5.2 Cosmic Shear
6.1.1 Projected Power Spectrum
the concentration-mass relation etc. Another advantage is that the results are comple-mentary to traditional probes that rely on luminous tracers of the mass distribution like measurements of the rotation curves of stars in spiral galaxies or the velocity dispersion in elliptical galaxies, which provide important evidence for dark matter halos around galaxies. However, with luminous tracers one can only probe the dark matter halo out to distances of roughly 100h−1kpc. On the other hand, GGL is sensitive to the total mass and allows for measurements beyond 1h−1Mpc. Furthermore, since GGL depends on the matter-galaxy cross-correlation function it can shed light on the scale-dependent galaxy bias. For example one can extract the galaxy bias by combining measurements of the cosmic shear and the GGL two-point correlation functions (Hoekstra et al. 2002).
The first attempt to measure the GGL signal failed due to the use of scans of photographic plates and the poor seeing (Tyson et al. 1984). We had to wait for more than 10 years until the breakthrough of the first detection of the GGL signal mostly due to the improvement of observational techniques (Brainerd et al. 1996). The advent of high-resolution telescopes and wide-field galaxy surveys greatly increased the number of useable foreground-background pairs thereby significantly reducing the statistical errors and improving the constraints on halo parameters. Current experiments can be divided into two main classes: shallow (low-redshift) and large-area surveys like the SDSS (Fischer et al. 2000; McKay et al. 2001) or relatively deep (higher-redshift) experiments like the RCS1 (Hoekstra et al. 2002). At the moment, the tightest constraints on halo profiles are provided by results from the SDSS since this survey consists of a large field with exquisite redshift information of fore- and background galaxies in five different filters (e.g., Mandelbaum et al. 2006a,c,b). The main drawback is that the relatively bad seeing prevents cosmic shear analyses in the SDSS. Hence, the best galaxy bias constraints from GGL come from the RCS survey where one can combine cosmic shear and GGL measurements (Hoekstra et al. 2002).
whereν(w) is a selection function accounting for the fraction of objects that are included in the galaxy sample. Furthermore, we introduce the mean number density on the sky:
N¯ = Z wH
0
dw fK2(w)ν(w)¯ng(w). (6.3) These definitions lead to the redshift distribution of foreground galaxies (denoted by the subscript “f”), or more precisely, their distribution in comoving distance
pf(w) = fK2(w)ν(w)¯ng(w)
N¯ , (6.4)
which is normalized by definition, such that R
dw pf(w) = 1.
Employing Eq. (6.1), we get a relation between the two- and the three-dimensional number density contrast
N(θ)/N¯ = 1 + Z wH
0
dw pf(w)δg[fK(w)θ, w]. (6.5) This allows us to define the fractional density contrast of the number density of fore-ground galaxies on the sky as
κg(θ)≡ N(θ)−N¯ N¯ =
Z wH
0
dw pf(w)δg[fK(w)θ, w], (6.6) where we used Eq. (6.5) in the last step.
We can now build cross- and auto-power spectra of the convergence κ(θ) and the fractional density contrast κg(θ) by using Limber’s equation (5.64). The projected cross-power spectrum is defined by the two-point correlator
h˜κ(l)˜κg(l0)i= (2π)2δD(l+l0)Pκg(l). (6.7) The possible combinations are the dark matter auto-power spectrum
Pκκ(l) = Z wH
0
dw G2(w)Pδδ l
fK(w);w
, (6.8)
the galaxy auto-power spectrum Pgg(l) =
Z wH
0
dw p2f(w) fK2(w)Pgg
l fK(w);w
, (6.9)
and the dark matter-galaxy cross-power spectrum Pκg(l) =
Z wH
0
dwG(w)pf(w) fK(w) Pδg
l fK(w);w
. (6.10)
0 1 2 3 4 5 6 7
0 0.1 0.2 0.3 0.4 0.5
pf(z)
z
m’min = 1011
m’min = 1012
m’min = 1013
m’min = 1014
Figure 6.1: Redshift distribution of foreground galaxies shown for four different minimal HOD masses as indicated in the figure, where we use m0min ≡mmin/(h−1M). We find that higher threshold masses lead to an enhancement of the probability for low redshifts and to a reduction for higher redshifts. Note that we employ a maximal redshift of zmax,f = 0.4 for the lenses. Moreover, we assume that all sources are located at a single redshift zs= 1.
Assuming that the distributions of foreground (lenses) and background galaxies (sources) are given by Dirac delta functions at redshift zl (pf(z) = δD(z−zl)) and at zs (ps(z) = δD(z−zs)), respectively, the projected cross-power spectrum (6.10) simplifies to
Pκg(l) = 3 2Ωm
H0 c
2
(1 +zl)w(zs)−w(zl) w(zl)w(zs) Pδg
l w(zl);zl
, (6.11)
where we inserted the weight function (5.69) valid for a single source redshift. Note that we assumed in a flat Universe in the derivation of Eq. (6.11). Then the projected spectrum is directly proportional to the three-dimensional spectrum times geometrical factors that describe the distances of the lensing system. Note that we cannot derive an expression of the projected galaxy power spectrum (6.9) considering a Dirac delta distribution for the foreground redshift distribution because of the occurrence of the factor p2f(w). The cause of this problem is that Limber’s approximation is not valid anymore for this product of weight functions (Schneider 1998).
Having laid out the concept of projected spectra, we want to adopt our halo model to predict their dependence on the Fourier mode l. First, we give the distribution of foreground galaxies (6.4) in terms of redshift using pf(z)dz=pf(w)dw. In this case we
find
pf(z) = c H(z)
w2(z)¯ng[w(z)]
N¯ Θ(zmax,f−z), (6.12)
where we assumed a flat Universe and the selection function ν(z) = Θ(zmax,f −z) for simplicity, and Θ(x) denotes the step function. Here zmax,f is the selected maximal redshift of a foreground galaxy sample. In addition, we could set a minimum redshift which is accessible in a potential experiment. The mean number density of galaxies
¯
ng is connected to the adopted form of the HOD by the completeness relation in Eq. (4.2). We depict the distribution in Fig. 6.1 for four different minimal masses mmin where we adopt the Kravtsov et al. (2004) parametrization of the HOD given in Eqs. (4.13) (central galaxies) and (4.14) (satellite galaxies). The corresponding angular number densities which define our galaxy samples are N¯ '1.66arcmin−2 for mmin = 1011h−1M, N¯ ' 0.19arcmin−2 for mmin = 1012h−1M, N¯ ' 0.019arcmin−2 for mmin = 1013h−1M and N¯ ' 9.71×10−4arcmin−2 for mmin = 1014h−1M. Note that we truncate the foreground distribution at a maximal redshift of zmax,f = 0.4. As one can see, the difference between the four curves is small. Since the distribution is normalized an enhancement (reduction) for small redshifts results in a reduction (enhancement) for large redshifts. We will employ this foreground redshift distribution for the following plots and assume that the background galaxies are located at a single redshift of zs = 1 (approximately twice as large as the maximal foreground redshift) unless otherwise stated2.
We define for the three introduced projected power spectra their reduced (dimension-less) form as
∆XY(l)≡ l2
2πPXY(l). (6.13)
Here the two subscripts can take the values (X,Y)∈ {κ,g}. The reduced spectra have a much weaker scale dependence than the spectra themselves. The three-dimensional spectra are calculated with our halo model implementation, in particular see Eqs.(3.103) and (3.106)for the dark matter, Eqs. (4.27)and (4.29) for the galaxy and Eqs.(4.36) and(4.37)for the cross-power spectrum. We depict them in Fig. 6.2 as a function ofl for mmin = 1012h−1M. Most notably, the galaxy and cross-spectrum can be approximately described by a single power law inl, whereas the convergence spectrum is first increasing and then decreasing for small scales. Hence, the form of the three-dimensional spectra is approximately maintained by the projections (compare with Fig. 4.3 and Fig. 4.6). On the other hand, at first sight it might be surprising that the (scale-dependent) difference between the galaxy, convergence and the cross-spectrum is very large (up to three to four orders of magnitude). To further analyze this difference, we define according to Eq. (4.53) the projected bias factor and theprojected correlation coefficient by
¯b(l)≡ s
Pgg(l)
Pκκ(l), ¯r(l) = Pκg(l) pPκκ(l)Pgg(l),
¯b(l)
¯
r(l) ≡ Pgg(l)
Pκg(l), (6.14)
2This is similar to the redshift distribution used by Simon et al. (2008) in the analysis of the RCS field with maximal foreground redshiftzf= 0.4 and mean background redshift of ¯zs≈0.85.
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101
101 102 103 104 105
1 10
100 1000
∆XY(l)
l θ=2π/l [arcmin]
Figure 6.2: Reduced projected power spectra (6.13) versus Fourier mode l for mmin = 1012h−1M. We show the convergence power spectrum (solid red line), the projected cross-power spectrum (dotted blue line) and the projected galaxy cross-power spectrum (dashed green line). In addition, we depict on the upper abscissa the correspondence to the real-space Fourier conjugate of l defined asθ= 2π/l which is given in units of arcmin.
where the last relation quantifies the difference of the galaxy and cross-spectrum. Note that these quantities additionally depend on the redshift distribution of foreground and background galaxies. We show the results in Fig. 6.3 for the same l-range as before but for four different minimal masses as indicated in the figure. Now we can compare the left-hand panel which shows the projected bias factor with the left-hand panel of Fig. 4.6, where we plot the three-dimensional scale-dependent bias between galaxy and dark matter clustering. We find that the shape of the curves is similar. However, the amplitude is clearly enhanced for the projected spectra. This is mainly due to the different weight functions used in Limber’s equation (compare the weightings in Eqs. 6.8 and 6.9). In addition, we note that¯b(l) does not converge to a constant on the largest depicted scales in contrast to the three-dimensional bias factor. Clearly, this is an effect of the redshift weighting of the projected spectra. On the right-hand panels of Fig. 6.3 and Fig. 4.6 we show the ratio¯b/¯r of the projected and b/r of the three-dimensional correlation parameter, respectively. In this case the scale dependence is strongly reduced compared to the bias since the galaxy and the cross-spectrum have a similar shape especially on small scales. Furthermore, the form of the curves are slightly different
101 102 103
101 102 103 104 105
1 10
100 1000
b- (l)
l θ=2π/l [arcmin]
m’min=1011 m’min=1012 m’min=1013 m’min=1014
101 102 103
101 102 103 104 105
1 10
100 1000
b- (l)/r - (l)
l θ=2π/l [arcmin]
m’min=1011 m’min=1012 m’min=1013 m’min=1014
Figure 6.3:Projected bias factor (left panel) and ratio of the projected bias to the correlation coefficient (right panel) versus Fourier mode l(see definitions in Eq. 6.14) for four different minimal masses as indicated in the figure.
compared to their three-dimensional counterparts because of the redshift weighting of the projection.
We showed in Sect. 4.3.3 that for the three-dimensional spectra the amplitudes of the bias and correlation coefficient are enhanced for larger minimal masses mmin. This is due to the fact that considering a larger minimal mass leads to a smaller number density of galaxies. Since the spectra are normalized by the mean number of galaxies a larger minimal mass results in an enhancement of the amplitudes. In Fig. 6.3 we see that this trend is preserved for the projected bias factor and correlation coefficient.