# The Merian Survey: Design, Construction, and Characterization of a Filter
Set Optimized to Find Dwarf Galaxies and Measure their Dark Matter Halo
Properties with Weak Lensing

Yifei Luo1, Alexie Leauthaud1, Jenny Greene2, Song Huang1,2,3, Erin Kado-
Fong2,4, Shany Danieli2, Ting S. Li5, Jiaxuan Li2, Diana Blanco1, Erik J.
Wasleske6, Joseph Wick1, Abby Mintz2, Runquan Guan1, Annika H. G.
Peter7,8,9,10, Vivienne Baldassare6, Alyson Brooks11,12, Arka Banerjee13, Joy
Bhattacharyya7,9, Zheng Cai3, Xinjun Chen1, Jim Gunn2, Sean D. Johnson14, Lee
S. Kelvin2, Mingyu Li3, Xiaojing Lin3, Robert Lupton2, Charlie Mace7,8,9,
Gustavo E. Medina5, Justin Read15, Rodrigo Córdova Rosado2, Allen Seifert1
1Department of Astronomy and Astrophysics, University of California, Santa
Cruz, 1156 High Street, Santa Cruz, CA 95064 USA
2Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane,
Princeton, NJ 08544, USA
3Department of Astronomy, Tsinghua University, Beijing 100084, People’s
Republic of China
4Physics Department, Yale Center for Astronomy & Astrophysics, PO Box 208120,
New Haven, CT 06520, USA
5Department of Astronomy and Astrophysics, University of Toronto, 50 St.
George Street, Toronto ON, M5S 3H4, Canada
6Department of Physics and Astronomy, Washington State University, Pullman, WA
99163, USA
7The Ohio State University, Department of Physics, Columbus, OH 43210, USA
8The Ohio State University, Department of Astronomy, Columbus, OH 43210, USA
9Center for Cosmology and Astroparticle Physics, 191 West Woodruff Avenue,
Columbus, OH 43210, USA
10School of Natural Sciences, Institute for Advanced Study, Princeton, NJ
08540, USA
11Department of Physics and Astronomy, Rutgers, The State University of New
Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA
12Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue,
New York, NY 10010, USA
13Department of Physics, Indian Institute of Science Education and Research,
Homi Bhabha Road, Pashan, Pune 411008, India
14Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
15Department of Physics, University of Surrey, Guildford, GU2 7XH, United
Kingdom
E-mail<EMAIL_ADDRESS>
###### Abstract

The Merian survey is mapping $\sim$ 850 degrees2 of the Hyper Suprime-Cam
Strategic Survey Program (HSC-SSP) wide layer with two medium-band filters on
the 4-meter Victor M. Blanco telescope at the Cerro Tololo Inter-American
Observatory, with the goal of carrying the first high signal-to-noise (S/N)
measurements of weak gravitational lensing around dwarf galaxies. This paper
presents the design of the Merian filter set: N708 ($\lambda_{c}=7080$Å,
$\Delta\lambda=275$Å) and N540 ($\lambda_{c}=5400$Å, $\Delta\lambda=210$Å).
The central wavelengths and filter widths of N708 and N540 were designed to
detect the $\rm H\alpha$ and $\rm[OIII]$ emission lines of galaxies in the
mass range $8<\rm\log M_{*}/M_{\odot}<9$ by comparing Merian fluxes with HSC
broad-band fluxes. Our filter design takes into account the weak lensing S/N
and photometric redshift performance. Our simulations predict that Merian will
yield a sample of $\sim$ 85,000 star-forming dwarf galaxies with a photometric
redshift accuracy of $\sigma_{\Delta z/(1+z)}\sim 0.01$ and an outlier
fraction of $\eta=2.8\%$ over the redshift range $0.058<z<0.10$. With 60 full
nights on the Blanco/Dark Energy Camera (DECam), the Merian survey is
predicted to measure the average weak lensing profile around dwarf galaxies
with lensing S/N $\sim 32$ within $r<0.5$ Mpc and lensing S/N $\sim 90$ within
$r<1.0$ Mpc. This unprecedented sample of star-forming dwarf galaxies will
allow for studies of the interplay between dark matter and stellar feedback
and their roles in the evolution of dwarf galaxies.

###### keywords:

cosmology: observations – gravitational lensing – dwarf galaxies

##  1 Introduction

Dwarf galaxies are unique probes of the nature of dark matter and baryonic
processes such as stellar feedback (e.g. Collins & Read 2022; Lelli 2022;
Battaglia & Nipoti 2022; Sales et al. 2022). Extensive studies of dwarf
galaxies in the Local Group and beyond have revealed a considerable scatter in
many of their fundamental properties (e.g. Bullock & Boylan-Kolchin 2017).
Reliably characterizing both the dark matter and baryonic components of these
galaxies is key to establishing a complete picture of dark matter on small
scales, understanding the threshold of galaxy formation, and studying dark
matter in dwarf galaxies.

For several decades, $\Lambda$CDM simulations have struggled to reproduce the
properties of observed dwarf galaxies (“Small-Scale Challenges”, Bullock &
Boylan-Kolchin 2017). These challenges include the “Core-Cusp” (e.g. Flores &
Primack 1994; Moore 1994), “Missing Satellites” (e.g. Klypin et al. 1999;
Moore et al. 1999), and “Too-Big-to-Fail” (e.g. Read et al. 2006; Boylan-
Kolchin et al. 2011) problems. In recent years, simulations have produced more
realistic dwarf galaxies due to improvements in resolution and more
physically-motivated treatments of stellar feedback (e.g., Brooks & Zolotov
2014; Wetzel et al. 2016; Brooks et al. 2017). However, it remains unclear
whether feedback prescriptions are realistic enough. Indeed, reproducing both
the distribution of star-formation rates, and the observed scatter in dwarf
rotation curves, has proven challenging for $\Lambda$CDM simulations. This is
known as the “diversity” problem (e.g., Oman et al. 2015; Santos-Santos et al.
2018; Roper et al. 2023). Non-CDM theories of dark matter such as self-
interacting, warm, and fuzzy dark matter (e.g. Fry et al. 2015; Kamada et al.
2017; Robles et al. 2017; Fitts et al. 2019; Nadler et al. 2021) have been
proposed as solutions to the diversity problem. Alternatively, it is possible
that the star formation feedback sub-grid models in simulations are not yet
accurate enough. Or, it could be that data themselves have been
misinterpreted. We know of at least one example, IC1613, where a combination
of stellar and HI gas kinematics show that its inclination is systematically
biased, driving an apparent scatter in the rotation curve (e.g. Oman et al.
2016; Read et al. 2016; Collins & Read 2022). It is not clear, however,
whether such effects can explain all the observed diversity.

Improved measurements of the stellar-to-halo mass relation (SHMR) for dwarf
galaxies are important not only to understand galaxy formation and evolution,
but also to disentangle baryonic effects from non-CDM theories of dark matter
(e.g. Nadler et al. 2020; Munshi et al. 2021; Danieli et al. 2022). The SHMR
is poorly constrained on dwarf galaxy scales, even for massive dwarf galaxies
(e.g., $8<\rm\log M_{*}/M_{\odot}<9$), which are easier to detect compared to
fainter (lower mass) dwarfs (e.g. Miller et al. 2014; Hudson et al. 2015; Read
et al. 2017; Posti et al. 2019; Read & Erkal 2019). The mass range $8<\rm\log
M_{*}/M_{\odot}<9$ is a “sweet spot” for detecting correlations between the
properties of a dwarf’s baryonic matter, the slope of its inner dark matter
halo, and its overall dark matter halo mass (e.g., Di Cintio et al. 2014a, b;
Relatores et al. 2019). Zoom-in hydrodynamical simulations, such as NIHAO and
FIRE-2, have revealed that baryonic feedback is able to efficiently produce
core-like density profiles for galaxies in the mass range $8<\rm\log
M_{*}/M_{\odot}<9$ (e.g. Mashchenko et al. 2008; Governato et al. 2010, 2012;
Pontzen & Governato 2012; Teyssier et al. 2013; Wetzel et al. 2016; Tollet et
al. 2016). We now have compelling observational evidence for such cusp-core
transformations occurring in nearby dwarf galaxies (Read et al. 2019; Bouché
et al. 2022; De Leo et al. 2023). Finally, $\Lambda$CDM simulations with pure
dark matter and observations also show differences in terms of the inner dark
matter halo kinematics at $\rm M_{*}<10^{9}M_{\odot}$ (Klypin et al. 2015;
Papastergis et al. 2015). Constraining the SHMR at these mass scales is
therefore of great interest.

Measurements of the dark matter halos of dwarf galaxies are key to
understanding the dwarf population. Rotation curves have been used to
constrain dark matter halos for decades (e.g., Rubin et al. 1978, 1980;
Boylan-Kolchin et al. 2012; Oh et al. 2015). However, rotation curves only
constrain the inner regions of dark matter halos, where most of the baryons
are located (e.g., stars, gas, globular clusters). A typical $\rm
M_{*}=10^{8.5}M_{\odot}$ dwarf galaxy has a light-weighted effective radius of
$\sim$ 1 kpc, if not smaller (Eigenthaler et al. 2018). The halo radius
R200m111The radius enclosing an overdensity of 200 times the mean matter
density of the universe. for a $\rm M_{*}=10^{8.5}M_{\odot}$ galaxy is
$90-150$ kpc in simulations (e.g., Kravtsov 2013). This means that the radial
scales probed by rotation curves are typically a factor of $\sim$20-30 smaller
than the R200m. Any halo mass estimate from rotation curves is therefore an
extrapolation that relies on assumptions about the shape of the dark matter
profile (e.g., Buckley & Peter 2018; McQuinn et al. 2022). For this reason,
methods for measuring the full dark matter halos of dwarfs will be of great
value.

A powerful way to directly probe total halo masses is via weak gravitational
lensing. In particular, the “galaxy-galaxy lensing” technique measures the
average weak lensing signal from background “source” galaxies around a sample
of foreground “lens” galaxies (e.g., Brainerd et al. 1996). Existing weak
lensing measurements have been limited to galaxies with $\rm
M_{*}>10^{9}M_{\odot}$ (e.g., Leauthaud et al. 2012; Hudson et al. 2015;
Dvornik et al. 2022). Leauthaud et al. (2020) recently predicted that lensing
surveys such as the HSC Strategic Survey Program (HSC-SSP, e.g. Aihara et al.
2018), $\rm Euclid$ (Laureijs et al. 2011), the $\rm Roman$ Space Telescope
(Spergel et al. 2013), and the Legacy Survey of Space and Time (LSST, Ivezić
et al. 2019) with the Rubin Observatory, will be sufficiently deep and wide to
provide large samples of dwarf galaxies out to $z\sim 0.2$, enabling
measurements of galaxy-galaxy lensing for dwarf galaxies with high
significance.

This paper focuses on a weak lensing survey for dwarf galaxies using the
existing HSC-SSP survey. The HSC-SSP wide layer is mapping $\sim$1000 deg2
down to i $\sim$ 26 mag in $grizy$ (Aihara et al. 2018). The $5\sigma$ depths
for $g$, $r$, $i$, $z$, and $y$ in the HSC-wide field are 26.5, 26.1, 25.9,
25.1, 24.4 mag, respectively (Aihara et al. 2018). The HSC-SSP survey has both
the required depth and area coverage to select large samples of low redshift
dwarfs (see more details in Section 2). Note that several other existing
surveys on the Dark Energy Camera (DECam) also have the ability to detect
large samples of dwarf galaxies for lensing. For example, the Dark Energy
Survey (DES, Abbott et al. 2021) covers $\sim$5000 deg2 down to $i\sim 23.8$
($10\sigma$ depth). The DECam Local Volume Exploration survey (DELVE, Drlica-
Wagner et al. 2022) covers $\sim$17,000 deg2 down to $i\sim 23.5$ ($5\sigma$
depth). Both of these two surveys could also form large samples of dwarf
galaxies below $z\sim 0.2$ that are comparable to HSC-SSP or even larger,
benefiting from their wide survey areas. However, our decision to utilize the
HSC-SSP survey in this work is due to several other reasons. First, the weak
lensing measurements require not only a large sample of lens galaxies, but
also a large sample of source galaxies. Accurate galaxy shape measurements of
these distant sources are crucial for lensing analysis. Imaging surveys with a
better Point Spread Function (PSF) have less smearing effect, which could
provide more precise shape measurements. Among the three existing deep and
wide surveys mentioned above, HSC-SSP has the best PSF (i-band median seeing
$\sim$0.61 arcsec) compared to DES (i-band median seeing $\sim$0.88 arcsec)
and DELVE (i-band median seeing $\sim$1.02 arcsec). Furthermore, HSC-SSP is
also deeper than the other two surveys, which means it can detect more source
galaxies at higher redshifts. The high number density of the source galaxies
in HSC-SSP will also help with the lensing measurements. Therefore, we choose
to use the HSC-SSP survey to compile our dwarf galaxy sample for lensing in
this work.

Although HSC-SSP is deep enough to detect a large sample of dwarf galaxies,
their broad-band photometric redshifts are not optimized for dwarf galaxies at
low redshifts. Weak lensing measurements usually need redshifts for lens
galaxies with an accuracy of $\Delta z<0.2$, but the typical photo-$z$
precision at $z\sim 0.1$ from broad-band surveys with $grizy$ bands (such as
HSC) is $\Delta z\sim 0.5$ (Tanaka et al. 2018; Speagle et al. 2019). This is
because the HSC filter set does not span the 4000Å break at low redshifts.
Furthermore, most dwarf galaxies in the mass range $8<\rm\log
M_{*}/M_{\odot}<9$ are star-forming (Geha et al. 2012; Mao et al. 2021) and do
not have a prominent 4000Å break. Thus, the inclusion of $u$-band data would
not guarantee accurate redshifts for star-forming dwarf galaxies. In some
cases, imaging-based machine learning methods can be used to improve redshift
estimates. For example, Darragh-Ford et al. (2022) trained a Convolutional
Neural Network (CNN) with broad-band images from the Dark Energy Camera Legacy
Survey (DECaLS, Dey et al. 2019) to select dwarf galaxies at $z<0.03$. This
method achieved a 90% completeness, but with only 20% efficiency. Although
their efficiency is higher than traditional photometric methods ($\sim$ 1%),
this method still requires spectroscopic follow-up observations.

Narrow and medium band filters have also been used with great success to
improve redshift estimates. This approach takes advantage of the fact that
narrow/medium-band filters can be used to isolate specific spectral features,
such as emission lines or absorption features, which provide additional
redshift information. By integrating these features into the photometric
redshift estimation procedure, redshift estimates can be significantly
improved, particularly for galaxies with strong emission lines. This technique
has been successfully applied in a number of imaging surveys, such as the
COSMOS survey (Ilbert et al. 2009; Laigle et al. 2016), the ALHAMBRA survey
(Moles et al. 2008), the PAU survey (Benítez et al. 2009), the J-PAS survey
(Benitez et al. 2014), the S-PLUS survey (Mendes de Oliveira et al. 2019), the
SILVERRUSH survey (Ouchi et al. 2018), the LAGER survey (Zheng et al. 2017,
2019), and the ODIN survey (Lee et al. 2024). In this paper, we present the
design and optimization of a dual filter system on DECam to detect the
emission lines of dwarf galaxies in the range $8<\rm\log M_{*}/M_{\odot}<9$ in
the HSC-SSP fields.

The Merian survey222Honoring 17th century Maria Sibylla Merian (1647, 1717),
one of the first female entomologists and naturalists. Her fascination with
the world of tiny things combined with unique observational skills led to a
number of important discoveries, including the previously unknown
metamorphosis of caterpillars into
butterflies.333https://merian.sites.ucsc.edu/ is an ongoing large imaging
survey with 60 nights on the Victor M. Blanco 4-m telescope at the Cerro
Tololo Inter-American Observatory (CTIO). Merian uses imaging in two medium-
band filters to build a sample of dwarf galaxies at $0.058<z<0.10$. Merian
will image $\sim$ 850 deg2 of the HSC-SSP survey and is expected to detect
$\sim$ 85,000 dwarf galaxies in the mass range $8<\rm\log M_{*}/M_{\odot}<9$.
This paper presents the design of the Merian filters. Merian will yield the
first high S/N weak lensing profile measurements for dwarf galaxies. HSC-SSP
imaging data will also provide exquisite measurements of dwarf sizes, shapes,
stellar masses, and star-formation rates. The Merian Survey design, observing
strategy, data reduction, and preliminary data products will be presented in a
companion paper (Danieli et al., in prep).

This paper is organized as follows. The data and sample used to design the
Merian filters are described in Section 2. The filter design methodology is
presented in Section 3. Section 4 shows the filter optimization results and
the expected photo-$z$ performance. Section 5 describes the construction of
the Merian filter set, and shows the performance of two Merian filters using
an early reduction of the Merian data. Section 6 presents a summary and our
conclusions. We adopt a flat $\Lambda$CDM cosmology with $H_{0}$ = 70 km $\rm
s^{-1}\ Mpc^{-1}$, $\Omega_{m}$ = 0.3 and $\Omega_{\Lambda}$ = 0.7. We use
physical units for the galaxy-galaxy lensing observable $\Delta\Sigma$. All
magnitudes are in the AB system (Oke & Gunn 1983).

##  2 Data Used for Filter Design

In this section, we introduce the data set used to design the Merian dual
filter system. We show the basic observational properties of a mass complete
sample of dwarf galaxies within a target mass and redshift range drawn from
the COSMOS2015 catalog. We also build a model SED library for our target dwarf
galaxies by fitting the 30-band photometry from the COSMOS2015 catalog to
predict the fluxes of these dwarf galaxies in different filter designs.

###  2.1 The COSMOS catalog

To predict the S/N of fluxes of dwarf galaxies in different filter designs, we
first need a sample of dwarf galaxies (our truth table) to fully understand
their observational properties. This truth table should contain extensive
photometric data, redshift measurements, and be deep enough to be complete in
the stellar mass and redshift range of interest ($8<\rm\log M_{*}/M_{\odot}<9$
and $z<0.2$).

We use the COSMOS2015 catalog as our baseline for understanding the properties
of dwarfs to inform our analysis. The COSMOS2015 catalog (Laigle et al. 2016)
offers a wealth of deep multi-wavelength photometric data over 2 deg2 in the
COSMOS field (Scoville et al. 2007) and contains nearly 500,000 objects. It
combines photometry from X-ray (Chandra, Civano et al. 2016), near-UV (GALEX,
Zamojski et al. 2007), optical (CFHT/MegaCam, Subaru COSMOS-20, and Hyper-
Suprime-Cam, Taniguchi et al. 2007, 2015; Miyazaki et al. 2018), near-IR
(UltraVISTA, McCracken et al. 2012), mid-IR (MIPS/Spitzer, Le Floc’h et al.
2009), and far-IR (PACS/Herschel and SPIRE/Herschel, Lutz et al. 2011; Oliver
et al. 2012). The UV and optical photometry achieve a 3$\sigma$ depth of at
least 25 mag. We obtain photometric redshifts ($z$PDF), stellar masses
(MASSMED), half-light radii, and star-formation rates (SFR) from the
COSMOS2015 catalog (Laigle et al. 2016). We start with galaxies (TYPE=0)
within the 2 deg2 COSMOS area from the COSMOS2015 catalog (FLAG_COSMOS=1), and
then select galaxies with reliable photometric redshifts in the UltraVISTA
area (FLAG_HJMCC=0), not influenced by artifacts around saturated objects
(FLAG_PETER=0). We obtain the half-light radii measured from high resolution
HST/ACS F814W COSMOS images (Leauthaud et al. 2007).

The COSMOS2015 catalog provides photometric redshift measurements with high
accuracy ($\sigma_{\Delta z/(1+z)}\sim 0.01$ for both star-forming and
quiescent galaxies with $i\sim 23$, and $\sigma_{\Delta z/(1+z)}\sim 0.02$ for
fainter objects with $23<i<24$), along with stellar mass and star-formation
rate estimates (Laigle et al. 2016). These properties are derived with
template fitting using the Spectral Energy Distribution (SED) fitting code
LEPHARE (Arnouts et al. 2002; Ilbert et al. 2006), assuming a Chabrier (2003)
Initial Mass Function (IMF), and a combination of exponentially declining and
delayed star-formation histories (SFH) ($\rm SFR\sim\tau^{-2}te^{-t/\tau}$
with e-folding time $\tau$). Figure 1 shows stellar mass vs. 30-band
photometric redshift for galaxies from the COSMOS2015 catalog.

We also obtained spectroscopic redshifts (spec-$z$’s) from the COSMOS team
(Salvato et al., in prep). This catalog includes spectroscopic redshifts from
several spectroscopic surveys, including the $z$COSMOS survey (Lilly et al.
2007), the VIMOS Ultra Deep Survey (VUDS, Le Fèvre et al. 2015), the Complete
Calibration of the Color–Redshift Relation Survey (C3R2, Masters et al. 2019),
the DEIMOS 10K Spectroscopic Survey (Hasinger et al. 2018), and the FMOS-
COSMOS Survey (Kashino et al. 2019). Approximately 30 percent of galaxies at
$z<0.2$ and $\rm\log M_{*}/M_{\odot}>8$ have spectroscopic redshift
measurements (indicated by red points in Figure 1). Figure 2 shows $i$-band
magnitude vs. redshift for dwarf galaxies in our target mass range $8<\rm\log
M_{*}/M_{\odot}<9$ color-coded according to their redshift type. As can be
seen, spec-$z$’s are incomplete for faint dwarf galaxies (e.g. $i>23$). The
selection functions of existing spec-$z$ samples are complicated, i.e. they
are not representative in colors/brightness, since most of the large
spectroscopic surveys in COSMOS targeted galaxies at higher redshifts. In the
remainder of this paper, when using the COSMOS2015 redshifts ($\rm
z_{COSMOS}$), we will use the spectroscopic redshift when available, and the
photometric redshift $z$PDF (median of the likelihood distribution) otherwise.

Figure 1: Stellar mass vs. redshift for galaxies in the COSMOS2015 catalog.
Red points indicate galaxies with spectroscopic redshifts and grey points
correspond to galaxies with COSMOS 30-band photometric redshifts. The black
box indicates the redshift and stellar mass range targeted by the Merian
survey and corresponds to the filter design presented in this paper (see
details in Section 4). Figure 2: $i$-band magnitude vs. redshift for dwarf
galaxies with $8<\rm\log M_{*}/M_{\odot}<9$ in the COSMOS2015 catalog. Red
points indicate galaxies with spectroscopic redshifts and grey points
correspond to galaxies with COSMOS 30-band photometric redshifts. Figure 3:
Distributions of four basic properties ($i$-band magnitude, $u-r$ color, half-
light radius, and specific star-formation rate) for dwarf galaxies with
$8<\rm\log M_{*}/M_{\odot}<9$ and $\rm z_{COSMOS}<0.2$ (our Initial Dwarf
Sample) from the COSMOS2015 catalog. Blue points (histograms) and orange
crosses (histograms) correspond respectively to star-forming and quiescent
dwarf galaxies in the Initial Dwarf Sample according to the classification in
the COSMOS2015 catalog.

###  2.2 Starting Parameter Space for Filter Design

Our goal is to work in the mass and redshift range where HSC is complete and
unaffected by surface brightness limits. Leauthaud et al. (2020) showed that
detections in both COSMOS and HSC would be primarily affected by surface
brightness effect for galaxies with $\rm\log M_{*}/M_{\odot}\sim 8$ at $z>0.2$
and that galaxies with $\rm\log M_{*}/M_{\odot}\sim 7.5$ are $\sim$90%
complete at $z=0.2$ (see Figure 1 in Leauthaud et al. 2020). Therefore, we
focus on galaxies in the target mass and redshift range: $8<\rm\log
M_{*}/M_{\odot}<9$ and $z<0.2$ where HSC should not be affected by surface
brightness limits.

As shown in Figure 2, dwarf galaxies with $8<\rm\log M_{*}/M_{\odot}<9$ at
$\rm z_{COSMOS}<0.2$ can reach magnitudes as faint as $i\sim 24$. It is
important to note that the final Merian survey target redshift range does not
extend to $z=0.2$, but rather spans $0.058<z<0.1$. Consequently, most of the
Merian target dwarfs are actually brighter than $i\sim 23$, as shown in Figure
2 (see other properties for the final Merian target dwarfs in Figure 9 in
Section 4). Nonetheless, for the filter design process, we started with dwarf
galaxies at $z<0.2$. For this, it was important to have a sample of dwarfs
with deep enough data (down to $i\sim 24$) to use as a truth table. In
reality, very few large imaging surveys are complete down to $i\sim 24$, and
none of the large spectroscopic surveys are deep enough to form a library of
SEDs for our target dwarf galaxies. For instance, the Sloan Digital Sky Survey
(SDSS) and Galaxy And Mass Assembly (GAMA) surveys are only complete down to
$r<17.77$ and $r<19.8$ mag (much shallower than $i\sim 24$), respectively
(York et al. 2000; Liske et al. 2015). Deeper spectroscopic surveys do exist,
such as VVDS ($I_{\rm AB}<22.5$), PRIMUS ($i_{\rm AB}<23.5$), and DEEP2
($R_{\rm AB}<24.1$) (Le Fèvre et al. 2013; Coil et al. 2011; Newman et al.
2013), but deeper surveys usually use specific color selections to target
high-redshift galaxies, thereby excluding some dwarf galaxies in our target
mass and redshift range. For this reason, we choose to use the COSMOS2015
catalog as our initial truth table and to build our dwarf SED library. We
select 1,822 dwarf galaxies with $8<\rm\log M_{*}/M_{\odot}<9$ at $\rm
z_{COSMOS}<0.2$ from the COSMOS2015 catalog, of which 639 have spectroscopic
redshifts. We refer to this sample as the “Initial Dwarf Sample” and the basic
properties of this sample are presented below.

###  2.3 Initial Dwarf Sample

Figure 3 shows the distributions of $i$-band apparent magnitude, $u-r$ color,
half-light radius in arcseconds, and specific star-formation rate (sSFR) for
star-forming and quiescent galaxies for the Initial Dwarf Sample. The star-
forming/quiescent classification used here is from the COSMOS2015 catalog, and
is based on absolute $\rm NUV-r$ and $\rm r-J$ colors (Laigle et al. 2016).
The $K$-correction in each band is estimated from the 30-band SED fitting.
Quiescent objects have $\rm NUV-r>3(r-J)+1$ and $\rm NUV-r>3.1$. According to
this classification, 95% of dwarf galaxies in the Initial Dwarf Sample are
star-forming (blue points/histograms in Figure 3). Similarly, nearly 95% of
dwarf galaxies in the Initial Dwarf Sample have an sSFR greater than 10-11
yr-1. Overall, these dwarf galaxies have an $i$-band median apparent magnitude
of $\overline{i}\sim 22$ mag and 99% have $i<24$ mag, which should easily be
detected in HSC broad-band imaging data. The Initial Dwarf Sample has a median
half-light radius (measured from the HST/ACS F814W catalog of Leauthaud et al.
2007) of 0.6 arcseconds. The median seeing in the HSC $i$-band is also 0.6
arcsec (Aihara et al. 2018). Therefore, many dwarfs in the Initial Dwarf
Sample will be extended objects in HSC catalogs. Most galaxies in our target
mass range $8<\rm\log M_{*}/M_{\odot}<9$ will have a disc like bulge or will
be entirely bulgeless (Dutton 2009; Fisher & Drory 2011) and can be treated as
exponential disks (Kado-Fong et al. 2020; Kado-Fong et al. 2021). For
simplicity, we use exponential light profiles when simulating dwarf images
given different filter designs (see Section 3.3).

###  2.4 SED library for the Initial Dwarf Sample

Figure 4: Model SEDs (normalized to 5500Å) for the Initial Dwarf Sample
generated with Prospector, color-coded by the $\rm H\alpha$ equivalent width.
Every galaxy in this dwarf SED library has a model SED with continuum and
emission lines, and are used to predict fluxes for different filter designs.

Although COSMOS2015 provides a wealth of photometry and data products from
their 30-band SED fitting (e.g., photo-$z$’s, stellar masses, and sSFR), it
does not include a best-fit SED with emission lines and continuum for every
galaxy, which is crucial for the Merian filter design. Therefore, we run our
own SED fitting to the Initial Dwarf Sample to build a dwarf SED library. We
now describe the photometric data used for this SED fitting process.

Our analysis of the Initial Dwarf Sample indicates that the photometric data
in most of the COSMOS2015 bands have sufficient depth for our goals, with a
few exceptions. Only 70% of the galaxies in the Initial Dwarf Sample have S/N
$>$ 3 in GALEX-FUV and GALEX-NUV, while over 95% of galaxies have S/N $>$ 3
for all other bands in the Subaru COSMOS 20 band survey. In addition, because
our dwarf galaxies have relatively small angular sizes, we need the imaging
PSF to be uniform through all the bands to avoid blending issues which could
affect the SED fitting. For these reasons, we choose to fit the SEDs only
using data from MegaCam-u∗, HSC-y, VIRCAM-Y/J/H/Ks and the 20 optical filters
from the Subaru COSMOS 20 band survey (Taniguchi et al. 2007, 2015). The
Subaru COSMOS 20 band survey contains photometry from 6 broad-bands (B, V, g,
r, i+, z++), 12 medium-bands (IA427, IA464, IA484, IA505, IA527, IA574, IA624,
IA679, IA709, IA738, IA767, IA827), and 2 narrow-bands (NB711, NB816). Note
that these medium and narrow band filters were for the now decommissioned
Subaru SuprimeCam and therefore cannot be used for Merian. All the bands that
we use for SED fitting have S/N$>$3 for more than 90% of dwarfs in the Initial
Dwarf Sample.

We perform SED fitting for the Initial Dwarf Sample using the code Prospector
(Johnson et al. 2021). Prospector is a modular galaxy stellar populations
inference code based on Bayesian forward modeling and Monte Carlo sampling of
the parameter space using stellar populations generated by the Flexible
Stellar Population Synthesis (FSPS) code (Conroy et al. 2009; Conroy & Gunn
2010). We use a parametric delayed-$\tau$ star formation history, a Calzetti
et al. (2000) attenuation curve, and free gas-phase and stellar metallicities
in the fitting process. We use the MIST stellar evolutionary isochrones and
the MILES spectral library in FSPS (Choi et al. 2016; Sánchez-Blázquez et al.
2006) for the stellar population. Nebular continuum and emission lines are
included using the pre-compiled CloudyFSPS models in Byler et al. (2017). Note
that the choice of the stellar isochrones may affect the emission line
luminosities due to the treatment of stellar rotation (Byler et al. 2017).
Isochrones which include stellar rotation (e.g. MIST) produce better matches
to emission line ratios in diagnostic diagrams such as the Baldwin-Phillips-
Terlevich (BPT) diagram (Baldwin et al. 1981), but they sometimes overestimate
the amount of ionized photons (Byler et al. 2017). There is a possibility that
the MIST isochrones overestimate the emission line luminosities in the medium-
band filters for some dwarf galaxies, which would affect their observed S/N.
However, we defer a more in-depth consideration of this effect for future
work.

During the SED fitting process, we fix the redshift to the best fit 30-band
photo-$z$ ($\rm z_{COSMOS}$) from the COSMOS2015 catalog. The spec-$z$ is used
when available. We use the dynamic nested sampling package dynesty to sample
the posteriors (Speagle 2020). Figure 4 shows the SED library (normalized to
5500Å) generated by Prospector for the Initial Dwarf Sample, color-coded by
the $\rm H\alpha$ equivalent width measured directly from the model SEDs. Each
model SED includes both continuum and emission lines and can be utilized to
predict fluxes for dwarf galaxies in any filter design.

###  2.5 Predicting Fluxes from SEDs

Figure 5: Comparison between predicted HSC $g-i$ and $g-z$ colors from our
model SEDs and the observed data from HSC-PDR3 for the Initial Dwarf Sample.
Galaxies with existing spec-$z$’s are highlighted as red points, while grey
open circles are galaxies with a COSMOS 30-band photo-$z$’s. The median $\mu$
and root mean square $\sigma$ of the y-axes for the whole sample (red and grey
points) are shown in the upper right corners. Our SED fitting process
accurately predicts HSC colors that were not included in our input data vector
for the Initial Dwarf Sample.

To verify that the SEDs we generated for the Initial Dwarf Sample in Section
2.4 produce reasonable fluxes in different filters, we predict fluxes in HSC
broad bands by convolving model SEDs with HSC transmission curves and compare
them with observed HSC fluxes. Since we care more about the relative colors
between different bands than absolute fluxes, we obtain the broad-band PSF-
matched aperture photometry (the “afterburner photometry”) with target 1.09
arcsec PSF and 1.5 arcsec apertures taken from HSC-PDR3 database (Aihara et
al. 2018). This is recommended for accurate colors and is the photometry used
for HSC photo-$z$ estimates (Tanaka et al. 2018).

Figure 5 shows the difference between the HSC $g-i$ (and $g-z$) colors from
our model SEDs and the observed data from HSC-PDR3 for the Initial Dwarf
Sample. The mean value ($\mu$) of the $g-i$ and $g-z$ difference for the
Initial Dwarf Sample is $0.009$ mag and 0.007 mag respectively, with a scatter
of $\sigma=0.058$ mag and $\sigma=0.06$ mag. Note that the HSC $g-i$ PSF color
has an accuracy of $\mu\sim 0.01$ and a precision of $\sigma=0.08$ mag for
objects with $i\sim 23$ mag (Huang et al. 2018). The precision of the colors
generated from our model SED library is within the HSC photometric precision,
which proves that our model SED library provides reliable colors. Galaxies
with 30-band photo-$z$’s (grey points) do not have significantly larger
scatter than galaxies with spec-$z$’s (red points), indicating that the
precision of the 30-band photo-$z$’s is comparable to spec-$z$’s when
generating model SEDs. For the Merian survey, we will use the two Merian
medium-band filters along with the five HSC broad-band data to improve the
photo-$z$’s. We will use the predicted HSC broad band photometry presented
here when characterizing the Merian photo-$z$ performance in Section 4.3.

To create a sample of dwarfs at any given redshift, we convert apparent sizes
in arcsec into physical sizes in kpc, and observed SEDs into rest-frame SEDs,
assuming no size and SED evolution below $z<0.2$. We then use the full Initial
Dwarf Sample to create mock dwarf samples and mock dwarf fluxes for different
filter designs.

##  3 Filter design methodology

In this section, we discuss the method used to design the filter set for the
Merian survey. The rational for using narrow/medium-band filters is to capture
specific spectral features (i.e. emission lines for our target dwarf galaxies)
to improve the precision of photo-$z$’s. Since most of the dwarfs in our
target mass range are star-forming (as shown in Figure 3), they will have
strong emission lines. $\rm H\alpha$ and $\rm[OIII]\lambda 5007$ (hereafter
$\rm[OIII]$) are often the strongest emission lines for galaxies in the
Initial Dwarf Sample. Although $\rm[OII]\lambda 3725$ and $\rm[OII]\lambda
3727$ are also strong emission lines in the Initial Dwarf Sample, their
wavelengths are below 4500Å at $z<0.2$. Since DECam is relatively inefficient
below 4500Å we only focus on $\rm H\alpha$ and $\rm[OIII]$ for the filter
design. We start with the optimization of the $\rm H\alpha$ filter, and then
design the $\rm[OIII]$ filter (which helps remove interlopers at higher
redshfits, see details in Section 4.2) based on the corresponding redshift
range given by the optimized $\rm H\alpha$ filter. We optimize the $\rm
H\alpha$ filter central wavelength ($\lambda_{c}$) and full width at half
maximum FWHM ($\Delta\lambda$) (that defines the filter bandpass) following
the optimization criteria and method described below.

###  3.1 Optimization Criteria

The primary goal of the Merian survey is to build a large photometric sample
of dwarf galaxies that can be used to measure the dark matter halos of dwarfs
using galaxy-galaxy lensing. For a fixed observing time, there is a trade-off
between survey volume and survey depth. More specifically, we design the
filter set based on the optimization of two competing requirements:

  * •

Sample Size: we aim to obtain a sample of dwarf galaxies in the mass range
$8<\rm\log M_{*}/M_{\odot}<9$ that is large enough to achieve a weak lensing
S/N from HSC-SSP greater than 30 within $r<0.5$ Mpc.. The S/N of the lensing
signal depends directly on the number of dwarf (lens) galaxies. A wider
redshift range (i.e. larger volume) contains more galaxies. For a specific
emission line, this means wider wavelength range (larger filter
$\Delta\lambda$). A larger $\lambda_{c}$ corresponds to a higher central
redshift which also yields a larger volume. Thus, this requirement prefers a
wider filter $\Delta\lambda$ and larger $\lambda_{c}$.

  * •

Sample Completeness: we want to ensure that the detection completeness of
star-forming dwarf galaxies in the target mass range achieves at least 80$\%$
in the Merian filters. This requirement prefers a narrower filter
$\Delta\lambda$ to ensure that the emission line strength can be distinguished
from the continuum. A larger $\lambda_{c}$, i.e. higher redshift for a given
emission line, results in fainter apparent magnitude, which is harder to
detect. Therefore this requirement also prefers low $\lambda_{c}$. In
addition, it is important to take into account sky emission lines at specific
wavelengths. In general, sky lines become stronger above 8000Å. Therefore,
this requirement prefers filters with narrow $\Delta\lambda$ and central
wavelengths below 8000Å.

Our final design represents a balance between these competing requirements.

###  3.2 Lensing Signal-to-Noise Ratio

To evaluate the weak gravitational lensing performance of different filter
designs we need to predict the uncertainties on the galaxy-galaxy lensing
signal for dwarf galaxies with survey parameters corresponding to a given
filter design. The galaxy-galaxy lensing signal that we predict is the excess
surface mass density $\Delta\Sigma$, which is defined as:

$\Delta\Sigma(r)\equiv\overline{\Sigma}(<r)-\overline{\Sigma}(r)=\Sigma_{c}\times\gamma_{t}(r),$
(1)

where $\overline{\Sigma}(<r)$ is the mean surface density within proper radius
r, $\overline{\Sigma}(r)$ is the azimuthally averaged surface density at
radius r, $\Sigma_{c}$ is the critical surface mass density, and
$\gamma_{t}(r)$ is the tangentially projected shear (e.g. Miralda-Escude 1991;
Wilson et al. 2001; Leauthaud et al. 2012). The lensing S/N depends on the
number of lens galaxies (in our case, these are the target dwarf galaxies) and
the number density of source galaxies. The number of lens galaxies is
determined by the area and volume of the medium band survey. The redshift
range of the survey is set by the wavelength range of the filter.

We use the HSC-SSP wide layer for the basis for our predictions. To predict
the number of lens galaxies, we assume a survey area that can take on a
maximum value of 1000 deg2. We use the COSMOS2015 stellar mass function to
compute the number density of dwarf galaxies given redshift range within the
mass range $8<\rm\log M_{*}/M_{\odot}<9$. The combination of the galaxy number
density and the survey volume gives the number of lens galaxies. For source
galaxies, we assume the HSC S16A source density and a mean redshift of
$\langle z_{s}\rangle=0.81$ (e.g. Hikage et al. 2019).

We compute the predicted lensing uncertainties for $\Delta\Sigma$ using the
same methodology described in Singh et al. (2017). Here we briefly summarize
the salient features of this computation and refer the reader to Singh et al.
(2017) for further details. The Gaussian covariance for the galaxy–galaxy
lensing measurement $\Delta\Sigma$ within a projected separation $r$ is given
by:

${\rm
Cov}(\Delta\Sigma(r),\Delta\Sigma(r^{\prime}))\approx\frac{\Sigma_{c}^{2}(\chi_{s},\chi_{g})}{V_{W}}\int
dk_{\perp}k_{\perp}J_{2}(k_{\perp}r)$
$J_{2}(k_{\perp}r^{\prime})\left[\left(P_{gg}(k_{\perp})+\frac{1}{n_{g}}\right)\left(P_{\kappa\kappa}(k_{\perp})+\frac{\sigma^{2}}{n_{s}}\right)+P_{g\kappa}^{2}(k_{\perp})\right],$
(2)

where $\Sigma_{c}$ is the critical surface density, $\chi_{g}$ and $\chi_{s}$
are the comoving distances to lens and source galaxies, $V_{W}$ is the
physical volume of the survey, $k_{\perp}$ is the wave vector perpendicular to
the line-of-sight, $J_{2}$ is the second-order spherical Bessel function,
$P_{gg}$ is the lens galaxy power spectrum, $\sigma$ is the shape noise,
$n_{s}$ is the source galaxy density, $n_{g}$ is the lens galaxy density,
$\frac{\sigma^{2}}{n_{s}}$ is the shape noise term, and $\frac{1}{n_{g}}$ is
the lens galaxy shot noise term. For the power spectrum, we use the HaloFit
nonlinear power spectrum (Smith et al. 2003; Takahashi et al. 2012). For the
lens galaxy power spectrum $P_{gg}$, we use linear galaxy bias with the non-
linear matter power spectrum. The galaxy-matter power spectrum
($\Sigma_{c}^{2}P_{g\kappa}$) is obtained by direct inverse Hankel transform
of $\Delta\Sigma$. The convergence power spectrum, $P_{\kappa\kappa}$ in units
of $P(k)$ is given by:

$P_{\kappa\kappa}(k_{\perp})=\int_{0}^{\chi_{s}}d\chi\frac{\overline{\rho}^{2}}{\Sigma_{c}^{2}(\chi_{s},\chi)}P_{mm}(k_{\perp}\frac{\chi_{g}}{\chi}),$
(3)

where $\overline{\rho}$ is the mean density and $P_{mm}$ is the matter power
spectrum.

These $\Delta\Sigma$ errors include all the terms needed to describe the
Gaussian covariance. We do not account for the effects due to selection
functions or complicated survey masks. We also ignore contributions from the
connected covariance which includes super sample covariance, as well as the
trispectrum between galaxies and shear. These ignored terms will result in an
overestimated lensing S/N by up to $\sim$25%, hence we apply a $\sim$25%
reduction in our lensing S/N estimates. The exact normalization of this
calculation is not important for this paper since the filter design uses
relative comparisons between S/N estimates. For further details, please see
Singh et al. (2017); Leauthaud et al. (2020).

###  3.3 Simulating Filter Performance

We now describe our method for predicting the performance of various $\rm
H\alpha$ filter designs and optimizing the filter central wavelength
($\lambda_{c}$) and full width at half maximum FWHM ($\Delta\lambda$) based on
the optimization criteria presented in Section 3.1. As discussed in Section
2.3, some dwarf galaxies in the Initial Dwarf Sample are extended sources in
HSC. In order to more realistically predict the fluxes and uncertainties for
dwarf galaxies in different filter designs, we generate mock images to run the
filter optimization. We generate dwarf mock images as follows.

  * •

For each $\rm H\alpha$ filter design (a combination of $\lambda_{c}$ and
$\Delta\lambda$), we re-assign physical properties for every dwarf galaxy in
the Initial Dwarf Sample. We consider the redshift range corresponding to a
given filter choice. The redshift range is derived from the wavelength range
given $\lambda_{c}$ and $\Delta\lambda$ for the $\rm H\alpha$ emission line.
We then assign a random redshift to every galaxy in the Initial Dwarf Sample
drawn uniformly from the redshift range of a given filter choice. We convert
physical sizes in kpc to apparent sizes in arcsec using size estimates from
the COSMOS ACS catalog. We then convert the SED from the rest-frame to the
observed-frame using this newly assigned redshift and then compute fluxes in
the HSC broad bands and the Merian $\rm H\alpha$ band.

  * •

We create dwarf galaxy images assuming that the light profiles of our target
dwarf galaxies follow circular exponential profiles with a face-on
orientation. The choice of orientation is a conservative assumption since it
corresponds to the lowest S/N of fluxes due to minimum surface brightness. We
create mock postage stamp images for these galaxies with a given exposure time
and the instrumental parameters of DECam using the galaxy image simulation
tool Galsim (Rowe et al. 2015). We assume a fixed 1.0 arcsec seeing and dark
time condition. The sky background is added using a model optical sky spectrum
for CTIO. Mock fluxes and S/N values were checked against the Exposure Time
Calculator (ETC) provided by DECam.

  * •

We generate dwarf galaxy images with and without emission lines (i.e.
continuum only) for each filter design, assuming the same spatial
distributions for the continuum and emission line regions. We then use the
photometry code SEP (Barbary et al. 2015) to measure the total fluxes of
galaxies. We use FLUX_AUTO from SEP as well as the associated flux
uncertainties (FLUXERR_AUTO). The flux with emission lines is noted $\rm
Flux_{EL}$ and the flux without emission lines is noted $\rm Flux_{CONT}$.

At this stage, we assume that galaxy centroids are known, since the position
of galaxies can be well-determined from the deeper broad-band survey data.
Indeed, Merian aims to take advantage of the deeper HSC photometry to perform
forced photometry on the shallower medium band images. We now need a criterion
to determine if the emission line of a dwarf galaxy has been detected in the
medium band filters. We consider an emission line to be detected if $\rm
Flux_{EL}-Flux_{CONT}>5\times Fluxerr_{EL}$ where $\rm Fluxerr_{EL}$ is the
FLUXERR_AUTO for images with emission lines.

We run image simulations for different filter designs with $\lambda_{c}$
corresponding to $\rm H\alpha$ at $0.02<z<0.2$ and $\Delta\lambda$ values
between 100Å and 400Å, and starting with 600-second exposures on DECam. The
upper limit of the total survey area is set to be $\rm 1000\ deg^{2}$ given
the size of the HSC-SSP wide layer. During the optimization, if the detected
dwarf fraction is below 80%, we increase the exposure time by 300s. Longer
exposure times result in smaller survey areas at a fixed total number of
nights. Note that the filter design does not depend on the total number of
nights, only the predicted lensing S/N depends on the number of nights. We
adopt 24 full nights for the $\rm H\alpha$ filter design, which is the actual
number of nights allocated to the $\rm H\alpha$ filter for Merian. In
practice, Merian has 60 full nights in total, where 36 nights are dedicated to
the $\rm[OIII]$ filter (see Section 4.1 for details).

We use the stellar mass function from Davidzon et al. (2017) to compute the
number of dwarf galaxies with $8<\rm\log M_{*}/M_{\odot}<9$ in the redshift
range corresponding to each filter design. We multiply the total number of
galaxies by 80% to roughly account for the expected sky area loss due to
bright stars (e.g., Leauthaud et al. 2007; Heymans et al. 2012; Coupon et al.
2018). For each filter design, we compute the total number of detected dwarf
galaxies, and the survey area assuming 24 full nights on DECam for the $\rm
H\alpha$ filter. The lensing S/N is then computed using the method described
in section 3.2 based on the corresponding redshift range and the total number
of dwarf galaxies (which serve as lens galaxies).

Figure 6: Filter width vs. filter central wavelength for different designs of
the $\rm H\alpha$ filter. Upper and lower x-axes show the filter central
wavelength in Å and the corresponding redshift for $\rm H\alpha$. Left and
right y-axes show the filter width in wavelength and the corresponding
redshift range. Colors in the left panel represent the predicted weak lensing
S/N at $r<0.5$ Mpc given a fixed observing time of 24 full nights on DECam
(for $\rm H\alpha$). The right panel shows the expected number of dwarf
galaxies (lenses) for each filter design. The dotted diagonal lines indicate
the location of the strong sky emission line at 5580Å for a matched
$\rm[OIII]$ filter. The star shows our final choice for the N708 $\rm H\alpha$
filter which is a compromise between predicted lensing S/N, the strength of
excess fluxes between medium-band and broad-band filters, and the desire to
avoid strong skylines in the matched $\rm[OIII]$ filter. Figure 7: Mock colors
between the HSC $i$-band and filters with central wavelength set to 7080Å and
varying filter widths for the Initial Dwarf Sample shifted within each filter
design. The red, blue, and grey shaded areas show the 66.7%, 80%, and 90%
width of the magnitude difference distribution, while the black curve shows
the median. The black vertical dashed line indicates the final Merian N708
filter width (275Å). Figure 8: Left panel: Relative transmission curves for
DECam N708 and N540 compared to HSC $grizy$ broad-band filters (grey shaded
area). Right panel: Zoom-in region for N708 and N540, with sky emission
(yellow) and an example dwarf galaxy spectrum at $z=0.079$ (black). The red
solid curve indicates the DECam response curve in this wavelength range.
Figure 9: Distributions of four basic properties ($i$-band magnitude, $u-r$
color, half-light radius, and specific star-formation rate) for the target
dwarf galaxies from the Initial Dwarf Sample corresponding to the Merian
filter set (N708 and N540). Blue points (histograms) and orange crosses
(histograms) correspond respectively to star-forming and quiescent dwarf
galaxies in the Merian target mass and redshift range ($8<\rm\log
M_{*}/M_{\odot}<9$, $\rm 0.058<z_{COSMOS}<0.1$) according to the
classification in the COSMOS2015 catalog. Figure 10: Mock noise-free $i-\rm
N708$ and $r-\rm N540$ color vs. redshift color-coded by mock $i$-band
magnitude for galaxies (all masses) in the COSMOS2015 catalog. Mock photometry
in $i$-band, N708 and N540 are computed from model SEDs generated following
the same procedure as our dwarf SED library for the Initial Dwarf Sample (see
details in Section 2.4). The upper x-axis indicates the corresponding rest-
frame wavelengths for N708 and N540 (i.e. 7080Å/(1+z) and 5400Å/(1+z)),
including the rest-frame wavelengths for $\rm H\alpha$, $\rm[OIII]$, $\rm
H\beta$, and $\rm[OII]$. The shaded areas show the target Merian redshift
range $0.058<z<0.1$. Table 1: Characteristics of the final Merian N708 and
N540 filters and Merian survey design.

|  N708 ($\rm H\alpha$) |  N540 ($\rm[OIII]$)
---|---|---
Central wavelength $\lambda_{c}$ (Å) |  7080 |  5400
Filter width $\Delta\lambda$ (Å) |  275 |  210
Exposure time (min) |  40 |  60
Nnights |  24 |  36

Redshift range |  $0.058<z<0.10$
---|---
Mass range |  $8<\rm\log M_{*}/M_{\odot}<9$
Survey area (deg2) |  864
Expected number of dwarfs |  85,447
Lensing S/N at $r<0.5$ Mpc |  32
Lensing S/N at $r<1.0$ Mpc |  90

##  4 Filter Optimization Results

Figure 11: HSC 5-band photo-$z$’s (left panel) and mock 7-band photo-$z$’s
(right panel) vs. $\rm z_{COSMOS}$, color-coded by $g-i$ color. Black dashed
lines show $\rm z_{phot}=\rm z_{COSMOS}$. The black boxes indicate our target
redshift range $0.058<z<0.1$. Note that we only show galaxies with HSC 7-band
photo-$z$ lower than 0.3 in the right panel since there are no outliers above
this cut, while the scatter of HSC 5-band photo-$z$’s extends to higher
redshifts. The 5-band HSC broad-band selection has a photo-$z$ accuracy of
$\sigma_{\Delta z/(1+z)}\sim 0.5$ and an outlier fraction of $\eta=60\%$, with
48% completeness and 12% purity at $z<0.1$. The 7-band selection has a
photo-$z$ accuracy of $\sigma_{\Delta z/(1+z)}\sim 0.01$ and an outlier
fraction of $\eta=2.8\%$, with 89% completeness and 90% purity at $z<0.1$.
Figure 12: Predicted line-of-sight distance localization precision $\rm\Delta
d$ for Merian 6 and 7-band photo-$z$’s using idealized photometry. The
predicted standard deviation of $\rm\Delta d$ for Merian 7-band photo-$z$
(blue) and 6-band photo-$z$ (yellow) is 44.5 Mpc and 72.2 Mpc, respectively.

In this section, we present the optimization of the $\rm H\alpha$ filter
following the methodology described in Section 3. We then explain the design
process for the matched $\rm[OIII]$ filter and for the final joint design. To
evaluate the dwarf identification success rate in the target redshift range,
we compute mock photometry in the final Merian $\rm H\alpha$ and $\rm[OIII]$
filter designs. We use mock photometry for Merian (medium band) and HSC (broad
band) to predict photo-$z$ performance.

###  4.1 Predicted lensing S/N and Merian filter design

We now present the results of the filter optimization. Figure 6 displays
$\Delta\lambda$ vs. $\lambda_{c}$ where $\lambda_{c}$ has been converted into
the corresponding redshift $z_{c}$ for $\rm H\alpha$. Colors in the left panel
indicate the predicted lensing S/N within 0.5 Mpc. Colors in the right panel
indicate the total number of detected dwarf galaxies for each filter design.
The predicted lensing S/N peaks at S/N $\sim$ 32 with a filter width of
$\Delta\lambda\sim$ 300Å and a filter center at $z_{c}\sim 0.08$ in units of
$\rm H\alpha$ redshift, assuming a fixed observing time of 24 full nights on
DECam.

In general, the lensing S/N decreases above $z_{c}\sim 0.1$. This is because
higher redshift galaxies need longer exposure times which results in a smaller
survey area, volume, and therefore also a smaller lens sample. We run the
optimization for $\rm H\alpha$ between $0.02<z_{c}<0.2$ but Figure 6 only
shows the results for $0.02<z_{c}<0.13$ because the lensing S/N continues to
decrease at $z_{c}>0.13$. The lensing S/N also decreases below $z_{c}\sim
0.05$. This is the result of the fixed area of the HSC-SSP wide layer. Indeed,
while lower redshift dwarf galaxies are easier to detect with shorter exposure
times, the volume becomes limited by the fact that HSC has a fixed area of
$\rm\sim 1000\ deg^{2}$. Note that a similar optimization applied to the Rubin
LSST survey might prefer lower redshifts due to the wider area that will be
available from LSST (Ivezić et al. 2019).

In the optimization procedure, for simplicity, we do not measure photo-z’s for
every filter design. Instead, we measure the fluxes for mock dwarfs with and
without emission lines (i.e. continuum only) and characterize the strength of
the detection based on the difference between the two. However, in practice we
do not have the fluxes for the continuum underlying $\rm H\alpha$, only the
HSC broad-band photometry. The closest broad-band filters would inevitably
also include emission lines, which means we do not have a clean continuum to
subtract. Therefore, we rely on the photo-$z$/SED fitting procedure to produce
a reliable model of $\rm H\alpha$ continuum to fit the $\rm H\alpha$ emission
correctly and also to determine redshifts. The reliability of the model
continuum from this fitting process depends on the photometric precision of
Merian and HSC, which is difficult to predict without a comprehensive
understanding of the real data. In general, it is not easy to achieve
accurate, unbiased, and consistent photometry at the 0.1 mag level for
galaxies at $i\sim 23$ in large imaging surveys (Huang et al. 2018).
Therefore, in practice, we need the magnitude difference between our medium-
band and broad-band filter to be large enough to not be affected by
systematics in the photometry.

Figure 7 shows the mock magnitude difference between $i$-band and a series of
filters with central wavelength set to 7080Å and different filter widths for
the Initial Dwarf Sample. We choose the central wavelength to be 7080Å for
Figure 7 because our final $\rm H\alpha$ filter design is centered at 7080Å.
The red, blue, and grey shaded areas in Figure 7 show the 66.7%, 80%, and 90%
width of the magnitude difference distribution, while the black curve shows
the median. Narrower filters include less continuum and hence correspond to
larger magnitude differences. As shown in Figure 7, the filter design
corresponding to the lensing S/N peak ($\Delta\lambda\sim$ 300Å, $z_{c}\sim
0.08$) in Figure 6 has a fairly large $i-\rm N708_{mock}$ color. 90% of the
dwarfs in this filter have $i-\rm N708_{mock}>-0.1$ and 50% of the dwarfs have
$i-\rm N708_{mock}>0.1$. As expected, the $i-\rm N708_{mock}$ color continues
to decrease as the filter width becomes wider because the continuum
contributes more to the total flux, thereby decreasing the detection of the
$\rm H\alpha$ emission line. Our final filter design for N708
($\lambda_{c}=7080$Å, $\Delta\lambda=275$Å) is shown as a vertical dashed line
in Figure 7 and with black stars in Figure 6. In practice, we adjusted the
filter width of the final Merian $\rm H\alpha$ filter design from 300Å to 275Å
because of the $\rm[OIII]$ matched filter. This is now discussed below.

The purpose of a second Merian filter (targeting $\rm[OIII]$) is to reduce
contamination and improve Merian redshifts. The combination of the $\rm
H\alpha$ and $\rm[OIII]$ matched filter set will help to limit interlopers
from higher redshifts. For example, with a single filter design, the
$\rm[OIII]$ emission line moving into the $\rm H\alpha$ filter will cause
contamination from galaxies at $z\sim 0.4$ (see Section 4.2 for more
discussion). The wavelength range of the $\rm[OIII]$ filter is set to match
the redshift range probed by the $\rm H\alpha$ filter. The predicted S/N of
the lensing peaks for $\rm H\alpha$ filter designs with $z_{c}\sim 0.08$,
$\Delta\lambda\sim 300$Å (6939Å $<\lambda<$ 7239Å), corresponding to
$0.057<z_{c}<0.103$. The corresponding wavelength range for the $\rm[OIII]$
filter is 5293Å $<\lambda<$ 5522Å. However, we wish to avoid a strong sky
emission line at 5580Å. Although our filters will have rough top-hat
transmission curves, wings in the transmission curves can extend the
wavelength range ($\pm$ 10Å). In addition, transmission curves can also shift
($\sim$ 50Å) between the filter center and the filter edge due to different
angles of incidence. This will be shown in Section 5 with real Merian filter
transmission curves. Therefore, we opt for a conservative design and reduce
the wavelength/redshift range and shift $\lambda_{c}$ slightly compared to the
optimum value from Figure 6. Table 1 summarizes the final design and
characteristics of the Merian dual filter set. The Merian $\rm H\alpha$ (N708)
filter has $\lambda_{c}=7080$Å and $\Delta\lambda=275$Å. The Merian
$\rm[OIII]$ (N540) filter has $\lambda_{c}=5400$Å and $\Delta\lambda=210$Å.
The optimized exposure times for N708 and N540 are 40 min and 60 min,
respectively.

The exposure time for the $\rm[OIII]$ filter is determined by the same image
simulator presented in Section 3.3. We assume the same seeing and moon phase
as in Section 3.3 and only change the filter transmission curve. On average,
the $\rm[OIII]$ emission line is slightly weaker than $\rm H\alpha$ (based on
our dwarf SED library). Furthermore, the DECam response efficiency at 5400Å is
lower than 7080Å (see the red solid curve in Figure 8). For these two reasons,
we need 60 min exposures for the $\rm[OIII]$ filter to achieve a similar depth
and survey area compared to N708 (40 min). Both exposure times are set to
achieve a 80% completeness limit in our image simulations, as described in
Section 3.3. In total, Merian has 60 full nights dedicated to the Merian wide
layer survey. These 60 nights are broken into 24 full nights for the $\rm
H\alpha$ filter and 36 full nights for the $\rm[OIII]$ filter. The detailed
Merian survey design will be presented in Danieli et al. in prep.

Based on our simulated estimates (see Section 3.3), we will obtain 85,447
dwarf galaxies over 864 deg2 with a total of 60 nights on DECam with Merian.
This sample will allow us to measure the weak lensing signal with a lensing
S/N = 32 within $r<0.5$ Mpc and lensing S/N = 90 within $r<1.0$ Mpc. The
Merian redshift range of $0.058<z<0.1$ is narrow and will allow for accurate
measurements of galaxy-galaxy lensing profiles with less smearing in the
radial profile compared to samples selected based on photo-$z$’s derived from
broad-band photometry. Figure 8 shows the final filter transmission curves for
N708 and N540 along with the HSC broad-band filters. The right panel of Figure
8 shows a zoom-in region of N708/N540 together with the sky spectrum of a mock
dwarf galaxy at $z=0.079$. The red solid line indicates the response curve of
DECam.

Figure 9 displays the properties of dwarf galaxies from the Initial Dwarf
Sample that are in the final Merian redshift and mass range ($8<\rm\log
M_{*}/M_{\odot}<9$ and $\rm 0.058<z_{COSMOS}<0.1$) . Final Merian target
galaxies have similar basic properties (color, size, sSFR) as the Initial
Dwarf Sample, but have brighter apparent magnitudes ($\sim$1 mag brighter)
because they are at lower redshifts. Roughly 95% of star-forming dwarfs in the
Merian target range have $i<23.3$ mag.

###  4.2 Mock Merian Photometry

We now aim to predict the photo-$z$ performance when we combine the HSC
photometry with our custom N708 and N540 medium-band filters. The exact
photo-$z$ performance will depend on the quality of the photometry, and
systematic issues such as deblending, which are not trivial to model in
simulations. Hence, the estimates presented here are theoretical scenarios. To
estimate photo-$z$ performance, we first generate mock Merian synthetic
photometry.

Following the same procedure as in Section 2.4, we run SED fitting for all
galaxies in our COSMOS2015 sample below $z<0.65$ and compute mock fluxes in
N708, N540, and HSC $grizy$ given the best-fit model SEDs. Figure 10 shows
mock $i-\rm N708$ and $r-\rm N540$ color vs. $\rm z_{COSMOS}$ color-coded by
mock $i$-band magnitude for all galaxies in this sample. Upper x-axes show the
rest-frame wavelength corresponding to the redshifts of $\rm H\alpha$,
$\rm[OIII]$, $\rm H\beta$ and $\rm[OII]$ in N708 and N540. Mock N708 (N540)
magnitudes are the sum of the fluxes from emission lines and the continuum in
N708 (N540), while the $i$-band ($r$-band) magnitude broadly traces the
continuum in N708 (N540).

The upper panel of Figure 10 shows that a simple $i-\rm N708$ color can be
used to select star-forming galaxies between $0.058<z<0.10$. For example, most
$\rm H\alpha$ emitting galaxies have $i-\rm N708>-0.1$. However, emission
lines with shorter wavelengths (e.g. [OIII]) can shift into N708 and will
create interlopers from galaxies at higher redshifts. In fact, the number of
higher redshift interlopers could outnumber Merian dwarfs, due to the larger
cosmological volume at higher redshifts. $\rm[OIII]\lambda 5007$,
$\rm[OIII]\lambda 4959$ and $\rm H\beta$ contribute most interlopers at $z\sim
0.44$. The lower panel in Figure 10 suggests that a simple $r-\rm N540$ color
cut (e.g. $r-\rm N540>-0.2$) could exclude a good fraction of the $z\sim 0.44$
interlopers. Removing these interlopers is a key motivation for the
$\rm[OIII]$ filter.

In our target redshift range $0.058<z<0.10$, the three emission lines
$\rm[OIII]\lambda 5007$, $\rm[OIII]\lambda 4959$ and $\rm H\beta$ can shift in
and out of N540 from $z=0.058$ to $z=0.10$. Fluxes in N540 will be brighter if
they include more than one emission line. Therefore N540 will provide extra
redshift information for galaxies in our target redshift range $0.058<z<0.10$.
We will show in the next section that photo-$z$’s based on 7-band photometry
(N708, N540, and $grizy$) can select dwarf galaxies in the range
$0.058<z<0.10$ and may also have some redshift discriminating precision within
the target redshift range.

Note that mock fluxes are generated directly from the best-fit model SED via
Prospector and are noise free. In practice, Merian photometry will have
systematic errors arising from various steps in the data reduction process and
the photometry measurement method, which could impact the redshift precision.
However, characterizing photometric uncertainties in Merian is beyond the
scope of this paper and will be presented in future work.

Figure 13: Filter transmission curves for N708 and N540 measured in a parallel
beam at AOI 5 degrees. There were measured by the filter manufacturer Asahi
Spectra Co. Blue, light blue, green, orange, and red solid curves are measured
at positions that are 0 mm, 150 mm, 200 mm, 250 mm, and 280 mm away from the
filter center. Blue and red dash-dotted lines show the central wavelength of
the transmission curves at the filter center (r=0 mm) and filter edge (r=280
mm).

###  4.3 Predicted photo-$z$ Performance

We now discuss how much redshift information can be obtained with photo-$z$
fitting for galaxies in $0.058<z<0.10$. We characterize the expected photo-$z$
performance of the Merian 7-band design by generating mock samples of dwarfs
and performing SED fitting on this mock sample. We focus on galaxies at $\rm
z_{COSMOS}<0.2$ in this section since higher redshift interlopers can be
removed by color-color cuts (e.g. $i-\rm N708>-0.1$ and $r-\rm N540>-0.2$), as
shown in the previous section. This is a first indication of the Merian data,
and the actual Merian selection will be studied and described in upcoming
Merian papers.

We fit the SEDs of mock galaxies using Prospector with the same model and
priors described in Section 2.4. We run the SED fitting to derive photo-$z$’s,
stellar masses, and star formation rates with 6-band ($grizy\rm N708$) and
7-band ($grizy\rm N708N540$) separately. Note that this is an optimistic
scenario since we use the same galaxy Prospector galaxy model in the
simulations and in the fitting process.

We now compare our results with HSC 5-band ($grizy$) photo-$z$’s. We use the
HSC 5-band photo-$z$ based on the Bayesian template fitting-code Mizuki for
comparison (Tanaka et al. 2018). Figure 11 shows the comparison between HSC
5-band photo-$z$’s, our 7-band photo-$z$’s and $\rm z_{COSMOS}$, color-coded
by HSC $g-i$ color as a star-formation rate indicator. The black boxes
indicate our target redshift range of $0.058<z<0.1$. The 5-band HSC broad-band
selection has a photo-$z$ accuracy of $\sigma_{\Delta z/(1+z)}\sim 0.5$ and an
outlier fraction ($|z_{phot}-z_{spec}|/(1+z_{spec})>0.15$) of $\eta=60\%$. The
5-band selection below $z<0.1$ has very poor completeness and purity (48% and
12%), where completeness stands for how many dwarfs with $z_{\rm true}<0.1$
have $z_{\rm phot}<0.1$ and purity stands for how many dwarfs with $z_{\rm
phot}<0.1$ are actually at $z_{\rm true}<0.1$. The 7-band design yields
$\sigma_{\Delta z/(1+z)}\sim 0.01$ and $\eta=2.8\%$ (this precision was also
tested using galaxies in COSMOS with spectroscopic redshifts) and excellent
completeness and purity (89% and 90%) in the target redshift range. The
precision of the 6-band selection (not shown in the figure) is in between the
5-band and 7-band design, with $\sigma_{\Delta z/(1+z)}\sim 0.02$.

The 7-band selection achieves a line-of-sight localization precision of $44.5$
Mpc (compared to 72.2 Mpc from the 6-band selection) within the $0.058<z<0.10$
window (Figure 12). Most star-forming galaxies in our target redshift range
(based on their $g-i$ color) have a 7-band photo-$z$ within the target
redshift range $0.058<z<0.1$. We think that the 7-filter design improves
redshifts over the 6-band design for two reasons: a) medium/narrow bands
placed at the intersection between broad bands help to improve photo-$z$’s in
general (Ilbert et al. 2009), and b) the N540 filter also provides redshift
information as the $\rm H\alpha$, $\rm[OIII]$ and H$\beta$ lines move through
the filters.

Our method is not optimized for redshift accuracy for quenched galaxies.
However, the quenched fraction of dwarf galaxies in our target mass range is
low. Geha et al. (2012) found a quenched fraction for SDSS field dwarf
galaxies with $8<\rm\log M_{*}/M_{\odot}<9$ of less than $0.06\%$. And
recently, with the data from the Satellites Around Galactic Analogs Survey
(SAGA), Mao et al. (2021) found a quenched fraction for satellite dwarf
galaxies around Milky Way analogs in this mass range ($8<\rm\log
M_{*}/M_{\odot}<9$) of $\sim 10\%$. Carlsten et al. (2022) report a higher
satellite quenched fraction (15% for $8<\rm\log M_{*}/M_{\odot}<9$) using the
Exploration of Local VolumE Satellites (ELVES) Survey. Since we are not
selecting dwarf galaxies by their environments, we expect that the quenched
fraction will be below $15\%$. In fact, 95% of dwarf galaxies between
$8<\rm\log M_{*}/M_{\odot}<9$ in our Initial Dwarf Sample sample are star-
forming, according to the COSMOS2015 quiescent/star-forming classification,
which is consistent with the quenched fraction found by Geha et al. (2012) and
Mao et al. (2021).

The improved photo-$z$ precision of Merian over HSC 5 band photometry is
important for selecting clean samples of dwarf galaxies and for measuring weak
gravitational lensing for dwarfs. However, it is also important to note that
this is an ideal simulation that does not account for photometric systematic
errors (e.g. blending, galaxy morphology, photometric calibration, etc.). The
same models were also used to generate the mock data and and to fit SEDs. The
actual Merian photometric redshifts may therefore not achieve this theoretical
performance.

##  5 Construction and Characterization of the Merian Dual Filter System

Figure 14: Merian observed $i-\rm N708$ and $r-\rm N540$ colors vs. redshift,
color-coded by integrated cModel i-band magnitude for galaxies with $16<i<23$
and with a spectroscopic redshift. The blue solid curves show how $i-\rm N708$
and $r-\rm N540$ change with redshift for a mock star-forming dwarf galaxy
drawn from our dwarf SED library. The upper x-axis indicates the restframe
wavelength probed by N708 and N540 (i.e. 7080Å/(1+z) and 5400Å/(1+z)) and
indicates when $\rm H\alpha 6563$ and $\rm[OIII]5007$ are in the bandpass.
Shaded areas show the target Merian redshift range $0.058<z<0.1$.

Here we present the construction of the Merian filter set and show the actual
performance of N708 and N540. We also present the first version of the Merian
photometry, and highlight Merian’s potential for finding dwarf galaxies at
$0.058<z<0.1$ with actual data.

###  5.1 Filter Construction and Performance

The N708 and N540 filters were manufactured by Asahi Spectra Co. in 2020 and
2021 and then shipped to Chile and installed on DECam. Asahi Spectra Co.
measured the filter transmission curves at various locations in the laboratory
before the filters were shipped. These transmission curves are presented in
Figure 13. Color solid curves are filter transmission curves at 0 mm (i.e.
center), 150 mm, 200 mm, 250 mm, and 280 mm (close to the edge and away from
the filter center). The dash-dotted lines show the central wavelength of the
transmission curves at the filter center ($r=0$ mm) and the filter edge
($r=280$ mm). The filter transmission curves shift by 19Å for N708 and 11Å for
N540 from the filter center to the filter edge, which inevitably would also
affect the corresponding redshift range. This effect will contribute to the
systematic uncertainties of the photometry and will also add uncertainties to
the final Merian photo-$z$.

###  5.2 Merian First Year Data

Merian observations have been ongoing since March 2021, and as such we can
also provide the reader with a first look at the realized performance of the
medium-band filters. We defer a full description of the observation and data
reduction process to Danieli et al., in prep.; here we will simply note that
the broadband and medium-band data were reduced using the same pipeline, and
that the $i-{\rm N708}$ and $r-{\rm N540}$ colors are corrected for
differences in seeing between the HSC and DECam data.

In Figure 14 we show the real and observed equivalent of the mock color-
redshift figure that we presented in Figure 10 for galaxies with spectroscpic
redshifts and $16<i<23$. Here, redshifts are from a cross-match with the GAMA
spectroscopic survey and the COSMOS spectroscopic redshift catalog of Salvato
et al., in prep. We see good agreement between the predicted and observed
distribution of medium-band to broadband colors as a function of spectroscopic
redshift. This is demonstrated visually by the blue curve in Figure 14, which
traces the mean path of observed color evolution as a function of redshift for
a star-forming dwarf from our mock sample. As discussed above, the presence of
other lines that move into the medium-bands at higher redshift (particularly
$\rm[OIII]\lambda$5007Å into N708 and the $\rm[OII]\lambda 3727\AA,\lambda
3729\AA$ into N540 at $z\approx 0.4$) necessitates the use of a medium-band-
assisted photometric redshift approach rather than a color cut.

Further analysis of the observed sample is beyond the scope of this work: here
we simply underscore that a first look at the observed Merian galaxies shows
that the behavior of our medium-band filters are indeed well-described by the
mock catalog presented in this work.

##  6 Summary and Conclusions

Upcoming deep and wide imaging surveys open up the possibility of direct
measurements of the dark matter halos of dwarf galaxies with weak
gravitational lensing (Leauthaud et al. 2020). However, the key challenge
towards achieving this goal is not the weak lensing measurement per se, but
rather the selection of a reliable sample of dwarf galaxies from broad-band
imaging data and the determination of accurate redshifts for these lenses. In
this paper, we have optimized the design of two medium band filters (N708 and
N540) to achieve this goal. The filter set designed in this paper forms the
basis of the Merian survey which has been allocated 60 nights on the Blanco
telescope to carry out an imaging survey of $\sim$ 850 degrees2 and will
perform the first high S/N measurements of gravitational lensing around dwarf
galaxies.

The Merian filter design was optimized using a sample of dwarf galaxies from
the COSMOS2015 catalog. We built a library of SEDs including emission lines
for a complete sample of dwarf galaxies with $8<\rm\log M_{*}/M_{\odot}<9$. We
simulated the fluxes for these targets for various $\rm H\alpha$ filter
designs, varying both filter width $\Delta\lambda$ and filter central
wavelength $\lambda_{c}$. Our design considers the predicted weak lensing S/N
for dwarf galaxies as well as the magnitude difference between the Merian
filters and HSC broad-band filters to measure photo-$z$’s. Our key design
elements are summarized as follows.

  * •

We first design our $\rm H\alpha$ filter by optimizing the predicted weak
lensing S/N. We force the detected fraction of dwarf galaxies to be more than
80$\%$, which sets the exposure time. The total number of dwarf galaxies used
for lensing is then computed using a fixed number of nights (24 full nights
for the $\rm H\alpha$ filter). The predicted weak lensing S/N peaks at
S/N$\sim$32 with a $\rm H\alpha$ filter design $\Delta\lambda\sim$ 300Å and
$z_{c}\sim 0.08$. For the filter central wavelength in $\rm H\alpha$ redshift
$z_{c}$, the predicted lensing S/N decreases above $z_{c}\sim 0.1$ because
more distant, fainter galaxies need longer exposure times, resulting in a
smaller survey area, cosmological volume, and lens sample. The lensing S/N
also decreases below $z\sim 0.05$ because of the fixed area of the HSC-SSP
wide layer ($\rm\sim 1000\ deg^{2}$).

  * •

Merian will use photo-$z$/SED fitting to produce a reliable model of continuum
in order to fit fluxes from emission lines, as well as to determine
photo-$z$’s. The reliability of the model continuum from this fitting will
depend on the photometric precision of Merian and HSC. We estimate the
magnitude difference between Merian and HSC magnitudes as a function of filter
width. Narrower filters have larger magnitude difference since they have less
continuum to smear the emission line fluxes. We design a matched $\rm[OIII]$
filter with the same corresponding redshift range as the $\rm H\alpha$ filter.
We adjust the filter design to avoid a strong sky line at 5580Å for the
$\rm[OIII]$ filter.

  * •

The optimization procedure led to the design of the Merian dual filter set:
N708 ($\lambda_{c}=7080$Å, $\Delta\lambda=275$Å) and N540
($\lambda_{c}=5400$Å, $\Delta\lambda=210$Å). The predicted lensing S/N for our
final filter design is 32 at $r<0.5$ Mpc and 90 at $r<1.0$ Mpc, assuming a
fixed observing time of 60 full nights (24 for $\rm H\alpha$ and 36 for
$\rm[OIII]$) on DECam. The exposure times for the $\rm H\alpha$ and
$\rm[OIII]$ filters are estimated by requiring the same depth according to our
image simulations.

  * •

Using mock data we predict that Merian may be able to measure redshifts using
7-band photometry with a photometric redshift accuracy of $\sigma_{\Delta
z/(1+z)}\sim 0.01$ and an outlier fraction of $\eta=2.8\%$ with excellent
completeness and purity (89% and 90%) for dwarf galaxies with $8<\rm\log
M_{*}/M_{\odot}<9$ at $0.058<z<0.10$.

In this paper we also show early Merian photometry from our first year
internal data release. We show that the emission lines are indeed detected in
Merian filters with trends that mimic those predicted by our simulations.
Science-ready photometry and photo-$z$ measurements are actively under
development and will be presented in future work.

The large sample of $\sim$ 85,000 star-forming dwarf galaxies that will be
discovered by Merian will also allow for a wealth of ancillary studies of the
properties of dwarf galaxies. This method of efficiently selecting low
redshift dwarf galaxies from large imaging surveys can also be considered for
other future imaging surveys, such as Euclid (Laureijs et al. 2011), Nancy
Grace Roman telescope (Spergel et al. 2013), and the Legacy Survey of Space
and Time (LSST) with the Rubin Observatory (Ivezić et al. 2019).

## Acknowledgements

This material is based upon work supported by the National Science Foundation
under Grant No. 2106839.

We acknowledge use of the lux supercomputer at UC Santa Cruz, funded by NSF
MRI grant AST 1828315. The authors are pleased to acknowledge that the work
reported on in this paper was substantially performed using the Princeton
Research Computing resources at Princeton University which is a consortium of
groups led by the Princeton Institute for Computational Science and
Engineering (PICSciE) and Office of Information Technology’s Research
Computing.

This research uses services or data provided by the Astro Data Lab at NSF’s
National Optical-Infrared Astronomy Research Laboratory. NOIRLab is operated
by the Association of Universities for Research in Astronomy (AURA), Inc.
under a cooperative agreement with the National Science Foundation.

This research makes use of software developed for the Vera C. Rubin
Observatory. We thank the Vera C. Rubin Observatory for making their code
available as free software at http://dm.lsstcorp.org.

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 National Astronomical
Observatory of Japan (NAOJ), Kavli Institute for the Physics and Mathematics
of the Universe (Kavli IPMU), University of Tokyo, High Energy Accelerator
Research Organization (KEK), Academia Sinica Institute for Astronomy and
Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was
contributed by the FIRST program from Japanese Cabinet Office, Ministry of
Education, Culture, Sports, Science and Technology (MEXT), Japan Society for
the Promotion of Science (JSPS), Japan Science and Technology Agency (JST),
Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton
University.

This research made use of: SciPy, an open source scientific tool for Python
(Virtanen et al. 2020); NumPy, a fundamental package for scientific computing
with Python (Harris et al. 2020); Matplotlib, a 2-D plotting library for
Python (Hunter 2007); Astropy, a community-developed core Python package for
astronomy (Astropy Collaboration et al. 2013, 2018, 2022); GalSim, a modular
galaxy image simulation toolkit in Python (Rowe et al. 2015). sep Source
Extraction and Photometry in Python (Bertin & Arnouts 1996; Barbary 2016);
dynesty, a dynamic nested sampling package for estimating Bayesian posteriors
and evidences (Speagle 2020); TOPCAT, a user-friendly graphical program for
viewing, analysis and editing of tables (Taylor 2005).

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Abbott et al. (2021) Abbott T. M. C., et al., 2021, ApJS, 255, 20
  * Aihara et al. (2018) Aihara H., et al., 2018, PASJ, 70, S4
  * Arnouts et al. (2002) Arnouts S., et al., 2002, MNRAS, 329, 355
  * Astropy Collaboration et al. (2013) Astropy Collaboration et al., 2013, A&A, 558, A33
  * Astropy Collaboration et al. (2018) Astropy Collaboration et al., 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2022) Astropy Collaboration et al., 2022, apj, 935, 167
  * Baldwin et al. (1981) Baldwin J. A., Phillips M. M., Terlevich R., 1981, PASP, 93, 5
  * Barbary (2016) Barbary K., 2016, Journal of Open Source Software, 1, 58
  * Barbary et al. (2015) Barbary K., Boone K., Deil C., 2015, sep: v0.3.0, doi:10.5281/zenodo.15669, http://dx.doi.org/10.5281/zenodo.15669
  * Battaglia & Nipoti (2022) Battaglia G., Nipoti C., 2022, Nature Astronomy, 6, 659
  * Benítez et al. (2009) Benítez N., et al., 2009, ApJ, 691, 241
  * Benitez et al. (2014) Benitez N., et al., 2014, arXiv e-prints, p. arXiv:1403.5237
  * Bertin & Arnouts (1996) Bertin E., Arnouts S., 1996, A&AS, 117, 393
  * Bouché et al. (2022) Bouché N. F., et al., 2022, A&A, 658, A76
  * Boylan-Kolchin et al. (2011) Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2011, MNRAS, 415, L40
  * Boylan-Kolchin et al. (2012) Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2012, MNRAS, 422, 1203
  * Brainerd et al. (1996) Brainerd T. G., Blandford R. D., Smail I., 1996, ApJ, 466, 623
  * Brooks & Zolotov (2014) Brooks A. M., Zolotov A., 2014, ApJ, 786, 87
  * Brooks et al. (2017) Brooks A. M., Papastergis E., Christensen C. R., Governato F., Stilp A., Quinn T. R., Wadsley J., 2017, ApJ, 850, 97
  * Buckley & Peter (2018) Buckley M. R., Peter A. H. G., 2018, Phys. Rep., 761, 1
  * Bullock & Boylan-Kolchin (2017) Bullock J. S., Boylan-Kolchin M., 2017, ARA&A, 55, 343
  * Byler et al. (2017) Byler N., Dalcanton J. J., Conroy C., Johnson B. D., 2017, ApJ, 840, 44
  * Calzetti et al. (2000) Calzetti D., Armus L., Bohlin R. C., Kinney A. L., Koornneef J., Storchi-Bergmann T., 2000, ApJ, 533, 682
  * Carlsten et al. (2022) Carlsten S. G., Greene J. E., Beaton R. L., Danieli S., Greco J. P., 2022, arXiv e-prints, p. arXiv:2203.00014
  * Chabrier (2003) Chabrier G., 2003, PASP, 115, 763
  * Choi et al. (2016) Choi J., Dotter A., Conroy C., Cantiello M., Paxton B., Johnson B. D., 2016, ApJ, 823, 102
  * Civano et al. (2016) Civano F., et al., 2016, ApJ, 819, 62
  * Coil et al. (2011) Coil A. L., et al., 2011, ApJ, 741, 8
  * Collins & Read (2022) Collins M. L. M., Read J. I., 2022, Nature Astronomy, 6, 647
  * Conroy & Gunn (2010) Conroy C., Gunn J. E., 2010, ApJ, 712, 833
  * Conroy et al. (2009) Conroy C., Gunn J. E., White M., 2009, ApJ, 699, 486
  * Coupon et al. (2018) Coupon J., Czakon N., Bosch J., Komiyama Y., Medezinski E., Miyazaki S., Oguri M., 2018, PASJ, 70, S7
  * Danieli et al. (2022) Danieli S., Greene J. E., Carlsten S., Jiang F., Beaton R., Goulding A. D., 2022, arXiv e-prints, p. arXiv:2210.14233
  * Darragh-Ford et al. (2022) Darragh-Ford E., et al., 2022, arXiv e-prints, p. arXiv:2212.07433
  * Davidzon et al. (2017) Davidzon I., et al., 2017, A&A, 605, A70
  * De Leo et al. (2023) De Leo M., Read J. I., Noel N. E. D., Erkal D., Massana P., Carrera R., 2023, arXiv e-prints, p. arXiv:2303.08838
  * Dey et al. (2019) Dey A., et al., 2019, AJ, 157, 168
  * Di Cintio et al. (2014a) Di Cintio A., Brook C. B., Macciò A. V., Stinson G. S., Knebe A., Dutton A. A., Wadsley J., 2014a, MNRAS, 437, 415
  * Di Cintio et al. (2014b) Di Cintio A., Brook C. B., Dutton A. A., Macciò A. V., Stinson G. S., Knebe A., 2014b, MNRAS, 441, 2986
  * Drlica-Wagner et al. (2022) Drlica-Wagner A., et al., 2022, ApJS, 261, 38
  * Dutton (2009) Dutton A. A., 2009, MNRAS, 396, 121
  * Dvornik et al. (2022) Dvornik A., et al., 2022, arXiv e-prints, p. arXiv:2210.03110
  * Eigenthaler et al. (2018) Eigenthaler P., et al., 2018, ApJ, 855, 142
  * Fisher & Drory (2011) Fisher D. B., Drory N., 2011, ApJ, 733, L47
  * Fitts et al. (2019) Fitts A., et al., 2019, MNRAS, 490, 962
  * Flores & Primack (1994) Flores R. A., Primack J. R., 1994, ApJ, 427, L1
  * Fry et al. (2015) Fry A. B., et al., 2015, MNRAS, 452, 1468
  * Geha et al. (2012) Geha M., Blanton M. R., Yan R., Tinker J. L., 2012, ApJ, 757, 85
  * Governato et al. (2010) Governato F., et al., 2010, Nature, 463, 203
  * Governato et al. (2012) Governato F., et al., 2012, MNRAS, 422, 1231
  * Harris et al. (2020) Harris C. R., et al., 2020, Nature, 585, 357
  * Hasinger et al. (2018) Hasinger G., et al., 2018, ApJ, 858, 77
  * Heymans et al. (2012) Heymans C., et al., 2012, MNRAS, 427, 146
  * Hikage et al. (2019) Hikage C., et al., 2019, PASJ, 71, 43
  * Huang et al. (2018) Huang S., et al., 2018, PASJ, 70, S6
  * Hudson et al. (2015) Hudson M. J., et al., 2015, MNRAS, 447, 298
  * Hunter (2007) Hunter J. D., 2007, Computing in Science & Engineering, 9, 90
  * Ilbert et al. (2006) Ilbert O., et al., 2006, A&A, 457, 841
  * Ilbert et al. (2009) Ilbert O., et al., 2009, ApJ, 690, 1236
  * Ivezić et al. (2019) Ivezić Ž., et al., 2019, ApJ, 873, 111
  * Johnson et al. (2021) Johnson B. D., Leja J., Conroy C., Speagle J. S., 2021, ApJS, 254, 22
  * Kado-Fong et al. (2020) Kado-Fong E., Greene J. E., Huang S., Beaton R., Goulding A. D., Komiyama Y., 2020, ApJ, 900, 163
  * Kado-Fong et al. (2021) Kado-Fong E., et al., 2021, ApJ, 920, 72
  * Kamada et al. (2017) Kamada A., Kaplinghat M., Pace A. B., Yu H.-B., 2017, Phys. Rev. Lett., 119, 111102
  * Kashino et al. (2019) Kashino D., et al., 2019, ApJS, 241, 10
  * Klypin et al. (1999) Klypin A., Kravtsov A. V., Valenzuela O., Prada F., 1999, ApJ, 522, 82
  * Klypin et al. (2015) Klypin A., Karachentsev I., Makarov D., Nasonova O., 2015, MNRAS, 454, 1798
  * Kravtsov (2013) Kravtsov A. V., 2013, ApJ, 764, L31
  * Laigle et al. (2016) Laigle C., et al., 2016, ApJS, 224, 24
  * Laureijs et al. (2011) Laureijs R., et al., 2011, arXiv e-prints, p. arXiv:1110.3193
  * Le Fèvre et al. (2013) Le Fèvre O., et al., 2013, A&A, 559, A14
  * Le Fèvre et al. (2015) Le Fèvre O., et al., 2015, A&A, 576, A79
  * Le Floc’h et al. (2009) Le Floc’h E., et al., 2009, ApJ, 703, 222
  * Leauthaud et al. (2007) Leauthaud A., et al., 2007, ApJS, 172, 219
  * Leauthaud et al. (2012) Leauthaud A., et al., 2012, ApJ, 744, 159
  * Leauthaud et al. (2020) Leauthaud A., Singh S., Luo Y., Ardila F., Greco J. P., Capak P., Greene J. E., Mayer L., 2020, Physics of the Dark Universe, 30, 100719
  * Lee et al. (2024) Lee K.-S., et al., 2024, ApJ, 962, 36
  * Lelli (2022) Lelli F., 2022, Nature Astronomy, 6, 35
  * Lilly et al. (2007) Lilly S. J., et al., 2007, ApJS, 172, 70
  * Liske et al. (2015) Liske J., et al., 2015, MNRAS, 452, 2087
  * Lutz et al. (2011) Lutz D., et al., 2011, A&A, 532, A90
  * Mao et al. (2021) Mao Y.-Y., Geha M., Wechsler R. H., Weiner B., Tollerud E. J., Nadler E. O., Kallivayalil N., 2021, ApJ, 907, 85
  * Mashchenko et al. (2008) Mashchenko S., Wadsley J., Couchman H. M. P., 2008, Science, 319, 174
  * Masters et al. (2019) Masters D. C., et al., 2019, ApJ, 877, 81
  * McCracken et al. (2012) McCracken H. J., et al., 2012, A&A, 544, A156
  * McQuinn et al. (2022) McQuinn K. B. W., et al., 2022, arXiv e-prints, p. arXiv:2203.10105
  * Mendes de Oliveira et al. (2019) Mendes de Oliveira C., et al., 2019, MNRAS, 489, 241
  * Miller et al. (2014) Miller S. H., Ellis R. S., Newman A. B., Benson A., 2014, ApJ, 782, 115
  * Miralda-Escude (1991) Miralda-Escude J., 1991, ApJ, 370, 1
  * Miyazaki et al. (2018) Miyazaki S., et al., 2018, PASJ, 70, S1
  * Moles et al. (2008) Moles M., et al., 2008, AJ, 136, 1325
  * Moore (1994) Moore B., 1994, Nature, 370, 629
  * Moore et al. (1999) Moore B., Ghigna S., Governato F., Lake G., Quinn T., Stadel J., Tozzi P., 1999, ApJ, 524, L19
  * Munshi et al. (2021) Munshi F., Brooks A. M., Applebaum E., Christensen C. R., Quinn T., Sligh S., 2021, ApJ, 923, 35
  * Nadler et al. (2020) Nadler E. O., et al., 2020, ApJ, 893, 48
  * Nadler et al. (2021) Nadler E. O., Banerjee A., Adhikari S., Mao Y.-Y., Wechsler R. H., 2021, ApJ, 920, L11
  * Newman et al. (2013) Newman J. A., et al., 2013, ApJS, 208, 5
  * Oh et al. (2015) Oh S.-H., et al., 2015, AJ, 149, 180
  * Oke & Gunn (1983) Oke J. B., Gunn J. E., 1983, ApJ, 266, 713
  * Oliver et al. (2012) Oliver S. J., et al., 2012, MNRAS, 424, 1614
  * Oman et al. (2015) Oman K. A., et al., 2015, MNRAS, 452, 3650
  * Oman et al. (2016) Oman K. A., Navarro J. F., Sales L. V., Fattahi A., Frenk C. S., Sawala T., Schaller M., White S. D. M., 2016, MNRAS, 460, 3610
  * Ouchi et al. (2018) Ouchi M., et al., 2018, PASJ, 70, S13
  * Papastergis et al. (2015) Papastergis E., Giovanelli R., Haynes M. P., Shankar F., 2015, A&A, 574, A113
  * Pontzen & Governato (2012) Pontzen A., Governato F., 2012, MNRAS, 421, 3464
  * Posti et al. (2019) Posti L., Marasco A., Fraternali F., Famaey B., 2019, A&A, 629, A59
  * Read & Erkal (2019) Read J. I., Erkal D., 2019, MNRAS, 487, 5799
  * Read et al. (2006) Read J. I., Wilkinson M. I., Evans N. W., Gilmore G., Kleyna J. T., 2006, MNRAS, 367, 387
  * Read et al. (2016) Read J. I., Iorio G., Agertz O., Fraternali F., 2016, MNRAS, 462, 3628
  * Read et al. (2017) Read J. I., Iorio G., Agertz O., Fraternali F., 2017, MNRAS, 467, 2019
  * Read et al. (2019) Read J. I., Walker M. G., Steger P., 2019, MNRAS, 484, 1401
  * Relatores et al. (2019) Relatores N. C., et al., 2019, ApJ, 887, 94
  * Robles et al. (2017) Robles V. H., et al., 2017, MNRAS, 472, 2945
  * Roper et al. (2023) Roper F. A., Oman K. A., Frenk C. S., Benítez-Llambay A., Navarro J. F., Santos-Santos I. M. E., 2023, MNRAS, 521, 1316
  * Rowe et al. (2015) Rowe B. T. P., et al., 2015, Astronomy and Computing, 10, 121
  * Rubin et al. (1978) Rubin V. C., Ford W. K. J., Thonnard N., 1978, ApJ, 225, L107
  * Rubin et al. (1980) Rubin V. C., Ford W. K. J., Thonnard N., 1980, ApJ, 238, 471
  * Sales et al. (2022) Sales L. V., Wetzel A., Fattahi A., 2022, Nature Astronomy, 6, 897
  * Sánchez-Blázquez et al. (2006) Sánchez-Blázquez P., et al., 2006, MNRAS, 371, 703
  * Santos-Santos et al. (2018) Santos-Santos I. M., Di Cintio A., Brook C. B., Macciò A., Dutton A., Domínguez-Tenreiro R., 2018, MNRAS, 473, 4392
  * Scoville et al. (2007) Scoville N., et al., 2007, ApJS, 172, 1
  * Singh et al. (2017) Singh S., Mandelbaum R., Seljak U., Slosar A., Vazquez Gonzalez J., 2017, MNRAS, 471, 3827
  * Smith et al. (2003) Smith R. E., et al., 2003, MNRAS, 341, 1311
  * Speagle (2020) Speagle J. S., 2020, MNRAS, 493, 3132
  * Speagle et al. (2019) Speagle J. S., et al., 2019, MNRAS, 490, 5658
  * Spergel et al. (2013) Spergel D., et al., 2013, arXiv e-prints, p. arXiv:1305.5425
  * Takahashi et al. (2012) Takahashi R., Sato M., Nishimichi T., Taruya A., Oguri M., 2012, ApJ, 761, 152
  * Tanaka et al. (2018) Tanaka M., et al., 2018, PASJ, 70, S9
  * Taniguchi et al. (2007) Taniguchi Y., et al., 2007, ApJS, 172, 9
  * Taniguchi et al. (2015) Taniguchi Y., et al., 2015, PASJ, 67, 104
  * Taylor (2005) Taylor M. B., 2005, in Shopbell P., Britton M., Ebert R., eds, Astronomical Society of the Pacific Conference Series Vol. 347, Astronomical Data Analysis Software and Systems XIV. p. 29
  * Teyssier et al. (2013) Teyssier R., Pontzen A., Dubois Y., Read J. I., 2013, MNRAS, 429, 3068
  * Tollet et al. (2016) Tollet E., et al., 2016, MNRAS, 456, 3542
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Wetzel et al. (2016) Wetzel A. R., Hopkins P. F., Kim J.-h., Faucher-Giguère C.-A., Kereš D., Quataert E., 2016, ApJ, 827, L23
  * Wilson et al. (2001) Wilson G., Kaiser N., Luppino G. A., Cowie L. L., 2001, ApJ, 555, 572
  * York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
  * Zamojski et al. (2007) Zamojski M. A., et al., 2007, ApJS, 172, 468
  * Zheng et al. (2017) Zheng Z.-Y., et al., 2017, ApJ, 842, L22
  * Zheng et al. (2019) Zheng Z.-Y., et al., 2019, PASP, 131, 074502