{'fraction_non_alphanumeric': 0.03571842596766922, 'fraction_numerical': 0.018973401172151735, 'mean_word_length': 4.545032497678737, 'pattern_counts': {'":': 0, '<': 22, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Topology of Luminous Red Galaxies from the Sloan Digital Sky Survey

We present measurements of the genus topology of luminous red galaxies (LRGs) from the Sloan Digital Sky Survey (SDSS) Data Release 7 catalog, with unprecedented statistical significance. To estimate the uncertainties in the measured genus, we construct 81 mock SDSS LRG surveys along the past light cone from the Horizon Run 3, one of the largest N-body simulations to date that evolved 7210^3 particles in a 10815 Mpc/h size box. After carefully modeling and removing all known systematic effects due to finite pixel size, survey boundary, radial and angular selection functions, shot noise and galaxy biasing, we find the observed genus amplitude to reach 272 at 22 Mpc/h smoothing scale with an uncertainty of 4.2%; the estimated error fully incorporates cosmic variance. This is the most accurate constraint of the genus amplitude to date, which significantly improves on our previous results. In particular, the shape of the genus curve agrees very well with the mean topology of the SDSS LRG mock surveys in the LCDM universe. However, comparison with simulations also shows small deviations of the observed genus curve from the theoretical expectation for Gaussian initial conditions. While these discrepancies are mainly driven by known systematic effects such as those of shot noise and redshift-space distortions, they do contain important cosmological information on the physical effects connected with galaxy formation, gravitational evolution and primordial non-Gaussianity. We address here the key role played by systematics on the genus curve, and show how to accurately correct for their effects to recover the topology of the underlying matter. In a forthcoming paper, we provide an interpretation of those deviations in the context of the local model of non-Gaussianity.


INTRODUCTION
The current standard cosmological scenario, supported by observations of the cosmic microwave background (CMB) and of the large-scale structure (LSS), appears to be consistent with the ΛCDM concordance model, where the Universe is dominated by cold dark matter (CDM) and its accelerating expansion driven by a cosmological constant Λ or dark energy (DE). A recent strong support of this paradigm has been presented by Park et al. (2012), who was able to prove that observed high-and low-density LSSs have the richness/volume and size distributions consistent with the ΛCDM universe.
In addition, the primordial density perturbations from which halos and galaxies form are assumed to be a Gaussian random field, as predicted by inflationary theories (Guth 1981;Linde 1982; Bardeen et al. 1986); state-ofthe-art data from the Wilkinson Microwave Anisotropy Probe (WMAP; Spergel et al. 2003, Komatsu et al. 2011, the Sloan Digital Sky Survey (SDSS; York et al. 2000;Stoughton et al. 2002;Abazajian et al. 2009) or the WiggleZ survey (Blake et al. 2011) are still favoring this hypothesis. However, some claims or hints of primordial non-Gaussianity have recently appeared in the literature (Jeong & Smoot 2007;Yadav & Wandelt 2008;Komatsu et al. 2009Komatsu et al. , 2011Slosar et al. 2008; Corresponding Author: Graziano Rossi (graziano.rossi@cea.fr) Smidt et al. 2010), and challenged the validity of the simplest inflationary paradigm. Indeed, if detected, primordial non-Gaussianity would indicate a structure formation scenario different from the concordance cosmological model, and force us to revise the physics of the very early Universe -along with several aspects of the LSS dynamics (but see also Hwang 2012 for a more general discussion on modern cosmology).
To this end, topology-related statistics offer a precious benchmark for testing the underlying Gaussianity of the initial density field, since topology can be regarded as an important physical property of the matter density that can be compared with predictions of the simplest inflationary models, where Gaussian random phase initial conditions are generated from quantum fluctuations of an inflaton field in the early Universe. In addition, topology measured at the present epoch should reflect that of the initial conditions on smoothing scales considerably larger than the correlation length, because fluctuations which are still in linear regime maintain their initial topology (see Gott et al. 1987, who confirmed this property with N -body simulations); this fact allows one to test directly the Gaussian paradigm, and permits to use topology as a cosmic standard ruler .
From the theoretical side, since the pioneering work of Gott et al. (1986), a variety of analytic and numerical tools to analyze observational and simulated data for measuring topology have long been developed -mainly using the genus statistics to quantify the topology of isodensity contours (Gott et al. , 1989Hamilton et al. 1986;Vogeley et al. 1994;Park et al. 2005aPark et al. , 2005b. In particular, the analytic prediction for the genus curve of a Gaussian field in linear regime is well-known , and its perturbative expression in the weakly nonlinear regime has also been obtained (Matsubara 1994); this lognormal model turned out to be a good empirical approximation in the strongly nonlinear regime (Hikage et al. 2002). Along with analytic tools, large-volume N -body simulations are routinely used to quantify several systematics which affect the genus curve, such as finite pixel size, sparse sampling, peculiar velocity distortions in redshift space or survey boundaries. The ability to correct for these effects is essential, as the remaining small deviations from the random phase curve give important information about the physics connected with galaxy formation, nonlinear gravitational clustering, and primordial non-Gaussianity -if any. In fact, on smaller scales nonlinear gravitational evolution and biased galaxy formation make the topology of the observed galaxy distribution deviate from the Gaussian form, even if the initial conditions were Gaussian-distributed. Using fractional volume rather than direct density threshold as the independent variable in topology analysis mitigates but does not fully eliminate these nonlinear and biasing effects Melott et al. 1988). Ultimately, all the secondary non-Gaussianities need to be disentangled from the primordial contribution, and this can now be done very accurately, without assuming any 'a priori' model for the underlying signal.
From the observational side, a long list of studies have been pursued on a variety of datasets (see for example Park, Gott, & da Costa 1992;Park et al. , 2005bPark, Gott, & Choi 2001;Moore et al. 1992;Vogeley et al. 1994;Rhoads et al. 1994;Protogeros & Weinberg 1997;Canavezes et al. 1998;Hoyle et al. 2002;Hikage et al. 2002Hikage et al. , 2003James et al. 2007James et al. , 2009Gott et al 2008Gott et al , 2009Choi et al. 2010). They focused on characterizing the three-dimensional topology, and showed that, depending on the considered scale, topology is useful in constraining both cosmological parameters and the galaxy formation mechanism. For example, Park et al. (2005b) characterized the topology of the SDSS Main galaxy sample from the NYU Value Added Galaxy Catalog (NYU VAGC; Blanton et al. 2005), which has similar sky coverage as the SDSS Data Release 4 (DR4; Adelman-McCarthy et al. 2006), and presented the first clear demonstration of luminosity dependence of galaxy clustering topology (i.e. brighter galaxies show a stronger signal of meatball topology); more recently, Gott et al. (2009) measured the three-dimensional LSS topology of the SDSS DR4plus LRG sample from the NYU VAGC (a subsample of the SDSS DR5; Adelman-McCarthy et al. 2007) and found strong consistency with Gaussianity of the primordial fluctuations. In the latter case, the large sample size available allowed topology to be an important tool for testing galaxy formation models. Also, Choi et al. (2010) measured the topology of the Main galaxy distribution using the SDSS DR7 (Abazajian et al. 2009;Choi, Han, & Kim 2010), studied the scale-dependent topology bias, and examined the dependence of galaxy clustering topology on galaxy properties (i.e. luminosity, morphology, color) at different smoothing scales. The large volume-limited sample enabled an unprecedented measurement of the genus curve, with an amplitude of G = 378 at 6 h −1 Mpc smoothing scale and an estimated uncertainty of 4.8%, including all systematics and cosmic variance. In addition, Choi et al. (2010) detected deviations of the genus curve from Gaussianity, interpreted as the fact that voids and superclusters are more connected and their sizes are larger than those expected for Gaussian fields. The same authors then used these results to test five different galaxy formation models, which indeed are tuned to reproduce the two-point correlation function and the luminosity function but not high-order statistics, and found significant discrepancies: none of the models could reproduce all the aspects of the observed clustering topology.
While the significant discrepancies in the SDSS DR7 galaxy clustering topology at nonlinear scales found by Choi et al. (2010) are mainly driven by the inaccuracy of galaxy formation models, a cleaner problem is to consider the topology of LRGs instead -which is the main focus of this paper. This is because the LRG sample covers a much larger and deeper volume (i.e. it allows one to observe topology at the largest scales), essentially in the linear regime. In addition, LRGs are particularly useful in refining cosmological parameters (Tegmark et al. 2006), and are expected to play an important role in characterizing DE through the ratio of DE pressure to energy density (see Bassett et al. 2005;Eisenstein et al. 2005;Percival et al. 2007). A number of previous analyses have addressed the clustering of LRGs, especially in relation to Baryon Acoustic Oscillation (BAO) science (see for instance Ross et al. 2012, and references therein). On the contrary, fewer studies have been based on topology. Among those, Gott et al. (2009) measured the three-dimensional genus topology of LRGs, using two volume-limited samples constructed from the SDSS DR4plus sample, i.e. a dense shallow sample with 21 h −1 Mpc smoothing, and a sparse deep sample with 36 h −1 Mpc smoothing. The amplitude of the genus curve was found to reach about 167 with a 4.1% uncertainty at 21 h −1 Mpc scale. A major result of their study was that topology of LRGs in the SDSS agrees very well with that of mock galaxies in the ΛCDM universe with the same cosmological parameters: small distortions in the genus curve, expected from nonlinear biasing and gravitational effects, are well explained by N -body simulations with a subhalo finding technique adopted to locate LRGs. This suggests that the formation of LRGs can be modeled well without any free-fitting parameters.
The main goal of this paper is to characterize the threedimensional genus topology of spectroscopic LRGs using the SDSS DR7 catalog, improving on the previous results presented by Gott et al. (2009). In particular, we strive to carefully model and remove all known systematics which affect the observed genus (i.e. finite pixel size, survey boundary, radial and angular selection function and shot noise), and estimate the uncertainties in the measured genus accurately. This is achieved by comparing our measurements with 81 mock SDSS LRG surveys along the past light cone constructed from the Horizon Run 3 (HR3; Kim et al. 2011), one of the largest Nbody simulations to date that evolved 7210 3 particles in Fig. 1.-Genus curve measured from the number density field of the SDSS DR7 volume-limited sample (red solid line). For illustrative purposes, the curve is compared to the one obtained by Gott et al. (2009) from the SDSS DR4plus SHALLOW sample (i.e. blue solid line). The main point of the figure is to show the dramatic increase in the genus amplitude over the two different datasets, due to the much larger volume spanned by the SDSS DR7. The observed genus curves in this plot have not been corrected for any systematic bias, since the figure is used just as an illustrative example. In addition, while the two LRG samples considered are defined in the same way, slightly different smoothing lengths and pixel sizes have been applied to their corresponding density fields (i.e. 21 h −1 Mpc and 7 h −1 Mpc for the DR4plus SHALLOW sample, and 22 h −1 Mpc and 3.7 h −1 Mpc for the DR7, respectively). Grey dashed lines are genus curves averaged over mock SDSS surveys in the simulated ΛCDM universe. We will return to this measurement in great detail in Section 5. a 10815 h −1 Mpc size box. Our main result for the observed genus curve is shown in Figure 1 (red solid line), and compared with a previous topology measurement on the SDSS DR4plus dataset (blue solid line; Gott et al. 2009). The main point of the figure is to show the dramatic increase in amplitude of the genus curve over the two different datasets (i.e. the SDSS DR4plus versus the SDSS DR7) in the same redshift range from 0.16 to 0.36 and rest-frame g-band absolute magnitudes of −23.2 < M g < −21.1, due to the much larger volume that is covered. In fact, we find the genus amplitude to reach 285 with an uncertainty of 4.0% at 22 h −1 Mpc Gaussian smoothing scale including cosmic variance (the most accurate measurement to date), while Gott et al. (2009) found the genus curve to reach about 167 with a 4.1% uncertainty at 21 h −1 Mpc smoothing scale; for comparison with a different galaxy population, Park et al. (2005b) and Choi et al. (2010) reported an uncertainty of 9.4% and 4.8% at 5h −1 Mpc and 6h −1 Mpc scales, re-spectively, for the genus obtained from the Main galaxy sample of the SDSS DR3 and DR7. We will return to this measurement in great detail in Section 5, while in a forthcoming publication we interpret the deviation of the genus curve from the expected Gaussian prediction in the context of primordial non-Gaussianity.
The layout of the paper is organized as follows. In Section 2, we present the main theoretical framework of this study; in particular, we review the genus statistics for Gaussian fields and discuss how to extend the formalism for non-Gaussian fields. In Section 3, we describe the SDSS LRG sample used for our measurements and the methodology applied to the observational data. In Section 4, we present the HR3 N -body simulation and explain the procedure adopted to construct the 81 mock LRG surveys. In Section 5, we show our results for the LRG genus statistics, compare measurements from SDSS DR7 data and simulations and quantify the non-Gaussian deviations of the genus curve with genus-related quantities. In Section 6, we discuss the effects of known systematics on the genus, and present the genus curve after corrections for systematics. We conclude in Section 7, and leave some more details on the genus curves in the Appendix.

THEORETICAL BACKGROUND
We begin by revisiting the basic theory of the genus for Gaussian fields and by introducing some genus-related statistics. We also briefly review the formalism for describing the non-Gaussian effect on the genus curve in the weakly nonlinear regime according to second-order perturbation theory, originally derived by Matsubara (1994Matsubara ( , 2003. We will later compare the theory outlined here with results from the SDSS DR7 LRG dataset, and with measurements from simulated LRG sample. The full extension to non-Gaussian fields with the inclusion of primordial non-Gaussianity will be presented and discussed in a forthcoming publication.

Genus statistics for Gaussian random fields
The genus is a measurement of the topology of isodensity contour surfaces in a smoothed galaxy field . In mathematical terms, it is defined as follows. Consider a three-dimensional Gaussian random field ρ ≡ ρ(x) with x the spatial coordinate, and measure the topology of the excursion regions where ρ is equal to, or is above a given threshold levelρ + νσ 0 . Hereρ is the mean of the field ρ, σ 0 its root mean square (r.m.s.) value, and ν = (ρ −ρ)/σ 0 . Denote with M the space which contains the set of the excursion regions (i.e. a 3-manifold subset), and indicate its boundaries with δM (i.e. a 2-manifold subset). For each component S i of δM , according to the Gauss-Bonnet theorem, the mathematical genus satisfies the following relation where χ(S i ) is the Euler characteristic of the surface of the three-dimensional excursion region (i.e. the integrated Gaussian curvature of the surface). Hence, the total genus of the boundary δM becomes where N is the number of components of δM -see Park et al. (2013) for a full derivation of the previous formula.
In cosmology, the standard definition of genus slightly differs from the previous mathematical one, since the genus is defined as the number of holes minus the number of isolated regions in the isodensity contour surfaces, at a given threshold level ν. Namely, G(ν) = Number of holes in contour surfaces − Number of isolated regions.
The relation between the two definitions is simply expressed by G = G − N , where N has been defined above. For further insights on topological invariants, and for the mathematical connection between the cosmological genus and the Betti numbers for excursion sets of Gaussian random fields, we refer the reader to Park et al. (2013).
In the case of Gaussian fields, the genus per unit volume g(ν) = G(ν)/V as a function of density threshold level ν is known (i.e. Doroshkevich 1970;Adler 1981;Hamilton, Gott, & Weinberg 1986): The amplitude g(0) is given by while the spectral moments of the fields, σ 2 j , are computed from In the previous relation, P (k, z) is the power spectrum smoothed on a scale R G by a window function W , where and P m (k, z) is the matter power spectrum. In particular, in this study we adopt a Gaussian smoothing of the form W (kR G ) = exp(−k 2 R 2 G /2). Note also that, for j = 0, σ j is the variance of the fluctuating field, while σ j is the variance of its derivative when j = 1.
To separate the variation in topology from the change of the one-point density distribution, in this work we also measure the genus as a function of the volume-fraction threshold ν f (as opposed to the direct density threshold ν). This parameter defines the density contour surface such that the volume fraction f in the high density region is the same as the volume fraction in a Gaussian random field contour surface having ν = ν f , namely: 2.2. Genus statistics and perturbation theory After correcting for known systematics, deviations of the observed genus curve from the Gaussian expectation (i.e. Eq. 4) are due to nonlinear gravitational evolution and non-Gaussianity of the primordial density field. A number of studies in the literature have already addressed the impact of non-Gaussianity on the genus curve (see for example Weinberg et al. 1987;Park & Gott 1991;Park, Kim, & Gott 2005a). In what follows, we briefly discuss the non-Gaussian effect on the genus curve caused by nonlinear gravitational evolution in the weakly nonlinear regime (which tends to distort the Gaussian expectation for the genus statistic), in the context of secondorder perturbation theory -along the lines of Matsubara (1994Matsubara ( , 2003. More details on the non-Gaussian modifications of the genus curve will be presented in Young-Rae Kim et al. (in preparation).
To first order in σ 0 , the nonlinear correction that one must apply to the genus curve due to gravitational evolution is an odd function of the threshold ν. Hence, this correction causes a shift and an asymmetry between high-and low-density regions, with no change in the amplitude at ν = 0.
In particular, when we use a threshold rescaled by a volume fraction of the smoothed field, ν f , the genus of the matter density field per unit volume -expanded to first order in mass variance σ 0 -can be written as a sum of a Gaussian term g G (ν f ) plus a non-Gaussian term g NG (ν f ), namely: g The Gaussian part is expressed by while the non-Gaussian term is given by (Young-Rae Kim et al., in preparation): In the previous equations, H n (ν f ) are Hermite polynomials, and in particular H 0 (ν f ) = 1, f + 15ν f . Also, the various S (a) , a = 0, 1, 2, are skewness parameters obtained by integrating the bispectrum B(k 1 , k 2 , k 3 , z) over k 1 and k 2 (see Eq. 61-64 in Matsubara 2003). In addition, the bispectrum can be given in terms of the nonlinear contributions from nonlinear gravitational evolution, and also primordial non-Gaussianity -an aspect that we do not consider here (but see Appendix B in Hikage et al. 2006). Note that the non-Gaussian part g NG (ν f ) only appears with terms of the form S (a) − S (0) . Assuming galaxy biasing local and deterministic in the weakly nonlinear regime, the skewness parameters of the galaxy bispectrum, S g )σ 0,g when the biased variance at first order is then given by σ 0,g = bσ 0 ; therefore, the non-Gaussian correction of the galaxy density field is exactly the same as the one of the unbiased mass density field in Equation 11 -hence independent of the bias parameter. This can be considered as an advantage of the volume fraction threshold, as opposed to the direct density threshold (see Matsubara 2003 for more details).

Genus-related statistics
The measured genus curve can be compared with predictions of the simplest inflationary models, which assume Gaussian random phase initial conditions (and so with Eq. 4). However, even if the initial conditions were perfectly Gaussian, small deviations from Gaussianity are expected because of systematic effects (for example shot noise or redshift-space distortions), and because of the physics connected with galaxy formation, nonlinear gravitational evolution, and primordial non-Gaussianity (if any). Therefore, it is important to quantify even small departures from Gaussianity from the observed genus. This is done by parametrizing the genus curve with several derived quantities. In what follows, we consider measurements as a function of the volume fraction threshold ν f , and introduce four genus-related statistics. The first quantity is simply the best-fit genus amplitude G fit (0), measured by a least-squares fit of the theoretical random phase curve to the data considering only the range −1 ≤ ν f ≤ 1. In principle, its value is given by Equation 5, but the measured one is always lower because of nonlinear clustering and biasing due to coalescence of structures (Park & Gott 1991;Vogeley et al. 1994;Canavezes et al. 1998;Gott et al. 2008).
The second quantity is the shift parameter ∆ν f , defined as where G o and G fit are the observed and best-fit Gaussian genus curves -both given by Equation 4, but in the former case with the observed amplitude G o (0) and in the latter with the best-fit one, G fit (0), as explained in . The parameter ∆ν f controls the horizontal shifts of the central part of the genus curve. For a density field dominated by voids, ∆ν f is positive and we say that the density field has a "bubble-like" topology. For a cluster-dominated field, ∆ν f is negative and we say that the field has a "meatball-like" topology. We then further introduce two additional quantities, A C and A V , which measure the abundances of clusters (C) and voids (V), respectively, relative to the expectations for a Gaussian random field. They are defined by the following relation where the integration intervals are +1.2 < ν f < +2.2 for A C , and −2.2 < ν f < −1.2 for A V (Park, Gott, & Choi 2001;Park, Kim, & Gott 2005a). These intervals are centered near the minima of the Gaussian genus curve (i.e. ν f = ± √ 3), far away from the thresholds where the genus is often affected by the shift phenomenon. The previous intervals also exclude extreme thresholds, where for low density regions ν f is very sensitive to the exact density value. These parameters are defined so that the condition A C > 1 (A V > 1) implies that more independent clusters (voids) are observed, with respect to those predicted by a Gaussian field at a fixed volume fraction. On the opposite, A C < 1 (A V < 1) means that fewer independent clusters (voids) are seen.
The effects of gravitational evolution, galaxy biasing, and cosmology dependence on the statistics defined by ∆ν f , A V , and A C -as a function of the smoothing scale -have been addressed in detail by Park, Kim, & Gott (2005a); we refer to their study for more details.

METHODOLOGY
In this section we briefly describe our LRG sample obtained from the SDSS DR7, along with the methodology applied to the observational dataset. The genus computed from the LRG sample and its related statistics will be presented later on, in Section 5.

The SDSS DR7 LRG sample
The SDSS is a successful ground-based survey, designed to explore the large-scale distribution of galaxies and quasars by using a dedicated 2.5 m telescope at Apache Point Observatory (see Gunn et al. 2006 for technical details). The photometric survey has imaged roughly π steradians of the Northern Galactic Cap in five photometric bandpasses denoted by u, g, r, i, and z, and centered at 3551, 4686, 6165, 7481 and 8931 Angstroms, respectively. The imaging camera used is to bottom, three representative density contours enclosing low-density regions, which occupy respectively 2.3% (ν f = −2.0), 6.7% (ν f = −1.5), and 50% (ν f = 0.0) of the sample volume. (Right) Three density contours enclosing now high-density regions, filling respectively 2.3% (ν f = 2.0), 6.7% (ν f = 1.5), and 50% (ν f = 0.0) of the sample volume -from top to bottom, in symmetry with the low-density cases. The Earth is located at the center of the x-y plane shown in the figure, and the size of each axis corresponds to a scale of 200 h −1 Mpc. equipped with 54 CCDs (Fukugita et al. 1996;Gunn et al. 1998). The limiting photometric magnitudes are 22.0, 22.2, 22.2, 21.3 and 20.5 in the previous five bandpasses, at a signal-to-noise ratio of 5 : 1. The median width of the PSF is 1.4 ′′ and the r.m.s. photometric uncertainties are at the 2% level (Abazajian et al. 2004). After image processing (Lupton et al. 2001;Stoughton et al. 2002;Pier et al. 2003) and calibration (Hogg et al. 2001;Smith et al. 2002), targets are selected for spectroscopic follow-up observations. The spectra are obtained with two dual fiber-fed CCD spectrographs at a spectral resolution λ/∆λ ≃ 1800, and r.m.s. uncertainty in redshift of ∼ 30 kms −1 . Because of mechanical constructions, two fibers cannot be placed closer than 55 ′′ on the same tile. The incompleteness percentage of the spectroscopic survey reaches about 6%. The SDSS spectroscopy yields three major samples: the Main galaxy sample (Strauss et al. 2002), the LRG sample (Eisenstein et al. 2001), and the quasar sample (Richards et al. 2002). In particular, the LRG sample considered here is part of the final data release of the SDSS-II, indicated as DR7, which yields 928, 567 galaxy spectra over the legacy spectroscopic coverage of 8032 deg 2 .
In this work, we made a volume-limited sample including 67, 385 LRGs in the redshift range from 0.16 to 0.36 and rest-frame g-band absolute magnitudes of −23.2 < M g < −21.1, passively evolved to z = 0.3 (see Zehavi et al. 2005;Eisenstein et al. 2005), by using the "DR7-Full" sample of Kazin et al. (2010). K-corrections have been applied to all the galaxies in the sample, assuming a fiducial ΛCDM model with Ω m = 0.26 and h = 1, not Ω m = 0.25 which was applied to the sample by Kazin et al. (2010). To maximize the volume-to- surface ratio, we trim the sample as in Choi et al. (2010) -see their Figure 1 for more details. Both the Southern Galactic Cap region and the Hubble Deep Field region are dropped. These cuts leave a total of 60, 466 LRGs over about 2.33 sr in the survey region with an angular selection function greater than 0.6. Figure 2 shows three-dimensional isodensity contours of the smoothed galaxy number density fields obtained from the SDSS LRG sample at ν f = ±2.0, ±1.5 and 0, of which the corresponding volume fractions are 2.3%, 6.7%, and 50%. A Gaussian smoothing is applied with R G = 22 h −1 Mpc. As expected, the asymmetry between high-and low-density regions of the observed genus curve shown in Figure 1 is also clearly seen in this visual comparison. The low-density regions (left upper panels) tend to be more connected and filamentary than the highdensity regions (right upper panels), where structures appear to be more isolated and rounder. As the volume fraction increases, the structures increase in size.

Construction of the galaxy mass and number
density fields from the observed LRG sample For an arbitrary large-scale galaxy survey, the sampling of galaxies as a function of redshift is usually not uniform. Moreover, typically the survey is designed so that not only the mean galaxy number density is not constant, but also the sampling in absolute magnitude is non-uniform.
An example is shown in Figure 3, for the semi-volumelimited LRG sample. The solid lines in the panel identify the volume-limited sample considered in this study. The high non-uniformity of the sample, as a function of redshift and absolute magnitude, is clearly visible.
In this situation, it would be incorrect to give a singlevalue weight to galaxies in each redshift bin, based only on the radial selection function; in fact, this simple scheme would over-weight the galaxies fully sampled, and under-weight those under-sampled. Galaxies with different luminosity are known to cluster differently   2013), and therefore they should get different weights if the sampling varies with luminosity -to avoid the clustering mismatch. The problem becomes more serious when galaxy luminosity or mass are used as weights, to obtain the galaxy luminosity density or the mass density field, respectively. In particular, when the sampling in luminosity or mass varies with redshift, the resulting luminosity or mass density will have different mean values across different redshift bins, even if the galaxy number density is matched. Therefore, one should also consider the radial density gradient.
For the construction of our galaxy mass and number density fields from the observed LRG sample, we devise a new weighting scheme (called 'luminosity function matching') which properly accounts for the sampling rate variations depending on the location of the galaxy both in redshift and absolute magnitude space, variations that are caused by the LRG target selection procedure. In what follows, we consider the case when there is no evolution of the luminosity function (LF) with redshift, and briefly summarize our procedure (see also Figs. 4, 5 and 6 below).
i. Select a reference redshift bin, and compute the ref- The LF determined in this (arbitrary) redshift interval will be used to match the LF in other redshift bins, as shown in Figure 4. For our study, we choose the interval 0.16 < z < 0.20 as the reference z-bin; Φ ref (M g ) computed in this interval is indicated with the thick solid line in Figure 4.
ii. Select bins in the two-dimensional plane defined by redshift versus absolute magnitude, and compute the LF for each pixel of the two-dimensional array (M g , z) above an absolute magnitude cut, M g,cut .
iii. For each pixel of the two-dimensional array (M g , z), Fig. 5.-Comoving number density of the SDSS LRGs sample as a function of redshift (thin solid line). The number density has been corrected for object-loss due to fiber collisions ) and spectroscopic completeness. Dotted line (green histogram) shows the radial distribution constructed by weighting each galaxy only with the radial selection function. Thick solid line (blue histogram) is the galaxy number density field constructed by weighting each galaxy with the new 'luminosity function matching' weighting procedure described in the main text -i.e. w(Mg, z), Equation 14.
calculate the proper weight w(M g , z) as From Figure 4, one can easily infer, just by looking in the magnitude range fainter than M g ∼ −21.9, that this weighting scheme will clearly weight galaxies at moderate redshifts more than those at low redshifts.
iv. Construct the galaxy number density field, weighting each galaxy by w(M g , z), which is linearly interpolated from the two-dimensional array (M g , z) computed as described in the previous steps. The number density field obtained in this way will be uniform both in redshift and luminosity space (see the thick blue histogram in Fig. 5), as opposed to the one constructed by weighting each galaxy with the radial selection function alone (dotted line in the same figure). In particular, the density field is calculated on a mesh with cubic pixels from a discrete particle distribution using the cloud-in-cell (CIC) mass assignment scheme.
v. Alternatively, construct the mass-weighted halo density field from the observed galaxy sample. The galaxy mass, M gal should be the halo mass M h corresponding to the g-band galaxy luminosity M g , i.e.
M h = f (M g ) (see point iii. in Section 4.2 for more details). Using the LRG cumulative LF measured at the reference redshift bin, and the halo cumulative mass function derived from a full cubic data snapshot of the HR3 at z = 0.2 (which is compatible with the reference redshift), we apply the halo-galaxy oneto-one correspondence model (HGC) of Kim, Park, & Choi (2008) and convert galaxy luminosities into halo masses, and vice-versa. Fig. 6.-Relation between galaxy luminosity (Mg) and halo mass (M h ) obtained from the HR3 with the halo-galaxy one-toone monotonic correspondence model (HGC) of Kim, Park, & Choi (2008). The mapping is used to compute the galaxy mass M gal (see the end of Sec. 3.2). Figure 6 shows the relation between galaxy luminosity and halo mass, used to determine M gal . The halo mass corresponding to the absolute magnitude cut, M g,cut , is M h,cut = 10 13.466 h −1 M ⊙ . To compute the galaxy mass density field, each galaxy is weighted by w(M g , z) × M h . The mass density field derived with this procedure is equivalent to the one obtained from uniformly selected LRGs.

METHODOLOGY
In this section we briefly describe the Horizon Run 3 N -body simulation, and the procedure to construct the SDSS DR7 mock LRG samples from the simulation output. We will then compare numerical results and measurements from data in Section 5. The mock surveys will also be used to quantify several nonlinearities due to systematics which affect the genus curve: correcting for these effects allows one to accurately recover the topology of the underlying matter, as we will present in Section 6.

The Horizon Run 3 N -body simulation
The Horizon Run 3 (HR3; Kim et al. 2011) is one of the largest N -body simulations to date, made using 7210 3 = 374 billion particles, spanning a volume of (10.815 h −1 Gpc) 3 -which is over 8800 times the volume of the Millennium Run (Springel et al. 2005). The particle mass is down to 2.44 × 10 11 h −1 M ⊙ , allowing to resolve galaxy-size halos with mean particle separation of 1.5 h −1 Mpc. The simulation is based on the ΛCDM cosmology, with parameters fixed by the WMAP 5-year data (Komatsu et al. 2009). The linear power spectrum used is obtained with the CAMB source (http://camb.info/sources), which provides a better measurement of the BAO scale. The simulation starts at z i = 27, and is evolved till the present epoch with 600 global time steps. The code used for the run is an improved version of the Grid-of-Oct-Trees-Particle-Mesh code (GOTPM), originally devised by Dubinski et al. (2004); a new procedure has been implemented, in order to describe more accurately the particle positions using single precision.
In the HR3, halos are first identified via a standard Friend-of-Friend (FoF) procedure. Then subhalos are found -out of FoF halos -with a subhalo finding technique developed by Kim & Park (2006) and Kim, Park, & Choi (2008). This method allows one to identify physically self-bound (PSB) dark matter subhalos not tidally disrupted by larger structures at the desired epoch. In particular, LRGs are identified as the most massive dark matter subhalos. To make the comparison with observational data, we saved the particle positions and velocities along the past light cone for 27 separated observers, and found subhalos in the past light cone surface from z = 0 to z = 0.7.
From each simulated light cone, we made 3 mock samples using exactly the same survey mask and angular selection function of the SDSS sample. In addition, we applied the same smoothing length as for the observational case. In total, we are able to obtain 81 non-overlapping mock samples, thanks to the enormous volume of the HR3. To this end, we note that the ability to simulate big volumes is essential (particularly for the LRG distribution), since larger volumes allow one to model more accurately the true power at large scales and the corresponding power spectrum. A large box size will guarantee small statistical errors in power spectrum estimates, so that the acoustic peak scale can be measured with an accuracy better than 1%, and the genus curve characterized with unprecedented statistical significance.

Construction of the mock LRG samples
A crucial step in our analysis is the construction of realistic mock LRG samples. This requires the ability to mimic all the observational biases, such as survey boundary, radial and angular selection function, redshift space distortions and so forth. To build the various simulated catalogs, 27 observers were placed in the HR3 box, each covering the redshift range 0 < z < 0.7 without overlaps: this means that the survey volumes are totally independent. In addition, the LRG mocks are made so that they span exactly the same range in absolute magnitude as the observational sample; hence, the number of galaxies in each mock is nearly equal to the observed one (at the percentage level accuracy). Moreover, the simulated galaxies should be observed in redshift space along the past light cone with the same radial and angular selection function of the observational sample, and also with the same selection function in absolute magnitude space as in the real observation. Figure 7 shows a three-dimensional example of the simulated LRG number density field obtained from the HR3, smoothed with a Gaussian filter at R G = 22 h −1 Mpc scale. The plot is the equivalent of Figure 2, but now for the LRG mock samples constructed from the HR3 simulation. Again, the left panels display three representative density contours enclosing low-density regions, which occupy respectively 2.3% (ν f = −2.0; top), 6.7% Since we apply identical techniques both to the SDSS LRG sample and to the LRG mock surveys, we expect the results of the analysis to be identical across datasets -within statistical variations -if the simulations are correctly modeling the distribution of galaxies. In what follows, we describe in more detail how to build the SDSS DR7 mock LRG samples from the HR3 simulation output. Results from our procedure confirm that we are correctly modeling the LRG distribution (see Figs. 8 and 9). The major steps of the construction process are summarized next.
i. Locate 27 observers in the HR3 simulation box, and save all dark halos along the past light cone of the observer during the simulation, in the redshift range 0 < z < 0.7. From each light cone data, make 3 mock surveys using exactly the same survey mask and angular selection function as for the SDSS volumelimited sample.
ii. Apply a proper correction to make the halo mass function uniform in redshift. In fact, in the HR3 simulation the minimum mass limit of subhalos that can have LRGs with constant observed number density varies as a function of redshift (see Fig. 6 in Kim et al. 2011). This leads to the following relation; f (z) = (−8. iii. Populate dark matter halos with galaxies using a suitable correspondence scheme. In essence, to connect galaxies with halos one needs to make an assumption on the relation between galaxy luminosity and halo mass. A widely-used approach is the subhalo abundance matching, where more luminous galaxies are assigned to more massive haloes Tasitsiomi et al. 2004;Vale & Ostriker 2006;Conroy & Wechsler 2009;Guo et al. 2010;Behroozi et al. 2010;Kim, Park & Choi 2008). This scheme assumes that halos with mass above a certain threshold and with a given mean number density correspond to galaxies with luminosity or mass above a certain threshold and having the same mean halo number density. For our mocks, we apply the halo-galaxy one-to-one monotonic correspondence model (HGC) of Kim, Park, & Choi (2008) which extends the subhalo abundance matching procedure: there is one and only one galaxy in each subhalo, and a more massive subhalo hosts a more luminous galaxy. The mapping M h = f (M g ) is shown in Figure 6. This correspondence scheme allows us to assign a luminosity to each LRG mock galaxy, and to compute galaxy masses (see also the end of Sec. 3.2).
iv. Account for the effects of the color-dependent luminosity cut imposed by the SDSS LRG volume-limited sample selection criteria, both in redshift and luminosity space, which reduces the sampling density (see again Sec. 3.2). In order to do so, we discard mock galaxies with a rejection probability given by 1/w, where w ≡ w(M g , z) is derived from the observed sample (see Eq. 14).
Our procedure successfully reproduces the dependence of the LRG sampling rate on luminosity and redshift as in the SDSS LRG sample. This is shown in Figures 8 and  9, the corresponding counterparts of Figures 4 and 5, respectively, obtained from simulated samples. In particular, Figure 8 displays the luminosity function computed at different redshift bins from one of our LRG mocks. To facilitate the comparison with the actual SDSS data, open circles in the figure are luminosity function measurements derived from the SDSS LRG sample in the reference redshift bin 0.16 < z < 0.20 -as in Figure 4, see Section 3.2. Clearly, the simulated results agree very well with observational measurements in terms of matching the luminosity function. To this end, Figure 9 shows the comoving number density of SDSS mock LRGs averaged over all the 81 mock samples, as a function of redshift (black dashed line). Solid line shows the radial distribution obtained by weighting each galaxy with w(M g , z). Colored thin lines show results of six arbitrary mock surveys. Filled and open circles in the figure are analogous measurements derived from the SDSS LRG sample -as in Figure 5. The weighting scheme used is explained in Section 3.2. Even in this case, the plot confirms the correctness of our modeling procedure: our mocks have the same sampling rate in redshift as the observed SDSS LRG sample.

MEASUREMENTS
In this section we present results for the genus measured from the SDSS DR7 LRG sample, and from our LRG mocks obtained from the HR3 LCDM simulation. In both cases, we compute the genus curves using the mass weighted density field and the number density field -although later on we will only use the number density. By contrasting observational results against mock measurements which assume Gaussian initial conditions, we detect significant non-Gaussian deviations of the observed genus curve from theoretical expectations. We then further quantify these discrepancies by introducing a new statistical test. A large part of the non-Gaussian deviations is caused by systematics, and we address their impact on the genus curve in Section 6. 5.1. Genus of SDSS LRGs from the mass weighted density field and from the number density field In Sections 3.2 and 4.2 we constructed the mass weighted density field and the number density field in Fig. 10.-Genus curves derived from the smoothed mass density and number density fields, respectively, as a function of the volume threshold ν f . Measurements are not corrected for systematics yet. A Gaussian smoothing of radius R G = 22h −1 Mpc has been applied, at a pixel size of p = 3.7h −1 Mpc. In both panels, red thick solid lines are used for the number density field, and black thin lines for the mass weighted density field. (Top) Measurements averaged from 81 mock LRG samples. The genus curve of the dark matter distribution in real space using the full simulation cube is also shown, with the dotted green line. (Bottom) Analogous measurements from the SDSS sample. See the main text for more details.
order to compute the genus. This is because Jee et al. (2012) found that the halo mass density has a much tighter (and simpler) relation with the underlying matter density than the halo number density. A similar conclusion was reached by , in relation to the gravitational shear field. To this end, Figure 10 shows the genus curves derived from the mass density and number density fields, as a function of the volume fraction, ν f .
No corrections for systematics are yet applied. We smooth our density field (either the mass weighted or the number density one) with a Gaussian filter of radius R G = 22h −1 Mpc at a pixel size of p = 3.7 h −1 Mpc. To minimize any nonlinearity introduced by the choice of the pixel dimension, we use the smallest possible pixel size we can afford. In the figure, red thick solid lines are used for the number density field, and black thin lines are for the mass weighted density field.
In particular, the top panel shows our measurements  Figure 10. averaged from 81 mock LRG samples, where we also plot the genus curve of the dark matter distribution in real space using the full simulation cube -with the dotted green line. The difference between halo and matter density field curves are mostly due to halo biasing and discrete sampling of the halo density field (i.e. shot noise). The genus amplitude measured from the halo mass weighted density field is lower than the one obtained from the halo number density field. This is consistent with the result of Seljak, Hamaus, & Desjacques (2009); namely, weighting halo galaxies by halo mass can significantly suppress shot noise. The bottom panel shows similar measurements from the SDSS LRG sample. To date, these are the best measurements of the genus curve from the SDSS survey catalog. In Table 4 of Appendix A, the genus values are given as a function of the volume-fraction threshold level. In particular, the genus amplitude obtained from the number density field is equal to G o = 285, with a ∼ 4.0% error including all systematic effects such as finite pixel size, survey boundary, radial and angular selection functions and sparse sampling (see also Fig. 1). The errorbars in the figure are the 1σ deviations computed from 81 independent mock samples.

5.2.
Genus-related statistics: quantifying the non-Gaussian deviations Although the shape of the genus curve does not depend on the weighting scheme as much as its amplitude, results from simulations reveal that the mass weighted density field tends to show more 'meatball'-shifted topology and more isolated clusters. In Table 1 we report the details of the genus-related statistics (see Sec. 2.3), for the genus measurements displayed in Figure 10, while in Appendix A we list the complete genus values relative to these measurements, as a function of the volume-fraction threshold ν f (see Tab. 4). We also provide similar measurements as a function of ν. The latter table may be useful for readers who wish to directly use galaxy clustering topology for cosmological applications.
From Figure 10 and from the results for the genusrelated statistics, it is evident that in the mock measurements low-density regions are definitely less affected by the weighting scheme. Instead, the weighting scheme Fig. 11.-Statistical test to estimate quantitatively the non-Gaussian discrepancies between the predicted and observed genus curve, for the SDSS sample -as explained in the main text. Histograms are integrals of the the differences between the genus curve of each individual mock sample and the curve obtained by averaging all our 81 mock LRG samples. Three different intervals in ν f are considered, as indicated in the various panels. Triangular symbols are measurements obtained from the SDSS sample at the corresponding threshold intervals. The discrepancies are seen near the mean density regions and in low-density regions (−3.0 ≤ ν f ≤ −1.7), with a significance level of 90%.
does not introduce any change in the genus amplitude derived from the observational sample, and the mass weighted density field still produces more isolated clusters. This finding clearly points towards the existence of a significant amount of scatter in the relation between galaxy observables and their underlying halos; hence, one needs to gain a better understanding of the connection between galaxies and their dark matter halos, and of the galaxy formation process in general. In this paper, hereafter we will adopt the use of number density instead of the mass weighted density, for simplicity.
Overall, the HGC galaxy assignment scheme of Kim, Park, & Choi (2008) is able to match well the observed amplitudes and shapes of the corresponding genus curves. In fact, from Table 1, one can notice that the values obtained for ∆ν f , A V and A C from the simulated LRG samples agree well -within the quoted uncertainties -with those measured from the observational sample. However, we detect some significant discrepancies between mocks and observations for A V in the lower density regions beyond the integration intervals quoted in Section 2.3 (i.e. ν f ≤ −1) -see indeed the difference between the red and grey lines in Figure 1. We also detect some discrepancies for the observed and predicted genus amplitudes.
To estimate quantitatively the statistical significance of these discrepancies, we use the following method. First, we calculate the differences between the genus curve of each individual mock sample and the curve obtained by averaging all the 81 mock samples; we do this at three different intervals, i.e. −3.0 ≤ ν f ≤ −1.7, −0.2 ≤ ν f ≤ 0.3, and 1.2 ≤ ν f ≤ 2.2. We then plot the integrals of these differences as histograms in Figure 11. Finally, we place in the same plot our measurements obtained from the SDSS sample at the corresponding threshold intervals, indicated with triangular symbols. Here we measured genus curves from the number density fields of the samples.
As one can infer from the figure (with a significance level of 90%), departures from Gaussianity are seen near the mean density regions, and in low-density regions (i.e. in the interval −3.0 ≤ ν f ≤ −1.7). The radical difference between the genus curves of the observational and simulated data in low-density regions shows that topology is highly sensitive to the connectivity of voids. In the next part, we will address the key role played by systematics on the genus curve which will explain some of the discrepancies, and show how to accurately correct for their effects to recover the topology of the underlying matter. In a forthcoming paper, we provide an interpretation of the remaining deviations (i.e. after correcting for known systematics) in the context of primordial non-Gaussianity.

GENUS TOPOLOGY OF LRGS: SYSTEMATICS
In this section we briefly discuss the known systematics which affect the genus curve. We then test and quantify their impact on the genus using the genus-related statistics presented in Section 2.3, with the help of mock LRG samples (Section 4.2), and show how to correct for their effects. By applying those corrections, we obtain the most accurate constraint on the genus amplitude to date, which significantly improves on our previous measurements. In particular, Figures 14 and 15 are among the most important results of our paper.
6.1. Impact of systematics on the genus curve As anticipated in Section 2.3, even if the initial conditions were perfectly Gaussian, small deviations from Gaussianity are expected because of systematics. Since systematics directly impact the shape and amplitude of the genus curve, it is imperative to be able to quantify and correct for their effects. This can now be done quite accurately, with the help of realistic mock catalogs such as those constructed from the HR3 (Section 4.2).
Broadly speaking, systematics that cause non-Gaussian deviations in the genus curve can be classified into three main classes: those due to the observational or analysis strategy, those due to statistics, and those of cosmological origin. Finite pixel size effects, survey boundary mask, radial and angular selection function, past light cone gradient, and initial conditions of the simulations belong to the first class. Shot noise or sparse sampling and cosmic variance are of statistical origin, while galaxy biasing, nonlinear gravitational evolution, and redshift-space distortions (RSD) are related to cosmology. Sometimes, but this depends on the chosen terminology, the last class is not considered as a systematic effect. Here, we broadly term all these three classes as systematic biases on the genus curve.
Clearly, several of the previously mentioned effects are connected, and so one needs to remove them simultaneously. In the absence of other known systematics, an eventual residual of non-Gaussianity (after applying the corrections mentioned above) has to be ascribed to a primordial origin. In what follows, we discuss in particular the nonlinear gravitational evolution, and the effects of galaxy bias and past light cone on the genus. More details on the full modeling of systematics in topology measurements will be presented in Young-Rae Kim et al. (in preparation).

Modeling and correcting for systematics
In this work we perform similar corrections as those applied by Choi et al. (2010), who studied the effects of systematics on the genus computed from the nearby Main galaxy sample of the SDSS DR7. The overall goal is to remove the nonlinear systematics in the observed sample step-by-step, as well as to estimate the genus curve of the underlying matter density field using a set of mock samples.
We first consider the effect of nonlinear gravitational evolution on the genus curve, measured from our mock LRG samples -along with survey mask and initial condition effects. For this purpose, we compute the genus curves of 81 dark matter density fields, both at the initial (z i = 27) and final (z f = 0) redshifts; these quantities, measured in real space, are indicated as G j m,r |z i and G j m,r |z f , respectively, where the index j refers to the particular light-cone mock survey considered, m stands for matter, and r for real space. Each density field is selected within the SDSS survey mask, at a particular region in the simulation so that the evolved density field has a one-to-one correspondence with the initial density field.
Panel (a) in Figure 12 shows the gravitational evolution effect on the genus curve, in real space. The black solid line is the genus curve averaged over all the 81 simulated initial matter density fields (at z i = 27), while the blue dashed line is the corresponding final one, at z f = 0, computed in the same way. The green dotted line is the predicted linear theory genus, relative to the entire survey volume. The SDSS survey mask is applied. The lower part in the same panel displays the ratio ∆ = [G(ν f ) − G LIN (ν f )]/G LIN (ν f = 0), as a function of the volume threshold ν f and for the two different redshifts , as a function of the volume threshold ν f and for the two different redshifts considered. See the main text for more details. (b) Genus curve of the matter distribution in real space at z = 0, using the full cubic data. The black solid line is used for the full simulation cube, while the dashed blue line is obtained with the SDSS LRG survey mask applied. The dotted red line is the second-order perturbation theory prediction of Matsubara (2003). The difference between the red and black lines shows the discrepancy between second-order perturbation theory prediction and N -body simulations.
considered. In particular, the shape of ∆(ν f ) at z i = 27 is affected by cosmic variance -which causes small deviations from Gaussianity (of statistical nature) in a finite volume sample -and bias, which arises from the initial conditions of the HR simulations obtained via the Zeldovich approximation. The genus curves averaged over all the matter density fields are relatively noisy (about 9%).
Panel (b) in Figure 12 shows instead the genus curve Fig. 13.-Impact of systematics on the genus curve: galaxy (halo) biasing, shot noise, past light cone gradient and RSD. A Gaussian smoothing length of R G = 22 h −1 Mpc is applied. (a) Effects of shot noise and galaxy biasing. The black solid line shows the genus obtained from the averaged matter density field, and the dashed blue line is used for the one obtained from an average of the halo density field. Similarly as in Figure 12, the middle part in the same panel displays the quantity ∆, which proves that the dark matter density field and the halo number density have very different topology. In the bottom part of the same panel, G j m,r |z f is the genus value averaged over all the 81 mocks in real space at the final epoch. (b) Effects of past light cone gradient and RSDs. The solid black and dotted red lines are the genus curves averaged over 81 past light cone mock SDSS surveys of the LRGs in real and redshift spaces, respectively. See the main text for more details. In the lower part of the same panel, G j h,r |z f is the genus value averaged over all the 81 halo density fields in real space at the final epoch.
of the matter distribution in real space at z = 0, using the full cubic data. The black solid line is used for the full simulation cube, while the dashed blue line is obtained with the SDSS LRG survey mask applied. The ratio ∆ν f gives information only about nonlinear gravitational evolution. We have attempted to fit the discrepancy with the second-order perturbation theory prediction of Matsubara (2003). The dotted red line in the lower panel of the same figure shows the result of this fit. Genus (2) gr = 3.695 for the WMAP five-year cosmology assumed. The difference between the dotted red curve and the solid black line shows the discrepancy between second-order perturbation theory prediction and N -body simulation measurements. Hence, we find that perturbation theory considerably disagrees with the numerical simulation result at the smoothing scale of R G = 22h −1 Mpc. The gravitational evolution produces a negative shift and decreases the genus amplitude by ∼ 3%.
The top part of Table 2 lists the genus-related statistics of the samples used in Figure 12 to quantify the deviations of the genus curves from the Gaussian expectation -due to the mentioned systematic effects.
We then consider the effect of galaxy (halo) biasing on the genus -along with shot noise, past light cone gradient and RSD. To this end, we made 81 mock samples from the snapshot halo full cubic data at z = 0, in exactly the same way as the past light cone of the LRG mock samples. We then compared the genus curve averaged over all the halo density fields, G j h,r |z f , where here h stands for halo, with the one obtained by averaging the 81 matter density fields at the same redshift, G j m,r |z f . Results are displayed in Figure 13(a), with the black solid line for the genus obtained from the averaged matter density field, and with the dashed blue line for the one obtained from an average of the halo density field in real space. Similarly as in Figure 12, the middle part in the same panel displays the quantity ∆ previously defined, which proves that the dark mat-ter density field and the halo number density have very different topology at 22 h −1 Mpc smoothing scale. Note however that the genus curve here includes shot noise due to discrete sampling of the galaxy density field as well as the galaxy biasing effect. Their combined effect has been presented by Hikage, Taruya, & Suto (2001 and Park, Kim, & Gott (2005a). As it can be seen from the scatter between the two curves, the combined effect of galaxy biasing and shot noise yields significantly larger non-Gaussianities than the nonlinear gravitational evolution effect. The bottom part of the same figure shows the combined effect. In particular, the combined effect increases the genus amplitude and strongly alters the skewness of the genus curve (see the value of G j h,r |z f in Table 2; shifted towards meatball topology, more percolated and thus larger void structures, and larger number of isolated clusters compared to those from the matter density fields, G j m,r |z f ). Past light cone effects and RSDs are quantified in panel (b) of Figure 13. This is achieved by comparing the genus curve measured from our 81 past light cone (LC) mock samples in real space, G j h,r |LC constructed as explained in Section 4.2, with the averaged one obtained from the halo density fields, G j h,r |z f . Similarly, we can quantify the effect of RSDs by computing the difference between G j h,r |LC and the average genus curve measured from the 81 past light cone mock samples but now in redshift space, G j h,z |LC -see also Table 2. All these curves are displayed in Figure 13. Again, the lower parts in the same panel clearly describes these effects: basically, the systematic effect introduced by the past light cone on the genus curve reaches up to 2% (see the black solid line in the bottom part) which is nearly as significant as the gravitational evolution effect -see the black solid line in the middle part of Fig. 13(a) -, while RSDs make the amplitude of the genus curve decrease, but do not alter its shape significantly.
With the help of our mocks, we are able to understand and quantify the nonlinear systematics involved in the observational sample. In particular, we find that a considerable portion of nonlinearity comes from the combined effects of galaxy biasing and shot noise. We can remove the nonlinearities due to those systematics, and thus estimate the genus curve of the underlying matter density field from the observed sample. The correction term that should apply to the observed data, derived from the j-th mock sample, is given by This correction removes all the systematic effects previously mentioned, including RSD, survey mask, shot noise and galaxy biasing, from the observed genus curve G o ; clearly, the underlying assumption is that our HGC assignment scheme is able to model correctly the relation between galaxies in our sample and the underlying halos. The genus of the observed (underlying) matter density field at z = 0 in real space, reconstructed by applying the correction term, is calculated as follows: where now N is the number of mock surveys. We also included the correction for finite pixel size effect, ∆G pix , and one for the bias of the j-th mock sample originated from the cosmic variation in the initial conditions of the HR3 simulation, ∆G j sim = G j m,r |z i −G LIN . The correction for the finite pixel size effect is given by the following equation: where A is the genus amplitude calculated from the variance and derivative (i.e. σ 0 , σ 1 ) of the matter density field on a mesh with vanishing pixel effect (where p ≃ 0), and the coefficients of each Hermite polynomial, a, b, c, and d, are 0.04794, 0.02337, 0.33146, and 0.03843, respectively. For the density field smoothed with a Gaussian filter R G = 22 h −1 Mpc and pixel size of p = 3.7 h −1 Mpc, this effect can be as large as about 1%, and should be taken into account. Full details of the modeling of the pixel effect can be found in Young-Rae Kim et al. (in preparation).
6.3. Genus curve after corrections for systematics Finally, we are able to obtain the genus curve of the underlying matter density field from the SDSS LRG sample with the effects of shot noise, galaxy bias, RSDs, survey boundary and finite pixel size all corrected. Our final result contains only the nonlinearity produced by nonlinear gravitational evolution, and a possible primordial non-Gaussian component -if any. Figure 14 shows the observed genus curve before the corrections for systematics (open circles), and after applying those corrections (grey thick solid line, with 1σ error bars). In the same figure, we also display the genus curve averaged over all the mock surveys after applying the same systematics correction (black thin solid line), and the linear theory prediction (green dashed line). Overall, the shape of the genus curve agrees very well with the mean topology of the SDSS LRG mock surveys in the ΛCDM Universe (see also the appendix for the tables of the genus curves, and for more details on the effect of these corrections on the mocks). However, comparison with simulations also shows small deviations of the observed genus curve from the theoretical expectation for Gaussian initial conditions. Figure 15 quantifies these deviations, by showing the difference between the genus curves for the SDSS (thick red line) -after the correction for systematic effects -and the simulated dark matter distribution at z = 0 (thin black line), and the linear analytical predictions G LN (normalized by the maximum value of G LN ). The shaded area indicates the 1σ limits calculated from the 81 HR3 halo mock surveys, after Fig. 15.-Difference between the genus curves (denoted by G) of the underlying matter distribution derived from the SDSS LRG distribution (thick red line) after the correction for systematic effects and the HR3 snapshot matter distribution at z = 0 in real space (black thin line), and the linear analytical prediction for the WMAP 5-year cosmology (i.e. G LN ) normalized by the amplitude from the linear prediction. The shaded area indicates the 1σ limits calculated from the 81 HR3 halo mock surveys after the same correction has been applied. The thin solid gray line shows the averaged deviation from the mock surveys. The dashed line shows the second-order perturbation expectation at the median redshift z = 0.28668 of the SDSS sample given by Matsubara (2003); clearly, the underlying matter distribution at the present epoch needs additional terms, compared to the second-order perturbation expectation.
the same correction has been applied. The thin solid gray line shows the averaged deviation from the mock surveys. Taking those noisy level-to-level variations into account, and given the uncertainties, the observed underlying matter distribution is in good agreement with the genus computed from the N -body simulation and the HGC galaxy formation model -except for the underdense regions below ν f = −1.8 filling 3.5% of the sample volume. The dashed line shows the second-order perturbation expectation at the median redshift z = 0.28668 of the SDSS sample given by Matsubara (2003). However, the genus curve measured from the gravitationally evolved matter density field in the HR3 simulation (solid line in Fig. 15) indicates that perturbation theory cannot model properly gravitational evolution effects. On the contrary, those effects are modeled well if we add extra terms in the second-order perturbation formula of Matsubara (2003), depending on H 0 and H 2 . To this end, the dotted line is a fit to the simulation using the following new perturbation formula: where G(0) = 298.5. The second-order perturbation theory predicts the power spectrum P (k, z) in Equation 6 as follows: P (k, z) = P LIN (k, z)W 2 (kR) + O (σ 4 0 ), where P LIN (k, z) is the linear power spectrum and σ 0 (up to the lowest order) is 0.15381 at the median red- shift z = 0.28668 of the SDSS sample. The additional coefficients A 0 and A 2 are −0.2226 and 1.4702, respectively. Table 3 lists the genus-related statistics for all the genus measurements relative to Figure 15.
Finally, in Figure 16 we present the genus-related statistics for the previous genus curves; this is helpful in order to understand systematic biases. The statistics for the random phase fluctuations are indicated by thick crosses. Triangles are statistics from the genus curve of the derived real-space dark matter particle distribution at the initial epoch of the simulation (G M,m,r |z pri and G o,m,r |z pri ), which includes only a primordial non-Gaussianity contribution for both the observation and simulation. The correction applied here has only the contribution of non-Gaussianity produced by nonlinear gravitational evolution, G j m,r |z i − G j m,r |z f in Table  2. The distribution of the matter density (circles) has a smaller overall amplitude of the genus curve, more voids, and fewer clusters, and is bubble shifted compared to that of the galaxy density (squares). Triangles are obtained from the real space genus curve of the dark matter particle distribution at the initial epoch of the simulation, which includes only a primordial non-Gaussian contribution. The difference between circles and triangles indicates the effect of nonlinear gravitational evolution. While the statistics of the initial matter density  TABLE 3 Genus-related statistics for the genus curves in Figure 16.  field of the ΛCDM N -body simulation with primordial Gaussianity (open triangles) are nearly the same as those of the random phase curve (as expected), the statistics of the initial matter density field obtained from the LRG sample (filled triangles) show still deviations from the Gaussian expectation; A V and ∆ν f are within about 1σ of the Gaussian values, and the amplitude and A C show a relatively large difference between the observation and the Gaussian prediction.

CONCLUSIONS
In this paper we presented measurements of the genus topology of LRGs from the SDSS DR7 catalog, with unprecedented statistical significance. We made a volumelimited sample in the redshift range 0.16 < z < 0.36 and and rest-frame g-band absolute magnitudes of −23.2 < M g < −21.1 using the DR7-Full sample of Kazin et al. (2010) and then imposed some additional cuts as in Choi et al. (2010), which leave a total of 60, 466 LRGs over about 2.33 sr. We constructed the galaxy mass and number density fields from the observed LRG sample, using a novel technique -called 'luminosity function matching' -outlined in Section 3, and computed the observed genus curve. We also produced 81 independent mock LRG samples from the HR3 (Kim et al. 2011), one of the largest N -body simulations currently available, that evolved 7210 3 particles in a 10815h −1 Mpc size box. The construction of simulated LRG catalogs required several subtle steps, explained in Section 4. In particular, we adopted the halo-galaxy one-to-one monotonic correspondence model (HGC) of Kim, Park, & Choi (2008) to populate dark matter halos with galaxies, and identified LRGs as the most massive subhalos.
Thanks to the unprecedented volume of the HR3, we were able to carefully model and study all the known systematics which affect the genus curve, such as finite pixel size, survey boundary mask, radial and angular selection function, past light cone gradient, initial conditions of the simulations, shot noise, cosmic variance, RSDs and galaxy biasing. Upon removal of all known systematics, our final genus curve (Section 6.3, Figure 14) contains only the nonlinearity produced by nonlinear gravitational evolution, and a possible primordial non-Gaussian component -if any. In particular, we find the observed genus amplitude to reach 285 with an uncertainty of 4.0% including cosmic variance (before the correction for the systematics): this is the most accurate constraint on the genus amplitude to date, which significantly improves on our previous measurement (Gott et al. 2009). Overall, the shape of the observed genus curve agrees very well with the mean topology of the SDSS LRG mock surveys in the ΛCDM universe, and this should be considered as a success of our large volume N -body simulation, as well as of our procedure to construct mock LRG samples from the HR3 (see Fig. 1).
However, comparison with simulations also shows small but significant deviations of the observed genus curve from the theoretical expectation for Gaussian initial conditions: Figures 15 and 16 show explicitly these deviations. We used genus-derived statistics (Section 2.3, Section 5.2 and Tables 1-3) to estimate and quantify departures from Gaussianity of the genus curve. While a consistent part of the non-Gaussian deviations is caused by systematics, and mainly driven by shot noise and biasing, removing their effects on the genus curve still leaves some discrepancies from the Gaussian expectations. This fact can be attributed to the nonlinearity produced by gravitational evolution, in addition to a possible primordial non-Gaussian component. We investigated here the role of nonlinear gravitational evolution on the genus curve, while in a forthcoming publication we will provide an interpretation of the remaining deviations in the context of primordial non-Gaussianity. In particular, in this study we found that the second-order perturbation theory prediction of Matsubara (2003) disagrees significantly with the genus curve measured from the gravitationally evolved matter density field in the HR3 simulation (solid line in Fig. 15). On the contrary, the nonlinear gravitational evolution effects are modeled well if we add extra terms in the second-order perturbation formula of Matsubara; to this end, we also provided a new secondorder perturbation formula (Eq. 18) which better fits our results.
In summary, the main achievements of this paper are as follows: i. We measured the genus amplitude from the SDSS LRG volume-limited sample, and found it to reach 285 with an uncertainty of 4.0% including cosmic variance; this is the most accurate constraint on the genus amplitude to date, which significantly improves on the results by Gott et al. (2009).
ii. The overall shape of the observed genus curve agrees very well with the mean topology of the SDSS LRG mock surveys in the ΛCDM universe, confirming the correctness of our large volume N -body simulation and procedure to construct mock LRG samples. This should also be seen as another strong support of the ΛCDM paradigm, similar to the one recently presented by Park et al. (2012), who was able to prove that observed high-and low-density LSSs have the richness/volume and size distributions consistent with the ΛCDM universe.
iii. Thanks to our unprecedented large volume simulation (HR3), we gained an excellent control of the numerous systematics which affect the genus curve, ranging from observational to statistical or cosmological effects which introduce non-Gaussianities in the genus shape. We were able to successfully model and remove all the known systematics.
iv. We proposed a new method to construct the galaxy mass and number density fields from the observed LRG sample (i.e. the 'luminosity function matching' technique), and an accurate procedure to construct LRG mock samples.
v. We proved that second-order perturbation theory (Hikage et al. 2002;Matsubara 2003) cannot model the genus curve measured from the gravitationally evolved matter density field in the HR3 simulation, and provided a new fitting formula which adds extra terms to the original second-order perturbation expression and matches better our results.
After removing all known systematics and modeling the nonlinear gravitational evolution more accurately, we are still left with some additional non-Gaussian signal. Clearly, since we were able for the first time to isolate and quantify a non-Gaussian contribution directly from the observed genus curve which is not due to systematics, and argued that even an improved second-order perturbation theory cannot explain all the non-Gaussian discrepancies, our next step is to interpret those deviations in the context of the local f NL -type model (Sa-lopek & Bond 1990;Gangui et al. 1994;Verde et al. 2000;Komatsu & Spergel 2001). This will allow us to constrain the standard non-Gaussianity parameter f NL directly from topological measurements of the LSS. Notes. 'Number' and 'Mass' stand for number density field and mass weighted density field, respectively.
THRESHOLD DENSITY VERSUS VOLUME FRACTION Figure 17 shows the SDSS and mock genus curves, plotted versus the direct density threshold ν. These curves deviate from the Gaussian predictions more than those obtained using the volume-fraction ν f shown in Figure 1. Table 4 from the previous Appendix A lists the genus values as a function of ν. By inspecting those values, we conclude that the direct density threshold parametrization is more sensitive to the skewness in the density probability distribution. However, given the vanishing genus and its large dispersion at threshold levels below ν ≃ −1.7 (see Figure 18), it is not appropriate to use this genus curve to inspect the non-Gaussian deviation of the observational sample from Notes. Go is the genus value of the SDSS LRG sample, and G M is the one averaged over all the 81 light cone LRG mock samples, G j h,z |LC . Go,m,r|z = 0 and G M,m,r |z = 0 are the genus values of the underlying dark matter distribution, derived from Go and G M , respectively, after correcting for the various systematics. Go,m,r|z pri and G M,m,r |z pri are the real space genus values of the dark matter distribution at the initial epoch of the simulation, which include only the contribution of primordial non-Gaussianity (both for the observation and simulation). Gm,r|z f is the genus value for matter distribution in real space at z = 0, using the full cubic data.
perturbation theory, at the smoothing scale adopted.