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A&A manuscript no. (will be inserted by hand later) Your thesaurus codes are: 02 (12.03.4; 12.04.1; 12.12.1; 11.07.1; 11.17.3)

ASTRONOMY AND ASTROPHYSICS

Two-point correlation functions on the light cone: testing theoretical predictions against N-body simulations

Takashi Hamana1 , St? ephane Colombi1,2, and Yasushi Suto3,4

1

arXiv:astro-ph/0010287v2 4 Dec 2000

2 3

4

Institut d’Astrophysique de Paris, CNRS, 98bis Boulevard Arago, F 75014 Paris, France email: hamana@iap.fr, colombi@iap.fr NIC (Numerical Investigations in Cosmology) Group, CNRS Department of Physics, University of Tokyo, Tokyo 113-0033, Japan email: suto@phys.s.u-tokyo.ac.jp Research Center for the Early Universe (RESCEU), School of Science, University of Tokyo, Tokyo 113-0033, Japan

Received July 3, 2000 ; accepted 30 November, 2000

Abstract. We examine the light-cone e?ect on the twopoint correlation functions using numerical simulations for the ?rst time. Speci?cally, we generate several sets of dark matter particle distributions on the light-cone up to z = 0.4 and z = 2 over the ?eld-of-view of π degree2 from cosmological N-body simulations. Then we apply the selection function to the dark matter distribution according to the galaxy and QSO luminosity functions. Finally we compute the two-point correlation functions on the lightcone both in real and in redshift spaces using the paircount estimator and compare with the theoretical predictions. We ?nd that the previous theoretical modeling for nonlinear gravitational evolution, linear and nonlinear redshift-distortion, and the light-cone e?ect including the selection function is in good agreement with our numerical results, and thus is an accurate and reliable description of the clustering in the universe on the light-cone. Key words: cosmology: theory – dark matter – largescale structure of universe – galaxies: general – quasars: general

in the redshift space, and presented various predictions in canonical cold dark matter (CDM) universes. Their predictions, however, have not yet been tested quantitatively, for instance, against numerical simulations. This is not surprising since it is fairly a demanding task to construct a reliable sample extending over the light-cone from the conventional simulation outputs at a speci?ed redshift, z . In the present paper, we examine, for the ?rst time, the validity and limitation of the above theoretical framework to describe the cosmological light-cone e?ect against the mock catalogues on the light-cone. Such catalogues from cosmological N -body simulations have been originally constructed for the study of the weak lensing statistics (Hamana et al. 2000, in preparation). Applying the same technique (§3.1), we generate a number of di?erent realizations for the light-cone samples up to z = 0.4 and z = 2, evaluate the two-point correlation functions directly, and compare with the theoretical predictions. 2. Predictions of two-point correlation functions on the light cone In order to predict quantitatively the two-point statistics of objects on the light cone, one must take account of (i) nonlinear gravitational evolution, (ii) linear redshiftspace distortion, (iii) nonlinear redshift-space distortion, (iv) weighted averaging over the light-cone, (v) cosmological redshift-space distortion due to the geometry of the universe, and (vi) object-dependent clustering bias. The e?ect (v) comes from our ignorance of the correct cosmological parameters, and (vi) is rather sensitive to the objects which one has in mind. Thus the latter two e?ects will be discussed in a separate paper, and we focus on the e?ects of (i) ? (iv) throughout the present paper. Nonlinear gravitational evolution of mass density ?uctuations is now well understood, at least for two-point statistics. In practice, we adopt an accurate ?tting for-

1. Introduction In the proper understanding of on-going redshift surveys of galaxies and quasars, in particular the Two-degree Field (2dF) and the Sloan Digital Sky Survey (SDSS), it is essential to establish a theory of cosmological statistics on the light cone. This project has been undertaken in a series of our previous papers (Matsubara, Suto, & Szapudi 1997; Yamamoto & Suto 1999; Nishioka & Yamamoto 1999; Suto et al. 1999; Yamamoto, Nishioka, & Suto 2000; Suto, Magira & Yamamoto 2000). Those papers have formulated the light-cone statistics in a rigorous manner, described approximations to model the clustering evolution

Send o?print requests to : T. Hamana

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Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

mula (Peacock & Dodds 1996) for the nonlinear power R spectrum Pnl (k, z ) in terms of its linear counterpart. Then the nonlinear power spectrum in redshift space is given as

S R Pnl (k, ?) = Pnl (k, z )[1 + β?2 ]2 Dvel [k?σP ],

(1)

where k is the comoving wavenumber, and ? is the direction cosine in k -space. The second factor in the righthand-side comes from the linear redshift-space distortion (Kaiser 1987), and the last factor is a phenomenological correction for non-linear velocity e?ect. In the above, we introduce β (z ) ≡ 1 d ln D(z ) , b(z ) d ln a (2)

and ΛCDM, whose parameters are summarized in Table 1, to test the accuracy of the ?tting formula of the pairwise velocity dispersion, eq. (9). The measured velocity dispersions at z = 0, 1 and 2 are shown in Figure 1. The dotted lines in Figure 1 indicate predictions of eq. (9) integrated over the wavenumbers existing in our N -body simulations. The analytical model predictions agree with our data within a 10% accuracy at the large separations. This level of agreement is as good as that found originally by Mo et al. (1997). Nevertheless since we are mainly interested in the scales around 1h?1 Mpc, we adopt the following ?tting formula throughout the analysis below which better approximates the small-scale dispersions in physical units: σP (z ) ? 740(1 + z )?1 km/s for SCDM model 650(1 + z )?0.8 km/s for ΛCDM model. (10)

where D(z ) is the gravitational growth rate of the linear density ?uctuations, a is the cosmic scale factor, and the density parameter, the cosmological constant, and the Hubble parameter at redshift z are related to their present values respectively as ?(z ) = λ(z ) = H0 H (z ) H0 H (z )

2

Integrating equation (1) over ?, one obtains the direction-averaged power spectrum in redshift space:

S 2 Pnl (k, z ) 1 = A(κ) + β (z )B (κ) + β 2 (z )C (κ) R 3 5 Pnl (k, z )

(1 + z )3 ?0 ,

2

(11)

(3) (4)

where λ0 , arctan(κ) , κ arctan(κ) 3 , B (κ) = 2 1 ? κ κ 3 3 arctan(κ) 5 1? 2 + . C (κ) = 2 3κ κ κ3 A(κ) = (12) (13) (14)

H (z ) = H0

?0 (1 + z )3 + (1 ? ?0 ? λ0 )(1 + z )2 + λ0 . (5)

We assume that the pair-wise velocity distribution in real space is approximated by √ 1 2|v12 | exp ? , (6) fv (v12 ) = √ σP 2σP with σP being the 1-dimensional pair-wise peculiar velocity dispersion. In this case the damping term in Fourier space, Dvel [k?σP ], is given by Dvel [k?σP ] = where κ(z ) = k (1 + z )σP (z ) √ . 2H (z ) (8) 1 , 1 + κ2 ?2 (7)

Adopting those approximations, the directionaveraged correlation functions on the light-cone are ?nally computed as dVc [φ(z )n0 (z )]2 ξ (xs ; z ) dz zmin , zmax dVc dz [φ(z )n0 (z )]2 dz zmin dz

zmax

ξ LC (xs ) =

(15)

where zmin and zmax denote the redshift range of the survey, and ξ (xs ; z ) ≡ 1 2π 2

∞ 0 S Pnl (k, z )

Note that this expression is equivalent to that in Magira et al. (2000) but written in terms of the physical velocity units. On large scales, σP (z ) can be well approximated by a ?tting formula proposed by Mo, Jing & B¨ orner (1997): σP,MJB (z )

2

sin kxs 2 k dk. kxs

(16)

Throughout the present analysis, we assume a standard Robertson – Walker metric of the form: ds2 = ?dt2 + a(t)2 {dχ2 + SK (χ)2 [dθ2 + sin2 θdφ2 ]}, (17) where SK (χ) is determined by the sign of the curvature K as ? √ √ (K > 0) ? sin ( Kχ)/ K SK (χ) = χ (18) ( K = 0) √ √ ? sinh ( ?Kχ)/ ?K (K < 0)

≡

∞ 1+z ?(z )H 2 (z ) D2 (z ′ ) 1? 2 dz ′ 2 (1 + z ) D (z ) z (1 + z ′ )2 ∞ dk ?2 NL (k, z ) . (9) × k k2 0

We compute the pairwise velocity dispersion of particles in N -body simulations (see §3.1) both for SCDM

Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

3

Fig. 1. Pairwise peculiar velocity dispersions of dark matter particles at z = 0, 1 and 2. Dashed lines indicate the values predicted from the formula of MJB (eq.[9]), while solid lines indicate our adopted ?t (eq.[10]). The radial comoving distance χ(z ) is computed by

t0

χ(z ) =

t

1 dt = a(t) a0

z 0

dz . H (z )

(19)

In our de?nition, K is not normalized to ±1 and 0, but rather written in terms of the scale factor at present, a0 , the Hubble constant, H0 , the density parameter, ?0 and the dimensionless cosmological constant, λ0 :

2 K = a2 0 H0 (?0 + λ0 ? 1).

(20)

The comoving angular diameter distance Dc (z ) at redshift z is equivalent to S ?1 (χ(z )), and, in the case of λ0 = 0, is explicitly given by Mattig’s formula: √ 1 1 + z + 1 + ?0 z z √ Dc (z ) = . (21) a0 H0 1 + z 1 + ?0 z/2 + 1 + ?0 z Then dVc /dz , the comoving volume element per unit solid angle, is explicitly given as dVc dχ 2 = SK (χ) dz dz = H0

els, SCDM and ΛCDM, adopting a scale-invariant primordial power spectral index of n = 1. Their cosmological parameters are listed in Table 1. While SCDM are known to have several problems in reproducing the recent observations (e.g., de Bernardis et al. 2000), this model is suitable for testing the theoretical formula since the clustering evolution on the light-cone is more signi?cant. We use a series of N -body simulations originally constructed for the study of weak lensing statistics (Hamana et al. 2000, in preparation). These simulations were generated with a vectorized PM code (Moutarde at al. 1991) modi?ed to run in parallel on several processors of a CRAY-98 (Hivon 1995). They use 2562 × 512 particles and the same number of force mesh in a periodic rectangular comoving box. We use both the small and large boxes (Table 1). The initial conditions are generated adopting the transfer function of Bond & Efstathiou (1984, see also Jenkins et al. 1998) with the shape parameter Γ = ?0 h. The amplitude of the power spectrum is normalized by the cluster abundance (Eke, Cole & Frenk 1996; Kitayama & Suto 1997). Using the above simulation data, we generated lightcone samples as follows; ?rst, we adopt a distance observer approximation and assume that the line-of-sight direction is parallel to Z -axis regardless with its (X, Y ) position (Fig.2). Second, we periodically duplicate the simulation box along the Z -direction so that at a redshift z , the position and velocity of those particles locating within an interval χ(z ) ± ?χ(z ) are dumped, where ?χ(z ) is determined by the output time-interval of the original N -

(22) . ?0 (1 + z )3 + (1 ? ?0 ? λ0 )(1 + z )2 + λ0

2 SK (χ)

3. Evaluating two-point correlation functions from N-body simulation data 3.1. Particle distribution on the light cone from N-body simulations We test the theoretical modeling against simulation results, we focus on two spatially-?at cold dark matter mod-

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Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

Table 1. Parameters in N -body simulations.

Model SCDM small box SCDM large box ΛCDM small box ΛCDM large box ?0 1 1 0.3 0.3 λ0 0 0 0.7 0.7 h 0.5 0.5 0.7 0.7 σ8 0.6 0.6 0.9 0.9 Box size LX × LY × LZ [h?3 Mpc3 ] 80 × 80 × 160 240 × 240 × 480 120 × 120 × 240 360 × 360 × 620 Force resolution [h?1 Mpc] 0.31 0.94 0.45 1.4 zmax 0.4 2 0.4 2

Table 2. Parameter values for the polynomial evolution model of Boyle et al. (2000).

?0 1 0.3 λ0 0 0.7 α 3.45 3.41 β 1.63 1.58

? MB ? 5 log h -20.59 -21.14

k1 1.31 1.36

k2 ?0.26 ?0.27

Φ? [h3 Mpc?3 mag?1 ] 0.80 × 10?5 2.88 × 10?6

Table 3. Summary of number of particles in each realization (real space/redshift space).

Model SCDM small box Realization 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Total 8193106 / 8216016 8291309 / 8311402 8448034 / 8479865 9181442 / 9250736 8263119 / 8324278 6253827 / 6254790 6321816 / 6319899 6346239 / 6342617 6423700 / 6417089 6298022 / 6300195 3253963 / 3224663 4326797 / 4341618 4429032 / 4423464 4859939 / 4842481 4993640 / 4988234 5358865 / 5370894 5277031 / 5286441 5625180 / 5631157 5630820 / 5631761 5606636 / 5612974 Random selection 10258 / 10282 10388 / 10413 10591 / 10627 11533 / 11618 10340 / 10434 10481 / 10482 10591 / 10582 10626 / 10623 10767 / 10757 10546 / 10552 7589 / 7512 10025 / 10050 10263 / 10258 11245 / 11201 11532 / 11517 9834 / 9863 9665 / 9681 10322 / 10326 10326 / 10326 10287 / 10300 LF based selection 125477 / 125289 168363 / 168999 165217 / 165192 175769 / 175773 178135 / 177348 2037314 / 2041146 2077216 / 2077310 2090246 / 2090222 2102122 / 2099505 2077854 / 2079776 43377 / 42960 48690 / 48581 62073 / 62274 59105 / 59022 72490 / 72674 1427660 / 1429062 1415712 / 1418226 1507183 / 1507424 1511963 / 1511565 1507176 / 1508490 LF based with random selection 8546 / 8540 11444 / 11497 11221 / 11218 11927 / 11927 12075 / 12025 10348 / 10363 10552 / 10552 10622 / 10622 10671 / 10664 10553 / 10564 8760 / 8666 9808 / 9791 12517 / 12553 11903 / 11890 14608 / 14635 9588 / 9593 9498 / 9516 10174 / 10175 10219 / 10218 10174 / 10187

SCDM large box

ΛCDM small box

ΛCDM large box

body simulation. Finally we extract ?ve independent (nonoverlapping) cone-shape samples with the angular radius of 1 degree (the ?eld-of-view of π degree2 ), each for small and large boxes as illustrated in Figure 2. In this manner, we have generated mock data samples on the light-cone continuously extending up to z = 0.4 (relevant for galaxy samples) and z = 2.0 (relevant for QSO samples), respectively from the small and large boxes. While the above procedure selects the same particle at several di?erent redshifts, this does not a?ect our conclusion below because we are mainly interested in scales much below the box size along the Z -direction, LZ . In practice, we apply the above procedure separately in real and redshift spaces by using zreal and zobs of each

particle (see eq.[25] below). The total numbers of particles in those realizations are listed in Table 3. 3.2. Pair counts in real and redshift spaces Two-point correlation function is estimated by the conventional pair-count adopting the estimator proposed by Landy & Szalay (1993): ξ (x) = DD(x) ? 2DR(x) + RR(x) . RR(x) (23)

For this purpose, we distribute the same number of particles over the light-cone in a completely random fashion. When the number of particles in a realization exceeds 106 , we randomly select 10,000 particles as center particles in

Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

5

240 Mpc/h 240 Mpc/h QSO sample

The comoving separation x12 of two objects located at z1 and z2 with an angular separation θ12 is given by

2 2 2 2 2 x2 12 = x1 + x2 ? Kx1 x2 (1 + cos θ12 )

?2x1 x2

1 ? Kx2 1

1 ? Kx2 2 cos θ12 ,

(24)

z=2.0

where x1 ≡ Dc (z1 ) and x2 ≡ Dc (z2 ). In redshift space, the observed redshift zobs for each object di?ers from the “real” one zreal due to the velocity distortion e?ect: zobs = zreal + (1 + zreal )vpec , (25)

where vpec is the line of sight relative peculiar velocity between the object and the observer in physical units. Then the comoving separation s12 of two objects in redshift space is computed as

2534 Mpc/h

2 2 2 2 2 s2 12 = s1 + s2 ? Ks1 s2 (1 + cos θ12 )

? 2 s1 s2

1 ? Ks2 1

1 ? Ks2 2 cos θ12 ,

(26)

where s1 ≡ Dc (zobs,1 ) and s2 ≡ Dc (zobs,2 ). 3.3. Selection functions In properly predicting the power spectra on the light cone, the selection function should be speci?ed. In this subsection, we describe the selection functions appropriate for galaxies and quasars samples. For galaxies, we adopt a B-band luminosity function of the APM galaxies (Loveday et al. 1992) ?tted to the Schechter function: φ(L)dL = φ? L L?

α

80 Mpc/h

80 Mpc/h galaxy sample

z=0.4

928Mpc/h

exp ?

L L d , L? L?

(27)

? with φ? = 1.40 × 10?2 h3 Mpc?3 , α = ?0.97, and MB = ?19.50 + 5 log10 h. Then the comoving number density of galaxies at z which are brighter than the limiting magnitude Blim is given by

z=0 observer

z=0 observer

∞

ngal (z, < Blim ) =

L(Blim ,z ) ?

φ(L)dL (28)

= φ Γ[(α + 1, x(Blim , z )], Fig. 2. Schematic geometry of our light-cone samples. The comoving distances denoted in the ?gure are for SCDM model. In the case of ΛCDM model, the redshifts are not changed but the radial comoving distance is 1086(3626)h?1Mpc and the side length is 120(360)h?1Mpc for the small(large) box. where x(Blim , z ) ≡ dL (z ) L(Blim , z ) = L? 1h?1 Mpc

2

102.2?0.4Blim , (29)

and Γ[ν, x] is the incomplete Gamma function. Figure 3 plots the selection function de?ned by φgal (< Blim , z ) ≡ with zmin = 0.01. ngal (z, < Blim ) ngal (zmin , < Blim ) (30)

counting the pairs. Otherwise we use all the particles in the pair counts.

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Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

1

B<19.0 B<18.0 B<17.0

B<21.0 0.1 B<20.0 B<19.0 B<18.0

1

0.2

0.4

0.6

0.8

1

2

Fig. 3. Selection function of galaxies in a case of SCDM model on the basis of the B-band luminosity function of APM galaxies (Loveday et al. 1992). For quasars, we adopt the B-band luminosity recently determined by Boyle et al. (2000) from the 2dF QSO survey data:

Fig. 4. Selection function of QSOs in a case of SCDM model on the basis of the 2dF QSO sample (Boyle et al. 2000).

In practice, we adopt the galaxy selection function φgal (< Blim , z ) with Blim = 19 and zmin = 0.01 for the small box realizations, while the QSO selection function φQSO (< Blim , z ) with Blim = 21 and zmin = 0.2 for the Φ? Φ(MB , z ) = 0.4(1?α)[M ?M ? (z)] (31) . large box realizations. We do not introduce the spatial bi? B B 10 + 100.4(1?β )[MB?MB (z)] asing between selected particles and the underlying dark matter, which will be discussed elsewhere. For comparison, In the case of the polynomial evolution model: we also select the similar number of particles randomly but ? ? MB (z ) = MB (0) ? 2.5(k1 z + k2 z 2 ), (32) independently of their redshifts. It should be emphasized here that our simulated data are constructed to match and we adopt the sets of their best-?t parameters listed the shape of the above selection functions but not the amin Table 2 for our SCDM and ΛCDM. plitudes of the number densities. The ?eld-of-view of our To compute the B-band apparent magnitude from a simulated data, π degree2 , is substantially smaller than quasar of absolute magnitude MB at z (with the luminos- those of 2dF and SDSS, and we sample particles much ity distance dL (z )), we applied the K-correction: more densely than the realistic number density. Since our main purpose of this paper is to test the reliability of B = MB + 5 log(dL (z )/10pc) ? 2.5(1 ? p) log(1 + z ) (33) the theoretical modeling described in section 2, and not ?p for the quasar energy spectrum Lν ∝ ν (we use p = 0.5). to present detailed predictions, this does not change our Then the comoving number density of QSOs at z which conclusions below. The numbers of the selected particles are brighter than the limiting magnitude Blim is given by in each realization are listed in Table 3. The averaged selection functions for our ?ve realizations in real space are M (Blim ,z ) plotted as histograms in Figures 3 and 4. nQSO (z, < Blim ) = Φ(MB , z )dMB . (34)

?∞

Figure 4 plots the selection function de?ned by φQSO (< Blim , z ) ≡ with zmin = 0.2. nQSO (z, < Blim ) nQSO (zmin , < Blim ) (35)

4. Results Consider ?rst the two-point correlation functions for particles on the light cone but without redshift-dependent selection. Figures 5 and 6 plot those correlations for z < 0.4 samples from small-box simulations (upper panels) and

Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

7

1

SCDM h=0.5

1

1

1

Fig. 5. Mass two-point correlation functions on the light cone without redshift-dependent selection functions in SCDM model. Upper: z < 0.4, Lower: 0 < z < 2.0. Left: all particles on the light cone, Right: randomly selected particles. for z < 2 from large-box ones (lower panels), for SCDM and ΛCDM, respectively. In these ?gure, we plot the averages over the ?ve realizations (Table 3) in open circles (real space) and in solid triangles (redshift space), and the quoted error-bars represent the standard deviation among them. If we use all particles from simulations (left panels), the agreement between the theoretical predictions (solid lines) and simulations (symbols) is quite good. The scales where the simulation data in real space become smaller than the corresponding theoretical predictions simply re?ect the force resolution of the simulations listed in Table 1. In order to examine the robustness of the estimates from the simulated data, we randomly selected N ? 104 particles from the entire light-cone volume (independently of their redshifts). The resulting correlation functions are plotted in the right panels. It is remarkable that the estimates on scales larger than ? 1h?1 Mpc are almost the same. This also indicates that the error-bars in our data are dominated by the sample-to-sample variation among the di?erent realizations. Next we examine the e?ect of selection functions. Figures 7 and 8 plot the two-point correlation functions in SCDM and ΛCDM, respectively, taking account of the selection functions described in the subsection 3.3. It is clear

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Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

1

h=0.7

1

1

1

Fig. 6. Same as Figure 5 but for ΛCDM model. that the simulation results and the predictions are in good agreement. It should be noted that the results shown in the upper-left panel (intended to correspond to galaxies) have substantially larger error-bars compared with the corresponding ones in Figures 5 and 6. This is an artifact to some extent because of the very small survey volume in our light-cone samples; if one applies the galaxy selection function which rapidly decreases as z (see, Fig.3), the resulting structure mainly probes the universe at z < 0.1 and thus large-scale nonlinearity or variation for the di?erent line-of-sight becomes signi?cant. If we are able to use the same number of particles but extending over the much larger volume, the sample-to-sample variations should be substantially smaller. This interpretation is supported by the upper-right panel where we randomly sample N ? 104 particles from those used in the upper-left panel. Despite the fact that the number of particles is only 5% (20%) of the original one for SCDM (ΛCDM) model, the resulting correlation functions and their error-bars remain almost unchanged. The lower panels corresponding to QSOs show the similar trend. 5. Conclusions and discussion We have presented detailed comparison between the theoretical modeling and the direct numerical results of the two-point correlation functions on the light-cone. In short, we have quantitatively shown that the previous theo-

Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

9

1

SCDM h=0.5

1

1

1

Fig. 7. Mass two-point correlation functions on the light cone for particles with redshift-dependent selection functions in SCDM model. Upper: z < 0.4, Lower: 0.2 < z < 2.0. Left: with selection function whose shape is the same as that of the B-band magnitude limit of 19 for galaxies (upper) and 21 for QSOs (lower). Right: randomly selected N ? 104 particles from the particles in the left results. retical models by Yamamoto & Suto (1999) and Yamamoto, Nishioka & Suto (1999) are quite accurate on scales 1h?1 Mpc < x < 20h?1 Mpc where the numerical simulations are reliable. It is also encouraging that this conclusion remains true even for the particle number of around 104 . In fact, the error-bars in our estimates of the two-point correlation functions are dominated by the sample-to-sample variance due to the limited angular-size (π -degree2 ) and thus the limited volume. In order for the more realistic evaluation of the statistical and systematic uncertainties, one needs mock lightcone data samples with a much wider sky coverage. More importantly such datasets enable one to access the e?ect of biasing on the two-point correlation functions on the lightcone. Since our present study indicated that all the physical e?ects except for the biasing are well described by the existing theoretical models, it is very interesting to examine in detail how to extract the e?ect of the galaxy/QSO biasing from the upcoming redshift survey on the basis of the above mock samples. We plan to come back to these issues with larger simulation datasets in near future.

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Hamana, Colombi, & Suto: Two-point correlation functions on the light cone

1

h=0.7

1

1

1

Fig. 8. Same as Figure 7 but for ΛCDM model.

Acknowledgements. This research was supported in part by the Direction de la Recherche du Minist` ere Fran? cais de la Recherche and the Grant-in-Aid by the Ministry of Education, Science, Sports and Culture of Japan (07CE2002) to RESCEU. The computational resources (CRAY-98) for the present numerical simulations were made available to us by the scienti?c council of the Institut du D? eveloppement et des Ressources en Informatique Scienti?que (IDRIS). Eke, V.R. Cole, S., & Frenk, C.S. 1996, MNRAS, 282, 263 Hivon, E., 1995, PhD thesis, University Paris XI Jenkins, A., et al., 1998, ApJ, 499, 20 Kaiser, N. 1987, 227, 1 Kitayama, T., & Suto, Y. 1997, ApJ, 490, 557 Landy, S.D., & Szalay, A.S., 1993, ApJ, 412, 64 Magira, H., Jing, Y. P., & Suto, Y. 2000, ApJ, 528, 30 Matarrese, S., Coles, P., Lucchin, F., & Moscardini, L. 1997, MNRAS, 286, 115 Matsubara, T., & Suto, Y. 1996, ApJ, 470, L1 Matsubara, T., Suto, Y., & Szapudi, I. 1997, ApJ, 491, L1 Mo, H. J., Jing, Y. P., & B¨ orner, G. 1997, MNRAS, 286, 979 (MJB) Moscardini, L., Coles, P., Lucchin, & F., Matarrese, S. 1998, MNRAS, 299, 95

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Hamana, Colombi, & Suto: Two-point correlation functions on the light cone Moutarde, F., Alimi, J. M., Bouchet, F. R., Pellat, R., & Ramani, A., 1991, ApJ, 382, 377 Nakamura, T. T., Matsubara, T., & Suto, Y. 1998, ApJ, 494, 13 Peacock, J.A., & Dodds, S.J. 1996, MNRAS, 280, L19 Suto, Y., Magira, H., Jing, Y. P., Matsubara, T., & Yamamoto, K. 1999, Prog.Theor.Phys.Suppl., 133, 183 Suto, Y., Magira, H., & Yamamoto, K. 2000, PASJ, 52, 249 Yamamoto, K., Nishioka, H., & Suto, Y. 1999, ApJ, 527, 488 Yamamoto, K., & Suto, Y. 1999, ApJ, 517, 1

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- The Functional Integration and the Two-Point Correlation Functions of the Trapped Bose Gas
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