3.1 Global quenching

Environmental influences on star formation are commonly quantified using the quenched fraction, defined here as $f_{{Q}} \equiv N_{\mathrm{quenched}}/N_{\mathrm{total}}$ within a given bin (e.g. Balogh et al. Reference Balogh2004; Peng et al. Reference Peng2010; Wetzel et al. Reference Wetzel, Tinker and Conroy2012; Knobel et al. Reference Knobel, Lilly and Woo2015; Fossati et al. Reference Fossati2017; Davies et al. Reference Davies2019). Throughout this analysis, galaxies are classified as quenched using the same specific star formation rate (sSFR) criterion adopted in Paper I: i.e. galaxies with $\log_{10}\mathrm{sSFR}\lt -11.0\,\mathrm{yr^{-1}}$ are considered quenched and all uncertainties and error bars reported throughout are derived via bootstrap resampling with 10 000 realisations, and are quoted using the 16th and 84th percentiles.

Figure 3.

Quenched fraction ( $f_{{Q}}$ ) as a function of halo mass and stellar mass for group and field galaxies in the SGP (black points) and Nexus (red points). Top panel: $f_{{Q}}$ as a function of halo mass for group galaxies. Middle panel: $f_{{Q}}$ as a function of stellar mass for group galaxies. Bottom panel: $f_{{Q}}$ as a function of stellar mass for field (non-grouped) galaxies. In each panel, points indicate the binned median $f_{{Q}}$ values with their associated uncertainties. For the halo mass panel, the best-fitting logistic model (Equation 2) is shown for the SGP (solid black curve) and Nexus (solid red curve). For the stellar mass panels, the best-fitting Gaussian plus logistic model (Equation 3) is shown for the SGP (solid black curve) and Nexus (solid red curve). In the two group galaxy panels (as functions of halo and stellar mass) we overlay the binned values from Davies et al. (Reference Davies2019) as a grey dashed line for comparison.

Following Paper I, SFR upper limits are treated in a manner that explicitly accounts for the mass- and distance-dependent sensitivity of the WISE W3 band (our main star-formation tracer). Upper limits arise in two distinct regimes: (i) at high stellar masses ( $\log M_{\mathrm{stellar}} \geq 10$ ), where galaxies typically exhibit intrinsically low or negligible star formation and (ii) at low stellar masses ( $\log M_{\mathrm{stellar}} \lt 10$ ), where the low surface brightness of galaxies can fall below the W3 detection threshold. For galaxies at $\log M_{\mathrm{stellar}} \geq 10$ and $z \lt 0.1$ , the W3 band is effectively complete for detecting star formation at the levels expected for the star-forming population. In this regime, an SFR upper limit therefore indicates a quenched system, and such galaxies are classified accordingly. However, at lower stellar masses, the detectability of star formation becomes strongly dependent on both surface brightness and distance. For each galaxy with an SFR upper limit in this regime, we therefore evaluate whether its expected W3 flux, given its redshift, lies above or below the survey detection threshold. If the expected flux exceeds the threshold, the upper limit is classified as quenched. Conversely, if the expected flux falls below the detection limit, the upper limit is deemed inconclusive, as the galaxy may host star formation below the sensitivity of the data. In these cases, we conservatively classify the galaxy as star-forming to avoid artificially inflating the quenched fraction.

Table 1.

Spearman’s rank correlation coefficient (r) and p-value for the relationship between $f_{{Q}}$ and either stellar mass ( $M_{\rm stellar}$ ) or halo mass ( $M_{\mathrm{halo}}$ ), for the global SGP and Nexus samples. Best-fit parameters and $1\sigma$ uncertainties for the logistic model (Equation 2) are reported for the halo mass relations, and for the Gaussian plus logistic model (Equation 3) for the stellar mass relations, where $\mu_{{g}}$ and $\sigma_{{g}}$ are the centre and width of the low-mass Gaussian component, $A_{{g}}$ is its amplitude, $M_{50}$ is the characteristic mass at which the logistic component reaches $f_{{Q}}=0.5$ , and k parametrises the steepness of the high-mass rise. Results are shown for both group and field environments.

Figure 3 presents the quenched fraction, $f_{{Q}}$ , measured across the SGP and Nexus samples, enabling a direct comparison between an average reference environment and a dynamically assembling superstructure. This comparison provides a baseline against which environmental effects on quenching can be assessed (Balogh et al. Reference Balogh2004; Wetzel et al. Reference Wetzel, Tinker and Conroy2012; Fossati et al. Reference Fossati2017; van der Burg et al. Reference van der Burg2018). The figure shows $f_{{Q}}$ as a function of halo mass in the top panel, and as a function of stellar mass in the lower two panels, with group galaxies shown in the top and middle panels and field galaxies in the bottom panel. Field galaxies are defined as systems not assigned to any identified group in the adopted SGP group catalogue (see Paper I). The data are binned in logarithmic mass intervals, with points representing the median $f_{{Q}}$ in each bin and associated uncertainties. To ensure statistical robustness while maintaining mass coverage, we include only bins containing at least 10 galaxies in the SGP and at least 5 galaxies in the Nexus. The corresponding binned values are listed in Table A1.

Figure 3 includes a comparison to the group galaxy $f_{{Q}}$ values from Davies et al. (Reference Davies2019), shown as grey dashed lines. The Davies et al. (Reference Davies2019) measurements utilise dynamically-derived halo masses from the GAMA group catalogue (Robotham et al. Reference Robotham2011), providing an independent dataset against which to benchmark our results. The comparison reveals interesting systematic differences between the two approaches, particularly as a function of halo mass. In the halo mass panel, the Davies et al. (Reference Davies2019) results show notably higher $f_{{Q}}$ values, especially at group scale halo masses ( $\log M_{\mathrm{halo}} \lt 14$ ). Conversely, the group stellar mass trends in the middle panel demonstrates a closer agreement between our measurements and those of Davies et al. (Reference Davies2019), with both datasets exhibiting the similar quantitative features within the mass range probed.

To quantify the overall monotonic trends within these binned relations, we report the Spearman rank correlation coefficients and associated p-values in Table 1. We emphasise that Spearman’s p-value measures the strength of a monotonic relationship, and remains a valid and well-defined statistic for nonlinear but monotonically increasing correlations such as those seen in the halo mass and group stellar mass panels. It is only in the presence of genuinely non-monotonic relations, most notably the low-mass bumps visible in the field stellar mass panels, that Spearman’s test is expected to underestimate the strength of correlation. In those cases, the low correlation coefficient and high p-value should therefore be interpreted as reflecting the non-monotonic character of the relation rather than an absence of correlation, and readers are cautioned against interpreting these values as evidence for a weak or absent trend.

As a function of halo mass, we fit the binned results with a logistic function in the form:

(2) \begin{equation} f_{{Q}} = \frac{1}{1 + \exp[\!-k(\!\log {M} - {M}_{50})]}, \end{equation}

where k parametrises the transition steepness and $M_{50}$ is the characteristic mass at which $f_{{Q}}=0.5$ (e.g. Woo et al. Reference Woo2013; Bluck et al. Reference Bluck2016). As a function of stellar mass, the logistic model provides a poor description of the low-mass end due to the prevalence of a low-mass bump (particularly noticeable with the field population). To better describe this behaviour, we instead fit a Gaussian plus logistic model:

(3) \begin{equation} f_{{Q}} = A \exp\left[\!-\frac{(\!\log M - \mu_{{g}})^2}{2\sigma_{{g}}^2}\right] + \frac{1}{1 + \exp[\!-k(\!\log M - M_{50})]}, \end{equation}

where the Gaussian component captures the low-mass bump and the logistic component, captures the high mass end. Here, $A_g$ is the amplitude of the low-mass Gaussian bump, $\mu_g$ is the stellar mass at which this excess peaks, and $\sigma_g$ characterises its width in log stellar mass space. The Gaussian amplitude is constrained to be non-negative, ensuring $f_Q \geq 0$ at all masses. Best-fit parameters and $1 \sigma$ uncertainties derived from the covariance matrix are reported in Table 1. The corresponding fitted curves are presented in Figure 3 and provide a compact summary for comparing the onset and sharpness of quenching between samples.

The correlation and fit summaries highlight several differences between environments and between the SGP and Nexus. For the group population, $f_{{Q}}$ correlates strongly with halo mass in the SGP (Table 1), whereas the corresponding halo mass trend in the Nexus is weaker and not statistically significant at the current sample size. We note that this difference is partly driven by the SGP extending to higher halo masses, which broadens the dynamic range of the correlation. In contrast, $f_{{Q}}$ shows a clear stellar mass dependence for group galaxies in both regions, with the Gaussian plus logistic fit indicating a lower characteristic transition mass ( $M_{50}$ ) and a steeper transition (larger k) in the Nexus group sample relative to the SGP group sample.

Figure 4.

Quenched fraction ( $f_{{Q}}$ ), for galaxies in the SGP (left panels) and Nexus (right panels). The top panels show the distribution of field galaxies as a function of stellar mass, colour-coded by the $f_{{Q}}$ . The bottom panels show $f_{{Q}}$ for group galaxies in the stellar–halo mass plane, derived using a smoothed kernel density estimate (KDE), with the underlying galaxy distribution shown by black points. The colour bar indicates the $f_{{Q}}$ .

For the field population, the Spearman coefficients are low and p-values large, but as noted above this reflects the non-monotonic shape of the relation rather than a genuinely weak trend. The Gaussian plus logistic fits reveal a clear low-mass bump followed by a high-mass rise in both the SGP and Nexus field populations, with the high-mass logistic transition occurring at higher $M_{50}$ than for the group population. This is consistent with quenching becoming common only at the high mass end in the absence of identified group-scale environments. We quantify these trends as clearly as possible to facilitate comparison with forthcoming studies, providing a physically motivated decomposition that separately examines the low-mass excess, as well as the onset and sharpness of the high-mass quenching transition, across field and group environments and between the SGP and Nexus.

We note that the SGP samples a broader range of halo masses than the Nexus, extending to higher $M_{\mathrm{halo}}$ and consequently higher $f_{{Q}}$ values at the massive end. This difference in dynamic range should be borne in mind when comparing the two samples in the halo mass panel; however, we have verified that this does not materially affect our conclusions. Excluding the highest-mass SGP bins from the fit leaves the best-fit parameters unchanged to within the reported uncertainties, indicating that the elevated SGP correlation is not an artefact of its extended mass baseline but rather reflects a genuinely stronger halo-mass dependence of quenching across the full SGP volume.

Figure 5.

Quenched fraction ( $f_{{Q}}$ ) for galaxies in the SGP (left panels) and Nexus (right panels). Top panels: $f_{{Q}}$ as a function of halo mass ( $M_{\mathrm{halo}} \,[\log\, M_\odot]$ ) shown for different stellar mass bins ( $M_{\mathrm{stellar}} \,[\log\, M_\odot]$ ) for group galaxies. Bottom panels: $f_{{Q}}$ as a function of stellar mass for different halo mass bins for group galaxies. In the bottom panels, the $f_{{Q}}$ of field galaxies is also shown for comparison. Points indicate binned median values with their associated uncertainties.

In summary, the binned trends of Figure 3, $f_{{Q}}$ increases with stellar mass in both group and field environments in both the SGP and Nexus, although the strength of the relation varies between the two. For group galaxies, $f_{{Q}}$ also increases with halo mass in both group and field environments, but with a stronger global halo mass dependence in the SGP than in the Nexus. These one-dimensional measurements provide an accessible summary of how quenching varies with the average mass in each region, but they do not control for the covariance between stellar mass and halo mass.

The scale and richness of our dataset enables us to explore the variation of the $f_{{Q}}$ in a two-dimensional plane between halo and stellar mass. Figure 4 primarily provides a two-dimensional qualitative view of how the $f_{{Q}}$ varies jointly with stellar mass and halo mass, and how this differs between the SGP and the Nexus. The top panels show the stellar mass distributions of field galaxies, with the bar colours indicating $f_{{Q}}$ in each stellar mass bin. The bottom panels show group galaxies in the stellar mass and halo mass plane, where the background colour indicates $f_{{Q}}$ evaluated on a regular grid. The underlying two-dimensional distribution is estimated using a kernel density estimate (KDE) with adaptive smoothing. The kernel full width at half maximum is chosen to be approximately twice the typical uncertainty in stellar mass (0.15 dex) and halo mass (0.3 dex). This choice is intended to reduce bias from mass measurement scatter in the smoothed density field, and to limit artificial sharpening of $f_{{Q}}$ gradients due to underestimated uncertainties. In each grid cell, $f_{{Q}}$ is computed as the ratio of the KDE weighted density of quenched galaxies to the total KDE weighted density. Individual galaxies are shown as black points to indicate sampling of the stellar mass and halo mass plane.

As qualitatively demonstrated in Figure 4, we need to control for the covariance between stellar mass and halo mass, Figure 5 presents the results from Figure 4 by binning $f_{{Q}}$ trends in controlled bins of the complementary variable. The left columns present the SGP and the right column shows the Nexus. In the top panels, we show $f_{{Q}}$ as a function of halo mass for several bins in stellar mass, and in the bottom panels we present $f_{{Q}}$ as a function of stellar mass for a range of halo mass bins. Included in the bottom panel stellar mass panels is the distribution of $f_{{Q}}$ for the field galaxies for comparison to non-grouped environments within the SGP and Nexus. For this binned analysis, a minimum of three galaxies per bin was adopted for both SGP and Nexus. This lower threshold was chosen to probe the structure of the trends at fixed stellar mass or fixed halo mass while pushing the current dataset to its practical limits. The trade off is increased statistical noise and a greater sensitivity to sampling variance, most evident in the low and high mass ranges (both stellar and halo) of the sample. The tabulated values corresponding to Figure 5 are provided in Tables A2 and A3.

Together, Figures 4 and 5 clarify how the joint stellar mass and halo mass distribution shapes the marginalised trends in Figure 3. The two-dimensional KDE maps show that high $f_{{Q}}$ values are concentrated where both stellar mass and halo mass are large, and they indicate that the SGP and Nexus populate the stellar–halo mass plane differently. Figure 5 therefore shows explicitly how the apparent dependence on halo mass changes across stellar mass bins, and how the apparent dependence on stellar mass changes across halo mass bins (Tables A2 and A3). This demonstrates why controlling for mass is necessary: one-dimensional relations of $f_{{Q}}$ versus halo mass (integrating over all stellar masses) or versus stellar mass (integrating over all halo masses) conflate coupled dependencies and can therefore obscure or exaggerate environmental differences between samples (e.g. Peng et al. Reference Peng2012; Wetzel et al. Reference Wetzel, Tinker and Conroy2012; Knobel et al. Reference Knobel, Lilly and Woo2015).

In summary, $f_{{Q}}$ increases with both stellar mass and halo mass in the marginalised one-dimensional relations, but Figures 4 and 5 show that these dependencies are coupled through the joint distribution of stellar mass and halo mass. Examining $f_{{Q}}$ as a function of halo mass within narrow bins of stellar mass, and conversely as a function of stellar mass within narrow bins of halo mass, provides the clearest evidence for how quenching depends jointly on stellar mass and halo mass in the SGP and Nexus.

Figure 6.

Binned mean sSFR (sSFR $[\log \, \mathrm{yr}^{-1}]$ ) as a function of halo mass and stellar mass for group and field galaxies, with the SGP sample shown as black points and the Nexus sample shown as red points. Top panel: Binned mean sSFR as a function of halo mass for group galaxies. Middle panel: Binned mean sSFR as a function of stellar mass for group galaxies. Bottom panel: Binned mean sSFR as a function of stellar mass for field (non-grouped) galaxies. In each panel, the underlying binned mean sSFR measurements are shown with their associated uncertainties. The best-fitting relations are outlined, with linear fits adopted as a function of halo mass trends a and power-law as a function of stellar mass (see Table 2 for fitting parameters and uncertainties).

Figure 6. Long description

Three line graphs depict the relationship between specific star formation rate (sSFR) and halo mass or stellar mass for group and field galaxies. Panel A: The top panel shows binned mean sSFR as a function of halo mass for group galaxies. The x-axis represents halo mass in logarithmic scale (log M☉) ranging from 11.5 to 14.5, and the y-axis represents sSFR in logarithmic scale (log yr⁻¹) ranging from -11.5 to -9.5. Black points represent the SGP sample, and red points represent the Nexus sample. The best-fitting linear relations are shown as solid black and dashed red lines. Panel B: The middle panel shows binned mean sSFR as a function of stellar mass for group galaxies. The x-axis represents stellar mass in logarithmic scale (log M☉) ranging from 8.0 to 11.0, and the y-axis represents sSFR in logarithmic scale (log yr⁻¹) ranging from -11.5 to -9.5. Black points represent the SGP sample, and red points represent the Nexus sample. The best-fitting power-law relations are shown as solid black and dashed red lines. Panel C: The bottom panel shows binned mean sSFR as a function of stellar mass for field (non-grouped) galaxies. The x-axis represents stellar mass in logarithmic scale (log M☉) ranging from 8.0 to 11.0, and the y-axis represents sSFR in logarithmic scale (log yr⁻¹) ranging from -11.5 to -9.5. Black points represent the SGP sample, and red points represent the Nexus sample. The best-fitting power-law relations are shown as solid black and dashed red lines.

3.2 Global star formation

To characterise how star formation is regulated across environment, we measure the mean and scatter of the sSFR distribution for the star-forming population of galaxies in the SGP and Nexus. Throughout this section, we analyse the sSFR ( $\log \mathrm{sSFR}\,[\log \mathrm{yr}^{-1}]$ ) for galaxies classified as star-forming using the same selection and quality cuts described in Paper I. SFR upper limits are excluded from this analysis; as discussed above, upper limits predominantly arise in systems with intrinsically low star formation activity, particularly at $\log M_{\mathrm{stellar}} \geq 10$ , where the WISE W3 band is complete and non-detections robustly indicate quenched galaxies. At lower stellar masses, upper limits can also occur due to surface brightness limitations. However, these cases are conservatively treated as star-forming only when the expected flux falls below the detection threshold. Consequently, the population of galaxies with SFR upper limits is dominated by quenched or weakly star-forming systems, and their removal preferentially excludes the quenched population from the sample. In addition, upper limits cannot be reliably incorporated into binned statistics of $\log \mathrm{sSFR}$ . Treating upper limits as detections would bias the mean towards artificially low values, we therefore restrict our analysis to galaxies with robust SFR measurements. Additionally, we require the SFR signal-to-noise ratio to be greater than 2, ensuring that only credible measurements contribute to the derived trends. This selection retains galaxies with sSFR below the canonical star-forming main sequence, including systems with $\log\,\mathrm{sSFR} \lt -11$ , thereby capturing galaxies with suppressed, but still detectable, star formation. This approach allows us to probe variations in star formation efficiency without contamination from poorly constrained or non-detected systems. Further details and motivation for these cuts are provided in Paper I.

Figure 6 presents the binned median of the mean $\log \mathrm{sSFR}$ against logarithmic mass bins for the SGP (black points) and the Nexus (red points) respectively. The top row shows the mean $\log \mathrm{sSFR}$ of star-forming group galaxies as a function of halo mass ( $M_{\mathrm{halo}}$ ), while the middle and bottom rows present the mean $\log \mathrm{sSFR}$ as a function of stellar mass ( $M_\mathrm{stellar}$ ) for star-forming group and field galaxies, respectively. In each panel, points show the binned median of the mean values, with associated uncertainties. For these global one-dimensional trends, only bins containing more than eight galaxies in the SGP and more than four galaxies in the Nexus are considered, to reduce sensitivity to small number statistics whilst attempting to probe the extremes of our mass ranges for the large SGP sample and the smaller Nexus sample. The binned values underlying Figure 6 are reported in Table A4.

Table 2.

Spearman’s rank correlation coefficient (r) and p-value for the relationship between mean $\log \mathrm{sSFR} \, (\log \mathrm{yr}^{-1})$ and either stellar mass ( $M_{\rm stellar}$ ) or halo mass ( $M_{\mathrm{halo}}$ ), for the global SGP and Nexus samples. The best-fit parameters and their uncertainties to the linear regression for the group galaxies as a function of halo mass and the power-law slope as a function of stellar mass for the group and field galaxies.

To quantify the overall trends in the binned relations, Spearman rank correlation coefficients and associated p values are reported in Table 2. For trends as a function of halo mass, a linear relation of the form $\log\mathrm{sSFR} = m\,\log M_{\mathrm{halo}} + c$ is fitted to the binned points. For trends as a function of stellar mass, we instead adopt a power-law motivated fit that allows for curvature in the $\log \mathrm{sSFR}$ – $\log M_{\star}$ relation, thereby better capturing the flattening of the star-forming sequence at low stellar masses (e.g. Noeske et al. Reference Noeske2007; Whitaker et al. Reference Whitaker, van Dokkum, Brammer and Franx2012; Speagle et al. Reference Speagle, Steinhardt, Capak and Silverman2014; Schreiber et al. Reference Schreiber2015). This fit is in the form:

(4) \begin{equation} \log_{10}\mathrm{sSFR} = a + b\log_{10}M_{\mathrm{stellar}} - \log_{10}\left[1 + \left(\frac{M_{\mathrm{stellar}}}{M_{0}}\right)^{c}\right], \end{equation}

where a is a normalisation constant, b is the low-mass power-law slope, $M_{0}$ is the characteristic turnover mass above which the sSFR begins to decline, and c controls the sharpness of that turnover.

The correlation and fit summaries provide a compact comparison of how strongly star formation is regulated by stellar mass and halo mass, and whether this differs significantly between the SGP and the Nexus. The mean $\log \mathrm{sSFR}$ shows very strong anti-correlations with stellar mass for both group and field galaxies in both regions (Table 2), with $r \leq -0.98$ in all cases except for the Nexus field sample ( $r=-0.92$ ), and all correlations remaining highly significant ( $p \ll 0.01$ ). The fitted power-law parameters indicate that the decline of $\log \mathrm{sSFR}$ with stellar mass is steeper for group galaxies than for field galaxies. In the SGP, the power-law slope parameter is $a=-10.90\pm0.71$ for group galaxies compared to $a=-9.66\pm0.31$ for field galaxies, while in the Nexus the corresponding values are $a=-8.61\pm1.62$ (group) and $a=-7.60\pm0.66$ (field), albeit with larger uncertainties.

For group galaxies, the binned mean $\log \mathrm{sSFR}$ also declines with halo mass in both regions, with Spearman coefficients of $r=-1.00$ in both the SGP and Nexus samples. The fitted linear slopes indicate a stronger halo mass dependence in the SGP ( $m=-0.34\pm0.02$ ) compared to the Nexus ( $m=-0.15\pm0.06$ ), although the latter is subject to larger uncertainties due to the smaller sample size. Interpreting these coefficients requires caution because the fitted relations are summary statistics of binned trends; however, they provide a uniform basis for comparing the relative steepness of the mean relations between stellar mass and halo mass, and between the SGP and Nexus environments.

Figure 7 provides a qualitative two-dimensional view of how star formation varies in the stellar–halo mass plane for group galaxies, complemented by stellar mass histograms for the field population. The top pair of panels presents a KDE smoothed mean sSFR (mean $\log \mathrm{sSFR}$ ) as a function of halo mass and stellar mass, whilst the bottom pair of panels shows the one standard deviation scatter of the KDE smoothed mean sSFR value ( $\sigma\,\log\,\mathrm{sSFR}$ ). The analysis of the KDE smoothed mean and one standard deviation maps use the same methodology as in Section 3.1. For the mean map, each grid cell is computed as the KDE weighted mean of $\log \mathrm{sSFR}$ for star-forming galaxies. For the standard deviation map, $\sigma$ , each grid cell is calculated from KDE weighted moments of the $\log \mathrm{sSFR}$ distribution, by combining the KDE weighted mean of $\log \mathrm{sSFR}$ with the KDE weighted mean of $(\!\log \mathrm{sSFR})^{2}$ to recover the cell variance and hence the standard deviation. These maps are intended as qualitative diagnostics of joint trends and of regions where the star-forming population exhibits elevated intrinsic diversity.

Figure 7.

Mean and scatter of sSFR in the stellar–halo mass plane for group and field galaxies in the SGP (left panels) and Nexus (right panels). The top two panels show the mean sSFR: the top histogram panels display the mean sSFR of field galaxies as a function of stellar mass, while the second row shows the mean sSFR of group galaxies as a function of stellar and halo mass, derived using a smoothed KDE. The bottom two panels show the scatter in sSFR, quantified as the standard deviation ( $\sigma \log \mathrm{sSFR}\,[\log\,\mathrm{yr}^{-1}]$ ). The bottom histogram panels show the sSFR scatter for field galaxies as a function of stellar mass, while the bottom row shows the sSFR scatter for group galaxies in the stellar–halo mass plane, also estimated using a smoothed KDE. In the KDE panels, the underlying galaxy distributions are indicated by black points.

Figure 7. Long description

Panel A: The top left histogram shows the mean specific star formation rate of field galaxies as a function of stellar mass in the SGP region. The x-axis represents stellar mass in logarithmic scale, and the y-axis represents the field count. Panel B: The top right histogram shows the same for the Nexus region. Panel C: The second row left scatter plot shows the mean specific star formation rate of group galaxies as a function of stellar and halo mass in the SGP region, derived using a smoothed kernel density estimate. The x-axis represents stellar mass in logarithmic scale, the y-axis represents halo mass in logarithmic scale, and the color bar indicates the mean log specific star formation rate. Black points indicate the underlying galaxy distributions. Panel D: The second row right scatter plot shows the same for the Nexus region. Panel E: The bottom left histogram shows the scatter in specific star formation rate for field galaxies as a function of stellar mass in the SGP region. The x-axis represents stellar mass in logarithmic scale, and the y-axis represents the field count. Panel F: The bottom right histogram shows the same for the Nexus region. Panel G: The bottom left scatter plot shows the scatter in specific star formation rate for group galaxies in the stellar-halo mass plane in the SGP region, estimated using a smoothed kernel density estimate. The x-axis represents stellar mass in logarithmic scale, the y-axis represents halo mass in logarithmic scale, and the color bar indicates the standard deviation of log specific star formation rate. Black points indicate the underlying galaxy distributions. Panel H: The bottom right scatter plot shows the same for the Nexus region.

To control for the covariance between stellar mass and halo mass in their relation with sSFR, as demonstrated in Figure 7, Figure 8 presents the mean $\log \mathrm{sSFR}$ binned by halo mass at fixed stellar mass, and by stellar mass at fixed halo mass. The left column shows SGP and the right column shows Nexus. The top panels show mean $\log \mathrm{sSFR}$ as a function of halo mass within stellar mass bins. The bottom panels show mean $\log \mathrm{sSFR}$ as a function of stellar mass within halo mass bins, with the field relation included for reference. For this analysis, a minimum of three galaxies per bin is adopted for both the SGP and the Nexus, with bins below this threshold excluded. The corresponding binned values are reported in Tables A5 and A6.

Figure 8.

Binned mean sSFR (sSFR $[\log \, {\rm yr}^{-1}]$ ) for galaxies in the SGP (left panels) and Nexus (right panels). Top panels: Binned mean sSFR fraction as a function of halo mass ( $M_{\mathrm{halo}} \,[\log\, {\rm M}_\odot]$ ) shown for different stellar mass bins ( $M_{\mathrm{stellar}} \,[\log\, {\rm M}_\odot]$ ) for group galaxies. Bottom panels: Binned mean sSFR as a function of stellar mass for different halo mass bins for group galaxies. In the bottom panels, the binned mean sSFR of field galaxies is also shown for comparison. Points indicate binned median values with their associated uncertainties.

Taken together, Figures 7 and 8 show that the aggregate halo mass trend in Figure 6 is not uniform across stellar mass. When the sample is split into stellar mass bins (Figure 8, top panels), the halo mass dependence varies in strength and, in some bins, becomes comparatively weak within the uncertainties. Conversely, when split by halo mass and examined as a function of stellar mass (Figure 8, bottom panels), the decline of mean $\log \mathrm{sSFR}$ with stellar mass remains evident across halo mass bins, indicating that the dominant driver of the mean trend is stellar mass, with halo mass providing an additional modulation that can be masked or amplified when the sample is marginalised. The two-dimensional mean maps in Figure 7 are consistent with this coupled behaviour, showing structure across the joint stellar mass and halo mass plane rather than a single separable dependence. The standard deviation maps add complementary information by showing that $\sigma\,\log\,\mathrm{sSFR}$ varies more strongly with stellar mass than with halo mass across the parameter space occupied by the star-forming group population.

As shown prior, controlling for mass is necessary when interpreting the one-dimensional binned trends in Figure 6. A relation of mean $\log \mathrm{sSFR}$ versus halo mass that integrates over all stellar masses mixes galaxies with systematically different typical sSFR, and a relation of mean $\log \mathrm{sSFR}$ versus stellar mass that integrates over all halo masses mixes environments with different levels of star formation regulation. The controlled comparisons in Figure 8, together with the joint view in Figure 7, therefore provide the clearest description of how mean star-forming activity and its diversity depend jointly on stellar mass and halo mass in the SGP and Nexus.

Overall, the global diagnostics provide a consistent empirical picture of star-forming activity across the two regions. The one-dimensional binned relations show that mean $\log \mathrm{sSFR}$ declines with increasing stellar mass in both group and field environments, and that an average decline with halo mass is also present for group galaxies. The controlled binning and two-dimensional maps show that these dependencies are coupled through the joint distribution of stellar mass and halo mass, with stellar mass setting the dominant decline of mean $\log \mathrm{sSFR}$ and halo mass acting as an additional modulation that varies across stellar mass. The standard deviation maps further indicate that the diversity of star-forming activity varies more strongly with stellar mass than with halo mass over the range probed here, and that the SGP measurements provide the higher signal to noise baseline against which the corresponding Nexus patterns can be compared.

3.3 Radial trends in the Nexus

The Nexus region provides a direct opportunity to examine how quenching and star formation vary with distance from the cosmic node and forming supercluster with A4038 at the centre. Motivated by the expectation that environmental processing changes systematically with local density, we analyse both the $f_{{Q}}$ and the star-forming population properties as a function of stellar mass and halo mass within the three radial zones defined in Section 2.4. These zones span the inner core (0–5 Mpc), an intermediate region (5–15 Mpc), and an outer region (15–25 Mpc). Given the construction described in Section 2.4, the zone assignments are expected to be highly robust for the purposes of this qualitative comparison, and we explicitly compare zones to highlight how the observed patterns vary with radius.

Figure 9 shows the distribution of the $f_{{Q}}$ within each radial zone. Each column corresponds to one zone, with the field population shown as a stellar mass histogram in the top row and the group population shown in the stellar mass and halo mass plane in the bottom row. In each zone, $f_{Q}$ is estimated using the same two-dimensional KDE approach adopted in Section 3.1, applied independently to the galaxies in that zone, with the background colour indicating the KDE based $f_{Q}$ and black points indicating the underlying galaxy distribution.

Figure 9.

Distribution of the Quenched fraction ( $f_{{Q}}$ ) in the stellar mass–halo mass plane for galaxies in the Nexus region. The left panels show galaxies within the 5 Mpc shell from A4038, the middle panels show galaxies between 5 and 15 Mpc, and the right panels show galaxies between 15 and 25 Mpc. The bottom panels indicate the colour-coded $f_{{Q}}$ for grouped galaxies as a function of both stellar mass ( $\log~M_{\rm stellar}~[{\rm M}_{\odot}]$ ) and halo mass ( $\log~M_{\rm halo}~[{\rm M}_{\odot}]$ ), with the colour scale ranging from blue (predominantly star-forming) to red (predominantly quenched). The underlying distribution is computed using a two-dimensional KDE (KDE) with adaptive smoothing, where the $f_{{Q}}$ represents the ratio of the KDE-weighted density of quenched galaxies to the total KDE-weighted density in each cell. Black points indicate the positions of individual galaxies within the sample. The top panels presents a coloured histogram showing the $f_{{Q}}$ distribution of field galaxies as a function of $\log~M_{\rm stellar}~[{\rm M}_{\odot}]$ .

Figure 9. Long description

The image contains six panels arranged in a 2x3 grid. Panel A, Panel B, and Panel C in the top row are vertical bar graphs showing the distribution of field galaxies as a function of stellar mass (Mstellar) in logarithmic scale (log M☉) for different distance ranges from the galaxy cluster A4038. Panel A covers distances less than 5 megaparsecs (Mpc), Panel B covers distances between 5 and 15 Mpc, and Panel C covers distances between 15 and 25 Mpc. The x-axis represents stellar mass (Mstellar) in logarithmic scale (log M☉), and the y-axis represents the field count. Panel D, Panel E, and Panel F in the bottom row are heat maps indicating the quenched fraction (fq) of group galaxies as a function of both stellar mass (Mstellar) and halo mass (Mhalo) in logarithmic scale (log M☉). The color scale ranges from blue (predominantly star-forming) to red (predominantly quenched). The underlying distribution is computed using a two-dimensional kernel density estimation (KDE) with adaptive smoothing. Black points indicate the positions of individual galaxies within the sample. The heat maps show how the quenched fraction varies with stellar mass and halo mass at different distances from the galaxy cluster.

Figure 10 presents the corresponding radial trends for the star-forming population, using the same Nexus zone definitions. For this analysis we use the same star-forming sample defined in Section 3.2, following the selection described in Paper I. The figure is arranged by column, with one column per zone, and it shows both the mean $\log \mathrm{sSFR}\,[\!\log \mathrm{yr}^{-1}]$ and the standard deviation, $\sigma$ , of $\log \mathrm{sSFR}$ .

Figure 10.

Mean (top row) and RMS scatter (bottom row) of $\log\mathrm{sSFR}$ in the stellar–halo mass plane for Nexus group galaxies, split by cluster-centric distance from A4038. Columns show $d\lt 5$ Mpc (left), $5 \lt d \lt 15$ Mpc (middle), and $15 \lt d \lt 25$ Mpc (right). In each panel, colours are computed with a two-dimensional adaptive KDE over $\log M_{\rm stellar}$ and $\log M_\mathrm{halo}$ ; black points mark individual group galaxies and cells with insufficient sampling are masked (white). Top row: colour-coded mean $\log\mathrm{sSFR}$ [yr $\pm^{-1}$ ], from blue (higher sSFR) to red (lower sSFR). Bottom row: colour-coded RMS of $\log\mathrm{sSFR}$ , from magenta (lower scatter, more uniform sSFR) to green (higher scatter, more varied sSFR). The histograms above each column show the field-galaxy counts as a function of $\log M_{\rm stellar}$ for the corresponding radial shell, providing a baseline independent of group environment.

The $f_{Q}$ and mean $\log \mathrm{sSFR}$ maps in Figures 9 and 10 show clear structure across the plane in all three zones, with lower $f_{Q}$ and mean $\log \mathrm{sSFR}$ values occurring preferentially towards higher stellar mass, and additional variation with halo mass at fixed stellar mass in parts of the plane. Differences between zones are evident in the range of stellar mass and halo mass that is populated and in the spatial extent of low $f_{Q}$ and mean $\log \mathrm{sSFR}$ regions ( $f_{Q} \lt 0.6$ , $\log \mathrm{sSFR} \lt -10.8$ ), with the inner zone showing the strongest masking and small scale incompleteness due to limited sampling. The standard deviation maps show a distinct qualitative behaviour, where the strongest gradients in $\sigma$ occur along the stellar mass direction, with comparatively weaker variation with halo mass at fixed stellar mass. The field histograms provide additional context by showing how the stellar mass distribution of star-forming field galaxies changes across the three zones, and by indicating that, at high stellar mass, field galaxies remain less quenched and more strongly star-forming than group galaxies at similar stellar masses, while the stellar mass dependence of $\sigma$ is broadly similar between the field and group populations over the stellar mass range where the field statistics are well constrained.

Apparent differences between zones are present in the group population, but these should not be interpreted as evidence for an intrinsic distance-driven trend. The occupied regions of the stellar mass–halo mass plane differ strongly between zones, with each zone sampling a largely distinct halo-mass range and only a small number of groups. Accordingly, the observed changes in $f_{{Q}}$ and mean/ $\sigma$ sSFR across zones can be explained by differences in the halo mass function probed in each zone, rather than by projected distance from A4038 alone. The field population provides complementary context, but the same sampling limitation applies when comparing zone-to-zone behaviour.

Taken together, Figures 9 and 10 demonstrate that the sampled stellar mass–halo mass domain differs between zones, and that this difference strongly conditions the inferred $f_{Q}$ and sSFR patterns. We therefore do not infer that zone-to-zone variation is driven by projected distance itself. With the current data, the apparent radial differences are plausibly dominated by the differing halo-mass ranges sampled in each zone, compounded by small-number statistics. Robustly testing whether there is an additional large-scale environmental dependence at fixed stellar mass and fixed halo mass will require combining multiple analogous structures and analysing substantially larger survey volumes; importantly, even the SGP reference volume remains limited in its sampling of the most massive halos.

3.4 A4038 projected phase–space

As a final, cluster-scale view that complements the Nexus-wide radial trends (Section 3.3), we use the projected phase–space (PPS) diagram of A4038 to relate quenching to orbital phase within a single system. We consider PPS coordinates of the ratio between the projected radius of cluster members and the virial radius of the cluster ( $R_\mathrm{proj}/R_\mathrm{vir}$ ) and the absolute line-of-sight velocity between the cluster and the cluster galaxies normalised by the cluster velocity dispersion ( $|V_\mathrm{ LOS}|/\sigma$ ), and assign galaxies to eight PPS regions following the calibration of Pasquali et al. (2019), which used the Yonsei Zoom-in Cluster Simulations (YZiCS; Choi & Yi Reference Choi and Yi2017) to map positions within the PPS to a mean infall time, $\overline{T}_\mathrm{inf}$ . The adopted zone boundaries and corresponding $\overline{T}_\mathrm{ inf}$ values are shown in Figure 11. In this scheme, zones 1–3 correspond to earlier infall (older typical $\overline{T}_\mathrm{ inf}\!\gtrsim\!4.5$ Gyr) and zones 6–8 correspond to recent or first infall (younger typical $\overline{T}_\mathrm{inf}\!\lesssim\!2.8$ Gyr).

The $f_{{Q}}$ is measured independently within each PPS zone (Figure 12) using the same quenched classification adopted throughout this analysis. In the top panel, $f_{{Q}}$ is calculated for each of the eight PPS zones as the fraction of quenched galaxies relative to the total population, with associated uncertainties. In the bottom panel, we extend this analysis to account for stellar mass dependence shown in Sections 3.1 and 3.2 while preserving sufficient number statistics by grouping the PPS zones into three infall phases (Late, Mid, and Early), and computing $f_{{Q}}$ separately for low-mass ( $\log M_\mathrm{ stellar} \lt 10$ ) and high-mass ( $\log M_\mathrm{stellar}\geq10$ ) galaxies. The mean infall times for each zone, along with the 16th–84th percentile ranges that capture projection effects and orbital diversity, are listed in Table 3.

The top panel of Figure 12 shows that $f_{{Q}}$ s are higher in regions of projected phase–space associated with earlier infall than in regions associated with more recent infall. In particular, zones 1–3, which lie predominantly at low $R_\mathrm{proj}/R_\mathrm{vir}$ and/or low $|V_\mathrm{LOS}|/\sigma$ and correspond to older typical accretion times in the Pasquali et al. (2019) calibration (Figure 11), exhibit elevated $f_{{Q}}$ s (50%, 29%, and 52%, respectively). In contrast, zones 4–8, which extend to larger projected radii and/or higher line-of-sight velocities and are associated with more recent or first infall in the same mapping, show lower quenched fractions overall (23%, 17%, 25%, 17%, and 17%, respectively). While there is scatter between adjacent zones, the lowest $f_{{Q}}$ s occur in the most recent-infall regions (zones 6–8), consistent with the visual impression in Figure 11 that star-forming systems are more prevalent at larger $R_\mathrm{ proj}/R_\mathrm{vir}$ and higher $|V_\mathrm{LOS}|/\sigma$ .

The per-zone number counts annotated in Figure 12 clarify the statistical weight behind these measurements and help interpret the varying size of the uncertainty intervals. The earlier-infall zones (1–3) contain moderate sample sizes (11, 19, and 26 galaxies, respectively), whereas several of the outer/high-velocity zones contain substantially fewer galaxies (e.g. 7 and 11 galaxies in zones 7 and 8). This limited sampling contributes to broader confidence intervals and increases sensitivity to individual objects, particularly in the highest-zone indices where the PPS selection corresponds to extreme projected radii and/or velocities. Accordingly, differences between some neighbouring recent-infall zones should be interpreted cautiously, while the contrast between the earlier-infall group (zones 1–3) and the more recent-infall regime (zones 6–8) is more robust at the level supported by the current counts.

However, the bottom panel of Figure 12 shows that the apparent relationship between $f_{{Q}}$ and PPS zone is strongly dependent on stellar mass. When the sample is split into low- and high-mass galaxies, $f_{{Q}}$ remains approximately constant for low-mass systems as they progress from early- to late-phase infall, suggesting that these galaxies may be largely unaffected by the environmental processes associated with cluster accretion, or that quenching occurs on timescales longer than their infall history. In contrast, high-mass galaxies exhibit a clear increase in $f_{{Q}}$ from early- to late-phase cluster infall regions, indicating that their star formation is more sensitive to environmental effects such as ram-pressure stripping or tidal interactions. This divergence between low- and high-mass systems highlights the importance of stellar mass in regulating galaxy quenching and suggests that environmental quenching is most effective for massive galaxies that are already predisposed to earlier shutdown of star formation.

These PPS-based results are primarily descriptive. The mapping from PPS position to a mean infall time, $\overline{T}_\mathrm{inf}$ , is probabilistic and subject to projection effects, redshift-space distortions, and orbital diversity, and A4038 may not perfectly match the dynamical assumptions implicit in the simulation-based calibration. In addition, the top-panel $f_{{Q}}$ measurements do not control for stellar mass, while the bottom-panel split uses only a coarse two-bin stellar-mass partition with limited counts in some bins. The available sample sizes in the outer/high-velocity zones limit the precision of zone-to-zone comparisons. Within these limitations, the PPS analysis provides an orbit-informed, cluster-specific view showing that the $f_{{Q}}$ varies across A4038’s projected phase–space in a manner consistent with the stratification of typical accretion history shown in Figure 11.

Table 3.

Mean PPS infall-times from Pasquali et al. (2019) and adopted cluster phases. Values are means with asymmetric uncertainties given by the 16th and 84th percentiles.

Figure 11.

Phase-space diagram of A4038, showing eight infall zones based on mean infall time, adapted from Pasquali et al. (2019). The x-axis represents the projected cluster-centric radius normalised to the virial radius, and the y-axis shows the absolute line-of-sight velocity normalised to the velocity dispersion. The background colour corresponds to the mean infall time in gigayears, as indicated by the colour bar. Blue stars represent star-forming galaxies, while red triangles indicate quenching galaxies.

Figure 12.

Quenched fraction as a function of mean infall time for galaxies in A4038. Top panel: $f_{Q}$ as a function of the eight infall zones defined in Figure 11. Bottom panel: $f_{Q}$ grouped by infall phase (Late, Mid, and Early), further subdivided into low-mass ( $\log M_\mathrm{stellar} \lt 10$ ; blue circles) and high-mass ( $\log M_\mathrm{stellar} \geq 10$ ; red squares) populations to illustrate mass-dependent quenching. In both panels, error bars follow the global uncertainty prescription introduced at the start of this section. The fractions indicated below indicate the number of quenched galaxies relative to the total number of galaxies in that specific bin.

Figure 12. Long description

The image contains two scatter plots. Panel A: The top scatter plot shows the quenched fraction as a function of the eight infall zones defined in Figure 11. The horizontal axis represents the infall zones, labeled from Zone 1 to Zone 8. The vertical axis represents the quenched fraction. Data points are color-coded and include error bars. The fractions indicated below each zone show the number of quenched galaxies relative to the total number of galaxies in that specific bin. Panel B: The bottom scatter plot groups the data by infall phase (Late, Mid, and Early), further subdivided into low-mass (log M_stellar 10, represented by blue circles) and high-mass (log M_stellar 10, represented by red squares) populations. The horizontal axis represents the infall phase, and the vertical axis represents the quenched fraction. Data points include error bars. The fractions indicated below each phase show the number of quenched galaxies relative to the total number of galaxies in that specific bin.