How Does Environment Affect the Morphology of Radio AGN?

Of the sample of 185 bent doubles used for this study, 102 were chosen from a larger sample of 882 galaxies identified in the FIRST survey by the automated image-recognition techniques of Proctor (2006), which categorized radio galaxies as bent based on their morphology in FIRST. From this sample, 261 sources had optical counterparts that overlapped with the FIRST source whose spectroscopic redshifts were measured in SDSS DR7. We select these galaxies because we know their location in redshift space very precisely, so that when we perform our FoF algorithm, we know exactly what redshift we should be comparing to.

From this sample, we visually identified bent doubles based on the radio morphology with respect to the host galaxy. Examples of galaxies that have been visually selected are shown in the left two panels of Figure 1. The radio contours are in white and the location of the host galaxy is circled in red. This visual identification left us with an initial sample of 160 visually confirmed bent doubles with measured spectroscopic redshifts. The 101 radio galaxies identified by Proctor (2006) that we left out of our sample either did not visually resemble bent doubles or were not associated with the SDSS source. While it can be difficult to determine the precise morphology for many of our bent sources, due to resolution limitations, we can say that at least 75% of the bent double sources in our sample are WATs, which is slightly higher than the percentages found by Bhukta et al. (2021; 71% WATs) and Pal & Kumari (2021; 65% WATs) and significantly higher than that of Mingo et al. (2019; 42% WATs).

For unbent AGN and the rest of our bent AGN sample, we used the ROGUE catalog (Kozieł-Wierzbowska et al. 2020), which morphologically classified radio counterparts in FIRST and NVSS for galaxies with observed spectra in SDSS DR7. From their sample, we looked only at FRI and FRII galaxies and selected 191 unbent AGN based on a visual confirmation of the radio contours in FIRST with respect to the optical host galaxy. Of these, 64 are FRI radio galaxies and 127 are FRII radio galaxies. The right images in Figure 1 are two examples from this sample. We also selected an additional 83 bent doubles by looking at galaxies they identified as WATs or NATs. These galaxies did not appear in the selection from Proctor (2006), possibly due to the fact that the ROGUE catalog relies not only on FIRST but also on NVSS to identify AGN. It is possible that, without these additional observations, Proctor (2006) was unable to identify these sources as bent AGN.

While both samples were visually classified, they were selected using slightly different radio criteria. The bent sample from Proctor (2006) only used FIRST contours, while galaxies selected from the ROGUE catalog used both FIRST and NVSS, which resulted in a slightly lower redshift distribution. In order to avoid redshift-dependent differences between the two samples, we limit the rest of this analysis to only those galaxies below a redshift of 0.5, beyond which there are no unbent AGN in our sample. This reduced our sample of 243 bent AGN to a total of 185 bent doubles. The FIRST Flux distributions can be found in Figure 2 and shows that these samples are similar in their fluxes. The redshift distributions of both samples can be found in Figure 3 and are fairly consistent with each other to minimize redshift-dependent differences between the two populations.

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DECaLS and unWISE are two surveys that provide deep (r < 23) photometric data across the sky in the g, r, and z bands for DECaLS and the W1 and W2 filters for unWISE (Dey et al. 2019). To determine halo membership, we use the work of Zou et al. (2019), which measures the k-corrected absolute magnitudes, stellar mass, and photometric redshifts for galaxies observed by DECaLS and unWISE.

Figure 4 plots the k-corrected absolute magnitudes of the unbent and bent double sample, as well as all of the group members identified by the FoF algorithm. For reference, the blue line represents an apparent magnitude of r = 21. While DECaLS and unWISE do reach r < 23, we find that the galaxies with low photometric errors tend to be the brighter galaxies, as can be seen in Figure 5. Therefore, when applying a photometric error cutoff, we tend to exclude many of the dimmer galaxies that fall between 21 < r < 23. It is also noteworthy that the radio galaxies are on the whole much brighter than the other galaxies occupying the same halos.

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We use a Friends of Friends algorithm to identify possible galactic neighbors using their positions in the sky and measured photometric redshifts from DECaLS and unWISE (Zou et al. 2019). This method has been used throughout the literature to identify potential galaxy groups, though it is usually only applied to galaxies with measured spectroscopic redshifts (Duarte & Mamon 2014; Tempel et al. 2014). Errors on photometric redshifts are typically larger than potential group or even cluster velocity dispersions, making them particularly susceptible to false positives (i.e., falsely identifying a galaxy as a group member). In order to minimize the likelihood of false positives, we use a probability-motivated approach when considering the locations of galaxies in redshift space, whereas we use the FoF method when considering galaxy positions in the plane of the sky.

We calibrated and tested the algorithm on mock images created using the Illustris TNG300 simulation, which simulated a box of cosmic space that is 300 Mpc long on each side. This allowed us to sample a wide variety of different environments and see how the FoF algorithm performed on each of them.

3.1.1. FoF: Plane of the Sky

A typical FoF algorithm starts at the position of an initial galaxy and searches within a predetermined radius, known as the linking length, for nearby neighboring galaxies. When a neighboring galaxy is found within one linking length of the original galaxy, another search is done for additional galaxies within one linking length of the neighboring galaxy. This process is repeated for all galaxies that fall within one linking length of another galaxy until no more have been detected. Our FoF algorithm starts at the bent double and only includes galaxies that fulfill the redshift criteria outlined in Section 3.1.2.

There are a variety of linking lengths in the literature (Duarte & Mamon 2014), but many of these are established using data sets that vary drastically from ours in their completeness, largely due to the fact that they rely on spectroscopic observations instead of photometric observations. Spectroscopic observations allow them to scale the linking length by the number density of galaxies, calculated precisely due to the fact that spectroscopic redshifts have much lower uncertainties than photometric redshifts. Because we are relying on photometric redshifts, such scaling becomes unreliable. Therefore, we have chosen a static value of 300 kpc, which is motivated by the distribution of the distance to the nearest group galaxy as given by Illustris (see Section 3.3.2). This linking length will include the majority of the galaxies in the same group as the bent double based on the plot in Figure 6. We also set a maximum distance from the bent double at 2 Mpc, as groups are unlikely to extend that far, and if the algorithm does attempt to go that far, it is likely due to a false-positive detection (Tempel et al. 2014; Silverstein et al. 2018).

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3.1.2. Probability-based: Redshift Space

Because typical photometric redshift uncertainties tend to be much larger than expected redshift variations due to galaxy group/cluster velocity dispersions, we could not apply the FoF algorithm in redshift space. Instead, we devised a method of measuring the relative probability that two galaxies are at the same redshift based upon their uncertainties and set a cutoff, where any galaxies that fell below this were not considered to be at the redshift of the bent double.

We first assumed that the probability density functions P could be described by normal distributions with the mean μ being the measured redshift and the standard deviation σ being the uncertainties on said redshift. By multiplying the probability density functions for the bent double and the potential neighbor and normalizing by the maximum probability, we found the probability that these two galaxies reside at the same redshift.

The maximum possible probability was calculated by multiplying the probability distributions, assuming they had the same redshift μ_o with their measured uncertainties σ₁ and σ₂. The choice of μ_o is less relevant than the choices of the uncertainties:

$$\begin{eqnarray*}&&{P}_{\max }=\int P({\mu }_{o},{\sigma }_{1})P({\mu }_{o},{\sigma }_{2}).\end{eqnarray*}$$

The final probability was taken by multiplying the real probability density functions using μ and σ as the measured redshift and redshift uncertainty and dividing by ${P}_{\max }$:

$$\begin{eqnarray*}&&P=\displaystyle \frac{\int P({\mu }_{1},{\sigma }_{1})P({\mu }_{2},{\sigma }_{2})}{{P}_{\max }}.\end{eqnarray*}$$

Spectroscopic measurements of the bent double were always used for P(μ₁, σ₁), because these redshifts were all more accurate than photometric redshifts. Therefore, the redshifts of potential neighbor galaxies were always compared to the redshift of the bent double, as opposed to being compared to one another. If the final probability that the potential neighbor was at the redshift of the bent double was less than 0.2 (motivated in Section 3.3.3), it was removed from the group in order to reduce the number of potential false positives as much as possible.

To understand how well our algorithm performs and to tune various parameters, we carried out a series of tests using mock data created from the Illustris TNG300 simulations, which evolve a cubic volume with 300 Mpc on every side (Nelson et al. 2019). We used catalogs of subhalos (potential galaxy analogs) and groups of dark matter subhalos produced by the Illustris team as our starting point for generating mock observational data. We then ran the FoF algorithm described above to tune the parameters and find the best balance between a high percentage of completeness for each individual group and a low percentage of false group member identifications.

We started this analysis by creating a mock catalog of galaxy positions and assigning photometric redshift measurements based on their positions in the simulation volume. Using the subhalo catalog from the z = 0.10 snapshot of the TNG300 simulation volume, we narrowed our search by selecting only subhalos with absolute r-band magnitudes <−16 in order to immediately eliminate any subhalos that would not be visible with DECaLS. The entire simulation volume was then projected along the z-axis such that the front end was assigned z = 0.10 and redshifts were calculated for each subhalo using their position in the simulation volume along the z-axis. From here, right ascensions and declinations were calculated for each galaxy based on their position in the xy plane of the simulation and their calculated redshift.

Our next goal was to make this idealized catalog seem more like real DECaLS data. We started by calculating apparent r-band magnitudes for each subhalo and then assigning a redshift uncertainty based on real uncertainties from DECaLS, which vary with r, as shown in Figure 5. Using a 2D histogram of DECaLS photometric redshift uncertainties and r-band magnitudes, we applied a Gaussian kernel density estimate to smooth the histogram (as seen in Figure 5) and then, for each simulation subhalo, sampled from a range of uncertainties based on their calculated apparent r-band magnitudes. We then added a small offset to the redshift, based on the randomly assigned uncertainty. After this, we applied an apparent magnitude cutoff of r = 21.6. While DECaLS technically goes as deep as r = 23, after applying a photometric redshift uncertainty cutoff of 0.04, very few galaxies fell above a magnitude of 21.6, due to the fact that fainter galaxies had higher redshift errors, as can be seen in Figure 5. In Figure 4, the absolute magnitudes are plotted for our real sample of FoF-selected group galaxies and the bent/unbent host galaxies as a function of redshift, with the blue line representing the absolute magnitude for a galaxy of apparent magnitude r = 21 at various redshifts.

Once a final catalog of simulation subhalos was created, we then updated the group catalog to only consider subhalos that were in our final catalog, and set another cutoff such that our FoF algorithm only considered groups in which there were at least four subhalos that fulfilled the criteria outlined above. This allowed us to exclude groups of three or fewer that may have had large spatial separations and thus would not have been picked up by any FoF algorithm.

The parameters that had to be tested and tuned for the group finding algorithm are: the linking length, the photometric redshift uncertainty cutoff, and the cutoff for the probability that two galaxies are at the same redshift. To tune these parameters, we carried out a series of tests in which, for each group in the Illustris group catalog, we selected a random group member and ran the group-finding algorithm starting at that galaxy. We then compared the results of the group-finding algorithm to the real group catalog, in order to determine how well the algorithm did.

It was essential to select the parameters that resulted in groups that minimize the number of interlopers and maximize the completeness. Therefore, we specified and measured a couple of quantities for each group. The first, dubbed the false-positive rate, is defined as the percentage of galaxies, out of those identified by the algorithm, that were not real group members. The second, the completeness, is defined as the percentage of galaxies, out of the real group members, that were correctly detected by the algorithm. These values tend to be correlated with one another since, if the algorithm considers more galaxies as group members, it will be able to correctly identify more group galaxies that may be on the edges of the group, but it will also likely identify more false positives as well. Therefore, it is important to strike a reasonable balance between these two effects when calibrating parameter values.

We ran the group-finding algorithm on the entire group catalog multiple times in order to understand how each parameter affects the accuracy of the group finding results. To begin with, we selected initial values for each parameter and measured the completeness and false-positive rate for each group. Then, we began to vary one parameter at a time and looked at how the distribution of completeness and the false-positive rate for all of the groups changed, selecting the one that had the best balance between the two. We would then update that parameter value, keep it static, and begin varying the next parameter. This process was repeated for each of the three parameters until we found a set of values that maximized completeness while minimizing false positives. Then, we started over again, only now using the updated parameters as initial static values and then varying one parameter at a time to see if there were any changes. Figure 7 shows how the completeness and false-positive rate changed as we varied each parameter, with the solid gold lines indicating the final choice of parameter.

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Each of the three parameters that we varied came with its own unique set of challenges, which are summarized in the subsections below.

3.3.1. Photometric Redshift Uncertainty

The biggest downside to performing an FoF algorithm on data that use photometric redshift measurements as opposed to spectroscopic ones is that these tend to have much higher uncertainties due to the fact that they rely heavily on SED fitting of broadband filters. Because of this, we imposed an uncertainty cutoff for our photometric redshift error measurements. Before we carried out tests with Illustris, however, we began by examining how different uncertainty cutoffs affect how many galaxies we exclude from this analysis. This helped us decide on a range of values over which we performed testing with Illustris.

Comparisons between DECALS photometric redshifts and SDSS spectroscopic redshifts can be found in Figure 8, where we can see the overall scatter increase at higher redshifts. We also looked at the distribution of photometric redshift uncertainties for DECaLS galaxies, which can be seen in Figure 9, to understand how imposing a redshift cutoff would impact our final sample. Ideally, we would only be using galaxies whose photometric redshift uncertainties were low, in order to minimize the number of false positives identified by the algorithm. However, too low of a cutoff would leave out a large percentage of galaxies in DECaLS. Therefore, we decided to run the FoF tests on Illustris while varying the uncertainty cutoff between 0.02 and 0.05.

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The results of the final test are summed up by Figure 7, which shows how these quantities vary over a range of uncertainty cutoffs. Figure 9 shows the cumulative distribution of photometric redshift uncertainties for galaxies observed in DECaLS. We selected a redshift cutoff of 0.04 in the end, due to the fact that it provides a good balance between accuracy of the FoF tests without excluding too many galaxies from our analysis. This ends up including roughly 70% of the galaxies in DECaLS.

3.3.2. Linking Length

3.3.3. Probability Metric

We look into the properties of bent and unbent environments by comparing how often the galaxies hosting AGN are alone or in pairs, if they are the brightest galaxy in their group with three or more galaxies, or if they are in a group with three or more galaxies and are not the brightest galaxy. This information is summarized in Table 1. We choose these three categories to first identify how many galaxies are in underdense environments, such as singles or pairs. We then separate groups of three or more galaxies into groups where the AGN is the brightest and groups where the AGN is not the brightest. If the AGN host is the Brightest Group Galaxy (BGG), it is very likely that it has undergone an evolution defined largely by hierarchical structure formation of merging galaxies (De Lucia & Blaizot 2007), where it has been able to grow more massive than a typical large elliptical galaxy by accreting satellite galaxies that fall into the gravitational potential of the group (Gozaliasl et al. 2019; Pasini et al. 2021).

One of the most significant differences between the bent and unbent populations shown in Table 1 is the percentage that are alone or in pairs versus in groups of three or more galaxies. We looked at pairs and singles because galaxies in these environments are likely not embedded in a dense IGM like those in groups of three or more (Garon et al. 2019). In our study, 16.6% of bent doubles are in pairs, while 27.8% of unbent doubles are in pairs. Therefore, while it is unlikely to find a bent AGN alone or in a pair, they do exist, just not to the extent that unbent AGN do.

While the percentage of groups where the radio galaxies are the brightest is similar between the bent and unbent samples, a larger difference arises when looking at how many AGN hosts are alone/in pairs, or are not the BGG. These numbers become more stark when only considering galaxies in groups of three or more members. In 25.3% of the groups with bent AGN, the AGN host galaxy was not the brightest group galaxy, while this number was reduced to 8.1% when considering unbent AGN. This shows that AGN with unbent morphologies are less likely than bent AGN to exist in groups of three or more galaxies without being the BGG.

We performed a Pearson's chi-squared test on the data in Table 1 in order to see if the differences noted above are real, or if they could have been observed by chance. We found that we can reject the null hypothesis that this was a chance observation of the same population with a confidence level of over 99.99% and a p-value of 10⁻⁵.

To summarize these comparisons, unbent AGN are more likely to exist alone or in a pair than bent AGN. When in groups of three or more galaxies, unbent AGN are very likely to be the brightest cluster galaxy. For bent AGN, while they are still fairly likely to be the BGG, there is still a large population of bent AGN that exist in groups without being the BGG.

When we refer to the size of the group, we are referring to the number of group members identified by the FoF algorithm. The distribution of group membership for groups of three or more galaxies can be seen for groups occupied by both bent and unbent radio galaxies in Figure 11. The overall shapes of these distributions is somewhat similar, but it is clear that the unbent sample is skewed toward lower membership, while the bent sample reaches higher membership. Using the k-sample Anderson Darling test, we find that the null hypothesis can be rejected at a confidence level of 99.99%, indicating that these are two distinct populations.

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We estimate the projected group radius using the equation below:

$$\begin{eqnarray*}&&{R}_{\mathrm{group}}^{2}=\displaystyle \frac{1}{n}\sum _{i=1}^{n}{r}_{i}^{2}.\end{eqnarray*}$$

This is the mean square of r_i , the comoving distance from a galaxy i to the group center calculated using astropy's cosmology package (Astropy Collaboration et al. 2013, 2018). This equation is similar to that used by Tempel et al. (2014) to estimate the group extent. Distributions of radii for groups hosting bent and unbent radio AGN can be found in Figure 12, which shows that bent AGN are more likely to exist in groups with larger radii. We performed an Anderson–Darling test to determine the likelihood that these two distributions could be derived from the same parent population and found that we could reject the null hypothesis to a confidence level of 3σ, obtaining a p-value of less than 0.002. It is worth noting that the actual difference in radius of these two samples is fairly small, with the average group hosting bent AGN having a radius of 332 kpc and the average group hosting an unbent AGN having a radius of 259 kpc. It is also worth noting that the majority of all groups in our sample have radii below 500 kpc, providing quantitative justification for using 500 kpc as the radius within which we calculate the group density.

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We use multiple proxies as tracers of the density in galaxy groups: the number density of galaxies projected within 0.5 Mpc, between 0.5 and 2 Mpc of the group center, and the distance to the nearest group member. These distances are measured in the plane of the sky and we only consider galaxies with photometric redshift measurements that meet the probability criterion outlined in Section 3.2. By using these static values as opposed to the distance to the nth nearest neighbor, we are better able to probe a wide range of environments, from pairs of galaxies to groups that are on the verge of being clusters. Additionally, these measurements are less affected by potential false positives that could be identified by our algorithm.

The maximum projected distance a group member can be found from the bent double is 2 Mpc, due to the constraints placed on the algorithm (outlined in Sections 3.1.1 and 4.3). However, there are very few groups that actually reach this cutoff, and most groups do not extend too far beyond 0.5 Mpc (Tempel et al. 2014; Riggs et al. 2021). Therefore, when measuring the galaxy density of the core of the group, we consider all galaxies within 0.5 Mpc of the group center. This is partially motivated by the richness metric used by various studies in which the number of galaxies below an absolute magnitude threshold within 0.5 Mpc of a galaxy cluster or group center are counted (Allington-Smith et al. 1993; Zirbel 1997; Wing & Blanton 2011). A more recent study of clustering in galaxy groups found that most galaxy groups reach peak clustering at a radius of 0.5 Mpc, beyond which the clustering falls off (Riggs et al. 2021). For the galaxy density of the surrounding environment, we consider all galaxies between 0.5 and 2 Mpc, as galaxies at these distances from the group center are less likely to be gravitationally coupled to the group.

Figure 13 shows the results of our group-finding algorithm for groups hosting both bent and unbent galaxies. When considering the density of the inner 0.5 Mpc of the group, there is a clear distinction between bent and unbent doubles: bent doubles are more likely to reside in halos that are denser than their unbent counterparts. We performed a series of statistical tests in order to ensure that these two histograms are not being drawn from the same sample. Using the Anderson–Darling k-sample test, we found that the null hypothesis that these two samples are drawn from the same population can be rejected with a statistical significance of 0.1%. We also used the Mann–Whitney–Wilcoxon Test, which tests the location of separate distributions and compares them to see how similar the distributions themselves are. This test rejected the null hypothesis that these two distributions are the same with a reported p-value of less than 0.01. These separate statistical tests give us a reasonable degree of confidence when stating that bent doubles are indeed found in more dense groups than unbent doubles.

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When considering the density of the larger environment, the difference between bent and unbent doubles is less stark, but still present. We can reject the null hypothesis using the Anderson–Darling test at roughly 1.1% significance and using the Mann–Whitney–Wilcoxon Test with a p value of less than 0.01. This shows that even the larger environments, out to 2 Mpc, are more dense for bent AGN, if only slightly.

We also examine the distances to the nearest neighbor for galaxies in groups. Measuring the distance to the nearest neighbor for every galaxy in the group is another way to learn about the group environment and how densely packed galaxies in this environment are to one another. This is a proxy for the number density of galaxies in these groups, but it considers every galaxy in the group and is not bound by the limitations of selecting a static aperture size within which the density is measured. This means that calculating the nearest neighbor for every galaxy in the group will include group members that exist beyond 0.5 Mpc from the center of the group, making it a proxy for the density of the group as a whole instead of just the core of the group where we expect it to be at its most dense. However, it is important to note that this metric is particularly sensitive to false positives and incompleteness, as a single false-positive or missed group galaxy can throw off the nearest neighbor distance for many group members. Therefore, while this metric can be another proxy for the group density, it is important to note that these caveats may lead to it not being the most accurate proxy in this study.

The results of measuring the distances to the nearest neighbor for the entire sample can be found in Figure 14. There is a great deal of overlap between the two distributions, and while both the mean and median distance to the nearest neighbor for the bent sample are lower than the unbent sample, it is typically by 10–20 kpc. However, according the the Mann–Whitney and Anderson–Darling tests, the null hypothesis can be rejected at a statistically significant level with a p-value less than 0.01 for the Mann–Whitney test and a significance of roughly 0.7% for Anderson–Darling. It is important to note that the significance level of this measurement is not as high as those of the group densities, mainly because this particular metric is very sensitive to false positives and incompleteness of the group sample. Therefore, it is important to take the results of this metric with a grain of salt. That being said, these results are consistent with the results from the densities discussed earlier: bent doubles tend to exist in groups that have higher number densities than do unbent doubles.

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Because we are working with less precise photometric redshifts, we are unable to precisely measure the group dynamics. However, we can use the r-band magnitude gap to understand the dynamical history and age of the groups (discussed further in Section 5.1). The r-band magnitude difference between the brightest and second-brightest group galaxies can reveal something about the merger history of the group and thus about its dynamical age. Because galaxy groups are formed via hierarchical structure formation, the BGG will tend to gain mass via mergers with satellites (D'Onghia et al. 2005; De Lucia & Blaizot 2007; Oogi et al. 2016). This evolution leads BGGs to be statistically more massive and thus brighter than the next brightest group galaxy, and it has been shown to correlate with the r-band magnitude gap (Yang et al. 2008; Shen et al. 2014; Solanes et al. 2016; Dalal et al. 2021). In this study, we use the r-band magnitude gap as a proxy for the dynamical ages of the galaxy groups in our sample.

In Figure 15, we plot the Gaussian Kernel Density Estimates (KDEs) of the r-band magnitude differences using the difference between the first and second (m₁₂), the first and third (m₁₃), and the first and fourth (m₁₄) brightest group galaxies. Our justification for using multiple metrics is both because this is what has been used in the literature (Solanes et al. 2016) and to help offset any issues introduced by false positives. For all three metrics, the median magnitude gap is higher for the unbent sample by roughly 0.4 mag, and after applying the Anderson–Darling test, we can reject the null hypothesis that these distributions are the same to a statistical significance level of roughly 0.2%. The Mann–Whitney test corroborates this with p-values of roughly 10⁻³. These results indicate that bent AGN tend to exist in groups with lower magnitude gaps than their unbent counterparts.

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One of the downsides to using an FoF algorithm is the fact that the linking length is generally a difficult parameter to tune, and can result in very different-looking environments, depending on the choice. Additionally, our group-finding algorithm requires that we set a cutoff for the photometric redshift and a cutoff for the relative probability that two galaxies are at the same redshift, for a total of three selectable parameters. While the parameters were chosen in a systematic manner to optimize the balance between completeness and false-positive rate, it is prudent to examine the robustness of the main results across a range of parameter values.

We performed the group-finding algorithm on the real data two additional times: once with all of the parameters increased and once with them decreased. We varied the linking length by 100 kpc, the z uncertainty cutoff by 0.01, and the probability cutoff by 0.15. Tuning these parameters to their extreme values should result in the largest differences between our test groups and the groups we get using the original parameter values. With these test groups, we carried out the same statistical analyses on distributions of group membership, group and environmental density, and magnitude differences.

For both tests, we found that, even with the parameters at different values, the overall results stay the same. The only difference is that the results are not quite as statistically significant. The largest difference is that the magnitude gap in the low parameter case goes from a significance level of 0.2% to 3.4%. So, even in the most extreme cases, the statistical significance only changes by a few percent, indicating that our results are robust.

Hierarchical structure formation drives much of the evolution of BCGs and BGGs, causing them to grow in mass and luminosity, especially compared to the next-brightest galaxies in the group (D'Onghia et al. 2005; De Lucia & Blaizot 2007; Oogi et al. 2016). This leads to the r-band magnitude gap between the brightest and the second-brightest galaxy growing over time as the BCG continues to accrete mass and luminosity (Yang et al. 2008; Shen et al. 2014; Solanes et al. 2016; Dalal et al. 2021). Therefore, the r-band magnitude gap can be used as a tracer for the dynamical age and the current dynamics of a cluster or group (Shen et al. 2014; Lopes et al. 2018), with larger magnitude gaps indicating groups that are more settled and whose BGGs have had more time to accrete mass. On the other hand, smaller magnitude gaps could indicate that a group has recently formed or merged with another group, which could lead to the group being less dynamically relaxed—and could potentially result in turbulence in the IGM or galaxies with higher velocities, making conditions ideal for AGN jets to bend. This could also indicate a recent merger may have caused the BGG to be offset from the group center (Sakelliou & Merrifield 2000). The fact that 62.3% of bent doubles are the BGG could indicate that they exist in recently merged systems and are moving through a dense, turbulent medium toward the new group center. It is worth noting, however, that the findings of Wing & Blanton (2013) indicate that there is not a significant difference in the substructure of clusters hosting bent and unbent doubles, supporting the idea that cluster–cluster mergers are not solely responsible for bending AGN jets.

We find that, using a variety of metrics, groups hosting AGN with bent jets have lower r-band magnitude gaps, implying that they exist in groups that have formed more recently and whose BGGs have not had the time to grow appreciably due to hierarchical structure formation, or that they are part of groups that have recently merged, causing the two brightest galaxies to be similar in luminosity.

A dense environment is likely to play an important role in bending the jets for the most part, but there are still a number of AGN with bent jets in our sample that seemingly exist outside of groups or clusters. This is consistent with with the results of Garon et al. (2019), who found that bent AGN are not necessarily always located near known clusters, but still tend to exist in overdensities of 28ρ_crit (highly bent) and 18ρ_crit (less bent) from 50 to 350 kpc. Wing & Blanton (2011) also found that, while most galaxies hosting bent AGN existed in overdense regions, there were still a few that existed in environments with very few nearby neighbors. Edwards et al. (2010) found a bent double located in a cosmic web filament and were able to use it to measure the density of gas in the filament, and de Vos et al. (2021) found that NAT radio sources can exist out to 10R₅₀₀ of a nearby cluster. It could be that many of these bent doubles in singles and pairs are on the outskirts of a larger group or cluster and are embedded in a filament similar to that found in Edwards et al. (2010) or those observed by de Vos et al. (2021). Deeper, more complete spectroscopic observations of isolated bent doubles and any potential neighboring galaxies could provide us with a more complete picture of their environments, helping to better reveal the causes of bending in very sparse environments.

We find that 5.9% of unbent AGN in groups are not the BCG/BGG. There is evidence that a radio galaxy's position in a group/cluster depends on aspects of the radio source itself, such as the radio luminosity (Croston et al. 2019), with fainter radio galaxies existing further from the group/cluster center. Studies have also found AGN with unbent jets outside of the brightest cluster galaxy, even in higher-redshift clusters (Moravec et al. 2020). The fact that these galaxies are not bent might appear surprising, as we would expect a group/cluster member that is not the brightest galaxy to likely be moving through the group or cluster and thus be susceptible to ram pressure from the surrounding medium that would lead to bending. It could be possible that these particular galaxies are in environments with lower IGM densities, or that the jets are powerful enough to overcome the influence of ram pressure from the surrounding medium. Additionally, bending should decrease at higher distances from the cluster center, so the fact that there are some galaxies that are unbent and not located at the center of the cluster is not surprising (Garon et al. 2019). There is also a chance that these galaxies could actually host bent jets that are moving directly toward or away from us, such that we are unable to see the bending.