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
$\sim 10$
%. 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
The AngloAustralian 2degree Field Galaxy Redshift (2dFGRS) [
89]
has recently been completed with redshifts for 230,000 galaxies selected from the APM catalogue December 2002) down to an extinction corrected magnitude limit of
${b}_{J}<19.45$
. The main survey regions are two declination strips, in the northern and southern Galactic hemispheres, and also 100 random fields, covering in total about
$1800de{g}^{2}$
(see Figures 20 and 21 ). The median redshift of the 2dFGRS is
$\overline{z}\sim 0.1$
(see [
11,
65]
for reviews).
Figure 20
: The 2dFGRS fields (small circles) superimposed on the APM catalogue area (dotted outlines of Sky Survey plates).
Figure 21
: The distribution of 63,000 2dFGRS galaxies in the NGP (left panel) and SGP (right panel) strips.
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
$8873$
˚ A. For further details of SDSS, see [
102,
80]
The latest map of the SDSS galaxy distribution, together with a typical slice, are shown in Figures 22 and 23 (see also [
32]
). The threedimensional map centered on us in the equatorial coordinate system is shown Figure 22 . Redshift slices of galaxies centered around the equatorial plane with various redshift limits and thicknesses of planes are shown in Figure 23 :
$z<0.05$
with thickness of
$10{h}^{1}Mpc$
centered around the equatorial plane in the upperleft panel;
$z<0.1$
with a thickness of
$15{h}^{1}Mpc$
in the upperright panel;
$z<0.2$
with a thickness of
$20{h}^{1}Mpc$
in the lower panel.
Figure 22
: 3D redshiftspace map centered on us, and its projection on the celestial sphere of SDSS galaxy subset, including the three main regions. (Figure taken from [
32]
.)
Figure 23
: Redshift slices of SDSS galaxy data around the equatorial plane. The redshift limits and the thickness of the planes are
$z<0.05$
and
$10{h}^{1}Mpc$
(upper panel),
$z<0.1$
and
$15{h}^{1}Mpc$
(middle panel), and
$z<0.2$
and
$20{h}^{1}Mpc$
(lower panel). The size of points has been adjusted. Note that the data for the Southern part are sparser than those for the Northern part, especially for thick slices. (Figure taken from [
32]
.)
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
$\sim 1000$
galaxies to a limit of
$I=24.5$
. Phase 2 of the DEEP project will use the new DEIMOS spectrograph to obtain spectra of
$\sim 65,000$
faint galaxies with redshifts
$z\sim 1$
. 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
$54\times 54arcse{c}^{2}$
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
$0.02hMp{c}^{1}<k<0.15hMp{c}^{1}$
, 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 ).
Figure 24
: The power spectrum of the 2dFGRS. The points with error bars show the measured 2dFGRS power spectrum measurements in redshift space, convolved with the window function. Also plotted are linear CDM models with neutrino contribution of
${\Omega}_{\nu}=0$
,
${\Omega}_{\nu}=0.01$
, and
${\Omega}_{\nu}=0.05$
(bottom to top lines). The other parameters are fixed to the concordance model. The good fit of the linear theory power spectrum at
$k>0.15hMp{c}^{1}$
is due to a conspiracy between the nonlinear gravitational growth and the fingerofGod smearing [
72]
. (Figure taken from [
20]
.)
If one fits the
$\Lambda $
CDM model predictions to the 2dFGRS power spectrum (see Figure 24 ) over the above range in
$k$
, one can constrain the cosmological parameters. For instance, assuming a Gaussian prior on the Hubble constant
$h=0.7\pm 0.07$
(from [
22]
), Percival et al. [
72]
obtained the 68 percent confidence limits on the shape parameter
${\Omega}_{m}h=0.20\pm 0.03$
, and a baryon fraction
${\Omega}_{b}/{\Omega}_{m}=0.15\pm 0.07$
. For a fixed set of cosmological parameters, i.e.,
$n=1$
,
${\Omega}_{m}=1{\Omega}_{\Lambda}=0.3$
,
${\Omega}_{b}{h}^{2}=0.02$
, and
$h=0.70$
, the r.m.s. mass fluctuation amplitude of 2dFGRS galaxies smoothed over a tophat radius of
$8{h}^{1}Mpc$
in redshift space turned out to be
${\sigma}_{8g}^{S}({L}_{s},{z}_{s})\approx 0.94$
.
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,
${\Omega}_{\nu}={m}_{\nu ,tot}{h}^{2}/(94eV)\approx 0.001$
.
Large scale structure data can put an upper limit on the ratio
${\Omega}_{\nu}/{\Omega}_{m}$
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
${\Omega}_{\nu}/{\Omega}_{m}<0.13$
(95% CL), or with concordance prior of
${\Omega}_{m}=0.3$
,
${\Omega}_{\nu}<0.04$
, or an upper limit of
$\sim 2eV$
on the total neutrino mass, assuming a prior of
$h\approx 0.7$
[
20,
19]
(see Figure 24 ). In order to minimize systematic effects due to biasing and nonlinear growth, the analysis was restricted to the range
$0.02<k<0.15hMp{c}^{1}$
. 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, The CMB fluctuations are commonly represented by the spherical harmonics
${C}_{\ell}$
. The connection between the harmonic
$\ell $
and
$k$
is roughly
$$\begin{array}{c}\ell \simeq k\frac{2c}{{H}_{0}{\Omega}_{m}^{0.4}}\end{array}$$ 
(171)

for a spatiallyflat Universe. For
${\Omega}_{m}=0.3$
, the 2dFGRS range
$0.02<k<0.15hMp{c}^{1}$
corresponds approximately to
$200<\ell <1500$
, which is well covered by the recent CMB experiments.
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
${\Omega}_{\Lambda}\sim 0.7$
(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
$h={0.71}_{0.03}^{+0.04}$
,
${\Omega}_{b}{h}^{2}=0.0224\pm 0.0009$
,
${\Omega}_{m}{h}^{2}={0.135}_{0.009}^{+0.008}$
,
${\sigma}_{8}=0.84\pm 0.04$
,
${\Omega}_{tot}=1.02\pm 0.02$
, and
$w<0.78$
(95% CL, assuming
$w\ge 1$
) [
79]
.
6.2.4 Redshiftspace distortion
An independent measurement of cosmological parameters on the basis of 2dFGRS comes from redshiftspace distortions on scales
$\lesssim 10{h}^{1}Mpc$
: a correlation function
$\xi (\pi ,\sigma )$
in parallel and transverse pair separations
$\pi $
and
$\sigma $
. As described in Section 5 , the distortion pattern is a combination of the coherent infall, parameterized by
$\beta ={\Omega}_{m}^{0.6}/b$
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
$\beta ({L}_{s}=0.17,{z}_{s}=1.9{L}_{*})=0.43\pm 0.07$
, and Hawkins et al. [
30]
obtained
$\beta ({L}_{s}=0.15,{z}_{s}=1.4{L}_{*})=0.49\pm 0.09$
and a velocity dispersion
${\sigma}_{P}=506\pm 52km{s}^{1}$
. Using the full 2dF+CMB likelihood function on the
$(b,{\Omega}_{m})$
plane, Lahav et al. [
42]
derived a slightly larger (but consistent within the quoted errorbars) value,
$\beta ({L}_{s}=0.17,{z}_{s}=1.9{L}_{*})\simeq 0.48\pm 0.06$
.
6.2.5 The bispectrum and higher moments
It is well established that important information on the nonlinear growth of structure is encoded at the high order moments, e.g., the skewness or its Fourier version, the bispectrum. Verde et al. [
95]
computed the bispectrum of 2dFGRS and used it to measure the bias parameter of the galaxies. They assumed a specific quadratic biasing model:
$$\begin{array}{c}{\delta}_{g}={b}_{1}{\delta}_{m}+\frac{1}{2}{b}_{2}{\delta}_{m}^{2}.\end{array}$$ 
(172)

By analysing 80 million triangle configurations in the wavenumber range
$0.1<k<0.5hMp{c}^{1}$
they found
${b}_{1}=1.04\pm 0.11$
and
${b}_{2}=0.054\pm 0.08$
, in support of no biasing on large scale.
This is a nontrivial result, as the analysis covers nonlinear scales. Baugh et al. [
5]
and Croton et al. [
14]
measured the moments of the galaxy count probability distribution function in 2dFGRG up to order
$p=6$
(order
$p=2$
is the variance,
$p=3$
is the skewness, etc.). They demonstrated the hierarchical scaling of the averaged
$p$
point galaxy correlation functions. However, they found that the higher moments are strongly affected by the presence of two massive superclusters in the 2dFGRS volume. This poses the question of whether 2dFGRS is a 'fair sample' for high order 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
$\lesssim 10{h}^{1}Mpc$
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
Madgwick et al. [
45]
applied the Principal Component Analysis to compress each galaxy spectrum into one quantity,
$\eta \approx 0.5p{c}_{1}+p{c}_{2}$
. Qualitatively,
$\eta $
is an indicator of the ratio of the present to the past star formation activity of each galaxy. This allows one to divide the 2dFGRS into
$\eta $
types, and to study, e.g., luminosity functions and clustering per type. Norberg et al. [
61]
showed that, at all luminosities, earlytype galaxies have a higher bias than latetype galaxies, and that the biasing parameter, defined here as the ratio of the galaxy to matter correlation function
$b\equiv \sqrt{{\xi}_{g}/{\xi}_{m}}$
varies as
$b/{b}_{*}=0.85+0.15L/{L}_{*}$
. Figure 25 indicates that for
${L}_{*}$
galaxies, the real space correlation function amplitude of
$\eta $
earlytype galaxies is
$\sim 50\%$
higher than that of latetype galaxies.
Figure 25
: The variation of the galaxy biasing parameter with luminosity, relative to an
${L}_{*}$
galaxy for the full sample and for subsamples of early and late spectra types. (Figure taken from [
61]
.)
Figure 26 shows the redshiftspace correlation function in terms of the lineofsight and perpendicular to the lineofsight separation
$\xi (\sigma ,\pi )$
. The correlation function calculated from the most passively (`red', for which the present rate of star formation is less than 10 % of its past averaged value) and actively (`blue') starforming galaxies. The clustering properties of the two samples are clearly distinct on scales
$\lesssim 10{h}^{1}Mpc$
. The `red' galaxies display a prominent fingerofGod effect and also have a higher overall normalization than the `blue' galaxies. This is a manifestation of the wellknown morphologydensity relation. By fitting
$\xi (\pi ,\sigma )$
over the separation range
$8\text{\u2013}20{h}^{1}Mpc$
for each class, it was found that
${\beta}_{active}=0.49\pm 0.13$
,
${\beta}_{passive}=0.48\pm 0.14$
and corresponding pairwise velocity dispersions
${\sigma}_{P}$
of
$416\pm 76km{s}^{1}$
and
$612\pm 92km{s}^{1}$
[
45]
.
At small separations, the real space clustering of passive galaxies is stronger than that of active galaxies: The slopes
$\gamma $
are respectively 1.93 and 1.50 (see Figure 27 ) and the relative bias between the two classes is a declining function of separation. On scales larger than
$10{h}^{1}Mpc$
the biasing ratio is approaching unity.
Figure 26
: The two point correlation function
$\xi (\sigma ,\pi )$
plotted for passively (left panel) and actively (right panel) starforming galaxies. The line contours levels show the bestfitting model. (Figure taken from [
45]
.)
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.
Figure 27
: The correlation function for early and late spectral types. The solid lines show bestfitting models, whereas the dashes lines are extrapolations of these lines. (Figure taken from [
45]
.)
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:
$\xi \left(r\right)=(r/6.1\pm 0.2{h}^{1}Mpc{)}^{1.75\pm 0.03}$
for
$0.1{h}^{1}Mpc\le r\le 16{h}^{1}Mpc$
. The galaxy pairwise velocity dispersion is
${\sigma}_{12}\approx 600\pm 100km{s}^{1}$
for projected separations
$0.15{h}^{1}Mpc\le {r}_{p}\le 5{h}^{1}Mpc$
. 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
$r\sim 10{h}^{1}Mpc$
. 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
$\approx 1.8$
with correlation lengths of approximately
$7.4{h}^{1}Mpc$
,
$6.3{h}^{1}Mpc$
, and
$4.7{h}^{1}Mpc$
, respectively.
Figures
27 and 28 pose an interesting challenge to the theory of galaxy formation, to explain why the correlation functions per luminosity bins have similar slope, while the slope for early type galaxies is steeper than for late type.
Figure 28
: The SDSS (EDR) projected correlation function for blue (squares), red (triangles) and the full sample, with bestfitting models over the range
$0.1<{r}_{p}<16{h}^{1}Mpc$
(upper panel), and the SDSS (EDR) projected correlation function for three volumelimited samples, with absolute magnitude and redshift ranges as indicated and bestfitting powerlaw models (lower panel). (Figure taken from [
104]
.)
6.3.3 SDSS: Threepoint correlation functions and the nonlinear biasing of galaxies per luminosity and color
Let us move next to the threepoint correlation functions (3PCF) of galaxies, which are the lowestorder unambiguous statistic to characterize nonGaussianities due to nonlinear gravitational evolution of dark matter density fields, formation of luminous galaxies, and their subsequent evolution. The determination of the 3PCF of galaxies was pioneered by Peebles and Groth [
70]
and Groth and Peebles [
27]
using the Lick and Zwicky angular catalogs of galaxies. They found that the 3PCF
$\zeta ({r}_{12},{r}_{23},{r}_{31})$
obeys the hierarchical relation :