Can anisotropy in the galaxy distribution tell the bias?
aa r X i v : . [ a s t r o - ph . C O ] A p r MNRAS , 1–9 (2016) Preprint 9 October 2018 Compiled using MNRAS L A TEX style file v3.0
Can anisotropy in the galaxy distribution tell the bias?
Biswajit Pandey ⋆ Department of Physics, Visva-Bharati University, Santiniketan, Birbhum, 731235, India
ABSTRACT
We use information entropy to analyze the anisotropy in the mock galaxy cata-logues from dark matter distribution and simulated biased galaxy distributions fromΛCDM N-body simulation. We show that one can recover the linear bias parameterof the simulated galaxy distributions by comparing the radial, polar and azimuthalanisotropies in the simulated galaxy distributions with that from the dark matterdistribution. This method for determination of the linear bias requires only O ( N )operations as compared to O ( N ) or at least O ( N log N ) operations required for themethods based on the two-point correlation function and the power spectrum. We ap-ply this method to determine the linear bias parameter for the galaxies in the 2MASSRedshift Survey (2MRS) and find that the 2MRS galaxies in the K s band have a linearbias of ∼ . Key words: methods: numerical - galaxies: statistics - cosmology: theory - largescale structure of the Universe.
The homogeneity and isotropy of the Universe on large scalesis a fundamental tenet of modern cosmology. Our currentunderstanding of the cosmos relies heavily on this princi-ple. Presently a large number of cosmological observationssuch as the temperature of the Cosmic Microwave Back-ground (CMBR) (Smoot et al. 1992; Fixsen et al. 1996), X-ray background (Wu et al. 1999; Scharf et al. 2000), angu-lar distributions of radio sources (Wilson & Penzias 1967;Blake & Wall 2002), Gamma-ray bursts (Meegan et al.1992; Briggs et al. 1996), supernovae (Gupta & Saini 2010;Lin et al. 2016), galaxies (Marinoni et al. 2012; Alonso et al.2015) and neutral hydrogen (Hazra & Shafieloo 2015) areknown to favour the assumption of statistical isotropy. Butthis assumption does not hold on small scales and theanisotropies present on these scales can tell us a lot aboutthe Universe. For example, the CMBR is not completelyisotropic and the anisotropies imprinted in the CMBR per-haps provide the richest source of information in cosmology(Planck Collaboration et al. 2016). In the current paradigm,the large scale structures in the Universe are believed toemerge from the gravitational amplification of the minisculedensity fluctuations generated in the early Universe. Theanisotropies in the CMBR shed light on the conditions thatprevailed in the early Universe whereas the anisotropies inthe present day mass distribution help us to unravel the ⋆ E-mail: [email protected] formation and evolution of the large scale structures in theUniverse.Currently there exist a wide variety of statistical tools toquantify the distribution of matter in the Universe. Besidesthere use in the study of CMBR anisotropies, the two-pointcorrelation function and the power spectrum also remainthe most popular choice for the study of clustering. Thetwo-point correlation function (Peebles 1980) measures theamplitude of galaxy clustering as a function of scale whereasthe shape and amplitude of the power spectrum also providethe information about the amount and nature of matter inthe Universe. The three-point correlation function and thebispectrum has been also widely used in the study of cluster-ing in the galaxy distribution. These statistics are popularas one can directly relate them to the theories of structureformation.The distribution of galaxies are believed to trace themass distribution on large scales, where the density fluc-tuations in galaxies and mass are assumed to be relatedlinearly (Kaiser 1984; Dekel & Rees 1987). In the linearbias assumption, both the two point correlation functionand power spectrum can be employed to determine the lin-ear bias between galaxies and mass (Norberg et al. 2001;Tegmark et al. 2004; Zehavi et al. 2011). The distribution ofthe galaxies are inferred from their redshifts. The peculiarvelocities induced by the density fluctuations perturb theirredshifts. This distorts the clustering pattern of galaxies inredshift space and cause the two-point correlation functionand power spectrum to be anisotropic. They are suppressed c (cid:13) Pandey, B. on small scales due to the motion of galaxies inside virial-ized structures and enhanced on large scales due to coherentflows into over dense regions and out of under dense regions.The anisotropies in the two-point correlation function andthe power spectrum can be decomposed into different angu-lar moments (Kaiser 1987; Hamilton 1992) and their ratioscan be used to determine the linear distortion parameter β ≈ Ω . m b where Ω m is the mass density parameter and b isthe linear bias parameter. This method has been used to de-termine the linear bias (Hawkins et al. 2003; Tegmark et al.2004). One can also use the three-point correlation func-tion and bispectrum (Feldman et al. 2001; Verde et al. 2002;Gazta˜naga et al. 2005) to measure the bias. It may be notedthat some sort of parameter degeneracies are involved inall these methods. Computing the correlation functions andthe poly spectra are also computationally expensive for verylarge data sets.The information entropy is related to the higher ordermoments of a distribution and hence captures more infor-mation about the distribution. Pandey (2016b) propose amethod based on the information entropy-mass variance re-lation to determine the large scale linear bias from galaxyredshift surveys. We investigate if this relation also holdsfor the anisotropy measure proposed in Pandey (2016a) andcan one exploit this relation to measure the linear bias bydirectly measuring the anisotropy in the galaxy distribution.An important advantage of this method is the fact that forany given data set, it is computationally less expensive thanthe methods which are based on the two-point correlationfunction and the power spectrum. The only disadvantage ofthe method is that the information entropy is sensitive tobinning and sampling. But this relative character of entropydoes not pose any problem provided the distributions arecompared with the same binning and sampling rate.The modern redshift surveys (SDSS, York et al. 2000;2dFGRS, Colles et al. 2001; 2MRS, Huchra et al. 2012) havenow mapped the galaxy distribution in the local Universewith unprecedented accuracy. The SDSS and 2dFGRS aredeeper than 2MRS but they only cover parts of the sky.Moreover the 2MASS redshift survey (2MRS) maps thegalaxies over nearly the entire sky ( ∼ µm whichmakes it less susceptible to extinction and stellar confusion.The old stellar populations which are otherwise missed bythe optical surveys are also retained in 2MRS due to its op-eration in the infrared window. The survey is 97% completedown to the limiting magnitude of K s = 11 .
75 which pro-vides a fair representation of the mass distribution in thelocal Universe. These advantages offered by the 2MRS overthe other surveys make it most suitable for the analysis inthe present work.We use a ΛCDM model with Ω m = 0 .
31, Ω Λ0 = 0 . h = 1 for converting redshifts to distances throughoutour analysis.A brief outline of the paper follows. In section 2 wedescribe the method of analysis followed by a description ofthe data in section 3. We present the results and conclusionsin section 4. The information entropy was first introduced by ClaudeShannon (Shannon 1948) to find the most efficient way totransmit information through a noisy communication chan-nel. It quantifies the uncertainty in the measurement of arandom variable. Given a probabilistic process with proba-bility distribution p ( x ) where the random variable x has n outcomes given by { x i : i = 1 , ....n } , the average amount ofinformation to describe the random variable x is given by, H ( x ) = − n X i =1 p ( x i ) log p ( x i ) (1)The quantity H ( x ) is known as the information entropy ofthe random variable x .Pandey (2016a) propose an anisotropy measure basedon the information entropy and carry out tests on variousisotropic and anisotropic distributions to find that it canefficiently recover various types of anisotropies inputted ina distribution. The method divides the entire sky into equalarea pixels by carrying out uniform binning of cos θ and φ .Here θ and φ are respectively the polar and azimuthal anglesin spherical polar co-ordinates. The entire sky is dividedinto m total = m θ m φ angular bins or pixels where m θ and m φ correspond to the number of bins used for binning cos θ and φ respectively. At any distance r , each of these pixelssubtend equal volumes. The method counts the number ofgalaxies inside each of these volume elements and define arandom variable X θφ with m total outcomes each given by, f i = n i ( 1, one can relate the entropy deficit tothe variance in number counts as,( H θφ ) max − H θφ ( r ) = σ r σ r = 1(2 π ) Z ∞ k P ( k ) f W ( kr ) dk (5)where, r is the size of the filter used for smoothing, P ( k ) isthe power spectrum and f W ( kr ) is the Fourier transform ofthe filter. The filter shape has to be specified which wouldthen determine f W ( kr ). We carry out our analysis in co-ordinate space where averaging kernels have exactly samevolume but somewhat different shapes. We do not expectthis small variation in the shapes to make a difference whenthe kernels have larger volumes.One can then use the entropy deficit ( H θφ ) max − H θφ ( r )of a distribution to determine its linear bias on large scaleswhere the density fluctuations are smaller. On large scales, P g ( k ) = b P m ( k ) and consequently the linear bias is givenby, b = s [( H θφ ) max − H θφ ( r )] g [( H θφ ) max − H θφ ( r )] m = s [ a θφ ( r )] g [ a θφ ( r )] m (6)where the subscripts g and m corresponds to galaxy andmass respectively.The same argument also holds for the polar and az-imuthal anisotropies and one can use them independentlyto measure the linear bias for a given galaxy distribution.We do not expect them to be different and it would be in-teresting to measure and compare them. We analyze theanisotropies in the galaxy distribution from the 2MRS fol-lowing the method outlined in this section and determinethe linear bias parameter. It may be noted that one can use any spherical co-ordinates for this analysis. In the present work, we use thegalactic co-ordinates ( l, b ). Accordingly we replace θ and φ in the previous definitions by b and l respectively. The Two Micron All Sky Redshift Survey (2MRS)(Huchra et al. 2012) is an all-sky redshift survey in the nearinfra-red wavelengths. The survey is 97 . 6% complete to alimiting magnitude of K s = 11 . 75 and covers 91% of the sky.It provides the spectroscopic redshifts of ∼ , 000 galaxiesin the nearby Universe. 2MRS selects the galaxies with ap-parent infrared magnitude K s ≤ . 75 and colour excess E ( B − V ) ≤ | b | ≥ ◦ for 30 ◦ ≤ l ≤ ◦ and | b | ≥ ◦ otherwise. Huchra et al. (2012) rejected thesources which are of galactic origin (multiple stars, plane-tary nebulae, HII regions) and discarded the sources whichare in regions of high stellar density and absorption. Thefinal 2MRS catalog by Huchra et al. (2012) contain 43 , z ≤ . 12 beyond whichthere are a very few galaxies. This redshift limit is used tosimulate the mock catalogues for the 2MRS. We use this2MRS flux limited sample which contains 43 , 305 galaxies.To construct mock catalogues for the 2MRS flux lim-ited sample we first model the redshift distribution using aparametrized fit (Erdoˇgdu et al. 2006a,b) given by, dN ( z ) dz = A z γ exp[ − (cid:0) zz c (cid:1) α ] (7)We calculate the redshift distribution in the 2MRS usinguniform bin size of 200 km/s and then fit it with Equation 7using the nonlinear least-squares method (Marquardt-Levenberg algorithm). Each point in the data are assignedequal weights. We find the values of the best fit parame-ters to be A = 116000 ± γ = 1 . ± . z c =0 . ± . 002 and α = 2 . ± . z .We would like to have a galaxy distribution over full-sky for our analysis. This requires us to artificially fill theZone of Avoidance (ZOA), the region near the Galactic planewhich is obscured due to the extinction by Galactic dustand stellar confusion. We randomly select galaxies from theunmasked region and then place them at random locations inthe masked area so as to have the same average density in themasked and unmasked region (Lynden-Bell et al. 1989). Weclone 4 , 375 galaxies to fill the ZOA and after carrying outthe cloning procedure, finally we have 47 , 680 galaxies in our2MRS sample. The distribution of the galactic coordinatesof galaxies in the 2MRS after filling the ZOA is shown in theright panel of Figure 1. We construct 30 jackknife samplesfrom the 2MRS data each containing 35 , 000 galaxies. MNRAS , 1–9 (2016) Pandey, B. Figure 1. The left panel shows the redshift histogram in the 2MASS redshift survey (2MRS) along with the best fit (Equation 7) toit. The right panel shows the distribution of galactic coordinates of the 2MRS galaxies after the zone of avoidance is filled with clonedgalaxies. Figure 2. Different panels of this plot show the redshift histograms for a simulated galaxy sample with different bias values along withthe best fit to the 2MRS galaxy sample. The linear bias values of the respective samples are indicated in each panel. We use the bestfit (Equation 7) to the 2MRS redshift distribution to simulate the mock galaxy catalogues for unbiased and biased distributions fromN-body simulation of the ΛCDM model. MNRAS , 1–9 (2016) ias from anisotropy Figure 3. This shows the galactic coordinates in a mock 2MRS galaxy sample with different bias values as indicated in each panel. We use a Particle-Mesh (PM) N-body code to simulate thepresent day distributions of dark matter in the ΛCDM modelin a comoving volume of [921 . h − Mpc] . We use 256 parti-cles on a 512 mesh and the following cosmological parame-ters: Ω m = 0 . 31, Ω Λ0 = 0 . h = 0 . σ = 0 . 81 and n s =0 . 96 (Planck Collaboration et al. 2016) are used in the sim-ulation. In the current paradigm, the galaxies are believed toform at the peaks of the density field. We implement a sim-ple biasing scheme (Cole, Hatton & Weinberg 1998) wherethe galaxies are allowed to form only in those peaks wherethe overdensity exceeds a certain density threshold. One canvary the threshold in this sharp cut-off biasing scheme togenerate galaxy distributions with different bias values. Wedetermine the linear bias parameter b for these samples as, b = s ξ g ( r ) ξ m ( r ) (8)where ξ g ( r ) and ξ m ( r ) are the two-point correlation func-tions for the galaxy and dark matter distribution respec- tively. We generate the distributions for three different biasvalues b = 1 . b = 2 and b = 2 . z = z c ( γα ) α . Substitutingthe best fit values of the parameters A , γ , z c and α , wefind that the maximum probability of finding a galaxy inthe 2MRS sample is at z max = 0 . P max from Equation 7 usingthe values of z max and the best fit parameters. To simu-late the 2MRS mock catalogues from the N-body simula-tion and the biased distributions, we treat the particles asgalaxies and place an observer at the center of the box. Wemap the galaxies to redshift space using their peculiar ve-locities. We randomly choose a galaxy within the redshiftrange 0 ≤ z ≤ . 12 and calculate the probability of detect-ing this galaxy using Equation 7. We also randomly choosea probability value in the range 0 ≤ P ( z ) ≤ P max . If thecalculated probability is larger than the randomly selectedprobability then we retain the randomly selected galaxy inour sample. This process is repeated until we have 47 , MNRAS , 1–9 (2016) Pandey, B. Figure 4. The top left, top right and bottom left panel show that the radial anisotropies in the simulated galaxy samples can beobtained by scaling the radial anisotropies in the dark matter distribution by b where b is the linear bias parameter of the simulatedgalaxy sample. The bottom right panel shows that the radial anisotropies in the 2MRS galaxy samples is well represented by the radialanisotropies expected in a galaxy distribution in ΛCDM model with linear bias b = 1 . 3. The radial anisotropies for the simulated samplesshown in each panel are the mean anisotropies obtained from 30 mocks in each case. The 1 σ errorbars shown in each panel are obtainedfrom 30 mock catalogues for the N-body simulations and 30 jackknife samples for the 2MRS data. galactic coordinates for a mock 2MRS sample with differentbias values in Figure 3. In Figure 4 we compare the radial anisotropies a lb ( r ) inthe simulated mock biased galaxy samples with that fromthe mock samples from the dark matter distribution in theΛCDM model. The top left panel, top right panel and thebottom left panel of Figure 4 show the comparisons for lin-ear bias values b = 1 . b = 2 and b = 2 . b where b is the linear bias parameter of the simulatedgalaxy sample, reproduces the actual radial anisotropies ob-served in the respective galaxy samples on large scales. How-ever this scaling shows a large deviation on small scaleswhich gradually decreases and finally merges with the ob-served radial anisotropies in the biased samples beyond alength scale of 90 h − Mpc. We do not expect the linear bi- asing to hold on small scales. On small scales, the differ-ences result from the non-linearities present on those scalesdue to the gravitational clustering. The contributions fromthe higher order moments of the probability distribution inEquation 3 are not negligible on smaller scales and the biasvalues obtained by using Equation 6 are expected to deviatefrom its actual value. Eventually the assumption of linearbias may prevail on some larger scale and the Equation 6can faithfully recover the linear bias values only on a scalewhere the non-linearity becomes negligible. In the bottomright panel of Figure 4 we compare the radial anisotropies inthe 2MRS galaxy sample with that expected from the unbi-ased ΛCDM model. Interestingly, when we scale the radialanisotropies in the unbiased ΛCDM model by 1 . we findthat it nicely represents the radial anisotropies observed inthe 2MRS galaxy sample for nearly the entire length scalesbeyond 20 h − Mpc. This indicates that the non-linearity be-comes less important in the 2MRS galaxy sample beyond alength scale of 20 h − Mpc. It is also interesting to note thatthough the radial anisotropy in all the biased galaxy distri- MNRAS , 1–9 (2016) ias from anisotropy Figure 5. Same as Figure 4 but for polar anisotropies. butions decreases with increasing length scales, they reacha plateau at different length scales. We note that for b = 1, b = 1 . b = 2 and b = 2 . h − Mpc, 130 h − Mpc, 150 h − Mpc and 170 h − Mpc re-spectively. This indicates that the signatures of anisotropymay persist up to different length scales depending on thebias of the galaxy distribution.Our scheme maintains equal area for all the pixels byuniformly binning cos θ and φ . This causes the shapes ofthe pixels to vary across different parts of the sky. Thesevariations may contribute to the anisotropies measured inour scheme. To assess this we carry out some tests withHEALPix (G´orski et al. 1999, 2005) which uses equal areaand nearly same shape for all the pixels. We calculate theradial anisotropy in the same datasets using NSide= 8 inHEALPix which provides a total 768 pixels on the sky.It may be noted that we use m b = 20 and m l = 40 inour scheme which results into a total 800 pixels. We findthat HEALPix pixelization gives exactly the same radialanisotropy as measured in our scheme (Pandey 2017).We compare the polar and azimuthal anisotropies inthe biased and unbiased samples in the top left, top rightand bottom left panels of Figure 5 and Figure 6 respectively.We notice that a scaling similar to Figure 4 also applies here despite the fact that a smaller number galaxies areused to compute the anisotropies at each b and l . It maybe noted that the peaks and troughs in the polar and az-imuthal anisotropy curves for the simulated samples appearnearly at the same l and b values as the biased distributionsare produced from the same unbiased distribution. But if wecompare these results with that from a galaxy distribution,we do not expect this to happen as they represent two differ-ent statistical realizations of the density field. We find thatthe Equation 6 can be also used effectively with polar andazimuthal anisotropies to recover the linear bias parameterof the biased galaxy samples (Table 1). We consider the po-lar and azimuthal anisotropies estimated from 30 samples ineach case to measure the linear bias values using Equation 6respectively at each latitude ( b ) and longitude ( l ). We esti-mate the average linear bias values and their standard errorsby combining the bias measurements over different latitudesand longitudes and list them in Table 1.In Table 1 we see that the linear bias values recoveredfor the simulated galaxy samples are quite close to theiractual bias values. When we apply the same method toestimate the linear bias of the 2MRS galaxy sample, weget b = 1 . 31 from polar anisotropy and b = 1 . 29 fromazimuthal anisotropy. It is interesting to note that we get MNRAS , 1–9 (2016) Pandey, B. Figure 6. Same as Figure 4 but for azimuthal anisotropies. Table 1. This shows the linear bias values estimated from the polar and azimuthal anisotropies for the simulated samples and the 2MRSsample. We calculate the linear bias values using Equation 6 but with the average polar and azimuthal anisotropies measured from the30 samples in each case. We average the bias measurements from polar and azimuthal anisotropies over different latitudes and longitudesrespectively. The errors quoted with the bias values in the table are the standard errors.Sample Bias from a l ( b ) Bias from a b ( l )ΛCDM, b = 1 1 1ΛCDM, b = 1 . . ± . 026 1 . ± . b = 2 1 . ± . 051 1 . ± . b = 2 . . ± . 089 2 . ± . . ± . 067 1 . ± . nearly the same bias value b ∼ . particles from 10 spherical regions ofradius 200 h − Mpc from each of the biased and unbiaseddistributions and repeated the analysis. We find that onecan recover the linear bias of the simulated galaxy samplesfollowing the same method presented in this work. This sug- gests that the same method can be applied to determine thelinear bias parameter of the volume limited sample from dif-ferent galaxy surveys.It may be noted that the computation of the two-pointcorrelation function and power spectrum scales as O ( N )where N is the number of galaxies in the sample. So thecomputational requirements scales very fast with the size ofthe sample. Use of tree algorithms or FFT can reduce thisscaling to O ( N log N ) (Szapudi & Szalay 1998; Pen 2003;Szapudi et al. 2005). Interestingly the method presented inthis work requires a scaling of only O ( N ) and hence it is MNRAS , 1–9 (2016) ias from anisotropy computationally least expensive among all the other existingmethods for the determination of linear bias.We finally note that a combined study of the radial,polar and azimuthal anisotropies in the galaxy distributionprovides a powerful new alternative to measure the linearbias parameter from galaxy distributions. The author thanks an anonymous reviewer for the valuablecomments and suggestions. The author would like to thankthe 2MRS team for making the data public. The authoracknowledges financial support from the SERB, DST, Gov-ernment of India through the project EMR/2015/001037. Iwould also like to acknowledge IUCAA, Pune and CTS, IIT,Kharagpur for providing support through associateship andvisitors programme respectively. REFERENCES Alonso, D., Salvador, A. I., S´anchez, F. J., et al. 2015, MNRAS,449, 670Blake, C., & Wall, J. 2002, Nature, 416, 150Briggs, M. S., Paciesas, W. S., Pendleton, G. N., et al. 1996, ApJ,459, 40Cole, S., Hatton, S., Weinberg, D. H., & Frenk, C. S. 1998, MN-RAS, 300, 945Colles, M. et al.(for 2dFGRS team) 2001,MNRAS,328,1039Dekel A., Rees M. J. 1987,Nature,326,455Erdoˇgdu, P., Lahav, O., Huchra, J. P., et al. 2006, MNRAS, 373,45Erdoˇgdu, P., Huchra, J. P., Lahav, O., et al. 2006, MNRAS, 368,1515Feldman, H. A., Frieman, J. A., Fry, J. 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