# A Tilt in the Dark Matter Halo of the Galaxy

Jiwon Jesse Han Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Rohan P. Naidu Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Charlie
Conroy Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Ana Bonaca The Observatories of the Carnegie
Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA
Dennis Zaritsky Steward Observatory, University of Arizona, 933 North Cherry
Avenue, Tucson, AZ 85721-0065, USA  Nelson Caldwell Center for Astrophysics
$|$ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Phillip
Cargile Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Benjamin D. Johnson Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Vedant
Chandra Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Joshua S. Speagle (沈佳士) Banting & Dunlap Fellow
David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto,
50 St. George Street, Toronto ON M5S 3H4, Canada  Dunlap Institute for
Astronomy and Astrophysics, University of Toronto, 50 St George Street,
Toronto, ON M5S 3H4, Canada  Department of Statistical Sciences, University of
Toronto, 100 St George St, Toronto, ON M5S 3G3, Canada  Yuan-Sen Ting (丁源森)
Research School of Astronomy & Astrophysics, Australian National University,
Cotter Road, Weston Creek, ACT 2611, Canberra, Australia  Research School of
Computer Science, Australian National University, Acton ACT 2601, Australia
Turner Woody Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA

###### Abstract

Recent observations of the stellar halo have uncovered the debris of an
ancient merger, Gaia-Sausage-Enceladus, estimated to have occurred $\gtrsim
8\text{Gyr}$ ago. Follow-up studies have associated GSE with a large-scale
tilt in the stellar halo that links two well-known stellar over-densities in
diagonally opposing octants of the Galaxy (the Hercules-Aquila Cloud and Virgo
Overdensity; HAC and VOD). In this paper, we study the plausibility of such
unmixed merger debris persisting over several Gyr in the Galactic halo. We
employ the simulated stellar halo from Naidu et al. (2021), which reproduces
several key properties of the merger remnant, including the large-scale tilt.
By integrating the orbits of these simulated stellar halo particles, we show
that adoption of a spherical halo potential results in rapid phase mixing of
the asymmetry. However, adopting a tilted halo potential preserves the initial
asymmetry in the stellar halo for many Gyr. The asymmetry is preserved even
when a realistic growing disk is added to the potential. These results suggest
that HAC and VOD are long-lived structures that are associated with GSE and
that the dark matter halo of the Galaxy is tilted with respect to the disk and
aligned in the direction of HAC-VOD. Such halo-disk misalignment is common in
modern cosmological simulations. Lastly, we study the relationship between the
local and global stellar halo in light of a tilted global halo comprised of
highly radial orbits. We find that the local halo offers a dynamically biased
view of the global halo due to its displacement from the Galactic Center.

Galaxy: halo

##  1 Introduction

Figure 1: 2D histograms of the dark matter (DM) distribution in the fiducial
simulation of N21. We adopt a greyscale with asinh stretch that is normalized
to include 98% of the sample. Overplotted in solid lines are equipotential
contours in logarithmic spacing, and dashed lines are corresponding contours
of a spherical potential to guide the eye. Top and middle panels show DM from
the simulated GSE at lookback time $t_{\text{lb}}=$ 5 Gyr and present day,
respectively. These panels demonstrate that the DM distribution of GSE is
tilted and stable over the last 5 Gyr. Bottom panels show the combined DM
distribution of the simulated GSE and MW.

Modern stellar surveys (e.g., 2MASS, SDSS, Gaia) have carried out a remarkably
successful archaeological excavation of our Galaxy’s fossil graveyard, the
stellar halo. Since the foundational works of Eggen et al. (1962) and Searle &
Zinn (1978), our picture of the Galactic halo formation history has evolved to
encompass both in-situ and accretion processes, some of which are continuing
to the present day as evident in the Sagittarius stream (Ibata et al., 1994;
Majewski et al., 2003) and the infalling Magellanic Clouds (Mathewson et al.,
1974; Besla et al., 2007; Zaritsky et al., 2020). Further back in the Galaxy’s
timeline, a striking discovery from the Gaia mission suggests that the bulk of
the stellar halo formed from a single merger event 8-10 billion years ago.
Referred to here as Gaia-Sausage-Enceladus (GSE; Belokurov et al. 2018, Helmi
et al. 2018), the debris of this merger comprises the majority of the inner
stellar halo (Galactocentric radius $r_{\text{gal}}\lesssim 30\text{kpc}$;
Iorio & Belokurov 2019, Naidu et al. 2020), providing compelling evidence for
hierarchical formation of the Galaxy as predicted by the $\Lambda$-CDM
standard cosmological model (Press & Schechter, 1974; Blumenthal et al., 1984;
Bullock & Johnston, 2005).

In light of this discovery, two major stellar overdensities in the halo, the
Hercules-Aquila Cloud (HAC, Belokurov et al. 2007) and the Virgo Overdensity
(VOD, Vivas et al. 2001), have been interpreted as apocentric pile-ups (Deason
et al., 2018; Simion et al., 2019; Naidu et al., 2021) of the GSE merger
debris. Supporting evidence of this association is that they occupy diagonally
opposing octants in the Galaxy (Iorio & Belokurov 2019), overlap in orbital
properties (Simion et al., 2019), and exhibit metallicity distributions
indistinguishable from that of GSE (Naidu et al., 2021, hereafter N21). In
addition, Han et al. (in prep) measure the global shape and orientation of the
stellar halo using a sample of giants from the H3 survey (Conroy et al., 2019)
equipped with 6D phase space, metallicity, and $[\alpha/\text{Fe}]$ abundance
measurements beyond distances $d\sim 50\text{kpc}$. Using a chemodynamical
selection to identify GSE giants, they find that the diffuse halo substructure
is best fit by a prolate spheroid, tilted off the Galactic plane pointing
towards VOD in the North, and HAC in the South. This is consistent with the
distribution of Gaia DR2 RR Lyrae in the halo reported by Iorio & Belokurov
(2019).

Figure 2: 2D histograms of the simulated GSE stellar distribution in N21 in
three projections, adopting an asinh stretch that is normalized to include 98%
of the sample. All following figures with 2D histograms adopt the same color
scheme. The top and bottom panels show the stellar distribution at lookback
time $t_{\text{lb}}=$ 5 Gyr and present day, respectively. These panels show
that the tilt in the stellar distribution of the simulated GSE persists over 5
Gyr.

Parallel to these efforts constraining the spatial extent of GSE, the timing
of the merger has also been investigated. Stellar ages estimated from various
techniques have been used to infer a timeline of the merger (Gallart et al.,
2019; Bonaca et al., 2020; Belokurov et al., 2020; Chaplin et al., 2020; Borre
et al., 2021; Grunblatt et al., 2021; Montalbán et al., 2021), building a
consensus that the merger completed $8-10\text{Gyr}$ ago. This merger timeline
is also supported by the kinematics of in-situ halo stars, which are
consistent with being heated from the thick (high-$\alpha$) disk due to GSE
(Zolotov et al., 2009; Purcell et al., 2010; Bonaca et al., 2017). In
simulations, Fattahi et al. (2019) identify halos in the Auriga project (Grand
et al., 2017) that host an accreted, highly radial stellar halo to find that
these halos are preferentially created from a single merger with a dwarf
galaxy $6-10\text{Gyr}$ ago. Furthermore, Grand et al. (2018) and Mackereth et
al. (2018) demonstrate using Auriga and EAGLE simulations that the bimodality
of the $[\alpha/\text{Fe}]$-$[\text{Fe/H}]$ sequence in the Galactic disk
could be the consequence of an early ($z>1$) star formation event triggered by
a gas-rich merger. This is once again consistent with the observationally
inferred timing of the GSE merger.

Figure 3: Stellar halo distribution in the Galactocentric YZ projection at
three representative time steps ($t_{lb}=\text{5Gyr}$, $t_{lb}=\text{4Gyr}$,
and present day). Top row shows N-body snapshots from N21, and middle and
bottom rows show orbital integrations in a tilted halo and a spherical halo.

While cosmological simulations have been helpful to constrain general
properties of the merger, a tailored simulation is required to study the
initial conditions of the merger and its subsequent evolution in detail. To
this end, N21 present a grid of $\sim 500$ N-body simulations from which they
identify a fiducial simulation that closely resembles GSE based on H3 data
(Conroy et al., 2019). This simulation is a 2.5:1 total mass ratio merger of
the Milky Way at $z=2$ with a dwarf galaxy on an inclined, retrograde orbit.
The resulting z-component of the angular momentum $L_{z}$ and galactocentric
distance $r_{\text{gal}}$ distributions of the merger match those of the GSE
sample from the H3 survey (Naidu et al., 2020). Remarkably, the simulation
also reproduces HAC/VOD-like stellar overdensities in the correct
galactocentric locations, which was not a condition used to select the GSE
analog. Furthermore, these HAC/VOD analogs persist over several Gyr, which is
consistent with the observationally inferred timeline of the GSE merger.
Lastly, the dark matter contribution from GSE is $\sim 20$% within
$r_{\text{gal}}<30\text{kpc}$ in this simulation, which is consistent with
hydrodynamical simulations of GSE-like major mergers in MW-like galaxies
(Fattahi et al., 2019; Dillamore et al., 2021).

Thus, a coherent picture emerges: the stellar halo out to $30\text{kpc}$ is
dominated by debris from an ancient ($\sim 8-10\text{Gyr}$) radial merger and
is globally asymmetric, bookended by the two diagonal apocenters at VOD (above
the disk) and HAC (below the disk).

Alternatives to this picture have been proposed. For example, Balbinot & Helmi
(2021) integrate the orbits of a local (heliocentric distance
$d<2.5\text{kpc}$) Gaia EDR3 6D sample to predict the global distribution of
halo substructures. After integrating GSE stars over $8\text{Gyr}$, they
report a symmetric distribution of orbits on the sky. From this, they argue
that the current distribution of GSE cannot be globally asymmetric, and thus
the observed stellar halo asymmetries in Iorio & Belokurov (2019) could be an
artifact of observational biases arising from the Gaia scanning pattern. Also
based on orbital integration, Donlon et al. (2019, 2020, 2021) conclude that
the radial debris observed in the stellar halo must have been deposited
recently ($\sim 2\text{Gyr}$) in order to explain the spatial coherence of
VOD. According to these authors, a large-scale asymmetry in the stellar halo
phase mixes too quickly to be compatible with the proposed merger epoch of
GSE.

Both of these studies assume a spherical dark matter (DM) halo to integrate
orbits of halo stars. Indeed, that is common practice: although observational
constraints on the shape of the DM halo are highly uncertain (e.g., Read,
2014; Bland-Hawthorn & Gerhard, 2016; de Salas & Widmark, 2021), a spherical
DM halo serves as the default model for the purpose of orbital integration.
More sophisticated models still consider an oblate (Vera-Ciro & Helmi, 2013;
Hattori et al., 2021), prolate (Bowden et al., 2016), or triaxial halos (Law &
Majewski, 2010) that are aligned with the Galactic disk. However, there are
well grounded motivations to consider a DM halo that is tilted with respect to
the disk. First, observations of the stellar halo suggest a large-scale tilt
on the relevant scales of $r_{\text{gal}}\sim 30\text{kpc}$, hinting that the
DM halo may embody a similar tilt. Second, cosmological simulations show that
it is common for DM halos in MW-like galaxies to exhibit non-spherical shapes
that are misaligned with the disk (e.g., Prada et al., 2019; Emami et al.,
2021; Dillamore et al., 2021) for a variety of reasons linked to their
accretion history. In the Galaxy, such a disk-halo misalignment has been
proposed to explain the stability of the Sagittarius stream (Debattista et
al., 2013) or the co-planar orbits of Galaxy’s satellites (Shao et al., 2021).

Inspired by these clues, in this paper we study the evolution of radial
stellar debris in a spherical halo and a tilted halo, and thus the
plausibility of unmixed merger debris such as HAC/VOD persisting over several
Gyr in the Galactic halo. The rest of the paper is organized as follows. In
section 2 we define the Galactic potentials of a spherical and tilted DM halo.
In section 3 we present the results of integrating stellar halo orbits in
these potentials. Finally, in section 4 we interpret these results in the
context of the Galaxy.

Figure 4: Present day distribution of simulated GSE stars evolved in a tilted,
light disk potential (top panels) and a tilted, growing disk potential (bottom
panels) in three projections. In the latter case, the disk grows up to six
times heavier than its initial mass. Despite the addition of this massive disk
to the potential, the tilt of GSE persists over 5 Gyr.

##  2 Galactic Potential

In this section, we construct the Galactic potentials used to study the
evolution of radial stellar debris in the halo.

We begin by defining the spherical and tilted DM halo models. The former is
the canonical picture of the Galactic halo and has been explored to great
extent. For this study, we choose the DM halo described by Bovy (2015),
following an NFW profile (Navarro et al., 1997) with scale radius $r_{s}=16$
kpc and virial mass $M_{\text{vir}}=8\times 10^{11}M_{\odot}$. The purpose of
a tilted halo model in this study is to explore the evolution of large scale
stellar halo asymmetries like HAC/VOD, which N21 successfully reproduces.
Thus, we adopt the N21 DM distribution as the fiducial tilted halo model.

Figure 1 shows the DM distribution in N21. Each panel is a 2D histogram of the
positions of DM particles, adopting a greyscale that is normalized to include
98% of the sample. Overplotted in solid lines are equipotential contours on a
logarithmic scale, and dotted lines mark corresponding contours of the
spherical halo from Bovy (2015) to guide the eye. We show the simulated GSE DM
distribution at $t_{\text{lb}}=\text{5Gyr}$ in the top panels, and at present
day in the middle panels. These two rows demonstrate that the GSE DM
distribution is tilted and stable over the last 5 Gyr. The bottom panels show
the combined DM distribution of the GSE and MW analogs at present day. By
fitting a triaxial ellipsoid to these distributions, we find that the GSE DM
is triaxial (axis ratio 1:0.9:0.57) and the MW DM is oblate (axis ratio
1:1:0.9), resulting in an overall oblate (axis ratio 1:0.96:0.8) DM
distribution that is tilted $\sim 30^{\circ}$ above the Galactic disk towards
the VOD in the North and HAC in the South.

To study the evolution of stellar orbits in these halos, we now calculate the
Galactic potential by combining models of the halo, disk, and bulge.

The potential generated by the spherical halo can be analytically derived from
the NFW profile (e.g., Łokas & Mamon, 2001). Meanwhile, the tilted DM halo
does not follow any particular analytical density function, so we employ the
self-consistent field method (SCF; Hernquist & Ostriker 1992, Lowing et al.
2011) implemented in the python package gala (Price-Whelan et al., 2020) to
approximate the potential. The advantage of this framework is its flexibility
to approximate non-spherical distributions to the desired level of accuracy.
After analyzing the level of variance in the terms of the SCF expansion, we
include expansion terms up to $n=20$ and $l=4$ to sufficiently reproduce the
potential while keeping computational costs manageable. The amplitude of the
SCF expansion is determined by matching the energies of orbits near the solar
neighborhood to that derived from the spherical potential.

The Galactic disk is another important component of the Galactic potential.
While N21 self-consistently describes the dynamics of the merging Galaxy at
$z=2$, its disk model mimics the thick disk of the Galaxy and stays at a fixed
mass. In the real Galaxy, the thin (low-$\alpha$) disk is $\sim 6$ times more
massive than the thick (high-$\alpha$) disk at the present day (Bland-Hawthorn
& Gerhard, 2016). Observations suggest that the bulk of the thin disk grew
steadily after $z\sim 1-2$ (e.g., Haywood et al., 2013; Fantin et al., 2019;
Bonaca et al., 2020), following a rapid star formation and quenching history
of the thick disk at $z\sim 2$. Over 5 Gyr, an average GSE star can cross the
disk up to $\sim 10$ times, so the significant growth of the thin disk would
clearly impact the dynamics of halo stars. Thus, to examine the effect of the
growing thin disk, we consider two models. We first consider a static thick
disk with fixed mass $6\times 10^{9}M_{\odot}$, scale length $2\text{kpc}$,
and scale height $900\text{pc}$. This model is analogous to the disk in N21
and the present day thick disk of the Galaxy. We then consider the thick disk
coupled with a linearly growing thin-disk with scale length $2.6\text{kpc}$
and scale height $300\text{pc}$ as described in Bland-Hawthorn & Gerhard
(2016). We increase the mass of the thin disk at every Myr, starting from $0$
and ending at $3.5\times 10^{10}M_{\odot}$. This model aims to capture the
growth of the thin disk in the Galaxy. Throughout this paper, we refer to the
static thick disk model as the “light disk” and the thick and thin disk model
as the “growing disk.”

Finally, for the bulge, we adopt a Hernquist profile (Hernquist, 1990) with
mass of $1.4\times 10^{10}M_{\odot}$ and scale length of $1.5\text{kpc}$
(Bland-Hawthorn & Gerhard, 2016).

We thus consider four Galactic potentials in this study: a tilted/spherical
halo with a light disk, and a tilted/spherical halo with a growing disk. In
these potentials, we integrate the orbits of stars from the
$t_{\text{lb}}=\text{5Gyr}$ snapshot in N21 to the present day. Figure 2 shows
the N-body snapshots of the stellar halo at these two epochs. We integrate
orbits using the Dormand & Prince (1978) integrator at $1\text{Myr}$ timestep,
implemented in the python package gala (Price-Whelan et al., 2020).

##  3 Orbits in the Stellar Halo

Here we present the results from integrating stellar halo orbits in the four
Galactic potentials. We outline results separately for the global stellar halo
and the local (solar neighborhood) halo, the latter being relevant to studies
that utilize local 6D data (e.g., the commonly used Gaia DR2 and EDR3 RVS
samples).

Figure 5: Excess fraction of stars in diagonally opposing octants of the
Galaxy as a function of time, defined in equation 1. This diagonal excess is
an indicator of the degree to which orbits are spatially mixed. In the N-body
and tilted halo models, the initial asymmetry persists beyond $5\text{Gyr}$,
while in the spherical halo models the asymmetry is erased by the first Gyr.
Figure 6: Relation between eccentricity and
$|\Phi_{\text{spherical}}-\Phi_{\text{tilted}}|$, where the latter is defined
to be the difference in the potential energy that a star experiences over 600
Myr in a tilted potential and a spherical potential as defined in equation 2.
There is a clear positive correlation in eccentricity and
$|\Phi_{\text{spherical}}-\Phi_{\text{tilted}}|$, demonstrating that stars on
eccentric (radial) orbits are more sensitive to its host potential compared to
stars on disky orbits. Figure 7: Origin of local halo stars in the N21
simulation. The top row shows the present-day stellar distribution, with local
halo stars marked in red. The middle row shows the distribution of local halo
stars $500\text{Myr}$ ago, and the bottom row shows the global halo
$500\text{Myr}$ ago. In the XY and XZ projections, the $t_{lb}=500\text{Myr}$
distributions of the local and global halo are notably different. This
demonstrates that the orbits of local halo stars do not directly trace the
spatial structure of the global halo. Meanwhile, in the YZ plane, the local
and global halo stars occupy similar distributions, suggesting that in some
projections the local halo orbits can reflect the spatial structure of the
global halo.

###  3.1 Global Halo

We show the evolution of the global distribution of halo stars in Figure 3.
The leftmost panel shows the $t_{\text{lb}}=\text{5Gyr}$ snapshot in N21,
followed by two columns showing the $t_{\text{lb}}=\text{4Gyr}$ and present
day distributions. In the top row we show the N-body simulations. In the
middle and bottom rows, we show the positions of stars derived from orbital
integrations at their respective $t_{\text{lb}}$ and the preceding 100 Myr. We
choose to display the Galactic YZ plane, and note that the XZ and XY planes
show the same trend.

Figure 3 showcases how the shape of the halo can greatly influence the
evolution of stellar orbits, and we highlight two important features. First,
the N-body snapshots and tilted halo orbits closely resemble each other,
demonstrating the sufficiency of the SCF approximation of the tilted halo
potential. Second, the spherical halo erases the tilt in the stellar halo
within the first Gyr, whereas the tilted halo preserves the tilt over
$5\text{Gyr}$. This is somewhat to be expected, since spherical potentials
host rosette orbits that can fill the entire available space, while triaxial
potentials yield more complex orbits.

Next, Figure 4 shows the effect of a growing disk on stellar halo orbits. We
show the present day stellar distribution from orbital integration in a tilted
halo and light disk in the top panels, and tilted halo and growing disk in the
bottom panels. Most notably, the tilt in the stellar halo with respect to the
disk is preserved in both potentials. As expected, orbits in the XZ and YZ
planes are somewhat concentrated onto the disk, but the spatial asymmetry is
still visibly clear. This indicates that the growth of the thin disk in the
Galaxy would not erase HAC/VOD like overdensities in the halo.

Figure 5 further quantifies the degree to which the stellar halo is spatially
mixed. We compute the “diagonal excess” of the stellar halo at each time step,
defined as

$\text{DiagonalExcess}\equiv\frac{f_{+\text{X}+\text{Y}-\text{Z}}+f_{-\text{X}-\text{Y}+\text{Z}}}{2}-\frac{f_{\text{other}}}{6}$
(1)

where $f_{+\text{X}+\text{Y}-\text{Z}}$ and $f_{-\text{X}-\text{Y}+\text{Z}}$
are the fraction of stars contained in two diagonal octants (respectively
containing the bulk of HAC/VOD in the Galaxy) compared to the fraction of
stars in the remaining six octants ($f_{\text{other}}$). For example, a
stellar halo that is tilted and aligned with HAC/VOD will show a positive
diagonal excess, while a spherical stellar halo will show a diagonal excess of
zero. Thus, if the GSE debris becomes fully mixed, the diagonal excess will
converge to zero, while if a large scale asymmetry persists, the diagonal
excess will be positive. Figure 5 clearly demonstrates this trend. While the
N-body simulation (black line) and tilted halo models (red lines) show mixing
timescales greater than $5\text{Gyr}$, the spherical halo (blue lines) fully
mixes its orbits by the first Gyr. The growing disk (dashed lines) acts to
catalyze the spatial mixing of orbits, but not strongly enough to erase global
stellar halo asymmetries evolving in a tilted DM halo. We note that when we
integrate orbits in a spherical halo only (excluding the disk and the bulge),
there is a remaining diagonal excess of $\sim 0.05$ by 5 Gyr. This
demonstrates that while the sphericity of the potential is the driving factor
in preserving or destroying the large scale asymmetry of GSE, the disk also
plays a role in breaking the spherical symmetry to catalyze the mixing of
orbits.

Lastly, Figure 6 provides insight into why GSE could be a sensitive probe of
the underlying potential. We plot the orbital eccentricity of each star in N21
against $|\Phi_{\text{spherical}}-\Phi_{\text{tilted}}|$, where the latter is
defined as

$|\Phi_{\text{spherical}}-\Phi_{\text{tilted}}|\equiv\left|\sum^{600\text{Myr}}_{i=0\text{Myr}}\phi_{\text{spherical},i}-\phi_{\text{tilted},i}\right|$
(2)

where $\phi_{\text{spherical,i}}$ and $\phi_{\text{tilted,i}}$ are the
spherical/tilted potentials evaluated at a star’s location at $i$ Myr. The 600
Myr summation timescale is chosen such that most stars would have had enough
time for at least one pericenter passage. The eccentricity is calculated in
the tilted potential, and we make note that the result does not change when
eccentricity is calculated in a spherical potential. There is a strong
positive correlation in eccentricity and
$|\Phi_{\text{spherical}}-\Phi_{\text{tilted}}|$, and the stars on disky
orbits show $|\Phi_{\text{spherical}}-\Phi_{\text{tilted}}|$ converging to
zero. This correlation indicates that eccentric orbits that probe the full
range of radii in the halo are more sensitive to the underlying potential than
disky orbits. Since GSE stars are predominantly on highly radial/eccentric
orbits, it is a sensitive tracer of the shape of the DM halo that its
progenitor deposited in the Galaxy.

###  3.2 Local Halo

We define the local (solar neighborhood) halo to be a sphere of $d\lesssim
3\text{kpc}$ centered on the Sun. The local halo sample has been historically
important as it contains the most complete 6D information of stars. Numerous
studies have extrapolated the motions of local halo stars in order to
understand the global halo (e.g., Carollo et al., 2010), and in this section
we assess how well the local halo sample represents properties of the global
halo.

In Figure 7 we identify local halo stars in N21 that satisfy $d<3\text{kpc}$
at the present day (top row, marked in red), and track where they were 500 Myr
ago (middle bottom rows). If the local halo is an unbiased sample of the
global halo, then we expect these two distributions to look similar. However,
in the XY and XZ projections, the $t_{lb}=500\text{Myr}$ distribution of the
global and local halo are notably different. This suggests that the orbits of
local halo stars do not reflect the spatial distribution of the global halo,
but are rather on more spatially mixed orbits. This is not surprising, since
the solar location is off-axis from the tilt of the global halo and off-
centered from the Galactic Center.

Examining the YZ projection in Figure 7, we see that the bottom two panels are
similar to each other. This suggests that the local halo stars retain memory
of the global halo in the YZ plane, since the solar location on the YZ plane
is not off-centered with respect to the Galactic Center. The exact
relationship between the local and global halo is a complicated product of the
spatial/dynamical distribution of the global halo and the solar position, and
cannot be decoupled from one’s assumptions about the global halo.

##  4 Discussion

In this study we have investigated the evolution of stellar debris from an
inclined merger remnant in the Galactic halo. Starting with a simulated
stellar halo from Naidu et al. (2021), we integrated stellar orbits in tilted
and spherical DM halos with both light and growing disks to present day. After
$5\text{Gyr}$, orbits in a spherical halo are fully phase mixed, while orbits
in a tilted halo preserve a large-scale tilt even in the influence of a
growing (massive) disk. We further quantified stellar halo asymmetries by
calculating the excess fraction of stars in the two octants containing
HAC/VOD, from which we evaluated the timescales of spatial mixing in the
Galactic halo. Finally, we examined the orbits of local halo stars, and showed
that these orbits to not trace the spatial distribution of the global halo.

We now interpret these results in the context of the Galactic halo. The
interpretation of HAC/VOD as apocenter pileups from GSE is supported by
compelling evidence from spatial, dynamical, and chemical considerations. A
key issue is whether one would expect HAC and VOD to be long-lived structures,
in light of the evidence that the GSE merger occurred $\sim 8-10\text{Gyr}$
ago. If the DM halo of the Galaxy is spherical (or in general is aligned with
the disk), then any large-scale asymmetries traced by radial merger debris
will be quickly erased. In this case it would be difficult to understand how
HAC and VOD could be associated with an ancient merger. However, a tilt in the
DM halo, as we have explored in this work, enables these over-densities to
persist for many Gyr. The evidence supporting an association between HAC/VOD
and GSE, combined with the estimated age of the merger, leads us to conclude
that the DM halo of the Galaxy is likely tilted with respect to the disk in
the direction of the HAC and VOD over-densities.

In a globally tilted halo, the local halo occupies a spatially and dynamically
biased location due to its offset from the Galactic Center. In particular, any
sample of stars on radial orbits in the local halo will preferentially contain
orbits that have been phase mixed in the Galaxy. In certain Galactocentric
projections (e.g., YZ) where the solar neighborhood is not offset from the
Galactic Center, these orbits can still retain information about the global
halo. In other projections, most of this information is lost. Inferring
properties of the global halo by extrapolating the local halo must therefore
be carried out with caution.

The non-trivial shape of DM halos has been the topic of many recent studies
using cosmological simulations. For example, Emami et al. (2021) report that
only 32% of Milky Way-like galaxies in the Illustris TNG50 simulation
(Pillepich et al., 2019; Nelson et al., 2019) show simple DM halos, and the
rest of the sample exhibit either twisted (gradually rotating) or stretched
(abruptly rotating) halos. Prada et al. (2019) find that 20% of Milky-Way like
galaxies in the Auriga project (Grand et al., 2017; Monachesi et al., 2019)
have twisted halos, while Dillamore et al. (2021) find that most galaxies in
the ARTEMIS simulations (Font et al., 2020) that show GSE-like features
exhibit a global change in orientation of their DM halo. In EAGLE simulations
(Schaye et al., 2015; Crain et al., 2015), Shao et al. (2021) find that the DM
halo of galaxies are well aligned with the common orbital plane of their
satellite galaxies. They notice that the common orbital plane of the Galaxy’s
satellites is misaligned with the disk, from which they argue that the DM halo
of the Galaxy must be twisted. In isolated simulations, Debattista et al.
(2013) predicted a halo-disk misalignment in the Galaxy based on models of the
Sagittarius stream. Thus, the halo-disk misalignment that we find in this
study is consistent with theoretical expectations.

An important caveat of this study is that we do not consider the mutual
interaction between the halo and its environment: the disk and the satellites
of the Galaxy. For example, the disk will pull the tilt of the halo closer to
the plane, while the halo may induce a warp in the disk at large radii. This
could mean that the initial tilt of the halo has to be larger than the current
day configuration to explain the spatial distribution of HAC/VOD. We note that
GSE contributes only $\sim 10$% of the dark matter within
$r_{\text{gal}}<10\text{kpc}$ in the N21 simulation, so where the disk gravity
is dominant the contribution from GSE is smaller than in the outer halo. Among
the satellites of the Galaxy, the Large Magellanic Cloud (LMC) has a
significant influence on the global shape of the halo (e.g., Garavito-Camargo
et al., 2019; Erkal et al., 2020; Conroy et al., 2021; Petersen & Peñarrubia,
2021). However, this effect is most prominent at larger distances
($r_{\text{gal}}>50\text{kpc}$) than what we consider here, and within
$r_{\text{gal}}<30\text{kpc}$ the influence of the LMC is likely to be much
smaller compared to GSE (though see Lucchini et al. 2021 for an alternative
scenario). Smaller satellites also contribute to the steady growth in the mass
of the halo, but this accretion again occurs mainly in the outer halo while
the inner halo is expected to remain quiescent after $t_{\text{lb}}\sim
8\text{Gyr}$ (e.g., Bullock & Johnston, 2005; Font et al., 2006; De Lucia &
Helmi, 2008; Cooper et al., 2010; Pillepich et al., 2014; Bland-Hawthorn &
Gerhard, 2016).

Future work will address these issues. Additional observations are necessary
to constrain the precise locations and extent of HAC and VOD and the overall
tilt of the stellar halo, which will inform the configuration of the DM halo.
In light of ongoing and upcoming spectroscopic surveys such as H3, DESI,
SDSS-V, WEAVE, and 4MOST, these are timely endeavors.

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# Cylindrical Films for Electronics in Low Background Physics Searches

E. Brown  K. Odgers, 11footnotetext: Corresponding author. M. Giordano  K.
Lewis,22footnotetext: Presently: Department of Physics and Astronomy, College
of Arts and Sciences, Howard University, Washington DC 20059, USA T. Berger
J. Freedberg

###### Abstract

A technique for manufacturing thin-film resistors on cylindrical substrates is
demonstrated. These devices are aimed for application in rare-event detectors
that must minimize radioactive backgrounds from trace impurities in electronic
components inside the detector. Cylindrical, conducting Ni films were created
via Electron Beam Deposition, using a mechanism that rotates the substrate, to
demonstrate proof of principle and measure the resistivity on axis and in
azimuth. These films are characterized by measurements using a facsimile of
the Van Der Pauw method combined with electrostatic simulations. In the two
cylindrical samples made we observe anisotropic electrical behavior with
resistivities of 1392.5, 888.5 $n\Omega m$ around the azimuth and of 81.9,
72.8 $n\Omega m$ along the axis of the sample. We show that this anisotropy is
not caused just by the electron beam evaporation by measuring a planar
rectangle sample made in the same process but without spinning which has
estimated resistivities of 66.5, and 71.9 $n\Omega m$ in both directions, and
calculated resistivity using the standard Van der Pauw equation of $66.1\pm
2.8$ $n\Omega m$. In spite of the anisotropy in the cylindrical samples, we
show that these films can be used as resistors.

##  1 Introduction

Thin film electronics with high radiopurity materials are an exciting prospect
for rare event physics, and other experiments requiring a low radioactive
background. In experiments measuring recoil energies from eV up to several
MeV, trace radioimpurities in detector components contribute a substantial
background for these experiments [1] [2]. To reach the low background needed
for physics searches, each material used in the detectors must have a minimum
of radio-contamination. The concentration of 238U and 232Th must be at or
below the ppt level 3331 ppt = $10^{-12}$mol/mol [1] [3][4][5].

One class of rare event detectors uses a time projection chamber (TPC) filled
with a cryogenic liquid [6]. There are two types of TPCs, either single or
dual phase. These correspond to whether the detecting material is entirely
liquid, or if it has liquid and gas segments. The TPC is based off a particle
interaction or decay in the liquid phase yielding two signals. This radiation
induces a prompt scintillation signal that is measured by photosensors, and an
ionization signal. The ionized electrons are drift through the cryogenic
liquid to an anode via a strong electric field where a charge signal is read
out. The time delay of the two signals and localization of the readout of the
ionization signal allow 3D position reconstruction.

To generate the uniform drift field in a liquid noble TPC, field shaping
electrodes line the outer edge of the detector with a cascading potential. The
cascade requires a resistor chain, and the shaping electrodes require
electrically insulating structural support. Even small resistors contribute
sizably to radioactive backgrounds, as do the macroscopic quantities of
plastics typically used to support the electrodes. Thus, future detectors are
focusing efforts to alleviate these sources of background.

One strategy for minimizing electronics near the fiducial region is to use
small resistive spacers to link the field shaping electrodes. These can be
realized by depositing a resistive film on the surface of insulating rods. The
high purity and small quantity of material deposited for a film on the order
of hundreds of nanometers allows the construction of highly radiopure support
rods. To this end, we developed a system for depositing films onto cylinders,
which can be used to build resistive electrode supports. We present here a
proof of concept for the production and characterization of these cylindrical
films.

Four point probe measurements are a common practice for measuring the
resistivity of planar sample, and techniques commonly use either a linear
series of probes, or probes attached to four corners of a rectangular sample
[7]. Probe locations analogous to this method were used to characterize the
cylindrical samples studied and results were compared to simulations of the
devices.

##  2 Film Production and Rotation Device

Films are prepared in a Temescal Electron Beam Evaporator. To produce
cylindrical films, a mechanism was built to hold and turn a cylindrical
substrate in the vacuum environment as seen in figure 1. The mechanism was
built to hold a hollow tube ranging from 2 to 10 centimeters in length in the
middle of the vacuum chamber, parallel to the bottom of the chamber. The extra
space on the mechanism allows mounting of additional planar samples that can
be used as a calibration for each deposition by measuring the thickness and
resistivity of planar samples produced simultaneously with the cylindrical
samples.

Figure 1: Left) The schematic of the motorized substrate holder with the arms,
motor body and substrate mount and mask. Center) Fully constructed device with
several different substrates attached prior to implementing the mask. Right)
Cylindrical sample after having the kapton tape removed showing the clean
edges and that the film being deposited and measured is far from where scrapes
and scores would occur.

When rotating the substrates during deposition, some corrections must be made
to interpret the thickness measurement of the in-situ quartz crystal resonance
instrument. The deposition covered a substrate area of $2\pi rl$ however the
cross sectional area of the substrate during the deposition was $2rl$. A
correction of $\frac{1}{\pi}$ for the film thickness on the cylinder compared
to that measured by the crystal and planar samples must be applied.
Additionally, a mask was installed to restrict the deposition width to 3 mm,
which further reduced the thickness of the cylindrical sample versus planar,
the exact factor of which depends on the outer diameter of the tube.

The cylindrical quartz substrate was cut from a long tube with an outer
diameter of 12.7 mm, with a substrate length on the order of several
centimeters. Cutting the tube left scoring near the top and bottom of the
sample, and uneven edges, which were addressed after cleaning. The cleaning
and preparation for the quartz cylinders, quartz slides, and silicon wafer was
a piranha etch of equal parts sulfuric acid and hydrogen peroxide. Afterwards,
the sample was rinsed with distilled water and dried with nitrogen. Kapton
tape was used on the cylindrical substrate to mitigate the jagged and scored
ends from the cutting process and ensure clean, parallel edges at the two ends
of the substrate, as can be seen in figure 1.

A nickel source was used for the deposition onto the bare quartz. For any
nickel film thicker than 40 nanometers the conductivity of the samples is too
high for use as a resistor, but for developing the process, choosing a metal
which would not have any Schottky barrier or diodic behavior was important.
Our conclusions, however, can be applied to more resistive materials. The
entire assembly was put into the deposition chamber and pumped down to
$5\times 10^{-7}$ Torr before deposition was started. After heating the nickel
source to the predetermined electron beam power for deposition, the
cylindrical substrate holder was set to 60 RPM and the shutter between the
nickel and substrate was opened. The deposition was maintained between 1-3
nm/s for the entire duration of the deposition as measured by the in-situ
crystal resonance instrument. The crystal monitor measures the total thickness
in-situ as well as deposition rate in angstroms per second for a planar
sample. Due to the planar to cylindrical correction factor discussed above,
the target thickness as read by the monitor was increased.

The first nickel sample examined was manufactured without a mask. Therefore,
half of the surface of the cylinder was exposed to the nickel source at any
given time during the deposition. A second sample was made with a mask that
had a 3mm cut out parallel to the axis of the substrate, so the film was
deposited onto the "flattest" portion of the substrate with the total
deposition thickness being varied as well.

##  3 Measurement Method

###  3.1 Flat Substrate Calibration

Measurements of the planar samples were made with a Keithley 2400 Source Meter
as a four point probe device. The probes were placed at four corners of a
small, rectangular cleaved portion of the nickel-on-wafer sample. Silver paint
was used to connect copper wires to the designated contact areas which
measured no more than 10 square millimeters. The current was applied between
all possible combinations of two adjacent contacts (in other words, not
including diagonals). The other two contacts not supplied with a current had
the potential difference between them measured. These are standard
measurements for the Van Der Pauw Method which uses equation 3.1 to calculate
resistivity from the currents and voltages [8]

$e^{\frac{-\pi cV_{H}}{I_{0}\rho}}+e^{\frac{-\pi cV_{V}}{I_{0}\rho}}=1$ (3.1)

In equation 3.1, we use the potential differences in the horizontal and
vertical edge directions ($V_{H},V_{V}$), the film thickness ($c$), and the
supplied current ($I_{0}$) to fit the resistivity ($\rho$). The thickness is
known from the crystal resonance measurement made during deposition of the
film. The last two variables, current and voltage, were set and measured by
the Keithley Source Meter.

Equation 3.1, can be modified and used for most samples which are
topologically equivalent (homeomorphic) to a rectangle. For topographically
different samples, modifications to the equation must be made as is discussed
in other papers [9]. The cylindrical samples fabricated for this study can be
described as topographically different from the closed surface required to use
the Van Der Pauw equation. When mapped to a flat surface, the cylinder would
be a surface with a hole in it. For these purposes, equation 3.1 is not used
in determining the resistivity of the samples and instead a different
technique is introduced in section 3.3.

###  3.2 Cylindrical Four Point Measurement

Similar to the Van Der Pauw method used for the planar sample, four probes
were attached to each cylindrical sample near what could be considered
corners. Two were placed at the top and bottom of the sample in a line
parallel with the axis of the cylinder, and on the other side (rotated $\pi$
radians) the other two were placed in a similar position. This can be seen in
figure 2. For both cylindrical samples, measurements similar to those made of
the planar sample were made. After room temperature measurements, one of the
sample cylinders was placed in a cryostat, and measurements were repeated as
temperatures were cycled from 100 K to 300 K as seen in figure 3.

Figure 2:  Geometry of Film on Glass and Probe/Source Locations

###  3.3 Electronic Simulations of Cylindrical Films

In conjunction with the construction and measurement of the films, a finite
element model of the film was made in COMSOL 5.1 with the Electric Current
Shell module (ECS) [10]. Simulations were made for both the planar and
cylindrical samples. For the planar model, two current sources (one positive
and one negative) were placed on the modeled film along with two voltage
probes as well. The current sources were set to 0.5 and -0.5 mA respectively.
The resistivity was left as a free parameter to be varied for each solution.
The difference between voltages probes was recorded for many iterations of the
simulation with resistivities spanning an order of magnitude above and below
what is expected for nickel. These values were used to find a function for
voltage difference given some resistivity using linear regression. These
functions were found for each orientation of probes (across the width and
length) and then inverted to obtain a formula for resistivity given a measured
voltage. Applying the inverted formula to the actual voltage measurements is
how the resistivities reported were determined.

The ECS module was used for the cylindrical film models as well, with a
similar process employed to construct a resistivity formula. A solid which
accurately reflected the dimensions of the cylindrical films was generated in
the software, with four regions defined at the points where the contacts were
painted on physical samples. The voltage at the two probe simulations was then
simulated at various values of resistivity. The same fitting and inverting was
applied to get the resistivity formulas for each probe orientation for each
cylindrical sample. Each of these formulas was applied to the corresponding
measured potential differences to determine the reported resistivities of the
samples.

##  4 Results

In table 1, we report measurements made for the two cylindrical samples as
well as the planar sample. The reported polar and axial voltages and
resistivities would be those where currents were passed from $1\rightarrow 3$
and $1\rightarrow 2$ as labelled in figure 2. The voltages reported are
averages of four separate but complimentary measurements in the same
orientation. For instance, the reported axial $\Delta V$ comes from averaging
$V_{1,2},V_{2,1},V_{3,4}$ and $V_{4,3}$, corresponding to a measurement, its
reversed polarity, its reciprocal, and its reciprocal in reversed polarity.
Averaging these voltage measurements is standard practice to reduce the effect
of unwanted voltage biases. The resistivities presented are those estimated
using the comparison to COMSOL as detailed in section 3.3 for the voltages
reported.

Table 1:  Voltages for orthogonal probe orientations with 1 mA source. For the cylindrical samples, Directions 1 (D1) and 2 (D2) correspond to polar and axial measurements of the sample respectively. For the planar, the directions correspond to measurements along the width and length of the rectangular sample. For both types of sample, C is the thickness of the sample. Film | R x L [mm] | C [nm] | D1 $\Delta V$[V] | D2 $\Delta V$[V] | D1 $\rho$ [$n\Omega m$] | D2 $\rho$ [$n\Omega m$]
---|---|---|---|---|---|---
Cyl 1 | 6.4 x 18 | 80 | 1.301 | 0.044 | 1392.5 | 81.9
Cyl 2 | 6.4 x 17 | 31 | 2.517 | 0.064 | 888.5 | 72.8
| WxL [mm] | C [nm] |
Planar | 6x15 | 120 | .449 | .014 | 66.5 | 71.9

The accepted resistivity of nickel depends on the purity of the nickel as well
as what contaminants are in the sample [11]. While some nickel samples can
have a resistivity as high as 167 $n\Omega m$ , we compare our measurements to
a value of 61.6 $n\Omega m$ as given by the National Physics Laboratory in the
UK [12]. The ratio of resistivities for directions D1 to D2 is nearly unity
for the planar sample (1.09), but is more than an order of magnitude larger
for both of the cylindrical samples(17.0 and 12.2). We interpret this as an
electrical anisotropy for the cylindrical samples as described in section
4.1.When the axial direction of the cylinder was probed the biggest relative
difference was 18% between measured and accepted resistivity. The polar
resistivities disagreed strongly with the accepted value with a relative error
of 1900%.

For the planar sample, equation 3.1 was used to calculate the bulk resistivity
as a check to show that the method of fitting to COMSOL simulations agreed
with accepted practices. When this standard analysis was performed, the
resistivity of the sample found was $66.1\pm 2.8$ $n\Omega m$. Equation 3.1,
returns a single resistivity for the sample, unlike our comparison to COMSOL,
but when we compare the calculated value to either of our fitted values, or to
the accepted resistivity of nickel we observe the values are all roughly in
agreement. The uncertainty on the resistivity calculated with equation 3.1 was
calculated by adding the uncertainty on voltage and thicknesses in quadrature.
Because of the transcendental nature of equation 3.1, numerical derivatives
were taking with a 5-point stencil for calculating the error.

Because of the nature of our fitting to simulation, uncertainties and errors
were left off for the fitted values. Possible sources of error could be a non-
uniform deposition, an incorrect correction for thickness on the cylinder,
contact resistance with the silver used for bonding the probes to the film, or
simulation errors. The low relative error for both planar measurements shows
that this method was successful within a small factor. This holds true for the
estimated resistivity for the axial probe orientation, but not for the polar
orientation. This discrepancy is discussed later.

The last measurement completed was the same four point measurement but
performed as the temperature warmed from 100 K to 300 K. In figure 3 it is
shown that the voltage difference of the substrate and thus resistance
increases as the temperature rises. This is consistent with what is expected
for a metal, and shows that electrical properties of these cylindrical films
can return to normal after heating and that the films survive cold
temperatures and thermal contractions and expansions of the quartz. This is a
critically important feature for applications in cryogenic liquid detectors.

Figure 3:  Low Temperature Performance of a Thin Cylindrical Film

###  4.1 Electrical Anisotropy

Table 1 shows that the resistivity found by axial and polar probe measurements
differed by more than an order of magnitude. The simulations used to estimate
the resistivity to voltage relationship used isotropic resistivities, so the
numbers in table 1 do not represent the physical vector components of the
resistivity. However, the resistivity definitively shows that there is an
anisotropy to the film, and that resistivity is greater going around the
circumference of the cylinder versus along the axial direction. The mechanism
of this anisotropy has not been explored but several possible explanations
have been formed. The most likely cause is that the middle of the substrate
during the deposition is perpendicular to the direction, but as you deviate
from the middle the angle between substrate and incoming particles becomes
oblique. Glancing Angle Deposition has been used to form meso and nano
structures, and may play a part in this [13]. It is also possible that grain
growth is easier along the axis of the cylinder, giving the electrical bias in
that direction.

While the source of the anisotropy observed in the resistivity is of interest,
for the purposes of constructing resistors from thin films it should have a
negligible impact if the currents are run along the axial direction of the
cylinder. This is due to the fact that a bulk resistivity is sufficient for
operating a thin film resistor in applications for liquid noble TPCs. We thus
leave precise investigations of the source, impact and potential use of these
effects to further studies.

##  5 Conclusion

We have successfully demonstrated the ability to produce thin, resistive films
on cylindrical surfaces for use in cryogenic applications. Our proof of
concept using nickel showed the reproducibility of our fabrication method, and
through comparison to COMSOL simulations we can characterize the bulk
resistivity of our samples using a four point probe method of measurement. For
the planar sample, calculation from equation 3.1 yielded a resisitvity of
$66.1\pm 2.8$ $n\Omega m$, while the COMSOL simulations yielded $66.5$ and
$71.9$ $n\Omega m$ for the two directions. This is in fair agreement with the
literature value of $61.6$ $n\Omega m$ [12]. In characterizing the cylindrical
samples through comparison to simulations with an isotropic resistivity, the
resistivity for the axial and polar probe orientations differed by more than
an order of magnitude. For the two samples, the respective resitivities were
81.9 and 1392.5 $n\Omega m$ and 72.8 and 888.5 $n\Omega m$. This difference
when fitting the resistivities leads to the conclusion that there is an
anistotropic resistivity for these cylindrical samples, and that using a mask
can affect this anisotropy. This hints to the cause of the anisotropy being
the glancing angles during deposition. Stable operation in the temperature
range of 77 to 300 K show promise for creating thin film resistors of this
type using other materials for use in cryogenic applications.

##  6 Acknowledgments

We would like to thank Poomirat Nawarat for his time and help in finalizing
this study.

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