# Reconciling the Results of the $z\sim 2$ MOSDEF and KBSS-MOSFIRE Surveys1

Jordan N. Runco,2 Naveen A. Reddy, 3 Alice E. Shapley,2 Charles C.
Steidel,4Ryan L. Sanders,5,6 Allison L. Strom,7 Alison L. Coil,8 Mariska
Kriek,9,10Bahram Mobasher,3 Max Pettini11 Gwen C. Rudie,12 Brian
Siana,3Michael W. Topping,2,13 Ryan F. Trainor,14 William R. Freeman,3 Irene
Shivaei,13Mojegan Azadi,15 Sedona H. Price,16 Gene C. K. Leung,17 Tara
Fetherolf,3Laura de Groot,18 Tom Zick,9 Francesca M. Fornasini,19 Guillermo
Barro20
1Based on data obtained at the W.M. Keck Observatory, which is operated as a
scientific partnership among the California Institute of
Technology, the University of California, and the National Aeronautics and
Space Administration, and was made possible by the generous
financial support of the W.M. Keck Foundation.
2Physics & Astronomy Department, University of California: Los Angeles, 430
Portola Plaza, Los Angeles, CA 90095, USA
3Department of Physics & Astronomy, University of California, Riverside, 900
University Avenue, Riverside, CA 92521, USA
4Cahill Center for Astronomy and Astrophysics, California Institute of
Technology, 1200 E California Blvd, MC249-17, Pasadena, CA 91125, USA
5Department of Physics, University of California, Davis, One Shields Ave,
Davis, CA 95616, USA
6Hubble Fellow
7Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University,
Princeton, NJ 08544, USA
8Center for Astrophysics and Space Sciences, University of California, San
Diego, 9500 Gilman Dr., La Jolla, CA 92093-0424, USA
9Astronomy Department, University of California, Berkeley, CA 94720, USA
10Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, The
Netherlands
11Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge,
CB3 0HA, UK
12The Observatories of the Carnegie Institution for Science, 813 Santa Barbara
Street, Pasadena, CA 91101, USA
13Department of Astronomy/Steward Observatory, 933 North Cherry Ave, Rm N204,
Tucson, AZ 85721-0065, USA
14Department of Physics and Astronomy, Franklin & Marshall College, 637
College Ave., Lancaster, PA 17603, USA
15Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge,
MA, 02138, USA
16Max-Planck-Institut für Extraterrestrische Physik, Postfach 1312, Garching,
85741, Germany
17Department of Astronomy, The University of Texas at Austin, 2515 Speedway
Blvd Stop C1400, Austin, TX 78712, USA
18Department of Physics, The College of Wooster, 1189 Beall Avenue, Wooster,
OH 44691, USA
19Department of Physics and Astronomy, Stonehill College, 320 Washington
Street, Easton, MA 02357, USA
20Department of Physics, University of the Pacific, 3601 Pacific Ave,
Stockton, CA 95211, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The combination of the MOSDEF and KBSS-MOSFIRE surveys represents the largest
joint investment of Keck/MOSFIRE time to date, with $\sim$3000 galaxies at
$1.4\lesssim z\lesssim 3.8$, roughly half of which are at $z\sim 2$. MOSDEF is
photometric- and spectroscopic-redshift selected with a rest-optical magnitude
limit, while KBSS-MOSFIRE is primarily selected based on rest-UV colors and a
rest-UV magnitude limit. Analyzing both surveys in a uniform manner with
consistent spectral-energy-distribution (SED) models, we find that the MOSDEF
$z\sim 2$ targeted sample has higher median $M_{\ast}$ and redder rest U$-$V
color than the KBSS-MOSFIRE $z\sim 2$ targeted sample, and smaller median SED-
based SFR and sSFR (SFR(SED) and sSFR(SED)). Specifically, MOSDEF targeted a
larger population of red galaxies with U$-$V and V$-$J $\geq$1.25, while KBSS-
MOSFIRE contains more young galaxies with intense star formation. Despite
these differences in the $z\sim 2$ targeted samples, the subsets of the
surveys with multiple emission lines detected and analyzed in previous work
are much more similar. All median host-galaxy properties with the exception of
stellar population age — i.e., $M_{\ast}$, SFR(SED), sSFR(SED), $A_{\rm{V}}$,
and UVJ colors — agree within the uncertainties. Additionally, when uniform
emission-line fitting and stellar Balmer absorption correction techniques are
applied, there is no significant offset between both samples in the [O
III]$\lambda$5008/H$\beta$ vs. [N II]$\lambda$6585/H$\alpha$ diagnostic
diagram, in contrast to previously-reported discrepancies. We can now combine
the MOSDEF and KBSS-MOSFIRE surveys to form the largest $z\sim 2$ sample with
moderate-resolution rest-optical spectra and construct the fundamental scaling
relations of star-forming galaxies during this important epoch.

###### keywords:

galaxies: evolution — galaxies: high-redshift — galaxies: ISM

††pubyear: 2021††pagerange: Reconciling the Results of the $z\sim 2$ MOSDEF
and KBSS-MOSFIRE Surveys1–14

##  1 Introduction

Rest-frame optical emission-line spectroscopy provides a wealth of information
about a galaxy, including its dust extinction, star-formation rate (SFR),
virial and non-virial dynamics (e.g., large-scale outflows), active galactic
nucleus (AGN) activity, and properties of the ionized interstellar medium
(ISM) such as the metallicity, electron density ($n_{\rm{e}}$), ionizing
spectrum, and ionization parameter ($U$; i.e., the ratio of ionizing photon
density to hydrogen, and therefore electron, density). Understanding the
emission-line properties of star-forming galaxies throughout cosmic history is
thus essential for characterizing the formation and evolution of the stellar
and gaseous contents of galaxies. One of the most important time periods for
studying rest-optical emission-line spectra of galaxies is at $z\sim 2$, which
hosts the peak level star formation in the universe (Madau & Dickinson, 2014).
During this epoch, the modern Hubble sequence was not yet fully established
and many key features of current galaxy patterns were being set, including the
bimodal distribution of galaxy colors and correlations between structural
parameters and stellar properties.

In the last decade, the commissioning of multi-object near-IR spectrographs on
large ground-based telescopes has given us the ability to characterize the
rest-optical emission-line properties of these high-redshift galaxies in large
statistical samples. Two surveys taking advantage of the MultiObject
Spectrometer For Infra-Red Exploration (MOSFIRE; McLean et al. 2012)
instrument on the 10 m Keck I telescope are the MOSFIRE Deep Evolution Field
(MOSDEF; Kriek et al. 2015) survey and Keck Baryonic Structure Survey (KBSS:
Steidel et al. 2014). Each of the two surveys contains $\sim$1500 galaxies
with deep MOSFIRE observations at $1.4\lesssim z\lesssim 3.8$, with roughly
half the galaxies in each survey at $z\sim 2$.

At $z\sim 2$, Keck/MOSFIRE can observe the full rest-optical wavelength range
from $3700-7000$ Å, including all of the rest-optical emission-lines needed to
track the location of $z\sim 2$ galaxies on the “BPT” diagrams. The [O
III]$\lambda$5008/H$\beta$ vs. [N II]$\lambda$6585/H$\alpha$ diagram (first
introduced by Baldwin et al. 1981 and which we refer to hereafter as the “[N
II] BPT diagram”), and the [O III]$\lambda$5008/H$\beta$ vs. [S
II]$\lambda\lambda$6718,6733/H$\alpha$ diagram (first introduced in Veilleux &
Osterbrock 1987 and which we refer to hereafter as the “[S II] BPT diagram”)
can both be used to distinguish between AGN and star formation activity as the
main ionizing source for the ISM of a galaxy. The ionizing spectrum shape as
well as the typical ionization parameter are different in gas photoionized by
hot stars as opposed to by an AGN. Therefore, star-forming galaxies and AGNs
occupy distinct regions within these rest-optical emission-line diagrams.

In early studies with Keck/NIRSPEC, it was found that star-forming galaxies at
$z>1$ show elevated [N II]$\lambda$6585/H$\alpha$ at fixed [O
III]$\lambda$5008/H$\beta$ (or vice versa; e.g. Shapley et al. 2005; Erb et
al. 2006b; Liu et al. 2008) on average compared to local galaxies in the Sloan
Digital Sky Survey (SDSS; York et al. 2000), indicating a redshift dependence
in the [N II] BPT diagram. Subsequent work based on larger samples collected
with MOSFIRE (e.g., Steidel et al. 2014; Shapley et al. 2015), has since
confirmed the systematic [N II] BPT offset. There have been numerous
explanations that have been proposed for this systematic offset, including
variations in the physical properties of galaxies such as H II region ionizing
spectra at fixed metallicities, H II region ionization parameter, gas-phase
N/O abundance ratio differences, H II region electron densities and density
structure, unresolved AGN activity, shocks, and galaxy selection effects
(e.g., Liu et al. 2008; Brinchmann et al. 2008; Wright et al. 2010; Kewley et
al. 2013; Yeh et al. 2013; Juneau et al. 2014; Masters et al. 2014; Steidel et
al. 2014; Steidel et al. 2016; Coil et al. 2015; Shapley et al. 2015, 2019;
Sanders et al. 2016; Strom et al. 2017, 2018; Freeman et al. 2019; Kashino et
al. 2019; Topping et al. 2020; Runco et al. 2021). Results based on both KBSS
and recent MOSDEF observations favor the idea that the main driver for the
observed offset is a harder ionizing spectrum at fixed nebular oxygen
abundance, which arises due to $\alpha$-enhancement in the massive stars in
star-forming galaxies at $z\sim 2$ (Steidel et al., 2014; Steidel et al.,
2016; Strom et al., 2017; Shapley et al., 2019; Topping et al., 2020; Reddy et
al., 2021; Runco et al., 2021).

Understanding this offset is vital because rest-optical emission-lines are
often used to infer many physical properties of the ISM. Locally, many
calibrations exist between rest-optical emission-lines and ISM properties
(e.g., gas-phase oxygen abundance; Pettini & Pagel 2004). Without a complete
understanding of the [N II] BPT offset, it will remain unclear if these local
calibrations can be used in the $z>1$ universe (Bian et al., 2018; Sanders et
al., 2020).

While the MOSDEF and KBSS studies have converged on the factors physically
driving the [N II] BPT offset, there are still differences in key results.
Most notable is that the two surveys have found quantitatively different
offsets between the emission-line sequences for $z\sim 2$ and local galaxies
on the [N II] BPT diagram, with the KBSS sample showing higher emission-line
ratios on average (Steidel et al., 2014; Shapley et al., 2015; Strom et al.,
2017; Runco et al., 2021). This discrepancy suggests that there might be
fundamental differences between galaxies contained in the two surveys.

To fully understand the driver of the [N II] BPT offset and the correlation
between $z\sim 2$ rest-optical emission-lines and galaxy properties (e.g.,
gas-phase oxygen abundance), a data set of unprecedented size is needed. To
achieve such a sample, this study represents the beginning of a collaboration
between the MOSDEF and KBSS teams. With over 100 nights of Keck time and
$\sim$3000 galaxies observed between $1.4\lesssim z\lesssim 3.8$ (about half
at $z\sim 2$), the combination of these two surveys represents the largest
total investment of Keck/MOSFIRE time. Combining the two data sets would yield
a sample with a statistical size capable of determining robust trends in
galaxy properties and relationships in the $z\sim 2$ universe analogous to
what exists locally.

However, because of differences in survey selection methods and results from
previous studies (primarily concerning the [N II] BPT diagram), a thorough
comparison of the two samples to search for possible biases is necessary
before they can be combined. Here we perform a uniform comparison of the two
samples to understand if any systematic differences between them exist, and if
so, how to merge the samples in a way that mitigates those differences for
future work. In this study, we analyze the MOSDEF and KBSS samples using
consistent stellar population synthesis modeling assumptions applied to
emission-line-corrected photometry, as well as the same methodology for
emission-line fitting. This controlled approach eliminates any systematic
biases stemming from different analysis methods, and isolates differences
between the samples themselves. From the consistent SED fitting, we compare
galaxy properties between the two samples. From the consistent emission-line
measurements, we compare the locations of MOSDEF and KBSS samples on the [N
II] and [S II] BPT diagrams and quantify the differences in their offsets from
the local sequence. The combination of these two analyses enables the
discovery of any systematic differences between the two samples of galaxies.

In Section 2, we present and compare the selection methods for the MOSDEF and
KBSS surveys and discuss the $z\sim 2$ samples from each survey that are
analyzed in this study. Section 3 describes the methodology for the SED and
emission-line fitting analyses, and Section 4 presents the results from these
fitting methods for the $z\sim 2$ spectroscopic samples with multiple
emission-line detections that have been analyzed in previous studies. Section
5 discusses the results for the $z\sim 2$ spectroscopic samples with multiple
emission-line detections and compares their SED properties with those of the
larger sample of $z\sim 2$ targeted galaxies from which they are drawn.
Section 6 presents a summary of key results and looks ahead to future work.
Appendix A provides SED fitting results for the MOSDEF and KBSS $z\sim 2$
samples assuming constant star formation (CSF) histories instead of the
delayed-$\tau$ star formation histories described in Section 3.1. Finally,
Appendix B provides examples of spectra to highlight differences between the
results of the two emission-line fitting methods discussed in this study. We
adopt the following abbreviations for emission-line ratios used frequently
throughout the paper.

$\rm{N2=[N~{}\textsc{II}]\lambda 6585/H\alpha}$ (1)
$\rm{S2=[S~{}\textsc{II}]\lambda\lambda 6718,6733/H\alpha}$ (2)
$\rm{O3=[O~{}\textsc{III}]\lambda 5008/H\beta}$ (3) $\rm{O3N2=O3/N2}$ (4)
$\rm{O3S2=O3/S2}$ (5)

All emission-line wavelengths are in vacuum. Throughout this paper, we adopt a
$\Lambda$-CDM cosmology with $H_{0}$ = 70 km s-1 Mpc-1, $\Omega_{\rm{m}}$ =
0.3, and $\Omega_{\Lambda}$ = 0.7. Also, we assume the solar abundance pattern
from Asplund et al. (2009).

##  2 Observations & Sample Selection

Here we provide overviews of the MOSDEF (Section 2.1) and KBSS (Section 2.2)
surveys. We describe the sample selection methods of the surveys and the
$z\sim 2$ spectroscopic samples that we adopt from previous studies (KBSS:
Strom et al. 2017; MOSDEF: Sanders et al. 2018). In addition, we highlight
some of the key differences between the surveys in Section 2.3.

Some notable similarities between the two surveys are that they both utilized
Keck/MOSFIRE and therefore both obtained moderate spectral resolution ($R=$
3000-3650) data, both surveys were conducted over roughly the same period
(2012-2016), and both contain approximately the same number of galaxies
(N$\sim$1500).

###  2.1 The MOSDEF Survey

The MOSDEF survey was a 48.5-night Keck/MOSFIRE observing program spanning
multiple years (2012-2016). Approximately 1500 galaxies were targeted across
three redshift ranges: $1.37\leq z\leq 1.70$, $2.09\leq z\leq 2.61$, and
$2.95\leq z\leq 3.80$. These ranges were selected to optimise the detection of
strong rest-optical emission-lines (e.g., [O II]$\lambda\lambda$3727,3730,
H$\beta$, [O III]$\lambda\lambda$4960,5008, H$\alpha$, [N II]$\lambda$6585,
and [S II]$\lambda\lambda$6718,6733) within windows of atmospheric
transmission. Roughly half of the galaxies in the survey are in the middle
redshift range at $z\sim 2.3$. In this redshift range, $\sim$1/3 of the
galaxies were targeted based on existing spectroscopic redshifts, while the
remaining $\sim$2/3 were targeted based on photometric redshifts drawn from
the 3D-HST survey (Momcheva et al., 2016). The galaxies were selected from
five well-studied extragalactic legacy fields covered by the CANDELS and
3D-HST surveys: AEGIS, COSMOS, GOODS-N, GOODS-S, and UDS (Grogin et al., 2011;
Koekemoer et al., 2011; Momcheva et al., 2016). The survey is $H$-band (F160W)
magnitude limited, with brightness limits of $H_{\rm{AB}}=$ 24.0, 24.5, and
25.0, respectively, for the lowest, middle and highest redshift ranges.
Galaxies were observed for 1.5 hours in each filter ($Y$, $J$, and $H$ for a
subset) in the lowest redshift bin $z\sim 1.5$. The observing time was
increased to 2 hours per filter for the middle ($z\sim 2.3$; $J$, $H$, and
$K_{\rm{s}}$ bands) and highest ($z\sim 3.4$; $H$ and $K_{\rm{s}}$ bands)
redshift bins. For a full description of MOSDEF observing details and sample
selection, see Kriek et al. (2015).

In this study, we use the MOSDEF subsample from Sanders et al. (2018), which
contains 260 star-forming galaxies at $1.9\leq z\leq 2.7$ ($z_{\rm{med}}$ =
2.29) with S/N of both H$\alpha$ and H$\beta$ $\geq$ 3\. Galaxies with sky
lines significantly contaminating H$\alpha$ or H$\beta$ were not included, nor
those showing evidence of AGN activity based on their X-ray luminosity, IR
colors, or if N2 $\geq$ 0.5 (Coil et al., 2015; Azadi et al., 2017; Leung et
al., 2017). Galaxies with $\log_{10}(M_{\ast}/M_{\odot})<9.0$ are rejected due
to lack of completeness below that mass cutoff (see Section 2.4 for more
details).

The current study, which utilizes updated SED and emission-line fitting
methods, finds small differences in stellar mass (i.e., $M_{\ast}$) and
emission-line S/N from the values used in Sanders et al. (2018). Nine of the
260 galaxies from Sanders et al. (2018) have updated
$\log_{10}$($M_{\ast}/M_{\odot}$) $<$ 9.0 values, and are removed from this
study. Additionally, one galaxy is not in the 3D-HST v4.1 catalog, which is
needed for broadband SED fitting, and is also removed from the sample.
Hereafter in this work, we define the resulting sample of 250 galaxies as the
“MOSDEF $z\sim 2$ spectroscopic sample.”

Figure 1 (top left panel) features a redshift histogram of all MOSDEF galaxies
at $1.9\leq z\leq 2.7$, which we refer to hereafter as the “MOSDEF $z\sim 2$
targeted sample”. These include galaxies targeted to lie at $2.09\leq z\leq
2.61$ (Kriek et al., 2015) based on existing photometric or spectroscopic
redshifts, and galaxies serendipitously detected on MOSDEF slits with
redshifts measured in the range $1.9\leq z\leq 2.7$. Blue indicates the
distribution of the 250 galaxies in the $z\sim 2$ spectroscopic sample that
meet the selection criteria from Sanders et al. (2018). Green indicates the
422 galaxies that have a confirmed MOSFIRE redshift, but do not meet all of
the selection criteria outlined above (i.e., S/N, AGN, and/or $M_{\ast}$
cuts). Purple indicates the 114 galaxies that do not have a confirmed MOSFIRE
redshift. The photometric redshift from the 3D-HST catalogs is shown for these
galaxies. The median redshift for all MOSDEF galaxies at $1.9\leq z\leq 2.7$
is the same as that for the 250 galaxy spectroscopic sample (i.e.,
$z_{\rm{med}}$ = 2.29).

There are 536 galaxies in the MOSDEF $z\sim 2$ targeted sample that were not
included in the spectroscopic sample described here. Therefore, there are 786
galaxies in the MOSDEF $z\sim 2$ targeted sample. Galaxies were removed due to
several different criteria, as listed above. There were 114 galaxies lacking a
robust MOSFIRE redshift. Of the remaining 422 galaxies, 97 galaxies were
flagged as AGNs and removed. Of the 325 non-AGNs, 61 galaxies met all of the
criteria to make it into the Sanders et al. (2018) sample, but were removed
due to sky-line contamination of H$\alpha$ and/or H$\beta$. This removal was
performed based on visual inspection of the spectra. In addition, 26 galaxies
were removed because they have $\log_{10}$($M_{\ast}/M_{\odot}$) $<$ 9.0; 98
have S/NHβ and S/NHα $<$ 3, and are not retained; finally, 140 galaxies were
removed because they have S/NHβ $<$ 3 and S/NHα $\geq$ 3\. Of the 325 galaxies
that have a robust MOSFIRE redshift and are not AGN, 73% are removed due to
S/NHβ $\leq 3$.

###  2.2 The KBSS Survey

The Keck Baryonic Structure Survey (KBSS) began as a redshift survey of star-
forming galaxies in 15 independent fields, subtending a total area of 0.25
square degrees. Each KBSS survey region is centered on a bright ($V\sim
16-17$) QSO with $z=2.7\pm 0.1$. Galaxies were selected for spectroscopy using
photometric pre-selection based on rest-frame far-UV color criteria that had
been described and tested in earlier work (Steidel et al., 2003, 2004;
Adelberger et al., 2004). The rest-UV spectroscopy was conducted using LRIS-B
(Steidel et al., 2004) over the years 2002-2009, deliberately focusing on the
redshift range $2\lesssim z\lesssim 2.7$ to maximize the sensitivity of the
background QSO sightline to the IGM and CGM within the survey volume (e.g.,
Rudie et al. 2012, 2019; Turner et al. 2014). Most of the spectroscopically-
observed galaxies satisfied the “BX” or “MD” color selection criteria,
yielding galaxy redshifts $\langle z\rangle=2.17\pm 0.33$ and $\langle
z\rangle=2.63\pm 0.28$, respectively. In addition to the UV color criteria,
the KBSS UV spectroscopic sample was limited to galaxies with apparent
magnitude ${\cal R_{\rm AB}}$[6830 Å]$\leq 25.5$ to ensure that the resulting
spectra would yield redshifts for galaxies both with and without observed
Ly$\alpha$ emission. The full KBSS UV spectroscopic sample consists of 2341
objects (2285 star-forming galaxies and 66 AGN) with $\langle
z\rangle=2.354\pm 0.432$. Of these, 1435 (61.3%) have $1.9\leq z\leq 2.7$, and
950 (40.6%) have $2.1\leq z\leq 2.6$. The latter sub-sample comprised the
highest priority targets for near-IR spectroscopy, for the reasons outlined
above.

The KBSS-MOSFIRE survey (Steidel et al., 2014; Strom et al., 2017) was
initiated soon after MOSFIRE was commissioned in 2012 April. The survey had
multiple goals, including obtaining redshifts for galaxies that had failed to
yield redshifts from previous LRIS spectroscopy, especially those within a few
arc minutes in projection from the central QSO and whose rest-UV colors
suggested redshifts in the broader range $1.8\lesssim z\lesssim 3.5$.
Additional photometric selection criteria were imposed to ensure that the
targeted sample at $z\sim 2$ would roughly uniformly span the full range of
star-forming galaxy properties: first, we used the galaxy ${\cal R}-K_{\rm s}$
color ($\sim 2000$ Å $-$ 6500 Å in the rest-frame) for galaxies with $z\sim
2.3$, with preference given to those with ${\cal R}-K_{\rm s}>2$. Also added
as potential MOSFIRE targets were galaxies satisfying ${\cal R}-K_{\rm s}>2$
that were drawn from a region in $U_{\rm n}G{\cal R}$ color space adjacent to
the BX and MD region, in an effort to include relatively massive galaxies
likely to be more heavily reddened by dust; these were given the name “RK”
galaxies (see Strom et al. 2017). None of the RK objects had UV-based
redshifts prior to their observation with MOSFIRE.

Unlike MOSDEF, KBSS-MOSFIRE attempted to obtain the complete set of strong
nebular lines only for galaxies in the range $z=2-2.7$. However, MOSFIRE
slitmasks also included galaxies without previous spectroscopic redshifts,
either because they had never been observed with LRIS, or when previous LRIS
observations did not result in a secure redshift. For objects without prior
redshifts measurements, once a redshift was measured from initial MOSFIRE
spectroscopy, the object was retained as a high priority for subsequent
MOSFIRE observations only if it fell in the range $2\leq z\leq 2.7$.

MOSFIRE mask design was done iteratively as the survey progressed. Initially,
the highest priority targets were objects with known redshifts $2.1\leq z\leq
2.6$ drawn from the KBSS rest-UV spectroscopic survey. Any remaining space on
each mask was assigned to galaxies in the following order of priority:

  1. 1.

Galaxies without previous redshifts, drawn from the BX, MD, and RK color
selection; BX and MD candidates without known redshifts were favored if they
also had ${\cal R}-K_{\rm s}>2$.

  2. 2.

Galaxies with UV-based redshifts falling outside the primary $2\leq z\leq 2.7$
range.

  3. 3.

Galaxies without UV-based redshifts drawn from color selection windows least
likely to fall in the primary redshift range of interest (i.e., “BM”, “M”, “C”
candidates).

Figure 1: Top: Redshift histograms. Shown in these panels are distributions of
galaxies at $1.9\leq z\leq 2.7$ for the MOSDEF (left) and KBSS (right)
surveys. Blue indicates the $z\sim 2$ spectrosopic samples discussed in
Sections 2.1 (MOSDEF) and 2.2 (KBSS). Green indicates galaxies that were
observed and have a MOSFIRE redshift, but were not included in the $z\sim 2$
spectrosopic samples due to the various sample selection criteria discussed in
Sections 2.1 and 2.2. Purple indicates galaxies that were targeted, but a
robust MOSFIRE redshift was not obtained. For MOSDEF, the purple redshifts
displayed are photometric redshifts. For KBSS, the purple redshifts displayed
were obtained by previous rest-UV observations with Keck/LRIS. The LRIS
redshifts were estimated from either Ly$\alpha$ or low-ionization interstellar
absorption lines. Bottom: Corner plots comparing SFR(SED) vs. $M_{\ast}$
(left) and sSFR(SED) vs. $M_{\ast}$ (right) for the MOSDEF (red) and KBSS
(blue) $z\sim 2$ targeted samples. The dashed lines on the 1D histograms mark
the median values for the distributions, and the contours on the diagonal
trace the 3$\sigma$ regions in 2D space. Individual data points are also shown
in the 2D distribution. The range of datapoints is limited by the fact that
$M_{\ast}$ and SFR(SED) are correlated and are both determined by the
normalization of the SPS model to the photometry. In addition, for the 2D
panels the data crowd in the upper-left portion of the distribution (left
panel) and at high sSFR(SED) (right panel) because of the lower age limit for
the stellar populations. In total, there are 786 MOSDEF and 850 KBSS galaxies.
Note that two MOSDEF galaxies with log10(SFR(SED)/M⊙/yr-1) $<$ $-$3.0 are
excluded from the left panel and one MOSDEF galaxy with log10(sSFR(SED)/yr-1)
$<$ $-$14.0 is excluded from the right panel to better show the data.

The status of all objects in a given survey field was evaluated after each
MOSFIRE observing run, and objects were re-prioritized based on the S/N
achieved for each emission line of interest; for $H$\- and $K_{\rm{s}}$-band
observations these included H$\beta$, [O III]$\lambda\lambda$4960,5008 in $H$
band, H$\alpha$ and [N II]$\lambda\lambda$6550,6585 in $K_{\rm{s}}$ band. High
priority targets (based on redshift) were retained for inclusion on new masks
until all relevant lines were successfully detected.

For the purposes of comparison with MOSDEF, we consider the KBSS-MOSFIRE
$z\sim 2$ targeted sample to include all MOSFIRE-observed galaxies whose
redshifts were known from prior rest-UV spectroscopy to be in the range
$1.9\leq z\leq 2.7$ (621 objects in total, of which 558 yielded MOSIFRE
redshifts and 63 did not), plus the sample of objects that had not been
identified prior to MOSFIRE observation that fall in the primary $1.9\leq
z\leq 2.7$ redshift range (237 objects). The $z\sim 2$ targeted sample thus
defined has 858 objects, of which 795 yielded MOSFIRE redshifts111The $z\sim
2$ targeted KBSS-MOSFIRE sample does not include objects without UV redshifts
for which no significant emission lines were detected in the MOSFIRE spectra.
Many are targets observed in only 1 MOSFIRE atmospheric band as filler, and
would be unlikely to be assigned to subsequent masks if initial spectroscopy
showed no sign of emission in the desired range.. We do not have the broadband
photometry required for SED fitting for 8/858 galaxies (seven have a MOSFIRE
redshift while one does not). Therefore, the final KBSS $z\sim 2$ targeted
sample to be used in this analysis consists of 850 galaxies, 788 of which have
a MOSFIRE redshift. In comparison, the MOSDEF $z\sim 2$ targeted sample
contains 786 galaxies, of which 672 have a MOSFIRE redshift.

KBSS did not dedicate a specific amount of time to observing each galaxy,
instead choosing to stay in a given field until all rest-optical emission-
lines were detected in the majority of galaxies in that field. The median
observing time for a galaxy in the $H$-band (covering H$\beta$ and [O
III]$\lambda\lambda$4960,5008) was 2.44 hours. The maximum time spent on a
galaxy was 10.64 hours. For the $K_{\rm s}$-band (covering H$\alpha$, [N
II]$\lambda$6585, and [S II]$\lambda\lambda$6718,6733), the median and maximum
observing times were, respectively, 2.78 and 12.13 hours. For full KBSS-
MOSFIRE observing details, see Steidel et al. (2014) and Strom et al. (2017).

In this study, we use the KBSS-MOSFIRE sample from Strom et al. (2017), which
contains 377 galaxies at $1.9<z<2.7$ ($z_{\rm{med}}$ = 2.30) with spectral
coverage of the H$\beta$, [O III]$\lambda$5008, H$\alpha$, and [N
II]$\lambda$6585 emission lines (i.e., the four required for the [N II] BPT
diagram). To be included in this sample, the S/N of H$\alpha$ must be $\geq$
5, the S/N of H$\beta$ and [O III]$\lambda$5008 must be $\geq$ 3, and [N
II]$\lambda$6585 must simply fall within the wavelength coverage.

Galaxies are identified as having an AGN based on X-ray luminosity, mid-IR
photometry, strong UV emission in high-ionization lines (e.g., C
IV$\lambda$1549, C III]$\lambda$1908, and N V$\lambda$1240; Reddy et al. 2008;
Hainline et al. 2011), broad emission-line features in the rest-optical
spectra, and/or N2 $\geq$ 0.5.

To be consistent with the MOSDEF sample, we remove eight KBSS galaxies from
the Strom et al. (2017) sample flagged in the sample as AGN. Hereafter in this
work, we define the resulting sample of 369 galaxies as the “KBSS $z\sim 2$
spectroscopic sample.”

Figure 1 (top right panel) includes a redshift histogram of all KBSS galaxies
at $1.9\leq z\leq 2.7$, which we refer to hereafter as the “KBSS $z\sim 2$
targeted sample”. Blue indicates the distribution of the 369 galaxies in the
$z\sim 2$ spectroscopic sample that meet the selection criteria from Strom et
al. (2017). Green indicates galaxies that have a confirmed MOSFIRE redshift,
but do not meet all of the selection criteria (i.e., emission-line S/N and
broadband photometry criteria) outlined above. Purple indicates galaxies that
do not have a confirmed MOSFIRE redshift. For these galaxies, the redshifts
shown are based on Keck/LRIS measurements of either the Ly$\alpha$ or low-
ionization interstellar absorption features. The median redshift for all KBSS
galaxies at $1.9\leq z\leq 2.7$ (i.e., $z_{\rm{med}}$ = 2.29) is slightly
lower than for the 369 galaxy spectroscopic sample (2.31). Out of these 369
galaxies, 357 (96.7%) are $U_{\rm{n}}G\mathcal{R}$ color selected and 12
(3.3%) are “RK” galaxies.

###  2.3 MOSDEF & KBSS Sample Comparison

The selection methods for the MOSDEF and KBSS surveys are distinct. MOSDEF is
selected based on photometric redshifts (or spectroscopic redshifts when
available), with a magnitude limit in the rest-optical ($H$ band), while KBSS-
MOSFIRE is a UV-color-selected sample with a magnitude limit in the
$\mathcal{R}$ band. As discussed in Shivaei et al. (2015), the MOSDEF
magnitude-limited selection method is incomplete in the low-mass regime. At
the same time, the KBSS rest-UV sample can be biased against high-mass or
dusty galaxies whose $U_{\rm{n}}G\mathcal{R}$ colors place them outside of the
nominal rest-UV selection windows (e.g., Reddy et al. 2005; Strom et al.
2017). This difference in the $M_{\ast}$ regimes probed by the two samples can
be seen in the bottom panels of Figure 1. This figure contains corner plots
showing the 1D histograms and 2D distributions of SFR(SED) vs. $M_{\ast}$
(bottom left) and sSFR(SED) vs. $M_{\ast}$ (bottom right) for both samples.
The sample medians are marked on the 1D histograms and the 3$\sigma$ regions
are traced by contours in the 2D space. It is clear that KBSS more heavily
targets the low mass regime while MOSDEF targets span a wider range of stellar
populations in the high mass regime. These features of the two samples suggest
that they are complementary to one another in the low- and high-mass extremes,
and that the combination of the MOSDEF and KBSS $z\sim 2$ surveys would be
complete across a wide $M_{\ast}$ range of $\sim 10^{8.5}-10^{11.5}$ M⊙.
Additionally, the MOSDEF and KBSS samples are complementary in sSFR(SED)
space, as KBSS is better sampled at $\log_{10}$(sSFR(SED)/yr-1) $\geq$ $-8.0$
(indicating more intense star formation), especially at the low-mass end,
while MOSDEF has more coverage at $\log_{10}$(sSFR(SED)/yr-1) $\leq$ $-9.0$
(indicating less intense star formation), especially at the high-mass end.
Between the low- and high-mass regimes (i.e., $\sim
10^{9.5}-10^{10.5}M_{\odot}$) the contour maps overlap for the two samples.
Our methods for measuring these galaxy stellar population properties are
presented in Section 3.1 and a detailed comparisons of these properties for
the two samples are discussed in Sections 4.1 and 5.2.

Despite the differences in selection of the MOSDEF and KBSS $z\sim 2$ targeted
catalogs, there are notable similarities between the two surveys. The surveys
have comparable $z\sim 2$ catalog sizes (786 galaxies for MOSDEF and 850 for
KBSS) and median redshifts (2.29 for both surveys). Additionally, the majority
of galaxies in each survey would meet the selection criteria of the other
survey. Most of the KBSS $z\sim 2$ targeted sample (817/850 galaxies or 96.1%)
has $H_{\rm{AB}}<$ 24.5 (i.e., the main MOSDEF selection criterion). Table 1
shows the percent of the MOSDEF sample that falls into each region of the
$U_{\rm{n}}G\mathcal{R}$ diagram as defined in Steidel et al. (2003, 2004);
Strom et al. (2017). For MOSDEF galaxies, we passed best-fit stellar
population model SEDs (see Section 3.1) through the $U_{\rm{n}}G\mathcal{R}$
filters to estimate synthetic magnitudes using the IRAF (Tody, 1986, 1993)
routine, sbands. The statistics for the MOSDEF sample are compared to those of
the KBSS sample. The distribution of MOSDEF and KBSS galaxies in the various
regions on the $U_{\rm{n}}G\mathcal{R}$ diagram is very similar. Most notable
is that MOSDEF has a similar percentage of BX galaxies (82.8%) as KBSS
(80.5%). The BX region on the $U_{\rm{n}}G\mathcal{R}$ diagram is
statistically where $z\sim 2$ galaxies are most likely to be (Steidel et al.,
2004). Also, 773/786 galaxies (98.3%) in the MOSDEF $z\sim 2$ targeted sample
have $\mathcal{R}<25.5$ and fall into one of the defined regions of
$U_{\rm{n}}G\mathcal{R}$ color space. Of the remaining 13 galaxies in the
“Other” category, 12 have $\mathcal{R}>25.5$ while the other one has
$\mathcal{R}<25.5$ but did not fall into any of the defined regions on the
$U_{\rm{n}}G\mathcal{R}$ diagram. Finally, as we detail in the next section,
the spectroscopic incompleteness of the MOSDEF survey leads to an even greater
similarity between the distributions of galaxy properties for the MOSDEF and
KBSS $z\sim 2$ spectroscopic samples that have been analyzed in, e.g., Sanders
et al. (2018) and Strom et al. (2017).

$U_{\rm{n}}G\mathcal{R}$ Diagram Statistics
---
$U_{\rm{n}}G\mathcal{R}$ Region | N/850 of KBSS | N/786 of MOSDEF
(1) | (2) | (3)
BX | 684 (80.5%) | 651 (82.8%)
BM | 10 (1.2%) | 4 (0.5%)
M/MD | 97 (11.4%) | 75 (9.5%)
C/D | 23 (2.7%) | 0
RK | 28 (3.3%) | 43 (5.5%)
Other | 8 (0.9%) | 13 (1.7%)
Table 1:  Col. (1): Region of $U_{\rm{n}}G\mathcal{R}$ space, as defined in
Steidel et al. (2003, 2004); Strom et al. (2017). Col. (2): Percent of the
KBSS sample in that region of $U_{\rm{n}}G\mathcal{R}$ space. Col. (3):
Percent of the MOSDEF sample in that region of $U_{\rm{n}}G\mathcal{R}$ space.
Note that to fall into one of the regions on the $U_{\rm{n}}G\mathcal{R}$
diagram, the galaxy must have $\mathcal{R}<25.5$. Some of the galaxies in the
“Other” category fall into one of the defined $U_{\rm{n}}G\mathcal{R}$
regions, but have $\mathcal{R}>25.5$.

###  2.4 MOSDEF Sample Incompleteness

For MOSDEF, it has been known since the design of the survey that both the
number of targets and the success rate drop below
$\log_{10}(M_{\ast}/M_{\odot})$ = 9.0 due to the $H$-band magnitude selection
limits (Kriek et al., 2015). Accordingly, Sanders et al. (2018), who analyze a
galaxy sample that our current MOSDEF sample is intended to represent, removed
any galaxies below a mass limit of $\log_{10}(M_{\ast}/M_{\odot})$ = 9.0. Our
new SED fitting methodology utilizing emission-line subtracted photometry
resulted in slightly different stellar masses from what was used in Sanders et
al. (2018), with a small fraction (3.5%) now below that mass limit. As stated
above, we remove these galaxies to be consistent with the well documented
MOSDEF incompleteness in this mass regime. KBSS does not have this
incompleteness at low mass. Since the goal of this study is to compare MOSDEF
and KBSS samples that have been analyzed in previous work (Strom et al., 2017;
Sanders et al., 2018), we do not remove the galaxies with
$\log_{10}(M_{\ast}/M_{\odot})$ $<$ 9.0 from the KBSS $z\sim 2$ spectroscopic
sample.

We note that the MOSDEF survey is also incomplete in the red, massive regime.
The success rate of obtaining a $z_{\rm{MOSFIRE}}$ declines for galaxies with
red rest-frame optical/near-IR colors, where rest-frame U$-$V and V$-$J are
both greater than $\sim$1.25 (we discuss the UVJ diagram and how U$-$V and
V$-$J colors are estimated in Section 3.1). Due to the rest-UV sample
selection methods (including the $\mathcal{R}\leq 25.5$ cut off), the KBSS
survey does not target galaxies in this regime. Most of the RK galaxies are
not in this regime as well. The spectroscopic incompleteness of the MOSDEF
survey makes the MOSDEF and KBSS $z\sim 2$ spectroscopic samples more similar
than the MOSDEF and KBSS $z\sim 2$ targeted samples are. We explore this topic
more in Section 5.2.

###  2.5 SDSS Comparison Sample

When discussing the [N II] and [S II] BPT diagrams, we compare the emission
lines from our high-redshift MOSDEF and KBSS samples with similar measurements
from galaxies in the local universe. We pull archival emission-line
measurements from the SDSS Data Release 7 (DR7; Abazajian et al. 2009),
specifically from the MPA-JHU DR7 release of spectrum
measurements222https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/. The SDSS sample
is restricted to a redshift range of $0.04\leq z\leq 0.10$, and AGN are
removed using equation 1 from Kauffmann et al. (2003) or if $\mbox{N2}>0.5$.
On the [N II] ([S II]) BPT diagrams, a S/N $\geq$ 3 is required for the
H$\beta$, [O III]$\lambda$5008, H$\alpha$, and [N II]$\lambda$6585 ([S
II]$\lambda\lambda$6718,6733) features. These criteria result in comparison
samples of 103,422 and 102,543 SDSS galaxies, respectively, on the [N II] and
[S II] BPT diagrams.

##  3 Methodology

For this study, it is essential to analyze the MOSDEF and KBSS samples using
the same methodology. Doing so eliminates any bias introduced from the
analysis process, and can therefore isolate any systematic intrinsic
differences between the two samples. Section 3.1 introduces how the host
galaxy properties are obtained using SED fitting, and Section 3.2 describes
the emission-line measurements.

###  3.1 SED Fitting

Both MOSDEF and KBSS samples are covered by multi-wavelength photometry
enabling robust SED fitting; however, the specific sources of the photometry
vary between surveys. For MOSDEF, the specific set of ground-based and HST
optical and near-IR photometric bands varies from field to field (AEGIS,
COSMOS, GOODS-N, GOODS-S, and UDS), though all have Spitzer/IRAC coverage
(Skelton et al., 2014). All KBSS galaxies have $U_{\rm{n}}$, $G$,
$\mathcal{R}$, $J$, and $K_{\rm{s}}$ imaging. Additionally, all 15 fields have
Spitzer/IRAC coverage (typically two channels per field; Reddy et al. 2012),
10 fields have at least one pointing from Hubble/WFC3-IR F160W (Law et al.,
2012; Mostardi et al., 2015), and 8 fields have $J$1, $J$2, $J$3, $H$1, $H$2
imaging using Magellan-FourStar (Persson et al., 2013). While both MOSDEF and
KBSS have broadband photometry spanning a similar wavelength range (rest-UV to
rest-IR), we note that MOSDEF has a denser coverage with 15 photometry points
on average compared to 7 for KBSS.

In this study, we apply the same SED fitting method to both the MOSDEF and
KBSS samples. We use the SED fitting code FAST (Kriek et al., 2009) to obtain
many of the key galaxy properties compared between the two galaxy samples. For
this modeling, we adopt the Flexible Stellar Population Synthesis (FSPS)
library from Conroy & Gunn (2010), assume a Chabrier (2003) stellar initial
mass function (IMF), a Calzetti et al. (2000) dust attenuation curve, and
delayed-$\tau$ star-formation histories where SFR(SED) $\propto$ $t\times
e^{-t/\tau}$. Here, $t$ is the time since the onset of star formation, and
$\tau$ is the characteristic star-formation timescale. Both $t$ and $\tau$ are
allowed to vary in the FAST modeling. In addition to $t$ and $\tau$, we allow
the amount of dust attenuation ($A_{\rm{V}}$) and stellar mass of the galaxy
($M_{\ast}$) to vary and set the metallicity to 0.019, which is defined to be
solar metallicity in the Conroy & Gunn (2010) library.

We use emission-line subtracted photometry, where the flux of the strong rest-
optical emission lines measured directly from the MOSFIRE spectra (e.g.
H$\beta$, [O III]$\lambda\lambda$4960,5008, and H$\alpha$) has been subtracted
from the photometric measurements. These strong lines could bias the shape of
the SED fit on the red side of the Balmer break, resulting in the fitting code
favoring older ages than it would based solely on the stellar continuum.

From the SED fitting, we compare estimates of $M_{\ast}$, SFR(SED), sSFR(SED),
$A_{\rm{V}}$, $t/\tau$ for the MOSDEF and KBSS samples. All SFR values are
those inferred from the SED fits. These values are listed in Section 4.1 for
the $z\sim 2$ spectroscopic samples and Section 5.2 for the $z\sim 2$ targeted
samples.

Using the best-fit SEDs obtained from FAST, we also estimate the rest-frame
UVJ colors of the MOSDEF and KBSS samples using the IRAF (Tody, 1986, 1993)
function sbands. The UVJ diagram is a useful tool because the combination of
the U$-$V and V$-$J colors breaks the degeneracy for age and reddening (i.e.,
distinguishing between red quiescent galaxies and dusty star-forming
galaxies). Additionally, at red (i.e., high) U$-$V values, increasing V$-$J
traces increasing sSFR (Williams et al., 2009). In the star-forming track of
the UVJ diagram extending from low U$-$V and V$-$J up towards higher U$-$V and
V$-$J, low mass galaxies populate the bottom left region (i.e., blue U$-$V and
V$-$J colors) while high mass star-forming galaxies occupy the upper right
region (i.e., red U$-$V and V$-$J colors). Quiescent galaxies lie in the upper
left portion of the diagram (Williams et al., 2009).

We note that we experimented with CSF models similar to those utilized in past
KBSS studies (e.g., Steidel et al. 2014; Steidel et al. 2016; Strom et al.
2017) and some MOSDEF studies (e.g., Du et al. 2018; Reddy et al. 2018).
Appendix A provides a description of both the CSF models and the results for
the $z\sim 2$ spectroscopic and targeted samples. The results from the FAST
delayed-$\tau$ and CSF models do not perfectly agree with each other; however,
the differences are minor (particularly for the spectroscopic samples) and the
key takeaways of this study are the same regardless of which SED fitting
method we use.

We also experiment with models where star formation increases with time
(hereafter referred to as $\tau$-rising models), as this star formation
history can provide a better fit to $z\sim 2-3$ galaxies assuming constant
star formation (Reddy et al., 2012). Similar to the CSF models, the
differences in results between the delayed-$\tau$ and $\tau$-rising models are
minor, and the key takeaways of this study would remain unchanged if we used
the results from the latter.

###  3.2 Emission-line Measurements

We performed measurements for the MOSDEF and KBSS spectra using a custom
emission-line fitting IDL code developed by the MOSDEF team that fits a single
Gaussian to each emission-line in the spectra (Reddy et al., 2015). The FWHM
estimates are allowed to vary independently for each emission-line, but with
the following restrictions. The lower limit for the FWHM is the instrumental
resolution, which is determined by sky lines. The upper limit is 0.5 Å larger
than the FWHM measured for the line with the highest S/N (usually H$\alpha$).
The best-fit FAST models obtained through SED fitting were used to correct
H$\alpha$ and H$\beta$ for the underlying contribution of stellar Balmer
absorption. Specifically, the continuum under each line was fixed to the best-
fit model from FAST. If the best-fit Gaussian model deviates significantly
from the data, the emission-line fitting code will pick the integrated
bandpass flux as the preferred estimate of the emission-line flux. This custom
emission-line fitting code has been used for past MOSDEF studies (e.g.,
Shapley et al. 2015, 2019; Sanders et al. 2016, 2018); however, it has been
since notably updated for more accurate estimates of the contribution of
stellar Balmer absorption. For the MOSDEF $z\sim 2$ spectroscopic sample, the
median decrease in Balmer absorption correction is 36.4%, when compared with
the previous iteration of the MOSDEF code, which results in a 4.1% median
decrease in H$\beta$ flux.

The current emission-line fitting code used in this study is different from
that used in earlier published work for both MOSDEF and KBSS, which results in
small variations in the emission-line S/N compared to past studies. For
MOSDEF, updates to the estimates of the stellar Balmer absorption correction
result in a H$\beta$ S/N $<$ 3 for three galaxies that previously had a higher
S/N and were thus included in the Sanders et al. (2018) sample. For KBSS, the
utilization of a different code compared to the one used in Strom et al.
(2017) results in 24 galaxies with H$\beta$ S/N $<$ 3, nine galaxies with [O
III]$\lambda$5008 S/N $<$ 3, and two additional galaxies with H$\beta$ and [O
III]$\lambda$5008 S/N $<$ 3\. Although their S/N falls below the original
sample selection criteria, we still include these galaxies in our study and
investigate their SED properties. However, we include only the subset of
galaxies with S/N $\geq$ 3 for all relevant emission-lines in our analysis of
the [N II] and [S II] BPT diagrams.

We note that one difference between the MOSDEF and KBSS data reduction
approaches lies in the 2D to 1D extraction methods. MOSDEF uses an “optimal”
(Freeman et al., 2019) extraction while the Strom et al. (2017) catalogs uses
a boxcar extraction. We tested how this difference affects the measured
emission-line flux by extracting MOSDEF spectra using the boxcar method and
refitting the emission lines. For the four emission lines on the [N II] BPT
diagram, we find that the boxcar method has a higher median flux of 1-3%
compared to the optimal extraction method. For the line ratios (i.e., [O
III]$\lambda$5008/H$\beta$ and [N II]$\lambda$6585/H$\alpha$), the difference
is less than 1%. Since the difference is negligible, we continue to use the
optimal extraction for MOSDEF and boxcar extraction for KBSS, as these are the
methods used for the previously published survey results.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Distribution of physical properties for the MOSDEF (red) and KBSS
(blue) $z\sim 2$ spectroscopic samples. The red and blue skeletal histograms
represent the full spectroscopic samples (i.e., 250 galaxies for MOSDEF and
369 galaxies for KBSS), while the filled regions identify the subset of the
spectroscopic samples that are on the [N II] BPT diagram (i.e., 143 galaxies
for MOSDEF and 213 galaxies for KBSS). The sample medians, with uncertainties
for are shown in yellow and green for the MOSDEF (KBSS) full spectroscopic
samples (i.e., the larger empty histograms), and are given in Table 2. The
following galaxy properties are shown: (a) $M_{\ast}$, (b) SFR(SED), (c)
sSFR(SED), (d) log10($t/\tau$) of the stellar population using a
delayed-$\tau$ star formation model, (e) $A_{\rm{V}}$, and (f) the UVJ
diagram. The quiescent galaxy region in the upper left is enclosed in a solid
curve, while star-forming galaxies occupy the bottom half and upper right-hand
corner of the diagram. We find that all KBSS and MOSDEF sample median
properties agree within the 1$\sigma$ uncertainty, except for $t/\tau$. The
x-axes for panels a-e and both axes for panel f are the same as in Figure 7 to
show how the distributions of the $z\sim 2$ spectroscopic sample compared to
those for the $z\sim 2$ targeted samples. Median Values for Physical
Properties of the MOSDEF and KBSS $z\sim 2$ Spectroscopic Samples
---
Physical Property | MOSDEF Median | KBSS Median | p-value | Statistical significance
(1) | (2) | (3) | (4) | (5)
log${}_{10}(M_{\ast}/M_{\odot}$) | 9.85 $\pm$ 0.03 | 9.81 $\pm$ 0.03 | 0.0061 | 2.74$\sigma$
log10($t/\tau$) | 0.30 $\pm$ 0.04 | 0.00 $\pm$ 0.12 | 8.9e$-05$ | 3.75$\sigma$
log10(SFR(SED)/$M_{\odot}$/yr-1) | 1.29 $\pm$ 0.04 | 1.32 $\pm$ 0.03 | 0.064 | 1.85$\sigma$
log10(sSFR(SED)/yr-1) |  $-$8.61 $\pm$ 0.04 |  $-$8.67 $\pm$ 0.07 | 8.7e$-10$ | 6.13$\sigma$
$A_{\rm{V}}$ | 0.60 $\pm$ 0.05 | 0.60 $\pm$ 0.04 | 5.5e$-05$ | 4.03$\sigma$
U$-$V | 0.69 $\pm$ 0.02 | 0.66 $\pm$ 0.02 | 0.059 | 1.89$\sigma$
V$-$J | 0.49 $\pm$ 0.02 | 0.48 $\pm$ 0.03 | 0.18 | 1.34$\sigma$
Table 2:  Col. (1): Physical property of the MOSDEF and KBSS $z\sim 2$
spectroscopic samples (i.e., the skeletal histograms) shown in Figure 2. Col.
(2): Median value with uncertainty of the MOSDEF $z\sim 2$ spectroscopic
sample. Col. (3): Median value with uncertainty of the KBSS $z\sim 2$
spectroscopic sample. Col. (4): Two-tailed p-value, based on the K-S test,
estimating the probability that the null hypothesis can be rejected. Col. (5):
Statistical significance (i.e., the $\sigma$ value) corresponding to the
p-value.  Figure 3: Corner plot comparing the distributions of U$-$V and V$-$J
colors for the MOSDEF (red) and KBSS (blue) $z\sim 2$ spectroscopic samples.
The dashed lines on the 1D histograms mark the median values for the
distributions of U$-$V and V$-$J colors given in Table 3, and the contours in
the 2D panel trace the 3$\sigma$ regions in UVJ color-color space. The
individual data points are also shown in the 2D UVJ distribution. It is shown
here that the MOSDEF and KBSS $z\sim 2$ spectroscopic samples occupy a very
similar region of UVJ space, in addition to having sample medians that agree
within the uncertainties.

Figure 4: Corner plots comparing SFR(SED) vs. $M_{\ast}$ (left) and sSFR(SED)
vs. $M_{\ast}$ (right) for the MOSDEF (red) and KBSS (blue) $z\sim 2$
spectroscopic samples. The dashed lines on the 1D histograms mark the median
values for the distributions, and the contours on the diagonal trace the
3$\sigma$ regions in 2D space. The individual data points are also shown in
the 2D distribution. As stated for Figure 1, the distribution of datapoints is
limited by the fact that both $M_{\ast}$ and SFR(SED) are determined by the
normalization of the SPS model to the photometry. Additionally, in 2D space
the datapoints crowd in the upper-left part of the distribution (left panel)
and at high sSFR(SED) (right panel) because of the lower age limit for the
stellar populations. We have the same axes limits as the bottom two panels in
Figure 1 so the distribution of galaxies in the $z\sim 2$ spectroscopic
samples can easily be compared with the $z\sim 2$ targeted samples. Figure 5:
Stacked SEDs for the MOSDEF (red) and KBSS (blue) $z\sim 2$ spectroscopic
samples. The SEDs are normalized at 4550 Å to show any differences in the
global spectral shape. The $1\sigma$ scatter of the average SEDs is shown with
shaded regions for both samples.

Figure 6: The MOSDEF (red) and KBSS (blue) $z\sim 2$ spectroscopic samples on
the [N II] BPT (left) and [S II] BPT (right) diagrams. The grayscale 2D
histogram indicates local SDSS galaxies. Galaxies must have S/N $\geq$ 3 for
the H$\beta$, [O III]$\lambda$5008, H$\alpha$, and [N II]$\lambda$6585 ([S
II]$\lambda\lambda$6718,6733) emission-lines to be included in the [N II] BPT
([S II] BPT) diagram. The yellow and green lines represent binned medians
where the sample is binned in log10(O3N2) (left) and log10(O3S2) (right) for
the MOSDEF and KBSS samples. The orange curve on the [N II] BPT diagram is a
fit to the $z\sim 0$ star-forming locus (Kewley et al., 2013).

##  4 The Physical Properties of the $z\sim 2$ Spectroscopic Samples

In this section, we compare the MOSDEF and KBSS $z\sim 2$ spectroscopic
samples introduced in Sections 2.1 (MOSDEF) and 2.2 (KBSS). Specifically, in
Section 4.1 we analyze the host galaxy properties of the samples derived from
SED fitting. In Sections 4.2 and 4.3, we investigate the locations of the
$z\sim 2$ spectroscopic samples on, respectively, the [N II] and [S II] BPT
diagrams.

###  4.1 SED Fitting Properties

Figure 2 (panels a-e) displays histograms of the galaxy properties for the
MOSDEF and KBSS $z\sim 2$ spectroscopic samples obtained through the FAST SED
fitting. The red and blue lines give the distributions for the full MOSDEF
(250 galaxies) and KBSS (369 galaxies) $z\sim 2$ spectroscopic samples. The
red and blue filled portions of the distributions indicate the subset of
galaxies from each sample (143 MOSDEF and 213 KBSS galaxies) that have S/N
$\geq$ 3 for H$\beta$, [O III]$\lambda$5008, H$\alpha$, and [N
II]$\lambda$6585 and are included on the [N II] BPT diagram. Sample medians
are displayed as vertical lines for every galaxy property, and we derive
1$\sigma$ uncertainties on the medians through bootstrap resampling. These
medians are estimated using all galaxies in the $z\sim 2$ spectroscopic
samples, not only the subsets of galaxies on the [N II] BPT diagram, however
the [N II] BPT subsamples yield sample medians that do not significantly
change the key results of this study. These values are given in Table 2. We
also list the probabilities (i.e., p-values), based on the Kolmogorov-Smirnov
(K-S) test, of the null hypothesis that the two samples are drawn from the
same distribution. The samples are also shown on the UVJ diagram (panel f). We
include the Williams et al. (2009) boundary between quiescent and star forming
galaxies on the UVJ diagram. Figure 3 provides a corner plot of the U$-$V and
V$-$J colors showing both 1D histograms of each parameter and the distribution
of data points in 2D space overlaid with 3$\sigma$ contours.

Overall, we find that the MOSDEF and KBSS $z\sim 2$ spectroscopic samples have
very similar galaxy property distributions and median values. The KBSS sample
has a smaller (i.e., younger) median $t/\tau$ compared to the MOSDEF sample.
The p-value corresponds to an offset of greater than 3$\sigma$ indicating that
this difference is significant. The median values for all other host galaxy
properties agree within the uncertainties. We note that despite the sample
medians for sSFR(SED) and $A_{\rm{V}}$ agreeing within the uncertainties, the
p-values correspond to a significance greater than 3$\sigma$ indicating that
the two samples do not appear to occupy the same parameter space for these
properties. Of the KBSS sample, 125 galaxies (33.9%) have
$\log_{10}$(sSFR(SED)/yr-1) $\geq$ $-8.0$ compared to only 27 galaxies (10.8%)
of the MOSDEF sample. Also, we find that 50 galaxies (13.6%) in the KBSS
sample have stellar population age estimates of $\log_{10}$($t/\tau$)
$\leq-2.0$. There is only one MOSDEF galaxy (0.4% of the sample) in this low
$t/\tau$ regime. The KBSS galaxies with $\log_{10}$($t/\tau$) $\leq$ $-$2.0
are the same galaxies that have $\log_{10}$(sSFR(SED)/yr-1) $\geq-$8.0. This
correspondence indicates that the KBSS sample has a subset of galaxies that is
very young with intense star formation, for which there is no analogous subset
of galaxies in the MOSDEF sample. For $A_{\rm{V}}$, we find that 158 MOSDEF
galaxies (63.2% of the sample) have $0.5\leq A_{\rm{V}}\leq 1.0$ compared to
161 KBSS galaxies (only 43.6% of the sample). Otherwise, the distributions of
galaxy properties between the two samples are very similar, indicated by
p-values at less than 3$\sigma$ significance.

Figure 4 shows corner plots of SFR(SED) vs. $M_{\ast}$ (left panel) and
sSFR(SED) vs. $M_{\ast}$ (right panel) for the MOSDEF and KBSS $z\sim 2$
spectroscopic samples. The sample medians from Table 2 are shown on the 1D
histogram, and the 3$\sigma$ contour for each sample is outlined in the 2D
distribution. These differences echo those observed for the $z\sim 2$ targeted
samples (see Figure 1). The difference at low mass is accentuated by the fact
that we implemented a mass cut at 109 M⊙ for MOSDEF and did not for KBSS.
However, the MOSDEF sample cut was implemented because of the known
incompleteness at low mass. We emphasize that our goal here is to compare the
properties of samples that have been analyzed in previously published work
(i.e., Sanders et al., 2018; Strom et al., 2017), and that these different
selections at the low-mass end simply reflect what was adopted in these
earlier papers. As for the difference in sSFR(SED), the fact that KBSS
contains more galaxies at high sSFR(SED) than MOSDEF matches our observation
when comparing the distributions in panel (c) of Figure 2.

The high $M_{\ast}$, low sSFR(SED) end of the 3$\sigma$ contours is
approximately the same for both samples. This is different from what we find
for the $z\sim 2$ targeted samples, where MOSDEF extends to higher mass and
lower sSFR(SED) compared to KBSS in Figure 1. The difference is due to the
MOSDEF sample, where the 3$\sigma$ contour for the $z\sim 2$ spectroscopic
sample does not extend to as high mass and as low sSFR(SED) as the $z\sim 2$
targeted sample. This again points out the MOSDEF incompleteness with respect
to the massive, low sSFR(SED) galaxies targeted in the survey (see Section 5.2
further discussion on this topic).

Figure 5 shows the stacked SEDs of the MOSDEF and KBSS $z\sim 2$ spectroscopic
samples. Similar to Figure 13 in Strom et al. (2017), we normalize each
spectrum by the flux at 4550 Å , sum the normalized spectra in each sample,
then divide by the sample size to account for the uneven sizes of the MOSDEF
and KBSS samples. We find that the stacked SEDs of the two samples are nearly
identical. It is reasonable that the differences in rest-frame flux of the UV,
optical, and near-IR regimes are negligible given that the sample medians for
$\log_{10}$($t/\tau$) and UVJ colors agree within the uncertainties. The
MOSDEF Balmer break is larger compared to KBSS; however, given the separation
in median $t/\tau$ between the two samples, it is unclear why the difference
in the Balmer breaks is not more pronounced.

###  4.2 The [N II] BPT Diagram

The left panel of Figure 6 shows the [N II] BPT diagram for the MOSDEF and
KBSS samples. We include median lines binned by $\log_{10}$(O3N2) for both
samples. There are equal numbers of galaxies in each bin. The binning scheme
was adopted because it divides the samples into subgroups segregated roughly
along the local star-forming sequence. Only galaxies with S/N $\geq$ 3 for the
H$\beta$, [O III]$\lambda$5008, H$\alpha$, [N II]$\lambda$6585 emission-lines
are included on the figure and in the binned medians. There are 143 MOSDEF and
213 KBSS galaxies that meet the S/N requirements to be included in this plot.

We find that the two binned medians for the MOSDEF and KBSS samples have an
almost identical offset from the local sequence. When discussing the [N II]
BPT diagram, we define the term “offset” as the orthogonal distance from the
Kewley et al. (2013) fit of the local star-forming locus. The MOSDEF and KBSS
bins have respectively,a median offset of 0.12 $\pm$ 0.02 and 0.10 $\pm$ 0.02
dex (0.02 $\pm$ 0.02 dex separation) perpendicular to the Kewley et al. (2013)
fit. The overlap of the offsets within the uncertainties is very different
from those presented in past MOSDEF (Shapley et al., 2015) and KBSS (Steidel
et al., 2014; Strom et al., 2017) work. These earlier studies have cited KBSS
as having a larger offset from the local sequence than MOSDEF by $\sim$0.1
dex. We credit the convergence of the [N II] BPT offsets to consistent
emission-line fitting between the two samples. Both samples underwent changes
to line ratios compared to past measurements.

Here we find that median O3 ratio for the MOSDEF sample is 0.04 dex higher
compared to past MOSDEF studies (e.g., Shapley et al., 2015; Sanders et al.,
2016). When restricting the $z\sim 2$ spectroscopic sample to galaxies
included on the [N II] BPT diagram, the updated (and more robust) methodology
for estimating stellar Balmer absorption results in a median decrease of 6.7%
in the H$\beta$ fluxes. This updated methodology has a negligible effect on
the H$\alpha$ flux and therefore results in a negligible change to the N2
ratio. The KBSS sample is characterized by a 0.07 dex lower N2 compared to
results presented in past KBSS studies. For galaxies with [N II]$\lambda$6585
S/N $\geq 3.0$ in both this study and past KBSS studies (Steidel et al., 2014;
Strom et al., 2017), we find a median decrease of 12.8% in [N II]$\lambda$6585
flux estimates. In summary, when using a uniform emission-line fitting and
Balmer absorption correction methodology we find that both higher O3 for the
MOSDEF sample and lower N2 for the KBSS sample combine to erase the apparent
differences between the two samples in the [N II] BPT diagram that serve to
make the median excitation sequences from each survey indistinguishable within
the uncertainties.

###  4.3 The [S II] BPT Diagram

The right panel of Figure 6 shows the [S II] BPT diagram for the MOSDEF and
KBSS samples. We include median lines binned in $\log_{10}$(O3S2) for both
samples. Each bin has an equal number of galaxies. Similar to the case for the
[N II] BPT diagram, the binning scheme was adopted because it divides the
samples into subgroups segregated roughly along the local star-forming
sequence. Also similar to the [N II] BPT diagram, only galaxies with S/N
$\geq$ 3 for the emission-lines relevant to the [S II] BPT diagram (i.e.,
H$\beta$, [O III]$\lambda$5008, H$\alpha$, and [S
II]$\lambda\lambda$6718,6733) are included in the figure and the binned
medians. There are 156 MOSDEF and 228 KBSS galaxies that meet the S/N
requirements to be included in this plot.

Aside from in the bin with the highest S2, the binned medians for the two
samples are very similar. Furthermore, the MOSDEF and KBSS median line ratios
are more similar than in previous studies (Steidel et al., 2014; Shapley et
al., 2015; Strom et al., 2017; Sanders et al., 2018). The MOSDEF sample shows
a 0.04 dex increase in O3 (similar to the [N II] BPT diagram), a result of the
updated method for estimating stellar Balmer absorption corrections. For KBSS,
there is an 0.04 dex decrease in the S2 ratio due to smaller flux estimates by
8.6% on average for the [S II]$\lambda\lambda$6718,6733 doublet than in the
methodology from Strom et al. (2017). This change is analogous to the effect
we observed with the [N II]$\lambda$6585 flux estimates, which will be
discussed in Section 5.1. Overall, the changes observed in the [S II] BPT
diagram reflect similar shifts in MOSDEF and KBSS to those we found in the [N
II] BPT diagram.

##  5 Discussion

This section begins with a discussion of the [N II] BPT offset. In Section
4.2, we have shown that the median offset from the local sequence for the
$z\sim 2$ MOSDEF and KBSS spectroscopic samples agree within the uncertainties
when using consistent analysis methods. Section 5.1 discusses the significance
of this result, compares the results here to previous MOSDEF and KBSS studies,
and investigates the bias introduced from utilizing different emission-line
fitting methodologies.

We then turn our attention to the MOSDEF and KBSS $z\sim 2$ targeted samples,
from which the $z\sim 2$ spectroscopic samples were drawn. Section 5.2
compares the host galaxy properties of the $z\sim 2$ targeted samples.
Additionally, we discuss differences found between the $z\sim 2$ targeted
samples (i.e., the full set of galaxies the MOSDEF and KBSS surveys intended
to observe) and the smaller $z\sim 2$ spectroscopic samples (i.e., the subsets
of the $z\sim 2$ targeted samples with high S/N spectra). This comparison is
significant because it reveals if the galaxies targeted by the MOSDEF and KBSS
surveys are fundamentally different from the subsets of galaxies with high S/N
analyzed in past publications (e.g., Steidel et al. 2014; Shapley et al. 2015;
Strom et al. 2017; Sanders et al. 2018).

###  5.1 The [N II] BPT Offset

We have shown that, when subject to identical analysis methods, the MOSDEF and
KBSS samples are much more similar than reported in previous work. As
discussed in Section 3.2, we used an updated version of the MOSDEF emission-
line fitting code for this study. Using the same code for both samples yielded
offsets of 0.12 $\pm$ 0.02 and 0.10 $\pm$ 0.02 dex from the Kewley et al.
(2013) fit (0.02 $\pm$ 0.02 dex separation) for, respectively, the MOSDEF and
KBSS samples.

As a consistency check, we fit the emission lines of both samples using the
same Gaussian fitting assumptions as the MOSPEC code that KBSS previously used
(Steidel et al., 2014; Strom et al., 2017). MOSPEC is a 1D spectral analysis
code written in IDL and developed to analyze the MOSFIRE spectra of faint
galaxies. Because “MOSPEC” and “MOSDEF” are very similar acronyms, we will
hereafter refer to the MOSPEC code used by the KBSS team as the “KBSS MOSPEC
code” for clarity. The key difference between the two codes is that the KBSS
MOSPEC code fixes the centroids of every emission-line in a given band to a
single redshift. Also, the velocity widths of all emission-lines are fixed to
be the same in each band (i.e., $H$ and $K_{\rm{s}}$). This restriction
assigns H$\beta$ and each line in the [O III]$\lambda\lambda$4960,5008 doublet
the same velocity widths in the H band, and H$\alpha$, [N II]$\lambda$6585,
and each line in the [S II]$\lambda\lambda$6718,6733 doublet the same velocity
width in the $K_{\rm{s}}$-band. As discussed in Section 3.2, the centroids and
widths are allowed more freedom to vary within the MOSDEF code (Kriek et al.,
2015; Reddy et al., 2015). Appendix B provides examples of the two codes
fitting the same set of spectra to highlight the similarities and differences
between them.

It is important to emphasize the motivations that led to the way these two
codes were written. For the MOSDEF code, more freedom was allowed in the
fitting (e.g., in the centroids and the FWHMs) to fully marginalize over
errors in these parameters for the highest S/N line, given that in some cases
these lines were relatively weak and/or did not have a Gaussian shape. For
KBSS, which observed the same galaxies many times at different PAs and for a
longer period, the line shape can be better quantified. Thus, it is more
likely that all lines will provide an accurate probe of the FWHM and redshift.

We find that using the KBSS MOSPEC code results in a $\sim$13% median increase
in [N II]$\lambda$6585 flux estimates for both the MOSDEF and KBSS samples
compared to when using the MOSDEF code. This difference corresponds to higher
N2 line ratio estimates for both samples as well. A similar effect is observed
for the [S II]$\lambda\lambda$6718,6733 doublet.

When using the KBSS MOSPEC code, the MOSDEF and KBSS samples, respectively,
have a median offset of 0.15 $\pm$ 0.02 and 0.12 $\pm$ 0.02 dex (0.03 $\pm$
0.02 dex separation) perpendicular to the Kewley et al. (2013) fit. The
separation of the two samples with the KBSS MOSPEC code is within the
uncertainties of that found with the MOSDEF code (i.e., 0.02 $\pm$ 0.02 dex).
Note that due to the changes to the N2 and O3 flux ratios, both the MOSDEF and
KBSS samples are more offset from the local SDSS sequence by 0.03 and 0.02
dex, respectively, when using the KBSS MOSPEC code compared to when the MOSDEF
code is used. As noted above, this shift arises primarily from the KBSS MOSPEC
code yielding larger [N II]$\lambda$6585 flux estimates compared to the MOSDEF
code. There is no significant shift in O3 regardless of which line fitting
code is adopted.

When compared to the [N II]$\lambda$6585 and [S II]$\lambda\lambda$6718,6733
doublet bandpass integrated line flux, we find that the MOSDEF Gaussian fits
generally estimates a lower flux on average while the KBSS Gaussian fits
estimates a higher flux on average. In other words, the MOSDEF code
underestimates the flux while the KBSS MOSPEC code overestimates the flux for
these weaker emission-lines compared to their integrated bandpass fluxes. The
goal of this paper is not to determine which fitting method is more correct,
but instead to show that different assumptions, both reasonable, when
implemented into emission-line fitting codes, can lead to different flux
measurements resulting in shifts of $\sim$0.02-0.03 dex on the [N II] BPT
diagram. These varying assumptions introduce inherent systematic uncertainty
in the flux measurements that are not captured within the statistical
uncertainties of the spectra. However, it is important to note that we find
nearly identical small relative offsets — 0.02 $\pm$ 0.02 dex using the MOSDEF
code and 0.03 $\pm$ 0.02 dex using the KBSS MOSPEC code — between the median
MOSDEF and KBSS emission-line sequences on the [N II] BPT diagram when the
same line fitting code is used. Therefore, the same conclusions regarding the
locus of $z\sim 2$ star-forming galaxies in the [N II] BPT diagram would be
reached regardless of which line fitting code is used for analysis.

We note that the $\sim$0.1 dex offset cited in previous MOSDEF (e.g., Shapley
et al. 2015) and KBSS (e.g. Steidel et al. 2014; Strom et al. 2017) works is
resolved based on updated (smaller) stellar Balmer absorption corrections for
the MOSDEF sample as well as consistent emission-line fitting methods.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Distribution of physical properties for the MOSDEF (red) and KBSS
(blue) $z\sim 2$ targeted samples. The sample medians, with uncertainties are
shown in yellow (green) for the MOSDEF (KBSS) samples, and are given in Table
3. The following galaxy properties are shown: (a) $M_{\ast}$, (b) SFR(SED),
(c) sSFR(SED), (d) log10($t/\tau$) of the stellar population using a
delayed-$\tau$ star formation model, (e) $A_{\rm{V}}$, and (f) the UVJ
diagram. We find that the MOSDEF $z\sim 2$ targeted sample has a higher
$M_{\ast}$, lower SFR(SED) and sSFR(SED), and a redder U$-$V color compared to
the KBSS $z\sim 2$ targeted sample. Note that two MOSDEF galaxies with
log10(SFR(SED)/M⊙/yr-1) $<$ $-$3.0 are excluded from the panel (b), and one
MOSDEF galaxy with log10(sSFR(SED)/yr-1) $<$ $-$14.0 is excluded from panel
(c) to better show the data. Median Values for Physical Properties of the
MOSDEF and KBSS $z\sim 2$ Targeted Samples
---
Physical Property | MOSDEF Median | KBSS Median | p-value | Statistical significance
(1) | (2) | (3) | (4) | (5)
log${}_{10}(M_{\ast}/M_{\odot}$) | 9.95 $\pm$ 0.03 | 9.84 $\pm$ 0.02 | 1.9e$-$7 | 5.21$\sigma$
log10($t/\tau$) | 0.40 $\pm$ 0.05 | 0.30 $\pm$ 0.09 | 7.5e$-$18 | 8.61$\sigma$
log10(SFR(SED)/M⊙/yr-1) | 1.13 $\pm$ 0.02 | 1.30 $\pm$ 0.02 | 4.2e$-$9 | 5.88$\sigma$
log10(sSFR(SED)/yr-1) |  $-$8.85 $\pm$ 0.04 |  $-$8.67 $\pm$ 0.01 | 1.1e$-$20 | 9.33$\sigma$
$A_{\rm{V}}$ | 0.60 $\pm$ 0.01 | 0.60 $\pm$ 0.03 | 6.2e$-$8 | 5.41$\sigma$
U$-$V | 0.78 $\pm$ 0.02 | 0.69 $\pm$ 0.01 | 3.6e$-$8 | 5.51$\sigma$
V$-$J | 0.51 $\pm$ 0.02 | 0.49 $\pm$ 0.01 | 3.9e$-$5 | 4.11$\sigma$
Table 3:  Col. (1): Physical property of the galaxies in the sample shown in
Figure 7. Col. (2): Median value with uncertainty of the MOSDEF $z\sim 2$
targeted sample. Col. (3): Median value with uncertainty of the KBSS $z\sim 2$
targeted sample. Col. (4): Two-tailed p-value, based on the K-S test,
estimating the probability that the null hypothesis can be rejected. Col. (5):
Statistical significance (i.e. the $\sigma$ value) that the p-value
corresponds to.  Figure 8: Corner plot comparing the distributions of U$-$V
and V$-$J colors for the MOSDEF (red) and KBSS (blue) $z\sim 2$ targeted
samples. The dashed lines on the 1D histograms mark the median values for the
distributions of U$-$V and V$-$J colors given in Table 3, and the contours in
the 2D panel trace the 3$\sigma$ regions in UVJ color-color space. The
individual data points are also shown in the 2D UVJ distribution. It is shown
here that MOSDEF targeted a population of red galaxies with high U$-$V color
that is not seen in the KBSS $z\sim 2$ targeted sample. Additionally, MOSDEF
(KBSS) extends to slightly redder (bluer) regions of V$-$J color space. Figure
9: Stacked SEDs for the MOSDEF (red) and KBSS (blue) $z\sim 2$ targeted
samples. The SEDs are normalized at 4550 Å to show any differences in the
global spectral shape. The scatter of the SEDs is shown with shaded regions
for both samples.

###  5.2 SED Properties of MOSDEF and KBSS $z\sim 2$ Targeted Samples

In Section 4.1, we compared the galaxy properties obtained through SED fitting
for the $z\sim 2$ spectroscopic MOSDEF and KBSS samples (i.e., the galaxies
shown in blue in Figure 1 and analyzed in previous studies Strom et al. 2017;
Sanders et al. 2018). Except for $t/\tau$ (i.e., age), the sample medians for
every other galaxy property investigated agree within the uncertainties. Here,
we expand our analysis and examine the SED properties of the full MOSDEF and
KBSS $z\sim 2$ targeted survey catalogs (i.e., all galaxies shown in Figure 1,
including the galaxies where a $z_{\rm{MOSFIRE}}$ was not measurable). The
goal of this analysis is to determine whether the different survey selection
methods used by MOSDEF and KBSS resulted in galaxies with systematically
different properties targeted in the surveys. As we discuss in Section 2.3,
most of the MOSDEF $z\sim 2$ galaxies would fall into the KBSS selection
criteria (and vice versa). The majority (96.1%) of the KBSS $z\sim 2$ targeted
sample satisfies the main selection criterion of the MOSDEF survey (i.e.,
$H_{\rm{AB}}<24.5$), while the distribution of the MOSDEF $z\sim 2$ targeted
sample on the $U_{\rm{n}}G\mathcal{R}$ diagram is very similar to KBSS with
98.5% of MOSDEF galaxies having $\mathcal{R}<25.5$. This analysis will help to
understand the biases of each survey with respect to the other, which is
necessary for future studies using the combined sample.

We consider the same galaxy properties as in Section 4.1: $M_{\ast}$,
SFR(SED), sSFR(SED), $\log_{10}$($t/\tau$) of the stellar population using a
delayed-$\tau$ star formation model, $A_{\rm{V}}$, UVJ colors, and stacked SED
shapes. The UVJ diagram along with histograms of these properties for MOSDEF
and KBSS samples are shown in Figure 7. We derive 1$\sigma$ uncertainties on
the medians through bootstrap resampling. Table 3 lists the medians with
uncertainties, two-tailed p-value based on the K-S test, and the level of
significance (the $\sigma$ value) of the p-value. Additionally, Figure 8
explores the UVJ colors in more detail by including both a 2D contour map in
UVJ space and 1D histrograms of the U$-$V and V$-$J distributions. Finally,
the stacked SEDs for the MOSDEF and KBSS $z\sim 2$ targeted samples are shown
in Figure 9. The stacked SEDs are made using the same methods described in
Section 4.1 (i.e., the SEDs are normalized at 4550 Å to best show any
variations in SED shape between the two samples).

Based on visual inspection of the stacked SEDs, it is clear that on average
the MOSDEF $z\sim 2$ targeted sample has a larger Balmer break and higher flux
redward of $\sim$5000 Å. These observed differences in SED shapes for the
$z\sim 2$ targeted samples are not seen in the stacked SEDs for the $z\sim 2$
spectroscopic samples, indicating that the MOSDEF and KBSS $z\sim 2$ targeted
samples are less similar than the $z\sim 2$ spectroscopic samples.
Additionally, we find that the MOSDEF $z\sim 2$ targeted sample has a larger
median $M_{\ast}$, smaller SFR(SED) and sSFR(SED) medians, and a redder U$-$V
compared to the KBSS $z\sim 2$ targeted sample. The medians for the remaining
properties (i.e., $t/\tau$, $A_{\rm{V}}$, and V$-$J color) all agree within
the uncertainties. The p-value from the K-S test for every galaxy property
investigated in Figure 7 has a significance of at least 4$\sigma$, indicating
that the shape of the distribution for all galaxy properties considered — even
the ones with overlapping sample medians — are significantly different.

Examining the distribution of the galaxies in Figure 7, we find clear
differences between the two surveys aside from the sample medians. In the KBSS
$z\sim 2$ targeted catalog, 281 galaxies (33.1% of the sample) have
$\log_{10}$(sSFR(SED)/yr-1) $\geq-8.0$ and 106 galaxies (12.5% of the sample)
have $\log_{10}(t/\tau)\leq-2.0$. All 106 of the galaxies in this low $t/\tau$
regime are in the high sSFR(SED) regime. The MOSDEF $z\sim 2$ targeted catalog
is not well represented in these regions of parameter space, with only 77
galaxies (9.8% of the sample) at $\log_{10}$(sSFR(SED)/yr-1) $\geq-8.0$ and 14
galaxies (1.8% of the sample) at $\log_{10}(t/\tau)\leq-2.0$. Similar to KBSS,
all 14 of the MOSDEF $z\sim 2$ targeted sample galaxies with low $t/\tau$ also
have a high sSFR(SED). It is apparent that the KBSS $z\sim 2$ sample contains
a population of very young galaxies with intense star formation that is mostly
absent from the MOSDEF $z\sim 2$ sample. At the same time, MOSDEF contains 59
galaxies (7.5% of the sample) with both U$-$V and V$-$J $\geq$ 1.25, whereas
KBSS comparatively only has 18 galaxies (2.1% of the sample) in that area of
the UVJ diagram. The 3$\sigma$ contours in UVJ space in Figure 8 also show
that the MOSDEF $z\sim 2$ targeted sample probes a more reddened part of UVJ
space than KBSS. Galaxies in this regime of the UVJ diagram tend to be more
heavily dust obscured or have very low SFR(SED). The distributions of the
$z\sim 2$ targeted samples with $A_{\rm{V}}\geq 1$ are similar (22.1% and
26.1% of the MOSDEF and KBSS samples), indicating that the reddest galaxies in
the MOSDEF $z\sim 2$ targeted sample — i.e., those that are not well
represented in the KBSS $z\sim 2$ targeted sample — are not dominated by
heavily dust obscured systems but rather those with low sSFR(SED). This
difference can be attributed to the original selection method of the KBSS
survey (i.e., rest-UV criteria based on $U_{\rm{n}}G\mathcal{R}$ colors), and
is one of the reasons why the new subset of “RK” galaxies Strom et al. (2017)
was added to the survey. However, it is important to note that the “RK”
galaxies are a small minority, only 28 galaxies corresponding to 3.3%, of the
850 galaxy KBSS $z\sim 2$ targeted sample.

Some of the results for the $z\sim 2$ targeted samples in Figure 7 are
different from what we find for the $z\sim 2$ spectroscopic samples in Section
4.1. The MOSDEF $z\sim 2$ targeted sample has a higher median $M_{\ast}$ and
$t/\tau$, a lower median SFR(SED) and sSFR(SED), and redder U$-$V color
compared to the MOSDEF $z\sim 2$ spectroscopic sample. The remaining host
galaxy properties (i.e., $A_{\rm{V}}$ and V$-$J) agree within the median
uncertainties. Additionally, the subset of the MOSDEF $z\sim 2$ targeted
sample with U$-$V and V$-$J colors $\geq$ 1.25 mentioned above are not in the
$z\sim 2$ spectroscopic sample (only 2 galaxies, 0.8% of the sample, are in
this regime). This indicates that the population of the MOSDEF survey that is
red, old, massive with a relatively low SFR is mostly left out of the $z\sim
2$ spectroscopic sample. For many of these galaxies, a $z_{\rm{MOSFIRE}}$ was
not obtained, indicating that this is an incomplete regime within the MOSDEF
survey that will need to be addressed with deeper spectroscopy. This region of
UVJ space is mostly absent from the KBSS $z\sim 2$ targeted sample as well.

The KBSS $z\sim 2$ targeted sample has a higher $t/\tau$ compared to the KBSS
$z\sim 2$ spectroscopic sample analyzed here. The remaining sample properties
agree within the median uncertainties, including $M_{\ast}$, SFR(SED),
sSFR(SED), $A_{\rm{V}}$, and UVJ colors. It does not appear that there is an
analogous incompleteness within the KBSS catalog to what is observed for
massive, low sSFR(SED) galaxies in MOSDEF. However, it is important to note
that the KBSS $z\sim 2$ targeted sample does not contain the red population
that MOSDEF targeted but was ultimately unsuccessful with. This could be
attributed to the differences in the MOSDEF and KBSS observing strategies.
MOSDEF observed each object for the equivalent of 2 hours per filter in clear
conditions in each band while KBSS retained objects for subsequent observation
if all lines had not been detected.

Despite the notable differences in the $z\sim 2$ targeted samples described
above, the MOSDEF and KBSS $z\sim 2$ spectroscopic samples analyzed in the BPT
diagrams have mostly similar host galaxy property distributions and median
values. Because the targeting strategies of the surveys are so different, the
similarity of the MOSDEF and KBSS $z\sim 2$ spectroscopic samples reveals the
challenges of detecting emission-lines for dusty and/or low sSFR galaxies at
$z\sim 2$. As the MOSDEF incompleteness is addressed with deeper spectroscopy
for massive galaxies, and therefore a MOSDEF $z\sim 2$ spectroscopic sample
that better reflects its $z\sim 2$ targeted sample is assembled, it is unclear
how the relative emission-line ratios of the MOSDEF and KBSS $z\sim 2$
spectroscopic samples will change at the massive end.

##  6 Summary

The combination of the MOSDEF and KBSS surveys represents the largest
investment of Keck/MOSFIRE observing time, with $\sim$3000 galaxies (about
half at $z\sim 2$) observed over $\sim$100 nights. The statistical weight of
the combined MOSDEF+KBSS sample is capable of determining robust trends in
galaxy properties and physical relationships in the $z\sim 2$ universe
analogous to what currently exists locally. To understand both how the
differences in the survey selection criteria affect the samples and how to
account for any biases between the samples when combining them, this study
presents a thorough comparison of the $1.9\leq z\leq 2.7$ MOSDEF and KBSS
samples. Using consistent broadband emission-line subtracted SED fitting and
emission-line fitting, we compared the physical and emission-line properties
of the previously published MOSDEF (Sanders et al., 2018) and KBSS (Strom et
al., 2017) samples. In addition, we investigate how well these samples
represent the $z\sim 2$ targeted samples from which they are drawn. Applying
the same analysis methodology to both samples eliminates biases stemming from
the analysis process, meaning any variations found between the samples are due
to systematic differences between the samples due to the survey selection
method.

The main results are as follows:

  1. 1.

Despite the differences in galaxy selection methods for the MOSDEF and KBSS
surveys, the $z\sim 2$ spectroscopic samples have very similar galaxy
properties. KBSS has a lower median $t/\tau$ of the stellar population
assuming a delayed-$\tau$ star formation model compared to MOSDEF, but the
medians for all other galaxy properties investigated ($M_{\ast}$, SFR(SED),
sSFR(SED), $A_{\rm{V}}$, and UVJ colors) agree within the uncertainties.
Additionally, the normalized stacked SEDs are very similar as well. The most
notable sample difference is that there are 50 KBSS galaxies (13.6% of the
sample) that are very young ($\log_{10}(t/\tau)\leq-2.0$) with intense star
formation ($\log_{10}$(sSFR(SED)/yr-1) $\geq-8.0$) compared to only 1 such
galaxy within the MOSDEF sample.

  2. 2.

The MOSDEF and KBSS $z\sim 2$ spectroscopic samples have offsets,
respectively, of 0.12 $\pm$ 0.02 and 0.10 $\pm$ 0.02 dex from the local SDSS
sequence on the [N II] BPT diagram using the emission-line fitting assumptions
from the MOSDEF code. These offsets change to 0.15 $\pm$ 0.02 and 0.12 $\pm$
0.02 dex, respectively, from the local SDSS sequence using the emission-line
fitting assumptions from the KBSS MOSPEC code. The binned medians are also
more similar on the [S II] BPT diagram compared to previous MOSDEF and KBSS
studies. Because the two samples have similar patterns in their galaxy
properties, this agreement on the [N II] BPT diagram can be attributed to the
utilization of consistent emission-line fitting. Updated methods for
estimating stellar Balmer absorption resulted in smaller H$\beta$ flux
estimates (therefore higher O3) for the MOSDEF sample compared to previous
studies. Fitting the KBSS sample with the MOSDEF emission-line fitting code
resulted in smaller [N II]$\lambda$6585 and [S II]$\lambda\lambda$6718,6733
flux estimates (therefore smaller N2 and S2) compared to past KBSS studies
that used the KBSS MOSPEC code.

  3. 3.

The MOSDEF $z\sim 2$ targeted sample has a larger median $M_{\ast}$, smaller
SFR(SED) and sSFR(SED) medians, and a redder U$-$V compared to the KBSS $z\sim
2$ targeted sample. These sample differences are highly robust ($>4\sigma$).
Additionally, each survey occupies regions of parameter spaces that are
complementary to those spanned by the other survey. In particular, 59 galaxies
(7.5%) in the MOSDEF $z\sim 2$ targeted sample are red, displaying U$-$V and
V$-$J colors $\geq$ 1.25 while only 18 galaxies (2.1%) in the KBSS $z\sim 2$
targeted sample are in this regime. On the other hand, 106 galaxies (12.5%) in
the KBSS $z\sim 2$ targeted sample have $\log_{10}(t/\tau)\leq-2.0$ and
$\log_{10}$(sSFR(SED)/yr-1) $\geq-8.0$ compared to only 14 galaxies (1.8%) in
the MODSEF $z\sim 2$ targeted sample. These differences indicate that the
MOSDEF $z\sim 2$ targeted sample contains more massive galaxies with low
sSFR(SED) and KBSS includes more young galaxies with intense star formation.

  4. 4.

While the KBSS $z\sim 2$ spectroscopic sample reflects its $z\sim 2$ targeted
sample very well, there are some differences between the MOSDEF $z\sim 2$
spectroscopic sample and its $z\sim 2$ targeted sample. There is a population
of the MOSDEF $z\sim 2$ targeted sample that is red, old, massive (i.e., large
U$-$V, V$-$J, $M_{\ast}$, and $t/\tau$) with a relatively low SFR that is not
present in the $z\sim 2$ spectroscopic sample. A $z_{\rm{MOSFIRE}}$ was not
obtained for many of these galaxies, indicating that this is an incomplete
regime in the MOSDEF survey. Deeper spectroscopy for these galaxies will be
required to fill in this gap.

The results of this study also highlight the systematic uncertainties that
arise based on how the measurements are performed on spectroscopic data of
faint galaxies. The fact that the previously established difference in [N II]
BPT offset between the MOSDEF and KBSS surveys can in large part be attributed
to how stellar Balmer absorption corrections and weak emission-line (e.g., [N
II]$\lambda$6585 and [S II]$\lambda\lambda$6718,6733) fitting are performed,
makes this abundantly clear. The different assumptions in the MOSDEF and KBSS
fitting codes were both reasonable given the S/N of the spectra for each
survey. However, the uncertainty that arises from minor differences in data
analysis methods can be extremely difficult to estimate. Until standard
methods to analyze basic quantities are adopted and commonly used (if that is
even possible given that one’s approach to fitting emission-lines depends on
the spectral S/N), it is important to consider how minor differences in the
data analysis process can affect results from different studies.

With the MOSDEF and KBSS samples now better understood, it is possible to
combine them to form a $z\sim 2$ sample of unprecedented size. Future work
will involve utilizing this joint sample to construct galaxy scaling relations
for the largest rest-optical spectroscopic sample at $z\sim 2$. Currently,
there is no consensus on the redshift dependent offset of the mass-metallicity
relationship (MZR), or if evolution exists between local and $z\sim 1-3$
galaxies in the “fundamental metallicity relationship” (FMR). In particular,
we will investigate the evolution of the mass-metallicity and fundamental
metallicity relationships (e.g., Mannucci et al., 2010; Sanders et al., 2021;
Strom et al., 2021) using the joint MOSDEF-KBSS sample, enabling the most
statistically robust characterization of these relationships at $z\sim 2$.

In this study, we have shown that for large samples spanning multiple orders
of magnitude in $M_{\ast}$ and SFR(SED), the median offset of $z\sim 2$
galaxies perpendicular to the local sequence is approximately $0.10-0.15$ dex.
A challenging systematic uncertainty to quantify and minimize moving forward
is that stemming from analysis methods, as discussed above. However, the
existence of the offset between $z\sim 2$ and $z\sim 0$ galaxies in the [N II]
BPT diagram persists regardless of the analysis method used, and therefore
must be reckoned with as we seek to describe the evolution of rest-optical
emission properties of galaxies at high redshift.

## Acknowledgements

We acknowledge support from NSF AAG grants AST1312780, 1312547, 1312764,
1313171, 1313472, 2009313, 2009085, and 2009278; grant AR-13907 from the Space
Telescope Science Institute; and grant NNX16AF54G from the NASA ADAP program.
We also acknowledge a NASA contract supporting the “WFIRST Extragalactic
Potential Observations (EXPO) Science Investigation Team” (15-WFIRST15-0004),
administered by GSFC. Support for this work was also provided through the NASA
Hubble Fellowship grant #HST-HF2-51469.001-A awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Incorporated, under NASA contract NAS5-26555. We thank
the 3D-HST collaboration, who provided spectroscopic and photometric catalogs
used to select MOSDEF targets and to derive stellar population parameters. We
acknowledge the First Carnegie Symposium in Honor of Leonard Searle for useful
information and discussions that benefited this work. This research made use
of Astropy,333http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018). Finally, we wish to
extend special thanks to those of Hawaiian ancestry on whose sacred mountain
we are privileged to be guests.

## Data Availability

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

Facilities: Keck/MOSFIRE, Keck/LRIS, SDSS

Software: Astropy (Astropy Collaboration et al., 2013, 2018), Corner (Foreman-
Mackey, 2016), IPython (Perez & Granger, 2007), IRAF (Tody, 1986, 1993),
Matplotlib (Hunter, 2007), NumPy (van der Walt et al., 2011; Harris et al.,
2020), Pandas (pandas development team, 2020), SciPy (Oliphant, 2007; Millman
& Aivazis, 2011; Virtanen et al., 2020)

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##  Appendix A CSF Modeling

In this Appendix, we investigate the galaxy properties of the MOSDEF and KBSS
$z\sim 2$ spectroscopic and $z\sim 2$ targeted samples assuming a CSF history
instead of a delayed-$\tau$ star formation history. The SEDs are fit using the
stellar population templates from Bruzual & Charlot (2003) assuming a Chabrier
(2003) IMF. The Bruzual & Charlot (2003) templates do not account for
emission-lines or nebular continuum emission. Following previous studies
(Reddy et al., 2018; Du et al., 2018), we adopt two combinations of extinction
curves and metallicity: a 1.4$Z_{\odot}$ metallicity with the Calzetti et al.
(2000) attenuation curve (hereafter referred to as “1.4 $Z_{\odot}$+Calzetti”,
and a 0.28$Z_{\odot}$ metallicity with the Small Magellanic Cloud (SMC)
attenuation curve “0.28 $Z_{\odot}$+SMC”. Additionally, the code allows
stellar population ages between 10 Myr and 5 Gyr and E(B-V) values between 0.0
and 0.6.

We employ several constraints on the models based on past studies of $z\sim 2$
galaxies. Following Du et al. (2018), we use the 1.4 $Z_{\odot}$+Calzetti grid
for galaxies with log${}_{10}(M_{\ast}/M_{\odot})\geq 10.45$ and the 0.28
$Z_{\odot}$+SMC grid for galaxies with log${}_{10}(M_{\ast}/M_{\odot})\leq
10.45$. This choice in assumed metallicity based on stellar mass is consistent
with the mass-metallicity relationship, MZR, (e.g., Tremonti et al. 2004; Erb
et al. 2006a; Andrews & Martini 2013; Steidel et al. 2014; Sanders et al.
2015). We set a minimum stellar population age of 50 Myr, which is based on
typical dynamical timescales at $z\sim 2$ (Reddy et al., 2012).

We probe the same set of galaxy properties as with the FAST delayed-$\tau$
models: $M_{\ast}$, SFR(SED), sSFR(SED), age of the stellar population (now
parameterized by $t$ instead of $t/\tau$), $A_{\rm{V}}$, and UVJ colors. Once
again, we used IRAF/sbands to estimate the UVJ colors from the best-fit SEDs.
Using the $M_{\ast}$ from this fitting method, 5/259 galaxies from the MOSDEF
$z\sim 2$ spectroscopic sample have $\log_{10}$($M_{\ast}/M_{\odot}$) $<$ 9.0
and are removed from this sample. This is close to the number of galaxies
removed from the MOSDEF $z\sim 2$ spectroscopic sample with the FAST fitting
methodology (9 galaxies). Therefore, the MOSDEF $z\sim 2$ spectroscopic sample
consists of 254 galaxies using the CSF models. The KBSS $z\sim 2$
spectroscopic sample and both the MOSDEF and KBSS $z\sim 2$ targeted samples
have the same number of galaxies as they did in the main analysis.

Figures 10 and 11 show the histograms and UVJ diagram for the galaxy
properties of the $z\sim 2$ spectroscopic and $z\sim 2$ targeted samples,
respectively. Tables 4 and 5 contain the median values of the distributions
for the $z\sim 2$ spectroscopic and $z\sim 2$ targeted samples, respectively.
Additionally, Figure 12 gives corner plots showing the 1D histograms and 2D
distribution with contour maps of the UVJ colors for the $z\sim 2$
spectroscopic and $z\sim 2$ targeted samples.

We find that the MOSDEF $z\sim 2$ spectroscopic sample has a slightly larger
median $t$ (i.e., older) and U$-$V color (i.e., redder) and slightly smaller
median SFR(SED), sSFR(SED), and $A_{\rm{V}}$ compared to the KBSS $z\sim 2$
spectroscopic sample. The remaining galaxy properties, $M_{\ast}$ and V$-$J
color, agree within the median uncertainties. The p-values estimated from the
KS test are greater than $3\sigma$ for $t$, sSFR(SED), and $A_{\rm{V}}$. The
result that the KBSS $z\sim 2$ spectroscopic sample is younger than its MOSDEF
counterpart is similar to the results using the FAST SED fitting from Section
4.1. Also similar is that the 3$\sigma$ contours occupy similar regions for
the UVJ colors in Figure 12. The differences in the median SFR(SED),
sSFR(SED), $A_{\rm{V}}$, and U$-$V color were not seen in Section 4.1. Similar
to the delayed-$\tau$ model results, there is a population (49/369 or 13.3%)
of young KBSS galaxies with intense star formation (i.e.,
log${}_{10}(t/$yr$)\leq 8.0$ and log10(sSFR(SED)/yr-1)$\geq-8.0$). There are
only 3/254 (1.2%) MOSDEF galaxies in this regime.

When looking at the $z\sim 2$ targeted samples, we find that MOSDEF has a
larger median $M_{\ast}$ and $t$, a smaller median sSFR(SED), and redder UVJ
colors compared to KBSS. The medians for SFR(SED) and $A_{\rm{V}}$ agree
within the uncertainties. These results are all highly significant, with
p-values greater than $4\sigma$ for all galaxy properties. The results from
the CSF models find differences in the median $t$ and U$-$V color of MOSDEF
and KBSS $z\sim 2$ targeted samples, which was not seen in the FAST fitting.
On the other hand, the results from the CSF models find no differences in the
median SFR(SED), as found in the results from FAST. Similar to the
delayed-$\tau$ models, 124/850 (14.6%) of KBSS galaxies are young
(log10($t$/yr)$\leq 8.0$) with intense star formation
(log10(sSFR(SED)/yr-1)$\geq-8.0$) compared to only 12/786 galaxies (1.5%) in
the MOSDEF sample. Meanwhile, 134/786 (17.0%) of MOSDEF galaxies are extremely
reddened (U$-$V and V$-$J $\geq$ 1.25) compared to 22/850 galaxies (2.6%) in
the KBSS sample. Additionally, the 3$\sigma$ contour map for MOSDEF extends to
redder UVJ colors compared to KBSS. We note here that the distribution of
inferred U-V and V-J colors for the MOSDEF $z\sim 2$ targeted sample changes
when assuming CSF models, relative to the results for the delayed-$\tau$
modeling presented in Section 5.2. Specifically, the CSF models do not
accurately capture the SED shapes of the most massive, low sSFR(SED) galaxies
in the MOSDEF $z\sim 2$ targeted sample, and therefore place them
(erroneously) outside of the quiescent region of the UVJ diagram.

Figure 13 contains corner plots showing both the 1D histograms and 2D contour
mapping for SFR(SED) vs. $M_{\ast}$ and sSFR(SED) vs. $M_{\ast}$ for the
MOSDEF and KBSS spectroscopic (top panels) and targeted (lower panels)
samples. We find mostly similar results to the delayed-$\tau$ FAST results in
the main body of the text (see Figures 1 and 13). KBSS extends to lower
$M_{\ast}$ and higher sSFR(SED) than MOSDEF in both the $z\sim 2$
spectroscopic and $z\sim 2$ targeted samples. The distributions for the KBSS
$z\sim 2$ spectroscopic and $z\sim 2$ targeted samples are very similar in
both SFR(SED) vs. $M_{\ast}$ and sSFR(SED) vs $M_{\ast}$ space. Meanwhile, we
find the same differences between the MOSDEF $z\sim 2$ spectroscopic and
$z\sim 2$ targeted samples. The 3$\sigma$ contour for the MOSDEF spectroscopic
sample reaches a similar upper mass, lower sSFR(SED) limit as the KBSS sample,
while the MOSDEF $z\sim 2$ targeted sample extends to higher $M_{\ast}$. This
once again shows that MOSDEF is incomplete in this high mass regime. As we
discuss previously in the paper, KBSS does not target the high mass galaxies
for which MOSDEF is not successful in obtaining high quality S/N spectra.

While the results for the delayed-$\tau$ and CSF models do not perfectly align
(i.e., the relative median values between the MOSDEF and KBSS $z\sim 2$
spectroscopic and $z\sim 2$ targeted samples sometimes vary based on which
fitting method is used, and the rest-frame UVJ colors of the most massive
galaxies in the MOSDEF $z\sim 2$ targeted sample are not captured by the CSF
models), the key takeaways of the study concerning the galaxy properties are
the same for both models.

  1. 1.

The MOSDEF and KBSS $z\sim 2$ spectroscopic samples are more similar to each
other than the $z\sim 2$ targeted galaxy samples.

  2. 2.

The KBSS spectroscopic and $z\sim 2$ targeted samples have a subset of young
galaxies with intense star formation that are mostly absent from the MOSDEF
samples.

  3. 3.

The MOSDEF $z\sim 2$ targeted sample contains extremely red (i.e., high UVJ
colors) galaxies that are mostly absent from the KBSS $z\sim 2$ targeted
sample.

  4. 4.

The subset of extremely red MOSDEF galaxies in the $z\sim 2$ targeted sample
are not represented in the spectroscopic sample, highlighting that MOSDEF is
incomplete in this regime.

As stated in Section 3.1, these key takeaways remain unchanged if we use
$\tau$-rising models instead of CSF or delayed-$\tau$ models.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10: Distribution of physical properties for the MOSDEF (red) and KBSS
(blue) $z\sim 2$ spectroscopic samples. The sample medians, with uncertainties
are shown in yellow (green) for the MOSDEF (KBSS) samples, and are given in
Table 4. The following galaxy properties are shown: (a) $M_{\ast}$, (b)
SFR(SED), (c) sSFR(SED), (d) log10($t$) of the stellar population using a
constant star formation model, (e) $A_{\rm{V}}$, and (f) the UVJ diagram. We
find that the MOSDEF spectroscopic sample has a higher age, lower SFR(SED),
lower sSFR(SED), and $A_{\rm{V}}$, and redder U$-$V color compared to the KBSS
spectroscopic sample. Median Values for Physical Properties of the MOSDEF and
KBSS $z\sim 2$ Spectroscopic Samples (CSF Models)
---
Physical Property | MOSDEF Median | KBSS Median | p-value | Statistical significance
(1) | (2) | (3) | (4) | (5)
log${}_{10}(M_{\ast}/M_{\odot}$) | 9.91 $\pm$ 0.04 | 9.83 $\pm$ 0.05 | 0.023 | 2.27$\sigma$
log10($t$/yr) | 8.81 $\pm$ 0.04 | 8.71 $\pm$ 0.04 | 0.00066 | 3.41$\sigma$
log10(SFR(SED)/M⊙/yr-1) | 1.04 $\pm$ 0.05 | 1.13 $\pm$ 0.03 | 0.026 | 2.23$\sigma$
log10(sSFR(SED)/yr-1) |  $-$8.81 $\pm$ 0.03 |  $-$8.71 $\pm$ 0.04 | 0.00056 | 3.45$\sigma$
$A_{\rm{V}}$ | 0.24 $\pm$ 0.01 | 0.29 $\pm$ 0.01 | 0.00019 | 3.73$\sigma$
U$-$V | 0.68 $\pm$ 0.02 | 0.63 $\pm$ 0.02 | 0.057 | 1.90$\sigma$
V$-$J | 0.37 $\pm$ 0.02 | 0.37 $\pm$ 0.02 | 0.46 | 0.74$\sigma$
Table 4:  Col. (1): Physical property of the galaxies in the sample shown in
Figure 10. Col. (2): Median value with uncertainty of the MOSDEF $z\sim 2$
spectroscopic sample. Col. (3): Median value with uncertainty of the KBSS
$z\sim 2$ spectroscopic sample. Col. (4): Two-tailed p-value, based on the K-S
test, estimating the probability that the null hypothesis can be rejected.
Col. (5): Statistical significance (i.e. the $\sigma$ value) that the p-value
corresponds to.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: Distribution of physical properties for the MOSDEF (red) and KBSS
(blue) $z\sim 2$ targeted samples. The sample medians, with uncertainties are
shown in yellow (green) for the MOSDEF (KBSS) samples, and are given in Table
5. The following galaxy properties are shown: (a) $M_{\ast}$, (b) SFR(SED),
(c) sSFR(SED), (d) log10($t$) of the stellar population using a constant star
formation model, (e) $A_{\rm{V}}$, and (f) the UVJ diagram. We find that the
MOSDEF $z\sim 2$ targeted sample has a higher $M_{\ast}$ and age, lower
SFR(SED) and lower sSFR(SED), and redder UVJ colors compared to the KBSS
$z\sim 2$ targeted sample. Median Values for Physical Properties of the MOSDEF
and KBSS $z\sim 2$ Targeted Samples (CSF Models)
---
Physical Property | MOSDEF Median | KBSS Median | p-value | Statistical significance
(1) | (2) | (3) | (4) | (5)
log${}_{10}(M_{\ast}/M_{\odot}$) | 10.04 $\pm$ 0.04 | 9.91 $\pm$ 0.03 | 2.5e$-$7 | 5.16$\sigma$
log10($t$/yr) | 9.01 $\pm$ 0.05 | 8.76 $\pm$ 0.05 | 7.8e$-$14 | 7.47$\sigma$
log10(SFR(SED)/M⊙/yr-1) | 1.07 $\pm$ 0.04 | 1.13 $\pm$ 0.02 | 7.2e$-$6 | 4.49$\sigma$
log10(sSFR(SED)/yr-1) |  $-$9.01 $\pm$ 0.03 |  $-$8.78 $\pm$ 0.04 | 1.6e$-$14 | 7.68$\sigma$
$A_{\rm{V}}$ | 0.27 $\pm$ 0.01 | 0.29 $\pm$ 0.01 | 3.5e$-$13 | 7.27$\sigma$
U$-$V | 0.78 $\pm$ 0.01 | 0.65 $\pm$ 0.02 | 1.2e$-$10 | 6.44$\sigma$
V$-$J | 0.48 $\pm$ 0.02 | 0.39 $\pm$ 0.01 | 7.3e$-$9 | 5.78$\sigma$
Table 5:  Col. (1): Physical property of the galaxies in the sample shown in
Figure 11. Col. (2): Median value with uncertainty of the MOSDEF $z\sim 2$
targeted sample. Col. (3): Median value with uncertainty of the KBSS $z\sim 2$
targeted sample. Col. (4): Two-tailed p-value, based on the K-S test,
estimating the probability that the null hypothesis can be rejected. Col. (5):
Statistical significance (i.e. the $\sigma$ value) that the p-value
corresponds to.

Figure 12: Corner plots comparing the distributions of U$-$V and V$-$J colors
for the $z\sim 2$ spectroscopic (left) and targeted (right) samples. MOSDEF
and KBSS are shown in red and blue, respectively. The dashed lines on the 1D
histograms mark the median values for the distributions of U$-$V and V$-$J
colors given in Tables 4 and 5, and the contours in the 2D panel trace the
3$\sigma$ regions in UVJ color-color space. The individual data points are
also shown in the 2D distribution. Similar to Figure 3 and 8, it is shown here
that while the $z\sim 2$ spectroscopic samples span a similar range in UVJ
space, the MOSDEF $z\sim 2$ targeted sample extends to redder UVJ colors than
the KBSS $z\sim 2$ targeted sample.

Figure 13: Top: $z\sim 2$ targeted samples. Bottom: $z\sim 2$ spectroscopic
samples. The left (right) column contains corner plots comparing SFR(SED) vs.
$M_{\ast}$ (sSFR(SED) vs. $M_{\ast}$). MOSDEF and KBSS are shown in red and
blue, respectively. The dashed lines on the 1D histograms mark the median
values for the distributions, and the contours on the diagonal trace the
3$\sigma$ regions in 2D space. The individual data points are also shown in
the 2D distribution. These plots utilize CSF fitting, and are analogous to
Figures 1 ($z\sim 2$ targeted sample) and 4 ($z\sim 2$ spectroscopic sample).

##  Appendix B Emission-line Fitting Examples

Here we provide examples of the MOSDEF and KBSS emission-line fitting. In
these examples, we focus on the fit of [N II]$\lambda$6585 because weaker
emission-lines exhibited the most flux variance when one compares the MOSDEF
and KBSS MOSPEC codes. The fits to H$\alpha$ and [N II]$\lambda$6550 (not
relevant to this study) are included in the plots due to their proximity to [N
II]$\lambda$6585\. Figure 14 shows the fits to four example spectra, two from
the MOSDEF $z\sim 2$ spectroscopic sample and two from the KBSS $z\sim 2$
spectroscopic sample. Each panel includes the spectra as well as the fits from
both the MOSDEF and KBSS MOSPEC codes.

As discussed in Section 5.1, the MOSDEF Gaussian fit to the [N
II]$\lambda$6585 line estimates a lower flux on average compared to the
integrated bandpass flux. Meanwhile, the KBSS Gaussian fit estimates a higher
flux on average compared to the integrated bandpass flux. The assumptions used
by the two methods are both reasonable, given the differences in typical S/N
of the MOSDEF and KBSS surveys. The goal of this paper has not been to
determine which fitting method is more correct, but instead to show that
different assumptions in the emission-line fitting can lead to small
systematic variations in measured flux.

(a)

(b)

(c)

(d)

Figure 14: Examples showing how the MOSDEF (red) and KBSS MOSPEC (blue) codes
fit the H$\alpha$ and [N II]$\lambda$6585 emission-lines. The codes also
simultaneously fit the [N II]$\lambda$6550; however, we did not use it for
this study. We pick the following galaxies to highlight the following details
about the code. (a) Q1603-BX379, for which we display the typical [N
II]$\lambda$6585 offset between the two codes (i.e., the KBSS MOSPEC code is
$\sim$13% larger). The MOSDEF code uses the Gaussian fit for the preferred
flux. (b) Q1549-BX223, for which we display similar [N II]$\lambda$6585
fluxes, where the MOSDEF code uses the Gaussian fit for the preferred flux.
(c) AEGIS 6377, for which we display the typical [N II]$\lambda$6585 offset
between the two codes where the MOSDEF code uses the Gaussian fit for the
preferred flux. (d) AEGIS 33172, for which we display similar [N
II]$\lambda$6585 fluxes, where the MOSDEF code uses the Gaussian flux for the
preferred flux. Panel (c) shows that the Gaussian peaks for [N
II]$\lambda$6585 do not always perfectly agree in the two codes.