The orientation of disk galaxies around large cosmic voids
Jesús Varela, Juan Betancort-Rijo, Ignacio Trujillo, Elena Ricciardelli
TThe orientation of disk galaxies around large cosmic voids.
Jes´us VarelaJuan Betancort-RijoIgnacio Trujillo and
Elena Ricciardelli
Instituto de Astrof´ısica de Canarias (IAC), E-38200 La Laguna, Tenerife, SpainDepto. Astrof´ısica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain
November 11, 2018
ABSTRACT
Using a large sample of galaxies from the SDSS-DR7, we have analysed the alignment of disk galax-ies around cosmic voids. We have constructed a complete sample of cosmic voids (devoid of galaxiesbrighter than M r − h = − .
17) with radii larger than 10 h − Mpc up to redshift 0.12. Disk galaxiesin shells around these voids have been used to look for particular alignments between the angular momen-tum of the galaxies and the radial direction of the voids. We find that disk galaxies around voids largerthan (cid:38) h − Mpc within distances not much larger than 5 h − Mpc from the surface of the voids presenta significant tendency to have their angular momenta aligned with the void’s radial direction with a sig-nificance (cid:38) .
8% against the null hypothesis. The strenght of this alignment is dependent on the void’sradius and for voids with (cid:46) h − Mpc the distribution of the orientation of the galaxies is compatiblewith a random distribution. Finally, we find that this trend observed in the alignment of galaxies is similarto the one observed for the minor axis of dark matter halos around cosmic voids found in cosmologicalsimulations, suggesting a possible link in the evolution of both components.
Subject headings:
Large Scale Structure: Voids; Galaxies : General
1. Introduction
The study of the alignment of galaxies with respectto the large scale structure is a recurrent topic still notfully settled. The first works studying the alignment ofgalaxies focused on clusters and superclusters. Stud-ies on the alignment of galaxies in clusters (Adamset al. 1980) and superclusters (Flin & Godlowski 1986;Kashikawa & Okamura 1992) claimed to find partic-ular alignments of the galaxies with respect to theirlocal large scale structure. On the other side, similarstudies did not find any particular alignment (Helou &Salpeter 1982; Dekel 1985; Garrido et al. 1993). More recently, Navarro et al. (2004) revisited the analysisdone by Flin & Godlowski (1986) on the alignmentof galaxies in the Local Supercluster (LSC) under thelight of the Tidal Torque Theory (for a recent reviewabout the Tidal Torque Theory or TTT, see Sch¨afer2009). The authors found a tendency of galaxies tohave their spin parallel to the plane of the LSC, alsoknown as supergalactic plane, that would support thepredictions from the TTT.However, the observational analysis is hindered bytwo main di ffi culties: the accurate determination of thedirection of the angular momentum of the galaxies andthe determination of the distribution of matter around1 a r X i v : . [ a s t r o - ph . C O ] S e p hem. The determination of the spin of disk galaxiescan be guessed by the shape of the galaxy, consideringthat galaxies spin around their minor axis. However,there is still an indetermination due to projection ef-fects since in most of the cases it is not possible toknow which half, of the two in which a galaxy is di-vided by its major axis, is closer to the observer. Thepresence of dust lanes or the use of kinematic data canhelp to solve this degeneracy but in most of the casesthis information is not available. To deal with thisproblem some authors have taken all the possibilitiesof the spin as independent ones (Kashikawa & Oka-mura 1992) while others have opted for taking just onepossibility (Lee & Erdogdu 2007).Regarding the accurate determination of the massdistribution around the galaxies, the main problemcomes from the e ff ects of the proper motion of galaxieswhich introduces uncertainties in the conversion fromredshift to distances.To overcome both problems, Trujillo et al. (2006,hereafter T06) proposed the use of spiral galaxies seenedge-on or face-on (so the direction of the spin vec-tor is better determined) located in the shells aroundcosmic voids. The advantage of the regions aroundlarge cosmic voids is that the direction of the gradientof density is strongly aligned with the radial directionwhich can be determined in a robust way despite theuncertainties of converting redshifts in distances. Us-ing this technique and data from the third data releaseof the Sloan Digital Sky Survey (SDSS-DR3) and the2dF Galaxy Redshift Survey (2dFGRS), T06 found atendency of galaxies around shells of voids to havetheir spin vector perpendicular to the radial direction.Cuesta et al. (2008) working on cosmological simula-tions of dark matter halos around voids found resultsin apparent agreement with those of T06. The simu-lations show that the angular momentum of the darkmatter halos tend to be also aligned to the perpendicu-lar direction. In both cases, the results were in agree-ment with the prediction done using the TTT (Lee &Pen 2000), that the angular momentum would tend tobe aligned with the intermediate axis of the tidal sheartensor, that in the surface of the voids is in the perpen-dicular direction.However, recently, Slosar & White (2009, hereafterS09) have redone a similar analysis, but using a largersample of galaxies from the SDSS-DR6, obtaining aresult that is consistent with a random distribution oforientations, in contrast with the previous results.In this work, we revisit the analysis of the alignment of galaxies around voids with two significant improve-ments with respect to those two previous works. First,we make use of the latest data release of the SDSS, i.e.SDSS-DR7, and we combine it with the morphologi-cal classification from the Galaxy Zoo project (Banerjiet al. 2010; Lintott et al. 2010) to select disk galaxies.Second, we have developed a statistical procedure topartially correct the indetermination in the spin direc-tion due to the projection e ff ect so we can obtain infor-mation also from galaxies that are not edge-on or face-on, increasing by a factor of 3 the e ff ective number ofgalaxies that are used in our analysis with respect tothe restriction to edge-on and face-on galaxies.The outline of this Paper is as follows. Section 2presents the data used for our analysis; Section 3 de-scribes the procedure to search for voids; Section 4 isdevoted to the selection of galaxies and the computa-tion of their alignments; Section 5 contains the finalresults; in Section 6 the results are discussed and com-pared with previous works and in Section 7 the sum-mary of the resuls are presented.Through this paper we assume a Λ CDM cos-mological model with Ω M = . Ω Λ = . H = h km s − Mpc − .
2. The data
On what follows we describe the data that we haveused to: a) create a sample of cosmic voids, and b)obtain a sample of galaxies in the shells surroundingthem to explore the orientation of galaxies in the largescale structure.Our main source of data has been the New YorkUniversity Value-Added Galaxy Catalog (NYU-VACG,Blanton et al. 2005; Padmanabhan et al. 2008; Adelman-McCarthy et al. 2008), which is based on the photo-metric and spectroscopic catalog of the SDSS-DR7 (Strausset al. 2002). The main characteristics of the NYU-VACG are: • Spectroscopically complete up to r ∼ . – Completeness ∼ – Success rate ∼ . • µ ( r − band ) ≤ . − http: // sdss.physics.nyu.edu / vagc / http: // cas.sdss.org / astrodr7 / en / ∼
90 targets / deg • Median(z) = We have established thresholds in absolute magni-tude, M limr , and redshift, z lim , in order to maximizethe amount of galaxies while keeping the final sam-ple complete in spectroscopy. The completeness of theinitial catalog is a basic requirement to avoid the de-tection of spurious voids.In Figure 1 it is plotted the number of galaxies with M r ≤ M limr ( z ) as a function of the redshift. The value of M limr ( z ) corresponds to completeness limit ( r = . z . From the peak of this distributionwe obtain the limits of our final sample: • z ≤ . • M r − h ≤ − . Although we focus our analysis in the largest con-tinuous volume of SDSS-DR7, the irregular limits ofthe volume still posed di ffi culties in the reliable detec-tion of voids. For this reason, we have defined new reg-ular limits minimizing the detection of spurious voidswhile still keeping ∼
90% of the original volume.Figure 2 shows the limits of our trimmed volume,projected onto the original distribution of galaxies, de-fined as follows: • δ > ◦ [Southern limit] • δ < − . α − ◦ ) [Western limit] • δ < . α − ◦ ) [Eastern limit] • δ < arcsin (cid:34) . α − . ◦ ) √ − [0 . α − . ◦ )] (cid:35) [Northernlimit]Table 1 summarizes the main properties of the ref-erence catalog that we have used in our search for cos-mic voids. We opted for no applying a k-correction due to the small redshiftrange probe and the high uncertainties in its determination.
Another important requirement of the galaxy cata-log to be suitable to search for voids is the homogein-ity. One common test of homogeinity is the (cid:104) V / V max (cid:105) test, for which a value of 0.5 is expected for an ho-mogenous distribution.To perform this test, first of all, for each galaxy it iscomputed the volume, V , of the sphere with radius thedistance along the line-of-sight to it. Then, the max-imum of all the volumes, V max , is found and the ratio V / V max is obtained for each galaxy. The final step is tocalculate the average value of these ratios, (cid:104) V / V max (cid:105) .For our galaxy catalog (cid:104) V / V max (cid:105) = .
3. Catalog of voids
With the catalog of galaxies described in the previ-ous section, we proceed to search for cosmic voids onit. In this section we describe the procedure followedto construct our catalog of voids.
First of all, we need to establish the definition of“void” that we use in our analysis. We have optedfor the simplest one: a spherical volume devoid ofany galaxy brighter than our completeness limit. Thisdefinition has been already used in other works suchas Patiri et al. (2006a), Patiri et al. (2006b), Trujilloet al. (2006), Brunino et al. (2007), and Cuesta et al.(2008). Cuesta et al. (2008) found that for dark matterhaloes in cosmological simulations, using ellipsoidalvoids instead of spherical ones does not a ff ect signifi-cantly their results. This gives us confidence in the useof spherical voids for our analysis.Apart from minor di ff erences, the procedure thatwe have followed is basically the HB V oid F inder de-scribed in Patiri et al. (2006a). These are the basicsteps:1. Random points are thrown within the volume ofthe catalog.2. For each trial point, the 4 closest galaxies arefound and the center and radius of the spheredefined by these 4 galaxies are computed andstored.3. Of the resulting spheres, those fulfilling any ofthe following criteria are rejected:3eference catalog NYU-VAGC (Galaxies)Spectroscopic completeness limit r ≤ . . < z < . M r − h ≤ − . . × π Total Volume 0.0276556 ( h − Gpc) Average density of galaxies 0.00514 ( h − Mpc) − Table 1: Summary of properties of the reference catalog. .
00 0 .
05 0 .
10 0 .
15 0 .
20 0 .
25 0 .
30 0 .
35 0 . z N ( < z ) [ M r ≤ M l i m r ( z ) ] − . − . − . − . − . − . − . M limr Fig. 1.— Number of galaxies brighter than M limr with z < z lim . M limr is the absolute magnitude correspondingto the spectroscopic limit ( r = .
8) at redshift z lim .The peak of the distribution it is used to establish thethresholds in M r and z of our initial catalog. Fig. 2.— Partial projection of galaxies from the SDSS-NYU catalog. The limits used in this work are over-plotted. • Not being empty. • Intersecting the border of the volume. • Having a radius smaller than 10 h − Mpc.Finally, to have well defined spherical voids, we im-pose that they can not overlap. In the case in whichseveral voids overlap, only the largest one is kept. Theorder in which the rejection of the voids is done a ff ectstheir final sample. Therefore, to ensure that our finalsample contains the largest possible voids, the processof rejection of overlapping voids it is done from thelargest void to the smallest one. Patiri et al. (2006a) used a di ff erent technique. They put artificialgalaxies in the limits of the survey, allowing voids to be defined bythree real galaxies and one fake one.
4n summary, the voids selected for our final catalogsfulfill the following conditions:1. They are empty of galaxies from the initial cata-log, i.e. M r < − . + h .2. Their radius is larger than 10 h − Mpc.3. They are completely inside the surveyed vol-ume.
4. Voids do not overlap, i.e. the distance betweenthe centers of two voids is larger than the sum oftheir radii.The power of this procedure to produce a completecatalog of voids depends critically on the relation be-tween the density of galaxies, the size of the voidsand the number of trial points used in the search. Wehave performed several tests and have found that using ∼ initial random points (corresponding to a den-sity of ∼
35 ( h − Mpc) − trial points) ensures that thecompleteness of our catalog of voids is > h − Mpc. We found a median radiusof 11 . h − Mpc and the average density of voids is32 . × − ( h − Mpc) − . Table 2 provides the main information of thesevoids. For each void, we include the position ofthe center, both in Cartesian coordinates ( X , Y , Z ), andequatorial coordinates ( α, δ ) and redshift z ; and the ra-dius ( R ). The Cartesian coordinates are computed asfollows: X = D ( z ) cos δ cos α Y = D ( z ) cos δ sin α Z = D ( z ) sin δ where D ( z ) is the comoving radial distance.
4. Catalog of galaxies in shells around voids
Using the previous catalog of voids, we extractthose galaxies within shells of 10 h − Mpc around eachof them. The morphology of the galaxies has been Be aware that this criterion implies that the e ff ective volume inwhich the center of voids can reside is smaller than the whole vol-ume and depends on the sizes of the voids. Density computed within the e ff ective volume in which the centersof the voids can be located. obtained from the Galaxy Zoo catalog (Banerji et al.2010; Lintott et al. 2010), which provides robust dis-tinction between elliptical and disk galaxies, although ∼
52% of the galaxies remain classified as “uncertain”.We select only those galaxies classified as “spiral” inthe Galaxy Zoo catalog ( ∼ The use of measurements ofgalaxies falling in more than one shell is justified be-cause the small uncertainties introduced are compen-sated by the increase in the size of the sample.In the following sections we describe in detail howthis alignment has been computed.
The advantage of studying galaxies around voids isthat, on average, the density increases radially. Thismakes the radial direction a good proxy for the distri-bution of matter around each galaxy. Therefore, weuse the minor axis of the galaxies to define the orien-tation of their angular momentum, and the radial di-rection of the voids to characterize the distribution ofmatter around them. Hence, our analysis is focused onthe angle between these two directions, θ . For prac-tical reasons, in the context of this work we will usethe expression “radial direction” to mean the directiondefined by a galaxy and the center of the correspond-ing void, and “perpendicular direction” to mean anydirection perpendicular to the radial direction.To compute the angle θ we need to define the di-rection of the angular momentum or spin, s , of eachgalaxy. We do this, first of all, by computing the incli-nation angle between the plane of the galaxy disk andthe line of sight, ζ . Following Haynes & Giovanelli(1984) and Lee & Erdogdu (2007) , we use a model ofthick disk with a projected minor-to-major axis ratio a / b and an intrinsic flatness f (i.e. the ratio betweenthe real minor axis and the real major axis). According These data is avalible upon request to the authors. A schematic illustration of this method is shown in Figure 1 of T06. Lee & Erdogdu (2007) used the angle i defined as the angle betweenthe plane of the galaxy and the projected plane of the sky, therefore ζ = π/ − i . Y Z R α δ z [ h − Mpc] [ h − Mpc] [ h − Mpc] [ h − Mpc] [deg] [deg]-193.463 -145.544 193.384 18.703 216.954492 38.617468 0.1059-227.081 26.372 140.238 18.205 173.375602 31.526741 0.0914-226.366 17.375 175.425 18.104 175.610699 37.692842 0.0979-86.105 93.444 136.020 17.606 132.659407 46.949188 0.0630Table 2: Catalog of voids (excerpt). Complete version in electronic form. See text for more details.to this model, ζ can be obtained applying the formula:sin ζ = ( b / a ) − f − f (1)For values of b / a < f , the angle is set to 0.The flatness f depends on the morphological typeof the galaxies and we use an average value of 0.14.From Equation (1) it is easy to see that to a singlevalue of b / a corresponds two values of ζ : ζ + = | ζ | and ζ − = −| ζ | . The indetermination is irrelevant for ζ = ζ = ± π/ s we followed the pre-scription by T06. According to this, if ( α , δ ) are theequatorial coordinates of a galaxy, ζ the inclination an-gle obtained from Equation (1) and φ is the positionangle of the galaxy increasing counterclockwise (i.e.from north to east in the plane of the sky), the compo-nents of s are: s x = cos α cos δ sin ζ + cos ζ (sin φ cos α sin δ − cos φ sin α ) (2) s y = sin α cos δ sin ζ + cos ζ (sin φ sin α sin δ + cos φ cos α ) (3) s z = sin δ sin ζ − cos ζ sin φ cos δ (4)Next, we compute the angle between the radial vec-tor that connects the center of the void, r void , with thecenter of the galaxy, r galaxy : r = r galaxy − r void . (5)Having obtained r and s , the angle between them, θ ,it is computed as: θ = arccos (cid:32) s · r | s || r | (cid:33) (6) θ We compute the angle θ for all the galaxies in oursample of galaxies around voids, obtaining a distribu-tion of θ . Betancort-Rijo & Trujillo (2009) provide ananalytical model for the distribution of the angle θ , orto be more precise, of | cos θ | , P ( | cos θ | ). From theoret-ical principles confirmed by simulations, the authorsfound that P ( | cos θ | ) is well described by the expres-sion: P ( µ ) = p du [1 + ( p − µ ] / ; µ ≡ | cos θ | , (7)where p is a free parameter that describes the overallshape of the probability distribution. An interestingproperty of this distribution is the relation between theparameter p and the average value of | cos θ | : p = (cid:104)| cos θ |(cid:105) − p are related with the exis-tence or absence of particular alignment between thevectors r and s according to the following criteria: p < . r and s tend to be parallel. p = . There is no particular alignment between r and s . p > . r and s tend to be perpendicular.It has been found that Equation (7) describes wellthe results from cosmological simulations (Bruninoet al. 2007; Cuesta et al. 2008).An alternative expression for the same distributiondescribed by Equation (7) it is provided by Lee (2004)and used in several works (eg., T06, Lee & Erdogdu2007, S09). This alternative expression is character-ized by a parameter c and, from the comparison be-tween Equation (9) of S09 and Equation (7) of the6resent work, it is possible to obtain the following ex-pression relating both characteristic parameters: p = (cid:115) + c − c ) (9) P ( | cos θ | )Di ff erent approaches have been used to deal withthe indetermination of the values of ζ . For example,Kashikawa & Okamura (1992) uses the two values of ζ independently. Another possibility is to use just onesign in the definition of ζ as done by Lee & Erdogdu(2007). However, these authors acknowledge that thisdecreases the strength of the measured alignment.On the other hand, T06 and S09 have overcome theproblem with the indetermination of the values of ζ us-ing only edge-on and face-on galaxies, for which thedirection of the spin is well determined. The main dis-advantage of this approach is that the number of galax-ies suitable for computing P ( | cos θ | ) is greatly reduced.For example, using the criteria from T06, the fractionof galaxies that can be used is ∼
22% of all the diskgalaxies.We opted for a statistical approach that allows tocompute a corrected distribution P c ( | cos θ | ) from thecombination of the distributions P ( | cos θ | ) obtained us-ing both signs.In Appendix A we describe in detail this procedure.We also show in this appendix the results from severalMonte Carlo simulations that show the ability of theprocedure to recover the correct values of p (Table 7).The fact that we actually do not know the real val-ues of ζ is reflected in the uncertainties of the proce-dure. Although the values of p are well recovered, theuncertainties measured from the simulations are largerthan those expected from considering just the size ofthe sample, N g . Of course, this is because we are notusing the real values of ζ . Nevertheless, from the sim-ulations we have obtained that our procedure has a pre-dictibility power equivalent to that of a sample 0 . N g with complete knowledge of the real values of ζ . Let’sremember that the common procedure of using onlyface-on or edge-on galaxies is restricted to ∼
20% ofthe total amount of spiral galaxies. This means thatour statistical procedure increases by a factor of 3 thee ff ective number of galaxies with respect to previousworks.
5. Results
Using the procedure described in Appendix A, wehave computed the corrected distribution P c ( | cos θ | )of the sample of galaxies around voids. Given thelarge size of our initial sample, we have also computed P c ( | cos θ | ) for di ff erent subsamples combining di ff er-ent sizes of voids and shells around them.An important point of our analysis has been to es-tablish the significance of our results in a robust way.This is done by comparing, in each case, the mea-sured signal with the standard deviation σ (cid:104)| cos θ |(cid:105) ofthe theoretical distribution in case of null signal, i.e. (cid:104)| cos θ |(cid:105) = . In the situation of complete knowl-edge of the real values of ζ , we would have σ (cid:104)| cos θ |(cid:105) = ( (cid:112) × N g ) − for a sample of N g measures. However,we have already shown that our procedure has uncer-tainties equivalent to a sample of size 0 . N g , therefore,the previous expression needs to be corrected to σ (cid:104)| cos θ |(cid:105) = (cid:112) × . × N g . (10)Knowing the value of σ (cid:104)| cos θ |(cid:105) , we can establish thesignal to noise ratio ( S NR ) of the signal of a subsampleof N g measurements as: S NR = . − (cid:104)| cos θ |(cid:105) corr σ (cid:104)| cos θ |(cid:105) , (11)where (cid:104)| cos θ |(cid:105) corr is obtained from the statisticalcorrection and σ (cid:104)| cos θ |(cid:105) from Equation (10). Note thatthe denominator is the signal, which corresponds to thedi ff erence between the observed value of (cid:104)| cos θ |(cid:105) corr and that of the random distribution which is 0.5. Forpractical reasons, the sign of S NR has been chosen sothat is positive for values of p > (cid:104)| cos θ |(cid:105) corr < . p < (cid:104)| cos θ |(cid:105) corr > . R minVoid )while, in Table 5, samples are constructed using voidswith radii in the ranges R Void ± . h − Mpc. Apartfrom this di ff erence in the definition of the first col-umn, both tables share the description of the rest of the In the analysis of the significance is more convenience the use of (cid:104)| cos θ |(cid:105) than that of p because the former has a gaussian distributionbut the latter has not. For convenience, we will refer the samples of the first table as “cu-mulative” samples and those of the second table as “di ff erential”samples. S W is the width of the innermost shell in h − Mpc; N is the number of measures of the sample; (cid:104)| cos θ |(cid:105) is the mean of the | cos θ | ; p is the characteris-tic parameter of Equation (7); and S NR is the signal tonoise ratio computed with Equation (11). From simu-lations it has been found that results obtained with lessthan ∼
100 measures are not reliable, therefore, sam-ples with less than this number have been flagged witha question mark beside the value of the
S NR .Errors in (cid:104)| cos θ |(cid:105) are computed using the standarddeviation of | cos θ | resulting from 10000 Monte Carlosimulations with no signal (see below) corrected usingEquations (B1-B3) assuming the following relation: σ (cid:104)| cos θ |(cid:105) ( p (cid:44) corr ) σ (cid:104)| cos θ |(cid:105) ( p (cid:44) theo ) = σ (cid:104)| cos θ |(cid:105) ( p = sim ) σ (cid:104)| cos θ |(cid:105) ( p = theo ) (12)where σ (cid:104)| cos θ |(cid:105) ( p ; theo ) is computed using the the-oretical expressions described in Equations (B1-B3), σ (cid:104)| cos θ |(cid:105) ( p = sim ) is computed from the simulationsand σ (cid:104)| cos θ |(cid:105) ( p (cid:44) corr ) is the final value used in Ta-bles 4-5.Since p does not follow a Gaussian distribution,for this parameter we provide the confidence levels at1 σ again correcting the theoretical values from Equa-tions (B5-B6) with the confidence levels measuredfrom the simulations with no signal.In Figures 3 and 4 are plotted the values of p (upperpanels) and S NR (lower panels) as a function of theradius of the voids, for the cumulative and di ff eren-tial samples, respectively. In the first figure the plottedradius is the minimum radius of each sample and inthe second the central radius of each bin. In di ff erentcolors are plotted the 10 shells widths that have beenexplored.The first result is that p < h − Mpc in shells of3 h − Mpc ( | S NR | > .
6) using 179 galaxies, but the | S NR | is higher than 3 increasing the width of the shellup to 7 h − Mpc and the sample size to 614 galaxies.Therefore, the further the galaxies are from the surfaceof the void, the lower is the signal although the in-crease in the size sample keeps the significance high. The relation between the strength of the alignmentand the radius of the void is clearly shown in Fig-ure 4 where the di ff erential samples are used. Forvoids smaller than ∼ h − Mpc these results are com-patible with a random distribution. It is for voids (cid:38) h − Mpc when it appears a signal of alignment,although given the smaller size of the samples thehighest significance reached is 2.96 for voids with16 h − Mpc ≤ R Void ≤ h − Mpc and a shell of6 h − Mpc.Figure 6 shows the corrected histograms of θ val-ues (see Equation A5) for the cases in which is reachedthe maximum S NR for the cumulative (left panel) andthe di ff erential samples (right panel). The continuousred line shows the analytical model described by Equa-tion (7) with the p values corresponding to these twomaxima.To compute in a more robust way the significancelevel of our results we need to compare them with acontrol sample with no signal. To construct this con-trol sample, we have run 10000 Monte Carlo simula-tions in which the spin direction of the galaxies (de-termined by their position angles and axial ratios) hasbeen shu ffl ed so that each galaxy it is assigned the spindirection of any other galaxy randomly selected. Thisprocedure has the advantage of ensuring the random-ness of the spin distribution and, therefore, the lack ofany alignment signal, while using real data.For each simulation we have repeated the analysisperformed in the real data using cumulative subsam-ples, and we have also computed 2 statistics used bothin the real data and in the simulations. These statisticswere computed as follows: for each bin in R Void , themean (median) value of the
S NR measured in the 10di ff erent shell widths was computed and the extremevalue (i.e. with highest absolute value) of each simu-lation was kept.The significance level from each statistic is com-puted as1 − f (extreme { S NR
S im } < extreme { S NR
Real } ),where f (extreme( S NR
S im ) < extreme { S NR
Real } ) isthe fraction of simulations with extreme values of themean (median) of SNR lower than the observed ones(-2.73 for the mean; -3.10 for the median). The factof multiplying by two the observed fraction takes intoaccount that we are considering only one side of thedistribution of the statistics.The results from this analysis are summarized inTable 3. Using the mean we obtain a significance of88 .
8% which improves slightly to 99 .
5% if the medianis used instead.Although this test ensures the existence of a globalsignal, we have also checked that the increase of the
S NR with the radius of the voids shown in Tables 4and 5 is not just a consequence of the variation in thesize of the samples. Figure 5 shows the dependence ofthe
S NR with the size of the samples ( N g ). For claritypurposes we have restricted the analysis to the voidslarger than 14 h − Mpc. Each line corresponds to dif-ferent limits in R Void and each point corresponds to ashell width. It can be seen that all subsamples show asimilar trend with the width of the shell, showing themaximum value of
S NR , in absolute value, at interme-diate shell widths. Also, for subsamples with similarsizes, the
S NR shows variations depending on the sizeof the voids and shells, rejecting the hypothesis thatthe variations of the
S NR are due only to variations ofthe size of the samples.Finally, to check the dependence of the signifi-cance of the signal with the distance of the galaxiesto the surface of the void, we have constructed sub-samples containing galaxies closer than 5 h − Mpc ofthe voids’ surface and galaxies between 5 h − Mpc and10 h − Mpc. With these two groups of galaxies wehave repeated the analysis (only cumulative). In Ta-ble 6 are presented the results of this analysis whichshow that galaxies at distances larger than 5 h − Mpcdo not present any significance alignment.
6. Discussion
We have analysed a large sample of galaxies around699 voids with radius larger than 10 h − Mpc up to z = .
12. We have found that for voids with ra-Criteria N S im %min {(cid:104)
S NR (cid:105)} < − .
73 58 98.8min { median( S NR ) } < − .
10 23 99.5Table 3: Results from 10000 simulations with reshuf-fling of position angle and axial ratio betweengalaxies, showing the number of simulations pre-senting values for two statistics smaller than theobserved values (extreme {(cid:104)
S NR (cid:105)} (Real Data) = -2.73;extreme { median( S NR ) } (Real data) = -3.10). The lastcolumn shows the significance of the result, whichtakes into account that we are considering only oneside of the distribution of the statistics. See text formore details. dius (cid:38) h − Mpc and within a shell not larger than ∼ h − Mpc, disk galaxies present a significant ten-dency to have their spin vectors aligned with the radialdirection of the void.The maximum | S NR | = .
62 is measured forvoids with R Void ≥ h − Mpc and a shell widthof 3 h − Mpc with a strength of the alignment p = . + . − . . However, this value gives an overestima-tion of the real strength since has been selected as thebest case out of many subsamples.In the next sections we compare our results withprevious empirical works and with results from numer-ical simulations. From the observational point of view, T06 and,more recently, S09 have performed a similar analysisto the one done here. T06 analysed 201 face-on andedge-on galaxies around voids with R > h − Mpcusing data from the SDSS-DR3 and the 2dFRGS. Theyfound a significant tendency of the spin of the galaxiesto be in the direction perpendicular to the radial direc-tion of the void. More recently, S09 using two samplesof 578 and 258 galaxies from the SDSS-DR6 with sim-ilar selection criteria found no statistical evidence fordeparture from random orientations.Using the same criteria on the size of the voids( R Void > h − Mpc) and on the width of the shell(4 h − Mpc) as in those previous works, we find no sig-nificance alignment ( p = . S NR = − .
15; seeTable 4). The size of the sample used to establish thisresult is of 11060 galaxies, which after applying thecorrection factor of 0.6, means an equivalent size of6636 galaxies. This number is 8 times larger the sizeused by S09.For a better comparison, we have computed the sig-nal of the alignment using criteria similar to that of S09regarding the selection of spiral galaxies ( g − r < . b / a < .
27) and face-on( b / a < .
96) galaxies . However, we keep our limitin M r instead of using M r > − + h as doneby S09 because otherwise the final number of galaxieswould be too small. After applying these criteria, wefinished with a sample of 252 face-on and edge-on spi-ral galaxies. The value of p obtained with this sampleis 0.993 with a S NR = . We used an slightly di ff erent way to compute the axial ratio b / a compared with S09, however, we do not consider this to have a sig-nificance e ff ect on our comparison. m i n V o i d S W N (cid:104) | c o s θ | (cid:105) p S N R . ± . . + . − . - .
157 11114970 . ± . . + . − . .
106 1218840 . ± . . + . − . .
211 1315030 . ± . . + . − . - .
026 1412430 . ± . . + . − . - .
260 1511280 . ± . . + . − . - .
341 161480 . ± . . + . − . - . ? . ± . . + . − . - . ? . ± . . + . − . . ? . ± . . + . − . - .
687 11229980 . ± . . + . − . - .
355 12217810 . ± . . + . − . - .
032 13210150 . ± . . + . − . - .
220 1424980 . ± . . + . − . - .
705 1522620 . ± . . + . − . - .
252 1621090 . ± . . + . − . - .
106 172610 . ± . . + . − . - . ? . ± . . + . − . - . ? . ± . . + . − . - .
185 11348770 . ± . . + . − . - .
827 12328880 . ± . . + . − . - .
751 13316630 . ± . . + . − . - .
979 1438260 . ± . . + . − . - .
703 1534290 . ± . . + . − . - .
819 1631790 . ± . . + . − . - .
620 1731020 . ± . . + . − . - .
747 183450 . ± . . + . − . - . ? . ± . . + . − . - .
148 11470830 . ± . . + . − . - .
521 12442320 . ± . . + . − . - .
137 13424630 . ± . . + . − . - .
265 14412290 . ± . . + . − . - .
974 1546320 . ± . . + . − . - .
021 1642560 . ± . . + . − . - .
459 1741480 . ± . . + . − . - .
141 184700 . ± . . + . − . - . ? . ± . . + . − . - .
053 11598450 . ± . . + . − . - .
365 12558650 . ± . . + . − . - .
864 13534310 . ± . . + . − . - .
492 14517120 . ± . . + . − . - .
814 1558790 . ± . . + . − . - .
297 1653550 . ± . . + . − . - .
473 1752010 . ± . . + . − . - .
141 185950 . ± . . + . − . - . ? R M i n V o i d S W N (cid:104) | c o s θ | (cid:105) p S N R . ± . . + . − . - .
193 116129970 . ± . . + . − . - .
750 12677770 . ± . . + . − . - .
355 13645380 . ± . . + . − . - .
702 14622830 . ± . . + . − . - .
256 15611730 . ± . . + . − . - .
746 1664880 . ± . . + . − . - .
332 1762740 . ± . . + . − . - .
727 1861290 . ± . . + . − . - .
094 107254730 . ± . . + . − . - .
436 117164890 . ± . . + . − . - .
249 12798070 . ± . . + . − . - .
925 13757400 . ± . . + . − . - .
016 14728780 . ± . . + . − . - .
534 15714710 . ± . . + . − . - .
628 1676140 . ± . . + . − . - .
092 1773390 . ± . . + . − . - .
684 1871610 . ± . . + . − . - .
002 108315600 . ± . . + . − . - .
192 118204280 . ± . . + . − . - .
991 128122390 . ± . . + . − . - .
565 13871290 . ± . . + . − . - .
074 14836090 . ± . . + . − . - .
260 15818330 . ± . . + . − . - .
157 1687560 . ± . . + . − . - .
468 1784230 . ± . . + . − . - .
180 1882030 . ± . . + . − . - .
209 109382730 . ± . . + . − . - .
500 119248480 . ± . . + . − . - .
991 129149410 . ± . . + . − . - .
599 13986840 . ± . . + . − . - .
038 14943680 . ± . . + . − . - .
747 15922030 . ± . . + . − . - .
373 1699040 . ± . . + . − . - .
736 1795190 . ± . . + . − . - .
703 1892470 . ± . . + . − . .
287 1010455220 . ± . . + . − . - .
260 1110296530 . ± . . + . − . - .
703 1210179290 . ± . . + . − . - .
723 1310104060 . ± . . + . − . - .
195 141052260 . ± . . + . − . - .
938 151025850 . ± . . + . − . - .
305 161010780 . ± . . + . − . - .
054 17106240 . ± . . + . − . - .
062 18102840 . ± . . + . − . - . T a b l e : S t a ti s ti c s f o r s ub s a m p l e s w it hd i ff e r e n t s i ze s o f vo i d s a nd s h e ll s . T h i s t a b l ec o rr e s pond s t o t h e“c u m u l a ti v e s a m p l e s ” i n w h i c h eac h s a m p l ec on t a i n s g a l a x i e s a r oundvo i d s w it h r a d i u s l a r g e r t h a n R M i n V o i d . R M i n V o i d : L o w e r li m it o f t h e vo i d ’ s r a d i u s i n eac h s a m p l e , i n h − M p c . S W : W i d t ho f t h e s h e ll , i n h − M p c . (cid:104) | c o s θ | (cid:105) : A v e r a g e o f t h e d i s t r i bu ti ono f | c o s θ | . p : R e s u lti ngv a l u e o f t h e p p a r a m e t e r o f E qu a ti on ( ) . S N R : T h e o r e ti ca l s i gn a lt ono i s e r a ti o c o m pu t e d w it h E qu a ti on ( ) . S a m p l e s w it h l e ss t h a n100g a l a x i e s h a v e b ee n fl a gg e d w it h a qu e s ti on m a r kn ea r by t h e S N R t o r e m a r k t h e l o w r e li a b ilit yo f t h e s e r e s u lt s . E rr o r s i n (cid:104) | c o s θ | (cid:105) a nd p a r ec o m pu t e du s i ng f o r m u l ae i n A pp e nd i x B i n c o m b i n a ti on w it h M on t e C a r l o s i m u l a ti on s . S ee t e x t f o r m o r e d e t a il s . V o i d S W N (cid:104) | c o s θ | (cid:105) p S N R . . ± . . + . − . - .
657 11 . . ± . . + . − . .
022 12 . . ± . . + . − . .
179 13 . . ± . . + . − . .
330 14 . . ± . . + . − . - .
092 15 . . ± . . + . − . . ? . . ± . . + . − . - . ? . . ± . . + . − . - . ? . . ± . . + . − . . ? . . ± . . + . − . - .
114 11 . . ± . . + . − . .
265 12 . . ± . . + . − . - .
181 13 . . ± . . + . − . - .
055 14 . . ± . . + . − . .
092 15 . . ± . . + . − . - .
121 16 . . ± . . + . − . - . ? . . ± . . + . − . - . ? . . ± . . + . − . - . ? . . ± . . + . − . .
851 11 . . ± . . + . − . .
550 12 . . ± . . + . − . - .
532 13 . . ± . . + . − . .
030 14 . . ± . . + . − . - .
902 15 . . ± . . + . − . - .
485 16 . . ± . . + . − . - . ? . . ± . . + . − . - . ? . . ± . . + . − . - . ? . . ± . . + . − . .
951 11 . . ± . . + . − . .
377 12 . . ± . . + . − . - .
349 13 . . ± . . + . − . .
440 14 . . ± . . + . − . - .
965 15 . . ± . . + . − . - .
962 16 . . ± . . + . − . - .
671 17 . . ± . . + . − . - . ? . . ± . . + . − . - . ? . . ± . . + . − . .
919 11 . . ± . . + . − . .
536 12 . . ± . . + . − . - .
782 13 . . ± . . + . − . .
041 14 . . ± . . + . − . - .
515 15 . . ± . . + . − . - .
238 16 . . ± . . + . − . - .
752 17 . . ± . . + . − . - .
249 18 . . ± . . + . − . - . ? R V o i d S W N | c o s θ | p S N R . . ± . . + . − . .
774 11 . . ± . . + . − . - .
013 12 . . ± . . + . − . - .
509 13 . . ± . . + . − . .
555 14 . . ± . . + . − . .
006 15 . . ± . . + . − . - .
645 16 . . ± . . + . − . - .
960 17 . . ± . . + . − . - .
354 18 . . ± . . + . − . - .
094 10 . . ± . . + . − . .
044 11 . . ± . . + . − . - .
178 12 . . ± . . + . − . - .
051 13 . . ± . . + . − . .
434 14 . . ± . . + . − . .
560 15 . . ± . . + . − . - .
859 16 . . ± . . + . − . - .
763 17 . . ± . . + . − . - .
356 18 . . ± . . + . − . - .
002 10 . . ± . . + . − . .
421 11 . . ± . . + . − . .
348 12 . . ± . . + . − . - .
153 13 . . ± . . + . − . - .
202 14 . . ± . . + . − . .
314 15 . . ± . . + . − . - .
674 16 . . ± . . + . − . - .
417 17 . . ± . . + . − . - .
300 18 . . ± . . + . − . - .
209 10 . . ± . . + . − . .
378 11 . . ± . . + . − . .
394 12 . . ± . . + . − . - .
795 13 . . ± . . + . − . .
250 14 . . ± . . + . − . .
243 15 . . ± . . + . − . - .
840 16 . . ± . . + . − . - .
720 17 . . ± . . + . − . - .
120 18 . . ± . . + . − . .
287 10 . . ± . . + . − . - .
173 11 . . ± . . + . − . .
013 12 . . ± . . + . − . - .
852 13 . . ± . . + . − . .
220 14 . . ± . . + . − . - .
424 15 . . ± . . + . − . - .
324 16 . . ± . . + . − . - .
035 17 . . ± . . + . − . - .
023 18 . . ± . . + . − . - . T a b l e : S t a ti s ti c s f o r s ub s a m p l e s w it hd i ff e r e n t s i ze o f vo i d s a nd s h e ll s . T h i s t a b l ec o rr e s pond s t o t h e“ d i ff e r e n ti a l s a m p l e s ” i n w h i c h eac h s a m p l ec on t a i n s g a l a x i e s a r oundvo i d s w it h r a d iii n t h e i n t e r v a l s ( R V o i d − . , R V o i d + . ) . R V o i d : M i dpo i n t o f eac h i n t e r v a l , i n h − M p c . S W : W i d t ho f t h e s h e ll , i n h − M p c . (cid:104) | c o s θ | (cid:105) : A v e r a g e o f t h e d i s t r i bu ti ono f | c o s θ | . p : R e s u lti ngv a l u e o f t h e p p a r a m e t e r o f E qu a ti on ( ) . S N R : T h e o r e ti ca l s i gn a lt ono i s e r a ti o c o m pu t e d w it h E qu a ti on ( ) . S a m p l e s w it h l e ss t h a n100g a l a x i e s h a v e b ee n fl a gg e d w it h a qu e s ti on m a r kn ea r by t h e S N R t o r e m a r k t h e l o w r e li a b ilit yo f t h e s e r e s u lt s . E rr o r s i n (cid:104) | c o s θ | (cid:105) a nd p a r ec o m pu t e du s i ng f o r m u l ae i n A pp e nd i x B i n c o m b i n a ti on w it h M on t e C a r l o s i m u l a ti on s . S ee t e x t f o r m o r e d e t a il s . . . . . . . . . p SW=1 SW=2 SW=3 SW=4 SW=5 SW=6 SW=7 SW=8 SW=9 SW=10
10 11 12 13 14 15 16 17 18 R Min [ h − Mpc] − − − − S N R Fig. 3.—
Upper panel:
Values of the parameter p for subsamples of galaxies in shells of width S W and voids withradius larger than R Min . Lower panel:
Signal to noise of the alignment found for each subsample. This gives thesignificance of rejecting the null hyphotesis of not existence of any alignment. R MinVoid h − Mpc < R S hell ≤ h − Mpc 5 h − Mpc < R S hell ≤ h − Mpc N (cid:104)| cos θ |(cid:105) p SNR N (cid:104)| cos θ |(cid:105) p SNR10 15289 0.500 ± . + . − . -0.053 30233 0.502 ± . + . − . -0.91511 9845 0.505 ± . + . − . -1.365 19808 0.501 ± . + . − . -0.37412 5865 0.509 ± . + . − . -1.864 12064 0.504 ± . + . − . -1.17013 3431 0.509 ± . + . − . -1.492 6975 0.503 ± . + . − . -0.69814 1712 0.525 ± . + . − . -2.814 3514 0.502 ± . + . − . -0.30415 879 0.541 ± . + . − . -3.297 1706 0.504 ± . + . − . -0.42816 355 0.569 ± . + . − . -3.473 723 0.495 ± . + . − . ± . + . − . -2.141 423 0.491 ± . + . − . ± . + . − . -0.516 189 0.502 ± . + . − . -0.080Table 6: Strength of the alignment for galaxies within a shell up to 5 h − Mpc (left) and for galaxies within a shellbetween 5 h − Mpc and 10 h − Mpc (right). Each line corresponds to samples of N galaxies around voids with radiuslarger than R MinVoid . Errors are computed using theoretical expressions in Appendix B.12 . . . . . . . . p SW=1 SW=2 SW=3 SW=4 SW=5 SW=6 SW=7 SW=8 SW=9 SW=10
10 11 12 13 14 15 16 17 18 19 R V oid [ h − Mpc] − − − − S N R Fig. 4.—
Upper panel:
Values of the parameter p for subsamples of galaxies in shells of di ff erent widths S W (in h − Mpc) and voids in bins of 1 h − Mpc in radius.
Lower panel:
Signal to noise of the alignment found for eachsubsample. This gives the significance of rejecting the null hyphotesis of not existence of any alignment.13 . . . . . . lg( N g ) − . − . − . − . − . − . − . − . . . S N R Cumulative
R > R > R > R > R > . . . . . . − . − . − . − . − . − . . . . . S N R Differential . < R < . . < R < . . < R < . . < R < . . < R < . Fig. 5.— Variation of the SNR as a function of the number of galaxies N g for di ff erent samples. Each line correspondsto a selection in void’s radius (di ff erential in the upper panel and cumulative in the lower one) and each numberindicates the width of the shell in h − Mpc. For practical purposes, only the samples of voids larger than 14 h − Mpcare shown. 14ig. 6.— Distribution of | cos θ | after applying the statistical correction for the two subsamples that reach the highest | S NR | in Table 4 (left) and Table 5 (right). The continuous line corresponds to the theoretical model described byEquation (7) for the measured value of p , shown in the upper left corners.with a random distribution.In relation to T06, it is important to note that inthat work, the signal presented corresponds to the peakof the signal found after exploring in shells of di ff er-ent widths. Consequently, to properly address the sig-nification of the signal in T06 we have reviewed thedata used in that paper taking into account this ex-ploration. To do this, we have run simulations withsimilar number of galaxies around similar voids ( R > h − Mpc) searching for the maximum of the | S NR | in shells of width from 3 h − Mpc to 7 h − Mpc in stepsof 0.1 h − Mpc. Then, it has been computed the frac-tion of simulations with max( | S NR | ) larger than theone found in T06 when only SDSS data was used(max( S NR T S DS S ) = . After doing our analy-sis we found that ∼
16% of the simulations showedmax( | S NR | ) >
2, decreasing the significance of the re-sult of T06 to ∼ > σ ) betweenthe direction of the spin and the intermediate princi-pal axis of the shear tidal tensor. Nevertheless, the In the process of conducting this analysis we noted some duplica-tions in a few galaxies that make the measured
S NR decreases from2.4 to 2. The authors obtained a value of c = . ± .
014 which corresponds comparison with our results is not direct since we donot use a direct measure of the orientation of the sheartidal tensor in the position of each galaxy and the ra-dial direction of the voids can be considered only as astatistical proxy for the direction of the major princi-pal axis of the shear tidal tensor. Given the di ff erencesin methodology, a meaningful comparison of the re-sults from both works would need an analysis that it isbeyond the scope of this paper. Another way to study the alignment of galaxieswith their local large scale environment is through nu-merical simulations (Porciani et al. 2002a,b; Navarroet al. 2004; Bailin & Steinmetz 2005; Bailin et al.2005; Altay et al. 2006; Patiri et al. 2006b; Bruninoet al. 2007; Arag´on-Calvo et al. 2007; Cuesta et al.2008; Zhang et al. 2009; Hahn et al. 2010; Bett et al.2010). Since the behaviour of the halos can be de-pendent on the environment or the large scale struc-ture in which they reside, to perform a meaningfulcomparison we have focused on the analysis done byPatiri et al. (2006b), Brunino et al. (2007) and Cuestaet al. (2008) in which it was studied the orientationof dark matter halos around cosmic voids using dif-ferent cosmological simulations. The criteria imposedto the dark matter halos and the procedure to detectvoids tried to match the criteria used in T06. All threeworks found that the minor axis and the major axis to p = . ± .
15f the halos have significant tendencies to be alignedwith the radial and the perpendicular directions, re-spectively. The results regarding the orientation ofthe angular momentum of the halos were less clear.Patiri et al. (2006b) did not find any particular orienta-tion for the angular momentum of the halos. Bruninoet al. (2007) also did not found any particular align-ment for the angular momentum in their full sample ofhalos although they detected a tendency for those haloswith a disc-dominated galaxy to have their angular mo-mentum perpendicular to the radial direction. Finally,Cuesta et al. (2008) measured a significant ( > σ ) ten-dency of the spin of the dark matter halos to lie inthe plane perpendicular to the radial direction. How-ever, the same authors found that the strength of thealignment is mainly produced by the outer regions ofthe DM halos and this would explain the discrepancieswith Brunino et al. (2007) were the inner regions of theDM were used to measure the alignment.How these results relate with ours is not straightfor-ward since we observe the collapsed baryonic matterand they studied the dark matter or the non-collapsedbaryonic matter. We can only point out the fact thatthe alignment that we find in the galaxies is shared bythe minor axis of the halos studied in the simulations,either dark matter or gas halos. This is suggestive toan interaction between the galaxy and the hosting ha-los around it (either of dark or baryonic matter) lead-ing to a tendency of the minor axis of the galaxy (and,therefore, its angular momentum) to be aligned withthe minor axis of the halo’s matter distribution.
7. Summary
Analysing a volume of ∼ × ( h − Mpc) fromthe SDSS-DR7 we have searched for cosmic voids de-void of galaxies brighter than M r − h = − . R Void > h − Mpc. We have found 699 nonoverlapping voids for which we provide positions andsizes.We have used this catalog of voids to search for diskgalaxies around them and study the alignment betweenthe direction of the angular momentum of these galax-ies and the radial direction with respect to the center ofthe voids.We have included two improvements with respectto previous similar works.First, we have used an updated version of the SDSSspectroscopic catalog (data release 7) and we havecombined it with the visual morphological classifica- tion from the Galaxy Zoo project to get a reliable sam-ple of disk galaxies.Second and more important, we have introduced astatistical procedure that has allowed us to overcomethe problem of the indetermination of the real incli-nation of galaxies computed from their apparent axialratio. We have performed extensive Monte Carlo sim-ulations to check the validity of this procedure. Weshow that the procedure recover the real signal withoutpractically any bias and its power in terms of capacityto reject the null hypothesis it is equivalent to the casein which it is used a sample with complete knowledgeof the real direction of the spin of the galaxies using60% the amount of galaxies. In comparison with thecommon procedure of selecting only edge-on and face-on galaxies, this procedure means an increase of abouta factor 3 in the amount of measurements used in theanalysis of the alignment.These improvements have allowed us to detect astatistically significant ( (cid:38) . R (cid:38) h − Mpc) to havetheir angular momentum align with the radial directionof the voids. However, for smaller voids this tendencydisappears and the results are consistent with no spe-cial alignment.We have also found that the strength of the align-ment depends on the distance of the galaxies to thesurface of the voids and for galaxies further than ∼ h − Mpc the distribution of the alignments is com-patible with a random distribution independent of thesize of the voids.Previous similar works found opposite alignment(T06) or no alignment (S09). However, these worksused too few galaxies around voids with R ≥ h − Mpcwhich, according to our work, could mask the signal.In fact, using the same criteria for the size of the voidsand the width of the shells as in those works, our datais compatible with a random distribution of spins with-out any particular alignment, as found by S09.The comparison with the results from cosmolog-ical simulations points to a possible connection be-tween the alignment of the halos (of dark matter andnon-collapse baryonic matter) and that of the galax-ies which could explain the similar orientation of bothcomponents observed in the simulations and in ourwork, respectively.
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In the analysis of the alignment of galaxies, one of the main sources of uncertainty is the indetermination in theinclination of the plane of a disk galaxy with respect to the line of sight ± ζ using exclusively the observed axial ratio b / a of the galaxies (see Equation 1). In other words, if a galaxy is divided in two halves separated by its major axis, itis not possible to know which of the two halves is the closest to the observer. This indetermination is negligible foredge-on ( b / a ∼
0) and face-on galaxies ( b / a ∼
1) but using only these galaxies reduces the sample size to ∼ / b / a , the increasing uncertainties in ζ will result in anincreasing degradation of any existing alignment but the statistics improve. The question is whether this improvementof the statistics can compensate for the increasing degradation of any possible signal. The answer is “yes”. Choosingalways the plus sign in the computation of ζ , or the minus sign, or any random assignment of signs, leads to the samestatistical results (i.e. they are equally powerful tests), which are better than those obtained with any limitation of therange of possible values of b / a . However, the estimate of the alignment obtained in this manner is biased towardssmaller values (the strength of the alignment is given by (1 − p ) (cid:39) − c / − p + (cid:39) . − p ) (A1)where p corresponds to the real alignment and p + is the value obtained using the plus sign for ζ .The main problem with the use of p + , or any other sign assignment, is that it introduces an artificial randomness thatincreases the scatter of the estimates. Using p + , we assign the correct sign to half of the galaxies, on average, whilethe other half gets the wrong sign, but the exact number of galaxies getting the correct sign fluctuates from sample tosample (with variance N g /
4, being N g the size of the sample) resulting in an increased error. Furthermore, since we donot take into account the other possible sign assignment, we do not know how large is the degradation of the alignmentimplied by those galaxies that get the wrong sign.To avoid these problems we propose a method that uses all the information in the data and does not introduceartificial randomness. To this end, we consider the two possible values of θ associated with every galaxy (one valuefor each possible sign of ζ ) and assume that only half of the values of θ falling in a given range are correct while theother half is incorrect. The correct values for the latter half of galaxies would be the conjugate of θ , θ (cid:48) , correspondingto the value of θ using the opposite sign for ζ . Thus, if the actual probability distribution for θ were: ¯ P (cos θ, p ) = p (1 + ( p −
1) cos θ ) / ) , (A2)the probability distribution that would be inferred from the 2 N g values of θ treating them as if they were independent, P (cos θ ), would be given, for the j − th bin, by: P (cos θ j ) =
12 ¯ P (cos θ j , p ) + l l (cid:88) i = ¯ P (cos θ (cid:48) j , p ) , (A3)where l is the number of θ j values in the j − th bin. The use of kinematic information or the presence of dust lanes can help to break this indetermination, however, in most of the cases this isinformation is not accessible. We are only interested in the direction of the alignment and therefore the analysis can be restricted to 0 ≤ θ ≤ π/ θ = | cos θ | . /
2, the probability density in the bin centered in θ j is givenby the real distribution ¯ P evaluated at θ j (correct sign assignment), while with probability 1 /
2, the probability densityat θ j is the averaged of the value of ¯ P over the conjugate values ( θ (cid:48) j ( i )) of the l values of θ falling in bin j .So, the factor: Q (cos θ j ) ≡ ¯ P (cos θ j ) ¯ P (cos θ j , p ) + l (cid:80) li = ¯ P (cos θ (cid:48) j ( i ) , p ) (A4)is an estimate of the ratio between the actual distribution, ¯ P (cos θ j ), and the first estimate, P (cos θ j ).Therefore, we have for the estimate of ¯ P (that we denote by P c ): P c (cos θ j ) = P (cos θ j ) Q (cos θ j ) (A5)When the alignment is very strong, the assumption that the two values θ, θ (cid:48) of a conjugate couple have the sameprobability can no longer be mantained. Instead, we should used:Prob( θ ) = P (cos θ ) P (cos θ ) + P (cos θ (cid:48) ) (A6)Prob( θ (cid:48) ) = P (cos θ (cid:48) ) P (cos θ ) + P (cos θ (cid:48) ) (A7)and modify the definition of Q consequently. However, this complication of the method is not worthy to ourpurpose. In fact, from Table 7, we can see that even for considerable alignment strengths, the bias implied by neglectingthis last refinement is small, and can be corrected by the following expression: p db = − (1 + . − p c ) )(1 − p c ) (A8)where p c is the value obtained using Equation (A5), and p db is the debiased value.From Table 7 we can also see that the relative error, σ p / | − p | , is always larger for p + or p − than for p c . Forweak alignments ( | − p | (cid:46) .
1) the former is ∼
20% larger than the latter, while for larger alignments the di ff erencediminishes.Finally, it must be noticed that the method that we have just described does not depend on the form of ¯ P (cos θ ). A.2. Description of the procedure
In this appendix we describe the actual implementation of the method presented above.The procedure is as follows:1. For each galaxy, we compute the two possible values of cos θ corresponding to the two alternatives signs of ζ and hence the two possible spin orientations.2. Then, we construct a normalized histogram assuming the two values of cos θ of each galaxy as independentvalues. The normalization is done dividing each bin by 2 times the total number of galaxies of the sample ( N g )and by the width of the bins. We call this non-corrected histogram P (cos θ ).3. Next, in each bin centered in cos θ j , we compute the value of the corrected histogram P c (cos θ ) using Equa-tions (A5) and (A4): P c (cos θ j ) = P (cos θ j ) Q (cos θ j ) (A9)remembering that 19 (cos θ j ) = ¯ P (cos θ j , p ) ¯ P (cos θ j , p ) + l (cid:80) li = ¯ P (cos θ (cid:48) j ( i ) , p )and that θ (cid:48) j ( i ) are the conjugate values of θ for those galaxies with θ j ( i ) within the interval | θ j ( i ) − θ j |≤ ∆ θ j / l is the total number of values within the bin, and¯ P (cos θ, p ) ≡ p (1 + ( p −
1) cos θ ) / . (A10)The final corrected value of p is computed numerically using its relationship with (cid:104) cos θ (cid:105) from Equation (8), whichgiven the distribution P c (cos θ j ) as a discrete distribution can be expressed as: (cid:80) nj = P c (cos θ j ) cos θ j (cid:80) nj = P c (cos θ j ) = + p (A11)with n the total number of bins in which the distribution is divided. A.3. Robustness of the statistical correction
To check the robustness of the statistical correction we have performed a series of Monte Carlo simulations. Inthese simulations, we use samples of fake galaxies in the position of the real ones but with spin directions assignedrandomly following a p − distribution (Equations A2) with a given p input . Then, the samples of fake galaxies areanalysed in the same manner of the real galaxies and a final p output value is obtained.We have run 2 sets of simulations using two samples with di ff erent number of galaxies ( N g ) to check the robustnessof the procedure also as a function of the sample size. These samples correspond to galaxies in shells of 4 h − Mpcand R Void > h − Mpc (Sample A, following the usual criteria used in previous works) and to galaxies in shellsof 3 h − Mpc and R Void > h − Mpc (Sample B, corresponding to our maximum
S NR ). For each sample we haverun 1000 Monte Carlo realizations with 7 di ff erent initial distributions of (cid:104)| cos θ |(cid:105) described by their corresponding p values ( p input ). These values covered the typical values of p that we have found in our analysis.Table 7 shows the results of this analysis. For each subset of 1000 realizations we give the size of the sample N , the input value p input , the mean value of p obtained if a fixed sign for ζ is used ( p + and p − , for plus and minorsign, respectively) and the mean value of p when applying our stastical correction, p output . The uncertainties showncorrespond to 1 σ of the distribution of the single values in the 1000 realizations.We found that for most of the cases the statistically corrected value is within 1 σ of the input value showing the highaccuracy of the procedure, especially when comparing with the cases in which a fixed sign is used.The results of these simulations have been used to compute the “e ff ective size” of the initial sample. This e ff ectivesize is defined as the size that a sample with complete knowledge of the real signs of ζ for each galaxy should have toshow the same uncertainties that we find in our simulations. On what follows, it is described how we have computedthe correction factor to be applied to our samples to obtain their e ff ective sizes.It can be proved theoretically that the value of the standard deviation of (cid:104)| cos θ |(cid:105) , σ (cid:104)| cos θ |(cid:105) , for the case in which thereis no preferential alignment ( p = σ (cid:104)| cos θ |(cid:105) = √ N , (A12)where N is the total number of galaxies used to compute (cid:104)| cos θ |(cid:105) . However, this theoretical expression assumesthe full knowledge of the values of θ for all the galaxies, while empirically we do not have such full informationbecause of the indetermination in the sign of ζ . Therefore, we have compared the standard deviation obtained from thesimulations with di ff erent values of N , with the theoretical value. From this comparison, we have obtained a correctionfactor to be applied to the total number of galaxies N equal to 0.6. This means that our statistical approximation carriedan uncertainty that is equivalent to the uncertainty of having ∼
60% of the galaxies with full information.20ample
N p input p + p − p output (cid:104) p + (cid:105) σ p + (cid:104) p − (cid:105) σ p − (cid:104) p output (cid:105) σ p output A 11060 0.50 0.664 0.008 0.665 0.008 0.437 0.010A 11060 0.75 0.841 0.010 0.842 0.009 0.746 0.012A 11060 0.90 0.938 0.010 0.938 0.010 0.900 0.013A 11060 1.00 1.000 0.011 1.000 0.011 1.000 0.014A 11060 1.10 1.060 0.012 1.060 0.012 1.102 0.016A 11060 1.25 1.143 0.012 1.142 0.012 1.256 0.018A 11060 1.50 1.272 0.013 1.272 0.014 1.542 0.025B 179 0.50 0.642 0.060 0.644 0.059 0.486 0.059B 179 0.75 0.833 0.071 0.830 0.075 0.753 0.082B 179 0.90 0.938 0.081 0.938 0.085 0.907 0.101B 179 1.00 0.999 0.088 1.004 0.086 1.004 0.107B 179 1.10 1.073 0.091 1.072 0.091 1.115 0.121B 179 1.25 1.162 0.101 1.155 0.100 1.258 0.140B 179 1.50 1.305 0.113 1.313 0.110 1.573 0.561Table 7: Results of several simulations to test the validity and robustness of our statistical correction. Two sampleswith di ff erent number of galaxies are shown: Sample A is made of galaxies in shells of 4 h − Mpc around voids with R > h − Mpc and Sample B is made of galaxies in shells of 3 h − Mpc around voids with R > h − Mpc. Each rowcorresponds to a set of 1000 realizations in which to each real galaxy a synthetic spin vector was assigned followingthe theoretical distribution given by Equation (7) with a p = p input . p + and p − are the values of p obtained when fixingthe sign of ζ . p output is the final value after applying the statistical correction. For each parameter ( p + , p − , p output ), themean and the standard deviation of the 1000 realizations are shown. See text for more details. B. Computation of the uncertainties in (cid:104)| cos θ |(cid:105) and p In this section we present the expressions used to compute the uncertainties in (cid:104)| cos θ |(cid:105) and p , in the general case.The standard deviation of (cid:104)| cos θ |(cid:105) , σ (cid:104)| cos θ |(cid:105) , is computed as σ | cos θ | / (cid:112) N g , where N g is the total number of galaxies. σ | cos θ | is the root mean square of | cos θ | for the distribution given by Equation 7. Computing σ | cos θ | analytically, wefind the following expressions for σ (cid:104)| cos θ |(cid:105) depending on the value of p = (cid:104)| cos θ |(cid:105) − −
1, : σ (cid:104)| cos θ |(cid:105) = (cid:112) N g (cid:115) p ( p − / ln( p + (cid:113) p − − p − − + p ; p > σ (cid:104)| cos θ |(cid:105) = (cid:112) N g √
12 ; p = σ (cid:104)| cos θ |(cid:105) = (cid:112) N g (cid:115) − p − p (1 − p ) / arcsin( (cid:113) − p ) − + p ; p < N g (see Table 7).Since the distribution of p is not Gaussian, we can compute the value p and the limits of the 1 σ confidence interval( p − σ , p + σ ) with the next expressions: p = (cid:104)| cos θ |(cid:105) − p − σ = (cid:104)| cos θ |(cid:105) + σ (cid:104)| cos θ |(cid:105) − + σ = (cid:104)| cos θ |(cid:105) − σ (cid:104)| cos θ |(cid:105) − // //