mi  and CROWS
Betz:2013dza . CAST Redondo:2008aa  and SHIP Schwarz:2015lqa  are both
helioscopes, setting bounds on DPs emitted by the Sun. Finally, TEXONO
Soma:2014zgm  is a reactor neutrino experiment, for which a low mass DP limit
was derived in Ref. Danilov:2018bks .

Dedicated direct detection bounds on the DP, specifically as a DM candidate,
are set by the following experiments: DAMIC Aguilar-Arevalo:2019wdi , Dark
E-field Radio Godfrey:2021tvs , DM Pathfinder Phipps:2019cqy , FUNK
Andrianavalomahefa:2020ucg , SENSEI Barak:2020fql , SHUKET Brun:2019kak ,
SuperCDMS Aralis:2019nfa , SQuAD Dixit:2020ymh , three Tokyo dish antennae
experiments Suzuki:2015sza ; Knirck:2018ojz ; Tomita:2020usq , WISPDMX
Nguyen:2019xuh , and XENON1T/XENON100 Bloch:2016sjj ; Aprile:2019xxb ;
Aprile:2020tmw ; Bloch:2020uzh ; Alonso-Alvarez:2020cdv ; An:2020bxd . Several
other underground DM detectors sensitive to keV-mass DPs have also set limits
Abgrall:2016tnn ; Armengaud:2018cuy ; She:2019skm ; Sato:2020ebe ;
Lasenby:2020goo ; GERDA:2020emj  that are less sensitive than XENON’s—we have
neglected these to reduce clutter.

One of the focuses of this work is on reinterpreting haloscope limits on
axions in the context of DPs. Those shown are ADMX Asztalos:2001jk ;
Asztalos:2009yp ; Du:2018uak ; Boutan:2018uoc ; Braine:2019fqb , HAYSTAC
Zhong:2018rsr ; Backes:2020ajv , CAPP Lee:2020cfj , and QUAX Alesini:2020vny .
Results from several well-known axion haloscopes DePanfilis:1987dk ;
Hagmann:1990tj ; McAllister:2017lkb ; Ouellet:2018beu ; Gramolin:2020ict  are
not shown because they used their $B$-field to test for potential (axion)
signals. In other words, a DP could have been observed, but its signal would
have been vetoed.

The upper limit of viable dark photon dark matter (DPDM), shown by a thick
white line, is taken from various references. Although we run the risk of
being overly-stringent, we adopt the most democratic approach of taking the
lower envelope of all published analyses, including: Arias et al. Arias:2012az
, Witte et al. McDermott:2019lch ; Witte:2020rvb , and Caputo et al.
Caputo:2020rnx ; Caputo:2020bdy , though we note that there are some
substantive disagreements between these analyses. Three astrophysical limits
also require DPDM: those based on the heating of the intergalactic medium
(IGM) Dubovsky:2015cca , the gas in the Leo T dwarf Wadekar:2019mpc , and the
gas cloud at the galactic centre G357.8-4.7-55 Bhoonah:2019eyo , and again,
there are also disagreements between these analyses. Also at these lightest
masses, a recent experimental bound was set on DPDM from an analysis
Fedderke:2021rrm ; Fedderke:2021aqo  of SuperMAG data—a global network of
magnetometers studying the geomagnetic field.

The astrophysical bounds at higher masses are those based on stellar cooling
arguments applied to the Sun, horizontal branch (HB) stars, and red giant (RG)
stars in Ref. Redondo:2013lna , and neutron stars in Ref. Hong:2020bxo . Note
that for the straight part of the solar bound below 10 eV we use the improved
limit from the solar global fit performed in Ref. Vinyoles:2015aba . These
bounds assume a non-dynamical generation of the DP mass: the Stueckelberg
case. However, if the DP mass originated via a Higgs mechanism, the stellar
bounds would be much stronger—plateauing at $\chi\sim 10^{-13}$ for
$m_{X}~{}\lesssim~{}100$ eV, down to arbitrarily small masses An:2013yua ;
An:2020bxd .

Another astrophysical bound was set using gamma rays from the Crab nebula
Zechlin:2008tj . The final cosmological bound is on $\gamma\rightarrow X$
happening in the early universe to the degree that it would generate spectral
distortions to the CMB, which are tightly constrained by COBE and FIRAS
Fixsen:1996nj . Several groups have derived these constraints in the past
Mirizzi:2009iz ; Caputo:2020rnx ; Caputo:2020bdy ; Garcia:2020qrp , with
broad, but not perfect, agreement. The one shown in Fig. 1 is from Ref.
Caputo:2020bdy . Lastly, we shade in grey the mass window $6.5\times
10^{-15}\,{\rm eV}<m_{X}<2.9\times 10^{-11}\,{\rm eV}$. If a DP existed in
that range, the field would spin down stellar mass black holes due to
superradiance Stott:2020gjj ; Ghosh:2021zuf ; Cardoso:2018tly .

Data for every bound shown in this figure can be downloaded individually at
https://cajohare.github.io/AxionLimits/docs/dp.html.

##  III Dark photon cosmology

Dark photons are a compelling candidate for new physics. As they are simply
the gauge boson of an additional U(1), they represent a very minimal extension
to the SM. Recently, several novel production mechanisms for DPDM have been
proposed, igniting interest even further. In this section we summarise some of
these mechanisms, focusing on the degree of polarisation they leave the relic
DPDM with. This point is often overlooked in the literature but is extremely
relevant for their subsequent detection. We also stress that, contrary to
previous claims found in the literature, the DP polarisation is different from
the isotropy of the stress-energy tensor associated to the field, and depends
on the production mechanism.

Arguably one of the simplest ways to produce DPDM is the misalignment
mechanism, which is the most popular mechanism used to generate axions
Preskill:1982cy ; Abbott:1982af ; Dine:1982ah . However, unlike axions, a
minimal coupling to gravity does not lead to the correct relic abundance—a
nonminimal coupling to the Ricci scalar needs to be invoked Arias:2012az ;
Graham:2015rva ; AlonsoAlvarez:2019cgw . This typically comes at the cost of
introducing instabilities in the longitudinal DP mode Himmetoglu:2008zp ;
Himmetoglu:2009qi ; Karciauskas:2010as . Therefore, some extra work is
required to make the theory consistent (though finding such a UV theory is not
the goal of this work). For our purposes, the most notable consequence of the
misalignment mechanism, however it may be constructed, is that it naturally
leads to relic DPDM with a fixed polarisation within the cosmological horizon.
We refer to this scenario later on as the _fixed polarisation scenario_ , and
it will have the most dramatic consequences for direct detection. A scenario
similar to the misalignment mechanism consists of the DPDM production via
quantum fluctuations during inflation Graham:2015rva  (see also Refs.
Kolb:2020fwh ; Ema:2019yrd ; Ahmed:2020fhc ; Nakai:2020cfw ). In contrast to
scalars and tensors, the vector is produced with a power spectrum peaked at
intermediate wavelengths, evading bounds from long-wavelength isocurvature
perturbations. Furthermore, this mechanism does not require a non-minimal
coupling to gravity.

Another scenario is one based on tachyonic instabilities555A similar
mechanism, based on a different instability, is through parametric resonance.
See e.g. Ref. Dror:2018pdh . which arise when the DP couples to a misaligned
axion Agrawal:2018vin ; Co:2018lka ; Bastero-Gil:2018uel . The energy density
is initially stored in the axion field, and then the axion’s zero mode
transfers to both transverse and longitudinal components of the DP. Any
production mechanism involving tachyonic instabilities dominantly produces a
specific DP helicity, and we would expect that the final relic will also carry
the same helicity—although later scatterings can deplete the degree of
polarisation Ratzinger:2020oct . However, there exist other scenarios
Co:2021rhi  where the degree of polarisation may be even more pronounced and
likely surviving scatterings. Lattice simulations of these models would be
very relevant to the experimental campaign.

DPDM could also be produced from the decay of topological defects, such as a
network of near-global, Abelian-Higgs cosmic strings Long:2019lwl . In this
scenario, the transversely polarised DP interaction is suppressed and the
radiation is dominated by the emission of longitudinally polarised DPs. These
modes would then come to constitute the DM. The evolution of the network is
complicated, and consists of both short loops and infinite strings. An
educated guess would be that long strings, with lengths of the order of the
Hubble horizon, may lead to some degree of polarisation in the DP field by
identifying a preferred direction. On the other hand, DPs would be produced
also from the collapse of smaller closed loops, so eventually the polarisation
alignment may be washed out. We refer to the case where the DP has no single
polarisation as the _random polarisation scenario_.

To summarise, there are many interesting DP production mechanisms that can
successfully produce the correct abundance of DM, several of them leading to
some level of coherence in the polarisation distribution of the relic DP
field, possibly over the entire Universe. Unfortunately, a more precise
statement than this is not possible at the moment. In fact, the situation
becomes even less clear when considering structure formation. It is not
readily apparent what effect the formation of DM halos would have, if any, on
the distribution of DP polarisations on the mpc-scales probed by an
experimental campaign. Although dedicated simulations will be needed to
resolve this issue, we can at least try to appreciate what impact gravity will
have on the DP polarisation with a simple back-of-the-envelope calculation.

Consider a particle with four-velocity $u^{\alpha}$ and polarisation
$S^{\alpha}$; we know that $u_{\alpha}S^{\alpha}=0$ should hold in any frame.
One can thus derive the precession of the polarisation according to the
equation of parallel transport,

$\frac{\textrm{d}S_{\alpha}}{\textrm{d}\tau}=\Gamma^{\lambda}_{\alpha\nu}S_{\lambda}\frac{\textrm{d}x^{\nu}}{\textrm{d}\tau}\,.$
(1)

where $\Gamma^{\lambda}_{\alpha\nu}$ are the Christoffel symbols and $\tau$ is
proper time. From here we can specialise to the motion in a gravitational
potential $\phi$. Following e.g. Ref. Weinberg:1972kfs , we can write the time
variation of the polarisation as,

$\displaystyle\frac{\textrm{d}\mathbf{S}}{\textrm{d}t}=-\mathbf{S}\frac{\partial\phi}{\partial
t}-2\mathbf{v}\cdot\mathbf{S}\nabla\phi-\mathbf{S}(\mathbf{v}\cdot\nabla\phi)$
$\displaystyle+\mathbf{v}(\mathbf{S}\cdot\nabla\phi)+\frac{1}{2}\mathbf{S}\times(\nabla\times\mathbf{S})\,.$
(2)

It is then useful to define a new vector

$\mathbf{S}_{1}=(1+\phi)\mathbf{S}-\frac{1}{2}\mathbf{v}(\mathbf{v}\cdot\mathbf{S})\,,$
(3)

whose evolution is governed by a spin-orbit equation,

$\frac{\textrm{d}\mathbf{S}_{1}}{\textrm{d}t}=\bm{\Omega}\times\mathbf{S}_{1},$
(4)

where
$\bm{\Omega}=-\frac{1}{2}\nabla\times\bm{\zeta}-\frac{3}{2}\mathbf{v}\times\nabla\phi$,
with $\bm{\zeta}$ the vector potential. We can then estimate the relative
polarization variation as

$\frac{\delta S}{S}\sim Tv\frac{\phi}{Rc^{2}}$ (5)

where $T$ and $R$ are the typical time and length scales of the problem, and
we have restored the factor of $c$ for clarity. We can then use the virial
theorem, $\phi\sim v^{2}$, and normalize everything to typical values of our
galaxy. We thus find

$\frac{\delta S}{S}\sim 4\times 10^{-3}\Big{(}\frac{v}{2\times
10^{-3}}\Big{)}^{3}\frac{T}{13\times 10^{9}\,{\rm yr}}\frac{8\,\rm kpc}{R},$

which shows how dark matter can easily preserve some degrees of its initial
polarisation over the lifetime of the galaxy. While a dedicated study is
needed to understand how a real halo of dark matter would be affected, this
estimate shows, at the very least, that a fixed polarisation over laboratory
scales is a plausible scenario.

In this work we will take a phenomenological approach and consider the two
extreme cases: fixed polarisation, and totally randomised polarisation. Our
results are such that any real scenarios will be bounded within these two
limits. The fixed polarisation scenario will be the focus of the majority of
our results since it leads to signals that require the most care to describe.
We emphasise though that the details of this study do not solely apply to the
extreme case where the DP polarisation is fixed over the entire Universe. They
also apply to cases where only a fraction of the field is coherently
polarised, or if the polarisation varies slowly relative to our measurements.

##  IV Dark photon electrodynamics

Working with the assumption that the DM distribution around the Earth is
comprised of a cold population of DPs, we now discuss how to detect them in
the laboratory. This requires us to first explain some aspects of
electrodynamics in the presence of DPs.

The low-energy effective Lagrangian due to the presence of a gauge boson $X$
of a dark U(1) that kinetically mixes Holdom:1985ag ; Arias:2012az ;
An:2014twa ; Fabbrichesi:2020wbt  with the visible photon $A$ reads,

$\displaystyle\mathcal{L}\supset$
$\displaystyle-\dfrac{1}{4}{F}_{\mu\nu}{F}^{\mu\nu}-\dfrac{1}{4}{X}_{\mu\nu}{X}^{\mu\nu}+\frac{\sin{\alpha}}{2}F^{\mu\nu}{X}_{\mu\nu}$
$\displaystyle+eJ_{\rm
EM}^{\mu}{A}_{\mu}+\dfrac{m_{X}^{2}\cos^{2}{\alpha}}{2}{X}^{\mu}{X}_{\mu}\,,$
(6)

where $F_{\mu\nu}$, $X_{\mu\nu}$ are the field strengths of the SM photon and
the DP, $J_{\rm EM}^{\mu}$ is the electromagnetic current, $m_{X}$ is the DP
mass and $\sin\alpha$ is the kinetic mixing parameter. Note that we neglect
terms $\mathcal{O}(\alpha^{2})$.

We can remove the kinetic mixing term by diagonalisation through
$\tilde{A}={A}\cos{\alpha},\tilde{X}=X-\sin{\alpha}A$. In the so-called
interaction basis, $(\tilde{A},\tilde{X})$, the effective Lagrangian is,

$\displaystyle\mathcal{L}\supset$
$\displaystyle-\dfrac{1}{4}{\tilde{F}}_{\mu\nu}\tilde{F}^{\mu\nu}-\dfrac{1}{4}\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}+\dfrac{e}{\cos{\alpha}}J_{\rm
EM}^{\mu}\tilde{A}_{\mu}$
$\displaystyle+\dfrac{m_{X}^{2}\cos^{2}{\alpha}}{2}\left(\tilde{X}^{\mu}\tilde{X}_{\mu}+2\chi\tilde{X}_{\mu}\tilde{A}^{\mu}+\chi^{2}\tilde{A}^{\mu}\tilde{A}_{\mu}\right)~{},$
(7)

where $\chi\equiv\tan{\alpha}$ and $\tilde{A}$, $\tilde{X}$ are interpreted as
the photon produced in electromagnetic interactions and the DP sterile state
respectively. These interaction states are the most relevant when discussing
DP detection.

In the interaction basis the electromagnetic coupling is renormalised to
$(e/\cos\alpha)$ and there are $\tilde{A}$-$\tilde{X}$ oscillations due to the
mass-mixing term. Due to decoherence, the DPDM will be in the massive
propagation eigenstate found instead by diagonalising the mass term in the DP
Lagrangian, Eq. (IV). Assuming the kinetic mixing $\chi$ to be small, the DM
is mostly aligned with the sterile eigenstate $\tilde{X}$.

Neglecting $\mathcal{O}(\chi^{2})$ terms and dropping the tildes, we obtain
the Lagrangian

$\displaystyle\mathcal{L}\supset$
$\displaystyle-\dfrac{1}{4}{{F}}_{\mu\nu}{F}^{\mu\nu}-\dfrac{1}{4}{X}_{\mu\nu}{X}^{\mu\nu}+eJ_{\rm
EM}^{\mu}{A}_{\mu}$
$\displaystyle+\dfrac{m_{X}^{2}}{2}\left({X}^{\mu}{X}_{\mu}+2\chi{X}_{\mu}{A}^{\mu}\right)\,,$
(8)

from which one finds the wave equation in momentum space

$\displaystyle-K^{2}{A}^{\nu}=\chi m_{X}^{2}{X}^{\nu}\,.$ (9)

Here, we defined the four momentum $K=(\omega,\mathbf{k})$ and used the
Fourier expansion for a free field with the energy
$\omega=+\sqrt{|\mathbf{k}|^{2}+m_{X}^{2}}$. In the following, we will treat
the fields as complex, $X^{\mu}_{c}(t,\mathbf{x})$, of which the actual fields
constitute the real part $X^{\mu}=$ Re$\\{X^{\mu}_{c}\\}$. Following Refs.
Knirck:2018knd ; Gelmini:2020kcu , we include a volume $V$ in the definition
of the Fourier transform,

$\displaystyle{X_{c}^{\mu}}(t,\mathbf{x})=\sqrt{V}\int\frac{\mathrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}$
$\displaystyle X^{\mu}(\mathbf{k})e^{-i(\omega
t-\mathbf{k}\mathbf{x}+\delta({\bf k}))}\,.$ (10)

In principle, one can have non-trivial information such as the clumpiness of
DM in the phases $\delta({\bf k})$ Knirck:2018knd , however for our purposes
we are neglecting such issues and assuming the density of DM remains constant
throughout measurements. Our use of a classical field description is justified
as the state occupation number required to make up a local DM density of
$\rho_{\rm DM}\sim 0.3$–$0.5$ GeV cm-3 deSalas:2020hbh  out of sub-eV DPs must
be very large. As the classical calculation gives the expectation value of the
measurement for bosonic two-level mixing Raffelt:1991ck , the large occupation
number means a large number of DPs will be involved, so that the overall
measurement simply gives the expectation value.666As with axion haloscopes,
there is no Bose-enhancement if the final state is occupied Ioannisian:2017srr
. Thus one would only need to worry about a non-classical state if higher
order correlations are measured.

We will now show that most of the DM energy is stored in the zero mode of the
DP field. The energy density of the DP field is given by

$\rho=\frac{1}{V}\int\frac{d^{3}\textbf{k}}{(2\pi)^{3}}\frac{\omega(\textbf{k})^{2}}{2}|\textbf{X}(\textbf{k})|^{2}.$
(11)

The space-averaged DP field can be written as

$\displaystyle\langle X_{c}^{\mu}(t)\rangle=\frac{1}{V}$
$\displaystyle\int\mathrm{d}^{3}\textbf{x}\,X^{\mu}_{c}(t,\textbf{x})$
$\displaystyle=\frac{X^{\mu}(\textbf{k}=0)}{\sqrt{V}}e^{-im_{X}t}\equiv
X^{\mu}_{0}e^{-im_{X}t}\,,$ (12)

where we have defined the amplitude of the plane wave
$X^{\mu}_{0}=\left(X^{0}_{0},\textbf{X}_{0}\right)=X^{\mu}(\textbf{k}=0)/\sqrt{V}$.
At this point, we can make contact between the field and the particle
description. Given the local velocity distribution in the laboratory rest
frame, $f_{\rm lab}(\textbf{v})$, the DM density is

$\rho=\rho\int\mathrm{d}^{3}\textbf{v}\,f_{\rm lab}(\textbf{v})\,,$ (13)

so the DM velocity distribution can be identified with

$f_{\rm
lab}(\textbf{v})=\frac{m_{X}^{3}\omega^{2}}{2(2\pi)^{3}\rho}|\textbf{X}(\textbf{k})|^{2}\,,$
(14)

where we assumed the DM to be non-relativistic, i.e.
$\textbf{k}=m_{X}\textbf{v}$. Neglecting the DM kinetic energy,
$\omega(\textbf{k})^{2}=m_{X}^{2}$, we can write

$\rho=\frac{m_{X}^{2}}{2}\langle|\textbf{X}_{c}(t)|^{2}\rangle\,,$ (15)

by taking a spatial average of the squared amplitude of the DP field. Finally,
if we also neglect velocity in the Fourier transform Eq. (10) (i.e. we assume
the velocity of the DM to be zero), we can write

$\rho\simeq\frac{m_{X}^{2}}{2}|\langle\textbf{X}_{c}(t)|\rangle|^{2}=\frac{m_{X}^{2}}{2}|\textbf{X}_{0}|^{2}\,.$
(16)

The quantity relevant to experiments is the ordinary electric field induced by
the DP field which acts like a source in the wave equation, Eq. (9), for the
interaction eigenstate. In the limit of classically oscillating fields,
Gauss’s law in a homogeneous, isotropic medium with no external sources reads
$\nabla\cdot\mathbf{D}=0$. The electric displacement field is related to the
electric field via $\mathbf{D}=\epsilon\mathbf{E}$ where $\epsilon$ is the
dielectric function of the medium ($\epsilon_{\rm vacuum}=1$).

Therefore, the electric field produced by the DP is,

$\left|\textbf{E}_{0}\right|=|\frac{\chi m_{X}}{\epsilon}\textbf{X}_{0}|\,.$
(17)

But if the experiment is sensitive to a particular component of the produced
electric field, for example in the direction $\hat{\mathbf{z}}$, then the
resulting electric field shall be written as

$\left|\textbf{E}_{0}\right|=\left|\frac{\chi
m_{X}}{\epsilon}\textbf{X}_{0}\cos\theta\right|\,,$ (18)

where
$\cos\theta=\hat{\mathbf{z}}\cdot\textbf{X}_{0}/|\textbf{X}_{0}|\equiv\hat{\mathbf{z}}\cdot\hat{\mathbf{X}}$.
In conclusion, we see that the DP signal depends upon the orientation of the
experiment with respect to the DP’s polarisation.

##  V Dark Photon detection

Experiment |  Magnetic field [T] |  Latitude [∘] |  Measurement time, $T$ |  Directionality |  $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$
---|---|---|---|---|---
Cavities | ADMX-1 |  Asztalos:2009yp  |  7.6 |  47.66 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.025
ADMX-2 |  Du:2018uak  |  6.8 |  47.66 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
ADMX-3 |  Braine:2019fqb  |  7.6 |  47.66 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
ADMX Sidecar |  Boutan:2018uoc  |  3.11777Run B of Ref. Boutan:2018uoc  started at 0.78 T before the magnetic field was ramped up to 2.55 T, so it cannot naively be reinterpreted as a DP limit. |  47.66 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
HAYSTAC-1 |  Zhong:2018rsr  |  9 |  41.32 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
HAYSTAC-2 |  Backes:2020ajv  |  9 |  41.32 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
CAPP-1 |  Lee:2020cfj  |  7.3 |  36.35 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
CAPP-2 |  Jeong:2020cwz  |  7.8 |  36.35 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
CAPP-3 |  Kwon:2020sav  |  7.2 and 7.9 |  36.35 |  $90~{}{\rm s}$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
CAPP-3 [KSVZ] |  Kwon:2020sav  |  7.2 |  36.35 |  $15~{}{\rm hr}$ |  $\hat{\mathcal{Z}}$-pointing |  0.20
QUAX-$\alpha\gamma$ |  Alesini:2020vny  |  8.1 |  45.35 |  4203 s |  $\hat{\mathcal{Z}}$-pointing |  0.023
†KLASH |  Alesini:2017ifp  |  0.6 |  41.80 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{Z}}$-pointing |  $\sim$0.019
RBF |  DePanfilis:1987dk  | Magnetic field veto
UF |  Hagmann:1990tj  | Magnetic field veto
ORGAN |  McAllister:2017lkb  | Magnetic field veto
| RADES |  Melcon:2021dyi  | Magnetic field veto
LC-circuits | ADMX SLIC-1 |  Crisosto:2019fcj  |  4.5 |  29.64 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{N}}/\hat{\mathcal{W}}$-facing |  $\sim$0.19
ADMX SLIC-2 |  Crisosto:2019fcj  |  5 |  29.64 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{N}}/\hat{\mathcal{W}}$-facing |  $\sim$0.19
ADMX SLIC-3 |  Crisosto:2019fcj  |  7 |  29.64 |  $\mathcal{O}({\rm min})$ |  $\hat{\mathcal{N}}/\hat{\mathcal{W}}$-facing |  $\sim$0.19
ABRACADABRA |  Ouellet:2018beu  | Magnetic field veto888The pickup geometry also does not allow for linearly polarised DP to be detected.
SHAFT |  Gramolin:2020ict  | Magnetic field veto
Plasmas |  †ALPHA |  Lawson:2019brd  |  10 |  Unknown |  $\mathcal{O}$(week) |  $\hat{\mathcal{Z}}$-pointing |  0.28–0.33
Dielectrics |  †MADMAX |  TheMADMAXWorkingGroup:2016hpc  |  10 |  53.57 |  $\mathcal{O}$(week) |  $\hat{\mathcal{Z}}$-pointing or $\hat{\mathcal{N}}/\hat{\mathcal{W}}$-facing |  0.26 or 0.62–0.66999MADMAX can sense the $E$-field parallel to its disks, which will be vertical. But whether MADMAX will be axial or planar depends on its antenna. The first number is if the antenna can sense only a single polarisation, and the second is if it can sense arbitrary polarisation, in which case the range reflects the unknown final horizontal orientation of the experiment.
†LAMPOST |  Baryakhtar:2018doz  |  10 |  Unknown |  $\mathcal{O}$(week) |  Any-facing |  0.61–0.66
†DALI |  DeMiguel-Hernandez:2020mon  |  9 |  28.49 |  $\mathcal{O}$(month) |  Any-facing101010The DALI experiment is designed to be on an altazimuth mount and so can be aligned as required. |  0.61–0.66
Dish antenna |  †BRASS |  BRASS  |  1 |  53.57 |  $\mathcal{O}$(100 days) |  Any-facing |  0.61–0.66
Topological insulators |  †TOORAD |  Schutte-Engel:2021bqm  |  10111111The $B$-field would be varied for tuning purposes |  Unknown |  $\mathcal{O}$(day) |  Any-pointing |  0.18–0.33
Table 1: Summary of axion haloscopes and their parameters relevant for recasting exclusion limits to DPs. We denote planned experiments with a “$\dagger$”. Note that some axion exclusions cannot be reinterpreted as DP exclusions if the experiment used a magnetic field to veto potential signals Gelmini:2020kcu . We assumed that, unless stated otherwise, all experiments did not use some form of magnetic field veto, though it is possible that one was used without explicit acknowledgement. In the final column we have put our estimate of $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ which is a geometric factor used to convert axion exclusion limits (with varying C.Ls) to 95% C.L. DP exclusion limits, accounting for the unknown polarisation. The factor $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ is defined in Eq.(34), and its calculation is the subject of Sec. VI. For experiments with unknown locations, we have taken the largest and smallest values in the range $\lambda_{\rm lab}\in[35^{\circ},55^{\circ}]$. For experiments with unknown orientations, we have taken the largest and smallest values over the range of possible orientations. Experiment | Latitude |  Measurement | Directionality |  Assumed |  $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$
---|---|---|---|---|---
|  |  | [∘] |  time, $T$ |  |  $\langle\cos^{2}{\theta}\rangle_{T}$ |
Cavities | WISPDMX |  Nguyen:2019xuh  | 46.14 |  $\mathcal{O}({\rm day})$ | ($0.92\hat{\mathcal{N}}+0.38\hat{\mathcal{W}}$)-pointing |  1/3 |  0.23
SQuAD |  Dixit:2020ymh  | 41.88 |  12.81 s | Unspecified |  1/3 |  0.019
Dielectrics |  $\dagger$NYU Abu Dhabi |  Manenti  | 24.45 |  ${\cal O}({\rm day})$ |  $\hat{\mathcal{Z}}$-facing |  N/A |  0.65
Dishantennae | Tokyo-1 |  Suzuki:2015sza  | 35.68 |  29 days121212This measurement alternated signal and background every 30s |  $\hat{\mathcal{W}}$-facing |  2/3 |  0.62
Tokyo-2 |  Knirck:2018ojz  | 36.06 |  $\mathcal{O}$(week) | Axial, $\hat{\mathcal{N}}$/$\hat{\mathcal{W}}$-pointing |  1/3 |  0.15–0.2
Tokyo-3 |  Tomita:2020usq  | 36.13 |  12 hr |  $\hat{\mathcal{N}}$/$\hat{\mathcal{W}}$-pointing or $\hat{\mathcal{Z}}$-facing |  Unspecified |  0.15 or 0.62
SHUKET |  Brun:2019kak  | 48.86 |  8000 s |  $\hat{\mathcal{Z}}$-pointing |  1/3 |  0.04
FUNK |  Andrianavalomahefa:2020ucg  | 49.10 |  $\mathcal{O}$(month) | ($-0.5\hat{\mathcal{N}}-0.87\hat{\mathcal{W}}+0.28\hat{\mathcal{Z}}$)-facing |  2/3 |  0.56
LC-circuits | DM Pathfinder |  Phipps:2019cqy  | 37.42 |  5.14 hr |  $\hat{\mathcal{Z}}$-pointing |  1131313Rather than assume a randomised angle, the DP current was assumed to be aligned with the device |  0.075
Dark E-field |  Godfrey:2021tvs  | 38.54 |  3.8 hr 141414These numbers include dead time, so should not be taken as a single continuous measurement |  $\hat{\mathcal{W}}$-pointing |  1/3 |  0.29
Dark E-field spots |  Godfrey:2021tvs  | 38.54 |  5.8 days151515These numbers include dead time, so should not be taken as a single continuous measurement |  $\hat{\mathcal{W}}$-pointing |  1/3 |  0.58
Table 2: Summary of dedicated DPDM experiments and some of their relevant
parameters. We mark planned experiments with a “$\dagger$”. The penultimate
column refers to the value of $\langle\cos^{2}{\theta}\rangle_{T}$ assumed by
the corresponding reference, whereas the last column is our best attempt to
estimate the value of $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ for the
fixed polarisation scenario, based on the published measurement times,
locations, and experimental orientations. This factor is defined in Eq.(34)
and describes how much a 95% C.L. exclusion limit on the DP is impacted by the
unknown DP polarisation angle with respect to the experiment. The calculation
of $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ factor is the subject of
Sec. VI. Figure 2: Schematic of the two categories of experiment that we
structure this study around. They are distinguished by their sensitivity to
the direction of the DP polarisation. On the left, “axial” experiments
sensitive to a single direction of polarisation are represented via a
cylindrical cavity. In the middle, “planar” experiments, sensitive to a plane
of polarisation, are shown using a simple sketch of a dish antenna. We express
the DP polarisation here in the lab-centered North-West-Zenith basis shown in
the far right image. To introduce some nomenclature that will be important
later: the cavity in this example would be sensitive to electric fields along
the $\hat{\mathcal{Z}}$ axis, so we refer to this as “Zenith-pointing”. On the
other hand, the dish antenna is facing North, so it is sensitive polarisations
in the $\hat{\mathcal{Z}}$-$\hat{\mathcal{W}}$ plane, which we refer to as
“North-facing”.

Most searches for DPs as dark matter use the mixing of the DP and visible
photon to induce $E$ or $B$ fields, which are then coupled to a detector. This
is similar to searches for the axion-photon coupling, which typically use an
external $B$-field to mix axions with photons. Hence there is substantial
overlap between experiments that can search for axions and those that can
search for DPs. All experiments that we will discuss are listed in Tables 1
and 2.

Types of experiment exploiting electromagnetic mixing include: cavities
Sikivie:1983ip ; Rybka:2014xca ; Woohyun:2016 ; Goryachev:2017wpw ;
Alesini:2017ifp ; Melcon:2018dba ; Melcon:2020xvj , dielectric disks
TheMADMAXWorkingGroup:2016hpc ; Baryakhtar:2018doz , dish antennae
Horns:2012jf ; Jaeckel:2013sqa ; Suzuki:2015sza ; Experiment:2017icw ; BRASS ,
plasmas Lawson:2019brd , LC circuits Sikivie:2013laa ; Chaudhuri:2014dla ;
Kahn:2016aff ; Silva-Feaver:2016qhh ; Crisosto:2018div , and electric-field
radios Godfrey:2021tvs . However, there are two key differences between DPs
and axions. The first is that the axion-induced $E$-field is always aligned
with the external magnetic field due to the ${\bf E}\cdot{\bf B}$ coupling of
the axion, whereas the DP can be polarised in any direction. The second is
that the mixing does not “turn off” in the absence of a $B$-field, which as
pointed out in Ref. Gelmini:2020kcu  is often not checked when reinterpreting
axion limits in the context of DPs. This behaviour both changes potential
noise vetos (such as demanding the signal vary as $B^{2}$) and the
requirements of shielding Arias:2014ela ; Chaudhuri:2014dla .

By design, the majority of these experiments are only sensitive to fields
induced in specific directions. Most commonly a single direction, e.g. a
single polarisation of light as in Eq. (18), but for some designs all
polarisations lying along a two-dimensional plane may be measurable. A
schematic of these two classes is shown in Fig. 2. We refer to experiments
that are sensitive to a single polarisation direction as _axial_ , and those
that can sense polarisations along a plane as _planar_. In the following
subsections we will describe which types of experiment fall into each of these
two categories. In this figure we also define our lab-centered coordinate
system basis that will become important when we discuss daily modulation:
$(\hat{\mathcal{N}},\hat{\mathcal{W}},\hat{\mathcal{Z}})$, which point towards
the North, West and Zenith respectively.

###  V.1 Cavity haloscopes

First introduced by Sikivie in Ref. Sikivie:1983ip , cavity haloscopes are the
prototypical axion (and DP) direct detection experiment. A cavity haloscope
uses the resonant enhancement of a cavity mode to increase the probability of
DM converting to photons over a narrow frequency range (given by the width of
the resonant mode, or quality factor, of the cavity). However, resonant
enhancement is only achieved if the overlap of the photon wavefunction and DM
wavefunction is nonzero Sikivie:1983ip ; Knirck:2018knd . Which cavity modes
achieve a nonzero overlap depends on the geometry of the cavity. For the
popular cylindrical cavity, only transverse magnetic (TM) modes have a nonzero
overlap—transverse electric (TE) modes, on the other hand, do not couple
Stern:2015kzo .161616More precisely, only TM0n0 modes have a significant
nonzero overlap, with the largest being the TM010 mode Stern:2015kzo . Thus,
cylindrical cavities are only sensitive to the component of the DP polarised
along the axis parallel to the magnetic field, which is usually aligned
vertically.

For a more complicated cavity, such as multicavity arrays Goryachev:2017wpw ;
Melcon:2018dba ; Melcon:2020xvj ; Jeong:2020cwz , dedicated mode analyses are
required to determine how the DP couples. Even if the DP couples to modes in
multiple directions, each mode will have a preferred $E$-field direction.
Depending on how the cavity is read out, as long as these modes are not
degenerate and have sufficient frequency separation to be resolved, then each
mode will provide information on a specific polarisation. If the modes are not
similar in their overall signal power, then the sensitivity will be
directionally sensitive in the same way as a cylindrical cavity. On the other
hand, if each mode is similar in power (for example, a cubic cavity), then the
experiment will be sensitive regardless of the DP polarisation. Cavities
designed to detect axions are usually not optimised for multiple polarisation
directions since they do not need to be.

So we consider cavities to be “axial” in the sense that they are sensitive to
a single polarisation direction. In these cases, the signal power is
suppressed by the angular factor introduced in Eq.(18), $\cos^{2}{\theta}$,
where $\theta$ the angle between $\hat{\mathbf{X}}$ and that direction. The
average of this factor over all possible DP polarisation axes is
$\langle\cos^{2}{\theta}\rangle=1/3$.

###  V.2 Dish antennae

In stark contrast to cavity experiments, dish antennae allow a broadband DM
search, relying on the nonresonant breaking of translation invariance
Horns:2012jf ; Jaeckel:2013sqa . As shown in Eq.(18) the mixing between dark
and ordinary photons depends on the medium. So by changing medium, the DP-
induced $E$-field also changes. Because parallel $E$ and $B$-fields must be
conserved across changing media, propagating photon waves are emitted to
compensate, which could then be detected. In other words, when the DPDM passes
through the dish the small electric field of the DP makes the electrons in the
dish oscillate, emitting an (almost) ordinary electromagnetic wave
perpendicular to the surface. Dish antennae can, at most, only be sensitive to
the component of $E$-field parallel to the interface (usually a metallic
mirror) Jaeckel:2013sqa . In contrast to cavities, we classify this kind of
experiment as “planar”, as displayed by the middle panel of Fig. 2. As a
consequence, the emitted $E$-field is suppressed by $\cos\theta$, where
$\theta$ is the angle between $\hat{\mathbf{X}}$ and the plane of the
interface.

Complicating matters is the issue that dish antennae are often, as the name
suggests, rounded. If the polarisation is spatially constant over scales
comparable to the radius of curvature, $\cos\theta$ will depend upon the
position on the dish. Because of this, the total power detected is
proportional to $\int\mathrm{d}A\cos^{2}\theta({\bf x)}$ Horns:2012jf , only
being trivial for planar or completely spherical systems. The details depend
on the exact geometry of the dish antenna, which can vary considerably between
experiments, so we leave the details for dedicated analyses. To provide a
rough guide, we make an estimate assuming a planar dish geometry. When
averaged over all possible DP polarisations,
$\langle\cos^{2}\theta\rangle=2/3$, for any dish antenna with a sufficiently
large radius of curvature.

Furthermore, depending on the antenna technology, some experiments may only be
sensitive to certain polarisations of light, e.g. some stages of the Tokyo
dish antennae Knirck:2018ojz ; Tomita:2020usq . This would introduce an
additional factor of $\cos\theta_{a}$, where $\theta_{a}$ is the angle between
the light emitted from the dish antenna and detector polarisation. For our
purposes we will consider dish antennae either planar or axial depending on
whether the detector is sensitive to two polarisations or just one.

###  V.3 Dielectric haloscopes

Dielectric haloscopes use the same principles as a dish antenna, but employ
many semi-transparent dielectric layers, arranged so that the waves emitted at
each add constructively. The constructive interference of the emitted waves
enhances the sensitivity at the expense of reduced bandwidth. This is the core
design principle of MADMAX TheMADMAXWorkingGroup:2016hpc ; Millar:2016cjp ,
but the concept can be implemented on much smaller scales/higher frequencies,
as proposed for LAMPOST Baryakhtar:2018doz . Dielectric haloscopes have the
same polarisation dependence as a planar dish antenna. Depending on whether
the antenna is polarisation-specific, a dielectric haloscope could be
sensitive to $E$-fields aligned with either the plane of the dielectric disks
or the polarisation of the antenna Jaeckel:2013eha ; Baryakhtar:2018doz ;
Brun:2019lyf .

The recent proposal of using a topological insulator as an axion detector,
TOORAD, is mathematically equivalent to a single-layer dielectric haloscope
with near-zero refractive index Schutte-Engel:2021bqm . However, the device
would only be sensitive to a single polarisation due to the very strong
anisotropy of the medium.

###  V.4 Plasma haloscopes

A more recent proposal, ALPHA, belongs to a distinct class of experiment known
as plasma haloscopes Lawson:2019brd . These experiments allow DM axions or DPs
to convert to photons by matching the photon mass (plasma frequency) to the DM
mass. While the principle holds for any plasma, a more specific proposal
allowing for tuneable, cryogenic plasmas at the GHz regime is to use thin wire
metamaterials, aligned in a single direction Lawson:2019brd . As pointed out
in Ref. Gelmini:2020kcu , the matching of the dispersion relation only occurs
in the direction of wire alignment. If the boundary of the plasma is a
conducting cylinder, the experiment behaves like a resonant cavity for non-
axial electric fields. Thus for transverse polarisations the experiment
behaves like a cylindrical cavity, that is, with transverse electric modes not
coupling to DM. Thus a plasma haloscope consisting of a cylindrical cavity
with wires in only one direction will only be sensitive to $E$-fields aligned
with the wires.171717However, this is a geometry-dependent statement. In
principle, one could design an isotropic plasma inside a geometry that led to
equal sensitivity to any polarisation.

###  V.5 LC circuits

Rather than measuring the $E$-field induced by DPDM, LC circuits (also
referred to as lumped element circuits) attempt to inductively measure a
$B$-field. This field can be generated directly from the DM Sikivie:2013laa ,
or more commonly, indirectly via the $E$-field causing a current in a
conductor Chaudhuri:2014dla ; Kahn:2016aff ; Silva-Feaver:2016qhh ;
Crisosto:2018div ; Ouellet:2018beu . In fact, the (tiny) electric field
generated by the DP will create a displacement current, $\textbf{J}_{X}$. This
in turn creates a magnetic field and an induced electric field, defined
respectively via

$\displaystyle\nabla\times\textbf{B}_{X}=\textbf{J}_{X}\,,$ (19a)
$\displaystyle\nabla\times\textbf{E}_{X}=-\frac{\partial\textbf{B}_{X}}{\partial
t}\,.$ (19b)

If the system size, $r$, is much smaller than a Compton wavelength,
$m^{-1}_{X}$, the $E$-field will be suppressed, however the $B$-field will
not, $|\textbf{E}_{X}|\sim m_{X}r|\textbf{B}_{X}|$ Chaudhuri:2014dla ;
Arias:2014ela .

As the induced current is in the direction of the DP-induced $E$-field, only
polarisations parallel to the conductor can induce a $B$-field Arias:2014ela .
As this $B$-field is read out by an inductive loop, the directionality is
further constrained to polarisations that generate a $B$-field correctly
aligned with the readout loop. For an example, DM Radio Silva-Feaver:2016qhh
has a closed toroidal conducting sheath and reads out azimuthal $B$-fields.
With such a geometry the read-out $B$-field is proportional to
$\cos\theta=\hat{\mathbf{z}}\cdot\hat{\mathbf{X}}$. Whereas in other
geometries, such as ADMX SLIC Crisosto:2019fcj , the sensitivity can instead
lie in a plane Arias:2014ela .

A similar idea (in fact, an electromagnetic dual) is to have a shielded room
much larger than the Compton wavelength, and simply place an antenna to read
out any DP-induced $E$-fields. Though there was an early proposal for such an
experiment to search for axions Smith:1987kz , it has only recently been
revived Godfrey:2021tvs . While in principle all polarisations could be read
out, the experiment operating currently, referred to as Dark $E$-field Radio
Godfrey:2021tvs , uses a polarisation sensitive antenna.

Thus we can see that a large class of the most developed and promising light
wave-like DM experiments are sensitive to the polarisation of DPs.

###  V.6 Signal-to-noise

While the exact methods for reading out a signal depend on the design of an
experiment, measurement always boils down to increasing the signal-to-noise
ratio ($S/N$) by integrating the signal over some measurement time, $T$.
Measurements at different times are often combined, either to increase the
signal-to-noise, or to test an excess for signal veracity (often referred to
as a “rescan”). This combination usually relies on the assumption that the
signal will remain constant with time, i.e. if a $3\sigma$ excess is measured
during an initial campaign, it should persist with subsequent interrogation.
However, this is not necessarily a safe assumption for DP signals.

For the experiments we are exploring, there are two main methods of detection:
linear amplification and single photon counting. Regardless of whether the $E$
or $B$-field is being measured, the fundamental sensitivity comes from the
power generated by the conversion of DPs, relative to the noise of the system.
In all cases, the power in the measuring device is determined by the dominant
field, either $E^{2}$ or $B^{2}$. As discussed above, projecting a DP
polarisation onto the experiment also results in a factor of $\cos\theta$,
where from now on $\theta$ is taken to be the angle between the DP
polarisation and the experimentally relevant geometry: either a plane or an
axis. This means we can assume that the DP signal power can be written as,

$P(t)=P_{\rm X}\cos^{2}\theta(t)$ (20)

While real experiments will design dedicated statistical analyses to test for
signals, we can understand what the generic consequences of this kind of
temporal variation would be by writing down some simple formulae for S/N.

  * •

Linear Amplification: The signal-to-noise of a linear amplifier with constant
signal and noise temperatures is easily written via Dicke’s radiometery
equation Dicke

$\dfrac{S}{N}=\dfrac{P}{T_{\rm sys}}\sqrt{\dfrac{T}{\Delta\nu_{\rm DP}}}\,,$
(21)

where $T_{\rm sys}$ is the system noise temperature and $\Delta\nu_{\rm DP}$
is the DP linewidth. Since most experiments relying on this formula are not
analysing the variation in the power within the measurement time, $T$, the
relevant power in this expression is, $P\to\int\mathrm{d}tP(t)/T$.

  * •

Photon Counting: At higher frequencies, rather than amplifying a measured
voltage, it is often more practical to use a photon counting device. Instead
of a noise temperature, the background is often better characterised by a dark
count (false counts). The signal-to-noise in such a system is

$\dfrac{S}{N}=2(\sqrt{n_{s}+n_{d}}-\sqrt{n_{d}})\,,$ (22)

where $n_{s}$ is the number of signal events and $n_{d}$ the number of dark
counts Bityukov:1998ju ; Bityukov:2000tt ; Arias:2010bh . The integrated
signal over a continuous measurement is given by

$n_{s}=\eta\int\mathrm{d}t\frac{P(t)}{\omega}\,,$ (23)

where $\eta$ is the detector efficiency.

So we can see quite generically that a simple analysis of the total $S/N$ for
a continuous measurement is determined by

$\frac{1}{T}\int\mathrm{d}tP(t)\equiv
P_{X}\langle\cos^{2}{\theta}\rangle_{T}\,.$ (24)

Crucially, the quantity $\langle\cos^{2}{\theta}\rangle_{T}$ will depend upon
the distribution of $\hat{\mathbf{X}}$ around the Earth. Recall that we
classified two scenarios in Sec. III: the random polarisation scenario, where
$\hat{\mathbf{X}}$ is randomly drawn in every coherence time; and the fixed
polarisation scenario where there is only a single $\hat{\mathbf{X}}$ for the
entire experimental campaign.

In the randomly polarised scenario, as long as the measurement covers many
coherence times, i.e.

$T\gg\tau=\frac{2\pi}{m_{X}v^{2}}\simeq
400\,\upmu\mathrm{s}\left(\frac{10\,\upmu\mathrm{eV}}{m_{X}}\right)\,,$ (25)

then $\langle\cos^{2}\theta\rangle_{T}$ is simply given by the average of
$1/3$ for a single polarisation, or $2/3$ for a plane. In this case the only
thing the experiment needs to do to account for the DP polarisation is to
apply this factor to their expected signal, e.g. $P=P_{X}/3$

The situation for the fixed polarisation scenario is more involved because
$\theta(t)$ varies over course of the measurement (or measurements). This can
have important consequences for the signal-to-noise. Take the simple case of
two measurements, each of equal length $T$ but separated by some time $T_{\rm
wait}$, with signals-to-noise $S_{1}/N_{1}$ and $S_{2}/N_{2}$. For simplicity,
we will assume the backgrounds are the same for both ($N_{1}=N_{2}$), and that
the measurement is not in a background-free regime. In this case, the combined
$S/N$ is given by,

$\displaystyle\frac{S}{N}$
$\displaystyle\simeq\frac{S_{1}+S_{2}}{\sqrt{2N_{1}}}$ (26)
$\displaystyle\propto\frac{P_{\rm
X}}{T}\int_{0}^{T}\mathrm{d}t\cos^{2}\theta(t)+\frac{P_{\rm X}}{T}\int_{T_{\rm
wait}}^{T_{\rm wait}+T}\mathrm{d}t\cos^{2}\theta(t)\,.$

Assuming the first time average of $\cos^{2}\theta(t)$ is equal to the second,
then when the original signal is real there would be a $\sqrt{2}$ increase in
$S/N$. But if the first result was a fluke, the second measurement would
reduce $S/N$ by $\sqrt{2}$. However, imagine if the case were,

$\int_{0}^{T}\mathrm{d}t\cos^{2}\theta(t)\gg\int_{T_{\rm wait}}^{T_{\rm
wait}+T}\mathrm{d}t\cos^{2}\theta(t)\,,$ (27)

i.e. the DP field was well aligned for the first measurement, but poorly
aligned for the second. In such a scenario, an analysis assuming a constant
$P(t)$ would then see a reduction in $S/N$ by a factor of $\sqrt{2}$, and
potentially conclude that the signal was a statistical fluke. While this is an
oversimplified case, it serves to illustrate that it can be dangerous to
reinterpret analyses that did not consider time varying signals. However, we
are not completely without handles on this time variation. The Earth rotates
in a predictable and unambiguous way. As we will discuss in the next section,
simply accounting for—or, even better, planning the experiment around—the
Earth’s rotation, one can greatly reduce the chances of encountering the sorts
of bad luck that led to the example described above.

Figure 3: A diagram of the coordinate systems used in this paper. On the left,
a geocentric equatorial coordinate system in which we fix the DP polarisation
vector, $\hat{\mathbf{X}}$, defined by angles $(\theta_{X},\phi_{X})$. The
$\hat{\mathbf{x}}_{e}$ direction is conventionally chosen to point towards the
vernal equinox, with $\hat{\mathbf{y}}_{e}$ pointing 90∘ of right ascension to
the East. The Earth rotates anticlockwise in the
$\hat{\mathbf{x}}_{e}$-$\hat{\mathbf{y}}_{e}$ plane. On the right, the
detector-centric Cartesian coordinate system using axes pointing towards the
North, West, and Zenith. We also show three planes in the coordinate system
which we will need when describing those experiments that are sensitive to the
component of $\hat{\mathbf{X}}$ projected onto a two-dimensional plane. The
Zenith-facing plane is horizontal, whereas the North-facing and West-facing
planes are both vertical in the lab.

###  V.7 Deriving limits on dark photons

We have listed all past, current, and planned axion haloscopes searching for
the axion-photon coupling in Table 1. This table includes all the relevant
information needed to convert their limits on axions to limits on DPs.

The crucial differences between axions and DPs come down to two factors: the
absence of $B$ in the DP case, and the non-trivial polarisation. The former
implies that vetos demanding that the signal vanish in the absence of a
magnetic field, as often used in axion searches, would also veto any DP
signals and so make an experiment incapable of discovering them. For clarity,
we have also listed the experiments for which this caveat applies in Table 1.

As for the DP polarisation, this issue requires more care and we must think
about what an exclusion limit means. Most haloscope analyses take the form of
a hypothesis test— testing for the presence of an axion signal above some
noise. In the absence of any convincing signal candidates, they can set an
exclusion limit on the coupling—the convention usually being at the 90 or 95%
C.L. In the axion case the signal is a fixed number and the exclusion is
straightforward, however in the DP case, there is an unknown parameter which
influences the signal strength: $\langle\cos^{2}{\theta}\rangle_{T}$.
Therefore, since the measured $\langle\cos^{2}{\theta}\rangle_{T}$ will be
drawn from some (known) distribution, to preserve the statistical meaning of
an axion limit when translating it to DPs, we should calculate some additional
factor $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ that accounts for
this. We define this factor to be ratio of the axion and DP power thresholds
that can be excluded at 95% C.L. in the absence of a signal. The quantity
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ (or more precisely, its
square root since $P\propto\chi^{2}$) will therefore encode how much the
exclusion limit on $\chi$ is weakened by the distribution of
$\langle\cos^{2}{\theta}\rangle_{T}$.

Doing this conversion accurately would require additional information we
cannot easily obtain. For example, we would need to know the noise
distribution of each experiment, as well as the local significance of the
measured power at each mass point. Nevertheless, we can attempt an estimate of
the limit one would obtain from a full data analysis by 1) assuming the noise
follows Gaussian distribution, and 2) assuming the measured signal equals the
median expected noise. The former assumption is very often the case for most
experiments, since they often employ a technique of data stacking which will
render most noise distributions Gaussian by the central limit theorem. The
latter assumption is a practical one that will make the resulting limit
slightly inaccurate and only an estimate of the median exclusion limit. We
emphasise here that our intention is not to set definitive limits on the DP
using axion haloscope data, but to lay out the recipe for doing so, and
predict what would be found.

Next, we adopt a stripped down toy model of the signal and noise distributions
to explain how we can relate the axion and DP cases. Let us define the signal
and noise in the axion case as,

$P=P_{a}+N\,$ (28)

where $P_{a}$ is the axion signal power and $N$ is the noise which follows a
Gaussian distribution. For simplicity we assume the noise has a mean of $0$
and variance $\sigma_{N}$. The expected (i.e. median) 95% C.L. exclusion limit
will therefore be an axion coupling that gives,

$\frac{P_{a}}{\sigma_{N}}>\Phi^{-1}[0.95]=1.64\,,$ (29)

where $\Phi^{-1}$ is the inverse cumulative distribution function of the
Gaussian noise.181818For experiments that quote 90% C.L. exclusions we use
1.28 here instead.

On the other hand, in the DP case, the signal is not a single value, but
follows a distribution given by,

$P=P_{X}+N\equiv P^{0}_{X}\langle\cos^{2}{\theta}\rangle_{T}+N\,.$ (30)

To get an equivalent 95% C.L. exclusion limit on $P^{0}_{X}$ we must evaluate
an inverse cumulative distribution function, but this time not just of the
normally distributed noise, but the joint distribution of the noise and our
nuisance parameter $\langle\cos^{2}{\theta}\rangle_{T}$. The cumulative
distribution function for a variable $z=x+y$ can be written in terms of the
distributions on $x$ and $y$, which, assuming they are independent variables,
is the following,

$\Phi(Z)=\mathbb{P}(z\leq
Z)=\int_{-\infty}^{\infty}\mathrm{~{}d}x\int_{-\infty}^{Z-x}\mathrm{d}y\,f(x)f(y)\,.$
(31)

For our problem we must therefore solve,

$\Phi[0]\equiv\int_{-\infty}^{+\infty}\textrm{d}P_{X}\int_{-\infty}^{0-P_{X}}\textrm{d}N\,f(P_{X})f(N)=1-0.95\,,$
(32)

where we integrate the noise up to $N=0-P_{X}$, because we want the cumulative
distribution function for $P<0$, which is the expected noise level—i.e. we are
demanding that 95% of measured signals would give a signal greater than the
observed noise. We can perform the integral over $f(N)$, which is just a
Gaussian, to find,

$\displaystyle\int_{0}^{1}\textrm{d}\langle\cos^{2}{\theta}\rangle_{T}\frac{f(\langle\cos^{2}{\theta}\rangle_{T})}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{-P^{0}_{X}\langle\cos^{2}{\theta}\rangle_{T}}{\sqrt{2}\sigma_{N}}\bigg{)}\bigg{]}$
$\displaystyle=1-0.95\,.$ (33)

where $f(\langle\cos^{2}{\theta}\rangle_{T})=f(P_{X})/P^{0}_{X}$. We then
solve this equation for $P^{0}_{X}/\sigma_{N}$, giving us the power threshold
required for 95% of all possible signals to be above the median noise. Then,
since we are matching this threshold to the axion’s one,
$P_{a}/\sigma_{N}=1.64$, we can take the ratio of these thresholds to define,

$\langle\cos^{2}{\theta}\rangle^{\rm
excl.}_{T}=\frac{1.64\,\sigma_{N}}{P^{0}_{X}}\,,$ (34)

which effectively describes how much the DP power threshold for a 95% C.L.
exclusion needs to be enhanced over the axion case. We will see in the next
section that this results in a value of $\langle\cos^{2}{\theta}\rangle^{\rm
excl.}_{T}$ that is typically in the 15–30th percentile of the distribution
$f(\langle\cos^{2}{\theta}\rangle_{T})$.

As a sanity check we can imagine what would happen if the distribution of
$\langle\cos^{2}{\theta}\rangle_{T}$ was a delta function:
$\delta(1-\langle\cos^{2}{\theta}\rangle_{T})$. In that case, the solution to
Eq.(V.7) is $P^{0}_{X}/\sigma_{N}=1.64$, giving
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}=1$. So the power threshold for
the DP exclusion limit is identical to the axion, as expected. This case is
similar to what we expect in the randomised polarisation scenario. If the DP
has a random $\cos^{2}\theta$ in every coherence time—and typically a single
measurement will be over a very large number of coherence times—the central
limit theorem will dictate that the resulting
$f(\langle\cos^{2}{\theta}\rangle_{T})$ for the stacked data will be very
tightly contained around its average of $1/3$ (axial) or $2/3$ (planar). This
means we can assume that $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ will
be equal to either $1/3$ or $2/3$ for the random polarisation scenario.

For the fixed polarisation scenario on the other hand, there is no single
factor we can use since $f(\langle\cos^{2}{\theta}\rangle_{T})$ depends upon
$T$, as well as the location, orientation and readout of the experiment. In
general though, we will find that for very short measurement times, the value
of $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ is $\sim$0.025 for axial
experiments and $\sim$0.37 for planar experiments191919Note that if the
original published exclusion limit was at 90% C.L. rather than our chosen
benchmark of 95%, then these factors are 0.019 and 0.29. For longer $T$, the
rotation of the Earth will restrict the distribution of possible
$\langle\cos^{2}{\theta}\rangle_{T}$ that are available to the experiment and
therefore $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ will increase. We
present our estimate of $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ for
axion haloscopes in the final column of Table 1, but the full mathematical
recipe for its calculation is the subject of the following section.

We note in passing that previous treatments of the fixed polarisation scenario
Arias:2012az ; Ghosh:2021ard  defined their conversion factor as the 5th
percentile of $f(\langle\cos^{2}{\theta}\rangle_{T})$, which gives a value of
$0.0025$ instead of 0.025 for very short $T$ axial experiments. Such a
prescription is more demanding of the signal and the resulting limit using
that conversion factor would turn out to be suppressed over a limit that was
truly at the 95% C.L.

For the sake of completion, we have also made a similar table for experiments
that are dedicated to searching only for DPs: Table 2. Despite the fact that
the directionality of the DP polarisation and its relationship to the device’s
geometry and orientation is of central importance to computing the signal in
these experiments, it is surprisingly not always explicitly stated how this
information was taken into account. In the cases where it was not possible to
glean the orientation or location of the experiment from the manuscript cited,
the authors were contacted for further information.

Finally, as well as the expected exclusion limits, we can use the same
technique as described above to define the conversion factor needed to find
the DP’s _discovery limit_ , $\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$.
We define “discovery” in this context to be the case when 95% of experiments
could reject the noise-only hypothesis at $5\sigma$ or more. To calculate this
we simply need to take Eq.(32) and replace the upper limit of the integral
over $N$ with $5\sigma_{N}-P_{X}$, so that we are imposing that 95% of the
signal’s distribution is $5\sigma_{N}$ away from the median noise. This leaves
us with solving,

$\displaystyle\int_{0}^{1}\frac{f(\langle\cos^{2}{\theta}\rangle_{T})}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{5\sigma_{N}-P^{0}_{X}\langle\cos^{2}{\theta}\rangle_{T}}{\sqrt{2}\sigma_{N}}\bigg{)}\bigg{]}\textrm{d}\langle\cos^{2}{\theta}\rangle_{T}$
$\displaystyle=1-0.95\,.$ (35)

If the solution to this equation is some value of $P^{0}_{X}/\sigma_{N}$ then

$\langle\cos^{2}{\theta}\rangle^{\rm
disc.}_{T}=\frac{(5+1.64)\sigma_{N}}{P^{0}_{X}}\,.$ (36)

We will use $\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$ when we want to
discuss ways of optimising the experiment to have the greatest chance of
_discovering_ the DP signal, not just setting exclusion limits based on its
absence.

##  VI Accounting for the rotation of the Earth

So far we have seen that because the DP possesses a polarisation, induced
electromagnetic signals of the DP will inherent the features of that
polarisation. In particular, since we observe the DM signal from a rotating
reference frame, any fixed DP polarisation on scales much larger than the mpc-
scales probed by experimental campaigns, will generate signals with a strong
preference on the experimental orientation, and consequently, a daily
modulation. In this section we will derive these signals.

We first need to lay down some coordinate systems. There are two relevant
ones: we need a coordinate system to define the DP polarisation which is fixed
relative to the rotating Earth, and a coordinate system to define our detector
that rotates with the surface of the Earth. These are both shown in Fig. 3.

Following the left-hand panel of Fig. 3, we set the first of these coordinate
systems to be the conventional geocentric equatorial frame defined by
$(\hat{\mathbf{x}}_{e},\hat{\mathbf{y}}_{e},\hat{\mathbf{z}}_{e})$. These axes
point, respectively, towards the vernal equinox, 90 degrees of right ascension
East from vernal equinox, and parallel to the Earth’s spin axis.202020We are
adopting the conventional definition of equatorial coordinates but the
orientation of $\hat{\mathbf{x}}_{e}$ and $\hat{\mathbf{y}}_{e}$ is irrelevant
when are assuming we have no knowledge of the polarisation axis of the DP and
we are only looking at differences in time. We refer the reader to the
Appendix of Ref. Mayet:2016zxu  for the full rotation matrices needed to
convert between laboratory and equatorial coordinates at a precise local time.
Next we define the detector-centric coordinate system, as displayed by the
three arrows in the right-hand panel of Fig. 3. This system is defined by axes
$(\hat{\mathcal{N}},\hat{\mathcal{W}},\hat{\mathcal{Z}})$ which point towards
the North, West, and the zenith respectively.

Figure 4: We define $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ in
Eq.(34) to parameterise how much a 95% C.L. DP exclusion limit is impacted by
the distribution of possible polarisation angles. This parameter is an
effective conversion factor for recasting prior exclusion limits into the
fixed polarisation scenario. Here, we show the latitude dependence of this
factor when $T$ is an integer number of sidereal days. The left-hand panel is
for axial (1D) experiments, and the right-hand panel is for planar (2D)
experiments. Each panel has three lines for the three possible orientations of
those experiments. This figure is to demonstrate the preferential latitudes
derived in Eqs.(VI.1) and (VI.2) for integer-day-long measurements. The factor
plotted here depends upon the shape of the distribution of
$\langle\cos^{2}{\theta}\rangle_{T}$ and is typically at the 15-30th
percentile.

Figure 5: Distributions of $\langle\cos^{2}{\theta(t)}\rangle_{T}$ after
sampling the DP polarisation $(\theta_{X},\phi_{X})$ isotropically across the
sky. For each value of $T$ we use the logarithmic colour-scale to show the
value of the distribution, normalised by its maximum value. The three columns
correspond to three different latitudes, $\lambda_{\rm
lab}=35^{\circ},45^{\circ}$, and $55^{\circ}$, from left to right. The two
rows correspond to North-pointing experiments (upper panels) and Zenith-
pointing experiments (lower panels). Notice that at integer values of $T$ in
sidereal days, the distribution approaches a single point at a value of $1/3$
in the upper right-most panel, and the lower left-most panel. These correspond
to the peaks of the North-pointing and Zenith-pointing lines shown in the
left-hand panel of Fig. 4. The white lines correspond to our two conversion
factors which we use to parameterise how much DP limits are impacted by the
distribution of $\langle\cos^{2}{\theta}\rangle_{T}$ in the fixed polarisation
scenario, as described in Sec. V.6. The dashed line is
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$, as defined in Eq.(34), and
can be used to rescale _exclusion_ limits. Whereas the dot-dashed line is
defined in Eq.(36), and can be used to rescale _discovery_ limits.

Figure 6: As in Fig. 5, but now for planar experiments, i.e. those where the
DP signal depends upon the angle between the DP polarisation and a plane. We
show the distributions of $\langle\cos^{2}{\theta(t)}\rangle_{T}$ after
sampling the DP polarisation $(\theta_{X},\phi_{X})$ isotropically across the
sky. For each value of $T$ we use the colour-scale to display the
distribution, which we have normalised by its maximum value. The three columns
correspond to three different latitudes, $\lambda_{\rm
lab}=35^{\circ},45^{\circ}$, and $55^{\circ}$, from left to right. The two
rows correspond to North-facing (upper panels) and Zenith-facing (lower
panels) experiments. Notice that at integer values of $T$ in sidereal days,
the distribution approaches a single point at a value of $2/3$ in the upper
right-most panel, and the lower left-most panel. These correspond to the peaks
of the North-facing and Zenith-facing lines in the right-hand panel of Fig. 4.

###  VI.1 Axial experiments

To see one example of how the daily modulation plays out, we first consider an
experiment in which the DP signal is proportional to the cosine squared of the
angle with respect to the vertical (as is the case for all experiments in
Tables 1 and 2 where the direction in the Directionality column is listed as
“Zenith-pointing”). This direction, expressed in our equatorial coordinate
system $(\hat{\mathbf{x}}_{e},\hat{\mathbf{y}}_{e},\hat{\mathbf{z}}_{e})$ is,
as a function of time,

$\hat{\mathcal{Z}}(t)=\left(\begin{array}[]{c}\cos{\lambda_{\rm
lab}}\cos{\omega_{\oplus}t}\\\ \cos{\lambda_{\rm
lab}}\sin{\omega_{\oplus}t}\\\ \sin{\lambda_{\rm lab}}\end{array}\right)\,$
(37)

where $\omega_{\oplus}=2\pi/{\rm 1\,day}$ is the angular frequency of the
Earth’s rotation.212121Note that these signals will have a period equal to one
sidereal day, which is 23 hours and 56 minutes long, as opposed to the solar
day which is 4 minutes longer. This nuance is unimportant for our analysis,
but it would be an important part of testing potential signals. Since we are
only going to take time-averages, we have neglected the temporal phase that
describes the longitude and the local time of observation.

If we now define a direction for the DP’s polarisation in equatorial
coordinates,

$\hat{\mathbf{X}}=\left(\begin{array}[]{c}\sin{\theta_{X}}\cos{\phi_{X}}\\\
\sin{\theta_{X}}\sin{\phi_{X}}\\\ \cos{\theta_{X}}\end{array}\right)\,$ (38)

then the angle we are interested in is,

$\cos^{2}{\theta(t)}=\big{(}\hat{\mathbf{X}}\cdot\hat{\mathcal{Z}}(t)\big{)}^{2}\,.$
(39)

The DP signal power accumulated over a measurement time, $T$, is proportional
to the time-average,

$\langle\cos^{2}{\theta(t)}\rangle_{T}\equiv\frac{1}{T}\int_{0}^{T}\cos^{2}{\theta(t)}\mathrm{d}t\,.$
(40)

As discussed in the previous section, the statistics of this quantity when
sampling ($\theta_{X},\phi_{X})$ across the sky will be what influences the
sensitivity of the experiment. It can be shown that,

$\frac{1}{4\pi}\int\langle\cos^{2}{\theta(t)}\rangle_{T}\,\mathrm{d}\cos{\theta_{X}}\,\mathrm{d}\phi=\frac{1}{3}$
(41)

for all $T$, meaning the squared polarisation component when averaged over the
sky is always $1/3$ regardless of how long the observation time is.

This result has implications for the exclusion and discovery thresholds that
we defined in Sec. V.7. The factors $\langle\cos^{2}{\theta}\rangle^{\rm
excl.}_{T}$ and $\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$ must be below
the mean of the distribution, as they describe how much the power threshold
needs to be enhanced due to the distribution of
$\langle\cos^{2}{\theta(t)}\rangle_{T}$. Therefore the maximum that
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ and
$\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$ can ever reach will be when
the distribution is very tightly contained around the mean. This means that
for axial experiments, $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ will
be at most 1/3, which happens to be the value for the random polarisation
scenario. So when we are optimising our experiment _the best we could hope to
do is to make the limit the same for both DP polarisation scenarios._

However, although the all-sky mean of $\langle\cos^{2}{\theta(t)}\rangle_{T}$
is independent of the location and $T$, its _distribution_ will vary greatly,
as we will now show. To simplify things further, let us consider
$\langle\cos^{2}{\theta(t)}\rangle_{T}$ when $T=1$ sidereal day (or any
integer number). Since in this case the detector axis has spun around $2\pi$
in the $(\hat{\mathbf{x}}_{e},\hat{\mathbf{y}}_{e})$ plane, we expect any
dependence on $\phi_{X}$ (which is the angle defined in this plane) to drop
out,

$\displaystyle\langle\cos^{2}{\theta(t)}$ $\displaystyle\rangle_{T=n\,{\rm
days}}=$ $\displaystyle\frac{1}{8}$
$\displaystyle\big{(}3+\cos{2\theta_{X}}-(1+3\cos{2\theta_{X}})\cos{2\lambda_{\rm
lab}}\big{)}\,.$ (42)

So $\langle\cos^{2}{\theta(t)}\rangle_{T=n\,{\rm days}}$ is bounded between
$(1~{}+~{}\cos{2\lambda_{\rm lab}})/4$ and $(1~{}-~{}\cos{2\lambda_{\rm
lab}})/2$ with some distribution. This distribution is straightforward to
compute numerically since $\cos{\theta_{X}}$ is drawn from a uniform
distribution between $[-1,1]$.

We can do the same thing if we were interested in West or North-pointing
experiments. For these cases we write the detector axes in equatorial
coordinates in a similar fashion,

$\hat{\mathcal{W}}(t)=\left(\begin{array}[]{c}\sin{\omega_{\oplus}t}\\\
-\cos{\omega_{\oplus}t}\\\ 0\end{array}\right)\,,$ (43)

and,

$\hat{\mathcal{N}}(t)=\left(\begin{array}[]{c}\sin{\lambda_{\rm
lab}}\cos{\omega_{\oplus}t}\\\ -\sin{\lambda_{\rm
lab}}\sin{\omega_{\oplus}t}\\\ \cos{\lambda_{\rm lab}}\end{array}\right)\,.$
(44)

Repeating the calculation for these two directions leaves us with three
expressions for $\langle\cos^{2}{\theta(t)}\rangle_{n-\mathrm{days}}$ that
hold whenever $T$ is an integer number of sidereal days:

$\langle\cos^{2}{\theta(t)}\rangle_{n-\mathrm{days}}=\left\\{\begin{array}[]{ll}\frac{1}{8}\big{(}3+\cos{2\lambda_{\rm
lab}}+(1+3\cos{2\lambda_{\rm lab}})\cos{2\theta_{X}}\big{)}&\text{\bf
North}\\\ \frac{\sin^{2}{\theta_{X}}}{2}&\text{\bf West}\\\
\frac{1}{8}\big{(}3+\cos{2\theta_{X}}-(1+3\cos{2\theta_{X}})\cos{2\lambda_{\rm
lab}}\big{)}&\text{\bf Zenith}\end{array}\right.\,.$ (45)

Interestingly, we notice that one can choose the value of $\lambda_{\rm lab}$
such that the $\theta_{X}$ dependence drops out. In these instances, the value
of $\langle\cos^{2}{\theta(t)}\rangle_{n-\mathrm{days}}$ is the same for every
possible DP polarisation. The distribution
$f(\langle\cos^{2}{\theta}\rangle_{T})$ is a delta function, so these cases
will be when $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ and
$\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$ are the largest possible.
Inspecting Eq.(45), we can see that this occurs when,

$\displaystyle{\bf North:}$ $\displaystyle\quad\lambda_{\rm
lab}=\frac{1}{2}\cos^{-1}{\bigg{(}-\frac{1}{3}\bigg{)}}\approx\pm
54.74^{\circ}\,,$ $\displaystyle{\bf West:}$ no latitude dependence, (46)
$\displaystyle{\bf Zenith:}$ $\displaystyle\quad\lambda_{\rm
lab}=\frac{1}{2}\cos^{-1}{\bigg{(}\frac{1}{3}\bigg{)}}\approx\pm
35.26^{\circ}\,.$

For experiments that use measurements that are on the order of days in length,
these latitudes will clearly be preferential.222222Conveniently, all the
experiments from Tables 1 and 2 are located within $\sim 10^{\circ}$ of one of
the latitudes that maximise the North and Zenith-pointing sensitivities Since
a West-pointing experiment always points in the direction that the Earth is
rotating, it will always do a $2\pi$ loop in space once every sidereal day,
independent of the latitude. It is worth remarking that the effect of being at
a different latitude can be replicated by tilting the device by the relevant
angle, however many experiments are unlikely to have this level of freedom.

We show the full latitude-dependence of the distribution of
$\langle\cos^{2}{\theta(t)}\rangle^{\rm excl.}_{n-\mathrm{days}}$ in the left-
hand panel of Fig. 4. We see that the preferred latitudes from Eq.(VI.1) are
the latitudes that maximise $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$.
Also note that when a Zenith-pointing experiment is located at the North/South
poles, or when a North-pointing experiment is located at the Equator, the
resulting value of $\langle\cos^{2}{\theta(t)}\rangle^{\rm excl.}_{T}$
converges the instantaneous value of 0.025, since those instruments will not
rotate with respect to $\hat{\mathbf{X}}$.

Next, we show the full distribution,
$f(\langle\cos^{2}{\theta(t)}\rangle_{T})$, as a function of $T$ by sampling
over all DP polarisations. In Fig. 5, we show these distributions for the
North and Zenith cases, and for three example latitudes. We can see that for
$\lambda_{\rm lab}\approx 55^{\circ}$ in the North-pointing case, and
$\lambda_{\rm lab}\approx 35^{\circ}$ in the Zenith-pointing case, the
distribution converges on a single value ($1/3$) for integer-day-long
experiments. This coincides with the preferential locations observed in Fig.
4.

###  VI.2 Planar experiments

We also wish to calculate the signal for experiments that are sensitive to any
polarisation component lying along a plane. So for example, to calculate the
angle $\theta$ with respect to the plane defined by the
$\hat{\mathcal{Z}}$-$\hat{\mathcal{W}}$ axes, we simply take the complement of
the angle with respect to the axis perpendicular to that plane i.e.,

$\cos{\theta(t)}=\sqrt{1-\big{(}\hat{\mathbf{X}}\cdot\hat{\mathcal{N}}(t)\big{)}^{2}}\,,$
(47)

To inspect how this angle behaves further we simplify in the same way as
before by taking the average over an integer number of sidereal days,

$\langle\cos^{2}{\theta(t)}\rangle_{n-\mathrm{days}}=\left\\{\begin{array}[]{ll}\frac{1}{8}\big{(}5-\cos{2\lambda_{\rm
lab}}-(1+3\cos{2\lambda_{\rm lab}})\cos{2\theta_{X}}\big{)}&\text{Zenith-West
\quad\, (perpendicular to floor)}\\\
\frac{1}{4}(3+\cos{2\theta_{X}})&\text{North-Zenith \quad(perpendicular to
floor)}\\\ \frac{1}{8}\big{(}5+\cos{2\theta_{X}}+(3\cos{2\lambda_{\rm
lab}}-1)\cos{2\theta_{X}}\big{)}&\text{North-West \quad\,\,\, (parallel to
floor)}\end{array}\right.$ (48)

To make discussing these planes less confusing, we refer to them as “North-
facing”, “West-facing”, and “Zenith-facing”, respectively, and the axial
experiments as “North-pointing” etc. We find that we have an identical
latitude preference as for the angles with respect to an axis,

North-facing $\displaystyle\quad\lambda_{\rm
lab}=\frac{1}{2}\cos^{-1}{\bigg{(}-\frac{1}{3}\bigg{)}}\approx\pm
54.74^{\circ}\,,$ West-facing: no preferred latitude, (49) Zenith-facing:
$\displaystyle\quad\lambda_{\rm
lab}=\frac{1}{2}\cos^{-1}{\bigg{(}\frac{1}{3}\bigg{)}}\approx\pm
35.26^{\circ}\,,$

We show the full latitude dependence of $\langle\cos^{2}{\theta}\rangle^{\rm
excl.}_{T}$ in the right-hand panel of Fig. 4. Notice that in this case the
average angle with respect to a plane at any one time is $2/3$, meaning that
at the preferential latitudes the entire distribution of
$\langle\cos^{2}{\theta(t)}\rangle_{n-\mathrm{days}}$ converges on this value.

As we did for the axial case, we also show the full distribution of
$\langle\cos^{2}{\theta(t)}\rangle_{T}$ as a function of $T$, and for three
latitudes, in Fig. 6. The behaviour is very similar, with the 35∘ and
$55^{\circ}$ latitude cases having singular points at integer values of $T$
for the Zenith-facing and North-facing planes respectively.

###  VI.3 Reinterpreting dark photon limits

Figure 7: Closeup of bounds on DPs in the radio-microwave frequency regime.
The limits set by axion cavity haloscopes are in shades of red, whereas the
limits from devoted DP experiments operating in the same range of frequencies
are shown in shades of green. We impose an upper limit on DPs as DM from the
bounds of Refs. Arias:2012az ; Witte:2020rvb  (the same upper limit as shown
in Fig. 1). All of the experimental bounds shown here have been rescaled from
the original sources. Firstly, they have been rescaled such that they all
assume the same DM density of $\rho_{0}=0.45$ GeV cm-3. Secondly, we have
rescaled them such that they all consider the fixed DP polarisation scenario.
This relies on the factor $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$,
which is different for each experiment. The method of deriving these factors
is detailed in Sec. V.7, and the result for each experiment is listed in the
final columns of Tables 1 and 2.

Figure 8: We display the improvement that can be made to both exclusion limits
(top row) and discovery limits (bottom row) on the DP’s kinetic mixing
parameter, $\chi$, as a function of the measurement time, $T$, and for several
experiment geometries and orientations. The left-hand panels correspond to
axial experiments, whereas the right-hand panels correspond to planar
experiments. In all cases we display the improvement made to the limit
relative to the most conservative possible assumption ($\chi_{\rm conserv.}$):
ignoring daily modulation and taking the instantaneous results for
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$ and
$\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$ when $T\rightarrow 0$. In the
scenario that the DP has a random polarisation every coherence time,
$\langle\cos^{2}{\theta}\rangle_{T}$ effectively takes on a single value of
$1/3$ (axial) or $2/3$ (planar), and is the best the experiment could do. What
this figure shows is that by properly accounting for the daily modulation and
how that changes the distribution of $\langle\cos^{2}{\theta}\rangle_{T}$, the
sensitivity in the fixed-polarisation scenario can be significantly
improved—and for certain cases it can even match the random-polarisation
scenario. The potential improvement could be up to a factor of $\sim$3.75 or
$\sim$1.36 in the exclusion limits, and $\sim$10.5 or $\sim$2.37 in the
discovery limits.

Now that we know how to calculate the distributions of
$\langle\cos^{2}{\theta}\rangle_{T}$, we can use this knowledge to estimate
what $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}.$ should be for past
experiments. The results are listed in the final columns of Tables 1 and 2.
This means we can now do what we hinted at in Sec. V, and reinterpret previous
exclusion limits in the context of the fixed polarisation scenario.

For a concrete example of how this is done, we consider the prototypical
axion/DPDM experiment: the cavity haloscope. The power is often written in
terms of the physical quantities of the cavity: volume $V$, quality factor
$Q$, external applied $B$-field ${\bf B}$, and coupling factor $\kappa$,

Dark photons: $\displaystyle\quad P_{\rm cav}=\kappa{\cal G}_{X}VQ\rho_{\rm
DM}\chi^{2}m_{X}\,,$ (50a) Axions: $\displaystyle\quad P_{\rm cav}=\kappa{\cal
G}_{a}V\frac{Q}{m_{a}}\rho_{\rm DM}g_{a\gamma}^{2}B^{2}\,,$ (50b)

where $g_{a\gamma}$ is the dimensionful axion-photon coupling, and $m_{a}$ is
the axion mass. The geometry factors $\mathcal{G}_{X,a}$ are given by

Dark photons: $\displaystyle\quad{\cal G}_{X}=\frac{\left(\int dV\,{\bf
E}_{\alpha}\cdot{\bf\hat{X}}\right)^{2}}{V\frac{1}{2}\int dV\,\epsilon({\bf
x}){\bf E}_{\alpha}^{2}+{\bf B}_{\alpha}^{2}},$ (51a) Axions:
$\displaystyle\quad{\cal G}_{a}=\frac{\left(\int dV\,{\bf E}_{\alpha}\cdot{\bf
B}\right)^{2}}{VB^{2}\frac{1}{2}\int dV\,\epsilon({\bf x}){\bf
E}_{\alpha}^{2}+{\bf B}_{\alpha}^{2}}.$ (51b)

Here we have denoted the field of the $\alpha$ mode of a cavity via ${\bf
E}_{\alpha},{\bf B}_{\alpha}$. It is simple to convert between the two
expressions for the cavity power. So to recast a limit on $g_{a\gamma}$ to one
on $\chi$ we just replace Arias:2012az ,

$\chi=g_{a\gamma}\frac{B}{m_{X}|\cos\theta|}\,,$ (52)

where $\cos\theta={\bf\hat{X}}\cdot{\bf\hat{B}}$.

In fact, this statement holds generally for all the experiments we consider,
as long as $\theta$ is defined with respect to the appropriate axis or plane.
One caveat however is that this conversion assumes a constant magnetic field
over the cavity, which may not always be the case. The differences in most
experiments may be small, but designs such as the original Orpheus proposal
Rybka:2014cya 232323Note that the name has been reused by the ADMX
collaboration for a dielectric loaded resonator Carosi:2020akt . would
actually be very insensitive to DPs, as it would have employed an oscillatory
magnetic field. We stress that deriving fully accurate limits requires
dedicated calculations of the geometry factors—relying on this simple
recasting may not always be sufficient.

Figure 7 shows limits on DPDM in the fixed polarisation scenario that are the
most accurate ones to date. We have zoomed in on the radio-microwave range
where the majority of the experimental activity takes place. Though these
limits are still not fully optimised for each experiment, they are consistent
in their assumptions, which was not true prior to this. As we mentioned in
Sec. V.7, the approach adopted for handling this scenario was to simply take
the 5th percentile of the distribution:
$\langle\cos^{2}{\theta}\rangle_{T\rightarrow 0}=0.0025$ or $0.0975$
Arias:2012az .242424We note that there was also subtle plotting error for the
reinterpreted axion limits presented in Ref. Arias:2012az , which has been
fixed in more recent work Gelmini:2020kcu ; Ghosh:2021ard . Looking at the
final columns of Tables 1 and 2, we can see that these values are certainly
overly conservative for many experiments, especially those integrating for
longer than a few hours.

The factors of $\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}=1/3$ (axial) or
$2/3$ (planar), obtained under the random polarisation scenario, would enhance
these bounds by $\sim 3.75$ or $1.36$ respectively. As we discussed in Sec.
III, this can only be a valid assumption for certain DP production mechanisms.
We stress therefore, that if these values are chosen, an accompanying
statement must be made about the requirement this places on the DP production
mechanism. Due to the large differences in these values, such a statement
would be more important than one about, say, the assumed DM density, which
also varies between publications but whose differences lead to discrepancies
of factors around $\sim$1.2.

In Fig. 7, we rescaled each limit so that they apply for $\rho_{\rm DM}=0.45$
GeV cm-3. This value is what the axion direct detection community has adopted
since Ref. Asztalos:2001jk  in 2002.252525This unusual choice seems to stem
from the fact that Ref. Asztalos:2001jk  presented their exclusion limits as a
function the DM density for specific QCD axion models, as opposed to the other
way around as is now convention. KSVZ axions were ruled as contributing more
than 0.45 GeV cm-3 in 2002 and that appears to have been adopted in subsequent
experimental analyses which presented limits for a fixed value of $\rho_{\rm
DM}$, despite the fact they were at different frequencies. However, we are not
certain if this interpretation of history is correct. However a value of 0.3
has been the standard over many years in other direct detection communities.
The other experiments shown here: Dark E-field, SHUKET, and WISPDMX all state
(or at least imply) that they have chosen a value of 0.3 (with the exception
of SQuAD who chose 0.4).

To see even more clearly the difference in sensitivity between the fixed
polarisation scenario, and the randomised polarisation scenario, we use Fig.
8. Since the sensitivity to $\chi$ scales as
$\sim(\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T})^{-1/2}$ we have plotted
this against the measurement time for both axial and planar experiments, but
rescaled relative to the instantaneous value
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T\rightarrow 0}=0.025$ or 0.37.
The difference between the most optimistic and most pessimistic assumptions is
around a factor 3.75 for the axial case, and around a factor of 1.36 for the
planar case. For the discovery limits however, because these factors are much
more sensitive to the low tails of $f(\langle\cos^{2}{\theta}\rangle$, we
reach improvement factors up to $10.5$ and $2.37$ for axial and planar
experiments respectively.

So in summary, simply using a certain value of $T$ can bring the fixed DP
scenario limits closer, and even equal to, the randomised case. In particular,
for the cases when the experiment is placed at the optimal latitude and
integrates over integer-day-long measurement times, the sensitivity under the
two scenarios is _the same_.

###  VI.4 Making multiple measurements

Figure 9: In each panel the colour-scale refers to the value of
$\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{3T}$ obtained from _three_
measurements, each of duration $T$, and each separated by time $T_{\rm wait}$.
The left column corresponds to axial experiments, and the right column to
planar experiments. The top row is for North-pointing/facing experiments
located at their optimal latitude of $\sim 55^{\circ}$, the middle row is for
West-facing/pointing experiments which can be located at any latitude, and
finally the bottom row is for Zenith-pointing/facing experiments at the
optimal latitude of $\sim 35^{\circ}$. Note that in all cases the distribution
is periodic in $T_{\rm wait}$ over one sidereal day, so there is no need to
extend the plots vertically. The orange contours with labels enclose the
region of the $T$-$T_{\rm wait}$ space where that choice of $T_{\rm wait}$
leads to an enhancement $\langle\cos^{2}{\theta}\rangle^{\rm
disc.}_{3T}/\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$ by at least the
amount shown.

As we hinted at in Sec. V.6, the sensitivity of a DP search can be greatly
improved by a strategic choice of the time at which to do a repeat
measurement, or rescan. For the same reason, the sensitivity can also be
greatly _harmed_ if the time of rescan is poorly chosen. Fortunately, the
rotation of the Earth is something we can plan for. While prior to now, no
strategy of this kind of strategy seems to have not been adopted in any direct
search for DPs. Nevertheless, we can detail some examples of better-practice
here, should a future experiment wish to adopt a more strategic measurement
schedule. Before we begin, it is worth emphasising that there are two distinct
possibilities that are relevant here. One is if the experiment is repeating
measurements to _test_ a candidate signal, and the other is if the repeat
measurements are simply to obtain more data to enhance the _sensitivity_. As
we will see these two cases require very different strategies to be put in
place.

We begin by picking up where the discussion in Sec. V.6 left off, namely the
case where we have made a measurement of duration $T$, and we wish to add
statistics to that measurement by repeating it, some time $T_{\rm wait}$
later. From what we have seen so far, there are scenarios where this
measurement could be disastrous for the detection of the DP. Say we have a
West-pointing experiment that happened to get very lucky and picked up the DP
polarisation in perfect alignment with the instrument. If the experimentalists
decided to check that measurement, say, 6 hours later, the polarisation would
now be anti-aligned with the instrument and the signal will have disappeared.
It would then be quite reasonable to chalk the event up to transient RF noise
and toss the Nobel-prize winning signal in the bin with all the other radio
stations.

Of course, this is an extreme case, but the point is that the DP signal has
large variations in time. These can benefit the experiment, but they can also
cause signals to disappear when the alignment is bad. For this issue though
the solution is simple: Test candidate DP signals at the same time of the
sidereal day as they were originally measured.

Now, if we want to talk about maximising the potential to detect signals in
the first place, we must flip this argument. The solution will be less
straightforward, because it relies on the range of $\cos{\theta}$ swept about
by a given experiment over a given time, but we can use our formulae derived
in the previous subsections. We want to determine the best strategy for
performing rescans in the _absence_ of any candidate signals. This issue is
very relevant for many axion experiments which enhance their potential $S/N$
by means of rescans and subsequently stacking their data. In the context of
DPs which are subject to strong daily modulations it can be even more
relevant. With a judicious choice of $T_{\rm wait}$, it is possible to use the
rotation of the Earth to game the statistics of
$\langle\cos^{2}{\theta}\rangle_{T}$ in the experiment’s favour. Note that
this need not involve a substantial increase in the total time spent measuring
a particular frequency.262626The only increase in total measurement time comes
from the additional time spent tuning. If such a time were short compared to
the time spent measuring, then there is no real cost. Even if the tuning time
is non-trivial, we will show that the gain is high enough that it is likely
worth the additional time. As long as the measurement is noise dominated
(which is the case for all experiments considered here), the $S/N$ remains the
same for any temporally constant signals, while the probability of a poor
alignment can be significantly reduced for the temporally varying signals.

Imagine the following scenario: An experiment makes three measurements, each
with a duration $T$. The start of each measurement is spaced out from the
start of the previous one by a time $T_{\rm wait}$. We have already calculated
the distribution $\langle\cos^{2}{\theta}\rangle_{T}$ for a single measurement
of time $T$ (these are shown in Figs. 5 and 6), but say we were to stack the
data from the three measurements together and take it is one measurement, what
is the distribution of $\langle\cos^{2}{\theta}\rangle_{T}$ then, and how does
it depend on $T$ and $T_{\rm wait}$? Figure 9 shows precisely this.

In Fig. 9 we display $T$ on a logarithmic scale to show both the cases
involving short sub-hour measurement times, as well as those lasting days.
However, we show $T_{\rm wait}$ between 0 and 1 days, and on a linear scale,
as these results are all periodic in that direction. This means that $T_{\rm
wait}$ should be interpreted as the delay in the local time from the original
measurement. For example if the original measurement took place at 00:00 then
a value of $T_{\rm wait}=0.5$ days, corresponds to any second measurement
taking place at 12:00 (and then a third, the same time afterwards). It does
not matter if that measurement is several days afterwards. Similarly, it does
not matter what the exact time of the original measurement was, $T_{\rm
wait}=0.5$ days just means that there was 12 hours on the clock between them.
Recall that we are considering sidereal days, i.e. a shift of four minutes per
calendar day is required to convert times of the sidereal day to the calendar
day.

Since we are interested now in optimising future experiments, we need to
ensure we are preparing them to have the best chance of discovering the DP,
not just setting exclusion limits. Our discovery limit conversion factor
$\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$ is much more sensitive to the
tails $f(\langle\cos^{2}{\theta}\rangle_{T})$, so we can appreciate the
effects of optimisation much more by focusing on this factor than
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$. The three panels on the
left-hand side are for axial experiments, whereas the three on the right-hand
side are for planar experiments. The colour-scale corresponds to the value of
$\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{3T}$. The orange contours enclose
values of a different, but related, distribution: The enhancement in the value
of $\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{3T}$ that is gained from doing
the repeat measurements. Keep in mind the quantity we are calculating is a
time average, so any enhancement originates solely from the rotation of the
Earth and not from the fact we are observing for longer.

Examining Fig. 9, we see that when $T\gtrsim 1$ sidereal day, it does not
matter when the next measurements start, since the experiment is already long
enough that it samples all the DP polarisations it can. On the other hand, for
very short $T$, the correct choice of $T_{\rm wait}$ makes a great difference.
For instance, looking at the top-right panel, if the original measurement of
the North-facing experiment was only 10 minutes, then simply choosing the next
measurement to be at a time that was $\sim 0.33$ or $\sim 0.66$ sidereal days
later would allow the power to be enhanced by a factor of 3, just from the
factor of $\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{3T}$ alone.

The most dramatic cases are the North and Zenith-pointing experiments where a
strategic timing of short measurements can lead to a power enhancement of over
a factor of 100, leading to an enhancement in sensitivity to $\chi$ of a
factor $\gtrsim$10\. This is roughly the factor difference between the fixed
and randomised polarisation scenarios that we saw in the lower panels of Fig.
8, which shows that three measurements is already enough to sample almost the
whole distribution of possible polarisations. Since this factor is purely
geometrical, the experiment gains an order of magnitude in sensitivity to
$\chi$, with only a factor of three increase in measurement time. As a point
of comparison, for an unmodulating DP signal, the sensitivity would scale as
$\chi^{-1}\propto T^{1/2}$. Note again that for short measurement times, where
such a technique is most appropriate, this could correspond to gaining a order
of magnitude in sensitivity simply by dividing the measurement in three.

Before moving on, we remark that we have also performed the same calculation
for two measurements rather than three.272727The equivalent figures for these
cases can be found in the GitHub repository linked to this paper. The
qualitative trends are the same, however the enhancement factors are lower,
gaining at most a factor of around 15 in $\langle\cos^{2}{\theta}\rangle^{\rm
disc.}_{T}$. In the West-pointing and West-facing cases, the optimum time to
do a rescan for sub-day-long measurements is around $\pm$6 hours, and does not
improve for three measurements rather than two. Calling back to the discussion
at the beginning of this subsection, this is the opposite strategy to what one
would do to test for a candidate signal, however the reasoning is the same.
With two short measurements 6 hours apart, the experiment is maximising its
potential to capture a range of possible polarisation angles, but that is
exactly what one should _not_ do to test for a signal seen in the first
measurement.

It is unsurprising that the more individual measurements we make, the closer
we can get to the optimal experiment which observes over the whole day and
captures all possible DP polarisations. However, we found that three is
already sufficient to get very close to the optimal case in the North and
Zenith-pointing experiments, which represents a dramatic potential improvement
in sensitivity for only a very minor reorganisation of the experimental data-
taking. For planar and West-pointing experiments however, there is little
difference between two and three measurements. A simple way to understand why
is to consider that, when defining a volume, one needs either three axes or
two planes. When an experiment is West-pointing, at most one can sweep out a
disk, which requires only two axes to define.

###  VI.5 Optimising future experiments

Figure 10: Projected DP discovery limits for planned experiments, and future
runs of existing experiments: DM-Radio DMRADIO , Dark E-field Radio
Godfrey:2021tvs , ALPHA Lawson:2019brd ; Gelmini:2020kcu , MADMAX
TheMADMAXWorkingGroup:2016hpc , LAMPOST Baryakhtar:2018doz , SuperCDMS
Bloch:2016sjj  (assuming a Ge target), and LZ Akerib:2021qbs . We have plotted
projections for two scenarios, the ‘standard’ projection is the most
pessimistic and assumes that no timing or directional information is taken
into account when running the experiment, or calculating the limit. Our
‘optimised’ projection would be obtained if the experiment followed some
simple changes to the data-taking that are outlined in the text of Sec. VI.5.
For the rest of the bounds, rather than show every existing limit individually
as in Fig. 1, we have combined the limits into three categories based on the
level of assumption involved. The red region encloses all constraints that are
based on photon$\rightarrow$DP transitions (e.g. light-shining through walls)
which in a sense are the least model-dependent. In green, we show bounds
invoking stellar cooling arguments, which also rely on photon-DP conversion,
but could be circumvented through similar model-dependent arguments as those
used for axions Jain:2005nh ; Masso:2005ym ; Jaeckel:2006xm ; Masso:2006gc ;
Budnik:2020nwz ; DeRocco:2020xdt ; Bloch:2020uzh . Then in blue we show the
bounds that rely on DPs comprising the majority of DM. In grey, we show the
direct detection bounds from devoted DP experiments, namely, SHUKET, WISPDMX,
and Dark E-field Radio. Note that the future projections vary in their levels
of optimism about the future, hence this is not the fairest comparison,
however it serves to demonstrate what might be possible.

Now that we have discussed the best strategies for setting limits on DPs, we
look towards the future and examine the extent to which these strategies could
impact the sensitivities of upcoming experiments. Again we focus on the fixed
polarisation scenario as this would require the greatest amount of care in
orienting the experiment and would leave the DP sensitivity in any other
polarisation scenario unchanged.

To do this, we first select a few of the major DP search proposals listed in
Tables 1 and 2 that aim to cover the presently unconstrained DP parameter
space. We then use their stated projections and apply scaling factors to
improve their sensitivity in the fixed polarisation scenario. Since there are
external factors that dictate what values of $T$ are possible, we will not
attempt to alter anything about the total measurement times. Rather we suggest
ways in which individual measurements could be divided, or in some cases, the
orientation of the experiment itself could be changed. The strategies for each
future experiment are as follows:

  * •

DM-Radio: Each frequency is integrated over three $\mathcal{O}$(minute)
measurements spaced 8 sidereal hours apart (the precise numbers of minutes
make only minor differences to the resulting
$\langle\cos^{2}{\theta}\rangle_{T}$).

  * •

Dark E-field Radio: The readout antenna is rotated so that it is Zenith-
pointing. The ${\cal O}({\rm month})$ measurement time is rounded to an
integer number of sidereal days.

  * •

ALPHA: The experiment is placed at a latitude as close to $\lambda_{\rm
lab}=\pm 35^{\circ}$ as possible. The $\mathcal{O}$(week)-long integration
times are rounded to an integer number of sidereal days.

  * •

MADMAX: A two-polarisation sensitive antenna is used. The experiment is then
aligned so that the disks face the North-South direction. The
$\mathcal{O}$(week)-long integration times are rounded to an integer number of
sidereal days.

  * •

LAMPOST: if a laboratory is available close to $\pm 35^{\circ}$ then the
experiment should be rotated so its dielectric layers are parallel to the
floor. If instead a latitude close to $\pm 55^{\circ}$ is more feasible then
the layers should face the North-South direction. The $\mathcal{O}$(week)-long
integration times are rounded to an integer number of sidereal days.

While integer-day-long measurements mildly improve the value of
$\langle\cos^{2}{\theta}\rangle^{\rm excl.}_{T}$, in practice being able to
measure for slightly longer may improve the total integrated power per cost of
running. We stick to integer-day-long measurements purely for simplicity.

In Fig. 10 we plot the rescaled experimental projections for the discovery of
the DP at 5$\sigma$ in 95% of experiments. We compare our optimised
projections against the estimate one would make without incorporating any
information about the timing or orientation of the experiment. This
pessimistic scenario is, as we have already mentioned, overly conservative,
and corresponds to setting our discovery conversion factor to the
instantaneous result $\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T\rightarrow
0}=0.0036$ or 0.13.

We neglect to include any projections for axion experiments in this figure,
whose DP sensitivity will likely rely on a post hoc data analysis (in the
style of Fig. 7). We instead focus on those experiments that plan on a
dedicated DP search. For comparison, we have also shown projections from
SuperCDMS Bloch:2016sjj  and LZ Akerib:2021qbs  however these experiments rely
on DP absorption and electronic emission, which is why their limits are
insensitive to the DP polarisation scenario. Our point here is that some DP
limits depend upon the production mechanism, and some do not—a fact that is
commonly brushed over, or ignored.

In all cases we find that these strategies are sufficient to raise the limits
in the fixed polarisation scenario up to a level very close to the randomised
polarisation scenario. All this requires is some slight adjustments to the
data-taking schedule, and a judicious choice in experimental alignment.

Much of this parameter space will therefore be constrained in the coming
decades. As well as upcoming DM searches, several strategies used to set the
bounds on Figs. 1 and 10 will also see improvements over the next few years.
For example, upgraded LSW experiments Bahre:2013ywa ; Spector:2016vwo ;
Inoue:2015cta ; Kim:2020ask , experiments using atomic transitions
Flambaum:2019cqi ; Bhoonah:2019eyo ; Jones:2019qny , or Aharanov-Bohm
experiments Boulware:1989up ; Arias:2016vxn  may improve the purely-laboratory
bounds on DPs. Whereas searches using X-ray Thorpe-Morgan:2020rwc  and radio
Lobanov:2012pt ; An:2020jmf  telescopes, fast radio burst timing
Landim:2020ked , or asteroseismology Ayala:2019isl , may improve upon existing
astrophysical bounds.

##  VII Using the daily modulation as a signal

The daily modulation of the DP signal in the fixed polarisation scenario
represents one of the clearest smoking gun signals of DM that an experiments
of this kind could hope to observe. This is fortunate, given that a DP signal
candidate cannot be confirmed or rejected via the tuning of the magnetic
field. At least one DP experimental collaboration has already suggested that
they already will employ the daily modulation as a final stage in their signal
confirmation decision trees Godfrey:2021tvs . It was also shown that daily
modulation can be measured with an array of directionally sensitive detectors
Kopylov:2021ndl . The most convincing property that the modulation will
exhibit for confirming its DM origin will be a period equal to one sidereal
day (23 hours and 56 minutes). If there were some way in which a source of
terrestrial noise would modulate on the timescale of a day (and it is already
hard to contrive of such a background), it would almost certainly have a
period equal to one solar day. The way this would manifest in practice is that
the phase of the daily modulation in local time would be seen to drift forward
slowly over the year. Around 6 months later the phase would be 12 hours ahead
of the time it was initially measured, but it would then cycle back around
over the next 6 months.

The daily modulation would be a striking signal in the fixed polarisation
scenario: it would dominate the signal when measured over timescales longer
than a few hours. Yet the only scenario in which the DP signal will _not_
sidereally modulate282828This statement only refers to the modulation due to
the DP polarisation. In fact all DP signals will exhibit an extremely small
($v\sim 10^{-6}$) daily modulation in the lineshape due to the relative motion
of the rotating Earth-bound laboratory and the DM halo. This is likely to be
unobservable for the main DM halo which has a typical dispersion of $v\sim
10^{-3}$. However it is conceivable that if the experiment encountered a
strongly coherent configuration of the field like a Proca star Brito:2015pxa ,
or ultracold stream Sikivie:1997ng ; Ling:2004aj ; Duffy:2006aa ;
Hoskins:2016svf , then the daily modulation could be of comparable size to the
signal’s width in frequency space. is when the polarisations are totally
randomised in every coherence spacetime patch. One can imagine scenarios where
only fraction of the DM is polarised to begin with, or the polarisation was
mixed somehow during structure formation and hence varied over phase space,
but in any of these scenarios a suppressed daily modulation would still be
present. Thus one must be careful using daily modulation as a veto on possible
DP signals as one could inadvertently exclude a potential discovery of a non-
modulating or slightly modulating scenario.

The natural question to ask is then: If we see a signal of DPs in one of the
experiments we have discussed, can the DP polarisation axis be measured, and
can this information be used to understand the production mechanism behind the
creation the DP dark matter. Just as in the case of axions, any further
scanning becomes unnecessary post-detection and the experiment can quickly
accumulate very high $S/N$. This permits the fine-grained study of the
spectral lineshape, in the process unravelling the velocity structure of the
local DM halo. Note that the velocity distribution is annually modulated by
the relative motions of the Earth and the halo, allowing the full _velocity_
-structure of $f(\mathbf{v})$ to be probed, as opposed to just the speed
distribution, $f(v)$ OHare:2017yze . However, one could also employ a
multiplexed network of phase-linked detectors that rely on interferometry to
further unravel this structure on much shorter timescales Foster:2020fln .
These kinds of measurements would allow the experiment to measure properties
about the DM halo and, via combination with simulations and galactic surveys,
would allow us to probe the merger history of the Milky Way’s halo
OHare:2018trr ; Evans:2018bqy ; OHare:2019qxc .

If the DP field is polarised in some way then the measurement of that
polarisation may therefore offer an even more distant window into the early
Universe. Without further intuition as to expected DP polarisation
distribution in galaxies generated by different production mechanisms, we
cannot be more precise than simply raising this as a tantalising possibility.
However, even if the DP polarisation is of no fundamental interest at all, its
measurement would be a crucial step in optimising the continuous study of the
DP lineshape.

So in a similar spirit to Refs. OHare:2017yze ; Foster:2017hbq ;
Knirck:2018knd ; Foster:2020fln , we wish to understand the extent to which an
experiment could reconstruct the true properties of the DP signal. In the case
of the polarisation though, because we are less certain of what the
distribution of $\hat{\mathbf{X}}$ even is, it is not clear what we should
assume the correct answer to be. As we have stressed, this subject is ripe for
further exploration, but we can sketch a simple example to show how such a
study would proceed.

###  VII.1 Measuring the daily modulation

We will set up a toy statistical test that, while not accurately reflecting
the details of a real experimental analysis, will at least resemble one, and
incorporate some of the main features. Firstly we assume that our toy
experiment reads in some time-series electromagnetic signal and takes the
Fourier transform of it to measure a power spectrum. We then assume that many
of these power spectra are stacked together, which eventually total the
measurement time $T$. This has the effect of both amplifying any signal
present, whilst mitigating against the exponential noise in the randomly drawn
DP signal amplitudes in each frequency bin, and making the expected noise
spectrum close to Gaussian.

First let us assume that in this stacked data we have a signal
$P_{X}\langle\cos^{2}{\theta}\rangle_{T}$ which is contained in the bandwidth
of the experiment $\Delta\nu$ and is distributed according to the
distributions shown in Fig. 5. Then we assume that we have some normally
distributed noise with mean $P_{N}$ and standard deviation
$\sigma_{N}=P_{N}/\sqrt{T\Delta\nu}$.

We adopt an Asimov dataset approach for analysing the daily modulation now. So
we shift our definition of $\langle\cos^{2}{\theta}\rangle^{\rm disc.}_{T}$
slightly since we are no longer interested in optimising an experiment for a
guaranteed discovery, but rather we want to know what the typical experiment
could do. In this case we wish to find the discovery threshold for when the
median experiment seeing a signal could reject the noise at $3\sigma$. This
can be straightforwardly calculated using the same technique as in Sec. V.7,

$P_{X}\langle\cos^{2}{\theta}\rangle^{\rm
disc.}_{T}>(3+\Phi^{-1}[0.5])\sigma_{N}\approx 3\sigma_{N}\,,$ (53)

where $P_{X}$ is the solution to292929We have removed the superscript ‘0’ to
reduce clutter.,

$\displaystyle\int_{0}^{1}\frac{f(\langle\cos^{2}{\theta}\rangle_{T})}{2}\bigg{[}1+\text{erf}\bigg{(}\frac{3\sigma_{N}-P_{X}\langle\cos^{2}{\theta}\rangle_{T}}{\sqrt{2}\sigma_{N}}\bigg{)}\bigg{]}\textrm{d}\langle\cos^{2}{\theta}\rangle_{T}$
$\displaystyle=0.5\,.$ (54)

Note that one can also arrive at the same results by writing down a profile
likelihood ratio test statistic comparing the signal+noise and noise
hypotheses, and this result will follow from the application of Chernhoff’s
theorem Chernoff:1954eli  (see Refs. Cowan:2010js ; Algeri:2019arh  for
further details).

Now we ask the following question: If the DP signal is strong enough that the
median experiment can exclude the noise at 3$\sigma$, how much more time will
it need to detect the daily modulation? To answer this, we need to write down
a likelihood ratio that compares the modulated hypothesis with an unmodulated
one,

$\displaystyle\Lambda(\theta_{X},\phi_{X})=2\bigg{[}\ln$
$\displaystyle\mathcal{L}\big{(}d|\mathcal{M}_{t},\\{\hat{P}_{X},\theta_{X},\phi_{X},\hat{P}_{N}\\}\big{)}$
$\displaystyle-\ln$
$\displaystyle\mathcal{L}\big{(}d|\mathcal{M}_{0},\\{\hat{\hat{P}}_{X},\hat{\hat{P}}_{N}\\}\big{)}\bigg{]}\,,$
(55)

where we use a single hat to refer to the maximum likelihood estimators (MLEs)
under the modulating model, $\mathcal{M}_{t}$, and double hats to refer to the
MLEs under the unmodulating model, $\mathcal{M}_{0}$.

Say that we observe some power, $P^{\rm obs}$, from $t=0$ to $t=T$, and split
the data up into temporal bins of width $\Delta t$ centered on times $t_{j}$.
Then the first likelihood in Eq.(VII.1) is,

$\displaystyle\ln\,\mathcal{L}(P^{\rm obs}|$
$\displaystyle\mathcal{M}_{t},\\{P^{0}_{X},\theta_{X},\phi_{X},P_{N}\\})=$
$\displaystyle-\frac{1}{2\sigma^{2}_{N}}$
$\displaystyle\sum_{j=1}^{N_{t}}\bigg{[}P_{j}^{\rm obs}-P_{X}\langle
c(\theta_{X},\phi_{X})\rangle_{j}-\frac{\Delta t}{T}P_{N}\bigg{]}^{2},$ (56)

where $N_{t}=T/\Delta t$, and we have used the shorthand,

$\langle
c(\theta_{X},\phi_{X})\rangle_{j}\equiv\langle\cos^{2}{\theta}\rangle_{j}=\frac{1}{\Delta
t}\int_{t_{j}-\Delta t/2}^{t_{j}+\Delta t/2}\cos^{2}{\theta}\,\mathrm{d}t\,.$
(57)

Then, the second likelihood in Eq.(VII.1) is for the unmodulating case where
we take the integrated signal over the full $T$, sacrificing any sensitivity
to $(\theta_{X},\phi_{X})$,

$\displaystyle\ln\mathcal{L}(P^{\rm obs}|\mathcal{M}_{t},\\{P_{X},P_{N}\\})=$
(58) $\displaystyle-\frac{1}{2\sigma^{2}_{N}}$ $\displaystyle\left(P^{\rm
obs}-P_{X}-P_{N}\right)^{2}\,.$

We can use the maximised ratio $\Lambda(\hat{\theta}_{X},\hat{\phi}_{X})$ as a
test statistic (TS) for determining the detectability of daily modulation, but
we can also use the unmaximised ratio to determine the accuracy with which the
parameters $(\theta_{X},\phi_{X})$ could be measured (which we will do in the
next subsection). Rather than doing a full Monte Carlo simulation of mock
experiments, we can make progress with minimal effort by simply asking what
the median experiment would be able to do. This can be answered with the
Asimov dataset Cowan:2010js , because profile likelihood ratio test statistics
obtained under this dataset often provide a very good estimate to the median
of their full asymptotic distributions. The Asimov dataset is the case where
we set the observation equal to the expectation for some set of “true”
parameters. In our case when,

$P_{j}^{\rm obs}=P_{j}^{\rm Asi}\equiv P_{X}\langle
c(\theta_{X},\phi_{X})\rangle_{j}+\frac{\Delta t}{T}P_{N}\,.$ (59)

for all $j$.

We now interpret $\Lambda$ as a test statistic for detecting modulations,
${\rm TS}_{\rm mod}$. Under the Asimov data, the first log-likelihood in
Eq.(VII.1) vanishes, leaving us with

${\rm TS}_{\rm
mod}(\theta_{X},\phi_{X})=\frac{1}{\sigma^{2}_{N}}\left(\sum_{j}P_{X}\langle
c(\theta_{X},\phi_{X})\rangle_{j}-\hat{\hat{P}}_{X}\right)^{2}.$ (60)

The value of $P_{X}$ that maximises the unmodulated signal model likelihood
will be $\hat{\hat{P}}_{X}=P_{X}\langle c\rangle_{T}$. Substituting that in
and manipulating the subsequent expression we can find,

$\displaystyle{\rm TS}_{\rm mod}(\theta_{X},\phi_{X})$
$\displaystyle=\left(\frac{P_{X}}{\sigma_{N}}\right)^{2}\left(\sum_{j}\langle
c\rangle_{j}-\langle c\rangle_{T}\right)^{2}$
$\displaystyle=\left(\frac{P_{X}}{\sigma_{N}}\right)^{2}\left(\frac{1}{\Delta
t}\int_{0}^{T}c\mathrm{d}t-\langle c\rangle_{T}\right)^{2}\,.$
$\displaystyle=\left(\frac{T\langle
c\rangle_{T}P_{X}}{\sigma_{N}}\right)^{2}\left(\frac{1}{\Delta
t}-\frac{1}{T}\right)^{2}$ $\displaystyle\approx\left(\frac{T\langle
c\rangle_{T}P_{X}}{\Delta t\,\sigma_{N}}\right)^{2}\,.$ (61)

In the first step we converted the sum over $t$-bins to an integral, and in
the final step we have assumed $\Delta t\ll T$. Note that we have suppressed
the dependence on the DP angles, but they enter in via the dependence on
$\langle c\rangle_{T}$.

If we assume that the signal in one bin of interval $\Delta t$ was already
large enough to detect, then we can substitute in our result from Eq.(53) for
the median 3$\sigma$ exclusion of the background,

${\rm TS}_{\rm mod}\approx\left(\frac{3T\langle c\rangle_{T}}{\Delta
t\langle\cos^{2}{\theta}\rangle_{\Delta t}^{\rm disc.}}\right)^{2}\,.$ (62)

Taking the typical value of $\langle c\rangle_{T}\sim 1/3$, we can see that if
we needed a time $\Delta t$ to discover the DP, then we would need
$(3\langle\cos^{2}{\theta}\rangle_{\Delta t}^{\rm disc.})^{-1}$ times that
much data to exclude a non-modulating signal at the same significance. In the
specific cases we discovered had $\langle\cos^{2}{\theta}\rangle_{\Delta
t}^{\rm disc.}=1/3$, (e.g. for Zenith-pointing experiments at $\lambda_{\rm
lat}=35^{\circ}$ observing for day-long measurement times) then this factor is
equal to one. Meaning a DP signal discovered at 3$\sigma$ is already enough to
make the same claim about the modulation. The results for the remaining
experiments and for various values of $\Delta t$ can be inferred from Fig. 8.

###  VII.2 Measuring the DP polarisation

We can also use the same likelihood ratio to estimate how well some true
values of the angles $(\theta^{\rm true}_{X},\phi^{\rm true}_{X})$ could be
measured. We do this by taking the likelihood ratio, Eq.(VII.1), under the
Asimov dataset for the true angles, and then examine the dependence on
$(\theta_{X},\phi_{X})$. The formula for this is very similar to the
modulation test statistic we just derived, but now the first likelihood does
not vanish since we are allowing $(\theta_{X},\phi_{X})$ to vary,

$\displaystyle\Lambda(\theta_{X},\phi_{X})=$
$\displaystyle\left(\frac{3}{\langle\cos^{2}{\theta}\rangle_{\Delta t}^{\rm
disc.}}\right)^{2}\bigg{[}\bigg{(}T\langle c_{\rm
true}\rangle_{T}\bigg{(}\frac{1}{\Delta t}-\frac{1}{T}\bigg{)}\bigg{)}^{2}$
$\displaystyle-\frac{1}{\Delta t}\int_{0}^{T}\bigg{(}c_{\rm
true}-c\bigg{)}^{2}\mathrm{d}t\bigg{]}\,,$ (63)

where we write $c_{\rm true}=\cos^{2}{\theta}(\theta^{\rm true}_{X},\phi^{\rm
true}_{X})$. This likelihood ratio will be asymptotically distributed