# Constraining $z\lesssim 2$ ultraviolet emission
with the upcoming ULTRASAT satellite

Sarah Libanore<EMAIL_ADDRESS>Department of Physics, Ben-Gurion
University of the Negev, Be’er Sheva 84105, Israel  Ely D. Kovetz
<EMAIL_ADDRESS>Department of Physics, Ben-Gurion University of the Negev,
Be’er Sheva 84105, Israel

###### Abstract

The Extragalactic Background Light (EBL) carries a huge astrophysical and
cosmological content: its frequency spectrum and redshift evolution are
determined by the integrated emission of unresolved sources, these being
galaxies, active galactic nuclei, or more exotic components. The near-UV
region of the EBL spectrum is currently not well constrained, yet a
significant improvement can be expected thanks to the soon-to-be launched
Ultraviolet Transient Astronomy Satellite (ULTRASAT). Intended to study
transient events in the 2300-2900 Å observed band, this detector will provide
wide field maps, tracing the UV intensity fluctuations on the largest scales.
In this paper, we suggest how to exploit ULTRASAT full-sky map, as well as its
low-cadence survey, to reconstruct the redshift evolution of the UV-EBL volume
emissivity. We build upon the work of Chiang et al. (2019), where the
Clustering-Based Redshift (CBR) technique was used to study diffuse light maps
from GALEX. Their results showed the capability of the cross correlation
between GALEX and SDSS spectroscopic catalogs in constraining the UV
emissivity, highlighting how CBR is sensitive only to the extragalactic
emissions, avoiding foregrounds and Galactic contributions. In our analysis,
we introduce a framework to forecast the CBR constraining power when applied
to ULTRASAT and GALEX in cross correlation with the 5-year DESI spectroscopic
survey. We show that these will yield a strong improvement in the measurement
of the UV-EBL volume emissivity. For $\lambda=1500\,$Å non-ionizing continuum
below $z\sim 2$, we forecast a $1\sigma$ uncertainty $\lesssim 26\%\,(9\%)$
with conservative (optimistic) bias priors using ULTRASAT full-sky map;
similar constraints can be obtained from its low-cadence survey, which will
provide a smaller but deeper map. We finally discuss how these results will
foster our understanding of UV-EBL models.

##  I Introduction

Across the whole electromagnetic spectrum, light from unresolved sources of
astrophysical and cosmological origins overlaps, producing the so-called
Extragalactic Background Light (EBL). Such integrated emission is observed and
constrained through different probes; reviews in Refs. Cooray (2016); Hill et
al. (2018) describe the main features of the EBL and highlight the degree of
accuracy reached in the various frequency bands. Some regimes have been widely
analyzed, for instance the microwave band where the cosmic microwave
background (CMB) is observed or the optical-infrared region that collects
redshifted emission from stars. Other frequency bands, instead, are still
largely unprobed: the near-ultraviolet (NUV) to far-ultraviolet (FUV) region
is one of these. Between $\nu_{\rm obs}\sim 1$ \- $3\cdot 10^{6}\,{\rm GHz}$
($\lambda_{\rm obs}\sim 3000$ \- $1000\,$Å) a lower EBL limit has been
estimated both using the integrated light from galaxy number counts and
through photometric measurements from the Voyager 1 and 2 missions Murthy et
al. (1999); Edelstein et al. (2000), Hubble Brown et al. (2000); Bernstein et
al. (2002) and the Galaxy Evolution Explorer (GALEX, Martin et al. (2005);
Morrissey et al. (2007); Murthy et al. (2010)), even if with large statistical
and systematic errors Cooray (2016). This lack of measurements is not only due
to technological issues: extragalactic photons in this regime get mostly
absorbed by neutral hydrogen in the intergalactic medium (IGM) and inside the
Milky Way. Moreover, Galactic light in this part of the spectrum provides a
strong foreground contribution. This problem also affects other wavebands, for
example the optical range; techniques have been developed to characterize and
remove the foregrounds (see, e.g., the measurements from the New Horizon
mission in Ref. Symons et al. (2023)).

The study of the UV background plays an important role in understanding
structure formation and star formation history (depicted e.g., in Refs. Madau
& Dickinson (2014); Bernstein et al. (2002) and references within). First of
all, its spectral energy distribution provides an unbiased estimate of the
energy produced by star formation. Moreover, the EBL can be used to study the
chemical enrichment history of the Universe, the total baryon fraction in
stars and the local mass density of metals. The background radiation in this
frequency band is highly correlated with the one in the infrared range Sasseen
et al. (1995); Schiminovich et al. (2001), which is consistent with an origin
related with dust-scattered stellar radiation. The UV-EBL can also contain
contributions of cosmological origin, such as black hole accretion in quasars
and active galactic nuclei (AGN), direct-collapse black hole formation or
decaying and annihilating dark matter (see e.g., Refs. Dwek et al. (1998);
Bond et al. (1986); Yue et al. (2013); Henry et al. (2015); Creque-Sarbinowski
& Kamionkowski (2018); Kalashev et al. (2019); Bernal et al. (2021); Carenza
et al. (2023)).

Improving our knowledge of the UV-EBL is therefore a goal that should be
pursued with upcoming instruments and surveys. In this context, a very
interesting opportunity will soon be offered by the Ultraviolet Transient
Astronomy Satellite (ULTRASAT, Sagiv et al. (2014); Ben-Ami et al. (2022);
Shvartzvald et al. (2023)), whose launch is scheduled for 2026. ULTRASAT
foresees as its primary goal the study of astrophysical transients in the NUV
band. In order to achieve this, part of the first six months of the mission
will be dedicated to the realization of a full sky map,111The map will use an
integration time of $15,000$ seconds at Galactic latitudes $|b|>30\,$deg. See
the ULTRASAT official website for more details. to be used as a reference
frame for subsequent observations. Such map will reach a limiting AB magnitude
of 23.5. Moreover, during its regular science operation, ULTRASAT will scan
repeatedly $\sim 40$ selected extragalactic fields, in the so called low-
cadence survey (LCS, Shvartzvald et al. (2023)).222In the low-cadence survey,
ULTRASAT plans to observe 40 extragalactic fields with 900s integration time
each; over the full duration of the mission, this will result in a theoretical
limiting magnitude $\sim 24.5$ \- $25.5$.  Preliminary studies on the PSF and
on the expected confusion noise, suggest that these images will provide a
$\sim 6800\,{\rm deg}^{2}$ map with magnitude limit $\sim 24.5$.

These outputs will provide a $\sim 10$ \- $100$ times sensitivity improvement
with respect to the currently existing benchmark, namely the diffuse UV light
map constructed in Ref. Murthy (2014).

This work used data from the All Sky and Medium Imaging Surveys performed by
GALEX, masking resolved sources and binning pixels in order to estimate the
combined contribution of the UV-EBL Sujatha et al. (2009, 2010) and
foregrounds, these being sourced by Galactic dust-scatter light, near-Earth
airglow, or other Hamden et al. (2013); Murthy (2013); Henry et al. (2015);
Akshaya et al. (2018); the origin of part of the foreground is yet
unidentified. The GALEX map provided important results on the study of dust
scattering emissions (e.g., Refs. Murthy (2016); Chiang & Ménard (2019)),
correlation between UV and IR emissions (e.g., Refs. Murthy (2014); Saikia et
al. (2017)), and extragalactic sources (e.g., Refs. Welch et al. (2020);
Chiang et al. (2019)). The frequencies probed by GALEX allowed to access the
UV-EBL between redshifts $z\\!\sim\\!0-1.5$; ULTRASAT will allow us to improve
the sensitivity around $z\sim 1$, pushing the observations towards slightly
higher $z$.

An innovative application has been presented by the authors of Chiang et al.
(2019) ( C19 throughout this paper). UV satellites as GALEX and ULTRASAT
observe a wide frequency range; therefore, radiation with various rest frame
wavelengths, emitted at different cosmological distances from the observer,
can be redshifted inside their observational band. The redshift dependence of
the integrated, observed emission can be reconstructed through broadband
tomography, which is based on the cross correlation between UV data and
spectroscopic galaxy samples. This technique, also referred to as Clustering-
Based Redshift (CBR), was originally developed in Refs. Newman (2008); McQuinn
& White (2013); Ménard et al. (2013), and applied in different contexts, for
instance to obtain a statistic estimate of the redshift of photometric samples
Rahman et al. (2015); Scottez et al. (2016); Morrison et al. (2017); Davis et
al. (2018); van den Busch et al. (2020), to study the cosmic infrared
background Schmidt et al. (2015); Cheng & Chang (2022), or the presence of
extragalactic sources in Galactic dust maps Chiang & Ménard (2019), and to
improve the constraining power of radio surveys on cosmological parameters
Kovetz et al. (2017a); Scelfo et al. (2022). Cross correlation between EBL and
galaxy surveys have also been used to test star formation models Sun et al.
(2023) and to the derive luminosity and mass functions for photometric galaxy
catalogs Bates et al. (2019).  C19 applied this technique to the GALEX map,
using catalogs from the Sloan Digital Sky Survey (SDSS, Blanton et al. (2005);
Reid et al. (2016)), to reconstruct the redshift evolution of the UV-EBL
volume emissivity. A follow-up of this work was realized in Scott et al.
(2022) ( S21 in the paper), where the authors applied a similar formalism to
forecast the constraining power of the Cosmological Advanced Survey Telescope
for Optical and UV Research (CASTOR, Cote et al. (2019)) in cross correlation
with Spectro-Photometer for the History of the Universe, Epoch of Reionization
and Ices Explorer (SPHEREx, Doré et al. (2014, 2016, 2018)).

The goal of our work is to build on the  C19 analysis, applying it to the
forecasted capabilities of ULTRASAT in cross correlation with upcoming
spectroscopic galaxy surveys, in particular the Dark Energy Spectroscopic
Instrument (DESI, Levi et al. (2013); Aghamousa et al. (2016a, b)). When the
full release of DESI data will be available, it will also be possible to cross
correlate its spectroscopic catalogs with the GALEX map from Ref. Murthy
(2014), and further improve the results from  C19. In our analysis, therefore,
we consider the possibility of combining together GALEX and ULTRASAT maps with
DESI, to perform a powerful CBR analysis, to access the UV-EBL particularly
around $z\sim 1$.

Our paper is structured as follows. We begin in Sect. II by presenting the
ULTRASAT satellite and giving some information about its specifications, in
particular with respect to the full-sky map and the estimate of their noises.
The approach we follow is inspired by line-intensity mapping science (see
e.g., Refs. Kovetz et al. (2017b); Bernal & Kovetz (2022) for review). We
proceed in Sect. III by modeling the signal that we aim to constrain, namely
the UV volume emissivity (III.1). We then build the observable of interest in
the context of the CBR technique, namely the angular cross correlation between
an intensity map and a spectroscopic galaxy survey (III.2 and III.3), for
which we also provide a noise estimate (III.4). Section IV presents our
forecast analysis and collects our results. In particular, we start by
reproducing the  C19 constraints for GALEX$\times$SDSS (IV.2), and then we
proceed by applying the technique to ULTRASAT$\times$DESI and
(GALEX$+$ULTRASAT)$\times$DESI, comparing our forecasted results for the
volume UV-EBL emissivity (IV.3) with various constraints available in the
literature. We show that this approach will yield a strong improvement in the
measurement of the UV-EBL volume emissivity. Our conclusions are presented and
discussed in Sect. V.

##  II The ULTRASAT satellite

The observational window of ULTRASAT will span between $\lambda_{\rm
min}=2300\,$Å, $\lambda_{\rm max}=2900\,$Å, which implies an observed
frequency range centered at $\nu_{\rm obs}=1.17\cdot 10^{6}\,{\rm GHz}$. To
help understanding which redshifts can be mapped using these wavelengths, we
consider Lyman-$\alpha$ (Ly$\alpha$) line emission with rest frame
$\lambda_{\rm Ly\alpha}=1216\,$Å: provided that $\lambda_{\rm
obs}=\lambda_{\rm rest}(1+z)$, ULTRASAT will observe $z_{\rm
Ly\alpha}\in[0.9,1.4]$.

In our analysis in Sect. IV, we assume to rely on the UV map obtained during
the first six months of the ULTRASAT mission. We assume that these will be
realized using a technique similar to the one Ref. Murthy (2014) applied to
GALEX data. The dataset we assume to analyze, hence consists of UV intensity
values measured in the pixels of each map; the map is characterized by an
intensity average value, which we also refer to as the monopole, and by
spatial fluctuations; this indeed makes its study analogous to what is done in
line-intensity mapping surveys Kovetz et al. (2017b); Bernal & Kovetz (2022).

For both ULTRASAT full-sky map and GALEX, the observed field is almost full-
sky. The pixel scale is $5.4^{\prime\prime}/{\rm pix}$ for ULTRASAT,
$5^{\prime\prime}/{\rm pix}$ for GALEX. The response function $R(\lambda_{\rm
obs})$ of the instruments is defined in terms of the quantum efficiency, that
is the number of electrons per incident photon depending on the frequency,
normalized through

$\int_{\lambda_{\rm obs}^{\rm min}}^{\lambda_{\rm obs}^{\rm max}}d\lambda_{\rm
obs}\frac{R(\lambda_{\rm obs})}{\lambda_{\rm obs}}=1.$ (1)

We compute the ULTRASAT response function based on information in Refs.
Bastian-Querner et al. (2021); Asif et al. (2021), and we show it in Fig. 1.
Here, we also show $R(\lambda_{\rm obs})$ for the two GALEX filters,
respectively centered in the NUV range $\lambda_{\rm obs}\in[1750,\,2800]\,$Å
and in the FUV range $\lambda_{\rm obs}\in[1350,\,1750]\,$Å.333GALEX response
functions are provided in the public repository http://svo2.cab.inta-
csic.es/svo/theory/fps3/. GALEX NUV filter observes Ly$\alpha$ in a similar
range with respect to ULTRASAT, while GALEX FUV refers to a lower redshift
regime, $z_{\rm Ly\alpha}\in[0.1,0.4]$. GALEX FUV and NUV filters, therefore,
are complementary in probing Ly$\alpha$ emissions in the local Universe;
ULTRASAT will improve the sensitivity mostly at $z\sim 1$.

The signal of interest for our analysis is modeled in the next section; here
instead we provide an estimate of the noise variance in the maps. As  C19
details, masking the pixels with flux above the detection limit, resolved
sources can be separated from the diffuse light contribution. This separation
on the map level helps to implement the algorithms required to mitigate the
foreground noise; on the other hand, a combined analysis of the two components
allows us to study the UV intensity field without selection effects. From now
on, we thus assume that a procedure similar to  C19 has been performed:
resolved sources and diffuse light are initially separated to create the maps
and estimate their noises, and then summed to perform the analysis.

The noise variance per pixel in the “overall” map is estimated based on the
$5\sigma$ AB magnitude limit $m_{\rm AB}$, weighted by the different exposure
times during the observations. In the case of GALEX NUV and FUV,  C19
indicates $m_{\rm AB}\sim 20.5$ \- $23.5$; we adopt the intermediate value
$m_{\rm AB}=22$.444The value is chosen consistently with analysis in  C19. We
thank Y. K. Chiang for discussion. For ULTRASAT reference map instead,
following Ref. Shvartzvald et al. (2023), we assume $m_{\rm AB}=23.5$, so to
get a $\times 10$ sensitivity improvement from GALEX.

The limiting magnitude characterizes the detection level above which sources
(with Galactic or extragalactic origin) cannot be resolved; for this reason,
we can use it to describe the ground level of the observed intensity inside a
region with comparable size to the point spread function. We convert $m_{\rm
AB}$ to flux per pixel via $m_{\rm AB}=8.90-2.5\log_{10}\mathcal{F}_{\nu}$,
and we rescale it to flux density inside a certain angular region as
$f_{\nu}^{\rm pix}=\mathcal{F}_{\nu}/A_{\rm pix}$. The pixel surface area is
computed as $A_{\rm pix}=L_{\rm pix}^{2}$, where $L_{\rm pix}$ is the pixel
scale. The value obtained corresponds to sources $5$ times brighter than the
noise level; hence, we get the total noise variance per pixel

$\sigma_{N}^{2}=\left[{f_{\nu}^{\rm pix}}/{5}\right]^{2}.$ (2)

To reduce the value of $\sigma_{N}^{2}$, it is always possible to combine the
pixels in groups, which we call “effective pixels”: this reduces their number,
and it averages noise fluctuation on the smallest scales. We will discuss in
Sect. III.4 that, in the context of the CBR analysis,  C19 used effective
pixels with $L_{\rm pix}^{\rm eff}=50\,^{\prime\prime}/{\rm pix}$; we will
show that instead, ULTRASAT will obtain good results already using $L_{\rm
pix}$. Note that, so far, we did not account for foreground contributions;
these will be described in Sect. III.4.

Table 1 collects the main survey specifications we discussed so far, and it
compares their estimated $\sigma_{N}^{2}$ value. The table also reports the
value of the observed monopole $J_{\nu_{\rm obs}}$, which we define in Sect.
III.2.

| $\lambda_{\rm obs}$ | $z_{\rm Ly\alpha}$ | $L_{\rm pix}^{\rm(eff)}$ | $J_{\nu_{\rm obs}}$ | $\sigma_{N}^{2}$
---|---|---|---|---|---
| [Å] | [“/px] | [Jy/sr] |  [${\rm Jy^{2}/sr^{2}}$]
ULTRASAT | [2300, 2900] | [0.9, 1.4] | 5.45 | 264 | 422
GALEX | [1350, 1750] | [0.1, 0.4] | 5 (50) | 237 | 1959
[1750, ,2800] | [0.4, 1.3] | 79
Table 1: Specs of the broadband surveys ULTRASAT Shvartzvald et al. (2023) and
GALEX Martin et al. (2005). The redshift range $z_{\rm Ly\alpha}$ is computed
for the Ly$\alpha$ line $\lambda_{\rm Ly\alpha}=1216\,$Å; other parts of the
UV spectrum are shifted to different ranges. We indicate both the pixel scale
$L_{\rm pix}$ and (in parenthesis) the effective scale of the pixels used in
the analysis in Sect. IV. The monopole $J_{\nu_{\rm obs}}$ and the variance
$\sigma_{N}^{2}$ are estimated using respectively Eqs. (10) and (2).

##  III Modeling the signal and noise

The broadband measurements ULTRASAT will realize integrate the radiation
emitted over a wide frequency and redshift range. Reconstructing the spectral
shape of the signal, as well as its evolution across cosmic time, is of
fundamental importance to disentangle cosmological dependencies and
astrophysical properties of the emitters. One way this goal can be pursued is
via the broadband tomography, or clustering-based redshift (CBR) technique
Newman (2008); McQuinn & White (2013); Ménard et al. (2013), namely the cross
correlation between one broadband survey (ULTRASAT or GALEX, in our case) and
a reference spectroscopic galaxy catalog. Using cross correlations allows us
to overcome the presence of Galactic foregrounds, ensuring that the measured
signal has extragalactic origin. The possibility of chance correlation is
further reduced by the fact that foregrounds fluctuate mainly at large scales,
while we are interested in cross correlations on small scales. Hence, in our
analysis, foregrounds only contribute to the noise budget.

The authors of  C19 applied CBR to GALEX data, in order to constrain the
redshift evolution of the UV-EBL volume emissivity. In their case, a subset of
the SDSS catalogs Blanton et al. (2005); Reid et al. (2016) was adopted as
reference. ULTRASAT, with its improved NUV capability, seems to be the perfect
heir for this kind of analysis. Considering its timeline, a bunch of state-of-
the-art galaxy surveys will be available to perform the cross correlation. As
we will detail in Sect. III.4, the best results are obtained for small
spectroscopic redshift uncertainties and large galaxy number densities in the
range under analysis. For this reason, we decided to rely on the DESI Levi et
al. (2013); Aghamousa et al. (2016a, b) forecasted 5 years capability, which
are depicted in Ref. Adame et al. (2024).

To forecast the constraining power ULTRASAT$\times$DESI will have compared to
GALEX$\times$SDSS, we introduce the quantities required to model the
observables.

In Sect. III.1, we follow the formalism from  C19 and  S21 and we describe the
UV-EBL volume emissivity that we aim to constrain. The emissivity is then
converted into observed intensity in Sect. III.2. We then review how CBR can
reconstruct its redshift evolution in Sect. III.3, and we estimate the
uncertainty in its measurement in Sect. III.4. Figure 1 summarizes the logic
and modeling of the analysis.

###  III.1 UV volume emissivity

The first ingredient we need in order to model the signal is the spatially
averaged comoving volume UV emissivity, $\epsilon(\nu,z)$ $[\rm
erg\,s^{-1}Hz^{-1}Mpc^{-3}]$. In analogy to  C19 and  S21, we parameterize its
frequency and redshift dependencies using a piecewise function, capable of
describing the different spectral contributions the EBL receives. This model
is in agreement with simulation results based on the study of the radiative
transfer of Lyman continuum photons in the IGM Haardt & Madau (2012).

We distinguish between the non-ionizing continuum at $\lambda_{\rm
rest}>1216\,$Å and between $912\,$Å$<\lambda_{\rm rest}\leq 1216\,$Å, the
Ly$\alpha$ line at $\lambda_{\rm rest}=1216\,$Å, and the ionizing continuum at
$\lambda_{\rm rest}\leq 912\,$Å, below the Lyman break. Above Ly$\alpha$, we
describe the frequency and redshift evolution of the non-ionizing continuum
with respect to a pivot value as

$\epsilon(\nu,z)=\epsilon_{1500}\left[\frac{\nu}{\nu_{1500}}\right]^{\alpha_{1500}},$
(3)

where $\nu_{1500}$ is the frequency at $\lambda_{\rm rest}=1500\,$Å.

Between the Ly$\alpha$ line and the Lyman break, we adopt

$\epsilon(\nu,z)=\epsilon_{1500}\left[\dfrac{\nu_{1216}}{\nu_{1500}}\right]^{\alpha_{1500}}\begin{cases}&\left[\dfrac{\nu}{\nu_{1216}}\right]^{\alpha_{1100}}\\\
&\dfrac{{\rm
EW}}{\nu_{1216}\delta}\dfrac{\nu^{2}}{c}+\left[\dfrac{\nu}{\nu_{1216}}\right]^{\alpha_{1100}}\end{cases}$
(4)

where the first line holds for $|\nu-\nu_{1216}|>\nu_{1216}\delta$, with
$\delta=0.005$ and line frequency $\nu_{1216}=2.5\times 10^{6}\,{\rm GHz}$.
Finally, in the ionizing range we set

$\epsilon(\nu,z)=\epsilon_{1500}\left[\frac{\nu_{1216}}{\nu_{1500}}\right]^{\alpha_{1500}}f_{\rm
LyC}\left[\frac{\nu_{912}}{\nu_{1216}}\right]^{\alpha_{1100}}\left[\frac{\nu}{\nu_{912}}\right]^{\alpha_{900}},$
(5)

where $f_{\rm LyC}$ represents the ionizing photons escape fraction, and it
determines the presence of the Lyman break and Lyman forest in the spectra of
galaxies and quasars Meiksin & White (2003); Khaire et al. (2019).

The redshift dependencies of Eqs. (3), (4), (5) are encapsulated in the
normalization and slope parameters of the continuum, namely

$\displaystyle\epsilon_{1500}=\epsilon_{1500}^{z=0}(1+z)^{\gamma_{1500}},$ (6)
$\displaystyle\alpha_{1500}=\alpha_{1500}^{z=0}+C_{1500}\log_{10}(1+z),$
$\displaystyle\alpha_{1100}=\alpha_{1100}^{z=0}+C_{1100}\log_{10}(1+z),$
$\displaystyle\alpha_{900}=\alpha_{900}^{z=0},$

as well as in the line equivalent width,555The equivalent width is usually
defined as the side of a rectangle whose height is equal to the continuum
emission and the area is the same as the line itself, namely ${\rm EW}=\int
d\lambda(1-J_{s}(\lambda)/J_{c}(\lambda))$, where $J_{c}(\lambda)$ is the
continuum intensity, while $J_{s}(\lambda)=J_{l}(\lambda)+J_{c}(\lambda)$ is
the spectrum intensity, including both the continuum and the line. In our
convention, a positive value of EW gives rise to an emission line, while a
negative value produces an absorption line.

$\displaystyle{\rm EW}=C_{\rm Ly\alpha}\log_{10}\left(\frac{1+z}{1+z_{\rm
EW1}}\right)+{\rm EW}^{z=z_{\rm EW1}},$ (7) $\displaystyle C_{\rm
Ly\alpha}=\frac{{\rm EW}^{z=z_{\rm EW2}}-{\rm EW}^{z=z_{\rm
EW1}}}{\log_{10}[(1+z_{\rm EW2})/(1+z_{\rm EW1})]}\,,$

and in the Lyman continuum escape fraction in the IGM

$\displaystyle\log_{10}f_{\rm LyC}=C_{\rm
LyC}\log_{10}\left(\frac{1+z}{1+z_{\rm C1}}\right)+\log_{10}f_{\rm
LyC}^{z=z_{\rm C1}}\,,$ (8) $\displaystyle C_{\rm LyC}=\frac{\log_{10}f_{\rm
LyC}^{z=z_{\rm C2}}-\log_{10}f_{\rm LyC}^{z=z_{\rm C1}}}{\log_{10}[(1+z_{\rm
C2})/(1+z_{\rm C1})]}\,.$

Overall, as Fig. 1 shows in the top panel, the amplitude of the volume
emissivity is set by $\epsilon_{1500}(z)$, while the characteristics of the
Ly$\alpha$ line are determined by ${\rm EW}(z)$, and the amplitude of the
Lyman break by $f_{\rm LyC}(z)$. At each $z$, the $\alpha_{l}(z)$ parameters
$(l=1500,1100,900)$ set the slopes of the frequency dependence in the
different regimes.

Figure 1: Summary of the modeling in our analysis. We characterize the UV
volume emissivity $\epsilon(\nu,z)$ based on Eqs. (3) to (6) (panel 1).
Emissions from different $z$ contribute to the EBL (panel 2, the dots show
$\lambda_{\rm rest}=1500\,$Å; the Ly$\alpha$ line is in absorption at low
$z$), which is weighted by the detector response function in the broadband
observation in Eq. (10) (panel 3). The CBR allows us to reconstruct
$b_{J}(z)dJ_{\nu_{\rm obs}}/dz$ in Eqs. (11), (13) (panel 4). Figure inspired
by Ref. Chiang et al. (2019).

The upper part of Table 2 collects the parameters through which we
parameterize the volume emissivity $\epsilon(\nu,z)$, together with their
fiducial values, which we choose to match the posterior results of  C19. As
for the $f_{\rm LyC}^{z_{\rm C1}},\,f_{\rm LyC}^{z_{\rm C2}}$ parameters, the
analysis in  C19 only provided 3$\sigma$ upper bounds; in the following, we
hence set these parameters to the largest values allowed by their results.
This is done to show the capability ULTRASAT will have in improving the
constraints on such upper limits; in Sect. IV.4 we discuss the implications in
the case they are set to smaller values.

To analyze both the GALEX NUV and FUV filters,  C19 sets the pivot points of
the EW in Eq. (7) to $\\{z_{\rm EW1}=0.3,\,z_{\rm EW2}=1\\}$, and the ones of
the escape fraction in Eq. (8) to $\\{z_{\rm C1}=1,\,z_{\rm C2}=2\\}$. In our
ULTRASAT analysis, we decided to keep the same pivot points for the escape
fraction, while we pivoted the EW at $\\{z_{\rm EW1}=1,\,z_{\rm EW2}=2\\}$:
the higher redshift has been chosen according to the ULTRASAT $\lambda_{\rm
obs}$ regime, which collects emissions from the Ly$\alpha$ line at $z_{\rm
Ly\alpha}\in[0.9,\,1.4]$ (compare with Table 1) and from the Lyman break at
$z_{912}\in[1.5,\,2.2]$. To get a continuous signal, we estimated ${\rm
EW}^{z=2}$ from Eq. (7), using the  C19 pivots points. Figure 1 shows
$\epsilon(\nu,z)$ at different $z$.

Parameter | Fiducial value | Parameter | Fiducial value
---|---|---|---
$\log_{10}\epsilon_{1500}^{z=0}$ | 25.63 | $\gamma_{1500}$ | 2.06
$\alpha_{1500}^{z=0}$ | -0.08 | $C_{1500}$ | 1.85
$\alpha_{1100}^{z=0}$ | -3.71 | $C_{1100}$ | 0.50
$\alpha_{900}^{z=0}$ | -1.5 | ${\rm EW}^{z=0.3}$ | -6.17 Å
${\rm EW}^{z=1}$ | 88.02 Å | ${\rm EW}^{z=2}$ | 176.7 Å
$\log_{10}f_{\rm LyC}^{z=1}$ | -0.53 | $\log_{10}f_{\rm LyC}^{z=2}$ | -0.84
$b_{1500}^{z=0}$ | 0.32 | $\gamma_{\nu}$ | -0.86
$\log_{10}[\epsilon b]_{1500}^{z=0}$ | 25.62 | $\gamma_{z}$ | 0.79
Table 2: Parameters to model the emissivity $\epsilon(\nu,z)$ in Eqs. (3),
(4), (5) and the bias in Eq. (12); in the analysis in Sect. IV, we use the
parameter $\log_{10}[\epsilon b]_{1500}^{z=0}$ to collect the degeneracy
between $\epsilon_{1500}^{z=0}$ and $b_{1500}^{z=0}$. Values from Ref. Chiang
et al. (2019).

###  III.2 Observed intensity

The volume emissivity we modeled so far describes the UV emission at the
source; broadband observations, however, have to deal with a combination of
effects to convert this quantity into the observed, integrated, specific
intensity $J_{\nu_{\rm obs}}\,[{\rm Jy/sr}]$.

First of all, the presence of HI clumps along the line-of-sight partially
absorbs the UV radiation before it is observed. The effective optical depth
$\tau(z)$ of this phenomenon has been originally estimated in Ref. Madau
(1995), accounting for a Poisson distribution of the absorbers as a function
of their column density and redshift. Absorption is efficient for radiation
with $\lambda_{\rm rest}$ either smaller than the ionization wavelength 912 Å
or comparable with Ly$\alpha$ and other lines in the Lyman series, which give
rise to the Lyman forest Gunn & Peterson (1965); Steidel & Sargent (1987);
Madau (1995). In our analysis, we rely on the improved semi-analytical model
in Ref. Inoue et al. (2014), described by a piecewise power-law in overall
agreement with Ref. Madau (1995).

The UV volume emissivity in our computation hence is given by
$\epsilon(\nu,z)e^{-\tau(\nu)}$, where $\tau(\nu)$ is the optical depth. By
using the radiative transfer function in an expanding Universe, it is possible
to convert this quantity into specific intensity Gnedin & Ostriker (1997),
through

$j(\nu_{\rm
obs})=\frac{c}{4\pi}\int_{0}^{\infty}\frac{dz}{H(z)(1+z)}\,\epsilon(\nu,z)\,e^{-\tau(\nu)}.$
(9)

where $\nu_{\rm obs}=\nu/(1+z)$, $c$ is the speed of light and $H(z)$ the
Hubble factor. In a broadband survey, such as ULTRASAT or GALEX, observations
are performed over a wide range of frequencies, weighted by the detector
response function. The observed monopole, hence, is

${J}_{\nu_{\rm obs}}=\int_{\nu_{\rm obs}^{\rm min}}^{\nu_{\rm obs}^{\rm
max}}\frac{d\nu_{\rm obs}}{\nu_{\rm obs}}\,j(\nu_{\rm obs}){R}(\nu_{\rm
obs})\,,$ (10)

where $R(\nu_{\rm obs})$ is the quantity in Eq. (1) once the wavelength is
converted into frequency. We estimate ${J}_{\nu_{\rm obs}}=237\,{\rm Jy/sr}$
for ULTRASAT, while ${J}_{\nu_{\rm obs}}=264\,{\rm Jy/sr},\,79\,{\rm Jy/sr}$
for GALEX NUV and FUV respectively, in agreement with measurements presented
in  C19.

Our modeling of $J_{\nu_{\rm obs}}$ only accounts for extragalactic
contributions, while real data include foregrounds. As we will discuss in the
next section, the CBR technique generally provides an unbiased estimator of
the extragalactic radiation field, which overcomes the foreground problem.

At this stage, we can thus safely rely on a signal model that only contains
extragalactic UV contributions, and include the foregrounds only in the noise
budget.

###  III.3 Spectral tagging via clustering-based redshifts

The band-averaged specific intensity in Eq. (10) contains the integrated
information about the redshift evolution of the UV volume emissivity. The CBR
technique Newman (2008); McQuinn & White (2013); Ménard et al. (2013) offers a
possible way to reconstruct the redshift dependence $dJ_{\nu_{\rm obs}}/dz$,
based on the correlation between the broadband survey and the spectroscopic
galaxy catalog that is used as reference sample.

On one side, the specific intensities $J_{\nu_{\rm obs}}$ measured in the
pixels of the broadband map provide a dataset with unknown redshifts, whose
spatial fluctuations are determined by the way UV emitters trace the
underlying large-scale structure (LSS). On the other side, galaxies in the
spectroscopic sample trace the LSS as well, and they can be divided in
consecutive slices of well-known redshift. The angular cross correlation
between the two datasets can then be exploited to remap $J_{\nu_{\rm obs}}$ in
terms of the redshift-dependent specific intensity

$\displaystyle\frac{dJ_{\nu}(z)}{dz}=$ $\displaystyle\frac{c}{4\pi
H(z)(1+z)}\,\,\times$ (11) $\displaystyle\times\int_{\nu_{\rm obs}^{\rm
min}}^{\nu_{\rm obs}^{\rm max}}\frac{d\nu_{\rm obs}}{\nu_{\rm obs}}R(\nu_{\rm
obs})\epsilon(\nu,z)e^{-\tau(\nu_{\rm obs},z)}\,.$

The amplitude of the correlation, in fact, will peak for intensity-galaxy
pairs who trace the LSS in a common redshift range. As  C19 shows, this
quantity can be constructed by using the absolute value of the intensity,
namely including foregrounds and Galactic emissions; even if this is the case,
the cross correlation is capable of isolating the extragalactic contribution.

The measurement obtained with CBR, however, will be degenerate with the bias
of the sources of the unknown-redshift sample, namely the UV emitters;
following  C19 and  S21, we parameterize it as

$b(\nu,z)=b_{1500}^{z=0}\left[\frac{\nu}{\nu_{1500}}\right]^{\gamma_{b_{\nu}}}(1+z)^{\gamma_{b_{z}}},$
(12)

where we normalized the frequency dependence to the value at 1500 Å,
analogously to the other parameters used to model the emissivity in Sect.
III.1; fiducial values are collected in Table 2. The value of $b_{1500}^{z=0}$
is degenerate with $\epsilon_{1500}^{z=0}$ in determining the amplitude of the
signal, as will become evident in the following.

In the data of the UV broadband survey, the information on the clustering
enters through an effective intensity-weighted bias $b_{J}(z)$, which is
computed from the source bias in Eq. (12) through

$b_{J}(z)=\frac{\int_{\nu_{\rm obs}^{\rm min}}^{\nu_{\rm obs}^{\rm
max}}{d\nu_{\rm obs}}{\nu_{\rm obs}^{-1}}\,{R}(\nu_{\rm
obs})\,b(\nu,z)\epsilon(\nu,z)\,e^{-\tau(\nu)}}{\int_{\nu_{\rm obs}^{\rm
min}}^{\nu_{\rm obs}^{\rm max}}{d\nu_{\rm obs}}{\nu_{\rm
obs}^{-1}}\,{R}(\nu_{\rm obs})\epsilon(\nu,z)\,e^{-\tau(\nu)}}.$ (13)

To model the CBR observable, we start by defining the fluctuations in the map
measurements in terms of the angular position $\phi$ as

$\Delta J_{\nu_{\rm obs}}(\phi)=J_{\nu_{\rm obs}}(\phi)-\langle J_{\nu_{\rm
obs}}\rangle,$ (14)

where $\langle J_{\nu_{\rm obs}}\rangle$ is the ensemble average. The galaxy
survey, instead, is used to trace the 3D overdensity field through

$\delta_{g}(\phi,z)=\frac{n_{g}(\phi,z)-\langle n_{g}(z)\rangle}{\langle
n_{g}(z)\rangle}=b_{g}(z)\delta_{m}(\phi,z),$ (15)

where $n_{g}(\phi,z)$ is the galaxy number density, $b_{g}(z)$ the galaxy bias
and $\delta_{m}(\phi,z)$ describes the LSS overdensities in the underlying
Dark Matter (DM) field.

The choice of the reference galaxy survey depends on the redshift range and
precision one wants to achieve.  C19 uses SDSS data from Refs. Blanton et al.
(2005); Reid et al. (2016), while we decided to adopt the forecasts for the
5-year results of DESI, described in Ref. Adame et al. (2024). As Table 3
shows, both surveys provide catalogs mapping different kind of sources; Fig. 2
shows the number of galaxies they contain in each redshift bin, namely

$N_{g,i}=\Delta z_{i}\Omega_{\rm survey}\frac{dN_{g}}{dzd\Omega}\,,$ (16)

where $dN_{g}/dzd\Omega$ is the number of galaxies per steradian per redshift
bin, $\Omega_{\rm survey}$ is the sky area observed by the survey, and $\Delta
z_{i}$ the width of the bin.

| $N_{g}$ | $\lambda/\Delta\lambda$ | $\Delta z_{i}$ | catalog | $z$
---|---|---|---|---|---
DESI | $\mathcal{O}(10^{7})$ | 0.001 | 0.1 | BGS | [0.1, 0.4]
LRG | [0.4, 1.1]
ELG | [1.1, 1.6]
QSO | [1.6, 2]
SDSS | $\mathcal{O}(10^{5})$ | 0.002 | 0.1 | NYU MAIN | [0.1, 0.2]
BOSS LOWZ | [0.1, 0.4]
BOSS CMASS | [0.4, 0.7]
DR14 QSO | [0.7, 2]
Table 3: Specs of the galaxy surveys (DESI, Adame et al. (2024), and SDSS,
Blanton et al. (2005); Reid et al. (2016)) considered in this work. The
forecasted sources in DESI are divided between bright galaxies (BGS), luminous
red galaxies (LRG), emission line galaxies (ELG) and quasars (QSO); SDSS
instead contains results from the galaxy NYU value-added catalog, the luminous
red galaxy samples in BOSS and the quasars in the DR14 release. The total
number $N_{g}$ is obtained by integrating $dN_{g}/dzd\Omega$ over the redshift
and observed sky area. $\lambda/\Delta\lambda$ defines the instrumental
resolution, which sets the best possible spectroscopic resolution that one can
reach, namely ${\rm min}(\delta z_{i})=\lambda/\Delta\lambda$. Finally,
$\Delta z_{i}$ describes the width of the redshift bins used in our forecast
analysis, see Sect. III.4.

The angular cross correlation between intensity pixels and galaxies that are
separated by an angle $\theta$ on the sky is finally defined as

$\omega_{Jg}(\theta,z)=\langle\Delta J_{\nu_{\rm
obs}}(\phi)\delta_{g}(\phi,z)\rangle,$ (17)

and then marginalized over $\theta$ in order to get

$\bar{\omega}_{Jg}(z)=\int_{\theta_{\rm min}}^{\theta_{\rm
max}}d\theta\,W(\theta)\omega_{Jg}(\theta,z)\,.$ (18)

This is the main observable we are interested in our analysis. The choice of
the window function $W(\theta)$ is arbitrary.

Figure 2: Number of galaxies per redshift bin with size $\Delta z=0.1$; values
for DESI are from Ref. Adame et al. (2024), while SDSS from Refs. Blanton et
al. (2005); Reid et al. (2016). Discontinuities are due to different catalogs.

In analogy with  C19 and  S21, we choose
$W(\theta)=\theta^{-0.8}/\int\theta^{-0.8}d\theta$. The minimum angular
distance above which the cross correlation is performed, is set to
$\theta_{\rm min}(z)=\arctan[r_{\rm min}/D_{A}(z)]$, where $D_{A}(z)$ is the
cosmological angular diameter distance and $r_{\rm min}=0.5\,{\rm Mpc}$ is a
physical scale chosen to avoid strong non-linear clustering on small scales;
to provide a more conservative result, in Sect. IV we also consider the case
in which $r_{\rm min}=1\,{\rm Mpc}^{-1}$. For $z\in[0.1,2]$ we get
$\theta_{\rm min}\in[4,0.97]\,{\rm arcmin}$, larger than $L_{\rm pix}$ in
Table 1. The choice of $\theta_{\rm max}$, instead, is set to avoid large
fluctuations in the calibration of the map. In our case, ULTRASAT has a wide
field-of-view (FoV) $\Omega_{\rm Fov}=204\,{\rm deg}^{2}$, across which
observational properties vary. Ref. Shvartzvald et al. (2023) shows that the
effective PSF is almost constant up to $\sim 4\,{\rm deg}$ from the center of
the FoV; therefore, we assume that inside this range calibrations are stable
and we run our analysis up to $\theta_{\rm max}=4\,{\rm deg}$. In the case of
GALEX, instead,  C19 chooses $\theta_{\rm max}(z)=\arctan[5{\,\rm
Mpc}/D_{A}(z)]\in[44,10]\,{\rm arcmin}=[0.7,0.2]\,{\rm deg}$. We adopt the
same value when dealing with GALEX.

The angular cross correlation $\bar{\omega}_{\tilde{J}{\rm g}}(z)$ defined in
Eq. (18) represents the CBR observable, and can then be used to infer the
redshift evolution of the specific intensity. In fact, we can re-express it as

$\bar{\omega}_{\tilde{J}{\rm g}}(z)=b_{J}(z)b_{g}(z)\frac{dJ_{\nu_{\rm
obs}}}{dz}\int_{\theta_{\rm min}(z)}^{\theta_{\rm
max}(z)}d\theta\,W(\theta)\omega_{m}(\theta,z),$ (19)

where $b_{J}(z)$ and $b_{g}(z)$ are the biases of the two LSS tracers, and
$\omega_{m}(\theta,z)$ is the angular two-point function of the underlying DM
field. This can be related to the non-linear matter power spectrum Maller et
al. (2005) via

$\omega_{m}(\theta,z)=\frac{1}{2\pi}\int_{0}^{\infty}dk\,kP_{m}(k,z)\mathcal{J}_{0}(k\theta\chi(z))\frac{dz}{d\chi},$
(20)

where $\mathcal{J}_{0}$ is the Bessel function of the first type, $\chi(z)$
the radial comoving distance and $dz/d\chi=H(z)/c$. We compute the
$P_{m}(k,z)$ DM power spectrum using the public library
CAMB666https://github.com/cmbant/CAMB Challinor & Lewis (2011), and we adopt
the halofit model prescription from Ref. Mead et al. (2020).

The observation of $\bar{\omega}_{\tilde{J}{\rm g}}(z)$ then leads to the
reconstruction of the redshift evolution of the intensity weighted by the
bias, $d\tilde{J}_{\nu_{\rm obs}}/dz=b_{J}(z)dJ_{\nu_{\rm obs}}/dz$. This in
turn represents a summary statistics of the UV emission and absorption across
space and time. We show $d\tilde{J}_{\nu_{\rm obs}}/dz$ in Fig. 1, for
ULTRASAT and GALEX (NUV, FUV), while Fig. 3 shows the CBR signal-to-noise
ratio of $\bar{\omega}_{\tilde{J}{\rm g}}(z)$ when their maps are cross
correlated with the galaxy reference catalogs from SDSS or DESI. These are the
quantities that enter our analysis in Sect. IV.

###  III.4 Noise

Refs. Newman (2008); Ménard et al. (2013) estimate analytically the
uncertainty on the CBR angular cross correlation $\bar{\omega}_{\tilde{J}{\rm
g}}(z)$ introduced in Eq. (19). We follow a similar reasoning to they do,
accounting for the fact that the broadband UV data are measured as intensity
in pixels rather than point sources, as is done instead in photometric galaxy
surveys.777 S21 computes the noise differently from us. In particular, their
forecast analysis is built directly on $d\tilde{J}_{\nu_{\rm obs}}/dz$ instead
of $\bar{\omega}_{\tilde{J}g}$, thus their noise estimate refers to this
quantity.

In the broadband survey, we model the monopole in each pixel by summing the
EBL from Eq. (10) with an estimate of the foreground. This provides an offset
between the expected and observed $J_{\nu_{\rm obs}}$, that has indeed been
observed. The authors of Ref. Akshaya et al. (2018) estimated the observed
surface brightness in photon units888In photon units, the EBL monopole is
$89\,{\rm ph\,cm^{-2}s^{-1}sr^{-1}}$Å-1 for FUV, $172\,{\rm
ph\,cm^{-2}s^{-1}sr^{-1}}$Å-1 for NUV. to be $\sim 250\,{\rm
ph\,cm^{-2}s^{-1}sr^{-1}}$Å-1 in the FUV band, and $\sim 550\,{\rm
ph\,cm^{-2}s^{-1}sr^{-1}}$Å-1 in the NUV band.  C19 explains the offset as due
to the presence of three main foregrounds: near-Earth airglow and zodiacal
light (the latter being relevant only in the NUV band, see Ref. Murthy
(2013)), Galactic dust, and a component whose origin is unknown. We account
for all these sources adding to the EBL monopole a fractional contribution
$\mathcal{A}_{\rm fg}J_{\nu_{\rm obs}}$, so that $J_{\nu}^{\rm
obs}=J_{\nu_{\rm obs}}(1+\mathcal{A}_{\rm fg})$. We set $\mathcal{A}_{\rm
fg}=1.8$ for GALEX FUV and $2.2$ for GALEX NUV and ULTRASAT, since they
observe similar bands. We assume data are Poisson-distributed over pixels
(thus we estimate the variance as the signal mean value), while fluctuations
are Gaussian with variance $\sigma_{N}^{2}$ from Eq. (2); the total noise
hence is

$\sigma_{J}^{2}=[J_{\nu_{\rm obs}}(1+\mathcal{A}_{\rm
fg})]^{2}+\sigma_{N}^{2}.$ (21)

As described in Sect. III.3, the CBR cross correlation is computed in patches
of area $\pi\theta_{\rm max}^{2}$, where $\theta_{\rm max}=4\,{\rm deg}$ for
ULTRASAT and $\theta_{\rm max}\sim[0.7,\,0.2]\,{\rm deg}$ for GALEX.
Therefore, to infer the noise we need to account for all the possible pairs of
pixels and reference galaxies per redshift bin that can be created inside the
patches. We rescale the total number of pixels $N_{\rm pix}^{\rm tot}$ inside
$\pi\theta_{\rm max}^{2}$ as

$N_{\rm pix}=N_{\rm pix}^{\rm tot}{\pi\theta_{\rm max}^{2}}/{\Omega_{\rm
sky}}={\pi\theta_{\rm max}^{2}}/{A_{\rm pix}},$ (22)

where we used $N_{\rm pix}^{\rm tot}=\Omega_{\rm sky}/A_{\rm pix}$,
$\Omega_{\rm sky}$ being the sky area observed by the broadband survey, while
$A_{\rm pix}=L_{\rm pix}^{2}$ the area of the pixels computed using Table 1.
Meanwhile, the number of galaxies per $z$ bin in the same area is

$N_{g,i}^{\theta_{\rm max}}=\frac{dN_{g}}{dzd\Omega}{\Delta
z_{i}\pi\theta_{\rm max}^{2}}=N_{g,i}\frac{\pi\theta_{\rm
max}^{2}}{\Omega_{\rm survey}},$ (23)

where $N_{g,i}=\Delta z_{i}\Omega_{\rm survey}dN_{\rm g}/dzd\Omega$ is the
number of reference galaxies in the observed area of the survey $\Omega_{\rm
survey}$ in the $i$-th redshift bin having size $\Delta z_{i}$ (see Fig. 2).

Finally, we estimate the number of pairs to be

$N_{\rm pairs}=N_{\rm pix}N_{g,i}^{\theta_{\rm max}}=\frac{N_{g,i}}{A_{\rm
pix}}\frac{(\pi\theta_{\rm max}^{2})^{2}}{\rm{min}[\Omega_{\rm
survey},\Omega_{\rm sky}]}\,.$ (24)

In the case of ULTRASAT$\times$DESI and GALEX$\times$DESI, the observed area
is set by the galaxy survey, namely $\Omega_{\rm survey}=14000\,{\rm deg}^{2}$
Adame et al. (2024). Following the procedure in  C19, instead, for
GALEX$\times$SDSS we set $\Omega_{\rm survey}=5500\,{\rm deg^{2}}$, which is
the size of the overlapping footprints. In analogy to  S21 and Ref. Chiang &
Ménard (2019), for SDSS we set $\Delta z_{i}=0.1$, hence we sample the $[0,2]$
redshift range with 20 bins. This width is larger than the uncertainty of the
spectroscopic redshift bins $\delta z_{i}$: in fact, as it has been shown in
Refs. Ménard et al. (2013); Rahman et al. (2015), CBR provides a measurement
of the redshift which is degenerate with the bias evolution in $z$. This
degrades the goodness of the CBR-redshift inference, and it can lead to the
overlap between contiguous bins when their width is too small. Moreover, using
too small redshift bins would increase the shot noise term encapsulated in
$N_{g,i}$. In the case of SDSS, the width $\Delta z_{i}=0.1$ allows us to
overcome this issue; in the case of DESI, we make the conservative choice of
using the same redshift bin width.

Figure 3: Noise analysis for ULTRASAT$\times$DESI (pink), GALEX
NUV$\times$SDSS and $\times$DESI (light blue, dashed and continuous) and GALEX
FUV$\times$SDSS and $\times$DESI (dark blue, dashed and continuous). Left:
$\mathcal{N}_{\rm CBR}$ from Eq. (25). Here, we consider different sizes for
the effective pixels; for each detector, the dot indicates the noise for
$L_{\rm pix}$ as in Table 1, while the cross indicates the value adopted in
the analysis. Right: SNR obtained from Eqs. (19) and (25).

Moreover, a further contribution to the CBR noise comes from the width $\delta
z_{i}$ of the spectroscopic bins. Assuming that matter clusters on scale
$\delta z_{c}$ and not beyond, a galaxy survey with either bins $\delta
z_{i}>>\delta z_{c}$ or $\delta z_{i}<\delta z_{c}$ would not be precise in
reconstructing the $dJ_{\nu_{\rm obs}}/dz$ redshift evolution based on cross
correlation information. The best strategy according to Ref. Ménard et al.
(2013) is to choose reference redshift bins with $\delta z_{i}\sim\delta
z_{c}\sim 10^{-3}$: in this way, the amplitude of the cross correlation
between the intensity and a reference galaxy is high only locally, that is
inside the redshift bin from where the intensity comes from. Following Ref.
Ménard et al. (2013), we account for this type of noise through a $\delta
z_{i}/\delta z_{c}$ factor, where we set $\delta z_{c}={\rm
max}[10^{-3},H(z)(1+z)r_{\rm min}]$ to avoid strong non-linear clustering and
account for redshift evolution in the clustering scale. The quantity $r_{\rm
min}$ is the same minimum physical distance adopted in Sect. III.3 to define
the size of the patches in which the cross correlation is computed. On the
other hand, we define the width of the spectroscopic bins $\delta z_{i}$ for
SDSS and DESI as the largest value between $\delta z_{c}$ and the instrumental
resolution $\lambda/\Delta\lambda$ in Table 3.

Finally, we need to account for the number of sky patches $N_{\theta}$, over
which the statistical analysis can be performed. This introduces an extra
$(N_{\theta})^{-1/2}=\sqrt{\pi\theta_{\rm max}^{2}/\Omega_{\rm survey}}$
factor. Collecting all the elements described up to this point, we estimate
the noise in the CBR procedure to be

$\displaystyle\mathcal{N}_{\rm CBR}$ $\displaystyle=\frac{\delta z_{i}}{\delta
z_{c}}\sqrt{\frac{\sigma_{J}^{2}}{N_{\rm pairs}N_{\theta}}}$ (25)
$\displaystyle=\frac{\delta z_{i}}{\delta z_{c}}\sqrt{\frac{A_{\rm
pix}{\bigl{[}[J_{\nu_{\rm obs}}(1+\mathcal{A}_{\rm
fg})]^{2}+\sigma_{N}^{2}\bigr{]}}}{\pi\theta_{\rm max}^{2}N_{g,i}}}.$

The CBR noise $\mathcal{N}_{\rm CBR}$ clearly depends on the properties of
both the broadband and galaxy surveys. To minimize it, on one side we need a
spectroscopic survey where the uncertainty $\delta z_{i}$ is small, but which
observes enough galaxies to guarantee a high $N_{g,i}$: the former allows us
to improve the quality of the clustering redshift estimation, while the latter
reduces the shot noise. On the other side, a sweet spot exists between using
small pixels and getting a not-too-large value for $\sigma_{N}^{2}$; in fact,
using smaller pixels would provide a larger number of pairs, hence a smaller
statistical noise; the flux density per pixel $f_{\nu}^{\rm pix}$ defined in
Sect. II, however, increases when the area of the pixels gets smaller, leading
to a larger $\sigma_{N}^{2}$. On the other hand, the uncertainty due to the
statistical Poissonian fluctuations encapsulated in $J_{\nu_{\rm
obs}}(1+\mathcal{A}_{\rm fg})$ determines a ground level for the error budget.
Since the noise variance cannot be lower than the monopole expected in the map
from the astrophysical contribution, the optimal value is reached when
$\sigma_{N}^{2}\sim[J_{\nu_{\rm obs}}(1+\mathcal{A}_{\rm fg})]^{2}$. By
looking at Table 1, it is evident that ULTRASAT already satisfies this
condition with $L_{\rm pix}$, while GALEX requires the use of larger effective
pixels. As Fig. 3 shows, this translates to two different prescriptions when
computing $\mathcal{N}_{\rm CBR}$: while for ULTRASAT the pixel scale $L_{\rm
pix}$ is already good enough to minimize the noise, for GALEX it is better to
group the pixels, in order to reach an effective scale $L_{\rm pix}^{\rm
eff}=50^{\prime\prime}/{\rm pix}$, which is indeed the one adopted in  C19. In
this case, the noise variance is $\sigma_{N}^{2}=78\,{\rm Jy/sr}$; finally,
the difference between NUV and FUV is due to the different $J_{\nu_{\rm
obs}}$. Accurate foreground cleaning and mitigation can reduce the noise with
respect to the value we estimated.

Under this choice of parameters, we estimate the noise $\mathcal{N}_{\rm CBR}$
to be used in the analysis in Sect. IV. Figure 3 compares the signal-to-noise
ratio ${\bar{\omega}_{\tilde{J}{\rm g}}(z)}/\mathcal{N}_{\rm CBR}$ (SNR)
estimated for all the surveys; the redshift dependence is evident in the
signal, while in the noise it only enters in the choice of $\theta_{\rm max}$
for GALEX and the estimate of $\delta z_{i}/\delta z_{c}$. By looking at GALEX
in this plot, it is clear that the cross correlation with DESI will lead to a
larger SNR: this is due both to the better spectroscopic redshift
determination and to the larger number of galaxies observed. The comparison
between GALEX NUV$\times$DESI and ULTRASAT$\times$DESI in this figure reflects
instead the smaller noise ULTRASAT has thanks to its smaller $A_{\rm pix}$ and
wider $\theta_{\rm max}$.

##  IV Forecasts

To constrain the parameters of the UV-EBL emissivity,  C19 analyses data from
GALEX$\times$SDSS. Since our final goal is to perform a forecast analysis, we
rely instead on the Fisher matrix formalism Vogeley & Szalay (1996); Tegmark
et al. (1997). For each detector-galaxy survey pair, we compute the Fisher
matrix by summing over the $i$-redshift bins of the CBR analysis as

$F_{\alpha\beta}=\sum_{i}\frac{\partial\bar{\omega}_{\tilde{J}g}(z_{i})}{\partial\vartheta_{\alpha}}\frac{\partial\bar{\omega}_{\tilde{J}g}(z_{i})}{\partial\vartheta_{\beta}}\frac{1}{\mathcal{N}_{\rm
CBR}^{2}(z_{i})},$ (26)

where the noise $\mathcal{N}_{\rm CBR}(z_{i})$ is estimated in Eq. (25).

The parameter set is

$\displaystyle\vartheta=$
$\displaystyle\\{\log_{10}[b\epsilon]_{1500}^{z=0},\,\gamma_{1500},\,\gamma_{\nu},\,\gamma_{z}$
(27)
$\displaystyle\alpha_{1500}^{z=0},\,C_{1500},\,\alpha_{1100}^{z=0},\,C_{1100},\,\alpha_{900}^{z=0},$
$\displaystyle{\rm EW}^{z=z_{\rm EW1}},\,{\rm EW}^{z=z_{\rm EW2}},$
$\displaystyle\log_{10}f_{\rm LyC}^{z=z_{\rm C1}},\,\log_{10}f_{\rm
LyC}^{z=z_{\rm C2}}\\},$

which collects the parameters introduced in Table 2. Whenever we want to
account for priors on certain parameters, we sum the Fisher matrix with the
prior matrix, which has $1/\sigma_{\rm prior}^{2}$ diagonal elements. The
marginalized error on each $\vartheta_{\alpha}$ can be finally estimated from
the inverse of the Fisher matrix (already summed with the prior matrix, if
needed) as $\sigma_{\alpha}=\sqrt{F^{-1}_{\alpha\alpha}}$; these forecasts can
then be propagated to estimate the uncertainty on the volume emissivity
$\epsilon(\nu,z)$ or on the ionizing photon escape fraction $f_{\rm LyC}$
using $\sigma_{\aleph}=\sqrt{F\mathcal{D}_{\aleph,\vartheta}}$, where $\aleph$
is the new parameter,
$\mathcal{D}_{\aleph,\vartheta}=\left(\partial\aleph/\partial\vartheta_{\alpha},...\right)$
the vector of its derivatives with respect to the old ones and $F$ the Fisher
matrix in Eq. (26).

Before proceeding, it is important to remember that, while being extremely
useful for order-of-magnitude estimates, the Fisher matrix formalism usually
provide optimistic forecasts. In particular, the marginalized errors it
obtains tend to underestimate the uncertainty in the presence of significant
degeneracies between the parameters; for comparison between forecasts obtained
using MCMC and the Fisher approach, see for example Ref. Wolz et al. (2012)
and references within.

###  IV.1 Breaking the bias degeneracy

| Conservative | Optimistic |  G$\times$S |  U$\times$D |  (G$+$U)$\times$D
---|---|---|---|---|---
$b_{1500}^{z=0}$ | 0.05 | 0.01 | $\checkmark$ | $\checkmark$ | $\checkmark$
$\gamma_{b_{\nu}}$ | 1.30 | 0.30 | $\checkmark$ | $\checkmark$ | $\checkmark$
$\gamma_{b_{z}}$ | 0.30 | 0.10 | $\checkmark$ | $\checkmark$ | $\checkmark$
$\gamma_{1500}$ | 0.30 | $-$ | $\checkmark$ | $\usym{2717}$ | $\usym{2717}$
$C_{1500}$ | 1.50 | $-$ | $\checkmark$ | $\usym{2717}$ | $\usym{2717}$
$C_{1100}$ | 1.50 | $-$ | $\checkmark$ | $\usym{2717}$ | $\usym{2717}$
Table 4: For all cases, we set priors on the bias parameters.
GALEX$\times$SDSS (G$\times$S) also has informative priors on
$\\{\gamma_{1500},C_{1500},C_{1100}\\}$, as in Ref. Chiang et al. (2019).

As discussed in Sect. III.2, the normalization value $\epsilon_{1500}^{z=0}$
and the local bias $b_{1500}^{z=0}$ are degenerate; for this reason, in our
analysis we collect them in a single parameter $\log_{10}[\epsilon
b]_{1500}^{z=0}$. Moreover, results in  C19 and  S21 show further degeneracies
between the parameters that determine the slope of the bias, namely
$\\{\gamma_{b_{\nu}},\gamma_{b_{z}}\\}$, and the ones that characterize how
the emissivity depends on the frequency and redshift.

To overcome these issues, we introduce Gaussian priors on
$\\{\gamma_{b_{\nu}},\gamma_{b_{z}}\\}$; in particular, we rely on  C19
results and set $\sigma_{\gamma b_{\nu}}=1.30,\,\sigma_{\gamma b_{z}}=0.30$.
These are indeed the largest errorbars  C19 obtained from the
GALEX$\times$SDSS CBR analysis and can thus be interpreted as a conservative
choice for our forecasts. To these priors we add the ones  C19 uses on
$\\{\gamma_{1500},C_{1500},C_{1100}\\}$. Imposing a prior $\sigma=1.3$ on
$\gamma_{1500}$, and wide Gaussian priors with $\sigma=1.5$ on
$C_{1500},C_{1100}$ is still needed in the case of GALEX$\times$SDSS, while
they can be removed when using DESI.

The motivation behind our prior choice relies on the fact that our main
targets are the parameters regulating the emissivity, while the bias can
potentially be constrained from different probes, such as the resolved sources
catalog of the broadband survey itself. In  C19, such possibility is explored
by estimating the bias of the GALEX resolved sources, and rescaling its value
to the EBL regime. To do so, they assume that the bias of the resolved sources
is larger than the bias of the EBL diffuse light map, due to the flux-limit
that allows us to resolve only of the brightest, and thus more clustered,
sources. The analysis is done accounting for the information on the GALEX
redshift-dependent luminosity threshold and the luminosity-dependent SDSS
galaxy bias in Ref. Zehavi et al. (2011), and it returns an estimated errorbar
$\sigma_{b_{1500}^{z=0}}=0.05$.

We assume that a similar procedure will be applied to ULTRASAT$\times$DESI,
and GALEX$\times$DESI; the estimated errorbars in that cases will reasonably
be $\leq 0.05$. Similarly to what  C19 discusses, this will make it possible
to break the degeneracy between $b_{1500}^{z=0}$ and $\epsilon_{1500}^{z=0}$.
Hence, in the following, we present results for $\epsilon(\nu,z)$; these are
obtained by marginalizing over
$\\{b_{1500}^{z=0},\gamma_{b_{\nu}},\gamma_{b_{z}}\\}$ with Gaussian priors
respectively $\\{0.05,0.30,1.30\\}$. We stress again that these values are
chosen according to GALEX$\times$SDSS results, therefore represent a
conservative choice in our forecast. To get a more optimistic result, we also
test the case $\\{0.01,0.10,0.30\\}$. Priors are summarized in Table 4.

###  IV.2 Validation and parameter forecast

To validate our results, first of all we apply the Fisher formalism to GALEX
FUV$\times$SDSS and NUV$\times$SDSS and we try to “post-dict” the results in
C19. We use the same reduced parameter set that they adopted, namely

$\displaystyle\vartheta_{\rm C19}=$ $\displaystyle\\{\log_{10}[\epsilon
b]_{1500}^{z=0},\,\gamma_{1500},\alpha_{1500}^{z=0},\,C_{1500},\,\alpha_{1100}^{z=0},$
(28) $\displaystyle C_{1100},\,{\rm EW}^{z=0.3},{\rm
EW}^{z=1},\,\gamma_{\nu},\,\gamma_{z}\\}.$

Here, we fixed all the parameters related with the ionizing continuum below
the Lyman break in Eq. (5), for which  C19 showed that GALEX$\times$SDSS has
no constraining power, and we adopt the priors in Table 4.

We run the analysis in Eq. (26) separately for the NUV and FUV filters, and we
assume that they are uncorrelated, so we can sum their matrices to improve the
constraints. The fact that the filters are sensitive to different ranges is
crucial to capture the shape of the redshift-dependent parameters in Eq. (6),
better reconstructing $\epsilon(\nu,z)$. A further confirmation of this can be
seen in the good results the authors of  S21 obtained combining the three
CASTOR filters.

| GALEX$\times$SDSS | ULTRASAT$\times$DESI
---|---|---
| C19 | This work | Full-sky
$\log_{10}[\epsilon b]_{1500}^{z=0}$ | $25.62_{-0.01}^{+0.01}$ | 0.02 | 0.03
$\gamma_{1500}$ | $2.06_{-0.30}^{+0.31}$ | 0.30 | 0.39
$\alpha_{1500}^{z=0}$ | $-0.08_{-0.84}^{+1.28}$ | 1.38 | 1.18
$C_{1500}$ | $1.85_{-1.28}^{+1.22}$ | 1.50 | 0.02
$\alpha_{1100}^{z=0}$ | $-3.71_{-0.98}^{+1.34}$ | 4.50 | 1.09
$C_{1100}$ | $0.50_{-1.44}^{+1.46}$ | 1.50 | 0.87
${\rm EW}^{z=0.3}$ | $-6.17_{-11.43}^{+12.63}$ | 89.8 | $-$
${\rm EW}^{z=1}$ | $88.02_{-48.87}^{+51.44}$ | 553 | 1.19
${\rm EW}^{z=2}$ | $176.7$ | $-$ | 10.2
Table 5: $1\sigma$ marginalized errors from our “post-diction” for
GALEX$\times$SDSS, compared with results in Ref. Chiang et al. (2019), and
with the prediction for ULTRASAT$\times$DESI. We use the $\vartheta_{\rm C19}$
set and we applied the priors described in Table 4.

Our results are collected in Table 5 and compared with the actual results of
C19: our Fisher forecasts are in the same ballpark. However, our analysis
lacks in accurately reproduce constraints on the equivalent width parameters
${\rm EW}^{z=1,2}$. The reason for this resides in the fact that these
parameters only affect a small portion of the spectrum described in Eq. (4),
hence they introduce a discontinuity in the analytical model that is more
difficult to capture in the computation of the numerical derivatives required
in Eq. (26).999See also the discussion on the practicality of the Fisher
matrix compared with MCMC in Ref. Wolz et al. (2012)). Despite this, the
equivalent width parameters are not degenerate with the others, thus they do
not alter the result on the other marginalized errors estimated from the
Fisher matrix computation.

Since our “post-diction” is reasonable, we proceed by applying it to the
ULTRASAT$\times$DESI scenario. As we discussed in Sect. III.1, in this case we
decided to pivot the evolution of the equivalent width EW at higher redshift,
namely $z_{\rm EW1}=1,\,z_{\rm EW2}=2$. Table 5 presents the results we found:
the constraining power is increased with respect to GALEX$\times$SDSS on all
the parameters, except $\log_{10}[\epsilon b]_{1500}^{z=0}$ for which GALEX is
helped by the wider redshift range probed by its two filters, NUV and FUV.
ULTRASAT sensitivity at $z\sim 1$, instead, leads to a large improvement in
the parameters determining the redshift evolution, as well as in the line
equivalent width. In the case of ULTRASAT$\times$DESI there is no need of
using priors on $\\{\gamma_{1500},C_{1500},C_{1100}\\}$. We checked that both
GALEX$\times$SDSS and ULTRASAT$\times$DESI have no constraining power on the
parameters $\\{\alpha_{900}^{z=0},\log_{10}f_{\rm LyC}^{z=z_{\rm
C1}},\log_{10}f_{\rm LyC}^{z=z_{\rm C2}}\\}$ of the ionizing continuum.
Moreover, including them in $\vartheta$ worsens the results on the other
parameters.

Having shown that ULTRASAT$\times$DESI will improve the constraints, we now
turn our attention to the full parameter set (GALEX$+$ULTRASAT)$\times$DESI,
which we consider as our baseline in the following sections. In this case, the
full set $\vartheta$ in Eq. (27) can be tested. As Table 6 shows, all the
parameters are indeed well constrained; the only degeneracy not yet solved is
the one between $\gamma_{1500}$ and $\gamma_{b_{z}}$; using a prior on one of
them, helps constraining the other. Results are obtained by computing
separately the Fisher matrices for GALEX NUV$\times$DESI, GALEX
FUV$\times$DESI and ULTRASAT$\times$DESI, then summing them. The full matrix
includes the EW pivot parameters at $z_{\rm EW}=\\{0.3,1,2\\}$, with $z_{\rm
EW}=1$ constrained by both the detectors, while the others either by GALEX or
ULTRASAT.

(GALEX + ULTRASAT)$\times$DESI
---
$\log_{10}[\epsilon b]_{1500}^{z=0}$ | $25.62\pm 0.005$ | $\gamma_{1500}$ | $2.06\pm 0.30$
$\alpha_{1500}^{z=0}$ | $-0.08\pm 0.25$ | $C_{1500}$ | $1.85\pm 0.02$
$\alpha_{1100}^{z=0}$ | $-3.71\pm 0.44$ | $C_{1100}$ | $0.50\pm 1.16$
$\alpha_{900}$ | $-1.5\pm 5.10$ | ${\rm EW}^{z=0.3}$ | $-6.17\pm 3.05$
${\rm EW}^{z=1}$ | $88.02\pm 0.51$ | ${\rm EW}^{z=2}$ | $176.7\pm 2.79$
$\log_{10}f_{\rm LyC}^{z=1}$ | $-0.53\pm 0.20$ | $\log{10}f_{\rm LyC}^{z=2}$ | $-0.84\pm 0.18$
Table 6: Forecasted $1\sigma$ marginalized errors from our Fisher analysis for
(GALEX+ULTRASAT)$\times$DESI. We consider the full set $\vartheta$ with priors
as in Table 4.

Figure 4: Forecasts at $1\sigma$ on the UV-EBL emissivity $\epsilon(\nu,z)$ at
different redshifts for (GALEX$+$ULTRASAT)$\times$DESI, compared with our
“post-diction” at $1\sigma$ for GALEX$\times$SDSS as in Ref. Chiang et al.
(2019) (compare with Fig. 6 in their work). We tested both the optimistic and
conservative bias priors described in Table 4. The gray shaded area delimits
the observational window of GALEX$+$ULTRASAT; the gray vertical line shows
where GALEX NUV would stop, if ULTRASAT was not included. The parts of the
spectrum inside the observed window are directly probed via the CBR, while the
others are model-based extrapolations.

Our results depend on the choice of $r_{\rm min}=0.5\,{\rm Mpc}$, which we
introduced in Sect. III.3 to avoid strong non-linear clustering. To test the
stability of our conclusions with respect to this parameter, we re-run the
Fisher analysis cutting the angular correlation at scales associated with
$r_{\rm min}=1\,{\rm Mpc}$. In this case, the forecasts in Table 6
deteriorate, and the marginalized errors almost double. Despite this, the
constraining power remains good even when less optimistic prescriptions are
considered. In the next sections, we therefore rely on the constraints
obtained in Table 6; other scenarios can be recovered straightforwardly.

###  IV.3 Forecasts on the volume emissivity

The parameters constrained in the previous section are combined in Eqs. (3),
(4), (5) to model the UV-EBL volume emissivity. As discussed in Sect. IV.1, we
can assume the local bias $b_{1500}^{z=0}$ is measured, and so convert the
$\log_{10}[\epsilon b]_{1500}^{z=0}$ constraints to $\epsilon_{1500}^{z=0}$
and consequently $\epsilon(\nu,z)$, after marginalizing all the other
parameters. We analyse results on $\epsilon(\nu,z)$ using this procedure and
applying both the conservative and optimistic bias priors in Table 4.

Figure 4 shows our $1\sigma$ forecast constraints on $\epsilon(\nu,z)$ for
(GALEX$+$ULTRASAT)$\times$DESI, at $z=\\{0,0.5,1,2\\}$, using the $\vartheta$
parameter set. Here, we also show how the GALEX$\times$SDSS results from the
previous section propagate to $\epsilon(\nu,z)$. Our $1\sigma$ forecasts in
this case are obtained using the reduced $\vartheta_{\rm C19}$ and including
priors from Table 4; they can be directly compared with Fig. 6 in  C19.

It is evident that (GALEX$+$ULTRASAT)$\times$DESI will provide very good
constraints on the UV-EBL volume emissivity reconstruction. Results could be
further improved with respect to our findings, if foreground mitigation was
taken into account. For example, setting $\mathcal{A}_{\rm fg}\,=\,0$ in Eq.
(21) leads to constraints on the emissivity parameters that are $\sim 0.5$ of
the ones in Table 6. Figure 4 allows us to further comment on an important
aspect of our analysis. Not all the redshifted wavelengths of the UV-EBL
emissivity fall in the observational windows of GALEX$+$ULTRASAT. The forecast
constraints on $\epsilon(\nu,z)$ that are found in the shaded areas are
extrapolated knowing the dependencies of the model parameters in Eqs. (3),
(4), (5). The way these have been chosen is agnostic regarding the physics or
the type of sources involved: they only require the UV emission to contain a
line (Ly$\alpha$), a break (the Lyman break) and a continuum, whose slope
varies in the different frequency ranges. In the following section, we discuss
the implications this will have in our knowledge of the UV-EBL sources.

###  IV.4 Forecasts on the UV-EBL sources

Figure 5: $1\sigma$ forecasts on the non-ionizing $\lambda=1500\,$Å continuum
for (GALEX$+$ULTRASAT)$\times$DESI (shaded area), with conservative and
optimistic priors on the bias. To get GALEX$\times$SDSS (C19) results, we
propagated our “post-diction” in Sect. IV.2. We compare with available results
in the literature. Ref. Schiminovich et al. (2005) data are based on the UV
luminosity function of the galaxies in GALEX, Ref. Dahlen et al. (2006) on the
GOODS survey from the Hubble Space Telescope, while Ref. Reddy et al. (2008)
use a compilation of data from Hubble. The green line reproduces the fitting
of Ref. Dominguez et al. (2011) over multiband data, while the dotted lines
are the semi-analytical models obtained in Ref. Gilmore et al. (2012) with
different dust prescriptions.

The shape of the continuum in the non-ionizing region is showed in  C19 to be
in good agreement with the model in Ref. Haardt & Madau (2012), which accounts
for galaxy emission and dust extinction. Ref. Haardt & Madau (2012) stresses
that the non-ionizing continuum for instance at 1500 Å, provides valuable
information on the star formation rate history (modeled e.g., in Ref. Madau
(1995)) and the metal-enrichment history (e.g., in Refs. Nagamine et al.
(2001); Kewley & Kobulnicky (2007)), to be tested against the UV luminosity
functions (e.g., Refs. Wyder et al. (2005); Reddy & Steidel (2009); Bouwens et
al. (2011)). Extra-contributions from AGN are negligible in this frequency
range, according to Ref. Faucher-Giguère (2020); on the contrary, the ionizing
continuum $\lambda_{\rm rest}<912\,$Å seems to be AGN-dominated at $z\lesssim
4$. The result, however, depends on the AGN luminosity function (e.g., Refs.
Hopkins et al. (2007); Kulkarni et al. (2019); Shen et al. (2020)) and on the
EBL ionizing photon escape fraction $f_{\rm LyC}$, this being largely
uncertain.  C19 manages to provide only upper bounds on $f_{\rm
LyC}^{z=1},\,f_{\rm LyC}^{z=2}$. In our analysis, we adopted these as our
fiducials; actual values of $f_{\rm LyC}$, however, may be smaller than this
limit Flury et al. (2022). We tested how the observable and forecasts in our
analysis change when $f_{\rm LyC}$ is lowered: since this parameter sets the
Lyman break depth (compare with Fig. 1), it mainly determines the shape of
$dJ_{\nu_{\rm obs}}/dz$ at high-$z$, where the signal drops. Therefore,
lowering its value only affects the forecasts on $f_{\rm LyC}^{z=z_{\rm
C1},z_{\rm C2}}$ and on $\alpha_{900}$: when the escape fraction becomes too
small, CBR can no longer constrain the EBL at small wavelengths.

As a final remark, we refer to Fig. 5 to show that
(GALEX$+$ULTRASAT)$\times$DESI will be able to determine the amplitude and
redshift evolution of the non-ionizing continuum at $\lambda=1500\,$Å up to
$z\sim 2$, with $1\sigma$ uncertainty $\lesssim 26\%\,(9\%)$ with conservative
(optimistic) bias priors, see Table 4. We compare our forecasts with the
constraints Ref. Schiminovich et al. (2005) obtained using the galaxy UV
luminosity function in GALEX, and the ones Refs. Dahlen et al. (2006); Reddy
et al. (2008) got from data compilations from the Hubble Space Telescope.
Moreover, we include the fitting Ref. Dominguez et al. (2011) realized over
multiband Spitzer data, and the semi-analytical models of Ref. Gilmore et al.
(2012), which account for different dust models.101010We follow the notation
in  C19, where $\epsilon(\nu,z)$ is the comoving specific emissivity;
differently from Ref. Haardt & Madau (2012), where this symbol indicates the
proper volume emissivity. Here, $\epsilon_{1500}(z)$ coincides with the
luminosity density, while in Refs. Haardt & Madau (2012); Schiminovich et al.
(2005), is $\rho_{1500}$.

The improved forecast constrains we obtained on the $\vartheta$ parameters,
therefore, can be used to constrain the astrophysical sources of the UV-EBL.
With such measurement, the combination of GALEX and ULTRASAT maps will also
open the possibility of disentangling any extra-contribution due to emissions
beyond galaxies and AGN, for example from decaying DM. We will analyze this
enticing possibility in an upcoming, dedicated work.

##  V Conclusion

Many satellites are planned to be launched and start operating over the coming
years. Their characteristics and goals are the most diverse, and it is crucial
to understand how to exploit their data as best as we can. New observables and
estimators will be needed, in order to probe the Universe at various redshifts
and scales, so as to deepen our understanding of cosmic evolution.

Among these space-borne observatories, the Ultraviolet Transient Astronomy
Satellite (ULTRASAT, Sagiv et al. (2014); Ben-Ami et al. (2022); Shvartzvald
et al. (2023)) will observe the near-UV (NUV) range between $2300\,$Å and
$2900$ Å: its main goal will be the study of transients, such as supernovae,
variable stars, AGN and electromagnetic counterparts to gravitational-wave
sources. To perform its intended analysis, ULTRASAT will first of all build a
reference full sky map, which itself will contain an enormous amount of
information. In addition, during the lifetime of the mission, its low-cadence
survey (LCS) will provide a even deeper map, with sensitivity $\sim 10$
better, covering an area of $\sim 6800\,{\rm deg}^{2}$.

Besides the emission from resolved galaxies, such map will collect diffuse
light of astrophysical and cosmological origin, produced for example by
unresolved galaxies, AGN, dust Madau & Dickinson (2014); Bernstein et al.
(2002); Haardt & Madau (2012) or more exotic components such as decaying or
annihilating dark matter, or direct-collapse black holes Dwek et al. (1998);
Bond et al. (1986); Yue et al. (2013); Creque-Sarbinowski & Kamionkowski
(2018); Kalashev et al. (2019); Bernal et al. (2021); Carenza et al. (2023).
The overlap of all these processes produce the extragalactic background light
(EBL), which in the UV regime is not yet well constrained.

The ULTRASAT full-sky map will be a good tool to boost our knowledge in this
field; to find different ways to exploit its potential, a possible way is to
look at the studies that were performed on the full-sky map realized with the
All Sky and Medium Imaging Surveys performed by the Galaxy Evolution Explorer
(GALEX, Martin et al. (2005); Morrissey et al. (2007)). ULTRASAT, similarly to
GALEX, will observe a broad frequency range, hence it will collect the
integrated UV emission over a wide redshift range, and map its intensity
fluctuations as function of sky position.

The GALEX diffuse light map has been analyzed by the authors of Ref. Chiang et
al. (2019) ( C19) via the clustering-based redshift technique (CBR, Refs.
Newman (2008); McQuinn & White (2013); Ménard et al. (2013)): by cross
correlating it with the spectroscopic galaxy catalogs from the Sloan Digital
Sky Survey (SDSS, Blanton et al. (2005); Reid et al. (2016)), they
reconstructed the redshift evolution of the comoving volume emissivity of the
UV-EBL, providing constraints on the parameters that describe the non-ionizing
continuum and the Ly$\alpha$ line. These, in turn, can be used to constrain
properties such as the star formation rate or metallicity history.

A very interesting aspect of CBR is that it is only sensitive to extragalactic
contributions; the presence of foregrounds does not alter its signal, while it
contributes to the overall noise budget. A similar study was performed in Ref.
Scott et al. (2022) ( S21) to forecast the constraining power of the
Cosmological Advanced Survey Telescope for Optical and UV Research (CASTOR,
Cote et al. (2019)). CASTOR has been proposed to have three filters,
collecting radiation from $\lambda_{\rm obs}\sim 1500$ Å to $\lambda_{\rm
obs}\sim 5500$ Å. This will make the analysis sensitive to the EBL in the
optical region, as well as to the UV-EBL sourced at higher redshift. The use
of three filters will further boost the capability of breaking internal
degeneracies between the parameters in the CBR analysis, leading to a better
understanding of the EBL cosmic evolution. Further improvements may also come
from future experiments that will improve the sensitivity in the FUV band
already covered by GALEX; an interesting possibility in this case will be
offered by the Ultraviolet Explorer (UVEX, Kulkarni et al. (2021)).

In this work, we studied how to build forecasts for the CBR technique when
this is applied to the cross correlation between a broadband UV survey and a
spectroscopic galaxy catalog. We summarized how to model the main observable
that is required in this context, namely the angular cross correlation between
intensity measurements in pixels and galaxies, and we derived an analytical
expression to estimate its noise. We then ran a Fisher forecast with respect
to the parameters that model the UV-EBL comoving volume emissivity.

To validate our method, we first of all reproduced the setup of the analysis
performed in  C19, where GALEX$\times$SDSS is considered: the constraints we
obtained are in the same ballpark of the actual results. Once we tested the
reliability of our analysis, we applied it to forecast the cross correlation
between ULTRASAT full-sky maps and galaxies in the spectroscopic bins of the
Dark Energy Spectroscopic Instrument (DESI, Levi et al. (2013); Aghamousa et
al. (2016a, b)). We verified that the smaller redshift uncertainty and larger
galaxy number density DESI has with respect to SDSS, together with the smaller
noise variance and larger field of view in the ULTRASAT map, will imply an
improvement in the ULTRASAT$\times$DESI constraints with respect to
GALEX$\times$SDSS, when applied to the same parameter set and under the same
conditions.

While in the main text we relied on the ULTRASAT full-sky map, we also
analyzed the case of the low-cadence survey. Its $\sim 10$ times better
magnitude limit (24.5 vs 23.5) and $\sim 1/2$ smaller field of view (14 000
deg2 vs 6 800 deg2), lead to forecasts that are $\sim 10\%$ worse than the
ones described in Table 6. We note however that the specifics we assumed for
the full-sky map are optimistic, since we do not model in detail the
calibration and foreground cleaning procedures. On the other side, the 24.5
magnitude cut for the low-cadence survey represents a conservative choice;
reasonably, therefore, the constraints that will be obtained from it will be
comparable with our forecasts in Table 6.

Driven by this result, we applied the forecasted CBR analysis also to the
(GALEX$+$ULTRASAT)$\times$DESI full setup. We showed that the large redshift
range and the improved sensitivity will allow us to constrain with good
accuracy all the parameters in the emissivity model. Once propagated to the
emissivity uncertainty, these will lead to the forecasts shown in Fig. 4,
where we accounted for different choices of priors on the bias parameters.

Our results implicitly account for an approximation, namely the uncorrelation
between datasets, that allows us to simply sum the Fisher matrices in Eq.
(26). Accounting for correlation between the different datasets is not
straightforward: these may arise either because we use the same spectroscopic
catalog as reference for the CBR, or because the UV-EBL observed by GALEX FUV,
GALEX NUV and ULTRASAT have common sources. To estimate how much this would
affect our considerations, we consider an extreme scenario: we assume to
separate the reference galaxy catalog in three groups, each of which is cross
correlated with only one diffuse light map. Similarly, we assume to mask in
each full-sky map $2/3$ of the voxels, to remove double counting of the same
sources among different maps. This leads to a factor $\sim 2$ worse
constraints with respect to Table 6.

To further explore this point, we remove from the analysis the Fisher matrix
associated with GALEX NUV. Since the FUV filter and ULTRASAT observe non-
overlapping bands (see Fig. 1), sources that are relevant for the UV-EBL in
one of them may not necessarily be relevant for the other. Moreover, using the
ULTRASAT map realized in the low-cadence survey map would require only half of
the DESI catalog; we can hence assume to cross correlate GALEX FUV with the
other half of its footprint. This would drastically reduce the correlation
between the datasets, and it allow us to safely use $F=F_{\rm FUV}+F_{\rm
LCS}$. Results are $\lesssim 50\%$ worse than the values collected in Table 6;
even in this case, our technique leads to a good improvement with respect to
current constraints on the UV-EBL.

We found that (GALEX$+$ULTRASAT)$\times$DESI will be able to constrain the
full UV emissivity frequency dependence in the redshift range $z\lesssim 2$.
In our work, we assumed that this is composed by different contributions: the
non-ionizing continuum, the Ly$\alpha$ line and the ionizing continuum above
the Lyman break. The amplitude and slope of each of these elements is
determined by the UV-EBL sources; therefore, this measurement will provide
interesting insights on the astrophysics. In Fig. 5, we showed our forecasts
on the non-ionizing EBL emissivity at $\lambda=1500\,$Å. Here, current data
show an overall agreement with the EBL models that accounts for galaxies, AGN
emissions and dust. The improvement (GALEX$+$ULTRASAT)$\times$DESI will bring
will foster our understanding of this regime, opening the window to the
detection of exotic cosmological emissions, if any.

To conclude, we stress once again how the study of the UV extragalactic
background light will be crucial in the upcoming years. ULTRASAT, with its
reference map, will offer the possibility of mapping its intensity
fluctuations across the full sky, offering a tool to probe the underlying
large scale structure in a novel, alternative way. To be able to exploit the
enormous amount of scientific information it will contain, it is important to
develop specific tools for its analysis; the cross correlation with galaxy
surveys, in particular in the context of the clustering-based redshift
analysis, is indeed one of them.

###### Acknowledgements.

The authors thank Yi-Kuan Chiang for his comments, which helped improving the
quality of the paper. A special thanks goes to José L. Bernal for his insights
and suggestions about the analysis, and to the anonymous referee for their
comments. We thank the ULTRASAT Collaboration for useful discussion about the
project, and Brice Ménard, Yossi Shvartzvald and Marek Kowalski for feedback
on the manuscript. SL thanks the Azrieli Foundation for support and the Padova
Cosmology group for hospitality during the tough times between October 7th and
November 2023. SL is supported by an Azrieli International Postdoctoral
Fellowship. EDK was supported by a Faculty Fellowship from the Azrieli
Foundation. EDK also acknowledges joint support from the U.S.-Israel Bi-
national Science Foundation (BSF, grant No. 2022743) and the U.S. National
Science Foundation (NSF, grant No. 2307354), and support from the ISF-NSFC
joint research program (grant No. 3156/23).

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