# New constraints on the dark matter density profiles of dwarf galaxies from
proper motions of globular cluster streams

Khyati Malhan Humboldt Fellow and IAU Gruber Fellow  The Oskar Klein Centre,
Department of Physics, Stockholm University, AlbaNova, SE-10691 Stockholm,
Sweden  Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117,
Heidelberg, Germany  Monica Valluri Department of Astronomy, University of
Michigan, Ann Arbor, MI, 48109, USA  Katherine Freese The Oskar Klein Centre,
Department of Physics, Stockholm University, AlbaNova, SE-10691 Stockholm,
Sweden  Theory Group, Department of Physics, The University of Texas at
Austin, 2515 Speedway, C1600, Austin, TX 78712-0264, USA  Rodrigo A. Ibata
Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR
7550, F-67000 Strasbourg, France

###### Abstract

The central density profiles in dwarf galaxy halos depend strongly on the
nature of dark matter. Recently, in Malhan et al. (2021), we employed N-body
simulations to show that the cuspy cold dark matter (CDM) subhalos predicted
by cosmological simulations can be differentiated from cored subhalos using
the properties of accreted globular cluster (GC) streams since these GCs
experience tidal stripping within their parent halos prior to accretion onto
the Milky Way. We previously found that clusters that are accreted within
cuspy subhalos produce streams with larger physical widths and higher
dispersions in line-of-sight velocity and angular momentum than streams that
are accreted within cored subhalos. Here, we use the same suite of simulations
to demonstrate that the dispersion in the tangential velocities of streams
(${\sigma_{v_{\mathrm{Tan}}}}$) is also sensitive to the central DM density
profiles of their parent dwarfs and GCs that were accreted from: cuspy
subhalos produce streams with larger ${\sigma_{v_{\mathrm{Tan}}}}$ than those
accreted inside cored subhalos. Using Gaia EDR3 observations of multiple GC
streams we compare their ${{\sigma_{v_{\mathrm{Tan}}}}}$ values with
simulations. The measured ${\sigma_{v_{\mathrm{Tan}}}}$ values are consistent
with both an “in situ” origin and with accretion inside cored subhalos of
$M\sim 10^{8-9}M_{\odot}$ (or very low-mass cuspy subhalos of mass $\sim
10^{8}M_{\odot}$). Despite the large current uncertainties in
${\sigma_{v_{\mathrm{Tan}}}}$, we find a low probability that any of the
progenitor GCs were accreted from cuspy subhalos of $M\gtrsim
10^{9}M_{\odot}$. The uncertainties on Gaia tangential velocity measurements
are expected to decrease in future and will allow for stronger constraints on
subhalo DM density profiles.

dark matter - Galaxy: halo - stars: kinematics and dynamics - globular
clusters - stellar streams

††journal: ApJL

##  1 Introduction

The true nature of dark matter (DM) is currently unknown (cf. Bertone et al.
2005) and our understanding about this mysterious particle is based primarily
on theoretical predictions from cosmological simulations and observations of
large scale structure (cf. Salucci 2019). While particle physicists have been
working for decades to set limits on the mass of the putative DM particle,
much is still unknown. For instance, the widely accepted cold dark matter
(CDM) theory hypothesizes that the DM particle is non-relativistic (“cold”),
collisionless and weakly interacting (White & Rees, 1978; Blumenthal et al.,
1984). The CDM framework predicts that galaxy halos (irrespective of their
sizes) should possess cuspy DM distributions, with very steeply rising inner
density profiles of the form $\rho_{\rm DM}\propto r^{-1}$ (Dubinski &
Carlberg, 1991; Navarro et al., 1997). Alternative theories which differ from
CDM in terms of the behaviour of their elementary particles (e.g. ultra-light
DM, a.k.a. fuzzy DM, Hui et al. 2017), interaction strength (e.g. self-
interacting DM, Spergel & Steinhardt 2000; Elbert et al. 2015), etc.
Interestingly, many of these alternative DM theories instead predict cored DM
distributions on galactic/sub-galactic scales, where central densities are
approximately constant. Therefore, measurements of the central DM densities in
dwarf galaxies provide a possible avenue to constrain the fundamental
properties of DM. It has also been previously suggested that the widths of
dwarf galaxy streams can also be a probe of their parent galaxy’s DM density
profiles (Errani et al., 2015).

Recently, in Malhan et al. (2021), we presented a new method of probing the
central DM densities in dwarf galaxies using globular cluster (GC) stellar
streams. Stellar streams are produced from the tidal stripping of a progenitor
(e.g. a GC) as it orbits in the potential of the host galaxy. In the Milky Way
(MW), nearly a 100 streams have been detected to date (Mateu, 2022). Among
this set, some of the progenitor GCs of these streams are suspected to have
been accreted; i.e., these GC streams originally evolved within their parent
dwarf galaxies and only later merged with the MW (e.g., Malhan et al. 2019b,
a; Bonaca et al. 2021; Malhan et al. 2022). Motivated by this scenario, we
asked in Malhan et al. (2021): can the present day physical properties of
accreted GC streams inform us about the DM density profiles inside their
parent dwarf galaxies? To explore this question, we ran several N-body
simulations and showed that GCs that accrete within cuspy CDM subhalos produce
streams that are substantially wider (physically) and dynamically hotter than
those streams that accrete inside cored subhalos. This difference occurs due
to the difference in the dynamical evolution of GCs inside two different
potential models – cuspy and cored – with the former causing larger tidal
stripping of the GC (inside the parent subhalo) than the latter. This implies
that the physical properties of accreted GC streams provides a means to probe
the DM density profiles inside their parent dwarfs.

In Malhan et al. (2021), the physical properties of the streams were
quantified in terms of their a) transverse physical widths ($w$), b)
dispersion in the line-of-sight (los) velocities
(${\sigma_{v_{\mathrm{los}}}}$), and c) dispersion in the z-component of
angular momenta (${\sigma_{L_{z}}}$). We found that these parameters differ in
in cuspy and cored halos (see Figure 7 of Malhan et al. 2021). In particular,
the parameters ${\sigma_{v_{\mathrm{los}}}}$ and ${\sigma_{L_{z}}}$ depend on
the spectroscopic los velocities, that we lack for a majority of stream stars.
However, with ESA/Gaia mission (Gaia Collaboration et al., 2016), we now
possess excellent proper motions and parallaxes for millions of halo stars,
and this data can be used to measure the tangential velocities
($v_{\mathrm{Tan}}$) of stream stars. Our aim in this work is to show that the
intrinsic dispersion in the tangential velocities of stream stars
(${\sigma_{v_{\mathrm{Tan}}}}$) can be used as an alternative parameter to
differentiate between the cusp/core scenario (at least for halos of
mass$\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}10^{9}{\rm\,M_{\odot}}$), and this
provides a new means to probe the central DM density profiles inside the dwarf
galaxies.

This article is arranged as follows. Section 2 details the computation of
${\sigma_{v_{\mathrm{Tan}}}}$ for the simulated stream models produced in
cuspy vs. cored halos. Section 3 describes the procedure to measure
${\sigma_{v_{\mathrm{Tan}}}}$ of the observed streams of the MW using Gaia
EDR3 dataset (Lindegren, Lennart et al., 2020). Finally, in Section 4, we
compare ${\sigma_{v_{\mathrm{Tan}}}}$ of the observations and the simulations
and provide our conclusions.

##  2 Tangential velocity dispersions of the N-body stream models

###  2.1 N-body simulations of accreted globular cluster streams

The N-body stream simulations in Malhan et al. (2021) were of two types: those
that were produced by in situ GCs and those that were produced accreted GCs.

The in situ GC streams arise from GCs that formed inside the MW, and whose
evolution has been primarily determined by the MW potential. In Malhan et al.
(2021), we simulated $n=5$ in situ GC streams. The progenitor GC were modeled
by King profiles (King, 1962) with masses ranging from $M_{\rm
GC}=[3-10]\times 10^{4}{\rm\,M_{\odot}}$, central potential ranging from
$W=1.5-3$ and tidal radius from $r_{t}=0.05-0.2{\rm\,kpc}$. This mass range
was motivated by previous studies on clusters and streams of the Milky Way
(e.g., Baumgardt 2016; Thomas et al. 2016). The star particles had individual
masses of $5{\rm\,M_{\odot}}$ and softenings of $2{\rm\,pc}$. To evolve these
N-body GC models in a host Galactic potential (that mimics the MW), we used
model #1 of Dehnen & Binney (1998). This is a static, axisymmetric potential
comprising of a thin disk, a thick disk, interstellar medium, bulge and DM
halo. The simulations were evolved for $T=8{\rm\,Gyr}$ using the collisionless
GyrfalcON integrator (Dehnen, 2002) from the NEMO package (Teuben, 1995).

To produce accreted GC streams, we tried a total of $4$ parent subhalos that
were constructed using the Dehnen model (Dehnen, 1993). The Dehnen model is
expressed as

$\rho(r)=\frac{(3-\gamma)M_{0}}{4\pi
r^{3}_{0}}\Big{(}\frac{r}{r_{0}}\Big{)}^{-\gamma}\Big{(}1+\frac{r}{r_{0}}\Big{)}^{\gamma-4},$
(1)

where $M_{0},r_{0},-\gamma$ are the mass, scale radius, and the logarithmic
slope of the inner density profile of the subhalo, respectively. Two of the
subhalos possess cuspy (NFW-like) profile and two possess cored density
profiles. These subhalos are described as 1) SCu (small/cuspy) model:
$\\{M_{0},r_{0},\gamma\\}$=$\\{10^{8}{\rm\,M_{\odot}},0.75{\rm\,kpc},1\\}$; 2)
SCo (small/cored) model:
$\\{M_{0},r_{0},\gamma\\}$=$\\{10^{8}{\rm\,M_{\odot}},0.75{\rm\,kpc},0\\}$; 3)
LCu (large/cuspy) model:
$\\{M_{0},r_{0},\gamma\\}$=$\\{10^{9}{\rm\,M_{\odot}},1.60{\rm\,kpc},1\\}$ and
4) LCo (large/cored) model: $\\{M_{0},r_{0},\gamma\\}$ =
$\\{10^{9}{\rm\,M_{\odot}},1.60{\rm\,kpc},0\\}$. This mass range was adopted
because it is similar to the masses of some of the dwarf galaxies that host
GCs (e.g., Forbes et al. 2018), and also similar to the mass of the
(hypothesized) parent dwarf galaxy of the “GD-1” stream (Malhan et al., 2019a,
b). The mass and softening parameters of the DM particles were
$750{\rm\,M_{\odot}}$ and $20{\rm\,pc}$, respectively111This choice of
resolution, for both the subhalos and the GCs, was based on several numerical
tests that we undertook in Malhan et al. (2021).. Each subhalo model was
populated with one GC model, and this GC was placed at an off-centre location
and was launched on an orbit inside the subhalo. At the same time, the subhalo
was launched on an orbit inside the host Galactic potential. The integration
time of every simulation was $T=8{\rm\,Gyr}$ and the GC spends $\sim
3-4{\rm\,Gyr}$ inside the parent subhalo before escaping into the host (see
Malhan et al. 2021).

We ran over $100$ N-body simulations of accreted GC streams, including many
different orbital configurations of GCs inside the subhalo (see Table 1 of
Malhan et al. 2021). The majority of orbits of the subhalos (hosting the GC)
within the MW, were circular (with galactocentric radius $\sim 60{\rm\,kpc}$),
and only a few were eccentric. Furthermore, while most of the simulations
employed subhalos that lacked an extended population of stars, we did
experiment with a few cases that included a stellar population (as expected
from dwarf galaxies). However, we found that in both the cases (with and
without the stellar population), the final morphologies of the accreted GC
streams were the same.

All of the GC stream models were transformed from the galactocentric Cartesian
coordinates to the heliocentric equatorial coordinates from which we measure
${\sigma_{v_{\mathrm{Tan}}}}$ of the simulated streams. This transformation
provided for every star particle its position ($\alpha,\delta$), heliocentric
distances ($d_{\odot}$) and proper motions
($\mu^{*}_{\alpha}\equiv\mu_{\alpha}{\rm cos}\delta,\mu_{\delta}$)222Naturally
we also obtained $v_{\rm vlos}$ for each star, but that is not used in this
study.. Here, we use all of these quantities to measure
${\sigma_{v_{\mathrm{Tan}}}}$ of streams. Note that these are the same
quantities that are provided by the Gaia dataset, except for $d_{\odot}$ (as
Gaia provides only parallaxes of stars). In Section 3 we discuss how we use
Gaia parallaxes to estimate the distances to stream stars.

Figure 1: Using ${\sigma_{v_{\mathrm{Tan}}}}$ of GC streams to probe the DM
density profiles inside their parent subhalos (or parent dwarf galaxies).
Upper panel: each red/black/gray point represents the tangential velocity
dispersion (${\sigma_{v_{\mathrm{Tan}}}}$) of a particular simulated stream,
and the corresponding error bar reflects the dispersion in the
${\sigma_{v_{\mathrm{Tan}}}}$ measurement along that stream. The $Y$ axis
denotes different simulations. The red points correspond to the in situ GC
stream models and the black/gray points correspond to streams that accreted
inside cuspy/cored subhalos (where small/large markers correspond to cases
where subhalos had mass of $M=10^{8}{\rm\,M_{\odot}}/10^{9}{\rm\,M_{\odot}}$).
The colored triangles are ${\sigma_{v_{\mathrm{Tan}}}}$ values we measure for
5 Milky Way streams, using Gaia EDR3 data. Lower panel: red/black/gray
Gaussians correspond respectively to the distribution of simulated
${\sigma_{v_{\mathrm{Tan}}}}$ values from the in situ/cuspy/cored scenarios
(including the scatter in the ${\sigma_{v_{\mathrm{Tan}}}}$ measurements).
Gaussian with thin/thick borders correspond to cases where subhalos had mass
of $M=10^{8}{\rm\,M_{\odot}}/10^{9}{\rm\,M_{\odot}}$). In summary, in situ GC
streams (red stars) possess extremely low values of
${\sigma_{v_{\mathrm{Tan}}}}$, GC streams accreted inside cuspy CDM subhalos
(black diamonds) possess very large values of ${\sigma_{v_{\mathrm{Tan}}}}$,
while streams accreted inside cored subhalos (gray circles) lie in between.

###  2.2 Computing tangential velocity dispersion
(${\sigma_{v_{\mathrm{Tan}}}}$) of simulated streams

To compute the dispersion in the tangential velocity of a given stream
(${\sigma_{v_{\mathrm{Tan}}}}$), we first compute tangential velocities of the
individual member stars ($v_{\mathrm{Tan}}$). Tangential velocity is defined
as $v_{\mathrm{Tan}}=k\times d_{\odot}\times\mu$; where
$k=4.7405{\rm\,km\,s^{-1}}{\rm\,kpc}^{-1}({\rm\,mas\,yr^{-1}})^{-1}$,
$\mu=\sqrt{\mu^{*\,2}_{\alpha}+\mu^{2}_{\delta}}$. Instead of computing
${\sigma_{v_{\mathrm{Tan}}}}$, one maybe tempted to directly compute the
dispersion in the proper motions; since it is the proper motion of stars that
are provided by the Gaia dataset. However, proper motions are distance
dependent, therefore we use the dispersion in tangential velocities which is
independent of distance.

To compute ${\sigma_{v_{\mathrm{Tan}}}}$ of simulated streams, we follow a
similar approach as that used in Malhan et al. (2021) to measure other
dynamical quantities. For a given stream, we first transform the positions of
its member stars from the equatorial ($\alpha,\delta$) coordinate system to
the ($\phi_{1},\phi_{2}$) coordinate system, where $\phi_{1}$ is the angle
that is aligned with the stream and $\phi_{2}$ is the angle perpendicular to
the stream. Next, we consider small segments along $\phi_{1}$ of length
$30^{\circ}$ and compute ${\sigma^{s}_{v_{\mathrm{Tan}}}}_{,i}$ independently
for each $i^{\rm th}$ segment (the reason for undertaking this “segment-wise”
calculation is described below). To compute
${\sigma^{s}_{v_{\mathrm{Tan}}}}_{,i}$ in a given segment we first fit
$v_{\mathrm{Tan}}$ of the star particles using a smooth function of the form

$v_{\mathrm{Tan}}(\phi_{1})=a_{1}+b_{1}\phi_{1}+c_{1}\phi^{2}_{1}\,,$ (2)

where $a_{1},b_{1},c_{1}$ are the fitting parameters to obtain the systemic
value of $v_{\mathrm{Tan}}(\phi_{1})$. After this, we subtract the fitted
$v_{\mathrm{Tan}}$ function from the $v_{\mathrm{Tan}}$ of star particles to
obtain the residual distribution. The standard deviation of this distribution
provides the tangential velocity dispersion for the $i^{\rm th}$ segment of
the stream (i.e., ${\sigma^{s}_{v_{\mathrm{Tan}}}}_{,i}$). This procedure is
iterated over all the segments in a given stream. Finally, the median and the
standard deviation of the $\sigma^{s}_{{\rm vtan},i}$ distribution provides
the ${\sigma_{v_{\mathrm{Tan}}}}$ measurement for the entire stream and the
dispersion on this measurement, respectively. We use this procedure to compute
${\sigma_{v_{\mathrm{Tan}}}}$ for all the N-body stream models. The reason we
compute ${\sigma_{v_{\mathrm{Tan}}}}$ independently for each segment of a
stream is that many accreted GC streams are long and highly complex in
structure (see Figures 1, 5, 6 of Malhan et al. 2021). Therefore, it is
difficult to approximate the entire stream with a single function.
Nonetheless, our procedure to obtain ${\sigma_{v_{\mathrm{Tan}}}}$ also
provides the dispersion on the ${\sigma_{v_{\mathrm{Tan}}}}$ measurements.

Figure 1 (upper panel) shows the ${\sigma_{v_{\mathrm{Tan}}}}$ measurements
for all the N-body stream models as red stars (in-situ), grey circles (cored)
and black diamonds (cuspy); the dispersions on their
${\sigma_{v_{\mathrm{Tan}}}}$ are shown with error-bars. A visual inspection
of this figure already indicates that streams produced in different scenarios
(i.e. in situ, cuspy and cored) possess quite different values of
${\sigma_{v_{\mathrm{Tan}}}}$. For a given in situ/cored/cuspy scenario, we
quantify the variance in ${\sigma_{v_{\mathrm{Tan}}}}$ distribution (denoted
as $\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle$) by modeling the
${\sigma_{v_{\mathrm{Tan}}}}$ measurements with a Gaussian function of mean
$\langle{x}\rangle$ and intrinsic dispersion $\sigma_{x}$. To this end, we use
the MCMC sampler emcee (Foreman-Mackey et al., 2013) and define the log-
likelihood function for every stream seqment $i$ as:

$\displaystyle\ln\mathcal{L}$
$\displaystyle=\sum_{i}^{n}\left[-\ln(\sqrt{2\pi}\sigma_{i})-0.5\dfrac{(x_{i}-\langle{x}\rangle)^{2}}{\sigma_{i}^{2}}\right]\,,$
(3) $\displaystyle{\rm with}\,\,\sigma^{2}_{i}$
$\displaystyle=\sigma^{2}_{x}+\delta^{2}_{i}.$

Here, $x_{i}={\sigma_{v_{\mathrm{Tan}}}}$ of the stream segment $i$ and
$\delta_{i}$ is the dispersion on ${\sigma_{v_{\mathrm{Tan}}}}$ of that
segment of the stream.

For the in situ GC streams, we find
$\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle=0.5\pm 0.1{\rm\,km\,s^{-1}}$,
implying that these streams are dynamically very cold. Furthermore, the cuspy
SCu (LCu) subhalo with mass $M_{0}=10^{8}(10^{9}){\rm\,M_{\odot}}$ produced GC
streams with value $\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle=3.7\pm
0.2\,(8.5\pm 0.4){\rm\,km\,s^{-1}}$. This implies that these streams are
dynamically very hot. For the cored SCo (LCo) subhalo we infer the value of
$\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle=1.6\pm 0.3\,(2.1\pm
1.0){\rm\,km\,s^{-1}}$. This difference in the
$\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle$ measurement of streams produced
under cuspy/cored subhalo implies that present day
${\sigma_{v_{\mathrm{Tan}}}}$ of streams are sensitive to the gravitational
potential of their parent subhalos (i.e., ${\sigma_{v_{\mathrm{Tan}}}}\propto
M_{0}/r_{0}$). Some degeneracy between subhalo mass and the presence of a cusp
is apparent in Figure 1. This issue is discussed further in Section 4.

Figure 2: Computing tangential velocity dispersion
(${\sigma_{v_{\mathrm{Tan}}}}$) of the Milky Way streams using Gaia EDR3. In a
given column of this plot, all the panels provide details of a particular
stream (the name of the stream is provided in the top panel). In a given
column, panel (a) shows positions of the stream stars in the rotated
($\phi_{1},\phi_{2}$) coordinate system, panel (b) shows the distance fit to
the stream obtained using Gaia EDR3 parallaxes and photometry (where the
fitted curves represent $100$ Monte Carlo representations), and panel (c)
shows the $v_{\mathrm{Tan}}$ fit to the stream (where “$v_{\rm Tan}$ (obs)” is
obtained by multiplying distance solutions of panel (b) with the Gaia EDR3
proper motion of stars). Panel (d) shows the residuals of the $v_{\rm Tan}$
(obs) after the mean trend has been subtracted off, and the “blue band”
represents the intrinsic dispersion. In panel (d), the quoted
${\sigma_{v_{\mathrm{Tan}}}}$ value represents the median and the
corresponding uncertainties reflect the 16th to 84th percentile range of the
distribution (see text). Specifically for “GD-1”, we compare our distance fit
with that of its orbit solution, only to ensure that our distance solutions
are reliable (the orbit solution is taken from Malhan & Ibata 2019).

Figure 3: Same as Figure 2, but for different streams.

##  3 Tangential velocity dispersions of the Milky Way streams

There is now mounting evidence that some of the GC streams that orbit the MW
halo were accreted from dwarf galaxies (e.g., Malhan et al. 2019b, a; Gialluca
et al. 2021; Bonaca et al. 2021). This implies that the
${\sigma_{v_{\mathrm{Tan}}}}$ measurement of these streams provide an
opportunity to test the prediction that we obtained above, and thus understand
whether the parent dwarfs of these streams possessed cuspy or cored DM
distribution.

Here, we measure ${\sigma_{v_{\mathrm{Tan}}}}$ of $n=5$ streams, namely
“GD-1”, “Phlegethon”, “Fjörm”, “Gjöll”, “Sylgr”. The reason for choosing these
particular streams is that (1) these are GC streams333This has been previously
established for GD-1 (Malhan & Ibata, 2019; Bonaca et al., 2020a), Fjörm
(Palau & Miralda-Escudé, 2019), Gjöll (Palau & Miralda-Escudé, 2021), and
tentatively for Sylgr (Roederer & Gnedin, 2019) and Phlegethon (Ibata et al.,
2018)., (2) these streams have been hypothesized to be of accreted origin
(e.g., Bonaca et al. 2021; Malhan et al. 2022), and (3) these are long streams
that also possess high stellar densities, and are thus suitable for performing
the intended analysis. All of these streams are quite metal poor (with their
[Fe/H] lying below $\sim-2$ dex, Malhan et al. 2022), and this further
supports the accretion scenario.

The member stars of these streams are taken from the Ibata et al. (2021)
catalogue. The streams in this catalogue were detected in the Gaia EDR3
dataset using the STREAMFINDER algorithm (Malhan & Ibata, 2018; Ibata et al.,
2019). In this catalogue, every star possesses Gaia EDR3 based position
($\alpha,\delta$), parallax ($\varpi$), proper motions
($\mu^{*}_{\alpha},\mu_{\delta}$) and photometry ($G,G_{\rm BP},G_{\rm RP}$),
along with the associated uncertainties. The photometric information is used
along with parallaxes to improve the distance estimates (see below). The
parallaxes are corrected for the global parallax zero-point in Gaia EDR3
(Lindegren, Lennart et al., 2020) and the photometry is corrected for
extinction Ibata et al. (2021). These streams are shown in Figures 2 and 3.

To measure ${\sigma_{v_{\mathrm{Tan}}}}$ of these streams, we follow a similar
procedure as described in Section 2, with slight modifications in order to
account for the observational errors. First we transform the positions of
stream stars from ($\alpha,\delta$) to ($\phi_{1},\phi_{2}$) coordinates
aligned with each stream. This is shown in panels (a) of Figures 2, 3. Next we
compute the distance of the stream as a function of $\phi_{1}$ (which can then
be multiplied with proper motions to obtain $v_{\mathrm{Tan}}$) (we do not
simply compute the average parallax of the stream since this can bias
${\sigma_{v_{\mathrm{Tan}}}}$ measurement for streams with distance
gradients). Therefore, to properly account for the possible distance
gradients, we follow a pragmatic approach. In a given stream, we consider
segments along $\phi_{1}$ of length $\approx 10^{\circ}$. This length allows
us to have a minimum of $15$ stars in every segment. For each segment we use
the stars to compute uncertainty-weighted average mean parallax value (along
with the uncertainty on this mean parallax). A reliable estimate of mean
parallax value requires high enough number of stars in a given segment. Taking
the inverse of this mean parallax provides the average heliocentric distance
($d_{\odot}$) of that segment (along with the uncertainty on $d_{\odot}$).
This $d_{\odot}$ value is computed at all the segments of the stream, that
provides a means to constrain the distance gradient of the entire stream
structure. These distance measurements are shown in panels (b) of Figures 2
and 3. The typical distance uncertainty (per segment) is $\approx
0.5{\rm\,kpc}$.

Next, for a given stream, we fit these $d_{\odot}$ measurements using a
similar function described by equation 2 (except this time we fit the entire
stream at once, and not in individual segments). This fitting is performed
using the emcee and it takes into account the uncertainties in $d_{\odot}$
measurements. The posterior on the parameters $a_{1},b_{1},c_{1}$ provides the
distance fit (as a function of $\phi_{1}$) and the spread on the posterior
provides the uncertainty on this distance fit. Effectively, this procedure
allows us to estimate the distance (and the uncertainty) for every star using
its $\phi_{1}$ value. For a given star, we can now multiply its distance with
its proper motion to obtain its ${\sigma_{v_{\mathrm{Tan}}}}$ (as explained
below).

In passing, we also note that the above distance fitting procedure is
augmented with the information on color magnitude diagram (CMD) of stars
([$G_{\rm BP}-G_{\rm RP},G$], that comes from Gaia EDR3); the CMD information
is used as a prior in our likelihood evaluation (see Appendix A). Since the
scatter in the CMDs of all the streams are reduced after this distance
correction step, it gives us confidence that the estimated distances are
reliable. This is because streams, in general, have distance gradients.
Therefore, their observed CMDs are slightly smeared out in apparent magnitude.
However, if the observed magnitude of each star is corrected by its “true”
distance value, then the corrected CMD should have a reduced scatter. Here, we
quantify the scatter in a stream’s CMD using the k-nearest neighbors algorithm
(implemented using NearestNeighbors module in sklearn package). For this, we
set the parameter n_neighbors=10 and metric=euclidean. In Figure 4, we compare
the distance corrected CMDs with the observed CMDs. Furthermore, we also note
that our fitted distance solutions are compatible with the distance
measurements of Bailer-Jones et al. (2021); as shown in Appendix B.

In a given stream, to obtain $v_{\mathrm{Tan}}$ measurements of the member
stars, we multiply the above distance solutions with the Gaia proper motions.
For a given star, the uncertainties on the distance solution and on the proper
motions provides the uncertainty on the $v_{\mathrm{Tan}}$ measurement. Using
these $v_{\mathrm{Tan}}$ measurements (along with the uncertainties), the
stream is fitted using equation 2; the entire stream structure is fitted at
once, and not in segments. The best fit solutions for $v_{\mathrm{Tan}}$ for
all the streams are shown in panels (c) of Figures 2 and 3. We further
highlight that our fitted $v_{\mathrm{Tan}}$ solutions are compatible with the
$v_{\mathrm{Tan}}$ measurements that one would derive by simply multiplying
Bailer-Jones et al. (2021) distances with Gaia’s proper motions (see Appendix
B).

Finally, to obtain ${\sigma_{v_{\mathrm{Tan}}}}$ measurement of a given
stream, we subtract-off the above $v_{\mathrm{Tan}}$-fit as the systemic
velocity of the stream from measured $v_{\mathrm{Tan}}$. Then we model the
residuals with a Gaussian distribution, including uncertainties on
$v_{\mathrm{Tan}}$, to derive the ${\sigma_{v_{\mathrm{Tan}}}}$ of the stream.
These residuals are shown in the panels (d) of the Figures 2 and 3. For the
resulting posterior distribution, its median and $16/84$ percentile provide
the ${\sigma_{v_{\mathrm{Tan}}}}$ of the stream and the uncertainty on
${\sigma_{v_{\mathrm{Tan}}}}$, respectively. These values are shown in panels
(e) of Figures 2 and 3 and they are also plotted in Figure 1.

In Appendix C we demonstrate that these ${\sigma_{v_{\mathrm{Tan}}}}$
measurements of the streams are robust.

Table 1 provides the $z$-score (for two-tailed hypothesis test) and the
corresponding $p$-value for the null hypothesis that an observed
${\sigma_{v_{\mathrm{Tan}}}}$ measurement (with its associated uncertainty) is
drawn from the Gaussian distribution for one of the 5 simulation scenarios
shown in the lower panel of Figure 1. For a given stream $s$ and a given
scenario $i$, the z-score is computed as:

$\displaystyle z$
$\displaystyle=({\sigma_{v_{\mathrm{Tan}}}}^{s}-{\sigma_{v_{\mathrm{Tan}}}}^{i})/\sigma\,,$
(4)

where ${\sigma_{v_{\mathrm{Tan}}}}^{s}$ is the ${\sigma_{v_{\mathrm{Tan}}}}$
measurement of the observed stream $s$, ${\sigma_{v_{\mathrm{Tan}}}}^{i}$
corresponds to that of the scenario $i$, and $\sigma$ is the sum in quadrature
of the uncertainties on these two quantities. A given $p$-value implies that
the probability that the observed stream was drawn from the population
describing a given simulation scenario can be rejected with confidence of
$(1-p)\times 100$% (e.g. $p=0.01$, implies the null hypothesis can be rejected
at the 99% level).

Table 1: z-scores (and p-values) for Milky Way stream streams being drawn from various simulated stream populations MW stream | in-situ | SCu | LCu | SCo | LCo
---|---|---|---|---|---
GD-1 | 0.361 ($0.718$) | -4.162 ($<10^{-3}$) | -9.712 ($<10^{-3}$) | -1.221 ($0.222$) | -1.172 ($0.241$)
Sylgr | 0.979 ($0.328$) | -1.148 ($0.251$) | -3.731 ($<10^{-3}$) | 0.017 ($0.986$) | -0.233 ($0.816$)
Phlegethon | 0.768 ($0.442$) | -2.046 ($0.041$) | -5.683 ($<10^{-3}$) | -0.355 ($0.722$) | -0.599 ($0.549$)
Fjorm | 1.122 ($0.262$) | -0.787 ($0.431$) | -3.284 ($0.001$) | 0.398 ($0.691$) | 0.066 ($0.948$)
Gjoll | 1.094 ($0.274$) | -0.632 ($0.528$) | -2.639 ($0.008$) | 0.384 ($0.701$) | 0.059 ($0.953$)

##  4 Conclusion and Discussion

We draw our main conclusions by inspecting Figure 1 and Table 1. They compare
the predicted values of ${\sigma_{v_{\mathrm{Tan}}}}$ (that we obtained by
analysing N-body GC stream models produced in different DM scenarios) with the
observations (coming from the MW streams). The bottom panel of Figure 1 shows
Gaussians that quantify the scatter in ${\sigma_{v_{\mathrm{Tan}}}}$
measurements of the simulated streams produced in in situ/cored/cuspy
scenarios. These Gaussians imply that: 1) in situ GC streams should possess
$\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle=0.5\pm 0.2{\rm\,km\,s^{-1}}$, 2) GC
streams accreted inside the cuspy SCu (LCu) subhalo with mass
$M_{0}=10^{8}(10^{9}){\rm\,M_{\odot}}$ should possess
$\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle=3.7\pm 0.2\,(8.5\pm
0.4){\rm\,km\,s^{-1}}$, and 3) GC streams accreted inside the cored SCo (LCo)
subhalo with mass $M_{0}=10^{8}(10^{9}){\rm\,M_{\odot}}$ should possess
$\langle{\sigma_{v_{\mathrm{Tan}}}}\rangle=1.6\pm 0.3\,(2.1\pm
1.0){\rm\,km\,s^{-1}}$. We summarize our main results below.

  * •

N-body simulations of GC tidal streams accreted from dwarf galaxies with
different central DM density profiles (cuspy vs. cored) show that there are
significant and measurable differences in the observed
${\sigma_{v_{\mathrm{Tan}}}}$ (the tangential velocity dispersion stars in the
stream) that reflect the nature of the central density profiles of their
parent dwarf galaxies.

  * •

Current Gaia EDR3 proper motions and parallaxes are used to determine
${\sigma_{v_{\mathrm{Tan}}}}$ for 5 GC streams (“GD-1”, “Phlegethon”, “Fjörm”,
“Gjöll”, “Sylgr”) studied in this work. It is not possible with current Gaia
observational uncertainties to reject the hypothesis that these streams were
formed in situ. Most of the observed GC streams in this study orbit at a
galactocentric distance of $\approx 20{\rm\,kpc}$, while the in situ stream
models in (Malhan et al., 2021) were simulated with orbital radii of
$60{\rm\,kpc}$. Therefore, for a fair comparison, we ran $5$ additional N-body
simulations of streams under the in situ framework, but this time adopting the
GC’s orbital radius as $\approx 20{\rm\,kpc}$. These additional streams can be
seen as the top $5$ red markers in Figure 1. These additional simulations do
not alter our conclusion on this point.

  * •

If however, the progenitor GCs of the MW streams analysed here were indeed
accreted as previously argued (see below), our ${\sigma_{v_{\mathrm{Tan}}}}$
measurements enable us to reject with high confidence the hypothesis that
their parent dwarf galaxies were cuspy with $M_{0}\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}10^{9}{\rm\,M_{\odot}}$. We can
also reject higher mass cuspy subhalos since GC streams from such dwarfs are
expected be even hotter. Also, it is not possible that these MW streams would
have originated from lower-mass cuspy subhalos because dwarfs with
$M_{0}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10^{8}{\rm\,M_{\odot}}$ are not
expected to host any GC population (e.g., Forbes et al. 2018). In view of
these arguments, our current analysis disfavours the cuspy CDM subhalos.

  * •

The Gaia uncertainties on proper motions and parallaxes are currently too
large to definitively determine whether the parent subhalos of these streams
were cored or cuspy with $M_{0}=10^{8}{\rm\,M_{\odot}}$.

  * •

Additionally, we recompute ${\sigma_{v_{\mathrm{Tan}}}}$ of 5 GC streams by
incorporating the “systematic errors” present in Gaia EDR3’s proper motions
and parallaxes (for details on these errors, see Lindegren, Lennart et al.
2020). This analysis is performed to examine the impact of these errors on the
${\sigma_{v_{\mathrm{Tan}}}}$ values that we measure in this work. As shown in
Appendix D, the inclusion of these systematic errors only minutely change the
${\sigma_{v_{\mathrm{Tan}}}}$ measurements, and they do not affect the final
conclusion of this work.

Although we are unable to definitively rule out (based on the kinematic
analysis done here) the possibility that the progenitors of these streams were
in situ GCs, there are other lines of evidence that indicate most of them have
an accreted origin. The in situ GC population is overall redder and more metal
rich than the accereted GC population (e.g. Kruijssen et al., 2019).
Furthermore, orbital action space clustering of GCs and halo stars and a
comparison of the metallicities of GCs and those same halo stars has been used
to assign many accreted GCs, including GD1 to previous merger events (Myeong
et al., 2019; Massari et al., 2019; Kruijssen et al., 2020; Bonaca et al.,
2021; Malhan et al., 2022). The metal rich in situ GC population has a slight
net prograde rotation, while the accreted GC population has no net rotation
but subsets associated with specific accretion events can be seen to be
clustered in angular-momentum (Massari et al., 2019). In addition to having a
nearly circular and retrograde orbit, GD1 is extremely metal poor with a mean
metallicity of $-2.2$ dex (Malhan & Ibata, 2019), much closer to the
metallicity of dwarf spheroidal satellites of the MW (Kirby et al., 2013) than
in situ GCs (Zinn, 1985).

In addition to ${\sigma_{v_{\mathrm{Tan}}}}$, the other two stream parameters
that are also useful to probe the DM density profiles inside dwarfs are:
transverse physical widths ($w$) and dispersion in the los velocity
(${\sigma_{v_{\mathrm{los}}}}$). Malhan et al. (2021) showed that in situ GCs
produce streams with $(\langle
w\rangle,\langle{\sigma_{v_{\mathrm{los}}}}\rangle)=(45\pm 15{\rm\,pc},0.7\pm
0.2{\rm\,km\,s^{-1}})$, GC streams accreted in cuspy subhalos produce with
$(\langle w\rangle,\langle{\sigma_{v_{\mathrm{los}}}}\rangle)\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}(650{\rm\,pc},4{\rm\,km\,s^{-1}})$,
and somewhat smaller widths $(\langle
w\rangle,\langle{\sigma_{v_{\mathrm{los}}}}\rangle)\sim(90-500{\rm\,pc},<4{\rm\,km\,s^{-1}})$
result when GCs accrete inside cored subhalos444These constrains are based on
the subhalo models with mass
$M_{0}=10^{8}{\rm\,M_{\odot}},10^{9}{\rm\,M_{\odot}}$. A combination of
multiple parameters could provide a stronger means to probe the DM density
profile inside the parent dwarf. For instance, we can in principle compare the
predicted $w$ values with the recent $w$ measurements of other MW streams
(e.g., Bonaca et al. 2020b; Ferguson et al. 2022; Tavangar et al. 2022) to
comment on their “accretion” origin. For GD-1, while its
${\sigma_{v_{\mathrm{Tan}}}}$ measurement appears to be more consistent with
the in-situ scenario (see Table 1), consideration of these additional
parameters: $w$ ($=130^{+30}_{-20}$ pc, Malhan et al. 2019b) and
${\sigma_{v_{\mathrm{los}}}}$ ($=2.1\pm 0.3{\rm\,km\,s^{-1}}$, Gialluca et al.
2021) suggest that GD-1 was likely accreted inside a cored subhalo.

In summary, our analysis indicates that 4 (out of 5) MW streams shown in
Figure 1 favor cored DM subhalos over cuspy CDM subhalos. Although this
inference is based on only two subhalo masses (i.e.,
$M_{0}=10^{8}{\rm\,M_{\odot}}$ and $10^{9}{\rm\,M_{\odot}}$), we argue that it
is unlikely that these streams could have accreted inside cuspy subhalos of
higher mass since such streams would be even hotter.

The origin of cored subhalos is still hotly debated. While cored subhalos are
favored by alternative DM candidates, hydrodynamical simulations have shown
that DM cores can result from erasure of DM cusps if the dwarf galaxy had a
sufficiently vigrous and episodic star formation phase (e.g. Pontzen &
Governato 2012). Under such a scenario, the resulting cored subhalo would
still be consistent with the CDM paradigm. Recent cosmological hydrodynamic
simulations predict that subhalos with $M_{0}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10^{10}{\rm\,M_{\odot}}$ would have
formed too few stars over their lifetimes, and the resulting baryonic feedback
is too weak to unbind their DM cusps (e.g., Lazar et al. 2020). If a
significant fraction of tidal streams from accreted GCs are found to be
dynamically consistent with having originated from cored subhalos with
$M_{0}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10^{10}{\rm\,M_{\odot}}$, then we
may be forced to move to models beyond CDM. However additional simulations
with a greater variety of dwarf galaxy properties and orbital initial
conditions are needed before firm conclusions can be drawn.

Cosmological hydrodynamical zoom-in simulations with different types of dark
matter: CDM, WDM, SIDM and mixed DM, e.g. WDM with self-interaction, (Fitts et
al., 2019) show that the addition of baryons substantially decrease
differences between the simulations with different types of DM. However
baryons decrease the sizes of cores in SIDM and WDM+SIDM subhalos compared to
SIDM-only simulations, but they have significantly lower central densities
than CDM-only halos. In future, it will be interesting to simulate a wider
variety of cored subhalo models (by varying their mass ranges, physical sizes,
core sizes and inner density slopes).

In Malhan et al. (2021), we showed that three observationally determinable
quantities for accreted GC streams: physical width $w$, line-of-sight velocity
dispersion $\sigma_{\mathrm{v}los}$ and dispersion in the $z$-component of
angular momentum $L_{z}$, were all sensitive probes of the degree of tidal
heating experienced by a GC stream in its parent dwarf galaxy and could enable
us to set constraints on the DM profiles of dwarf galaxies. In this work we
have shown, in addition, that ${\sigma_{v_{\mathrm{Tan}}}}$ is able to provide
similar discrimination.

In the future, we will consider additional heating arising from passage of the
stream through the disk or interactions with molecular clouds (Amorisco et
al., 2016) or the bar (Pearson et al., 2017). Furthermore,we will assess
whether all 6 phase space coordinates when combined may yield stronger
constraints on DM. In practice however stream membership is difficult to
assess in the absence of spectroscopy and accurate Gaia proper motions,
especially for distant streams. While radial velocities have the smallest
uncertainties – e.g. $1-2{\rm\,km\,s^{-1}}$ uncertainty for $G\lesssim 19$
with current large multiobject spectrographs like DESI (DESI Collaboration et
al., 2016a, b; Allende Prieto et al., 2020; Cooper et al., 2022) – the fact
that tidal streams generally extend over tens of degrees on the sky make it
extremely expensive observationally to obtain the large numbers of
${{v_{\mathrm{los}}}}$ measurements needed to reliably compute
${\sigma_{v_{\mathrm{los}}}}$ for many streams. While Gaia DR3 released
${{v_{\mathrm{los}}}}$ for over 30 million stars brighter than $G=14$, Figure
4 shows that most of the stars of interest here are fainter than this
magnitude limit.

The metric we study in this work, ${\sigma_{v_{\mathrm{Tan}}}}$, depends on
accurate measurements of both the proper motions of stream stars and their
distances. We obtained both quantities in this work from Gaia EDR3
observations. Future Gaia data releases are expected to decrease the
uncertainties on both the measured proper motions and parallaxes by around 50%
for each quantity relative to EDR3 uncertainties (see, Gaia Collaboration et
al., 2021, and the Gaia-ESA
website555https://www.cosmos.esa.int/web/gaia/science-performance) resulting
in a net decrease in the uncertainty on ${\sigma_{v_{\mathrm{Tan}}}}$ of $\sim
60-65$% for the streams we consider here. If both
${\sigma_{v_{\mathrm{los}}}}$ and ${\sigma_{v_{\mathrm{Tan}}}}$ are available
for a significant sample of stars, one might combine them to obtain a 3D
velocity dispersion, but currently adequate numbers of $v_{\mathrm{l}os}$
measurements do not exist for the streams considered here. At the present time
and for the foreseeable future, Gaia proper motions and parallaxes, being the
most abundantly measured quantities, offer the best way to quantify the
velocity dispersions of GC tidal streams.

## ACKNOWLEDGEMENTS

We thank our referee for their constructive and valuable comments. KM and KF
acknowledge support from the $\rm{Vetenskapsr\mathring{a}de}$t (Swedish
Research Council) through contract No. 638-2013-8993 and the Oskar Klein
Centre for Cosmoparticle Physics. KM acknowledges support from the Alexander
von Humboldt Foundation at Max-Planck-Institut für Astronomie, Heidelberg. KM
is also grateful to the IAU’s Gruber Foundation Fellowship Programme for their
finanacial support. MV is supported by NASA-ATP award 80NSSC20K0509. KF
gratefully acknowledges support from the Jeff and Gail Kodosky Endowed Chair
in Physics at the University of Texas, Austin; the U.S. Department of Energy,
Office of Science, Office of High Energy Physics program under Award Number
DE-SC0022021 at the University of Texas, Austin; the DoE grant DE- SC007859 at
the University of Michigan; and the Leinweber Center for Theoretical Physics
at the University of Michigan. RI acknowledges funding from the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation programme (grant agreement No. 834148).

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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##  Appendix A Comparing the observed and distance-corrected CMDs of streams

In Section 3, we perform distance fitting to the streams as a function of
their $\phi_{1}$ coordinate. For this distance fitting, we use Gaia’s
parallaxes and also Gaia’s photometry ($G_{\rm BP}-G_{\rm RP},G$). The reason
for using the photometry information can be explained as follows. Stellar
streams generally possess distance gradients along their lengths, and
therefore their observed CMDs are slightly smeared out in apparent magnitude
(here, $G$ magnitude). However, if the photometry of each star is corrected by
its “true” distance value, then the corrected CMD will have less scatter.
Therefore, this additional information on the “CMD scatter” provides a means
to better constrain the streams’ distance solutions. For this, during our
distance fitting procedure, we impose a (constant) prior condition in
likelihood evaluation that – the resulting distance solution should be such
that produces a CMD with less scatter than the observed CMD.

The corresponding result is shown in Figure 4, that compares the “observed”
and “distance corrected” CMDs of all the streams. The scatter in these CMDs
are quantified using the NearestNeighbors module, and this confirms that the
distance corrected CMDs have less scatter than the observed CMDs. This result
can also be discerned by visually inspecting Figure 4. This implies that our
distance solutions are reliable.

Figure 4: Color Magnitude Diagrams (CMDs) for 5 observed stellar streams (as
indicated by labels) constructed from Gaia photometry. In each panel the
observed, extinction corrected CMD (left, red points) and distance corrected
CMD (right, blue points) are shown. The $knn$ values in the legend of each
panel estimate the scatter in each CMD (smaller $knn$ values imply less
scatter).

##  Appendix B Examining the accuracy of our fitted solutions for $d_{\odot}$
and $v_{\mathrm{Tan}}$

We assess the reliability of our fitted solutions for the distances
($d_{\odot}$) and the tangential velocities ($v_{\mathrm{Tan}}$) of the stream
stars (shown in panels b and d of Figures 2 and 3) as follows.

We compare our fitted $d_{\odot}$ solutions (based on the Gaia EDR3
parallaxes, see main text) with the $d_{\odot}$ measurements from the Bailer-
Jones et al. (2021) catalogue. This comparison is shown in Figure 5b for the
GD-1 stream. Based on the visual inspection, we conclude that our solutions
are consistent with these measurements. We also note that uncertainties on
distances of the individuals stars from Bailer-Jones et al. (2021) are very
large ($\sim 8{\rm\,kpc}$). We repeated this exercise for other streams as
well and found similar consistency.

Finally, we compare our fitted $v_{\mathrm{Tan}}$ solutions with those derived
by simply multiplying $d_{\odot}$ from Bailer-Jones et al. (2021) and proper
motions from Gaia EDR3. This comparison is shown in Figure 5c for GD-1. Based
on the visual inspection, we conclude that our solutions are consistent with
these measurements; although the uncertainties on the measurements of the
individual stars are very large ($\sim 550{\rm\,km\,s^{-1}}$). We repeated
this exercise for other streams as well and found similar consistency, leading
us to conclude that our fitted $d_{\odot}$ and $v_{\mathrm{Tan}}$ solutions
are reliable.

Figure 5: Comparing our fitted $d_{\odot}$ and $v_{\mathrm{Tan}}$ solutions
with those obtained from Bailer-Jones et al. (2021) and Gaia EDR3. This plot
corresponds to the “GD-1” stream.

##  Appendix C Examining the robustness of the measured
${\sigma_{v_{\mathrm{Tan}}}}$ of the Milky Way streams

###  C.1 Examining the robustness due to possible mis-estimates of the
observational uncertainties on $v_{\mathrm{Tan}}$

For the Milky Way streams, we note that their member stars possess quite large
observational uncertainties on $v_{\mathrm{Tan}}$ (of the order of $\sim
20{\rm\,km\,s^{-1}}$, see panels ‘d’ in Figures 2 and 3). However, we
constrain the ${\sigma_{v_{\mathrm{Tan}}}}$ to the order of $\sim
1-2{\rm\,km\,s^{-1}}$. In this appendix we demonstrate that even though the
$v_{\mathrm{Tan}}$ uncertainties on tangential velocity measurements for
individual stars are large our method is sensitive to the changes in these
large uncertainties and able to measure intrinsic velocity dispersions that
are much smaller than the current uncertainties.

To illustrate this we take the Phlegethon stream and artificially modify the
$v_{\mathrm{Tan}}$ uncertainties on individual stars and recompute the
${\sigma_{v_{\mathrm{Tan}}}}$, always keeping the $v_{\mathrm{Tan}}$
measurements unchanged (i.e. only modifying the uncertainties). In the first
case, we set these uncertainties to $0{\rm\,km\,s^{-1}}$ which results in
${\sigma_{v_{\mathrm{Tan}}}}=17.19{\rm\,km\,s^{-1}}$ by applying equivalent of
eq. 4 (see Section 3). This value is much larger than the value mentioned
above for Phlegethon, but this is expected because now the uncertainty-term
(in eq. 4) attributes the entire spread in the “observed $v_{\mathrm{Tan}}$
(obs) – $v_{\mathrm{Tan}}$ (fit)” distribution (i.e., residuals shown in
panels ‘d’ in Figure 2) to the internal dispersion of the stream
${\sigma_{v_{\mathrm{Tan}}}}$. In the second case, we decrease the
$v_{\mathrm{Tan}}$ uncertainty to half of the actual values and measure
${\sigma_{v_{\mathrm{Tan}}}}=12.25{\rm\,km\,s^{-1}}$. Note that this value is
smaller than the one computed in the first case because now the spread in the
residual distribution is being shared by the uncertainty-term (that are finite
and non-zero) and the internal dispersion ${\sigma_{v_{\mathrm{Tan}}}}$. In
the third case we decrease the uncertainties to $80\%$ of the actual values
and measure ${\sigma_{v_{\mathrm{Tan}}}}=1.29{\rm\,km\,s^{-1}}$. As expected
${\sigma_{v_{\mathrm{Tan}}}}$ decreases further because now the velocity
uncertainties absorb a larger share of the residual distribution. This
explains both why the measured intrinsic dispersion is so much smaller than
the observed dispersion and why we assert that a decrease in the uncertainties
on $v_{\mathrm{Tan}}$ expected from future Gaia data releases will improve
these ${\sigma_{v_{\mathrm{Tan}}}}$ measurements.

###  C.2 Determining the effects of correlations

We assess the effects of correlations between uncertainties in proper motions
and parallax in the following way. First, we take the Phlegethon stream and
shuffle the proper motion uncertainties of its stars while keeping the
parallax uncertainties unchanged (i.e., we randomly reassign the proper motion
uncertainty of star $j$ to star $k$ and star $k$ to $i$, and so on). We do
this $10$ times to examine whether the resulting ${\sigma_{v_{\mathrm{Tan}}}}$
(on an average) is same as what we report above. Based on this we find that
${\sigma_{v_{\mathrm{Tan}}}}$ (on an average) changes by only $+1.5\%$. We
repeat the above exercise, except this time we shuffle the parallax
uncertainties between stars while keeping the proper motion uncertainties
unchanged. In this case we find that the resulting
${\sigma_{v_{\mathrm{Tan}}}}$ (on an average) changes by only $-2.7\%$.
Finally, we repeat the above exercise with a few other streams and find
similarly small changes in our estimated ${\sigma_{v_{\mathrm{Tan}}}}$
measurements. This suggests that the correlations should have minor effects on
the reported ${\sigma_{v_{\mathrm{Tan}}}}$ values of the streams.

##  Appendix D Examining the impact of systematic errors on the measured
${\sigma_{v_{\mathrm{Tan}}}}$ of the Milky Way streams

We recompute ${\sigma_{v_{\mathrm{Tan}}}}$ of 5 GC streams by incorporating
the “systematic errors” present in Gaia EDR3’s proper motions and parallaxes.
These errors are provided in Section 5.6 of Lindegren, Lennart et al. (2020)
as $0.0108$ mas in $\varpi$, $0.0112{\rm\,mas\,yr^{-1}}$ in $\mu^{*}_{\alpha}$
and and $0.0107{\rm\,mas\,yr^{-1}}$ in $\mu_{\delta}$. These values
essentially put a floor on the precision with which parallaxes and proper
motions are measured.

To recompute ${\sigma_{v_{\mathrm{Tan}}}}$, we do the following. For a given
stream, we consider the individual stars, and to these we add the above errors
(in quadrature) to the observed Gaia uncertainties in parallaxes and proper
motions. This essentially inflates the uncertainties of every star. Then, we
compute ${\sigma_{v_{\mathrm{Tan}}}}$ by following the same procedure as
described in Section 3. The final ${\sigma_{v_{\mathrm{Tan}}}}$ values are
provided in Table 2. Table 2 also provides the $p-$values for the null
hypothesis that these new ${\sigma_{v_{\mathrm{Tan}}}}$ values (with their
associated uncertainties) are drawn from the counterpart
${\sigma_{v_{\mathrm{Tan}}}}$ measurements that we computed in Section 3
without the inclusion of systematic errors. To compute these $p-$values, we
follow the same method described in Section 3. The fact that these $p-$values
are $\sim 1$ indicate that, for a given stream, the two types of
${\sigma_{v_{\mathrm{Tan}}}}$ measurements are similar.

Table 3 is similar to Table 1, except this time produced using the new
${\sigma_{v_{\mathrm{Tan}}}}$ measurements. The fact that values in Table 3
are qualitatively similar to those present in Table 1 suggests that inclusion
of systematic errors do not affect our final conclusion in regard to the
cusp/core scenario of the parent subhalos.

Table 2: ${\sigma_{v_{\mathrm{Tan}}}}$ of Milky Way streams (in ${\rm\,mas\,yr^{-1}}$) computed by including the systematic errors. The values in the brackets provide the $p-$values of these new ${\sigma_{v_{\mathrm{Tan}}}}$ measurements being drawn from their counterpart streams whose ${\sigma_{v_{\mathrm{Tan}}}}$ were measured without including the systematic errors. GD-1 | Sylgr | Phlegethon | Fjörm | Gjöll
---|---|---|---|---
$0.63^{+0.67}_{-0.45}\,(0.963)$ | $1.76^{+1.83}_{-1.23}\,(0.95)$ | $1.12^{+1.23}_{-0.78}\,(0.984)$ | $1.86^{+1.66}_{-1.29}\,(0.874)$ | $2.19^{+2.25}_{-1.53}\,(0.994)$
Table 3: Same as Table 1, but using the ${\sigma_{v_{\mathrm{Tan}}}}$ values computed by including the systematic errors. MW stream | in-situ | SCu | LCu | SCo | LCo
---|---|---|---|---|---
GD-1 | 0.289 ($0.773$) | -4.37 ($<10^{-3}$) | -10.05 ($<10^{-3}$) | -1.314 ($0.189$) | -1.218 ($0.223$)
Fjorm | 1.053 ($0.293$) | -1.1 ($0.271$) | -3.891 ($<10^{-3}$) | 0.198 ($0.843$) | -0.122 ($0.903$)
Phlegethon | 0.791 ($0.429$) | -2.068 ($0.039$) | -5.701 ($<10^{-3}$) | -0.378 ($0.705$) | -0.617 ($0.537$)
Sylgr | 1.023 ($0.306$) | -1.055 ($0.291$) | -3.598 ($<10^{-3}$) | 0.124 ($0.901$) | -0.164 ($0.869$)
Gjoll | 1.104 ($0.27$) | -0.668 ($0.504$) | -2.761 ($0.006$) | 0.379 ($0.704$) | 0.05 ($0.96$)