Large-Scale Structure of the Universe and Cosmological ...

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Fig. 58. The bispectrum Q3 vs. θ for the PSCz catalog for triangles with 0.2 ≤ k1 ≤ 0.4 h/Mpc and with two sides of ratio k2/k1 = 0.4 − 0.6 separated by angle θ. The solid curve shows Q3 in redshift space averaged over many 2LPT realizations of the ΛCDM model. Symbols show results from the PSCz survey for bands in k1: filled triangles, k1 = 0.20–0.24h/Mpc; filled squares 0.24–0.28; filled circles 0.28-0.32; open circles 0.32–0.36; and open squares 0.36–0.42. The dashed curve shows the 2LPT prediction for ΛCDM with the best-fit bias parameters 1/b = 1.20, b2/b 2 = −0.42. Taken from [211]. QDOT were marginal, due to the very sparse sampling (one galaxy every six) Q3 was only shown to be of order unity without any discernible dependence on configuration. The results from 1.9Jy and 1.2Jy showed a systematic shape dependence similar to that predicted by gravitational instability. These results were considerably extended with the analysis of the PSCz bispectrum [211]. Figure 58 shows the PSCz reduced bispectrum Q3 as a function of the angle θ between k1 and k2 for triangles with k1/k2 ≈ 2 and different scales as described in the figure caption [211]. The configuration dependence predicted by gravitational instability [232,313] (solid lines for an unbiased distribution, predicted by 2LPT, see e.g. Fig. 48) is clearly seen in the data. This is also the case for all triangles, not just those shown in Fig. 58, see Fig. 1 in [211]. Implications of these results for galaxy biasing and primordial non-Gaussianity are discussed in Sect. 8.3.5. given in [669]. 252
Table 19 Some measurements of S3 and S4 in redshift catalogs. In most cases, only the mean values over a range of scales were published. In cases where measurements of the individual values for each scale are reported in the literature, we quote the actual range of estimates over the corresponding range of scales. In most cases error bars should be considered only as rough estimates, see text for discussion. S3 S4 Scales Sample Year Ref 2 ± 1 − 6 ± 4 — 5 − 20 QDOT 1991 [549] 1.5 ± 0.5 4.4 ± 3.7 0.1-50 IRAS 1.2Jy 1992 [88],[92] 1.9 ± 0.1 4.1 ± 0.6 2-22 CfA 1992 [246] 2.0 ± 0.1 5.0 ± 0.9 2-22 SSRS 1992 [246] 2.1 ± 0.3 7.5 ± 2.1 3-10 IRAS 1.9Jy 1994 [224] 2.4 ± 0.3 13 ± 2 2-10 PPS 1996 [267] 2.8 ± 0.1 6.9 ± 0.7 8-32 IRAS 1.2Jy 1998 [377] 1.8 ± 0.1 5.5 ± 1 1-10 SSRS2 1999 [35] 1.9 ± 0.6 7 ± 4 1-30 PSCz 2000 [632] 1.82 ± 0.21 ∼ 3 12.6 Durham/UKST 2000 [319] 2.24 ± 0.29 ∼ 8 18.2 Stromlo/APM 2000 [319] 8.3.4 Skewness, Kurtosis and Higher-Order Cumulants Table 19 shows different estimates for S3 = ξ3/ξ 2 2 and S4 = ξ4/ξ 3 2 , the ratios of the cumulants ξN obtained by counts-in-cells. The shape of the cells correspond to top-hat spheres, unless stated otherwise. The QDOT results by [549] were obtained from counts-in-cells with a Gaussian window. The errors, from a minimum variance scheme, are quite large but they suggest a hierarchical scaling ξ3 ≃ ξ 2 2 , with a value of S3 consistent with gravity from Gaussian initial conditions, as argued in [137]. Figure 59 displays the 1.2 Jy IRAS results ([88], [92], left panel) and CfA-SSRS results ([246], right panel). There is a convincing evidence for the hierarchical scaling in ξ3 and ξ4 (denoted by straight lines) but the resulting S3 and S4 amplitudes are probably affected by sampling biases (see discussion below). Note that the scaling is preserved well into the non-linear regime, this is in agreement with expectations from N-body simulations which show that in redshift space the growth of Sp parameters towards the non-linear regime is suppressed by velocity dispersion from virialized regions ([391,437], see e.g. Fig. 49). In their analysis of higher-order moments in the CfA, SSRS, and IRAS 1.9 Jy catalogs, [224] studied the sensitivity of Sp to redshift distortions, by calculating moments in spherical cells and conical cells. The latter were argued to be less sensitive to the redshift space mapping that acts along the line of 253
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arXiv:astro-ph/0112551v1 27 Dec 200
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4.1.1 Emergence of Non-Gaussianity
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6.11.1 Maximum Likelihood Estimates
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1 Introduction and Notation Underst
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with N-point functions, whereas Cha
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Table 3 Notation for the Cosmic Fie
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2 Dynamics of Gravitational Instabi
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In the following we will only use c
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∂u(x, τ) + H(τ) u(x, τ) = −
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Eq. (17) we can write the vorticity
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θn(k) = d 3 q1 . . . d 3 qn δD(k
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2.4.3 Cosmology Dependence of Non-L
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approximation f(Ωm, ΩΛ) = Ω3
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The somewhat complicated expression
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where Φ denotes the gravitational
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or more precisely D2(τ) ≈ − 3
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ever turning around, washing out st
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2.9.2 Direct Summation Also known a
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(e.g., [314,532]). Finally, it is w
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such initial conditions are likely
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3.2.1 Statistical Homogeneity and I
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δ 1 δ δ 2 3 c = δ1 c = 00000 11
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3.2.5 Probabilities and Correlation
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3.3.3 Generating Functions It is co
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values of y are then also of the or
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4 From Dynamics to Statistics: N-Po
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Figures 5 and 6 show the tree diagr
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e approximated by a fitting functio
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Fig. 10. The tree-level three-point
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4 2 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0
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n P13/(πA 2 a 4 ) P22/(πA 2 a 4 )
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Fig. 13. The power spectrum for n =
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where: B222 ≡ 8 d 3 qPL(q, τ)F
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Fig. 16. The left panel shows the o
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them in Sect. 5.6. It is worth emph
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Fig. 17. The reduced bispectrum ˜
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(1) There are no characteristic tim
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velocity exactly cancels the Hubble
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A simple generalization of this arg
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growth factor has been written as D
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function in the stable clustering l
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S sat 4 (n) = 16 Qsat 4 (n) = 8 54
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For the reasons discussed in Sect.
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δ δ Evolution of an initially und
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obtained by expansion about Ωm =
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y the orthogonality relation betwee
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σ = 2 ; ν σ = ; 4 2 ν σ = 6 3
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(that plays a role similar to the v
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Fig. 26. The predicted Sp parameter
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Table 6 Tree-level and one-loop cor
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expressed in terms of the linear de
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give to non-Gaussian initial condit
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+ −4 + 8 3 SG 3 − 1 S 6 G 2 3
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the initial conditions. In Sect. 2.
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Fig. 30. The ratio of the tree-leve
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5.8 The Density PDF Up to now, we h
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Fig. 32. Comparison between predict
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9 5 p(δ) = 3/2 4π Ns(1 + δ) 3 σ
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Table 10 Parameters of the singular
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5.10.2 The Shape of the PDF The abo
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Fig. 35. Example of a joint PDF of
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Table 11 The coefficients a1,... ,a
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originally in previous work in the
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Table 12 Parameters used in fit (35
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6 From Theory to Observations: Esti
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the size of the catalog and optimiz
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The cosmic error is most useful whe
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expectation number ¯ N = ¯ngv, P
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1 〈δn(k1)δn(k2)δn(k3)〉 = N2
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catalog, the latter being equivalen
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size. In this regime, where ξ(r) i
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The expressions (395) and (400) can
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The techniques developed to measure
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At smaller scales, in the regime k
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Fig. 38. The top panel shows the me
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If ¯ng is determined with arbitrar
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Factorial moments thus verify Fk =
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To second order the cosmic bias [Eq
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The finite-volume error comes from
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6.7.5 Cosmic Error and Cosmic Bias
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factorial moment correlators [620]
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The generalization of Eq. (422) rea
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in [152]. Similarly to Eq. (472), t
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constrain theories with observation
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From this simple result, we see tha
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Fig. 41. The cosmic distribution fu
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of functions of the data ˆx. The p
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6.11.2 Quadratic Estimators In real
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is positive and compact in Fourier
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6.12 Measurements in N-Body Simulat
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7 Applications to Observations 7.1
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properties of the matter distributi
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deterministic bias results hold for
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effects due to the gravitational dy
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0000 1111 0000 1111 ϕ (y)= 0000 11
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where the volume V = 4πR3 /3 is re
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in Sect. 7.1.3, Eq. (555), plus Eqs
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In the following we first review th
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if the 3D correlation function is
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Table 13 Projection factors for dif
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Page 203 and 204: Fig. 47. Tree-level PT predictions
Page 205 and 206: 7.3 Weak Gravitational Lensing The
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Page 209 and 210: 7.4 Redshift Distortions In order t
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Page 215 and 216: Fig. 48. The left panel shows the b
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Page 223 and 224: Table 14 Angular Catalogs. The firs
Page 225 and 226: The DeepRange Catalog ([530] 1998)
Page 227 and 228: Fig. 50. The two-point angular corr
Page 229 and 230: Fig. 51. The APM 3D power spectrum
Page 231 and 232: linearization first done in [289] a
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Page 235 and 236: and Rb = 4.3 ± 1.2 [226]. These re
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Page 239 and 240: Table 16 The reduced skewness and k
Page 241 and 242: estimations are split in its 6 × 6
Page 243 and 244: the near future. An early applicati
Page 245 and 246: The Stromlo-APM redshift survey ([4
Page 247 and 248: A recent linear analysis of the LCR
Page 249 and 250: at the non-linear scale, and no sig
Page 251: Fig. 57. The redshift-space reduced
Page 255 and 256: Fig. 60. The redshift-space skewnes
Page 257 and 258: such as the abundance of massive cl
Page 259 and 260: the weakly non-linear regime is qui
Page 261 and 262: the mock catalogs. The resulting 3D
Page 263 and 264: few technical issues that need more
Page 265 and 266: A The Spherical Collapse Dynamics T
Page 267 and 268: More specifically we define ϕ(y) a
Page 269 and 270: ∞ (−τ1) τ2 =ξ y1 νp p=1 p
Page 271 and 272: This result writes as a kind of com
Page 273 and 274: E PDF Construction from Cumulant Ge
Page 275 and 276: E.3 Approximate Forms for P(ρ) whe
Page 277 and 278: A simple change of variable, t 1−
Page 279 and 280: It is then easy to calculate cross-
Page 281 and 282: References [1] S.J. Aarseth, E.L. T
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[547] R.F. Sanford, 1917, Lick Obse
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[597] M. Snethlage, Metrica, 49, (1
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[644] A.N. Taylor, P.I.R. Watts, MN
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[692] M.B. Wise, in The Early Unive
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