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

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5.9 Two-Dimensional Dynamics 115 5.10 The Velocity Divergence PDF 116 5.10.1 The Velocity Divergence Cumulants Hierarchy 117 5.10.2 The Shape of the PDF 119 5.10.3 Comparison with N-Body Simulations 119 5.11 The Velocity-Density Relation 121 5.12 The Two-Point Density PDF 124 5.13 Extended Perturbation Theories 126 6 From Theory to Observations: Estimators and Errors 129 6.1 Introduction 129 6.2 Basic Concepts 132 6.2.1 Cosmic Bias and Cosmic Error 132 6.2.2 The Covariance Matrix 133 6.3 Fair Sample Hypothesis and Local Poisson Approximation 134 6.3.1 The Fair Sample Hypothesis 134 6.3.2 Poisson Realization of a Continuous Field 134 6.4 The Two-Point Correlation Function 137 6.4.1 Estimators 137 6.4.2 Cosmic Bias and Integral Constraint of the LS Estimator 139 6.4.3 Cosmic Error of the LS Estimator 141 6.4.4 The Covariance Matrix 143 6.4.5 Recipes for Error Calculations 144 6.5 The Power Spectrum 144 6.5.1 Simple Estimators 145 6.5.2 Cosmic Bias and Integral Constraint 146 6.5.3 The Cosmic Error 147 6.5.4 The Covariance Matrix 148 6.6 Generalization to Higher-Order Correlation Functions 150 6.7 One-Point Distributions: Counts-in-Cells 152 6.7.1 Definitions: 152 6.7.2 Estimators 153 6.7.3 Error Propagation: Cosmic Bias vs. Cosmic Error 154 6.7.4 Cosmic Error and Cross-Correlations of Factorial Moments 155 6.7.5 Cosmic Error and Cosmic Bias of Cumulants 159 6.8 Multivariate Count-in-Cells 160 6.9 Optimal Weighting 162 6.10 Cosmic Distribution Function and Cross-Correlations 166 6.10.1 Cosmic Distribution Function and Likelihood 166 6.10.2 Cross-Correlations Between Different Statistics 168 6.10.3 Validity of the Gaussian Approximation 169 6.11 Optimal Techniques for Gaussian Random Fields 170 4
6.11.1 Maximum Likelihood Estimates 172 6.11.2 Quadratic Estimators 175 6.11.3 Uncorrelated Error Bars 176 6.11.4 Data Compression and the Karhunen-Loève Transform 177 6.12 Measurements in N-Body Simulations 179 6.12.1 Cosmic Error and Cosmic Bias in Simulations 179 6.12.2 N-Body Relaxation and Force Softening 180 7 Applications to Observations 181 7.1 The Problem of Galaxy Biasing 181 7.1.1 Some General Results 182 7.1.2 Halo Clustering in the Tree Hierarchical Model 187 7.1.3 Halo Clustering in the Extended Press-Schechter Approach 190 7.1.4 Galaxy Clustering 191 7.2 Projection Effects 194 7.2.1 The Projected Density Contrast 195 7.2.2 The Small-Angle Approximation 196 7.2.3 Projection in the Hierarchical Model 197 7.2.4 The Correlation Hierarchy for the Projected Density 199 7.2.5 Comparison with Numerical Simulations 201 7.2.6 Reconstructing the PDF of the Projected Density 204 7.3 Weak Gravitational Lensing 205 7.3.1 The Convergence Field as a Projected Mass Map 205 7.3.2 Statistical Properties 207 7.3.3 Next to Leading Order Effects 207 7.3.4 Biasing from Weak Gravitational Lensing 208 7.4 Redshift Distortions 209 7.4.1 The Density Field in Redshift Space 209 7.4.2 The Redshift-Space Power Spectrum 212 7.4.3 The Redshift-Space Bispectrum 214 7.4.4 The Higher-Order Moments in Redshift Space 217 7.4.5 Cosmological Distortions 217 8 Results from Galaxy Surveys 220 8.1 Galaxies as Cosmological Tracers 220 8.2 Results from Angular Galaxy Surveys 222 8.2.1 Angular Catalogs 222 8.2.2 The Angular Correlation Function and Power Spectrum 225 8.2.3 Inversion from Angular to 3D Clustering 229 8.2.4 Three-Point Statistics and Higher Order 232 8.2.5 Skewness, Kurtosis and Higher-Order Cumulants 238 8.2.6 Constraints on Biasing and Primordial Non-Gaussianity 241 8.3 Results from Redshift Galaxy Surveys 243 5
Page 1 and 2: arXiv:astro-ph/0112551v1 27 Dec 200
Page 3: 4.1.1 Emergence of Non-Gaussianity
Page 7 and 8: 1 Introduction and Notation Underst
Page 9 and 10: with N-point functions, whereas Cha
Page 11 and 12: Table 3 Notation for the Cosmic Fie
Page 13 and 14: 2 Dynamics of Gravitational Instabi
Page 15 and 16: In the following we will only use c
Page 17 and 18: ∂u(x, τ) + H(τ) u(x, τ) = −
Page 19 and 20: Eq. (17) we can write the vorticity
Page 21 and 22: θn(k) = d 3 q1 . . . d 3 qn δD(k
Page 23 and 24: 2.4.3 Cosmology Dependence of Non-L
Page 25 and 26: approximation f(Ωm, ΩΛ) = Ω3
Page 27 and 28: The somewhat complicated expression
Page 29 and 30: where Φ denotes the gravitational
Page 31 and 32: or more precisely D2(τ) ≈ − 3
Page 33 and 34: ever turning around, washing out st
Page 35 and 36: 2.9.2 Direct Summation Also known a
Page 37 and 38: (e.g., [314,532]). Finally, it is w
Page 39 and 40: such initial conditions are likely
Page 41 and 42: 3.2.1 Statistical Homogeneity and I
Page 43 and 44: δ 1 δ δ 2 3 c = δ1 c = 00000 11
Page 45 and 46: 3.2.5 Probabilities and Correlation
Page 47 and 48: 3.3.3 Generating Functions It is co
Page 49 and 50: values of y are then also of the or
Page 51 and 52: 4 From Dynamics to Statistics: N-Po
Page 53 and 54: Figures 5 and 6 show the tree diagr
Page 55 and 56:
e approximated by a fitting functio
Page 57 and 58:
Fig. 10. The tree-level three-point
Page 59 and 60:
4 2 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0
Page 61 and 62:
n P13/(πA 2 a 4 ) P22/(πA 2 a 4 )
Page 63 and 64:
Fig. 13. The power spectrum for n =
Page 65 and 66:
where: B222 ≡ 8 d 3 qPL(q, τ)F
Page 67 and 68:
Fig. 16. The left panel shows the o
Page 69 and 70:
them in Sect. 5.6. It is worth emph
Page 71 and 72:
Fig. 17. The reduced bispectrum ˜
Page 73 and 74:
(1) There are no characteristic tim
Page 75 and 76:
velocity exactly cancels the Hubble
Page 77 and 78:
A simple generalization of this arg
Page 79 and 80:
growth factor has been written as D
Page 81 and 82:
function in the stable clustering l
Page 83 and 84:
S sat 4 (n) = 16 Qsat 4 (n) = 8 54
Page 85 and 86:
For the reasons discussed in Sect.
Page 87 and 88:
δ δ Evolution of an initially und
Page 89 and 90:
obtained by expansion about Ωm =
Page 91 and 92:
y the orthogonality relation betwee
Page 93 and 94:
σ = 2 ; ν σ = ; 4 2 ν σ = 6 3
Page 95 and 96:
(that plays a role similar to the v
Page 97 and 98:
Fig. 26. The predicted Sp parameter
Page 99 and 100:
Table 6 Tree-level and one-loop cor
Page 101 and 102:
expressed in terms of the linear de
Page 103 and 104:
give to non-Gaussian initial condit
Page 105 and 106:
+ −4 + 8 3 SG 3 − 1 S 6 G 2 3
Page 107 and 108:
the initial conditions. In Sect. 2.
Page 109 and 110:
Fig. 30. The ratio of the tree-leve
Page 111 and 112:
5.8 The Density PDF Up to now, we h
Page 113 and 114:
Fig. 32. Comparison between predict
Page 115 and 116:
9 5 p(δ) = 3/2 4π Ns(1 + δ) 3 σ
Page 117 and 118:
Table 10 Parameters of the singular
Page 119 and 120:
5.10.2 The Shape of the PDF The abo
Page 121 and 122:
Fig. 35. Example of a joint PDF of
Page 123 and 124:
Table 11 The coefficients a1,... ,a
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originally in previous work in the
Page 127 and 128:
Table 12 Parameters used in fit (35
Page 129 and 130:
6 From Theory to Observations: Esti
Page 131 and 132:
the size of the catalog and optimiz
Page 133 and 134:
The cosmic error is most useful whe
Page 135 and 136:
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
Page 143 and 144:
The expressions (395) and (400) can
Page 145 and 146:
The techniques developed to measure
Page 147 and 148:
At smaller scales, in the regime k
Page 149 and 150:
Fig. 38. The top panel shows the me
Page 151 and 152:
If ¯ng is determined with arbitrar
Page 153 and 154:
Factorial moments thus verify Fk =
Page 155 and 156:
To second order the cosmic bias [Eq
Page 157 and 158:
The finite-volume error comes from
Page 159 and 160:
6.7.5 Cosmic Error and Cosmic Bias
Page 161 and 162:
factorial moment correlators [620]
Page 163 and 164:
The generalization of Eq. (422) rea
Page 165 and 166:
in [152]. Similarly to Eq. (472), t
Page 167 and 168:
constrain theories with observation
Page 169 and 170:
From this simple result, we see tha
Page 171 and 172:
Fig. 41. The cosmic distribution fu
Page 173 and 174:
of functions of the data ˆx. The p
Page 175 and 176:
6.11.2 Quadratic Estimators In real
Page 177 and 178:
is positive and compact in Fourier
Page 179 and 180:
6.12 Measurements in N-Body Simulat
Page 181 and 182:
7 Applications to Observations 7.1
Page 183 and 184:
properties of the matter distributi
Page 185 and 186:
deterministic bias results hold for
Page 187 and 188:
effects due to the gravitational dy
Page 189 and 190:
0000 1111 0000 1111 ϕ (y)= 0000 11
Page 191 and 192:
where the volume V = 4πR3 /3 is re
Page 193 and 194:
in Sect. 7.1.3, Eq. (555), plus Eqs
Page 195 and 196:
In the following we first review th
Page 197 and 198:
if the 3D correlation function is
Page 199 and 200:
Table 13 Projection factors for dif
Page 201 and 202:
Note that the rp coefficients are v
Page 203 and 204:
Fig. 47. Tree-level PT predictions
Page 205 and 206:
7.3 Weak Gravitational Lensing The
Page 207 and 208:
elation (598) is then entirely dime
Page 209 and 210:
7.4 Redshift Distortions In order t
Page 211 and 212:
where [δD]n ≡ δD(k − k1 −
Page 213 and 214:
proximation. In fact, Eq. (616) is
Page 215 and 216:
Fig. 48. The left panel shows the b
Page 217 and 218:
They obtained analogous results to
Page 219 and 220:
that for wide surveys such as 2dFGR
Page 221 and 222:
Machine, [374]) and COSMOS [421] mi
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
Page 233 and 234:
most of the measurements only probe
Page 235 and 236:
and Rb = 4.3 ± 1.2 [226]. These re
Page 237 and 238:
3 2 1 3 2 1 0 -1 10 5 0 -5 -10 0 90
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 and 252:
Fig. 57. The redshift-space reduced
Page 253 and 254:
Table 19 Some measurements of S3 an
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
Page 283 and 284:
[54] F. Bernardeau, R. Van De Weyga
Page 285 and 286:
[100] A. Buchalter, M. Kamionkowski
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[150] S. Colombi, F.R. Bouchet, L.
Page 289 and 290:
[196] G. Efstathiou, in Cosmology a
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[245] A. Gangui, Phys. Rev. D, 62,
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[299] A.J.S. Hamilton, M. Tegmark,
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[352] R. Jeannerot, Phys. Rev. D, 5
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[399] A.R. Liddle, D.H. Lyth, Phys.
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[448] P. McDonald, J. Miralda-Escud
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[497] J.A. Peacock, S. Cole, P. Nor
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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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