The distance to a galaxy is determined from its redshift through a cosmological model, follow- ing equation (2.3). Therefore the determination of separations can be affected by a mixture of physical effects, e.g. gravitational redshift , Doppler redshift, and observational effects such as photometric redshift errors. In a galaxy redshift survey, distortions on large scales are associated with the motion of the hosting dark matter halo towards matter overdensities, while on small scales, the distortions are mainly due to virialized motion of galaxies in dark matter haloes (the so called “finger-of-god” effect). Thus the observed distance to a galaxy with measured redshiftz is the contribution of the distance corresponding to its cosmologi- cal redshiftz0, plus the deviation along the line of sight due to its peculiar velocity in that
componentu(r):
s=r(z0)+
ur
H(z0)
,
Following Dodelson (2004), in order to measure the peculiar velocity of a galaxy, we must have an estimate of its comoving distance r, which must be given by some redshift independent measurement (such like the determination of light-curves of Cepheid stars or Supernovae). Typical velocities of galaxies during their coherent in-fall towards clusters are of the order of
∼500km/s, leading to∆r=v/H(z)∼5E(z)−1 Mpch−1. This also shows that the peculiar velocity
field induces distortions in the distances of ∆r ∝ v/H(z). In order to measure the peculiar motion of a galaxy, an experiment determining its comoving distance must generate results with a precision such that it is possible to distinguish the motion due to overdensities from the motion due to the Hubble expansion. The limiting precision the distance of a galaxy at some distancermust have in order to make such differentiation is σr∼100(∆r/r)%=(500/r)E(z)−1%, such that the more distant the galaxy is, the higher the relative precision the experiment must achieve in order to decouple the two effects. A galaxy at redshift z=0.05 implies σr ≈3%, rising up to ∼ 16% for a galaxy at z = 0.01. In other words, given some precision from a redshift independent experimentσ, only galaxies located at distances r ≤huE(z)−1/σMpc h−1
can be used to measure peculiar velocities.
When no measurement of peculiar velocities are available, the statistical analysis must be developed directly in redshift space. Transforming the basic quantities from real to redshift space requires some approximations. Assuming galaxy number conservation to leading order
18 2. Cosmological model and structure formation
in the peculiar velocity u(r), it can be shown that, on large scales, the matter fluctuation in redshift space can be written to leading order in the peculiar velocity as (Kaiser, 1987; Hamilton, 1992):
δ[s(r)]≈δ(r)−∂u(r)
∂r .
When transformed to Fourier space, the redshift space power spectrum can be written as (e.g. Kaiser, 1987; Heavens et al., 1998)
Ps(k)=Ps(k, µk)=
1+βµ(k)22Pr(k), (2.14)
whereµk≡kk/k=ˆr·ˆkis the cosine of the angle between the wave vector and the line-of-sight direction,kk is the projection of the wave vector in the line-of-sight direction (k=(k2
k +k
2
⊥)1/2)
and β≡ f (z)/b(z), with f (z) ≡d lnδ/d ln a as the growth index. Hence, in redshift space, the three dimensional power spectrum acquires an enhancement with respect to the power in real space, which depends on the direction of the wavenumber (note that the real power spectrumP(k)is isotropic). For parallel directions andβ=1, one has Ps(k)=4Pr(k). Note also that Ps(k⊥ = k,kk =0,z) =P(k,z) i.e, the real power spectrum can be obtained by measuring
the redshift power spectrum in the transverse direction, as can be seen in the Kaiser effect equation (2.14) by setting µk = 0 (Hamilton and Tegmark, 2000; Tegmark et al., 2004). Nevertheless, it is not clear how determine, in Fourier space, what the tangential or the parallel direction with respect to the line-of- sight are.
It is interesting to note that the source of redshift distortions, mainly the peculiar velocity field, satisfies the linearized equation∇·u∝ −˙aδ. Only in the context of General Relativity, the proportionality constant isβ= f/b (e.g. Hamilton, 1992). For a matter dominated universe, it has been written as f (z)= Ωcdm(z)0.6 Peebles (1980), fitted for a ΛCDM universe by Carroll
et al. (1992) and extended by Wang (2000) for constant dark energy equation of state:
f (a)= Ωmat(a)γ(z) γ(a)=3(1−ωx)
5−6ωx + 3 2 (2−3wx)(1−wx) (5−6wx)3 (1−Ωmat(a)), (2.15)
with a1%error with respect to the exact solution. This leads to a growth index for theΛCDM cosmological model ofγ(z)=0.55+0.011Ωx(z), leading a valueγ=0.5580forwx =−1. Given the
current estimates of the dark energy equation of state, the variation of the exponentγ under variations of its value 0.55 is justδγ=0.154δwx. A 5%result in the measurement of wx leads
to a deviation of 10−4 in the value γ=0.55. Therefore a deviation from the value 0.55 in a
measurement of the exponentγ constitutes a smoking gun for alternative theories of gravity (e.g. Linder, 2008).
The small angle approximation provides a simple frame to treat redshift distortions. For wide angle surveys, this approximations does not hold anymore and a complete treatment must be carried out as developed by Szalay et al. (1998). Also, redshift distortions can affect the clustering pattern encoded in the angular correlation function, as explored by Nock et al. (2010).
Multipole expansion
In general one can write the galaxy power spectrum Pggs,r(k, µk) as a linear combination of Legendre Polynomials Pl(µ) by developing a power expansion with respect to µk of the form
Ps(k) = P
2.4 Distortions in the linear clustering pattern 19
(ℓ=0), quadrupole (ℓ=2) and a hexadecapole contribution (ℓ=4). The coefficients are given by f0(β)=1+ 2 3β+ 1 5β 2, f 2(β)= 4 3β+ 4 7β 2, f 4(β)= 8 35β 2,
The azimuthally-averaged power spectrum in redshift space is given by the monopole contri- butionPs(k)=P(k) f0(β)which is only a function ofβ. Hence on large scales the increase in the
power spectrum is scale-independent, providing a way to measure the parameterβ(Hamilton, 1998). Note that when one wants achieve such measurement from the monopole contribution to the redshift power spectrum Ps(k), one has to have a previous knowledge of the underly- ing real space power spectrum. Instead, what was usually done was to use the information contained in all the multipoles by measuring the three dimensional power spectrumPs(k)and use for instance the quadrupole-to-monopole ratio Q = Ps
2(k)/P
s
0(k), which only depends on
β. Translating the multipole expansion to configuration space is almost straightforward (e.g Hawkins et al., 2003).
Small-scale distortions
On small scales, the main distortions in redshift space are due to the motion of galaxies in virialized dark matter haloes. These distortions are modeled via the streaming model where the correlation function in real space is convolved with the pairwise velocity distribution function of galaxies (Peebles, 1980). Numerical simulations have shown how a Gaussian distribution or a Lorentzian distribution for the pair-wise velocity is not a good description of galaxies in dark matter haloes, though these are the two typical models that lead to analytical expressions (e.g. Scoccimarro, 2004). The first of these approximations describes galaxies with velocities drawn from a Maxwellian distribution function, such that the best one-dimensional pairwise velocity distribution function is an exponential function (Landy, 2002; Jing et al., 2002; Peebles, 1980). Measurements of the two-dimensional correlation function can thus give constraints on the parameter β and estimates of pair-wise velocity dispersions (e.g Guzzo et al., 2008; Hawkins et al., 2003). When translated to Fourier space, the streaming model generates a three dimensional power spectrum written as
Ps(k,z)=Ps(k, µk,z)=P(k,z) 1+µ2kβ(z)2D(µk,k,z), (2.16) wherek2=k2 k+k 2
⊥ and the factor D(µk,k,z)is the Fourier transform of the velocity distribution function. Analytical results can be obtained for the function D(µk,k,z) with Gaussian and exponential distributions (e.g. Lahav and Suto, 2004).