# Synergies between Vera C. Rubin Observatory, Nancy Grace Roman Space
Telescope, and Euclid Mission: Constraining Dark Energy with Type Ia
Supernovae

B. M. Rose Department of Physics, Duke University, 120 Science Drive, Durham,
NC, 27708, USA<EMAIL_ADDRESS>G. Aldering  Physics Division, E.O.
Lawrence Berkeley National Laboratory, 1 Cyclotron Rd., Berkeley, CA, 94720,
USA  M. Dai Department of Physics and Astronomy, Johns Hopkins University,
Baltimore, MD 21218, USA  S. Deustua Space Telescope Science Institute, 3700
San Martin Drive Baltimore, MD 21218  R. J. Foley Department of Astronomy and
Astrophysics, University of California, Santa Cruz, CA 95064, USA  E. Gangler
Laboratoire de Physique de Clermont, IN2P3/CNRS, Université Clermont Auvergne,
F-63000 Clermont-Ferrand, France  Ph. Gris Laboratoire de Physique de
Clermont, IN2P3/CNRS, Université Clermont Auvergne, F-63000 Clermont-Ferrand,
France  I. M. Hook Department of Physics, Lancaster University, Lancaster, LA1
4YB, U.K.  R. Kessler  Kavli Institute for Cosmological Physics, University of
Chicago, Chicago, IL 60637, USA  Department of Astronomy and Astrophysics,
University of Chicago, Chicago, IL 60637, USA  G. Narayan University of
Illinois at Urbana-Champaign, 1003 W. Green St., IL 61801, USA  Centre for
Astrophysical Surveys, National Centre for Supercomputing Applications,
Urbana, IL 61801, USA  P. Nugent Physics Division, E.O. Lawrence Berkeley
National Laboratory, 1 Cyclotron Rd., Berkeley, CA, 94720, USA  Department of
Astronomy, University of California, Berkeley, CA 94720-3411, USA  S.
Perlmutter  Physics Division, E.O. Lawrence Berkeley National Laboratory, 1
Cyclotron Rd., Berkeley, CA, 94720, USA  Department of Physics, University of
California Berkeley, 366 LeConte Hall MC 7300, Berkeley, CA, 94720, USA  K. A.
Ponder SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA
94025, USA  B. Racine Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
D. Rubin Physics and Astronomy Department, University of Hawaii, Honolulu, HI
96822, USA  Physics Division, E.O. Lawrence Berkeley National Laboratory, 1
Cyclotron Rd., Berkeley, CA, 94720, USA  B. O. Sánchez Department of Physics,
Duke University, 120 Science Drive, Durham, NC, 27708, USA;
<EMAIL_ADDRESS>D. M. Scolnic Department of Physics, Duke University,
120 Science Drive, Durham, NC, 27708, USA<EMAIL_ADDRESS>W. M. Wood-
Vasey Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT
PACC). Physics and Astronomy Department, University of Pittsburgh, Pittsburgh,
PA 15260, USA  D. Brout NASA Einstein Fellow  Center for Astrophysics, Harvard
& Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  A. Cikota European
Southern Observatory, Alonso de Cordova 3107, Vitacura, Casilla 19001,
Santiago de Chile, Chile  Physics Division, E.O. Lawrence Berkeley National
Laboratory, 1 Cyclotron Road, Berkeley, CA, 94720, USA  D. Fouchez  Aix
Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France  P. M. Garnavich
University of Notre Dame, Center for Astrophysics, Notre Dame, IN 46556, USA
R. Hounsell University of Maryland, Baltimore County, Baltimore, MD 21250, USA
NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  M. Sako Department
of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street,
Philadelphia, PA 19104  C. Tao  Aix Marseille Univ, CNRS/IN2P3, CPPM,
Marseille, France  S. W. Jha Department of Physics and Astronomy, Rutgers, the
State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854,
USA  D. O. Jones NASA Einstein Fellow  Department of Astronomy and
Astrophysics, University of California, Santa Cruz, CA 95064, USA  L. Strolger
Space Telescope Science Institute, 3700 San Martin Drive Baltimore, MD 21218
H. Qu Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104, USA

(March 8, 2021)

###### Abstract

We review the needs of the supernova community for improvements in survey
coordination and data sharing that would significantly boost the constraints
on dark energy using samples of Type Ia supernovae from the Vera C. Rubin
Observatories, the Nancy Grace Roman Space Telescope, and the Euclid Mission.
We discuss improvements to both statistical and systematic precision that the
combination of observations from these experiments will enable. For example,
coordination will result in improved photometric calibration, redshift
measurements, as well as supernova distances. We also discuss what teams and
plans should be put in place now to start preparing for these combined data
sets. Specifically we request coordinated efforts in field-selection and
survey operations, photometric calibration, spectroscopic follow-up, pixel-
level processing, and computing. These efforts will benefit not only
experiments with Type Ia supernovae, but all time-domain studies, and
cosmology with multi-messenger astrophysics.

DOE/NASA Request for Information: Focus Area 3

##  1 Introduction

Type Ia Supernovae (SNe Ia) are a key probe for measuring dark energy, and
currently hold a critical place in two of the three large surveys: the Legacy
Survey of Space and Time (LSST) with the Vera C. Rubin Observatories and Nancy
Grace Roman Space Telescope. Both LSST and the Roman SN Ia surveys will be
unprecedented in their statistical sample size. The LSST optical sample at
redshift $0<z<1$ will be a factor of $300\times$ larger than the cumulative
sample of SNe Ia today and the Roman NIR sample at $0.3<z<3.0$ will discover
and measure $50\times$ more SNe Ia at $z>1$ than in our sample today and
increase the sample of NIR light-curves by a similar magnitude. While Euclid
does not have a planned SN Ia survey in its baseline plan (see here for a
proposed strategy: Astier et al. 2014), it will revisit deep fields with a
regular cadence, which can produce SN Ia detections and be combined with the
LSST sample.

The synergy between combined data sets is enormous — especially if the surveys
coordinate observations. Combining data sets allows increases in redshift
range and wavelength range, boosting the statistical precision and the
systematic control. Furthermore, as discussed in other responses to this call,
the SN Ia programs will immensely benefit from cross-survey data products to
improve calibration and photometric redshifts. Particularly due to the time-
dependent nature of SN Ia surveys, planning must be done now for survey
coordination and combined analyses. In this response to the DOE/NASA Request
for Information, we separate aspects that need decisions now versus those
where planning should begin soon to be considereded for future synergies.

For the SN Ia programs, it is particularly important that the multiple science
agencies who have issued this RFI, continue to work together because the
cosmological analyses will be built on SN Ia data that must be obtained using
large time allocations from facilities that are run by these (and other)
agencies. The quality of the results will also strongly depend on the
development of techniques and knowledge that individually may go beyond each
different agency’s traditional focus areas. To maximize the science impact of
each agency’s investment, scientists must have broad support, including for
ways that allow them to work together across agencies’ traditional boundaries.
A scientist should only need to apply to a single agency for funding to fully
participate in all aspects of the problem.

##  2 Science Enhancements

> a. What are the key dark energy science areas that will be enhanced by these
> activities? What level of scientific enhancement is expected by carrying
> them out after the datasets are public?

The LSST and Roman Space Telescope surveys will produce SN Ia samples with
significantly different, but overlapping redshift ranges. LSST will discover
an enormous ($>$$300,000$) number of SNe around $z\approx 0.3$–$0.4$ in its
wide survey, and a smaller ($>$10,000) but more cosmologically-constraining
number in deep fields out to $z\approx 0.9$–$1.0$. Roman will discover a
smaller number ($>$$10,000$) but reaching to much higher redshifts ($z>2.0$).
Combining SN Ia samples from LSST and Roman improves the dark energy figure of
merit (Albrecht et al., 2006) by up to a factor of two, over independent
analyses. In addition, there will be hundreds of strongly gravitationally
lensed SNe Ia from these surveys in which joint analysis will greatly improve
their cosmological utility in a similar manner as it will for the normal SN Ia
population (Goldstein et al., 2019).

With such high statistical precision, analyses will leverage systematic
uncertainty ‘self calibration’ (Kim & Miquel, 2006; Rubin et al., 2015; Brout
et al., 2020), but will nonetheless be limited by systematic uncertainties.
Combining data from Rubin, Roman, and Euclid — as well as complementary
spectroscopic follow-up — will greatly improve control of certain systematic
uncertainties. There are multiple reasons for this improvement due to both
better instrumental calibration and improved knowledge of SN Ia physics. A
brief list of the systematic uncertainties that can be improved are:

  * •

Photometric calibration A key source of systematic uncertainty for measuring
the dark energy equation-of-state is the lack of a common, accurate, chromatic
scale referenced to the physics-based International System (SI) The
uncertainty in the flux standards, and hence, uncertainty in the photometric
zero points are a large contributor to the systematic error budget (Hounsell
et al., 2018; Battaglieri et al., 2017). Work to establish such calibration is
a focus of several groups — for example, STarDICe, SCALA, NISTStars,
Collimated Beam Projector, ORCASat, ORCAS — that are not-accidentally
dominated by SN Ia researchers (Lombardo et al., 2017; Aldering et al., 2021;
Coughlin et al., 2018). Measurements of dark energy, in particular SN Ia
cosmology, require an external calibration error budget, so we must rely on
more than one of these paths to be successful. Between them, Roman, Euclid,
and LSST span the optical and near-infrared wavelengths, presenting a special
challenge. They require not only joint observations of SI-referenced optical-
NIR spectrophotometric standard stars, but also must ensure photometric
consistency between these experiments through coordinated observations and
measurement algorithms for agreed-upon calibration fields. One key activity
for achieving the desired cross-mission accuracy is for each
mission/experiment to commit the necessary resources towards understanding and
tracking instrument performance over time, in addition to wavelength.

  * •

Spectroscopic follow-up Combining data from Rubin, Roman, Euclid, and
complementary spectroscopic follow-up will greatly improve control of certain
systematics. Systematic biases in SN Ia distances can be controlled (or in
some cases eliminated) with spectroscopic observations of the supernovae
themselves. These observations can remove residual non-Ia contamination and
mis-identified redshifts, minimize host-galaxy SN-luminosity dependence, test
for population evolutionary drift, and significantly improve the statistical
uncertainty of the standardization (Jones et al., 2018; Boone et al., 2021).

  * •

Photometric redshift measurements are improved with a broader wavelength
coverage (Capak et al., 2019; Rhodes et al., 2019). Photometric redshifts will
be an important piece of the overall strategy to obtain SN Ia redshifts. The
training of photometric redshift algorithms will also be improved by synergies
between observatories, see the response to this RFI “Photometric Redshifts for
the Next Generation of Weak Lensing Surveys” by Daniel Masters, et al. for
details.

  * •

Host-galaxy redshifts In order to measure dark energy with SNe Ia, redshifts
are needed. These can come from the SN or its host galaxy. LSST-DESC has
baselined obtaining redshifts from host-galaxies due to the multiplex
advantage of this approach, and discussions are ongoing to partner with 4MOST
for this purpose. But 4MOST will not reach sufficient depth in the LSST Deep-
Drilling Fields (DDF), where most of the DESC SN Ia cosmology will be
performed. Roman will estimate a fraction of its redshifts from the spectra
obtained with its prism and grism surveys. Roman also has access to 100 nights
of time on Subaru, where the PFS instrument could be used to obtain these
redshifts for Roman SN fields north of declination $-30^{\circ}$. Simulations
by the Roman SN SITs show that more than 100 nights on Subaru with PFS could
be productively consumed by the Roman SN program alone, because there will be
a sizable population of intrinsically faint host galaxies of high-redshift
Roman SNe Ia. Therefore, there is certain to be an unmet need for additional
SN host redshifts. A coordinated effort would result in a more efficient
follow-up campaign.

  * •

SN Ia standardization Systematic biases in SN Ia distances are better
understood when combining light-curve data across broad wavelengths, such as
LSST+ NIR from Rubin and Euclid (Ponder et al., 2020; Uddin et al., 2020).
These observations would need to be coordinated so all surveys could observe
the same transients. While rest-frame optical (and rest-frame NIR) bands are
highly correlated among themselves, optical and NIR bands are relatively
uncorrelated, indicating unique information when observing across this wide
wavelength range (Mandel et al., 2011). Additionally, a broader wavelength
range provides a larger lever arm to constrain dust reddening and perhaps
disentangle the effects of intervening dust from intrinsic color (Brout &
Scolnic, 2020; Mandel et al., 2020). In addition, SNe Ia have a lot of
diversity in the rest-frame UV, which can help to perform better
standardization if it can be exploited (Ellis et al., 2008; Cooke et al.,
2011; Foley et al., 2012; Léget et al., 2020; Boone et al., 2021). The
complimentary observed wavelengths of Roman and Rubin allow for improved SNe
Ia standardization at a broader redshift, resulting in better constraints on
dark energy. Furthermore, many studies have shown that properties of SNe
correlate with properties of the host-galaxy properties, and these
correlations can be used to better standardize SNe Ia. Combined photometry
from the different missions will allow for greater wavelength range to
determine host-galaxy properties and significantly improve these studies.

  * •

Selection functions When combining surveys that target different redshifts,
one can better characterize selection effects and faint source photometry by
comparing photometry of different surveys with different depths (Rubin et al.,
2015; Kessler & Scolnic, 2017).

> b. What is the scope of work required, as well as the opportunities and
> costs?

There are a number of work items that fall under the theme of “survey
coordination.” Due to the timeline of LSST, much of this needs to be done now.
A TAG survey coordination task-force is already in place, though their charge
has been mainly limited to sky-area selection (deep and wide fields). While
there are currently groups in place to discuss survey strategy coordination, a
key obstacle is different timelines between when decisions need to be made by
each survey. What needs to be coordinated now:

  * •

A coordinated selection of overlapping deep field location, time, season
length, and filters.

  * •

Establishment of an all-sky calibration network: to make sure all surveys can
observe standard stars from same network. This includes ensuring that SI-
traceable, calibrated standard stars are available at multiple locations
around the sky, as well as establishing a set of “touchstone” fields suitable
for survey facilities. Stable stars in these fields should be identified, and
if possible, subjected to the same vetting process as the traditional,
individual flux standards, as in the CALSPEC (Bohlin et al., 2014) data base.
There are have been a few independent efforts, the most recent and extensive
being faint, northern white dwarf standards placed on the CALSPEC system
(Narayan et al., 2019; Calamida et al., 2019).

What needs to be planned soon:

  * •

Forced-position photometry on subtracted images from surveys with faint
detection.

  * •

Prioritized spectroscopic follow-up for supernovae and host-galaxies.

  * •

Artificial source injection on real images for measuring detection efficiency
and validating photometric pipeline.

Forced-position photometry on subtracted images. In terms of forced-position
photometry, while all surveys have plans for forced photometry at specific
locations, there is no plan in place for coordination across surveys. There
should be a system to share lists of SN positions and peak dates (for building
templates). Furthermore, it is unclear if Euclid plans to perform image
subtraction, and a separate agency/survey may be necessary to provide the
software necessary to perform this task. Archive access, between science
analysis platforms, is needed to allow reprocessing for consistency and
homogeneity will be critical, particularly for transient detection and
classifications. This would enable, for example, using SN locations determined
from one survey to perform forced-position photometry on data from a second
survey.

Consistent observation and modeling of SNe Ia in their host galaxies will
require forced-position photometry or scene modeling of pixel data based on
recorded positions of detected supernovae. Specifically, SNe detected by Roman
should be photometered in the Rubin data, even in cases where the SN is below
the detection threshold.

Follow-up spectroscopy Due to the time-sensitive nature of transient science,
combining information across surveys must be done quickly. This is
particularly important if one wants to include interesting transients or
observe SNe Ia early in their evolution. For spectroscopic programs, there are
multiple options including Subaru/PFS and The Dark Energy Spectroscopic
Instrument (DESI). Priority should be given to targets in fields from multiple
telescopes. DOE’s DESI instrument could help fulfill this need for any Roman
fields or LSST DDF fields it can reach. For any blocks of the Roman wider-
field tier that are contiguous over more than 7 sq. deg, DESI will be as
efficient as PFS (in terms of host redshifts per hour). Trial observations
with DESI – requiring total exposure on a field in the neighborhood of 25 hr –
could test the power of DESI for this application. The result could influence
the selection of the sky regions for the Roman SN program. Field selection
will take place over the next few years.

To facilitate the coordination of spectroscopic follow-up, multiple ‘Broker’
groups are studying optimal prioritization. This includes a joint effort
between DESC and the Cosmostatistics Initiative (COIN) to develop a
Recommendation System for Spectroscopic Follow-up (RESSPECT; Kennamer et al.,
2020), which will optimize follow-up taking into account multiple metrics
including photometric classification uncertainty and availability of telescope
time for spectroscopic follow-up. With collaboration between the three
experiments, we can construct different metrics to optimize combined science
goals.

Artificial source injection Artificial source injection on real images is used
to measure single-visit detection efficiency vs. S/N in each bandpass (Pain et
al., 1996; Kessler et al., 2019), measure efficiency vs. S/N for machine-
learning used to reject subtraction artifacts (Bailey et al., 2007; Goldstein
et al., 2015), and to characterize anomalous noise on bright galaxies (Sec.
6.4 of Kessler et al. 2019; Sec. 4.4 of Brout et al. 2019b). The resulting
efficiency and anomalous-noise maps are used to measure biases in simulated
data to correct for expected biases in real data. Artificial source magnitudes
should span from near saturation to $\sim 2~{}\text{mag}$ beyond the detection
limit. Artificial sources need not be associated with realistic light curves,
but should be placed near galaxies following their light profile. Ultimately,
artificial light-curve injection is useful to check the entire cosmology
analysis as shown in Sec. 6 of Brout et al. (2019a). From experience with this
data set, artificial SN light curves provide an adequate flux range to measure
a single-visit detection efficiency.

Rubin and Roman will have the bulk of their SNe at different redshifts.
Analyses that are sensitive to potential SN Ia population drift and detection
efficiency will be greatly improved and less total work if done consistently
for both Rubin and Roman. This can be done both on the real data once
available, but should also be done on consistent, simulated images. In
particular, the part of the pixel processing that matters for SN sensitivity
is the distribution of SN light on top of different host galaxy types and
local surface brightness. An effort to do this correctly and consistently
across Euclid, Rubin, and Roman will (1) be much simpler to understand and
compare across analyses; and (2) be less total effort to do jointly than for
each team to re-do and then have to re-interpret across different teams.

> c. What are key obstacles, impediments, or bottlenecks to advancing
> development of these plans?

The key operational obstacles are communication and timeliness. Since SNe Ia
reach peak only a few weeks after discovery, communication must be done both
proactively and in real time. Speed is even more important for ancillary
science with rare transients where their evolution is unknown. Furthermore, it
is still unclear which data can be shared and with what latency. This will
complicate efforts to optimally yield the best cosmological constraints from
joint observations and joint analyses. The key structural challenge is giving
teams authority to work on topics across projects. Many endeavors will work
best when a team working on a topic under one project is encouraged to make
the connection to complementary projects. However, this will requires agencies
to be willing and able to coordinate.

There are also some potential challenges with the proprietary nature of the
Rubin Observatory LSST data. The Roman data are fully public, while the LSST
data have a 2-year proprietary period. The Roman science collaborations will
be international, and unlikely to precisely match up with international data
rights in LSST. Thus if one wanted to do a joint Roman+LSST analysis of the
first year of Roman data, the LSST data on those same supernovae would not be
available to the full Roman community until 2 years later. However, the LSST
data rights community could do a full analysis of the LSST + Roman data.

> d. Are there other science topics besides dark energy that drive the
> requirements for joint data processing or analysis?

Simultaneous observations with Rubin and Roman will result in a large sample
of transients beyond SNe Ia. Combined optical/NIR observations will constrain
dust properties of extinguished supernovae and reveal other exotic transients
whose emission peaks at $\sim$1 $\mu$m such as luminous red novae and
kilonovae. Core-collapse supernova rates, which constrain the cosmic star-
formation rate, can be more accurately determined without requiring a dust-
obscured correction.

The hundreds of strongly gravitationally lensed SNe Ia observed from these
surveys, though not a part of typical cosmological analyses, would benefit
from joint analysis and result in a significant improvement to their
cosmological utility (Goldstein et al., 2019). Furthermore, multi-wavelength
follow-up of gravitational wave counterparts is necessary to constrain the
composition of the ejected material.

##  3 Collaboration and Partnerships

> k. What cooperation or partnerships between DOE and NASA could further the
> scientific and technology advances?

  * •

While a TAG Survey Coordination group is already formed (led by Dan Scolnic
and Daniel Stern), communication between this group and the collaborations
should be formalized and the charge expanded to improve survey operations to
further coordination between survey planners, specifically with the authority
and ability to tweak the strategies before and during the multiple surveys. We
advocate that all survey strategies be reviewed and adjusted periodically
during operations, in order to further optimize synergies based on what is
learned from initial data sets, and based on what new observing facilities
become available.

  * •

The creation of a joint survey calibration task force to establish
communication across the missions, identify common algorithms, and ensure
coordination so that uncertainties in photometric calibration are not the
limiting factor in achieving Stage IV dark energy figure of merit (Albrecht et
al., 2006).

  * •

Spectroscopic follow-up task force to ensure that access to sufficient follow-
up spectroscopy resources is obtained, through different TACs, MoUs with
external groups in exchange for data rights, and through the construction of
new facilities and instruments. There will be multiple types of spectroscopic
follow up to achieve the different goals of SN identification, redshift
determination, and statistical and systematic error constraints, and to follow
SN targets that are visible in different hemispheres at different times.
Moreover, systems (and pipelines) will need to be planned, organized, and
developed to ensure rapid and efficient use of all available spectroscopic
resources. As an example of a prioritization issue that this planning should
address: if a SN is observed by multiple surveys, it should be preferentially
allocated appropriate follow-up resources, while avoiding duplication of
observations to maximize science return.

  * •

Expansion of in-place joint pixel-processing task force to generate a plan for
all surveys to grant access of pixel-level data associated with transients, in
less than one day turn round, and with oversight of the development of the
necessary infrastructure to realize this goal. Many images will be crowded and
confusion-limited, making catalog-only data of limited utility relative to the
previous generation of ground-based surveys. A joint pixel-processing facility
will be necessary to provide postage stamps generated from the images taken
from the different surveys, together with a probabilistic source separation,
and probability density functions of photometric redshift. The same facility
will be useful for understanding the host environments of transients of all
varieties, and in particular how galaxy evolution impacts transient
explosions. Deep multi-survey images of the host will also be useful for early
classification, and to detect rare strongly-lensed supernovae which will need
prioritized followup.

Joint pixel-processing of data from different surveys will also be necessary
for multi-messenger astrophysics (e.g. detecting the optical counterparts of
events seen in future surveys such as CMB-S4). Services and standards to
deliver alerts for detections in multi-messenger missions such as HopSkotch
are being developed by the SCiMMA group111https://scimma.org/, and
coordination between these groups would be a major responsibility of such a
joint processing task force. A joint pixel processing facility is also crucial
to several other probes of dark energy, detailed in a response to this RFI
from LSST DESC in Annis et al., “Response from the Rubin Observatory LSST Dark
Energy Science Collaboration: Extending LSST Studies of Dark Energy and Dark
Matter with Roman and Euclid”.

  * •

A real-time forced-position photometry task force to recover measurements for
SN detected in other surveys. As missions detect SNe, it can be very valuable
to recover sub-threshold measurements from other surveys. It is possible to
study the earliest phases of a potentially strongly-lensed SNe detected in
LSST several days after explosion if it is possible to extract forced
photometry at the location of other lensed images from Roman observations. A
forced photometry service should also be capable of working together with a
pixel processing facility to allow reprocessing of data, as well as fake
source injection to model the joint survey selection function. This will be
essential to understand the demographics of novel and rare classes of
transients such as pair-instability supernovae, as well as to propagate
measurements from each of these missions into joint cosmological constraints
on the properties of dark energy.

  * •

Joint computational task force to manage shared computing of these multi-
mission datasets, including common simulations (e.g. simulating the multi-
mission sky to include the effect of a novel dark energy model, to ensure its
effects are correctly propagated through all three surveys), and tools for
data access and processing (e.g. a shared JupyterHub environment that
scientists working on any of these missions can use). Computing is the central
nexus between observers, theorists, and instrumentalists, and now encompasses
astrostatistics and data science, in addition to computational physics. A
research platform facility that colocates (or provides transparent access to)
data holdings and provides computing will also be essential for multi-
messenger astrophysics, which has a similiar need to combine observations from
several different observatories (e.g. LIGO-VIRGO-Kagra, IceCube and ngVLA)
together with Euclid, Roman and Rubin.

> l. What mix of institutions or collaboration models could best carry out the
> envisioned research and/or development?

We advocate the use of cross-survey (and agency) task forces to plan and
coordinate the above synergistic activities. These task forces should have
both sufficient scientific expertise and authority to be able to address
specific issues, such as synergistic survey strategies, calibration networks,
and spectroscopic follow-up.

> m. What resources, capabilities and infrastructure at DOE National
> Laboratories or the NASA Centers would be beneficial for and could
> accelerate or facilitate research in this topic?

DOE National Laboratories and the NASA Centers can jointly support mutually
beneficial resources such as long-term computing infrastructure personnel, who
have different skill sets and career paths than typical academics. This can be
important to ensure the access to technically focused long-term joint
processing and simulation efforts. Storage and computational time in the Rubin
and Roman analysis pipelines and user spaces to do these tasks. Many of the
previously mentioned task forces can be coordinated out of one or more DOE
National Lab or NASA Center.

> n. Are there other factors, not addressed by the questions above, which
> should be considered in planning HEP and APD activities in this subject
> area?

No answer.

##  4 Conclusion

Type Ia supernovae (SNe Ia) are a key cosmological probe, and the supernovae
that the three great observatories - Euclid, Nancy Grace Roman Space Telescope
and the Vera C. Rubin Observatory - will discover promise to revolutionize our
understanding of the nature of dark energy. These transients are already key
science drivers for two of the three large surveys of the 2020s: LSST and
Roman, which have large and active communities involved in all aspects of
these missions. Combining data sets allows increases in redshift range and
wavelength range, boosting the statistical precision and the systematic
control. We discussed improvements to both statistical and systematic
precision that combinations of data sets will enable such as improved
photometric calibration, SN Ia standardization, and redshift measurements.
Other science cases will also benefit from these synergies. In particular, the
photometric redshift improvements will benefit cosmological measurements from
weak lensing. Also the discovery and confirmation of strong-lensing systems,
galaxy clusters, and gravitational wave counterparts.

While the community will benefit from combining data from these observatories,
orchestrating them to act in concert with one another will yield the greatest
scientific benefits. Particularly due to the time-dependent nature of SN
surveys, planning must be done both prior to and during the surveys. We have
discussed what teams and plans should be put in place now to start preparing
for these combined sets by setting up cross-agency task forces. These task
forces will allow for the needed communication to improve survey operations,
calibration, spectroscopic follow-up, joint pixel-processing, and analysis. By
working together, the DOE and NASA, can improve their investments with an
increase of scientific output, and in particular, a deeper understanding of
the nature of dark energy and our Universe.

The Roman Supernova Science Investigation Teams and the LSST DESC Supernova
Working Group endorse this response.

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11institutetext:  Istituto ricerche solari Aldo e Cele Daccò (IRSOL), Faculty
of Informatics, Università della Svizzera italiana (USI), CH-6605 Locarno,
Switzerland 22institutetext: Euler Institute, Università della Svizzera
italiana (USI), CH-6900 Lugano, Switzerland
22email<EMAIL_ADDRESS>33institutetext: Leibniz-Institut für
Sonnenphysik (KIS), D-79104 Freiburg i. Br., Germany

# Diagnostic potential of wavelength-integrated scattering polarisation
signals of the solar He ii Ly-$\alpha$ line

Fabio Riva 1122 Gioele Janett 1122 Luca Belluzzi 112233

(Received xxx; accepted yyy)

###### Abstract

Aims.  The main goal of this work is to study the potential of He ii
Ly-$\alpha$ wavelength-integrated scattering polarisation for probing the
magnetism of the solar upper chromosphere. Meanwhile, the suitability of
different modelling approximations is investigated.

Methods.  Radiative transfer calculations are performed in semi-empirical 1D
solar atmospheres, out of local thermodynamic equilibrium, considering a two-
term atomic model and accounting for the Hanle, Zeeman, and magneto-optical
effects. The problem is suitably linearised and discretised, and the resulting
numerical system is solved with a matrix-free iterative method. The results
obtained modelling scattering processes with three different descriptions,
namely in the limit of complete frequency redistribution (CRD), and accounting
for partial frequency redistribution (PRD) effects under the angle-averaged
(AA) approximation and in the general angle-dependent (AD) formulation, are
compared.

Results.  In the line-core, the synthetic Stokes profiles resulting from CRD,
PRD–AA, and PRD–AD calculations show a very good agreement. On the other hand,
relevant differences are observed in $Q/I$ outside the line-core region.
Besides, the precise structure of the atmospheric model does not noticeably
affect the line-core profiles, but it strongly impacts the $Q/I$ signals
outside the line-core. As most of the He ii Ly-$\alpha$ photons originate in
the core region, it turns out that wavelength-integrated linear polarisation
signals are almost insensitive to both the scattering description and the
atmospheric model. Appreciable wavelength-integrated
$\overline{U}/\overline{I}$ signals, showing observable sensitivity to
horizontal magnetic fields in the range 0-1000 G, are also found, particularly
near the limb. It turns out that, while the integration time required to
detect magnetic fields in the quiet chromosphere with this line is too long
for sounding rocket missions, magnetic fields corresponding to typical plage
areas would produce detectable signals, especially near the limb.

Conclusions.  These results, to be confirmed by 3D calculations that will
include the impact of horizontal inhomogeneities and bulk velocity gradients,
show that filter-polarimetry in the He ii Ly-$\alpha$ line has a promising
potential for chromospheric magnetic field diagnostics. In near-limb plage
regions, this could already be assessed through sounding rocket experiments.

###### Key Words.:

Radiative transfer – Scattering – Polarization – Sun: atmosphere – Methods:
numerical

##  1 Introduction

The dynamics of the outer layers of the solar atmosphere is largely governed
by the presence of magnetic fields that, however, are notoriously hard to
measure. The most promising state-of-the-art remote sensing approach to
overcome this difficulty and explore the magnetism of the solar upper
chromosphere and transition region is the interpretation of the polarisation
signals produced by both the Zeeman effect and scattering processes in strong
UV resonance lines, such as Mg ii $h\&k$, H i Ly-$\alpha$, or He ii
Ly-$\alpha$ (e.g. Trujillo Bueno & del Pino Alemán 2022). Unfortunately, the
Zeeman effect is not very effective at short wavelengths and in hot plasmas.
Thus, the signals produced by the weak magnetic fields of the quiet solar
chromosphere through this mechanism can hardly be detected in far and extreme
UV lines. On the other hand, all the aforementioned resonance lines show
measurable linear scattering polarization signals, with an interesting
magnetic sensitivity via the combined action of the Hanle and magneto-optical
(MO) effects.

Thanks to its sensitivity to chromospheric magnetic fields, the H i
Ly-$\alpha$ line at 1216 Å attracted increasing attention during the last few
years. Modelling scattering processes in the limit of complete frequency
redistribution (CRD) and neglecting $J$-state interference, Trujillo Bueno et
al. (2011) showed that the line-core scattering polarisation signal is
sensitive, via the Hanle effect, to the presence of magnetic fields of
strengths between 10 G and 100 G in the upper chromosphere. Štěpán et al.
(2012) and Štěpán et al. (2015) extended the investigation of the Hanle
sensitivity of the H i Ly-$\alpha$ line by considering realistic 2D and 3D
atmospheric models resulting from a state-of-the-art radiation magneto-
hydrodynamic simulation. A very interesting outcome of these investigations is
that the centre-to-limb variation (CLV) of the spatially averaged line-core
$Q/I$ signal is qualitatively similar to that found considering semi-empirical
1D atmospheric models. Moreover, Belluzzi et al. (2012) showed that the joint
action of partial frequency redistribution (PRD) and $J$-state interference
effects produces conspicuous wing scattering polarisation signals, which are
very sensitive to the chromospheric thermal structure. These results motivated
the development of the Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP)
sounding rocket experiment (Kobayashi et al. 2012; Kano et al. 2012), which
paved the way to the observation of the intensity and linear polarisation of
the solar H i Ly-$\alpha$ line. The CLASP data showed a remarkably good
qualitative agreement with the theoretical predictions (Kano et al. 2017) and
have been exploited to get new constraints on the magnetization and the degree
of corrugation of the chromosphere-corona transition region (Trujillo Bueno et
al. 2018). More recently, Alsina Ballester et al. (2018) and Alsina Ballester
et al. (2023) proved that the wavelength-integrated linear polarisation
signals of this line are strongly sensitive to magnetic fields of strengths of
about 50 G in the middle–upper chromosphere via MO effects.

By contrast, a reduced number of investigations has been dedicated to the
scattering polarisation signals of the He ii Ly-$\alpha$ line at 304 Å and
especially to its magnetic sensitivity. This for two reasons. First, the He ii
Ly-$\alpha$ line is much less sensitive to the Hanle effect than the H i
Ly-$\alpha$ line. Indeed, having a much larger Einstein coefficient for
spontaneous emission, the Hanle critical field of the He ii Ly-$\alpha$ line
is $B_{\text{H}}\approx 850$ G, while the H i Ly-$\alpha$ line has
$B_{\text{H}}\approx 53$ G. By applying the CRD limit, Trujillo Bueno et al.
(2012) confirmed that the He ii Ly-$\alpha$ line-core $Q/I$ signal is nearly
immune to magnetic fields weaker than 100 G, thus proposing to use it as a
reference signal for a differential Hanle effect technique. Second, although
the He ii Ly-$\alpha$ line shows large fractional linear polarisation signals,
the required exposure time to reach any given polarimetric sensitivity is
significantly larger than for the H i Ly-$\alpha$ line (e.g. Trujillo Bueno &
del Pino Alemán 2022).

The scattering polarisation signals of the He ii Ly-$\alpha$ line in
unmagnetised atmospheres were further explored by Belluzzi et al. (2012), who
showed that the joint action of PRD and $J$-state interference effects
produces complex scattering polarisation $Q/I$ profiles with sharp peaks of
large amplitude outside the line-core region. Although these peaks are largely
insensitive to the Hanle effect, in principle they could be impacted by
magnetic fields through the action of MO effects (e.g. Alsina Ballester et al.
2016; del Pino Alemán et al. 2016). This potential, yet unexplored, magnetic
sensitivity of the He ii Ly-$\alpha$ line motivates the work in the present
paper. Moreover, we note that, for computational simplicity, the PRD
calculations of Belluzzi et al. (2012) were performed within the angle-
averaged (AA) approximation (e.g. Mihalas 1978; Leenaarts et al. 2012;
Sampoorna et al. 2017; Alsina Ballester et al. 2017). The suitability of such
an approximation in the modelling of He ii Ly-$\alpha$ line polarisation
signals has yet to be proven.

Motivated by the studies of Trujillo Bueno et al. (2012) and Belluzzi et al.
(2012), the main goal of this work is to investigate the potential of He ii
Ly-$\alpha$ wavelength-integrated scattering polarisation for probing the
magnetism of the upper solar chromosphere. Concurrently, we aim to assess the
suitability of the CRD and PRD–AA approximations to model the polarization
profiles and the wavelength-integrated polarization signals of this line. This
is done through a comparison with general angle-dependent (AD) PRD
calculations, which are more accurate but, at the same time, also imply a
significantly higher computational cost.

The present article is organised as follows. Section 2 exposes the considered
non-equilibrium transfer problem for polarised radiation, as well as its
linearization, discretization, and algebraic formulation. Section 3 presents
the adopted numerical solution strategy and the numerical set-up. In Sect. 4,
we report and analyse the synthetic emergent Stokes profiles of the He ii
Ly-$\alpha$ line, comparing CRD, PRD–AA, and PRD–AD calculations and
investigating the impact that the magnetic field and the atmospheric
parameters have on the numerical results. Finally, Sect. 5 discusses the main
results and their implications, and provides some final considerations.

##  2 Formulation of the problem

Aiming to model the scattering polarisation signal of the He ii Ly-$\alpha$
line at 304 Å, we solved the non-equilibrium transfer problem for polarised
radiation in 1D models of the solar atmosphere. To correctly model this
signal, a two-term atomic model, which includes quantum interference between
different fine-structure (FS) levels of the same term ($J$-state interference,
see Landi Degl’Innocenti & Landolfi 2004) has to be considered (Belluzzi et
al. 2012). Scattering processes are modelled considering three different
descriptions of increasing computational complexity: (i) the CRD limit; and
including PRD effects (ii) within the AA approximation; or (iii) in their
general AD formulation. We also consider the impact of magnetic fields of
arbitrary strength and orientation, accounting for the Hanle, Zeeman, and MO
effects. The theoretical framework is presented in the following.

###  2.1 Two-term atomic model

The He ii Ly-$\alpha$ line results from two distinct FS transitions between
the ground level of ionized helium $1s\,^{2}\mathrm{S}_{1/2}$ and the excited
levels $2p\,^{2}\mathrm{P}_{1/2}^{\mathrm{\,o}}$ and
$2p\,^{2}\mathrm{P}_{3/2}^{\mathrm{\,o}}$. These two FS components are
separated by 5 mÅ only, and are thus completely blended. To take into account
both these transitions, as well as the $J$-state interference between the two
upper levels, we consider a two-term ${}^{2}\mathrm{S}$ –
${}^{2}\mathrm{P}^{\mathrm{\,o}}$ atomic model. Noticing that the lower level
has $J=1/2$ and therefore cannot carry atomic alignment (see Landi
Degl’Innocenti & Landolfi 2004), and that it has a very long lifetime, we can
assume that it is unpolarized and infinitely sharp. Neglecting stimulated
emission, which is generally a good assumption in the solar atmosphere, the
statistical equilibrium equations for a two-term atom with unpolarized and
infinitely-sharp lower term have an analytic solution. Thus, the scattering
contribution to the line emissivity can be expressed through the
redistribution matrix formalism, which is a very convienient formulation for
modelling PRD effects. In this work, we used the redistribution matrix for the
aforementioned atomic system as derived by Bommier (2017) (see Sect. 2.4). For
the energies of the upper levels ${}^{2}\mathrm{P}_{1/2}^{\mathrm{\,o}}$ and
${}^{2}\mathrm{P}_{3/2}^{\mathrm{\,o}}$, relative to the ground state, we
considered $E_{u}=329179.29\ \text{cm}^{-1}$ and $E_{u}=329185.15\
\text{cm}^{-1}$, respectively, while for the Einstein coefficient for
spontaneous emission from upper to lower term, we used $A_{ul}=1.0029\cdot
10^{10}\ \text{s}^{-1}$. The splitting of the magnetic sublevels in the
presence of a magnetic field was calculated considering the Paschen-Back
effect. The theoretical values of the Landé factors were used. We finally note
that only the transition from the upper level
${}^{2}\mathrm{P}_{3/2}^{\mathrm{\,o}}$ contributes to the emergent linear
scattering polarization (Trujillo Bueno et al. 2011).

###  2.2 Semi-empirical 1D atmospheric models

The calculations presented in the following were obtained with the plane-
parallel semi-empirical 1D models of Fontenla et al. (1993, hereafter FAL
models), making use of the microturbulent velocities as determined in Fontenla
et al. (1991). In particular, we considered three quiet Sun atmospheric
models, representing a faint inter-network region (FAL-A), a bright network
region (FAL-F), and an average region (FAL-C), as well as a model that mimics
a plage area (FAL-P), that is a region that appears particularly bright in the
chromospheric hydrogen H-$\alpha$ line, typically found next to active solar
regions. Despite their simplicity, these atmospheric models are realistic
enough to assess the potential of He ii Ly-$\alpha$ wavelength-integrated
scattering polarisation for investigating chromospheric magnetic fields.

###  2.3 Radiative transfer equation

The intensity and polarisation of a beam of radiation are fully described by
the Stokes vector $\bm{I}\in\mathbb{R}^{4}$. Considering a Cartesian reference
system with the $z$-axis (quantization axis for total angular momentum)
directed along the vertical, the transfer of partially polarised radiation is
described by the following differential equation

$\cos(\theta)\frac{\rm d}{{\rm
d}z}\bm{I}(z,\mathbf{\Omega},\nu)=-K(z,\mathbf{\Omega},\nu)\bm{I}(z,\mathbf{\Omega},\nu)+\bm{\varepsilon}(z,\mathbf{\Omega},\nu),$
(1)

where the vector $\@vec{\Omega}=(\theta,\chi)\in[0,\pi]\times[0,2\pi)$ and the
scalar $\nu$ specify the direction and the frequency, respectively, of the
considered radiation beam. The inclination $\theta$ is measured with respect
to the $z$-axis and corresponds to the heliocentric angle (generally specified
by $\mu=\cos{\theta}\in[-1,1]$) of the observed point. The azimuth $\chi$ is
measured from the $x$-axis counter-clockwise for an observer at positive $z$.
The propagation matrix $K\in\mathbb{R}^{4\times 4}$ describes how the medium
differentially absorbs the polarised radiation (dichroism) and how it couples
the different Stokes parameters (MO effects). The emission vector
$\bm{\varepsilon}\in\mathbb{R}^{4}$ describes the polarised radiation emitted
by the solar plasma.

In the following, we consider the contributions to $K$ and
$\@vec{\varepsilon}$ due to both line and continuum processes. These
contributions, labelled with the superscripts $\ell$ and $c$, respectively,
simply sum up. Moreover, both $\@vec{\varepsilon}^{\ell}$ and
$\@vec{\varepsilon}^{c}$ can in turn be written as the sum of two terms,
describing the contributions from scattering (label “sc”) and thermal (label
“th”) processes, respectively. Hence, we have

$K(z,\mathbf{\Omega},\nu)=K^{c}(\@vec{r},\mathbf{\Omega},\nu)+K^{\ell}(z,\mathbf{\Omega},\nu),$

and

$\displaystyle\bm{\varepsilon}(z,\mathbf{\Omega},\nu)=$
$\displaystyle\bm{\varepsilon}^{c,\text{th}}(z,\mathbf{\Omega},\nu)+\bm{\varepsilon}^{c,\text{sc}}(z,\mathbf{\Omega},\nu)$
(2) $\displaystyle+$
$\displaystyle\bm{\varepsilon}^{\ell,\text{th}}(z,\mathbf{\Omega},\nu)+\bm{\varepsilon}^{\ell,\text{sc}}(z,\mathbf{\Omega},\nu).$

The explicit expressions for $K^{c}$ and $K^{\ell}$ are given by Landi
Degl’Innocenti & Landolfi (2004), while we refer to the work of Alsina
Ballester et al. (2022) for $\bm{\varepsilon}^{c,\text{th}}$,
$\bm{\varepsilon}^{c,\text{sc}}$, and $\bm{\varepsilon}^{\ell,\text{th}}$. The
line scattering contribution to emissivity,
$\bm{\varepsilon}^{\ell,\text{sc}}$, is described in the following section.

###  2.4 Line scattering emissivity

Using the redistribution matrix formalism and following the convention that
primed and unprimed quantities refer to the incident and scattered radiation,
respectively, the line scattering term can be written through the scattering
integral

$\bm{\varepsilon}^{\ell,\text{sc}}(z,\bm{\Omega},\nu)\\!=\\!k_{M}(z)\\!\\!\int\\!\\!{\rm
d}\nu^{\prime}\\!\oint\\!\frac{{\rm
d}\bm{\Omega}^{\prime}}{4\pi}R(z,\bm{\Omega}^{\prime},\bm{\Omega},\nu^{\prime},\nu)\bm{I}(z,\bm{\Omega}^{\prime},\nu^{\prime}),$
(3)

where the factor $k_{M}$ is the wavelength-integrated absorption coefficient
(e.g. Alsina Ballester et al. 2022) and $R\in\mathbb{R}^{4\times 4}$ is the
redistribution matrix for the considered two-term atomic model (see Bommier
2017). We recall that $R$ encodes the statistical equilibrium (SE) equations,
which determine the state of the atomic system.

The redistribution matrix can be expressed as the sum of two terms,
$R=R^{\scriptscriptstyle\rm II}+R^{\scriptscriptstyle\rm III}$, which describe
scattering processes that are coherent ($\rm II$) and completely incoherent
($\rm III$) in the atomic reference frame. As far as $R^{\scriptscriptstyle\rm
III}$ is concerned, we considered its simplified expression under the
assumption of CRD in the observer’s frame (see Bommier 1997; Sampoorna et al.
2017; Alsina Ballester et al. 2017; Riva et al. 2023, for a discussion about
its suitability). For conciseness, we do not provide the explicit expressions
of $R^{\scriptscriptstyle\rm II}$ and $R^{\scriptscriptstyle\rm III}$.
Instead, we refer to the work of Alsina Ballester et al. (2022) for the
analytical form of $R^{\scriptscriptstyle\rm II}$ in the observer’s frame,
both in its general AD formulation and under the AA approximation, as well as
for the expression of $R^{\scriptscriptstyle\rm III}$, under the
aforementioned assumption of CRD in the observer’s frame. Moreover, we note
that the limit of CRD is obtained by artificially increasing the rate of
elastic collisions in the branching ratios of $R^{\scriptscriptstyle\rm II}$
and $R^{\scriptscriptstyle\rm III}$ so that the contribution of the former
vanishes (see Alsina Ballester et al. 2022). In this respect, we recall that
the considered redistribution matrix does not account for the depolarizing
effect of elastic collisions with neutral perturbers, because the SE equations
for a two-term atom in the presence of magnetic fields cannot be solved
analytically when such effect is included (see Bommier 2017, for more
details). Noticing that Alsina Ballester et al. (2021) verified that this
effect can be safely ignored for the Na i D lines, we can argue that the same
conclusion also holds for the He ii Ly-$\alpha$ line, which forms
significantly higher in the solar atmosphere.

###  2.5 Linearization, discretization, and algebraic formulation

The radiative transfer (RT) problem consists in finding a self-consistent
solution of the RT equation for the radiation field $\bm{I}$ and of the SE
equations for the atomic system, whose state determines $K$ and
$\bm{\varepsilon}$. The problem is in general integro-differential, nonlocal,
and nonlinear. In our formulation, based on the redistribution matrix
formalism, the nonlinearity lies in the dependence of both $K$ and
$\bm{\varepsilon}$ on the coefficient $k_{M}$. This coefficient is
proportional to the population of the lower term, which in turn nonlinearly
depends on $\bm{I}$ through the SE equations (see e.g. Landi Degl’Innocenti &
Landolfi 2004). However, the RT problem can be suitably linearised by fixing a
priori the coefficient $k_{M}$, which corresponds to assuming a fixed lower
term population (see e.g. Belluzzi & Trujillo Bueno 2014; Sampoorna et al.
2017; Alsina Ballester et al. 2017; Janett et al. 2021; Benedusi et al. 2021,
2022). This assumption makes the propagation matrix $K$ independent of
$\bm{I}$, whereas $\bm{\varepsilon}$ depends on $\bm{I}$ only linearly. In
this case, the whole problem consists in finding a self-consistent solution of
Eqs. (1)-(3) and it is linear in $\bm{I}$.

To numerically solve the resulting linearised RT problem, we first need to
discretise the continuous variables $z$, $\mathbf{\Omega}$, and $\nu$. The
spatial discretization is done by using $N_{r}$ unevenly spaced grid nodes.
The angular grid, which samples the unit sphere with $N_{\Omega}$ angular
nodes, usually depends on the quadrature chosen to evaluate the angular
integral in Eq. (3) (see Sect. 3.1). Finally, we consider a finite spectral
interval $[\nu_{\min},\nu_{\max}]$ for $\nu$, discretised with $N_{\nu}$
unevenly spaced nodes. More details on the numerical parameters used for the
present work are given in Sect. 3.4.

The discretised radiation field and emissivity vectors are thus represented by
$\mathbf{I}\in\mathbb{R}^{N}$ and $\bm{\varepsilon}\in\mathbb{R}^{N}$, with
$N=4N_{r}N_{\nu}N_{\Omega}$ the total number of degrees of freedom. Hence, we
can express the transfer equation, Eq. (1), and the calculation of the
emission coefficient, Eq. (2), as

$\displaystyle\mathbf{I}$
$\displaystyle=\Lambda\boldsymbol{\varepsilon}+\mathbf{t},$ (4)
$\displaystyle\boldsymbol{\varepsilon}$
$\displaystyle=\Sigma\mathbf{I}+\boldsymbol{\varepsilon}^{\text{th}},$ (5)

respectively. Here, the transfer operator $\Lambda\in\mathbb{R}^{N\times N}$
and the scattering operator $\Sigma\in\mathbb{R}^{N\times N}$ are linear, the
vector $\mathbf{t}\in\mathbb{R}^{N}$ represents the radiation transmitted from
the boundaries, while $\boldsymbol{\varepsilon}^{\text{th}}\in\mathbb{R}^{N}$
represents the thermal contributions to the emissivity. The evaluation of the
term $\Lambda\boldsymbol{\varepsilon}+\mathbf{t}$ is commonly known as formal
solution and consists in the numerical solution of Eq. (1) for $\mathbf{I}$,
provided $\boldsymbol{\varepsilon}$, $K$, and appropriate boundary conditions.
The term $\Sigma\mathbf{I}$ corresponds to the numerical evaluation of the
scattering contribution to the emission vector, given $\mathbf{I}$. This
requires approximating the integral in Eq. (3) via suitable quadrature rules
(see Sect. 3.1). Equations (4) and (5) can then be combined into a matrix
equation of the form $A\mathbf{x}=\mathbf{b}$, namely,

$(Id-\Lambda\Sigma)\mathbf{I}=\Lambda\bm{\varepsilon}^{\text{th}}+\mathbf{t},$
(6)

being $Id\in\mathbb{R}^{N\times N}$ the identity matrix (see e.g. Janett et
al. 2021; Benedusi et al. 2022).

##  3 Methods and setting

We now expose the numerical methods employed to solve Eq. (6) and the physical
and numerical parameters used in the calculations.

###  3.1 Angular and spectral quadratures

The numerical integration of Eq. (3) requires a proper choice of quadrature
rules. For the angular integral, we resorted on a tensor product of a
trapezoidal quadrature in the azimuthal interval $[0,2\pi)$ for $\chi$ and two
Gauss-Legendre quadratures in each inclination subinterval $(-1,0)$ and
$(0,1)$ for $\mu=\cos(\theta)$. For the integral in $\nu^{\prime}$, we
selected a set of spectral points distributed ad hoc to approximately follow
the shape of the redistribution matrix $R$. More precisely, we used a Gauss-
Hermite quadrature rule in spectral regions where $R$ is similar to a Gaussian
function, while we used a Gauss-Legendre quadrature rule in the rest of the
parameter domain.

###  3.2 Formal solution

For a better agreement with the results by Belluzzi et al. (2012), we applied
the DELOPAR formal solver to evaluate the $\Lambda$ operator in Eq. (6) (for
details, see Trujillo Bueno 2003; Janett et al. 2017; Janett & Paganini 2018;
Janett et al. 2018), using a trapezoidal quadrature for the conversion to
optical depth. We note that the use of the $L$-stable DELO-linear method only
affects the amplitude of the emergent Stokes profiles, but leads to the same
general conclusions.

###  3.3 Iterative solver

The whole transfer problem has been recast in the linear system given by Eq.
(6), for which we have to find a solution. Given the size of the system, with
$N\sim 10^{6}$ (see Sect. 3.4), the use of direct matrix inversion solvers is
definitely unfeasible. Thus, the application of an iterative method was
preferred. Moreover, it is also too expensive to explicitly assemble the dense
operator $Id-\Lambda\Sigma$ in Eq. (6). Therefore, the action of the operator
$Id-\Lambda\Sigma$ was encoded in a routine and an iterative method was
applied in a matrix-free context.

In this work, we applied a Krylov solver, namely the generalised minimal
residual (GMRES) method, which proved to be highly effective in terms of
convergence rate and time-to-solution for this kind of problem (Benedusi et
al. 2021, 2022, 2023). For the PRD calculations, we additionally exploited the
innovative approach to designing efficient physics-based preconditioners and
initial guesses extensively described in Janett et al. (2023) and we monitored
the convergence of the GMRES iterative method by considering the
preconditioned relative residual

${\rm
res}=\frac{\|P^{-1}(\Lambda\bm{\varepsilon}^{\text{th}}+\mathbf{t}-(Id-\Lambda\Sigma)\mathbf{I})\|_{2}}{\|P^{-1}(\Lambda\bm{\varepsilon}^{\text{th}}+\mathbf{t})\|_{2}},$

with $P$ the preconditioner, taking as a stopping criterion ${\rm res}<{\rm
tol}=10^{-9}$. In the calculations presented in this paper, both CRD- and
PRD–AA-based preconditioners have a noticeable impact, ensuring a solution of
the problem in 4-8 iterations. Moreover, the inclusion of polarisation in the
preconditioner has small impact (with respect to the unpolarised version) and
it is thus neglected.

###  3.4 Physical and numerical parameters

The lower term population was provided by the FAL atmospheric models (Fontenla
et al. 1993). The RH code by Uitenbroek (2001) was used to calculate the
continuum total opacity, scattering opacity, and thermal emissivity, the
damping parameter, as well as the collisional rates, except for the one for
inelastic de-exciting collisions, which was evaluated as $C_{ul}=N_{e}Q_{ul}$,
with $N_{e}$ the electron density and

$Q_{ul}=8.63\cdot 10^{-6}\cdot\upsilon/(g_{u}\sqrt{T}),$

where $g_{u}=6$ is the degeneracy of the upper term, $T$ is the local
temperature, and $\upsilon$ (which depends on $T$) is taken from Janev et al.
(1987). For the formal solution, we assumed the following boundary conditions

$\displaystyle I(z_{\min},\theta,\chi,\nu)$ $\displaystyle=B_{P},\qquad$
$\displaystyle\text{ for }\;\theta\in[0,\pi/2),\forall\chi,\forall\nu,$
$\displaystyle I(z_{\max},\theta,\chi,\nu)$ $\displaystyle=0,\qquad$
$\displaystyle\text{ for }\;\theta\in(\pi/2,\pi],\forall\chi,\forall\nu,$

where $B_{P}$ is the line-centre frequency Planck function at the local
temperature.

The wavelength interval
$[\lambda_{\min},\lambda_{\max}]=[303.01\text{\,\AA},304.55\text{\,\AA}]$ was
discretised with ${N_{\nu}=141}$ frequency nodes, approximately uniformly
spaced in the line core and logarithmically distributed in the wings. The
spatial grid was provided by the considered semi-empirical 1D atmospheric
models. However, since the He ii Ly-$\alpha$ line forms in a very narrow
region in the outermost layers of the solar chromosphere (e.g. Trujillo Bueno
et al. 2012; Belluzzi et al. 2012), we decided to cut out the layers below
$z_{\text{min}}=1375$, 1475, 1475, and 1495 km for the FAL-A, FAL-C, FAL-F,
and FAL-P models, respectively, as their contribution to the emergent
radiation field is negligible.111We repeated part of the calculations with the
full FAL models, finding undetectable differences. Conversely, we doubled the
spatial resolution in the uppermost layers. This resulted in 61-76 unevenly
distributed spatial nodes, depending on the considered atmospheric model. For
the angular discretization, we used two 6-nodes Gauss-Legendre grids for the
inclination and a 9-nodes grid for the azimuth.

##  4 Results

Figure 1: Stokes $I$ (_top panels_) and $Q/I$ (_bottom panels_) profiles for
the He ii Ly-$\alpha$ line as a function of wavelength. The results are
obtained with the unmagnetised FAL-C atmospheric model and for two LOSs, near
the limb ($\mu=0.1$, _left panels_) and near the disk centre ($\mu=0.9$,
_right panels_), respectively. Blue, red, and yellow lines correspond to CRD,
PRD–AA, and PRD–AD calculations, respectively. The intensity is in units of
number of photons per unit of surface, time, wavelength, and solid angle. The
intensity profiles for PRD–AA and PRD–AD calculations overlap. Figure 2:
Stokes $Q/I$ (_top panels_), $U/I$ (_middle panels_), and $V/I$ (_bottom
panels_) profiles for the He ii Ly-$\alpha$ line as a function of wavelength.
The results are obtained with the FAL-C atmospheric model, with a uniform
horizontal ($\theta_{B}=\pi/2$ and $\chi_{B}=0$) magnetic field of 100 G, and
for two LOSs, near the limb ($\mu=0.1$, _left panels_) and near the disk
centre ($\mu=0.9$, _right panels_), respectively. Blue, red, and yellow lines
correspond to CRD, PRD–AA, and PRD–AD calculations, respectively. The
unmagnetic CRD and PRD–AD cases are given for reference as black dash-dotted
and solid lines, respectively.

The physical problem discussed in Sect. 2 and discretised as outlined in Sect.
3 was implemented as a _MATLAB_ (MATLAB 2023) routine and solved with the
_MATLAB_ _gmres_ solver. In this section, we present the numerical results
obtained for the He ii Ly-$\alpha$ line with two main goals. First, we aim at
assessing the suitability of the CRD and PRD–AA modellings through comparisons
with the general PRD–AD calculations, both in the absence and presence of
magnetic fields. Second, we investigate the potential of the wavelength-
integrated scattering polarisation signal of this line for probing the
magnetism of the solar chromosphere.

###  4.1 Impact of PRD–AD effects

The impact of the AD treatment on the emergent Stokes profiles is first
analysed considering the FAL-C atmospheric model and two lines of sight
(LOSs), with $\mu=0.1$ and $\mu=0.9$. Figures 1 and 2 expose the synthetic
emergent Stokes profiles in the absence and in the presence of a magnetic
field, respectively. For the latter case, as an example of interest, we
considered a uniform horizontal ($\theta_{B}=\pi/2$ and $\chi_{B}=0$) magnetic
field of strength $B=100$ G. In the unmagnetic case, the $U/I$ and $V/I$
profiles vanish and are consequently not shown. In addition, as the $I$
profile is not affected by magnetic fields of practical interest, it is only
displayed for the unmagnetic case in Fig. 1.

In general, we found a good agreement between the CRD, PRD–AA, and PRD–AD
calculations in the core of the line for all Stokes profiles and all magnetic
conditions. Reasonably, the CRD modelling is however not able to reproduce the
large peaks outside the line-core region of the $Q/I$ profiles. Noticeable
discrepancies are also present near the disk centre ($\mu=0.9$) in the $Q/I$
peaks between AA and AD calculations. This is not true near the limb
($\mu=0.1$), where we found an excellent agreement between AA and AD
calculations. We also note that our $Q/I$ profiles are in good agreement with
those shown in Fig. 3 of Belluzzi et al. (2012). This proves the robustness of
our results, considering that the two works were carried out applying
completely different solution strategies.222We recall that the results of
Belluzzi et al. (2012) were obtained applying the redistribution matrix of
Belluzzi & Trujillo Bueno (2014). This differs from the one of Bommier (2017)
in the $R^{\scriptscriptstyle\rm III}$ term, which was derived from heuristic
arguments. This difference is, however, irrelevant in the modeling of He ii
Ly-$\alpha$ because the contribution from
$R^{\scriptscriptstyle\scriptscriptstyle\rm III}$ is completely negligible at
the height where this line is formed (see left panel of Fig. 1 of Belluzzi et
al. 2012).

Figure 2 also shows that, as expected, the $Q/I$ signal is barely sensitive to
the Hanle effect for the considered magnetic field of 100 G (cf. the colour
with the black lines). However, a significant Hanle signal appears in the
$U/I$ profile, particularly near the limb (see the central left panel).
Moreover, as expected because of the reduced effectiveness of the Zeeman
effect at these wavelengths, the circular polarization signal is much weaker
than the linear ones. Notably, the amplitudes of the line-core $Q/I$ and $U/I$
signals are in excellent agreement with the CRD results shown in the left
panel of Fig. 5 of Trujillo Bueno et al. (2012). In addition, the MO effects
seem to play a marginal role for the considered cases. In confirmation of
this, we performed the same calculations by forcing to zero the MO term of the
propagation matrix (see Alsina Ballester et al. 2017), obtaining almost
overlapping emergent $Q/I$ and $U/I$ signals. For the sake of completeness, we
also carried out calculations analogous to Fig. 2, but for vertical magnetic
fields (not shown here for conciseness). For magnetic field strengths typical
of the quiet solar chromosphere, the $Q/I$ signals did not display any
noticeable variation with respect to the unmagnetic case, while the $U/I$
signals were extremely weak and therefore of reduced diagnostic interest.

A striking difference between the CRD and PRD results of Figs. 1 and 2 appears
in the $I$, $Q/I$, and $U/I$ profiles far away from the line centre. In
particular, the amplitude of the fractional polarisation signals at distances
larger than 0.1 Å from the line centre is much larger for CRD than for PRD
calculations. A detailed investigation of these results, not reported here for
conciseness, revealed that this different behaviour is not due to continuum
contributions, which in this spectral region are extremely low. The difference
is rather due to the strong redistribution effects resulting from the slowly
decreasing wings of the Voigt emission profile entering the
$R^{\scriptscriptstyle\rm III}$ redistribution matrix describing the CRD
limit, combined to the very large and sharp emission peak of this line (moving
away from the line centre, the intensity decreases by more than four orders of
magnitude in about 0.1 Å).333We repeated part of the above calculations
considering a broader spectral domain
$[\lambda_{\min},\lambda_{\max}]=[266.04\text{\,\AA},354.00\text{\,\AA}]$ and
we discretised it with 381 spectral nodes. We verified that the continuum
intensity and the $Q/I$ continuum polarisation obtained for the CRD and PRD
treatments are the same, whereas $U/I$ goes to zero moving away from the line
centre, irrespectively of the scattering description. This leads to Stokes
profiles that are much broader for the CRD than for the PRD case.

###  4.2 Sensitivity of Stokes profiles to the atmospheric model

Figure 3: Stokes $Q/I$ (_left panel_), $U/I$ (_middle panel_), and $V/I$
(_right panel_) profiles for the He ii Ly-$\alpha$ line as a function of
wavelength. The results are obtained performing PRD-AD calculations for a LOS
near the limb ($\mu=0.1$ and $\chi=0$) and with a uniform horizontal
($\theta_{B}=\pi/2$ and $\chi_{B}=0$) magnetic field of 100 G. Blue, red,
yellow, and purple lines correspond to profiles obtained with the FAL-A,
FAL-C, FAL-F, and FAL-P atmospheric models, respectively.

To analyse the sensitivity of the He ii Ly-$\alpha$ emergent Stokes profiles
to atmospheric conditions, we considered the four plane-parallel semi-
empirical 1D FAL-A, FAL-C, FAL-F, and FAL-P atmospheric models and we solved
the RT problem in the presence of a uniform horizontal ($\theta_{B}=\pi/2$ and
$\chi_{B}=0$) magnetic field of 100 G. Figure 3 shows the emergent $Q/I$,
$U/I$, and $V/I$ profiles for the aforementioned atmospheric models and a LOS
with $\mu=0.1$ and $\chi=0$. We found a very good agreement in the line core
of the $Q/I$ profile. On the other hand, some relevant differences appear in
the peaks just outside the line-core region. Moreover, the $U/I$ and $V/I$
signals almost overlap for all the considered atmospheric models. We note
that, although we only showed the $\mu=0.1$ case, good agreement was found for
all LOSs. We also note that, although the present $Q/I$ profiles were obtained
in a magnetised atmosphere, these are in excellent qualitative agreement with
those in Fig. 5 by Belluzzi et al. (2012).

###  4.3 Wavelength-integrated linear polarisation signals

Figure 4: Centre-to-limb variation of $\overline{Q}/\overline{I}$ (_left
panel_) and $\overline{U}/\overline{I}$ (_right panel_) signals as a function
of $\mu=\cos(\theta)$. The results are obtained for $\chi=0$ and with the
FAL-C atmospheric model, considering uniform horizontal ($\theta_{B}=\pi/2$
and $\chi_{B}=0$) magnetic fields of different strengths (see the legend).
Dash-dotted, dashed, and solid lines correspond to CRD, PRD–AA, and PRD–AD
calculations, respectively. Figure 5: Plots of $\overline{Q}/\overline{I}$
(_left panel_) and $\overline{U}/\overline{I}$ (_right panel_) as a function
of $B$. The results are obtained performing PRD-AD calculations for three LOSs
with $\mu=0.1,\,0.6,$ and 0.9 (solid, dashed, and dash-dotted lines,
respectively), $\chi=0$, and considering uniform horizontal
($\theta_{B}=\pi/2$ and $\chi_{B}=0$) magnetic fields of different amplitudes.
Blue, red, yellow, and purple lines correspond to profiles obtained with the
FAL-A, FAL-C, FAL-F, and FAL-P atmospheric model, respectively.

We now focus on the He ii Ly-$\alpha$ linear polarisation signals obtained
using narrowband filters. Synthetic narrowband signals can be obtained by
integrating over wavelength the numerical emergent Stokes profiles, weighted
by a Gaussian that mimics the action of a narrowband filter. Namely, (Alsina
Ballester et al. 2023)

$\overline{X}(\mu)=\frac{1}{\sqrt{2\pi}\sigma_{\lambda}}\int_{\lambda_{0}-\Delta\lambda}^{\lambda_{0}+\Delta\lambda}{\rm
d}\lambda\exp\left(-\frac{(\lambda-\lambda_{0})^{2}}{2\sigma_{\lambda}^{2}}\right)X(z_{\max},\mu,\lambda),$
(7)

where $X=I,Q,U$, $\lambda_{0}=303.784$ Å is the He ii Ly-$\alpha$ line centre
wavelength in vacuum, and $\sigma_{\lambda}$ is the standard deviation of the
Gaussian weighting function, which corresponds to a full width at half maximum
$\text{FWHM}=2\sqrt{2\log(2)}\sigma_{\lambda}$ (here we do not consider the
circular polarization signal as its profile is symmetric and thus $\bar{V}$
vanishes). We note that the choice of FWHM affects the amplitude of the
narrowband signals $\overline{I}$, $\overline{Q}$, and $\overline{U}$, but it
does not impact the ratios $\overline{Q}/\overline{I}$ and
$\overline{U}/\overline{I}$ and their dependency on $\mu$. In addition, any
choice with $\Delta\lambda\geq 0.5$ Å has no impact on the results, because
both the intensity and the scattering polarisation signals quickly drop to
zero moving away from the line centre. Thus, in the calculations, we adopted a
FWHM of 1 Å and we used $\Delta\lambda=1$ Å.

Figure 4 displays the CLV of the narrowband $\overline{Q}/\overline{I}$ and
$\overline{U}/\overline{I}$ signals for LOSs with $\mu\in[0.1,0.9]$,
considering the FAL-C atmospheric model in the presence of uniform horizontal
magnetic fields of strength between 0 and 1000 G. For $B\leq 200$ G, the
amplitude of the $\overline{Q}/\overline{I}$ and $\overline{U}/\overline{I}$
signals increases from the disk centre to the limb for all the scattering
descriptions, reaching a maximum close to $\mu=0.2$, and then slightly
decreases towards $\mu=0.1$. Moreover, the $\overline{Q}/\overline{I}$ signal
proves to be almost insensitive to magnetic fields of strengths $B\leq 50$ G,
while its amplitude decreases for $200\leavevmode\nobreak\ \mathrm{G}\geq
B\geq 50$ G due to the Hanle effect. Conversely, the amplitude of the
$\overline{U}/\overline{I}$ signal increases almost linearly with $B$ for all
the considered inclinations. For $B>200$ G, the amplitude of
$\overline{U}/\overline{I}$ reaches a maximum near $B=500$ G, and then
decreases for stronger magnetic fields due to Hanle rotation. Furthermore, we
note that $\bar{Q}/\bar{I}$ signals become positive approaching the disk
centre, with a larger amplitude for stronger magnetic fields. This is due to
the forward-scattering Hanle effect (Trujillo Bueno et al. 2002). In general,
we found a very good agreement between CRD, PRD–AA, and PRD–AD calculations,
although the CRD description slightly underestimates the amplitude of the PRD
results, in particular for $\mu<0.5$. Noticeably, the agreement between PRD–AA
and PRD–AD calculations decreases near the disk centre for $B\geq 500$ G.

To investigate the dependence of the synthetic narrowband signals on
atmospheric parameters, in Fig. 5 we present $\overline{Q}/\overline{I}$ and
$\overline{U}/\overline{I}$ as a function of the strength of the horizontal
magnetic field for the three different LOSs $\mu=0.1,\,0.6,$ and 0.9 and for
the four aforementioned FAL atmospheric models. The dependence of all signals
on the considered atmospheric models is negligible. Moreover, the left panel
of Fig. 5 shows that $\overline{Q}/\overline{I}$ is almost flat for $B<50$ G,
while it reveals a slight Hanle depolarisation for stronger magnetic fields,
in agreement with the results presented in Fig. 4. In addition, in the right
panel of Fig. 5 we see that the amplitude of the appreciable
$\overline{U}/\overline{I}$ signal increases almost linearly with $B$, also in
agreement with the results in Fig. 4.

##  5 Discussion and conclusions

In Sect. 4.3, we showed that synthetic narrowband signals of the He ii
Ly-$\alpha$ line are insensitive to the considered atmospheric models. At the
same time, a study of the impact of bulk velocities on the He ii Ly-$\alpha$
line, not reported here for conciseness, revealed that
$\overline{Q}/\overline{I}$ and $\overline{U}/\overline{I}$ are also
insensitive to typical chromospheric plasma bulk velocities.444We repeated
part of the calculations presented above for a vertical bulk velocity with a
uniform gradient of 0.009 1/s. The results do not display any noticeable
difference. We also showed that, although the Stokes profiles are nearly
immune to MO effects, $\overline{U}/\overline{I}$ displays an almost linear
dependence on $B$, via the Hanle effect, for magnetic strengths typical of the
solar chromosphere, and that $\overline{Q}/\overline{I}$ is almost insensitive
to horizontal magnetic fields of strength weaker than 50 G. Consequently,
since $|\overline{U}/\overline{I}|>0.3\%$ near the solar limb for a horizontal
magnetic field of 100 G, He ii Ly-$\alpha$ linear polarisation signals
obtained using narrowband filters may represent an interesting novel
capability for inferring chromospheric magnetic fields, provided that a
sufficient signal-to-noise ratio can be achieved.

It is thus important to assess what is the integration time, $t_{\text{int}}$,
necessary to be sensitive to chromospheric magnetic fields for a given
instrument. In this respect, we applied Eq. (56) in Trujillo Bueno et al.
(2017) considering: a near-limb LOS, for which $|\overline{U}/\overline{I}|$
and $|\overline{Q}/\overline{I}|$ are maximal; a polarimetric sensitivity of
$0.05\,\%$ that, in the present geometry, means a sensitivity to horizontal
magnetic fields of about 10 G; an instrumental efficiency of $1\,\%$; a
spatial resolution of 2 arcsec; a diameter of the telescope of 30 cm, similar
to the instrument installed on the CLASP experiment (Kobayashi et al. 2012);
and a FWHM of the narrowband filter of 1 Å. Given these constraints, we found
that the integration time necessary to be sensitive to horizontal
chromospheric magnetic fields of 10 G is $t_{\text{int}}=885$, 447, 157, and
22 s for the atmospheric models FAL-A, FAL-C, FAL-F, and FAL-P, respectively.
We note that, while $\overline{Q}/\overline{I}$ and
$\overline{U}/\overline{I}$ are independent of the atmospheric model,
$t_{\text{int}}$ is not, as the intensity signal, and thus the number of
photons reaching the telescope, strongly depend on the atmospheric temperature
structure.

Concerning quite-solar conditions (i.e. models FAL-A, FAL-C, and FAL-F), the
resulting $t_{\text{int}}$ are too large to be reached through sounding rocket
experiments. On the other hand, the integration time for the FAL-P model,
which corresponds to a typical plage area, is realistic for present days’
rockets (e.g. CLASP). Overall, valuable He ii Ly-$\alpha$ polarization signals
could also be measured in quiet regions, across the whole solar disk, with
spacecraft missions, as these allow for considerably larger telescopes and
longer exposure times than sounding rocket experiments.

Another interesting outcome of the work is the very good agreement of
wavelength integrated He ii Ly-$\alpha$ fractional linear polarisation signals
obtained from CRD, PRD–AA, and PRD–AD calculations, especially for magnetic
fields up to 200 G. This means that a complete PRD–AD description of
scattering processes seems not to be strictly necessary to accurately model
the impact of such magnetic fields on these signals. The CRD formulation could
be enough. This would drastically reduce the computational cost of inferring
magnetic fields from future He ii Ly-$\alpha$ observations.

We remind that the results in the present paper were obtained using a semi-
empirical 1D model. However, the solar atmosphere is definitely not 1D, but it
has large horizontal inhomogeneities as well as bulk velocities with strong
gradients, which can strongly impact the amplitude and shape of scattering
polarisation signals (e.g. Manso Sainz & Trujillo Bueno 2011; Štěpán &
Trujillo Bueno 2016; Jaume Bestard et al. 2021). This is particularly true in
the upper chromosphere. Consequently, in the future it will be essential to
assess the impact of these effects on the polarisation signals of the He ii
Ly-$\alpha$ line through full 3D RT calculations in state-of-the-art models of
the solar atmosphere. The same 3D calculations, to be performed both in CRD
(e.g. using the PORTA code of Štěpán & Trujillo Bueno 2013) and in PRD (e.g.
using a code based on the solution strategy outlined in Benedusi et al. 2023),
would confirm whether also in the presence of such effects, the limit of CRD
remains a good approximation for modeling the narrowband signals of this line.

This work highlights the diagnostic potential of filter-polarimetry in the He
ii Ly-$\alpha$ line. Our 1D results, to be confirmed by more accurate 3D
calculations, suggest that in near-limb plage regions the sensitivity of
narrowband He ii Ly-$\alpha$ linear polarisation signals to chromospheric
magnetic fields can already be assessed through sounding rocket experiments.
Given its minor sensitivity to horizontal magnetic fields,
$\overline{Q}/\overline{I}$ could be used as a reference, while
$\overline{U}/\overline{I}$ would be exploited to infer information on the
magnetic field in the upper chromospheric layers. Notably, given the very
narrow region in which the He ii Ly-$\alpha$ line forms (Trujillo Bueno et al.
2012; Belluzzi et al. 2012), we would also have strong constraints on the
precise location of the estimated magnetic fields.

###### Acknowledgements.

The financial support by the Swiss National Science Foundation (SNSF) through
grant CRSII5_180238 is gratefully acknowledged.

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