# Searching for eV-mass Axion-like Particles with
Cross Correlations between Line Intensity and Weak Lensing Maps

Masato Shirasaki<EMAIL_ADDRESS>National Astronomical Observatory
of Japan (NAOJ), Mitaka, Tokyo 181-8588, Japan  The Institute of Statistical
Mathematics, Tachikawa, Tokyo 190-8562, Japan

###### Abstract

We study cross correlations between line intensity and weak lensing maps to
search for axion-like particles (ALPs). Radiative decay of eV-mass ALPs can
contribute to cosmic background emissions at optical and infrared wavelengths.
Line intensity mapping is a unique means of measuring the background emission
at a given photon frequency. If ALPs constitute the abundance of cosmic dark
matter, line intensity maps can correlate with large-scale structures probed
by weak gravitational lensing effects in galaxy imaging surveys. We develop a
theoretical framework to predict the cross correlation. We then explore
potentiality of probing ALPs with the cross correlation in upcoming galaxy-
imaging and spectral surveys. Assuming SPHEREx and the Vera Rubin
Observatory’s Legacy Survey of Space and Time (LSST), we find that the cross
correlation by the ALP decay can be greater than the astrophysical-line
counterparts at wavelength of $\sim 3000\,{\rm nm}$ for ALPs with a particle
mass of $m_{a}\sim 1\,\mathrm{eV}$ and a particle-to-two-photons coupling of
$g_{a\gamma\gamma}\sim 1\times 10^{-11}\,\mathrm{GeV}^{-1}$. We also predict
that a null detection of the cross correlation can place a $2\sigma$-level
upper bound of $g_{a\gamma\gamma}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10^{-11}\,\mathrm{GeV}^{-1}$ for
eV-mass ALPs, improving the current constraint by a factor of $\sim 10$. We
then make a forecast of expected constraints on ALP parameters in SPHEREx and
LSST by Fisher analysis, providing a guideline of searching for the ALP decay
with the large-scale structure data.

##  I Introduction

The nature of cosmic dark matter is one of the long-standing mysteries in
modern astronomy and physics. Dark matter accounts for about 80% of the
average mass density in our universe [1], but can not be explained by the
Standard Model of particle physics. The QCD axion is a well-motivated
candidate of dark matter [2, 3, 4], while it is originally proprosed as a
solution to the strong CP problem [5, 6, 7, 8]. The QCD axion is a
pseudoscalar boson with an approximate shift symmetry. More general light
pseudoscalars are known as axion-like particles, or ALPs. ALPs are pseudo-
Nambu Goldstone bosons produced through the breaking of global $U(1)$
symmetries, or zero modes of higher dimensional gauge fields [9, 10, 11, 12].
Because ALPs are not associated with the strong CP problem, they can exhibit a
wider range of couplings and masses. Hence, ALPs are considered to be a
compelling candidate of dark matter [13].

In this paper, we are interested in the search for ALPs with astronomical
datasets. In particular, we pay special attention to radiative decay of ALPs
with their particle mass of about 1 eV. The eV-mass ALPs can decay into two
photons with photon frequencies of $\sim 10^{14}\,\mathrm{Hz}$, corresponding
to optical and infrared wavelenghs. Galaxy surveys at optical and infrared
bands are about to reach their peak. Upcoming large surveys include the Vera
Rubin Observatory’s Legacy Survey of Space and Time
(LSST)111https://www.lsst.org/, The Nancy Roman Space
Telescope222https://roman.gsfc.nasa.gov/,
Euclid333https://sci.esa.int/web/euclid, and
SPHEREx444https://www.jpl.nasa.gov/missions/spherex/.

Among the upcoming surveys, SPHEREx provides a unique opportunity of studying
cumulative background emissions at optical and infrared bands with line-
intensity mapping techniques. The line-intensity mapping measures photon
intensity at a narrow range of frequency along a given line-of-sight
direction, allowing us to explore faint astrophysical sources that are
difficult to be detected individually. If a fraction of cosmic dark matter
would consist of eV-mass ALPs, ALP-decay photons contribute to the line
intensity maps obtained by SPHEREx. At the same time, ALPs are at work as
building blocks of large-scale structures in the universe and they distort
shapes of distant galaxies by weak gravitational lensing effects (see, e.g.
Ref. [14] for a review). Hence, ALPs can affect two different observables of
line intensity maps and weak lensing effects, introducing spatial cross
correlations between the two. Weak gravitational lensing effects are now
recognized as a main science driver in galaxy imaging surveys and most
upcoming imaging surveys are aimed at measuring the effects with unprecedented
accuracy. It is thus strongly motivated to study expected cross correlations
between the line intensity maps and weak lensing effects in galaxy imaging
surveys for the search for ALPs, enhancing the science returns from large
galaxy surveys.

It would be worth noting that Ref. [15] has already proposed the ALP search by
using cross correlations between line intensity maps and other large-scale
structure data. Nevertheless, they mainly focus on a biased tracer of large
scale structures such as galaxies, while we consider the weak lensing effects
to extract unbiased information about underlying cosmic mass density fields.
We also study possible astrophysical contributions to the cross correlation
analysis, which are ignored in Ref. [15]. Recently, Ref. [16] proposed a
statistical approach to constrain decaying dark matter with line-intensity
maps alone. They studied a joint analysis of auto correlation functions and
one-point probability distributions in line intensity maps. Our approach is
thought to be complementary to that in Ref. [16].

The rest of the present paper is organized as follows. In Section II, we
describe observables of interest and relevant physical effects to the
observables. In Section III, we summarize a theoretical model to predict cross
correlation functions between line intensity maps and weak lensing effects in
galaxy shapes. We present the results in Section IV. Concluding remarks and
discussions are given in Section V. Throughout, we consider a flat universe
with the standard cosmological parameters $H_{0}=100h\,{\rm km}\,{\rm s}^{-1}$
with $h=0.68$, the average matter density $\Omega_{\rm m0}=0.315$, the baryon
density $\Omega_{\rm b0}=0.0491$, the cosmological constant
$\Omega_{\Lambda}=0.685$, the spectral index of initial curvature fluctuations
$n_{s}=0.97$, and the amplitude of matter density fluctuations within
$8\,h^{-1}\,{\rm Mpc}$, $\sigma_{8}=0.83$. Note that we adopt natural units
with $c=\hbar=1$, where $c$ is the speed of light and $\hbar$ is the reduced
Planck constant.

##  II Observables

In this section, we introduce two observables of interest, the line-intensity
and weak-lensing maps. The former provides the cumulative emission from
sources fainter than detection limits in spectroscopic surveys, while the
latter is the projected mass density distribution obtained in galaxy imaging
surveys.

###  II.1 Line Intensity

The observed intensity $I$ at photon frequency $\nu$ along a given direction
$\theta$ is expressed as

$\displaystyle
I(\mbox{\boldmath$\theta$}\,|\,\nu)=\frac{1}{4\pi}\int\frac{\mathrm{d}\chi}{1+z}\,\epsilon(\mbox{\boldmath$r$},\nu^{\prime},z),$
(1)

where $\nu^{\prime}=\nu(1+z)$ is the photon frequency emitted at the source
redshift $z$, $\chi$ is the comoving distance to the source, and
$\epsilon(\mbox{\boldmath$r$},\nu^{\prime},z)$ represents the volume
emissivity (i.e., the photon energy emitted per unit volume, time, and energy
range) at the three-dimensional position of
$\mbox{\boldmath$r$}=(\chi(z)\,\mbox{\boldmath$\theta$},\chi(z))$ and the
frequency $\nu^{\prime}$. Note that we define the volume emissivity in the
comoving coordinate in Eq. (1). In line intensity mapping measurements, we are
interested in any emission lines which can be observed at the frequency $\nu$.
Because the line width is usually negligible compared to the spectroscopic
resolution in the instrument555 The virial velocity dispersion in typical
galaxy-sized halos with $M=10^{12-13}\,M_{\odot}$ can be estimated as $\sim
200\,\mathrm{km}/\mathrm{s}$. Hence, we can ignore the velocity dispersion
effect in the analysis as long as setting the frequency resolution to $\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}100$., we assume that the volume
emissivity is given by

$\displaystyle\epsilon(\mbox{\boldmath$r$},\nu,z)=\rho_{L}(\mbox{\boldmath$r$},z)\,\delta^{(1)}_{\mathrm{D}}(\nu-\nu_{0}),$
(2)

where $\rho_{L}$ is the luminosity density of sources,
$\delta^{(n)}_{\mathrm{D}}$ is the $n$-dimensional Dirac delta function,
$\nu_{0}$ is the rest-frame line frequency of interest. Throughout this paper,
we define the intensity in units of
$\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}/\mathrm{Hz}/\mathrm{str}$. In the
following, we briefly describe observable lines at optical and infrared bands,
associated with the ALP decay and some relevant astrophysical sources.

####  II.1.1 ALP decay

Axion-like particles (ALPs) are light, weakly-interacting pseudo-scalar
particles. A common feature among ALP models is that they can decay into
photons with their decay rate $\Gamma$ being

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{a\gamma\gamma}\,m^{3}_{a}}{64\pi}$ (3)
$\displaystyle=$ $\displaystyle 7.556\times
10^{-26}\,\mathrm{s}^{-1}\,\left(\frac{g_{a\gamma\gamma}}{10^{-10}\,\mathrm{GeV}^{-1}}\right)^{2}\,\left(\frac{m_{a}}{1\,\mathrm{eV}}\right)^{3},$
(4)

where $m_{a}$ and $g_{a\gamma\gamma}$ are the mass and the particle-to-two-
photon coupling constant of ALPs, respectively. In this decay, the emitted
photon frequency is given by $\nu_{a}=m_{a}/(4\pi)=1.21\times
10^{14}\,\mathrm{Hz}\,(m_{a}/1\,\mathrm{eV})$.

Suppose that ALPs can constitute a part of dark matter, we can express the
luminosity density due to the ALP decay as

$\displaystyle\rho_{L,a}(\mbox{\boldmath$r$},z)=\Gamma\,f_{a}\,\rho_{\mathrm{DM}}(\mbox{\boldmath$r$},z),$
(5)

where $f_{a}$ is the fraction of ALPs in the dark matter and
$\rho_{\mathrm{DM}}(\mbox{\boldmath$r$},z)$ represents the dark matter density
at redshift $z$. In the following, we consider the case of $f_{a}=1$ for
simplicity666Our parameter forecasts can be easily generalized for $f_{a}<1$
if one replaces $g_{a\gamma\gamma}$ with $g_{a\gamma\gamma}\,f_{a}^{1/2}$..
Using Eqs. (1), (2), and (5), one can find

$\displaystyle I_{a}(\mbox{\boldmath$\theta$}\,|\,\nu)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}\chi\,W_{\mathrm{DM}}(z|\nu)\,\left[1+\delta_{\mathrm{DM}}(\mbox{\boldmath$r$},z)\right],$
(6) $\displaystyle W_{\mathrm{DM}}(z|\nu)$ $\displaystyle=$
$\displaystyle\frac{\Gamma}{4\pi}\frac{\Omega_{\mathrm{DM}}\rho_{\mathrm{crit},0}}{\nu_{a}\,H(z)}\,\delta^{(1)}_{\mathrm{D}}(\chi-\chi_{a}),$
(7)

where $\Omega_{\mathrm{DM}}=\Omega_{\rm m0}-\Omega_{\rm b0}\sim 0.26$ is the
dimensionless dark matter density, $\rho_{\mathrm{crit,0}}$ is the critical
density at present, $\delta_{\mathrm{DM}}$ is the dark matter density
contrast, $H(z)$ is the Hubble expansion parameter at redshift $z$, $\chi_{a}$
is the comoving distance to the redshift $z_{a}$ which is defined by
$z_{a}\equiv\nu_{a}/\nu-1$.

####  II.1.2 Astrophysical sources

The $\mathrm{H}_{\alpha}$ line is a Balmer line that corresponds to a
transition between energy levels $n=3$ to $n=2$ of neutral hydrogen. It is
commonly used to study star-forming galaxies and quasars at $z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2$ so far. The rest-frame
wavelength of the $\mathrm{H}_{\alpha}$ line is given by 656.28 $\mathrm{nm}$,
which is suitable for studying faint galaxies as well as diffuse emissions
from intergalactic media in ongoing line-intensity mapping surveys.

It is known that the strength of the $\mathrm{H}_{\alpha}$ line shows a tight
correlation with the star-formation rate (SFR) of galaxies [17]. The commonly
used relation between the SFR and the intrinsic $\mathrm{H}_{\alpha}$
luminosity is provided as

$\displaystyle L_{\mathrm{H}_{\alpha}}=1.26\times
10^{41}\,(\mathrm{erg}/\mathrm{s})\,\left(\frac{\mathrm{SFR}}{1\,M_{\odot}/\mathrm{yr}}\right),$
(8)

where $L_{\mathrm{H}_{\alpha}}$ is the intrinsic luminosity. The observed
$\mathrm{H}_{\alpha}$ luminosity $L^{\mathrm{obs}}_{\mathrm{H}_{\alpha}}$ is
subject to dust extinction. We obtain the observed luminosity by accounting
for the extinction with a magnitude of $A_{\mathrm{H}_{\alpha}}$,

$\displaystyle
L^{\mathrm{obs}}_{\mathrm{H}_{\alpha}}=10^{-A_{\mathrm{H}_{\alpha}}/2.5}\,L_{\mathrm{H}_{\alpha}}.$
(9)

In this paper, we adopt $A_{\mathrm{H}_{\alpha}}=1\,\mathrm{mag}$ which is a
typical value in large $\mathrm{H}_{\alpha}$ surveys [18].

To evaluate the $\mathrm{H}_{\alpha}$ luminosity density, we assume that any
star formation processes occur at the centers of single dark matter halos and
SFRs can be determined by a function of halo masses and redshifts. In this
halo-based model, we can express the $\mathrm{H}_{\alpha}$ luminosity density
$\rho_{L,\mathrm{H}_{\alpha}}$ as

$\displaystyle\rho_{L,\mathrm{H}_{\alpha}}(\mbox{\boldmath$r$},z)$
$\displaystyle=$
$\displaystyle\sum_{i}\,L^{\mathrm{obs}}_{\mathrm{H}_{\alpha}}(M_{i},z)\,\delta^{(3)}_{\mathrm{D}}(\mbox{\boldmath$r$}-\mbox{\boldmath$r$}_{i})$
(10) $\displaystyle=$ $\displaystyle{\cal
L}_{\mathrm{H}_{\alpha}}\sum_{i}\dot{M}_{*}(M_{i},z)\delta^{(3)}_{\mathrm{D}}(\mbox{\boldmath$r$}-\mbox{\boldmath$r$}_{i}),$
(11)

where $\dot{M}_{*}$ is the SFR in units of $M_{\odot}/\mathrm{yr}$, $M_{i}$
and $\mbox{\boldmath$r$}_{i}$ represent a halo mass and spatial coordinate of
the $i$-th halo, the index $i$ runs over all the halos at redshift $z$ in a
unit volume, and we introduce ${\cal L}_{\mathrm{H}_{\alpha}}=1.26\times
10^{41}\,(\mathrm{erg}/\mathrm{s})\times 10^{-A_{\mathrm{H}_{\alpha}}/2.5}$.

We start from computing the average luminosity density based on Eq. (10). It
is given by

$\displaystyle\bar{\rho}_{L,\mathrm{H}_{\alpha}}(z)=10^{-A_{\mathrm{H}_{\alpha}}/2.5}\,\int\mathrm{d}L_{\mathrm{H}_{\alpha}}\,\frac{\mathrm{d}n}{\mathrm{d}L_{\mathrm{H}_{\alpha}}}\,L_{\mathrm{H}_{\alpha}},$
(12)

where $\mathrm{d}n/\mathrm{d}L_{\mathrm{H}_{\alpha}}$ is the
$\mathrm{H}_{\alpha}$ luminosity function (LF)777Note that one can express
$\bar{\rho}_{L,\mathrm{H}_{\alpha}}(z)$ by integrating the halo mass function
and the SFR-to-halo-mass relation as well. Nevertheless, we think Eq. (12)
would give a more reasonable estimate, because the SFR-to-halo-mass relation
is poorly constrained at present.. The mean intensity of the
$\mathrm{H}_{\alpha}$ emission is then computed as

$\displaystyle\bar{I}_{\mathrm{H}_{\alpha}}(z)=\frac{1}{4\pi}\frac{1}{\nu_{\mathrm{H}_{\alpha}}H(z)}\bar{\rho}_{L,\mathrm{H}_{\alpha}}(z),$
(13)

where $\nu_{\mathrm{H}_{\alpha}}$ is the rest-frame frequency of the
$\mathrm{H}_{\alpha}$ line. Ref. [19] provides a phenomenological model of Eq.
(13) using Eq. (12) and the current constraint of LFs at different redshifts.
Their model is expressed as

$\displaystyle\bar{I}_{\mathrm{H}_{\alpha}}(z)=3.326\times
10^{-9}\,\frac{0.027+0.28z}{1+(z/4.8)^{5.3}}\,10^{-A_{\mathrm{H}_{\alpha}}/2.5}$
$\displaystyle\qquad\times\left(\frac{\nu^{-1}_{\mathrm{H}_{\alpha}}\,H^{-1}(z)}{h^{-1}\,\mathrm{Mpc}\,\mathrm{Hz}^{-1}}\right)\,\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}/\mathrm{Hz}/\mathrm{str}.$
(14)

Finally, we express the observed $\mathrm{H}_{\alpha}$ intensity as

$\displaystyle I_{\mathrm{H}_{\alpha}}(\mbox{\boldmath$\theta$}\,|\,\nu)$
$\displaystyle=$
$\displaystyle\int\mathrm{d}\chi\,W_{\mathrm{\mathrm{H}_{\alpha}}}(z|\nu)\,\left[1+\delta_{\mathrm{\mathrm{H}_{\alpha}}}(\mbox{\boldmath$r$},z)\right],$
(15) $\displaystyle W_{\mathrm{\mathrm{H}_{\alpha}}}(z|\nu)$ $\displaystyle=$
$\displaystyle\bar{I}_{\mathrm{H}_{\alpha}}(z)\,\delta^{(1)}_{\mathrm{D}}(\chi-\chi_{\mathrm{H}_{\alpha}}),$
(16)

where $\chi_{\mathrm{H}_{\alpha}}$ is the comoving distance to the redshift
$z_{\mathrm{H}_{\alpha}}$ which is defined by
$z_{\mathrm{H}_{\alpha}}\equiv\nu_{\mathrm{H}_{\alpha}}/\nu-1$, and
$\delta_{\mathrm{\mathrm{H}_{\alpha}}}$ is the density contrast in the
$\mathrm{H}_{\alpha}$ luminosity density.

In actual observations, several lines can contaminate the observed
$\mathrm{H}_{\alpha}$ intensity maps because the measurement relies on
spectroscopy at single wavelengths. The contamination lines include ionized
oxygen $[\mathrm{O}_{\mathrm{II}}]$ 372.7 nm and $[\mathrm{O}_{\mathrm{III}}]$
500.7 nm lines, the hydrogen $\mathrm{H}_{\beta}$ 486.1 nm, and Lyman $\alpha$
(Ly$\alpha$) 121.6 nm lines888 In principle, $[\mathrm{N}_{\mathrm{II}}]$
658.3 nm/654.8 nm and $[\mathrm{S}_{\mathrm{II}}]$ 671.7 nm/673.1 nm doublet
lines also contribute to the $\mathrm{H}_{\alpha}$ intensity maps.
Nevertheless, the SPHEREx is not able to distinguish the $\mathrm{H}_{\alpha}$
line between these doublet lines in frequency spaces. In this paper, we
include the contribution from these doublet lines as follows in Ref. [19]. We
assume that the $\mathrm{N}_{\mathrm{II}}$ line contributes 22% of the sum the
$\mathrm{H}_{\alpha}$ and $\mathrm{N}_{\mathrm{II}}$ line intensities [20],
while the $\mathrm{S}_{\mathrm{II}}$ doublet line intensity is set to 12% of
the $\mathrm{H}_{\alpha}+\mathrm{N}_{\mathrm{II}}+\mathrm{S}_{\mathrm{II}}$
line intensity [21]. . To take into account these interlopers, we assume the
published relations between line luminosity and the SFR. For a given line
denoted as $Q$, we express the luminosity-SFR relation as

$\displaystyle L^{\mathrm{obs}}_{Q}={\cal
L}_{Q}\left(\frac{\mathrm{SFR}}{1\,M_{\odot}/\mathrm{yr}}\right),$ (17)

where $L^{\mathrm{obs}}_{Q}$ is the observed luminosity for the line $Q$, and
${\cal L}_{Q}$ is the scale luminosity in units of $\mathrm{erg}/\mathrm{s}$.
Note that the term ${\cal L}_{Q}$ includes the dust attenuation effect as
$10^{-A_{Q}/2.5}$ where $A_{Q}$ is the extinction for the line $Q$. Using Eqs.
(8), (9), (12) and (17), one can find

$\displaystyle\bar{\rho}_{L,Q}=\frac{{\cal L}_{Q}}{{\cal
L}_{\mathrm{H}_{\alpha}}}\bar{\rho}_{L,\mathrm{H}_{\alpha}},$ (18)

where $\bar{\rho}_{L,Q}$ is the luminosity density for the line $Q$. Table 1
summarizes the model of ${\cal L}_{Q}$ in this paper.

Line |  $\lambda_{Q}$ (nm) | ${\cal L}_{Q}\,(\mathrm{erg/s})$ | $A_{Q}\,(\mathrm{mag})$
---|---|---|---
$\mathrm{H}_{\alpha}$ | 656.28 |  $1.26\times 10^{41}\times 10^{-A_{\mathrm{H}_{\alpha}}/2.5}$ (111Ref. [17].) | 1 (222Ref. [18].)
$\mathrm{O}_{\mathrm{III}}$ | 500.7 |  $1.32\times 10^{41}\times 10^{-A_{\mathrm{O}_{\mathrm{III}}}/2.5}$ (333Ref. [22].) | 1.35 (444Ref. [23].)
$\mathrm{H}_{\beta}$ | 486.1 |  $4.43\times 10^{40}\times 10^{-A_{\mathrm{H}_{\beta}}/2.5}$ (555Using Eq. (8) and the recombination emission line ratios.) | 1.35 (666Refs. [24, 25].)
$\mathrm{O}_{\mathrm{II}}$ | 372.7 |  $7.18\times 10^{40}\times 10^{-A_{\mathrm{O}_{\mathrm{II}}}/2.5}$ (777Using a ratio of $0.57$ between the $\mathrm{O}_{\mathrm{II}}$ and $\mathrm{H}_{\alpha}$ fluxes observed in local galaxies [17].) | 0.62 (888Ref. [26].)
Ly$\alpha$ | 121.6 |  $1.10\times 10^{42}\times 10^{-A_{\mathrm{Ly}\alpha}/2.5}$ (999Ref. [17].) | 0
Table 1:  Characteristics of astrophysical lines in this paper. For a given
line $Q$, $\lambda_{Q}$ represents the rest-frame wavelength, ${\cal L}_{Q}$
is the scale luminosity in the luminosity-SFR relation in Eq. (17), and
$A_{Q}$ is the extinction.

Hence, the total intensity from astrophysical sources is written as

$\displaystyle I_{\mathrm{astro}}(\mbox{\boldmath$\theta$}\,|\,\nu)$
$\displaystyle=$
$\displaystyle\sum_{Q}\int\mathrm{d}\chi\,W_{Q}(z|\nu)\,\left[1+\delta_{\mathrm{astro}}(\mbox{\boldmath$r$},z)\right],$
(19) $\displaystyle W_{Q}(z|\nu)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\frac{\bar{\rho}_{Q}}{\nu_{Q}\,H(z)}\,\delta^{(1)}_{\mathrm{D}}(\chi-\chi_{Q})$
(20) $\displaystyle=$ $\displaystyle\frac{{\cal L}_{Q}}{{\cal
L}_{\mathrm{H}_{\alpha}}}\frac{\nu_{\mathrm{H}_{\alpha}}}{\nu_{Q}}\bar{I}_{\mathrm{H}_{\alpha}}(z)\,\delta^{(1)}_{\mathrm{D}}(\chi-\chi_{Q})$

where
$Q=\\{\mathrm{H}_{\alpha},[\mathrm{O}_{\mathrm{II}}],[\mathrm{O}_{\mathrm{III}}],\mathrm{H}_{\beta},\mathrm{Ly}\alpha\\}$,
$\nu_{Q}$ is the rest-frame frequency for the line $Q$, $\chi_{Q}$ is the
comoving distance to the redshift $z_{Q}$ which is defined by
$z_{Q}\equiv\nu_{Q}/\nu-1$, $\delta_{\mathrm{astro}}$ is the density contrast
in the luminosity density of astrophysical sources. Because the luminosity
density is always expressed as in Eq. (11) within our framework, statistics of
$\delta_{\mathrm{astro}}$ are determined by the SFR alone, not depending on
kinds of lines. To be specific, the field $\delta_{\mathrm{astro}}$ is given
by

$\displaystyle 1+\delta_{\mathrm{astro}}(\mbox{\boldmath$r$},z)$
$\displaystyle=$ $\displaystyle{\cal
N}^{-1}_{\mathrm{astro}}(z)\sum_{i}\dot{M}_{*}(M_{i},z)$ (21)
$\displaystyle\qquad\qquad\qquad\times\delta^{(3)}_{\mathrm{D}}(\mbox{\boldmath$r$}-\mbox{\boldmath$r$}_{i})$
$\displaystyle{\cal N}_{\mathrm{astro}}(z)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}M\,\frac{\mathrm{d}n}{\mathrm{d}M}\,\dot{M}_{*}(M_{i},z),$
(22)

where $\mathrm{d}n/\mathrm{d}M$ is the halo mass function.

###  II.2 Weak Lensing

####  II.2.1 Cosmic shear

Weak gravitational lensing effect is usually characterized by the distortion
of image of a source object by the following 2D matrix:

$A_{ij}=\frac{\partial\beta^{i}}{\partial\theta^{j}}\equiv\left(\begin{array}[]{cc}1-\kappa-\gamma_{1}&-\gamma_{2}\\\
-\gamma_{2}&1-\kappa+\gamma_{1}\\\ \end{array}\right),$ (23)

where $\theta$ represents the observed position of a source object, $\beta$ is
the true position, $\kappa$ is the convergence, and $\gamma$ is the shear. In
the weak lensing regime (i.e., $\kappa,\gamma\ll 1$), each component of
$A_{ij}$ can be related to the second derivative of the gravitational
potential $\Phi$ [14]. Using the Poisson equation and the Born approximation,
one can express the weak lensing convergence field as the weighted integral of
matter overdensity field $\delta_{\rm m}(\mbox{\boldmath$x$})$:

$\kappa(\mbox{\boldmath$\theta$})=\int_{0}^{\chi_{H}}{\rm d}\chi\
W_{\kappa}(\chi)\,\delta_{\rm m}(\mbox{\boldmath$r$},z),$ (24)

where $\delta_{\rm m}$ is the matter overdensity field, $\chi_{H}$ is the
comoving distance up to $z\rightarrow\infty$ and $W_{\kappa}(\chi)$ is called
lensing kernel. For a given redshift distribution of source galaxies, the
lensing kernel is expressed as

$W_{\kappa}(\chi)=\frac{3}{2}\,\Omega_{\rm
m0}H^{2}_{0}\,(1+z(\chi))\int_{\chi}^{\chi_{H}}{\rm
d}\chi^{\prime}p(\chi^{\prime})\frac{\chi(\chi^{\prime}-\chi)}{\chi^{\prime}},$
(25)

where $p(\chi)$ represents the redshift distribution of source galaxies
normalized to $\int_{0}^{\chi_{H}}{\rm d}\chi\,p(\chi)=1$. In this paper, we
assume that $p(\chi)$ is expressed as

$\displaystyle
p(\chi)\propto\left(\frac{\mathrm{d}\chi}{\mathrm{d}z}\right)^{-1}\,z^{\alpha}\,\exp\left[-\left(\frac{z}{z_{0}}\right)^{\beta}\right],$
(26)

where we adopt $\alpha=1.27$, $\beta=1.02$, and $z_{0}=0.50$. These parameters
are expected in a realistic situation in the LSST [27]. Note that the median
source redshift in this model is given by 0.83.

####  II.2.2 Intrinsic alignments of galaxies

Galaxies are thought to be formed at high density regions in the universe, or
gravitationally-bound dark matter halos. Such dark matter halos are grown by
anisotropic mass accretion through cosmic filaments. Because filamentary
structures can introduce tidal interaction of dark matter halos and
surrounding density fields, orientation of halo shapes can not be randomly
distributed, but aligned with the filaments [28, 29].

Possible large-scale alignments of galaxies’ shape even in the absence of
gravitational lensing, referred to as intrinsic alignments (IAs), can
introduce a systematic uncertainty in any cross correlation analyses based on
weak lensing [30]. To account for the IA effect on the cross correlation
between line intensity and weak lensing maps, we work with the linear
alignment model [31, 32]. In this paper, we set a relevant IA field to lensing
convergence so that it can reproduce the E-mode power spectrum of shapes in
the linear alignment model [33]. The IA field $\kappa_{\rm IA}$ is then given
by

$\displaystyle\kappa_{\rm IA}(\mbox{\boldmath$\theta$})$ $\displaystyle=$
$\displaystyle\int\mathrm{d}\chi\,W_{\rm IA}(\chi)\,\delta_{\rm
m}(\mbox{\boldmath$r$},z),$ (27) $\displaystyle W_{\rm IA}(\chi)$
$\displaystyle=$ $\displaystyle-{\cal A}_{\rm IA}C_{1}\frac{\Omega_{\rm
m0}\rho_{{\rm crit},0}}{D(z)}\left(\frac{1+z}{1+z_{0}}\right)^{\eta_{\rm
IA}}p(\chi),$ (28)

where ${\cal A}_{\rm IA}$ controls the IA amplitude, $\eta_{\rm IA}$ is a free
parameter to characterize redshift evolution, $C_{1}=5\times
10^{-14}\,h^{-2}\,M_{\odot}\,{\rm Mpc}^{3}$ is a normalization constant,
$z_{0}=0.62$ is the pivot redshift, and $D(z)$ is the normalized linear growth
factor, so that $D(0)=1$. We adopt an estimate of $\eta_{\rm IA}=3$ based on
observations of SuperCOSMOS Sky Survey [34] and Subaru Hyper Suprime Cam [35],
while we set ${\cal A}_{\rm IA}=1$ for our fiducial model. Note that some
recent weak-lensing measurements reported a marginal non-zero ${\cal A}_{\rm
IA}$ [36, 35, 37], but the exact value should depend on the selection of
source galaxies used in weak lensing analyses.

###  II.3 Relevant redshift ranges

Figure 1:  The relation between observable wavelengths and line redshifts in
the SPHEREx. The gray filled region shows the range of wavelengths in the
SPHEREx, while the dashed and dotted lines represent the wavelength-redshift
relation for $\mathrm{H}_{\alpha}$ and $\mathrm{O}_{\rm III}$ lines,
respectively. The colored solid lines show the relation for ALP decay lines
with various ALP masses $m_{a}$. ALPs with smaller $m_{a}$ emit photons with a
longer wavelentgh as shown in redder lines in this figure.

We here describe effective redshift ranges to the cross correlation analysis
between line intensity and weak lensing maps. For the line intensity maps, we
consider the SPHEREx survey observing an all sky in optical and near-infrared
wavelengths. The SPHEREx will provide line intensity maps at wavelengths of
$750-4100$ nm with a frequency resolution being $R=41.5$ [38].

Figure 1 shows the relation between observed wavelengths
$\lambda_{\mathrm{obs}}$ and line redshifts in the SPHEREx. Different colored
solid lines represent line redshifts by ALP decays with a variety of ALP
masses $m_{a}$. For the wavelength range in the SPHEREx, we expect that ALP
decays with $0.5\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}m_{a}\,[\mathrm{eV}]\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}3.5$ would be relevant to the cross
correlation. For $m_{a}\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}3.5\,\mathrm{eV}$, the wavelength
by ALP-decay line becomes shorter than those of astrophysical lines such as
$\mathrm{H}_{\alpha}$ and $\mathrm{O}_{\rm III}$. Because the weak lensing can
probe large-scale structures at $z\sim 0.5$ in modern galaxy surveys, the ALP
decay with $m_{a}\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}3.5\,\mathrm{eV}$ is not expected
to induce a significant correlation between the lensing and the SPHEREx
intensity maps. On the other hand, the wavelength coverage in the SPHEREx does
not allow us to detect the line emission from the ALP decay with $m_{a}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.5\,\mathrm{eV}$.

Figure 2:  Relevant redshift ranges in the line intensity and weak lensing
maps. In this figure, we assume an LSST-like imaging survey for the lensing
map. The blue solid line shows the lensing kernel function $W_{\kappa}$ for
the LSST-like survey, while the green dotted line represents our fiducial
model of the mean intensity for $\mathrm{H}_{\alpha}$ line. The orange dashed
line is the mean intensity as a function of redshift $z$ for ALPs with a mass
of $m_{a}=1\,\mathrm{eV}$ and a particle-to-photons coupling of
$g_{a\gamma\gamma}=1\times 10^{-11}\,\mathrm{GeV}^{-1}$. Note that the mean
intensity is defined in units of
$\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}/\mathrm{Hz}/\mathrm{str}$. Different
line segments at the middle in this figure represent relevant redshifts to
line-intensity measurements for given observed wavelentghs
$\lambda_{\mathrm{obs}}$. The right color bar covers the range of
$\lambda_{\mathrm{obs}}$ in SPHEREx. For instance, when using a line intensity
map at $\lambda_{\mathrm{obs}}\sim 2500\,\mathrm{nm}$, one will probe large-
scale structures at $z\sim 0.1$ by the ALP decay, while the
$\mathrm{H}_{\alpha}$ line provides astrophysical information at $z\sim 3$.
Because the lensing is efficient to extract the information about structures
at $z\sim 0.3$, the cross correlation with the line intensity and weak lensing
maps allows us to separate ALP-decay signals from the total line line
intensity when $\lambda_{\mathrm{obs}}$ is limited to be greater than $\sim
2500\,\mathrm{nm}$.

Figure 2 summarizes the relevant redshift range to the cross correlation with
line-intensity and weak-lensing maps. In this figure, the blue solid line
represents the lensing kernel defined in Eq. (25) for the expected source-
redshift distribution in the LSST. The LSST can be the most efficient to probe
the large-scale cosmic mass density at $z\simeq 0.3-0.6$. For comparison, the
orange dashed line shows the function of

$\displaystyle\bar{I}_{\rm
DM}(z)=\frac{\Gamma}{4\pi}\frac{\Omega_{\mathrm{DM}}\rho_{\mathrm{crit},0}}{\nu_{a}\,H(z)},$
(29)

where this provides the redshift dependence on the mean line intensity by the
ALP decay. In Figure 2, we consider a representative example as ALPs with
$m_{a}=1\,\mathrm{eV}$ and $g_{a\gamma\gamma}=1\times
10^{-11}\,\mathrm{GeV}^{-1}$. In addition, the green dotted line shows the
mean ${\rm H}_{\alpha}$ line intensity defined in Eq. (13). For the line
intensity mapping, we measure the cumulative emission from any photon sources
at a given wavelegth $\lambda_{\rm obs}$. The line segments at the middle of
Figure 2 highlight effective redshifts in the line-intensity map as a function
of $\lambda_{\rm obs}$. According to the relation between line redshifts and
$\lambda_{\rm obs}$ in Figure 1, the ALP-decay line can always provide lower-
redshift information than the ${\rm H}_{\alpha}$ line as long as we limit the
particle mass to be $m_{a}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}3.5\,\mathrm{eV}$. Because the
lensing measurement mainly carries the information at $z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1$ in the LSST, the cross
correlation between the SPHEREx line intensity and the LSST weak lensing
allows us to extract the ALP-decay signal alone in an efficient way when one
measures the correlation as a function of $\lambda_{\rm obs}$.

##  III Statistics

We here define the statistic of interest, i.e. the cross correlation between
the line intensity and weak lensing maps.

###  III.1 Cross Correlation Functions

For a given set of two different random fields $X(\mbox{\boldmath$\theta$})$
and $Y(\mbox{\boldmath$\theta$})$, the cross correlation function is defined
as

$\displaystyle\xi_{XY}(\theta)=\langle
X(\mbox{\boldmath$\phi$})Y(\mbox{\boldmath$\phi$}+\mbox{\boldmath$\theta$})\rangle,$
(30)

where $\langle\cdots\rangle$ represents an ensemble average. The Fourier
transform of Eq. (30) is known as the cross power spectrum and it is given by

$\displaystyle\langle\tilde{X}(\mbox{\boldmath$\ell$})\tilde{Y}(\mbox{\boldmath$\ell$}^{\prime})\rangle=(2\pi)^{2}\,C_{XY}(\ell)\,\delta_{\mathrm{D}}^{(2)}(\mbox{\boldmath$\ell$}-\mbox{\boldmath$\ell$}^{\prime}),$
(31)

where $C_{XY}$ is the cross power spectrum, $\tilde{X}$ represents the Fourier
transform of the field $X$, and so on. In this paper, we use the cross power
spectrum to study the cross correlation between the line intensity and weak
lensing maps.

As shown in Section II, the observed maps are decomposed into

$\displaystyle I_{\mathrm{obs}}(\mbox{\boldmath$\theta$}\,|\,\nu)$
$\displaystyle=$ $\displaystyle
I_{a}(\mbox{\boldmath$\theta$}\,|\,\nu)+I_{\mathrm{astro}}(\mbox{\boldmath$\theta$}\,|\,\nu)+I_{\mathrm{N}}(\mbox{\boldmath$\theta$}\,|\,\nu),$
(32) $\displaystyle\kappa_{\mathrm{obs}}(\mbox{\boldmath$\theta$})$
$\displaystyle=$
$\displaystyle\kappa(\mbox{\boldmath$\theta$})+\kappa_{\mathrm{IA}}(\mbox{\boldmath$\theta$})+\kappa_{\mathrm{N}}(\mbox{\boldmath$\theta$}),$
(33)

where we denote $I_{\rm obs}$ and $\kappa_{\rm obs}$ as the observed line
intensity and weak lensing maps, respectively. Apart from the components as
described in Section II, we include observational noise terms in Eqs. (32) and
(33). The noise in the line intensity map $I_{\rm N}$ will be dominated by
photo noise from zodiacal light in the SPHEREx [38], which is diffuse and
nearly uniform across the field of view. On the other hand, the noise in the
weak lensing map $\kappa_{\mathrm{N}}$ mainly arises from the intrinsic
scatter in galaxy shapes.

Hence, the cross power spectrum between $I_{\rm obs}$ and $\kappa_{\rm obs}$
has four different contributions as

$\displaystyle C^{(\mathrm{obs})}_{\mathrm{LIM}-\kappa}(\ell|\nu)$
$\displaystyle=$ $\displaystyle
C_{\mathrm{ALP}-\kappa}(\ell|\nu)+C_{\mathrm{ALP}-\mathrm{IA}}(\ell|\nu)$ (34)
$\displaystyle\,\,+C_{\mathrm{astro}-\kappa}(\ell|\nu)+C_{\mathrm{astro}-\mathrm{IA}}(\ell|\nu),$

where $C_{\mathrm{ALP}-\kappa}$ is the cross correlation between the ALP-decay
line and cosmic shear, $C_{\mathrm{ALP}-\mathrm{IA}}$ is the cross correlation
between the ALP-decay line and IA terms, $C_{\mathrm{astro}-\kappa}$ is the
cross correlation between the astrophysical lines and cosmic shear, and
$C_{\mathrm{astro}-\mathrm{IA}}$ is the cross correlation between the
astrophysical lines and IA terms. Note that the noise terms $I_{\mathrm{N}}$
and $\kappa_{\mathrm{N}}$ are assumed to be uncorrelated with other fields. In
the following, we evaluate each term in the right hand side of Eq. (34).

###  III.2 Model

####  III.2.1 ALP decay line and cosmic shear

The term $C_{\mathrm{ALP}-\kappa}$ represents the cross correlation between
the ALP decay line and cosmic shear. Note that the ALP decay line exactly
traces the dark matter distribution in the universe (see Eq. [5]), while
cosmic shear, or lensing convergence $\kappa$ provides an unbiased estimate of
projected total mass density with some weight function along a line of sight
(see Eq. [24]). Using the Limber approximation [39], we can express the cross
power spectrum as

$\displaystyle C_{\mathrm{ALP}-\kappa}(\ell|\nu)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}\chi\frac{W_{\mathrm{DM}}(z|\nu)\,W_{\kappa}(z)}{\chi^{2}}P_{\mathrm{DM-m}}\left(\frac{\ell}{\chi},z(\chi)\right)$
(35) $\displaystyle=$
$\displaystyle\frac{\bar{I}_{\mathrm{DM}}(\chi_{a})\,W_{\kappa}(\chi_{a})}{\chi^{2}_{a}}\,P_{\mathrm{DM-m}}\left(\frac{\ell}{\chi_{a}},z_{a}\right),$

where $P_{\mathrm{DM}-m}(k,z)$ is the three-dimensional cross power spectrum
between the dark matter and total mass density fields. Throughout this paper,
we assume that the dark-matter overdensity field is expressed as
$(\Omega_{\mathrm{DM}}/\Omega_{\mathrm{m0}})\,\delta_{\rm m}$. Then, the cross
power spectrum is given by

$\displaystyle
P_{\mathrm{DM-m}}(k,z)=\frac{\Omega_{\mathrm{DM}}}{\Omega_{\mathrm{m0}}}\,P_{\mathrm{m}}(k,z),$
(36)

where $P_{\mathrm{m}}(k,z)$ is the three-dimensional non-linear matter power
spectrum. In the following, we adopt a simulation-calibrated fitting formula
[40] to compute $P_{\mathrm{m}}(k,z)$. It would be worth noting that we ignore
possible baryonic effects on $P_{\mathrm{DM}-m}(k,z)$, while it may induce a
$\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}30\%$ level systematic
error in our forecasts on the constraint of $\Gamma$ (see Ref. [41] for a
review of baryonic effects on large-scale structures). Since the decay rate
$\Gamma$ is proportional to $g_{a\gamma\gamma}^{2}$, we expect that the
constraint of $g_{a\gamma\gamma}$ can be affected by the baryonic effects with
a level of $\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}15\%$ at most.
One needs to account for this systematic error in real measurements, but it
does not play a central role in our forecasts.

####  III.2.2 ALP decay line and intrinsic alignments

Similarly, the cross correlation between the ALP decay line and IA terms is
given by

$\displaystyle C_{\mathrm{ALP}-\mathrm{IA}}(\ell|\nu)$ $\displaystyle=$
$\displaystyle\frac{\bar{I}_{\mathrm{DM}}(\chi_{a})\,W_{\mathrm{IA}}(\chi_{a})}{\chi^{2}_{a}}$
(37) $\displaystyle\qquad\times
P_{\mathrm{DM-m}}\left(\frac{\ell}{\chi_{a}},z_{a}\right),$

where this term is strongly degenerate with
$C_{\mathrm{ALP}-\kappa}(\ell|\nu)$ at a given $\nu$, because ether is
proportional to $P_{\mathrm{DM-m}}$ at the same redshift of $z_{a}$.
Nevertheless, one can constrain an unknown IA parameter ${\cal
A}_{\mathrm{IA}}$ in the function of $W_{\mathrm{IA}}$ and break the
degeneracy when using the frequency-dependence on
$C_{\mathrm{ALP}-\mathrm{IA}}(\ell|\nu)$. The frequency dependence on
$C_{\mathrm{ALP}-\mathrm{IA}}(\ell|\nu)$ can not be same as that of
$C_{\mathrm{ALP}-\kappa}(\ell|\nu)$, because the kernel functions $W_{\kappa}$
and $W_{\mathrm{IA}}$ exhibit the different redshift-dependence.

####  III.2.3 Astrophysical lines and cosmic shear

Under the Limber approximation with Eqs. (19) and (24), we can write the term
$C_{\mathrm{astro}-\kappa}$ as

$\displaystyle C_{\mathrm{astro}-\kappa}(\ell|\nu)$ $\displaystyle=$
$\displaystyle\sum_{Q}\,\frac{{\cal L}_{Q}}{{\cal
L}_{\mathrm{H}_{\alpha}}}\frac{\nu_{\mathrm{H}_{\alpha}}}{\nu_{Q}}\,\frac{\bar{I}_{\mathrm{H}_{\alpha}}(z_{Q})\,W_{\kappa}(z_{Q})}{\chi^{2}_{Q}}$
(38) $\displaystyle\qquad\times
P_{\mathrm{astro-m}}\left(\frac{\ell}{\chi_{Q}},z_{Q}\right),$

where $P_{\mathrm{astro-m}}(k,z)$ is the three-dimensional power spectrum
between two fields of $\delta_{\mathrm{astro}}$ and $\delta_{\mathrm{m}}$ (see
Eq. [21] for our definition of $\delta_{\mathrm{astro}}$).

In this paper, we compute $P_{\mathrm{astro-m}}(k,z)$ based on a halo-model
approach [42]. Within the halo model, the power spectrum can be decomposed
into two terms. One is the one-halo term arising from two-point correlations
in single dark matter halos, and another is the two-halo term given by two-
point correlations between neighboring halos. Each term can be expressed as

$\displaystyle
P_{\mathrm{astro-m}}(k,z)=P^{\mathrm{1h}}_{\mathrm{astro-m}}(k,z)+P^{\mathrm{2h}}_{\mathrm{astro-m}}(k,z),$
(39)

where $P^{\mathrm{1h}}_{\mathrm{astro-m}}$ and
$P^{\mathrm{2h}}_{\mathrm{astro-m}}$ are the one-halo and the two-halo terms,
respectively, and

$\displaystyle P^{\mathrm{1h}}_{\mathrm{astro-m}}(k,z)$ $\displaystyle=$
$\displaystyle{\scriptstyle{\cal
N}^{-1}_{\mathrm{astro}}(z)}\int\mathrm{d}M\frac{\mathrm{d}n}{\mathrm{d}M}\frac{\rho_{h}(k|M,z)}{\bar{\rho}_{\mathrm{m}}}$
(40) $\displaystyle\qquad\qquad\qquad\qquad\times\dot{M}_{*}(M,z),$
$\displaystyle P^{\mathrm{2h}}_{\mathrm{astro-m}}(k,z)$ $\displaystyle=$
$\displaystyle\left(\int\mathrm{d}M\frac{\mathrm{d}n}{\mathrm{d}M}b_{h}(M,z)\frac{\rho_{h}(k|M,z)}{\bar{\rho}_{\mathrm{m}}}\right)$
(41) $\displaystyle\times$ $\displaystyle\left({\scriptstyle{\cal
N}^{-1}_{\mathrm{astro}}(z)}\int\mathrm{d}M\frac{\mathrm{d}n}{\mathrm{d}M}b_{h}(M,z)\dot{M}_{*}(M,z)\right)$
$\displaystyle\times$ $\displaystyle P_{L}(k,z),$

where $\rho_{h}(k|M,z)$ is the Fourier transform of a spherical halo density
profile, $\bar{\rho}_{\mathrm{m}}=\rho_{\mathrm{crit,0}}\Omega_{\mathrm{m0}}$,
$b_{h}$ is the linear halo bias and $P_{L}$ is the linear matter power
spectrum. In the halo-model computations, we define the halo mass as the mass
of a spherical overdensity with 200-times the mean density of the universe.
For the halo density profile, we adopt the the analytical Navarro-Frenk-White
(NFW) profile [43], where we use the concentration-mass-redshift relation in
Ref. [44]. We also use the fitting formulas of the halo mass function [45] and
the linear halo bias [46]. The key ingredient in the computation of Eq. (39)
is the SFR as a function of halo masses and redshifts, $\dot{M}_{*}(M,z)$.
Although this quantity is still uncertain, we adopt a simulation-based fitting
formula as in Ref. [19] for our baseline model. Their model of
$\dot{M}_{*}(M,z)$ has a form of

$\displaystyle\dot{M}_{*}(M,z)=10^{a(z)}\left(\frac{M}{M_{1}}\right)^{b(z)}\left(1+\frac{M}{M_{2}(z)}\right)^{c(z)},$
(42)

where $M_{1}=10^{8}\,M_{\odot}$ and there exist four parameters of
$a(z),b(z),c(z)$ and $M_{2}(z)$ in the model. Ref. [19] have provided the
parameters at $z=0,0.8,1,2.2$ and $4.8$ by analyzing semi-analytic galaxy
catalogs in Ref. [47]. We infer $a(z),b(z),c(z)$ and $M_{2}(z)$ at a given $z$
by employing a linear interpolation of the fitting results at $z=0,0.8,1,2.2$
and $4.8$. Note that $b\sim 2.6$, $c\sim-3$ and $M_{2}$ ranges from
$10^{11-12}\,M_{\odot}$ over redshifts. Hence, a typical halo mass in the SFR
activity (i.e. the mass of the halo which most efficiently produces stars) is
set to be $M\sim 10^{12-13}\,M_{\odot}$ in this model. Although our baseline
model would provide an order-of-magnitude estimate for $P_{\mathrm{astro-m}}$,
it is based on the semi-analytic model of galaxy formation and poorly
constrained by observations. To account for theoretical uncertainties in our
model, we introduce two dimensionless parameters ${\cal A}_{\mathrm{astro}}$
and $\eta_{\mathrm{astro}}$, which control the overall amplitude and redshift
evolution in $P_{\mathrm{astro-m}}$. To be specific, we set

$\displaystyle P_{\mathrm{astro-m}}(k,z)\rightarrow{\cal
A}_{\mathrm{astro}}(1+z)^{\eta_{\mathrm{astro}}}\,P_{\mathrm{astro-m}}(k,z),$
(43)

when computing $C_{\mathrm{astro}-\kappa}(\ell|\nu)$. For our fiducial model,
we adopt ${\cal A}_{\mathrm{astro}}=1$ and $\eta_{\mathrm{astro}}=0$.

####  III.2.4 Astrophysical lines and intrinsic alignments

The cross correlation between the astrophysical lines and the IA field is then
expressed as

$\displaystyle C_{\mathrm{astro}-\mathrm{IA}}(\ell|\nu)$ $\displaystyle=$
$\displaystyle\sum_{Q}\,\frac{{\cal L}_{Q}}{{\cal
L}_{\mathrm{H}_{\alpha}}}\frac{\nu_{\mathrm{H}_{\alpha}}}{\nu_{Q}}\,\frac{\bar{I}_{\mathrm{H}_{\alpha}}(z_{Q})\,W_{\mathrm{IA}}(z_{Q})}{\chi^{2}_{Q}}$
(44) $\displaystyle\quad\times{\cal A}_{\rm
astro}\,(1+z_{Q})^{\eta_{\mathrm{astro}}}$ $\displaystyle\quad\times
P_{\mathrm{astro-m}}\left(\frac{\ell}{\chi_{Q}},z_{Q}\right).$

###  III.3 Survey specifications and statistical errors

Figure 3:  Expected signals of the cross-correlation function with the line
intensity and weak lensing maps as well as the auto correlation of the line
intensity map. In this figure, we assume an LSST-like imaging survey for
lensing and an SPHEREx-like spectroscopic survey at wavelength
$\lambda_{\mathrm{obs}}=3000\,\mathrm{nm}$. Details of the survey
specification are found in Section III.3. The left panels show the cross power
spectra and its cumulative signal-to-noise ratio. In the left top, blue points
with error bars show the ALP-decay signal for an ALP mass of
$m_{a}=1\,\mathrm{eV}$ and a particle-to-photons coupling of
$g_{a\gamma\gamma}=3\times 10^{-11}\,\mathrm{GeV}^{-1}$. For comparison, the
orange line shows our fiducial model of astrophysical contributions to the
cross power spectrum. The left bottom panel represents the cumulative signal-
to-noise ratio of the ALP-decay signal as a function of the maximum multipole
$\ell_{\mathrm{max}}$. The black dashed line in the bottom indicates a
$3\sigma$-level detection. The right panels are similar to the left, but those
are for the auto power spectrum of the intensity map. This figure highlights
that the cross power spectrum with the line intensity and weak lensing maps
can be dominated by the ALP-decay signal, while the auto power spectrum of the
intensity maps will be largely determined by the astrophysical sources.

We here describe observational parameters of line intensity and weak lensing
surveys in our analysis and summarize how to set expected statistical
uncertainties of Eq. (34).

For line intensity maps, we assume a hypothetical SPHEREx-like survey covering
a full sky. Among several observational modes in SPHEREx, we consider the all-
sky survey mode with an angular resolution of 0.1 arcmin and a $5\sigma$-level
point-source sensitivity of 18.5 AB magnitude per pixel [38]. Note that there
exists the “deep” survey mode in SPHEREx and it will has a much greater point-
source sensitivity. Hence, the deep survey mode would allow us to perform
robust line-intensity mapping measurements, while the sky coverage is planed
to be $200\,\mathrm{deg}^{2}$ only. Because the statistical uncertainty in our
cross correlation analysis scales with the inverse of the survey area, the
deep survey mode is not suitable to search for large-scale cross correlation
signals. We also assume the frequency resolution of $R=41.5$ at wavelengths of
$750-4100$ nm.

For weak lensing maps, we assume a LSST-like imaging survey covering a sky of
18000 $\mathrm{deg}^{2}$. In weak lensing measurements, we assume the source
number density of $26\,\mathrm{arcmin}^{-2}$, the intrinsic scatter of galaxy
ellipticity per components being 0.26, and the source redshift distribution as
in Eq. (26). These survey parameters have been investigated in Ref. [27].

Given survey configurations, we can evaluate the statistical uncertainty of
Eq. (34) by using covariance matrices. Assuming that the observable fields
follow Gaussian, we can write the covariance matrix as

$\displaystyle\mbox{\boldmath$C$}(\ell,\nu|\ell^{\prime},\nu^{\prime})$
$\displaystyle\equiv$
$\displaystyle\mathrm{Cov}\left[C^{(\mathrm{obs})}_{\mathrm{LIM}-\kappa}(\ell|\nu),C^{(\mathrm{obs})}_{\mathrm{LIM}-\kappa}(\ell^{\prime}|\nu^{\prime})\right]$
(45) $\displaystyle=$
$\displaystyle\frac{\delta^{K}_{\ell\ell^{\prime}}}{{\color[rgb]{0,0,0}2\ell}\,\Delta\ell\,f_{\mathrm{sky}}}\Bigl{[}C^{(\mathrm{obs})}_{\mathrm{LIM-
LIM}}(\ell|\nu,\nu^{\prime})C^{(\mathrm{obs})}_{\kappa-\kappa}(\ell)$
$\displaystyle+C^{(\mathrm{obs})}_{\mathrm{LIM}-\kappa}(\ell|\nu)C^{(\mathrm{obs})}_{\mathrm{LIM}-\kappa}(\ell^{\prime}|\nu^{\prime})\Bigr{]},$

where $f_{\mathrm{sky}}$ is the fraction of sky coverage used in the cross
correlation analysis, $\delta^{K}_{ij}$ is the Kronecker delta symbol,
$C^{(\mathrm{obs})}_{\mathrm{LIM-LIM}}(\ell|\nu,\nu^{\prime})$ represents the
cross power spectrum between two intensity maps with different frequency
bands, and $C^{(\mathrm{obs})}_{\kappa-\kappa}(\ell)$ is the auto power
spectrum of the observed convergence field (Eq. [33]).  The derivation of Eq.
(45) is found in Appendix A. Note that $f_{\mathrm{sky}}$ is set to be
$18,000/41,252.96=0.436$ for our case. In Eq. (45), we employ a binning in the
multipole $\ell$ for the measurements of cross power spectra with the bin
width being $\Delta\ell$. In this setup, the covariance matrix becomes non-
zero only when two different $\ell$ bins have an identical value. We summarize
our model of $C^{(\mathrm{obs})}_{\mathrm{LIM-LIM}}(\ell|\nu,\nu^{\prime})$
and $C^{(\mathrm{obs})}_{\kappa-\kappa}(\ell)$ in Appendix A.

##  IV Results

In this section, we show our main results in this paper. When computing power
spectra, we employ a logarithmic binning in the range of $10<\ell<3000$ with
the number of bins being 20. Also, we compute all power spectra with respect
to $\nu\,I(\mbox{\boldmath$\theta$}|\nu)$ in units of
$\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}/\mathrm{str}$.

###  IV.1 Expected Signal

Let us first consider expected signals in the cross correlation analysis. As a
representative example, we assume ALPs with a mass of $m_{a}=1\,\mathrm{eV}$
and a particle-to-photons coupling of $g_{a\gamma\gamma}=3\times
10^{-11}\,\mathrm{GeV}^{-1}$. The top left panel in Fig. 3 shows our baseline
model for the cross power spectrum between line intensity maps and weak
lensing convergence. In the figure, we suppose the line intensity map at
$\lambda_{\mathrm{obs}}=3000\,\mathrm{nm}$. The blue points with error bars
represent the ALP-decay signal $C_{\mathrm{ALP}-\kappa}$, while the orange
line stands for the astrophysical contribution $C_{\mathrm{astro}-\kappa}$. In
the left bottom, we characterize the detectability of the signal
$C_{\mathrm{ALP}-\kappa}$ by introducing the cumulative signal-to-noise ratio.
The signal-to-noise ratio at a given photon frequency is defined as

$\displaystyle\left(S/N\right)^{2}(\ell_{\mathrm{max}})$ $\displaystyle=$
$\displaystyle\sum_{\ell_{i},\ell_{j}\leq\ell_{\mathrm{max}}}C_{\mathrm{ALP}-\kappa}(\ell_{i}|\nu)\,\mbox{\boldmath$C$}^{-1}_{\rm
null}(\ell_{i},\nu|\ell_{j},\nu)$ (46) $\displaystyle\qquad\qquad\times
C_{\mathrm{ALP}-\kappa}(\ell_{j}|\nu),$

where $\mbox{\boldmath$C$}_{\rm null}$ is the covariance matrix when we set
the decay rate to be $\Gamma=0$. Hence, Eq. (46) can be used to compute the
significance of the ALP-decay signal over a null detection hypothesis. The
blue points in the left bottom panel in Fig. 3 shows the signal-to-noise ratio
as a function of maximum multipoles $\ell_{\mathrm{max}}$ in the analysis,
while the dashed line represents a $3\sigma$-level significance.

On the other hand, the right panels in Fig. 3 summarize the results for the
auto power spectrum in the intensity map. We compute the expected auto
correlation signal as in Appendix A. In the top right panel, the orange line
shows the auto power spectrum arising from clustering of astrophysical lines
(referred to as $C_{\mathrm{astro-astro}}$ in Appendix A). The blue points
with error bars are the counterpart of the ALP decay line, $C_{\mathrm{ALP-
ALP}}$ as defined in Eq. (63). The error bars in the top right panel are
computed as

$\displaystyle\mathrm{Cov}\left[C^{(\mathrm{obs})}_{\mathrm{LIM-
LIM}}(\ell|\nu),C^{(\mathrm{obs})}_{\mathrm{LIM-
LIM}}(\ell^{\prime}|\nu)\right]$
$\displaystyle\qquad=\frac{\delta^{K}_{\ell\ell^{\prime}}}{\ell\Delta\ell\,f_{\mathrm{sky}}}C^{(\mathrm{obs})}_{\mathrm{LIM-
LIM}}(\ell|\nu)C^{(\mathrm{obs})}_{\kappa-\kappa}(\ell),$ (47)

where details of $C^{(\mathrm{obs})}_{\mathrm{LIM-LIM}}$ and
$C^{(\mathrm{obs})}_{\kappa-\kappa}$ are given in Appendix A. The signal-to-
noise ratio of $C_{\mathrm{ALP-ALP}}$ can be computed in a similar way to Eq.
(46) and it is shown in the bottom right panel in Fig. 3.

The comparisons between left and right panels in Fig. 3 clarify that the ALP
decay can induce a significant correlation signal compared to the counterpart
of astrophysical lines in the cross correlation analysis, but not in the auto
correlation of line intensity maps. This can be understood as the cross
correlation between line intensity and weak lensing maps can probe lower-
redshift large-scale structures and the ALP decay line mainly comes from those
lower-redshift structures. As shown in Fig. 2, the auto correlation function
will be dominated by astrophysical lines and the astrophysical lines
effectively probe higher redshifts at $z\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}2$ as long as the observed
wavelength is set to be $\lambda_{\mathrm{obs}}\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}2000\,\mathrm{nm}$.

Figure 4:  Separated contributions to the observed cross power spectrum
between line intensity and weak lensing maps. We here suppose the LSST-like
source galaxy distribution for lensing and the line intensity map at
$\lambda_{\mathrm{obs}}=3000\,\mathrm{nm}$. For the ALP decay, we consider the
ALPs with a mass of $m_{a}=1\,\mathrm{eV}$ and a particle-to-photons coupling
of $g_{a\gamma\gamma}=3\times 10^{-11}\,\mathrm{GeV}^{-1}$. The blue thick
line shows the cross correlation between the ALP decay line and weak lensing
convergence, while the orange thin line stands for the counterpart of
astrophysical lines. The green dotted and pink dashed lines represent the IA-
induced signals arising from the ALP decay and astrophysical lines,
respectively. Note that the IA-induced signal becomes negative over the
angular scales of interest.

Apart from gravitational lensing effects, we will have the cross correlation
arising from the IA effects of galaxy shapes. Fig. 4 summarizes our baseline
model of the cross power spectrum between line intensity and weak lensing
maps. Different colored lines in the figure show four separated terms in the
observed power spectrum as in Eq. (34). Note that the IA-induced terms
($C_{\mathrm{astro-IA}}$ and $C_{\mathrm{ALP-IA}}$) will show a negative value
because major axes in galaxy shapes are expected to align in the radial
direction toward high density regions [31, 32]. Our baseline model shows that
the cross correlation between astrophysical lines and intrinsic galaxy shapes
can dominate the ALP-decay-induced lensing signal at $\ell\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}1000$, while the term
$C_{\mathrm{ALP}-\kappa}$ still has a significant contribution to the observed
signal at small angular scales of $\ell\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}1000$. We explore the detectability
of $C_{\mathrm{ALP}-\kappa}$ in the presence of the IA effects in Section
IV.3.

###  IV.2 Signal-to-noise ratios

The lensing kernel function is relatively broad in redshifts and the effective
redshift for the ALP decay can vary as we change frequency bands in the
intensity map (see Fig. 2). Hence, combining the cross power spectra among
multi-frequency bands can further improve the detectability of the ALP decay
line. The total signal-to-noise ratio of the ALP-decay signal in the range of
observed frequency $\nu_{\mathrm{min}}\leq\nu\leq\nu_{\mathrm{max}}$ can be
defined as

$\displaystyle(S/N)^{2}_{\mathrm{tot}}(\ell_{\mathrm{max}}|\nu_{\mathrm{min}},\nu_{\mathrm{max}})=\sum_{\nu_{\alpha},\nu_{\beta}}\sum_{\ell_{i},\ell_{j}\leq\ell_{\mathrm{max}}}C_{\mathrm{ALP}-\kappa}(\ell_{i}|\nu_{\alpha})$
$\displaystyle\qquad\qquad\times\mbox{\boldmath$C$}^{-1}_{\rm
null}(\ell_{i},\nu_{\alpha}|\ell_{j},\nu_{\beta})\,C_{\mathrm{ALP}-\kappa}(\ell_{j}|\nu_{\beta}),$
(48)

where we impose
$\nu_{\mathrm{min}}\leq\nu_{\alpha},\nu_{\beta}\leq\nu_{\mathrm{max}}$ to take
the sum. In our case, we assume the SPHEREx-like situation with the wavelength
coverage of $750-4100$ nm, corresponding to
$\nu_{\mathrm{min}}=73.1\,\mathrm{THz}$ and
$\nu_{\mathrm{max}}=399.7\,\mathrm{THz}$. When setting the frequency
resolution being $R=41.5$, we find that 70 intensity maps are available in
total.

For a given $m_{a}$, we can express the total signal-to-noise ratio as

$\displaystyle(S/N)_{\mathrm{tot}}(\ell_{\mathrm{max}})$ $\displaystyle=$
$\displaystyle\left[\frac{\Gamma}{\Gamma_{\mathrm{upp}}(\ell_{\mathrm{max}})}\right]$
(49) $\displaystyle=$
$\displaystyle\left[\frac{g_{a\gamma\gamma}}{g_{a\gamma\gamma,\mathrm{upp}}(\ell_{\mathrm{max}})}\right]^{2}.$
(50)

Fig. 5 shows $\Gamma_{\mathrm{upp}}$ and $g_{a\gamma\gamma,\mathrm{upp}}$ for
ALPs with a particle mass of $m_{a}=1\,\mathrm{eV}$. We find that combining
the cross power spectra at multi-frequency bands allows us to probe the
particle-to-photon coupling down to $g_{a\gamma\gamma}\sim 1\times
10^{-11}\,\mathrm{GeV}^{-1}$ if we can have a precise model of the cross power
spectra at $\ell\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}800$.

Figure 5:  Total signal-to-noise ratio of the cross power spectrum between the
line intensity and weak lensing maps for ALPs with a particle mass of
$m_{a}=1\,\mathrm{eV}$. For a fixed mass $m_{a}$, the total signal-to-noise
ratio is given by
$(S/N)_{\mathrm{tot}}=(g_{a\gamma\gamma}/g_{a\gamma\gamma,\mathrm{upp}})^{2}=\Gamma/\Gamma_{\rm
upp}$, where $g_{a\gamma\gamma}$ is the particle-to-photons coupling constant
for ALPs and $\Gamma$ is the ALP decay rate. The solid line and dashed lines
show $g_{a\gamma\gamma,\mathrm{upp}}$ and $\Gamma_{\rm upp}$ as a function of
the maximum multipole $\ell_{\rm max}$ in the cross-correlation analysis,
respectively. In this figure, we combine the cross power spectra defined with
multiple-wavelength intensity maps in the range of $\lambda_{\rm
obs}=750-4100\,\mathrm{nm}$. Also, we assume that a sky of 18,000 ${\rm
deg}^{2}$ is available for the analysis.

Fig. 6 summarizes expected $2\sigma$ upper limits of $g_{a\gamma\gamma}$ for
different particle masses $m_{a}$ if the cross correlation analysis in LSST
and SPHEREx is consistent with a null detection. Note that we set
$\ell_{\mathrm{max}}=2000$ in Fig. 6. In the figure, we also show several
observational constraints in the literature for comparison. Those include the
study of stellar evolution of horizontal branch (HB) to red giants in globular
clusters [48], “axion helioscope” observations by the CERN Axion Solar
Telescope (CAST) [49], spectroscopic observations of the dwarf spheroidal
galaxy Leo T by the Multi Unit Spectroscopic Explorer (MUSE) [50]. The blue
filled region in Fig. 6 represents the expected constraint by the LSST-SPHEREx
cross correlation analysis, enabling us to improve the existing constraint at
$m_{a}=1-2.7\,\mathrm{eV}$ by a factor of $6-10$.  For the interested readers,
we also provide a forecast in terms of the dark matter lifetime $\Gamma$ in
Appendix B.

Figure 6:  $2\sigma$-level upper limits of the ALP-to-photons coupling
$g_{a\gamma\gamma}$ as a function of ALP mass $m_{a}$. The blue region can be
excluded by the cross correlation analysis with SPHEREx line intensity maps
and LSST lensing. For comparison, we show existing constraints in the
literature; the gray, orange, and green regions have been excluded by the
study of stellar evolution of horizontal branch (HB) to red giants in globular
clusters [48], the CERN Axion Solar Telescope (CAST) [49], spectroscopic
observations of the dwarf spheroidal galaxy Leo T by the Multi Unit
Spectroscopic Explorer (MUSE) [50], respectively.

###  IV.3 Forecast of ALP parameter constraints

Figure 7:  Forecast of parameter constraints on ALP. In this figure, we assume
the cross correlation analysis in hypothetical LSST-like and SPHEREx-like
surveys covering 18000 ${\rm deg}^{2}$. In each panel, the solid error circle
represents a 68% confidence level when one uses the cross power spectra at
$\ell\leq\ell_{\rm max}=1000$. The orange dashed and green dotted lines show
the confidence level for $\ell_{\rm max}=2000$ and 500, respectively.

In actual observations, we have several expected components in the cross
correlation between line intensity and weak lensing maps as in Eq. (34). As
shown in Fig. 4, the cross correlation caused by the astrophysical lines and
the IA effect of galaxy shapes $C_{\mathrm{astro-IA}}$ is expected to prevent
us from inferring the ALP parameters from the observed power spectra. To study
impacts of $C_{\mathrm{astro-IA}}$ on constraints of the ALP parameters, we
perform a Fisher analysis for cross correlation analyses in LSST and SPHEREx.
The Fisher matrix is commonly used to assess expected constraints of
parameters of interest with given observables. In our case, it is computed as

$\displaystyle F_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\sum_{i,j,p,q}\frac{\partial C^{(\rm
obs)}_{\mathrm{LIM}-\kappa}(\ell_{i}|\nu_{p})}{\partial
s_{\alpha}}\,\mbox{\boldmath$C$}^{-1}(\ell_{i},\nu_{p}|\ell_{j},\nu_{q})\,$
(51) $\displaystyle\qquad\qquad\qquad\qquad\times\frac{\partial C^{(\rm
obs)}_{\mathrm{LIM}-\kappa}(\ell_{j}|\nu_{q})}{\partial s_{\beta}},$

where $C^{(\rm obs)}_{\mathrm{LIM}-\kappa}(\ell_{i}|\nu_{p})$ represents the
observed power spectrum at the $i$-th $\ell$ bin for the intensity map with
the observed frequency being $\nu_{p}$, $C$ is the covariance matrix of
$C^{(\rm obs)}_{\mathrm{LIM}-\kappa}$ as in Eq. (45), and $s_{\alpha}$ is the
$\alpha$-th parameter. The inverse of the Fisher matrix provides an estimate
of the error covariance for two parameters as

$\displaystyle F^{-1}_{\alpha\beta}=\langle\Delta s_{\alpha}\Delta
s_{\beta}\rangle,$ (52)

where $\Delta s_{\alpha}$ is the statistical uncertainty of parameter
$s_{\alpha}$.

| Optimistic ($\ell_{\mathrm{max}}=1000$) | Optimistic ($\ell_{\mathrm{max}}=2000$) | Pessimistic ($\ell_{\mathrm{max}}=1000$) | Pessimistic ($\ell_{\mathrm{max}}=2000$)
---|---|---|---|---
$\Delta m_{a}\,(\mathrm{eV})$ | 0.150 (0.146) | 0.112 (0.107) | 0.183 (0.146) | 0.146 (0.107)
$\Delta g_{a\gamma\gamma}\,(10^{-11}\,\mathrm{GeV}^{-1})$ | 0.423 (0.416) | 0.302 (0.292) | 0.672 (0.416) | 0.552 (0.292)
$\Delta{\cal A}_{\mathrm{astro}}\times 10^{3}$ | 6.15 (5.93) | 4.92 (4.74) | 10.4 (5.93) | 8.27 (4.74)
$\Delta\eta_{\mathrm{astro}}$ | - | - | 0.0301 (0.0171) | 0.0191 (0.00904)
$\Delta{\cal A}_{\mathrm{IA}}\times 10^{2}$ | 1.32 (1.26) | 1.04 (0.993) | 8.32 (1.26) | 6.99 (0.993)
$\Delta\eta_{\mathrm{IA}}$ | - | - | 0.110 (0.0169) | 0.0924 (0.0131)

Table 2:  Summary of the Fisher forecast of the ALP and astrophysics by the
cross power spectra between line intensity and weak lensing maps. We assume
the effective survey area to be $18,000\,\mathrm{deg}^{2}$. We consider the
LSST-like source galaxy distribution (Eq. [26]) and the intensity maps at
$\lambda_{\mathrm{obs}}=750-4100\,\mathrm{nm}$ with the frequency resolution
being 41.5. We refer the “optimistic” scenario when the redshift evolution in
the astrophysical line intensity and the IA effects are precisely known in
advance (i.e. $\eta_{\mathrm{astro}}$ and $\eta_{\mathrm{IA}}$ are fixed). On
the other hand, the pessimistic scenario assumes that $\eta_{\mathrm{astro}}$
and $\eta_{\mathrm{IA}}$ are unknown. In each table cell, the number without
brackets show the $1\sigma$ constraint of single parameter when we marginalize
other parameters, while the one in brackets is the un-marginalized
counterpart. Note that the fiducial parameters are set to
$(m_{a},g_{a\gamma\gamma},{\cal
A}_{\mathrm{astro}},\eta_{\mathrm{astro}},{\cal
A}_{\mathrm{IA}},\eta_{\mathrm{IA}})=(1\,\mathrm{eV},1\times
10^{-11}\,\mathrm{GeV}^{-1},1,0,1,3)$.

In our Fisher analysis, we consider the following parameters to vary:
$\mbox{\boldmath$s$}=\\{m_{a},g_{a\gamma\gamma},{\cal
A}_{\mathrm{IA}},\eta_{\mathrm{IA}},{\cal
A}_{\mathrm{astro}},\eta_{\mathrm{astro}}\\}$. The fiducial values of our
parameters is set to
$\mbox{\boldmath$s$}_{\mathrm{fid}}=\\{1\,\mathrm{eV},1\times
10^{-11}\,\mathrm{GeV}^{-1},1,3,1,0\\}$. When computing the Fisher matrix, we
limit the range of multipoles as $10<\ell\leq\ell_{\mathrm{max}}$ and examine
three cases of $\ell_{\mathrm{max}}=500,1000$, and 2000. Changing
$\ell_{\mathrm{max}}$ in our Fisher matrix tells us how smaller-scale
information will help to improve the ALP constraints999 It would be worth
noting that our theoretical model of cross power spectra is still
phenomenological and model uncertainties at small scales are not fully
understood yet. Hence, the analysis with larger $\ell_{\mathrm{max}}$ will
provide more stringent constraints, while it may induce some biased
estimations of the ALP parameters in actual observations.. For reference, our
cosmological model predicts

$\displaystyle k_{\mathrm{max}}$ $\displaystyle=$
$\displaystyle\frac{\ell_{\mathrm{max}}}{\chi(z)}$ (53) $\displaystyle\simeq$
$\displaystyle\left(\frac{\ell_{\mathrm{max}}}{1000}\right)\,\left(\frac{z}{0.3}\right)^{-1}\left(1+\frac{z}{2}\right)\,h\mathrm{Mpc}^{-1},$

where $k^{-1}_{\mathrm{max}}$ gives an estimate of the smallest physical scale
at a fixed $\ell_{\mathrm{max}}$ and redshift $z$. Eq. (53) has a 10%-level
accuracy in the range of $0.05\leq z\leq 2$. Hence, the power spectrum with
$\ell_{\mathrm{max}}=1000$ can extract the information of structures at
$z=0.3-0.4$ with their size of $\lower
2.15277pt\hbox{$\;\buildrel>\over{\sim}\;$}\mathrm{Mpc}$.

In our analysis, we use multi-frequency information of line intensity maps,
allowing us to study the redshift evolution in our cross power spectra.
Because the effective redshift in the ALP-decay signal is different from the
astrophysical counterparts, the multi-frequency information plays an essential
role in breaking degeneracies among our cross power spectra. Nevertheless, we
expect that the efficiency of multi-frequency cross correlation depends on our
prior knowledge of the redshift evolution in astrophysics-related parts. In
our case, the redshift evolution in the astrophysical line intensity and the
IA effects of galaxy shapes controls how the multi-frequency analysis works.
To see impacts of prior knowledge in the redshift evolution, we consider two
scenarios. One is the optimistic scenario when we assume that the parameters
of $\eta_{\mathrm{astro}}$ and $\eta_{\mathrm{IA}}$ are perfectly known, and
another is the pessimistic scenario making these two parameters unknown. In
realistic situations, we will have some informative prior of these two
parameters from other analyses. For instance, tomographic cosmic shear
analyses can place reasonable constraints of $\eta_{\mathrm{IA}}$, while
analyses of line intensity maps with the SPHEREx deep survey mode provide
cleaner information of astrophysical sources and allow us to put constraints
of $\eta_{\mathrm{astro}}$. Hence, our Fisher analysis sets two extreme cases
in the cross correlation analysis and gives some sense of how tight
constraints of the ALP parameters will be obtained in more realistic
situations.

Fig. 7 summarizes an expected constraint in two-dimensional planes for the
optimistic scenario with different $\ell_{\mathrm{max}}$. We find that the
multi-frequency information can successfully break degeneracies among the ALP-
decay and astrophysical signals if we precisely know the redshift evolution in
the astrophysical signals. Note that cross correlation analyses at single
frequencies can not separate the terms of $C_{\mathrm{ALP}-\kappa}$ and
$C_{\mathrm{ALP-IA}}$ in practice. The multi-frequency information can give
tight constraints of ${\cal A}_{\mathrm{IA}}$ as well as ${\cal
A}_{\mathrm{astro}}$, improving the ALP constraints. Note that the
astrophysical-line terms will probe higher-redshift structures than the ALP-
decay terms do. This can be seen from the difference in $\ell$-dependence of
each term. Using larger $\ell_{\mathrm{max}}$ will help to constrain the
$\ell$-dependence of the observed power spectra, making the parameter
constraints tighter.

For the pessimistic scenario, the parameter degeneracy between ${\cal
A}_{\mathrm{IA}}$ and $\eta_{\mathrm{IA}}$ is significant as shown in Appendix
C (see Fig. 9). This strong degeneracy can degrade the constraints of the ALP
parameters by a factor of $\sim 1.5$. We expect that the degeneracy between
${\cal A}_{\mathrm{IA}}$ and $\eta_{\mathrm{IA}}$ can be partly broken if we
further apply a tomographic analysis in lensing observables [51], while we
leave it for future studies. Table 2 provides a summary of our Fisher
analysis.

##  V Conclusion and Discussions

In this paper, we have studied cross correlation functions between line
intensity and weak lensing maps to search for decaying axion-like particles
(ALPs). The ALP decay introduces an emission line with its frequency given by
$\nu_{a}=1.21\times 10^{14}\,(m_{a}/1\,\mathrm{eV})\,\mathrm{Hz}$, where
$m_{a}$ is the particle mass of ALPs. If a fraction of cosmic dark matter is
made up of the ALPs, higher mass-density regions in the universe can emit
larger line intensity by the ALP decay, as well as induce a more prominent
distortion of background galaxy shapes by gravitational lensing effects. We
explored correlation signals caused by the ALP decay and weak gravitational
lensing effects in large scale structures. For this purpose, we developed a
theoretical framework to predict the cross correlation functions by taking
into account several astrophysical lines and intrinsic alignments (IAs) of
galaxy shapes. Suppose that the ALP explains the whole abundance of dark
matter, our findings are summarized as follows.

  * •

Our baseline model shows that the cross correlation with an eV-mass ALP decay
and weak lensing effects can dominate the astrophysical-line counterparts when
observed wavelengths are set to $\lambda_{\mathrm{obs}}\sim
2000-3000\,\mathrm{nm}$. This is because weak lensing effects in modern galaxy
surveys can probe the large-scale structures at redshifts of $z=0.3-0.6$, but
astrophysical line sources populate higher-$z$ structures and can not
introduce significant correlations.

  * •

Assuming upcoming line intensity measurements by SPHEREx and weak lensing
measurements by the Large Survey of Space and Time (LSST), we found that a
null detection of the cross correlation can place stringent upper limits of
ALP-to-two-photons coupling $g_{a\gamma\gamma}\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10^{-11}\,\mathrm{GeV}^{-1}$ at
$m_{a}=1-3.5\,\mathrm{eV}$. The expected upper limits improve existing
constraints by the study of stellar evolution of horizontal branch to red
giants [48] by a factor of $\sim 6-10$.

  * •

Main systematic effects in our ALP search arise from the correlation between
astrophysical lines and the IA effect. This makes the observed correlation
signals negative depending on the IA amplitude. We performed a Fisher analysis
to study detectability of the ALP-induced signal in the presence of the
correlations between IA effects and astrophysical lines. Our Fisher analysis
showed that the ALP decay with $m_{a}=1\,\mathrm{eV}$ and
$g_{a\gamma\gamma}=10^{-11}\,\mathrm{GeV}^{-1}$ can be constrained with a 68%
confidence level of $\Delta m_{a}\sim 0.1-0.2\,\mathrm{eV}$ and $\Delta
g_{a\gamma\gamma}\sim 0.3-0.7\times 10^{-11}\,\mathrm{GeV}^{-1}$ by SPHEREx
and LSST. The constraint depends on maximum multipoles in the analysis and
prior information about the redshift evolution in astrophysical effects.

The cross correlation between line intensity and weak lensing maps can test
various hints of ALP signals reported so far. For example, Ref. [48] analyzed
39 galactic globular clusters and reported that the observed number ratio of
stars in horizontal over red giant branch of old stellar clusters differs from
an astronomical prediction with a $2\sigma$ level. This discrepancy can be
interpreted as the existence of the ALP decay with
$g_{a\gamma\gamma}=0.45^{+0.12}_{-0.16}\times 10^{-10}\,\mathrm{GeV}^{-1}$,
where the error represents a 68% confidence level. Recently, Ref. [52] studied
the ALP decay as a potential origin of the anisotropy of the near-infrared
background intensity and found that the ALP decay with
$m_{a}=2.3-3\,\mathrm{eV}$ and $g_{a\gamma\gamma}=1.1-1.6\times
10^{-10}\,\mathrm{GeV}^{-1}$ can explain the excess in the observed power
spectrum of background intensity at sub-arcmin scales. These hints can be
robustly examined by the cross correlation analysis with SPHEREx and LSST.

We expect that our results will provide a guideline of searching for the ALP
decay with large-scale structure data. Weak lensing analyses with several
source redshift bins can improve constraints of the IA amplitude and offer
further improvements of the ALP constraints. Apart from weak lensing
measurements in galaxy imaging surveys, there exist several tracers of large
scale structures in the universe. Those include number density of galaxies and
secondary anisotropies in the cosmic microwave background (CMB) [53].
Spectroscopic observations of galaxies enable us to map the large-scale
structure in three-dimensional space, while such maps are known to be a biased
tracer of underlying cosmic dark matter [54] and are subject to redshift space
distortion effects [55, 56, 57]. Among the secondary effects in CMB,
gravitational lensing effects in CMB, referred to as CMB lensing, can provide
cosmological information about higher-redshift matter density distributions at
$z=2-4$ [58]. The cross correlation with line intensity maps and CMB lensing
allows us to place meaningful constraints of astrophysics but may be less
sensitive to the ALP decay. Cross correlations among line intensity maps and
large-scale structures would be interesting not only for the ALP search but
also constraints of cosmological parameters. We leave cosmological
applications for our future study.

###### Acknowledgements.

We thank Naoki Yoshida and Kana Moriwaki for useful comments. This work is
supported by MEXT KAKENHI Grant Numbers of 19K14767 and 20H05861. Numerical
computations were carried out on Cray XC50 at the Center for Computational
Astrophysics in NAOJ.

##  Appendix A Auto power spectra in line intensity and weak lensing maps

###  A.1 Gaussian covariance of binned power spectra

Let $A$, $B$, $X$, and $Y$ be two-dimensional Gaussian random fields. The
estimator of the cross power spectrum between two fields $A$ and $B$ is given
by

$\displaystyle\hat{C}_{AB}(\ell_{i})=\frac{1}{N_{\mathrm{mode}}(\ell_{i})}\sum_{\mbox{\boldmath$\ell$};\ell\in\ell_{i}}\,\tilde{A}(\mbox{\boldmath$\ell$})\tilde{B}(\mbox{\boldmath$\ell$}),$
(54)

where we employ a linear binning in multipoles $\ell$ with the bin width being
$\Delta\ell$, $\ell_{i}$ represents the $i$-th bin of $\ell$, and
$N_{\mathrm{mode}}(\ell_{i})$ is the number of Fourier modes used for the
power spectrum estimation at the $i$-th bin. The number of Fourier modes can
be computed as

$\displaystyle N_{\mathrm{mode}}(\ell_{i})$ $\displaystyle=$
$\displaystyle\sum_{\mbox{\boldmath$\ell$};\ell\in\ell_{i}}1$ (55)
$\displaystyle\simeq$
$\displaystyle\frac{2\pi\ell_{i}\,\Delta\ell}{(2\pi)^{2}/\Omega_{S}}=2\ell_{i}\,\Delta\ell\,f_{\mathrm{sky}},$

where $\Omega_{S}$ is the survey area and
$f_{\mathrm{sky}}=\Omega_{S}/(4\pi)$. The ensemble average of Eq. (54) is
equal to its expected value $C_{AB}(\ell_{i})$.

We then consider the covariance matrix between two estimators of
$\hat{C}_{AB}(\ell_{i})$ and $\hat{C}_{XY}(\ell_{j})$. The covariance is
defined by

$\displaystyle\mathrm{Cov}\left[\hat{C}_{AB}(\ell_{i}),\hat{C}_{XY}(\ell_{j})\right]$
$\displaystyle=$
$\displaystyle\langle\hat{C}_{AB}(\ell_{i})\hat{C}_{XY}(\ell_{j})\rangle$ (56)
$\displaystyle-C_{AB}(\ell_{i})C_{XY}(\ell_{j}).$

For the Gaussian fields, it holds that

$\displaystyle\langle\tilde{A}(\mbox{\boldmath$\ell$})\tilde{B}(\mbox{\boldmath$\ell$})\tilde{X}(\mbox{\boldmath$\ell$}^{\prime})\tilde{Y}(\mbox{\boldmath$\ell$}^{\prime})\rangle=\langle\tilde{A}(\mbox{\boldmath$\ell$})\tilde{B}(\mbox{\boldmath$\ell$})\rangle\langle\tilde{X}(\mbox{\boldmath$\ell$}^{\prime})\tilde{Y}(\mbox{\boldmath$\ell$}^{\prime})\rangle$
$\displaystyle\qquad\qquad\qquad\qquad+\langle\tilde{A}(\mbox{\boldmath$\ell$})\tilde{X}(\mbox{\boldmath$\ell$}^{\prime})\rangle\langle\tilde{B}(\mbox{\boldmath$\ell$})\tilde{Y}(\mbox{\boldmath$\ell$}^{\prime})\rangle$
$\displaystyle\qquad\qquad\qquad\qquad+\langle\tilde{B}(\mbox{\boldmath$\ell$})\tilde{X}(\mbox{\boldmath$\ell$}^{\prime})\rangle\langle\tilde{A}(\mbox{\boldmath$\ell$})\tilde{Y}(\mbox{\boldmath$\ell$}^{\prime})\rangle.$
(57)

Note that

$\displaystyle\sum_{\mbox{\boldmath$\ell$};\ell\in\ell_{i}}\sum_{\mbox{\boldmath$\ell$}^{\prime};\ell^{\prime}\in\ell_{j}}\langle\tilde{A}(\mbox{\boldmath$\ell$})\tilde{X}(\mbox{\boldmath$\ell$}^{\prime})\rangle\langle\tilde{B}(\mbox{\boldmath$\ell$})\tilde{Y}(\mbox{\boldmath$\ell$}^{\prime})\rangle$
(58) $\displaystyle=$
$\displaystyle\delta^{K}_{ij}\sum_{\mbox{\boldmath$\ell$};\ell\in\ell_{i}}\langle\tilde{A}(\mbox{\boldmath$\ell$})\tilde{X}(\mbox{\boldmath$\ell$})\rangle\langle\tilde{B}(\mbox{\boldmath$\ell$})\tilde{Y}(\mbox{\boldmath$\ell$})\rangle$
$\displaystyle=$
$\displaystyle\delta^{K}_{ij}\sum_{\mbox{\boldmath$\ell$};\ell\in\ell_{i}}C_{AX}(\ell_{i})\,C_{BY}(\ell_{i})$
$\displaystyle=$
$\displaystyle\delta^{K}_{ij}\,N_{\mathrm{mode}}(\ell_{i})\,C_{AX}(\ell_{i})\,C_{BY}(\ell_{i}),$

where $\delta^{K}_{ij}$ is the Kronecker delta symbol. Hence, we arrive at

$\displaystyle\langle\hat{C}_{AB}(\ell_{i})\hat{C}_{XY}(\ell_{j})\rangle-
C_{AB}(\ell_{i})C_{XY}(\ell_{j})$ (59) $\displaystyle=$
$\displaystyle\frac{\delta^{K}_{ij}}{N_{\mathrm{mode}}(\ell_{i})}\Bigg{[}C_{AX}(\ell_{i})C_{BY}(\ell_{i})$
$\displaystyle\qquad\qquad\qquad\qquad+C_{BX}(\ell_{i})C_{AY}(\ell_{i})\Biggr{]}.$

If one sets $A(\mbox{\boldmath$\theta$})=I(\mbox{\boldmath$\theta$}|\nu)$,
$X(\mbox{\boldmath$\theta$})=I(\mbox{\boldmath$\theta$}|\nu^{\prime})$,
$B(\mbox{\boldmath$\theta$})=Y(\mbox{\boldmath$\theta$})=\kappa(\mbox{\boldmath$\theta$})$
where $I(\mbox{\boldmath$\theta$}|\nu)$ is the observed intensity at the
frequency $\nu$ and $\kappa(\mbox{\boldmath$\theta$})$ is the lensing
convergence, Eq. (59) leads to Eq. (45).

###  A.2 Each component in Eq. (45)

As in Eq. (45), the covariance matrix for the cross power spectrum
$C^{(\mathrm{obs})}_{\mathrm{LIM}-\kappa}(\ell|\nu)$ includes two components
of the cross power spectrum between two different intensity maps
$C^{(\mathrm{obs})}_{\mathrm{LIM-LIM}}(\ell|\nu,\nu^{\prime})$ and the auto
power spectrum of the observed lensing field
$C^{(\mathrm{obs})}_{\kappa-\kappa}(\ell)$. We here describe our model of
these two power spectra.

The cross power spectrum between two different intensity maps can be computed
within the halo model as introduced in Section III.2.3. Using the Limber
approximation, we formally write

$\displaystyle C^{(\mathrm{obs})}_{\mathrm{LIM-
LIM}}(\ell|\nu,\nu^{\prime})=C_{\mathrm{astro-astro}}(\ell|\nu,\nu^{\prime})$
$\displaystyle\qquad\qquad+C_{\mathrm{astro-ALP}}(\ell|\nu,\nu^{\prime})$
$\displaystyle\qquad\qquad+C_{\mathrm{ALP-
ALP}}(\ell|\nu,\nu^{\prime})+C_{\mathrm{N}}(\ell|\nu,\nu^{\prime}),$ (60)

and each term is given by

$\displaystyle C_{\mathrm{astro-
astro}}(\ell|\nu,\nu^{\prime})=\sum_{Q,Q^{\prime}}\int\mathrm{d}\chi\,\frac{W_{Q}(z|\nu)\,W_{Q^{\prime}}(z|\nu^{\prime})}{\chi^{2}}$
$\displaystyle\qquad\qquad\times{\cal
A}^{2}_{\mathrm{asto}}\,(1+z)^{2\eta_{\mathrm{astro}}}P_{\mathrm{astro-
astro}}\left(\frac{\ell}{\chi},z\right),$ (61) $\displaystyle
C_{\mathrm{astro-
ALP}}(\ell|\nu,\nu^{\prime})=\sum_{Q}\int\mathrm{d}\chi\,\Biggl{(}\frac{W_{Q}(z|\nu)\,W_{\mathrm{DM}}(z|\nu^{\prime})}{\chi^{2}}$
$\displaystyle\qquad\qquad\qquad+\frac{W_{Q}(z|\nu^{\prime})\,W_{\mathrm{DM}}(z|\nu)}{\chi^{2}}\Biggr{)}{\cal
A}_{\mathrm{asto}}\,(1+z)^{\eta_{\mathrm{astro}}}$
$\displaystyle\qquad\qquad\qquad\times\left(\frac{\Omega_{\mathrm{DM}}}{\Omega_{\mathrm{m0}}}\right)P_{\mathrm{astro-m}}\left(\frac{\ell}{\chi},z\right),$
(62) $\displaystyle C_{\mathrm{ALP-
ALP}}(\ell|\nu,\nu^{\prime})=\int\mathrm{d}\chi\,\frac{W_{\mathrm{DM}}(z|\nu)\,W_{\mathrm{DM}}(z|\nu^{\prime})}{\chi^{2}}$
$\displaystyle\qquad\qquad\qquad\times\left(\frac{\Omega_{\mathrm{DM}}}{\Omega_{\mathrm{m0}}}\right)^{2}\,P_{\mathrm{m}}\left(\frac{\ell}{\chi},z\right),$
(63) $\displaystyle
C_{\mathrm{N}}(\ell|\nu,\nu^{\prime})=\delta^{K}_{\nu\nu^{\prime}}\frac{\sigma^{2}_{n}}{n_{\mathrm{pix}}},$
(64)

where
$Q=\\{\mathrm{H}_{\alpha},[\mathrm{O}_{\mathrm{II}}],[\mathrm{O}_{\mathrm{III}}],\mathrm{H}_{\beta},\mathrm{Ly}\alpha\\}$,
$P_{\mathrm{astro-astro}}$ is the three-dimensional auto power spectrum of the
field $\delta_{\mathrm{astro}}$ (see Eq. [21]), $\sigma_{n}$ is the
observational noise in line intensity maps in units of
$\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}/\mathrm{Hz}/\mathrm{str}$, and
$n_{\mathrm{pix}}$ is the number density of pixels in intensity maps. For a
given $5\sigma$-limiting AB magnitude $m_{\mathrm{lim}}$ per each pixel,
$\sigma_{n}$ is given by

$\displaystyle\sigma_{n}=\frac{10^{-(m_{\mathrm{lim}}+48.60)/2.5}}{5\theta^{2}_{\mathrm{pix}}}\,[\mathrm{erg}/\mathrm{s}/\mathrm{cm}^{2}/\mathrm{Hz}/\mathrm{str}],$
(65)

where $\theta_{\rm pix}$ is the angular size of each pixel. Note that
$n_{\mathrm{pix}}=\theta^{-2}_{\mathrm{pix}}$. Within the halo model, we can
compute $P_{\mathrm{astro-astro}}$ as

$\displaystyle P_{\mathrm{astro-astro}}(k,z)=P^{\mathrm{1h}}_{\mathrm{astro-
astro}}(k,z)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+P^{\mathrm{2h}}_{\mathrm{astro-
astro}}(k,z),$ (66) $\displaystyle P^{\mathrm{1h}}_{\mathrm{astro-
astro}}(k,z)={\scriptstyle{\cal
N}^{-2}_{\mathrm{astro}}(z)}\int\mathrm{d}M\frac{\mathrm{d}n}{\mathrm{d}M}\,\dot{M}^{2}_{*}(M,z),$
(67) $\displaystyle P^{\mathrm{2h}}_{\mathrm{astro-
astro}}(k,z)=\left({\scriptstyle{\cal
N}^{-1}_{\mathrm{astro}}(z)}\int\mathrm{d}M\frac{\mathrm{d}n}{\mathrm{d}M}b_{h}(M,z)\dot{M}_{*}(M,z)\right)^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times P_{L}(k,z).$ (68)

In practice, the integrand of Eq. (60) includes the product of delta
functions, leading a divergence. Hence, we approximate the integral over
$\chi$ by a discrete sum such as

$\displaystyle\int\mathrm{d}\chi\delta^{(1)}_{\mathrm{D}}(\chi-\chi_{Q})\delta^{(1)}_{\mathrm{D}}(\chi-\chi_{Q^{\prime}})\cdots$
$\displaystyle\rightarrow\sum^{\infty}_{i=0}\Delta\chi\,\Theta(\chi_{i}-\chi_{Q}|\Delta\chi)\Theta(\chi_{i}-\chi_{Q^{\prime}}|\Delta\chi)\cdots,$
(69)

where $\chi_{i}=(i+1/2)\,\Delta\chi$, and the function of
$\Theta(\chi-\chi_{Q}|\Delta\chi)$ returns $1/\Delta\chi$ for
$|\chi-\chi_{Q}|\leq\Delta\chi/2$ and zero otherwise. The width of comoving
distance $\Delta\chi$ is set by

$\displaystyle\Delta\chi$ $\displaystyle=$
$\displaystyle\Biggl{[}\left(\frac{\mathrm{d}\chi}{\mathrm{d}z}\right)\Bigg{|}_{z_{Q}}\frac{1+z_{Q}}{\nu}\Delta\nu$
(70)
$\displaystyle\qquad\times\left(\frac{\mathrm{d}\chi}{\mathrm{d}z}\right)\Bigg{|}_{z_{Q^{\prime}}}\frac{1+z_{Q^{\prime}}}{\nu^{\prime}}\Delta\nu^{\prime}\Biggr{]}^{1/2},$

where $\Delta\nu=\nu/R$, $\Delta\nu^{\prime}=\nu^{\prime}/R$,
$z_{Q}=\nu_{Q}/\nu-1$, and $z_{Q^{\prime}}=\nu_{Q^{\prime}}/\nu^{\prime}-1$.
Note that $\nu_{Q}$ is the rest-frame frequency for the line $Q$ and $R$ is
the frequency resolution in line intensity mapping measurements.

The auto power spectrum of the observed convergence field,
$C^{(\mathrm{obs})}_{\kappa-\kappa}(\ell)$, is then given by

$\displaystyle C^{(\mathrm{obs})}_{\kappa-\kappa}(\ell)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}\chi\,\frac{W^{2}_{\kappa}(z)}{\chi^{2}}\,P_{\mathrm{m}}\left(\frac{\ell}{\chi},z\right)$
(71) $\displaystyle+\int\mathrm{d}\chi\,\frac{2W_{\kappa}(z)W_{\rm
IA}(z)}{\chi^{2}}\,P_{\mathrm{m}}\left(\frac{\ell}{\chi},z\right)$
$\displaystyle+\int\mathrm{d}\chi\,\frac{W^{2}_{\rm
IA}(z)}{\chi^{2}}\,P_{\mathrm{m}}\left(\frac{\ell}{\chi},z\right)$
$\displaystyle+\frac{\sigma^{2}_{e}}{n_{\mathrm{g}}},$

where $\sigma_{e}$ is the intrinsic scatter of galaxy ellipticies per
components, and $n_{\mathrm{g}}$ is the mean number density of source
galaxies.

##  Appendix B Expected upper limits of dark matter lifetime

Figure 8:  $2\sigma$-level upper limits of the lifetime of decaying dark
matter (DM) $\Gamma$ as a function of DM mass $m_{\mathrm{DM}}$. The orange
shaded region can be excluded by the cross correlation analysis with SPHEREx
line intensity maps and LSST lensing when one uses the power spectra with
$\ell\leq\ell_{\mathrm{max}}=1000$. The blue line shows the limits when one
uses the information from smaller angular scales with
$\ell_{\mathrm{max}}=2000$.

Fig. 8 provides an expected $2\sigma$ upper limit of the dark matter lifetime
$\Gamma$ by the cross power spectrum between line intensity and weak lensing
maps in LSST and SPHEREx.

##  Appendix C Full two-dimensional error circles in our Fisher analysis

Fig. 9 provides an expected 68%-level confidence level of our six parameters
by the cross power spectrum between line intensity and weak lensing maps in
LSST and SPHEREx.

Figure 9:  Similar to Fig. 7, but we include uncertainties of redshift
evolution in astrophysical line intensity and intrinsic alignments of galaxy
shapes.

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