MAGNIFICATION BY GALAXY GROUP DARK MATTER HALOS

The lenses in this study consist of X-ray-selected galaxy groups in the COSMOS field. See Leauthaud et al. (2010) for the detailed properties of these groups. From the full sample of 206 groups investigated in the aforementioned study, we use the shear-calibrated mass estimates to construct the most massive subsample of groups for this magnification study. Here masses are characterized by the parameter M₂₀₀, the total mass interior to a sphere of radius R₂₀₀, within which the average density is 200 times the critical.

Any groups that have less than four member galaxies that appear to be undergoing mergers, have uncertain centroids, or that raise concerns about projection effects are excluded from the analysis. These restrictions follow from the group catalog requirement FLAG_INCLUDE=1, discussed in George et al. (2011). The remaining 44 most massive groups have shear determined masses in the range 3.56 × 10¹³ ⩽ M₂₀₀/M_☉ ⩽ 1.70 × 10¹⁴, and we employ stacking to increase the S/N of the magnification measurement. The redshift range of the groups is 0.32 ⩽ z ⩽ 0.98. Figure 1 displays these lens properties.

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Choosing an optimal lens centroid about which to construct angular bins is an area of ongoing research, and common choices include the brightest central galaxy or the X-ray emission peak. If the location of the dark matter density peak were known a priori, then it would obviously be the ideal choice, but instead we must rely upon some combination of observables to approximate this position. In this paper, we define lensing mass centers by the location of the group galaxy with the highest stellar mass (MMGG_scale) lying within a distance (R_s + σ_x) of the X-ray center, where R_s is the group scale radius and σ_x is the uncertainty in the X-ray center position (George et al. 2011). In order to be very confident about the locations of group centers, we exclude groups for which this galaxy is not the most massive member of the group. This choice of centroid has been shown to accurately trace the centers of halos in this sample by optimizing the shear signal on small scales (George et al. 2012).

Background sources are LBGs, a type of high-redshift star-forming galaxy that has been used successfully in previous magnification studies (see Hildebrandt et al. 2009b, 2011). These LBGs were selected using the typical three-color dropout technique. For the U-, G-, and R-dropouts the selections described in Hildebrandt et al. (2009a) were used; however, the COSMOS Subaru g⁺ and r⁺ data were used instead of the CFHT-LS g* and r* data (see Capak et al. 2007 for the filter definitions). For the B-dropouts the selection from Ouchi et al. (2004) was used.

The appeal of using LBGs for magnification is rooted in the fact that their luminosity functions (LFs) have been extensively studied and their redshift distributions are fairly narrow and accurate. After all quality cuts and image masking, we are left with 45,132 LBGs in total. The four distinct sets are comprised of 12,980 U-, 22,520 G-, 4870 B-, and 4762 R-dropouts, located at redshifts of ∼3.1, 3.8, 4.0, and 4.8, respectively.

We first test our data selection by cross-correlating the foreground groups with LBGs separated into discrete magnitude bins. Here we use the basic Landy & Szalay (1993) estimator,

$$\begin{equation} w(\theta)=\frac{{\rm D}_1 {\rm D}_2 - {\rm D}_1 {\rm R} - {\rm D}_2 {\rm R} + {\rm RR}}{{\rm RR}}, \end{equation} \tag{ 6 }$$

to simply compute cross-correlations between groups and background sources. D₁ and D₂ represent the data sets of lenses and sources, and R are the random objects from a mock catalog we create, containing points uniformly distributed throughout the COSMOS survey area. Each product of terms is the number of pairs of those objects found to lie within some angular bin, normalized by the total number of pairs found at all angular separations. This cross-correlation estimator has been shown to be both robust and unbiased (Kerscher et al. 2000).

In any lensing study, care must be taken to ensure that regions of an image containing artifacts such as saturated pixels, satellite tracks, or other spurious effects are masked out of the investigation. We consistently apply the same masks to the group, source, and random catalogs, prior to the correlation analysis. Using a large number (584,586) of objects in this random catalog serves to reduce shot noise.

We expect that the faintest (brightest) magnitude bins should yield a negative (positive) cross-correlation with the group centers, and this is exactly what we find. Figure 2 displays this anticipated result, where we simply use a number count weighted average to combine the signal of the distinct LBG samples. As discussed in Hildebrandt et al. (2009b), this negative correlation is one of the strongest verifications that no redshift overlap exists between lens and source populations, for no viable reason other than lensing magnification can be given for such a signal to exist. Redshift overlap between samples must be avoided in magnification studies, as positive cross-correlations due to physical clustering would overwhelm any lensing-induced signal.

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Along with the mass of the lens itself, the slope of the source number counts as a function of magnitude, parameterized by the quantity α ≡ α(m), controls the amplitude and sign of the expected magnification signal. To interpret the correlations that we measure, and to implement an optimally weighted procedure, we must determine the value of this quantity for every source galaxy that we intend to use in the measurement. Fortunately, LBGs have been extensively studied and many measurements of their LFs have been published.

For the U-, G-, and R-dropouts, we use the recent measurements by van der Burg et al. (2010). For the B-dropouts we use the results of Sawicki & Thompson (2006). These two sets of measurements both involved fitting a Schechter function (Schechter 1976) to their galaxy number counts, and their best-fit parameters that we use here are displayed in Table 1. The Schechter Function is given by

$$\begin{equation} \Phi (M)=0.4\ln (10)\Phi ^\ast 10^{0.4(\alpha _{{\rm LF}}+1)(M^\ast -M)} \exp [-10^{0.4(M^\ast -M)}], \end{equation} \tag{ 7 }$$

where Φ*, M*, and α_LF are the normalization, characteristic magnitude, and faint-end slope of the LF. Note that the α(m) that we want to calculate is not the same as the constant parameter α_LF, but approaches it in the limit of very faint magnitudes.

Solving this equation for α(m), we obtain

$$\begin{eqnarray} \alpha (m) &=& 2.5 \frac{d}{ d m} \log n_0(m)= 2.5 \frac{ d}{d M} \log \Phi (M) \nonumber\\ &=& 10^{0.4(M^{*} -M)}-\alpha _{{\rm LF}}-1. \end{eqnarray} \tag{ 8 }$$

We convert the observed apparent magnitudes m of the LBGs to absolute magnitudes M via the relationship M = m − DM + 2.5log (1 + z), where DM and z are the distance modulus and redshift of the galaxy in question. Since we select apparent magnitudes in the r, i, and z bands for the U-, G- and B-, and R-dropouts, we probe very similar rest-frame wavelengths and the K-correction between the samples is negligible. Thus we ignore it here. Using the LF parameters in Table 1, combined with the conversion to absolute magnitudes, we then obtain a measure of α(m) for every LBG in the sample.

It is important to assess how uncertainties in the quantity α(m) can affect the interpretation of the magnification measurement. Since we will rely on the quantity (α − 1) as a weight factor in this analysis, using a wrong α could potentially lead to a bias in the mass measurement. For very faint objects the observed magnitudes become less certain due to shot noise. We propagate these magnitude errors through Equation (8) to obtain an uncertainty on α(m), which is used to find a magnitude-based cut on the sources. We find that cutting ∼10% of the very faintest sources, largely R-dropouts, gives us a good balance between removing the most uncertain α values, but still retaining a significant number of sources for the analysis.

Here we consider two possible sources of systematic error, which are combined in quadrature to yield the total systematic error, reported in Section 4.3. The first source is uncertainty on the LF parameters (see Table 1), which includes the effects of cosmic variance. We repeat the composite-halo fit, detailed in Section 4.3, for the range of permitted values of M* and α_LF, finding a maximum variation in the mass measurement of up to ∼40%. Second, we consider the possibility for a small photometric offset to exist between the various surveys used in this work. Assuming a maximum offset of ±0.05 magnitudes between surveys, we vary all observed source magnitudes uniformly by offsets in the range −0.5 ⩽ δm ⩽ 0.5, and find the maximum effect on the mass measurement to be ∼15%.

We implement a modified version of the Landy & Szalay (1993) estimator for the angular cross-correlation function, in which pair counts are weighted by their expectations from the differential source number counts as a function of magnitude. This weighted correlation function has been shown to optimally boost the magnification signal (Menard et al. 2003):

$$\begin{equation} w(\theta)_{{\rm optimal}}=\frac{{\rm S}^{\alpha -1} {\rm L} - {\rm S}^{\alpha -1} {\rm R} - \langle \alpha -1 \rangle {\rm LR}}{{\rm RR}} + \langle \alpha -1 \rangle. \end{equation} \tag{ 9 }$$

Optimal-weighting was first implemented by Scranton et al. (2005) and, apart from notation, this equation is identical to the estimator used in Hildebrandt et al. (2009b). As with the original basic estimator, each term represents the number of pairs of objects found in a given angular θ bin, normalized by the total number of pairs at all angular separations.

S stands for the sources, or background lensed galaxies, L are the lenses, or X-ray groups, and once again R are the random objects. The superscript (α − 1) on the S indicates that pair counts involving sources are to be weighted by this factor. After removing masked objects from the catalogs, and satisfying the above selection criteria, we are left with 39,710 LBG sources, 44 X-ray group lenses, and 584,586 random objects for the analysis.

The brightest source galaxies, which are observationally found to lie in the steepest part of the LF, are expected to be positively correlated with the group centers, have the largest value of (α − 1), and so receive a relatively large weight in this correlation study. In contrast, the faintest background galaxies are expected to be anti-correlated, on average, with the group positions, because the effects of magnification dilution should be greater than the amplification of flux can compensate for, and these galaxies thus receive a negative weight. Sources for which (α − 1) ≈ 0 ought to have the effects of dilution and amplification cancel out overall, and receive very little to no weight in this analysis (Scranton et al. 2005).

The optimally weighted correlation function is given in Figure 3, and shows the measured radial profiles for this stack of massive galaxy groups. Error bars are 1σ uncertainties, obtained by jackknife resampling of the source population. To do this, we create 50 jackknife samples of data, each with a different 1/50 of sources removed from it. Then we measure the optimal correlation function for each, and from these estimate the covariance matrix through

$$\begin{eqnarray} C(\theta _1, \theta _2) &=& \left(\frac{N}{N-1} \right)^2 \times \sum _{j=1}^N [w_j(\theta _1)-\bar{w}(\theta _1)] \nonumber \\ && \times [w_j(\theta _2)-\bar{w}(\theta _2)], \end{eqnarray} \tag{ 10 }$$

where the index j runs over the N = 50 jackknife measurements.

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Measuring the magnification-induced effects on source number counts behind massive lenses allows one to estimate properties of the lens, such as the mass profile. In this paper, we use a composite-halo approach which allows us to fit for the full range of both group masses and redshifts. The horizontal axes in Figures 2 and 3 are therefore actual transverse distances obtained by taking account of the angular diameter size at each unique group redshift. We incorporate both the singular isothermal sphere (SIS) and the Navarro–Frenk–White (NFW; Navarro et al. 1997) density profiles into the composite-halo modeling.

The magnification contrast is δμ(θ) ≡ μ(θ) − 1, and for an SIS halo it is simply given by

$$\begin{equation} \delta \mu _{{\rm SIS}}(\theta)=\frac{\theta _{{\rm E}}}{\theta -\theta _{{\rm E}}}, \end{equation} \tag{ 11 }$$

where θ_E = 4π(σ_v/c)²D_ls/D_s is the Einstein radius of the lens, and D_ls and D_s are angular diameter distances between lens and source, and observer and source, respectively. The velocity dispersion of the lens, σ_v, can be expressed in terms of the mass and critical energy density of the universe at lens redshift z:

$$\begin{equation} \sigma _{v}=\left[ \frac{\pi }{6}200\rho _{{\rm crit}}(z)M_{200}^2 G^3 \right]^\frac{1}{6}. \end{equation} \tag{ 12 }$$

For the NFW halo, the magnification contrast takes a slightly more complicated form. From Equation (2), we have

$$\begin{equation} \delta \mu _{{\rm NFW}}(\theta)=[ (1-\kappa _{{\rm NFW}})^2 - \left|\gamma _{{\rm NFW}}\right|^2]^{-1} -1. \end{equation} \tag{ 13 }$$

We use the analytical NFW expressions for κ and γ derived in Wright & Brainerd (2000) to evaluate δμ_NFW for every lens–source pair in the study. The two NFW fit parameters are the scale radius r_s and the concentration c, which together determine the mass

$$\begin{equation} M_{200}=\frac{4\pi }{3}(200)\rho _{{\rm crit}}(z)c^3r_{{\rm s}}^3. \end{equation} \tag{ 14 }$$

As we do not find this magnification measurement precise enough to provide meaningful two-parameter constraints, we use the mass–concentration relation given in Muñoz-Cuartas et al. (2011) to reduce this to a single-parameter fit.

We perform a composite-halo fit for both lens models, similar to the multi-SIS used in Hildebrandt et al. (2011). This allows us to fit for a range of masses and redshifts, thereby avoiding any biases that would be introduced by simply fitting to a stacked average lens profile. The optimally weighted correlation function is related to the magnification contrast through

$$\begin{equation} w(a)_{{\rm optimal}}= \frac{1}{N_{l}}\sum _{i=1}^{N_{l}} \langle (\alpha -1)^2 \rangle _i \delta \mu (z_i, aM_{{\rm shear},i}), \end{equation} \tag{ 15 }$$

where i runs over all lenses. Here the fit parameter a characterizes the scaling relation between the M₂₀₀ previously measured from the shear and the best-fit M₂₀₀ from magnification, so that a ≡ M_{magnification}/M_shear.

We use the generalized minimum-χ² method to fit the composite-halo profiles to the magnification measurements (see Figure 3), using the full unbiased inverse covariance matrix, according to the prescription in Hartlap et al. (2007). The χ²/dof is 1.5 for the composite-SIS and 0.8 for the composite-NFW (χ²_SIS = 7.5, χ²_NFW = 4.2, dof = 5 in both cases). For the composite-SIS, we obtain a best-fit value of a = 1.2 ± 0.4 ± 0.4^sys, and with the composite-NFW we get a = 1.8 ± 0.5 ± 0.4^sys. These results indicate consistency with the previous shear mass measurements, albeit with large uncertainties.