Cosmological Parameter Estimation from the Two-Dimensional Genus Topology -- Measuring the Expansion History using the Genus Amplitude as a Standard Ruler
Stephen Appleby, Changbom Park, Sungwook E. Hong, Ho Seong Hwang, Juhan Kim, Motonari Tonegawa
DDraft version February 3, 2021
Preprint typeset using L A TEX style emulateapj v. 12/16/11
COSMOLOGICAL PARAMETER ESTIMATION FROM THE TWO-DIMENSIONAL GENUS TOPOLOGY -MEASURING THE EXPANSION HISTORY USING THE GENUS AMPLITUDE AS A STANDARD RULER
Stephen Appleby
Asia Pacific Center for Theoretical Physics, Pohang, 37673, Korea andQuantum 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 ( 홍 성 욱 ) Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Korea andNatural 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
Draft version February 3, 2021
ABSTRACTWe measure the genus of the galaxy distribution in two-dimensional slices of the SDSS-III BOSScatalog to constrain the cosmological parameters governing the expansion history of the Universe. TheBOSS catalogs are divided into twelve concentric shells over the redshift range 0 . < z < . z < .
12. We combine the low- and high–redshift measurements, finding the cosmological model which minimizes the redshift evolution of thegenus amplitude, using the fact that this quantity should be conserved. Being a distance measure, thetest is sensitive to the matter density parameter (Ω m ) and equation of state of dark energy ( w de ). Wefind a constraint of w de = − . +0 . − . , Ω m = 0 . ± .
036 after combining the high- and low–redshiftmeasurements and combining with Planck CMB data. Higher redshift data and combining data setsat low redshift will allow for stronger constraints. INTRODUCTION
Topological statistics have a long history of use withincosmology Gott et al. (1990); Park & Gott (1991); Meckeet al. (1994); Schmalzing & Buchert (1997); Schmalz-ing & Gorski (1998); Hikage et al. (2006); Ducout et al.(2013); Melott et al. (1989); Park et al. (1992); Gottet al. (1992). Theoretical studies of random fields havebeen undertaken; both Gaussian (Doroshkevich 1970;Adler 1981; Gott et al. 1986; Hamilton et al. 1986; Ry-den 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; Matsub-ara & 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 thatgo beyond the standard Minkowski functional analysis; [email protected]
Minkowski tensors Beisbart et al. (2001b,a); Ganesan &Chingangbam (2017); Chingangbam et al. (2017); Ka-pahtia et al. (2019); Appleby et al. (2018a,b); Kapahtiaet 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 Sous-bie et al. (2011); Codis et al. (2018); Kraljic et al. (2020).Previous application of the Minkowski functionals to var-ious 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 cos-mological scale, large scale structure data has allowedmeasurements of the higher point functions induced by a r X i v : . [ a s t r o - ph . C O ] F e b gravitational collapse Wiegand et al. (2014); Wiegand &Eisenstein (2017); Buchert et al. (2017); Sullivan et al.(2019) that would be difficult to extract using conven-tional N -point methods.The genus belongs to the family of Minkowski Func-tionals. The genus of the matter density field, as tracedby galaxies, can be used as a cosmological probe. Bymeasuring the genus curve at different redshifts, one canextract information regarding the parameters governingthe expansion history of the Universe. The redshift de-pendence of the genus amplitude was originally proposedas a standard ruler in Park & Kim (2010); Zunckel et al.(2011). For the ΛCDM model, the amplitude of the genuscurve is related to the slope of the linear matter powerspectrum, which does not evolve with redshift. By com-paring this quantity at high and low redshift, we shoulddetect no evolution. However, if we select an incorrectcosmological model to infer the distance-redshift rela-tion, then comoving smoothing scales and volumes be-come systematically incorrect with increasing redshift.This will generate a spurious evolution in the statistic.Hence, by measuring the genus using different cosmo-logical models to infer distance scales, one can find theexpansion history that conserves this statistic.This cosmological test was first proposed in Park &Kim (2010). More recently, the authors have revisitedthis possibility and applied the method to projectedtwo dimensional galaxy density fields, using all-sky mockgalaxy lightcone data (Appleby et al. 2017, 2018c). Theanalysis presented here provides a conclusion of theseworks, as we apply the methodology to a combinationof low- and high-redshift galaxy catalogs to obtain aconstraint on the cosmological parameters Ω m and darkenergy equation of state w de . This test was pursued inBlake et al. (2014), with the first direct application ofthe method to galaxy data (specifically the WiggleZ sur-vey Blake et al. (2011)). Competitive distance measure-ments were obtained from three-dimensional Minkowskifunctional measurements, and issues associated with thismeasurement (principally sparse sampling) were high-lighted. The conclusion of the work was that topologyis potentially competitive with Baryon Acoustic Oscilla-tions (BAO) as a standard ruler, however the physics andassumptions that go into the analysis are more involved,as the Minkowski functionals measure the shape of thefull extent of the power spectrum in an integrated sense.In this work we measure the genus of both theBOSS, 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 MGSdata provides a robust measure of the genus amplitudeat low redshift, practically insensitive to the distance-redshift relation. In contrast, the higher redshift BOSSdata will be sensitive to our choice of cosmological pa-rameters when inferring distances. If we select an incor-rect distance-redshift relation, the genus amplitude ex-tracted from the BOSS data will systematically evolve,relative to the low redshift measurement. The reason forthis effect is that an incorrect choice of comoving dis-tance will cause us to select erroneous smoothing scalesand effective areas, meaning that we will be measuringthe slope of the matter power spectrum at different scalesas a function of redshift. As the matter power spectrumis not scale invariant, this will manifest as an evolving genus amplitude.The principal challenge when using the genus as a stan-dard ruler is that we must compare high redshift mea-surements to low redshift counterparts. However, the lowredshift Universe is restricted in volume and the statisti-cal uncertainty provides the dominant limitation on pa-rameter constraints. To mitigate this problem, we mea-sure the genus of the full three-dimensional field at lowredshift. We then convert the three-dimensional measure-ment into a constraint on the theoretical expectation ofthe two-dimensional genus amplitude.The paper will proceed as follows. In Section 2 we dis-cuss some of the issues associated with using the genusamplitude as a standard ruler, and our method of ex-tracting this quantity from galaxy data. We briefly re-view the extraction of the genus from two-dimensionalshells of BOSS data in Section 3. In Section 4 we de-tail the data, mask, mock catalogs and systematics as-sociated with SDSS MGS measurement of the three-dimensional genus. The conversion from three dimen-sional measured genus to the theoretical expectationvalue of the two-dimensional genus amplitude is ex-plained in Section 5. Finally in Section 6 we place con-straints on cosmological parameters, then close with adiscussion 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 powerspectrum. We discuss the relation between the two ap-proaches in Section 7. 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) g ( ν A ) = A (2D)G e − ν / (cid:20) H ( ν A ) + (cid:20) (cid:16) S (1) − S (0) (cid:17) × H ( ν A ) + 13 (cid:16) S (2) − S (0) (cid:17) H ( ν A ) (cid:21) σ + O ( σ ) (cid:21) , (1)where A (2D)G is the amplitude A (2D)G ≡ π ) / σ σ , (2)and the skewness parameters S (0) , S (1) , S (2) are relatedto the three point cumulants and will not be used here. σ and σ are defined as integrals over the power spectrum,in this work smoothed with Gaussian kernels of comovingscale R G σ = 1(2 π ) (cid:90) d k ⊥ e − k ⊥ R P ( k ⊥ , z ) , (3) σ = 1(2 π ) (cid:90) d k ⊥ k ⊥ e − k ⊥ R P ( k ⊥ , z ) , (4)and the projected two-dimensional power spectrum P ( k ⊥ , z ) is related to its full three-dimensional coun-terpart according to P ( k ⊥ , z ) = 2 π (cid:90) dk (cid:107) P ( k, z ) sin [ k (cid:107) ∆] k (cid:107) ∆ , (5)where ∆ is the comoving thickness of the two-dimensional slices of the field. (cid:126)k ⊥ and k (cid:107) are the wavenumbers perpendicular and parallel to the line of sightrespectively. The three-dimensional power spectrum ofthe density field that is traced by galaxies is the sum ofthe redshift-space distorted matter field and a shot noisecontribution P ( k, k (cid:107) , z ) = b (cid:32) β k (cid:107) k (cid:33) P m ( z, k ) + P SN , (6)where P m ( z, k ) is the matter power spectrum at redshift z , P SN is the shot noise power spectrum P SN = 1 / ¯ n ,where ¯ n is the number density of galaxies. We introduce β = f /b , b is the linear galaxy bias and f is the growthfactor. The quantity ν A is the density threshold such thatthe excursion set has the same area fraction as a corre-sponding Gaussian field - f A = 1 √ π (cid:90) ∞ ν A e − t / dt, (7)where f A is the fractional area of the field above ν A .This choice of ν 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 amplitudeis a measure of the shape of the linear matter powerspectrum P m ( z, k ), which is a conserved quantity forthe ΛCDM model and certain generalisations (such as w CDM, assuming dark energy perturbations are negligi-ble). If we use an incorrect cosmological model to inferthe distance-redshift relation, then we get the smoothingscale R G and volume occupied by galaxy data system-atically wrong at different redshifts. Hence we will mea-sure the shape of the power spectrum at different scaleswhen using an incorrect expansion history. As a result,the genus amplitude that we extract from the data willspuriously evolve with redshift if we get the expansionhistory wrong. A low redshift measurement will representthe ‘true’ genus amplitude having little dependence onthe cosmology adopted, against which high redshift mea-surements can be compared. This effect was predicted in(Park & Kim 2010) and explicitly measured using mockgalaxies in Appleby et al. (2018c).In reality a number of small systematic effects arepresent in real galaxy data that generate redshift evo-lution of this statistic. The primary sources of contami-nation are as follows, listed in order of severity1. We bin galaxies into redshift shells and apply amass cut to fix the number density of tracers ateach redshift to be constant, thus fixing a constantshot noise power spectrum P SN in each shell. Incontrast, the amplitude of the matter power spec-trum P m ( z, k ) decreases with redshift. It follows that the relative importance of the shot noise con-tribution in (6) will increase with redshift, whichwill manifest as an increasing genus amplitude athigher z . This effect depends on R G relative tothe mean galaxy separation ¯ r , and is negligible for R G (cid:29) ¯ r .2. Linear redshift space distortion decreases the am-plitude of the two-dimensional genus by around ∼ < z < .
7. However, it also introduces a mild ∼
1% redshift dependent evolution, decreasing theamplitude with increasing redshift. This is due tothe redshift dependence of β ( z ) in equation (6).3. Non-linear gravitational evolution will typically actto decrease the genus amplitude with decreasingredshift, which is an O ( σ ) effect (so-called grav-itational smoothing Melott et al. (1989); Park &Gott (1991); Park et al. (2005)).The magnitude of each of these effects depends on thenumber density of galaxies, the smoothing scales perpen-dicular and parallel to the line of sight ( R G and ∆) andthe area of the data. In Appendix A we use mock galaxylightcone data to examine these effects in isolation, andargue that for the data and smoothing scales used inthis work, no significant redshift evolution of the genusamplitude will be induced.To briefly summarise the results in Appendix A : Wetake constant comoving scale R G = 20Mpc to Gaussiansmooth the data perpendicular to the line of sight, andcomoving slice thickness ∆ = 80 Mpc along the line ofsight. At these scales, the redshift space distortion andshot noise effects both introduce an evolution of the genusamplitude of order ∼
1% over the redshift range 0 < z < .
7. Shot noise/redshift space distortion causes the genusamplitude to increase/decrease with increasing z . Thetwo competing effects effectively cancel for the particulargalaxy sample considered in this work. Furthermore, themean galaxy separation of the two-dimensional projectedfields is approximately ¯ r (cid:39)
15 Mpc, smaller than R G =20 Mpc. This makes the non-Gaussianity of the shot noisecontribution small.The non-Gaussian gravitational corrections to the am-plitude are small. We quantify this statement by measur-ing the next-to-leading-order correction term a H ( ν A ),finding it to be ∼ O (1%) at the scales probed. Non-Gaussian corrections are suppressed when the area frac-tion threshold ν A is used to define the excursion set asopposed to the standard threshold ν . We find no evidenceof evolution of a over the range 0 . < z < . | ν | are more difficultto be sampled in a smaller area. Whenever the excursionset is poorly sampled, the genus amplitude will generi-cally be biased high. To eliminate this bias, we must onlymeasure the genus curve over a range of threshold values − ν < ν < ν for which the excursion set is well sam-pled at all redshifts. We vary the threshold limit ν tocheck that the data provides an unbiased measurementof the genus curve. The range | ν A | < . LOWZ CMASS0 . < z ≤ .
271 0 . < z ≤ . . < z ≤ .
292 0 . < z ≤ . . < z ≤ .
313 0 . < z ≤ . . < z ≤ .
334 0 . < z ≤ . . < z ≤ .
356 0 . < z ≤ . . < z ≤ .
378 0 . < z ≤ . TABLE 1The redshift limits of the LOWZ and CMASS shells usedin this work. sented within our shells, so we measure the genus curveover this range. OBSERVATIONAL DATA 0 . < z < . Our treatment of the high redshift data – SDSS-IIIBaryon Oscillation Spectroscopic Survey (BOSS) (Yorket al. 2000) – has been described in detail in Applebyet al. (2020). To briefly review, we bin the galaxies into N z = 12 shells of comoving thickness ∆ = 80 Mpc, 6/6from the LOWZ and CMASS data, over the range 0 . 6. We apply a mass cut to fix the number densityas ¯ n = 6 . × − (Mpc) − within each shell. With thischoice, the shot noise contribution to the field is largebut the clustering signal is dominant for the smoothingscales adopted in this work. The galaxies are weightedto account for observational systematics. Specifically, thefollowing weight was applied to each galaxy in the LOWZand CMASS sample w tot = w systot ( w cp + w noz − 1) (8)where w cp is the correction factor to account for the sub-sample of galaxies that are not assigned a spectroscopicfibre, w noz is for the failure in the pipeline to assign red-shifts due to certain galaxies, and w systot represents non-cosmological fluctuations in the CMASS target densitydue to stellar density and seeing.The redshift bin limits are presented in Table 1; thesewere derived using the Planck cosmological parameters w de = − 1, Ω m = 0 . 307 to define slices of constant comov-ing thickness ∆ = 80 Mpc. We should vary these limitsand re-bin the galaxies each time we vary the cosmologyin the distance redshift relation. However, because thegenus amplitude is insensitive to ∆ for thick slices, wecan fix these limits throughout without biasing our re-sults. We provide evidence to support this statement inAppendix B.HEALPix (Gorski et al. 2005) is used to bin thegalaxies into pixels on the unit sphere. A galaxy num-ber density field δ i,j ≡ ( n i,j − ¯ n j ) / ¯ n j is defined, where1 ≤ j ≤ N z denotes the redshift bin (of which thereare N z = 12 in total) and 1 ≤ i ≤ N pix is the pixelidentifier on the unit sphere. ¯ n j is the mean number ofgalaxies 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 pix = 12 × pixels. Thesurvey geometry and veto masks Reid et al. (2016) werethen used to generate a binary healpix map : Θ i = 1 ifthe survey angular selection function in the i th pixel islarger than some cutoff Θ cut = 0 . i = 0 otherwise, http://healpix.sourceforge.net where i runs over N pix pixels. The Θ i mask was appliedto the galaxy field δ i,j .We smooth the two-dimensional density fields, andthe Θ i mask, in each shell using angular scale θ G = R G /d cm ( z j , Ω m , w de ), where R G = 20Mpc is the comov-ing smoothing scale and d cm ( z j , Ω m , w de ) is the comovingdistance to the center of the j th redshift shell. Defining˜Θ i,j and ˜ δ i,j as the smoothed mask and density fields,we re-define ˜ δ i,j = 0 if ˜Θ i,j < Θ cut and ˜ δ i,j → ˜ δ i,j / ˜Θ i,j otherwise. Finally, we re-apply the original unsmoothedΘ i mask. This procedure eliminates regions close to theboundary, where the field may not be well reconstructed.In Appendix C of Appleby et al. (2020) we explicitly showthat our masking procedure, and method of genus extrac-tion, provides an unbiased estimate of the genus, and wedirect the reader to this paper for further details. Theimportant underlying point is that we are extracting thegenus per unit area, which is a local quantity and hencecan be estimated in an unbiased manner from a cut-skygalaxy sample.Finally we divide the genus by the total area of thedata A j = 4 πf sky d ( z j , Ω m , w de ), where f sky is the frac-tional area of the data on the sky. The genus is recon-structed using the method described in Schmalzing &Gorski (1998); Appleby et al. (2018c), which provides anunbiased estimate of the full sky genus from an observedpatch.We measure the genus for 200 values of the thresh-old ν A , equi-spaced over the range − . < ν A < . N ν A = 50 measurements. We label the measured values g nj , where j runs over the redshift shells and 1 ≤ n ≤ N ν A over the N ν A = 50 thresholds. We then extract the genusamplitudes A (2D) j by minimizing the following χ func-tions at each redshift – χ j = N ν A (cid:88) n =1 N ν A (cid:88) m =1 ∆ g nj Σ − n,m ( z j )∆ g mj , (9)with respect to the parameters A (2D) j , a ,j , a ,j , a ,j ,where∆ g nj = g nj − A (2D) j e − ν ,n / [ a ,j H ( ν A ,n )+ H ( ν A ,n ) + a ,j H ( ν A ,n ) + a ,j H ( ν A ,n )] , (10)and Σ n,m ( z j ) are the covariance matrices associatedwith ∆ g nj . Σ n,m ( z j ) are obtained using the patchy mockgalaxy catalogs (Kitaura et al. 2016; Rodr´ıguez-Torreset al. 2016; Kitaura et al. 2014; Kitaura et al. 2015) –further information on the covariance matrices used inour analysis can be found in Appleby et al. (2020).The measured genus values g nj are functions of thedistance-redshift relation, and hence the cosmologicalparameters (Ω m , w de ). This parameter sensitivity entersin the definition of the angular smoothing scale θ G = R G /d cm ( z j , Ω m , w de ) and the area occupied by the data A j = 4 πf sky d ( z j , Ω m , w de ). We repeat our measure-ment of g nj and minimization of (10) for each cosmolog-ical parameter set. We fix h = 0 . 677 to its Planck valuethroughout, where H = 100 h km s − Mpc − . Parameter Fiducial ValueΩ m . h . w de − 1∆ 80Mpc R G TABLE 2Fiducial parameters used to fix the slice thickness, andthe fiducial parameters used to calculate the genus inthis work. ∆ is the thickness of the two dimensional slicesof the density field, and R G is the Gaussian smoothingscale used in the two-dimensional planes perpendicular tothe line of sight. Redshift A × Mpc − χ / DoF0.26 4 . ± . 16 1.490.28 4 . ± . 15 1.120.30 5 . ± . 13 1.530.32 5 . ± . 13 1.580.35 5 . ± . 13 1.020.37 4 . ± . 12 1.200.46 5 . ± . 09 1.340.49 5 . ± . 10 1.080.51 4 . ± . 09 1.630.54 5 . ± . 08 0.960.56 5 . ± . 08 1.150.59 5 . ± . 08 1.41 TABLE 3The mean and σ uncertainty of the genus amplitudesextracted from the six LOWZ and CMASS shells. Thethird column is the reduced χ value of the fit (46degrees of freedom). Figure 1 exhibits the two-dimensional genus curves[top panel] and the corresponding amplitudes A (2D) j [bot-tom panel] extracted from the N z = 12 LOWZ andCMASS data shells (Appleby et al. 2020). The genuscurves and amplitudes are functions of the assumed cos-mological model, and in this figure we have taken aΛCDM model with parameters given in Table 2. Thegenus amplitude is reconstructed to accuracy ∼ 3% and ∼ . 5% in the LOWZ/CMASS data respectively, and wepresent the best fit amplitudes, 1 σ error bars and re-duced χ values in Table 3. Three of the redshift binspresent relatively poor fits with a χ per degree of free-dom > . A (2 D ) data points, assumingthe data are uncorrelated. We find a p-value of p = 0 . . < z < . 6. This is expected from theo-retical arguments, but provides an important consistencycheck on our analysis. LOW REDSHIFT DATA 0 < z < . To test the expansion history, we also require an ac-curate measurement of the genus at low redshift, whichshould be practically insensitive to the distance-redshiftrelation. This would provide an anchor, a measurementof the shape of the linear matter power spectrum againstwhich high redshift genus curves can be compared.However, two-dimensional slices at low redshifts havevery small areas and suffer from curvature effects. Toovercome this limitation we use the three-dimensionallocal galaxy distribution in the SDSS MGS and apply aGaussian smoothing over a smaller scale. The measuredthree-dimensional genus will be used to estimate the two-dimensional genus amplitude. In the following sections,we describe in detail our method – the theory underly-ing the three-dimensional genus, the galaxy data used,the mask, how we remove systematics from the genusamplitude using mock galaxy catalogs and how we in-fer the two-dimensional genus amplitude from the three-dimensional data. − − ν A − g D ( M p c − ) × − . . . . z . . . . . . . A ( D ) ( M p c ) − × − Fig. 1.— [Top panel] Twelve, two-dimensional genus curves ob-tained from the BOSS LOWZ and CMASS data, as a functionof ν A . [Bottom panel] Two-dimensional genus amplitude measure-ments derived from the N z = 12 genus curves presented in the toppanel. The same color scheme is applied in both panels. Theory – Expectation value of three-dimensionalgenus The genus per unit volume of a three dimensional,Gaussian random field as a function of threshold ν isgiven by (Tomita 1986; Adler 1981; Gott et al. 1986;Hamilton et al. 1986) g ( ν ) = 14 π (cid:18) Σ (cid:19) / (cid:0) − ν (cid:1) e − ν / , (11)Σ = (cid:104) δ (cid:105) , Σ = (cid:104)|∇ δ | (cid:105) , where Σ , are the two-point cumulants of the three-dimensional field, related to the power spectrum asΣ = (cid:90) d ke − k Λ P ( k ) , (12)Σ = (cid:90) d ke − k Λ k P ( k ) , (13)(14)where we have smoothed with a Gaussian kernel of widthΛ G . The genus amplitude is given by A (3D)G = 14 π (cid:18) Σ (cid:19) / . (15)The leading order non-Gaussian expansion of the genus,in terms of the ν A threshold convention, is given by (Mat-subara 1994b, 2000; Pogosyan et al. 2009; Gay et al. 2012;Codis et al. 2013) g ( ν A ) = A (3D)G e − ν / (cid:104) H ( ν A ) + (cid:104)(cid:16) S (1) − S (0) (cid:17) × H ( ν A ) + (cid:16) S (2) − S (0) (cid:17) H ( ν A ) (cid:105) Σ + O (Σ ) (cid:105) . (16)As for the two-dimensional genus, the amplitude (coeffi-cient of H Hermite polynomial) is not modified by thenon-Gaussian effect of gravitational collapse to linear or-der in the Σ expansion (16). Data To extract the genus of the low redshift matter den-sity, we use the seventh data release of the main galaxycatalog of the SDSS DR7 (Abazajian et al. 2009). Specif-ically, we adopt the Korea Institute for Advanced StudyValue Added Galaxy Catalog (KIAS VAGC) (Choi et al.2010a; Blanton et al. 2005; Padmanabhan et al. 2008).The KIAS catalog supplements redshifts from other ex-isting galaxy redshift catalogs – the updated Zwicky cat-alog (Falco et al. 1999), the IRAS Point Source CatalogRedshift Survey Saunders et al. (2000), the Third Refer-ence 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 SDSSmain galaxies in the r -band Petrosian magnitude range10 < r p < . 6. Details of the selection criteria, classi-fication schemes and angular selection functions can befound in (Choi et al. 2010a). To maximize the area toboundary ratio of the data, we remove the three south-ern stripes and Hubble deep field region. The catalog provides angular positions, redshifts andabsolute, r -band magnitudes normalised to the z = 0 . E ( z ) = 1 . z − . 1) (Tegmarket al. 2004). All magnitudes and colors are corrected tothe redshift z = 0 . M r < − . 19 + 5 log h to gen-erate a volume limited sample over the redshift range0 . < z < . r gal = ¯ n − / = 8 . n gal is the mean galaxynumber density within the volume. The redshift rangewas selected to ensure a maximal number of galaxies areused in the analysis. The galaxies are presented as a func-tion of redshift and absolute magnitude in Figure 2 (toppanel), and the angular distribution of all galaxies usedin this work are presented in the bottom panel. Note thatin this Figure and in what follows the factor 5 log h willbe dropped in the expression of M r .To convert the galaxy catalog into a three-dimensionaldensity field, we construct a regular three-dimensional . 000 0 . 025 0 . 050 0 . 075 0 . 100 0 . 125 0 . z − − − − − − − M r D E C ( d e g r ee s ) Fig. 2.— [Top panel] The absolute, r -band magnitude of theSDSS MGS galaxies as a function of redshift. The solid red linesindicate the boundaries of our volume limited sample with 0 . 116 and M r < − . 19. [Bottom panel] The angular distribu-tion of the volume limited sample of galaxies on the sky; declinationvs right ascension (in degrees). N = 512 pixel lattice in a cube of side length L box =750Mpc and use cosmological parameters given in Ta-ble 2 to infer the distance-redshift relation. We bin thegalaxies into pixels using the Cloud-in-Cell scheme, gen-erating a three-dimensional number density field δ ijk =( n ijk − ¯ n ) / ¯ n , where ¯ n is the average number of galaxieswithin the unmasked pixels and 1 ≤ i, j, k ≤ N pix sub-scripts are pixel labels. The galaxies are weighted via theangular selection function during this binning procedure.The angular selection function constitutes a set ofweights as a function of angular position on the sky - w ( θ, φ ). It is defined in this work as w = 0 when outsidethe survey geometry or inside a bright star mask and0 < w ≤ w into a binary field with w = 1 if w > w cut and w = 0 oth-erwise, where w cut was selected as w cut = 0 . 8. Using thefiducial distance-redshift relation, we define w ijk as theprojection of the angular selection function into a 512 ,three-dimensional pixel cube of the same dimensions as δ ijk .We smooth the three-dimensional density field δ ijk with a Gaussian kernel of width Λ G = 8 . R G = 6Mpc /h , following Choi et al. (2010b)),and also smooth the projected selection function w ijk with the same kernel, defining the smoothed counterpartsas ˜ δ ijk and ˜ w ijk . We then redefine ˜ δ ijk = 0 for all pixelsin which ˜ w ijk < . δ ijk = ˜ δ ijk / ˜ w ijk if ˜ w ijk ≥ . δ ijk = 0 if w ijk = 0. This eliminates all data in the vicinity of thesurvey boundary.From the masked field we reconstruct the three dimen-sional genus, by generating iso-field triangulated meshesand calculating the Gaussian curvature at the trianglevertices. Details of the method can be found in (Applebyet al. 2018b). We calculate the genus as a function of ν A ,where ν A is the threshold chosen to match the volumefraction of a Gaussian random field. We select 200, ν A threshold values over the range − . < ν A < . 5, thentake the average of every four values to obtain the genusat N ν A = 50, ν A values equi-spaced over this range. Theresulting measured genus values are presented in Figure3 (top panel, red points).To extract the genus amplitude from the measure-ments, we fit a Hermite polynomial expansion to the datapoints by minimizing the following χ function χ = ∆ g T Γ − ∆ g, (17)where∆ g i = g i − A (3D) e − ν ,i / [ a H ( ν A ,i ) + a H ( ν A ,i )+ H ( ν A ,i ) + a H ( ν A ,i ) + a H ( ν A ,i )] . (18) A (3D) , a , a , a , a are free parameters to be constrainedvia the minimization of (17), the i subscript denotes the i th , ν A threshold bin and g i are the measured genus val-ues. In the fitting procedure we include the leading order a , a Hermite polynomial coefficients and the next-to-leading order even Hermite polynomial contributions a , a . Introducing additional Hermite polynomials does notsignificantly modify the fit.The covariance matrix Γ ij is obtained from mockgalaxy catalogs, as described in the following section. Mock Galaxy Catalogs Mock galaxy catalogs are generated using Horizon Run4 (HR4) Kim et al. (2015). Horizon Run 4 is a cosmo-logical scale dark matter simulation in which N = 6300 particles in a volume V = (3150Mpc /h ) are evolved us-ing a modified GOTPM scheme . The initial conditionsare obtained using second order Lagrangian perturba-tion theory L’Huillier et al. (2014), and the cosmologicalparameters used are h = 0 . n s = 0 . 96, Ω m = 0 . b = 0 . z = 0 snapshot box to cre-ate mock galaxy catalogs, using the HR4 cosmologicalparameters to infer distances. Details of the numericalimplementation, and the method by which mock galax-ies are constructed can be found in Hong et al. (2016).The mock galaxies are defined using the most bound haloparticle galaxy correspondence scheme, and the survivaltime of satellite galaxies post merger is estimated via themerger timescale model described in Jiang et al. (2008).The snapshot box is decomposed into N r = 360 non-overlapping volumes, and mock galaxy catalogs are con-structed from each region, with the same number density,redshift range and survey geometry as the data.From each mock sample we repeat our analysis ; binthe galaxies into a regular cubic pixel lattice, smooththe resulting number density field with a Gaussian ofscale Λ G = 8 . w ijk thenunsmoothed binary mask w ijk , then extract the genusfrom ˜ δ ijk .The result is a set of genus measurements g (3D) i,m , where1 ≤ i ≤ N ν A runs over N ν A = 50, ν A bins uniformlysampled in the range − . < ν A < . ≤ m ≤ N r runs over the randomly sampled realisations. We mea-sured the genus for 200 values and averaged every fourthpoint to arrive at the N ν A = 50 values in each mocksample. The covariance matrix is constructed asΓ ij = 1 N r − N r (cid:88) m =1 (cid:16) g (3D) i,m − (cid:104) g (3D) i (cid:105) (cid:17) (cid:16) g (3D) j,m − (cid:104) g (3D) j (cid:105) (cid:17) , (19)where (cid:104) g (3D) i (cid:105) is the average value of the genus in the i th threshold bin. In Figure 3 (bottom panel) we exhibitthe covariance matrix Γ ij extracted from the mock re-alisations. We note the strong correlation between genusvalues measured at different thresholds. A similar covari-ance matrix was numerically extracted from mock datain Blake et al. (2014). Results – Three-Dimensional Genus of SDSS MGS In Figure 3 (top panel) we exhibit the genus mea-sured from the SDSS MGS (red points), and the best-fitcurve reconstruction g (th) ( ν A ) (black solid line). The er-ror bars are the square root of the diagonal elements For a description of the original GOTPM code, pleasesee Dubinski et al. (2004). A description of the modifica-tions introduced in the Horizon Run project can be found athttps://astro.kias.re.kr/ kjhan/GOTPM/index.html. − − ν A − g D ( M p c − ) × − − − ν A − − ν A − . − . − . − . . . . . . × − Fig. 3.— [Top panel] The genus curve measured from the SDSSMGS using a Gaussian smoothing length of Λ G = 8 . ij . Bins separated by ∆ ν A < . 25 are strongly correlated(red), and bins at larger separations present anti-correlation (blue). of Γ ij . After minimizing the χ function (17), in Ta-ble 4 (first row) we present the best fit and uncertaintyon the ( A (3D) , a , a , a , a ) parameters. We also presenta fit including just ( A (3D) , a , a ), and ( A (3D) ) only forcomparison. If we regard equation (18) as an expansionin σ , then a , a should be of order a , ∼ O ( σ ) and a , a ∼ O ( σ ). At the scales adopted in this work, thehigher order terms are large, which indicates that thefield is non-linear. In spite of this, all three amplitudemeasurements are consistent. However, the second andthird rows yield a significantly worse χ .The genus amplitude presented in the first row of Ta-ble 4; A (3D) = 4 . × − Mpc − , and uncertainty∆ A (3D) = 0 . × − Mpc − , will be used as the lowredshift genus amplitude measurement from the SDSSMGS. This low redshift data point will be used to com-plement the higher redshift, two-dimensional BOSS mea-surements.The measured amplitude A (3D) of the SDSS MGSis effectively insensitive to cosmological parameters. (0 . , − 1) (0 . , − 1) (0 . , − 1) (0 . , − . 5) (0 . , − . (Ω m , w de ) . . . . . A ( D ) ( M p c − ) × − Fig. 4.— The genus amplitude as measured from the SDSS MGS,assuming five different cosmological models to infer the distanceredshift relation. The measured amplitude of the low redshift sam-ple is effectively insensitive to our choice. We select a fiducial cos-mology (Ω m , w de ) = (0 . , − 1) to infer A (3D) (red diamond). We confirm that reasonable variation of cosmologicalparameters does not affect the measured value of A (3D) in Figure 4. We select five parameter sets (Ω m , w de ) =(0 . , − , (0 . , − , (0 . , − , (0 . , − . , (0 . , − . A (3D) by minimizing the χ function(17). The resulting amplitudes and uncertainties arepresented in Figure 4. We find no significant changein the measured genus amplitude if we use differentcosmological parameters to infer the distance redshiftrelation, as expected at low redshift z < . 12. For thisreason, we fix A (3D) = 4 . ± . × − Mpc − ,corresponding to the red data point in Figure 4. THREE- TO TWO-DIMENSIONAL GENUS AMPLITUDE Our intention is to combine the SDSS MGS and BOSSgenus measurements, and find the cosmology that mini-mizes the evolution of the two-dimensional genus ampli-tude. However, to directly compare these results, we mustconvert the three dimensional genus amplitude measure-ment from the SDSS MGS to a corresponding effectivetwo-dimensional amplitude. To do so, we perform the fol-lowing steps –1. Correct the measured three-dimensional genus am-plitude for gravitational smoothing and non-linearredshift space distortion with a correction factorobtained from simulations.2. Using the now corrected, real space amplitude, per-form a cosmological parameter search by compar-ing this value to its Gaussian expectation value.The result is a set of parameter constraints on(Ω c h , n s ) which determine the shape of the lin-ear power spectrum.3. Use the best fit cosmological parameters (Ω c h , n s )to infer the two-dimensional theoretical expecta-tion of the genus amplitude A (3D) × (Mpc − ) a a a a χ . ± . 197 0 . ± . − . ± . 025 0 . ± . − . ± . 014 63 . . ± . 135 - − . ± . 026 0 . ± . 018 - 117 . . ± . 130 - - - - 124 . TABLE 4Best fit Hermite polynomial coefficients for the three-dimensional genus curve extracted from the SDSS MGS. The toprow is the full fitting function used in this work. In the second row we set a = a = 0 and in the third row we fix a = a = a = a = 0 , and fit a Gaussian curve to the points. A (2D)G = 12(2 π ) / (cid:82) k ⊥ e − k ⊥ R P ( k ⊥ ) dk ⊥ (cid:82) k ⊥ e − k ⊥ R P ( k ⊥ ) dk ⊥ . (20)Where the two-dimensional power spectrum P is related to the three dimensional matter powerspectrum according to equation (5), and we usethe three-dimensional power spectrum (6) with¯ n = 6 . × − Mpc − , b = 2, R G = 20Mpc,∆ = 80Mpc; the values relevant to the BOSS data.The end result is an inferred two-dimensional genusamplitude, based on the SDSS MGS data.In the following subsections we discuss each point in turn. Systematics removal Before comparing A (3D) to its Gaussian expectationvalue, we must account for non-linear effects. The mostsignificant systematics that must be corrected are non-linear gravitational evolution and redshift space distor-tion. At the smoothing scale Λ G = 8 . m and w de are varied (Shin et al. 2017; Park et al. 2019;Hong et al. 2020; Tonegawa et al. 2020). Since our lowredshift genus measurement will be practically insensi-tive to the value of w de , we use three of the simulationswith cosmological parameters (Ω m , w de ) = (0 . , − . , − 1) and (0 . , − 1) with all other cosmological pa-rameters fixed as Ω b = 0 . h = 0 . n s = 0 . 96. Eachsimulation comprises N p = 2048 dark matter particlesin a 1024 h − Mpc = 1422 Mpc box, gravitationallyevolved using a modified GOTPM code which uses thePoisson equation ∇ Ψ = 4 πGa ¯ ρ m δ m (cid:18) D de D m Ω de ( a )Ω m ( a ) (cid:19) . (21)The same random number sequence was used to gener-ate the initial condition for each simulation at z = 99,to eliminate cosmic variance when comparing differentmodels. The power spectrum was normalised such that the rms of the matter fluctuation, smoothed with top hat8 h − Mpc and linearly evolved to z = 0, is σ = 0 . z = 0 snapshotboxes in real and redshift space are presented in the toppanel of Figure 5, for three different cosmological modelsΩ m = 0 . , . , . 31. We use the entire box with peri-odic boundary conditions to make these measurements –that is we apply no mask in this subsection. For each sim-ulation, we fix the number density of the mock galaxiessuch that the mean separation is ¯ r = ¯ n − / = 8 . 33 Mpc,by applying a mass cut. The solid/dashed lines corre-spond to real/redshift space mock galaxy catalogs, andgreen/red/blue corresponds to Ω m = 0 . , . , . 31 re-spectively. In all cases one can observe an amplitude dropdue to the effect of redshift space distortion.In the bottom panel, we exhibit amplitude mea-surements extracted from the genus curves in the toppanel. The green/red/blue color scheme is the same asfor the top panel, and diamonds/stars correspond toreal/redshift space measurements of the genus ampli-tude. We denote the real/redshift space genus amplitudesas A (3D)real and A (3D)rsd respectively. The fractional differencebetween the redshift and real space amplitude measure-ments – a (3D)rsd ≡ A (3D)rsd /A (3D)real – is a (3D)rsd = 0 . , . , . m = 0 . , . , . 31 respectively. The effect of red-shift space distortion is a ∼ 10% effect on the genus am-plitude at these scales, and is only weakly dependent oncosmological parameters. Specifically, a (3D)rsd exhibits nosignificant, systematic dependence on Ω m . We use theΩ m = 0 . 26 simulation and take a (3D)rsd = 0 . 92 in whatfollows, correcting the measured A (3D) amplitude by afactor of (1 − ∆ rsd ) − with ∆ rsd = 0 . 08. This factor con-verts A (3D) to real space.To account for non-linear gravitational evolution, wecompare the measurement of the three-dimensional genusamplitude of the multiverse simulations in real space tothe Gaussian expectation value (15), where we use thelinear matter power spectrum plus shot noise P ( k ) = b P m ( k ) + P SN , sdss , (22)to generate the cumulants Σ , . We use the SDSS MGSnumber density ¯ n = 1 . × − Mpc − for the shotnoise power spectrum P SN , sdss = 1 / ¯ n and galaxy bias b sdss = 1 . A (3D) to real space using the cor-rection factor ∆ rsd .In Figure 5 (bottom panel) we exhibit the genus am-plitude of the z = 0, real space Multiverse simulationsnapshot boxes (green, red and blue diamonds), and thecorresponding Gaussian expectation value (15) with thesame cosmological parameters (yellow squares, labeled0 − − ν A − g D ( M p c − ) × − Ω m = 0 . m = 0 . m = 0 . . 21 0 . 26 0 . Ω m . . . . . A ( D ) ( M p c − ) × − Mock galaxies, Real spaceMock galaxies, Redshift spaceGaussian random fields Fig. 5.— [Top panel] Measured genus curves as a function of ν A for three multiverse simulations with Ω m = 0 . , . , . 31 (green,red, blue lines). The solid lines are real space measurements, dashedare redshift space. [Bottom panel] The genus amplitudes extractedfrom the top panel. The diamonds/stars represent the real/redshiftspace measurements respectively. The yellow squares represent theprediction for a Gaussian field for the given Ω m . The real space,mock galaxy amplitudes are lower than the Gaussian predictiondue to gravitational smoothing, and the redshift space values arestill lower due to the effect of redshift space distortion. ‘GRF’).The Gaussian expectation values are systematicallyhigher than the genus measured from each simulationbox – this highlights the ‘gravitational smoothing’ ef-fect of non-linear gravitational collapse. The effect is a gr ≡ A (3D)real /A (3D)G = 0 . , . , . 92 for the Ω m =0 . , . , . 31 simulations respectively. To directly com-pare the measured genus amplitude from the SDSS MGSto the corresponding Gaussian expectation value, we cor-rect A (3D)G by a factor of (1 − ∆ gr ) − with ∆ gr = 0 . A (3D) to account for non-linear redshift space distortion andgravitational smoothing, the next step is to compare A (3D) to the expectation value (15) to obtain a set ofparameter constraints. We minimize the simple χ func-tion . 04 0 . 08 0 . 12 0 . 16 0 . 20 0 . Ω c h . . . . . . . . . . n s Fig. 6.— The 1 , − σ contours in the n s -Ω c h plane obtainedby minimizing the chi square function (23). The black star is thePlanck best fit value of these parameters and green square thecosmological model used to infer the non-linear corrections to themeasured genus curve. χ = [(1 − ∆ rsd − ∆ gr ) − A (3D) − A (3D)G (Ω c h , n s )] σ , (23)where A (3D)G is the Gaussian expectation value of thethree-dimensional genus curve (15), and is sensitive toΩ c h , n s and weakly to Ω b h . As the dependence onΩ b h is very weak, we fix this parameter to its Planckbest fit value Ω b h = 0 . σ = 0 . × − Mpc − is the statistical uncertaintyon A (3D) .In Figure 6 we present the two-dimensional 1 , − σ contours in the n s , Ω c h plane, obtained by performingan MCMC parameter search, minimizing (23). We ob-serve a strong degeneracy between n s and Ω c h , as bothcan vary the degree of small scale power and hence in-crease/decrease the genus amplitude. The Planck best fitis shown as a black star, and the cosmological model ofthe Multiverse simulation used to make the non-linearredshift space distortion and gravitational smoothingcorrections ∆ rsd , ∆ gr is presented as a green square. Bothare within the 1 − σ contour.In the next step of our analysis, we convert these con-straints to a measure of the two-dimensional genus am-plitude A (2D)G . Conversion from cosmological parameters to A (2D)G Finally, we transform from the cosmological parame-ters Ω c h , n s to a prediction for the two dimensionalgenus amplitude. To do so, we transform each parame-ter set and corresponding χ value (Ω c h , n s , χ ) fromthe previous section to ( A (2D)G , χ ) by inserting Ω c h , n s into the definition of the theoretical expectation of thetwo-dimensional genus amplitude (2) using (3 − b = 2 and ¯ n = 6 . × − Mpc − suitable for theBOSS galaxy sample, and smoothing scales ∆ = 80 Mpc, R G = 20 Mpc. The result is a one-dimensional probabil-ity distribution function for A (2D)G , as inferred from thethree-dimensional measurement. We present the result-1 . . . . . . . . . A (2D)G (Mpc) − × − . . . . . . P D F . . . . . . z . . . . . . . A ( D ) ( M p c ) − × − SDSS MGS Fig. 7.— [Top panel] The probability distribution of the ampli-tude of the two-dimensional genus A (2D)G obtained by minimizingthe χ function (23). [Bottom panel] The two-dimensional genusmeasurement A (2D)G inferred from the low redshift SDSS MGS ispresented as a silver star, along with the BOSS data points overthe redshift range 0 . < z < . 6. The BOSS points are derivedassuming a Planck cosmology. ing probability distribution in Figure 7 (top panel). Fromthis we infer the best fit and 1 σ uncertainty on the two-dimensional genus amplitude as A (2D)G = 5 . ± . × − (Mpc) − .To review, the A (3D) measurement provides a con-straint of the shape of the linear matter power spec-trum. We have used the best fit values and uncertain-ties on the parameters Ω c h and n s to infer the bestfit and uncertainty on the theoretical expectation valueof A (2D)G at low redshift. In the bottom panel of Fig-ure 7 we present the two-dimensional genus measure-ment inferred from the SDSS MGS (silver star) and thosedirectly measured from the BOSS data (multi-coloureddata points). We take the SDSS MGS measurement tolie at z = 0 . 1. We assume that the low redshift measure-ment of A (2D)G is insensitive to variations of the expan-sion history in what follows, and treat it as a constant A (2D)G = 5 . ± . × − (Mpc) − . In this section, we have pursued a rather complex pathto extracting A (2D)G from the low-redshift data. How-ever, the reasoning behind our method lies in maximiz-ing the constraining power of the data. For our methodto provide a reasonable constraint, we must minimizethe statistical uncertainty of the low-redshift measure-ment as far as possible, and this required us to selectsmaller smoothing scales than the high redshift data. Thesmallest smoothing scales that we can adopt at low andhigh-redshift are fixed by the mean galaxy separationof the SDSS MGS and BOSS catalogs; ¯ r sdss ∼ . r BOSS ∼ G = 8 . /h ) and∆ = 80Mpc, R G = 20Mpc.Given the different number densities, bias factors andsmoothing scales used in the analysis of the SDSS MGSand BOSS data, the only logical approach to relate thetwo is to infer the theoretical expectation A (2D)G using oneof the data sets, and proceed to compare this value to thesecond. To apply this method, one must carefully correctfor any non-linear effects such as gravitational collapseusing simulations. PARAMETER CONSTRAINTS We are now able to combine the low- and high-redshiftmeasurements to constrain the distance-redshift relation.To do so, we minimize the following χ function χ = N z (cid:88) k =1 N z (cid:88) j =1 p j cov − jk p k , (24)where p j = A (2D) j /A (2D)G − jk = ( σ ,j + σ ) δ jk ; the covariance matrix is the sum of the sta-tistical uncertainties on the BOSS and SDSS measure-ments. We have used a diagonal covariance matrix forour analysis, assuming that A (2D)G represents the unbi-ased, central theoretical expectation value of the genusamplitude to which we compare our measured genus val-ues to. A direct comparison between measured values ofany statistic at high and low redshift would introducecorrelation between p j components. However, our anal-ysis has used the measured low redshift data to infer thetheoretical expectation value of the genus amplitude. Adirect comparison of A (2D)G posterior probability distri-butions inferred from the SDSS data and twelve BOSSshells, for each parameter set used to infer the distanceredshift relation, would provide a more rigorous statisti-cal comparison. However, such a procedure is computa-tionally intractable so we make the simplifying assump-tion that the high redshift shells are drawn from a PDFwith central value given by the SDSS LRG value of A (2D)G .The second implicit assumption with our choice of di-agonal covariance matrix is that we have neglected large-wavelength correlations between the SDSS LRG andBOSS galaxy samples. When generating the covariancematrices for the two-dimensional genus measurements ofthe BOSS data, we found no statistically significant cor-relation between neighbouring shells. This indicates thatthe cross correlation of the genus measurements is negli-2 Data Ω m w de Genus (BOSS+MGS) 0 . +0 . − . − . +1 . − . Genus (BOSS+MGS)+ Planck (2018) 0 . ± . − . +0 . − . TABLE 5Parameter best fit and − σ uncertainties, obtained byminimizing the χ function ( ). After combining our genuslikelihood with Planck 2018 temperature data, we obtainthe second row. gible.We fix h = 0 . 677 and vary Ω m , w de . For each param-eter set Ω m , w de , we estimate the distance to the cen-ters of the j redshift shells using d cm ( z j , Ω m , w de ) andreconstruct the genus curves using angular smoothingscales θ G ,j = R G /d cm ( z j , Ω m , w de ) and effective area ofthe data A j = 4 πf sky d ( z j , Ω m , w de ). After measuringthe genus curves for the given expansion history, we cal-culate the χ function (24). The low redshift measure-ment A (2D)G is assumed to be independent of input cos-mological model, as elucidated in section 4.4. Performinga MCMC exploration of the two-dimensional parameterspace, the resulting 1 , − σ contours (blue) are presentedin Figure 8. The tan contour is the w CDM parameterconstraint obtained from the Planck 2018 temperaturedata. If we combine the two data sets, we obtain a com-bined constraint on Ω m and w de (pink contours) . Themarginalised parameter constraints for Ω m , w de are pre-sented in Table 5.The degeneracy between Ω m and w de , exhibited in theblue contour in Figure 8, has been found previously (Park& Kim 2010; Appleby et al. 2018c). The Planck temper-ature data presents an almost orthogonal contour in the w de -Ω m plane, so by combining these two data sets wecan obtain a ∼ 15% constraint on the equation of stateof dark energy, w de = − . +0 . − . .The sensitivity of our test to Ω m and w de is relativelyweak as we are restricted to redshifts z < . 6. A higherredshift measurement will improve the constraints con-siderably. Although the constraint is modest, the ΛCDMexpansion history is consistent with the data over theredshift range considered. The constraint from the genusarises almost entirely from the combination of SDSSMGS low-redshift and CMASS data points; the LOWZdata have error bars that are too large and lie at a red-shift that is too low to make a strong contribution.The results are not sensitive to the absolute value ofthe genus amplitude – we are extracting information fromthe difference between different redshift bins. The abso-lute value also contains information, related to the shapeof the matter power spectrum, as discussed further in acompanion paper Appleby et al. (2020).The derived parameter constraints have been obtainedunder the assumption that the genus amplitude is a con-served quantity. For non-standard gravity or dark mat-ter models, the matter power spectrum can possess red-shift and scale dependent corrections. Similarly, we haveassumed that dark energy perturbations do not signifi-cantly affect the shape of the matter power spectrum at Specifically, we used publicly available w CDM, MCMC chainsfrom the Planck collaboration (Aghanim et al. 2020), combining(24) and the Planck MCMC likelihoods in quadrature. . . . . . Ω m − . − . − . − . − . . w d e BOSS DR12 (Genus)Planck (2018)Combined − . − . − . − . − . . w de Fig. 8.— [Top panel] Two-dimensional 68 , 95% contours in the(Ω m , w de ) plane obtained by minimizing the χ function (24) andusing the genus amplitude as a standard ruler (blue contours). Thetan contours are the marginalised constraints in the w de -Ω m planeobtained from Planck temperature data (Aghanim et al. 2020), andthe pink contours are the result of combining the genus and Planck χ functions in quadrature. The black star is the ΛCDM Planckbest fit, and grey circle the best fit of the combined genus + Planck(pink) contour. [Bottom panel] The marginalised one-dimensionalprobability distribution functions of w de . The colour scheme is thesame as in the top panel. low redshift. DISCUSSION In this work, we have obtained constraints on w de and Ω m from the tomographic analysis of the two-dimensional slices of observed large-scale galaxy distri-bution. The amplitudes of the two-dimensional genuscurves are measured in a series of concentric slices ofdensity fields derived from the SDSS BOSS data. Theamplitude at low-redshift is derived from the three-dimensional genus of the SDSS MGS data, and combinedto find the cosmological parameters minimizing the red-shift evolution of the genus. In doing so, we arrive at aconstraint of w de = − . +1 . − . , or w de = − . +0 . − . ,Ω m = 0 . ± . 036 if we combine our analysis withPlanck temperature data (Aghanim et al. 2020). The pa-rameter constraints arising solely from the genus statisticare particularly weak; this is due to the strong degener-acy between parameters and also the limited statisticalpower that we are able to employ. The presence of shot3noise fundamentally restricts our ability to reconstructthe density field from the galaxy point distribution, aswe must smooth on scales of at least the mean galaxyseparation Kim et al. (2014); Blake et al. (2014); Ap-pleby et al. (2017). In contrast, methods such as theAlcock-Paczynski (AP) test Li et al. (2016) (see alsoLi et al. (2017); Zhang et al. (2019a); Li et al. (2018);Park et al. (2019); Zhang et al. (2019b)) employ informa-tion from very small scales, eliminating non-perturbative,non-linear systematics using simulations. In addition, theAP test does not require the application of mass cuts togenerate uniform data samples with redshift, as we areforced to. As a result, Li et al. (2016) were able to ob-tain tight parameter constraints on w de and Ω m usingthe same BOSS data. For the genus to be competitivewith other statistics, we must first learn how to modeland remove observational systematics.Beyond sampling noise, another dominant limitationof the method is in the comparison of high- and low-redshift measurements, as the low-redshift data are sub-ject to large statistical uncertainty. This is the dominantcontribution to the parameter uncertainties. The onlyway to evade this issue is to smooth the data on smallerscales, but in doing so we are increasingly exposed tonon-linear physics. In this work we corrected the low-redshift, three-dimensional genus amplitude by factorsof ∆ rsd = 0 . 08 and ∆ gr = 0 . 10 to account for redshiftspace distortion and gravitational collapse. These valueswere inferred from simulations. Better theoretical under-standing of the non-linear regime and its impact on thegenus curve will be necessary in the future to improveour analysis. Similarly, a better understanding of the ef-fect of shot noise will allow us to probe smaller scales;in the current work we regard the mean galaxy separa-tion ¯ r of a catalog to be a hard limit below which we aresubjected to unknown non-Gaussian corrections. As thelow redshift SDSS MGS is more dense than the BOSScatalog, we were able to smooth the former on smallerscales and thus extract more information.In a companion paper Appleby et al. (2020), we mea-sured the genus curves of two-dimensional shells of theBOSS data and directly compared their amplitudes tothe Gaussian expectation value. As we smooth the BOSSdata with large scales ∆ = 80 Mpc, R G = 20 Mpc, we didnot apply any non-linear correction factors to our mea-surements, and were able to use the Kaiser formula toestimate the effect of redshift space distortion. In Ap-pleby et al. (2020) we placed constraints on cosmologicalparameters that determine the shape of the linear matterpower spectrum; Ω c h and n s . The information extractedin that work came from the absolute value of the genusamplitude. In the present analysis, we measure the red-shift evolution of the genus amplitude, irrespective of itsabsolute value. One can interpret the two approaches as ameasure of the initial condition/transfer function of thedark matter perturbations and a test of the expansionhistory respectively. In Appleby et al. (2020), we fixedthe distance-redshift relation using the Planck 2018 bestfit cosmology Aghanim et al. (2020) and measured thegenus curves of the BOSS data a single time. We wereable to do this as we restricted our analysis to the ΛCDMmodel, and the constraints obtained in this work are con-siderably 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 toΩ m and the equation of state of dark energy.To improve the parameter constraints, a number of av-enues remain open. We can combine different low-redshiftdata sets, increasing the effective volume and reducingthe statistical uncertainty. We can calculate analyticallythe non-linear corrections due to gravitational smoothingand redshift space distortion, which will provide a bet-ter understanding of the non-linear effects that we mustaccount for on small scales. In addition, we can applyour method to high-redshift data, such as Lyman breakgalaxies. We expect that a high redshift data point willprovide a significantly improved constraint on the expan-sion history. As the distance between observer and dataincreases, the effect of choosing an incorrect cosmologybecomes more pronounced.Finally, the three-dimensional Minkowski Functionalscontain more information than their two-dimensionalcounterparts, and a complete analysis of the three-dimensional field will be forthcoming. In this work, andthroughout a series of papers (Appleby et al. 2017, 2018c,2020), we have focused on the two-dimensional genus, ex-tracted from shells of the three-dimensional galaxy dis-tribution. The reasoning behind this choice is two-fold.First, the BOSS galaxy catalog is relatively sparse, andwe mitigate this issue by taking thick slices along theline of sight. Binning galaxies in this way is a smoothingchoice, so we can interpret our approach as anisotropicsmoothing perpendicular and parallel to the line of sight.Smoothing on larger scales parallel to the line of sightallows us to use linear redshift space distortion physics,which is important as non-linear redshift space distor-tion effects on topological statistics are not yet well un-derstood. Second, in future work we intend to compareour results with higher redshift photometric redshift cat-alogs, which will require galaxies to be binned into thickshells. An understanding of how photometric redshift un-certainty modifies our analysis must be further exploredbefore this comparison can be made. ACKNOWLEDGEMENT APPENDIX A – SYSTEMATIC EFFECTS In Section 2 we listed three effects that can introduce asmall evolution in the genus amplitude. In the Appendixwe consider each point in turn and in isolation, to confirmthat we have all known systematics under control.We use all sky, mock galaxy lightcone data from theHorizon Run 4 dark matter simulation project to per-form these tests. We direct the reader to Kim et al.(2015); Hong et al. (2016) for information on the sim-ulation and mock galaxy catalogs. We use the lightconedata over the range 0 . < z < . 7, creating N = 20shells and applying mass cuts to generate constant num-ber density samples, exactly as we did for the BOSS data.We bin the galaxies into shells of thickness ∆ = 80Mpcand smooth perpendicular to the plane with comovingscale R G Mpc. The simulation was performed using a flatΛCDM cosmology with parameters h = 0 . 72, Ω m = 0 . σ = 0 . n s = 0 . Shot noise is the single largest systematic associatedwith information extraction using the genus curve Kimet al. (2014). There are two issues associated with thisphenomenon – it is non-Gaussian and it can potentiallyintroduce a redshift evolution of the genus amplitude.First regarding the non-Gaussianity. As a simple ap-proximation, we have corrected for shot noise by addinga constant white noise contribution to the total powerspectrum; P SN = 1 / ¯ n . In reality, the noise is a Poissonprocess (roughly speaking), but when writing the genusin terms of a Hermite polynomial expansion as in (1) wehave implicitly assumed that the field is drawn from a perturbatively Gaussian distribution. As a Poisson distri-bution possesses a different moment generating functioncompared to a Gaussian, we can expect that shot noisewill introduce modifications to the shape of the genuscurve. This was observed in both Kim et al. (2014) andAppleby et al. (2018c). If shot noise becomes significant,then the shape of the genus will not be well representedby the first few Hermite polynomials and we lose theinterpretation of the genus amplitude as the ratio of sec-ond order cumulants of the perturbatively Gaussian fieldthat we are trying to measure. In short, the field thatwe measure is non-Gaussian due to both gravitationalcollapse and the nature in which it is sampled. However,only the gravitational non-Gaussianity is treated in theexpansion (1). This issue is suppressed if we smooth onscales larger than the mean galaxy separation, in whichcase the shot noise effect can be approximately repre-sented by the white noise term P SN . This remains animperfect approximation except in the limit R G (cid:29) ¯ r .To present the non-Gaussianity induced by the shotnoise sampling, we measure the genus of two-dimensionalshells of the Horizon Run 4 mock galaxy lightcone inreal space, fixing the smoothing scales ∆ = 80Mpc, R G = 20Mpc and applying three mass cuts to thedata to fix the number density of our galaxy sampleas ¯ n = 3 . × − Mpc − , ¯ n = 7 . × − Mpc − and ¯ n = 3 . × − Mpc − . We assume that the mostdense sample has a shot noise contribution that is sup-pressed, as the mean galaxy separation is much lowerthan the smoothing scale R G = 20Mpc. We label thegenus curves g (1)2D ( ν A ), g (2)2D ( ν A ) and g (3)2D ( ν A ) respectively,where 1 , , n , , . We repeat our measurement for twenty concentricnon-overlapping shells and take the average genus curveto show the effect of shot noise. In Figure 9 (top panel)we exhibit the average genus curves for ¯ n (black), ¯ n (red) and ¯ n (blue). Below we also present the differencebetween the genus curves ∆ g (2)2D = g (2)2D − g (1)2D (blue) and∆ g (3)2D = g (3)2D − g (1)2D (red) respectively. Clearly the differ-ence between the genus curves is not simply an amplitudeshift – the shape of the residual curve is both shifted anddistorted. This is due to the non-Gaussian nature of thesampling, and is most significant in the most sparse sam-ple ¯ n . For the case ¯ n , these effects are less pronouncedbut still present. This indicates that our treatment ofshot noise using the white noise contribution P SN is im-perfect. This effect will be further studied by the authorsin the future.However, even if the effect of shot noise can be repre-sented by a white noise term P SN = 1 / ¯ n , it can still gen-erate a redshift evolution in the genus amplitude. This isbecause we are fixing P SN to be constant at each redshift,but the matter power spectrum has a decreasing ampli-tude with increasing redshift. Hence the shot noise termincreases in significance to the past, and will manifest asan increasing genus amplitude with increasing z .To show the hypothetical redshift evolution of thegenus amplitude, in Figure 9 (middle panel) we presentthe theoretical expectation A (2D)G as a function of red-shift, using (2) with power spectrum (6) and taking thereal space power spectrum (that is, setting β = 0 inequation (6)). We plot the amplitude assuming negligi-5 − . − . . . . g D × ( M p c − ) ¯ n = 3 . × − Mpc − ¯ n = 7 . × − Mpc − ¯ n = 3 . × − Mpc − − − − − ν A − ∆ g D × ( M p c − ) . . . . . . z . . . . . . . A ( D ) G ( M p c − ) × − ¯ n = 6 . × − Mpc − ¯ n = 2 × Mpc − . . . . . . z . . . . . . . . A ( D ) ( M p c − ) × − SparseDense Fig. 9.— [Top panel] The average genus curves extracted fromall-sky mock galaxy lightcone data, taking three mass cuts to fixa constant number density in each shell ¯ n = 3 . × − Mpc − (black), ¯ n = 7 . × − Mpc − (red), ¯ n = 3 . × − Mpc − (blue). We also exhibit the difference ∆ g between the sparse sam-ples and the dense catalog. [Middle panel] The Gaussian predictionfor the genus amplitude in real space, for a model with no shot noise(grey) and with the fiducial number density ¯ n = 6 . × − Mpc − (yellow). Shot noise introduces significant evolution in the genusamplitude. [Bottom panel] Genus amplitudes extracted from two-dimensional slices of dark matter particle snapshot boxes for asparse N = 64 (gold points) and dense N = 512 (grey points)sample. The solid lines are the Gaussian prediction. ble shot noise, setting an arbitrarily high hypotheticalnumber density ¯ n = 2 × Mpc − (grey line) and thefiducial number density of the BOSS catalog used in thiswork ¯ n = 6 . × − Mpc − (yellow line), both with con-stant linear galaxy bias b = 2. The grey line representsthe idealised case and as expected is constant; in this in-stance the genus is a measure of the shape of the linearmatter power spectrum. The yellow curve represents ahypothetical sparse galaxy catalog with constant largescale galaxy bias – the shot noise contribution causes thegenus amplitude to evolve with redshift.To confirm this behaviour, we take simulated dark mat-ter particle snapshot boxes of volume V = (1024Mpc /h ) at z = 0 , . , (dense) and 64 (sparse) particles randomly. We then construct flat,two-dimensional slices of thickness ∆ = 80 Mpc andsmoothed in the plane with R G = 20 Mpc Gaussian ker-nel. The we extract the genus from these fields and thenthe genus amplitudes. The results are presented in thebottom panel of Figure 9. The points/error bars are themean and standard deviation of 15 slices of the snap-shot boxes, with the yellow/grey points correspondingto the sparse and dense samples respectively. The solidyellow/grey lines are the Gaussian expectation value forthe given number density (and bias factor b = 1, as weare using dark matter particles). The behaviour of themiddle panel is reproduced, the sparse sample exhibits asystematic evolution in redshift, but the dense sample isconserved.In this work we have fixed the galaxy bias of the BOSSgalaxies to be constant, b = 2. If the bias evolves withredshift, then this must also be taken into account whenassessing the effect of shot noise. The net effect dependson the relative amplitude of the galaxy power spectrum– hence b ( z ) D ( z ) A s – and the shot noise term P SN ,where D ( z ) is the growth rate and A s is the primordialamplitude. The effect of redshift space distortion is to decreasethe genus amplitude by ∼ A (2D)G redshift andreal ( β = 0) space, obtained from the theoretical expec-tation (2) assuming negligible shot noise (that is, fix-ing ¯ n = 2 × Mpc − ) and different galaxy bias val-ues. The green solid line is the fiducial, constant galaxybias used in this work b = b . We also exhibit a rsd for different linear galaxy bias models b ( z ) = b + b z ;( b , b ) = (1 . , , (1 . , . , (1 . , 1) (yellow solid, blackdash-dot and red dashed lines respectively). We alsopresent the values of a rsd inferred from the Horizon Run4 all-sky mock galaxy shells as pale red points. For ourfiducial choice b = 2, the genus amplitude decreases by ∼ O (8%) and decreases with increasing redshift whenmeasured in redshift space (green line). For different biasfactors, the redshift dependence of a rsd can change sig-nificantly (cf yellow, black, red lines).In the middle panel of the figure, we plot A (2D)G inredshift space for two different number densities – ¯ n =2 × Mpc − (red) and fiducial number density used inthis work ¯ n = 6 . × − Mpc − (blue), fixing b = 2. In6the absence of shot noise, the genus amplitude decreaseswith redshift (red line), however as shown in the previoussection shot noise acts to increase the genus amplitudewith z . The net effect is that redshift space distortiondecreases and shot noise increases A (2D)G , with the resultthat the measured genus amplitude of the galaxy catalogsshould remain approximately constant over the redshiftrange considered in this work 0 . < z < . 7. Specifi-cally, the measured genus amplitude should follow theblue curve in the bottom panel. We stress however, thatthis argument is sensitive to the galaxy sample. Differ-ent bias factors, number densities and redshift ranges willnot necessarily yield a genus amplitude that is conservedwith redshift.We again confirm our hypothesis that redshift spacedistortion introduces a mild dependence of the genus am-plitude on redshift by extracting A (2D) from slices of darkmatter particle data from our simulation. We take z =0 , . , V = (1024Mpc /h ) ,sub-sample 512 particles from the full data then perturbthe particles along the x direction according to their ve-locities to create plane, parallel redshift space distortedslices. We then extract the two-dimensional genus am-plitude from slices using ∆ = 80Mpc, R G = 20Mpc asbefore. The results are exhibited in the bottom panel ofFigure 10. The grey/red points and error bars are themean and standard deviation of 15 slices of the snapshotboxes in real/redshift space respectively. We observe noevolution in real space, but a systematic decrease in thegenus amplitude in redshift space.Future dense and high redshift galaxy catalogs will notsuffer from the many of the issues discussed in this work.For these data, the shot noise contribution will be signif-icantly reduced. In this case, the correct course of actionwould be to correct the measured genus amplitudes by amultiplicative factor to convert them to real space Mat-subara (1996), after which they should be conserved withredshift. This procedure was undertaken for mock galax-ies in Appleby et al. (2018b). It is well known that higher order corrections in thenon-Gaussian expansion of the genus curve ∼ O ( σ ) willmodify the genus amplitude ; empirically it has been ob-served that the genus amplitude decreases on small scalescompared to the Gaussian expectation value when mea-sured from galaxy catalogs Melott et al. (1989); Park &Gott (1991); Park et al. (2005). To test the magnitude ofthis effect, we measure the coefficient of the a Hermitepolynomial expansion of the genus curve – g (cid:39) a e − ν / [ a H + H + a H + a H ] . (25)According to the non-Gaussian perturbative expansionof the genus Matsubara (1994b, 2000); Pogosyan et al.(2009); Gay et al. (2012); Codis et al. (2013), a is thegenus amplitude, a , are the first order corrections oforder a , ∼ O ( σ ) and we can expect a will be inducedat second order a ∼ O ( σ ). We therefore use this termas a proxy to estimate the magnitude of higher ordercorrections to the genus amplitude.We extract a , a , a from the twenty all-sky lightconeshells of Horizon Run 4, in redshift space, by integrating . . . . . . z . . . . . a r s d b , b = 1 . , b , b = 1 . , . b , b = 1 . , . b , b = 2 . , . . . . . . z . . . . . A ( D ) G ( M p c − ) × − ¯ n = 6 . × − Mpc − ¯ n = 2 × Mpc − . . . . . . z . . . . . . A ( D ) ( M p c − ) × − Dense, real spaceDense, redshift space Fig. 10.— [Top panel] The ratio of genus amplitudes A (2D)G as measured in redshift and real space, for different bias models b ( z ) = b + b z . The green line corresponds to the fiducial val-ues used in this work b = 2, b = 0 in the galaxy bias model b ( z ) = b + b z . The red points are the values of a rsd inferred fromthe mock galaxies from the Horizon Run 4 lightcone in real and red-shift space. Generally, the effect of linear redshift space distortionis to decrease the genus amplitude by ∼ 8% and introduce a weakredshift dependence. [Middle panel] The expectation value of thegenus amplitude in redshift space, taking b = 2, ¯ n = 2 × Mpc − (red) and ¯ n = 6 . × − Mpc − (blue). The measured genus am-plitude from the BOSS data should trace the blue curve. [Bottompanel] The genus amplitude extracted from dark matter snapshotboxes for a dense sample 512 in real (grey) and redshift (red)space. . . . . z − . . . . . . . a n a a a Fig. 11.— The Hermite polynomial coefficients a , a , a (grey,blue, red) obtained from twenty shells of all-sky mock galaxy data.There is no strong evidence of evolution of a . the genus curve using a n = 1 n ! (cid:82) − dν A g ( ν A ) H n ( ν A ) (cid:82) − dν A g ( ν A ) H ( ν A ) , (26)taking ν = 4. In Figure 11 we present a , a , a (grey,blue, red). The red curve is the next to leading ordercorrection term a . There is some suggestion that a isincreasing with decreasing redshift, from ∼ . 02 at z =0 . 25 to ∼ . 01 at z = 0 . 6. Although the effect is smalland the statistical uncertainty large, the higher order,non-linear corrections require further study. The a termis present at the 1% level at the scales probed.The a , a coefficients are perturbatively small at thescales studied in this work. These terms can be inter-preted as integrals over the Bispectrum, and containcomplementary information to the amplitude studied inthis work. a exhibits some evidence of evolution overthe redshift range under consideration. The first three issues described above are physical ef-fects. The fourth is a purely spurious systematic that canbe introduced into the analysis if we improperly select the ν A threshold range. Specifically, one can observe evolu-tion of the genus amplitude with redshift if we measurethe genus over threshold values that are too high. Thereason for this lies in the relation between ν A and ν . The ν A parameterisation of the genus curve selects thresholdsthat have the same area fraction as a Gaussian randomfield. However, since the galaxy catalogs occupy a finitearea, high threshold peaks will not be represented withinthe observed domain, and the area fraction will be sys-tematically under-represented compared to a hypothet-ical Gaussian random field of arbitrarily large extent.This leads to an increase in the genus curve in the high ν A tails, which increases the genus amplitude. This canintroduce spurious redshift evolution because the area ofthe data at low redshift is smaller than at high redshift,and so the low- z regime will contain fewer high thresholdpeaks.We can eliminate this effect by restricting our analy-sis to ν A threshold values that are well sampled at eachredshift. To present the effect, we take the twenty all- sky lightcone mock galaxy shells from the Horizon Run4 simulation in real space, smooth them and then applya mask, only keeping a spherical cap of data of radius θ cap = π/ (2 √ 2) rad. We select this value as the areafraction of such a cap roughly matches the area of theBOSS mask. We then measure the genus of this subsetof data over the threshold ranges − < ν A < − . < ν A < . 5. As a proxy for the genus amplitude,we use the following integral A (2D) (cid:39) √ π (cid:90) ν − ν g ν A dν A , (27)with ν = 2 . , 4. As ν → ∞ , the integral (27) approachesthe exact genus curve amplitude. In Figure 12 we present A (2D) for ν = 4 (blue points) and ν = 2 . A (2D) is not relevant to our discussion, theimportant point is the clear redshift evolution in the bluepoints, which is due to selecting a large value ν = 4. Forthe more conservative choice ν = 2 . 5, no redshift evolu-tion is detected relative to the mean value (red points).This indicates that peaks in the range − . < ν A < . . < z < . − . < ν A < . APPENDIX B – VARIATION OF ∆ Finally, we check that the genus amplitudes extractedfrom the data are insensitive to the small variations inshell thickness ∆ induced by selecting different cosmo-logical models to infer the distance-redshift relation. Al-though the genus is a function of the thickness ∆, we willargue that for large ∆ this sensitivity is low and can beneglected.To show this, we take the Horizon Run 4 all-skymock galaxy lightcone, and use four different cosmo-logical models to fix the redshift boundaries of theshells. For each cosmology we select redshift limits ofthe shells z min , z max such that the comoving distance . . . . . z . . . . . . A ( D ) ( M p c − ) × − − < ν A < − . < ν A < . Fig. 12.— The amplitude proxy A (2D) defined in equation (27),measured from the twenty shells of lightcone data. The red/bluepoints correspond to ν = 2 . ν = 4 respectively. The solid hori-zontal lines are the mean values of the respective points. One canclearly observe a systematic evolution in the blue points, due tothe lack of high threshold maxima/minima at low- z . Model ˜Ω m ˜ w de . − 1I 0 . − . . − . . − . − TABLE 6The four models used in Appendix B to test the effect ofvariable ∆ slice thickness on the genus amplitude. The model is the fiducial model of the simulation, and yields aconstant ∆ = 80Mpc slice thickness. d cm ( z max , ˜Ω m , ˜ w de ) − d cm ( z min , ˜Ω m , ˜ w de ) = ˜∆ = 80Mpc,where tildes indicate incorrect cosmological parametersthat are presented in Table 6, with model 0 beingthe correct, fiducial cosmology of the simulation. Thetrue values of the slice thicknesses are given by ∆ = d cm ( z max , Ω m , w de ) − d cm ( z min , Ω m , w de ) with Ω m = 0 . w de = − 1. In Figure 13 (top panel) we present ∆( z ) asa function of z for each of the cosmological models usedto infer the distance redshift relations of the shells. Foreach cosmological model we have selected redshift limitssuch that ˜∆ = 80Mpc, independent of redshift, but thetrue value of ∆ (obtained by using the true cosmology)is evolving.After fixing the redshift shell limits using the incorrectcosmological models, we proceed to calculate the genusin the twenty data shells using the correct cosmologicalmodel. We do this as we wish to isolate the effect of a sys-tematically evolving ∆ thickness. We measure the genuscurves and extract the amplitudes. In Figure 13 (bot-tom panel) we present the genus amplitude A (2D) ( ˜∆).For clarity we plot the average and standard deviationof every four shells. 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