3.1. Net Ly $\alpha$ EW distribution and spectral type classification

The profile of Ly $\alpha$ in the spectrum of high-redshift LBGs manifests in absorption, emission, or a combination of both. In the $z\sim3$ LBG sample of Shapley et al. (Reference Shapley, Steidel, Pettini and Adelberger2003, hereafter S03), for example, the distribution of net Ly $\alpha$ EWs is centred near zero and varies from $\lesssim$ $-$ 50 Å to $\gtrsim$ $+$ 200 Å. Net Ly $\alpha$ EW values for our 557 $z\sim2$ LBGs span a similar range ( $-$ 85.0 Å to $+$ 108.7 Å) and, like the S03 sample, are asymmetrically dispersed toward higher net Ly $\alpha$ EWs around a median near zero ( $-$ 4.42 Å at $z\sim2$ and $+$ 0.56 Å at $z\sim3$ ). These similarities, however, belie a change in the shape of the distribution that is evidenced by a shift in the mean net Ly $\alpha$ EW for the respective full samples from $+$ 10.3 Å at $z\sim3$ to $-$ 2.2 Å at $z\sim2$ .

Figure 1. Normalised histograms showing the distribution of net Ly $\alpha$ EWs for $z\sim3$ (green) and $z\sim2$ (gold) LBG samples. Inset: The same distributions plotted with a logarithmic ordinate axis to accentuate the ’tails’ of the Ly $\alpha$ EW distributions. Net Ly $\alpha$ EWs less than zero are essentially identical between the two populations, while the $z\sim3$ sample has a significantly larger fraction of net Ly $\alpha$ -emitters (see Table 1).

The changing shape of the net Ly $\alpha$ EW distribution with redshift is readily apparent in Fig. 1 that shows normalised histogram plots for the $z\sim2$ and $z\sim3$ samples. Consistent with the result of Reddy et al. (Reference Reddy2008), we find that the two distributions are very similar at net Ly $\alpha$ EWs $\lesssim$ 0 Å, but there is a sharp drop off at $z\sim2$ toward higher values of net Ly $\alpha$ EW that is largely responsible for the difference in overall mean net Ly $\alpha$ EW between the two samples.

Table 1. Statistics for sub-samples of $z\sim2$ and $z\sim3$ LBGs divided on the basis of net Ly $\alpha$ EW.

^a $z\sim3$ LBGs from Shapley et al. (Reference Shapley, Steidel, Pettini and Adelberger2003).

^bOur $z\sim2$ sample divided into numerical quartiles.

^cOur $z\sim2$ sample divided into aLBG, $\rm{G_a}$ , $\rm{G_e}$ and eLBG Ly $\alpha$ spectral types as per the definitions given in Section 3.1.

^dNumber of galaxies.

^eFraction of full sample ( $N_{tot}$ ) in sub-sample (N).

^fMean net Ly $\alpha$ EW for each (sub)sample.

The utility of dividing a population of rest-frame UV-colour selected galaxies into sub-samples based on observed net Ly $\alpha$ EW was first demonstrated by S03, and it continues to be a useful approach in the study of relationships between Ly $\alpha$ and the physical and spectral properties of LBGs (e.g., Du et al. Reference Du2018; Pahl et al. Reference Pahl2020). In their discovery of the broadband photometric segregation versus Ly $\alpha$ EW in the S03 sample, C09 exploited the same approach to derive the method of photometric Ly $\alpha$ spectral-type classification (see Section 3.2).

Table 1 shows a comparison of the statistics for our $z\sim2$ and $z\sim3$ LBG samples divided into numerical quartiles on the basis of net Ly $\alpha$ EW. There is a shift toward more negative mean net Ly $\alpha$ EW ( $\Delta \sim -5$ Å) of the most absorbing quartile at $z\sim2$ compared to the same quartile at $z\sim3$ . The two $z\sim2$ quartiles that span the more Ly $\alpha$ -emitting end of the distribution show a larger (and increasing) shift to lower average net Ly $\alpha$ EW compared to the analogous quartiles of S03 ( $\Delta -9.0$ Å and $\Delta -32.5$ Å for q3 and q4, respectively).

Motivated by these observations, and our results showing a relationship between net Ly $\alpha$ EW and nebular emission-line kinematics (Foran et al. 2023b, submitted), we applied to our $z\sim2$ sample the same net Ly $\alpha$ EW cuts used by S03 to generate numerical quartiles at $z\sim3$ . We define the most absorbing fraction of galaxies with net Ly $\alpha$ EW $\leq$ $-$ 10.0 Å as ‘aLBGs’, and the most strongly emitting fraction with net Ly $\alpha$ EW $\geq$ $+$ 20.0 Å as ‘eLBGs’. We further divide the remaining LBGs into $\rm{G_a}$ and $\rm{G_e}$ spectral types with net Ly $\alpha$ EWs $-$ 10.0 Å $<$ net Ly $\alpha$ EW $<$ 0.0 Å and 0.0 Å $<$ net Ly $\alpha$ EW $<$ $+$ 20.0 Å, respectively. Table 1 summarises the population statistics of the $z\sim2$ sample and our Ly $\alpha$ spectral types compared to the $z\sim3$ LBGs of S03.

Given that we have defined our spectral types using the same net Ly $\alpha$ EW cuts as S03, it is not surprising that the central ( $\rm{G_a}$ and $\rm{G_e}$ ) spectral types have mean net Ly $\alpha$ EWs similar ( $\Delta \sim -1$ Å) to the equivalent quartiles (q2 and q3) in the $z\sim3$ sample. It is noteworthy, however, that despite the overall shift in mean net Ly $\alpha$ EW of $\sim -12.5$ Å between the $z\sim3$ and $z\sim2$ samples, the mean net Ly $\alpha$ EW of the $z\sim2$ aLBGs is similarly only $\sim -2$ Å more negative than the equivalent S03 quartile, indicative of a compression of the LBG population toward the Ly $\alpha$ -absorbing end of the distribution. This behaviour is likely due to the fact that the measured net Ly $\alpha$ EW becomes insensitive to the total absorption once the Ly $\alpha$ absorption feature is saturated. That is, beyond that point, any further decrease in measured EW would reflect only the contribution of the damping wings, and depend weakly on increasing HI column density (see Section 3.2 for manifestation of this effect in the broadband imaging data).

Figure 2. Illustration of the origin of the colour separation of Lyman break galaxy (LBG) Ly $\alpha$ spectral types. Left: Plotted are the G and $\mathcal{R}$ filter transmission curves (Steidel et al. Reference Steidel2003) in green and orange, respectively, shifted to the $z\sim3$ rest-frame. Overlaid are the (smoothed) quartile 1 (red, representative of aLBGs) and quartile 4 (blue, representative of eLBGs) composite spectra of Shapley et al. (Reference Shapley, Steidel, Pettini and Adelberger2003). The composite spectra consist of $\sim$ 200 $z\sim3$ LBG spectra with similar Ly $\alpha$ EW, with the mean values indicated in the legend. The spectra are shown normalised over the G filter to help illustrate the ( $G - \mathcal{R}$ ) colour difference in the two spectral types for a given G magnitude. The origin of the Ly $\alpha$ spectral type photometric segregation on the $(G-\mathcal{R})$ vs $\mathcal{R}$ CMD results from their colour differences based on the UV continuum slope relationship with spectral type and a small (and inverse) contribution from the Ly $\alpha$ emission/absorption feature and the magnitude differences in spectral type, in that $z\sim3$ aLBGs are brighter on average than eLBGs. Right: Similar to the left plot, but for LBGs at z $\sim$ 2. The composite spectra are shown normalised over the $U_n$ filter (violet, see text). Note: the composite spectra and the normalisation are shown for illustrative purposes and extend to 2000Å, rest-frame. However, the UV continuum slopes of quartiles 1 (red) and 4 (blue) maintain a significant difference in $\mathcal{R}$ that is sufficient to separate aLBG and eLBG spectral types in ( $U_n - \mathcal{R}$ ) colour and $\mathcal{R}$ magnitude on the CMD. Depending on the redshift of z $\sim$ 2 LBGs, the Ly $\alpha$ feature may fall in or out of the $U_n$ filter (see Section 3.4).

Conversely, the mean net Ly $\alpha$ EW of the eLBG spectral type sub-sample at $z\sim2$ is $\sim -14$ Å more negative than the analogous (most strongly Ly $\alpha$ -emitting) S03 quartile, and the relative fraction of eLBGs at $z\sim2$ is 0.08 compared to 0.25 at $z\sim3$ – a clear reflection of the lower relative abundance of net Ly $\alpha$ emitting LBGs in the universe and/or within the LBG selection function at $z\sim2$ compared to $z\sim3$ .

3.2. Segregation of $z\sim$ 2 LBGs in colour-magnitude space

C09 discovered that over the redshift path $z \sim 3.0 \pm 0.3$ , the relationship between rest-frame UV continuum slope and net Ly $\alpha$ EW leads to a photometric dispersion of LBGs, and an ability to separate LBG spectral types on a broadband colour-magnitude plane based on their net Ly $\alpha$ EW. At $z\sim3$ , aLBGs are (on average) brighter in $\mathcal{R}$ magnitude and redder than eLBGs. They separate in $(G-\mathcal{R})$ colour as a result of the redder UV continuum slopes of aLBGs, combined with an additional small red enhancement as a result of the Ly $\alpha$ absorption in the G-band, as compared to eLBGs with bluer UV continuum slopes, combined with an additional blue enhancement from the Ly $\alpha$ emission in the G-band. Together, these behaviours enable a statistical segregation of the two populations on a $(G-\mathcal{R})$ vs $\mathcal{R}$ colour-magnitude diagram (CMD), with subsets containing pure samples of each spectral type. Fig. 2 illustrates the origin of the $z\sim3$ broadband imaging segregation of Ly $\alpha$ -absorbing and Ly $\alpha$ -emitting LBGs that enables the determination of photometric Ly $\alpha$ spectral types.

To test whether a similar relationship between broadband photometry and net Ly $\alpha$ EW might exist at $z\sim2$ , we use $(U_n-\mathcal{R})$ colours and $\mathcal{R}$ -band magnitudes to construct a CMD for our sample of 557 $z \sim2$ LBGs. We use $(U_n-\mathcal{R})$ rather than $(U_n-G)$ , to sample the rest-frame UV continuum farther redward of the Ly $\alpha$ feature so as to increase the segregation between the redder-sloped aLBGs and the bluer-sloped eLBGs, and to avoid any possibility of contamination of our redward filter by Ly $\alpha$ emission. The separation in wavelengths probed by the $U_n$ and G filters at $z\sim2$ is smaller than the separation of the G and $\mathcal{R}$ filters at $z\sim3$ (see Fig. 2).

The left panel of Fig. 3 shows the spectroscopic $z\sim2$ LBG sample dispersed in colour $(U_n-\mathcal{R})$ and magnitude $\mathcal{R}$ , with symbols colour-coded on a red-to-blue gradient according to their measured net Ly $\alpha$ EW. For visualisation purposes the colour table is scaled to map the range $-$ 35.0 Å $<$ net Ly $\alpha$ EW $<$ $+$ 40.0 Å which encompasses $\gtrsim$ 95% of the galaxies in our sample. Plotting the data in this way (i.e., downplaying the colour effect of the few extreme net Ly $\alpha$ EW cases), the bulk of the LBG sample manifests on the CMD as a visible colour gradient from red to blue moving diagonally from roughly the top left to bottom right. The galaxies in our sample with the most negative net Ly $\alpha$ EW (dark red symbols) do not lie at the extreme end of the colour gradient direction as might be expected for a simple monotonic relationship. While they are certainly well within the ‘absorbing’ half of the CMD, they lie toward the centre of the distribution, and approximately along a line orthogonal to the underlying trend. This apparently anomalous behaviour of the most absorbing galaxies in our sample notwithstanding, the overall trend is confirmed by the points labelled s1 to s6 on the colour gradient plot, that indicate the positions of the magnitude and colour distribution means for the numerical sextiles (of $\sim$ 93 galaxies each) grouped according to their net Ly $\alpha$ EW (see Table 2 for a summary of the sextile statistics). The more positive net Ly $\alpha$ EW LBGs (weaker absorption and more emission) show an overall trend toward fainter $\mathcal{R}$ -band magnitudes and bluer ( $U_n-\mathcal{R}$ ) colours. Indeed, only the most absorbing sextile (s1 in Fig. 3) does not follow this monotonic trend. That being said, the colours and magnitudes of the $z\sim2$ sextiles converge with increasing Ly $\alpha$ absorption strength (i.e., from s6 to s1), unlike at $z\sim3$ , where the mean colours (magnitudes) continue to redden (brighten) monotonically (cf. C09).

Table 2. Statistics for the dispersion of $z\sim2$ LBGs in colour ( $U_n-\mathcal{R}$ )—magnitude ( $\mathcal{R}$ ) space divided into numerical sextiles based on net Ly $\alpha$ EW.

^aFull $z\sim2$ spectroscopic LBG sample (557 galaxies) divided into sextiles of $\sim$ 93 galaxies each.

We can speculate that the ‘off trend’ positions of the strongest Ly $\alpha$ absorbers on the CMD is a manifestation of the environmental effect proposed by C13 over their intrinsic net Ly $\alpha$ EWs. In this scenario, the most negative net Ly $\alpha$ EWs observed, with otherwise typical aLBG colour and magnitude, may be a result of their environment near the cores of groups and proto-clusters and the presence of larger column densities of intra-group/cluster neutral gas at or near the systemic velocity of the LBGs along the line of sight (e.g., Muldrew et al. Reference Muldrew, Hatch and Cooke2015; Toshikawa et al. Reference Toshikawa2016; Lemaux et al. Reference Lemaux2018). More prosaically, it is also plausible that this behaviour, and the grouping of the three most absorbing sextiles (s1–s3) on the CMD, is a reflection of the compression of the $z\sim2$ sample towards the more negative end of the net Ly $\alpha$ EW distribution (as described in Section 3.1), combined with the inherently greater uncertainties (25–50%) associated with the net Ly $\alpha$ EW measurements for the most absorbing systems (S03) and the inherent photometric scatter (see Appendix 1).

Figure 3. Rest-frame UV colour $(U_n-\mathcal{R})$ –magnitude ( $\mathcal{R}$ ) diagrams (CMDs) for Lyman break galaxies (LBGs) in the redshift range $1.7<z<2.5$ , and with magnitude cuts of $\mathcal{R}$ $<$ 25.5 and $U_n$ $<$ 26.5. Left: $z\sim2$ LBGs dispersed in colour-magnitude space with symbols colour-coded on a red-blue gradient according to their measured net Ly $\alpha$ EW. The colour table maps the range $-$ 35.0 Å $<$ net Ly $\alpha$ EW $<$ $+$ 40.0 Å, which encompasses $\gtrsim$ 95% of the sample. Points labelled s1 to s6 indicate the colour and magnitude distribution means of the numerical sextiles of the LBG sample divided on the basis of net Ly $\alpha$ EW. Right: Grey plus (+) marks denote the 557 galaxies in the $z\sim2$ spectroscopic sample. Galaxies with net Ly $\alpha$ EW $\leq -10.0$ Å (aLBGs) are overlaid with red squares, and those with net Ly $\alpha$ EW $\geq +20.0$ Å (eLBGs) are overlaid with blue triangles. The mean value for each distribution is marked with a black cross (X), with aLBG mean indicated by the upper cross and eLBG mean by the lower. The dotted-dashed blue and dashed red lines indicate a 1.5 $\sigma$ dispersion in colour from the primary cut (green line) that divides the aLBG and eLBG distributions, respectively (see text).

The association between net Ly $\alpha$ EW and the photometric properties of $z\sim2$ LBGs suggested by the trend shown in the left panel of Fig. 3, prompts a statistical examination using the Ly $\alpha$ spectral type classification scheme described in Section 3.1 and the method demonstrated by C09. In the right panel of Fig. 3 we plot on the CMD the LBG spectral types as described in Section 3.1, i.e., aLBGs with net Ly $\alpha$ EW $\le-10$ Å and eLBGs with net Ly $\alpha$ EW $\ge +20$ Å, and show that they segregate into two cohesive, albeit overlapping, distributions.

We define a primary cut (solid green line) that passes through the midpoint between the mean colour and magnitude values of the aLBG and eLBG distributions (black crosses), and has slope that maximises the difference in mean net Ly $\alpha$ EW and spectral-type purity between the sub-samples that lie above and below the broken blue and red lines, respectively. These dashed (red) and dotted-dashed (blue) lines indicate an offset of 1.5 $\sigma$ in colour dispersion from the primary cut for the aLBG and eLBG distributions, respectively (see Section 3.3.1). Statistics for the segregation of the aLBG and eLBG spectral types shown in the right panel of Fig. 3 are summarised in Table 3 together with (for comparison) the segregation statistics for the $z\sim3$ sample of C09.

Table 3. Statistics for the photometric segregation of Ly $\alpha$ -absorbing and Ly $\alpha$ -emitting spectral types in $z\sim$ 2 and $z\sim$ 3 LBGs.

Although the slope and intercept for the $z\sim2$ segregation quoted in Table 3 are the values that give the maximum difference in mean net Ly $\alpha$ EW between the photometrically selected sub-samples, the maximum is shallow, asymmetric, and relatively insensitive to the choice of slope. For example, in the optimal case where $c_{\sigma} = 1.25$ (see Section 3.3.1), the maximum difference in mean net Ly $\alpha$ EW is 17.1 Å at a slope of 0.31. We note, however, that the difference in mean net Ly $\alpha$ EW is greater than 16.0 Å for all slopes between 0.17 and 0.40. Thus we might quote an uncertainty (or ‘range of confidence’) of slope = $0.31^{+0.09}_{-0.13}$ . within which any choice of slope would result in photometrically selected sub-samples with a difference in mean net Ly $\alpha$ EW that is within $\sim$ 5% of the maximum. Constraining the primary cut to pass through the mid-point of the aLBG and eLBG distribution means similarly gives intercept values in the range $-6.62^{+3.43}_{-2.30}$ .

3.3. Photometric Ly $\alpha$ spectral type selection and sub-sample purity

3.3.1. $z\sim2$ LBGs

Following the method used by C09 at $z\sim3$ , we use the parameters of the segregated aLBG and eLBG distributions shown in Fig. 3 to isolate sub-samples with Ly $\alpha$ dominant in absorption (‘photometric’ aLBGs, or p-aLBGs) and with Ly $\alpha$ dominant in emission (‘photometric’ eLBGs, or p-eLBGs) from the parent $z\sim2$ LBG sample. Invoking the primary cut slope and intercept values from Table 3, and the supplied broadband photometry, the following relationships can be used to extract sub-samples of the desired photometric Ly $\alpha$ spectral type.

For p-aLBGs,

(1) \begin{equation}\mbox{($U_n - \mathcal{R}$)} \, \ge \,\mbox{0.3091} \cdot \mathcal{R} - \mbox{6.6208} \, \mbox{+ c$_\sigma$} \cdot\mbox{$\sigma_e$}\end{equation}

and for p-eLBGs,

(2) \begin{equation}\mbox{($U_n - \mathcal{R}$)} \, \le \,\mbox{0.3091} \cdot \mathcal{R} - \mbox{6.6208} \,\mbox{- c$_\sigma$} \cdot \mbox{$\sigma_a$}\end{equation}

where $c_{\sigma}$ is the coefficient of colour standard deviation by which boundaries used to isolate the photometric spectral type sub-samples are offset from the primary cut on the CMD, and ${\sigma}_{a}$ (0.3509) and ${\sigma}_{e}$ (0.3558) are the 1 $\sigma$ standard deviations of the $(U_n-\mathcal{R})$ colour distributions for the aLBG and eLBG subsets, respectively.

We use ${\sigma}_a$ and ${\sigma}_e$ to estimate the density of aLBGs and eLBGs on the CMD. This approach implies that any $\sigma$ should extend around a distribution mean in some circular (or similar) contour. The primary cut we make between the aLBG and eLBG distribution means (and its use as the basis for estimating photometric spectral type purity) is a line for which our assumptions only formally apply at the point of closest approach (tangent to a circular contour) of our lines to the respective distribution means. Thus, the multiples of $c_{\sigma}$ (1.5 in Fig. 3) applied to ${\sigma}_a$ and ${\sigma}_e$ , plus the fraction of $\sigma$ by which the primary cut is removed from the respective distribution mean positions ( $\sim 0.5 {\sigma}_a$ and $\sim 0.5 {\sigma}_e$ ), represent a minimum coefficient of $\sigma$ that can be used to estimate the extent and purity of different Ly $\alpha$ spectral types on the CMD. For example, cuts on the CMD for which $c_{\sigma} = 1.5$ along the same slope as the primary cut approximate (for Gaussian distributions) criteria for selecting Ly $\alpha$ spectral type sub-samples $\gtrsim$ 2 $\sigma$ from the mean value of the opposite distribution.

In theory, the above criteria can be made stricter (or relaxed) by varying the value of $c_{\sigma}$ , thereby trading sub-sample size for sub-sample purity according to the properties of the parent sample, and the requirements of the intended application. In practice, the range of $c_{\sigma}$ values that can be meaningfully employed is limited by the degree to which the aLBG and eLBG distributions deviate from Gaussian behaviour, and by small-number statistics at higher values of $c_{\sigma}$ – especially for eLBGs which are $\sim$ 4 times less abundant than aLBGs in our $z\sim2$ sample. Table 4 summarises the statistics for p-aLBG and p-eLBG Ly $\alpha$ spectral type sub-samples selected from the parent $z\sim2$ LBGs using the selection criteria given in Equations (1) and (2), respectively, and a range of $c_{\sigma}$ values.

Table 4. Statistics for photometric sub-samples with Ly $\alpha$ dominant in absorption (p-aLBGs) and Ly $\alpha$ dominant in emission (p-eLBGs) selected from the parent $z\sim2$ LBG sample using Equations (1) & (2) and different values of $c_{\sigma}$ .

^aCoefficient of colour standard deviations ( $\sigma_a$ & $\sigma_e$ ) by which boundaries used to isolate the photometric spectral type sub-samples are offset from the CMD primary cut.

^bNumber of galaxies in the photometric sub-samples.

^cFor p-aLBGs: Percent purity with respect to eLBG and (eLBG + $\rm{G_e}$ ) spectral types. For p-eLBGs: Percent purity with respect to aLBG and (aLBG + $\rm{G_a}$ ) spectral types.

^dMean net Ly $\alpha$ EW for each sub-sample.

We estimate the purity of each photometric spectral type sub-sample by calculating the degree to which they exclude galaxies with opposite spectral type as determined by their measured net Ly $\alpha$ EW and our classification scheme described in Section 3.1. That is, for example, for each p-aLBG sub-sample selected using a different value of $c_{\sigma}$ , we calculate the contamination fraction of spectroscopic eLBGs and $\rm{eLBG} + \rm{G_e}$ spectral types. The purity of the p-aLBG sub-sample thus determined is quoted as a percentage with respect to eLBGs and with respect to $\rm{eLBG}$ + $\rm{G_e}$ spectral types (parenthesised) in Table 4. The mean net Ly $\alpha$ EW of the p-aLBG and p-eLBG sub-samples (also listed in Table 4) is a further measure of the quality of the broadband photometric segregation, and the average properties of the respective sub-samples.

Across a wide range of $c_{\sigma}$ values, we select high-purity p-aLBG sub-samples, particularly with respect to contamination by spectroscopic eLBGs. Indeed, even using the primary cut between the aLBG and eLBG spectral types (i.e., $c_{\sigma} = 0.0$ ), results in a large sub-sample of 311 p-aLBGs ( $\gtrsim$ 55% of total LBGs) that is $\gtrsim$ 97% free of eLBGs and $\gtrsim$ 70% pure with respect to galaxies with any detectable net Ly $\alpha$ emission. The practical upper limit of $c_{\sigma}$ for the selection of p-aLBGs appears to be restricted only by the diminishing return of smaller sub-sample sizes. As a result, large broadband photometric samples can greatly benefit from stricter cuts. The optimal coefficient for the dataset here of $\sigma_e$ ( $c_{\sigma} \approx 1.0 -1.25$ ) selects $\sim$ 100–140 photometric aLBGs that are $\gtrsim$ 97% and $\gtrsim$ 79% pure with respect to spectroscopic eLBGs, and $\rm{eLBG}$ + $\rm{G_e}$ spectral types respectively, and for which the mean net Ly $\alpha$ EW is $\sim$ $-$ 8 Å.

With $c_{\sigma} \approx 1.0 - 1.25$ we select a sample of $\sim$ 50–70 LBGs with Ly $\alpha$ dominant in emission (p-eLBGs) that are $\gtrsim$ 85% and $\gtrsim$ 65% pure with respect to spectroscopic aLBGs and $\rm{aLBG}$ + $\rm{G_a}$ spectral types, respectively, with a mean net Ly $\alpha$ EW of $\sim$ +8 Å. These purities represent a significant enhancement over the native (full sample) abundances of eLBGs ( $\sim$ 8%) and the sum of eLBG and $\rm{G_e}$ spectral types ( $\sim$ 40%).

The segregation versus net Ly $\alpha$ EW of photometrically selected $z\sim2$ p-aLBG and p-eLBG sub-samples with $c_{\sigma} = 1.0$ is plotted in the top panel of Fig. 4 compared to the distribution versus net Ly $\alpha$ EW of the parent $z\sim2$ LBG sample. In the optimal case, we select sub-samples with a desired Ly $\alpha$ spectral type at $z\sim2$ that are for p-aLBGs, comparable to, and for p-eLBGs $\sim$ 10% less pure than, the optimised $z\sim3$ result of C09 (see Section 3.3.2). The lower optimised purity of p-eLBGs at $z\sim2$ is attributable to the intrinsic overlap of the aLBG and eLBG distributions, and the relatively lower fraction of Ly $\alpha$ -emitting LBGs selected at this redshift. That is, the ratio of aLBGs to eLBGs has increased from around 1:1 at $z\sim$ 3 to more than 4:1 at $z\sim$ 2 when comparing samples to the same absolute magnitude. This is not specific to our sample. On the contrary, a reduced fraction of Ly $\alpha$ -emitting galaxies with decreasing redshift is expected from the findings of Stark et al. (Reference Stark, Ellis, Chiu, Ouchi and Bunker2010), Stark, Ellis, & Ouchi (Reference Stark, Ellis and Ouchi2011), Mallery et al. (Reference Mallery2012), Cassata et al. (Reference Cassata2015) who consistently report an evolutionary decrease in the fraction of LBGs with Ly $\alpha$ in emission from $z\sim6$ to $z\sim2$ at fixed luminosity.

Figure 4. Histograms of p-aLBGs and p-eLBGs are multiplied by 4 for clarity.

Histograms versus net Ly $\alpha$ EW of $z\sim2$ and $z\sim3$ ‘photometric’ aLBG (p-aLBG) and ‘photometric’ eLBG (p-eLBG) spectral type sub-samples overlaid on the distribution versus net Ly $\alpha$ EW of their respective parent samples shown in grey. Vertical dashed lines indicate the net Ly $\alpha$ EW thresholds used here to divide the spectroscopic sample into aLBG, $\rm{G_a}$ , $\rm{G_e}$ and eLBG Ly $\alpha$ spectral types. Red and blue shaded regions indicate aLBGs and eLBGs, respectively. Top: $z\sim2$ p-aLBGs and p-eLBGs selected from the parent sample of 557 $z\sim2$ LBGs using the selection criteria given in Equations (1) & (2) with $c_{\sigma}=1.0$ . Histograms of p-aLBGs and p-eLBGs are multiplied by 2 for clarity. Bottom: $z\sim3$ p-aLBGs and p-eLBGs selected from the parent sample of 775 $z\sim3$ LBGs using the selection criteria given in Equations (3) & (4) with $c_{\sigma}=1.5$ .

3.3.2. $z\sim3$ LBGs

For the purposes of reference and direct comparison, we present here the Ly $\alpha$ spectral type photometric selection results for $z\sim3$ LBGs analysed and presented in the same format as the $z\sim2$ result above.

Parameters for the photometric segregation of $z\sim3$ LBG Ly $\alpha$ -absorbing and Ly $\alpha$ -emitting spectral types in $(G-\mathcal{R})$ vs $\mathcal{R}$ colour-magnitude space as determined by C09 are listed in Table 3, and we re-produce in Equations (3) & (4) criteria for the photometric selection of p-aLBG and p-eLBG spectral type sub-samples at $z\sim3$ .

For p-aLBGs,

(3) \begin{equation}\mbox{($G - \mathcal{R}$)} \, \ge \,\mbox{0.4047} \cdot \mathcal{R} - \mbox{9.3760} \, \mbox{+ c$_\sigma$} \cdot\mbox{$\sigma_e$}\end{equation}

and for p-eLBGs,

(4) \begin{equation}\mbox{($G - \mathcal{R}$)} \, \le \, \mbox{0.4047} \cdot\mathcal{R} - \mbox{9.3760} \, \mbox{- c$_\sigma$} \cdot \mbox{$\sigma_a$}\end{equation}

where for the parent sample of $z\sim3$ LBGs, $\sigma_a = 0.2392$ and $\sigma_e = 0.3095$ .

Table 5 summarises the statistics for p-aLBG and p-eLBG Ly $\alpha$ spectral type sub-samples selected from a parent sample of 775 $z\sim3$ LBGs using the above relationships and a range of $c_{\sigma}$ values. The bottom panel of Fig. 4 shows the segregation versus net Ly $\alpha$ EW of $z\sim3$ p-aLBG and p-eLBG sub-samples selected with $c_{\sigma} = 1.5$ compared to the distribution versus net Ly $\alpha$ EW of the parent $z\sim3$ LBG sample.

Table 5. Statistics for photometric sub-samples with Ly $\alpha$ dominant in absorption (p-aLBGs) and Ly $\alpha$ dominant in emission (p-eLBGs) selected from the parent sample of 775 $z\sim3$ LBGs using the spectral type criteria of C09^a and different values of $c_{\sigma}$ .

^aFor the purposes of determining photometric segregation criteria, C09 defined aLBGs and eLBGs as having net Ly $\alpha$ EW $\leq -12.0$ and $\geq +26.5$ Å respectively.

^bCoefficient of colour standard deviations ( $\sigma_a$ & $\sigma_e$ ) by which boundaries used to isolate the photometric spectral type sub-samples are offset from the CMD primary cut.

^cNumber of galaxies in the photometric sub-samples.

^dFor p-aLBGs: Percent purity with respect to eLBG and (eLBG + $\rm{G_e}$ ) spectral types. For p-eLBGs: Percent purity with respect to aLBG and (aLBG + $\rm{G_a}$ ) spectral types. Mean net Ly $\alpha$ EW for each sub-sample

The optimal coefficient of $\sigma_e$ ( $c_{\sigma} \approx 1.5$ ) selects $\sim$ 120 photometric aLBGs that are $\gtrsim$ 96% and $\gtrsim$ 76% pure with respect to spectroscopic eLBGs, and $\rm{eLBG}$ + $\rm{G_e}$ spectral types respectively, and for which the mean net Ly $\alpha$ EW is $\sim$ $-$ 5 Å.

With any coefficient of $\sigma_a \gtrsim 1.0$ , we select large samples of photometric eLBGs that are $\sim$ 94–98% pure with respect to spectroscopic aLBGs. Over the range $c_{\sigma} = 1.0-2.5$ , the purity of the p-eLBG sample with respect to all net Ly $\alpha$ -absorbers ( $\rm{aLBG}$ + $\rm{G_a}$ spectral types) increases monotonically from $\sim$ 74% to $\sim$ 93%, with a commensurate increase in mean net Ly $\alpha$ EW from $\sim+27$ to $\sim+$ 51 Å.

3.4. The contribution of Ly $\alpha$

The segregation of $z\sim3$ aLBGs and eLBGs on the CMD is enhanced by the contribution of the Ly $\alpha$ feature itself when it falls within the bandpass of the G filter (see Section 3.2 and Fig. 2). The contribution of Ly $\alpha$ to the observed luminosities was estimated by C13 to be $\sim$ -0.1 mags for aLBGs, $\sim$ +0.1 mags for eLBGs, and negligible for LBGs with net Ly $\alpha$ EW near zero. A similar effect might be anticipated at $z\sim2$ when Ly $\alpha$ falls within the bandpass of the relevant ( $U_n$ ) filter.

Unlike the $z\sim3$ case where the Ly $\alpha$ spectral feature lies within the G-band filter across the full redshift range of the sample ( $2.5<z<3.5$ ), about half ( $\sim$ 52%) of the $z\sim2$ LBG sample is in the redshift range $2.17 \leq z \leq 2.50$ , where the Ly $\alpha$ feature lies outside the half-power bandpass limits of the $U_n$ filter. Reddy et al. (Reference Reddy2008) showed that the ratio of strong emitters to absorbers for LBGs at redshifts $2.17 \leq z \leq 2.48$ is approximately the same as for those selected by the same set of colour criteria at $z<2.17$ (see the respective population statistics in Table 6). Thus, there is no underlying selection bias of aLBGs versus eLBGs that could affect the segregation properties in the different redshift ranges. This result does not, however, preclude the possibility that the segregation statistics across the full z-range of the sample may be variably affected by the contribution of Ly $\alpha$ to the measured $U_n$ -band photometry. For this reason – and because the measured segregation at $z\sim2$ is less well resolved than at $z\sim3$ – we look to quantify the effect of Ly $\alpha$ on the observed broadband segregation for galaxies in the $z\sim2$ sample in different redshift ranges and with different Ly $\alpha$ spectral type.

Table 6. Statistics for the segregation of $z\sim2$ LBGs in colour ( $U_n-\mathcal{R}$ )—magnitude ( $\mathcal{R}$ ) space over different redshift ranges.

To this end, we divide the $z\sim2$ LBG sample into two subsets: one containing only galaxies in the range $1.7<z<2.17$ where Ly $\alpha$ falls within the bandpass of the $U_n$ filter (3250–3850 Å), and another comprising galaxies in the $2.17<z<2.5$ range for which Ly $\alpha$ lies beyond the red half-power bandpass limit of the same filter (see Fig. 2). We then optimise the primary cut slope in each redshift bin in the same manner as for the sample as a whole (see Section 3.2), and compare the segregation statistics for the two subsets with each other, and with those for the full z-range sample (see Table 6).

We find a significantly stronger segregation in the $2.17<z<2.5$ sample, most apparent in the greater average colour segregation between aLBGs and eLBGs in this redshift range (0.52) compared to that in the lower redshift bin (0.24). This effect is likely due to the larger contribution of the Ly $\alpha$ forest to the $U_n$ -band of the $2.17<z<2.5$ sample, leading to redder $(U_n-\mathcal{R})$ colours on average. There is also a larger colour dispersion of eLBGs ( $\sigma = 0.41$ ) in the lower redshift range that blurs the photometric segregation.

This difference in the degree of segregation between aLBGs and eLBGs translates into the purity of p-aLBG and p-eLBG sub-samples that can be selected from the two redshift ranges. Using the same methodology as was applied to the full $z\sim2$ and $z\sim3$ samples in Section 3.3, we determined the purity of p-aLBG and p-eLBG sub-samples selected from each redshift bin using the segregation parameters listed in Table 6, and a range of $c_{\sigma}$ values (see Table 7). The optimised purity of the p-aLBGs and p-eLBGs selected from the $2.17<z<2.5$ sample ( $\sim$ 98% and $\sim$ 94%, respectively) is significantly better than can be achieved from the lower redshift bin ( $\sim$ 94% for p-aLBGs and $\sim$ 84% for p-eLBGs). In fact, with $c_{\sigma}$ values of 1.0–1.5, the p-aLBG and p-eLBG sub-samples in the $2.17<z<2.5$ range have purities that are essentially indistinguishable from those achievable in the $z\sim3$ sample, and comparably high $\Delta_{Ly\alpha\ EW}$ between them, indicating that any direct contribution of the Ly $\alpha$ feature to the segregation properties of aLBGs and eLBGs is dominated by other redshift-dependent spectrophotometric effects such as the one described above.

Table 7. Statistics for p-aLBG and p-eLBG sub-samples photometrically selected from the parent $z\sim2$ LBG sample using segregation parameters optimised in different redshift ranges.

^aCoefficient of colour standard deviations by which boundaries used to isolate the photometric spectral type sub-samples are offset from the primary cut in each redshift range.

^bNumber of galaxies in the photometric sub-samples.

^cFor p-aLBGs: Percent purity with respect to eLBG and (eLBG + $\rm{G_e}$ ) spectral types. For p-eLBGs: Percent purity with respect to aLBG and (aLBG + $\rm{G_a}$ ) spectral types.

^dMean net Ly $\alpha$ EW for each sub-sample.

^eDifference in mean net Ly $\alpha$ EW between the p-aLBG and p-eLBG sub-samples. In the case of Full $_{bins}$ , ${\Delta}_{Ly\alpha\ EW}$ is the weighted average of $\Delta_{Ly\alpha\ EW}$ values for each magnitude bin.

3.5. LSST photometric selection criteria for $z\sim3$ LBG Ly $\alpha$ spectral types

A key objective of this work is to develop a method that can be applied to large samples of $z\sim2-6$ LBGs identified from current and future large-area and all-sky photometric campaigns. As a first step toward this goal, we adapt the spectrophotometric method of C13 to model photometric selection criteria by which populations of $z\sim3$ LBGs with Ly $\alpha$ dominant in absorption and Ly $\alpha$ dominant in emission might be selected from the broadband ugri photometric data of the Vera Ruben Observatory Legacy Survey of Space and Time (VRO/LSST).

In order to model the segregation statistics for different Ly $\alpha$ spectral types in the LSST ugri photometric system, it is first necessary to calculate ugri magnitudes for each galaxy in our $z\sim3$ sample. Our photometric segregation method relies on the fact that LBGs with different net Ly $\alpha$ EW have different spectral properties – in particular rest-frame UV continuum slope – that give rise to different rest-frame UV colours depending on their Ly $\alpha$ absorbing/emitting properties (see Fig. 2). Thus, in order to convert from $U_nG\mathcal{R}$ to ugri magnitudes via spectrophotometry, we must be able to assign to each galaxy in our sample, an appropriate spectrum corresponding to its Ly $\alpha$ spectral type. For this purpose, we make use of the four composite spectra of Shapley et al. (Reference Shapley, Steidel, Pettini and Adelberger2003), derived from the $z\sim3$ $U_nG\mathcal{R}$ LBGs described in Sections 2 & 3.1 divided into quartiles on the basis of net Ly $\alpha$ EW. C13 showed that spectrophotometry of these composite spectra accurately reproduces the magnitude and colour means and dispersions of each of the four net Ly $\alpha$ EW quartile samples, as well as the full distribution of $z\sim3$ LBGs on the $(G-\mathcal{R})$ /G CMD when combined. Thus, although the colours and magnitudes of individual galaxies vary within each quartile, the composite spectra can be used to compute net Ly $\alpha$ EW means and dispersions on the CMD for our $z\sim3$ LBG sample when viewed through the VRO/LSST ugri filtersFootnote ^a . Fig. 5 shows our $z\sim3$ $U_nG\mathcal{R}$ LBG sample dispersed in VRO/LSST $(g-r)$ vs r colour-magnitude space with photometry thus derived from spectrophotometry of the net Ly $\alpha$ EW quartile composite spectra.

Following the method described in Section 3.3, we use the parameters of the segregated aLBG and eLBG distributions shown in Fig. 5 and tabulated in Table 8 to determine photometric selection criteria by which pure populations of $z\sim3$ LBGs with Ly $\alpha$ dominant in absorption (p-aLBGs; Equation (5)) and Ly $\alpha$ dominant in emission (p-eLBGs; Equation (6)), might be selected from VRO/LSST LBG data dispersed in $(g-r)$ vs r space.

Figure 5. Rest-frame UV $(g-r)$ vs r colour-magnitude diagram (CMD) for Steidel et al. (Reference Steidel2003) $z\sim3$ LBGs, constructed with VRO/LSST ugri photometry derived from the net Ly $\alpha$ EW quartile composite spectra of Shapley et al. (Reference Shapley, Steidel, Pettini and Adelberger2003). The CMD shows the segregation with net Ly $\alpha$ EW of galaxies with Ly $\alpha$ dominant in absorption (aLBGs, red squares) and Ly $\alpha$ dominant in emission (eLBGs, blue triangles). Grey plus (+) symbols denote galaxies with intermediate values of net Ly $\alpha$ EW. Black crosses mark the mean positions of the aLBG and eLBG distributions. The dashed red and dotted-dashed blue lines indicate a $1.5\sigma$ dispersion in colour from the primary cut (green line) for the aLBG and eLBG distributions, respectively.

Table 8. Statistics for the segregation of LSST $z\sim3$ LBG Ly $\alpha$ spectral types in $(g-r)$ /r colour-magnitude space.

^aConsistent with C09, we define aLBGs and eLBGs as having net Ly $\alpha$ EW $\leq -12.0$ and $\geq +26.5$ Å respectively, for the purposes of constructing the CMD and determining photometric segregation criteria.

Specifically, for p-aLBGs:

(5) \begin{equation}(g - r) \ge 0.25 \cdot r - 5.72 + c_{\sigma} \cdot \sigma_e\end{equation}

and for p-eLBGs:

(6) \begin{equation}(g - r) \le 0.25 \cdot r - 5.72 - c_\sigma \cdot \sigma_a\end{equation}

where $c_{\sigma}$ , ${\sigma}_{a}$ , and ${\sigma}_{e}$ are the coefficient and respective standard deviations of colour dispersion as described in Section 3.3.1.

These selection criteria provide a useful starting point for the isolation of p-aLBG and p-eLBG populations from LSST photometry. They will be confirmed and/or refined, and selection criteria in other redshift ranges added – especially at $z\sim2$ – once LSST data of sufficient depth has been measured in fields within which Ly $\alpha$ spectroscopic data are available.