# Proto-Strange Quark Star Structure

Gholam Hossein Bordbar1,2 111Corresponding author.
<EMAIL_ADDRESS>Fatemeh Sadeghi1, Fatemeh Kayanikhoo 3 and
Ahmad Poostforush1 1 Department of Physics, Shiraz University, Shiraz 71454,
Iran222Permanent address
2 Department of Physics and Astronomy, University of Waterloo, 200 University
Avenue West, Waterloo, Ontario, N2L 3G1, Canada
3 Department of Physics, University of Birjand, Birjand, Iran

###### Abstract

Abstract: In this paper, we investigate the newborn strange quark stars with
constant entropy. We also use the MIT bag model to calculate the thermodynamic
properties in two cases; the density-dependent bag constant and the fixed bag
constant ($B=90$ $MeV$). We show that the equation of state becomes stiffer by
using the density-dependent bag constant and by increasing the entropy.
Furthermore, we show that the adiabatic index of the system reaches to
$\frac{4}{3}$ at high densities. Later, we calculate the structure of a
strange quark star using the equation of state and the general relativistic
equations of hydrostatic equilibrium, the Tolman-Oppenheimer-Volkoff (TOV)
equations. We show that the gravitational mass of the star decreases by
increasing the entropy and the maximum gravitational mass is larger when we
use the density-dependent bag constant at fixed central energy density. It is
shown that the mass-radius relation for this system obeys $M$ $\propto$
$R^{3}$ for different cases of the calculations. Finally, we show that for a
given stellar mass considering the fixed bag constant, the maximum
gravitational redshift of a strange quark star occurs at larger values of
entropy.

Keywords: Strange quark matter; strange quark star; structure; mass; radius,
adiabatic index

PACS number: 14.65.-q; 26.30.-k;12.39.Ba

## 1\. Introduction

Compact stars are created at the end of the life of massive stars. Study of
compact objects has always been an important subject in physics and
astrophysics since strong conditions such as high densities and temperatures
in the core of these objects can be investigated. Study of compact objects was
started with the discovery of white dwarfs and the investigation of their
thermodynamic properties using Fermi-Dirac statistics rk1 ; rk2 ; rk2a .
Strange quark stars (SQS) are examples of compact objects, the possibility of
their formation was raised by Ivanenko and Kurdgelaidze in 1965 rk3 ; rk4 .
Witten in 1989, suggested that the base state of hadrons could be strange
matter rk5 .

When nuclear matter is at high enough density (more than $10^{15}$ $g/cm^{3}$)
in the core of the star, the nucleons overlap and it is expected that nucleons
convert into quarks under a phase transition rk5a ; rk5b ; rk5c  when the
nuclear matter changes to the quark matter. So, there are two possible cases
for these stars: a compact object which completely consists of quarks (a pure
SQS) or a neutron star with a core consisting of quark matter (a hybrid star);
Itoh calculated the maximum gravitational mass of the SQS in 1970 rk6 , he
ignored the weak interaction of quarks in his calculations. In 1976, after
presentation of the MIT bag model, Brecher and Caporaso calculated the maximum
gravitational mass of the hybrid star rk7 .

A pure SQS which is made of strange quark matter (SQM) that includes three
flavors of quarks (up, down and strange), is more stable than nuclear matter,
and its gravitational mass is proportional to the third power of radius ($M$
$\propto$ $R^{3}$). The maximum gravitational mass is almost twice the mass of
the sun and has a radius of about $10$$-$$12$ $km$ rk8 . The electric field of
an SQS is about $10^{18}$$-$$10^{19}$ $V/cm$, and it has a strong magnetic
field about $10^{15}$$-$$10^{19}$ $G$ rk9 ; rk10 ; rk11 . The objects,
RX-J185635-3754 ($M=1.7$$M_{\odot}$ and $R=11$ $km$) and 3C58 are candidates
for SQS rk12 .

As we know, in the core of massive stars nuclear processes continue in such a
way that the final element is ${}^{56}Fe$. At this point, the degenerate
pressure is not enough to keep the star from gravitational collapse, and the
star’s life comes to an end. During the collapse, it is expected that a thin
shell of the star is driven out by transition of neutrinos and electromagnetic
waves while the central compact core makes a compact object. In the stage of
collapsing, neutrinos are trapped so the collapse is adiabatic. In other
words, the newborn quark star is opaque in the emission of neutrinos with
constant entropy rk13 . Investigation of the protoneutron star matter has been
reported in 2002 rk13a . Before that Bethe et al. had calculated the equation
of state in the gravitational collapse of stars rk14 . Also, Fischer et.al.
have studied core-collapse supenovae explosions triggered by a quark-hadron
phase transition rk14a . Also, the new-born quark stars have been investigated
using NJL model by Sadin and Blaschke rk14b . The temperature evaluation of
the new-born compact stars have been studied by Wong and Chu rk14c . In
another paper Dexheimer et.al. have calculated the properties and structure of
the proto-quark stars using both NJL model and MIT bag model rk14d .

We have published different papers regarding strange quark stars where the
structural properties of the SQS at zero and finite temperatures have been
computed. These include the structure of an SQS using the MIT bag model with
the fixed and the density-dependent bag constant in the presence and the
absence of magnetic fields rk15 ; rk16 ; rk17 ; rk18 ; rk19 . We have also
computed the maximum gravitational mass and other structural properties of a
neutron star with a quark core at zero temperature and finite temperatures
rk20 ; rk21 . Also, we have studied the effect of dynamical quark mass in the
calculation of SQS structure using MIT bag model and NJL model at zero
temperature rk22 .

In this paper, we investigate the thermodynamic properties and the structure
of a proto-strange quark star. In section $II$, we calculate the thermodynamic
properties of SQM using MIT bag model with the constant entropy for different
densities. Also, we investigate the behavior of quarks at high densities. In
section $III$, we compute the maximum gravitational mass and radius of the
proto-SQS using the equation of state (EOS) and the Tolman-Oppenheimer-Volkoff
(TOV) equations. Also we calculate the gravitational redshift of the SQS in
the last section.

## 2\. Calculation of the thermodynamic properties of SQM with constant
entropy

We calculate the thermodynamic parameters of an adiabatic system such as the
chemical potential, total energy, and the equation of state (EOS) with the
variable temperature.

First, we calculate the temperature and chemical potential when entropy is
constant. Then we calculate the total energy of the system to find the EOS.
Also, we use the EOS to calculate the adiabatic index of the system.

In this paper, we consider the pure SQS containing up, down and strange
quarks. In our calculations, the masses of strange quark is considered to be
$150$$MeV/c^{2}$, while the mass of up and down quarks are ignored.

### 2.1. Temperature and chemical potential

We use the density and entropy equations of Fermions to calculate the
temperature and the chemical potential. Fermi-Dirac distribution function is,

$f_{i}(p)=\frac{1}{e^{\beta{(\epsilon_{i}(p)-\mu_{i})}}+1},$ (1)

where $\mu_{i}$ is the chemical potential of quark $i$, $\beta$ is equal to
$1/(k_{B}T)$ with $k_{B}$ as the Boltzmann constant. Also $\epsilon$ is the
energy of the single-particle,

$\epsilon=(\hbar^{2}k^{2}c^{2}+m^{2}c^{4})^{1/2}-mc^{2}.$ (2)

The number density of quark $i$ is,

$n_{i}=\frac{N_{i}}{V_{i}}=\frac{6}{(2\pi)^{3}}\int_{0}^{\infty}f(n_{i},k_{i},T)d^{3}k_{i},$
(3)

where $d^{3}k_{i}=4\pi k^{2}dk$ and $i=u,d,s$. Also the baryonic density is
defined in the following form,

$n_{B}=\frac{1}{3}(n_{u}+n_{d}+n_{s}).$ (4)

The entropy per baryon for each type of quark $i$ is given by,

$\displaystyle s_{i}={}$
$\displaystyle-\frac{3}{n_{B}\pi^{3}}\int_{0}^{\infty}k^{2}(f(n_{i},k_{i},T)\ln{f(n_{i},k_{i},T)}$
$\displaystyle+((1-f(n_{i},k_{i},T))\ln{(1-f(n_{i},k_{i},T))})dk,$

and the total entropy per baryon of SQM is,

$S_{tot}=s_{u}+s_{d}+s_{s}.$ (6)

We have shown the variation of temperature versus density for different
entropies in Fig. 1 where it can be seen that the temperature increases by
increasing entropy. When entropy is fixed, temperature increases as a function
of density.

Figure 1: Temperature versus baryonic density in different entropies.

The chemical potential of quarks versus density has been plotted in Fig. 2.
Since, the chemical potential of down and strange quarks are equal rk3 , the
plot for down quark is not shown. It can be seen that for a fixed density, the
chemical potential decreases by increasing entropy. Also, the chemical
potential is a rising function of density whenever entropy is fixed.

Figure 2: The chemical potential of up and strange quarks versus density in
different entropies. Thin curves (thick curves) have been used for up quarks
(strange quarks).

### 2.2. Energy of SQM

We use the MIT bag model to calculate the total energy of SQM. The MIT bag
model is a model describing quarks bounded in the hadrons. In this model,
quarks are considered like a gas in a bag with pressure of $B_{bag}$. The
total energy of SQM ($\epsilon_{tot}$) is the kinetic energy of quarks plus
the bag constant. We consider two cases for the bag constant: $A$$)$ fixed bag
constant equal to $90$ $MeV/fm^{3}$ and $B$$)$ and density-dependent bag
constant which is defined by a Gaussian equation using the experimental data
of CERN rk13 ; rk16 ; rk23 ,

$B_{n}=B_{\infty}+(B_{0}+B_{\infty})e^{-\beta(\frac{n}{n_{0}})^{2}},$ (7)

where $\beta$ is the numerical parameter equal to the density of normal
nuclear matter ($n_{0}=0.17$), and $B_{0}=B(n=0)$ is equal to $400$
$MeV/fm^{3}$. Also, the value of the parameter $B_{\infty}$ depends only on
the value of the parameter $B_{0}$ and is obtained by the LOCV cluster
expansion. We found that $B_{\infty}$ is $8.99$$MeV/fm^{3}$ where the energy
density of quark matter is equal to that of the nuclear matter rk19 .

The total energy of SQM using the MIT bag model is defined as follows,

$\epsilon_{tot}=\varepsilon_{tot}+B_{bag}.$ (8)

The kinetic energy density $\varepsilon_{i}$ is,

$\varepsilon_{i}=\frac{3}{\pi^{2}}\int_{0}^{\infty}\epsilon(k_{i})f(n_{i},k_{i},T)k_{i}^{2}dk_{i},$
(9)

and the total kinetic energy of SQM is,

$\varepsilon_{tot}=\varepsilon_{u}+\varepsilon_{d}+\varepsilon_{s}.$ (10)

In Fig. 3, the energy density of SQM, by considering fixed bag constant
($B=90MeV/fm^{3}$) and density-dependent bag constant in different entropies
has been plotted. It can be seen that in both cases ($B=90MeV/fm^{3}$ and
density-dependent bag constant), the energy density increases by increasing
density in the fixed entropy case. Also, it is shown that in the fixed density
case, the energy density increases by increasing entropy. This behavior is due
to the increase in temperature and chemical potential of SQM. By considering
$B=90MeV/fm^{3}$, the energy density curve is almost linear but by considering
the density-dependent bag constant, the curve has a minimum point of energy
density at the density of $0.4$$fm^{-3}$. The minimum in energy density is a
sign of a stable state. Furthermore, it can be seen that when we use the
density-dependent bag constant, the energy is larger than the case when we use
the fixed bag constant.

Figure 3: The energy density versus density of system in different entropies
using $B=90MeV/fm^{3}$ (thin curves) and density-dependent bag constant (thick
curves).

### 2.3. The equation of state

The equation of state is given by the following relation,

$P=n\frac{d\epsilon_{tot}}{dn}-\epsilon_{tot},$ (11)

where $n$ is the density and $\epsilon_{tot}$ is the total energy density. We
have calculated the EOS of the system using the Eqs. 8 and 11.

Fig. 4 shows pressure of the system versus density in different entropies
using fixed bag constant and density-dependent bag constant. In both cases,
the pressure increases as a function of density. Also, we see that the
pressure increases by increasing the entropy at each value of the given
density for the case of $B=90MeV/fm^{3}$, while for case of density-dependent
bag constant, this behavior is seen after a particular value of density. As it
is shown in Fig. 4, in the case of $B=90MeV/fm^{3}$, for densities greater
than $0.8fm^{-3}$, the equation of state is softer than the case of density-
dependent bag constant. These behaviors are due to the change of bag constant
versus the density in the case of density dependent bag constant (Eq. 7).
Also, it is clear that each pressure curve for $B=90MeV/fm^{3}$ is almost a
linear function of density while for density-dependent bag constant this is
not correct. The results for the case of density-dependent bag constant is
more similar to that we expected for the fermions.

$\begin{array}[]{cc}\includegraphics[width=130.88284pt]{peC.eps}\includegraphics[width=130.88284pt]{zoompeC.eps}\end{array}$

Figure 4: The pressure versus the energy density in different entropies using
$B=90MeV/fm^{3}$ (thin curves) and density-dependent bag constant (thick
curves).

### 2.4. Calculation of the adiabatic index using the polytropic behavior

The polytropic behavior of a system is one of the important topics in
thermodynamics governed by an equation of the form,

$P\propto n^{\gamma},$ (12)

where $P$ is pressure, $n$ is density, and $\gamma$ is the polytropic factor.
For adiabatic systems, $\gamma$ is the adiabatic constant. The adiabatic
constant is the parameter which shows the stiffness of system and is defined
as a function of density,

$\gamma=\frac{dln(P)}{dln(n)}.$ (13)

The logarithmic variations of pressure versus density has been plotted in Fig.
5 for both cases, $B=90MeV/fm^{3}$ and density-dependent bag constant with
different entropies. It is seen that in both cases, the curves are rising.
Also, at fixed density, the logarithmic variations of pressure increases by
increasing entropy, furthermore, at lower densities, the gradient of the graph
is steeper. The logarithmic variations of pressure using the density-dependent
bag constant is larger at densities higher than $0.8fm^{-3}$.

Figure 5: The logarithmic variations of pressure versus density in different
entropies using $B=90MeV/fm^{3}$ (thin curves) and density-dependent bag
constant (thick curves).

We have plotted the adiabatic index as a function of density in Fig. 6 with
different entropies for both $B=90MeV/fm^{3}$ and density-dependent bag
constant. It is clear that the adiabatic index decreases to $\frac{4}{3}$ by
increasing density in both cases of bag constants. Since by increasing the
density, interaction of quarks decrease, it is excepted that the system
behaves as a free Fermi gas at high densities so the adiabatic index reaches
$\frac{4}{3}$. It is also seen that the adiabatic index decreases by
increasing the entropy at a given density, since the equation of state becomes
stiffer as the entropy increases. This behavior occurs in both cases
considered for bag constant.

Figure 6: The adiabatic index versus density in different entropies using
$B=90MeV/fm^{3}$ (thin curves) and density-dependent bag constant (thick
curves).

## 3\. The structure of the proto-strange quark star

Quark stars are relativistic objects, so we should use general relativity to
calculate their structure. To investigate the structure of the SQS one needs
to calculate the mass and radius of the star. The mass- radius relation can be
obtained using Tolman-Oppenheimer-Volkoff (TOV) equations,

$\frac{dP}{dr}=-\frac{G\left[\varepsilon(r)+\frac{P(r)}{c^{2}}\right]\left[m(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}\right]}{r^{2}\left[1-\frac{2Gm(r)}{rc^{2}}\right]},$ (14)
$\frac{dm}{dr}=4\pi r^{2}\varepsilon(r).$ (15)

Using the equation of state obtained in the previous section and the boundary
conditions ($P(r=0)=P_{c}$, $P(r=R)=0$, $m(r=0)=0$ and $m(r=R)=M_{max}$) we
integrate the TOV equations to compute the structure of the SQS rk24 ; rk25 .
$M_{max}$ is the maximum gravitational mass of the star and the corresponding
radius (the radius of the star) is shown as $R$. The maximum gravitational
mass is important because it shows the star’s stability against becoming a
black hole.

The changes in gravitational mass versus the central energy density is shown
in Fig. 7 for both cases of bag constant in different entropies. It can be
seen that the gravitational mass increases as a function of the central energy
density until the maximum mass is reached. It is also clear that in both cases
considered for bag constat, the maximum gravitational mass decreases by
increasing the entropy at a fixed central energy density. The maximum
gravitational mass has a larger value when we have used the density-dependent
bag constant.

Figure 7: The gravitational mass versus central energy density in different
entropies using $B=90MeV/fm^{3}$ (thin curves) and density-dependent bag
constant (thick curves).

In Fig. 8, the radius of the SQS versus the central energy density has been
plotted for both cases of bag constant in different entropies. It is shown
that the radius increases by increasing the central energy density until it
reaches the maximum value of the radius. It is also shown that for each given
central energy density, the maximum value of radius decreases by increasing
the entropy. Furthermore, the radius of the SQS has a larger value when we
consider the density-dependent bag constant.

Figure 8: The radius versus central energy density in different entropies
using $B=90MeV/fm^{3}$ (thin curves) and density-dependent bag constant (thick
curves). Figure 9: The gravitational mass versus radius in different entropies
using $B=90MeV/fm^{3}$ (thin curves) and density-dependent bag constant (thick
curves).

In Fig. 9, the gravitational mass is also shown as a function of the radius
with the same conditions as of the previous figures. It can be seen that the
larger gravitational mass for any fixed radius is related to the larger
entropy for each case of the bag constant, and values of mass and radius are
larger where we have used the density-dependent bag constant. Also, in the
case of fixed entropy, the gravitational mass increases by increasing the
radius, but after reaching the maximum gravitational mass the radius
decreases. In fact according to our calculations, the star is more stable for
the case of density-dependent bag constant. Increasing radius of SQS as a
function of the gravitational mass is different from that of a neutron star.
For the SQS, our structure results show that the relation of mass-radius
follows $M$ $\propto$ $R^{3}$ in each considered case.

### 3.1. Gravitational redshift

The gravitational redshift $Z$ of stars is defined by the following relation,

$Z=(1-\frac{2M}{Rc^{2}})^{-1/2}-1.$ (16)

The gravitational redshift of SQS has been plotted in Fig. 10 with the same
conditions as those of the previous figures. It is clear that the
gravitational redshift increases by increasing entropy. Also, the
gravitational redshift has a larger value by considering the density-dependent
bag constant. The maximum value of the gravitational redshift for SQS,
$Z=0.48$, is related to a system with the density-dependent bag constant with
the entropy $S=2.5k_{B}$, and the minimum value, $Z=0.4$, is related to the
fixed bag constant with the entropy $S=1.5k_{B}$.

Figure 10: The gravitational redshift versus the gravitational mass in
different entropies using $B=90MeV/fm^{3}$ (thin curves) and density-dependent
bag constant (thick curves).

## 4\. Conclusions

In this paper, we have investigated thermodynamic properties and structure of
proto-strange quark star (SQS) in which the neutrinos are trapped, and
therefore the entropy is constant. In section $II$, we calculated the
temperature and chemical potential of strange quark matter (SQM) with fixed
entropy. Next, we calculated the free energy of SQM in different entropy
cases. We have used the MIT bag model with two cases of bag constants (fixed
bag constant $B=90$ $MeV/fm^{3}$ and density-dependent bag constant). Using
the free energy we have calculated the equation of state (EOS) of the SQM. We
showed that the EOS of the system becomes stiffer by considering the density-
dependent bag constant. Also we showd that the EOS is softer in the lower
entropies. Furthermore, we investigated the adiabatic index of the SQM where
we have shown that the adiabatic index has reached $\frac{4}{3}$ at high
densities for both cases of the bag constants. Actually the SQM behaves like a
free fermi gas at higher densities because the bag constant decreases in
comparison with the energy scale. In section $III$, we used the EOS and TOV
equations to calculate the mass and radius of the SQS. we showed that the
maximum gravitational mass and the radius of SQS decrease by increasing
entropy. In general, we have found that the larger values of entropy lead to
the stiffer equation of state, the smaller adiabatic index, and the smaller
values of the maximum gravitational mass and radius for the SQS at a given
central energy density. Also, it has been shown that the maximum gravitational
mass and radius of the SQS are larger when we use the density-dependent bag
constant. Furthermore, we plotted the gravitational redshift versus the
gravitational mass where we have shown that the maximum gravitational redshift
occurs for $S=2.5k_{B}$ considering the fixed bag constant. Finally, an $M$
$\propto$ $R^{3}$ relation was found for the mass and radius of SQS for all
relevant calculations.

###### Acknowledgements.

We wish to thank Shiraz University Research Council.

## References

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11institutetext: Yaroslav-the-Wise Novgorod State University, Velikiy Novgorod
173001, Russia 11email<EMAIL_ADDRESS>
# Stochastic differential equations for limiting description of UCB rule for
Gaussian multi-armed bandits††thanks: The reported study was funded by RFBR,
project number 20-01-00062.

Sergey Garbar 11 0000-0002-5205-5252

###### Abstract

We consider the upper confidence bound strategy for Gaussian multi-armed
bandits with known control horizon sizes $N$ and build its limiting
description with a system of stochastic differential equations and ordinary
differential equations. Rewards for the arms are assumed to have unknown
expected values and known variances. A set of Monte-Carlo simulations was
performed for the case of close distributions of rewards, when mean rewards
differ by the magnitude of order $N^{\textendash 1/2}$, as it yields the
highest normalized regret, to verify the validity of the obtained description.
The minimal size of the control horizon when the normalized regret is not
noticeably larger than maximum possible was estimated.

###### Keywords:

Gaussian multi-armed bandit UCB stochactic differential equations limiting
description

##  1 Introduction

We consider a multi-armed bandit (MAB) problem. MAB can be viewed as a slot
machine with $J$ arms. Each one of the arms can be selected for play, which
yields some random income (reward). It is suggested that a number of times
that a gambler can play is known beforehand. This number $N$ is called the
control horizon size. Formally MAB is a controlled stochastic process
$\xi(n),n=1,2,…,N$. Value $\xi(n)$ only depends on the chosen arm. Gambler’s
task is to maximize the total cumulative rewards during the time of control.

MABs and corresponding algorithms are thoroughly analyzed in [12]. This is a
reinforcement learning problem that exemplifies the exploration–exploitation
tradeoff dilemma, so it is also studied in machine learning [1][3]. This
problem is also known as the problem of adaptive control in a random
environment [14] and the problem of expedient behavior [15]. MABs have also
been used to model problems such as managing research projects in a large
organization like a science foundation or a pharmaceutical company [8][2].

Further we consider a Gaussian MAB, when rewards are normally distributed with
probability density functions

$f_{l}\left(x\right)=\left(2\pi
D_{l}\right)^{-1/2}e^{-\dfrac{\left(x-m_{l}\right)^{2}}{2D_{l}}},\
l=1,2,\ldots,J.$

We assume that mean rewards $m_{1},m_{2},…,m_{J}$ are unknown. Variances of
rewards $D_{1},D_{2},…,D_{J}$ are assumed to be known. Therefore, the
considered MAB can be described with vector parameter
$\theta=(m_{1},…,m_{J})$.

In [10, 6] it is shown that incorrect values for variances only slightly
affect overall rewards, therefore they can be estimated during the initial
steps of control. That means that the requirement to know the variances can be
removed.

Gaussian MAB can be useful in a batch processing setting [13, 4, 9]. In that
scenario gambler can only change the choice of the arm after he or she has
used it for a given number of times (a batch). According to the central limit
theorem, the sum of rewards for a batch will have a normal distribution (if
batch size is relatively large) for a wide range of distributions of single-
step rewards (if variances of rewards are finite). Batch processing can be
also performed in parallel.

When some control strategy $\sigma$ is used, the expected cumulative reward
will be lower than the maximally possible. That happens due to the fact that
some number of steps will be spent to explore the distributions of arms’
rewards. Maximal cumulative reward would be obtained if the most lucrative arm
was known and used on every step of the control. The expected difference
between the maximal possible and the obtained rewards is called the regret and
can be found as

$L_{N}(\sigma,\theta)=E_{\sigma,\theta}\left(\sum_{n=1}^{N}\left(max(m_{1},…,m_{J})-\xi_{n}\right)\right).$

$E_{\sigma,\theta}(\cdot)$ denotes the expected value calculated with respect
to measure generated by strategy $\sigma$ and parameter $\theta$. Therefore,
the goal of control is to minimize the regret.

Regret is dependent on the horizon size. To compare the strategies for
different horizon sizes it is reasonable to consider the scaled regret

$\hat{L}_{N}(\sigma,\theta)=(DN)^{-1/2}L_{N}(\sigma,\theta),$

for $D=\max_{J}{D_{J}}$.

UCB strategy (as described by Lai [11]) with the parameter $a$ is defined as
selection of arm maximizing the value

$U_{l}(n)=\dfrac{X_{l}(n)}{n_{l}}+\dfrac{\sqrt{2D_{l}log(n/n_{l}})}{\sqrt{n_{l}}},l=1,2,…,J;n=1,2,…,N.$

Strategy prescribes to use each of the arms once in the beginning of control
to make the initial estimations regarding the mean rewards.

Presence of slowly growing term $\sqrt{2D_{l}log(n/n_{l})/n_{l}}$ ensures the
necessity to use every arm from time to time even if the estimated mean reward
$X_{l}(n_{l})/n_{l}$ from its usage so far was lower than from the other arms
due to some reason. That is the way the UCB strategy negotiates the
exploration–exploitation dilemma: the gambler strives to maximize his or her
reward by using the arm with the highest estimated mean, but also collects the
information about the less explored arms.

Single-step income on step $n$ can be expressed as

$\xi(n)=m_{l}+\sqrt{D_{l}}\eta,$

where $\eta$ is a standard normal random variable, and $l=1,2,…,J$.

Then cumulative reward for using $l$-th arm on step $n$ can be expressed as

$X_{l}(n)=n_{l}m_{l}+\sqrt{n_{l}D_{l}}\eta,$

where $n_{l}$ is the number of times the $l$-th arm was used,
$l=1,2,\ldots,J$; $n_{1}+\ldots+n_{J}=n$.

We aim to build a limiting description (with a system of stochastic
differential equations) of UCB strategy for a Gaussian multi-armed bandit when
$N\to\infty$.

##  2 System of stochastic differential equations to describe UCB strategy for
Gaussian two-armed bandit

First, we give a limiting description for the case of two-armed bandit.

In [5, 7] an invariant description of a UCB strategy for UCB strategy for a
MAB was given. Invariant description scales the control horizon to a unit size
by scaling the step number to smaller values (proportional to $N$). If
$N\rightarrow\infty$, then step value is infinitely small.

Differentials of cumulative reward for using $l$-th arm can be expressed as

$dX_{l}=m_{l}dt_{l}+\sqrt{D_{l}}dW_{l},\ l=1,2.$

Here $W_{1},W_{2}$ are independent Wiener processes, $t_{l}$ is a continuous
variable that corresponds to the usage ratio of each arm. Each of those
equations correspond to a Wiener process with a constant drift. At each moment
only one of values $t_{1},t_{2}$ can be increased as only one of the arms is
used. Therefore, we can consider two areas: in the first $t_{1}$ increases
with time $t$, in the second $t_{2}$ increases. We’ll use indicators
$I_{1}\left(t\right),I_{2}\left(t\right)$ for convenience.

If the first arm is chosen, then

$\begin{cases}dt_{1}=dt,\\\ dt_{2}=0,\end{cases}$ and
$\begin{cases}I_{1}(t)=1,\\\ I_{2}(t)=0.\end{cases}$

If the second arm is chosen, then

$\begin{cases}dt_{1}=0,\\\ dt_{2}=dt,\end{cases}$ and
$\begin{cases}I_{1}(t)=0,\\\ I_{2}(t)=1.\end{cases}$

We’ll take notice that $I_{2}\left(t\right)=1-I_{1}\left(t\right)$ and
$dt_{2}=dt-dt_{1}$. Also remember that $t_{2}=t-t_{1}$: one and only one arm
is chosen at every moment.

Each of the areas is determined according to the chosen strategy. UCB rule can
be rewritten as

$U_{l}\left(t_{1},t_{2}\right)=\dfrac{X_{l}\left(t_{1}\right)}{t_{l}}+\dfrac{\sqrt{aD_{l}\log{\dfrac{t}{t_{l}}}}}{\sqrt{t_{l}}},\
l=1,2.$

The area where the first arm is chosen will be defined by the condition
$U_{1}\left(t_{1},t_{2}\right)>U_{2}\left(t_{1},t_{2}\right)$. The second arm
is chosen when $U_{1}\left(t_{1},t_{2}\right)\leq
U_{2}\left(t_{1},t_{2}\right)$. To write this condition formally we use the
Heaviside step function $H(\cdot)$ notation. Its value is zero for negative
arguments and one for positive arguments.

For the UCB rule the indicator $I_{1}\left(t\right)$ will be the function of
arguments $t,t_{1},X_{1},X_{2}$:

$I_{1}\left(t,t_{1},X_{1},X_{2}\right)=H\left(\dfrac{X_{1}}{t_{1}}+\dfrac{\sqrt{aD_{1}\log{\dfrac{t}{t_{1}}}}}{\sqrt{t_{1}}}-\dfrac{X_{2}}{t-t_{1}}-\dfrac{\sqrt{aD_{2}\log{\dfrac{t}{t-t_{1}}}}}{\sqrt{t-t_{1}}}\right).$

Using this notation, we write the system of two Itô stochastic differential
equations and one ordinary differential equation equations to describe the UCB
strategy for Gaussian two-armed bandit as

$\begin{cases}dX1=I_{1}m_{1}dt+D_{1}dW_{1},\\\
dX_{2}=1-I_{1}m_{2}d_{t}+D_{2}dW_{2},\\\ dt_{1}=I_{1}dt.\end{cases}$

##  3 Simulation results

A series of Monte-Carlo simulations was performed to verify how the obtained
system of equations describes the behavior of the Gaussian two-armed bandit
when the UCB strategy is used. Without the loss of generality, we consider a
Gaussian two-armed bandit with the zero mean reward for the first arm
$m_{1}=0$. We observe how the normalized regret depends on the value of the
mean reward for the second arm. Also, we assume $D_{1}=D_{2}=D=1$: in [7] it
is shown that the maximum normalized regret is observed when the variances for
the arms are equal. Also, we set $a=1$.

Further we consider the case of “close” distributions of rewards as the
highest values of regret are observed [16]. Its means are defined by the
difference of mean rewards of order $N^{-1/2}$:

$\left\\{m_{l}=m+c_{l}\sqrt{D/N};m\in{R},\left|c_{l}\right|\leq
C<\infty,l=1,2\right\\}.$

For the Gaussian multi-armed bandit simulations, we consider the cases of
horizon sizes $N=200,\ 400$. As we have the invariant description of the
strategy, we can state that normalized regret will be identical for different
fairly large horizon sizes. Figure 1 contains the plots of normalized regret
vs different values of $c_{2}$. Data for the plots are averaged over 10000
simulations.

For numerical solutions of stochastic differential equations, we use the
Euler–Maruyama method with step values of ${10}^{-3}$ and $5\cdot{10}^{-4}$.
Figure 1 shows the normalized regrets calculated over these solutions for
different values of $c_{2}$, averaged over 10000 simulations.

Figure 1: Normalized regret for different differences of mean rewards
(parametrized by $c_{2}$) obtained by direct simulations and as solution of
system of stochastic differential equations

We see that (as the invariant description would suggest) normalized regret is
the same for different horizon sizes. Also, the obtained system of equations
gives the fitting description of the UCB strategy. We can note that maximal
normalized regret (approximately 0.73) is observed when $c_{2}\approx 3.6$.

Having the limiting description allows to answer the question what is the
minimum horizon size $N$ when the normalized regret is not noticeably larger
than when $N\rightarrow\infty$. This question is of importance as the batch
version of the algorithm (reported in [5, 7]) allows to vary the batch sizes.
Parallel processing is possible in that case, and it can result in shorter
processing times.

Figure 2 shows normalized regret vs the control horizon size (left plot shows
$N\in\left[3,100\right]$, right plot shows $N\in\left[36,100\right]$). Data is
averaged over 10000 simulations.

Figure 2: Normalized regret vs the control horizon size $N$

We can see that after $N=45$ the normalized regret exceeds the maximum
(determined before as 0.73) no more than 2%. That means that if we use the
parallel processing, the batch size might be chosen as large as the horizon
size is fairly small (e.g. $N=50$).

##  4 System of stochastic differential equations to describe UCB strategy for
Gaussian MAB

Further we obtain a limiting description with a system of stochastic
differential equations for the UCB strategy for Gaussian multi-armed bandit
with $J$ arms using the similar reasoning as before. Considered MAB has
unknown mean rewards $m_{1},\ldots,m_{J}$ and known variances of rewards
$D_{1},\ldots,D_{J}$.

First, we can express $t_{J}$ as

$t_{J}=t-\sum_{i=1}^{J-1}t_{i}.$

Indicator for the arm will be equal to one if the corresponding upper
confidence bound is greater that the confidence bounds for all other arms. We
can express it as the product of step functions

$\displaystyle I_{l}(t,t_{1},$
$\displaystyle\ldots,t_{J-1},X_{1},\ldots,X_{J})=$
$\displaystyle=\prod_{\begin{subarray}{c}i=1\\\ i\neq
l\end{subarray}}^{J}{H\left(\dfrac{X_{l}}{t_{l}}+\dfrac{\sqrt{aD_{l}\log\dfrac{t}{t_{l}}}}{\sqrt{t_{l}}}-\dfrac{X_{i}}{t_{i}}-\dfrac{\sqrt{aD_{i}\log\dfrac{t}{t_{i}}}}{\sqrt{t_{i}}}\right)},$

for $l=1,…,J-1$.

$I_{J}\left(t,t_{1},\ldots,t_{J-1},X_{1},\ldots,X_{J}\right)=1-\sum_{i=1}^{J-1}{I_{i}\left(t,t_{1},\ldots,t_{J-1},X_{1},\ldots,X_{J}\right)}.$

Then the system of $J$ stochastic differential equations and $J-1$ ordinary
differential equations that describes the UCB strategy takes form

$\begin{cases}dX_{1}=I_{1}m_{1}dt+D_{1}dW_{1},\\\ …\\\
dX_{J-1}=I_{J-1}m_{J-1}dt+D_{J-1}dW_{J-1},\\\
dX_{J}=\left(1-\sum_{i=1}^{J-1}I_{i}\right)m_{J}dt+D_{J}dW_{J},\\\
dt_{1}=I_{1}dt,\\\ …\\\ dt_{J-1}=I_{J-1}dt.\end{cases}$

##  5 Conclusion

We built and examined a limiting description (for $N\rightarrow\infty$) with a
system of stochastic differential equations for the UCB strategy for the
Gaussian multi-armed bandit.

A series of Monte-Carlo simulations was performed to verify the validity of
obtained results for the case of two-armed bandit in the case of “close”
distributions of rewards. Maximum normalized regret for the case
$N\rightarrow\infty$ was determined. The minimal size of the control horizon
when the normalized regret is not noticeably larger than maximum possible was
estimated.

###  5.0.1 Acknowledgements

The reported study was funded by RFBR, project number 20-01-00062.

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11institutetext: Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333
CA Leiden, the Netherlands
11email<EMAIL_ADDRESS>
# The effect of spiral arms on the Sérsic photometry of galaxies

Alessandro Sonnenfeld 11

###### Abstract

Context.  The Sérsic profile is a widely used model for describing the surface
brightness distribution of galaxies. Spiral galaxies, however, are
qualitatively different from a Sérsic model.

Aims.  The goal of this study is to assess how accurately the total flux and
half-light radius of a galaxy with spiral arms can be recovered when fitted
with a Sérsic profile.

Methods.  I selected a sample of bulge-dominated galaxies with spiral arms.
Using photometric data from the Hyper Suprime-Cam survey, I estimated the
contribution of the spiral arms to their total flux. Then I generated
simulated images of galaxies with similar characteristics, fitted them with a
Sérsic model, and quantified the error on the determination of the total flux
and half-light radius.

Results.  Spiral arms can introduce biases on the photometry of galaxies in a
way that depends on the underlying smooth surface brightness profile, the
location of the arms, and the depth of the photometric data. A set of spiral
arms accounting for $10\%$ of the flux of a bulge-dominated galaxy typically
causes the total flux and the half-light radius to be overestimated by 15% and
30%, respectively. This bias, however, is much smaller if the galaxy is disk-
dominated.

Conclusions.  Galaxies with a prominent bulge and a non-zero contribution from
spiral arms are the most susceptible to biases in the total flux and half-
light radius when fitted with a Sérsic profile. If photometric measurements
with high accuracy are required, then measurements over finite apertures are
to be preferred over global estimates of the flux.

###### Key Words.:

galaxies: spiral – galaxies: fundamental parameters

##  1 Introduction

Two widely used quantities to describe the global properties of a galaxy are
the total flux and the radius enclosing half of that flux, that is, the half-
light radius. When the distance of a galaxy is known and an estimate of its
stellar mass-to-light ratio is available (for example, from stellar population
synthesis modelling), the total flux and the half-light radius can be
converted into measurements of the total stellar mass and half-mass radius.
These are fundamental properties of a galaxy, the knowledge of which is
crucial for understanding the baryon cycle in the Universe and galaxy
evolution. Stellar mass measurements can be used, for example, to constrain
the efficiency of the process of star formation across cosmic time (Moster et
al., 2018; Behroozi et al., 2019), while the half-mass radius of a galaxy is
sensitive to the angular momentum (Mo et al., 1998) and the merger history of
its host dark matter halo (Naab et al., 2009; Hilz et al., 2013; Shankar et
al., 2014).

Photometric and spectroscopic surveys such as the Sloan Digital Sky Survey
(SDSS; York et al., 2000), the Kilo-Degree Survey (KiDS; de Jong et al.,
2013), the Galaxy And Mass Assembly (GAMA) survey (Driver et al., 2011), and
the Hyper Suprime-Cam (HSC) Subaru Strategic Program (HSC SSP, or HSC survey;
Aihara et al., 2018) have enabled the measurement of the fluxes and
sizes111Throughout this paper, I use the terms ‘size’ and ‘half-light radius’
interchangeably. of millions of galaxies (Simard et al., 2011; Kelvin et al.,
2012; Meert et al., 2015; Kawinwanichakij et al., 2021). Such a large amount
of data makes it possible to determine galaxy scaling relations, such as the
stellar mass-size relation, with extremely high statistical precision. When
statistical uncertainties are very small, however, inferences are generally
limited by systematic errors. This work assesses the impact of one possible
source of systematic error, which is related to the method with which fluxes
and sizes are determined.

The standard approach to measuring the total flux and half-light radius of a
galaxy is by fitting its image with a simply parameterised model for its
surface brightness distribution. The most commonly used model is the Sérsic
profile (Sersic, 1968). The Sérsic model consists of a smooth surface
brightness distribution with three degrees of freedom in the radial direction.
The success of this model owes to the fact that it is able to approximate, on
average, the surface brightness distribution of a broad range of galaxies,
from massive ellipticals to disks. Most galaxies, however, exhibit
substructure that deviates from a smooth description of the surface brightness
distribution, such as spiral arms, rings, bars, or shells. For these galaxies,
a Sérsic model is unable to reproduce their images down to the noise level.
These bad fits are prone to biases.

In this work, I investigate the possible bias introduced by fitting a Sérsic
model to galaxies with spiral arms. I focus on a particular class of objects,
that of bulge-dominated spirals, also known as red spirals. The reason for
this choice is that these are objects for which it is easier to separate the
spiral arms from the underlying smooth component of the surface brightness
profile. In the first part of this study, I select a sample of bulge-dominated
spiral galaxies and quantify the fraction of their optical flux that is due to
spiral arms. Subsequently, I simulate images of spiral galaxies with similar
characteristics to the observed sample, fit them with a Sérsic model, and
measure the error on the inferred flux and half-light radius.

The structure of this work is the following. In Sect. 2 I show estimates of
the fraction of flux in spiral arms for a select sample of galaxies. In Sect.
3 I show the results of the Sérsic profile fit to the simulated galaxies. I
discuss the results and draw conclusions in Sect. 4.

##  2 Spiral arm contribution

For the purpose of gaining insight into the typical photometric properties of
spiral arms, I obtained a sample of $53$ bulge-dominated spiral galaxies and
carried out a decomposition of their light in a smooth surface brightness
component and a set of spiral arms. Section 2.1 introduces the sample of
galaxies used for this analysis, and Sect. 2.2 explains how the contribution
of the spiral arms to the total surface brightness distribution was estimated.

###  2.1 The sample

I selected galaxies on the basis of their colour and morphology, as follows. I
considered objects from the main galaxy sample of the SDSS (Strauss et al.,
2002) as the starting point. I first defined a narrow redshift slice,
$0.19<z<0.21$: this cut allows colours to be easily measured in a homogeneous
way, without the need to apply K-corrections. Then, I considered their
observed-frame $g-r$ colours defined using ‘model’ magnitudes from data
release 16 of the SDSS (Ahumada et al., 2020) and applied the following cut:

$g-r>1.2.$ (1)

Finally, I restricted the analysis to objects with available photometric data
from the second public data release (PDR2) of the HSC survey (Aihara et al.,
2019). I obtained $201\times 201$ pixel cutouts (33.8” per side) of
background-subtracted, co-added images in the $g$, $r$, and $i$ bands from the
HSC PDR2, made colour-composite images, and visually inspected them to select
galaxies with spiral arms. The resulting $53$ red spiral galaxies are shown in
Fig. 1: these are the main subject of this section.

Figure 1:  Colour-composite HSC images in the $g$, $r$, and $i$ bands of 53
galaxies with observed-frame $g-r$ colour larger than $1.2$ and spiral
morphology. For each image the following morphological classification scores
from Galaxy Zoo are shown: ${\rm P}_{\mathrm{el}}$ is the de-biased
probability of the galaxy being elliptical, and ${\rm P}_{\mathrm{sp}}$ is the
de-biased probability of it being a spiral.

For the sake of comparing the HSC-based visual classification of this work
with the literature, Fig. 1 also shows morphological classification scores
from Galaxy Zoo (Lintott et al., 2008, 2011), which are based on SDSS images.
Interestingly, 39 out of the 53 galaxies have a higher probability of being
elliptical than of being a spiral, according to Galaxy Zoo. This dramatic
difference in the morphological classification of this set of objects is
probably due to the lower depth and lower resolution of SDSS images, compared
to HSC data, which makes it difficult to see the spiral arms.

###  2.2 Photometric decomposition

I define as a spiral arm any positive excess flux that cannot be accounted for
by a smooth model for the surface brightness distribution of a galaxy. In
principle, this definition leads to the inclusion of features that are not
spiral arms in a strict sense, such as rings or shells. In practice, as I
discuss in Sect. 4, the effect of such features on the fit of a Sérsic profile
is qualitatively similar to that of proper spiral arms. Therefore, I treat any
flux excess associated with the main galaxy as part of a spiral arm. With this
definition, I proceeded as follows. I fitted a Sérsic surface brightness
profile to the HSC $r$-band images of the $53$ red spiral galaxies. The $r$
band is the one that is most commonly used in photometric studies of galaxies
with SDSS data. The Sérsic model surface brightness is defined as follows:

$I(x,y)=I_{0}\exp{\left\\{-b(n)\left(\frac{R}{R_{\mathrm{e}}}\right)^{1/n}\right\\}},$
(2)

where $x$ and $y$ are Cartesian coordinates with origin at the galaxy centre
and aligned with the galaxy major and minor axes, $R$ is the circularised
radius

$R^{2}\equiv qx^{2}+\frac{y^{2}}{q},$ (3)

$q$ is the axis ratio, $n$ is the Sérsic index, and $b(n)$ is a numerical
constant defined in such a way that the isophote of radius $R=R_{\mathrm{e}}$
encloses half of the total flux (see Ciotti & Bertin, 1999).

This model is meant to describe only the smooth component of the surface
brightness profile. For this reason, I manually masked out the region over
which the spiral arms dominate the surface brightness when doing the fit.
Without doing so, the relative contributions of the smooth component and of
the spiral arms to the total flux would be respectively overestimated and
underestimated. Examples of spiral arm masks are shown in the third column of
Fig. 2.

For each galaxy, I found the set of model parameters that minimises the
following $\chi^{2}$:

$\chi^{2}=\sum_{j}\frac{\left(I_{j}^{\mathrm{(mod)}}-I_{j}^{\mathrm{(obs)}}\right)^{2}}{\sigma_{j}^{2}},$
(4)

where $I_{j}^{\mathrm{(mod)}}$ is the seeing-convolved Sérsic model in the $r$
band evaluated at pixel $j$, $I_{j}^{(\mathrm{obs})}$ is the observed surface
brightness at that pixel, and $\sigma_{j}$ is the observational uncertainty. I
assumed flat priors on all of the model parameters. For the Sérsic index, I
imposed lower and upper limits at $0.5<n<8$. I sampled the parameter space of
the structural parameters with a Markov chain Monte Carlo. At each step of the
chain I found the value of the amplitude of the model (parameter $I_{0}$ in
Eq. 2) that minimises the $\chi^{2}$. Figure 2 shows fits for three example
galaxies.

Figure 2:  Fits of a Sérsic profile to the smooth component of the surface
brightness distribution of three example galaxies. Column 1: Colour-composite
image of the galaxy in the HSC $g,r,$ and $i$ bands. Column 2: Grey-scale
image of the galaxy in the HSC $r$ band. Column 3: Area over which the spiral
arms dominate the surface brightness (green regions) and contaminants (red
regions). Both are masked out from the fit. Column 4: Best-fit model of the
smooth component in the $r$ band. Column 5: Residual in the $r$ band. The
value of the $r$-band spiral-to-total flux ratio is shown in the bottom-right
corner.

After finding the minimum-$\chi^{2}$ set of parameters, I subtracted the best-
fit model from the data. I then estimated the spiral arm flux by taking the
sum of the residual flux in the masked region. I summarised the properties of
the spiral arms with two numbers. The first is the $r$-band spiral-arm-to-
total ratio, $f_{\mathrm{spi}}$, defined as the ratio between the flux
associated with the spiral arms and the total flux. The second quantity is the
light-weighted radius, defined as

$R_{\mathrm{spi,lw}}=\dfrac{\sum_{i\in{\mathrm{spi}}}I_{i}\cdot
R_{i}}{\sum_{i\in{\mathrm{spi}}}I_{i}},$ (5)

where $I_{i}$ is the surface brightness of the spiral arm at the $i$-th pixel,
$R_{i}$ is the circularised radius, defined with respect to the smooth
component, at the same pixel, and the sum is taken over all pixels in the
spiral arm mask with non-negative flux. Figure 3 shows the distribution in
$f_{\mathrm{spi}}$, $R_{\mathrm{spi,lw}}$, total (smooth component plus spiral
arms) $r$-band magnitude, and half-light radius and Sérsic index of the smooth
component.

Figure 3:  Distribution in spiral-arm-to-total ratio, $f_{\mathrm{spi}}$,
spiral arm light-weighted radius, $R_{\mathrm{spi,lw}}$, total r-band
magnitude, $m_{r,\mathrm{tot}}$, half-light radius of the smooth component,
$R_{\mathrm{e,smo}}$, and Sérsic index of the smooth component,
$n_{\mathrm{smo}}$, of the 53 red spiral galaxies in the sample.

The smooth component covers a broad range in values of the Sérsic index, with
most of the galaxies having $n_{\mathrm{smo}}>4$, which is typical of bulges.
The median value of the spiral-arm-to-total ratio is $f_{\mathrm{spi}}=0.11$,
while the median value of the ratio between the light-weighted radius and the
half-light radius of the smooth component,
$R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}$, is approximately $1.5$.

##  3 Tests on simulated galaxies

In this section I quantify how the presence of spiral arms affects the
accuracy of the inference of the total flux and half-light radius of a galaxy
when these are measured by means of the fit of a Sérsic model. This problem is
a high-dimensional one: the answer can depend, in principle, on many different
features of the surface brightness distribution of the galaxy, but also on the
details of the data used for the fit, as well as on aspects of the fitting
procedure (for instance, the choice of priors). In this work, I focus on the
following properties: the spiral-arm-to-total ratio, $f_{\mathrm{spi}}$, the
ratio between the light-weighted radius of the spiral arm and the half-light
radius of the smooth surface brightness component,
$R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}$, the Sérsic index of the smooth
component, and the depth of the photometric data.

In order to isolate the effect that each of these aspects has on the
inference, I simulated different sets of simulated images, varying only one
property at a time. Section 3.1 describes the general characteristics of the
simulations. The ensuing sections describe each set of simulations and the
results obtained by fitting a Sérsic model to them.

###  3.1 Simulations. General characteristics

Each simulated galaxy consists of a smooth surface brightness component,
described by a Sérsic profile, and a set of spiral arms. Throughout the
experiment, I fixed the half-light radius of the smooth component to
$R_{\mathrm{e,smo}}=2.52^{\prime\prime}$ ($15$ HSC pixels), its $r$-band
magnitude to $m_{r,\mathrm{smo}}=17.5$, and its axis ratio to one.

The shape of the spiral arms is described by the following function
(Ringermacher & Mead, 2009):

$R_{\mathrm{spi}}(\phi)=-\frac{A}{\ln{\left(B\,\rm{tan}\frac{\phi}{2N}\right)}},$
(6)

where $R$ and $\phi$ are polar coordinates, and $A$, $B,$ and $N$ are free
parameters. This model describes a single spiral arm originating at angle
$\phi=0$. A second spiral arm can be obtained by rotating the coordinate frame
by $\pi$. The arm extends along positive values of $\phi$ as long as both of
the following conditions are met:

$\displaystyle B\,\rm{tan}\frac{\phi}{2N}<1,$ (7) $\displaystyle
R(\phi)<R_{\mathrm{max}},$ (8)

where $R_{\mathrm{max}}$ is a parameter describing the maximum allowed
distance from the galaxy. The first condition ensures that the logarithm in
Eq. 6 returns a negative value, so that $R$ is positive, while the second
condition is added in order to avoid producing excessively long spiral arms.
For all simulated galaxies, I fixed the value of $R_{\mathrm{max}}$ to $0.5A$.

Equation 6 describes a spiral arm with a pitch angle222The pitch angle of a
spiral arm is the angle between its tangent and a circle centred on the galaxy
centre. that decreases as a function of radius. The parameter $A$ sets the
scale of the arm, while the parameters $B$ and $N$ affect how rapidly the
pitch angle varies as a function of $\phi$.

Equation 6 defines the position of the core of each arm. I then assumed that
the contribution of the arm to the surface brightness distribution of the
galaxy is given by the following exponential function,

$I_{\mathrm{spi}}(R,\phi)=I_{\mathrm{spi},0}\exp{\left\\{-\frac{|R-R_{\mathrm{spi}}(\phi)|}{h}\right\\}},$
(9)

where the parameter $h$ sets the width of the arm. I fixed the value of $h$ to
two HSC pixels (0.336”) for all cases.

Given a set of model parameters that describe a spiral galaxy, I generated
HSC-like images by convolving the model surface brightness distribution with
the HSC point spread function. I added to each pixel a Gaussian noise with
dispersion $\sigma_{\mathrm{sky}}$ equal to that observed in HSC data, as well
as Poisson noise, assuming an exposure time of $10$ minutes (the median
exposure time of the HSC PDR2 in the $r$ band). As in Sect. 2, the analysis is
based on $r$-band images. Nevertheless, for the purpose of creating colour-
composite images, I generated data in the $g$\- and $i$-band data as well. In
all simulations, the colours of the smooth component are fixed to $g-r=1.2$
and $r-i=0.5$, while those of the spiral arms are fixed to $g-r=0.5$ and
$r-i=0.1$.

I generated four sets of simulated galaxies, each aimed at testing the effect
of a given property on the bias on the Sérsic photometry. The characteristics
of each set are summarised in Table 1.

Table 1: Simulation properties

Set | $f_{\mathrm{spi}}$ | $n_{\mathrm{smo}}$ | $A$ | $B$ | $N/B$ | $\sigma_{\mathrm{sky}}$
---|---|---|---|---|---|---
A | $0.1$ | $4$ | $[4^{\prime\prime},8^{\prime\prime},12^{\prime\prime},16^{\prime\prime},20^{\prime\prime}]$ | $[0.1,1]$ | $[5,10,20]$ | $\sigma_{\mathrm{HSC}}$
B | $0.1$ | $[1,3,5]$ | $[4^{\prime\prime},8^{\prime\prime},12^{\prime\prime},16^{\prime\prime},20^{\prime\prime}]$ | $1$ | $10$ | $\sigma_{\mathrm{HSC}}$
C | $[0.01,0.05,0.1,0.2]$ | $4$ | $12^{\prime\prime}$ | $1$ | $10$ | $\sigma_{\mathrm{HSC}}$
D | $0.1$ | $4$ | $12^{\prime\prime}$ | $1$ | $10$ | $[0.2,0.5,1,2,10,20,50]\times\sigma_{\mathrm{HSC}}$

333Simulation parameters for each set. Column (2): Spiral-arm-to-total flux
ratio in the $r$ band. Column (3): Sérsic index of the smooth component.
Column (4): Scale parameter of the spiral arms. Column (5): Parameter $B$ of
the spiral arms. Column (6): Ratio $N/B$ of the spiral arms. Column (7): Sky
background fluctuation level. The quantity $\sigma_{\mathrm{HSC}}$ indicates
the sky background noise level of HSC data.

###  3.2 Set A. Effect of the shape and position of the spiral arms

The first set of simulations, labelled as set A, consists of $30$ galaxies
with a fixed smooth component and varying properties of the spiral arms. Its
purpose is to check the effect that the shape and the position of the spiral
arms has on the inferred flux and size of the galaxy. The Sérsic index of the
smooth component is fixed to $n_{\mathrm{smo}}=4$. The spiral arm parameters
cover a grid of five values of $A$, two values of $B,$ and three values of the
ratio $N/B$, as indicated in the first row of Table 1. In all cases, the
$r$-band spiral-arm-to-total ratio is fixed to $f_{\mathrm{spi}}=0.1$. The
value of the resulting total $r$-band magnitude is $m_{r,\mathrm{tot}}=17.40$.
The half-light radius, which I define here as the radius of the circle that
encloses half of the total flux in the $r$ band, varies depending on the
location of the spiral arms: it increases with increasing
$R_{\mathrm{spi,lw}}$. Figure 4 shows colour-composite images of these
simulated galaxies and reports the values of $R_{\mathrm{spi,lw}}$ and
$R_{\mathrm{e}}$ of each galaxy.

Figure 4:  Colour-composite images in HSC $g,r,i$ of red spiral galaxies from
simulation set A. Simulations consist of the sum of a smooth component, a
Sérsic profile with $n=4$ and $R_{\mathrm{e}}=2.52^{\prime\prime}$, and a set
of spiral arms, accounting for $10\%$ of the total $r$-band flux. The spiral
arm parameters cover a grid of values, as indicated in the panels. The values
of the $r$-band half-light radius and the light-weighted radius of the spiral
arm are shown in the bottom right of each panel.

I fitted the $r$-band image of each simulated galaxy with a Sérsic profile
using the same procedure described in Sect. 2 but without masking the spiral
arms. Figure 5 shows the results of the fit in an example case. Both the total
flux and the half-light radius are overestimated. The residuals show over-
subtraction in the region enclosed within the spiral arms and positive flux on
the spiral arms themselves.

Figure 6 offers a more quantitative view of this result, as it shows the
azimuthally averaged surface brightness profiles of the simulated galaxy and
of the best-fit Sérsic model, as well as the difference in the enclosed flux
profile between the model and the truth. The spiral arms create a bump in the
surface brightness profile at $\approx 6^{\prime\prime}$. The model is unable
to fit both this bump and the sharp decline in surface brightness at larger
radii. As a result, the best-fit model has a shallower surface brightness
profile than the truth at large radii. Although the bias in the integrated
flux is small within the region constrained by the data (only $0.02$ mag for
$R<15^{\prime\prime}$, as shown by the bottom panel of Fig. 6), the
extrapolation to the large radii of the model results in biases of $0.16$ mag
on the total flux and $26\%$ on the half-light radius.

Figure 5:  Fit of a Sérsic profile to the $r$-band image of a simulated
galaxy. Left panel: $r$-band image. Middle panel: Image of the
minimum-$\chi^{2}$ model. Right panel: Residual. The galaxy is taken from
simulation set A, with spiral arm parameters $A=16^{\prime\prime}$, $B=1$,
$N=10,$ and $f_{\mathrm{spi}}=0.1$ and smooth component parameters
$n_{\mathrm{smo}}=4$, $R_{\mathrm{e}}=2.52^{\prime\prime}$, and
$m_{r,\mathrm{smo}}=17.5$. The true values of the total $r$-band magnitude and
half-light radius are $m_{r,\mathrm{tot}}=17.40$ and
$R_{\mathrm{e}}=3.06^{\prime\prime}$. The parameters of the best-fit Sérsic
model are reported in the top-middle panel.  Figure 6:  Quantitative analysis
of the fit of Figure 5. Top: Azimuthally averaged surface brightness profile
of the galaxy and best-fit Sérsic model. The horizontal dashed line marks the
sky background root mean square surface brightness fluctuation for a single
pixel. Bottom: Difference between the magnitude within radius $R$ of the model
and that of the true galaxy image, as a function of $R$.

Figure 7 shows the bias on the inferred $r$-band magnitude and half-light
radius for simulated set A (red points). These biases are plotted as a
function of the ratio between the light-weighted radius of the spiral arms and
the half-light radius of the smooth component,
$R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}$. We see a very clear trend between
$R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}$ and the bias. Flux and size are
underestimated in galaxies with very small spiral arm radii and overestimated
for $R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}>1$. Remarkably, there is very
little scatter around the mean trend: for this simulated set, the bias appears
to be entirely described by the position of the spiral arms, while the shape
of the arms themselves only produces minimal deviations. The maximum bias
occurs at a value of $R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}$ between $1.5$
and $2.0$. Interestingly, this is close to the median value of the real sample
of red spirals measured in Sect. 2. For galaxies of this kind, the inferred
magnitude is smaller than the true value by $0.18$, which is equivalent to the
flux being overestimated by $0.06$ dex (15%). The half-light radius is
overestimated by $30\%$.

Figure 7:  Bias on the $r$-band magnitude (bottom panel) and half-light radius
(top panel) from the fit of a Sérsic profile to the $r$-band images of the set
of simulated galaxies shown in Fig. 4, as a function of the ratio between the
light-weighted radius of the spiral arms and the half-light radius of the
smooth surface brightness component. All of the simulated galaxies have a
spiral-arm-to-total ratio of $f_{\mathrm{spi}}=0.10$ and a smooth component
with $n=4$ and $R_{\mathrm{e,smo}}=2.52^{\prime\prime}$, and the images have
HSC-like properties.

###  3.3 Set B. Effect of the Sérsic index of the smooth component

Simulation set B consists of 15 galaxies generated by varying the Sérsic index
of the smooth component, $n_{\mathrm{smo}}$, and the scale parameter, $A$, of
the spiral arms, as described in Table 1. The goal of these simulations is to
check to what extent the results obtained in the previous section depend on
the choice of $n_{\mathrm{smo}}$. The results of the Sérsic profile fit are
shown in Fig. 7.

At fixed $n_{\mathrm{smo}}$, galaxies from this set also show a trend between
the bias and the ratio $R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}$, with the
maximum bias occurring at $R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}\approx 1.8$.
The amplitude of the bias, however, appears to be a strong function of the
Sérsic index of the smooth surface brightness component. While galaxies with
$n_{\mathrm{smo}}=5$ can have a bias on the total magnitude as large as $0.24$
mag, the largest bias observed in galaxies with $n_{\mathrm{smo}}=1$ and
$R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}>1$ is only $0.04$ mag. Simulations
with smaller values of the ratio $R_{\mathrm{spi,lw}}/R_{\mathrm{e,smo}}$ can
produce larger biases, but such galaxies are not very realistic.

The reason for this difference lies in the large-scale behaviour of the
surface brightness profile as a function of $n_{\mathrm{smo}}$. The higher the
Sérsic index, the larger the fraction of the total flux that is associated
with the outer regions of the galaxy. For example, for a Sérsic profile with
$n=1$, a circular aperture with radius equal to two half-light radii encloses
$85\%$ of the total flux. The same aperture for a Sérsic profile with $n=4$,
however, encloses only 69% of the total flux. The presence of the spiral arms
causes the best-fit Sérsic model to deviate from the surface brightness
profile of the smooth component at large radii (as shown in Fig. 6). If the
value of $n_{\mathrm{smo}}$ is small, this deviation does not introduce a
large bias, because most of the total flux is enclosed in a region that is
well constrained by the data. If $n_{\mathrm{smo}}$ is large, however,
differences in the large-scale behaviour between the true surface brightness
profile and the model can have a higher impact.

In the case shown in Fig. 6, for example, the bias on the flux enclosed within
an aperture of $16^{\prime\prime}$ (the size of the cutout) is as small as
$0.02$ mag. However, since the best-fit Sérsic profile has an index $n=4.97$
and a half-light radius of $R_{\mathrm{e}}=3.86^{\prime\prime}$, regions at
$R>16^{\prime\prime}$ still contribute 17% of the model total flux (the
difference between the total magnitude and the $16^{\prime\prime}$ aperture
magnitude is $0.20$), and the extrapolation to large radii can therefore
introduce a larger bias. Increasing the size of the cutout does not offer any
help in this case, because the surface brightness of the galaxy already falls
below the sky background fluctuation level at $R\approx 10^{\prime\prime}$.

###  3.4 Set C. Trends with $f_{\mathrm{spi}}$

Simulation set C consists of four galaxies with a fixed smooth component,
fixed spiral arm shape parameters, and different values of the spiral-arm-to-
total flux ratio, $f_{\mathrm{spi}}$. Its purpose is to understand how the
amplitude of the bias scales with $f_{\mathrm{spi}}$. The results of the
Sérsic fits to these galaxies are shown in the left-hand column of Fig. 8.
When $f_{\mathrm{spi}}=0.01$, the bias is very small: this is a sanity check,
showing that, in the limit in which the spiral arms are very faint, the Sérsic
model is able to recover the true magnitude and size of the galaxy (which is
dominated by the smooth component). The other data points show that the
amplitude of the bias increases approximately linearly with
$f_{\mathrm{spi}}$.

Figure 8:  Bias on simulation sets C and D. Left: Bias on the half-light
radius (top) and total magnitude (bottom) as a function of the spiral-arm-to-
total flux ratio, obtained by fitting a Sérsic profile to images from
simulation set C. Right: Bias on half-light radius (top) and total magnitude
(bottom) as a function of the ratio $R_{\mathrm{sky}}/R_{\mathrm{e,smo}}$ for
simulation set D. The quantity $R_{\mathrm{sky}}$ is defined as the radius at
which the surface brightness of the smooth component equals the amplitude of
the sky background fluctuation for a single pixel.

###  3.5 Set D. Effect of the depth of the data

The last test is aimed at checking how the bias depends on the depth of the
data used for the fit. For this reason, simulation set D consists of images of
the same galaxy that differ only by the amplitude of the background noise. In
particular, indicating with $\sigma_{\mathrm{HSC}}$ the $r$-band sky
background fluctuation of the HSC data, I generated images with sky background
fluctuation amplitude equal to $\sigma_{\mathrm{sky}}=f\sigma_{\mathrm{HSC}}$,
where $f$ varies from $0.2$ (highest depth) to $50$ (lowest depth). The
results of fits to these images are shown in the right-hand column of Fig. 8.
The bias on the magnitude and half-light radius are plotted as a function of
the quantity $R_{\mathrm{sky}}/R_{\mathrm{e,smo}}$, where $R_{\mathrm{sky}}$
is the radius at which the surface brightness of the smooth component is equal
to the amplitude of the background noise fluctuation for a single pixel.
Roughly speaking, the galaxy is detected on pixels that lie within a circle of
radius $R_{\mathrm{sky}}$ and are undetected outside it. Higher values of
$R_{\mathrm{sky}}$, then, correspond to deeper data. Figure 8 shows that the
accuracy of the fit improves with increasing depth of the data: as information
on the outer profile of a galaxy becomes available, the model is able to
better recover its total flux.

##  4 Discussion and summary

The experiment carried out in Sect. 3 shows that the presence of spiral arms
can cause a bias in the determination of the total flux and half-light radius
of a galaxy when these are determined by fitting a Sérsic profile to its
surface brightness distribution. Given a galaxy consisting of the sum of a
smooth surface brightness component, following a Sérsic profile, and a set of
spiral arms, the amplitude of the bias is most sensitive to the relative
contribution of the arms to the total flux, the position of the arms relative
to the half-light radius of the smooth component, the Sérsic index of the
smooth component, and the depth of the data. In particular, for a $z=0.2$
galaxy with a Sérsic index of the smooth component $n_{\mathrm{smo}}=4$, a set
of spiral arms accounting for 10% of the total flux (which is typical,
according to the analysis of Sect. 2) causes the total flux to be
overestimated by $0.06$ dex (15%) and the half-light radius to be
overestimated by $30\%$. Conversely, the bias is very small if the smooth
component of the surface brightness profile has a Sérsic index close to $1$.

These results were obtained on simulated images and, strictly speaking, only
apply to galaxies with properties that match exactly those of the simulations.
In order to make sure that they can be generalised to real cases, I carried
out the following test. I took the 53 red spiral galaxies introduced in Sect.
2.1 and compared the total $r$-band flux measured in Sect. 2.2, which was
obtained by separating out the spiral arms and the smooth component, with that
obtained by fitting a Sérsic profile to the full data (without masking the
spiral arms). Figure 9 shows the difference in the total magnitude obtained
with these two methods as a function of the $r$-band spiral arm fraction,
$f_{\mathrm{spi}}$.

Figure 9:  Difference in the $r$-band magnitude of the $53$ red spiral
galaxies of Sect. 2 obtained by fitting a Sérsic profile and with a two-
component description (smooth Sérsic component plus spiral arms) as a function
of the spiral arm fraction, $f_{\mathrm{spi}}$. The points are colour-coded by
the Sérsic index of the smooth component.

For all galaxies but two, the single Sérsic profile fit produces $r$-band
fluxes that are larger than that those obtained with the two-component
description. The difference correlates with the spiral arm fraction and also
with the Sérsic index, qualitatively confirming the findings of Sect. 3. The
median difference in magnitude between the two-component and single Sérsic fit
is $-0.11$ mag. This test shows that real galaxies behave in a similar way to
the ones simulated in Sect. 3 with respect to the effect of spiral arms on the
fit of Sérsic fits. It does not imply, however, that the fluxes obtained with
the two-component fit are perfectly accurate: these measurements are based on
the assumption of a Sérsic profile for the smooth component and therefore
could still be biased if this assumption breaks down at large radii. This is
particularly important for fits with a large Sérsic index of the smooth
component, since a relatively large fraction of their total flux is due to the
extrapolation of the profile to large radii. As I discuss below, the problem
of determining accurate total fluxes of spiral galaxies is an open one.

In light of these results, one can expect that the galaxies that are most
susceptible to biases are those with a dominant bulge component, that is, a
smooth surface brightness profile with a large ($n>2.5$) Sérsic index, such as
the red spiral galaxies studied in the first part of this paper. Red spiral
galaxies are relatively rare: they make up only 14% of the total of red
galaxies in the sample of Sect. 2. However, the population of spiral galaxies
with a prominent bulge extends well into the blue galaxy cloud. Of the SDSS
main sample galaxies with $g-r<1.2,$ 40% have a Sérsic index $n>2.5$, as
measured by Meert et al. (2015). Upon visual inspection, 60% of them have
spiral arms (24% of the total of blue galaxies).

In general, this bias is more important for samples of galaxies in which there
is a comparable mixture of disks and bulges. At redshift zero, this occurs
roughly at stellar masses $\log{M_{*}/M_{\odot}}\sim 10.5$-$11.0$, also known
as the knee of the galaxy stellar mass function. The overlap between the late-
type and early-type galaxy populations at this stellar mass is well
illustrated by Taylor et al. (2020) in their Fig. 1, which shows the
distribution in various properties of a large sample of galaxies in a narrow
stellar mass range around $\log{M_{*}/M_{\odot}}\sim 10.5$. Their data show
the well-known bimodality in colour, with red and blue galaxies accounting for
similar fractions of the full sample. Interestingly, the blue galaxy
population spans a broad range in Sérsic index, with approximately 20% of them
above the limit of $n=2.5$ that is often used to separate disks from bulges.

The spiral arm bias could have an impact on measurements aimed at detecting
differences between the red and blue galaxy populations at fixed stellar mass,
such as the studies of the average dark matter halo mass of these two galaxy
types (for example, More et al., 2011; Zehavi et al., 2011; Tinker et al.,
2013; Hudson et al., 2015; Mandelbaum et al., 2016). If the total flux is
overestimated for a set of galaxies, so is their stellar mass. Since halo mass
correlates strongly with stellar mass, that would lead to a biased
interpretation of the results. Assuming that the test of Fig. 9 is
representative of the general galaxy population, then the typical bias due to
spiral arms is $\sim 0.1$ mag on the total flux. Based on the arguments made
above in this section, one can expect that only 20% of the galaxies, those
with both a prominent bulge and spiral arms, are affected by this bias.
Therefore, given the current observational uncertainties of the stellar-to-
halo mass relation, the bias is probably not significant. Nevertheless,
systematic errors related to the photometric measurements can become important
as precision improves.

This work focused only on the effect of the spiral arms on the results of
Sérsic profile fits: throughout the paper, the bias was defined with respect
to the ideal case in which the true surface brightness profile of a galaxy
consists of a smooth component that follows a Sérsic profile and a set of
spiral arms. If the true surface brightness profile of the smooth component of
a galaxy deviates from a Sérsic model, however, the measurement will suffer
from additional biases. The tests carried out in this work, therefore,
represent a best-case scenario. Additionally, real spiral galaxies often show
ring-like features (see for example the third row in Fig. 2), which are not
directly captured by the model of Eq. 6. Nevertheless, I carried out tests on
simulated galaxies consisting of the sum of a Sérsic and a ring-like
component, using the ring model of Sonnenfeld et al. (2018), and found very
similar results to those shown in Sect. 3, at fixed $f_{\mathrm{spi}}$.

The problem of determining the total flux of a galaxy is a difficult one. The
results of Sect. 3.5 show that increasing the depth of the data can help
greatly in reducing the spiral arm bias. However, it is important to point out
that all of the other tests carried out in this work are based on galaxies at
relatively low redshift, $z=0.2$, observed with HSC-like data. While upcoming
surveys such as the Legacy Survey of Space and Time
(LSST444https://www.lsst.org/) will go approximately one magnitude deeper than
HSC, this will still be insufficient to probe the outskirts of higher redshift
galaxies.

A method that is often adopted to fit galaxies with complex morphology is that
of performing a bulge-disk decomposition. This is usually done by describing
the total surface brightness as the sum of a de Vaucouleurs profile (a Sérsic
profile with $n=4$) for the bulge and an exponential profile ($n=1$) for the
disk. Unfortunately, there are limitations with this approach, as recently
pointed out by Pap++22. The disk is generally associated with the younger
component of the stellar population of a galaxy. This young component,
however, is often missing in the inner regions of galaxies, where the star
formation activity has stopped, causing the true surface brightness profile to
deviate from a simple exponential model. Moreover, the assumption of a de
Vaucouleurs profile for the bulge component might also lead to biases if the
true surface brightness profile deviates from an $n=4$ model at large radii.

A better solution would be to increase the complexity of the model used to
describe a galaxy. Although there exists software that allows complex spiral
arm structure to be fit (for example Galfit; Peng et al., 2002), this approach
typically requires human interaction and does not scale well to very large
samples of galaxies. A powerful alternative is the use of neural networks
trained on a large set of simulated galaxy images (see for example Smith et
al., 2021; Li et al., 2021). While neural networks are very efficient at
producing fits to galaxies with complex morphology, their output depends
critically on the simulations used to train them. If the training set images
are biased with respect to the true surface brightness distribution of
galaxies, for instance at large radii, then they will introduce biases on the
total flux.

Given how challenging it is to accurately estimate the total flux of a galaxy,
an alternative is to instead quantify galaxy fluxes over finite apertures. One
example is the description in terms of the total flux within $10$ kpc and
light-weighted surface brightness slope in the same aperture, introduced by
Sonnenfeld (2020) for elliptical galaxies. As the bottom panel of Fig. 6
shows, the Sérsic model is able to correctly measure the total flux within a
finite aperture with a precision of $\sim 0.02$ mag. This precision could be
further improved with the use of non-parametric fitting methods, such as the
isophote-fitting tool AutoProf (Stone et al., 2021), or the segmentation map-
based software PROFOUND (Robotham et al., 2018).

In conclusion, while the Sérsic profile is a very useful model for
approximating the surface brightness distribution of a large set of galaxies,
it can introduce systematic errors on the total flux and half-light radius of
galaxies with spiral arms. These errors are small in the inner regions of
galaxies but can grow to tens of percent as a consequence of the extrapolation
to the very large radii of the Sérsic model, especially when the derived
Sérsic index is large. If highly precise photometric measurements are
required, a description based on the surface brightness distribution within a
finite aperture should be preferred over an estimate of the total flux and
half-light radius.

###### Acknowledgements.

I thank Henk Hoekstra for useful discussions. The Hyper Suprime-Cam (HSC)
collaboration includes the astronomical communities of Japan and Taiwan, and
Princeton University. The HSC instrumentation and software were developed by
the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for
the Physics and Mathematics of the Universe (Kavli IPMU), the University of
Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia
Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and
Princeton University. Funding was contributed by the FIRST program from the
Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science
and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS),
Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ,
Kavli IPMU, KEK, ASIAA, and Princeton University. This paper makes use of
software developed for the Large Synoptic Survey Telescope. We thank the LSST
Project for making their code available as free software at http://dm.lsst.org
This paper is based [in part] on data collected at the Subaru Telescope and
retrieved from the HSC data archive system, which is operated by the Subaru
Telescope and Astronomy Data Center (ADC) at National Astronomical Observatory
of Japan. Data analysis was in part carried out with the cooperation of Center
for Computational Astrophysics (CfCA), National Astronomical Observatory of
Japan. The Subaru Telescope is honored and grateful for the opportunity of
observing the Universe from Maunakea, which has the cultural, historical and
natural significance in Hawaii. Funding for the SDSS and SDSS-II has been
provided by the Alfred P. Sloan Foundation, the Participating Institutions,
the National Science Foundation, the U.S. Department of Energy, the National
Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max
Planck Society, and the Higher Education Funding Council for England. The SDSS
Web Site is http://www.sdss.org/. The SDSS is managed by the Astrophysical
Research Consortium for the Participating Institutions. The Participating
Institutions are the American Museum of Natural History, Astrophysical
Institute Potsdam, University of Basel, University of Cambridge, Case Western
Reserve University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns Hopkins
University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute
for Particle Astrophysics and Cosmology, the Korean Scientist Group, the
Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-
Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for
Astrophysics (MPA), New Mexico State University, Ohio State University,
University of Pittsburgh, University of Portsmouth, Princeton University, the
United States Naval Observatory, and the University of Washington.

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