THE SLOAN LENS ACS SURVEY. XI. BEYOND HUBBLE RESOLUTION: SIZE, LUMINOSITY, AND STELLAR MASS OF COMPACT LENSED GALAXIES AT INTERMEDIATE REDSHIFT

We use a subset of lenses from the SLACS survey (Bolton et al. 2008) as our sample. We select those lenses which were classified as "definitely a lens" and were imaged in the V_HST, I₈₁₄, and H₁₆₀ bands. The instruments and filters these refer to are as follows: for the V_HST-band, ACS/WFC F555W or WFPC2 F606W, for the I₈₁₄-band, ACS/WFC or WFPC2 F814W, and for the H₁₆₀-band, NICMOS/NIC2 F160W. For the majority of the observations a full orbit's exposure (t_exp ∼ 2000 s) is available, except for 15 lenses that have only HST Snapshot images in ACS/WFC F814W (t_exp = 420 s). For details on the observations and data analysis, as well as full object coordinates, see Paper IX. All images were drizzled onto a pixel scale of 005. This multi-filter sample comprises 46 lensed galaxies.

When performing lens modeling, we attempt to fit the often-faint lensed images, which can be hidden by or confused with light from the lens (foreground) galaxy; thus, it is necessary to remove the light of the lens galaxy prior to modeling. As the SLACS survey preferentially selects bright lens galaxies this is particularly important for our work. We use the radial B-spline technique, first introduced by Bolton et al. (2006, in the appendix), for this purpose. We refer the reader to Paper V for a full discussion of this method, but provide a brief summary here. First, zero-weight pixels, neighboring objects, and potential source galaxy features are masked. Second, the data are fit using only monopole, dipole, and quadrupole terms in the angular structure. The lensed features are masked based on the residual image from this initial fit; the image is then re-fitted using higher order multipole terms as necessary. Marshall et al. (2007) estimated that the systematic errors in the source size and brightness due to the subtraction of lens galaxy light by this method are approximately 2% (0.01 kpc for a 0.6 kpc source at z_s = 0.6) and 0.10 mag, respectively. This is an important but usually not dominant source of systematic error. Similarly, subtraction of the lens galaxy may introduce a systematic uncertainty on the Sérsic index, similar to those introduced by improper sky subtraction; based on the detailed work by Marshall et al. (2007), we estimate this systematic uncertainty to be of order 0.1–0.2. This is larger than random errors, but does not effect the results of this paper.

We have modeled the SLACS lenses in three filters; the data and model lens planes for the primary modeling filter are shown in Figure 1. In order to obtain robust and easy-to-interpret constraints on the source galaxies, we make three fairly standard assumptions, for which we give both a priori justification and a posteriori validation.

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Assumption 1. As in the testing program described in the Appendix, we use only one source galaxy, even though in a few cases the source is evidently more complicated (see Bolton et al. 2008). Without this assumption, we would be unable to interpret the size and stellar mass of the galaxy. The alternative would be to use a more complex source model, perhaps defined on a pixelated grid (e.g., Warren & Dye 2003; Treu & Koopmans 2004; Brewer & Lewis 2006; Suyu et al. 2006; Vegetti & Koopmans 2009), and then derive the size and magnitude of the source from this complex model. However, imposing a simple, one-component Sérsic model is standard in the study of faint galaxies, even though they may have irregular morphologies or multiple knots of star formation (e.g., Ferguson et al. 2004). If we are to compare our sample with others in the literature, we need to derive the same parameters as were studied by previous authors. A potential concern is that oversimplification in the source surface brightness distribution may lead to small biases in the mass model. However, as we discuss in Section 3.2, comparison of our mass model parameters with those inferred using complex multicomponent source distributions shows that, at least for the SLACS sources, this is a negligible source of error. In conclusion, considering that using a simple analytic source model greatly speeds up computation time (∼15 minutes in our approach compared with several hours for a typical pixel-based source reconstruction, with a standard desktop CPU), we adopt this procedure.

Assumption 2. We require that the mass model be identical across the different filters, in order not to bias our source reconstruction. Although ideally this would be avoided naturally because there is only one deflector potential, lens galaxy subtraction and differences in contrast and signal-to-noise ratio (S/N) can result in small differences, which we quantify as follows. Using the I₈₁₄ band as reference, we found the difference in σ_SIE to be, on average, 0.6 km s⁻¹ with an rms scatter of 13.5 km s⁻¹ in the H₁₆₀ band, and −2.0 km s⁻¹ with an rms scatter of 11.5 km s⁻¹ in the V_HST band. Likewise, the difference in mass distribution axis ratio q was, on average, −0.007 with an rms scatter of 0.15 in the H₁₆₀ band and −0.03 with an rms scatter of 0.12 in the V_HST band. Two or three outliers (defined as having |Δσ_SIE| > 50 km s⁻¹ or |Δq| > 0.6) were excluded from each of these calculations. Fixing the mass model to that of the highest S/N reconstruction ensures that we do not introduce any unnecessary extra scatter in the properties of the source.

Assumption 3. We fix the source morphology parameters (position angle, inclination, size, and Sérsic index) to the best-fit values from the filter with highest S/N. (For two-thirds of galaxies this filter is F814W; for the remainder, only snapshot images were available in F814W and the filter with the highest S/N is F606W. Those objects with only snapshot images in F814W are indicated in Table 1.) This is necessary because the S/N sometimes differs significantly between filters, which can cause low-surface brightness features to be missed and result in overly small sizes and Sérsic indices, which in turn affects the inferred magnitude. This procedure, analogous to the SDSS model magnitudes (Abazajian et al. 2009 and references therein), is effectively equivalent to measuring colors within fixed aperture and is widely adopted in order to obtain robust colors when the S/N varies significantly between filters.

Inferred, unlensed angular sizes and apparent magnitudes for the source galaxies are given in Table 1, along with their lens and source redshifts, and a flag indicating whether the V_HST band magnitude refers to WFPC2/F606W or ACS/F555W. We also give the total magnification μ of each system. This quantity varies widely, between 2.4 and 44.3; the median magnification is 8.8^+6.7 _{− 5.1}, where the error bars correspond to the 16th and 84th percentiles. The lens galaxy properties—mass axis ratio, inclination, and the velocity dispersion—are not presented here, as our concern is with the source galaxies. For a thorough treatment of the SLACS lens galaxies, we refer the reader to Paper V and Paper IX.

The F814W images of the lenses in our sample have been modeled by Bolton et al. (2008) and Auger et al. (2009) using multiple objects in the source plane. This gives us the opportunity to test klens' ability to model real lenses and estimate systematic (modeling) errors. Although we cannot compare source models (given the different definitions), we are able to look at the effects of using only one source object on inferred mass model parameters. The lens mass model is described by the mass axis ratio (q), the inclination, the velocity dispersion of the best-fitting singular isothermal ellipsoid (σ_SIE), and the mass centroid. We compare mass axis ratios and velocity dispersions and find that they agree to within levels expected from systematic errors: the average offset between our fits and those from Paper V is −0.015  ±  0.002 for q and 2.0  ±  0.2 km s⁻¹ for σ_SIE. The comparison for velocity dispersion is shown in Figure 2.

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We can use the scatter of these relations to estimate total errors, including systematics: we calculate 0.07 for the error on q and 1.78% for the error on σ_SIE. These are slightly larger than the errors adopted in Paper V (0.05 on q and 1.0% on σ_SIE), as expected because our models are less flexible. As discussed by Marshall et al. (2007) and in the summary at the end of the Appendix, if the potential is not perfectly described by an SIE (e.g., Koopmans et al. 2009; Auger et al. 2010) then additional errors will apply to the source galaxy properties. These are taken into account in the analysis of the source population.

We note that one of the systems, J0737+3216, has a published source size, magnitude, and stellar mass (Marshall et al. 2007). Even with a completely different lens modeling code, we find lens parameters, source size, and source Sérsic index for this system that are consistent with those in this previous work, providing further confirmation of the robustness of our approach. However, we infer a source that is 1.5 mag fainter uniformly in all bands. This was traced to an error in the modeling code used by Marshall et al. (2007), which summed the flux in twice-subsampled pixels rather than averaging it: this factor of four in flux translates to a magnitude difference of 1.5. After correcting for this "bug" the magnitudes are also in excellent agreement. The results here supercede those published in Marshall et al. (2007).

These comparisons show that our models are reliable and give answers consistent with those obtained by other methods once systematic errors are taken into account.

We now use our multi-band photometry of the SLACS sources to infer rest frame luminosity in the B and V (Johnson Vega) bands as well as stellar masses (M_*). This analysis is based on our models of the unlensed source galaxies, where modeling has been carried out in three separate bands. For this purpose, we use the Bayesian code developed by Auger et al. (2009) to fit stellar population synthesis (SPS) models to our multi-band source galaxy photometry. The code computes, for each galaxy, the likelihood of SPS models as a function of stellar mass, age, metallicity, star formation history, and dust content. In combination with a prior on each of these parameters, the likelihood then gives a posterior PDF for each model parameter. The same models and posterior can also be used to generate self-consistent rest-frame luminosities. For this application we adopt SPS models by Bruzual & Charlot (2003) and uniform priors on the logarithm of stellar mass, age, metallicity, and timescale of the exponential star formation history, as is appropriate for the situation where we do not know the order of magnitude of these quantities.

As discussed by Auger et al. (2009 and references therein), although parameters such as age and metallicity are often degenerate, stellar masses and luminosities in the range of wavelength covered by the data can be derived quite accurately for a given initial mass function (IMF). Typical errors on the transformations to rest frame luminosities are of order 0.05–0.1 mag, while typical errors on stellar mass are of order 0.1–0.2 dex (see Table 2 for details). For simplicity we neglect the impact of emission lines on broadband photometry, which is estimated to be of order a few percent for the typical Hα fluxes (the strongest line in the wavelength range of interest) of a few 10⁻¹⁶ erg s⁻¹ cm⁻². Typically, no strong emission line is present in the H₁₆₀-band filter, which is providing the bulk of the information for stellar mass estimate. The main residual source of uncertainty is the normalization of the IMF. In this paper, we adopt the Kroupa (2001) normalization of the IMF, to facilitate comparisons with the GEMS work (Barden et al. 2005). The uncertainty in the lens mass density profile slope is an additional source of error (Marshall et al. 2007).

As a sanity check we compared our estimated stellar masses with those inferred by applying the "standard" recipe by Bell et al. (2003) to our inferred colors. For the same IMF the stellar masses agree very well, with an average offset of 0.02 ± 0.02 dex (rms scatter 0.15 dex). Our synthetic rest frame photometry agrees well with that inferred by standard "K-correction" procedures (for example, by comparing with the V-band magnitudes inferred via an Sbc template we find an average offset of 0.06 ± 0.06 mag). We conclude that our stellar mass and synthetic photometry are robust and unbiased within the errors.

For the first comparison sample, we follow Barden et al. (2005) and select, from the publicly available GEMS catalog,⁸ galaxies with successful galfit structural fits (GEMS_FLAG = 4) matched within 05 of a COMBO-17 object with successful photometric redshift estimate (COMBO_FLAG = 3). We did not select galaxies based on their size or Sérsic index (other than to reject objects with large size or Sérsic index uncertainties, or that reached the galfit boundary conditions, as did Barden et al. 2005), but we did reject objects not detected in both F606W and F850LP filters. The resulting GEMS sample comprised 6999 galaxies with measured absolute magnitudes and sizes. We adopt the F850LP-measured sizes, since these differ from the rest-frame V-band sizes by typically only 3% (Barden et al. 2005). We then select galaxies in the redshift range 0.4 < z < 0.8 to approximately match the redshift distribution of SLACS sources, resulting in 3369 galaxies. These are shown as small black points or hatched blue histograms in Figures 3–8. We note that this GEMS subsample represents the brighter part of the GEMS full HST catalog with surface photometry, owing to the shallower COMBO-17 multiband photometry used to determine photometric redshifts (completeness limit of F850LP ∼ 23.5 versus F850LP ∼ 24.5 for the HST catalog; Barden et al. 2005).

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We computed stellar masses for all objects in the GEMS sample by applying the recipe by Bell et al. (2003, for a Kroupa IMF) to the rest frame photometry provided in the GEMS catalog. As discussed before, for the SLACS sample the Auger et al. (2009) stellar masses are in excellent agreement with the ones obtained using the recipe by Bell et al. (2003), ensuring that we are able to make this comparison.

For the second comparison sample, we use the galaxy catalog⁹ from the HUDF analyzed by Coe et al. (2006). We selected galaxies in a manner similar to the process used for GEMS. Galaxies included in our sample are those: (1) with good galfit fits, (2) that are matched to Beckwith et al. (2003) ACS objects within 05, (3) with successful Bayesian photometric redshifts, and (4) which are detected in B, V, i', and z'. Our criterion for successful galfit and redshift fits is that the χ² be within 4.5σ of the mean. We also restrict our sample to galaxies in the redshift range 0.4 < z < 0.8 to match the redshifts of the GEMS and SLACS source samples, for a final tally of 841 galaxies. These are shown as black points or hatched red histograms in Figures 3–8.

We use kcorrect from Blanton & Roweis (2007) to compute rest-frame luminosities of the HUDF comparison sample. We then use the Bell et al. (2003) algorithm to calculate stellar masses, as described in Section 4.1.

We plot the distributions of source Sérsic index and apparent F814W source magnitude in Figure 3, showing both the SLACS sources and the GEMS sample. The vertical dotted lines show the magnitudes tested by our simulations, demonstrating that we are in a regime where source properties can be reliably measured by klens. In this figure and throughout this work, the quoted source galaxy magnitudes are for the unlensed galaxy, i.e., corrected for magnification.

When considering apparent magnitudes, the SLACS sources lie squarely between the two regimes sampled by the GEMS and HUDF surveys. Most SLACS source galaxies fall in the range 22 < F814W < 26, peaking at magnitudes fainter than the completeness limit of the GEMS survey but brighter than most galaxies in the HUDF sample. As mentioned earlier, the drop-off of the GEMS catalog at ∼23.5 is mostly due to the COMBO-17 completeness limit. However, a substantial fraction of the SLACS sources are fainter than even the GEMS HST completeness limit, suggesting that we are indeed exploring a different population of intrinsically fainter objects. The HUDF survey is much deeper than the SLACS survey (144 HST orbits versus one in I band), so it is not surprising that these objects are fainter than the SLACS sources. The average magnitudes for the SLACS, GEMS, and HUDF surveys are 24.3, 22.0, and 27.5, respectively.

In contrast, the distribution of Sérsic indices is essentially the same for the three samples—peaked at n ∼ 1 generally interpreted as dominated by faint disks or compact galaxies—even though the magnitude ranges probed by the three surveys are very different. The median for both SLACS and HUDF is 1.1; for GEMS it is 1.3.

The difference between the SLACS sources and the GEMS galaxies is even more pronounced in terms of the size distribution, shown in Figure 4 with the same notation as Figure 3. The size distribution of GEMS sources is much broader and peaks at 078, petering off below 02. In contrast, Figure 4 highlights the similarity of the SLACS and HUDF galaxies. The overall size distributions for the two samples are remarkably similar, though the HUDF sample extends to much larger sizes, beyond the plotted region. This long tail pushes the average size up to 035 (2.3 kpc) for HUDF, compared to 019 (1.24 kpc) for the SLACS sources. The median sizes, however, are more similar: 012 for HUDF and 014 for SLACS, corresponding to 0.8 kpc for both samples.

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The complementarity of the SLACS, GEMS, and HUDF samples is further illustrated by Figure 5 where sizes are plotted against apparent magnitude. With one exception, SLACS sources have comparable surface brightness to that of the GEMS sources, but extend much further down in magnitude and size. A few of the brightest SLACS sources have magnitudes comparable to those of the GEMS sources, but have sizes at the compact end of the distribution. In contrast, the SLACS sources have significantly higher surface brightness on average than the HUDF galaxies.

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In Figure 5, crosses represent galaxies in the low-redshift bin (0.4 < z < 0.6) while squares indicate the higher redshift galaxies (0.6 < z < 0.8). The galaxies in all three samples are not strongly segregated by redshift, consistent with a broad distribution in intrinsic luminosity.

The comparison makes it apparent that the SLACS source population is significantly different from the one studied by typical HST broadband imaging surveys such as GEMS. These sources are most likely excluded by HST field surveys because they are too faint and/or too compact to be detected and identified as galaxies (as opposed perhaps to stars). Furthermore, even if detected, they are generally too faint and compact to determine their surface brightness profile, size, and stellar mass. The HUDF survey relies on its extremely large exposure times to detect and model such sources. Conversely, SLACS identifies them as galaxies and measures their redshifts via their emission lines (even though they are detected in all three broad bands), and therefore their intrinsic faintness and compactness does not represent an obstacle to detection, while magnification helps in determining their structural parameters. In fact, the distribution of SLACS sources and sizes is in agreement with a generic feature of lensing survey. Owing to magnification bias, faint and compact sources tend to dominate the source population when their number density decreases sharply with luminosity and size (e.g., Treu 2010 and references therein). We will return briefly to the lensing selection function in the next section.

We now explore the distribution in size, luminosity, color, and stellar mass for the SLACS source galaxies. We use two-dimensional plots to examine bivariate distributions in the space of well-known correlations representing physical mechanisms: the size–luminosity relation, the color–magnitude diagram, and the size–stellar-mass relation. Again we use the GEMS and HUDF samples as a comparison.

We begin with the size–luminosity relation in Figure 6. We fit a size–luminosity relation of $\log _{10} r_{\rm eff}\,\rm (kpc) = -0.12 \times {\textit {V}} \,{-}\, 2.25$. However, this result should be interpreted with a care, keeping in mind the small number of objects in our sample and the uncertain selection function. A linear extrapolation of the GEMS size–luminosity relation shows little offset between the GEMS and HUDF galaxies, but the SLACS sources are offset toward smaller sizes by 0.6 kpc or ∼30%. This offset may indicate that there is an extended tail of ultracompact galaxies of which the SLACS sources are a part, similar to the population of compact ELGs (Koo et al. 1995; Phillips et al. 1997; Drozdovsky et al. 2005; Straughn et al. 2009).

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We note that, as expected for a flux limited sample, the intrinsically fainter objects are only detected at the lowest redshifts (in Figures 6–8, crosses represent galaxies between 0.4 ≲ z ≲ 0.6 and squares those between 0.6 ≲ z ≲ 0.8). Other than this luminosity segregation, no evolution is apparent within our data set.

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The color–magnitude diagram shown in Figure 7 paints a similar picture of the properties of the SLACS sources. They span the range of blue colors typical of star-forming galaxies, and of the GEMS and HUDF samples, even with their different intrinsic luminosities. Again, other than luminosity segregation, no evolution is apparent. Our data (and GEMS) are also consistent with no relation between color and magnitude, in contrast to the sample of HUDF galaxies we have selected for comparison, which do show bluer galaxies at higher redshift. This perhaps surprising result is consistent with those of Pirzkal et al. (2006), who looked at ELGs in the HUDF, with −14 < M_B < −22 and rest-frame B − V colors from ∼0 to ∼1. As we discuss below, the SLACS sources may be best represented as ELGs, which could be the cause of this similarity.

Finally, the offset in size with respect to GEMS and HUDF is pronounced even in the size–stellar-mass plane (Figure 8), where the SLACS sources seem to define a sequence offset to smaller radii by 0.5 kpc or ∼30%. This indicates that the offset in the size–luminosity relation is not due to abnormally low mass-to-light ratios but to an intrinsic compactness of the population. For the SLACS sources, we find a size–mass relation of log₁₀r_eff (kpc) = 0.24 × log₁₀M_*/M_☉ − 2.20, although we again caution the reader as to the small sample size and uncertain selection function.

We note that the stellar masses of the SLACS sources are remarkably low for a population at cosmological distances, extending below 10⁸ M_☉. Three systems in particular have remarkably low stellar mass: J1538+5817 (z_s = 0.531, M_* = 10^7.03 M_☉, L_V = 10^7.9 L_☉), J1402+6321 (z_s = 0.481, M_* = 10^6.99 M_☉, L_V = 10^7.8 L_☉), J1719+2939 (z_s = 0.578, M_* = 10^6.86 M_☉, L_V = 10^7.7 L_☉).

Their V-band luminosities are just 0.5–0.8 dex higher than those of the brighter Milky Way dwarf satellites, Sagittarius and Fornax, and considerably fainter than the Large Magellanic Cloud (3 × 10⁹ L_☉; van der Marel et al. 2002). As can be seen in Figure 1, these three sources are being lensed into high magnification partial Einstein rings (their total magnifications are the three highest: 44.3, 31.2, and 40.7, respectively). The J1538+5817 source may be a low luminosity satellite of a brighter companion, visible just outside the ring.

Although the comparison between our galaxies and GEMS or HUDF is useful—it allows us to compare the physical properties of our galaxies, including stellar mass, to large samples with the same redshift distribution—the selection algorithm for SLACS is quite different from that of GEMS or HUDF. The SLACS sample was selected based on the detection of source galaxy emission lines (Bolton et al. 2004); GEMS and HUDF are both imaging surveys. It is therefore useful to compare the SLACS sample with emission line selected galaxies (ELGs). SLACS is sensitive to observed fluxes as low as ∼6 × 10⁻¹⁷ erg s⁻¹ cm⁻² (Bolton et al. 2004; Dobler et al. 2008). Taking into account the effects of magnification the equivalent depth is typically ∼7 × 10⁻¹⁸ erg s⁻¹ cm⁻², but can be higher by a factor of a few for the most extreme objects.

ELGs identified by HST grism surveys (e.g., Drozdovsky et al. 2005; Straughn et al. 2009) reach comparable depths and provide a good benchmark. Those samples span similar broadband magnitude ranges to the SLACS sources and have comparable sizes. For example, the size distribution of galaxies in the sample of Drozdovsky et al. (2005) is limited to small sizes (r_eff < 05) and peaks at r_eff ∼ 01–02. Magnitudes peak at F814W ∼ 24, reaching magnitudes as faint as 27. These are quite similar distributions to those for the SLACS galaxies.

The gravitational lenses in our sample are known to be well approximated by simple, SIE mass distributions (Koopmans et al. 2006). We therefore use an SIE model to describe the mass distribution of the lens galaxy. This model has five parameters: the velocity dispersion σ_SIE, elliptical axis ratio q = b/a, orientation angle θ_d, and position (x_d, y_d) (see, e.g., Kormann et al. 1994, for details). For the source galaxy surface brightness, we use an elliptically symmetric distribution with Sérsic profile. The source model has seven parameters: the position (x_s, y_s), orientation angle θ_s, inclination angle i (more often referred to be cos i, which equals the apparent ellipse axis ratio), effective radius r_eff, AB apparent magnitude m, and Sérsic index n. A Sérsic index of n = 0.5 corresponds to a Gaussian, n = 1 to an exponential disk, and n = 4 to a de Vaucouleurs profile (see, e.g., Peng et al. 2002 for details).

In Paper V, we used multiple Sérsic profile components to describe the source galaxy. Because our goal here is to measure the physical properties of the source such as size, magnitude, and mass and compare them with corresponding measurements of non-lensed galaxies in the literature, we choose instead to describe the source as a single object. As a result, we do not expect to be able to fit all our sources to the same level of detail as we did in Paper V. One option would be to remove all lenses with complex sources from our sample; however this would bias us toward smaller and simpler sources, which is undesirable. This same single-component simplification was used in, for example, the GEMS and SDSS size–mass relation studies; restricting ourselves to single-component sources allows us to make direct comparisons with these non-lensing surveys.

Asserting that the assumption of an SIE model for the lens galaxy mass profile is the most significant source of systematic uncertainty, Marshall et al. (2007) estimated the systematic errors in the source size and brightness to be approximately 12% (0.07 kpc for a 0.6 kpc source at z_s = 0.6) and 0.26 mag, respectively. However, we note that these errors will be common to all observations of a given lens.

To ease the exploration of the model parameter space, we work with hidden parameters constrained to vary uniformly on a hypercube, such that their typical values are around one. These hidden parameters are then transformed to the physical parameters described in the previous section before the predicted image is generated. This transformation need not be linear: in fact, it is used to encode our prior knowledge about the model parameters. For example, a uniform prior PDF is implemented as a linear transformation, while exponentiating a hidden parameter corresponds to assigning a scale-free ("Jeffreys," 1/x) prior. Truncation of the prior PDFs is achieved where necessary by setting the log likelihood to be very large and negative outside the specified limits. The parameters of our model are listed in Table A1, along with the prior PDFs assigned to them, unit transformation used, and the range over which they are allowed to vary.

At each point in parameter space visited we compute the pixel value likelihood function. We assume Gaussian errors on the pixel values, given by the weight (inverse variance) map produced during data reduction (Paper IX). The log likelihood is then just minus half the image χ². The predicted image, required for this statistic, is computed by mapping each pixel's position back to the source plane via the lens equation, and looking up the value of the model source surface brightness. We oversample the image plane to make sure we resolve the source. The lensed images are then convolved with a model PSF; we use Tiny Tim (Krist 2003) to generate this PSF. Note that we do not model lens galaxy light, only the source surface brightness. We therefore assume perfection subtraction of lens galaxy light, an assumption which incurs the systematic error discussed at the end of Section 2.

Since the prior PDF is implemented by parameter transformation, optimization of the posterior is equivalent to maximizing the log likelihood with respect to the hidden parameters. For this optimization we use the IDL routine mpfit (Markwardt 2009), which is a generalized implementation of the Levenberg–Marquardt nonlinear least-squares algorithm.

The likelihood function is typically quite sharply peaked, with multiple local maxima to be avoided. We find that a three-step process is required for the global peak to be successfully located. First, we make an initial estimate of the source galaxy parameters using a best-guess mass model: the image pixels are traced through the mass model using the lens equation and flux is accumulated on a grid in the source plane. Where multiple image pixels map to the same source plane pixel, we take their simple average. We then take the initial source galaxy position to be the first moment of this resulting light distribution, while summing all the source plane pixel values provides an initial approximation to the source magnitude (this is usually an overestimate). We gain no knowledge of the source inclination, size, position angle, or Sérsic index in this first stage; the source position and magnitude are the most important for finding a good fit to the data, in the sense that small deviations in these parameters can lead to very low likelihood models.

Next, we refine our initial estimates by running mpfit, using our initial estimates of the source position and magnitude as its starting point (we initialize the other parameters at typical values). This preliminary run is performed without image plane subsampling, for speed. We also use a deflated weight image (the square root of the weight) in this second stage; this smooths the likelihood function, making it easier for the optimizer to locate the global maximum. Finally, in the third step we re-start mpfit at the position found in step two to get final maximum posterior model parameters for the mass and source. For this final run, we use an image plane oversampling of four (along each axis), and the normal weight image. Convolution with the PSF is done on the oversampled grid; we then use neighborhood averaging to return to the original size.

In order to verify that the klens code works properly, we performed tests on (1) simulated non-lensed galaxies, (2) real non-lensed galaxies, and (3) simulated lenses. All simulated data were created using HST/ACS I₈₁₄-band (F814W) image parameters and PSFs, while the real galaxies were selected from HST/ACS I₈₁₄-band images.

A.4.1. Testing klens on Simulated Non-lensed Galaxies

We first test klens on simulated non-lensed galaxies. Each mock galaxy is taken to be a single elliptically symmetric Sérsic profile component, to allow us to investigate straightforward parameter recovery by the code. The sizes, magnitudes, and Sérsic indices of these galaxies were varied over a 3 by 3 by 3 grid: we chose sizes from (015, 03, 06), magnitudes from (20.5, 21.5, 22.5), and Sérsic indices from (0.5, 1, 2). All position angles were set to θ_s = 1.75 and inclinations at cos (i) = q = 0.6. The simulated galaxies were placed at the center of 42 × 42 pixel images. Magnified by a typical lens, this size is approximately equivalent to the 122 × 122 pixel cutouts used for the lenses. We generated ten independent noise realizations for each galaxy, giving a total of 270 simulated observations. The whole sample were then modeled with both klens (using the priors given in Table A1) and galfit.

The results of the parameter recovery are given in Table A2, which lists the mean (inferred − truth) difference and associated rms scatter values. We look at the results more closely in Figure A1. Overall klens and galfit give very similar results, although the rms scatter is larger with klens. For both modeling programs, the largest galaxies contribute most of the bias, which is in the sense that klens tends to slightly underestimate sizes, while galfit tends to slightly overestimate them, even though the residual images are consistent with simulated noise. This is because the two programs are sampling different parts of the covariance between size, magnitude, and Sérsic index. To illustrate this point, we plot the best-fit Sérsic indices against the best-fit sizes in Figure A2, again grouped by true parameter value.

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Table A2. Mean Parameter Difference (Inferred − True) and Associated rms Scatter for Simulated Non-lensed Galaxies

Parameter	klens Models	galfit Models
〈Δm〉	(118 ± 1) × 10−4	(− 76.0 ± 0.6) × 10−4
σm	0.027	0.016
〈Δreff〉	(− 730'' ± 7) × 10−5	(546'' ± 4) × 10−5
σr	0.019	0.012
〈Δn〉	(− 350 ± 3) × 10−4	(159 ± 2) × 10−4
σn	0.079	0.041

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A.4.2. Test on Real Non-lensed Galaxies

Next, we tested klens on real non-lensed galaxies. We selected a sample of 981 galaxies, detected in a subset of the SLACS fields, to match the nine magnitude and size bins used in the previous test: the sample consists of all objects in those fields that have SExtractor magnitudes 20 < m < 23 and 01 < r < 065 that are not stars or image defects. We then ran klens on each test image, with the priors given in Table A1; we also ran galfit on each data set.

For this sample we do not know the true values for each object's size, magnitude, and Sérsic index; we can only compare galfit and klens results. When comparing Sérsic indices, we noted significant systematic differences between the two programs, especially at n > 1.5. This scatter is, as discussed in Appendix A.4.1, a result of klens and galfit sampling different parts of the covariance between Sérsic index, size, and magnitude. We compare the two codes' best-fit magnitudes in Figure A3. In this plot we highlight those galaxies for which best-fit Sérsic indices differ by less than 20%; 408 objects meet this requirement. The magnitudes and sizes for these galaxies were found to be in reasonable agreement: 〈Δm〉 = 0.147  ±  0.008 with rms scatter 0.170. A similar analysis for size gives 〈Δr_eff〉 = −0.174  ±  0.009 with rms scatter 0.175. As we decrease the allowable difference in Sérsic index, the bias and scatter both decrease.

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galfit occasionally returns very low magnitudes relative to the SExtractor values, and that these are associated with Sérsic indices very different from the klens values. We note that, for example, Barden et al. (2005) reject objects with galfit magnitudes differing from their SExtractor magnitudes by more than 0.6 mag.

A.4.3. Test on a Simulated Lens Sample

Finally, we tested the ability of klens to recover the structural parameters of lensed source galaxies. We can do this to a limited extent by comparing our klens mass models to the original SLACS lens models (Section 3.2). However, for the source parameters we need to use simulated galaxies, as we did in Section A.4.1 in the non-lensed case. We generated a sample of lensed galaxies to be representative of the possible lensed image configurations and sources. The exact properties of the lens itself are less important: we used the same mass profile (σ_SIE = 270 km s⁻¹, q = 0.77, and θ_d = 1.75 rad, typical parameters for a SLACS lens) for each synthetic system. We also fixed the source galaxy orientation at θ_s = 1.75 rad and its inclination at cos (i) = q = 0.6, again typical values.

The source position, effective radius, magnitude, and Sérsic index were varied, to give 108 possible lenses. We chose four source positions such that the four main lens morphologies (double, caustic, cusp, and quad) were represented. The source effective radius was chosen from (005, 01, 02), the magnitude from (23, 24, 25), and the Sérsic index from (0.5, 1.0, 2.0). Taking into account magnification due to lensing, these sizes and magnitudes are roughly equivalent to those used in the non-lensed galaxy tests. For each mock system we simulated pixelated images, again with no lens galaxy light. We generated 10 noise realizations for each lens, to give a total of 1080 mock observations. We then ran klens on each mock data set, with the priors given in Table A1.

Parameters for which the same true value was used in all simulations (i.e., the lens parameters, and the source inclination and orientation angle) are well modeled, with only small biases well below the rms scatter (see Table A3). We divide the sample into bins to consider the inferences of the remaining parameters, their means and scatters across the sample; these results are presented in Table A4. In Figure A4 we look at the performance of klens in recovering the input effective radius and magnitude, binned by their true values.

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Table A3. klens Parameter Recovery in a Simulated Lens Sample—Common Lens and Source Parameters

Parameter	True Value	Inferred Value	rms Scatter
q	0.77	0.76044 ± 0.00006	0.063
σSIE (km s−1)	270	269.961 ± 0.007	7.294
θd (rad)	1.75	1.7368 ± 0.0002	0.221
θs (rad)	1.75	1.7288 ± 0.0003	0.293
cos (i)	0.6	0.58685 ± 0.00004	0.044

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Table A4. klens Parameter Recovery in a Simulated Lens Sample—Variable Source Parameters

Parameter	True Value	Inferred Value	rms Scatter
Effective radius	005	00543 ± 0.0006	0011
	01	01023 ± 0.0009	0016
	02	0195 ± 0.001	0022
Magnitude	23	23.01 ± 0.02	0.107
	24	24.00 ± 0.01	0.062
	25	24.95 ± 0.02	0.265
Sérsic index	0.5	0.62 ± 0.02	0.354
	1.0	1.05 ± 0.01	0.247
	2.0	1.98 ± 0.02	0.291

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We find that the inferred parameters are consistent with the input values, albeit with larger scatter than was seen in the non-lensed simulation case. This is partly due to our use of fainter galaxies, which are harder to measure when the image configuration is of lower magnification. (We note that the scatter is smaller for high surface brightness galaxies, indicating that—as might be expected—it is easier to model such galaxies.) It may also be due to the way the information in the data is being used to simultaneously constrain the lens model—to some extent changes in the source parameters can be balanced by changes in the lens model.

A.4.4. Summary