according to a $\chi^{2}_{2}$ distribution since the models differ by two
parameters Wilks:1938dza . Therefore we can draw contours for
$\Lambda(\theta_{X},\phi_{X})-\Lambda(\theta^{\rm true}_{X},\phi^{\rm
true}_{X})=-6.17$ to show the typical size of a 2$\sigma$ measurement of the
DP polarisation axis. This is what is shown in Fig. 11.

Figure 11: Three mock examples of the reconstruction accuracy of the DP
polarisation axis. The true polarisation axis is given by the black stars. In
each case we assume that the experiment has been able to discriminate the
signal from background at the 95% C.L. in one day, and the value of $T$
corresponds to the _additional_ amount of data used to test for the daily
modulation. The size of each contour corresponds to the expected 2$\sigma$
contour around $(\theta_{X},\phi_{X})$ that the experiment could set given the
measurement time $T$. We display these angles in galactic longitude and
latitude $(l,b)$ with the galactic plane running horizontally through $b=0$,
and the galactic centre at $(0,0)$. The celestial equator is shown with a
purple line, and the projected directions of the North and South poles are
shown with purple circles.

The darker shading in the three panels of Fig. 11 correspond to increasing the
value of $T$. We assume that $\Delta t=1$ day, i.e. the experiment required 1
day of observation to detect the DP signal. Then $T$ here corresponds to how
many _additional_ days of observation are being assumed.

To make this result more evocative of a real scenario we have switched from
the equatorial coordinates we defined $\hat{\mathbf{X}}$ in, to galactic
coordinates. We define the galactic longitude and latitude, $(l,b)$, using the
coordinate rotation,

$\left(\begin{array}[]{c}\sin{\theta_{X}}\cos{\phi_{X}}\\\
\sin{\theta_{X}}\sin{\phi_{X}}\\\ \cos{\theta_{X}}\end{array}\right)=R_{\rm
gal}\begin{pmatrix}\cos l\cos b\\\ \sin l\cos b\\\ \sin b\end{pmatrix}\,,$
(64)

where,

$R_{\rm gal}=\begin{pmatrix}-0.05487556&+0.49410943&-0.86766615\\\
-0.87343709&-0.44482963&-0.19807637\\\
-0.48383502&+0.74698225&+0.45598378\end{pmatrix}\,,$

with values assuming the International Celestial Reference System convention
for the right ascension and declination of the North Galactic Pole,
$(\alpha_{\rm GP},\delta_{\rm GP})=(192^{\circ}.85948,\,+27^{\circ}.12825)$ as
well as the longitude of the North Celestial Pole $l_{\rm CP}=122^{\circ}.932$
BinneyGalacticAstronomy .

We can see from Fig. 11 that the North and Zenith-pointing experiments require
a smaller amount of _additional_ data than the West-pointing experiments to
obtain similar sized contours around the true polarisation axis. Unlike West-
pointing experiments, the North and Zenith pointing experiments sample the
polarisation in three dimensions, so they can eventually refine their
measurement of $(\theta_{X},\phi_{X})$ to a single axis. This can be seen by
the fact that the contours on the top and bottom panels of Fig. 11 are centred
around the black stars whereas the middle panel’s contours are centered around
four locations. In the West-pointing case we find that there are two
degenerate axes that the experiment is unable to tell apart. These two axes
are separated by $180^{\circ}$ of rotation around the Earth’s spin axis. This
phenomenon would persist even with indefinite data-taking—a rotation of the
experiment’s antennae would be needed to lift the degeneracy.

Therefore, we conclude that the polarisation axis of the DP should be
measurable very soon after the signal is detected. This is good because an
experiment designed to _study_ the DP could then be rotated by a suitable
amount to maximise the signal.

##  VIII Summary

We have highlighted some of the difficulties involved in setting limits on
dark photons (DPs) as a dark matter (DM) candidate, focusing on the fact that
the DP polarisation distribution around the Earth is unknown. The polarisation
state of the DPDM is initiated by its production mechanism. However, as of
now, the precise polarisation distributions generated by the several proposed
production mechanisms, and the subsequent effects of structure formation, have
not been rigorously studied. If it can be shown that structure formation
completely randomises the polarisation then direct detection limits on DPs can
be strengthened, sometimes significantly. However, a serious possibility that
appears to be true for several mechanisms is for the DP to have a single
polarisation over scales much larger than those probed by terrestrial
experiments. This latter scenario presents the greatest challenge for
detection and should be used as a conservative baseline. Most experiments are
sensitive to either one or two (but not all) DP polarisations at a time. Thus
a limit must take into account the probability of the experiment being
misaligned with the DP polarisation. This probability varies by over two
orders of magnitude depending on the measurement time, location, and alignment
of the experiment.

We began our discussion by explaining how to reinterpret limits from axion
haloscopes in the context of DPs. The first issue to reiterate is that the
common approach of vetoing candidate axion signals by testing for their
disappearance when the magnetic field is switched off means that a DP could
have been discovered but would have been discarded as noise. This makes the
RBF and UF axion haloscope bounds unsuitable for reinterpretation. For other
haloscopes, a lack of information about the exact magnetic field employed for
a given measurement also forbids a concrete reinterpretation in terms of DPs.

The second issue is related again to the DP polarisation. Accurately
accounting for the variation in the signal due to the rotation of the Earth
requires precise timing and orientation information. In both axion and
dedicated DP experiments, this information is usually not made available. Such
omissions make it impossible to set robust DP bounds. Instead, we have
demonstrated what would need to be done, by performing a conservative
calculation based on the information that is available. The resulting bounds
are shown in Fig. 7. As an example of what could be achieved in the future if
the experimental data-taking were more strategic—and we have laid out the
steps for doing so in this work—we have displayed some projections in Fig. 10.

To provide clarity for future experiments, we have outlined a recipe for
calculating the DP polarisation geometrical factor $\cos^{2}{\theta}$ for any
arbitrary measurement time, polarisation angle, and experiment orientation.
This is detailed in Sec. VI. We then outlined the best practices to both
maximise the sensitivity, and to avoid accidentally discarding DP signals. In
general, by measuring for times close to an integer number of sidereal days,
with well-aligned experiments, one can obtain close to (or even equal) to the
best possible scenario. This represents a gain of over an order of magnitude
in coupling for experiments polarised in a single direction, and a factor of
three for experiments sensitive to a plane of polarisation. This result can be
seen in Fig. 8. Even for experiments where measurement times are short,
improvements in sensitivity of an order of magnitude are possible simply by
splitting measurements into shorter ones spaced several hours apart—as is
shown in Fig. 9. We have also found that the location of the experiment also
impacts the sensitivity—as shown by Fig. 4—with $\pm 35^{\circ}$ and $\pm
55^{\circ}$ representing the optimal latitudes in most cases.

With the recent increase in the number of axion and DP-sensitive haloscopes,
it is more important than ever to be sure that we are extracting robust and
instructive limits. Rather than relying on others reinterpreting data with
limited information, we strongly encourage experimental collaborations to
perform dedicated DP analyses taking care of the role of magnetic field vetos,
as well as orientation and timing information. Further, without any reduction
to the experiment’s axion sensitivity or incurring any additional run time,
simple changes can result in substantial improvements in DP sensitivity. If
all experiments adopted such strategies it would maximise our chances of
discovering DPDM or, at the very least, would allow us to rule out vast
swathes of unexplored parameter space.

## Acknowledgements

The authors thank Pierre Brun, Woohyun Chung, Akash Dixit, Stefan Knirck,
Laura Manenti, Le Hoang Nguyen, Jonathan Ouellet, Tony Tyson, Darko Veberič
and SungWoo Youn for providing information about their experiments, and
Gonzalo Alonso-Àlvarez, Raymond Co, Keisuke Harigaya, and Andrew Long for
useful discussions about production mechanisms. CAJO thanks Masha Baryakhtar
and Sam Witte for helpful information that improved Fig. 1. We also thank Amit
Bhoonah, Priscilla Cushman, Divya Sachdeva, and Chang Sub Shin, for useful
comments. AC is supported by the Foreign Postdoctoral Fellowship Program of
the Israel Academy of Sciences and Humanities. CAJO is supported by The
University of Sydney. AC also acknowledges support from the Israel Science
Foundation (Grant 1302/19), the US-Israeli BSF (Grant 2018236) and the German-
Israeli GIF (Grant I-2524-303.7). AM is supported by the European Research
Council under Grant No. 742104 and by the Swedish Research Council (VR) under
Dnr 2019-02337 “Detecting Axion Dark Matter In The Sky And In The Lab
(AxionDM)”. The work of EV was supported in part by the U.S. Department of
Energy (DOE) Grant No. DE-SC0009937. AM dedicates this paper to Rosie Millar,
born this day. May she enjoy all the wonders of the Universe.

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