6.2 Galaxy-Galaxy-Galaxy Lensing
6.2.2 Projected Cross-Bispectra
Similar to the two-point shear correlation functions, we can write the GGGL correlation functions as a weighted integral over the corresponding projected bispectra. For a detailed derivation of these relations we refer to Schneider & Watts (2005). We analyze here the behavior of the two cross-bispectra which are probed by GGGL surveys and
10-1 100 101 102 103
101 102 103 104 105 1 10
100 1000
Qeq XYZ(l)
l θ=2π/l [arcmin]
κκκ κκg ggκ ggg
10-6 10-5 10-4 10-3 10-2 10-1 100 101
101 102 103 104 105 1 10 100
1000
∆eq XYZ(l)
l θ=2π/l [arcmin]
Figure 6.7: Reduced projected bispectra as a function of Fourier model for an equilateral configuration (see text for details). We show the four possible auto- and cross-bispectra which are the galaxy bispectrum (dotted magenta line), the dark matter bispectrum (solid red line) and the two cross-bispectra, namelyQκκg (long dashed green line) andQggκ(short dashed blue line). We use a minimal mass mmin = 1012h−1M for the HOD. A higher (lower) threshold mass results in a reduction (enhancement) of the scale-dependence of the spectra including galaxies and the maximum is shifted to lower (higher) l. The left- and right-hand panels show two different definitions of the dimensionless bispectrum (see text for details).
compare the results with the two auto-bispectra probed in galaxy and in cosmic shear surveys. We show predictions for all of these spectra with our halo model implementation.
The four projected auto- and cross-bispectra are defined by the following connected three-point correlators:
h˜κ(l1)˜κ(l2)˜κ(l3)ic= (2π)2δD(l123)Bκκκ(l1,l2,l3), (6.34) h˜κ(l1)˜κ(l2)˜κg(l3)ic= (2π)2δD(l123)Bκκg(l1,l2;l3), (6.35) h˜κg(l1)˜κg(l2)˜κ(l3)ic= (2π)2δD(l123)Bggκ(l1,l2;l3), (6.36) h˜κg(l1)˜κg(l2)˜κg(l3)ic= (2π)2δD(l123)Bggg(l1,l2,l3), (6.37) with l123 ≡ l1 +l2 +l3. Applying Limber’s approximation (5.65) to the two three-dimensional cross-spectra in Eqs. (6.35)and (6.36)yields the corresponding projected cross-spectra:
Bκκg(l1,l2;l3) = Z wH
0
dwG2(w)pf(w) w2 Bδδg
l1 w,l2
w;l3 w;w
, (6.38)
Bggκ(l1,l2;l3) = Z wH
0
dwG(w)p2f(w) w3 Bggδ
l1 w,l2
w;l3 w;w
, (6.39)
and the projected galaxy bispectrum Bggg(l1,l2;l3) =
Z wH
0
dwp3f(w) w4 Bggg
l1
w,l2
w;l3
w;w
. (6.40)
Note that the expression for the convergence bispectrum is given in the previous chapter in Eq.(5.71). If all foreground galaxies are located at a single redshiftzland background galaxies at zs the first projected bispectrum (6.38) simplifies to
Bκκg(l1,l2;l3) = 9 4Ω2m
H0 c
4
(1 +zl)2(ws−wl)2 ws2wl2 Bδδg
l1 wl, l2
wl; l3 wl;zl
, (6.41) wherewl=w(zl) andws =w(zs). Note that we cannot derive such an expression for the galaxy and the galaxy-galaxy-mass bispectrum since Limber’s approximation is not valid in this case.
We define the two reduced projected cross-bispectra according to the three-dimensional reduced bispectra introduced in Eqs. (4.72) and (4.73):
Qggκ(l1,l2;l3)≡ Bggκ(l1,l2;l3)
Pκg(l1)Pκg(l2) +Pgg(l1)Pκg(l3) +Pgg(l2)Pκg(l3), (6.42) Qκκg(l1,l2;l3)≡ Bκκg(l1,l2;l3)
Pκg(l1)Pκg(l2) +Pκκ(l1)Pκg(l3) +Pκκ(l2)Pκg(l3). (6.43) Accordingly, we define the two reduced projected auto-bispectra following Eq. (4.71).
Note that we parametrize in the following the reduced bispectra by the length of the three sides, l1, l2 and l3 that build a closed triangle in Fourier space.
In Fig. 6.7 we depict the four reduced bispectra as a function of Fourier model for an equilateral configuration defined such that QeqXYZ(l)≡QXYZ(l, l, l) where (X,Y,Z)∈ {κ,g} (left panel). Furthermore, we show another definition of the dimensionless bispectrum given by ∆eqXYZ(l)≡(l2/2π)p
BXYZ(l, l, l) (right panel). As already seen for the projected power spectra, the difference between the projected bispectra is much larger than their non-projected three-dimensional counterparts that are depicted in Fig. 4.7. This can be easily explained by the different weight functions used in the projections. The shape of the three-dimensional bispectra is roughly conserved by the projections. In particular, we find that the reduced convergence bispectrum is increasing for small scales, whereas the bispectra including galaxy correlations are decreasing for small scales. For the three-dimensional spectra we pointed out that the different behavior on small scales stems from the dependence of the bispectra on the density profile. More specifically, the bispectra including galaxy correlations are on small scales dominated by central galaxy correlations which are only weighted by two density profiles.
Projected Bispectrum Bias
To clarify this issue, we define analogous to the three-dimensional bispectrum bias factors (see Eqs. 4.77, 4.78 and 4.79), the projected bispectrum bias factors:
¯b3 =
Bggg Bκκκ
1/3
, R¯2 ≡ ¯b3
¯
r2 = Bggg
Bggκ , R¯1 ≡ ¯b3
√r¯1 = s
Bggg
Bκκg , (6.44)
¯
r1 = Bκκg Bκκκ
Bκκκ Bggg
1/3
, r¯2 = Bggκ Bκκκ
Bκκκ Bggg
2/3
, (6.45)
where the functions depend on the triangle sides l1, l2 and l3. We illustrate ¯beq3 (l) ≡
¯b3(l, l, l) in Fig. 6.8 for equilateral configurations as a function of l for three different minimal masses (left panel). Furthermore, we depict R¯eq1 (l) ≡ R¯1(l, l, l) (left panel) and R¯eq2 (l)≡R¯2(l, l, l) (right panel) in Fig. 6.9. We see that for smalll the functions do not converge to a constant value in contrast to the three-dimensional functions (compare with Fig. 4.8 and Fig. 4.10) and are rather decreasing. On small scales ¯beq3 becomes highly scale-dependent, whereas R¯eq1 and R¯eq2 approximately converge to a constant. In addition, all curves have a bump feature at l≈60 which is not seen for the three-dimensional spectra. To analyze the origin of the bump feature, we employed an approximation of the galaxy and convergence bispectra which is composed of the one-halo terms and the large-scale limit of the three-one-halo terms (e.g., the projected tree-level bispectrum for the convergence bispectrum). We find that the bump is still persistent for this approximation of ¯b3 which means that it cannot be explained by our lack of modeling halo exclusion (which affects the two- and three-halo terms). In the right-hand panel of Fig. 6.8 we study the dependence of the bump feature by varying specific input parameters of the halo model keeping the minimal mass mmin = 1013h−1M fixed.
Enhancing (reducing) Ωm and σ8 results in a reduced (enhanced) amplitude of the bias with a stronger effect for variations of Ωm. The position of the bump is approximately
0 20 40 60 80 100 120 140
101 102 103 104 105 1 10
100 1000
b- 3eq (l)
l θ=2π/l [arcmin]
m’min=1011 m’min=1012 m’min=1013
0 20 40 60 80 100 120 140
101 102 103 104
10 100
1000
b- 3eq (l)
l θ=2π/l [arcmin]
SDSS-like
MAX-like
Figure 6.8: Equilateral configuration of the projected bias factor ¯beq3 (see Eq. 6.44) as a function of lfor three different minimal masses (left panel). The dotted magenta line gives the result for mmin = 1013h−1M when approximating the bispectra by a sum of their one-halo contribution and the large-scale limit of the three-halo terms. We showed the corresponding three-dimensional bispectrum bias factor in Fig. 4.10. In the right-hand panel we study the dependence of the bump feature at small l by varying specific input parameters of the halo model keepingmmin = 1013h−1M fixed. We analyze the dependence on the redshift distribution of fore- and background galaxies depicting the results for an SDSS-like survey withzmax,f = 0.2 andzs = 0.4 (thick long-dashed line) and for a future deep survey (MAX) withzmax,f = 0.6 and zs = 1.5 (thin long-dashed line). Additionally, we show the influence on the cosmological parameters Ωm andσ8, namely Ωm= 0.2 (thin dotted line) and Ωm= 0.4 (thick dotted line), andσ8= 0.8 (thin dot-dashed line) and σ8 = 1 (thick dot-dashed line).
25 30 35 40 45 50 55 60 65 70
101 102 103 104 105
1 10
100 1000
R- 1eq (l)
l θ=2π/l [arcmin]
m’min=1011 m’min=1012 m’min=1013
25 30 35 40 45 50 55 60 65 70
101 102 103 104 105
1 10
100 1000
R- 2eq (l)
l θ=2π/l [arcmin]
Figure 6.9: Square root of the ratio of the projected galaxy to the convergence-convergence-galaxy bispectrum (left panel), and ratio of the projected convergence-convergence-galaxy to the convergence-convergence-galaxy-convergence-convergence-galaxy- galaxy-galaxy-convergence bispectrum (right panel) (see Eq. 6.44). In particular we depict the ratios in equilateral configuration as a function ofl for three different minimal masses as indicated in the figure.
invariant under these variations. However, changing the redshift distribution of sources and lenses (here we study an SDSS-like survey and a potential future deep survey which we termed MAX-like) results in a change of the amplitude and slope of the curves and an off-set of the position of the bump. For a larger lens and source redshift the bump feature is strongly reduced and shifted to larger values of l, whereas for smaller redshifts it is strongly enhanced. Hence, we conclude that the bump mainly depends on the adopted redshift distributions. The difference between the three-dimensional and the projected bias is then due to the redshift-dependent weight factors in Limber’s approximation.
In Fig. 6.10 we give the results for the two projected correlation coefficients¯r1eq andr¯2eq as a function of l. On large scales both quantities converge to a constant value, whereas they are increasing on small scales. In addition, the amplitude of r¯1eq is larger than
¯
req2 on small scales which is the behavior we already found for their three-dimensional counterparts (see Fig. 4.9). However, on large scales the projected coefficients do not converge to 1 due to the redshift weighting of the projections.
Configuration Dependence
The configuration dependence of the reduced projected bispectra is shown in Fig. 6.11 as a function of the three sides of the triangle l1, l2 and l3. We keep l2 fixed to a value indicated in each panel, whereas l1 and l3 vary from 50 to 2×105. We show four
0 2 4 6 8 10
101 102 103 104 105
1 10
100 1000
r- 1eq (l)
l θ=2π/l [arcmin]
m’min=1011 m’min=1012 m’min=1013
0 2 4 6 8 10
101 102 103 104 105
1 10
100 1000
r- 2eq (l)
l θ=2π/l [arcmin]
Figure 6.10: Projected correlation coefficientsr¯eq1 (left panel) and¯req2 (right panel) as defined in Eq.(6.45). In particular, we depict the coefficients in equilateral configuration as a function of lfor three different minimal masses as indicated in the figure.
columns which illustrate the results for Qggg(l1, l2, l3),Qggκ(l1, l2;l3),Qκκg(l1, l2;l3) and Qκκκ(l1, l2, l3) going from left to right. Each column consists of three panels where we fixed l2 = 10, l2 = 103 andl2 = 105 going from bottom to top. Note that we chose these combination of parameters to study the amount of asymmetry when we interchange l1 and l3 in the two cross-spectra. Moreover, we show contour lines for the amplitudes 102,104,106,108,1010,1012 and 1014in each panel. First of all we notice that the reduced bispectra cover a large range of scales. We see that the amplitude of Q is enhanced if we go from the left panels to the right panels which is best visible for large l1 andl3. If we compare the bottom to the top panels of each row the yellow region that corresponds to small values of Q is extended and shifted to larger l. Moreover, in the two middle columns we can study the asymmetry inherent in the cross-spectra. The asymmetry amounts to a factor of 2 for large l1 and l3 (and l2 = 103,105) and is reduced for small l. In all the plots (also for the cross-spectra) we have to keep in mind that the plots for different l2 carry not completely independent information because of the symmetry properties of the functions.
In Fig. (6.12) we compare the results of the configuration dependence of the conver-gence bispectrum obtained with tree-level perturbation theory (left panel) and the full halo model (right panel). We find that the halo model result is in accordance with perturbation theory on large scales, whereas the perturbative result underestimates the result of the halo model on small scales as expected.
l1 l3
102 103 104 105 102
103 104
105 l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l1
102 103 104 105 l2=101
l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l3
102 103 104 105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l3
Qggg(l1,l2,l3)
102 103 104 105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105 Qggκ(l1,l2;l3)
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l1
102 103 104 105 l2=101
l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105 Qκκg(l1,l2;l3)
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l1
102 103 104 105 l2=101
l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
10-2 100 102 104 106 108 1010 1012 1014
Qκκκ(l1,l2,l3)
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
l2=101 l2=103 l2=105
Figure 6.11: Configuration dependence of the reduced bispectra shown as a contour plot. We varyl1 andl3 by keepingl2 fixed as indicated in each panel. The different rows show the four different bispectra. Note that the two-cross spectra are not symmetric under interchangingl1
and l3. We refer to the text for detailed explanations.
l1 l3
QκκκPT (l1,l2,l3)
102 103 104 105
102 103 104 105
10-2 100 102 104 106 108 1010 1012 1014
l1 Qκκκ(l1,l2,l3)
102 103 104 105
Figure 6.12: Configuration dependence of the reduced convergence bispectrum shown as a contour plot. We varyl1 and l3 and setl2 = 103. The left-hand panel depicts the convergence bispectrum using only tree-level perturbation theory and the right-hand panel shows the full halo model result of the bispectrum.