# The Universe SPHEREx Will See: Empirically Based Galaxy Simulations and
Redshift Predictions

Richard M. Feder California Institute of Technology, 1200 E California Blvd,
91125, USA  Daniel C. Masters IPAC, California Institute of Technology, 1200 E
California Blvd, 91125, USA  Bomee Lee Korea Astronomy and Space Science
Institue (KASI), Daejeon, Korea  James J. Bock California Institute of
Technology, 1200 E California Blvd, 91125, USA  Yi-Kuan Chiang Institute of
Astronomy & Astrophysics, Academia Sinica, Taiwan  Ami Choi NASA Goddard Space
Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA  Olivier Doré Jet
Propulsion Laboratory, California Institute of Technology, Pasadena,
California 91109, USA  California Institute of Technology, 1200 E California
Blvd, 91125, USA  Shoubaneh Hemmati IPAC, California Institute of Technology,
1200 E California Blvd, 91125, USA  Olivier Ilbert Aix Marseille Univ, CNRS,
CNES, LAM, Marseille, France

###### Abstract

We simulate galaxy properties and redshift estimation for SPHEREx, the next
NASA Medium Class Explorer. To make robust models of the galaxy population and
test spectro-photometric redshift performance for SPHEREx, we develop a set of
synthetic spectral energy distributions based on detailed fits to COSMOS2020
photometry spanning $0.1-8$ $\mu$m. Given that SPHEREx obtains low-resolution
spectra, emission lines will be important for some fraction of galaxies. Here
we expand on previous work, using better photometry and photometric redshifts
from COSMOS2020, and tight empirical relations to predict robust emission line
strengths and ratios. A second galaxy catalog derived from the GAMA survey is
generated to ensure the bright ($m_{AB}<18$ in the $i$-band) sample is
representative over larger areas. Using template fitting to estimate
photometric continuum redshifts, we forecast recovery of 19 million galaxies
over 30000 deg2 with redshifts better than $\sigma_{z}<0.003(1+z)$, 445
million with $\sigma_{z}<0.1(1+z)$ and 810 million with $\sigma_{z}<0.2(1+z)$.
We also find through idealized tests that emission line information from
spectrally dithered flux measurements can yield redshifts with accuracy beyond
that implied by the naive SPHEREx channel resolution, motivating the
development of a hybrid continuum-line redshift estimation approach.

Cosmology (343), Spectrophotometry (1556), Redshift surveys (1378), Emission
line galaxies (459), Large-scale structure of the universe (902)

††journal: AJ

##  1 Introduction

The Spectro-Photometer for the History of the Universe, Epoch of Reionization,
and Ices Explorer (SPHEREx) is the next NASA Medium Class Explorer mission
which is planned for launch in early 2025. Using a wide-field 20 cm diameter
telescope with an instantaneous field of view of $3.5^{\circ}\times
11.3^{\circ}$ for each of two $1\times 3$ detector mosaics, SPHEREx will
conduct the first all-sky spectrophotometric survey in the near infrared (NIR)
at wavelengths between 0.75 $\mu$m and 5 $\mu$m through four consecutive
surveys over the nominal two year mission.

SPHEREx and other cosmology missions generally need to build and test
methodologies using the best possible simulated data. SPHEREx aims to
constrain the large-scale distribution of galaxies in order to put constraints
on primordial non-Gaussianity (Doré et al., 2014). This measurement will need
excellent redshifts for hundreds of millions of galaxies, with robust control
and understanding of statistical and systematic errors. In the case of
SPHEREx, this is particularly challenging as the properties galaxies will
display at infrared wavelengths ($0.75-5$ $\mu$m) with low-resolution ($R\sim
35-130$) spectroscopy are currently poorly constrained.

Stickley et al. (2016) illustrated the use of synthetic SPHEREx
spectrophotometry to measure redshifts of a large sample of bright galaxies to
high accuracy ($\gtrsim 10^{7}$ galaxies over the full sky with redshift
accuracy $\sigma_{z}<0.003(1+z)$, and many more with accuracy at the $\sim
1-10\%$ level). These simulations were performed on model spectra inferred
from the Cosmological Evolution Survey field (COSMOS; Scoville et al., 2007),
for which the complexity of the galaxy population is well constrained through
deep, 30-band photometry spanning $0.1-8$ $\mu$m.

Given the spectral resolution and infrared coverage of SPHEREx, emission line
galaxies (ELGs) are an important population to consider. Nebular emission
lines, typically rest-frame optical and UV (H$\alpha$, [OII], [OIII],
Ly$\alpha$), are the targets of numerous existing and upcoming surveys such as
the extended Baryon Acoustic Oscillations Spectroscopic Survey (eBOSS;
Raichoor et al., 2021), the _Euclid_ Wide and Deep surveys (using slitless
spectroscopy, Laureijs et al., 2011), the Dark Energy Spectroscopic Instrument
ELG survey (DESI; Raichoor et al., 2022), the _Roman_ High Latitude
Spectroscopic Survey (HLSS; Wang et al., 2022), the Prime Focus Spectrograph
Galaxy Evolution Survey (PFS; Greene et al., 2022), the Physics of the
Accelerating Universe Survey (PAUS; Eriksen et al., 2019; Alarcon et al.,
2021), and the Javalambre-Physics of the Accelerating Universe Astrophysical
Survey (J-PAS; Benítez et al., 2009), among others. These surveys promise to
deliver emission line measurements for tens of millions of galaxies,
increasing the size of existing samples by an order of magnitude.

Properly identified emission lines serve as anchors for precise redshift
measurements, making ELGs the target of modern large scale structure studies
probing cosmic expansion and acceleration (Raichoor et al., 2022; Zhai et al.,
2021). Emission line strengths and ratios are also commonly used as
observational proxies for intrinsic galaxy properties such as the star
formation rate (SFR). Upcoming ELG surveys will chart galaxy evolution and
formation in unprecedented detail, enabling studies of galaxy properties both
across a broad range of cosmic history and as a function of local environment
(Pharo et al., 2020; Scoville et al., 2013).

SPHEREx is unique in its potential to deliver precise redshifts using low-
resolution spectroscopy. Both active and quiescent galaxies can yield high
quality redshifts through accurate modeling of their continuua. SPHEREx will
help in obtaining precise redshifts for a large number of sources by extending
source flux measurements into the NIR. In the case where emission lines are
used to improve redshifts for continuum-selected galaxies, there may be a
large number of sources where proper modeling of low-significance emission
lines is important. This places requirements on the accuracy of continuum
modeling, however correlations between lines and continuua measured by SPHEREx
can potentially break degeneracies of single-line identifications that plague
existing spectroscopic surveys (Kirby et al., 2007; Garilli et al., 2014;
Okada et al., 2016). Because SPHEREx is an imaging survey, no pre-selection is
required to isolate ELG targets. This distinguishes SPHEREx from fiber-based
spectroscopic measurements that are optimized with pre-selected targets. In
this sense SPHEREx’s survey strategy reduces the impact of target selection on
ELG samples and opens up the prospect of blind searches for emission lines
over large portions of the sky (e.g., Mary et al., 2020).

The goal of this work is to create and use simulated galaxies to demonstrate
the accurate measurement of redshifts using SPHEREx data. Our current best
constraints on the distribution of galaxy spectra over the SPHEREx bandpass
come from COSMOS2020 photometry. However, the resolution is poorly matched to
SPHEREx because COSMOS photometry cannot precisely constrain the emission line
properties of sources over a range of redshifts. To tackle this issue, we use
empirical (Brown et al., 2014, hereafter B14) and model-based COSMOS templates
to fit the COSMOS2020 photometry, which yields a realistic distribution of SED
continua. To model emission lines, we turn to tight empirical relations for
how line strengths and ratios scale with redshift, stellar mass, and spectral
type. After developing the line prediction method, we test it through
comparisons at the population level (line luminosity functions, line ratio
distributions, line equivalent widths) and with individual source comparisons
using spectroscopic catalogs covering the COSMOS field.

The catalog we develop is therefore a faithful representation of the galaxy
population as constrained by COSMOS2020 (Weaver et al., 2022a) and numerous
published studies on emission lines. Our catalog is similar in qualities to
the synthetic Emission Line COSMOS catalog (EL-COSMOS, Saito et al., 2020),
which is derived from COSMOS2015 photometry (Laigle et al., 2016). Emission
lines in that catalog are modeled though a prescription for H$\beta$ based on
estimated star formation rates and assumed metallicities $Z>0.2Z_{\odot}$ to
obtain line ratios (which are effectively fixed aside from dust attenuation).
The predictions in this work are similar to those in Saito et al. (2020),
however our model differs by 1) using a combination of empirically measured
and model-based templates, rather than a large grid of continuum models, and
2) combining empirical scaling relations for lines with locally calibrated
($z\sim 0$) emission line equivalent width (EW) measurements.

Because SPHEREx is an all-sky mission, we complement the COSMOS catalog with
wider survey data from the Galaxy and Mass Assembly survey (GAMA, Driver et
al., 2022) and associated multi-wavelength photometry to ensure that the
distribution of bright galaxies in our simulated sample is representative of
the full sky (our COSMOS catalog only covers $1.27$ deg${}^{2}=3.9\times
10^{-4}$ sr, while the GAMA footprint is 217 deg2).

Our procedure for painting emission lines onto continuum estimates from
template fits, implemented in the tool Conditional LIne Painting on Synthetic
Spectra (CLIPonSS), is described in §2, and the synthetic line catalog is
validated against several measurements in the literature in §3. The details of
converting these SEDs into synthetic SPHEREx observations are presented in §4,
where we also consider the coverage of SPHEREx sources by external catalogs.
In §5 we forecast recovery of galaxy continuum redshifts by running the
photometric redshift template fitting code from Stickley et al. (2016) on
synthetic photometry, also showing initial demonstrations of redshift
estimation that utilize low-resolution, spectrally dithered line flux
measurements.

All apparent magnitudes are specified in the AB magnitude system (Oke & Gunn,
1983).

##  2 Synthetic Spectral Library

Our approach for predicting galaxy emission-line properties is to combine
observations from the COSMOS survey with empirical models. The COSMOS
photometry has sufficient depth and wavelength coverage to constrain galaxy
spectral energy distributions (SEDs) over the SPHEREx bandpass. Our model then
predicts emission lines expected for galaxies based on their stellar masses,
redshifts, and best-fit spectral templates. Predicting emission line strengths
is feasible because of the empirical observed correlations between the lines
and other galaxy properties arising from the evolution of the mass-metallicity
relation (Tremonti et al., 2004), the galaxy main sequence (Daddi et al.,
2007; Whitaker et al., 2012; Speagle et al., 2014) and the global star
formation evolution (Connolly et al., 1997; Madau & Dickinson, 2014).

The procedure for assigning emission lines to galaxy continuua is illustrated
in Fig. 1. Unlike the COSMOS templates, which lack emission lines, many Brown
et al. (2014) templates have lines which we rescale when painting new lines
onto the synthetic SEDs. In this way, the line strengths in our empirical
model are calibrated to observed galaxies at low redshift ($z<0.1$).

Figure 1: Track of synthetic galaxy SEDs as a function of redshift
demonstrating our model procedure. For a given source, we start with the best-
fit measured spectrum (blue) fit as a template to broad band photometry. We
then subtract the existing lines (black), and insert new lines according to
our empirical prescription. This is shown for three redshifts: $z=0.5$
(orange); 1.2 (green); and 2.0 (red), for a fixed stellar mass
$\log(M_{\star}/M_{\odot})$= 9.5. Spectra are shifted vertically for visual
purposes. The right hand panels show the three lines/line complexes at higher
resolution for the three redshifts. [OII] is modeled with a single Gaussian
rather than as a doublet.

###  2.1 Multi-wavelength photometry

####  2.1.1 COSMOS2020

The galaxy simulations rely on multi-wavelength photometry from COSMOS2020,
which comprises two catalogs with 30-band photometry spanning $0.1-8$ $\mu$m
(Weaver et al., 2022a). In particular we use the COSMOS2020 catalog derived
from The Farmer, a profile-fitting tool for multi-wavelength photometry, along
with associated photometric redshifts.

We select on classified galaxies with $i<25$ and flux measurements in at least
one NIR band (UltraVISTA $J$ and $H$ bands). We further select on sources with
consistent redshift measurements from both sets of COSMOS2020 photometry
($\hat{\sigma_{z}}<0.2(1+z)$,
$|\hat{z}_{\textrm{Classic}}-\hat{z}_{\textrm{Farmer}}|/(1+\hat{z}_{\textrm{Farmer}})<0.15$),
selecting galaxies as classified by LePhare (see §5.1 in Weaver et al.,
2022a). The majority of sources with this selection have accurate redshifts
($\sigma_{z}<0.02(1+z)$) and sufficiently precise photometry to estimate
stellar masses. In total our selection yields a sample of 166,014 galaxies
covering an effective area of 1.27 deg2.

####  2.1.2 Bright, low-redshift galaxies from the GAMA survey

There are limitations in building a representative synthetic catalog from
COSMOS2020 alone. Given the size of the COSMOS field, the diversity of spectra
will be underestimated due to cosmic variance. Likewise, the limited volume
probed by COSMOS suggests a limited low redshift sample, especially for
massive galaxies. The COSMOS2020 galaxy catalog is limited to sources with
$i>18$, with any sources brighter than this classified as stars.

For these reasons we supplement the COSMOS2020 catalog using a combination of
spectroscopic measurements from the Galaxy and Mass Assembly (GAMA) survey
(Driver et al., 2022) and corresponding twelve-band photometry ranging from
the far UV to the infrared. We select sources with $i<18$ and a designated
science class (SC) of 8. This is the selection for the primary science catalog
used by GAMA – the catalog has spectroscopic completeness of 98 per cent down
to $r<19.5$ (Bellstedt et al., 2020). In total we obtain 44,135 sources across
four fields with effective areas of 55, 57, 57 and 48 square degrees. We fit
the same template library described in §2.3 to the available broad-band
photometry with fixed redshifts.

###  2.2 Galaxy Template SEDs

In total we utilize 160 templates to fit observed galaxy photometry (§2.3) and
then generate synthetic SEDs and SPHEREx spectrophotometry (§4). The two sets
of templates described in this section complement each other in terms of
reproducing observed colors and galaxy types.

####  2.2.1 B14 templates

The collection of 129 measured galaxy SEDs from Brown et al. (2014) comprise a
broad range of galaxy archetypes from different stages of evolution and
environment. The SEDs are constructed from a combination of optical (Moustakas
& Kennicutt, 2006; Moustakas et al., 2010) and infrared spectroscopy from the
_Spitzer_ Space Telescope Infrared Spectrograph (IRS) for $5.3-38$ $\mu$m
(Werner et al., 2004; Houck et al., 2004) and _Akari_ (Murakami et al., 2007)
Infrared Camera (IRC) spectroscopy for $2.5-5$ $\mu$m (when available),
spanning beyond the wavelength range of SPHEREx observations. Regions of the
SEDs without coverage are interpolated using model spectra fit using the
MAGPHYS model (da Cunha et al., 2008), which utilizes a stellar population
synthesis (SPS) model (derived from the same set of BC03 templates described
in the next sub-section) and a self-consistent prescription for dust emission,
absorption and polycyclic aromatic hydrocarbon (PAH) emission in the infrared.
While the measured spectra have little to no coverage in the near infrared,
the sections of interpolated spectra are calibrated against existing broadband
photometry covering the range from 2MASS (Skrutskie et al., 2006), _Spitzer_
and the Wide-field Infrared Space Explorer (WISE; Wright et al., 2010). The
B14 galaxy SEDs constitute a diverse template basis for fitting galaxy
photometry and reproducing observed galaxy colors. Many of the galaxies in the
B14 sample have well measured optical emission lines, which we use to
calibrate our line model locally (i.e., $z\sim 0$) before extrapolating to
higher redshifts.

####  2.2.2 COSMOS templates

Thirty one of the templates are SEDs used in Ilbert et al. (2009), which
include templates from Polletta et al. (2007) and twelve model-based templates
made from Bruzual & Charlot (2003) (BC03 templates, hereafter), which are
generated from SPS models along the starburst track and for passive elliptical
galaxies. These templates were initially used in Weaver et al. (2022a) to fit
the photometric redshifts of COSMOS2020 galaxies. They complement the B14
templates, as the B14 templates come from low-redshift galaxies that may not
be representative of higher redshift populations.

###  2.3 Template fits to multi-band photometry

The templates described in the previous section are fit to multi-wavelength
photometry from the COSMOS and GAMA extragalactic surveys. We use the SED
fitter Fitcat, used previously in Stickley et al. (2016), to derive continuum
models and galaxy properties. These properties include stellar mass
$\log(M_{\star}/M_{\odot})$, dust extinction E$(B-V)$, dust law and index of
the best-fit galaxy template. Note that the photometric redshifts from
COSMOS2020 are taken as fixed within Fitcat.

Figure 2 shows SED fits to COSMOS2020 photometry for six example galaxies. As
can be seen, the SED templates are able to capture the properties of both
star-forming and quiescent galaxies in our sample. The median reduced chi-
squared of the fits is $\chi^{2}_{red}=2.2$, with less than 1% of objects
having $\chi^{2}_{red}>10$. The higher than expected median $\chi^{2}$ is
driven in part by the inability of the model templates to capture the
diversity of galaxy properties, in particular at long wavelengths where
emission from PAHs is difficult to model. In addition, the flux uncertainties
for bright sources may tend to be underestimated (Weaver et al., 2022a) and
certain correlated uncertainties across wavelengths are neglected. A large
fraction of catalog sources are best fit by a small subset ($<10$) of the 160
templates, where the “best-fit template” refers to the template corresponding
to the smallest model $\chi^{2}$ compared with the data. The fraction of
sources best fit by the COSMOS templates increases from thirty per cent at
$i=18$ to forty percent at $i=25$. Likewise, for the GAMA sample, the fraction
of sources best fit by COSMOS templates ranges from 35% to 55% across
$13<i<18$. These trends are consistent with our expectations, as the B14
templates are calibrated to (local) bright objects.

Figure 2: A selection of six spectral energy distribution (SED) fits to multi-
band COSMOS2020 photometry for galaxies of various redshifts and stellar
masses, indexed by Tractor IDs from the COSMOS2020 Farmer catalog. In some
cases the COSMOS2020 photometry are sensitive to strong emission lines, for
example in the top and bottom left panels.

###  2.4 Emission line model

The method used to assign emission lines to template continua that are fit to
COSMOS2020 and GAMA sources consists of:

  * •

Inferred H$\alpha$ and [OII] equivalent widths for each galaxy based on its
best-fit spectral template and/or UV continuum;

  * •

Observed scaling relations of [NII]/H$\alpha$ and [OIII]/H$\beta$ with
redshift and stellar mass; and

  * •

The (redshift-dependent) nebular extinction correction from Saito et al.
(2020).

With these measured empirical relations we derive realistic emission lines
expected for the COSMOS galaxies, which are then painted onto the best-fit
template continua.

The rest-frame optical lines in the SPHEREx wavelength range include: Balmer
series lines H$\alpha$ and H$\beta$; singly- and doubly-ionized oxygen [OII]
[$\lambda$3728] and [OIII] [$\lambda$5007]; nitrogen [NII] [$\lambda$6584];
and the sulfur doublet SII [$\lambda\lambda$6716,6731]. Given SPHEREx NIR
spectral coverage, we also include the less probed Paschen-$\alpha$ line at
$\lambda_{rest}=1.87$ $\mu$m. Mid-infrared emission from PAHs will also be
present, in particular the rest frame 3.3$\mu$m bump will be detectable for
redshifts $z<0.5$. As we lack a realistic model of PAH emission in galaxies,
we choose to leave any observed PAH emission from the B14 templates in our
synthetic SEDs.

We follow a procedure similar to Speagle & Eisenstein (2017) to extract
emission line properties from the set of B14 templates. In regions of the
observed SEDs where relevant SPHEREx emission lines are present, we fit a
simple line + continuum model directly to the spectra. For each template, a
small region centered on each line (widths varying from $\pm 50-100$ Å for
rest frame optical lines) is extracted and fit using a variable number of
Gaussians depending on the number of spectral features, while the continuum is
modeled locally with an offset and slope. From these models we estimate the
flux and local continuum level associated with each line, from which we
compute the equivalent width. We skip this step for any galaxies classified as
passive using the Baldwin, Phillips & Terlevich diagram (BPT diagram; Baldwin
et al., 1981) in B14.

Line |  $\lambda_{\textrm{air}}$ [Å] | SPHEREx coverage
---|---|---
H$\alpha$ | 6562.8 | $0.15<z<6.6$
$\left[\textrm{NII}\right]$ | 6583.5, 6548.0 | same as H$\alpha$
H$\beta$ | 4861.4 | $z>0.55$
$\left[\textrm{OIII}\right]$ | 5006.8, 4958.9 | $z>0.5$
[OII] | 3728.8 | $z>1.0$
$\left[\textrm{SII}\right]$ | 6716.4, 6730.8 | $z>0.1$
Paschen-$\alpha$ | 18750.9 | $0<z<1.6$
Table 1: Emission lines modeled in this work with rest-frame wavelengths in
air. The third column indicates the redshift coverage for each observed line
given the full SPHEREx bandpass.

####  2.4.1 Predicting emission line strengths

Our procedure for generating sets of emission lines relies on tight empirical
relations of lines and line ratios as a function of redshift and stellar mass.
All of the empirical relations used in this section are obtained from
measurements that have been corrected for dust extinction. Once the intrinsic
line fluxes are computed, we then apply stellar and nebular extinction
corrections to each source.

To predict H$\alpha$ line fluxes, we use direct measurements of H$\alpha$ from
the B14 templates (described in §2.1) to determine each “local” (low-redshift)
equivalent width. For the set of active COSMOS templates, we compute the
H$\alpha$ equivalent width using scaling relations between the mean UV
luminosity $L_{\textrm{UV}}$ and the star formation rate (SFR)

$\textrm{SFR}(M_{\odot}\textrm{yr}^{-1})=1.4\times
10^{-28}L_{\textrm{UV}}\quad(\textrm{erg s}^{-1}\textrm{Hz}^{-1}),$ (1)

which can then be used to predict the H$\alpha$ line luminosity
$L_{\textrm{H}\alpha}$ (Kennicutt, 1998);

$\textrm{SFR}(M_{\odot}\textrm{yr}^{-1})=7.9\times
10^{-42}L_{\textrm{H}\alpha}\quad(\textrm{erg s}^{-1}).$ (2)

For B14 templates with detectable H$\alpha$, the rest-frame equivalent width
of H$\alpha$, denoted EW(H$\alpha$), is computed and scaled to higher
redshifts using the relation of Fig. 6 from Brinchmann et al. (2008) between
$\Delta\textrm{EW(H}\alpha)$ and
$\Delta\log\left([\textrm{OIII}]/\textrm{H}\beta\right)$, i.e., the deviations
of equivalent width and line flux ratio from $z=0$. We model the redshift
dependence of $\log\left([\textrm{OIII}]/\textrm{H}\beta\right)$ using Eq. 1
from Kewley et al. (2013)

$\log\left(\frac{[\textrm{OIII}]}{\textrm{H}\beta}\right)=\frac{0.61}{\log\left(\textrm{[NII]}/\textrm{H}\alpha\right)-\gamma}+\eta,$
(3)

where $\gamma=0.02+0.1833z$ and $\eta=1.2+0.03z$. We calculate the flux ratio
[NII]/H$\alpha$ as a function of $\log(M_{\star}/M_{\odot})$ and redshift
using an interpolation of Table 1 from Faisst et al. (2018), which
approximates the stellar mass vs. gas phase metallicity relation as a function
of redshift. The line ratio NII$[\lambda 6585]$/NII$[\lambda 6549]=3$ is
applied to each [NII] doublet.

Once the H$\alpha$-[NII] complex is calculated, we apply the intrinsic ratios
H$\alpha$/H$\beta$=2.86 and P$\alpha/H\alpha=0.123$ to obtain H$\beta$ and
P$\alpha$ line fluxes. We then use the extrapolated log[OIII]/H$\beta$ from
(3) and OIII$[\lambda 5007]$/OIII$[\lambda 4959]=3$ to obtain the two OIII
line fluxes. The sulfur doublet [SII] is calculated using a best fit parabola
to the local relation from SDSS DR12 of the O3S2 BPT diagram (see Fig. 6 of
Masters et al., 2016)

$y=-0.44-0.3x-0.66x^{2},$ (4)

where $y=$log([SII]/H$\alpha$) and $x=$log([OIII]/H$\beta$).

We follow a similar procedure to predict [OII] as for H$\alpha$. When
available, we use [OII] equivalent widths measured directly from the B14
templates. For COSMOS templates we use the mean SFR-L[OII] calibration from
Kewley et al. (2004), which assumes a fixed (de-reddened) [OII]/H$\alpha$
ratio, based on measurements from the Near Field Galaxy Survey (NFGS)

$\textrm{SFR}[\textrm{M}_{\odot}\textrm{yr}^{-1}]=(6.58\pm 1.65)\times
10^{-42}\textrm{L}_{[OII]}.$ (5)

####  2.4.2 Dust extinction

When fitting the set of galaxy templates to photometry of COSMOS2020 and GAMA
sources, we apply dust attenuation using a grid of extinction curves, namely
those derived in Prevot et al. (1984); Calzetti et al. (2000); Seaton (1979);
Allen (1976); Fitzpatrick & Massa (1986). Although the template fits constrain
each galaxy’s stellar extinction well, it is known from measurements of the
H$\alpha$/H$\beta$ Balmer decrement that the dust extinction in nebular
regions differs from that of the stellar continuum and tends to be more
pronounced at lower redshifts (Puglisi et al., 2016; Calzetti et al., 1994;
Izquierdo-Villalba et al., 2019; Reddy et al., 2015; Kashino et al., 2019;
Faisst et al., 2019). To account for this differential nebular attenuation we
scale the Fitcat-derived stellar extinction E$(B-V)$ by the redshift-dependent
differential extinction $f(z)$ derived in Saito et al. (2020), which is
parameterized by a linear function capped at unity

$f(z)=\frac{E(B-V)_{star}}{E(B-V)_{neb}}=\textrm{min}(1,0.44+0.2z).$ (6)

For all sources we apply nebular attenuation assuming the extinction curve
from Calzetti et al. (2000).

##  3 Line model Validation

To test the fidelity of the empirical line model, we make a number of
comparisons to existing line measurements. These include population-level
comparisons of line equivalent widths (H$\alpha$+[NII]), luminosity functions
(H$\alpha$, [OII] and [OIII]) and line ratios as a function of stellar mass
and redshift. Where external measurements in the COSMOS field are available,
we also make direct, cross-matched comparisons of emission line fluxes.

###  3.1 H-alpha+[NII] Line equivalent widths

We begin by comparing the predicted equivalent width of the H$\alpha$ \+ [NII]
complex as a function of stellar mass with measurements from Fumagalli et al.
(2012) using the VIMOS VLT Deep Survey at $0.2<z<0.4$ (VVDS; Le Fèvre et al.,
2005), 3D-HST ($0.8<z<1.5$) and Erb et al. (2006) ($2.0<z<2.6$). The evolution
of EW(H$\alpha$) with redshift is often measured as an observational proxy for
the specific star formation rate (sSFR)-redshift relation (Khostovan et al.,
2021). The comparison between our catalog and existing measurements is shown
in Fig. 3. To compare with the actively star-forming samples (SF), we select
all sources with EW(H$\alpha$+[NII])$>3$ Å, with the caveat that the samples
we compare with come from surveys with varying selections and sensitivities.
Nonetheless, the average equivalent widths within each stellar mass bin from
our model are in close agreement with measurements.

It is understood that more massive galaxies undergo less vigorous star
formation (Juneau et al., 2005; Zheng et al., 2007), leading to smaller
H$\alpha$ equivalent widths (at fixed redshift). This trend is captured by
previous measurements and by our synthetic line catalog. Our catalog also
captures the shift to higher equivalent widths at higher redshifts (Brinchmann
et al., 2008). For galaxies with $10\leq$$\log(M_{\star}/M_{\odot})$$\leq
10.5$, the mean line equivalent width increases from
$\langle$EW(H$\alpha$+[NII])$\rangle=$ 70 Å for $z=0.2-0.4$ up to $\sim 200$ Å
for $z=2.0-2.6$. For the largest mass bin considered
($11\leq$$\log(M_{\star}/M_{\odot})$$\leq 11.5$) the mean equivalent width
goes from $\sim 15$ Å to nearly 100 Å over the same redshift range.

Figure 3: Predicted H$\alpha$ \+ [NII] equivalent widths of our synthetic line
catalog as a function of stellar mass and redshift (grey points), with the
binned average of the synthetic catalog plotted in red. These relations agree
with previous spectroscopic measurements in the literature presented in
Fumagalli et al. (2012) (light blue) from VVDS ($0.2<z<0.4$), 3D-HST
($0.8<z<1.5$) and Erb et al. (2006) (dark blue, $2.0<z<2.6$).

###  3.2 Line luminosity functions

From the catalog of line fluxes and redshifts we compute line luminosity
functions (LFs). For each LF we select all catalog sources with estimated
redshifts within 0.05 of $\overline{z}$, where $\overline{z}$ denotes the
central redshift of the externally measured LFs. The width of the redshift
bins is chosen to be small enough that redshift evolution is negligible within
the bins but large enough to obtain good sample statistics. We use the total
comoving volume within each redshift slice (i.e., within
$[\overline{z}-0.05,\overline{z}+0.05]$ with $A_{eff}=1.27$ deg2) to normalize
the LFs. The synthetic LFs presented are not corrected for the Eddington
biases sourced by flux uncertainties in the photometric catalogs (c.f. Saito
et al., 2020), and the LFs include the effects of intrinsic dust attenuation.

####  3.2.1 H-alpha

We evaluate the synthetic H$\alpha$ LF for redshift bins centered at
$\overline{z}=\\{0.4,0.84,1.47,2.23\\}$. Figure 4 shows our derived H$\alpha$
LFs compared with measurements and best-fit Schechter functions from Sobral et
al. (2013), in which H$\alpha$ measurements are corrected for [NII]
contamination following the relation from Sobral et al. (2012). On the bright
end, our model is in close agreement with measurements in all redshift bins
for $L\geq L_{*}$. Our LFs tend to fall steeply at fainter luminosities,
underestimating $\phi$ for $L<L_{*}$. Unlike the LFs from Sobral et al.
(2013), our derived LFs are uncorrected for luminosity incompleteness. The
underprediction of $\phi$ is explained primarily by our selection on the
COSMOS2020 catalog, rather than by the line flux modelling. Note that this
behavior is common to all lines in our catalog. As the COSMOS catalog goes
considerably deeper than SPHEREx photometry, this faint-source incompleteness
starts to set in after the SPHEREx faint-source limit is reached, and as such
is acceptable for modeling SPHEREx observations. Our line luminosities have a
lower limit that is fainter than measurements in the Sobral et al. (2013)
sample (with the exception of the $z=2.23$ bin), which we understand as
reflecting the fainter population probed by the COSMOS2020 catalog through
broad-band continuum sensitivity.

Figure 4: H$\alpha$ line luminosity function evaluated at $z=$ 0.4 (purple),
0.84 (green), 1.47 (red) and 2.23 (blue), comparing best-fit Schechter
functions from Sobral et al. (2013) with empirically drawn fluxes from our
model. The synthetic line LFs shown have not been corrected for catalog
incompleteness.

####  3.2.2 [OII] doublet

The [OII] line will fall in SPHEREx’s bandpass for galaxies with $z>1$. While
only the brightest [OII] lines will be detectable at SPHEREx full-sky depth
(see Section 4.4), there are near-future telescopes that will measure the
doublet through optical spectroscopy at lower redshift, for example _Euclid_
(Laureijs et al., 2011) and _Roman_ (Wang et al., 2022). For this reason we
validate [OII] across a range of redshifts.

To validate [OII] for lower redshifts we compare our synthetic LFs with
measurements from the FORS2 instrument at the Very Large Telescope (VLT) and
the SDSS-III/BOSS spectrograph, along with measurements from GAMA, zCOSMOS and
VVDS (Comparat et al., 2015). We plot LFs derived from our line catalog
calculated in six redshift bins spanning $0<z<1.3$ alongside these
measurements in Fig. 5. We also include LF measurements from the HETDEX pilot
survey for $0<z<0.4$ (Ciardullo et al., 2013) and from the Deep Extragalactic
Evolutionary Probe 2 (DEEP2) galaxy survey for $0.7<z<1.3$ (Zhu et al., 2009).
For redshift bins centered on $z=0.15$ and $z=0.3$, our LFs agree down to
$L\sim 10^{40}$ erg s-1. For $z>0.5$ the bright end LF is consistent with Zhu
et al. (2009), though both are higher than measurements from Comparat et al.
(2015).

Figure 5: [OII] luminosity function evaluated in six redshift bins between
$z=0.1$ and $z=1.3$. Line fluxes from our model are compared with a
combination of emission line measurements from Comparat et al. (2015) (black
points), Zhu et al. (2009) (blue) and Ciardullo et al. (2013) (green). The
synthetic line LFs shown have not been corrected for catalog incompleteness.

In Fig. 6 we compare [OII] LFs with measured constraints from the HiZELS
survey (Khostovan et al., 2015). The bright-end LFs broadly agree with one
another aside from in the highest redshift bins, where the synthetic
COSMOS2020 catalog contains few sources. In general these results indicate
that [OII] is captured well at the population level.

Figure 6: [OII] luminosity functions evaluated from our synthetic line catalog
at $\overline{z}=$ 1.47 (red), 2.25 (blue), 3.34 (brown) and 4.69 (yellow),
compared with measurements from HiZELS (Khostovan et al., 2015, black points)
and their best-fit Schechter functions (grey lines). The incompleteness in the
synthetic catalog becomes more pronounced at high redshifts, especially in the
$z=4.69$ bin.

####  3.2.3 [OIII]+H-beta

The [OIII] + H$\beta$ complex will be detectable by SPHEREx for galaxies with
$z>0.55$ and can potentially aid redshift measurements if measured in addition
to H$\alpha$ and/or [OII]. We show line LFs for [OIII] and [OIII]+H$\beta$ in
Fig. 7. Our predicted LFs are higher on average than those derived from the
HiZELS survey (Khostovan et al., 2015). However, our predictions on the bright
end are in close agreement with Colbert et al. (2013) and a recent study using
$1.2<z<1.9$ emission line galaxies identified on the 3D-HST grism (Nagaraj et
al., 2022). Given the disagreement in measurements between Khostovan et al.
(2015) and Colbert et al. (2013), it is difficult to evaluate the significance
of the model disagreement with Khostovan et al. (2015).

Figure 7: [OIII] (solid) and [OIII]+H$\beta$ (dashed) luminosity function
evaluated from our synthetic line catalog at in four redshift bins spanning
$z=0.84$ (left, green) and $z=3.24$ (right, brown). The black line and grey
shaded regions show LF constraints from Khostovan et al. (2015). Similar
constraints for [OIII] are shown from Colbert et al. (2013) (yellow) and
Nagaraj et al. (2022) (blue). The synthetic line LFs shown have not been
corrected for catalog incompleteness.

####  3.2.4 Paschen-alpha

The broad spectral coverage and resolving power offered by SPHEREx in the
near-infrared enable detection of the 1.87 $\mu$m Paschen-$\alpha$ line.
Figure 8 shows LF predictions for Paschen-$\alpha$ in four redshift bins
between $0<z<1.6$, taking the sum of COSMOS2020- and GAMA-derived LFs for
$z=0.2$ and $z=0.4$. The bright end LF evolves mildly for redshifts $z>0.5$.

Figure 8: Paschen-$\alpha$ line luminosity function predicted from the
synthetic catalog in four redshift bins spanning $0<z<1.6$, i.e., the redshift
coverage for Paschen-$\alpha$ by SPHEREx. The synthetic line LFs shown have
not been corrected for catalog incompleteness.

###  3.3 Direct line flux comparison in the COSMOS field

We complement validation of our line model at the population level with direct
comparisons to existing line flux measurements. One-to-one comparisons of
cross-matched sources in the COSMOS field allow us to quantify any consistent
biases as a function of line flux while controlling for the properties of the
galaxies (assuming they are well constrained by one or both surveys).

The H$\alpha$ sample from Sobral et al. (2013) overlaps partially with the
COSMOS field. The measured and predicted line fluxes are plotted in Fig. 9, in
which no extinction correction is applied. Positional cross matches are
performed for galaxies in the same redshift bins as Fig. 4, however there are
many sources for which the estimated redshift from Weaver et al. (2022a)
differs by more than $|\Delta z|=0.2$ from the nominal redshift bin (indicated
by open circles). These discrepancies are most pronounced for fainter line
fluxes, and could be caused by outliers in the COSMOS2020 catalog. The
synthetic line fluxes are relatively unbiased for $F_{H\alpha}>10^{-16}$ erg
cm-2 s-1 and at low redshift, with a slight positive bias for higher redshift
bins.

Figure 9: Direct H$\alpha$ line fluxes compared with measurements from Sobral
et al. (2013) for bins with $\overline{z}=$ 0.4 (purple), 0.84 (green), 1.47
(red) and 2.23 (blue). Open circles indicate galaxies where the COSMOS2020
photometric redshift differs from that of the nominal Sobral et al. (2013)
redshift bin by more than 0.2.

We compare our model fluxes with two other sets of emission line measurements
in the COSMOS field, namely zCOSMOS-Bright (Saito et al., 2020; Lilly et al.,
2007) and 3D-HST (Brammer et al., 2012; Momcheva et al., 2016). These surveys
have obtained spectroscopic redshifts in the COSMOS field that are magnitude-
limited to $i_{AB}<22.5$. From zCOSMOS and 3D-HST, H$\alpha$ is measured for
redshifts $z\leq 0.46$ and $0.67\leq z\leq 1.6$, respectively, while for [OII]
they cover $0.47\leq z\leq 1.57$ and $1.95\leq z\leq 3.56$. [OIII] is measured
in the redshift ranges $0.11\leq z\leq 0.92$ and $1.19\leq z\leq 2.39$,
respectively. The line fluxes from zCOSMOS are aperture corrected following
the procedure from Lamareille et al. (2009), which uses the measured sizes of
the galaxies to estimate the fraction of total flux falling within the one
arcsecond slit. Saito et al. (2020) estimates the line flux completenesses of
zCOSMOS and 3D-HST to be $\log(F_{line}/$erg cm-2 s${}^{-1})=-15.8$ and
$\log(F_{line}/$erg cm-2 s${}^{-1})=-16.5$.

Figure 10 shows the ratio of predicted and observed fluxes for H$\alpha$,
[OII] and [OIII], with the mean and scatter (binned by log-flux) plotted in
black. There is strong agreement with zCOSMOS-measured fluxes for all three
lines, though for [OIII] our model fluxes overestimates fluxes with
$F_{[OIII]}<2\times 10^{-16}$ erg cm-2 s-1. This bias is consistent with
trends seen in Saito et al. (2020) and our direct comparisons with HiZELS (see
Fig. 9. For the 3D-HST sample, which contains more high redshift sources, our
model H$\alpha$ fluxes in close agreement with measured fluxes on average.
However, for [OII] and [OIII] our model fluxes tend to underestimate measured
fluxes above $F\sim 1\times 10^{-16}$ erg cm-2 s-1.

We consider several potential explanations for why our model [OII] and [OIII]
fluxes are low relative to the observed sample:

  * •

The predicted quiescent fraction may be too high for this selection of sources
when in reality they are star forming. The fraction of cross-matched zCOSMOS
sources above the completeness limit (dash-dotted lines) labeled as passive by
our model (i.e., $F_{line}=0$) is 3.0%, 2.0% and 1.3% for H$\alpha$, [OII] and
[OIII] respectively, and for the 3D-HST sample the corresponding fractions are
4.6%, 9.4% and 10.5%.

  * •

Incorrect redshift assignments could bias the fluxes of the sources low. Only
a handful of COSMOS2020 sources have discrepant redshifts when compared
against spec-zs from either zCOSMOS or 3D-HST. Removing sources with $\delta
z/(1+z)>0.1$ from the comparison does not ameliorate the discrepancies seen in
[OII] for the $1.95\leq z\leq 3.56$ sample. Momcheva et al. (2016) quotes a
redshift scatter of $\sim 10\%$ for $z\sim 2$ galaxies when compared against
duplicate measurements and follow-up observations from the MOSDEF survey
(Kriek et al., 2015) for $JH_{IR}<24$, which is the case for the galaxies in
question with $F_{line}^{3DHST}>10^{-16}$ erg cm-2 s-1.

  * •

Likewise, the discrepancy could be present if the 3D-HST fluxes for these
objects are biased high. There is evidence for a positive bias in [OII] and
[OIII] on average for 3D-HST fluxes for $F_{line}^{3DHST}\lesssim 10^{-16}$
erg cm-2 s-1 however the paucity of direct comparisons prohibits us from
assessing whether this fully explains the observed differences.

  * •

An incomplete model of nebular attenuation may impact our [OII] and [OIII]
predictions, in particular at higher redshifts.

  * •

If there is a non-negligible AGN fraction in the high-redshift sample it may
explain the higher line fluxes compared to our model fluxes. Because the WFC3
G141 grism has moderate resolution ($R=130$) it is not possible to resolve the
H$\alpha$+[NII] complex, meaning we cannot distinguish between star forming
galaxies and AGN through the BPT diagram (e.g., Fig. 12). AGN have been
identified and removed in the COSMOS2020 catalog using morphological and SED
criteria, however it is likely there is additional AGN contamination in
particular at the higher redshifts considered.

Figure 10: One-to-one line comparisons of H$\alpha$ (left), [OII] (middle) and
[OIII] (right) between model predicted fluxes and direct flux measurements
from zCOSMOS (top row) and 3D-HST (bottom row). The black errorbars show the
mean and 68 percentile range of the colored points in bins of equally spaced
log-fluxes. The grey dash-dot lines indicate the completeness limits of the
respective surveys as estimated in Saito et al. (2020) (see Fig. 2 from that
work).

At the population level, our line model captures the scatter in line strengths
seen in the two measured line catalogs. Figure 11 compares the distribution of
line fluxes from our model with those from zCOSMOS and 3D-HST as a function of
stellar mass. For the low redshift sample we place an additional cut on
$i<22.5$ to match the zCOSMOS selection and find that our model fluxes
adequately cover the range of measured fluxes. When compared against 3D-HST,
our model fluxes match both the core and tails of the measured flux
distributions. While we use mean trends to predict line strengths and ratios,
our synthetic line catalog are conditioned on estimates of redshift, stellar
mass and dust attenuation, all of which contribute to the observed flux
scatter. Reproducing this scatter in our redshift predictions is important for
obtaining realistic forecasts on the number of SPHEREx sources with line
detections (see §4.4).

Figure 11: Log-density plot of H$\alpha$ line fluxes and stellar masses, for
synthetic fluxes from our COSMOS2020 catalog (black) and for observed samples
from zCOSMOS (blue) and 3D-HST (red). The two columns show the COSMOS2020
catalog with redshift cuts matching the H$\alpha$ ranges of zCOSMOS and
3D-HST. Density contours for the spectroscopic catalogs are overlaid on the
synthetic flux distributions for comparison.

###  3.4 Line ratios/trends

Evolution of the line ratios as a function of redshift can be seen by placing
galaxy line ratios on the O3N2 BPT diagram. This is shown in Fig. 12, with
synthetic line ratios color-coded by redshift. While the population of low-
redshift galaxies resides in a locus centered on the relation from Brinchmann
et al. (2008) (red, dashed), moving to higher redshift has the effect of
shifting this locus upward on the BPT diagram. This trend is consistent with
measurements from MOSFIRE (Steidel et al., 2014) of a sample of 251 galaxies
between $z=1.5$ and $z=3.5$ ($\langle z\rangle\sim 2.3$).

Figure 12: O3N2 BPT diagram with line ratios from the synthetic line catalog,
plotted against measured trends of local ($z\sim 0$, Brinchmann et al. (2008))
and higher redshift ($\langle z\rangle=2.3$, Steidel et al. 2014) galaxies.
Plots show line ratios of the same sources, colored by redshift (top) and
stellar mass (bottom).

###  3.5 Quiescent fraction

The fraction of passive galaxies is important to quantify as a check that our
model does not overproduce star-forming galaxies with strong emission lines.
Using our synthetic line catalog, we find that 7% of objects are either best
fit by passive galaxy templates (see §2.3) or have H$\alpha$ equivalent width
less than 5 Å. At low redshifts ($z<1$), the quiescent fractions of our
synthetic catalog for mass bins $\log(M_{\star}/M_{\odot})$ $\in\ (10,10.5]$,
$\log(M_{\star}/M_{\odot})$ $\in\ (10.5,11.0]$ and $\log(M_{\star}/M_{\odot})$
$\in\ (11,11.5]$ are 20%, 34% and 44%, respectively. We also compute quiescent
fractions the more standard separation using the $UVJ$ diagram (Williams et
al., 2009; Daddi et al., 2004). We use the rest frame ($NUV-r$) vs. ($r-J$)
selection from Ilbert et al. (2013) ($NUVrJ$ in shorthand):

$(NUV-r)>3(r-J)+1\textrm{ and }(NUV-r)>3.1.$ (7)

The absolute magnitudes are calculated in the Farmer catalog from Weaver et
al. (2022a) using the best-fit photo-z solutions. Using this classification,
we find the same selections of galaxies have quiescent fractions of 29%, 41%
and 53%, respectively. While our quiescent galaxy classification using
EW(H$\alpha$) is more conservative than using $NUVrJ$, the trends of both
classifications match the expectation of an increasing fraction of quiescent
galaxies with higher stellar mass. Our $NUVrJ$ classification yields quiescent
fractions consistent with other studies using COSMOS2020 (c.f. Fig. 9 of
Weaver et al., 2022b). These quiescent fractions are lower than those
determined from measurements using UltraVISTA and 3D-HST which range from 25%
up to 70% across the same range of $\log(M_{\star}/M_{\odot})$ (e.g., Fig. 2
of Martis et al., 2016).

##  4 Predicting SPHEREx Photometry

We now use the empirical model detailed above to generate synthetic SPHEREx
spectrophotometry with realistic colors. In this section we describe the
unique spectral scan strategy employed by SPHEREx and detail the noise
properties of the full-sky and deep surveys.

###  4.1 SPHEREx

SPHEREx uses six HAWAII-2RG (H2RG) detector arrays arranged in two mosaics,
separated by a dichroic beam splitter that allows the focal plane to be
simultaneously imaged (details on the instrument configuration can be found in
Korngut et al., 2018). A set of linear variable filters (LVFs) sit above the
focal planes, which function as bandpass filters whose central wavelength
varies linearly with detector position. The spectrum for a source can thus be
obtained by modulating its position across the field of view in a series of
exposures.

Over its nominal two-year mission, SPHEREx will complete four full-sky
surveys, where each survey comprises measurements in 102 spectral channels on
each sky position. Observations are shifted by half a spectral channel between
the first/third and second/fourth surveys to Nyquist sample the response
function. Each spectral channel is defined in steps of
$\Delta\lambda=\lambda/R$ across each of the six detectors (detectors in this
context are also referred to as “bands”). The resolution of each LVF is fixed
between $R=35-130$, where

$R=\frac{\lambda_{c}T(\lambda_{c})}{\int T(\lambda)d\lambda},$ (8)

$\lambda_{c}$ is the central wavelength and $T(\lambda)$ is the filter
transmission which varies continuously as a function of detector position.
This will be done through a scan strategy that involves a combination of large
and small slews as the spacecraft follows a low-earth sun-synchronous orbit,
observing near great circles which precess over six months. As a result of
this observing strategy, SPHEREx will scan the northern and southern ecliptic
caps (NEP and SEP, respectively) with much higher cadence, leading to an total
area of 200 deg2 with $\sim 200$ measurements per spectral channel after two
years. The two survey depths are distinguished as “shallow” (or “full-sky”)
and “deep” throughout this work.

Figure 13 shows coverage maps of both NEP and SEP for the simulated deep field
survey. The presence of the Small Magellanic Cloud near the SEP motivates an
avoidance strategy for SPHEREx which leads to slightly shallower coverage
compared with NEP, along with some asymmetric structure. In these regions
SPHEREx obtains considerably more measurements than of the full sky. However
in the deep field regions the number of complete spectra (defined as having
one measurement per channel) varies considerably as a function of ecliptic
latitude, from roughly fifty complete spectra per line of sight in the
outskirts of the field up to over four hundred spectra ($\sim 40000$ dithered
measurements per source) in the deepest parts of the NEP field. This will
yield galaxy spectro-images that are oversampled both in the spatial and
spectral domains.

Figure 13: Coverage maps for the SPHEREx deep fields, located near the south
(left) and north (right) ecliptic poles. The southern field is slightly offset
from the SEP in order to avoid contamination from the Small Magellanic Cloud
(SMC). Each of the two coverage maps shown covers $\sim 100$ deg2, with the
central red core of each map spanning a diameter of 3.5 degrees.

We compute observed fluxes by integrating the set of noiseless SEDs over our
nominal set of 102 SPHEREx channel bandpasses. Once this is done we then add
observational noise consistent with SPHEREx’s expected sensitivity. While the
Zodiacal light (scattered sunlight and thermal emission by interplanetary dust
grains; ZL) varies in intensity as a function of celestial position and time,
for simplicity we use the conservative maximum expected value (MEV) estimates
for point source sensitivity in each channel. These estimates assume a sky-
averaged ZL surface brightness level informed by measurements from DIRBE
(Kelsall et al., 1998). Many of the galaxies at full-sky sensitivity will have
fluxes that are photon noise-limited (or limited by confusion noise), however
we also add Poisson noise which primarily impacts the brightest galaxies. We
plot the SPHEREx point source flux sensitivities used throughout this work in
Fig. 14. At full-sky depth, the average MEV 5$\sigma$ per channel point source
depths varies from 19.3 at 0.75 $\mu$m to 19.7 at 3.8 $\mu$m (bands 1-4), with
reduced sensitivity for $\lambda>3.8$ $\mu$m (bands 5 and 6). The deep fields
will push roughly two magnitudes deeper in point source sensitivity, with a
dependence on celestial position (see Fig. 13).

Figure 14: 1$\sigma$ MEV point source sensitivity, per spectral channel. The
curves represent the average sensitivity for the SPHEREx full-sky (blue) and
deep (red) field depths after the nominal two-year mission.

The synthetic COSMOS catalog used in this work goes several magnitudes deeper
than the SPHEREx full-sky sensitivity. This means the catalog can be used to
make predictions for the majority of sources for which SPHEREx can measure
redshifts. The catalog can also be used to simulate realistic distributions of
fainter sources that contribute in the form of confusion noise, however we do
not simulate source confusion in this work.

Figure 15 highlights two sample galaxies from the COSMOS2020 sample with
simulated photometry; in the upper panel is a massive, quiescent galaxy at low
redshift ($z=0.25$), while the lower panel shows an ELG observed at $z=1.14$.
Like in the example shown, many quiescent galaxies have well-resolved, rest-
frame 1.6$\mu$m bumps driven by a minimum in H- opacity along with PAH
features at longer wavelengths. These continuum features can help derive
precise redshifts for luminous red galaxies (Sawicki, 2002; Simpson &
Eisenhardt, 1999). In contrast, star-forming galaxies are typically less
massive but contain several emission lines/line complexes which are detectable
by SPHEREx, namely H$\alpha$+[NII], [OIII]+H$\beta$ and Paschen-$\alpha$.

Figure 15: Mock photometry of two galaxies: a massive, quiescent galaxy at
$z=0.25$ (top) and an active emission line galaxy at $z=1.14$ (bottom).
Simulated flux measurement uncertainties are based on the current SPHEREx all-
sky MEV point source sensitivity for 102 spectral channels. The grey lines
show the underlying SEDs generated through our emprirical model, while black
points show existing COSMOS photometry of the two sources. Both galaxies have
photometry simulated at full-sky (blue) and deep (red) survey depths. The
spectrum in the top panel shows signatures for both the rest-frame 1.6 $\mu$m
bump (red shaded region; Simpson & Eisenhardt, 1999) and PAH emission in the
black shaded region. In the bottom panel, the galaxy is detected at low
significance at full sky depth while deep field photometry of the same source
is sensitive to H$\alpha$+[NII] ($\lambda_{obs}\sim 1.4$ $\mu$m, green dashed
line), [OIII]+H$\beta$ ($\lambda_{obs}\sim 1.4$ $\mu$m, blue) in the rest
frame optical and Paschen$-\alpha$ in the near infrared ($\lambda_{obs}\sim
4.0$ $\mu$m, orange).

###  4.2 Comparison with existing/near-future surveys

Ancillary catalog data from surveys in the optical and the infrared are
important for defining SPHEREx’s “reference catalog”, the list of sources
SPHEREx will measure the spectra for using forced photometry. The instrumental
point spread function (PSF) for SPHEREx varies as a function of wavelength,
with a core PSF full width at half maximum (FWHM) ranging from 2″ at short
wavelengths to a diffraction-limited 7″ at longer wavelengths. This does not
include additional spread from pointing jitter, however this is expected to be
controlled at the $\sigma_{RMS}\lesssim 1\arcsec$ level. The SPHEREx PSF will
in general be undersampled due to the relatively large 6.2″ $\times$ 6.2″
pixel size, which makes deblending adjacent sources difficult. This motivates
the use of forced photometry at the locations of reference catalog sources,
and is effective when the positional errors from the reference catalog are
small compared to the SPHEREx pixel scale (generally the case for the
ancillary catalogs mentioned in this section). Symons et al. (2021)
demonstrate PSF estimation and photometry in this limit on mock SPHEREx
exposures. Redshifts derived from these sources will form the basis for
downstream cosmology measurements.

At optical wavelengths, several broad-band photometric surveys have been
undertaken over large portions of the sky. The PanSTARRS 3$\pi$ survey has a
nominal 5$\sigma$ depth of $i=23.1$ (Chambers et al., 2016). The DESI Legacy
Imaging Surveys combines optical data from three separate surveys (the
Beijing-Arizona Sky Survey (BASS), the Dark Energy Camera Legacy Survey
(DeCALS) and the Mayall $z$-band Legacy Survey (MzLS)), covering $\sim 14,000$
deg2 and reaching median $grz$ 5$\sigma$ depths of $m_{AB}=24.0$, 23.4 and
22.8, respectively (Dey et al., 2019). Looking ahead, the _Rubin_ observatory
LSST will obtain optical photometry in six bands ($ugrizy$) across 18000 deg2
in the southern sky (Ivezić et al., 2019). Catalogs from the first data
release are expected to reach a 5$\sigma$ co-added depth of $i\sim 25$ after
one year of observations, while after ten years the depth is predicted to be
$i\sim 26.8$. These optical catalogs will resolve sources with considerably
finer angular resolution than SPHEREx.

In the infrared, catalogs from full sky WISE imaging detect large numbers of
extragalactic sources in its two broad bands centered at 3.4 and 4.5 $\mu$m
(W1 and W2, respectively). The infrared catalog from 5-year WISE co-added
images reaches depths of $m_{AB}\sim 20.7$ and 19.9 in W1 and W2 band
respectively (Schlafly et al., 2019), which is slightly deeper than the
typical SPHEREx single-channel full-sky sensitivity. By the time of SPHEREx’s
launch WISE will have catalogs derived from eight years of imaging. When we
cross-match WISE sources against known galaxy positions in the COSMOS field,
we find a positional accuracy ranging from 0.2″ on the bright end up to
$0.5-1\arcsec$ for $W1\sim 20-21$.

###  4.3 Predictions from synthetic photometry

The set of high resolution model SEDs are convolved with LSST $i$-band and
WISE W1 filters to obtain predicted magnitudes. These can then be compared
with the nominal external survey depths to quantify the coverage afforded by
the reference catalog for measurable SPHEREx sources. Figure 16 shows the
distribution of SPHEREx sources and synthetic _Rubin_ /WISE counterparts. For
$i<18$ we include a subset of galaxies from our GAMA catalog matching the
effective area of the COSMOS catalog.

SPHEREx will be effective at probing the population of infrared bright,
optically faint galaxies. We inspect the population of sources in our
synthetic COSMOS catalog with $24<i<25$ and SPHEREx $20<m_{[2.2\mu m]}<21$,
which are measurable at SPHEREx full-sky depths. These are shown in Fig. 17.
As implied by our selection, these sources are very red ($i-[2.2$ $\mu$m$]>3$)
and are typically massive ($\log(M_{\star}/M_{\odot})$ $>10.5$), quiescent
(EW(H$\alpha)<50$ Å) galaxies with redshifts $z\sim 1-2$. Higher-redshift
populations like this are important for SPHEREx cosmology measurements, which
rely on precise measurement of long wavelength modes to constrain local
primordial non-Gaussianity (pNG; Dalal et al., 2008).

Figure 16: Magnitude-magnitude diagrams of simulated COSMOS2020 SEDs, for
SPHEREx 2.2 $\mu$m magnitudes (x-axis) and LSST $i$-band (top) and WISE W1
(bottom). Also shown are expected 5$\sigma$ point source sensitivities for the
three photometric surveys. The SPHEREx MEV 5$\sigma$ single channel depths are
indicated by the red dashed lines for 2.2 $\mu$m (vertical) and 3.4 $\mu$m
(horizontal). The SPHEREx deep field depths do not include confusion noise.
Figure 17: Distribution of galaxy redshifts and stellar masses after applying
a cut on $i$\- and $K$-band magnitudes. The points are colored by their
H$\alpha$ (log-) equivalent widths.

From estimated survey depths of the ancillary catalogs, the redshift
distribution of sources selected by each catalog can be computed. This is
shown in Fig. 18 for two cuts $K<20$ and $K<22$. For $K<20$, the optical and
infrared external catalogs are complete out to $z\sim 1$. For redshifts $z>1$,
completeness using PanSTARRS and DECaLS falls considerably, while the WISE-
selected catalog remains largely complete. For the $K<22$ cut, incompleteness
of the external catalogs is more severe. The union of optical/infrared
catalogs (e.g., WISE+DECaLS) complements each individual catalog (low
redshifts covered by optical, higher redshifts covered by infrared), however
no combination of these catalogs is complete down to SPHEREx’s deep field
sensitivity across the 200 deg2 covering the ecliptic poles.

Figure 18: Comparison of the redshift distribution of our synthetic
COSMOS2020+GAMA catalog (grey), along with that obtained using PanSTARRS
(green), DECaLS (orange) and WISE (red) reference catalogs.

Another consideration is that the source density of the varies significantly
between the optical and infrared catalogs that comprise the reference sample.
This is seen clearly in Fig. 19, in which synthetic LSST $i$-band and WISE 3.4
$\mu$m densities are plotted for a range of magnitudes near each catalog’s
expected limiting depth. As one approaches $W1=21$ in the WISE catalog, the
source density approaches 0.06 pixel-1, or roughly one source per 16 SPHEREx
pixels. In contrast, the LSST source density for all sources with $i<25$ is
$\sim 0.4$ pixel-1, i.e., one source per 2.5 SPHEREx pixels. This has
implications for strategies that utilize deeper reference catalogs to define
SPHEREx targets in forced photometry, and will require some parsimony in the
effective number of sources that are fit simultaneously.

Figure 19: Source density as a function of reference catalog depth, selecting
on LSST $i$-band (blue, top axis) and WISE W1 at 3.4 $\mu$m (red, bottom
axis). These number densities are calculated assuming the COSMOS2020 footprint
of 1.27 deg2 and assuming 6.2″ $\times$ 6.2″ pixels.

###  4.4 Prevalence of detectable emission lines in SPHEREx observations

Quantifying the fraction of galaxies where emission lines are present is
important in forecasting redshift constraints. It has been shown using COSMOS
30-band photometry that accounting for emission lines in photo-z measurements
treatment can increase redshift accuracy by over a factor of two (Ilbert et
al., 2009), however it is an open question what impact emission lines will
have for the SPHEREx sample. The number of detected lines and their equivalent
widths will determine the redshift precision of the ELG sample. In addition,
accounting for emission lines, for example H$\alpha$, can improve SFR and dust
opacity estimates for many galaxies (Smit et al., 2016).

To compute SPHEREx line flux sensitivity we start by assuming the total flux
from an emission line falls into an individual channel, i.e.,
$\Delta\lambda\gg\sigma_{line}$, where $\Delta\lambda$ is the channel width.
For a given flux $F_{line}$ in erg cm-2 s-1, the flux density averaged across
channel $i$ is given by

$\Delta S_{\lambda_{i}}^{line}[\mu
Jy]=\frac{10^{29}\lambda_{i}^{2}}{c}\times\frac{F_{line}}{\Delta\lambda_{i}},$
(9)

where $\lambda_{i}$ is the central wavelength of the bandpass in Å and
$c=3.0\times 10^{18}$Å s-1. This expression allows us to compute the
significance of detecting a signal from the line in the presence of noise. We
compute this at both full-sky and deep survey depths over the full set of 102
nominal channels, assuming $R=[41,41,41,35,110,130]$ across the six SPHEREx
bands, where each band corresponds to a separate detector (Condon et al.,
2022). This calculation does not take into account the spectral dithering with
which SPHEREx will sample emission lines, nor the details of line-continuum
separation, which may introduce additional errors and covariances.

The resulting flux sensitivities are shown in Fig. 20. $F_{line}^{lim}$
depends on a combination of the wavelength dependent point source sensitivity
and the spectral resolution across the six LVFs. These sensitivity estimates
exclude effects of confusion noise. Despite the larger instrumental noise
expected at longer wavelengths, the spectral resolution is $>3$ times higher
than at short wavelengths, leading to a minimum in line sensitivity around 4
$\mu$m that coincides with the minimum in Zodiacal light intensity. At full-
sky depth, the 3$\sigma$ line flux sensitivity ranges from $3\times 10^{-15}$
erg cm-2 s-1 at $\lambda\sim 1$ $\mu$m down to a minimum of $6\times 10^{-16}$
erg cm-2 s-1 at $\lambda\sim 4$ $\mu$m. At the same reference wavelengths the
3$\sigma$ deep survey sensitivities range from 4$\times 10^{-16}$ erg cm-2 s-1
down to $\sim 9\times 10^{-17}$ erg cm-2 s-1. This approaches the line
sensitivity expected for the _Euclid_ and _Roman_ grisms and should complement
these surveys through coverage beyond 2 $\mu$m.

Figure 20: Line flux sensitivity as a function of observed wavelength, for
SPHEREx full-sky (blue) and deep (red) surveys. The sensitivities are
calculated for 2$\sigma$ (solid), 3$\sigma$ (dashed) or 5$\sigma$ (dash-dot)
detections, ignoring errors from line-continuum separation and other confusion
noise.

Using these sensitivity estimates we can predict, given a catalog of emission
line fluxes and redshifts, how many sources are detectable by SPHEREx. This is
done for H$\alpha$+[NII], [OIII]+H$\beta$, [OII] and Paschen-$\alpha$. The
number of $>2\sigma$ line detections at both full-sky and deep survey depth
are shown as a function of redshift in Fig. 21. Combining line detections from
both COSMOS2020 and GAMA catalogs (weighted by their effective areas), we
predict full-sky number densities of $\overline{n}\sim 770$ (260) deg-2 for
H$\alpha$ detectable at 2$\sigma$ (3$\sigma$). The next most prevalent line is
Paschen-$\alpha$, with $\overline{n}\sim$ 250 (140) deg-2. Despite the small
intrinsic line flux ratio $P\alpha/H\alpha=0.12$ (for case B recombination),
Paschen-$\alpha$ is more immune to dust extinction than H$\alpha$ and is
observed at longer wavelengths where SPHEREx has better line flux sensitivity.
SPHEREx is less sensitive to rest-frame optical lines blueward of H$\alpha$,
due to a combination of intrinsic line ratios, more severe dust attenuation
and poorer sensitivity in the blue end of the SPHEREx bandpass ($\lambda<2$
$\mu$m). At full-sky depth, the [OIII]+H$\beta$ complex is detectable at
$>2\sigma$ for 110 COSMOS2020 catalog sources (85 deg-2), however this drops
to only 15 sources detected at $>3\sigma$. The situation is worse for [OII],
for which no lines are predicted to be detectable at full-sky depth. [OII]
line emission is expected to be primarily detected by SPHEREx in the deep
fields, for individual sources or in aggregate through line intensity mapping
(e.g., Cheng et al., 2020).

Detecting multiple lines simultaneously with SPHEREx will enable precise and
robust redshift measurements across a broad range of distances. At full-sky
depth, our synthetic catalog predicts 12% and 15% of H$\alpha$ lines will have
a Paschen-$\alpha$ counterpart in which both lines are detected at $>2\sigma$
and 3$\sigma$ respectively. The fraction of H$\alpha$ detections with
[OIII]+H$\beta$ are 11% and 5% with the same criteria, though most
[OIII]+H$\beta$ detections should have a H$\alpha$ detection (red dashed
histogram in top panel of Fig. 21). Simultaneous detection of H$\alpha$,
Paschen-$\alpha$ and [OIII] is rare due to limited redshift overlap.

Our deep field predictions paint a much richer picture for the putative
emission line sample near the ecliptic poles, with multiple-line detections
extending from $z=0.17$ to $z=4$ and beyond. Because SPHEREx will probe the
bright end of each line LF (which varies strongly with luminosity), the
predicted number densities are highly sensitive to observing depth. Indeed,
while the mean sensitivity of the deep survey is $\sim 10\times$ that of the
full-sky, the predicted number densities are larger by factors ranging from
$\mathcal{O}(10)$ up to $\mathcal{O}(1000)$. We summarize the implied ELG
number densities at full-sky and deep survey depths in Table 2. The number
density estimates for the deep fields may be conservative in the sense that
our initial cut on $i<25$ has a larger impact on the deep field forecasts. Due
to this and our incomplete knowledge of higher-redshift line populations, we
caution over-interpretation of the deep field predictions.

Figure 21: Redshift distribution of galaxies with detectable lines (above 2$\sigma$ significance) at full-sky (top) and deep (bottom) survey depth. This is shown for individual lines/line complexes (blue, green, orange) as well as for simultaneous line detections satisfying $>2\sigma$ in both lines. Line(s) | 2$\sigma$ (full sky) | 3$\sigma$ (full sky) | 4$\sigma$ (full sky) | 5$\sigma$ (full sky) | 3$\sigma$ (deep) | 5$\sigma$ (deep)
---|---|---|---|---|---|---
| [deg-2] | [deg-2] | [deg-2] | [deg-2] | [deg-2] | [deg-2]
H$\alpha$ \+ [NII] | 770 | 260 | 115 | 65 | 17800 | 8000
P$\alpha$ | 250 | 140 | 100 | 75 | 3800 | 1600
(H$\alpha$ \+ [NII]) $\wedge$ P$\alpha$ | 90 | 40 | 20 | 10 | 3100 | 1100
[OIII] + H$\beta$ | 85 | 12 | $<10$ | $<10$ | 10500 | 3600
(H$\alpha$ \+ [NII]) $\wedge$ ([OIII] + H$\beta$) | 74 | 12 | $<10$ | $<10$ | 7900 | 2600
[OII] | $<1$ | $<1$ | $<1$ | $<1$ | 30 | $<10$
Table 2: Predicted number densities for galaxies with lines detectable through
SPHEREx spectrophotometry. Number densities for combinations of lines
correspond to sources where both lines/line complexes are detected above the
noted significance.

##  5 Redshift recovery

We test redshift recovery using the photometric redshift estimation code
implemented in Stickley et al. (2016), which is similar in spirit to the
widely used template fitting code LePhare (Arnouts et al., 1999; Ilbert et
al., 2006). The code performs a $\chi^{2}$ minimization across a pre-specified
grid of models, which we construct from the same underlying set of 160
templates used in §2.3 to generate our synthetic observations. For each
template, we deploy a grid of models with $E(B-V)=0-1$ in steps of $\delta
E(B-V)=0.05$ for three dust extinction laws (Prevot, Calzetti and Allen) with
redshifts spanning $z=0-3$ with $\delta z=0.002$. We assume flat priors over
these parameters and the set of templates. In §5.1.2 we test reducing the set
of templates used in redshift estimation as a measure of robustness for our
results.

While sufficient for broad band photometric redshift measurements, the
emission line model implementations of these codes have shortcomings with
stronger implications for intermediate spectral resolution SPHEREx
measurements. As a result we choose in this work to assess the redshift
information from continua and lines separately, which can be combined in a
hybrid line-continuum redshift estimation approach that will be the subject of
a future publication.

To emulate the selection of SPHEREx target galaxies, we evaluate redshift
recovery for galaxies pre-selected using optical and infrared ancillary
catalogs. In particular, we select any galaxies detected by DECaLS ($g<24.0$,
$r<23.4$ or $z<22.8$) or WISE ($W1<20.5$ or $W2<20.5$), which constitutes
fifty four thousand galaxies of the full 160K simulation catalog. We include
synthetic DECaLS $grz$ and WISE $W1/W2$ photometry with representative noise
in the fits, adding a 1% noise floor to capture additional photometric errors
in the external catalogs.

###  5.1 Continuum redshift results

####  5.1.1 COSMOS2020

We calculate the mean $\chi^{2}$ statistic of the fits to be 101.7,
corresponding to a reduced chi-squared of $\chi^{2}_{red}\approx 0.99$
assuming four model parameters (redshift, $E(B-V)$, template scale, template
index) and 107 total bands (SPHEREx + external). We confirm that the
distribution of best fits follows a $\chi^{2}$ distribution, with few galaxies
having $\chi^{2}_{red}>1.5$. Given the use of the same galaxy templates used
to fit the photometry as used to generate the SEDs, this level of agreement
indicates that our fits are well behaved.

We plot a random selection of redshift PDFs with increasing $\sigma_{z}$ in
Figure 22. Each redshift point estimate is computed from the first moment of
the redshift $p(z)$, which roughly coincides with the maximum _a posteriori_
estimate for unimodal distributions. It can be seen that many of the redshift
PDF estimates have non-Gaussian structure, including heavy tails and often
more than one local $\chi^{2}$ minimum. This is to be expected given the
complexity of the template set, for which several templates may be degenerate,
along with other parameters in the model space. Motivated by this, we include
two-sided redshift uncertainty estimates derived from the highest (posterior)
density interval (HPDI), defined as the shortest interval on a posterior
density for some given confidence level. In some cases where the uncertainty
is comparable to the redshift step size ($\sigma_{z}\sim\delta z=0.002$, e.g.,
top left panel of Fig. 22), discretization effects may impact the redshift
estimates, suggesting that for the highest accuracy sample some refinement of
$p(z)$ will be necessary.

Figure 22: Normalized redshift probability density functions (PDFs) for a
collection of examples from the COSMOS2020 sample. For each source, the true
redshift, the reported redshift estimate and 68%/95% credible intervals are
indicated.

Using these estimates we plot redshift error distributions for the COSMOS
sample in Fig. 23 relative to the true redshifts of the samples. These are
binned by reported fractional redshift uncertainties,
$\hat{\sigma}_{z/(1+\hat{z})}$, which can be compared with the true errors to
assess the fidelity of the redshift estimates. For this comparison we compute
redshift uncertainties corresponding to one-half the width of the 68% credible
interval (i.e., $\hat{\sigma}_{z}^{68}$). To quantify redshift errors in each
bin, we calculate the normalized median absolute deviation (NMAD), a measure
of dispersion that is robust to outliers. We also quantify the outlier
fraction $\eta$, which is defined as the fraction of $3\sigma$ outliers given
$\hat{\sigma}_{z}$ and the true error. These results demonstrate that SPHEREx
will measure a wide range of high- and low-accuracy galaxy redshifts. In this
test configuration the reported uncertainties closely match the true errors
(which can also be evaluated in terms of the z-score distribution, see
Appendix B). The 3$\sigma$ outlier fraction remains at the few percent level
for all redshift uncertainty bins. We note that there is a mild negative bias
that becomes larger for the lower-accuracy samples, which upon further
inspection is largely driven by the quiescent galaxy samples. This motivates
further investigation into the parameter degeneracies within our template
fitting, which may lead to multi-modal redshift solutions for some fits.

Figure 23: Redshift error distributions for COSMOS sources, binned by the
reported redshift uncertainty $\hat{\sigma}_{z/(1+z)}$. The dashed Gaussian
curves have widths that correspond to the median reported uncertainties within
each redshift uncertainty bin, $\tilde{\sigma}_{z/(1+z)}$ and are normalized
to the peaks of the histograms. For each sample we report
$\tilde{\sigma}_{z/(1+z)}$, the mean redshift bias, the normalized median
absolute deviation (NMAD) and the 3$\sigma$ outlier fraction $\eta$. These
results do not account for effects of source confusion.

To further understand the redshift results we also plot the input and
recovered redshifts for the COSMOS2020 sample as a function of $W1$ magnitude
in Fig. 24. Within each $W1$ bin there is a wider distribution of redshift
uncertainties. Nonetheless there is a clear trend between the redshift
accuracy and $W1$, along with for the mean bias and outlier fraction. Redshift
measurements in the bin $20.5<W1<21.5$, for which long-wavelength SPHEREx data
are largely uninformative, show a clear bias toward lower redshift values
which warrants further investigation.

Figure 24: Recovered redshifts for synthetic SPHEREx photometry derived from
the COSMOS2020 catalog from sources selected in the $W1$ band. For each
magnitude bin the galaxy density $\overline{n}$, normalized median absolute
deviation (NMAD), median redshift uncertainty $\bar{\sigma}_{z/(1+z)}$, mean
bias and 3$\sigma$ outlier fraction $\eta$ are included. The colormap
corresponds to the (log-) number density of galaxies in each bin.

By combining the recovered number densities from COSMOS2020 and GAMA (see
Appendix 2.1.2 for redshift recovery results of the GAMA sample) we can
forecast the number of galaxy redshifts accessible to the SPHEREx full-sky
cosmology sample. Assuming an effective area of 3$\times 10^{4}$ deg2 and
removing 3$\sigma$ outliers, our results imply a sample of 19 million galaxies
with $\delta z<0.003(1+z)$, which primarily occupy redshifts $z\lesssim 1$.
There are many more intermediate and low-accuracy redshifts; we forecast a
sample of 445 million galaxies with $\delta z<0.1(1+z)$, and this grows to 810
million galaxies with $\delta z<0.2(1+z)$. While the loss of information on
$f_{NL}$ from galaxies becomes more significant beyond $\sigma(z)=0.1(1+z)$
(de Putter & Doré, 2017), the number density of galaxies grows significantly
across this range, meaning these low-accuracy galaxies do still provide useful
information for pNG constraints. Other studies may also benefit from such low-
accuracy samples, for example cross correlations with CMB lensing (Krolewski
et al., 2021; Farren et al., 2023).

####  5.1.2 Sensitivity to choice of template library

Thus far we have assumed the same set of templates used to fit the redshifts
as were used to generate galaxy SEDs. However in practice a reduced set of
templates may suffice, both for recovering reliable redshifts and for
computational performance. To test the effects of different template sets on
the recovered redshifts, we perform similar fits using the empirical Brown
templates (129 in total) and the model-based COSMOS templates (31 in total)
separately. These results are shown in Figure 25. Looking at sources with
$\hat{\sigma}_{z/(1+z)}<0.1$ (black points), both the Brown-only and COSMOS-
only template sets perform reasonably well compared to the full template set,
albeit with slightly higher outlier fractions and NMAD. Interestingly, the
mean bias from using the 31 model-based templates is much smaller than the
other two cases. In contrast, the high-accuracy results
($\hat{\sigma}_{z/(1+z)}<0.01$, red points) are more sensitive to template
coverage. Compared to the full template set results, those using B14 templates
alone have a 5$\times$ higher outlier fraction, a $10\times$ larger bias, and
a NMAD that is much larger than the reported uncertainty. The high-accuracy
results degrade further when using just the 31 model-based templates, with a
3$\sigma$ outlier fraction that rises to 25.3% and a NMAD that is nearly twice
as large as the reported uncertainties. These results confirm our intuition
that high SNR fits are much more sensitive to template coverage than the
lower-accuracy samples.

Figure 25: Input vs. recovered redshifts on the COSMOS2020 sample, using three
different template library sets: the full 160 template set used to generate
the photometry (left), the 129 empirical B14 templates (middle) and the 31
model-based templates used in Weaver et al. (2022a) (right). We distinguish
high- and low-accuracy (red, black) samples whose redshift recovery statistics
are noted as well.

###  5.2 Redshifts from spectrally dithered emission line measurements

Our line detection and redshift results thus far have utilized photometry in
102 homogenized spectral bandpasses, corresponding to the 102 SPHEREx
“channels”. However in practice, each observed SPHEREx source will have flux
measurements sampled at a unique set of sub-channel positions which roughly
Nyquist sample the spectral response function (see §4). This additional
complexity comes with an opportunity. In this section we demonstrate that by
modeling emission lines with the native flux measurements it is possible to go
beyond the naive redshift accuracy implied by the per-channel resolution.

To illustrate the potential of SPHEREx’s low-resolution spectroscopy we
simulate spectrally dithered line flux measurements consistent with the
nominal full-sky survey strategy. We focus in this work on the H$\alpha$+[NII]
complex and Paschen-$\alpha$, using line fluxes from our GAMA catalog. For
each source we simulate four measurements per channel, and assume the filters
are separated at twice the channel resolution. We consider an idealized setup
in which the continuum is perfectly subtracted and the continuum measurements
constrains the positions of emission lines with a redshift accuracy $\lesssim
10\%$, i.e., the position of each lines are known to within a few SPHEREx
channels. Within this range we assume a uniform prior over redshift.

We employ a $\chi^{2}$ minimization to fit the flux measurements from one or
several lines, evaluated over a grid of redshifts using Gaussian line
profiles. To model the H$\alpha$+[NII] complex we use a fixed prior on the
line ratio [NII]/H$\alpha$$=0.35$, which is informed by the distribution of
detectable lines at full-sky depth. At each redshift, we marginalize over the
amplitude of the line(s) (denoted $\\{A_{i}\\}$) to obtain the conditional
maximum _a posteriori_ (MAP) estimate:

$\displaystyle\\{\hat{A_{i}}\\}^{MAP}$ $\displaystyle=\max\left[\ln
p(\vec{F}|\\{{A_{i}}\\})+\ln p(\\{{A_{i}}\\}|z)\right]$ (10)
$\displaystyle\approx\max\left[\ln p(\vec{F}|\\{{A_{i}}\\})\right].$ (11)

We do not impose a prior on the line amplitudes (beyond a fixed
H$\alpha$/[NII] line ratio), i.e., we do not enforce positive solutions for
the line amplitudes. Once the model is evaluated over the pre-determined
redshift range we compute the global MAP estimate and 68% credible interval of
the 1D redshift PDF $p(z|\vec{F})$.

Figure 26 shows the line fitting results for three $z\sim 0.2$ galaxies with
varying levels of detectable H$\alpha$ and Paschen-$\alpha$ emission. The
simulated flux measurements use the nominal 102 channel filters with central
wavelengths spaced at twice the channel resolution, which approximates Nyquist
sampling of the spectral response function. In reality, the observations will
have more dispersion in filter locations that depend on the sub-channel
(pixel) positions of the sources and the overall survey strategy. The redshift
estimates become more precise as the total line SNR increases, with
uncertainties ranging from $\sigma_{z}=0.01$ for our faint example down to
$\sigma_{z}=0.0015$ for the brightest example.

Figure 26: Line fits and redshift PDFs for simulated SPHEREx measurements of
three isolated, low-redshift galaxies. The left column shows the derived line
redshift PDFs – the green and red dashed lines indicate the input and
recovered redshifts for each case, while the solid red lines bound the 68%
credible interval of each PDF. The middle and right columns show the synthetic
fluxes (blue) and best-fit model photometry (red) for measurements near
H$\alpha$ and Paschen-$\alpha$, respectively.

Figure 27 shows redshift errors plotted as a function of line flux for one-
and two-line fits. Our redshift estimates are unbiased with redshift errors
that decrease from $\delta z/(1+z)\sim 0.01$ down to sub-percent precision for
brighter lines. In each case there is a flux limit below which the redshift
errors effectively revert to the original redshift range considered, i.e., the
lines are uninformative. These thresholds roughly correspond to the
2$\sigma$-3$\sigma$ line flux sensitivities at 0.8 and 2.0$\mu$m. Despite
SPHEREx’s coarse spectral resolution, in our tests we found that fitting a
single Gaussian to the H$\alpha$+[NII] complex resulted in a mild bias $\delta
z/(1+z)=-0.001$. This motivates the use of line models that account for the
full line complex, as done for H$\alpha$+[NII] in this work, and will be
relevant for other cases such as the [OIII]+H$\beta$ complex.

To prevent spurious effects of overfitting we calculate the improvement in
$\chi^{2}$ from the best line model relative to the null model case (i.e., no
lines) as a test statistic to place an example cut on the sources with line
fits. Assuming a likelihood ratio $\Lambda$ and invoking Wilks’ theorem with
$n_{p}=2$ and 3 model parameters for the one- and two-line cases,
respectively, the test statistic $-2\ln\Lambda$ should be
$\chi^{2}$-distributed with $n_{p}$ degrees of freedom. We identify the subset
of lines with best fit models above the 95th percentile of their expected
$\chi^{2}$ distributions (plotted in red), while those below are plotted in
blue. These criteria are flexible in the sense that we can specify the desired
likelihood ratio threshold, however in general the cut is effective at
separating line fits with high accuracy from those unconstrained by the
photometry.

Table 3 summarizes the redshift errors as a function of H$\alpha$ and
Paschen-$\alpha$ line fluxes after making cuts on $\Delta\chi^{2}$. As seen by
eye in Fig. 27, the errors decrease monotonically with increasing line flux.
The single-line redshift errors for H$\alpha$ and Paschen-$\alpha$ are of
similar size at fixed line SNR. When H$\alpha$ and Paschen-$\alpha$ are fit
together, the dispersion of redshift errors is $\sim 30\%$ smaller than that
from fitting H$\alpha$ alone to the same set of sources. Broadly speaking,
these results demonstrate that when one or several lines are detectable and
correctly identified, it is possible to recover highly accurate redshifts.

Figure 27: Fractional redshift errors as a function of line flux, for different one- and two-line fits. The red and blue points indicate galaxies with and without “high-quality fits”, respectively, where high-quality fits have a sufficient improvement in the delta log-likelihood compared to a model with no lines ($\Delta\chi^{2}>6$ for single line (complex) fits, $\Delta\chi^{2}>7.8$ for two lines). The mean redshift uncertainty and NMAD are calculated for each high-quality fit sample. The solid and dashed black lines indicate the 2$\sigma$ and 3$\sigma$ line flux sensitivities from Fig. 20, respectively, at observed wavelengths of 0.8 $\mu$m for H$\alpha$ and 2.0 $\mu$m for Paschen-$\alpha$. We assume a fixed ratio [NII]/H$\alpha$ when fitting the native photometry, though the true line ratios in the simulated lines vary. $F_{H\alpha}$ [$\times 10^{-15}$ erg cm-2 s-1] |  | $[2,3)$ | $[3,5)$ | $[5,7)$ | $[7,10)$ | $[10,50)$ | $[50,100)$ | $[100,1000)$
---|---|---|---|---|---|---|---|---
NMAD ($H\alpha$ only) |  | 0.0048 | 0.0041 | 0.0028 | 0.0022 | 0.0011 | 0.0003 | 0.0001
NMAD ($H\alpha$+P$\alpha$) |  | 0.0049 | 0.0038 | 0.0023 | 0.0014 | 0.0005 | 0.0001 | $<0.0001$
$F_{P\alpha}$ [$\times 10^{-15}$ erg cm-2 s-1] | $[0.8,2)$ | $[2,3)$ | $[3,5)$ | $[5,7)$ | $[7,10)$ | $[10,50)$ | $[50,100)$ | $[100,1000)$
NMAD (P$\alpha$ only) | 0.0038 | 0.0027 | 0.0018 | 0.0012 | 0.0008 | 0.0004 | 0.0001 | 0.0001
NMAD ($H\alpha$+P$\alpha$) | 0.0020 | 0.0014 | 0.0010 | 0.0007 | 0.0006 | 0.0003 | 0.0001 | $<0.0001$
Table 3: Normalized median absolute deviation (NMAD) of redshift errors
$(\hat{z}-z_{in})/(1+\hat{z})$ for the GAMA sample, separated as a function of
$H\alpha$ and P$\alpha$ line fluxes. These galaxies come from our quality cuts
based on the line fits ($\Delta\chi^{2}>6$ for single lines,
$\Delta\chi^{2}>7.8$ for two lines).

Isolating the emission line measurements from the full SED fits enables
redshift estimates that are more robust to the details of the SED model,
however this technique relies on some prior estimate for the redshift and thus
the line center(s). The quality of the continuum redshift and errors in the
continuum model near the lines will primarily impact the purity of the
recovered line redshifts. Using continuum-driven line ratio priors can
potentially improve redshift estimates for many ELGs, however they may not be
appropriate in all cases. This procedure is being considered by the flight
software pipeline team to account for as-observed spectrophotometry, however a
detailed investigation of the technique and its limits are left to future
work.

###  5.3 Redshift validation

While our use of synthetic data allows us to directly quantify redshift
errors, spectroscopic data from existing surveys will be used in practice to
validate SPHEREx redshifts. This approach has been used to perform redshift
validation for existing surveys including the Dark Energy Survey (DES; Myles
et al., 2021) and near-future surveys conducted with _Euclid_ (Naidoo et al.,
2023). Such a procedure relies on a set of independent, high-resolution
spectroscopic measurements that span the spectro-photometric color space
occupied by SPHEREx galaxies, and may motivate targeted spectroscopic surveys
such as the Complete Calibration of the Color-Redshift Relation survey (C3R2;
Masters et al., 2019) to fill in observational gaps.

##  6 Conclusion

In this work we have presented a set of galaxy SEDs that combine multi-
wavelength fits to existing photometry with an emission line prescription
rooted in empirical scaling relations. After validating that the line model is
consistent with a number of existing measurements, we generate mock photometry
from the synthetic SEDs observed across the SPHEREx bandpass using existing
sensitivity estimates. We then demonstrate that precise, accurate redshifts
can be obtained using continuum and line information from simulated full-sky
depth photometry. These redshift simulations will form the basis for
downstream clustering measurements through the power spectrum and bispectrum.

Due to the SPHEREx survey strategy, the observing depth in the NEP and SEP
(100 deg2 each) will be considerably deeper than that of the full sky survey,
with measurements along each line of sight that are both spatially and
spectrally dithered, oversampling the SPHEREx response function. Given the
dearth of existing NIR spectroscopic measurements, galaxy spectra from the
deep fields will be useful for refining the galaxy templates used for the full
sky survey (the results of this work assume by construction that the templates
are representative of the SPHEREx photometry). Forecasts on emission lines
will also become more refined with observations from upcoming spectroscopic
surveys such as DESI and PFS and at higher redshift by JWST. However,
obtaining reliable spectra will rely on a proper treatment of source
confusion, which will be much more pronounced. This is not addressed in this
work as we do not directly perform forced photometry on mock observations. A
dedicated study of deep field photometry and implications for calibrating the
full sky survey are left to future work.

We show (under idealized conditions) that by directly fitting the spectrally
dithered flux measurements and assuming a mild redshift prior derived from the
galaxy continuum, it is possible to recover reliable redshifts with accuracy
approaching that of higher-resolution spectroscopic surveys. More work is
needed to implement this technique for more realistic cases that incorporate
errors in line-continuum separation, errors due to line interlopers and
confusion noise from sources along the same line of sight. Despite these
additional complexities, it should be possible to obtain accurate redshifts
when lines are detected at moderate significance. These results may also be
improved in the limit of more dithered measurements (as in the deep fields).

The synthetic catalogs from this work do not include other objects such as
stars and active galactic nuclei (AGN). As we only simulate galaxies in this
work, our redshift predictions assume that source classes are properly
separated. A similar empirical approach to this work may be used with a set of
star and/or AGN templates (e.g., Brown et al., 2019) to generate synthetic
SEDs. While not explored in this work, SPHEREx has an advantage for star-
galaxy-AGN separation because of its broad spectral coverage and intermediate
spectral resolution, however external information may be needed for certain
cases.

Recent studies have shown that the assumption of universality in the halo mass
function is a poor description for relating the pNG bias $b_{\phi}$ to linear
galaxy biases $b_{1}$ (Barreira et al., 2020). However, by relaxing the
universality assumption and choosing informed galaxy sub-samples, it may be
possible to improve constraints on $f_{NL}$ beyond those currently assumed
(Barreira & Krause, 2023; Sullivan et al., 2023). The diversity of SPHEREx
galaxy types provides opportunities to isolate samples with different linear
biases $b_{1}$ and non-Gaussian biases $b_{\phi}$ for multi-tracer analyses
using the power spectrum and potentially higher-order statistics such as the
bispectrum.

We release our full synthetic COSMOS and GAMA catalogs to the public (these
along with high-resolution SEDs are available upon request from the
corresponding author). While these simulations will form the basis for tests
of the SPHEREx redshift pipeline, the synthetic line catalogs and mock spectra
should also be useful for surveys beyond SPHEREx. Our modeling framework is
implemented in the tool CLIPonSS. The tool is flexible and can be tailored to
other use cases which require consistent modeling of line features and
continuum properties.

While more work is needed to match the realism of the full SPHEREx survey and
to soon process observed samples, we have demonstrated a redshift procedure
which is effective and meets the SPHEREx science requirements with margin,
laying the groundwork for measurements of galaxy clustering on both small and
large scales.

## acknowledgements

Part of this work was done at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration. We also acknowledge support from the SPHEREx project
under a contract from the NASA/GODDARD Space Flight Center to the California
Institute of Technology. We thank Sylvain de la Torre for providing cross-
matched versions of the zCOSMOS/3D-HST catalogs with our synthetic catalog for
direct source comparison. We also thank Sean Bryan and the SPHEREx survey team
for providing the deep field coverage maps shown in this work. More
information on the COSMOS survey is available at
https://cosmos.astro.caltech.edu. GAMA is a joint European-Australasian
project based around a spectroscopic campaign using the Anglo-Australian
Telescope. The GAMA input catalogue is based on data taken from the Sloan
Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary
imaging of the GAMA regions is being obtained by a number of in-dependent
survey programmes including GALEX MIS, VST KiDS, VISTA VIKING, WISE,
HerschelATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded
by the STFC (UK), the ARC (Australia), the AAO, and the participating
institutions. The GAMA website is http://www.gama-survey.org/.

##  Appendix A SED fits of GAMA sources and redshift recovery

In this section we include example SED fits to multiband photometry of GAMA
galaxies described in §2.1.2. Each galaxy in the catalog we use has
spectroscopic redshifts and cross-matched photometry from GALEX (FUV/NUV), the
VST Kilo-Degree Survey (KiDS, $ugri$), the near-infrared VISTA VIKING survey
(ZYJH$K_{s}$) and WISE all-sky infrared data (W1 and W2) (Bellstedt et al.,
2020). A sample of our fits is shown in Fig. 28.

Figure 28: Six example fits to 12-band photometry of GAMA sources with $i<18$
(blue points), with best fit SEDs shown in grey. These sources will be
measured by SPHEREx at high SNR across many channels.

The synthetic GAMA catalog represents a bright, low-redshift population for
which SPHEREx will measure many precise redshifts. We perform template fitting
on synthetic SPHEREx 102-band continuua for ten thousand galaxies of the 44K
GAMA sample, also including WISE W1/W2 and DECaLS $grz$ photometry. To account
for photometric errors of bright sources we include a 1% uncertainty floor on
the external photometry. In addition, given the GAMA sample is comprised of
low-redshift galaxies, we deploy a finer grid from $z=0$ to $z=1$ with $\delta
z=0.001$. We show the results of this test in Fig. 29. The majority of sources
from this sample (85%) have redshift uncertainties $\sigma_{z/(1+z)}<0.003$,
contributing an additional $\bar{n}=170$ deg-2 to our high accuracy sample.
This sample has a similar NMAD to our COSMOS sample, with a slightly lower
outlier fraction to our COSMOS sample but a considerably smaller NMAD, which
is expected since this comprises the bright end distribution of the SPHEREx
galaxy sample. There is a small $-0.3\sigma$ bias in the sample which warrants
further investigation.

Figure 29: GAMA bright source catalog redshift results. The redshift error
distribution is shown on the left, with the black dashed curve following a
Gaussian whose width matches $\bar{\sigma}_{z/(1+z)}$. The right panel shows
the distribution of reported uncertainties for the same sample.

##  Appendix B Validation of redshift estimates using z-scores

To test the reliability of photometric redshift uncertainties we calculate the
z-score distribution for our redshift results. The z-score for an individual
estimate is given by $Z=(\hat{z}-z_{in})/\hat{\sigma}_{z}$ (with capital $Z$
indicating a z-score, not a redshift) and should be unit Gaussian-distributed
if the uncertainties statistically match the true errors. In Figure 30 we show
the z-score distributions for the nominal COSMOS2020 results in §5.1.1,
grouped in the same redshift uncertainty bins. This distribution is
qualitatively similar in shape to the redshift error distributions, with a
negative tail of outliers for the medium- and low-accuracy samples. For the
first three uncertainty bins, the distribution of z-scores indicates
consistency between reported uncertainties and true errors within 5%. For the
two lowest accuracy samples, the widths of the z-score distributions suggest
that the reported uncertainties are $\sim 10\%$ overconfident, though this may
also be driven by the larger mean biases and outlier tails.

Figure 30: Distribution of z-scores for COSMOS2020 continuum redshift
estimates, binned in redshift uncertainty $\hat{\sigma}_{z/(1+z)}$. The black
dashed curves indicate a Gaussian with unit variance, i.e., the desired
z-score distribution. The $(\mu,\sigma)$ labels indicate the mean and standard
deviation of the z-score histograms.

Stricter tests, such as those utilizing the probability integral transform
(PIT), can be used to test the reliability of the full $p(z)$ distribution,
for example near the tails of the distribution. These metrics will be
important for assessing the reliable $p(z)$ information that gets passed
downstream to clustering measurements.

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