# Prevalence of Compact Nuclear Radio Emission in Post-Merger Galaxies and its
Origin

Gregory Walsh Department of Physics and Astronomy, West Virginia University,
Morgantown, WV 26506, USA  Center for Gravitational Waves and Cosmology, West
Virginia University, Chestnut Ridge Research Building, Morgantown, WV 26506,
USA  Sarah Burke-Spolaor Department of Physics and Astronomy, West Virginia
University, Morgantown, WV 26506, USA  Center for Gravitational Waves and
Cosmology, West Virginia University, Chestnut Ridge Research Building,
Morgantown, WV 26506, USA Gregory Walsh<EMAIL_ADDRESS>
###### Abstract

Post-merger galaxies are unique laboratories to study the triggering and
interplay of star-formation and AGN activity. Combining new, high resolution,
10 GHz Jansky Very Large Array (VLA) observations with archival radio surveys,
we have examined the radio properties of 28 spheroidal post-merger galaxies.
We find a general lack of extended emission at (sub-)kiloparsec scales,
indicating the prevalence of compact, nuclear radio emission in these post-
merger galaxies, with the majority (16/18; 89%) being radio-quiet at 10 GHz.
Using multi-wavelength data, we determine the origin of the radio emission,
discovering 14 new radio AGN and 4 post-mergers dominated by emission from a
population of supernova remnants. Among the radio AGN, almost all are radio-
quiet (12/14; 86%). We discover a new dual AGN (DAGN) candidate, J1511+0417,
and investigate the radio properties of the DAGN candidate J0843+3549. 4 of
these radio AGN are hosted by SF emission-line galaxies, suggesting that radio
AGN activity may be present during periods of SF activity in post-mergers. The
low jet powers and compact morphologies of these radio AGN also point to a
scenario in which AGN feedback may be efficient in this sample of post-
mergers. Lastly, we present simulated, multi-frequency observations of the 14
radio AGN with the Very Long Baseline Array (VLBA) and the VLBI capabilities
of the Next Generation Very Large Array (ngVLA) to assess the feasibility of
these instruments in searches for supermassive black hole binaries (SMBHBs).

Galaxy mergers (608); Active galactic nuclei (16); Radio active galactic
nuclei (2134); Radio jets (1347); Supernova remnants (1667); Evolution of
galaxies (594)

††facilities: VLA, LOFAR, ASKAP, Galex, WISE††software:  CASA (THE CASA TEAM
et al., 2022), Numpy (van der Walt et al., 2011), Scipy (Virtanen et al.,
2020), Matplotlib (Hunter, 2007), Astropy (Astropy Collaboration et al., 2018)

##  1 Introduction

Theoretical studies predict that galaxy mergers are the main contributor to
the buildup of stellar mass in galaxies and to the formation of bulges and
massive elliptical galaxies (Springel, 2000; Cox et al., 2008; Di Matteo et
al., 2008; Torrey et al., 2012). Integral to this evolution is what role
galaxy mergers have in the triggering of an active galactic nucleus (AGN)
and/or intense starburst activity. Early studies of ultra-luminous infrared
galaxies (ULIRGs), which are at least partially powered by a heavily obscured
AGN (Lonsdale et al., 2006), found a nearly ubiquitous fraction hosted by
interacting galaxy systems (Murphy et al., 1996; Veilleux et al., 2002). These
early works suggested that ULIRGs are triggered by galaxy merger-induced
processes. Likewise, intense starburst activity has been observed in merging
systems (Tacconi et al., 2008). For both cases, the triggering of these
phenomenon are caused by nuclear inflows of gas produced by gravitational
torques during the merger process (Hopkins et al., 2006), linking the growth
of supermassive black holes (SMBHs) and their host galaxies. Indeed, observed
correlations between the SMBH and host galaxy properties confirm their co-
evolution (Kormendy & Richstone, 1995; Magorrian et al., 1998; Ferrarese &
Merritt, 2000; Gebhardt et al., 2000; Tremaine et al., 2002; Gültekin et al.,
2009; McConnell & Ma, 2013; Sahu et al., 2019). Thus, detailed studies of
galaxy mergers at various stages of evolution are needed to fully realize the
astrophysical processes governing these phenomena.

Observational studies are in conflict with one another over the role mergers
play in triggering an AGN. While many have found either an increased AGN
fraction in merging systems (Ellison et al., 2011; Satyapal et al., 2014a;
Donley et al., 2018; Goulding et al., 2018) or an increased merger fraction in
AGN hosts (Chiaberge et al., 2015; Fan et al., 2016; Gao et al., 2020; Marian
et al., 2019; Breiding et al., 2023), others have found no such connection
between AGN and mergers (Grogin et al., 2005; Cisternas et al., 2011; Böhm et
al., 2013; Villforth et al., 2017; Lambrides et al., 2021). Selection biases
almost certainly contribute to this dissonance. Different AGN selection
criterion, e. g., mid-infrared (e. g., Satyapal et al., 2014b; Donley et al.,
2018; Goulding et al., 2018), X-ray (e. g., Grogin et al., 2005; Villforth et
al., 2017), optical (e. g., Böhm et al., 2013), and radio (e. g., Chiaberge et
al., 2015; Breiding et al., 2023), and selection of mergers at various stages
of their evolution also necessarily select different astrophysical scenarios
(e. g., Sanders et al., 1988).

Among the evolutionary stages of merger systems, post-merger galaxies, those
in which the stellar nuclei have coalesced, perhaps present the most unique
laboratories to study these triggering mechanisms and the effects of AGN
feedback on star formation as a result of the advanced stage of the merger.
Small samples of post-merger systems have found hints at enhancement in the
star-formation rate (Ellison et al., 2013) and AGN incidence over galaxies in
close pairs (Carpineti et al., 2012; Bickley et al., 2023). The effectiveness
of AGN feedback, however, is questioned when examining post-merger galaxies.
Post-mergers appear to host a significantly higher fraction of post-starburst
galaxies (Ellison et al., 2022; Li et al., 2023), characterized as having
recently experienced intense star-formation activity that was rapidly
truncated (Couch & Sharples, 1987), although this itself does not imply that
an AGN is the main driver of this quenching. Indeed, this low efficiency
scenario is corroborated by the works of Kaviraj et al. (2015) and Shabala et
al. (2017). Both of these studies determined that the onset of merger-
triggered AGN activity is delayed with respect to the peak of starburst
activity, significantly limiting its ability to impact the star-formation rate
of the host galaxy. Expanding the overall population of post-merger galaxies
for which we can study these evolutionary effects is important towards
understanding the general trends in their behavior.

Post-merger galaxies are also ideal targets in which to search for
supermassive black hole binaries (SMBHBs). As all massive galaxies are
believed to harbor a SMBH (Kormendy & Ho, 2013), a major galaxy merger should
lead to the formation of a SMBHB. At the initial stages of SMBHB evolution,
dynamical friction is the dominant mechanism through which the SMBHs lose
energy and momentum (Begelman et al., 1980), eventually settling into the
gravitational center of the merger remnant. Simulations of galaxy mergers have
found that this phase of the evolution may be as short as 1 Gyr (Dosopoulou &
Antonini, 2017; Kelley et al., 2017), shorter than the timescale over which
the bulges will centralize. Thus, by the time the stellar nuclei have merged,
the resident SMBHs are likely to already reside in the gravitational center of
the merger remnant (Cf. Dvorkin & Barausse, 2017; Kelley et al., 2017). At
parsec-scale separations, the SMBHs will form a gravitationally-bound SMBHB.
Here, several different processes, of varying efficiency, are hypothesized to
contribute to the shrinking of the binary’s orbit; the so-called ‘last parsec’
problem. If the SMBHB is able to overcome the last parsec, it will reach sub-
pc orbital separation, where the emission of low-frequency gravitational waves
will efficiently bring the binary to merger. Thus, establishing a population
of observed SMBHBs at various orbital separations is key towards our
understanding of the nHz gravitational wave population, which will soon be
probed by pulsar timing arrays (Agazie et al., 2023a, b). Critical to this
aspect, however, is the poorly understood evolution of SMBHBs themselves. Low
efficiency at pc-scale can create a scenario in which binaries at these
orbital separations are still present in the post-merger host galaxy.
Observations of galaxy mergers at this post-merger phase must be taken to
better understand SMBHB evolution.

In this paper we present a multi-wavelength analysis of 30 galaxies identified
as post-mergers in Galaxy Zoo to study their emission mechanisms. The paper is
organized as follows. Section 2 presents the post-merger sample. Section 3
describes new 10 GHz observations taken with the Karl G. Jansky Very Large
Array (VLA) of this post-merger sample, while in Section 4, we describe the
multi-frequency data obtained via archival radio surveys. The optical
emission-line classifications of each post-merger galaxy, and the radio
luminosities, morphologies and spectra of these sources are presented in
Section 5. In Section 6, we present analyses to determine the origin of the
radio emission in our 10 GHz-detected post-merger galaxies. We then discuss
the prevalence of radio-quiet emission in these post-mergers, the impact of
AGN feedback in radio AGN hosts, and the properties of the SF-dominated radio
sources in Section 7. Lastly, in Section 8, we present simulated, multi-
frequency observations of the radio AGN with the Very Long Baseline Array
(VLBA) and Next Generation VLA (ngVLA) to assess the feasibility of SMBHB
searches for these post-merger galaxies. Our results are summarized in Section
9.

Throughout this paper, we have adopted a $\Lambda$CDM cosmology with
$H_{0}=67.4$ km s-1 Mpc-1 and $\Omega_{m}=0.315$ (Planck Collaboration et al.,
2020). We use the radio spectral index convention
$S_{\nu}\propto\nu^{\alpha}$.

##  2 Sample

Our sample consists of the 30 spheroidal post-merger (SPM) galaxies presented
by Carpineti et al. (2012, hereafter C12). The C12 sample was selected from
the larger sample of Darg et al. (2010). These authors constructed a catalog
of 3003 local (0.005 $<z<$ 0.1) galaxy merger systems identified through
visual inspection via the Galaxy Zoo Project (Lintott et al., 2008). Of these
3003, 370 of these merging systems were considered strongly perturbed, e. g.,
the presence of strong tidal tails, but could not be clearly divided into a
pair of two interacting galaxies. From the parent sample of 370 late-stage
merger systems, C12 selected their sample of 30 SPM galaxies via the distinct
visual characteristics of SPMs: SPM galaxies are defined as a single galaxy
that displays morphological disturbances associated with a recent merger
event, e. g., tidal tails, and contain only a single dominant bulge, making
them the likely progenitors of early-type galaxies. As a consistency check,
after visual inspection of each SPM candidate, C12 utilized the SDSS parameter
`fracdev` in the optical $r$-band for a quantitative representation of the
bulge-dominant nature of each system. `fracdev` represents the likelihood of
the surface brightness distribution to be fit by a de Vaucoulerus profile:
pure bulge systems have a value of 1; pure exponential or disc-like
distributions have a value of 0. All of the 30 SPM galaxies have `fracdev`
$>0.5$, signifying they are bulge dominated, with most having `fracdev` $\geq
0.8$. Further, the 30 SPM systems are all of high stellar mass (10.3
$\leq\mathrm{log}(M_{*})\leq$ 11.76), typical of early-type galaxies.
Additionally, C12 found that these 30 SPM systems are diverse in their large-
scale environments. Using the environment parameter $\rho_{g}$ defined by
Schawinski et al. (2007b), C12 found two SPM systems which inhabit a cluster
environment, 19 in a group environment, and 9 in a field environment.

##  3 Observations and Data Calibration

High resolution, Karl G. Jansky Very Large Array (VLA) observations of the 30
SPM galaxies were taken from 2016 to 2022. J1018+3613 (catalog WISEA
J101833.64+361326.7) was observed on May 26, 2016 (PI: S. Burke-Spolaor) using
the S- (2-4 GHz) and X-band (8-12 GHz) receivers of the VLA while in B
configuration. 3C 186 was used for bandpass calibration and J1018+3542 was
used to perform phase referencing. J0206$-$0017 (catalog Mrk 1018), J1445+5134
(catalog VV 713), J1511+0417 (catalog IC 1102), J1511+2309 (catalog CGCG
135-026) and J1655+2639 (catalog IC 4630) were observed in two observing
programs on November 15, 2020 and December 12, 2020 (PI: P. Breiding) using
the S- and X-band receivers of the VLA while in BnA and A configuration. 3C
286 and 3C 138 were used for bandpass calibration and a nearby phase reference
calibrator for phase calibration of each target. We obtained observations for
the remaining 24 SPM galaxies on May 19, and May 28, 2022 using the X-band
receiver of the VLA with 4 GHz bandwidth while in A configuration. We observed
3C 147 and 3C 286 for bandpass calibration and a nearby phase reference
calibrator for phase calibration of each target. Our observations were
designed to reach a nominal sensitivity of $15\,\mu$Jy for each target, with a
$3\sigma$ detection threshold of $45\,\mu$Jy. Two of the SPM galaxies,
J0908+1407 and J0933+1048, were removed from the analysis due to technical
issues during the observing session. For that reason, only the remaining 28
SPM galaxies are included in this analysis.

The data sets were calibrated either using the VLA calibration
pipeline111https://science.nrao.edu/facilities/vla/data-processing/pipeline or
following standard data calibration techniques. The VLA calibration pipeline
could not be used for the observing session that contained technical issues.
To check for calibration consistency, we calibrated the data set without
technical errors using both the VLA calibration pipeline and our manual
calibration routine. We achieved the same results using both calibration
methods for this data set.

The data were inspected, flagged, and imaged in the Common Astronomy Software
Applications (CASA; THE CASA TEAM et al. 2022) package. To account for the
large fractional bandwidths at each band, $\sim$ 60% and $\sim$ 40%
respectively for S- and X-band, we used multi-Taylor, multi-frequency
synthesis deconvolution (MTMFS; Rau & Cornwell, 2011) when cleaning our
images. Because of the limited $uv$-coverage from our observations, we
utilized a Briggs weighting scheme (Briggs, 1995) with a robust parameter of
0.7 to suppress the sidelobes present in images of moderately strong sources
($S_{\nu}>1$ mJy). Otherwise, we used a natural weighting scheme when imaging.
Where applicable, we performed standard self-calibration techniques on the
target data to improve the image quality.

##  4 Radio Surveys

We wish to construct a broadband radio SED for each of the 28 SPM galaxies in
our survey. In this section, we describe the surveys used to construct each
SED.

It is important to note that each survey, observed at a different frequency
and with different angular resolutions, is, by nature, sensitive to different
forms of radio emission and may or may not suffer from source confusion. Low
frequency ($<1$ GHz) surveys are more sensitive to diffuse emission, likely
associated with star formation, and are more likely to suffer from source
confusion due to their larger resolution elements. High frequency surveys, in
contrast, generally resolve out this same extended, diffuse emission if it is
of low surface brightness, making them good identifiers of compact radio jets
and cores, features associated with an AGN. This can create an artificial
steepening of the radio spectrum due to the high frequency surveys missing
flux density information recovered at lower frequency for diffuse emission, or
confusion from background source blending.

We attempted to mitigate these issues by visually inspecting all of the survey
images used in our analysis. In particular, we are interested in the spectral
index measurement of the nuclear radio emission, which will be obtained
through the 3 and 10 GHz flux density measurements, or their limits (see
Section 5.4). Background source blending is not an issue because of the high
angular resolution at these frequencies and the low redshift ($z<0.1$) of
these SPMs. For lower frequency observations, visual inspection showed no
complex structure in the vast majority of the radio maps. For those select few
that do have complex structure, we describe our procedures for their flux
density measurements in Section 5.3. This is likewise true for any high
frequency image that showed complex structure.

For each radio survey, we considered a source to be detected if it was found
in an available source catalog, e. g., LoTSS, RACS, and FIRST, or the signal-
to-noise (S/N) ratio in the respective survey image is $>5\sigma$, where
$\sigma$ is the local image RMS noise. In some cases, a source was identified
at only a $3\sigma$ significance. We considered the $3\sigma$ source a true
detection if it was spatially coincident with a source of $\geq 5\sigma$
detection in any of the other radio surveys. Each image was inspected visually
to assure that no sources were missed. It is important to distinguish this to
properly account for the difference in sensitivities between the various radio
surveys used. If we only classified sources at the $5\sigma$ level and
greater, our multi-frequency analyses would be incomplete and not truly
representative of the SPM sample radio population within the limits of each
survey. The 10 GHz map of J1015+3914 (catalog KUG 1013+394), presented in
Figure 14, illustrates this point. The diffuse, $3\sigma$ emission at 10 GHz
would not by itself be substantial for a detection. However, J1015+3514 is
detected at high signal-to-noise ratio in all other surveys that observed it,
and this diffuse 10 GHz emission is spatially coincident with those
detections. For this reason, we consider the 10 GHz source a true detection
and include it as part of our analyses.

For $3\sigma$ sources, we determined the flux density by performing a 2D
Gaussian fit to the observed radio emission using the task `IMFIT` in CASA.

###  4.1 LoTSS

The LOw Frequency ARray (LOFAR; van Haarlem et al., 2013) Two Meter Sky Survey
(LoTSS; Shimwell et al., 2022) is an on-going survey covering the northern sky
above $+34\arcdeg$ conducted at a central observing frequency of 144 MHz. For
our analysis, we use the second data release222https://lofar-
surveys.org/dr2_release.html, which covers 27% of the northern sky with a
resolution of 6$\arcsec$ and median RMS sensitivity of 83 $\mu$Jy beam-1. 16
of the 28 SPM galaxies in our survey fall within the LoTSS DR2 sky coverage,
of which 12 were detected.

###  4.2 RACS

The Rapid ASKAP Continuum Survey (RACS; McConnell et al., 2020; Hale et al.,
2021) is the first large-area survey completed using the Australian Square
Kilometer Array Pathfinder (ASKAP; Hotan et al., 2021). RACS covered the
entire southern sky up to a declination $+41\arcdeg$ with a median field RMS
sensitivity of 250 $\mu$Jy beam-1. RACS-low, as part of the RACS
DR1333https://research.csiro.au/casda/the-rapid-askap-continuum-survey-stokes-
i-source-catalogue-data-release-1/, was observed at a central frequency of
887.5 MHz with a resolution of $15\arcsec$. 14 of the 28 SPM galaxies in our
survey fall within the RACS sky coverage, of which 7 were detected.

###  4.3 FIRST

The Faint Images of the Radio Sky at Twenty Centimeters (FIRST; Helfand et
al., 2015) survey was a VLA survey conducted at 1.4 GHz and observed the
entire sky north of $+10\arcdeg$ and south of $+65\arcdeg$, covering 10,575
deg2. The survey resolution is given at 5$\arcsec$ with a typical RMS
sensitivity of 150 $\mu$Jy beam-1. 27 of the 28 SPM galaxies in our survey
fall within the FIRST sky coverage, of which 14 were detected. We used the
flux density and RMS values listed for each source from the catalog of Helfand
et al. (2015), except for 3$\sigma$ sources, which were not included in this
catalog.

###  4.4 NVSS

The NRAO VLA Sky Survey (NVSS; Condon et al., 1998) was a VLA survey conducted
at 1.4 GHz and observed the entire sky north of $-40\arcdeg$. The nominal
resolution of the survey is 45$\arcsec$ with a typical RMS sensitivity of 450
$\mu$Jy beam-1. Because of the large angular resolution, we used the NVSS
catalog data for only one source, J1304+6520 (catalog WISEA
J130414.50+652044.2), which was not part of the FIRST sky coverage, but was
observed and detected by NVSS.

###  4.5 VLASS

The VLA Sky Survey (VLASS; Lacy et al., 2020) is an on-going VLA survey
conducted at S-band, covering the frequency range 2-4 GHz, which will cover
the whole sky observable by the VLA ($\delta>-40\arcdeg$) over three observing
epochs. Each observing epoch is designed to reach a nominal RMS sensitivity of
120 $\mu$Jy beam-1 with a resolution of 2.5$\arcsec$. VLASS has currently
completed two observing epochs, with raw and calibrated data sets and Quick
Look images available for both epochs 1 and 2. The flux density accuracy of
Quick Look sources in the first campaign of the first epoch of VLASS
(VLASS1.1) were affected by antenna pointing errors, giving systematically
lower flux density measurements of 10% with a scatter of $\pm 8$% for flux
densities below $\approx 1$ Jy (see VLASS Memo
13444https://library.nrao.edu/public/memos/vla/vlass/VLASS_013.pdf for more
detail). For this reason, we used only the campaigns from the second epoch of
VLASS (VLASS2.1 and VLASS2.2) for the S-band flux density of sources of
interest. As mentioned in Section 3, we observed 6 of the SPM galaxies in
separate VLA observing campaigns at S-band. We used these 3 GHz VLA
observations for these sources to derive source parameters instead of any
corresponding VLASS detections for them. The remaining 22 SPM galaxies have
all been observed in the second VLASS campaign, of which 7 were detected. To
extract the flux density of the detected sources, we used the CASA task
`IMFIT` to fit a two-dimensional Gaussian to the source in each Quick Look
image.

##  5 Source Properties

Following the detection criterion of Section 4, 75% (12/16) of the sources
with available LoTSS data were detected; 50% (7/14) with available RACS data
were detected; 54% (15/28) with available 1.4 GHz data, from either FIRST or
NVSS, were detected; 32% (7/22) with available VLASS data were detected, with
a 100% detection rate for the remaining 6 with separate 3 GHz VLA
observations; and 67% (18/28) were detected by our 10 GHz VLA observations.

###  5.1 Emission-Line Activity

Figure 1: The BPT diagnostic diagrams for the emission-line galaxies in the
C12 SPM sample. Each point is colored by its $u-r$ color, with the colorbar
indicating the range of values in the scale. Even for the most blue SPMs, the
SPMs are still predominantly red in color. The dashed line in the
[NII]$\lambda 6583$/H$\alpha$ diagram (left panel) is the empirical SF line of
Kauffmann et al. (2003) and the straight line that divides Seyferts and LINERs
is from Schawinski et al. (2007a). Seyferts and LINERs are divided in the
[SII]$\lambda 6717$/H$\alpha$ (middle panel) and [OI]$\lambda 6300$/H$\alpha$
(right panel) diagrams from the line of Kewley et al. (2006). In all diagrams,
the solid line is the theoretical maximum from the starburst models of Kewley
et al. (2001). Galaxies that fall between the lines of Kauffmann et al. (2003)
and Kewley et al. (2001) are SF-AGN composite, while those below Kauffmann et
al. (2003) are purely SF.

Emission-line diagnostics are a powerful tool to probe the dominant ionization
mechanism in a galaxy. To examine the emission-line behavior of the 30 SPM
galaxies, we have used the OSSY catalog (Oh et al., 2011) to obtain the
intrinsic fluxes of the H$\beta$, [OIII]$\lambda 5007$, H$\alpha$,
[NII]$\lambda 6583$, [SII]$\lambda 6717$, and [OI]$\lambda 6300$ emission
lines. Oh et al. (2011) determined these values by performing a spectral
fitting routine to the SDSS DR7 spectrum of each source. If the signal-to-
noise (S/N) ratio of the H$\beta$, [OIII]$\lambda 5007$, H$\alpha$ or
[NII]$\lambda 6583$ lines was $<3$, we classified the galaxy as quiescent. For
the remaining galaxies, we followed the standard BPT diagram analysis (Baldwin
et al., 1981). For the [NII]/H$\alpha$ diagnostic, we used the demarcation of
Kauffmann et al. (2003) to distinguish between pure star-forming (SF) and SF-
AGN composite galaxies. Composite galaxies and AGN are divided using the
theoretical maximum starburst model from Kewley et al. (2001). AGN are then
subdivided between Seyferts and LINERs (Low-Ionization Nuclear Emission-line
Regions) by the division of Schawinski et al. (2007a). The best indication of
Seyfert or LINER behavior is achieved by using the [OI]$\lambda 6300$ emission
line (Schawinski et al., 2007a). However, the [OI]$\lambda 6300$ line is
typically weaker than any of the other lines used and we only employed this
diagnostic if the [OI]$\lambda 6300$ line was detected with a S/N ratio $\geq
3$. Otherwise, we employed the [SII]$\lambda 6717$ diagnostic to distinguish
between Seyfert AGN and LINERs. For these two diagnostics, we used the
Seyfert-LINER demarcation lines of Kewley et al. (2001). If neither line was
detected, we used the [NII] diagnostic to distinguish between Seyferts and
LINERs.

The results of our BPT analysis are presented in Figure 1, where each data
point colored by its $u-r$ color. The emission-line classification of each SPM
galaxy is listed in Table 1. It should be noted that for even the bluest of
the SPM galaxies in the C12 sample, their overall $u-r$ color is still
predominantly red. This is expected, as C12 found that the $u-r$ colors of
this SPM sample is indicative of a recent star-formation episode, e. g,. bluer
than an early-type control sample, but one that peaked prior to the merger
coalescence, e. g., redder than a sample of ongoing mergers (see Figure 5 of
C12).

The BPT diagnostic for J0206$-$0017 deserves special attention. The middle
panel of Figure 1 shows only 16 of the 17 identified active galaxies. This is
because the data point for J0206$-$0017 has log([SII]$\lambda
6717$/H$\alpha$)=-1.27. In comparison to the much larger sample of active SDSS
galaxies used by Kewley et al. (2006), there are no galaxies which approach
this value of J0206$-$0017\. This is most likely attributable to the fact that
J0206$-$0017 is a known changing-look AGN with asymmetric broad-line emission
(Cohen et al., 1986; McElroy et al., 2016). The prescription used by OSSY to
determine the line fluxes would not have accounted for the extremely broad
nature of the H$\alpha$ and H$\beta$ lines for this source, and we would most
likely need to perform our own spectral fitting routine to extract a reliable
flux value for the narrow emission-line components to these broad lines.
Because of this, we have classified J0206$-$0017 as a Seyfert AGN instead of
as a star-forming galaxy as would be determined by its BPT diagnostics.

We also note that the spectra of J0908+1407, J1511+2309, and J1655+2639 all
contain H$\beta$ absorption. In all of these cases, the H$\beta$ absorption
appears to be of stellar origin. Through visual inspection, there does not
appear to be a significant blueshift in the H$\beta$ absorption, which would
be representative of an AGN-related outflow (e. g., Williams et al. 2017). For
any H$\beta$ emission present in these sources, the S/N ratio of the emission
line was $<3$. Although the emission lines of [OIII]$\lambda 5007$, H$\alpha$,
and [NII]$\lambda 6583$ are all detected with S/N ratio $\geq 3$ in these
spectra, for consistency, we classified them as quiescent because of the weak
H$\beta$ emission.

In total, we found that 43% (13/30) of the C12 SPM galaxies were classified as
quiescent. The remaining $\approx$57% (17/30) were classified as either purely
SF (10%; 3/30), SF-AGN composite ($\approx$13%; 4/30), Seyfert AGN
($\approx$13%; 4/30), or LINER (20%; 6/30) from their BPT diagnostics. In
comparison to the emission-line diagnostics performed by C12, our analysis
finds a higher percentage of quiescent galaxies ($16\%\pm 6\%$ to 43%), a
lower percentage of Seyfert AGN ($42\%\pm 6\%$ to 13%), and a similar
percentage of LINERs ($26\%\pm 6\%$ to 20%) and star-forming galaxies
($16\%\pm 6\%$ to 10%). Direct comparison is somewhat ambiguous though, since
C12 did not use the SF-AGN composite classification for their BPT analysis. It
is unclear where the composite systems we identified would fall in the
analysis of C12. It is interesting, however, that we arrive at different
conclusions for the number of quiescent galaxies considering both the OSSY
catalog and C12 used the `gandalf` code (Sarzi et al., 2006) to perform
emission-line fitting of the spectra. We would expect, then, that the S/N
ratio of the requisite emission lines would not change between the two
analyses. Even if the 3 H$\beta$ absorption spectra are considered as active
galaxies by C12, this only marginally reduces the percentage of quiescent
galaxies we have identified from 43% to 30%, which is still a factor of 2
greater than what was found by C12.

Table 1: Spheroidal Post-Merger Sample & BPT Classification

Source | RA | Dec | $z$ | BPT
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
J0206$-$0017 | 31.567 | -0.291 | 0.043 | AGN
J0759+2750 | 119.952 | 27.839 | 0.067 | Composite
J0833+1523 (catalog WISEA J083309.47+152353.4) | 128.289 | 15.398 | 0.076 | Quiescent
J0843+3549 | 130.937 | 35.828 | 0.054 | AGN
J0851+4050 | 132.978 | 40.836 | 0.029 | LINER
J0908+1407 | 137.156 | 14.122 | 0.088 | Quiescent
J0916+4542 | 139.212 | 45.700 | 0.026 | Composite
J0933+1048 | 143.447 | 10.811 | 0.085 | Quiescent
J1015+3914 | 153.992 | 39.243 | 0.063 | Starforming
J1018+3613 | 154.640 | 36.224 | 0.054 | AGN
J1041+1105 | 160.266 | 11.096 | 0.053 | LINER
J1056+1245 (catalog WISEA J105646.98+124544.9) | 164.196 | 12.762 | 0.092 | Quiescent
J1113+2714 | 168.419 | 27.241 | 0.037 | Starforming
J1117+3757 (catalog WISEA J111732.56+375746.9) | 169.385 | 37.963 | 0.096 | LINER
J1124+3005 | 171.142 | 30.095 | 0.055 | LINER
J1135+2913 | 173.781 | 29.891 | 0.046 | Starforming
J1144+2309 (catalog SPRC 027) | 176.183 | 23.162 | 0.048 | Quiescent
J1230+1146 (catalog WISEA J123013.16+114613.3) | 187.554 | 11.770 | 0.089 | Quiescent
J1253+3944 (catalog WISEA J125350.03+394416.9) | 193.458 | 39.738 | 0.092 | Quiescent
J1304+6520 | 196.060 | 65.346 | 0.083 | AGN
J1314+2607 (catalog ARP 196 NED01) | 198.656 | 26.123 | 0.074 | Quiescent
J1326+5653 (catalog WISEA J132654.49+565320.7) | 201.726 | 56.889 | 0.090 | Quiescent
J1405+4001 (catalog 2MASX J14053947+4001556) | 211.414 | 40.032 | 0.084 | Quiescent
J1433+3444 | 218.327 | 34.735 | 0.034 | LINER
J1445+5134 | 221.438 | 51.581 | 0.030 | Composite
J1511+0417 | 227.771 | 4.294 | 0.042 | LINER
J1511+2309 | 227.964 | 23.151 | 0.052 | Quiescent
J1517+0409 | 229.454 | 4.162 | 0.037 | Quiescent
J1617+2512 | 244.426 | 25.206 | 0.031 | Composite
J1655+2639 | 253.790 | 26.663 | 0.035 | Quiescent

Note. — Column 1: Source name. Column 2: Right Ascension. Column 3:
Declination. Column 4: Spectroscopic redshift from SDSS. Column 5: BPT
classification.

###  5.2 Radio Flux Densities and Luminosities

Flux density measurements were obtained either from survey catalog entries or
from the CASA task `IMFIT` when reported values were not available. The
integrated flux density measurements and their associated errors for each
source are summarized in Table 2. For sources identified in the LoTSS, RACS,
and FIRST/NVSS catalogs, the measurement error in Table 2 is the error quoted
by each catalog summed in quadrature with a 5% uncertainty in the absolute
flux scale. For VLASS and $3\sigma$ detections in any of the archival radio
surveys, the error is the RMS image noise and 5% uncertainty in the absolute
flux scale. For sources detected by the 3 and 10 GHz VLA observations, the
errors are the image RMS and a 3% uncertainty in the absolute flux scale
(Perley & Butler, 2017).

Table 2: Integrated Flux Density Values

Source |  $S_{\mathrm{LoTSS}}$ (mJy) |  $S_{\mathrm{RACS}}$ (mJy) |  $S_{\mathrm{1.4GHz}}$ (mJy) |  $S_{\mathrm{3GHz}}$ (mJy) |  $S_{\mathrm{10GHz}}$ (mJy)
---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6)
J0206$-$0017 | – | $5.38\pm 0.81$ | $3.36\pm 0.18$ | $1.91\pm 0.07$ | $1.64\pm 0.06$
J0759+2750 | $11.3\pm 0.6$ | $4.23\pm 0.52$ | $3.45\pm 0.21$ | $1.93\pm 0.15$ | $0.786\pm 0.016$
J0833+1523 | – | $<1.65$ | $<0.41$ | $<0.48$ | $<0.06$
J0843+3549* | $29.7\pm 0.8$ | – | $3.77\pm 0.17$ | $2.79\pm 0.13$ | $0.539\pm 0.017$
J0851+4050 | $1.0\pm 0.2$ | – | $<0.43$ | $<0.37$ | $0.226\pm 0.015$
J0916+4542 | $2.6\pm 0.2$ | – | $0.782\pm 0.13$ | $<0.33$ | $0.11\pm 0.013$
J1015+3914** | $11.5\pm 0.4$ | – | $1.36\pm 0.19$ | $1.13\pm 0.13$ | $0.526\pm 0.024$
J1018+3613 | $23.4\pm 0.9$ | – | $16.2\pm 0.8$ | $8.75\pm 0.25$ | $2.72\pm 0.08$
J1041+1105 | – | $3.3\pm 0.5$ | $<0.39$ | $<0.45$ | $0.08\pm 0.012$
J1056+1245 | – | $<1.2$ | $<0.41$ | $<0.47$ | $<0.04$
J1113+2714 | – | $1.08\pm 0.30$ | $1.32\pm 0.14$ | $<0.35$ | $0.063\pm 0.012$
J1117+3757 | $2.0\pm 0.2$ | – | $<0.43$ | $<0.33$ | $<0.04$
J1124+3005 | $0.9\pm 0.2$ | – | $<0.44$ | $<0.33$ | $<0.03$
J1135+2913 | $15.2\pm 0.6$ | – | $3.38\pm 0.22$ | $2.89\pm 0.14$ | $0.355\pm 0.013$
J1144+2309 | – | $<1.38$ | $<0.45$ | $<0.34$ | $<0.04$
J1230+1146 | – | $<16$ | $<0.65$ | $<0.65$ | $<0.05$
J1253+3944 | $<0.30$ | – | $<0.39$ | $<0.41$ | $<0.04$
J1304+6520 | $12.3\pm 0.5$ | – | $2.41\pm 0.24$ | $2.95\pm 0.22$ | $0.99\pm 0.29$
J1314+2607 | $<0.30$ | $<0.78$ | $<0.41$ | $<0.40$ | $<0.04$
J1326+5653 | $<0.30$ | – | $<0.46$ | $<0.56$ | $<0.04$
J1405+4001 | $<0.30$ | – | $<0.41$ | $<0.50$ | $<0.04$
J1433+3444 | $15.6\pm 0.7$ | – | $2.69\pm 0.19$ | $2.02\pm 0.16$ | $1.15\pm 0.036$
J1445+5134 | $28.2\pm 1.0$ | – | $11.9\pm 0.58$ | $6.06\pm 0.14$ | $1.05\pm 0.049$
J1511+0417* | – | $<2.3$ | $1.55\pm 0.17$ | $1.068\pm 0.032$ | $1.342\pm 0.039$
J1511+2309 | – | $11.8\pm 1.7$ | $1.32\pm 0.19$ | $0.495\pm 0.031$ | $0.249\pm 0.017$
J1517+0409 | – | $<1.8$ | $<0.45$ | $<0.41$ | $0.12\pm 0.013$
J1617+2512 | – | $1.47\pm 0.46$ | $1.41\pm 0.15$ | $1.06\pm 0.13$ | $0.22\pm 0.012$
J1655+2639 | – | $7.8\pm 0.59$ | $4.70\pm 0.23$ | $2.50\pm 0.07$ | $0.77\pm 0.02$

Note. — Column 1: Source name. Column 2: LoTSS (144 MHz) flux density and
error (Shimwell et al., 2022). Upper limits indicate a $3\sigma$ non-
detection, whereas no entry means the source was not included in the survey
field. Column 3: RACS (888 MHz) flux density and error (McConnell et al.,
2020; Hale et al., 2021). Column 4: 1.4 GHz flux density and error, reported
from either FIRST (27/28; Helfand et al., 2015) or NVSS (1/28; Condon et al.,
1998). Column 5: 3 GHz flux density and error, reported from either VLASS
(22/28; Lacy et al., 2020) or for the first time from VLA observations (6/28).
Column 6: 10 GHZ flux density and error, reported from our VLA observations.
*: These flux density measurements are reported for the dominant component
when the source is resolved into a multi-component morphology. **: The flux
density at 10 GHz was found after applying a uv-taper to the image plane.

The observed radio flux densities span $0.90-30$ mJy, with a median of 12 mJy,
for the 12 LoTSS detections; $1.1-12$ mJy, with a median of 4.2 mJy, for the 7
RACS detections; $0.78-16$ mJy, with a median of 2.7 mJy, for the 15
FIRST/NVSS detections; $0.50-8.8$ mJy, with a median of 2.0 mJy, for the 13
VLASS/3 GHz VLA detections; and $0.06-2.7$ mJy, with a median of 0.50 mJy, for
the 18 with a 10 GHz VLA detection.

Each SPM galaxy has an associated redshift measurement from SDSS. We used this
and the flux density measurement to calculate a luminosity, in units of Watts
(W), at each observing frequency for all of the detected radio sources. We
show the luminosity distributions for the $\nu<1$ GHz (LoTSS, RACS) and
$\nu>1$ GHz (FIRST/NVSS, VLASS, VLA) surveys in Figure 2. After calculating
the luminosities for each source, we compared these to the standard radio-loud
demarcation spectral luminosity $\nu L_{\nu}\approx 10^{32}$ W. A luminosity
value above this demarcation is considered radio-loud; radio AGN dominate the
radio luminosity function above this demarcation (Condon et al., 2002; Kimball
et al., 2011). We find that J1018+3613 is radio-loud at GHz frequencies, and
J1304+6520 is radio-loud at 3 and 10 GHz. The remaining radio sources are all
radio-quiet objects.

Figure 2: Left: Luminosity ($\nu L_{\nu}$) distribution of the radio sources
associated with each spheroidal post-merger (SPM) galaxy from the radio
surveys observed in the MHz regime: LoTSS (solid blue) and RACS (hatched
green). Some of the SPM galaxies were not observed in each survey. The dashed
vertical line represents the demarcation between radio-loud and radio-quiet
objects at 1.4 GHz, or $\nu L_{\nu}=1.4\times 10^{32}$ W. Right: Same as left
but for radio surveys observed in the GHz regime: FIRST or NVSS (dotted red),
VLASS or 3 GHz VLA (solid purple), and 10 GHz VLA (hatched).

###  5.3 Radio Morphology

We describe the bulk radio morphology properties of the detected sources in
each of the radio surveys. Each individual source and its intensity maps are
discussed and presented in Appendix A.

For each source, we categorize the morphology into one of the following
classifications:

  1. 1.

Unresolved: The peak-to-integrated flux density ratio is unity within
$3\sigma$ uncertainty and the source is a single Gaussian component that does
not exhibit any flux beyond the synthesized beam. 0/12 sources are unresolved
by LoTSS; 3/7 sources by RACS; 12/15 sources by FIRST/NVSS; 5/13 sources by
VLASS or 3 GHz VLA; and 6/18 sources by our 10 GHz VLA observations.

  2. 2.

Marginally Resolved: The peak-to-integrated flux density ratio is unity within
$3\sigma$ uncertainty and the source is marginally extended along one axis of
the synthesized beam. 1/12 sources are marginally resolved by LoTSS; 2/7
sources by RACS; 2/15 sources by FIRST/NVSS; 2/13 sources by VLASS or 3 GHz
VLA; and 1/18 sources by our 10 GHz VLA observations.

  3. 3.

Resolved: The peak-to-integrated flux density ratio of the source is
significantly less than unity, and the deconvolved major and minor axes have
non-zero size. 11/12 sources are resolved by LoTSS; 2/7 sources by RACS; 1/15
sources by FIRST/NVSS; 6/13 sources by VLASS or 3 GHz VLA; and 9/18 sources by
our 10 GHz VLA observations.

  4. 4.

Multi-component: The intensity map of the radio source shows two or more
distinct radio components common to one central engine. 0/12 sources are
multi-component in LoTSS; 0/7 in RACS; 0/15 in FIRST/NVSS; 1/13 in VLASS or 3
GHz VLA; and 2/18 in our 10 GHz VLA observations.

Visual inspection of the FIRST intensity map of J1511+2309 source (Figure 23)
shows two distinct radio components separated by $\sim 10\arcsec$, or 10 kpc
at the redshift of the source. The central component is compact and spatially
coincident with the optical nucleus of the host galaxy. The second component
is extended in morphology and has no optical counterpart. However, there is no
clearly definable jet axis which extends from the primary component in any of
the 1.4, 3, or 10 GHz maps that may link the two radio components to a common
central engine. The VLASS map, which is closer in resolution to FIRST than the
3 GHz VLA observations of this source, further resolves this second component
into two more distinct components, neither of which lie along a preferential
axis and appear to be unassociated with one another or the primary component.
This is similarly true upon examination of the 3 GHz, higher resolution map
produced from VLA observations of J1511+2309. For these reasons, we concluded
that the second component detected by FIRST is unassociated with this SPM and
did not classify the radio emission associated with this galaxy as multi-
component. Similarly, the 144 MHz LoTSS map of J0843+3549 (catalog WISEA
J084344.98+354942.3) shows two distinct radio components separated by
$27.3\arcsec$. The second radio source, observed to the southwest of the
primary source, is associated with the galaxy cluster GMBCG
J130.93151+35.82210 (Hao et al., 2010) at a redshift of $z=0.475$ (Rozo et
al., 2015). We did not classify the 144 MHz morphology of J0843+3549 as multi-
component because of this.

Figure 3: Left: Distribution of peak-to-integrated flux density ratios of the
10 GHz-detected SPM galaxies in the LoTSS (solid blue) and RACS (hatched
green). Some of the SPM galaxies were not observed in each survey. More
negative values indicate resolved structure of the radio source. Right: Same
as left but for FIRST or NVSS (dotted red), VLASS or 3 GHZ VLA (solid purple),
and 10 GHz VLA (hatched).

Figure 3 shows the logarithm of the peak-to-integrated flux density ratios for
each of the detected sources. It is clear that at the lowest frequency, 144
MHz, each of the radio sources has an extended emission component. For most of
these sources, this extended emission is diffuse and non-axisymmetric, meaning
it is unlikely from an AGN. Each of the LoTSS sources, however, still displays
an unresolved component that is spatially coincident with the optical center
of the host galaxy. J1433+3444 (catalog UGC 09367) is the only source with
collimated emission. We discuss this in Section A.12. For the peak-to-
integrated flux density ratio of this source, we determined the total
integrated flux density of the unresolved, nuclear emission plus the diffuse
component by applying a mask to the region of interest in the intensity map
within the CASA task `VIEWER`. However, the 144 MHz flux density reported in
Table 2 and used in Section 5.4 is only that of the unresolved emission. We
did this to mitigate the effects of artificial steepening of the radio
spectrum of the nuclear emission, which is the main emission region of
interest for our study. For the 7 sources with a detection by RACS, we find a
mix of unresolved, resolved, and moderately resolved sources.

At 1.4 GHz, the majority of sources become unresolved, with the unresolved
emission being spatially coincident with the optical nucleus of the host
galaxy. This shows that compact, nuclear emission is prevalent in the radio-
detected SPM galaxies. However, higher resolution observations at 3 GHz reveal
that extended emission is indeed prevalent in the majority of sources (8/13),
and the unresolved emission of J1511+0417 resolves into two components.
Likewise, our 10 GHz observations further reveal extended emission of these
nuclear radio sources. With the high resolution of our observations, many of
the formerly unresolved sources at lower frequencies now show a diffuse, non-
axisymmetric component to the nuclear radio source (J0759+2750 (catalog WISEA
J075948.40+275019.4), J1015+3914, J1135+2953, J1304+6520 and J1617+2512
(catalog WISEA J161742.15+251220.5)). Like J1511+0417 at 3 GHz, J0843+3549
resolves into two components at 10 GHz.

###  5.4 Radio Spectra

The radio spectrum is a useful tool for interpreting the underlying physical
characteristics of the radio source, including the dominant production
mechanism for the observed emission. For AGN, the radio emission is dominated
by synchrotron emission from a distribution of relativistic electrons,
creating a distinctive non-thermal power law spectrum. For older jetted and
lobe structures, the highest energy electrons in the distribution will radiate
away the fastest, causing a break in the power law at higher frequencies and
creating a steep radio spectrum with a power law slope $\alpha<-0.5$. Radio
cores, which are associated with the region of emission closest to the active
SMBH itself, are actively injected with fresh high energy electrons, creating
a flat spectrum with a power law slope $\alpha>-0.5$. For HII regions, thermal
emission is dominant at rest-frame frequencies of $\nu>10$ GHz, and is
characterized by a power law slope $\alpha\approx-0.1$. At $\nu<10$ GHz, the
non-thermal emission from supernova remnants (SNR) dominates, with varying
power law slopes. Both mechanisms of emission can be self-absorbed at low
frequency, causing a characteristic spectral turnover and inverted slope
$\alpha>0$. Identifying and quantifying the power law slope $\alpha$, as well
as the curvature and peak frequency, if present, can greatly aid in the
interpretation of the radio source.

To explore this parameter space, we constructed a radio SED using the multi-
frequency flux density measurements we have tabulated for each of the 18 radio
sources we detected with our 10 GHz observations. We have chosen not to
include J1117+3757 and J1124+3005 (catalog WISEA J112434.22+300544.7), since
these two SPMs were only detected by LoTSS. 12 of these 18 radio sources were
observed and detected in 4 or more of the radio surveys we have used. We
considered these 12 to be well-sampled radio spectra for SED analysis.

To constrain the overall shape of the radio SED for these 12 well-sampled
radio sources, we have performed a two-fold fitting procedure. First, the
spectrum is fit by a simple power law of the form

$S_{\nu}=A\nu^{\alpha},$ (1)

where $S_{\nu}$ is the flux density in mJy, $\nu$ is the observing frequency
in GHz, and $\alpha$ is the spectral index value. The second fit describes a
parabola in log space and accounts for curvature in the overall shape of the
radio spectrum:

$S_{\nu}=A\nu^{\alpha}\mathrm{e}^{q(\ln\nu)^{2}}\,,$ (2)

where $S_{\nu}$, $\nu$, and $\alpha$ are identical to Eqn. 1, and $q$ gives
the spectral curvature. For cases of significant curvature, e. g., $|q|\geq
0.2$ (Duffy & Blundell, 2012), $\alpha$ and $q$ lead to a peak frequency
$\nu_{peak}$ of

$\nu_{peak}=\mathrm{e}^{-\alpha/2q}\,.$ (3)

Here, $q$ is strictly phenomenological. Physically-motivated synchrotron self-
absorption or free-free absorption models, or models with multiple electron
populations, would require the use of more free parameters, e. g., more flux
density measurements at distinct frequencies, than were available for this
analysis (Tingay et al., 2015). However, $q$ is still an important constraint
to describe the overall shape of the radio spectrum and can hint at the
underlying physical mechanism of the radio emission (Callingham et al., 2017;
Nyland et al., 2020; Patil et al., 2022).

For the remaining 6 sources without well-sampled spectra, the maximum number
of detections for a single source across all the surveys used is 3. Then, we
could not perform the curved power law fit to these spectra given the paucity
of data. In addition, the spectral index values determined by our two-fold
fitting procedure are often difficult to interpret for such a wide frequency
range, spanning approximately 2 decades in frequency for some sources. To
obtain a representative spectral index value, we performed a linear fit to the
3 and 10 GHz flux density values for each of the eighteen 10 GHz-detected
radio sources. For those sources without a 3 GHz detection, this estimate
provides a lower limit to the actual spectral index value. We chose to use the
3 and 10 GHz flux density values because our ultimate goal is to characterize
the nuclear radio emission detected by our high resolution VLA observations.
These observing frequencies have the highest angular resolution among the
surveys used for our analysis, giving us the best approximation to the true
spectral index value of the nuclear radio emission.

The two-point spectral index $\alpha_{3}^{10}$ is given by

$\alpha_{3}^{10}=\frac{\mathrm{log}(S_{3}/S_{10})}{\rm{log}(3/10)}\,.$ (4)

with an associated error of

$\sigma_{\alpha}=\frac{1}{\rm{ln}(10/3)}\sqrt{\left(\frac{\sigma_{S_{3}}}{S_{3}}\right)^{2}+\left(\frac{\sigma_{S_{10}}}{S_{10}}\right)^{2}}\,,$
(5)

given by standard propagation of errors.

Figure 4: Broadband radio SED for each of the 12 well-sampled radio sources
detected by our 10 GHz observations. Radio surveys used are labeled by
different markers and colors, with the key in the upper right of each SED.
1$\sigma$ errors are plotted for each flux density measurement. The best-fit
power law and curved power law are plotted as the dashed and solid black
lines, respectively. Figure 5: Broadband radio SED for each of the 6 radio
sources detected by our high resolution, 10 GHz observations that were a non-
detection by 1 or more of the radio surveys used. Radio surveys are labeled by
different markers and colors, with the key in the upper right of each SED.
Unfilled markers indicate a $3\sigma$ non-detection in that survey. The two-
point spectral index $\alpha_{3}^{10}$ determined from a linear fit to the 3
and 10 GHz fluxes, or their limits, is plotted as the dot-dashed black line.

Figure 4 and Figure 5 show each radio SED and the results of our fitting
analyses for the eighteen 10 GHz-detected radio sources. Table 3 lists the
reduced $\chi^{2}$ values of the power law and curved power law fits for the
12 well-sampled radio spectra. For each of these, the spectral curvature
parameter $q$ is provided for those spectra that are better fit by a curved
power law than a simple power law
($\chi^{2}_{\mathrm{red,PL}}>\chi^{2}_{\mathrm{red,CPL}})$. We also list the
peak frequency $\nu_{\mathrm{peak}}$ for the 3 sources that have $q\leq-0.2$.
Table 3 also lists the two-point spectal index value $\alpha_{3}^{10}$, or its
lower limit, for all 18 sources.

We find that 5 of the 12 well-sampled radio SEDs show evidence of significant
curvature: J0206$-$0017, J1018+3613, J1445+5134, J1511+2309 and J1617+2512. Of
these, J1018+3613, J1445+5134 and J1617+2512 are all peaked spectrum objects,
while J0206$-$0017 and J1511+2309 have $q>0.2$, indicative of an inverted
spectrum. Visual inspection of the radio SEDs for these two sources show that
their actual nature is most likely not inverted. Instead, it’s probable that
larger-scale emission components with steeper spectra are being resolved out
by the higher frequency, higher resolution observations, leaving only the most
compact features as the sole contributor to the recovered flux density and
causing the spectrum to flatten. Indeed, the flat-spectrum nature of
J0206$-$0017 was confirmed by Walsh et al. (2023). We expect that higher
frequency observations of J1511+2309 would confirm the presence of a flat-
spectrum object in this source. We did not find evidence of significant
curvature in the remaining 7 well-sampled radio SEDs.

The two-point spectral index values $\alpha_{3}^{10}$ span a range
$0.19\geq\alpha_{3}^{10}\geq-1.74$. The majority of sources (10/13) have a
steep spectral index, e. g., $\alpha<-0.5$; J0206$-$0017 and J1433+3444 have a
flat spectral index, e. g., $-0.5\leq\alpha\leq 0$; and J1511+0417 has an
inverted spectral index value of $\alpha_{3}^{10}>0$. The spectrum of
J1511+0417 is likely truly inverted, unlike the spectra of J0206$-$0017 and
J1511+2309, though more flux density measurements are required to confirm
this. Of those sources with lower limits on $\alpha_{3}^{10}$, J0851+4050
(catalog UGC 04635) likely has a flat spectral index given that its
$\alpha_{3}^{10}>-0.41$. The limits for the remaining 4 radio sources leave
their radio spectral class ambiguous.

Table 3: Radio Spectral Fitting Parameters

Source | $\chi^{2}_{\rm PL}$ | $\chi^{2}_{\rm CPL}$ | $\alpha_{3}^{10}$ | $q$ |  $\nu_{\rm peak}$ (MHz)
---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6)
J0206$-$0017 | 76.0 | 0.23 | -0.13 $\pm$ 0.04 | 0.29 $\pm$ 0.01 | –
J0759+2750 | 113 | 1.52 | -0.75 $\pm$ 0.07 | -0.04 $\pm$ 0.02 | –
J0843+3549* | 95.0 | 531 | -1.37 $\pm$ 0.05 | – | –
J0851+4050 | – | – | $>-0.41$ | – | –
J0916+4542 | – | – | $>-0.91$ | – | –
J1015+3914 | 7164 | 5673 | -0.64 $\pm$ 0.10 | 0.10 $\pm$ 0.07 | –
J1018+3613 | 1048 | 21.8 | -0.97 $\pm$ 0.03 | -0.19 $\pm$ 0.02 | 290
J1041+1105 | – | – | $>-1.43$ | – | –
J1113+2714 | – | – | $>-1.42$ | – | –
J1135+2913 | 1348 | 1538 | -1.74 $\pm$ 0.05 | – | –
J1304+6520 | 13.9 | 186 | -0.91 $\pm$ 0.07 | – | –
J1433+3444 | 64.7 | 4.27 | -0.47 $\pm$ 0.07 | 0.08 $\pm$ 0.02 | –
J1445+5134 | 1458 | 19.3 | -1.46 $\pm$ 0.04 | -0.18 $\pm$ 0.02 | 155
J1511+0417* | – | – | 0.19 $\pm$ 0.03 | – | –
J1511+2309 | 217 | 114 | -0.57 $\pm$ 0.08 | 1.61 $\pm$ 0.36 | –
J1517+0409 | – | – | $>-1.02$ | – | –
J1617+2512 | 216 | 4.92 | -1.31 $\pm$ 0.11 | -0.37 $\pm$ 0.08 | 1114
J1655+2639 | 5.80 | 19.5 | -0.98 $\pm$ 0.03 | – | –

Note. — Column 1: Source name. Column 2: Reduced $\chi^{2}$ value for a power
law fit to the radio SED. Only reported for sources with 4 or more survey
detections. Column 3: Reduced $\chi^{2}$ value for a curved power law fit to
the radio SED. Only reported for souces with 4 or more survey detections.
Column 4: Spectral index value, and its error, for the optically-thin
emission, or its lower limit. This was estimated by using the 3 and 10 GHz
flux density values, or the upper limit on the 3 GHz flux for non-detections
at this frequency. Column 5: Spectral curvature parameter determined by the
curved power law fit. Only reported if $\chi^{2}_{\rm red,\,PL}>\chi^{2}_{\rm
red,\,CPL}$. Column 6: Peak frequency of the radio SED. Only reported if
$\chi^{2}_{\rm red,\,PL}>\chi^{2}_{\rm red,\,CPL}$ and $q<0$.

Note: J0843+3549 and J1511+0417 are resolved into two distinct components at
10 GHz. For both, we used the dominant, southern component to calculate
$\alpha_{3}^{10}$. See Figure 11 and Figure 22 for the 10 GHz contour maps of
J0843+3549 and J1511+0417, respectively.

The radio SED of J0843+3549, J1015+3914, J1135+2953, and J1304+6520 each show
a significant deviation from their best-fit simple and curved power laws at
1.4 GHz. This is reflected in the abnormally high reduced $\chi^{2}$ values
for the fits of these sources. This can be interpreted in two ways. First, the
significant deviation at 1.4 GHz may be explained by intrinsic variability of
the radio source. The observations at 144 MHz (LoTSS), 888 MHz (RACS), 3 GHz
(VLASS/VLA), and 10 GHz (VLA) are quasi-contemporaneous; at most, the
observations were taken within 5 years of one another. Yet, the 1.4 GHz
observations, by either FIRST or NVSS, were conducted well over a decade ago,
at least, at the time of this analysis. If the source underwent significant
variability over a years-long timescale prior to its most recent observation
at 1.4 GHz, all of the flux density measurements at frequencies besides 1.4
GHz would reflect this. It is possible, then, that these sources have
naturally varied over this intervening time span and are no longer well fit to
the quasi-contemporaneous data points of the other surveys. The corresponding
variability amplitudes at 1.4 GHz, assuming a power law fit to the spectrum
without the 1.4 GHz flux density properly describes the spectral shape, range
from 12%-61%. These amplitudes are certainly plausible, given that some
sources have been found to reach variability amplitudes higher than 2400% at
this observing frequency (Nyland et al., 2020). Alternatively, the flux
densities at lower frequencies are representative of a separate electron
population. For example, LoTSS sources in the local universe will be dominated
by diffuse emission associated with star-formation processes. Using higher
frequency and higher spatial resolution observations, this diffuse emission
will be resolved out. If there is a second, distinct population of electrons
producing a radio source that is more compact and hosted by the same SPM, this
will become apparent by a break in the broadband radio spectrum. Essentially,
the two electron populations, both of which are located within the same host
galaxy, are confused with each other at low frequency, and only the high
frequency observations we have used are truly representative of the second,
more compact, nuclear source. Follow-up observations at high angular
resolution with better frequency sampling, including 1.4 GHz, are required to
distinguish between the two methods we have outlined that may produce this
observed break in the radio spectrum. The nature of these sources is further
discussed in Section 6.

##  6 Origin of Radio Emission

In this section, we assess the origin of the radio emission in our SPM sample.
Our analysis will make use of mid-IR fluxes available from the ALLWISE source
catalog to perform a dust correction to the far-UV (FUV) flux for each SPM,
for which we wish to calculate the star-formation rate (SFR). In this
paradigm, active star formation heats dust grains in the surrounding
interstellar medium that re-radiate this energy as thermal emission in the
mid-IR. However, it is possible for an AGN to also assume this heat engine
role and this will introduce systematic effects into our calculation of the
SFR using the FUV fluxes. As such, we first identify if any of our sources are
AGN by mid-IR selection criterion. To do this, we utilize a WISE color-color
diagram to search for mid-IR AGN using the selection criterion of Jarrett et
al. (2011) for each of the eighteen 10 GHz-detected radio sources. These
results are shown in Figure 6. We find that J0843+3549 and J0206$-$0017 are
within the mid-IR AGN box of Jarrett et al. (2011), and J1018+3613 is
consistent with a mid-IR AGN within errors. The remaining 15 sources are well
outside of the AGN region and occupy the region of star-forming galaxies, e.
g., spiral galaxies, luminous infrared galaxies (LIRGS) and
starburst/ultraluminous infrared galaxies (ULIRGS) (Wright et al., 2010).
J0206$-$0017 (Osterbrock, 1981; Cohen et al., 1986; McElroy et al., 2016;
Walsh et al., 2023), J0843+3549 (Véron-Cetty & Véron, 2001; Stern & Laor,
2012; Koss et al., 2018), and J1018+3613 (Stern & Laor 2012, Walsh et al. in
prep) are all known AGN, in addition to being identified as Seyfert AGN via
their emission-line ratios as discussed in Section 5.1. To mitigate the impact
of potential systematics introduced by the mid-IR AGN to our SFR calculation,
we conclude that the radio emission in each of these sources is associated
with the AGN and remove them from the analyses described in this section. For
each of the remaining fifteen 10 GHz-detected radio sources, we consider the
following origin scenarios for their radio emission: thermal emission from
star-forming regions, synchrotron emission from individual radio supernova
(RSN) or a population of supernova remnants (SNRs), or an AGN.

Figure 6: WISE color-color diagram for the eighteen 10 GHz-detected SPM
galaxies in our sample. The dashed black lines define the region of mid-IR AGN
taken from the selection criterion of Jarrett et al. (2011). Two of our
sources, J0206$-$0017 and J0843+3549, are selected as mid-IR AGN, while a
third, J1018+3613, falls in this region within $1\sigma$ error. Each of these
has multi-wavelength evidence for an AGN. We conclude that their radio
emission is associated with the AGN and remove these sources from further
analyses to avoid possible introduction of systematics by the AGN to the SFR
calculation of each host SPM.

###  6.1 Radio Excess

We begin by searching for excess radio emission in each of the remaining 15
SPM radio sources. To do this, we predict what the SFR for each of the SPMs
would be from their 1.4 GHz luminosity and compare this radio-predicted SFR to
that calculated from the FUV emission of the host galaxy. Sources that have an
over-prediction of the SFR from their radio luminosity are called radio
excess. These radio excess sources cannot be explained from SF processes
alone, while those that do not show radio excess can be, though aren’t
necessarily.

We first predict the radio-based SFR using the 1.4 GHz luminosity. To do this,
we use Equation 17 of Murphy et al. (2011):

$\left(\frac{\mathrm{SFR}_{\mathrm{1.4GHz}}}{M_{\odot}\,\mathrm{yr}^{-1}}\right)=6.35\times
10^{-29}\left(\frac{L_{\mathrm{1.4GHz}}}{\mathrm{erg\,s^{-1}\,Hz^{-1}}}\right)\,.$
(6)

This SFR is based on the FIR-radio correlation, which relates the galactic FIR
properties to the galactic radio continuum properties. Murphy et al. (2011)
note that the expected contribution to the total radio emission from non-
thermal processes is negligible for some cases in which the emission is co-
spatial with an active HII region. Condon & Yin (1990) argue that this is the
case only for small HII regions, for which stars with $M>8M_{\odot}$ can
escape before exceeding their lifetime of $<3\times 10^{7}$ yr. For our
sources that are identified as SF or SF-AGN composite galaxies via their
optical emission-line ratios, the SDSS spectroscopic fiber has a diameter of
$3\arcsec$. We only know that within the galactic region covered by the
spectroscopic fiber, an active HII region is, at least partially for composite
galaxies, contributing to the ionization. The area covered by the SDSS fiber
is much larger than the synthesized beamwidth of our 10 GHz VLA observations
($\sim 0.2\arcsec$). Because none of our sources show features comparable in
angular size to the SDSS fiber, for the consistency of the analysis, we
continued under the assumption that the HII region is large enough such that
there could be significant non-thermal radio emission spatially coincident
with it.

To estimate the radio-based SFR, we use the 1.4 GHz luminosity values derived
from the FIRST or NVSS, for J1304+6520, catalog entry for each source. We do
this instead of extrapolating to the 1.4 GHz luminosity using
$\alpha_{3}^{10}$ because, as mentioned in Section 5.4, some sources exhibit
clear breaks from their best-fit power law at 1.4 GHz (J0843+3549, J1015+3914,
J1135+2953, and J1304+6520), and the synthesized beams of FIRST and NVSS,
$5\arcsec$ and $45\arcsec$, respectively, are better matched to galaxy-scale
properties than the synthesized beam of our 10 GHz observations. It is
important to most accurately trace the galaxy-scale radio emission because the
radio-FIR correlation has been shown to deviate from a linear correlation for
regions of radio emission with low thermal fractions (Hughes et al., 2006). At
1.4 GHz, the expected thermal contribution to the total radio emission for any
radio source is approximately 5 to 10% (Condon, 1992; Murphy, 2013). This is
true even for starburst systems, for which Murphy (2013) found a thermal
fraction of 5% in a sample of 31 local starburst galaxies. We assume, then,
that the radio sources we have detected are not extraordinary in this regard,
and have low thermal fractions at 1.4 GHz. However, the spatial scale range
for which Hughes et al. (2006) found the radio-FIR correlation to deviate from
a linear correlation is from 50-250 pc for their low thermal fraction radio
sources. At $5\arcsec$ resolution, the smallest spatial scale probed for our
18 source sample is approximately 3 kpc, or an order of magnitude larger than
what was found by Hughes et al. (2006). Because of this, we do not expect any
deviations from the standard radio-FIR correlation using the 1.4 GHz
luminosity for our sources.

We now calculate the host galaxy SFRs for 13 SPMs that had FUV measurements
available from GALEX. The remaining 2 did not have GALEX measurements
available and we describe the calculation of their SFRs later on in this
section. We first correct the FUV luminosity for dust absorption by using the
25$\mu$m WISE luminosity for each source, following Hao et al. (2011):

$L(\mathrm{FUV})_{\mathrm{corr}}=L(\mathrm{FUV})_{\mathrm{obs}}+3.89L(25\mu\mathrm{m})\,,$
(7)

where all luminosity values are in units of erg s-1. Here, we have used the
available WISE 22$\mu$m flux density as a proxy for the 25$\mu$m luminosity,
since the flux density ratio between these two values is expected to be unity
for early-type galaxies (Jarrett et al., 2013). After calculating the WISE-
corrected FUV luminosity, we find the host galaxy SFR following Table 1 of
Kennicutt & Evans (2012) for the FUV band:

$\left(\frac{\mathrm{SFR}}{M_{\odot}\,\mathrm{yr}^{-1}}\right)=4.5\times
10^{-44}L(\mathrm{FUV})_{\mathrm{corr}}\,.$ (8)

Using this method, the 1$\sigma$ uncertainty on the SFR is 0.13 dex (Hao et
al., 2011). For the two SPM detections without available GALEX FUV
measurements, J1304+6520 and J1511+2309, we use the H$\alpha$ luminosity to
calculate the host galaxy SFR. Kennicutt et al. (2009) provide a dust-
attenuated correction to the H$\alpha$ luminosity using the 25 $\mu$m
luminosity:

$L(\mathrm{H}\alpha)_{\mathrm{corr}}=L(\mathrm{H}\alpha)_{\mathrm{obs}}+0.020L(25\mu\mathrm{m})\,.$
(9)

As before, all luminosity values are in erg s-1. For
$L(\mathrm{H}\alpha)_{\mathrm{obs}}$, we use the values provided by the OSSY
catalog (Oh et al., 2011) for each of the two optical spectra. We again follow
Table 1 of Kennicutt & Evans (2012) to calculate the host galaxy SFR using the
dust-corrected H$\alpha$ luminosity:

$\mathrm{SFR}=5.37\times
10^{-42}\left(\frac{L(\mathrm{H}\alpha)_{corr}}{\mathrm{erg\,s^{-1}}}\right)\,.$
(10)

The 1$\sigma$ uncertainty for the H$\alpha$ method is 0.4 dex (Kennicutt et
al., 2009).

The radio-based SFR is plotted against the galaxy-based SFR for each of our 10
GHz-detected SPM sources in Figure 7. The host galaxy SFRs are in the range
$0.2\,\mathrm{M}_{\odot}\,\mathrm{yr}^{-1}\leq$ SFR $\leq
17\,\mathrm{M}_{\odot}\,\mathrm{yr}^{-1}$. We find 4 radio sources that do not
have excess radio emission, indicating that their radio emission could be
explained by star-formation processes alone. These are: J1015+3914, J1041+1105
(catalog VIII Zw 101), J1445+5134, and J1617+2512. We note that J0851+4050,
J1041+1105 and J1517+0409 (catalog UGC 09804 NED01) are non-detections at 1.4
GHz, and thus their radio-based SFRs are upper limits. 9 of our sources fall
on either side of the radio-excess line within their 1$\sigma$ errors,
including all of those sources that do not show excess radio emission. Only 4
sources have excess radio emission above a factor of $3\sigma$ from what is
expected by star-formation process. These are: J0759+2750, J1304+6520,
J1433+3444, and J1655+2639. Now that we have identified which radio sources do
or do not show excess radio emission, we seek to answer what physical process
is the dominant means of radio emission production for each.

Figure 7: Comparison of the 1.4 GHz, radio-based SFRs to the host galaxy SFRs
for 15 of the 10 GHz-detected radio sources in our sample of SPMs.
J0206$-$0017, J0843+3549, and J1018+3613 were removed from this analysis due
to the presence of an IR AGN. Host galaxy SFRs were determined using either
the IR-corrected FUV (purple circles) or H$\alpha$ luminosity (black squares).
Unfilled data points are non-detections at 1.4 GHz, and their radio-based SFRs
are upper limits determined using a $3\sigma$ detection threshold from FIRST.
Points above the dashed line exhibit radio excess, while those below do not.

###  6.2 Thermal Emission from Star Formation

The first consideration for the origin of the radio emission is that of
thermal bremsstrahlung (free-free) emission produced by ionized hydrogen in
active star-forming regions. In the optically-thin regime, radio emission that
is dominated by a free-free component is characterized by a flat spectral
index of $\alpha=-0.1$ (Condon, 1992; Murphy et al., 2011; Klein et al.,
2018). Each of the 4 non-radio-excess sources are in the optically-thin regime
at GHz frequencies, as indicated by their broadband radio SEDs (see Figure 4
for J1015+3914, J1445+5134, and J1617+2512, and Figure 5 for J1041+1105).
However, their optically-thin spectral index values, $\alpha_{3}^{10}$, range
from -1.46 to -0.64, with J1041+1105 having a lower limit of -1.43. Although
there may be a contribution from free-free emission in each of these radio
sources, it is clear from their spectral index values that free-free emission
is not the dominant radio emission mechanism for any. This is not unexpected,
as free-free emission from HII regions does not usually dominate the radio SED
for $\nu<10$ GHz (Condon, 1992; Murphy, 2013).

###  6.3 Radio Supernova and Supernova Remnants

Our second consideration for the origin of the nuclear radio emission in our
SPM sources is from non-thermal emission produced by either an individual
radio supernova (RSN) or a population of supernova remnants (SNRs). We first
consider an individual RSN as the progenitor for the radio emission. RSN are
morphologically compact radio sources that span a range of radio luminosity
(Weiler et al., 2002) and spectral index values (Bendo et al., 2016; Klein et
al., 2018; Galvin et al., 2018; Emig et al., 2020). The radio emission
associated with a RSN is powered by synchrotron processes. Generally, for
star-forming galaxies, this synchrotron emission is diffuse, tracing the host
galaxy’s morphology. In the optically thin regime, RSN associated with a Type
Ib/c event have $\alpha<-1$ ($S_{\nu}\propto\nu^{\alpha}$), while those
associated with a Type II event have a shallower spectral index $\alpha>-1$
(Weiler et al., 2002). The radio luminosity of an individual RSN will peak a
few 100s of days after the initial SN explosion, reaching a maximum 5 GHz
luminosity of $L_{\mathrm{6cm\,peak}}\approx 1.3\times 10^{27}$ erg s-1 Hz-1
(Weiler et al., 2002). However, two of the most luminous RSN, SN1998bw
(Kulkarni et al., 1998) and PTF11qcj (Corsi et al., 2014; Palliyaguru et al.,
2019), have a peak luminosity value as high as $10^{29}$ erg s-1 Hz-1 at 5
GHz. We take this luminosity to be the upper limit to what a RSN can achieve
and compare this to the extrapolated 5 GHz luminosity for each of our SPM
detections.

To extrapolate to the 5 GHz luminosity, we use $\alpha_{3}^{10}$ for each
source (Table 3, column 4) determined by performing a linear fit to the log of
the 3 and 10 GHz flux density values. After extrapolation, 2 of the 15
detections have a 5 GHz luminosity greater than $10^{29}$ erg s-1 Hz-1:
J0759+2750, and J1304+6520. All of the sources with a lower limit to
$\alpha_{3}^{10}$ are below this luminosity. The remaining 13 have a median
luminosity value of $4.3\times 10^{28}$ erg s-1 Hz-1, which is only a factor
of 2.3 lower than this RSN luminosity limit. While the spectral index and
luminosity values do not rule out the individual RSN origin for these 15
sources, it is extremely unlikely that each radio source is associated with an
individual, extremely luminous, nuclear RSN. Nonetheless, we pursue a more
robust argument to rule out this scenario.

Chomiuk & Wilcots (2009) determined an expression that relates the maximum 1.4
GHz luminosity of a RSN to the SFR of its host galaxy:

$L_{\mathrm{1.4\,GHz}}^{\mathrm{max}}=\left(95^{+32}_{-23}\right)\mathrm{SFR}^{0.98\pm
0.12}\,,$ (11)

where the 1.4 GHz luminosity is in units of 1024 erg s-1 Hz-1 and the SFR is
measured in M⊙ yr-1.

For a given SFR, we first use this relation to determine the maximum 1.4 GHz
luminosity of a RSN, then extrapolate this to a 10 GHz luminosity to compare
to our VLA sources. There is some freedom here in which value to choose for
the spectral index. Chomiuk & Wilcots (2009) use $\alpha=-0.5$ when deriving
the synchrotron emission from a RSN. This comes from the assumption that the
cosmic ray (CR) energy spectrum is a power law of the form E-2, which gives a
synchrotron spectral index of $\alpha_{\mathrm{syn}}=-0.5$. However, focusing
on the most luminous RSNs, SN1998bw has a steeper spectral index of
$\alpha_{\mathrm{syn}}=-0.75$ (Chevalier & Fransson, 2006), and PTF11qcj has a
varying late-time spectral index $\alpha_{\mathrm{syn}}\gtrsim-1$ (Corsi et
al., 2014). Björnsson (2013) note that the spectral index should approach a
value of $\alpha_{\mathrm{syn}}=-1$ in the optically-thin regime, and, indeed,
this is in agreement with those values listed in Table 1 of Chevalier &
Fransson (2006). For our analysis, we have used a spectral index value of
$\alpha_{\mathrm{syn}}=-0.5$, as is done in Chomiuk & Wilcots (2009). We chose
this spectral index value because it is the shallowest among those discussed.
Thus, if any of our sources lie above the extrapolated RSN luminosity using a
spectral index value of $\alpha_{\mathrm{syn}}=-0.5$, they will certainly do
the same for a steeper spectral index value. Using this method, we find that
the observed 10 GHz luminosity of each radio source is greater than the
expected luminosity of an individual RSN by at least a factor of 80. It is
evident that an individual, luminous RSN is not responsible for the radio
emission in these SPM galaxies.

For the 4 radio sources that do not show excess radio emission, it is likely,
then, that their radio emission is produced by a population of SNRs. Only one
of these radio sources (J1015+3914) is hosted by a star-forming galaxy as
determined by its optical emission-line ratios. Two are identified as SF-AGN
composite galaxies (J1445+5134, J1617+2512), and the remaining is a LINER
(J1041+1105).

The median spectral luminosity at 10 GHz of these 4 radio sources is
$1.4\times 10^{28}$ erg s-1 Hz-1. For comparison, the nuclear starbursts
identified by Song et al. (2022) in sample of 63 local (U)LIRGS with SFRs in
the range of $0.14-13\,\mathrm{M_{\odot}}\,\mathrm{yr}^{-1}$ have a median
spectral luminosity of $5.8\times 10^{27}$ erg s-1 Hz-1, or about a factor of
2 lower than what we have found for our radio SF sources. However, higher
luminosity radio SF sources do exist: NGC 4945, a powerful, local starburst
with a SFR of 1.5 $\mathrm{M_{\odot}}\,\mathrm{yr}^{-1}$, has a higher
spectral luminosity at 10 GHz of $2.4\times 10^{28}$ erg s-1 Hz-1 (Lenc &
Tingay, 2009). To emphasize, only J1015+3914 may be powered by SF processes
alone, as determined by its emission-line ratios. The other three may have
contributions from another ionization process, e. g., an AGN. Interestingly,
this source has the highest 10 GHz luminosity ($4.3\times 10^{28}$ erg s-1
Hz-1) of any of the non-excess sources.

Among these sources with a detection in the FIRST catalog, none are resolved
at the spatial scales probed by the $5\arcsec$ FIRST beam. That is, they do
not display the diffuse emission morphology that is characteristic of
synchrotron emission from SNRs. This morphology does become more apparent at
low frequency (LoTSS or RACS), and at higher spatial resolution for a few
sources, e. g., J1015+3914 (Figure 14), though remains elusive at GHz
frequency in others, e. g., J1041+1105 (Figure 16) and J1617+2512 (Figure 25).
This is discussed in more detail for each individual source in Appendix A.
When examining the ratio of peak flux to integrated flux density however, all
of these sources become resolved at 10 GHz (see Figure 3). Deeper observations
at 1.4 and 3 GHz may reveal lower surface brightness emission indicative of
this characteristically diffuse nature.

###  6.4 AGN

For the 11 radio-excess sources, it is likely that their radio emission is
dominated by an AGN component. We did not find a one-to-one match between the
radio AGN classification and that derived from our BPT analysis (Section 5.1).
Instead, the radio AGN are found to occupy all categories of the BPT
classifications: J1113+2714 (catalog WISEA J111340.54+271428.3) and J1135+2913
(catalog UGC 06557) are SF; J0759+2750, J0916+4542 (catalog UGC 04892) are SF-
AGN composites; J1304+6520, J1433+3444 are Seyfert AGN; J0851+4050, J1511+0417
are LINERs; and J1511+2309, J1517+0409, J1655+2639 are hosted by quiescent
systems. In addition to these 11, we have already concluded that the radio
emission in J0206$-$0017, J0843+3549, and J1018+3613 is each associated with
an AGN. From the emission-line perspective alone, this indicates that radio
AGN activity may be present during an ongoing or recent stage of star
formation activity in the host SPM. In total, we find that the nuclear radio
emission for 14 of the 18 sources at 10 GHz, or 78%, is dominated by a radio
AGN.

We begin by discussing the three radio sources hosted in quiescent emission-
line galaxies: J1511+2309, J1517+0409, and J1655+2639. As noted before in
Section 5.1, the spectra of J1511+2309 and J1655+2639 contain H$\beta$
absorption that is most likely from a stellar origin due to its absorption
trough being centered on the rest-frame H$\beta$ wavelength. Because there is
not a significant detection of any H$\beta$ emission, as reported by OSSY (Oh
et al., 2011), in these two spectra, we have classified them as quiescent,
even though the [OIII]$\lambda 5007$, H$\alpha$, and [NII]$\lambda 6583$ lines
are detected with a S/N ratio $>3$. The radio emission properties, however,
indicate the clear presence of radio AGN activity, as we discuss further in
this section. J1517+0409 is unique among these quiescent emission-line SPMs
and radio AGN. The optical spectrum for this SPM is largely featureless: only
the [NII]$\lambda 6583$ line is detected with a S/N ratio $>3$. Additionally,
it was a non-detection by each of the archival radio surveys that observed it
(RACS, FIRST, VLASS), making it the only 10 GHz-detected source with no
additional radio detection(s). The radio-based SFR of J1517+0409 is within the
radio-excess area shown in Figure 7, though this was determined by using a
3$\sigma$ detection limit to the FIRST image of this source. With more
sensitive 1.4 GHz observations, it is likely that J1517+0409 would not show
radio-excess, and the origin of its radio emission would need to be re-
examined. However, considering the data available during this analysis, we
count J1517+0409 to be among the likely radio AGN discovered by our
observations.

####  6.4.1 Radio AGN Morphologies

For these radio AGN, almost all of them display only a single-component
morphology except for J0843+3549, J1433+3444, and J1511+0417. The LoTSS map of
J1433+3444 alone displays the extended morphology characteristic of AGN
(Figure 20.) The majority of these radio AGN do not have collimated jets
and/or extended lobe emission that is easily and clearly identifiable in any
of their intensity maps (see Appendix A for all intensity maps). This is
perhaps unsurprising given that almost all of our 10 GHz sources are radio-
quiet, whereas radio-loud AGN are nearly ubiquitously associated with highly
relativistic emission arising from radio jets. Yet, the majority of our
sources are also not dominated by flat-spectrum radio cores. These objects are
identified via their unresolved radio emission, flat spectral index
($\alpha\geq-0.5$), and high brightness temperature, indicative of non-thermal
emission, and are almost always associated with an AGN. J0206$-$0017,
J1433+3444 and J1511+0417 are the most likely sources to contain a dominant
radio core when considering their unresolved morphology and flat spectral
index values (Table 3). Indeed, this is the case for J0206$-$0017, as VLBI
observations by Walsh et al. (2023) revealed that the radio emission remains
compact down to pc scale, retains its flat spectral index, and shows chromatic
variation in its position, confirming its radio-core nature. J0851+4050,
J0916+4542, J1113+2714 and J1517+0409 only have lower limits to their spectral
index value so the true nature of their radio AGN emission is ambiguous.

It is evident from the ratio of peak-to-integrated flux density at 10 GHz that
some of our single-component sources are moderately resolved (Figure 3). Since
there is little evidence for kpc-scale radio jets or lobes at GHz frequencies,
aside from J0843+3549 and J1511+0417, this moderately resolved nature of the
nuclear emission could be an indication of young or frustrated radio jets,
which are confined to only the central, sub-kpc region of their host galaxy.
J1655+2639 is the most identifiable example of this scenario. The radio
emission is moderately resolved along a linear feature (VLA X panel of Figure
26) that has a steep spectral index of $\alpha_{3}^{10}=-0.97\pm 0.03$.

Alternatively, the moderately resolved nature of some of these radio AGN
sources can be explained by the presence of non-thermal emission associated
with star formation. As noted, 2 of our radio AGN sources are hosted by SF
emission-line galaxies, and 2 more are classified as SF-AGN composite.
J0759+2750, hosted by a SF-AGN composite galaxy, is perhaps an archetypal
source for SF and AGN activity because its 1.4 GHz radio luminosity is
$>3\sigma$ higher than what is expected from star formation alone, but the 10
GHz morphology shows both a diffuse, non-linear component and an unresolved
component (VLA X panel of Figure 10). J0916+4542, also hosted by a SF-AGN
composite galaxy, is similarly resolved by its peak-to-integrated flux density
ratio at 10 GHz, although the point-like morphology makes the nature of the
diffuse emission unclear (Figure 13).

Like J0759+2750, J1135+2913, hosted by a SF galaxy, shows low surface
brightness emission in its 10 GHz map (VLA X panel of Figure 18), though its
radio emission is only a factor of 2.2$\sigma$ higher than expected from star
formation. However, the diffuse emission in J1135+2913 forms a linear feature.
This is particularly evident once this source is imaged with a uv-taper to
create a lower resolution map at 10 GHz. It is likely that this feature is a
radio jet associated with the radio AGN, like J1655+2639, though we cannot
rule out that there is no contribution to the diffuse radio emission from
star-formation processes.

The diffuse emission of J1304+6520 is more difficult to interpret (Figure 19).
Like J0759+2750, J1304+6520’s 1.4 GHz radio luminosity is $>3\sigma$ higher
than what is expected from star formation alone. However, unlike J0759+2750,
J1304+6520 is hosted by a Seyfert AGN emission-line galaxy. So, if this
diffuse emission is from SF processes, it is not evident that such would be
the case from its optical emission-lines. To test if this diffuse emission
would resolve into a jet-like feature, we created 10 GHz maps using different
weighting schemes. However, these maps did not provide clear evidence of a
radio jet. Further observations are required to ascertain the nature of this
radio emission. For this analysis, we cannot conclusively determine the
emission is from a jet, and do not count it as such as a result.

Lastly, we discover a potential precessing radio jet in J0843+3549. The
emission of the compact feature is moderately resolved in all of its intensity
maps, and the position angle (PA) of this moderately resolved feature changes
from a range of 131$\arcdeg$-151$\arcdeg$ at kpc scales to $-18\arcdeg$ in our
10 GHz intensity map, nearly aligning with the second radio component (Figure
11). The different observing frequencies probe different periods in the AGN’s
evolution because of their differing resolution elements. Thus, the change in
PA from the largest spatial scales ($2.5\arcsec-6\arcsec$ resolution) to the
smallest ($\sim 0.25\arcsec$) is indicative of time evolution in the PA of the
radio jet. The near-alignment of the second radio component with the
moderately-resolved structure of the primary points towards a common origin
for both emission features. To confirm this precessing jet, further
observations are needed over a wide frequency and spatial resolution coverage.

Then, 4 of the 14 radio AGN in our sample (29%) show evidence for a compact
radio jet from their morphology.

####  6.4.2 Dual AGN Candidates

J0843+3549 and J1511+0417 are both radio doubles; that is, they show two
morphologically distinct radio components (see Figure 11 and Figure 22). For
both, the radio AGN is likely hosted by the southern, dominant component in
each source. The two point spectral index value $\alpha_{3}^{10}$ for these
dominant components is $-1.37\pm 0.05$ and $0.19\pm 0.03$ for J0843+3549 and
J1511+0417, respectively. J0843+3549 is thus a candidate compact steep
spectrum (CSS) object, which are compact radio sources less than 20 kpc in
linear size and have $\alpha\leq-0.5$ (O’Dea & Saikia, 2021). J1511+0417 is
host to a radio core, as is evident by the inverted spectral index and
unresolved morphology of the southern component. The two components are
separated by 1.6 kpc for J0843+3549, and 2.1 kpc for J1511+0417. For both
systems, there is no clear radio emission which connects the dominant source
to the weaker one. J1511+0417 is particularly noteworthy because of the co-
location of the northern radio component with a second optical nucleus.
Although this galaxy merger is classified as a post-merger system, which are
defined as containing only a single optical nucleus, it’s clear through the
visual identification of distinct optical nuclei that this system is in an
earlier stage of galaxy merger evolution, prior to the merging of the stellar
nuclei. Indeed, this is corroborated by the GAIA photometric catalog
containing an additional source identification at the location of the northern
optical nucleus. The northern radio component has a two point spectral index
of $-0.60\pm 0.04$, also making it a candidate CSS object. These
characteristics make J1511+0417 a candidate dual AGN.

J0843+3549 has also been identified as a dual AGN candidate by previous work.
Using deep NIR imaging, Koss et al. (2018) revealed a population of hidden
nuclear mergers in a sample of heavily obscured, hard X-ray selected AGN. They
identified a second IR source in the central kpcs of this optically selected
post-merger galaxy that was blended into the optical nucleus in its low-
resolution SDSS image. We identified two radio components in J0843+3549, the
dominant of which is co-spatial with the central IR/optical component and the
second component is found to the north of this. However, Koss et al. (2018)
identified the second IR nucleus located 2.9 kpc to the east of the dominant
component. If indeed the second IR nucleus of Koss et al. (2018) is associated
with this galaxy merger and not a chance projection, we find no evidence of
radio emission associated with it in our 10 GHz map, down to a $3\sigma$
luminosity limit of $2.7\times 10^{20}$ W Hz-1. We require further
observations of J0843+3549 to confirm the nature of the northern radio source,
since the two components are blended in the VLASS Quick Look image and we do
not have lower frequency observations of higher resolution such as for
J1511+0417.

##  7 Discussion

###  7.1 Prevalence of RQ Emission

Much attention has been given in recent years to studying the connection
between merging galaxy systems and the triggering of a radio-loud or radio-
quiet AGN (Ramos Almeida et al., 2011; Bessiere et al., 2012; Ramos Almeida et
al., 2012; Chiaberge et al., 2015; Kozieł-Wierzbowska et al., 2017; Pierce et
al., 2022). The general finding is that radio-loud AGN are associated with
merging galaxy systems, while the fraction of radio-quiet AGN hosted by
merging galaxies is often indistinguishable from non-merging systems
(Chiaberge et al., 2015), suggesting that the radio-quiet phase is ubiquitous
in the formation of all early-type galaxies (Ramos Almeida et al., 2013). This
is particularly striking given that 89% (16/18) of the radio emission we
detected is radio-quiet, whether its origin be from an AGN, SF, or a
combination of both; and 86% (12/14) of the radio AGN are radio-quiet. From
the overall sample, this translates into 57% (16/28) of our post-mergers
hosting radio-quiet emission, 43% (12/28) hosting a radio-quiet AGN, and 7%
hosting a radio-loud AGN.

We emphasize important distinctions between our study and those focusing on
the incidence of galaxy mergers in radio-loud/radio-quiet systems. These
studies (e g., Chiaberge et al., 2015; Breiding et al., 2023) placed
constraints on the merger fraction of AGN host galaxies; they began by
selecting for a sample of luminous quasars and examined the host galaxy
morphology for signs of ongoing or recent gravitational interaction. Our
sample, however, is of known merging galaxies that were selected only because
of their host galaxy morphology and the presence of a single optical nucleus
(Carpineti et al., 2012). Then, the merger fraction for our radio-quiet and
radio-loud AGN is unity by definition; that is, 100% of the radio-AGN are
hosted by mergers. We cannot make direct comparisons to similar, though
distinct, AGN-merger studies because of the contrasting selection criterion
used for the different samples. Additionally, this means that the C12 sample
is unbiased towards AGN activity, whereas previous studies either favored, or
outright required, the identification of a radio AGN. The nature of our study
is to holistically examine the radio properties of these post-merger galaxies
and constrain the progenitor of their radio emission, including SF-related
activity. This is not to say that previous works have found that all merging
galaxies will produce a radio-loud AGN; such a case is clearly unrealistic by
the existence of inactive merging galaxy systems, as is the case for a number
of the SPMs in both their radio emission and optical emission-line activity
presented in this work.

The deep nature of our observations may also play a key role in the high
fraction of radio-quiet AGN in our sample. The sensitivity limits from ongoing
and past GHz-frequency radio surveys are close to, if not more than, an order
of magnitude less sensitive than what we achieved with our 10 GHz
observations. Without the high-significance 10 GHz detection revealed by our
observations, a number of these radio sources would be classified as non-
detections by standard survey selection criterion, which are often $\geq
5\sigma$, and thus would be excluded from such a study examining the link
between merging galaxies and the incidence of radio-loud/radio-quiet emission.
However, because of the 10 GHz detection that is co-spatial with the low-
significance radio emission revealed by these surveys, we are confident that
this emission is of astrophysical origin and should be examined as such.
Again, the nature of our study makes it impractical to compare our detection
rates to the merger fraction of AGN hosts from previous studies. Nonetheless,
it is expected that with deeper, more sensitive observations, the number of
radio-quiet AGNs hosted by galaxy mergers will increase. Such observations are
required to attain a full representation of the radio-quiet AGN population, as
is evidenced by the high fraction of radio-quiet AGN in our post-merger
sample.

####  7.1.1 The Role of Mergers in RQ AGN

The predominance of radio-quiet AGN among our radio AGN sample points to a
scenario in which the majority of the SMBHs powering these AGN have low black
hole spin values. This comes from the framework first proposed by Blandford &
Znajek (1977), in which the energy extracted from a spinning BH via a highly
magnetized accretion disk results in the launching of a jet. The radio-
quiet/radio-loud dichotomy can then be explained by the SMBH possessing either
a low or high spin (Wilson & Colbert, 1995).

Because these radio AGN are hosted exclusively by post-merger galaxies, we can
explore this SMBH spin paradigm from an interesting perspective: namely, the
coalescence of a supermassive black hole binary (SMBHB). Wilson & Colbert
(1995) proposed that the mass ratio of the SMBHB imposes significant
evolutionary effects on the radio-loud nature of the coalesced SMBH. Only for
SMBHBs with a mass ratio of order unity, with each SMBH of high mass ($\geq
10^{8}M_{\odot}$), will the resultant coalesced SMBH be highly spinning and
thus able to form a radio-loud AGN. This event is intrinsically rare, since
the mass function of SMBHs declines for high mass values (McLure & Dunlop,
2004; Hopkins et al., 2006; Gültekin et al., 2009), making the formation of
such a SMBHB rare as well. Leaving out J1511+0417, which is not a post-merger
system and likely would not yet have formed a SMBHB, the radio-loud AGN are
outnumbered by the radio-quiet AGN in our sample by a factor of 5.5, which is
consistent with the broad expectations of the Wilson & Colbert (1995)
framework.

There is observational evidence to support the merger-spin framework to
explain the radio-loud/radio-quiet dichotomy. de Ruiter et al. (2005) and
Capetti & Balmaverde (2006) have found that radio-loud AGN are ubiquitously
hosted by cored galaxies, i. e., those that show a flattening of their
brightness distribution towards the optical nucleus. Cored galaxies themselves
are products of SMBHB evolution. Once the SMBHB reaches pc separation, it will
scatter stars whose orbits form close encounters with itself, creating the
cored brightness distribution profile of the post-merger galaxy (Merritt &
Milosavljević, 2005). Then, at least a subset of cored post-merger galaxies
will host a coalesced SMBH, some of which may form a radio-quiet AGN and a
rare few a radio-loud AGN. This gives a self-consistent model for the
triggering and high relative fraction of radio-quiet AGN emission we have
discovered in our sample of post-merger galaxies. Follow-up radio and optical
observations would be needed to test this hypothesis for our sample of post-
mergers. Very Long Baseline Interferometry (VLBI) is needed to probe the state
of any SMBHB in these radio AGN. In Section 8, we present the results of
simulated observations using the current and next-generation VLBI instruments
to ascertain the feasibility of these studies. Space-based or adaptive optics
optical observations would be needed to model the brightness distribution
profiles of these post-mergers to assess if they are truly cored galaxies.

Alternatively, accretion can lead to a spinning up of the SMBH. As discussed
in Volonteri et al. (2013) and Chiaberge et al. (2015), coherent accretion, i.
e., the accreted material has a constant angular momentum axis, will lead to a
spinning up of the single SMBH. However, this naturally requires the accretion
to be constant over a significant time evolution, which, in turn, requires a
large gaseous reservoir for the SMBH to reside in. For gas-rich galaxy
mergers, it is well established that such a reservoir can be created via tidal
torquing of the gas, which drives it towards the nucleus where it forms a
circumbinary disk (Di Matteo et al., 2005). By examining the colors of the SPM
sample, C12 concluded that at least 55% of these SPMs are the product of a
galaxy merger consisting of one or more late-type, e. g., gas-rich, galaxies.
This accretion-driven spin up of the SMBH may be a plausible origin for the
radio emission in a subsample of these SPMs. However, it should be noted that
this framework can only produce radio-loud AGN; the radio-quiet AGN require an
alternative explanation.

###  7.2 Impact of AGN Feedback

Important to the overall discussion of post-merger evolution is an AGN’s
ability to either trigger or cut-off star formation in post-mergers through
AGN feedback. The centralization of gas that occurs during a gas-rich galaxy
merger can both fuel accretion to power an AGN and also act as a catalyst for
triggering starburst activity. The chronology of these two processes is
crucial: if the AGN activity is not prompt during the starburst period, AGN
feedback will have little to no effect on the SFR of the host galaxy.

Kaviraj et al. (2015) found that the host galaxies for a sample of VLBI-
detected, merger-triggered radio AGN were a factor of 3 more likely than
stellar mass- and redshift-matched inactive early-type galaxies to lie on the
UV-optical red sequence. Using this and timescale arguments, the authors argue
that these merger-triggered radio AGN are inefficient at regulating the SF in
the host galaxy. Shabala et al. (2017) found a similar result for VLBI-
detected radio AGN hosted by gas-rich minor mergers. By reconstructing the SF
history of the host galaxies, they found that none of the radio AGN were
triggered within 400 Myr of the onset of the starburst activity in the host,
limiting their ability to impact the overall SFR. Carpineti et al. (2012) also
found that the fraction of star-forming galaxies peaks in ongoing mergers, but
the fraction of optical AGN peaks in post-merger systems, indicating different
triggering times for these two processes during the merger evolution.

Our radio-detected sources span a broad expanse of radio luminosity and
dominant emission mechanisms. As shown in Section 6.3, at least 4 of these
sources are likely dominated by emission from SNRs associated with past or
ongoing SF. 3 of these 4 radio sources are hosted by LINER or SF-AGN composite
emission-line galaxies. These 3 may represent the very earliest stages of AGN
feedback occurring in the post-mergers, as there is evidence for both an AGN
and recent SF activity, or at least ambiguously for the LINER. On the other
hand, 4 of the likely radio AGN are hosted by either SF or SF-AGN emission-
line galaxies. These may represent the next stage of AGN feedback, as the AGN
is now the dominant radio emission mechanism, outshining in the radio band the
total emission from the supernova remnants associated with the SF. There is
multi-wavelength evidence for potential feedback processes occurring in at
least these 8 post-mergers based on their radio and emission-line activity,
though, as shown in Section 6.1, almost all of the post-merger galaxies appear
to be forming stars at a rate $\geq 1\,\mathrm{M_{\odot}}\,\mathrm{yr}^{-1}$.
Although for the SF-dominated radio sources we, naturally, cannot estimate the
effect of an AGN-powered jet on feedback, we can discuss how our population of
radio AGN fits within the realm of AGN feedback.

Establishing the total AGN jet power from its radio luminosity is not
straightforward (see Godfrey & Shabala 2016). However, the capability of the
jet to substantially impact the surrounding interstellar medium (ISM),
creating feedback, can be broadly interpreted even from an order of magnitude
estimation. We used the relation of Cavagnolo et al. (2010) to estimate the
jet power for each of the radio AGN in our sample. We found a range of jet
powers spanning $10^{41}-10^{43}$ erg s-1, making these low power radio jets.
These jet powers are actually favorable for feedback process: low power jets,
confined to the central kpcs of the host, are more effective at large,
sustained disruption of the surrounding ISM compared to high power jets that
easily puncture through the dense circumnuclear ISM and remain collimated at
large distances from the launching region (Mukherjee et al., 2016). Indeed,
the compact morphology of the radio sources favors this scenario, as none of
the radio AGN show collimated jets beyond the nuclear region of their host
galaxy. In conjunction with the substantial SFRs of the host post-merger
galaxies, this makes these sources good candidates to study the impact of AGN
feedback in post-merger galaxies by searching for AGN-triggered bursts of SF
activity (positive feedback) and/or multiphase outflows (negative feedback).

###  7.3 Spectral Index of SF-Related Emission

In the GHz regime, radio emission related to the shocks propagated by SNRs is
optically thin, with a canonical spectral index value $\alpha\sim-0.8$
(Condon, 1992). By comparing the radio-based SFR to that derived from host
galaxy properties, we have identified 4 compact, nuclear, 10 GHz radio sources
that are likely dominated by SF-related processes. Among these, we have placed
strict constraints on the optically-thin spectral index for 3 of these sources
using their 3 and 10 GHz flux densities. These 3 radio sources have a median
optically-thin spectral index $\alpha_{3}^{10}=-1.14\pm 0.08$, significantly
steeper than the canonical value of $\alpha\sim-0.8$. Such a difference may be
cause for concern. However, recent work by Klein et al. (2018) has provided a
more in-depth analysis of the broadband radio spectra of integrated SNR
populations. By using nearly 2 decades of frequency coverage, these authors
found that for 14 SF galaxies, the synchrotron spectral index at low
frequency, i. e., $\nu<$ 1 GHz, was similar to the canonical value. However,
in the 1-12 GHz frequency range, the spectra required either a break or an
exponential decline, indicating a steepening of the radio spectra. These
features would be caused either by significant synchrotron or inverse-Compton
loses to the high-energy electrons produced by the SN shocks, although Klein
et al. (2018) explain that a cutoff in the synchrotron spectrum is difficult
to explain without inducing a single-injection scenario, which is unlikely
when considering the integrated properties of a SNR population.

This situation is alleviated somewhat by the emission-line activity for each
of the SF-dominated radio sources. Only J1015+3614 is identified as a purely
SF galaxy from its emission-line ratios, and its radio source has the flattest
optically-thin spectral index of these sources of interest with
$\alpha_{3}^{10}=-0.64\pm 0.10$. This is not dissimilar to the spectral index
values found for other SF galaxies (Chyży et al., 2018; An et al., 2021), and
while slightly flatter than the canonical $\alpha\sim-0.8$, this can be
explained by the ratio of CR electrons favoring a younger, more energetic
population, e. g., due to recent supernova activity. J1445+5134 and J1617+2512
have been identified as SF-AGN composites by their emission-line ratios, and
have much steeper optically-thin spectral index values, $-1.46\pm 0.04$ and
$-1.31\pm 0.11$, respectively. These steep spectral index values for SF
galaxies have been observed before: using a sample of 41 6 GHz-detected
submillimeter galaxies (SMGs), Thomson et al. (2019) found that the median
1.4-6 GHz spectral index for a sub-sample of bright SMGs to be $-1.35\pm
0.24$. Additionally, the radio spectra of the bright SMGs showed a steepening
at these GHz frequencies when compared to the same spectra at MHz frequencies.
Thomson et al. (2019) discussed that such a phenomenon may be caused by the
mixing of distinct electron populations accelerated by decoupled processes
that dominate at different frequencies and at different spatial scales. We
have presented this hypothesis to explain the spectral breaking (Section 5.4)
and radio morphology (Section 6.4.1) for a number of different radio sources
in our sample. We further extend this idea to the two SF-dominated radio
sources hosted by SF-AGN composite emission-line galaxies. If there is a
contribution from both a steep-spectrum radio AGN component and SNR population
to the observed radio emission, such a mixing may cause a steepening of the
radio spectral index. This would most likely arise from resolution effects, i.
e., the diffuse emission from SNRs is resolved out at higher frequency,
leaving only the steep-spectrum, compact AGN component. A scenario in which
the diffuse emission from SNRs is cutoff at higher frequency is unlikely, as
discussed previously (Klein et al., 2018). Multi-band SED modeling of these
sources is needed to better understand the fractional contribution of the AGN
(Dietrich et al., 2018; Ramos Padilla et al., 2020).

The remaining SF-dominated radio source is J1041+1105. Unlike for the other
SF-dominated radio sources, we could only place a lower limit on the
optically-thin spectral index of J1041+1105 ($\alpha_{3}^{10}>-1.43$) due to
the non-detection of any radio emission at 3 GHz. Like J1445+5134 and
J1617+2512, $\alpha_{3}^{10}$ for this source may also be steeper than
expected, given its SF-dominated nature. However, when considering the 1.4 GHz
flux density limit, which is more sensitive than the 3 GHz limit, the lower
limit to the spectral index becomes -0.81. We know, then, that J1041+1105 does
not show the same type of highly steep spectral index at GHz frequencies that
J1445+5134 and J1617+2512 do. This may be indicative of the more canonical
radio emission associated with SF, like J1015+3614. However, unlike
J1015+3614, the host galaxy of J1041+1105 is a LINER. Although the exact
nature of LINER emission is ambiguous, the surface brightness profiles of the
low-ionization emission lines found in LINER hosts though integral field
spectroscopy favor a scenario in which the ionization is powered by post-
asymptotic giant branch stars (Yan & Blanton, 2012; Singh et al., 2013). Such
a scenario is also favorable due to the ubiquity of these stars in all
galaxies, especially those with little active star formation. Radio
observations, however, seemingly favor the presence of an AGN over pure SF-
related processes to produce the observed radio emission (Filho et al., 2004;
Singh et al., 2015). Although J1041+1105 may possibly have an optically-thin
spectral index close to the canonical SF value of -0.8, more sensitive 1.4 and
3 GHz observations are required to better understand the association of this
nuclear radio source to its LINER emission-line host galaxy.

##  8 SMBHBs in SPM Systems

###  8.1 Do Our SPMs Host SMBHBs?

Radio observations are a powerful tool to probe both the kpc- and pc-scale
environment to search for and confirm SMBHB candidates. At kpc scales, the
morphology of the extended jets or lobes may hold signatures of SMBHB
evolution: an X-shaped morphology due to a coalesced SMBHB (Begelman et al.,
1980; Merritt & Ekers, 2002), or an S-shaped (helical) morphology due to jet
precession caused by a SMBHB (e.g., Rubinur et al. 2017). These radio
structures have steep spectral index measurements ($\alpha\leq-0.5$), making
low frequency observations particularly advantageous towards their
identification. After visual inspection, we find no evidence for any of these
kpc-scale, SMBHB evolutionary signatures in the radio morphology of these SPM
galaxies at any observing frequency (see Appendix A for multi-frequency radio
maps for each SPM). However, the absence of these S- or X-shaped morphologies
does not dismiss that these SPMs may harbor a SMBHB.

Likewise, only 2 of these SPMs (J0841+3549 and J1511+0417), show any evidence
of DAGN behavior, as discussed in Section 6.4.2. Our 10 GHz observations would
be the most adept at identifying DAGN candidates because of their sensitivity
(nominal image RMS $\sim 15$ $\mu$Jy) and high angular resolution, which would
be able to resolve potential blended radio cores in lower resolution images.
Then, we find no evidence for secondary radio emission in the remaining 12
radio AGN down to a range of limiting $3\sigma$ luminosities $L_{\nu}=8\times
10^{19}-7\times 10^{20}$ W Hz-1. A second AGN in these systems would be
extremely radio-faint and require ultra-deep sensitivities to detect. This is
also true for the SF-dominated sources: we find no evidence of multi-component
radio emission in any of these systems above a 10 GHz luminosity of
$L_{\nu}=3.7\times 10^{20}$ W Hz-1.

Gaia’s superb astrometric precision has enabled a number of searches for DAGN
and SMBHBs candidates that utilize astrometric variability induced by
photocenter pseudo-motion of the unresolved SMBH pair, or varstrometry (Hwang
et al., 2020; Shen et al., 2021; Chen et al., 2022; Schwartzman et al., 2023).
This technique is particularly powerful for systems with $z>0.5$, as Gaia’s
astrometry is optimized for compact sources (Makarov & Secrest, 2022). Below
this redshift, extended features in the host galaxy, e. g., tidal tails, will
induce false astrometic noise (Souchay et al., 2022). Because our sample of
SPMs were selected to have extended, tidal features and $z<0.1$, any analysis
using the photometric center from Gaia is unreliable, including searching for
offsets in the radio and optical photometric centers.

Then, for the majority of the 10 GHz-detected SPM galaxies in our sample
(16/18; 89%), we find no evidence for SMBHB evolution at the (sub-)kpc scale.
We emphasize, however, that the lack of evidence does not preclude that any of
these SPMs may host a SMBHB system.

###  8.2 Searches with Very Long Baseline Interferometry

Very Long Baseline Interferometry (VLBI) offers a plethora of direct and
indirect methods to identify SMBHB candidates. Among these, the identification
of dual, flat-spectrum radio cores at pc-scale separation provides the most
compelling evidence of any SMBHB, a technique that is only feasible because of
VLBI’s unique milliarcsecond angular resolution. Indeed, this technique has so
far provided the best evidence for a SMBHB, hosted in the elliptical galaxy
0402+379, through imaging (Rodriguez et al., 2006) and proper motion
constraints (Bansal et al., 2017). VLBI observations can also provide
corroboratory evidence of SMBHB candidates through indirect methods.
Significant position angle differences between the pc- and kpc-scale radio jet
may be indicative of binary evolution (e. g., Mooley et al. 2018), as the jet
opening angle widens as the binary loses energy due to the emission of
gravitational waves (Kulkarni & Loeb, 2016). Periodic variability observed in
both the radio luminosity and radio core position may be indicative of SMBHB-
induced precession of the radio jet of sub-pc SMBHB candidates (e. g., Sudou
et al. 2003; Stirling et al. 2003; Kun et al. 2014).

We are interested in establishing the feasibility of performing VLBI
observations to search for a SMBHB in each of the 14 radio AGN we have
discovered with our 10 GHz VLA observations. We did this by simulating two
different VLBI observatories: the Very Long Baseline Array (VLBA), and the
Next Generation Very Large Array (ngVLA). For each frequency band of each
array, we calculated the expected continuum RMS image sensitivity of a 1-hour
long integration and a 10-hour long integration. Here, we are using
integration to represent the total time on-source for each target of interest.
Each observation, encompassing this on-source time, is then necessarily longer
than 1 hour and 10 hours to account for overheads. In actual targeted
observations for searches of SMBHBs, Walsh et al. (in prep) found that roughly
75% of their total observation time was on-source, with the remaining 25%
dedicated to scans of amplitude and phase reference calibrators. So, we can
expect an increase of approximately 25% for each integration time, or
observations which are approximately 1.3 and 13.3 hours for the 1-hour and
10-hour integration, respectively. For the VLBA, we have assumed an efficiency
factor $\eta_{s}$ of 0.8, and that all 10 antennas, thus 90 baselines, are
included for each simulated integration. Our simulated integrations use the
L-band (21 cm), S-band (13 cm), C-band (6 cm and 5 cm), X-band (4 cm), Ku-band
(2 cm), K-band (1.4 cm), Ka-band (1.25 cm), and Q-band (0.7 cm) receivers for
the VLBA. For each frequency band, we have assumed the maximum possible data
rate (2048 Mbps for L- and S-band, 4096 Mbps for all others) and used the SEFD
provided for the
VLBA555https://science.nrao.edu/facilities/vlba/docs/manuals/oss/bands-perf.
For the simulated ngVLA integrations, we used the ngVLA sensitivity calculator
Python script666https://gitlab.nrao.edu/vrosero/ngvla-sensitivity-calculator
to calculate the expected continuum RMS image sensitivity. We did this for
each of the central frequencies listed for the VLBA, since the larger
bandwidths of the ngVLA receivers encapsulate multiple VLBA receiver frequency
ranges. We simulated the 1-hour and 10-hour long integrations at the first 5
bands of the ngVLA for this analysis, with central frequencies at 2.4, 8, 16,
27, and 41 GHz. We have not taken into account the RFI environment for any of
these simulated integrations. This is especially prevalent at lower
frequencies, where up to half of the bandwidth may be unusable due to the
persistent, dominating presence of RFI. Each of these simulated integrations
is an ideal case and represents a lower limit to what is achievable in actual
observations.

Figure 8: VLBI-scale flux density estimates of the 14 newly-discovered radio
AGN from our sample of SPM galaxies plotted with the simulated, $5\sigma$
sensitivity curves of a 1-hour and 10-hour integration with the Very Long
Baseline Array (VLBA) and the Long Baseline Array (LBA) of the Next Generation
Very Large Array (ngVLA). These data points represent the potential flux
density of a supermassive black hole binary (SMBHB), which can only be
resolved with VLBI. Points above a $5\sigma$ sensitivity curve indicate a
detection. At lower frequency ($\nu<10$ GHz), the VLBA observations may
already reach the desired sensitivity to achieve a significant detection of
milliarcsecond-scale radio emission. The notable difference is at higher
frequency ($\nu>10$ GHz), where a 1-hour integration with the ngVLA vastly
improves the probability of detection compared to a 10-hour VLBA integration.

To calculate the expected milliarcsecond-scale radio flux density at each
frequency, we used the 10 GHz flux density value (Table 2, column 6) and the
$\alpha_{3}^{10}$ spectral index value (Table 3, column 4) for each of the
radio AGN. For these radio AGN with more than one resolved component
(J0843+3549 and J1511+0417), we used the 10 GHz flux density value of the
dominant radio component. We began by extrapolating the 10 GHz flux density to
each of the central frequencies listed above. Here, we have assumed that the
broader radio SED for each of these radio components follows a simple power
law $S_{\nu}\propto\nu^{\alpha}$, where $\alpha$ is $\alpha_{3}^{10}$. We have
used the 10 GHz flux density values because this represents the best
approximation to the radio flux density we would expect from a dominant radio
core. All other flux density measurements were taken at lower frequency and
angular resolution. Because of this, those flux density measurements are more
likely to have contributions from non-core-related phenomena, such as steep-
spectrum features, e. g., a radio jet, or star formation, especially for the
case of the 144 MHz LoTSS data. Indeed, only J0206$-$0017 and J1511+0417 have
a flat spectral index ($\alpha>-0.5$) in the optically-thin regime, which is
expected if the dominant contributor to the radio flux density were from a
radio core. We do note that J1433+3444 may also be a flat spectrum object
since $\alpha_{3}^{10}=-0.47\pm 0.07$. The same could also be true for all
source for which we could only place a lower limit on $\alpha_{3}^{10}$
(J0851+4050, J0916+4542, J1113+2714, and J1517+0409). This is critical to
establishing the expected population of detected radio sources for each of our
simulated integrations. If we systematically overestimate the expected
milliarcsecond-scale flux density, we will also overestimate the number of
significant detections achievable in each of our simulated integrations. We
have also assumed that $\alpha_{3}^{10}$ also represents the dominant VLBI
component. This certainly does not need to be the case, as even unresolved
features at sub-arcsecond scale may be resolved out at the mas-scale probed by
VLBI observations, possibly revealing a dominant radio core at mas scales.
However, we are using these values since they best represent the physical
situation as we can currently determine. Once again, we only wish to estimate
what the VLBI-scale emission properties are; only through actual observation
could these flux density values be determined.

After extrapolating the 10 GHz flux density to the designated frequency
values, we apply a factor of 0.3 in converting from sub-arcsecond-scale flux
density to mas-scale flux density. This value was chosen from the analysis of
Deller & Middelberg (2014). In their analysis, these authors determined the
ratio of peak VLBI flux density at 1.4 GHz to peak FIRST flux density for a
large sample of VLBI sources detected in the mJy Imaging VLBA Exploration at
20 cm (mJIVE-20) survey. Overall, they found that 30%-35% of all sources have
compact VLBI emission in which the majority of the FIRST flux is recovered,
with this trend increasing towards lower FIRST flux density. Indeed, for FIRST
sources with a flux density measuring from 1-2 mJy that are detected with
VLBI, all recover at least 32% of the FIRST flux density value at VLBI scales,
with about 25% of sources having greater than 64% of this value recovered. We
acknowledge that this extrapolation was determined only for 1.4 GHz
observations, whereas our analysis uses the flux density determined at 10 GHz.
Our factor of 0.3, then, may even underestimate the VLBI-scale flux density
for each source. Though, for the sake of this analysis, this is preferred to
an overestimation.

We now have estimations for the VLBI-scale flux density for each of the SPM
sources detected with our 10 GHz observations. Figure 8 plots these estimated
VLBI-scale flux density values with the simulated continuum image RMS
sensitivities of a 1-hour long and 10-hour long integration with the VLBA and
ngVLA. For each simulated integration, the sensitivity curve represents a
5$\sigma$ detection threshold. Points that fall above these curves represent a
detection, while those below are not expected to be detected. Notably, lower
frequency integrations ($\nu<10$ GHz) with the VLBA may already achieve a
sensitivity to reach a significant detection of VLBI-scale radio emission.
While these frequencies may not be optimal for isolating the core emission,
the flux density information they provide is nonetheless critical to
establishing the spectral index value of the potential radio core associated
with each binary constituent. The significant improvement in detection
threshold appears in the higher frequency integrations ($\nu>10$ GHz). At
these frequencies, we expect that the vast majority of sources would not be
detected even with a 10-hour integration time using the VLBA. However, with
the ngVLA, we find that a 1-hour integration time is sufficient to detect all
of the sources at 15 GHz, and the majority of sources at 22, 23, and 43 GHz.
These higher frequencies observations are best at isolating the radio core by
resolving out larger-scale emission and provide high astrometric precision due
to their high angular resolution.

##  9 Summary

In this paper, we have analyzed the emission properties of a sample of 30
local post-merger galaxies from Carpineti et al. (2012) to search for star
formation- and AGN-related activity. Our main results are as follows:

  1. 1.

Diverse Emission-Line Activity: Using the optical emission-line flux ratios
derived from the OSSY catalog (Oh et al., 2011), and standard BPT diagram
analyses, 43% of the post-mergers are optically quiescent, 10% are dominated
by SF, 13% by a combination of SF and AGN, 13% by Seyfert AGN, and 20% by
LINER activity.

  2. 2.

Low-luminosity Radio Emission: Of those with detectable radio emission,
through both archival radio surveys and new, high resolution, 10 GHz
observations with the VLA, the vast majority are radio-quiet, with only 2
reaching the spectral luminosity threshold ($\nu L_{\nu}>10^{32}$ W) to be
classified as radio-loud. We discovered a number of nuclear radio sources at
high significance ($\geq 5\sigma$) with our 10 GHz observations that were
otherwise non-detections by archival radio surveys, emphasizing the importance
of deep observations to reveal the full population of radio-quiet systems.

  3. 3.

Prevalence of Compact Radio Emission: At the largest spatial extents, sampled
by 144 MHz LoTSS observations, all of the detected radio sources have a
diffuse or extended emission component. Only J0843+3549 and J1433+3444 display
an AGN-like morphology among these sources. The nuclear emission is unresolved
in all but one of the 15 sources at 1.4 GHz ($5\arcsec$ resolution),
indicating that compact, nuclear emission is prevalent in these post-mergers.
At the most compact scales, sampled by our 10 GHz observations ($\approx
0.2\arcsec$) the sources show a variety of AGN- and SF-related morphologies.

  4. 4.

Radio Spectra and Spectral Index Measurements: Of the 12 radio sources with
well-sampled (4 or more detections) spectra, we found 3 that showed evidence
of significant curvature ($|q|\geq 0.2$) in their broadband spectrum, with one
being a Gigahertz peaked spectrum (GPS) source (J1617+2512). Two spectra were
found to be inverted ($q\geq 0.2$), though we believe this is likely due to an
overall flattening of the spectrum at high frequency and not a true inversion.
The spectra of J0843+3549, J1015+3914, J1135+2953, and 1304+6520 are poorly
fit by both a simple and curved power law, as indicated by the
$\chi_{red}^{2}$ value for each fit. This is either due to variability at one
or more flux values, or a blending of two distinct electron populations at low
angular resolution. We also calculated the spectral index value
$\alpha_{3}^{10}$ to the compact radio emission using the 3 and 10 GHz flux
values, or its lower limit for sources without a 3 GHz detection. These
$\alpha_{3}^{10}$ values range from $0.19\geq\alpha\geq-1.74$, though the
majority have a steep spectral index $\alpha<-0.5$.

  5. 5.

SF Activity in Post-Mergers: The 1.4 GHz luminosity of 4 of the radio sources
(J1015+3914, J1041+1105, J1445+5134, and J1617+2512) can be explained by SF
processes alone. For each, we have determined that their emission is most
likely due to a population of supernova remnants, as the expected luminosity
from an individual, luminous radio supernova is at least a factor of 80 dimmer
than each of their observed 10 GHz luminosity. Further, these 4 radio sources
are hosted by either SF, SF-AGN composite, or LINER emission-line galaxies,
providing corroboratory evidence of ongoing or recent SF activity in each of
these post-mergers. It is notable that the spectral index value for 3 of these
4 is significantly steeper than the canonical value $\alpha\sim-0.8$ for
shock-dominated sources (Condon, 1992). However, these spectral index values
are not dissimilar to those of other shock-dominated sources (Chyży et al.,
2018; Thomson et al., 2019; An et al., 2021, e. g.,), and may be indicative of
an older CR electron population due to evolved supernova activity.
Alternatively, as two of these sources are hosted by SF-AGN composite
emission-line galaxies, the steep spectral index may be due to resolution
effects in which the diffuse synchrotron is resolved out at higher angular
resolution, revealing compact, steep-spectrum AGN emission.

  6. 6.

Discovery of Radio AGN: We have discovered 14 likely radio AGN in these post-
mergers: 3 because of their association with a known AGN and 11 for which we
found excess radio emission compared to SF predictors. 86% (12/14) of these
radio AGN are radio-quiet, with only 14% (2/14) being radio-loud. The post-
merger hosts are found to occupy all regions of the BPT diagrams, indicating
that radio AGN activity may be present even during stages of star-forming
activity of the post-merger evolution. We also report on the discovery of a
precessing jet in the dual AGN candidate J0843+3549 (Koss et al., 2018), and
discover a new dual AGN candidate, J1511+0417.

  7. 7.

The Origin of Radio AGN Activity in Mergers: The prevalence of radio-quiet AGN
among our radio AGN population lends itself to a scenario in which radio-quiet
AGN in ongoing or recent galaxy mergers may be more populous than previously
believed. Because our sample is comprised of late-stage and post-merger
systems, the high fraction of radio-quiet AGN can be explained by SMBH spin up
due to the coalescence of a supermassive black hole binary (SMBHB; Wilson &
Colbert (1995)). In this framework, radio-loud AGN are only produced for the
most massive binary systems, which are intrinsically rare due to the sharp
decline at high mass of the SMBH mass function (McLure & Dunlop, 2004; Hopkins
et al., 2006; Gültekin et al., 2009). Indeed, we have found that the radio-
quiet AGN outnumber the radio-loud AGN by a factor of 5.5. Alternatively, gas-
rich mergers may produce radio-loud AGN if the SMBH sustains coherent
accretion for an extended period of time. Both scenarios need further
observations to test rigorously.

  8. 8.

Jet-ISM Feedback: We estimated the total power of the jets for our sample of
radio AGN. The jet powers span a range of $10^{41}-10^{43}$ erg s-1, making
them low power. The majority of the post-mergers have a SFR $\geq 1$ M⊙ yr-1,
indicating that the AGN may play an important role in providing positive or
negative feedback. Importantly, these low power jets, confined to the central
kpcs of the host, are more effective at large, sustained disruption of the
surrounding ISM compared to high power jets that easily puncture through the
dense circumnuclear ISM and remain collimated at large distances from the
launching region (Mukherjee et al., 2016). Indeed, the compact morphology of
these radio sources agrees with this scenario. These radio AGN are then good
candidates to study the impact of AGN feedback in post-merger systems by
searching for signatures of multi-phase gas outflows.

  9. 9.

Next-generation Searches of SMBHBs: Lastly, we simulated 1- and 10-hour
integrations at multiple frequencies with the Very Long Baseline Array (VLBA)
and the VLBI capabilities of the Next Generation VLA (ngVLA). These
simulations present the necessary time commitment for each instrument to reach
the deep sensitivities required to perform robust searches for a SMBHB in each
of these radio AGN hosted by a post-merger galaxy. We estimated the
milliarcsecond-scale flux density of the radio source using the spectral index
value $\alpha_{3}^{10}$ for each radio AGN and additional factors from the
literature (e. g., Deller & Middelberg (2014)). We found that at low frequency
($\nu<10$ GHz), the VLBA can already perform these robust searches, though the
low frequency, milliarcsecond environment of radio AGN will often be dominated
by extended, steep spectrum emission, making radio core identification
difficult. The ngVLA will be a particularly powerful instrument for searches
of SMBHBs at high frequency ($\nu>10$ GHz), where the dual, flat-spectrum
cores of the SMBHB are the dominant emission signature. These high frequency,
high angular resolution observations also offer significantly better
astrometric precision than low frequency observations, which will be important
for constraining proper motion measurements of the SMBHB constituents.

Our study of the multi-wavelength emission properties of 30 post-merger
galaxies has discovered a number of exciting phenomena and individual sources.
Future work on this topic will expand the sample population of post-merger
galaxies, include better frequency coverage, and examine the radio emission of
galaxy mergers at various stages of their evolution. This work will further
our understanding of the astrophysical processes occurring during the merger
sequence, the impact of AGN feedback, and establish new radio sources for
which to follow-up with VLBI with the hope of detecting individual SMBHBs.

## Acknowledgments

GW and SBS were supported in this work by NSF award grant #1815664. We thank
Amy Reines for the discussions on multi-wavelength analyses to determine the
origin of radio emission and Julie Comerford for recommendation of the OSSY
catalog to perform our optical spectral analysis. The National Radio Astronomy
Observatory is a facility of the National Science Foundation operated under
cooperative agreement by Associated Universities, Inc. The NANOGrav
collaboration, which funded some components associated with this research,
receives support from National Science Foundation (NSF) Physics Frontiers
Center award #1430284 and #2020265. This research has made use of NASA’s
Astrophysics Data System Bibliographic Services.

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