Investigation of Homogeneity and Matter Distribution on Large Scales
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Eur. Phys. J. C manuscript No. (will be inserted by the editor) Investigation of homogeneity and matter distribution on large scales using large quasar groups Ming-Hua Lia,1, Zhi-Bing Lib 1School of Physics and Engineering, Sun Yat-Sen University, Guangzhou 510275, China Received: date / Revised version: date Abstract We use 12 large quasar group (LQG) samples to as well as the results and observations of other groups are investigate the homogeneity of 0.5 . z . 2 Universe (z de- presented. notes the redshift). We calculate the bias factor b and the PACS 98.54.-h Quasars 98.65.Dx Large-scale structure two-point correlation function ξ for such groupsfor three LQG of the Universe 98.80.Es· Observational cosmology different density profiles of the LQG dark matter halos, i.e. · the isothermal profile, the Navarro-Frenk-White(NFW) pro- file, and the (gravitational) lensing profile. We consider the 1 Introduction ΛCDM concordancemodel of our Universe with Ωm = 0.28, Ω 1 1 Λ = 0.72, the Hubble constant H0 = 100h km s− Mpc− Large galaxy redshift surveys have long been carried out with h = 0.72 in our calculations. Dividing the samples into to record the angular positions and redshifts of the galax- three redshift bins, we find that the LQGs with higher red- ies. Its primary aim is to measure the matter distribution of shift are more biased and correlated than those with lower our Universe and to determine possible structures on large redshift. The redshift-increasingLQG correlation amplitudes scales. Since the discovery of CfA2 Great Wall of galaxies we find is incompatible with that predicted by the standard [1], several large scale structures (LSS) have been reported theory of structure growth. The homogeneity scale RH of in the past two decades. The Sloan Great Wall (SGW) [2] the LQG distribution is also deduced. It is defined as the identified in the Sloan Digital Sky Survey (SDSS; [3]) is a comoving radius of the sphere inside which the number of filamentary structure of galaxies, of which the characteristic 3 LQGs N(< r) is proportional to r within 1%, or equiva- size (volume1/3) is about 128 Mpc (unless otherwise stated, lently above which the correlation dimension of the sam- all scales presented in this paper are comoving). The U1.11 ple D2 is within 1% of D2 = 3. For the NFW dark matter [4] and the U1.28 (or CCLQG) [5] are both connected units halo profile, the homogeneity scales of the LQG distribution of quasars, respectively spans about 380 and 350 Mpc. 1 1 are RH 224 h− Mpc for 0.5 < z 1, RH 316 h− Mpc According to the standard structure formation theory, all ≃ ≤1 ≃ for 1 < z 1.5, and RH 390 h− Mpc for 1.5 < z . 2. the structures like galaxies and clusters in our Universe to- ≤ ≃ These values are above the characteristic sizes of the LQG day have their origins in the primordial Gaussian random arXiv:1412.2998v1 [astro-ph.CO] 9 Dec 2014 samples in each bin, implying the validity of the cosmo- phase density fluctuations that was generated by the cos- logical principle on the LQG scale, i.e. a length range of mic inflation. The fluctuations of (dark) matter density, once 1 14 200 400 h− Mpc and a mass scale of 10 M . The entered the horizon, were amplified by gravity and eventu- ∼ ∼ ⊙ possibilities of a top-down structure formation process as ally turned into the rich structures we see today. The valid- was predicted by the hot/warm dark matter (WDM) scenar- ity of this theory has well been supported by a number of ios and the redshift evolution of bias factor b and correlation observations and N-body simulations. One of the inevitable ξ amplitude LQG of the LQGs as a consequence of the cos- outcomes of this structure formation scenario is the ‘hierar- mic expansion are both discussed. Theoretical implications chical clustering’, i.e small systems collapsed first at high redshift in the early Universe (since the wave lengths of the corresponding density fluctuation modes are short and thus ae-mail: [email protected] entered the horizon at a time earlier than those of long wave be-mail: [email protected] lengths), and then grew and merged together to form larger 2 structures like clusters and halos. At the top of this hierarchi- In this paper, we extend the investigation of large-scale cal pyramid are the superclusters or the filaments of galax- homogeneity to the scale of LQGs, using the 12 LQGs sam- ies/clusters. Combining with the fact that the Universe has ples in [24]. These groups were identified to have reliable a definite age (as 13.8 billion years in the ΛCDM con- redshifts within the range of 0.5 . z . 2 and thus provide ≃ cordance model), one can assume that the structures in our good candidates to explore the mass distribution in the ear- Universe today have a finite size. Yadav et al. [6] gaveanup- lier Universe. The fact that quasars being biased tracers of 1 per limit of this size as 260 h− Mpc (the Hubble constant the mass distribution is implemented in that the correlation ∼ 1 1 is given as H0 = 100h km s− Mpc− ). They called this the amplitude of quasars is related to that of the underlying dark 2 ‘scale of homogeneity’, above which the fractal dimension matter by a bias factor b, i.e. ξqua = b ξmass. We calculate of the galaxy or matter distribution is equal to the ambient this factor for the LQGs and study its behavior as a func- 1 dimension of the space . tion of z and the halo mass MDMH for three different density The scale of homogeneityof matter distribution has been profiles of the LQG dark matter halos, i.e. the isothermal studied for more than one decade. The results are quite scat- profile, the NFW profile, and the lensing profile. We con- tered. Hogg et al. [7] once investigated the enormous Lu- sider the ΛCDM concordance model of our Universe with minous Red Galaxy sample [8] of SDSS [9] and presented a Ωm = 0.28, ΩΛ = 0.72, the Hubble constant H0 = 100h km 1 1 1 homogeneity scale as RH 70 h Mpc. This result was later s− Mpc− with h = 0.72 in our calculations. The corre- ∼ − supported by Sarkar et al. [10] and Scrimgeour et al. [11], lation function ξ of the LQGs and the homogeneity scale who carried out a multifractal analysis of the distribution RH are also estimated from the theory. It is based on the of over 200,000 blue galaxies in the WiggleZ survey [12]. ‘counts-in-spheres’ measurement N(< r), i.e. the number Contrary to these coincidences, Labini et al. [13] claimed to of LQGs inside a sphere of radius r centered at each LQG, 1 find a homogeneity scale above 100 h− Mpc. Homogeneity which is averaged over the whole sample. For an ideal ho- on large scales is one of the cornerstones of modern cos- mogeneous distribution of the LQGs, this number scales as ∝ 3 ρ 4πr3 ρ mological theory. It assumes that over some large smooth- r (for N(< r)= ¯ 3 , where ¯ is now a universal con- ing scale, the distribution of matter is Ôstatistically homoge- stant). The homogeneity scale RH is defined to be the radius neousÕ. Small-scale inhomogeneities occur as perturbations above which this scaling behavior holds with accuracy of of the density field with an isotropic Gaussian distribution. 99%. This definition of RH together with the relevant fractal The large-scale homogeneity in the very early Universe (at analysis were widely used in the studies of large-scale ho- redshift z 1100) was well supported by the high degree of mogeneity in the galaxy surveys [6, 7, 10, 11, 13] andthusis ≃ isotropy of the cosmic microwave background (CMB) radia- used in our investigation of LQGs. The results we obtained tion power spectrum [14] and PLANCK’s primary data [15]. are compared with those of others and possible explanations But that does not guarantee the homogeneity of the present are given. Universe. Recently, large quasar groups (LQG) that exceed the up- The rest of the paper is organized as follows. In Section 1 per limit of homogeneity scale 260 h− Mpc presented by 2, we justify the use of the LQG sample in our study. In ∼ Yadav et al. [6] were reported discovered [16, 17]. Quasars, Section 3, we calculate the correlation function ξ (r) and the like galaxies and galaxy clusters, are biased tracers of the bias factor b of the LQG samples from the theory. In Sec- mass distribution [18]. Unlike galaxies which were mostly tion 3.1, we introduce three different models of bias b for formed at z < 1, most quasars have redshifts z > 1 and thus quasars. In Section 3.2, the total halo masses of the LQGs are suitable candidates to probe the matter distribution of are estimated respectively in three different density profiles the high-redshiftUniverse. The clustering of quasars and the of dark matter. In Section 3.3, we give the calculation results underlying dark matter halos have long been studied [19– of ξ (r) and b. The redshift-evolution of them are discussed. 23]. Unlike the fruitful results about quasars, the clustering The results are plotted. Section 4 are dedicated to study the of LQGs are rarely discussed in the references. That is be- homogeneity of the LQG distribution. In Section 4.1, we cause the discovery of a LQG and the determination of their briefly review the definition of fractal dimension and its re- spatial distribution are highly constrained by the insufficient lation with the homogeneity study.