# Cosmological Parameter Estimation from the Two-Dimensional Genus Topology -
Measuring the Expansion History using the Genus Amplitude as a Standard Ruler

Stephen Appleby<EMAIL_ADDRESS>Asia Pacific Center for Theoretical
Physics, Pohang, 37673, Korea  Quantum Universe Center, Korea Institute for
Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea  Changbom Park
School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-
gu, Seoul, 02455, Korea  Sungwook E. Hong (홍성욱)  Korea Astronomy and Space
Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Korea
Natural Science Research Institute, University of Seoul, 163 Seoulsiripdaero,
Dongdaemun-gu, Seoul, 02504, Korea  Ho Seong Hwang  Korea Astronomy and Space
Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Korea  Juhan
Kim  Center for Advanced Computation, Korea Institute for Advanced Study, 85
Hoegiro, Dongdaemun-gu, Seoul, 02455, Korea  Motonari Tonegawa  School of
Physics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul,
02455, Korea

###### Abstract

We measure the genus of the galaxy distribution in two-dimensional slices of
the SDSS-III BOSS catalog to constrain the cosmological parameters governing
the expansion history of the Universe. The BOSS catalogs are divided into
twelve concentric shells over the redshift range $0.25<z<0.6$ and we
repeatedly measure the genus from the two-dimensional galaxy density fields,
each time varying the cosmological parameters used to infer the distance-
redshift relation to the shells. We also indirectly reconstruct the two-
dimensional genus amplitude using the three-dimensional genus measured from
SDSS Main Galaxy Sample with galaxies at low redshift $z<0.12$. We combine the
low- and high–redshift measurements, finding the cosmological model which
minimizes the redshift evolution of the genus amplitude, using the fact that
this quantity should be conserved. Being a distance measure, the test is
sensitive to the matter density parameter ($\Omega_{\rm m}$) and equation of
state of dark energy ($w_{\rm de}$). We find a constraint of $w_{\rm
de}=-1.05^{+0.13}_{-0.12}$, $\Omega_{\rm m}=0.303\pm 0.036$ after combining
the high- and low–redshift measurements and combining with Planck CMB data.
Higher redshift data and combining data sets at low redshift will allow for
stronger constraints.

##  1\. Introduction

Topological statistics have a long history of use within cosmology Gott et al.
(1990); Park & Gott (1991); Mecke et al. (1994); Schmalzing & Buchert (1997);
Schmalzing & Gorski (1998); Hikage et al. (2006); Ducout et al. (2013); Melott
et al. (1989); Park et al. (1992); Gott et al. (1992). Theoretical studies of
random fields have been undertaken; both Gaussian (Doroshkevich, 1970; Adler,
1981; Gott et al., 1986; Hamilton et al., 1986; Ryden et al., 1989; Gott et
al., 1987; Weinberg et al., 1987) and perturbatively non-Gaussian (Matsubara,
1994a, b; Matsubara & Suto, 1996; Melott et al., 1988; Matsubara & Yokoyama,
1996; Matsubara, 2000; Hikage et al., 2008; Pogosyan et al., 2009; Gay et al.,
2012; Codis et al., 2013), and modern techniques are being developed that go
beyond the standard Minkowski functional analysis; Minkowski tensors Beisbart
et al. (2001b, a); Ganesan & Chingangbam (2017); Chingangbam et al. (2017);
Kapahtia et al. (2019); Appleby et al. (2018a, b); Kapahtia et al. (2018);
Joby et al. (2019), Betti numbers Park et al. (2013); Feldbrugge et al.
(2019); Pranav et al. (2019a, b, 2017); Shivshankar et al. (2015); van de
Weygaert et al. (2011) and multi-scale analyses of the cosmic web Sousbie et
al. (2011); Codis et al. (2018); Kraljic et al. (2020). Previous application
of the Minkowski functionals to various modern data sets can be found in (Park
et al., 2001; Hikage et al., 2002, 2003; Park et al., 2005; James et al.,
2009; Gott et al., 2009; Choi et al., 2010b; Zhang et al., 2010; Petri et al.,
2013; Blake et al., 2014; Wiegand et al., 2014; Parihar et al., 2014; Wang et
al., 2015; Wiegand & Eisenstein, 2017; Buchert et al., 2017; Sullivan et al.,
2019; Hikage et al., 2001; Gott et al., 2008). The advent of cosmological
scale, large scale structure data has allowed measurements of the higher point
functions induced by gravitational collapse Wiegand et al. (2014); Wiegand &
Eisenstein (2017); Buchert et al. (2017); Sullivan et al. (2019) that would be
difficult to extract using conventional $N$-point methods.

The genus belongs to the family of Minkowski Functionals. The genus of the
matter density field, as traced by galaxies, can be used as a cosmological
probe. By measuring the genus curve at different redshifts, one can extract
information regarding the parameters governing the expansion history of the
Universe. The redshift dependence of the genus amplitude was originally
proposed as a standard ruler in Park & Kim (2010); Zunckel et al. (2011). For
the $\Lambda$CDM model, the amplitude of the genus curve is related to the
slope of the linear matter power spectrum, which does not evolve with
redshift. By comparing this quantity at high and low redshift, we should
detect no evolution. However, if we select an incorrect cosmological model to
infer the distance-redshift relation, then comoving smoothing scales and
volumes become systematically incorrect with increasing redshift. This will
generate a spurious evolution in the statistic. Hence, by measuring the genus
using different cosmological models to infer distance scales, one can find the
expansion history that conserves this statistic.

This cosmological test was first proposed in Park & Kim (2010). More recently,
the authors have revisited this possibility and applied the method to
projected two dimensional galaxy density fields, using all-sky mock galaxy
lightcone data (Appleby et al., 2017, 2018c). The analysis presented here
provides a conclusion of these works, as we apply the methodology to a
combination of low- and high-redshift galaxy catalogs to obtain a constraint
on the cosmological parameters $\Omega_{\rm m}$ and dark energy equation of
state $w_{\rm de}$. This test was pursued in Blake et al. (2014), with the
first direct application of the method to galaxy data (specifically the
WiggleZ survey Blake et al. (2011)). Competitive distance measurements were
obtained from three-dimensional Minkowski functional measurements, and issues
associated with this measurement (principally sparse sampling) were
highlighted. The conclusion of the work was that topology is potentially
competitive with Baryon Acoustic Oscillations (BAO) as a standard ruler,
however the physics and assumptions that go into the analysis are more
involved, as the Minkowski functionals measure the shape of the full extent of
the power spectrum in an integrated sense.

In this work we measure the genus of both the BOSS, LOWZ and CMASS galaxy
catalogs (Alam et al., 2015) and the SDSS Main Galaxy Sample (SDSS MGS)
(Abazajian et al., 2009). The low redshift SDSS MGS data provides a robust
measure of the genus amplitude at low redshift, practically insensitive to the
distance-redshift relation. In contrast, the higher redshift BOSS data will be
sensitive to our choice of cosmological parameters when inferring distances.
If we select an incorrect distance-redshift relation, the genus amplitude
extracted from the BOSS data will systematically evolve, relative to the low
redshift measurement. The reason for this effect is that an incorrect choice
of comoving distance will cause us to select erroneous smoothing scales and
effective areas, meaning that we will be measuring the slope of the matter
power spectrum at different scales as a function of redshift. As the matter
power spectrum is not scale invariant, this will manifest as an evolving genus
amplitude.

The principal challenge when using the genus as a standard ruler is that we
must compare high redshift measurements to low redshift counterparts. However,
the low redshift Universe is restricted in volume and the statistical
uncertainty provides the dominant limitation on parameter constraints. To
mitigate this problem, we measure the genus of the full three-dimensional
field at low redshift. We then convert the three-dimensional measurement into
a constraint on the theoretical expectation of the two-dimensional genus
amplitude.

The paper will proceed as follows. In Section 2 we discuss some of the issues
associated with using the genus amplitude as a standard ruler, and our method
of extracting this quantity from galaxy data. We briefly review the extraction
of the genus from two-dimensional shells of BOSS data in Section 3. In Section
4 we detail the data, mask, mock catalogs and systematics associated with SDSS
MGS measurement of the three-dimensional genus. The conversion from three
dimensional measured genus to the theoretical expectation value of the two-
dimensional genus amplitude is explained in Section 5. Finally in Section 6 we
place constraints on cosmological parameters, then close with a discussion in
Section 7.

This work is a companion to Appleby et al. (2020), which uses the absolute
value of the genus amplitude (rather than its evolution with redshift, as in
this work) to place constraints on the shape of the matter power spectrum. We
discuss the relation between the two approaches in Section 7.

##  2\. Genus amplitude as a standard ruler

The two-dimensional genus of a perturbatively non-Gaussian field without
boundary is given by the so-called Edgeworth expansion Matsubara (1994b,
2000); Pogosyan et al. (2009); Gay et al. (2012); Codis et al. (2013)

$\displaystyle g_{\rm 2D}(\nu_{\rm A})=A_{\rm G}^{(\rm 2D)}e^{-\nu_{\rm
A}^{2}/2}\left[H_{1}(\nu_{\rm A})+\left[{2\over
3}\left(S^{(1)}-S^{(0)}\right)\times\right.\right.$
$\displaystyle\quad\left.\left.H_{2}(\nu_{\rm A})+{1\over
3}\left(S^{(2)}-S^{(0)}\right)H_{0}(\nu_{\rm A})\right]\sigma_{0}+{\cal
O}(\sigma_{0}^{2})\right],$ (1)

where $A_{\rm G}^{(\rm 2D)}$ is the amplitude

$\displaystyle A_{\rm G}^{(\rm 2D)}\equiv{1\over
2(2\pi)^{3/2}}{\sigma_{1}^{2}\over\sigma_{0}^{2}},$ (2)

and the skewness parameters $S^{(0)},S^{(1)},S^{(2)}$ are related to the three
point cumulants and will not be used here. $\sigma_{1}$ and $\sigma_{0}$ are
defined as integrals over the power spectrum, in this work smoothed with
Gaussian kernels of comoving scale $R_{\rm G}$

$\displaystyle\sigma_{0}^{2}={1\over(2\pi)^{2}}\int
d^{2}k_{\perp}e^{-k_{\perp}^{2}R_{\rm G}^{2}}P_{\rm 2D}(k_{\perp},z),$ (3)
$\displaystyle\sigma_{1}^{2}={1\over(2\pi)^{2}}\int
d^{2}k_{\perp}k_{\perp}^{2}e^{-k_{\perp}^{2}R_{\rm G}^{2}}P_{\rm
2D}(k_{\perp},z),$ (4)

and the projected two-dimensional power spectrum $P_{\rm 2D}(k_{\perp},z)$ is
related to its full three-dimensional counterpart according to

$P_{\rm 2D}(k_{\perp},z)={2\over\pi}\int dk_{\mathbin{\|}}P_{\rm
3D}\left(k,z\right){\sin^{2}[k_{\mathbin{\|}}\Delta]\over
k_{\mathbin{\|}}^{2}\Delta^{2}},$ (5)

where $\Delta$ is the comoving thickness of the two-dimensional slices of the
field. ${\vec{k}}_{\perp}$ and $k_{\mathbin{\|}}$ are the wave numbers
perpendicular and parallel to the line of sight respectively. The three-
dimensional power spectrum of the density field that is traced by galaxies is
the sum of the redshift-space distorted matter field and a shot noise
contribution

$P_{\rm 3D}(k,k_{\mathbin{\|}},z)=b^{2}\left(1+\beta{k_{\mathbin{\|}}^{2}\over
k^{2}}\right)^{2}P_{\rm m}(z,k)+P_{\rm SN},$ (6)

where $P_{\rm m}(z,k)$ is the matter power spectrum at redshift $z$, $P_{\rm
SN}$ is the shot noise power spectrum $P_{\rm SN}=1/\bar{n}$, where $\bar{n}$
is the number density of galaxies. We introduce $\beta=f/b$, $b$ is the linear
galaxy bias and $f$ is the growth factor. The quantity $\nu_{A}$ is the
density threshold such that the excursion set has the same area fraction as a
corresponding Gaussian field -

$f_{A}={1\over\sqrt{2\pi}}\int^{\infty}_{\nu_{A}}e^{-t^{2}/2}dt,$ (7)

where $f_{A}$ is the fractional area of the field above $\nu_{A}$. This choice
of $\nu_{\rm A}$ parameterization eliminates the non-Gaussianity in the one-
point function (Gott et al., 1987; Weinberg et al., 1987; Melott et al.,
1988).

For the case of a Gaussian field, the genus amplitude is a measure of the
shape of the linear matter power spectrum $P_{\rm m}(z,k)$, which is a
conserved quantity for the $\Lambda$CDM model and certain generalisations
(such as $w$CDM, assuming dark energy perturbations are negligible). If we use
an incorrect cosmological model to infer the distance-redshift relation, then
we get the smoothing scale $R_{\rm G}$ and volume occupied by galaxy data
systematically wrong at different redshifts. Hence we will measure the shape
of the power spectrum at different scales when using an incorrect expansion
history. As a result, the genus amplitude that we extract from the data will
spuriously evolve with redshift if we get the expansion history wrong. A low
redshift measurement will represent the ‘true’ genus amplitude having little
dependence on the cosmology adopted, against which high redshift measurements
can be compared. This effect was predicted in (Park & Kim, 2010) and
explicitly measured using mock galaxies in Appleby et al. (2018c).

In reality a number of small systematic effects are present in real galaxy
data that generate redshift evolution of this statistic. The primary sources
of contamination are as follows, listed in order of severity

  1. 1.

We bin galaxies into redshift shells and apply a mass cut to fix the number
density of tracers at each redshift to be constant, thus fixing a constant
shot noise power spectrum $P_{\rm SN}$ in each shell. In contrast, the
amplitude of the matter power spectrum $P_{\rm m}(z,k)$ decreases with
redshift. It follows that the relative importance of the shot noise
contribution in (6) will increase with redshift, which will manifest as an
increasing genus amplitude at higher $z$. This effect depends on $R_{\rm G}$
relative to the mean galaxy separation $\bar{r}$, and is negligible for
$R_{\rm G}\gg\bar{r}$.

  2. 2.

Linear redshift space distortion decreases the amplitude of the two-
dimensional genus by around $\sim 9\%$, roughly constant over the redshift
range $0<z<0.7$. However, it also introduces a mild $\sim 1\%$ redshift
dependent evolution, decreasing the amplitude with increasing redshift. This
is due to the redshift dependence of $\beta(z)$ in equation (6).

  3. 3.

Non-linear gravitational evolution will typically act to decrease the genus
amplitude with decreasing redshift, which is an ${\cal O}(\sigma_{0}^{2})$
effect (so-called gravitational smoothing Melott et al. (1989); Park & Gott
(1991); Park et al. (2005)).

The magnitude of each of these effects depends on the number density of
galaxies, the smoothing scales perpendicular and parallel to the line of sight
($R_{\rm G}$ and $\Delta$) and the area of the data. In Appendix A we use mock
galaxy lightcone data to examine these effects in isolation, and argue that
for the data and smoothing scales used in this work, no significant redshift
evolution of the genus amplitude will be induced.

To briefly summarise the results in Appendix A : We take constant comoving
scale $R_{\rm G}=20{\rm Mpc}$ to Gaussian smooth the data perpendicular to the
line of sight, and comoving slice thickness $\Delta=80\,{\rm Mpc}$ along the
line of sight. At these scales, the redshift space distortion and shot noise
effects both introduce an evolution of the genus amplitude of order $\sim 1\%$
over the redshift range $0<z<0.7$. Shot noise/redshift space distortion causes
the genus amplitude to increase/decrease with increasing $z$. The two
competing effects effectively cancel for the particular galaxy sample
considered in this work. Furthermore, the mean galaxy separation of the two-
dimensional projected fields is approximately $\bar{r}\simeq 15\,{\rm Mpc}$,
smaller than $R_{\rm G}=20\,{\rm Mpc}$. This makes the non-Gaussianity of the
shot noise contribution small.

The non-Gaussian gravitational corrections to the amplitude are small. We
quantify this statement by measuring the next-to-leading-order correction term
$a_{3}H_{3}(\nu_{\rm A})$, finding it to be $\sim{\cal O}(1\%)$ at the scales
probed. Non-Gaussian corrections are suppressed when the area fraction
threshold $\nu_{\rm A}$ is used to define the excursion set as opposed to the
standard threshold $\nu$. We find no evidence of evolution of $a_{3}$ over the
range $0.25<z<0.6$ relevant to the BOSS data.

Numerical systematic effects also exist. The area of our data slices decreases
at low redshifts for fixed solid angle, and the excursion set regions at high
$|\nu|$ are more difficult to be sampled in a smaller area. Whenever the
excursion set is poorly sampled, the genus amplitude will generically be
biased high. To eliminate this bias, we must only measure the genus curve over
a range of threshold values $-\nu_{0}<\nu<\nu_{0}$ for which the excursion set
is well sampled at all redshifts. We vary the threshold limit $\nu_{0}$ to
check that the data provides an unbiased measurement of the genus curve. The
range $|\nu_{A}|<2.5$ is well represented within our shells, so we measure the
genus curve over this range.

##  3\. Observational Data $0.25<z<0.6$

Our treatment of the high redshift data – SDSS-III Baryon Oscillation
Spectroscopic Survey (BOSS) (York et al., 2000) – has been described in detail
in Appleby et al. (2020). To briefly review, we bin the galaxies into $N_{\rm
z}=12$ shells of comoving thickness $\Delta=80\,{\rm Mpc}$, 6/6 from the LOWZ
and CMASS data, over the range $0.25<z<0.6$. We apply a mass cut to fix the
number density as $\bar{n}=6.25\times 10^{-5}\,({\rm Mpc})^{-3}$ within each
shell. With this choice, the shot noise contribution to the field is large but
the clustering signal is dominant for the smoothing scales adopted in this
work. The galaxies are weighted to account for observational systematics.
Specifically, the following weight was applied to each galaxy in the LOWZ and
CMASS sample

$w_{\rm tot}=w_{\rm systot}\left(w_{\rm cp}+w_{\rm noz}-1\right)$ (8)

where $w_{\rm cp}$ is the correction factor to account for the subsample of
galaxies that are not assigned a spectroscopic fibre, $w_{\rm noz}$ is for the
failure in the pipeline to assign redshifts due to certain galaxies, and
$w_{\rm systot}$ represents non-cosmological fluctuations in the CMASS target
density due to stellar density and seeing.

The redshift bin limits are presented in Table 1; these were derived using the
Planck cosmological parameters $w_{\rm de}=-1$, $\Omega_{\rm m}=0.307$ to
define slices of constant comoving thickness $\Delta=80\,{\rm Mpc}$. We should
vary these limits and re-bin the galaxies each time we vary the cosmology in
the distance redshift relation. However, because the genus amplitude is
insensitive to $\Delta$ for thick slices, we can fix these limits throughout
without biasing our results. We provide evidence to support this statement in
Appendix B.

LOWZ | CMASS
---|---
$0.250<z\leq 0.271$ |  $0.453<z\leq 0.476$
$0.271<z\leq 0.292$ |  $0.476<z\leq 0.500$
$0.292<z\leq 0.313$ |  $0.500<z\leq 0.524$
$0.313<z\leq 0.334$ |  $0.524<z\leq 0.548$
$0.334<z\leq 0.356$ |  $0.548<z\leq 0.573$
$0.356<z\leq 0.378$ |  $0.573<z\leq 0.598$
Table 1 The redshift limits of the LOWZ and CMASS shells used in this work.

HEALPix111http://healpix.sourceforge.net (Gorski et al., 2005) is used to bin
the galaxies into pixels on the unit sphere. A galaxy number density field
$\delta_{i,j}\equiv(n_{i,j}-\bar{n}_{j})/\bar{n}_{j}$ is defined, where $1\leq
j\leq N_{\rm z}$ denotes the redshift bin (of which there are $N_{\rm z}=12$
in total) and $1\leq i\leq N_{\rm pix}$ is the pixel identifier on the unit
sphere. $\bar{n}_{j}$ is the mean number of galaxies contained within a pixel
at each redshift shell, and $n_{i,j}$ is the number of galaxies contained
within pixel $i$ in redshift slice $j$. We use $N_{\rm pix}=12\times 512^{2}$
pixels. The survey geometry and veto masks Reid et al. (2016) were then used
to generate a binary healpix map : $\Theta_{i}=1$ if the survey angular
selection function in the $i^{\rm th}$ pixel is larger than some cutoff
$\Theta_{\rm cut}=0.8$ and $\Theta_{i}=0$ otherwise, where $i$ runs over
$N_{\rm pix}$ pixels. The $\Theta_{i}$ mask was applied to the galaxy field
$\delta_{i,j}$.

We smooth the two-dimensional density fields, and the $\Theta_{i}$ mask, in
each shell using angular scale $\theta_{\rm G}=R_{\rm G}/d_{\rm
cm}(z_{j},\Omega_{\rm m},w_{\rm de})$, where $R_{\rm G}=20{\rm Mpc}$ is the
comoving smoothing scale and $d_{\rm cm}(z_{j},\Omega_{\rm m},w_{\rm de})$ is
the comoving distance to the center of the $j^{\rm th}$ redshift shell.
Defining $\tilde{\Theta}_{i,j}$ and $\tilde{\delta}_{i,j}$ as the smoothed
mask and density fields, we re-define $\tilde{\delta}_{i,j}=0$ if
$\tilde{\Theta}_{i,j}<\Theta_{\rm cut}$ and
$\tilde{\delta}_{i,j}\to\tilde{\delta}_{i,j}/\tilde{\Theta}_{i,j}$ otherwise.
Finally, we re-apply the original unsmoothed $\Theta_{i}$ mask. This procedure
eliminates regions close to the boundary, where the field may not be well
reconstructed. In Appendix C of Appleby et al. (2020) we explicitly show that
our masking procedure, and method of genus extraction, provides an unbiased
estimate of the genus, and we direct the reader to this paper for further
details. The important underlying point is that we are extracting the genus
per unit area, which is a local quantity and hence can be estimated in an
unbiased manner from a cut-sky galaxy sample.

Finally we divide the genus by the total area of the data $A_{j}=4\pi f_{\rm
sky}d_{\rm cm}^{2}(z_{j},\Omega_{\rm m},w_{\rm de})$, where $f_{\rm sky}$ is
the fractional area of the data on the sky. The genus is reconstructed using
the method described in Schmalzing & Gorski (1998); Appleby et al. (2018c),
which provides an unbiased estimate of the full sky genus from an observed
patch.

We measure the genus for $200$ values of the threshold $\nu_{\rm A}$, equi-
spaced over the range $-2.5<\nu_{\rm A}<2.5$, then take the average over every
four values to obtain $N_{\nu_{\rm A}}=50$ measurements. We label the measured
values $g_{j}^{n}$, where $j$ runs over the redshift shells and $1\leq n\leq
N_{\nu_{\rm A}}$ over the $N_{\nu_{\rm A}}=50$ thresholds. We then extract the
genus amplitudes $A^{(\rm 2D)}_{j}$ by minimizing the following $\chi^{2}$
functions at each redshift –

$\chi^{2}_{j}=\sum_{n=1}^{N_{\nu_{\rm A}}}\sum_{m=1}^{N_{\nu_{\rm A}}}\Delta
g^{n}_{j}\Sigma_{n,m}^{-1}(z_{j})\Delta g^{m}_{j},$ (9)

with respect to the parameters $A_{j}^{\rm(2D)},a_{0,j},a_{2,j},a_{3,j}$,
where

$\displaystyle\Delta g^{n}_{j}=g_{j}^{n}-A_{j}^{\rm(2D)}e^{-\nu_{{\rm
A},n}^{2}/2}\left[a_{0,j}H_{0}(\nu_{{\rm A},n})+\right.$
$\displaystyle\qquad\left.H_{1}(\nu_{{\rm A},n})+a_{2,j}H_{2}(\nu_{{\rm
A},n})+a_{3,j}H_{3}(\nu_{{\rm A},n})\right],$ (10)

and $\Sigma_{n,m}(z_{j})$ are the covariance matrices associated with $\Delta
g^{n}_{j}$. $\Sigma_{n,m}(z_{j})$ are obtained using the patchy mock galaxy
catalogs (Kitaura et al., 2016; Rodríguez-Torres et al., 2016; Kitaura et al.,
2014, 2015) – further information on the covariance matrices used in our
analysis can be found in Appleby et al. (2020).

The measured genus values $g_{j}^{n}$ are functions of the distance-redshift
relation, and hence the cosmological parameters $(\Omega_{\rm m},w_{\rm de})$.
This parameter sensitivity enters in the definition of the angular smoothing
scale $\theta_{\rm G}=R_{\rm G}/d_{\rm cm}(z_{j},\Omega_{\rm m},w_{\rm de})$
and the area occupied by the data $A_{j}=4\pi f_{\rm sky}d_{\rm
cm}^{2}(z_{j},\Omega_{\rm m},w_{\rm de})$. We repeat our measurement of
$g_{j}^{n}$ and minimization of (10) for each cosmological parameter set. We
fix $h=0.677$ to its Planck value throughout, where $H_{0}=100h\,{\rm
km}\,{\rm s^{-1}}\,{\rm Mpc}^{-1}$.

Parameter | Fiducial Value
---|---
$\Omega_{\rm m}$ | $0.307$
$h$ | $0.677$
$w_{\rm de}$ | $-1$
$\Delta$ | $80{\rm Mpc}$
$R_{\rm G}$ | $20{\rm Mpc}$
Table 2Fiducial parameters used to fix the slice thickness, and the fiducial
parameters used to calculate the genus in this work. $\Delta$ is the thickness
of the two dimensional slices of the density field, and $R_{\rm G}$ is the
Gaussian smoothing scale used in the two-dimensional planes perpendicular to
the line of sight.

Figure 1 exhibits the two-dimensional genus curves [top panel] and the
corresponding amplitudes $A^{(\rm 2D)}_{j}$ [bottom panel] extracted from the
$N_{\rm z}=12$ LOWZ and CMASS data shells (Appleby et al., 2020). The genus
curves and amplitudes are functions of the assumed cosmological model, and in
this figure we have taken a $\Lambda$CDM model with parameters given in Table
2. The genus amplitude is reconstructed to accuracy $\sim 3\%$ and $\sim
1.5\%$ in the LOWZ/CMASS data respectively, and we present the best fit
amplitudes, $1\,\sigma$ error bars and reduced $\chi^{2}$ values in Table 3.
Three of the redshift bins present relatively poor fits with a $\chi^{2}$ per
degree of freedom $>1.5$ : two in the LOWZ data and one CMASS slice. This
could indicate that the mocks are under-predicting the true statistical
uncertainty, possibly lacking cosmic variance. A theoretical understanding of
the statistical uncertainty of the Minkowski functionals is currently lacking,
as no prediction for their covariance is available.

However, the amplitudes extracted from the slices are all consistent. We test
this by performing a simple linear regression to the best fit $A^{(2D)}$ data
points, assuming the data are uncorrelated. We find a p-value of $p=0.83$ for
the null hypothesis that the slope of the linear fit is consistent with zero,
indicating no statistically significant redshift evolution of the genus
amplitude over the redshift range $0.25<z<0.6$. This is expected from
theoretical arguments, but provides an important consistency check on our
analysis.

Figure 1.— [Top panel] Twelve, two-dimensional genus curves obtained from the BOSS LOWZ and CMASS data, as a function of $\nu_{\rm A}$. [Bottom panel] Two-dimensional genus amplitude measurements derived from the $N_{\rm z}=12$ genus curves presented in the top panel. The same color scheme is applied in both panels.  Redshift | $A_{\rm 2D}\times 10^{5}{\rm Mpc}^{-2}$ | $\chi^{2}/{\rm DoF}$
---|---|---
0.26 | $4.97\pm 0.16$ | 1.49
0.28 | $4.95\pm 0.15$ | 1.12
0.30 | $5.16\pm 0.13$ | 1.53
0.32 | $5.19\pm 0.13$ | 1.58
0.35 | $5.14\pm 0.13$ | 1.02
0.37 | $4.82\pm 0.12$ | 1.20
0.46 | $5.15\pm 0.09$ | 1.34
0.49 | $5.11\pm 0.10$ | 1.08
0.51 | $4.92\pm 0.09$ | 1.63
0.54 | $5.04\pm 0.08$ | 0.96
0.56 | $5.03\pm 0.08$ | 1.15
0.59 | $5.14\pm 0.08$ | 1.41
Table 3The mean and $1\,\sigma$ uncertainty of the genus amplitudes extracted
from the six LOWZ and CMASS shells. The third column is the reduced $\chi^{2}$
value of the fit (46 degrees of freedom).

##  4\. Low Redshift Data $0<z<0.12$

To test the expansion history, we also require an accurate measurement of the
genus at low redshift, which should be practically insensitive to the
distance-redshift relation. This would provide an anchor, a measurement of the
shape of the linear matter power spectrum against which high redshift genus
curves can be compared.

However, two-dimensional slices at low redshifts have very small areas and
suffer from curvature effects. To overcome this limitation we use the three-
dimensional local galaxy distribution in the SDSS MGS and apply a Gaussian
smoothing over a smaller scale. The measured three-dimensional genus will be
used to estimate the two-dimensional genus amplitude. In the following
sections, we describe in detail our method – the theory underlying the three-
dimensional genus, the galaxy data used, the mask, how we remove systematics
from the genus amplitude using mock galaxy catalogs and how we infer the two-
dimensional genus amplitude from the three-dimensional data.

###  4.1. Theory – Expectation value of three-dimensional genus

The genus per unit volume of a three dimensional, Gaussian random field as a
function of threshold $\nu$ is given by (Tomita, 1986; Adler, 1981; Gott et
al., 1986; Hamilton et al., 1986)

$\displaystyle g_{\rm 3D}(\nu)={1\over 4\pi^{2}}\left({\Sigma_{1}^{2}\over
3\Sigma_{0}^{2}}\right)^{3/2}\left(1-\nu^{2}\right)e^{-\nu^{2}/2},$ (11)
$\displaystyle\Sigma_{0}^{2}=\langle\delta^{2}_{\rm
3D}\rangle,\qquad\Sigma_{1}^{2}=\langle|\nabla\delta_{\rm 3D}|^{2}\rangle,$

where $\Sigma_{0,1}$ are the two-point cumulants of the three-dimensional
field, related to the power spectrum as

$\displaystyle\Sigma_{0}^{2}=\int d^{3}ke^{-k^{2}\Lambda_{\rm G}^{2}}P_{\rm
3D}(k),$ (12) $\displaystyle\Sigma_{1}^{2}=\int d^{3}ke^{-k^{2}\Lambda_{\rm
G}^{2}}k^{2}P_{\rm 3D}(k),$ (13)

where we have smoothed with a Gaussian kernel of width $\Lambda_{\rm G}$. The
genus amplitude is given by

$A_{\rm G}^{(\rm 3D)}={1\over 4\pi^{2}}\left({\Sigma_{1}^{2}\over
3\Sigma_{0}^{2}}\right)^{3/2}.$ (15)

The leading order non-Gaussian expansion of the genus, in terms of the
$\nu_{\rm A}$ threshold convention, is given by (Matsubara, 1994b, 2000;
Pogosyan et al., 2009; Gay et al., 2012; Codis et al., 2013)

$\displaystyle g_{\rm 3D}(\nu_{\rm A})=A_{\rm G}^{(\rm 3D)}e^{-\nu_{\rm
A}^{2}/2}\left[H_{2}(\nu_{\rm
A})+\left[\left(S^{(1)}-S^{(0)}\right)\times\right.\right.$
$\displaystyle\quad\left.\left.H_{3}(\nu_{\rm
A})+\left(S^{(2)}-S^{(0)}\right)H_{1}(\nu_{\rm A})\right]\Sigma_{0}+{\cal
O}(\Sigma_{0}^{2})\right].$ (16)

As for the two-dimensional genus, the amplitude (coefficient of $H_{2}$
Hermite polynomial) is not modified by the non-Gaussian effect of
gravitational collapse to linear order in the $\Sigma_{0}$ expansion (16).

###  4.2. Data

To extract the genus of the low redshift matter density, we use the seventh
data release of the main galaxy catalog of the SDSS DR7 (Abazajian et al.,
2009). Specifically, we adopt the Korea Institute for Advanced Study Value
Added Galaxy Catalog (KIAS VAGC) (Choi et al., 2010a; Blanton et al., 2005;
Padmanabhan et al., 2008). The KIAS catalog supplements redshifts from other
existing galaxy redshift catalogs – the updated Zwicky catalog (Falco et al.,
1999), the IRAS Point Source Catalog Redshift Survey Saunders et al. (2000),
the Third Reference Catalogue of Bright Galaxies (de Vaucouleurs et al.,
1991), and the Two Degree Field Galaxy Redshift Survey (Colless et al., 2001).

The KIAS VAGC contains $593,514$ redshifts of SDSS main galaxies in the
$r$-band Petrosian magnitude range $10<r_{\rm p}<17.6$. Details of the
selection criteria, classification schemes and angular selection functions can
be found in (Choi et al., 2010a). To maximize the area to boundary ratio of
the data, we remove the three southern stripes and Hubble deep field region.

The catalog provides angular positions, redshifts and absolute, $r$-band
magnitudes normalised to the $z=0.1$ epoch, calculated from extinction
corrected AB fluxes and an evolution correction $E(z)=1.6(z-0.1)$ (Tegmark et
al., 2004). All magnitudes and colors are corrected to the redshift $z=0.1$
epoch. Following Choi et al. (2010b), we apply a magnitude cut $M_{\rm
r}<-20.19+5\log h$ to generate a volume limited sample over the redshift range
$0.02<z<0.116$, with a mean galaxy separation of $r_{\rm gal}=\bar{n}_{\rm
gal}^{-1/3}=8.3{\rm Mpc}$, where $\bar{n}_{\rm gal}$ is the mean galaxy number
density within the volume. The redshift range was selected to ensure a maximal
number of galaxies are used in the analysis. The galaxies are presented as a
function of redshift and absolute magnitude in Figure 2 (top panel), and the
angular distribution of all galaxies used in this work are presented in the
bottom panel. Note that in this Figure and in what follows the factor $5\log
h$ will be dropped in the expression of $M_{\rm r}$.

Figure 2.— [Top panel] The absolute, $r$-band magnitude of the SDSS MGS
galaxies as a function of redshift. The solid red lines indicate the
boundaries of our volume limited sample with $0.02<z<0.116$ and $M_{\rm
r}<-20.19$. [Bottom panel] The angular distribution of the volume limited
sample of galaxies on the sky; declination vs right ascension (in degrees).

To convert the galaxy catalog into a three-dimensional density field, we
construct a regular three-dimensional $N_{\rm pix}^{3}=512^{3}$ pixel lattice
in a cube of side length $L_{\rm box}=750{\rm Mpc}$ and use cosmological
parameters given in Table 2 to infer the distance-redshift relation. We bin
the galaxies into pixels using the Cloud-in-Cell scheme, generating a three-
dimensional number density field $\delta_{ijk}=(n_{ijk}-\bar{n})/\bar{n}$,
where $\bar{n}$ is the average number of galaxies within the unmasked pixels
and $1\leq i,j,k\leq N_{\rm pix}$ subscripts are pixel labels. The galaxies
are weighted via the angular selection function during this binning procedure.

The angular selection function constitutes a set of weights as a function of
angular position on the sky - $w(\theta,\phi)$. It is defined in this work as
$w=0$ when outside the survey geometry or inside a bright star mask and
$0<w\leq 1$ when inside the survey geometry. This function represents the
survey completeness as a function of position on the sky. Because we weight
the galaxies according to the angular selection function, we convert $w$ into
a binary field with $w=1$ if $w>w_{\rm cut}$ and $w=0$ otherwise, where
$w_{\rm cut}$ was selected as $w_{\rm cut}=0.8$. Using the fiducial distance-
redshift relation, we define $w_{ijk}$ as the projection of the angular
selection function into a $512^{3}$, three-dimensional pixel cube of the same
dimensions as $\delta_{ijk}$.

We smooth the three-dimensional density field $\delta_{ijk}$ with a Gaussian
kernel of width $\Lambda_{\rm G}=8.86{\rm Mpc}$ (this value is $R_{G}=6{\rm
Mpc}/h$, following Choi et al. (2010b)), and also smooth the projected
selection function $w_{ijk}$ with the same kernel, defining the smoothed
counterparts as $\tilde{\delta}_{ijk}$ and $\tilde{w}_{ijk}$. We then redefine
$\tilde{\delta}_{ijk}=0$ for all pixels in which $\tilde{w}_{ijk}<0.9$ and
$\tilde{\delta}_{ijk}=\tilde{\delta}_{ijk}/\tilde{w}_{ijk}$ if
$\tilde{w}_{ijk}\geq 0.9$. We then re-apply the original mask and set
$\tilde{\delta}_{ijk}=0$ if $w_{ijk}=0$. This eliminates all data in the
vicinity of the survey boundary.

From the masked field we reconstruct the three dimensional genus, by
generating iso-field triangulated meshes and calculating the Gaussian
curvature at the triangle vertices. Details of the method can be found in
(Appleby et al., 2018b). We calculate the genus as a function of $\nu_{\rm
A}$, where $\nu_{\rm A}$ is the threshold chosen to match the volume fraction
of a Gaussian random field. We select $200$, $\nu_{\rm A}$ threshold values
over the range $-2.5<\nu_{\rm A}<2.5$, then take the average of every four
values to obtain the genus at $N_{\nu_{\rm A}}=50$, $\nu_{\rm A}$ values equi-
spaced over this range. The resulting measured genus values are presented in
Figure 3 (top panel, red points).

To extract the genus amplitude from the measurements, we fit a Hermite
polynomial expansion to the data points by minimizing the following $\chi^{2}$
function

$\chi^{2}=\Delta g^{\rm T}\Gamma^{-1}\Delta g,$ (17)

where

$\displaystyle\Delta g_{i}=g_{i}-A^{(\rm 3D)}e^{-\nu_{{\rm
A},i}^{2}/2}\left[a_{0}H_{0}(\nu_{{\rm A},i})+a_{1}H_{1}(\nu_{{\rm
A},i})+\right.$ $\displaystyle\qquad\qquad\left.H_{2}(\nu_{{\rm
A},i})+a_{3}H_{3}(\nu_{{\rm A},i})+a_{4}H_{4}(\nu_{{\rm A},i})\right].$ (18)

$A^{(\rm 3D)},a_{0},a_{1},a_{3},a_{4}$ are free parameters to be constrained
via the minimization of (17), the $i$ subscript denotes the $i^{\rm th}$,
$\nu_{\rm A}$ threshold bin and $g_{i}$ are the measured genus values. In the
fitting procedure we include the leading order $a_{1},a_{3}$ Hermite
polynomial coefficients and the next-to-leading order even Hermite polynomial
contributions $a_{0}$, $a_{4}$. Introducing additional Hermite polynomials
does not significantly modify the fit.

The covariance matrix $\Gamma_{ij}$ is obtained from mock galaxy catalogs, as
described in the following section.

###  4.3. Mock Galaxy Catalogs

Mock galaxy catalogs are generated using Horizon Run 4 (HR4) Kim et al.
(2015). Horizon Run 4 is a cosmological scale dark matter simulation in which
$N=6300^{3}$ particles in a volume $V=(3150{\rm Mpc}/h)^{3}$ are evolved using
a modified GOTPM scheme222For a description of the original GOTPM code, please
see Dubinski et al. (2004). A description of the modifications introduced in
the Horizon Run project can be found at https://astro.kias.re.kr/
kjhan/GOTPM/index.html.. The initial conditions are obtained using second
order Lagrangian perturbation theory L’Huillier et al. (2014), and the
cosmological parameters used are $h=0.72$, $n_{\rm s}=0.96$, $\Omega_{\rm
m}=0.26$, $\Omega_{\rm b}=0.048$. We use the $z=0$ snapshot box to create mock
galaxy catalogs, using the HR4 cosmological parameters to infer distances.
Details of the numerical implementation, and the method by which mock galaxies
are constructed can be found in Hong et al. (2016). The mock galaxies are
defined using the most bound halo particle galaxy correspondence scheme, and
the survival time of satellite galaxies post merger is estimated via the
merger timescale model described in Jiang et al. (2008). The snapshot box is
decomposed into $N_{\rm r}=360$ non-overlapping volumes, and mock galaxy
catalogs are constructed from each region, with the same number density,
redshift range and survey geometry as the data.

From each mock sample we repeat our analysis ; bin the galaxies into a regular
cubic pixel lattice, smooth the resulting number density field with a Gaussian
of scale $\Lambda_{\rm G}=8.86{\rm Mpc}$, apply the smoothed mask
$\tilde{w}_{ijk}$ then unsmoothed binary mask $w_{ijk}$, then extract the
genus from $\tilde{\delta}_{ijk}$.

The result is a set of genus measurements $g^{(\rm 3D)}_{i,m}$, where $1\leq
i\leq N_{\nu_{\rm A}}$ runs over $N_{\nu_{\rm A}}=50$, $\nu_{\rm A}$ bins
uniformly sampled in the range $-2.5<\nu_{\rm A}<2.5$ and $1\leq m\leq N_{\rm
r}$ runs over the randomly sampled realisations. We measured the genus for
$200$ values and averaged every fourth point to arrive at the $N_{\nu_{\rm
A}}=50$ values in each mock sample. The covariance matrix is constructed as

$\Gamma_{ij}={1\over N_{\rm r}-1}\sum_{m=1}^{N_{\rm r}}\left(g^{(\rm
3D)}_{i,m}-\langle g^{(\rm 3D)}_{i}\rangle\right)\left(g^{(\rm
3D)}_{j,m}-\langle g^{(\rm 3D)}_{j}\rangle\right),$ (19)

where $\langle g_{i}^{(\rm 3D)}\rangle$ is the average value of the genus in
the $i^{\rm th}$ threshold bin. In Figure 3 (bottom panel) we exhibit the
covariance matrix $\Gamma_{ij}$ extracted from the mock realisations. We note
the strong correlation between genus values measured at different thresholds.
A similar covariance matrix was numerically extracted from mock data in Blake
et al. (2014).

###  4.4. Results – Three-Dimensional Genus of SDSS MGS

In Figure 3 (top panel) we exhibit the genus measured from the SDSS MGS (red
points), and the best-fit curve reconstruction $g^{(\rm th)}(\nu_{\rm A})$
(black solid line). The error bars are the square root of the diagonal
elements of $\Gamma_{ij}$. After minimizing the $\chi^{2}$ function (17), in
Table 4 (first row) we present the best fit and uncertainty on the $(A^{(\rm
3D)},a_{0},a_{1},a_{3},a_{4})$ parameters. We also present a fit including
just $(A^{(\rm 3D)},a_{1},a_{3})$, and $(A^{(\rm 3D)})$ only for comparison.
If we regard equation (18) as an expansion in $\sigma_{0}$, then $a_{1},a_{3}$
should be of order $a_{1,3}\sim{\cal O}(\sigma_{0})$ and $a_{0},a_{4}\sim{\cal
O}(\sigma_{0}^{2})$. At the scales adopted in this work, the higher order
terms are large, which indicates that the field is non-linear. In spite of
this, all three amplitude measurements are consistent. However, the second and
third rows yield a significantly worse $\chi^{2}$.

The genus amplitude presented in the first row of Table 4; $A^{(\rm
3D)}=4.040\times 10^{-6}{\rm Mpc}^{-3}$, and uncertainty $\Delta A^{(\rm
3D)}=0.197\times 10^{-6}{\rm Mpc}^{-3}$, will be used as the low redshift
genus amplitude measurement from the SDSS MGS. This low redshift data point
will be used to complement the higher redshift, two-dimensional BOSS
measurements.

$A^{(\rm 3D)}\times 10^{6}(\rm Mpc^{-3})$ | $a_{0}$ | $a_{1}$ | $a_{3}$ | $a_{4}$ | $\chi^{2}$
---|---|---|---|---|---
$4.040\pm 0.197$ | $0.095\pm 0.016$ | $-0.009\pm 0.025$ | $0.042\pm 0.019$ | $-0.006\pm 0.014$ | $63.9$
$4.167\pm 0.135$ | - | $-0.015\pm 0.026$ | $0.026\pm 0.018$ | - | $117.6$
$4.084\pm 0.130$ | - | - | - | - | $124.0$
Table 4Best fit Hermite polynomial coefficients for the three-dimensional
genus curve extracted from the SDSS MGS. The top row is the full fitting
function used in this work. In the second row we set $a_{0}=a_{4}=0$ and in
the third row we fix $a_{0}=a_{1}=a_{3}=a_{4}=0$, and fit a Gaussian curve to
the points.

Figure 3.— [Top panel] The genus curve measured from the SDSS MGS using a
Gaussian smoothing length of $\Lambda_{\rm G}=8.86{\rm Mpc}$. The red points
correspond to measured values, and the error bars are from the diagonal
components of the covariance matrix (19). The black solid line is the best fit
curve reconstruction (18) with parameters given in the first row of table 4.
[Bottom panel] The covariance matrix $\Gamma_{ij}$. Bins separated by
$\Delta\nu_{\rm A}<0.25$ are strongly correlated (red), and bins at larger
separations present anti-correlation (blue).

The measured amplitude $A^{(\rm 3D)}$ of the SDSS MGS is effectively
insensitive to cosmological parameters. We confirm that reasonable variation
of cosmological parameters does not affect the measured value of $A^{(\rm
3D)}$ in Figure 4. We select five parameter sets $(\Omega_{\rm m},w_{\rm
de})=(0.21,-1),(0.31,-1),(0.38,-1),(0.31,-0.5),(0.31,-1.5)$ to infer the
distance redshift relation, construct the density field from the galaxy
positions and measure the genus amplitude $A^{(\rm 3D)}$ by minimizing the
$\chi^{2}$ function (17). The resulting amplitudes and uncertainties are
presented in Figure 4. We find no significant change in the measured genus
amplitude if we use different cosmological parameters to infer the distance
redshift relation, as expected at low redshift $z<0.12$. For this reason, we
fix $A^{(\rm 3D)}=4.040\pm 0.197\times 10^{-6}{\rm Mpc}^{-3}$, corresponding
to the red data point in Figure 4.

Figure 4.— The genus amplitude as measured from the SDSS MGS, assuming five
different cosmological models to infer the distance redshift relation. The
measured amplitude of the low redshift sample is effectively insensitive to
our choice. We select a fiducial cosmology $(\Omega_{\rm m},w_{\rm
de})=(0.307,-1)$ to infer $A^{(\rm 3D)}$ (red diamond).

##  5\. Three- to Two-dimensional Genus Amplitude

Our intention is to combine the SDSS MGS and BOSS genus measurements, and find
the cosmology that minimizes the evolution of the two-dimensional genus
amplitude. However, to directly compare these results, we must convert the
three dimensional genus amplitude measurement from the SDSS MGS to a
corresponding effective two-dimensional amplitude. To do so, we perform the
following steps –

  1. 1.

Correct the measured three-dimensional genus amplitude for gravitational
smoothing and non-linear redshift space distortion with a correction factor
obtained from simulations.

  2. 2.

Using the now corrected, real space amplitude, perform a cosmological
parameter search by comparing this value to its Gaussian expectation value.
The result is a set of parameter constraints on $(\Omega_{\rm c}h^{2}$,
$n_{\rm s})$ which determine the shape of the linear power spectrum.

  3. 3.

Use the best fit cosmological parameters $(\Omega_{\rm c}h^{2}$, $n_{\rm s})$
to infer the two-dimensional theoretical expectation of the genus amplitude

$A^{(\rm 2D)}_{{\rm G}}={1\over 2(2\pi)^{3/2}}{\int
k_{\perp}^{3}e^{-k_{\perp}^{2}R_{\rm G}^{2}}P_{\rm
2D}(k_{\perp})dk_{\perp}\over\int k_{\perp}e^{-k_{\perp}^{2}R_{\rm
G}^{2}}P_{\rm 2D}(k_{\perp})dk_{\perp}}.$ (20)

Where the two-dimensional power spectrum $P_{\rm 2D}$ is related to the three
dimensional matter power spectrum according to equation (5), and we use the
three-dimensional power spectrum (6) with $\bar{n}=6.25\times 10^{-5}\,{\rm
Mpc}^{-3}$, $b=2$, $R_{\rm G}=20{\rm Mpc}$, $\Delta=80{\rm Mpc}$; the values
relevant to the BOSS data. The end result is an inferred two-dimensional genus
amplitude, based on the SDSS MGS data.

In the following subsections we discuss each point in turn.

###  5.1. Systematics removal

Before comparing $A^{(\rm 3D)}$ to its Gaussian expectation value, we must
account for non-linear effects. The most significant systematics that must be
corrected are non-linear gravitational evolution and redshift space
distortion. At the smoothing scale $\Lambda_{\rm G}=8.86{\rm Mpc}$, the effect
of redshift space distortion on the genus amplitude will not be well
approximated by the linear Kaiser approximation Choi et al. (2013). Therefore,
to eliminate its effect we directly compare the three-dimensional genus
measured from the simulations in real and redshift space, and use the
difference between these measurements as a correction factor to be applied to
the SDSS genus amplitude measurement.

To model these systematic corrections, we use the KIAS Multiverse simulations;
a set of five cosmological scale, dark matter only simulations. Each is
generated from a different cosmological model in which $\Omega_{\rm m}$ and
$w_{\rm de}$ are varied (Shin et al., 2017; Park et al., 2019; Hong et al.,
2020; Tonegawa et al., 2020). Since our low redshift genus measurement will be
practically insensitive to the value of $w_{\rm de}$, we use three of the
simulations with cosmological parameters $(\Omega_{\rm m},w_{\rm
de})=(0.21,-1)$, $(0.26,-1)$ and ($0.31,-1)$ with all other cosmological
parameters fixed as $\Omega_{\rm b}=0.044$, $h=0.72$, $n_{\rm s}=0.96$. Each
simulation comprises $N_{p}=2048^{3}$ dark matter particles in a
$1024^{3}h^{-3}{\rm Mpc}^{3}=1422^{3}{\rm Mpc}^{3}$ box, gravitationally
evolved using a modified GOTPM code which uses the Poisson equation

$\nabla^{2}\Psi=4\pi Ga^{2}\bar{\rho}_{\rm m}\delta_{\rm m}\left(1+{D_{\rm
de}\over D_{\rm m}}{\Omega_{\rm de}(a)\over\Omega_{\rm m}(a)}\right).$ (21)

The same random number sequence was used to generate the initial condition for
each simulation at $z=99$, to eliminate cosmic variance when comparing
different models. The power spectrum was normalised such that the rms of the
matter fluctuation, smoothed with top hat $8h^{-1}{\rm Mpc}$ and linearly
evolved to $z=0$, is $\sigma_{8}=0.794$.

The genus curves of the Multiverse $z=0$ snapshot boxes in real and redshift
space are presented in the top panel of Figure 5, for three different
cosmological models $\Omega_{\rm m}=0.21,0.26,0.31$. We use the entire box
with periodic boundary conditions to make these measurements – that is we
apply no mask in this subsection. For each simulation, we fix the number
density of the mock galaxies such that the mean separation is
$\bar{r}=\bar{n}^{-1/3}_{\rm gal}=8.33\,{\rm Mpc}$, by applying a mass cut.
The solid/dashed lines correspond to real/redshift space mock galaxy catalogs,
and green/red/blue corresponds to $\Omega_{\rm m}=0.21,0.26,0.31$
respectively. In all cases one can observe an amplitude drop due to the effect
of redshift space distortion.

In the bottom panel, we exhibit amplitude measurements extracted from the
genus curves in the top panel. The green/red/blue color scheme is the same as
for the top panel, and diamonds/stars correspond to real/redshift space
measurements of the genus amplitude. We denote the real/redshift space genus
amplitudes as $A^{(\rm 3D)}_{\rm real}$ and $A^{(\rm 3D)}_{\rm rsd}$
respectively. The fractional difference between the redshift and real space
amplitude measurements – $a_{\rm rsd}^{(\rm 3D)}\equiv A^{(\rm 3D)}_{\rm
rsd}/A^{(\rm 3D)}_{\rm real}$ – is $a_{\rm rsd}^{(\rm 3D)}=0.89,0.92,0.91$ for
$\Omega_{\rm m}=0.21,0.26,0.31$ respectively. The effect of redshift space
distortion is a $\sim 10\%$ effect on the genus amplitude at these scales, and
is only weakly dependent on cosmological parameters. Specifically, $a_{\rm
rsd}^{(\rm 3D)}$ exhibits no significant, systematic dependence on
$\Omega_{\rm m}$. We use the $\Omega_{\rm m}=0.26$ simulation and take $a_{\rm
rsd}^{(\rm 3D)}=0.92$ in what follows, correcting the measured $A^{(\rm 3D)}$
amplitude by a factor of $(1-\Delta_{\rm rsd})^{-1}$ with $\Delta_{\rm
rsd}=0.08$. This factor converts $A^{(\rm 3D)}$ to real space.

Figure 5.— [Top panel] Measured genus curves as a function of $\nu_{\rm A}$
for three multiverse simulations with $\Omega_{\rm m}=0.21,0.26,0.31$ (green,
red, blue lines). The solid lines are real space measurements, dashed are
redshift space. [Bottom panel] The genus amplitudes extracted from the top
panel. The diamonds/stars represent the real/redshift space measurements
respectively. The yellow squares represent the prediction for a Gaussian field
for the given $\Omega_{\rm m}$. The real space, mock galaxy amplitudes are
lower than the Gaussian prediction due to gravitational smoothing, and the
redshift space values are still lower due to the effect of redshift space
distortion.

To account for non-linear gravitational evolution, we compare the measurement
of the three-dimensional genus amplitude of the multiverse simulations in real
space to the Gaussian expectation value (15), where we use the linear matter
power spectrum plus shot noise

$P_{\rm 3D}(k)=b_{\rm sdss}^{2}P_{\rm m}(k)+P_{\rm SN,sdss},$ (22)

to generate the cumulants $\Sigma_{0,1}$. We use the SDSS MGS number density
$\bar{n}=1.7\times 10^{-3}{\rm Mpc}^{-3}$ for the shot noise power spectrum
$P_{\rm SN,sdss}=1/\bar{n}$ and galaxy bias $b_{\rm sdss}=1.5$ (Howlett et
al., 2015; Ross et al., 2015). We have already converted $A^{(\rm 3D)}$ to
real space using the correction factor $\Delta_{\rm rsd}$.

In Figure 5 (bottom panel) we exhibit the genus amplitude of the $z=0$, real
space Multiverse simulation snapshot boxes (green, red and blue diamonds), and
the corresponding Gaussian expectation value (15) with the same cosmological
parameters (yellow squares, labeled ‘GRF’).

The Gaussian expectation values are systematically higher than the genus
measured from each simulation box – this highlights the ‘gravitational
smoothing’ effect of non-linear gravitational collapse. The effect is $a_{\rm
gr}\equiv A^{(\rm 3D)}_{\rm real}/A^{(\rm 3D)}_{\rm G}=0.88,0.90,0.92$ for the
$\Omega_{\rm m}=0.21,0.26,0.31$ simulations respectively. To directly compare
the measured genus amplitude from the SDSS MGS to the corresponding Gaussian
expectation value, we correct $A_{\rm G}^{(\rm 3D)}$ by a factor of
$(1-\Delta_{\rm gr})^{-1}$ with $\Delta_{\rm gr}=0.10$.

After correcting the measured genus amplitude $A^{(\rm 3D)}$ to account for
non-linear redshift space distortion and gravitational smoothing, the next
step is to compare $A^{(\rm 3D)}$ to the expectation value (15) to obtain a
set of parameter constraints. We minimize the simple $\chi^{2}$ function

$\chi^{2}={[(1-\Delta_{\rm rsd}-\Delta_{\rm gr})^{-1}A^{(\rm 3D)}-A^{(\rm
3D)}_{\rm G}(\Omega_{\rm c}h^{2},n_{\rm s})]^{2}\over\sigma_{\rm 3D}^{2}},$
(23)

where $A^{(\rm 3D)}_{\rm G}$ is the Gaussian expectation value of the three-
dimensional genus curve (15), and is sensitive to $\Omega_{\rm c}h^{2}$,
$n_{\rm s}$ and weakly to $\Omega_{\rm b}h^{2}$. As the dependence on
$\Omega_{\rm b}h^{2}$ is very weak, we fix this parameter to its Planck best
fit value $\Omega_{\rm b}h^{2}=0.0222$ Aghanim et al. (2020). $\sigma_{\rm
3D}=0.197\times 10^{-6}{\rm Mpc}^{-3}$ is the statistical uncertainty on
$A^{(\rm 3D)}$.

In Figure 6 we present the two-dimensional $1,2-\sigma$ contours in the
$n_{\rm s}$, $\Omega_{\rm c}h^{2}$ plane, obtained by performing an MCMC
parameter search, minimizing (23). We observe a strong degeneracy between
$n_{\rm s}$ and $\Omega_{\rm c}h^{2}$, as both can vary the degree of small
scale power and hence increase/decrease the genus amplitude. The Planck best
fit is shown as a black star, and the cosmological model of the Multiverse
simulation used to make the non-linear redshift space distortion and
gravitational smoothing corrections $\Delta_{\rm rsd}$, $\Delta_{\rm gr}$ is
presented as a green square. Both are within the $1-\sigma$ contour.

Figure 6.— The $1,2-\sigma$ contours in the $n_{\rm s}$-$\Omega_{\rm c}h^{2}$
plane obtained by minimizing the chi square function (23). The black star is
the Planck best fit value of these parameters and green square the
cosmological model used to infer the non-linear corrections to the measured
genus curve.

In the next step of our analysis, we convert these constraints to a measure of
the two-dimensional genus amplitude $A^{(\rm 2D)}_{\rm G}$.

###  5.2. Conversion from cosmological parameters to $A^{(\rm 2D)}_{\rm G}$

Finally, we transform from the cosmological parameters $\Omega_{\rm c}h^{2}$,
$n_{\rm s}$ to a prediction for the two dimensional genus amplitude. To do so,
we transform each parameter set and corresponding $\chi^{2}$ value
$(\Omega_{\rm c}h^{2},n_{\rm s},\chi^{2})$ from the previous section to
$(A^{(\rm 2D)}_{\rm G},\chi^{2})$ by inserting $\Omega_{\rm c}h^{2}$, $n_{\rm
s}$ into the definition of the theoretical expectation of the two-dimensional
genus amplitude (2) using ($\ref{eq:s02}-\ref{eq:p3df}$), taking $b=2$ and
$\bar{n}=6.25\times 10^{-5}\,{\rm Mpc}^{-3}$ suitable for the BOSS galaxy
sample, and smoothing scales $\Delta=80\,{\rm Mpc}$, $R_{\rm G}=20\,{\rm
Mpc}$. The result is a one-dimensional probability distribution function for
$A^{(\rm 2D)}_{\rm G}$, as inferred from the three-dimensional measurement. We
present the resulting probability distribution in Figure 7 (top panel). From
this we infer the best fit and $1\sigma$ uncertainty on the two-dimensional
genus amplitude as $A^{(\rm 2D)}_{\rm G}=5.084\pm 0.087\times 10^{-5}\,({\rm
Mpc})^{-2}$.

To review, the $A^{(\rm 3D)}$ measurement provides a constraint of the shape
of the linear matter power spectrum. We have used the best fit values and
uncertainties on the parameters $\Omega_{\rm c}h^{2}$ and $n_{\rm s}$ to infer
the best fit and uncertainty on the theoretical expectation value of $A^{(\rm
2D)}_{\rm G}$ at low redshift. In the bottom panel of Figure 7 we present the
two-dimensional genus measurement inferred from the SDSS MGS (silver star) and
those directly measured from the BOSS data (multi-coloured data points). We
take the SDSS MGS measurement to lie at $z=0.1$. We assume that the low
redshift measurement of $A^{(\rm 2D)}_{\rm G}$ is insensitive to variations of
the expansion history in what follows, and treat it as a constant $A^{(\rm
2D)}_{\rm G}=5.084\pm 0.087\times 10^{-5}\,({\rm Mpc})^{-2}$.

Figure 7.— [Top panel] The probability distribution of the amplitude of the
two-dimensional genus $A_{\rm G}^{(\rm 2D)}$ obtained by minimizing the
$\chi^{2}$ function (23). [Bottom panel] The two-dimensional genus measurement
$A_{\rm G}^{(\rm 2D)}$ inferred from the low redshift SDSS MGS is presented as
a silver star, along with the BOSS data points over the redshift range
$0.25<z<0.6$. The BOSS points are derived assuming a Planck cosmology.

In this section, we have pursued a rather complex path to extracting $A^{(\rm
2D)}_{\rm G}$ from the low-redshift data. However, the reasoning behind our
method lies in maximizing the constraining power of the data. For our method
to provide a reasonable constraint, we must minimize the statistical
uncertainty of the low-redshift measurement as far as possible, and this
required us to select smaller smoothing scales than the high redshift data.
The smallest smoothing scales that we can adopt at low and high-redshift are
fixed by the mean galaxy separation of the SDSS MGS and BOSS catalogs;
$\bar{r}_{\rm sdss}\sim 8.3{\rm Mpc}$ and $\bar{r}_{\rm BOSS}\sim 25{\rm Mpc}$
respectively. We cannot smooth below these scales without introducing unknown
non-Gaussian systematics due to shot noise. This philosophy motivated our
choice of $\Lambda_{\rm G}=8.86{\rm Mpc}$ ($=6\,{\rm Mpc}/h$) and
$\Delta=80{\rm Mpc}$, $R_{\rm G}=20{\rm Mpc}$.

Given the different number densities, bias factors and smoothing scales used
in the analysis of the SDSS MGS and BOSS data, the only logical approach to
relate the two is to infer the theoretical expectation $A^{(\rm 2D)}_{\rm G}$
using one of the data sets, and proceed to compare this value to the second.
To apply this method, one must carefully correct for any non-linear effects
such as gravitational collapse using simulations.

##  6\. Parameter Constraints

We are now able to combine the low- and high-redshift measurements to
constrain the distance-redshift relation. To do so, we minimize the following
$\chi^{2}$ function

$\chi^{2}=\sum_{k=1}^{N_{\rm z}}\sum_{j=1}^{N_{\rm z}}{\bf p}_{j}{\rm
cov^{-1}}_{jk}{\bf p}_{k},$ (24)

where ${\bf p}_{j}=A^{(\rm 2D)}_{j}/A^{(\rm 2D)}_{\rm G}-1$ and ${\rm
cov}_{jk}=(\sigma_{{\rm BOSS},j}^{2}+\sigma_{\rm SDSS}^{2})\delta_{jk}$; the
covariance matrix is the sum of the statistical uncertainties on the BOSS and
SDSS measurements. We have used a diagonal covariance matrix for our analysis,
assuming that $A^{\rm(2D)}_{\rm G}$ represents the unbiased, central
theoretical expectation value of the genus amplitude to which we compare our
measured genus values to. A direct comparison between measured values of any
statistic at high and low redshift would introduce correlation between ${\bf
p}_{j}$ components. However, our analysis has used the measured low redshift
data to infer the theoretical expectation value of the genus amplitude. A
direct comparison of $A^{(\rm 2D)}_{\rm G}$ posterior probability
distributions inferred from the SDSS data and twelve BOSS shells, for each
parameter set used to infer the distance redshift relation, would provide a
more rigorous statistical comparison. However, such a procedure is
computationally intractable so we make the simplifying assumption that the
high redshift shells are drawn from a PDF with central value given by the SDSS
LRG value of $A^{\rm(2D)}_{\rm G}$.

The second implicit assumption with our choice of diagonal covariance matrix
is that we have neglected large-wavelength correlations between the SDSS LRG
and BOSS galaxy samples. When generating the covariance matrices for the two-
dimensional genus measurements of the BOSS data, we found no statistically
significant correlation between neighbouring shells. This indicates that the
cross correlation of the genus measurements is negligible.

We fix $h=0.677$ and vary $\Omega_{\rm m}$, $w_{\rm de}$. For each parameter
set $\Omega_{\rm m}$, $w_{\rm de}$, we estimate the distance to the centers of
the $j$ redshift shells using $d_{\rm cm}(z_{j},\Omega_{\rm m},w_{\rm de})$
and reconstruct the genus curves using angular smoothing scales $\theta_{{\rm
G},j}=R_{\rm G}/d_{\rm cm}(z_{j},\Omega_{\rm m},w_{\rm de})$ and effective
area of the data $A_{j}=4\pi f_{\rm sky}d_{\rm cm}^{2}(z_{j},\Omega_{\rm
m},w_{\rm de})$. After measuring the genus curves for the given expansion
history, we calculate the $\chi^{2}$ function (24). The low redshift
measurement $A_{\rm G}^{(\rm 2D)}$ is assumed to be independent of input
cosmological model, as elucidated in section 4.4. Performing a MCMC
exploration of the two-dimensional parameter space, the resulting $1,2-\sigma$
contours (blue) are presented in Figure 8. The tan contour is the $w$CDM
parameter constraint obtained from the Planck 2018 temperature data. If we
combine the two data sets, we obtain a combined constraint on $\Omega_{\rm m}$
and $w_{\rm de}$ (pink contours)333Specifically, we used publicly available
$w$CDM, MCMC chains from the Planck collaboration (Aghanim et al., 2020),
combining (24) and the Planck MCMC likelihoods in quadrature.. The
marginalised parameter constraints for $\Omega_{\rm m},w_{\rm de}$ are
presented in Table 5.

The degeneracy between $\Omega_{\rm m}$ and $w_{\rm de}$, exhibited in the
blue contour in Figure 8, has been found previously (Park & Kim, 2010; Appleby
et al., 2018c). The Planck temperature data presents an almost orthogonal
contour in the $w_{\rm de}$-$\Omega_{\rm m}$ plane, so by combining these two
data sets we can obtain a $\sim 15\%$ constraint on the equation of state of
dark energy, $w_{\rm de}=-1.05^{+0.13}_{-0.12}$.

The sensitivity of our test to $\Omega_{\rm m}$ and $w_{\rm de}$ is relatively
weak as we are restricted to redshifts $z<0.6$. A higher redshift measurement
will improve the constraints considerably. Although the constraint is modest,
the $\Lambda$CDM expansion history is consistent with the data over the
redshift range considered. The constraint from the genus arises almost
entirely from the combination of SDSS MGS low-redshift and CMASS data points;
the LOWZ data have error bars that are too large and lie at a redshift that is
too low to make a strong contribution.

The results are not sensitive to the absolute value of the genus amplitude –
we are extracting information from the difference between different redshift
bins. The absolute value also contains information, related to the shape of
the matter power spectrum, as discussed further in a companion paper Appleby
et al. (2020).

The derived parameter constraints have been obtained under the assumption that
the genus amplitude is a conserved quantity. For non-standard gravity or dark
matter models, the matter power spectrum can possess redshift and scale
dependent corrections. Similarly, we have assumed that dark energy
perturbations do not significantly affect the shape of the matter power
spectrum at low redshift.

Data |  $\Omega_{\rm m}$ |  $w_{\rm de}$
---|---|---
Genus (BOSS+MGS) |  $0.507^{+0.104}_{-0.126}$ |  $-2.24^{+1.07}_{-1.14}$
| Genus (BOSS+MGS)
---
\+ Planck (2018)
$0.303\pm 0.036$ |  $-1.05^{+0.13}_{-0.12}$
Table 5Parameter best fit and $1-\sigma$ uncertainties, obtained by minimizing
the $\chi^{2}$ function (24). After combining our genus likelihood with Planck
2018 temperature data, we obtain the second row.

Figure 8.— [Top panel] Two-dimensional $68,95\%$ contours in the $(\Omega_{\rm
m},w_{\rm de})$ plane obtained by minimizing the $\chi^{2}$ function (24) and
using the genus amplitude as a standard ruler (blue contours). The tan
contours are the marginalised constraints in the $w_{\rm de}$-$\Omega_{\rm m}$
plane obtained from Planck temperature data (Aghanim et al., 2020), and the
pink contours are the result of combining the genus and Planck $\chi^{2}$
functions in quadrature. The black star is the $\Lambda$CDM Planck best fit,
and grey circle the best fit of the combined genus + Planck (pink) contour.
[Bottom panel] The marginalised one-dimensional probability distribution
functions of $w_{\rm de}$. The colour scheme is the same as in the top panel.

##  7\. Discussion

In this work, we have obtained constraints on $w_{\rm de}$ and $\Omega_{\rm
m}$ from the tomographic analysis of the two-dimensional slices of observed
large-scale galaxy distribution. The amplitudes of the two-dimensional genus
curves are measured in a series of concentric slices of density fields derived
from the SDSS BOSS data. The amplitude at low-redshift is derived from the
three-dimensional genus of the SDSS MGS data, and combined to find the
cosmological parameters minimizing the redshift evolution of the genus. In
doing so, we arrive at a constraint of $w_{\rm de}=-2.24^{+1.07}_{-1.14}$, or
$w_{\rm de}=-1.05^{+0.13}_{-0.12}$, $\Omega_{\rm m}=0.303\pm 0.036$ if we
combine our analysis with Planck temperature data (Aghanim et al., 2020). The
parameter constraints arising solely from the genus statistic are particularly
weak; this is due to the strong degeneracy between parameters and also the
limited statistical power that we are able to employ. The presence of shot
noise fundamentally restricts our ability to reconstruct the density field
from the galaxy point distribution, as we must smooth on scales of at least
the mean galaxy separation Kim et al. (2014); Blake et al. (2014); Appleby et
al. (2017). In contrast, methods such as the Alcock-Paczynski (AP) test Li et
al. (2016) (see also Li et al. (2017); Zhang et al. (2019a); Li et al. (2018);
Park et al. (2019); Zhang et al. (2019b)) employ information from very small
scales, eliminating non-perturbative, non-linear systematics using
simulations. In addition, the AP test does not require the application of mass
cuts to generate uniform data samples with redshift, as we are forced to. As a
result, Li et al. (2016) were able to obtain tight parameter constraints on
$w_{\rm de}$ and $\Omega_{\rm m}$ using the same BOSS data. For the genus to
be competitive with other statistics, we must first learn how to model and
remove observational systematics.

Beyond sampling noise, another dominant limitation of the method is in the
comparison of high- and low-redshift measurements, as the low-redshift data
are subject to large statistical uncertainty. This is the dominant
contribution to the parameter uncertainties. The only way to evade this issue
is to smooth the data on smaller scales, but in doing so we are increasingly
exposed to non-linear physics. In this work we corrected the low-redshift,
three-dimensional genus amplitude by factors of $\Delta_{\rm rsd}=0.08$ and
$\Delta_{\rm gr}=0.10$ to account for redshift space distortion and
gravitational collapse. These values were inferred from simulations. Better
theoretical understanding of the non-linear regime and its impact on the genus
curve will be necessary in the future to improve our analysis. Similarly, a
better understanding of the effect of shot noise will allow us to probe
smaller scales; in the current work we regard the mean galaxy separation
$\bar{r}$ of a catalog to be a hard limit below which we are subjected to
unknown non-Gaussian corrections. As the low redshift SDSS MGS is more dense
than the BOSS catalog, we were able to smooth the former on smaller scales and
thus extract more information.

In a companion paper Appleby et al. (2020), we measured the genus curves of
two-dimensional shells of the BOSS data and directly compared their amplitudes
to the Gaussian expectation value. As we smooth the BOSS data with large
scales $\Delta=80\,{\rm Mpc}$, $R_{\rm G}=20\,{\rm Mpc}$, we did not apply any
non-linear correction factors to our measurements, and were able to use the
Kaiser formula to estimate the effect of redshift space distortion. In Appleby
et al. (2020) we placed constraints on cosmological parameters that determine
the shape of the linear matter power spectrum; $\Omega_{\rm c}h^{2}$ and
$n_{\rm s}$. The information extracted in that work came from the absolute
value of the genus amplitude. In the present analysis, we measure the redshift
evolution of the genus amplitude, irrespective of its absolute value. One can
interpret the two approaches as a measure of the initial condition/transfer
function of the dark matter perturbations and a test of the expansion history
respectively. In Appleby et al. (2020), we fixed the distance-redshift
relation using the Planck 2018 best fit cosmology Aghanim et al. (2020) and
measured the genus curves of the BOSS data a single time. We were able to do
this as we restricted our analysis to the $\Lambda$CDM model, and the
constraints obtained in this work are considerably weaker than those obtained
in Appleby et al. (2020). The redshift evolution test considered here is a
measure of distance, and hence is principally sensitive to $\Omega_{\rm m}$
and the equation of state of dark energy.

To improve the parameter constraints, a number of avenues remain open. We can
combine different low-redshift data sets, increasing the effective volume and
reducing the statistical uncertainty. We can calculate analytically the non-
linear corrections due to gravitational smoothing and redshift space
distortion, which will provide a better understanding of the non-linear
effects that we must account for on small scales. In addition, we can apply
our method to high-redshift data, such as Lyman break galaxies. We expect that
a high redshift data point will provide a significantly improved constraint on
the expansion history. As the distance between observer and data increases,
the effect of choosing an incorrect cosmology becomes more pronounced.

Finally, the three-dimensional Minkowski Functionals contain more information
than their two-dimensional counterparts, and a complete analysis of the three-
dimensional field will be forthcoming. In this work, and throughout a series
of papers (Appleby et al., 2017, 2018c, 2020), we have focused on the two-
dimensional genus, extracted from shells of the three-dimensional galaxy
distribution. The reasoning behind this choice is two-fold. First, the BOSS
galaxy catalog is relatively sparse, and we mitigate this issue by taking
thick slices along the line of sight. Binning galaxies in this way is a
smoothing choice, so we can interpret our approach as anisotropic smoothing
perpendicular and parallel to the line of sight. Smoothing on larger scales
parallel to the line of sight allows us to use linear redshift space
distortion physics, which is important as non-linear redshift space distortion
effects on topological statistics are not yet well understood. Second, in
future work we intend to compare our results with higher redshift photometric
redshift catalogs, which will require galaxies to be binned into thick shells.
An understanding of how photometric redshift uncertainty modifies our analysis
must be further explored before this comparison can be made.

## Acknowledgement

SAA is supported by an appointment to the JRG Program at the APCTP through the
Science and Technology Promotion Fund and Lottery Fund of the Korean
Government, the Korean Local Governments in Gyeongsangbuk-do Province and
Pohang City and by a KIAS Individual Grant QP055701 via the Quantum Universe
Center at Korea Institute for Advanced Study. SEH was supported by Basic
Science Research Program through the National Research Foundation of Korea
funded by the Ministry of Education (2018R1A6A1A06024977). We thank the Korea
Institute for Advanced Study for providing computing resources (KIAS Center
for Advanced Computation Linux Cluster System).

Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, and the U.S.
Department of Energy Office of Science. The SDSS-III web site is
http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS-III Collaboration
including the University of Arizona, the Brazilian Participation Group,
Brookhaven National Laboratory, Carnegie Mellon University, University of
Florida, the French Participation Group, the German Participation Group,
Harvard University, the Instituto de Astrofisica de Canarias, the Michigan
State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence
Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max
Planck Institute for Extraterrestrial Physics, New Mexico State University,
New York University, Ohio State University, Pennsylvania State University,
University of Portsmouth, Princeton University, the Spanish Participation
Group, University of Tokyo, University of Utah, Vanderbilt University,
University of Virginia, University of Washington, and Yale University.

The massive production of all MultiDark-Patchy mocks for the BOSS Final Data
Release has been performed at the BSC Marenostrum supercomputer, the Hydra
cluster at the Instituto de Fısica Teorica UAM/CSIC, and NERSC at the Lawrence
Berkeley National Laboratory. We acknowledge support from the Spanish MICINNs
Consolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064, MINECO
Centro de Excelencia Severo Ochoa Programme under grant SEV- 2012-0249, and
grant AYA2014-60641-C2-1-P. The MultiDark-Patchy mocks was an effort led from
the IFT UAM-CSIC by F. Prada’s group (C.-H. Chuang, S. Rodriguez-Torres and C.
Scoccola) in collaboration with C. Zhao (Tsinghua U.), F.-S. Kitaura (AIP), A.
Klypin (NMSU), G. Yepes (UAM), and the BOSS galaxy clustering working group.

Some of the results in this paper have been derived using the healpy and
HEALPix package

## Appendix A – Systematic Effects

In Section 2 we listed three effects that can introduce a small evolution in
the genus amplitude. In the Appendix we consider each point in turn and in
isolation, to confirm that we have all known systematics under control.

We use all sky, mock galaxy lightcone data from the Horizon Run 4 dark matter
simulation project to perform these tests. We direct the reader to Kim et al.
(2015); Hong et al. (2016) for information on the simulation and mock galaxy
catalogs. We use the lightcone data over the range $0.15<z<0.7$, creating
$N=20$ shells and applying mass cuts to generate constant number density
samples, exactly as we did for the BOSS data. We bin the galaxies into shells
of thickness $\Delta=80{\rm Mpc}$ and smooth perpendicular to the plane with
comoving scale $R_{\rm G}\,{\rm Mpc}$. The simulation was performed using a
flat $\Lambda$CDM cosmology with parameters $h=0.72$, $\Omega_{\rm m}=0.26$,
$\sigma_{8}=0.794$, $n_{\rm s}=0.96$.

### 1 - Shot noise

Shot noise is the single largest systematic associated with information
extraction using the genus curve Kim et al. (2014). There are two issues
associated with this phenomenon – it is non-Gaussian and it can potentially
introduce a redshift evolution of the genus amplitude.

First regarding the non-Gaussianity. As a simple approximation, we have
corrected for shot noise by adding a constant white noise contribution to the
total power spectrum; $P_{\rm SN}=1/\bar{n}$. In reality, the noise is a
Poisson process (roughly speaking), but when writing the genus in terms of a
Hermite polynomial expansion as in (1) we have implicitly assumed that the
field is drawn from a perturbatively Gaussian distribution. As a Poisson
distribution possesses a different moment generating function compared to a
Gaussian, we can expect that shot noise will introduce modifications to the
shape of the genus curve. This was observed in both Kim et al. (2014) and
Appleby et al. (2018c). If shot noise becomes significant, then the shape of
the genus will not be well represented by the first few Hermite polynomials
and we lose the interpretation of the genus amplitude as the ratio of second
order cumulants of the perturbatively Gaussian field that we are trying to
measure. In short, the field that we measure is non-Gaussian due to both
gravitational collapse and the nature in which it is sampled. However, only
the gravitational non-Gaussianity is treated in the expansion (1). This issue
is suppressed if we smooth on scales larger than the mean galaxy separation,
in which case the shot noise effect can be approximately represented by the
white noise term $P_{\rm SN}$. This remains an imperfect approximation except
in the limit $R_{\rm G}\gg\bar{r}$.

To present the non-Gaussianity induced by the shot noise sampling, we measure
the genus of two-dimensional shells of the Horizon Run 4 mock galaxy lightcone
in real space, fixing the smoothing scales $\Delta=80{\rm Mpc}$, $R_{\rm
G}=20{\rm Mpc}$ and applying three mass cuts to the data to fix the number
density of our galaxy sample as $\bar{n}_{1}=3.7\times 10^{-4}{\rm Mpc}^{-3}$,
$\bar{n}_{2}=7.4\times 10^{-5}{\rm Mpc}^{-3}$ and $\bar{n}_{3}=3.7\times
10^{-5}{\rm Mpc}^{-3}$. We assume that the most dense sample has a shot noise
contribution that is suppressed, as the mean galaxy separation is much lower
than the smoothing scale $R_{\rm G}=20{\rm Mpc}$. We label the genus curves
$g^{(1)}_{\rm 2D}(\nu_{\rm A})$, $g^{(2)}_{\rm 2D}(\nu_{\rm A})$ and
$g^{(3)}_{\rm 2D}(\nu_{\rm A})$ respectively, where $1,2,3$ superscripts
denote the number density cuts $\bar{n}_{1,2,3}$. We repeat our measurement
for twenty concentric non-overlapping shells and take the average genus curve
to show the effect of shot noise. In Figure 9 (top panel) we exhibit the
average genus curves for $\bar{n}_{1}$ (black), $\bar{n}_{2}$ (red) and
$\bar{n}_{3}$ (blue). Below we also present the difference between the genus
curves $\Delta g_{\rm 2D}^{(2)}=g^{(2)}_{\rm 2D}-g^{(1)}_{\rm 2D}$ (blue) and
$\Delta g_{\rm 2D}^{(3)}=g^{(3)}_{\rm 2D}-g^{(1)}_{\rm 2D}$ (red)
respectively. Clearly the difference between the genus curves is not simply an
amplitude shift – the shape of the residual curve is both shifted and
distorted. This is due to the non-Gaussian nature of the sampling, and is most
significant in the most sparse sample $\bar{n}_{3}$. For the case
$\bar{n}_{2}$, these effects are less pronounced but still present. This
indicates that our treatment of shot noise using the white noise contribution
$P_{\rm SN}$ is imperfect. This effect will be further studied by the authors
in the future.

However, even if the effect of shot noise can be represented by a white noise
term $P_{\rm SN}=1/\bar{n}$, it can still generate a redshift evolution in the
genus amplitude. This is because we are fixing $P_{\rm SN}$ to be constant at
each redshift, but the matter power spectrum has a decreasing amplitude with
increasing redshift. Hence the shot noise term increases in significance to
the past, and will manifest as an increasing genus amplitude with increasing
$z$.

Figure 9.— [Top panel] The average genus curves extracted from all-sky mock
galaxy lightcone data, taking three mass cuts to fix a constant number density
in each shell $\bar{n}_{1}=3.7\times 10^{-4}{\rm Mpc}^{-3}$ (black),
$\bar{n}_{2}=7.4\times 10^{-5}{\rm Mpc}^{-3}$ (red), $\bar{n}_{3}=3.7\times
10^{-5}{\rm Mpc}^{-3}$ (blue). We also exhibit the difference $\Delta g_{\rm
2D}$ between the sparse samples and the dense catalog. [Middle panel] The
Gaussian prediction for the genus amplitude in real space, for a model with no
shot noise (grey) and with the fiducial number density $\bar{n}=6.25\times
10^{-5}{\rm Mpc}^{-3}$ (yellow). Shot noise introduces significant evolution
in the genus amplitude. [Bottom panel] Genus amplitudes extracted from two-
dimensional slices of dark matter particle snapshot boxes for a sparse
$N=64^{3}$ (gold points) and dense $N=512^{3}$ (grey points) sample. The solid
lines are the Gaussian prediction.

To show the hypothetical redshift evolution of the genus amplitude, in Figure
9 (middle panel) we present the theoretical expectation $A_{\rm G}^{(\rm 2D)}$
as a function of redshift, using (2) with power spectrum (6) and taking the
real space power spectrum (that is, setting $\beta=0$ in equation (6)). We
plot the amplitude assuming negligible shot noise, setting an arbitrarily high
hypothetical number density $\bar{n}=2\times 10^{8}{\rm Mpc}^{-3}$ (grey line)
and the fiducial number density of the BOSS catalog used in this work
$\bar{n}=6.25\times 10^{-5}{\rm Mpc}^{-3}$ (yellow line), both with constant
linear galaxy bias $b=2$. The grey line represents the idealised case and as
expected is constant; in this instance the genus is a measure of the shape of
the linear matter power spectrum. The yellow curve represents a hypothetical
sparse galaxy catalog with constant large scale galaxy bias – the shot noise
contribution causes the genus amplitude to evolve with redshift.

To confirm this behaviour, we take simulated dark matter particle snapshot
boxes of volume $V=(1024{\rm Mpc}/h)^{3}$ at $z=0,0.5,1$ and sampled $512^{3}$
(dense) and $64^{3}$ (sparse) particles randomly. We then construct flat, two-
dimensional slices of thickness $\Delta=80\,{\rm Mpc}$ and smoothed in the
plane with $R_{\rm G}=20\,{\rm Mpc}$ Gaussian kernel. The we extract the genus
from these fields and then the genus amplitudes. The results are presented in
the bottom panel of Figure 9. The points/error bars are the mean and standard
deviation of $15$ slices of the snapshot boxes, with the yellow/grey points
corresponding to the sparse and dense samples respectively. The solid
yellow/grey lines are the Gaussian expectation value for the given number
density (and bias factor $b=1$, as we are using dark matter particles). The
behaviour of the middle panel is reproduced, the sparse sample exhibits a
systematic evolution in redshift, but the dense sample is conserved.

In this work we have fixed the galaxy bias of the BOSS galaxies to be
constant, $b=2$. If the bias evolves with redshift, then this must also be
taken into account when assessing the effect of shot noise. The net effect
depends on the relative amplitude of the galaxy power spectrum – hence
$b^{2}(z)D^{2}(z)A_{\rm s}$ – and the shot noise term $P_{\rm SN}$, where
$D^{2}(z)$ is the growth rate and $A_{\rm s}$ is the primordial amplitude.

### 2 - Redshift space distortion

The effect of redshift space distortion is to decrease the genus amplitude by
$\sim 8\%$, and introduce a mild redshift dependence. To show this effect, in
Figure 10 (top panel) we present the ratio of $A_{\rm G}^{(\rm 2D)}$ redshift
and real ($\beta=0$) space, obtained from the theoretical expectation (2)
assuming negligible shot noise (that is, fixing $\bar{n}=2\times 10^{8}{\rm
Mpc}^{-3}$) and different galaxy bias values. The green solid line is the
fiducial, constant galaxy bias used in this work $b=b_{0}$. We also exhibit
$a_{\rm rsd}$ for different linear galaxy bias models $b(z)=b_{0}+b_{1}z$;
$(b_{0},b_{1})=(1.8,0),(1.8,0.5),(1.8,1)$ (yellow solid, black dash-dot and
red dashed lines respectively). We also present the values of $a_{\rm rsd}$
inferred from the Horizon Run 4 all-sky mock galaxy shells as pale red points.
For our fiducial choice $b=2$, the genus amplitude decreases by $\sim{\cal
O}(8\%)$ and decreases with increasing redshift when measured in redshift
space (green line). For different bias factors, the redshift dependence of
$a_{\rm rsd}$ can change significantly (cf yellow, black, red lines).

In the middle panel of the figure, we plot $A_{\rm G}^{(\rm 2D)}$ in redshift
space for two different number densities – $\bar{n}=2\times 10^{8}{\rm
Mpc}^{-3}$ (red) and fiducial number density used in this work
$\bar{n}=6.25\times 10^{-5}{\rm Mpc}^{-3}$ (blue), fixing $b=2$. In the
absence of shot noise, the genus amplitude decreases with redshift (red line),
however as shown in the previous section shot noise acts to increase the genus
amplitude with $z$. The net effect is that redshift space distortion decreases
and shot noise increases $A_{\rm G}^{(\rm 2D)}$, with the result that the
measured genus amplitude of the galaxy catalogs should remain approximately
constant over the redshift range considered in this work $0.1<z<0.7$.
Specifically, the measured genus amplitude should follow the blue curve in the
bottom panel. We stress however, that this argument is sensitive to the galaxy
sample. Different bias factors, number densities and redshift ranges will not
necessarily yield a genus amplitude that is conserved with redshift.

We again confirm our hypothesis that redshift space distortion introduces a
mild dependence of the genus amplitude on redshift by extracting $A^{(\rm
2D)}$ from slices of dark matter particle data from our simulation. We take
$z=0,0.5,1$ snapshot boxes of volume $V=(1024{\rm Mpc}/h)^{3}$, sub-sample
$512^{3}$ particles from the full data then perturb the particles along the
$x_{3}$ direction according to their velocities to create plane, parallel
redshift space distorted slices. We then extract the two-dimensional genus
amplitude from slices using $\Delta=80{\rm Mpc}$, $R_{\rm G}=20{\rm Mpc}$ as
before. The results are exhibited in the bottom panel of Figure 10. The
grey/red points and error bars are the mean and standard deviation of $15$
slices of the snapshot boxes in real/redshift space respectively. We observe
no evolution in real space, but a systematic decrease in the genus amplitude
in redshift space.

Figure 10.— [Top panel] The ratio of genus amplitudes $A_{\rm G}^{(\rm 2D)}$
as measured in redshift and real space, for different bias models
$b(z)=b_{0}+b_{1}z$. The green line corresponds to the fiducial values used in
this work $b_{0}=2$, $b_{1}=0$ in the galaxy bias model $b(z)=b_{0}+b_{1}z$.
The red points are the values of $a_{\rm rsd}$ inferred from the mock galaxies
from the Horizon Run 4 lightcone in real and redshift space. Generally, the
effect of linear redshift space distortion is to decrease the genus amplitude
by $\sim 8\%$ and introduce a weak redshift dependence. [Middle panel] The
expectation value of the genus amplitude in redshift space, taking $b=2$,
$\bar{n}=2\times 10^{8}{\rm Mpc}^{-3}$ (red) and $\bar{n}=6.25\times
10^{-5}{\rm Mpc}^{-3}$ (blue). The measured genus amplitude from the BOSS data
should trace the blue curve. [Bottom panel] The genus amplitude extracted from
dark matter snapshot boxes for a dense sample $512^{3}$ in real (grey) and
redshift (red) space.

Future dense and high redshift galaxy catalogs will not suffer from the many
of the issues discussed in this work. For these data, the shot noise
contribution will be significantly reduced. In this case, the correct course
of action would be to correct the measured genus amplitudes by a
multiplicative factor to convert them to real space Matsubara (1996), after
which they should be conserved with redshift. This procedure was undertaken
for mock galaxies in Appleby et al. (2018b).

### 3 - Gravitational smoothing

It is well known that higher order corrections in the non-Gaussian expansion
of the genus curve $\sim{\cal O}(\sigma_{0}^{2})$ will modify the genus
amplitude ; empirically it has been observed that the genus amplitude
decreases on small scales compared to the Gaussian expectation value when
measured from galaxy catalogs Melott et al. (1989); Park & Gott (1991); Park
et al. (2005). To test the magnitude of this effect, we measure the
coefficient of the $a_{3}$ Hermite polynomial expansion of the genus curve –

$g_{\rm 2D}\simeq a_{1}e^{-\nu_{\rm
A}^{2}/2}\left[a_{0}H_{0}+H_{1}+a_{2}H_{2}+a_{3}H_{3}\right].$ (25)

According to the non-Gaussian perturbative expansion of the genus Matsubara
(1994b, 2000); Pogosyan et al. (2009); Gay et al. (2012); Codis et al. (2013),
$a_{1}$ is the genus amplitude, $a_{0,2}$ are the first order corrections of
order $a_{0,2}\sim{\cal O}(\sigma_{0})$ and we can expect $a_{3}$ will be
induced at second order $a_{3}\sim{\cal O}(\sigma_{0}^{2})$. We therefore use
this term as a proxy to estimate the magnitude of higher order corrections to
the genus amplitude.

We extract $a_{0},a_{2},a_{3}$ from the twenty all-sky lightcone shells of
Horizon Run 4, in redshift space, by integrating the genus curve using

$a_{n}={1\over n!}{\int_{-4}^{4}d\nu_{\rm A}g_{\rm 2D}(\nu_{\rm
A})H_{n}(\nu_{\rm A})\over\int_{-4}^{4}d\nu_{\rm A}g_{\rm 2D}(\nu_{\rm
A})H_{1}(\nu_{\rm A})},$ (26)

taking $\nu_{0}=4$. In Figure 11 we present $a_{0},a_{2},a_{3}$ (grey, blue,
red). The red curve is the next to leading order correction term $a_{3}$.
There is some suggestion that $a_{3}$ is increasing with decreasing redshift,
from $\sim 0.02$ at $z=0.25$ to $\sim 0.01$ at $z=0.6$. Although the effect is
small and the statistical uncertainty large, the higher order, non-linear
corrections require further study. The $a_{3}$ term is present at the $1\%$
level at the scales probed.

The $a_{0},a_{2}$ coefficients are perturbatively small at the scales studied
in this work. These terms can be interpreted as integrals over the Bispectrum,
and contain complementary information to the amplitude studied in this work.
$a_{2}$ exhibits some evidence of evolution over the redshift range under
consideration.

Figure 11.— The Hermite polynomial coefficients $a_{0},a_{2},a_{3}$ (grey,
blue, red) obtained from twenty shells of all-sky mock galaxy data. There is
no strong evidence of evolution of $a_{3}$.

### 4 - Lack of high threshold critical points

The first three issues described above are physical effects. The fourth is a
purely spurious systematic that can be introduced into the analysis if we
improperly select the $\nu_{\rm A}$ threshold range. Specifically, one can
observe evolution of the genus amplitude with redshift if we measure the genus
over threshold values that are too high. The reason for this lies in the
relation between $\nu_{\rm A}$ and $\nu$. The $\nu_{\rm A}$ parameterisation
of the genus curve selects thresholds that have the same area fraction as a
Gaussian random field. However, since the galaxy catalogs occupy a finite
area, high threshold peaks will not be represented within the observed domain,
and the area fraction will be systematically under-represented compared to a
hypothetical Gaussian random field of arbitrarily large extent. This leads to
an increase in the genus curve in the high $\nu_{\rm A}$ tails, which
increases the genus amplitude. This can introduce spurious redshift evolution
because the area of the data at low redshift is smaller than at high redshift,
and so the low-$z$ regime will contain fewer high threshold peaks.

We can eliminate this effect by restricting our analysis to $\nu_{\rm A}$
threshold values that are well sampled at each redshift. To present the
effect, we take the twenty all-sky lightcone mock galaxy shells from the
Horizon Run 4 simulation in real space, smooth them and then apply a mask,
only keeping a spherical cap of data of radius $\theta_{\rm
cap}=\pi/(2\sqrt{2})\,{\rm rad}$. We select this value as the area fraction of
such a cap roughly matches the area of the BOSS mask. We then measure the
genus of this subset of data over the threshold ranges $-4<\nu_{\rm A}<4$ and
$-2.5<\nu_{\rm A}<2.5$. As a proxy for the genus amplitude, we use the
following integral

$A^{(\rm 2D)}\simeq{1\over\sqrt{2\pi}}\int_{-\nu_{0}}^{\nu_{0}}g_{\rm
2D}\nu_{\rm A}d\nu_{\rm A},$ (27)

with $\nu_{0}=2.5,4$. As $\nu_{0}\to\infty$, the integral (27) approaches the
exact genus curve amplitude. In Figure 12 we present $A^{(\rm 2D)}$ for
$\nu_{0}=4$ (blue points) and $\nu_{0}=2.5$ (red squares) from the twenty
slices. We also show the mean value of the points as similarly coloured
horizontal lines. The exact value of $A^{(\rm 2D)}$ is not relevant to our
discussion, the important point is the clear redshift evolution in the blue
points, which is due to selecting a large value $\nu_{0}=4$. For the more
conservative choice $\nu_{0}=2.5$, no redshift evolution is detected relative
to the mean value (red points). This indicates that peaks in the range
$-2.5<\nu_{\rm A}<2.5$ are suitably well represented over the range
$0.2<z<0.6$ and motivates our choice $-2.5<\nu_{\rm A}<2.5$ in the main body
of the text.

Figure 12.— The amplitude proxy $A^{(\rm 2D)}$ defined in equation (27),
measured from the twenty shells of lightcone data. The red/blue points
correspond to $\nu_{0}=2.5$, $\nu_{0}=4$ respectively. The solid horizontal
lines are the mean values of the respective points. One can clearly observe a
systematic evolution in the blue points, due to the lack of high threshold
maxima/minima at low-$z$.

## Appendix B – variation of $\Delta$

Finally, we check that the genus amplitudes extracted from the data are
insensitive to the small variations in shell thickness $\Delta$ induced by
selecting different cosmological models to infer the distance-redshift
relation. Although the genus is a function of the thickness $\Delta$, we will
argue that for large $\Delta$ this sensitivity is low and can be neglected.

To show this, we take the Horizon Run 4 all-sky mock galaxy lightcone, and use
four different cosmological models to fix the redshift boundaries of the
shells. For each cosmology we select redshift limits of the shells $z_{\rm
min}$, $z_{\rm max}$ such that the comoving distance $d_{\rm cm}(z_{\rm
max},\tilde{\Omega}_{\rm m},\tilde{w}_{\rm de})-d_{\rm cm}(z_{\rm
min},\tilde{\Omega}_{\rm m},\tilde{w}_{\rm de})=\tilde{\Delta}=80{\rm Mpc}$,
where tildes indicate incorrect cosmological parameters that are presented in
Table 6, with model $0$ being the correct, fiducial cosmology of the
simulation. The true values of the slice thicknesses are given by
$\Delta=d_{\rm cm}(z_{\rm max},\Omega_{\rm m},w_{\rm de})-d_{\rm cm}(z_{\rm
min},\Omega_{\rm m},w_{\rm de})$ with $\Omega_{\rm m}=0.26$, $w_{\rm de}=-1$.
In Figure 13 (top panel) we present $\Delta(z)$ as a function of $z$ for each
of the cosmological models used to infer the distance redshift relations of
the shells. For each cosmological model we have selected redshift limits such
that $\tilde{\Delta}=80{\rm Mpc}$, independent of redshift, but the true value
of $\Delta$ (obtained by using the true cosmology) is evolving.

After fixing the redshift shell limits using the incorrect cosmological
models, we proceed to calculate the genus in the twenty data shells using the
correct cosmological model. We do this as we wish to isolate the effect of a
systematically evolving $\Delta$ thickness. We measure the genus curves and
extract the amplitudes. In Figure 13 (bottom panel) we present the genus
amplitude $A^{(\rm 2D)}(\tilde{\Delta})$. For clarity we plot the average and
standard deviation of every four shells. One can observe no systematic
evolution with redshift for any of the cosmological models selected, and the
statistical uncertainty is dominant. This insensitivity is because we are
using relatively thick slices $\Delta\sim 80{\rm Mpc}$; thinner slices will
exhibit stronger cosmological parameter sensitivity.

Model |  $\tilde{\Omega}_{\rm m}$ |  $\tilde{w}_{\rm de}$
---|---|---
0 |  $0.26$ |  $-1$
I |  $0.26$ |  $-1.2$
II |  $0.26$ |  $-0.8$
III |  $0.2$ |  $-1$
IV |  $0.32$ |  $-1$
Table 6The four models used in Appendix B to test the effect of variable
$\Delta$ slice thickness on the genus amplitude. The $0$ model is the fiducial
model of the simulation, and yields a constant $\Delta=80{\rm Mpc}$ slice
thickness.

Figure 13.— [Top panel] The redshift evolution of the shell thickness
$\Delta$, if we use an incorrect cosmological model to infer the redshift
limits of the shells. The $0$ model is the fiducial (correct) model of the
simulation, and I-IV are incorrect models with parameters given in Table 6.
[Bottom panel] The measured genus amplitudes of all-sky lightcone shells of
the mock galaxy data, using the shell widths presented in the top panel. No
systematic evolution of the genus amplitude is found as a result of selecting
an incorrect slice thickness.

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