# Directional detection of meV dark photons with Dandelion

C. Beaufort  M. Bastero-Gil  A. Catalano  D-S. Erfani-Harami  O. Guillaudin
D. Santos  S. Savorgnano  and F. Vezzu

###### Abstract

This paper presents Dandelion, a new dish antenna experiment searching for
dark photons (DPs) with masses around the meV that will start acquiring data
by the end of 2023. A spherical mirror acts as a conversion surface between
DPs and standard photons that converge to a matrix of $418$ Kinetic Inductance
Detectors cooled down to $150~{}\rm{mK}$. A tilt of the mirror at
$1~{}\rm{Hz}$ moves the expected signal over the pixels thus enabling a
continuous background measurement. The expected signal has two modulations: a
spatial modulation providing a directional signature for the unambiguous
discovery of a DP, and an intensity modulation allowing the determination of
the polarization of the DP. For masses near the meV, the inflationary
production of longitudinal and transverse DPs are mutually excluded, thus the
polarization determination by Dandelion could shed a new light on the
inflation phase of the early universe. A first Dandelion prototype operating
for 30 days would improve by more than one order of magnitude the current
exclusion limits on DPs at the meV mass scale and would probe this region with
an unprecedented discovery potential based on directional detection.

##  1 Introduction

The direct detection of Dark Matter (DM) particles relies on complementary
detection strategies to explore the "galaxy" of postulated DM candidates [1,
2]. While the observation of a DM-like signal in a detector would represent an
important forward step, the proper demonstration that such a signal is
actually associated with local DM remains an additional challenge and is often
left for the future. Directional detection overcomes this limitation by
designing experiments able to measure unambiguous DM signatures
distinguishable from any background. The directional strategy consists in
correlating the measured signal to the detector’s motion through the galactic
DM halo [3, 4].

The dark photon (DP) is a hypothetical massive spin-1 boson predicted in
multiple extensions of the Standard Model (SM) and arising from a new $U(1)$
symmetry under which SM fields are uncharged [5, 6, 7]. There is a large
variety of cosmological production mechanisms, thermal and non-thermal, for
the DP as a suitable cold DM candidate [8, 9, 10, 11, 12, 13]. Besides this
compelling property, the interest in the dark photon is also enhanced as being
an appropriate vector portal to the dark sector [14, 15]. Since the DP is the
gauge boson of an Abelian $U(1)$ symmetry, it interacts with the standard
photon via a kinetic mixing [16]:

${\cal
L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{\chi}{2}F_{\mu\nu}X^{\mu\nu}+\frac{1}{2}m_{X}^{2}X_{\mu}X^{\mu}+eJ_{\mu}^{em}A^{\mu}\,,$
(1.1)

where $A^{\mu}$ is the standard photon with field strength $F^{\mu\nu}$,
$X^{\mu}$ is the DP with field strength $X^{\mu\nu}$, and $J_{\mu}^{em}$ is
the electromagnetic current. The kinetic mixing, which is quantified by the
dimensionless parameter $\chi$, opens the window for detection by the
conversion of a DP into a standard photon.

DPs are constrained experimentally by numerous experiments; we refer to [17]
for a comprehensive review. The kinetic mixing parameter is however poorly
constrained for a DP mass around $1~{}\rm{meV}$. We will see that a DP with a
mass near $1~{}\rm{meV}$ would provide a clear signature for distinguishing
between production mechanisms, hence serving as a probe for early universe
physics. These two main motivations, in parallel with developments of new
detectors, result in an increasing experimental interest for meV DPs. Multiple
experiments with diversified strategies of detection have set limits on DPs
with masses close to the meV [18, 19, 20, 21, 22, 23] and efforts will
continue to explore this region in the near future. In particular, the BREAD
collaboration [24] aims to build a detector with significantly improved
sensitivity compared to existing limits.

In comparison with cited experiments, the main contribution of our detector
lies in its directional feature for identification of the DP111Note that the
BREAD collaboration considers directional detection as a future upgrade [24]..
We rely on the concept of dish antenna for DP searches that was proposed in
[25] and refined in [26, 27, 28] for directional detection. A spherical
metallic mirror acts as a conversion surface from DPs to ordinary photons by
conservation of the parallel electric field at the surface. The energy of the
emitted photons can be considered equal to the mass of the DPs. The DP-induced
photons are emitted almost perpendicularly to the mirror surface and thus
concentrate in a spot at a distance $R$ from the mirror, with $R$ the
curvature radius, where one can place a photon detector. The emission deviates
from the normal of the mirror by an angle $\psi=|\mathbf{p}_{||}|/m$ where
$\mathbf{p}_{||}$ is the DP momentum component parallel to the surface of the
mirror. This deviation results in a spatial modulation of the signal which is
correlated to the detector’s motion through the galactic DM halo, thus
providing a directional signature for an unambiguous DP detection [26].
Assuming that all the local DM is made of DP, the DP-induced photon power at
time $t$ and a position $(x,y)$ from the center of a photon detector placed at
a distance $R$ from the mirror is given by [25]:

$P(x,y,t;E_{\gamma})=\chi^{2}\,\rho_{\mathrm{CDM}}\,\eta(E_{\gamma})\,A_{\mathrm{mirr.}}\,I\big{(}x,y,t)\,\cos^{2}\alpha(t)\,,$
(1.2)

with $\rho_{\mathrm{CDM}}$ the local DM density, $\eta(E_{\gamma})$ the
detection efficiency for a photon of energy $E_{\gamma}$, $A_{\mathrm{mirr.}}$
the surface of the mirror, $I\big{(}x,y,t)$ the normalized signal spread on
the detector, and $\alpha(t)$ the angle between the DP polarization and the
tangent of the mirror.

Figure 1: Schematics of the working principles of the Dandelion experiment.

In this paper, we present Dandelion, a directional dish antenna experiment
being installed at the LPSC (Grenoble, France) scheduled by the end of 2023.
The DP-induced photons are detected on a 2D-matrix of Lumped Element Kinetic
Inductance Detectors[29, 30], hereafter referred as KIDs, cooled down to
$150~{}\rm{mK}$ by means of a cryostat. The working principles of the
Dandelion experiment are presented in Figure 1. Dandelion benefits from an
extensive experience in the detection of meV photons with KIDs from the
experiences NIKA2 [31], KISS [32], and CONCERTO [33] and uses a matrix of KIDs
and a cryostat initially developed for astrophysical purposes. Theoretical
motivation for meV DP as a probe for the early universe is presented in
Section 2. The Dandelion experiment is described in Section 3 with a focus on
the background. Section 4 is dedicated to the modelling of the expected signal
and its modulations. We derive the expected sensitivities in Section 5, both
for exclusion limits and discovery potentials.

##  2 meV dark photons as a window towards the early universe

A population of dark photons that can account for the DM abundace can be
produced through different non-thermal mechanisms as: gravitational production
of longitudinal massive vectors during inflation [9, 34, 35, 36]; parametric
resonance and/or tachyonic instability due to an axionic scalar coupling [10,
11, 12]; production from the dark Higgs dynamics [37, 38], or from the cosmic
strings associated with the spontaneous breaking of the Dark $U(1)$ symmetry
[39, 40]. All of them would allow for a dark matter candidate in the meV range
for the appropriate choices of model parameters, but with different
cosmological implications. As an example we focus on inflationary production,
comparing that of longitudinal and transverse DPs.

Inflationary production of longitudinal modes requires the DP to be a light
massive field during inflation, i.e. $0<m_{X}<H_{I}$, $H_{I}$ being the
typical inflationary scale. Assuming instant or fast reheating, i.e.,
$m_{X}<H_{RH}$, $H_{RH}$ being the Hubble parameter at the end of reheating,
their present abundance $\Omega_{L}$ normalized by the DM abundance today
$\Omega_{DM}$ is given by [9]:

$\frac{\Omega_{L}}{\Omega_{DM}}\simeq\sqrt{\frac{m_{X}}{\rm{meV}}}\left(\frac{H_{I}}{2.7\times
10^{13}\,\rm{GeV}}\right)^{2}\,,$ (2.1)

where the subscript “$L$” refers to the longitudinal polarization. On the
other hand, in the case of a late reheating, i.e., $m_{X}>H_{RH}$, the
abundance depends on the effective equation of state during reheating $w$ [34,
35, 36]:

$\frac{\Omega_{L}}{\Omega_{DM}}\simeq\left(\frac{m_{X}}{0.33~{}\rm{GeV}}\right)^{2w/(1+w)}\left(\frac{H_{I}}{3.3\times
10^{10}\,\rm{GeV}}\right)^{2}\left(\frac{2.1\times
10^{9}~{}\rm{GeV}}{T_{RH}}\right)^{(3w-1)/(1+w)}\,,$ (2.2)

where $T_{RH}$ is the reheating temperature222We have used
$H_{RH}=(\pi^{2}g_{*}/90)^{1/2}T_{RH}^{2}/m_{P}$, with $g_{*}=100$., and one
recovers Eq. (2.1) with $w=1/3$. For $m_{X}\simeq 1$ meV, reheating can affect
the final abundance only for temperatures $T_{RH}\lesssim 840~{}\rm{GeV}$. In
general, for a meV DP, this production mechanism requires large scale
inflation, $H_{I}\sim O(10^{13}-10^{14})$ GeV, in order to explain the present
DM abundance, which in turn may be in tension with the current limit on the
tensor-to-scalar ratio [41] $r\simeq 0.032$, and the upper bound it sets on
the scale of inflation $H_{I}\simeq 4.44\times 10^{13}$ GeV. For example for a
matter dominated late reheating, $w=0$, the final abundance is independent of
the dark photon mass and given by:

$\frac{\Omega_{L}(w=0)}{\Omega_{DM}}\simeq\left(\frac{H_{I}}{5.2\times
10^{13}\,\rm{GeV}}\right)^{2}\left(\frac{T_{RH}}{840~{}\rm{GeV}}\right)\,.$
(2.3)

Having $\Omega_{L}=\Omega_{DM}$ then implies $r\simeq 0.04$, already excluded.
Instant reheating or $m_{X}<T_{RH}$, i.e. $w=1/3$, leads to $r\simeq 0.01$,
while an effective $w=1$ requires $H_{I}\simeq 1.2\times 10^{13}$ GeV and then
$r\simeq 2\times 10^{-3}$. Therefore, this mechanism for longitudinal DP
production implies a tensor-to-scalar ratio that could be determined by future
experiments [42, 43, 44].

On the other hand, inflationary production of transverse modes can be
efficient for smaller values of the inflationary scale, but requires an extra
coupling of the inflaton field to the DP:

${\cal L}_{\rm inf}=-\frac{\alpha_{V}}{f}\phi
X_{\mu\nu}{\tilde{X}}^{\mu\nu}\,,$ (2.4)

where $\phi$ is the inflaton field, $f$ the scale of symmetry breaking, and
${\tilde{F}}^{\mu\nu}=\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2$. This
kind of couplings leads to the tachyonic production of one of the transverse
components of the DP. The final abundance $\Omega_{T}$ depends on the value of
the coupling $\alpha_{V}m_{P}/f=\sqrt{2}\xi_{end}$, and the details of the
reheating period after inflation [10, 45]:

$\frac{\Omega_{T}}{\Omega_{DM}}\simeq 2\times
10^{-13}\left(\frac{m}{\rm{meV}}\right)\left(\frac{H_{end}}{10^{12}\,\rm{GeV}}\right)^{3/2}\left(\frac{10^{-1}}{\epsilon_{R}}\right)^{3}\frac{e^{2\pi\xi_{end}}}{\xi_{end}^{3}}\,,$
(2.5)

where now the subscript “$T$” refers to transverse modes, and the parameter
$\epsilon_{R}=H_{RH}/H_{end}$ models the decrease in the energy density during
reheating. The small pre-factor in Eq.(2.5) can be compensated by an efficient
tachyonic production, i.e., a relative large value $\xi_{end}\gtrsim O(6)$.
For example we get $\Omega_{T}=\Omega_{DM}$ with $\xi_{end}\simeq 6.5$,
$H_{end}\simeq 1.9\times 10^{10}$ GeV, and an efficient reheating
$\epsilon_{R}=0.1$.

The mechanism for transverse DP production works even in the massless limit
during inflation, the symmetry being broken later. But either case, massive or
massless during inflation, we have to impose that indeed the DP becomes non-
relativistic before matter-radiation equality [10, 45], which translates into
an upper bound for the Hubble parameter at the end of inflation:

$H_{end}\lesssim 1.3\times
10^{12}~{}\epsilon_{R}^{2}\left(\frac{m_{X}}{\rm{meV}}\right)^{2}~{}\rm{GeV}\,.$
(2.6)

Therefore this production mechanism for an meV DP leads to a negligible
tensor-to-scalar ratio, $r\lesssim O(10^{-5})$, contrary to the inflationary
longitudinal production.

On the other hand, transverse DP modes will source the tensor components of
tensor perturbations after inflation ends. And given that tachyonic production
of transverse DP only efficiently excites one of the transverse components of
the DP, they lead to the production of parity violating gravitational waves
(GW) [46, 47, 48, 49]. Whether or not this GW spectrum can be detected by
present and near future experiments is still an open question. Large scale
inflationary models, say $H_{end}\simeq O(10^{13})$ GeV, typically give rise
to a GW spectrum peaked at large frequencies, $O({\rm MHz})$, outside present
capabilities of detection. Inflationary models with lower $H_{end}$ may lead
to a smaller frequency peak, but also to smaller amplitudes.

To summarize, as an example of possible production mechanism for a DP with a
mass near the meV, we have compared the inflationary production of
longitudinal and transverse photons. Once we fix the mass parameter, these
mechanisms are mutually excluded, and we cannot have simultaneously both
polarizations at production in our DM population. They will be also related to
different cosmological signatures: longitudinal production leads to a tensor-
to-scalar ratio within the reach of near future proposed missions, whereas
this signature is negligible for transverse DP; but the latter may induce a
signal in the GW spectrum at much larger frequencies. We will show in Section
5 that Dandelion could determine the polarization of the DP, hence allowing to
rule out one of the two inflationary production mechanisms. However,
cosmological evolution of our population of DPs may lead to the conversion of
longitudinal-transverse polarizations, giving rise to a random polarized
background of DPs today. To study how efficient might be the conversion,
depending on the production mechanism, is beyond the scope of this work, and
we will just consider the two limiting cases in the following: either a fixed
polarization, or a random polarization.

##  3 The Dandelion experiment

Dandelion stands for DArk photoN DirEctionaL detectION. Just as the pappus on
a dandelion flies away according to the direction of the wind, Dandelion could
trace its measurements back to the direction of the DP wind due to the motion
of the Earth through the DM halo.

The conversion surface between a DP and a standard photon is a spherical
aluminium mirror. While a large surface would be suitable for improving the
expected DP-induced photon flux on the detector, we here operate a prototype
to experimentally validate the strategy of detection. We consider a spherical
mirror with a curvature radius $R=5~{}\rm{m}$ and an aperture diameter of
$D=50~{}\rm{cm}$. For a spherical mirror, the incident photons parallel to the
optical axis converge to $R/2$, as it is the case for astrophysical
measurements, whereas the photons emitted perpendicularly to the surface focus
at $R$. In other words, the DP-induced photons and the reflected ambient
background photons do not converge to the same positions. We thus place the
detector at a distance $R$ from the mirror. The experiment is located at
Grenoble, France, at a latitude of $45.1^{\circ}$ and a longitude of
$5.7^{\circ}$. The mirror is facing the direction North-West with an angle of
$48^{\circ}$ with respect to the North. These coordinates will play a role in
Section 4 to model the directional signal.

The detector consists of a matrix of KIDs that forms a disk of diameter
$6.5~{}\rm{cm}$ made of $418$ pixels of dimensions $2.8\times
2.8~{}\rm{mm^{2}}$. This matrix, the cryostat to cool it down to
$150~{}\rm{mK}$, as well as the related acquisition soft were previously
developed for other research projects [50]. The detection efficiency of the
KIDs, $\eta(E_{\gamma})$, has been measured; it shows a maximum at
$250~{}\rm{GHz}$ (corresponding to $1~{}\rm{meV}$) and exceeds $50\%$ between
$200$ and $300~{}\rm{GHz}$. Some optics ensure an optimized focus on the KIDs.
The alignment between the detector and the mirror is controlled every day by
means of an optical laser located at the center of the mirror in which a hole
has been drilled. To reduce alignment uncertainties, the mirror is
mechanically tight to the cryostat. Figure 2 presents a CAD model of the setup
and a picture of the matrix of KIDs.

Figure 2: Left: Computer-Aided Design (CAD) of the experimental setup. The
cryostat hosting the detector is tight to the mirror by a mechanical
structure. Right: the mounted matrix made of $418$ LEKID pixels.

The main source of background is the thermal background. The background
contribution is minimized by the optics placed in front of the matrix of KIDs
such as the input pupil is smaller than the mirror at a distance of $R$, as
detailed in Figure 3. The reflected signal on the surface of the mirror is
then at $150~{}\rm{mK}$, i.e. the cryostat’s temperature. From Planck’s law,
one can derive the thermal background, from emission and reflection, in one
single pixel of the detector and thus determine the optical Noise Equivalent
Power (NEP) defined as the signal power required to get a unit signal-to-noise
ratio (SNR) in one Hertz bandwidth. We theoretically obtain a mean NEP of
$3\times 10^{-17}~{}\rm{W/\sqrt{Hz}}$ on any of the $418$ pixels. Sources of
$1/f$-like noise are suppressed by a synchronous modulation technique: the
mirror is tilted by $5~{}\rm{mrad}$ at a frequency of $1~{}\rm{Hz}$ around its
vertical axis (perpendicular to the main axis of the mirror) by means of a
motor, resulting in a shift of $2.5~{}\rm{cm}$ of the expected signal spot.
Such a shift is sufficient to move the signal out of the central pixels while
letting it focused on some other pixels of the matrix of KIDs. The modulation
of the expected signal over the pixels happens at a fast enough frequency so
the background can be considered constant, hence leading to a continuous
measurement of the noise for each pixel which is then subtracted for
extracting a signal.

Figure 3: Dandelion optical design (Zemax software) with details of the cold
optics inside the cryostat. Inverse ray tracing simulation from detectors to
the $50~{}\rm{cm}$ diameter mirror. Detectors illumination is constrained
inside the mirror in order to minimize the thermal loading.

##  4 Signal modelling

In this section, we investigate the features of the expected signal and its
spatial and intensity modulations. Some compromises must be determined in the
design of the experiment to optimize the detection efficiency. For instance, a
larger diameter for the mirror increases the power of the signal while it
deteriorates the spatial resolution depending on where the DP-induced photons
are emitted on the mirror [26]. Following [27], we obtain that such optical
aberrations are negligible for the Dandelion geometry since they result in a
maximal spread of the signal of $0.3~{}\rm{mm}$.

###  4.1 Diffraction

The main limitation for a dish antenna searching for DPs with masses typically
below $100~{}\rm{meV}$ comes from the diffraction. After the emission of a DP-
induced photon, the mirror acts as a source of secondary wavelets which
interfere with the primary wave. According to the Huygens-Fresnel principle,
the electric field on the matrix of KIDs, $E(x,y,R)$, is related to the
electric field emitted at the surface of the mirror,
$E(x^{\prime},y^{\prime},0)$, by [51]:

$E(x,y,R)=\frac{R}{i\lambda}\,\int_{\rm{mirr.}}dx^{\prime}\,dy^{\prime}\,E(x^{\prime},y^{\prime},0)\,\frac{e^{-ikr}}{r^{2}}\,,$
(4.1)

where $k=m/(\hbar c)$ is the wavenumber and $r$ is the distance between a
point with coordinate $(x^{\prime},y^{\prime})$ on the mirror and a point with
coordinate $(x,y)$ on the readout plane, which can be expressed as:

$r=R\bigg{\\{}1+\frac{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}{2R^{2}}-\frac{x^{\prime
2}+y^{\prime
2}}{2R^{2}}+o\Big{(}\big{(}\frac{x}{R}\big{)}^{4}\Big{)}\bigg{\\}}\,.$ (4.2)

Since we are in a near-field configuration, we apply the Fresnel approximation
to obtain:

$E(x,y,R)\simeq\frac{e^{-ikR}}{i\lambda
R}\int_{\rm{mirr.}}dx^{\prime}dy^{\prime}E(x^{\prime},y^{\prime},0)\,\exp\bigg{(}-\frac{ik}{2R}\Big{(}(x-x^{\prime})^{2}+(y-y^{\prime})^{2}\Big{)}\Bigg{)}\exp\bigg{(}\frac{ik(x^{\prime
2}+y^{\prime 2})}{2R}\Bigg{)}\,.$ (4.3)

The last exponential term is an additional phase compared to the expression of
the Fresnel diffraction in textbooks that treat the case of a circular
aperture whereas here we consider a spherical surface. This phase
significantly reduces the influence of the diffraction. The spread of the
signal due to the diffraction is represented in Figure 4 from a numerical
computation of Eq. (4.3) and is comparable to the results of the Zemax
simulation. This spread limits the spatial resolution of the detection: $1/3$
of the signal is contained in the 5 central pixels; and $2/3$ of the signal in
the 13 central pixels.

Figure 4: The diffraction of the signal at the readout plane. Left:
representation 2D. Right: projection along the radial component.

###  4.2 Spatial modulation: the directional signature

Let us now derive the spatial modulation of the signal which will be compared
in Section 5 to the spatial resolution to evaluate the discovery potential of
the detector. The spatial modulation is related to the out-of-normal photon
emission with an angle $\psi=|\mathbf{p}_{||}(t)|/m$ that we determine in two
steps. First, we express $-\mathbf{v}_{\rm{lab}}$, the DP velocity vector in
the mirror frame, accounting for multiple motions: the galactic rotation at
the position of the Sun, the peculiar velocity of the Sun in the galaxy, the
revolution of the Earth around the Sun, and the rotation of the Earth. Second,
we project $-\mathbf{v}_{\rm{lab}}$ along the tangent to the mirror. We deeply
rely on [52] for this analysis. Appendix A extrapolates some equations of
their paper to our configuration and we provide the expression for the spatial
modulation in Eq. (A.11).

A typical spatial modulation is presented in Figure 5 which exhibits the
tendencies that are observed for any input day: the offset variations in X and
Y cover between one et two pixels; the signal modulates with a period of one
day, reaching an extrema every 12h; there is a delay of about 2h between the
extrema in X and in Y. These features are intrinsically derived from the
hypothetical existence of a DM halo surrounding the galaxy, and they cannot be
reproduced by the background. While the detection sensitivity is studied in
Section 5, we can already say that the power induced on a single pixel varies
up to a factor of 4 between two extrema once accounting for the diffraction.

Figure 5: The expected spatial modulation for a 48h-long measurement starting
on the 1st of December 2023. Based on the modelling of [52].

###  4.3 Intensity modulation

The detectable signal modulates in intensity proportionally to the cosine
squared of the angle between the polarization of the DP and the tangent of the
mirror, i.e. the $\alpha(t)$ angle of Eq. (1.2). For the data analysis, we
will integrate the measured signal over a period $T$, so the quantity of
interest becomes $\langle\cos^{2}\alpha(t)\rangle_{T}$.

An important contribution to the modelling of $\alpha(t)$ is presented in
[17]. We decide however to approach the modelling from another point of view
based on the description of the relative motion of the mirror in the galactic
halo, as detailed at the end of Appendix A. This approach simplifies the
connection between the spatial and intensity modulations in our work.
Moreover, we express the DP polarization in the galactic frame whereas it is
expressed in the geocentric frame in [17] as an approximation.

Since the DP polarization is unknown, we will consider the two scenarios
presented in Section 2: the random polarization case and the fixed
polarization case. Figure 6 presents the time evolution of
$\langle\cos^{2}\alpha(t)\rangle_{T}$ and $\cos^{2}\alpha(t)$ for our geometry
based on $10^{5}$ random DP polarizations and computed from Eq. (A.13). In the
figure, the modelling starts on the 1st of December 2023. By systematically
varying the initial time, we can establish the following points:

  * •

While the most probable values of the distribution of
$\langle\cos^{2}\alpha(t)\rangle_{T}$ depend on the initial time, the mean
value only feebly fluctuates. One can conservatively assume
$\langle\cos^{2}\alpha(t)\rangle_{T}\simeq 2/3$ for all $T$ larger than one
hour in the random scenario.

  * •

The instantaneous value of $\cos^{2}\alpha(t)$ presents two minima and two
maxima in $24~{}\rm{h}$. The maxima always correspond to
$\cos^{2}\alpha(t)=1$.

  * •

In the random scenario, the values of the minima feebly evolve depending on
the initial time: one near $0$ and the other near $0.7$.

  * •

In the fixed scenario, the values of the minima significantly depend on the
polarization components.

The intensity modulation presents a detectable signature that, in case of
detection, could enable the identification of the polarization of the DP.
However, since the polarization of the DP is unknown, the intensity modulation
is not a robust enough signature for the directional detection compared to the
spatial modulation.

Figure 6: Time evolution of the probability distribution of the time-averaged
$\langle\cos^{2}\alpha(t)\rangle_{T}$ (left) and of $\cos^{2}\alpha(t)$
(right), where $\alpha(t)$ is the angle between the DP polarization and the
tangent of the mirror. The probability distributions are scaled such that
their maxima are equal to unity. The plots are constructed from $10^{5}$ DP
polarizations that are randomly drawn from an isotropic distribution in the
galactic frame. In the left plot, the distribution of the mean value at each
time is represented as a white curve.

##  5 Expected sensitivities

We now have all the required elements to evaluate the power in Eq. (1.2). We
use $\rho_{\rm{CDM}}=0.45~{}\rm{GeV\,cm^{-3}}$ for consistency with other DP
searches [17]. The signal spread $I(x,y,t)$ embeds both the diffraction and
the spatial modulation and is normalized such that $\int dxdy\,I(x,y,t)=1$ for
each $t$. This quantity and $\cos^{2}\alpha(t)$ are computed numerically to
account for the detector’s motion through the galactic halo.

We rely on a simple and conservative approach for the derivation of the
sensitivities. Finer limits could be obtained in the future, for instance
based on dedicated methods for directional detection [53, 54, 4]. For an
integration time $T$, the DP-induced signal on the pixel of coordinate $(i,j)$
is given by:

$S_{ij}(E_{\gamma})=\frac{1}{T}\int_{0}^{T}dt\int_{\rm{pixel}}dx\,dy\,P(x,y,t;E_{\gamma})\,,$
(5.1)

where the spatial integral only runs over the dimension of the pixel $(i,j)$.
The mirror is tilted by $5~{}\rm{mrad}$ at a frequency of $1~{}\rm{Hz}$ so the
tilt is small enough to keep the expected DP signal inside the matrix of the
$418$ KIDs. In other words, the entire experimental time is used both for
signal and background measurements. Since this paper precedes the data
acquisition, we assume that all pixels experience a constant noise equal to
the NEP, so the background on the pixel $(i,j)$ after an integration time $T$
is:

$B_{ij}\simeq\frac{\rm{NEP}}{\sqrt{T}}\,.$ (5.2)

We define a region of interest (ROI) of the 13 pixels receiving the maximal
expected power. Increasing or reducing the diameter of the ROI, up to a factor
of 2, has negligible influence on the results. A DP-like signal would be
detected if the following requirement is fulfilled:

$\frac{\sum_{i,j}S_{ij}(E_{\gamma})}{\sum_{i,j}B_{ij}}>5\,,$ (5.3)

where $(i,j)$ are located inside the ROI. This condition defines the Dandelion
exclusion sensitivity.

The discovery potential requires an analysis of the spatial modulation. We
define two signal regions corresponding to the extreme deciles of the radius
of the spatial modulation: $S_{ij}^{10}$ integrates the power in pixel $(i,j)$
only for the times that fulfil the condition
$r(t)<0.1(r_{\rm{max}}-r_{\rm{min}})+r_{\rm{min}}$, and $S_{ij}^{90}$ only for
$r(t)>0.9(r_{\rm{max}}-r_{\rm{min}})+r_{\rm{min}}$. Similarly, $B_{ij}^{10}$
and $B_{ij}^{90}$ are the background measurements during the same time
periods. For determining the discovery potential, we only keep the pixels in
which the SNR>5 and we require an unambiguous signal difference between the
two extreme deciles:

$\sum_{i^{*},j^{*}}\bigg{|}1-\frac{S_{i^{*}j^{*}}^{90}}{S_{i^{*}j^{*}}^{10}}\bigg{|}>5\hskip
17.07182pt\rm{with}~{}(i^{*},j^{*})~{}\rm{such~{}that}\hskip
8.5359pt\frac{S^{10}_{i^{*}j^{*}}}{B^{10}_{i^{*}j^{*}}}>5\hskip
8.5359pt\rm{and}\hskip
8.5359pt\frac{S^{90}_{i^{*}j^{*}}}{B^{90}_{i^{*}j^{*}}}>5\,.$ (5.4)

If DPs are detected, we could go one step further by determining if the
polarization is random or fixed. The $S^{90}_{i^{*}j^{*}}$ signal covers a
time period of about 6 hours in which the $\cos^{2}\alpha(t)$ goes from one
maximum to one minimum. We cut this period in 5 timesteps, noted $ts$, such
that in the random scenario $\langle\cos^{2}\alpha(t)\rangle_{ts}\simeq 2/3$.
In the fixed scenario, the evolution of $\langle\cos^{2}\alpha(t)\rangle_{ts}$
depends on the polarization although in most cases it varies by about a factor
of 5 so the intensity modulation might be observed from one timestep to
another. A conservative limit is placed by requiring that the signal
difference between two timesteps exceeds twice the background. This condition
is added to the requirements for the discovery of the DP.

The projected sensitivities of Dandelion for a test campaign of 30 days are
presented in Figure 7. The prototype described in this paper would improve by
more than one order of magnitude the existing limits around the meV DP mass.
It would also provide an unprecedented discovery potential and could
discriminate between two polarization scenarios. Dandelion would probe a
significantly larger parameter space than Tokyo-2 [21] which also exploits the
dish antenna principle without the directional information.

Figure 7: The expected sensitivities for the Dandelion prototype running for
30 days. We show the exclusion limits, the discovery potential, and the
discovery potential discriminating between the polarization scenarios. Figure
adapted from the Github of Ciaran O’Hare [55].

##  6 Conclusion

The directional detection of a DM particle would represent a momentous and
exciting discovery. The dark photon, besides being a suitable DM candidate,
can serve as a portal to a dark sector. While the mass of a DP can cover more
than twenty decades, the meV region is of particular interest since it has few
experimental constraints and that the polarization of a meV DP provides
information about the primordial universe.

In this paper, we present the Dandelion experiment that is currently being
installed at the LPSC (Grenoble, France). Dandelion is an experiment that
benefits from an already operational matrix of KIDs, cooled down to
$150~{}\rm{mK}$ by a cryostat, to detect DP-induced photons. We have shown
that the experiment would be sensitive both to the spatial and intensity
modulations of the signal related to the motion of the detector through the
galactic halo. A small Dandelion prototype and a testing measurement campaign
of 30 days would improve by more than one order of magnitude the exclusion
limits near the meV. If a signal is detected, the directional signature would
check without ambiguity the galactic origin of the signal, hence demonstrating
or not the discovery of a DM particle.

The first campaign of the Dandelion prototype is planned for the end of 2023.
Some near-term updates could improve by several orders of magnitude the
sensitivity of the detector: increasing the measurement period, building a
larger mirror, using multiple matrices of KIDs to cover a larger parameter
space, and improving the statistical treatment of the data. If a signal is
detected, we might determine the DP polarization from the intensity modulation
but also measure it by means of a Martin-Puplett interferometer with only few
modifications to the setup since it was previously used with the matrix of
KIDs at LPSC. Another future upgrade relies on the use of tunable filters to
reduce the probed bandpass. While the KIDs cover a bandwidth $\Delta\nu$, one
can reduce the thermal background while not altering the signal, whose energy
is given by the DP mass, by performing $n$ runs of $\Delta\nu/n$ bandwidths.
Spring is just around the corner for the dandelion to bloom.

## Acknowledgments

The authors thank Christopher Smith, Jérémie Quevillon, François-Xavier Désert
and Fabrice Naraghi for the fruitful discussions at the first steps of the
project. The authors are deeply grateful to Julien Bounmy and Damien Tourres
for their support with the electronics. The authors acknowledge funding from
the French Programme d’investissements d’avenir through the Enigmass Labex.

##  Appendix A Mirror’s motion through the galactic DP halo

We entirely rely on [52] for modelling the motion of the mirror with respect
to the galactic DP halo. This appendix transposes their main results to our
experimental setup. Let us first introduce some coordinate systems. We define
a reference frame for the mirror with unit vectors
$(\hat{\mathbf{X}},\hat{\mathbf{Y}},\hat{\mathbf{Z}})$ where
$\hat{\mathbf{X}}$ and $\hat{\mathbf{Y}}$ lie in the plane of the mirror and
$\hat{\mathbf{Z}}$ is perpendicular to the tangent of the mirror. The origin
of the coordinate system is located at the center of the mirror and we assume
the coordinate system to be right-handed.

We now define a reference frame fixed to the laboratory with unit vectors
$(\hat{\mathcal{N}},\hat{\mathcal{W}},\hat{\mathcal{Z}})$ for which
$\hat{\mathcal{N}}$ points to the North, $\hat{\mathcal{W}}$ to the West, and
$\hat{\mathcal{Z}}$ to the Zenith. The relation between the mirror frame and
the laboratory frame is described by the following coefficients:

$\displaystyle\hat{\mathbf{X}}$ $\displaystyle=$
$\displaystyle\alpha_{X}~{}\hat{\mathbf{\mathcal{N}}}+\beta_{X}~{}\hat{\mathbf{\mathcal{W}}}+\gamma_{X}~{}\hat{\mathbf{\mathcal{Z}}}$
$\displaystyle\hat{\mathbf{Y}}$ $\displaystyle=$
$\displaystyle\alpha_{Y}~{}\hat{\mathbf{\mathcal{N}}}+\beta_{Y}~{}\hat{\mathbf{\mathcal{W}}}+\gamma_{Y}~{}\hat{\mathbf{\mathcal{Z}}}$
(A.1) $\displaystyle\hat{\mathbf{Z}}$ $\displaystyle=$
$\displaystyle\alpha_{Z}~{}\hat{\mathbf{\mathcal{N}}}+\beta_{Z}~{}\hat{\mathbf{\mathcal{W}}}+\gamma_{Z}~{}\hat{\mathbf{\mathcal{Z}}}$

The main axis of the experimental setup, the $\hat{\mathbf{Z}}$ axis, is
perpendicular to the wall of the experimental room and we approximate the
mirror as a disk instead of a spherical surface. In such a case,
$\hat{\mathbf{Y}}$ is oriented along $\hat{\mathbf{\mathcal{Z}}}$. The mirror
is oriented N-W with an angle of $\theta_{NW}=48^{\circ}$ from the North. We
can thus approximate the coefficients:

$\displaystyle\alpha_{X}$ $\displaystyle\simeq-\sin(\theta_{NW})$
$\displaystyle\beta_{X}$ $\displaystyle\simeq\cos(\theta_{NW})$
$\displaystyle\gamma_{X}$ $\displaystyle\simeq 0$ $\displaystyle\alpha_{Y}$
$\displaystyle\simeq 0$ $\displaystyle\beta_{Y}$ $\displaystyle\simeq 0$
$\displaystyle\gamma_{Y}$ $\displaystyle\simeq 1$ (A.2)
$\displaystyle\alpha_{Z}$ $\displaystyle\simeq\cos(\theta_{NW})$
$\displaystyle\beta_{Z}$ $\displaystyle\simeq\sin(\theta_{NW})$
$\displaystyle\gamma_{Z}$ $\displaystyle\simeq 0$

We want to determine $\mathbf{v}_{||}(t)$, the DP-velocity which is parallel
to the surface of the mirror as a function of time. To do so, we define
$\hat{\mathbf{q}}$ a unit vector in the mirror frame that is used to project
$-\mathbf{v}_{\rm{lab}}$ along any direction, $-\mathbf{v}_{\rm{lab}}$ being
the DP velocity vector in the mirror frame. According to [52], one has:

$\displaystyle-\hat{\mathbf{q}}\cdot{\bf v}_{{\rm lab}}=$
$\displaystyle\bigg{\\{}\bigg{[}\cos(t^{\circ}_{\rm
lab})~{}A(t)-\sin(t^{\circ}_{\rm lab})~{}B(t)\bigg{]}\sin\lambda_{\rm
lab}-C(t)~{}\cos\lambda_{\rm
lab}\bigg{\\}}\left(\alpha_{X}q_{X}+\alpha_{Y}q_{Y}+\alpha_{Z}q_{Z}\right)$
$\displaystyle-\bigg{\\{}\sin(t^{\circ}_{\rm lab})~{}A(t)+\cos(t^{\circ}_{\rm
lab})~{}B(t)-0.465\cos\lambda_{\rm
lab}\bigg{\\}}\left(\beta_{X}q_{X}+\beta_{Y}q_{Y}+\beta_{Z}q_{Z}\right)$
$\displaystyle-\bigg{\\{}\bigg{[}\cos(t^{\circ}_{\rm
lab})~{}A(t)-\sin(t^{\circ}_{\rm lab})~{}B(t)\bigg{]}\cos\lambda_{\rm
lab}+C(t)~{}\sin\lambda_{\rm
lab}\bigg{\\}}\left(\gamma_{X}q_{X}+\gamma_{Y}q_{Y}+\gamma_{Z}q_{Z}\right)$
(A.3)

where

$\displaystyle A(t)$
$\displaystyle=0.4927~{}V_{\rm{GalRot}}-1.066~{}\textrm{km/s}+(V_{\oplus}(\lambda(t)){\cal
A}(t)$ $\displaystyle B(t)$
$\displaystyle=0.4503~{}V_{\rm{GalRot}}+16.56~{}\textrm{km/s}-(V_{\oplus}(\lambda(t)){\cal
B}(t)$ $\displaystyle C(t)$
$\displaystyle=0.7445~{}V_{\rm{GalRot}}+7.077~{}\textrm{km/s}+(V_{\oplus}(\lambda(t)){\cal
C}(t)$ (A.4)

and where we defined multiple quantities:

  * •

$V_{\rm{GalRot}}$ is the velocity of the rotation of the galaxy taken as
$V_{\rm{GalRot}}=220~{}\rm{km/s}$;

  * •

$\lambda_{\rm{lab}}=45.1^{\circ}$ is the latitude of the experiment;

  * •

$l_{\rm{lab}}=5.7^{\circ}$ is the longitude of the experiment;

  * •

$t^{\circ}_{\rm lab}$ is the local apparent sidereal time converted into
degrees. In [52], the authors parametrize this quantity as:

$t^{\circ}_{\rm
lab}~{}=~{}101.0308~{}+~{}36000.770\,T_{0}~{}+~{}15.04107\,\mathrm{UT}~{}+~{}l_{\rm{lab}}$
(A.5)

We derive alternate expressions for $T_{0}$ and $\mathrm{UT}$ based on the
Unix time given in seconds:

$\begin{cases}T_{0}=3.1688\times 10^{-10}t_{\mathrm{Unix}}~{}-~{}0.4\\\
\mathrm{UT}=\mathrm{mod}\big{(}t_{\mathrm{Unix}},86400\big{)}/3600\end{cases}$
(A.6)

  * •

$V_{\oplus}(\lambda(t))=V_{\oplus}\Big{[}1-e\sin\big{(}\lambda(t)-\lambda_{0}\big{)}\Big{]}$
with $V_{\oplus}=29.8~{}\rm{km/s}$ the orbital speed of the Earth,
$e=0.016722$ the ellipticity of the Earth’s orbit,
$\lambda_{0}=13^{\circ}+1^{\circ}$ the ecliptic longitude of the orbit’s minor
axis, and

$\lambda(t)~{}=~{}L+\big{(}1^{\circ}.915-0^{\circ}.0048\,T_{0}\big{)}\sin
g+0^{\circ}.020\sin 2g$ (A.7)

where

$\begin{cases}L=281^{\circ}.0298+36000^{\circ}.77T_{0}+0^{\circ}.04107\,\rm{UT}\\\
g=357^{\circ}.9258+35999^{\circ}.05T_{0}+0^{\circ}.04107\rm{UT}\end{cases}$
(A.8)

are, respectively, the mean longitude of the Sun corrected for aberration and
the mean anomaly.

  * •

Finally :

$\displaystyle{\cal A}(t)=$
$\displaystyle-0.06699\cos\beta_{x}\sin(\lambda(t)-\lambda_{x})+0.4927\cos\beta_{y}\sin(\lambda(t)-\lambda_{y})-0.8676\cos\beta_{z}\sin(\lambda(t)-\lambda_{z})$
$\displaystyle{\cal B}(t)=$
$\displaystyle-0.8728\cos\beta_{x}\sin(\lambda(t)-\lambda_{x})-0.4503\cos\beta_{y}\sin(\lambda(t)-\lambda_{y})-0.1883\cos\beta_{z}\sin(\lambda(t)-\lambda_{z})$
$\displaystyle{\cal C}(t)=$
$\displaystyle-0.4835\cos\beta_{x}\sin(\lambda(t)-\lambda_{x})+0.7446\cos\beta_{y}\sin(\lambda(t)-\lambda_{y})+0.4602\cos\beta_{z}\sin(\lambda(t)-\lambda_{z})$
(A.9)

where

$\begin{cases}\beta_{i}=\Big{(}-5^{\circ}.5303,~{}59^{\circ}.575,~{}29^{\circ}.812\Big{)}\\\
\lambda_{i}=\Big{(}266^{\circ}.141,~{}-13^{\circ}.3485,~{}179^{\circ}.3212\Big{)}\end{cases}$
(A.10)

The coordinates $\big{(}x(t),y(t)\big{)}$ at which the signal is formed on the
array of KIDs, with $(0,0)$ the center of the array, can be determined by
projecting $-\mathbf{v}_{\rm{lab}}$ to the X and Y axes of the mirror frame
and multiplying by the curvature radius $R$ [26]. Using the coefficients of
Eq. (A.2), we get:

$\displaystyle x(t)$ $\displaystyle\simeq$ $\displaystyle
R\sin(\theta_{NW})\,\bigg{\\{}\bigg{[}-\cos(t^{\circ}_{\rm
lab})~{}A(t)+\sin(t^{\circ}_{\rm lab})~{}B(t)\bigg{]}\sin\lambda_{\rm
lab}+C(t)~{}\cos\lambda_{\rm lab}\bigg{\\}}$
$\displaystyle-R\cos(\theta_{NW})\,\bigg{\\{}\sin(t^{\circ}_{\rm
lab})~{}A(t)+\cos(t^{\circ}_{\rm lab})~{}B(t)-0.465\cos\lambda_{\rm
lab}\bigg{\\}}$ $\displaystyle y(t)$ $\displaystyle\simeq$ $\displaystyle
R\bigg{\\{}\bigg{[}-\cos(t^{\circ}_{\rm lab})~{}A(t)+\sin(t^{\circ}_{\rm
lab})~{}B(t)\bigg{]}\cos\lambda_{\rm lab}-C(t)~{}\sin\lambda_{\rm
lab}\bigg{\\}}$ (A.11)

In subsection 4.3, the angle between the polarization of the DP and the
tangent of the mirror must be evaluated. Since the DP polarization is
expressed in the galactic frame, one needs to transform it to the mirror
frame. Once again, we rely on [52] to do so. We here only give the expression
for the projection of the galactic frame
$(\hat{\mathbf{x}_{g}},\hat{\mathbf{y}_{g}},\hat{\mathbf{z}_{g}})$ along
$\hat{\mathbf{Z}}$ that is required for our analysis:

$\displaystyle\hat{\mathbf{x}}_{g}\cdot\hat{\mathbf{Z}}$ $\displaystyle\simeq$
$\displaystyle-0.06699\Big{\\{}-\cos\theta_{NW}\cos t^{\circ}_{\rm
lab}\sin\lambda_{\rm lab}+\sin\theta_{NW}\sin t^{\circ}_{\rm lab}\Big{\\}}$
$\displaystyle+0.8728\Big{\\{}\sin\theta_{NW}\sin t^{\circ}_{\rm
lab}\sin\lambda_{\rm lab}+\sin\theta_{NW}\cos t^{\circ}_{\rm
lab}\Big{\\}}~{}~{}-0.4835\cos\theta_{NW}\cos\lambda_{\rm lab}$
$\displaystyle\hat{\mathbf{y}}_{g}\cdot\hat{\mathbf{Z}}$ $\displaystyle\simeq$
$\displaystyle 0.4927\Big{\\{}-\cos\theta_{NW}\cos t^{\circ}_{\rm
lab}\sin\lambda_{\rm lab}+\sin\theta_{NW}\sin t^{\circ}_{\rm lab}\Big{\\}}$
$\displaystyle+0.4503\Big{\\{}\sin\theta_{NW}\sin t^{\circ}_{\rm
lab}\sin\lambda_{\rm lab}+\sin\theta_{NW}\cos t^{\circ}_{\rm
lab}\Big{\\}}~{}~{}+0.7446\cos\theta_{NW}\cos\lambda_{\rm lab}$
$\displaystyle\hat{\mathbf{z}}_{g}\cdot\hat{\mathbf{Z}}$ $\displaystyle\simeq$
$\displaystyle-0.8676\Big{\\{}-\cos\theta_{NW}\cos t^{\circ}_{\rm
lab}\sin\lambda_{\rm lab}+\sin\theta_{NW}\sin t^{\circ}_{\rm lab}\Big{\\}}$
$\displaystyle+0.1883\Big{\\{}\sin\theta_{NW}\sin t^{\circ}_{\rm
lab}\sin\lambda_{\rm lab}+\sin\theta_{NW}\cos t^{\circ}_{\rm
lab}\Big{\\}}~{}~{}+0.4602\cos\theta_{NW}\cos\lambda_{\rm lab}$

Let us consider
$\hat{\mathbf{X}}_{\rm{DP}}=\Big{(}\sin\theta\cos\phi,\,\sin\theta\sin\phi,\,\cos\theta\Big{)}$
a unit vector of the DP polarization in the galactic frame. The angle $\alpha$
between the DP polarization and the tangent of mirror is the complement of the
angle with respect to $\hat{\mathbf{Z}}$:

$\cos^{2}\alpha=1-\Big{(}\hat{\mathbf{X}}_{\rm{DP}}\cdot\hat{\mathbf{Z}}\Big{)}^{2}$
(A.13)

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