Measuring our Universe from Galaxy Redshift Surveys
Abstract
Galaxy redshift surveys have achieved significant progress over the last couple of decades. Those surveys tell us in the most straightforward way what our local Universe looks like. While the galaxy distribution traces the bright side of the Universe, detailed quantitative analyses of the data have even revealed the dark side of the Universe dominated by nonbaryonic dark matter as well as more mysterious dark energy (or Einstein’s cosmological constant). We describe several methodologies of using galaxy redshift surveys as cosmological probes, and then summarize the recent results from the existing surveys. Finally we present our views on the future of redshift surveys in the era of precision cosmology.
Keywords
Dark Matter Cosmic Microwave Background Cosmological Parameter Galaxy Formation Dark Matter Halo1 Introduction
Nowadays the exploration of the Universe can be performed by a variety of observational probes and methods over a wide range of the wavelengths: the temperature anisotropy map of the cosmic microwave background (CMB), the Hubble diagrams of nearby galaxies and distant Type Ia supernovae, widefield photometric and spectroscopic surveys of galaxies, the power spectrum and abundances of galaxy clusters in optical and Xray bands combined with the radio observation through the SunyaevZel’dovich effect, deep surveys of galaxies in submm, infrared, and optical bands, quasar surveys in radio and optical, strong and weak lensing of distant galaxies and quasars, highenergy cosmic rays, and so on. Undoubtedly gammarays, neutrinos, and gravitational radiation will join the above already crowded list.

Redshift surveys have unprecedented quantity and quality:
The numbers of galaxies and quasars in the spectroscopic sample of Two Degree Field (2dF) are ∼ 250, 000 and ∼ 30, 000, and will reach ∼ 800,000 and 100, 000 upon completion of the ongoing Sloan Digital Sky Survey (SDSS). These unprecedented numbers of the objects as well as the homogeneous selection criteria enable the precise statistical analysis of their distribution.

The Universe at z ≈ 1000 is well specified:
The firstyear WMAP (Wilkinson Microwave Anisotropy Probe) data [6] among others have established a set of cosmological parameters. This may be taken as the initial condition of the Universe from the pointofview of the structure evolution toward z = 0. In a sense, the origin of the Universe at z ≈ 1000 and the evolution of the Universe after the epoch are now equally important, but they constitute well separable questions that particle and observational cosmologists focus on, respectively.

Gravitational growth of dark matter component is well understood:
In addition, extensive numerical simulations of structure formation in the Universe has significantly advanced our understanding of the gravitational evolution of the dark matter component in the standard gravitational instability picture. In fact, we even have very accurate and useful analytic formulae to describe the evolution deep in its nonlinear regime. Thus we can now directly address the evolution of visible objects from the analysis of their redshift surveys separately from the nonlinear growth of the underlying dark matter gravitational potentials.

Formation and evolution of galaxies:
In the era of precision cosmology among others, the scientific goals of research using galaxy redshift surveys are gradually shifting from inferring a set of values of cosmological parameters using galaxy as their probes to understanding the origin and evolution of galaxy distribution given a set of parameters accurately determined by the other probes like CMB and supernovae.
2 Clustering in the Expanding Universe
2.1 The cosmological principle
Our current Universe exhibits a wealth of nonlinear structures, but the zeroth order description of our Universe is based on the assumption that the Universe is homogeneous and isotropic smoothed over sufficiently large scales. This statement is usually referred to as the cosmological principle. In fact, the cosmological principle was first adopted when observational cosmology was in its infancy; it was then little more than a conjecture, embodying’ Occam’s razor’ for the simplest possible model.

The ancient Indian cosmological principle:
The Universe is infinite in space and time and is infinitely heterogeneous.

The ancient Greek cosmological principle:
Our Earth is the natural center of the Universe.

The Copernican cosmological principle:
The Universe as observed from any planet looks much the same.

The (generalized) cosmological principle:
The Universe is (roughly) homogeneous and isotropic.

The perfect cosmological principle:
The Universe is (roughly) homogeneous in space and time, and is isotropic in space.

The anthropic principle:
A human being, as he/she is, can exist only in the Universe as it is.
Like with any other idea about the physical world, we cannot prove a model, but only falsify it. Proving the homogeneity of the Universe is particularly difficult as we observe the Universe from one point in space, and we can only deduce isotropy indirectly. The practical methodology we adopt is to assume homogeneity and to assess the level of fluctuations relative to the mean, and hence to test for consistency with the underlying hypothesis. If the assumption of homogeneity turns out to be wrong, then there are numerous possibilities for inhomogeneous models, and each of them must be tested against the observations.

CMB fluctuations
Ehlers, Garen, and Sachs [18] showed that by combining the CMB isotropy with the Copernican principle one can deduce homogeneity. More formally their theorem (based on the Liouville theorem) states that “If the fundamental observers in a dust spacetime see an isotropic radiation field, then the spacetime is locally given by the FriedmanRobertsonWalker (FRW) metric”. The COBE (COsmic Background Explorer) measurements of temperature fluctuations (ΔT/T = 10^{−5} on scales of 10°) give via the SachsWolfe effect \((\Delta T/T = {1 \over 3}\Delta \phi /{c^2})\) and the Poisson equation r.m.s. density fluctuations of δρ/ρ ∼ 10^{−4} on 1000 h^{−1} Mpc (see, e.g., [99]), which implies that the deviations from a smooth Universe are tiny.

Galaxy redshift surveys
The distribution of galaxies in local redshift surveys is highly clumpy, with the Supergalactic Plane seen in full glory. However, deeper surveys like 2dF and SDSS (see Section 6) show that the fluctuations decline as the lengthscales increase. Peebles [69] has shown that the angular correlation functions for the Lick and APM (Automatic Plate Measuring) surveys scale with magnitude as expected in a Universe which approaches homogeneity on large scales. While redshift surveys can provide interesting estimates of the fluctuations on intermediate scales (see, e.g., [72]), the problems of biasing, evolution, and Kcorrection would limit the ability of those redshift surveys to ‘prove’ the cosmological principle. Despite these worries the measurement of the power spectrum of galaxies derived on the assumption of an underlying FRW metric shows good agreement with the ACDM (cold dark matter) model.

Radio sources
Radio sources in surveys have a typical median redshift of \(\bar z \sim 1\), and hence are useful probes of clustering at high redshift. Unfortunately, it is difficult to obtain distance information from these surveys: The radio luminosity function is very broad, and it is difficult to measure optical redshifts of distant radio sources. Earlier studies claimed that the distribution of radio sources supports the cosmological principle. However, the wide range in intrinsic luminosities of radio sources would dilute any clustering when projected on the sky. Recent analyses of new deep radio surveys suggest that radio sources are actually clustered at least as strongly as local optical galaxies. Nevertheless, on very large scales the distribution of radio sources seems nearly isotropic.

Xray background
The Xray background (XRB) is likely to be due to sources at high redshift. The XRB sources are probably located at redshift z < 5, making them convenient tracers of the mass distribution on scales intermediate between those in the CMB as probed by COBE, and those probed by optical and IRAS redshift surveys. The interpretation of the results depends somewhat on the nature of the Xray sources and their evolution. By comparing the predicted multipoles to those observed by HEAO1, Scharf et al. [75] estimate the amplitude of fluctuations for an assumed shape of the density fluctuations. The observed fluctuations in the XRB are roughly as expected from interpolating between the local galaxy surveys and the COBE and other CMB experiments. The r.m.s. fluctuations δρ/ρ on a scale of ∼ 600 h^{−1} Mpc are less than 0.2%.
The rest of the current section is devoted to a brief review of the homogeneous and isotropic cosmological model. Further details may be easily found in standard cosmology textbooks [96, 62, 69, 64, 10, 63].
2.2 From the Einstein equation to the Friedmann equation
2.3 Expansion law and age of the Universe
 Einsteinde Sitter model \(({\Omega _{\rm{m}}} = 1,{\Omega _\Lambda} = 0)\):$$a(t) = {\left({{t \over {{t_0}}}} \right)^{2/3}},\quad\quad{t_0} = {2 \over {3{H_0}}}.$$(19)
 Open model with vanishing cosmological constant (Ω_{m} < 1, Ω_{Λ} = 0):$$a = {{{\Omega _{\rm{m}}}} \over {2(1  {\Omega _{\rm{m}}})}}(\cosh \theta  1),$$(20)$${H_0}t = {{{\Omega _{\rm{m}}}} \over {2{{(1  {\Omega _{\rm{m}}})}^{3/2}}}}(\sinh \theta  \theta)$$(21)$${H_0}{t_0} = {1 \over {1  {\Omega _{\rm{m}}}}}  {{{\Omega _{\rm{m}}}} \over {2{{(1  {\Omega _{\rm{m}}})}^{3/2}}}}\ln {{2  {\Omega _{\rm{m}}} + 2\sqrt {1  {\Omega _{\rm{m}}}}} \over {{\Omega _{\rm{m}}}}}.$$(22)
 Spatiallyflat model with cosmological constant (Ω_{m} < 1, Ω_{Λ} = 1 − Ω_{m}:$$a(t) = {\left({{{{\Omega _{\rm{m}}}} \over {1  {\Omega _{\rm{m}}}}}} \right)^{1/3}}{\left[ {\sinh {{3\sqrt {1  {\Omega _{\rm{m}}}}} \over 2}{H_0}t} \right]^{2/3}},$$(23)$${H_0}{t_0} = {1 \over {3\sqrt {1  {\Omega _{\rm{m}}}}}}\ln {{2  {\Omega _{\rm{m}}} + 2\sqrt {1  {\Omega _{\rm{m}}}}} \over {{\Omega _{\rm{m}}}}}.$$(24)
The present age of the Universe in units of (h/0.7)^{−1} Gyr.
Ω_{m}  Open model (Ω_{Λ} = 0)  Spatiallyflat model (Ω_{Λ} = 1 − Ω_{m}) 

1.0  9.3  9.3 
0.5  10.5  11.6 
0.3  11.3  13.5 
0.1  12.5  17.8 
0.01  13.9  28.0 
2.4 Einstein’s static model and Lemaître’s model
So far we have shown that solutions of the Einstein equation are dynamical in general, i.e., the scale factor a is timedependent. As a digression, let us examine why Einstein once introduced the Λterm to obtain a static cosmological solution. This is mainly important for historical reasons, but is also interesting to observe how the operationally identical parameter (the Λterm, the cosmological constant, the vacuum energy, the dark energy) shows up in completely different contexts in the course of the development of cosmological physics.
2.5 Vacuum energy as an effective cosmological constant
If the Λterm is introduced in the l.h.s., it should be constant to satisfy the energymomentum conservation T_{ μν };^{ ν }. Once it is regarded as a sort of matter field in the r.h.s., however, it does not have to be constant. In fact, the above example shows that the equation of state for the field has w = −1 only in special cases. This is why recent literature refers to the field as dark energy instead of the cosmological constant.
2.6 Gravitational instability
2.7 Linear growth rate of the density fluctuation
 Einsteinde Sitter model (Ω_{m} = 1, Ω_{Λ} = 0):$$D(z) = {1 \over {1 + z}}.$$(64)
 Open model with vanishing cosmological constant (Ω_{m} < 1, Ω_{Λ} = 0):$$D(z) \propto 1 + {3 \over x} + 3\sqrt {{{1 + x} \over {{x^3}}}} \ln (\sqrt {1 + x}  \sqrt x),\quad \quad x \equiv {{1  {\Omega _{\rm{m}}}} \over {{\Omega _{\rm{m}}}(1 + z)}}.$$(65)
 Spatiallyflat model with cosmological constant (Ω_{m} < 1, Ω_{Λ} = 1 − Ω_{m}):$$D(z) \propto \sqrt {1 + {2 \over {{x^3}}}} \int\nolimits_0^x {{{\left({{u \over {2 + {u^3}}}} \right)}^{3/2}}du,} \quad \quad x \equiv {{{2^{1/3}}{{(\Omega _{\rm{m}}^{ 1}  1)}^{1/3}}} \over {1 + z}}.$$(66)
3 Statistics of Cosmological Density Fluctuations
3.1 Gaussian random field
The Gaussian nature of the primordial density field is preserved in its linear evolution stage, but this is not the case in the nonlinear stage. This is clear even from the definition of the Gaussian distribution: Equation (71) formally assumes that the density contrast distributes symmetrically in the range of −∞ < δ_{ i } < ∞, but in the real density field δ_{ i } cannot be less than −1. This assumption does not make any practical difference as long as the fluctuations are (infinitesimally) small, but it is invalid in the nonlinear regime where the typical amplitude of the fluctuations exceeds unity.
Incidentally the onepoint phase distribution turns out to be essentially uniform even in a strongly nonGaussian field [81, 21]. Thus it is unlikely to extract useful information directly out of it mainly due to the cyclic property of the phase. Very recently, however, Matsubara [51] and Hikage et al. [31] succeeded in detecting a signature of phase correlations in Fourier modes of mass density fields induced by nonlinear gravitational clustering using the distribution function of the phase sum of the Fourier modes for triangle wavevectors. Several different statistics which carry the phase information have been also proposed in cosmology, including the void probability function [97], the genus statistics [26], and the Minkowski functionals [57, 76].
3.2 Lognormal distribution
A probability distribution function (PDF) of the cosmological density fluctuations is the most fundamental statistic characterizing the largescale structure of the Universe. As long as the density fluctuations are in the linear regime, their PDF remains Gaussian. Once they reach the nonlinear stage, however, their PDF significantly deviates from the initial Gaussian shape due to the strong nonlinear modecoupling and the nonlocality of the gravitational dynamics. The functional form for the resulting PDFs in nonlinear regimes are not known exactly, and a variety of phenomenological models have been proposed [34, 74, 9, 25].
At this point, the transformation (85) is nothing but a mathematical procedure to relate the Gaussian and the lognormal functions. Thus there is no physical reason to believe that the new field δ should be regarded as a nonlinear density field evolved from g even in an approximate sense. In fact it is physically unacceptable since the relation, if taken at face value, implies that the nonlinear density field is completely determined by its linear counterpart locally. We know, on the other hand, that the nonlinear gravitational evolution of cosmological density fluctuations proceeds in a quite nonlocal manner, and is sensitive to the surrounding mass distribution. Nevertheless the fact that the lognormal PDF provides a good fit to the simulation data, empirically implies that the transformation (85) somehow captures an important aspect of the nonlinear evolution in the real Universe.
3.3 Higherorder correlation functions
3.4 Genus statistics
3.5 Minkowski functionals
In fact, genus is one of the complete sets of N + 1 quantities, known as the Minkowski functionals (MFs), which determine the morphological properties of a pattern in Ndimensional space. In the analysis of galaxy redshift survey data, one considers isodensity contours from the threedimensional density contrast field δ by taking its excursion set F_{ ν }, i.e., the set of all points where the density contrast δ exceeds the threshold level ν as was the case in the case of genus described in the above subsection.
The above MFs can be indeed interpreted as wellknown geometric quantities: the volume fraction V_{0}(ν), the total surface area V_{1}(ν), the integral mean curvature V_{2}(ν), and the integral Gaussian curvature, i.e., the Euler characteristic V_{3}(ν). In our current definitions (see Equations (101, 108), or Equations (102, 115)), one can easily show that V_{3}(ν) reduces simply to −G(ν). The MFs were first introduced to cosmological studies by Mecke et al. [57], and further details may be found in [57, 32]. Analytic expressions of MFs in weakly nonGaussian fields are derived in [52].
4 Galaxy Biasing
4.1 Concepts and definitions of biasing
As discussed above, luminous objects, such as galaxies and quasars, are not direct tracers of the mass in the Universe. In fact, a difference of the spatial distribution between luminous objects and dark matter, or a bias, has been indicated from a variety of observations. Galaxy biasing clearly exists. The fact that galaxies of different types cluster differently (see, e.g., [16]) implies that not all of them are exact tracers of the underlying mass distribution (see also Section 6).
The above deterministic linear biasing is not based on a reasonable physical motivation. If b > 1, it must break down in deep voids because values of δ_{g} below −1 are forbidden by definition. Even in the simple case of no evolution in comoving galaxy number density, the linear biasing relation is not preserved during the course of fluctuation growth. Nonlinear biasing, where b varies with δ_{m}, is inevitable.
Indeed, an analytical model for biasing of halos on the basis of the extended PressSchechter approximation [59] predicts that the biasing is nonlinear and provides a useful approximation for its behavior as a function of scale, time, and mass threshold. Nbody simulations provide a more accurate description of the nonlinearity of the halo biasing confirming the validity of the Mo and White model [35, 103].
4.2 Modeling biasing
Biasing is likely to be stochastic, not deterministic [15]. An obvious part of this stochasticity can be attributed to the discrete sampling of the density field by galaxies, i.e., the shot noise. In addition, a statistical, physical scatter in the efficiency of galaxy formation as a function of δ_{m} is inevitable in any realistic scenario. For example, the random variations in the density on smaller scales is likely to be reflected in the efficiency of galaxy formation. As another example, the local geometry of the background structure, via the deformation tensor, must play a role too. Such ‘hidden variables’ would show up as physical scatter in the densitydensity relation [87].
 a local stochastic nonlinear bias,$${B_{{\rm{obj}}}}(x,z\vert R) = b_{{\rm{obj}}}^{({\rm{sn}})}[x,z,R,{\delta _{\rm{m}}}(x,z\vert R),\vec{\mathcal{A}}(x,z\vert R), \ldots ],$$(120)
 a local deterministic nonlinear bias,and$${B_{{\rm{obj}}}}(x,z\vert R) = b_{{\rm{obj}}}^{({\rm{dn}})}[z,R,{\delta _{\rm{m}}}(x,z\vert R)],$$(121)
 a local deterministic linear bias,$${B_{{\rm{obj}}}}(x,z\vert R) = {b_{{\rm{obj}}}}(z,R).$$(122)
4.3 Density peaks and dark matter halos as toy models for galaxy biasing
Let us illustrate the biasing from numerical simulations by considering two specific and popular models: primordial density peaks and dark matter halos [86]. We use the Nbody simulation data of L = 100 h^{−1} Mpc again for this purpose [36]. We select density peaks with the threshold of the peak height ν_{th} = 1.0, 2.0, and 3.0. As for the dark matter halos, these are identified using the standard friendoffriend algorithm with a linking length of 0.2 in units of the mean particle separation. We select halos of mass larger than the threshold M_{th} = 2.0 × 10^{12} h^{−1} M_{⊙}, M_{th} = 5.0 × 10^{12} h^{−1} M_{⊙}, and M_{th} = 1.0 × 10^{13} h^{−1} M_{⊙}.
We use twopoint correlation functions to quantify stochasticity and nonlinearity in biasing of peaks and halos, and explore the signature of the redshiftspace distortion. Since we are interested in the relation of the biased objects and the dark matter, we introduce three different correlation functions: the autocorrelation functions of dark matter and the objects, ξ_{mm} and ξ_{oo}, and their crosscorrelation function ξ_{om}. In the present case, the subscript o refers to either h (halos) or ν (peaks). We also use the superscripts R and S to distinguish quantities defined in real and redshift spaces, respectively. We estimate those correlation functions using the standard paircount method. The correlation function ξ^{(S)} is evaluated under the distantobserver approximation.
4.4 Biasing of galaxies in cosmological hydrodynamic simulations
Popular models of the biasing based on the peak or the dark halos are successful in capturing some essential features of biasing. None of the existing models of bias, however, seems to be sophisticated enough for the coming precision cosmology era. The development of a more detailed theoretical model of bias is needed. A straightforward next step is to resort to numerical simulations which take account of galaxy formation even if phenomenological at this point. We show an example of such approaches from Yoshikawa et al. [103] who apply cosmological smoothed particle hydrodynamic (SPH) simulations in the LCDM model with particular attention to the comparison of the biasing of dark halos and simulated galaxies (see also [78]).
Galaxies in their simulations are identified as clumps of cold and dense gas particles which satisfy the Jeans condition and have the SPH density more than 100 times the mean baryon density at each redshift. Dark halos are identified with a standard friendoffriend algorithm; the linking length is 0.164 times the mean separation of dark matter particles, for instance, at z = 0. In addition, they identify the surviving highdensity substructures in dark halos, DM cores (see [103] for further details).
The correlation functions of galaxies are almost unchanged with redshift, and the correlation functions of dark halos only slightly evolve between z = 0 and 2. By contrast, the amplitude of the dark matter correlation functions evolve rapidly by a factor of ∼ 10 from z = 2 to z = 0. The biasing parameter b_{ξ,g} is larger at a higher redshift, for example, b_{ξ,g} ≃ 2–2.5 at z = 2. The biasing parameter b_{ξ,h} for dark halos is systematically lower than that of galaxies and DM cores again due to the volume exclusion effect. At z = 0, galaxies and DM cores are slightly antibiased relative to dark matter at r ≃ 1 ^{h−1} Mpc. In lower panels, we also plot the onepoint biasing parameter b_{var,i} ≡ σ_{ i }/σ_{m} at r = R_{s} for comparison. In general we find that b_{ξ,i} is very close to b_{var,i} at z ∼ 0, but systematically lower than b_{var,i} at higher redshifts.
For each galaxy identified at z = 0, we define its formation redshift z_{f} by the epoch when half of its cooled gas particles satisfy our criteria of galaxy formation. Roughly speaking, z_{f} corresponds to the median formation redshift of stars in the presentday galaxies. We divide all simulated galaxies at z = 0 into two populations (the young population with z_{f} < 1.7 and the old population with z_{f} > 1.7) so as to approximate the observed number ratio of 3/1 for latetype and earlytype galaxies.
4.5 Halo occupation function approach for galaxy biasing
Since the clustering of dark matter halos is well understood now, one can describe the galaxy biasing if the halo model is combined with the relation between the halos and luminous objects. This is another approach to galaxy biasing, halo occupation function (HOF), which has become very popular recently. Indeed the basic idea behind HOF has a long history, but the model predictions have been significantly improved with the recent accurate models for the mass function, the biasing and the density profile of dark matter halos. We refer the readers to an extensive review on the HOF by Cooray and Sheth [13]. Here we briefly outline this approach.
The halo occupation formalism, although simple, provides a useful framework in deriving constraints on galaxy formation models from large data sets of the upcoming galaxy redshift surveys. For example, Zehavi et al. [105] used the halo occupation formalism to model departures from a power law in the SDSS galaxy correlation function. They demonstrated that this is due to the transition from a largescale regime dominated by galaxy pairs in different halos to a smallscale regime dominated by those in the same halo. Magliocchetti and Porciani [47] applied the halo occupation formalism to the 2dFGRS clustering results per spectral type of Madgwick et al. [45]. This provides constraints on the distribution of latetype and earlytype galaxies within the dark matter halos of different mass.
5 Relativistic Effects Observable in Clustering at High Redshifts
Redshift surveys of galaxies definitely serve as the central database for observational cosmology. In addition to the existing shallower surveys (z < 0.2), clustering in the Universe in the range z = 1–3 has been partially revealed by, for instance, the Lymanbreak galaxies and Xray selected AGNs. In particular, the 2dF and SDSS QSO redshift surveys promise to extend the observable scale of the Universe by an order of magnitude, up to a few Gpc. A proper interpretation of such redshift surveys in terms of the clustering evolution, however, requires an understanding of many cosmological effects which can be neglected for z ≪ 1 and thus have not been considered seriously so far. These cosmological contaminations include linear redshiftspace (velocity) distortion, nonlinear redshiftspace (velocity) distortion, cosmological redshiftspace (geometrical) distortion, and the cosmological lightcone effect.
We describe a theoretical formalism to incorporate those effects, in particular the cosmological redshiftdistortion and lightcone effects, and present several specific predictions in CDM models. The details of the material presented in this section may be found in [83, 101, 100, 46, 28, 29].
5.1 Cosmological lightcone effect on the twopoint correlation functions
Observing a distant patch of the Universe is equivalent to observing the past. Due to the finite light velocity, a lineofsight direction of a redshift survey is along the time, as well as spatial, coordinate axis. Therefore the entire sample does not consist of objects on a constanttime hypersurface, but rather on a lightcone, i.e., a null hypersurface defined by observers at z = 0. This implies that many properties of the objects change across the depth of the survey volume, including the mean density, the amplitude of spatial clustering of dark matter, the bias of luminous objects with respect to mass, and the intrinsic evolution of the absolute magnitude and spectral energy distribution. These aspects should be properly taken into account in order to extract cosmological information from observed samples of redshift surveys.
 1.
nonlinear gravitational evolution,
 2.
linear redshiftspace distortion,
 3.
nonlinear redshiftspace distortion,
 4.
weighted averaging over the lightcone,
 5.
cosmological redshiftspace distortion due to the geometry of the Universe, and
 6.
objectdependent clustering bias.
5.2 Evaluating twopoint correlation functions from Nbody simulation data
The theoretical modeling described above was tested against simulation results by Hamana, Colombi, and Suto [28]. Using cosmological Nbody simulations in SCDM and ΛCDM models, they generated lightcone samples as follows: First, they adopt a distance observer approximation and assume that the lineofsight direction is parallel to the Zaxis regardless of its (X, Y) position. Second, they periodically duplicate the simulation box along the Zdirection so that at a redshift z, the position and velocity of those particles locating within an interval χ(z) ± Δχ(z) are dumped, where Δχ(z) is determined by the output timeinterval of the original Nbody simulation. Finally they extract five independent (nonoverlapping) coneshape samples with the angular radius of 1 degree (the fieldofview of π degree^{2}). In this manner, they have generated mock data samples on the lightcone continuously extending up to z = 0.4 (relevant for galaxy samples) and z = 2.0 (relevant for QSO samples) from the small and large boxes, respectively.
5.3 Cosmological redshiftspace distortion
Consider a spherical object at high redshift. If the wrong cosmology is assumed in interpreting the distanceredshift relation along the line of sight and in the transverse direction, the sphere will appear distorted. Alcock and Paczynski [2] pointed out that this curvature effect could be used to estimate the cosmological constant. Matsubara and Suto [54] and Ballinger, Peacock, and Heavens [3] developed a theoretical framework to describe the geometrical distortion effect (cosmological redshift distortion) in the twopoint correlation function and the power spectrum of distant objects, respectively. Certain studies were less optimistic than others about the possibility of measuring this AlcockPaczynski effect. For example, Ballinger, Peacock, and Heavens [3] argued that the geometrical distortion could be confused with the dynamical redshift distortions caused by peculiar velocities and characterized by the linear theory parameter \(\beta \equiv \Omega _{\rm{m}}^{0.6}/b\). Matsubara and Szalay [55, 56] showed that the typical SDSS and 2dF samples of normal galaxies at low redshift (z ∼ 0.1) have sufficiently low signaltonoise, but they are too shallow to detect the AlcockPaczynski effect. On the other hand, the quasar SDSS and 2dFGRS surveys are at a useful redshift, but they are too sparse. A more promising sample is the SDSS Luminous Red Galaxies survey (out to redshift z ∼ 0.5) which turns out to be optimal in terms of both depth and density.
While this analysis is promising, it remains to be tested if nonlinear clustering and complicated biasing (which is quite plausible for red galaxies) would not ‘contaminate’ the measurement of the equation of state. Even if the AlcockPaczynski test turns out to be less accurate than other cosmological tests (e.g., CMB and SN Ia), the effect itself is an interesting and important ingredient in analyzing the clustering pattern of galaxies at high redshifts. We shall now present the formalism for this effect.
5.4 Twopoint clustering statistics on a lightcone in cosmological redshift space
Note that k_{s} and x_{s}, defined in \(P_l^{({\rm{CRD}})}({k_{\rm{s}}};z)\) and \(\xi _l^{{\rm{CRD}}}({x_{\rm{s}}};z)\), are related to their comoving counterparts at z through Equations (158) and (154), while those in \(P_l^{({\rm{LC,CRD}})}({k_{\rm{s}}})\) and \(\xi _l^{({\rm{LC,CRD}})}({x_{\rm{s}}})\) are not specifically related to any comoving wavenumber and separation. Rather, they correspond to the quantities averaged over the range of z satisfying the observable conditions \({x_{\rm{s}}} = (c/{H_0})\sqrt {\delta {z^2} + {z^2}\delta {\theta ^2}}\) and k_{s} = 2π/x_{s}.
Let us show specific examples of the twopoint clustering statistics on a lightcone in cosmological redshift space. We consider SCDM and LCDM models, and take into account the selection functions relevant to the upcoming SDSS spectroscopic samples of galaxies and quasars by adopting the Bband limiting magnitudes of 19 and 20, respectively.
The upper and lower panels correspond to magnitudelimited samples of galaxies (B < 19 in 0 < z < z_{max} = 0.2; no bias model) and QSOs (B < 20 in 0 < z < z_{max} = 5; Fry’s linear bias model), respectively. We present the results normalized by the realspace power spectrum in linear theory P^{(R,lin)}(k; z) [4], and \(P_0^{({\rm{S}})}(k;z = 0),P_0^{({\rm{S}})}(k;z = {z_{\max}}),P_0^{({\rm{CRD}})}({k_{\rm{s}}};z = {z_{\max}})\) and \(P_0^{({\rm{LC,CRD}})}({k_{\rm{s}}})\) are computed using the nonlinear power spectrum [67].
Consider first the results for the galaxy sample (upper panels). On linear scales (k <; 0.1 h Mpc−1), \(P_0^{({\rm{S}})}(k;z = 0)\) plotted in dashed lines is enhanced relative to that in real space, mainly due to a linear redshiftspace distortion (the Kaiser factor in Equation (131)). For nonlinear scales, the nonlinear gravitational evolution increases the power spectrum in real space, while the fingerofGod effect suppresses that in redshift space. Thus, the net result is sensitive to the shape and the amplitude of the fluctuation spectrum, itself; in the LCDM model that we adopted, the nonlinear gravitational growth in real space is stronger than the suppression due to the fingerofGod effect. Thus, \(P_0^{({\rm{S}})}(k;z = 0)\) becomes larger than its realspace counterpart in linear theory. In the SCDM model, however, this is opposite and \(P_0^{({\rm{S}})}(k;z = 0)\) becomes smaller.
The power spectra at z = 0.2 (dashdotted lines) are smaller than those at z = 0 by the corresponding growth factor of the fluctuations, and one might expect that the amplitude of the power spectra on the lightcone (solid lines) would be inbetween the two. While this is correct, if we use the comoving wavenumber, the actual observation on the lightcone in the cosmological redshift space should be expressed in terms of k_{s} (see Equation (158)). If we plot the power spectra at z = 0.2 taking into account the geometrical distortion, \(P_0^{({\rm{CRD}})}({k_{\rm{s}}};z = 0.2)\) in the dotted lines becomes significantly larger than \(P_0^{({\rm{S}})}(k;z = 0.2)\). Therefore, \(P_0^{({\rm{LC,CRD)}}}({k_{\rm{s}}})\) should take a value between those of \(P_0^{({\rm{CRD)}}}({k_{\rm{s}}};z = 0) = P_0^{({\rm{S}})}(k;z = 0)\). This explains the qualitative features shown in the upper panels of Figure 18. As a result, both the cosmological redshiftspace distortion and the lightcone effect substantially change the predicted shape and amplitude of the power spectra, even for the galaxy sample [60]. The results for the QSO sample can be basically understood in a similar manner, except that the evolution of the bias makes a significant difference, since the sample extends to much higher redshifts.
In fact, since the resulting predictions are sensitive to the bias, which is unlikely to quantitatively be specified by theory, the present methodology will find two completely different applications. For relatively shallower catalogues, like galaxy samples, the evolution of bias is not supposed to be so strong. Thus, one may estimate the cosmological parameters from the observed degree of the redshift distortion, as has been conducted conventionally. Most importantly, we can correct for the systematics due to the lightcone and geometrical distortion effects, which affect the estimate of the parameters by ∼ 10%. Alternatively, for deeper catalogues like highredshift quasar samples, one can extract information on the objectdependent bias only by correcting the observed data on the basis of our formulae.
In a sense, the former approach uses the lightcone and geometrical distortion effects as real cosmological signals, while the latter regards them as inevitable, but physically removable, noise. In both cases, the present methodology is important in properly interpreting the observations of the Universe at high redshifts.
6 Recent Results from 2dF and SDSS
6.1 The latest galaxy redshift surveys
Redshifts surveys in the 1980s and the 1990s (e.g., the CfA, IRAS, and Las campanas surveys) measured thousands to tens of thousands galaxy redshifts. Multifibre technology now allows us to measure redshifts of millions of galaxies. Below we summarize briefly the properties of the main new surveys 2dFGRS, SDSS, 6dF, VIRMOS, DEEP2, and we discuss key results from 2dFGRS and SDSS. Further analysis of these surveys is currently underway.
6.1.1 The 2dF galaxy redshift survey
6.1.2 The SDSS galaxy redshift survey
The SDSS (Sloan Digital Sky Survey) is a U.S.JapanGermany joint project to image a quarter of the Celestial Sphere at high Galactic latitude as well as to obtain spectra of galaxies and quasars from the imaging data[93]. The dedicated 2.5 meter telescope at Apache Point Observatory is equipped with a multiCCD camera with five broad bands centered at 3561, 4676, 6176, 7494, and 8873 Å. For further details of SDSS, see [102, 80]
6.1.3 The 6dF galaxy redshift survey
The 6dF (6degree Field) [91] is a survey of redshifts and peculiar velocities of galaxies selected primarily in the Near Infrared from the new 2MASS (Two Micron All Sky Survey) catalogue[90]. One goal is to measure redshifts of more than 170,000 galaxies over nearly the entire Southern sky. Another exciting aim of the survey is to measure peculiar velocities (using 2MASS photometry and 6dF velocity dispersions) of about 15,000 galaxies out to 150 h^{−1} Mpc. The high quality data of this survey could revive peculiar velocities as a cosmological probe (which was very popular about 10–15 years ago). Observations have so far obtained nearly 40,000 redshifts and completion is expected in 2005.
6.1.4 The DEEP galaxy redshift survey
The DEEP survey is a twophased project using the Keck telescopes to study the properties and distribution of high redshift galaxies [92]. Phase 1 used the LRIS spectrograph to study a sample of ∼ 1000 galaxies to a limit of I = 24.5. Phase 2 of the DEEP project will use the new DEIMOS spectrograph to obtain spectra of ∼ 65,000 faint galaxies with redshifts z ∼ 1. The scientific goals are to study the evolution of properties of galaxies and the evolution of the clustering of galaxies compared to samples at low redshift. The survey is designed to have the fidelity of local redshift surveys such as the LCRS survey, and to be complementary to ongoing large redshift surveys such as the SDSS project and the 2dF survey. The DEIMOS/DEEP or DEEP2 survey will be executed with resolution R 4000, and we therefore expect to measure linewidths and rotation curves for a substantial fraction of the target galaxies. DEEP2 will thus also be complementary to the VLT/VIRMOS project, which will survey more galaxies in a larger region of the sky, but with much lower spectral resolution and with fewer objects at high redshift.
6.1.5 The VIRMOS galaxy redshift survey
The ongoing FrancoItalian VIRMOS project[94] has delivered the VIMOS spectrograph for the European Southern Observatory Very Large Telescope (ESOVLT). VIMOS is a VIsible imaging MultiObject Spectrograph with outstanding multiplex capabilities: With 10 arcsec slits, spectra can be taken of 600 objects simultaneously. In integral field mode, a 6400fibre Integral Field Unit (IFU) provides spectroscopy for all objects covering a 54 × 54 arcsec^{2} area. VIMOS therefore provides unsurpassed efficiency for large surveys. The VIRMOS project consists of: construction of VIMOS, and a Mask Manufacturing Unit for the ESOVLT. The VIRMOSVLT Deep Survey (VVDS), a comprehensive imaging and redshift survey of the deep Universe based on more than 150,000 redshifts in four 4 squaredegree fields.
6.2 Cosmological parameters from 2dFGRS
6.2.1 The power spectrum of 2dF Galaxies on large scales
An initial estimate of the convolved, redshiftspace power spectrum of the 2dFGRS was determined by Percival et al. [72] for a sample of 160,000 redshifts. On scales 0.02 h Mpc^{−1} < k < 0.15 h Mpc^{−1}, the data are fairly robust and the shape of the power spectrum is not significantly affected by redshiftspace distortion or nonlinear effects, while its overall amplitude is increased due to the linear redshiftspace distortion effect (see Section 5).
6.2.2 An upper limit on neutrino masses
The recent results of atmospheric and solar neutrino oscillations [24, 1] imply nonzero masssquared differences of the three neutrino flavours. While these oscillation experiments do not directly determine the absolute neutrino masses, a simple assumption of the neutrino mass hierarchy suggests a lower limit on the neutrino mass density parameter, Ω_{ ν } = m_{ν,tot}h^{−2}/(94 eV) ≈ 0.001. Large scale structure data can put an upper limit on the ratio Ω_{ ν }/Ω_{m} due to the neutrino’ free streaming’ effect [33]. By comparing the 2dF galaxy powerspectrum of fluctuations with a fourcomponent model (baryons, cold dark matter, a cosmological constant, and massive neutrinos) it was estimated that Ω_{ ν }/Ω_{m} < 0.13 (95% CL), or with concordance prior of Ω_{m} = 0.3, Ω_{ ν } < 0.04, or an upper limit of ∼ 2 eV on the total neutrino mass, assuming a prior of h ≈ 0.7 [20, 19] (see Figure 24). In order to minimize systematic effects due to biasing and nonlinear growth, the analysis was restricted to the range 0.02 < k < 0.15 h Mpc^{−1}. Additional cosmological data sets bring down this upper limit by a factor of two [79].
6.2.3 Combining 2dFGRS and CMB
While the CMB probes the fluctuations in matter, the galaxy redshift surveys measure the perturbations in the light distribution of particular tracer (e.g., galaxies of certain type). Therefore, for a fixed set of cosmological parameters, a combination of the two can better constrain cosmological parameters, and it can also provide important information on the way galaxies are ‘biased’ relative to the mass fluctuations,
Recent CMB measurements have been used in combination with the 2dF power spectrum. Efstathiou et al. [17] showed that 2dFGRS+CMB provide evidence for a positive cosmological constant Ω_{Λ} ∼ 0.7 (assuming w = −1), independently of the studies of supernovae Ia. As explained in [72], the shapes of the CMB and the 2dFGRS power spectra are insensitive to Dark Energy. The main important effect of the dark energy is to alter the angular diameter distance to the last scattering, and thus the position of the first acoustic peak. Indeed, the latest result from a combination of WMAP with 2dFGRS and other probes gives \(h = 0.71_{ 0.03}^{+ 0.04},\,\,{\Omega _{\rm{b}}}{h^2} = 0.0224 \pm 0.0009,\,\>{\Omega _{\rm{m}}}{h^2} = 0.135_{ 0.009}^{+ 0.008}\), σ_{8} = 0.84 ± 0.04, Ω_{tot} = 1.02 ± 0.02, and w < −0.78 (95% CL, assuming w ≥ −1) [79].
6.2.4 Redshiftspace distortion
An independent measurement of cosmological parameters on the basis of 2dFGRS comes from redshiftspace distortions on scales ≲ 10 h^{−1} Mpc: a correlation function ζ(π, σ) in parallel and transverse pair separations π and σ. As described in Section 5, the distortion pattern is a combination of the coherent infall, parameterized by \(\beta = \Omega _{\rm{m}}^{0.6}/b\) and random motions modelled by an exponential velocity distribution function (see Equation (133)). This methodology has been applied by many authors. For instance, Peacock et al. [66] derived β(L_{s} = 0.17, z_{s} = 1.9L_{*}) = 0.43 ± 0.07, and Hawkins et al. [30] obtained β(L_{s} = 0.15, z_{s} = 1.4L_{*}) = 0.49 ± 0.09 and a velocity dispersion σ_{P} = 506 ± 52 km s^{−1}. Using the full 2dF+CMB likelihood function on the (b, Ω_{m}) plane, Lahav et al. [42] derived a slightly larger (but consistent within the quoted errorbars) value, β(L_{s} = 0.17, z_{s} = 1.9L_{*}) ≃ 0.48 ± 0.06.
6.2.5 The bispectrum and higher moments
6.3 Luminosity and spectraltype dependence of galaxy clustering
Although biasing was commonly neglected until the early 1980s, it has become evident observationally that on scales ≲ 10 h^{−1} Mpc different galaxy populations exhibit different clustering amplitudes, the socalled morphologydensity relation [16]. As discussed in Section 4, galaxy biasing is naturally predicted from a variety of theoretical considerations as well as direct numerical simulations [37, 59, 15, 87, 86, 103]. Thus, in this Section we summarize the extent to which the galaxy clustering is dependent on the luminosity, spectraltype, and color of the galaxy sample from the 2dFGRS and SDSS.
6.3.1 2dFGRS: Clustering per luminosity and spectral type
Another statistic was applied recently by Wild et al. [98] and Conway et al. [12], of a joint countsincells on 2dFGRS galaxies, classified by both color and spectral type. Exact linear bias is ruled out on all scales. The counts are better fitted to a bivariate lognormal distribution. On small scales there is evidence for stochasticity. Further investigation of galaxy formation models is required to understand the origin of the stochasticity.
6.3.2 SDSS: Twopoint correlation functions per luminosity and color
Zehavi et al. [104] analyzed the Early Data Release (EDR) sample of the SDSS 30,000 galaxies to explore the clustering of per luminosity and color. The inferred realspace correlation function is well described by a single powerlaw: ζ(r) = (r/6.1 ± 0.2 h^{−1} Mpc)^{−1.75±0.03} for 0.1 h^{−1} Mpc ≤ r ≤ 16 h^{−1} Mpc. The galaxy pairwise velocity dispersion is σ_{12} ≈ 600 ± 100 km s^{−1} for projected separations 0.15 h^{−1} Mpc ≤ r_{ p } ≤ 5 h^{−1} Mpc. When divided by color, the red galaxies exhibit a stronger and steeper realspace correlation function and a higher pairwise velocity dispersion than do the blue galaxies. In agreement with 2dFGRS there is clear evidence for a scaleindependent luminosity bias at r ∼ 10 h^{−1} Mpc. Subsamples with absolute magnitude ranges centered on M_{*} − 1.5,
M_{*}, and M_{*} + 1.5 have realspace correlation functions that are parallel power laws of slope ≈ −1.8 with correlation lengths of approximately 7.4 h^{−1} Mpc, 6.3 h^{−1} Mpc, and 4.7 h^{−1} Mpc, respectively.
6.3.3 SDSS: Threepoint correlation functions and the nonlinear biasing of galaxies per luminosity and color
As we have seen in Section 6.3.2, galaxy clustering is sensitive to the intrinsic properties of the galaxy samples under consideration, including their morphological types, colors, and luminosities. Nevertheless the previous analyses were not able to examine those dependences of 3PCFs because of the limited number of galaxies. Indeed Kayo et al. [39] were the first to perform the detailed analysis of 3PCFs explicitly taking account of the morphology, color, and luminosity dependence. They constructed volumelimited samples from a subset of the SDSS galaxy redshift data, ‘Largescale Structure Sample 12’. Specifically they divided each volume limited sample into color subsamples of red (blue) galaxies, which consist of 7949 (8329), 8930 (8155), and 3706 (3829) galaxies for −22 < M_{r} − 5 log h < −21, −21 < M_{r} − 5 log h < −20, and −20 < M_{r} − 5 log h < −19, respectively.
Such behavior is unlikely to be explained by any simple model inspired by the perturbative expansion like Equation (176). Rather it indeed points to a kind of regularity or universality of the clustering hierarchy behind galaxy formation and evolution processes. Thus the galaxy biasing seems much more complex than the simple deterministic and linear model. More precise measurements of 3PCFs and even higherorder statistics with future SDSS datasets would be indeed valuable to gain more specific insights into the empirical biasing model.
6.4 Topology of the Universe: Analysis of SDSS galaxies in terms of Minkowski functionals
All the observational results presented in the preceding Sections 6.1, 6.2, and 6.3 were restricted to the twopoint statistics. As emphasized in Section 3, the clustering pattern of galaxies has much richer content than the twopoint statistics can probe. Historically the primary goal of the topological analysis of galaxy catalogues was to test Gaussianity of the primordial density fluctuations. Although the major role for that goal has been superseded by the CMB map analysis [41], the proper characterization of the morphology of largescale structure beyond the twopoint statistics is of fundamental importance in cosmology. In order to illustrate a possibility to explore the topology of the Universe by utilizing the new large surveys, we summarize the results of the Minkowski Functionals (MF) analysis of SDSS galaxy data [32].
In an apparentmagnitude limited catalogue of galaxies, the average number density of galaxies decreases with distance because only increasingly bright galaxies are included in the sample at larger distance. With the large redshift surveys it is possible to avoid this systematic change in both density and galaxy luminosity by constructing volumelimited samples of galaxies, with cuts on both absolutemagnitude and redshift. This is in particular useful for analyses such as MF and was carried out in the analysis shown here.
The good match between the observed MFs and the mock predictions based on the LCDM model with the initial randomGaussianity, as illustrated in Figure 31, might be interpreted to imply that the primordial Gaussianity is confirmed. A more conservative interpretation is that, given the size of the estimated uncertainties, these data do not provide evidence for initial nonGaussianity, i.e., the data are consistent with primordial Gaussianity. Unfortunately, due to the statistical limitation of the current SDSS data, it is not easy to put a more quantitative statement concerning the initial Gaussianity. Moreover, in order to go further and place more quantitative constraints on primordial Gaussianity with upcoming data, one needs a more precise and reliable theoretical model for the MFs, which properly describes the nonlinear gravitational effect possibly as well as galaxy biasing beyond the simple mapping on the basis of the volume fraction. In fact, galaxy biasing is a major source of uncertainty for relating the observed MFs to those obtained from the mock samples for dark matter distributions. If LCDM is the correct cosmological model, the good match of the MFs for mock samples from the LCDM simulations to the observed SDSS MFs may indicate that nonlinearity in the galaxy biasing is relatively small, at least small enough that it does not significantly affect the MFs (the MFs as a function of ν_{ σ } remain unchanged for the linear biasing).
6.5 Other statistical measures
In this section, we have presented the results on the basis of particular statistical measures including the twopoint correlation functions, power spectrum, redshift distortion, and Minkowski functionals. Of course there are other useful approaches in analysing redshift surveys: the void probability function, countsincells, Voronoi cells, percolation, and minimal spanning trees. Another area not covered here is optimal reconstruction of density field (e.g, using the Wiener filtering). The reader is referred to a good summary of those and other methodology in the book by Martinez and Saar [48].
Admittedly the results that we presented here are rather observational and phenomenological, and far from being wellunderstood theoretically. It is quite likely that when other ongoing and future surveys are being analysed in great detail, the nature of galaxy clustering will be revealed in a much more quantitative manner. They are supposed to act as a bridge between cosmological framework and galaxy formation operating in the Universe. While the proper understanding of physics of galaxy formation is still far away, the future redshift survey data will present interesting challenges for constructing models of galaxy formation.
7 Discussion
As a classical probe, galaxy redshift surveys still remain an important tool for studying cosmology and galaxy formation. On large scales (> 10 h^{−1} Mpc or so) they nicely complement the cosmic microwave background, supernovae Ia, and gravitational lensing in quantifying in detail the cosmological model. On small scales (< 10 h^{−1} Mpc) the clustering patterns of different galaxy types (defined by structural or spectral properties) provide important constraints on models of biased galaxy formation.
The redshift surveys mainly constrain Ω_{m} via both redshift distortion (which also depends on biasing) and the shape of the ΛCDM power spectrum, which depends on the primordial spectrum, the product Ω_{m}h, and also the baryon density Ω_{b}. Redshift surveys at a given epoch are not sensitive to the Dark Energy (or the cosmological constant, in a specific case), but combined with the CMB they can constrain the cosmic equation of state.
Symbol  WMAP alone  WMAP+SDSS  Description 

Ω_{Λ}  \(0.75_{ 0.10}^{+ 0.10}\)  \(0.699_{ 0.045}^{+ 0.042}\)  Dimensionless cosmological constant 
Ω_{b}h^{2}  \(0.0245_{ 0.0019}^{+ 0.0050}\)  \(0.0232_{ 0.0010}^{+ 0.0013}\)  Baryon density parameter 
Ω_{m}h^{2}  \(0.140_{ 0.018}^{+ 0.020}\)  \(0.1454_{ 0.0082}^{+ 0.0091}\)  Total matter density parameter 
σ _{8}  \(0.99_{ 0.14}^{+ 0.19}\)  \(0.917_{ 0.072}^{+ 0.090}\)  Mass fluctuation amplitude at 8 h^{−1} Mpc sphere (linear theory) 
n _{s}  \(1.02_{ 0.06}^{+ 0.16}\)  \(0.977_{ 0.025}^{+ 0.039}\)  Primordial scalar spectral index at k = 0.05 Mpc^{−1} 
τ  \(0.21_{ 0.11}^{+ 0.24}\)  \(0.124_{ 0.057}^{+ 0.083}\)  Reionization optical depth 

Both components of the model, Λ and CDM, have not been directly measured. Are they ‘real’ entities or just ‘epicycles’? Historically epicycles were actually quite useful in forcing observers to improve their measurements and theoreticians to think about better models!

‘The Old Cosmological Constant problem’: Why is Ω_{Λ} at present so small relative to what is expected from Early Universe physics?

‘The New Cosmological Constant problem’: Why is Ω_{m} ∼ Ω_{Λ} at the presentepoch? Why is w ∼ −1? Do we need to introduce new physics or to invoke the anthropic principle to explain it?

There are still open problems in ΛCDM on the small scales, e.g., galaxy profiles and satellites.

Could other (yet unknown) models fit the data equally well?

Where does the field go from here? Should the activity focus on refinement of the cosmological parameters within ΛCDM, or on introducing entirely new paradigms?
Notes
Acknowledgements
We thank Joachim Wambsganss and Bernard Schutz for inviting us to write the present review article. O.L. thanks members of the 2dFGRS team and the Leverhulme Quantitative Cosmology group for helpful discussions. Y.S. thanks all his students and collaborators for over many years, in particular Thomas Buchert, Takashi Hamana, Chiaki Hikage, Yipeng Jing, Issha Kayo, Hiromitsu Magira, Takahiko Matsubara, Hiroaki Nishioka, Jens Schmalzing, Atsushi Taruya, Kazuhiro Yamamoto, and Kohji Yoshikawa among others, for enjoyable and fruitful collaborations whose results form indeed the important elements in this review. Y.S. is also grateful for the hospitality at the Institute of Astronomy, University of Cambridge, where the most of present review was put together and written up for completion. O.L. acknowledges a PPARC Senior Research Fellowship. We also thank Idit Zehavi for permitting us to use Figure 28.
Numerical simulations were carried out at the ADAC (Astronomical Data Analysis Center) of the National Astronomical Observatory, Japan (project ID: mys02, yys08a). This research was also supported in part by the GrantsinAid from MonbuKagakusho and the Japan Society of Promotion of Science.
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