AN HST/WFC3-IR MORPHOLOGICAL SURVEY OF GALAXIES AT z = 1.5–3.6. I. SURVEY DESCRIPTION AND MORPHOLOGICAL PROPERTIES OF STAR-FORMING GALAXIES^*

Data were obtained using the WFC3/IR camera on board HST (HST-WFC3) as part of the Cycle 17 program GO-11694 (PI: D. R. Law). This program was comprised of 42 orbits using the F160W filter (λ_eff = 15369 Å, which traces rest-frame 5123/3824 Å at z = 2/3, respectively), divided among 14 pointings in 10 different survey fields (see Table 1) for a combined sky coverage of ∼65 arcmin² centered on lines of sight to bright (V ∼ 17) background QSOs.⁶ Each pointing had a total integration time of 8100 s composed of nine 900 s exposures dithered using a custom nine-point sub-pixel offset pattern designed to uniformly sample the point-spread function (PSF).

The data were reduced using the MultiDrizzle (Koekemoer et al. 2002) software package to clean, sky subtract, distortion correct, and combine the individual frames. The raw WFC3 frames are undersampled with a pixel scale of 0.128 arcsec; these frames were drizzled to a pixel scale of 0.08 arcsec pixel⁻¹ using a pixel droplet fraction (pixfrac) of 0.7. This combination of parameters was found to give the cleanest, narrowest PSF while ensuring that the rms variation of the final weight map was less than ∼7% across the 136 × 123 arcsec field of view. Using nine isolated and unsaturated stars in the Q1623+26 field we estimate that the FWHM of the PSF is 0.18  ±  0.01 arcsec (i.e., Nyquist sampled by the 0.08 arcsec drizzled pixels), varying by less than 4% across the detector and from field to field.

Our 14 individual pointings are located within 10 survey fields centered on lines of sight to bright background QSOs (z_QSO ∼ 2.7). In the present contribution, we focus on actively star-forming galaxies drawn from rest-UV color-selected catalogs of z ∼ 1.5–3.5 star-forming galaxy candidates constructed according to the methods described by Steidel et al. (2003, 2004) and Adelberger et al. (2004). These catalogs are based on deep ground-based imaging and therefore select galaxies with ${\cal R} \lesssim 27$ independent of morphology or surface brightness (since even the largest galaxies are nearly unresolved in these seeing-limited images). Extensive ancillary information is available in these survey fields. In addition to deep ground-based $U_n G {\cal R}$ optical imaging and rest-UV spectroscopy, many of the fields also have deep ground-based J/K_s imaging, Spitzer Infrared Array Camera (IRAC)/Multiband Imaging Photometer photometry, and for Q1549+19/Q1700+64, respectively, spatially resolved HST/WFC3-ultraviolet/visible and HST/ACS rest-UV imaging. All galaxy candidates in these catalogs are detected with WFC3 at >10σ down to ∼27.5 AB.

Rather than relying on photometric redshifts, which typically have large uncertainties (Δz/(1 + z) ≳ 0.06 at z > 1.5; van Dokkum et al. 2009), we restrict our attention to the subsample of galaxies with ${\cal R} \le 25.5$ that have been spectroscopically confirmed using Keck/Low Resolution Imaging Spectrometer rest-UV spectra to lie in the redshift range 1.5 < z < 3.6; i.e., the "BM" (〈z〉 = 1.70  ±  0.34), "BX" (〈z〉 = 2.20  ±  0.32), and "LBG" or U_n-dropout (2.7 < z < 3.6) samples defined by Steidel et al. (2003, 2004). Systemic redshifts for the majority of our galaxies were derived from rest-UV absorption/emission-line centroids using the prescriptions of Steidel et al. (2010); for 51 galaxies that have been successfully observed to date with either long-slit (Erb et al. 2006b) and/or IFU spectroscopy (13 galaxies; Förster Schreiber et al. 2009; Law et al. 2009; Wright et al. 2009) systemic redshifts were derived from rest-optical nebular emission lines (e.g., Hα, [O iii]).⁷

Additionally, we omit from our sample any galaxies that lie within ∼1.5 arcsec of the edge of the WFC3-IR detector (where our dither coverage is incomplete), or which are known to contain AGNs on the basis of rest-UV spectroscopy (24 systems; 12 bright QSOs with H₁₆₀ < 19 AB and 12 faint AGNs with H₁₆₀ > 19 AB). We discuss the morphological properties of these AGNs in detail in a forthcoming contribution (D. R. Law et al. 2012, in preparation). The redshift and F160W magnitude distribution of the final sample of 306 galaxies are shown in Figure 1. As detailed in Table 1, the galaxies are roughly evenly distributed among the ten fields (with additional pointings in Q1623+26 and Q2343+12). Motivated by the redshift ranges of the photometric selection criteria we loosely divide our galaxies into the three redshift ranges z = 1.5–2.0, z = 2.0–2.5, and z = 2.5–3.6, containing 72/127/107 galaxies, respectively. Although we include galaxies up to z = 3.6 in our analysis we note that the galaxy sample is very sparse for z > 3.2, as shown in Figure 1.

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Reduced HST/WFC3 images were registered to the same world coordinate system (WCS) as our deep ground-based optical/near-IR data using ∼10–15 stars per pointing. Source Extractor (Bertin & Arnouts 1996) was then used to perform automated object detection (with no smoothing kernel) and produce an initial segmentation map in which each source is assigned a unique identifier. We set the source detection threshold to 1.5σ with a required minimum of 10 pixels above threshold for analysis, 32 deblending thresholds, and a minimum deblending contrast of 1%.⁸ We adopt an rms map proportional to the inverse square root of the weight map produced by MultiDrizzle, scaling by a correction factor F_A = 0.3933 (see discussion by Casertano et al. 2000) to account for the fact that the MultiDrizzle process introduces correlation in the pixel-to-pixel noise.

The initial segmentation map was manually inspected for each galaxy in our sample to ensure both that no spurious pixels were assigned to the galaxies and that each galaxy was not artificially broken into multiple objects. Since galaxies in the redshift range z ∼ 2–3 are well known to be clumpy (e.g., Conselice et al. 2005; Law et al. 2007b, and references therein), this latter goal is non-trivial and Source Extractor frequently classifies multi-component galaxies as separate sources (see, e.g., Colley et al. 1996). While some neighboring clumps are likely to be physically associated with each other (if, for instance, they are embedded in a common envelope of low surface brightness emission), it is not always obvious which clumps are part of the target source and which are unassociated low- or high-redshift interlopers along the line of sight. Generally, we assume that all clumps that lie within a 1.5 arcsec (∼12 kpc at z ∼ 2–3) radius about the ${\cal R}$-band centroid (i.e., the original detection image) are physically associated with a given galaxy unless there is evidence to the contrary (e.g., different spectroscopic redshifts or dramatically different ${\it U_n G {\cal R} J H K}$ colors), and combine them under a single identifier. We discuss the validity of this method with respect to the incidence of genuine versus apparent pairs in Section 6.1.2.

In Figure 2, we present postage-stamp images of the galaxy sample (all pixels identified with sources other than the target 1.5 < z < 3.6 galaxy sample have been cosmetically masked out by Gaussian random noise matched to the noise characteristics of the background sky). As expected on the basis of previous rest-UV morphological studies there is considerable diversity among the morphologies, which range from compact isolated sources to multi-component systems with extended regions of diffuse emission. While this initial segmentation map is adequate for estimating total source magnitudes and constructing postage-stamp images, it is inadequate for calculating quantitative morphologies; we discuss construction of second-pass segmentation maps in Section 3.8.

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We photometrically calibrated our data using the zero-point magnitude of 25.96 AB given for the F160W filter in the HST/WFC3 data handbook. Masking all pixels identified with luminous sources using Source Extractor (Bertin & Arnouts 1996), we use a 3σ-clipped mean to estimate that our drizzled images typically reach a limiting depth of 27.9 AB for a 5σ detection within a 0.2 arcsec radius aperture, or 3σ surface brightness sensitivity of 25.1 AB arcsec⁻².⁹

Initial estimates of the F160W magnitudes of the galaxies are obtained from the Source Extractor corrected isophotal magnitudes (MAG_ISOCOR), which are consistent to within 0.04 mag with estimates obtained from matched-aperture photometry from images smoothed to the angular resolution of the ground-based ${\cal R}$-band survey images for well-defined, isolated sources. We perform Monte Carlo tests of the statistical uncertainty and photometric biases in these magnitudes by inserting 1000 artificial galaxy models with known total magnitudes into randomly selected blank-field regions of the images and calculating the accuracy with which their magnitudes are recovered using Source Extractor. The galaxy models are constructed using GALFIT (see Section 3.2) to model the light profiles of five real galaxies in the Q1700+64 field that span a wide range of effective radii and Sérsic index. These tests are performed for 0.5 mag bins spanning the range H₁₆₀ = 22–25 AB of the galaxy sample, and suggest (Table 2) that MAG_ISOCOR systematically underestimates the brightness of objects by ΔH_ZP = 0.05–0.08 mag. After correcting for this systematic offset, we find that the magnitudes of ten isolated, bright (H₁₆₀ ∼ 15–16 AB), unsaturated stars in our target fields all agree with values published in the Two Micron All Sky Survey (2MASS) point-source catalog (Skrutskie et al. 2006) to within the photometric uncertainty of the catalog.

Stellar masses, ages, and SFRs were calculated by fitting the broadband spectral energy distribution (SED) of the galaxies with stellar population synthesis models using a customized IDL code (N. A. Reddy et al. 2012, in preparation). In addition to the HST/F160W and ground-based $U_n G {\cal R}$ photometry many galaxy models also incorporate J/K_s-band data, and in some cases Spitzer IRAC photometry. The SED fitting process is described in detail by Shapley et al. (2001, 2005), Erb et al. (2006c), and Reddy et al. (2006, 2010); in brief, we use S. Charlot & G. Bruzual's (2012, in preparation) population synthesis models, a Chabrier (2003) initial mass function (IMF), and a constant (τ = ∞) star formation history.

Although the statistical uncertainty of the H₁₆₀ magnitudes is small (see Table 2), the true uncertainty in the continuum magnitudes H_cont is significantly larger due to the uncertain contribution from nebular line emission that falls within the F160W bandpass (λλ14028–16711). In order to ensure that the H₁₆₀ magnitudes do not unduly influence the SED fit with their small formal uncertainties (see also discussion by McLure et al. 2011) we attempt to quantify the additional uncertainty due to nebular emission in a physically motivated manner by bootstrapping approximate line fluxes from broadband scaling laws and typical nebular line ratios. We use the ground-based $U_n G {\cal R}$ magnitudes to estimate the rest-frame monochromatic luminosity L_ν at 1500 Å, and convert this to a UV SFR using the Kennicutt (1998) relation. This UV SFR is corrected for extinction by estimating the UV slope β from the $U_n G {\cal R}$ photometry, and converting to an estimated extinction E(B − V) using the Meurer et al. (1999) relation in combination with a Calzetti et al. (2000) attenuation law (motivated by comparison with direct indicators of the dust emission at 24 μm). We then assume that the extinction-corrected UV SFR is equal to the Hα SFR (see discussion by Erb et al. 2006b), and use the Kennicutt (1998) relation to estimate the corresponding Hα nebular emission-line flux. Based on standard atomic physics and the observations of Maiolino et al. (2008) and Erb et al. (2006a), we assume that the other strong rest-optical nebular emission lines have typical flux ratios given by: Hα/Hβ = 2.9, $\hbox{[O\,{\sc iii}]}\,\lambda 5007/{\rm H}\beta = 4.6$, $\hbox{[O\,{\sc iii}]}\,\lambda 5007/\hbox{[O\,{\sc iii}]}\,\lambda 4959 = 3.0$, $\hbox{[O\,{\sc ii}]}\,\lambda 3727/{\rm H}\beta = 1.5$, $\hbox{[N\,{\sc ii}]}\,\lambda 6585/{\rm H}\alpha = 0.16$. All of these estimated emission-line fluxes are converted to observed values using the extinction coefficients described above.

The combined flux ΔH_line of emission lines that fall within the F160W bandpass at the redshift of each galaxy is added to the photometric bias-corrected H₁₆₀ magnitude to obtain an estimate of the continuum magnitude H_cont = H₁₆₀ + ΔH_line. There are significant uncertainties associated with almost every step of our estimate of the nebular line-emission correction described above, not least of which is the strong variation in line flux ratios (e.g., $\hbox{[O\,{\sc iii}]}\,\lambda 5007/{\rm H}\beta$) with metallicity. We therefore conservatively estimate the uncertainty in the continuum magnitudes as $\sigma _{\rm cont} = \ssty\sqrt{\sigma _{H}^2 + \Delta H_{\rm line}^2}$. Typical values of this uncertainty are generally in the range 0.1 < σ_cont < 0.3 mag, but values as high as 0.5 mag can occur in 10% of cases (and 1.0 mag in 1% of cases). Due to the downweighting of the WFC3 data point when σ_cont ≈ 0.5, derived stellar masses in such cases differ by only 1% on average from stellar masses derived by omitting the WFC3 data point from the SED fit entirely.

Our first morphological classification groups galaxies visually based on the apparent nucleation of their light profiles and the number of distinct components. As illustrated in Figure 3, we group galaxies from Figure 2 into three general classes.

• Type I. Single, nucleated source with no evidence for multiple luminous components or extended low surface brightness features. One hundred and twenty-seven galaxies in our sample.
• Type II. Two or more distinct nucleated sources of comparable magnitude, with little to no evidence for extended low surface brightness features. Fifty-six galaxies in our sample.
• Type III. Highly irregular objects with evidence of non-axisymmetric, extended, low surface brightness features. One hundred and twenty-three galaxies in our sample.

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Type I galaxies appear consistent with being regular and isolated systems, while Type II galaxies may represent either early-stage mergers between two such formerly isolated systems or intrinsically clumpy systems with little continuum emission between the clumps. Type III galaxies in contrast may represent later-stage mergers with bright tidally induces disturbances, or clumpy concentrations within a single extended system (e.g., Bournaud et al. 2007). Of course, there is significant overlap between the three classes, and degeneracy in the classes to which a given galaxy may be assigned. Galaxies with identical luminosity profile but different surface brightness may, for instance, be assigned to either Type I or Type III depending on whether the low surface brightness features are above or below the limiting surface brightness of the data, and the division between Types II and III is similarly unclear. The goal of these visual classifications is not to provide decisive quantitative divisions however, but simply as a reference point to describe the general qualitative appearance of galaxies throughout the following discussion.

In the local universe the surface brightness profiles of galaxies can often by well fit by Sérsic (1963) models over a large dynamic range in luminosities (e.g., Kormendy et al. 2009). While regular ellipsoidal models are clearly an incomplete description of the irregular galaxy morphologies illustrated in Figure 2, such models nonetheless provide a useful description of the characteristic sizes and surface brightness profiles of the major individual clumps. We therefore use GALFIT 3.0 (Peng et al. 2002, 2010) to fit the galaxy sample with two-dimensional (2D) Sérsic profiles described by the functional form

$$\begin{equation} \Sigma (r) = \Sigma _e {\hbox{exp}} \left[ -\kappa \left( \left(\frac{r}{r_{1/2}}\right) ^{1/n} -1 \right) \right] \end{equation} \tag{ 1 }$$

convolved with the observational PSF. These models are characterized by the effective half-light radius r_1/2 and the radial index n of the profile. GALFIT actually calculates the effective half-light radius along the semimajor axis (r); following a common practice in the literature (e.g., Shen et al. 2003; Trujillo et al. 2007; Toft et al. 2009) we convert this to a circularized effective radius $r_{\rm e} = r \sqrt{b/a}$, where b/a is the minor/major axis ratio. As described in Peng et al. (2002, see their Figure 1), two of the most commonly observed values of the radial index in the nearby universe are n = 1, which corresponds to the exponential disk profile, and n = 4, which corresponds to a classical de Vaucouleurs profile with steep central core and relatively flat outer wings typical of elliptical galaxies and galactic bulges.

We use a median-combined stack of isolated, bright (H₁₆₀ < 20 AB), unsaturated stars from across our WFC3 imaging fields to define the PSF model. While the structure of the PSF varies slightly across a given field, and from field to field with the HST-WFC3 roll angle, we find the details of our PSF model have little effect on the derived physical properties of our faint and extended galaxies (see also discussion by Szomoru et al. 2010). Since GALFIT convolves physical models with the observational PSF it is able to determine effective radii down to extremely small spatial scales. Following the method described by Toft et al. (2007), we use a variety of stellar point sources as PSF models to fit Sérsic models to 11 stars in our WFC3 imaging fields, finding that the mean estimated size of known point sources is 0.16  ±  0.25 pixels. We therefore adopt a 3σ limit for unresolved point sources of 0.16 + 3 × 0.25 = 0.91 pixels, or 0.073 arcsec, corresponding to 0.62 kpc at redshift z = 2.0.

Our procedure for fitting Sérsic models to individual galaxies is as follows. We used the Source Extractor segmentation map to mask out all objects not associated with the target galaxies, replacing these pixels with Gaussian random noise matched to the noise characteristics of the image. We then cut out a 5 × 5 arcsec region surrounding each galaxy and subtracted from it a "local sky" estimated from the median of pixels excluded from the segmentation map. GALFIT is then used to fit the minimum number of axisymmetric (we do not introduce bending or Fourier modes) components required to satisfactorily reproduce the observed light distribution. For the majority of galaxies shown in Figure 2 we use a single component, unless there are clearly multiple spatially distinct clumps or significant asymmetry in the light distribution. All GALFIT models were inspected by two of us (D.R.L. and S.R.N.) in order to verify that a consistent approach was taken throughout the galaxy sample.

Unlike the non-parametric morphological statistics (which represent an integrated quantity over the entire light distribution of a galaxy), r_e and n can formally be multi-valued for galaxies fit by multiple Sérsic components (i.e., Type II and some Type III galaxies). We adopt the convention of describing such multi-component galaxies by the r_e and n of the brightest individual component; as we discuss in Section 5.4, this assumption does not significantly bias our conclusions. For the few cases for which a reasonable model cannot be obtained with ≲ 5 Sérsic components (e.g., Q1009-MD28, which is in close physical proximity to the bright Q1009 QSO and turns out to be a Lyα blob based on recent narrowband imaging) we consider r_e and n to be undefined.

The Gini coefficient (G; Gini 1912) was introduced into the astronomical literature by Abraham et al. (2003) and further developed by Lotz et al. (2004). G measures the cumulative flux distribution of a "population" of pixels and is insensitive to the actual spatial distribution of the individual pixels.

Formally G is defined (Glasser 1962) in the range G = 0–1 as

$$\begin{equation} G = \frac{1}{\bar{f} N (N-1)}\sum _{i=1}^N(2i -N -1)f_i, \end{equation} \tag{ 2 }$$

where $\bar{f}$ is the average flux and the f_i pixel fluxes are sorted in increasing order before the summation over all N pixels in the segmentation map. High values of G represent the majority of the total flux being concentrated in a small number of pixels, while low values represent a more uniform distribution of flux.

The spatial distribution of the light may be quantified via the second-order moment of the light distribution, M₂₀, introduced in this context by Lotz et al. (2004). M₂₀ is defined as the second-order moment of the brightest pixels that constitute 20% of the total flux in the segmentation map, normalized by the second-order moment of all of the pixels in the segmentation map. Mathematically,

$$\begin{equation} M_{20} = \hbox{log} \left(\frac{\sum _i M_i}{M_{\rm tot}}\right), \hbox{while} \sum _i f_i < 0.2 f_{\rm tot}, \end{equation} \tag{ 3 }$$

where

$$\begin{equation} M_{\rm tot} = \sum _i^N M_i = \sum _i^N f_i [(x_i - x_c)^2 + (y_i - y_c)^2]. \end{equation} \tag{ 4 }$$

Following Lotz et al. (2004, 2006) we adopt the position that minimizes M_tot as the center (x_c, y_c) of the light distribution.

Typical values of M₂₀ range from ∼ − 1 (most irregular, often with multiple clumps) to ∼ − 2 (most regular).

The concentration index C (Kent 1985; Abraham et al. 1994; Bershady et al. 2000; Conselice 2003) measures the concentration of flux about a central point in the galaxy. While slightly different versions have been introduced by various authors, we adopt the "C₂₈" standard:

$$\begin{equation} C = 5 \hbox{log} \left(\frac{r_{80}}{r_{20}}\right), \end{equation} \tag{ 5 }$$

where r₂₀/r₈₀ are the circular radii containing 20%/80%, respectively, of the total galaxy flux within the segmentation map. Following Conselice et al. (2008), we adopt the flux-weighted centroid of the segmentation map as the center for the concentration calculation. While in many cases this corresponds naturally to a peak in the flux distribution, it is not necessarily the case for extremely irregular galaxies without well-defined central flux concentrations.

Typical concentration values range from ∼1 (least compact) to ∼5 (most compact). We note, however, that galaxies with two or more clumps (e.g., Type II galaxies) that are each individually compact are not generally compact in a global sense.

The asymmetry A (Schade et al. 1995; Conselice et al. 2000) quantifies the 180° rotational asymmetry of a galaxy. Mathematically, A is calculated by differencing the original galaxy image with a rotated copy:¹¹

$$\begin{equation} A = \hbox{min} \left(\frac{\sum | f_{0,i} - f_{180,i}|}{\sum | f_{0,i} |}\right) - \hbox{min} \left(\frac{\sum | B_{0,i} - B_{180,i}|}{\sum | f_{0,i} |}\right), \end{equation} \tag{ 6 }$$

where f_{0, i} represents flux in the original image pixels and f_{180, i} flux in the rotated image pixels. Following Conselice et al. (2000, 2008) we determine the rotation center iteratively by allowing it to walk about an adaptively spaced grid with 0.1 pixel resolution until converging on the point that minimizes Σ|f_{0, i} − f_{180, i}|. The B_{0, i} and B_{180, i} terms represent fluxes in nearby background pixels to which we have applied an identical segmentation map, and are included to subtract the contribution of noise to the total galaxy asymmetry. As discussed by Conselice et al. (2008), the background sum is minimized similarly to the original image.

Typical values of A range from 0 for the most symmetric galaxies to 1 for galaxies with the strongest 180° rotational asymmetry.

The multiplicity coefficient, introduced by Law et al. (2007b), calculates the effective "potential energy" of the light distribution:

$$\begin{equation} \psi _{\rm actual} = \sum _{i=1}^{N}\sum _{j=i+1}^{N}\frac{f_i f_j}{r_{ij}}, \end{equation} \tag{ 7 }$$

where f_i and f_j are the fluxes in pixels i/j, respectively, r_ij is the separation between pixels i and j, and where the sum runs over all of the N(N − 1)/2 i – j pixel pairs.¹² This is compared to the most compact possible rearrangement of pixel fluxes that by analogy with a gravitational system would require the most "work" to pull apart. This compact map is constructed by rearranging the positions of all N galaxy pixels so that the brightest pixel is located in the center of the distribution, and the surrounding pixel fluxes decrease monotonically with increasing radius. Calling r'_ij the distance between pixels i and j in this compact map,

$$\begin{equation} \psi _{\rm compact} = \sum _{i=1}^{N}\sum _{j=i+1}^{N}\frac{f_i f_j}{r^{\prime }_{ij}}. \end{equation} \tag{ 8 }$$

The multiplicity coefficient Ψ measures the degree to which the actual distribution of pixel fluxes differs from the most compact possible arrangement, i.e.,

$$\begin{equation} \Psi = 100 \times \hbox{log}_{10} \left(\frac{\psi _{\rm compact}}{\psi _{\rm actual}}\right). \end{equation} \tag{ 9 }$$

As discussed by Law et al. (2007b), values for Ψ can range from 0 (i.e., for which the galaxy pixels are already in the most compact possible arrangement) to ≳ 10 for extremely irregular sources. Generally, we find that isolated, regular galaxies in our sample may be described by Ψ ⩽ 1, galaxies with some morphological irregularities by 1 < Ψ < 2, and galaxies with strong morphological irregularities or multiple components by Ψ ⩾ 2.

The preliminary segmentation maps constructed in Section 2.3 above assign pixels to a given galaxy based on a constant surface brightness threshold tied to the noise characteristics of the WFC3 data. While such a segmentation map is sufficient for estimating total source magnitudes, it is inadequate for calculating quantitative morphologies using the non-parametric statistics defined in Sections 3.3–3.7 since surface-brightness-based pixel selection produces results that vary with total source luminosity, redshift, and limiting survey magnitude. Multiple methods have been adopted in the literature for defining robust segmentation maps; in Appendix A.1. we discuss four such methods (Conselice et al. 2000; Lotz et al. 2004; Abraham et al. 2007; Law et al. 2007b) and calculate values for G, M₂₀, C, A, and Ψ in each.

In part, Appendix A.1. is provided so that our results can be directly translated to the readers preferred choice of segmentation map, but it is also instructive to consider how the calculated values of the morphological parameters depend upon this choice. While we find that the values of G, M₂₀, C, A, and Ψ are well correlated between different segmentation maps, there can be significant systematic offsets in dynamic range (particularly for G; see also Lisker 2008) between the systems. We discuss the implications of such offsets in Section 6 below.

Throughout the following analysis, we choose to calculate our baseline morphologies using the Abraham et al. (2007) quasi-Petrosian method with isophotal threshold η = 0.3 as this method is arguably most well suited to the irregular morphologies of our target galaxies. Using the transformation relations presented in Appendix A.1. however, we convert our values to the Lotz et al. (2004, 2006) systematic reference frame in order to compare both recent observational results and numerical simulations (e.g., Lotz et al. 2010a, 2010b).

The robustness of our morphological indices has been discussed in the literature many times before (e.g., Bershady et al. 2000; Lotz et al. 2006; Lisker 2008; Gray et al. 2009). Generally speaking, such work suggests that morphological statistics are relatively robust for large, bright galaxies but that they can become unreliable at faint magnitudes and for galaxies that are small with respect to the observational PSF. Most of these previous studies are tailored to the analysis of deep HST/ACS imaging in public survey fields however, and in order to understand the effects of systematic biases on our WFC3 imaging data (and on our specific galaxies) it is necessary to perform many robustness tests anew.

The details of our analysis exploring the robustness of each of the five quantitative morphological statistics, and the Sérsic parameters r_e and n, to total source magnitude H₁₆₀, the size of the observational PSF, and our choice of pixel scale are presented in Appendices A.2.–A.4.. In brief, we find the following.

• 1.  
  The derived values of six of the seven indices are fairly robust for galaxies with magnitudes H₁₆₀ ⩽ 24.0 (roughly corresponding to total signal-to-noise ratio (S/N) >100), but become less reliable at fainter magnitudes. The exception is the concentration parameter C, for which the small sizes of many of our galaxies cause the inner 20% flux isophote to be unresolved at all magnitudes and therefore C to be unreliable (see also Bershady et al. 2000). We therefore omit C from detailed discussion, and restrict our analyses in the following sections (except where indicated) to the subsample of galaxies with H₁₆₀ ⩽ 24.0, resulting in a sample of 206 galaxies, 59/95/52 in the z = 1.5–2.0, z = 2.0–2.5, and z = 2.5–3.6 redshift bins, respectively. The physical implications of this self-imposed apparent magnitude limit, and of systematic variations with the observational PSF, are discussed in the relevant sections below.
• 2.  
  Six of the seven indices (except C) are robust to our choice of a 0.08 arcsec pixel scale; our conclusions would be unchanged if we had drizzled our data to 0.06 arcsec or 0.1 arcsec pixels instead.
• 3.  
  Given the small size of many of our galaxies, the non-parametric statistics G, M₂₀, C, A, and Ψ can vary systematically with the observational PSF as morphological features become more or less well resolved (see also discussion by Lotz et al. 2004, 2008b). In particular, these five statistics will have less dynamic range to their values than in high-resolution imaging as it becomes progressively more difficult to distinguish them from point sources. This complicates quantitative comparisons to data obtained at different wavelengths or local comparison samples, but is less significant for comparisons within the z ∼ 1.5–3.6 population. In contrast, the Sérsic parameters r_e and n are relatively robust to the PSF because the modeling process convolves theoretical models with the observational PSF.
• 4.  
  The uncertainty in each of the seven indices is calculated via Monte Carlo simulations placing GALFIT model galaxies atop different blank-field regions of the WFC3 footprint in order to compare different realizations of the noise statistics. This uncertainty varies as a function of both source magnitude and morphological type; averaged over these considerations, typical uncertainties are 3% in G, 4% in M₂₀, 11% in C, 22% in A, 21% in Ψ, 2% in r_e, and 15% in n.

As illustrated by Figures 4 and 5, typical galaxies are best described by an n ∼ 1 Sérsic profile. Indeed, only six galaxies have n > 2.5, of which four have estimated radii less than our 3σ resolution estimate, suggesting that these galaxies are simply too small to robustly determine their structure. Focusing our attention on the 200 galaxies with n < 2.5 we find a mean 〈n〉 = 0.63 with standard deviation of 0.39, corresponding to flat inner regions intermediate between a Gaussian (n = 0.5) and an exponential profile (n = 1.0), and a steeply declining profile at larger radii (similar to previous results by, e.g., Ravindranath et al. 2006; Conselice et al. 2011b; Förster Schreiber et al. 2011).

In order to investigate the faint extended structure of our star-forming galaxy sample we create a composite stack of galaxies (irrespective of redshift and total H₁₆₀ magnitude). We cut out 5 × 5 arcsec regions around each galaxy, align all of the flux-weighted image centroids using sub-pixel bilinear interpolation, and stack the individual images together using a 3σ-clipped mean algorithm. The resulting stack for our 127 galaxies of morphological Type I (i.e., those galaxies whose morphologies are most regular and well defined) reaches a 3σ limiting surface brightness of 27.8 AB arcsec⁻² (31.2 AB for a 5σ detection in a 0.2 arcsec radius aperture).

As illustrated in Figure 6 the stacked radial profile is well described by an n = 1.38 Sérsic model with effective radius r_e = 1.36 kpc. The profile is a good match to the Sérsic model out to at least 6 kpc (>4r_e) and deviates only moderately from the model out to the detection limit at r ∼ 15 kpc. As expected, including the Type III galaxies (which, by definition, are more extended) in the stack results in a slightly larger characteristic effective radius r_e = 2.04 kpc but a similarly good match to an n ∼ 1 Sérsic model.

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We caution, however, that while Figure 6 confirms that n ∼ 1 models are a fairly good representation of z = 1.5–3.6 star-forming galaxies, the stacked profile does not account for variability in the size or orientation of its component galaxies. By effectively discarding information about the projected ellipticity the stack overestimates the mean effective radius of the sample by a factor ${\sim}\sqrt{\langle b/a \rangle }$. We discuss the characteristic sizes of the star-forming galaxies in detail in Section 5.

In Figure 7 we plot a histogram of b/a for galaxies with H₁₆₀ < 24.0, n < 2.5, and both major and minor axis lengths well resolved (a total sample of 164 galaxies).¹³ The distribution¹⁴ is strongly peaked about (b/a)_peak ≈ 0.6 with tails extending to both extremes b/a = 0 and b/a = 1. As we demonstrate below, such a distribution is strongly inconsistent with a population of thick exponential disks as is commonly assumed in the literature (e.g., Genzel et al. 2008) and much more consistent with a population of triaxial ellipsoids.

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As discussed by Padilla & Strauss (2008, and references therein) for a large sample of local galaxies drawn from the Sloan Digital Sky Survey (SDSS), a population of spiral galaxies with random orientations defines a distribution in b/a that is relatively flat above some minimum value corresponding to the edge-on thickness of the disks. Taking i to be the inclination of such a disk to the line of sight (where i = 0° represents a disk viewed face-on), the observed axial ratio (b/a) of a flattened axisymmetric system is given by (see, e.g., Hubble 1926; Tully & Fisher 1977)

$$\begin{equation} {\rm cos}^2 i = \frac{(b/a)^2-r_0^2}{1-r_0^2}, \end{equation} \tag{ 10 }$$

where r₀ is the intrinsic minor/major axis ratio for a perfectly edge-on system. In the thin-disk approximation r₀ = 0 and Equation (10) reduces to the familiar b/a = cos i. In the local universe, typical values for r₀ range from r₀ ∼ 0.20 for Sa-type to ∼0.08 for Sd-type galaxies (Guthrie 1992; Ryden 2006), although variations can also occur with wavelength (e.g., Dalcanton & Bernstein 2002). At redshifts z > 1.5 however, star-forming galaxies are known to have significant vertical velocity dispersion (e.g., Law et al. 2009; Förster Schreiber et al. 2009), and analysis of five of the most disk-like objects (based on velocity maps derived from integral-field spectroscopy) indicates that the median r₀ ≈ 0.34 (Genzel et al. 2008). For more typical dispersion-dominated galaxies (v/σ ⩽ 1) r₀ might be expected to be even larger.

We perform Monte Carlo tests in which we artificially observe a sample of 10⁶ flattened axisymmetric disks from a random distribution of inclinations.¹⁵ Formally, we quantify the difference between the observational data and the model by the statistic

$$\begin{equation} \chi ^2 = \frac{1}{\nu } \sum _{i=1}^{B} \frac{(N_{\rm model}-N_{\rm obs})^2}{N_{\rm obs}}, \end{equation} \tag{ 11 }$$

where N_obs is the number of galaxies observed in each of our B = 10 bins in b/a, ν = 8, and N_model is the number of galaxies expected in each bin according to the assumed model. We overplot the distribution of b/a obtained using such flattened axisymmetric disk models on the observational data in Figure 7. Regardless of the value of r₀ adopted, it is not possible to satisfactorily explain the observed distribution of b/a; r₀ = 0.0, 0.2, and 0.4 models have χ² = 43.0, 21.5, and 16.9, respectively.

In contrast, the peaked distribution of b/a is exactly the form expected for a population of randomly oriented triaxial ellipsoids such as that found by van den Bergh (1988) for a sample of local irregular galaxies. We therefore repeat our Monte Carlo analysis assuming that the galaxies can be characterized as triaxial ellipsoids with axis lengths r_x, r_y, r_z. Calculating the projected minor/major axis ratio b/a of a triaxial ellipsoidal surface viewed in an arbitrary orientation is an interesting problem in its own right, and we discuss the details of this calculation in Appendix B. Since we are only interested in axial ratios rather than the absolute lengths we set r_z = 1 and consider a grid of values in the range r_x, r_y = 0.1, 0.2, ..., 1.0.

In Figure 8 we show a surface plot of χ² as a function of r_x and r_y. The best agreement between model and observations clearly occurs for a well-defined region around (r_x, r_y = 0.7, 0.3). The expected distribution of b/a for (r_x, r_y = 0.7, 0.3) is shown in Figure 7. While this is clearly a better description of the observations than the axisymmetric disk model (particularly in the expected number of systems with b/a > 0.8) it is still imperfect (χ² = 6.2; ν = 7), predicting no galaxies with b/a < 0.3 and a large excess with b/a ∼ 0.35–0.45. These remaining imperfections likely reflect the intrinsic range of morphologies within the galaxy sample—rather than every galaxy having an identical shape there is undoubtedly some range about these values. Permitting a more realistic distribution of axis ratios (i.e., picking r_x and r_y at random from Gaussian distributions with mean 0.7 and 0.3, and 1σ width 0.1 and 0.2, respectively), it is possible to reproduce the observed distribution of b/a extremely well (solid red line in Figure 7; χ² = 1.2 with ν = 5).

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At present, it is meaningless to distinguish between minor/major axis ratios of 0.2 versus 0.3, or to state with certainty that a Gaussian distribution of intrinsic axis ratios is appropriate. Our fundamental conclusion, however, is that the majority of z = 1.5–3.6 star-forming galaxies are best represented by triaxial systems rather than geometrically thick disks (as previously discussed by Ravindranath et al. 2006 and Elmegreen et al. 2005) and it is worth asking what this means in a physical sense.

Given the overall similarity between the rest-UV and rest-optical morphology (Section 4.3) it may simply be that light from clumpy (and asymmetrically distributed) star-forming regions within galaxies (e.g., Bournaud & Elmegreen 2009) dominates the emergent flux at both UV and optical wavelengths. Alternatively, since we derived b/a for the brightest subcomponent of each galaxy we may be measuring the intrinsic shape distribution of individual giant star-forming clumps. However, the peaked distribution of b/a persists if we restrict our attention to the most regular single-component systems (i.e., Type I galaxies) suggesting that we are observing galaxy scale structures with characteristic radii ∼1–3 kpc. Similarly, the distribution of b/a persists for galaxies with stellar masses greater than 10¹⁰ M_☉ in which stellar continuum emission should be well detected in the WFC3 imaging data, suggesting that the stellar mass distribution itself is strongly asymmetric.

Combining our morphological results with observations (e.g., Law et al. 2007a, 2009; Förster Schreiber et al. 2009) that typical z ∼ 2–3 star-forming galaxies have large gas fractions, high velocity dispersions >50 km s⁻¹, and velocity fields that are in many cases inconsistent with rotationally supported disk models (especially at lower stellar masses; see discussion by Law et al. 2009), we suggest that the distribution of stars and gas in these rapidly star-forming galaxies may be inherently triaxial rather than residing largely in a geometrically thick disk. Such a distribution of gas would be gravitationally unstable, suggesting that the life cycle of z ∼ 2–3 star-forming galaxies may be continually passing in and out of dynamical equilibrium (e.g., Ceverino et al. 2010). In such a scenario, gas disks may be only short-lived and continuously forming from recently accreted gas (whether acquired from mergers or hot/cold-mode accretion; e.g., Dekel et al. 2009a; Kereš et al. 2009), rapidly becoming disrupted, and reforming again until the triaxial stellar component (perhaps a precursor of modern-day bulges) acquires sufficient mass to stabilize the growth of a long-lived and extended gas disk (e.g., Martig & Bournaud 2010). We discuss additional observational support for such a scenario based on low-ionization gas-phase kinematics in a companion paper (D. R. Law et al. 2012, in preparation).

We note that both our results and conclusions are qualitatively consistent with those of Ravindranath et al. (2006),¹⁶ who used HST/ACS imaging in the GOODS fields to demonstrate that the rest-UV morphologies of star-forming galaxies at z ∼ 3–4 also have a peaked distribution of ellipticities. While we found the distribution for rest-optical morphologies to be peaked about (b/a)_peak ∼ 0.6 however, Ravindranath et al. (2006) found (b/a)_peak ∼ 0.5/0.3 for galaxies at z = 3/4, respectively, as seen in the rest-UV. This difference may be explained in part by the difference in rest-frame wavelength probed by the two studies; it is perhaps unsurprising that the ellipticity of star-forming galaxies in the young universe changes slightly from rest-frame 2000 Å (tracing the regions of most recent star formation) to rest-frame 5000 Å (tracing the older stellar population). In contrast, van der Wel et al. (2011) observed a relatively flat distribution of b/a (above b/a ∼ 0.5) for a sample of 14 massive (M_* > 8 × 10¹⁰ M_☉) compact quiescent galaxies at z ∼ 2. While this may represent a fundamental structural difference between the star-forming and quiescent galaxy samples, we caution that the quiescent galaxies are 10 times more massive than the typical star-forming galaxy in our survey, and note that if increasing stellar mass stabilizes the formation of disks then star-forming galaxies of similarly high mass may prove to have similarly disk-like ellipticities.

One of our fields (Q1700+64) was imaged previously using HST/ACS with the F814W filter (GO-10581; PI: A. E. Shapley). This filter (λ_eff = 8332 Å) traces rest-UV wavelengths ranging from 2000 to 3000 Å, depending on the redshift of the target galaxy. The detailed morphologies resulting from this rest-UV imaging program have already been discussed elsewhere (Peter et al. 2007). Here we compare the rest-optical and rest-UV morphologies of galaxies overlapping with our WFC3/IR imaging. For consistency we re-reduce the raw observational data from GO-10581, drizzling them to a 0.08 arcsec pixel scale and smoothing them to an FWHM of 0.18 arcsec in order to match the observational characteristics of our WFC3/IR imaging data. We calculate that the F814W image reaches a limiting depth of 28.7 AB for a 5σ detection within a 0.2 arcsec radius aperture, or ∼1 mag deeper than our WFC3/IR imaging data.

We show the morphologies of the 18 star-forming galaxies that overlap between the two samples in Figure 9. Qualitatively, we note that the morphologies of most galaxies are similar in both rest-UV and rest-optical bandpasses; morphological irregularities or multiple components visible in one bandpass are similarly visible in the other, resulting in a small morphological k-correction (see discussion by Conselice et al. 2011b). The smallest variation is exhibited by galaxies of low stellar mass (for which the light from young stars might reasonably be expected to dominate both the rest-UV and rest-optical light of the galaxy), while high-mass galaxies exhibit greater differences consistent with the establishment of an evolved stellar population. In particular, the galaxies that were observed to be extremely low surface brightness, red (${\cal R}-K_s \sim 3$ AB), "wispy" systems in the rest-UV tend to be high-mass systems that are much brighter and well nucleated in the rest-optical (e.g., Q1700-MD103, Q1700-BX767). This result is similar to that found by Toft et al. (2005) for a population of red star-forming galaxies.

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We quantify this morphological difference by calculating the internal color dispersion ξ (Papovich et al. 2005) after carefully aligning the ACS and WFC3 images using the measured centroids of ten stars:

$$\begin{equation} \xi (I_1,I_2) = \frac{\sum (I_2 - \alpha I_1 - \beta )^2-\sum (B_2 - \alpha B_1)^2}{\sum (I_2-\beta )^2 - \sum (B_2 - \alpha B_1)^2}, \end{equation} \tag{ 12 }$$

where I₁ and I₂ are the pixel fluxes in the F814W and F160W bandpasses, α = ∑(I₁ I₂)/∑(I²₁) is a scaling factor describing the overall color of the galaxy, β adjusts for the variable background level, and B₁ and B₂ represent blank background sky regions in each image. α is set by minimizing the sum ∑(I₂ − αI₁)², i.e., α = ∑(I₁I₂)/∑(I²₁). The sum is performed over all pixels in the F160W segmentation map. The background sums were done by adopting the mean from the calculation performed on the segmentation map grafted onto 1000 different regions of blank sky.

Values for ξ calculated for each galaxy are quoted in Figure 9, and confirm our visual impression that the UV and optical morphologies differ more greatly for high-mass (alternatively, red) galaxies. At the low-mass end (M_* < 10¹⁰ M_☉) 〈ξ〉 = 0.02, while for galaxies with M_* > 10¹⁰ M_☉ we find 〈ξ〉 = 0.09, peaking at ξ = 0.28 for the highest-mass galaxy Q1700-BX767 which displays a red core with a surrounding blue ring. Similar trends were noted by Labbé et al. (2003), who found significant rest-UV to rest-optical morphological differences for a sample of six K-bright z ∼ 1.4–3 disk galaxies, and Papovich et al. (2005), who noted that their galaxies with the highest values of ξ were those with the reddest colors. We caution that there are relatively few galaxies in our sample however, and recent work by Bond et al. (2011) looking at the rest-optical versus rest-UV morphologies of 117 (1.4 < z < 2.9) star-forming galaxies in the GOODS-S field found a similar mean ξ = 0.02 but no evidence for a correlation with galaxy color. In the near future we anticipate that the relation between rest-UV and rest-optical morphology will be greatly refined by the large-area and multi-band CANDELS survey (Grogin et al. 2011; Koekemoer et al. 2011).

It is also possible to compare the effective radii r_e derived in each of the two bandpasses. Similarly to Dutton et al. (2011) and Barden et al. (2005), we find (Figure 10) that the rest-UV sizes of these galaxies are 21% ± 2% larger on average than their optical sizes, although there is increasing scatter in the relation at large radii (i.e., large mass) in part because these galaxies are red (${\cal R}-K_s \sim 3$ AB) and poorly defined in the F814W data. This relation is largely unchanged if the Sérsic index n of the radial profile is kept fixed between the F160W and F814W data.

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In Figure 11 we plot the effective circularized radius r_e as a function of stellar mass for all galaxies with H_AB ⩽ 24.0 and Sérsic index n < 2.5, constituting a sample of 59/93/50 galaxies in the z = 1.5–2.0/2.0–2.5/2.5–3.6 redshift ranges, respectively. Of these 202 galaxies, 9 (∼4%) have effective radii consistent with an unresolved point source, and may represent either the compact end of the galaxy distribution or faint AGNs (albeit with no obvious signature in the UV spectra or broadband SED out to ∼7000 Å rest frame).

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Figure 11 indicates that galaxies occupy a large range of effective radii at all redshifts z = 1.5–3.6 and stellar masses M_* = 10⁹–10¹¹ M_☉ with the 1σ standard deviation of the distribution ∼0.2 dex comparable to the scatter in the local star-forming galaxy relation (e.g., Shen et al. 2003). Despite the large width of the distribution in r_e, however, there is a mean mass–radius relation in place at early as z ∼ 3 that evolves with decreasing redshift. Binning our sample by redshift and stellar mass we calculate¹⁷ that 〈r_e〉 = 1.29  ±  0.11 (1.65  ±  0.18) kpc for galaxies in the mass range M_* ⩽ 10¹⁰ (>10¹⁰) M_☉, respectively, at redshift z = 2.5–3.6, increasing with cosmic time to 〈r_e〉 = 1.34  ±  0.07 (1.84  ±  0.13) kpc by z = 2.0–2.5, and to 〈r_e〉 = 1.56  ±  0.11 (2.33  ±  0.20) kpc by z = 1.5–2.0 (see summary in Table 3). These results are consistent with the early values calculated for a subset of our sample by Nagy et al. (2011) to within the estimated uncertainty; the strongest evolution in effective radius with redshift occurs for higher-mass galaxies M_* > 10¹⁰ M_☉. Parameterizing the stellar mass–radius relation as r_e ∼ M^α_* we find that the best-fit value of the power-law index α = 0.22  ±  0.05, 0.13 ± 0.05, and 0.09 ± 0.06 for the redshift z = 1.5–2.0, 2.0–2.5, and 2.5–3.6 intervals, respectively.

As indicated by the top histogram in Figure 11 the three redshift samples each probe galaxies with a slightly different range of stellar masses, and it is therefore useful to calculate a normalized quantity r_e/r_SDSS for each galaxy, where r_SDSS as a function of M_* (solid black line in Figure 11) is the mean effective circularized radius for late-type (i.e., n < 2.5) low-redshift galaxies in the SDSS (Shen et al. 2003). As indicated by Figure 12, typical star-forming galaxies at fixed stellar mass were significantly smaller at z > 1.5 than in the nearby universe, with 〈r_e/r_SDSS〉 = 0.70 ± 0.04, 0.59 ± 0.03, and 0.45 ± 0.02 for the z = 1.5–2.0, 2.0–2.5, and 2.5–3.6 samples, respectively.¹⁸ If galaxies at fixed stellar mass in the range M_* = 10⁹–10¹¹ M_☉ can be assumed to grow with redshift as r_e ∼ (1 + z)^γ, a linear least-squares fit to the data indicates that γ = −1.07  ±  0.28 between z = 3.6 and z = 1.5 (solid line in Figure 12). This is consistent with similar determinations γ = −1.3 and γ = −1.11 found for massive star-forming galaxies by van Dokkum et al. (2010) and Mosleh et al. (2011), respectively. Extrapolation of this power law suggests that actively star-forming galaxies in the young universe may evolve onto the local late-type mass–radius relation by z ∼ 1 (although see Section 5.4), consistent with recent evidence that the mass–radius relation for star-forming galaxies evolves only weakly in the redshift interval z = 0–1 (Barden et al. 2005).

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Individual star-forming galaxies, however, grow in both stellar mass and radius simultaneously and eventually evolve into typical ∼L* galaxies by the present day as indicated by clustering analyses (e.g., Conroy et al. 2008). Given the shallow observed mass–radius relation for star-forming galaxies at z = 1.5–3.6, it is clearly not possible for individual galaxies to evolve along this relation to match the local sample. Rather, galaxies need to add mass at large radii via steeper growth of the form r ∼ M or r ∼ M² as illustrated in Figure 13 (see also Figure 8 of van Dokkum et al. 2010). Such growth may be consistent with expectations for major and minor mergers, respectively (e.g., Bezanson et al. 2009, Naab et al. 2009 for early-type galaxies).

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In Figure 13 we plot the best-fit power-law model of the stellar mass–radius relation for our z = 2.0–2.5 galaxy sample against a variety of previous observational samples available in the literature.¹⁹ Our results are generally consistent at the 1–2σ level with previous studies that, due to observational limitations, have typically been conducted for galaxies with high stellar masses M_* > 10¹⁰ M_☉ (e.g., Franx et al. 2008; Toft et al. 2009; Williams et al. 2010; Targett et al. 2011) and extend these previous results down to M_* ∼ 10⁹ M_☉.

The most direct comparison can be made to Mosleh et al. (2011), who used deep ground-based K-band imaging across the GOODS-N field to measure the characteristic sizes of 41 massive (M_* = 10¹⁰–10¹¹ M_☉) BM/BX star-forming galaxies in the redshift range z = 1.4–2.7 and 4 LBGs in the redshift range z = 2.7–3.5 for which spectroscopic redshifts have been made publicly available by Reddy et al. (2006). Although these target galaxies were selected and spectroscopically confirmed in a manner identical to our own sample, we find significant disagreement with respect to the mean r_e as a function of stellar mass. As given in their Table 3, the Mosleh et al. (2011) BM/BX galaxy sample has a median mass of log(M_*/M_☉) = 10.4 and median radius of r_e = 2.68 ± 0.19 kpc, and the LBG sample a median mass of log(M_*/M_☉) = 10.3 and median radius of r_e = 2.22 ± 0.61 kpc. Within the same ranges of redshift and stellar mass, our BM/BX and LBG samples have 2.5σ-clipped mean radii of r_{e, c} = 1.91 ± 0.10 kpc and r_e = 1.62 ± 0.28 kpc, respectively. Although our WFC3 imaging data are significantly deeper (∼2.5 mag) and better resolved (0.18 arcsec versus ∼0.5 arcsec) than the ground-based K-band imaging, our experience with the robustness of r_e (see Appendix A) and tests degrading our images to the quality of the ground-based data do not suggest an obvious instrumental reason for the large ∼4σ difference in the mean values.²⁰

A particularly valuable comparison can also be made to Förster Schreiber et al. (2011), who used HST/NICMOS F160W (PSF FWHM ∼0.14 arcsec) to study the rest-optical morphologies of six massive star-forming galaxies at z = 2.0–2.5 selected from the SINS Hα integral-field kinematics survey (Förster Schreiber et al. 2009). Each of these six galaxies are plotted as colored circles in Figure 13; we convert their measurements to circularized effective radii by multiplying by $\sqrt{b/a}$ as tabulated in their Table 4.²¹ Two of these six galaxies were also observed as part of our HST/WFC3 imaging program: Q1623-BX528 and Q2343-BX389. While our measured effective radii for Q1623-BX528 differ by ∼30% due to a different number of morphological components used to fit the complicated light distribution (Förster Schreiber et al. 2011 used two components, while we used three), our radii for the single-component Q2343-BX389 agree to within 1%, suggesting that there is negligible systematic difference between the radii calculated by the two surveys. Except for the multi-component Q1623-BX528 (which Förster Schreiber et al. 2009 classify as a merger on the basis of kinematic data and multi-component rest-frame optical continuum morphology), all of the galaxies studied by Förster Schreiber et al. (2011) have radii roughly twice the mean size of their parent color-selected, spectroscopically confirmed galaxy population at a given stellar mass and lie in the top 5% of the r_e distribution for our observed sample of BM/BX galaxies at z = 2.0–2.5. This suggests that the subset of galaxies observed by Förster Schreiber et al. (2011), the majority of which were selected to be the most disk-like within the SINS z ∼ 2 sample, falls among the high r_e extreme of the galaxy population in the stellar mass range M_* ∼ 10¹⁰–10¹¹ M_☉ (see also discussion by Law et al. 2009; Dutton et al. 2011), while following some of the general trends observed at this redshift between size, specific SFR, and stellar mass surface density (Franx et al. 2008). We expand upon this discussion by relating the morphologies of these galaxies (plus 12 additional galaxies from the OSIRIS and/or SINS kinematic surveys that fell within our WFC3 imaging fields) to their ionized-gas kinematics in a forthcoming contribution (D. R. Law et al. 2012, in preparation).

Although theoretical simulations of z ∼ 2–3 star-forming galaxies are still in their infancy, the sizes predicted by such simulations are in rough agreement with our observed values. In Figure 13 (dotted and dashed lines) we illustrate the results of two such models from Sales et al. (2010) and Dutton et al. (2011), respectively.

Sales et al. (2010) use cosmological N-body/smoothed particle hydrodynamics (SPH) simulations to model the growth of baryonic structures in galaxies for four different feedback prescriptions. Of these four prescriptions, their "WF2Dec" model most closely matches both our observations and our physical understanding of these galaxies; in this model relatively strong feedback from star-forming regions results in the efficient removal of gas from galaxies via an outflowing wind with velocity ∼600 km s⁻¹. Such peak outflow velocities are generally consistent with observations for our BM/BX/LBG galaxy sample (see, e.g., Steidel et al. 2010). As discussed by Sales et al. (2010), as feedback strength increases it suppresses star formation so that galaxies of a given stellar mass tend to inhabit larger halos and can thus have correspondingly larger characteristic sizes. Assuming that their stellar half-mass radii roughly correspond to visible-band half-light radii, and converting to circularized values by multiplying by $\langle \sqrt{b/a} \rangle \approx 0.77$, we plot their predicted stellar mass–radius relation in Figure 13 (dotted line). The model is generally consistent with our observations, although it slightly underpredicts the typical galaxy size by ∼0.1 dex. In contrast, in models with no feedback the majority of stars form in dense systems at early times, resulting in mean circularized half-light radii ∼0.5 kpc at M_* ∼ 10¹⁰ M_☉ that disagree strongly with our observations.

Dutton et al. (2011) also study the evolution of scaling relationships with redshift using a series of semi-analytic models that roughly reproduce the velocity–mass–radius relations at z = 0. In particular, they focus on the evolution of the zero-point calibration of these relations, predicting that the evolution from z = 2 to z = 0 shifts the mass–radius relation upward in radius by ∼0.3 dex. As illustrated in Figure 13 (dashed line), the magnitude of this zero-point shift is consistent with our observations at M_* ∼ 10¹⁰ M_☉ (i.e., the mass at which the models also overlap the observed local relation).²²

We close by discussing a few of the caveats and complications that can affect the mass–radius relation that we have derived.

First, the galaxies in the z = 2.5–3.6 subsample have fainter H₁₆₀ magnitudes than galaxies in the lower redshift bins (Figure 1), and Figure 22 demonstrated (see discussion in Appendix A.2) that the recovered value of r_e can vary as a function of total source magnitude. However, this effect does not significantly influence our conclusions. First, r_e is extremely stable for galaxies with isolated morphologies and small radii characteristic of much of the observational sample. While r_e is less robust to H₁₆₀ magnitude for larger and more irregular galaxies, the majority of the variation occurs for magnitudes H₁₆₀ > 24.0 which we deliberately exclude from our analysis. The mean observed magnitudes of our 1.5 ⩽ z < 2.0, 2.0 ⩽ z < 2.5, and 2.5 ⩽ z < 3.6 samples are 〈H₁₆₀〉 = 23.1, 23.2, 23.5, respectively. Across such a small range ΔH₁₆₀ = 0.4 mag the change in radius for all morphological types is ≲ 4%, comparable to the statistical uncertainty in the quoted 〈r_e/r_SDSS〉. Indeed, even were we to include faint galaxies with H₁₆₀ > 24.0 in our analysis we find that the mean values of 〈r_e/r_SDSS〉 change by ≲ 1σ.

It is also possible that our results may be biased due to our assumption that the radius r_e of a multi-component system may be characterized by the radius of the brightest individual component, while our stellar masses (derived from seeing-limited ground-based photometry and similarly confused Spitzer/IRAC photometry) represent the integral over the light of all of the components. If we repeat our previous analyses instead assuming that the stellar mass of these systems is proportional to the fraction of the H₁₆₀ flux in the primary component, or simply omitting galaxies with multiple well-defined individual components from our analysis, we find that values for 〈r_e/r_SDSS〉 in each of the three redshift bins are consistent with their previously calculated values to within ∼1σ. We are therefore confident that our results are not significantly affected by our assumption of how to define r_e for multi-component systems.

Some of the apparent evolution in characteristic radius at fixed stellar mass from z ∼ 3 to z ⩽ 2 may also be due to the variable K-correction in our fixed observational bandpass. With an effective wavelength of λ_eff = 15369 Å, the F160W filter probes rest-frame 5548, 4758, and 4044 Å emission at the mean redshift of three samples (〈z〉 = 1.77, 2.23, 2.80). However, we note that the effective radii derived for our galaxies in the Q1700+64 field varied by only ∼20% from rest-frame 5000 Å to rest-frame 2500 Å; linear interpolation suggests that the change from 5000 Å to 4000 Å would be much smaller, ≲ 8%. Similarly, Dutton et al. (2011) make theoretical predictions for the difference in effective radius between a variety of optical/NIR bandpasses; interpolating their results suggests that we might expect a systematic increase of 0.04 ± 0.03 dex in log(r_e) from the lowest to highest redshift sample (i.e., sizes measured at longer wavelengths are smaller than those measured at shorter wavelengths, corresponding to inside-out disk growth) due to such bandshifting. This is comparable to the formal uncertainty on our measured 〈r_e/r_SDSS〉 in each of the three redshift bins, and would represent only a minor correction. Likewise, the results of Barden et al. (2005; see their Figure 2) suggest that the correction factor would be ≲ 2%, which is much smaller than our ∼5%–10% uncertainty on 〈r_e〉 in each of our redshift bins.

Finally, we caution that the precise values derived for the size evolution of galaxies compared to their low-redshift counterparts at similar stellar mass is complicated by uncertainties in the local relation. Although we adopted the Shen et al. (2003) estimate of the local mass–radius relation for late-type galaxies, we note that numerous authors (e.g., Barden et al. 2005; Trujillo et al. 2006; Guo et al. 2009) find that Shen et al. (2003) underestimate their effective radii. This discrepancy is due in part to systematic differences in analysis techniques (GALFIT modeling versus one-dimensional radial profile fitting), definition of early- versus late-type galaxies (n < 3.5 versus n < 2.5), and effective wavelength (r versus z band) of the observations. Although the measured discrepancy among radii is less pronounced for low Sérsic indices similar to those of our galaxy sample (Guo et al. 2009), these varied effects may considerably complicate interpretations of the evolution of the high-redshift mass–radius relation to the present day.

6.1.1. Quantitative Morphologies

One common way of identifying mergers is to use their morphological asymmetry A, as discussed extensively in the literature by, e.g., Conselice et al. (2000, 2003, 2008, 2009), Lotz et al. (2008b, 2010a, 2010b), Papovich et al. (2005), and Scarlata et al. (2007). In Figure 14 we plot C versus A for our magnitude-limited sample of galaxies (H₁₆₀ < 24.0, left panels) and for a mass-limited subsample (M_* > 10¹⁰ M_☉, right panels). At low redshifts ongoing mergers have typically been identified by the criterion A > 0.35 (e.g., Conselice et al. 2003), although for less well resolved, lower surface brightness galaxies similar to those of our sample Lotz et al. (2008b) find that A > 0.30 is more appropriate. Adopting the A > 0.30 criterion, we find that the merger fraction (for M_* > 10¹⁰ M_☉) is 0.32 ± 0.06, 0.43 ± 0.04, 0.41 ± 0.07 in the 1.5 ⩽ z < 2.0, 2.0 ⩽ z < 2.5, and 2.5 ⩽ z < 3.6 samples, respectively.²³

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Another common method of identifying mergers is by their location in G–M₂₀ space, as originally defined by Lotz et al. (2004, 2006). In Figure 15 we plot G versus M₂₀ for our magnitude-limited sample of galaxies (H₁₆₀ < 24.0, left panels) and for a mass-limited subsample (M_* > 10¹⁰ M_☉, right panels). The merger criterion defined by Lotz et al. (2008b) for high-redshift galaxies²⁴

$$\begin{equation} G > -0.14 M_{20} + 0.33 \end{equation} \tag{ 13 }$$

gives a merger fraction of 0.14  ±  0.04, 0.23  ±  0.03, 0.24  ±  0.05 at 1.5 ⩽ z < 2.0, 2.0 ⩽ z < 2.5, and 2.5 ⩽ z < 3.6, respectively.

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Figures 14 and 15 demonstrate the necessity for caution when estimating the merger fraction using different segmentation maps: if we had calculated the morphological statistics using a segmentation map modeled on the methods of Conselice et al. (2009), typical points in these figures would be offset in the direction indicated by the green arrows. While the effect in the C − A plane is fairly minimal, values of G can change drastically, pushing a large number of points over the merger/non-merger dividing line and resulting in a wildly different derived merger fraction if the merger/non-merger division is not made appropriately. As discussed by Lisker (2008), this offset may in large part account for the discrepancy in the number of mergers identified in similar observational samples at z ∼ 1 using the G–M₂₀ technique by Lotz et al. (2008a; see their Figure 10) and Conselice et al. (2008; see their Figure 8).

6.1.2. Nearby Pairs

Another method of identifying mergers is to count the number of systems with close physical pairs. In practice, we consider systems with multiple distinct clumps of comparable H₁₆₀ flux (∼3:1–1:1) in their light profiles, colors consistent with the rest-UV selection criteria, and well-defined separations in the range 5 <r < 16 kpc (i.e., are classified as Type II galaxies) as physical pair candidates.²⁵ For galaxies with H₁₆₀ < 24.0 and M_* > 10¹⁰ M_☉ we find that the fraction of pairs is 0.14^+0.10_−0.05, 0.23^+0.07_−0.06, and 0.24^+0.12_−0.08 at 1.5 ⩽ z < 2.0, 2.0 ⩽ z < 2.5, and 2.5 ⩽ z < 3.6, respectively.²⁶ Some fraction of these candidates will not be physical pairs however, but simply projected angular pairs of galaxies with different redshifts and no physical association.

One effort to constrain the incidence rate of false pairs can be made by extrapolating the false pair fraction observed at larger distances for which spectroscopic redshifts can be obtained for individual objects. Considering the 2874 galaxies (across 19 different fields) in our catalog with spectroscopic redshifts in the range 1.5 < z < 3.5, we count the number of distinct angular pairs as a function of separation in comparison to the number of genuine physical pairs whose spectroscopic redshifts lie within Δz = 0.01 of each other. As illustrated by Figure 16, extrapolation of this relation to the radii probed by our WFC3 data suggests that ∼50% of our observed pairs should correspond to genuine physical pairs.

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Alternatively, we can also estimate the false pair fraction based on the statistical distribution of objects in the WFC3 imaging fields. Using our Source Extractor catalogs, we evaluate the number of unique pairs with primary magnitudes in the range H₁₆₀ = 22.0–24.0 and secondary magnitudes within 1 mag of the primary as a function of their separation radius. Assuming that the majority of such pairs in the WFC3 fields are false pairs, we estimate that 7% ± 1% of galaxies have false pairs within r < 16 kpc. Subtracting this 0.07 false pair fraction from the angular pair fraction calculated above, we obtain the true physical pair fractions 0.07^+0.10_−0.05, 0.16^+0.07_−0.06, and 0.17^+0.12_−0.08 at 1.5  ⩽  z < 2.0, 2.0  ⩽  z < 2.5, and 2.5  ⩽  z < 3.6, respectively, for separations in the range 5 kpc <r < 16 kpc. We note that these values are consistent to with observational uncertainty with what would be derived had we simply assumed that 50% of angular pairs were false pairs.

As detailed above, estimates of the merger fraction f_merg derived from all three methods are roughly constant across our three redshift ranges, albeit with mild evidence (at the ∼1–2σ level) for a decline in the merger fraction at z < 2 (see Figure 17, left-hand panel). In order to construct the merger rate from the merger fraction it is necessary to combine the merger fractions with the estimated timescale T for visibility and the comoving space density n(z) of the target sample (see, e.g., Lotz et al. 2008a):

$$\begin{equation} N_{\rm merg} = n(z) \, f_{\rm merg} / T. \end{equation} \tag{ 14 }$$

Estimating the comoving space densities by integrating the mass functions for the star-forming galaxy sample given by Reddy & Steidel (2009) above M_* = 10¹⁰ M_☉, and adopting T_GM20 = 0.24  ±  0.14 Gyr, T_A = 0.76 ± 0.16 Gyr, and T_pair = 0.20  ±  0.38 Gyr (see discussion in Section 6.3), we obtain estimates of the merger rate as shown in Figure 17 (right-hand panel). Clearly the actual merger rate of our galaxies is highly uncertain, and for the small number of galaxies observed in the present sample it is not possible to comment meaningfully on the evolution of the merger fraction with redshift (although our results are consistent with those derived for similar populations of galaxies in other studies; see, e.g., Conselice et al. 2011a, and references therein). Even for significantly larger galaxy samples (e.g., Koekemoer et al. 2011; Grogin et al. 2011) it may prove difficult to constrain the merger rate given the large uncertainty in observability timescales that require numerical simulations to constrain.

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It is not obvious whether it is meaningful from a physical sense to identify galaxies as mergers on the basis of their rest-frame optical morphology. As argued by some authors (e.g., Bournaud et al. 2008; Genzel et al. 2011) irregular morphologies may instead arise from dynamical instabilities within gas-rich systems. Additionally, as we demonstrated in Law et al. (2007b) and expand upon below, merger-like morphologies are poorly correlated with other physical observables.

There are significant differences between the subsamples of M_* > 10¹⁰ M_☉ galaxies from our survey selected as mergers at z = 2.0–2.5 according to different criteria. 43% ± 4% of such galaxies are identified as mergers on the basis of their morphological asymmetry, while 23% ± 3% are identified using the G–M₂₀ selection criterion, and 16⁺⁷₋₆% using pair statistics. As illustrated in Figures 14 and 15, 76% of galaxies selected as mergers according to G–M₂₀ are also selected as mergers using A > 0.30, but only 39% of galaxies selected as mergers using A > 0.30 are also selected as mergers according to G–M₂₀. Similarly, 59% (45%) of mergers identified by G–M₂₀ (asymmetry) are also identified as mergers based on the presence of a nearby angular pair. Clearly, while there is a significant overlap between the galaxy samples, there are also a significant number of galaxies uniquely selected by each technique.

This difference is unsurprising given that the various morphological selection criteria may isolate mergers with different mass ratios and in a different range of evolutionary phases. Lotz et al. (2008b, 2010a, 2010b) performed a series of hydrodynamic simulations to explore the timescales and visibility of disk galaxy mergers as a function of morphological selection criterion, mass ratio, and gas content. Dividing their mergers into six stages (pre-merger, first-pass, maximal separation, final merger, post-merger, remnant) these authors found that the observability of mergers at rest-frame 4686 Å can vary dramatically from stage to stage. For the "G3gf1" model,²⁷ Lotz et al. (2010b) find that G–M₂₀ and pair criteria tend to have observability timescales T_GM20 = 0.24  ±  0.14 Gyr and T_pair = 0.20  ±  0.38 Gyr (predominantly identifying first-passage mergers) while the A > 0.30 criterion has a longer observability timescale T_A = 0.76  ±  0.16 (identifying both first-passage and final mergers; see also Conselice 2006). The greater fraction of galaxies that we identify as mergers based on their asymmetry than by the other two methods (Figure 17) may therefore simply reflect this large difference in observability timescales. Further, Lotz et al. (2010a) find that while A is most sensitive to major mergers like those identified using our pair selection criteria (∼3: 1–1: 1H₁₆₀ flux ratio), G–M₂₀ detects both major and minor mergers, potentially explaining why we identify more mergers using G–M₂₀ than with pair selection.

In Figure 18 we plot histograms of various physical properties for galaxies classified as mergers/non-mergers according to the G–M₂₀, A, and pair criteria and use a K-S test to evaluate the significance of the null hypothesis that both sets of galaxies (mergers and non-mergers) were drawn from the same distribution. We conclude that for almost all physical parameters (stellar mass, SFR, rest-frame U − B color,²⁸ etc.) there is no significant difference (confidence in the null hypothesis >5%) between putative mergers and non-mergers. Similarly, there is no obvious difference in the gas-phase kinematics between mergers and non-mergers, although our sample size of 35 galaxies with systemic Hα redshifts and high-quality UV spectra is too small to conclusively rule out association. The one notable exception is that galaxies identified as mergers via the G–M₂₀ or pair classification schemes have significantly smaller radii and correspondingly higher Σ_SFR than non-mergers. This may suggest either that Σ_SFR peaks around the first-passage during a major merger event, or that the G–M₂₀ and pair classification schemes are simply effective at finding galaxies with small radii.

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The lack of correlation observed between morphology and these physical observables may be unsurprising in light of both numerical uncertainties in our morphologies (i.e., exactly where the dividing line between mergers and non-mergers lies) and expectations (e.g., Lotz et al. 2010a, 2010b) that star formation may typically peak after the major morphological disturbances have subsided. Regardless, it is unclear whether it is physically meaningful to classify z ∼ 2–3 galaxies as mergers on the basis of morphology alone given that there appears to be little to distinguish these systems (whether observed in the rest-optical or the rest-UV; see discussion by Law et al. 2007b, see also Swinbank et al. 2010 for a similar discussion of submillimeter galaxies) from their non-merging counterparts. Rather, it may simply be that most z ∼ 2–3 star-forming galaxies are dynamically unstable systems driven by the accretion of large quantities of gas, whether this gas is acquired through mergers, cold-mode, or hot-mode accretion processes.

Three methods have been generally adopted in the literature for defining robust segmentation maps.

• 1.  
  Law et al. (2007b) and Peter et al. (2007) used a scaled surface brightness method to select galaxy pixels whose flux is at least nσ, where σ is the standard deviation of the sky pixels and n scales with source redshift as n = 3(1 + z/1 + z_max)⁻³, where z_max = 3.0.²⁹ This method is independent of galaxy morphology and compensates for cosmological surface brightness dimming (which scales as (1 + z)⁻³ for a fixed observational bandpass), therefore giving consistent results across a given redshift interval. However, it is explicitly tied to the noise characteristics of the observational data, and can yield morphological parameters that vary systematically with total flux for galaxies with identical morphological profiles but different total luminosities (see, e.g., Figure 9 of Law et al. 2007b; Lotz et al. 2008b).
• 2.  
  Many authors (e.g., Conselice et al. 2000, 2008; Lotz et al. 2004, 2006) start with simple Source Extractor segmentation maps to either pre-select galaxy pixels or mask foreground/background objects, and apply either a circular or elliptical Petrosian (1976) selection technique to select pixels independent of total galaxy flux or background noise characteristics. Conselice et al. (2000, 2008) include in their segmentation map all pixels within 1.5 Petrosian radii (r_P, i.e., the radius at which the surface brightness is some fraction η of the enclosed surface brightness), while Lotz et al. (2004, 2006) include only pixels with flux greater than the surface brightness at the Petrosian radius (but following the potentially irregular isophotal contours). While robust to total source magnitude, cosmological dimming, and observational noise characteristics, these method can sometimes yield suboptimal results when applied to the often-irregular morphologies of z ∼ 2 galaxies (e.g., Figure 2) because of their ill-defined Petrosian radii.
• 3.  
  Abraham et al. (2007) generalized the Petrosian pixel selection method to work equally well for galaxies of arbitrary shapes whose flux components are not necessarily contiguous. As outlined by Abraham et al. (2007), all pixels in the preliminary segmentation map calculated using Source Extractor are sorted in decreasing order of flux into the array f_i, which is then used to construct the cumulative flux array F_i = ∑ⁱ_{j = 1}f_j. The quasi-Petrosian isophote is set by determining the pixel index i at which f_i = η(F_i/i) where F_i/i is the cumulative mean surface brightness. This quasi-Petrosian segmentation map preserves the advantages of Petrosian-based methods (i.e., robustness to source magnitude, cosmological dimming, and observational noise) while being applicable to arbitrary morphology.

We adopt the quasi-Petrosian method of Abraham et al. (2007) as our baseline segmentation method. As described by these authors, the failure mode of this approach is graceful in that, if the isophotal Petrosian threshold η is below the surface brightness threshold of the initial Source Extractor segmentation map, it simply defaults to the initial map (Section 2.3). It is not desirable for this to occur frequently however, since it eliminates the advantages of the Petrosian pixel selection. The isophotal threshold η must therefore be set sufficiently high that it is more restrictive than a simple 1.5σ surface brightness cut, but sufficiently low that it rejects as little information (i.e., pixels) as possible from the final segmentation maps.

In Figure 19, we plot the critical surface brightness threshold η_crit at which the pseudo-Petrosian algorithm produces a segmentation map that is more restrictive than the initial Source Extractor 1.5σ surface brightness segmentation map for each of our 306 galaxies. Intuitively, there is a strong correlation with mean apparent surface brightness μ_H (defined as the H₁₆₀ magnitude divided by the Source Extractor segmentation map area). We note that for the lowest mean surface brightness objects (disproportionately galaxies of Type III and/or at redshifts z > 2.5) η_crit is relatively high; that is, the galaxy surface brightness decreases only slightly to ∼40% of its mean value before reaching the 1.5σ sky background. In contrast, for higher surface brightness objects the dynamic range of the galaxy is greater and can decrease to ∼20% or less of its mean value before reaching the sky background. Given these results, a traditional choice of η = 0.2 would result in an unsatisfactorily high ∼60% of our galaxies defaulting to a simple 1.5σ isophotal pixel selection. We therefore take η = 0.3 instead, for which only ∼23% of galaxies default to the surface-brightness-limited segmentation map. This fraction decreases to ∼15% when we reject from consideration galaxies with H₁₆₀ > 24.0, for which we find that quantitative morphological statistics are not robust regardless of segmentation map (see Appendix A.2).

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In Figures 20 and 21 we compare morphological statistics calculated using the following five segmentation maps.

• 1.  
  ("QP3"). Quasi-Petrosian segmentation map with threshold η = 0.3, this is the default segmentation map. We denote statistics calculated using this map with subscripts of the form, e.g., G_QP3.
• 2.  
  ("QP2"). Quasi-Petrosian segmentation map with threshold η = 0.2. We denote statistics calculated using this map with subscripts of the form, e.g., G_QP2.
• 3.  
  ("CPL"). Elliptical Petrosian segmentation map with threshold η = 0.2 that includes all pixels with flux greater than the surface brightness at the Petrosian radius (Lotz et al. 2004, 2006) but following the potentially irregular isophotal contours. We denote statistics calculated using this map with subscripts of the form, e.g., G_CPL.
• 4.  
  ("CPC"). Circular Petrosian segmentation map with threshold η = 0.2 that includes all pixels within 1.5 Petrosian radii irrespective of flux (Conselice et al. 2000, 2008). We denote statistics calculated using this map with subscripts of the form, e.g., G_CPC.
• 5.  
  ("SB"). Scaled isophotal (surface brightness) segmentation map of the form adopted by Law et al. (2007b) and Peter et al. (2007). We denote statistics calculated using this map with subscripts of the form, e.g., G_SB.

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There is generally good correlation among the non-parametric statistics derived using each of these segmentation maps, especially when restricting our attention to the higher surface brightness systems for which the η = 0.2 threshold is well defined (blue, black, and red points). In order to aid comparison between morphological properties derived by different groups in the literature, we present below a series of transformations that relate values calculated using different segmentation maps. In determining these relations we consider only those galaxies for which the η = 0.2 threshold is well defined and the morphologies robust to statistical uncertainties (i.e., we require H₁₆₀ < 24.0 and η_crit ⩽ 0.2), and perform a linear least-squares fit with uniform uncertainties in both quantities.

The greatest variation occurs in the Gini parameter G, which is extremely sensitive to the pixels included in the segmentation map (see also a previous analysis by Lisker 2008). As illustrated by Figure 20 (bottom left-hand panel), simply adopting a Petrosian threshold of η = 0.2 significantly increases G over the η = 0.3 case by including more low-flux pixels in the segmentation map (note that this effect is not as noticeable for the green-colored points, for which the dynamic range of the galaxy surface brightness did not permit the Petrosian algorithm to reach the 20% flux threshold, and defaulted instead to the initial Source Extractor 1.5σ isophotal segmentation map). Similarly, the CPL map also results in systematically higher values of G than calculated by our default η = 0.3 quasi-Petrosian algorithm. This effect is even more noticeable in the CPC segmentation map; this map increases the mean value of G significantly by including many more low-surface-brightness pixels than the other segmentation maps, and also compresses the dynamic range of G among the galaxy sample. In contrast, the scaled surface brightness selection technique (SB) stretches the dynamic range of G, but corresponds poorly to estimates obtained using other segmentation maps. We find that the key transformations between these segmentation maps are given on average by

$$\begin{equation} G_{\rm CPL} = 1.78 G_{\rm QP3} - 0.19 \end{equation} \tag{ A1 }$$

$$\begin{equation} G_{\rm CPL} = 2.02 G_{\rm CPC} -0.87. \end{equation} \tag{ A2 }$$

The second-order moment of the light distribution (M₂₀) is tightly correlated among all five segmentation maps, although values calculated using the CPC segmentation map lie at systematically lower values due to the inclusion of additional low-surface-brightness pixels. The key transformations are described by

$$\begin{equation} M_{\rm 20, CPL} = 1.04 M_{\rm 20, QP3} -0.03 \end{equation} \tag{ A3 }$$

$$\begin{equation} M_{\rm 20, CPL} = 0.96 M_{\rm 20, CPC} + 0.06. \end{equation} \tag{ A4 }$$

The concentration parameter C exhibits minimal systematic offsets between segmentation maps, but considerably more scatter than the other morphological statistics. As discussed in Appendix A.3 this is primarily due to the poor sampling of the inner 20% of the light profile in the z ∼ 2–3 galaxies, which are small and poorly resolved in comparison to nearby galaxy samples (see discussion by Bershady et al. 2000). The key transformations are described by

$$\begin{equation} C_{\rm CPL} = 0.84 C_{\rm QP3} + 0.47 \end{equation} \tag{ A5 }$$

$$\begin{equation} C_{\rm CPL} = 1.11 C_{\rm CPC} - 0.51. \end{equation} \tag{ A6 }$$

The asymmetry parameter A is also well correlated between different segmentation maps, with relatively little scatter and no significant systemic shifts among four of the five segmentation maps. There is more scatter and a systematic offset however when comparing estimates to the CPC segmentation map, with A_CPC systematically lower compared to the other four segmentation maps. We find that the mean relations are governed by the equations

$$\begin{equation} A_{\rm CPL} = 0.80 A_{\rm QP3} + 0.03 \end{equation} \tag{ A7 }$$

$$\begin{equation} A_{\rm CPL} = 1.15 A_{\rm CPC} + 0.03. \end{equation} \tag{ A8 }$$

The tightest correlation is found for Ψ, for which all segmentation maps produce nearly identical values with only minimal scatter about the 1–1 relation.

We perform Monte Carlo tests to quantify the mean and standard deviation of the morphological statistics for sources of fixed structure with different total magnitudes. We choose five sources (four galaxies and reference star; see top row of Figure 22) that are roughly representative of the range of morphologies found within our galaxy sample, and construct morphological models of them using GALFIT. We scale the total flux of these models to H₁₆₀ = 22.0 AB, and insert ten copies of each into randomly selected blank-field regions of the WFC3 images in order to obtain multiple realizations of the background noise. For each copy, we compute the segmentation maps and morphological parameters as described above in Section 3. This exercise is repeated every 0.5 mag in the range H₁₆₀ = 22–25 AB.

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Figure 22 suggests that while the morphological parameters are relatively robust for H₁₆₀ ⩽ 24 AB, at H₁₆₀ > 24 AB most start to break down either in the sense that their mean values deviate significantly from the mean values derived at brighter magnitudes, or the variance among different realizations of the background noise becomes large relative to the mean. This behavior for each statistic can be summarized as follows.

• 1.  
  M₂₀, C, and A exhibit ∼1%/2%/15% uncertainty (averaged over the four star-forming galaxy models) at the bright end of the sample (H₁₆₀ ∼ 22 AB), increasing to 13%/15%/45% at the faint end (H₁₆₀ ∼ 25 AB). Mean uncertainty at the average magnitude of the H₁₆₀ ⩽ 24 AB sample is 4%/11%/22%, respectively; in the case of C this is sufficient to confuse the relative ordering between galaxies of different morphologies. There are no systematic variations with magnitude.
• 2.  
  G has a bright-end uncertainty ∼2%, a faint-end uncertainty ∼13%, and a mean uncertainty at the average magnitude of the H₁₆₀ ⩽ 24 AB sample of 3%. While the mean value of G remains relatively constant down to H₁₆₀ ∼ 24 AB, it decreases systematically at lower magnitudes. This systematic decline is because the η = 30% surface brightness threshold for the quasi-Petrosian pixel decreases below the 1.5σ Source Extractor threshold, resulting in effective loss of the lowest-flux pixels from the segmentation map.
• 3.  
  Ψ has a bright-end uncertainty ∼7%, a faint-end uncertainty ∼41%, and a mean uncertainty at the average magnitude of the H₁₆₀ > 24 AB sample of 21%. Ψ systematically increases for H₁₆₀ > 23 AB, most noticeably for H₁₆₀ > 24 AB.
• 4.  
  The Sérsic index n and effective circularized radius r_e have mean bright-end uncertainties of ∼5%/1%, faint-end uncertainties of ∼54%/11%, and mean uncertainties at the average magnitude of the H₁₆₀ ⩽ 24 AB sample of 15%/2%, respectively. Both r_e and n decline systematically with magnitude as it becomes progressively more difficult to distinguish faint outer regions of the galaxies from the background sky (see also Gray et al. 2009). These effects are particularly pronounced for H₁₆₀ > 24 AB. The significance of the decline in r_e varies as a function of morphology; regular symmetric objects show negligible variation across the full range H₁₆₀ = 22–25 AB, while more irregular multi-component galaxies may decline by as much as 30%.

In the interests of measuring physically meaningful morphological statistics, we therefore impose an apparent magnitude cut on our sample of galaxies at H₁₆₀ ⩽ 24 AB, corresponding closely to a signal-to-noise ratio cut S/N > 110 (see Figure 23). This is consistent with the analyses of Conselice et al. (2000) and Lisker (2008), who found, respectively, that A became dominated by the background noise and that G becomes less robust below S/N ∼ 100.

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It is also worth investigating the dependence of the calculated morphological statistics on the pixel scale that we selected to drizzle our WFC3 data onto. We therefore inserted the GALFIT galaxy models (normalized to H_AB ∼ 22) from Appendix A.2 into random blank regions of the WFC3 fields, and rebinned the data using linear interpolation (using the IDL routine CONGRID, with conservation of total flux) to a variety of pixel scales that we could realistically have chosen. As illustrated by Figure 24, the recovered morphologies are extremely robust to variations ∼ a factor of two in pixel scale, with the exception of the stellar source model (for A, C, M₂₀, and G) and the concentration parameter C (for the stellar source, Q1009-BX146, and Q1700-BX691). That is, all of the morphological parameters can vary unsatisfactorily with choice of sampling scale for unresolved objects, or (in the case of C) for objects in which the innermost region containing 20% of the light is poorly sampled. Indeed, we note that while C is not robust for Q1009-BX146 and Q1700-BX691, it is more so for Q1700-BX759 and Q1217-MD20 because these two galaxies are significantly more spatially extended and the central region containing 20% of the total light correspondingly better sampled. This resolution dependence of C for the poorly sampled inner radius r₂₀ is well known in the literature (see, e.g., Figure 9 of Bershady et al. 2000).

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Given the general robustness of the morphological parameters to the choice of angular sampling scale, we do not make any corrections to the measured morphologies due to the small (<10%) change in angular size subtended by a physical kpc across the redshift interval 1.5 < z < 3.6, but we choose not to use the concentration parameter C in our analyses.

We also explore the robustness of the morphological statistics to the width of the observational PSF, which affects the degree to which spatial structures are resolved. In order to reliably trace structures resolved on scales smaller than the WFC3/IR PSF we repeat the analysis from Appendix A.2, but using GALFIT models of five sources in the Q1700+64 field observed with HST/ACS F814W as part of program GO-10581 (PI: A. E. Shapley; see description in Peter et al. 2007). These models are convolved with 2D Gaussian profiles (using the IDL routine FILTER_IMAGE) to mimic observations with PSF FWHM ranging from 0.1 (i.e., native resolution for the F814W imaging data) to 0.3 arcsec.

As illustrated in Figure 25, there is little change in the statistical uncertainty of the morphological measurements with PSF FWHM, but most exhibit systematic variations (as noted previously by, e.g., Lotz et al. 2004, 2008b). As discussed in Appendix A.3, C is poorly behaved since the inner 20% of the light profile is poorly sampled. As PSF FWHM increases, the dynamic range of A, M₂₀, G, and Ψ decreases, approaching the limit that when the PSF is large compared to the size of the galaxies all objects will be unresolved and have indistinguishable morphologies. The compression of the dynamical range is less pronounced for objects such as ACS/BX1031, which have two well-separated components that require a more substantial change in the PSF to lose information about the double morphological structure. Similarly, the parametric statistics r_e and n are generally quite stable to variations in the PSF since GALFIT incorporates the observational PSF in its fitting algorithm, although n declines by a few percent from FWHM ∼0.1 arcsec to ∼0.3 arcsec. This effect was previously noted by Buitrago et al. (2008) who found that Sérsic indices measured in the infrared with NICMOS were 13% ± 12% smaller than measured in the optical with ACS.

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Since all of our galaxies have been observed with uniform coverage and a PSF that varies by less than ∼4%, the trends illustrated in Figure 25 will be unimportant for internal comparisons between the morphologies of galaxies in our sample. These trends will be important, however, to keep in mind when comparing any of our galaxies to low-redshift samples, or to similar z ∼ 2 galaxies observed with HST in bandpasses tracing the rest-frame UV.