# Not So Round: VLA Observations of the Starless Dark Matter Halo Candidate
Cloud-9

Alejandro Benítez-Llambay Dipartimento di Fisica G. Occhialini, Università
degli Studi di Milano Bicocca, Piazza della Scienza, 3 I-20126 Milano MI,
Italy  Rajeshwari Dutta IUCAA, Postbag 4, Ganeshkind, Pune 411007, India
Michele Fumagalli Dipartimento di Fisica G. Occhialini, Università degli Studi
di Milano Bicocca, Piazza della Scienza, 3 I-20126 Milano MI, Italy  INAF –
Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143 Trieste,
Italy  Julio F. Navarro Department of Physics and Astronomy, University of
Victoria, Victoria, BC V8P 5C2, Canada

###### Abstract

Observations with FAST recently detected H i 21-cm emission near M94,
revealing an intriguing object, Cloud-9, without an optical counterpart.
Subsequent analysis suggests Cloud-9 is consistent with a gas-rich ($M_{\rm
HI}\approx 10^{6}\ M_{\odot}$), starless dark matter (DM) halo of mass
$M_{200}\approx 5\times 10^{9}\ M_{\odot}$. Using VLA in D-array
configuration, we present interferometric observations of Cloud-9 revealing it
as a dynamically cold ($W_{50}\approx 12\rm\ km\ s^{-1}$), non-rotating, and
spatially-asymmetric system, exhibiting gas compression on one side and a
tail-like structure towards the other, features likely originating from ram
pressure. Our observations suggest Cloud-9 is consistent with a starless
$\Lambda$CDM dark matter halo if the gas is largely isothermal. If interpreted
as a faint dwarf, Cloud-9 is similar to Leo T, a nearby gas-rich galaxy that
would fall below current optical detection limits at Cloud-9’s distance
($d\approx 5\rm\ Mpc$). Further observations with HST reaching magnitudes
$m_{g}\approx 30$ would help identify such a galaxy or dramatically lower
current limits to its stellar mass ($M_{\rm gal}\lesssim 10^{5}\ M_{\odot}$).
Cloud-9 thus stands as the firmest starless DM halo candidate to date or the
faintest galaxy known at its distance.

Dark matter(353) — Cosmology(343) — Reionization(1383) —

##  1 Introduction

The existence of myriads of starless dark matter halos is a basic tenet of the
$\Lambda$ cold dark matter ($\Lambda$CDM) model of structure formation
(Blumenthal et al., 1984; Hoeft et al., 2006; Okamoto et al., 2008; Okamoto &
Frenk, 2009). Indeed, reconciling the abundance of collapsed dark matter halos
with that of faint galaxies requires stars to form only in a fraction of the
halos, particularly those whose mass exceeds a redshift-dependent critical
mass, $M_{\rm crit,z}$ (e.g., Bullock et al., 2000; Benson et al., 2002).
Within this picture, the only halos that can harbor luminous galaxies in their
center today are those whose mass exceeded $M_{\rm crit}$ at some point in the
past. On the other hand, halos with masses consistently below this threshold
at all times remain dark (Benitez-Llambay & Frenk, 2020, and references
therein; hereafter BLF20).

Theoretical models and results from numerical simulations suggest a present-
day value for the critical mass, $M_{\rm crit,0}\approx 5\times 10^{9}\
M_{\odot}$ (; ), indicating that all dark matter halos with virial mass,
$M_{200}>M_{\rm crit,0}$, should host luminous galaxies today. On the other
hand, halos with masses below this value may or may not host stars, depending
on their particular mass assembly history and on whether they exceeded $M_{\rm
crit}$ in the past.

Using high-resolution hydrodynamical cosmological simulations, Benítez-Llambay
et al. (2017, hereafter BL17) studied the origin and detailed properties of
starless halos. They found that the most massive systems, with masses
comparable to $M_{\rm crit}$ at present, still retain gas confined to their
center. Despite significant gas loss due to the impact of the external
ultraviolet background (UVB) radiation field responsible for reionizing the
universe, these halos maintain a small gas reservoir in or near hydrostatic
equilibrium with the dark matter halo and in thermal equilibrium with the UVB.
The relatively high density and low temperature of the gas in these systems
facilitate the recombination of hydrogen, resulting in the formation of a
prominent neutral hydrogen (H i) core in the center. This is why we call these
massive H i-rich starless halos Reionization-Limited-HI-Clouds (RELHICs).

RELHICs should display distinctive properties: they should contain a central
nearly-spherical H i core in hydrostatic equilibrium and thermal equilibrium
with the UVB, making them detectable through their narrow 21-cm emission line.
These properties make RELHICs remarkable tests of the $\Lambda$CDM model on
the smallest scales because their detailed gaseous structure offers a direct
probe into the underlying structure of their pristine dark matter halos.

Recent observations by Zhou et al. (2023, hereafter, Z23) with the Five-
hundred-metre Aperture Spherical Telescope (FAST) detected marginally resolved
21-cm emission near the M94 galaxy, separated by $\approx 50^{\prime}$ in
projection, without an optical counterpart. Unlike previous candidate “dark” H
i clouds (e.g. Adams et al., 2013; Xu et al., 2023), all known properties of
this source (Cloud-9 hereafter) appear broadly consistent with RELHICs, as
pointed out by Z23 and recently discussed by Benitez-Llambay & Navarro (2023,
hereafter, BLN23). Subsequent unresolved observations with the Green Bank
Telescope (GBT) confirmed the detection of this radio source (Karunakaran &
Spekkens, 2024, hereafter KS24) but with a slightly broader and more
asymmetric line profile than that reported by Z23. Although the observations
of KS24 suggest Cloud-9 has a more complex structure than anticipated, they
could not exclude Cloud-9 from being a RELHIC. Assuming a distance to the
object, $d=4.66\rm\ Mpc$111We refer the reader to the work of BLN23 for a
detailed discussion on the distance estimate for Cloud-9., FAST and GBT
observations bracket the total H i mass of the system in the range, $1\times
10^{6}\lesssim M_{\rm HI}/M_{\odot}\lesssim 2\times 10^{6}$. Moreover, at the
adopted distance, Cloud-9 must be dark matter dominated, since its inferred
dynamical mass is much higher than its H i mass (BLN23).

Figure 1: VLA-D spectrum of Cloud-9 centered at the H i 21-cm line (blue solid
histogram), compared with the spectra obtained by FAST (orange solid line) and
GBT (green solid line). The red dashed line shows a Gaussian fit to the VLA-D
line profile centered at $v_{0}\approx 304\rm\ km\ s^{-1}$, and full width at
half maximum, $W_{\rm 50}\approx 12\ \rm km\ s^{-1}$. Both FAST and GBT data
were taken from the work of KS24.

Although Cloud-9 is overall consistent with a RELHIC, the main doubts about
its true nature concern primarily the low angular resolution of the FAST
observations (half-power beamwidth $\approx 2.9\rm\ arcmin$ at 1.4 GHz). FAST
observations thus offer limited insight into the system’s central H i column
density, making it difficult to impose tight constraints on its size and
projected shape. In addition, Cloud-9’s detailed column density profile
appears slightly, but systematically, more extended than expected for RELHICs.
This mismatch could originate for several reasons ranging from uncertainties
in the FAST primary beam’s characterization to the system being closer to us,
to even departures from the $\Lambda$CDM model (BLN23). Characterizing the
system better thus requires higher-resolution radio observations.

On the other hand, using the surface brightness limits from the DESI Legacy
Imaging Survey (DESI LS) estimated by Martínez-Delgado et al. (2023) –$29.15$,
$28.73$, and $27.52$ mag arcsec-2 in the $g$, $r$, and $z$ bands,
respectively–, Z23 derived a large upper limit to the stellar mass of any
luminous counterpart (galaxy mass, $M_{\rm gal}\lesssim 10^{5}\ M_{\odot}$).
This particularly high galaxy mass opens the possibility that Cloud-9 is an
analog of some of the faint Local Group dwarf galaxies but at the M94
distance. Elucidating whether or not Cloud-9 has a luminous counterpart thus
requires deeper observations in the optical.

In this paper, we aim to mitigate current limitations by considering
interferometric observations of Cloud-9 in radio wavelengths. To this end, we
use the Karl G. Jansky Very Large Array (VLA) in the D-array configuration
(hereafter VLA-D). This configuration provides higher spatial resolution than
FAST and GBT and allows resolving the central parts of the system better in
comparison. We start by discussing the observations and data reduction in
Section 2. We then present the results and discussions in Section 3 and
Section 5, respectively, and summarize our main results and conclusions in
Section 6.

##  2 Observations and data reduction

###  2.1 Data acquisition

We observed Cloud-9 with the L-band receivers of VLA in the D-array
configuration between 1 and 13 December 2023 (Proposal ID: VLA/23B-313; PI: A.
Benitez-Llambay). The observations were centered at RA, Dec (J2000) = 12h 51m
52s, +40d 17m 29s. The total on-source integration time was $\approx 3.2$ h,
which was split into four separate observing runs. The primary calibrator,
3C286, was observed once in each observing run for calibration of flux density
and bandpass. In addition, a secondary calibrator, J1227+3635, was observed at
regular intervals during the observing runs for complex gain calibration. The
observations used the WIDAR correlator with the 8-bit samplers, two
polarizations, and a single 8 MHz spectral window that was centered on the
redshifted H i 21-cm line. This configuration yielded a velocity coverage of
$\approx$1700 km s-1 at Cloud-9’s redshift, adequate to cover the H i 21-cm
emission detected by FAST, and to provide line-free channels for continuum
subtraction. The spectral window was split into 1024 channels, resulting in a
spectral resolution of 7.8 kHz per channel or $\approx 1.7$ km s-1, similar to
that of the FAST spectrum reported by Z23.

###  2.2 Data reduction

The data were reduced using the Common Astronomy Software Applications package
(casa, version 6.6; McMullin et al., 2007), following standard procedures. The
dataset of each observing run was calibrated independently for the flux
density scale, bandpass, and complex gain, after flagging the bad data due to
non-working antennae and radio frequency interference. The continuum was
subtracted in the $uv$-plane using a first-order polynomial fit to the line-
free channels (excluding edge channels). The continuum-subtracted visibilities
of each observing run were combined to produce the H i image cube with the
routine TCLEAN in CASA, using a cell size of 9 arcsec and Natural weighting.
The cube was “cleaned” during the imaging process using auto-masking by
multiple thresholds for the deconvolution. Any residual continuum was
subtracted from the image cube using a first-order polynomial fit to the line-
free channels (excluding edge channels). The resulting synthesized beam of the
final cube has a size $65.2^{\prime\prime}\times 53.7^{\prime\prime}$ with a
position angle of $\approx-76^{\rm o}$, and an $rms$ uncertainty of $\approx
1.3$ mJy beam-1 per velocity channel of 1.7 km s-1.

##  3 Results

###  3.1 Cloud-9’s VLA-D Spectrum

We show Cloud-9’s H i 21-cm spectrum, specifically the flux density as a
function of the velocity channel, in Figure 1 (blue line), extracted by
integrating, for each channel, the flux contained within the outermost contour
shown in Figure 2. For comparison, we also include the spectra obtained from
single-dish radio observations carried out with FAST (orange line) and GBT
(green line), both taken from Figure 1 of KS24.

Our VLA-D observations confirm the detection of emission from H i 21-cm at
heliocentric velocities of approximately $v_{\rm helio}\approx 300\rm\ km\
s^{-1}$, consistent with the results of Z23 and KS24. However, unlike the GBT
results, our measured line profile appears fairly symmetric and is well
described by a Gaussian function. The red dashed line shows the best-fit
Gaussian model to the VLA-D line profile and is consistent with a mean
heliocentric recessional velocity of $v_{\rm helio,VLA}\approx(304\pm 2)\rm\
km\ s^{-1}$ and a full width at half maximum of $W_{\rm 50,VLA}\approx(12\pm
1)\rm\ km\ s^{-1}$. Although the recessional velocity agrees with results from
FAST and GBT, our estimate of the line broadening is roughly a factor of two
smaller than that obtained by these instruments.222We note that Z23 report a
line broadening of approximately $\approx 20\rm\ km\ s^{-1}$ for Cloud-9, but
a subsequent data release as part of the FASHI catalogue (Zhang et al., 2024)
indicates that Cloud-9’s line broadening, as observed by FAST, is consistent
with GBT results. This issue was pointed out by KS24.

Figure 2: Cloud-9’s H i column density isocontours superimposed on an image
cutout of the DESI Imaging Legacy Survey. The ellipse at the bottom left
indicates the “clean” beam. The isocontours start at $3\sigma_{\rm N_{\rm
HI}}=5.45\times 10^{18}\ \rm cm^{-2}$, where $\sigma_{\rm N_{\rm HI}}$ is the
rms uncertainty, and they increase in steps of $\sqrt{2}$ towards the centre.
The maximum column density reached in the centre is $N_{\rm HI,0}=1.6\times
10^{19}\rm\ cm^{-2}$. The image is centred at $(\alpha,\delta)_{\rm
J2000}=(192.9642^{o},40.30139^{o})$.

###  3.2 Cloud-9’s H i Column Density Map

To describe the spatial distribution of Cloud-9, we estimate the H i column
density map, assuming, as usual, that the source is optically thin. Under this
assumption, the relation between the H i column density, $N_{\rm HI}$, and the
measured surface brightness temperature, $T_{\rm B}$, becomes (e.g., Dickey,
1990; McClure-Griffiths et al., 2023):

$N_{\rm HI}=(1.823\times 10^{18}\rm\
cm^{-2})\displaystyle\int{\left(\frac{T_{B}}{K}\right)\left(\displaystyle\frac{dv}{\rm
km\ s^{-1}}\right)},$ (1)

in which the integration is carried out over the velocity channels, and
$T_{B}$ is the brightness temperature, related to the observed flux density,
$I_{\nu}$, at frequency $\nu$, through (e.g., Wilson et al., 2009):

$T_{B}=\left(1.68\times 10^{2}\rm\
mK\right)\left(\displaystyle\frac{I_{\nu}}{\rm Jy\ beam^{-1}}\right)\\\
\left(\displaystyle\frac{\nu}{1.42\ \rm
GHz}\right)^{-2}\left(\displaystyle\frac{\theta_{m}}{60^{\prime\prime}}\right)^{-1}\left(\displaystyle\frac{\theta_{M}}{60^{\prime\prime}}\right)^{-1}.$
(2)

The angles $\theta_{m}$ and $\theta_{M}$ correspond to the full width at half
maximum of the “clean” beam along the minor and major axes, respectively. For
reference, with the adopted VLA-D configuration and data reduction pipeline
(see Sec. 2), the mean “clean” beam, averaged over the velocity channels, is
well described by
$(\theta_{m},\theta_{M})\approx(53.7^{\prime\prime},65.2^{\prime\prime})$, and
a position angle, $\rm\beta\approx-76^{o}$ (as shown by the dashed ellipse in
the bottom left of Figure 2).

We select the velocity integration range to maximize the signal-to-noise ratio
($\rm S/N$) while ensuring convergence of the maximum column density value
across the map. In practice, these criteria lead to an optimal integration
range in the interval $(293-314)\rm\ km\ s^{-1}$ (12 channels), resulting in a
column density rms uncertainty, $\sigma_{\rm N_{HI}}\approx 1.8\times
10^{18}\rm\ cm^{-2}$, and a peak signal-to-noise ratio of approximately
$\rm(S/N)\approx 9$. We employ a similar procedure to calculate the source’s
total flux.

We show the resulting column density map in Figure 2. The white solid lines
display four H i column density isocontours, starting at $3\sigma_{\rm
N_{HI}}$ in the outer parts and increasing in steps of $\sqrt{2}$ towards the
center. The origin of the map corresponds to the location where the system
reaches the peak column density, $N_{\rm HI,0}\approx 1.63\times 10^{19}\rm\
cm^{-2}$, with coordinates, $(\alpha,\delta)_{\rm J_{2000}}=(12^{\rm h}51^{\rm
m}51.3^{\rm s},+40^{o}18^{\prime}5^{\prime\prime})$.

Although the central parts of the system are arguably unresolved given the
extent of the “clean” VLA-D beam, the higher spatial resolution of our
observations reveals a more complex structure than anticipated from the lower
resolution FAST dataset (we refer the reader to Figure 1 of BLN23 for
comparison with the FAST H i column density map, which looks fairly round and
symmetric).

Unlike the single-dish FAST observations, our interferometric VLA-D data
reveal Cloud-9 as a system with a slightly distorted morphology relative to
its peak emission. The system is extended towards the east, compact towards
the west, and does not entirely resemble a sphere in projection.
Interestingly, hints of this asymmetry, particularly the extended emission
towards the far east side, were already present in the FAST data.

As we will discuss later in Section 5, the perturbed morphology of Cloud-9
could originate from interactions with the environment, particularly ram
pressure, which is expected to compress the gas in one direction and leave
behind a tail-like structure, features consistent with the observed gas
distribution.

###  3.3 Cloud-9’s Total H i Flux

The total VLA-D flux density of the source within the $3\sigma_{\rm N_{HI}}$
region is $S_{21}\approx(0.13\pm 0.02)\ \rm mJy\ km\ s^{-1}$. We have verified
that this flux value is robust, as measurements over a smoothed image yield
comparable results. Specifically, when applying a natural weighting scheme
with a Gaussian taper, producing a larger synthesized beam of size
$(\theta_{m,t},\theta_{M,t})\approx(105^{\prime\prime},86^{\prime\prime})$ at
PA = -80 degrees, results in a lower flux value of $S_{21,t}\approx(0.1\pm
0.1)\ \rm mJy\ km\ s^{-1}$.

Our measurement of the flux density value is thus systematically lower than
those measured by Z23 using FAST, $(0.24\pm 0.01)\rm\ mJy\ km\ s^{-1}$, and
KS24 using GBT, $(0.28\pm 0.04)\rm\ mJy\ km\ s^{-1}$. At Cloud-9’s fiducial
distance, $d=4.66\rm\ Mpc$, the VLA-D integrated flux density corresponds to a
total H i mass, $M_{\rm HI}=\left(7\pm 1\right)\times 10^{5}\ M_{\odot}$,
which we obtain from the usual relation (e.g., Roberts, 1962):

$M_{\rm HI}=2.36\times 10^{5}\ M_{\odot}\left(\displaystyle\frac{d}{1\rm\
Mpc}\right)^{2}\\\ \displaystyle\int{\left(\displaystyle\frac{I_{\nu}}{\rm
mJy}\right)\ \left(\displaystyle\frac{dv}{\rm km\ s^{-1}}\right).}$ (3)

The overall VLA-D flux (H i mass) deficit compared with FAST and GBT, together
with the discrepancy in the shape of the line profile, may originate from
either the presence of diffuse emission missed in our VLA-D interferometric
observations or contamination of the GBT field by emission we do not sample;
note that the largest angular scale limit imposed by the L-band VLA-D
configuration is $\approx 16$ arcmin. Additionally, the relatively lower
surface brightness sensitivity and incompleteness in the $uv$-plane coverage,
particularly at short baselines, may preclude adequate sampling of large-scale
diffuse emissions on scales smaller than this limit. The systematic difference
among the datasets obtained with GBT, FAST, and VLA-D suggests that all these
issues may be contributing factors (see, e.g., Wang et al., 2024, for a recent
comparison between FAST and VLA observations showing similar discrepancies).
Nevertheless, our observations compellingly demonstrate the presence of a
prominent neutral phase with a fairly narrow velocity profile.

Figure 3: Cloud-9’s intensity-weighted velocity map, relative to the inferred
heliocentric velocity of the line, $v_{\rm helio,VLA}=304\rm\ km\ s^{-1}$. The
dashed ellipse in the bottom left indicates the extent of the beam, while the
black lines show the same column density isocontours in Figure 2. The lack of
a velocity gradient suggests the absence of ordered rotation.

###  3.4 Cloud-9’s H i Intensity-weighted Velocity Map

To quantify the system’s degree of ordered rotation, we consider the
intensity-weighted velocity map of the spectral line in Figure 3, in which we
only display the regions with column density above $3\sigma_{\rm N_{HI}}$. The
color map indicates the velocity relative to the overall inferred heliocentric
velocity, $v_{\rm helio,VLA}=304\rm\ km\ s^{-1}$. The lack of coherent
rotation over the sampled scales is readily apparent, demonstrating that the
sampled gas cloud is not supported by circular motion. This is consistent with
FAST results, which also reveal Cloud-9 as a non-rotating system.

We thus conclude that the broad properties of Cloud-9, as derived from our
VLA-D observations, are still consistent with those expected for a RELHIC,
except for its shape. We will discuss the potential origin of this distorted
morphology in Section 5.

Figure 4: Cloud-9’s column density profile as a function of the effective
radius, $R_{\rm eff}=\sqrt{ab}$, in which $a$ and $b$ are the minor and major
semiaxes of each ellipse shown in the inset, and are in identical proportion
to those of the beam. The reported column density corresponds to the mean
column density of the pixels intersected by each ellipse. The uncertainties
represent the $rms$ scatter along the ellipse and coincide with the typical
dispersion found by measuring the profile along independent directions. The
vertical dashed line indicates the radial extent of the beam.

###  3.5 Cloud-9’s Detailed H i Column Density Profile

We now turn our attention to Cloud-9’s detailed column density profile. Given
the asymmetry of the H i isocontours and the high ellipticity of the VLA-D
beam, we measure Cloud-9’s detailed column density profile within ellipses
centered at the peak. For each ellipse, we calculate the effective radius as
the geometric mean of the minor and major semiaxes, $R_{\rm eff}=\sqrt{ab}$,
which are in proportion to those of the beam (see the inset panel in Figure
4). We then estimate the column density at $R_{\rm eff}$ as the mean column
density of the pixels intersected by each ellipse and the scatter as the
standard deviation. We have verified that the scatter obtained in this manner
effectively captures the dispersion obtained by measuring the column density
profile along various independent directions. For example, measuring the
profile from the center towards the west produces a profile consistent with
the upper bound. Similarly, measurements towards the east produce a profile
consistent with the lower bound. The resulting column density profile together
with the associated rms is shown in Figure 4, in which the vertical dashed
line indicates the radial extent of the beam.

##  4 The Nature of Cloud-9

Having presented Cloud-9’s properties as observed by VLA-D, we now analyze two
possible interpretations for the system. One interpretation is that Cloud-9 is
an H i-rich starless dark matter halo (RELHIC), as predicted by state-of-the-
art numerical hydrodynamical simulations, an idea explored qualitatively by
Z23 and quantitatively by BLN23. The other interpretation is that Cloud-9 is
an H i-rich dwarf galaxy that is too faint to be detected by current optical
observations of this field. These possibilities are justified by the fact
that, as discussed next, a large amount of mass other than neutral hydrogen is
required for the system to be largely in equilibrium.

If Cloud-9 is in or near hydrostatic equilibrium in the central regions, the
free-fall time should be comparable to the sound crossing time within the
observed range. Defining the free-fall time as, $t_{\rm
ff}=\left(3\pi/16G\bar{\rho}_{\rm HI}\right)^{1/2}$, and the sound-crossing
time as $t_{\rm sc}=\left(r/c_{s}\right)$, with $G$ being Newton’s
gravitational constant, $\bar{\rho}_{\rm HI}\sim 3M_{\rm HI}/4\pi r^{3}$ the
mean enclosed neutral hydrogen density within a sphere of radius $r$, and
$c_{s}$ the sound speed of the gas, the identity $t_{\rm ff}\sim t_{\rm sc}$
yields:

$r\sim\left(8.5\ {\rm pc}\right)\left(\displaystyle\frac{M_{\rm HI}}{7\times
10^{5}M_{\odot}}\right)\left(\displaystyle\frac{c_{s}}{12\ {\rm km\
s^{-1}}}\right)^{-2}.$ (4)

At Cloud-9’s distance, $8.5\rm\ pc$ corresponds to a projected size of
approximately $0.4\ \rm arcsec$, which is orders of magnitude smaller than
observed. This mismatch implies the necessity for a substantial amount of
gravitational mass other than neutral hydrogen to sustain equilibrium over the
observed scales.

###  4.1 Cloud-9 as a Starless $\Lambda$CDM RELHIC Candidate

To test the RELHIC hypothesis for Cloud-9 we compare its detailed column
density profile with that of RELHICs. For this comparison, we use analytic
RELHIC models constructed as described by BLN23, a procedure we explain below.

For a given Navarro-Frenk-White (Navarro et al., 1996, hereafter NFW) dark
matter halo of virial mass, $M_{200}$, and concentration, $c_{\rm NFW}$, we
calculate its detailed gas density profile, assuming the gas is in hydrostatic
equilibrium with the dark matter halo and in thermal equilibrium with the
external UVB, whose spectrum corresponds to that of Haardt & Madau (2012). To
integrate the hydrostatic equilibrium equation we assume a boundary condition
such that the system’s pressure at infinity matches that of the intergalactic
medium at the mean density of the universe. This approach, which has no free
parameters other than the present-day mean baryon density, the mass of the
halo, and the UVB spectrum, has been shown to reproduce results from
hydrodynamical simulations (e.g., ; , ). We shall refer to this model as the
BL17 RELHIC model.

Figure 5: The top panel shows Cloud-9’s H i column density profile (grey
points with error bars), compared with the mock-observed profile of
$\Lambda$CDM RELHICs modeled following the BL17 model whose central column
density matches Cloud-9’s (green shaded region), and isothermal RELHICs with
sound speed matching Cloud-9’s observed 21-cm line broadening (red dashed
lines). These RELHICs inhabit dark matter halos with virial masses and
concentrations consistent with the $\Lambda$CDM mass-concentration relation.
The blue dot-dashed line represents a BL17 RELHIC in a dark matter halo with
$M_{200}\approx 9\times 10^{10}\ M_{\odot}$ and concentration $c_{\rm NFW}=2$,
which is incompatible with $\Lambda$CDM expectations. The bottom panel shows
the $\Lambda$CDM mass-concentration relation with $1\sigma$ scatter (black
line with shaded region), along with the combinations of halo mass and
concentration that yield RELHIC column density profiles matching Cloud-9’s
central column density (black dashed line). The thick green/red lines indicate
the mass and concentration range producing the green/red profiles in the top
panel, while the blue triangle marks the location of the RELHIC shown by the
blue line in the top panel.

To convert the gas density profile into an H i density profile, we use the
results from Rahmati et al. (2013), which provide the H i fraction as a
function of total hydrogen gas density and temperature. We then integrate the
resulting H i density profile along the line of sight to obtain the intrinsic
H i column density profile. We place this intrinsic profile at Cloud-9’s
distance and convolve it with the VLA-D “clean” beam, resulting in a mock H i
column density profile that we “observe” and measure along the same ellipses
defined before.

We note that Cloud-9’s distorted morphology, particularly in the outer
regions, may indicate that the system is being perturbed. For example, if the
system were moving within a gaseous background associated with M94 and
experiencing ram pressure, it would display similar features as observed and
would likely tend to expand due to the removal of gas in the outer parts.
However, whether this gas becomes photoionized by the external photoionizing
background as it expands depends on the system’s intrinsic gas density. In
addition, there is no guarantee that the local intensity of the UVB at
Cloud-9’s location is that of Haardt & Madau (2012).

As shown by BL17 and Sykes et al. (2019), the neutral phase of RELHICs can
reach temperatures $\lesssim 10^{4}\ \text{K}$, which roughly corresponds to a
sound speed of about $c_{s}\approx 9\ \rm km\ s^{-1}$, assuming a mean
molecular weight of $\mu=1.2$. The broadening of Cloud-9’s H i 21-cm line, as
inferred from our observations, is approximately $12\rm\ km\ s^{-1}$, implying
a temperature for Cloud-9 exceeding by more than a factor of 2 the temperature
expected for RELHICs. Additionally, the higher H i 21-cm line broadening
measured by FAST (Z23) and GTB (KS24), together with Cloud-9’s asymmetric
morphology, may indicate the presence of a perturbing mechanism that could
also be responsible for thermalizing the system to a higher temperature than
expected for RELHICs.

Because of these reasons, we also explore an alternative “isothermal” model,
assuming the gas is fully isothermal with sound speed $c_{s}=12\rm\ km\
s^{-1}$, a value matching Cloud-9’s observed H i 21-cm line broadening. For
reference, this sound speed corresponds to a gas temperature of about $\approx
2\times 10^{4}\rm\ K$, assuming again a mean molecular weight of $\mu=1.2$. We
shall refer to this model as the “isothermal” RELHIC model.

Similar to Figure 4, the top panel of Figure 5 shows Cloud-9’s detailed column
density profile and its associated scatter (grey circles). The green shaded
region represents the range of profiles allowed for the BL17 $\Lambda$CDM
RELHICs, matching Cloud-9’s central maximum column density when placed at
Cloud-9’s distance and convolved with the VLA-D beam. The bottom panel’s green
line indicates that these RELHICs reside in halos with virial masses in the
range $4\lesssim M_{200}/10^{9}\ M_{\odot}\lesssim 6$, and concentrations in
the range $17\gtrsim c_{\rm NFW}\gtrsim 10$, all consistent with the
$\Lambda$CDM mass-concentration relation (e.g., Ludlow et al., 2016, grey
shaded region in the bottom panel).

The comparison carried out in Figure 5 reveals that Cloud-9’s column density
profile is overall inconsistent with the BL17 RELHIC models. In other words,
Cloud-9 cannot be described as a gas cloud in hydrostatic equilibrium with
$\Lambda$CDM halos in an otherwise unperturbed background illuminated by the
Haardt & Madau (2012) UVB. Specifically, Cloud-9’s H i distribution is too
extended to align with this model fully. This inconsistency may partially stem
from Cloud-9’s distorted morphology, which suggests the system is not truly in
isolation. Alternatively, the mismatch may be ascribed to an intrinsic
difference in the density or temperature structure of Cloud-9 compared to the
BL17 models, or even departures from $\Lambda$CDM, as pointed out by BLN23. We
discuss these issues further below and in Sec. 5.

Figure 6: Total (top left) and neutral hydrogen density (top right), gas
temperature (bottom left), and H i fraction (bottom right), as a function of
radius. Different lines correspond to the RELHICs shown in Figure 5,
specifically: a BL17 RELHIC inhabiting a $\Lambda$CDM halo of mass,
$M_{200}\approx 5\times 10^{9}\ M_{\odot}$, matching Cloud-9’s central column
density (green solid line), and an isothermal RELHIC consistent with
observations (red solid line), with mass, $M_{200}\approx 4\times 10^{9}\
M_{\odot}$, both with concentration, $c_{\rm NFW}\approx 13$.

Following BLN23, we now take the discrepancy between Coud-9’s column density
profile and the BL17 RELHIC model at face value and explore whether it is
possible to alleviate the issue by considering a dark matter halo that departs
from $\Lambda$CDM expectations. In agreement with BLN23, we find that the
discrepancy between the observed column density profile and RELHIC models is
resolved if we allow a RELHIC to occupy a dark matter halo with extremely low
concentration, so low that it becomes incompatible with $\Lambda$CDM. For
instance, the RELHIC represented by the blue dot-dashed line in the top panel
of Figure 5, residing in a dark matter halo of mass, $M_{200}\approx 9\times
10^{9}\ M_{\odot}$, and concentration, $c_{\rm NFW}\approx 2$, aligns with
observations. However, this RELHIC inhabits a dark matter halo that
significantly deviates from $\Lambda$CDM expectations, as indicated by the
blue triangle in the bottom panel, which lies by more than $3\sigma$ away from
the Ludlow et al. (2016) $\Lambda$CDM mass-concentration relation.

Interestingly, by adopting a gas temperature of $2\times 10^{4}\ K$
($c_{s}\approx 12\rm\ km\ s^{-1}$), it is possible to find a family of
“isothermal” RELHICs inhabiting $\Lambda$CDM halos that match Cloud-9’s H i
column density profile. Indeed, the red dashed lines in the top panel of
Figure 5 show three isothermal models with masses in the range $3.9\lesssim
M_{200}/10^{9}\ M_{\odot}\lesssim 4.3$, and concentration matching the
$\Lambda$CDM mass-concentration relation (see the red line in the bottom
panel). Remarkably, these RELHICs not only bracket Cloud-9’s column density
profile but also have a similar shape. Moreover, it is possible to find one
model that reproduces observations.

To understand where the differences between the BL17 and isothermal RELHICs
originate, we consider, in Figure 6, the intrinsic profiles of two models that
reproduce Cloud-9’s central column density. These are the same models
presented in Figure 5: a BL17 RELHIC inhabiting a $\Lambda$CDM dark matter
halo (green solid line) and an isothermal model (red dashed line), with sound
speed matching the broadening of the VLA-D H i 21-cm line. Both models reach
the mean density of the universe at infinity.

The reason why the isothermal model, as opposed to the BL17 model, reproduces
observations is readily apparent in the top right panel. The isothermal RELHIC
matches Cloud-9’s column density largely because its neutral hydrogen profile
is intrinsically more extended than that of the BL17 model. Although the
central H i density of the isothermal RELHIC is almost an order of magnitude
lower than that of the BL17 model, the larger physical extent of the neutral
gas in the isothermal model compensates for this deficit, resulting in a
similar central column density profile for both models. Furthermore, the more
extended H i distribution in the isothermal RELHIC leads to a more extended H
i column density profile, thus favoring a good match with observations.

The larger radial extent of the H i profile compared to the BL17 model arises
from intrinsic differences between the two models in both temperature and
density, as demonstrated in the left panels of Figure 6. Notably, compared to
the adopted isothermal model, BL17 RELHICs are colder in the center and hotter
in the outskirts, favoring a steeper density gradient and a sharper decline of
the H i fraction with radius (see bottom right panel). Also, although the
isothermal model has a lower neutral hydrogen fraction in the center compared
to the BL17 model, this deficit is compensated by the larger inner hydrogen
density the system develops in comparison.

Finally, we note that Figure 6 explicitly shows that the BL17 RELHICs matching
Cloud-9’s inner column density profile are colder than observed. It is thus
not entirely surprising that these models fail to reproduce Cloud-9’s column
density profile in detail if the broadening of the H i line traces the gas
temperature of the neutral phase.

Despite our exercise highlighting the sensitivity of the assumed equation of
state on the RELHICs’ H i density profile, it is remarkable to see that a
simple model, with parameters motivated by observations, can reproduce
Cloud-9’s column density profile.

Hence, we conclude that Cloud-9’s detailed column density profile is
consistent with an isothermal $\Lambda$CDM RELHIC whose temperature matches
observations. However, we stress that taken at face value, the system appears
to be in tension with the RELHIC model introduced by BL17. This mismatch may
simply reflect the fact that Cloud-9 is not truly in isolation, localized
departures from the adopted UVB, or departures from the equation of state for
the gas.

Until further optical observations confirm the existence of a luminous
counterpart associated with the system, our VLA-D observations do not exclude
the possibility that Cloud-9 is a RELHIC.

###  4.2 Cloud-9 as a Local Group Dwarf Galaxy Analog

We now turn our attention to the hypothesis that Cloud-9 is a Local Group
dwarf galaxy counterpart at the M94 distance, a case that rests on solid
grounds for at least two reasons.

Firstly, results from hydrodynamical simulations and analytic considerations
indicate that the majority of systems with dark matter mass similar to that
inferred for Cloud-9 host luminous galaxies in their center. Indeed, BL17 (see
also ; , and references therein) finds that more than $90\%$ of the halos with
present-day masses, $M_{200}>5\times 10^{9}M_{\odot}$, contain stars, making
it likely that Cloud-9 is a luminous galaxy rather than a RELHIC. This idea is
reinforced by the fact that quiescent low–mass galaxies today in cosmological
hydrodynamical simulations display gaseous halos and total H i masses similar
to those inferred for Cloud-9 and RELHICs (e.g., Pereira-Wilson et al., 2023;
Herzog et al., 2023).

Secondly, current photometric limits do not rule out the presence of a faint
stellar counterpart at Cloud-9’s location. For example, the surface brightness
limits of the DESI Legacy Survey (LS) reported by Martínez-Delgado et al.
(2023), namely $29.15$, $28.73$, and $27.52$ $\rm mag\ arcsec^{-2}$ in the
$g,r$, and $z$ bands, respectively, place large upper limits to the stellar
mass of any luminous component associated with Cloud-9, as discussed by Z23.
In particular, these authors estimate that Cloud-9’s luminous counterpart
would not be more massive than $\approx 10^{5}\ M_{\odot}$. This maximum mass
is well above the minimum stellar mass of some faint dwarfs observed in the
Local Group (see, e.g., Simon, 2019, for a recent review on the faintest
nearby galaxies), which makes us wonder whether there is any known nearby
faint dwarf galaxy with properties similar to those inferred for Cloud-9 that
might be missed in current observations of this field.

####  4.2.1 Cloud-9 as a Leo T Lookalike

Interestingly, Leo T –a local dwarf galaxy lying at the outskirts of the Milky
Way at a distance $d_{\rm LeoT}\approx 420\rm\ kpc$– shares striking
similarities with the properties derived for Cloud-9. The stellar mass of Leo
T is $M_{\rm gal}\approx 10^{5}\ M_{\odot}$ (Irwin et al., 2007). The galaxy
displays old and young stellar populations but no signs of ongoing star
formation or H ii regions at present despite having a total H i mass, $M_{\rm
HI}=4.1\times 10^{5}\ M_{\odot}$ (Irwin et al., 2007; Weisz et al., 2012;
Adams & Oosterloo, 2018, and references therein). Moreover, as we explore
next, it would be difficult to detect its stellar counterpart at the fiducial
distance of Cloud-9 in the DESI footprint.

To determine whether it is sensible to assume that Cloud-9 has a luminous
counterpart similar to or fainter than Leo T, we consider a random realization
of Leo T placed at Cloud-9’s fiducial distance. To construct the photometric
model, we use Leo T’s structural parameters and the photometry for 182
individual stars reported by Irwin et al. (2007), who find that stars with
colors, $(g-r)<1$, and magnitudes, $m_{g}<25$, are distributed in projection
according to the following Plummer profile:333These parameters correspond to
those listed in Table 1 of Irwin et al. (2007), and reproduce the Plummer
profile depicted in Figure 5 of that paper. However, we note that the central
density value quoted by Irwin et al. (2007) is erroneously expressed in
magnitudes instead of $\rm arcsec^{-2}$.

$N(R)=26.9{\rm\ arcsec^{-2}}\left[1+\left(\displaystyle\frac{R}{1.4\rm\
arcmin}\right)^{2}\right],$ (5)

Figure 7: The top left panel shows the two innermost Cloud-9’s H i column
density isocontours superimposed on a DESI LS cutout in the $g$ band. The top
right panel shows the same plus a random realization of Leo T’s best Plummer
fit at Cloud-9’s fiducial distance and centered at the peak column density
(white dots). The bottom left panel shows this Plummer realization injected
into the DESI LS data, in which we assigned magnitudes to the individual stars
from the CMD shown in the bottom right panel and convolved with the DESI LS
PSF of this field. For reference, the white open circles indicate the location
of the eight brightest stars redder than $(g-r)=0.5$ (red dots in the CMD
panel).

One realization is shown in the top right panel of Figure 6, in which the
white dots display the location of the stars for which we have photometric
information, randomly sampled from the Plummer profile that best fits Leo T.
The image in the background corresponds to a DESI LS cutout in the $g$ band.
For reference, the panels are $2\times 2$ arcmin, and the solid lines indicate
the innermost column density isocontours shown in Figure 2. We place Leo T’s
random realization at Cloud-9’s distance and assign magnitudes to the stars
from the color-magnitude diagram (CMD) displayed in the bottom right panel of
the same figure. The CMD data was also taken from the work of Irwin et al.
(2007) and shifted from a distance of $420\rm\ kpc$ to Cloud-9’s distance,
$d=4.66\rm\ Mpc$.444Although Irwin et al. (2007) report that Leo T’s CMD
corresponds to the stars in the inner $1.8$ arcmin from the center, we find
that this assumption results in a wrong normalization for the reported Plummer
profile. Therefore, we do not enforce a distance constraint to populate our
Plummer realization, which results in a perfect match to Irwin et al. (2007)’s
results and a self-consistent normalization for the profile.

Detecting Leo T’s tip of the red giant branch at Cloud-9’s distance would thus
require photometric observations deeper than $m_{g}\gtrsim 26$ magnitudes in
the $g$ band, above which distinguishing stars from the expected large number
of unresolved external galaxies becomes challenging with ground-based
facilities (see, e.g., Soumagnac et al., 2015, and references therein).

To further highlight the difficulty of observing the stellar counterpart of a
Leo T-type dwarf at Cloud-9’s distance, we inject our random realization into
the DESI LS cutout centered at Cloud-9’s location and convolve it with the
corresponding point-spread-function (PSF). The top left panel of Figure 6
shows the original cutout in the g-band whereas the bottom left panel shows
the same field after injecting Leo T’s model. A quick visual inspection and
comparison with the original cutout demonstrates that the brightest stars (we
indicate the location of the eight brightest stars redder than $(g-r)>0.5$
with red symbols in the right-hand panels) are too faint to be seen in DESI.

This exercise demonstrates that detecting a Leo T-like galaxy at Cloud-9’s
location would be challenging with DESI imaging. Indeed, our exercise suggests
that none of the Leo T stars can be detected with current data. Furthermore,
the inability to see a potential Leo-T candidate in this cutout agrees with
our measurement of the $5\sigma$ PSF magnitude detection limit at Cloud-9’s
location of approximately 25.6 in the g band.

Hence, we conclude that should Cloud-9 have an optical counterpart centered at
its peak column density, its luminosity should be either comparable to or
fainter than Leo T if the galaxy has a similar extent. This argument places a
robust upper limit to the total stellar mass of any luminous counterpart
associated with Cloud-9, $M_{\rm gal,CL9}\lesssim 10^{5}\ M_{\odot}$, which
aligns with previous estimates.

How does Leo T’s neutral hydrogen distribution compare with that of Cloud-9?
To answer this question and carry out a meaningful comparison we mock-observe
Leo T with the VLA-D “clean” beam, assuming it is at Cloud-9’s distance. To
this end, we first derive Leo T’s intrinsic H i column density profile by
heuristically “deconvolving” the total averaged profile observed with the
Westerbork Synthesis Radio Telescope (WSRT) (Adams & Oosterloo, 2018).

The top panel of Figure 7 shows that the following model reproduces Adams &
Oosterloo (2018) detailed observations after convolving it with the reported
WSRT beam:555The WSRT beam for this observations is $(\theta_{M},\theta_{m})$
= $(57.3^{\prime\prime},15.7^{\prime\prime})$ (see Sec. 2 of Adams &
Oosterloo, 2018).

$N_{\rm HI}=3.1\times 10^{20}{\rm\
cm^{-2}}\exp\left[-\left(\displaystyle\frac{R}{0.21\rm\
kpc}\right)^{3/2}\right],$ (6)

We use this model to place Leo T’s H i component at Cloud-9’s distance, after
which we convolve it with the VLA-D “clean” beam.

Comparing the mock-observed Leo T VLA-D column density profile (orange line in
the bottom panel) with that of Cloud-9 (black symbols with shaded regions
displaying $1\sigma$ uncertainties) reveals that both systems share a similar
central column density value but Cloud-9 is more extended than Leo T, possibly
due to Cloud-9’s perturbed morphology.

Figure 8: The top panel shows Leo T’s total averaged H i column density
profile measured by Adams & Oosterloo (2018) (grey triangles) together with
our adopted model (Eq. 6) placed at Leo T’s distance and convolved with the
WRST beam (orange solid line). The bottom panel shows the observed Cloud-9’s
VLA-D H i profile with $1-\sigma$ uncertainties (black circles and shaded
region, respectively), together with our Leo T model placed at Cloud-9’s
distance, $d=4.66\rm\ Mpc$, convolved with the VLA-D beam.

The broad similarities between the gaseous components of Leo T and Cloud-9,
combined with the limits of current optical surveys at Cloud-9’s location,
make it difficult to rule out the presence of a luminous galaxy counterpart
with luminosity comparable to or fainter than Leo T. Moreover, the high
expected frequency of luminous galaxies in halos with masses similar to that
derived for Cloud-9 suggests that Cloud-9 is more likely an analog of Leo T
rather than a starless RELHIC. Deep photometric observations of this field
with the Hubble Space Telescope could help resolve individual stars fainter
than 26 magnitudes, revealing an associated faint galaxy or tightening current
constraints on the stellar mass of any luminous counterpart associated with
Cloud-9.

##  5 Discussion

While Cloud-9 appears round and symmetric in FAST observations, our new VLA-D
data reveal that the system has a more complex morphology than previously
anticipated. As discussed in Section 3, Cloud-9’s neutral hydrogen reservoir
appears perturbed compared to a spherical distribution centered at the peak
column density, showing compression towards the west and expansion towards the
east. Additionally, taken at face value, Cloud-9’s detailed H i column density
profile is more extended than expected for the BL17 RELHICs matching its
central column density value, and also more extended than Leo T, a local dwarf
galaxy with similar inferred properties. However, these features do not
immediately rule out the RELHIC or Leo T-like counterpart interpretation for
the system.

Firstly, if Cloud-9 were a RELHIC, its perturbed morphology could simply be
ascribed to ram pressure caused by its motion relative to a gaseous background
—a well-known process affecting galaxies that fall onto massive companions
(e.g. Gunn & Gott, 1972; Abadi et al., 1999, and references therein), and low-
mass systems in isolation as they interact with the cosmic web (Benítez-
Llambay et al., 2013; Herzog et al., 2023). Moreover, the H i content of some
relatively isolated dwarf galaxies within our Local Group also displays signs
of ram pressure (see, e.g., McConnachie et al., 2007).

The relative proximity of Cloud-9 to the massive galaxy M94 ($52$ arcmin in
projection, corresponding to a projected distance of about $\approx 70$ kpc)
makes this interpretation plausible.

Given these issues, it is then not entirely surprising that the BL17 model
–designed to describe the gas distribution of RELHICs in an otherwise
unperturbed background– fails to describe Cloud-9’s gas distribution. Whether
or not ram pressure can give rise to a BL17 RELHIC resembling Cloud-9 is,
however, uncertain. There are currently no theoretical predictions regarding
the detailed distribution of the neutral hydrogen content of RELHICs
undergoing ram pressure stripping.

Ram pressure-stripped starless halos have been identified in simulations by
BL17, who noted that some of these systems, even when not necessarily
satellites of a massive companion, lose their gas through cosmic web
stripping. This process arises from the interaction between the small gas
reservoir of RELHICs and the gas in the filaments and sheets of the cosmic
web. These authors also observed that RELHICs efficiently lose their loosely
bound gas when encountering even mild gas overdensities in the field. This
results in a dichotomy where the gas content of starless halos is either in
hydrostatic equilibrium with their host dark matter halo or is lost
completely. However, this dichotomy between RELHICs and “gas-free” halos may
stem from the limited resolution of these simulations666 BL17’s simulations
cannot reliably track the gas content of systems with masses, $M_{\rm
gas}<<10^{5}\ M_{\odot}$., making it unclear whether or not the central H i
structure of ram pressure-stripped RELHICs, as modeled by BL17, would resemble
that of Cloud-9. Further high-resolution numerical simulations, tailored to
resolve the gas-stripping process in RELHICs, may shed light on these
uncertainties.

On the other hand, the intrinsic properties of the BL17 RELHICs shown in
Figure 6 demonstrate that these systems are colder in the central regions
compared to the inferred temperature for Cloud-9. Given the sensitivity of the
models to the adopted temperature, as highlighted in Figure 6, it is not
entirely surprising that the BL17 model fails to match observations. Within
the RELHIC interpretation, the different temperature for Cloud-9 compared to
the expectations from the BL17 model may originate from either variations in
the local intensity of the UVB, or perturbations induced by ram pressure, or
both. Moreover, FAST and GBT observations suggest the presence of diffuse
emission missed in our VLA-D observations, displaying a larger broadening.
These facts, together with Cloud-9’s perturbed morphology, may be signaling
the presence of a mechanism responsible for increasing the temperature of the
system compared to what is naively expected.

Modeling Cloud-9 as an isothermal RELHIC, with a temperature matching our
VLD-D observations, has proven successful. Specifically, we have demonstrated
that Cloud-9’s column density profile can be accurately modeled by assuming an
underlying dark matter distribution consistent with that expected for
$\Lambda$CDM halos. This success supports the idea that Cloud-9 is a starless
dark matter halo in near hydrostatic equilibrium with its dark matter
component. The success of this model in matching Cloud-9’s column density
profile stems from the hotter inner temperature assumed compared to the BL17
RELHIC model. Understanding in detail the mechanisms that could increase the
inner temperature of RELHICs and make them depart from the BL17 equation of
state requires, however, more advanced modeling.

Further progress and a deeper understanding of this system in the context of
RELHICs may be achieved through very high-resolution simulations of individual
RELHICs inhabiting $\Lambda$CDM halos, with sufficient resolution to trace
their inner H i reservoir as it becomes stripped by external ambient gas
moving relative to the system, modeling also the effect of the UVB self-
consistently using radiative transfer calculations to estimate the ionization
state of the gas as it expands. Understanding the telltale signatures of ram-
pressure stripping in RELHICs, particularly regarding the central distribution
of the neutral hydrogen content and resulting temperature structure, will
undoubtedly provide robust theoretical foundations to better interpret our
observations.

Additionally, understanding Cloud-9’s outer morphology with deeper radio
observations may prove useful in assessing its external structure, whereas
high-resolution radio observations will help constrain its inner morphology
and assess whether the system is in equilibrium.

If Cloud-9 were a luminous galaxy similar to Leo T, but with its gas being
stripped by ram pressure, our analysis suggests that deeper observations might
detect its stars. The best chances of identifying a galaxy associated with
Cloud-9 would be through deep HST observations that can resolve faint
individual stars and distinguish them from background sources. The galaxy
could be detected and characterized by reaching $5\sigma$ PSF magnitudes,
$m_{g}\approx 30$. For instance, observations reaching $m_{g}\lesssim 29$
would detect Leo T’s 20 brightest stars at Cloud-9’s distance and reveal the
tip of the red giant branch. Observations reaching $m_{g}\lesssim 30$ would
identify approximately 30 stars.

Finally, detecting and resolving the stars of a galaxy at Cloud-9’s location
would be remarkable, as this would become the faintest galaxy known at that
distance. In turn, characterizing its stellar populations would help place
robust constraints on the distance of the system and benchmark current
estimates based on the redshift of the H i 21-cm line.

##  6 Conclusions

We have presented VLA-D interferometric observations of a recently discovered
H i cloud (Cloud-9) located on the outskirts of M94 and lacking a luminous
counterpart. Our analysis follows up on the discovery of the system by Z23 and
the subsequent analysis by BLN23, who showed that the system is consistent
with being a $\Lambda$CDM starless dark matter halo filled with gas in
hydrostatic equilibrium and in thermal equilibrium with the external UVB. The
main results of our work may be summarized as follows:

  * •

Although FAST observations by Z23 reveal Cloud-9 as a largely spherical
system, our new interferometric VLA-D observations indicate a more complex
morphology. We find that the system appears perturbed, exhibiting features
consistent with compression of its H i content towards the west and expansion
towards the east. We have discussed the possibility that these features
originate from ram pressure between the inner gas content of the system and
ambient gas associated with M94.

  * •

Unlike KS24, we find that Cloud-9’s 21-cm H i line appears symmetric and
Gaussian, with a very narrow broadening, $W_{50}=12\pm 1\rm\ km\ s^{-1}$. This
value sharply contrasts with that derived by Z23 and KS24 using the FAST and
GBT instruments, respectively. We speculate that the differences may arise
from our interferometric observations missing some diffuse emission component.
However, our observations compellingly demonstrate the presence of a
prominent, dynamically cold, neutral hydrogen component.

  * •

Cloud-9 does not exhibit signs of ordered rotation within the sampled scales.
The lack of rotation suggests that if the system is in equilibrium, it is the
gas pressure that maintains this balance. Assuming equilibrium, the extent of
Cloud-9’s gas distribution and the narrow H i 21-cm line profile indicates the
presence of a large amount of gravitational mass that is not neutral hydrogen,
suggesting the system is either a dark matter-dominated galaxy or a starless
RELHIC.

  * •

Taken at face value, Cloud-9’s H i column density profile is more extended
than the BL17$+\Lambda$CDM RELHIC models. As noted in earlier work by BLN23
and confirmed here, the system appears much more extended in projection than
RELHICs. If the system is indeed in equilibrium with a dark matter halo and
thermal equilibrium with the Haardt & Madau (2012) UVB, this would imply halo
properties distinct from those predicted by $\Lambda$CDM. However, this
discrepancy may also stem from Cloud-9’s outer perturbed morphology or
discrepancies in the temperature of the system compared to the modeling.
Assessing these scenarios requires further theoretical work to understand the
impact of ram-pressure stripping on the gaseous structure of RELHICs,
particularly on the centrally concentrated H i component and its temperature.

  * •

Despite the difficulties of the BL17 model in describing Cloud-9’s column
density profile, we find that a model consisting of an isothermal gaseous
sphere with a sound speed of $\approx 12\ \rm km\ s^{-1}$ matching the
broadening of the 21-cm H i line successfully reproduces Cloud-9’s column
density profile. This is achievable if the cloud is in hydrostatic equilibrium
with $\Lambda$CDM halos of mass $3.9\lesssim M_{200}/10^{9}\ M_{\odot}\lesssim
4.3$. This model is arguably better suited for Cloud-9 given observational
constraints and the uncertain intensity of the UVB at Cloud-9’s location.

  * •

The high expected frequency of luminous galaxies in halos as massive as
inferred for Cloud-9 ($M_{200}>3\times 10^{9}\ M_{\odot}$) suggests that
Cloud-9 could be a luminous galaxy. Moreover, we find that Cloud-9’s gas
properties resemble those of Leo T, a nearby H i-rich dwarf galaxy in the
Local Group that is not currently forming stars. Specifically, their similar
inner H i column density suggests that Cloud-9 may be a Leo T-like galaxy.
This possibility is consistent with current imaging DESI LS data in which it
is challenging to detect a Leo T analog. Our analysis places a realistic upper
limit to the stellar mass of a luminous counterpart associated with Cloud-9 of
about $M_{\rm gal,CL9}\lesssim 10^{5}\ M_{\odot}$.

To make progress in the characterization of Cloud-9, we recommend further
observations in both optical and radio wavelengths. Given the high likelihood
of Cloud-9 hosting a luminous galaxy, we suggest deep photometric observations
with either the Hubble Space Telescope or the James Web Space Telescope,
targeting PSF magnitudes $m_{g}>30$. The ability of these observations to
sample the tip of the red giant branch of a Leo T analog at Cloud-9’s distance
would allow us to detect the galaxy or significantly lower the limits on the
stellar mass of such a counterpart if no stars are detected. Given the
importance of RELHICs as a test for the $\Lambda$CDM model, we believe these
efforts are well justified. Additionally, our analysis indicates that any
galaxy associated with Cloud-9 will likely become the faintest known to date
at that distance, and certainly, not the first in the local volume discovered
through follow-up observations of 21-cm emission (see, e.g., Giovanelli et
al., 2013, for the discovery of the faint Leo P galaxy).

Secondly, to better understand the gaseous structure of the system, we
recommend efforts on two fronts. First, further radio observations should aim
to resolve both larger and smaller scales, potentially helping constrain the
temperature profile of the system. Observing the system with VLA in the B or
C-array configuration would help decrease the beam’s impact and better resolve
the central regions, crucial for assessing Cloud-9’s internal equilibrium.
Second, additional observations with FAST would allow us to study the outer
diffuse emission further and understand the true extent of Cloud-9, helping
the assessment of departures from equilibrium or signs of coherent rotation on
large scales.

Finally, further theoretical work tailored to understand the gas-stripping
process in RELHICs, particularly on the central H i reservoir and its
temperature, may prove useful and help better interpret Cloud-9’s morphology.
Current simulations that resolve the gas content of RELHICs lack sufficient
resolution to characterize the structure of affected systems in detail. Until
these analyses are available, it will be difficult to determine whether the
tension between Cloud-9 and $\Lambda$CDM RELHICs, as modeled by BL17,
challenges $\Lambda$CDM or simply reflects Cloud-9’s perturbed morphology.

In conclusion, Cloud-9 still proves to be a fascinating system that,
regardless of its nature, likely probes the faint edge of galaxy formation. We
strongly advocate for deep optical and radio observations, alongside high-
resolution simulations that can resolve the H i stripping process of RELHICs
and faint dwarfs. These efforts will not only clarify the nature of this
system but also provide valuable insights into the understanding of the
clustering of dark matter and the process of galaxy formation at the smallest
scales.

The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. A.B.L. acknowledges support from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (GA
101026328). This study is supported by the Italian Ministry for Research and
University (MUR) under Grant “Progetto Dipartimenti di Eccellenza 2023-2027”
(BiCoQ).

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Ilias S. Kotsireas, Arne Winterhof

11institutetext: CARGO Lab, Wilfrid Laurier University, Waterloo, ON N2L 3C5,
Canada,
11email<EMAIL_ADDRESS>RICAM, Austrian Academy of Sciences,
Altenbergerstr. 69, 4040 Linz, Austria,
22email<EMAIL_ADDRESS>
# Quaternary Legendre Pairs

Ilias S. Kotsireas 11 Arne Winterhof 22

###### Abstract

We introduce quaternary Legendre pairs of length $\ell$. In contrast to binary
Legendre pairs they can exist for even $\ell$ as well. First we show that they
are pertinent to the construction of quaternary Hadamard matrices of order
$2\ell+2$ and thus of binary Hadamard matrices of order $4\ell+4$. Then for a
prime $p>2$ we present a construction of a pair of sequences of length $p$
from which we can derive quaternary Legendre pairs of length $\ell=2p$ by
decompression for $p=3,5,7,13,19,31,41$. Moreover, we give also constructions
of Legendre pairs of length $\ell$ for all remaining even $\ell\leq 24$.

###### keywords:

quaternary Legendre pairs, discrete Fourier transform, compression,
autocorrelation, quaternary Hadamard matrices

##  1 Introduction

Let

$A=[a_{0},a_{1},\ldots,a_{\ell-1}]\in\mathbb{C}^{\ell}$

be a finite complex sequence of length $\ell$ periodically continued with
period $\ell$, that is,

$a_{j+\ell}=a_{j},\quad j=0,1,\ldots$

The periodic autocorrelation function $PAF(A,s)$ of $A$ at lag $s$ is

$PAF(A,s)=\sum_{j=0}^{\ell-1}a_{j}\overline{a_{j+s}},\quad
s=0,1,\ldots,\ell-1,$

where $\overline{a}$ is the complex conjugate of $a\in\mathbb{C}$. Two
sequences

$A=[a_{0},a_{1},\ldots,a_{\ell-1}]\quad\mbox{and}\quad
B=[b_{0},b_{1},\ldots,b_{\ell-1}]$

of length $\ell$ form a Legendre pair $(A,B)$ if

$PAF(A,s)+PAF(B,s)=-2,\quad
s=1,2,\ldots,\left\lfloor\frac{\ell}{2}\right\rfloor.$ (1)

Note that

$PAF(A,\ell-s)=\overline{PAF(A,s)},\quad
s=1,2,\ldots,\left\lfloor\frac{\ell}{2}\right\rfloor,$

and we may extend the range of $s$ in (1) to $s=1,2,\ldots,\ell-1$.

A Legendre pair $(A,B)$ is binary if both sequences $A$ and $B$ are binary,
that is, $A,B\in\\{-1,1\\}^{\ell}$. Binary Legendre pairs of length $\ell$ are
pertinent to the construction of Hadamard matrices of order $2\ell+2$, see for
example [10, Theorem 3], [13, Theorem 1] and [1, Theorem 2]. Since the order
of an Hadamard matrix is either $2$ or divisible by $4$, there is no binary
Legendre pair $(A,B)$ of even length $\ell$. Binary Legendre pairs of length
$\ell$ are known if

  * •

$\ell>2$ is a prime (Legendre sequences), see [10, Section 3],

  * •

$2\ell+1$ is a prime power (characteristic sequences of Szekeres difference
sets), see [18, Theorem 5.18],

  * •

$\ell=2^{r}-1$, $r\geq 2$ ($m$-sequences), see for example [11, Property 5.5],

  * •

$\ell=p(p+2)$ where $p$ and $p+2$ are twin primes (twin-prime sequence), see
for example [2, 17].

Several other constructions for a finite set of $\ell$ are known [3, 14, 19].
The list of integers less than $200$ for which the question of existence of
binary Legendre pairs remains open is

$\ell=115,145,147,159,161,169,175,177,185,187,195,$

see [3].

Note that for all these $\ell<200$ there exists an Hadamard matrix of order
$2\ell+2$ constructed without using Legendre pairs of order $\ell$. The
smallest orders for which the existence of an Hadamard matrix is still open
are

$2\ell+2=668,716,892,1132,1244,1388,1436,1676,1772,1916,1948,1964,$

see [7].

A Legendre pair $(A,B)$ is quaternary if both sequences $A$ and $B$ are
quaternary, that is, $A,B\in\\{-1,1,-i,i\\}^{\ell}$, where $i=\sqrt{-1}$.
Quaternary Legendre pairs exist also for even $\ell$. For example, for
$\ell=2$ we take

$A=[1,-1]\quad\mbox{and}\quad B=[1,i],$

that is,

$PAF(A,1)=-2\quad\mbox{and}\quad PAF(B,1)=0.$

A quaternary (or complex) Hadamard matrix is an $n\times n$-matrix with
entries in $\\{-1,1,-i,i\\}$ which satisfies

$H\overline{H}^{T}=nI_{n},$

where $I_{n}$ denotes the $n\times n$ identity matrix. The order $n$ of a
quaternary Hadamard matrix must be even and it is conjectured that there is a
quaternary Hadamard matrix for any even order. The smallest undecided order is
$n=94$, see [6, 9].

In Section 2 we construct quaternary Hadamard matrices of order $2\ell+2$ from
quaternary Legendre pairs of length $\ell$ (and thus binary Hadamard matrices
of order $4\ell+4$).

In Section 3, for a prime $p>2$ we provide a construction of a ‘seed’ pair of
length $p$ from which we can derive a quaternary Legendre pair of length

$\ell=2p\quad\mbox{for}\quad p=3,5,7,13,19,31,41.$

The smallest prime for which this method fails is $p=11$. However, we also
provide a quaternary Legendre pair of length $22$ as well as pairs for the
remaining even $\ell\leq 24$ in Section 4. The main idea to reduce the search
space is a modified PSD-criterion that certain PSDs (Power Spectral Densities)
are integers which are sums of two integer squares. We conjecture that there
is a quaternary Legendre pair for any even length $\ell$, which is stronger
than the (complex) Hadamard conjecture. The smallest open case is $\ell=28$.

##  2 Quaternary Legendre pairs and (quaternary) Hadamard matrices

First note that quaternary Legendre pairs are (almost) balanced.

######  Lemma 2.1.

Write

$A=[a_{0},a_{1},\ldots,a_{\ell-1}],\quad B=[b_{0},b_{1},\ldots,b_{\ell-1}]$

and let $(A,B)$ be a quaternary Legendre pair of length $\ell$. Put

$\alpha=\sum_{j=0}^{\ell-1}a_{j}\quad\mbox{and}\quad\beta=\sum_{j=0}^{\ell-1}b_{j}.$

Then we have

$|\alpha|^{2}+|\beta|^{2}=2,$ $\alpha,\beta\in\\{-1,1,-i,i\\}\quad\mbox{if
$\ell$ is odd}$

and

$\\{\alpha,\beta\\}\in\\{\\{0,1+i\\},\\{0,1-i\\},\\{0,-1+i\\},\\{0,-1-i\\}\\}\quad\mbox{if
$\ell$ is even}.$

Proof. We have

$|\alpha|^{2}+|\beta|^{2}=\sum_{s=0}^{\ell-1}(PAF(A,s)+PAF(B,s))=2\ell+(\ell-1)(-2)=2.$

For odd $\ell$ we have $\alpha\beta\neq 0$ and since
$\alpha,\beta\in\mathbb{Z}[i]$ we get $|\alpha|=|\beta|=1$ and the result
follows. For even $\ell$ we have either $\alpha=0$ and $|\beta|^{2}=2$ or
$|\alpha|^{2}=2$ and $\beta=0$ and the result follows. $\Box$

For odd $\ell$ we may assume $\alpha=\beta=1$ since otherwise we multiply the
sequence $A$ element-wise by $\overline{\alpha}$ and $B$ element-wise by
$\overline{\beta}$ without changing the PAFs. For even $\ell$ we may assume

$\alpha=0\quad\mbox{and}\quad\beta=1+i$ (2)

since otherwise we may interchange $A$ and $B$ if $\beta=0$ to get $\alpha=0$
and $|\beta|^{2}=2$, then we may multiply $B$ element-wise by $-1$ if
$\beta=-1\pm i$ to get $\beta=1\mp i$, and take the complex conjugates of $A$
and $B$ if $\beta=1-i$.

We use the notation

$C(A)=\left(\begin{array}[]{ccccc}a_{0}&a_{1}&\ldots&a_{\ell-2}&a_{\ell-1}\\\
a_{\ell-1}&a_{0}&\ldots&a_{\ell-3}&a_{\ell-2}\\\
\vdots&\vdots&&\vdots&\vdots\\\ a_{2}&a_{3}&\ldots&a_{0}&a_{1}\\\
a_{1}&a_{2}&\ldots&a_{\ell-1}&a_{0}\end{array}\right),\quad
A=[a_{0},a_{1},\ldots,a_{\ell-1}],$

and recall that $C(A)C(B)=C(B)C(A)$, see for example [5, Theorem 3.2.4]. Then
the following construction of quaternary Hadamard matrices from quaternary
Legendre pairs can be easily verified.

######  Theorem 2.2.

Let $(A,B)$ be a quaternary Legendre pair of odd length $\ell$ with

$\alpha=\beta=1.$

Then

$\left(\begin{array}[]{cc|cccccc}-1&-1&1&\ldots&1&1&\ldots&1\\\
-1&1&1&\ldots&1&-1&\ldots&-1\\\ \hline\cr 1&1&&&&&&\\\
\vdots&\vdots&&C(A)&&&C(B)&\\\ 1&1&&&&&&\\\ \hline\cr 1&-1&&&&&&\\\
\vdots&\vdots&&C(\overline{B})^{T}&&&-C(\overline{A})^{T}&\\\
1&-1&&&&&&\end{array}\right)$

is a quaternary Hadamard matrix of order $2\ell+2$.

Let $(A,B)$ be a quaternary Legendre pair of even length $\ell$ with

$\alpha=0\quad\mbox{and}\quad\beta=1+i.$

Then

$\left(\begin{array}[]{cc|cccccc}-1&i&1&\ldots&1&1&\ldots&1\\\
-i&1&1&\ldots&1&-1&\ldots&-1\\\ \hline\cr 1&1&&&&&&\\\
\vdots&\vdots&&C(A)&&&C(B)&\\\ 1&1&&&&&&\\\ \hline\cr 1&-1&&&&&&\\\
\vdots&\vdots&&C(\overline{B})^{T}&&&-C(\overline{A})^{T}&\\\
1&-1&&&&&&\end{array}\right)$

is a quaternary Hadamard matrix of order $2\ell+2$.

Note that a binary Hadamard matrix of order $4\ell+4$ can be constructed from
a quaternary Hadamard matrix of order $2\ell+2$, see [4, Theorem 1], that is,
from a quaternary Legendre pair of length $\ell$.

##  3 $2$-compressed pairs and ‘seeds’

For binary sequences the concept of compression was introduced in [8]. We
adjust it here to quaternary sequences of length $\ell=2p$. Write

$A=[a_{0},a_{1},\ldots,a_{2p-1}],\quad B=[b_{0},b_{1},\ldots,b_{2p-1}]$

and let $(A,B)\in\\{-1,1,-i,i\\}^{4p}$ be a quaternary Legendre pair of length
$2p$.

The $2$-compression of $(A,B)$ is the pair $(A_{p},B_{p})$ of sequences of
length $p$ with

$A_{p}=\left[a_{0}^{(p)},a_{1}^{(p)},\ldots,a_{p-1}^{(p)}\right],\quad
B_{p}=\left[b_{0}^{(p)},b_{1}^{(p)},\ldots,b_{p-1}^{(p)}\right]$

defined by

$a_{j}^{(p)}=a_{j}+a_{j+p},b_{j}^{(p)}=b_{j}+b_{j+p}\in\\{-2,0,2,-2i,2i,-1-i,-1+i,1-i,1+i\\}$

for $j=0,1,\ldots,p-1$. For prime $p>2$, we start with a theoretic
construction of a ‘seed’ pair $(A_{p},B_{p})$ and ’hope’ to be able to
decompress it to a quaternary Legendre pair $(A,B)$. In particular, the
decompression $A$ of $A_{p}$ is essentially unique (up to four choices of
$(a_{0},a_{p})$) and can be explicitly described. So we are only left with
finding a decompression $B$ of $B_{p}$.

######  Definition 3.1.

For a prime $p>2$ let two ‘seed’ sequences

$A_{p}=\left[a_{0}^{(p)},a_{1}^{(p)},\ldots,a_{p-1}^{(p)}\right]\quad\mbox{and}\quad
B_{p}=\left[b_{0}^{(p)},b_{1}^{(p)},\ldots,b_{p-1}^{(p)}\right]$

be defined by

$a_{j}^{(p)}=\left\\{\begin{array}[]{ll}0,&j=0,\\\
2\left(\frac{j}{p}\right),&j=1,2,\ldots,p-1,\end{array}\right.$

and

$b_{j}^{(p)}=\left\\{\begin{array}[]{ll}1+i,&j=0,\\\
0,&j=1,2,\ldots,p-1,\end{array}\right.$

periodically continued with period $p$, where $\left(\frac{.}{.}\right)$ is
the Legendre symbol.

Note that the definition of $A_{p}$ is inspired by the construction of binary
Legendre pairs of length $p$ which can be also considered the core of Paley’s
construction of Hadamard matrices.

The discrete Fourier transform $DFT(A,s)$ of $A$ at lag $s$ is

$DFT(A,s)=\sum_{j=0}^{\ell-1}a_{j}\xi_{\ell}^{js},\quad
s=0,1,2,\ldots,\ell-1,$

where $\xi_{\ell}$ is the primitive $\ell$th root of unity

$\xi_{\ell}=\exp\left(\frac{2\pi i}{\ell}\right).$

The power spectral density $PSD(A,s)$ of $A$ at $s$ is

$PSD(A,s)=\left|DFT(A,s)\right|^{2}=\sum_{j=0}^{\ell-1}PAF(A,j)\xi_{\ell}^{-js}=\sum_{j=0}^{\ell-1}\overline{PAF(A,j)}\xi_{\ell}^{js}.$

For a Legendre pair $(A,B)$ of length $\ell$ we immediately get

$PSD(A,s)+PSD(B,s)=2\ell+2,\quad
s=1,2,\ldots,\left\lfloor\frac{\ell}{2}\right\rfloor.$

The PSD-criterion to search for (binary) Legendre pairs $(A,B)$ was introduced
in [10] and discards all candidate sequences $B$ with

$PSD(B,s)>2\ell+2\quad\mbox{for some
}s\in\left\\{1,2,\ldots,\left\lfloor\frac{\ell}{2}\right\rfloor\right\\}$ (3)

from the search. Here we can use a much stronger PSD-criterion since we know
all values $PSD(B,s)$, $s=1,2,\ldots,\left\lfloor\frac{\ell}{2}\right\rfloor$.

In the following theorem we determine the four decompressions $A$ of $A_{p}$
and mention many properties of possible decompressions $B_{p}$ of $B$
including all of their PSDs.

######  Theorem 3.2.

The construction of Definition 3.1 satisfies:

  1. 1.

$PAF(A_{p},0)=4(p-1),\quad PAF(B_{p},0)=2,$ $PAF(A_{p},s)=-4,\quad
PAF(B_{p},s)=0,\quad s=1,2,\ldots,p-1.$

  2. 2.

$DFT(A_{p},s)=\left\\{\begin{array}[]{cc}2\left(\frac{s}{p}\right)p^{1/2},&p\equiv
1\bmod 4,\\\ 2i\left(\frac{s}{p}\right)p^{1/2},&p\equiv 3\bmod
4,\end{array}\right.$ $DFT(B_{p},s)=1+i,\quad s=1,2,\ldots,p-1,$
$PSD(A_{p},s)=4p,\quad PSD(B_{p},s)=2,\quad s=1,2,\ldots,p-1,$
$PSD(A_{p},s)+PSD(B_{p},s)=2(2p+1),\quad s=1,2,\ldots,p-1.$

  3. 3.

There are exactly four $2$-decompressions $A=[a_{0},a_{1},\ldots,a_{2p-1}]$ of
$A_{p}$ defined by

$a_{p}=-a_{0},\quad a_{0}\in\\{-1,1,-i,i\\}$

and

$a_{j}=a_{j+p}=\left(\frac{j}{p}\right),\quad j=1,2,\ldots,p-1.$

They satisfy

$PAF(A,0)=2p,\quad PAF(A,p)=2p-4,$ $PAF(A,s)=PAF(A,2p-s)=-2,\quad
s=1,2,\ldots,p-1,$
$DFT(A,2s)=2\left(\frac{s}{p}\right)\left\\{\begin{array}[]{cc}p^{1/2},&p\equiv
1\bmod 4,\\\ ip^{1/2},&p\equiv 3\bmod 4,\end{array}\right.$
$PSD(A,2s)=4p,\quad s=1,2,\ldots,p-1,$ $DFT(A,2s-1)=2a_{0},\quad
PSD(A,2s-1)=4,\quad s=1,2,\ldots,p.$

  4. 4.

Assume that $B$ is a $2$-decompression of $B_{p}$. Then we have

$b_{0}+b_{p}=1+i\quad\mbox{and}\quad b_{j+p}=-b_{j},\quad j=1,2,\ldots,p-1,$
$PAF(B,0)=2p,\quad PAF(B,p)=-2p+2,$ $DFT(B,2s)=1+i,\quad PSD(B,2s)=2,\quad
s=1,2,\ldots,p-1.$

  5. 5.

In addition, if $(A,B)$ is a Legendre pair, we have

$PAF(B,s)=PAF(B,2p-s)=0,\quad s=1,2,\ldots,p-1,$ $PSD(B,2s-1)=4p-2,\quad
s=1,2,\ldots,p,$ $DFT(B,p)=a+ib\quad\mbox{with integers $a,b$ satisfying
}a^{2}+b^{2}=4p-2$

and

$a\equiv b\equiv 1\bmod 2.$

Proof. 1\. We have

$PAF(A_{p},s)=4\sum_{j=0}^{p-1}\left(\frac{j}{p}\right)\left(\frac{j+s}{p}\right)=\left\\{\begin{array}[]{cc}4(p-1),&s=0,\\\
-4,&s=1,\ldots,p-1,\end{array}\right.$

see for example [16, Lemma 6.4.7]. The values of $PSD(B_{p},s)$ are trivial.

2\. Next we have

$DFT(A,2s)=DFT(A_{p},s)=2\sum_{j=1}^{p-1}\left(\frac{j}{p}\right)\xi_{p}^{js}=2G_{s},$

where $G_{s}$ is a Gauss sum, that is,

$G_{s}=\left\\{\begin{array}[]{cc}\left(\frac{s}{p}\right)p^{1/2},&p\equiv
1\bmod 4,\\\ i\left(\frac{s}{p}\right)p^{1/2},&p\equiv 3\bmod
4,\end{array}\right.$

by [15, Theorems 5.12 and 5.15]. The rest of 2. is trivial.

3\. We only verify

$DFT(A,2s-1)=\sum_{j=0}^{p-1}(a_{j}-a_{j+p})\xi_{2p}^{(2s-1)j}=2a_{0}$

and the rest is straightforward.

4\. is trivial.

5\. Now assume that $(A,B)$ is a Legendre pair, that is in particular,

$PSD(A,p)+PSD(B,p)=4p+2$

and thus $PSD(B,p)=4p-2$. Since

$DFT(B,p)\in\mathbb{Z}[i]$

and

$DFT(B,p)=\sum_{j=0}^{p-1}(b_{j}-b_{j+p})=b_{0}-b_{p}+2\sum_{j=1}^{p-1}b_{j}\equiv
1+i\bmod 2,$

that is $DFT(B,p)=a+ib$ for some integers $a\equiv b\equiv 1\bmod 2$, we have
$4p-2=PSD(B,p)=|DFT(B,p)|^{2}=a^{2}+b^{2}$ and the rest is trivial. $\Box$

Remarks. 1\. If we assume that besides $b_{j+p}=-b_{j}$, $j=1,2,\ldots,p-1$,
we also have

$b_{p-j}=-b_{j},\quad j=1,2,\ldots,\frac{p-1}{2},$ (4)

then we get

$b_{2p-j}=b_{j},\quad j=1,2,\ldots,p-1,$

and thus

$DFT(B,s)=b_{0}+(-1)^{s}b_{p}+\sum_{j=1}^{p-1}b_{j}\left(\xi_{2p}^{js}+\xi_{2p}^{-js}\right),\quad
s=1,2,\ldots,p-1,$

that is,

$DFT(B,s)=DFT(B,2p-s),\quad s=1,2,\ldots,p-1,$

and

$PSD(B,s)=PSD(B,2p-s),\quad s=1,2,\ldots,p-1.$ (5)

(Note that without this condition we only have

$DFT(B,s)=\overline{DFT(\overline{B},2p-s)},\quad s=1,2,\ldots,p-1,$

and

$PSD(B,s)=PSD(\overline{B},2p-s),\quad s=1,2,\ldots,p-1.)$

This has been a suitable condition to reduce the search space. In addition we
may choose

$b_{0}=1\quad\mbox{and}\quad b_{p}=i.$ (6)

In this case we have

$DFT(B,p)=1-i+4\sum_{j=1}^{(p-1)/2}(-1)^{j}b_{j}=a+ib\quad\mbox{with
}a^{2}+b^{2}=4p-2,$

and thus $a\equiv-b\equiv 1\bmod 4$, which further reduces the number of
possible $b_{1},b_{2},\ldots,b_{(p-1)/2}\in\\{-1,1,-i,i\\}$.

2\. Note that $4p-2$ cannot be a sum of two integer squares if $2p-1$ is
divisible by an odd power of a prime equivalent to $3$ modulo $4$ by the sum
of two squares theorem. Hence, our first method fails for
$p=11,17,29,47,\ldots$

However, it was successful for $p=3,5,7,13,19,31,41$.

The decompressed sequences $B=[b_{0},b_{1},\ldots,b_{2p-1}]$ which lead to
quaternary Legendre pairs $(A,B)$ are defined by

$b_{0}=1,b_{p}=i,\quad b_{j+p}=b_{p-j}=-b_{j},\quad j=1,2,\ldots,p-1,$

and $[b_{1},b_{2},\ldots,b_{(p-1)/2}]$ given in the following table:

$\begin{array}[]{c|c}p&[b_{1},b_{2},\ldots,b_{(p-1)/2}]\\\ \hline\cr 3&[1]\\\
5&[1,i]\\\ 7&[1,-i,-1]\\\ 13&[1,1,-1,i,1,i]\\\ 19&[1,-1,1,-i,-1,-i,-i,1,1]\\\
31&[1,-1,1,-1,-1,-i,-1,-i,-i,-1,i,i,-i,-1,-1]\\\
41&[1,1,-i,1,-1,-i,-1,i,1,i,1,-i,i,i,i,-i,1,1,-1,-i]\end{array}$

##  4 The remaining even $\ell\leq 24$

Again we may assume (2) and denote by

$A_{k}=\left[a_{0}^{(k)},a_{1}^{(k)},\ldots,a_{k-1}^{(k)}\right]$

the $\frac{\ell}{k}$-compression of $A$, that is,

$a_{j}^{(k)}=\sum_{n=0}^{\ell/k-1}a_{kn+j},\quad j=0,1,\ldots,k-1.$

The alphabet of the $\frac{\ell}{k}$-compressions $A_{k}$ of
$A\in\\{-1,1,-i,i\\}^{\ell}$ has $\left(\frac{\ell}{k}+1\right)^{2}$ possible
elements, namely those $a+ib$ with

$|a|+|b|\leq\frac{\ell}{k}\quad\mbox{and}\quad a+b\equiv\frac{\ell}{k}\bmod
2,$

that is, $a_{j}^{(k)}$, $j=0,1,\ldots,k-1$, is of the form

$a+i\left(\frac{\ell}{k}-a-2j\right),\quad
a=-j,-j+1,\ldots,\frac{\ell}{k}-j,\quad j=0,1,\ldots,\frac{\ell}{k}.$

Recall

$PSD\left(A,\frac{s\ell}{k}\right)=PSD(A_{k},s),\quad s=1,2,\ldots,k-1.$

We state now several restrictions on the PSDs which reduce our search space
for Legendre pairs.

######  Theorem 4.1.

Let $(A,B)$ be a quaternary Legendre pair of length $\ell$ with (2).
1\. For $\ell\equiv 0\bmod 2$, $PSD(B_{2},1)$ is an integer with

$PSD(B_{2},1)\equiv 2\bmod 8$

and the square-free parts of both

$PSD(B_{2},1)\quad\mbox{and}\quad PSD(A_{2},1)=2\ell+2-PSD(B_{2},1)$

are not divisible by a prime $\equiv 3\bmod 4.$
2\. For $\ell\equiv 0\bmod 4$, $PSD(B_{4},1)$ is an integer with

$PSD(B_{4},1)\equiv 2\bmod 8$

and the square-free parts of both

$PSD(B_{4},1)\quad\mbox{and}\quad PSD(A_{4},1)=2\ell+2-PSD(B_{4},1)$

are not divisible by a prime $\equiv 3\bmod 4.$

Proof. 1\. We have

$DFT(B,\ell/2)=\sum_{j=0}^{\ell-1}b_{j}(-1)^{j}=a+ib\in\mathbb{Z}[i]$

for some integers $a,b$. Since

$1+i=\sum_{j=0}^{\ell-1}b_{j}\equiv a+ib\bmod 2$

we must have

$a\equiv b\equiv 1\bmod 2$

and thus

$PSD(B_{2},1)=PSD(B,\ell/2)=a^{2}+b^{2}\equiv 2\bmod 8.$

Similarly, we get

$PSD(A_{2},1)=PSD(A,\ell/2)=c^{2}+d^{2},\quad c,d\in\mathbb{Z}.$

$PSD(A_{2},1)+PSD(B_{2},1)=2\ell+2$ and the sum of two squares theorem
finishes the proof of the first part.

2\. First note that $DFT(A,\ell/4),DFT(B,\ell/4)\in\mathbb{Z}[i]$. Then we
have

$DFT(B,\ell/4)=\sum_{j=0}^{\ell-1}b_{j}i^{j}\equiv\sum_{j=0}^{\ell-1}b_{j}(-1)^{j}\equiv
DFT(B,\ell/2)\equiv 1+i\bmod(1+i)$

since $DFT(B,\ell/2)\equiv 1+i\bmod 2$ and $1+i$ divides $2=(1+i)(1-i)$ in
$\mathbb{Z}[i]$. Moreover, we get

$DFT(B,\ell/4)\equiv\sum_{j=0}^{\ell-1}b_{j}\equiv 1+i\bmod(1-i)$

and thus

$DFT(B,\ell/4)\equiv 1+i\bmod 2$

since $1+i$ and $1-i$ are primes in $\mathbb{Z}[i]$. Now the second result
follows analogously to the first one. $\Box$

The above condition on $PSD(A,\ell/2)$ and $PSD(B,\ell/2)$ reduces our search
to sequences with the following values of these PSDs:

$\begin{array}[]{c|c}\ell&(PSD(A,\ell/2),PSD(B,\ell/2))\\\ \hline\cr
2&{\bf(4,2)}\\\ 4&{\bf(0,10)},(8,2)\\\ 8&(0,18),(8,10),{\bf(16,2)}\\\
12&(0,26),(8,18),{\bf(16,10)}\\\ 16&(0,34),(8,26),(16,18),{\bf(32,2)}\\\
18&(4,34),{\bf(20,18)},(36,2)\\\ 20&(8,34),{\bf(16,26)},(32,10),(40,2)\\\
22&(20,26),{\bf(36,10)}\\\ 24&{\bf(0,50)},(16,34),(32,18),(40,10)\\\
\end{array}$

The boldfaced pairs correspond to the Legendre pairs $(A,B)$ given in the
following table:

$\begin{array}[]{c|c}\ell&A\\\ \hline\cr 2&[1,-1]\\\ 4&[1,1,-1,-1]\\\
8&[1,1,-1,-i,-1,1,-1,i]\\\ 12&[1,1,1,-1,i,-i,1,-1,-1,-1,-i,i]\\\
16&[1,1,1,-1,-i,-1,i,1,i,-1,i,-1,-i,-i,i,-i]\\\
18&[1,1,-1,i,-1,-1,-i,1,-1,i,-i,1,-1,-i,1,-i,i,i]\\\
20&[1,1,1,-1,i,-i,-i,i,i,1,i,-1,-1,-i,-1,1,-i,-i,i,-1]\\\
22&[1,1,1,1,-1,-1,-1,1,-1,1,-1,-i,i,1,-1,-1,-i,i,-i,-i,i,i]\\\
24&[1,1,1,1,1,-1,1,1,-1,-1,1,-1,1,-1,-1,1,-1,1,-1,-1,-1,1,-1,-1]\end{array}$
$\begin{array}[]{c|c}\ell&B\\\ \hline\cr 2&[1,i]\\\ 4&[1,i,1,-1]\\\
8&[1,1,1,i,-1,1,-1,-1]\\\ 12&[1,1,-1,-1,1,-1,1,-1,-1,1,1,i]\\\
16&[1,1,1,-1,1,1,-1,1,-1,i,-i,-i,-1,i,i,-1]\\\
18&[1,1,-1,i,-i,-1,1,-1,1,i,-1,-1,i,i,1,1,-i,-i]\\\
20&[1,1,1,-i,-i,-i,i,-1,-i,i,-i,i,1,-1,i,1,i,-1,i,-1]\\\
22&[1,1,1,1,-1,1,1,-1,-i,1,-i,i,i,i,-1,-i,-1,i,-i,-1,i,-1]\\\
24&[1,1,1,1,1,i,-1,-1,-i,1,-1,i,1,-1,-i,-1,1,i,-1,1,-i,-1,-1,i]\end{array}$

For $\ell\equiv 0\bmod 4$ the set of possible pairs of values
$(PSD(A,\ell/4),PSD(B,\ell/4))$ is the same as for
$(PSD(A,\ell/2),PSD(B,\ell/2))$ but the values can be different. Here is a
list of these values for the Legendre pairs in the above table:

$\begin{array}[]{c|cccccc}\ell&4&8&12&16&20&24\\\
\hline\cr(PSD(A,\ell/4),PSD(B,\ell/4))&(8,2)&(8,10)&(16,10)&(16,18)&(32,10)&(0,50)\end{array}$

Remarks. 1\. For $\ell\equiv 0\bmod 6$, if we assume that the
$\frac{\ell}{3}$-compression $A_{3}$ satisfies

$a_{0}^{(3)}-a_{2}^{(3)},a_{1}^{(3)}-a_{2}^{(3)}\in\mathbb{Z},$

then

$PSD(A_{3},1)\in\mathbb{Z},$

since
$DFT(A_{3},1)=(a_{0}^{(3)}-a_{2}^{(3)})+(a_{1}^{(3)}-a_{2}^{(3)})\xi_{3}$. In
this case $PSD(A,\ell/3)=PSD(A_{3},1)=PSD(A_{6},2)$ is of the form
$(a+b\xi_{3})(a+b\xi_{3}^{2})=a^{2}-ab+b^{2}$, $a,b\in\mathbb{Z}$, and thus
its square-free part is not divisible by a prime $\equiv 2\bmod 3$, see for
example [12, Chapter 9.1]. Moreover, if $A_{6}$ satisfies

$a_{0}^{(6)}-a_{3}^{(6)}+a_{5}^{(6)}-a_{2}^{(6)},a_{4}^{(6)}-a_{1}^{(6)}+a_{5}^{(6)}-a_{2}^{(6)}\in\mathbb{Z},$

then the square-free part of $PSD(A,\ell/6)=PSD(A_{6},1)$ is not divisible by
a prime $\equiv 2\bmod 3$.

Note that both conditions on $A_{3}$ and $A_{6}$ are fulfilled if $A_{6}$ is
integral. For example, our constructions of $A$ for $\ell=6$, that is,
$A=A_{6}=[1,1,-1,-1,1,-1]$, for $\ell=12$, that is, $A_{6}=[2,0,0,-2,0,0]$ and
$\ell=24$, that is, $A_{6}=[2,0,-2,2,0,-2]$, are all integral and thus satisfy
these conditions. In particular, we have

$(PSD(A_{6},1),PSD(A_{6},2))=(4,12)\mbox{ for }\ell=6,$
$PSD(A_{6},1)=PSD(A_{6},2)=16\mbox{ for }\ell=12,$

and

$(PSD(A_{6},1),PSD(A_{6},2))=(0,48)\mbox{ for }\ell=24.$

2\. If $a_{2}^{(3)}=a_{1}^{(3)}$, then $DFT(A_{3},1)\in\mathbb{Z}[i]$ and
$PSD(A_{3},1)=PSD(A_{6},2)$ is a sum of two squares, that is, its square-free
part is not divisible by a prime $p\equiv 3\bmod 4$. If
$a_{4}^{(6)}+a_{5}^{(6)}=a_{1}^{(6)}+a_{2}^{(6)}$, then also $PSD(A_{6},1)$ is
not divisible by a prime $p\equiv 3\bmod 4$.

Both conditions are fulfilled for our solution for $\ell=12$. The second
condition is also fulfilled for our solutions for $\ell=6$ and $\ell=24$.

3\. For $\ell\equiv 0\bmod 6$, there are further very promising ‘seeds’ of the
form $A_{3}=[0,a+ib,-(a+ib)]$ with integers $a$ and $b$ satisfying
$|a|+|b|\leq\ell/3$, $a^{2}+b^{2}\leq(2\ell+2)/3$ and
$|a|+|b|\equiv\ell/3\bmod 2$. In this case we have

$DFT(A,\ell/3)=DFT(A_{3},1)=(a+ib)(\xi_{3}-\xi_{3}^{2})=(a+ib)\sqrt{-3}$

and thus

$PSD(A_{3},1)=3(a^{2}+b^{2}).$

For example, our quaternary Legendre pair for $\ell=18$ satisfies

$A_{3}=[0,1-i,-1+i]\quad\mbox{and}\quad PSD(A_{3},1)=6.$

##  5 Computational and programming details

In order to conduct a systematic investigation of 2-decompression for
quaternary Legendre pairs using the pairs of ‘seed’ sequences of Definition
3.1, we wrote a Maple meta-program that produces efficient C code that
implements the 2-decompression for ‘seed’ sequences with elements from
$\\{-2,0,+2,1+i,-1-i\\}$. Using this meta-program we found the examples in
Section 3. To give some idea of the complexity involved in these computations,
we mention the relevant information for the pattern of Definition 3.1, namely:

$A_{p}=\left[0,2\left(\frac{1}{p}\right),2\left(\frac{2}{p}\right),\ldots,2\left(\frac{p-1}{p}\right)\right],\quad
B_{p}=[1+i,\underbrace{0\ldots 0}_{p-1\mbox{ terms}}].$ (7)

There are exactly four $2$-decompressions of $A_{p}$ and $2\cdot
4^{p-1}=2^{2p-1}$ possible $2$-decompressions of $B_{p}$, see Theorem 3.2. All
four $2$-decompressions of $A_{p}$ give the same PAFs and PSDs. We restricted
ourselves to the cases (4) and (6) to reduce the search space for
decompressions to $4^{(p-1)/2}=2^{p-1}$ possible choices of
$b_{1},b_{2},\ldots,b_{(p-1)/2}\in\\{-1,1,-i,i\\}$.

Now by (5) and since $PSD(B,2s)=2$, $s=1,2,\ldots,p-1$, for any decompression
$B$ of $B_{p}$, for any
$(b_{1},b_{2},\ldots,b_{(p-1)/2})\in\\{-1,1,-i,i\\}^{(p-1)/2}$ we have to
check the condition

$PSD(B,2s-1)=4p-2,\quad s=1,2,\ldots,\frac{p+1}{2},$

where

$PSD(B,2s-1)=|DFT(B,2s-1)|^{2}=\left|1-i+4\sum_{j=1}^{(p-1)/2}b_{j}\cos\left(\frac{(2s-1)j\pi}{p}\right)\right|^{2}.$

(Note that this procedure can only succeed if the square-free part of $2p-1$
is not divisible by a prime $\equiv 3\bmod 4$ since otherwise $PSD(B,p)=4p-2$
is not possible.)

For the sequences found in Section 4, instead of using the classical PSD-test
(3), we use a much more precisely targeted PSD test, given that we know
explicitly all the PSD values of the desirable decompressed sequence.

First we determined all eligible pairs

$(PSD(A_{2},1),PSD(B_{2},1)).$

For $\ell\equiv 0\bmod 4$ the same values are eligible for
$(PSD(A_{4},1),PSD(B_{4},1))$. Then we determined all compressions satisfying
these conditions and decompressed them. Some additional ideas mentioned in the
remarks also helped to reduce the search space.

Finally, note that in contrast to the case $\ell=24$ the case $\ell=26$ was
settled with the more efficient method of Section 3. Compared to $\ell=24$, it
seems to be natural to assume that the complexity of finding a Legendre pair
for $\ell=28$ increases by the factor $256$. This may be in reach with some
more programming effort such as parallelization or using faster computers.
However, we stopped here and prefer to search for suitable ‘seeds’ which may
cover more cases, say, of the form $\ell=4p$.

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