CORRECTING THE z ∼ 8 GALAXY LUMINOSITY FUNCTION FOR GRAVITATIONAL LENSING MAGNIFICATION BIAS

The ongoing BoRG survey is a pure-parallel imaging program with the HST WFC3. The current survey covers ∼350 arcmin² divided into 71 independent fields located randomly on the sky. This reduces cosmic variance below the level of statistical noise (Trenti & Stiavelli 2008; Bradley et al. 2012). The photometry is in the visual and near-infrared, primarily using the four HST WFC3 filters F606W, F098M, F125W, and F160W (commonly referred to as V-, Y-, J-, and H-bands respectively). The z ∼ 8 BoRG survey consisted mainly of HST programs GO/PAR 11700 and GO/PAR 12572 (PI: Trenti) and includes a small additional number of coordinated parallels from the Cosmic Origins Spectrograph (COS)-GTO. 53 core BoRG fields are complemented by other archival data including 8 fields from GO/PAR 11702 (PI: Yan, Yan et al. 2011) and 10 COS-GTO fields, which used the F600LP-band instead of the F606W-band. The BoRG survey is the largest current survey of Y-band dropouts by solid angle.

The z ∼ 8 galaxy candidates were identified from Y-band dropouts; full details of the selection criteria used to find dropouts are described in Schmidt et al. (2014a). The BoRG survey detected 38 Lyman break galaxy (LBG) candidates at z ∼ 8 with signal-to-noise ratio (S/N) > 5 in the J-band, of which 10 have S/N > 8 (Bradley et al. 2012; Schmidt et al. 2014a). We use the 5σ sample of objects in this work. Throughout this work we will assume 42% of the selected BoRG dropouts are contaminants (usually z ∼ 2 interlopers, see e.g., Hayes et al. 2012; Bouwens et al. 2013). This is the fiducial contamination fraction for the BoRG sample and was shown to be robust in the estimation of the LF by Bradley et al. (2012) and Schmidt et al. (2014a). By definition we cannot determine which specific sources are contaminants without further photometry and spectroscopy, but our rigorous Bayesian method to determine the LF allows us to accurately estimate the LF parameters accounting for the presence of random contaminants (Schmidt et al. 2014a).

In Section 4.2 we estimate the velocity dispersions of strong lens candidates in the BoRG fields by comparing their photometry with similar early-type galaxies which have both HST photometry and spectroscopically determined velocity dispersions (Treu et al. 2005; Belli et al. 2014a, 2014b). We divided the galaxy samples into three large redshift bins in order to account for the position of the 4000 Å break in the filters at higher redshifts.

In the range z < 1 we used a sample of 165 spheroidal galaxies from Treu et al. (2005) with photometry from the Great Observatories Origins Deep Survey North (GOODS-N, Bundy et al. 2005). For z > 1 we use a sample of 66 massive quiescent galaxies, presented by Belli et al. (2014a, 2014b), which were selected from HST photometric catalogs of objects in the COSMOS, GOODS and Extended Groth Strip fields (Grogin et al. 2011; Koekemoer et al. 2011; Windhorst et al. 2011). We used an aperture correction to rescale observed velocity dispersions, σ_obs, to σ_e, the velocity dispersion within one effective radius, R_e.

We follow Belli et al. (2014a) and used the model of van de Sande et al. (2013) which proposes a constant rescaling:

$$\begin{eqnarray}&&{{\sigma }_{e}}=1.05{{\sigma }_{{\rm obs}}}\end{eqnarray} \tag{ 1 }$$

For galaxies at z < 1 (the Treu et al. 2005 sample), we used the model of Cappellari et al. (2006):

$$\begin{eqnarray}&&{{\sigma }_{e}}={{\left( \frac{{{R}_{e}}}{R} \right)}^{-0.066}}{{\sigma }_{{\rm obs}}}\end{eqnarray} \tag{ 2 }$$

where the slit size, R, is the 1'' aperture on Keck DEIMOS (Treu et al. 2005).

The reference photometry used for the individual samples differ. As listed in Table 1 we use HST F606W for galaxies at z < 0.5, HST F850LP from Treu et al. (2005) (converted to F098M through linear interpolation) for galaxies at 0.5 < z < 1.0, and HST F160W for galaxies at z > 1.

In Section 5 we describe our method to generate weak lensing probability density functions (PDFs) by reconstructing simulation data along the LOS to z ∼ 8. Due to the very high redshift of our sources, it was necessary to use simulation data containing halos out to redshifts above 5.

We used 24 1.4 × 1.4 square degree simulated lightcones built by Henriques et al. (2012) from the Millennium Simulation (Springel et al. 2005) which contain halos out to z ∼ 12. While the Millennium Simulation contains halos from very high redshift, it has a box length of only 500 Mpc h⁻¹. The comoving distance in the universe to z = 1 is 2390 Mpc h⁻¹, so it is necessary to build lightcones with the galaxies correctly distributed in comoving volumes (see Blaizot et al. 2005 and Kitzbichler & White 2007 for a thorough discussion of generating mock lightcones).

These lightcones were generated using the semi-analytical galaxy formation model of Guo et al. (2011), and photometric properties were calculated using the stellar population synthesis code by Maraston (2005) which can be applied at high redshift.

When a simply parametrized form is needed, we describe the LF by a Schechter function (Schechter 1976):

$$\begin{eqnarray}&&{\Psi }(L)=\frac{{{{\Psi }}^{\star }}}{{{L}^{\star }}}{{\left( \frac{L}{{{L}^{\star }}} \right)}^{\alpha }}{\rm exp} \left( -\frac{L}{{{L}^{\star }}} \right)\end{eqnarray} \tag{ 3 }$$

where ${{L}^{\star }}$ marks the characteristic break in the LF, ${{{\Psi }}^{\star }}$ is the characteristic density at that luminosity and α is the power-law exponent slope of the faint end.

If the LOS to a background source is closely aligned with a massive foreground object, e.g., a cluster or single massive galaxy, gravitational lensing can produce multiple observed images of the source (Schneider et al. 1992, 2006). Multiple imaging signifies the regime of strong gravitational lensing.

3.2.1. Singular Isothermal Sphere

Strong gravitational lenses are commonly modeled as Singular Isothermal Spheres (SIS), which provides a convenient analytic form to describe the mass profiles of massive galaxies (e.g., Treu 2010, and references therein). The scale of image separation is characterized by the Einstein radius of the lens:

$$\begin{eqnarray}&&{{\theta }_{{\rm ER}}}(\sigma ,z)=4\pi \frac{{{D}_{{\rm ls}}}}{{{D}_{s}}}{{\left( \frac{\sigma }{c} \right)}^{2}}\end{eqnarray} \tag{ 4 }$$

where D_ls and D_s are the angular diameter distances between the lens and source, and from the observer to the source respectively, σ is the velocity dispersion of the lens galaxy, and c is the speed of light. Velocity dispersion is the most important property for determining the strength of a strong gravitational lens as it scales with the mass of the dark matter in the system (Turner et al. 1984; Schneider et al. 2006; Treu 2010).

The magnification, μ, due to an SIS lens is given by:

$$\begin{eqnarray}&&\mu =\frac{|\theta |}{|\theta |-{{\theta }_{{\rm ER}}}}\end{eqnarray} \tag{ 5 }$$

where θ is the distance between the lens and the source in the image plane. An SIS lens can produce two images, with the brighter one having magnification μ > 2, or one image with magnification μ < 2. The case of multiple imaging is referred to here as strong lensing. In this paper we refer to images with 1.4 < μ < 2 as intermediate lensing.

3.2.2. Multiple Image Optical Depth

The optical depth τ_m is the cross-section for a galaxy at redshift z_S to be multiply imaged (i.e., strongly lensed) by a foreground galaxy at z_L: it is the fraction of the sky covered by the Einstein radii of all intervening deflectors at redshifts z_L. Following standard practice and assuming SIS deflectors, Wyithe et al. (2011) defines it as:

$$\begin{eqnarray}&&{{\tau }_{m}}=\int _{0}^{{{z}_{S}}}d{{z}_{L}}\int d\sigma {\Phi }(\sigma ,{{z}_{L}}){{(1+{{z}_{L}})}^{3}}c\frac{dt}{d{{z}_{L}}}\pi D_{L}^{2}\theta _{{\rm ER}}^{2}(\sigma ,{{z}_{L}})\quad \end{eqnarray} \tag{ 6 }$$

where Φ(σ, z_L) is the velocity dispersion function of the deflectors, D_L is the angular diameter distance to z_L, and t is time. Without the magnification bias, the optical depth gives the probability of a high-redshift source being multiply imaged.

Weak gravitational lensing is the deflection of light that causes the magnification and distortion of an observed source, but without producing multiple images. There are no empty LOS in the universe, so all light traveling to us has been deflected some amount by intervening mass (Hilbert et al. 2007). While it is impossible to determine the exact effect on individual observed sources, it can be done in a statistical sense and is important to quantify this effect for our high-redshift sources.

The lens equation can be constructed for an arbitrary number of lens planes due to an ensemble of deflectors along the LOS (Hilbert et al. 2009; McCully et al. 2014). The magnification of a source in a multiplane system is a function of the total convergence and total shear experienced. Hilbert et al. (2009) showed to first order that the total convergence and shear are the sum of the individual contributions from each object along the LOS:

$$\begin{eqnarray}&&\mu =\frac{1}{{{(1-{{\sum }_{i}}{{\kappa }_{i}})}^{2}}-|{{\sum }_{i}}{{\gamma }_{i}}{{|}^{2}}}\end{eqnarray} \tag{ 7 }$$

The convergence, κ_i, and shear, γ_i, of each object are determined by the lens model.

The gravitational lensing of a source with luminosity L in a solid angle Ω of sky has two effects. The observed luminosity is magnified by a factor μ and sources are now distributed over a magnified solid angle μΩ. In a flux-limited sample intrinsically low luminosity sources can be magnified above the survey limit, while the number density of sources can decrease for a given observed solid angle.

Since the faint end of the LF of high-redshift LBG galaxies is so steep, in regions around large low-redshift deflectors we may observe an excess of intrinsically faint high-redshift sources. These effects are known as the magnification bias and will affect our inferences about the population and LF of high-redshift galaxies.

If it were possible to observe all galaxies in the universe without the magnification bias the probability of a high-redshift galaxy being strongly lensed would be purely given by the optical depth, τ_m (Section 3.2.2). However, magnification of more numerous intrinsically faint sources into our surveys implies that we do not observe the true population of galaxies with luminosity. The magnification bias increases the probability that a sample of observed high-redshift sources have been gravitationally lensed.

The magnification bias for sources with observed luminosities above L_lim in a flux-limited sample is given by:

$$\begin{eqnarray}&&B=\frac{\int _{{{\mu }_{{\rm min} }}}^{{{\mu }_{{\rm max} }}}d\mu \;p(\mu )N\left( \gt \frac{{{L}_{{\rm lim} }}}{\mu } \right)}{N(\gt {{L}_{{\rm lim} }})}\end{eqnarray} \tag{ 8 }$$

assuming that each source could be magnified between μ_min and μ_max. Here p(μ) is the probability distribution for magnification of a source and N(>L_lim) is the integrated galaxy LF (Wyithe et al. 2011).

The true probability of a high-redshift source being multiply imaged is Bτ_m. Therefore, using B it is possible to find the fraction of galaxies at a given redshift in a flux-limited sample that are multiply imaged:

$$\begin{eqnarray}&&{{F}_{{\rm mult}}}=\frac{B{{\tau }_{m}}}{B{{\tau }_{m}}+B^{\prime} (1-{{\tau }_{m}})}\end{eqnarray} \tag{ 9 }$$

We assume that B', the bias for galaxies to not be multiply imaged, is close to unity.

If the survey limit is brighter than the characteristic apparent magnitude of the observed sample the magnification bias is expected to be large, as a large fraction of the observed sources are likely to be intrinsically fainter sources magnified above the detection threshold of the survey.

We can compute the gravitationally lensed LF, including strong and weak gravitational lensing:

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{{\Psi }}_{{\rm mod} }}(L) & = & (1-{{\tau }_{m}})\frac{1}{{{\mu }_{{\rm demag}}}}{\Psi }\left( \frac{L}{{{\mu }_{{\rm demag}}}} \right) \\ {} & {} & +\;{{\tau }_{m}}\int _{0}^{\infty }d\mu \frac{1}{\mu }p(\mu ){\Psi }\left( \frac{L}{\mu } \right) \\ \end{array}\end{eqnarray} \tag{ 10 }$$

where μ_demag < 1 is introduced such that the mean magnification over the entire sky is unity (Pei 1995; Wyithe et al. 2011) and p(μ) is the full probability density for magnification of a high-redshift source, as above. For a Schechter LF, the gravitationally lensed LF is predicted to exhibit a 'kick' in the bright end (e.g., Wyithe et al. 2011) due to a pile-up of brightened galaxies, whereas at the faint end the magnification of flux is balanced by the loss of number density (for faint-end slope α ∼ −2, Blandford & Narayan 1992) so there is no distortion, even if many strongly lensed faint sources are observed.

In order to compute the strong lensing optical depth and multiple image probability, we follow Wyithe et al. (2011) and use a simple SIS lensing model (see Section 3.2.1) with a flat cosmology. Strong lenses are assumed to be uniformly distributed in the universe and we can calculate the probability of encountering a strong lens along the LOS to a high-redshift source, i.e., the lensing optical depth (see Section 3.2.2). By considering the number of galaxies observed above a certain flux limit we can calculate the magnification bias factor, B, from Equation (8), assuming a Schechter LF (Equation (3)). For these calculations we use the z ∼ 8 LF inferred by Schmidt et al. (2014a), with a characteristic magnitude of ${{M}^{\star }}=-20.15_{-0.38}^{+0.29}$, faint-end slope of $\alpha =-1.87_{-0.26}^{+0.26}$, and number density of ${{{\rm log} }_{10}}{{{\Psi }}^{\star }}({\rm Mp}{{{\rm c}}^{-3}})=-3.24_{-0.24}^{+0.25}$. We marginalize over the entire Markov chain Monte Carlo (MCMC) chain for each of the Schechter parameters.

In their calculation of the optical depth Wyithe et al. (2011) used the local velocity dispersion function as measured by SDSS (Choi et al. 2007). As most strong lenses occur at $z\lesssim 1.5$ (Fassnacht et al. 2004; Treu 2010), Wyithe et al. (2011) assumed that the velocity dispersion function does not evolve with redshift for massive galaxies. This is consistent with studies of the velocity dispersion function out to z ∼ 1 (e.g., Chae 2010; Bezanson et al. 2012). However, significant galaxy growth and evolution is observed from z > 1 as structure forms (van de Sande et al. 2013; Belli et al. 2014a), and we can improve the accuracy of the model by allowing the parameters of the velocity dispersion function for massive ellipticals to evolve with redshift. Introducing redshift evolution is expected to reduce the optical depth (Barkana & Loeb 2000).

The dashed blue line in the left panel of Figure 3 shows the probability that the source has been multiply imaged as a function of lens redshift for a source at z ∼ 8, calculated using Equation (6). The distribution is strongly peaked at z_L ∼ 1, but there is a significant probability that z_L > 1.5. Only 48% of the contribution to the optical depth for strong lensing occurs at z_L < 1.5. We find that 90% of lensing occurs within a lens redshift of ${{z}_{L}}\lesssim 3.5$. Therefore, in order to account for most of the optical depth we need to find the form of the velocity dispersion function out to z ∼ 3–4 where the galaxy population is significantly different from recent times (Bundy et al. 2005; Muzzin et al. 2013; van de Sande et al. 2014).

Several studies have investigated the evolution of the velocity dispersion function out to z ∼ 1.5 (e.g., Chae 2010; Bezanson et al. 2011, 2012, 2013). These works are consistent with no evolution, but have large uncertainties. Measurements of velocity dispersion beyond z > 2 are very difficult as the brightest emission lines fall within near-IR atmospheric absorption regions (Kriek et al. 2006; Belli et al. 2014b).

Therefore, we estimate the evolution of the velocity dispersion function at high redshift based on the evolution of the stellar mass function, a related quantity that has been well measured at z > 2. We convert the stellar mass function into the velocity dispersion function by means of the well-known correlation between stellar velocity dispersion (σ) and stellar mass (M_stell) taken from Auger et al. (2010): ${\rm log} (\sigma [{\rm km}\;{{{\rm s}}^{-1}}])=p\bar{M}-11p+q$, where $p=0.24\pm 0.02$, q = 2.34 ± 0.01 and $\bar{M}={\rm log} ({{M}_{{\rm stell}}}/{{M}_{\odot }})$. This relation was derived for massive lens galaxies with high velocity dispersions, which will be the strongest contribution to the optical depth as τ ∼ σ⁴.

High-redshift galaxies are observed to have higher velocity dispersions at fixed mass than in the local universe (e.g., van de Sande et al. 2013; Belli et al. 2014a; Bezanson et al. 2015). Thus the stellar mass–velocity dispersion relation is expected to evolve with redshift. Following van de Sande et al. (2013) we expect evolution of the form $(\sigma /{{\sigma }_{0}})\propto {{(1+z)}^{\beta }}$, where σ₀ is the expected velocity dispersion at z ∼ 0. In Figure 1 we plot publicly available data from van der Wel et al. (2008), van Dokkum et al. (2009), Newman et al. (2010), Toft et al. (2012), Bezanson et al. (2013), van de Sande et al. (2013), Belli et al. (2014a, 2014b) and fit a relation of this form for all galaxies with estimated stellar masses between $10.8\lt {\rm log} ({{M}_{{\rm stell}}}/{{M}_{\odot }})\lt 12.0$, and measured velocity dispersion σ > 200 km s⁻¹ as this was the region where the Auger et al. (2010) relation was derived. We find β = 0.20 ± 0.07. Our result is lower than the result from van de Sande et al. (2013) because we use the Auger et al. (2010) stellar mass–velocity dispersion relation for massive lens galaxies as σ₀, whereas van de Sande et al. (2013) compare to a dynamical mass–velocity dispersion relation. As demonstrated in van de Sande et al. (2013) M_stell/M_dyn increases with redshift, so will reduce the evolution we find compared to that in van de Sande et al. (2013). If we consider the same galaxy sample and fit both our relation derived from stellar masses and the van de Sande et al. (2013) dynamical mass relation, and include the evolution in M_stell/M_dyn, our results are consistent. We note that because the optical depth depends on velocity dispersion to the fourth power, the form of the velocity dispersion function at z > 2 is the greatest source of uncertainty in the calculation of optical depths.

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The stellar mass function can be described by a Schechter function (e.g., Muzzin et al. 2013):

$$\begin{eqnarray}&&{{{\Phi }}_{S}}(\bar{M})=({\rm ln} 10)\;{\Phi }_{S}^{*}\;{{10}^{\left( \bar{M}-\bar{M}_{S}^{*} \right)(1+{{\alpha }_{S}})}}\;{\rm exp} \left[ -{{10}^{\bar{M}-\bar{M}_{S}^{*}}} \right]\end{eqnarray} \tag{ 11 }$$

The characteristic stellar mass is given by $\bar{M}_{S}^{*}={\rm log} (M_{{\rm stell}}^{*}/{{M}_{\odot }})$, ${\Phi }_{S}^{*}$ is the characteristic density normalization, and α_S is the low-mass-end slope.

In order to model the redshift evolution of the stellar mass function, we use publicly available data on quiescent galaxies at z ≤ 4 from the COSMOS/UltraVISTA Survey (Muzzin et al. 2013). They derive the best-fit single Schechter function parameters for the stellar mass function as a function of redshift. Their stellar mass function parameters for quiescent galaxies, allowing for evolution of α_S, are plotted as a function of redshift in Figure 2. We assumed the redshift evolution $X={{X}_{0}}{{(1+z)}^{a}}$, where X represents the stellar mass function Schechter parameters and X₀ represents the values at z = 0.

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We used a Bayesian MCMC linear fitting method to fit this functional form to the data, and plot the mean and one standard deviation confidence fits in Figure 2. There is significant evolution in ${\Phi }_{M}^{*}$. However, there is also large uncertainty in the evolution of ${\Phi }_{M}^{*}$ due to the spread of the data. We ignore evolution in the low-mass-end slope, since the lensing effect is dominated by the most massive galaxies. We also ignore evolution in $\bar{M}_{S}^{*}$, for which the evolution appears non-negligible but it has little effect on Equation (11). The redshift-dependent velocity dispersion function obtained in this way becomes

$$\begin{eqnarray}&&{\Phi }(\sigma ,z)={{p}^{-1}}\frac{{\Phi }_{S}^{*}(z)}{\sigma {{(1+z)}^{\beta }}}{{\left( \frac{\sigma }{{{\sigma }^{*}}} \right)}^{{{p}^{-1}}(1+{{\alpha }_{S}})}}{\rm exp} \left[ -{{\left( \frac{\sigma }{{{\sigma }^{*}}} \right)}^{{{p}^{-1}}}} \right]\end{eqnarray} \tag{ 12 }$$

with p = 0.24 ± 0.02, β = 0.20 ± 0.07 (obtained from the evolution of velocity dispersion in Figure 1), ${\Phi }_{S}^{*}(z)=3.75\pm 2.99\times {{10}^{-3}}{{(1+z)}^{-2.46\pm 0.53}}$ Mpc⁻³, α_S = −0.54 ± 0.32 and σ* = 216 ± 18 km s⁻¹. This was derived using the stellar mass–velocity dispersion relation above (Auger et al. 2010), including the scatter in the relation. At z = 0 recent well-measured velocity dispersion functions (e.g., Sheth et al. 2003; Choi et al. 2007) are within the uncertainties of this redshift-evolving relation, showing that our inferred evolution is consistent with direct measurements where they overlap.

Using this redshift-dependent velocity dispersion function we compute the optical depth for strong lensing, and the distribution of the optical depth with lens redshift. In the left panel of Figure 3, now using the redshift evolving deflector population from Equations (6) and (12), we see that the majority of the contribution to the optical depth is from lens galaxies at $z\lesssim 1.5$, which agrees with current observations of lensed high-redshift dropouts (Barone-Nugent et al. 2013; Schmidt et al. 2014b; Atek et al. 2015). In the right panel of Figure 3 we plot the optical depth as a function of source redshift and find that including the redshift evolution of the deflector population reduces the optical depth at high redshift compared with the work in Wyithe et al. (2011) as expected by theoretical predictions (Barkana & Loeb 2000), and it appears to start to flatten by z_S ∼ 10.

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Our estimated optical depth at z < 8 is in good agreement with values derived by an independent method by Barone-Nugent et al. (2015), and consistent with Wyithe et al. (2011) for $z\lesssim 8$. We note the optical depths presented in Barone-Nugent et al. (2015) are marginally higher than the results of this paper, but we can recover their optical depth using a steeper evolution of σ(z). It is clear that the uncertainty in the evolution of velocity dispersion, which is the best indicator of the mass of lens galaxies, provides the largest uncertainty in determining the optical depth.

Finally, we compute the probability that high-redshift galaxies in a flux-limited sample have been multiply imaged. This is shown in Figure 4 as a function of limiting magnitude for each of the BoRG fields. As expected, the probability that a source in each field is multiply imaged, F_mult (Equation (9)), increases with the survey limiting magnitude, owing to the magnification bias. We estimate 3–15% of observed sources brighter than ${{M}^{\star }}$ have been strongly lensed; this is consistent with the results of Barone-Nugent et al. (2015) who use an independent method to infer the lensed fraction.

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While all the fields are subject to weak lensing, it is necessary to establish which of the individual sources experience multiple imaging (μ > 2), or are close enough to a deflector to experience an intermediate magnification (1.4 < μ < 2). We expect strong lensing evens to be rare, but possible given the size of the BoRG survey. Among the BoRG sources, Barone-Nugent et al. (2013) presented a candidate strongly lensed system in borg_0440–5244 (for naming conventions see Bradley et al. 2012). The candidate appears to be lensed by a foreground group with an Einstein radius of 149, corresponding to a velocity dispersion of ∼300 km s⁻¹, producing a magnification of 3.7 ± 0.2 of the dropout. In this section we describe a method to identify other potentially lensed sources in the catalogs and illustrate how to account for them systematically when estimating the LF.

For computational speed, we considered as potential deflectors only z < 3 objects within 18 arcsec of the z ∼ 8 dropouts in each field (the typical Einstein radius is of order 1–2 arcsec for massive galaxies). The key quantity that we need to estimate the lensing strength is the velocity dispersion (Turner et al. 1984; Treu 2010). Thus for every galaxy sufficiently close to a dropout, we estimate their velocity dispersions by comparing their photometry with that of samples of similar objects with spectroscopically determined velocity dispersions. We selected galaxy samples with HST photometry in bands used in BoRG in order to estimate velocity dispersion based on our own photometry. As a comparison sample, we used data from Treu et al. (2005) and Belli et al. (2014a, 2014b), as described in Section 2.2.

As described in Section 4.1, the velocity dispersion–stellar mass relation is believed to evolve weakly with redshift since z ∼ 2 (e.g., van de Sande et al. 2013; Belli et al. 2014b; Bezanson et al. 2015), and galaxies will be intrinsically brighter at higher redshift due to younger stellar populations (e.g., Treu et al. 2005). We account for this by fitting an evolving Faber & Jackson (1976) relation to the comparison sample of the form L ∝ σ⁴(1 + z)^β.

In practice, we bin the data in redshift, and fit a function of the form ${\rm log} \;\sigma =-0.1\;m+a{\rm log} (1+z)+b$ using a Bayesian MCMC estimation where σ is the velocity dispersion in km s⁻¹, m is apparent magnitude in a given band, z is galaxy redshift, and a and b are constants. We restrict our fit to galaxies with a measured velocity dispersion of at least 200 km s⁻¹, where samples are less affected by incompleteness and selection effects. We present the estimated parameters in Table 1 and fits to the data are shown in Figure 5.

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The posterior probability distribution functions of Einstein radii for each object are found using Equation (4), sampling over the full MCMC chain for the velocity dispersion. The redshifts of the objects were determined using the Bayesian Photometric Redshifts (BPZ) code, using a flat prior and the default parameters and templates (Benitez et al. 2004; Coe et al. 2006). All photometric redshifts for relevant foreground galaxies are well fit by BPZ and have uncertainties in photometric redshift <15%. The PDF for magnification, p(μ), is found by computing the magnification, μ (Equation (5)), at the position of the dropout given the distribution of Einstein radii found for each foreground object using the distribution for its velocity dispersion, σ_inf, estimated from the fits in Table 1. The greatest source of error in this procedure is the magnitude–velocity dispersion–redshift relation: uncertainties in magnitude and redshift determination have small effects on the magnification PDFs in comparison to the uncertainty in velocity dispersion.

When the mean magnification produced by such a foreground object exceeds μ = 1.4 we use the magnification PDF derived from the above procedure and treat the dropout as described in Section 6.2.2 in our calculations of the LF.

Using this method, we find one of the dropouts (borg_0436–5259_1233, presented in Bradley et al. 2012) has a magnification probability distribution consistent with strong lensing. This dropout is shown in the top left panel of Figure 6 and its estimated lensing properties are given in Table 2. The dropout appears to be magnified by a large galaxy at z ∼ 0.40 with estimated velocity dispersion 294 ± 47 km s⁻¹ (estimated from photometry via the empirical relation presented in Table 1). We estimate its magnification to be μ = 2.05 ± 0.52; the large uncertainty is due to the uncertainty in the relationship between apparent magnitude and velocity dispersion. If the dropout is indeed strongly lensed the counter image would be almost directly behind the center of the lens galaxy, and will be demagnified according to Equation (5), unfortunately making it impossible to detect. The dropout is very faint (m_J = 27.0 ± 0.2) and no significant elongation is detected in any of the observed bands but this dropout would be an excellent object for further investigation.

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Three of the dropouts (borg_1301+0000_160, borg_1408+5503, and borg_2155–4411_341) experience mean magnification >1.4. Postage stamps of these dropouts are shown in Figure 6 and their lensing properties are presented in Table 2. As described in Section 2.1 the fiducial BoRG contamination fraction is 42% (Bradley et al. 2012; Schmidt et al. 2014a), meaning that some of the sources presented here may be lower redshift interlopers (e.g., Hayes et al. 2012; Bouwens et al. 2013). Without further photometry and/or spectroscopy we cannot identify which of the sources are interlopers, but we note that the photometric redshift PDFs for these four sources (obtained from BPZ) all have strong peaks at z ∼ 8, suggesting a higher probability than the average (58%) for these particular objects to be true z ∼ 8 sources. Interestingly, borg_1301+0000_160 is the brightest dropout in the survey, with m_J = 25.5 ± 0.2, and appears tangentially elongated in the J-band image (middle panel of Figure 6). This object is also a very interesting target for further imaging and spectroscopic follow-up.

We note that our method assigns a significantly lower velocity dispersion to the potential strong lens (borg_0440–5244_647) than the one estimated by Barone-Nugent et al. (2013) in their presentation of this object. They estimated the velocity dispersion of the deflector to be σ ∼ 300 km s⁻¹, whereas our method estimates a mean velocity dispersion of ∼170 ± 33 km s⁻¹. This is likely to be because our method does not account for lensing by groups and clusters, while Barone-Nugent et al. (2013) suggest that this dropout is lensed by a group of at least two objects at z ∼ 1.8, of which borg_0440–5244_647 is the largest. They estimated velocity dispersions of the deflector galaxies by using an abundance matching relation between mass and luminosity, derived from Cooray (2005), and measuring the angular size of the lensing objects. However, when using a redshift-dependent Faber & Jackson (1976) relation (Barone-Nugent et al. 2015) similar to ours (Table 1) they estimate the velocity dispersion of this single galaxy to be ∼180 ± 46 km s⁻¹ (via private communication), which agrees with our result. Neglecting group-scale lensing is a potential limitation of our method, which may underestimate magnification in a few cases. However the impact on the overall estimation of the LF inference is negligible since the phenomenon is so rare.

The weak lensing reconstruction model developed by Collett et al. (2013) takes simulation halo catalogs and places halos in a three-dimensional grid, with each halo contributing convergence κ_i and shear γ_i along a LOS to a source at a given redshift. Halos are modeled as truncated Navarro–Frenk–White (NFW) profiles (Baltz et al. 2009):

$$\begin{eqnarray}&&\rho (r)=\frac{{{\rho }_{{\rm NFW}}}(r)}{1+{{\left( \frac{r}{{{r}_{t}}} \right)}^{2}}}\end{eqnarray} \tag{ 13 }$$

where we used the truncation radius ${{r}_{t}}=5{{r}_{200}}$, shown to be robust by Collett et al. (2013). Here r₂₀₀ is the radius at which the mass density falls to 200 times the critical mass density of the universe. The convergence and shear derived from this profile are given in Baltz et al. (2009). Magnification due to all intervening deflectors along a LOS is given by Equation (7).

We built PDFs for all lensing parameters by sampling over 10³ LOS. As described in Section 2.3 we used lightcones built from the Millennium Simulation (Springel et al. 2005; Henriques et al. 2012).

The simulated catalogs provide a list of halos with associated galaxies, but they do not include other dark structure, clumped in filaments and absent in voids. This missing matter will affect the overall density of the universe so it is necessary to take this into account when estimating κ and μ. We account for this by subtracting convergence from redshift slices so that the mean convergence along all LOS in the catalogs to a given redshift equals zero, and the mean magnification is unity, as they should be.

Following work by Suyu et al. (2010) and Greene et al. (2013), we compare LOS in the BoRG fields with simulation data based on relative density of objects. We define the overdensity parameter

$$\begin{eqnarray}&&\xi =\frac{{{n}_{i}}}{{{n}_{{\rm tot}}}}\end{eqnarray} \tag{ 14 }$$

where n_i is the number of objects per unit area in each lightcone (or real field) and n_tot is the total number of objects divided by the total survey area. Given that the simulation catalogs are ∼500× larger than the total BoRG survey area, we expect them to give representative results.

We then calculate the number of objects per square arcsecond brighter than m = 24 in the J-band in each of the BoRG fields compared to the total number of objects above this flux limit in the whole survey. Similarly we calculate the overdensity of objects above the same limit in the simulated lightcones. Henriques et al. (2012) include mock photometry based on stellar population synthesis codes by Maraston (2005) which include J-band magnitudes. As shown in Figure 7, the distribution of overdensities for the observed data is within the range of that for simulated data. Finally, to generate magnification PDFs for a given BoRG field, we combine the magnifications from all simulation LOS which are within ±2% in overdensity of the observed value.

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In Figure 8 we plot the magnification PDFs for a source at various redshifts over all LOS. As the source redshift increases, the peak of the distribution shifts to lower magnification, but the high-magnification tail becomes more important, such that the mean magnification over all LOS remains unity. We match results for z < 6 from Hilbert et al. (2007) well. It is clear that there is little change in the distribution between z_S = 6 and z_S = 8, as there are negligible numbers of large halos above z > 5.

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In Figure 9 we plot the magnification PDFs for a variety of overdensities. The more overdense LOS produce a higher mean magnification, as expected, but also have a greater variance than the distributions for underdense LOS. This agrees well with the estimates at lower redshift by Greene et al. (2013).

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The kernel density estimates (Rosenblatt 1956; Parzen 1962) fit to the magnification PDFs for all the BoRG fields are shown in Figure 10. As expected, the BoRG fields do not have significant over- or underdensities, but are rather typical of blank fields at z ∼ 8, as shown in Figure 8.

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There is significant motivation for the magnification PDFs to take a log-normal form. The 3D matter density distribution of the universe is well described by a log-normal random field (Coles & Jones 1991), and weak lensing probability distributions arise directly from the mass distribution. However, when accounting for the magnification bias in individual fields to infer the LF from the dropout sample (see Section 6) it was necessary to express the magnification distributions in a form that could easily convolve analytically with a Gaussian distribution (for more details see the appendix). For this we used a Bayesian MCMC approach to fit the distributions of magnification for each field as a linear sum of Gaussian functions.

As in Schmidt et al. (2014a), we assume that the intrinsic LF is modeled by the Schechter function in Equation (3). In order to facilitate comparison with our previous work we use the sample of 38 BoRG Y-band dropouts and 59 additional fainter dropouts from the Hubble Ultra-Deep Field (HUDF) and ERS programs (Bouwens et al. 2011).

Bayesian statistics allows us to express the posterior probability that the LF is fit by a Schechter function with parameters $\theta =(\alpha ,{{L}^{\star }},{{{\Psi }}^{\star }})$ given the observed luminosity ${{L}_{{\rm J},{\rm obs}}}$ of the dropouts in the J-band, and the non-detections in the V-band (I_V = 0), as the product of the prior on the Schechter parameters and the likelihood:

$$\begin{eqnarray}&&p(\theta \;|\;{{L}_{{\rm J},{\rm obs}}},{{I}_{V}}=0)\propto p(\theta )\times p({{L}_{{\rm J},{\rm obs}}},{{I}_{V}}=0\;|\;\theta )\end{eqnarray} \tag{ 15 }$$

This posterior probability can be expressed (see the appendix for full details) as:

$$\begin{eqnarray}&&\begin{array}{ccccccccccccccc} p(\theta \;|\;{{L}_{{\rm J},{\rm obs}}},\;{{I}_{{\rm V}}}=0)\propto \;p(\theta )\times C_{(1-f)n}^{{{N}_{z}}}\times C_{fn}^{\frac{f}{1-f}{{N}_{z}}} \\ \quad \times \mathop{\prod }\limits_{l}^{\mathcal{C}}{{\left[ 1-{{\frac{{{A}_{l}}}{{{{\bar{\mu }}}_{l}}{{A}_{{\rm sky}}}}}^{{}}}\;p(I=1\;|\;\theta ) \right]}^{\frac{{{N}_{z}}-(1-{{f}_{l}}){{c}_{l}}}{1-{{f}_{l}}}}} \\ \quad \times \mathop{\prod }\limits_{i}^{n}p({{L}_{{\rm J},{\rm obs},{\rm i}}}\;|\;\theta ) \\ \end{array}\end{eqnarray} \tag{ 16 }$$

where we iterate over l fields with i z ∼ 8 candidates. Here N_z is the number of high-z dropouts in the surveyed comoving cosmological volume, A_l is the area of the individual $\mathcal{C}$ fields in Schmidt et al. (2014a), which each contain c_l high redshift candidates ($n=\sum _{l}^{\mathcal{C}}{{c}_{l}}$). Each candidate has an assumed contamination fraction of f_l. We use a fiducial value for the contamination of 42% for the BoRG sample; the contamination fractions for the HUDF/ERS samples are included in the selection function (see the appendix), as described in Oesch et al. (2012), Bradley et al. (2012), and Schmidt et al. (2014a). Changing the contamination value in the range f = 0–0.60 affects the characteristic magnitude and the number density of the LF by less than their estimated 1σ uncertainties, and the change in the faint-end slope is comparable to its 1σ uncertainty (Bradley et al. 2012; Schmidt et al. 2014a). The Bayesian framework allows us to accurately estimate the LF parameters accounting for contamination. A_sky is the area of the full sky. The C^a_b factors are binomial coefficients which are the fully correct method of modeling source counts.

We assume uniform priors on α, ${{{\rm log} }_{10}}{{L}^{\star }}$, and ${{{\rm log} }_{10}}{{N}_{z}}$. $p(I=1\;|\;\theta )$ is the probability distribution of an object making it into the dropout sample based on the photometric selection described in Schmidt et al. (2014a). $p({{L}_{{\rm J},{\rm obs},{\rm i}}}\;|\;\theta )$ is the likelihood function for the observed J-band luminosity of the ith object in the sample.

The last term includes marginalization over the magnification PDF:

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} p({{L}_{{\rm J},{\rm obs}}}\;|\;\theta ) & = & \int \int p(\mu )\;p({{L}_{{\rm J},{\rm obs}}}\;|\;\mu {{L}_{{\rm J},{\rm true}}}) \\ {} & {} & \times \;p({{L}_{{\rm J},{\rm true}}}\;|\;\theta )\;d{{L}_{{\rm J},{\rm true}}}\;d\mu \\ \end{array}\end{eqnarray} \tag{ 17 }$$

In the appendix we give the expanded expression of the posterior distribution from Equation (16) used when performing the LF parameter inference and describe the derivation and motivation for Equation (17). We refer to the appendix and Schmidt et al. (2014a) for further details.

6.2.1. Analytic Form for Magnification PDFs

6.2.2. Combining Lensing Regimes

The simple SIS strong lensing model described in Section 4.1 predicts the probability of z ∼ 8 sources in the BoRG survey being multiply imaged to be ∼ 3–15%, increasing as the field limiting magnitude becomes brighter than ${{M}^{\star }}$. The majority of the BoRG fields have a multiple-image probability for high-redshift sources of <10% (see Figure 4). We predict that 1–2 of the 38 BoRG Y-band dropouts may be strongly lensed.

One candidate strong lens system in BoRG was presented by Barone-Nugent et al. (2013); a rigorous search for strong lenses in all 71 BoRG fields as part of this work revealed one more candidate. Additionally, this search revealed three candidate intermediate lens systems, with μ > 1.4. These candidates are presented in Figure 6 and Table 2. While strong lensing creates larger magnification, the probability of encountering a strong lens along the LOS is low: as shown in Figure 3 the optical depth is roughly τ ≈ 0.31% for a source at z = 8. The optical depth for intermediate lensing is much higher: for an object to experience intermediate lensing it must be within 3.5θ_ER of the foreground deflector, resulting in τ ≈ 4% for a source at z = 8. Thus, intermediate lensing offers an additional boost to the flux of high-redshift galaxies, and must be correctly accounted for in estimations of the LF.

We estimate the z ∼ 8 LF from the sample of 97 LBG described in Section 2, including the 38 S/N_J > 5 objects from the BoRG survey, including the effects of magnification bias. We sample the posterior distribution function for the Schechter function parameters with an MCMC chain of 40,000 steps.

The results of the estimated LF are shown in Figure 11, and the correlations between the Schechter function parameters and their PDFs are shown in Figure 12. We plot the results of Schmidt et al. (2014a) for comparison in both figures. We see a small deviation from the uncorrected LF of ∼0.15 mag at the limit of the brightest BoRG source, and there is negligible difference between the LFs at M > −21. The Schechter function parameters for the new LF are within the uncertainties of the estimation by Schmidt et al. (2014a), though we find a slightly fainter value of ${{M}^{\star }}$ and higher value of ${{{\Psi }}^{\star }}$ than Schmidt et al. (2014a). This is expected because of the slight deviation at the bright end of the LF, and there is a strong correlation between these parameters, as shown in Figure 12. It is clear that magnification bias is not a significant effect at this redshift and the luminosity range of the BoRG sources. This also demonstrates that although we predict 3–15% of the BoRG sources are strongly lensed this does not affect the LF within the survey limits, as predicted in Section 3.4.

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Our results are in good agreement with those of Fialkov & Loeb (2015) who use an independent semi-analytic method to show that the effect of magnification bias is small below M > −21.5. Fialkov & Loeb (2015) predict that if the brightest observed galaxy has absolute magnitude ${{M}_{{\rm uv}}}=-24.5$—lying in the significantly distorted tail of the magnified LF (Wyithe et al. 2011)—there is a ∼ 13.3% discrepancy in the normalization of a Schechter LF at z ∼ 8 for sampled galaxies with μ_max = 2 (i.e., only weak and intermediate lensing effects) compared to the intrinsic LF. While this upper limit is several orders of magnitude brighter than currently observed, this demonstrates that it will be important to include the effects of magnification bias from weak and intermediate lensing in surveys that find extremely bright galaxies.

Table 3 summarizes the estimated Schechter function parameters for this LF in comparison with other recent LF estimates from the literature. We find that our fit parameters are in good agreement with the recent literature, demonstrating that magnification bias is not affecting current z ∼ 8 LF observations. Note that our results have significantly smaller error bars than those of Finkelstein et al. (2014), because their sample contains only three z ∼ 8 galaxies brighter than M = −21, making their fit less well-constrained at the bright end.

Table 3.  Comparison of z ∼ 8 Schechter LF Parameters

Reference	${{M}^{\star }}$	α	${{{\rm log} }_{10}}{{{\Psi }}^{\star }}$ (Mpc−3)
This work	$-19.85_{-0.35}^{+0.30}$	$-1.72_{-0.29}^{+0.30}$	$-3.00_{-0.31}^{+0.23}$
Finkelstein et al. (2014)	$-20.89_{-1.08}^{+0.74}$	$-2.36_{-0.40}^{+0.54}$	$-4.14_{-1.01}^{+0.65}$
Bouwens et al. (2014)	−20.63 ± 0.36	−2.02 ± 0.23	−3.68 ± 0.32
Schmidt et al. (2014a) $5\sigma$	$-20.15_{-0.38}^{+0.29}$	$-1.87_{-0.26}^{+0.26}$	$-3.24_{-0.34}^{+0.25}$
Schmidt et al. (2014a) $8\sigma$	$-20.40_{-0.55}^{+0.39}$	$-2.08_{-0.29}^{+0.30}$	$-3.51_{-0.52}^{+0.36}$
McLure et al. (2013)	$-20.12_{-0.48}^{+0.37}$	$-2.02_{-0.23}^{+0.22}$	$-3.35_{-0.47}^{+0.28}$
Schenker et al. (2013)	$-20.44_{-0.35}^{+0.47}$	$-1.94_{-0.24}^{+0.21}$	$-3.50_{-0.32}^{+0.35}$

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Our results show that magnification bias does not affect current estimates of the LF at $z\lesssim 8$ and therefore cannot explain the apparent flattening of the bright end of the LF recently observed by Bowler et al. (2014a, 2014b) and Finkelstein et al. (2014) at z ∼ 7–8. Bowler et al. (2014a, 2014b) accounted for strong lensing of bright sources, but they still find a deviation of ∼0.4 mag from a Schechter fit at M = −22. We predict a lensed fraction of ∼3–15% for bright galaxies (Figure 4) from the BoRG survey which is essentially free of cosmic variance (Trenti & Stiavelli 2008); so providing that cosmic variance and contamination by lower redshift interlopers (Bradley et al. 2012; Hayes et al. 2012; Bouwens et al. 2013; Schmidt et al. 2014a) were correctly accounted for in the work of Bowler et al. (2014a, 2014b) and Finkelstein et al. (2014), we expect the magnification bias to be negligible in the bright end of these LFs. This lends credence to the interpretation that these observations may be the result of the changing intrinsic properties of galaxies at $z\gtrsim 7$, possibly due to changing dust fractions (Cai et al. 2014) and/or feedback processes (Somerville et al. 2012).

There is clear evolution in the LF for z < 8 (e.g., Bouwens et al. 2007, 2014; van der Burg et al. 2010; Bowler et al. 2014a; Finkelstein et al. 2014), and this is expected to continue to higher redshifts. However, the processes which drive this evolution are not well understood: the evolution is thought to follow hierarchical structure formation and the evolution of the halo mass function (Vale & Ostriker 2004), but there are also important quenching processes that may reduce star formation in massive galaxies (Schneider et al. 2006; Somerville et al. 2012), and changes in the amount of dust present in galaxies will affect the attenuation of flux. Thus there are a multitude of theoretical models for the evolution of the LF.

The gravitationally lensed LF (Equation (10)) exhibits a significant "kick" in the bright-end tail for M ≲ −22 at z ∼ 8. This is just beyond the brightest BoRG objects, so it is unlikely that the BoRG survey observes the regime of magnification bias at the bright end. This is in agreement with theoretical studies by Wyithe et al. (2011) and Fialkov & Loeb (2015). However, in upcoming wide-area surveys magnification bias presents a useful tool to test LF evolution models because it allows us to probe the bright end, where there are large theoretical uncertainties and the evolution is expected to be fast (Bowler et al. 2014a).

In order to explore the range of possible scenarios, in Figure 13 we plot the predicted intrinsic (dashed lines) and observed (solid lines) LFs for a range of redshifts, comparing a variety of evolution models. We assume these models are the intrinsic LFs at a given redshift and used Equation (10) to estimate the observed LF. We plot the BoRG z ∼ 8 LF (Schmidt et al. 2014a) for comparison. Additionally, we mark the comoving volumes and magnitude ranges accessible to future high-redshift surveys.

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The top left panel shows the LF model from Bouwens et al. (2014) which is an extrapolation from observations at z < 10. The top right panel shows the LF model from Finkelstein et al. (2014) which is an extrapolation from observations at 4 < z < 8. The bottom left panel shows the model developed by Muñoz (2012) which follows the evolution of the halo mass function, and includes dust attenuation. The bottom right panel is a model from Behroozi & Silk (2015) constructed from a comparison of the specific star formation rate to the specific halo mass accretion rate, and including dust models from Charlot & Fall (2000). The four models have significantly different behaviors at the bright end. While the Bouwens et al. (2014) model has by construction a bright end that is very similar to that measured at lower redshifts, the Muñoz (2012) model has a very shallow bright end, and the Finkelstein et al. (2014) and Behroozi & Silk (2015) models are in-between. As a result, the effects of magnification bias (which are stronger for the steeper LF) are very different: negligible in the Muñoz (2012) case and appreciable in the three other cases. However, the bright end of the Muñoz (2012) model is the easier one to test observationally, within reach of a James Webb Space Telescope medium depth, medium width survey (e.g., JWST MD, Windhorst et al. 2006).

Except in the case of a very shallow bright end, we do not expect the magnification bias to be significant in our upcoming BoRG z ∼ 9, 10 survey (HST Cycle 22, PI Trenti). In all cases, it is clear that surveys covering >100 deg², e.g., Euclid and WFIRST, should find many bright z > 8 LBGs. We expect the observed high-redshift galaxy samples will be dominated by magnification bias in these surveys. We predict almost all z ∼ 8 sources in Euclid will have been strongly lensed. The framework developed in this work will be crucial for determining the intrinsic luminosity of high-redshift sources found in such surveys.

Our results confirm the suggestion by Wyithe et al. (2011) that magnification bias will be important to probe the bright end of the LF at high redshift. However, we find that the magnitude of the effect is less pronounced than in that study, owing mostly to our accounting for the redshift evolution of the deflector population.