profiles and thus the estimated power-law slopes are steeper, which was also found before by Auger et al. (2010). This is well in agreement with our conclusions from simulations, however, since obser-vations have shown spheroidals to be more compact than their present-day counterparts, this would implicate that the slopes at higher redshifts should also be steeper.
We calculated the stellar mass density for our simulated halos, using the stellar half-mass radius instead of the effective radius in Eq. 4.2. There is a clear correlation between the stellar mass density and the total density slope (see upper right panel of Fig. 4.7), which is the same for both our simulation samples and for the observations, albeit the scatter in the observed total density slopesγtotis larger than the scatter found in the simulations at a given stellar mass density. The METGs again match the observations successfully, while most of the CETGs are much more concentrated than the METGs and the observations. This is most likely again due to the missing AGN feedback in the CETG simulations.
For completeness, we plot in the lower left panel of Fig. 4.7 the stellar mass densityΣ∗versus the central dark matter fraction fDM(R1/2). As expected, there is a clear tendency for spheroidals with larger fDM(R1/2) to be less compact, and while this is supported by both simulations, METGs and CETGs, there is a clear offset between the actual values of both simulations. In this case, the simulations and observations do not match too well, albeit the match again is much worse for the CETGs than for the METGs.
Fig. 4.8 shows the relations between the total density slopesγtotand the stellar mass densitiesΣ∗
(left columns) and central dark matter fractions fDM(R1/2) (right columns), at four different redshifts z = 0, 0.5, 1 and 2 from top to bottom. There is a clear evolution trend found for the simulations, namely that the central dark matter fractions increase with redshift, while the stellar central concen-tration is decreasing. These evolution trends are seen in both simulation samples, and support our idea that, after aboutz=2, the evolution of spheroidals is dominated by merger events, which enhance the central dark matter fractions, lead to stronger growth in size than in mass, and evolve the total density slope towards an isothermal solution through dynamical friction and violent relaxation.
4.5. MOCKING THE SLOPES: “OBSERVING” OUR SIMULATIONS 103
Figure 4.8:Same as Fig. 4.7 but split into different redshift bins.Left panels: Total density slopesγtotversus stellar mass densitiesΣ∗.Right panels: Total density slopesγtotversus central dark matter fractions fDM(R1/2).
Rows: From top to bottom: z=0, z=0.5, z=1, z=2.
by their method as an input information are: The effective radiusreff of the lens galaxy, the Einstein radius of the lensrEin, the projected line-of-sight velocity dispersion σLOS within 0.5Reff, and the total massMtotwithin the lens area. To mock observations from our simulations, we use the following inputs to meet the requirements:
• The effective radiusreffof the lens galaxy is, as discussed before, approximated by the half-mass radiusr1/2.
• For the Einstein radius of the lens we assumerEin = 1.5r1/2, according to the ratios between rEin and reff which have been found by Sonnenfeld et al. (2013a) and Ruff et al. (2011) for the SL2S survey. The ratios of the lenses studied in the SLACS (and BELLS) survey were usually smaller, but the ratios for the LSD survey were of similar order. This choice, however, should not strongly influence the results, since Sonnenfeld et al. (2013b) showed that the ratio between effective radius and Einstein radius does not change the resulting slopes significantly.
The influence of changes in this ratio on the resulting total density slopes were smaller than the error of the measurements. This is important since there is a tendency for lenses at higher redshifts to have larger ratios betweenrEinandreff due to geometrical reasons.
• To mimic the projected line-of-sight velocity dispersion σLOS within 0.5Reff, we rotate our spheroidals on both the face-on and edge-on projection, and calculate the line-of-sight velocity dispersion within half of the half-mass radius for both projections separately. In the follow-ing study we will always consider both projections, as they basically are the maximum and minimum values which can be found.
• For the total mass within the lens area we include all star, gas and dark matter particles within the given projected radius ofrEin=1.5r1/2, for both projections.
The results are shown in Fig. 4.9, for the edge-on (left panels) and face-on (right panels) view, for all galaxies at the four different redshift bins considered in this work (z = 0, 0.5, 1, 2). CETGs are again shown as red circles, METGs as blue circles. As can clearly be seen, there is a strong discrepancy between the resulting density slopes taken directly from the simulations,γsim, and the mocked density slopesγmock. In an ideal case, the resulting values should have been ordered along the dash-dotted line which marks the 1:1 ratio whereγmock=γsim. However, in both the edge-on and the face-on view cases, the mocked slopes are closely scattered aroundγmock =−2.2 for the edge-on andγmock = −1.9 for the face-on projections, while the intrinsic slopes from the simulations vary fromγsim≈ −3 toγsim≈ −1.6. This is the case for both simulation samples, CETGs and METGs, but the METGs show even flatter mock slopes for the face-on view than the CETGs.
Therefore, we clearly see that there is a discrepancy between the “real” total density slopes and the mocked slopes, which is most prominent at the steeper-slope end. Indications for such a discrepancy between the simulated and observed slopes has already been presented in Sonnenfeld et al. (2014) using a comparison sample of major merger simulations, however, the difference is much larger for our cosmological simulations where we find significantly steeper slopes at high redshifts. In addition, as shown for the major merger sample studied in Chap. 3 (Remus et al., 2013), with the usual present-day configuration for disk galaxies, the initial slopes are also close to isothermal, which is why the effect of the discrepancy between the mocked and intrinsic slopes are less pronounced in those cases.
Since the line-of-sight velocity dispersion is one of the major input parameters to calculate the mocked slopes, and actually the most error-prone one, we tested whether the difference between the mocked
4.5. MOCKING THE SLOPES: “OBSERVING” OUR SIMULATIONS 105
Figure 4.9:Upper panels: total density slopes from mock observations of our simulated galaxies (γmock) ver-sus the total density slope calculated directly from the simulations (γsim), for CETGs (red circles) and METGs (blue circles). The dash-dotted line shows the 1:1 ratio.Lower panels: Line-of-sight velocity dispersion calcu-lated for our spheroidals versus the difference between the two density slopes from mocked observation s and simulations (γsim−γmock). Left and right panels show theγmock calculated from edge-on and face-on view of the simulated spheroidal, respectively.
Figure 4.10: Evolution of the total density slope obtained from the mock observations of our simulated spheroidals, γmock, with redshift. Results for the METGs are shown as blue lines, while those obtained for the CETGs are shown as red lines. Dashed (dash-dotted) lines show the average density slopehγmockiat each redshift bin for the face-on (edge-on) projections, with the according variance shown as error bars. The solid lines show the mean of the intrinsic slopes obtained directly from the simulations, for comparison, with the error bars showing the variance. For comparison, the mean value of the observed slopes for each of the four comparison samples is sown as large symbols, including the variance error bars, with the LSD data from Treu
&Koopmans (2004) (triangle), the SL2S data from Ruffet al. (2011) (circle), the SL2S II data from Sonnenfeld
et al. (2013b) (square) and the SLACS data from Auger et al. (2010) (star).
and the intrinsic slopes correlate with the line-of-sight velocity dispersions of our simulated sample.
As can be seen in the lower panels of Fig. 4.9, there is no correlation between the deviations of the two slopes andσLOS. On a side notice, we also see that the velocity dispersions obtained for the METGs are significantly lower and more realistically spread than those obtained for the CETGs. This is also due to the improved numerical schemes used in the Magneticum simulation sets.
To conclude the results of our study and analysis of the origin of the differences between the evolu-tion tendencies of the total density slopes of spheroidal galaxies seen in simulaevolu-tions and observaevolu-tions, we again plot the evolution of the slopes for simulations and observations, as done in Fig. 4.5, but this time we use the total density slopes obtained from the mock observations. As can clearly be seen in Fig. 4.10, the discrepancy between the observed and the simulated density slopes has vanished for the METGs and is significantly smaller for the CETGs. The dashed lines show the median values obtained for the face-on mocks, the dash-dotted lines show the median values obtained for the edge-on views