Measurement of the Tangential Shear

so-called cross-shearγc, also called B-mode. Gravity being a conservative force, i.e., not producing curls, the cross-shear signal is supposed to vanish. This can easily be checked. Vanishing of the cross-shear does not absolutely guarantee the absence of systematics, but on the other hand its presence is a strong indicator for remaining systematic effects.

Fig. 6.1 shows the tangential shear γt(R)as a function of the projected distance R between lens and source, averaging over all galaxies and over red and blue galaxies separately. For this analysis we only consider lenses with−20≤Mr^′≤ −24. As we can see the B-modes of all three lens samples are well consistent with zero. We estimate the velocity dispersion for the three galaxy lens samples assuming an SIS profile, only considering data points representing smaller separations than R=200 h⁻¹ kpc for two reasons. Firstly, an SIS profile is not a reasonable physical assumption for larger distances as the integrated mass does not converge and secondly, by rejecting larger separations we reduce the contamination by shear contributions induced by secondary halos. In order to convert the tangential shear signal amplitude into a velocity dispersion we need to define the effective distance ratio D_ds/Ds for the individual lens subsamples. We estimate this quantity by calculating the weighted mean of the individual distance ratios of all lens-source pairs, using the weight defined in equation (6.2). For the inner regions the fits follow very well the profile of an SIS out to a scale of R∼200 h⁻¹kpc, showing different amplitudes for the three lens samples. For the combined sample we find a velocity dispersion ofσ=117±1 km s⁻¹, the red galaxy sample shows a velocity dispersion ofσred=148±2 km s⁻¹ and the blue galaxy sample a value ofσblue=99±2 km s⁻¹. The higher value for the red and the lower value for the blue sample are partially explained by different mean rest-frame luminosities as the combined sample has an effective luminosity ofhM_r′i=−21.0 while the red galaxy sample has hM_r′i=−21.3 and the blue lens samplehM_r′i=−20.9. However, the more important reason for the observed amplitude difference is given by the higher mass of elliptical galaxies compared to spiral galaxies with same luminosity. Looking at larger scales, the tangential shear profile for the complete and for the blue lens sample are still consistent within the predictions of an SIS, while the shear profile for the red galaxy sample clearly exceeds the expectation of the SIS profile. This deviation is assumed to be most likely induced by secondary halos, more strongly affecting the red galaxies’

shear profile. This mirrors that the red (mainly early type) galaxies are more strongly correlated with each other and in general more often populate denser regions as galaxy groups or clusters than galaxies wither bluer colors. In order to estimate the expected signal strength, we compare the observational data to the predictions of a simulated lensing survey. The details of the simulations are described in Section 3.4.4. Fig. 6.2, shows that the tangential shear expectationsγt, assuming a BBS profile and assuming an NFW profile, respectively. Both describe observational tangential fairly well.

Extracting the theoretical values for the velocity dispersion from the simulated shear profiles, we obtainσ =114 km s⁻¹ for the BBS combined sample andσ=115 km s⁻¹ for the NFW combined sample (the observational value wasσ =117±1 km s⁻¹),σred=152 km s⁻¹for the BBS red sample andσred=151 km s⁻¹for the NFW red sample (the observational value wasσred=148±2 km s⁻¹) and finally a value ofσblue=92 km s⁻¹ for the blue BBS sample andσblue=94 km s⁻¹for the blue NFW sample, with σblue=99±2 km s⁻¹ being the observational value. A summary is shown in Table 6.1. In particular we confirm the results of Brainerd (2010), that multiple deflection effects lead to an excess in the measured shear signal, observing that especially the simulated red galaxy shear signal significantly exceeds the predictions of an SIS on larger scales. In contrast, looking at small separations, where the ‘main’ lens still dominates the signal and for spiral galaxies also for larger separations, a shear excess is not observed. We will further discuss the dependence of the shear signal with respect to the environment in the later sections, then also investigating the excess surface mass

M_r′ hM_r′i σtrue[km s⁻¹] σsim,BBS[km s⁻¹] σsim,NFW[km s⁻¹] Main Lens Sample

−24≤M_r′≤ −20 -21.0 117±1 114 115

−24≤Mr^′≤ −23 -23.3 240±4 219 239

−23≤M_r′≤ −22 -22.4 164±4 166 175

−22≤M_r′≤ −21 -21.4 124±2 123 123

−21≤Mr^′≤ −20 -20.5 93±2 91 87

Red Lens Sample

−24≤Mr^′≤ −20 -21.3 148±2 152 151

−24≤M_r′≤ −23 -23.3 255±5 237 259

−23≤M_r′≤ −22 -22.4 182±7 186 195

−22≤M_r′≤ −21 -21.5 147±3 150 147

−21≤M_r′≤ −20 -20.5 116±4 125 113

Blue Lens Sample

−24≤M_r′≤ −20 -20.9 99±2 92 94

−24≤M_r′≤ −23 -23.3 205±10 175 190

−23≤Mr^′≤ −22 -22.4 138±4 135 144

−22≤M_r′≤ −21 -21.4 109±4 103 106

−21≤M_r′≤ −20 -20.4 87±3 80 80

Table 6.1: Fit values for the velocity dispersionσ considering several luminosity bins, for observational data and BBS and NFW simulations, respectively.

density∆Σ.

We further investigate the tangential shear for different luminosities, splitting all three consid-ered lens samples into four magnitude intervals for M_r′ between -24 and -20 of one magnitude width.

Also in this case the observed B-modes are consistent with zero. Measuring the velocity dispersions of each individual luminosity bin, the observed decrease in velocity dispersion for fainter and therefore less massive lenses agrees well with the results of Faber & Jackson (1976) or Tully & Fisher (1977). The values for the fitted velocity dispersion σ in the considered combined lens luminosity bins are shown in Table 6.1. Further, considering the subsamples of different galaxy types but same luminosity, the conclusion of red galaxies being more massive than average or blue galaxies of the same luminosity is confirmed, as the velocity dispersions of red galaxies in all luminosity bins significantly exceed the values of their blue counterparts. Considering the combined lens sample the values for the velocity dispersions are, as expected, lower than for red galaxies, but higher than for the blue ones. The tangential shear profiles, discriminating the individual luminosity bins for all galaxies are shown in Fig. 6.3, for the red lens sample in Fig. 6.5 and finally for the blue lens sample in Fig. 6.7. We append the corresponding tangential shear plots based on BBS and NFW simulations for the individual luminosity bins for comparison in Figs. 6.4 (combined lens sample), 6.6 (red lens sample) and 6.8 (blue lens sample). The values for the shear amplitudes in the simulations mostly agree fairly well with the observational data, see Table 6.1.

Finally, we also consider the characteristics of the tangential shear profile for lenses populating

Fig. 6.1: Tangential shearγt for a lens sample with−24≤i^′≤ −20, fitting an SIS profile for the inner part out to a scale of R∼200 h⁻¹ kpc, considering all galaxy types (black circles and solid fit-line), red (red triangles and dashed fit-line) and blue galaxies (blue squares and dotted fit-line) individually.

The green dashed line indicates the 1-σ-level for remaining systematics. The signal amplitude is highest for red galaxies, exceeding the expectation for an SIS at scales larger than R=200 h⁻¹ kpc. The blue galaxy sample shows the lowest tangential shear amplitude, not deviating from an SIS profile for larger separations, as spiral are mostly found in environments of lower density than cluster environment. The combined galaxy sample shows a shear profile lying between elliptical and spiral sample. The values for the velocity dispersion σ, derived by fitting an SIS out to a distance of∼200 h⁻¹ kpc, are shown in Table 6.1.

Fig. 6.2: Simulated tangential shear profile (see Section 3.4.4) for lenses with−24≤i^′≤ −20. The upper panel shows the results based on the BBS simulation, the lower panel shows the results based on the NFW simulation. The tangential shear signals, based on either of both simulations, agree well with the actually observed profile (see Fig. 6.1). The values for the velocity dispersionσ, derived by fitting an SIS out to a distance of∼200 h⁻¹kpc, are shown in Table 6.1.

Fig. 6.3: Tangential shear profiles for the complete lens sample, discriminating four luminosity bins for

−24≤i^′≤ −23 in magenta (crosses, dashed-dotted fit-line), −23≤i^′≤ −22 in red (triangles, dashed fit-line),−22≤i^′≤ −21 in blue (squares, dotted fit-line) and finally−21≤i^′≤ −20 in green (circles, solid fit-line). The estimated values for the velocity dispersions decrease with decreasing luminosity, as predicted by the Faber-Jackson or Tully-Fisher relation. The values for the velocity dispersionσ^{, derived} by fitting an SIS out to a distance of∼200 h⁻¹kpc, are shown in Table 6.1.

Fig. 6.4: Simulated tangential shearγtfor all lenses, showing the profiles for individual luminosity bins with−24≤i^′≤ −20. The upper panel shows the results of the BBS simulation, the lower panel of the NFW simulation. The simulations are widely consistent with the observational data. The values for the velocity dispersionσ, derived by fitting an SIS out to a distance of∼200 h⁻¹kpc, are shown in Table 6.1.

Fig. 6.5: Tangential shear profiles for the red lens sample, discriminating four luminosity bins for−24≤ i^′≤ −23 in magenta (crosses, dashed-dotted fit-line),−23≤i^′≤ −22 in red (triangles, dashed fit-line),

−22≤i^′≤ −21 in blue (squares, dotted fit-line) and finally−21≤i^′≤ −20 in green (circles, solid fit-line). The estimated values for the velocity dispersions decrease with decreasing luminosity, as predicted by the Faber-Jackson. All values forσ are higher than for the combined lens sample. The values for the velocity dispersionσ, derived by fitting an SIS out to a distance of∼200 h⁻¹kpc, are shown in Table 6.1.

Fig. 6.6: Simulated tangential shearγtfor red lenses, showing the profiles for individual luminosity bins with−24≤i^′≤ −20. The upper panel shows the results of the BBS simulation, the lower panel of the NFW simulation. The simulations are widely consistent with the observational data. The values for the velocity dispersionσ, derived by fitting an SIS out to a distance of∼200 h⁻¹kpc, are shown in Table 6.1.

Fig. 6.7: Tangential shear profiles for the blue lens sample, discriminating four luminosity bins for−24≤ i^′≤ −23 in magenta (crosses, dashed-dotted fit-line),−23≤i^′≤ −22 in red (triangles, dashed fit-line),

−22≤i^′≤ −21 in blue (squares, dotted fit-line) and finally−21≤i^′≤ −20 in green (circles, solid fit-line). The estimated values for the velocity dispersions decrease with decreasing luminosity, as predicted by the Tully-Fisher relation. All values for σ are lower than for the combined lens sample. The values for the velocity dispersionσ, derived by fitting an SIS out to a distance of∼200 h⁻¹kpc, are shown in Table 6.1.

Fig. 6.8: Simulated tangential shearγtfor blue lenses, showing the profiles for individual luminosity bins with−24≤i^′≤ −20. The upper panel shows the results of the BBS simulation, the lower panel of the NFW simulation. The simulations are widely consistent with the observational data. The values for the velocity dispersionσ, derived by fitting an SIS out to a distance of∼200 h⁻¹kpc, are shown in Table 6.1.

Fig. 6.9: Tangential shear profile for the combined lens sample with−24≤i^′≤ −17, distinguishing be-tween environment of different density (see Section 5.5.2 for the exact definition). The SIS fits are obtained within a projected separation of R=200 h⁻¹kpc. The combined lens sample, consisting of all lenses in all environments, is shown with black circles and black solid fit-line as reference (see also Fig. 6.1). Blue (squares, dashed-dotted fit-line) and green (diamonds, solid fit-line) show lenses in environment with low and very low density, red (triangles, dashed fit-line) and magenta (crosses, dotted fit-line) show lenses in high and very high density environment. We see that the amplitude increases with environment density.

Further the contribution of the secondary halos significantly increases with environment density. This is negligible for low density samples, but strongly enhances the shear signal on large scales in high density environments, even leading to an almost constant signal in the very high density environment.

environments of different density. Following the definition in Section 5.5.2 we distinguish between lenses in high, low, very high and very low density environments. As Fig. 6.9 shows, the observed tangential profiles in the different environments significantly differ in amplitude and in large scale behavior. The lowest signal is observed for lenses populating the very low density environment, not only showing the lowest amplitude, but dropping to zero very soon and even showing a constantly negative E-mode for scales R>400 h⁻¹kpc. This indicates that the average convergence at the edge of the considered circle is higher than the mean convergence enclosed by this circle (see equa-tion 3.19). For the low density lens sample we find that the amplitude is higher than for the very low density case, the profile nicely follows an SIS on shorter scales, then dropping down to zero but not showing negative values. In both low density cases there is hardly any impact of nearby halos visible in the signal (as expected when the environment is poor). The mean density lens sample consisting of all lenses in all environments follows nicely an SIS out to a scale of R=200−300 h⁻¹kpc, even showing a small excess inγt. This effect is even stronger considering the high density lens sample, which shows a further increase in shear amplitude, exceeding the predictions of an SIS already for projected separations R>150−200 h⁻¹kpc at a higher level. The very high density sample finally hardly shows any dependence of shear on projected separation. On one hand the profile shows the highest amplitude of all considered environment subsamples and on the other hand the amplitude remains approximately constant on all considered scale out to a distance of R=700 h⁻¹kpc. This flat behavior is also confirmed in our 3D-LOS-projected lensing signal simulations (see Fig. 6.43 in Section 6.5), where we see that this flatness originates in the multiple gravitational deflections on brighter nearby galaxies in the close environment.