# The Supersonic Project: The eccentricity and rotational support of SIGOs and
DM GHOSts

Claire E. Williams Department of Physics and Astronomy, UCLA, Los Angeles, CA
90095  Mani L. Bhaumik Institute for Theoretical Physics, Department of
Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA
Smadar Naoz Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095, USA
William Lake Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095, USA
Yeou S. Chiou Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095, USA
Blakesley Burkhart Department of Physics and Astronomy, Rutgers, The State
University of New Jersey, 136 Frelinghuysen Rd, Piscataway, NJ 08854, USA
Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue,
New York, NY 10010, USA
Federico Marinacci Department of Physics & Astronomy “Augusto Righi”,
University of Bologna, via Gobetti 93/2, 40129 Bologna, Italy
Mark Vogelsberger Department of Physics and Kavli Institute for Astrophysics
and Space Research, Massachusetts Institute of Technology, Cambridge, MA
02139, USA
Gen Chiaki Astronomical Institute, Tohoku University, 6-3, Aramaki, Aoba-ku,
Sendai, Miyagi 980-8578, Japan  Yurina Nakazato Department of Physics, The
University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan  Naoki Yoshida
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo
113-0033, Japan  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), UT Institute for Advanced Study, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan  Research Center for the Early Universe, School
of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033,
Japan

###### Abstract

A supersonic relative velocity between dark matter (DM) and baryons (the
stream velocity) at the time of recombination induces the formation of low
mass objects with anomalous properties in the early Universe. We widen the
scope of the ‘Supersonic Project’ paper series to include objects we term Dark
Matter + Gas Halos Offset by Streaming (DM GHOSts)– diffuse, DM-enriched
structures formed because of a physical offset between the centers of mass of
DM and baryonic overdensities. We present an updated numerical investigation
of DM GHOSts and Supersonically Induced Gas Objects (SIGOs), including the
effects of molecular cooling, in high resolution hydrodynamic simulations
using the AREPO code. Supplemented by an analytical understanding of their
ellipsoidal gravitational potentials, we study the population-level properties
of these objects, characterizing their morphology, spin, radial mass, and
velocity distributions in comparison to classical structures in non-streaming
regions. The stream velocity causes deviations from sphericity in both the gas
and DM components and lends greater rotational support to the gas. Low mass
($\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{5.5}$ M⊙) objects in regions of streaming demonstrate
core-like rotation and mass profiles. Anomalies in the rotation and morphology
of DM GHOSts could represent an early Universe analogue to observed ultra-
faint dwarf galaxies with variations in DM content and unusual rotation
curves.

##  1 Introduction

According to the standard $\Lambda$CDM model of structure formation, small
overdensities seeded by quantum fluctuations in the homogeneous matter fields
of the early Universe grew through gravitational collapse into structures.
Prior to Recombination ($z\sim 1100$), overdensities of baryonic matter were
prevented from growing by the strong coupling between the baryonic and
photonic fields. Dark matter (DM) overdensities, however, were free to
collapse. By the time of Recombination, when baryons decoupled from radiation,
DM overdensities had grown to five orders of magnitude larger than the
baryonic overdensities (e.g., Naoz & Barkana, 2005). Once decoupled, baryons
collapsed into the significantly larger DM potential wells, resulting in the
formation of structures with a central baryon component inside a larger DM
halo (e.g., Wechsler & Tinker, 2018).

This $\Lambda$CDM picture of structure formation is very successful on large
scales (e.g., Springel, 2005; Vogelsberger et al., 2014a, b, 2020; Schaye et
al., 2015). Uncertainties and tensions remain, however, especially on the
scales of faint dwarf galaxies (e.g, Bullock & Boylan-Kolchin, 2017; Simon,
2019; Perivolaropoulos & Skara, 2022). From uncertainties such as the core-
cusp challenge (e.g., Flores & Primack, 1994; Moore, 1994) to serious tensions
such as the observed diversity of rotation curves compared to simulations
(e.g., Oman et al., 2015, 2019), challenges to $\Lambda$CDM at low masses
include not only tensions with observations (e.g., Webb et al., 2022) but also
discrepancies between different state-of-the-art cosmological simulations (see
Sales et al., 2022, for a review). The ultra-faint dwarf regime is thus
expected to be one of the most sensitive probes of models and simulations of
structure formation that succeed at the scales of Milky Way-like galaxies and
larger mass dwarf galaxies. A precise description of the morphologies,
dynamical histories, and star-formation histories of ultra-faint galaxies
under $\Lambda$CDM (and other models) will be central to resolving these
tensions.

In an effort to refine the physical understanding of $\Lambda$CDM,
Tseliakhovich & Hirata (2010) pointed out that previous work neglected the
highly supersonic relative velocity ($v_{\rm bc}$) between DM and baryonic
overdensities stemming from their five orders of magnitude difference in
density. At Recombination, the root-mean-square (rms) value of the relative
velocity ($\sigma_{vbc}$) was 30 km s-1, five times the speed of sound of the
baryons at the time. This velocity has important consequences for structure
formation at small scales in the early Universe. It is coherent over a few Mpc
(Tseliakhovich & Hirata, 2010), and on those scales it can be modeled as a
stream velocity of a single value.

Subsequent works further explored the early-Universe implications of structure
formation in the presence of the stream velocity. For example, the stream
velocity was shown to delay the formation of Pop III stars (e.g., Stacy et
al., 2011; Greif et al., 2011; Schauer et al., 2017a) with impacts on
reionization and the 21-cm signal (e.g., Visbal et al., 2012; McQuinn &
O’Leary, 2012; Muñoz, 2019; Park et al., 2021). It also suppresses halo
abundance and generates “empty” halos with low gas content (e.g., Naoz et al.,
2012; Asaba et al., 2016), generating large scale inhomogeneities of galaxies
(e.g., Fialkov et al., 2012) and affecting the minimum halo mass that holds
most of its baryons (e.g., Naoz et al., 2013). Furthermore, in regions with a
large relative velocity, gas accretion onto star-forming dwarf halos is
affected – the gas falls downwind of halos, and has very low densities (e.g.,
O’Leary & McQuinn, 2012). The stream velocity was shown to be responsible for
reducing the number of low mass, luminous satellite galaxies expected in
$\Lambda$CDM, somewhat resolving an existing tension with observation at the
time (e.g., Bovy & Dvorkin, 2013). Low mass galaxies in the stream velocity
also have colder, more compact radial profiles (e.g., Richardson et al.,
2013). Beyond galaxies, the stream velocity was suggested to enhance massive
black hole formation (e.g., Tanaka et al., 2013; Tanaka & Li, 2014; Latif et
al., 2014; Hirano et al., 2017; Schauer et al., 2017b). In addition, the
stream velocity produces supersonic turbulence, which can assist with the
generation of early magnetic fields in the Universe (Naoz & Narayan, 2013).

Intriguingly, the stream velocity effect is also expected to induce the
formation of objects with anomalous properties in patches of the Universe with
non-zero values of $v_{bc}$. Naoz & Narayan (2014) showed that the stream
velocity introduces a phase shift between DM and baryon overdensities, which
translates to a physical separation between the two components. Two
interesting classes of objects arise from this effect that differ from
classical $\Lambda$CDM objects at the same scales. First, for objects at a
range of low masses ($\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}{\rm few}\times 10^{6}$ M⊙), the
spatial offset is so large that the baryonic component collapses outside the
virial radius of its parent DM halo entirely, potentially surviving as a DM-
deficient bound object. Naoz & Narayan (2014) proposed that these
Supersonically Induced Gas Objects (SIGOs) may be the progenitors of globular
clusters (e.g., Naoz & Narayan, 2014; Popa et al., 2016; Chiou et al., 2018,
2019, 2021; Lake et al., 2021; Nakazato et al., 2022; Lake et al., 2022).

Second, for a range of slightly higher mass objects ($\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{8}$ M⊙), the spatial offset is such that the centers of
mass of the baryonic component and the parent DM halo are offset, but the
majority of the gas remains inside the DM virial radius (Naoz & Narayan,
2014). We term these objects Dark Matter + Gas Halos Offset by Streaming (DM
GHOSts). These structures consist of both a DM and gas component, unlike
SIGOs, which are almost entirely gas. Compared to their classical $\Lambda$CDM
analogues, DM GHOSts are enriched in DM and highly diffuse, because the stream
velocity advects a portion of their gas component out of the halo. Naoz &
Narayan (2014) suggested that these objects may be the progenitors of ultra
faint or dark-satellite galaxies.

The Supersonic Project was introduced to investigate the supersonic stream
velocity-induced objects and their ties to observed structures. Previous
studies focused on the formation and evolution of SIGOs (e.g., Popa et al.,
2016; Chiou et al., 2018, 2019, 2021; Lake et al., 2021; Nakazato et al.,
2022; Lake et al., 2022). These simulations attempted to confirm the existence
of SIGOs and investigate their connection to globular clusters using only
adiabatic and sometimes atomic cooling. All except Schauer et al. (2021);
Nakazato et al. (2022) and Lake et al. (2022) neglected the effects of
molecular hydrogen cooling. Popa et al. (2016) and Chiou et al. (2019) placed
early constraints on the rotational properties of SIGOs, showing that they are
highly elongated structures with seemingly greater rotational support than
both DM GHOSts and “classical” analogs–objects of the same mass in regions
without streaming. Chiou et al. (2018, 2021) and Lake et al. (2022) focused on
the potential for SIGOs to be sites of star formation. In a semi-analytic
study, Chiou et al. (2019) found that SIGOs occupy a similar part of
magnitude-radius space today as the population of observed globular clusters
(e.g., McConnachie, 2012). Lake et al. (2021) extrapolated the large-scale
variation of SIGO abundances across the sky, predicting anisotropies in the
distribution of gas-rich objects at low masses that could be observed by the
James Webb Space Telescope (JWST) and binary black hole gravitational-wave
sources detectable by gravitational-wave detectors.

Several recent studies indicate that molecular cooling may play an important
role in the evolution of SIGOs and other objects in the stream velocity.
Glover (2013) and Schauer et al. (2021) indicate that molecular cooling
affects the abundance of gas objects in the early Universe, and Nakazato et
al. (2022) found that SIGOs became more filamentary in their molecular cooling
simulations. Lake et al. (2022) studied the collapse of SIGOs in the context
of molecular cooling, drawing an analogy to giant molecular clouds, and found
that SIGOs should form stars outside DM halos. Studies have neither
investigated DM GHOSts in detail nor constrained the rotational and
morphological properties of the supersonically-induced objects with molecular
cooling.

Here, we present an updated analysis of the morphology, rotation, rotational
curves, and mass distribution of both SIGOs and DM-GHOSts using molecular
hydrogen cooling numerical simulations supplemented by an analytical
perspective. We characterize the population-level properties of these
elongated objects in the context of ellipsoid potentials, and quantify their
total angular momentum and rotational support. We find that the DM component
deviates from a spherical configuration in the presence of the stream
velocity. We also present the first rotation curves for these objects, finding
a bifurcation in rotation curve shape according to mass. This may serve as an
early universe analog to the rotational curve diversity observed in dwarf
galaxies (e.g., Sales et al., 2022).

The paper is organized as follows: § 2 describes the numerical simulations
used in the study and the classification criteria for SIGOs and DM GHOSts. §
3.1 is devoted to the analytical and numerical results of the study. In §
3.1.1, we present the analytical ellipsoid potentials used to understand
supersonically-induced objects, and we show the population level morphological
properties of SIGOs and DM GHOSts from our simulations in § 3.1.2. In § 3.2,
we discuss the rotational support and angular momentum of these objects. In §
3.3, we present density profiles and rotation curves of DM GHOSts. A summary
and discussion of the results is given in § 4. The appendices explain the
choice of cutoff gas fraction used to define a SIGO (App. A), a full
derivation of the potential and total mass from § 3.1.1 (App. B), and
supplemental morphological data, including comparisons to NFW profiles (App.
C) .

In this study we assume a $\Lambda$CDM cosmology, with
$\Omega_{\rm\Lambda}=0.73$, $\Omega_{\rm m}=0.27$, $\Omega_{\rm b}=0.044$,
$\sigma_{8}=1.7$, and $h=0.71$.

##  2 Numerical Set Up

In a similar manner to previous studies by the Supersonic Project (e.g., Chiou
et al., 2018, 2019, 2021), we perform hydrodynamical simulations using the
AREPO code (Springel, 2010). AREPO is a moving-mesh code that allows for high
resolution studies of structure formation with an accurate picture of the
stream velocity up to $z\sim 20$.

###  2.1 Simulation and Initial Conditions

We use a modified CMBFAST code (Seljak & Zaldarriaga, 1996), as presented in
Popa et al. (2016), to include the first-order correction of scale-dependent
temperature fluctuations on the initial conditions and their transfer
functions, following Naoz & Barkana (2005). This is necessary as the
corrections detemine gas fraction in halos at higher redshift (e.g., Naoz et
al., 2009, 2011, 2013).

Tseliakhovich & Hirata (2010) showed that the supersonic relative velocity is
coherent on scales of $\sim$few Mpc, so following Popa et al. (2016), we
choose a box size of 2 comoving Mpc, such that the relative velocity can be
modeled as a stream velocity. Evolution of the stream velocity, a second order
correction (Tseliakhovich & Hirata, 2010), is also included in the transfer
functions. The simulations begin at $z=200$, when a $2\sigma$ fluctuation in
the stream velocity corresponds to 11.8 km s-1. The stream velocity is thus
implemented as a boost of 11.8 km s-1 to all baryon particles in the
$+x$-direction. The box of 2 comoving Mpc contains 5123 DM particles with a
mass resolution of $m_{\rm DM}=1.9\times 10^{3}\text{ M}_{\odot}$ and 5123
Voronoi mesh cells representing the gas component, with a mass resolution of
$m_{\rm gas}=360\text{ M}_{\odot}$. Our results are presented at the end of
the simulations, $z=20$.

To investigate the effect of the stream velocity, we perform two runs without
the stream velocity (i.e., runs in a region of space with a $0\sigma_{v_{bc}}$
fluctuation in the velocity field) and two runs with a value of
$v_{bc}=2\sigma_{v_{bc}}$. For each set of two runs (with and without the
stream velocity), we include molecular (H2) cooling in one and only adiabatic
cooling in the other. The inclusion of molecular cooling is described in §
refsec:molecular below.

Run | $v_{bc}$ | H2 Cooling
---|---|---
$0$v | $0$ | No
$2$v | $2\sigma_{v_{bc}}$ | No
$0$vH$2$ | $0$ | Yes
$2$vH$2$ | $2\sigma_{v_{bc}}$ | Yes
Table 1: Simulation Parameters

###  2.2 Molecular Cooling

To understand the effect of molecular cooling, we perform two runs for each
value of the stream velocity ($0\sigma_{v_{bc}}$ and $2\sigma_{v_{bc}}$), one
with adiabatic cooling only and one with molecular cooling included. We denote
the H2 cooling runs with “H2”. The 0vH2 and 2vH2 runs were also used in Lake
et al. (2022). A summary of the runs in this work is given in Tab. 1.

As in Nakazato et al. (2022) and Lake et al. (2022), we explicitly account for
nonequilibrium chemical reactions and radiative cooling in the gas, using
GRACKLE, a chemistry and cooling library (Smith et al., 2017; Chiaki & Wise,
2019). The 0vH2 and 2vH2 runs include H2 and HD molecular cooling. The
radiative cooling rate of the former includes both rotational and vibrational
transitions (Chiaki & Wise, 2019). Chemistry for the following 15 primordial
species is included in H2 runs: e-, H, H+, He, He+, He++, H-, H2,
H${}_{2}^{+}$, D, D+, HD, HeH+, D-, and HD+. We do not include star formation.

###  2.3 Object Classification

We are interested in gas-rich structures, including SIGOs, which have somewhat
low statistical power in these small box simulations. Thus, following Popa et
al. (2016); Chiou et al. (2018); Chiaki & Wise (2019); Chiou et al. (2021);
Lake et al. (2022, 2021); Nakazato et al. (2022) we choose $\sigma_{8}=1.7$,
which will increase the statistical power. This represents a region of the
Universe where structure forms early, such as in the Virgo cluster (e.g., Naoz
& Barkana, 2007). These results can then be scaled to other regions
accordingly (e.g., Park et al., 2020).

To identify structures, we search for two object classes using a friends-of-
friends (FOF) algorithm (see e.g, Popa et al., 2016; Chiou et al., 2018).

  1. 1.

DM-primary/Gas-secondary (DM/G) objects are found using the FOF algorithm on
DM particles first. Gas cells in the same object are associated with the DM
groups at a secondary stage. We require DM/G objects to have at least 300 DM
particles, to avoid numerical artifacts.

  2. 2.

Gas-primary (GP) objects are found using the FOF algorithm only on gas cells.
This allows us to find objects such as SIGOs in the simulation that have
little or no DM component. We require GP objects to have at least 100 gas
cells, again in order to avoid non-physical numerical effects.

Run | # GP | # SIGOs | # DM GHOSts
---|---|---|---
$0$v | $2557$ | - | -
$2$v | $759$ | $25$ | $734$
$0$vH$2$ | $5823$ | - | -
$2$vH$2$ | $1406$ | $69$ | $1337$
Table 2: Summary of the number of gas primary (GP) objects and subclasses
found in the four runs used in this study. Only objects containing over 100
gas particles are included. SIGOs and DM GHOSts do not exist in regions with 0
stream velocity, so they are not tabulated for the 0v and 0vH2 runs, but see
App. A for an investigation of false identification of SIGOs in molecular
cooling runs.

The choice to cut off DM/G and GP objects at 300 particles and 100 cells
respectively gives us a minimum structure mass resolution of $5.7\times
10^{5}$ M⊙ for DM/G and $3.6\times 10^{4}$ M⊙ for GP objects.

Popa et al. (2016) and Chiou et al. (2018) found that GP objects are often
filamentary in nature, and thus a spherical fitting algorithm is not an
optimal choice, as it does not reflect the actual morphology of these
structures. We therefore employ the same fitting algorithm of these works,
which is based on a triaxial ellipsoid fit. We keep the axis ratio of a
triaxial ellipsoid with $N_{0}$ gas particles and maximum radius $R_{\rm
max,0}$ around the GP object constant and shrink it in increments of 0.5
percent until the condition $R_{\rm max,n}/R_{\rm max,0}>N_{\rm n}/N_{0}$ is
met, or until $N_{\rm n}/N_{0}<0.8$, where $R_{\rm max,n}$ and $N_{\rm n}$ are
the maximum ellipsoid radius and number of gas particles of the $n$th
iteration.

Figure 1: Projected gas (left) and dark matter (DM) (right) density around
several DM GHOSts and a SIGO in a region $5$ physical kpc on a side. The SIGO
is bounded by the white ellipse, located in a region relatively devoid of dark
matter, and contains no DM component. It is embedded in a stream of gas. The
DM GHOSts (A, B, C, and D) each contain a gaseous and a DM component. The gas
components of the DM GHOSts are shown in orange, whereas the DM components of
the DM GHOSts are shown in green. Note that the DM components are not entirely
spherical. The centers of mass of the gas components of A, C, and D are offset
from the centers of mass of the DM component, whereas B has had time for the
gas component to fall back into the center of the DM potential. Several DM
halos with no associated gas components also lie in this region–depicted in
pink. One of these may be the “parent” halo of the SIGO.

The GP FOF algorithm is performed to identify SIGOs, gas-rich objects that
form outside the virial radius of the parent DM halos. However, many of the GP
objects are located inside DM halos, being the gas component of the DM/G
structures. These structures are also of interest to this study. In order to
clarify the difference between structures formed via classical $\Lambda$CDM
and these dark matter and gas structures formed in regions with the stream
velocity, we term the DM/G objects in regions of streaming as Dark Matter +
Gas Halos Offset by Streaming (DM GHOSts). In previous papers, these were
referred to as “DM/G”. Having formed offset from the center of mass of their
parent DM halo, these structures display different morphological and dynamical
properties than those that formed in regions of the Universe with no relative
velocity (i.e., a patch with a $0\sigma_{v_{b}c}$ fluctuation), even though
many are no longer offset by the redshifts considered here due to dynamical
processes (such as the DM GHOSt labeled “B” in Fig 1.)

We follow the convention in Nakazato et al. (2022), where SIGOs are defined as
GP objects which meet the following two conditions:

  1. 1.

They are located outside the virial radius of their parent DM halo.

  2. 2.

They contain a gas fraction,

$f_{g}=\frac{M_{\rm g}}{M_{\rm DM}+M_{\rm g}}>0.6\ ,$ (1)

where $M_{g}$ is the total mass of gas in the object and $M_{DM}$ is the total
mass of DM in the object.

Similar criteria were used in Popa et al. (2016); Chiou et al. (2018, 2019,
2021); Lake et al. (2021). The gas fraction cutoff in those works was chosen
rather arbitrarily to be $0.4$. This value was implemented because those
studies were interested specifically in the gas rich structures in connection
with observed DM-deficient objects such as globular clusters. Our cutoff gas
fraction of $0.6$ is higher. Nakazato et al. (2022) found that choosing a
smaller cutoff gas fraction in runs with molecular cooling leads to the
identification of filamentary structure as SIGOs, such that without the stream
velocity many SIGOs are misidentified. We find similar behavior in our
molecular cooling run, and this choice of gas fraction is discussed further in
App. A. GP objects in runs with stream velocity that do not meet the SIGO
criteria above are classified as the baryonic component of DM GHOSts.

A DM GHOSt therefore contains two components, a DM component, identified by
the DM/G FOF algorithm, and a gaseous component, identified by the GP FOF. For
DM GHOSts, the GP FOF often identifies the gas structure within the DM-primary
object. Figure 1 shows the projected density of DM (left) and gas (right) in a
region of the simulation box with two DM GHOSts and a SIGO. The SIGO contains
only a component identified by the GP FOF, which can be clearly seen in the
plot of the gas density. The DM GHOSts are found in both particle FOF types
and have two overlapping (but offset) components.

In Chiou et al. (2019) and subsequent papers, a spherical overdensity
calculation was used to obtain the virial radius of the DM halos. However, in
this study we sought to explore the morphology of the diffuse DM GHOSts and
their dark matter component. Thus, we also perform an ellipsoid fit as
described above to the DM/G objects to explore whether they deviate from a
spherical morphology. So, as before, a triaxial ellipsoid with fixed axis
ratios is fit to the DM/G object, shrinking in $0.5\%$ increments until the
axis ratio is less than the ratio of the number of particles in the original
object to the shrunken ellipsoid, or $20\%$ of particles are removed.

A table of the GP objects found using the FOF algorithm described here is
presented in Tab. 2. The probability density distributions in § 3 are
calculated from this set of objects, with a Gaussian kernel density function
using a Scott bandwidth (Scott, 2010).

##  3 Physical properties and Analytical Description

In combination with an analytical understanding, this section describes the
morphological and rotational properties of the population of numerically
simulated structures from the four simulation runs described above.

###  3.1 Morphology

Historically, spherical overdensity calculations have been used to understand
the gravitational potentials of the Universe’s first structures (see Barkana &
Loeb, 2001, for a review). However, both the stream velocity and molecular
cooling were shown to induce gaseous filaments and elongate structures. For
this reason, we introduce the eccentricity as a measure of an object’s
deviation from an idealized spherical configuration, and present an analytical
potential for SIGOs and DM GHOSts in terms of their eccentricity. A full
derivation of the potential and other relevant equations is given in App. B.

####  3.1.1 Analytical ellipsoid potential of SIGOs and DM GHOSts

Figure 2: Choice of coordinates adopted in this work. The ellipsoids are
arranged such that the primary axes of the ellipsoid are aligned along the
cartesian coordinate directions, with $R_{\rm max}$, the polar radius of a
prolate spheroid, along the $z$-axis. In the prolate approximation, $R_{\rm
min}\sim R_{\rm max}$. Cylindrical coordinates are used in § 3.1.1 as the
natural choice for prolate ellipsoid potentials.

In order to analytically explore the role of the eccentricity in the
gravitational potential, we approximate SIGOs and DM GHOSts as prolate
ellipsoids, with $R_{\rm max}>R_{\rm mid}\sim R_{\rm min}$. We show in Sec.
3.1.2 that this approximation is consistent with the structures found in the
simulation.

Figure 3:  Left: Probability density distribution of $\log(1-e)$, where $e$ is
the eccentricity (Eq. (3)), for gas primary (GP) objects. Distributions are
separated into the object classes listed in Tab. 2 and calculated using a
Gaussian kernel density. The orange distributions show the gas component of DM
GHOSts, the grey distributions show the gas component of classical halos
(without $v_{\rm bc}$), and the blue distributions show SIGOs. The darker
lines denote runs with H2 cooling, while the lighter dashed lines denote no
cooling. Right: Scatter plot of $R_{\rm mid}/R_{\rm max}$ versus $R_{\rm
min}/R_{\rm mid}$ for gas-primary (GP) objects in H2 cooling runs. The color
bar is $R_{\rm min}/R_{\rm max}$. The left column has no stream velocity and
the right column is from the $v_{bc}=2\sigma_{v_{bc}}$ runs (See Tab. 2).
Stars represent SIGOs, as defined in Sec. 2.3.

In cylindrical coordinates ($R,z$), the gravitational potential ($\Phi$) of a
prolate ellipsoid can be written as:

$\Phi(\textbf{x})=\\\ -2^{3/2}\frac{4\pi GR_{\rm max}^{4}\rho(R_{\rm
max}^{2})\cos^{-1}(\sqrt{1-e^{2}})}{e\sqrt{1+(1-e^{2})\left(\frac{R^{2}}{R_{\rm
min}^{2}}+\frac{z^{2}}{R_{\rm max}^{2}}\right)}}\ ,$ (2)

where $G$ is the gravitational constant, $R_{\rm max}$ is half the length of
the maximum axis of the ellipsoid, $R_{\rm min}$ is half the length of the
minimum axis of the ellipsoid (see Fig. 2), $\rho(R_{\rm max}^{2})$ is the
density at $R_{\rm max}$, and $e$ is the eccentricity. See App. B for a
derivation of Eq. (2). The eccentricity is a measure of the ellipsoid
elongation, defined (following the convention used in Binney & Tremaine, 2008)
as:

$e\equiv\sqrt{1-\left(\frac{R_{\rm min}}{R_{\rm max}}\right)^{2}}\ .$ (3)

This parameter resembles the 2D ellipse eccentricity, and varies from 0
(spherical) to 1 (radial). Previous works by the Supersonic Project (Chiou et
al., 2018) used the prolateness factor ($\xi$) to characterize the shape of GP
objects:

$\xi=\frac{R_{\rm max}}{R_{\rm min}},$ (4)

The relation between the eccentricity and the prolateness factor is:

$e=\sqrt{1-\xi^{-2}}\ .$ (5)

In deriving the potential in Eq. (2), we have assumed a prolate spheroidal
density profile given by:

$\rho(m^{2})=\rho_{0}\left[1+\left(\frac{m}{R_{\rm
max}}\right)^{2}\right]^{-\frac{3}{2}}\ ,$ (6)

where $m$ is defined in cylindrical coordinates as:

$m^{2}\equiv\frac{R^{2}}{1-e^{2}}+z^{2}\ ,$ (7)

(Binney & Tremaine, 2008). Note that $0\leq m\leq R_{\rm max}$. In the above
formalism, we calibrate the density such that $\rho_{0}=2^{3/2}\rho(R_{\rm
max})$.

From here, we find the dependence of the total mass on eccentricity. Once
again, a complete derivation can be found in App. B. The total mass of the
ellipsoid is found by integrating over a set of similar ellipsoids from the
center to the outer edge of the object (i.e., $m=0$ to $m=R_{\rm max}$). Thus,
for an object with density given by Eq. (6), we find the total mass of the
object $M$:

$\displaystyle M$ $\displaystyle=$ $\displaystyle 4\pi(2^{3/2}\rho_{\rm
max})(1-e^{2})R_{\rm max}^{3}\left(\sinh^{-1}{(1)}-\frac{1}{\sqrt{2}}\right)$
(8) $\displaystyle\approx$ $\displaystyle 6.19\rho_{\rm max}(1-e^{2})R_{\rm
max}^{3}.$

Below we use the eccentricity parameter to estimate the prolateness of the
SIGOs and DM-GHOSts. We also compare the eccentricity inferred by the
analytical ellipsoid potential (Eq. (8)) for an average object with our
simulated objects’ eccentricity versus the mass enclosed within the ellipsoid
bounding each object, finding agreement between the analytic and numeric
results for eccentric objects.

Figure 4: Left: Probability density distribution of $\log(1-e)$ of dark matter
(DM)-primary objects, where $e$ is the eccentricity (see Eq. (3)). The orange
distributions include the DM component of DM GHOSts and the grey distributions
show the gas component of classical halos without $v_{bc}$. SIGOs, which have
little to no DM component, are not plotted. The darker lines denote runs with
H2 cooling, while the lighter dashed lines denote no cooling. Right: Scatter
plot of $R_{\rm mid}/R_{\rm max}$ versus $R_{\rm min}/R_{\rm mid}$ for DM-
primary (DM/G) objects. The color bar is $R_{\rm min}/R_{\rm max}$. The left
column has no stream velocity and the right column is from the
$v_{bc}=2\sigma_{v_{bc}}$ runs (See Tab. 2). SIGOs are not included because
they have no DM component. These results imply that the DM component for the
majority of objects is non-spherical, and the stream velocity induces further
elongation.

####  3.1.2 Morphology of numerically simulated objects

Figure 5: Top: Scatter plot of $\log(1-e^{2})$ versus $M_{\rm tot}$ for gas-
primary (GP) objects. SIGOs are denoted by stars. Bottom: Scatter plot of
$\log(1-e^{2})$ versus $M_{\rm DM}$ for DM-primary (DM/G) objects (we include
both DM components of DM GHOSts and other DM halos in the box that have no
associated GP component) In both plots, the top two panels show the H2 cooling
runs, and the bottom two panels show runs without cooling. The left column has
no stream velocity and the right column is from the $v_{bc}=2\sigma_{v_{bc}}$
runs. Stars represent SIGOs, as defined in Sec. 2.3. The color bar is the gas
fraction (Eq. (1)). The red overplotted line is the expected relationship from
Eq. (8) for an example object with the average density and maximum radius of
objects in the H2 cooling runs ($\bar{\rho}_{Rmax}=1.8\times 10^{8}$M⊙ kpc-3,
$\bar{R}_{max}=0.134$ kpc).

In the previous section we assumed a prolate relation between ellipsoid axes (
$R_{\rm max}>R_{\rm mid}\sim R_{\rm min}$). Interestingly, we find that both
the gas component and the DM component of structures become prolate in the
presence of the stream velocity as depicted in Figure 3.

The left panel of Fig. 3 shows probability density distributions for the
eccentricity of GP objects in all four runs. The gas components of classical
halos (i.e., no stream velocity), SIGOs, and DM GHOSts comprise three distinct
populations in eccentricity. The stream velocity induces elongation of
objects, with objects in 0v and 0vH2 runs being the most spherical. Among the
supersonically-induced objects, SIGOs are more elongated and prolate than DM
GHOSts. The average SIGO eccentricity of $0.977$ corresponds to an object
whose $R_{\rm min}$ is only around 20% of its $R_{\rm max}$, whereas for a
classical object with molecular cooling the average ratio is around 60% (See
Tab. 3 of App. C for the means of morphological parameters). The stream
velocity effect dominates runs with and without molecular cooling, but in the
no stream velocity case, molecular cooling also slightly elongates the gas
component.

The stream velocity also affects the shape of the DM component of DM GHOSts,
resulting in elongated DM structures. The left panel of Fig. 4 shows the
distribution of eccentricities of the DM-primary objects in the 0v and 0vH2
runs and DM GHOSts with and without cooling. Including H2 cooling, the DM
component of DM GHOSts tends to be less spherical than the classical DM halos.

The eccentricity is only a measure of the difference between the minimum and
maximum radii of the ellipsoid. Therefore, to justify the prolate
approximation, we show the ratios of all three axes of the gas (Fig. 3, right
panel) and DM (Fig. 4, right panel) component of the ellipsoids. The parameter
space is divided into spherical, triaxial, oblate, and prolate objects. Even
without a stream velocity, there is a range of morphologies among both the DM
and gas components of structures. The probability density distributions of all
the ratios are given in App. C. As seen in the Figures, the majority of the DM
components are spherical in nature, and those that deviate from sphericity
tend to be prolate.

For the gas components (top right panel in Figure 3), as expected, the
majority of the gas component of classical object is spherical, with
preference toward prolate configuration. The stream velocity elongates objects
into more extreme axis ratios. In fact, there are a scarce few truly spherical
objects in the runs with stream velocity. SIGOs, shown as stars in the figure,
have not only the most extreme eccentricities overall, but also tend towards
the triaxial region of the figure.

In Fig. 5, we plot the eccentricity of the objects’ gas component as a
function of mass. We overlay the expected relation from Eq. (8) for object of
average density and scale (solid line). Recall that this equation represents
the relationship between the mass and the elongation for an ellipsoid
potential. Thus, for the no stream velocity case, where the majority of the
structure is spherical (i.e., $\log(1-e^{2})=0$) most of the objects are
concentrated at low eccentricity. However, more elongated objects in runs with
the stream velocity follow the trend outlined by Eq. (8). In particular, for
the no cooling case, the plot shows a more elongated structure for smaller
mass systems. However, cooling, even in the presence of stream velocity, tends
to assist with collapse, thus resulting in a deviation compared to the
analytical prediction. Note that in general, cooling still tends to create
more elongated structures (this was also highlighted in Nakazato et al., 2022;
Lake et al., 2021). In the bottom plot of Fig. 5, we plot the same relation
for the DM/G objects. Again, those that are more eccentric in nature follow
the trend derived from the ellipsoid potential, while many halos, especially
classical halos, are concentrated towards circularity.

Figure 6: Probability density distribution of $\lambda_{\rm g}$ (left) and
$\lambda_{\rm DM}$ (right). These are calculated via Eqns. (11) and (9),
respectively. Distributions are separated into the object classes listed in
Tab. 2 and calculated using a Gaussian kernel density. The orange
distributions include the gas component of DM GHOSts, the grey distributions
show the gas component of classical halos without $v_{bc}$, and the blue
distributions show SIGOs. We do not include $\lambda_{\rm DM}$ for SIGOs since
they are dominated by gas and the DM does not contribute significantly to the
angular momentum of the system. The darker lines denote runs with H2 cooling,
while the lighter dashed lines denote no cooling.

###  3.2 The Spin Parameter

The angular momentum of galaxies has long been understood to be closely tied
to their formation and evolution (e.g., Peebles, 1969; Fall & Efstathiou,
1980). In particular, the relationship between the angular momentum of the DM
halo and the gas seems critical in shaping the final galactic morphology and
spin parameter (e.g., Bullock et al., 2001; Maller & Dekel, 2002; Danovich et
al., 2015; Rodriguez-Gomez et al., 2017; Wechsler & Tinker, 2018; Kurapati et
al., 2021; Yang et al., 2021; Rohr et al., 2022; Cadiou et al., 2022;
Ebrahimian & Abolhasani, 2022; Rodriguez-Gomez et al., 2022; Hegde et al.,
2022). DM halo spin in simulations follows a lognormal distribution (e.g.,
Bullock et al., 2001; Zjupa & Springel, 2017), and the spin of the cold gas of
galaxies seems to follow a similar distribution in observations and
simulations (e.g., Danovich et al., 2015; Burkert et al., 2016). Models that
conserve angular momentum suggest that the structure, size, and morphology of
galaxies follow the mass and angular momentum of their host halos (e.g.,
Somerville et al., 2008; Guo et al., 2011; Benson, 2012; Somerville & Davé,
2015). Initially, simulations struggled to replicate observed properties of
galaxies such as the large spin of the baryonic component compared to the DM
and the shape of angular momentum of galaxies, but it was recognized that
baryonic processes, including feedback, can explain the evolution of the
angular momentum of the baryonic component (e.g., Maller & Dekel, 2002; Teklu
et al., 2015; Zjupa & Springel, 2017; El-Badry et al., 2018; Rohr et al.,
2022). Furthermore, some recent work (e.g., Sales et al., 2012; Danovich et
al., 2015; Jiang et al., 2019) suggests that galaxies’ spins are not
correlated with the spins of their host halos at all, and observed scaling
relations must be explained via other mechanisms. Persistent uncertainties in
the relationship between angular momentum, morphology, and galaxy structure
remain, particularly at low masses (e.g., Nguyen et al., 2022; Ebrahimian &
Abolhasani, 2022). Including $v_{bc}$, which affects both the velocity and the
configuration of the baryonic component, has already been shown to affect the
spin at low masses (e.g., Chiou et al., 2018), and thus we continue with an
investigation of the angular momentum of our sample of structures.

To quantify the rotation and angular momentum of objects, we follow the
analytical formulation from Chiou et al. (2018). The total angular momentum,
denoted by the spin vector ($\mathbf{J}_{\rm sp}$) of a set of $N$ particles
each of mass $m_{i}$ is

$\mathbf{J}_{\rm sp}=\sum_{i=1}^{N}m_{i}\mathbf{r}_{i}\times\mathbf{v}_{i},$
(9)

where $\mathbf{r}_{i}$ and $\mathbf{v}_{i}$ are the particles’ position and
velocity vectors from the center of mass. For DM primary objects, we estimate
the angular momentum of the entire halo using the spin parameter (e.g.,
Peebles, 1969) as defined in Bullock et al. (2000):

$\lambda_{\rm DM}=\frac{J_{\rm sp}}{\sqrt{2}M_{200}v_{200}R_{200}}\ .$ (10)

Here $M_{200}$ is the virial mass of the object, $v_{200}=\sqrt{GM_{200}/R}$
(Barkana & Loeb, 2001), and $J_{\rm sp}=|\mathbf{J_{\rm sp}}|$. Chiou et al.
(2018) showed that the DM/G spin follows a lognormal distribution consistent
with Bullock et al. (2001).

Following Chiou et al. (2018), in order to account for the more ellipsoidal
nature of the gas component, we calculate the spin parameter for gas primary
objects using:

$\lambda_{\rm g}=\frac{J_{\rm g}}{6\sqrt{2}M_{\rm g}v_{\rm GP}R_{\rm max}},$
(11)

where $M_{\rm g}$ is the total gas mass, $v_{\rm GP}$ is the circular velocity
of the gas primary object at a distance $R_{\rm max}$, and $J_{\rm g}$ is
calculated from Eq. (9) for gas particles only.

Figure 6 shows the probability density distributions of the spin parameter of
gas primary objects and DM primary objects. The stream velocity induces higher
total spin for the gas component of all runs. However, for classical gas
objects, molecular cooling serves to lower the total angular momentum by
condensing gas inward, thus allowing for a smaller spin parameter. For
classical objects, the DM components have a larger spin parameter magnitude
overall: because DM constitutes most of the mass, it contributes most of the
total angular momentum. On the other hand, the stream velocity boosts the gas
spin for both SIGOs and the gas component of DM GHOSts, thus increasing the
total angular momentum of the system despite H2 cooling. SIGOs and DM GHOSts
thus have gas spin parameters an order of magnitude larger than the no stream
velocity gas (see Tab. 3).

In Fig. 8, we plot the spin parameter against the total mass (left) and
eccentricity (right) of objects. More eccentric objects tend to have higher
spins in all runs. The vector sum in the definition of $\mathbf{J}_{\rm sp}$
(Eq. (9)) means that this parameter encodes not only the magnitude of the
total angular momentum but also the alignment of particles’ rotation. Thus,
the trend on the right of Fig. 8 is consistent with spherical configurations
corresponding to an isotropic distribution of the particles’ orbits. Further,
it is consistent with prolate systems having a preferred directionality to the
angular momentum or ordered distribution of particle orbits.

Figure 7:  An ellipsoid in spherical coordinates for an arbitrary spin vector
direction. Coordinates are chosen such that the primary axes of the ellipsoid
are aligned along the cartesian coordinate directions, with $R_{\rm max}$, the
polar radius of the prolate spheroid, along the $z$-axis. The spin vector (Eq.
(9)) can be aligned in any direction with respect to the ellipsoid axes, and
its relative alignment with respect to these axes is described by the usual
spherical angular coordinates $\theta$ and $\phi$.

Furthermore, the spin of the objects in the no stream velocity case roughly
follows a $\lambda\sim M^{-2/3}$ slope, see bottom left panel in Fig. 8. This
relation is expected from Eq. (10) for mostly circular orbits. However, the
trend dissipates in the presence of the stream velocity and cooling, where the
objects deviate from spherical symmetry and the combined effects introduce a
preferred directionality for the angular momentum, almost regardless of the
mass. We attribute this to the turbulent and filamentary nature of these
structures in the presence of the stream velocity and molecular cooling (e.g.,
Nakazato et al., 2022; Lake et al., 2022). Note that the cut-off in the low-
mass regime is due to our resolution limit of $300$ particles minimum
(corresponding to a mass of $5.7\times 10^{5}$ M⊙), while at the high mass
regime we are limited by Poisson fluctuations of small number statistics at
this high redshift and small box size.

The question of whether these larger spin parameters imply greater overall
angular momentum or more ordered rotation leads us to an investigation the
connection between the morphology of the objects and their rotational support.
An investigation of the ellipsoids’ alignment with respect to the stream
velocity direction revealed that SIGOs are not always aligned with $R_{max}$
in the direction of the stream velocity. More frequently, they are embedded in
a stream of gas that is infalling towards a larger DM halo (see Fig. 1, for
example), and their longest axis aligns with this stream of gas. Thus, to
check how ordered these objects rotation is, we test whether the rotation axis
is aligned with any of the three primary ellipsoid axes.

Figure 8: Scatter plot $\lambda_{\rm g}$ (left) versus the dark matter mass
($M_{\rm tot}$) (left) and $\lambda_{\rm g}$ versus $\log(1-e)$ (right) of GP
objects. The top row includes H2 cooling. The left column denotes the runs
without the inclusion of the stream velocity, and the right column contains
runs with the stream velocity. A line corresponding to $\lambda_{g}\sim
M^{-2/3}$ is overplotted in red on the right hand side. This is the expected
relation from Eq. (11). The vertical cut-off at low masses is due to our
resolution limit of $300$ particles minimum (corresponding to a mass of
$5.7\times 10^{5}$ M⊙), while at high mass we are limited by Poisson
fluctuations of small number statistics at this high redshift and small box
size.

To describe the directionality of the angular momentum we utilize spherical
coordinate notation. With the maximum radius aligned with the $z$-axis and the
minimum radius aligned with the $x$-axis, we calculate the spherical $\theta$
and $\phi$ components of the spin vector of both the DM and gas of objects.
Fig. 7 shows an illustration of the orientation of the ellipsoid with respect
to the spin vector.

Figure 9: Top panels: Left: Probability density distribution of
$\cos{(\phi_{\rm g})}$ the angle between $R_{\rm min}$ and the spin vector of
the gas component of GP objects (See Fig. 2). Right: Probability density
distribution of $|\cos{(\theta_{g})}|$, the angle between $R_{\rm max}$ and
the spin vector. The darker lines denote runs with H2 cooling, while the
lighter dashed lines denote no cooling. Distributions are separated into the
object classes listed in Tab. 2 and calculated using a Gaussian kernel
density. The orange distributions include the gas component of DM GHOSts, the
grey distributions show the gas component of classical halos without $v_{bc}$,
and the blue distributions show SIGOs. Middle panels: Left Probability density
distribution of $\cos{(\phi_{\rm dm})}$ the angle between $R_{\rm min}$ and
the spin vector of the DM component of objects (See Fig. 2). Right:
Probability density distribution of $|\cos{(\theta_{\rm dm})}|$, the angle
between $R_{\rm max}$ and the spin vector. The darker lines denote runs with
H2 cooling, while the lighter dashed lines denote no cooling. Distributions
are separated into the object classes listed in Tab. 2 and calculated using a
Gaussian kernel density. The orange distributions include the gas component of
DM GHOSts, the grey distributions show the gas component of classical halos
without $v_{bc}$. SIGOs, which have little to no DM component, are not shown.
Bottom panels: Cartoon depiction of the orientation of the spin vector with
respect to the various object classes, classical halos (left), DM GHOSts
(middle) and SIGOs (right). The large, dark arrow indicates the most common
orientation, while the other faint arrows indicate a spread of other common
alignments for each object type according to the distributions in the top and
middle panels.

Figure 9 shows distributions of the misalignment between the primary ellipsoid
axes and the spin vector for both the gas component (top row) and the DM
component (middle row). As depicted, the classical halos (both for DM and gas
components) are preferentially spinning in alignment with their minimum axis,
and do not show a preference with respect to their maximum axis. These
classical halos were the most oblate group overall, thus the lack of
preference for alignment with the maximum axis could be due to the fact that
for an oblate spheroid, $R_{\rm max}\sim R_{\rm mid}$. In other words, they
are consistent with puffy disks. The bottom row of Fig. 9 shows a cartoon
depiction of the range of preferred rotation of classical objects here.

DM GHOSts, to a lesser degree, are rotating in alignment with their minimum
axis, but they also show a preference towards the maximum axis. This “spinning
top” type of behaviour seems to be consistent when considering their formation
(see Fig. 3 in Lake et al., 2022). The gas component of DM GHOSts (similar to
classical objects) is accreted in a stream onto the halo, but the stream
velocity induces a velocity gradient in a preferential direction in that
region perpendicular to the infall stream. This results in spinning-top
rotator behavior, depicted also in a cartoon at the bottom of Fig. 9.

SIGOs, however, exhibit a weak bimodal distribution of alignment with $R_{\rm
max}$. The majority are preferentially misaligned with the maximum axis, while
some demonstrate alignment as in the case of the DM GHOSts described above.
Considering an idealized growth scenario, SIGOs are embedded in the gas
stream, which is normally in the process of accreting onto a DM halo. This
configuration often yields an $R_{\rm max}$ in alignment with the accretion
stream, (as is the case in the example in Figure 1). As described above with
DM GHOSts, the stream velocity induces a velocity gradient (in our case
towards the $x-$direction) which may be perpendicular to the infall stream
moving in the $y-$ or $z-$directions. (For example, see Fig. 3 in Lake et al.,
2022, where a SIGO is embedded in a stream of gas infalling towards a larger
halo. All the gas in the region is moving towards this stream, however greater
velocities are found on the $+x$ side, a gradient induced by the original
stream velocity in the $+x$-direction.) This perpendicular accretion mode
causes objects with alignment between $\mathbf{J}_{\rm sp}$ and
$\mathbf{R}_{\rm max}$. However, this picture is idealized, and in practice
the SIGOs represent a density perturbation within the stream that results in
gas accretion that is not necessarily aligned with the object’s $R_{\rm max}$.
Those SIGOs that are preferentially misaligned respect to $R_{\rm max}$ show a
variety of alignments with respect to the $R_{\rm min}$ . This is potentially
due to the fact that (as opposed to the oblate case described above) for
prolate spheroids, the system’s symmetry is that $R_{\rm min}\sim R_{\rm
mid}$.

Figure 10: Probability density distribution of the cosine of the misalignment
between the spin parameter of the gas and dark matter components of individual
GP objects. (See Eq. (12)) The grey distributions are classical halos, the
orange distributions are DM GHOSts, and the blue dashed distributions show
SIGOs. The darker lines denote no cooling, while the lighter dashed lines show
the inclusion of molecular cooling.

The similarity between the DM and gas components in Fig. 9 is further
investigated below. Specifically, we calculate the misalignment angle between
the angular momentum of the DM component and the gas component:

$\cos{(\theta_{\rm g,DM})}=\frac{\mathbf{J}_{\rm DM}\cdot\mathbf{J}_{\rm
g}}{|J_{\rm DM}||J_{\rm g}|}\ .$ (12)

Note that for SIGOs the DM component is negligible, thus we only consider the
classical objects and DM GHOSts in this analysis. Figure 10 shows the
probability distributions of $\cos{(\theta_{\rm g,DM})}$. Consistent with
previous studies (e.g., Chiou et al., 2018), the classical halos are have a
strong alignment between the gas and DM components. On the other hand, the
alignment between gas and DM spin is weaker for the DM GHOSts, with a long
tail of nearly isotropic configurations. This result is consistent between
molecular and atomic cooling. In future work it may also be relevant to
examine the effects of feedback on this distribution. This may especially be
relevant for star-forming SIGOs.

###  3.3 Rotation Curves and Mass Distribution of DM GHOSts

Because the stream velocity affects the angular momentum and morphological
configuration of structures, we expect a possible effect on the density
distribution and rotational curves. In particular, since rotation curves
contain signatures of the DM component, we expect that both SIGOs and DM
GHOSts will deviate from the classical profiles.

Figure 11: Density of gas in DM GHOSts as a function of radius, normalized to
the $R_{\rm max}$ of the ellipsoid, for the molecular cooling run with
$v_{bc}=2\sigma_{v_{bc}}$ (top panel) compared to classical halos with
$v_{bc}=0\sigma_{v_{bc}}$ and molecular cooling (bottom panel). The density is
calculated in 50 ellipsoidal shells moving out from the center of the object.
We split objects by mass–the average rotation curve for objects above
$10^{5.5}$ M⊙ is shown in solid blue, while those below this cutoff are
plotted in solid pink. Low mass objects display a deviation from the classical
rotation curves, with less of a cusp. The blue shaded region shows $1\sigma$
away from the curve for objects above $10^{5.5}$ M⊙, while the purple hatched
region shows $1\sigma$ away from the curve for objects below $10^{5.5}$ M⊙. An
NFW profile for a $10^{5}$ M⊙ halo is shown for comparison as the dashed line.
Eq. (6) is plotted for a gas object with average density in solid blue.

In particular, in this section we focus our analysis to a comparison between
runs with and without stream velocity (0vH2 and 2vH2). Figure 11 shows the
density of gas of DM GHOSts (top panel) and classical halos (bottom panel) as
a function of radius from the center of mass, with an NFW (e.g., Navarro et
al., 1996a, b, 1997) halo profile overplotted (dashed line). The stream
velocity serves to reduce densities across the structure, as expected.
Physically, this is due to the advection of gas from the halo and the spatial
separation of the two components. In addition, at low masses ($\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{5.5}$ M⊙), we observe a deviation from the NFW shape, with
a constant, core profile, rather than a cusp. Using the prolate density
profile, Eq. (6), we see that a core-like structure is expected for these
ellipsoids (solid line). In § 3.1.2, we demonstrated that low mass objects had
high eccentricities. Since the classical NFW formulation assumes spherical
overdensities (e.g., Navarro et al., 1996a, b, 1997), another reason for the
deviation may be the extreme eccentricities of very low mass objects in stream
velocity simulations.

Figure 12: Rotation curves of DM GHOSts as a function of radius, calculated in
ellipsoidal shells going outward, for the molecular cooling run with
$v_{bc}=2\sigma_{v_{bc}}$ (top panel) and $v_{bc}=0\sigma_{v_{bc}}$ (bottom
panel). The average velocity of the gas in an ellipsoidal shell at each radius
is normalized by the $v_{circ}$, the circular radius at $R_{\rm max}$ of the
ellipsoid. The objects are colored by the mass of their gas component. The
average velocity of the gas in an ellipsoidal shell at each radius is
normalized by the $v_{circ}$, the circular radius at $R_{\rm max}$ of the
ellipsoid. We split objects by mass–the average rotation curve for objects
above $10^{5.5}$ M⊙ is shown in solid blue, while those below this cutoff are
plotted in solid pink. Low mass objects display a deviation from the classical
rotation curves, with less of a cusp. The blue shaded region shows $1\sigma$
away from the curve for objects above $10^{5.5}$ M⊙, while the purple hatched
region shows $1\sigma$ away from the curve for objects below $10^{5.5}$ M⊙.
The dashed lines show the minimum and maximum rotation curves of all the
objects by velocity.

Figure 12 shows the rotation curves of DM GHOSts (top panel) and classical
halos (bottom panel). The rotation curves as a function of radius for DM
GHOSts have two behaviours separated by mass. In particular, the core-type
density distribution of low-mass objects in the stream also means that their
rotational velocity does not climb as fast.

Figure 13:  Scatter plot of the velocity at $R_{\rm max}$ ($v_{Rmax}$) of
objects as a function of total mass ($M_{tot}$). The top two panels show the
molecular cooling runs, and the bottom two panels show no cooling. The left
column has no stream velocity and the right column is from the
$v_{bc}=2\sigma_{v_{bc}}$ Stars represent SIGOs, as defined in Sec. 2.3. The
color bar is the eccentricity (Eq. (3)). The line shows the expected value for
an NFW profile. Figure 14:  Scatter plot of $R_{\rm max}$ of objects as a
function of total mass ($M_{tot}$). The top two panels show the molecular
cooling runs, and the bottom two panels show no cooling. The left column has
no stream velocity and the right column is from the $v_{bc}=2\sigma_{v_{bc}}$
Stars represent SIGOs, as defined in Sec. 2.3. There are a few objects
misclassified as SIGOs in the runs with no stream velocity–see App. A for a
discussion. The color bar is the eccentricity (Eq. (3)) The line shows the
expected value for an NFW profile. For the 0vH2 run, 22% of objects fall above
the expected NFW line. For the 2vH2 run, this fraction rises to 58% of objects
located above the line.

DM GHOSts are more diffuse and rotationally supported than classical halos,
having received a boost from the stream velocity. In Figure 13, we show the
velocity at $R_{\rm max}$ as a function of the total mass. We compare this to
the nominal NFW expectation, following Navarro et al. (1997), taking the
maximum of the NFW circular velocity:

$v_{\rm
circ}^{2}=\frac{1}{x}\frac{\ln{(1+cx)}-(cx)/(1+cx)}{\ln(1+c)-c/(1+c)},$ (13)

where $x=r/r_{200}$, and $c$ is the halo concentration. In regions of
streaming, the velocity at $R_{\rm max}$ reaches or exceeds the expected
values. This behaviour is expected based on their larger overall spin
parameters than in the classical case as seen in Fig. 6.

We note that the radii of DM GHOSts and SIGOs are larger than the expected
from classical considerations. Specifically, in Fig. 14, the maximum ellipsoid
radius is plotted against the total mass of the object, with an NFW expected
relationship overplotted. H2 cooling runs show objects which have condensed to
smaller maximum radii (top left panel). We calculate the fraction of objects
in Fig. 14 above and below the NFW line, and find that the majority ($\sim
80\%$) of classical halos condense to smaller radii than the NFW $R_{\rm max}$
with H2 cooling in regions of no stream velocity. In the presence of stream
velocity, however, the velocity boost overall yields a larger radii (top right
panel). In the 2vH2 run, 60% of all objects lie $above$ the line, having
larger than expected maximum radius. Interestingly, SIGOs tend to have higher
$R_{\rm max}$ than NFW in all cases. Again, we suggest that eccentricity plays
a central role in giving objects much greater $R_{\rm max}$ than would be
possible in the spherical case.

These results illustrate the combined effects of the stream velocity and
molecular cooling that cause DM GHOSts to be more diffuse and rotationally
supported than their classical counterparts of similar masses.

##  4  Discussion

In this work, we investigate the spin, rotational and morphological properties
of structures in the presence of stream velocity at $z=20$ using high
resolution numerical simulations in AREPO. For the first time, molecular
cooling is included in a detailed study of these dynamical properties. We
focus on a class of objects that we term DM GHOSts, structures where the
baryonic component is offset from the dark matter halo, but does not fully
escape the virial radius (as with SIGOs, which were previously the focus of
studies by the Supersonic Project). As in Figure 1, we emphasize that as time
goes by, the gas sinks to the center of the DM halo, but carries the signature
of its unique formation channel. Using molecular cooling simulations, we are
able to more precisely constrain the properties of SIGOs and DM GHOSts in
comparison to classical low mass objects than was possible in previous studies
(e.g., Chiou et al., 2018).

We considered the following physical properties of DM GHOSts, comparing them
to classical objects and SIGOs.

  * •

Morphology: We show that SIGOs are the most elongated class of objects,
followed by DM GHOSts, for both gas and DM components (as depicted in Fig. 3
and Fig. 4). We note that the DM component, of DM GHOSts is significantly
elongated compared to the classical objects. Frequently, SIGOs and DM GHOSts
tend to be prolate ellipsoidal, and we present an analytical expression of
their gravitational potential. While the gas morphology deviates from
spherical symmetry, star formation takes place at density peaks, which end up
as less elongated ellipsoids (Lake et al. in prep.).

Interestingly, we find that the DM component of DM GHOSts is elongated as
well, unlike the classical (no stream velocity) counterparts. This prediction
may be observable with gravitational lensing models that allow for deviation
from spherical symmetry (e.g., Kneib & Natarajan, 2011). Note that while there
is no direct correlation between the stream velocity large scale distribution
and the density field, the stream velocity divergence relates to the density
field via the continuity equation (e.g., Tseliakhovich & Hirata, 2010;
Tseliakhovich et al., 2011). Thus, high density $\sigma$ peaks are weakly
correlated with large stream velocity patch e.g., Fialkov (2014). The box
considered here has an increased $\sigma_{8}$ compared to the average. Thus,
it roughly corresponds to a high redshift progenitor of a patch of the
Universe within a density peak such as the Virgo cluster (e.g., Naoz &
Barkana, 2007). Thus, because of the above weak correlation, we expect that
galaxy clusters are likely to host elongated DM substructures. Thus, given the
right alignments, they may be detected using strong lensing (e.g., Mahler et
al., 2022). We emphasize that about $40\%$ of the Universe has a stream
velocity larger that $1\sigma_{\rm vbc}$, and therefore, DM GOHSts with
elongated gas and DM components should be common regardless of large scale
density fluctuations.

  * •

Spin Parameter: The stream velocity serves to increase the total angular
momentum and thus rotational support of SIGOs and DM GHOSts. As shown in Fig.
6, the DM GHOSts have higher gas spin parameter compared to classical objects.
Less spherical objects (more eccentric objects) have greater angular momentum,
see Fig. 8.

As expected, the spin vectors of classical gas objects are aligned with those
objects’ minimum radius, forming a puffy-disk-like configuration at high
redshift, consistent with lower redshift analysis for larger objects (e.g.,
Jesseit, 2004; Kautsch et al., 2006; Wheeler et al., 2017; El-Badry et al.,
2018). DM GHOSts, on the other hand, demonstrate spin vectors that are often
aligned with their maximum axis (similar to a “spinning top,” see Fig. 9).
Lastly, SIGOs’ total gas angular momenta exhibit a weak bifurcation. Most are
misaligned with the maximum radius without a preference for alignment with the
minimum radius, while another group are aligned towards the maximum radius
(similarly to DM GHOSts, as shown in Fig. 9).

Additionally, the DM and gas components’ spins in classical halos are almost
always aligned. However, DM GHOSts, as shown in Fig. 10, have a weak
preference for alignment between the DM and gas’s spin, with a long tail of
nearly isotropic configurations.

  * •

Mass distribution: Classical objects are expected to have a cusp-like mass
distribution (e.g., Navarro et al., 1996a) which are often reproduced in
simulations (e.g., Delos & White, 2022). The stream velocity reduces the
density of objects and increases their size, causing them to be puffier and
more diffuse than classical objects and the theoretical NFW profile. The
ellipsoid-like configuration of low mass DM GHOSts yields a core-like profile
(see Fig. 11). As expected, SIGOs that follow an ellipsoid profile have a
core-like mass density, with a nearly constant density (see Fig 20 in App.C).
This behaviour for SIGOs is consistent with the suggestion that SIGOs are
giant molecular cloud analogs (Lake et al., 2022).

  * •

Rotation curves The stream velocity affects not only the spin parameter, but
also the rotational velocity curves of structures. Objects formed by streaming
have a higher maximum rotational velocity than those formed without for a
given mass (See Figs. 12 and 13). Furthermore, the bifurcation between high
and low mass objects seen in the radial mass distributions for DM GHOSts is
also reflected in their velocity profiles. Low mass ($\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}10^{5.5}$ M⊙) objects, which have cores, do not reach high
rotational velocities at their inner radii. The inclusion of molecular cooling
increases the velocity at the maximum radius and decreases the maximum radius
by condensing rotationally supported material inward.

We note that rotational curve anomalies have been observed for slightly larger
objects in the local Universe (e.g., Sales et al., 2022). We speculate that
anomalous rotation curves produced by the stream velocity at high redshift may
persist to low redshift structures. This may be related to the observed
“diversity of rotation curves” problem for ultra faint dwarf galaxies.

The combined effects of molecular cooling and the stream velocity give the
most accurate picture to date of the morphological and rotational properties
of DM GHOSts. We characterize these objects as highly diffuse, rotationally
supported dwarf structures with large radii and high eccentricities. Based on
these anomalous properties, we speculate that at low redshift, DM GHOSts may
evolve to form some ultra faint dwarf galaxies or anomalous dwarf galaxies. In
particular, some dwarf galaxies exhibit similar properties, including a
diffuse structure and atypical rotation curves (e.g., Bullock & Boylan-
Kolchin, 2017; Sales et al., 2022). Thus, while observed ultra faint dwarf
galaxies and dwarf galaxies may be more massive than DM GHOSts we find they
share similar characteristics at these high redshifts. We expect DM GHOSts to
grow over time according to the natural hierarchical growth of structure, and
may be the progenitors of some faint dwarf galaxies in regions of the Universe
with a highly supersonic stream velocity at early times.

## Acknowledgements

The authors would like to thank Sahil Hegde, Bao-Minh Hoang, and Keren Sharon
for constructive conversations. C.E.W., W.L., S.N., Y.S.C, B.B., F.M., and
M.V. thank the support of NASA grant No. 80NSSC20K0500 and the XSEDE AST180056
allocation, as well as the Simons Foundation Center for Computational
Astrophysics and the UCLA cluster Hoffman2 for computational resources. C.E.W.
also thanks the UCLA Competitive Edge program. S.N thanks Howard and Astrid
Preston for their generous support. Y.S.C thanks the partial support from UCLA
dissertation year fellowship. B.B. also thanks the the Alfred P. Sloan
Foundation and the Packard Foundation for support. MV acknowledges support
through NASA ATP grants 16-ATP16-0167, 19-ATP19-0019, 19-ATP19-0020,
19-ATP19-0167, and NSF grants AST-1814053, AST-1814259, AST-1909831 and
AST-2007355. NY acknowledges financial support from JST AIP Acceleration
Research JP20317829.

##  Appendix A Choice of Cutoff Gas Fraction

In the first papers by the Supersonic Project that included only adiabatic or
atomic cooling (e.g., Popa et al., 2016; Chiou et al., 2018, 2019, 2021; Lake
et al., 2021), a cutoff gas fraction of $f_{g}=0.4$ was chosen for the
definition of SIGOs. Those studies’ statistics for SIGO abundances and
properties were thus calculated for objects that were located outside of the
virial radius of their parent DM halo and had $f_{g}=0.4$ within the bounds of
the ellipsoid fit described in § 2.3. This choice of gas fraction was a
somewhat arbitrary choice, motivated by the fact that it was above the cosmic
baryon fraction and close to the stellar fraction of globular clusters (Chiou
et al., 2018). Nakazato et al. (2022) found that in molecular cooling
simulations, this choice was too lenient, and resulted in the identification
of SIGOs in runs without the stream velocity.

Figure 15: The same as Figure 5 (Scatter plot of $\log(1-e^{2})$ versus
$M_{\rm tot}$ for gas-primary (GP) objects,), with the definition of SIGOs
calculated using gas fractions of $0.4$ (left), $0.5$ (center) and $0.6$
(right). Significantly more SIGOs are found in the molecular cooling runs
without stream velocity (top left of each panel) for $f_{g}=0.4$ and
$f_{g}=0.5$, as was also shown in Nakazato et al. (2022). As in Fig. 5, the
top two panels show the H2 cooling runs, and the bottom two panels show runs
without cooling. The left column has no stream velocity and the right column
is from the $v_{bc}=2\sigma_{v_{bc}}$ runs. Stars represent SIGOs, as defined
in Sec. 2.3. The color bar is the gas fraction (Eq. (1)). The red overplotted
line is the expected relationship from Eq. (8) for an example object with the
average density and maximum radius of objects in the H2 cooling runs
($\bar{\rho}_{Rmax}=1.8\times 10^{8}$M⊙ kpc-3, $\bar{R}_{max}=0.134$ kpc).

We also find that a choice of $f_{g}=0.4$ results in an unacceptable number of
objects being identified as SIGOs in molecular cooling simulations. For
example, Fig. 15 shows the eccentricity versus mass of objects as in Fig. 5,
with a gas fraction of 0.4 (left), 0.5 (center), and 0.6 (right). The top left
panel shows the molecular cooling run with no stream velocity, and stars
represent SIGOs. With $f_{g}=0.4$ and $f_{g}=0.5$, there are many objects
identified as SIGOs by the algorithm. While these gas rich structures may be
interesting, they are obviously not the result of a large stream velocity. In
order to exclude as many of these false SIGOs as possible while still having
plenty of objects in the $2$v runs to study, we follow Nakazato et al. (2022)
and choose $f_{g}=0.6$ for this work.

Figure 16: Probability density distributions as shown in Fig. 3 ($\log(1-e)$,
where $e$ is the eccentricity (Eq. (3)), for gas primary (GP) objects) and
Fig. 6 ($\lambda_{gas}$ for GP objects), with the definition of SIGOs
calculated using gas fractions of $0.4$ (top row), $0.5$ (center row) and
$0.6$ (bottom row). As seen in Fig. 15, more SIGOs are found in the molecular
cooling runs without stream velocity for $f_{g}=0.4$ and $f_{g}=0.5$, as was
also shown in Nakazato et al. (2022), which motivates us to choose a gas
fraction in this work of $0.6$. However, the results are broadly consistent
despite variation in gas fraction, and the effects are only seen in the
distribution of SIGOs. As in Figs. 5 and 6, distributions are separated into
the object classes listed in Tab. 2 and calculated using a Gaussian kernel
density. The orange distributions include the gas component of DM GHOSts, the
grey distributions show the gas component of classical halos without $v_{bc}$,
and the blue distributions show SIGOs.

For completeness, Figs. 16 and 17 show the probability density distributions
for GP objects from this work (as in Figs. 3, 6, 9, and 10) with varying gas
fraction from the previous value of $0.4$ to the value of $0.6$ adopted in
this work. The results are generally consistent despite changing the gas
fraction cutoff.

Figure 17: Probability density distributions as shown in Fig. 9 (Left:
$\cos{\phi_{g}}$ the angle between rmin and the spin vector of the gas
component of GP objects (See Fig. 2); Center: $|\cos{(\theta_{g})}|$, the
angle between $R_{\rm max}$ and the spin vector) and Fig. 10 (Right:
$|\cos{\theta_{\rm g,dm}}|$ the misalignment between the spin parameter of the
gas and dark matter components of individual objects), with the definition of
SIGOs calculated using gas fractions of $0.4$ (top row), $0.5$ (center row)
and $0.6$ (bottom row). As seen in Fig. 15, more SIGOs are found in the
molecular cooling runs without stream velocity for $f_{g}=0.4$ and
$f_{g}=0.5$, as was also shown in Nakazato et al. (2022), which motivates us
to choose in this work a gas fraction of $0.6$. However, the results are
broadly consistent despite variation in gas fraction, and the effects are only
seen in the distribution of SIGOs. As before, the darker lines denote runs
with H2 cooling, while the lighter dashed lines denote no cooling.
Distributions are separated into the object classes listed in Tab. 2 and
calculated using a Gaussian kernel density. The orange distributions include
the gas component of DM GHOSts, the grey distributions show the gas component
of classical halos without $v_{bc}$, and the blue distributions show SIGOs.

##  Appendix B Derivation of Prolate Ellipsoidal Potential

Run | 0v | 2v | 0vH2 | 2vH2
---|---|---|---|---
Objects | All | SIGOs | DM GHOSts | All | All | SIGOs | DM GHOSts | All
Gas Eccentricity | 7.75E-01 | 9.82E-01 | 8.88E-01 | 8.92E-01 | 8.06E-01 | 9.77E-01 | 9.11E-01 | 9.15E-01
DM Eccentricity | 8.22E-01 | X | 8.19E-01 | X | 7.69E-01 | X | 8.12E-01 | X
Gas Spin | 9.12E-02 | 1.21E-01 | 1.53E-01 | 1.52E-01 | 6.44E-02 | 1.80E-01 | 1.26E-01 | 1.31E-01
DM spin | 3.25E-01 | X | 2.33E+00 | X | 4.25E-01 | X | 1.29E+00 | X
Total mass (M⊙) | 9.91E+05 | 3.61E+04 | 1.75E+06 | 1.69E+06 | 5.39E+05 | 1.98E+05 | 1.35E+06 | 1.29E+06
Gas Fraction | 1.20E-01 | 8.26E-01 | 1.47E-01 | 1.72E-01 | 2.70E-01 | 7.72E-01 | 2.30E-01 | 2.60E-01
Table 3: Mean value of selected parameters presented in this work for the four
runs. For the $2$v and $2v$H$2$ runs, means are given also for the populations
of SIGOs and DM GHOSts separately.

In this Appendix, we present a derivation of the gravitational potential and
total mass of prolate spheroids. Binney & Tremaine (2008) give the general
formulae for potentials of various ellipsoidal bodies in their Table 2.1. The
following equations apply to any inhomogeneous ellipsoid with axes $a_{1}$,
$a_{2}$ and $a_{3}$. The potential is

$\Phi(\mathbf{x})=-\pi
G\frac{a_{2}a_{3}}{a_{1}}\int_{0}^{\infty}\frac{\text{d}\tau}{\Delta}\\{\varphi(\infty)-\varphi[m(\tau,\mathbf{x})]\\},$
(B1)

where

$\Delta^{2}(\tau)\equiv\prod_{i=1}^{3}(a_{i}^{2}+\tau),$ (B2)
$m^{2}(\tau,\mathbf{x})\equiv
a_{1}^{2}\sum_{i=1}^{3}\frac{x_{i}^{2}}{a_{i}^{2}+\tau}$ (B3)

and

$\varphi(m)\equiv\int_{0}^{m^{2}}\rho(\mathbf{x})\text{d}m^{2}(0,\mathbf{x}).$
(B4)

For the prolate spheroidal case, we have

$a_{1}=a_{2}=R_{\rm min}$ (B5)

and

$a_{3}=R_{\rm max}.$ (B6)

Thus, in cylindrical coordinates $(R,z)$,

$m^{2}(\tau,\mathbf{x})=R_{\rm min}^{2}\left[\frac{R^{2}}{R_{\rm
min}^{2}+\tau}+\frac{z^{2}}{R_{\rm max}^{2}+\tau}\right],$ (B7)

and from Eq. (B2),

$\Delta^{2}(\tau)=(R_{\rm min}^{2}+\tau)^{2}(R_{\rm max}^{2}+\tau).$ (B8)

Eq. (B4) requires a density distribution, and here we will use the following
prolate spheroidal density distribution:

$\rho(m^{2})=\rho_{0}\left(1+\left(\frac{m}{a_{0}}\right)^{2}\right)^{-\frac{3}{2}},$
(B9)

where $\rho_{0}$ and $a_{0}$ are constants.

Plugging the density from Eq. B9 into Eq. B4 gives:

$\varphi(m)=\int_{0}^{m^{2}}\rho_{0}\left[1+\left(\frac{m}{a_{0}}\right)^{2}\right]^{-3/2}\text{d}m^{2}(0,\mathbf{x})$
(B10)
$=-2a_{0}^{2}\rho_{0}\left[1+\left(\frac{m(0,\mathbf{x})}{a_{0}}\right)^{2}\right]^{-1/2}+2a_{0}^{2}\rho_{0}.$
(B11)

Additionally,

$\varphi(\infty)=2a_{0}^{2}\rho_{0}$ (B12)

From in Eqs. (B1), (B7), (B8), (B11), and (B12), the prolate potential is

$\Phi(\textbf{x})=-2\pi GR_{\rm
max}a_{0}^{2}\rho_{0}\int_{0}^{\infty}\frac{\text{d}\tau}{(R_{\rm
min}^{2}+\tau)\sqrt{R_{\rm
max}^{2}+\tau}}\left[1+\left(\frac{m(0,\mathbf{x})}{a_{0}}\right)^{2}\right]^{-1/2}$
(B13)

And with Eq. (B7), we have Equation (B1) thus evaluates to

$\Phi(\textbf{x})=-2\pi GR_{\rm
max}a_{0}^{2}\rho_{0}\int_{0}^{\infty}\frac{\text{d}\tau}{(R_{\rm
min}^{2}+\tau)\sqrt{R_{\rm max}^{2}+\tau}}\left[1+\left(\frac{R_{\rm
min}^{2}}{a_{0}^{2}}\left[\frac{R^{2}}{R_{\rm min}^{2}}+\frac{z^{2}}{R_{\rm
max}^{2}}\right]\right)\right]^{-1/2}$ (B14)

Evaluating the integral, we get:

$\Phi(\textbf{x})=-4\pi GR_{\rm
max}^{2}a_{0}^{3}\rho_{0}\left(\frac{\cos^{-1}\left(\frac{R_{\rm max}}{R_{\rm
min}}\right)}{\sqrt{R_{\rm max}^{2}-R_{\rm
min}^{2}}}\right)\frac{1}{\sqrt{1+\left(\frac{R_{\rm
min}^{2}}{a_{0}^{2}}\left[\frac{R^{2}}{R_{\rm min}^{2}}+\frac{z^{2}}{R_{\rm
max}^{2}}\right]\right)}}$ (B15)

Taking $a_{0}=R_{\rm max}$:

$\Phi(\textbf{x})=-\frac{4\pi GR_{\rm
max}^{4}\rho_{0}\cos^{-1}(\sqrt{1-e^{2}})}{e\sqrt{1+(1-e^{2})\left(\frac{R^{2}}{R_{\rm
min}^{2}}+\frac{z^{2}}{R_{\rm max}^{2}}\right)}}.$ (B16)

Now, we find the dependence of the total mass on eccentricity, starting with a
similar argument to that presented in Binney & Tremaine (2008) for the
potential of oblate spheroids. In cylindrical coordinates a prolate spheroidal
shell with axes $\beta R_{\rm max}$ and $\beta R_{\rm min}$ is given by:

$\frac{R^{2}}{R_{\rm min}^{2}}+\frac{z^{2}}{R_{\rm max}^{2}}=\beta^{2}.$ (B17)

Here, the $z$-axis is aligned with the polar radius ($R_{\rm max}$) and the
$R$ coordinate points in the direction of the equatorial radius ($R_{\rm
min}$). The volume enclosed inside this shell is given by

$V=\frac{4}{3}\pi R_{\rm max}R_{\rm min}^{2}\beta^{3}$ (B18) $=\frac{4}{3}\pi
R_{\rm max}^{3}\beta^{3}(1-e^{2})$ (B19)

Thus, assuming a constant surface density, the mass enclosed between two
shells $\beta$ and $\beta+\delta\beta$ is:

$\delta M=4\pi\rho R_{\rm max}^{3}(1-e^{2})\beta^{2}\delta\beta$ (B20)

The full mass of the ellipsoid is found by integrating over a set of similar
spheroids from the center to the outer edge of the object. Using the notation
of Binney & Tremaine (2008), this set is given by all the spheroids for which:

$\text{constant}=m^{2}\equiv\frac{R^{2}}{1-e^{2}}+z^{2}.$ (B21)

This constant $m=\beta R_{\rm max}$. Thus, for some density function
$\rho(m^{2})$, according to Eq. (B20),

$\delta M=4\pi\rho(m^{2})(1-e^{2})m^{2}\delta m.$ (B22)

Integrating this equation over the ellipsoid gives the total mass:

$M=4\pi(1-e^{2})\int_{0}^{R_{\rm max}}\rho(m^{2})m^{2}dm.$ (B23)

Once again, we assume the density distribution of Eq. (6), and plugging into
Eq. (B23), we solve

$M=4\pi(1-e^{2})\int_{0}^{R_{\rm
max}}\left(1+\left(\frac{m}{a_{0}}\right)^{2}\right)^{-\frac{3}{2}}m^{2}dm.$
(B24)

to obtain

$M=4\pi\rho_{0}(1-e^{2})a_{0}^{3}\left[\sinh^{-1}{\left(\frac{R_{\rm
max}}{a_{0}}\right)}-\frac{R_{\rm max}}{\sqrt{a_{0}^{2}+R_{\rm
max}^{2}}}\right].$ (B25)

Taking $a_{0}=R_{\rm max}$ as above gives:

$M=4\pi\rho_{0}(1-e^{2})R_{\rm
max}^{3}\left(\sinh^{-1}{(1)}-\frac{1}{\sqrt{2}}\right)$ (B26) $\approx
2.19\rho_{0}(1-e^{2})R_{\rm max}^{3}.$

##  Appendix C Morphological Investigation

In this Appendix, we include several supporting Figures relating to our
morphological and rotational investigation above. Table 3 lists the means of
selected distributions from this work.

Figure 18: Probability density distribution of the ratios of GP objects.
Following the same convention as in Fig. 3, the various distributions
demonstrate runs with and without the stream velocity and cooling. The orange
distributions show objects with a gas fraction $f$ of less than 0.6 in the
runs without a stream velocity ($v_{bc}=0\sigma_{v_{bc}}$), the grey
distributions show the classical equivalent of $f<0.6$ objects in the
$v_{bc}=0\sigma_{v_{bc}}$ run, and the blue dashed distributions show SIGOs,
which are only found in the $v_{bc}=2\sigma_{v_{bc}}$ run. The darker lines
denote no cooling, while the lighter dashed lines show the inclusion of
molecular cooling.  Figure 19: Scatter plot of $R_{\rm max}$ ($v_{Rmax}$)
versus $R_{200}$ of GP objects. The top two panels show the molecular cooling
runs, and the bottom two panels show no cooling. The left column has no stream
velocity and the right column is from the $v_{bc}=2\sigma_{v_{bc}}$. Stars
represent SIGOs, as defined in Sec. 2.3. The color bar is the gas fraction
(Eq. (1)) The orange line shows $R_{\rm max}=R_{200}$. Colored points are GP
objects, and grey points are DM/G objects.

In Fig 18, we plot the probability density distributions of the three axis
ratios plotted on the axes of Fig. 3. The classical halos tendency towards
sphericity ($R_{\rm min}/R_{\rm mid}\sim R_{\rm min}/R_{\rm max}\sim R_{\rm
mid}/R_{\rm max}\sim 1$) is clearly seen here, as well as the distinct
deviation of SIGOs and DM GHOSts away from sphericity. For SIGOs especially,
the distributions show evidence of triaxiality (($R_{\rm min}/R_{\rm mid}\neq
R_{\rm min}/R_{\rm max}\neq R_{\rm mid}/R_{\rm max}$). Figure 18 also shows
the axes ratios for DM primary objects (bottom row). Here, we see that the DM
components of DM GHOSts are not only more eccentric, having a tail of small
axes ratios as in Fig. 3, but also show prolate shapes when $R_{\rm mid}$ is
taken into account. In the center bottom panel, the ratio $R_{\rm min}/R_{\rm
mid}$ for the DM component is close to one, whereas the ratio $R_{\rm
mid}/R_{\rm max}$ has a tail of small values. This is an indication of
prolateness ($R_{\rm min}\sim R_{\rm mid}<R_{\rm max}$).

Figure 20: Top panel: Density of gas in SIGOs as a function of radius,
normalized to the $R_{\rm max}$ of the ellipsoid, for the molecular cooling
run with $v_{bc}=2\sigma_{v_{bc}}$. The density is calculated in 50
ellipsoidal shells moving out from the center of the object. The objects are
colored by the mass of their gas component. An NFW profile for a $10^{5}$ M⊙
halo is shown for comparison as the dashed line. Eq. (6) is plotted for a SIGO
with average density in solid blue. Bottom panel: Rotation curves of SIGOs as
a function of radius, calculated in ellipsoidal shells going outward, for the
molecular cooling run with $v_{bc}=2\sigma_{v_{bc}}$. The average velocity of
the gas in an ellipsoidal shell at each radius is normalized by the
$v_{circ}$, the circular radius at $R_{\rm max}$ of the ellipsoid. The objects
are colored by the mass of their gas component. The average velocity of the
gas in an ellipsoidal shell at each radius is normalized by the $v_{circ}$,
the circular radius at $R_{\rm max}$ of the ellipsoid. The objects are colored
by the mass of their gas component. The average SIGO has $\bar{\rho}_{\rm
Rmax}=4.14\times 10^{7}$ M⊙ kpc-3 and $\bar{R}_{\rm max}=0.240$ kpc

In Fig. 19, we plot the maximum gas ellipsoid radius against the $R_{200}$ of
its parent halo. The DM halos, in general, are much larger than the gas–this
is expected. The stream velocity (as mentioned in § 3.1.2) drives more extreme
eccentricity, leading to large $R_{\rm max}$.

The DM maximum ellipsoid radius is also shown as dark points in Fig. 19 as a
function of the $R_{200}$ found from a spherical overdensity calculation. The
orange line corresponds to $R_{\rm max,DM}=R_{200}$ In Fig. 5, it was shown
that low mass objects have higher eccentricity. This is reflected in the fact
that the DM distribution deviated from the $1:1$ line at low masses, whereas
most DM objects fall on the line at higher masses.

Finally, in Fig. 20, we show the radial gas density and rotation curve of all
the SIGOs from the 0vH2 run, with an NFW profile and Eq. (6) overplotted. As
with the low mass DM GHOSts, the NFW is not a good fit. SIGOs seem to have a
core, rather than a cusp.

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