# Mitigating the impact of fiber assignment on the measurement of galaxy-
lensing cross correlation

Ryu Makiya  and Tomomi Sunayama

###### Abstract

We examine the impact of fiber assignment on the measurement of galaxy
clustering and its cross correlation with weak lensing fields. Unlike the past
spectroscopic galaxy surveys such as Baryon Oscillation Spectroscopic Survey
(BOSS), currently ongoing spectroscopic galaxy surveys such as Prime Focus
Spectrograph (PFS) and Dark Energy Spectroscopic Instrument (DESI) suffer from
the fiber assignment artifacts more severely because there are more target
galaxies than available fibers. The previous studies found that the fiber
assignment suppresses the amplitude of the galaxy power spectrum at all
scales. We newly find that the fiber assignment introduces the artificial
correlation of structure at different redshifts, which suppresses the
amplitude of the galaxy-lensing cross power spectrum. We show that the fiber
assignment effects on the cross power spectrum can be mitigated at all scales
with accuracy better than $\sim 1\%$, by up-weighting observed galaxies with
the probability to be observed. This is not the case for the galaxy power
spectrum, which is not fully corrected at $k\gtrsim 0.2$ [$h$/Mpc]. We find
that the galaxy-lensing cross power spectrum is not affected by the pairwise
probability of galaxies to be observed, and thus the correction method based
on the individual probability is sufficient at all scales.

##  1 Introduction

Spectroscopic galaxy surveys yield three-dimensional distributions of galaxies
in redshift space, which enable us to constrain the cosmological models
through the measurement of, e.g., baryon acoustic oscillation (BAO) and
redshift space distortion [1]. On-going and near-future spectroscopic galaxy
surveys such as the Subaru Prime Focus Spectrograph (PFS) [2], the Dark Energy
Survey Instrument (DESI) [3, 4], HETDEX [5], Euclid [6], Roman space telescope
[7], and Legacy Survey of Space and Time by Vera C. Rubin Observatory[8], as
well as future surveys such as MegaMapper [9], will continue to expand the
survey volume with higher galaxy number density. The large imaging surveys
such as the Subaru Hyper Suprime Cam (HSC) [10, 11], Kilo-Degree Survey (KiDS)
[12], and Dark Energy Survey (DES) [13, 14] provide complementary information
through the weak-lensing observation and its cross-correlation with
spectroscopic galaxies. These surveys will dramatically increase the
statistical precision of cosmological parameters, which in turn requires more
careful handling of systematic uncertainties.

In most spectroscopic galaxy surveys, the fiber-fed multi-object spectrograph
system is used to maximize the redshift accuracy. In these systems, galaxy
spectra are taken by placing the optical fibers, which are connected to the
spectrograph, on target galaxies. The spectroscopic survey can only observe a
fraction of the parent targets due to finite observing times, which results in
the complicated spatial pattern of missing objects and the non-trivial bias in
the observed galaxy clustering. One example is the so-called “fiber collision”
effect: two objects which are too close to place two fibers cannot be observed
and the angular clustering at the scale smaller than the physical size of
fibers is suppressed.

One approach taken in the Sloan Digital Sky Survey (SDSS) is to simply assign
the same redshift to the missing objects with the nearest target object [e.g.,
15, 16], assuming that these galaxies are likely to be physically associated.
This standard approach is so-called the“nearest neighbor” method [17]. Another
approach is to weight observed galaxies using the angular correlation function
of the parent photometric catalog [18, 19, 20, 21]. Several other methods to
correct for missing targets are proposed in the literature. For example,
ref.[22, 23] reduce the fiber effects by nulling the angular clustering using
random catalogs of the same (RA, Dec) as the observed galaxies. Ref.[24, 25,
26, 27] up-weight targets by the inverse of the probability that a galaxy or a
pair of galaxies is observed and apply the methods to DESI mock catalogs.
Ref.[28] and [29] improve the nearest neighbor method by modeling the line-of-
sight (LOS) distribution of fiber-collided objects and by considering the
multiple fiber-collided objects, respectively. We refer the reader to ref.[24]
for comparison of the different methods.

In future/on-going deep fiber-fed spectroscopic surveys such as the PFS and
DESI, the situation is more complicated than SDSS/BOSS for two reasons. First,
fibers can only move within the small patrol areas of robotic arms to place
fibers in the PFS and DESI, while the fibers can be placed anywhere unless
they do not collide with other fibers in SDSS/BOSS. This leads to a non-random
selection of galaxies and suppresses the clustering amplitude on all scales
[30]. We can achieve random sampling if we feed fewer parent photometric
galaxies than targets, but in that case, the fiber allocation efficiency (the
fraction of fibers which can find target), hence the number of observed
galaxies, decreases. Second, the wider redshift coverage of PFS ($0.6<z<2.4$)
and DESI ($0.6<z<1.6$ for emission line galaxies) than SDSS/BOSS complicates
the correction of fiber allocation effects, because the assumptions used in
most of the correction methods are violated. For example, the nearest neighbor
method assumes that two galaxies close in angular space are physically close,
but it is less likely in the PFS and DESI. The correction method using the
angular clustering of the parent photometric galaxies is also not usable,
because the observed angular clustering is almost homogeneous and the true
clustering is obscured due to the wide redshift range of these surveys.

Ref.[30] examined the effects of fiber assignment on the galaxy clustering in
configuration space (i.e., galaxy correlation function), focusing on the PFS
cosmology survey. In this paper, we extend our analysis to the galaxy
clustering in Fourier space (i.e., galaxy power spectrum) and the cross
correlation between galaxies and the weak-lensing field. The PFS cosmology
survey will perform the spectroscopic galaxy survey using the photometric
galaxies of Subaru HSC. The PFS and HSC surveys provide not only the
clustering measurements of [OII] emission-line galaxies (ELGs) but also its
cross-correlation with the weak lensing field [31]. To maximize the scientific
outcome of the PFS cosmology program in tandem with the HSC survey, it is
crucial to investigate the effect of the fiber assignment on the galaxy-
lensing cross power spectrum. We show how the fiber assignment affects the
measurement of the power spectrum and cross power spectrum, and how we can
mitigate them. We newly find that the fiber assignment creates artificial
correlations between different redshifts and results in a significant
suppression of the cross power spectrum.

This paper is organized as follows. In section 2, we describe the
specifications of the PFS cosmology survey, including the construction of the
mock catalog and the details of fiber assignment algorithm. In section 3, we
show the effect of fiber assignment on the galaxy power spectrum and the
galaxy-lensing cross power spectrum, and how we can mitigate them. We
summarize and conclude in section 4.

##  2 Specifications of the PFS cosmology survey

The Subaru PFS is a multiplexed fiber-fed optical and near-infrared
spectrometer mounted at the prime focus of the Subaru Telescope[32], which is
expected to be ready by the fall of 2023. The spectrograph covers a wide range
of wavelengths from 380 to 1260 nm in a single exposure, which enables the PFS
cosmology program to map the three-dimensional distribution of [OII] ELGs at
$0.6<z<2.4$ over 1200 ${\rm deg}^{2}$ [2]. The focal plane of the Subaru PFS
is equipped with 2394 robotically reconfigurable fibers distributed over the
1.25 degree wide hexagonal field-of-view (FoV) as illustrated in figure 1.
With this large FoV, the Subaru PFS can efficiently observe roughly 4 million
ELGs within 100 nights.

In this section, we provide an overview of the PFS cosmology survey design and
summarize details of the fiber assignment in the PFS cosmology as well as the
galaxy and lensing mock catalogs used in this paper. We omit some details of
the instrumental design specific to the Subaru PFS, which is discussed in
ref.[30]. While we use the fiber assignment for the Subaru PFS cosmology
survey as an example, the nature of the problem is common to any fiber-fed
spectroscopic galaxy surveys.

###  2.1 Survey design: tiling and fiber assignment

The PFS cosmology program aims to observe ELGs at $0.6<z<2.4$ within 100
nights covering the area of 1200 ${\rm deg}^{2}$ using a hexagonal plate. The
bottom panel of figure 1 illustrates the survey strategy: the hexagonal plates
are uniformly placed and the center of the plate is dithered by the radius of
the hexagon between the first and second visits. Throughout this paper, we
call each hexagonal region in the bottom panel of figure 1 a “tile”.

To achieve the survey with a reasonably high number density of galaxies over
the wide redshift range, maximizing the efficiency of fiber usage is crucial.
However, it results in a low completeness, which is the ratio of the number of
observed galaxies111In this paper we assume 100% success rate of redshift
determination for simplicity, i.e., the number of observed galaxies is equal
to the number of fiber-assigned galaxies. to the number of targets. In the
nominal two-visit survey of PFS cosmology, we prepare roughly 8000 targets per
field, i.e. $8000/2394\simeq 3.3$ targets per fiber, to achieve the 90% of
fiber usage efficiency [2] and 54% completeness. Having such low completeness
suppresses the amplitude of galaxy clustering due to the loss of long-
wavelength fluctuations, whose wavelength is larger than the tile size. This
is because the fluctuations in the number of target galaxies per tile are lost
by a fixed number of fibers. This was not the problem for the previous
spectroscopic surveys such as BOSS because the number of target galaxies was
less than the number of available fibers. The details of the problem and the
mitigation scheme are discussed in section 4.1 of ref.[30].

Another difference between the PFS and the previous spectroscopic surveys is
that each fiber positioner of PFS has a finite “patrol” area within which the
fiber can move to the desired target as shown in the top panel of figure 1. A
finite patrol area together with low completeness results in a non-uniform
sampling of galaxies. When a positioner has more than two targets within its
patrol area, only two galaxies are observed in the nominal two-visit survey.
This causes a non-random sampling of target galaxies, which further suppresses
the clustering amplitude on small scales.

Figure 1:  Top: An illustration of the configuration of fiber positioners on
the focal plane of PFS (taken from ref.[33]). The gray shaded hexagons in the
left panel show the 1.25 ${\rm deg}^{2}$ FoV of the PFS. Each fiber is moved
by two motors to cover its patrol area, as shown in the upper left panel.
Right panels show the configurations of the patrol area of fibers (black
circles) at the center and the edge of FoV, respectively. The gray circles
show the “dead” region, where we can not observe targets. Bottom: The tiling
strategy of PFS cosmology survey. Solid and dashed lines show the tiling of
the first and second visits, respectively. The blue dots show the mock target
galaxies at $0.6<z<2.4$.

###  2.2 Light-cone mock catalogs for galaxies and lensing map

To investigate the fiber assignment artifacts on the galaxy power spectrum and
the galaxy-lensing cross power spectrum, we generate the mock galaxy catalog
and weak lensing map by using the public code lognormal_lens
[31]222https://wwwmpa.mpa-garching.mpg.de/~komatsu/codes.html..

The code first generates a series of three-dimensional matter density fields
on a regular grid from the input power spectra assuming that density fields
follow a log-normal probability density function. To obtain the density
fluctuation field $\delta({\bm{x}})$, the code first generates a log-
transformed density field $\ln{[1+\delta({\bm{x}})]}$ which is a Gaussian
random field characterized by $\xi^{G}$ [34],

$\xi^{G}(r)=\ln{[1+\xi(r)]},$ (2.1)

where $\xi(r)$ is the input correlation function. The code then generates
$\delta({\bm{x}})$ by exponentiating the log-transformed density field.

Next, the code constructs a single light-cone from a series of three-
dimensional matter density fields and ray-traces a light-cone to obtain the
weak lensing convergence field. Ray-tracing is done by the public code RAYTRIX
[35]. First, we project a three-dimensional density field $\delta$ into a two-
dimensional field using the plane-parallel approximation,

$\delta^{\rm proj}(x,y)=\frac{1}{N_{{\rm
grid},z}}\sum_{z_{i}}\delta(x,y,z_{i}),$ (2.2)

where $N_{{\rm grid},z}$ is the grid number along the LOS. We shall call this
projected density field a “mass sheet”. Next, the two-dimensional deflection
potential $\Psi$ is computed from the mass sheet as

$\nabla^{2}\Psi(x,y)=\frac{3\Omega_{m}H_{0}^{2}}{c^{2}}\delta^{\rm
proj}(x,y),$ (2.3)

where $\Omega_{m}$ is the matter density parameter, $H_{0}$ is the Hubble
parameter and $c$ is the speed of light. From the deflection potential, a
Jacobian of lensed to unlensed coordinates, $A_{n}$, is computed as

$A_{n}=I-\sum_{i=1}^{n-1}\frac{\chi_{i}(\chi_{n}-\chi_{i})}{a(\chi_{i})\chi_{n}}U_{i}A_{i},$
(2.4)

and

$U_{i}=\begin{pmatrix}\displaystyle\frac{\partial^{2}\Psi^{i}}{\partial
x_{1}\partial x_{1}}&\displaystyle\frac{\partial^{2}\Psi^{i}}{\partial
x_{1}\partial x_{2}}\\\ &\\\ \displaystyle\frac{\partial^{2}\Psi^{i}}{\partial
x_{2}\partial x_{1}}&\displaystyle\frac{\partial^{2}\Psi^{i}}{\partial
x_{2}\partial x_{2}}\\\ \end{pmatrix},$ (2.5)

where $I$ is the identity matrix, $\Psi^{i}$ is the deflection potential of
the $i$-th mass sheet, $n$ is the total number of mass sheets, $a$ is the
scale factor, and $\chi_{n}$ and $\chi_{i}$ are the comoving distances to the
source plane and the $i$-th mass sheet, respectively. The source plane is
located at the $n$-th mass sheet. The Jacobian $A_{n}$ is related to the
lensing convergence $\kappa$ and shear field $\gamma=(\gamma_{1},\gamma_{2})$
as

$A_{s}=\begin{pmatrix}1-\kappa-\gamma_{1}&&-\gamma_{2}\\\
-\gamma_{2}&&1-\kappa+\gamma_{1}\\\ \end{pmatrix}.$ (2.6)

The convergence $\kappa$ and shear $\gamma$ characterize magnification and
stretching of the source image around the lens, respectively. In the
following, we focus on $\kappa$.

The code also generates galaxies by Poisson sampling the underlying galaxy
density field, which is also generated from the input galaxy power spectrum
assuming that the galaxy density field follows the log-normal distribution. We
use the same initial random seed for matter and galaxy density fields, to
ensure that the matter and galaxy fields correlate. We also generate the
peculiar velocity field of galaxies from the underlying matter density field
by using the linear continuity equation. The galaxy density field is generated
on the regular grid. We randomly populate galaxies within each grid cell,
while assigning the same velocity to galaxies within the same grid. We refer
the readers to ref.[31] and ref.[36] for further details of our lognormal
simulations.

The input power spectra are generated by the public code CLASS [37, 38],
assuming a flat $\Lambda$CDM model with the parameters of Planck 2015
‘TT,TE,EE+lowP’: $\Omega_{b}h^{2}=0.02225$, $\Omega_{c}h^{2}=0.1198$,
$n_{s}=0.9645$, $\ln(10^{10}A_{s})=3.094$ and $h=0.67021$ with the minimum
neutrino mass of $M_{\nu}=0.06$ [eV] [39]. For simplicity, we use the linear
matter power spectrum and assume the linear galaxy bias. Following ref.[2], we
assume that the linear galaxy bias is given by $b(z)=0.9+0.4z$. The source
galaxies of the weak lensing convergence field are assumed to be located at
$z=3.0$. We ignore the shape noise in the lensing measurement because it is
not affected by the fiber assignment.

The configuration of matter density fields and light-cone is shown in figure
2. The FoV of the simulation (shown in the black dotted lines in the figure 2)
is $25\times 25=625\;{\rm deg}^{2}$, similar to the “Fall equatorial field” of
the PFS cosmology, $630\;{\rm deg^{2}}$. The configuration of three-
dimensional boxes is determined as follows. First we prepare boxes in which
the PFS spectroscopic galaxy samples are generated (the red boxes in figure 2)
with the redshift interval of $\Delta z$ = 0.2 for $0.6<z<1.6$ and $\Delta
z=0.4$ for $1.6<z<2.4$, following the fiducial PFS survey parameters given in
ref.[2]. Next, we prepare other foreground and background boxes (the black
boxes in figure 2). The number of boxes is determined so that each box has a
similar length along the LOS, $L_{z}$. The maximum redshift $z_{\rm max}=3.2$
is chosen to match the assumed redshift distribution of source galaxies of the
HSC survey. The box side-lengths perpendicular to the LOS, $L_{x}$ and
$L_{y}$, are chosen to match the opening angle of the simulation times
comoving distances (the horizontal axis) to the box centers.

Each lognormal box is generated with the three-dimensional grid-number of
$N_{\rm grid,3D}=256$ for the axis perpendicular to LOS, while the grid number
along the LOS is determined so that each grid has a cubic shape. The number of
two-dimensional angular grids for the lensing map is also set to $N_{\rm
grid,2D}=256$, i.e., the grid size is $25.0/256=0.098$ degrees on a side.

The box geometry, galaxy number density, and the linear bias parameter of our
simulation are summarized in figure 2 and table 1.

Figure 2: Configuration of the mock light cone. The matter density field is generated in each solid box. The red boxes also contain galaxies mocking the spectroscopic sample of PFS cosmology survey. The black dotted lines show the $25\times 25$ ${\rm deg}^{2}$ FoV. The side lengths of each box are summarized in table 1. redshift | $L_{x,y}$ | $L_{z}$ | $\bar{n}_{g}^{\rm true}$ | $\bar{n}_{g}^{\rm obs}$ | $b_{g}$
---|---|---|---|---|---
| [Mpc/$h$] | [Mpc/$h$] | [$10^{-4}h^{3}{\rm Mpc^{-3}}$] | [$10^{-4}h^{3}{\rm Mpc^{-3}}$] |
$0.0<z<0.09$ | 56.7 | 255.6 | – | – | –
$0.09<z<0.18$ | 170.0 | 255.6 | – | – | –
$0.18<z<0.27$ | 283.3 | 255.6 | – | – | –
$0.27<z<0.38$ | 396.6 | 255.6 | – | – | –
$0.38<z<0.48$ | 510.0 | 255.6 | – | – | –
$0.48<z<0.6$ | 623.3 | 255.6 | – | – | –
$0.6<z<0.8$ | 768.8 | 400.6 | 5.4 | 2.7 | 1.18
$0.8<z<1.0$ | 937.2 | 354.7 | 17.2 | 8.4 | 1.26
$1.0<z<1.2$ | 1084.8 | 315.2 | 16.6 | 8.2 | 1.34
$1.2<z<1.4$ | 1217.0 | 281.4 | 22.4 | 10.9 | 1.42
$1.4<z<1.6$ | 1335.4 | 252.5 | 15.8 | 7.7 | 1.5
$1.6<z<2.0$ | 1487.7 | 434.5 | 7.2 | 3.5 | 1.62
$2.0<z<2.4$ | 1664.0 | 360.8 | 7.8 | 3.8 | 1.78
$2.4<z<2.8$ | 1811.6 | 305.1 | – | – | –
$2.8<z<3.2$ | 1937.3 | 262.0 | – | – | –
Table 1: The details of the simulation boxes generated to construct the mock
light-cone. $L_{x,y}$ is the comoving side lengths of the box perpendicular to
the LOS direction, while $L_{z}$ is the length along the LOS.
$\bar{n}_{g}^{\rm true}$ and $\bar{n}_{g}^{\rm obs}$ denote the mean number
density of target galaxies before and after the fiber assignment,
respectively. $b_{g}$ is the linear galaxy bias.

###  2.3 Exposure Targeting Software (ETS)

To simulate the fiber assignment scheme in the PFS cosmology survey, we use
the Exposure Targeting Software (ETS)333https://github.com/Subaru-
PFS/ets_fiberalloc. which is developed for the PFS [33].

As illustrated in figure 1, PFS has 2394 fibers within a hexagonal FoV of
$\sim 1.25{\rm deg}^{2}$. The fiber positioner, “Cobra”, consists of two
rotational stages to cover a circular patrol area per fiber. These positions
on the focal plane are fixed. The ETS takes angular positions (RA, Dec) and
priorities of target galaxies as inputs, and finds targets in the patrol area
of each fiber. We adopt the plane-parallel approximation for each simulation
box when projecting the three-dimensional galaxy distribution to the (RA, Dec)
space. If there are multiple galaxies in the same patrol area, the ETS selects
a galaxy based on the given priority and the position of galaxies within the
patrol area. In this paper, we employ a “naive” algorithm, in which the fiber
is simply assigned to a target galaxy with the highest priority. If there are
multiple galaxies with the highest priority, the algorithm randomly selects
one of them. Here we assign the same priority to all target galaxies, which is
appropriate for the PFS cosmology program. As shown in the bottom panel of
figure 1, we place the hexagonal tiles uniformly, while separating the tile
centers for the first and second visit by the radius of the hexagonal tile.

We generate 500 realizations of lognormal_lens simulation and run the ETS on
each mock. In the following, we call the galaxy catalog without fiber
assignment the “true” sample, while those with fiber assignment the “observed”
sample. The mean number density of galaxies in the observed sample is given in
table 1.

##  3 Fiber assignment artifacts

In this section, we present the impact of fiber assignment on the galaxy power
spectrum and the galaxy-lensing cross power spectrum as well as the mitigation
scheme.

We show that the observed galaxy power spectrum is suppressed at all scales in
the same manner as for the correlation function in configuration space [30].
We will also show that the suppression is localized at the $\mu_{k}=0$ modes,
where $\mu_{k}$ is the cosine between the wave vector and the LOS direction.

Next, we show that the observed galaxy-lensing cross power spectrum is also
suppressed and even becomes negative at high redshifts. This is due to the
artificial correlation between galaxies at different redshift slices
introduced by fiber assignment.

To mitigate the fiber assignment artifacts, we apply the so-called
“Individual-Inverse Probability” (IIP) method [24] in which observed galaxies
are up-weighted by the inverse probability to be observed. We find that the
IIP method recovers the true power spectrum at accuracy better than 1% up to
$k=0.2$ [$h$/Mpc]. If we further take into account the pairwise probability of
galaxies to be observed, we recover the true spectrum at $<1\%$ accuracy up to
the Nyquist frequency. On the other hand, the IIP method recovers the galaxy-
lensing cross power spectrum at an accuracy better than 1% up to the Nyquist
frequency. The IIP method is sufficient for the cross power spectrum since the
pairwise probability does not enter in this statistics.

###  3.1 Galaxy power spectrum in real space

First, we investigate the galaxy power spectrum in real space (i.e., ignoring
the peculiar velocity of galaxies). In the following, all of the power
spectrum and cross power spectrum measurements are done by using the Fast
Fourier transform (FFT). We use the FFTW library for Fourier transform [40].
The nearest grid point (NGP) assignment is adopted to compute the galaxy
number density contrast, $\delta_{g}({\bm{x}})=n_{g}({\bm{x}})/\bar{n}_{g}-1$,
where $n_{g}({\bm{x}})$ is the galaxy number density estimated within the
cubic grid at a three-dimensional position ${\bm{x}}$. The grid size is the
same with lognormal realizations of galaxies. The mean galaxy number density
$\bar{n}_{g}$ is defined as $\bar{n}_{g}=N_{g}/V$, where $N_{g}$ is the total
number of galaxies within survey volume $V$.

We estimate the power spectrum as

$P(k,\mu_{k})=\frac{1}{N_{\rm
k}}\sum_{i=1}^{N_{k}}\left[\tilde{\delta}_{g}({\bm{k}_{i}})\tilde{\delta}_{g}^{*}({\bm{k}_{i}})-P_{\rm
shot}\right],$ (3.1)

where $\tilde{\delta}_{g}({\bm{k}})$ is the Fourier transformed galaxy number
density contrast and $N_{k}$ is the number of Fourier modes within a given
$(k,\mu_{k})$ bin. The cosine of the angle between ${\bm{k}}$ and the LOS
($z$-direction), $\mu_{k}$, is defined as $\mu_{k}\equiv k_{z}/k$. Here we
assume that the LOS vector is the same for all galaxies (i.e., global plane-
parallel approximation). The shot noise power spectrum $P_{\rm shot}$ is given
by $P_{\rm shot}=1/\bar{n}_{g}$. Since the galaxy density fields are generated
to satisfy the periodic boundary condition, we do not need to care about the
window effect of the survey boundary.

The top-left panel of figure 3 shows the ratio of the “observed” to “true”
galaxy power spectra at $z=1.3$. Here we show the angle averaged power
spectrum as well as the power spectra at $\mu_{k}=0$ and $0<\mu_{k}\leq 1$.
The wave vector $k$ is binned in the interval of the fundamental frequency,
$\Delta k=0.022$ [$h$/Mpc]. Figure 3 clearly shows that the most of the fiber
assignment effect is localized at $\mu_{k}=0$. A similar trend is also seen in
the fiber-assigned power spectrum of the DESI-like survey [23]. This result
can be understood as follows. The $k_{z}=0$ ($\mu_{k}=0$) component of the
Fourier-transformed density field $\tilde{\delta}$ is expressed as

$\displaystyle\tilde{\delta}(k_{x},k_{y},k_{z}=0)$ $\displaystyle=$
$\displaystyle\int\int\int\delta(x,y,z)e^{-i(k_{x}x+k_{y}y)}\;{\rm d}x\;{\rm
d}y\;{\rm d}z$ (3.2) $\displaystyle=$
$\displaystyle\int\int\left[\int\delta(x,y,z)\;{\rm
d}z\right]e^{-i(k_{x}x+k_{y}y)}{\rm d}x\;{\rm d}y.$

This equation shows that the $\mu_{k}=0$ mode of $\delta(k,\mu_{k})$
corresponds to the projected density field along the LOS. Since the fiber
assignment is based on the angular positions of galaxies, the mode
corresponding to the projected density field is affected the most.

In ref.[30] we found that the suppression at large scales is caused by the
“tiling effect”. Since the number of fibers within each tile is fixed, the
correlation amplitude at a scale larger than a tile is suppressed when the
number of target galaxies is much larger than the number of fibers. To correct
for this tiling effect, ref. [30] used the weight given by

$w_{i}^{\rm tile}=\frac{N_{{\rm target},i}}{N_{{\rm obs},i}},$ (3.3)

where $N_{{\rm target},i}$ and $N_{{\rm obs},i}$ are the number of target and
observed galaxies within each tile, respectively. The top-right panel of
figure 3 shows the power spectra after applying the weights. We can see that
the weight recovers the true power spectrum at large scales.

The suppression at small scales is caused by the under-sampling of over-dense
regions due to the fiber assignment, which is also seen in ref.[30]. Since the
fibers are uniformly distributed within the FoV but the target galaxies are
not, we miss a certain fraction of targets in the dense region while we
observe most of the targets in the under-dense region. The PFS cosmology
program is designed to have a high number density of target galaxies to
achieve a high fiber usage efficiency. This results in a high probability of
having many galaxies in a single patrol area, hence the significant
suppression of the observed power spectrum.

To correct for the fiber assignment effects, we apply the IIP method following
ref.[30]. First, we run the ETS 100 times for each realization of the mock and
estimate the probability of the $i$-th galaxy to be observed, $p_{i}$, for all
galaxies. Next we up-weight each observed galaxy by $w_{i}=1/p_{i}$. In this
case, the galaxy number density contrast is computed as
$\delta_{g}({\bm{x}})=n_{g,{\rm IIP}}({\bm{x}})/\bar{n}_{g,{\rm IIP}}-1$,
where $n_{g,{\rm IIP}}({\bm{x}})$ is the local mean of $w_{i}$ within the grid
at ${\bm{x}}$, and $\bar{n}_{g,{\rm IIP}}=\sum_{i=1}^{N_{g}}w_{i}/V$ is the
global mean of $w_{i}$. Note that the IIP method also corrects the tiling
effect, since the tile-to-tile variation of the number of target galaxies is
naturally included in the variation of $w_{\rm IIP}$. Following ref.[41, 42,
25], the shot noise of weighted galaxies is computed as

$P_{\rm shot,IIP}=\frac{\sum_{i=1}^{N_{g}}w_{i}^{2}}{\bar{n}_{g,{\rm
IIP}}^{2}V}=\frac{\sigma[w]^{2}+\bar{w}^{2}}{\bar{w}^{2}}P_{\rm shot},$ (3.4)

where $\bar{w}$ and $\sigma[w]$ are defined as $\bar{w}=\sum_{i}w_{i}/N_{g}$
and $\sigma[w]^{2}=\sum_{i}w_{i}^{2}/N_{g}-\bar{w}_{i}^{2}$, respectively.
This equation shows that the shot noise is enhanced in the case of non-uniform
sampling, i.e. $\sigma[w_{\rm IIP}]>0$. The bottom panels of figure 3 show the
ratio of the observed to true galaxy power spectra for the IIP weighting. As
in the case of the galaxy correlation function discussed in ref.[30], the IIP
weighting scheme recovers the true power spectrum at an accuracy better than
$1\%$ up to $k=0.2$ [$h$/Mpc].

The spike-like features seen in the observed power spectra are caused by the
way we place the hexagonal tiles. Since the fibers cannot be placed at the
edge of the tile, there are small gaps between tiles where galaxies are not
observed at all and these gaps generate artificial clustering feature. As
shown in figure 1, the angular separation of the gaps is roughly the same with
the angular radius of tile ($\sim 0.7$ deg), which corresponds to the comoving
length of $\sim 34.0$ [Mpc/$h$] and the wave number of $k\sim 0.18$ [$h$/Mpc])
at $z=1.3$. The positions of the spike, $k\sim 0.2$ and $\sim 0.4$ [$h$/Mpc],
roughly match that wavelength and their integer multiples, respectively. The
effect of such a non-uniform survey window can be mitigated by using, e.g, a
random catalog [e.g., 43].

Figure 3: (Top-Left) The ratio of the “observed” to “true” galaxy power
spectra. The blue solid line is the total power spectrum, while the orange
dotted line and the green dashed line are the spectra with $\mu_{k}=0$ and
$0.0<|\mu_{k}|\leq 1.0$, respectively. The data points are the mean of 500
realizations. The error bars are the standard deviation of mean, i.e. the
standard deviation divided by the square root of the number of realizations.
(Top-Right) The same as Top-Left, but the tile weighting is applied. (Bottom-
Left) The same as Top-Left, but the IIP weighting is applied. (Bottom-Right)
Zoom-in view of Bottom-Left panel.

The IIP correction does not fully recover the “true” power spectrum at small
scales. This is because the IIP estimator is not unbiased at a small scale
where the correlation between $p_{i}$ is not negligible. Several studies [e.g.
24, 25] show that the so-called “Pairwise-Inverse Probability” (PIP) method,
in which galaxy pairs are up-weighted by the pairwise probability to be
observed, can unbias the galaxy spectrum multipoles at smaller scales than the
IIP method. In the left panel of figure 4 we show the ratio of the PIP
corrected to the IIP corrected galaxy pair count, as a function of the
separation perpendicular to the LOS, $r_{\perp}$. It is clearly shown in the
figure that the ratio deviates from unity at $r_{\perp}\lesssim 2.0\;[{\rm
Mpc}/h]$. Following ref.[24, 25], we can express the PIP corrected power
spectrum as

$\displaystyle P^{\rm PIP}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{I}\int\frac{d\Omega_{k}}{4\pi}\left[\sum_{ij}(w_{ij}-w_{i}w_{j})e^{i{\bm{k}}({\bm{x}}_{i}-{\bm{x}}_{j})}\right]+P^{\rm
IIP}(k)$ (3.5) $\displaystyle=$
$\displaystyle\frac{1}{I}\sum_{ij}(w_{ij}-w_{i}w_{j})j_{0}(k|{\bm{x}}_{i}-{\bm{x}}_{j}|)+P^{\rm
IIP}(k),$

where $I=\bar{n}^{2}_{g,{\rm IIP}}V$ is the normalization constant,
$d\Omega_{k}$ is the solid angle element in $k$-space, and $w_{ij}=1/p_{ij}$
is the pairwise weight for a pair of the $i$-th and $j$-th galaxies. In the
second raw of the equation, the angular integration in $k$-space is reduced to
the spherical Bessel functions of the first kind, $j_{\ell}$. The right panel
of figure 4 shows the power spectrum corrected by the PIP method. To reduce
the computational cost, we assume that $w_{ij}-w_{i}w_{j}=0$ at
$r_{\perp}>3.0\;[{\rm Mpc}/h]$. Figure 4 clearly shows that the PIP method
recovers the true power spectrum at an accuracy better than $1\%$ at all
scales.

Figure 4: (Left) The ratio of the galaxy pair count corrected by the PIP
weights, to that corrected by the IIP weights, as a function of the separation
perpendicular to the LOS, $r_{\perp}$, at $z=1.3$. The data points show the
mean and standard deviation of 10 realizations. (Right) The ratio of the
“observed” to “true” power spectra at $z=1.3$, corrected by the IIP and PIP
weights, respectively. The data points show the mean and standard deviation of
100 realizations.

Figure 5 shows the ratio of the observed to true galaxy power spectra in real
space for all redshift bins, as well as those corrected by the IIP weighting.
We also show the results when we remove the $\mu_{k}=0$ mode, which is a major
cause of suppression at small scales. We find that the IIP weighting recovers
the true power spectra at $\sim 1\%$ accuracy up to $k=0.2$ [$h$/Mpc] for all
redshifts. This accuracy improves to better than $1\%$ if we remove the
$\mu_{k}=0$ mode. The figure shows that the fiber effect is not strong at the
lowest redshift bin, $0.6<z<0.8$. This is due to its low number density of
target galaxies. When the small number of galaxies is considered as targets,
most of the fibers have less than two targets in the patrol area. In that
case, the selection of galaxies becomes closer to random and therefore the
fiber assignment has little effect.

Figure 5: The ratio of the “observed” to “true” galaxy power spectra for all
redshift bins of the PFS survey. The blue solid lines show the results without
any correction, while the orange dashed lines show the results corrected by
the IIP weights. The green dotted lines show the corrected by the IIP weights
and the $\mu_{k}=0$ mode is removed.

###  3.2 Galaxy power spectrum in redshift space

Here we investigate the galaxy auto power spectrum in redshift space. First we
displace the observed galaxies by the LOS component of their peculiar velocity
using the plane-parallel approximation, and then compute the multipole
expansion of the power spectrum $P_{\ell}(k)$ defined as

$P(k,\mu_{k})=\sum_{\ell=0}^{\infty}P_{\ell}(k)\mathcal{L}_{\ell}(\mu_{k}),$
(3.6)

where $\mathcal{L}_{\ell}$ is the $\ell$-th order Legendre polynomial.

Figure 6 and 7 show the ratio of the observed to true galaxy power spectra for
monopole ($\ell=0$) and quadrupole ($\ell=2$), respectively. Since the fiber
assignment introduces the $\mu_{k}$-dependent anisotropy as shown in figure 3,
the observed quadrupole power spectrum becomes larger than the true power
spectrum. As in the case of the real-space power spectra, the IIP weighting
recovers the true spectra up to $k\sim 0.2$ [$h$/Mpc].

Figure 6:  The same as figure 5, but for the monopole power spectra in
redshift space. Figure 7: The same as figure 5, but for the quadrupole power
spectra in redshift space.

As eq. (3.4) shows, the fiber assignment and the IIP weighting increase the
shot noise, hence the fractional error of the power spectrum. Figure 8 shows
the fractional error of the galaxy monopole and quadrupole at $z=1.3$. These
relative errors are estimated from 500 realizations of the simulation. We also
show the fractional error of the power spectrum estimated from the simulation
with the same galaxy number density with the “observed” sample but without the
fiber assignment. We find that the fiber assignment and the IIP weighting
increase the fractional error by $\sim 5\%$ at $k=0.05$ [$h$/Mpc] and $\sim
30\%$ at $k=0.2$ [$h$/Mpc] compared to the case without the fiber assignment.

Figure 8:  Fiber assignment effect on the fractional error of the galaxy power
spectrum monopole (left panel) and the quadrupole (right panel) at $z=1.3$.
The blue solid lines are the fiducial simulation, where the fiber assignment
is applied and corrected by the IIP weighting. The orange dashed lines show
the fractional error of the galaxy power spectrum without the fiber assignment
effects.

###  3.3 Galaxy-lensing cross power spectrum

We now investigate the fiber assignment effects on the galaxy-lensing cross
power spectrum. We measure the cross power spectrum as

$C^{g\kappa}({\ell})=\frac{1}{L^{2}}\left[\frac{1}{N_{\ell}}\sum_{\bm{\ell}}^{\ell\in\ell_{b}}\tilde{\delta}^{\rm
2D}_{g}({\bm{\ell}})\tilde{\delta}_{\kappa}^{*}({\bm{\ell}})\right],$ (3.7)

where $L$ is the side-length of the square-shape simulation field in radian,
$N_{\ell}$ is the number of modes within a given multipole bin, and
$\delta_{\kappa}$ is the Fourier transformed convergence field. The projected
galaxy density field $\delta^{\rm 2D}_{g}({\bm{x}})$ is obtained from the
three-dimensional galaxy density field in the same manner as for the
projection of matter density field, eq.(2.2).

Figure 9 shows the ratio of the “observed” to “true” cross power spectra for
all redshift bins, as well as those corrected by the IIP weighting. We find
that the observed cross power spectra are largely suppressed at all redshifts
and even become negative at $z>1.8$. Nevertheless, the IIP weighting recovers
the true cross power spectra at an accuracy better than $1\%$ at all scales
and redshift bins. The strong suppression in the observed spectra is due to
the correlation between the different redshift slices due to the fiber
assignment. Since the selection of one galaxy affects the selection of all the
neighboring galaxies in angular space, the fiber assignment creates the
artificial correlation of galaxies and underlying matter density fields
between different redshift slices, which are not physically associated. Since
the weak lensing field is the integration of the matter density field along
the LOS, this artificial correlation matters to the galaxy-lensing cross
correlation.

Figure 9: The ratio of the “observed” to “true” galaxy-lensing cross power
spectrum for all redshift bins. The blue solid lines show the results without
any correction, while the orange dashed lines show that corrected by the IIP
weighting.

To investigate this artificial correlation between galaxies and matter density
fields at different redshifts, we compute the galaxy-matter cross power
spectra. Figure 10 shows the projected galaxy-matter cross power spectra,
where the matter density fields are at the different redshifts from galaxies.
We show the results with the galaxies at $z=0.7,1.1,1.5,$ and $2.2$ as an
example. The negative matter-galaxy correlation is clearly appeared in the
figure. While the total amplitude of this artificial galaxy-matter anti-
correlation does not depend on the redshift of galaxies, the amplitude of
intrinsic galaxy-matter cross power spectrum is smaller at a higher redshift;
thus the artificial anti-correlation dominates the observed galaxy-lensing
cross power spectrum at high redshifts. These negative signals are corrected
by the IIP weighting, as shown in figure 11.

Figure 12 shows the ratio of the “observed” to “true” galaxy-matter cross
power spectra for all redshift bins, where galaxies and matter density fields
are at the same redshift. The figure shows that the IIP weighting recovers the
true spectrum up to the Nyquist frequency (the rightmost data points in the
figure). This is not the case for the galaxy auto power spectra as discussed
in section 3.1. This is because that the pairwise probability of galaxies does
not enter in the estimator of galaxy-lensing cross power spectrum.

Figure 10:  The cross power spectrum of the observed galaxy density field and
the matter density field at the redshift different from galaxies. Data points
show the mean of 500 realizations. The error of the mean is smaller than the
size of the square data points. We only show the results for the galaxies at
$z=0.7,1.1,1.5,2.2$ for the clarity of the figure. Other redshifts show
similar trends.  Figure 11:  The same as figure 10, but for the results
corrected by the IIP weighting.  Figure 12:  The ratio of the “observed” to
“true” galaxy-matter cross power spectra for all redshift bins, where the
matter density fields are at the same redshift with galaxies. The blue solid
lines show the results without any correction, while the orange dashed lines
show that corrected by the IIP weighting.

###  3.4 Required accuracy of the IIP weights

To estimate the IIP weights, we implicitly assume that all the galaxies in the
“true” sample are our target galaxies. In the real observation, however, non-
target objects are included in the catalog of target galaxies. This can
systematically bias the IIP weights. The contamination fraction is determined
by the quality of the photometric data used to select target galaxies; thus,
the bias in the IIP weights can vary with the spatial scale of the FoV of the
HSC survey for the case of the PFS Cosmology program, which corresponds to
$\sim 1\;{\rm deg^{2}}$. To mimic this field-to-field variation of the HSC
data quality, we first divide the entire simulation FoV into the $1\;{\rm
deg^{2}}$ sub-regions. Next, we add the random Gaussian fluctuations to the
individual probability to be observed, $p_{i}$, while keeping the galaxies in
the same sub-region have the same fluctuation. We force $p_{i}$ does not
exceed the natural range, $0<p_{i}\leq 1.0$.

The upper panels of figure 13 show the ratio of the “observed” to “true”
galaxy power spectra at $z=1.3$, varying the fluctuation on the IIP weights.
We find that the 3% (5%) fluctuations in the IIP weights with 1 deg scale is
translated to $\sim$1% ($\sim$4%) and $\sim$3% ($\sim$10%) systematic offsets
in the monopole and quadrupole of power spectrum at large scales,
respectively. With a 3% error of the IIP weights, the recovered galaxy power
spectra are consistent with the true spectra within the statistical errors.
This result indicates that the imaging survey used for the target selection is
required to achieve $<3\%$ precision in target number density to correct for
the galaxy power spectra by the IIP weighting. For the galaxy-lensing cross
power spectrum, shown in the lower panel of figure 13, the systematic offsets
are negligible even with the 5% fluctuations in the IIP weights. This is
because that the correlation of the IIP weights, which causes the systematic
offsets in the auto power spectra, does not enter in the cross power spectra.

Figure 13:  Effects of the systematic fluctuation in the IIP weights on the
measurement of the galaxy power spectrum multipoles (upper panels) and the
galaxy-lensing cross power spectrum (lower panel). The data points show the
ratio of the “observed” to “true” power spectra for the galaxy sample at
$z=1.3$. The blue solid lines are the case without any systematic fluctuation
in the IIP weights. The orange dashed lines (green dotted lines) show the case
when we divide the galaxy sample into 1 ${\rm deg}^{2}$ grids and add the 3%
(5%) Gaussian fluctuations keeping that the galaxies in the same grid have the
same fluctuation.

##  4 Summary and Conclusions

In this paper, we have investigated the fiber assignment effect on the galaxy
power spectrum and the galaxy-lensing cross power spectrum. We have shown that
the fiber assignment suppresses the amplitude of galaxy power spectrum at all
scales, as in the case of the galaxy clustering in configuration space [30].
We newly found that the amplitude of the galaxy-lensing cross power spectrum
is also suppressed by the fiber assignment, which even become negative at high
redshifts. This is because the fiber assignment introduces the artificial
anti-correlation of galaxies and matter density fields at different redshifts.

To mitigate the fiber assignment effect, we have tested the weighting method
using the inverse probability of the galaxies to be observed. The weighting
method recovers the galaxy power spectrum multipoles better than $\sim 1\%$
accuracy up to $k\sim 0.2$ [$h$/Mpc], but we need to consider the pairwise
probability of galaxy pairs to be observed to correct for the galaxy power
spectrum at $k>0.2$ [$h$/Mpc]. The galaxy-lensing cross power spectrum is
recovered better than $\sim 1\%$ up to the Nyquist frequency by the IIP
weighting. This is because the pairwise probability does not enter in this
statistics.

As we demonstrated in this paper, the galaxy power spectrum and the galaxy-
lensing cross power spectrum have the different responses to the fiber
assignment and the different correction methods are required for these
measurements. To maximize the scientific outcomes of the photometric and
spectroscopic surveys, the joint analysis of the power spectrum and the
galaxy-lensing cross power spectrum is crucial; thus, it is important to
understand how the fiber assignment affects these measurements and how the
effects are mitigated.

## Acknowledgments

We thank Eiichiro Komatsu and Shun Saito for their careful comments. We also
thank the PFS collaboration, especially the cosmology working group for
discussions. RM is supported by JSPS KAKENHI Grant Number 20K14515. TS is
supported by Grant-in-Aid for JSPS Fellows 20J01600 and JSPS KAKENHI Grant
Number 20H05855.

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