(a)Impact ofσ (b)Impact ofβ
(c)Impact ofMth (d)Impact ofα
(e)Impact ofM0 (f)Impact off
Figure 5.2:Impact of halo model parameters onhN N Mifor unmixed lens pairs. In each subfigure, one parameter is varied by adding/subtracting 25% of its fiducial value from Table 5.1, while keeping all other parameters fixed. Solid lines indicate the totalhN N Mi, while dashed lines show the 1-halo, dotted lines the 2-halo, and dash-dotted lines the 3-halo term.
5.4 Results
To estimate the sensitivity of G3L to the cross-correlation of lens populations, we show in Fig. 5.3 the predicted hN N Mifor two different lens populations which are fully correlated (𝑟 =1), not correlated (𝑟 =0) or anticorrelated (𝑟 =−1). We have set one parameter for each population to a value either higher or lower than the fiducial one while keeping the other parameters to the fiducial model. For comparison, we also show the auto-correlation signal for both populations.
The figure shows that thehN N Mifor independent populations is at the geometric mean of the auto-correlations. This finding is independent of which parameter we vary between the populations. Consequently, for𝑟 =0, the halo model prediction coincides with those of a linear deterministic galaxy bias (see Sect. 2.4.4 and Sect. 4.4).
Furthermore, the aperture statistics for𝑟 =−1, 0, and 1 coincide at large scales but differ below that for all parameters. The signal for anti-correlated lens populations is consistently smaller than for uncorrelated and positively correlated lenses. The amount of variation between the hN N Mi is almost independent of the varied parameter and is detectable even for different 𝜎.
A more detailed look at Fig. 5.3 reveals that𝑟 impacts small, but not large scales because it primarily affects the 1-halo term. The 2-halo term is less dependent on𝑟, and the 3-halo term is entirely independent of it. This observation is not surprising, as the 3-halo term only depends on galaxies from different halos and is therefore not sensitive to the correlation of galaxies in the same halo. For the same reason, the 2-halo term is less affected than the 1-halo term.
Another reason for the stronger dependence ofhN N Mion𝑟 at smaller scales is that smaller halos with lower masses dominate these scales. Therefore, the ratio𝜎sat/h𝑁sat|𝑚iis large and the term proportional to𝑟 in Eq. (5.9) contributes significantly. At larger scales, the 1-halo term is dominated by halos with larger masses and higher
𝑁sat|𝑚
. Therefore, the correlation term loses importance, which explains why the 1-halo terms for different values of𝑟 converge for large scales.
5.4.2 Results of fitting halo model to observations
Next, we give the results of fitting our halo model to the observations in the KV450×GAMA data. Figure 5.4 shows the measured G3L aperture statistics. The measurement has a lower S/N than the measurement in Sect. 4.4 because we did not weigh lens galaxy pairs according to their redshift differences. Consequently, the signal is lower by a factor of approximately 2, while the noise is lower by only 40%, compared to Fig. B.1. Nevertheless, a clear, non-zero signal can be detected, in particular for the red-red lens pairs in Fig. 5.4a.
The best fit of our halo model is also shown in Fig. 5.4, together with its decomposition into the 1-, 2-, and 3-halo terms. In all three cases, the fit agrees well with the measurement. For
(a)Impact ofσ (b)Impact ofβ
(c)Impact ofMth (d)Impact ofα
(e)Impact ofM0 (f)Impact off
Figure 5.3:Impact of halo model parameters onhN N Mifor mixed lens pairs. In each subfigure, one parameter is varied, while keeping the others at the fiducial values from Table 5.1. Lenses are either correlated (r=1, green lines), uncorrelated (r=0, blue lines) or anti-correlated (r=−1, red lines). Also shown are the auto-correlations in grey. Solid lines indicate the total aperture statistics, dashed lines the 1-halo, dotted lines the 2-halo, and dash-dotted lines the 3-halo term.
5.4 Results
(a)Red-red lens pairs (b)Blue-blue lens pairs
(c)Red-blue lens pairs
Figure 5.4:G3L measurement in KV450×GAMA (points) and best fitting halo model (lines). Solid lines indicate the total aperture statistics, dashed lines the 1-halo, dotted lines the 2-halo, and dash-dotted lines the 3-halo term of the fit. 5.4a shows the result for red-red galaxy pairs, 5.4b shows the result for blue-blue galaxy pairs, and 5.4c shows the result for red-blue mixed pairs.
Table 5.2:Best fitting values for halo model parameters for KV450×GAMA.
Parameter Best fitting value Parameter Best fitting value
α(red) 0.389±0.057 α(blue) 0.138±0.032
σ(red) 0.24±0.31 σ(blue) 0.25±0.33
Mth(red) (1.66±0.59)×1012M Mth(blue) (1.25±0.64)×1011M
β(red) 0.85±0.18 β(blue) 0.51±0.15
M0(red) (3.50±0.63)×1013M M0(blue) (1.82±0.73)×1014M
f(red) 1.35±0.52 f(blue) 0.83±0.32
r 0.88±0.47
this fit, the 𝜒2, as defined by Eq. (5.26) is𝜒2 =61.5859. Since the fit has 90−13=77 degrees of freedom, the reduced𝜒2is
𝜒2
redu = 𝜒2
d.o.f =0.799 . (5.38)
This indicates that the best-fit halo model agrees with the measurement at the 95% CL.
The aperture statistics for red-red lenses are dominated by the 1-halo term in the whole range from[0.01 : 200]. For blue-blue lens pairs, though, the signal is dominated by the 3-halo term for 𝜃 >3.07. For mixed lens pairs, the 3-halo term dominates for𝜃 >100.
The best-fitting parameter values are shown in Table 5.2. They indicate that red and blue galaxies need to be described by different HODs, as 𝑀(red)
th and 𝑀(blue)
th , 𝑀0(red) and 𝑀0(blue), as well as 𝛽(red) and 𝛽(blue) differ significantly. The threshold halo mass for red galaxies is 𝑀(red)
th = (1.66±0.59) ×1012M, while for blue galaxies 𝑀(blue)
th = (1.24±0.64) ×1011M. Consequently, halos need to be ten times as massive to host red than blue galaxies. However, because 𝑀0(red) < 𝑀0(blue) and 𝛽(red) > 𝛽(blue), as soon as the mass of a halo exceeds the threshold for red galaxies, it contains more red than blue galaxies.
The spatial distribution of galaxies inside a halo is consistent with unity for both red and blue galaxies, which indicates that their distribution follows the dark matter density profile.
The parameter 𝑓(red) for red galaxies is larger than for blue galaxies, but this difference is not significant.
We find that the cross-correlation of red and blue galaxies is positive (𝑟 =0.88±0.47). Con-sequently, red and blue galaxies are positively correlated.
As expected, we could not constrain the parameter 𝜎 with G3L. The 1𝜎 interval for this parameter corresponds to the whole prior range, both for red and blue galaxies.