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

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tribution of luminous matter compared to that of the underlying dark matter, in particular at small scales: such ‘galaxy biasing’ can be probed with the techniques reviewed in this work. Inevitably, we had to make some decisions in the choice of topics to be covered. Our presentation is definitely focused on the density field, with much less coverage on peculiar velocities. This choice is in particular motivated by the comparatively still preliminary stage of cosmic velocity fields, at least from an observational point of view (see however [607,160] for a review). On the other hand, note that since velocity field results are often obtained by identical techniques to those used for the density field, we mention some of these results but without giving them their due importance. In order to fully characterize the density field, we choose to follow the traditional approach of using statistical methods, in particular, N-point correlation functions [508]. Alternative methods include morphological descriptors such as Minkowski functionals (of which the genus is perhaps the most widely known), percolation analysis, etc. Unlike correlation functions, however, these other statistics are not as directly linked to dynamics as correlation functions, and thus are not as easy to predict from theoretical models. Furthermore, applications of PT to make predictions of these quantities is still in its infancy (see e.g [441] and references therein for recent work). Given that PT is an approximate method to solve the dynamics of gravitational clustering, it is desirable to test the validity of the results with other techniques. In particular, we resort to numerical simulations, which involve different approximations in solving the equations of motion that are not restricted to the weakly non-linear regime. There is a strong and healthy interplay between PT and N-body simulations which we extensively illustrate throughout this review. At large scales PT can be used to test quantitatively for spurious effects in numerical simulations (e.g. finite volume effects, transients from initial conditions), whereas at smaller, non-linear scales N-body simulations can be used to investigate the regime of validity of PT predictions. Although reviewing the current understanding of clustering at small scales is beyond the scope of this review, we have also included a discussion of the predictions of non-linear clustering amplitudes because connections between PT and strongly non-linear behavior have been suggested in the literature. We also include a discussion about stable clustering at small scales which, when coupled with self-similarity, leads to a connection between the large and small-scale scaling behavior of correlations functions. This review is structured so that different chapters can be read independently, although there are inevitable relations. Chapter 2 deals with the basic equations of motion and their solution in PT, including a brief summary of numerical simulations. Chapter 3 is a review of the basics of statistics; we have made it as succinct as possible to swiftly introduce the reader to the core of the review. For a more in-depth treatment we refer the reader to [609,61]. The next two chapters represent the main theoretical results; Chapter 4 deals 8
with N-point functions, whereas Chapter 5 reviews results for the smoothed one-point moments and PDF’s. These two chapters heavily rely on material covered in Chapters 2 and 3. In Chapter 6 we describe in detail the standard theory of estimators and errors for application to galaxy surveys, with particular attention to the issue of cosmic bias and errors of estimators of the two-point correlation function, power spectrum, and higher-order moments such as the skewness. Chapter 7 deals with theoretical issues related to surveys, such as redshift distortions, projection effects, galaxy biasing and weak gravitational lensing. Chapter 8 presents the current observational status of galaxy clustering, including future prospects in upcoming surveys, with particular emphasis on higher-order statistics. Chapter 9 contains our conclusions and outlook. A number of appendices extend the material in the main text for those interested in carrying out detailed calculations. Finally, to help the reader, Tables 1–4 list the main abbreviations and notations used for various cosmological variables, fields and statistics. 9
Page 1 and 2: arXiv:astro-ph/0112551v1 27 Dec 200
Page 3 and 4: 4.1.1 Emergence of Non-Gaussianity
Page 5 and 6: 6.11.1 Maximum Likelihood Estimates
Page 7: 1 Introduction and Notation Underst
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
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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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Note that the rp coefficients are v
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Fig. 47. Tree-level PT predictions
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7.3 Weak Gravitational Lensing The
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elation (598) is then entirely dime
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7.4 Redshift Distortions In order t
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where [δD]n ≡ δD(k − k1 −
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proximation. In fact, Eq. (616) is
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Fig. 48. The left panel shows the b
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They obtained analogous results to
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that for wide surveys such as 2dFGR
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Machine, [374]) and COSMOS [421] mi
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Table 14 Angular Catalogs. The firs
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The DeepRange Catalog ([530] 1998)
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Fig. 50. The two-point angular corr
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Fig. 51. The APM 3D power spectrum
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linearization first done in [289] a
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most of the measurements only probe
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and Rb = 4.3 ± 1.2 [226]. These re
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3 2 1 3 2 1 0 -1 10 5 0 -5 -10 0 90
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Table 16 The reduced skewness and k
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estimations are split in its 6 × 6
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the near future. An early applicati
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The Stromlo-APM redshift survey ([4
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A recent linear analysis of the LCR
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at the non-linear scale, and no sig
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Fig. 57. The redshift-space reduced
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Table 19 Some measurements of S3 an
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Fig. 60. The redshift-space skewnes
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such as the abundance of massive cl
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the weakly non-linear regime is qui
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the mock catalogs. The resulting 3D
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few technical issues that need more
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A The Spherical Collapse Dynamics T
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More specifically we define ϕ(y) a
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∞ (−τ1) τ2 =ξ y1 νp p=1 p
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This result writes as a kind of com
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E PDF Construction from Cumulant Ge
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E.3 Approximate Forms for P(ρ) whe
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A simple change of variable, t 1−
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It is then easy to calculate cross-
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References [1] S.J. Aarseth, E.L. T
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[54] F. Bernardeau, R. Van De Weyga
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[100] A. Buchalter, M. Kamionkowski
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[150] S. Colombi, F.R. Bouchet, L.
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[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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[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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