# The Sensitivity of the Redshift Distribution to Galaxy Demographics

Philipp Sudek1, Lucia F. de la Bella1, Adam Amara1, and William G. Hartley2
1Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth,
PO1 3FX, UK
2Department of Astronomy, University of Geneva, ch. d’Ecogia 16, CH-1290
Versoix, Switzerland
E-mail<EMAIL_ADDRESS>
(Accepted 2022 August 12. Received 2022 August 12; in original form 2022
January 11)

###### Abstract

Photometric redshifts are commonly used to measure the distribution of
galaxies in large surveys. However, the demands of ongoing and future large-
scale cosmology surveys place very stringent limits on the redshift
performance that are difficult to meet. A new approach to meet this precision
need is forward modelling, which is underpinned by realistic simulations. In
the work presented here, we use simulations to study the sensitivity of
redshift distributions to the underlying galaxy population demographics. We do
this by varying the redshift evolving parameters of the Schechter function for
two galaxy populations, star-forming and quenched galaxies. Each population is
characterised by eight parameters. We find that the redshift distribution of
shallow surveys, such as SDSS, is mainly sensitive to the parameters for
quenched galaxies. However, for deeper surveys such as DES and HSC, the star-
forming parameters have a stronger impact on the redshift distribution.
Specifically, the slope of the characteristic magnitude, $a_{\mathrm{M}}$, for
star-forming galaxies has overall the strongest impact on the redshift
distribution. Decreasing $a_{\mathrm{M}}$ by 148 per cent (its given
uncertainty) shifts the mean redshift by ${\sim}45$ per cent. We explore which
combination of colour and magnitude measurements are most sensitive to
$a_{\mathrm{M}}$ and we find that each colour-magnitude pair studied is
similarly affected by a modification of $a_{\mathrm{M}}$.

###### keywords:

galaxies: distances and redshifts – galaxies: luminosity function – software:
simulation

††pubyear: 2022††pagerange: The Sensitivity of the Redshift Distribution to
Galaxy Demographics–9

##  1 Introduction

The current standard model of cosmology, the $\Lambda$ cold dark matter
($\Lambda$CDM) model, describes a universe consisting of dark energy, cold
dark matter and baryonic matter evolving with time. At the present time, the
energy density of the Universe is mainly determined by the dark energy and the
dark matter components. However, these components are not fully understood yet
and have to be better constrained (e.g. Clifton et al., 2012; Capela et al.,
2013; Spergel, 2015).

With different cosmological probes, we can shed some light on the dark parts
of cosmology and constrain the parameters of dark energy and dark matter. Type
Ia supernovae, the cosmic microwave background (CMB) and weak gravitational
lensing (e.g. Albrecht et al., 2006; Nicola et al., 2017) are important
probes. For example, constraining the equation-of-state parameter is an
important objective of cosmic shear surveys. Most of these probes depend on
the accurate prediction of the redshift distribution of galaxies, $n(z)$.
Using spectroscopy is the most precise way to determine the redshifts of
galaxies. By identifying absorption and emission lines, one can calculate the
redshift of the corresponding object. Although, these so-called
‘spectroscopic’ redshifts are not a suitable choice for current and upcoming
experiments such as the Dark Energy Survey111https://www.darkenergysurvey.org;
the Hyper Suprime-Cam222https://hsc.mtk.nao.ac.jp/ssp/ of the Subaru
Telescope; the Sloan Digital Sky Survey333https://www.sdss.org; the Rubin
Observatory Legacy Survey of Space and Time444https://www.lsst.org (LSST) and
Euclid555https://sci.esa.int/web/euclid/. The big areas and depths observed by
these surveys make performing the required spectroscopy extremely time-
consuming and cost-intensive (Amara & Réfrégier, 2007; Abdalla et al., 2008).

As an example, the Dark Energy Survey (DES, The Dark Energy Survey
Collaboration, 2005) is a ground-based telescope observing 5000 deg2 of the
Southern Sky in the visible spectrum. It uses the Dark Energy Camera (DECam,
Honscheid & DePoy, 2008; Flaugher et al., 2015) to observe more than 300
million galaxies. The DES standard bandpasses are the DECam $g$, $r$, $i$, $z$
and $Y$ filters666http://www.ctio.noao.edu/noao/node/13140 spanning a
wavelength range from 400 to 1065 nm with a depth of $i=23.44$ (in Data
Release 1, Abbott et al., 2018).

Another example is the Hyper Suprime-Cam of the Subaru
Telescope777https://subarutelescope.org/en/ (HSC, Miyazaki et al., 2012). It
covers 1400 deg2 in the wide field, 27 deg2 in the deep field and 3.5 deg2 in
the ultra-deep field. HSC consists of five broad-band filters and four narrow-
band filters (Kawanomoto et al., 2018). The broad-band filters, $g$, $r$, $i$,
$z$ and $y$, cover a wavelength range of ${\sim}400$ nm to ${\sim}1100$ nm.
The $i$-band limiting magnitude is $26.2$ in the wide layer, $27.1$ in the
deep layer and $27.7$ in the ultra-deep layer (Aihara et al., 2018).

The Sloan Digital Sky Survey (SDSS, York et al., 2000) has a lower depth.
While taking data in the $u$, $g$, $r$, $i$ and $z$ filter bands888Note that
although we used the same notation for some of the DES, HSC and SDSS filter
bands, we mean the filters specific to the surveys. For example, the $g$-band
of DES, HSC and SDSS are different. the $i$-band limiting magnitude is $21.3$
(Stoughton et al., 2002). The covered area is 14555 deg2. Therefore, SDSS is
measuring a much larger part of the sky compared to DES and HSC. The filters
of SDSS cover a wavelength range of $300$ nm to $1100$
nm999https://www.sdss.org/instruments/camera/.

For such big surveys, photometry is an alternative to spectroscopy.
Photometric redshifts can be determined in two ways. First, one can fit the
spectral energy distribution (SED) in different broad-band filters. The
approach is to use different SED templates and fit them to the observed
fluxes. Through this process, one gains information about the galaxies’
redshifts (Bolzonella et al., 2000; Benitez, 2000). Alternatively, photometric
redshifts can be estimated by using machine learning methods (Carrasco Kind &
Brunner, 2013; Sadeh et al., 2016). This would require a sample of known
redshifts to train the algorithm that determines the redshifts of the
investigated sample. With template fitting and machine learning methods, we
are able to get redshift estimates even for deep surveys. The problem is, the
constraining power of cosmological probes that require redshift distributions
is decreased by the lack of precision in the photometric redshifts (Bordoloi
et al., 2010, 2012; Salvato et al., 2019).

To solve the problems of spectroscopic and photometric redshifts, Herbel et
al. (2017) developed a method to simulate the redshift distribution directly.
The authors used Approximate Bayesian Computation with a forward modelling
approach to determine a full posterior of $n(z)$ without measuring the
individual redshifts and which can be used in measurements. Thus, one does not
rely on the technically challenging spectroscopic redshifts for big surveys.
Furthermore, this approach delivers a more precise estimate of the redshift
distribution than photometry.

In this paper, our main goal is to study the sensitivity of the redshift
distribution of sets of galaxies to the underlying galaxy population
properties. The motivation is to better understand which features of galaxy
populations need to be well understood to make the next generation of
precision cosmological measurements.

To achieve our goal, we simulate galaxy catalogues solely with Herbel et al.
(2017)’s forward modelling approach based on redshift-dependent Schechter
luminosity functions, which describe the populations of different galaxy
types. We analyse how changes of the Schechter luminosity function impact the
redshift distribution for different survey types. Precisely, we identify the
model parameters (Schechter parameters) that affect the simulations of the
redshift distribution for the different survey types the most. Hence, this
work demonstrates which parameters need better constraints to improve the
precision of the simulated redshift distribution. This information will be
important for the analysis of next generation precision measurements.

Furthermore, we investigate how the Schechter parameters affect the galaxy
observables (apparent magnitude and colour distributions). As the magnitude
and colour distributions are directly measurable quantities, this helps us to
understand how the model parameters impact observations. In addition to this,
this analysis identifies the observables that are most sensitive to the
Schechter parameters and could help to constrain these parameters in the
future.

In our analysis, we considered two different galaxy populations, a star-
forming and a quenched model. Additionally, we chose the same parametric model
as Herbel et al. (2017) to assign spectra to the galaxies. These spectra are
used to calculate the apparent magnitudes of the galaxies. We simulated
observed galaxy catalogues for DES-, SDSS- and HSC-like surveys. The
catalogues contain the redshift values, absolute Johnson $B$-band magnitudes,
intrinsic101010Observational error not considered apparent magnitudes in
different filter bands, observed111111The word ”observed” indicates that the
magnitudes include an observational error. We describe the simulation of the
error in section 3.3.1. apparent magnitudes in different filter bands and the
errors of the apparent magnitudes.

To perform the simulations, we used SkyPy (SkyPy Collaboration, 2020; Amara et
al., 2021). SkyPy is a publicly available python package for simulating the
astrophysical sky and includes physical and empirical models for different
observables. Therefore, SkyPy can be used to perform end-to-end simulations of
these observables. We extended the code to account for observational errors
and to obtain the observed magnitudes. A detailed description of this is
summarised in section 3.

Furthermore, to account for signal-to-noise and saturation effects, we applied
survey specific magnitude cuts in the $i$-band that remove too bright and too
faint galaxies. We investigated the sensitivity of the redshift, colour and
magnitude distributions to the different Schechter parameters by repeating the
simulation after changing one parameter value by the range allowed by its
uncertainty. The resulting catalogue was used as our test catalogue and
compared to the fiducial catalogue121212Catalogue that was simulated with the
default set of Schechter luminosity function parameters. See sections 3 and
4..

This paper is structured as follows. In section 2, we summarise the
mathematical models of our simulations. Section 3 presents our simulation set-
up including the fiducial Schechter parameters, a detailed description of our
error model and how we perform the magnitude cuts. In section 4, we describe
the methodology of our simulations and the comparison of fiducial and test
catalogue. Our results are presented in section 5 and section 6 includes our
conclusions. Throughout this work, we use a standard $\Lambda$CDM cosmology
with $h=0.7$, $\Omega_{\mathrm{m}}=0.3$ and $\Omega_{\Lambda}=0.7$.

##  2 Forward Modelling Galaxy Catalogues

In this section, we describe our model for simulating galaxy catalogues
including redshifts, magnitudes and colours. Our simulations follow a forward
modelling approach to draw galaxies from the Schechter luminosity function, as
described by Herbel et al. (2017). We assign the redshifts and absolute
$B$-band magnitudes that follow the Schechter luminosity function to our
objects. With these quantities, we calculate the apparent magnitudes in
different filter bands by simulating the spectra that are based on template
coefficients. A deeper investigation and motivation of this model will be the
scope for future work.

###  2.1 Luminosity Function and Galaxy Numbers

Generating intrinsic galaxy catalogues requires a model of the expected galaxy
number in a certain volume of the sky. The luminosity function $\Phi$
describes the number of galaxies $N$ per comoving volume $V$ and absolute
magnitude $M$ as a function of redshift $z$,

$\Phi(z,M)=\frac{\mathrm{d}N}{\mathrm{d}V\,\mathrm{d}M}\;.$ (1)

Different observations (Loveday et al., 2012; López-Sanjuan et al., 2017) have
supported the work from Schechter (1976) that described the functional form of
the luminosity function. The Schechter luminosity function is given as

$\begin{split}\Phi(z,M)=\;&0.4\,\ln(10)\,\Phi_{*}(z)\,10^{0.4(M_{*}(z)-M)(\alpha(z)+1)}\\\
&\cdot\exp\left(-10^{0.4(M_{*}(z)-M)}\right)\>,\end{split}$ (2)

where the faint-end-slope $\alpha$, the characteristic magnitude $M_{*}$ and
the amplitude $\Phi_{*}$ are free-fitting parameters. These parameters depend
on the type of the galaxy population and on the redshift.

The evolution of the luminosity function with redshift can be empirically
motivated (Herbel et al., 2017). Keeping $\alpha$ constant for each galaxy
population, the characteristic magnitude and the amplitude are parametrised as

$\displaystyle M_{*}(z)$ $\displaystyle=a_{\mathrm{M}}\,z+b_{\mathrm{M}}\,,$
(3) $\displaystyle\Phi_{*}(z)$ $\displaystyle=b_{\phi}\exp(a_{\phi}z)\;.$ (4)

Combined each population is described by five parameters. However, we assume
that each type evolves according to the same functional form. In this paper,
we refer to the parameters $a_{\mathrm{M}}$, $b_{\mathrm{M}}$, $a_{\phi}$ and
$b_{\phi}$ as the “Schechter parameters” as these are the model parameters we
investigated.

Considering the volume element as a light cone, we can write the comoving
element in terms of the solid angle $\Omega$ and redshift $z$ as

$\mathrm{d}V=\frac{d_{\mathrm{H}}d_{\mathrm{M}}^{2}}{E(z)}\mathrm{d}\Omega\,\mathrm{d}z\;,$
(5)

where $d_{\mathrm{H}}$ is the Hubble distance, $d_{M}$ is the transverse
comoving distance and
$E(z)=(\Omega_{\mathrm{m}}(1+z)^{3}+\Omega_{\mathrm{k}}(1+z)^{2}+\Omega_{\Lambda})^{0.5}$.
Plugging this into equation 1, we get the number of galaxies per absolute
magnitude, redshift and solid angle

$\phi(z,M)=\frac{\mathrm{d}N}{\mathrm{d}M\,\mathrm{d}\Omega\,\mathrm{d}z}=\frac{d_{\mathrm{H}}d_{\mathrm{M}}^{2}}{E(z)}\Phi(z,M)\;.$
(6)

This function is the number density which can be used to draw samples of
galaxies. These samples contain redshift values and absolute magnitudes that
follow the distribution $\phi$. Additionally, by integrating equation 6, one
obtains the number of galaxies that would be observed in a light cone, spanned
by solid angle $\Omega$ with redshifts between $z_{1}$ and $z_{2}$ and
magnitudes between $M_{1}$ and $M_{2}$,

$N(z_{1},z_{2},M_{1},M_{2},\Omega)=\Omega\int_{M_{1}}^{M_{2}}\int_{z_{1}}^{z_{2}}\phi(z,M)\,\mathrm{d}z\,\mathrm{d}M\;.$
(7)

###  2.2 Galaxy Colours

The colour of a galaxy is defined as the difference in its magnitudes in two
different filter bands. Given the spectrum $f(\lambda)$ of a galaxy,
calculating the magnitude in a particular band is straightforward.

First, the absolute magnitude in this band has to be calculated. In the AB
magnitude system, this is given as (Blanton et al., 2003)

$M_{i}=-2.41-2.5\log_{10}\left[\frac{\int_{0}^{\infty}\mathrm{d}\lambda_{\mathrm{o}}\lambda
f(\lambda)R(\lambda_{\mathrm{o}})}{\int_{0}^{\infty}\mathrm{d}\lambda_{\mathrm{o}}\lambda^{-1}R(\lambda_{\mathrm{o}})}\right]\,,$
(8)

where $R(\lambda)$ is the filter response giving the fraction of photons that
are included in the signal. The subscript ‘o’ indicates the observed frame and
the observed and emitted wavelengths are related through the redshift relation
$\lambda_{o}=(1+z)\lambda$. The units of the spectrum $f(\lambda)$ are erg s-1
cm-2 Å-1 in this equation. To transform this into an apparent magnitude, we
employ the distance modulus,
$\mathrm{DM}(z)=5\log_{10}(d_{\mathrm{L}}(z)/10\,\mathrm{pc})$, where
$d_{\mathrm{L}}$ is the luminosity distance. Thus, the apparent magnitude in
band $i$ is given by

$m_{i}=M_{j}+\mathrm{DM}(z)+K_{ji}(z)\;,$ (9)

with $K_{ji}(z)$ accounting for the $K$ correction (Hogg et al., 2002; Blanton
& Roweis, 2007), which is relating the emitted rest-frame magnitude in broad
photometric bandpass j to the observed apparent magnitude in band i.

###  2.3 Galaxy Spectra

Simulating the magnitudes of the galaxies requires the galaxies’ spectra. To
model the spectrum, we again follow the approach of Herbel et al. (2017),
which is based on the work by Blanton & Roweis (2007). The flux density as it
would be seen if the galaxy is at a distance of 10 pc is given as a linear
combination of five template spectra $t_{i}(\lambda)$,

$f(\lambda)=\sum_{i}c_{i}t_{i}(\lambda)\,,$ (10)

where the units of $f(\lambda)$ are erg s-1 cm-2 Å-1 per solar mass. Herbel et
al. (2017) showed that after re-weighting the coefficients $c_{i}$ by a weight
$w_{i}$, the coefficients follow a Dirichlet distribution of order five so
that $\sum_{i}\tilde{c}_{i}=1$ with $\tilde{c}_{i}=c_{i}/w_{i}$. A Dirichlet
distribution (Kotz et al., 2019) of order $K\geq 2$ is defined as

$\mathrm{Dir}(x_{1},...,x_{K};\alpha_{1},...,\alpha_{K})=\frac{1}{\mathrm{B}(\bm{\alpha})}\,\prod_{i=1}^{K}x_{i}^{\alpha_{i}-1}\;,$
(11)

where $\bm{\alpha}=(\alpha_{1},...,\alpha_{K})$ with $\alpha_{i}>0$ are
parameters characterising the Dirichlet distribution and
$\mathrm{B}(\bm{\alpha})$ is the multivariate beta function.

Therefore a Dirichlet distribution of order five is characterised by five
parameters $\alpha_{i}$. These parameters are redshift dependent in general.
Once again according to Herbel et al. (2017), we assume that the evolution of
the parameters is described as

$\alpha_{i}(z)=(\alpha_{i,0})^{1-z/z_{1}}\cdot(\alpha_{i,1})^{z/z_{1}}\;.$
(12)

The two parameters $\alpha_{i,0}$ and $\alpha_{i,1}$ describe the galaxy
population at redshift $z=0$ and $z=z_{1}>0$, respectively. They are also
different for each galaxy type.

Note that the K-correct templates $t_{i}(\lambda)$ are intended to fit broad-
band photometry, which is suitable for our case. In this work, we decided to
build on the studies presented in Herbel et al. (2017), Fagioli et al. (2018);
Fagioli et al. (2020) and Tortorelli et al. (2018); Tortorelli et al. (2020);
Tortorelli et al. (2021), which show that the K-correct templates can be used
in our analysis, especially at low redshifts. We are also revisiting the
quality of the templates for higher redshifts in a new study (Hartley et al.
in prep), which will show that this model predicts colour and apparent
magnitude distributions even for samples with higher redshifts.

##  3 Simulation Setup

In this section, we describe the setup of our simulations. We summarise how
the fiducial catalogues of each survey were generated based on the model
described in section 2. That includes the Schechter parameters, the Dirichlet
parameters for generating the spectra, the simulated sky area and filter bands
of each survey type.

Furthermore, we describe how the intrinsic catalogues are transformed into
observed galaxy catalogues for each survey. Additionally, we explain our
choice of magnitude cuts to mock real observations.

###  3.1 Schechter Parameters

In our work, we used the parameters from Tortorelli et al. (2021) following a
model consisting of star-forming and quenched galaxies. Table 1 summarises the
values of the Schechter parameters together with their errors.

As aforementioned, each galaxy population can be simulated by drawing from the
same distribution with different parameters. The redshift dependency of our
model is given by the faint-end slope $\alpha$, the characteristic magnitude
$M_{*}$ and the amplitude of the Schechter function $\Phi_{*}$. In equations 3
and 4, we parametrise $M_{*}$ as a linear function and $\Phi_{*}$ as an
exponential function of redshift $z$. Therefore, our model is described by
five parameters, $\alpha$, $a_{\mathrm{M}}$, $b_{\mathrm{M}}$, $a_{\phi}$ and
$b_{\phi}$.

In our investigation, $\alpha$ was considered redshift independent such that
only four parameters characterised the evolution with redshift. We focused on
this set of parameters and investigated the sensitivity of the redshift
distribution to a change in these parameters for the different surveys. Since
the parameters have different values according to the galaxy type, we had one
set of four parameters for the star-forming galaxies and another set of four
parameters for the quenched galaxies.

Table 1: Schechter Parameters for our simulations. Values and errors are taken from Tortorelli et al. (2021). The positive and negative errors correspond to the 84th percentile and 50th percentile values, respectively. Parameter | Star-Forming | Quenched
---|---|---
$\alpha$ | $-1.3$ | $-0.5$
$a_{\mathrm{M}}$ | $-0.439^{+0.535}_{-0.652}$ | $-0.697^{+0.698}_{-0.729}$
$b_{\mathrm{M}}$ | $-20.623^{+0.417}_{-0.425}$ | $-20.372^{+0.513}_{-0.466}$
$a_{\phi}$ | $-0.088^{+0.297}_{-0.277}$ | $-0.836^{+0.812}_{-0.733}$
$b_{\phi}$ | $0.004245^{+0.001337}_{-0.001452}$ | $0.005169^{+0.003112}_{-0.003515}$

###  3.2 Dirichlet Parameters

Table 2 summarises the Dirichlet parameters from Tortorelli et al. (2018) and
the weights from Herbel et al. (2017).

As we explained in section 2, we simulated the spectra based on template
coefficients that are drawn from a Dirichlet distribution. We considered a
model independent of redshift such that $\alpha_{i}(z)$ from equation 12 is a
constant (just denoted as $\alpha_{i}$ from here on).

To account for the different galaxy types, we also introduced weights for each
coefficient (more details in Herbel et al. (2017)). Through the distinction
between star-forming and quenched galaxies, we had a total of ten parameters
and their corresponding weights, respectively.

Table 2: Dirichlet parameters $\alpha_{i}$ and weights $w_{i}$ for our simulations. The Dirichlet parameters are taken from Tortorelli et al. (2018) and the weights are from Herbel et al. (2017). Parameter | Star-Forming | Quenched
---|---|---
$\alpha_{1}$ | $1.9946549$ | $1.62158197$
$\alpha_{2}$ | $1.99469164$ | $1.62137391$
$\alpha_{3}$ | $1.99461187$ | $1.62175061$
$\alpha_{4}$ | $1.9946589$ | $1.62159144$
$\alpha_{5}$ | $1.99463069$ | $1.62165971$
$w_{1}$ | $3.47\cdot 10^{9}$ | $3.84\cdot 10^{9}$
$w_{2}$ | $3.31\cdot 10^{6}$ | $1.57\cdot 10^{6}$
$w_{3}$ | $2.13\cdot 10^{9}$ | $3.91\cdot 10^{8}$
$w_{4}$ | $1.64\cdot 10^{10}$ | $4.66\cdot 10^{10}$
$w_{5}$ | $1.01\cdot 10^{9}$ | $3.03\cdot 10^{7}$

###  3.3 Surveys and Observed Catalogues

We identified the Schechter parameters that have the biggest impact on the
simulated redshift distribution of different surveys. Within the context of
the Herbel et al. (2017) forward modelling approach, having more precise
parameter values directly translates into better cosmological constraints that
are based on the redshift distribution.

Moreover, we analysed the effect of the Schechter parameters on three survey
types, based on their depths and fields of view. We investigated a DES-like
survey, a deeper but narrower HSC deep-field-like survey and a shallower but
wider SDSS-like survey. These surveys have different sets of filter bands,
which we needed to reflect in the simulations.

With the SkyPy python package, we applied these filters directly. We chose to
simulate the DES-like survey with the DECam $g$, $r$, $i$ and $z$ filters. In
the case of HSC, we used the $g$, $r$, $i$, $z$ and $y$ filters. To simulate
the SDSS-like catalogue, we used the SDSS $u$, $g$, $r$, $i$ and $z$ filters.

Since the leading order effect of precision measurements is usually the number
of galaxies, we simulated different fields of view for each survey. Thus, we
had approximately the same number of galaxies in each fiducial catalogue and
could explore the trends beyond pure number counts. We chose a simulated area
of 10 deg2 for DES, 1.5 deg2 in the case of HSC and 135 deg2 for SDSS. That
resulted in ${\sim}58500$ galaxies in the DES-like fiducial observed
catalogue, ${\sim}59300$ in the SDSS-like catalogue and ${\sim}58400$ galaxies
in the HSC-like catalogue. A summary of the filter bands, sky area and the
resulting number of galaxies of our fiducial catalogues is given in table 3.

Table 3: Summary of the investigated surveys including the simulated filter bands, sky area and the resulting number of galaxies of each fiducial catalogue. Survey | Filter Bands | Sky Area | Number of Galaxies
---|---|---|---
DES |  $g$, $r$, $i$, $z$ | 10 deg2 | ${\sim}58500$
SDSS |  $u$ ,$g$, $r$, $i$, $z$ | 135 deg2 | ${\sim}59300$
HSC |  $g$, $r$, $i$, $z$, $y$ | 1.75 deg2 | ${\sim}58400$

To make the intrinsic catalogues of the surveys observed catalogues, we
performed two additional steps in our simulations:

  1. 1.

Adding survey specific errors to the simulated intrinsic magnitudes

  2. 2.

Applying survey specific magnitude cuts

####  3.3.1 Modelling Magnitude Uncertainties

In sections 3.1 and 3.2, we described the setup to simulate intrinsic
catalogues. However, making realistic simulations of galaxy catalogues implies
modelling the magnitude uncertainties appropriately. These uncertainties must
be specific to the survey. We used Rykoff et al. (2015)’s model to simulate
the uncertainty, $\sigma_{m}$, of the apparent magnitudes in each filter band
for every survey,

$\sigma_{m}(F;F_{\mathrm{noise}},t_{\mathrm{eff}})=\frac{2.5}{\ln{10}}\left[\frac{1}{Ft_{\mathrm{eff}}}\left(1+\frac{F_{\mathrm{noise}}}{F}\right)\right]^{1/2}\;,$
(13)

where

$F=10^{-0.4(m-m_{\mathrm{ZP}})}$ (14)

is the galaxy’s flux,

$F_{\mathrm{noise}}=\frac{F_{\mathrm{lim}}^{2}t_{\mathrm{eff}}}{10^{2}}-F_{\mathrm{lim}}$
(15)

is the effective noise flux and $t_{\mathrm{eff}}$ is the effective exposure
time131313Note that we absorbed the normalisation constant $k$, which is
mentioned in Rykoff et al. (2015), in the definition of $t_{\mathrm{eff}}$..
Furthermore, $m$ is the galaxy’s magnitude, $m_{\mathrm{ZP}}$ is the zero-
point magnitude of the filter band and $F_{\mathrm{lim}}$ is the $10\sigma$
limiting flux.

Equations 13, 14 and 15 then define the uncertainty of the magnitude,
$\sigma_{m}(m;m_{\mathrm{lim}},t_{\mathrm{eff}})$, depending on magnitude $m$,
limiting magnitude $m_{\mathrm{lim}}$ (magnitude associated with
$F_{\mathrm{lim}}$) and effective exposure time $t_{\mathrm{eff}}$. We also
used the same model as Rykoff et al. (2015) for the effective exposure time,

$\ln{t_{\mathrm{eff}}}=a+b(m_{\mathrm{lim}}-21)\;,$ (16)

where $a$ and $b$ are free parameters.

We fitted $\sigma_{m}(m)$ (i.e. fitting $a$ and $b$) to measured data. In this
way, we got the magnitude uncertainties specific to the survey. For the zero-
point magnitudes $m_{\mathrm{ZP}}$, we used a value of 30 for each filter band
and every survey. That is justified because this model is only an
approximation and has initially been developed for SDSS data only. The model
is empirically motivated and does not include any deeper analysis of signal-
to-noise effects that exist during observations.

Figures 1, 2 and 3 show the fitted magnitude uncertainties in each filter band
and compare them to the data for every survey type, respectively. We see that
the fits agree very well with the data in the regions of high galaxy density
for each survey type. In the case of DES- and HSC-like surveys, in figures 1
and 3, we observe that there are minor deviations of model and data in the
region of small magnitudes and errors. Since the error values are very small
compared to the magnitude values in this region, we do not expect these
deviations to impact our simulations and analysis majorly. In the case of
SDSS-like surveys in figure 2, the fits are less good compared to DES- and
HSC-like surveys. However, we used the results presented in Rykoff et al.
(2015) and did not perform our own fits. The largest deviations of fit and
data are again in the region of small error such that they do not impact our
analysis majorly. Further, please note that our goal was not to simulate the
magnitude-error correlation perfectly, but to have a model that describes the
relationship sufficiently. In the plots, we can see that we achieved this goal
although the given limiting magnitudes of SDSS and HSC do not correspond to
$10\sigma$ magnitude limits.

Tables 4, 5 and 6 summarise the fit results and the used magnitude limits. As
mentioned, we used the fit results from Rykoff et al. (2015) in the case of
the SDSS-like survey. For the DES-like survey, we performed the fit to the Y1
results and used the corresponding limiting magnitudes as described by Drlica-
Wagner et al. (2018). For SDSS, we compared the fit to DR16 data (Ahumada et
al., 2020) and used the limiting magnitudes described in Stoughton et al.
(2002). Regarding HSC, we fitted DR2 data and used the magnitude limits of the
deep survey described in Aihara et al. (2019).

The modelled uncertainties were used to assign an error to the intrinsic
magnitudes. We drew Gaussian random variables with mean zero and the standard
deviations being the modelled uncertainties. We added the random values to the
intrinsic magnitudes and generated the observed catalogue.

Table 4: Fit results of the magnitude uncertainty and used magnitudes limits of each filter band for a DES-like survey. The parameters $a$ and $b$ describe the effective exposure time in equation 16. The fit was performed to DES Y1 (Drlica-Wagner et al., 2018) data and the corresponding magnitudes limits were taken from Drlica-Wagner et al. (2018). Filter Band | a | b | $m_{\mathrm{lim}}$
---|---|---|---
$g$ | 1.541275 | -1.000737 | 23.4
$r$ | -0.743079 | -0.116539 | 23.2
$i$ | 0.846085 | -1.660630 | 22.5
$z$ | -0.044378 | -2.774705 | 21.8
Table 5: Fit results of the magnitude uncertainty and used magnitudes limits of each filter band for an SDSS-like survey. The parameters $a$ and $b$ describe the effective exposure time in equation 16. The fit results were taken from Rykoff et al. (2015) and the corresponding magnitudes limits were taken from Stoughton et al. (2002). Filter Band | a | b | $m_{\mathrm{lim}}$
---|---|---|---
$u$ | 3.41 | 1.15 | 22
$g$ | 4.27 | 0.85 | 22.2
$r$ | 4.53 | 0.91 | 22.2
$i$ | 4.56 | 1.00 | 21.3
$z$ | 4.39 | 1.34 | 20.5
Table 6: Fit results of the magnitude uncertainty and used magnitudes limits of each filter band for an HSC-like survey. The parameters $a$ and $b$ describe the effective exposure time in equation 16. The fit was performed to HSC DR2 (Aihara et al., 2019) data and the corresponding magnitudes limits were taken from Aihara et al. (2019). Filter Band | a | b | $m_{\mathrm{lim}}$
---|---|---|---
$g$ | 135.095407 | 20.425863 | 27.8
$r$ | -2.270331 | 0.924841 | 27.4
$i$ | -0.272460 | 0.563884 | 27.1
$z$ | -0.707218 | 0.618222 | 26.6
$y$ | 61.117607 | 13.634286 | 25.6

Figure 1: Magnitude uncertainties of a DES-like survey. The binned data points

show the DES Y1 measurements (Drlica-Wagner et al., 2018) of the apparent
magnitudes and its corresponding uncertainties, $\sigma_{m}(m)$, in the $g$,
$r$, $i$ and $z$ filter bands. The green lines show the respective fits to the
Rykoff et al. (2015) error model. We see that model and data generally match.

Figure 2: Magnitude uncertainties of an SDSS-like survey. The binned data
points show the SDSS DR16 measurements (Ahumada et al., 2020) of the apparent
magnitudes and its corresponding uncertainties, $\sigma_{m}(m)$, in the $u$,
$g$, $r$, $i$ and $z$ filter bands. The green lines show the respective fits
to the Rykoff et al. (2015) error model. We see that model and data generally
match for most of the data points. Figure 3: Magnitude uncertainties of a HSC-
like survey. The binned data points show the HSC DR2 measurements (Aihara et
al., 2019) of the apparent magnitudes and its corresponding uncertainties,
$\sigma_{m}(m)$, in the $g$, $r$, $i$, $z$ and $y$ filter bands. The green
lines show the respective fits to the Rykoff et al. (2015) error model. We see
that model and data generally match .

####  3.3.2 Magnitude Cuts

To generate mock observed catalogues, we needed to account for further
observational effects. Besides environmental factors like weather, which we
did not include, signal-to-noise effects had to be addressed in our
simulations. Taking observations, only measurements that exceed a certain
signal-to-noise threshold are kept, which guarantees the quality of the data.
Fainter sources are more affected than brighter objects because the noise does
not linearly scale with the brightness. The signal-to-noise ratio defines how
deep a survey is and sets a lower limit on the brightness of an object to be
observed.

However, signal-to-noise selection is a complex process and mocking it is not
straightforward. Simulating realistic signal-to-noise for galaxies needs to
account for large sets of galaxy properties (such as luminosity, size, profile
and ellipticity) and how these correlate with the point-spread-function
properties (size and shape) at the positions of galaxies (Bergé et al., 2013;
Herbel et al., 2017). This means that we would need to implement realistic
image simulations to fully account for signal-to-noise selection effects,
which exceeds the purpose of our paper. Therefore, we have focused on a fixed
magnitude cut where the sample should be complete.

In our work, we chose a cut in the $i$-band. That means all objects whose
$i$-band magnitude is fainter than a certain cut-off point
$i_{\mathrm{upper}}$ were removed. As the investigated surveys have varying
depths, the respective upper-cut magnitudes were different. In the case of a
DES-like survey, we chose $i_{\mathrm{upper,DES}}=24$ and for an HSC-like
survey, we used $i_{\mathrm{upper,HSC}}=27$. In the case of SDSS,
$i_{\mathrm{upper,SDSS}}$ were 21.

The saturation of pixels is another crucial effect that occurs during
observations. If a source is too bright, too many photons are detected by the
CCD and pixels start to saturate. To account for this, we introduced a
magnitude cut for bright objects. Again, we used a cut in the $i$-band.
However, the cut was the same for each survey. We removed all objects with
$i$-band magnitudes brighter than $i_{\mathrm{lower}}=18$. Table 7 summarises
all cuts that were used in our work.

Table 7: The upper and lower magnitude limits that were used in our analysis. Only objects with $i$-band magnitudes between $i_{\mathrm{lower}}$ and $i_{\mathrm{upper}}$ were kept in the catalogues. Survey | $i_{\mathrm{upper}}$ | $i_{\mathrm{lower}}$
---|---|---
DES | 24 | 18
SDSS | 21 | 18
HSC | 27 | 18

##  4 Method

In this section, we describe the method that we used to investigate the
sensitivity of the redshift distribution to the Schechter parameters and the
impact of those parameters to the observables.

First, we simulated the intrinsic fiducial galaxy catalogues for all surveys.
To do this, we used the SkyPy Python package, which enables us to perform the
simulations according to the model described in section 2. The intrinsic
catalogues contained redshift values, absolute Johnson B-band magnitudes and
apparent magnitudes in different filter bands specific to the survey. We
summarised the fiducial values of the parameters in sections 3.1 and 3.2.

Afterwards, the intrinsic catalogues were transformed into observed
catalogues. We added an observational error to the apparent magnitudes and
performed magnitude cuts to remove too faint and too bright objects. We refer
to section 3.3 for further details.

To analyse the sensitivity of the surveys to the different Schechter
parameters, we changed the value of one of the parameters by adding and
subtracting its error and repeated the whole simulation process as described
in the previous paragraphs. The positive and negative errors correspond to the
84th and 50th percentile values, respectively. Note, the sensitivity is
relative to the ability of modern datasets to constrain the parameters (see
table 2). Modifying the parameter changed the luminosity function and the
simulation resulted in a galaxy catalogue with different redshift, magnitude
and colour distributions. We call this catalogue the “test” catalogue.
Remember that the sample is defined by the Schechter parameters and the
Dirichlet coefficients. We concentrated on the effect of the Schechter
parameters on the redshift distribution. The correlation between the
parameters and coefficients, if there is one, has not been measured in the
literature and determining this correlation is beyond the scope of our paper.

Comparing the redshift distributions of fiducial and test catalogue allowed us
to identify the sensitivity of the survey’s redshift distribution to the
modified parameter. By comparing the colour and magnitude distributions of
fiducial and test catalogue, we diagnosed different observables (magnitude or
colour distribution) that are sensitive to a change in the parameter value
and, thus, could be used to constrain the parameters.

In this context, “sensitive to a Schechter parameter” means that the redshift
distribution of the galaxy catalogue changes after modifying the parameter.
Furthermore, a greater change indicates a higher sensitivity of this survey to
the parameter. Accordingly, an observable has better constraining ability if
it is more affected by the parameter change.

###  4.1 Investigating Redshift Sensitivity

Since the mean of the redshift distribution is the leading order term in weak
lensing analyses (Tessore & Harrison, 2020), we concentrated on the change of
the mean redshift. Comparing the mean redshift of the fiducial and the test
redshift distributions is beneficial in several ways. First, one can
investigate the absolute change that informs about the required precision of
the parameter. Second, the relative change reveals the parameter that has the
strongest impact on the redshift distribution.

We define the relative change as

$\delta z_{\mathrm{mean}}=\frac{|\Delta
z_{\mathrm{mean}}|}{\overline{z}_{\mathrm{fid}}}\;,$ (17)

where $\Delta
z_{\mathrm{mean}}=\overline{z}_{\mathrm{test}}-\overline{z}_{\mathrm{fid}}$ is
the absolute change of the mean redshift, $\overline{z}_{\mathrm{test}}$ is
the mean redshift of the test catalogue and $\overline{z}_{\mathrm{fid}}$ is
the mean redshift of the fiducial catalogue.

As the simulations are affected by statistical fluctuations, we made sure that
the change of the mean redshift did not arise from statistical effects. To
account for this, we generated 100 realisations of the fiducial and test
catalogues. We then compared the mean redshifts by looking at the mean of
$\delta z_{\mathrm{mean}}$ and $\Delta z_{\mathrm{mean}}$ over the 100
simulations. Hence, $\overline{\delta z}_{\mathrm{mean}}$ and
$\overline{\Delta z}_{\mathrm{mean}}$ were more appropriate to quantify the
change in the redshift distribution.

###  4.2 Important Observables

After identifying the parameters with the strongest impact on the redshift
distribution, we focused on how the Schechter parameters affect the
observables. Thus, we could find the observables that are most sensitive to
the parameters and that could help constraining these parameters.

In our analysis, we compared the magnitude distributions and colour
distributions of fiducial and test catalogues by using the Anderson-Darling
(AD) test (Scholz & Stephens, 1987; Feigelson & Babu, 2012). For the test, we
used the assumption that the two tested samples are drawn from the same
distribution as the null hypothesis. The AD test is a statistical test
measuring the sum of the squared differences of the distributions,
$d_{\mathrm{AD}}$. The sensitivity of the corresponding observable to the
parameter increases with growing value of $d_{\mathrm{AD}}$. The relating
p-value describes the probability to obtain a test statistic $d_{\mathrm{AD}}$
at least as extreme as measured under the assumption that the null hypothesis
is true.

As described in section 4.1, we ran 100 simulations to account for numerical
noise. We calculated the 95-th percentiles of the 100 d-values,
$d_{\mathrm{AD},95}$, and p-values, $p_{\mathrm{AD},95}$, for each observable.
These values then indicated the change of the observable after changing the
Schechter parameter. Furthermore, we looked for large values of
$d_{\mathrm{AD},95}$ and small values of $p_{\mathrm{AD},95}$.

We repeated the whole process of simulating the test catalogues and comparing
them to the fiducial catalogues for all eight Schechter parameters and all
surveys. We performed the analysis after changing each parameter value by its
given positive and negative error. That enabled us to identify which survey is
sensitive to which parameters. In addition to this, we could determine which
observables and which surveys show higher sensitivity to the specific
parameters and can, therefore, be more useful in constraining the parameters.

##  5 Results

This section describes the results of our analysis. We start with identifying
the parameters that have the strongest effect on the redshift distribution of
each survey in section 5.1. We finalise with the discussion about the impact
of the Schechter parameters on the different observables in section 5.2.

###  5.1 Redshift Sensitivity

As described in section 4, we compared the redshift distributions of fiducial
and test catalogue using sets of 100 simulations. The mean of the relative
change, $\overline{\delta z}_{\mathrm{mean}}$, and the mean of the absolute
change, $\overline{\Delta z}_{\mathrm{mean}}$, of the mean redshift indicated
the parameters with the highest impact on each survey. Tables 8, 9 and 10
summarise our results of the relative and absolute mean redshift change. They
show the impact on the redshift distribution after updating each parameter in
positive and negative directions.

Table 8: Relative change, $\overline{\delta z}_{\mathrm{mean}}$, and absolute change, $\overline{\Delta z}_{\mathrm{mean}}$, of the mean redshift for a DES-like survey. The values correspond to the mean of 100 realisations. They indicate the change in the redshift distribution after increasing (Positive) and decreasing (Negative) the corresponding parameter by its given errors. Higher values of $\overline{\delta z}_{\mathrm{mean}}$ indicate stronger sensitivity to the corresponding parameter. $\overline{\Delta z}_{\mathrm{mean}}$ is a measure for the required precision of the mean redshift. Parameter | Positive | Negative
---|---|---
|  $\overline{\delta z}_{\mathrm{mean}}$ in % | $\overline{\Delta z}_{\mathrm{mean}}$ |  $\overline{\delta z}_{\mathrm{mean}}$ in % | $\overline{\Delta z}_{\mathrm{mean}}$
$a_{\mathrm{M,SF}}$ | 10.5 | -0.071 | 44.9 | 0.304
$a_{\mathrm{M,Q}}$ | 3.8 | -0.026 | 13.5 | 0.091
$a_{\phi,\mathrm{SF}}$ | 4.9 | 0.033 | 3.6 | -0.024
$a_{\phi,\mathrm{Q}}$ | 8.7 | 0.059 | 2.8 | -0.019
$b_{\mathrm{M,SF}}$ | 5.5 | -0.037 | 7.7 | 0.053
$b_{\mathrm{M,Q}}$ | 2.7 | -0.018 | 3.5 | 0.024
$b_{\phi,\mathrm{SF}}$ | 0.5 | -0.004 | 1.0 | 0.007
$b_{\phi,\mathrm{Q}}$ | 1.1 | 0.008 | 1.7 | -0.012

Table 8 summarises the sensitivity of DES-like surveys. We found that a DES-
like survey is most sensitive to $a_{\mathrm{M,SF}}$. Decreasing
$a_{\mathrm{M,SF}}$ increases the mean redshift by ${\sim}45$ per cent. This
relative change corresponds to an absolute change of 0.304. The rest of the
parameters have less impact on the mean redshift. However, $a_{\mathrm{M,Q}}$,
$a_{\phi,\mathrm{Q}}$ and $b_{\mathrm{M,SF}}$ can also have a significant
effect on the redshift distribution. Updating these parameters causes relative
changes of the mean redshift up to 13.5 per cent, which corresponds to an
absolute change of 0.091.

Table 9: Relative change, $\overline{\delta z}_{\mathrm{mean}}$, and absolute change, $\overline{\Delta z}_{\mathrm{mean}}$, of the mean redshift for an HSC-like survey. The values correspond to the mean of 100 realisations. They indicate the change in the redshift distribution after increasing (Positive) and decreasing (Negative) the corresponding parameter by its given errors. Higher values of $\overline{\delta z}_{\mathrm{mean}}$ indicate stronger sensitivity to the corresponding parameter. $\overline{\Delta z}_{\mathrm{mean}}$ is a measure for the required precision of the mean redshift. Parameter | Positive | Negative
---|---|---
|  $\overline{\delta z}_{\mathrm{mean}}$ in % | $\overline{\Delta z}_{\mathrm{mean}}$ |  $\overline{\delta z}_{\mathrm{mean}}$ in % | $\overline{\Delta z}_{\mathrm{mean}}$
$a_{\mathrm{M,SF}}$ | 17.63 | -0.2053 | 22.69 | 0.2643
$a_{\mathrm{M,Q}}$ | 1.50 | -0.0175 | 1.74 | 0.0203
$a_{\phi,\mathrm{SF}}$ | 12.37 | 0.1441 | 9.29 | -0.1082
$a_{\phi,\mathrm{Q}}$ | 8.68 | 0.1011 | 1.18 | -0.0138
$b_{\mathrm{M,SF}}$ | 4.19 | -0.0488 | 3.84 | 0.0447
$b_{\mathrm{M,Q}}$ | 0.55 | -0.0064 | 0.51 | 0.0059
$b_{\phi,\mathrm{SF}}$ | 0.05 | -0.0006 | 0.06 | 0.0007
$b_{\phi,\mathrm{Q}}$ | 0.06 | 0.0007 | 0.13 | -0.0015

Table 9 shows the results for an HSC-like survey. Again, $a_{\mathrm{M,SF}}$
has the strongest impact on the redshift distribution. The mean redshift of an
HSC-like survey changes by ${\sim}23$ per cent after decreasing
$a_{\mathrm{M,SF}}$ by its negative error. Therefore, the uncertainty of
$a_{\mathrm{M,SF}}$ shifts the simulated mean redshift by up to ${\sim}0.26$.
Furthermore, we found that the parameters $a_{\phi,\mathrm{SF}}$ and
$a_{\phi,\mathrm{Q}}$ also have strong impact on the mean redshift. Increasing
these parameters causes relative changes of ${\sim}12.4$ per cent and
${\sim}8.7$ per cent, which corresponds to absolute changes of about 0.14 and
0.10, respectively.

Table 10: Relative change, $\overline{\delta z}_{\mathrm{mean}}$, and absolute change, $\overline{\Delta z}_{\mathrm{mean}}$, of the mean redshift for an SDSS-like survey. The values correspond to the mean of 100 realisations. They indicate the change in the redshift distribution after increasing (Positive) and decreasing (Negative) the corresponding parameter by its given errors. Higher values of $\overline{\delta z}_{\mathrm{mean}}$ indicate stronger sensitivity to the corresponding parameter. $\overline{\Delta z}_{\mathrm{mean}}$ is a measure for the required precision of the mean redshift. Parameter | Positive | Negative
---|---|---
|  $\overline{\delta z}_{\mathrm{mean}}$ in % | $\overline{\Delta z}_{\mathrm{mean}}$ |  $\overline{\delta z}_{\mathrm{mean}}$ in % | $\overline{\Delta z}_{\mathrm{mean}}$
$a_{\mathrm{M,SF}}$ | 3.8 | -0.013 | 9.1 | 0.031
$a_{\mathrm{M,Q}}$ | 5.8 | -0.020 | 10.5 | 0.036
$a_{\phi,\mathrm{SF}}$ | 0.9 | 0.003 | 0.7 | -0.002
$a_{\phi,\mathrm{Q}}$ | 4.8 | 0.016 | 3.3 | -0.011
$b_{\mathrm{M,SF}}$ | 2.7 | -0.009 | 5.9 | 0.020
$b_{\mathrm{M,Q}}$ | 7.1 | -0.024 | 9.0 | 0.031
$b_{\phi,\mathrm{SF}}$ | 1.5 | -0.005 | 2.4 | 0.008
$b_{\phi,\mathrm{Q}}$ | 2.7 | 0.009 | 5.4 | -0.019

Finally, we focused on the redshift sensitivity of SDSS-like surveys. Table 10
presents the impact of the Schechter parameters on an SDSS-like redshift
distribution. The parameter $a_{\mathrm{M,Q}}$ has the strongest effect on the
redshift distribution. The mean redshift changes by 10.5 per cent after
decreasing $a_{\mathrm{M,Q}}$. The corresponding absolute change of 0.036
illustrates that SDSS-like surveys are not strongly affected by the parameter
uncertainties.

Figures 7, 8 and 9 in appendix A illustrate the differences in the fiducial
and test redshift distributions for all survey types. The plots show the
redshift distributions of which the summary statistics are given in tables 8,
9 and 10. Larger values of $\overline{\delta z}_{\mathrm{mean}}$ correspond to
visually larger differences in the distributions.

####  5.1.1 Survey Comparison

Figure 4: Comparison of the normalised simulated fiducial (black) and test
redshift distributions after increasing (red) and decreasing (blue)
$a_{\mathrm{M,SF}}$, $a_{\phi,\mathrm{Q}}$ and $b_{\mathrm{M,SF}}$ by
$1\sigma$. Note that the distributions are normalised such that the integral
over the redshift range $z\in[0,3]$ is one. The columns correspond to the
different survey types. The first column shows the results of SDSS-like
surveys. The second and third columns display the results of DES- and HSC-like
surveys, respectively. The rows correspond to the different parameters.
Furthermore, we indicate the number of galaxies per square arcminute in the
legends of each plot. We see that DES- and HSC-like surveys are more sensitive
to $a_{\mathrm{M,SF}}$, $a_{\phi,\mathrm{Q}}$ and $b_{\mathrm{M,SF}}$.
Furthermore, $a_{\mathrm{M,SF}}$ has the strongest impact.

Figure 4 compares the fiducial and test redshift distributions for all survey
types after modifying different parameters. Each plot shows the fiducial
redshift distribution in black, the test redshift distribution after
decreasing the corresponding parameter by $1\sigma$ in blue and the test
redshift distribution after increasing the parameter by $1\sigma$ in red. The
columns correspond to the different surveys, the first column shows the
results for an SDSS-like survey, the second for a DES-like and the third for
an HSC-like survey. Moreover, we compare the effect of three different
parameters in the individual rows. The first row shows the impact of
$a_{\mathrm{M,SF}}$, the second of $a_{\phi,\mathrm{Q}}$ and the third of
$b_{\mathrm{M,SF}}$. In addition, we indicate the number of galaxies per
square arcminute in the legend of each plot.

The first row highlights that $a_{\mathrm{M,SF}}$ has a higher impact on DES-
and HSC-like surveys than on SDSS-like surveys. Decreasing $a_{\mathrm{M,SF}}$
increases the mean redshift. Increasing $a_{\mathrm{M,SF}}$ decreases the mean
redshift. This effect also correlates with the galaxy number density. The
second row shows similar characteristics. DES- and HSC-like surveys are more
sensitive to $a_{\phi,\mathrm{Q}}$ and increasing the parameter has a stronger
effect than decreasing it. Especially, the plots indicate that a change of the
parameters that increases the mean redshift has generally a stronger effect
than a change that decreases the mean redshift. This attribute is independent
of the direction of the parameter change and follows from the fact that fewer
high-redshift than low-redshift galaxies are measured. The change that
increases the mean redshift is also always accompanied by an increase in the
number density. Hence, constraining the direction of change that is causing
the increase in the mean redshift is more important than the direction that
decreases the mean redshift.

Looking at the model of the characteristic magnitude (eq. 3), the above-
mentioned observations agree with theoretical expectations. As
$a_{\mathrm{M}}$ describes the redshift evolution of the characteristic
magnitude, variations of $a_{\mathrm{M}}$ directly change the bright end of
the luminosity function. Since galaxy observations are in general limited by
the flux, the changes in the bright end influence the abundance of measured
high redshift objects. Although $b_{\mathrm{M}}$ is also affecting the
characteristic magnitude it does not affect the redshift evolution directly
and is expected to have less impact on the redshift distribution than
$a_{\mathrm{M}}$.

The different survey types also have distinct sensitivity to the parameters of
the two galaxy populations. Tables 8, 9 and 10 show that DES- and HSC-like
surveys tend to be more sensitive to parameters of the star-forming
population. On the other hand, SDSS-like surveys are more sensitive to the
parameters of quenched galaxies. For DES- and HSC-like surveys,
$a_{\mathrm{M,SF}}$ has the strongest impact on the redshift distribution.
Conversely, the mean redshift of an SDSS-like survey is most sensitive to
$a_{\mathrm{M,Q}}$ and $b_{\mathrm{M,Q}}$. The difference in the sensitivities
follows from the variety in the depths of the surveys. An SDSS-like survey has
a lower magnitude limit than the other surveys and measures mostly closer
galaxies, which have lower redshift values. These galaxies had more time to
get quenched and the ratio of star-forming to quenched galaxies is smaller
than for DES- and HSC-like surveys. DES- and HSC-like surveys are more
sensitive to parameters of the star-forming galaxies because these survey
types also measure younger galaxies. However, the majority of galaxies
measured by these surveys still have low redshifts. That explains why
parameters of the quenched galaxies can have a strong impact on the redshift
distribution of DES- and HSC-like surveys, for example, the 13.5 per cent
change of the DES-like mean redshift or the 17.5 per cent change in the case
of an HSC-like survey after decreasing $a_{\mathrm{M,Q}}$.

To summarise the analysis of the sensitivity to the Schechter parameters, we
found that $a_{\mathrm{M,SF}}$ has the highest impact on the redshift
distribution, although the sensitivity depends on the survey type. We conclude
that $a_{\mathrm{M,SF}}$ is the most important parameter to constrain. Within
the given uncertainty of $a_{\mathrm{M,SF}}$ the modelled mean redshift can
change by up to 0.3. This can cause strong bias in the cosmological
constraints when using weak lensing observations and the forward modelled
redshift distribution. We present the most sensitive observables in the
following section.

###  5.2 Colour and Magnitude Distributions

Figure 5: Corner plot of the $g-r$, $g-i$, $g-z$ colour and $i$\- and $z$-band
magnitude distributions for a DES-like survey. Black lines and contours
correspond to the distributions in the fiducial catalogue. The red and blue
colours represent the test catalogues after increasing and decreasing
$a_{\mathrm{M,SF}}$ by $3\sigma$, respectively. The colour distributions are
more affected by a change of $a_{\mathrm{M,SF}}$. A combination of colour and
magnitude is most suitable to constrain $a_{\mathrm{M,SF}}$.

To investigate the effect of the parameter changes on the observables, we
compared the magnitude and colour distributions of the fiducial and test
catalogues. We focused on DES-like surveys because their depths are similar to
those of future and upcoming surveys as previously mentioned. We investigated
the effect on all observables (apparent magnitudes in the different bands and
the corresponding colours) but concentrated on the five most sensitive
observables to $a_{\mathrm{M,SF}}$, which are the $g-r$, $g-i$, $g-z$ colour
and $i$\- and $z$-band magnitude distributions.

We focused on the results for $a_{\mathrm{M,SF}}$ for two reasons. First, we
showed that $a_{\mathrm{M,SF}}$ causes the strongest effect and is most
important to constrain. Second, we found that the to $a_{\mathrm{M,SF}}$ most
sensitive observables are generally sensitive to the other parameters as well.
Especially, the colour distributions are very sensitive to changes in most of
the parameters141414That is even true for the other survey types..

We found that the colour distributions $g-r$, $g-i$, $g-z$ and the magnitude
distributions in the $i$\- and $z$-band are the five most sensitive
observables to $a_{\mathrm{M,SF}}$ and have the highest potential to help
constraining its value in the future. As described earlier, we compared the
colour and magnitude distributions of fiducial and test catalogues
quantitatively using the Anderson-Darling test. In the case of the colour
distributions $g-r$, $g-i$ and $g-z$, we obtained d-values of the order
$10^{4}$ after decreasing the parameter value. In the case of the magnitude
distributions in the $i$\- and $z$-bands, the d-values were of the order
$10^{3}$. Hence, the colour distributions are more sensitive to a modification
of $a_{\mathrm{M,SF}}$ than the magnitude distributions. The d-value after
decreasing the value of $a_{\mathrm{M,SF}}$ by its error for the $g-r$ colour
distributions was ${\sim}21,300$. This value was the largest in our analysis.

Figure 5 shows a corner plot comparing these five observables. The plot
contains the one-dimensional distributions and contours of all observables.
The black lines and contours correspond to the fiducial catalogue. The colours
blue and red represent the results of the test catalogue after decreasing and
increasing $a_{\mathrm{M,SF}}$ by $3\sigma$. Again, blue corresponds to a
negative and red to a positive change.

The one-dimensional distributions in figure 5 verify the results indicated by
the AD test that the colours are more sensitive to a change of
$a_{\mathrm{M,SF}}$ than the magnitudes. Once more, we observed that the
negative change, which increases the mean redshift, has a stronger impact on
the distributions than a positive change. The figure also shows that modifying
the parameter affects the shape and the peak of the colour distributions.
However, in the case of the magnitude distributions, the shapes are affected
by the parameter change but the peaks stay the same. For different colour and
magnitude choices in this comparison, the conclusion might be different
because of the distinct sensitivity in the single bands.

The contours show that a combination of colour and magnitude is generally most
sensitive to the parameters and might be the best choice to constrain them.
The differences in the fiducial and test contours appear larger in colour-
magnitude space. In addition, the results indicate that the magnitude-
magnitude distributions are less sensitive to the parameters than the
distributions in colour-colour space. However, the sensitivity of the colour-
colour distributions is still smaller than in colour-magnitude space.
Additionally, the choice of colour-magnitude combination does not influence
the sensitivity to the parameter change because all test contours feature
similar deviations from the fiducial contours. Note that our analysis showed
that these colour-magnitude pairs are sensitive to the rest of the parameters
as well and not only $a_{\mathrm{M,SF}}$. The AD tests indicated high
sensitivity of these observables to all parameters. In general, the contour
plots for the other parameters are similar to figure 5 and only small
deviations exist. Therefore, these pairs might also be able to help
constraining the other parameters. Furthermore, our analysis has shown the
same results if we chose colour-magnitude pairs that are not included in the
figure (e.g $i-z$ or $r-i$). But, as mentioned, we included the colours and
magnitudes with the highest sensitivity.

Figure 6: Corner plot of the $g-r$, $g-i$, $g-z$ colour and $i$\- and $z$-band
magnitude distributions for an HSC-like survey. Black lines and contours
correspond to the distributions in the fiducial catalogue. The red and blue
colours represent the test catalogues after increasing and decreasing
$a_{\mathrm{M,SF}}$ by $3\sigma$, respectively. Comparing this plot with
figure 5, we see that an HSC-like survey is strongly affected by the parameter
change than a DES-like survey.

We also investigated the effect of modifying the Schechter parameters to HSC-
like measurements. Figure 6 shows the same plot as figure 5 but for an HSC-
like survey. Comparing the two plots illustrates that a deep HSC-like survey
has higher sensitivity and therefore better constraining ability than a DES-
like survey. The HSC-like contours of fiducial and test catalogues have a
larger deviation than in the case of DES-like surveys. Furthermore, the one-
dimensional colour and magnitude distributions are more responsive to the
parameter modification. Although the d-values for a negative modification of
the parameter are similar to the results of a DES-like survey, the d-values
after adding the parameter error to its value are larger in the case of HSC-
like measurements. The d-values of the colour distributions $g-r$, $g-i$ and
$g-z$ are ${\sim}6\cdot 10^{3}$ in the case of HSC-like survey and, hence,
about six times larger than in the case of DES-like surveys.

In summary, we suggest that a combination of colour-magnitude measurements is
more sensitive to changes to the Schechter parameters and can be more useful
to constrain them. Thereby, the exact selection of colour-magnitude pair has a
minor role because the sensitivity is in general similar for all combinations.
Additionally, HSC-like measurements can improve the constraining power
compared to DES-like measurements.

##  6 Conclusions

Different cosmological probes are used to constrain the parameters of dark
energy as well as dark matter and, therefore, to improve the cosmological
model. One of these probes is weak gravitational lensing. Cosmic shear
measurements rely on knowing the redshift distribution $n(z)$ of the galaxies
under study. However, measuring $n(z)$ with sufficient precision is difficult.
Spectroscopic redshifts are most accurate but current and upcoming experiments
are too large to measure redshifts efficiently using spectroscopy. On the
other hand, photometric redshifts do not have adequate precision.

Herbel et al. (2017) used a forward modelling approach to get a full posterior
of $n(z)$ without having information about each individual redshift. Their
model is based on redshift-dependent Schechter luminosity functions (see eq.
2). In this model, the characteristic magnitude is a linear function of
redshift $z$ and the amplitude has an exponential relation to $z$. The model
includes two different galaxy types, star-forming (SF) galaxies and quenched
(Q) galaxies. Therefore, the number of galaxies per comoving volume and
absolute magnitude depends on the eight Schechter luminosity function
parameters $a_{\mathrm{M,SF}}$, $a_{\mathrm{M,Q}}$, $a_{\phi,\mathrm{SF}}$,
$a_{\phi,\mathrm{Q}}$, $b_{\mathrm{M,SF}}$, $b_{\mathrm{M,Q}}$,
$b_{\phi,\mathrm{SF}}$ and $b_{\phi,\mathrm{Q}}$.

Throughout this paper, we investigated the sensitivity of the redshift
distributions from DES-, HSC- and SDSS-like surveys to these parameters with
the goal to identify which galaxy population features need to be well
understood for future precision measurements. As the result of this and within
the context of our forward model, we identified which Schechter parameters are
most important to constrain such that the uncertainties of shear measurements
are reduced. Additionally, we explored how the Schechter parameters affected
the galaxy observables (magnitudes and colours). In this way, we identified
the observables that are most sensitive to the Schechter parameters and that
could help to constrain the parameters. We refer to section 4 for more details
about our methodology.

We found that constraining $a_{\mathrm{M,SF}}$ has the highest priority for
simulating the redshift distribution of DES-like surveys. Since upcoming
surveys like Euclid and LSST have similar properties as DES, having better
constraints of $a_{\mathrm{M,SF}}$ is especially important for future studies
that are based on simulated redshift distributions. The uncertainty of
$a_{\mathrm{M,SF}}$ can affect the mean redshift by up to ${\sim}45$ %, which
corresponds to an absolute change of ${\sim}0.3$. The other parameters have
less effect on the simulations but the results suggest that
$a_{\mathrm{M,Q}}$, $a_{\phi,\mathrm{SF}}$, $a_{\phi,\mathrm{Q}}$ and
$b_{\mathrm{M,SF}}$ require better constraints as well.

Furthermore, we found that the two-dimensional distributions in colour-
magnitude space are generally more sensitive to the parameters than the
distributions in colour-colour and magnitude-magnitude space. Therefore, the
colour-magnitude distributions can help constraining the parameters,
especially $a_{\mathrm{M,SF}}$, the best. The results also show that the
choice of the colour-magnitude combination has a minor role. In the case of a
DES-like survey, we found that the $\mathit{(g-r)-i}$, $\mathit{(g-r)-z}$,
$\mathit{(g-i)-i}$ and $\mathit{(g-i)-z}$ colour-magnitude combinations are
the best options. However, HSC-like measurements might be a better choice to
constrain the parameters because they have higher sensitivity to the Schechter
parameters.

Finally, we will include the error model that is described in section 3.3.1 to
the `skypy.utils` module of the `SkyPy` library (SkyPy Collaboration, 2020;
Amara et al., 2021). Additionally, we aim to improve the constraints of the
parameters in future projects.

## Acknowledgements

We would like to thank all our colleagues in the SkyPy Collaboration. We
especially acknowledge N. Tessore and R. P. Rollins for many useful
discussions in the early stages of this paper.

The preparation of this manuscript was made possible by a number of software
packages: `NumPy` (Harris et al., 2020), `SciPy` (Virtanen et al., 2020),
`Astropy` (Price-Whelan et al., 2018), `Matplotlib` (Hunter, 2007),
`IPython/Jupyter` (Perez & Granger, 2007) and `GetDist` (Lewis, 2019). This
research partly developed and made use of `SkyPy`, a Python package for
forward modelling astronomical surveys (SkyPy Collaboration, 2020; Amara et
al., 2021).

PS and AA acknowledge support from a Royal Society Wolfson Fellowship grant.

## Data Availability

The data produced in this work will be available as the corresponding
configuration files for the simulations in the `SkyPy` GitHub repository,
https://github.com/skypyproject/skypy and
https://doi.org/10.5281/zenodo.4071945.

The public data used in figures 1, 2 and 3 were obtained from Drlica-Wagner et
al. (2018) at https://des.ncsa.illinois.edu/releases/y1a1/gold, Ahumada et al.
(2020) at http://skyserver.sdss.org/CasJobs/ and from Aihara et al. (2019) at
https://hsc-release.mtk.nao.ac.jp/datasearch/, respectively.

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##  Appendix A Plots Redshift Distributions

In this part of the paper, we visually compare the sensitivity of the redshift
distributions to all Schechter parameters for the three survey types. The
individual plots show the fiducial redshift distributions in black. The
redshift distributions of the test catalogues after increasing and decreasing
the parameters by their given errors are shown in red and blue, respectively.
The legends of each plot contain the galaxy number density for each catalogue.
Figure 7 shows the distributions of a DES-like survey. Figures 8 and 9
correspond to an HSC- and an SDSS-like survey, respectively. The plots are all
in agreement with the results in section 5.1.

Figure 7: Comparison of the normalised simulated fiducial (black) and test
redshift distributions of a DES-like survey. The red and blue lines correspond
to increasing and decreasing the Schechter parameter by $1\sigma$,
respectively. The number of galaxies per square arcminute is indicated in the
legend of each plot. The plots are in agreement with the results in section
5.1. Figure 8: Same as 7 but for an HSC-like survey. Figure 9: Same as 7 but
for an SDSS-like survey.