††thanks: deceased††thanks: deceased

# The daily modulations and broadband strategy in axion searches. An
application with CAST-CAPP detector

C. M. Adair  Department of Physics and Astronomy, University of British
Columbia, V6T 1Z1 Vancouver, Canada  K. Altenmüller  Centro de Astropartículas
y Física de Altas Energías (CAPA), Universidad de Zaragoza, 50009 Zaragoza,
Spain  V. Anastassopoulos  Physics Department, University of Patras, 26504
Patras, Greece  S. Arguedas Cuendis  European Organization for Nuclear
Research (CERN), CH-1211 Genève, Switzerland  J. Baier  Physikalisches
Institut, Albert-Ludwigs-Universität Freiburg, 79104 Freiburg, Germany  K.
Barth  European Organization for Nuclear Research (CERN), CH-1211 Genève,
Switzerland  A. Belov  Author affiliated with an institute covered by a
cooperation agreement with CERN  D. Bozicevic  University of Rijeka, Faculty
of Engineering, 51000 Rijeka, Croatia  H. Bräuninger  Max-Planck-Institut für
Extraterrestrische Physik, D-85741 Garching, Germany  G. Cantatore  Istituto
Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, 34127 Trieste, Italy
Università di Trieste, 34127 Trieste, Italy  F. Caspers  European Organization
for Nuclear Research (CERN), CH-1211 Genève, Switzerland  European Scientific
Institute (ESI), 74160 Archamps, France  J. F. Castel  Centro de
Astropartículas y Física de Altas Energías (CAPA), Universidad de Zaragoza,
50009 Zaragoza, Spain  S. A. Çetin  Istinye University, Department of Basic
Sciences, 34396 Sariyer, Istanbul, Türkiye  W. Chung  Center for Axion and
Precision Physics Research, Institute for Basic Science (IBS), Daejeon 34141,
Republic of Korea  H. Choi  Department of Physics, Korea Advanced Institute of
Science and Technology (KAIST), Daejeon 34141, Republic of Korea  J. Choi
Center for Axion and Precision Physics Research, Institute for Basic Science
(IBS), Daejeon 34141, Republic of Korea  T. Dafni  Centro de Astropartículas y
Física de Altas Energías (CAPA), Universidad de Zaragoza, 50009 Zaragoza,
Spain  M. Davenport  European Organization for Nuclear Research (CERN),
CH-1211 Genève, Switzerland  A. Dermenev  Author affiliated with an institute
covered by a cooperation agreement with CERN  K. Desch  Physikalisches
Institut, University of Bonn, 53115 Bonn, Germany  B. Döbrich  Max-Planck-
Institut für Physik (Werner-Heisenberg-Institut), Boltzmannstr. 8, 85748
Garching bei München, Germany  H. Fischer  Physikalisches Institut, Albert-
Ludwigs-Universität Freiburg, 79104 Freiburg, Germany  W. Funk  European
Organization for Nuclear Research (CERN), CH-1211 Genève, Switzerland  J.
Galan  Centro de Astropartículas y Física de Altas Energías (CAPA),
Universidad de Zaragoza, 50009 Zaragoza, Spain  A. Gardikiotis  Istituto
Nazionale di Fisica Nucleare (INFN), Sezione di Padova, 35131 Padova, Italy
S. Gninenko  Author affiliated with an institute covered by a cooperation
agreement with CERN  J. Golm  European Organization for Nuclear Research
(CERN), CH-1211 Genève, Switzerland  Institute for Optics and Quantum
Electronics, Friedrich Schiller University Jena, 07743 Jena, Germany  M. D.
Hasinoff  Department of Physics and Astronomy, University of British Columbia,
V6T 1Z1 Vancouver, Canada  D. H. H. Hoffmann  Xi’An Jiaotong University,
School of Science, Xi’An, 710049, China  D. Díez Ibáñez  Centro de
Astropartículas y Física de Altas Energías (CAPA), Universidad de Zaragoza,
50009 Zaragoza, Spain  I. G. Irastorza  Centro de Astropartículas y Física de
Altas Energías (CAPA), Universidad de Zaragoza, 50009 Zaragoza, Spain  K.
Jakovčić  Rudjer Bošković Institute, 10000 Zagreb, Croatia  J. Kaminski
Physikalisches Institut, University of Bonn, 53115 Bonn, Germany  M. Karuza
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, 34127
Trieste, Italy  University of Rijeka, Faculty of Physics, 51000 Rijeka,
Croatia  University of Rijeka, Photonics and Quantum Optics Unit, Center of
Excellence for Advanced Materials and Sensing Devices, and Centre for Micro
and Nano Sciences and Technologies, 51000 Rijeka, Croatia  C. Krieger
Institute of Experimental Physics, University of Hamburg, 22761 Hamburg,
Germany  Ç. Kutlu  Center for Axion and Precision Physics Research, Institute
for Basic Science (IBS), Daejeon 34141, Republic of Korea  Department of
Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon
34141, Republic of Korea  B. Lakić  Rudjer Bošković Institute, 10000 Zagreb,
Croatia  J. M. Laurent  European Organization for Nuclear Research (CERN),
CH-1211 Genève, Switzerland  J. Lee  Department of Physics, Korea Advanced
Institute of Science and Technology (KAIST), Daejeon 34141, Republic of Korea
S. Lee  Center for Axion and Precision Physics Research, Institute for Basic
Science (IBS), Daejeon 34141, Republic of Korea  G. Luzón  Centro de
Astropartículas y Física de Altas Energías (CAPA), Universidad de Zaragoza,
50009 Zaragoza, Spain  C. Margalejo  Centro de Astropartículas y Física de
Altas Energías (CAPA), Universidad de Zaragoza, 50009 Zaragoza, Spain  M.
Maroudas<EMAIL_ADDRESS>Institute of Experimental Physics,
University of Hamburg, 22761 Hamburg, Germany  L. Miceli  Center for Axion and
Precision Physics Research, Institute for Basic Science (IBS), Daejeon 34141,
Republic of Korea  H. Mirallas  Centro de Astropartículas y Física de Altas
Energías (CAPA), Universidad de Zaragoza, 50009 Zaragoza, Spain  L. Obis
Centro de Astropartículas y Física de Altas Energías (CAPA), Universidad de
Zaragoza, 50009 Zaragoza, Spain  A. Özbey  Department of Mechanical
Engineering, Istanbul University-Cerrahpaşa, Istanbul, Türkiye  K. Özbozduman
<EMAIL_ADDRESS>Bogazici University, Physics Department, 34342 Bebek,
Istanbul, Türkiye  M. J. Pivovaroff  Lawrence Livermore National Laboratory,
Livermore, CA 94550, USA  M. Rosu  Extreme Light Infrastructure - Nuclear
Physics (ELI-NP), 077125 Magurele, Romania  J. Ruz  Lawrence Livermore
National Laboratory, Livermore, CA 94550, USA  E. Ruiz-Chóliz  Institut für
Physik, Johannes Gutenberg Universität Mainz, 55128 Mainz, Germany  S. Schmidt
Physikalisches Institut, University of Bonn, 53115 Bonn, Germany  Y. K.
Semertzidis  Center for Axion and Precision Physics Research, Institute for
Basic Science (IBS), Daejeon 34141, Republic of Korea  Department of Physics,
Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141,
Republic of Korea  S. K. Solanki  Max-Planck-Institut für
Sonnensystemforschung, 37077 Göttingen, Germany  L. Stewart  European
Organization for Nuclear Research (CERN), CH-1211 Genève, Switzerland  I.
Tsagris  Physics Department, University of Patras, 26504 Patras, Greece  T.
Vafeiadis  European Organization for Nuclear Research (CERN), CH-1211 Genève,
Switzerland  J. K. Vogel  Centro de Astropartículas y Física de Altas Energías
(CAPA), Universidad de Zaragoza, 50009 Zaragoza, Spain  Lawrence Livermore
National Laboratory, Livermore, CA 94550, USA  M. Vretenar  University of
Rijeka, Faculty of Physics, 51000 Rijeka, Croatia  S. Youn  Center for Axion
and Precision Physics Research, Institute for Basic Science (IBS), Daejeon
34141, Republic of Korea  A. Zhitnitsky<EMAIL_ADDRESS>Department of Physics
and Astronomy, University of British Columbia, V6T 1Z1 Vancouver, Canada  K.
Zioutas  Physics Department, University of Patras, 26504 Patras, Greece
European Organization for Nuclear Research (CERN), CH-1211 Genève, Switzerland

###### Abstract

It has been previously advocated that the presence of the daily and annual
modulations of the axion flux on the Earth’s surface may dramatically change
the strategy of the axion searches. The arguments were based on the so-called
Axion Quark Nugget (AQN) dark matter model which was originally put forward to
explain the similarity of the dark and visible cosmological matter densities
$\Omega_{\rm dark}\sim\Omega_{\rm visible}$. In this framework, the population
of galactic axions with mass $10^{-6}{\rm eV}\lesssim m_{a}\lesssim
10^{-3}{\rm eV}$ and velocity $\langle v_{a}\rangle\sim 10^{-3}c$ will be
accompanied by axions with typical velocities $\langle v_{a}\rangle\sim 0.6c$
emitted by AQNs. Furthermore, in this framework, it has also been argued that
the AQN-induced axion daily modulation (in contrast with the conventional WIMP
paradigm) could be as large as $(10-20)\%$, which represents the main
motivation for the present investigation. We argue that the daily modulations
along with the broadband detection strategy can be very useful tools for the
discovery of such relativistic axions. The data from the CAST-CAPP detector
have been used following such arguments. Unfortunately, due to the dependence
of the amplifier chain on temperature-dependent gain drifts and other factors,
we could not conclusively show the presence or absence of a dark sector-
originated daily modulation. However, this proof of principle analysis
procedure can serve as a reference for future studies.

Axion Quark Nuggets, daily modulation, haloscope, CAST-CAPP

##  I Introduction and Motivation

The Peccei-Quinn mechanism, accompanied by axions, remains the most compelling
resolution of the strong $\cal{CP}$ problem; see original papers [1, 2, 3, 4,
5, 6, 7] and recent reviews [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. The
conventional idea for the production of the dark matter (DM) axions is either
by the misalignment mechanism when the cosmological field $\theta(t)$
oscillates and emits cold axions before it settles at a minimum, or via the
decay of topological objects, see recent reviews [8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18]. In both cases, the resulting axion is non-relativistic with
typical velocities $\langle v_{a}\rangle\sim 10^{-3}c$.

In addition to these well-established mechanisms, a fundamentally novel
mechanism for axion production was studied in recent papers [19, 20, 21, 22,
23]. This mechanism is rooted in the so-called axion quark nugget (AQN) DM
model [24], which will be briefly reviewed in Sect. II. For this introduction,
we just need to make several short comments on this model. First, the AQN
construction in many respects is similar to the original quark-nugget model
suggested by Witten [25]. This type of DM is “cosmologically dark” not because
of the weakness of the AQN interactions, but due to their small cross-section-
to-mass ratio, which scales down many observable consequences of an otherwise
strongly-interacting DM candidate, see Sect. II for more details. The second,
and most important feature for this work, is that the AQNs will emit
relativistic axions $\langle v_{a}\rangle\sim 0.6c$ in contrast with
conventional mechanisms mentioned above where the axions assume typical
galactic velocities $\langle v_{a}\rangle\sim 10^{-3}c$, see Sect. III for
details.

This distinct feature dramatically changes the entire axion detection strategy
as it must be a broadband search. Many new elements can be implemented in such
broadband searches, such as analysis for the daily modulation, see Sect. IV
for details.

The production of the relativistic axions has been discussed in the
literature, outside the AQN framework, see [26] where several mechanisms for
the production of the relativistic axions had been proposed and coined as the
CaB (cosmic axion background). In this sense, our proposal for broadband
searches has much more generic applicability, not limited to the AQN model.

However, there are a few differences between the AQN-induced axions and the
mechanisms of production of the CaB axions considered in [26]. The AQN-induced
axions are produced when an AQN interacts with the Earth’s material, and
therefore, are mostly localized on the Earth’s (or other stars or planets)
surface, while the axions considered in [26] are cosmological in nature, and
distributed uniformly in the Universe. As a result, the dominant portion of
the relativistic axions (which is the topic of the present work) should be
related to the AQN-induced events with a well-defined estimate for the axion
density on the Earth’s surface as reviewed in Sect. III.

The most distinct feature relevant to the present study, however, is as
follows. The study of the daily modulations of the axions produced by the CaB
mechanism considered in [26] requires the detector to be sensitive to the
directionality.

The daily modulations emerge due to the change in the relative orientation of
the Earth’s axis of rotation with respect to the galactic wind. If an
instrument is sensitive to the directionality then the daily modulations can
be studied by such an instrument.

The daily modulations of the relativistic axions can be, in principle, studied
(though not very efficiently) by the presently available instruments which
were not originally designed to measure directionality. In fact, the recent
analysis [27] uses ADMX data to study the daily modulations due to CaB. For
the AQN-induced axions, in addition to this effect, there will be an extra
contribution to the daily modulations that emerges as a result of the
difference in mass (size) of the AQNs when they enter or exit Earth as we
discuss in Sect. III. Therefore, even if the instrument is not sensitive to
the directionality it will be still capable of studying the daily modulations
as a result of the asymmetry between AQNs entering and exiting the Earth’s
surface as discussed in Sect. III.

We expect that the density of the AQN-induced axions on the Earth’s surface
should be much larger than the density of the CaB axions being constrained by
the CMB photon density, see Sect. III with the estimates. As a result, we
expect that the time modulation effects, which is the topic of the present
studies will be dominated by the AQN-induced axions111There is an enhancement
factor such as the modification of the local density of DM particles close to
the Earth’s surface. The corresponding enhancement factor of the DM density is
translated into a corresponding significant modification of the AQN-induced
density of the relativistic axions on the Earth’s surface. This effect is very
important for the AQN-induced axions and completely irrelevant for CaB axions
as the cosmic relativistic axions are not related to the DM density and cannot
be trapped by the Earth’s gravitational field.

There is another feature of the AQN-induced axions that is not shared by the
CaB mechanisms considered in [26]. The AQNs are very massive objects.
Therefore, the AQN flux on Earth is highly non-uniform, in contrast with
conventional DM models, and in contrast with CaB flux. As a result of this
aspect, there will be some “local flashes” which may enhance by several orders
of magnitude the axion count for a short period of time, see Sect. III with
details. It happens when the AQN hits the Earth’s surface in the vicinity
($100\text{\,}\mathrm{km}$ or less) of the detector, see next Sect. II with
specific estimates.

Our presentation is organized as follows. In the next Sect. II we briefly
overview the basic ideas of the AQN framework. In Sect. III and IV we overview
the main characteristics of the AQN-induced axions relevant to the present
work, including their spectrum and the intensity on the Earth’s surface. Sect.
V is devoted to our estimations for the daily modulation, including analysis
of possible tests to discriminate the genuine signal from spurious signals and
noise. In Sect. VI we apply all these generic features of the broadband
analysis to study possible daily modulations by using the previously recorded
CAST-CAPP dataset. We then conclude with Sect. VII providing some suggestions
for future studies.

##  II The AQN dark matter model. The basics

The AQN construction in many respects is similar to Witten’s quark nuggets,
see [25, 28, 29]. This type of DM is “cosmologically dark” as a result of the
smallness of the parameter relevant for cosmology, which is the cross-section-
to-mass ratio of the DM particles. This numerically small ratio scales down
many observable consequences of an otherwise strongly-interacting DM candidate
in the form of the AQN nuggets.

There are several additional elements in the AQN model in comparison with the
older well-known and well-studied theoretical constructions [25, 28, 29].
First, there is an additional stabilization factor for the nuggets provided by
the axion domain walls which are copiously produced during the QCD transition.
This additional element helps to alleviate a number of problems with Witten’s
original model 222In particular, a first-order phase transition is not a
required feature for nugget formation as the axion domain wall (with internal
QCD substructure) plays the role of the squeezer. Another problem of the old
construction [25, 28, 29] is that nuggets likely evaporate on the Hubble time
scale. For the AQN model, this is not the case because the vacuum-ground-state
energies inside (the color-superconducting phase) and outside the nugget (the
hadronic phase) are drastically different. Therefore, these two systems can
coexist only in the presence of an external pressure, provided by the axion
domain wall. This should be contrasted with the original model [25, 28, 29],
which is assumed to be stable at zero external pressure. This difference has
major observational consequences- the Witten’s nugget will turn a neutron star
(NS) into a quark star if it hits the NS. In contrast, a matter type AQN will
not turn an entire star into a new quark phase because the quark matter in the
AQNs is supported by external axion domain wall pressure, and therefore, can
be extended only to relatively small distance $\sim m_{a}^{-1}$, which is much
shorter than the NS size.. Secondly, the nuggets can be made of matter as well
as antimatter during the QCD transition.

Property |  |  | Typical value or feature
---
AQN’s mass $[M_{N}]$ |  |  $M_{N}\approx 16\,g\,(B/10^{25})$ [30]
baryon charge constraints $[B]$ |  |  $B\geq 3\cdot 10^{24}$ [30]
annihilation cross section $[\sigma]$ |  | $\sigma\approx\kappa\pi R^{2}\simeq 1.5\cdot 10^{-9}{\rm cm^{2}}\cdot\kappa(R/2.2\cdot 10^{-5}\rm cm)^{2}$
density of AQNs $[n_{\rm AQN}]$ |  |  $n_{\rm AQN}\sim 0.3\cdot 10^{-25}{\rm cm^{-3}}(10^{25}/B)$ [30]
survival pattern during BBN |  |  $\Delta B/B\ll 1$ [31, 32, 33, 34]
survival pattern during CMB |  |  $\Delta B/B\ll 1$ [31, 35, 33]
survival pattern during post-recombination |  |  $\Delta B/B\ll 1$ [36]
Table 1: Basic properties of the AQNs

The presence of antimatter nuggets in the AQN framework is an inevitable and
direct consequence of the $\cal{CP}$ violating axion field which is present in
the system during the QCD time. As a result of this feature the DM density,
$\Omega_{\rm DM}$, and the visible density, $\Omega_{\rm visible}$, will
automatically assume the same order of magnitude densities $\Omega_{\rm
DM}\sim\Omega_{\rm visible}$ irrespectively to the parameters of the model,
such as the axion mass $m_{a}$. This feature represents a generic property of
the construction [24] as both components, the visible, and the dark are
proportional to the same fundamental dimensional constant of the theory, the
$\Lambda_{\rm QCD}$.

We refer to the original papers [37, 38, 39, 36] devoted to the specific
questions related to the nugget’s formation, generation of the baryon
asymmetry, and survival pattern of the nuggets during the evolution in early
Universe with its unfriendly environment. We also refer to a recent brief
review article [30] which explains several subtle points on the formation
mechanism, and survival pattern of the AQNs during the early stages of the
evolution, including the Cosmic Microwave Background (CMB) Big Bang
Nucleosynthesis (BBN), and recombination epochs.

In this work we take the agnostic viewpoint and assume that such nuggets made
of antimatter are present in our Universe today and they were present at
earlier times, irrespective of their formation mechanism. This assumption is
consistent with all presently available cosmological, astrophysical, and
terrestrial constraints as long as the average baryon charge of the nuggets is
sufficiently large as we review below.

The strongest direct detection limit333Non-detection of etching tracks in
ancient mica gives another indirect constraint on the flux of DM nuggets with
mass $M>55$g [40]. This constraint is based on the assumption that all nuggets
have the same mass, which is not the case for the AQN model. is set by the
IceCube Observatory’s, see Appendix A in [21]:

$\displaystyle\langle B\rangle>3\cdot 10^{24}~{}~{}~{}[{\rm
direct~{}(non)detection~{}constraint]}.$ (1)

Similar limits are also obtainable from the ANITA and from geothermal
constraints which are also consistent with Eq. 1 as estimated in [41].

The authors of Ref. [33] considered a generic constraint for the nuggets made
of antimatter (ignoring all essential specifics of the AQN model such as quark
matter CS phase of the nugget’s core). Our constraints (Eq. 1) are consistent
with their findings including the CMB and BBN, and others, except the
constraints derived from the so-called “Human Detectors”. As explained in [42]
the corresponding estimates of Ref. [33] are oversimplified and do not have
the same status as those derived from CMB or BBN constraints.

We now present the rate at which AQNs hit the Earth assuming the local DM
density of $\rho_{\rm DM}\simeq 0.3\,{\rm{GeV}{cm^{-3}}}$. Assuming the
conventional halo model, one arrives at [21]:

$\displaystyle\frac{\langle\dot{N}\rangle}{4\pi R_{\oplus}^{2}}$
$\displaystyle\simeq$ $\displaystyle\frac{4\cdot 10^{-2}}{\rm
km^{2}~{}yr}\left(\frac{\rho_{\rm
DM}}{0.3{\rm\frac{GeV}{cm^{3}}}}\right)\left(\frac{\langle v_{\rm
AQN}\rangle}{220~{}{\rm\frac{km}{s}}}\right)\left(\frac{10^{25}}{\langle
B\rangle}\right),$ $\displaystyle\langle\dot{N}\rangle$ $\displaystyle\simeq$
$\displaystyle 0.67\,{\rm s}^{-1}\left(\frac{10^{25}}{\langle
B\rangle}\right)\simeq 2.1\cdot 10^{7}{\rm
yr}^{-1}\left(\frac{10^{25}}{\langle B\rangle}\right).$ (2)

In obtaining Eq. II it was assumed that $\langle\dot{N}\rangle\approx 4\pi
b^{2}n_{\rm AQN}v_{\rm AQN}$, where $b\approx 1.0013R_{\oplus}$ is the impact
parameter for Earth, and we averaged over incident angles of impacting AQNs,
see Appendix C in [21] with details. We also assume that the AQNs represent
the dominant portion of the DM. Numerical expressions for Eq. II were derived
in [21] using full-scale Monte Carlo simulations which account for all types
of AQN trajectories with different AQN masses $M_{N}\simeq m_{p}|B|$,
different incident angles, different initial velocities and size
distributions. It was shown that none of these factors significantly affect
our estimate. The result in Eq. II suggests that AQNs with a typical size
$B\approx 10^{25}$ hit the Earth’s surface with a frequency on the order of
once a day (i.e., hundreds per year) per $100\times 100\,{\rm km^{2}}$.

We conclude this brief review subsection with Tab. 1 which summarizes the
basic features and parameters of the AQNs. The parameter $\kappa$ in Tab. 1 is
introduced to account for the fact that not all matter striking the nugget
will annihilate and not all of the energy released by annihilation will be
thermalized in the nuggets. The ratio $\Delta B/B\ll 1$ in the Table implies
that only a small portion $\Delta B$ of the total (anti)baryon charge $B$
hidden in form of the AQNs get annihilated during big-bang nucleosynthesis
(BBN), Cosmic Microwave Background (CMB), or post-recombination epochs
(including the galaxy and star formation), while the dominant portion of the
baryon charge survives until the present time. Independent analysis [34] and
[33] also support our original claims, as cited in Tab. 1, that the anti-quark
nuggets survive the BBN and CMB epochs.

Finally, one should mention here that the AQN model with the same set of
parameters may explain several other puzzling observations in different
environments. In particular, it may explain the primordial Li puzzle [32]
during the early Universe evolution. On a different, galactic scale, it may
explain the observed excess of the UV radiation [43]. The excess of radiation
is observed in many different frequency bands, from UV to optical, IR, and
even radio frequency bands. It is normally expressed as excess in the Extra-
galactic Background Light (EBL), see e.g. review [44]. The energy injection
due to the AQN-induced processes may explain a portion or even entire observed
excesses in different frequency bands.

It may also explain some observed inconsistencies on small galactic scales
(such as the Core-Cusp problem [45, 46], the Too-Big-to-Fail problem [47],
etc) which are hard to explain within the conventional CDM paradigm [48]. It
may also shed some light on the long-standing solar corona heating problem
[49, 50], which could be the manifestation of the AQN’s processes at different
scales (outside of the galactic scale where the DM is expected to manifest
itself). Finally, the AQNs may also explain some mysterious events observed on
the Earth scale, such as the anomalous events recorded by ANITA [51].

##  III The AQN-induced axions

As we already mentioned the AQNs propagating in the Earth’s atmosphere and
interior will be losing their baryon charge as a result of annihilation
processes. These processes will also lead to the axion emission as suggested
in [19]. This new mechanism of axion production can be briefly explained as
follows. The total energy of an AQN finds its equilibrium minimum when the
axion domain wall contributes about 1/3 of its total mass [39]. This
configuration in the equilibrium does not emit any axions due to a purely
kinematical constraint: the static axion domain wall can be thought of as a
superposition of off-shell non-propagating axions. However, this static
picture drastically changes when some baryon charge from the AQN gets
annihilated as a result of interaction with the environment when the time-
dependent perturbations change this equilibrium configuration. In other words,
the configuration becomes unstable with respect to the emission of the axions
since the AQN is no longer at its minimum energy configuration with fewer
baryon charges in the quark nugget core. As a result of these annihilation
processes, the AQN starts to lose its mass, and consequentially the axion
domain surrounding the quark matter wall starts to oscillate and shrink. These
oscillations of the domain wall generate excitation modes and ultimately lead
to radiation of the propagating axions. The spectrum of the corresponding AQN-
induced axions has been computed in [20].

Several important features of this spectrum deserve to be mentioned here.
First of all, typical values $\langle v_{\rm axion}^{\rm AQN}\rangle\simeq
0.6c$ are very large in comparison with the conventional galactic non-
relativistic axions $\langle v_{\rm axion}\rangle\sim 10^{-3}c$. The
corresponding Monte Carlo simulations have been carried out in [22], leading
to the following estimate for the density of these relativistic axions on the
Earth’s surface:

$\displaystyle\langle\rho_{\mathrm{a}}^{\rm AQN}(R_{\oplus})\rangle\sim 5\cdot
10^{-6}A(t){\rm\frac{GeV}{cm^{3}}},~{}\langle v_{\mathrm{a}}^{\rm
AQN}(R_{\oplus})\rangle\simeq 0.6c,~{}~{}~{}~{}$ (3)

where $A(t)$ is the modulation/amplification time-dependent factor. The factor
$A$ for the daily and annual modulations is discussed in the next Sect. IV
below and is given by Eq. 6 and Eq. 7, correspondingly. In both cases, the
factor $A$ does not deviate from the average value $\langle A\rangle=1$ by
more than $10\%$.

However, one should emphasize that this normalization assumes the Standard
Halo Model (SHM) DM distribution with $\rho_{\rm DM}\approx{\rm
0.3~{}{GeV}\cdot{cm^{-3}}}$. The local DM density on the Earth’s surface could
be drastically different from this conventional value due to the capturing DM
particles by the stars and planets, see e.g. [52] and refs therein. In fact,
it was the main motivation of ref. [53] to critically analyse within the AQN
framework the basic SHM assumption by performing some specific computations.
In this and the next sections, we keep the conventional SHM value $\langle
A\rangle=1$ to avoid confusion with other effects that may modify the
normalization $A(t)$ locally for a very short period of time (see also below).

The AQN-induced axions are mostly produced in the Earth’s deep interior where
the density of surrounding material is the highest. The energy density of the
axions (Eq. 3) as well as the energy flux (Eq. 4) are not very sensitive to
the axion mass $m_{a}$ as explained in [22], in contrast with galactic non-
relativistic axions assuming that DM is saturated by the galactic axions444The
assumption on saturation of the DM density by the galactic axions cannot be
satisfied in the entire window of $10^{-6}{\rm eV}\lesssim m_{a}\lesssim
10^{-3}{\rm eV}$ as the conventional contribution is highly sensitive to
$m_{a}$ as $\rho_{a}\sim m_{a}^{-7/6}$ and may saturate the DM density at
$m_{a}\lesssim 10^{-5}{\rm eV}$, depending on additional assumptions on
production mechanism. This, in particular, implies that for $m_{a}\gtrsim
10^{-4}{\rm eV}$ the conventional galactic axions contribute very little to
$\Omega_{\rm DM}$, while the AQNs may be the dominant contributor to the DM
density irrespective to the axion mass $m_{a}$..

It is instructive to compare the axion energy density (Eq. 3) of the
relativistic axions with CMB photons because both components are relativistic
particles. For the CMB photons, the energy density is well known and is given
by $\rho_{\gamma}\approx 0.26~{}{\rm{eV}/{cm^{3}}}$, which is almost 4 orders
of magnitude below the corresponding expression for AQN-induced axions on the
Earth’s surface (Eq. 3). The CaB axion density considered in [26] and
mentioned in the Introduction must satisfy the condition $\rho_{\rm
CaB}\ll\rho_{\gamma}$. This implies that the density of the CaB axions must be
at least 4 orders of magnitude below the AQN-induced axions on the Earth’s
surface (Eq. 3).

There is a big difference in the distributions between the CMB photons and the
axion energy density: the CMB photons are (almost) uniformly distributed over
the entire Universe, while the axion density (Eq. 3) is localized around Earth
and quickly vanishes at larger distances at $r\gg R_{\oplus}$. Similar
estimates can be easily obtained for other stars and planets. Unfortunately,
the axions do not couple with our visible material as strongly as CMB photons,
which represents the major problem for their discovery despite their larger
density (Eq. 3) in comparison with the CMB photons at the Earth’s surface.

The resulting number density $n_{a}^{\rm AQN}\simeq\rho_{\mathrm{a}}^{\rm
AQN}/m_{a}$ is approximately 5 orders of magnitude smaller than the
conventional galactic axion number density assuming that the galactic non-
relativistic axions saturate the DM today. However, the flux
$(v_{\mathrm{a}}^{\rm AQN}\cdot n_{a}^{\rm AQN})$ related to relativistic
axions (Eq. 3) is only 2 orders of magnitude smaller than the conventional
flux of non-relativistic axion. Indeed, the AQN-induced axion energy flux on
the Earth’s surface can be estimated as follows [22]:

$\displaystyle\langle E_{a}\cdot\Phi^{\rm AQN}_{a}\rangle\simeq
10^{14}A(t)\left[{\rm\frac{eV}{cm^{2}\cdot s}}\right],~{}~{}~{}\langle
E_{a}\rangle\simeq 1.3\,m_{a}.$ (4)

It is instructive to compare the AQN-related flux (Eq. 4) with the flux
computed from the assumption that the galactic axions saturate the DM density
$\rho_{DM}\sim 0.3\,{\rm GeV\cdot cm^{-3}}$ today, see footnote 4. In this
case, the numerical value for the flux (Eq. 4) is approximately two orders of
magnitude below the value $10^{16}{\rm{eV}/({cm^{2}\cdot s})}$ computed for
the conventional galactic non-relativistic axions assuming these axions
saturate the DM density. One should emphasize once again that the density (Eq.
3) as well as the flux (Eq. 4) do not strongly depend on the axion mass
$m_{a}$ nor the initial misalignment angle $\theta_{0}$ in contrast with
conventional expression for the galactic axions, see footnote 4.

It is noted that the factor $A(t)$ could be numerically very large for a very
short period in case of rare bursts-like events, the so-called “local flashes”
in the terminology of Ref. [22]. These short bursts (with a duration time of
the order of a second for $A\simeq 10^{2}$) result from the interaction of the
AQN hitting the Earth in the close vicinity of a detector, see Tab. 2.

Table 2: Estimations of Local flashes for different $A$ as defined by Eq. 4. The corresponding event rate and the time duration $\tau$ depend on the factor $A$, which is determined by the shortest distance from the nugget’s trajectory to the detector. The table is adopted from [22]:  $A$ |  $\tau$ (time span) | event rate
---|---|---
1 | 10 s | 0.3 $\rm min^{-1}$
$10$ | 3 s | 0.5 $\rm hr^{-1}$
$10^{2}$ | 1 s | 0.4 $\rm day^{-1}$
$10^{3}$ | 0.3 s | 5 $\rm yr^{-1}$
$10^{4}$ | 0.1 s | 0.2 $\rm yr^{-1}$

Another important feature of the AQN-induced axions is the presence of the
daily modulations, which is not a common feature of a generic DM model when a
detector is not sensitive to the directionality, which is normally the case
for most modern DM detectors. Rather, it is a very specific property of the
AQN-induced production mechanism. The daily modulation plays a key role in our
arguments. Therefore, we would like to explain the physical origin of this
novel effect here.

The source of the daily modulation is the AQN’s size difference between the
moment of entry and moment of exit as a result of annihilation processes in
the Earth’s interior555On average a typical nugget loses approximately $30\%$
of its baryon charge when crosses the Earth. The axion domain wall shrinks
correspondingly, which eventually generates the axion density (Eq. 3) at the
Earth’s surface from all AQNs trans-passing the Earth’s interior.. Such
effects do not exist for any fundamental particles such as WIMPs because the
mass of any fundamental particle is the same at entry and exit points. The
average size difference between the AQNs entering and exiting the Earth’s
surface may generate a large effect $\sim 10\%$ for the daily modulations as
computed in [22] and explained below.

The key point here is the presence of relative orientation of the Earth’s axis
of rotation and the DM galactic wind. This relative orientation is an
irrelevant parameter for conventional WIMPs if the detector is not designed to
measure the directionality as already mentioned. For AQNs the flux is
sensitive to this orientation. This effect eventually generates the flux
asymmetry as a function of the latitude $\theta$, which consequently leads to
the daily modulation. This feature of the axion flux is expressed in terms of
parameter $P_{a}(\theta)$ which is normalized to one, i.e. $\int
P_{a}(\theta)\sin\theta d\theta=1$. In the absence of the daily modulations
parameter $P_{a}(\theta)=0.5$ is independent of the latitude $\theta$. Our
numerical Monte Carlo simulations shown in Fig. 1 unambiguously demonstrate
the presence of the $\theta$ dependence, as explained above. This $\theta$
dependence generates the daily modulation.

Figure 1: The result of the Monte Carlo numerical simulations for the
azimuthal distribution of axion flux $P_{a}(\theta)$ on the surface of the
Earth, adopted from [22]. A relative orientation of the Earth’s axis of
rotation and DM galactic wind generates the flux asymmetry sensitive to the
latitude $\theta$, which consequently leads to the daily modulation effect.

The production rate of the low energy AQN-induced axions with $v_{a}\ll c$ is
strongly suppressed as explained above. However, the axions that are produced
with extremely low velocities $v_{a}\lesssim 11\rm~{}{km}/{s}$ will be trapped
by the Earth’s gravitational field. These axions will be orbiting the Earth
indefinitely; therefore, they will be accumulated around the Earth during its
lifetime of 4.5 billion years.

The corresponding Monte Carlo simulations have been performed in [21] with the
following estimate for the gravitationally-bound axions:

$\displaystyle\rho_{\mathrm{a}}^{\rm bound}(R_{\oplus})\sim
10^{-4}{\rm\frac{GeV}{cm^{3}}},~{}~{}~{}\langle
v_{\mathrm{a}}(R_{\oplus})\rangle\simeq 8\rm\frac{km}{s}.$ (5)

The number density of the bound axions is at least 2 orders of magnitude
smaller than conventional axion number density assuming that the galactic non-
relativistic axions saturate the DM density. However, the corresponding
wavelength $\lambda_{\mathrm{a}}\sim{\hbar}/({m_{\mathrm{a}}v_{\mathrm{a}}})$
of the gravitationally bound axions is approximately $30$ times greater than
for galactic axions, which have a typical velocity of about $\sim 250$ km/s.
Therefore, coherent effects can be maintained for a longer time-period
compared to those for conventional galactic axion searches. One may hope that
the feature of having a large coherence length, $\lambda_{\mathrm{a}}\sim
v_{\mathrm{a}}^{-1}$, could play a key role in the design of instruments,
capable of discovering such gravitationally trapped axions.

One should also mention that these AQN-induced axions (relativistic as well as
gravitationally bound) can be treated as a classical field because the number
of the AQN-induced axions (Eq. 3) accommodated by a single de-Broglie volume
is very large even though the de-Broglie wavelength $\lambda$ for relativistic
AQN-induced axions is much shorter than for galactic axions,

$\displaystyle n_{a}^{\rm
AQN}\lambda^{3}\sim\frac{\langle\rho_{\mathrm{a}}^{\rm
AQN}\rangle}{m_{a}}\left(\frac{\hbar}{m_{a}v_{a}}\right)^{3}\sim
10^{6}\left(\frac{10^{-4}{\rm eV}}{m_{a}}\right)^{4}\gg 1.$

##  IV New strategy: detection of broadband axions

The large average velocities $\langle v_{a}\rangle\simeq 0.6c$ of the emitted
axions by AQNs change the entire strategy of axion searches since they are
characterized by a broad distribution with $m_{a}\lesssim\omega_{a}\lesssim
1.8~{}m_{a}$ as discussed in the previous subsection. Therefore, the
corresponding axion detectors must be some kind of broadband instrument. The
cavity-type experiments such as ADMX are to date the only ones to probe the
parameter space of the conventional QCD axions with $\langle v_{a}\rangle\sim
10^{-3}c$, while we are interested in the detection of the relativistic axions
with $\langle v_{a}\rangle\sim 0.6c$. This requires a different type of
instruments and drastically different search strategies. We assume that some
kind of broadband instruments can be designed and built, see reviews [14, 15,
16, 17, 18] with description of possible detectors.

With this assumption in mind, a new strategy to probe the QCD axion can be
formulated as follows [23]. It has been known since [54] that the DM flux
shows annual modulation due to the differences in relative orientations of the
DM wind and the direction of the Earth’s motion around the Sun. The
corresponding effect for AQN-induced axions was computed in [22]. The daily
modulation which is a very specific feature of the AQN model, as explained in
the previous section, was also computed in the same paper [22]. The broadband
strategy is to separate a large frequency band into a number of smaller
frequency bins with the width $\Delta\nu\sim\nu$ according to the axion
dispersion relation as discussed above.

The time-dependent signal in each frequency bin $\Delta\nu_{i}$ has to be
fitted according to the expected modulation pattern, daily, or annual. For
example, the daily modulation should be fitted according to the following
formula

$\displaystyle\rho_{\mathrm{a}}^{\rm AQN}(t)\equiv
A_{\rm(d)}(t)\langle\rho_{\mathrm{a}}^{\rm AQN}(R_{\oplus})\rangle$ (6)
$\displaystyle
A_{\rm(d)}(t)\equiv[1+\kappa_{\rm(d)}\cos(\Omega_{d}t-\phi_{0})],$

where the density $\langle\rho_{\mathrm{a}}^{\rm AQN}(R_{\oplus})\rangle$ is
estimated according to Eq. 3. In this formula $\Omega_{d}=2\pi\,\rm day^{-1}$
is the angular frequency of the daily modulation, while $\phi_{0}$ is the
phase shift. It can be assumed to be constant on the scale of days. However,
it slowly changes during the annual seasons due to the variation of the
direction of DM wind with respect to the Earth. The modulation factor
$A_{\rm(d)}(t)$ is normalized to unity, $\langle A_{\rm(d)}(t)\rangle=1$ when
one averages over a large number of daily cycles. The same modulation factor
$A_{\rm(d)}(t)$ with the same $\kappa_{\rm(d)}$ also enters the estimation for
the flux (Eq. 4).

A similar formula holds for the annual modulation defined as follows:

$\displaystyle\rho_{\mathrm{a}}^{\rm AQN}(t)\equiv
A_{\rm(a)}(t)\langle\rho_{\mathrm{a}}^{\rm AQN}(R_{\oplus})\rangle$ (7)
$\displaystyle A_{\rm(a)}(t)\equiv[1+\kappa_{\rm(a)}\cos\Omega_{a}(t-t_{0})],$

where $\Omega_{a}=2\pi\,\rm yr^{-1}$ is the angular frequency of the annual
modulation and label $``a"$ in $\Omega_{a}$ stands for annual. The
$\Omega_{a}t_{0}$ is the phase shift corresponding to the maximum on June 1
and minimum on December 1 for the standard galactic DM distribution, see [54,
55].

Several tests can unambiguously discriminate a true genuine signal from
spurious signals and noise. These tests will be discussed in detail in Sect.
V.2 after some simplified estimates for the expected signal are performed in
Sect. V.1.

##  V Expected Signals from the AQN-induced axions

The goal of this section is twofold. First, in Sect. V.1 we want to highlight
the basic differences in conversion rate between the AQN-induced relativistic
axions and the conventional galactic axions which normally enter the well-
known formulae. Secondly, in Sect. V.2 we formulate the key element of this
work: despite the very low rate of axion-photon conversion the daily
modulation effect reviewed in the previous Sect. IV effectively allows us to
discriminate the genuine signal from the spurious signals.

###  V.1 Estimates for the axion to photon conversion

The starting point for our analysis is the computation of the conversion rate
which is determined by the well-known formula, see e.g.review [18]:

$\displaystyle
P_{a\rightarrow\gamma}=\frac{1}{v_{a}}\left(\frac{gB}{q}\right)^{2}\sin^{2}\left(\frac{qL}{2}\right),$
(8) $\displaystyle q=\omega-\sqrt{\omega^{2}-m_{a}^{2}}$

where $L$ is a typical distance where the magnetic field $B$ is present and
$g\equiv g_{a\gamma\gamma}$ is defined as usual

$\displaystyle
g\simeq\frac{g_{\gamma}\alpha}{f_{a}\pi},~{}~{}g_{\gamma}=\frac{1}{2}\left(\frac{N_{e}}{N}-\frac{5}{3}-\frac{m_{d}-m_{u}}{m_{u}+m_{d}}\right)$
(9)

where parameter $N_{e}/N=0$ for KSVZ model, and $N_{e}/N=8/3$ for DFSZ model
such that $g_{\gamma}\approx-0.97$ for KSVZ model and $g_{\gamma}\approx 0.36$
for DFSZ model. The parameters $f_{a}$ and $m_{a}$ are not independent, but
tightly linked in conventional QCD axion models:

$\displaystyle m_{a}\approx 6\cdot
10^{-4}\left(\frac{10^{10}\rm~{}GeV}{f_{a}}\right){\rm eV}.$ (10)

In case of the resonant conversion when combination $(qL)\ll 1$ could be very
small the conversion rate assumes the usual form
$P_{a\rightarrow\gamma}=v_{a}^{-1}(gBL/2)^{2}$. The dependence
$P_{a\rightarrow\gamma}\propto L^{2}$ should be interpreted as the presence of
a coherent phenomenon when the amplitude of the transition is linearly
proportional to the coherence length $\propto L$, while the probability
$P_{a\rightarrow\gamma}$ shows the quadratic dependence
$P_{a\rightarrow\gamma}\propto L^{2}$ as stated above.

For non-relativistic axions, which are normally assumed to be the case for
haloscope studies, one can approximate $q\simeq m_{a}$. If one multiplies Eq.
8 with the axion flux $(\rho_{a}v_{a})/m_{a}$ one arrives at the following
expression for the flux $\Phi_{\gamma}$ which counts the number of microwave
photons emerging from the magnetic field region per unit time and unit area:

$\displaystyle\Phi_{\gamma}=\frac{(\rho_{a}v_{a})}{m_{a}}P_{a\rightarrow\gamma}\approx\frac{\rho_{a}}{m_{a}}\left[\frac{gB}{m_{a}}\sin\left(\frac{m_{a}L}{2}\right)\right]^{2}.$
(11)

This expression agrees with the formula of ref. [56] which was derived using a
very different approach. It is important to note that Eq. 11 does not depend
on the axion’s velocity $v_{a}$ in the non-relativistic limit
$v_{a}\rightarrow 0$. This is because Eq. 8 for the conversion is proportional
to $v_{a}^{-1}$ while computation of the $\Phi_{\gamma}$ requires
multiplication of the conversion $P_{a\rightarrow\gamma}$ to the axion flux
which itself is proportional to $v_{a}$ such that the dependence on $v_{a}$
cancels out at small $v_{a}/c\ll 1$.

We now turn to our relativistic case for the AQN-induced axions. In this case,
the axion density and the flux are determined by Eq. 3 and Eq. 4
correspondingly. Furthermore, the time formation of a photon is very short
$\sim m_{a}^{-1}$. The corresponding length scale is determined by the De
Broglie length $\lambda\sim m_{a}^{-1}$ which is also much shorter than the
size of the system $L$. Therefore, the conversion process can be treated as it
happens in an infinite volume system.

As a result, one can estimate the flux $\Phi^{\rm AQN}_{\gamma}$ which counts
the number of microwave photons emerging from the magnetic field region per
unit time and unit area as a result of a single elementary process as follows:

$\displaystyle\Phi^{\rm AQN}_{\gamma}\approx\Phi^{\rm
AQN}_{a}P_{a\rightarrow\gamma}\approx\Phi^{\rm
AQN}_{a}\left[\frac{gB}{m_{a}}\sin\left(\frac{m_{a}L}{2}\right)\right]^{2},~{}~{}$
(12)

where we ignored all numerical factors of the order of one. In particular, we
put for simplicity $\langle E_{a}\rangle\simeq m_{a},~{}\langle
v_{\mathrm{a}}^{\rm AQN}\rangle\simeq 1$. We also ignore the complications
related to the relative orientation of the magnetic field and axion momentum.
We also ignore the differences between $L_{x},L_{y},L_{z}$ characterizing the
geometry of the magnetic field, etc. All these order of one effects can be
ignored for our qualitative analysis in the present work. In principle, one
should integrate over the axion spectrum computed previously with a specific
geometry of the instrument when $E_{a}$ approximately varies in the range
$E_{a}\in(m_{a}-1.8m_{a})$. The non-resonant Primakoff conversion generates a
complicated photon’s broadband spectrum with $\nu\in(m_{a}-1.8m_{a})$.
However, for our qualitative estimates when we want to study the parametrical
dependence of the rate as a function of the key parameters such as the axion
mass $m_{a}$, magnetic field $B$, the basic size of the detector $L$ all these
numerical factors of the order of one are irrelevant, and will be ignored in
what follows.

Several key elements lead to the different treatments of the non-relativistic
galactic axions in comparison with relativistic AQN-induced axions. First of
all, the conventional treatment of the problem assumes the coherent conversion
such that $P_{a\rightarrow\gamma}\propto L^{2}$ as explained above. This
$L^{2}$ scaling holds as long as $L$ is smaller than the De Broglie wavelength
$\lambda$ where conversion effectively occurs. If coherence persists up to
distance $l<L$ the enhancement is much smaller, $L^{2}\rightarrow Ll$ [18]. It
should be contrasted with the AQN induced axions when $\Delta\nu/\nu\sim 1$
and one should average over many cycles such that
$\sin^{2}(\frac{qL}{2})\rightarrow 1/2$ in Eq. 12.

The second important element is as follows. Eq. 12 describes the conversion as
a result of a single elementary process when a typical formation length is of
order the Compton length $m_{a}^{-1}$. One should sum over the number of
effective “layers” along the axion’s path which is of order $(Lm_{a})\gg 1$.
This factor represents a strong enhancement factor. One should note that this
factor $(Lm_{a})\gg 1$ agrees with formula $Ll$ presented in [18] when the
coherence length $l$ should be replaced to the Compton length, i.e.
$l\rightarrow m_{a}^{-1}$.

Therefore, based on the above, we arrive at the following estimate for the
total flux $\Phi^{\rm AQN}_{\gamma}$ which counts the number of microwave
photons emerging from the magnetic field region per unit time and unit area:

$\displaystyle\Phi^{\rm AQN}_{\gamma}({\rm tot.})\approx\frac{A}{2}\Phi^{\rm
AQN}_{a}\cdot\left[\frac{gB}{m_{a}}\right]^{2}\cdot(m_{a}L).$ (13)

Numerically, this estimate can be presented as follows 666We use the
conversion factor for the magnetic field when $\rm Gauss\approx 1.95\cdot
10^{-2}{\rm eV^{2}}$ in Heaviside-Lorentz units when
$\alpha^{2}=e^{2}/(4\pi)$, see e.g. Appendix A in [18].:

$\displaystyle\Phi^{\rm AQN}_{\gamma}({\rm tot.})\approx
0.1A\left(\frac{g_{\gamma}}{0.97}\right)^{2}\left(\frac{B}{10T}\right)^{2}\left(\frac{L}{m}\right){\rm\frac{ph.}{m^{2}day}},~{}~{}~{}~{}$
(14)

which implies that a detector with area $\sim 10~{}m^{2}$ could see a few
microwave photons a day. This rate, of course, is quite low. However, recent
technological advances suggest that such photons can be, in principle,
detected using e.g. single photon counters. We refer to a recent talk by
Sonnenschein at the axion workshop [57] on broadband technology in axion
searches.

What is important here is that this rate could get a significant boost for a
short period of time when factor $A(t)$ could be much larger (than an average
value $\langle A\rangle=1$) resulting from the interaction of the AQN hitting
the Earth in close vicinity of a detector, see Tab. 2 with the expected
frequency of possible enhancements. For example, one should expect a daily
burst with enhancement $A\sim 10^{2}$ which lasts for a second as shown in
Tab. 2. The rarer bursts are shorter in duration but could be much more
powerful and pronounced.

Another point here is that the average rate is low, however, no detector
tuning is required in contrast with conventional resonance-type experiments.
Furthermore, as we already mentioned, the AQN-induced axion DM flux (Eq. 4)
does not depend on the axion mass $m_{a}$ in contrast with the conventional
axion production mechanisms when $\rho_{\rm DM}\sim m_{a}^{-7/6}$ (see
footnote 4). As a result, the rate in Eq. 14 also does not depend on $m_{a}$
and this holds for the entire allowed axion window
$m_{a}\in(10^{-6}-10^{-3})~{}\rm eV$.

The conversion rate in Eq. 8 allows us to estimate the excess of power due to
the emitted microwave photons. First, we have to multiply the axion density
(Eq. 3) by the conversion rate in Eq. 8. Secondly, one should average over
many cycles such that $\sin^{2}(\frac{qL}{2})\rightarrow 1/2$ and insert the
enhancement factor $(Lm_{a})\gg 1$ as explained after Eq. 12. Third, to
estimate the released energy per unit of time one should divide the obtained
expression to a typical formation time of the elementary process, $\Delta
t\sim m_{a}^{-1}$. Collecting all these factors together we arrive at the
following estimate for the power excess $P^{\rm AQN}_{\gamma}({\rm tot.})$ due
to the emitted microwave photons in unit volume:

$\displaystyle P^{\rm AQN}_{\gamma}({\rm tot.})\approx
A(t)\left(\frac{\langle\rho^{\rm AQN}_{a}\rangle
m_{a}}{2}\right)\left(\frac{gB}{m_{a}}\right)^{2}(m_{a}L).~{}~{}~{}~{}$ (15)

This estimate can also be represented in a conventional way as an excess of
power in unit volume measured in $W/m^{3}$:

$\displaystyle P^{\rm AQN}_{\gamma}({\rm tot})$ $\displaystyle\approx$
$\displaystyle
10^{-26}A(t)\left({\rm\frac{Watt}{m^{3}}}\right)\left(\frac{g_{\gamma}}{0.97}\right)^{2}$
$\displaystyle\times$ $\displaystyle\left(\frac{B}{10~{}\rm
Tesla}\right)^{2}\left(\frac{L}{m}\right)\left(\frac{m_{a}}{10^{-4}\rm
eV}\right)^{2}.$

This estimate can be compared with the conventional computations for the power
excess $P_{a}$ in cavity resonance-designed detectors when $P_{a}\propto
10^{-26}Q_{\alpha}\rm(W/m^{3})$, see [18] for a review. Numerically, the value
presented in [18] is similar to our estimate (Eq. V.1) though the conversion
rates and the magnitudes for the axion densities of the AQN-induced and the
galactic axions are very different. This numerical similarity will be used in
what follows for illustrative purposes in our comparisons of the broadband
strategy with conventional resonance-type searches.

The key element in resonance searches is the so-called “quality factor”
$Q_{\alpha}\sim 10^{6}$ which increases the likelihood of the detection of the
axion signal. Indeed, the power excess $P_{a}$ in the conventional analysis is
normally compared with the noise power $P_{\rm noise}$ such that the signal-
to-noise ratio is determined by the formula

$\displaystyle\frac{S}{N}\approx\frac{\sqrt{N}P_{a}}{P_{\rm
noise}}\approx\frac{P_{a}}{T}\sqrt{\frac{t}{\Delta\nu}},~{}P_{\rm
noise}=T\Delta\nu,~{}N\approx t\Delta\nu$

where $T$ is the total system noise temperature, $t$ is the integration time
and the $\Delta\nu$ is the bandwidth of the axion signal. The “quality factor”
$Q_{\alpha}\sim 10^{6}$ plays a crucial role in this approach. The basic idea
here is to detect the narrow resonance line with $\Delta\nu\simeq 10^{-6}\nu$
with sufficiently long integration time by scanning each narrow frequency band
to increase the signal-to-noise ratio.

However, the formula for the power excess $P^{\rm AQN}_{\gamma}({\rm tot})$,
as given by Eq. V.1, does not carry the enhancement factor $Q_{\alpha}\sim
10^{6}$ for the case of non-resonance conversion of the relativistic axions
with $\Delta\nu\sim\nu$. Therefore, the entire logic of collecting the signal
and discriminating it from the noise is very different from the conventional
treatment of the cavity-type experiments briefly reviewed above. The key point
for our studies is that there is a well-defined modulation factor $A(t)$ which
allows us to discriminate the signal from the spurious events and noise by
collecting the signal over entire frequency bands instead of scanning in a
single narrow frequency band in cavity-type experiments. In particular, one
should expect the daily modulation as discussed in the previous section.

The absence of the quality factor $Q_{\alpha}\sim 10^{6}$ in Eq. V.1 implies
that the broadband search is not particularly sensitive to the axion mass,
except for a basic normalization factor $\sim m_{a}^{2}$ resulted from a large
number of “layers” along the path of the relativistic axion, as explained
above. This should be contrasted with conventional resonance type searches
when the cavity is designed to be highly sensitive to a very specific narrow
frequency band expressed in terms of $Q_{\alpha}$. This feature (absence of
the resonance sensitivity to $m_{a}$) is of course a direct manifestation of
the non-resonance axion-photon conversion in the magnetic field when the
resonance features of the cavity do not manifest themselves in expression for
the power excess as given by Eq. V.1.

Therefore, the only option we are aware of to determine the axion mass $m_{a}$
in the broadband setting is to determine the frequency of a single photon
recorded by a non-resonant axion photon converter. As mentioned after Eq. 14
such technologies that could detect a single photon in this frequency band may
indeed become available soon.

We would like to illustrate the arguments presented above with the following
numerical estimates. In conventional resonance type searches the factor
$N\approx t\Delta\nu\approx 10^{5}$ for $\Delta\nu\simeq 10^{-6}\nu$ and
$\nu\sim\rm 1~{}GHz$ such that $\delta P_{\rm noise}\approx P_{\rm
noise}/\sqrt{N}\approx 0.3\cdot 10^{-2}P_{\rm noise}$ for $t\approx 1$ min
which is a typical integration time in resonance searches.

It should be contrasted with broadband strategy when one collects a signal in
the entire frequency band $\Delta\nu\sim\nu\sim\rm 1~{}GHz$ during the entire
year (separately for each hour during the day). In this case the integration
time $t\approx 10^{6}$ s and $N\approx t\Delta\nu\approx 10^{15}$ which is
more than sufficient to overcome the absence of the quality factor $Q$ in the
broadband Eq. 13, Eq. 15, and Eq. V.1.

There are a few additional important elements that may increase the likelihood
of the axion discovery expressed in terms of parameter $A(t)$ which could be
several orders of magnitude larger than its average value for a very short
period. Furthermore, the estimates in Eq. 13 and Eq. 15 explicitly show the
presence of enhancement factor $(Lm_{a})\gg 1$ which counts the number of
“layers” along the relativistic axion’s path, which is a very distinct element
in comparison with conventional searches for non-relativistic axions. This
feature implies that if an instrument is asymmetric, e.g. $L_{x}\gg L_{y}$ it
automatically becomes sensitive to the direction of the axion’s momentum, i.e.
the instrument potentially becomes directionally sensitive to the AQN-induced
axion flux. Therefore, potentially such an instrument allows us to study the
DM distribution locally, in our galaxy beyond the conventional Standard Halo
Model (SHM) paradigm. A hope is that these novel elements in data analysis can
overcome the absence of large numerical factor $Q_{\alpha}\sim 10^{6}$ which
is normally present in the cavity resonance type studies.

###  V.2 The reasoning behind the expected daily modulation

Now we turn our attention to the key element of the proposal of this work.
AQN-induced axions manifest themselves with the systematic effect of possible
daily modulations as given by Eq. 6. This implies that the count of the
emitted microwave photons as given by Eq. 14 or an excess of power as given by
Eq. V.1 can be accumulated during a long period of time (measured in years).
The presence of a signal can be analyzed by fitting to the daily modulation
Eq. 6 by summation over all days over the entire season during a specific hour
and fitting the resulting power excess as a function of time (measured in
hours) during 24 hours period. Any other observables generated by the axions
such as induced currents or induced voltage in the pickup loop (overviewed in
previous Sect. IV) must also show the same daily modulations irrespectively to
a specific recording method of a signal as the expression for $A_{\rm(d)}(t)$
given by Eq. 6 remains the same for all observables.

While the rate of emission (or excess of power) is very low the total number
of events collected during several years could be sufficiently high to see the
daily modulations if they are present in the data. What is more important is
that the corresponding studies can easily discriminate a genuine signal from
spurious signals and noise.

The simplest test for such discrimination is to check that the phase
$\phi_{0}$ in the fitting Eq. 6 does not stay the same during the year.
Instead, it must slowly vary during the annual cycle. In particular, the phase
$\phi_{0}$ should assume the value $\phi_{0}\simeq\pi$ for daily modulations
for two opposite seasons (e.g. summer vs winter) when the maximum becomes the
minimum and vice versa. This test, based on the comparison of the frequency
dependence in daily and annual modulation data, obviously requires much longer
observation time, measured in years. However, it is a highly nontrivial test
because all systematic errors are vastly different for the daily and annual
modulations.

Another simple test is to analyse the data sets with
$B=$0\text{\,}\mathrm{T}$$. The corresponding analysis for $B=0$ must generate
very different from $B\neq$0\text{\,}\mathrm{T}$$ results. The corresponding
$B=$0\text{\,}\mathrm{T}$$ and $B\neq$0\text{\,}\mathrm{T}$$ results should be
very different to discriminate a true genuine signal from noise.

The same daily modulations can be analysed by performing the Fourier
transform, in which case one should see a single well-defined peak at
$\Omega_{d}=2\pi\,\rm day^{-1}$ with a large number of possible smaller peaks
with randomly distributed frequencies representing the noise and spurious
effects.

Repeating the same procedure of the Fourier transform at $B=0$ should not
generate a well-defined peak at $\Omega_{d}=2\pi\,\rm day^{-1}$. Instead, the
analysis of the $B=$0\text{\,}\mathrm{T}$$ data sets should generate a
randomly distributed mountain range, rather than a well-defined single peak.

A more powerful test to exclude a spurious signal is based on the idea of
using some kind of network of synchronized instruments to study correlated
signals, see the original papers [23, 53] for the details.

There may be many background systematic fluctuations (due to temperature
fluctuations, tides, atmospheric phenomena, human-related activities, etc)
that could mimic the daily modulations studied above. The corresponding
spurious signals could be in principle discriminated from the genuine axion
signals by analysing the sets with $B=0$ and the sets at a different season to
observe the phase shift $\phi_{0}$ in the fitting Eq. 6. It is very hard to
imagine that any spurious signals and noise could generate all the features
similar to the AQN-induced axions described above.

Our previous comments essentially imply that the cavity-type experiments which
are to date the only experiments to probe the particularly interesting region
of parameter space corresponding to standard QCD axion models with
$10^{-6}{\rm eV}\lesssim m_{a}\lesssim 10^{-3}{\rm eV}$, can be used to study
the daily modulation as described above. These instruments can be viewed as
non-resonance axion-photon converters in the background of a large magnetic
field.

One should also add that any axion search instrument presently operating can,
in principle, analyze the daily modulation along the lines described above.
This might as well include all previously recorded data sets. In other words,
such studies can be carried out by any haloscope irrespective of the
conventional searches based on resonance scanning as the basic idea is to
combine entire data sets (let us say collected during a specific month when
phase $\phi_{0}$ from Eq. 6 can be assumed to be constant) for each given hour
to see if the data show any daily modulation.

In the present work, we implement this idea by analyzing the data sets
collected by the CAST-CAPP detector from Sept. 2019 to Jun. 2021. At this
stage, it should be considered as a proof of principle demonstration.

##  VI Data analysis

CAST-CAPP is an axion haloscope that operated from Sept. 2019 to Jun. 2021 at
CERN in Geneva, Switzerland. It consists of four tuneable $23\times 25\times
390$$\mathrm{mm}$ rectangular cavities, each with a volume of
$V=$224\text{\,}{\mathrm{cm}}^{3}$$ which were placed inside one of the two
bores of CAST’s superconducting dipole magnet of $8.8\text{\,}\mathrm{T}$.
CAST-CAPP introduced two novel mechanisms to increase its sensitivity to DM
axions but also to transient events:

  * •

The phase-matching technique combines the power outputs from the four
identical cavities coherently to increase the SNR linearly with the number of
cavities

  * •

The state-of-the-art fast-tuning mechanism with a speed of
$10\text{\,}\mathrm{MHz}\text{/}\mathrm{min}$ and a resolution of
$100\text{\,}\mathrm{Hz}$, together with wideband electronics to allow for a
search for transient events such as axion streams and axion miniclusters.

CAST-CAPP acquired 172 days of data using both single and phase-matched
cavities in data-taking conditions. The total covered frequency range extended
from $4.77\text{\,}\mathrm{GHz}5.43\text{\,}\mathrm{GHz}$ covering a parameter
phase space of $\sim$$660\text{\,}\mathrm{MHz}$ corresponding to axion masses
between $19.74\text{\,}\mathrm{\SIUnitSymbolMicro
eV}22.47\text{\,}\mathrm{\SIUnitSymbolMicro eV}$. This allowed CAST-CAPP to
exclude axion-photon couplings for virialized galactic axions down to
$g_{a\gamma\gamma}=8\times 10^{-14}\,${\mathrm{GeV}}^{-1}$$ at $90\%$
confidence level [58].

The design and the data-taking scheme of the CAST-CAPP experiment were not
originally optimized for a search of daily modulated signals. However, the
large acquired bandwidth of $5\text{\,}\mathrm{MHz}$ compared to the resonance
width of about $200\text{\,}\mathrm{kHz}$, together with the wide frequency
range of $660\text{\,}\mathrm{MHz}$, allows a first proof of principle
demonstration of such a novel analysis procedure.

###  VI.1 Simulation

The AQN signal search is simulated using artificially modulated signals
incorporated into simulated spectra. These spectra are composed of randomly
generated noise superimposed onto authentic spectral baselines acquired from
CAST-CAPP. The CAST-CAPP data are recorded in $1\text{\,}\mathrm{min}$ bins
which are then converted into frequency-domain using a Fast Fourier
Transformation (FFT). Each spectra carry a unique shape originating from the
$TE_{101}$ resonance mode of the cavities and the amplifier gain profile. To
preserve this unique shape in the simulated spectra, we produce spectral
baselines using a low-pass filter. Each simulated spectrum is comprised of
artificial Nyquist noise multiplied by a randomly selected spectral baseline.
The artificial noise has a Gaussian distribution of $\mu=1$ and
$\sigma=1/\sqrt{\text{resolution bandwidth}\cdot\text{spectrum time
length}}=0.018$. Then, the artificial daily modulated signal is simulated
using a sine function of tuned amplitude added on top of the artificial noise
spectrum.

We performed multiple simulations with the data-taking time ranging from 1 day
to 20 days at a single frequency and an AQN signal amplitude ranging from
$0.01\%$ to $3.5\%$ of the mean spectral power. Fig 2 shows the combinations
of simulation parameters i.e. number of data taking days and AQN amplitude
percentage with respect to the mean spectral power on a 2D space while the
color denotes if that specific combination yielded a significant signal that
can be detected on a Lomb Scargle periodogram. The confidence level for the
significance is set to be $90\%$. In Fig. 3, a periodogram for a significant
AQN signal is shown for 5 days of data taking. In this case, the power of the
AQN signal is adjusted to be $0.8\%$ of the mean spectrum power.

Figure 2: Simulation of AQN signals with different power percentages as a
function of data taking time. The red points indicate discoveries while the
blue ones denote insignificant signals. A decision boundary fit is denoted by
the black dashed line. Figure 3: Periodogram created using the Lomb Scargle
method with 5 days of simulated data taking time (5 days / 1 min = 7200
simulated spectra). The observed signal is above 90% CL.

###  VI.2 Data selection

During its $\sim$$2$ years of operation, CAST-CAPP recorded 64.3 days of data
using phase-matched cavities and 107.5 days of data using a single cavity. At
the same time, a total of 16.4 days of data were recorded with
$B=$0\text{\,}\mathrm{T}$$. The total frequency range of CAST-CAPP is
$644\text{\,}\mathrm{MHz}$ between
$4.79\text{\,}\mathrm{MHz}5.43\text{\,}\mathrm{MHz}$.

The strategy for quick identification of parasitic environmental background
electromagnetic signals involved the usage of an external omnidirectional
antenna that was placed outside the CAST magnet and was connected to a
spectrum analyzer recording data simultaneously with the cavities. This
EMI/EMC identification system was introduced in Nov. 2020 and collected $89.2$
d of data.

For the search for a daily modulation with the CAST-CAPP data, only the data
acquired from single cavities are used. The reason is that for the phase-
matching procedure, a series of programmable phase-independent attenuators
were used to adjust the amplitude of the resonance of the individual cavities
with $0.25\text{\,}\mathrm{dB}$ to match the resonance peaks of the phase-
matched cavities. This introduced additional attenuation although it is not
expected to follow a daily rhythm, it could introduce unnecessary artifacts in
the whole procedure.

Additionally, the data that were recorded during the usage of the fast-
frequency scanning search approach are omitted from the analysis due to the
inherent frequency-dependency of the amplitude of the resonance peak of the
$TE_{101}$ mode.

Therefore, as a first approximation only data taken from single cavities at
fixed frequencies are used. Due to the high mechanical $Q$-factor of the
sapphire strips of the tuning mechanisms, which were accidentally acting as a
mechanical tuning fork, a frequency modulation of the electromagnetic mode of
interest was observed in certain frequencies. The data that were dominated by
such mechanical vibrations as well as other undesired non-systematic effects
or possible EMI/EMC interferences were excluded from further consideration
using a series of selection criteria.

###  VI.3 Data treatment

Each CAST-CAPP spectrum that fulfills the aforementioned cuts is averaged
within itself to get the mean spectral power, then a timestamp is attached to
it. This way, a 2D array of mean spectral powers vs corresponding acquisition
timestamps is created. This array is then rebinned hourly to have mean
spectral powers for each hour of data acquisition.

The baseline on which the daily modulation in the CAST-CAPP data is
investigated is expected to fluctuate due to weekly environmental temperature
changes as well as amplifier gain drifts. To remove this effect before the
daily modulation analysis, the data are detrended. For this purpose, a
Savitzky Golay filter is used with a window length of two days and a
polynomial order zero. Fig. 4 shows an example of the mean spectral power
time-series belonging to cavity 3 between 13/08/2020 and 20/09/2020 with
$B=$8.8\text{\,}\mathrm{T}$$ before and after the detrending procedure. The
parameters of the low-pass filter are selected such that the long-range
fluctuations are removed while the investigated daily variations are
conserved. After detrending, the mean spectral variations are around a
baseline that equals 1.

Figure 4: Mean hourly spectral power (black) of cavity 3 between Aug 13 - Sep
20. The blue dashed curve denotes the trend of the data created by the low-
pass filter. The detrended power (red) is calculated by dividing data (black)
by its trend (blue).

###  VI.4 Systematics

CAST-CAPP’s experimental setup consists of a three-stage signal amplification
using HEMT amplifiers. The first amplification for each cavity was performed
close to the critically coupled signal output port at the
$4\text{\,}\mathrm{K}$ stage using a HEMPT low-noise amplifier (LNA) from Low
Noise Factory (LNF-LNC4_8D) providing about $39\text{\,}\mathrm{dB}$
amplification with $2\text{\,}\mathrm{K}$ noise for the frequency of CAST-CAPP
at data-taking conditions. Then, an external amplification of about
$22\text{\,}\mathrm{dB}$ for each of the four cavities is performed using four
room-temperature ZX60-83LN-S+ LNAs. Finally, a third stage amplification is
performed at the output signal after the signal combination using a Miteq
AFD3-0208-40-ST LNA providing about $30\text{\,}\mathrm{dB}$ gain.

Figure 5: Time series (left) and Lomb Scargle periodograms (right) for data
taken between 13/08/2020 and 20/09/2020 with cavity 3 and
$B=$8.8\text{\,}\mathrm{T}$$. The results for the ambient environmental
temperature as well as the cavity temperature are also shown.

Two main parameters that could influence the performance of these amplifiers
are the provided bias voltage as well as the temperature. Regarding the bias
voltage of the room-temperature amplifiers, this was provided using a Hameg
HM7044 power supply. The accuracy of this power supply is on the level of
$10\text{\,}\mathrm{mV}$ which translates into a tiny gain variation of about
$0.007\text{\,}\mathrm{dB}$ or $\sim$$0.2\%$ in terms of power for the second
and third stage amplification. For the first stage amplification, a LNF-PS_3
was used for the biasing ($0.25\text{\,}\mathrm{V}$ and
$7\text{\,}\mathrm{mA}$) for each one of the four LNAs corresponding to the
four cavities. The biasing conditions of the LNAs were optimized for minimal
heat dissipation and minimum noise figure. A current variation of about
$0.02\text{\,}\mathrm{mA}$ was observed which would translate into a variation
of the gain on the level of $0.02\text{\,}\mathrm{dB}$ or $\sim 0.5\%$ in
terms of power.

The temperature of the first stage LNA was monitored through Cernox
temperature sensors to be varied between
$6\text{\,}\mathrm{K}10\text{\,}\mathrm{K}$. For the specific bias of about
$0.25\text{\,}\mathrm{V}$ and $7\text{\,}\mathrm{mA}$ at a frequency of
$\sim$$5\text{\,}\mathrm{GHz}$ the gain of LNF-LNC4_8D has a linear dependency
on temperature, increasing the gain of about $0.25\text{\,}\mathrm{dB}$ and an
increase of their noise temperature $0.42\text{\,}\mathrm{K}$ for the
aforementioned temperature range [59]. This translates into a $\sim 5.9\%$
variation of the gain in terms of power.

The temperature and humidity of the experimental area of CAST were monitored
and kept stable via a central ventilation and temperature control system.
However, small changes of a few degrees were observed. More specifically, the
temperature variation observed in the room-temperature electronics of CAST-
CAPP during the whole data-acquisition period was measured using PT100 RTD
temperature sensors to be between
$19\text{\,}\mathrm{\SIUnitSymbolCelsius}26\text{\,}\mathrm{\SIUnitSymbolCelsius}$.
These small temperature changes result in a linear gain dependency of the
room-temperature amplifiers. For the second stage amplification a temperature
change between
$19\text{\,}\mathrm{\SIUnitSymbolCelsius}26\text{\,}\mathrm{\SIUnitSymbolCelsius}$
results in a decrease of the gain of about $0.078\text{\,}\mathrm{dB}$ or
about $1.8\%$ decrease in power [60]. Then, the third stage amplifier is
expected also to have a decrease of its gain with increasing temperature of
about $0.02\text{\,}\mathrm{dB}\text{/}\mathrm{\SIUnitSymbolCelsius}$ [61].
Therefore, for
$19\text{\,}\mathrm{\SIUnitSymbolCelsius}26\text{\,}\mathrm{\SIUnitSymbolCelsius}$,
the third-stage room-temperature amplifier would exhibit a decrease of
$\sim$0.14\text{\,}\mathrm{dB}$$ or about $3.3\%$ decrease in terms of power.

It is noted that the gain stability of the amplifiers over time is expected to
have a much smaller variation than the changes in temperature, voltage, or
other ambient conditions. An overview of the aforementioned systematics and
their effect on the gain of the various amplifiers of CAST-CAPP is shown in
Tab. 3.

| 1st stage | 2nd stage | 3rd stage
---|---|---|---
Temperature | $+5.9\%$ | $-1.8\%$ | $-3.3\%$
Bias | $-0.5\%$ | $+0.2\%$ | $+0.2\%$
Table 3: Systematic effects on the variation of power from the various
amplification stages of CAST-CAPP. The $\pm$ signs indicate a positive or a
negative correlation with the corresponding parameter.

Finally, the magnetic field of CAST’s dipole magnet was measured to have a
stability of $0.1\text{\,}\mathrm{mT}$ at the level of
$8.8\text{\,}\mathrm{T}$, translating into an insignificant $0.001\%$
variation.

Figure 6: Time series (left) and Lomb Scargle periodograms (right) for data
taken between 13/07/2020 and 13/08/2020 with cavity 2 and
$B=$0\text{\,}\mathrm{T}$$. The results for the ambient environmental
temperature as well as the cavity temperature are also shown.

###  VI.5 Results

####  VI.5.1 Lomb Scargle periodogram

The first step for a search for daily periodicities is performed using a Lomb
Scargle periodogram in the detrended data. This way periodicities around
$24\text{\,}\mathrm{h}$ can be investigated. Fig. 5 shows the time series (top
left) as well as the periodogram (top right) for $B=$8.8\text{\,}\mathrm{T}$$,
where the time-period from 13/08/2020 to 20/09/2020 is selected for single
cavity 3. At the same time, using the same period, the environmental
temperature from the PT100 RTD temperature sensors (Fig. 5 middle row) as well
as the cavity temperature from the Cernox sensors (Fig. 5 bottom row) are
shown. Tab. 4 gives a detailed account of the data statistics. The observed
gaps in the time series for both the cavity power and the cavity temperature
data are due to CAST-CAPP’s data acquisition scheme not being optimized for a
daily modulation search, i.e. CAST-CAPP acquired single-cavity data for a
limited and non-continuous amount of time. As the time series data were
detrended and normalized, the y-axes are given in arbitrary units. In the
periodograms (Fig. 5 right) that show the power spectral densities of the
corresponding time series, the red dashed vertical line denotes the daily
periodicity whereas the black dashed horizontal line shows the threshold for a
significant signal with a confidence level of $90\%$.

|  $B=$8.8\text{\,}\mathrm{T}$$ (Fig. 5) |  $B=$0\text{\,}\mathrm{T}$$ (Fig. 6)
---|---|---
Interval | Aug 13 - Sep 20 | Jul 13 - Aug 13
Data amount (1-min files) | 14400 | 11500
Total hours | 240 | 192
Hours per day | 8.0 | 5.2
Frequency range [$\mathrm{GHz}$] | 5.20 - 5.33 | 5.32 - 5.41
Table 4: Data statistics for Fig. 5 and Fig. 6.

As expected the environmental temperature follows a daily rhythm while the
cavity, and therefore the cryogenic LNA temperature, does not. Interestingly,
also the detrended cavity data for $B=$8.8\text{\,}\mathrm{T}$$ show a
$24\text{\,}\mathrm{h}$ periodicity above $90\%$ confidence level. However,
the ambient temperature change can be the leading cause behind the observed
daily variation due to its direct effect on the receiver chain, as mentioned
in the previous section. This is due to the room-temperature setup not being
sufficiently isolated from temperature changes.

For comparison reasons also $B=$0\text{\,}\mathrm{T}$$ data are analyzed. For
cavity 2, these are shown in Fig. 6 with the selected period ranging from
13/07/2020 to 13/08/2020. For practical reasons, this period could not be the
same as the period for the $B=$8.8\text{\,}\mathrm{T}$$, and therefore, a
perfect comparison between the two can not be made. The time difference
between the two is about 1 month. As can be seen, these data with
$B=$0\text{\,}\mathrm{T}$$ also indicate a daily modulation being less
pronounced than the $B=$8.8\text{\,}\mathrm{T}$$ data, which may be attributed
to the $B=$0\text{\,}\mathrm{T}$$ data being more sparse compared to
$B=$8.8\text{\,}\mathrm{T}$$ data. Therefore, we conclude that the appearance
of similar daily modulation in the detrended power spectra of Figs. 5
($B=$8.8\text{\,}\mathrm{T}$$) and 6 ($B=$0\text{\,}\mathrm{T}$$), does not
allow to exclude systematics for the hint in Fig. 5 but strengthens the
assumption of a temperature-dependent amplifier gain effect. Of course, there
is always the possible dark photon contribution with no magnetic field
dependence.

####  VI.5.2 Hourly distributions

Next, the hourly distributions of both $B=$8.8\text{\,}\mathrm{T}$$ and
$B=$0\text{\,}\mathrm{T}$$ data are shown in Fig. 7 which could provide more
information on the observed daily modulation. These distributions are derived
for the same time intervals as in Fig. 5 and Fig. 6, and are shown in the form
of box plots in Fig. 7a, top and bottom, respectively. A sine function is then
fitted to the data. The maxima of the fitted sinus shapes are found around
05:58 and 05:19 local hour, for $B=$8.8\text{\,}\mathrm{T}$$ and
$B=$0\text{\,}\mathrm{T}$$ respectively.

Figure 7: Box plots showing the hourly distributions of the normalized power
variation (left) and normalized environmental temperature variation (right)
for $B=$8.8\text{\,}\mathrm{T}$$ (top) and $B=$0\text{\,}\mathrm{T}$$
(bottom). A sine function with a daily periodicity is fitted to the data. The
gap in the upper left plot is due to the CAST-CAPP data acquisition schedule
in that time interval of data taken.

Even assuming the sine-wave fit of $8.1\%\pm 0.6\%$ and $6.0\%\pm 0.5\%$ for
the mean daily distributions of $B=$8.8\text{\,}\mathrm{T}$$ and
$B=$0\text{\,}\mathrm{T}$$ respectively, the CAST-CAPP data do not allow for a
significant conclusion in favor of the AQN-related axion signal.

The hourly temperature distributions corresponding to Fig. 7a are shown in
Fig. 7b where we observe that the temperature is maximum at around 18:30 CET.
This corresponds approximately to half a day phase shift between the mean
spectral power and the ambient temperature variations shown in Fig. 5 and can
be explained by the argumentation in Sect. VI.4.

Tab. 5 gives an overview of the sine fit characteristics for the box plots
shown in Fig. 7. The sine fits, as shown by chi-squared tests, are
significantly more suitable to the data compared to linear fits, confirming
the daily periodicity observed. However, it is apparent that the daily change
in the environmental temperature of the CAST hall does not allow for a
positive conclusion.

| Cavity | Environmental
---|---|---
| power | temperature
| Fluctuation | 8.1% $\pm$ 0.6% | 3.7% $\pm$ 0.3%
B=8.8T | Max hour | 5:58$\pm$3:05 CET | 18:37$\pm$1:15 CET
|  $\chi^{2}$/dof | 7.2(sine), 19.2(line) | 2.6
| Fluctuation | 6.0% $\pm$ 0.5% | 3.4% $\pm$ 0.3%
B=0T | Max hour | 5:19$\pm$2:33 CET | 18:21$\pm$0:23 CET
|  $\chi^{2}$/dof | 0.2(sine), 8.3(line) | 0.21
Table 5: Data statistics for Fig. 7.

Finally, as described in [58], an external quasi-omnidirectional antenna was
placed next to the CAST magnet and operated simultaneously with the SA
connected to the cavities and at the same frequency band. This was used as a
fast veto procedure where a candidate axion signal could be easily discarded
if it appeared both in the cavities and in the witness channel at the same
frequency. Therefore as an additional comparison, the hourly distribution of
the external antenna data is shown in Fig. 8. As expected due to the lack of
any temperature isolation, the mean daily spectral variation observed is about
$21\%$, larger compared to both $B=$8.8\text{\,}\mathrm{T}$$ and
$B=$0\text{\,}\mathrm{T}$$ data acquired by a receiver chain that is partially
temperature isolated.

Figure 8: Box plot of the external antenna spectral power.

####  VI.5.3 Seasonal phase shift

Due to the limited amount of data taken with CAST-CAPP, the expected seasonal
phase shift (see Sect. V.2) has been searched for, but it could not be
conclusively identified. The periods 13/08/2020 - 20/09/2020 and 18/11/2020 -
02/12/2020 have been analyzed, where the expected phase shift is only of the
order of $0.3\pi$ or 3 hours and 40 minutes, however, due to the limited
statistics, this number could not be determined.

##  VII Conclusion and future studies

In this work, as a proof of principle study, we have re-analyzed the CAST-CAPP
data already published in [58], focusing on the daily modulation of a
potential axion signal due to DM axions or from AQNs as it has been presented
in Sect. V. We have observed daily variations, but at the same time also for
$B=$0\text{\,}\mathrm{T}$$ (see Fig. 7 bottom) and with the external antenna
(see Fig. 8). Since a similar dependence is expected for the electronic gain
of the amplifier chain of the DAQ system, we could not conclusively determine
an axion-related signal. Additional tests to discriminate a true genuine
signal from the noise or spurious signals cannot be carried out as the CAST-
CAPP detector is no longer operational. Thus, our measurements do not allow us
to conclusively determine a DM axion signal (whatever its origin: from streams
or AQNs), or a signal from the hidden sector. Although the data in Fig. 7 does
show some daily oscillation, we conclude that this could be due to temperature
effects. However, we can formulate some lessons to be learned which can be
useful for future studies and different instruments, yet to be built.

The unavoidable fluctuations of the electronic gain of the amplifier chain, in
radiometers, have led to the invention of the Dicke switch to eliminate such
gain fluctuations of the entire chain, and not only the front-end like for the
noise figure. The inherent switching noise added by this type of radiometer,
can then be mitigated by the use of the correlation-type radiometer [62, 63].
Virtually every modern microwave remote-sensing satellite is based on this
correlation technique, and therefore axion DM experiments can also benefit
from this approach and replace the total power radiometer that is currently
widely used. This approach can in the future allow for a conclusive search for
daily modulations.

Additionally, the best option to discriminate the true genuine signal from the
noise or spurious signals is to analyze the correlations in two or more
different instruments representing a network, as suggested in [23, 53]. The
basic idea is to search for some spikes that occur at (almost) the same
instant in the different instruments of a network of axion detectors. The
typical rate for such correlated events depends on the strength of the spikes
as given in Tab. 2.

Finally, to determine the mass and other parameters of the axion (assuming it
is responsible for the daily modulation yet to be discovered) one should use a
truly broadband instrument.

If future experiments conclude that the AQN-induced axions are responsible for
daily modulation, it would have profound consequences. It would imply that two
long-standing puzzles in cosmology, the nature of the DM and the matter-
antimatter asymmetry of our Universe are simultaneously resolved. These two
puzzles are intimately linked within the AQN framework and an observed signal
cannot be explained by any other model since such a signal could be only
generated by the relativistic AQN-induced axions (which are mostly localized
on the Earth’s surface in contrast with CaB777As we mentioned in Sect. I the
local density of the AQN-induced axions on the Earth’s surface should be at
least four orders of magnitude higher than the density of the cosmologically-
produced CaB axions considered in [26], see paragraph above Eq. 4 for an
estimate. This is because the CMB photon density strongly constrains CaB
density. The spectral features of the AQN-induced axions are also vastly
different from the CaB axions.).

## Acknowledgments

This work has been supported by the Greek General Secretariat for Research
Innovation (GSRI), the National Science and Engineering Research Council of
Canada, the Institute for Basic Science (IBS) under Project Code No.
IBS-R017-D1 of the Republic of Korea, the Spanish Agencia Estatal de
Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under
project FPA-2016-76978-C3-2-P and PID2019-108122GB-C33, by the CERN Doctoral
Studentship programme, the European Research Council (ERC), the DFG Research
Training Group Programme 2044 “Mass and Symmetry after the Discovery of the
Higgs Particle at LHC”, MSE (Croatia) and Natural Sciences and Engineering
Research Council of Canada. We acknowledge support through the ERC under grant
ERC2018-StG-802836 (AxScale project) and ERC-2017-AdG-788781 (IAXO+ project).
Part of this work was performed under the auspices of the US Department of
Energy by Lawrence Livermore National Laboratory under Contract No. DE-
AC52-07NA27344.

## References

  * Peccei and Quinn [1977] R. D. Peccei and H. R. Quinn, Constraints imposed by CP conservation in the presence of pseudoparticles, Phys. Rev. D 16, 1791 (1977).
  * Weinberg [1978] S. Weinberg, A new light boson?, Phys. Rev. Lett. 40, 223 (1978).
  * Wilczek [1978] F. Wilczek, Problem of strong P and T invariance in the presence of instantons, Phys. Rev. Lett. 40, 279 (1978).
  * Kim [1979] J. E. Kim, Weak-interaction singlet and strong CP invariance, Phys. Rev. Lett. 43, 103 (1979).
  * Shifman  _et al._ [1980] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, Can confinement ensure natural CP invariance of strong interactions?, Nucl. Phys. B 166, 493 (1980).
  * Dine  _et al._ [1981] M. Dine, W. Fischler, and M. Srednicki, A simple solution to the strong CP problem with a harmless axion, Phys. Lett. B 104, 199 (1981).
  * Zhitnitsky [1980] A. R. Zhitnitsky, On Possible Suppression of the Axion Hadron Interactions. (In Russian), Sov. J. Nucl. Phys. 31, 260 (1980), [Yad. Fiz.31,497(1980)].
  * Van Bibber and Rosenberg [2006] K. Van Bibber and L. J. Rosenberg, Ultrasensitive Searches for the Axion, Physics Today 59, 30 (2006).
  * Asztalos  _et al._ [2006] S. J. Asztalos, L. J. Rosenberg, K. van Bibber, P. Sikivie, and K. Zioutas, Searches for Astrophysical and Cosmological Axions, Annual Review of Nuclear and Particle Science 56, 293 (2006).
  * Sikivie [2008] P. Sikivie, Axion Cosmology, in  _Axions_ , Lecture Notes in Physics, Berlin Springer Verlag, Vol. 741, edited by M. Kuster, G. Raffelt, and B. Beltrán (2008) p. 19, astro-ph/0610440 .
  * Raffelt [2008] G. G. Raffelt, Astrophysical Axion Bounds, in  _Axions_ , Lecture Notes in Physics, Berlin Springer Verlag, Vol. 741, edited by M. Kuster, G. Raffelt, and B. Beltrán (2008) p. 51, hep-ph/0611350 .
  * Sikivie [2010] P. Sikivie, Dark Matter Axions, International Journal of Modern Physics A 25, 554 (2010), arXiv:0909.0949 [hep-ph] .
  * Rosenberg [2015] L. J. Rosenberg, Dark-matter QCD-axion searches, Proceedings of the National Academy of Science 112, 12278 (2015).
  * Marsh [2016] D. J. E. Marsh, Axion cosmology, Phys. Rep. 643, 1 (2016), arXiv:1510.07633 .
  * Graham  _et al._ [2015] P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner, and K. A. van Bibber, Experimental Searches for the Axion and Axion-Like Particles, Annual Review of Nuclear and Particle Science 65, 485 (2015), arXiv:1602.00039 [hep-ex] .
  * Battesti  _et al._ [2018] R. Battesti _et al._ , High magnetic fields for fundamental physics, Phys. Rept. 765-766, 1 (2018), arXiv:1803.07547 [physics.ins-det] .
  * Irastorza and Redondo [2018] I. G. Irastorza and J. Redondo, New experimental approaches in the search for axion-like particles, Prog. Part. Nucl. Phys. 102, 89 (2018), arXiv:1801.08127 [hep-ph] .
  * Sikivie [2021] P. Sikivie, Invisible Axion Search Methods, Rev. Mod. Phys. 93, 015004 (2021), arXiv:2003.02206 [hep-ph] .
  * Fischer  _et al._ [2018] H. Fischer, X. Liang, Y. Semertzidis, A. Zhitnitsky, and K. Zioutas, New mechanism producing axions in the AQN model and how the CAST can discover them, Phys. Rev. D98, 043013 (2018), arXiv:1805.05184 [hep-ph] .
  * Liang and Zhitnitsky [2019] X. Liang and A. Zhitnitsky, Gravitationally bound axions and how one can discover them, Phys. Rev. D99, 023015 (2019), arXiv:1810.00673 [hep-ph] .
  * Lawson  _et al._ [2019] K. Lawson, X. Liang, A. Mead, M. S. R. Siddiqui, L. Van Waerbeke, and A. Zhitnitsky, Gravitationally trapped axions on Earth, Phys. Rev. D100, 043531 (2019), arXiv:1905.00022 [astro-ph.CO] .
  * Liang  _et al._ [2020] X. Liang, A. Mead, M. S. R. Siddiqui, L. Van Waerbeke, and A. Zhitnitsky, Axion Quark Nugget Dark Matter: Time Modulations and Amplifications, Phys. Rev. D 101, 043512 (2020), arXiv:1908.04675 [astro-ph.CO] .
  * Budker  _et al._ [2020] D. Budker, V. V. Flambaum, X. Liang, and A. Zhitnitsky, Axion Quark Nuggets and how a Global Network can discover them, Phys. Rev. D 101, 043012 (2020), arXiv:1909.09475 [hep-ph] .
  * Zhitnitsky [2003] A. R. Zhitnitsky, ‘Nonbaryonic’ dark matter as baryonic colour superconductor, JCAP 10, 010 (2003), hep-ph/0202161 .
  * Witten [1984] E. Witten, Cosmic Separation of Phases, Phys. Rev. D30, 272 (1984).
  * Dror  _et al._ [2021] J. A. Dror, H. Murayama, and N. L. Rodd, Cosmic axion background, Phys. Rev. D 103, 115004 (2021), [Erratum: Phys.Rev.D 106, 119902 (2022)], arXiv:2101.09287 [hep-ph] .
  * Nitta  _et al._ [2023] T. Nitta _et al._ (ADMX), Search for a Dark-Matter-Induced Cosmic Axion Background with ADMX, Phys. Rev. Lett. 131, 101002 (2023), arXiv:2303.06282 [hep-ex] .
  * Farhi and Jaffe [1984] E. Farhi and R. L. Jaffe, Strange matter, Phys. Rev. D 30, 2379 (1984).
  * De Rujula and Glashow [1984] A. De Rujula and S. L. Glashow, Nuclearites - A novel form of cosmic radiation, Nature (London) 312, 734 (1984).
  * Zhitnitsky [2021] A. Zhitnitsky, Axion quark nuggets. Dark matter and matter–antimatter asymmetry: Theory, observations and future experiments, Mod. Phys. Lett. A 36, 2130017 (2021), arXiv:2105.08719 [hep-ph] .
  * Zhitnitsky [2006] A. Zhitnitsky, Cold dark matter as compact composite objects, Phys. Rev. D74, 043515 (2006), arXiv:astro-ph/0603064 [astro-ph] .
  * Flambaum and Zhitnitsky [2019] V. V. Flambaum and A. R. Zhitnitsky, Primordial Lithium Puzzle and the Axion Quark Nugget Dark Matter Model, Phys. Rev. D99, 023517 (2019), arXiv:1811.01965 [hep-ph] .
  * Singh Sidhu  _et al._ [2020] J. Singh Sidhu, R. J. Scherrer, and G. Starkman, Antimatter as Macroscopic Dark Matter, Phys. Lett. B 807, 135574 (2020), arXiv:2006.01200 [astro-ph.CO] .
  * Santillán and Morano [2021] O. P. Santillán and A. Morano, Neutrino emission and initial evolution of axionic quark nuggets, Phys. Rev. D 104, 083530 (2021), arXiv:2011.06747 [hep-ph] .
  * Lawson and Zhitnitsky [2019] K. Lawson and A. R. Zhitnitsky, The 21cm Absorption Line and Axion Quark Nugget Dark Matter Model, Phys. Dark Univ. 24, 100295 (2019), arXiv:1804.07340 [hep-ph] .
  * Ge  _et al._ [2019] S. Ge, K. Lawson, and A. Zhitnitsky, The Axion Quark Nugget Dark Matter Model: Size Distribution and Survival Pattern, Phys. Rev. D99, 116017 (2019), arXiv:1903.05090 [hep-ph] .
  * Liang and Zhitnitsky [2016] X. Liang and A. Zhitnitsky, Axion field and the quark nugget’s formation at the QCD phase transition, Phys. Rev. D 94, 083502 (2016), arXiv:1606.00435 [hep-ph] .
  * Ge  _et al._ [2017] S. Ge, X. Liang, and A. Zhitnitsky, Cosmological C P -odd axion field as the coherent Berry’s phase of the Universe, Phys. Rev. D 96, 063514 (2017), arXiv:1702.04354 [hep-ph] .
  * Ge  _et al._ [2018] S. Ge, X. Liang, and A. Zhitnitsky, Cosmological axion and a quark nugget dark matter model, Phys. Rev. D 97, 043008 (2018), arXiv:1711.06271 [hep-ph] .
  * Jacobs  _et al._ [2015] D. M. Jacobs, G. D. Starkman, and B. W. Lynn, Macro Dark Matter, Mon. Not. Roy. Astron. Soc. 450, 3418 (2015), arXiv:1410.2236 [astro-ph.CO] .
  * Gorham [2012] P. Gorham, Antiquark nuggets as dark matter: New constraints and detection prospects, Phys. Rev. D86, 123005 (2012), arXiv:1208.3697 [astro-ph.CO] .
  * Ge  _et al._ [2020] S. Ge, M. S. R. Siddiqui, L. Van Waerbeke, and A. Zhitnitsky, Radio impulsive events in quiet solar corona and Axion Quark Nugget Dark Matter, Phys. Rev. D102, 123021 (2020), arXiv:2009.00004 [astro-ph.HE] .
  * Zhitnitsky [2022] A. Zhitnitsky, The mysterious diffuse UV radiation and axion quark nugget dark matter model, Phys. Lett. B 828, 137015 (2022), arXiv:2110.05489 [hep-ph] .
  * Mattila and Väisänen [2019] K. Mattila and P. Väisänen, Extragalactic Background Light: Inventory of light throughout the cosmic history, Contemp. Phys. 60, 23 (2019), arXiv:1905.08825 [astro-ph.GA] .
  * Flores and Primack [1994] R. A. Flores and J. R. Primack, Observational and Theoretical Constraints on Singular Dark Matter Halos, APJL 427, L1 (1994), arXiv:astro-ph/9402004 [astro-ph] .
  * Moore [1994] B. Moore, Evidence against dissipation-less dark matter from observations of galaxy haloes, Nature (London) 370, 629 (1994).
  * Boylan-Kolchin  _et al._ [2011] M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat, Too big to fail? The puzzling darkness of massive Milky Way subhaloes, Mon. Not. R. Astron. Soc. 415, L40 (2011), arXiv:1103.0007 [astro-ph.CO] .
  * Zhitnitsky [2023] A. Zhitnitsky, Structure formation paradigm and axion quark nugget dark matter model, Phys. Dark Univ. 40, 101217 (2023), arXiv:2302.00010 [hep-ph] .
  * Zhitnitsky [2017] A. Zhitnitsky, Solar Extreme UV radiation and quark nugget dark matter model, JCAP 10, 050 (2017), arXiv:1707.03400 [astro-ph.SR] .
  * Raza  _et al._ [2018] N. Raza, L. Van Waerbeke, and A. Zhitnitsky, Solar Corona Heating by the Axion Quark Nugget Dark Matter, Phys. Rev. D 98, 103527 (2018), arXiv:1805.01897 [astro-ph.SR] .
  * Liang and Zhitnitsky [2022] X. Liang and A. Zhitnitsky, ANITA anomalous events and axion quark nuggets, Phys. Rev. D 106, 063022 (2022), arXiv:2105.01668 [hep-ph] .
  * Leane and Smirnov [2023] R. K. Leane and J. Smirnov, Floating dark matter in celestial bodies, JCAP 10, 057, arXiv:2209.09834 [hep-ph] .
  * Liang  _et al._ [2021] X. Liang, E. Peshkov, L. Van Waerbeke, and A. Zhitnitsky, Proposed network to detect axion quark nugget dark matter, Phys. Rev. D 103, 096001 (2021), arXiv:2012.00765 [hep-ph] .
  * Freese  _et al._ [1988] K. Freese, J. A. Frieman, and A. Gould, Signal Modulation in Cold Dark Matter Detection, Phys. Rev. D37, 3388 (1988).
  * Freese  _et al._ [2013] K. Freese, M. Lisanti, and C. Savage, Colloquium: Annual modulation of dark matter, Rev. Mod. Phys. 85, 1561 (2013), arXiv:1209.3339 [astro-ph.CO] .
  * Ioannisian  _et al._ [2017] A. N. Ioannisian, N. Kazarian, A. J. Millar, and G. G. Raffelt, Axion-photon conversion caused by dielectric interfaces: quantum field calculation, JCAP 09, 005, arXiv:1707.00701 [hep-ph] .
  * [57] A. Sonnenschein, Broadband axion searches with coaxial dish antennas, Axions beyond Gen 2, Seattle, January 2021 .
  * Adair  _et al._ [2022] C. M. Adair _et al._ , Search for Dark Matter Axions with CAST-CAPP, Nat. Commun. 13, 10.1038/s41467-022-33913-6 (2022).
  * [59] Low Noise Factory, Private communication.
  * [60] Mini Circuits, Private communication.
  * [61] Narda-MITEQ, Private communication.
  * Tanner  _et al._ [2022] A. Tanner, J. Bosch-Lluis, and P. Kangaslahti, Applications of the pseudo-correlation microwave radiometer, in  _IGARSS 2022 - 2022 IEEE International Geoscience and Remote Sensing Symposium_ (2022) pp. 7241–7244.
  * Menon  _et al._ [2019] A. Menon, M. D. Grady, and T. Weller, A generalized radiometer system equation that includes temperature-dependent system losses, in  _2019 IEEE 20th Wireless and Microwave Technology Conference (WAMICON)_ (2019) pp. 1–4.