THE REDSHIFT DISTRIBUTION OF GIANT ARCS IN THE SLOAN GIANT ARCS SURVEY

The lensed background sources discussed in this Letter are located near the cores of foreground massive galaxy clusters with a median redshift, z_l = 0.435, and a median dynamical mass, M_Vir = 7.84 × 10¹⁴ M_☉ h⁻¹_0.7 (Bayliss et al. 2010).

We take published spectroscopic redshifts from Bayliss et al. (2010), and use optical colors combined with the absence of specific spectral features to identify upper and lower redshift bounds for some arcs that lack precise spectroscopic redshifts. Given the spectral coverage of the Gemini/GMOS spectroscopy presented in Bayliss et al. (2010) we expect to observe one or more prominent emission lines (e.g., O[II] λ3727, H-βλ4861, O[III] λ4959, 5007, and H-αλ6563) for galaxies at z ≲ 1.5 that are actively star forming, with some slight variation in this redshift limit that depends on the exact spectral coverage for each source. For strongly lensed sources at 1.5 ≲ z ≲ 3.3 we must rely on rest-frame UV features to measure redshifts, but the strength of these features can vary significantly depending on the physical properties of the individual galaxies. At redshifts of z ≳ 3.3 we expect to see a broadband flux decrement that would be indicative of the Lyα Forest and Lyman Limit being redshifted well into the g band, as well as strong absorption or emission from Lyαλ1216 redshifted into our data for sources with spectral coverage extending sufficiently blueward.

Using the criteria outlined above we identify six arcs in the Bayliss et al. (2010) Gemini/GMOS data for which we can confidently place lower and upper limits on the redshifts. These six arcs all have blue colors in the available photometry, indicating that they are actively star forming, and their optical spectra consist of blue continuum emission with no strong absorption or emission features. The precise redshift constraints vary slightly from arc to arc, depending on the exact limits of the wavelength coverage, and are given in Table 1.

Arcs in our primary giant arc sample meet two criteria: (1) they were visually identified in the original SDSS imaging data and (2) they were observed spectroscopically with Gemini/GMOS-North. Our primary giant arcs span a range in integrated g-band magnitude of 20.69 ⩽ g ⩽ 22.81 (Figure 1). This sample includes all sources identified as primary giant arcs in Bayliss et al. (2010), plus four additional arcs with redshift desert constraints as described in Section 2.1 of this Letter ("Primary" in Table 1). We are spectroscopically complete for the entire primary sample.

Zoom In Zoom Out Reset image size

Our sample of secondary background sources includes objects that appear to be strongly lensed but lack the brightness and/or morphology to be identified in the visual selection processes that produced our primary giant arc sample. Most of these sources were identified in the Gemini/GMOS pre-imaging data described in Bayliss et al. (2010) and are designated as "secondary strongly lensed sources" in that paper. The secondary sample selection is not as uniform as the primary sample, and is spectroscopically incomplete, as there were many of these candidate strongly lensed sources that were targeted in the Gemini observations of Bayliss et al. (2010) but for which the data do not provide a redshift.

Efforts to produce theoretical predictions for the abundance of giant arcs have modeled the population of background source galaxies in a variety of ways. It is known that the redshift distribution of background sources can have a dramatic impact on the giant arc statistics produced for a given cosmology (Hamana & Futamase 1997; Oguri et al. 2003; Wambsganss et al. 2004). Even more problematic is the fact that the uncertainty in the redshift distribution of background sources available to be lensed is effectively degenerate with the integrated lensing cross-section of the foreground halo population, and therefore with key cosmological parameters that strongly impact the properties of halos. This degeneracy arises because the Einstein radius, θ_E, of a lens at a fixed redshift increases as the distance to the source plane increases through its dependence on the angular diameter distance to the source.

The simplest background source model is one that places all background sources at a common redshift (e.g., Bartelmann et al. 1998; Meneghetti et al. 2010), for example z_s = 1 or z_s = 2, with some number density on the sky. Efforts by some groups (Wambsganss et al. 2004; Dalal et al. 2004; Hennawi et al. 2007; Puchwein & Hilbert 2009; Oguri & Blandford 2009) used a more advanced approach with background galaxies placed at several discrete source planes, incorporating empirical measurement of the density of galaxies in different redshift bins in order to capture the evolution of galaxy counts. The galaxy counts are typically taken from deep pencil-beam surveys, and Hennawi et al. (2007) point out that cosmic variance in such survey fields can be as large as 50%, with measurements of the galaxy density on the sky in different surveys differing by more than a factor of two in the same redshift bin. This discretized source plane model is an improvement relative to the single source plane approach, but still falls short of being physically realistic. Horesh et al. (2005) simulated giant arc statistics using the same clusters in Bartelmann et al. (1998) and use a background source distribution that is constructed entirely from photometric redshifts in the Hubble Deep Field (HDF). This model is encouraging in its use of a smooth redshift distribution, but the HDF is only 5 arcmin² and suffers from dramatic cosmic variance uncertainties (Hennawi et al. 2007).

We construct a new model that attempts to predict the redshift distribution of giant arcs as a function of the limiting intrinsic brightness of background sources that are distorted into detectable giant arcs. This is equivalent to setting a limiting magnitude in giant arc apparent brightness of and assuming a uniform magnification boost for all sources. In reality magnification factors vary for different sources depending on the details of each lens–source interaction. Assuming typical total magnifications of ∼10–25, and noting an approximate integrated limiting magnitude of g ≲ 22.5 for our primary giant arc sample, we produce model giant arc redshift distributions for two cases: (1) an intrinsic limiting source magnitude, g_lim < 25 and (2) g_lim < 26. We also produce a model redshift distribution for an intrinsic limiting source magnitude of g_lim < 24, which has a peak in the probability distribution closer to z_s = 1.

Our model is calculated as follows: for the arc cross-section, σ_arc, we adopt the scaling relation derived by Fedeli et al. (2010), which is based on large volume N-body simulations. The Fedeli et al. (2010) cross-section is defined from simulated arcs with length-to-width ratio, l/w ≳ 7.5, all of which are produced from sources at a redshift, z_s = 2. We derive the arc cross-sections for different source redshifts assuming a universal matter density profile slope, α, in the region around the Einstein radii for cluster lenses. The Einstein radius is always located in the very center of the lensing cluster potential where we expect a profile slope of ∼−1.5 (Moore et al. 1999). In order to account for some uncertainty in the exact slope of the density profile in the core we produce models evaluated for a small range of values for the slope, α = (−1.7, − 1.5, − 1.3).

The redshift distribution of source galaxies for a given magnitude limit, dn/dz_s, is estimated using the photometric redshift catalog in the 2 deg² COSMOS field (Ilbert et al. 2009). We then compute the expected redshift distribution of giant arcs by the equation

$$$ \frac{dp_{\rm arc}}{dz_{s}} = \frac{\sigma _{\rm arc} \frac{dn}{dz_{s}}}{\int\! dz_{s} \sigma _{\rm arc} \frac{dn}{dz_{s}}}. $$$

The length-to-width limit used to define the arc cross-section scaling relation in Fedeli et al. (2010) is consistent with the length-to-width lower limits on our sample of giant arcs as measured from ground-based imaging data (Bayliss et al. 2010).

The redshift distribution for our primary giant arc sample is shown in Figure 2, along with three curves which demonstrate the kinds of giant arc redshift distributions predicted by the model described above. Figure 2 also includes a plot of our primary giant arc redshift sample combined with our spectroscopically incomplete sample of secondary source redshifts. We use the Kolmogorov–Smirnov (K-S) test to quantify the agreement between our measured redshift distributions and our proposed models. Conducting K-S tests on our data requires that we correctly account for the giant arcs in our sample with constraints that place them in the redshift desert. To do this we use the information that we have about the distribution of giant arcs within the redshift desert: the precise redshifts that we measure for 13–14 giant arcs that are located within the redshift desert (the number depends on the exact redshift desert constraints).

Zoom In Zoom Out Reset image size

We produce many Monte Carlo realizations of our redshift distribution by assigning a precise redshift for each redshift desert arc that is drawn randomly from our sample of giant arcs that do have precise redshifts within the corresponding upper and lower redshift bounds. This method assumes that our giant arcs with precise redshifts between 1.45 ≲ z ≲ 3.3 are distributed within that range in the same way as our giant arcs that have only redshift desert constraints. We have no strong evidence to contradict this assumption and we consider it to be the most robust method for including the giant arcs with upper and lower redshift bounds in the K-S tests. Realizations that draw precise redshifts at random from within the upper and lower bounds for a given redshift desert arc produce distributions that are indistinguishable from the realizations that draw redshifts from our sample within those bounds, indicating that giant arcs with precise redshifts that fall within the redshift desert, 1.5 ≲ z ≲ 3.3, are evenly distributed within that range.

We also explore alternative ways of handling those arcs with redshift desert constraints in order to explore how the two extreme possible deviations from the Monte Carlo realization method described above affect the results of the K-S tests. The first extreme corresponds to the case where all redshift desert arcs have true redshifts at or near their lower redshift bound, and the opposite extreme where all redshift desert arcs have true redshifts that are at or near their upper redshift bound. This gives us three different giant arc redshift samples that span the range of possible true redshift distributions for our complete primary giant arc sample.

The realizations for our primary giant arc sample are tested against our models to produce average K-S probabilities of the likelihood that our redshift data are drawn from the model distributions. We perform another series of K-S tests on the combined primary giant arc plus secondary source samples, where the two secondary arcs with redshift desert constraints are handled in exactly the same three ways as the redshift desert primary giant arcs. Results from all K-S tests are shown in Table 2, and from these results we infer that the redshift distribution of our giant arc sample is reasonably well described by the our simple arc redshift distribution model, with values for the slope of the density profile, α that agree with the cold dark matter (CDM) paradigm and with an intrinsic limiting magnitude in the g band that corresponds to a reasonable average magnification for each arc, ∼10–25×.

The best agreement between our giant arc redshift sample and the model redshift distributions occurs when the arcs with redshift desert constraints are assumed to have true redshifts at or near their lower redshift bound; in this case our data have a ∼74% chance of being drawn from the same probability distribution as the model corresponding to a limiting intrinsic source magnitude of g_lim < 25 and a density profile with slope α = −1.5. It is reasonable to expect our arcs in the redshift desert to be more likely to have true redshifts closer to the lower end of the allowable range because of the spectral features that fall into our observed wavelength range. For example, at 1.6 ≲ z ≲ 2.3 we must rely on lines such as Mg ii λ2796, 2803, and Fe ii λ2344, 2372, 2384, 2586, 2600, which tend to be relatively weak compared to bluer features, such as C iv λ1448, 1551, Si ii λ1260, 1527, and Si iv λ1394, 1403. The combined primary + secondary strongly lensed source sample is never in good agreement with our model redshift distributions, which underscores the importance of using well-selected and spectroscopically complete redshift samples to study giant arc redshift distributions.

It is possible that our giant arc redshift distribution could be biased high relative to the true redshift distribution of SGAS giant arcs due to a systematic selection effect. Targets were selected for Gemini spectroscopy with a preference toward systems with larger apparent arc radii, R_arc, which is measured as the average angular separation between an arc and the center of the cluster lens—typically coincident with the brightest cluster galaxy. A selection based on large R_arc could bias our giant arcs toward having higher redshifts than if we had selected targets completely at random. However, our spectroscopic target selection was not based purely on R_arc, and, as discussed in Bayliss et al. (2010), we expect that it does not have a strong impact on our results.