# Variable Chaplygin Gas: Contraining parameters using FRBs

Geetanjali Sethi
Associate Professor,Department of Physics
St Stephen’s College
Delhi
<EMAIL_ADDRESS>
&Udish Sharma
Department of Physics
St Stephen’s College
Delhi
<EMAIL_ADDRESS>
&Nadia Makhijani
Department of Physics
St Stephens College
Delhi
<EMAIL_ADDRESS>

###### Abstract

We investigate the cosmological constraints on the Variable Chaplygin gas
model with the latest observational data from the Fast Radio Bursts and SNeIa
(Pantheon+SHOES). The Variable Chaplygin gas model is shown to be compatible
with Type Ia Supernovae and Fast Radio Burts. We have obtained tighter
constraints on cosmological parameters $B_{s}$ and $n$, using the the FRB data
set. By using the Markov chain Monte Carlo (MCMC) method on the SNeIa data
set, we obtain $B_{s}$=$0.1956\pm 0.1047$ , n= $1.3581\pm 1.1678$ and $H_{0}$=
$70.3902\pm 0.6381$ and we obtain $B_{s}$= $0.1780\pm 0.1069$ , n= $1.2633\pm
1.2433$ and $H_{0}$=70.3976$\pm 0.6597$ from the FRB data set. We are able to
get a good agreement between the $H_{0}$ values from the two data sets.

_K_ eywords Variable Chaplygin Gas $\cdot$ Fast Radio Bursts $\cdot$ SNe Ia
$\cdot$ Cosmology

##  1 Introduction

Over the years, through various developments in the field of observational
astronomy and astrophysics, cosmologists have been able to provide, with
constant improvements, many frameworks which explain the evolution of our
Universe on great timescales. The homogeneous and isotropic nature of the
Universe on large length scales($\approx 100Mpc$) has led to the development
of the FLRW Metric:

$ds^{2}=c^{2}dt^{2}-a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right]$

where k=$\pm 1,0$ defines the curvature of our Universe. Observationally, it
has been found that a $k=0$ universe is viable. This however has been proven
to be inadequate. Through advancements in Astronomy with the measurements of
galactic rotation curves which led to the missing mass problem and later
through the measurements of SNe Ia light curves ( [1] and [2]) which for the
first time established the long theorised expanding nature of the Universe
through observations, shortcomings in just the FLRW metric have been shown to
be present.

This led to the the emergence of the $\Lambda-CDM$ model of Cosmology. This
model introduced the idea of the presence of dark matter and dark energy, two
unique quantities with properties wildly different from Baryonic matter. Dark
matter is theorised to exist in Halos surrounding galaxies and is said to
undergo only gravitational interactions. Dark energy has more varied
interpretations but the idea that it exerts a negative pressure driving the
accelerated expansion of the universe has been established. The $\Lambda-CDM$
model, also now known as the general cosmological model, has been able to
predict accurately many significant observations. The model has been
successful in explaining the Cosmic microwave background and its temperature
fluctuations, the polarisation and the abundance of light matter in our
Universe. It also explains the phenomena of gravitational lensing with the
distribution that the Universe is composed of $68\%$ dark energy, $27\%$ dark
matter and $5\%$ ordinary matter. It explains the uneven distribution of
lensing and greater lensing around galactic clusters due to the higher
presence of matter. The model has also been able to explain the filament-like
structures[3] which exists in regards to the existence of galactic clusters.
Another important success that has been associated with the General
Cosmological model is its ability to explain the redshift-distance relation.
Redshift refers to the displacement of the spectrum of an astronomical object
towards longer wavelengths. The model takes into account the contributions
from baryonic matter, dark matter and dark energy and successfully puts into
picture the redshift of various astronomical sources moving away from us. This
is consistent with the general overview of an accelerated expanding Universe.
In spite of the success of the $\Lambda-CDM$ model making key predictions that
fall well in line with observed phenomena, there have been a lot of
shortcomings in the model.
The fine tuning problem refers to the model’s requirement of extremely tight
constraints on values of many key parameters. If these constraints were not
adhered to, the Universe would not exist the way we know it to be. The ratio
of the gravitational force and the electromagnetic force is one such
constraint which allows for the formation of the stars and their long life
times. A deviation from the currently established ratio would result in a
Universe just filled with hot dense gas. The Hydrogen-Helium fusion reactions
in the core of the stars can only be sustained by a very tight bound on the
mass of a proton. A lighter proton would result in too rapid of a reaction
leading the stars to burn out in short spans of time while a heavier proton
would lead to no reaction at all. The evolution of the Universe also as we
know it requires an extremely fine tuned value of the Hubble parameter and the
debate about the coincidence of the occurrence of these precise values which
support the model and its predictions has been on for quite some time. Another
significant objection to the model is Hubble tension[4]. Hubble tension refers
to the discrepancy in the measurement of the Hubble constant from the two
different methods of measurement. When calculated through observations of the
Cosmic Microwave background, the value of Hubble constant at the current epoch
is found to be $H_{0}=67Km/s/Mpc$ and when measured through SNe Ia is found to
be $H_{0}=73Km/s/Mpc$. This is a difference of the order of 5 $\sigma$ which
is very substantial and has pushed researchers forward to look for a unified
theory which is able to explain both the sets of observational data. Moreover,
the rather mysterious nature of Dark Matter and Dark energy themselves and the
fact that there has been no direct detection of dark matter further cast
doubts on the viability of something like the $\Lambda-CDM$ model.

There have been alternates such as the Quintessence dominated models([5])
which propose the existence of a scalar field with negative pressure as a
candidate for dark energy. These models have been able to explain the
accelerated expansion of the universe and also do away from the existence of a
cosmological constant $\Lambda$ which stays consistent over different epochs.
Despite the fact that the quintessence dominated models describe in great
effect the observations being made,they have their own share of issues. Little
to no concrete explanation is available for the scalar field which encompasses
dark energy. The models also suffer from their own set of fine tuning
problems.
As an alternative hence, in this paper, we provide a study of one of the
Chaplygin gas models. The generalisation to the original Chaplygin gas [6]
model by Sergei Chaplygin was introduced in 2003 [7], these models propose the
existence of fluid with an exotic equation of state in order to explain the
accelerated expansion of the Universe. A mathematical introduction to the
model is provided in section 2 followed by our analysis and estimation of the
model parameters and the Hubble constant in the sections that follow.
For this analysis, we have used SNe Ia data which is known to be a very
effective cosmological probe. We have alongside SNe Ia attempted to use Fast
radio bursts as another cosmic probe. FRBs are short duration transient bursts
that occur in the radio wavelengths. They have a well established DM- redshift
relation which allows the study of their properties in details. FRBs have been
shown to be good candidates for cosmological probes from the already done
studies. [8]

##  2 Variable Chaplygin Gas Model

The Chaplygin gas can be expressed by the equation of state
$P=\frac{-A}{\rho}$. Here A is a positive constant. Generalised Chaplygin gas
model is characterised by the equation of state:

$P_{ch}=-\bigg{(}\frac{A}{\rho_{ch}^{\alpha}}\bigg{)}$ (1)

where $\alpha$ is a constant such that $0<\alpha\leq 1$.
If we take the time component of the Energy-Momentum conservation equation,
$T_{\mu}^{\mu\nu}=0$, we obtain the continuity equation:

$\frac{\partial\rho}{\partial t}+3H(p+\rho)=0$ (2)

here H is the Hubble parameter, $H=\frac{\dot{a}}{a}$, $\rho$ = total density
of Universe. Using equation (2), the evolution of energy density of the
Chaplygin gas model is given as[9] :

$\rho{}_{ch}=\bigg{(}A+\frac{B}{a^{3(1+\alpha)}}\bigg{)}^{\frac{1}{1+\alpha}}$
(3)

Here $a$ is the scale of the universe in its current epoch and B is a constant
of integration. For the earlier epochs $a<<1$. Taking this into account,
equation (3) transforms into $\rho\propto a^{-}3$. Thus in this case, the
Chaplygin gas shows behaviour like Cold Dark Matter(CDM). In the late epochs,
when the Universe Scale factor $a>>1$, equation (3) transforms to
$p=-\rho=constant$. Hence, the behaviour of Chaplygin gas is like the
cosmological constant at later times therefore implying the accelerated
expansion of the Universe.
The Chaplygin gas equation of state results in a component that acts as non-
relativistic matter at the early stage of the universe and acts as
Cosmological constant equal to $8\pi GA^{\frac{1}{1+a}}$ at later stages. The
Chaplygin gas based models have not withered down in terms of interest among
researchers for the promise they have shown in the past.
However, the shortcoming with the Chaplygin gas model is it produces
oscillations or an exponential blowup of matter power spectrums which are
inconsistent with observations [10]. To avoid this instability, the combined
effect of shear and rotation can be taken into consideration. This slows down
the collapse with respect to the simple spherical collapse model[11]. Hence
,Variable Chaplygin Gas(VCG) has been proposed as a modification [12],[13]. In
the VCG model, the equation of state for the Chaplygin gas across the epochs
is free-flowing . The VCG model comes from the dynamics of a generalised
d-brane in a (d+1,1) space-time and can be described by a complex scalar field
$\phi$. The action can be expressed in the form of a generalised Born-Infield
action[9]. Considering a Born-Infield Lagrangian [14]

$\mathcal{L}_{\mathrm{BI}}=V(\phi)\sqrt{1+g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi}$
(4)

Here $V(\phi)$ is the scalar potential. When considering a spatially flat FLRW
universe, the energy density and pressure can be defined as: $\
\rho=V(\phi)\left(1-\dot{\phi}^{2}\right)^{-1/2}$ and
$P=-V(\phi)\left(1-\dot{\phi}^{2}\right)^{1/2}$. Correspondingly, the equation
of state for Chaplygin gas can be written as:

$P=-\frac{V^{2}(\phi)}{\rho}$ (5)

The self interaction potential can be rewritten as a function of the scale
factor: $V^{2}(\phi)=A(a)$. The equation of state for VCG thus becomes:

$P{}_{ch}=-\frac{A(a)}{\rho_{ch}}$ (6)

We can define $A(a)=A_{0}a^{-n}$ which is a positive function of the
cosmological scale factor $a$ and $A_{0}$ and $n$ are constants. When we use
the energy conservation equation (6) in a flat Friedmann-Robertson-Walker
Universe, the evolution of the VCG density is given as :

$\rho{}_{ch}=\sqrt{\frac{6}{6-n}\frac{A{}_{0}}{a{}^{n}}+\frac{B}{a{}^{6}}}$
(7)

where B is the constant of integration.
Adding to this Einstein’s Field equations $G_{\alpha\beta}=8\pi
GT_{\alpha\beta}$, and the FLRW metric, the Friedmann equations can be
obtained:

$\displaystyle H^{2}$ $\displaystyle=\frac{8\pi G}{3}\rho-\frac{k}{a^{2}}$ (8)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=-\frac{4\pi G}{3}(\rho+3p)$

Here $\rho$ is the baryonic matter density. Taking the dark energy component
into consideration, the rate of expansion of the Universe can be expressed in
terms of matter and radiation density $\rho$, curvature $k$, and the
cosmological constant, $\Lambda$ as :

$H^{2}\equiv\bigg{(}\frac{\dot{a}}{a}\bigg{)}^{2}=\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}+\frac{\Lambda}{3}$ (9)

Taking the critical density $\rho_{c}=\frac{3{H^{2}}}{8{\pi}G}$ and density
parameter $\Omega=\frac{\rho}{\rho_{c}}$, the first Friedmann equation can be
written as :

$\Omega_{b}(a)+\Omega_{k}(a)=1$ (10)

We can extend this relation directly to other models with more components.
When baryonic matter and radiation are taken into account, The Friedmann
equations is written as :

$H^{2}=H_{0}^{2}(\Omega_{m,0}a^{-3}+\Omega_{\gamma,0}a^{-4}+\Omega_{k,0}a^{-2})$
(11)

where $\Omega_{m,0}+\Omega_{\gamma,0}+\Omega_{k,0}=1$. All other components
can be added if their behaviour with the scale factor is known. Reducing
equation (9) to the case of a spatially flat Universe:

$H{}^{2}=\frac{8\pi G}{3}\rho$ (12)

$H\equiv\dot{a}/a$ is the Hubble parameter. Hence, for the case of VCG model
from equation (7) corresponding to the VCG density, the acceleration condition
$\ddot{a}>0$ is equivalent to:

$\left(\frac{12-3n}{6-n}\right)a^{6-n}>\frac{B}{A_{0}}$ (13)

To allow for accelerated expansion of the Universe, the necessary condition is
$n<4$. Taking $n=0$ restores the original Chaplygin gas behavior. Initially,
the gas is like dust like matter ($\rho_{ch}\propto a^{-}3$) and later acts as
a cosmological constant ($p=-\rho=constant$). In the current state of VCG, the
universe tends to be quintessence-dominated( $n>0$)[15][16] or phantom-
dominated($n<0$)[17]. The constant equation of state parameter
$w\equiv-1+\frac{n}{6}$. The first term of the right hand side of Eq (7) is
negligible initially so the equation can be written as $\rho\sim a^{-}3$. This
corresponds to a dust like matter dominated Universe. At present, the value of
Energy density of VCG is:

$\rho_{cho}=\sqrt{\frac{6}{6-n}A_{0}+B}$ (14)

Here, $a_{0}=1$. We can define a parameter $B_{s}$:

$B_{s}=\frac{B}{6A_{0}/(6-n)+B}$ (15)

The energy density can be written in terms of $B_{s}$ and $n$ as :

$\rho_{ch}(a)=\rho_{ch0}\left[\frac{B_{s}}{a^{6}}+\frac{1-B_{s}}{a^{n}}\right]^{1/2}$
(16)

###  2.1 Model

The Freidmann equation in terms of the redshift z, taking $a=\frac{1}{1+z}$
and using equation (16) becomes:

$\begin{split}H^{2}=&\frac{8\pi
G}{3}\bigg{\\{}\rho_{r0}(1+z)^{4}+\rho_{b0}(1+z)^{3}+\\\
&\rho_{ch0}\Big{[}B_{s}(1+z)^{6}+(1-B_{s})(1+z)^{n}\Big{]}^{1/2}\bigg{\\}}\end{split}$
(17)

where $\rho_{r0}$ and $\rho_{b0}$ are the present values of energy densities
of radiation and baryons, respectively. 111We have used the fact that for a
flat Universe, $\Omega_{b0}+\Omega_{r0}+\Omega_{ch0}=1$, i.e the total matter
density sums up to unity.
Using (1)

$\frac{\rho_{r0}}{\rho_{ch0}}=\frac{\Omega_{r0}}{\Omega_{ch0}}=\frac{\Omega_{r0}}{1-\Omega_{r0}-\Omega_{b0}}$
(18)

and

$\frac{\rho_{b0}}{\rho_{ch0}}=\frac{\Omega_{b0}}{\Omega_{ch0}}=\frac{\Omega_{b0}}{1-\Omega_{r0}-\Omega_{b0}},$
(19)

Equation (17) becomes:

$H^{2}=\Omega_{ch0}H_{0}^{2}a^{-4}X^{2}(a),$ (20)

where

$\begin{split}X^{2}(a)=&\frac{\Omega_{r0}}{1-\Omega_{r0}-\Omega_{b0}}+\\\
&\frac{\Omega_{b0}a}{1-\Omega_{r0}-\Omega_{b0}}+a^{4}\bigg{(}\frac{B_{s}}{a^{6}}+\frac{1-B_{s}}{a^{n}}\bigg{)}^{1/2}\end{split}$
(21)

##  3 Statistical Methodologies

The Pantheon + SH0ES supernova data set[18] includes 1550 distinct type Ia
supernovae and also includes observations from the host galaxies of 42 type Ia
supernovae.The redshift of SNe Ia range from $z=0.001$ to $z=2.26$.This is the
most recently compiled data set of SNe Ia. This data set provides us with the
distance modulus of all the observation points along with their covariance
matrix.
For FRB, data of the localised FRBs has been compiled from the respective
publications. Table1 lists the data points used for analysis.

NAME |  REDSHIFT (z) |  $DM_{obs}$ ($pc$ $cm^{-3}$) |  REFERENCES
---|---|---|---
rFRB 20121102A |  0.19273 |  557 $\pm$ 2 |  Tendulkar et al. (2017) [19]
FRB 20171020A |  0.0087 |  114.1 $\pm$ 0.2 |  Mahony et al. (2018) [20]
rFRB 20180301A |  0.3304 |  522$\pm$ 0.2 |  Luo et al. (2020b) [21]
rFRB 20180916B |  0.0337 |  349.349 $\pm$ 0.005 |  Marcote et al. (2020) [22]
rFRB 20180924C |  0.3214 |  361.42 $\pm$ 0.06 |  Bannister et al. (2019) [23]
FRB 20181030A |  0.0039 |  103.396 $\pm$ 0.005 |  Bhandari et al. (2022) [24]
FRB 20181112A |  0.4755 |  589.27 $\pm$ 0.03 |  Prochaska et al. (2019) [25]
FRB 20190102C |  0.2913 |  363.6 $\pm$ 0.3 |  Macquart et al. (2020) [26]
FRB 20190520B |  0.241 |  1202 $\pm$ 10 |  Niu et al. (2021) [27]
FRB 20190523A |  0.66 |  760.8 $\pm$ 0.6 |  Ravi et al. (2019) [28]
FRB 20190608B |  0.1178 |  338.7 $\pm$ 0.5 |  Macquart et al. (2020) [26]
FRB 20190611B |  0.3778 |  321.4 $\pm$ 0.2 |  Macquart et al. (2020) [26]
FRB 20190614D |  0.6 |  959.2 $\pm$ 0.5 |  Law et al. (2020) [29]
rFRB 20190711A |  0.522 |  593.1 $\pm$ 0.4 |  Macquart et al. (2020) [26]
FRB 20190714A |  0.2365 |  504 $\pm$ 2 |  Bhandari et al. (2022) [24]
FRB 20191001A |  0.234 |  506.92 $\pm$ 0.04 |  Bhandari et al. (2022) [24]
FRB 20191228A |  0.2432 |  297.5 $\pm$ 0.05 |  Bhandari et al. (2022) [24]
rFRB 20200120E |  0.0008 |  88.96 $\pm$ 1.62 |  Kirsten et al. (2022) [30]
FRB 20200430A |  0.1608 |  380.1 $\pm$ 0.4 |  Bhandari et al. (2022) [24]
FRB 20200906A |  0.3688 |  577.8 $\pm$ 0.02 |  Bhandari et al. (2022) [24]
rFRB 20201124A |  0.0979 |  413.52 $\pm$ 0.05 |  Ravi et al. (2022) [31]
Table 1: Compiled data set of localised FRBs

The dispersive measure or DM of an FRB has contributions from a multitude of
sources that can be encapsulated in the equation:

$DM_{obs}=DM_{MW}+DM_{IGM}+\frac{DM_{host}(z)}{1+z}$ (22)

Here $DM_{obs}$ is the observed DM. $DM_{MW}$ includes contributions from both
the interstellar medium and the Milky Way halo, and can be written as:

$DM_{MW}=DM_{Halo}+DM_{ISM}$ (23)

The value of $DM_{halo}$ has been calculated to be between $50\sim 80$ $pc\
cm^{-3}$ up to 200 KPc from the sun[32] Thus a normal distribution with a mean
of $65\ pc\ cm^{-3}$ with a standard deviation of $15\ pc\ cm^{-3}$ has been
assumed[33].

The NE2001[34] model takes into account galactic coordinates and has been used
to derive the contribution from $DM_{ism}$.
The estimates for $DM_{host}$ can be defined by :

$DM_{host}=A(1+z)^{\alpha}$ (24)

Where $A$ and $\alpha$ are constants defined by [35] with their respective
error limits. Thus $DM_{igm}$ can be found using (22) and is what gets used in
the calculation of cosmological distances since the contribution is entirely
from the intergalactic medium.
The uncertainty in $DM_{igm}$ is defined by the relation:

$\sigma_{IGM}(z)=\sqrt{\sigma_{obs}(z)^{2}+\sigma_{MW}^{2}+\left(\frac{\sigma_{host}(z)}{1+z}\right)^{2}}$
(25)

We have the $\sigma_{obs}$ for each observation and we can take the combined
contribution of uncertainties in $\sigma_{halo}$ and $\sigma_{ism}$ to be $30\
pc\ cm^{-3}$.
As for the uncertainty, $\sigma_{host}$, we use ([35]) to define the value.
The higher error limit and lower error limit on the constants $A$ and $a$ are
used to define the uncertainty of $\sigma_{host}$.
Having now obtained both $DM_{igm}$ and $\sigma_{igm}$, we can make use of
these values in the distance modulus equations.
The distance modulus and luminosity distance have shown explicit dependence on
the Variable Chaplygin gas model.

For SNe Ia, since we have the values of distance modulus$(\mu_{obs})$ and
redshift available in the data set, we consider distance modulus defined by
the equation :

$\mu=5log_{10}(\frac{d_{L}}{Mpc})+25$ (26)

Also, we define luminosity distance $d_{L}$ using the Friedmann equation in a
flat universe by:

$d_{L}=c(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z)}$ (27)

Taking $H(z)$ from the equation (20).This gives the definition of $d_{L}$ as :

$d_{L}=\frac{c(1+z)}{H_{o}}\int_{0}^{z}\frac{dz^{\prime}}{\Omega^{1/2}_{ch0}X(z)}$
(28)

We substitute the obtained value of $d_{L}$ in 26 to find the theoretical
value of the distance modulus, $\mu_{th}$.
From $DM_{igm}$ and $z$ data set, using the upper limit of the following
relation[36] :

$d_{L}<\left(\frac{DM}{500pccm^{-3}}\right)\left[\left(\frac{DM}{1000pccm^{-3}}\right)+2.4\right]\
Gpc$ (29)

we get $d_{L\ obs}$. A comparison with the formula for $redshift-d_{L}$
relation[36] :

$d_{L}\approx 2z(z+2.4)Gpc$

shows that this is not a bad assumption. $d_{L\ obs}$ acts as our
observational value for the luminosity distance. The theoretical value of
luminosity distance for FRBs is calculated using the relation 28.

In equation (28), we fit for the parameters $H_{0}$, $n$ and $B_{s}$. $X(z)$
comes from equation (21) where $a$ can be written as $a=\frac{1}{1+z}$. From
the table (1) above, we have not made the use of FRBs with $z<=0.01$ due to
their low values of DM. Similarly, from the Pantheon data set, SNe with
$z<=0.01$ have not been considered.

###  3.1 Monte Carlo Likelihood Analysis

Markov chain Monte Carlo algorithms are used to do parameter estimation. These
algorithms start with a set of parameters and initial proposal functions for
all those respective parameters. Newer values are sampled for the parameters
from all over the parameter space in accordance to the probabilities defined
by the proposal function about the previous value and are accepted if they
result in a higher probability. This way, the chain infallibly moves towards
the set of parameters with highest probability as the acceptance rate for new
proposed parameters decreases. For $i^{th}$ data-point, given a certain set of
parameters, we define its individual Gaussian likelihood as :

$P_{i}=\frac{1}{\sqrt{2\pi}}exp\left(-\frac{(x_{i}-\mu_{i})^{2}}{2\sigma_{i}^{2}}\right)$
(30)

In this equation, $\mu_{i}$ acts as the theoretical value corresponding to
that data-point while $\sigma_{i}$ is the error bar associated with it. The
combined probability of the entire data set can hence be defined as :

$P_{total}=\prod_{i=1}^{n}P_{i}(x_{i}|k)$ (31)

Where $k$ represents the set of parameters being sampled for. To get the to
ideal value of $P_{total}$, we intend to find the optimal value for the set of
parameters $k$.
Writing (31) with $log(P_{total})$ as $\Theta$, we get to the relation:

$\Theta(k)=-\frac{1}{2}\sum_{i}\frac{(x_{th}^{i}-x_{obs}^{i})^{2}}{2\sigma_{i}^{2}}$
(32)

For FRBs, we can write the same as :

$\Theta(H_{0},B_{s},n)=-\frac{1}{2}\sum_{i}\frac{(dl_{th}^{i}-dl_{obs}^{i})^{2}}{2\sigma_{i}^{2}}$
(33)

Doing a similar analysis, we can define the likelihood for the SNe Ia data set
as :

$\Theta(H_{0},B_{s},n)=-\frac{1}{2}(\mu_{th}-\mu_{obs})C^{-1}(\mu_{th}-\mu_{obs})$
(34)

Here $C$ refers to our covariance matrix that is obtained from the data set.
For the actual sampling, Algorithms like Metrapolis Hastings and Gibbs
sampling are employed. We make use of Cobaya [37][38]( Code for Bayesian
Analysis) which is a framework designed for finding the posterior
distributions of a set of parameters given their priors and their limits. The
Cobaya samplers which include MCMC sampler and Polychord Sampler allow for a
more efficient and faster sampling process. The functionalities of these
samplers include oversampling of fast parameters allowing the more steps to be
taken along the axis where conditional distributions are easier to navigate.
In our case, in sampling for the parameters $H_{0}$, $B_{s}$ and $n$, our aim
is to maximise the value of $\Theta$ with respect to the set of these 3
parameters. We use our likelihoods as defined in equations (34) and (33) for
Supernova and FRBs respectively.

##  4 Results and Analysis

We have used a Gaussian prior for sampling of the parameters in both cases,
the Pantheon data set and the FRB data set. The means of prior distributions
for $H_{0}$, $B_{s}$ and $n$ are taken to be $70.4$, $0.11$ and $1.2$
respectively. These are the values from previous constraints obtained using
the Pantheon data set and doing a $\chi^{2}$ analysis. The estimations for the
parameters of the VCG model using Cobaya with the likelihood defined by
equation (34) for the SNe Ia dataset gives the distribution given in Figure
(1):

Figure 1: Distributions for the parameters $B_{s}$, $n$ and $H_{o}$ for the
SNe Ia Data set using VCG model

The estimation for the parameters of VCG model using the FRB dataset with the
likelihood defined by equation (33) is depicted in Figure (2):

Figure 2: Distributions for the parameters $B_{s}$, $n$ and $H_{o}$ for the
FRB Data set using VCG model

The results with $1\sigma$ bounds were shown in Table (2):

Data set |  $B_{s}$ |  $n$ |  $H_{o}$
---|---|---|---
SNe Ia |  $0.1956\pm 0.1047$ |  $1.3581\pm 1.1678$ |  $70.3902\pm 0.6381$
FRB |  $0.1780\pm 0.1069$ |  $1.2633\pm 1.2433$ |  $70.3976\pm 0.6597$
Table 2: The constrained parameters $B_{s}$,$n$ and $H_{o}$ of VCG model
obtained from Supernova and FRB data

The constraints on the parameter values with $1\sigma$ bounds from the
Pantheon+Shoes and FRB data sets are listed in Table (2). The value of the
Hubble parameter obtained from the two results is $70.3902$ Km/s/Mpc and
$70.3976$ Km/s/Mpc respectively. The agreement is extremely good. The fact
that VCG is a good fit can further be established from the $1\sigma$ and
$2\sigma$ overlaps in $n$ and $B_{s}$ values as shown in Figure (3). The value
of $H_{0}$ obtained by one of the authors in a recent paper [39] using the GRB
data was $70.41\pm 0.67$ km/s/Mpc is again in a good agreement with values
obtained in this work.

Figure 3: 1 $\sigma$ and 2 $\sigma$ overlaps of $n$ and $B_{s}$ obtained from
SNe Ia and FRB data

##  5 Conclusion

The Variable Chaplygin gas model is indeed able to explain the accelerated
evolution of the universe. As the VCG Model speculates, the Chaplygin gas is
expected to behave like a non-relativistic entity and later evolves to account
for the accelerated expansion observed in the current epoch of the Universe.
The best fit values obtained for the VCG model with the FRB data sets lies
within confidence levels obtained in previously conducted analysis [13] and
[39]. The value of $H_{0}$ obtained using FRB data set is in good agreement
with value obtained using the SNeIa data.

## Acknowledgments

The authors are grateful to the Principal, St. Stephen’s College for his
support and encouragement towards this research. We also thank Dr. Akshay
Rana, St. Stephen’s College for his inputs and suggestions.

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11institutetext:  Kapteyn Astronomical Institute, University of Groningen,
Landleven 12, 9747 AD Groningen, The Netherlands
11email<EMAIL_ADDRESS>22institutetext: School of Natural Sciences,
Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA
33institutetext: National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan 44institutetext: Key Lab of Optical Astronomy,
National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Road, Chaoyang, Beijing 100101, China 55institutetext: Université de
Strasbourg, CNRS, Observatoire Astronomique de Strasbourg, UMR 7550, F-67000
Strasbourg, France 66institutetext: Institute of Statistical Mathematics, 10-3
Midoricho, Tachikawa, Tokyo 190-0014, Japan

# High-precision chemical abundances of Galactic building blocks. II.

Revisiting the chemical distinctness of the Helmi streams

Tadafumi Matsuno 11 Emma Dodd 11 Helmer H. Koppelman 22 Amina Helmi 11 Miho N.
Ishigaki 33 Wako Aoki 33 Jingkun Zhao 44 Zhen Yuan 55 Kohei Hattori 3366

###### Abstract

Context.  The Helmi streams are kinematic substructures whose progenitor is
likely a dwarf galaxy. Although 20 years have past since their discovery, it
is still unclear whether their members are chemically distinguishable from
other halo stars in the Milky Way.

Aims.  We aim to precisely characterize the chemical properties of the Helmi
streams.

Methods.  We analysed high-resolution, high-signal-to-noise ratio spectra for
eleven Helmi streams’ stars through a line-by-line abundance analysis. We
compared the derived abundances to homogenized literature abundances of the
other halo stars, including those belonging to other kinematic substructures,
such as Gaia-Enceladus and Sequoia.

Results.  Compared to typical halo stars, the Helmi streams’ members clearly
show low values of [X/Fe] in elements produced by massive stars, such as Na
and $\alpha$-elements. This tendency is seen down to metallicities of at least
$[\mathrm{Fe/H}]\sim-2.2$, suggesting type Ia supernovae already started to
contribute to the chemical evolution at this metallicity. We found that the
[$\alpha$/Fe] ratio does not evolve much with metallicity, making the Helmi
streams’ stars less distinguishable from Gaia-Enceladus stars at
$[\mathrm{Fe/H}]\gtrsim-1.5$. The almost constant but low value of
[$\alpha$/Fe] might be indicative of quiescent star formation with low
efficiency at the beginning and bursty star formation at later times. We also
found extremely low values of [Y/Fe] at low metallicity, providing further
support that light neutron capture-elements are deficient in Helmi streams.
While Zn is deficient at low metallicity, it shows a large spread at high
metallicity. The origin of the extremely low Y abundances and Zn variations
remain unclear.

Conclusions.  The Helmi streams’ stars are distinguishable from the majority
of the halo stars if homogeneously derived abundances are compared.

##  1 Introduction

Table 1: Summary of the data

Object | Gaia EDR3 source id | $S/N_{4500\,\mathrm{\AA}}$ | $S/N_{5533\,\mathrm{\AA}}$ | $S/N_{6370\,\mathrm{\AA}}$
---|---|---|---|---
2447_5952 | 2447968154259005952 | 80 | 108 | 173
4998_5552 | 4998741805354135552 | 72 | 83 | 78
6170_9904 | 6170808423037019904 | 87 | 88 | 77
6615_9776 | 6615661065172699776 | 97 | 161 | 122
6914_3008 | 6914409197757803008 | 73 | 86 | 120
J1306+4154 | 1527475951701753984 | 50 | 65 | 53
J1436+0929 | 1176187720407158912 | 134 | 110 | 91
J1553+3909 | 1376687518318241536 | 84 | 73 | 55
J1730+5309 | 1416077522383596160 | 84 | 63 | 42
J1642+2041 | 4564066449004092928 | 63 | 60 | 86
LP894-3 | 2891152566675457280 | 97 | 150 | 143
111We obtained new high-resolution spectra except for LP894-3. The signal-to-
noise ratio are converted to per $0.01\,\mathrm{\AA}$.

Table 2: Property of the targets

Object | $\pi$ | $\sigma(\pi)$ | $G$ | $B_{p}-R_{p}$ | $E(B-V)$ | RV
---|---|---|---|---|---|---
| (mas) | (mas) | (mag) | (mag) | (mag) | ($\mathrm{km\,s^{-1}}$)
2447_5952 | 2.817 | 0.021 | 11.963 | 0.677 | 0.096 | 145.5
4998_5552 | 3.463 | 0.018 | 12.561 | 0.784 | 0.000aa$a$This object is not covered by Green et al. (2019). Since Schlegel et al. (1998) provides $E(B-V)=0.01$ and since this object is nearby, we assumed $E(B-V)=0.0$ for this object. | -187.6
6170_9904 | 3.743 | 0.018 | 10.573 | 0.728 | 0.064 | -122.4
6615_9776 | 4.984 | 0.014 | 12.064 | 0.768 | 0.010 | -276.9
6914_3008 | 3.195 | 0.017 | 12.622 | 0.720 | 0.015 | 113.9
J1306+4154 | 1.470 | 0.013 | 13.499 | 0.668 | 0.046 | -294.7
J1436+0929 | 3.294 | 0.019 | 12.172 | 0.671 | 0.006 | -301.4
J1553+3909 | 1.944 | 0.013 | 13.619 | 0.741 | 0.018 | -288.3
J1642+2041 | 0.997 | 0.013 | 13.479 | 0.765 | 0.070 | -246.3
J1730+5309 | 0.751 | 0.011 | 13.967 | 0.817 | 0.082 | 70.9
LP894-3 | 5.347 | 0.038 | 11.090 | 0.710 | 0.010 | 303.0bb$b$From Ishigaki et al. (2012).
222

The Helmi streams (Helmi et al., 1999) provided some of the first evidence for
the hierarchical formation of the Milky Way expected from the standard
cosmology. The streams were discovered as an overdensity in the plane of
angular momenta in a sample of $<100$ low-metallicity stars. Later studies
also confirmed their existence (Chiba & Beers, 2000; Kepley et al., 2007;
Myeong et al., 2018; Koppelman et al., 2018). Their progenitor was likely a
dwarf galaxy with a stellar mass of $\sim 10^{8}\,\mathrm{M_{\odot}}$ and a
mean metallicity of $[\mathrm{Fe/H}]\sim-1.5$ that was accreted to the Milky
Way around $5-8\,\mathrm{Gyr}$ ago (Kepley et al., 2007; Koppelman et al.,
2019).

Despite the fact that the streams were discovered more than 20 years ago,
their chemical properties have not received much attention until recently.
Roederer et al. (2010) studied chemical abundances of 12 stars in the stream,
concluding that the stars have similar abundance ratios as the rest of halo
stars. In recent years, Limberg et al. (2021), Gull et al. (2021), Aguado et
al. (2021), and Nissen et al. (2021) conducted chemical analysis for the Helmi
streams’ members. While Gull et al. (2021) reached the same conclusion as
Roederer et al. (2010), some of these studies have hinted possible chemical
peculiarities of the streams. For example, the Sr abundance seems to be much
lower than typical of halo stars at $[\mathrm{Fe/H}]\lesssim-2$ (Aguado et
al., 2021); two stars in a binary system in the Helmi streams, namely G112-43
and G112-44, have enhanced ratios of [Mn/Fe], [Ni/Fe], [Cu/Fe], and [Zn/Fe]
compared to other accreted stars in the Milky Way (Nissen et al., 2021); and
the [$\alpha$/Fe] ratio seems to be decreasing at $[\mathrm{Fe/H}]\gtrsim-2$
(Limberg et al., 2021; Aguado et al., 2021).

Many of the previous comparisons made use of heterogeneous literature
compilation. They were therefore limited by systematic uncertainties, which
can be as large as 0.4 dex in [Fe/H] and 0.2 dex in [X/Fe]. As we will see
later in this paper, this can easily smear out or falsely create abundance
peculiarities. Another complication arises from the fact that the literature
sample was often a mixture of accreted and in situ stars. As (Nissen &
Schuster, 2010, hereafter, NS10) showed from their homogeneously analyzed
samples, the Milky Way halo contains two major populations, accreted and in
situ stars, which differ in abundance ratios. Without understanding the origin
of the comparison sample, it remains unclear if the chemical peculiarities of
the Helmi stream stars, if any, indicate just that they are accreted or that
the progenitor had a unique evolution. Although Nissen et al. (2021) is the
only exception in the sense that they compared Helmi streams’ members with
accreted stars using homogeneously derived chemical abundance, larger samples
can also help in understanding the interesting abundance pattern reported in
the two stars in a binary system.

Throughout this study, we aim to characterize chemical abundances of the Helmi
streams’ stars precisely and to compare them with homogeneous abundances of
other components in the Milky Way halo, including Gaia-Enceladus, in situ
stars (NS10; Reggiani et al. 2017, hereafter R17), and Sequoia (Matsuno et
al., 2021, hereafter, Paper I). Not only do the precise chemical abundance
ratios tell us about the progenitor’s properties, but also chemical
distinctness of kinematic substructures, if found, enables us to “chemically-
tag” stars originating from the same progenitor. In Paper I, we showed that a
combination of a line-by-line abundance analysis for a carefully selected
sample and homogenization of literature abundance using standard stars is a
powerful way to precisely characterize chemical property of kinematic
substructures.

We studied chemical abundance of the Helmi streams’ stars in the same way as
in Paper I. Namely, we analyzed high signal-to-noise ratio spectra for 11
stars in the Helmi streams, measured their chemical abundance through a line-
by-line analysis with careful treatment of uncertainty, and compared them with
other halo stars in NS10 and R17 after homogenizing all the abundances. We
describe the observation, data reduction, targets, and stars in literature in
Section 2. In Section 3, after briefly explaining our approach for the
chemical abundance measurement and homogenization, we present chemical
abundance ratios of the Helmi streams’ stars. After providing discussions in
Section 4, we summarise our findings in Section 5.

##  2 Data

###  2.1 Observation

Table 3: Kinematics of the targets

Object | $v_{z}$ | $\sigma(v_{z})$ | $L_{z}$ | $\sigma(L_{z})$ | $L_{\rm perp}$ | $\sigma(L_{\rm perp})$ | $E/10^{5}$ | $\sigma(E)/10^{5}$ | subclassaa$a$Subclass defined by Dodd et al. (2021). The hiL for the clump with higher $L_{\rm perp}$ and loL for the one with lower $L_{\rm perp}$.
---|---|---|---|---|---|---|---|---|---
| ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{kpc\,km\,s^{-1}}$) | ($\mathrm{kpc\,km\,s^{-1}}$) | ($\mathrm{kpc\,km\,s^{-1}}$) | ($\mathrm{kpc\,km\,s^{-1}}$) | ($\mathrm{km^{2}\,s^{-2}}$) | ($\mathrm{km^{2}\,s^{-2}}$) |
2447_5952 | -216.7 | 1.1 | 1278 | 11 | 1872 | 9 | -1.022 | 0.007 | …
4998_5552 | 230.1 | 1.1 | 1020 | 6 | 1846 | 9 | -1.429 | 0.002 | …
6170_9904 | -272.8 | 1.3 | 1200 | 9 | 2201 | 11 | -1.342 | 0.003 | hiL
6615_9776 | 213.8 | 0.9 | 1345 | 2 | 1757 | 8 | -1.339 | 0.003 | loL
6914_3008 | -283.9 | 1.5 | 1437 | 8 | 2279 | 12 | -1.275 | 0.004 | hiL
J1306+4154 | -281.3 | 0.9 | 1288 | 2 | 2363 | 7 | -1.293 | 0.002 | hiL
J1436+0929 | -249.4 | 0.9 | 1226 | 4 | 2050 | 7 | -1.312 | 0.003 | hiL
J1553+3909 | -268.6 | 0.8 | 1304 | 6 | 2208 | 7 | -1.299 | 0.002 | hiL
J1642+2041 | -217.0 | 1.2 | 1339 | 7 | 1737 | 9 | -1.392 | 0.003 | loL
J1730+5309 | 238.3 | 2.4 | 1423 | 14 | 1965 | 19 | -1.384 | 0.004 | loL
LP894-3 | -219.8 | 0.7 | 1015 | 6 | 1813 | 6 | -1.283 | 0.003 | …

333

Figure 1: Distribution of stars in kinematic spaces, namely $E-L_{z}$, $L_{\rm
perp}-L_{z}$, and $v_{y}-v_{z}$. The insets in the left two panels show better
distribution of the stars that are part of the Helmi streams. The Helmi
streams’ selection from Koppelman et al. (2019) is shown as a dashed box in
the inset of the middle panel. Stars are hatched according to the subgroups
found by Dodd et al. (2021). Three stars (2447_5952, 4998_5552, and LP894-3)
are not associated to either of the subgroups. Among them, an open symbol is
assigned only to 2447_5952 since it has much larger energy compared to the
other Helmi streams’ stars. We used filled symbols for stars from NS10 and R17
if they satisfy the kinematic selection for Gaia-Enceladus (see Paper I, ).
Figure 2: Location of the stars in the color-magnitude diagram. The symbols
follow those of Figure 1. The four isochrones are from the PARSEC tracks with
the age of 12 Gyr and
$[\mathrm{Fe/H}]=-2.0,\,-1.5,\,-1.0,\,\mathrm{and}\,-0.5$.

We obtained high-resolution spectra for ten stars with the High Dispersion
Spectrograph (Noguchi et al., 2002) on the Subaru telescope444Gaia EDR3
1909569058536197760 was also observed but not analysed since it is a double-
lined spectroscopic binary.. The selection and properties of the targets are
described in the next subsection. The observations were conducted on August
9-10th, 2020 with the standard setup of the HDS (StdYd), which yields a
wavelength coverage from $\sim 4000\,\mathrm{\AA}$ to $\sim
6800\,\mathrm{\AA}$. We used the image slicer #2 (Tajitsu et al., 2012), which
provides a resolving power of $R\sim 80,000$. The data reduction were
performed using an IRAF script555IRAF is distributed by the National Optical
Astronomy Observatory, which is operated by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation,
hdsql666http://www.subarutelescope.org/Observing/Instruments/HDS/hdsql-e.html,
including CCD linearity correction, scattered light subtraction, aperture
extraction, flat-fielding, wavelength calibration, and heliocentric velocity
correction. The information about the spectra is summarised in Table 1.

We also searched archives for high signal-to-noise ratio, high-resolution
spectra for stars that satisfy our selection criteria described in the next
subsection. We found that LP894-3 satisfies our selection and has spectra in
the Subaru telescope archive. We therefore analysed the archival spectrum of
this star. The spectrum used is the same as that used by Ishigaki et al.
(2012).

###  2.2 Targets

Targets were selected based on the orbital angular momenta of the stars
following Koppelman et al. (2019). Namely, the selection box was defined with
the component of the angular momentum along the $z$-axis of the Milky Way
($L_{z}$) and $L_{\rm perp}$, which is given by $\sqrt{L_{x}^{2}+L_{y}^{2}}$.
We also restricted our sample to those around the turn-off region using Gaia
DR2 photometry. The exact selection criteria were
$1000<L_{z}/\mathrm{kpc\,km\,s^{-1}}<1500$ and $1750<L_{\rm
perp}/\mathrm{kpc\,km\,s^{-1}}<2600$ (the kinematic selection, corresponding
to the box A of Koppelman et al., 2019), and $0.5<G_{BP}-G_{RP}<0.95$ and
$3.2<G_{\rm abs}<5.8$ (the color-magnitude selection), where $G_{BP}$,
$G_{RP}$, and $G_{\rm abs}$ are Gaia $BP$ and $RP$ magnitude, and the absolute
magnitude in the Gaia $G$ band, respectively.

Initially for the target selection, We used Gaia DR2 astrometry, photometry,
and radial velocity (Gaia Collaboration et al., 2018), and Large Sky Area
Multi-Object Fiber Spectroscopic Telescope (LAMOST) survey DR5 radial
velocity. We assumed $U_{\odot}=-11.1\,\mathrm{km\,s^{-1}}$,
$W_{\odot}=7.3\,\mathrm{km\,s^{-1}}$ (Schönrich et al., 2010),
$R_{0}=8.21\,\mathrm{kpc}$ (McMillan, 2017), and $z_{0}=20.8\,\mathrm{pc}$
(Bennett & Bovy, 2019) for the solar velocity and position. Adopting the
proper motion measurement for Sgr A∗ (Reid & Brunthaler, 2004), we assumed the
sun is moving at $245.3\,\mathrm{km\,s^{-1}}$ in the direction of the Galactic
rotation. When calculating orbital energies of the stars, we adopted the Milky
Way potential of McMillan (2017).

The information about the kinematic and photometric properies of the targets
is summarised in Tables 2 and 3, and visualized in Figures 1 and 2. We note
that we updated the kinematic and photometric information with Gaia EDR3
photometry and astrometry (Gaia Collaboration et al., 2020), and the radial
velocity measured from our high-resolution spectra, which was measured by
comparing the observed wavelengths of neutral Fe lines with those from
laboratory experiments.

Table 3 also includes information about subgroups among the Helmi streams’
stars. We here consider two ways for subdivision of the Helmi streams’ stars.
One is by the sign of $v_{z}$; Kepley et al. (2007) and Koppelman et al.
(2019) used the asymmetry in the number of stars in each subgroup to estimate
the accretion time of the progenitor of the Helmi streams. Another subdivision
follows that by Dodd et al. (2021) in the space of $L_{z}$ and $L_{\rm perp}$.
They showed that these two groups of stars with slightly different kinematics
reflect an effect of orbital resonance in the Milky Way potential. In either
definition, the two subgroups are considered to be parts of the same
substructure, the Helmi streams. In all figures, symbols are hatched according
to the subdivision by Dodd et al. (2021).

###  2.3 Literature

We compared the chemical abundance of the Helmi streams’ stars with those in
literature. Since our approach described in Section 3 is basically identical
as that used in Paper I, we can naturally compare our abundance in the present
study with Gaia-Enceladus and in situ stars from NS10 and R17 and with Sequoia
stars from Paper I. Since abundances from these studies were homogenized using
standard stars (see Paper I, ), precise comparisons are possible. We refer to
Paper I for more details.

Among the stars studied by NS10 and R17, we found two stars that satisfy our
kinematic selection criteria for the Helmi streams. These two stars are
G112-43 and G112-44 and were studied in detail by Nissen et al. (2021) as part
of the Helmi streams. Although they are considered to form a binary system,
their motions within the system do not affect the estimated orbital properties
within the Milky Way since the separation between the two stars is large
($>2000\,\mathrm{au}$).

##  3 Chemical abundances

Table 4: Stellar parameters Object | $T_{\rm eff}$ | $\sigma(T_{\rm eff})$ | $\log g$ | $\sigma(\log g)$ | $v_{t}$ | $\sigma(v_{t})$ | $[\mathrm{Fe/H}]$ | $\sigma([\mathrm{Fe/H}])$ | $\rho_{T_{\rm eff},\log g}$ | $\rho_{T_{\rm eff},v_{t}}$ | $\rho_{T_{\rm eff},[\mathrm{Fe/H}]}$ | $\rho_{\log g,v_{t}}$ | $\rho_{\log g,[\mathrm{Fe/H}]}$ | $\rho_{v_{t},[\mathrm{Fe/H}]}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| (K) | (K) | (dex) | (dex) | ($\mathrm{km\,s^{-1}}$) | ($\mathrm{km\,s^{-1}}$) | (dex) | (dex) |  |  |  |  |  |
2447_5952 | 6275 | 84 | 4.195 | 0.040 | 1.371 | 0.101 | -1.416 | 0.028 | 0.498 | 0.547 | 0.263 | 0.058 | 0.613 | -0.337
4998_5552 | 5562 | 57 | 4.413 | 0.058 | 1.022 | 0.154 | -1.282 | 0.040 | 0.071 | 0.695 | -0.305 | -0.268 | 0.605 | -0.579
6170_9904 | 5946 | 72 | 3.789 | 0.043 | 1.251 | 0.102 | -1.706 | 0.033 | 0.425 | 0.605 | 0.003 | 0.138 | 0.389 | -0.395
6615_9776 | 5490 | 61 | 4.481 | 0.033 | 0.752 | 0.226 | -2.175 | 0.026 | 0.440 | 0.836 | -0.540 | 0.239 | 0.195 | -0.782
6914_3008 | 5706 | 69 | 4.400 | 0.034 | 0.821 | 0.157 | -1.730 | 0.025 | 0.399 | 0.836 | -0.343 | 0.128 | 0.323 | -0.500
J1306+4154 | 6175 | 108 | 4.261 | 0.051 | 1.272 | 0.114 | -1.211 | 0.029 | 0.479 | 0.696 | 0.094 | 0.180 | 0.583 | -0.221
J1436+0929 | 6109 | 55 | 4.374 | 0.031 | 1.370 | 0.104 | -1.839 | 0.038 | 0.428 | 0.634 | 0.012 | 0.166 | 0.249 | -0.244
J1553+3909 | 5623 | 60 | 4.346 | 0.036 | 0.871 | 0.184 | -1.409 | 0.022 | 0.405 | 0.745 | -0.526 | 0.175 | 0.339 | -0.749
J1642+2041 | 5840 | 70 | 3.784 | 0.089 | 1.293 | 0.083 | -1.275 | 0.043 | -0.037 | 0.762 | -0.143 | -0.446 | 0.827 | -0.501
J1730+5309 | 5778 | 59 | 3.706 | 0.071 | 1.245 | 0.067 | -1.661 | 0.039 | -0.024 | 0.587 | -0.021 | -0.541 | 0.776 | -0.502
LP894-3 | 6035 | 83 | 4.341 | 0.038 | 1.321 | 0.142 | -1.505 | 0.034 | 0.561 | 0.856 | 0.154 | 0.387 | 0.370 | 0.021

Table 5: Linelist and line-by-line abundance

Object | species | $\lambda$ | $\chi$ | $\log gf$ | $EW$ | $\sigma(EW)$ | $A(X)$
---|---|---|---|---|---|---|---
|  | ($\mathrm{\AA}$) | (eV) | (dex) | ($\mathrm{m\AA}$) | ($\mathrm{m\AA}$) | (dex)
2447_5952 | NaI | 5682.633 | 2.102 | -0.706 | 5.1 | 0.5 | 4.675
2447_5952 | NaI | 5889.959 | 0.000 | -0.193 | 172.0 | 8.0 | 4.558
2447_5952 | NaI | 5895.910 | 0.000 | -0.575 | 151.6 | 7.0 | 4.505
2447_5952 | MgI | 4167.271 | 4.346 | -0.746 | 82.4 | 3.8 | 6.385
2447_5952 | MgI | 4730.040 | 4.340 | -2.379 | 5.3 | 0.6 | 6.472
$\sigma(A)_{T_{\rm eff}}$ | $\sigma(A)_{\log g}$ | $\sigma(A)_{v_{t}}$ | $\sigma(A)_{[\mathrm{Fe/H}]}$ | $\sigma(A)_{EW}$ | $s_{X}$ | weight |
(dex) | (dex) | (dex) | (dex) | (dex) | (dex) |  |
0.028 | 0.000 | 0.000 | 0.001 | 0.049 | 0.000 | 285.037 |
0.084 | -0.017 | -0.015 | 0.003 | 0.098 | 0.000 | 22.804 |
0.084 | -0.014 | -0.020 | 0.003 | 0.103 | 0.000 | 22.571 |
0.041 | -0.009 | -0.007 | 0.001 | 0.061 | 0.000 | 124.161 |
0.030 | 0.000 | 0.000 | 0.001 | 0.055 | 0.000 | 170.647 |
777The full table is available online at CDS and only a portion of the table
is shown here.

###  3.1 Abundance analysis

Figure 3: [Na/Fe] and [Mg/Fe] abundance ratios of the Helmi streams’ stars
(red symbols). Also plotted are halo stars from NS10 and R17, with Gaia-
Enceladus stars shown with filled symbols. G112-43 and G112-44 from NS10 are
shown with red simbols since their kinematics are consistent with the Helmi
streams. We also show kinematically selected Sequoia stars from Paper I.

We provide here a brief description of our analysis and readers may refer to
Paper I for a full and detailed explanation. The abundance measurements were
based on equivalent width measurements, where we assumed a Voigt profile for
the line shape. For Si, Mn, Zn, and Y, we applied spectral synthesis.
Hyperfine structure splitting was taken into consideration for Na, Mn, and Ba.
Abundances were estimated based on the assumption of local thermodynamic
equilibrium (LTE) using MOOG (Sneden, 1973) and based on one-dimensional plane
parallel MARCS atmosphere models (Gustafsson et al., 2008). We applied non-LTE
corrections of Lind et al. (2011) to the obtained Na abundance through the
INSPECT database888 Data obtained from the INSPECT database, version 1.0
(www.inspect-stars.net).

The effective temperature ($T_{\rm eff}$) was determined by minimizing the
correlation between the excitation potentials and abundances derived from
individual neutral iron lines. The microturbulent velocity ($v_{t}$) was
determined in a similar manner now by minimizing the correlation between
reduced equivalent widths and abundances. The surface gravity ($\log g$) was
determined from temperature, luminosity, and stellar mass, where the
luminosity was obtained by applying a bolometric correction of Casagrande &
VandenBerg (2018) to the absolute magnitude in the Gaia $G$ band and mass was
derived from isochrone fitting in the color magnitude diagram. We took
correlated uncertainties into consideration for the estimates of
uncertainties. The derived parameters are presented in Table 4.

The elemental abundances are the weighted average of the abundances derived
from the individual lines. The weights were determined following the
prescription by Ji et al. (2020). The weights on individual lines as well as
line-by-line abundances and their sensitivities to stellar parameters are
summarised in Table 5. When considering [X/Fe], we used the Fe abundance from
the same ionization stage as the species X.

All the analysis was conducted relative to HD59392, for which we adopted
$T_{\rm eff}=6012\,\mathrm{K}$, $\log g=3.954$,
$v_{t}=1.4\,\mathrm{km\,s^{-1}}$, and $[\mathrm{Fe/H}]_{\rm sp}=-1.6$ (NS10;
Paper I). The elemental abundance of this star was taken from NS10 and Nissen
& Schuster (2011). In Paper I, we validated our results using a few standard
stars and made sure that our abundances were on the same scale as NS10, Nissen
& Schuster (2011) and R17. We note that one of the standard stars was G112-43
in the Helmi streams, for which Nissen & Schuster (2011) and Nissen et al.
(2021) reported high Zn abundance. Since we did not find any systematic offset
in the derived abundance for this star when comparing with NS10 and Nissen &
Schuster (2011), our analysis also confirmed its high Zn abundance.

In case no lines were detected for a species, we provided 1-$\sigma$ and
3-$\sigma$ upper limits on the abundance, which were estimated from the
expected uncertainty on equivalent widths from the equation in Cayrel (1988).
We used the Si I line at $6237\,\mathrm{\AA}$ and the Y II line at
$4884\,\mathrm{\AA}$ to place upper limits. The derived abundances and upper
limits are presented in Table LABEL:tab:abundance.

As an additional test, we investigated the effect of slightly different
atmospheric structures due to different $\alpha$-element abundances in model
atmospheres since the sample includes stars showing different [$\alpha$/Fe]
compared to typical halo stars. We used two MARCS model atmospheres with
$T_{\rm eff}=6000\,\mathrm{K}$, $\log g=4.0$,
$v_{t}=1.5\,\mathrm{km\,s^{-1}}$, and $[\mathrm{Fe/H}]=-1.5$ differing in
$[\mathrm{\alpha/Fe}]$; one of which has the “standard composition” at this
metallicity with $[\mathrm{\alpha/Fe}]=+0.4$ and the other has a solar
$[\mathrm{\alpha/Fe}]$. Assuming the chemical composition of HD59392, we first
computed equivalent widths of absorption lines using the standard
$\alpha$-enhanced model atmosphere, and then derived abundances with the other
model from the computed equivalent widths. The maximum difference between the
assumed and derived abundances was found in strong lines to be $\sim
0.02\,\mathrm{dex}$. Since the variation in $[\mathrm{\alpha/Fe}]$ among our
sample is $\sim 0.2$ dex and since every line is affected in the same
direction, the effect of different $[\mathrm{\alpha/Fe}]$ in model atmospheres
is at most $0.01\,\mathrm{dex}$ when discussing [X/H] and even smaller when
discussing [X/Fe].

###  3.2 Results

Figure 4: $\alpha$-element abundances of the Helmi streams’ stars. The symbols
follow those of Figure 3. When a star does not have any detectable lines for
an element, we indicate the abundances corresponding to the 1-$\sigma$ and
3-$\sigma$ upper limits on the equivalent widths, which are, respectively,
shown with the location of the symbols and the upper end of the error bars.
Figure 5: Abundances of Cr, Mn, Ni and Zn. Symbols follow Figure 3.. Figure 6:
Abundances of Y and Ba. Symbols follow Figure 3.

Figure 3 shows [Na/Fe] and [Mg/Fe] ratios of the Helmi streams stars together
with the halo stars from NS10 and R17, and the Sequoia stars from Paper I, all
of which are on the same abundance scale. We note that we indicate Gaia-
Enceladus stars in the literature with filled symbols in figures. At
$-1.8\lesssim[\mathrm{Fe/H}]\lesssim-1.6$, the Helmi streams’ stars show
similar Na and Mg abundances to Sequoia stars, having clearly lower [Na/Fe]
and [Mg/Fe] values compared to other halo stars including Gaia-Enceladus
stars. The Helmi streams’ stars with $[\mathrm{Fe/H}]\gtrsim-1.5$ also have
lower values in these abundance ratios than in situ halo stars. Although they
are less separated from Gaia-Enceladus stars at this high metallicity, they
still tend to show lower [Mg/Fe] values compared to Gaia-Enceladus stars. The
most metal-poor star 6615_9776 also seems to show lower abundance of Na and Mg
compared to other halo stars. We, however, note that this star is in the
metallicity range, $[\mathrm{Fe/H}]<-2.1$, where R17 measured abundance of
stars using a standard star that is different from what they used for stars
with $[\mathrm{Fe/H}]>-2.1$, and hence there is no guarantee that the
abundance of 6615_9976 and R17 stars with similar metallicity are on the same
scale. Readers may refer to Paper I for the effect of different standard stars
(see the results of abundance comparison for HIP28104 in Section 3 of Paper
I).

Figure 4 shows Si, Ca and Ti abundances, which are also known to be different
between stars with different origins (NS10; Paper I). The behavior of the
Helmi streams’ stars in Ca and Ti is similar to that in Na and Mg; the stars
show clearly lower abundance at low metallicity than most halo stars while
they become more indistinguishable from Gaia-Enceladus stars at higher
metallicity. For Si, the picture is unclear because the Si lines become too
weak at $[\mathrm{Fe/H}]\lesssim-1.5$.

The observed trends in the $\alpha$-elements confirm the results of Limberg et
al. (2021) and Aguado et al. (2021) in the sense that the Helmi streams’ stars
have lower $\alpha$-element abundance compared to general Milky Way stars. We
here added another important result; the $\alpha$-element abundances of the
Helmi streams’ stars are even lower than another accreted population, the
Gaia-Enceladus stars, especially at $[\mathrm{Fe/H}]\lesssim-1.5$. Moreover,
contrary to the conclusion of Limberg et al. (2021) and Aguado et al. (2021)
that the $[\mathrm{\alpha/Fe}]$ ratios decrease with metallicity, the
evolution in $\alpha$-elements (Mg, Ca, and Ti) with metallicity observed in
the present study is rather flat from $[\mathrm{Fe/H}]\sim-2.1$ to
$[\mathrm{Fe/H}]\sim-1.2$. The low $\alpha$ abundance of the Helmi streams’
stars is also in contrast with Roederer et al. (2010), who concluded that the
Helmi streams’ stars have similar abundance as other halo stars. We consider
that our precise and homogeneous abundance comparisons enables us to detect
the abundance difference between the Helmi streams’ stars and Gaia-Enceladus
stars, and to precisely depict the chemical evolutionary track followed by of
the Helmi streams. We will further discuss the importance of the homogeneous
abundance analysis in Section 4.2.

The abundances of elements near iron-peak, namely Cr, Mn, Ni, and Zn, are
shown in Figure 5. It has been reported that Cr, Ni and Zn abundances are
different between accreted stars and in situ stars (Nissen & Schuster, 2011).
We therefore expect that these abundances be different among the various
kinematic substructures with different $\alpha$-element abundances. We,
however, do not see clear abundance differences between the Helmi streams’
stars and the other halo stars in Cr and Ni at our measurement uncertainty. On
the other hand, Zn seems to be depleted in the Helmi streams at low
metallicity while there is a large variation at high metallicity. The Helmi
streams’ stars tend to show low value of [Mn/Fe] at
$[\mathrm{Fe/H}]\lesssim-1.5$ while they seem to be more or less consistent
with typical Gaia-Enceladus stars at higher metallicity.

The high Zn abundance of G112-43 and G112-44 was reported in Nissen & Schuster
(2011) and confirmed in Nissen et al. (2021). It is also supported for G112-43
by our analysis in Paper I as described in Section 3.1. The other Helmi
streams’ stars in the present study, on the other hand, do not share a similar
Zn enhancement. We try to interpret the Zn abundance evolution of the Helmi
streams in Section 4.1.

Abundances of neutron capture elements are shown in Figure 6. The Helmi
streams’ stars are clearly deficient in Y at low metallicity. This behavior of
Y resembles that of Sr reported by Aguado et al. (2021), which is expected
since these two elements usually show similar trend with [Fe/H]. It is hard to
conclude if Ba abundance evolution of the Helmi streams is distinct from that
of Gaia-Enceladus or other halo stars because of large scatter. However, we
remark that the Helmi streams’ stars tend to be on lower side in [Ba/Fe] at
given [Fe/H] compared to other halo stars.

As described in Section 2.2, symbols in Figures 3 to 6 are hatched according
to the subgroups discussed by Dodd et al. (2021). It is clear that both
subgroups show similar abundance peculiarities compared to other halo stars
(lower abundances of $\alpha$-elements and Y). This supports that both
subgroups can be regarded as a part of the Helmi streams. We have made similar
inspection between stars having positive and negative $v_{z}$, and reached the
same conclusion.

On the other hand, the only star with an exceptionally high value of $E_{n}$
(2447_5952) has slightly different elemental abundance compared to the other
Helmi streams stars. This is especially prominent in Ca, Ti and neutron-
capture elements (Figures 4 and 6). This might indicate that the Helmi streams
do not extend toward high $E_{n}$, which is consistent with recent
identification of Helmi streams’ members based on clustering analysis of halo
stars (Lövdal et al., 2022; Ruiz-Lara et al., 2022).

##  4 Discussion

###  4.1 Properties of the Helmi streams’ progenitor

The chemical properties of the Helmi streams can be summarised as follows. The
abundances of elements that are usually produced by explosions of massive
stars (e.g., Na and $\alpha$-elements) are generally lower than the majority
of the other halo stars, including Gaia-Enceladus stars. While the difference
to Gaia-Enceladus stars becomes less clear at $[\mathrm{Fe/H}]\gtrsim-1.5$,
the Helmi streams’ stars still tend to be on lower side of the distribution in
[Mg/Fe], [Ca/Fe], and [Ti/Fe]. Zn and Y also show similar trends as these
elements, suggesting they originated in massive stars. This chemical abundance
pattern is similar to that seen for Sequoia stars (Paper I).

The low abundance of $\alpha$-elements and Na are usually attributed to
chemical enrichments from type Ia supernovae (SNe Ia). Interestingly, SNe Ia
seem to have started to operate in the Helmi streams’ progenitor already at
very low metallicity ($[\mathrm{Fe/H}]<-2.2$) since even the lowest
metallicity star in the sample (6615_9776) has low abundances of
$\alpha$-elements and Na. Assuming that every system has the same abundance
ratios among Na, $\alpha$-elements, and Fe before the onset of SNe Ia, we can
tentatively provide an upper limit for the “knee” metallicity as
$[\mathrm{Fe/H}]_{\rm knee}\lesssim-2.4$ from the $\sim 0.2\,\mathrm{dex}$
lower [Mg/Fe] of 6615_9776 than the halo stars from R17.

Another interesting feature in $\alpha$-elements is that there is no apparent
decreasing trend in $[\mathrm{\alpha/Fe}]$ with metallicity. Moreover, a few
stars at $[\mathrm{Fe/H}]\sim-1.2$ seem to have higher $[\mathrm{\alpha/Fe}]$
than the stars at $[\mathrm{Fe/H}]\sim-1.8$, although it remains to be seen if
they are actually the Helmi streams’ members rather than contaminants from
heated thick disk or other accreted populations. If this almost flat or
slightly increasing trend is confirmed with a larger sample of homogeneous
abundances, it would resemble the observed trend in the Large Magellanic Could
(LMC) by Nidever et al. (2020). These authors suggested that such a chemical
evolution trend is due to low star formation efficiency at early phase and
bursty star formation at later stage. Although the progenitor of the Helmi
streams might have a similar star formation history, the old age of the
majority of the Helmi streams’ stars (Koppelman et al., 2019; Ruiz-Lara et
al., 2022) suggest that the transition from quiescent to bursty star formation
occurred in a shorter timescale than that in LMC. A future chemodynamical
study with precise stellar age as well as chemical evolution modelling would
be necessary to confirm this scenario.

The Y abundance of the Helmi streams’ stars is particularly low at
$[\mathrm{Fe/H}]\lesssim-1.8$. Together with the results of Aguado et al.
(2021) on Sr abundance, this suggests that the Helmi streams stars have very
low abundance of light neutron capture elements. Such a low light neutron
capture element abundance is not common among the Milky Way stars but is seen
in low-mass dwarf galaxies such as Draco and Ursa Minor dwarf spheroids, and
most of ultra faint dwarf galaxies (e.g., Frebel et al., 2015). One possible
explanation is that, as a result of low stellar mass of the galaxy, the
progenitor did not experience rare $r$-process nucleosynthesis events, such as
neutron star mergers, electron capture supernovae, and magneto-rotational
supernovae, which might be producing most of light neutron capture elements in
the early Milky Way (e.g., Kobayashi et al., 2020). In this case, a small
amount of light neutron-capture elements could be produced by rapidly rotating
massive stars (Hirai et al., 2019; Tarumi et al., 2021). Another explanation
is that the progenitor dwarf galaxy had a small number of rotating massive
stars, which can also be an important source of neutron-capture elements at
low metallicity (Prantzos et al., 2018). The small number of rotating massive
stars might be a result of top-light initial mass function in dwarf galaxies
(Weidner & Kroupa, 2005), or different distribution of initial rotation
velocity of stars. The observational indication by Gull et al. (2021) that
metal-poor stars of the Helmi streams show $r$-process abundance pattern in
neutron capture elements heavier than Ba might favor the second possibility.
However, it is necessary to investigate the abundance pattern of light
neutron-capture elements as well to understand the cause of the low light
neutron-capture element abundance of the Helmi streams. A larger sample of
low-metallicity Helmi streams’ stars with neutron-capture element abundances
would also be welcomed as they would enable us to constrain the property of
the nucleosynthesis processes, such as their event rates, by studying how
neutron-capture elements were enriched as a function of metallicity (e.g.,
Tsujimoto et al., 2017).

The [Zn/Fe] values observed at high metallicity are also noteworthy. While the
binary pair G112-43 and G112-44 have high Zn abundance (Nissen et al., 2021),
other Helmi streams’ stars do not share such high abundance, resulting in a
large star-to-star variation in [Zn/Fe]. A significant dispersion in [Zn/Fe]
at high metallicity is also reported in the Sculptor dwarf spheroidal galaxy
(Skúladóttir et al., 2017). While Hirai et al. (2018) provided a theoretical
calculation of Zn enrichments in dwarf galaxies assuming electron capture
supernovae as one of the sources of Zn, they found it difficult to produce a
large scatter at high metallicity. Nissen et al. (2021) suggested pure helium
detonation type Ia supernovae could be a promising nucleosynthesis event
producing large amount of Zn. However, it remains to be seen whether chemical
evolution models including this type of SNe Ia are able to explain the Zn
abundance variation. A larger sample of stars with precise multi-elemental
abundances would be helpful to establish, e.g., if the binary pair is a
chemical outlier or a tail of [Zn/Fe] distribution in the system and if the Zn
abundance correlates with abundances of other elements.

###  4.2 Helmi streams in the literature

Figure 7: Chemical abundance of the Helmi streams’ stars from literature,
namely Roederer et al. (R10; 2010), Aguado et al. (A21; 2021), and Gull et al.
(G21; 2021). We also include Helmi streams’ stars in the present study and in
NS10. The background grey points are from the SAGA database (Suda et al.,
2008, 2011; Yamada et al., 2013). The UVES spectra of three stars from Aguado
et al. (2021) were reanalysed and abundances from the reanalysis are included
in the figure. Abundances from our reanalysis and those from Aguado et al.
(2021) are connected with a solid line for each star.

In this section, we highlight the importance of homogeneous chemical abundance
through a comparison of our results with those presented in the literature, by
Roederer et al. (2010), by Limberg et al. (2021), by Gull et al. (2021), and
by Aguado et al. (2021)999We consider five stars observed with high-resolution
spectrographs (HORuS and UVES) for the Aguado et al. (2021) sample.. All but
Limberg et al. (2021) conducted new observations of the Helmi streams’ stars,
while Limberg et al. (2021) used the data from GALAH DR3. All, including
Limberg et al. (2021), compared the chemical abundance of the Helmi streams’
stars with a literature compilation without homogenizing the chemical
abundances.

Figure 7 compares the abundances of the Helmi streams’ stars from those
literature studies and from our present work with those of stars in the Milky
Way from various literature sources (SAGA database; Suda et al., 2008, 2011;
Yamada et al., 2013). To confirm the membership to the Helmi streams, we
recomputed angular momenta of the stars in Roederer et al. (2010), Gull et al.
(2021), and Aguado et al. (2021) in a consistent way and re-selected the Helmi
streams’ members. As there are no report of radial velocity in Aguado et al.
(2021), we adopt radial velocity measurements by SDSS or LAMOST for their
stars. For the stars in common between Roederer et al. (2010) and Gull et al.
(2021), we prioritize the measurements by the latter study over the former. We
also include in the figure abundances which we obtained for three stars
studied by Aguado et al. (2021) by reanalysing their UVES spectra. For this
reanalysis, we redetermined stellar parameters following exactly the same
procedure as described in Paper I and estimated the abundances using these
parameters.

We first emphasize the importance of homogeneity in abundance by using the
three stars from Aguado et al. (2021) that we reanalyzed. As can be seen in
Figure 7, stars from Aguado et al. (2021) seem to have higher
$\alpha$-elements abundances compared to the Helmi streams’ stars in the
present study if we adopt the reported abundances. On the other hand, our
reanalysis shows that the three stars actually have similar abundance as the
sample in the present study, although the uncertainties are larger due to
different spectral quality. To investigate the effect of different stellar
parameters, we fixed $T_{\rm eff}$ and $\log g$ to the values adopted by
Aguado et al. (2021), determined $v_{t}$ and metallicity, and then measured
abundances, instead of redetermining the four stellar parameters. The results
do not change significantly compared to what we obtained from the full
reanalysis. The source of difference thus could be due to different atomic
data, line selection, and/or spectral synthesis software.

The different abundance from Aguado et al. (2021) and our reanalysis is not
too surprising since it has been known that systematic uncertainty can be
significant when abundances from different studies are compared without
homogenization. For example, G112-43 is a well studied star and its abundance
is available in a number of literature (Ryan et al., 2001; Charbonnel &
Primas, 2005; Zhang et al., 2009; Ishigaki et al., 2010; Nissen & Schuster,
2010, 2011; Ishigaki et al., 2012, 2013; Yan et al., 2016)101010The data are
collected through the SAGA database.. The reported metallicity
($[\mathrm{Fe/H}]_{\mathrm{II}}$) and [Mg/Fe] of this star respectively range
from $-1.51$ (Ishigaki et al., 2010) to $-1.14$ (Zhang et al., 2009), and from
0.16 (Zhang et al., 2009) to 0.37 (Ishigaki et al., 2010). These ranges are
significantly larger than the typical reported uncertainty. We note that we
derive $[\mathrm{Fe/H}]_{\mathrm{II}}=-1.256$ and $[\mathrm{Mg/Fe}]=0.179$ in
Paper I and NS10 (adopted for figures) derive
$[\mathrm{Fe/H}]_{\mathrm{II}}=-1.25$ and $[\mathrm{Mg/Fe}]=0.21$.

Figure 7 demonstrates that our sample has the smallest uncertainty, which was
made possible thanks to the high quality of the spectra. However, even with
the high precision, the chemical distinctness of the Helmi streams’ stars is
much less clear than what we see in Figures 3–6. This is because the
comparison sample from the SAGA database is not necessarily on the same
abundance scale as ours, because its distribution is broadened due to
systematic uncertainty, and because it can contain poor quality of data.

In conclusion, systematic uncertainties have hampered a clear chemical
characterization of the Helmi streams’ stars in previous studies. Thanks to
our approach of abundance analysis and homogenization of abundances from
literature (see Paper I, ), we are able to detect clear but small abundance
difference between the Helmi streams’ stars and other halo stars that include
Gaia-Enceladus stars.

##  5 Summary

Through a differential abundance analysis, we showed that the Helmi streams’
stars clearly depict lower [X/Fe] in Na, Mg, Ca, Ti, Zn, and Y, compared to
the other halo stars, including stars accreted from Gaia-Enceladus. While the
distinction from the Gaia-Enceladus stars is clearer at
$[\mathrm{Fe/H}]\lesssim-1.5$, a part of the Helmi streams’ stars start to
overlap with Gaia-Enceladus stars in elemental abundance ratios at higher
metallicity.

This is the first time that we see the chemical distinctness of the Helmi
streams’ stars over a wide metallicity range. This was made possible thanks to
the precise abundances derived in this study and to the homogenization of
abundances across NS10, R17 and the present study. As in Paper I, the present
investigation on the Helmi streams highlights the importance of homogenized
abundances for chemical identification of individual building blocks of the
Milky Way.

The observed low $\alpha$-element abundances of the Helmi streams are likely
due to the chemical enrichments from SNe Ia at low metallicity. There seems to
be a contribution of SNe Ia to the chemical evolution of the progenitor
already at very low metallicity ($[\mathrm{Fe/H}]\lesssim-2.4$), indicating
that the star formation proceeded with a rather low efficiency in early times.
Star formation at later times might have been bursty since [$\alpha$/Fe]
remains almost constant or slightly increases toward higher metallicity
according to (Nidever et al., 2020); such a feature may require a late star
burst as seen in the LMC.

The origin of the extremely low Y abundances at low metallicity remains
unclear. Our results, as well as those of Aguado et al. (2021) on Sr, strongly
suggest that the Helmi streams have very low abundance of light neutron-
capture element abundance at low metallicity. It would be interesting to
obtain detailed abundance patterns over light neutron-capture elements in
order to constrain the nucleosynthesis processes that produce this feature.

Further observations would also be able to understand the Zn abundance
variation at high metallicity. The binary pair G112-43 and G112-44 are the
only stars with clear Zn enhancements, and all the other Helmi stream’s stars
at $[\mathrm{Fe/H}]\gtrsim-1.5$ have Zn abundances comparable to or lower than
Gaia-Enceladus stars. It would be interesting to investigate the detailed
[Zn/Fe] distribution with a lager number of stars.

###### Acknowledgements.

We thank David Aguado for sharing their UVES spectra with us. This research
has been supported by a Spinoza Grant from the Dutch Research Council (NWO).
WA and MNI were supported by JSPS KAKENHI Grant Number 21H04499. MNI is
supported by JSPS KAKENHI Grant Number 20H05855. This research is based in
part on data collected at Subaru Telescope, which is operated by the National
Astronomical Observatory of Japan. We are honored and grateful for the
opportunity of observing the Universe from Maunakea, which has the cultural,
historical and natural significance in Hawaii. Part of the data were retrieved
from the JVO portal (http://jvo.nao.ac.jp/portal/) operated by ADC/NAOJ.

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##  Appendix A Additional table

1]

Table 6: Abundances of the Helmi stream stars | 2447_5952 |  | 4998_5552 |  | 6170_9904
---|---|---|---|---|---
| $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  | $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  | $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$
FeI | 103 | -1.399 | 0.032 | … | … |  | 88 | -1.278 | 0.031 | … | … |  | 93 | -1.772 | 0.034 | … | …
FeII | 12 | -1.420 | 0.025 | … | … |  | 13 | -1.268 | 0.042 | … | … |  | 11 | -1.733 | 0.025 | … | …
NaI | 3 | -1.585 | 0.055 | -0.186 | 0.054 |  | 2 | -1.430 | 0.041 | -0.152 | 0.039 |  | 2 | -2.222 | 0.093 | -0.450 | 0.089
MgI | 7 | -1.202 | 0.041 | 0.196 | 0.042 |  | 4 | -1.119 | 0.040 | 0.159 | 0.038 |  | 5 | -1.706 | 0.052 | 0.066 | 0.050
SiI | 2 | -1.077 | 0.051 | 0.322 | 0.054 |  | 6 | -0.973 | 0.046 | 0.305 | 0.052 |  | 0 |  $<$-1.727(-1.517) |  $<$0.045(0.255)
CaI | 19 | -1.010 | 0.041 | 0.389 | 0.038 |  | 18 | -1.051 | 0.044 | 0.227 | 0.040 |  | 19 | -1.555 | 0.038 | 0.217 | 0.035
TiI | 11 | -1.063 | 0.067 | 0.336 | 0.058 |  | 15 | -1.184 | 0.053 | 0.094 | 0.044 |  | 8 | -1.528 | 0.066 | 0.244 | 0.056
TiII | 13 | -1.102 | 0.041 | 0.319 | 0.033 |  | 8 | -1.064 | 0.043 | 0.205 | 0.039 |  | 12 | -1.604 | 0.035 | 0.129 | 0.033
CrI | 4 | -1.334 | 0.087 | 0.065 | 0.080 |  | 2 | -1.193 | 0.070 | 0.085 | 0.061 |  | 3 | -1.796 | 0.106 | -0.023 | 0.100
MnI | 5 | -1.686 | 0.060 | -0.287 | 0.053 |  | 7 | -1.712 | 0.068 | -0.434 | 0.062 |  | 5 | -2.212 | 0.071 | -0.440 | 0.063
NiI | 14 | -1.441 | 0.045 | -0.042 | 0.040 |  | 22 | -1.327 | 0.031 | -0.049 | 0.031 |  | 8 | -1.810 | 0.047 | -0.037 | 0.042
ZnI | 2 | -1.412 | 0.047 | -0.014 | 0.043 |  | 2 | -1.242 | 0.037 | 0.036 | 0.045 |  | 2 | -1.790 | 0.052 | -0.018 | 0.050
YII | 2 | -1.516 | 0.069 | -0.096 | 0.064 |  | 2 | -1.556 | 0.047 | -0.288 | 0.049 |  | 1 | -2.174 | 0.069 | -0.441 | 0.068
BaII | 4 | -1.610 | 0.058 | -0.189 | 0.048 |  | 4 | -1.627 | 0.049 | -0.358 | 0.048 |  | 4 | -2.351 | 0.055 | -0.618 | 0.053
| 6615_9776 |  | 6914_3008 |  | J1306+4154
| $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  | $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  | $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$
FeI | 92 | -2.153 | 0.027 | … | … |  | 95 | -1.810 | 0.032 | … | … |  | 102 | -1.141 | 0.037 | … | …
FeII | 5 | -2.183 | 0.030 | … | … |  | 11 | -1.737 | 0.027 | … | … |  | 13 | -1.214 | 0.030 | … | …
NaI | 2 | -2.654 | 0.083 | -0.501 | 0.078 |  | 3 | -2.208 | 0.116 | -0.398 | 0.114 |  | 3 | -1.424 | 0.059 | -0.283 | 0.059
MgI | 5 | -2.048 | 0.046 | 0.106 | 0.044 |  | 7 | -1.750 | 0.042 | 0.061 | 0.042 |  | 7 | -1.140 | 0.059 | 0.002 | 0.059
SiI | 0 |  $<$-1.894(-1.653) |  $<$0.258(0.499) |  | 1 | -1.334 | 0.079 | 0.476 | 0.082 |  | 3 | -1.007 | 0.087 | 0.134 | 0.090
CaI | 18 | -1.988 | 0.034 | 0.166 | 0.033 |  | 19 | -1.626 | 0.039 | 0.184 | 0.036 |  | 17 | -1.020 | 0.051 | 0.122 | 0.049
TiI | 8 | -2.117 | 0.064 | 0.037 | 0.056 |  | 10 | -1.769 | 0.069 | 0.041 | 0.059 |  | 11 | -1.027 | 0.082 | 0.115 | 0.075
TiII | 10 | -2.107 | 0.034 | 0.076 | 0.042 |  | 9 | -1.684 | 0.029 | 0.052 | 0.036 |  | 11 | -1.129 | 0.042 | 0.085 | 0.041
CrI | 2 | -2.308 | 0.090 | -0.155 | 0.085 |  | 3 | -2.018 | 0.073 | -0.208 | 0.064 |  | 5 | -1.020 | 0.089 | 0.122 | 0.080
MnI | 2 | -2.623 | 0.056 | -0.469 | 0.050 |  | 4 | -2.313 | 0.062 | -0.502 | 0.054 |  | 6 | -1.556 | 0.083 | -0.414 | 0.075
NiI | 5 | -2.298 | 0.081 | -0.145 | 0.079 |  | 12 | -1.900 | 0.042 | -0.090 | 0.039 |  | 13 | -1.231 | 0.060 | -0.090 | 0.056
ZnI | 2 | -2.203 | 0.052 | -0.050 | 0.055 |  | 2 | -1.889 | 0.040 | -0.079 | 0.044 |  | 2 | -1.263 | 0.057 | -0.122 | 0.054
YII | 0 |  $<$-2.729(-2.433) |  $<$-0.546(-0.250) |  | 1 | -2.294 | 0.095 | -0.557 | 0.098 |  | 2 | -1.462 | 0.074 | -0.248 | 0.074
BaII | 2 | -2.887 | 0.062 | -0.704 | 0.068 |  | 4 | -2.228 | 0.043 | -0.491 | 0.049 |  | 4 | -1.362 | 0.069 | -0.148 | 0.067
| J1436+0929 |  | J1553+3909 |  | J1642+2041
| $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  | $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  | $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$
FeI | 88 | -1.828 | 0.026 | … | … |  | 74 | -1.462 | 0.026 | … | … |  | 103 | -1.276 | 0.029 | … | …
FeII | 8 | -1.822 | 0.038 | … | … |  | 9 | -1.410 | 0.027 | … | … |  | 15 | -1.249 | 0.044 | … | …
NaI | 2 | -2.296 | 0.082 | -0.468 | 0.077 |  | 4 | -1.700 | 0.068 | -0.238 | 0.066 |  | 3 | -1.495 | 0.049 | -0.219 | 0.048
MgI | 6 | -1.742 | 0.047 | 0.086 | 0.044 |  | 4 | -1.298 | 0.041 | 0.164 | 0.039 |  | 6 | -1.216 | 0.045 | 0.060 | 0.044
SiI | 0 |  $<$-1.355(-1.203) |  $<$0.472(0.624) |  | 2 | -1.128 | 0.051 | 0.334 | 0.054 |  | 3 | -1.242 | 0.056 | 0.034 | 0.057
CaI | 13 | -1.691 | 0.034 | 0.137 | 0.031 |  | 12 | -1.241 | 0.033 | 0.220 | 0.030 |  | 16 | -1.080 | 0.038 | 0.196 | 0.036
TiI | 7 | -1.609 | 0.050 | 0.219 | 0.041 |  | 14 | -1.361 | 0.054 | 0.101 | 0.046 |  | 10 | -1.262 | 0.064 | 0.014 | 0.056
TiII | 12 | -1.704 | 0.039 | 0.118 | 0.049 |  | 2 | -1.175 | 0.039 | 0.235 | 0.041 |  | 10 | -1.202 | 0.051 | 0.047 | 0.041
CrI | 2 | -1.915 | 0.084 | -0.087 | 0.079 |  | 3 | -1.535 | 0.065 | -0.073 | 0.058 |  | 2 | -1.301 | 0.142 | -0.025 | 0.139
MnI | 3 | -2.221 | 0.049 | -0.393 | 0.043 |  | 4 | -1.856 | 0.056 | -0.394 | 0.051 |  | 5 | -1.698 | 0.077 | -0.422 | 0.072
NiI | 5 | -1.790 | 0.053 | 0.038 | 0.051 |  | 18 | -1.509 | 0.038 | -0.047 | 0.035 |  | 12 | -1.395 | 0.041 | -0.119 | 0.038
ZnI | 2 | -1.912 | 0.045 | -0.084 | 0.044 |  | 2 | -1.333 | 0.041 | 0.128 | 0.046 |  | 2 | -1.436 | 0.042 | -0.160 | 0.042
YII | 0 |  $<$-2.531(-2.238) |  $<$-0.709(-0.416) |  | 2 | -1.711 | 0.042 | -0.301 | 0.046 |  | 2 | -1.601 | 0.056 | -0.352 | 0.051
BaII | 3 | -2.332 | 0.052 | -0.510 | 0.061 |  | 4 | -1.783 | 0.043 | -0.373 | 0.044 |  | 4 | -1.609 | 0.057 | -0.360 | 0.050
| J1730+5309 |  | LP894-3 |  |  |  |  |  |
| $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  | $N$ | [X/H] | $\sigma$ | [X/Fe] | $\sigma$ |  |  |  |  |  |
FeI | 84 | -1.631 | 0.029 | … | … |  | 98 | -1.422 | 0.029 | … | … |  |  |  |  |  |
FeII | 12 | -1.643 | 0.039 | … | … |  | 14 | -1.488 | 0.033 | … | … |  |  |  |  |  |
NaI | 3 | -2.029 | 0.062 | -0.398 | 0.059 |  | 3 | -1.807 | 0.043 | -0.385 | 0.043 |  |  |  |  |  |
MgI | 6 | -1.508 | 0.057 | 0.123 | 0.054 |  | 6 | -1.329 | 0.055 | 0.093 | 0.052 |  |  |  |  |  |
SiI | 0 |  $<$-1.497(-1.352) |  $<$0.133(0.278) |  | 4 | -1.243 | 0.062 | 0.179 | 0.063 |  |  |  |  |  |
CaI | 16 | -1.429 | 0.037 | 0.202 | 0.034 |  | 18 | -1.257 | 0.039 | 0.165 | 0.036 |  |  |  |  |  |
TiI | 8 | -1.523 | 0.055 | 0.108 | 0.045 |  | 10 | -1.348 | 0.066 | 0.074 | 0.058 |  |  |  |  |  |
TiII | 9 | -1.546 | 0.049 | 0.097 | 0.042 |  | 7 | -1.376 | 0.037 | 0.112 | 0.042 |  |  |  |  |  |
CrI | 2 | -1.689 | 0.121 | -0.058 | 0.116 |  | 4 | -1.482 | 0.066 | -0.060 | 0.058 |  |  |  |  |  |
MnI | 7 | -2.161 | 0.078 | -0.530 | 0.073 |  | 3 | -1.730 | 0.074 | -0.308 | 0.070 |  |  |  |  |  |
NiI | 9 | -1.698 | 0.047 | -0.067 | 0.045 |  | 16 | -1.522 | 0.046 | -0.100 | 0.043 |  |  |  |  |  |
ZnI | 2 | -1.825 | 0.045 | -0.194 | 0.045 |  | 2 | -1.527 | 0.040 | -0.105 | 0.039 |  |  |  |  |  |
YII | 2 | -2.018 | 0.057 | -0.375 | 0.053 |  | 2 | -1.865 | 0.055 | -0.377 | 0.057 |  |  |  |  |  |
BaII | 3 | -1.914 | 0.057 | -0.271 | 0.052 |  | 4 | -1.929 | 0.054 | -0.442 | 0.055 |  |  |  |  |  |
Table 6: continued. 111111For upper limits, we provide both 1-$\sigma$ and
3-$\sigma$ upper limits. The values in parenthesis are for the 3-$\sigma$
upper limits.