# High-Precision Redshifts for Type Ia Supernovae with the _Nancy Grace Roman
Space Telescope_ P127 Prism

Bhavin A. Joshi Corresponding Author<EMAIL_ADDRESS>Department of Physics
and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA  Louis-
Gregory Strolger Department of Physics and Astronomy, Johns Hopkins
University, Baltimore, MD 21218, USA  Space Telescope Science Institute, 3700
San Martin Drive, Baltimore, MD 21218, USA  Russell E. Ryan, Jr  Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Alexei V. Filippenko Department of Astronomy, University of California,
Berkeley, CA 94720-3411, USA  Rebekah Hounsell University of Maryland,
Baltimore County, Baltimore, MD 21250, USA  NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA  Patrick L. Kelly  School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA  Richard Kessler  Kavli
Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637,
USA  Phillip Macias  Department of Astronomy and Astrophysics, University of
California, Santa Cruz, CA 95064, USA  Benjamin Rose Department of Physics,
Duke University, 120 Science Drive, Durham, NC, 27708, USA  Daniel Scolnic
Department of Physics, Duke University, 120 Science Drive, Durham, NC, 27708,
USA

(Received 2022)

###### Abstract

We present results from simulating slitless spectroscopic observations with
the Nancy Grace Roman Space Telescope’s (Roman) Wide-Field Instrument (WFI)
P127 prism spanning 0.75 $\mu m$ to 1.8 $\mu m$. We quantify the efficiency of
recovered Type Ia supernovae (SNe Ia) redshifts, as a function of P127 prism
exposure time, to guide planning for future observing programs with the
_Roman_ prism. Generating the two-dimensional dispersed images and extracting
one-dimensional spectra is done with the slitless spectroscopy package
pyLINEAR along with custom-written software. From the analysis of 1698
simulated SN Ia P127 prism spectra, we show the efficiency of recovering SN
redshifts to $z\lesssim 3.0$, highlighting the exceptional sensitivity of the
_Roman_ P127 prism. Redshift recovery is assessed by setting a requirement of
$\sigma_{z}=(\left|z-z_{\mathrm{true}}\right|)/(1+z)\leq 0.01$. We find that 3
hr exposures are sufficient for meeting this requirement, for $\gtrsim 50\%$
of the sample of mock SNe Ia at $z\approx 2$ and within $\pm 5$ days of rest-
frame maximum light in the optical. We also show that a 1 hr integration of
_Roman_ can achieve the same precision in completeness to a depth of $24.4\pm
0.06$ AB mag (or $z\lesssim 1$). Implications for cosmological studies with
_Roman_ P127 prism spectra of SNe Ia are also discussed.

##  1 Introduction

The Nancy Grace Roman Space Telescope (Roman) will, through its core community
High Latitude Time Domain Survey (HLTDS), observe on the order of $10^{4}$
Type Ia supernovae (SNe Ia; e.g., Hounsell et al., 2018; Akeson et al., 2019).
Most of these SNe will be within redshifts $0.5<z<2$, and are required to be
observed with sufficient depth per epoch to enable accurate measurements of
dark energy properties that significantly improve the dark energy figure of
merit (FoM; Spergel et al., 2015; Hounsell et al., 2018). To accomplish these
goals, the two Supernova Science Investigation Teams (SITs) have provided a
jointly recommended reference survey in Rose et al. (2021).

An important assumption in Rose et al. (2021) is that the redshifts for all
SNe Ia (or host galaxies) are well measured, i.e., on the order of a tenth of
a percent, and it is assumed that _Roman_ spectroscopic elements will be used
as a part of the Roman HTLDS strategy to obtain those redshifts. The _Roman_
Wide-Field Instrument (WFI), with a field-of-view of 0.281 deg2, is expected
to have two slitless-spectroscopy spectral elements: a low-resolution prism
(P127; $R\approx 70$–170; $0.75\lesssim\lambda\,[\mu m]\lesssim 1.8$) and a
slightly higher-resolution grism (G150; $R\approx 435$–865;
$1.0\lesssim\lambda\,[\mu m]\lesssim 1.93$).111
https://roman.gsfc.nasa.gov/science/WFI_technical.html provides more complete
details on the current _Roman_ WFI design. Compared to the grism, the _Roman_
prism, while lower resolution, is $\sim 3$ mag more sensitive than the _Roman_
grism for a 1 hr exposure making it better suited for analysis of fainter SNe
with broad absorption features (Table 1).

As part of the SIT effort, the primary objective of this work is to evaluate
the utility of the prism spectra, particularly the influence of the exposure
time on the signal-to-noise ratio (S/N) and redshift accuracy. These results
are needed to optimize the spectroscopic component of the HLTDS and future
observations with the _Roman_ prism. Current estimates for the HLTDS design
involve some (as yet undecided) combination of prism-to-imaging time, along
with coaddition of spectra taken from multiple visits, as well as time series
analysis (Saunders et al., 2018). Based on estimates of current _Roman_
WFI/P127 capabilities, we provide here a base platform from which future
considerations of prism exposures for the HLTDS can continue, while also
giving a broader context for other programs that involve the _Roman_ prism. We
analyze the redshift recovery for a single long-exposure visit (which can
still include multiple position angles) where all data for a given SN Ia are
taken such that spectral or luminosity variations can be neglected (over the
timescale of the visit).

With simulated broadband filter images of a single _Roman_ HLTDS pointing
(hereafter referred to as direct images) in the WFI/F106 filter as input (Wang
et al., 2022), our approach involves simulating two-dimensional (2D)
spectroscopic images (dispersed images) through the _Roman_ prism, and
analyzing the extracted one-dimensional (1D) spectra, as will be done for real
_Roman_ data. This paper focuses primarily on the recovery of SN Ia redshifts,
although our fitting pipeline also attempts to constrain SN Ia phase (relative
to peak) and dust extinction along the line of sight. The simulation products
and the associated codes will be publicly available at the IPAC-hosted
simulations page222https://roman.ipac.caltech.edu/sims/Simulations_csv.html
and GitHub333https://github.com/bajoshi/roman-slitless, respectively. Future
papers will focus on recovery of (a) other SN Ia properties (e.g., Si II
velocities) while also including coadded shorter exposures from multiple
visits at different phases such that the SN brightness and spectral evolution
are measured, (b) deblending of SN Ia and host galaxy spectra, and (c)
(host-)galaxy stellar-population parameters from their prism spectra.

This work is structured as follows: § 2 goes into details of the simulation
and analysis methods, § 2.1 discusses the models used for SNe Ia, galaxies,
and stars; § 2.2 describes how host galaxy light and SN light are blended for
SNe that overlap with their host galaxy in our simulation; § 2.3 and § 3.1
provide an overview of the pyLINEAR package used to simulate the 2D dispersed
images and extract the 1D spectra; and § 3.2 describes the fitting process
applied to the extracted 1D spectra. Finally, § 4 discusses the results
obtained from our fitting pipeline, particularly with regards to the
efficiency of recovering redshifts. The cosmological model assumed in this
paper is flat with $\Omega_{\mathrm{M}}=1-\Omega_{\Lambda}=0.3$ and
H${}_{0}=70$ km s-1 Mpc-1, and all magnitudes are in the AB system (Oke &
Gunn, 1983).

##  2 Simulation and Analysis Methods

Redshifts and distance moduli are the two independently-measured quantities
for SNe Ia that allow astronomers to constrain cosmological models. Our goal
here is to analyze the accuracy of redshifts recovered from simulated 1D P127
spectra of SNe Ia, while keeping the simulation as realistic as possible. A
schematic of the simulation pipeline, as well as the inputs and products, is
shown in Figure 1.

Figure 1: Schematic representation of the steps in our simulation pipeline. A
short description of the details in each step is also given. Each box also
provides, if applicable, a section/table number where it is described in
detail in the text.

Our process consists of the following steps which includes an end-to-end
simulation (steps i and ii) and analysis of the simulation products (steps iii
and iv):

  1. (i)

insert SNe as point sources into simulated direct images and assign SNe and
(host-)galaxies spectra (§ 2.1 and § 2.2),

  2. (ii)

generate dispersed images at required roll angles and exposure times for the
_Roman_ WFI/P127 prism, (§ 2.3)

  3. (iii)

extract 1D slitless spectra of all detected sources (§ 3.1), and

  4. (iv)

recovery of SNe Ia and galaxy stellar population parameters through a Markov
Chain Monte Carlo (MCMC; § 3.2) fitting of the “observed” slitless
spectroscopy.

The simulation is based on one pointing, of all 18 WFI detectors, generated at
three roll angles with total exposure times of 1200, 3600, and 10,800 s. The
roll angles we use are 0∘, 70∘, and 140∘, measured east of north. The exposure
times are built up by adding repeated observations assuming that the longest
individual exposure is limited to 900 seconds. Given that detector-specific
details are not yet known for Roman, our code runs the simulation on a single
detector and repeats the simulation 18 times with a different scene on the sky
each time (where “scene” is defined by the shapes, sizes, locations, and
brightness for all objects within the field-of-view). We also limit the number
of randomly-inserted SNe into each detector to a range of $90\leq N\leq 100$
(i.e., $<5$% of total sources which are typically $\sim 2000$), so as to not
artificially introduce excessive spectral overlap. In total, our simulation
contains 1698 SNe Ia.

Figure 2: Estimate of the sensitivity of Roman P127 prism, along with the HST
Wide Field Camera 3 IR grisms G102 and G141, in the +1 order. Table 1:
Comparison of HST and _Roman_ slitless elements

Optics/Detector characteristic | HST WFC3/IR grisms | Roman WFI
---|---|---
| G102 | G141 | P127 | G150
Dark current [$e^{-}$/pix/s] | 0.05 | 0.05 | 0.0015 | 0.0015
Read noise [$e^{-}$ rms] | $\sim 12$ | $\sim 12$ | $\sim 8$ | $\sim 8$
Zodiacal background [$e^{-}$/pix/s] aaThe zodiacal light estimates for the HST grisms vary primarily with Sun angle and time of year (Pirzkal & Ryan, 2020). We therefore use median values for the distribution of zodiacal light levels for the HST grisms. Assumes a 1.1$\times$ factor for zodiacal light for _Roman_ (private communication, Jeffrey Kruk) which is expected to depend on Galactic latitude. | 0.48 | 1.04 | 1.047 | 1.047
Thermal background [$e^{-}$/pix/s] | 0.02 | 0.02 | 0.064 | 0.064
1 hr limiting magbbFor the HST grisms, limiting magnitude is defined as 5$\sigma$ per continuum pixel in the $J$ and $H$ bands for G102 and G141, respectively (Kuntschner et al., 2010). For the _Roman_ prism and grism, limiting magnitude is defined as $10\sigma$ per 2 pixel resolution element in the continuum, at 1.2 $\mu$m, for a point source. | 22.6 | 22.9 | 23.5 | 20.5
Resolution [$\Delta\lambda/\lambda$] | $\sim 210$ | $\sim 130$ | 70–170 | 435–865
Wavelength coverage [nm] | 800–1150 | 1075–1700 | 750–1800 | 1000–1930

Note. — A comparison of relevant properties of the HST WFC3 IR grisms and the
_Roman_ prism and grism, some of which factor into the increased sensitivity
of the _Roman_ prism. See
https://roman.gsfc.nasa.gov/science/WFI_technical.html and
https://wfirst.ipac.caltech.edu/sims/Param_db.html, for details on WFI
detector performance. See
https://www.stsci.edu/hst/instrumentation/wfc3/documentation/grism-resources
for details on the HST WFC3 grisms. Since the grisms have multiple orders,
relevant quantities are quoted here for the +1 order.

Simulating the photoelectron count rate for a given source observed through
the _Roman_ prism requires use of the sensitivity curve (Figure 2) along with
(i) the counts for each pixel associated with the source in the direct image,
(ii) a description of the spectral trace for the _Roman_ prism, (iii) a
spectrum associated with each source pixel, and (iv) the noise characteristics
of the instrumentation (see noise budget components in Table 1). The spectral
trace is defined as the location of the centroid of the spectrum along the
spatial direction (see e.g., Kümmel et al., 2009; Pirzkal et al., 2016). Table
1 shows a comparison of properties between the Hubble Space Telescope (HST)
Wide Field Camera 3 (WFC3) infrared (IR) grisms and the _Roman_ prism and
grism. Table 1 also shows the noise terms included in our simulation. However,
it does not include IR detector-specific effects, such as flux-dependent
sensitivity and interpixel capacitance, which we expect to have a smaller
effect on the prism spectra compared to the terms already included. This table
also shows that the primary reason for preferring the _Roman_ prism over the
grism for observing SNe Ia is the $\sim 3$ mag deeper sensitivity of the
prism, which is particularly helpful for estimating redshifts to fainter SNe
Ia given the broad absorption features typical of SN Ia spectra. Another
significant reason for the increased sensitivity of the _Roman_ prism is the
lack of dispersion orders. Since the grisms disperse light into multiple
orders and the prism disperses light into a single order, the grisms lose a
significant fraction of the available light; typically only the +1 order
spectrum from a grism is analyzed given that gratings are blazed to maximize
throughput to the +1 order.

Figure 2 shows a comparison between the sensitivities of the _Roman_ prism and
HST WFC3 IR grisms G102 and G141. According to current best estimates of
sensitivity, the _Roman_ prism is a factor of $\gtrsim 20$ more sensitive than
HST/WFC3 IR grisms at all wavelengths covered by the HST grisms. The prism’s
performance is expected to result in $10\sigma$ per pixel in a 1 hr exposure
of a point source with $m_{F106}$ = 23.5 mag
AB444https://roman.gsfc.nasa.gov/instruments_and_capabilities.html (although
we recover $\sim 6\sigma$ per pixel from analyzing our simulations).

###  2.1 Stellar Population and SALT2 Models

Prior to dispersing the simulated direct images we assign each source in the
direct images a model spectrum. Briefly, the simulated direct images contain
stars and galaxies and while they also contain SNe Ia, the rates employed
result in too few SNe for our analysis (for more details we refer the reader
to Wang et al. 2022 and references therein).

Therefore, we begin by overlaying additional SNe into the direct images as
point sources. Typical SNe Ia simulations begin by selecting a random redshift
from a rate model, and then computing SN Ia model magnitudes at each phase. To
simulate an appropriate magnitude range, we simulate with a different approach
in which random magnitudes are chosen in the range $22.4\leq
m_{F106}\,[\mathrm{AB\ mag}]\leq 28.7$, and the corresponding redshift is
determined for each magnitude (corresponding to SNe Ia approximately within
$0.5\lesssim z\lesssim 3.0$). We overlay the SNe into the direct images with a
synthetic apparent magnitude chosen first instead of choosing a redshift first
and then letting the apparent magnitude be decided by the cosmology, SN Ia
phase, and line of sight dust attenuation. This approach is motivated by the
need to acquire a large enough sample of fainter SNe, because we expect the
accuracy of inferred redshifts to deteriorate for fainter SNe, while still
completing the simulations within a realistic time-frame, owing to the
computationally expensive nature of the simulations. Choosing the apparent
magnitude first allows us to have a well-known magnitude distribution for the
overlaid SNe Ia. For the SN Ia model template, we employ an extension of the
SALT2 SN Ia model that extends into the near-IR (Guy et al., 2007; Pierel et
al., 2018). This model provides rest-frame SN Ia spectra from 1700 Å to 2.5
$\mu$m, and with phases $-19$ days to $+50$ days relative to optical peak.

The probability distribution function (PDF) for the magnitude distribution
follows a power law ($P(x,a)=ax^{a-1}$ with $a=1.2$), ensuring a distribution
of SN magnitudes that allows for a slightly larger number of fainter SNe for
analysis. To obtain a statistically significant sample of SNe Ia for analysis
we insert a relatively large number of SNe into each detector (as previously
mentioned $\sim 95$ SNe per detector). While this rate model is not consistent
with the number density of SNe that will be observed in a single visit, our
decision is again driven by the need to obtain a large number of SN prism
spectra for analysis. Each SN is assigned a spectrum with a randomly chosen
phase within $\pm 5$ days from peak. Next, we apply dust extinction $A_{V}$
following the Calzetti et al. (2000) law, chosen from $0.0\leq A_{V}\leq 3.0$
mag, with an exponentially declining PDF weighted in favor of lower $A_{V}$
values ($P(x,\lambda)=\lambda\,e^{-\lambda x}$ with $\lambda=2$), in order to
simulate realistic dust extinction from local environments of SNe, following
Jha et al. (1999) and Holwerda et al. (2015). Finally, we apply cosmological
redshift to the inserted SN spectrum by redshifting the wavelength grid and
scaling the spectrum flux so that the synthetic magnitude matches the already
assigned SN magnitude.

For (host-) galaxies we use the Bruzual & Charlot (2003) GALAXEV library of
stellar-population models. Each galaxy is assigned a composite stellar
population with a delayed-$\tau$ star-formation-rate (SFR) model (SFR
$\propto\,t\,\mathrm{exp}(-t/\tau)$) for the star-formation history, along
with a stellar mass of $9.0\leq\mathrm{log(M_{s}\,[M_{\odot}])}\leq 12.0$, a
redshift of $0<z\leq 3.0$ (if it is an SN host galaxy, then the SN and host
are assigned the same redshift), and finally a dust extinction with $A_{V}$
randomly chosen from $0.0\leq A_{V}\leq 5.0$ mag. For Galactic stellar point
sources in the images we assign spectra from the Pickles stellar library, that
provide stellar spectra from 1150 Å to 25,000 Å (Pickles, 1998). The Pickles
library provides 131 flux-calibrated stellar templates for stellar types
within the range O5–M2, with the stellar type being chosen randomly in our
simulations.

###  2.2 Blending of host galaxy and Supernova Light

A significant advantage of slitless spectroscopy is the ability to obtain a
spectrum for all objects within the field-of-view (FoV). This also introduces
significant data-reduction challenges, particularly with disentangling spectra
from objects that are very close to each other and/or lie along the dispersion
direction such that their spectra overlap. In traditional slitless
spectroscopy parlance, the term contamination usually refers to the unwanted
overlap of one or more spectra on the spectrum of the object of interest.
Generally this effect can be mitigated by taking data at multiple roll angles
so that there is at least one roll angle within which the spectrum of interest
contains little to no contamination.

Figure 3: Examples of SNe Ia prism spectra that are blended with host galaxy
spectra. The input high-resolution template spectra are in blue for the SN and
red for the host galaxy. Extracted SN spectra are in black with gray errors.
Host galaxy and SN redshifts, as well as F106 magnitudes, are shown in the
legend. From top to bottom the extent of blending increases. Panel (a) shows
an example of mild blending — SN Ia features are easily identifiable and host
galaxy features are largely absent in the extracted SN spectrum. Panels (b)
and (c) show moderate blending — both SN Ia and host galaxy features can be
identified in the SN spectrum. Panel (d) shows extreme blending between SN and
host galaxy — SN Ia features are largely absent and host galaxy features
dominate in the SN spectrum. This “classification” of the extent of blending
(as mild, moderate, or extreme) is qualitative — we do not attempt here to
quantify the extent of the blending.

In the specific case of slitless spectroscopic observations of an SN Ia and
its host galaxy, an additional concern is the blending of light from the host
galaxy and the SN — that is, within pixels that overlap between the SN and its
host galaxy, the total flux (and therefore the spectrum from that pixel) is a
combination of host and SN. This introduces a minimum floor of contamination
(referred to as blending below to distinguish from the traditional notion of
contamination) that will be present in all observations of the SN Ia
regardless of the roll angle. This situation is analogous to that of taking
slitless spectra of a galaxy containing one or more star-forming knots — in
this case, the underlying galaxy spectrum is blended with the emission-line
spectrum of the star-forming region.

In order to represent this effect within our simulations, for each SN source
that has an overlap between the host galaxy and SN we carry out the following
steps.
(1) For each overlapping pixel $(x,y)$, the effective spectrum from the pixel
is given by

$\mathcal{O}=a_{\rm H}\,f^{\rm H}_{\lambda}+a_{\rm SN}\,f^{\rm SN}_{\lambda},$
(1)

where $a_{\rm H}$ and $a_{\rm SN}$ denote linear combination coefficients, and
$f^{\rm H}_{\lambda}$ and $f^{\rm SN}_{\lambda}$ are the spectra of the host
galaxy and SN, respectively. The coefficients $a_{\rm H}$ and $a_{\rm SN}$ are
fractions of the total counts within a pixel that belong to the host galaxy
and SN, respectively. We also make the simplifying assumption (for all
sources) that the input spectrum remains constant over the spatial extent of
the source.
(2) The final spectrum of the SN containing overlap with the host is given by
a weighted sum of the spectra from overlapping pixels (i.e., blended SN and
host spectra) and non-overlapping pixels (i.e., pure SN Ia spectra):

$\mathcal{S}=\Sigma_{i}a_{i}\mathcal{O}_{i}+\Sigma_{j}b_{j}\mathcal{N}_{j}\,,$
(2)

where $\mathcal{S}$ is the final blended spectrum of the SN and
($\mathcal{N}$)$\mathcal{O}$ denote (non-)overlapping pixels. The indices, i
and j, sum over SN pixels that overlap and do not overlap with the host
galaxy, respectively. The weights $a_{i}$ and $b_{i}$ are computed to ensure
that the relative flux contribution from each pixel toward the “effective”
source count, which includes both host galaxy and SN counts, is reflected in
the final spectrum. This final blended spectrum becomes the effective 1D
spectral template for the SN, i.e., each pixel associated with the SN, in the
segmentation map, is assigned the spectrum $\mathcal{S}$. The effective 1D
spectral template is then passed to pyLINEAR for simulating 2D dispersed
images as described in § 2.3.

Figure 3 shows examples of blended SN Ia and host galaxy spectra. Extracted
spectra (the extraction process is described in § 3.1) are effectively the
weighted sum (as described above) of the SN and host input templates. The
extent of blending depends on the number of pixels that overlap between the SN
and host galaxy, the exposure time, and their brightness relative to each
other. Without any attempt to deblend the host galaxy spectrum from the SN
spectrum, our automated fitting pipeline (described in § 3.2) fails to recover
the SN redshift in each of the cases shown here.

Figure 4: Dispersed prism image cutouts and extracted spectra for an example
pair of SN and its host galaxy. The set of panels in (a) show a portion of the
direct image (for a single detector, $\approx$ 600 $\times$ 450 pixels) and
corresponding dispersed images at our three position angles. The dispersed
images also highlight the regions used by pyLINEAR to extract the SN and host
galaxy spectra. Panel (b) shows the extracted SN spectrum along with the input
templates for the SN and host galaxy. The exposure time for the 2D dispersed
images and the 1D extracted spectra shown is 1200 seconds. For this particular
example the pixels between the SN and host galaxy do not overlap within the
segmentation map.

###  2.3 Simulation of 2D dispersed images

Here we provide a brief overview of the process to simulate 2D dispersed
images carried out by pyLINEAR. For a detailed explanation of the pyLINEAR
algorithm, the reader is referred to Ryan et al. (2018a, b).

The driving motivations for the algorithm behind pyLINEAR are to employ
dispersed data from multiple roll angles to optimally handle spectral
contamination due to overlapping spectra, and also to extract spectra with the
optimal spectral resolution without sacrificing the S/N. pyLINEAR simulates 2D
dispersed images by dispersing 1D high-resolution model spectral templates for
each pixel associated with each source in the direct image. It requires a
description of the scene on the sky and a description of the spectral trace
and sensitivity of the dispersing element being considered.

We modify pyLINEAR defaults to allow the simulation of prism exposures in
which the dispersion is strongly dependent on wavelength. Briefly, traditional
slitless spectroscopy routines (e.g., aXe, Kümmel et al. 2009; pyLINEAR)
describe the spectral trace with a polynomial that is parameterized by the
path length along the trace. For describing the spectral trace of the Roman
prism, we build on the foundation of the generalized 2D polynomials introduced
by Pirzkal & Ryan (2017). These polynomials are a function of the $x$ and $y$
coordinates on the dispersed image and another parameter $t$ (which spans
$0\leq t\leq 1$ and can generally, although not always, be identified with
wavelength coverage). One of the quantities required to be computed in
simulating 2D spectra is the wavelength $\lambda=f_{\lambda}(x,y,t)$. For most
grisms, this is a first-order equation in $t$. To allow for a variable
dispersion with wavelength relation, however, we had to change this to a
second-order equation in $t$. This change is nontrivial because the polynomial
relation must also be inverted to solve for the coefficients of $t$ that give
the known dispersion vs. wavelength relation. For readers interested in
simulating _Roman_ prism spectra, the configuration file for aXe and pyLINEAR
is available from the corresponding author. Figure 4 shows an example of
simulated 2D dispersed images through the _Roman_ prism and the extracted SN
spectrum from a SN and its host galaxy.

##  3 Analysis of simulation products

###  3.1 Extraction of 1D spectra

The extraction of 1D spectra, which involves solving for the flux at each
wavelength for each source, is an overconstrained problem that has more data
values than unknown parameters — that is, the set containing observed
information (direct and dispersed images) is far larger than the set of
information to be solved for (fluxed spectra for all objects). pyLINEAR
attempts to solve for the set of spectra that simultaneously satisfy all the
available dispersed data (i.e., all roll angles and dithers). pyLINEAR defines
the system of linear equations that connect the observed data and the unknown
fluxes and recasts to a known linear-algebra problem with sparse matrices,
$\mathbf{Ax=B}$. Roughly, elements in the matrices $\mathbf{A}$ and
$\mathbf{B}$ represent the observed counts and the weighting coefficients that
transform them into the fluxed spectra, and $\mathbf{x}$ represents the
unknown vector of fluxed spectra for all objects. This amounts to finding a
minimum in the equation,

$\chi^{2}=\Sigma||\mathbf{Ax-B}||^{2}.$ (3)

However, if the matrix $\mathbf{B}$ has noise (which real data does), then the
solution to equation 3 is numerically unstable. Therefore, the actual equation
to be solved is modified to the following regularized least squares equation,

$\chi^{2}_{\mathrm{true}}=\Sigma||\mathbf{Ax-B}||^{2}+\mathcal{L}\,\Sigma||\mathbf{x}||^{2},$
(4)

where, $\mathcal{L}$ is the regularization parameter (see Ryan et al. 2018a
for more details on the regularization). pyLINEAR employs the LSQR (Paige &
Saunders, 1982) algorithm for solving the above equation with large sparse
matrices.

In contrast to aXe (Kümmel et al., 2009), which provides a separate spectrum
per object for each roll angle, while leaving the combining of spectra at
different roll angles to the user, pyLINEAR provides a single extracted 1D
spectrum per object given its algorithmic philosophy of solving for the
optimal flux vector that satisfies all the available data. Effectively, the
combining of the spectra at different roll angles is done internally within
pyLINEAR. This approach naturally handles the problem of overlapping spectra
without leaving the contamination subtraction as a separate analysis task.
Additionally, pyLINEAR has the advantage of achieving better spectral
resolution without sacrificing the S/N in the extracted spectrum.

Figure 5: The recovery of redshift from our fitting pipeline. Left column
shows the recovery results for all the SNe Ia in our simulation, whereas the
right column is restricted to only those SNe whose spectra contain no blended
light from the host galaxy. The error bars have been suppressed for clarity.
The top panels show the inferred redshift vs. the true redshift and the bottom
panels show the residuals defined by
$(z_{\mathrm{inferred}}-z_{\mathrm{true}})/(1+z_{\mathrm{true}})$. The colors
of the points denote the different exposure times as shown in the legend.

###  3.2 Parameter Recovery through Bayesian Inference

Extracted 1D spectra are fit for redshift, phase, and dust extinction. The
fitting is done with the Python package emcee (Foreman-Mackey et al., 2013,
2019), which samples the 3-dimensional (3D) posterior space for SNe models.
Our automated fitting pipeline uses 500 walkers and 1000 steps. The walkers
use the default “stretch-move” within emcee for sampling the posterior. We
start the walkers at the optimal position indicated by the approximate global
minimum. The minima for the walkers’ starting location is found by doing a
brute force $\chi^{2}$ evaluation for each model on a coarse grid of redshift
($\Delta z=0.01$ within [0.01, 3.0]), phase ($\Delta\mathrm{Phase}=1$ within
[$-19$, 50]), and dust extinction ($\Delta A_{V}=0.5$ mag within [0.0, 3.0]
mag). We employ flat priors on SN phase and dust extinction ($A_{V}$) with the
following ranges: $-19\leq\mathrm{SN\ Phase\ [Day\ rel.\ to\ peak]}\leq 50$
and $0.0\leq A_{V}\leq 5.0$. For the SN redshift we employ a broad Gaussian
prior with a mean of 1.2 and a standard deviation of 0.7. The broad Gaussian
redshift prior allows down-weighting of very low-$z$ and high-$z$ SNe (while
not rejecting them outright). Finally, we require S/N$>$3 for the extracted 1D
spectrum prior to fitting. This S/N is computed for the entire prism spectrum
using the DER-SNR algorithm of Stoehr et al. (2008).

Figure 5 shows the recovery of SN Ia redshifts for all SNe in the simulation
(left column) and for the subset without host galaxy light (right column). The
recovery plots for phases and dust attenuation are shown in the appendix
(Figure 8). As expected, longer exposure times provide the most accurate
inferred quantities. Most of the failed estimates of redshift are caused by
host galaxy light blending with SN light. As shown in Figure 5, the inferred
redshifts are systematically underestimated for SNe whose spectra contain
light from the host galaxy, given that these failures are absent in the plot
that ignores SNe which contain blended host galaxy light. The failures
generally come from three modes. First, the continuum slope of the host galaxy
spectrum, when blended with the typically fainter SN spectrum, causes a
mismatch between SNe Ia absorption features. This mismatch is driven by the
continuum slope of the host galaxy at the red end which dominates the
$\chi^{2}$ in the SED fitting. The necessity for the fitting pipeline to match
the host galaxy continuum slope (instead of the SN continuum slope) pushes the
pipeline estimate to underestimated redshifts (and also younger phases, Figure
8). Second, strong host galaxy absorption features (e.g., 4000 Å/Balmer
breaks) are misidentified by the fitting pipeline with SN absorption features.
This is shown by the clear linear feature below the 1:1 line. Lastly, the
cluster of failures at the bottom of the figure are cases where host galaxy
light dominates the extracted 1D spectrum. However, it is worth recalling that
the redshift estimates in Figure 5 come from fitting only the SNe spectra, and
that a host galaxy spectrum can always be observed after the SN light has
faded allowing for a clean subtraction (and also allowing for a redshift
estimate directly from the host galaxy spectrum in some cases).

In this work, we do not attempt to deblend host galaxy and SN light. Our
fitting pipeline proceeds on the assumption that there is no host galaxy light
blended into the SN Ia spectrum, and it fits the SN Ia spectrum without any
attempt to subtract the host galaxy spectrum. We leave this deblending to a
future work which will likely involve marginalizing over nuisance parameters
that describe the host galaxy spectrum.

Figure 6: Fitted parameter residuals vs. S/N of observed spectrum for redshift
(top), phase (middle), and AV (bottom). Shaded exponential curves enclose
$\pm$1$\sigma$ of the residuals. For clarity, only a small subset of the SNe
unaffected by blending are shown. The S/N is computed for the entire prism
spectrum using the DER-SNR algorithm of Stoehr et al. (2008). The colors are
the same as those used in Figure 5.

##  4 Redshift Efficiency Dependence on Exposure Time

In this work, we assess the utility of _Roman_ prism spectra primarily with
regard to estimating redshifts. The metric most commonly employed in the
literature is to consider statistics of the redshift residuals defined by
$\sigma_{z}=(z_{\mathrm{inferred}}-z_{\mathrm{true}})/(1+z_{\mathrm{true}})$.

Figure 6 shows the dependence of the residuals of the recovered parameters on
the S/N of the simulated spectrum for different exposure times. For clarity,
this plot only shows those SNe from our simulation which do not contain any
blending with host galaxy light. For redshift residuals (top row) from low to
high S/N, three somewhat distinct S/N regimes can be observed — S/N $<5$,
$5\leq$ S/N $\leq 15$, and S/N $>15$ — which correspond to increasing accuracy
of recovered parameters. Interestingly, while the redshift residuals and the
dust-extinction residuals do not exhibit any appreciable bias in their
measurements relative to S/N, the recovered phase shows a clear bias toward
younger phases. We do not yet know the reason for this systematic phase bias.

Figure 7 displays the efficiency of recovering redshifts as a function of SN
Ia magnitude for our entire simulated sample of 1698 SNe Ia sources. Within
each magnitude bin the figure shows the fraction of spectra that achieve a
redshift accuracy of $\leq 0.01$, measured through the absolute value of the
redshift residuals $|\sigma_{z}|$. These redshifts are recovered solely from
P127 spectroscopy of the SN Ia without any reliance on emission or absorption
features in the host galaxy spectrum. While there is an implicit assumption
here that the SN Ia classification efficiency from P127 spectra is 100% at all
redshifts, this is a simplifying assumption we make given that any GO program
on _Roman_ will likely have classified a given transient as an SN Ia from
photometric observations, prior to requesting prism spectroscopic
observations. From preliminary work (done for HLTDS observations) we know that
SN Ia classification efficiency is $\gtrsim 90$% for SNe Ia within
$0.3\lesssim z\lesssim 1.75$, $\sim 65$% within $1.75\lesssim z\lesssim 2.0$,
and $\sim 20$% for $z>2$ (Steven Rodney, private communication, using
simulated P127 spectra of SNe Ia which are fit with SNID, Blondin & Tonry
2007). Therefore, while our completeness numbers here should be considered as
upper limits, the ability to classify a P127 spectrum as belonging to an SN Ia
only really impacts observations of SNe Ia at $z>2$ or SNe encountering
extreme dust extinction.

The longest exposure time of 3 hr in our simulated observations achieves 50%
efficiency for sources as faint as $\sim$25.5, and has much higher efficiency
at brighter SN magnitudes and lower redshifts. Our shorter exposure times of
20 min and 1 hr achieve $>50$% completeness for SNe Ia brighter than 24 and
24.4 mag AB, respectively. Particularly for $z<0.5$ SNe Ia, this implies that
the _Roman_ P127 is quick and efficient at yielding high-S/N spectra and
accurate redshifts. Note the gradual flattening of the curves as the exposure
time is increased owing to increased numbers of fainter SN spectra being fit
successfully, while of course the spectra of the brightest SNe are always fit
successfully. For the longest exposure time considered, the gradual transition
from high to low completeness also means that the 50% completeness cutoff
(denoted by $m_{c}$ in the figure legend) is not sharp and the completeness
drops to $<25$% beyond 27.5 mag AB.

We expect a 3 hr exposure time to reliably achieve 50% redshift completeness
for all SN Ia redshifts in the range $1\leq z\leq 2$. These SNe Ia are
expected to contribute most within cosmological analyses toward improving the
dark energy FoM (Albrecht et al., 2006). Cosmological analyses usually assume
that redshift errors have negligible effect on the uncertainty budget in their
analysis (e.g., Betoule et al., 2014; Scolnic et al., 2018). We show in this
work that this assumption should remain valid with cosmological analyses done
using SN Ia redshifts inferred from _Roman_ prism spectra. We find that the
median redshift accuracy, median($|\sigma_{z}|$), is on the order of a few
tenths of a percent for any exposure time that provides S/N $>5$ (Figure 6).
How the uncertainty in redshift propagates to the dark energy FoM is part of a
separate investigation (Macias et al., in prep.).

SNe Ia for which a host galaxy cannot be identified (e.g., Gupta et al.,
2016), “hostless” SNe, are susceptible to being dropped from cosmological
analyses which employ only host galaxy observations to estimate redshifts
(e.g., Dark Energy Survey, Abbott et al. 2019; Pan-STARRS, Jones et al. 2018).
Within the framework presented in this paper, i.e., SNe Ia without host galaxy
blended light, we can recover accurate redshifts for _Roman_ prism
observations of hostless SNe (right column of Figure 5); therefore, allowing
SNe Ia to be included in cosmological analyses which would otherwise be
ignored.

While the primary goal of this work is not to optimize the ratio of P127 to
imaging time for the HLTDS, we show a promising outlook for redshift inference
from P127 spectra especially for SNe Ia at $z>1$. Rose et al. (2021) presented
several survey strategies with the ratio of prism to imaging time being one of
the key unknown variables; the primary reference survey has 25% prism time
amounting to 7.5 hr per visit split between a wide and a deep tier (900 s and
3600 s per object from the wide and deep tiers, respectively). Our 20 min and
1 hr exposure times are similar to the exposure time of a single HLTDS
pointing, and achieve $>50$% completeness for SNe Ia brighter than 24 and 24.4
mag AB, respectively. However, if spectra from multiple visits are coadded,
then the completeness will improve further. Figure 5 shows that the main
problem hindering higher levels of redshift completeness is the blending of
host galaxy and SN light causing confusion for fitting routines. In general
for any rolling survey, disentangling the SN spectrum from the host galaxy
spectrum will be somewhat easier given that the host galaxies will be
repeatedly observed, allowing a subtraction of an uncontaminated host spectrum
from a host+SN spectrum, as long as data are taken with a wide enough baseline
such that SN light has faded.

Finally, we mention three caveats for the work presented here. Firstly, while
the simulated spectra in the 2D dispersed images have the appropriate variable
dispersion vs. wavelength for the _Roman_ prism, the extracted 1D spectra do
not yet have a variable dispersion. The 1D spectra have a constant wavelength
sampling of 55 Å per pixel, corresponding to the approximately average
dispersion for the _Roman_ prism – allowing for the variable dispersion in the
1D spectra requires significant restructuring of the pyLINEAR code base and is
a work in progress. We note that Figure 5 shows that the constant wavelength
sampling for the 1D spectra does not hinder our ability to infer accurate
redshifts, due to the broad absorption features in SN Ia spectra.
Additionally, because our goal is to evaluate redshift accuracies for SNe Ia
from _Roman_ P127 spectra, we do not include other classes of transients in
our simulation (e.g., core-collapse SNe), and lastly, we do not yet
incorporate complex dust geometries in our simulation.

Figure 7: Efficiency of recovering redshifts as a function of SN magnitude in
F106 for the entire simulated prism spectroscopic sample of 1698 SNe Ia. The
left ordinate denotes what we refer to as the redshift completeness — this is
the fraction of objects within a given magnitude bin that achieve a redshift
accuracy of $\sigma_{z}=\left|\Delta z/(1+z)\right|\leq 0.01$. The top
abscissa provides a guide to the reader as to the approximate redshift for the
SN F106 magnitude; the SN redshift shown is derived under the assumption of
peak phase and no intervening dust. The background histogram in gray shows the
magnitude distribution of the simulated SNe Ia with the number of objects
indicated on the right-hand ordinate. The points joined by dashed lines show
the measured redshift completeness from our simulation (compiled from
redshifts measured for individual SNe), with colors corresponding to different
exposure times as shown in the legend. The solid lines are sigmoid function
fits to the measured redshift completeness points. The 50% redshift
completeness magnitude for a given exposure time is denoted by $m_{c}$ in the
legend.

##  5 Summary

We have reported results from the analysis of 1698 simulated _Roman_ P127
spectra of SN Ia with exposure times of 1200, 3600, and 10,800 s. The _Roman_
P127 prism is exceptionally efficient at delivering high-S/N spectra, and
therefore is particularly useful for inferring accurate redshifts based on
absorption features in the continuum. We recover limiting magnitudes, i.e.,
$>$50% completeness, of 24.0, 24.4, and 25.3, for our three exposure times
(although see preceding discussion about the steepness of completeness curves)
under the requirement of
$\sigma_{z}=(\left|z-z_{\mathrm{true}}\right|)/(1+z)\leq 0.01$. The procedure
to employ pyLINEAR for simulating and extracting spectra, particularly for the
case of the _Roman_ P127 prism, is presented. The entire set of extracted 1D
spectra, and simulated and recovered SN properties in tabular form, are
available upon request.

B.A.J. is grateful for helpful discussions with members of the _Roman_ Science
Investigation Team (SIT) for supernova cosmology, in particular with Steven
Rodney, Ryan Foley, and Kevin Wang. A.V.F. received financial support from the
Christopher R. Redlich Fund and many individual donors. R.H. acknowledges work
supported by NASA under award number 80GSFC21M0002. The following software
packages were used in the analysis presented in this work: numpy (Harris et
al., 2020), matplotlib (Hunter, 2007), scipy (Virtanen et al., 2020), astropy
(The Astropy Collaboration et al., 2018), emcee (Foreman-Mackey et al., 2013,
2019), corner (Foreman-Mackey, 2016), pyLINEAR (Ryan et al., 2018a),
SExtractor (Bertin & Arnouts, 1996). This research has made use of NASA’s
Astrophysics Data System.

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##  Appendix A Examples of Parameter Recovery

Figure 8: The recovery from our fitting pipeline for phase (top row) and dust
extinction (bottom row). Similar to Figure 5 the left column shows the
recovery results for all the SNe Ia in our simulation, whereas the right
column is restricted to only those SNe whose spectra contain no blended light
from the host galaxy. The panels show the inferred value vs. the true value on
the top and the difference between the inferred and true values on the bottom.

Figure 8 shows the recovery of phase and dust attenuation by our fitting
pipeline. We also show two representative examples of parameter recovery by
our fitting pipeline. Example 1 in Figure 9 shows the results of a successful
fit, whereas example 2 in Figure 10 shows a common failure mode due to
misidentification of absorption features within SNe Ia spectra that are caused
by the red end continuum slope of the host galaxy (§ 3.2). Similar to the
examples of blending shown in Figure 3 the fitting fails for the extracted
spectrum in Figure 10 due to the $\chi^{2}$ being dominated by the addition of
the host galaxy continuum slope, even though it is clear that the extracted
spectrum has the SN absorption features at the correct wavelengths as seen in
the input template.

Figure 9: An example of a successful fit to a simulated SN Ia at $z\approx
2.2$ that does not include any blended host galaxy light. Panel (a): Corner
plot for the sampled parameter space, where the red lines indicate truth
values. The values and uncertainties on top of the panels indicate the
estimates from the MCMC sampling. Panel (b): extracted spectrum in black with
gray errors and 200 randomly drawn models, in magenta, from the posterior
distribution within $\pm 1\sigma$ of the inferred estimates. The input
template for the SN is shown in blue. Panel (c): trace plot for the MCMC
walkers, where red lines again indicate truth values. From top to bottom the
sub-panels are redshift, phase, and $A_{V}$. The exposure time is 1 hr for the
extracted 1D spectrum. Figure 10: Same as Figure 9 but for an unsuccessful fit
due to blended host galaxy light. The extracted 1D spectrum for the host
galaxy is also shown (red line) as a reference for the spectrum that is
blended with the SN input template. The true values are
$z_{\mathrm{true}}=0.5288$, $\mathrm{phase_{true}}=3$ days, and
$A^{\mathrm{true}}_{V}=0.02$ mag. The inferred values are $z=0.038$,
$\mathrm{phase}=3.91$ days, and $A_{V}=3.57$ mag (also indicated on the corner
plot).