Void alignment and density profile applied to measuring cosmological parameters
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Void alignment and density profile applied to measuringcosmological parameters
De-Chang Dai ⋆ Institute of Natural Sciences, Shanghai Key Lab for Particle Physics and Cosmology,and Center for Astrophysics and Astronomy, Department of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China
ABSTRACT
We study the orientation and density profiles of the cosmological voids with SDSS10data. Using voids to test Alcock-Paczynski effect has been proposed and tested inboth simulations and actual SDSS data. Previous observations imply that there existan empirical stretching factor which plays an important role in the voids’ orientation.Simulations indicate that this empirical stretching factor is caused by the void galax-ies’ peculiar velocities. Recently Hamaus et al. found that voids’ density profiles areuniversal and their average velocities satisfy linear theory very well. In this articlewe first confirm that the stretching effect exists using independent analysis. We thenapply the universal density profile to measure the cosmological parameters. We findthat the void density profile can be a tool to measure the cosmological parameters.
Key words: void, redshift distortion
Observations have shown that the universe is isotropic andhomogeneous at large scales. At the same time, the uni-verse is highly anisotropic at smaller scales and builds upa hierarchical structure of matter from galaxies to superclusters. While these dense structures are forming, it isunavoidable to create underdense regions which are calledvoids. Like the overdense regions, the voids are also highlyaffected by the evolution of the universe’s energy densitycomponents and therefore could be a powerful cosmologi-cal probe(Thompson & Gregory 2011). For example, theirshapes and sizes are sensitive to the nature of the darkenergy (Biswas et al. 2010; Bos et al. 2012; Jennings et al.2013; Li et al. 2012) and their internal dynamics may repre-sent the fifth force in the modified gravity theory(Cai et al.2015; Clampitt et al. 2013; Li & Zhao 2009).Unlike luminous overdense regions, the voids can notbe found directly, partly because they are generally faint(because of the lack of galaxies) or even completely dark.Thus, they are identified through mapping of the overdenseareas first and then by using void finders(Sutter et al. 2015)to pick up the voids. At large scales, the voids’ shapes arestatically isotropic and therefore stacking could make thevoids look like standard spheres. This is the basic require-ment to precede the Alcock-Paczynski test(Alcock et al.1979), and several such tests have been done (Sutter et al. ⋆ E-mail: [email protected] c (cid:13) RAS
De-Chang Dai
Table 1.
The void number in different sectionSample Name z min z max redshift comoving N void N void dr72dim1 0.0 0.05 110 102dr72dim2 0.05 0.1 186 184dr72bright1 0.1 0.15 186 188dr72bright2 0.15 0.2 112 96dr10lowz2 0.2 0.3 138 136dr10lowz3 0.3 0.4 215 198dr10lowz4 0.4 0.45 71 90dr10cmass1 0.45 0.5 227 229dr10cmass2 0.5 0.6 694 696dr10cmass3 0.6 0.7 386 × peculiar velocity effect. From the χ and likelihood analysischange with Ω Λ , we find that the density profile can serveas the cosmological probe, though the current void numbermight not be fully sufficient yet for very precise results and aprior function must be known to study the confidence level. Recently, several void catalogues were completed andreleased(Nadathur & Hotchkiss 2014; Sutter et al. 2012a).In this study, we use the voids in the Public Cosmic Voidcatalogue(Sutter et al. 2012a, 2014a,c), because this cata-logue includes the most recently results. This Catalogue putsthe voids and void galaxies in the following coordinates x = D c ( z ) cos(Dec) cos(Ra) x = D c ( z ) cos(Dec) sin(Ra) (1) x = D c ( z ) sin(Dec)Here z is redshift. D c ( z ) is the comoving distance fromthe center. Ra is Right ascension and Dec is Declination inthe equatorial coordinate system. Since the data set is within z <
1, the radial component can be ignored, and D c ( z ) iswritten as D c = cH Z z dz ′ E ( z ′ ) (2) E ( z ) = p Ω m (1 + z ) + Ω Λ (3)Here, we assume the universe is flat. Ω Λ is the darkenergy component and Ω m is the matter component. Ω Λ +Ω m = 1. The catalogue includes two sub-catalogues (co-moving and redshift) according to the coordinate in whichit is searched for the voids. Voids in comoving catalogueare found out through assuming dark matter componentΩ Λ = 0 .
73 and Voids in redshift catalogue is found outthrough assuming dark matter component Ω Λ = 1 . . Thevoid number and redshift range in each section are listed intable 1. We use only the ”central” voids which are selectedto avoid the boundary or mask effect(Sutter et al. 2012b,2014a,c). The void alignment depends on three sources: initial con-ditions, void packing and tidal forces(Platen et al. 2008).These three sources should be local effects and should notcontribute much at large scales. Therefore the distributionof the orientation of the cosmological voids should be a ran-dom distribution. Based on this, it was suggested that thevoid orientation with respect to the line of sight could bea measure of the redshift distortion(Foster & Nelson 2009;Ryden & Melott 1996).If void orientation is random, the angle, θ , between itsorientation direction and line-of-sight satisfies the distribu-tion P ( θ ) dθ = sin θdθ (4)or P (cos θ ) d cos θ = d cos θ (5)To examine this assumption, we calculate the orienta-tion of voids through ellipsoid approximation. We computethe location of the void’s center through the volume weightas ~x c = P ~x i V i P V i (6)Here, ~x i is the position of the void’s i-th galaxy’s and V i is its Voronoi cell’s volume. We then compute the shapetensor, S ij , S ij = X k ( x ki − x ci )( x kj − x cj ) m k (7) k is the void galaxy’s index. i and j are the three po-sition’s indexes. m k is the mass weight. Here we choose m k = 1 for all the cases. The three eigenvectors arethe orientation of the ellipsoid’s axes. The three eigen-values are their axes’ length square. We calculate theorientations under Ω Λ = 0 .
73 to compare with recentWMAP and Planck’s measurement(Komatsu et al. 2011;Planck Collaboration et al. 2014). We use voids in the co-moving catalogue and show the result in figure 1. At firstglimpse, one finds that the long axes have values larger than1 for small cos θ . That means the long axes are anti-alignedto the line-of-sight. It is not very clear whether median andshort axes have any preferred direction without a detailedanalysis.There are at least two reasons to this anti-alignmentphenomenon. One is Alcock-Paczynski effect(Alcock et al.1979) and the other is peculiar velocity. We will discuss themin the next section. It is known that Alcock-Paczynski effect could cause redshiftdistortion. At the same time it also distorts the orientation ofa void. For our purpose, to study the Alcock-Paczynski effecton the orientation, we can’t approximate the void shape witha sphere, because of the voids’ irregular shapes and theirtracers’ peculiar velocities. To simplify the argument, we c (cid:13) RAS, MNRAS , 1–8 oid alignment and density profile applied to measuring cosmological parameters Long axis Median axis Short axis dn / d c o s () cos( ) Figure 1.
The three axes’ orientation distribution. We set Ω Λ =0 .
73 for the figure. We use voids in the comoving catalogue inthe 9 sections. The orientation of the long axes are slightly anti-aligned to the line-of-sight. The bin size of cos θ is 1/30. consider the void to be a one dimensional rod. The rods’directions are the orientation of long axes. The real situationis more complicated, but our approximation can at least tellus what is going on when considering the Alcock-Paczyskieffect on the void’s orientation.Let’s consider a one dimensional void. Its angular extentis δθ v and redshift extent is δz v . In the comoving coordinatesystem, δr c = δθ v D c (8) δd c = δz v ∂ z D c (9) δd c void’s comoving size in line-of-sight direction and δr c is the void comoving size in angular direction. Therefore,the angle between the void’s long axis and line-of-sight istan θ c = D c ∂ z D c δθ v δz v (10) θ c follows a random distribution if D c is the real co-moving distance. If one uses the other factor, D n , ( D n = czH in general discussion), one will have mistaken the angle be-tween the void’s orientation and line-of-sight astan θ n = D n ∂ z D n δθ v δz v (11)In the measurement, δz v and δθ v are measured. How-ever, the comoving distance, D c , is not known directly. Onemay use D n instead of D c to study the voids’ orientation.The relation between θ n and θ c istan θ n = f ( z ) tan θ c (12) f ( z ) = ∂ z ln D c ∂ z ln D n (13)The distribution P ( θ n ) = P ( θ c ) dθ c dθ n (14)In the actual isotropic coordinate, the void’s long axis’sare random distributions and therefore P ( θ c ) = sin θ c . Fromequation 12, one finds dn / d c o s () cos( ) f=1.1 f=1 f=0.9 Figure 2. cos( θ n ) distribution. It aligns to the line of sight as f > P ( θ n ) = sin θ n f (cid:16) cos θ n + sin θ n f (cid:17) − (15)This is the angular distribution function for a coordi-nate different from the comoving coordinate system. Fromfigure 2, one finds it is anti-aligned to the line of sight as f > f < Λ from 0 . .
0. Figure 3shows that the voids’ orientation changes from anti-aligningto aligning to the line-of-sight as we go from Ω Λ = 0 toΩ Λ = 1 . Based on this, one expects that one could find thedark energy and dark matter component (Ω Λ and Ω m ) bysearching for P ( θ n ) which best satisfies a random distribu-tion. For a pure random distribution P ( θ ), < cos θ > = 0 . < cos θ > tocheck whether the distribution follows a random distri-bution. Figure 4 shows how < cos θ > changes with re-spect to Ω Λ . The figure shows that Ω Λ = 0 . ± .
02 in1 σ interval. However, we must point out that an empiri-cal stretching factor(Sutter et al. 2014c; Lavaux & Wandelt2012), which may be caused by the peculiar velocity, mustbe included. The same effect has been noticed in the SDSSDR5(Foster & Nelson 2009). They found that the averageorientation angle increases to higher θ values as z from 0 . .
16. If the empirical stretching factor does exist, then1 /f = 1 . ± .
04 in a comoving coordinate(Sutter et al.2014c; Lavaux & Wandelt 2012) and it will not be isotropiceven in the correct comoving coordinate. We can test the em-pirical stretch factor by calculating the average < cos θ > . < cos θ > = Z π/ P ( θ ) cos θdθ (16)For 1 /f = 1 . < cos θ > = 0 .
46. We plot all < cos θ > in Ω Λ = 0 .
73 case. Figures 5 shows that most of < cos θ > are less than 0 .
5, which is the value of a pure random distri-bution. This confirms the existence of the extra stretchingfactor. This extra stretch is from the void galaxies’ pecu-liar velocities(Sutter et al. 2014c; Lavaux & Wandelt 2012).However, the redshift type of a void and comoving type of avoid do not give exactly the same result. This confirms thatthe void galaxies’ quantities highly depend on the coordi- c (cid:13) RAS, MNRAS , 1–8
De-Chang Dai =0.0 =0.5 =1.0 dn / d c o s () cos( ) Figure 3.
The probability function changes with different Ω Λ .Here we use dr10cmass3 dataset. < c o s () > Figure 4. < cos θ > in different Ω Λ . The black line is the ex-pected < cos θ > in the isotropic case. The vertical dash lines arethe 1 σ interval of possible Ω Λ . The data point are redshift typesin dr10cmass3 section. nate which is used in the void finder(Nadathur & Hotchkiss2014). The previous section has confirmed that the voids’ orienta-tion depends not only on redshift effect, but also on an em-pirical factor. According to simulations, the empirical factoris caused by the galaxies’s peculiar velocities(Sutter et al.2014c; Lavaux & Wandelt 2012). In general, it is very dif-ficult to study peculiar velocity without any other assump-tion. However, it has been found that voids’ density profileis universal (Hamaus et al. 2014; Lavaux & Wandelt 2012;Colberg et al. 2005; Padilla et al. 2005; Ricciardelli et al.2013, 2014) and its average velocity fit the linear theoryvery well(Hamaus et al. 2014; Paz et al. 2013). Therefore,we could study the peculiar velocity from this universal den-sity profile.Here we adopt the density profile form from paper redshift comoving < c o s () > z Figure 5. < cos θ > in different catalogue section. The black oneis from the redshift type of the catalogue, and the red one is fromthe comoving type of the catalogue. Ω Λ = 0 .
73 in the calculation.The black line represents the random distribution result and thedash line represents a stretch factor 1 /f = 1 . (Hamaus et al. 2014). ρ v ( r )¯ ρ = 1 + δ c − ( r/r s ) α r/r v ) β (17) v v = −
13 Ω γm Hr ∆( r ) (18)∆( r ) = 3 r Z r (cid:16) ρ v ( q )¯ ρ − (cid:17) q dq (19)Here, γ = 0 . r = p x + x + x . v v is velocityfrom the linear theory(Peebles 1976). It has been foundthat α and β at z = 0 are related to r s /r v in the follow-ing form(Hamaus et al. 2014). α = − r s /r v −
2) (20) β = (cid:26) . r s /r v − . r s /r v < . − . r s /r v + 18 . r s /r v > .
91 (21)However, we study voids in z = 0 case and thereforewe should treat them as two free parameters. We hope that α and β dependence on the redshift could be found in thefuture studies. So far, x can not be obtained directly fromredshift, and one must consider the distortion effect fromthe galaxies’ peculiar velocities. Therefore, x = Z − v/H (22) Z = cH δz (23)where v is the galaxy’s velocity and δz is the redshift differ-ent between galaxy and void center. Since the galaxy’s ve-locity is involved in the measurement, one needs a velocitymarginal function to find x . The simplest marginal functionwill be a Gaussian distribution with the center at v v . How-ever, since we rescale the void according to its effect radius,( R v = ( V π ) / , V is the void’s volume) while stacking voids,the velocity will be rescaled too. Therefore whether Gaus- c (cid:13) RAS, MNRAS , 1–8 oid alignment and density profile applied to measuring cosmological parameters sian distribution is a velocity distribution function is un-clear. We hope future N-body simulation could also providethe form of marginal function. Right now we use Gaussiandistribution. f g ( η ) = 1 √ πv exp( − η v ) (24) v is the root mean square of the velocity. Exponentialdistribution is also a popular distribution in two-body cor-relation. We had tested it and found the result is similarto Gaussian distribution one, therefore we are going to giveonly Gaussian distribution’s result. Of course v should bea r dependent function. Finding its actual form will rely onthe future studies, and here we will treat it as a constant.The average density in ( x , x , Z ) is ρ o ( x , x , Z ; Ω Λ ) dZ = Z ρ v ( x , x , x )¯ ρ f ( w − x r v v ) δ ( Z − x − x r v v H ) dx dw (25)where δ ( η ) is a Dirac Delta function. Ω Λ appears in theformula, because Hubble constant and position are functionsof Ω Λ . Different Ω Λ s give different mass density. We firstrescale the voids given in comoving coordinates according totheir effect radius ( R v ) and then stack them together. Weexpect a single galaxy’s peculiar velocity could be severalhundred to thousand km/s . This will give several to dozensMpc of distortion in line-of-sight ( Z ) direction. Therefore wefocus only on voids bigger than 10Mpc. Low redshift voidseither do not satisfy or the void number is very low (lessthan one hundred). Considering the void number, we choosedr10mass2 to study the effect. We use voids in comovingcoordinates to minimize the void finder’s selection effect.In order to find the density profile, one needs the av-erage galaxy density. This however is not easy, because theaverage galaxy density depends on survey’s capability andis redshift and location dependent. Figure 6 shows how thegalaxy density changes with respect to redshift in the co-moving space. It is clear the galaxy density highly dependson the redshift. At this moment we treat this density as aver-age density (¯ ρ m ( z )) at each redshift(Nadathur & Hotchkiss2014; Nadathur et al. 2014). We then stack the voids accord-ing to void radius, R v . The density profile is calculated inthe following way¯ ρ a ( Z, d v ; Ω Λ ) = 1 N X i ρ i ( Z, d v ; Ω Λ )¯ ρ m ( z i ) (26) ρ i ( Z, d v ; Ω Λ ) is the galaxy density of i th void at ( Z, d v )respected to Ω Λ . Here z i is the redshift of the bin. d v = p x + x and N is the number of voids in the stacking.The error of each bin is calculated from σ ( Z, d v ; Ω Λ ) = 1 N ( N − X i ( ρ i ( Z, d v ; Ω Λ )¯ ρ m ( z i ) − ¯ ρ a ) (27)Figure 7 shows the stacking density profile accordingto different void size. We stack 175 voids for R v > > R v > > R v > × -4 -4 -4 -4 -4 m z Figure 6.
The galaxy density according with respect to redshift:The density is calculated according to Ω Λ = 0 .
73. The densitydepends highly on redshift and can not be treated as a constant.The density unit is Mpc − v <30 30 The voids are separated into three different radiusrange (20Mpc < R v < < R v < < R v ) stacking according to their radius. We rescale allthe voids to 30Mpc and stack them together. The bin size is3Mpc × Z, d v ) coordinate. different void has different volume and there are differentmasked regions to avoid. The smaller voids’ profile has ahigher density right outside the voids’ radius. Apparentlythe density profile depends on the voids’ sizes. Therefore itis unlikely to stack each void without considering its radius.At the following we study void density profile according tothese three stacking void profiles. Also we choose δ c = − ρ a ( Z, d v )) and error( σ ( Z, d v )). Near R v = 40Mpc, there is a point with muchlower ¯ ρ a ( Z, d v ) and its σ ( Z, d v ) is much less than its neigh-borhood. This shows that if the average density (¯ ρ a ( Z, d v ))is much smaller than average galaxy density, then σ ( Z, d v ) c (cid:13) RAS, MNRAS , 1–8 De-Chang Dai v <30 a R Figure 8. The density distribution with error bar: The stackingvoids’ effect radius are from 20Mpc to 30Mpc. Ω Λ = 0 . 5. Since thelimitation of dataset, sometimes the error bar is underestimated.In this case, a point near R v = 40Mpc is much smaller than itsneighborhood. This causes that its error bar is underestimated. 20 40 60204060 Z d v Figure 9. The dr10mass2 density distribution: we stack 400voids. The voids’ effect radius are from 30Mpc to 40Mpc. Thebin size is 3 × in ( Z, d v ) coordinate. Here Ω Λ = 0 . 73. Wechoose R e = 30Mpc in the figure. is underestimated. If these underestimated points are in-cluded in the calculation the increase in χ will be signif-icant. Since these abnormal points appear at small d v andare caused by the limitation of the dataset, we therefore re-move d v < σ ( Z, d v ).Figure 9 shows the dr10mass2’s average density profilein Z − d v plane. (Here we show Ω λ = 0 . 73) One finds exten-sion in Z-direction at small radius( ∼ ∼ 20 40 60204060 Z d v Figure 10. The density profile by fitting empirical density func-tion. The other parameters are the same as figure 9. Therefore we include only σ ( Z, d v ; Ω Λ ) to calculate χ . χ is calculated according to χ (Ω Λ , α, β, v , r s ) = X Z,d v (¯ ρ a ( Z, d v ) − ρ o ( Z, d v )) σ ( Z, d v ; Ω Λ ) (28)We use data between 10Mpc to 60Mpc to calculate χ .Figure 10 is the fit of the density profile from the empiricaldensity profile. The result shows similar density stretching,extension and distortion as in figure 9.One of our goals is to find the cosmological dark energycomponent, Ω Λ . Figure 11 shows the minimum χ changeswith respect to Ω Λ (The other parameters are not fixed). Wefit 291 points with 5 parameters. Therefore a reasonable χ value is ∼ ± √ χ at small Ω Λ is farther away fromthis region and can be ruled out. In 30Mpc < R v < < R v case, the χ minimums is still a little toolarge and their best fit parameters are not reasonable ( v lessthan 100 km/s which is unreasonable because the bin size,3Mpc, at least induces 300 km/s uncertainty on velocities).This implies a further study on the form of universal densityprofile is needed. At the same time, we assume all the pointsare independent and the likelihood is L (Ω Λ , α, β, v , r s ) = Y i πσ i ) / exp( − χ / 2) (29)The probability for a particular Ω Λ is calculated accord-ing to the following marginalization P (Ω Λ ) = Z L (Ω Λ , α, β, v , r s ) M (Ω Λ , α, β, v , r s ) dαdβdv dr s (30)Here M is a prior probability function. It depends onseveral parameters, but in general M is assumed to be aconstant. Figure 12 shows the likelihood ratio under this as-sumption. Again the small Ω Λ s are ruled out immediately.But we must emphasize that the minimum χ s appearing in30Mpc < R v < < R v cases are not rea-sonable and therefore M cannot be taken as only a constant.A further study of the prior probability is needed. c (cid:13) RAS, MNRAS , 1–8 oid alignment and density profile applied to measuring cosmological parameters v <30 30 The χ vs. Ω Λ . Here α , β , v and r s are not fixed. v <30 v <40 v l n ( P / P ) Figure 12. The probability ration vs. Ω Λ . We marginalize α , β , v and r s parameters, and assume the prior probability functionis a constant. P is the maximum probability. In this paper, we studied the properties (orientation anddensity profiles) of the cosmological voids using the SDSS10data. We first confirmed that the voids’ orientation is notpurely random as one might naively expect. This effecthas been first noticed by Foster etc(Foster & Nelson 2009).Later, Sutter etc. have found a constant empirical stretch-ing factor using numerical simulations, and applied thestretching factor to SDSS DR10 data(Sutter et al. 2014c;Lavaux & Wandelt 2012). They found that the voids’ ori-entation can be used as a dark energy probe. Along theselines, a constant stretching factor will make the study ofAlcick-Paczynski effect relatively straightforward. However,as illustrated in figure 4, we showed that the stretching fac-tor highly depends on the coordinate which is used in thevoid finder(Nadathur & Hotchkiss 2014). This is not surpris-ing, because different void finders give different void galax-ies. Because of this discrepancy, it is very difficult to studythe Alcick-Paczynski effect directly through the void orien-tation. To reduce this discrepancy, we included the galaxiesoutside voids into our analysis. Since the stretching effect is caused by the peculiar velocity, it is more realistic to apply amodel which can quantify the peculiar velocity effect. It hasbeen pointed out by Hamaus etal. that voids’ density profilesare universal both inside and outside voids and their aver-age velocities satisfy linear models very well(Hamaus et al.2014). Therefore we apply Hamaus’s density profile to thevoids density profile in SDSS 10DR.Although it is ultimately better to stack all the avail-able data at once, figure 7 shows that the density profilealso depends on the void’s effective radius. This reduces theamount of voids that can be stacked together at once. Inour study we separates the voids into three different effec-tive radius intervals and fit the void density profiles with theempirical density distribution form. The result is consistentwith the χ test and it implies that the empirical densityprofile form we obtained is reasonable. However, this is notenough to apply the result to calculate the confidence leveldirectly, since the prior functions must be known. Unfortu-nately, in this case assuming a constant prior function is notsuitable. It is unclear if this is caused by the limitation ifthe voids’ number or by the voids’ density profiles. This in-dicate that further analysis is warranted in order to betterunderstand the voids’ statistics and associated phenomena. ACKNOWLEDGMENTS D.C Dai was supported by the National Science Founda-tion of China (Grant No. 11433001), National Basic Re-search Program of China (973 Program 2015CB857001),No.14ZR1423200 from the Office of Science and Technologyin Shanghai Municipal Government and the key laboratorygrant from the Office of Science and Technology in ShanghaiMunicipal Government (No. 11DZ2260700). REFERENCES Alcock C. and Paczynski B. 1979, nat, 281, 358Biswas, R., Alizadeh, E., & Wandelt, B. D. 2010, prd, 82,023002Bos, E. G. 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