An Algorithm to locate the centers of Baryon Acoustic Oscillations
AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2020September 1, 2020
An Algorithm to locate the centers of Baryon Acoustic Oscillations
Z. Brown, G. Mishtaku, R. Demina, Y. Liu, and C. Popik
Department of Physics and Astronomy, University of Rochester,500 Joseph C. Wilson Boulevard, Rochester, NY 14627, USAe-mail: [email protected]
Received XX XX, XXXX; accepted YY YY, YYYY
ABSTRACT
Context.
The cosmic structure formed from Baryon Acoustic Oscillations (BAO) in the early universe is imprinted in the galaxydistribution observable in large scale surveys, and is used as a standard ruler in contemporary cosmology. BAO are typically detectedas a preferential length scale in two point statistics, which gives little information about the location of the BAO structures in realspace.
Aims.
The aim of the algorithm described in this paper is to find probable centers of BAO in the cosmic matter distribution.
Methods.
The algorithm convolves the three dimensional distribution of matter density with a spherical shell kernel of variable radiusplaced at di ff erent locations. The locations that correspond to the highest values of the convolution correspond to the probable centersof BAO. This method is realized in an open-source, computationally e ffi cient algorithm. Results.
We describe the algorithm and present the results of applying it to the SDSS DR9 CMASS survey and associated mockcatalogs.
Conclusions.
A detailed performance study demonstrates the algorithm’s ability to locate BAO centers, and in doing so presents anovel detection of the BAO scale in galaxy surveys.
Key words. cosmology: observations – large-scale structure of Universe – dark energy – dark matter
1. Introduction
Baryon acoustic oscillations (BAO) are density waves whichformed in the photon-baryon plasma in the primordial universe(Sunyaev & Zeldovich 1970; Peebles 1973; Eisenstein & Hu1998; Bassett & Hlozek 2009). They are the result of the compe-tition between the gravitational attraction pulling matter (mostlydark matter) into regions of high local density and radiation pres-sure pushing baryonic matter away from regions of high density.The resulting density waves, propagating at the sound speed ofthe plasma, produced ‘bubbles’ with high-density centers, rel-atively underdense interiors, and overdense spherical ’shells.’At the time of recombination when photons and matter fell outof thermal equilibrium, the bubbles ‘froze’ into the matter dis-tribution, with the centers remaining enriched in dark matterand shells enriched in baryonic matter (Eisenstein et al. 2007;Tansella 2018). After recombination the BAO centers and shellsbecame the seeds of galaxy formation.At late times the BAO can be observed as a preferred lengthscale in the distribution of galaxies using the two-point correla-tion function (2pcf) and its Fourier transform, the power spec-trum (Eisenstein et al. 2005; Percival et al. 2007), as well as thethree-point correlation function (3pcf) and its Fourier transform,the bispectrum (Slepian et al. 2017a,b). Since the BAO signal isa small perturbation in the large-scale clustering of galaxies, it isobserved on a statistical basis. The positions of individual BAOcenters and shells are not typically identified in large galaxy sur-veys. However, the identification of BAO centers would be ofsignificant cosmological and astrophysical interest. For exam-ple, the cross-correlations between BAO centers and other trac-ers of large-scale structure, such as galaxies, voids, and dark mat-ter tracers found via weak lensing, could probe di ff erent models of structure formation. Moreover, the identification of individ-ual BAO centers and shells can address questions not currentlyanswerable with the two and three-point statistics, such as theaverage number of galaxies in the shells.The construction of higher-order correlation functions iscomputationally expensive, so we propose a method called CenterFinder to locate the centers of BAO bubbles and num-ber of galaxies N displaced from the center by a distance R BAO .The
CenterFinder algorithm is inspired by a template track-finding algorithm originally suggested in Hough (1962) andgeneralized in Ballard (1981), which is typically used in par-ticle physics (see e.g., Demina et al. (2004)). In the follow-ing section, we describe the design and free parameters of the
CenterFinder algorithm. In Section 3 we study the perfor-mance of
CenterFinder using mock galaxy catalogs and theSDSS DR9 CMASS galaxy survey. We discuss the results inSection 4 and conclude in Section 5.
2. The CenterFinder Algorithm
CenterFinder locates BAO clustering centers by convolving3D spherical kernels of adjustable radii with tracers of large-scale structure. In this section we describe the inputs to the algo-rithm (Section 2.1), the estimator of the local density field (Sec-tion 2.2), the definition of the kernel (Section 2.3), the convolu-tion step (Section 2.4), and the output (Section 2.5). The source code may be downloaded from https://github.com/mishtak00/centerfinder
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CenterFinder takes as its input catalogs of various matter trac-ers from surveys or simulations. For each tracer, its right as-cension, α g , declination, δ g and redshift, z g are required. Tracerweights, w g may be used, but are not essential. Within the algo-rithm, celestial coordinates are converted to 3D Cartesian coor-dinates. The relationship between the redshift and the comovingradial distance, r g depends on the assumed cosmology: r g ( z g ) = cH (cid:90) z g dz (cid:48) (cid:112) Ω M ( z (cid:48) + + Ω k ( z (cid:48) + + Ω Λ , (1)where Ω M , Ω k , and Ω Λ are the present-day values of the rela-tive densities of dark matter, spatial curvature, and dark energy,respectively. H is the present day Hubble’s constant and c isthe speed of light. The integral in Eq. 1 is evaluated numericallyin CenterFinder . The Cartesian coordinates of each tracer arethen evaluated according to: X g = r g cos( δ g ) cos( α g ) , (2) Y g = r g cos( δ g ) sin( α g ) , (3) Z g = r g sin( δ g ) . (4)To prevent confusion with the redshift, z , Cartesian coordinatesare given as capital letters, X , Y , Z . Several methods of estimating the density field may be used inthis algorithm. The details are provided in the associated
README document. These methods rely on either raw or weighted galaxyhistograms, as well as the galaxy over-density with respect tothe survey mean density. Here we describe the option, which isrealized in this study. We start by defining a grid with spacing d c , such that the volume of each grid cell is d c . X i , Y j , Z k denotethe Cartesian coordinates of the i , j , k -th cell. On this grid wedefine 3-dimensional histograms N wtd ( X i , Y j , Z k ), which denotesa number count of tracers, and R ( X i , Y j , Z k ), which representsa number count of randomly distributed points within the samefiducial volume: R ( X i , Y j , Z k ) = R ( α i , δ j , z k ) dXdYdZdV sph , (5)where dXdYdZ and dV sph give the volume of a particular cellin Cartesian and spherical sky coordinates. The distribution R ( α i , δ j , z k ) is inferred from the input tracer catalog, and is gen-erated by assuming that the tracers’ angular, P ang ( α i , δ j ) andredshift, P z ( z k ) probability density distributions are factorizable(Demina et al. (2018)): R ( α i , δ j , z k ) = N tot [ P ang ( α i , δ j ) × P z ( z k )] , (6)where N tot is the number of tracers in the input catalog. Ifweights are available in the input catalogs, both N wtd and R couldbe weighted. A 3-dimensional local density di ff erence histogram M ( X i , Y j , Z k ) is then defined on the grid to represent the di ff er-ence in density between N wtd and R : M ( X i , Y j , Z k ) = N wtd ( X i , Y j , Z k ) − R ( X i , Y j , Z k ) . (7) Fig. 1.
A 2D representation of the tracer density histogram (left), beingconvolved with the spherical kernel (right). In this example the tensorproduct is 7.
To probe for BAO bubbles, we begin by creating a spherical ker-nel, K ( X i , Y j , Z k ), or a template to roughly model the distributionof matter expected around BAO centers, given a hypothesizedBAO scale, R . The kernel is constructed on a cube of the size of2 R in all directions, just large enough to encompass a sphere ofradius R . The grid defined on this cube has the same spacing, d c ,as the one used to construct M ( X i , Y j , Z k ). Grid cells intersectedby a sphere of radius R centered on the center of the cube, areassigned a value of 1. All other cells are 0. At this step we construct a 3-dimensional histogram C ( X i , Y j , Z k ), with entries that quantify the likelihood thata particular point in space ( X i , Y j , Z k ) hosts a BAO center. Thekernel K is first centered on some cell i , j , k and the value of C ( X i , Y j , Z k ) is calculated as the scalar product of the kernel andthe density map M . The kernel is then moved to a new cell,and the process is repeated until the whole surveyed volume iscovered. Thus, C ( X i , Y j , Z k ) is a result of the convolution of thematter density map M with the kernel, K : C ( X , Y , Z ) ≡ M ( X , Y , Z ) ∗ ∗ ∗ K ( X , Y , Z ) , (8)where ∗ ∗ ∗ denotes a 3-dimensional discrete convolution. Onestep in this process is visualized in Fig. 1.If the original density histogram were the raw count of trac-ers, the value of C would have a simple meaning: it gives thenumber of tracers “voting” for a particular cell to host a center.A larger number of “voters” indicates a higher likelihood of aBAO center. While C does not directly correspond to “voters”when using the density estimate in eq. 7, large values of C stillcorrespond to increased likelihood of hosting a BAO center at aparticular location.This highly vectorized convolutional algorithm, called CenterFinder , is quite e ffi cient; the runtime is set by the num-ber of cells in the surveyed volume, which is in turn set by theone dimensional grid length. Shown in Fig. 2, it decreases withthe grid length until it saturates at around 5 s. Prior to the sat-uration, the runtime scales as a power law, approximately d − c In our performance study (§ 3), a grid length of d c = . h − Mpc yields a runtime of approximately 240 s. This study applies
CenterFinder to the northern galactic cap (NGC) catalog ofthe SDSS DR9 CMASS survey using a personal computer witha 2.3 GHz Intel Core i7 processor and 8 GB of memory.
Article number, page 2 of 8. Brown et al.: An Algorithm to locate the centers of Baryon Acoustic Oscillations
Fig. 2.
The runtime of
CenterFinder , given as a function of the gridlength, for one of the CMASS mocks used in this study (blue squares).The red star indicates the value of d c chosen in this analysis. The timingdata is fit to a log power law of the form A ( d c / d scale ) α + β log( d c / d scale ) + C (grey dashed line). In the above fit, the value of the parameter α isapproximately − Once the 3-dimensional histogram C is generated, it is con-verted into a catalog of probable BAO centers by applying auser-specified threshold, C min , on C : C ( X i , Y j , Z k ) ≥ C min . (9)The output catalog in astropy FITS format provides the loca-tions (right ascensions, declinations and redshifts) of the proba-ble centers, and weights corresponding to the values of C . Theredshifts are calculated using the inverse of the relation in Eq. 1.Redshifts, rather than radial distances, are included since mostalgorithms for probing cosmological structure are designed tobe applied to redshift catalogs.
3. Performance Study
The performance of
CenterFinder , i.e. its ability to locateBAO centers, is tested on the SDSS DR9 CMASS galaxy survey(Ahn et al. 2012; Padmanabhan et al. 2012) and on an ensembleof associated mock and random catalogs. We limit our analysisto galaxies in the survey’s north galactic cap. Details regardingthe mock catalogs may be found in Manera et al. (2013). Themock ensemble consists of 20 catalogs. The random ensemblealso consists of 20 catalogs of density equal to that of the mocks,appropriately sampled from the several large random catalogs.
In this study we used the following values for cosmological pa-rameters: c = / s, H = h km / s / Mpc, Ω M = . Ω Λ = . Ω k =
0. Before analyzing the catalogswith
CenterFinder , we investigate the clustering behavior ofthe DR9 galaxies and mocks with the two point correlation func-tion (2pcf), ξ ( s ). We compute ξ ( s ) using the estimator of Landyand Szalay (Landy & Szalay 1993; Hamilton 1993), Fig. 3.
The 2pcf, ˆ ξ ( s ), of SDSS DR9 galaxies (grey circles) and 20mock catalogs (blue lines) described in § 3.1, as a function of the co-moving separation, s . The average of the mock ensemble is given bythe solid navy line. Kernel sizes (hypothesized BAO scales) used by CenterFinder in this analysis are shown by vertical grey dashed lines. ˆ ξ ( s ) = DD ( s ) − DR ( s ) + RR ( s ) RR ( s ) , (10)where DD , RR , and DR are the normalized distributions of thepairwise combinations of galaxies from “data”, D and “random”, R catalogs (plus cross terms) at given distances s from one an-other. The 2pcf calculations are done using the algorithm de-scribed in Demina et al. (2018).The 2pcf calculated using bin width ∆ s = . h − Mpc isshown for data and mocks in Fig. 3. In both, there is evidenceof clustering at small scales and a prominent BAO signal around109 h − Mpc, consistent with previous analyses on the same datasets (Anderson et al. 2012; Ross et al. 2012). We note that bothat small scales and near the BAO scale, the magnitude of ˆ ξ is no-ticeably smaller in the average over mocks compared to the DR9survey data. Additionally, the variance between mocks in thisensemble is considerable at separations approaching the BAOscale. To identify a BAO signal using
CenterFinder , we apply it toSDSS DR9 CMASS data, mock and random catalogs. Randomcatalogs match both the fiducial volume and the galaxy densityof the survey. The cosmology used is identical to the one used inthe calculation of the 2pcf.We generate a density map M using the method described in§ 2.2, with a grid spacing, d c = . h − Mpc. The radius of thekernel, R , is varied from 80 . − h − Mpc in 4 . R , the kernel is convolved with the densitymap M to generate a map of weights, C ( R ), where larger valuesindicate a higher probability of BAO centers being present at thatlocation in space. The distribution of the values of C ( R ) from arandom catalog is shown in Fig. 4 for di ff erent kernel sizes.These distributions have three prominent features. First, thevast majority of cells in the grid are empty or nearly empty, andtheir convolution with the kernel yields small values of C , asshown by the sharp spike at low weights. It is followed by grad- Article number, page 3 of 8 & A proofs: manuscript no. main
Fig. 4.
The distribution of center weights, for kernels of di ff erent sizes,applied to a random catalog with the same fiducial volume and densityas the SDSS DR9 CMASS NGC survey. The dark blue curve corre-sponds to R = . h − Mpc and yellow corresponds to R = h − Mpc. Intermediate colors correspond to the intermediate kernel sizesused in this analysis. The red stars indicate the threshold values, C min used to create catalogs of probable centers at each kernel size. Fig. 5.
The threshold values, C min , applied to our center catalogs, as afunction of the kernel size (blue squares). The points are fit to a secondorder polynomial (dashed grey line). ual falling o ff , which extends to higher values for larger kernels.Finally, there is a sharp drop in counts as the weights increase.To reduce the size of the output catalog we keep only thecells with values larger than a certain threshold, C min , which ischosen to keep the count of centers approximately constant fordi ff erent values of R . Since the number of counts on a spher-ical shell is proportional to the radius squared, we expect C min to be approximately proportional to R . We choose values of C min (shown by red stars in Fig. 4), which corresponds to thebeginning of a sharp drop o ff . We apply a threshold selection at C min =
150 for the smallest kernel size and at C min =
300 for thelargest. The dependence of the chosen thresholds on the kernelsize shown in Fig. 5. As expected, this dependence is well fit bythe second order polynomial.Finally, we generate FITS catalogs of the probable centersbased on 20 mock, 20 random, and DR9 data catalogs for severalvalues of the kernel size.
Fig. 6.
The angular distribution of the probable centers for the redshiftrange 0 . < z < .
52 (color scale), generated from the R = . h − Mpc kernel on a portion of the SDSS DR9 data. Galaxies locations areshown as red stars. The dashed circles reflect the angles subtended bythe BAO spheres at z = . Using a sample of galaxies from the SDSS DR9 survey as an ex-ample, we illustrate the output of
CenterFinder . In Fig. 6 weshow a slice in the red-shift region from z = .
51 to z = . R = . h − Mpc. The color of the background rep-resents the weights C , which characterize the probability that aparticular point is a location of a BAO center with more intenseyellow being the most probable. Red points show the locationof the galaxies from the SDSS DR9 catalog from the same fidu-cial volume. Dashed lines are circles of 109 . h − Mpc radiusaround the probable BAO centers. Even in this 2-dimensionalslice, we observe significant overlap between the circles denot-ing the BAO shells, and the galaxies from the survey. Since wechose a fairly low value for the threshold selection of BAO cen-ters there are multiple circles around the same location with highprobability to host the BAO center. It is also noticeable that theselocations typically host several galaxies, supporting the hypoth-esis that dark matter enriched BAO centers are likely to be seedsfor galaxy formation.
Since potential centers correspond to primordial over-dense re-gions, we expect the galaxies to have been preferentially formedin these regions of space. We test this hypothesis by cross corre-lating the potential BAO centers, D C , with galaxies, D G drawnfrom data or mock catalogs. For each catalog of centers, gener-ated using a kernel of size R , we calculate a cross correlationfunction, ˆ ξ GC ( s , R ):ˆ ξ GC ( s , R ) = D G D C − D G R C − D C R G + R G R C R G R C , (11)where R C and R G are random catalogs corresponding to the cen-ters and galaxy catalogs respectively. All pairwise distributions Article number, page 4 of 8. Brown et al.: An Algorithm to locate the centers of Baryon Acoustic Oscillations
Fig. 7.
The cross correlation of probable center locations with SDSSDR9 galaxies as a function of the objects’ separation, s , and the hy-pothesized BAO scale of CenterFinder used to generate the centercatalog, R . The cross correlation is adjusted for clarity (divided by s ).The dashed line (red) marks an estimate of the BAO scale observed inthe 2pcf. The dot-dash line (red) marks s = R . Fig. 8.
Same as in Fig. 7 for mock galaxies. are functions of the distance between a center and a galaxy, s andthe hypothesized BAO scale, R . R G corresponds to the random galaxies used in the evaluationof the 2pcf (§ 3.2). R C is a catalog of N CR randomly distributedcenters. The ratio of N CR to the number of found centers is cho-sen to be similar to the ratio between the number of galaxies indata D and the size of the random catalog R . R C is generatedby applying CenterFinder to several random galaxy catalogs.The angular and redshift distributions of R C match those of theprobable centers, D C as detailed in in App. A.ˆ ξ GC ( s , R ) is evaluated using nbodykit , which is describedin Hand et al. (2018). Figs. 7, & 8 and 9 show the result ofthe center-galaxy cross correlation in data, mock and randomcatalogs respectively. The distributions are divided into s binsof width ∆ s = . h − Mpc and R bins reflecting the samplingwhen applying CenterFinder . Fig. 9.
Same as in Fig. 7 for random galaxies.
Two clustering features are present in Figs. 7 & 8. First, a re-gion of higher values of ˆ ξ GC which follows s = R is observed inboth plots, with roughly equal magnitude across mock and SDSSDR9 data. This feature is an artifact of the CenterFinder . Thealgorithm is designed to find centers of densely populated spher-ical shells of a given radius. The observed excess is simply thecorrelation between the galaxies in these shells with the corre-sponding found centers. This feature confirms the most basicfunction of the algorithm, but is not physically informative.The second feature in both mock and DR9 survey data isthe observed increase in ˆ ξ GC at low s , which is enhanced whenthe hypothized BAO radius is near the true BAO scale (horizon-tal dashed line) (Anderson et al. 2012; Ross et al. 2012), anddiminished at larger and smaller radii. This behavior confirmsour original hypothesis that galaxies cluster around the potentialBAO centers. The first (diagonal) feature corresponding to thekernel radius is observed in random catalogs (Fig. 9), but thereis no excess at the small scales. This qualitatively confirms thatthe signal at low s seen in the mocks and in the data is due tolarge scale structures and is not a random fluctuation.We note that the BAO signal (increase in ˆ ξ GC at low s ) islarger in magnitude for SDSS DR9 data (Fig 7) compared tothe average over the mock catalogs (Fig 8). This is consistentwith the fact that the mock catalogs showed on average lessoverall galaxy clustering in the 2pcf than the SDSS DR9 data(Fig.3). This is also observed in the distribution of probable cen-ter weights, C , shown in Fig. 10. The distribution in Fig. 10 areshown prior to applying a threshold, C min .The SDSS DR9 galaxies show higher counts of probablecenters with larger weights, or values of C . This suggests ahigher degree of galaxy density and clustering on the BAOshells. This observation is consistent with the increased clus-tering behavior seen in the 2pcf. Furthermore, the distributionsof probable center weights in Fig. 10 show that the SDSS DR9galaxies and CMASS mock significantly deviate from the dis-tribution extracted from randoms at large weights. This indi-cates the presence of BAO shells imprinted in the mock and datagalaxy distributions. Article number, page 5 of 8 & A proofs: manuscript no. main
Fig. 10.
The distribution of center weights, for a kernel with a radiusreflecting the approximate BAO scale, extracted from the 2pcf, R = . h − for SDSS DR9 galaxies (black solid line), one of the CMASSmocks (blue dashed line), and one of the CMASS randoms (red dot dashline). Fig. 11.
Signal strength as a function of kernel size. Each mock isshown individually (blue), as is DR9 data (black), while randoms (red)are represented by a mean of all values at each kernel size, with a shadedregion given by the standard deviations.
To quantify the strength of the presumed BAO signal we sum thevalues of ˆ ξ GC in the first 10 s -bins (from s = .
72 to s = . h − Mpc) for every R value. The results are shown in Fig. 11for the SDSS DR9 data, mock, and random catalogs. The distri-bution over signal strengths for mocks and randoms, as well asthe value for SDSS DR9 data at R = . h − Mpc is shown inFig. 12. There is a large spread in signal strength for the mocksat every kernel size which is in agreement with the behavior ob-served in the two-point statistics (Fig. 3). The signal strength de-rived from random catalogs on the other hand is centered around0 and shows a much smaller spread. At the kernel size corre-sponding to the BAO scale, R = . h − Mpc, we find thatthe SDSS DR9 data di ff ers from the mean of the CMASS ran-doms by 5 . Fig. 12.
A normalized histogram of the signal strengths at R = . h − Mpc for the CMASS Randoms (red) and CMASS Mocks(blue). The SDSS DR9 data (black) is shown as a vertical line at 0 . . − .
4. Discussion
An approach similar to this work was suggested in Arnalte-Muret al. (2012). In this study the kernel shape is more complex,emulating a spherical wavelet template. Our algorithm also con-tains several di ff erent optional kernel shapes, including a simi-larly shaped wavelet.Another di ff erence is that CenterFinder scans over the en-tire surveyed volume identifying BAO centers that may or maynot be associated with other galaxies, while the algorithm de-scribed in Arnalte-Mur et al. (2012) uses Luminous Red Galax-ies as seeds to search for spherical shells. Hence the output of
CenterFinder can be cross correlated with other matter tracers,such as Lyman- α forest, and weak lensing dark matter maps.
5. Conclusions
In this paper we present the algorithm
CenterFinder designedto locate centers of spherical shells generated by Baryon Acous-tic Oscillations. So far the BAO signature was observed as astatistical feature in the CMB power spectrum, and in the twopoint correlation function of galaxy distributions. This algorithmis computationally e ffi cient and can be applied to a variety oftracer catalogs. A performance study using SDSS DR9 surveyand mock catalogs yielded a novel method to detect the BAOscale, and to generate catalogs of probable BAO center locationsto study in future analyses. Using these catalogs of centers, crosscorrelations between them and other tracers of the cosmic websuch as clusters, voids, Lyman- α forest, and weak lensing mapsmay be studied. Acknowledgements.
The authors would like to thank S. BenZvi, K. Douglassand S. Gontcho A Gontcho for useful discussions and insightful questions. RD.thanks D. Bianchi, L. Samushia and Z. Slepian for their interest and help-ful comments. The authors acknowledge support from the Department of En-ergy under the grant DE-SC0008475.0. Funding for SDSS-III has been pro-vided by the Alfred P. Sloan Foundation, the Participating Institutions, the Na-tional Science Foundation, and the U.S. Department of Energy O ffi ce of Sci-ence. The SDSS-III web site is . SDSS-III is man-aged by the Astrophysical Research Consortium for the Participating Institutionsof the SDSS-III Collaboration including the University of Arizona, the Brazil-ian Participation Group, Brookhaven National Laboratory, Carnegie Mellon Uni-versity, University of Florida, the French Participation Group, the German Par-ticipation Group, Harvard University, the Instituto de Astrofisica de Canarias,the Michigan State / Notre Dame / JINA Participation Group, Johns Hopkins Uni-versity, Lawrence Berkeley National Laboratory, Max Planck Institute for As-
Article number, page 6 of 8. Brown et al.: An Algorithm to locate the centers of Baryon Acoustic Oscillations trophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico StateUniversity, New York University, Ohio State University, Pennsylvania State Uni-versity, University of Portsmouth, Princeton University, the Spanish ParticipationGroup, University of Tokyo, University of Utah, Vanderbilt University, Univer-sity of Virginia, University of Washington, and Yale University.
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Fig. A.1.
For the SDSS DR9 CMASS galaxies (blue) and random galax-ies (orange), we plot the distribution over the right ascension (top), dec-lination (middle), and redshift (bottom).
Appendix A: Galaxy, Random, & CenterDistributions
In § 3.5, we cross correlate objects from two catalogs, galaxiesand probable centers. In addition, they are compared to randomdistributions, R G and R G . In Figs. A.1 & A.2, we show the dis-tributions of these tracers over their coordinates.The probable centers are clearly biased towards regions ofthe survey with high galaxy density. While the method outlinedin § 2.2 corrects for some of this, Figs. A.1 & A.2 demonstratethat the e ff ect is not entirely removed. The di ff erence in the dis-tribution over the redshift especially justifies use of an additionalrandom catalog, R C , in the evaluation of the galaxy to probablecenter cross correlation. Fig. A.2.
For the probable centers at R = . h −1